Linear Ray and Wave Optics in Phase Space: Bridging Ray and Wave Optics via the Wigner Phase-Space Picture

Linear Ray and Wave Optics in Phase Space Bridging Ray and Wave Optics via the Wigner Phase-Space Picture

For the cover design: 9 Ren6 Magritte, La corde sensible, BY SIAE 2005

Linear Ray and Wave Optics in Phase Space B r i d g i n g Ray and W a v e O p t i c s via the W i g n e r Phase-Space Picture

Amalia Torre

ENEA-UTS Tecnologie Fisiche A vanzate Frasca ti (Rome), Italy

G

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To my m o t h e r whose caress still lingers on my cheek. To my father whose smile still shines into my eyes. To my country whose colours feed my avid love of life.

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Preface Amo i colori, tempi di un anelito inquieto, irresolvibile, vitale spiegazione umilissima e sovrana dei cosmici " p e r c h , " del mio respiro. . . .

A. Merini, Colori (from A. Merini, Fiore di poesia, Einaudi, Torino, 1998)

Ray, wave and quantum concepts are central to diverse and seemingly incompatible models of light. Each model particularizes a specific "manifestation" of light, and then corresponds to adequate physical assumptions and formal approxilnations, whose domain of applicability is well established. Accordingly each model comprises its own set of geometrical and dynamical postulates with the pertinent mathematical means. Geometrical optics models the light field as an aggregate of incoherent light rays, naively perceived as the trace of the motion of the "luminous corpuscles", which, emitted by the source, move through spa.ce in obedience to the usual laws of mechanics. It treats light rays as lines in 3-space dimensions and is accordingly concerned with the dynamical laws establishing how the rays bend when propagating in inhomogeneous media described by the refractive index function. Geometrical optics is not suited to explain interference, diffraction and quantum coherence effects, but, in contrast, it provides a particularly convenient mea,ns for the design of optical systems, which is based on the purely geometrical rules of ray tracing. Geometrical optics has developed its own mathematical framework, which can remarkably be brought into correspondence with that of the Hamiltonian mechanics of point-particles, with "time" corresponding to the arc-length along the ray path and the mechanical "potential" to the refractive index of the optical medium. Wave optics accounts for the wave characteristics of light. Originating directly from the classical electromagnetic theory, it shares with this theory the same system of theoretical principles and methods, which can notably be put in correspondence with those proper to relativistic quantum mechanics. Then, the geometry of light rays is replaced by the geometry of "luminous" waves, whose propagation is geometrically pictured as the transfer of the interference shaped vibrations from one portion of the medium to the contiguous one. Wave optics treats the light waves as complex functions of position in 3space dimensions and is accordingly concerned with the dynamical laws establishing how the wave function changes as the optical wave propagates through inhomogeneous media.

viii

Linear Ray and Wave Optics in Phase Space

Q u a n t u m optics recovers the grainy view of geometrical optics, picturing the light ray as a stream of particle-like entities, the photons. Whereas geometrical optics deals with the influence at a macroscopic level of the medium on the trajectory of tile photon streams, q u a n t u m optics is typically concerned with the wave-like question relevant to the coherence properties of tim t)hoton 1)earns and to the relevance of those properties on the interactioll ()f ligllt with llla,ttcr, whi(:h (:an (:orrcspondingly t)(; tr('~a,t('d qllantllln m('~r Cohere;lit an(1 stpu;r state.s of light arc th(; lmilding (:on(:('pts of (tlm,llt11~ll ()t)ti(:s. Wigner optics 1)ri(tges t)etween ray an(1 wa.ve ()I)tics. It ofli;rs th('~ optical t)ha,se sl)a,(:('~ a,s th(; amt)i(;nc('~ a,n(1 tlm Wign(;r flliwtion |)ascd t(;(:lllli(tll(; a.s the nla, thelllati(:al lna.('llilmry to a,('(:t)llllll()(la,t(~ |)(~tw(~ell tim two ()t)t)()sit,e extr(mms ()f ligtlt r(~I)res('~llta,tioll: th(~ l()(~a.liz(:(l ray ()f g('~()Ill(~tri(:a,1 ot)ti(:s a.ll(1 tll(~ lnll()ca,liz(~(t wa,ve flln('ti()n ()f wave ()l)ti(;s. N()ta.|)ly (tllald, Ulll ()t)tics till(Is a, t:()llV(~llient f()rllnlla,tion ill tl~e ])l'()])(~.r I)hase st)a,ce with tlm (:()~se(tll(mt g(~()~ml,ri(:al vi('.w ()f ('.()ll(~I'(~llt an(l s(t~mez(~d states a,s circles a,nd ellil)s(~s. Tim Wig~mr fll~(',l,ion ~(:l,l~()(ls ('a,n suita,|)ly lm applied t,() (t~m,~tl~ ()l)t,i(:s as w(;ll t()(:~a.t)l(; (@~('t,iv(~ al~a.lyti('a,1 ~n('m~s f()l" ('a.l('.~fla,ting ext)e(:l, atiol~ va,lu(~s a,n(1 tra.~sil,i(n~ l)r()l)a,|)ilities f()r t,lm a,f()r(nn(~ld,i(nm(1 states. Tim l)UI'I)()se ()f t,lm 1)()()k ix t.() i~t,l'()(lu(:(~ tim r(m.(l(~r t() t,l~(~ ()t)t,i('al I)ha.sesl)a,('e a,n(1 t() tim al)l)r()aclms 1,()()l)ti(:s 1)a.se(1 ()~ 1,1~(~Wig~mr (lisl,ril)~l,i()n fltn(:l,i(n~, tt~a.t ha.v(~ 1)(:(,l~ (hw(4ot)e(1 ()v(:r tlm t)ast 25 years or st) i~ s(w(~ral s('ient, ific titles. These yiel(l tlm f()n~m.1 (:()ll|,(~Xt, where (:()~(:(~t)ts m~(t lnetl~t)(ls ()f 1)ott~ ray an(t wave ot)ti(:s (:()alesc(~ into a ~ i f v i n g formalis~n. In this reslmct, elnt)hasis is givei~ to the Li(~ alge|)ra r(~t)r(~seId,ation of ()i)ti(:a,1 systems m~(l a,(:(:()rdingly to the Lie algebra view of light I)r()t)agation thr()~gh oI)tical s y s t ( ~ s . The book is nm(le as self-c()~ta.in(~(1 as t)ossi|)le. Chapter 1 1)rcs(mts tile Halniltonian equation,s of motions, wl~i(:h are basic to the (tevelot)~mld, of t)oth the tra,nsfer matrix fonna,lism, a,i~t)roi)riate to t)araxial ray ot)ti(:s (Chapters 2 and 3), and the trai~sfer operator formalism, suited to para,xial wave optics (Chapters 4 a.nd 5). The re,la.tio~ of both formalisms to the Lie algebra inethods is gently (tisplayed. Chapter 6 introduces tile Wigner distribution flmction, elucidating its origin taken in q u a n t u m mechanics and illustrating its properties. A host of diverse optical signals are considered and the relevant Wigner distribution functions arc analytically evaluated and graphically shown to help the intuitive perception of the siinultaneous account of the signal in the space and spatial frequency domains, conveyed by tile Wigner distribution function. Chapter 7 frames tile Wigner distribution function within the broad reahn of tile phasespace signal representations, and illustrates the procedure, and the relevant optical architectures, for displaying the Wigner distribution function of a given signal. In Chapter 8 the laws for the transfer of the Wigner distribution function through linear optical systems are derived. Attention is drawn to the

Preface

ix

relation between the Wigner distribution function and the fractional Fourier transform, which is a revealing and effective tool for the space-frequency representation of signals (optical or not). Chapter 9 is concerned with the moments of the Wigner distribution function and their propagation laws. The Wigner representation is presented on the fascinating border-line between quantum mechanics and signal theory. Chapters are made as self-consistent as possible. Indeed, the Introduction to each chapter is conceived as a summary of the basic results of previous chapters, which are central to those that are going to be presented. A basic role is assigned to the diagrams, which illustrate the syllabus of each chapter, and the figures, which confer physical reality to conceptual architectures. A wide bibliography is given in relation to topics both carefully investigated and briefly mentioned. Throughout the book the calculations are kept at an accessible level; most ina,thematical steps are justified. Difficulties Inight be encountered in connection with the algebra of operators, which do not obey the familiar rules of the algebra, of scalars. Careful and illustrative comments on the peculiar behavior of operators are provided in w 1.4.1 in order to help the readers who are not acquainted with the operator algebra. It is my hope to give the flavour of the fascinating feature of optics that enables a visible account of abstra,ct inathenmtical entities, like, for instance, symplectic ma.trices and inetaplectic operators, represented through integral transforms. Symplectic matrices and integral transforms, which essentially provide the formal structures for the considerations developed in Chapters 1 to 5, are intimately related, being indeed different representations of the same Sp(2, N) ~ Mp(2, IR) group element. Firstly recognized within a purely quantum mechanical context, this relation has been applied in optics in connection with the fractional Fourier transform. The link between ray matrices a,nd transfer operators from the alternative viewpoint of linear ca,nonical transformations and relevant representations, is elucidated in w5.6. This is an example of those parallel paths, that, explicitly illustrated or implicitly suggested in the text or in the problems, are intended to improve the feeling for the specific topic under consideration and to gain some insight and intuition for unforeseen correspondences and analogies between totally different physical problems. I am pleased to express my deep gratitude to Professor W.A.B. Evans, whose stimulating discussions, critical comments and technical suggestions have been precious to the completion of tile book. I am greatly indebted to Dr. A. De Angelis for his enlightening suggestions, and to Professor A. Reale and Professor A. Scafati for their helpful comments. It is dutiful of me to thank Dr. G. Dattoli, who introduced me to the Lie algebra theory during the

x

Linear Ray and Wave Optics in Phase Space

stage of our collaboration on the quantum picture of the Free Electron Laser dynamics. I am grateful to Dr. S. Bollanti, Dr. F. Flora and Dr. L. Mezi for their useful comments, and to Mrs. G. Gili, Mr. S. Lupini, Mrs. G. Martoriati, Mrs. M.T. Paolini, Mrs. L. Santonato, Dr. S. Palmerio, Dr. B. Robouch, Dr. N. Sa(:(:hetti and Dr. V. Violante for their c o n s t a n t and invah> able sympathy. I t h a n k our librarians, Mrs. C. De Palo and Mrs. M. Liberati, wtlo at certain t)erio(ls have t)atiently a(:(:epte(t the roh; of "nly" librarians. It is a t)lea.sllre to t,tm,~lk the ()t)ti(:al Society of A~imri(:a for kin(tly givi~lg me the t)ennissi(m t() r(:I)r()(t11(:('. Ilm.l,('.rial fl'()~li At)plied ()t)ti(:s and ()l)ti('s Letters, and Einmuli for t)ernfit, ting m e t() ret)r()(hl(:e the lines fr(),n Merilfi's t)()enl, which ()t)en(',(1 this Prcfa(:(;, lily literal tra,nslati()n ()f whi(:h ll()w (:h)s(;s it (t)elow). I (;xt)ress xlly at)t)I'(;('iatioIl t() l)I'()fess()r A. L()lnlm,llll f()," lfis t)r(nlll)t ml(t kill(t ,'cst)(nlse to my re(llu;st ()f ret)r()(llu'ing nmteria,1 fl'()m t)at)ers |)y tlinls(;lf a.n(t his ('()w()rk('rs, whi('tl at)I)(;ar(;(1 ill ()I)ti('s C()~lmnl~fi(:ati()Ii. I a lll I)h;as(:(l l,()(;xI)i'css lily gratitu(tc t() l,ll('. F()ll(tal, i(nl Magritt('. for alh)wiIlg l,ll(', r(~I)r()(lll(:l,i()ll ()f the (w()(:atively e~(~l,i(n~a,1 Magritt(; I)ai~ti~g La (:ordu. s('.nsibl(' f(n" 1,1~; (:(~ver. I ain also i~(l(;|)t(;(t t,() l,l~e Els(;vier l)r()(l~(:ti(m teal~ f()I" exl)ertly i~l)le~e~d, ing n~y i(h;as in relati(n~ I,,) l,h(; (:()v(:t'. A j()yflfl "Tl~m~k y(n~, s()relli~a" is (tirecte(t t() l)r. 1". M~u'('i, f(n" (;~th~siasti(:ally list(u~il~g 1,() l,l~e (h's(:ril)l,i(n~ ()f my "(:(n~(:et)t~ml (:asl,h's". A 1)()w is for 1,t~(; fl'i(;n(ls wh(~ sire,r(; ~ y t)assi(n~ f(n" the th(;a.l,r('~, fin" f()rgivil~g n~y at)scn(:es fl'()~ the t)relil~s, l)eil~g f()l'getfl~lly e~rat)ture(t i~ "1~i(; a.(l()rat(: f()n~n~linc". It is with intense elll()ti()iI tlm,t lily t h a n k s g()es als() t() zia Ai(ta a~(t Maria, wh() (:()lfl(t not see the (:()l~l)l('~l,i()li ()f the book.

I love colours, times of a yearning restless, irresolvable, vital, very lmmble and supreme explanation of the cosmic "why" of my breath. . . .

Colours

Contents

I. Hamiltonian Picture of Light Optics. First-Order Ray Optics 1.1 Introduction 1.2 Hamiltonian picture of light-ray propagation 1.3 Halniltonian picture of light-ray propagation: formal settings 1.4 Hamilton's equations for the light-ray 1.5 Lie transformations in the optical phase space 1.6 Linear ray optics and quadratic Hamiltonian functions 1.7 Planar model of first-order optical systems 1.8 ABCD matrix and focal, principal and nodal planes 1.9 Summary Problems References

1 3 9 19 24 30 36 44 53 53 55

2. First-Order Optical Systems: The Ray-Transfer Matrix 2.1 Introduction 2.2 Ray-ensemble description of light propagation 2.3 Quadratic monomials and symplectic matrices 2.4 Quadratic monomials and first-order optical systems 2.5 Quadratic nlonomials in phase space 2.6 Summary Problems References

59 62 88 93 99 105 106 107

3. The Group of 1D First-Order Optical Systems 3.1 3.2 3.3 3.4

Introduction Ray matrix of composite optical systems The subgroup of free propagation and thin lens matrices Optical matrices factorized in terms of free-medium sections and thin lenses 3.5 Wei-Norman representation of optical elements: LST synthesis 3.6 Rotations and squeezes in the phase plane 3.7 Iwasawa representation of optical elements: LSF ~ synthesis 3.8 Canonical and noncanonical representations of symplectic matrices 3.9 Integrating the equation for the ray transfer matrix 3.10 Summary Problems References

111 113 115 120 131 134 151 153 156 162 162 164

4. Wave-Optical Picture of First-Order Optical Systems 4.1 Introduction 4.2 Essentials of the scalar wave model of light. The paraxial wave equation in a quadratic medium

167 169

xii

Linear Ray and Wave Optics in Phase Space

4.3 Ray and wave optics 4.4 From the ray-optical matrix to the wave-optical opera,tor 4.5 Eigenfunctions of ~ and ~: point-like and spatial ha,rmolfiC waveforms 4.6 Spatial Fourier ret)resentation of optical wave, fields 4.7 Summary Prol)lems I/ef(;ren(:es

5. 1D First-Order Optical Systems: The Huygens-Fresnel Integral 5.1 IId,r()(lu(',ti()ll 5.2 Qlm,(lrati(: Ha lililtr a ll(1 lllr Li(', alg(',l)ra 5.3 Wav(~-()I)ti(:al I I'a,IlSf(w r(4ati()xls fi)r a Il A B ( : I ) syst,tull 5.4 Tim ()I)ti(',al F()lll'ier t l'allS[t)rlll 5.5 R(~(:overing t ll(' ray-()l)ti('al (h's('riI)ti()II 5.6 Wave-ot)ti(',al t)r()l)agatt)rs as llllitary ret)resr162 ()f linear ca Il()llical t,l'allSfl)rlllal,i()llS 5.7 Sunlmary Problems Referen(;es 6. The Wigner Distribution Function: Analytical Evaluation 6.1 Introducti(m 6.2 The optical Wigll~'~r (tisl,ril)111i()ll flnl(:tion: ])asi(: (:()ll(:~l)lS 6.3 The WigIler (listril)llti()ll f'llll(:t,i()ll: t)a,si(: t)roI)(;rti(;s 6.4 The Wigner r fllll(:t,i()ll ()f light sig2mls: flll'l]l(',r r 6.5 Summary Prot)leInS 1-{eferences 7. The Wigner Distribution Function: Optical Production 7.1 Introducti(m 7.2 The sliding-wimh)w Fourier transform 7.3 The Wigner distribution flm(:tion and the general (:lass of space-frequency signal representations 7.4 The ambiguity flm(:tion 7.5 Understanding tile Wigner and ambiguity functions from the the viewpoint of the mutual intensity function 7 . 6 0 p t i c a J production of the Wigner distribution function: general considerations 7.7 Wigner processor for 1D real signals: basic configurations 7.8 Wigner processor for 1D complex signals: basic configurations 7.9 The smoothed Wigner distribution function and the cross-ambiguity function: optical production 7.10 Summary Problems References

174 186 194 198 214 215 216

221 224 234 242 257 261 266 267 268

271 277 282

303 333 333 335

341 343 354 358 369 379 384 394 398 400 400 403

Contents

8. 1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 8.1 Introduction 8.2 From the wave fimction to the phase-space representation 8.3 First-order optical systems: propaga, tion law for the Wigner distribution function 8.4 The Wigner distribution function and the optical Fourier transform: linking Fourier optics to Wigner optics 8.5 Transport equation for the Wigner distribution function 8.6 Summary Problems References 9. 1D First-Order Optical Systems: Moments of the Wigner Distribution Function 9.1 Introduction 9.2 Basic notions on moments 9.3 Preliminaries to the calculation of the moments of the Wigner distribution function 9.4 Wigner distribution function: local and global moments 9.5 Gaussian Wigner distribution functions: the variance matrix and its evolution 9.6 Propagation laws for the moments of the Wigner distribution function in first-order optical systems 9.7 Higher-order moments of the Wigner distribution function 9.8 Summary Problems References

xiii

409 411 424 438 451 456 457 458

463 466 472 477 492 499 512 514 515 516

A. Lie algebras and Lie groups: basic notions

519

Index

523

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1 Hamiltonian Picture of Light Optics. First-Order Ray Optics

1.1

Introduction

The phase space representation of light optics naturally arises from the Hamiltonian formulation of geometrical optics. Geometrical optics gives a, simple model for light behaviour, in which the wave character of light is ignored. It is valid whenever light waves propagate through or around objects which are very large compa, red to the wavelength of the light and when we do not examine too closely what is happening in the proximity of shadows or foci. Accordingly, it does not account for diffraction, interference or polarization effects. Geometrical optics employs the concept of light ray [1], which we may give the naive view as a,n infinitesimally thin beam of light. Several formal definitions of light ray have been elaborated within both the corpuscular and wave theory to accommodate geometrical abstraction and physical observability. All definitions work well in certain situations, but in others are confronted with intrinsically physical difficulties. Thus, for instance, the corpuscular view of rays as trajectories of "luminous" corpuscles confronts with the problem that the energy density may become infinite. Likewise the wave-like view of rays as orthogonal trajectories to the phase fronts of the light wave confronts with the difficulty of individualizing a defined wave front in the two-wave overlap distribution. Indeed, the ray must be thought of as a convenient and successful model which supports our perception, and hence facilitates the formal description, of a wide class of light phenomena. Geometrical optics establishes the geometrical rules governing the propagation of light rays through optical systems. The analogy of geometrical optics of light rays to Hamiltonian mechanics of material particles is well established and effectively exploited. The Hamiltonian formalism was originally developed by Hamilton for optics in his 1828 paper Theory of Systems of Rays and in subsequent papers and brief notes, published during the years from 1830 to 1837 [2.1]. In his papers, Hamilton

2

Linear Ray and Wave Optics in Phase Space

formulates the problem of s t u d y i n g the g e o m e t r y of light rays as they pass t h r o u g h optical systems in terms of well-defined relations between the local coordinates of the rays entering and emerging from the system, specified with respect to the optical axis and prot)erly chosen planes across tile axis. He shows that, if the ray coordina.tes a,rc suitably defined, the i n p u t - o u t p u t rcla; tions configure a,s symplectic tra, nsfornmtions, generated by a, function of the ray varia,t)lcs, the charact('~ri,~tic function, whose flm(:tiona.1 form is determined s()lely |)y tlle ()t)ti('al I)rot)erties ()f t,tw, syste~ll. Im.t(;r, tlmlfilt(m realiz(~(1 tha.t the Sa,lll(; nlet, h()(t (:olfl(t t)(~ a,I)t)lie(1 lmt:hange(1 t() nl(;('lm,lfi(:a,1 systems, rct)la(:ing the ot)ti(:al axis t)y the ti~llC a,xis, the light rays 1)y tim t)arti(:h; traj(;(:tories and the ray-(:()()r(linatcs t)y l,ll(; lll(;(:lm,ni(:a,1 I)lm,s(;-sI)a(:(; va,ria,lfics [2.2]o T h e t)hase st)a(:(; rct)r('~s(;ld, a.ti()ll is a, fa.llfilia,r nlct, ll()(t wil, llill tll(; Hallfilt(mia,n fornmlation ()f ('lassi(',a,1 nwclm~fi(:s, wlfi(:ll dcscril)(;s t,tlc (lylm,llfics ()f a. Ill(;(:lm,ni(:a.1 syst,(;nl witll 'll~ (l(~.grt',cs ()f fl'(',c(l()lll ill t(;rlllS ()f '/l~ g('.l~('.ralizc(t ill(l(;I)entttmt ('()()r(tinat(',s (q~, q~, ..., (1,,,) aal(l t,l~c sa~(', ~ f i ) ( ' . r ()f ('aI~()~i(:a,lly ('()~.j~ga,t(', va,rial)lcs (p, ,p~, ...,p,,,) [3]. TI~(: l~(;(:l~a,l~i(:al l)lm,s(; st)a(:c is t,l~(; Cart(;sim~ st)a.(:c of tl~(;se 2,ttt, (:()()l'(li~m.t(,.s. F()r ('Xa,liq)lc., ill(; st, at(; ()f a fl'(',e t)a,rti(:h' at a, (:crta.in t,in~c is r(~.t)r('.s(;nt('.(t in tim. i)r()t)cr 6 D t)hasc st)a,(:(; 1)y a 't'(:pr(:,s('.'nlativ(: point, st)e(:ific(t l)y t,h('. Ca rt('.sia~ ('()()r(li~mt(;s q = (q.,,, %, q:) m~(l tl~(', r(;l(;vant m()~(',l~l,a. p = (p:,,, p,/, p:). ~l'l~(' ~()l,i(n~ ()f tl~(', t)a.rt,i(:l(; i~ r(;al st)a.(:e (:()rl'(;st)()n(is t() a, t,ra,j(~.(:t()ry in 1)lm,se sI)a(:(;. Then, l,l~(~,st, a.t,('. ()f an (ms(',~fl)l('. ()f i(l(;~d,i(:al an(t n()nint, era(:ti~g t)a,rti(:l(',s a,l, a givc~ t,ilil(', (:()rI'(',sI)()ll(ts t() a, s(;l, ()f I)()iI~t,s i~ t,l~(; 6D t)has(' st)a('(:. Tl~(' (l()~min ()('('~q)i(;(l t)y this set ()f t)()i~t,s ~()v(:s t.lm)~gh i)has(; sI)a,(:e a.s th(', I)a.rti(:l(',s ~()v('. ii~ i'(',al sl)a,(:c, tt()w(wer, a,s t,l~(', t()ta,l ~n~fi)cr ()f t)a,rti(:le.s r(',l~m,i~ (:()nsta,nt, s() will the t()tal mmfi)(;r ()f I)has(; sI)a,(:(; I)()ints. Evi(lently a, rca,1 (t(;~sity can t)(; ass()(:ia.t(;(1 with the r(;t)I'(',S(;llta,I,iv(; l)()ilits in phase st)a,(:c, a,n(t (:()rre, sI)()~(tingly a. (tistrit)~d,i()I~ flln(:ti(m ()f (l(;~sity p(q, p, t) (:a.n be (lefin(.~(1 so tha,t p(q, p, t)dV st)e('ifies the m~nfl)cr of rct)r('.S(;ld;ativ(; t)()ints in the element of v()luIne dV in ttw, vicinity ()f the t)()int (q, p). Li()uville's theorem sta,tes the. i~w~,rian('e of the (lensity of ret)resentativ(', t)()i~d,s a,l()~g I,lm trajectory of any t)()int, and a(:('or(tingly ()f th('. volmne of the I)hasc spa,('.e domain, even though its sha.pc may (:l~a,ng(; (:onsi(t(;rably (luring the motion. Likewise the geometri(:-ot)tica,1 t)ha,se st)ace is the 4D C a r t e s i a n space of the ray position and m o m e n t u m coordinates (qz, qy, pz, py). However, tile phase st)aces of classical mechanics and geometrical optics are globally different. The t)a,rticle m o m e n t u m of classical mechanics is not restricted in value, whilst the ray m o m e n t u m of geometrical optics is confined within a circle determined by the local refra,ctive index t h r o u g h the inherent form of the optical Hamiltonian. In the lin(;ar a p p r o x i m a t i o n the ray m o m e n t u m is assumed to range w(;ll below its na,tural limit, which then is ignored. T h u s the geometric-optical phase space of linear optics comes to be similar to the mechanical phase space. Section 1.2 reviews the H a m i l t o n i a n formulation of geometrical optics and

Hamiltonian Picture of Light Optics. First-Order Ray Optics Lagrangian picture

Hamilton's principle

Fermat's principle

Solves Lagrange's equations for the

3

Particle trajectory in real space

Legendre transformation Hamiltonian picture

Solves Hamilton's equations for the

Particle trajectory in phase space

Lagrangian picture

Solves Lagrange's equations for the9 9

Ray trajectory in real space

9

.-

Legendre transformation Hamiltonian picture

Solves Hamilum's equations for th~

Ray trajectory in phase space

F I G U R E 1.1. T h e F e r m a t e x t r e m a l p r i n c i p l e b a s e d f o r m u l a t i o n of g e o m e t r i c a l optics m i r r o r s t h a t of classical m e c h a n i c s , b a s e d on t h e H a m i l t o n m i n i m a l principle.

introduces the related concept of geometric-optical phase space. Section 1.3 emphasizes the symplectic nature of ray propagation, and details the suited mathematical settings (Poisson brackets and Lie operators) to approach the integration of Hamilton's equations for the light ray. In Sect. 1.4 the ray-transfer operator is introduced and the relevant Lie-transformation based formalism is describe(t. Illustrative examples of phase-st)ace transformations are given in Sect. 1.5. Sections 1.6 and 1.7 illustrate the linear approximation to light-ray propagation, which naturally yields the ray-transfer matrix formalism. Finally, Sect. 1.8 clarifies the link between the ray-matrix approach and the cardinal point (and planes) method.

1.2

Hamiltonian picture of light-ray propagation

We will give a brief account of the Hamiltonian formulation of geometrical optics in order to fix the notations we adopt and to trace the conceptual path towards the I)hase space representation and the inherent geometry. Hamiltonian optics develops from Fermat's principle of extremal optical path, which is the optical analog of Hamilton's principle of least action (Fig. 1.1). From Hamilton's principle one can derive both the Lagrangian and Hamiltonian mechanics, related through the Legendre transformation [3]. Likewise from Fermat's principle one can develop the Lagrangian as well as the Hamiltonian formulation of optics [4]. The former yields the equations for the ray variables in real space, while the latter the equations for the ray variables in phase space. We will cursorily illustrate the basic steps leading to both

Linear Ray and Wave Optics in Phase Space

4

x

\

s

r (s) = ( x , y , : )

n (x, y,:) Z

FIGURE 1.2. Ge()Iimtri('al ()I)tics (h;scril)cs t,tle Ine(li~In by tt,e refl'm'tive i,l(lex flu,ction 'n(x, y, z) m~(l t,l,e ligld, rays 1)y l,lie 3-vc(:t()r ()f f,l,,(:ti(),~s r(s) = (:r(.~), y(.s), z(.s)) ()f the arc le~gt|~ .s Ineas~m;(l alo~g l,|~e ray I)al,]~. t)i(:t, lu'es, a,(t(h'essilig t,li(' I'(~a,(h;r t() [4] fl)r a lii()I'e (t(~taih;(t t,r(;atili(',lit. As a , m t l m d s(:(~iiari() f()r ilitr()(lli(:illg F(;nlmt's t)rill(:it)h; [4, 5], w(; (:()llsi(ter a,n inh()nl()g(m(;()lts 111(;(lilnll, ()(:(:lH)yillg a, (:(',rl,a,ill regi()n ill 1,11(; 3 D Sl)a(:e , w h e r e w(; sllt)l)()s(; a (~,a.rl,(;sia,ll sysI,(;lli ()f (:()()r(lilml,(;s (x, y, z) |)(; a ssiglle(l. 'I'11(; ()t)ti(:a,1 i)i'()t)ert, i(;s ()f tim lll(',tilull a,r(; tyI)i('a,lly (t(;s(:i'il)e(1 |)y t,ll(' r(',fl'a,(:tiv(; in(tex 'n,(:r, y, z), giv('~Ii as a. s(~ala,r fllll(:t,i()II ()f sire(:(; (''). A light ray is I)r()t)aga,ting in l,h(; in(;(lilnn a,h)ng s()~ll(', I,raj(;(:t()ry. R,(;gar(h',l a,s a, line in l,ll(; 3 D st)a,(:(;, the ray (:a,,, a,(:(:oi'(tiIlgly t)e (h;s('rit)(;(1 l)y t,tl(; l)()sit, i()11 ve(;t~()r r ( s ) ---- (.r(.s),y(,s),z(s)) for t)oint~s ()n 1,11(; ray, w i t h th(; (:()()r(lina,t,;s t)eing flm(:t,i()ns ()f l,ll(; a,I'(: length ,ll(;a,slu'e(t al()llg 1,11(; ray t)a,tll witll rest)e(:t, t() a (:hosen t)()ild, (Fig. 1.2). Fernm, t,'s t)I'ill(:it)h; (:()1111)ill(;s 1,11(;ge()nlet~ri(:a,1 an(1 t)hysi(:al a,sl)(;(:ts ()f the ray t)rot)a, gatioll t,tlr()llgh the, ('()n(:(',i)t, ()f optical path. W e recall t h a t give,l tw()l)()ild, s P, a,n(t P~ an(t a (:llrve C ('(),ln(;(:tillg t h e m , the geometrical t)a,t,tl lengttl 12 (C) from P~ t,() P2 along C is (letille(1 as the l e n g t h of C a,n(t hen(:(; is f()nna,lly given by t h e line integrM P2

11 (C) --

d,s,

(1.2.1)

1

t)erformed a l o n g C f r o m P1 tO P2; s d e n o t e s tile arc l e n g t h m e a s u r e d along t h e p a t h a n d ds = v / a x 2+ay2+dz 2 is the infinitesimal arc length. C o r r e s p o n d i n g l y , t h e optical p a t h l e n g t h s (C) a l o n g t h e ray t r a j e c t o r y C We will consider only linear spatially nondispersive isotropic media, whose refractive index fllnction is accordingly dependent on position and independent of direction. Hence we will distinguish only between homogeneous and inhomogeneous media, according to whether the scalar index function is uniform or changes from point to point within the medium. a

Hamiltonian Picture of Light Optics. First-Order Ray Optics

5

from P1 to P2 is defined as the line integral along C of the refractive index"

s (C) -

n(x, y, z)ds.

(1.2.2)

1

If the medium is homogeneous, so that n(x, y, z) - no, the optical path length is the geometrical path length multiplied by the refractive index: s - n 0 s Slightly correcting tile originM formulation, as given by Fermat in his (Evreus (1891) [5.1]" "Je reconnois premi~rement...la vdritd de ce principe, que la nature agit toujours par les voies les plus courtes," Fermat's principle states that, among all the possible paths C connecting the points P1 and P2, the light ray would follow the path C whose optical path length /2 (C) is an extremum. Therefore it may be a minimum, which is the most frequent case, a maximum, or it may be stationary with respect to the optical path lengths of other paths closely adjacent to C. In mathematical terms, the actual ray path C is identified as the extremal of the variational problem ~;z; (C) - ~

n(x, y, z ) d s - O,

(1.2.3)

1

where the ~ variation is intended for small deviations with respect to C of the integration path between the two fixed endpoints P1 and P2. The formal correspondence of Fermat's principle (1.2.3) with Hamilton's principle becomes apparent, once changing the integration variable from s to one of the Cartesian coor(tinates, say z. Then, evi(lencillg the (tifferential element dz in the line element ds, we may cast Fermat's principle (1.2.3) exactly in the same form as Hamilton's principle, i.e.,

(~

{~(X, y, Z)v/l+x'2+y'2}dz=O,

(1.2.4)

1

where the integrand function is naturally identified as the optical Lagrangian

L ( x , y , x ' y' z ) -

n(x y z)v/l+x'2+y '2

(1.2.5)

with

x~

dx dz '

y~

dy d--~"

(1.2.6)

The Lagrangian system of equations plainly follow as

d (OL_=__:.) OL dz

~"

d OL dz (~y--u

O~

)

OL

Oy

O, (1.2.7)

o,

6

Linear Ray and Wave Optics in Phase Space

and, on account of (1.2.5) and (1.2.6), can be rearranged into a, single vectorial differential equation of second order, known as the ray equation, d (ndr ds (-~s) - Vn,

(1.2.8)

relating the ray tra,jcctory t() the ot)tica,1 t)rot)erties ()f the ~ne(tilml [6]. Indeed, _d~ is the tang(ult t(~ the ra,y t)a.tl~, a,n(1 V n - (~~ o,,,,o:,j o,,,,o~) h~t(;restingly, Eq. (1.2.8) shows a noti(:oat)l('~ resellfl)lall(:e t() the e(tlm.ti(m ()f (:lassi(:al rclativisti(', nle(:ha,Ili(:s F - ~dt ~ tlmt I'llh',s the (lynanfi(:s ()f a I)(fillt-t)a,rti('le, th(; f()rc(; F (tetenllilfi~lg the rate of va,riati()ll i1~ t i ~ e ()f the t)a.rticlc lll()llle, lltlllll p. W i t h time ret)la.(',e(t l)y t,l~e ar(: ltu~gtl~ s a.~(t the ~tm(:lm,ifi(:al t)()to, ntia,1 })y tlw, ro,fra(:tive index '~,, tl~e gra(lie~t V'n ~imy l)c ild,(;rt)r(;t,e(1 as t,l~(; "f()r(:e", a,('ti~g ahmg the gra.(ti(nd, ()f the t)(~t(ud, ial 'n(ir, y, z), a,~(t ' n dr ~ a,s the l~(n~O,l~t~ll~ (ff tim tm,I'ti(:h'Y, whose rate ()f va,ria,ti(~l 1)ci~lg (teter~lfi~m(t t)y tlie fi~r(:c V~, [7]. Tlie t,ralisiti(ni if(nlx tlie Im,gra,~lgia,ii t() the IIa,xiiilt(niia,li l)i(:tt~re is then t)urmm(t ~n(t(;r Legend(Ire tra,~sf()r~m,ti()n, whM~ takes .r' a,~l(t y' t() the ray 'm,omr p:~: m~(l p:~ t,lir()~gti d,N

~

OL P:"

~

"

.

OL

0.," '

P:'/

(1.2.9)

Oy"

an(1 a,('(:or(lingly l,tl('~ Im,grmlgiml fllll('ti()xl t() the (q)ti(:a,1 tIa,lllill,()Iliml a,s ~

9

(1.2.10)

By (1.2.5) the gelleraliz(;(l nlOIllcnta p:,. and p , ext)lit:itly write a.s (til;

.r/.

PX --II(;U, ill, Z) x/l_F:~2+y/2 y' p:,j - '.(:r, y, z) v h + x ' ~ + . r

~

--

I t - -

---

l t , ~

its dy ds'

(1.2.11)

in accordance with tile nl(;(:hani(:al analogy suggeste(t above,. As we see, px and py equal the local value ()f the refractive index times the direction cosines (d.~ --d;s, ~ds) of the light ray relative to x and y axis, respectively. Hence, they are also termed optical direction cosines of the ray. We express the function (1.2.10) in terms of its proper variables to obtain the Hamiltonian of the ray in the form y,

-

-

y,

-

-

(1.2.12)

which is just the negative of the optical direction cosine of the ray along the z direction.

Hamiltonian Picture of Light Optics. First-OrderRay Optics

7

Finally, in analogy with mechanics we may write Hamilton's equations for the light ray as dqx OH dqv OH dz Opx ' dz Opy _

~

_

.

_

_

dpx

OH

dpy

OH

dz

Oqx '

dz

Oqy '

(1.2.13)

thus completing tile Hamiltonian formulation of geometrical optics. We changed x --+ qx and y ~ qy for purpose of future use. Evidently, within the mechanical analogy the light propagation can be assimilated to a dynamical problem with two degrees of freedom, the length coordinate z being given the distinguished role as the independent (evolution) variable. We emphasize that the z direction can be arbitrarily fixed, and hence conveniently chosen to coincide with some privileged direction of the system. Ordinary optical systems are commonly designed to have a plane of symmetry, where an ideal ray can be identified, which represents tile path of a ray through tile system corresponding to adequately assigned input conditions. This ideal ray is understood as the optical axiso We restrict our attention to centered optical systems, where all of the optical components are aligned with their optical axes lying on the same stra,ight line, which then may be designa,ted as the optical axis for the overall system; the z axis may be chosen a,s this common axis. The optica,1 axis of a system of coaxial lenses, for instance, is just tile COllltilOil axis of tile lenses and can conveniently be taken as z axis. Throughout the book we refer to the optical axis as the reference z axis. Equations (1.2.13) provide tile formal answer to the basic problem of geolnetrical optics of determining the final variables of the ray after passing through an optical system, once assigned the ray variables before propagation and specified the optical properties of the system. In fact, given the va,lues of the position and direction coordinates of the ray at some zi, i.e., the initial values for the equations of motion (1.2.13), it is possible in principle to solve (1.2.13) for the ray position and direction coordinates at any other z. Following again the suggestion of Hamiltonian mechanics, we can construct the geometrical-optical phase space as the Cartesian space of the four rayvariables (qx,qy,px,py). Notably, according to (1.2.11), the refractive index n represents the normalization factor to turn the geometrical variables dqx/ds and dqv/ds into the phase-space variables px and pv. The corresponding step in mechanics is to replace velocity with momentum, the mass being the proper normalization factor. In the geometrical optical phase space the light ray is individualized by the trajectory of the point whose z-evolving coordinates (q~, qy,Pz,Pv) from the specified initial values are determined by the equations of motion. A collection of rays distributed over a range of possible positions and directions

8

Linear Ray and Wave Optics in Phase Space

fills a, certain region in phase space, with which we can associate a density d i s t r i b u t i o n function p(q.~,qy,px,py, z). T h e change in the ray bundle as it p r o p a g a t e s t h r o u g h optical m e d i a reflects into the m o t i o n of the representative points as t h e y move t h r o u g h phase space in a,ccord with the equations of m o t i o n (1.2.13). While the exact phase-space m o t i o n of each representative t)oint (i.e., ca,oh ray in the t)lm(th~) is uni(tucly dctcrmin(',t by the initia,1 con(tit, i()llS, it, is r a t h e r illlt)i'a,(:t,i(:a,|)le to (:a,l(:lfla,t,e a,n exa,(:t s()lllt, i()~l f()r t,tle whole 1)un(llc of rays. It is thcref()rc convenient to 1)rovide a, sta,tisti(:a,1 des(',rit)tion of the bchavioln" of tllc a.sscllfl)ly of ra,ys, rt'ga,I'dCd a,s a,ll (uls(Ulfl)le ()f i(lentica,1 syst, cnls (tiffcI'i~lg ()ver a, rmlge ()f initial (:()II(tit,i()IlS, anti a(:(:()r(tingly t() follow the cvolld, ion ()f t,h(~ (tensity (listril)llti(m flm(:ti()n p. (~c(mlctric ()I)tics is t)ri~lmrily ('o~lcer~l(;(l with the t)ehavi(nlr ()f a, sillglc ray when it, t)a,sscs ttlr()llgtl ot)ti(:a,l nmdia,. We will firstly ta,t'kle tim g(ulera,l t)rol)lcm ()f the cvolld, i()ll lln(tcr the Itallfiltoniml (1.2.12), intro(tlu:illg tlle 1)asi(: tools ()f tim Ha,miltoll-Lie f()nlm.lislll. Tlmll we will f()t:l~s r att(ud,i()~ ()~ li~ma,r ray-ot)tics a.nd hen(:e on the rela,tcd ray-~na, trix forma,lisn~. We will ilh~stratc the a,|)ility ()f the ~m,trix n~ct,l~()(t a,s a, trm:ii~g t()()l ft)r |)()tl~ a, si~gl(' I)a,raxia.1 ray a,n(t a,n (u~stu~fl)le ()f t)a,raxia,1 rays, tln~s tm.ralleling tlm sil~glt~Im,rtit'le an(t l)arti(:lt~-(;nstu~flfle l)i(:t~res ()f (:la.ssit:a.1 sta,tistit:al met:tm,nit:s. I~ later (~lmt)ters, we will illustrat(~ t,l,e s~fl)sta,**tia]ly (liffeI'(u,t t()()ls of linear wavc-()t)ti(:s, (lcalil,g with light (tistri|)~,ti()**s i**st('a(1 ()f |)~m(ll(~s ()f rays. Th(~rc we will i**tr()(l~,(:(~ tim 'wave-optical pha.s'(', ,s'pa('(:, wl~()s(', <x)()r(tina.t(~s arc (:~lst()nm.rily set a,s I,l~(; st)a,tial (:()()r(tina,tc a,n(t the Sl)a,tia,1 frc(t~u~**(:y. T h e g('.onmtl'i(:- a.n(1 wavc-()l)ti(:a,1 t)hasc st)a,ccs are ba,si(:a, lly (liffcr(ud,, si**(:c a, "t)()i,d," in t)ha,sc sI)a,'e, i(l(ud, itic(t |)y a, (t('finit, e set, of all t)lmse-st)a(:(', (:()()I'(linat(~.s, has no t)t,ysi(:al **ma,**i**gi** wave()I)tics, due to tim ~n~(:(~rta,i,d,y r(fa,ti()n ()f F()~,rier analysis. Tim wav(~ ()t)tical phase space is in a, sense " g r a n u l a r " , there being a m i n i m u m t)()ssi})lc t)hase st)ace volume asso(:ia,t(xt with a, s()~,r('c ()r a,n ()t)tica,1 system in ('()nscqueIme of the wave t)r()t)erties of light. In this Cha,t)ter we (tea,1 with tim ge(),netrica,1optical phase spa,ce. Finally we evidence a basic feature of the optical phase space, whict, globally differentiates from the mechanical pha, se space. In fact, the functional form (1.2.12) of the ot)tical H a m i l t o n i a n liinits the optical m o m e n t a accor(ting to p~ + P~ _< n ~ . Thus, the r a y - m o m e n t u m coordinates px and py arc confined within a circle, whereas the ray-position coordinates q, and % are b o t h allowed to range over all the real line. As a consequence, the geometrical-optical phase space configures as the product of a circle for the R2-plane: C~ x R ~, in contrast with the mechanical phase space being the well known ~ 4 __ i~2 X I~ 2. As we will see, first-order optics ignores the above limit on the ray inomenta; it assumes t h a t b o t h the ray coordinates and m o m e n t a r e m a i n small t h r o u g h the p r o p a g a t i o n , so that p , and p~ are well inside their natural range, whose limits can therefore be ignored.

Hamiltonian Picture of Light Optics. First-Order Ray Optics

9

o'! ~-

!

~.~ o~*~

9-"" gli i! . , . " ' 9 Input plane i .--"" Ty; !! s 0~

.s

.s"

.~"

[Io

i. I

~...-.~ Output planei -.

Yo

i

s . S"

;

,'"'"

i

i

z.

i

i

i

.1o

i

(qx~176

i i

,.'" o,~s~

I

i

9 ,,. i . . , , ' ' ~

i

~"

s" s"

i.' .." ."'~qx, ' q Yt )

l

i!

~ .S~

F I G U R E 1.3. An o p t i c a l s y s t e m a n d t h e relevant i n p u t a n d o u t p u t planes, to which t h e c o o r d i n a t e s of the i n c o m i n g a n d o u t g o i n g rays are referred.

1.3

Hamiltonian picture of light-ray propagation" formal

settings

We consider a light ray propagating along an optica.1 system, which we suppose to be completely characterized by the refractive index function, given at each point before and after the system and in the system itself. The optical z-axis is commonly oriented as the direction of light propaga, tion, assumed to occur from left to right. The light rays therefore enter the system at the left and emerge from it at the right. It is natural to describe the evolution of the ray along the axis by determining a,t any a,xia,1 position the intersection point of the ray with a plane orthogonal to the axis, a,nd the optical direction cosines of the ray at that point. Thus, we fix two planes, IIi and IIo, across the optical axis, located at positions zi and Zo, respectively at the left and the right of the system (Fig. 1.3). We will refer to Hi as the in p u t plane and to IIo as the output plane. Also, we will equip each plane with a coordinate system. In the following, unless otherwise specified, rectangular coordinate systems will be taken in both the reference planes. The orientation and the position of the axes can be chosen arbitrarily; it is a bare matter of convenience to choose tile origins lying on tile optical z-axis and the coordinate axes parallel, and equally scaled, from one to the other plane. The incoming ray is unambiguously identified by the coordinates (q,~, qw) of the intersection point with the input plane IIi, and by the optical direction cosines (Px~, Py~) at that point. Explicitly, we can write Pxi -- ni sin ~x~ , py~ -- ni sin ~y~,

(1.3.1)

Linear Ray and Wave Optics in Phase 5pace

10

qx

q"l

ray s e c t i o n

"~ i

"'"'""

.

(a)

sect i o n

~..,o

..".... ,..".....

"'"'"

(b)

F I G U R E 1.4. T h e aIlglcs (a) r~. al~(l (b) rh,, clltc.rillg tile (lcfillitioxl ()f tile ()ptical l n o m e n t a p,. an(l p.~,, a r c t h e ('ox**plc**m,d,s to 2 of tim m,glcs ()f tim r a y to tim ;r a,,(l !/;txis, r e s p e c t i v e l y .

whcr(, 'n,~ ~_ 'n,(q.,:~, q:,~,, z i) is tl~(: l()(:al va,l~t(; ()f tl~(; rcfl'a(:tiv(; i~r at the intcrsc(:tion I)oint m~(I ((Lr,, (~'U,) axe l,l~(; a.~gh;s, tl~(; i~(:()~i~g ray (lcfil~(;s with the assign(;(l t)la,n(;s yi-z a l~(l .ri-z (Fig. 1.4). Sinfilarly, th(; ()~tl,g()i~g ray is st)c(:ificd |)y the coor(li~m,tt;s (q.r,,, q.q,,) of l,|~(; i~d,crsccl, i{)~ with I,l~(; t)~l,l)~t, t)la~(; IIo anti l)y I,l~(; ()I)l,i(:al (lirt;('l,i{)~ ('.{)si~cs (p.,., p~/,,) a,t l,l~is I)lmm: P.r,, - - 1}o S i l l ~kr,,

,

py,,

- - lt o S i l l ( ~ y , ,

(1.3.2)

whcl'C '~.o ~ 'n(q,:,,, qv,,, z,,) a.~r (,~:,,,,, ~:,/,) arc tl~(; a.I~gl(;s ()f tll(: ra,y I'clativc to the t)lancs yo-z a ll
p,,o = p.,:~, (q,,,~, qy,, p,:,, p:,j,; z o ) , Puo = p,.,/. (q.,.~ , qv~ , P:,:.~ , Pv.~ ; z , ) ,

which relate in a (tcsira,|)lc cxt)li(:it form tile t)osition and dire(:ti(m (:()ordinates of the incoming ray at zi to those of tile correst)onding outgoing ray a,t Zo. Tile i n p u t - o u t p u t relations (1.3.3) describe a transformation in the opti(:a,1 phase-spa,ce of tile representa,tive point Pi --= (q:r,, qy,, Pz~, Py,~) of the input ray to tlle representative point Po ---- (qzo, qyo, PZo, Pyo) of the o u t p u t ray. Evidently, the t r a n s f o r m a t i o n occurs t h r o u g h a trajectory from Pi to Po, ruled by the equations of motion (1.2.13) with the inherent Hamiltonian (1.2.12). The transformation ~ > Po is symplectic.

Hamiltonian Picture of Light Optics. First-Order Ray Optics

1.3.1

11

Ray propagation as a symplectic transformation

We clarify the formal a,nd physical significance of the ray p r o p a g a t i o n as a symplectic t r a n s f o r m a t i o n [3, 9]. To this end we firstly note that, like the mechanical phase space, the optical phase space naturally configures as a 4D linear vector space, the 4-vectors u being formed by the z-depending ray-coordinates (qx, qy,Px,Py). Similarly it can conveniently be equipped with the euclidean metric defined as usual by the scalar product of vectors" 4

ujvj - u T - v ,

(u, v) - ~

(1.3.4)

j=l

where u n- is the transpose vector and, for notational convenience, the vector c o m p o n e n t s are indexed by the integers from 1 to 4. Conforming with the c o m m o n practice, by "vectors" we mean column vectors; then, row vectors are identified by transposition, i.e., u - (qx, %,p~,py)-7 _ (Ul, u2, u3, u4) r . Exploiting the vector notation, we can write H a m i l t o n ' s equations (1.2.13) in one form duj 4 OH dz = ~ ,ljl Oul' j - 1, .., 4, (1.3.5) l=1

with the ai(t as well of tile suitably (tefined coefficients Jjl, which a,re allowed to be only 1, - 1 or 0 according to their own indices"

,]jl

-

~il, j + 2 ,

j -

,]jl

1, 2;

-

-~l,j-2,

j -

3, 4.

(1.3.6)

Evidently, Jjl's ca,n conveniently be a,rra,nged into the 4 x 4 a,n t i s y m m e t r i c m a t r i x J as

J-

(0I) -I

0

'

(1.3.7)

where I denotes the 2 x 2 unit m a t r i x as 0 tile 2 x 2 null m a t r i x (b) . The m a t r i x J is known a,s the symplcctic unit matrix. It has the nice properties j-1 _ jr_

_j,

j2 _ - I ,

detJ -1,

(1.3.8)

j T denoting the transpose matrix. We suppose now to take the vector u to the vector v - (Vl, v2, Va, v4) under the t r a n s f o r m a t i o n Vj -- Vj(Ul,...

, U4) ,

j

-

1,

.., 4 ,

(1.3.9)

b I t is a c o m m o n c o n v e n t i o n t o r e p r e s e n t 2 m • 2 m m a t r i c e s as p a r t i t i o n e d i n t o f o u r m • m s u b m a t r i c e s [8].

12

Linear Ray and Wave Optics in Phase Space

identified by the associated J a c o b i a n m a t r i x S, whose entries Sii = O v i / O u j (tet)en(t in general on uj, j = 1, .., 4. It is ca.sy to prove t h a t v obeys equa.tions dvj = dz

S:)~,,I~,~;Sha O H

,

j - 1, .., 4,

(1.3.10)

Ovh

sig~ls ()v(w l"(:I)(;a,l,(;(1 i~l(ti(:(:s t)(:i~g (mfittc(t. W(; r(;(p~ir(; that th(: tra~lsf()rllmti()~l (1.3.9) h;av(;s tll(: f()n~l ()f Hanfill,()n's (;(t~ati()ns inva,rimd,, i.(:., tlmt v ()t)(:ys th(; san~(: (:(t~m,ti()IlS (1.3.5) a,s u, na,m(;ly tll( ~, s ~ n ~ m t i ( ) ~

d~,j _ .lj O H dz i)~,t '

j - 1,.., 4.

(1.3.11)

Witll tlfis r(;(t~u:st al)t)li(;(1 t()(1.3.1()), w(: (;~(1 ~1t) witll tll(: ~mtrix r(:la,ti(nl: S J S m -- J.

(1.3.12)

It r(~t)r(~s(~llts tll(' ('()ll(liti(nl f()r S t.() 1)(~ a syllll)l('('ti(: llmtrix a.t rely l)()illt in I)llas(: sI)a(:(; [9]. As ll()t(:(l, ill fa(:t, S lllay (l(:t)(;ll(l ill g(',21(;ral ()zi 1,11(; v(:(:t()r u. Ill ('()ld,ra,s|,, tll(: I)l'()(lll('t S J S T r(:slllts lilt()tll(; lllatI'iX J, ill(l(:l)(:ll(l(:~d, ()f U. |';vi(le.~d,ly tll(: llllit llmtrix I m~(l ill(: llmtrix J a r(: sylllt)l(:(:ti(' as w(:ll. W(: r(:(:a,ll tlla.t ill g(;l~(:ral ll(nlsi~lgllla.r s(t~m,I'(: ~m.tri(:(:s fl)r~ a. gr()~ 1) ~nl(l(;r t|l(: r()w-(:()l~lnlll ~mtrix ~11~ltit)li(:a, ti()~l, tl~(: (:xist(:~(:(: ()f (,11(: i(l(:~til,y a n(l the ~nli(t~(: i~v(;rs(: as w(:ll as tl~(: ass()('iativity 1)(:i~g a, I)lah~ (:(n~s(:(l~uul('(: ()f th(; ~m,trix ~n~ltit)li(:al,i()~ l)r()t)(:rti(:s. I~ t)a,rti(:~la.r, l,l~(: ~la.tri(:(:s ()l)(;yi~g (1.3.12) (1() f()rnl a. gr()~q)(1~(; t() tl~(: (:l()m~r('. I)r()t)crty ()f tl~(', sy~q)le,(:ti(: (:(n~(li|,i()ll, whi(:h ix I)rcs(;rv(:(l |)y ~ml,rix ~n~ltiI)li('al,i()~; i~ flu:t, if S l a~(t S~ ()|)(;y (1.3.12), s() will th(; I)r()(t~mt S~ S~. W(: i~vit(: tll(: rca,(lcr t() v(:rify that, if S is sy~q)h;('tic, the ii~v(:rsc S - 1 exists a,n(t is syn~t)h;(:tic a,s well a,s the tra,nst)()s(: S T. Ttl(: group of the 4 x 4 sy~I)l(:(:ti(: r(',al ~mt, i'i('(;s ix (t(;~()t(;(t a.s S p ( 4 , R). Since (let J = 1 m~(t (tot S = (t('.t S T, t)y (1.3.12) it follows t h a t (let S = +1.

(1.3.13)

It can be shown that actually foI" a, symplectic m a t r i x it is always (let S = +1 [.().2]. Remarkably, in tile 2 x 2 case tile necessary and sufficient condition for a m a t r i x to be symplcctic is t h a t it has d e t e r m i n a n t +1. Consequent to the symplectic condition (1.3.12), we also find t h a t the 16 entries of the m a t r i x S are not all independent; only 10 can be arbitrarily chosen, the others being fixed by (1.3.12). This reflects the general p r o p e r t y of the 2m x 2 m symplectic matrices, for which only m ( 2 m + 1) of the 4 m 2 elements can arbitrarily be assigned ( P r o b l e m 1).

Hamiltonian Picture of Light Optics. First-Order Ray Optics

13

Interestingly the Jacobian matrix S relates the infinitesimal vectors dv and du (see also Problem 2). Indeed, being 4

dvi - ~

OVi

~

4

duj - E Sij duj,

j=l

(1.3.14)

j=l

we can concisely write dv - S du.

(1.3.15)

It is therefore a direct consequence of the symplectic condition (1.3.12) that the quantity ( d U l , J d u 2), for any choice of du 1 and du2, remains unchanged under the symplectic transformation (1.3.15), which takes u to v and accordingly dul to d v 1 and du 2 to dv 2. Explicitly, we can easily prove that

(1.3.16)

( d v 1 , Jdv2) = ( d U l , Jdu2).

In particular, if the new and old vectors, v and u, arc linked by a linear transformation as vi - ~ = 1 aijuj, i - 1, .., 4, the entries of S are just the uindependent coefficients of the transformation, i.e., S/j = aij. Hence, relations (1.3.15) and (1.3.16) hold true for the finite vectors as well. Explicitly, if

v=Su,

(1.3.17)

with S symplcctic and u-independent, then (V 1 , J v 2 ) = ( U l ,

(1.3.18)

Ju2).

It is evident that if the transformation which takes the vector u to v is z-dependent: Vj - - V j ( U l , ..., U4; Z), j = 1, .., 4, (1.3.19) the simple derivation we have sketched for the symplectic condition (1.3.12) no longer holds. But, indeed, this is just the case of our main interest. In fact, as wc said earlier, the propagation of light rays through optical systems is formally described by the solution to Hamilton's equations (1.3.5), which, for assigned input conditions, yield the ray variables, and so the relevant phase space vector u(z), at each z along the optical axis through the axial interval we are interested in, say from zi to Zo. Therefore, what we must verify is that the ray propagation from zi to Zo, we may picture as a transformation of the initial ray-vector u(zi) to the final ray-vector U(Zo):

u(zg)

, U(Zo) = u(z~ + Az),

Az =

Zo -

ruled by Hamilton's equations (1.3.5), be symplectic.

z~,

(1.3.20)

14

Linear Ray and Wave Optics in Phase 5pace

To prove this, it is enough to demonstrate that any infinitesimal transformation ahmg the path (1.3.20) is symplectic, i.e., the relevant Ja.cobian matrix obeys relation (1.3.12). The continuous evolution of the ray vector u(z) along the finite space interval Az can be viewed as a succession of infinitesima,1 symt)lecti(: transformations, which, in virtue of the pro(hlct t)reserva.tion t)r()t)crty menti(nic(t a,t)ovc, result into a. single finite symt)lc(:ti(: transformatioil. Theli, let ~lS (:o~si(l('x a,n iI~finit(;si~m,1 tra,nsf()r~m,ti()~ a,long the (;v()h~tion t)a,tl~ (1.3.20) l,aki~g th(; i~fitial ray v(;(:t()r u(z,~) at zs t()tl~(; ray vc(:t()r u(zi + d z ) at the illfinit(;si~mlly (:l()sc axia,1 t)()siti(m zi + dz. F()r (:()nv(;nien(:(;, w(: set w-

u( i) v-u(zi+dz)-w+

~du I:, dz

(1.3.21)

wh(;rc, ()f (:()In's(;, ()nly tirst-()r(lcr tcrnls in the (;l('~nlclltal (list)la.(:('.nl(;nt dz have })(;(;n r('.taill('~(l. In a(:(:()r(l witll Ha,lllilt(m's (;(tlmti()llS (1.3.5), ()ll('~ ll~LS duj dz

--,ljl

0ti

(1.3.22) Zi

wlficll, (hi(:(; ills('rt,(',(t illt,() tll(' s(;(:()~l(l ()f (1.3.2(}), yi('l(ls OH

dz.

(1.3.23)

~. "7

Tll(,~ ,Ja,(:()])ia,]l llmtrix (:(rams t() t)(;

Ot,j Ou~j 0 2H Sjl ..... O'u'! ..... O'u'! _of ,]jl,: OulOUl,: where the se(:ond or(tot derivatives of tile Hamiltonian flmction, H[~. -

(1.3.24)

o,,,ouk z~

which fornl tim s()-(:a,lh;(1 Hcssia,n matrix H " - (H[~)~k, a,rc cva,lua,tcd a,t u(zi) and zi. Dlu; to the asslHncd interchangeability of the derivative order with respect to the vector components in the Hessiai~ entries: Hi'~ - H ~ , relation (1.3.24) (:an finally be cast in the matrix form S

-

I + JH"(w,zi)dz.

(1.3.25)

In order to assess the symplecticity of S we must take the product S J S qwhich, consistently to first order, is easily proved to be: S J S T - (I + J H " ( w , z i ) d z ) J ( I H"(w,zi)Jdz) = J + J H " ( w , z i ) J d z - J H " ( w , z i ) J d z - J.

(1.3.26)

Hamiltonian Picture of Light Optics. First-Order Ray Optics

15

The symplecticity of the infinitesimal transformation mapping the initial ray vector u(z~) to the infinitesima,lly displaced vector u(z~ + dz) along the finite evolution path (1.3.20) governed by Hamilton's equation (1.3.5) is then demonstrated. Accordingly, we may think of the ray propagation as a continuous sequence of symplectic transformations, under which the ray vectors in phase space arc taken one to the other, the axial coordinate z being the stepping parameter. We may say that optical systems operate symplectic transformations on the ray variables q and p [10]. It is worth emphasizing that the foregoing discussion, we have developed with specific reference to the light ray propagation, frames within the general issue of Hamiltonian mechanics [3], stating that any canonical transformation (i.e., such to map phase-space vectors obeying equations of motion of Hamiltonian form into phase-space vectors obeying as well equations of motion of Hamiltonian form) is symplcctic, whether it depends or not on the evolution variable inherent in the dynamical problem under examination. Readers interested in a rigorous treatment of symplectic groups are directed to [9]. Additional comments are given in Chapters 2 and 3. The Appendix provides an elementary exposition of the basic notions pertaining to Lie algebras and Lie groups.

1.3.2

Poisson brackets and Lie operators

Central to our approach to the geometrical optics problem of deterinining the transformation (1.3.3) of the light-ray coordinates, generated by tile Hamiltonian (1.2.10) through Eqs. (1.2.13), are the strictly related concepts of Poisson bracket and Lie operator. Poisson brackets and Lie operators represent the formal tools to introduce the Hamiltonian equations of motion into the appropriate Lie algebraic ambience, where their intrinsic symmetry can more clearly be evidenced and effectively exploited. The definitions and main properties of both the Poisson brackets and Lie operators are recalled below. The discussion is developed at a general level. Thus, (q, p) denotes a set of canonically conjugate variables for some HaIniltonian system and z the indepeildent variable, independently of their specific meaning in ray optics.

Poisson brackets Let f and 9 be any two continuously differentiable functions of the canonical variables (q,p). The Poisson bracket of f and g, denoted by {f, g}, is still function of (q,p), formed according to [3, 10.2, 11]

{f' g} =~-~ OL

Of Og Oq~ Op~

Of Og "~ Op,~ Oq~ J '

(1.3.27)

16

Linear Ray and Wave Optics in Phase 5pace

where tile sum runs over all tile pairs of conjugate variables (q~,p~) for the problem at hand. Since both the functions f and g may depend on the independent variable z, the Poisson bracket {f, g} may depend on z as well. Clearly the Poisson bra('ket ot)era,tion creates a binary relation: (f, 9) {f, g} of every pair of dyim,mi(:al varia.bles to one dynamical variable, identified thro~@l the rule (1.3.27). In parti(:~flar, the relatioi~s

{ q,,, p/~ } = (5~,~;~,

(1.3.28)

(:hara(:terize the (:m~()ni(:a.lly (:(m.j~ga.te wtriat)les (q,~, p,,), as the f()ll()wing { q"' "f } -

i) f

i~'p,~ '

{P"' "f } -

O.f

Oq,, '

(1.3.29)

esta.t)lish tll(; rlll(~s I,() tin'ill l,lle P()iss(m t)ra(:kets ()f aaly tm.ir ()f (:(nljllga.tc varia.1)h's (q,,,p,,) witl~ a fi~t:l,i(n~ .f ()f (q,p). In t)a,rtit:~llar, if .f is ta.ke~ as the tIa,llfilto~ia,~ fl~xmti(m H, tl~c l)(fisso~ t)ra,(:kets (1.3.29) rcI)r()(l~(:e tt~e right lm,~(t si(les of Ha,~ilt,()~'s ('.(l~m,t,i()~s, wl~i('t~ a,(:(:()I'(li~gly write also a.s

dq,,

dz = { q'' ' I I } ,

dp,, dz = { p ' ' ' H } '

(1.3.30)

f~)r ('very t)air ()f (:()Iljllgat(' varial)l(;s (q,,, p,,). Evi(h;ntly the P()iss(m t)ra.(:k(;t ()f .f and th(, v(,(:tor-va.llw(l fll~l(:t,it)n g (.q, (q,p), ..,g .... (q,p))Y is tile ve(:t()r fllll(:(,i()II { f , g } = ({f,.q, }, .., {f,.q,,, })T The Poiss(m |)ra<:k(~t ()t)(~ra,ti(nl llas tlle f(nlr r(;llm,rka.t)l(; l)I'()t)(;rti('.s:

antisymmet'ry: linearity : derivation p'mpcrty: ,lacobi identity:

{.f, .q} - _ {.q, f}

{.f, aft + bh } = a {.f, .q} + b {.f, h } {.f , gh } - {.f , .q } h + .q {.f , h}

(1.3.31)

{f, {.q, h}} -- {{f, 9} ,h,} -5 {fl, {f, h}}

holding f()r a,ny flulctions f, g, h a,IM c()~lq)lex nlmfl)ers a, b. Ttmse properties directly follow from definiti(m (1.3.27), exploiting the linea,rity a,n(t the Leibnitz rule for diffcre~tia,tion. The a,n t i s y m m e t r y implies the identity: {f, f} = 0. Interestingly, the Poisson bracket operation behaves with respect to the ordinary and Poisson product of functions in the similar fashion as the derivative operation, i.e., through a Leibnitz-like rule. Let us consider now the set { f ( q , p ) } of all the continuously differentiable functions of the phase space variables (q,p). It is naturally structured as a linear vector space, since any linear combination of functions in the set yields a function in the set. Further, as remarked before, the Poisson bracket takes any pair of functions in the set to a function in the set. Thereby, observing the properties (1.3.31), it becomes clear t h a t the set { f ( q , p ) } configures as a Lie algebra with the Poisson bracket acting as the Lie product. In particular, as the

Hamiltonian Picture of Light Optics. First-Order Ray Optics

17

reader may easily verify, the quadratic homogeneous polynomials of the type aq 2 +bp 2 +cqp are closed under Poisson brackets and so they span a Lie algebra. The quadratic homogeneous polynomials identify a particularly interesting subset within the overall space of dynamical variables { f ( q , p)}, since they are basic to the representation of linear optical systems in both geometrical and wave optics. In fact, as we will see, in the linear approximation the optical Hamiltonian (1.2.12) turns into a quadratic polynolnial. We also note that the subset of the inhomogeneous linear polynomials of the type aq + bp + c does form a Lie algebra as well; in particular, the Poisson bracket operation images this subset onto that of constant functions. In Chapter 4 we will see that by the "quantization" procedure the Lie algebra of the inhomogeneous linear polynomials maps to the Heisenberg-Weyl algebra, which will appear to be the basic building unit of wave optics, in accordance with the claimed analogy of wave optics to quantum mechanics. Also, the quadratic Hamiltonian functions of ray optics will map to the quadratic Hamiltonian operators of wave optics and correspondingly the Lie group of the symplectic matrices of ray optics will map to the metaplectic group of the Fresnel-integral operators of wave optics. Finally, we establish a further characterizatioi~ of the sytnplectic transformations in terms of the Poisson brackets. We refer to the 4D t)hase st)ace of geometrical optics and gather the ray variables into the 4-vectors u, a,s in w 1.3.1. It is easily seen that the symplectic unit matrix J directly relates to the fundalnental Poisson brackets (1.3.28), which synthesize in one form { ui, uj } = Jij,

(1.3.32)

as well as to the general Poisson bracket (1.3.27), which writes concisely as Of Og {.f , .q} - ~ Jij' OUj 9

(1.3.33)

Composing the d e r i v a t i v e s (Of)i= 1,.. ,4 a n d ( ~ ) .i = 1 ,'" ,4 into the 4-vectors O,~f 01Z i and Ou9, the right-hand side of (1.3.33) can be interpreted as the scalar product of the vectors O~f and JO~g. In symbols we write { f , g} = (O~f , JO~g).

(1.3.34)

Let the phase space vector u transform to v under a symplectic transformation, whose associated Jacobian matrix S obeys thereby the relation (1.3.12). By the chain rule for derivatives, one has that 0~ = (S-r)-10,. Then, regarding f and g as functions of the new variables v, we find (O,f, JOv9) - (Ouf, s - l j ( s T ) - l O u g )

- (Ouf, JOug),

(1.3.35)

which shows that the Poisson bracket {f, g} has not changed under the symplectic transformation u---, v(u). Conversely, if the identity (1.3.35) holds, then the matrix S of the transformation u ~ v(u) is symplcctic.

18

Linear Ray and Wave Optics in Phase Space

C o n s e q u e n t l y the necessary and sufficient condition for a t r a n s f o r m a t i o n to be symplectic is that it preserves the Poisson brackets (1.3.27) of dynamical variables and in p a r t i c u l a r the flm(lamental Poisson brackets (1.3.28). T h e inva,riance of the Poisson brackets u n d e r symplectic t r a n s f o r m a t i o n s conforms with the inva.riance of the skew p r o d u c t a.s sta.ted in (1.3.16).

Lie operators

Lr

f ( q , p , z) 1)e s()liie (:()lit, ilul~)llsly (lifferr flul(:ti~)li ~)f t,ll~; (:~)ll.i~lgate varial)les q,p. We a~s()(:ia,l,r witii f l,hr Li<' Opf','FfL~,07" L/ t)y the lulle [11, 4]

Lf(q, p, z) - {., f } =~-~,

i~q
i)p,, 0(/<,

'

(1.3.36)

(1/

wllere as ill (1.3.27) tlle sluil rlllls ~w~r a,ll tlle I)airs (~f c:~ltjllga, t~', wwia,bles (q<~, p<,) t)ertiuellt re) the 1)l'~l~hull a.t lla,n~l (~'). Thc~u th~ ~.Li~ ~l~era.t~r L / g e u e r -

Lz.q - {.q, f } .

(t.:~.a7)

F(w ll(~ta,ti(~lial (:~ll(:iseliess, ill tll~', f~ll~wilig w~', will ~lliit, t~l sll~w tll~ (tet)en(loll(:e of L/ ~)li its ~wli argliliiclii,s q, p a,ll~l z.

Tl~e t)()w(;rs (~f L/ are ~l)l,a,ilie(l l)y r(;1)(;alx;(lly t)~iss~li l)ra,('kel,ilig, s() ttm,t the j - t h t)ow~r ~f L/uiea.lis .)-f(~l(l I~iss~li |~ra,(:kets witli f a.(',(:~r(lilig to

k}' - Z, L) - {., f }, - { {', I/, I },.. i'j - {{{..{.,.fI..i,.fI,.f}, ...

(1.3.38)

j |,lilies

()f (:ourse, tll~ zero) t)ower yields tlie i(telitity ~)t)era,tor. Clearly, L/ ca,u act also ~m a vector-wdued fiuwtion g - (.ql,.., .q,,,)T thus giving the w;ctor-wd~led fimction L I g - {g, f } - ({.q,, f } ,.., {.q..... f } ) T . T h e t)r(~tmrties ~f the Lie (~t)era, tors tm,ra.llel those of the Poisson t)rackets. In fact, as a c~mscqllence ~f (1.3.31) we fiu~t"

antisymmetry" lincarity" derivation propert;q: bracket property"

k f 9 - - L~ f k / ( a g + bh) - a k / g + b k / h Lf(.qh,) - (Lf.q) It -Jr-.q (Lf]l,) Lf {g, tt} -- {Lfg, h} + {g, Lfh}. /...

(1.3.39)

A

<: In the literature, the Lie operator, variously denoted as" f" [12.1], f [12.5], or {f, o} [11], is commonly defined by the left Poisson bracket {f,-}, whilst the operator (1.3.36) defined by tile right Poisson bracket is reported as vector field [3.2, 10.2]. Evidently tile operators { f , . } and {-, f } differ by a minus sign. We prefer to introduce the Lie operator through the "opposite" definition (1.3.36), in order to avoid the cumbersome presence of the minus sign in the equations, we will encounter in the following.

Hamiltonian Picture of Light Optics. First-Order Ray Optics

19

Like the Poisson bracket, the Lie operator acts on the ordinary and Poisson product of functions in the similar manner as the derivative operator, and hence through a Leibnitz-like rule. We consider now the set of dynamical variables {f(q, p)}, which, as previously remarked, configures as a Lie algebra under the Poisson bracket operation. Evidently, by (1.3.36) we can individualize a correspondence of {f(q, p)} to the set of the associated Lie operators {ki}" A

A

{f(q,p)}

~

{kf}.

(1.3.40)

This set can be structured as a linear space; in fact, exploiting the linearity of the Poisson bracket operation, we define linear combinations of Lie operators by the rule aLl + bkg - kaf+bg -- {', a f + bg}, (1.3.41) A

A

A

to obtain yet a Lie operator. Notably, the operators {kl} span a Lic algebra with respect to the commutation between operators. We recall that the commutator of two operators F and G is defined by the antisyinmetric product A

A

A

A

A

A

IF, G]

-

A

A

FG -

A

GF.

(1.3.42) A

The direct calculation shows that the commutator of two Lie operators, Lf and kg, is equal to the Lie operator associated with the Poisson bracket of the generating functions g and f; namely A

A

A

ILl,

L,]

A

-

- {., {f,g}}

-

A

- k{:,,}

-

k{g,f}.

(1.3.43)

The cominutator of two Lie operators is therefore a Lie operator as well. This is just the basic requisite for the linear space {k/} be a Lie algebra under the commutator product (1.3.42). In addition, relation (1.3.43) states that the correspondence (1.3.40) is an homomorphism [9.6], as the algebra-generating operation between elements of {f(q, p)}, i.e., the Poisson bracket {f, 9}, is imaged to the algebra-generating operation between the corresponding elements of {kf}, i.e., the commutator bracket [kf, kg ]. A

A

A

As an example ' with f = q2 2,g--

s-+

1.4

+

{s,

A

~2 and hence {f,g} -

qp, we find:

W1

Hamilton's equations for the light-ray

Hamilton's equations can be cast in terms of Poisson brackets. In fact, on account of (1.3.30), we may rewrite (1.2.13) in the suitable vector form d u(z)dz

{u(z) H ( q , p , z ) } ' '

u(zi)-

ui

(1.4 1)

Linear Ray and Wave Optics in Phase Space

20

the ray-vector u(z) containing the z-dependent ray variables, whose known values ~t the entrance plane a,t zi are grouped into the vector ui. Then, introducing the Lie opera,tor associated with the optical H a m i l t o n i a n a c c o r d i n g to (1.3.37); namely,

(OH O Op~, Oq,

--

L, (z) - {., H i - ~_~ ( )z - - : s

1,1

OH O ) Oq,, Op,,

'

(1.4.2)

t

l';(Is. (I.,I.I) at(: recast i~t() tl~(: (:()~(:is(: m~(l (:l(:ga.nt f()I'Iii dd- ~ U ( Z ) -

AL . (z) u ( z ) ,

u(zi) - u i

(1 4.3)

wlli(:ll (:()lllt)int;S t)t)t, ll t,llt'~ vt;('t,()r a ll(l Lit'~-t)t)t;ra.t()r ~l()tati()ll. Ft)r xl()t,a,titmal (:(m('ist;Iless, (rely tile t)()ssit)lc (tt;I)(nl(ttnl(:(; ()f k , (hi z is ('xl)li(:itly sll()wn. As a,ll'ea,tty rt;nm,rkt;(t, tll(; st)hltitnl ()f Eft. (1.4.3) is (,ll(', mlswer 1,() tll('~ flm(talllental t)I'()t)lt'~ni ()f gt;()metrit:al ()t)tit:s: 1,()(l(;tel'nline tile illt)ll(,-()ll(,t)ll(, rela,ti(ms f~n" a. giv(;~ ()t)ti(:al (lcvi(:c, wl~i(:l~ all()w ~s t() fin(1 ()~t tl~c lineal vc('t()rs u,, f()r tlmt the ray trm~sfer rela,ti(n~s ui --~ uo have s(m~(; (l(~sire(l 1)r()t)er(,ics. F;(llla.ti()liS ()f tile tyt)e (1.4.3) a,t)l)ear i~ sev(;ra,1 a~(l (litE'ri~g ti(:l(ls ()f l)l~ysi(:s. A la.rge wtriety ()f t)()tl~ a.~a.lyti(: a~(l ~n~(:ri(:a.1 n~el,l~()(ls ()f i~d,(:gra.ti(n~ lm.s |)een (levise(1, whi(:h t)r()vi(l(; exl)li(:it ext)r(:ssi(n~s f()r exa(:t ()r a l)I)r()xi~mt(' a nalyti(" s()h~tions a.~(t (t('~fi~(; gestural I)r()(:e(l~res f()r m~mcri(:al a,lg()ril,ln~s witl~ (tefinite (legree ()f a('(:~wa,(:y [~2, ~:t]. I~(lee(t, tl~('~ i~tegrati(n~ ()f (1.4.3) w(n~l(l (leserve a, l)r()t)er trea,t~(;~d, ()f its ()w~. Ih're w('~ will give a little i~sigld, i~t() tl~('~ l)r()lfle~n, will l)c l)~It ()n tlm Ol)(:ra(,()r lla,tlll'(: ()f (1.4.3). T h e rea
The ray transfer

operator'

The concise form (1.4.3) of the equations of motion resembles that of a homogeneous first-order differential equation, which writes in general as d y ( z ) / d z = a(z)y(z). The solution is well known to be the exponential function y(z) a(z) z p(zi)e" z~ , which simplifies into y(z) - y(zi) e a(z-z~) in the case of a zindependent coefficient: a(z) = a. Clearly the similarity is purely formal, as

Hamiltonian Picture of Light Optics. First-Order Ray Optics

21

the 4-component ray vector u(z) enters (1.4.3) in place of the one function y(z) and correspondingly the z-dependent Lie operator k , (z) in place of the ordinary z-function a(z). As is well known, operators behave in a substantially different way than c-functions, i.e., functions that send scalars to scalars. On approaching the integration of Eq. (1.4.3) we are unavoidably confronted with the operator nature of kz , which iilay also depend on the axial paralneter z. When the Lie operator I_H is z-independent, the similarity with the ordinary differential equation may be brought a step further, since, as we will see, the solution to (1.4.3) takes an exponential-like form. In fact, when the Hamiltonian function H does not depend explicitly on the axial variable z, as in the case of propagation in a homogeneous medium where n(x, y, z) - no, the associated Lie operator I_H is z-independent as well. The sought solution can then be given the transparent exponential form

u(z)-

f

c(z-z~) L- ui.

(1.4.4)

The propagation i of the light ray is seen as the effect of the action of the operator c(Z-Z~)'L,i, which in phase space transfers the point ui to the point u(z), and correspondingly in real space the ray at zi to the ray at z. Since the exponential function is defined for operators exactly like for scalars, the operator c(z-z~)L, is understood as the power series

e(z-zi), ~" kti _ ~ (z-zi )JLJ j! H" j=0

(1.4.5)

Exponential series of Lie operators, like (1.4.5), are in general known as Lie transformations [11]. The theory of Lie transformations develops quite naturally from the theory of Lie operators. In fact, the Lie transformations {ctLf;t C IR} constitute the Lie group associated with the Lie algebra {Lf} of the Lie operators; roughly speaking, we may say t h a t this group is obtained by exponentiating the corresponding Lie algebra through a real parameter. It is a common language in the theory of groups to refer to {L f} as the algebra of the generators of the group of transformations {ctLf; t E R}. Additional comments on Lie transformations are given in w 2.2.2. It is easy to prove t h a t the vector (1.4.4) is a solution of Hamilton's equations (1.4.3) corresponding to the initial values ui. Differentiating both sides of (1.4.4) with respect to z and summing back to u(z), we end up with the desired proof __

(

d z U . Z.

_

V~176( z - z i ) J - l ' k j u i _ z~, (j-I)' , j=l

"LH 9 e (z-z~)~H u i -

L

,

u(z)

"

(1.4.6)

22

Linear Ray and Wave Optics in Phase Space

However, the above vector provides only the formal answer to the problem of ray evolution, which actually is solved once the action of the opera,tor

c(~-z~)'L- on the initia,1 vc(:tor ui is explicitly defined. The functiona,1 form of is obviously determined by the Hamiltonian flmction H ( q , p ) , which in turn rcflc('ts tl~c geometry a,nd optical fca,turcs of the system. In other words, expression (1.4.4) is t)asi('ally a series cxpa,nsion, whose terms result from ret)ca, tc(tly P(fisscn~ 1)ra,(:kcti~g H with ui. It, is the ttamiltonian flln(:ti()n that (l('~t('rnfi**t's tlw ft)n** ()f tlw va,ri(n,s t('xn,s i** tl,('~ series m,(t lw**('(" th(' t)()ssi|)ility ()f S~lmlni~g l,l~(;~ 1,() a. (:l()s('(l ('xt)li(:it f()n~. A(l(lil,i(n~a,1 (:()llllll(~ll|,S (l(;scl'V(;S the ca,s(; when tim ttanfiltonian (lct)(;n(ls cxI)li(:itly ()~ z, m~r s()a,ls() k~ (lo(;s. In tlm,t (:asr t,l~(; r('la,t,i()~ l)(;t,w('('~ the ray vm'ial)lt,s at tl~(; l)lm~(;s zi m~(l z, a.s (;xI)I't;ss(;r i~ (1.4.4), is ll()t l)r()l)t;rly (:orr(;('t. Alt, l~()~gl~ m~ ()I)(:rat()r-t't)rm r(;t)r(;s(;~d,a ti()n f()r t,l~(; (;v()hd, i(n~ ()f tim ray va,ria,l)l('s (:m~ still |)(' g~u;ss(;(l, it ('mm()t |)('~ giv('~ i~ t,l~(' f()rlll ()f t,t~(' si~gl(' Lie c (z-z~)t.

t,ra.nsf()rnm,ti()l~ r i.(;., a.s t,h(; r (~f tl~c Lic ()t)(;rat()r k , over the wh()h; t)r()I)aga,ti(n~ l(;~gt, l~ fl'(m~ zi t,() z. W(' ~ a y i~ g(;~('ral r(;lat(' tt~(' ray wtI'ialfl('s at zi m~(l z i~ tl~(' ()l)('rat(~r f()rn~ A

u(,)

(,,

-

(1.4.7)

rcftc(:tillg l,ll~' vi~'w ~f l,ll('~ i~litial w;('tor u i l~Cillg l,ra,xlsfln'lll~;(l ild,~ u(z) t~y tllc otx'ra.t()r 9 J l ( z , z,i), wllir a,c(:~)r(tillgly is calh'(1 tll(' r a y t'ra'n,@','t" ope,'t'ato't" As t)r()ver Imf(~re, wlwll ttw~ Ha,miltoIlian is z-in(tel)en(l(;nt, tt~c tra.nsfer ~I)(;ra,t()r is ~'xa,~:tly l'el)res~;~d,e(t t~y the Lic tra,nsf~r~m,ti~ (1.4.5)" A

@ (z,

(1.4.8)

-

If the ttalllilt~nliml ~lel~llr (m Z, this ret)rcscnta, ti(nl n~ longt'.r h~l(ts. This is due to t,tlc fa,(:t t h a t ttw, a,rithinetic f()r ot)cra,tors is not the sa,nle as for s(:a,la,rs. ()I)era,t(~rs are r that may ~l()t obey the (:onmmta,tive law under multipli(:a,tion, a,IM hen(:c it, is not generally true tha,t AB - BA for opera,tors, whilst it is always trim tha,t ab - ba for s(:tfla,rs. Accordingly, (;xt)()ncntial operators may not obey the scmigroup property; thus it is not generally true A

A

A

A

t h a t c n . c s -- c n+B for operators, whilst it is always true that c ~ . c b - e ~+b for

scalars. To ga,in a feeling for this peculiar aspect of the arithmetic of operators, we a,rrangc Eq. (1.4.3) as A

du(

) -

Substituting for du(z) - u(z + d z ) -

(1.4.9)

L

u(z), we obtain A

u(z + d z ) -

[1 + L. (z) dz] u(z) - cLI-I (z)dz U(Z),

(1.4.10)

Hamiltonian Picture of Light Optics. First-Order Ray Optics

23

where the exponential representation on the right-hand side is in accord with the power series (1.4.5) for infinitesimal dz's. The above describes the ray propaga,tion by tile infinitesimal length from z to z + dz. As noted in w 1.3.1, the propagation by a finite length can be thought of as an infinite chain of infinitesimal propagations like (1.4.10), sequentially ordered according to the axial variable. Thus, for instance, we have A

u[(z+dz)+dz]-

A

c LH(zNdz) dz

u(z+dz)-

eL,

A

(z+dz) dz

(?LH(z) dz U(Z). (1.4.11)

The two exponential operators above multiply to one exponential operator

c[C,(z+az)+cii (z)]az only under the condition that the exponents LH(z) and ki~ (z + dz) coinmute with each other, i.e., k/~ ( z ) k t (z + dz) - k (z + dz)k (z). In that case, sequentially concatenating propagations by all the infinitesimal segments dz's along the desired distance from zi to z, we end up with the exponential representation

(z, zi) - c "fz~CI~ (z)az,

(1.4.12)

u

a(z)dz

in a certain analogy with tile solution y(z) -- y(zi)Y(Zi)CJz~ of tile scalar homogeneous differential equation. This is not surprising as the commutativity makes operators closer to scalars. Also, we regain the simple form (1.4.8) since, for a z-independent generator k , fzz k/i dz - ( z - zi)k u. Evidently the ray evolution is found once tile operator (1.4.12) is left to act oil tile input data to yield the explicit ray-coordinates at z. In general, as the Lie operators k , ( z ) ' s at different z do not commute with each other, tile sexnigroup property of exponential scalars is not recovered within the operator context. We are then confronted with the problem of seekingfor a proper representation, exact or approximate, for the transfer operator 9)t(z, zi), i.e., with the problem to give the infinite product of z-ordered infinitesimal transformations cc, (z)dz an explicit, possibly closed, functional form. Readers interested in the detailed aspects concerning the integration of equations of the type (1.4.3) are addressed to [12, 13]. In the forthcoming sections we will consider the ray propagation through linear (or first-order) optical systems, which can effectively be described by ray-parameter independent 4 • 4 matrices. More specifically, we will consider systems, whose symmetry properties allow to reduce the dimensionality of the problem from the 4D to the 2D optical phase space, so that we will need only a 2 x 2 matrix to describe the ray propagation. We will see that linear optical systems arise from Hamiltonians that are, or are approximated as, quadratic polynomials in the phase space variables; departures from the linearity are identified as aberrations [14]. Thus, second-degree polynomials

Linear Ray and Wave Optics in Phase Space

24

are seen to play a distinguished role within geometrical optics; indeed, linear optical systems s t a n d a,s the concrete realiza,tions of Hamiltonia,n dynamics, genera te(t |)y polynomials q u a d r a t i c in the phase spa,ce variables.

1.5

Lie transformations in the optical phase space

The ttm~fill,~n>I,i~' flWl~mlis~, {les(:ril){'~l ix~ t,hc Im'viCnts se(:ti~ms, alh~ws ~s to i(lc~tify a tm,tl~ t{~ ild,(:gra.t,(, I I m ~ i l t ~ ' s ~;~t~m.ti{n~s fin" tl~{', light ray t{~ the corrcst)ol~(li~g syl~l)le(:l,i(: l,ral~sf~)r~m,t, iol~ i~ tl~(; gt'~()~cl, ri(:a,l-()t)l, ica,1 t)]m,sc st)a,ce: II

Ell

Poisson bracket operation

...... Exponentiation

~'(z''-zi)Lll .

(1.5.1)

T h e r~'s~ll, il~g t)lm.s{: st)a,c~, l,rm~sf{)r~na.l,i()~: A

1[][(Zi)

by,, ( . . . . i)L//'

U(Zo)-

( '(:-z')'~ll U(Z.i),

(1.5.2)

l"Cfl(~,(:|,S [,llO, ew~llll,i(~ll ~f tilt; ray a l()llg l,ll(', ~l)l,i(:al z-axis. It, is (;vi~l~;ld, l,lm,t, as excllq)lili(',~l ill (1.5.1) fi)r ~t)l,i(:a,1 Ita,l~dlt,(~nialls, we (:an g~',ll~',ral,(; 1)lms~'-Sl)aC~' l,ra,]lsf()rxl~ml,i()]ls, lmvillg ~r ll~t, a (tir(~,('t, lillk with the Z-ev~)lllt,i~nl ~)f tll~' ligllt, ray. We llmy ~:~)l~si~l~'x" m~y ~n~t,ianu)usly ~liff~'rcntial)Ic, I)tias~;-slm.~:~' varia,l~l~; .[(q,p) a,~t t,lxe~ f~n'~ t,l~e ~:~n'rest)~n~ling Lie l,rm>for~na,t,ion, wl~s(', "wdi(lity" as a I)has(',-spa,(:(; l,ra,nsfi)r~m,l,i~)~ is (;sl,at)lisll(;(l a(:(:or(ling 1,() wl~(;l,l~er il, ('~wre('l, ly ~m,1)s l,l~(; g(;(~el,ri(>()I)l,it'a,1 l)t~a,s(;-st)a,('(; i~d,(~ itself. We recall i~ flu:t, tim,l, tim ()l)tica.1 I)tm,se st)a,ce lm.s t,l~(', l)(W~flia.r 672 • IR2 st, ru('talre, in C()lll,rasl, wil, l~ t,l~: IR4 str~u:t~rc ~)f l,l~e n~e(:ha,ni~:a,1 l)hasc st)a('e. Wc will giw; ,SOl~e cxm~t)lcs of Lic tra,x~sf(wl~m.t,i~n> in the ot)t, ical t)hasc st)a(:(',, h~ l)a,rl,i(:~lar, i~ w 1.,5.1 an(t 1.5.2 w(; will (:tmsidcr a,s I)lm,sc-st)a.ce fim(:ti~n~s .f(q,p) lixma.r a n~t q u a d r a t i c t)olynomiMs in q and p. In tha.t case, we cm~ ~;a.sily ~)|)tail~ tl~c explicit expressions of the tra,nsformed plmsc space veer(w, si~l)ly ~six~g l,l~e ext)onentia,1 series exI)a,nsio~ (1.4.5). T h e n in w 1.5.3 we will ])rielty ~les(:ril)e the factorization-ba,se(t mcth{)d widely used to deal with the Lie transfor~nations generated by higher-order polynomials. To simplify ~otatioI~s we will consider the single pair of conjugate variables (q, p), and hence we will a,ccount for the proper I - n , n] x IR s t r u c t u r e of the relevant 2 D phase-spa.ce. We will denote by ( the real p a r a m e t e r of ..-.. the expo-

nentiation from the Lie o p e r a t o r L, to the Lie t r a n s f o r m a t i o n eCLf. We will not detail the physical m e a n i n g of ~, whict~ depends, of course, on t h a t of the varia,ble f(q, p) in order t h a t the exponent, ~ k, be dimensionless. Thus, if f ( q , p) can be interpreted as an optical Hamiltonian, ~ may acquire the meaning a,s the evolution variable, and accordingly the relevant Lie transformation that Ks t,he ray-transfer operator.

Hamiltonian Picture of Light Optics. First-Order Ray Optics

25

qT ("

;

t,,,, I .,"i p.

:: ................. "-

r Z

(a)

(b)

F I G U R E 1.5. As the tip of the r a y - m o m e n t u m vector n = (p, pz) is confined on the circle of radius n, (a) the " m o m e n t u m translation" is a rotation of the vector, with the tip slipping along the circle, which a m o u n t s to (b) a variation of the direction of the ray with respect to the q-z reference axes.

1.5.1

L i n e a r p o l y n o m i a l s in q a n d p

The momentum variable p generates a shift of the position coordinate of the phase-space vector, without affecting the relative momentuni coordinate. In fact, being (1.5.3)

kp = Oq '

the pair (q,p) is turned into

( ) q(<)

()

__ e~'c)/cgq q

]'(r

__ ~

~ oJ

j=l

P

() ( ) q

__

OqJ p

q +

p

'

(1.5.4)

which may correspond to a shift of the ray position along the reference axis. Likewise, the position variable q, the associated Lie operator being A I_q

0 Op'

(1.5.5)

would generate a shift in tile momentum coordinate of tile phase-space vector as

-- ~

(q(~)'~ - - e -~O/Op t p(~))

\P

j=l

(-~)~

-OPJ

(1 5.6) -- ~

"

"

Since by (1.5.6) the m o m e n t u m coordinate could be carried out of its proper optical range: p2 _< n 2, the Lie transformation c - r 1 7 6 1 7does 6 not yield in general a proper transformation in the optical phase-space [ - n , n ] x R, which could not be properly mapped into itself.

Linear Ray and Wave Optics in Phase Space

26

Actually, the " r a y - m o m e n t u m translation" should signify a variation of the ray direction, which evidently can be realized by a r o t a t i o n of the ray about its position on the reference axis. As shown in Fig. 1.5, we denote by ~) the angle tile ray forms with tile optical axis. Hence, tile transverse and l(mgitu(tinal (:on~t)o~ents ()f the ray momentun~, p and pz, write a.s p - 'n sin 0, pz - ~. (:()s '0.

(1.5.7)

the, ray 1)y l,l~e a,~gl(; ~ a l)o~t its I)()sit,i()l~ ()~ tim referen(:e axis q, the (:()nlt)()n(;lltS of tt~e ray l~()~ent~l~ })(K:()IIIO I{()ta,tix~g

r

I-3 -

'n s i ~ ( ' O

-

r~) -

p cr

r

-

p~. s i ~ r~,

(1.5.8)

As w(' will (tis(:llss i~ (t('tail i~ S('('t. 1.6, li~ea,r g('~()~(;lri('al (~l)l.i('s assignees t h a t the (lire('ti()I~ ()f tim ray is ah~()st Imrallel 1,() l,l~(; ()I)ti('al z-axis, sr that the firsl,-()r(ter a.l)t)r()xi~mti()~ ()f rela,l,i()~s (1.5.7) a~r (1.5.8), we ()])l,ai~ t-5 "-" p -

'Item,

I--)= "" 'It,

(1.5.9)

being as well p ~ i~0. Evir wil,hin the linear a t)t)r()xillmti()ll r firstr ()I)ti(:s, a, wl,I'iati()li ()f tll(; ray (liI'e(:ti(m may 1)(; I)i(:tllI'('~r in l,ll('~ ()t)tica,1 t)hase-st)a,ce as a, sliifl r tim lll()IliC~lt,lllli cooI'dilmte, wlii(:li sa.f('ly r('lliaills well insith; its t)roI)er ()l)ti(:a,1 rallg(' Ih;llt:(;, in l,he lint'~a,r ()l)tit',s r(;gil~(; the Lie tra,nsfr c -<'~176 gelmrat(;(t ])y the t)()siti()n va,ria,|)le q, yi(;hls a, t)rot)er transf()r~lla,ti()ll ill I,]l(; ge( )l l l(;l,ri(:al-()l)ti('al l)tm,se-sI)a.(:e, a,(:tlmlly i~lv()lving a. small region arolm(t tll(; ()rigin well illsi(te the inherent striI) [-'n, n] • IR.

1.5.2

Quadratic polynomials in q and p

We consi~ler now the Lie transfornmtiolis, generated ])y qua(|ratic 1)~flynomials, and des(:ribe their effect on the t)hase-st)ace vectors. We disregar(t the question linked to the proper optical range for tlle ray m o m e n t u n l , assllming to work in the linear a p p r o x i m a t i o n according to which, as said earlier, the rays propagate 2 + qy2 << 1). in the close proximity of the optical axis (p2 + py2 << 7~2 and q.T In other words, we suppose t h a t only small regions in the neighbourhood of the phase-space origin are involved in the t r a n s f o r m a t i o n s we consider. We are therefore enabled to deal with the optical phase space as if it were the R4-hyperplane or the IRe-plane, when considering, as we will do below, only one transverse coordinate. We will see t h a t the dynamics generated by quadratic polynomials is depicted in phase plane by portions of (nondegenerate or degenerate) conics. The discussion anticipates the more accurate analysis we will develop in C h a p t e r s 2 and 3.

Hamiltonian Picture of Light Optics. First-Order Ray Optics

We consider firstly f(q,p)

lp2 ,

(1.5.10)

-~

_

27

which within the analogy with mechanics corresponds to the Hamiltonian of a free particle. The associated Lie operator is

Lp2/2 - - P O q '

--

0

(1.5.11)

which yields the phase plane transformation

q(~)~ (p(~)J

__ ~'pO/Oq (q) __ (q) _4_ ~ p~q (q) @ i

i

i

(1.5.12)

_~1~2 (po~)(po~)(q)i _4_ . . . _ (q~) ~_ ~ (~) __ (qiW~Pi) " Pi describing at "free" evolution with the m o m e n t u m coordinate being unchanged: p ( ~ ) = pi. It is easily verified that {q(C),P(~)} = { q i , p i } = 1.

Correspondingly, with f(q,p)

_

1 2 ~q ,

(1.5.13)

which may be put in relation with the potential part of the Hamiltonian of a classical particle experiencing a,n elastic force, we obtain ~-

0

(1.5.14)

kq2 /2 = -q-@p ,

and hence q(~)) _ ~--~qO/Op(q) _ (q) _ ~ q~p (q) __~ p(r i i i _~_~1~2 (qffp)(qffp)(q)i -~- "'" --(q~) -~- ~ (qO) 9 _ (piq~qi) .

(1.5.15)

In contrast with (1.5.12), the above transformation leaves unchanged the position coordinate q(~) = qi. The mixed polynomial f ( q , p ) = qp, (1.5.16) generates the Lie operator A 0 0 kqp - q--~q - P Op'

(1.5.17)

A

whose powers k j affect the phase-plane vector according to qP

A

L qP

~P

(_)jp

, ...

(1.5.18)

Linear Ray and Wave Optics in Phase Space

28

T h e n , s u n n n i n g up all tile terlns of the exponential series we end up with the transforInation (q(r

C ~'[qp

9

(1.,5.19)

T h e conjugate variables arc scaled by inverse fa('tors so t h a t q ( ( ) p ( ( ) = q@i. T h e t)ha,se-t)la.ne trajectories relevant to the "(ty~m.~lfi(:s" (tescrit)ed t)y (1.5.12), (1.5.15) a**(t (1.5.19) a,r(; (:l(;arly I)()rti(ms of tl~(; stra,igld, lines p = P i an(t q -- q i , t)()tll ()rigina,tillg fl'()lll the i~fit,ial t)()int ( q i , P i ) , all(| ()f 1,11C l)ra,n(:h of (,h(: e(lllila,tera,l hyl)(:rl)()la, (,In'(nlgll (,h(: l)()int (qi, Pi), Im,villg (,h(: q ml(l p axes as a,sy1111)t()tes. As a, fiu'(,ll(:r (:xmlII)h:, w(: (:()llsi(h:r the (llm,(Ira(,i(: I)()lyll()lllia,l

.f, ..... ( q , , p )

_ ~p 1 2 _f_ ~q2 1

(1.5.20)

whi(:h (:all 1)(: r(:ga,r(l(:(l as 1,11(: l)asi(: f()l'nl ()f (,h(: a,l,l,ra(:tiv(: ()s(:illa,t()r lla,lniltoniaIl. Tll(: a,ss()(:ia,(,(:(l IA(: ()l)(:ra, t()r is 0 L ...... -

t'~

-

0 q Op'

(1.5.21)

wll()s(: (:It'(:(:(, ()I1 tll(: l)llas(: l)la.ll(: v(:('.(,()r is (:asily (l(:(,(:nllilm(l t() I)(:

......

t,

-q'

7r (,lm(, is a, (:h)(:kwis(; r()(,a,(,i()ll 1)y 5" W(; (:a,ll tll(;r('l)y gll('Ss (,() r(~(:()V(:l" (,ll(; ilfitial ve(:t()r a,t tll(; forttl t)()w(;r, I)eil~g ~ - 27r. R(;a,(ters (:a,n (:l~(;(:k (,l~is I)y direct (:a,lc~fla,ti()n ()f the wu'i(nls t)()wers, which sum finally to give

9

[P(()

'

- q i sin r + Pi ('()s ( ]

~t'/i

(1.5.23)

Evidently, tl~c relatiw; traje(:t()ry in phase plane lies on the (:ir(:le q2 _f_ p2 _ q/~ + p/~, in a,nalogy with the well known phase-spa(:e t)i('t~re ()f the ~ot, ion of the mechanical h a r m o n i c os(:illator. It, is interesting to c o m p a r e the " d y n a m i c s " (1.5.23) relative to the phaseplane fimction (1.5.20) with t h a t g e n e r a t e d by the p o l y n o m i a l .f~.... (q,p)

_

lp2 lq2 ~ - ~ ,

(1.5.24)

which can be r e g a r d e d as the basic form of the Hamiltoxfian of a repulsive oscillator. T h e associated Lie o p e r a t o r is -0 0 Lr.~ -- p -~q --k q Op '

(1.5.25)

Hamiltonian Picture of Light Optics. First-Order Ray Optics

29

and the corresponding Lie transformation affects the phase plane vector a.s

((t(~)~ --e('*s .... (;) _ (qicosh~q-pisinh~) P(()J

i

qi sinh ( + pi cosh

"

(1.5.26)

The reader may easily see that the phase-plane trajectory relative to the "motion" (1.5.26) lies on the branch of the equilateral hyperbola q2_ p2 _ q~_ p~ through the point (qi,Pi), with the lines p =-+-q being the asymptotes. In Chapters 2 and 3 we will illustrate in details the phase plane view of the dynamics generated by all the above considered quadratic dynamical functions, with explicit reference to the optical systems they represent. Finally, we can consider the polynomial

f (q, p) - l aq2 + cqp + l bp2,

(1.5.27)

with constant coefficients a, b and c, whose Lie operator is .L, - . akq~/2 . + .Ckqp + bkp2/2 - (cq + bv)-~q 0 - (aq + cp) Op" 0

(1.5.28)

The reader can verify by direct calculation that the single-exponential form r is less comfortably managed when applied to the starting data to determine the ray-coordinates (q((),p(()) ill an explicit gild closed form. An appropriate disentanglement technique may usefully be exploited to turn the single exponential operator into a more convenient product of exponential operators, each being suitably determined and placed in order that the inherent calculations become more feasible. Among the possible multiple-exponential forms of the Lie transformation g([(cq+bp)O/Oq-(aq+cp)O/Op], we report for illustrative purposes the following ordered product A

A

A

A

C~{aLq2/2-t-cLqpq-bLp2/2}-- Ca(~). Lq2/2

A

g c (().L

A

qP(?b((). Lp2/2.

(1.5.29)

The explicit expressions of the ~-depending functions a (~), b (() a,nd c(() in terms of the coeflqcients a, b and c can be found in the literature [13]. We invite the reader to work out the phase plane vector u(() generated by (1.5.29) from the input data ui, on account of relations (1.5.12), (1.5.15) and (1.5.19). Also, we suggest to compare the explicit expressions of the coefficients a ((), b (() and c ((), and then to determine the range of ( from (i = 0, through which the multiple-exponential representation (1.5.29) may properly be used. As a conclusion, we evidence a significant feature common to all the rayvariable transformations, generated by second-degree polynomials. As it clearly emerges from the above results, the ray variables at any value of the stepping parameter ( are determined by the input data through a linear relation with coefficients depending on (. The linearity of the mathematical relation between the ray variables during propagation is the distinctive mark of first-order rayoptics, as we will see in Sect. 1.7.

30 1.5.3

Linear Ray and Wave Optics in Phase Space

Beyond the quadratic polynomials

As noted in the previous paragraph, the homogeneous quadratic polynomials provktc the basic m a t h e m a t i c a l models for first-order optica,1 systems. Notably they rot)resent the only Hamiltonia, n flm(:tions, for whi(:h the corresponding tra,nsfcr ot)cra,tors (:an be given a,n cxa,ct a,nalyti(: representation, typica,lly in the f()nll ()f a, finite t)r()(tll(:t, of cxt)onential ()t)crat()rs. This is tI"ll(; for both z-in(lct)(ul(t(;nt a,n(l z-(h;l)(;n(lcnt IIa,nfilto~fia~ls. Prc(:iscly, it, is t)r()vcd tha,t the ra,y-t, ra,ilsf(w ot)era,t()r , we ~imy (teil()te ])y 9Yl,~ (z, zi), a,ss()(:iate(t with (111a,(tra,t,i(: Ha,nfilt()lliallS is exa,('tly (;qlm,1, i]la lleiglfl)()llrll()()(l ()f t,ll(', i(lentity, t() t,ll(; t)ro(1ll('t, ()f th(; Lie transf()rnm,ti(nls, g(;ll(;ra,t,(;(t ])y the l,llr(;(; ])a,si(: m()ll(mfia,ls q2 p! a.n(t qp. Exi)li(:itly w(; may writ,(; I...._

,~ ( z,z.~ ).

,? (z,z,).L

fl ( z,z, )

where tim z-(h',I)(u~(ti~g (:r e~ (z, zi), /~ (z, zi) a,~(l 7 (z, zi),)lw, y a set of Ilcnllilmar first-()r(h;r r l e(l~m,ti()~s, (tet, ei'~i~m(t |)y t,t~(; Sl)(;(:ifi(: tim(:tional f()r~ of tl~(; Ita~ilt(nfim~ as a (t~a,(lra,ti(' fin~(:ti()~ ()f q a.~(t p. I)(;via,ti()~s fr(n~ li~ca,rity, sig~ifi(;(t t)y I)()w(;rs ()f q ()r p lfigher tlm,l~ two) in the ot)l,i('al tIa,~filt(n~ia.~, ('a~ l)(; a,(:(:()~()(ta,l,(;(1 ])y a~ a l)l)r()xi~m.te r~;1)res~;ntation, whi(:l~, t)i'e,scrvi~g I,t~' fit(:l,(w(',(l i)i'()(1~(:1, sl, rll(:l, llr(; ~f l,l~(' l,ra,nsfer ()l~,_)(;ral,()r, a,(t(ts fa,(:t()rs t() th(; l)asi(: (t~m,(lrati(:-tla]~ilt()nia,n g('~mratc(t ()t)(;ra,t()r 93l~ a,s [a2.1]

9Y~ (z, zi) -..-9Y~,,, (z, z ~ ) . . . 9Y~:,(z. z~)gY~ (z, z~),

m > 2,

(1.5.31)

thus yichtfi~g a I~()(t~flar, easily nm, nagca, t)h~., cxt)rcssi(m fi)r th(' tra,nsfer operator, in which ea,(:h a,(:(:~ra(:y-il~t)r()ving fa,('tor links to the t)rcvi(~usly and i~dct)cn(h'~tly esta, t)lished ot)cra.t()r nio(lulus. Each ad(led fm:t(~r, 93l.... (z,z~), 'ttt,- 3, 4,.., i~ tt~c (:o~(:ct)tually ixdilfitc pro(tuct (1.5.3 1) is in the f()rl~ of a, Lie tra,nsfl)r~m,tion ge~era, t,e(t t)y a, t)()lynomial flmt:ti()n of m - t h (legree. Clearly the Illllllt)or a,n(1 the st)e(:ifi(: cxt)rcssions of the, a,(tdcd fa,(:tors axe (tctcrmined by the required (t(;grec ()f a,(:(',ura(:y a,n(t the flmcti()Im,1 form of the, higher-order terms in the Ha,nfiltonia,n. The analytical a,nd nulnerica,1 proce(t~res to elabora,tc the rct)rcscntatiox~ (1.5.31) for 9)l (z, zi) a,t the desired a,bcrration order m are well d o c u m e n t e d in the literature [12, 14].

1.6

Linear ray optics and quadratic Hamiltonian functions

First-order (or linear) ray optics is marked by a linear relationship between the ray coordinates at the assigned end-planes. It is concerned with the behaviour of the paraxial rays, whose distances from the axis and inclination angles relative to the axis arc supposed to be small. First-order ray optics assumes that such off-axis distances and inclination angles are both small enough to

Hamiltonian Picture of Light Optics. First-Order Ray Optics

31

justify the lower order a p p r o x i m a t i o n for all the quantities p e r t a i n i n g to the ray p r o p a g a t i o n with respect to t h e m (a). The terms paraxial ray optics and first-order ray optics are frequently taken as equivalent. Thus we can replace the arc element ds along the ray by the straight element dz along the optical axis- ds ~ dz. T h e ray m o m e n t a (1.2.11) can therefore be written as dq.~ dqy ~ nqy, (1.6.1) P x -- n --~s "" n q ~ , p y -- n --~s _ i.e., in t e r m s of the slopes q~ and qy of the ray to the z axis which are further a p p r o x i m a t e d by the angles ~).~ and ~)y or equivalently c<~ and C~y (Fig. 1.4). As a consequence of (1.6.1), we m a y establish a further characterization of the paraxial p r o p a g a t i o n in terms of the ray m o m e n t u m according to 2 _ ?%2 t2 t2) ?%2 P~ + Py (qx + qy << ,

(1.6.2)

thus suggesting the quadratic a p p r o x i m a t i o n of the optical H a m i l t o n i a n as

H ( q , p, z)

1 --

2n(q,z)

(p2 _~_ 2 py) -- n(q, z).

(1.6.3)

We see t h a t the H a m i l t o n i a n of a paraxial ray has the same form as the H a m i l t o n i a n of a non relativistic particle with the well recognizable kinetic and potential parts [7]. Accordingly, the phase space of paraxial optics can be t h o u g h t of as similar to the phase space of classical mechanics; in other words, paraxial optics "lives well within" the proper restriction IPl << n (see Sect. 1.2), which then comes to represent no more a restriction but the " s t a t u s " . Evidently first-order ray optics is concerned with optical systems whose refractive index varies gently in space. In fact, if the refractive index varies drastically in space, we are likely to find rays with large slopes and off-axis distances. We m a y therefore e x p a n d the function n(qx, qy, z) with respect to the profile coordinates and retain only the constant, linear and q u a d r a t i c t e r m s to obtain

n(qx, qy, z) - n o (z)

--

Six

( )qx

(z)q.~

-

T~ly

(Z)qy

(1.6.4)

_

-

where no(Z ) - n(O, 0, z) denotes the refractive index along the optical axis. Needless to say, this a p p r o x i m a t i o n is limited to regions across the axis comprising values of the transverse coordinates such t h a t n(qx, qy, z) > 1. The optical H a m i l t o n i a n becomes then q u a d r a t i c in the ray-variables

H(q,p,z)

1 (p x + Pv) + n ~ ( Z ) q x + n ~ , ( z ) q v 2%(z) + in

(z)q2y +

(1.6.5)

(Z)qxqy - no(Z),

d The circular functions of angles, for instance, admit the first-order approximation, i.e., tan 0 ~ sin 0 ~ 0 and cos 0 ~ 1, for angles roughly estimated as 01 -< 15~ -- 0.262 rad. In fact, sin 15~ -- 0.259, tan 15~ -- 0.268 and cos 15~ -- 0.966.

32

Linear Ray and Wave Optics in Phase Space

with the same formal appearance as the H a m i l t o n i a n of the classical motion under a, weak potential [a.l, 7]. Of course, the zero-order term, -no(Z), is unessentia,1 as its Lie operator is zero. On the other hand, the first-order terms are rela,ted to possible tilts and misalignements with respect to the ot)ti(:a,1 a,xis; they vanish in centered systems as those we consider here. As anticipated in ~ 1.5.2, the ray prot)agation under a, qlm(trati(: Hamiltonia,n is descri|)e(t |~y a liimar relati~)ll |~etwe(ul the ray va,ria.|)les a,h~Ilg tile ot)ti(:al axis; in ()tllcr w()rds, (tua,(lrati(: IIa,nfilt(mia,ns genera,t(; lilmar synlt)lc('ti(: tra,nsf()rnmti(nls ill t)tm,s(; st)a,(:c. ~lb,nlls ()f ()r(ler lligll(;r l,]m,ll tw() iIl tile optical Hamiltonia,n (i(tcntific(1 a,s aberrations) generate nonlinca,r trmlsformations i~l t)hase st)a,t:e (Pr()|)l(;~ll 4). At)eI'rati()~ls (:a,n |)(; el(;gmd, ly (:la,ssiti(',(1 anti (;fl'e(:tiv(;ly (:()lllI)lll,(;(l |)y llS(; of a,I)t)rt)t)ria,te ga'()lq)-th(;or(;ti('a,1 l,()()ls [12, 13]. The nlo(hflar s(:henlc (l.5.a 1) is a,,~ (;x(;llll)la,ry reslflt ()f the gr()llt)-tll(',()reti(:al trca,t~mnt ()f a|)crra,ti~g syst(u~s. As ~()t(;(1, i~ it (t(;via,ti(n~s ()f tl~(; syst(u~ rcst)(ms(; fron~ li~ma,rity are rcI)r(;seI~te(t a,t every a,l)erra,ti~)~ ()r(ler t)y sy~I)h;(:ti(: x~m,t)s s~fit,a,|)ly (:()~(:a,t,(;Imt(;(t ()I1(~, t() t,l~(; ()l,l~(;r t,()(:a,l(:~fla,tc a,|)(;rra.ti()~s ~t) l,() th(; (t(;sire(t ()i'(t(;r. TI~(; (:(n~(:t;l)t~lal aa~(l ('(n~I)~ta, tir a,(lvaa~tag(;s ()f gr()~I)-tl~,;()r(;ti(:a,1 tr(;a.t,~mI~ts a,r(; l,l~t;~ evi(tent. We (:(msi(l(;r l~()w ()t)t,i(:al (l(;vi(:es, wht)se gl;()lll('~t,ry ena.1)h;s ~lS t() fl~rther si~t)lify cxt)r(;ssi(n~ (1.6.5) ()f l,h(; ()l)ti('al IIa.l~filt(mian. Tln~s, i~ orth,ogo'n,al systems l,hc li~u;a,r q:,,, % a,n(t ~ix(;(t q:,,qy l,(;r~s ix~ (1.6.5) (:m~ t)(; (lr()t)I)(;(t out; also, in a:rially ,symmc.tv'ic syst(;n~s the tra, nsverse (tet)eI~(tcnce (:a~ t)e ext)resscd l,hr()ugh tim ra,(lia,1 (lisl,a,n(:('. q~ + q~ fl'(m~ l,hc ()I)l,i(:al axis. I~ l)()l,h (:a.scs, the ray t)rot)a, ga,ti(n~ (:m~ t)(; t,rca,tc(1 a,s a. dylmn~i(:al t)ro|)lcx~ witl~ (nm-(lcgrcc of freedom, dcnmn(li~g then for a, 2D t)ha, sc sI)acc. Figure 1.6 sh()ws the possible realization of a~l orthogona,1 m~(t a,n axisynm~etric system; m~ exm~t)le of a nonorthogona,1 sysl,(;~ is also (t(;I)i(:t(;(t. The presenta,ti(n~ |)(;low is taken on a cursory and elementary level; it serves to justify the simplification from a 2D to a 1D model. The interested reader is dircctc(t to [4.4] f()r a,II exhaustive inv(;stigati(m of the role of "synm~ctrics" in the Hamiltonia,n formula, tion, whose power indeed "lies in the ability to yield

over-r'iding r'es'ult,s gover'ning the behaviou'v of types of systems defined ,solely with respect to the symmetries they possess." In this connection, we emphasize that light propagation occurs inherently in the real 3D space. The expression "2D (or 1D) model" accords with the mechanical-like view, where the evolution variable is the "time" coordinate, but contrasts with the real optical view (for stationary media), where the evolution variable is itself a spatial coordinate. Clearly, when speaking of "2D" or "1D" picture, we properly refer to the n u m b e r of profile coordinates suited to describe the propagation possibly in accord with the s y m m e t r y properties of the system.

Hamiltonian Picture of Light Optics. First-Order Ray Optics

33

y :

y

z

FIGURE 1.6. Examples of (a) a n orthogonal optical system, (b) an axially symmetric optical system and (c) a nonorthogonal optical system.

1.6.1

Orthogonal s y s t e m s

The configuration of Fig. 1.6.a) is a basic e x a m p l e of an o r t h o g o n a l system, being in general characterized by two m u t u a l l y o r t h o g o n a l planes of s y m m e t r y intersecting at the optical axis. T h e system sketched in the figure consists, in particular, of two identical cylindrical lenses oriented w i t h their axes orthogonal to one a n o t h e r (e). Clearly the response of such a s y s t e m to the incoming ra,y ca,n be investigated by considering s e p a r a t e l y w h a t h a p p e n s in the two m u t u a l l y p e r p e n d i c u l a r planes of s y m m e t r y , conveniently a r r a n g e d to be the planes x-z and y-z. Systems which are s y m m e t r i c a b o u t two o r t h o g o n a l planes of s y m m e t r y intersecting at the optical axis are n a m e d simple astigmatic, whereas systems having no planes of s y m m e t r y are n a m e d generally astigmatic [4.2]. Anticipating the m e t h o d s we will i n t r o d u c e later, we m e n t i o n t h a t simple astigmatic linear systems are described by two 2 x 2 matrices, A cylindrical lens has one or two cylindrical surfaces, their axes being strictly parallel. If the cylinder axis of the lens is aligned with the y axis, as the first in Fig. 1.6.a), the lens has the designed focal length for rays in the x-z plane, and no focusing power for rays in the y-z plane. In fact, a side view of the lens gives the appearance of a plano convex lens, while a top view shows a slab of transparent material. Cylindrical lenses are used to design anamorphic systems, which have different power or magnification between the two ordinarily orthogonal planes of symmetry; one can produce, for instance, a square image of a rectangular object.

Linear Ray and Wave Optics in Phase Space

34

accounting for the ray propagation in the two s y m m e t r y planes. As an example of a nonorthogona,1 system, we may think of a sequence of two cylindrical lenses oriented at an angle of, say, 45 ~ to one another (Fig. 1.6.c)). Dropt)ing out the linear and mixed terms, the optical Hamiltonian (1.6.5) is seen to t)e the slml of two flmctions, separately depending on the ray coordina.tes (q:,., p:,:) a.n(t (qy, p:~j) and in the same formal fashion, namely 1

2

2

1

2

i

(g)

H(q,p, z) - 2.,,.,,(=)(P.,, § P,,) + ~""2x(z)q:, 9+ ~"%~ =- H.,,(q:r, p.r, z) + ll~,(qy, py, z),

2

qy

(1.6.6)

with tllc I D Ilmllilt(nfians H.,, a.n(1 Hy,

H~ (q~, p~, z)

2

2,,,,(:)P~ + ~I~2~

.,

_

,

(1 6.7)

ruling t,llc ray (;wflllti(nl ill tll(; two t)la,II(;s ()f syllnlletry .r-z a11(t !l-z. E(tua,tions ()f nl()ti()ll (1.4.3) s(;t)aratc tll(nl int() tw() iIl(h;t)(;n(lexlt s(;ts ()f (;(tlla,ti()llS for th(; ray va,riat)h;s (q~,, p:,,) ml(1 (q,, Pu)" d

--

dzU((Z) - L , , ( u e ( z ) ,

ue(zi) - u ( , ,

wll(;r(; 1,]1(; 2-(:()I11[)()11('~11|, v(;(:|,()r II((Z)(:()lll])ris(;s

~ -:r,:q,

(1.6.8)

|,ll(; r a y (:()()r(tilmt(;s (q.,,,p:,,)or

(%, Pv) ml(l L~( ix l,ll(; I~i(; ()l)(;ra.t()r a.ss()('ia.te(1 witll H.,, ()r H u"

L,,(

"

i i) ,,,,(:)t,( i)q (

0 ,,,.,((z)q( /)p(

c - . r , !/.

(1.6.9)

In (xm('hlsi(nl w(; llmy say l, tm, t in 1,11(; ('a,sc ()f lill(;a,r sei)a,ra,t)h; systems the g(;nera.1 2D t)r()l)l(ull ()f tirsl,-()r(l(;r ot)ti(:s, rill(;(1 l)y l,h(; Ha,lnilt()lfia,ll (1.6.5) with the ()ne set ()f E(ls. (1.4.3) for 1,11(; overa, ll ray-nl()ti()ii, tllrns into two 1D t)rot)l(;~11s, rift(;(1 t)y tll(; tta,miltonia, ns H:,, a,n(t Hy with the two sets of Eqs. (1.6.8) for tll(; "I)a,rtial" ray-i~()tions in tlm two sy~lm~(;t,ry planes of 1,11(;system.

1.6.2

Azially symmetric

systems

Tile henfi-sph(;ri(:a,1 lens i~l Fig. 1.6.b) is a,n example of an optical system, which is symmetric under rotations around the ot)tical axis. Systems which are circularly synnnetri(: about the optical axis are n a m e d stigmatic [4.2]. In general, the refractive index in an axially symmetric device, with the optical axis conveniently taken as z axis, is a fllnction of the axial coordinate z and the distance p =v/q2-.i-q2 from the axis: n(q.~, qy, z) -- n(p, z) [4.2]. It is convenient to transform the rectangular coordinates qx and qy to the polar coordinates p and 4) in the transverse plane, with q, - p c o s $ ,

qy - psin 4).

(1.6.10)

Hamiltonian Picture of Light Optics. First-Order Ray Optics

35

The optical Hamiltonian (1.2.12) writes therefore

H(p, r pp, pc, z) - - I n 2 ( p , z ) - p~

Poi.

(1.6.11)

The optical m o m e n t a Po and pc are obtained from the proper Lagrangian function L(p, r p', r z) - n(p, Z)v/l+p'2+p2r by (1.2.9), and hence

p,

pp(Z) -- ~(D,Z) v/l+pt2+p2dy2,

p2r

p c ( z ) -- ~'~(D,Z) v/l+pt2+p2dp,2.

(1.6.12)

Reflecting the axial s y m m e t r y of the system, the Hamiltonian (1.6.11) does not explicitly depend on the angle r and hence is invariant under rotations of the coordinate system in the transverse plane. In the language of classical mechanics r is said a cyclic coordinate. The conjugate m o m e n t u m p+ is then a constant of the motion: pc(z) - pc(z~), determined by the initial values. The corresponding coordinate r is simply obtained integrating the second of (1.6.12), once the radial motion p(z) and pp(z) has been solved from the relevant Hamilton's equations. It is useful to introduce some definitions. A plane through the optical axis is referred to as a meridional plane or a meridian. A ray lying on a meridian is said a meridional ray, whilst a, ray t h a t does not lie on a meridian is a skew ray. The skewness is defined by the twisted product -

qxpy - qyPz,

(1.6.13)

whose formal analogy with the angular m o m e n t u m of a classical particle in a 2D motion is evident. The skewness just equals the ray m o m e n t u m PC" PC - q~Py- qyPx, and so the invariance of pc in axisymmetric systems implies the invariance of as well. Analogously in the motion of a particle under a central force, the angular m o m e n t is invariant. Also, as pc relates to the sine of the angle of the ray with the meridian at the ray position, for a mcridional ray we have t h a t pr - 0, and hence pc(z) - 0 at any other axial position. It is a property of axisymmetric systems t h a t a meridian ray remains a meridian ray t h r o u g h o u t the entire system, as a skew ray remains a skew ray. Incidentally, we note t h a t meridional and skew rays "feel" different refractive index functions. In fact, we may recast (1.6.11) in terms of a sort of an effective refractive index n s s (p, z) as

H(p, pp, pc, z) -- -v/n2,s (p, z) - p~.

(1.6.14)

For a meridional ray n s s is just the refractive index n(p, z)of the medium: neff

(D, z)

--

n(p, z),

meridional ray

(1.6.15)

Linear Ray and Wave Optics in Phase Space

36

whilst fi)r a skew ray n f f (p, z ) - -

In2(p, z) P'~~:'), skewr a y

(1.6.16)

(iue t() the n()n vanishing (:()nsta,nt wdlm po(zi) ()f the skewness. It is evi(teilt tha,t the t)roi)aga,tion ()f a, ineri(lioiml ray can t)e (tes(:ri|)ed by two wtria,t)les (rely. In fa,(:t, t)e(:a.use the ray rema,ins (:()t)la,na.r with the ot)tical axis, w(', 1flay tra.(:(', ()lily tw()()f the f()l~r ray va.ria,t)l(',s, f()ll()wing the, ray t)ath in its ()w~ ~l~(M(li()~ml I)la~(',, i(t(u~tifi(~(t t)y tl,', i~t(ws(;(:I,i(~ ()f I,t~(~,ray witt~ th(', int)~t plane. Tim, tra,~sfer t)r()I)erties ()f a,xisy~n~e, tri(" ()I)ti(:a,1 systen~s (:a,~ etfi(:a,(:io~My |)e (l(',t(w~fi~e(11)y tra(:i~g t)r()t)erly (:l~()s(m ~m,ri(ti()~m,1 rays. Th(', tra,(:i~g ()f skew rays is l)(~,rf()r~(',(t il~ tl~(~,(tesig~ ()f syst(u~s witl~ large a.t)(wt~tre, s, tlm.t il~ a. s(u~se, (:ontrasts witl~ l,l~e l~yt)()t,l~esis inh(w(u~t in the tirst-()r(ler a,I)t)r()xi~m,ti()n [17.~]. A(:('()r(li~gly, a,s fa.s as ~mri(li()~a,1 rays are ('()~('er~(~,(l, we are aga.i~ (;o~fr()nte(l witl~ tlw, l l) l)r()l)l(u~ ()f tra.(:il~g t,l~e ray l)atl~ i~ a, l)laa~(~ tl~r()~gl~ the ()t)ti(:al axis. M()r(~,()ver, a.t ttw, l()wer ()r(ler in the ray (x)()r(li~m.tes p al~(| pp, the Ha,nfill,()~ial~ (1.6.14) l)(;(:(n~(~s (t~m(lrati(',, i.(~,.,

u(,,j,,,,~)

l 2 ._,.,,,,(~)p, + 89

-

'

(1.6.17)

in t11(; llm,ri(li()lla,1 1)lml( ~, ill(livi(llm,liz(~,(l t)y t,ll( ~, illl)llt ray. Tile ()t)ti('s ()f lll(wi(li()lla,1 rays in lillea,r a,xisyllu~letri(" systenls is kn()wn a,s Ga'u,,,'sian ()t)ti(:s, in tl()II()r ()f K.F. (;a,llss [15], wtl() a,I)I)r()a,(:h(xl tim, a,lm.lysis ()f a r(~,fi'a,(:l,illg sl)lm, re 1)y lisillg a, t)()wer s(wies exl)a,llSi()]l ill l,lle ()if-axis (lista,n(:es an(t a,ngh~,s. T h e l,(~,rlll Ga, llssia,n ()I)ti(:s is fl'(~(tll(Ull,ly llse(t in the lit(~,ra,ture as e(tuiva.hult t() t)ara.xia,l ()I)ti(:s ()r first-()r(ter ()t)ti(:s.

1.7

Planar model of first-order optical systems

In a,(:(:()r(1 witll t,h(~ (:()nsi(t(wa,tions of the t)revi()us se(:tion, the t)r()t)a,ga.tion of para,xia,1 (Ilmri(ti()lm,1) ra,ys t h r o u g h simt)le astignmti(: and stigma,ti(: optical devi(:es can t)e trea,te(t as a, 1D d y n a m i c a l prot)lem, with the "plane of the motion" t)eing respe(:tively any one of the two pla,nes of s y m m e t r y of the s y s t e m ()r any ()ne of the infinite meridional pla, nes. In b o t h cases, the relevant Hamilt()nia,n is qua, dratic,

1 p2

H ( q , p, z) -- 2n0(z)

+ 89 2

(z)q2

,

(1.7.~)

A

with the associated Lie o p e r a t o r A

/ H

1

being t h e r e b y i9

i9

(1 7.2)

Hamiltonian Picture of Light Optics. First-Order Ray Optics

-Io

1-I i

7

I

merging ray

,, q,

Incoming ray

37

__~ ..........

Optical System

....-q-~ --I

Input plane

Zo

Z

}"

Output plane

F I G U R E 1.7. The ray coordinates in the linear approximation for a 2D optical system specify the height of the ray and the angle it forms relative to the optical axis.

and the ray coordinates q and p having the geometrical meaning as illustrated in Fig. 1.7. At any axial position z, q(z) specifies the ray displacement relative to the axis and p(z) = no(Z)~)(z ) the reduced ray slope, given as the inclination angle ~ of the ray to the axis, multiplied by the on-axis refractive index n o (z). Following the usual practice, q is taken positive upward from the axis and negative downward as t9 is reckoned positive if a counterclockwise rotation takes us from axis to ray. Interestingly, Hamiltonian (1.7.1) resembles that of the classical oscillator with varying ma,ss and frequency. The equations for the ray variables have, in fact, the oscillator-like form

d z q - no z)P'

(1.7.3)

d =

=

w,

where qi and pi specify the starting data. A closer look a,t Eqs. (1.7.3) reveals tile possibility of rewriting them into the nice matrix form d

dzU(Z)- H(z)u(z),

u(zi) - ui,

(1.7.4)

involving the phase-plane vector u ( z ) = (q(z),p(z)) q- and the 2 • 2 matrix H(z) --

0 _n2(z)

1~no(Z)) 0 '

(1.7.5) A

which can be regarded as the matrix representative of the Lie operator k , . We discuss this aspect within the proper forma,1 context.

Linear Ray and Wave Optics in Phase Space

38

1.7.1

Quadratic polynomials and matrix representatives

As a general property, any Poisson bracket based operator associated with a qua,(tratic polynomia,1 defined on a 2m-dimensional phase space can be given a, 271t • 271t matrix ret)resentation. We (:onsi(ter a, 2D t)ha,sc st)a,(:e a,t)t)r()pria,te to the t)resent 1D ()pti(:al inodel. l~et A(q, p) t)e a (llm,(trati(: ll()nl()g('ale()lls p()lyn()~lfial ill tlle ('(mjllga,te wwia.l)les q a.n(1 p, i.e., A (q, p) - 7,. l aq 2 + -~ 1 t)i)2 + cqp, (1 7.6) witll a,i't)itra,ry real (:()(;ffi('iellts a, b a,n(| c. Here, tll(; t)()ssil)h; (t(;t)ell(h;ll(:e of A a.n(t hen(:(; ()f th(; ('()eifi(:i(;nts a, b a,ll(t c ()n th(; axial wtria,1)h; is lm(;ss(;ntial. As note(1 in w 1.3.2, 1,1~(;s(;(, {A(q, p)} st)a,ns a, I~i(; a,lg(;l)ra, ~m(l(;r l)()iss(m 1)ra(:ket. Tlle P()iss()xl t)ra(:kets ()f tim (:()ll,juga,t(; wi,rial)h;s q a,ll(t p witll A are linea,r flm(:l,i()lls ()f q a.ll(l p. 111 fa.(:t, w(: Iin(l

\p/

, A(q,p)

-

"k,

wh(;r(,~ (:xi)li('itly is "k a - (cq + b p ) ~o in(livi(tlmliz(;s a, (:()rr('sl)(nl(l('~ll(:(~

-a,

P

_

(aq

{A(q,p)} <

+

cp) ~i~.

, {A}

-

~:

P

Evi(h,a~tly, r('~la,(,i()n (1.7.7)

(1.7.8)

()f the (tua,(tra,ti(: h(mu)g('al(;()lls t)()lyll()mia,ls A(q, p), an(t h(;lu:(; ()f tim a,ss()cia,ted Lie ()t)era,t()rs LA, |,() tll(; 2 x 2 llm,tri(:es

whose entries are given l)y the c()efli(:icnts of the (tua(trati(: m(m(mfia,ls in A. The nmtI'ix A (:a,n be rega,r(te(1 a,s tile matrix ret)resenta, tive of the quadratic form (1.7.6), an(t a(:('or(lingly ()f the corresponding Lie ot)era,tor ka- A has null trace being T r ( A ) = c - c = 0. Note that the matrix representative (1.7.5) of the qua,(tratic Hanfiltonian (1.7.1) is traceless. The linear space of nonsingula,r square matrices configures a,s a, Lie algebra with respect to the Lie bra,('ket [A, B] = A B

- BA.

(1.7.10)

In particular, since T r ( [ A , B ] ) = 0, even the subspace of traceless square matrices is a Lie algebra with respect to the commutation product (1.7.10). Tile Lie algebra {A(q, p)} of the homogeneous quadratic polynomials is isomorphic to the Lie algebra of the traceless matrices {A}, formed by the rule

Hamiltonian Picture of Light Optics. First-Order Ray Optics

39

(1.7.7). The isomorphism between two Lie algebras demands for a one-to-one mapping, which preserves the Lie products associated with the given algebraic structures [9.6]. It is not difficult to prove that if by (1.7.7) the matrices A and B correspond respectively to the quadratic homogeneous polynomials A and B, then the commutator C = [A, B] corresponds to the Poisson-bracket generated polynomial {A, [/}. Hence, the correspondence (1.7.8), generated through (1.7.7), preserves the Lie products of the two algebras, i.e., the Poisson bracket for the quadratic homogeneous polynomials { A ( q , p ) } and the Lie bracket for the 2 x 2 traceless matrices {A}. In addition, the polynomial A(q,p) can be recovered from the matrix (1.7.9), forming the scalar product l(Au, Ju)-

A(q,p)-

l ( q , p ) A T J (q),

(1.7.11)

where J is the 2 x 2 symplectic unit matrix and A T is the transpose of A. The mapping (1.7.8)is then an isomorphism of { A ( q , p ) } onto {A}. 1.7.2

I n p u t - o u t p u t relation f o r f i r s t - o r d e r optical s y s t e m s

Consider the matrix form (1.7.4) of the linearized Hamilton equations (1.7.3). Suppose to divide the space interval Z o - zi from the input to the output plane into a number N of segments A z j , j = 1, .., N appropriately chosen so that in each the Hamiltonian matrix H bc almost constant, i.e., H(z)- H(zj)-

Hj,

Zj ~ Z < Zj_t_I,

(1.7.12)

j = 1, .., N,

with z~ =_ zi and ZN_t_1 ~ ZO. Clearly in each subinterval the solution to (1.7.4) is obtained by the exponential mapping of the respective matrix Hj as U(Z) -- ~Hj.

U(Zj),

1, .., N. (1.7.13) The transfer of the ray from zi to Zo, and hence of the phase space vector from u(z~) to U(Zo), is then completed by concatenating the N exponential mappings (1.7.13) in the proper order according to Azj

U(ZO ) __

Azj

-- Z j + l -- z j ,

eHN'AZN.~

zj ~ z < Zj+l,

eH2./~z2 ell1. AZl U(Zi),

j --

(1.7.14)

and correctly taking the limit for N -+ oc, that is for infinitesimal elements A z j ---+dz, for which condition (1.7.12) can safely be kept. The exponential function is defined for matrices exactly as for scalars. Thus, each exponential in (1.7.14) means the power series oo

eHj.Azj Z ~

h=0

(Azj)h h h! H j ,

(1.7.15)

Linear Ray and Wave Optics in Phase Space

40

and so is seen to be a 2 x 2 matrix, as each power in the series is a 2 • 2 matrix.

Therefore we can ultimately express the i n p u t - o u t p u t relation from zi to zo in the explicit inatrix forln

po/

\Pi) '

(1.7.16)

l,lu~s esl, at)lishing the linear relatir 1)el,ween the ray varial)les at, the assigne(t t)la.nes, we (:la.inm(1 as ttw, (listin(:tive feat~n'e r first-r162 r 1.7.3

The ','ay t','ansfi;'," mat','iz

Tll(' 2 x 2 matrix ill (1.7.16), wlli(:tl r162 the ilq)llt-t()-()lltlnlt relati(m fi)r the ray t)rot)agati()ll thr(nlgll a "lillear" (tevicr betweell tw() assigllr reference s(:reens, is kn(~w~ as I,h~; 't'a,y t't'a,'n,.@:'~"(r A B C D ) 'nt, at'ri.r [16]. It is ~,vi(lent tlm,t the (tia,gonal elelnents A a,ll~l D are (tinlellsi~nfless, wllih; tlle ~fl'-~tiagonal B a.ll(l C tlave (tix~w~si(n~s ~f h'~gt, l~ m~(t i~w'rse h'x~gt,t~, resI)ectiw'~ly. M is .j~st the ,la(:()|)ian matrix (,f tl,e trm~sf()rl,mti()x, \](,~'~Ii'~---, --\]{q"],n,(h;rp,,tl,(; IIanfilt(mian (1.7.1); it, is thel~ sy~l)h;(:ti(: al~(l sr162 (h;t M .... 1 (s,;(; Prr 6). The ray tra,xlsfer ~ml,rix is 1,1~(;lmsi(: to{)l fl)r 1)(~tl~ l,l~(; (liagl~sti(: m~(l the (lesig~ {)f first-or(ter {)l)ti(:al sysl,el~s, l"irst-{)rr ()t)ti(:s ix i~ fa,r x~m,i~ly (:(n~r witl~ the {h;ternfi~ati(n~ {~f l,l~; ~I)ti(:al x~m,tri(:es, wt~i(:t~ c()~I)h;tr (lesr the t)araxial ray t,ra~sf~)r~m.l,i~x~s fl'{)~ l,lm int)~l, l,~) 1,t~r {)~11,t)~11,I)lmm ~)f l,l~e sysl, em. T~rn t)ack a,ga.ix~ t,~) E(t. (1.7.4). Following t,lm s~ggesl, icm r the transfer eq~lation (1.7.16), we ext)ress t,l~e general sol~tion t{, (1.7.4) in the fi,rm u(z) = M(z,

zi)u(zi),

M ( z , zi) --

A(z, z i) C(z, zi)

zi) )

D(z, zi)

(1.7.17)

letting the ray-matrix, i.e., its e11tries, vary contimumsly with tim axial variable z, the assigned initial positioll Zi entering as a fixe(1 parameter. Then, replacing for u(z) in (1.7.4), we obtain the equation obeye(t by M ( z , zi) as d

d-TM(z, zi) -

H(z)M(z,

zi), M(zi, zi)

- I.

(1.7.18)

The transfer m a t r i x in (1.7.16) is a definite deterInination of M ( z , z~) for the specific vaJue Zo of the evolution variable z; as the latter allows us to follow the ray evolution continuously along the propagation distance, the former catches two "instants" of this propagation at the assigned reference planes. If one is not interested in the deta, ils of the ray evolution along the optical system, once the relevant transfer m a t r i x has been determined, the real optical system t h a t fills the space between the end-planes becomes unessential; all we need to know about the t r a n s f o r m a t i o n the system may act on the light rays is

Hamiltonian Picture of Light Optics. First-Order Ray Optics [-[ i

41

l-] o

field ray

I ~i=0 ',

qi=l

'

~

axial ray

Z O

Input ptane

Z

Output ptane

FIGURE 1.8. Cos-like and sine-like solutions of the linearized Hamilton's equations for the light ray.

stored in that matrix. The knowledge of the transfer matrix may be gained by integration of (1.7.18) or by other sort of considerations, as we will see later. With the form (1.7.17) for the ray matrix M(z, zi), Eq. (1.7.18) yields the set of first order differential equations for the entries A, B, C and D as dA

C

dz dC

no

dz

=-n2A,

A(zi)- 1 C(zi)-

O.

dB dz dD dz

-

D no

=-n

,

B(z ) = o

2 B,

D(zi) = 1.

(1.7.19)

Both pairs (A, C) and (B, D ) o b e y Eqs. (1.7.3). They represent two particular rays evolving from the input data (1, 0) and (0, 1), respectively. In fact, the ray with coordinates (A, C) is a paraxial ray leaving the input plane parallel to the axis at the unitary height: qi = 1; it is called field ray. On the other hand, the vector (B, D) describes a paraxial ray having null height and slope angle Oi = 1 / n o at the entrance plane (Fig. 1.8); it is called axial ray [4.2]. Equation (1.7.17) suggests therefore a view of the ray propagation as resulting from the superposition of the axial and field ray evolutions with weights fixed by the initial values (qi, pi); clearly, superposition is a by-product of linearity. Eventually we are confronted back with Hamilton's equations (1.7.3). But, we search now for two particular solutions, which, originating from specific sample rays, allow us to define the ray propagation matrix M(z, zi). It is a significant consequence of linearity that the knowledge of the behaviour of the axial and field rays yields a global characterization of the ray propagation through the optical system (see Problem 7). In conclusion we may say that the 1D ray-optics of first-order systems is wholly contained into Eqs. (1.7.16) and (1.7.18), the former picturing any first-order optical system as a "black box" fully characterized by its own raytransfer matrix M, and the latter tracing the procedure to determine M, once

42

Linear Ray and Wave Optics in Phase Space

specified the geometry and the optical properties of the system through H, the m a t r i x image of the relevant H a m i l t o n i a n function. Alterna.tive a,pproa,ches to the integration of Eq. (1.7.4) or (1.7.18) make use of the Lie algebra, based m e t h o d s [13]. Matrices beha,ve like operators rather tha,n like scalars; they may not obey the commuta.tive la,w of multit)lica, tion, and hence the rules of ca,l(:~fl~s for scalars (to not apply in genera,1 to ma,tri(:es a,s th(;y (t() ~()t at)t)ly it~ general t() ()t)erat()rs. ()n integrating E(t. (1.7.4) ()r (1.7.18), w(: are, (:()~ffr()~m;(t witl~ tl~e, ()l)(;rat()r-likc ~mt~m', ()f tl~e ~mtrix H. TI~(;~, l,l~e (:()~si(lera,ti()~s (lev('l()t)(',(l ill ,~ 1.4.1 r(;ga.r(li~g tl~(; i~t(;gra,ti()~ ()f (1.4.3), a,t)t)ly lmre as well. We sha,ll ~()t, (t(;ta,il the st)(;(:ifi(: asl)(;(:ts ()f tl~e t)rol)len~. W(' ()nly ~('.~l,i()x~ that, as i~ tl~('~ (:a,s('~ ()f tl~(; trm~sf('x ()l)(;ra,t()r, when the Ha,**filt()**ia,** **mtrix H (t()(;s ,~()t ('xI)li(:itly (t(;I)('**(t ()** tl,(', axia,1 w~,ria,1)le z, w(' a.re (;~m.l)l(;(l t,() rel)r(;se~d, M ( z , zi) i~ tim (:l()s('~(l f()x'~ as tl~(,~ si~gl('~-(',xt)()~('~tia,1 ()f H thr(n~gl~ l,l~('~wl~()l(; t)r()t)a,gati(~ l(;~gtl~ fr(n~ z,i t() z: M ( z , zi) - ,,H.(=-:,).

(1.7.20)

Rea,(lers lllay {'lle(:k tlla,l, (1.7.2{)) s(flw;s (1.7.18) |)y (lil'e(:l, (:al{:lllal,i(,ll. Als(), we 1nay llote tlmt all tll(; illfillitcsillm,1 trmlsfor~im, ti()lls ()f tim tYt)C ~,H,l: in the ixlfillitc Se(tlU:ll('(; ()l)l,aille(l fl'(nll (1.7.14) f()l" N + oc (:()lira I,() lla,w~ t,lle same r ill 1,11(',r ()f a. z-ill
1.7.4

The Lagrange i n v a r i a n t

Let (ql ( z ) , P l ( z ) ) and ( % ( z ) , p 2 ( z ) ) denote any pair of paraxial rays evolving according to Eqs. (1.7.3). It immediately follows that d dz (qlP2 - q2Pl) - O,

(1.7.21)

Hamiltonian Picture of Light Optics. First-Order Ray Optics

43

and accordingly qlP2

--q2Pl

-- qliP2 ~ --q2iPl~

-- const.

(1.7.22)

Equations (1.7.21) and (1.7.22) establish the invariance of the cross product

A -- qlP2 - % P l

(1.7.23)

for any determination of the ray variables (q, p) along the ray trajectory. A is called Lagrange invariant [4.2]. Within the phase-plane picture, the modulus of the cross product qlP2 - - q2P~ is the area of the parallelogram in the (q,p) plane, delimited by the vectors with components (ql, Pl ) and (q2, P2 )- Equation (1.7.22) states then the inva,riance of the area of such a parallelogram, thus expressing Liouville's theorem within the context of 1D linear optics. The invariance of the skew-product (1.7.23), discussed here within the specific context of 1D first-order optics, characterizes in general any Hamiltonian system under the linear approximation. In fact, the above result relates directly to that established in w 1.3.1 about the invariance of the product (Ul, U 2 ) ~-- (Ul, J U 2 ),

(1.7.24)

under the linear symplectic transformation (1.3.17). Within the formalism of classical mechanics, = (u~, u 2) is cMled a 2-form [3.2, 3.3]. It establishes a correspondence of pairs of pha, se space vectors to rea,1 numbers; in particular, when applied to infinitesimal vectors, it produces the so called fundamental 2-form (due, du2) = (dUl,Jdu2) , which, as shown in w 1.3.1, rema,ins unchanged under any symplectic transformation. Evidently the form (1.7.24)is antisymmetric: w (Ul, u~)= - = (u2, u~). It reduces to the La,grange invariant (1.7.23) in the case of a 2D phase space. Notably, the Poisson bracket of dynamical variables can be understood as 2-form (Eq. (1.3.34)). Some properties of v~ (Ul, u2) are listed in Problem 9. As a conclusion we may say that linear ray-optics deals with a 4D (or 2D for stigmatic and simple astigmatic optical systems) linear vector space, where the antisymmetric and nondegenerate form (1.7.24) is defined. First-order optical systems generate symplectic transformations of the geometrical-optical phase space, that preserve the 2-form (1.7.24).

1.7.5

Square law medium

We briefly detail the second-order coefficient n2(z ) in the series expansion (1.6.4) of the refractive index n(q, z). The quadratic profile (1.6.4) is a typical model for the square-law (also referred to as lens-like or graded-index) medimn. We consider the surfaces n(q~, %, z) = const., which intersect the optical axis at right angles. In particular, the refracting surface that intersects the

Linear Ray and Wave Optics in Phase Space

44

axis at the point Zo is represented by the equation (1.7.25)

n ( q , , qy, z) -- n(O, O, % ) -- ',t o (Zo).

Ext)anding the left hand si(t(; at)ovc at the lower or(ter of qx, qy amt z, in the assu~nt)tion of rotational synmletry, we lmve ,,(q,,,qy,,)

~

, .,,,,(~o)+,,,,(,,,)(,

- ~,,)

, 2 + q2) , ~,,~(~,,)(q.,:

(1.7.26)

the' t)rinle (len()l, ing a.s llslm.1 (lit[erential, i()n with rest)cot 1,() z. ('~()mt)a.ring with (1.7.25), we ()l)t,a.i~l t.ll(', Slu'fa.('.('~ e.(llla.t,i()ll z - z,,

n 2 (z,,)

')

2

2.,,i)(~.,,) (q.; + q,) - ~

1

9

2

(q.7.. + q,),

(1.7.27)

whi(:ll r(:l)l'(:s(:lll.s a. l)ara.l)()h)i(l llavillg a l. (.If(: v(:rl.(:x z,, (.If(: ('llrwtl.llr(: ra.(lills

1<,, "i'(~") ,,~(~,,).

(1.7.2~)

The refra(:(.illg po',)('r 79 ()f a sln'fa.('(: is (l(:Iill(:(l as (.If(: ra(.i()()f (.If(: ()n-axis wu'ia.ti(nl ()f (.If(: r(:fi'a.(:tiv(: i~(h:x A)). (.l~r()~gl~ (.l~(: slli'[a.(:(: (.() (.l~(: (~lll'V/,l.|,llI'( ~. ra.(li~s R ()f t.l~(: s~rfa.(:(:, i.(;., 79 ---. A ~ / l ? . 'l'l~(:r(:f()r(: l)y (1.7.28) w(' (:a~ r(qa.t.(; t.l~(: (:()(:Ili(:i(:~d, 't,.2(z,, ) (.()(.I~(: s,~rfa(:(: i)()w(:r 79 (z,,)at, z,, as

',,..e(z,,)dz = 79 (z,,),

(1.7.29)

wlfi(:l~ (xn~w,.rs(;ly s~gg(;sts t,()(;xl)r(;ss 79 (z,,) a.s (.l~(; (tifl'~;r(;~tia.1 ()f tl~(; a r(~a, ~m(l(;r (,]~e (:~rw; 'he (z), ~ma~ely ~, (7.,,) - d

1.8

,,~(~')d~'.

(1.7.:~o)

ABCD matrix and focal, principal and nodal planes

In the theory ()f imag(;-fornfing (teviccs, the ot)ti(:a,1 systems arc c()mmonly characterized t)y the c a r d i n a l p o i n t s and their associated p l a n e s [17]. In this sc('tion we outline the relation between the r a y - m a t r i x entries A, B, C, D of the optical systcm, referenced to a.ssigned input and o u t p u t planes, and the location and role of the relative cardinal points or planes. In this respect, it is ('onvenicnt to recall the usually adopted nomenclature and sign conventions. 1.8.1

Object space and image space

Given an optical system, we take as usual the optical axis as z-axis, oriented from left to right according to the assumed direction of ray-propagation. Supposing the system occupies a certain region of the zyz-space, extending on the

Hamiltonian Picture of Light Optics. First-Order Ray Optics

45

axis from z o to zi, w(] call the space z < z o the object space and the space z > z~ the image space. It is a cotnmon situation that the optica,1 system separates two holnogeneous media. We assign the refractive indices n01 to the object space and n% to the image space. The light rays in the object and image spaces are sections of straight lines and the optical system determines a transformation of the straight lines of the object space into the straight lines of the image space. The portions of these straight lines that as light rays have a physical reality are called real parts of the ray, whilst their extrapolations, which have no physical reality, are called the virtual parts of the ray. Referring to the object and image spaces, it is evident that the half space z > z o is the virtual object space, as the half space z < z I is the virtual image space. A bundle of rays incoming toward the optical device is said to belong to a real or virtual object point, if it originates at or converge to a point located in the real or virtual object space. Similarly, real and virtual image points are understood as the points of intersection of the real and virtual parts of a bundle of rays emerging from the system; outgoing rays then converge to real image points, whilst diverge from virtual image points. In order to specify tile character (real or virtual) of the object and image points appropriate reference frames are arranged in the object and image spaces. Denoting indeed by II o and II~ the planes normal to the axis at z o and z,, in the object reference frame the axial distances are assumed positive when they are measured to the left, of II o and negative when they are measured to the right. Likewise in the image reference frame, the axial distances are positive when measured to the right of II~ and negative when to the left. In both cases, the transverse distances are positive upward from the axis and negative downward. In the following, the terms object and image spaces will be used to comprise the relative real as well as virtual half-spaces. Finally, we recall that a point P in the object space and a point P' in the ilnage space are called conjugate if all the rays through P are brought to P'. Due to the principle of the reversibility of light rays, it is evident that all the rays through P' are brought to P; in other words, P and P' as object and image points are interchangeable. The planes through P and P' perpendicular to the optical axis are called conjugate planes. It is worth stressing that the expression "all the rays through P" must be understood as all the rays of a finite bundle, whose transverse sizes are determined by the optical system and the approximations inherent in the adopted formalism. Then, let us assign the input and output planes IIi and IIo, whose location along the optical axis can arbitrarily be chosen; in fact, the ray-variables q and p refer to the real as well as the virtual parts of the ray. Clearly, a formal simplification is gained if one assumes the object and image planes ri o and II~ as input and output planes, respectively.

46 1.8.2

Linear Ray and Wave Optics in Phase Space

Ray-coordinate and m o m e n t u m magnification

The ray-coordinate (or linear) magnification of the optical system fl4q, relative to the assigned reference planes, is the ra,tio between the heights of the emerging and incoming rays, namely AAq = qo. qi

(1.8.1)

Sillfila.rly, tile 'l'ay-',wmr 'm,agn'tificatio'n, is (lcfi~le(l ill t,cr~llS ()f t,llc ()t)ti(',a.1 ~n()nlenta ()f t,ll(; i~l(:(~lllillg a.ll~l (~11t,g(fing rays a.t:(:(~r(lillg t() All v = P'~. Pi

(1.8.2)

Illt, enllS of tim clltri(;s ~f tim ray tra.llsfer ltm.trix (~f tll~; systelll, .A,qq ml(1 .A//p a,l'(; (;xl)r(;sse,(1 a.s A4q - A + I:~p~, .hAp - 1) + ( : q i . (1.8.3) q,i Pi F(~r a, fiel(t ray, l,ll(; ray-r162 ma,gnifi(:at, i{nl takes tim siIIll~l(', t'(n'lll .Mq = A ,

(tir

r a y : pi = 0)

(1.8.4)

a.n(t (:()rr(;st)(~(li~gly fl~r m~ axial ray tl~(; r a , y - ~ { ~ ( ' ~ d , ~ ~m,g~ifi(:ati(n~ is M,,--

I),

(axial r a y : qi = ())

(1.8.5)

t,hl~s yiehling fin" tim t;lll,l'i(;s A a,n(l D l,h(~ n~(;a,~i~g a,s l,h(; ray (:o()r(li~m,t(~ and In()lll(~llt, lllll I~mg~fili(:a,ti()~s f()r tim tiel(1 a,n(t axia,1 rays. I~terestingly we (:aa~ i'(~la,t(; t,]l(; ang~fla,r a,l)ertllre of a, fall of rays (;l~m,na,ting frolll all ot)ject t)()i~t, to t,l~a,t, ()f l,tm sa.ille fall ()f rays (:()~mi~rgi~g t() l,lm inm,ge 1)oint after tl'a,versi~g a I)(:rfi;t:t inmging syst(;~. Sut)t)()s(; tha, t Pi m~(1 Po are tw() (:()njugat(; i)()i~fl,s i~ l,h(; ()l)je(:t an(t ilnag(; st)a,(:(~ ()f tim system. All the rays fl'om Pi int, ers('~(:t a,t Po irrest)e(:tive of their a,~gles to t,h(; oI)tical axis. We may (:onsi(ter, for i~lsta, n(:(~, t,w() rays ema,nating frOlll Pi with a,ngles l~)l,i a,nd 02~ to t,l~e axis, whi(:t~ arc i~a,g(;(1 into Po with angles l~lo and ~2o. According to the phase-plane area preserving rule, which for t)araxia,1 t)rot)agation takes the form of the Lagrange invariance (1.7.22), we (:an write

ltol qi,/k~)i -- no,2 qoAzgo,

(1.8.6)

where qi and qo axe the heights to the axis of the object and image points, whilst Atgi = t92~- l~lli and A~9o = t92o -t91 o represent the angular separations of the fan of rays before and after propagation through the imaging system. Equation (1.8.6) establishes a clear relation between the linear and the angular magnification of a bundle of rays, whose spatial extension can be magnified only at the expense of the angular divergence, that is by having an image beam with a smaller angular divergence than that of the object beam.

Hamiltonian Picture of Light Optics. First-Order Ray Optics

HF

FIN

FIH

Hi

no

Optical

N fl

-~

~

st:

device

IN'

SI I

I !~

H N, H H,

A B

H

SN, -,

-! S

N

~

47

r

I

Zo

Object space

H'

L

["

i

f

f~ S F,

Image space

F I G U R E 1.9. Cardinal points and associated planes for an optical system represented by the ray matrix (~ ~ ) . The figure is drawn for the case where C < 0, @ > 0 and - ~ > 0.

1.8.3

Focal, principal and nodal p l a n e s

Consider ilOW light rays incoming parallel to tile optical axis" Pi O. Tile output ray variables qo a,nd po depend on qi only, thus yielding for the ratio qo/po the qi-in(tependent value -

qo A = Po C"

-

(1.8.7)

Since qo/po - - s 2 / n o 2, where s 2 specifics the axial coordinate in the image space of the intersection of the outgoing ray with the optical axis, the above relation states t h a t a bundle of parallel rays, incident oil the system, is transformed into a bundle of rays, whose real (virtual) part is directed toward (diverges from) the back focal point F ~, located on the axis at s~,

_

A -n02 ~,

(1.8.8)

from 11o. The axial coordinate s r, of F' is named back focal length and the plane II F, through F' orthogonal to the axis is the back focal plane (Fig. 1.9). The opposite situation corresponds to a ray emerging from the system with zero slope, po - Cqi + Dpi - 0, and arbitrary qo thus implying q~

D -

Pi

(1.8.9)

C"

Being qi/pi --Sx/Tt01 , where s I specifies the axial coordinate in the object space of the intersection of the incident ray with the optical axis, the real -

48

Linear Ray and Wave Optics in Phase Space

(virtual) p a r t of the incident ray diverges from (is directed toward) the front focal point F, located on the optical axis at s~

_

D

-n

0 1C --

(1.8.10)

~

from Hi, which is ret)orted as fl'ont focal len.qth. Tile plane II F t h r o u g h F norma,1 to the ot)tica] axis is tim front focal plane. (Fig. 1.9). E v i d e n t l y the t)rinm.ry a.~l(t se(:on(ta.ry f()(:al t)()ixlt,s are (:(mjugat,(; t() t)oillt, s at all infinitely great (lisl, a,n(:c fr(nll l,llc sysl,(nll r(;st)(;(:l,ivcly ill l,ll('~ i11m,g(; a.n(1 ()l).i(',(:l, st)a(:c. (',(nlsi(l('.r a,ga.i~l a,~l i~l(:i(l('~llt, ray, (liv(u'gi~lg fl'(nll tlle fl'()llt fi)(:al l)()illt F. T h e ray (:()()r(tina,t{;s (qi,Pi) a,r(; r(;la.tc(t a,s qi/Pi - - I ) / ( 7 , a,Ii(l tim ray (;merges fi'(nll tile (nltI)llt t)lml('~ II,, witll % -- --pi/C ml(l z(;r()sl()I)('.. Ext(ul(ling the i~l(:i(l(ult ml(t r(;fl'a(:t(:(t rays t,() tl~(;ir I)(fi~t ()f i~t(;rse(:ti(n~, w(; (t(;fi~e the t)la,ne II,, tl~r(n@~ tiffs l)()i~t ~()r~a.1 t() tl~(; axis a,s tl~('~ primary principal planeo Si~filarly, 1,1~(; s('(:o'n,dary tn'incipal plan('. II,., is i(l(:~titi('.(1 l)y l,l~(', i~fl,('.rs(;(:tion of tim r('a.1 ()r virt~m.1 imrl, ()f a l)aralh',l i~l)~l, ray m~(t l,lm (:()rr(:sl)(n~(li]~g (n~ti)~]t ray, (:(n~v(;rgii~g l,() lit(; tm(:k f()('al l)()i~t F' (% - Aqi, p,, - (;qi). 'l'l~('~ illl,('rs(;(:ti()n I)()i~l,s It a,~(l It' ()f II,, a,~(l II~,, wil,l~ l,h(; ()l)l,i(:a,l axis a,r(' 1,1~(' pri'm, ary a,n(t s('co'n,dary p'rin(:ipai points, l()(:a,t(;(l at '~ll

~

11[) ,

1 -(, I) .

s,,,

--'~,,~

I -A ('

'

(1.8.11)

r(;sl)(',(:l, iw;ly in t,h(: ()l).i(;('l, a.n(l i]~m,ge sl)a,(:('.. F~rt, l~('r]~()re, 1,1~(: l)()siti(n~ ()f 1,1~(; fi'(n~t f()(:a,1 i)()i~l, F wil,l~ r~'.Sl){:('l, t,() II~ is sI)(;(:ili(:(t t)y t,l~(: s()('alh:(l p,'i'mary .ritual h"n~.lth, .f,, given as qo .f, -

n,,

1to

--

--

'Pi

'

(1.8.12)

C '

and (:()rr(;st)(mdingly th(; t)()siti(n~ ()f F' with rcst)(;(:t to II,~, is given |)y

qi L -

'11'02

-"',,,2po -

C '

(1.8.13)

ret)ortc(t a,s the secondary focal length. Notat)ly, the t)rimary a,n(t se(:(m(tary focal lengths are in the s a m e ratio as the. refractive indices of the nm(tia on the two sides of the s y s t e m

f2 .fl

7?'02 --. rto~

(1.8.14)

In particular, if tile m e d i u m is tile same on b o t h sides: it01 - rt02 , then fl - fu. Conventionally, the ratio of the p r i m a r y and s e c o n d a r y focal length to the respective refractive indices, d e n o t e d as f

fl T/'01

=

L , TL02

(1.8.15)

49

Hamiltonian Picture of Light Optics. First-Order Ray Optics

is simply referred to as the ( r e d u c e d ) focal length of the system. The inverse of f defines the (focusing) p o w e r of the system: 1

7) -- --;, J

(1.8.16)

which is typically measured in diopters (D) if f is given in meters. As the focal points F and F' are both real ( f > 0) or virtual ( f < 0), the sign of f allows a characterization of the system as a converging or diverging device. When f > 0, rays diverging from F are imaged into parallel rays and correspondingly parallel rays are imaged into rays converging to F'. Conversely, when f < 0, the system turns rays converging to F into parallel rays and parallel rays into rays diverging from F'. Also, on account of (1.8.12) and (1.8.13), the lower-left entry C of the ray inatrix silnply relates to f as 1

C -

P.

(1.8.17)

f Remarkably, if the locations of the focal and principal points (or planes) relative to the assigned input and output planes are known, the behaviour of the system with respect, to these planes is completely determined, being the relevant ray matrix known by (1.8.8), (1.8.10)and (1.8.11). Thus, tile system can effectively be modeled by its two focal and principal planes. In tile next section, we discuss a characterization of tile optical systems, for which the focal or principa.1 planes are a,ssumed as reference planes. Let us establish now the condition under which an incident ray with slope ~9i - p i / rt01 at Hi emerges from Ho with tile same slope ~9o -- P o / no 2 -- tgi. According to the expression (1.8.3) for the ray-momentum magnification Adp, it is evident that the input ray coordinates must satisfy the relation .A4p - D + C qi Pi

n~ .

(1.8.18)

~Ol

Thereby, any incident ray, whose real (virtual) part originates at (is directed toward) the point N, located in the object space at

SN ---

~'-~01 ( ~'~'02 -- D)

C

--

--fl (f2

__ D)

fl

S01

(1.8.10)

'

is transformed into a ray with the same slope" tgo - ~9i, whose real (virtual) part is directed toward (originates at) the point N', located at

SNt in the image space.

•02 ( •01 C Tb02

A)

-

-L(-7-

fl--A) J2

,

(1.8.20)

Linear Ray and Wave Optics in Phase Space

50

T h e points N and N' are called nodal points and accordingly the nodal planes II N and II~, are tile planes t h r o u g h N an(t N' n o r m a l to tile axis. It is clear t h a t N and N' are conjugate points. If the m e d i u m on b o t h sides of the s y s t e m is tile same, the no(tal t)oints (:oinci(te with the t)rincit)al t)oints: N - H an(t N' - H'. In general, wh(m 'n ol r 'n02,, the two set of t)()ints are set)arat(;, a,lth(n@l th(: (tista,n(:(; t)etw(;(m the no(ta.1 t)oints is exa,ctly ('qual to l,l~(; (tista.n(:(; l)ctw(;('n the t)ri~:(:it)a,1 t)()i~ts: NN' = Ittt'. In a,(t(titi()n, tl~(', fl~rther r(;la.ti(n~s: F N = F'tt' a,~(1 F ' N ' = FIt t)('tw('~(;~ tl:(: f()(:al a,~(t ~()(tal I)()i~:ts hol(t. ()tl~(;r (:a,r(li~ml I)()i~l,s a,r(', (l(:ti~(',(l wl~i(:l~ a,r(; ()f s(n~(; int(;r(;sl,, as th(', 'negative ~:a'~'di'n,al a.n(1 nodal point,s, r(,st)(;(:tiv(;ly (:ha.ra,(:t(:riz(;(1 t)y a, ~l~il, ~(;gal,iv(; lat(:ral m~(1 m~g~fla,r ~mg~fifi(:a,l,i()x~ (s(;(; t)r()t)l(;l~ 11). Finally, t() fln'th(;r (',vi(h;l~(',e tl~(; a,na,h)g~V 1)(:l,w(;(:n the (:hara,(:l,(;riza.ti()n ()f th(; ilnagi:~g t)r()l)(;rti(;s ()f m~ ()t)l,i(:al sysl,(;n~ t)y tl~(', (:a.r(li~ml t)()ild,S (a.l~(l t)la,:~(;s) a.~i(t tlm.t t)y tl~(; ra.y l~mtrix, w(; (l(;(l~u:(; fl'()~ (1.7.1(i) l,l~(; relati()l~

[

'll,() l

1-1) ] + [ ,

'll,() 2

I-A ]

2

~,

_ -(7,

(1.8.21)

It(), 2

wl~(;r(; a,gai~ s~ m~(l s~ give l,lm axial ('()()r(ti~mt('s ()f th('~ t)()il:ts P~ m~(t P2 a,t whi(:l~ tl~(; i~('i(l('~l, m~(! r(;fi'a,'l,('(l rays i~l,('rs('('l, l,l~(; ()l)ti('al axis r(;sl)(,('l,ively in tl~(' ()l).i(;(:t m~(l i~a.g(; Sl)a.(:(;. ()~ a.('(:(n~t ()f (l.8.11) an(l (1.8.15), tl~(; al)()v(; relal,i()x~ (:m~ I)(, (:ast i~ 1,11(; fi)r~ ()f l,t~(; well k~()w~ thin lens fo'~"m,'u,la +

-

-

:~, ;v~ .f' with 741a,n(1 7~ (t(;ll()tillg tll(; axial (:()(n'(lilm,tt:s ()f tll(', t)()illts P l a,:l(t I),,. r(:fi;:'en(:ed t,) tile I)rinr l)lmms I-I~ a ll(t II,,,. By a, filrtlmr ~Im.lfitnflati()ll, (1.8.21) (:a,xl t)(; giv(;ll the Nc'wtonian form (1.8.23) or nlore c(nl(:is(~ly ,~, ~2 - f, L ,

(1.8.24)

where the axia,1 c o o r d i n a t e s '~1 and '~2 of the points P1 and P2 are referenced to the focal planes II~, and IIv, , respectively. 1.8.4

Special Optical Systems

T h e ray m a t r i x takes specific forms w h e n the focal or the principal planes are t a k e n as input or o u t p u t planes [16, 17]. In fact, w h e n tile principal planes IIH, IIii , are t a k e n as input a n d o u t p u t planes tile front and p r i m a r y focal points as well as the back a n d s e c o n d a r y focal points coincide: s~ = fl and s~, = f2T h e ray m a t r i x from the p r i m a r y to the s e c o n d a r y principal plane is therefore

Hamiltonian Picture of Light Optics. First-Order Ray Optics Hi

l-[o

I

Ilo

[]i

Optica[ System

qi I

51

Optica[ System

B=O

D=0

qi I r

(b)

(a) VIo

/@p

Hi

1-lo

Optical System

Optica[ System

A=0

C=0 r

FIGURE

1.10. O p t i c a l s y s t e m s h a v i n g one v a n i s h i n g e n t r y in their ray t r a n s f e r m a t r i x . (a) s y s t e m s ( D -- 0), (c) focusing s y s t e m s (A -- 0)

imaging s y s t e m s (B -- 0), (b) collimating a n d (d) telescopic s y s t e m s ( C - 0).

M. ,.,-

(1 0) C

1

'

(1.8.25)

where by (1.8.17) C = - 1 / f . The form of MH_+H, evidences the property of the principal planes to be conjugate, points of II H being imaged into points of IIH,. In particular, (1.8.25) produces a unit positive image magnification for any input ray. In fact, the principal planes are also defined as conjugate planes for which Adq = 1, and accordingly called unit planes. Matrix (1.8.25) is a particular form of the most general optical matrix Mimagi"g - -

(A 0) C

1/A

'

(1.8.26)

which, having B = 0, relates conjugate planes. In fact, as qo = Aqi, the position of the image ray depends only on the position of the object ray; thus, all the rays through the point at qi on IIi are brought to the point at qo on IIo (Fig. 1.10.a)). Evidently, AJq = A. Optical systems described by matrices having B = 0 are called imaging systems (or, magnifiers). Special systems are also obtained when the focal planes are taken as reference planes. In particular, the ray matrix for the propagation from the front

52

Linear Ray and Wave Optics in Phase Space

foca,1 pla,ne to a, given o u t p u t plane comes to have D = 0, n a m e l y Mc~

--

C

0

(1.8.27)

'

T h e sign of C = - 1 / f establishes if the object a,nd image distributions are real ( f > 0) or virt~m,1 (.[ < 0). ()ptica,1 systems, (tescrit)e(t by the ra~ m a t r i x (1.8.27), into,go a, t)()int int() a tnn~(tlc ()f tm,rallcl ra.ys (Fig. 1.1().1))) sim'e Po (Tqi f()r all Pi. 'I'tl(;y are r(;f(,rr(,(l 1,() as (:olli'mo, ti~t,fl syste~lls. C(nlvt'~rst;ly, (,lit' t)r()l)a.ga,t,i(nl fl'(nll a, givtul illt)ll(, t)l~nl(; 1,()(,ll('~ lm,(:k f()(:a.1 t)la,nc is (:ha,ra,(:tcrizc(t l)y A = (), a,ll(t ht;n(:(; t)y th(; ray l~m,trix Mr'"""~"~ =

C

D

"

(1.8.28)

Syst(:l~s (1.8.28) i~mg(: a t)~(ll(, ()f I)arall(:l rays i~t()~,. I)()i~t (Fig. 1.1().(:)) as qo = - P i / ( 7 f()r all qi. Tll(:y at(: (:alh:(l .fiw'tt.si,~,yl sysl,(:l~S. As w(: will s(:(: la,t(:i', tt~(: (:()1~(li(,i()l~ A = () is 1,1~(: r(:(l~isil,(: f()r (,1~(: ()l)ti(:a.1 sysl,(u~ (,() l)(:rf()n~ tt~(: i'mlw.'~fc<'t s(:al(;(l F()~n'i(;r l,l'a,llsf()rlll ()f (,l~(; i~l)~(, siglml. N()l,e (,]la.(, (,1~(; l)ositi(n~s ()f tl~(: ()~t(,t)~l, a,~(l l,l~(: il~l)~(, lfla.~(: f(n 1,1~(: sysl,(u~s (1.8.27) a~(l (1.8.28) (:a,l~ 1)c (l('tt;r~il~('(t fr(n~ t,l~(' r('lativ(' l)()sit, i(n~s (1.8.11) ()f t,l~(' l)ri~tt'ilml l)()i~t,s. Als(), if 1)()1,1~ 1,1~(,fl)(',a,l 1)lm~('s a.r(' (,a.k(;l~ as r('fi;r(;l~(:(' 1)lm~t's, 1,1~(' ray-l,ra,l~sf('~r 1~ml,rix ff()m (,h(: fr()lll, l,() (,I~(: l)a.(:k f()(:al l)Im~(: wril,(:s as M,., .,,,,

C

()

(1.8.29)

'

(,ln~s (:()~nl)i~d~g (,l~(: f(:a,(,~I'(:s ()f ])()(,l~ (1.8.27) m~(l (1.8.28). Sysl,(u~s lik(: (1.8.2(3) t)(:rfon~ tlm pc'Cfi'ct s(:ale(t F()~ri(:r (,rm~sf()r~ ()f tim i~t)~(, sig~al; a(:('()I'(lingly th(:y a,r(: l"(;l)ort(:(l as l)(:rf(:(:t fbttT'i('.'~' t'~'(m,,sfoT"nt(:'t',,~. Anotl~er iI~tcrcsting situa,ti(n~ a,riscs in (,h(: (:as(' wh(:r(: C = (). h~ (,hat case, po = Dpi, and h(:n(:(: the dirc(:tion of the ima,g(: ray (tcpentls only on the direction of the ()|).jcct ra,y; tln~s, a, t)~m(th: of tmralh:l rays is (,~u'~(:(t into an o t h e r t)~mdlc of tmra, llcl rays, as in a teles(:()t)(: f()(:~lsc(t to infinity (Fig. 1.10.d)). T h e ray m a t r i x takes the form M~f~

-

0

1/A

'

(1.8.30)

the c o r r e s p o n d i n g optical s y s t e m s being called afocal (or telescopic). Finally, the pure magr~ifier combines the features of the s y s t e m s (1.8.26) a n d (1.8.30), being b o t h imaging a n d telescopic. T h e relative ray m a t r i x is

M

.....

gnifier

( 0) 0

1/A

"

(1.8.31)

Hamiltonian Picture of Light Optics. First-Order Ray Optics

1.9

53

Summary

Within the Hamiltonian formulation of geometrical optics, light rays arc described a,s points in the 4D geometrical-optical phase space, while evolving along the optical axis z of the system. At every z = coast plane, the phase space variables (q,p) specify tile position and tile optical cosines of the ray with respect to the assigned coordinate system in the plane. As a consequence of Format's principle, the optical Hamiltonian function turns out to be H(q, p, z) = - v / n 2 ( q , z) - p2, with n(q, z) being the refractive index function. The phase-space dynamics, which parallels the ray propagation in physical space, is determined by the Hamiltonian set of ordinary differential equations. The ray propagation may be seen as a continuous sequence of symplectic transformations, which map points in phase space into points in phase space as the ray proceeds along the z direction. In this respect, Poisson brackets and Lie operators have a decisive role to form the ray transfer operator, which carries the input ray coordinates to the corresponding values at the desired z. First-order ray optics, which arises from the quadratic approximation of the optical Hamiltonian in both the position and direction ray variables, describes linear transformations in phase space. The ray tra,nsfer matrix M, i.e., the 4 x 4 symplectic matrix containing tile ray-variables independent coefficients of the linear relation of the output to the input ray data, is the mathematical tool to describe the transformation of the ray while going through first-order optical systeins. Notably, in the case of stigmatic and simple astigmatic optical systems, the 2D Hamiltonian model of ray propagation, which deinands for a 4D optical phase space, can be reduced to a 1D picture, demanding for a 2D optical phase space, and hence for a 2 x 2 optical matrix. In the next Chapters we will consider the basic optical systems within the framework of the 1D model of linear optics. Wc will derive the relevant raymatrices and analyse their effect on single light rays in physical space as well as in phase space. The effect on a collection of rays will also be visualized in the phase plane in the case of bundles of rays emitted by idealized light sources, which accordingly are amenable to simple phase-space representations both before and a,fter the propagation through the system. Problems 1. Consider a 2 m • 2 m m a t r i x M -- ( ca B ) , where A , B, C a n d D are m • m s u b m a t r i c e s . F i n d t h e relations o b e y e d by the s u b m a t r i c e s A , B, C a n d D w h e n M is s y m p l e c t i c . (b) Prove t h a t as a consequence of such relations only rn(2m + 1) entries of M can be a r b i t r a r i l y given. (c) Show t h a t , in the case of a 2 • 2 m a t r i x , t h e s y m p l e c t i c relation t u r n s into t h e condition t h a t det M = 1. 2. (a) P r o v e t h a t t h e J a c o b i m a t r i x S of the H a m i l t o n i a n evolution from ui to u ( z ) o b e y s t h e e q u a t i o n of evolution

(a)

d

dz

S -

JH"S

Linear Ray and Wave Optics in Phase Space

54

involving the Hessian matrix, introduced in w 1.3.1, H " - (H'at,)o, " , H;•" : 0 2 H/OuoOu,. (Hint: Slightly changing the initial vector to u~. + du;., which evidently evolves into u + d u with d u = Sdu~, find the equation obeyed by du). (b) Verify t h a t when the Hamiltonian is q u a d r a t i c in the phase space variables (Eq. (1.7.1)) the m a t r i x J H " = H (Eq. (1.7.5)), thus proving t h a t in t h a t case S obeys the same equation as u. (c) Prove t h a t in general the representative matrix A, assignable to any q u a d r a t i c phase space observable A(q,p) by (1.7.7), is also obtaiilable as A = JA", where A " is tile Hessiall matrix of the quadratic form A(q, p). 3. Using tim power series expmlsioll (1.4.5) for tim Lie transforillatioll, prove that the phase-plane motion ,m(ler poly,lomial (1.5.27)is (lescri|)e(l t)y tim expressions q(z) = [cos((,z)7L (:siIl(~z)/e~]qi + [bsin(,~z)/~]pi ml(l p(z) = [cos(<~z)- csin(~z)/r - [asiIl(<~z)/,~]qi, wlmre 4. A fre(:-pr()pagati()ll systelll is (tescribe(t by tile Ilmnilt()lliall fllIwti()ll t t ( q , p , z ) = _ p2, lt,, b('iIlg ttle COllStallt wthm of tile r('fractive ill(h'.x. ExpressiIlg tt ;~s a series of powers of tim ray lnolnent, llill p, (letermine the correctioIls t() tim paraxial propagation, essentially (h'scrit)e(! t)y (1.5.12), which arise froln powers big|mr t|mli two, like f()r instance /)4, /)6, ml(1 s()()ll (gelmrally ref(:rre(l to ;m splmrical al)erratiolls). 5. A skew ray el~ters a~l axially sy~mnetric ()l)tical syste~n, i~ltersecl, i~lg tim x axis at q,, with ot)ti(:al (tire('ti()I~ ('()simms (p,,, p:~, ). Determi~m the relati()~ l)etwe(,.~ tim s~fl)seq~mnt i~tersectio~ls q , ( z ) of ttm ray witl~ tim x axis (t~ri~g t,lm propagatio~ l.l~ro~gl~ ttm systt'an and the corresI)o~(li~lg opti(:al (lirectio~ cosines at those interse(:tio~ls. 6. Prove tim ~fi~n()(h~larity ()f the ray ~natrix M fr()I~ tim req~fir(;~ne~t ttmt q,,(qi, pi) an(t po (q.i, Pi) art; cmu)~fically ('()llj,lgat(' varial)h's, i.e., {q,,, p,,} = {q,, t', } = 1. 7. &) Fi,,(l tim fiehl-ray ,'o()r(li,mtes (A(z), C(z)) fi'o,n tl,e ~ssig,,e,l axial ray-,',)()r(linates ( B ( z ) , D ( z ) ) m~(! vi('(; versa. (b) Sl~ow ttmt it is possible to ti~(l a~y paraxial ray (q(z),p(z)) if tim fiehl-ray ()r t,lw axial-ray ('()()r(li~mt(:s are k~()w~. 8. Let M mul H !)(: two 2m • 2~. ~mtrices, li~k(;(l t,llr()l~gl~ tlw gr()~q) ()p(.'rati()~ M = exp {<,H }, witl~ ~ arl)itrary real l)armneter. (a) Pr()ve that if M ix sy~l)l(~'(:t,i(:, l,t~(;~ H satisfies tile relatioI~: H J + J H ~ = (). (b) Show t h a t T r ( H ) =(). (c) Sh()w tl~at H J is a sy~mnetric t,mtrix. (d)I)r()ve tl~at, l)artiti(),~i,~g H as M i~l Prol)h;~n l, i.e., H - - ( L F ) , t,l~(, s, fl)~latrices _ 41t2

F an(t G arc sy~n~n(,tri(" ml(t L = - N T. (e) Verify t h a t only "m(2"m -~ 1) entries of H arc iiulependent. 9. Prove that: (a) w ( a u , , bu2) = a b w ( u , , u2) for a,,y co, nph, x ,,,,,nbers a and b; (b)

~ ( u ~ , u, ) =

-~(,,,,,,~); 6:)

~ ( u , , , ) = 0;

(d)

~(u,

+ u~, v) = ~ ( , , , , v) + ~ ( u ~ , v);

(~)

~7(v, u I + u ~ ) = ~7(v, u, ) + w ( v, u._,); (f) w ( u 2 , M u , ) = w ( M ~u,, u2) for m~y sy~nplectic ~natrix M . 10. Show tllat t,lm i~ltegratio~l of the two li~marize(t eq~mtio~ls (1.7.3) is eq~ivale~lt to tile i~tegration of ()~m li,mar (lifferm,tial equation of ttm' R,iccati type: (a) s,, - n.~.s,. 2 _ 1/n() - - 0 .2 for the functio,, .s, (z) = q(z)/p(z), or (b) s 2! + s.~/n,, + u 2 - 0 for s2(z ) = p(z)/q(z). (c) Find the expressioi, of q(z) and p(z) ill terms of both .s, (z) and .% (z). (d) Give a geometrical interpretatio, l of s I (z) a,l(1 .%(z) i,, tile (q,p) plane. 6-') Express tim z-evoh,tio,, of Sl and s= in terms of tim field an(l axial ray coordinates.

11. Find the positions of (a) the negative cardinal points, (tefine(t as conjugate points for which .Allv = - 1 an(t (b) the negative nodal points, for which A/I v = - 1 .

Hamiltonian Picture of Light Optics. First-Order Ray Optics

55

References [1]

Definitions, comments and investigations pertaining to the concept of light ray from the mechanical and electromagnetic views can be found in [4.2J, [4.5], and in

[1.1]

M. Kline and I.W. Kay, Electromagnetic Theory and Geometrical @tics (Interscience Publishers, New York, 1965).

[1.2] G.V. Rozenberg, "The light ray (contribution to the theory of the light field)", Sov. Phys. Usp. 20, 55-79 (1977). [1.3] D.S. Goodman, "General Principles of Geometric Optics", in Handbook of Optics, Vol. I, M. Bass, E.W. Vail Stryland, D.R. Williams and W.L. Wolfe (eds.) (Mc-Graw Hill, New York, 2nd ed., 1995), ch. 1; and references therein.

[2]

The original papers by W. R. Hamilton concerning optics and mechanics are collected in [2.1]

[2.2]

[3]

The Mathematical Papers of Sir William Rowan Hamilton, Vol. I: Geometrical Optics, A.W. Conway and J.L. Synge (eds.) (Cambridge Univ. Press, London, 1931). Also in http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Optics.html The Mathematical Paper's of Sir" William Rowan Hamilton, Vol. II: Dynamics, A.W. Conway and J.L. Synge (eds.)(Cambridge Univ. Press, London, 1940).

The Hamiltonian formulation of classical mechalfiCS with the related formalism of phase space, and Liouville's theorem, is presented in

[3.11 H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, 2~a ed., 1980). [3.21 V.I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, New York, 1978).

[3.3] [3.4]

[4]

R. Abraham and J.E. Marsden, Foundations of Mechanics (Addison-Wesley, Redwood City, 2 "a ed., 1978). J. Liouville, "Sur la th6orie de la variation des constantes arbitraires", Journ. de Math. Pures et Appl. 3, 342-349 (1838).

As to the Hamiltonian formulation of geometrical optics we quote below a few basic textbooks and papers, in some of which investigations in the optical phase space are also illustrated at various extents. In particular, we address the reader to the book by Wolf [4.11], where through "the tactics of Lie algebra.s and groups" a complete description of "the geometric model of light" is presented, comprising the paraxial, metaxial and globa.1 regimes.

[4.1] J.L. Synge, Geometrical Optics (Cambridge Univ. Press, Cambridge, 1937). [4.2] R.K. Luneburg, Mathematical Theory of Optics (California Univ. Press, Berkeley, 1964). [4.3] M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1965). [4.4] H.A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge Univ. Press, Cambridge, 1970).

Linear Ray and Wave Optics in Phase 5pace

56

[4.5] O.N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic Press, New York, 1972). [4.6] D. Marcuse, Light Transmission Optics (Van Nostrai~(|-Rcinhold, Princeton, 1972). [4.7] A.K. Ghatak and K. Thyagarajan, Contemporary Optics (PlcnuIn Press, New York, 1978). [4.8] S. SoliIneno, B. Crosignani and P. Di Porto, Guiding, DiffTuction and Confinement of Optical Radiation (Aca(lcInic Press, Orlan(to, 1985). [4.9] ll.M. Ozatk~s, Z. Zalcvsky all(1 M.A. Kllt,ay, The Fractional t%'uricr 7~'ansform with Applications in Oplics and Signal l)rocc.s.si'ng (Wiley, New Y()rk, 2()()1). [4.10] V. I,akslHnilmrayalm.ll, A.K. Gtm.tak ml(l K. Ttlyaga.rajail, Lagrangian Optics (Klllwcr, Bost.()ll, 2(}02). [4.11] K.B. W(,lf, G,'mn,:tT~ic Optic.s on l'h,a.sr Space (SI)ri,~g(,r, B(;rli~, 2()()4). [4.12] l{..], l)cgis, "Tim lll()(l(;rll (lev(;l()l)lllCllt ()f llaInilt, olfiml Opti(:s", l)'rog'rcss in Optics, E. W()lf ((;(1.) (N()rtl~-ll()llm~(l, A~st,(,r(la~, 1.()(i3), V()l. l, (:!~. l, pp. 1-2.(). [4.13] .].A. Arlmlul, "llmIliltolliall tlmory of l)(;aiil propagati()ll", lu162 in Optics, E. W()lf ((;(l.) (N()rtll-lh)llml(l, Alllstcr(lalll, 1.()73), V()l. Xl, ell. 6, l)l). 247-304. [4.1,l] T. Sckig, mtfi ml(l K.l~. Wolf, "Tim llmlfilt, o~ia~ fon~ndatio~ of optics", AII~..]. Pt~ys. 55, s:~()-sas (~.(}ST). [4.15] M.A. Ahms() m~(I (;.W. F()rl)cs, "(;(;~mralizatio~t of IIm~filt()~'s f()r~mlis~ for gco~(;tri('a.l ()pti('s", J. Opt. S()('. AI~. A 12, 2744-2752 (I.().()5). A fas('i~at i~g s~rv('y ()f variat i()~al l)ri~('il)h;s a~(l limit r(;lat i()~ t() (ly~al~i(:s a1~(t (l~m~l~t~ th(;()ry is giv(,~ i~ [n.(;]. 'I'1~(' l)ri~iliv(; f()r~lali()~ ()f 11~(' l"(,r~mt l)rin(:it)l(', a~(l t l~(; l)(;r! i~('~! (;lai)()ra! i()~s al|(l i~v(;sliga! i()~s (,xl(;~(li~g t l~(' l)ri~('iI)le t() [5.1] [5.2] [5.3] [5.4]

I). (h; Fer~mt, (l~v'n;u.~, ((;m~tl~i(;r-Villars, l)aris, 18.91), I). 354. tI.II, lh)l)ki~s, "A~ (.'xtc~si()~ of l"(;r~mt's tl~eore~", Ol)t. Acta, 17, 223-225 (1970). W.A. Ncwco~fi), "(~,('~(;raliz(~'(l t4'.r~mt l)ri~mit)lc", A~..l. l)l~ys. 51, :~38-34() (1983). A.L. Rivcra, S.M. (,,lnn~mkov m~(l K.B. Wolf, "Ha~nilto~dm~ for~n~lati()n of geometrical anisotropic optics", .l. ()pt. Soc. Am. A12, 138()-138.() (1!).95). [5.5] W. Shcn, .J. Zhang, S. Wm~g aI~(l S. Zhu, "Format's principle, tim gmmral cikonal cq~mtion, m~(l space gconmtry i~ a static aIfisotropi(: ~nc(li~I~", .l. Opt. Soc. Am. h14, 2850-2854 (1997). [5.6] W. Yourgrau a~(1 S. Man(lclstam, Va~iational Principles in Dynamics and Quantum Thuo~T/ (Dover P~fl)l., New York, 1979).

[G]

A wi(l(; a I~(t (tetaih'~(l i~v(;stigation ()f the ray equati()~ i~ ii~h()n~()gxm()~s Inedia for various refractive in(tcx profiles is t)rcscnted in [4.10], and in [6.1] E.W. Marchan(l, Gradient Index Optics (Academic Press, New York, 1078). [6.2] S. Cornbleet, Microwave and Geometrical Optics (Aca(temic Press, London, 1994).

[r]

The optical-mechanical analogy is investigated in [4.2], [4.3], [4.13], and in [7.1] J.A. Arnaud, "Analogy between optical rays and nonrelativistic particle trajectories:

a comment", Am. J. Phys. 44, 1067-1069 (1976). [7.2] W.K. Kahn and S. Yang, "Hamiltonian analysis of beams in an optical slab guide", J. Opt. Soc. Am. 73, 684-690 (1983).

[7.3] J. Evans and M. Rosenquist, " ' F = m a ' optics", Am. J. Phys. 54, 876-883 (1986). [7.4] R.J. Black and A. Ankiewicz, "Fiber-optic analogies with mechanics", Am. J. Phys. 53, 554-563 (1985).

Hamiltonian Picture of Light Optics. First-Order Ray Optics

57

[7.5] N. Chako and B. B. Blank, "Electron Optics", in [4.2], Suppl. Note ~ 1, pp. 373-410. [8]

A.S. Deif, Advanced Matrix Theory for Scientists and Engineers (Abacus Press, Turnbridge Wells, 1982).

[9]

For the general theory of Lie algebras and Lie groups we may suggest among the great variety of textbooks the following: [9.1] H. Weyl, The Classical Groups (Princeton Univ. Press, Princeton, 1946). [9.2] M. Hamermesh, Group Theory and its Applications to Physical Problems (AddisonWesley, Reading, 1962). [9.3] S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York, 1962). [9.4] M. Hausner and J.T. Schwartz, Lie Groups; Lie Algebras (Gordon and Breach, New York, 1968). [9.5] W. Miller, Jr., Symmetry Groups and Their Applications (Academic Press, New York, 1972). [9.6] R. Gilmore, Lie Groups, Lie Algebras and Some of Their Applications (John Wiley & Sons, New York, 1974).

[10] Examples of applications of the concepts of symplecticity to physics, and in particular to optics, are in [10.1] Y.S. Kiin and M.E. Noz, "Illustrative examples of the symplectic group", Am. J. Phys. 51, 368-37s (1983). [10.2] v. Guillemin and S. Sternberg, Symplectic Techniques in Physics (Cambridge Univ. Press, New York, 1990). [11] S. Steinberg, "Lie series, Lie transformations, and their applications", in Lie Methods in Optics, J. SSnchez Mondragdn and K.B. Wolf (eds.) (Springer-Verlag, Berlin, 1986), ch. 3, pp. 45-104. [12] Conceptual and applicative analyses of Lie theoretic methods within light and magnetic optics of both linear and nonlinear systems are elaborated in [12.1] A.J. Dragt and J.M. Finn, "Lie series and invariant functions for analytic symplectic maps", J. Math. Phys. 17, 2215-2227 (1976). [12.2] A.J. Dragt, "Lectures on nonlinear orbit dynamics", in Physics of High Energy Particle Accelerators, AIP Conf. Proc. 87, R. A. Carrigan, F. R. Huson and M. Month (eds.) (AIP, New York, 1982), pp. 147-313; Lie Methods for Nonlinear Dynamics with Applications to Accelerator Physics, Univ. Maryland Phys. Dept. Rep. (1998). [12.3] A.J. Dragt, "Lie algebraic theory of geometrical optics and optical aberrations", J. Opt. Soc. Am. 72,372-378 (1982); Lie Algebraic Methods for Ray and Wave Optics, Univ. Maryland Phys. Dept. Rep. (1998). [12.4] A.J. Dragt, E. Forrest and K.B. Wolf, "Foundations of a Lie algebraic theory of geometrical optics", in Lie Methods in Optics, J. Sgnchez Mondragon and K.B. Wolf (eds.) (Springer-Verlag, Berlin, 1986), ch. 4, pp. 105-157. [12.5] K.B. Wolf, "Symmetry in Lie optics", Ann. Phys. 172, 1-25 (1986). [12.6] M. Navarro Saad and K.B. Wolf, "The group-theoretical treament of aberrating systems. I. Aligned lens systems in third aberration order", J. Math. Phys. 27, 1449-1457 (1986).

58

Linear Ray and Wave Optics in Phase Space [12.7] K.B. Wolf, "The group-theoretical treament of aberrating systems. II. Axissyinmctric inhomogcncous systems and fiber optics in third aberration order", J. Math. Phys. 27, 1458-1465 (1986); III. The classification of asymmetric aberrations", J. Math. Phys. 28, 2498-2507 (1987). [12.8] K.B. Wolf, "Symmetry-adapted classification of aberrations", J. Opt. Soc. Am. A 5, 1226-1232 (1988). [12.9] A..J. Dragt, "Computation of maps for particle and ligtlt optics by scaliIlg, splitting, anr sqlmring", Phys. Rcv. Lctt. 75, 1946-1948 (1995).

[13] ]Id('gratit)II xlicth~)~ls ~f ~t)crat()r (tiffer(',Iltial-cttllati~)lls arc well illllstrat~'r in [13.1] .J. Wei ml~l E. Norlnall, "Lie algebraic sohltioll of lillcar ~liffcrcntial cqllations", J. Matl~ l'hys. 4, 575-581 (1963). [13.2] ]{.M. Wilcox, "Exp-tt~coretic trcat~nent of Gm~ssim~ optics a~r 3rr aberrations", Lecture Note Phys. 201, 133-136 (1984). [14.6] M. Navarro Saa(t an(l K.B. Wolf, "Applications of a factorization tl~corcxn for 9thor~lei" aberration optics", J. Symb. Comput. 1, 235-239 (1985). [14.7] K.B. Wolf and G. Krotzsch, "Gro~p-classified polynoInials of phased-space in higherorder aberration expansions", J. SyInb. Comput. 12, 673-693 (1991). [15] K.F. Gauss, "Diot)tris(:l~c Untersuchugen", Trans. R()y. Soc. S(:i. at G6ttingen, w)l. I (1843). [16] The ray m a t r i x apt)roach to linear optics is extensively presented in many textbooks and papers. Scc, for instance, [12] in ch. 2. [17] The clm racterization of optical systems by cardinal planes is illustrated in [17.1] F.A. Jenkins and H.E. White, Fundamentals of Optics (McGraw-Hill, New York, 4 t~h cd., 1976). [17.2] W.J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966). [17.3] R. Zingslake, Optical System Design (Academic Press, New York, 1983).

2 1D First-Order Optical Systems: The Ray-Transfer Matrix

2.1

Introduction

We apply the strategy devised in Chapter 1 to construct the optical matrices, describing the propagation of light rays through linear stigmatic and simple astigmatic optical systems. The basic tools of the strategy arc the optical Hamiltonian H(q.~, %,px,py, z) and the associated Hamilton's equations for the evolution of the light ray as propagating through the system a,long the optica,1 z axis. The system is described by the refra, ctive index function n(q,, qy, z), as the ray is described by the canonically conjugate va,riables (qx, %) and (p:~,py), specifying the cartesian coordinates and the direction cosines of the ray at its crossing point with the reference screen transversal to the optical axis. More precisely, the optical momenta, (px, Py) are the direction cosines of the ray scaled by the local va.lue of the refractive index. The Cartesian space spanned by the ray position and m o m e n t u m coordinates is the geometrical optical phase space. The ray is represented by points in the 4D optical phase space a,nd accordingly the ray path in rea,1 space corresponds to a trajectory of the ray representative point in phase space [1]. The optical Hamiltonian has the peculiar form of the square-root flmction H(q, p, z) = -v/n2(q, z) - p2, having some resemblance to that of a relativistic particle. The associated Lie operator k , ( q , p, z), obtained from H by Poisson bracket: L/~ - {-, H}, rules the evolution of the ray variables q(z) and p(z), as functions of the propagation parameter z, according to Hamilton's equations

d (;((z)) _ -~u(q,p,z)(q(z)'~

-

-

(2.11) 9

zi denoting the axial position where the ray coordinates are known; q and p here are handy abbreviations of (q,, qy) and (px, py), respectively.

60

Linear Ray and Wave Optics in Phase Space

In particular, if p r o p a g a t i o n in the close proximity of the a,xis is concerned, the optical H a m i l t o n | a n simplifies into the form y-~ - n(q ' z) ' resembling the 2n H a m i l t o n i a n of a non relativistic particle moving under the potential -n(q, z) [2]. T h e opt|ca,1 momenta, p:r and py come to be directly linked to the inclination angles of the ray in x and y directions. In addition, as the refractive index is a sm()ot,}~ flm(:tion of st)a,(:e, it may be des(:ril)e(t by a, qua(Ira|i(: fllnct, i(m of the ray ('()()r(lina.t(;s (q:r,%) ()v(;r t,l~(; t)(;rt, i~(;~lt, sIlm,ll i'(;gi()ii a.(:ross t,l~(; axis. The Hamilt()nian fim(:t,i()n l)c(:()mcs then (t~m,(tra.|,i(: in |)()t,h q a,t~(l p (:()or(tinat, cs, t,h~ls yi('l(li~lg t,lw liIwa,riz('(l f()r~11 ()f Ha,~filI,o~ e(l~mti()~s (2.1.1) as

where 1,11eray-t)a,I-a,lllel,er i~l(let)(;Ii(h;ld, llm,l,rix H ( z ) is ()t)l,a,iIm(1 fl'()111 t,ll(' Ita,nfilt,()nia.n H(q, p, z) I)y I)()iss()ll I)rat:k('l, illg wil,ll tim <:()njllgat(' wtria,1)h,s q a,n(1 p. A(:c()r(lingly l,lm l)r()l)aga,t,i()ll lllallift;sl,s as a, S(;(llW,llCe ()f lilmar trmlsfornmt, i()ns ()f the ray va.riat)h;s t'r()lli tile ilfil,ial wdlms (qi,Pi), as il, is s~tita.|)ly ext)resse(1 t)y 1,11(' (:(nlt:is(' llla,l,rix f()rIll r 1,11<;i11|)111,-()111,1)11|, rt;lati,)~Mlil)

p,,]

\p,i

Tll(' ray wtrial)h,s (hi tll(' assiglle(1 illt)ld, ml(l ()llt,t)llt, t)la.lleS, (l('lilllit, illg the system a,t t)ositi(ms z,i a.ll(l zo ah)llg the ()l)t,i(:a,1 axis, arc r(;lat(;(1 ]~y I,h(; 4 • 4 raytra,llsfer 11m.t,rix M(zo, zi), wlli(:ll is the llm.t,llellm, t,i('a,1 re(:it)ield, f()r t,lle ot)ti(:al featm'es of the sysl,(;lli an(1 h(;llt:(; fin" its rest)(nlse t,o the ill(:()lllillg illlulfination. Th(' ()t)ti(:al 11m,t,rix is tll(, ('xlmllstiv(; (t('st:rit)t()r ()f ray t)r()t)agati()~l in geolneti'i(:al first-()r(h;r ()t)ti(:s, sill(:(; its st)(;('~ifi(:ati()n t)r()vi(h;s a (:()lilt)l('|(; a.(:count of the ingoi~lg-l,()-()ld, g()iIlg ray l,ra,ilsformat, i()ll ()l)ei'ate(t t)y t,ll(; ()t)I,i(:a,1 system. As discussed in w 1.7.;t, the ret)resentative Ha,re|It, o n | a n m a t r i x H ( z ) and the ray transfer m a t r i x M(z, zi) arc closely linke(t, the flmctiona,1 f()rlllS of the entries of M |)eing (tetermin(,(t |)y the flln('l,i(mal forms of those of H. In the particular case that the Hamiltonia, n flmction does not d e p e n d explicitly on the axia,1 p a r a m e t e r z, the ray-transfer m a t r i x M(zo, Zi) is simply obtained by e x p o n e n t i a t i n g the consta,nt m a t r i x H t h r o u g h the axial dista,nce Zo- zi between the re%rence e n t r a n c e a,nd exit pla,nes as

M(zo, zi)

- e I-I (Zo-Z,:).

(2.1.4)

Additional simplifications are attained when the propagation occurs through stigmatic or simple astigmatic systems. In the first case, the system is circularly symmetric about the optical axis; then, if the input ray lies on one of the infinite symmetry planes through the axis, we may track only two of the four

1D First-Order Optical Systems: The Ray- Transfer Matrix

61

Single-particle description conveyed by the canonica|ty conjusate variabtes q(c) and p(c) through Ham i[ton's equations

C l a s s i c a l Mechanics

G e o m e t r i c a l Optics

Ensemble statistical description conveyed by the phase-space density distribution p(q,p;z) through Liouvitte's equation

F I G U R E 2.1. Exemplificative scheme of the single-particle and ensemble descriptions of a classical dynamical system, e.g. a point particle, which are mirrored into the geometrical-optics approach to light propagation in terms of the single-ray variables q(z), p(z) and the ray-bundle density p(q, p; z) in phase space.

ray parameters, following tile ray path in that plane, where the propagating ray keeps on lying. The problem becomes one-dimensional, as the motion of the ray representative point takes place in a 2D subspace of the 4D optical phase space, and accordingly H and M come to be 2 x 2 matrices. Likewise, tile propagation through simple astigmatic systems, which arc symmetric about two orthogonal planes intersecting the optical axis, can be accounted for by 2 x 2 matrices, being the motion in the optical phase space decomposable into two independent motions in tile two 2D subspaces. We will liInit our discussion to first-order stigmatic and simple astigmatic optical systems, which, being describable within the 1D model, will shortly be referred to as 1D systems. The conceptual path, developed in Chapter 1 and briefly synthesized here, pertains to tile single-ray picture of light propagation. Evidently within tile ray-picture of geometrical optics a light distribution may be seen as an ensemble of uncorrelated rays. As noted in w 1.2, account of the propagation of the light distribution by integration of Hamilton's equations for each ray in the ensemblc is, of course, impracticable. An alternative strategy must be envisaged. We may presumably guess that, as the single-ray description of light propagation parallels the Hamiltonian formulation of the single particle dynamics, thus a ray-ensemble description might be elaborated paralleling the statistical formulation of the dynamics of an ensemble of particles [3, 4]. In other words, referring to Fig. 2.1, we may say that, as we have delineated the correspondence between classical mechanics and geometrical optics through the upper path, so we delineate here such a correspondence through the lower path. Indeed, Liouville's equation for the phase-space density distribution p(q, p; z) is

62

Linear Ray and Wave Optics in Phase Space

derived in Sect. 2.2, where are also examined some basic properties of the Lie transformations. The Lie operator I_, and actor(tingly the ray matrix M confirm as the basic tools to describe the light b e a m t)ropagation within the ray-m()(tel of gcometri(:al optics. Some considerations aimed at relating the geometrical and t)hysical interpretations of p (:lose the section. Th(u~, inst)ired t)y th(: flm(:ti(mal f()nn ()f th(: first-()r(t(:r ()t)ti(:s Hamilt()nian, being (t~a(lratic i~l th(: ray varia])h:s, w(: star(, t() (:()~lstr~l(:t the ~m,(,rix ()t)ti(:s, gen(:ra,t(:(t |)y the l ~ 1 l (pm,(h'a,ti(' n~()n()nfia,ls ~t? }q~ a,n(I ~qp, (,I~(: (:()rr(:sl)(m(ling ()l)ti('a,l systems l)eing th(u~ i(tentifi(:(t a.s a, s(:(:ti()n ()f l~()~()g(:n(:()~s n~(:(ti~m~, a refra.(:ti~g surface an(1 a,1~ i(teal ~a,gj~iii(:r, r(:st)e(:tiv(:ly (Se(:t. 2.4). Fiscally, Se(:t. 2.5 ilh~strates the t)l~as(:-st)a,(:e t)i(:(,~r(: ()f tl~(: tra,i~sf()r~m.t,i()~s (:fi'(:('t(:(t t)y tt~(: a,t)()v(: (t~()te(t syst,(:~s ()~ t)()(,l~ a, sil~gl(: ray a,~(1 s()~(: ex(:I~I)lary I)~(ll(:s ()f rays. h

2.2

Ray-ensembledescription of light propagation

The (:(>llsi(lera.ti()lls
1D First-Order Optical Systems: The Ray- Transfer Matrix

1-I

1-[ i I

I

I

I

I

i

............. i, i

I

t

qi

63

' I

"4

i I

i F

!

%'%'%w%% *%.%

I

i

i i

I

I

= (q,.,o)

"~*".

qi

po =(o,- 7)

.

.

.

".~.

i i

q

.

i

(a) I-I i

. . . . . . . . . .

(b) l--l o

I

I

i i

i i

i

!

p

i

.40

r,.:_-.~. -

q

I

I

(c)

(d)

F I G U R E 2.2. Naive view of tile phase-plane transformation of tile representative point of a single ray and the representative region of a ray-bundle, effected by an optical system. (a) A ray propagates to tile back focal point of a positive lens and (b) tile corresponding representative point in the phase plane "moves" from Pi to Po. Likewise, (c) a beam of parallel rays is focused to the lens back focus and (d) the corresponding representative segment P1P2, lying on the q-axis, changes to the segment PAP4, lying on the p-axis.

position z, the number of representative points dN'(q, p; z) in the volume element dqdp around the point (q,p) in phase space is given by the product dN'(q, p; z) = p(q, p; z)dqdp. Clearly, p(q, p; z) takes on values throughout the beam volume 12 in phase space, and vanishes outside )2. As an example, we may refer to the simple situations illustrated in Fig. 2.2, inspired by [a.,]. A paraxial ray, incoming parallel to the optical axis of a focusing lens, is deviated to pass through the lens back focus F' (Fig. 2.2.a)). The input-ray representative point Pi, that lies on the q-axis of the phase plane, "moves" to the output-ray representative point Po, that in contrast lies on the p-axis (Fig. 2.2.b)). Correspondingly, a beam of parallel rays hinging on the lens is focussed to a spot in the back focus. The incoming beam is represented in the phase plane by a segment, lying on the q-axis, of extension

64

Linear Ray and Wave Optics in Phase Space

equal to the spatial width 2w of the beam (Fig. 2.2.c)). Likewise the fan of the focussed rays passing t h r o u g h F' identifies a segment, lying on the p-axis, of extension 2~ (Fig. 2.2.(t)). It is evident that the region in the opti(:al phase plane representing the b e a m inoved and changed its shat)e; actually, in the simple (:as('. ()f the 1D model t)ertaining to Fig. 2.2, only the extensi(m of the zero-area, t)ha,se-t)lane region has been affc(:ted by the lcils f(l(:llsillg effect. Tile (tensity (if rcI)resentative t)(lillts i~l t)ha,se t)lan(' (',haIlgc(t as well. It is thcr(;flir~; na.tllral t() i]l(tuire whether it, is l){issitfle t~l cllvisa,gc, a, law for t,h~; ewllllti~m ~lf tll~', lflm.s~'~-sI)a,(:~; (lensity fl(q, p; z), (hi a.(:('tnnd, (if tha,t the evohlti~m ~f every singl(; ray ~/f t,ll(; CllS(;nfl/lt; is gt~W;nl('~l ]ly Ha,llfilt~nl's ('(tlla,tions. W~'~ will fimt that a.ll "ev(flllt, i~nf' e(tlm,tion fi~r p(q,p; z) (:a.~l t)(' written (t()wll, ml(l zl(ita,llly ill a, [()1"111 strictly simila,r t,() Ilmllilt(nl'S C(llm,ti(/l~s flir the ra,y (:()()r(lilmtes. M(ir(;(iV(;l', we will sec l,l~a.t l,lle t)lm.se sIla,(:e (le~sity p(q, p; z) is ev(/lv(;(l lw tl~e ray l,ral~sf(;r ()t)cra,l,t/r ()r, wl~e~ 1,l~(; li~l(;ar al)t)rt)xi~m.l, itm is concenm(l, 1)y the ray tra, l~sfer ~m.trix a,~ld tln~s we will rega, il~ tl~e sil~gle-ra,y t)i(:t~lre as l)asi(: t,(/l,l~(; ray-e~|s(;~fl)l(: I)i('.t,~r(; a,s well. Tl~e (lis('~ssi(n~ l)r(;se~t,e(t in tt~(; ~(;xt, Imragra,tfl~s lm,si(:ally l)aralMs t,lle (l(;riw~,ti(/l~ (if I,i(l~vilh;'s tlle()l'elll in statisti('al ~e('lm,~i(:s [:~, 41. l)lm,se-Sl)a,(,.e t)a,se(t I~etl~o(ts are (ir(li~a.l'y tools in ~a.g~(;ti(: (lliti(:s. I~(1(;(;(1, as is w(;ll k~()Wl~, tl~(; <:l~art4(;(l-t)(;m~ ,ll)ti(:s ~a,~fifests a, cl()s(; mml(igy witl~ tl~(; (tI)ti('s (if ligl~t rays. S~('I~ m~ mml(igy will fl'(;(l~uu~tly |)e cmt)l~asizc(l l,l~ro~gl~ tl~c 1)(l(ik, (;v(;~l l,llough ~lot Sl)C(:ifica,lly m~alysc(l; i~ this c(mne(:tio1~, wc sl~gg(;st s(n~(; tith;s ill [3]. Bef()r(; g()i~g (Ill, WC IltIl,(; t,t~a.t Figs. 2.t)) a,n(t 2.(t) mlti('.itm.t(; ill a, s(;~ls(; s()ine a,st)c(:l,s of l,h(; f()rl,l~(:(n~fillg (lis(:~ssi(n~. Ill fa(:t, hyI)(itheti('a,1 l,raj(;(:t(iries have |)(;en tra.('.e(l i~l the tigllI'e for l,l~e I'et)resentative t)oinl,s to (:arry t,l~(; i~fitia.1 point or region t(i the final t)oint (it regi(m. Thus, the t)a,ssage thro~@~ the lens is pictur(~'(t a,s a q-(:OllSta,llt tm,th (:(/mle(:ting tile zer(~-moment~ml t)(lints representiIlg tile ilmonfi1~g rays t() the ll()llzero-Ill()Illelltlllll points ret)resc~ting tile exiting rays. In a, like ma,nner, the propagation from tile lens t(i the ])a,(:k focal point is gra,t)tfi(:a,lly (let)i(:te(t a,s a, p-(:onstant t)ath to carry the ret)resentative points of ttl('~ redirected rays to the zero off-axis dist)laceme~lt t)oints on tile p-axis. S~u:h gra,phi(: views of tile quoted transformations (tra,nsfer by a, lens and prot)agation t h r o u g h a homogeneous medium) will be justified by the later considerations (Sects. 2.4 a,nd 2.5). We will see t h a t the prot)agation through a homogeneous m e d i u m can be depicted as a c o n t i n u u m horizontal segment (a q-shear), signifying indeed a continuous t r a n s f o r m a t i o n of the ray coordinate. In contrast, the passage through a lens produces a discontinuity of the ray direction, due to the redaction at the shaping surfaces of the lens at which the refractive index function abruptly changes from one side to the other. A continuous path picture of the redaction is a convenient, even though not strictly correct, visualization of the process, which thus is viewed as a p-shear.

1D First-Order Optical Systems: The Ray-Transfer Matrix

2.2.1

65

Liouville 's equation for the ray-ensemble

We consider the variation of the density function p(q, p; z) with the propagation variable z. It is mathematically given by the total derivative of p(q, p; z) with respect to z, namely

d 0 Op dq Op dp dzp(q, p; z) - -~zp(q, p; z) + Oq dz ~ Op dz'

(2.2.1)

which, of course, displays contributions from both the explicit and implicit dependence on z, manifested respectively through the partial derivative Op/Oz and the derivative-chain terms, involving the ray coordinates q and p (a). For brevity, we use again the abbreviation (q, p) for (qx, qy, px, py). Replacing for dq/dz and dp/dz the respective expressions from Hamilton's equations, the chain-like terms simplify into the Poisson bracket of p and H:

Op dq

Oq

Op dp O;

Op OH

Oq Op

Op OH Op Oq = {p' H},

(2.2.2)

by which the equation for p(q, p; z) follows in the concise form d

d p-

0

p(q, p;

{p, H}.

(2.2.3)

In general the z-evolution of any dynamicM v~riablc pertinent to a system, whose dynamics is governed by Hamilton-like equa, tions of evolution, occurs according to Eq. (2.2.3). On account of the physicM meaning of p(q, p; z), we can further specify its variation with z. In fact, we may observe that the total number of representative points in phase space remains constant, even though the shape of the region containing them may presumably change as a consequence of the individual motions of the enclosed points. In particular, let us consider a fixed volume )2 in phase space and denote by S the closed surface delimiting it. The number of ray-representative points inside ]2 at a certain z is obtained integrating the density p(q, p; z) over the volume itself, namely

.N"v (z) - Iv p(q' p; z)dV.

(2.2.4)

Evidently, N"v (z) depends on 12 and z. Being 12 fixed, N"v (z) changes only with z, due to the motion of the representative points reflecting the propagation of the corresponding rays in the beam. As no rays are created or destroyed, a The derivative ~ yields the z-derivative of p(q, p; z) in a frame moving with the point (q, p) in phase space. In contrast, o yields the z-derivative of p(q, p; z) in a fixed frame.

66

Linear Ray and Wave Optics in Phase Space

the net rate at which A/"v (z) varies ~ (~Z must equal the net rate of points crossing the boundary surface S. So we write

Oz

v(z) - -.

p u . nd$.

(2.2.5)

tim sm'fa(:c illtegral I'(~I)r('~s(mting the rat(; at whi(:h I)()ints (:ross tim sllrfa,(:e S ()lltwa,r(1. In fa('.t, in (:()lff()l'mity with the stan(tar(| (:()liventi()ns, n is the (),,twa,r(1-(h'awn unit n()rmal t,()tl,(; s,,rfa,(:e (;h;m(;nt dS, a.,,(1 u = (dq/dz, d p / d z ) T is the vclo(:ity vc(:tor in t)ha.s(; st)a.('(~, st)(x:ifying t,h(; rat(; 1)y whi(:h the ray a,t tl~e t)lm.se st)a.('c t)()i~t (q, p) ~()v(,s t() tt~(; t)()i~t (q + dq, p + dp). Tl~c mimes sigIl a,(:('olmts for tllc fa,('t tlmt, if t)()ints lea,re th(~ v()hl~(~ )2, Afv (l(x'rca,scs. A('(:or(ti~g to Gm~ss' (liv('.I'g(;~l(:c tlm()rc~ [5], tlm s~lrfa.(:c i~tcgra.1 ~lay bc t~lr~lc(t il~t,() t,h(~ v()hn~l(~ ixit,(~gral a,s .Is t' u . n d S -

jv V . (p U ) d r ,

(2.2.(~)

whcr(, V-( )(l(~n()t(~s tln~ (liv(,rg(,,,(:(, (,l)(,rat, i()~, an(11,(,n(:c V. ( ) = O/Oq+O/Op. Illt(~restingly w(; fill(l

i) dq 0 dp X7. (p U ) -- ,-7- (P-;-) + ,-:7-(P-;-) - {P, II }. oq az op az

(2.2.7)

By ~sc of ('xl)rcssi(),~ (2.2..1), givi,~g N"v as t.l~e v()l,n,~,: i,~t('gx'al ,,f p, a,~,l (2.2.6) in (2.2.5), w(~ cn(t 11t) wittl

{~p+

v.(pu

)} ~tv - ().

(2.2.s)

whi(:h, being the volmnc 12 arl)itra.rily ('h()s(m, yiehts tllc ('ontimfity equation f()r the t)ha.s(;-st)a(:(' (h;ilsity fl(q, p; z) a.s 0

o p + v. (p.)

- o.

(2.2.9)

Moreover, using the at)()ve relation into Eq. (2.2 93) for tile derivative dp on dz ~ a.(:count of (2.2.7), we find that the continuity equation is equivalent to d

dzp-

o.

(2.2.1o)

Equation (2.2.10) is tlle desired equation for the z-evolution of the phasespace density p(q, p; z). We rewrite it in the convenient form

Oz p -- - { p , H} - --kHP ,

(2.2.11)

1D First-Order Optical Systems: The Ray- Transfer Matrix

67

with the initial d a t a being conveyed through p(q,p; zi) - pi(q,p). We refer to (2.2.11) as Liouville's equation for the ray-ensemble, as it directly follows by Liouville theorem (2.2.10). It is evident its similarity to Hamilton's equations (2.1.1) for the light ray. Evidently, as the phase-space vector u ( z ) = ( q ( z ) , p ( z ) ) r may be regarded as the descriptor of the single ray, the phase space density p(q,p, z) may be ragarded as the descriptor of the ray-bundle. It is of value that the z-evolution of both u and p is essentially governed by the Lie operator LH, even though with opposite signs due to the invariance of p(q, p, z) along the ray path, on which we will comment later. As a conclusion, we may say that Hamiltonian optics deals with evolution equations of the type t-.

0 ~0---~v - ?/v,

(2.2.12)

where ~ may denote the phase-space vector (q(z),p(z)) r or the.. phase-space density p(q, p; z) and accordingly tile operator ?-t comes to be ?-t - +L H. The formal similarity of Eq. (2.2.11) to (2.1.1) makes evident that the considerations we developed in w 1.4.1 about the integration of (2.1.1) apply to the integration of (2.2.11) as well. The crucial point of those considerations was concerned with the operator nature of k H and its possible dependence on the a,xial parameter z. We found that the simplest case pertains to a zindependent Hamiltonian, in which case the solution to (2.2.11) takes the form

p(q, p; z) - c -(z-z~) k, p(q, p; zi),

(2.2.13)

where p(q, p; zi) denotes the phase-space density of the beam at the input reference screen. Expression (2.2.13) reflects the same view as Eq. (1.4.4); thus, the phase-space density p(q,p; z) at z originates from the assigned function

p(q, p; zi) at the input zi-plane by way of the Lie transformation c -(z-z~) kH. In Section 1.4 we considered Lie transformations acting directly on the ray variables (q, p), and, as specific examples in w 1.5.2 we described the Lie transforinations generated by quadratic fimctions of q and p. Here, the Lie transformation acts on the phase-space density p(q,p; z), which is a function of q and p. To see what happens we need to use some properties of the Lie transformations, on which we will briefly comment in the next paragraph. Firstly we will elaborate a geometrical and physical interpretation of Eq. (2.2.10). Geometry and Physics related with L iouville's theorem Equation (2.2.10) shows that the density of phase space points does not change by propagation of the b e a m through the optical medium. As d/dz represents the "mobile" derivative, we can give (2.2.10) a nice pictorial view in phase space. Suppose to individualize a "point" P = (q, p) in phase-space, i.e., a ray,

Linear Ray and Wave Optics in Phase Space

68

3/,,

p

-

p

-

F I G U R E 2.3. Tl~c I)lu~sC-Sl)aCc rcgio~ (stuulc(l arc~us) occ~pic(t 1)y tl~c ~ssc,nl)ly of the ray-1)~u~(tlc rcI)rcsc~tativc l)oi~d,s clmx~gcs as t,lie 1)emli I)rocccr alo~lg tlic z axis, lnd, the rclcvm~t v o h ~ c rc~nai~s (:o~stm~t: ]2i = ]2(z) = 12,,.

by some mark. Let p(P; z i) l)e the t)lm,se st)a,(:e (lensity a,t P for a, (:crtain zi. Tlle t)oint P ~les(:ril)es a traje(:t()ry ill I)lla.se sI)a,(:e witli i~l('reasillg z > zi. For ea,(:h z > z,i w(; (:all (',wdlla,t(; 1,11('~(l('~llsity p(P; Z) at P, lll()villg witll it, a h)llt4 its own t)ath. W('~ till(t tlmt it (l()(;s 11()1,(:lla.~lg(;, i.e., p(P; z) = p(I'; z~) V z > zi. T h e nl()ti()ll ()f tll('~ ra,y-('~lls('nfl)l(,~ r('l)l'(;s('~ld,ativ('~ 1)()ild,s ill t)lms(' sl)a(:(; is 1,11(~11lik(; ttm,t ()f a,I1 ill('(nllI)r(;ssit)l('~ tllli(l ill l'cal Slm(:e. Ill ()tiler w()r(ls, p(q, p; z) t)clmves like a "t)r()t)(;rty" ()f th(; t)()illl, in t)llas(; sI)a,(:(;, ml(t a,(:(:()I'(tiligly it, is tra,nst)oI'tt;d t)y the t)()ilit, a,l(nig its ()w~l traje(:t()ry wll(;~l it, is r(;t)r(;se~lta,tiv(; ()f a ligtit ray. Tl~c i~va,riml(:(' of 1,11(;1)llasc-st)a,('.(; (lc~lsity p(q, p; z) inlt)lics 1,11(;i~ma,l'ian(:(; of the t)ha,s(;-st)a(:(; v()l~tnle 12 ()f 1,11(;regi()ll (m(:lt)sillg tim ray-l)~ultlle r(,.t)i'escnta,tiv(; t)()ints. In fi~.(:t, s~t)t)()st; t() s(;l('(:t at a (:(;rtah~ z = zi a ('(;rt, ai~ ~nmfl)(;r ()f ret)rescntative I)oix~ts aroml(t a (:hose~ t)()int,, througtl wlli(:ll tllc (l(;~lsity takes ()I1 t h e COllStallt va,l~lc Pi. Let Ni l)(; the volcanic cn(:l()sing the (:h()s(;n t)oints. T h e tota,1 Inlnfl)(;r ()f t)oi~lts in Ni evi(lt;~ltly is Ni = piN,:.

(2.2.14)

Following the points as t h e y move t h r o u g h t)ha,sc-spacc, we see tha,t the boundaries of the enclosing volmne change with z > zi, but by definition the total n u m b e r of points contained inside it does not va,ry: N(z) = Ni. Also, the density of points keeps on the value p ( z ) = Pi by virtue of (2.2.10). Thus, for any z > zi we can write =

which by comparison with (2.2.14) immediately leads to V(z) = V i .

(2.2.16)

The volume enclosing a fixed n u m b e r of representative points in phase space remains constant in z, although, due to the motion of the enclosed points,

1D First-Order Optical Systems: The Ray-Transfer Matrix

69

its shape may sensibly change. So, the total phase-space vohlme 12 of tile beam remains constant in z, regardless of the motion of each ray and the corresponding motion of the representative points through phase space: 12o V(z) = lPi (Fig. 2.3). This result is rather intuitive if we think of a flowing "fluid". If tile total quantity of fluid and the density of the fluid do not change through the motion, the volume occupied by the fluid remains constant as well, even though it may deform significantly during the fluid's flow. The invariance of the phase space volume applies obviously also to the volume elements dip 's: d12o = d~)i along any ray path. In w 2.4 we will elaborate a geometrical interpretation of the volume element dV and its invariance along the ray path in connection with the conceptual definition of the intuitive perception of the elementary bundle of rays. Finally, we note that the ray transformation from the phase-space point P~(q~,p~) to the point Po(qo, po) is accompanied by a transformation of the phase-space volume elements from dl)i to dl2o (b). We known that if S denotes the Jacobian matrix associated with the transformation P i --+ P o (see w 1.3.1), the relevant volumes dl)i and dl)o are linked by the determinant of S as dl2o det S dlPi. The invariance of the volume elements under ray propagation implies therefore that det S = 1, thus solving between the two possibilities offered by Eq. (1.3.13) for the determinant of the symplectic matrix associated with the ray transfer. Viceversa, the unimodularity of the symplectic Jacobian matrix associated with the ray transformation implies the invariance of the phase space volume. This offers a clear geometrical interpretatioil of the syInplectic nature of ray propagation, signifying that elementary volumes along the ray path in phase-space remain constant. Evidently, areas play the role as volumes within a 1D picture, we are mainly concerned with (see w 1.7.4). 2.2.2

Basics of Lie transformations

As noted in w 1.3.2, given a dynamical variable f(q, p) we can generate the Lie operator k s by Poisson bracket operation" kf - {-, f},

(2.2.17)

and the Lie transformation TI by computing the power series" t-

cx~ ~j Aj

Ts (~) -- ~

-~. Ls

_

_

e~f

~

~ e R,

(2.2.18)

j=O

In the present discussion the concept of volume in phase space has been left on the same intuitive footing as t h a t in physical space. Actually, phase space is a m a t h e m a t i c a l space in which indeed the infinitesimal volume is defined in t e r m s of the linear a n t i s y m m e t r i c nondegenerate 2n-form, and finite volume are accordingly obtained from it by integration over the domain of interest. For an accurate analysis the reader is directed to [3.3].

Linear Ray and Wave Optics in Phase Space

70

A

where ~ is a real p a r a m e t e r and the j - t h power of Lf is obviously Aj L/ -

{{{..{-,f}..},

f}, f}.

(2.2.19)

j times

As evidenced |)y the cxponc~I~tia,1 nota,tion on tile right-ha,nd side of (2.2.18), the power series is n o t h i n g trot tile cxt)()~lcnl, ia,1 flm('.tion of tile ot)crator ( k f . Evi(h;,~tly, 'Y/(())is t,h('~ i(]cld, il,y ()t)('~ra,t()l': "Y/(()) - I [6]. Lie tra,nsf()rmatio~ls ha,v(; ninny i~lI,(;r(;sting a,n(] ~s(;fl~l t)r()l)crl, i(;s, whi(:h dire(:t, ly follow K()~ t,t~(; t)rot)eI'ties of t,l~c Li(; ()t)cra, t,ors a,~(l t11(; cxt)()~m1~tiation. We shall ll()I, give a. (tct, aih,(t a.t:(:()ll~d, ()f a,ll tim t)r()t)(;rt, i(;s ()f 1,1~(;Li('~ tra,lisforlna,tio~s, for wlfi(:t~ tim r('a(h,r is (lirc(:t(;(l t,() [(;], t)~I, slm.ll ( ' ( ) ~ ( , ~ t , ill Im,rti(:ula,r ()~l I,h()s(; l,hal, arc ~s(;fifl l,() ga.iIl insight int() t,tl(; })(;tmvi()~r ()f l,h('~ l)lms(;-st)a,ce (hqlsity fl(q,p; z) tl~r()~gl~ E(t. (2.2.13). Wc ('l~t)lm,sizc tim, I, I,l~(; ~ml, l~('~m,tica.1 setti~g }1()I'(~ is ~()l, t)r()I)erly (:()~t)h;t(;; (t~mst, i()l~s r(;h;va,~d,, f()r i~sl, m~(:('~, t() tile a lm,lyti(:ity ()r t,lm ra,(Ii~s ()f (:()llV(;l'~(;ll(:O ()f l,ll(; t)()wcr s(;I'io,s (;Xl)m~si()I~S ()r of ()t)era.l,()r i(h;ntiti('s, we will ]~(' (:()l~(x,rll(;(l wit, l~, will n()t 1)(; aI)t)roa.(:tm(t. 77w. trl'od'm~t tnY;.se'l"tTatimt, tn'ope'l'ty We t)r()w; tile pt'od'tt,:t p'l','.sr

tio'u p'rope'l'ty, wllic:ll t,lu'()ltgll l,ll(; r(;la,ti()ll

~:<': h.:/ - (,~<': h.) (,.<'::/),

(2.2.20)

sta,t(;s tha.t tll(; I,i(; tra,llsf()nlm,ti()ll a,(:t,illg ()ll t,ll(; l)r()(hlt:t ()f fiU~(:ti()llS l)rotluccs the sa,nl(; r(;slflt a.s the i)r()(111(:t ()f the Lie tra,nsf()rIlm,ti()IlS a,(:ting s(;i)arately (m tlm sillgl(; fin~('t,i(ms ii~ t,h(~ I)r(~(111(:t. 'Fh(',ii, (:()nsi(h~.r tll(; I)r(~(lll('t of two (lyna.~fi(:al wuiaI~l('~s h(q,p) an(l g(q, p) for which the Poiss(m ])ra,('k(;ts with .f(q,p) a,re lll(;a,lliI~gflfl: v(q, p) -- h,(q, p)g(q, p). (2.2.21)

We wish t,o d(;t,('~i'inin(~ the (;flb~c:t (,f the Lie t,ransforma,tion ~ ( 4 ) ( ) I I v(q,p). ()bvi()llsly, w('~ arc fa,(:c(t with tim ('Na,hm,ti()n of rct)eated Poisson |)ra,ckets of f a,nd v. In fact, iIl accord with tim power s('rics rcprcsenta,tion (2.2.18), one has cCLfv(q'P) -- E

~. LsV(q'P) -- v + C{v, f} + -~-{{v, f}, f } + ...,

(2.2.22)

j=0

We recall from w 1.3.2 t h a t the Lie operator acts on the p r o d u c t of fimctions in the similar way as differentiation. Higher powers of the Lie operator satisfy an identity analogous to the Liebniz rule for derivatives, and hence J

"LJ / hg - ~ (J~) L/ h L/ g. l--O

Al

~j--l

(2.2.23)

1D First-Order Optical Systems: The Ray-Transfer Matrix

71

We invite the reader to prove this relation by iteration. With this expression inserted in (2.2.22), we obtain

v -

E

(j_~), ,, L~ h L~ g - [ E ~

j=0 l=0

"

/=0

C~

z=0

O0

hi[ Lr

.~.j

~"

"~"

j! Lcg] - ( ec

L/

~-~ (j-l)! Lr

.

h) ( cc

(2.2.24)

g)

j=0

thus proving the desired property (2.2.20). In the second step of (2.2.24), we have made use of the permutation co

j

oc

EE

oc

(2.2.25)

E Ej=l

j = 0 l=O

l=O

Accordingly, the Lie transformation of powers of q and p is A

A

A

eCLcqmp n -- (eCLf q) m (eCLr

n,

(2.2.26)

as can easily be proved by applying repeatedly the rule (2.2.24). We consider now a dynamical variable g(q, p), and suppose that it can be expanded in a power series with respect to the conjugate variables q and p around a, certain point in phase-space, which for brevity we may asssume to be the origin: (q,p)= (0, 0). We write therefore (X)

(X)

~(q,p)- ~ 52g ..... qmp,.

(2.2.27)

m=0 n=O A

We apply T/(r to (2.2.27) and use the linearity of the Lie transformation to pass the exponential operator to each term of the series. Explicitly, we have (X)

OO

(2.2.28) m=O n=O

Then, using the product preserving property (2.2.20) to pass tile exponential operator into each variable, we finally obtain (2.2.29) m--0 n=0

It is evident that we can recompose the function g according to the series A

A

expansion (2.2.27) at the phase-space point (cr162q, CCLr A

eCLf g(q,p) __ g(e r

A

namely

A

q,er

p).

(2.2.30)

72

Linear Ray and Wave Optics in Phase Space

T h e action of a Lie t r a n s f o r m a t i o n on a t)hase-st)ace flmction is then to perform a, Lie t r a n s f o r m a t i o n on its a r g u m e n t s . In other words, the above result establishes a parMlel between the t r a n s f o r m a t i o n experienced by the phasest)a('e ve(:tor u = (q,p)T a,nd tha, t ext)erienced t)y the phase-spa,ce flmction

.q(q, p) = . q ( ~ ) , i.~., phase-space vector . . . . . . u ~ ,,~ ~ u v -

>

-

phase-space function ~~/(uV) ~ ,.~'~ ~(uV) .q(vV) -

(2.2.31)

-

T h e (:()i~l)()siti()l~ I)r()I)crty (2.2.3()) ()f tt~(; I,i(; tra.~sfi)Hlm.ti()~s is ('ss(;~tia,1 for the la.ter (:~llsi(lcra.ti(n~s rcgar(li~g tt~(; ~;v(~l~lt,i()~ (ff tl~c l)l~asc-st)a(:(; (lc~sity. The t~oi.~so'n-b'racket p'r~:.scrvatio'n p'~'operty

An()thcr re~mrkal)le I)r(~l)crty (~f the Lie. trm~sf()rnmti(n~s is the l~ois,so'n b'~'ack('.t preser't~ing property, wl~i(:l~ i~ a sc~se ret)ro(l~l(:e.s (2.2.2()) witl~ t,l~('. (w(ti~m,l'y t)ro(h~('t h.q i)eing I'et)la~:c(l ])y t,h(; P()iss()~ia.n I)r()(t~('t { h, g}, i.e.,

T h e Lic l,rallSfi)rllm, l,i()ll a,(:l,illg (m 1,11~',P(~iss(~ll l~ra,('k~'~t (~f fllll(:t,i(nlS gives the sa.llw, reslllt as tile t~(~iss~ll })ra,(:k(;t ()f tll~' IAe i,ra.llsfiwllm.l,i()lls a(:l,illg s~'.l)arately ()111;t1(; sillgl(; fllll(:ti()llS i~v(~lv(;(l in. N(;e,(tl(;ss t,() say, t,lw. P()iss()x~ t)ra~:ket t)r(;s(;rw~,tio~ t)r()I)(;rty lms tt~(; sa.~m l)r()()f as (2.2.20), t)(;i~g |)as(;(t ()11 tl~(; I~('~it)~dz rill(; fi)r 1,11(; ldgher t)()w(;rs (ff Lie (~I)(;rat()rs a,(:ting ( ~ a. P(fiss(m t)ra,(:ket: J

-.j

L,{/,,:~} - Z

,-/

~j-/

(~){ L, t,,

,

:~},

(2.2.aa)

I=0

which, pa,ra,llcliilg (2.2.23), follows fronl the deriva, tion-like bchavi()ur of I_/ with rcst)c(:t to tim' Poissoxl t)ra, cket nnlltit)li(:a, tion. Evi(tclltly tllc t)r()t)erty (2.2.32) esta,t)lishcs that every Lic tra, nsformation is symplc(:ti(:. In fa,(:t, the Lic transforma,tion T / ( ( ) gcncra,tes a, ma,pping which takes the (:onjuga,te variables q and p to the new variables ~ and 7) a,s A

q ~ 0 - e~L/^ q, p ---+ ~fi -- e C t-f p.

(2.2.34)

T h e Poissoll bracket of h a,nd g as functions of ~ and p is then A

{h(q,p),9(q,~)}-

{eCkfh(q,p),

A

er

{h, 9}(~,~) ,

(2.2.35)

thus evidencing the preservation of the Poisson bracket under the m a p p i n g (2.2.34); use has been done of the composition p r o p e r t y (2.2.30) to pass the

1D First-Order Optical Systems: The Ray-Transfer Matrix

73

A

operator e~kf from tile arguments to tile respective functions h and g, and viceversa, after applying (2.2.32), from the flmction {h,.q} to the respective arguments. In particular, (2.2.36) In w 1.3.2 we proved that the preservation of the Poisson bracket of any two dynamicM variables is a necessary and sufficient condition for a transformation be symplectic. Lie transformations as a Lie group

We now take up the question of the group structure of the set of Lie transformations {c;Lf; ~ E N} generated by the dynamical variable f ( q , p ) . Accordingly we will linger on some properties of Lie algebras and Lie groups to help the reader to gain some familiarity with these concepts (see also Appendix A). Although the discussion below will be concerned with the Lie algebra of the operators (2.2.17) and the corresponding Lie group of the transformations (2.2.18), it has a general validity, being relevant to tile link between the concept of Lic algebra and that of the associated Lie group [7]. As remarked in w 1.3.2, the set of Lie operators {k I}, generated by the dynamical variables { f ( q , p ) } through (2.2.17), do form a Lie algebra with respect to the Lie bracket [, ] of operators [A,

-

-

(2.2.37)

which, e~s proved by (1.3.43), defines a binary operation taking two Lie operators to a Lie operator as well: [,]" (Lh,L9) E {LI) • {Ls) ~ [Lh,Lg] - L{g,h} C {LI)"

(2.2.38)

Evidently, if {g, h} = 0, the corresponding Lie operators commute with each i

-

-

,

.

.

1

other" ]Lh,Lg] - - 0 . The notion of group as that of algebra implies the existence of a binary operation sending pairs of elements of the set into elements of the set. When dealing with operators the binary operation to structure a set as a Lie algebra or a Lie group is just the commutation product (2.2.37) between elements in the algebra and the sequential application, i.e. tile multiplication, of operators in the group. It is a notable property of Lie algebras that by exponentiating the elements of the algebra through a real parameter C we may form the associated group. An element 7" of a Lie group is said to be generated by the element L of a Lie algebra in the sense that for an infinitesimal value d~ of the group L

.1

A

"~"

Linear Ray and Wave Optics in Phase Space

74 A

A

A

parameter we have T ( d ~ ) - I + d~L. Thus, the Lie algebra of a group is a description of that group near the identity element. Sequential nmltiplications of infinitesimal-step elements T ( d ~ ) lead by induction to finite-step elements T ( ( ) , for which the exponential represe.ntation in terms of the generator L is consequently obtained. In fa,(:t, it is proved by in(hmtion that A

A

A

t~

~

T ( ( + d ( ) - T ( ( ) T ( d ~ ) - T(()(I +

whi(:l~ ~ea,ns T(() + dT-

T(() +

d(L), A

d(LT((),

(2.2.39) (2.2.40)

ttn~s yie](li~g tl~e fiI'st-(w(h:r (liff(:l'(:~tia] e(t~m.t,i()~ fi)r 7-(()"

'Z,-/-(() _ Cr

(2.2.41)

d(

"l'l~(: (:Xl)()~(:xd,ia,l.r(:l)r(:s(:~d,a,(,i()~ ()f (,l~(' gI'()iiI) (:h:lll(:id, 'T(() i~ (,(:n~s of the alg(:|)ra, (:I(:~(:~(, k i~(:(lia,(,(:ly f()lh)ws: A

q-(() _ ,.r k,

(2.2.42)

witll 1)eillg T(()) - I fin" (;w;ry L ill (,ll(' a lg(;lwa.. Tlle~l, ew'~ry ()l)(;rat(n" ill tile a.lgel)ra gellera,tes a sllt)gr()lll) ()f exl)(nlelltia.1tyt)e ()I)era,t(ws {('( L; ( E R}, wlli(:ll is (:(nltilnu)lMy l)armll(;triz(;(1 1)y the tea,1 t)araI~eter (. ()n(: t,yI)i(:a.lly speaks ()f a onc-param, cter s~d)gr()~l), t';ach value of the t)a,ra,n~cter ( i~(livi(l~m,lizes ()11(~. ()t)erat()r i~ the s~fl)gr()~l). ()t)era,tors ('orrest)()~(til~g 1,()(lifli;rel~t val~u;s ()f 1,11e t)ara,~eter t'(n~n~ml,e witl~ (;a(:t~ other; na,n~ely 7 ( ( , )7((:~) - 7-((2)7-((,) , (2.2.43) which (lire(:tly follows fr()~l~ (,l~e t)a,si(: I'ela,tio~ (2.2.39) (:lea,rly i~l)lyil~g in finite terms that - r

+ G) -

+ q) - "-2(G)r

(2.2.44)

A

The subgroup {c~k; ~ E ]~} is therefore abelian. Also, (2.2.43) allows us to identify the inverse to T ( ( ) a,s the operator individualized by the opposite value - ( of the parameter, namely

~'-1(~) __ r

__ G-~L.

(2.2.45)

In the light of these general considerations, the Lie transformations for a given Lie operator k s are seen to form the one-parameter subgroup { ~ ( ~ ) A

c r ks; ~ E R}, for which

(2.2.46)

1D First-Order Optical Systems: The Ray-Transfer Matrix

7'5

and

_ r

(2.2.47)

Each operator in the subgroup produces a symplectic mapping of the conjugate variables q and p to the new variables ~ and p as given in (2.2.34), i.e.,

{q,P}

{-0 - c~LI q, p -

c~LI p} 9

(2.2.48)

Clearly, successive Lie transformations produce a symplectic transformation A

as well. Thus, the composite operator e (1 kf e~ 2 Ig generates the two-step symplectic mapping A

A

A

A

I~l Lfq-. ~ _ e~1Lf/,l~}

{q,P} ~ {q-- er Lgq, iO-- er L~P} ~ { q -

"

(2.2.49)

However it cannot in general be associated with a single Lie transformation, unless, according to (2.2.46), g = f, or more in general, as will be proved below, {.q,h} = O. Notably, near the identity, i.e., for sufficiently small values of the parameters, Lie transformations multiply into Lie transformations. Namely, we have z~f(~l)Z~g(~2)

-- 7"~h(~'),

(2.2.50)

for ~1 and ~2 sufficiently small. The explicit expression of the closure relation (2.2.50) for Lie tra.nsformations, i.e., for exponential operators, within a sufficicntly small neighborhood of the identity is known as the Baker-CampbellHausdorff (BCH) formula. It dictates the conditions and the rules to unit two exponential operators into a single exponential operator [8]. The exponents of the factors on one side combine into a single exponent of the product on the other, which is ultimately determined by the underlying algebraic structure, directly involving the Lie products of the exponents of the factors. Here we will prove (2.2.50) in a rather naYve way, exploiting the power series expansion (2.2.18). Firstly we consider the interesting case that {f,g} = 0 and so [L/, Lg] - 0. In that case, in fact, the closure relation (2.2.50) of Lie _l I_ transformations holds for finite values of the parameters ~, and ~2, with the multiplication law turning into the additivity of the exponents as ~f(~l)z~g(~2)

-

-

"~lf+~2g(1)

-

-

'Tg(~2)~f(~l)~

which further becomes merely additive with respect to the parameters ~2 if also g = f, i.e., ~f(~l)'Tf(~2) t-..

-- ~ f ( ~ l

-Jl-~2) -- z ~ f ( ~ 2 ) ' T f ( ~ l ) ,

thus recovering the L/-generated subgroup relation (2.2.46).

(2.2.51)

~1 and (2.2.52)

Linear Ray and Wave Optics in Phase 5pace

7'6

A

It will be evident t h a t tile hypothesized c o n u n u t a t i v i t y of tile operators Lf a,nd L.q is crucia,1 to extend the additivity of the exponents from exponentiM scalars to exponential ()perators, thus yiehting the s('a,lar-like relation (2.2.51). We consider the exponentia,1 ot)erator on the right of (2.2.51), which explicitly writes a,s ~

~

~

1

,TC,/~<~:,(1) _ ,,<, L,+(:~ L:, _ Z

j --()

^

)j

.~. ((, kf + (2"L.q .

(2.2.53)

a,n(t ll(;Xl(:e (h;llmll(ls (;vahlat, illg t)()w(;rs ()f the ()t)erat()r (~ LI + (~ L,/. Tll('~ S(llm,r('~ is, fl)r illsl,ml(:(;,

(~, L~ + ~ L,,)" - (r L~ + C~ L,,)(C, L~ + ~ L,,) 1. 2

.

.

.

.

.

.,

')

(2 Ls + (, (2 L~ L:, + (., (, L:, L s + q;. .., L~. "" L;. - (2 "L 2s + 2(s (2 ^L.r ^L:I + (,~ It sll()lll(1 I)(~ ll()t,(',l ill tile lnlfi)l(ler (~xl)r(~ssi~nl a.1)r t,ln' two) 1)rr LI L.q a,u(1 k:/L I, wl~i(:l~ a r~', i~ g(',~nwal ~lifli~r~',~l,. thn~('~', t,l~y (:m~ 1~(~s~n~n~(l I,~ 2 k~ L.q only ~ ( l ( w l,ln~ a,ss~l)ti()~ ()f tl~(~ (:()~nlta,tivil,y ()f k z toni L:t. l~ l,lm.t (:a.s(;, t,l~(: S(l~mr(', ~f t,ln' (~l)(Wat,~w (, LI ~ (~ L:~ (:()~('s t,~ l~aw' t,l~(' stain, ('xl)r(;ssi()n a,s tl~a,t (~f t,l~(, s(l~m,r~' ~f a I~y~n~ia,1 ~f s(:alm's. A(:('~r(li~gly, tin' j-tl~ I)(~wer ((1 LI + (~ L.,~)J (:m~ 1~', (:al(:~lla,t(~! l)y 1,1~(~well k~)w~ fin'~n~la, fi~r tln~ j-tl~ l)()wer ~f a, t)y~r ..

..

(C, L / + (~ L:,)9~ - ~

J

A j_!

(1)([ L// (.j-' L:/

(2.2.55)

l=()

UI)()n s~fl)stit~ti~g (2.2.55) i~d,(~ tlm t)()w(;r series (2.2.53), w(; ~4)l,ai~ ^

A

("~1 Lf-[-Q2 L'q----

o~

j

j--()~ l--0 ~

= Z ~(I /=0

(J-l)!l'Ql" ks(j-z k.~ 1

L/ j /

l ---I

Aj-I

L.q

(j-z)!

.,

thus t)r()vi~g relation (2.2.51). We (;~t)ha,size again tha,t (2.2.51) grounds ()n the ext)r(;ssi()~ (2.2.55) for powers ()f a, bynomial of ot)era,tor functions, which reproduces tha,t for powers of a, byn()nfial of scaJar fim(:tions only for c o m m u t i n g operators ( P r o b l e m 3). In general, to prove (2.2.50) we exploit the power series ext)ansion (2.2.18). We sut)t)ose now t h a t b o t h the rea,1 n u m b e r s (~ and (~, are sufficiently small to justify the t r u n c a t i o n of the series expansions of b o t h Tf((l ) a,nd T~((~) at the first order of (1 and (~. We write then A

'Tf(~" 1_

A

~

A

O) -- -Jl-t.. [ ~,..k~'

% ( q ~ o) - I +

Lg,

(2.2.57)

77'

1D First-Order Optical Systems: The Ray- Transfer Matrix A

A

so that the product of TIff1 ) and Tg((2)is

TI(r A

- I + r kf + <2 Lg,

A

A

A

<1, ~2

A

~

0

(2.2.58)

where, of course, only the first order terms in the parameters ~1 and .(2 have been retained. Resorting again to the power series expansion, we can recover the exponential representation as A

1 . .

t - .

_

GkClf+~2g ~(1/+~9(1) '

A

Tf (~1)~9(<2) c(lkf+~2kg _

A

_

_

_

_

(2.2.59)

_

for r and r sufficiently small, and hence near the identity ]'. As previously remarked, the validity of the multiplication law (2.2.51) for Lie transforlnations near the identity grounds on tile BCH theoreln, which provides the rule to combine the exponents of tile factors on the left as A

A

C<1 kf C<~. Lg __

A

A

A

e kh((~l'~2' kf 'kg),

(2.2.60)

giving a complete analytic expression of the exponent on the right as an infinite series of iterated Lie brackets of ~1 kf and ~2 kg. Explicitly,

[L.,[g]]+

[Lg,[.]]}+

3

(2.2.61)

and the series may in general converge only for sufficiently small ~1 and ~2 [8]. Tile BCH formula confirms ill precise terms that, as noted a number of times, the additivity of exponents does not apply in general to the product of exponential operators, being applicable only to commuting operators. In fact, we see from (2.2.61) that, if k I and kg commute with each other, the infinite summation on the right-hand side simply reduces to the sum <1 kf + ~2 kg, thus

exactly yielding

e ~1 kf e< 2 kg _ e~l kf +<2 kg jUSt aS for s c a l a r s . R e l a t i o n

w e proved in the case that [LI, Lg] - 0

(2.2.51),

using the power series expansions of the operators on both sides, is evidently in accord with the BCH formula. It is needless to emphasize the relevance of the BCH formula within the group theoretical framework, as it establishes a direct relationship between the structure of the Lie algebra of a group (exemplified by the Lie brackets in (2.2.61)) and the local structure of that group (exemplified by the multiplication of operators). Finally we wish to comment on the physical meaning of the parameter entering the algebra-to-group operation of exponentiation, of which (2.2.18) is an example relatively to the algebra of Lie operators (2.2.17). As remarked also in w 1.5, the meaning of r depends on the context we are working in as well as on the meaning of the phase space variable that generates the Lie transformation. i_

_1

Linear Ray and Wave Optics in Phase 5pace

78

geometrical optics is concerned and the Lie transformation is associated with the Hamiltoifian flow, the exponentiation p a r a m e t e r ~ takes the role as the axial distance Zo- zi by which the light ray propagates and accordingly the Lie tra, nsformation is the optical "prot)aga,tor" of the ray from z,i to Zo. We note that, a,s the ot)ti(:a,1 Hainiltonia.x~ is ~mitless, tile (:orrest)on(ting Lie ~)I)crator tins the (timensi(m ()f lei~gth -1. When

2.2.3

Ray-bu'n, dh: trmpagation: th, c ray t'ra'n,sfcr operator and matri:r~

Al)t)lyi~g tl~(; r(;s~lt (2.2.3()) t() E(t. (2.2.13) we ii~ul tlm.l, p(q, p; z) (;v()lv(;s fi'om 1,11(; ilfitia.l (lisl,ril)~l,i()~ p(q,p; zi) a.(:(:()r(til~g 1,()

t,(q, p; z) - t,(, ,-(~-~,) t.,, q,`-(:-:,) ,,, p; z~).

(2.~.~)

We re(:all l,l~a,t i~ .~ 1.4.1 wl~e~ i~lvestiga,ti~g the z-ev~fll~l,i~n~ ~f tln; l)osition a,nd (tire('l,i{)l~ (x)~n'{li~m,tes ~)f tim ray, w~: i~d,Ix)~tlu'(:~l l,lw, ra,y-l.ra.l~sf(;r ~)l)eI'a,tor 9Yl(z, zi) a~{t l)r{)wxt tl~a.t i~ tim case (~f a z-iluh;l)~;~{te~t Ila~ilt~n~ia,~ it takes l,h(; exi)()~n;~d,ia.l,e(t fi)rx~ A

~l(z, zi )

- ,'(~'- :~ ) L,,.

(2.2.63)

A(:(:()r(li~g t(~ (2.2.,17), tin: ()t)(;ra,t()r (:~t(:ri~g tl~(; a r g ~ ( ; ~ t s (~f p i~ (2.2.62) a.t)I)(;a.rs t() I)(: tl~(; i~v(;rs(; ()f tl~(; ra,y-tra,~sfi;r ()l)(;rat()r (2.2.63), ~m~l(;ly A

, -(~-~.,) t_,, _

.~-~(z, zi).

(2.2.~4)

In fa,(:t, if tl,(; ray w,.ria,l)h;s u(z) - (q(z),p(z)) T (;v()lv(: fix),,, tl,(; i,,I),,t (ta,ta, u(zi) t)y 9Yt(z, zi), (:()l~v(;rs.(;ly, th(; int)~lt (la,ta u(zi)(:a,~ t)(; r(;(:()v(;r(;(t fr()m u(z) 1)y the i~lv(;rs(; ()t)era,tor 9)1 - l (z, zi)" u ( ~ ) - @ -~ (~, ~ ) u ( ~ ) .

(2.2.(~5)

Rcla,ti~m (2.2.62) s~ggests therefore the; following t)ict~re. Let (q, p) be the coordinates of a ray at z, tlmn ( c - ( z - ~ ' ) e - q , c -(~-~~) e, p) ma,y t)e interpreted as the starting da, ta for the ('t~osen ray (q, p) consistently with the Hamiltonian dynamics, identified by the relevant Lie operator [ , . Evidently, this is a manifestation of the previously established invariance of the ptmse-space density Mong the phase-space trajectory. A

Remarkably, according to (2.2.62), the Lie transformation c(z-z~) e - , which is basic to the single-ray tracing, becomes as well basic to the r~y-ensemble picture of light propagation ruling the evolution of the phase-space dm~sity. In w 1.5.3 we mentioned that in genera.1 the ray transfer operator 0Jr(z, zi) can be represented in the factored form as the product of Lie transformations. Then, applying property (2.2.30) to each t e r m in the product or exploiting the

1D First-Order Optical Systems: The Ray-Transfer Matrix

7'9

group closure property (2.2.50) of Lie transformations, we may restate relation (2.2.62) in the general form

fl(q,p; Z) --

p(@ -I(z,

zi)q,

@ -I(z,

zi)p; Zi) --

pi(@-I(z,

zi)q,

@ -I(z,

Zi)p),

(2.2.66) involving the ray-transfer operator, of which the single-exponential form (2.2.63) is a specific determination, suited to a z-independent Hamiltonian. In particular, since in tile linear apt)roxiInation a 4 x 4 or 2 x 2 matrix representation is allowed for the ray transfer operator, property (2.2.62) ca.n be formulated in terms of the ray-transfer matrix M(z, zi) as well. Thus, using the handy vector notation, we can write

p(uT; Z)- p ( v T ; z i ) -

pi(vT),

V - M - l ( z , zi)u,

(2.2.67)

which in particular shows that p(0, 0; z) = p~(0, 0),

(2.2.68)

being the phase plane origin mapped into itself by M. Equation (2.2.66) and the corresponding l i n e a r model form (2.2.67) signify the relevance of the ray-transfer operator 92~(z, zi) and the ray-transfer matrix M for both the sinlge-ray and ray-ensemble pictures of light propagation t)ertinently to the geometri(:al-ot)tics view. Equations (2.2.66) and (2.2.67) clearly show, in fact, that any distribution function is propagated along the ray-trajectories in phase space. In particular, the single-ray picture is easily recovered from (2.2.66) by noting that if the phase-space distribution is initially represented by a single ray, and so mathematically modelled by a (5 function, p(q,p; zi) = (5(q - qi)rS(p - pi),

(2.2.69)

it remains a (5-fimction at any z > zi, evolving into p(q, p; z) = gi(q - q(z))rS(p - p ( z ) ) .

(2.2.70)

Here, q(z) and p(z) are the ray variables at z propagated from the input data (q~, p~) through the transfer operator 9)l (z, z~) or, in the linear model, through the transfer matrix M(z, z~) (see also Problem 4). Accordingly, throughout the remaining of the chapter we will work within the single-ray picture of linear geometrical optics. In due account of the established significance of the ray-transfer matrix, in the next sections we will proceed deriving the ray matrices for assigned z-independent model Hamiltonians. Their correspondence to recognizable optical systems will then be investigated. Before closing this section we wish to illustrate the relation of the phase space density p(q,p; z) to those measurable quantities that the traditional radiometry uses to characterize the light distribution and its propagation in connection with the radiative properties of the source. A

80

2.2.4

Linear Ray and Wave Optics in Phase Space Phase space density and radiance

As a n n o u n c e d , in this pa, ragra, ph we elucidate tile connecting link between the ba,sic q u a n t i t y of the phase spa,(:(; represent~tion of gcometrica,1 optics, i.e., tile t)hase st)a,('e density p(q,p; z), a,n(t the basic q u a n t i t y of radiometry, i.e., tim radian(:e (~). T h e (tisr is strictly aimc(t a,t giving a, feeling for the t)hysi(:a,1 con(:rcteness of the t)ha,sc spa(:c density. Accordingly we merely derive the f()rnm,1 relati()~l of p t() tim ra,(timlcc fll~l(:t,ir a(l(lressing tim tea(let to [9] for a fllller a(:(:()llld, ()f ])()tllt, tl(; ra(tio~lletri(" lll()(l(;1 a,ll(l tll(; relatively re('.cntly inv(;stiga,te(t lillk wit, tl tilt, statisti(:al wave; I,tl(;(~ry. W(; ('lllI)tlasiz(; tim.t, a,s g(;()Ill(;tri(:al ()t)ti(:s, ra(tit)lllt;tI'y a n(t the r(;la.tc(t theory ()f ra.(lia.tive (ulergy l,ra,llSl)t)rt are |)a,se(l ()ll 1,11(' (:(nlt:el)t ()f t,lm light ray, lm(t(;rst()()(t a,s tll(' gt;(nll('tri(:al |,l'aj(;(ti,()ry a l(nlg wlli(:ll t,ll(' ()l)ti('al t;llt;I'gy, ra(tia,te(t |)y tile s(nlr(:e, I)rt)lm,gates [l()]. Tlnls, ar t() I,lm ra(li()metl'i(: nl()(tel, tlle rar (;lmrgy is l()('alize(I |)()t,l~ i~ sI)a('(; a~(l i~ (lir(;('ti()~. W(; will s('(; tlmt the rar is (h;fil~(;(l as a fim('ti(m ()f |)()t,l~ l)()sil,i()l~ a,l~(l (lire(:ti()~. P()sit, i()n a,i~(t r162 a,r(; tl~(; ln~il(li~g Wtl'ial)lt;s ()f tl~(, g(;()~('tri(:al-()I)tir I)lm,s(' sI)a,(:e, whi(:h tht;n s(;(;~s i,()()f[t;r 1,1~(;lmt~ral fi'an~(; ft)r tim ra.(litn~ml,ri(' th(:()ry a.s well. I{a,r t~as I)(;liev(;(l I,()t)r()t)(;l'ly (h;st'ril)(; l,l~e I)el~avi()r ()f ra,
1D First-Order Optical Systems: The Ray-Transfer Matrix

81

wave-optical phase-space description of light fields relate to the generalized radiance functions pertaining to r a d i o m e t r y with partially coherent sources. Moreover, b o t h the generalized radiance flmctions of r a d i o m e t r y and the spacefrequency distribution flmctions of wave optics do not retain in general all the properties expected for t h e m in traditional r a d i o m e t r y and in the o r d i n a r y probability theory, respectively. This fact indicates t h a t neither the formers neither the latters can in general formulated in a m a n n e r t h a t is strictly similar to the traditional radiometric and geometric-optical description of light fields [9.9]. However, it was shown t h a t all the generalized radiance functions regain the properties p o s t u l a t e d for the radiance in conventional radiometry, in the case of p l a n a r secondary quasi-homogeneous sources (a) and in the limit of short wavelengths. This result unequivocally establishes the place and the role of classical r a d i o m e t r y within the statistical wave t h e o r y of optics [9.11, 9.12].

Radiance: basic definition The radiance (or brightness) is the area and solid angle density of the radiant power. It depends in general on the point in space at which it is evaluated and on the direction t h r o u g h t h a t point. In order to visualize the definition of radiance we consider a planar radiating surface S in a homogeneous m e d i u m of refractive index n 0. S m a y be the surfa,ce of a real source or of an optical element like a mirror, a lens or a limiting aperture; also, it :nay be a, fictitious surfa,ce situated everywhere in the radiation field. We m a y locate S in the plane z - 0 , and suppose it radiates into the half-space z > 0. Then, for any point Ps on S, specified by the position vector q - (qx, qy) in the source plane, we denote by dS tile element of a,rea around Ps, and by d ~ tile element of solid angle a r o u n d a direction specified by tile unit vector v originating from Ps (Fig. 2.4.a,)). It is assumed in r a d i o m e t r y t h a t tile power dP [W] r a d i a t e d by the source element dS within the solid angle element dr2 is obtainable according to the elementary law dP - / 3 ( % v ) c o s OdSdf~. (2.2.71) Here 0 is tile angle t h a t the v-direction makes with the positive z axis; evidently, 0 < 0 < 7:/2. Thus, the p r o d u c t dA - d S c o s 0 yields the projected eleinent of area perpendicular to v. T h e function B(q, v) is known as tile rad The optical intensity across a planar quasihornogeneous source changes so slowly with position in the source plane that is remains essentially constant through distances comparable with the correlation length of the radiation field across the source plane. Also, it is supposed that the linear dimensions of the source are large compared with both the mean wavelength of the radiated light and the correlation length. Evidently quasihoinogeneous sources may be spatially coherent and incoherent in the "local" sense, but are always spatially rather incoherent in the "global" sense [9.8, 9.4]. Finally we recall that a secondary source radiates after direct or indirect (i.e., through an optical system) illumination by a primary source.

Linear Ray and Wave Optics in Phase Space

82

qx

qx

J

dr)

dS

s

q.l'

q)

(a)

(b)

FIGUII,I" 2.4. I~a.sic elei~eld, S ill~strati~g tim (h;li~fi(,ioI~ ,)f rar (a) Tim l)lalmr so~rce S i~ tl~e pirate z = () ra(liati~g il~to tim half-l)lmm z > () m~(l (b) tim g(:o~mtrical q~mi~titics i~fl~crei~(, iI~ Eq. (2.2.71), i.e., tlw area el(~lt'~H, d,S' aro~n~(l tt~e s(n~rce l)()i~l, l's(q ) aim the element d ~ of soli(l m~gh', ar()~n~(l tlw ray-(lirt:ctio~ v.

(limit:(: ()r lwigl, llllCSs ()f l,ll(: s()llrc(:. It, rt:l)rt:st:lll,s 1,11(: ral,(: a l, wlli(:ll ellt:rgy is ra,ttiatcd in(,() 1,11(: lllli(, s()li(l a.llgh; a r()111l(l l,ll(: st)(:(:ifi(:(1 (tir(:t:l,i(nl v t)y (,hi: s()lxr(:(: (:l(:llleld, ()f 1111i1 l)r()j(:(:(,(:(l a r(:a (t)(;rl)(ql(li('lllar 1,() v ) a l'()llll(l Ill(: Sl)('('iIi(:(l I)Oild, -'> - I (,,) P s ( q ) ; ll('~ll(:e i(, is lll(;a,SlU'(;(l il~ I V . ~ - . . s t A(:(:()r(li~g 1,() its l)lWsi(:a,l sig~lifi('m1(:(:, (,h(: ra(lia,n(:(: is ~m(:(:ssarily ~()mmga,-

(,iv(:, B(q, v) > ()

(2.2.72)

fin" all l)()ssi])I(: wd~(:s ()f its a rg~m~(:~(,s, m1(l s() fi)r a,ll l)()si(,i()ll v(:(:t()rs q = (q:~, % ) i~i t,h(: s()~rc(: l)la,~1(: m~(l (lir(:c(,io~is v l)()i~ti~g i~(;() t,h(: Im.lf-sl)a,(:(: z > (). Also, it, vanish(:s a.(, l)()ints q i~ (,h(: s()~n'c(: l)la,n(: l y i n g ()~It, si(l(: (,I~(: s()~r(:(: a,r(:a, S, i.e., /3(q, v ) -- ()

if q ~ S,

(2.2.73)

f()r a,ll tll(: t)()ssit)l(: (lir(:('tillr(:(: surfa,(:(: S. Evi(t(:n(,ly, (,t1(: tota.1 t)()w(:r ra,(tiate(l |)y t h e s o u r c e is

p-

co.. 0 s

v).S,

74)

e Oil account of the spectral content of the emitted radiation, it is custoinary to consider the contributions to the radiated power for each frequency component separately. The symbol dP~ [W-rim-1] is typically used with the meaning as the limiting vahle of the ratio AP/ACe for freqlxency interval Ace being made infinitesimally small; A P is clearly intended as tile amount of radiant power contained within Aw around the specific frequency co. Needless to say, dP~ obeys the law (2.2.71) in terms of the spectral radiance B ~ ( q , p ) [W. m -2 sr-1 . rim-l]. We will ignore here these frequency-related aspects of the radiometric model; accordingly we will not consider the possible frequency-dependence of the refractive index.

1D First-Order Optical Systems: The Ray-Transfer Matrix

83

the integrations extending over the source area S and over the 27r-solid angle formed by all the directions v pointing into the half-space z > 0. The radiance as a phase-space f u n c t i o n

Being a function of both position in space and direction from that position, the brightness comes naturally to be a kind of phase space function. Evidently the pair (q, v) identifies a ray emerging from the source point Ps (q) into the v direction (Fig. 2.4.b)), the ray m o m e n t u m p being obtained just scaling the unit vector v by the refractive index value n o of the medium. Thus the radiance can as well be understood as a flmction of the proper ray-variables (q,p): B(q, v) ~ , B ( q , p ) . Resorting to the phase-space picture of light distributions in terms of the density of rays in phase space, it is evident that the radiating surface S corresponds to a phase-space density function, we may denote as Ps (q, P)- Likewise the sample ray through the source point Ps(q) extending in the v direction corresponds to the point (q,n0v) in phase space. Then, the rays emitted by the surface element dS a,t Ps within the solid angle df~ around v individualize the element dl; of volume in phase space, which encloses the sample ray. According to the geometric-optical picture of uncorrelated rays, giving independent energy contributions, we can say that ea,ch ray radiating from the surfa, ce c~I'ries a certa,in amount of power independently of the other emitted rays. Evidently, dP is the power carried by the phase-space volume dI;, and so it is also obtain~tble as dP = (5Pps (q, p)dl2,

(2.2.75)

Here (SP roughly denotes the average power carried by each ra,y radia,ted by the surface S. The product ~PPs(q,P) acquires the meaning as the optical power density in phase space of the light field across the ra,dia,ting surface. It is worth noting that the linear addition of power represented by the integral (2.2.74) for the total power is consistent with the view of incoherent energy contributions emerging from all elements of the source into a.ll directions, thus implying that there are no interference effects present. We consider in some detail the phase-space volume d12 = dqxdqydpxdpy, which we may rewrite as d)2 = d S d p x d p y , (2.2.76) in terms of the area element dS = dqxdqy enclosing the chosen point Ps (q.~, %) in the source plane. Recalling from Eq. (1.3.1) the expressions of the optical momenta pz, py: Px = no sin ctx,

dpy = n o sin C~y = n o cos c~x sin ~)y,

(2.2.77)

Linear Ray and Wave Optics in Phase Space

84

qx

Z

FI(;[JI/E 2.5. (h~(n,~ctry of tl}c miglcs rclcva~t to Eqs. (2.2.77). wher(~ 1,1}(~. g(~.()l~el, ry ()f l,ll(~ i~v()lve(l a.l~gl(~s is (:la,rifi('~(l ill Fig. 2.5, tim (lilli;renrials (tp.,, a,l~(l dp,.q wri(,(~ as

dp.r Ac(:()r(li~gly,

-

'It() ('()S

-

py

(t.rd(t.,:,

--

' I t o (:(),'-; ( l / : r

(:()S

l)y S~l(:('(:ssiv(: sl,(;l)S w(: (:~-l.ll (:lal)()ral,(:

d F - I~)dS ('()s- (~.,: (:()s,)u

.,,

f()r d F t,11(; (;xl)r(:ssi()]~

')

.

'l)yd't):q.

.

'2

(2.2.79) a.ft(:r I'(:a,lizi]lg l, lm.t (:()s~.,, (:()s'0 u -- (:()s 0 ml(t t i m (:h:lll(:]d,a.l s()li(l migl(: a,r()lm(t

(tt~

I,h(: v - ( l i r ( : ( ' l , i ( n l

is

E(t]m.ti()ll

(2.2.79)

= (:()s

(~:,:d(~:,,(t'Oy.

(:stal)lish(:s

a.ll illl,(:r(:sl, i n g l i n k

t)(;l,w(;(:n I,ll(; I)|ms(;-st)a,(:(:

(t~ i-l,ll(t S ( ) l l l O g(~()lll(q,ri(',a,1 (tlla.lll, il,i(~,s ill t)hysi(:a.1 Sl)a(:e, lik(' ~I,I'(~I-IN ~1,I1(| s()li(t a,ngl(~s. Inscrti~g (2.2.79) i~lt() (2.2.75) a,n(t (:()]~t)a,ring with (2.2.71), w(~ a,r(; (;md)hxl to relat(~ t,tle ra,(tianc(~ B(q, p) t() tile ()I)ti(:al t)()w(~r (t(nlsity i1~ l)lmsc space; t)reciscly, we (:an write v()llllll( ~,

/3(q, p) --

'n~bPps

(q, p).

(2.2.s0)

The dei)en(tent:e, on z of t)oth B~ and Ps above ca,n ta,citly be asslmled beca,use, as noted, the radia, ting s~lrfa('c may be a,rbitrarily located in space. E q u a t i o n (2.2.80) provides the key to turn from the real spa,('e representation to the pha,se space picture. A mca,sura,ble qua,ntity associated with the optical field in real space, i.e., the distribution of radiant energy at a given plane z, relates to the corresponding density of rays in phase space, thus reflecting as well the intuitive view t h a t the ray density in phase space reproduces the spatial and angula,r energy distribution of the optical beam. In this connection, the concept of emittance domain in phase space arises quite naturally as the region t h r o u g h the representative phase-space volume of the ray

1D First-Order Optical Systems: The Ray- Transfer Matrix

8,5

bundle, which encloses most of the beam energy, say 86% or 99% according to tile adopted criterion. By Liouville's theorein, this region, or more specifically its boundary, can loosely be taken as a representative region, and accordingly as a representative contour, to provide through its shape evolution a pictorial view in phase space of the propagation of the ray bundle in real space. Remarkably, by (2.2.80) tile invariance of the phase space density Ps along the ray trajectory implies the invariane of the quantity ~ along the ray path ( /

in a homogeneous lossless and gainless medium (see also Problem 7); this is a well established result of traditional radiometry [9.1]-[9.3]. As already noted, relation (2.2.80) applies to the classical context of geometrical optics and spatially incoherent sources. Under these conditions, in fact, we can loosely define an energy density distribution in phase space, and interpret it in terms of the radiance of the classical theory of radiometry. In the modern theory various expressions for the radiance have been elaborated, which account for coherent and partially coherent sources, being linearly expressed in terms of various second-order correlation functions of the optical distribution across the source. However, they cannot be interpreted the same as the radiance of the classical theory, i.e., as the radiated power distribution in position and direction. Likewise, wave-optical analogs of the phase space density have been defined in order to combine in a single view, comprising both position and direction, tile two wave-optical description of light signals, separately pertaining to the spatial and spatial frequency (i.e., angular) domains. As an example of such phase-space (i.e., space-frequency) distributions, in Chapter 6 we will consider the Wigner distribution function. We will see that it cannot strictly be interpreted as an energy density distribution in phase space, even though a direct relation with a generalized radiance function can be established, thus resembling the classical relation (2.2.80). For completeness' sake, we report as well the definitions of other two measurable quantities, which are obtained separately integrating the brightness 13 over the source area S or over the 27r-solid angle formed by all the possible ray-directions p. Thus, we obtain the radiant intensity J ( p ) [ W - s r -1] in a given direction by the integral J ( p ) -- cos 0 fs B(q, p)dS,

(2.2.81)

which specifies the angular distribution of the power radiated by the source. Correspondingly the irradiance g(q) [ W - m - 2 ] , specifying the radiated power by the unit area of the source at any given point q, results from the integral g(q) -- f J2

B(q, p) cos Odft.

(2.2.82)

71"

As the radiance, the radiant intensity and the irradiance are typically referred to a specific interval of frequency. Also, as the radiance, they were

Linear Ray and Wave Optics in Phase Space

86

origina,lly defined for r a t h e r incoherent thermal sources and later generalized to sources of any sta,te of coherence. T h e generalization preserves the forreal expressions (2.2.81) and (2.2.82) tn~t in terms of the generalized radiance flm(:tion. Thus, as the ra.diance, the generalized irra,(tia, n(:e does not t)reserve the I)hysi(:a,1 meaning a,s the (:orrest)on(tiI~g (p~a,ntity of tra,ditiona,1 ra(tiometry. In contrast, the generalized radiant intensity retains its t)hysi(:al n~ca,ning a,s r(;t)r(;s(;~t,i~g t,t~(; a.x~glflar (tistri|)~ti()~ r tl~c ra,(tia,tr162 t)()wcr [:).:), :).~0, .~.4].

Ele'me'ntar;q beam of '~'ays Before; r

tllis t)a,ra,gl'a,t)ll wc 1)ricfty r162

()ll tile ra,(ti(mlctric r

of

elcm, e,'ntar!l beam of light. ExI)l(fitiIlg S~Xll~; ge~)xlletri(:a,1 a,rglnllcnts a,ll~l rela,ted inwl,riml(:(:s, it is l)r t,~ giw~ a,ll "~)I)~ra,tiw~" (lefilfiti~nl ~f tl~e at)stra,(:t ('(m('(;I)t ()f ray, (:(n~r i~g t,lw, ~m,tlw,~m,tical i(lca, ()f t,t~('~ ray as a tl~i~ t)cncil r rar witt~ va.~dshing s~li~t a,ngh; m~l ('r(~ss-sc(:ti~m,1 area., a.nr lm~(:c not (:at)a.1)h; ()f (:a,rryi~g a ti~it,(; a,~(ntl~t ()f c~crgy, a,n(l 1,1~(; l)l~ysi('a.1 ~(;('~(1 f()r an "clarity" t'a,l)a/)h;, ill C()lltI'ast, ()f ca,rryi~g a. fi~fite a,ll(1 n~c.a.s~md)lc,, CVel~ tl~ough i~di~fit(;si~m.lly s~all, a~l~(nl~d, ()f c~mrgy. Rest)rti~g t() t,l~(; t)a.sir g(',()Ill(',l,I'y ()f Fig. 2.4.1)), w(: r(;(:()l~si(h:r t,l~(: (I)r fictit, i(n~s) ('l('l~('l~t,a.l s~rfi~.('t' dS~ ('cl~t,('~r(;(i ar(nn~r l)()i~t, P ~ m~(t tl~('~ light ray ('~liicrgilig fl'()lii l ) iill,() l,ll(; (lil'(;(:t,i()li v (Fig. 2.6) I~ct (iS 2 l)(; |,lit' (I)()ssi|)ly fi(:titi(nls) ch;nl('~ltal sln'fi~,('c, tha,t, r a,I'()lnl(l l)()i~lt P2 at (lista,n(:r d fl'()In I ) al()~lg v, r162 a,ll tllr r a y s ()rigi~m,t,ixig fl'()~li t)()i~lts ()f dS l The cl<',n~c~tal soli(l m @ c d t ~ s~fl)te~th'(l |)y dS~ at I)~ is dt~.~ - dA~/d 2, where dA~ - d S , (:()s0~is the t)r()jr a.rca (;h;~nc~t. (;orr(;st)r the (:lcmcntal soli(t m@(', (tt~ s~l)t(;~(t('~(l |)y dS2 at P~ is dtt I - dA.~/d ~, with dA~ - dS~ (:os 0~. It is ea,sy t,() s('~(~, tim,l, dA 1dt~ l - dA.,dt~.~, (2.2.83) l

I

*

'

9

"

tlnls i1111)lyiIlg tim ixwa,rian(:r along the, ray t)ath of tile t)ro(tuct of the projected a,rea, clc~Imllt dA 1 a,ll(1 tile solid angle ( t ~ sul)tenr at this element 1)y a,nother area, (;l(',lll(',~lt a l(nlg th(; ray, which collects all tile rays (:oming from dA 1. Rcla,ti(m (2.2.83) sllgg(;sts to (tcfinc a,n ele'menta'~'y beam of ~nys as t h a t including all tile rays t)assing through a,ny two elements of area, along a segment of some central ray path, whose length be very large coinpa,red with tile linea, r dimensions of either area, element (Fig. 2.6) [9.5, 9.2, 9.a]. The inva,riant projected-area solid-angle product in (2.2.83) is said the geometrical extent of the beam; in a sense it may be regarded a,s the product of w h a t we m a y consider the spatial and angular extents of the beam. Interestingly, by (2.2.79), the element of volume dY in phase spa,ce enclosing the ray of interest directly relates to the geometrical extent of the corresponding beaxn by the constant refractive index of the m e d i u m through which propagation occurs. It is therefore evident that Eq. (2.2.83) is another form to express

1D First-Order Optical Systems: The Ray- Transfer Matrix d~

O~ dSl

( ~pi,,

"'"

'-< " . . . . . . .

'.

87

dA2 Centrat ray

d S2

.~'""

d~2

Ray-path sesment of [ength d

F I G U R E 2.6. Illustrating the definition of the geometrical extent of an elementary beam of rays and its invariance under propagation through a homogeneous isotropic medium.

Liouvillc's invariancc of phase-space volumes along the ray path. Also, by (2.2.71), dP can be interpreted as the power carried by tile elementary beam radiated by the source around the ray emerging from the specified point Ps (q) into the specified direction v. Hence, by virtue of the conservation of power we recover the already established invariance of the quantity B / n o2 along the ray path, i.e, within the corresponding elementary beam.

2.2.5

An example

As an illustrative example of the application of Liouville equation (2.2.11), we determine the z-evolution under paraxiM propagation through a free medium of a,n initially gaussian phase-space density distribution. We take __~_ __EL p~ (q, p; zi)

1 g 2~q~p

2~

2~

(2.2.84)

where the positive quantities O-q and ap specify the "extension" of the ray density in phase plane, respectively, along the q and p-~xis at the input plane zi. We note that p~ is normMized to unity. Gaussian distributions like (2.2.84) are of noticeable relevance since, as we will see, the Wigner distribution function of a Gaussian signal for which 1 , k ~ __ 27r O'qO'p -2k ~ ~ being the vacuum wave number, basically takes the same form as (2.2.84), being Gaussian both in the space and spatial frequency domains. Notably, this is the only case where the Wigner distribution function is nonnegative everywhere through the phase space, and so it is interpretable as an energy density distribution. When propagating through a homogeneous medium of refractive index no, the linear approximation gives for the optical Hamiltonian the free-particle

Linear Ray and Wave Optics in Phase Space

88

like form H -

~I

p2.

Accordingly, Liouville's e q u a t i o n (2.2.11) gives 0 o ~ , ( q , p; ~) -

p 0 ',,... Oq ~(q' p; ~),

(~.~.s~)

wh()se g(~(~z'al s()lutioz~ is :,(u p; ~) - :,(q '~

",

Tt 0

~

t,; ,~)

(2.2.s~)

9

Ill l)arti(:lllar, tile i]ll)llt, (le]lsity (listril)llti()]l (2.2.84) I)r()l)a~a,t,(:s iIlt() _ ('I-v-'/"u

/, . (q, p., z) - 2,~,,~, ~ ('

)~ _ '_z~_

~"~

'~

(2.2.87)

Evi~h;litly, it, r~;Ilm,i~ls Gallssia.~l llll(l(;I" fl'(;(; t)r~>t)a.~a.ti()ll, wliose ~;fI'(;(:t in the t)hase l)lml~' is l)i(:t, lu'er as a shorn" ~)f the initial ~list,z'it)ld, i()ll a,l~)llg t,ll(; q-axis, as will sI)e('ifir 1)(: (;vi(h;ll('e(t ill !i 2.5. Ill g(;llcz'a,1, wc will s('~(; t,lla.t, (]allssia,n (listril)llt,i()]ls l'eilla,ill (]allssiaxi lnl(h'~r l)rr t]n'()llgli a,lly [irst-()r(tcr ot)tical syst,(:lli, i.(;., syst,(;~is t,]la.t at(' (h's<:ril)e(l ])y IJe tra,]lsf()i']lm,ti()]~s g(;]leratex| |)y i)()ly~l()]lfia.ls (ttm,(lrat,i( 9i~l tlic ()l>ti(:al (:(>()I'r q ml(l p. Fi]la,lly, it, is w()rt,]l (;la,l)()l'ati]lg a.~l exa,]llt)le ()f l,]lc gC()liiet,ri(:a.l ]>(;a]li-cxt,t;nt syliilil('~t, rir a,l'()llll(]

tll(: l >r( )l )a,ga,t i( )l i z-axis. As is w('ll klit)Wli, t,]l('~ ])(;a,lll ilitclisity tra liSV('rsally (](;(a'(:as('.s t,lir()ll~h a. ga~lssia~l Ira)tile l,() ,,-2 ~ ().135 ()f it,s (m-axis va,l~(; at t,ll(; l)ca]li z'a,(lil~s, tts~la,lly (h;~lr t)y u~(z). Als(>, the l)ca,ni (tivcrgcs a,s it, l)rr a,h)nt4 t,iie z-a,xis with an ill('r(;asc ()f the |)(~a,ni ra.(li~ts, whi(:h ta,k('s (>~l tl~(; ~l~i~fi~lnnii wd~(; u~o at tl~c waist, m~(l tl~cn fill" fl'()~l tlw~ waist i.e., fi)r r(;h;wud, Fr(;s~l(:l ]nulfl)(;rs 7ru,12/Az << 1 - fi)lh)ws t,l~c sin~I)h'~ la.w witt~ z:

Az ,,,(z) ~_ ~ ,

(2.~.ss)

7T'l l ~o

,~ t)ci~g tlic wa,vch'~xigtli ill the Ille(lilllll [13]. It is ]la,t~md t(> take a,s a,n cxenq)lary a,z'ca, A, tile (:ross se(:ti(>li ()f the b e a m at the waist" A l - 7rw~, and a(x:ordingly a.s soli(t angle t~ ttm t s u b t e n d e d at the waist by the (:ross section of the b e a m at some z far fr()m the waistf~ - 7rw2(z)/z 2 - A2/Trw~o. T h e I)roduct A~ t~ is then the z-indet)cndent value

A1~'~1

2.3

-- ,~2.

(2.2.89)

Quadratic monomials and symplectic matrices

As briefly recalled in the i n t r o d u c t o r y notes, the optical Hamiltonian, suited to the 1D picture of geometrical linear optics, takes the q u a d r a t i c form (w 1.7)

H(q, p) _ 2__~0 2 p12 + _~q2,

(2.3.1)

1D First-Order Optical Systems: The Ray- Transfer Matr&

Quadratic monomiats:

A

Matrix representatives:

N

p2/2, q2/2, qp/2

K_, K + , K3 ~ ,V~(2, P, )

<1==5

89

Ray-matrices of feasible optical systems 9 Sp ( 2, P, )

by e x p o n e n t i a t i o n to the group t h r o u g h a real p a r a m e t e r

b.v Poisson b r a c k e t i n g with q and p

FIGURE 2.7. Conceptual and formal path to single out the basic optical systems pertaining to the linear approximation. where n o denotes the on-axis value of the refractive index and n2 is linked to the medium refracting power (see w 1.7.5). We have also recalled from Sect. 1.7 the steps to proceed from the optical Hamiltonian H to the corresponding ray-transfer matrix M : H

~

H

~

M,

(2.3.2)

with M being locally or globally expressed as the exponential mapping of the matrix representative H of H by the pertinent (i.e., infinitesimal or finite) propagation distance Az: M(zi + A z , z i ) -- c z x z n .

(2.3.3)

Tile 2 x 2 traceless matrix H correst)onds to tile optical Hamiltonian (2.3.1) under the Poissonian bracket based rule (1.7.7), and so H-(

0

l/:J~ ) .

(2.3.4)

~71, 2

Evidently, (2.3.2) retraces the path (1.5.1) within the purely matrix formalism appropriate to first-order r~y-optics. Here wc proceed investigating the optical vMcnce of the m~trices generated, through the path (2.3.2), by homogeneous quadratic polynomials in the optical coordinates q and p (Fig. 2.7). Since any quadratic homogeneous function of q and p can be expressed as a linear combination of the quadratic monomials p2, q2 and qp, we will firstly derive the matrix representatives of these monomials, and then we will exponentiate them through proper real parameters to form the corresponding transformation matrices. Finally, we will analyse the effects that such matrices produce on the ray variables in order to verify whether they may be interpreted as the ray-transfer matrices of constructible optical systems. We will be able to identify the p2_ and q2-generated optical systems as a section of homogeneous medium and a refracting surface (or, equivalently, a thin lens). In the next chapter we will prove that every symplectic real matrix as well as every first-order optical system can be synthesized by an

90

Linear Ray and Wave Optics in Phase Space

appropriately designed sequence of homogeneous medium sections and thin lenses. This will confer concreteness to the qp-generated optical system, which, identifie(t as a pure magnifier, will then t)e use(t as a, basic corot)orient for more (:omt)a(:t (as well as conceptually and t)ractically interesting) syntheses of symt)lectic matrices and accordingly first-order optical systems. The Lie operators associated with the monomia,ls p2/2, q2/2 a,n(t qp/2, we st)(;('ifi(:ally (t(;~l()t(; by K_, K+ ml(t K:~, wI'it('~ as (see als() .~ 1.5.2) t..

*"

A

A

1

K

-{',5P2}

_

K*"+

1 q2 } _ -- {., _~

^

1

K:, - { . , S q p }

0

pOq, i) , _ q ~,p

(2.3.5)

1 (i)

, o)

- -~ q N - P N

"

Corresl)(m(lillgly, t)y Poiss(nl t)ra,('keting witll q an(t p a~:('()r(ling t() (1.7.7) the rcst)(~,ctiv(~ llmtI'ix r(~tn'(;s(~Ilta,tiv(~s a,l't~,()l)tailm,(t a,s

,

l(,p

K)

) K:

_(00) --1 0

'

I (:) ()I) "

It is (~,vi(t(~,ld, tlmt th(; llm,trix r(;I)r(;s(;lml.tiv('~ H ()f th(; IImllill,()lliaxl (2.3.1) (:an lmi(pu;ly I)e (;xi)ress(~,(t a,s a, lin(;a,r (:()ml)ina,ti()n ()f the a,t)()v(; llm,tri(:(;s: H _ __1 K

+ 'Jt.2K ~.

(2.3.7)

?}'o Before l)r()(:(;eding fllrttmr, w(; briefly (:(mmmnt on nmtri(:(;s (2.3.6). Again we keep the (tiscllssion ()~l a, rela.tively (;l(;nl(;lltary level; w('~slmll n()t (teta,il, in fact, lit)on sonm g('iminely nmtheInatical pe(:lfliariti(;s of th(; gr(mt) Sp(2, N) (and the relevant algt;bra ,sp(2, IR)), for wt~ich the reader is (tir(;(:t(;(t t()[11]. ()ur main interest is ill evidencing the relation ()f the symplecti(: a,lg(;l)ras a,n(t groups to the context of geometrical linear optics we are dealing with. The algebra may be seen as providing the "seed" for the evolution, being linked to the equation ruling tim ray propagation. Correspondingly, the group may be seen a,s providing the "agent" of tile evolution, being linked to the ingoing-to-outgoing ray relationshit)s. Thus, as the algebra comprises the Hamiltonian-related matrices, the group comprises the propagation describing matrices. We noted before that the structures of the algebra and the associated group may be related by the exponential mapping. Notably, the group Sp(2, R) is not of exponential type; namely, the elements comprised by all the one-parameter subgroups in Sp(2, R), generated by the elements of the relevant algebra sp(2, R), do not exhaust the entire group but leave a good portion out [11.2, 11.5].

1D First-Order Optical Systems: The Ray- Transfer Matrix

2.3.1

91

The symplectic Lie algebra sp(2, IR) and Lie group Sp(2, N)

Firstly we note that K_, K+ and K 3 obey the commutator brackets [K+,K_] - 2K3,

(2.3.8)

[K3, K+] - ~=K+.

The set {K_, K+, Ka} is then closed with respect to the Lie product, and hence, the linear space, it spans under the ordinary addition of matrices and multiplication of a matrix by (real) scalars, configures as a Lie algebra. Precisely, the commutation relations (2.3.8) are constitutive of the symplectic algebra sp(2, R) [7, 11]. Every matrix A in the algebra can uniquely be expressed in terms of the basis set {K_, K+, K a } through arbitrary real coefficients as

A-a

K - +a+K+ +a3K3- ( a3/2 -a a- /2 )

"

(2.3.9)

Then, using the general expression above, it is an easy matter to prove that every element in the algebra obeys the relations Tr(A) - 0,

A J + J A r - 0,

(2.3.10)

(see also Problem 1.8), where J denotes the antisymmetric matrix

J-(-011)0 '

(2.311)

introduced in w 1.3.1 and further exploited in w 1.7.1. Evidently, tile matrix representative H of the optical Hatmiltonia.n (2.3.1) is an element of sp(2, R). It was earlier remarked that in general an algebra generates the corresponding group by means of the algebra-to-group operation, i.e., by the exponentiation of the aJgebra, elements through arbitrary parameters. On account of the operator nature of matrices, we may reiterate the considerations developed in w 2.2.2 about the algebra of Lie operators and the corresponding group of Lie transformations. Hence we may say that every matrix A in the symplectic algebrasp(2, R) uniquely determines the one-parameter subgroup of reaJ matrices {TA(C) -- etA; ~ E R} in the symplectic group Sp(2, R), the binary operation being the additive composition with respect to the parameter, i.e., A

A

'~A(~I)'J"A(~2)

A

-- 'TA(r

A

A

-~- ~2) -- '~A(~2)'~r'A(~I)

9

(2.3.12)

Every subgroup comprises both the identity, as TA(0) -- I, and the unique inverse . ~ 1 ( ~ ) _ "YA(--~). The symplectic group Sp(2, N) is therefore formed (even though not exclusively) by real matrices S's obtained by computing the power series expansions

~-'--=.~~JA j - c cA j-o

A E sp(2, R), ~ C R

(2.3 13)

92

Linear Ray and Wave Optics in Phase Space

with A element in the algebra and ,( real number. Multiplication of grout) elements within a sufficiently small neighborhood of the identity m a t r i x I is ruled by the BCH formula, (see w 2.2.2), and hence t,~IA(,(2B

__ c C ( ( 1 , ( 2 , A , B ) .

(2.3.14)

The nla,trix C writes i~ t(;rn~s of the nmti'ices (~ A a,n(t (2B i~1 tim si~lfilar way as k,, i~x (2.2.61) writes in l,(;rlns ()f 4~ L/ an(1 4~ k.q, a.ntl st) it, is an infinite s~ml ()f l,(,,r~l~s inv()lving (nfly tl~r nmt,ri(:es (~A a,n(t (2B an(t tl~r ~ulltit)le (:onmnmmws:

C(.(l,(2, A, B ) - ( , A + (~B + C,-@'[A, BI + ~ {<2(~[A,[A, B)]] + <,42[B, [B, A]]} + O((:)(~, (2(2, (, (~).

(2.3.15)

151 gelleral tim, series nmy (:(nlverge, (rely fi)r sldti(:iently mlmll (1 ml(l (.,. It is a, xl()tal)h; ast)(;(:t ()f Sp(2, IR) tlm.t it, is ll()t, ()f (;Xl)(nmlitial l,yI)e. Na,in(;ly, Sp(2, IR) llla,t,ri(:(',s exist tlla,l, (h) 11()I, arise (lire(:tly i)y a. lllli(tll(; exl)(nmntia,1 mat)t)ing Kr 1,t1(; a,lgel)ra sp(2, IR), 1)lit, iIl (:()llti'ast 1)y the, (:(nlq)()siti()n of a t)t)r()i)ria.l,(: eXl)()ll(:~ll,ia.l-tYl)(,~ eh:llleld,S ()f l,ll(', grolq). Tlle rea,(h',r ild,eresl,ed in t,lfis rellm,rka,l)h;, ml(l r(;t:(;ld,ly iliv(;sl,igal,(;(t, f(',a.l,lwe ()f l,ll(: Sp(2, IR) lllay (:()Ilslllt the r(;h'vald, refs. [11.2, ll.5], lie, re, we will 11()I, flirt, her (h;ta,il l,his l,()l)i(:. As a, (liI'(;('l, ('()ns(;(pu;~l(:r ()f 1,11(; r nm,I)I)illg (2.3.13) fl'(ml the algcl)ra lira,trices, o|)eyi~lg (2.3.1()), t,() t,ll(; gr()~ll) ~liat,l'i('(:s, t,ll('~ ~l~at,ri('t'.s ill ('very a,lgcl)ra,-g(;~lcra,l,r s~l)gr()~l) r 1)r t)r()vr I,()()l)r l,ll(; rela, l,i()l~

SJS q- - J,

(2.3.16)

whi(:h (~f (:(nn'se extell(ts t,(~ tll(~ ll(m-t;xI)(mential tyI)e (',lenlents as well, t)eing preserve(t lm(ter matrix t)r(~(tlu:t. Rela,ti(m (2.3.16) is tlle i(tentifyillg t)roperty of the synll)le(:ti(" ma, tri(:es (see also ~ 1.a.1). The rea,(le, r is suggeste(1 to prove identity (2.3.16), t)y writing it in the han(ty form J S q- - S - l J, and then using the series ext)a,nsions f()r S T a,n(t S -1 a,n(t the secon(t ()f relati(nls (2.3.10). As a, filrther identifying feature of the symt)lectic ilm, trices, we can say that, as a,lrea,dy e,sta.blished in w 1.3.1, they define linear tra,nsformations of a 2D vector spa,(:e.( u - ( U l , '~t2)T } under which the a,ntisymmetric bilinear form

(u, Jv)

-

~/~1 V2 - - U 2 V l '

(2.3.17)

(which in the optical context is just the Lagrange invaria,nt A of w 1.7.4), is invariant. In fact, if u --+ u ' - - S u and v --+ v' - Sv, by (2.3.16) we find t h a t

(u', Jv') - (u', S -rJSv') - (u, Jv).

(2.3.18)

1D First-Order Optical Systems: The Ray- Transfer Matrix

93

On the other hand, the skew product (2.3.17) can be obtained as the determinant of the 2 x 2 matrix formed by the components of the involved vectors u and v; namely /

(u, Jv)

--UlV

2 --U2V 1

\

-- d o t ( U l V l } \ , / 2~ tV2

Under the linear transformation u --~ u ~ = Su and v --~ v ~ = nant in (2.3.19) is multiplied by det S, and hence

(2.3.19)

Sv, the determi-

(u', J v ' ) = det S.(u, J v ) = (u, J v ) ,

(2.3.20)

thus establishing that the symplectic matrices are unimodular(/) det S - 1.

(2.3.21)

Consequently, only three of the four entries of S are linearly independent. The symplectic group Sp(2, R) is, in fact, 3-dimensional. One generally speaks of Sp(2, IR) as a three-parameter Lie group. It is evident that the generating algebra sp(2, R) is 3-dimensional as well, as unequivocally displayed by the general expression (2.3.9) for elements A in terms of the three algebra generating matrices K _ , K+ and K a (see also Problem 5 in ch.1). Thus, the path n~'ively traced in (2.3.2) can formally be interpreted as developing from the Lie algebra of the quadratic homogeneous polynomials in both q and p coordinates, and then proceeding through the Lie algebra of the corresponding traceless matrices, formed by Poisson bracket operation with q and p, up to the symplectic Lie group of the optical ray matrices, formed by exponentiation of that matrix algebra,, as sketched in Fig. 2.7.

2.4

Quadratic monomials and first-order optical systems

We form now the exponential mappings of the matrices (2.3.6) and then inspect the transformations they act on the ray variables, in order to identify the q2 and l qp. We will see optical systems associable with the monomials ~i p2 , 89 that p2 describes a section of homogeneous medium as q2 an interface between two media. The mixed monomial q p will be seen to produce an overall scaling of the ray variables, whose optical realization will be clarified in the next chapter. To still preserve the link between the matrix exponentiating procedure, which may appear somewhat "abstract", and the Hamilton equations solving I The proof of the unimodularity of the Sp(2, R) matrices, given here, is based on that presented in [7.1] (already quoted in w 1.3.1), where the general case of n-dimensional symplectic groups and accordingly n-dimensional vector spaces is considered. Evidently, it can as well be interpreted in the light of Liouville's invariance of areas, which transform through det S under the transformation of the relevant vector space by S.

Linear Ray and Wave Optics in Phase Space

94

procedure, we will resort to both procedures when examining the monomials p2 and q2, that directly relate to the model Hamiltonian (2.3.1). It is worth noting that the matrices K _ , K+ and K a ha,ve b e e n formed by 1 2, -~ lq2 and -~qp, 1 rule (1.7.7) from t h e g e n e r i c monomials -5p that have not b e e n given an explicit interpretation as optical Hamiltonian fllnctions, for which indeed at)prot)riatc normalizing factors, possibly for the t)osition variable q, sho~fld t)e n(~e(te(t. The e~tries of K _ , K+ an(t K:~ ha.ve theref()re (tcfinite ~mits in a,(:(:()r(t witt~ th()se ()f tim ra,y (:()()r(lix~a,tes q [le~gtl~] a,~(t p [~nfitless]. A(:('()r(li~gly, the t)ara,nmt(~r ( ()f the cxt)()~m~tia,1 ~m,t)t)i~g fl'()~ tlm a,lgebI'a ~m, trices { K ,K+,K:~} t() the grolq) ~m.tri(:es (:mn~()t a. t)riori 1)e given the role as a, "lex~gtl~" a,s i~ (2.3.3); its n~ea,~i~g (:()nf()r~s witl~ tl~c nmtrix at hand i~ or(ter t,l~at tlm I'es~lltil~g ext)onentia,te(l f()rIll 1)O. i~t('xI)I'(~ta,|)le as i.l,ll ot)ti('.a,1 ma.trix, fl)r wlfi(:l~, a.s we k~()w, the (tia.g()~m,1 cntri(~s ar(~ (til~(~si()nless, whilst the off(tiag()~ml lmve ullits ()f lcngtl~ a~(t l(u~gtl~-~. Finally wc e~l)l~asize tha.t, ()n (wa,h~a,tix~g tlm. exI)()~m.~tia,1 fin~(:ti()I~S of the ~mti'i(:(~s K , K+ m~(1 K:~, we (:onq)ute, m~(1 tlw,~ s u ~ ~t), tlm t)()weI'S in the series (2.3.13). This ('an easily 1)c (:arrie(l ()l~t, (lue t() tim si~nt)le f()nn ()f the inv()lve(t ~mtri(:es, fi)r wl~i('l~ in fa(:t we ti~(t K 2 - K 2 - 0, _ +

K 2j-1 1 :~ -- 22(J_1 ) K:~, _

K~

2j _ 1 - 22ji,

j-

1,2,3...

(2.4.1)

Then, wc are elmt)h;{t I,() clisrcgard the (tlu;sl, i()ns r162 with the converg(;n('e ml(t tim (,vallmti()Ii ()f nm,t,rix flnl('t.i()IlS (l(,fiIle(t |)y infi~lil,e series like (2.3.13), f()r whi(:l, w(; ,,,ay a(t(tress th(' r('a(t(;r t()[14]. 2.4.1

Propagation along a section of homogeneous medium

As ('arli('r ll()tc(t, tlm l)araxial t)rot)agati()ll ill a h(mlogelle(nls nle(tilml of refractive in(lex n o is mo(telled by the quadratic monomial H(q,

p)

-

1 ,

P

9

(2.4.2)

In fact, the equations for the ray variables write as

r

(2.4.a)

showing that, oil propagating froIIl tile input plane Hi to tile output IIo, w h i c h b o u n d t h e m e d i u m r e s p e c t i v e l y at zi a n d Zo, t h e ray e x p e r i e n c e s t h e transformation

(qo)

(lp

0

p

,244,

1D First-Order Optical Systems: The Ray-Transfer Matrix I-I i

I-] i : I - ] 0

1-] o

[-.:i')'__"

no

. . . . . .

~

'9~

= ~9i

qo =qi +

--"

/701

F/O 2

9

Pi

z

!--

Az

Zi

95

z

"i Zo

(a)

F I G U R E 2.8. T h e optical systems g e n e r a t e d by the m o n o m i a l s 71p2 a n d 1q2 2 are respectively identified as ( a ) a section of homogeneous m e d i u m and ( b ) a refracting surface.

The position coordinate changes by ~ q - -%o ZXzPi, whilst the optical m o m e n t u m remains constant. The ray travels in the medium by the axial distance Az along a straight line in the incoming direction of motion (Fig. 2.8.a)). Likewise the exponentiation of the matrix representative H = K / n 0 of (2.4.2) yields the matrix on the right of eq. (2.4.4). Consequently, K may be regarded as the generator of the free propagation, and the exponentiated form

0

1

'

d-

az

(2.4.5)

n o

may be interpreted as the ray matrix of a homogeneous medium of refractive index n o extending over the axial length Az. The ratio d - A___~of AZ to n 0 is n o

usually reported as the reduced length [12]. Note that tile positions of the reference planes Hi and IIo enter the ray matrix (2.4.5) through the relative separation distance Z o - zi. The system is in fact inva,riant under translations along the z axis, provided tha,t the light ray is equally translated. Clearly this invariance reflects the medium homogeneity. The above results are of course in accord with those of w 1.5.2, concerning the Lie transformation generated by the operator Lp2/2 - K and its effect on the phase-space variables. Equations (2.4.4) and (1.5.12) are in fact identical, both describing a translation in the unperturbed initial direction of motion. 2.4.2

R e f r a c t i o n at the i n t e r f a c e between two m e d i a

Consider the quadratic monomiM 1 q2 H(q, p) -- -~n 2 ,

(2.4.6)

Linear Ray and Wave Optics in Phase Space

96

which can be

roughly regarded as obtained from the model Hamiltonia,n (2.3.1) in the case that the free-propagation term p 2 / n o remains negligible through the evolution length of interest. Tile releva, nt Hamilton's equations { ~q-0,

(2.4.7)

r

-

-

are easily iIlt('gra,te(t to tl~e tra,nsf()r~lmti(nl

(qo)_ ( Po

qi

)_ (

Pi -- 7t2 / k z q i

O) (qi) Az--zo-zi

1 --n 2Az

1

Pi

(2.4.8)

'

which leav(;s llll(:tm, ng(;(t tile t)()sition co()r(tina, te q o - qi, whilst ('ha, xlging the optica,1 m()nl(;ntllnl |)y A p -- --Tt2/kzqi. DII(; t() tilt', lumv()i(ta,|)h; (litfra.ction ()f light thr(nlgll a. fixlit(: (listml(:(;, tll(; a.t)()v(; tra.xlsf()nlm.ti()Ii is a.ctlm,l)l(; ()111y if the input ml(t ()lltI)llt I)lmles are intillit(;ly (:h)s(; t() (:a.(:ll ()tiler: A z ~ d z . Thus, the trallsf()riIm, t,i()Ii (2.4.8) r(;t)r(;s('llts tll(; i'nstantaneo'u,s variati()Ii ()f th(; (tir(;(:tion ()f l)r()l)aga.ti()ll ()f l,ll(; ray, a.s tlm.t ()(:(:lu'rillg ill 1,11(;refra,(:t,i()ll a,t the boun(ta,ry sllrfil,:(; I)(;t,w(;(;ll tw() regions ()f (lifft;r(ud, r(;fra,('tiv(: izl(ti('(:s, with t,ll(; tw() refer(~Ii(:(~ l)lml(',s t)eillg l)la(',e(1 inmle,(liat(',ly t() I,ll(', l(',ft ml(1 illlIIl(',(lia,t(4y t() t,ll(; right ()f tl~(, s~n'fiu.,(, (Fig. 2.8.1))). I~ flu:t, a(:(:()r(li~g t() t,l~(', ixg(;I'I)r(;ta,ti()xl ()f n,~ giv(;~l i~ w 1.7.5, tile 11()11 z(w()()t[-(liag()xm,1 entry i~1 (2.4.8) is nfim~s th(; refl'a,(:tixlg I)()w(;r ()f 1,t1(; s~u'fa,(:c 7) - ix,,,, if An, r(;t)res(;nts 1,11(; (lit[(;ren(:c R ' between the re,fra,('tive in(ti(:es a,t the outl)~it and int)~t t)lmms a,n(t I{ st)(;(:ifi(~s the local (:~wva,ture ra(ti~s ()f the |)()un(ta,ry s~rfa,c(;. We re(:all ttm,t the ~sual (:onvcnti()ns r(;(t~fir(; t() take R t)()sitive ()r xlcga,tiv(; a,(:(:()r(ling t() wh(;ther the (:enter of (:urw~,t~we is to the right or left, ()f the surfa,('c, witll tim ligld, l)cing a,s usual assumc(l to t)rot)a,gate from left to right (scc w 1.8 for 1,11(;sign conventions in the ot)ti(:s ()f i~m,ge-fi)r~li~lg illstrunlellts). By cxp(m(;ntia, ting the x~m,trix representative H - n 2 K + ()f (2.4.6), wc obt,a,in the refra,(:ti(m matrix i~ (2.4.8). The, refore, K+ can |)(; intert)r(;t(;d as the generator ()f the r(;fracti(m a,t the interface between tw() (tifferent media, an(t accordingly tile nmt, rix for the, corresponding ray tra,nsforma,tion is cpK+ = ( 1_7)

0)1 '

(2.4.9)

with 7) [(length) -~] being the surface refracting power. The reference pla, nes a,rc taken at the left and right of the surface. On account of the results of w 1.8.3 concerning the lower-left entry C of the ray-matrix of a,n arbitrary optical system (see Eq. 1.8.17), the power 7) can be expressed in terms of the reduced focal length f of the refracting surface as

7)

1 f~

f

fl

f~,

TL0~

TL02

(2.4.10)

97

1D First-Order Optical S y s t e m s : The Ray- Transfer Matrix

1-I s

1-I i

f,

n~

~/ ./. ....".... ~9i

./"~o

)

YlO 2

qo

Z Az zi

zx

F I G U R E 2.9. T h e refraction at the interface between two m e d i a transforms the input ray variables (q~, no1 sin 0~), referred to the chosen z~-plane, to the ray variables (qs, no2 sin 0o), referred to the profile screen at the intersection point Ps on the surface, which are t h e n projected back to the input plane into the final ray variables (qo, no 2 sin 0o). W i t h A z --~ 0 and [q << 1 we recover the i n s t a n t a n e o u s view of Fig. 2.8.b).

where fl and f2 denote the primary and secondary focal lengths of the refracting surface, a s ltOl and n % tile on-axis values of tile refractive indices of tile media on the two sides of the surface. In particular, the refraction matrix for a planar interface (R ~ oc) turns into the identity matrix, with the input-output relations being then qo -

qi,

(').zt.ll~

\,-.,..,--,-/

Po -- Pi.

Interestingly the second relation above expresses the conservation law of the optical momentum before and after the refraction at the planar interface. Also, it may be regarded as the paraxial form of the Snell law of refraction, explicitly writing as n o 2 i ) o -- nOlt9 i. (2.4.12) Picture (2.4.8) conforms with the result expressed by Eq. (1.5.15), dcscribA A ing the canonical transformation generated by the Lie operator L q 2 / 2 - K+. Before closing this paragraph, we briefly evidence the conceptual difficulties arising when dealing with refracting surface-transformations within the Hamiltonian formalism. Indeed, since the Hamiltonian formalism would not easily accoInodate light refraction, the symplecticity of the transformation is not directly assured. By its own nature, the Hamiltonian view to optical propagation would imply that each ray have a unique evolution once specified the refractive index function of the medium through which propagation occurs. In fact,

98

Linear Ray and Wave Optics in Phase Space

once assigned the initial values of the ray coordinates, i.e, position and angle, the first order differential e q u a t i o n s (2.1.1), or their linearized form (2.1.2), lmi(tlmly d e t e r m i n e the " f u t u r e " values of the ray ('oordinates, i.e., position a,nd angle at every s u b s e q u e n t determinatioI1 of the prot)agation variable. In (:ontrast, at the refracting surfa(:e s e p a r a t i n g two different m e d i a the incident ray ordinarily t)rodu('cs botll a refracted and a reflected ray. Moreover, the differential c(tlla, tion f()rIlmlat, i()ii ()f Hanfiltonia,Ii ()t)tit:s rr a mr with a (liffert'~nl,iat)le refl'at't, iv(; in(lex. Evittenl, ly, l,he inl,(;rfa(:('~ int, rt)(lll(:t;s a, (list'ontinuity of the intt(;x flm(:t, ioll, i.(;., "t,l~e t)ot(mtial", entering ttalldlttnl's e(tlla,tions (2.1.1) (()r ('(t~fiwdcnt, ly, t,l~(: ray (;(t~m.ti()n (1.2.8)). I)is(:(mti~nu)~ls-il~(lex sys(,(;his sl~(',h as r(;fl'a,t't, ing s~lrfa,(:(;s have t)(;(m (,r(;a.l,e(t (,l~r(n~gl~ a, (:(m~I)l(;l~cnt, ary (l(;s(:riI)l,i(n~, r('~s()rl,il~g (,() m~il,a})l(; g(;()~l~(;t,ri(:al (:()llt,rll(:l,i()llS a11(| I)l~ysi(:al (:onsitlt:I'ati()~ls a,I)()~t t,ll(; (ty~la.lxiit:s ()f t,li(; I)I'()(:(:ss I'atlit;r t,limi t,() tll(; (tiff(;I'(;ntial ettlla,l,i()li f()rlllalislii; it, lias ])(;ell t)r()ve(t t,lla|, I'efi'at'l,i~lg sllrfat:e l,rallsforniati(ms fa.t:l,()riz(: inl,() l,w() r()t)t, sy~t)l(;(:l,i(: tra.l~sf()r~m.t, iol~s, ea.('l~ (t(:l)e~ltling oi1 1,1~(; t)r()t)t;rties of ()~(; of tim t,w() ~ e ( l i a s(;t)aral,(;(l l)y t,l~(; i~d,erfitt:e (Fig. 2.9, i~sl)ire(l t)y [~.5.2]). 'l'l~e rt;a~ler i~l,(;r(;sl,(;(l i~ t,l~is r(;fi'a,(:ti()l>relal,(~(l asl)(;t:t ()f the tlan~ill,()l>lik(; l,l~(;()ry ()f ligld, l)r()t)a.ga, t,i()~ I~my (:()llslllI, [15]. Wil, l~i~ tim linear aI)t)I'()ximal,i(n~ w(; a.r(; (:(nl(:(:nie(l wil,l~, w(; will ~ts(; (~xI)ressi()l~ (2.4.9) f()r the rt;Kat:l,itn~ ray-~m.l,rix, i~ll)li(:il, ly a s s ~ i ~ g l,]~al, 1,1~(;relt;vmd, r(;fl'at:l,i~g s~rfa,ce lm.s 1)(;(;~ at)I)r()t)ria.l,t;ly t,rt:al,(:(l 1,() rt:(l~u'(: t,lm ligld, r(;flt;t:t,i()~ l,()tim ~i~lil~nln~. 2.4.3

Positive ,scale tran.@)'rm of tit(: ra.!l-coor'diT~atcs

As a Conlt)lctiti()ll of tlle t)r()gl'alll skett:he(l ill Fig. 2.7, w(; ll()W (:()llsi(t(;r tile ()t)t, it:a,1 t,ra,llsf()rllm.l,i(nl gell(;ratc(l 1)y I,he tllir(1 ()f nm.trit:(;s (2.3.6), i.e., K3. A (liter:t, (:alt:lllati(m ()f the (;Xl)()lmld, ial fllllt:|,i()ll r wil, ll ,s 1)(;illg a (limensionless real t)ara~llet, er, yiel(ls the scaling relati()ll t)(,tw(;(;ll the I)()sition and lll()lll(~,llt,lllll c()()r(tinat,(;s of t,]l(; ilmomiIlg anti ollt,g()i~lg rays, t,tm.t, is -

l)o

,

()

7tl,-1

' m

-

\Pi

,

(2.4.13)

in accor(t as well with the results of w 1.5.2 (Eq. (1.5.19)). T h e system represented by the m a t r i x S(m)-

(m 0

0 )

m -1

is t h e n recognized as an ideal magnifier (see tive m a g n i f i c a t i o n factor m > 0. T h e s y s t e m for m > 1, p r o d u c i n g a m a g n i f i c a t i o n of the a d e m a g n i f i c a t i o n by the s a m e a m o u n t of the s p o n d i n g l y as a demagnifier for m < 1.

(2.4 14)

Eq. (1.8.31)), having a posi(2.4.14) works as a magnifier r a y - c o o r d i n a t e and conversely ray r e d u c e d slope, and corre-

1D First-Order Optical Systems: The Ray-Transfer Matrix

99

Accordingly, qopo = qipi,

(2.4.15)

and hence the product of the spatial and angular ray-coordinates remains invariant under propagation along the system (2.4.14). The product qp is known as the Smith-Hehnholtz invariant, and (2.4.15) is the paraxial form of the Abbe's sine condition holding in general for perfect imaging systems [9.1, 3.4]. The optical realizations of the scale transforming system (2.4.13) will be illustrated in the next chapter.

2.5

Quadratic monomials in phase space

We recall that, according to the linear approximation, small regions across the optical axis in the reference screens as well as small inclination angles with respect to the axis are implicitly assumed to be involved under the propagation of the ray through the system. Therefore, as already noted, the geometricaloptical phase space, which would properly have the [-no, no] x ]R structure, can safely be viewed as the R 2 phase-plane. As stated earlier, the variables we are dealing with have different units. Therefore, in order that the expressions we shall obtain be dimensionally meaningflfl, the phase plane variables q and p Inust suitably be scaled (or thought of as being suitably scaled), for instance, by some characteristic length for the problem under examination, so that they come to have the same dimensionality. This is implied in the discussion and all the graphs below, even when, for simplicity's sake, the ordina,ry symbols q and p are still displayed. Tr~msform~tions (2.4.4) and (2.4.8) have an immediate pictorial view in the optica,1 pha,se-plane, both describing a, tra,nslation of the ray representativc point along directions parallel to the coordinate axes. In fact, by (2.4.4) the representative point Pi - ( q i , p i ) of the input ra,y is tra,nsla, ted to the representative point P o - (q~ + P ~ A Z / n o , P ~ ) of the output ray along the line p - p~, as shown in Fig. 2.10.a). Likewise, the tra, nsformation (2.4.8) can conveniently be visualized as a translation of the point Pi - (qi, Pi) to the point 1'o - (qi,Pi - 7)qi) along a segment parallel to the p axis (Fig. 2.11.a)). It is interesting to see the effect of the transformations (2.4.4) and (2.4.8) on a collection of rays. We will consider some idealized bundle of rays, whose phase-space representative distributions can easily be depicted. Needless to say, the 1D picture, we are working in, confers further astraction to the descriptions to follow; a propagating spherical wave, for instance, is visualized by a longitudinal section of a cone of rays as a laser spot by a segment line. Firstly, let us consider a fan of rays uniformly emanating from a point source placed at qi in the plane zi, which is physically obtainable, for instance, by focusing a He-Ne laser beam. It is represented by a segment in the phase-

Linear Ray and Wave Optics in Phase 5pace

100

plane parallel to the p-axis, whoso e x t e n t A p i is d e t e r m i n e d by the angular w i d t h of the b e a m . As a result of the free p r o p a g a t i o n from zi to Zo, the reprosenta, tive segment is horizonta,lly sheared by the angle/3, w i t h t a n ~ = A z / n o. Accordingly, the st)atia,l (;xtcnt of the [)cain is changed to A q o = / k p i / k z / 1 t o , whilst the a,ngular wi(tth remains inva,ria,nt (Fig. 2.10.b)).

P, p~ = p , ,

I 5q=

P,,

. . . . . . .+,.. . . . . . . . . . . . . . . . . . .

:

-~

;

I

p,

i

;

'+'~: . p ,

= I)+,

I

i

i

q,

q,,

= q,

+

A:

. p,

q

i

i

q,

%

.

q

I10

(b)

(a)

A: 9 9

5q I~

/)!

--

/)o

=

An o

i

I> T

i,< .

D'q

~ " /

= tl o

~ I

!

i

q~

% (c~

,'

~

:'

t~

inp~l +

i

i

I . . j %

P,

ill

,

/]

i

. p,

i

I', i

I

q

q,

%

q

(d)

FIGURE 2.1(). IAgtlt I)r()t)agati()ll l,llr(ntgtl a tl()tn()gelmolts Ine(littltl n,, ()ver tim (listance Az. Scheinatic of |,]l(~ pluLse plane trmlsfortnations of (a) the representativ(, poild, of a single ray, aim of the represetltative (tistriblttions of (b) a t)lln(tl(~ of rays ('tnmmtitlg fr()tn tim saine point, (c) a btln(lle of parallel rays (listribttte(l over a certain spatial exteilt, an(l (d) a bundle of rays uniforInly (tistributcd over different positions an(t directions. We m a y ()l)s(;rv(; t h a t a t)oint source, gcn(;ra.tcs a st)tlcric:al way(;, which is describe(t in the linear at)tm)ximation t)y a quadrati(:-t)ha,sc wa,v('f()rnl. T h e sheared se,gnmnt of Fig. 2.10.b), representing the fan of rays after t)r()pagating by d from the source, can therefore be i n t e r p r e t e d as tile representative phasespace d i s t r i b u t i o n of a spherical wave w i t h c u r v a t u r e radius j u s t equal to d. Accordingly, we m a y say ttmt a spherical wave with c u r v a t u r e radius R is represented in tile phase plane by tile line p - ~ . W i t h increa,sing d the r e p r e s e n t a t i v e segment is sheared more a n d more within the horizontal strip, d e t e r m i n e d by the initial angular a p e r t u r e of the wave, g e t t i n g parallel to the q axis when d ---+ oe. At infinite distance from tile source, in fact, a spherical wave is fairly like a plane one. A b e a m of parallel rays uniformly d i s t r i b u t e d over a certain spot in the

1D First-Order Optical Systems: The Ray-Transfer Matrix

101

zi-plane, obtainable for instance by collimating a He-Ne laser beam into a given direction, may be represented by a segment parallel to the q-axis, whose length Aqi reproduces the spatial extent of the beam. Effected by (2.4.4), the representative segment is shifted parallel to the q-axis by ~q -- -~oPi , the spatial and angular extents of the beam remaining unchanged (Fig. 2.10.c)). Finally, we consider a bundle of rays uniformly distributed over different positions and angles about a certain phase-plane point. A line source of given spatial extent emitting rays uniformly within an angular width At~i around a fixed direction iilgy be a model source for such a bundle of rays, practically realizable by a thermal source, followed by an appropriate set of long and narrow slits for collimating purposes. The representative distribution in the phase plane covers a rectangular region, whose width and height are determined respectively by the spatial extent and angular divergence of the beam. As sketched in Fig. 2.10.d), the free propagation produces an overall translation and, most importantly, a horizontal shearing of the ray-distribution by the angle/3 - ta,n-1 (ZXz ~-0 )" The angular width of the beam remains constant a,s the rays propagate over the homogeneous section from zi to Zo: A p i = Apo, but the spatial extent is increased from Aqi to A q o - /Nqi-k Api Az. n o

It is evident that the transformation (2.4.4) preserves the area of the phaseplane representative distributions: Ai = No, in accord with the Liouville's theorem. In fact, we see that the zero-area distributions in Figs. 2.10.b) and 2.10.c) are turned into zero-area distributions by (2.4.4); likewise, the regions of Fig. 2.10.d), representing the ray bundle before and after the propagation over Az, have tile same area: Ai = Ao =/Nqi/NPi. Correspondingly, the effect of the transforma, tion (2.4.8) on the above considered bundles of rays is shown in Fig. 2.11.b)-d). The vertical segment in Fig. 2.11.b), representing a, bundle of rays emanating from one point within an angular width A0i, is translated along the p-axis by @ = - 7 ) qi, the spatial and angular extents of the bundle being unchanged. Likewise, the horizontal segment in Fig. 2.11.c), representative of a beam of parallel rays, is vertically shifted by @ = - 7 ) qi and sheared by fl, being tan fl = - 7 ). The angular extent of the beam increases to Apo = zx qi. In accordance with the previously developed considerations concerning the phase spa,ce representation of a spherical wave, it is evident that Fig. 2.11.c) shows the transformation of a beam of parallel rays into a spherical wave with curvature radius R = - 1 / 7 ) due to the refraction at a non planar surface. Conversely, due to the invertibility of the ray path, we may say that a spherical wave may be transformed into a bundle of parallel rays by matching the curvature radius of the wave to the refracting power, and so the focal length 1 (Eq. (2.4.10)), of the surface such that R ~. Finally, the rectangular region representing a beam of rays uniformly dis-

102

Linear Ray and Wave Optics in Phase Space

P

-~ P,

..........

PI

.....

P'

i

i t P,, = P t - 72q,

72 qi

I I

..........

p.

"1' P,,

= p , - 72q,

I

q, = q .

iii

q

q, = q,,

q

(a)

[}l

(b)

P I

!! I I

]'-

'~P -

P,

y, q ,

bp

p,, = p , - 7~q,

p. q~ = q,, (c)

q

= p,

- ~-q,

N q' = q"

q(d)

output

FIGUI{E 2.11. l{eKactioll at all illtc;rface I)r twr162 witll rcfl'actillg power 7) (ill the figllres 7 > > 0). Scllellmt,ic of tim plm.se l)lmm trmlsforIimt,i()lls of (a) tlm reI)rt,'scld,ative point of a sillgle ray, ml(l of tim rr seld, ative (listributio~s of a I)~(lle of rays witl~ (b) angular extent only, (c) Sl)atial ext,o~t o~fly a.~M (d) m~g~flar m~(l Sl)atial cxt,c~ts. tri|)lltcd over r l)<~sit,i<)llS a,llr (]ir(',(:ti{)lls is t,ra,llslatc~t a,ll(t p-slma,rcd by the a,nglc/~t = -t,a,l~-~ 7) (Fig. 2.11.(1)). T h e spatia.1 e x t e n t of the t~ca,m is not (:ha,ngc(t t)y the refra,r t,tlr~llgh the interfa,(:r Aqo = Aqi, wlfilst t,lm, angula,r w i d t h is incrcasc 0) or in the second a n d forth q u a d r a n t (qipi < 0). In tile case illustrated in tile figure, qiPi > 0. T h e r e p r e s e n t a t i v e point Pi = (qi,Pi) of the incoming ray slips along the relevant h y p e r b o l a b r a n c h to tile r e p r e s e n t a t i v e point Po - (mqi, ~m ) of the o u t g o i n g ray.

1D First-OrderOptical Systems: The Ray-Transfer Matrix

-. !~ "~~

103

~"x input

Pi

"\

.,"

.s'po

~9

. ~..

".

~..

~"'~",~ I

q

q

output

q p = q, P,

(a)

FIGURE 2.12. The scaling matrix in the phase plane. The transformations experienced by (a) a single ray and (b) a spatially and angularly extended bundle of rays. In Fig. 2.12.b) we show the transformation acted by (2.4.13) on the phase plane rectangular region, representing a bundle of rays uniformly distributed over position and direction through the respective ranges Aqi and Api.The spatial and angular extents of the beam are changed to Aqo = m A q i and Apo = A p i / m . So, it is possible to reduce the spatial extent of the bundle but only producing a spread in the directions of the rays in the bundle. The behaviour of the system (2.4.13) gives signs, in geometrical optics, of the "uncertainty" principle. Mutuing the wording from the q u a n t u m mechanical context, we may say that the pure magnifier (2.4.13) produces a squeeze (g) in the phase plane contracting the ray-bundle representative area, in one direction and enlarging it in the orthogonal direction. Finally, we consider the effects of the ray-transformations (2.4.4), (2.4.8) and (2.4.13) on the phase-plane Gaussian distributions of the type (2.2.84). Specifically, we show the deformations experienced by the relative contours of constant density, which are clearly exemplified by the ellipses q2

p2 +

= r

(2.5.16)

for assigned values of the positive number c. The value of c determines the fraction of the total energy enclosed within the relevant elliptical contour; thus, for instance, the contour corresponding to c = 2 comprises about 86.5% of the optical energy of the beam. Thus, in Figs. 2.13.b) and 2.13.c) we m a y respectively see the q- and p-shears of the phase-plane contours (being circular 9 The term squeeze is largely used in quantum mechanics to indicate that some of the quantum fluctuations are squeezed out of one observable and into the second noncommuting observable.

104

Linear Ray and Wave Optics in Phase Space

FIGURE 2.13. ~iYmlsforlnatiolls of the plla.sc-l)lalle contollrs of tile (]mlssia, l ray-density 2~,,,~,, (' ' ' ( a , ) ' I ' t ~ e circ~lar cont(),~rs of the initial ~listrib~tio~ are ( b ) q-sl~eared by the propagation thrcJtlgh a ho~nogeIleolls medilml, ( c ) p-sheared by t.lm refraction at a two-metlia iIlterface, an~l (c) sqlleezetl ill ttle p tlirection by a scaling trm~sfi)r~n.

a,t, the int)111, plane) r162 t() t,lu', t)rot)a, ga,ti()n thr<)llgll a ll()lll<)geneous mediunl and the refra,ctioll at an interface. Simila,rly the squeeze along the p-axis due to the t)rot)agatioll through a magnifier with ma, gxlifi(:ation m > 1 is shown ill Fig. 2.13.~t), the initial (:ircles being t,ra,nsformer into the ellipses with the m a j o r axes oI1 the q-axis. We emphasize t h a t circles axl(t ellipses are the pictorial representations a d o p t e d in q u a n t u m optics for coherent and squeezed states. It is evi(tent from the a,bove figures t h a t segments are t u r n e d into segments, romboidal contours a,re t u r n e d into romboidal contours as finally elliptical contours are t u r n e d into elliptical contours [3.10, 3.7, a.g] (see Problem10). Indeed, the above figures make evident the general p r o p e r t y t h a t the functional form of the phase-plane ray distribution is preserved by any linear r a y - t r a n s f o r m a t i o n generated by q u a d r a t i c functions in q and p. This straightforwardly follows from relation (2.2.67) implying t h a t the a r g u m e n t s of the input ray distribu-

1D First-Order Optical Systems: The Ray-Transfer Matrix

10,5

tion are replaced by their linear combination according to the specific optical transformation at hand. As a conclusion, we note that within the framework of charged particle beam dynamics, the area of a closed domain over the phase-plane pertaining to the beam, chosen possibly in conformity with definite criteria, is usually referred to as the beam (transverse) emittance [3.10]-[3.13], apart from multiplicative constant factors differing among the various slightly different definitions existing in the literature. The emittance remains unchanged if the beam propagates through a non dissipative medium and hence it is an intrinsic parameter of the beam. It corresponds in optics to the second-moment-based M 2 beam quality parameter. In full analogy with particle beam optics, a phase-plane domain can as well be associated with optical systems on account of their geometry and optical performance; it represents the largest region in phase plane within which all the incoming rays can propagate through the system reaching then the output plane, and is referred as the systeln acceptance. The investigation of the propagation properties of the beam emittance in relation to the system acceptance is evidently basic to the optimal design of the system in view of the desired optical performance.

2.6

Summary

We have derived Liouville's equation for the phase-space density p(q, p; z) appropriate to a "collective" description of light ray propagation. Then, exploiting the properties of Lie transformations we have showed that the phase-space ray density is propagated through the individual phase space ray-trajectories. We proceeded considering the matrix optics generated by the homogeneous quadratic monolnials lp2, lq2 and 89 the associated Lie matrices being the generators of the symplectic group Sp(2, N). The optical systems, generated by 89 and 89 have been recognized as performing the elementary optical operations as propagation through a homogeneous medium and refraction at 1 the interface between two media, whereas the optical system generated by -~qp acts as a positive scaler, which may enlarge the spatial extent of the beam at the expense of the angular spread. Finally the phase-plane picture have allowed us to recognize the transformations acted by the aforementioned optical systems on light beam as a horizontal shear, a vertical shear and an overall scale (or, "squeezing"). In the next chapter we will prove that every linear optical system can be understood as a sequence of a certain number of translations and refractions. We will establish two interesting decompositions of optical matrices, realizable under specific conditions. One of these decompositions involves the optical systems generated by the algebra basis matrices associated with the monomials

106

Linear Ray and Wave Optics in Phase Space

89 lq2 and -~qp. Accordingly, optical systems may be described in phase t)lane by a sequence of a horizontal shear, a vertical shear and a, squeezing. The other decomposition involves the optical systems genera,ted by the algebra ba,sis ma,trices associated with the polynomiaJs ~q 1 2 + ~p 1 2, ~q2 1 a.n(t lqp, the f()rmer |)eing t)rove(t t() genera,re r()tati()ns in t)ha,s(~-t)la,ne. Problems 1. Prove that througl~ the sy~nplectic ~nappii~g (2.2.34) geimrate(l by the Lie trm~sforination e ~ r associate(t witt~ t,he (ty~ainical variable f ( q , p ) , f is a~ invarim~t fl~IwtioI~, na~ncly f(-~, :f) - f(q, p) over all rax~ge ()f t,|m evol,~tio~ parmner ( i~ (2.2.34). Verify tlmt tl~e results of w 1.5.2 are in accor(l witt~ tl~is i~wai'im~ce. 2. Co~si(ler a perfi;ct i~mgi~g syst(un. Applyi~g tlm i~warim~ce of tim t)lmse-I)la~m are~s, prove the si~e con(litio~ n,), Aq, si~ Oi - 'n.(,.,_Aq,, siI~ 0,,, where Aqi a~d Aq,, (le1~()te tim tr~.ulsvcrse ext, eimio~s ()f tim ot)ject ;u~(l t|m i~nage, whilst '3; ai~(l '~?, represei~t tim li~nitiI~g m~gles (to tlm optical axis) of tim fm~ ()f rays respectively accepte(l m~(l t r a n s ~ i t t e ( t t)y l,lm s y s t c ~ . 3. Verify t,lmt ( .~" ) ~,~):~ .a':~ ) :~.~"~ ,,"] ) :~.4 ,,"~2 ) ~']:~ :~A'[ A". ~1 :~~";I.~'. i;I ) I .~. I .~. ~'~11 :~I~ I~, .'~1. A

A

which clearly evi(leilce t,lle role of tim ('()lIllmltator [A, B] to estal)lisll tim (liifercllce of the algebra ()f ()perat,()rs t() t,lm.t ()f scalars. i

4. Write (lowll tim ('(nlll)()siti()ll rllle fi)r tile l)r()(llwt r

~,!~ ()f tw() ('Xl)()lmlltial ()I)erators, t.,.

~..

whose exl)oImllts (:oIil,,ulte to a scalar: [A,/3] - ,'. Verify tlmt r ,.,~ is 1,or, c(),,llntltative. 5. Prove tlmt illsertizlg tim forzn (2.2.7()) for tlm r~\v (h'.llsity (listril)llt, ioll ill ttm Liollville equation (2.2.11) yiehls Halnilt(nl's eqlmti(nls for the light ray. 6. Exl)loiti~g tim i~variml('e of t,l~e [)lm.'~e-sl)ace vol~une m~(l tim relati(n~ I)etwee~ the volume eleInent dV ag~(t tim ge()~mtrical exteI~t dAdl~ ()f tl~e ele~m~tary l)~(lle ()f rays, (le(l~ce the tra~sfor~mti(n~ law ()f l,h(~' geo~netrical extend, of tim eleI~mId,ary beam under refractio~ at tt~e ii~terface between two ~m(lia. 7. Exploitii~g th(, invaria~ce ()f t.lm ph;m(,-space vol~mm, (le(]~w(' the trm~sfor~mtioi~ law of the radiance ~m(ter refractio~ at the ii~terfacc t)etween two ~m(lia, o~ a c ( ' o ~ t of tim relation between the vol~Hne eh~me.nt d~) m~(! t,|m geoinetrical fa('tor dScosOd~ as well as of the conservation of the ra(tiant energy. 8. Prove that iI~ the case of propagation thro~lgh a ho~nog(umo~s ~ne(ti~un, wlfich is described by the Hamiltonian flu~ction H - - v / n 2 - p 2 , the exact ray evoh~tion obtains Pt

q ( z ) - q~ + p(z)

v/,,~_

~ ( z - z~),

- p~,

where tile term v,,"/~-P~ is tile tangent of tile angle of tile input ray to tile axis. 9. Analyse the effect of tile spherical aberrations, i.e., deviations from tile paraxial propagation, oil the Gaussian density distribution (2.2.84), using the results of Problem 4 in ch. 1. Verify tile accord with the general result of Problem 8. 10. Verify t h a t (a) straight lines are m a p p e d into straight lines under the linear rayoptics transformation (2.1.3); in particular, verify that parallel straight lines are turned into parallel straight lines, the relevant angular coefficients q being related by the hyperbolic MSbius transformation

qo

Dq~ § C Aq~ + B '

and that (b) ellipses are m a p p e d into ellipses with the same area.

1D First-Order Optical Systems: The Ray-Transfer Matrix

107

References [i] [2] [3]

For the Hamiltonian formulation of geometrical optics see [4] in ch. 1. See [7] in ch.1 for a discussion of the analogy between geometrical optics and classical, and relativistic, mechanics of a point-particle. The light-magnetic optics analogy is as well evidenced in [3.10], [3.11], [3.13]. In relation with the Liouville's theorem we quote the original paper by Liouville, and the classical textbooks on mechanics by Arnold [3.3] and Goldstein [3.2], or on statistical mechanics by Tolman [4.1], ch. 3, pp. 48-52, Pathria [4.2], ch. 2, pp. 34-36, Balescu [4.3], ch. 2, pp. 37-44, and Toda [4.4], ch. 1, 17-22. A derivation of Liouville's theorem with an enlightening discussion of its implications in geometrical optics is presented by Marcuse [3.4]. Examples of the applicative implications of Liouville's theorem are in the papers [a.s]-[a.s]. For a discussion of the phase space method, and in particular of Liouville's theorem, within the context of the charged-beam optics, we suggest the books by Lichtenberg [3.10], Wollnik [3.11], and Hawkes and Kasper [3.13]. The latter exhaustively covers the whole of electron optics from both the geometrical and wave optics viewpoints. The paper by Lejeune and Aubert [3.11] provides one of tile first systematic accounts of tile multivariate issues concerned with the concepts of emittance and brightness in magnetic optics. See also the recent title by Dragoman and Dragoman [3.14].

[3.q

J. Liouville, "Sur la th~orie de la variation des constantes arbitraires", Journ. de Math. Pure et Appl. 3, 342-349 (1838). [3.2] H. Goldstein, Classical Mechanics (Addison Wesley, Reading, 2"d ed., 1980), ch. 9, pp. 426-428. [3.3] V.I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, New York, 4 th ed., 1984), ch. 3, pp. 68-74. [3.4] D. Marcuse, Light Transmission Optics (Van Nostrand-Reinhold, Princeton, New Jersey, 1972), ch. 3, pp. 112-124. [3.5] D. Marcuse, "Compression of a bundle of light rays", Appl. Opt. 10,494-497 (1971). [3.6] M.C. Hudson, "Calculation of the maximum optical coupling efficiency into multimode optical waveguides", Appl. Opt. 13, 1029-1033 (1974). [3.7] G. Nemes, I.E. Teodorescu and M. Nemes, "Phase space treatment of optical beams", Preprint LOP-43-1984, CIP, Bucharest (1984). [3.8] W.A.E. Goethals, "Laser beam analysis by geometrical optics", in The Physics and Technology of Laser Resonators, D.R. Hall and P.E. Jackson (eds.)(Adam Hilger, Bristol, 1989), pp. 143-153. [3.9] G. Nemes, "A phase space approach to beam-pulse representations and measurements", Preprint LOP-76-1990, IAP, Bucharest (1990). [3.10] A.J. Lichtenberg, Phase-Space Dynamics of Particles (J. Wiley & Sons, New York,

1969).

Linear Ray and Wave Optics in Phase Space

108

[3.11] C. Lejeune and J. Aubert, "Emittance and Brightness: Definitions and Measurenmnts" in Applied Charged Particle Optics, Adv. Electron. & Electr. Physics, Suppl. 13A, A. Septicr (ed.) (Academic Press, New York, 1980), pp. 159-259. [3.11] H. Wolhfik, Optics of Charged Particles (Academic Press, Orlando, Florida, 1987). [3.13] P.W. Hawkes and E. Kasper, Principles of Electron Optics, 3 vols. (Academic Press, LoIulon, Vol. 1-2, 1989, Vol. 3, 1994). [3.14] D. Dragoinail a11(t M. Dragoman, Quantum-Classical Analogies (Spriilger, Berlin, 20()4), ell. 6, pp. 143-186. [4]

As ]msi{' lextl)~)r

()11 st alisli{'al lller

We (llt{)t~'

[4.1] R.C. "l'olnmll, 7'/u: l"J+nciplc.s of Slatisl'ical ltlcHum, ir (()xfor~l tlldv, l'rcss, New Y<~rk, 1938). [4.2] II.K. l)at]lria, Statistical Mechanics (Pr Press, ()xfor{l, 1.972). [4.3]

li. l~alesml, Igq'uilibri'wm and Noncquilibriwm, Statistical Mcch,anics (.lr Wiley &: Soils, New York, 1!}75). IIere, Lie operators anr Lie trallsforllmt, iCnls are intror162 toni exl)lr also, bot, i~ l,lle (:lassi('al ~tlltl tile qllalll, lllll pll~mCbSlmCe pit:tllres are l~rese,ld,e~l, l,}le latter lmillg r tl~ro~g]~ l,}~e Wig~n;r ~lisl,ril~tio~ fl~cl, io~.

[4.,1] M. 'I'o
G.('. ()ales, "Ve{:l<w A~alysis", il~ lhmdbook of Appli~'d Malhr - S~'Iccled '~'~'sulls a'mt 'm~'thod.s. ('. E. i~'ars~n~ (e~l.) (Va~ N~)slra~l-l{ei~l~l{l ('~)~l)a~y, New Y~)rk, 1983), r 3, 1)1). 129-178.

[6]

S. Slei~l)erg, "l,ie s~,rie, l,ie tra~sfiw~ali~)~s, a~r their at)l)lir162 il~ Lie Moth,otis in ()pile.s, .l. Sfi~r Xl<~,~lrag;5~ a~r K.B. VGflf (e~ls.) (Sl)ri~ger-Verlag, Berli~, 1r ell. :1~, I~1). ,15-1().1.

[7]

F,,r l lw g, umral tlm,,ry ,,r Lie algcl,ras a~,t gr,,~t)s we a,hlrcss tin: r<;a,h;r to [9] i~ (:t~. 1. SI)ecifically, we (l~U)te I)cl~w tin; lextl){~{)k t)y t t a ~ e r ~ e s ] ~ l)ei~g ~>f concern to tim t)I'{){)f ~)f tim ~ i ~ u ~ ( l ~ l a r i t y ~>f l lw sy~I)lccti(: ~ a l r i c c s aff{w{h:r i~ w 2.3.1. [7.1] M. Hamer~nesh, Group Theory and ils Applications to Ph!lsical Problems (AddisonWesley, Rea~liI~g, MA, 1962), pp. 403-406.

[8]

We list 1)clow the ,~rigiI~al I)ai)ers 1)y Baker Is.a], a h ~ g with a few relate~l titles f i ' ~ t,he on the B C H fornmla are give~ in [6], wlmre derivation of the B C H fornmla can t)c fimnd

Is.2], Ca~i)t)ell [s.~], a~r ttausdorff wide reh;vaI~t, literature. C~mmmnts a list of references is provided. The also in the t e x t b o o k s [8.7] and [8.8].

[8.1] J.E. Campbell, "On a law of combination of operators", Proc. London Math. Soc. 29, 14-32 (1897). [8.2] H.F. Baker, "On the exponential theorem for a simply transitive continuous group, and the calculation of the finite equations from constants of structure," Proc. London Math. Soc. 3, 91-129 (1901). [8.3] F. Hausdorff, "Die symbolische exponentialformel in der gruppentheorie", Ber. Verh. Saechs. Akad. Wiss. Leipzig Math.-Phys. Klasse 58, 19-48 (1906).

[8.4] W. Magnus, "On the exponential solution of differential equations for a linear operator", Comm. Pure Appl. Math. 7, 649-673 (1954).

1D First-Order Optical Systems: The Ray-Transfer Matrix

109

[8.5] R.D. Richtmeyer and S. Greenspan, "Espansions of the Campbell-Baker-Hausdorff formula by computer", Comm. Pure Appl. Math. 18, 107-108 (1965). [8.6] R. M. Wilcox, "Exponential operators and parameter differentiation in quantum physics", J. Math. Phys. 8, 962-982 (1967). [8.7] S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York, 1962), pp. 36-40. [8.8] M. Hausner and J.T. Schwartz, Lie Groups; Lie Algebras (Gordon and Breach, New York, 1968), pp. 65-71. [9]

The basic concepts of radiometry and radiative transfer theory, and their connection to partial coherence theory are illustrated in a variety of titles; a few are quoted below. See [3.11] and [3.1a], ch. 47, pp. 971-988, ch. 78-79, pp. 1719-1767, for a discussion of the concept of brightness, and coherence and brightness from the viewpoint of electron optics; the close analogy to light optics is enlightened and effectively supported by numerous references. [9.1] M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), pp.181-183. [9.2] F. Grum and R.J. Becherer, Optical Radiation Measurements. L Radiometry (Academic Press, New York, 1979), chs. 1-3, pp. 1-97. [9.3] W.R. McCluney, Introduction to Radiometry and Photometry (Artech House, Boston, 1994). [9.4] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995), ch. 5, pp. 287-307 and references therein. [9.5] F. Nicodemus, "Radiance", Am. J. Phys. 31, 368-377 (1963). [9.6] A. Walther, "Radiometry and Coherence", J. Opt. Soc. Am. 58, 1256-1259 (1968). [9.7] A. Walther, "Radiometry and Coherence", J. Opt. Soc. Am. 63, 1622-1623 (1973). [9.8] E. Wolf, "Coherence and Radiometry", J. Opt. Soc. Am. 68, 6-17 (1978). [9.9] A . T . Priberg, "On the existence of a radiance fimction for finite planar sources of arbitrary statcs of coherence", J. Opt. Soc. Am. 69, 192-197 (1979). [9.10] A. T. Fribcrg, "Phase-space methods for partially cohcrcnt wavcficlds", in Optics in four dimensions, M.A. Machado and L.M. Narducci (eds.) (Amer. Inst. Phys., New York, 1981), pp. 313-330. [9.11] L.A. Apresyan and Yu.A. Kravtsov, "Photometry and coherence: wave aspects of the theory of radiation transport", Sov. Phys. Usp. 27, 301-313 (1984). [9.12] J.T. Foley and E. Wolf, "Radiometry as a short-wavelength limit of statistical wave theory with globally incoherent sources", Opt. Comm. 55, 236-241 (1985). [9.13] G.S. Agarwal, J.T. Foley and E. Wolf, "The radiance and phase-space representations of the cross-spectral density operator", Opt. Comm. 62, 67-72 (1987). [9.14] R.G. Littlejohn and R. Winston, "Corrections to classical radiometry", J. Opt. Soc. Am. A 10, 2024-2037 (1993).

[10] A reading list about the concept of light ray is provided in [1] of ch. 1. [11] Specific investigations about the symplectic group, in relation as well to geometrical optics, are presented in [11.1] V. Guillemin and S. Sternberg, Symplectic Techniques in Physics (Univ. Press, New York, 1990). [11.2] R. Simon and N. Mukunda, "The two-dimensional symplectic and metaplectic groups and their universal cover", in Symmetries in Science VI, B. Gruber (ed.) (Plenum Press, New York, 1993), pp. 659-689.

110

Linear Ray and Wave Optics in Phase Space

[11.3] Arvind, B. Dutta, N. Mukunda and R. Simon, "The Real symplectic groups in quantum mechanics and optics", Pramana 45, 471-513 (1995); also in arXiv:quantph/9509002 v3. [11.4] J.F. Carifiena and J. Nasarre, "On the symplectic structures arising in geometric optics", Fortschr. Phys. 44, 181-198 (1996). [11.5] R. Simon and K.B. Wolf, "Structllrc of the set of paraxial optical systems", J. Opt. Soc. Am. 17, 342-355 (2000). [11.6] K.B. Wolf, Geometric Optics on Phase Space (Springer, Berlin, 2004). [12] As t() tl~c ray ~mtI'ix a l)tmmr tr ()I)ti('s we sllgg('st [13.3], [13.5] all(1 [12.1] W. BI'ollwer, Matrix Mchods in Optical ht..st'uumcnt Design (BeIljmlfiIl, New York, [12.2] A. Nussbmml ml(1 R.A. Plfillips, Conte'mpom77! Optics fin" Scientists and Enginee~;s (PrcIlticc-Ilall, Ellglcwoo~l Cliffs, 1976). [,2.3] M.V. Klci~ m~d T. bhu'tak, Optics (Joh~ Wiley & Sons, New York, 2 ''d e~l., 1986). [12.4] K. ttall~acll, "Matrix rcl)resr atioll of Gmlssiml optics", Aiil..]. Pllys. 32, 90-1()8 (1:)(i4). [12.5] H. Kogel~lik, "llnagiilg of optical Inor I/csolmtors witll iiltcrnal lcllses", Bell Systcln 'Fcctl..I. 44, 455-494 (1965). [13] I)~'.taih'd r162162 ~f tim im)l)crtics of Gaussia~ I)ca~s arc l)rcsc~tcr i~ [3.4], [13.1] A. Yariv, Quantum l';lcl~vnic.s (,]ol~ Wiley & So~s, New York, 2 ''d ctl, 1975), oh. 6, pp. 99-129. [13.2] ,I.A. Ar~m.~i, lh'am and l"ibcr Optic.s (Aca~h;~,fic l'rcss, I~,~,lcn~, 1.q76), ('1~. 2, pp. 54-79. [13.3] A.E. Sieg~nm~, Lascns (Univ. Book, Mill Valley, 1.98(i), tits. 16-17, pp. (i2(i-(i97. [13.4] P.W. Milo~fi m~r .].H. Eberly, Lasers (.lol~ Wiley & So~s, New York, 1.q88), pp. 484-5O8. [13.5] B.E.A. Salch m~r M.C. Tcicl~, Fundamentals of l'hotonics (,Iol~ Wiley & Sons, New York, 1991), ell. 3, t)P. 80-107. [13.6] N. Ho(lgso~ m~(l tl. Wt4)cr, Optical Resonators - Fundamentals, Advanced Concepts and Applications (Spri~gcr-Vcrlag, Lon(to~, 1997), cir. 2, pp. 76-104. [14] A. S. Dcif, Ad't,a'm'ed Matrix Theory for Scie'nti.sts and Enginec'rs (Al)a(:us Press, ~mfl)ri(tgc Wells, 1982), oh. 5; in particular, for the methods to compute the exponential fin~ction of matrices see pp. 176-182. [15] For an invcst igati()I~ of t]~(', refractiI~g surface transf()ri:nations we address the reader t,() [11.6] , Ctl. 4, t)t). 47-56 and to [15.1] H. Raszillier and W. Scheinpp, "Fourier optics from the perspective of the Heisenberg group", in Lie Methods in Optics, J. Ss Mondrag6n and K.B. Wolf (eds.) (Springcr-Verlag, Berlin, 1986), oh. 2, pp. 29-43. [15.2] M. Navarro-Saad and K.B. Wolf, "Factorization of the phase-space transformation produced by all arbitrary refracting surface", J. Opt. Soc. Am. A 3, 340-346 (1986). [15.3] T. Sekiguchi aim K.B. Wolf, "The Hamiltonian formulation of optics", Am. J. Phys. 55, 830-835 (1987). [15.4] K.B. Wolf, "Symmetry in Lie optics", Ann. Phys. 172, 1-25 (1986). [15.5] K.B. Wolf and G. Kr6tzsch, "Geometry m~d dynamics in refracting systems", Eur. J. Phys. 16, 14-20 (1995). [15.6] O.N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic Press, New York, 1972).

3 The Group of 1D First-Order Optical Systems

3.1

Introduction

In the previous chapter we have traced a line of approach for investigating the relation between the ray-optics description of paraxial propagation through 1D optical systems and the symplectic Lie group of 2 x 2 real matrices, associated with the quadratic polynomials in the conjugate va,ria,bles q and p under Poisson bracket operation and matrix exponentiation. When considering rays with small inclination angles to the axis (p2 << p~ ~ n 2) and accordingly objects and images extending across the ~xis well within all the pupil-apertures of the system (Iq[ << all system pupil-apertures), the optica,1 Hamiltonia, n becomes quadratic in both q and p, i.e.,

H(q, p, z)

_

1 p2

~o

.~..q2

+ 2

,

(3.1.1)

showing separate "kinetic" and " potential" -like contributions. Hamiltonian (3.1.1) is a model function in several fields of physics (Fig. 3.1), both in the classical-like context, where q and p are interpreted as canonically conjugate variables for which {q,p} = 1, and in the quantum-like context, where q and p are interpreted as noncommuting operators for which [q, p] = Ri, R signifying the "quantization constant" inherent in the specific framework. Accordingly, we have examined the transformations of the position and q2 and momentum ray-variables, generated by the quadratic monomials ~lp2 , 89 1 -~qp in the real space as well as in the optical phase-space, in order to identify the possible optical systems that may produce such transformations. We have drawn in fact a definite correspondence of each monomial to a traceless matrix in the symplectic Lie algebra sp(2, R) and then to a unimodular transformation 1 2 1 matrix in the corresponding symplectic Lie group @(2, IR). The ~p _,~q2_ and i2 qp-generated optical systems are thereby recognized as a free-medium section, a refracting interface and a positive scaler, as sketched in the scheme:

Linear Ray and Wave Optics in Phase Space

112

X'+/s

_#/e~/#'" l~ r ~

I tt ( q. l) ) = I q

2

+

I

FIGIJI/.t'; 3.1. T i m tlanIl()lfi(: (>s('illat()r-likc ttalnill,()Ifiml pl~ysi('al systc~ns i~ ~m~W l)rm~(:lms ()f l>twsics.

generating function free- reed iu m section

~t>=

thin lens

1 ~q2

positive magnifier

1 ')

tt(q,p)

generating matrix

--+

K

--+

K, -(_(~

I qp ---+ 2

K:,

--

1 )2 + 5q 1 ~ Ii~()(lcls several ~1

transformation matrix

- ( I )) ~))

__ }(~, 1

(~)) -()l

)

( , -f

t

__

(,,) 11

--/f

__+

(,.sK3

_

l

(~..~/2 1) '~ \() c .~/~} "

(3.1.2)

Als<), we lm,ve i::-sll(;ar a,~l
The Group of the 1D First-Order Optical Systems

113

is established. Section 3.6 illustrates the link of the attractive and repulsise oscillator-like Hamiltonian dynamics with fractional Fourier transformers and beam expanders (or reductors), which individualize in the (q, p) plane scaled rotations and squeezes. Two interesting realizations of optical matrices are described. One realization, which applies locally near the group identity, involves the lqp_ and lp2-generated optical systems, and accordingly is visualized in the optical phase plane as a q-shear followed by a scaling and then by a p-shear (w 3.5). The other realization, which in contrast applies to the entire 1 1 group, involves the ~i q2_ , -~qp-, and ~(p2 + q2 )-generated optical systems; it is pictured in the (q, p) plane as a rotation followed by a scaling and a vertical shear (w 3.7). Conclusive considerations about the link between quadratic polynomials and paraxial ray matrices are given in Sect. 3.8. Finally, Sect. 3.9 describes the method to integrate Hamilton's equation for the optical ray matrix, detailing separately focusing and defocusing quadratic index media.

~1q2-,

3.2

Ray matrix of composite optical systems

Optical systems, consisting of several components connected together in cascade, can be handled symply by multiplying the ray matrices of the individual optical dements, arranged in the reverse order [1]. In fact, each optical d e m e n t has its own input and output t)lanes and acts on the ray incident on its input plane in a certain way to produce the ray at its output plane. Thus the ray propagation through a sequence of N optical elements with ray-matrices M1, M.~, ..., M N is accounted for by the following chain of matrix relations U 1

-- MlU0,

U2 -

M2ul,

(3.2.1)

oo~

U N -- MNUN_I,

where as usual uj denotes the ray-coordinate vector uj - (qj,pj)T at the fixed reference plane IIj, j = 0, .., N (Fig. 3.2). The intermediate planes IIj, j = 1, .., N - 1 play the double role as the output and input planes respectively for the j - t h and (j + 1)-th optical element, whilst H 0 and H N are the input and output planes for the first and last clement as well as for the overall system. Combining the input-output relations (3.2.1), we obtain the transfer law for the ray coordinates from H o to H N in the form u N -- M N M N _ 1 - . - M 2 M lr o = M u 0.

(3.2.2)

Accordingly, the ray matrix M of the overall system comes to be the product M = MNMN_I--" M2M 1

(3.2.3)

114

Linear Ray and Wave Optics in Phase 5pace FI

lli = H0

1-12

FIN_2

HN-I

HN = FIo

ii.0 I n c o m i n g ray

i

9

i

M2

MN-I

MN

" ~ ' ~ N-2

UN-I

Input plane

i

i E m e r g i n g ray

Output plane

FI(ll~]I1E 3.2. Propagatio~l of a ligl~t ray tln'o~gl~ a seq~mncc of optical clc~ncnts. The optical ~natrix ()f tim ()vcrall s y s t c ~ is tim I)ro(l~ct of the i~t(livi(l~ml ray I~mtriccs fro~ l l,, to H N .

with tlm ~ml, ri(:(;s M~, .., M x l)(;ing a rra~ge(l i~ tlle illW;l'S(', ()l'(l(;r fr()~ t h a t in wlfi(:l~ t,l~(; (~orresl)()~(li~g (,h:l~(',~fl,s are t)la.(~(;(l ()~ t|m axis wit,t~ r('st)(:ct Lo t,hc (lir('~('ti()~ ()f t,l~(; i~(:()~i~g ray. 3.2.1

77~,ic/,: a'n,d th, m lc, nsc,s'

As all (;xalllI)h', ()f a (:()llll)()llllr r ,";yst,clll, WC r162 a th, it'k h:lls, wlli(:ll l)asi(~ally (:()llsists ()f tw() sl)ll(;ri(:al sllrfil(~(;s S(:l)arat(;(l t)y 1,1,11at)I)r(:('ial)h; (tisl, ml(:(;. ~1'11r r a y 1)r()l)a,ga,ti()ll l,]lrr a th, i~:k lolls llla,y l)(; s(;(;ll a,s ('()llq)rising l, ll(; f()ll()wi]lg s(;(lll(;ll(:(; ()f ray-l,l'a,~lsf()rlIlal, i()lls: 1) r(:fl'a,(:l,i()~l a l, l,ll(; fl'()lll, sllrfa,ce w i t l l r('fra('l,ilw4 1)()w(:r ~ ,

2) fl'cc t)r()l)aga,l,i()l~ l hr()~gl~ l,hc lc,~s ()f a x i a l t,hi(:k-

~(;ss (t m~(l 3) r(;fra('ti()~ al, t,lm t)a,(~k s~lrfa(~(; witt~ t)()w(;r 7)2 (Fig. 3.:l.a,)). T h e thi(:k l(;~s ~m,t,rix is titan ()l)tain(;(l D(m~ tim ()r(h;r(;(l t)r()(t~(:t, ()f r(;fl'a('~tion-K(;(; i)l'()t)a,ga,l,i()]~-rt;fra('ti()~ l~m,tri('cs, wl~i(~l~ ti~m,lly gives M t h i c k lens --

( 1-07~' --P

i)

1 -- 07)2

)

'

(3.2.4)

wt~(;r(; t,t~r fo(:al t)()w(;r 7) ()f the, lens is 7) - 7)~ + 7)2 - i)7)~7)2, a,n(t i) denotes the axia,l ttfi(:k~(;ss ()f tl~(; l(;~s (tivi(Ie(t |)y th(; r(;la.tiv(; re,fra,(:tiv(; in(t(;x. As a fl~rther (~,xa,~q)h;, wc consider a, t h i n lens, for~m.lly define,(] a,s consisting of t,w() refracting s~rfa.(:(;s with negligil)le set)a.ra,tion in bet, ween, so t,hat, a ray incident height

a,t a, give, n p o i n t fr()m the, o t h e r

on one surface emerges

s~rfa,ce. T h e

reference

a , p t ) r o x i m a , t e l y a,t t h e s a m e

planes

IIi an(t Iio a,re, t a k e n

to

(:()in('i(h~ with t,h(; t)la,nc ()f tt~e lens, a,lt,h o u g h Hi rela,tes to incidenL rays whilst, Iio relates to e m e r g i n g rays (Fig. 3.3.b)). T h e ray m a t r i x of the thin lens is then obtaine(t as the p r o d u c t of two refraction m a t r i c e s relevant to the two lens interfaces. T h e resulUng m a t r i x looks like a, refraction m a t r i x , i.e., M t h i n lens --

_']'9

1

'

"

115

The Group of the 1D First-Order Optical Systems l-[i= ]-] o

",%,," 1-1o I i

//01

n01

//0 2

(q~ ,p

I

~"

! d-~O .;_ . . . . . . . . . . . . . .

k-- ----J i

",

~--

I

d

(a)

,.

i

i

i i

n02

i~(qo.po)

:

."

i i i9 i. i9 i.

---~

(b)

.t

FIGURE 3.3. (a) Propagating through a thick lens, the light ray experiences the refraction at the front surface, the propagation through the lens and another refraction at the back surface. (b) Being the refracting surfaces of a thin lens separated by a negligible distance, the light ray experiences only the refractions at the lens front and back surfaces. tile focal power 7) of tile lens being tile SUlil of the refracting powers of the two surfa, ces: 7) = P l -31-P 2 . The lens power 7) is c o m m o n l y expressed in t e r m s of the reduced focal length f a,s 7) - 1/f, with f - f1/7~01 - f2/~/~02, ~,01 and n % d e n o t i n g the refractive indices of the media, in the object a,nd image spaces. T h e formal analogy between the refraction and thin lens matrices establishes the optical equivalence between a refracting interface with a, given refracting power 7) and a thin lens with the same focusing power. Disregarding the real design, in the following we will refer to a thin lens a,s the optical element described by the ray m a t r i x (3.2.5). Also, a,ccording to the usual practice, we will refer to converging, or positive, lenses, when f > 0, and to diverging, or negative, lenses, when f < 0. As is well known, positive lenses cause converging spherical waves to converge more rapidly, and diverging spherical waves to diverge less rapidly or to converge; in contrast, nega, tive lenses cause converging spherical wa,ves to converge less rapidly or to diverge, and diverging waves to diverge more rapidly.

3.3

The subgroup of free propagation and thin lens matrices

Free-medium sections a,nd thin lenses are the basic c o m p o n e n t s of every firstorder optical system. As it is proved, in fact, every optical t r a n s f o r m a t i o n can be realized by a finite sequence of free p r o p a g a t i o n s and lens operations, or equivalently every 2 x 2 real symplectic m a t r i x can optically be synthesized properly arranging free p r o p a g a t i o n sections and thin lenses.

116

Linear Ray and Wave Optics in Phase Space

We denote by T(d) and L(f) the optical matrices relevant respectively to portions of a uniform refractive index medium, T(d)--cdK---

( 10 d)l '

(33.1)

a,n(1 ttli~l l(;::S(;S, L(.f)-

(,~K,

__

(

1

-1/.f

0) 1

"

(3.3.2)

W(; r('(:all ttm.t et ml(t .f (l(;~(,t(, th(; s(,(.ti()ll l('llgt, ll a.ll(t ttl(; l('lls f()(:a.1 l('llgt, ll, t)()t,h s(:a,l(',(l 1)y 1,11(;r(',l(wa,ld, I'(',fra,(:tiv(', ill(|(',x. Als(), thin l(',lls llla,l,l'i(:(~,s a,l'(', (',(tllivalelltly (tcn()t(;(l t)y L ( . f ) ( ) r L(7) ), 7) 1)(;i~g tl~(~ f()(:a,1 t)()w(;r: 7) = l/f. Tl~(', lil~fit ('a,s(;s ()f (t --, () (se(:ti()~ ()f ~(;gligit)l(; l(;~gtl~) m~(t f --~ 4-oo (l(;~s ()f ~(~gligit)l(~ f()(:al t)()w(',r) i(leld,ify tlm llllit,y nmtrix I: T(()) = I a,n(1 L(-t-oc) = I. Inl,(~r(~stillgly, T a.n(l L a r(~ lmilll()(llllar ml(l, r(;st)(;(:tively, lq)t)er- ml(l l()wertria,llgllla,r llm,tri(:(',s. 'l'll(~y ar(', tll(', r(',I)res(~ld, a,tiv(~s ()f l,ll(; tw() AI)(;limi ()i~(~t)a,ra.~(;t(;r s~fi)gr()~q)s ()f tt~(', sy~q)l(;(:ti(: gr(nq) 5~p(2, IR), g(;~(;ra,t(;(l r(;sI)(',(:tiv(;ly t)y t,l~(~ a lg(4)ra ('~l(;~(;~d,s K _ m~(l K + as ilh~strat(;(l il~ (ll~a.l)t(;r 2 (!i!i 2.3, 2.4.1 an(t 2.4.2) [2]. Tlm s~fi)gr()~q) a,(l(litivity l)r()t)(~rty (2.3.12) witl~ r(;sl)(;(:t t() the i)a,l'a.lll(',l,(',r ()f l,II(; alg(:l)ra-t()-gr()~q) (;Xl)()~(;~tiati(n~ writ(;s as T(d~ )T(d~) = T ( d ~ ) T ( d I ) = T(d),

(3.3.3)

d - d~ 4- d~,

(3.3.4)

L(.f, )L(.f,) = L(./~ )L(.fl ) = L(.f ),

(3.3.5)

witl~

a.n(1 sin~ila,rly witl~

1

1

1

117 ~,fl +.7.,'

'"

Th(~ a,t)()v(~ r(4a,ti(n~s ext)ress in f()rnm,1 temps tim well kn()wn t)r()tmrty tha, t, the se(lllell(:(', ()f two free st)a,ce sections d 1 an(t d~ is equivalent t()()lie section of length d = d I + d~ and, similarly, two casca,de(t lenses f~ a,n(t .f~ (with no fu sepa,ra,ti()n i~ between) a,re equiva,lent to the lens f - f~f~+f~. As a,n inune(tiate consequence of (3.3.3) and (3.3.5), we also ha,ve T(d)T(-d)

= T(-d)T(d)-

I,

(3.3.7)

L(f )L(-f)

= L(-f

) L ( f ) = I,

(3.3.8)

and which sta, te the existence and unicity of the inverse to every T and L-type matrix a,ccording to [T(d)] -1 - W ( - d ) - e - d K (3.3.9)

The Group of the 1D First-Order Optical Systems

117

and

[L(f )]-1

_ L(-f)

- e - ~ K+.

(3.3.10)

Formally, since both T ( - d ) and L ( - f ) reproduce the typology of the matrices (3.3.1) and (3.3.2), relations (3.3.9) and (3.3.10) confirm that T and L-type matrices do form two proper one-parameter subgroups of Sp(2, R), when regarded as real upper and lower triangular matrices with unit diagonal entries. Optically, relation (3.3.10) establishes ttmt the inverse of a lens system should be a lens with opposite value of the focal length. This is a physically realizable system; relation (3.3.8) can therefore be implemented by the simple sequence of two lenses with opposite focal lengths. As the lens f alters the direction of propagation of the ingoing ray, the lens - f acts oppositely on the intermediate ray, which then regains the original direction of propagation. Hence matrices (3.3.2) do form a subgroup even when interpreted as the optical matrices of thin lenses. On the contrary, by (3.3.9) the inverse of the free propagation by the distance d should be the free propagation by the distance - d , which in principle may a,ppea, r not physically realizable since distances occur naturally as positive quantities. Therefore inatrices (3.3.1), when interpreted as optical inatrices for propagation through portions of a homogeneous medium, do not strictly form a, subgroup. In the next pa,ragra, ph we will see that the propagation corresponding to "negative" values of d can optically be realized by a, suitable sequence of free propaga, tion sections and thin lenses. The inverse of the free propaga,tion matrix, i.e. the matrix T ( - d ) , acquires then a physical concreteness, although unlike T(d) it does not correspond to a single and unique optical element. Accordingly, relation (3.3.7) becomes optically synthesizable, the effect of the free propagation by d being completely vanished by the propagation through the system T ( - d ) , as the ray eventually recovers the initial position and direction. Apart from their physical rea,liza,bility, "negative" free propagations, when framed within the colnInon conventions rega,rding the optical systems, can be given a practical interpretation which relates to the character, real or virtual, of the rays we are dealing with. Let us write d in terms of the axial coordinate as d = Zo - zi (the refractive index being omitted for simplicity), so that T - l ( z o - zi) = T ( - ( Z o - zi)) = T ( z i -

Zo).

(3.3.11)

Suppose that the ray coordinates (qo, po) are known at the plane Zo. Evidently the ray coordinates (qi, pi) at the preceding plane zi may be calculated by the inverse matrix T -1 (Zo - zi) = T(zi - Zo). In symbols, we have

(q:)

.o)(qo),

<0

(..1.)

If the portion of space from zi to Zo is not free, but occupied by some optical system, the ray at zi, we might obtain through T ( z i - Zo), is not the actual ray

118

Linear Ray and Wave Optics in Phase Space

at zi, t)ut the virtual ray (see w 1.8.1), namely the ray t h a t would produce the given ray at Zo, if the intervening space between zi and Zo were free. E x e m p l a r y to this view are some optical systems, we will describe below as, for instance, the Fourier tra.nsforming configura,tion realized by (tivcrging thin le~ses. The a.l)ove discussion gives a t)hysi(:a.1 concreteness to free t)rot)agation and thin lens matri(:es, T(d) and L ( f ), with negative va,lu('~s ()f d a,n(t .f. The present t r e a t m e n t a,(:(p~ires in(lee(t a, g(;n(;ral vali(tity, (:Oral)rising within the same for~m,lisl~ tl~(; (:a,s(;s ()f (liv(;rgil~g l(;~scs, virtual ()t)jc(:ts al~(1 virt~ml i~m,g(;s.

3.3.1

f~}'ee prvpagation th',vugh "negative" distartces

Fr(;(; t)r()I)a,ga, ti()lls (:()rr('~st)()ll(liIlg t() n(;gativ(; va,llu;s ()f l,ll(; (lisl,ml(:(; (:aal t)(; rea,1ize(t t)y ltsing a tillit(; lnmd)cr ()f l,liill l(,ns('~s s(;I)a.ra,t(;(l I)y tilfil,('~ (listalm(;s. This lnm(tlfiv()(:a,lly (;sta,t)lisll(;s l,ll(: l)llysi(:al r(;a,liza,t)ility ()f tlm ilw(;rs(; ()f free t)rot)aga.ti()ll l)y a. ('.(nlstru(:til)h'. ()l)ti(:al syst(nll, ml(t ll(;ll(:(; (:()llq)()S(;S tlu; (:()lltra.st t)ctwe('.ll tll(; "llmtlleIlmti(:al" ml(l "()I)ti(:al" t)etm,vi()ln" ()ftll(; s(;t, ()f Iim,tri(:cs (3.3.1) in fa.v(nlr ()f tim gr(nlI) strll(:tllre ()f tim set ()f lill('ar ()l)ti(:a.l syst(;nls. Arra,ngi~lg l,llre(; l(uls(;s al)l)r()I)riat(4y (:h()s(ul ml(l Sl)a~:(;(t ()ira fl'()l~ tim ()thers ()11(; lllay sy~tl~('.siz(', a "~wgativ(:" fr(;('. I)r()I)agati()n. I~ fa(:t, tl~('~ f()r~ml i(t(;l~tity (A ~ ) _ L(.f a )T(d.~)L(f.e)T(d ~)L(.f,),

(3.3.13)

(:a.~ |)(; s()lv(;(t fi)r tll(: l)ara,~l(:t(;rs of tlm ()I)ti(:a,1 a,rra,~g(uim~d, ()~ tll(: right, i.e. tl~e f()('al l(n@d,s .f~, f.~, fa a~(t t,l~e (lista~('(:s d,, d.,, in ()r(h'r t() I)r()(tu(:c a "n(;ga,tive" free t)r()I)a.gati()~ nmtrix, fi)r whi(:h in(tee(l we nn~st l~a,ve

A=D=

1, ( : = 0 ,

B<0.

(3.3.14)

S~l)stituting l,h(; va,,'i()~s T an(l L ~na.tri(:cs ent(;ring (3.3.13), w(; ()l)ta,in for the e~d,ri(;s of tim nm, trix (n~ tlu; left, tim expressi()~s

AC :

.

1

1

f2

1- ~

B

f2

d2

f3 + f 2 f a

fl' fDl '

B-(t~+(t~(1-~) D -- 1 + dl

(

1

f2 '

1

f2

f'*

.

qt..

d__~) f2 fa,

(3.3.15) --

f'~' .

which by (3.3.14) provide the relations

1 fl

--

1

1

f2

f3 + f2f3'

d2

dI

d2

fl -- f3 '

1

1

1

f2 > ~ + d--~"

(3.3.16)

It is immediately evident tha,t f2 must be positive. Also, it can easily be checked t h a t f~ and fa as well must be positive. Therefore, identity (3.3.13) satisfying requirements (3.3.14) is concevaible with all the three lenses positive:

L, s

> o.

(3.3.17)

The Group of the 1D First-Order Optical Systems

119

....- < ........................................................ < ............................................!

__2,!

d>0

no

,

~o

!

<

FI,

>

i

!

1.

d 1

]

n

,

0

]

i

9

I 13o

i.f !A ill

.f

.1~ A!

A ii

If!

!

I

i

d
"

'

31

31

'

1........................ :..: .............................. :.:................................................ j

FIGURE 3.4. A possible three-lens configuration implementing the free propagation by a "negative" distance. Clearly the choice of the p a r a m e t e r s , i.e., focal lenghts and distances, satisfying the above relations is not unique. However, in order to prove the existence of some optical configuration being in accord with (3.3.16), we choose f2 - fa - f.

(3.3.18)

Consequently, we have 1 fl =

2 d2 f + Y > 0,

(3.3.19)

which can be satisfied, for instance, by setting d 2 - 3f.

(3.3.20)

It follows t h a t fl - f,

dl - 3 f -

d2,

(3.3.21)

thus yielding a s y m m e t r i c configuration with three identical lenses equally spaced from each other (Fig. 3.4). T h e B - e n t r y of the ray m a t r i x comes to be B-

-3f

< O.

(3.3.22)

T h e value of f will be fixed according to some specific request on the value of B. Thus, from B - - d , d > 0, we obtain f - 5" a T h e reader m a y work out other configurations satisfying (3.3.16).

120

3.4

Linear Ray and Wave Optics in Phase Space

Optical matrices factorized in terms of free propagation sections and thin lenses

E v e r y 2 x 2 real symt)h;ctic m a t r i x can be u n d e r s t o o d a,s a ray m a t r i x reprcs(',~d,i~lg a. first-or(ler ot)ti(:al syst,(',nl. Ih',n(:e, first-()r(t(;r ()t)tica.1 sysI, elIIS Call |)e rega.r(te(t a,s ot)tit:a,1 mm,l()gs ()f I)hysit:a,1 t)ro(:ess(',s (t(',s(:ril)e(t |)y 2 x 2 real synlt)lc(:(,i(: maI, ri(:(',s. I(, (:a,n 1)(; l)r()vc(t, in fact, tlm,t (',very 2 x 2 real symt)le(:ti(: llmtrix is exI)r(',ssit)l(; as tlle I)r()(lll('t ()f a fillit(: llluld)('r ()f T ml(l L-like mal, ii(',(:s, (,tHis a.ls() I)l'()vi~lg (,Ira,l, every ()I)tic:a.1 sys(,(~,lll (:a.ll 1)(; r(',a,lize,(1 as a,n a.i)t)r()i)ria(,(', ([i~fit(',) s(',(t~t(;~(:(', ()f fl'(;(', t)r()t)a,ga.l,i(n~ s(',(:ti(ms a,II(] (,lfin l(',nses [3 ]. th',re w(', (;xmlli11(', tll('~ r('a,lizal)ility ()f (',v(',ry Hp(2, N) syst(',lll |)y ml ()t)tic:al il,l'l'il,llg(~,lll(~,llI, ()f tllill l(',llS(',S ml(l fr(',(',-~ll(',(liln~l S(',('t,i(nlS. W(', slla,ll 11()1, (twell ()n tll(', (tll(;sti(nl r(;ga,r(ting tll(: lllilli]lnnil ]nu~fl)(',r ()f h',~s(',s ~(;(',(1(',(1 1,() r(;aliz(~ a i)a,rti(:~llar systeI~l. Tl~is (lll(',s(,i()li is a~alyz(;(1 i~ (l(;I,ail i~ [a.n]; tl~(;l'(:, as a, res~flt ()f a.II a,(:(:lll'a,(,(: i~w(:stiga,(,i()~ ()f tlm va.ri(n~s r(:gi()~s ()f Hp(2,1R), (,lm.(, (:a.lt 1)(: r(:a.(:h(:(l l)y (n~(:, (,w() an(l (,l~r(:(: ](:~s (:()nligura,(,i(nls, it is sl~()w~ tlmt (:very (:I(:~(:l~(, i~ (,I~(: gr()~I) (:a,~ l)(: r(:aliz(:(l l)y an ()I)ti(:al (:()l~lig~m~(,i()l~ i~w()Iving n() l~i()r(: t]m.ll tlu'(:(: h:llS(:s. W(: firs(,ly (:()~si(l(:r ()l)(,i('al (:()l~Iig~tra,ti()~s, (l(:~m.l~(li~g f()r (,In'(:(: l)a,rani(:(,crs (,() I)(: (l(:li~(:(l, m~(l l)r()v(: tlm,t t,l~(:y yi(;l(l sld(,a.l)l(: ()l)(,i('al sy~(,l~(:s(:s f()r (:l(:~(:I~ts in Sl)(:(:ili(: r(:gi(n~s ()f (,I~(: Hp(2, lR) gr(nq). ~l'h(:l~, ~()r(: (:(n~l)l(:x a.rrm~g(:n~(:nts wi(,l~ il~(:r(:a,si~g ~nl~l)(:rs ()f l(:~s(:s m~(l fr(:(: l)r(>l)a,ga,(,i()~ s(:(:(,i()~s ar(; (:()nfigurc(l i~ ()r(l('r t() sl~()w (lint ~n~i~()(l~lar l)~r('ly (liag()iml ~mt, ri('('s at(' ()l)(i('ally realiza,l)l(:. Tlfis is l)a.si(: t() (,I~(: ('()~(:h~siv(: l)r()()f a,s (:la,l)()I'a.t(:(l i~ ,~ 3.4.4, (:()~(:(;rning (,I~(: I)()ssil)ili(,y ()f r(:I)r(:s(:~(,i~g (:v(:ry r(:a,l sy~q)l(:(:(,i(: ~m,(,l'ix wi(,l~i~ the c()n~(:(:(,(:(I i(l(:~(,i(,y (:()~l)()~(:n(, ()f (,l~(: gr()~l) in tlm f()rn~ a,s (,h(: fa.(:t()r(:(l l)ro(lu(:t of T, L a~(l I)~r(:ly (liag()~m,1 ~mt, ri('(;s, h~ the f()rtl~('()~d~g (tis(:~ssi()~ tl~e ma,trix T ( d ) is ~n~(t(;rs(,()()(l a,s (t(;s(:ril)i~g (,r~fly fl'(;(; t)r()l)a,ga,(,i(n~ s(;(:(,i()l~S, a,~(t hence the t)a,ram(;ter d is limite(t t() n()mmgative values. This is in or(ler t() evidence the rcleva,n(:e of tlm inh(:r(:nt ,scmigr()ul) (inst(:a.(l ()f gr()~H)) s(,r~u:t,~rc of the set of the free,-t)r()t)aga,ti()~ nm,ti'i(:('~s to I)revent (,1~(', sinq)h; thre(; t)arameter configura,tions, c()mt)rising one or two lenses, from e x h a u s t i n g all the threeparameter Sp(2, IR) systems. The, restriction on the m a t r i x T(d) is relaxed in w 3.4.4, w h e r e " n e g a t i v e " (tistan(:es are allowed.

3.4.1

TL T and L TL configurations

The simplest realization of a 2 x 2 symplectic matrix by means of thin lenses and free propagation sections, one may envisage, should involve an arrangement allowing for three parameters, which then might be two distances and one focal length or conversely two focal lengths and one distance. The resulting configurations will be denoted as TLT and LTL, respectively.

The Group of the 1D First-Order Optical Systems

121

/A

Then, let M (~ ~)) be a syInplectic inatrix. Supposing M has non null entries, the following sequence of optical elements can be hypothesized \

/

(~ BD) -- T ( d 2 ) L ( f ) T ( d i )

(8.4.1)

having M as tile relevant ray matrix. The lengths dl, d 2 and f are determined according to the relations 1 -

d-z

A

s

,

1 -

~

s

1

D,

C,

s

dI + d2

-

dl d2 s

-

B.

(3.4,2)

As f may range from - o o to +oc, whilst d I and d 2 are thought as nonnegative quantities, not all the possible values of the matrix entries A, B, C, D can be reached. In fact, only the two families of symplectic matrices"

['A>I \C >0 B>0 D>I

and

B ) \/'A
(3.4.3)

are synthesizable by the one lens configuration (3.4.1), according to whether the central lens is chosen as negative or positive. The complementary two lens configuration -- L ( f 2 ) T ( d ) L ( f I )

(3.4.4)

is also possible, with the choice 1

d _ A, 1 fl

--

d f2

D, d -

B,

1

fl

1

d

f2 ~ fl f2

C,

(3.4.5)

which, on account of the restriction d _> 0, allows to implement matrices having

(3.4.6) As a simple example for the optical realization (3.4.4), we may consider a thick lens. On account of the equivalence between a refracting surface and a thin lens, having the same, respectively, refracting and focal power 7), it is evident that a thick lens can be realized by a pair of thin lenses spaced by the lens thickness; the ray matrix (3.2.4) can then be factorized as in (3.4.4).

3.4.2

Special s y s t e m s as T L T and L T L configurations

It is interesting to investigate the potential of the previously described single and double lens configurations to implement matrices with one or both vanishing off-diagonal elements, of course, within the inherent restrictions. So doing, we gain a useful visualization of the special systems, which, abstractly introduced in w 1.8.4 as corresponding to ray matrices with one vanishing entry, are now seen to be realizable by thin lenses and free-propagation sections.

122

Linear Ray and Wave Optics in Phase 5pace

Telescopic and imaging systems

T h e LTL configuration (3.4.4) is a p p r o p r i a t e to synthesize symplectic matrices having C = 0. In fact, the identity

((a 2>O) _ L ( L ) T ( d ) L ( f ~ )

(3.4.7)

is a,(:t~lal)le with d = .f, + f2 = B a,n(t A = - . f 2 / f l . We re(:a,ll fi'o~ !i 1.8.4 tl~a,t ()t)ti(:al systel~S (les(:riber t)y ray-~m,tri(:es with (7 = () are lla,Ille(l afocal ()r tele,scop'ic syst, e ~ s , as |,l~e r ()f the c ~ e r g e n t ray (lel)c~(ls ()lily ()~l t,l~(; (]ire(:l,i(m of |,t~e in(:i(lent ray. T l ~ s any ray (:oming into t,l~(' sysl,ex~ l)a,ra,lh'l 1,() the ()t)t,i(:al axis (;~(;rg(;s t)a.rallel to the axis a.s well. If I~ - (), 1,1~(; si~tzh; h;~s ('()~fiN~ra,t,i()~ (3.4.1) ca,~ 1)(', (lesigl~e(1 witl~ 1/.f - ( 7 , d.~/d, = - A a.~,(l l / f -= l / d I + 1/d.~ (focal r i()l,)i~, ()r(ler 1,()give A<0

()

c: <() A ,) - T ( d ' 2 ) L ( f ) T ( d , )"

(3.4.8)

limite(1, ~f ('(~r,sc, t,r negatiw; wdlles of t,t~c eiltries A a n(l (~. A(:(:or~ling to tile f~r rela,t,i~ll, tll~ i11terllw,~tia,te lellS is I)~sitiw~ a,ll~l r t~e llll~h;rst~o~l as (:Olllt)Ose
Collimating and focusing systcms Symplectic matrices having one zero along the diagonal, when regarded as optical matrices, are representative of collimating and focusing systems. In general, matrices with D = 0 describe the p r o p a g a t i o n from the prima, ry focal plane of the system to the plane l o c a t e d by the axial position of the relative secondary principal plane i.e. sH' = 1-A Correspondingly, m a t r i c e s n02

C

"

The Group of the 1D First-Order Optical Systems Hi !

f=_l

d= A-I(,

123

I-I o !

I-Ii !

i

i

i

i

i i , i i i i

i i , , i t i

i i , ; a i i

i i , i , i i

f

~-- f

(o-V_~ [c ~cl

-"

f=

Q--->

I d=D-! -~ --U

d--i

i--~- d

(a)

17 0 !

f

f

(b)

FIGURE 3.5. Optical realization of a symplectic matrix having (a) D -- 0 ( c o l l i m a t i n g system) and (b) A - 0 ( i m p e r f e c t F o u r i e r transforming system).

with A - 0 account for the propagation from the plane located by the axial 1-D position of the primary principal plane as su c to the secondary focal It01

plane of the system (w 1.8.4). Evidently, a D - 0 matrix can be realized by the one lens setup (~ -C-1

o

) - T(d)L(f)T(f)

(3.4.9)

with taking 1 c,

f

d

A-1 c,

(3.4.10)

which then demand for C < 0 and A < 1 in order that both T(d) and T ( f ) represent positive free propagation sections. Likewise, to produce an A = 0 matrix, the reverse configuration can be arranged according to (~ -C-1

D ) -- T ( f ) L ( f ) T ( d )

(3.4.11)

with f

1 c,

d

- D -c1 ,

( 3 . 4 12)

thus requiring that C < 0 and D < 1. The single lens decompositions (3.4.9) and (3.4.11) establish, within the specified restrictions, the equivalence of collimating and focusing systems to thin lenses, whose input and output planes be properly placed. In (3.4.9) the input is taken at the primary focal plane of the lens as the output at distance d A-1 from the secondary principal plane, which for a thin lens is the lens plane (Fig. 3.5.a)). Reversely, in (3.4.11) the input is taken at the distance d - o-1 C from the lens plane and the output at the secondary focal plane (Fig. 3.5.b)). Focusing systems perform the imperfect Fourier transform of the --

C

124

Linear Ray and Wave Optics in Phase Space

VI i

Vlo

i i

' i

I:" - YoCl

i

,_

~'

i

:,

,

i

.t

.f .--..i

'x) ,

I-I i

~I o

i i

i i

i

i

i

,

T

~

i ~

.t --.-----

(w

.t ~

(b)

FI(]I~I~E 3.6. Si~glc le~s rcalizatio~ of a~ m~ti(liag()~ml sy~I)l(:ctic ~natrix (pc)fi:ct Fourier transfi)r~i~g syste~). The "2f" sysl,c~n witl~ (a) a p()sitivc l(:~s m~(l (b) a ~mgativc lens. i~I)~(, sig~ml, (,1~(: a lX~l)li(,~(l(: (lis(,ril)~(,i()~x ()f tile ()~1,t)~(, sig~m,l t)(:i~g i~(,(:rt)r(:t(:(t a.s (,1~(: l:()~ri(:r (,rm~sfi)r~xl ()f 1,1~(: inl)~(, ~n~l(,il)li(:(1 1)y a (l~m.(lra.(,i(: t)lxa,s(: t(:rm. Tll(: fi~r(,l~('r r(:(t~(:s(, A = ()i~ (3.4..9), ()r D - () i1~ (3.4.11), yi(:l(ls d -- .f, (,ll~s t)r()vi(li~xg fi)r (,Ix(: pe'~:fect F()~n'i(:r t,ra~|sfi)rlxxil~g sys(,(:l~l t,l~(: si~tzl(: l(:~xs r(:aliza,tion F(f)

--

() - 1 / ()( ' ) (7

_ T(f)L(f)T(f)

(3.4.13)

l Th(: a.ll(,i(liag()xml llm.(,rix a.l)()v(: (l(:s(a'il)(:s ill g(:n(:ra.l (,h(: ()l)tiwi(,h f - c" (:a,l l)r()Im.ga,(,i()ii fl'()lll (,If(: l)rinmry fl)(:a,l l)lan(: ()f th(; syst,(:Ixl (,() (,If(: s(:(:(m(lary (m(:, ()ll wlli(~ll th(: l)(:rf(:(:(, F()In'i(:r (,railsf()rnl ()f ill(: illg()illg signa,l is f()I'mc(l (,~ 1.8.4). A('(~()r(lillg t() (3.4.13), tlx(: silxxl)l(:st r(:a.liza,ti()Ix ()f s11(:ll a syst(:lll (:(resists ()f a l)()sitiv(; tllill l(:lls, wit, If tlx(: ligllt rays |)(:illg l)r()l)agat(:(l fr()i11 tll(: fr(mt t() th(: ])a(~k ft)t~al I)lml(: (s(:(: tll(: s(:ll(:xxx(: in Fig. 3.6.a) ft)r t,lx(: (:x(:nq)lary (:a.s(: of a bi-('()llv(:x l(:ns). Ill t)rilwiI)l(:, th(: T L T (l(:sign (3.4.13) sh()lfl(t s(,ri(:tly a d m i t only I)()sitiv(: l(:ns(:s, ('()rr(:sI)()ll(|ing t() ll(:ga,tiv(: va,hms ()f C in th(: given ma, trix. H()w(:v(:r, lx(:gativ(: wfllms ()f f arc a,lh)w(:(t as w(:ll. Tlx(: r(:slflting ()t)(,i(:al configura,ti()ll is (l(:I)i(:t(:(t ixx Fig. 3.6.t)) f()r ttl(: illustrativ(: (:a,s(: ()f a, t)i-(:()~l(:a,ve lens. T a k i n g iIlt() a,(:(:ollnt the m(:aning of the sign of th(: fo(:al length f , which, in a.ccor(t with the (:ommon (:onventions (w167 1.8.1 and 3.3), esta,blishes the character, r(:al ()r virtual, of the object a n d image rays, the reader m a y easily verify t h a t the forma.1 r e p r e s e n t a t i o n for the opti('al setup in the figure turns to be .illst the T L T p r o d u c t (3.4.13), with f retaining its own sign in the T - m a t r i c e s . Thus, llnimo(tular antidiagonal m a t r i c e s can be realized by use of one lens of f()cal length f c1, placed m i d w a y b e t w e e n two reference planes s e p a r a t e d by 2 Ifl. This configuration is r e p o r t e d as the 2 f s y s t e m . Finally we note t h a t the double lens configuration (3.4.4) can also be valued to c o r r e s p o n d to a ray m a t r i x with D = 0, A = 0, or A = D = 0. In particular,

125

The Group of the 1D First-Order Optical Systems

[1

17 o

[-[i

,

f

io

7j

ff

1-1o

~,

~!

7

X

\

~=

f

(a)

(b)

FIGURE 3.7. Double lens realization of an antidiagonal symplectic matrix (perfect Fourier transforming system). The Fourier tube with (a) positive lenses and (b) negative lenses.

in Fig. 3.7 we show the double lens setups (also known as Fourier tubes), suited to perform the perfect Fourier transform of the object distribution.

3.4.3

Pure magnifiers

The basic one and two lens configura,tions (3.4.1) and (3.4.4) are not suited to realize unimodular purely diagonal matrices, which within the optical context are interpreted as the ray transfer matrices of pure magnifiers (w 1.8.4). Either additional thin lenses or additional free propagation sections or both must be added to the optica,1 arrangements (3.4.1) and (3.4.4), thus introducing further parameters, which can be adjusted in order that both the off-diagonal elements in the resulting ray matrix may vanish. We consider the optical identity (3.4.8) for the imaging matrix, which for convenience is rewritten as A<0 C<0

with

1 fl

=

0 a-l) - T(d,)L(L)T(dl),

1

+

dl

1

- -C,

d2

(3.4.14)

d,, A - ---.

(3.4.15)

dl

Concatenating the TLT sequence (3.4.14) with a thin lens on the left or right, the lower-left entry of the resulting optical matrix may vanish. Accordingly, we may realize the pure magnifier matrix by the optical arrangement 0

M -1

-

L(f2)T(d2)L(II)T(dI)

,

(3.4.16)

the magnification M being determined by the parameters f and d as M

d2

f2

dl --

fl'

1 1 1 fl = d l + d2

(3.4.17)

Linear Ray and Wave Optics in Phase Space

126

Flo

Vii ,

/ '21< >'

f,

f2i

0

M

------

&

-

(I~

------

f, h

i

I

-

I

;-

(I:

" *

FIGURE 3.8. 'I'wo-lc~s rcalizatio~ of a I)~rc ~nag~filier. whi,:h also yM(t d~ = .f, + f2 (Fig. 3.8). Th(; nmgnifi(:a,ti()n .hd is (:lea,rly nega.tivc, ttnls signifyillg an inv(;rsi(nl ()f the ray t)()sitioll wittl r(;st)('~(:t t() tlm ()t)ti(:al a,xis. W(; (:ml ()])taill a l)()sitiv(; ina.gnifi(:ati()n t)y (:as(:a~lillg tw() llla,gllifi(;rs like (3.4.16). Ill fa,(:t, l)(;illg a striking t)r()l)(;rty ()f (tiag()lm.1 llml, ri(:('s tllat tlmy llnflt, it)ly ill|,() (liag()lml llml,ri(:(;s a,s

()

-~ M l

()

M~-'

-

()

M

~

(3.4.18)

with 3d - ,hd~3d 2, w(; ('ml lmv(; 3d > () witll 1)()ill 3d~ < () ml(1 3/[ 2 < (). E(t~m.ti()~s (3.4.16) a l~(l (3.4.18) arc t)a.si(: t()()~r t)~rt)()s(;s, 1)(;('m~s(', th(;y rei)resent tlw, l)r()()f tlla,t Inlr(;ly (lia,g()na,l synq)h;(:ti(: nm, tri(:(;s (:ml t)e syld, hesiz(;d t)y (',()I~(:a,te~m,ti~g tl~i~ l(;~ses m~(1 free t)r()l)aga,ti()~ se(:ti()ns. The rea,(t(;r (:an easily verify t h a t tlm sa,~n(; ot)ti(:al (:()~fig~ra,ti()n as in (3.4.16) inay t)e ()l)tai~(;(1 a,(l(li~g a, fr(;(; t)r()t)a,gati()I~ s(;(:ti()~ t() tlm right of the (h)uble lens rcalizati()n (3.4.7) ()f a,n a,fi)ca,1 nm, trix.

Pure magnifiers and Fbu'ri~;'r t'innsforming systt~ms Rcma,rkably, a, purely magnifying configuration can also b(; r(;a,(:hed t)y enlarging the two h;ns (:onfigura,tion (3.4.16) to (:()nlprise a,n a,(htitiona,1 fl'ee propa,gation section, thus y M d i n g

D(.A4)- T(d2)L(.f2)T(d)L(.fl)T(dl ) ,

(3.4.19)

where the involved lengths are related through AA - _12 fl

1 --

d fl

d + dl.Ad -1- d-2- -- 0 ~

.A/I

(3.4.20) "

Preserving the restriction of 0 _< dx, d~ d 2 ~ C<) in order t h a t the T-matrices strictly correspond to free p r o p a g a t i o n sections, the magnification A/I, satisfying (3.4.20), comes to be negative and hence the focal lenghts fl and f2 are required to be b o t h positive or b o t h negative.

127

The Group of the 1D First-Order Optical Systems rli i i

.ti

i i

i

1-]o

Pli

I

a

.IS.

rio ./I

i

l:*

i

i

i

i

!

i i

. i

~ :

i

!

i

i= '

.tl

,u=

.fi

?_ '

.ti

_v_

/2

i

_!

I

-'

' [i

i

.fl

'

(a)

.12.

(h)

F I G U R E 3.9. Optical realization of a pure magnifier with .hal < 0 by cascading two Fourier t r a n s f o r m i n g systems. (a) I.hd < 1 (b) .hd I > 1.

We observe that the second of (3.4.20) may be satisfied by setting d I -fl, d2 - f2 and accordingly d - f l + f2- The resulting two lens configuration turns to be the sequence of two 2 f systems, described in w 3.4.2 and pictured in Fig. 3.6.a). Therefore wc write D(A//) - F ( f 2 ) F ( f l )

.A/[ - - [ z

fl

"

(3.4.21)

On account of the considerations of w 3.4.2 concerning the interpretation of the sign of the focal lengths in the implementation of the Fourier transform by thin lenses, the focal lengths fl a,nd f2 above ca.n be let to be a,rbitrarily positive or negative, thus allowing for both negative and positive magnifications. According to (3.4.21), a pure magnifier can be realized by using two lenses, spaced so as to have a common focal point, and embraced by two free-medium sections, whose lengths are adjusted according to the focal length of the corresponding a,djacent lens. Thus, in the whole system both the effective length and the optical power vanish. As all example, Fig. 3.9 shows two possible realizations of an M < 0 configuration by concatenating two Fourier transformers, both having positive lenses; the respective performances are pictured for an incoming bundle of parallel rays. The propagation through both systems manifests an inversion of the ray-coordinate with respect to the optical axis along with a magnifying or dcmagnifying effect according to whether ~ > 1 or < 1. In particular, we emphasize that two identical Fourier transformers in sequence produce a pure inversion of the ray coordinates, in accord with the general property of the Fourier transform. Then, a ray emerging from a soquence of four identical Fourier transforming systems has the same coordinates (position and inclination) as when entering to (see also w 3.6.1). In contrast, f14 > 0 configurations demand for lenses of opposite signs. Figure 3.10 shows two possible realizations. The path of an ingoing bundle of

Linear Ray and Wave Optics in Phase Space

128 Fli

Vii

l-lo

El o i

:i

1

l0

i i i ) i

<;==5

I

34>0

.-4.

0,)

a0

FIGURE 3.10. Optical rcalizatioI~ of a pure magnifier with M > () by casca(li~g two Fo~ricr trm~sfor~ni~g systelnS. (a) M < l (b) M > I. t)a,ra,lM rays is t,ra(:(~(t, sll()willg tim (:()llt,ra,sl, illg (~tt'(~(:ts ()f tim tw() systcnls, tim first t)r()(llu:illg a, re(llu:ti()ll (.A4 < 1) ()f the traxlsversc size ()f t,tle tnnl(lle, the SO,(:()ll(l, ()ll t,ll(} (:()ld,ra,ry, a,ll eXl)a,llsi()ll (.A4 > 1). Tim ot)ti(:al (:()~lfigurati()Ils t)i(:tllrc(l ill Figs. 3.9 ml(t 3.10 nmy tlmll 1)e lnl(lcrst()()(t as tll(~ ()t)ti(:al mlal()gs ()f I)llysi(:a,1 I)r()('esses, (les(:ril)e(1 t)y 2 x 2 sylllI)l(~(:t,i(: t)ln'ely (tiag()na,1 llm.tri(:es with ll(~ga,tive all(l I)()sitiv(~ (~t,ries, r(~sI)(~(:tiv(~ly. Also, as a, t)()sitiv(; ~a,g~ifi(w ('i.1,111)(~,a,rra,~g'(~(t })y (:a,s(:a~li~g tw() n(;gat,iv(~ l~mgnifying systems, f(n~r I)()sitive F(n~rier traaM'()rnfing (:()nfig~mtti()ns (:a,n |)e (:()n(:a,t(;na,t(;(t to sy~thesize the ()I)ti(:a,1 mm, l()g ()f m~y I)l~ysi(:a,l I)r()('ess (tes(:ri|)e(l t)y a, 2 x 2 synq)le('ti(: (tiag(nm,1 I)()sitive ~na,trix. Finally, we n()te tha,t the (:()nfig~mtti()n (3.4.16) (:a,n als() t)e (mla,rge(t to (-omprisc a,n a,(t(titi()~m,1 thin le~s thus lca,(li~g t() tim thr(~(~-leI~S sct~t) (3.4.22)

D(.A4) - L ( . f , ~ ) T ( d , ~ ) L ( f ) T ( d I )L(.fl ).

We invite the reader to find out the explicit expressions linking the involved t)arameters d I , d 2, fl, f2 and f. Also, wc suggest to prove the rcalizability of the (:onfigura,tioIl with fl - di, f2 - d2, 7I -_ ~1 + ~ 1 a,n(t accordingly A/I

_12

which is interpreted as the sequence of two Fourier tubes.

The subgroup o f pure m a g n i f i e r s

Tile ideal magnifiers form a proper subgroup of the set of 1D linear optical systems. This straightforwardly follows from the properties of the unimodular diagonal matrices, we have denoted by D ( 3 / ) . In fact, as already shown, the p r o d u c t of two matrices in the set { D ( 3 d ) , Ad real} belongs to the set as weli, a,nd hence just rewriting (3.4.18) we have D(.A41)D(.M2) - D(.A/[2)D(.M1) - D(Ad),

~4

-

.A/~IM

2 .

(3.4.23)

The Group of the 1D First-Order Optical Systems

129

The inverse of a pure magnifier is the magnifier with reciprocal magnification: [D(.s

-1 - D ( . / ~ - I ) ,

(3.4.24)

and the unit matrix I can be interpreted as the optical matrix of the magnifier with A/l = 1: D(1) = I. (3.4.25) We note that, according to (3.3.7) and (3.3.8), the unit matrix can be optically synthesized by cascading two, positive and "negative", free propagation sections or two thin lenses having opposite focal lenghts. Due to (3.4.25), the identity matrix can also be implemented by cascading two positive or two negative pure magnifiers with A42 = M1I ' and hence it can be realized as two, positive and negative, Fourier tubes nested one into the other or as four positive Fourier transforming systems concatenated one to the other with the relative focal distances being related by flf3 = f~ f4.

Pure magnifiers without inversion Positive ideal magnifiers constitute a proper subgroup of the group of unimodular purely diagonal matrices {D(AJ), ,k4 real}, because cascading two positive magnifiers produces a positive magnifier. In w2.4.3 we have elaborated for the pure magnifiers without inversion the exponential representation, rewritten in (3.1.2), involving the traceless matrix K3, representative of the mixed quadratic monomial lpq. As we will see later, under specific restrictions the scale matrix S(m) plays a crucial role as a building "block" for the representation of optical matrices in a factored product form involving as well the T and L matrices. Therefore, we further coinInent here on the system S(m), suggesting in particular some possible optical realizations of it. We may rephrase relatively to the scaling matrix S(m)

_

m-1 ) ~

?It _

r

(3.4.26)

the formal considerations developed in w 3.3 in respect to the free propagation section and thin lens inatrices, T(d) and L ( f ) , thus completing the view of the three one parameter subgroups of @(2, IR), generated by the algebra basis {K_, K+, K3} by exponentiation through a real parameter. Evidently, S-like matrices constitute a proper subgroup of Sp(2, IR); more precisely, the set {S(m), m > 0} is included into the subgroup {D(A/I), A/I real} of unimodular diagonal matrices D(A/I). Then, as particular cases of (3.4.23) and (3.4.24), we can state the closure property S(T~I)S(~2) = 8(~2)8(~1)

= 8(77-~),

m -- m l m 2 > 0

(3.4.27)

130

Linear Ray and Wave Optics in Phase Space

for ma,trices (3.4.26), and hence the existence and unicity of the inverse being

[S(I~,.)] -1 -- S(lrt -1).

(3.4.28)

In a,ddition, the previously developed considerations concerning the optical realiza,tion of pure-magnification devices with ,A/I > 0 apt)ly to the system (3.4.26) as well. Thus, tim symplectic diagonal ma,tric(',s, generated by the a,lg('])ra, nmt, rix Ka, (:an ()t)ti(:a,lly t)e realiz(;(t |)y (:()n(:a,t(',~m,ting tw() systems like (3.4.16)()r two Fouri(',r tra,~sformers (3.4.13), the relewml, fo(:a,1 l(;ngt, hs |)eing related t() th(; nm.gnifi('a,ti()~ '~, a,s

.f~ = -,,,,,.f,.

(3.4.29)

An (;xa,~q)le is giv(;~ i~ Fig. 3.10. Also, we ('m~ (:as(:a,(l(; tw() ~mgat,iv(; ~m,gnifiers t() sy~tl~(',siz(', tim ~m,t,rix (3.4.26), tim resull, i~g (:(mfig~n'a,ti()~ nmy t)e regar(te(t a.s ()t)ti(:a.1 a,I~al()gs ()f tl~(', ~a.trix ~,,sKa, tim, i~v()lv(;(t f()(:al l(;~gt,l~s fl, f2, .f:~, .f~ 1)(;ing s~ita|)ly (:l~()s('a~ t() satisfy tim r(;la,ti(m

f.e.f4 - (,,~/2.f, f,a. 3.4.4

(3.4.30)

Sympler:tic mat's'ices as optical mat's'ices

At)t)r{)t)ria.te (:()ll.jllr {)f T, L a.ll{t D-like ilm.l,rir {:a,Ii 1)e a.rrmlged to be eqlliva,lent t() ally giv(',ll sylllI)lectic llmtrix, (;x(:llutiIlg tim a.ld,iclia,g()lm,1 nm,t,ri(:es, for whi(:h, a,s (lis(:llsse(1 ill .~ 3.4.2, the 2.f sysl, ellls a.ll(t t,lm F(nn'ier tut)es t)rovide the slfit, a,t)le ()l)l,i(:al rea,liza,ti(ms. In tiffs (:()llll('~(:l,i()ll, il, is w()rt,h emphasizing once a,ga.iil tim (litf'erence, relatively t{) tim gr{nlI) I)r()perty, of t,he matrices T ' s a.(:(:or(ting to whether they are rega,r{te(l as In~r('ly ~m,thematical entities or interI)reted a.s fi)rnm.1 l,ools for n~o{telling ot)tir syst, e~ns. Ma,thematica,lly, the 2 x 2 ~nin~o{lular real upper triangular ma,trir (3.3.1) do form a, subgroup of tl~c synq)lcctic grout) Sp(2, I~). In co~trast, {)ptically ttmy represent ra,y-t)roI)aga,ti(m througl~ I)ortions of a, l~(m~ogeimous nm(timn, a,~(t hence do not form a. subgroup because the inverse of a, fr(;e-propagation section cannot be synthesized a,s a free-t)ropagation section as well; it should demand in fact for sections of "negatiw~" length and thereby is not physically realizable. In the following we rega,rd T(d) just as the upper triangula,r matrix ot)tically synthcsizabh; by portions of a homogeneous medium or by more complex arrangements of lenses separated by uniform medium sections according to whether the parameter d takes on a positive or negative value. It is easily proved t h a t every symplectic matrix with non vanishing upperleft entry can be realized as the ray matrix of an optical system, composed by the sequence of a free propagation, a pure magnifier and a thin lens, namely

(Ac#0 DB) -- L ( f ) D ( . M ) T ( d )

(3.4.31)

The Group of the 1D First-Order Optical Systems

131

with the parameters d, 3/[ and f being related to the entries of the matrix as

d

_B A ' M-A

'

f=

A C"

(3.4.32)

Needless to say, the optical arrangement on the right of (3.4.31) is basically composed by thin lenses and free propagation sections. Interestingly, since D represents an optical system with vanishing effective length and focal power, the presence of the matrices T and L in the synthesis (3.4.31) directly relates to the entries B and C, respectively. Thus, if B = 0 or C = 0, correspondingly T or L disappears from (3.4.31). If A = 0, and hence D ~ 0, the reader ma,y easily verify that the reverse ordering TDL may be effectively applied:

,

1.

(3.4.33)

However, following the suggestion put forward in w 3.4.2 we can verify the feasibility of the optical equivalence

(~ --D C-1 ) -- F ( - ~

)T(~),

(3.4.34)

which unlike (3.4.33) coInprises tile case D = 0 as well. Oil account of tile optical realization of F ( f ) by a 2 f system, it is evident that the product in (3.4.34) describes tile propagation from an input plane, placed at distance d from the primary focal plane of the lens, to the secondary focal plane. The presence of T ( d ) reveals in fact that system (3.4.34) performs the imperfect Fourier transform of tile input signal, and so ma.rks tile difference with respect to a perfect Fourier transforming system, corresponding to d = 0, i.e. D = 0. Equation (3.4.31) and all others, obtainable changing the order of the optical components in the product configuration on the right, represent the conclusive step of the investigation presented in this section. We may accordingly state that every 2 x 2 symplectic matrix can arise as an optical matrix; in other words, every Sp(2, R) system can be realized a,s an optical system composed by a suitably arranged and characterized set of thin lenses separated by (positive) free propagation sections. This establishes the homomorphism between the symplectic group Sp(2, R) and the group of the 1D linea,r optical systems.

3.5

Wei-Norman representation of optical elements: LST synthesis

In the previous section it has been shown that every real symplectic matrix with non vanishing diagonal is expressible in a product form involving the matrices T, L and D. In general, the matrix T is to be considered from a purely

132

Linear Ray and Wave Optics in Phase 5pace

mathematical viewpoint as a real unimodular u p p e r - t r i a n g u l a r matrix, admitting the exponential reprcsenta, tion in t e r m s of the algebra, m a t r i x K , and hen(:e describing from the optical viewpoint a single free-propagation section or a suitable a,rrangement of free-ine(tilm~ sections and thin lenses. Here we will see that, resorting to the well known W e i - N o r m a n t h e o r e m of Lie group theory [4.1], sy~t)le(:ti(: nmtrices having a positive A or D entry a,d~fit a,n ordered t)ro(tu(:t forn~ rel)rese~tati()~ in t e n ~ s of T, L a,n(t S-like ~lm,tI'i('(;s; Im,~ll(;ly i~l I,('r~ls ()f a l~t)t)(;r trimitL~la,r ~lm,trix witli ~mit (tia,g(mal entl'i(;s, a, ~lninl()(l~fla,r t)()sitiv('-(h'finitc (lia,g()nal matrix, ml(l a h)wer triangular IIm,ti'ix with uliit (lia,g()lial (;~ltI'ies. In fa,('t, tll('~ ()t)ti(:a,1 a,rI'a,llgelllellt (~ B)

_

L(7))S(.,,,.)T(d )

(3.5.1)

(h;~xm,~(ls for tll(; I)ara,~(;t(;rs P , 't~, m~(1 d 1)(; rela,te(l t() tl~(; ~m,tl'ix eh;l~ents t)y 7'-

_c. ,

',,,-

A ,

,t-

B_.4'

(;3.5.2)

.j~st a,s (3.4.32) witl~ M rel)la(:(;(l t)y 'ttt. As 't~t > (), i(h;~d,ity (3.5.1) is a,(l~fissit)h; ()nly f()r llmtri('.es witll t)()sitive (liagoilal (;IlI,ry A, wllih; D is ill general a,lh~w('~(l t() rmlg~'~ fr(~lll --C~ t,(~ OO. Tll(; rea,(l(;r iimy w;rify tlmt th(' fa,('.t()riza,t,i(~ils wllere t,tl('~ limt, rix L is (~ll tll(; l(;ft ()f tll('~ xlmtrix T (i.e., t,llc I~ST, SLT a,n(1 LTS (:(mfigllra,ti(ms) are sllital)le t,() syntllesize sYnll)h;(:ti(: xlm,tri(:(;s having A > () a,~(t - ~ < I ) < c~. I~ (:~)ld,ra,st, t,tle fa,(:t()riza,ti~)~ls wll('xo, tlm nm,trix L is (m th(; right (~f T (wtli('ll lea,(l t,(~ t tlo TSL, STI~ a~l(t TI~S ('(~Ilfigl~rati(~s) (:a,n a,(:(:()~nd, for D > () a,x~(l A a,rlfitra,rily ra,~gi~g t,l~r(~gl~ t,l~(; real line. T h e eq~fiva,h;~t:e (;xt)I't,,ss(:(t i~l (3.5.1) a,s w(;ll a,s in all tl~e ()tl~er fa,(:t()I'e(t forms, obta,ina,|)h; chm~gi~g tl~e ()rtter r the (:omt)onent ~m,t,ri(:es in the t)r()(l~lct on the right, sta,tes the I)ossil)ility of representing every Sp(2, I~) element with t)ositive ut)t)er-h;fl, or lower-right entry a,s the or(lere(1 1)ro(tuct of elements (trawn from the on(~t)a,ra,~neter subgrout)s generated |)y the corresponding a,lgel)ra, ma,tri('es K , K+, and K 3. Thus, on ac('ou~t of the ext)onential ret)resentatioI~s of the ma,tri(:es involved in the t)ro(hlct, we (:a,n write

with 7~, s and d defined t)y (3.5.2); namely, 7~ cA, s - - 2 1 n A , d - ~ . Accordingly, every 2 x 2 symplectic matrix with positive upper-left or lowerright entry can be interpreted as an optical matrix arising from the appropriate conjuctions of (positive or negative) free-propagations, thin lenses and recipro(:al s(:aling systems. W i t h i n tile (:ontext of 1D linear optics tile decoInposition (3.5.3) acquires an utmost relevance when a p p r o a c h i n g p r o p a g a t i o n problems. In that case the initial values of the ray matrix entries are set in order to reproduce the identity matrix. The continuity of the process assures that in a

The Group of the 1D First-Order Optical Systems

133

Hi i

r

no I

i

i i

!

I

i

I

0

AD-B('=I

i

i

Q.

I I

i

0

i

""

d free-propagation

m

section

positive magnifier

f

thin lens

(positive or negative) .m

>

O JE

AD-BC=I lower-triangular

El. :~

unit-diagonal matrix

LO

unit-diagonal matrix

(5'p(2,R) parabolic subgroup)

O~

upper-triangular

(,,;p(2,R)hyperbolic subgroup) unit-diagonal matrix (5;p(2,R) parabolic subgroup)

-y, - ~9

( (,

q-

$ A~-.~'--I t~ t"

........ / - q

q

] q-shear

overall scaling

p-shear

F I G U R E 3.11. W e i - N o r m a n synthesis of an optical m a t r i x .

neighbourhood of the initial position, i.e. in a neighbourhood of the identity, the entries A and D remain greater than zero. The representation (3.5.3) is therefore "locally" allowed in that neighbourhood, where we can write M(z, zi) - c p(z)K+ cs(z)Kac d(z)K- ,

M(zi, zi) - I,

(3.5.4)

thus decomposing the overall problem of determining the transfer matrix M(z, zi) into the "partial" problems of determining the parameters 7)(z), s(z) and d(z). Notably the factors in (3.5.4) directly relate to the "dynamics" associated respectively with the quadratic monomials l q2 , lqp and ~l p 2 , which can easily be investigated and possibly expressed through definite functional forms of the relevant characteristic parameters P(z), s(z) and d(z) (see w167 1.5.2 and 2.4). As an example, when considering the propagation in a parabolic index profile medium, we will see that the differential equations for the matrix ele-

Linear Ray and Wave Optics in Phase Space

134

ments A, B, C, D turn into differential equations for the parameters 7), s, d of the decomposition (3.5.4), which, offering an alternative parametrization of the problem, may provide a deeper feeling for the behaviour of the system. Figure 3.11 sumnm,rizes the vari(ms i n t e r p r e t a t i o n s of the LST de(:omposition (3.5.1) within the grmlt) theorcti(:a,1 context as well as within the optica,1 context in terms of b o t h optical system a,rrangeme~lts a,Ii(t t)hase-t)la,Im transf()rnm,ti()n sequen(:es. M()reover, as in tile lilmar a t)I)r()xinmtion tim ray tA,ra,nsfer m a t r i x M is the Iim,i,rix r('4)n'st'lltativ('~ ()f t,tl('~ ray transfer ()t)cra,t()r 9)l, relat, i(m (3.5.4) a.lso sta,tes t]lat t]le ray tI'a,llS[(W ()i)(;ra,tt)r ()f a,]ly ()]It tirst-t)I'(h'x 1D ()t)ti(:al systenl is fa,(:tt)rizal)h'~ in a ll('~iglfl)t)llrh()()(t ()f the (;ntra,ll('c t)la,nc illl,() tlw, t)r()(tu(:t of tile Lie transfornmti(ms g('al('xa.te(1 |)y tlm lll()n(nllials ~2 ml(l a,s ~ ( z , z~) - (,P(~)K+ r

9~l(Z~,Zi)

-

I,

(3.5.5)

where K , K~ ml(l K:~ are tile Lic ()l)cra, t()rs ass()(:ia, te(t witll tlm (tll()tc(l mono,,rials (see .~ 1.5.2 an(l 2.3) [4]. Rt'~stri(:tly to tl~(: li~mar al)t)r()xi~m, tiol~ the ~m.trix an(l ot)eI'a.t()r i(lentities, (:].5.4) /-I,11(| (:].5.5), tn'(wi(h: tlm a,~swer t,(, l,l~(; I,,(,],1(',~ (,f z-,w(h,ri,~g, whi(.h lma,v()i(lal)ly arises wl~(;~ al)l)I'(m(:l~i~g tim i~t, egrati,m ()f ttm~ilt,()~'s e(l~ations in tl~e (:a,se of a, z-(h;t)el~(li~g IIa~ilt,()~im~, (t~(; t,() 1,1~(' i~ gel~(;ra.1 ~()~(:()~mn~tatiw'~ nat,~tre ()f ~m,tri('es a,n(t ()I)erat()rs. In w 1.5.3 w(; n~(;~iti()I~('(t tha,t the fa(~l,()rizati()~i ()f 9/1 ca~ 1)(, (~()l~t,i~nt(;(t t,() i~(:l~(h, higher or(h'r t(,rn~s, a(:(',()unti~g fl)r t,lic a.1)crra,ti()~s. Ea~'l~ ,~f 1,1~(' a,(htit, i,)~m,1 fi~:t()rs tak(;s tl~(; fi)n~ of a, Lie tra,nsfi)n~m,tio~, a,ltl~()~lgl~ it, ('a,~()t t)e giwu~ a 2 x 2 ~m,trix ret)rcse~ta,tion.

3.6

Rotations and squeezes in the phase plane

In w 1.5.2, when discussing the t)hase-st)ace d y n a m i c s g e n e r a t e d by the Lie tI'ansfornmtions ass()('iate(t with (tynami(:al flm(:tions qua(h'ati(: in tile t)osition an(t m ( ) m e n t u m ray varia|)les, we have (:onsi(tere(t the t)olynomials 1(p2 + q2) q2). We hav(~ sh()we(t t h a t they give rise to (tefinitely different and ff1 (p2 motions in the phase plane, the corresponding trajectories lying respectively along a circle and a branch of hyt)erbola. Here we reconsider in detail the above bynomials in order to identify the optical systems they may describe. -

3.6.1

Attractive oscillator, phase-plane rotation and fractional Fourier transform

We denote as Ha.o. the quadratic homogeneous polynomial Hao 1 .

.

2 + ~_~12q2 ,

"Js

(3.6.1)

The Group of the 1D First-Order Optical Systems

135

which typically models the attractive oscillator-like dynamics. The space coordinate q has been explicitly scaled by a,n a p p r o p r i a t e length factor f s ' whose optical interpretation is clarified below. The corresponding Lie o p e r a t o r ka.o. is A

~0 L~.o. - p Oq

q 0

f 2 Op'

(3.6.2)

t..

~..

which we rewrite in terms of the operators K_ and K+, defined by (2.3.5), in the convenient form 1" __ 1 _ 1 K+) (3 .6 .3 ) L~.o. - fs ( L K + ~ 9

Accordingly tile m a t r i x representative Ha.o. linearly relates to the sp(2,]R) basis matrices K _ and K+ as

.ao. . ( -1/fs2 ~

1

__

~s_sUl(fs),

(3.6.4)

where we have denoted by K~ (fs) the algebra element K 1 (fs) - l ( f s K

+ ~1K + ) -

-

- 1 ( - 1 /0f ~

f s. ) 0

(3.6.5) "

Tile convenience of such a, seemingly odd definition will become a p p a r e n t later. In order to identify the optical system described by the harmonic oscillatorlike Hamiltonian (3.6.1), we exponentiate the traceless m a t r i x (3.6.4) t h r o u g h the axial distance A z -- z - zi. As a result, we obtain the "transfer matrix" for the ray propagation from zi to z in the form (a)

F~(fs) . .cAzHa . . . _c2~Kl(fs) ~ -- ( - ~ c~162 1 sin r

.fssinr cos r

'

(3.6.6)

having interpreted the dimensionless a r g u m e n t ZXz of the circular functions as tile angle 6, c o m m o n l y measured in unity of ~ t h r o u g h the t)araIneter (~: r

~

Z--

fs

Zi

7r

-- c~(z)~.

(3.6.7)

Note that, as fs has the dimension of length, the off-diagonal entries in (3.6.6) have the dimensions of length and (length) -1 respectively, as it should be for an optical m a t r i x in the ray-coordinate and m o m e n t u m representation. a As in the case of the matrices K_, K+ and K a (see w 2.4), we may compute the exponential function of the matrix K 1(fs) (as well of K2(fs ) below) by simply applying the exponential power series, since powers of g 1(fs) (and g 2(fs)) are easily computed. The interested reader may consult [14] in ch. 2 for a general account of the methods of computing exponential functions of matrices.

136

Linear Ray and Wave Optics in Phase 5pace

- - - " - ~ P (z)

FIGURE 3.12. I)hasc-t)lane (ly~mmic generate(l by the attractive oscillator Hmniltonian (3.6.1). The ray represe~tative poi~t slips alo~g the ellipse (leter~ni~e(1 by tim i~fitial data and the frostily parm~mter fs.

The tra~sfl)r~m.ti(n~ a('t(;(l 1)y tim ~xm.trix F " ( f s ) ()~ th(; t)lm.s(,'-l)la,n(; wtria|)h;s (q,p) fr()~ l,lm i~itia.l (la.ta (q,s,Pi) is (l(;s(:ril)e(l 1)y

q(z) ..... qi (:()s q5 + Pi.fs sil~ (/), p( z ) -

1

- ~ qi si~ 4) + pi (:()s 05,

whicll is ()f (:(Jllrs(; ill a(:(:(n'r wittl rcslllt (1.5.23). The (:y(:li(: lm.t,llre (ff tim l)llas(; l)la,n(; lll()ti(m (3.6.8) is al)l)ar(;llt. As tile ray I)r()t)agal,(;s (,llr()llgll (,ll(; "()l)(,i(:al" sys(,(;Ill (3.6.6), (,ll(; r(;I)r(;s(;~d,ativ(; I)()i~t P in tim q-p t)lan(; l~()v(;s al()l~g (,l~e (;llit)s(;

q2 2 e - - ~ + Pi'

q(fz2)2, s + P ( z ) 2 - ~'

(3.6.9)

The consta,~t e is (t(;t(;rn~i~m(t t)y the t)aran~etcr fs a,n(l the initial Wd~leS; r specifies fi)r ea,(:tl va,hu', ()f tllc axial t)arameter z tllc a,xlgle of tlm rotation of the (:orrcst)on(li~g representative t)oint P(q(z),p(z)) with rest)cot to the initial t)()int P~(q~,p~) (Fig. 3.12). The rotation is ('h)(:kwise or (:o~nter(:lockwisc accor(tiIlg to the sigll of fsEvi(tcntly the nmtrix F'~(fs) relates to a rotation by r ct~ along a I l in general cllipti(:al contour in the phase-plane. In fact, as will be clarified later, it can t)asically be interpreted as a phase-plane rotator. In particular, with fs = 1, F'~(1) takes the appearance of a pure rotation matrix by the angle r - sin r K

cos r

where K 1 -- Z 1 (1) = =2+K+ , the unit a r g u m e n t being omitted. Accordingly the representative point Inoves along a circle centered around the origin with radius v(, - v/q] + p~.

The Group of the 1D First-Order Optical Systems

Interestingly, with r form matrix (3.4.13),

137

~ (i.e. c ~ - 1) we recover the perfect Fourier trans-

F(fs) -

0 --1/f

fS ) -- (~TrKl(f S ) S

0

(3.6 11) '

"

(the unit order being omitted as well), which then is seen to relate to the phase plane rotation by rr/2. The matrix F~(fs) is reported in the literature as the fractional Fourier transfoTwz matrix of order c~ and family parameter fs" The term "fractional" evokes the charming vision of the "extension to continuum" of a process that occurs through finite-size steps. It is in fact in conformity with this "idea of fragmentation" that tile fractiona,1 Fourier trailsform has been introduced within different contexts of both mathematics and physics. Condon, for instance, investigated about a continuous group of functional transformations isomorphic with the group of rotations of a plane about a fixed point by multiples of an arbitrary angle, thus generalizing the property of the Fourier transform group, which in fact corresponds to rotations by multiples of tile right a,ngle [5.1]. Likewise, Namias ela,bora,ted tile functional form of Fourier-like operators, a,dlnitting, as the Fourier tra,nsform, the HerlniteGauss eigenfunctions, but with eigenvalues evenly separa,ted by a fraction of the imaginary unit [5.2]. The definition of the Fourier transform of fractional order within the optical context has been inspired by the performance of an optical system perceived as obtainable by fragmenting the Fourier transform configuration into a larger and larger number of shorter and shorter free-space sections interleaved with weaker and weaker lenses [5.a], or equivalently as capable of extending to arbitrary angles the property of the Fourier tra,nsforming system to rotate by a right angle the Wigner chart in the inherent Wigner phase plane [5.4]. The rotation by a right angle in the optical phase plane is the underlying canonical transformation acted by the Fourier operation on the conjugate variables q and p. Letting the rotation angle vary continuously leads to the fractional Fourier operation. Thus, the finite-step transformations marked by multiples of the right angle, corresponding to Fourier transform systems possibly concatenated one to the other, become specific events within the smooth evolution of the process governed by the continuously varying angle qS(z) over the 2re range, and so by the continuously varying axial parameter z. Such a phase plane rotation by a continuously varying angle may be achieved by sectioning the basic Fourier transforming configurations into an increasing number of similar configurations appropriately designed, as we will see later. The relation, and the relevant visualization in the optical phase plane, between the Fourier transform and the Wigner distribution function is clarified in w 8.4.1.

Linear Ray and Wave Optics in Phase Space

138

After its i n t r o d u c t i o n into the field of optics, the (:oncept of fractional Fourier t r a n s f o r m and the relevant forma, lism gave rise to a huge variety of applications, investigations and new formulations in an increasingly enriched ()t)tics scenario. In the forthcoming paragrat)hs we will simply cla,rify the con(:cI)t of fractional Fourier t r a n s f o r m at a very basic l(;v(;1, a,(t(lr(;ssing the rea,(lcr to the related lit,era, t~re for a wider mid (teet)er t,rcatnie~lt [5].

FT"actional t:owrit'.'r t'ra'n.@~Twt matT"let's as a sy'mplt'ctic subgT"oup Rcnmrkat)ly, fin" a fixc(l wdlu; ()f th(' fanlily Im,ra.nlcter fs, tllc ma.tri(:cs F ~ span a t)rot)cr Ab(;liall slfl)grr162 the symplc(:tic gr()llt) Sp(2, R) with rcst)c('t to the order (~, wtfi('tl Ilmy ra.llge ill t)riIl(:it)lc tllr(nlgll all tile r(;a,1 lilm. In fact, with (~ - 0 w(: (fi)ta,in the i(lcntity nm,trix; lla,llmly, F~

I.

(3.6.12)

Also, l)y th(: a,(l(li(,i()ll f()nlnllas ()f ill(: (:ir(:lfla.r flult:i,i()llS, w(: lmv(: F'~' ( f s ) F ' ~ ( f s )

-


fs sin(01 +0:~1 ) (3.6 13)

~o.~(r +%)

= F",+"~(.fs ) - F,,'~(.fs)F,,,(fs) wlfi(:h r(,flc(:ts tll(', w('ll kll()wl~ I)r()I)('rty tl~a,t (,l~(' ('()~I)()siti(n~ ()f tw() r()tati()ns l)y th(: a,ngl(:s 4>, m~(l ~is (:(l~fiva,h:nt (,() th(: r()I,a,(,i()ll })y tl~(: a,llgl(: (/5 -- ('/>, + ~ . T h e ad(litivi(,y t)r()l)('rty (3.(i. 13) with r(;sI)('~(:t (,() tim ()r(ler (~ lea(Is t()i(tcntify tt~('~ i~v('xsc t() F " (fs) a,s (,Ira ~m,t,rix (:()I'r(;st)()~(li~g t() (,lm ()I)l)()si(,('~ vahm ()f (~, [F" (.fs)]-'

-

F-')(fs ) -

((:()s
-'fssi"4>) (:i)s 4>

"

(3.614)

wlfi(:h then r(:la,t(:s t() (,l~(: r()(,a(,i()n l)y tl~(: mlgl(: c/> i~i t,l~(: I'(~V(~l'S(~(lir(:(:tion. Also, the n,a,trix F~'(/s),'.a.,, t,e interpret,',| a,s tim <~-th I,,,wcr of Fl(.fs), i.e., of the "()r(linary" F()~ri(;r (,ra,nsf()rm: F"(L ) -[F(.5)]".

(3.6.15)

In P r o b l e m 5 we suggest a, (tircct proof of the a,bovc rcla,tion in the case of a - ~1, 7n -- +1, • ... Wc invite the reader to interpret the exponential reprcscnta,tion (3.6.6) of F'~(.fs) from the viewpoint of the power relation (3.6.15). Interestingly, with 4) - 2jTr for any integer value j - 0, 4-1, • .., we recover the identity matrix: FnJ(fs) -- F ~ -- I. (3.6.16) T h e cyclic b e h a v i o u r of the fractional Fourier t r a n s f o r m i n g systems with respect to the order a is then easily established in the general form F~+4J(fs) -- F ~ ( f s ) ,

(3.6.17)

The Group of the 1D First-Order Optical Systems I

i i i

~

I

I

i i i

, i i

i i i

i , i i

(a)

i , i

139 I

J~

i i i

! i ,

i

i

(b)

FIGURE 3.13. Lohmann type I optical realization of the fractional Fourier transform matrix F~.(a) Positive lens. (b) Negative lens. which is in accord of course with the well known periodicity of the ordinary Fourier transforming system. Hence the angle r can conveniently be limited to anyone interval of amplitude 27r, as r E (-Tr, 7r] for instance, and accordingly the order a to the interval c~ c ( - 2 , 2].

Fragmenting the optical Fourier transforming system. The type I and type H optical setups We investigate now the possibility of optically synthesizing the fractional Fourier transform matrix F~(fs). Taking as model tile single lens geometry of the Fourier transform, described in w 3.4.2, we may think of the F ~ system as formed by a thin lens of focal length .[~ placed between two free-medium sections of the same length d~. It is easy to verify that the symmetric single lens geometry F~(fs) = T(d~)L(f~)T(d~), (3.6.18) can be designed with the involved parameters d~ and f~ being determined by the angle r and the family parameter fs according to the relations d~ - fs tan(C/2),

fs r f~ - sin

-7c < r < 7r.

(3.6.19)

Positive values of r in the inherent range, namely 0 < r < 7r, guarantee t h a t both d~ and f~ are positive, thus yielding a one-lens configuration like that pictured in Fig. 3.13.a) with a bi-convex lens. In contrast, negative values of r -7c < r < 0, yield negative values of d~ and f~ as well; the optical configuration demands therefore for a negative lens as that sketched in Fig. 3.13.b), virtual objects and virtual images being involved. The single lens realization (3.6.18) of the F ~ system is reported as Lohmann's type I geometry [5.4]. Likewise, a double lens geometry can be designed to optically implement the F ~ system, consisting of two identical lenses of focal length f~ spaced by

Linear Ray and Wave Optics in Phase Space

140

J~,

~,

=

d,,

f,,

:

f,

=

el,,

(a)

FIGURE

F".(a)

3.14. I x ) l ~ m ~ l ) o s i t i v e le~ses.

--_

(h)

tyI)e lI ()l)ti('al r(;alizati()~ ()f tl~e fra('ti()~al F o ~ r i e r t r m ~ s f l ) r ~ ~ n a t r i x N e g a t i v e Ictuses.

(b)

a fl'(;(;-t)r()t)a.gati()~ s(',('t,i()~ ()f l(;l~gl,l~ d~. Exl)li(:itly, w(', ()t)ta.i~ F'~(L) : L(L~)T(d~)L(.f~),

(3.(i.2())

wil,l~ l,l~(; (t(:Ii~i~g r('lal,i()~s fl)r tl~(; r(4(:vm~t l)ara~(q,(;rs ct~ a~(l .f~ ct~ - f s

si,,0,

'f" -

ta,~(O/2)'

- 7 r < 0 < 7r.

(3.6.21)

Tlle ()I)ti(:al (~()xlfigln'ati()~ls at(: sll()Wll ill Fig. 3.14. l)()sitive l(;lls(;s, axl(l s() r(;al ()l).j(;(:t an(1 illm,g(; rays, at(; illv()lv(;(l f()r l)()sitiv(~ vahl('~s ()f th(; r()tati()li allglc 0, () < 0 < 7r, wllilst liegativ(~ l(;lls(;s, ml(l ll(;ll(:('~ virtlm,1 ()|).je(:t ml(l illm,g('~ rays, are ixiv()lve(t f()r ll(:gativ(: va.hws ()f 0, -~r < 4) < (). The (t(nll)h; lens realizati()n (3.6.2()) ()f t,l~(; F '~ syst,(;~ is r(;l)()rt(;(1 as Loh,'m,a'n,'n,',s' t y p e I I 9eo'm, ct'~'y [~.4]. Tile, ()I)ti('a,1 (:()llfigllra,ti()llS (3.6.18) an(1 (3.6.2()) are, (les(a'il)(;(l a.s well l)y st)e(:ifi(: ra,y-llmtri('(;s i11v()lvillg l,ll(; r(',sI)(;(:tiv(; 1)a.ra,lll(;t(;rs d~, .f~ a.~l(1 d~z , ,f~, an(| ()f (:o~n's(; tll(', ()r(h;r c~ tllr()llgh th(; mlgl(; 0, illst,(;a(t ()ftll(; g(;ll(;ral raym a t r i x (3.6.6) illv()lving the fancily t ) a r a ~ e t e r fs- The r(~a(ler is a.skc(1 to write down tt~(; ray ~m.t,ri(:('~s fi)r tl~e fl'a.(:ti()~al t,yt)e I a~(t tyt)e II ~u~it,s. Evi(hmtly tim valuta 0 = 7r (:a,m~ot t)e a(:(:()~n~te(t for t)y the ot)ti(:al representati()i~s (3.6.18) m~(t (3.6.20), as the relevant relations (3.6.19) an(t (3.6.21) become un(tefin(;(t in t h a t case. Ih)wevcr, with 4~ = 7~ the rotation matrix has the forn~ of the scaling ~mtrix with ma,gnification factor M = - 1 , namely

F2(L)-

D(-1)

- -I,

(3.6.22)

thus signifying reflection with respect to the optical axis, whatever be the value of fs- According to the results of w 3.4.3, the sequence of two identical (i.e., with lenses having the same focal distances) Fourier transforming systems produces a 7v-rotation of the ray vector in the q-p plane. This is clearly in accord with

The Group of the 1D First-Order Optical Systems

141

the additivity property (3.6.13), by which we can write F ( f s ) F ( f s ) - F2(fs).

(3.6.23)

With 0 - ~ (i.e., (~ - 1) we regain f r o I I l (3.6.18) and (3.6.20) the one and two lens configurations of the perfect Fourier transforming system. With decreasing values of a, [a[ < 1, we can "continuously" fragment the single or double lens Fourier transforming configuration into an increasing number, Ict[-1, of similar single or double lens configurations with the relevant free-propagation sections and thin lenses, respectively, becoming increasingly shorter and weaker. With (~ - 89 for instance, the one-lens geometry (3.6.18) obtains with di

-

(v/~

-

1)fs,

f~ - V~fs.

(3.6.24)

The relevant ray matrix is

F-} ('f~ ) -

(

l/v~ -1/ x~ f s

fs/x~ ) 1/ x~ '

(3.6.25)

whose square is readily verified to yield F(fs); namely __1

1

F2 (fs)" F2 (fs) - F(fs)-

(3.6.26)

! In other words, F 2 (fs) represents the optical system that operates the square root of the Fourier transform. Thus, with the order c~ ranging within the interval ( - 2 , 2], we obtain a "family" of optical systems with the geometries depicted in Figs. 3.13 and 3.14. The family is uniquely individualized by the real parameter fs, also reported as the standard focal length. It represents the scale of the ordinary Fourier transform, which corresponding to ct = 1 results by appropriately cascading fractional transforms of lower orders belonging to the same family fsIn other words, the family parameter fs identifies the basic Fourier transform unit, which can be fragmented and reconstructed by the opposite operations of sectioning into fractional units down to the required c~-th root, and of cascading such fractional units up to the required c~-th power. It is clear that both the fragmentation and the reconstruction of the basic Fourier transforming unit can be achieved with fractional configurations of different orders which, of course, sum up to unity. In Fig. 3.15, for instance, we show three possible realizations of an imaging system producing a perfect inverted image of the object (c~ = 2), by cascading fractional order units, the relevant orders summing up to 2 in all cases. Evidently the fragmentation of a system is not unique.

Linear Ray and Wave Optics in Phase Space

142

li

f 2/.~

........

1213

I !

||

.......~. 9

..b..-..

i

2: ......

.....

_-..-~._

"

Ho

12/3 ~.,

.......

'

__

!

-.-

......

(u)

121~

I!o

t41~

9

ill

r

(h)

!

t4/~

l_,/~

0

f4/~ .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

iiiiiiiiiiiiiiiiiiiiiii iiiii!i ii .

.

.

.

.

4

(c)

FIGURE

3.15. P e r f e c t i m a g i n g s y s t e m ((~ - 2) o b t a i l m ( l b y c a s c a ( l i l l g f r a c t i o l l a l F()llrier l m i t s 2. (b) t w o ,lIlit, s w i t l l (~ -- 52 a n d (a) t h r e ( , llIlit.s w i t h (~ -- (~2 -- (~:~ _ .~,

of (tifferent or(lers.

(~.2

~ (f2/:, - f,/:,); ((:)three

,mits with ~,

.~, ( ~

-.~

.

-

-7-2/3)

Tile (:on(:eI)l, ()f family, wtmre tile fra,(:tional traalsfc)rnts witll the sa.me stan(ta,r(t focal length fs group together, naturally arises frolll tlle rule, by which the systems (3.6.6) axe (:a,scaded. As displa,yed by the s(;migroup property (a.O.la), only cas(-a,diiLg fractional units having tile same a,bsolute value of fs, tile overall systeln is aga,in a fractiona,1 Fourier transform, and also belongs to tile same fa,mily. Notably, as already evidenced in (3.6.10), with fs = 1 we obtain the family of the pure rotations in tile q-p plane: F

~

-

R(0),

0-

c~.

(3.6.27)

Finally, we wish to comment on the optical "performance" of a thin lens as it emerges from the considerations here developed. A lens is ab]e to perform

The Group of the 1D First-Order Optical Systems

143

two basic operations in optical processing: imaging and Fourier transform. The concept of fractional Fourier transform allows to frame both operations within the same context, where the imaging can be seen as a fractional Fourier transformation and correspondingly the ordinary Fourier transform as a particular realization of the fractional transformation. We know, in fact, that the elementary optical system consisting of a thin lens of focal length f placed midway between two reference planes separated by 2d can operate as an imaging or a Fourier transforming systeIn according to tile relative values of f and d. For d = 2f the systems acts as an imaging system, whilst with d = f it operates as a Fourier transformer. It is evident that in both the type I and type II geometries one can range from one operation to the other through the intermediate steps of fractional Fourier transformation by continuously adjusting the order parameter c~, and accordingly the involved lengths d I and f~, or d~ and f~. It may appear however that the claimed "fractionalization to the continuum" does not have a practical consistency due to tile ixnpossibility of realizing an infinitely short free-propagation section as an infinitely weak thin lens. We will see in w 3.9.1 that a quadratic graded-index focusing medium behaves in respect to the (paraxial) ray propagation as a fractional Fourier transforming system, thus offering the possibility of physically defining the process of a continuous fractionalization of the Fourier transforming operation.

Rotations and squeezes in the fractional Fourier transform operation It is evident that the phase plane operation of the F ~ system results from rota.tions and scaling transformations, appropriately designed and concatenated one to the other. In fact, as said earlier, with fs - 1 the fractional Fourier transform matrix takes the form as that of a pure rotation matrix by the associated angle r c ~ . Then we are naturally led to decompose the fractional Fourier transform matrix F(~(f s) into scale and rotation matrices according to the sinfilarity transformation F~(fs)-

S(f:/2)F"S(fsl/2 ).

(3.6.28)

The pure rotation by 4) - c ~ is sandwiched between two scale transformations with reciprocal magnifications, f s 1/2 and f:/2. Note that S(fsl/2 ) -

[S(fs--1/2)]-l. Of course, the off-diagonal entries in the pure rotation matrix F ~ above are understood having their own dimensions of length and (length) -1, although not evidenced by a visible parameter. Also, the magnification factors fs- 1/2 and f 1/2 of the scaling matrices must be regarded as numerical values, s as those obtainable, for instance, by cascading two Fourier transforming systerns as described in w 3.4.3. In particular, with c~ - I we obtain on one side the ordinary Fourier transform whilst on the other the pertinent decomposition in

144

Linear Ray and Wave Optics in Phase Space

t e r m s of scaling and rotations by tile right angle, i.e.,

( 0 fs)- ( ~ -1/f s

0

0

1/~

0 )(01 1)(1/~/~s 0) s

-

0

0

~S

"

(3.6.29)

Th(; ab()ve (tis(:llssi()n ('(mfirms th(; relation ()f the Follrier t r a n s f o r m to tile t)has(:-t)la~l(: s(:a,h:(1 r()tati()ns })y th(: right angle and (:orr(:st)()n(tingly of the fi'a,(:ti()lml F()uri(:r trmlsfi)r111 t() s(',ah:(l r()tati(ms 1)y arl)itra.ry, a,~l(t (',oIltimlously a,(tjllstal)h:, fra,('l,i()llS ()f 1,t1(: I'igtlt a.11~1(:, hi a('(:()r(t witll (t(:('(nllI)()siti()~l (3.6.28) w(: ~11a,y r(:gar(l 1,11(: fl'a,(:l,i()lm.1 F()llri(:r (,ransf()rnl(:rs a,s ph,a,s'e-plane 'lvtato'r,s. R(:s()rt, illg t() (,lm ()l)t,i(:al (:()llfigllra,ti()n, (h:s(:rit)(:(t ii1 1,11(: l)r(:vi()us t)ara,grat)ll, w(: llmy l,ll(:r(:f()r(: say l,lml, 1,11(: fra,gill(:lll,a,ti()~l ()f 1,11(: F()llri(:r tra,nsf()rllli~lg (',()~lfigllra,ti(nl i~ll,() shllilar (:()nfigllrati()lls witll ill(:r(:a,sillgly sh()rt(:r fl'(~(~-i)r()l)aga,ti()ll s(~('l,i(nls i11t(;rl('a,v(',(l t)y i~l(:r('a,sillgly w('a,k(;r (,llill I(Uls('~s (:()rrcst)(n~(ls i~1 1,11(; ()l)ti('a,1 t)l~as(; t)la,~m t() a, fl'a,g~ll(;~ltati()~l ()f tlm r()ta,ti()~l a,~lp;lc int() i~(:r(;a,singly s~m.ll(;r Ka(:i,i()ns ()f th(~ right a,~lgl(;. Fil~a,lly, l)a,ralh;li~lg 1,1~(:si~gl(; l)()i~t t)lm,s(;-l)la,l~(; vi(',w ()f Fig. 3.12, it is i~lt,('x(:sl, i~Ig 1,() vis~mliz(: 1,11(:(:II'(:('t ()f (,l~(: fra,(:l,i()~lal F()~u'i(:r (,ra,l~sf()r~ ()l)(:ra(,i()~1 ()n a, ray (tis(,ril)~ti()~l i~1 l)lms(:-l)la.~m. W(: (:()xlsi(h:r 1,11(: Ga,~ssia,~l (i(:x~si(,y (listril)~tion 2,~,,~,,

~,

'

,

(3.6.30)

a.lr(;a(ly ~s(:(1 f()r s()~(: ill~sl,ra(,iv(: (:xm~t)l(:s i~ (~lm,t)(,(:r 2; as ~()(,(:(1 (,l~(:I'(:, (,h(: t)()sitiv(: (t~mx~ti(,i(:s c~q a,~(l a~, sl)(:('ify th(: rills (:xt(:llsi()ns ()f (,l~(: (l(:~sity (listrit)~(,i()~ in (,11(: t)has(: l)lan(: al()~lg (,l~(: q a,n(t p-axis, r(:st)(:(',tiv(:ly. ()~ a,(:(',()~l(, ()f r(:la(,i()~l (2.2.67), it is (:a,sy (,()(:la|)()ra,t(: (,l~(: a~a,ly(,i(:a,1 (:xt)r(:ssion fi)r ttl(: (:fli:(:t I)r()(t~(:(:(t ()~ tl)(: (tistril)~ti()~ (3.6.3()) ])y (,t~(: ()I)(,i(:a,1 F()~ri(:r tra,nsfi)rnm,(,i()n (3.(i.8). H()w(:v(:r, it is h(:lt)flfl t() giv(: a t)i(:(,()ria,1 vi(:w. Figure 3.16 sllows th(: tra.~lsf()rmati()~l a.(:t(:(t by F~/3( 1~) ()n th(: r(:I)r(:s(:nta,tiv(: phaseplan(: a,rca of the (tistril)~tioI~ (3.6.30), id(:ntific(t by any ()~1(: ()f tll(: elliptical contours (2.5.16), as, for ins(,an(:(:, t h a t corrcst)on(ting to th(: wdu(: ~ = 2. The proper s(:quen('(: of s(:aling, r()(,ation and scaling, to whi(-h a(:(:()r(ti~lg to (3.6.28) the (tesired fractional Fouri(;r t r a n s f o r m a t i o n , i.e., F2/3(~), can be thought as equivalent is exemt)lifie(t " In the graphs the scaled variables qO-q and -p-have O-p

1

been us('d, thus o b t a i n i n g as initial phase plane ar(;as circles of radius r = v ~ . 3.6.2

Repulsive oscillator, phase-plane squeeze and Lor'cntz transformations

We consider now the q u a d r a t i c h o m o g e n e o u s polynomial

Hr 9~. __ -~P 1

2 _

~ 1

2 f ~ . q 2,

(3.6.31)

145

The Group of the 1D First-Order Optical Systems

FIGURE 3.16. Phase-plane view of the decomposition (3.6.28) of the fractional Fourier 1 into the proper sequence of scaling, rotation and scaling, through transform matrix F2/3(7) the relevant effects produced on the Gaussian ray density (3.6.30).

which typically models the repulsive oscillator dynamics. The Lie operator Lr.o. is

--

0

q O

(3.6.32)

L r . o . - p -~q -t- f : Op ' t--

A

A

which, in the similar way as k~.o., relates to the operators K_ and K+ as A __ 1 _ 1 ~+) Lr.o. fs ( f s ~ - ~ "

(3.6.33)

Likewise, the matrix ret)resentative Hr.o.writes in terms of tile matrices K and K+ according to 0 Hr.o. _ ( 1/f~

10) _

~2 K 2 ( f s,) '

(3.6.34)

where K 2(fs) is the algebra element K 2 ( f s)

_ _ 1~(fs

K-

__ ~1 K

+)__1

0

1If s

fs )

(3.6.35)

0

Exponentiating the traceless matrix (3.6.34) through the real parameter Az -z - zi, we obtain the "transfer matrix" for the ray propagation from zi to z as B(T, fs) -- c A z H

....

2TK2(fs) 7 (- ~1 -COSh sinh T

s )cosh sinhT

'

(3.6.36)

having denoted the dimensionless argument Az of the hyperbolic sine and cosine functions by -

fs

"

(3.6.37)

146

Linear Ray and Wave Optics in Phase Space

The t r a n s f o r m a t i o n acted by the m a t r i x B(~-, fs) on the phase-plane varia,bles (q,p) from the initial data (qi,pi) is then

q(z) =

qi cosh 7- + pifs sinh 7-,

(3.6.38)

p( z) - -Tff1 qi sinh 7- + pi cosh T,

i~ a,(:(:()r(t witl~ E( t. (1.5.26). Evidently, the t)hase-t)la.lm lllOti()ll is no lllOr(; cyclic as i~ the (:a,se r the systen~ (3.6.(i). In fa,('t, as th(: ray t)r~)t)agates thr(n~gh the "oI)ti(:al" syst~;~ (3.6.36), tlm ret)rescn~ta,tiw; l)(fi~d, i~ th~'~ (q, p) t)laue moves a,l(n~g ~n~(; of tim lwm~('l~es ~f tl~e hyt)erl)()la q(z) 2

.ri-'

- p(z) ~ -

O,

0 -

q~

fi;

,,

pT,

(3.6.3,9)

wll~s~'~ ch'lillit~', slla,l)~'~ is ~h'~t~;rllli~e(l l~y 1,1~r il~itial ~la.ta,, wl~i(:l~ al~n~g with the t)a.ra,~r fs tix tl~e wd,~; of tl~e (:r b. F~)r a, givcu~ va.l~u'~ ,~f fs, the' ~la,tri('r B(7-, fs) fin'l~ al~ Al)elia~ s~fl)gro~lt)~)f tl~e sy~I)lecti(: ga'~n~l) Sp(2, N) witl~ rr t,~) tim I)al'a,lllett'.r 7-. I~ fa(:t, tl~e a,~htiti~)~ fln'~n~lae fin" l,t~,; l~yt)crt~)li(: fl~(:l,i~n~s yM~l tim se~igr~n~l) l)r~t)erty (a.~.40)

B ( r, , .[s ) B ( r.2 , .[s ) = B(7-, + r.~,./s)-- B ( r.~ , .[s ) B ( 7-, , .[s ) ,

a,c(:~)r(lillg t~) wlfi('ll we a.ls() lla,ve

B(r, A ) B ( - T , fs) = B ( - T , A ) B ( r , .fs) = I,

(3.0.41)

tl~e i~hu~tity ~m.trix I~ci~g i~lce~t I = B(0, .fs), wl~a.tew;r t~e .fs. Tlle illW;l'Se t,~ B(7-, fs) is therel)y tlle llmtrix /B(T, fs)] -1 - B ( - T , fs) --

(

- ~ 1 sinh r

('ostl 7-

'

(3.6.42)

The lm.tllre ()f the', lfim.sc t)la,ne dynamics (tescribed by (3.6.38) is ba,sically (teter,lli,led by the t)rescnce of the hyperbolic fimctions so as the circular functions entering (3.6.8) deter,nine the cyclic behaviour of the fractional Fourier tra,,lsform gcnerate(t (tymtmics. Then, we set fs = 1 in (3.6.36) in order to evidem:e the intrinsic (:hal'acter of the B-generated pha,se-pla,ne dynamics.

B-genes'rated phase-plane dynamics: wtations and squeezes W i t h fs = 1, the ma, ti'ix (3.6.36) takes tile symmetric form

(ohsink c2 K2co ) h

/3643t

The Group of the 1D First-Order Optical Systems

147

paralleling in t e r m s of the m a t r i x K~ K_-K+ 2 , and so of the hyperbolic functions, expression (3.6.10) for the fractional Fourier t r a n s f o r m m a t r i x F ~ relative to a unit value of the family p a r a m e t e r . Notably, F (~ is an o r t h o g o n a l m a t r i x : [FC~]-1 = [F~] n-, whereas B(T) is s y m m e t r i c and positive definite (b). E q u a t i o n (3.6.39) identifies in the q-p plane the equilateral h y p e r b o l a q(z) 2 - P(Z) 2 - b,

b - q2 _ p/2,

(3.6.44)

whose branches intersect the q or p-axis according to w h e t h e r the initial values (qi ,Pi) a.re such t h a t b > 0 or b < 0. E i t h e r cases, the a,symptotes are the straight lines p = +q. T h e d y n a m i c s of the ray vector in the phase plane under the action of B(~-) is simila, r to t h a t under S ( m ) , which, a,s noted in w 2.5, identifies the h y p e r b o l a qp = qiPi. It is evident, in fact, t h a t the h y p e r b o l a s qp = qiPi a,nd q2 _ p2 q~ _ p/2 can be t u r n e d one into the other by a p r o p e r r o t a t i o n by 4" Then, as phase-plane r o t a t i o n s are a c t u a t e d by fractional Fourier transfornfing systems, we m a y guess for B(T) a t h r e e - s t e p realization in t e r m s of pure r o t a t i o n s and squeezes as B(T) -- F -1/2 S ( c ~ ) F 1/2. (3.6.45) Notably, (3.6.45) a,llows us to identify a,n optical realization of the symplcctic m a t r i x B(T), on account of the optical realizations of the fl'a(:tional Fourier t r a n s f o r m i n g and pure magnifying s y s t e m s discussed in w 3.6.1 and w 3.4.3. We invite the reader to prove t h a t a. s y m m e t r i c single-lens realization of m a t r i x (3.6.45) is actuabh', with d - t a n h ( ~ ) a n d f - - s i n h - l ( T ) when 7- > 0, whilst a more complex configuration d e m a n d i n g for more t h a n two lenses is needed to i m p l e m e n t negative values of T. In order to clearly show t h a t the m a t r i x B(T) o p e r a t e s in the phase plane as a scale t r a n s f o r m a t i o n in a similar fashion as S ( m ) , but w i t h respect to a reference direction Inaking an angle 0 - ~ w i t h the q-axis, we consider again the Gaussian d i s t r i b u t i o n (3.6.30) ~nd the relevant r e p r e s e n t a t i v e c = 2 phaseplane ellipse. For illustrative purposes, we set Crq = crp = or, and accordingly consider the circle C of radius r = 2or centered a r o u n d the origin, q2 + p 2 _

r2,

(3.6.46)

s A matrix A is positive definite if the scalar product (u, Au) is positive for any vector u in the linear space acted by A. It is evident that in the case of matrix (3.6.43) we have (p2 + q2) cosh ~- + 2pq sinh T > 0 on account of the inequalities [sinhr[ < cosh r and 2pq <_ p2 + q2 holding through all the domain of the real parameter r as well as of the ray variables q and p.

148

Linear Ray and Wave Optics in Phase Space

F I G I I I I E 3.17. l'l~ase-plal~e view of tl~e
A(.l,i~lg (nl 1,11(~1)ea,,~ 1)y S(,)~) (w(; ,l~ay (,ake t~ = ('T), (,lie r(;l)r(~s(n~i,aiiw; (:ir(:le C is (.rm~sf(n'~(:(l i~(,()tlw ellil)s('~ ( , - 2 r q 2 + (,2rp2 =

r2,

(3.6.47)

syllllll('l,ri(:ally l)()si(,i()ll(;(l ill 1,11(;q-p l)lmte wi(,ll s(;llliaxis a = ~'rr a,ll(l b = ('-Tr. Tlw.ll, Slll)l)()sillg r > 1), w(: llmy S(;e (,lm,(, (,lie ])('.a,lll-l,l'a.llsf()rllla,(,i()ll a.t:l,e,(t 1)y 1,11(; ()l)ti(:al lllat,rix S(tl~) t)r()(lll(:('s all (~l()llga.(,i()ll ()f 1,11(' q-siz(' ()f (,ll(' (:ir(:le t)y (,w a,ll(l (',()l'l'(:sl)()ll(tillgly a ('()~l(,ra('l,i(nl ()f 1,11(:p-siz(: l)y 1,11(:sallle fiu'l,()r. Ill ()l,tl(:l" w()r(ls, l,tl(: ])(:roll r(:l)r(:s(:llt,ai,iv(: ('iI'('l(: C is s(tll(:(:z(:(l a.l(mg 1,11(:p axis (s(:(: t,hir(t gra,t)ll ill Fig. 3.17). rI'll(: a,r(:a, ()f (:llil)s(: (3.6.47) is ()l)vi()llsly 1,11(: sa,111(: as I,haI, ()f tll('~ ()rigi~lal ('ir(:l('~: M = 471(72. Likewis(;, (,ra.llsfi:rrillg 1,11('.t)('.a.~11 fl'(nll zi 1,() z (,hr()llgll B(T), 1,11('~l)lmse-t)la.ne (:ir(:le C is tra,nsf()rlll('~(l ild,() (,lm ellit)s(; (',()sll(2r) q2 + ( : ( ) s l l ( 2 r ) t ?

- 2 si~lll(2r) p q = r 2.

(:).o.4s)

T h e llm.j()r axis is ill tile ~71" ()r --~71" (tirecti(m a(:(:or(ting to w h e t h e r r is t)()sitive or negat, ive. If 7- > 0, f()r inst, a.ii(:e, t,he xlm,jor axis (:(nlies t,o lie a,l()ng l,ll(: (lirect, ion m a k i n g a, { angle wil, h the q-a,xis (see t,he la,st, grat)h in Fig. 3.17), st)ecifically being a = cTr an(t b = ('.-Tr. T h e init, ial t)ha,se-plane area, is then stretched up to the factor m - c r in the dire(:tion at { to the q-axis and orthogonally squeezed down t)y the same factor. T h e system B(T) is indeed referred to as h y p e r b o l i c ezpa'r~,der ()r 'l'eductor a,(:t:()r(ting to w h e t h e r r > 0 or r < 0. Figure 3.17 shows the r o t a t i o n - s c a l i n g - r o t a t i o n sequence, whose overall effect is the e x p e c t e d squeeze of the phase t)lane area in the direction at ~ to tile q-axis. As in the graphs of Fig. 3.16, the scaled variables qO-q and ~cr have been used. It is possible to p r o d u c e squeeze in any direction 0, com[~ining scaling and r o t a t i o n m a t r i c e s t h r o u g h a similarity t r a n s f o r m a t i o n analogous to (3.6.45) with F 1/2 replaced by F ~. In fact, the reader can easily verify t h a t the m a t r i x

The Group of the 1D First-Order Optical Systems

B(r, 0) - F -~ S(cr

149

~ - ( cosh r + cos(20)sinh 7sin(20) sinh r ) sin(20) sinh r cosh r - cos(20)sinh r / ' k

(3.6.49)

produces a scale transformation with respect to the direction making the angle 0 - c, 2 with the q-axis. Like (3.6.43), the matrix B(r, 0) is symmetric and positive definite. Also, it is characterized by two parameters specifying the "entity" and the "direction" of the squeeze. In accord with the previously adopted symbology, we identify the specific determinations of matrix (3.6.49) corresponding to 0 - 0 and 0 - { as S(m) and B ( r ) respectively; namely, B(r, 0) - S(m), m - e ~ and B(r, {) - B ( r ) . Interestingly, although the matrices S and F ~ separately form proper subgroups of the group Sp(2, R), the matrices B ( r , 0) do not span a subgroup. In fact, successive squeeze transformations along different directions do not compose into one scale transformation, but into a squeeze preceded or followed by a rotation (see Problem 10). It is worth evidencing the formal similarity between the squeeze matrix B ( r ) and the Lorentz transformation matrix in the Cartesian system of z and ct, positive and negative values of r respectively corresponding to boosts and antiboosts. Likewise, the pure magnifier matrix S(m) has the form of the Lorentz transforInation matrix in the Dirac's light-cone coordinate system, which is rotated in the Euclidean plane (z, ct) by -~. A(:(:()rdingly, we may note that as successive ilOIl collinear squeezes do not combine into one squeeze, Lorentz boosts in different directions do not multiply into one Lorentz boost, but into a Lorentz transformation preceded or followed by the Wigner rotation. The Lie algebra sp(2, R) is in fact isomorphic to the underlying algebra so(2, 1) of the group S0(2, 1) of the Lorentz transformations in three (two space and one time) dimensions. This isomorphisIn, a.nd in particular the analogy in the behaviour of optical scale transformations and Lorentz boosts, has suggested optical experiments to test the abstract properties of the Lorentz group [6]. Finally, we briefly comment on the factored representation of the squeeze matrix (3.6.36), which is naturally suggested by (3.6.45) in the form B(T, fs) -- F - 1 / 2 ( f s ) S ( e r ) F 1 / 2 ( f s ) -- S ( f : / 2 ) B ( T ) S ( f ( 1 / 2 ) ,

(3.6.50)

paralleling expression (3.6.28) for the fractional Fourier transform matrix F ~ ( f s ) . The parameter fs retains its role as the focal length of the Fourier transform unit associated with the fractional Fourier transforming systems with c~ - -t-1, entering (3.6.50). The transformation acted by B(r, fs) produces therefore the further scalings by f s 1/2 and f l / 2 with respect to that acted by B(~-).

Linear Ray and Wave Optics in Phase Space

150 3.6.3

Generator matrices of pure rotations and squeezes

We frame the matrices S 1 and K,~ within the proper algebraic context. Defined a,ccording to relati(ms (3.6.5) and (3.6.35) for a unit value of the length t)a.rameter .fs, the ma.tri(:es K1 a.nd K~ ha,ve been re('ognized as the generators of pure r()ta,tions a,nd S(l~W.ezes in the t)ha,se plane, and a.sso(:ia.ted with optical systems (tist)la,ying r(~st)(;('tiv(@ a (tefinit( ~,f()(:~si~g a,n(t (tefi)(:l~sing })ehaviour. I~(~t l~s rewrite t,l~e l)erti~(nd, (;xl)ressi()~s, i.e. K, - ~~ ( K

+-K, ) ,

K~ - ~(U

-K,),

(3.6.51)

witl~ tl~e (;xI)li('it ~mtrix f<)r~s <)f K a~
89 1 2_

-~p

generating matrix

K

--~ 1

~q

2

~

- ~(() ..

1[0

---+ U ~ - ~ t .

-l

transformation matrix

~(,)

F"

1~

o~

(,20 K~

__ /(cos0 si~0 ) ~,- sit~ 4) cos 0

B (r) - (

si,,h r

c(,sl,

'

wlficll alh)w us t,()(:llara~'t(Mz(~ tile 89 (p2 + q2)_~elle, ra.t(~(l syst(~lllS 1)y (n'tll()~onal lmilll<)
[K+, K ] - 2K:,,

[K• K:,] - •

(3.6.53)

hi a like mam~(;r, the set ()f l~mtri(:e,s {KI, K~, K:~} a,ls()('h)se l~ll(ler (:()mnmta,ti(m a(x:or(ting t()

[K1,K~] - U:,,

[K,,K:,] - - K ~ ,

[K~,K:~] - - K ~ ,

(3.6.54)

Theref()r(~, like {K_, K~, U:, }, tlm set {U~, U~, K:~ } is a t)asis f()r ,~p(2, R) as well; it is, in flu:t, easy t() verify that a,ny matrix in the algebra, can lmiquely be written a,s a, linear rea,1 (:()~nt)i~m,tion of K~, K~ a,nd K 3. Finally we mention that the ()ne-pa, rameter subgroups generated by K~ and Ka, formed rest)ectively t)y the matrices B(T) a,n(t S(m), are reported in the group theory as hyperbolic subgroups, whilst that genera,ted by K1, formed by the pure rotation matrices F '', is called elliptic subgroup. Also, the subgroups generated by U and U + , respectively formed by the m~trices T(d) and L(f), are referred to as pa~ubolic s'ubg'rvups. Tile reader Ilmy gain a feeling for the difference between matrices in these subgroups, by noting that, unlike for the elliptic subgroup (see Eq. (3.6.15)), powers of the matrices in the hyperbolic and parabolic subgroups grow indefinitely and never reproduce the identity; the hyt)erbolic and parabolic subgroups are in fact noncompact [2].

The Group of the 1D First-Order Optical Systems

3.6.4

151

Rotations and squeezes as building blocks of any symplectic transformation

As a conclusion of the present investigation of the rotation and squeeze generating matrices, we prove tha,t any optical (symplectic) matrix M E Sp(2, R) is uniquely expressible as the product of a B- and an F~-matrix. In symbols, M

B(T, 0)F ~,

-

(3.6.55)

with the entries At B, C, D on one side being related to the two squeezeparameters 7-, 0, and the rotation angle r c~7c/2 on the other side as cosh z ~t'-"~'jfant'9A% _

89v / A 2 + B 2 + C 2 --t- D 2 + 2

2(BD+AC) A2+Bg._Cg_D 9.

tan

(3.6.56)

(t:) - - B - C -A+D

The inverse-order factorization, namely M - F~'B(T ', 0'),

(3.6.57)

is as well a,ctuable with ' -

~,

r

-

r ,

0' -

0 + r

(3.~.5s)

Equations (3.6.55) and (3.6.57) yield the polar decomposition of symplectic matrices, in general involving a real symmetric positive-definite unimodula,r matrix and a real skew-symmetric unimodular matrix, corresponding to B(T, O) and F ~ in our case [3.6].

3.7

Iwasawa representation of optical elements

LSF

~

synthesis

In w 3.5 we have described the representation of optical matrices as ordered products of elements drawn from the K+-, K a- and K_-generated subgroups. Inspired by the original suggestion by Wei and Norman [4.1], the LST decomposition is widely exploited in quantum mechanics in connection with the solution of the Schr6dinger equation with an underlying SU(2) or SU(1, 1), i.e. angular m o m e n t u m and harmonic oscillator-like, symmetry. However, as remarked, the LST optical arrangement is not global as it is not applicable to every optical matrix, although it is definitely suitable to propagation problems. In contrast, the representation of optical matrices as ordered products of three elements drawn from the K 1 - , K a- and K+-generated subgroups is global in the sense that it applies to every symplectic matrix and is unique in the sense that the parameters of the representation are uniquely determined in terms

152

Linear Ray and Wave Optics in Phase Space

of tile entries of the given matrix. In fact, the LSF (~ a r r a n g e m e n t involving thin lenses, pure magnifiers a,nd pha,se-t)lane rota, tors, we will describe here, resort to the Iwa,sa,wa de(:omposition theorem, whi(:h is a, basic reslllt of the Lie gr(nq) theory valid for every non(:onq)a.ct semisimtfle Lie grollp. For a, st)e(:ifi(: (tis(:ussion ()f th(; Iwa,sa,wa de(:omt)osition t,he()relll within the general (:()llt,(;xt, of tim Lie grolq) t,lmory, we address t,h(; rea(h;r to [7]. We will sll()w tierc l,tl(~ fcasil)ilit,y ()f l,lm Iwasawa r(;t)r(;s(;llt, at, i()ll ill l,llc (:asc ()f interest, I,() 11s, i.(;., f()r t,lm ()t)ti('.al llm,tl'i('.('~S l)(;hnlgillg l,() t,lie sylllt)l(;(:l,i(: gr(nlt) b~p(2, R). Ev(~ry ray-transfer llml, rix ('a,ll t)(; (h;('.()lllI)()s(;(t illl,() tllc fa,(:t,()r(',(l 1)r()(tllct, ()f a, leIls, a. t)()sitivc-(t(;filfit(; I)llr(; llm.gllitier a,ll(l a, tflm,s('~-l)lalm l'()l,a,t,()r a.s

(~}, l/))

_ L(T))S(,,,,,)F,,"

(3.7.1)

Tl~e I'(~lati()lls l)(~tw(;(:II tll(~ vari()lls l)a.rml~(~t(~rs ()l)ta,ill ill t,llc fi)l'll~ A - ',,~ ('()s (/,, B - ',,~ sin r ' (:()s r (~' - - ' , , ~ 7 ) ('()s r - •I l l si,~ 4), D - - , , ~ 7) si,~ 4) + ,,--;

(3.7.2)

whi(:l~ art; st)lv(:(l f()r 7 ), ,,~ m~(l (/) l.()giv(: a

+

,,.,,

II, l~;I.y l)(; (:()llV(;lli(;lll, I,()('~l~l)l()y a ~()~ ~u~it Wd~(; ()f 1,1~('sta~(taI'(l f()('al l(;l~gth fs i~ tl~(; fl'a.(:ti()~al Kn~ri('r trm~sf()r~ ~ a t r i x e~teri~g (3.7.1). I~ tlmt (:as(:, .fs ~la,y t)e left, as a fl'(;(; l)aral~mt(;r, wl~()s(; ~m.g~fit~t(l(; (:an fl'(;(;ly l)e ('l~()s(;I~ a(:(:()r(ling t,() ()l~r (:(n~v(u~i(n~(:(: i'(:lativ(:ly t() t,l~('~Sl)(;(:iti(: t)r()t)l(u~ ~ ( t ( ; r ('xm~i~m.ti()n Is]. ~I~,ki~g (l~m a,('('()~t ()f t,lm ext)()lm~tia.1 r(~I)I'eSe~tat,i()~s ()f t,l~(~ ~mtri('.(~s L, S a.n(1 F '~, w(~ (:a,n Cxl)r(:ss th(~ I)r()(l~(:t i~ (3.7.1) int() tlm fiu~(:ti()nal f()nn (A //)D _ , "PK, (~.~K:,,~.~OK,,

',,, _ , .~/2.

(3.7.4)

E(t~m.ti()~ (3.7.1) exI)r(~sscs in tlm ()t)ti(:a.1 (:ontext tim g(u~eral I)r()I)erty of every ~o~('()l~q)a.(:t s(,~isin~t)lc Lie gr(),q)(in t)arti(:~llar, ()f Sp(2, R)), whose eh;ments (:m~ ~mi(tuely })c (h~('()Int)()sc(t int() the pro(tu(:t ()f tier(;(; fa.(:t()rs, taken ea,ch ()~e fr(n~ a, nmxinm,1 ~filt)()tent gr()ut) N', a, lnaxi~m,1 At)elia,~ s~fl)group A and the m a x i m a l ('()mt)a,(:t subgrout) ]C. For Sp(2, R) the subgrout)/C is formed by the ()rtt~()gonal ma.tri(:es F (~, whilst N" and A can be chosen to comprise respectively the lower-triangular matrices L with unit entries on the diagonal and the mutua, lly c o n m m t i n g t)ositive (tiagona,1 matrices S. Hence, the "A/A/C' decomt)osition takes the form (3.7.1). We note t h a t the three factors belong respe(:tively to the parabolic, hyperbolic and elliptic s u b g r o u p of Sp(2, R). Figure 3.18 s u m m a r i z e s the group-theoretical and optical content of the LSF" (te(:omt)ositioI~ (3.7.1), inirroring Fig. 3.11 relative to the LST decomposition. Notably, as evidenced in the figure, the ray t r a n s f o r m a t i o n acted by any first order optical system can be interpreted as a definite sequence of a phase-plane rotation, an overall scaling and a p-shear.

153

The Group of the 1D First-Order Optical Systems

I3o

Fl i i i I

~

I

i

>

i

i

:

n

.o 4~

I

I

AD-BC= I

i

i

i

i

,

0

i

m

fractional Fourier transformer

positive magnifier

thin lens

m 9

r lower-triangular unit-diagonal matrix

O_ "~ 0 L

unit-diagonal matrix (5;p(2,P,) h y p e r b o l i c s u b g r o u p )

anti-symmetric matrix (5'/,(2,R) elliptic s u b g r o u p )

(,~;p(2,R) p a r a b o l i c s u b g r o u p )

o~

._o > r

o-/..o.)<

~.

C

d)

AD-B('=

r" O.

D

I rotation

overall scaling

p-shear

F I G U R E 3.18. I w a s a w a s y n t h e s i s of a n o p t i c a l m a t r i x .

3.8

Canonical and noncanonical representations of symplectic matrices

We complete here tile investigation started in tile previous chapter concerning the relation between quadratic polynomials and optical systems. We consider indeed the general quadratic function in q and p (see also w167 1.5.2 and 2.3) 1 1 Q(q, p) - -~bp 2 + cqp + -~aq 2.

(3.8.1)

The representative traceless matrix is

--a

--c

(3.8.2)

Linear Ray and Wave Optics in Phase Space

154

which is expressible in t e r m s of the algebra basis { K

Q-aK++2cK

a+bK

, K + , K a } as

.

(3.8.3)

It is easy t() (:omt)~te the ext)o~mntia,1 flmction of Q, tlms (:(mq)rising all the (:orrest)onden(:es establishe(t in (3.1.2) and (3.6.52) into one fon~" generating fu nction

51 t)t)2 + (:qp + 21aq2 a,b,c,

---+

generating matrix

transformation matrix

Q-

.

,

,.

~)(<)

9

( ~ R

(:).s.4) Tl~e ead,ries ()f t,l~(; gl'{)ll t) eh;lll(;llt, ~'< Q,
A(<) - r + ~ si,,1,(A<), C(() - - ~ sil,l,(A(), I)(()

--

/3(() -- -~ sil,l,(A(), t:()sl,(A()- ~ si,,l,(A().

(3.8.5)

wllere A is t,llt: Stllla.rt'~ rt)t)l, t)f (let Q: A = v/r .2 - ab. Ill tlle gr(nlI)l,lw()ry tlle real ~n~fi)ers (a, b. r are k~()w~ as tim "t'()(n'(li~mte" ()f 1,t~(; ca'n.o'n.i(:a/ r(;l)r(:s(:~l,al,i(n~ ()f gl'()ll I) ('.l(u~(;~d,s ()f ('Xl)(n~(;~l,ia,l l,yl)(;, i.e., rel)rese~ll, a.1)h; i~l 1,11('~fin'~l ~'(q wil,li l,ll('~ g(;~l(;rat, ing ~lml, rix Q ])ei~lg (;xl)ressed i~ t,erI~lS ()f tl~(,, a.lg('J)ra ])asis { K , K ~ , K a }, a.s sl~()w~ i~ (3.8.3). As ~(;~ti()ned i~ w 2.3.1, l,l~('r(; a.r(; I)(wl, i(n~s ()f 1,t~(; sy~l)h;(:l,i(: grr Sp(2, IR), w|~()s(' ('~l('~x~t;~l,s (:a,n ~()I, ])(; r(;l)r(;s(ud,(;(l as (;Xl)()n(ud,ia,1 l~ml)t)ings ()f ('~l(;l~(ud,s il~ its a lg(;|)ra ()~ t,t~e ()tl~(;r l~a,~(l, i~ w 3.5 we lmw; (h;scril)e(l (,1~{; I,S'I' r{;l)r('se~d,at, io~ of symI)le(:ti(: nmtri(:es, wl~ir h)(:ally a I)I)li('s t() })()th exI)()~(u~tial an(t n(m ext)(mential tyl)(; ~m,l,ri(:es. It inw)lw;s three fa,{:t()rs ta,ke~ (;a,{:l~ fl'(n~ the ()net ) a r a m e t e r sul)gr(n~I)S ge~mrate(1 |)y t,l~(; a,lge|)ra, basis elen~e~d,s { K , K + , K a }. In the gr(nH) tlm()ry, th(; LST-lik(; r(;alizati(m is kn()wn as a noncanonical r(;pr e s e n t a t i o n ()f symt)le(:ti(: l~m.trices. T h e relevant t)aran~e,t(;rs (7~, '~t,d) can therefore |)e intert)rete(t a,s the (:oor(tinates ()f the (local) n()n (:a~()nica,1 L S T representa,ti(m of the ext)onential-tyt)e grout) element. Evidently, the tra(:eless m a t r i x Q is equally expressible in t e r m s of the a,lgebra elements {K+, K a , K 1 } , which p e r t a i n to the I w a s a w a r e p r e s e n t a t i o n discussed in w 3.7. In fact, one has

Q = aK+ + 2oK a + 2bK1,

(3.8.6)

with, of course, a=a-b,

b=b,

c=c.

(3.8.7)

T h e n paralleling the relation between the coefficients (a, b, c) and the p a r a m eters (7), m, d) of the Wei-Norman representation, the coefficients (a, b, c) and

The Group of the 1D First-Order Optical Systems

155

the parameters (7), m, r of the global LSF ~ representation (3.7.1) are similarly interpretable as canonical and noncanonical (global) coordinates of the exponential-type group elements. The reader may deduce the relations between the canonical and noncanonical coordinates (a, b, c) and (P, m, d), as well as (a, b, r and (7), m, r using (3.8.5), (3.5.2) and (3.7.2) [4.3, 4.6]. 3.8.1

A bit on the non-exponential type elements of Sp(2,IR)

Let us summarize the basic results we have estabished up to now. The symplectic group Sp(2, R) is a three-parameter Lie group, which consists of all 2 x 2 real unimodular matrices S. The unimodularity of elements in Sp(2, R) is equivalent to the symplectic condition SJS T -a,

(3.8.8)

where a - (_~1 {)) is the unit symplectic matrix. The Lie algebra sp(2, IR) consists of all 2 x 2 real traceless matrices. It is the algebra of Sp(2, R) in the sense that the exponential of any matrix in sp(2, IR) is a matrix of Sp(2, IR); namely E=c r QESp(2,R)

VQEsp(2,~)

and

(ER.

(3.8.9)

However, not all d e m e n t s in Sp(2, R) can be interprete(t as exponential of el(;ments in the relevant algebra sp(2, IR). We may convince ourselves of this by a simple reasoning. Let us consider the trace of any exponential-type matrix in Sp(2, IR). Being

--Z7 j=0 ,

(3.8.10) ,

the linearity of the trace allows us to write TrE-

(X)

(X)

j=o

j=o

~ ~Trqr - ~ r ~ Trq L(2j)!

2r + (2j+l)! ~2jq-1 TrQ2J+I],

(3.8.11)

where tile even and odd power terms have been separated. Noting that Q2j _ (_ det Q)JI,

Q2j+I _ ( _ det Q)JQ,

j - 0, 1, 2, ...

(3.8.12)

on due account of that TrI = 2 and TrQ =0, we end up with (X)

TrE - 2 ~ (-)J X ~ (det Q)J = 2 cos(C v/det Q). (2j)! j=0

(3.8.13)

Linear Ray and Wave Optics in Phase Space

156

Tt~en comprising the case that (let Q > 0, yielding the circular cosine ( - 1 < cos x _< 1), as well as the case t h a t det Q < 0, yielding the hyperbolic cosine (cosh:r _> 1), we can (:onclude that all the single-exponential type inatrices E in Sp(2, IR) satisfy the relation

T,. E _> - 2 .

(~.s.14)

Evi(lr sy~I)h;(:l,i(: ~mtrir witl~ trar < -2 r m~r sr syl~t)h;r nmI,l"i(:cs l,]m,l, (|() ll()[, a,l'is(; fr()l~l (;Xl)()]le~d,ial l~ml)t)i]~gs ()f ch;~l('~ld,S in l,]l(; a,lgel)ra exist. As 1)asi(' (;xamt)lcs, wc lnay ret)()rt tll(; llla,t,l'i(:(;s (set; als() Prt)|)le~n 11)

(,()

_

(, ,,)

a,llIl

(," ~-()

-1

(3.8.15)

'

wld<'.l~ i~ fi~r (:m~ l)(; ()l)tail~e(l as tim l)r()(l~zc:t, ()f t,w()eXl)()~e~l,ia.1 l,yt)(; ~m.trices" ~ _~

3.9

- r

'

'.

(3.8.16)

Integrating the equation for the ray transfer matrix

W~; r~',~'~nlsi~h;r ll~;r~; t,ll~'~Im)l)hnll <)f s<)lvillg llalllilt
fin" I,IH: <)l)ti(:a.1

( A(z, zi) I~(z, zi) )

M(z, zi) -

C(z,

z.i)

(3.9.1)

l)(z, zi)

'

fin" t,lie l)r~l)a,gal,i~ni lln~ler l,ll~; (tlm.(lra.l,ir IImiiill,~)nimi (3.1.1), where l,ll~; lne(lillm t)ara,,,eters 't,,, (z) a,,r 't,~ ( z ) a r e in gr z-(lel)e,,,le,~l, fimc:ti,),,s. It, lms ])~'~'ll I)r~)v~'~l ill .~ 1.7.3 thal, 1,11r eigries A, B, C, D ~)f M ( z . zi) are {)t)ta.ine{l as s~)ll~ticn~s t{) tl~{; set ~)f first-r162 {tiffi'xentia.1 eq~m,ti{ms A' =

c~ ",,

A(z,~, z,~) -

1,

B' -

'

C' - -'t~,.~A, C(z.i, z.i) =

D '",,

'

B(zi,

zi) --

D' = -,~,.~B, D(zi,

0;

O,

zi) = 1;

(~.9.2)

the t)rix~e ~le~{~t,ing ~teriva,tive wit, h rest)cot to the axia,1 coor(timtte z. Interestingly, the ot)tic:al equivah'~nces established in the previous sections, in partic:~flar the LST parametrization, suggest alternative, and possibly further enlightening and effective, integration approaches to the set (3.9.2). Since A(z~,z~) = 1 and D(z~,z~) = 1, the contimfity of the elements of M ( z , zi) a,s functions of z, assures t h a t m(z, zi) > 0 and similarly D(z, z~) > 0 in a neighl)ourhood of the initial position zi. We can therefore resort to the factoriza.tion (3.5.1) in terms of the matrices L, S and T, thus expressing M(z, zi) in terms of the z-dependent parameters 7), s and d as M ( z , zi)

-

~ 7~(z)K+

cs(z)K3ed(z)K-.

(3.9.3)

The Group of the 1D First-Order Optkal Systems

157

As prescribed by relations (3.5.2), the magnification factor re(z) - c s(z)/2, the focal power P ( z ) and the reduced distance d(z) are linked to the matrix entries A, B, C, D through A - e */2,

C-

--7) e s/2

,

B -

de ~12.

(3.9.4)

Therefore, exploiting Eqs. (3.9.2) it is easy to obtain the set of differential equations for 7), s and d as 7),

- ~ 1 7)2 - n ~ - o , 2

~' + ~

p(z~)-o,

- o,

~(~)

d' - l e-* - 0

- o,

(3.9.5)

d(zi) - O.

n0

Equation (3.9.4) represents the link between the paralnetrizations (3.9.1) and (3.9.3) of the optical matrix M. Indeed, once the explicit expressions for the overall matrix entries are known from (3.9.2), the decomposition (3.9.3) is uniquely deterufined through the algebraic relations (3.9.4). Conversely, solving (3.9.5) for 7), s, d allows us to evaluate the matrix entries by (3.9.4). Moreover, limiting ourselves to consider in (3.9.2) the equations for (A, C), as the entries (B, D) ()bey formally similar set of equations, we see that they yield two independent homogeneous linear equations of the second order for A and C; namely,

A" + ~'~ n0

+ 21~,0Z A -

c"-

+ ~c

~c' /~'2

n0

O,

A(zi)-

- o,

1~ A ' ( z i ) -

0 (3.9.6)

c ( ~ ) - o c ' ( ~ ) - - . . . (~) ~

The entries B and D obey the same equations as A and C, respectively, with 1 ) and D ( z i ) - - 1 the proper initial values being B(z~) - O, B'(z~) no(Z~ D'(z~) = 0. Solving, for instance, the first of (3.9.6) for A and B, allows us to obtain C and D simply by integration of the relevant equations in (3.9.2). Likewise, inspecting Eqs. (3.9.5), it is evident that solving for 7) allows us to integrate for s and then for d. In this connection, the equation for 7) deserves some comments. It is structured as the generalized Riccati equation [9] dw dz

= a~

-q- a l ( Z ) W

whic, in fact, with ao(Z ) - n 2 ( z ) , equation for 7) . Setting

-[-

al(z )

a2(z)w 2, --0

w(zi) --

gild a ~ ( z ) -

wi.

(3.9.7)

1 turns into the no(Zi)

Vt

-no(Z)--, (3.9.8) v the equation for 7) is transformed into the homogeneous linear equation of the second order for the function v(z) 7)

--

158

Linear Ray and Wave Optics in Phase Space t

/t2

7t o

7t0

'1/'+ n--~~ + ~ v -

0,

(3.9.9)

the s a m e a,s t h e e q u a t i o n f o r A in (3.9.6). T h e a,t)ovc cxt)rcsscs a genera,1 p r o t ) c r t y of the Riccati cqlm,tio~l (3.9.7), which is tlmlc(t int() a, h()m()gcn(~()lls linca,r (~(tlm,ti()n of the sc(:(m(t ()r(tcr 1)y the subst, it, lit, i()11 w ",uv' (~),,,. C()llV('~rsely, . a,ny liIlea.r s(;c(nl(t ()r(h;r (;(tlm,ti(m (:a,n bc giv(;11 1,11(; I/i(:(:a,ti f()I'lll l)y a sllital)h; (,l'a,llsf()l'llla,(,i()ll ()f 1,11(; (h;l)Cll(h;llt wlrial)lc. As a, (:()n('lllsi()11, we llmy say t t m t tll(; tra,('ki~lg ()f a l)araxial ray I)r()t)a,ga,t, iIlg (,111"()11gll a,]l illl1()n1()gtU~(:()l~S ~(:(li~lnl (I(:I)(U~(Is ()~I (,lie s()l~ll, i(n~ ()f a, ]~()~1()g(:~1(:(n~s lin(:ar (:(t~m,(,i()~l ()f (,11(: s(:('.(nl(1 ()r(l(:r ((3.9.6) ()r (:).9.0)). Fr()lll i(, a.ll (,lm 1)a,ram(:t(:rs, whi(:h (:()~lq)l(:(,(:ly ('lm.ra,(:t(:riz(: (,11(: r(:l(:wu~t ()t)ti(:a.1 ~m,(,rix, i.(:., (,lie (u~tri(:s A, H, ('~, D ()r (,t1(: flu~('(,i(n~s 7), s, d ()f tll(: I)()ssil)ih: fa,(:t()riza,ti()~l i~ (,(:rills ()f the ~lm(,ri(:(:s L, S, T , (:a.~ s(,ra,ig(,l~fi)rwa.r(lly l)(: ()l)(,ailm(l. W(: (:l~ll)llasiz(: y(:(, a.ga,in tha,t th(: I)ara.~1(:(,(:rs P , ,~, d l)(:l"(,a.in (,() a, h)(:a.l r(:l)r(:s(ulta,(,i()~1 ()f (,11(: ()l)(,i(:al n m t r i x M ( z , z;), whilst (,11(: (:lH,ri(:s A, H, (:, D (,() a gh)l)al r(:l)r(:s(:~lta,ti()n. I~ tll(: ~l(:xt 1)ara,gral)ll w(: (lis(:~ss i~1 (h:tail t, ll(: si~l)l(: (:as(: wl~(:I'(: tl~(: ~ll(:(li~m l)ara,n~(:t(;rs ~t()(z) a,~1(l ~,~ (z) a,r(: z-in(h:l)(;l~(h:n(,. In (,Ira(, (:a,s(:, th(: s()h~(,i()ns to l';(IS. (3.9.6) (:m~ I)(: giv(u~ i~ (:xI)li(:it f()ni~. 3.9.1

Ray-tT'acing in a q'uad'l'atic gradcd-ind(',:r mc, d i u m

'l'llc ray t)rt)I)agati(nl al(nlg a (tlla,(trati(: grath',(1-ill(lt;x lll('~(Iilllll, (l(;s(',i'i|)(;(t |)y th(; I)a,ra,|)()li(: I)r(Jfih; ()f 1,11(; rcfl'a,(:tivc in(h;x [10] 'n(q, z ) -

'n() -

l =_,r~.~q 2

(3.9.10)

whcr(~ the ()ll-axis va,llu; n o ml(t the trallsv(;rs(', (tcriva, tiv(', 'It,2 a,rc l)()(,ll (:()nsta,nt w i t h the axial ('()()r(lilm.t(: z, is cxt)li(:itly s()lwdflc("). In fa,(:t, wit, t1 z-ill(tot)trident me(titan t)a,ra, Inct(;rs, E(ts. (3.9.6) tak(; th(; form

dz2y + - - y - (), '/t 0

(3.9.11)

w i t h the initial va,lues t)cillg assigned a,ccording to the specific entry, A or C, we are solving for. T h e cha,racter of t h e s o l u t i o n of the at)ove e q u a t i o n , a,nd hence the beh a v i o u r of t h e light ray in tim m e d i u m , (tepen(ts on t h e sign of n e. Accordingly, w(; m a y d i s t i n g u i s h bctw(;(;n focusing a n d d c f o c u s i n g m e d i a . As already noted, the i n d e x flmction (3.9.10) is valid over a limited range of the transverse c o o r d i n a t e q, a p p r o p r i a t e to the linear a p p r o x i m a t i o n of our concern, as it b e c o m e s u n p h y s i c a l if n b e c o m e s less t h a n the refractive i n d e x in the v a c u u m , generally taken as unity.

159

The Group of the 1D First-Order Optical Systems Focusing quadratic medium

A focusing index profile, for which n 2 > 0, exhibits a maximum on the optical axis and decreases radially from it. The general solution of Eq. (3.9.11) writes in terms of the harmonic functions as y(z)

:

C,

C 1 COS ~ -31- C2 s i n

(3.9.12)

where c 1 and c 2 denote two arbitrary constants, determined by the initial values, and ( is the dimensionless axial coordinate ~ ( z ) - ~-n-Z(z%

(3.9.13)

- zi).

Determining c I and c 2 according to the initial values appropriate to the entries A, B, C or D, we obtain A -- cos ~,

~

__

sin ~,

1

(3.9.14) C = - v / n 0 n 2 sin ~,

D = cos ~.

It is apparent that the field ray corresponds to the cosine solution of Eq. (3.9.11) as the axial ray to the sine solution. The optical ma,trix M(z, zi) writes therefore

(

Mfo r ( z, zi ) --

cos~

~

1

- v/no n e sin ~

siIl~ )

cos ;

(3.9.15) '

and the ray variables change with the a,xial coordinate according to q(z) -

qi c o s ~ +

P~

sin~, (3.9.16)

p(z) = - qi v/n0 n2 sin ~ + Pi cos ~. The ray oscillates back and forth ~cross the axis with the spatial period Z-

27r~,

(3.9.17)

known as the p i t c h (Fig. 3.19). The maximum transverse excursion of the ray to

qm x

+

m

xim.m

ray

forms with the axis along its path is / ) m a x ~ P m a x / n o = V / n 2 / n o q m a x . If the width of the material is greater than 2qmax, the ray remains confined within the medium, which then acts as a light guide. The matrix (3.9.18) has the same form as the fractional Fourier transform matrix (3.6.6). In fact, we can write Mfoc(Z, zi ) - F~(Z)(fs) ,

(3.9.18)

Linear Ray and Wave Optics in Phase Space

160

fli

flo

i (qi,Pi

~

i~ \

i

/ \

(qo,Po)

_

r

FI(~,URE 3.19. Ray t r a j e c t o r y i~ a foc~mi~g ~ n e ( l i ~ .

where the stm~(la,r(t f()ca,1 hu~gtl~ .fs is i(tentifier a,s fs ~

a,n(t tim or(ter ~ ra,I~ges r a,(:(:()r(ting t() relati(n~ (3.6.7), wlfir (:(z)

-

(3.9.19)

1

witl~ t,l~e axial gives tl~en ~

z--z

(:r

te z

as exI)e(:te(t

i

As alrea,tly ll()l,e(l, ill fat:t, a, (tlm,tlra,tic gra,tlett-i11(lex fi)t:llsi~lg llle(lilllll t)rovitles the ()t)tit'al sysl,eIll I,() iIllI)lement the fl'at:titmal Fr tra,nsf()rlning ot)era,tion with a. (:(mtimit)llsly ranging ortter. T h e r ()f the ra,y I)a.th a,(:ross the ot)tit'al a,xis (:orI'eSl)()ii(l to the (:()IltilnlOllS rota,titnl of tlw, ret)resent,a,tive I)oint ill l,lle l)tm,se, l)lmle, wtli('ll dest:rit)es tim ellit)se (3.6.9) (letermine(t t)y the mc'(ti11111 Imra,meters 'n,0 a,n(t 'n2, a,n(t the initial (ta,ta,. As the ()rr ~ is r lillke(t to the t)rr wt,ria,t)l(; z, tim system nmu a,('t a,s a, F()lu'ier tra, nsf()rmillg s y s t e m a,t the t)r()t)er (tist, a,ll(:e L F fl't)m the input screen. In fa(:t, if L v is sllch t h a t a ( L v + z~) - 1, a,n(t hen(:e Lv

Z -

2 ~/~,,, V "

(3.9.20)

4'

the p r o p a g a t i o n t h r o u g h th(; m e d i u m over the distance L v from the input pla, ne results in the o r d i n a r y Fourier t r a n s f o r m a t i o n with foca,1 lenth fs- This establishes the possibility of optically realizing a Fourier t r a n s f o r m a t i o n as well by a piece of a quadra,ti(: m e d i u m of proper length L~. At distances c~Lv from z i , tile s y s t e m behaves as a Fourier transformer of fractional order c~. T h e n we can imagine the g r a d e d - i n d e x m e d i u m as formed by a c o n t i n u u m of planes from zi along the optical axis; on each plane at z we z Lv - z ~ of the object can observe the fractional Fourier t r a n s f o r m of order (~ _

at zi. At planes regularly placed from the i n p u t plane at distances LF, 2L~,

The Group of the 1D First-Order Optical Systems l-I 4(h

]7411 [1

rl4hll

I

/ -

161

4h~

2

]7 4h,

i 1}

3

~x/7770n 2

Z

[F(.l.)l,lh,

:

[V(./)]

4h

=

i

I =

,

I

,

e

I

4

F(./)

[F(.f)]4h,'~

:

[F(.f)]

,

e 4

4

hl~ 2

=-

.__._[___ '

I

z

= F(-.f) ',

[F(.f)

,

--M-.--

4

=

I

z

I

4

FIGURE 3.20. A quadratic focusing medium is like a sequence of Fourier transformers. 3Lr, and so on, we may detect the successive powers of the Fourier transform of the object; specifically, at distance j L ~ we observe the j - t h power of the Fourier transform of the object (Fig. 3.20). For j = 4h, we recover the identity matrix and hence the input signal, according to the periodicity of the Fourier transform operation (a). The overall medium can be understood as a sequence of identical Fourier transforming configurations, L~ fixing the z-size of the Fourier transform units, by which the medium may be thought as composed. D e f o c u s i n g quadratic m e d i u m

For a, defocusing index profile: n 2 < 0, which exhibits a, minitnum on the a,xis a,nd increa,ses fa,r from it, Eq. (3.9.11) is solved by hyperbolic sine a,nd cosine functions. Hence the optical matrix writes as ( M d e r ( Z ' Z i ) --

cosh ~ V/--rt0•2 sinh{

1 % sinh~) v,,_,~o

,

~ ( z ) - i _2a. (z - zi)

cosh{

%

(3.9.21) which has evidently the same form as the squeeze matrix (3.6.36), being indeed Mdef(Z , Zi)

-

-

B(~, fs),

.fs -

1 v/_% ~2 9

(3.922)

As the height of the ray with respect to the axis increases while propagating from the input plane, the representative point in the phase plane moves along the branch of the hyperbola (3.6.39). The propagation along a defocusing quadratic medium physically realizes the three-step phase-plane transformation (3.6.50) involving squeezes and scaled rotations. a More precisely, as it will be shown in w 5.4, the Fourier transformed signal is affected by a phase factor, which obviously cannot be accounted for within the ray-optics description of the performance of the optical system.

Linear Ray and Wave Optics in Phase Space

162

B e i n g (:oshsc > 0, t h e W e i - N o r m a n d e c o m p o s i t i o n c a n safely b e a p p l i e d t h r o u g h t h e w h o l e { (i.e., z) d o m a i n . T h i s is left as a n e x e r c i s e to t h e r e a d e r .

3.10

Summary

It ha,s |)cell t)r()vc(1 tha, t e v e r y first-()r(ter ()t)ti(:al systcxll (:a,ll t)c l u l ( l c r s t o o d a.s a s('xtll(',~l(:(', ()f tllill l(',~lscs s(',l)a,ra,t(;(l l)y ff(;('~-I)r()t)aAati()ll s('~(:ti(ms, visllaliza.t)lc a S('qll('~ll(:('~ ()f v(;rti(:a,1 a ll(l ll()l'iz()Id, a.1 stl(;a.rs in tilt; t)llas(; i)lmle view. Tile ()t)ti(:al syst,enls, ~(',ll(;ra,te(l l)y t,]l(', at,tra,(:tiv(; a,]l(l l'(;1)lllsiv(', ()s(:illa,tof like ttmlfilt()llia,llS 1721)2+ ~q21' a,ll(l :jp I 2 _ }1q2 , lm.vc l)(',(~,ll (',Xa,lllillc(1. q'lw, f()rnlcr is rc(:()glfiZ(',(t as I)erf()nldllg s(:al(',(t r()t,a.ti()lls ill l,ll(', I)lm.s(; l)la,~(;, wl~(;r(',a,s t h e l a t t e r a,(:ts as a s(p~(',czing trm~sf()rnm,ti(m. T w o i~t,cI'eStil~g rc, a liza.ti()~s (~f ()I)ti(:al ~m,ti'i(:e,s l~a.w', 1)e,(;~ (lcs(:ril)e,(l i~ (:or-

,'(,st)o,,(l(;1,(:(, wit,l, t,l,c tw,)il.lg,~,},I'a. ],a.sis { K

, K,,

K:,} a.,,(l { K

, K , , K:,}.

( ) h e realizati()~l, wlii(:ti al)l)li(;s l()(:a,lly Ii(',a,r tll(; (',ld,ra~l(:(; s(:re(;]l, syld, ltcsises t,l~e ()l)ti('a.l ~ m t r i x as a s(;(ll~(;~(:(; ()f a. (1)()sitiv(; ()r ~l(;gativ(;) fl'(;(~l)r()l)a,ga,ti()n s(;(:ti()n, a. I)()sitiv(;-(l(;linit(; l)llr(; ~m,guili(;r m~(l a, t h i ~ l(;~s, a,u(l a(x:()r(li~g;ly is vislm,liz(;(l i~ t,l~e ()I)ti(:al 1)lm,s(; I)la.~(; a,s a l~()riz()~d,a,1 sl~(',ar f()ll()wc(l I)y a s(:a,ling; a,n(1 a, v(',l'l,i(:al sl~(m,l'. TI~(; ()t,l~(',r r(;a.lizati()~ all()ws ~s t,() (l(',s(:ril)(', t,l~(; syste~n a.s a, s(',(t~(;l~(:(: ()f a fl'a,(:ti()~a,1 F()~rier I,l'a,]~sf()rl~i]~g~ sysl,(;l~, a, I)()sitiv('-(l(;fi~fitc t)urc lna,g;l~ilier a,l~(t a, t,l~i~ lel~S, m~(t h(',ll(:e is t)i(:t~rc(I i~ tl~c t)ha,s(', l)la,n(; a,s a, r()t,a,ti()~ f()ll()w(;(l |)y a, s ( : a l i ~ m~(1 l,t~(',~ t)y a v('rti('a.1 st~(',a,r. In t,ll(', ~(;xt, (;lmt)t,(',r, tl~(', ligl~t, t)r()t)a,ga,ti()l~ will 1)(', r(;(:()~si(tcrc(l witlfin t h e fram(',w()rk ()f wav(; ()t)ti(:s. W e will i~v(~,st,ig;a,t,(~ t,h(', ra,y-wa,ve (:(mne(:ti()]~, w()rking ()~d, t h e ()I)(;rat()r r(;t)r(',s(;~ta,ti()l~ ()f tt~(', litmar 1D ()t)ti(:al systcn~s tl~r()ugh a s()rt ()f (t~m,~xtiza,ti()~ t)r()('(',(il~re, sit~filar t() tl~a,t a,t)t)lic(i t,() (:la,ssi(:a,1 ~x~(',(:ha,ni(:s t() yicl(t (l~m~d,~m~ m(',(:tm,~i('s. R a y vc(:t()rs will t)e rct)la(:(',(t |)y w a v e fim(:tions a,n(t t,h(', ~mi~()(lula,r nmtri(:(;s ()f t,h(', symI)lc(:ti(: g;r()ut) ~p(2, ~) will t)c rcpla, ced t)y t h e mfita,ry ()t)(',rators ()f t,l~c I~C,tapl(',(:ti(: grout) AJp(2, ]~).

Problems 1. Co~si(ler a thick lens, co~sisfing of two refracting surfaces with curvature radius/~t mid /-~2 separ~rte(t by a distance d on the optical axis. Deilote by 'n(,~ and 'rt(,~ the refractive

indices in the object aim image space, and by n the refractive index of the lens material. Take the reference planes imincdiately at the left and at the right of the lens (Fig. 3.3.a)). (a) Find tt~c positions of the principal, focal aim nodal points of a the lens. (b) Express the ttfick lens matrix (3.2.4) in terms of the coordinates of the principal points s H and SH,. (C) Write down the explicit expression of the reduced focal length of a thin lens in terms of the relevant parmncters, i.e., n(,1, n,,.e, n, R~ and R~. 2. Enlarge configuration (3.3.13) by (a) an additional free-propagation section placed on the right or the left, and (b) two additional free-propagation sections placed on both sides. Write down in both cases the relations for the relevant parameters and individualize some configuration in accord with the obtained relations. 3. Specialize the parameters of the LTL configurations, which may synthesize symplectic matrices having (a) D = 0, (b) A = 0, and (c) A = D = 0. Specify in every case the

The Group of the 1D First-Order Optical Systems

163

restrictions on the non null matrix elements. 4. Specify the parameters 7), m and d for the "dual" decomposition: (A ~) _ T ( d ) S ( m ) L ( 7 ) ) ,

D > 0,

which reproduces what in the group theory is called the antinormal ordering, hi turn, the LST realization (3.5.1) reproduces the so called normal ordering. 5. Prove relation (3.6.15) for c ~ - 1~.n ~ m - 1 , 2, 9.(Hint: Rewrite (3.6.15) as [F~(fs)] 1/~ -F ( f s) and use Sylvester's theorem according to which

(A)ml(Asin(mO)-sin(m-1)O BD

- - sin(~nO)

B sin(rn, O) Dsin(rn~)--sin(m-1)O

Csin(rnO)

A+D

) ,

COS(~ -- ----~.)

6. Decompose the fractional Fourier transforming matrix F ~ ( f s ) into the form S-1Fc~S for negative values of fs. 7. Write down the relation between the matrix entries (A, B, C, D) and the the parameters (7), rn, r of the K:M.hf decomposition: (A B) _ F ~ S ( m ) L ( 7 ) ) . 8. Consider two symplectic matrices M 1 and M2, whose relative N'.AK; decompositions are given with the appropriate parameters being 7)1, m l , (751 and 7)2, rn2, r Prove t h a t the parameters 7), m, r of tile LSF ~ synthesis of the product matrix M 2 M 1 -- L ( 7 ) ) S ( m ) F ~ are

m-mm~,

where

r 1 6 2 +r

P-P~ +~,

~-~2 __ , H 2 COS 2 (~)2 -t- (?Tt.12 -~- 1H.127) 2) SiIl 21~)2 - - 77L127)1 S i I l ( 2 0 2 ) ,

--7) __ 2_._~_~ .1 --

[2rn~7), cos(24)2)+ (m~ - m72 - rn~P~2) sin(202)], t a n 4) 2

tan r -- ,,~ ( l _ p I tan 4)2)"

(Hint" Write the Iwasawa decomposition of tile internal sequence F~2L(7), )S(rn, ) -- L(7)) S ( ' ~ ) F ~ in tile product M 2 M 1 ; then combine tile resulting sequential rotations, lenses and magnifiers). 9. Identify the parameters (7), m, 4)) of the A/'MK~ decomposition of the squeeze matrix (3.6.49)" B ( r , 0) -- L(7))S(rn)F ~, in terms of the squeeze parameters r, 0. 10. Prove that two squeezes into different directions do not multiply into a single squeeze. (Hint: Consider as an example the product B ( % , 0)B(%,01); identify the parameters of the relevant N'AK; decomposition; then, verify t h a t they do not reproduce the results of Problem 9, thus proving that B ( % , 0 ) B ( % , 01) = B ( r , 0)N ;~. It is also possible to express (r, 0, ~p --/32 ) in terms of (%, %, 01 ).) 11. Verify t h a t the matrices -- cos 4) -- sin 4)

sin 4) ~ --cos 4)]

and

( -- cosh r sinh T sinh r - - c o s h z-]

,

are not of exponential type. (Hint: Write down the above matrices in the neighbourhood of the identity, i.e., respectively as I+(~Q 1-t-O((~ 2) and I + r Q 2 +O(r2). Verify t h a t Q1 and Q2 do not belong to sp(2, R).) 12. Infer the equations for the parameters 7), rn and 4) relevant to the Iwasawa decomposition of the optical matrix M ( z , zi) of the parabolic index medium.

164

Linear Ray and Wave Optics in Phase Space

References [1]

For the ray m a t r i x apt)roach to linear ot)tics see [12] of oh. 2. For tl~cr a n d at)t)licative (:onsiderations on the symt)lectic Lie algebra and gr()~ll) , see [9] alitt [1()] ii~ t']~. 1. Nit)re re('ent ilWestigatit)I~S, ii~ relatioil a,s well to get)~etrit:al ()t)tit:s, t:a~ 1)r finn~(1 i~ [11] ()f t:t~. 2. T h e ~()~ eXl)()~le~t, ial ~mt~u'e ()f

the syn~t)lc(:lir gr()~t) S p ( 2 , IR) is ana lyse(l in [2.1] R. Si~o,~ m~(l N. Ni~k~(la, "Tl~e l,wo-
"~all~c~al,i(:s" a~(l "r

()f ll~t, 2 • 2 real sy~t)lt;(:ti(:

lllal, rix is i~vcsl,igal,t'sI i~

[:~.~] [a. 2]

[aa] [a.4] [:~.,~] [:~q [at]

H. H. Arse~m~flt, "(lm~eralizatio~ of tl~e pri~mipal plm~e ('o~mept i~ ~natrix optics", A~n..I. l'l~ys. 48, 397-39.9 (198()). M. Nazaratl~y a ~ l .1. Slam,fir, "First-or~ler ~pti(:s- a ('m~>~icai operat~r represe~tatioga: l~ssless systemics", .1. Opt. Soc. A~,. 72, 356-364 (1982). II. I I. Arse~m~llt mul I~. Ma('~kow, "l,'~wt,orizatio~ of t,t~e t,rm~sfer ~mtrix for sy~u~etrical <~ptical systm~ls", .l. ()pt. Soc. A~n. 73, 1350-1359 (1983). l~. Nlac~kr a ~ i I I. l l. Arse~mt~lt, "Matrix r fi)r ~ s y ~ m l , rical ol)tical sysl,elllS", .l. ()l)t. St)('. A,~. 73, 136()-136(i (1!)83). E.(?.(~. S~(larst~m~, N. lk,l~k~n~(la m~(l 1{. Si~o~, "I{ealizatio~ of tirst or(ler optical sysl,e~s ~si~g l,lfil~ Ictuses", ()pti(:a A(:ta 32, 855-872 (1985). R. Si~r m~l K.I~. Wolf, "Str~cl,~re ~)f tim set of paraxial ~)l)tical systemics", .I. Opt. So('. A~n. A 17, 342- 355 (2()()()). K.B. Wolf, Gcomc.tric Optic,s on Ph,asc S'pac~'. (Springer, Berli~, 2()()4), (:1~. 1(), pp. 179-2()4. A

T h e {~t)crat~w (~r lllatrix) tlt;(:r

ill refills r

tile a lgcl~ra gicilc~ratr

K +,

A

K _ a nr K:~ (t)r K , ,

K

, K:~) is illllstratcd in several Iml)crs ill relation with

b o t h tilt, ilnilary t;v()lltti()ll ()t)t;rat()rs ass~t:iatett with tllmlllllll~ systen~s anti the symt)let:t it' IIla t)s a sst)t:iate(t with Ha Illiltt)Iliax~ systems. We (l~)tt',

[4.1] [4.2]

[4.a] [4.4] [4.5]

,1. Wei all(1 E. Normall, "Lie algebraic solution of linear differential equations", J. Math I~hys. 4, 575-581 (1963). R. M. Wilcox, "Exponelltial operators mid parameter differentiation ill quantum physics", J. Math. Phys. 8, 962-982 (1967). D.T. Truax, "Baker-Cainpbell-Haus(torff relations and unitarity of SU(2) and SU(1, 1)squeeze operators", Phys. Rev. D 31, 1988-1991 (1985). G. Dattoli, J. C. Gallardo and A. Torre, "An algebraic view to the operatorial ordering and its applications to optics", 1Riv. Nuovo Cim. 11, 11, 1-79 (1988). G. Dattoli and A. Torre, "A general view to Lie algebraic methods in applied mathematics, optics and transport systems for charged beam accelerators", in Dynamical Sgrrrmet'ries and Chaotic Behaviour" in Physical Systems, G. Maino, L. Fronzoni and M. Pettini (eds.) (World Sc. Publ., Singapore, 1991), pp. 21-89.

The Group of the 1D First-Order Optical Systems

165

[4.6] M. Ban, "Decomposition formulas for SU(1, 1) and SU(2) Lie algebras and their applications in quantuin optics", J. Opt. Soc. Am. B 10, 1347-1359 (1993). [4.7] A. Torre, "Lie algebra treatment of two and three-mode interaction Hamiltonians", Recent Res. Devel. Quantum Electronics, 2, 111-131 (2000). [4.8] A.J. Dragt and J.M. Finn, "Lie series and invariant functions for analytic symplectic maps", J. Math. Phys. 17, 2215-2227 (1976). [4.9] A.J. Dragt, "Lectures on nonlinear orbit dynamics", Physics of High Energy Particle Accelerators, AIP Conference Proceedings, R. A. Carrigan, F. R. Huson and M. Month (eds.) 87 (Amer. Inst. Phys., New York, 1982), pp. 147-313. [4.10] A. J. Dragt, "Lie algebraic theory of geometrical optics and optical aberrations", J. Opt. Soc. Am. 72,372-378 (1982). [4.11] A. J. Dragt, E. Forrest and K. B. Wolf, "Foundations of a Lie algebraic theory of geometrical optics", Lie Methods in @tics, J. S~nchez Mondragon and K.B. Wolf (eds.) (Springer-Verlag, Berlin, 1986), ch. 4, pp. 105-157.

[5]

The bibliography about the fractional Fourier transform and its various applications in optics is extremely wide. Here we quote the seminal papers by Condon and Namias as well as the basic papers by Mendlovic and Ozakta.s, and by Lohmann, through which the concept of fractional Fourier t r a n s f o r m has been introduced in optics. A wider view on the subject can be found in the review papers [5.7]-[5.9], and in the COlnprehensive book [5.10]. Fractional Fourier transformers are also investigated in [3.7] (see, in particular, ch. 15, pp. 341-353).

[5.1]

E. U. Condon, "Immersion of the Fourier transform in a contimlous group of functional transformations", Proc. Natl. Acad. Sci. USA 23, 158-164 (1937).

[5.2]

V. Namias, "The fractional order Fourier transform and its application to quantum mechanics", J. Inst. Maths. Applics. 25, 241-265 (1980).

[5.3] D. Mendlovic and H.M. Ozaktas, "Fourier transforms of fractional order and their optical interpretation: I", J. Opt. Soc. Am. A 10, 1875-1881 (1993); II, ibid., 25222531. [5.4] A.W. Lohmann, "hnage rotation, Wigner rotation, and the fractional order Fourier transform", J. Opt. Soc. Am. A 10, 2181-2186 (1993). [5.5] D. Mendlovic, H.M. Ozaktas and A.W. Lohmann, "Graded-index fibers, Wigner distribution functions, and the fractional Fourier transform", Appl. Opt. 33, 61886193 (1994). [5.6] G.S. Agarwal and R. Simon, "A simple realization of fractional Fourier transformation and relation to harmonic oscillator Green's function", Opt. Commun. 110, 23-26 (1994). [5.r] A.W. Lohmann, D. Mendlovic and Z. Zalevsky, "Fractional transformations in optics", in Progress in Optics, Vol. XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1997), ch. 4, pp. 263-342. [5.s] H. M. Ozaktas, M.A. Kutay and D. Mendlovic, "Introduction to the fractional Fourier transform and its applications", in Advances in Imaging and Electron Physics, Vol. 106, P.W. gawkes (ed.) (Academic Press, San Diego, 1999), ch. 4, pp. 239-291. [5.9] A. Torre, "The fractional Fourier transform and some of its applications to optics", in Progress in Optics, Vol. 43, E. Wolf (ed.) (Elsevier, Amsterdam, 2002), ch. 7, pp. 521-586. [5.10] H.M. Ozatkas, Z. Zalevsky and M.A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).

Linear Ray and Wave Optics in Phase Space

166

[6] The link between the symplectic group Sp(2, IR) and the little Lorentz group SO(2, 1) is investigated in [3.7], ch. 5, pp. 69-90, and in [6.1] Y.S. Kim and M.E. Noz, "Dirac's light-cone coor(tinate system", Am. J. Phys. 50, 721-724 (1982). [6.2] Y.S. Kim an(t M.E. Noz, "Ilhlstrative exainples of the syInplectic group", Am. J. Phys. 51, 368-375 (1983). [6.:3] D. Hml, Y.S. Kiill mltl M.E. Noz, "Lillear cmloIlical trmlsforlilatiolls of cotlerent an(t sqlmeze(t states ill tile Wigller plmse space", Plws. R ev. A 37, 807-814 (1.088);II Qlmld, itat,iw,~ mlalysis", Pllys. Rev. A 40, .()()2-!)12 (1.989). [6.4] Y.S. Killl ml(1 M.E. Noz, l'h,a,sc Space l~ict,'wn: of Qua'n,t,u'm, M~'chanics (Worl(1 So. Plfl)l., Sillgat)ore, 1991). [6.5] P.K. Aravill(1, "SiIIullatillg tile Wigiler a,llgle wittl a parmlletric mllplifier", Phys. I{ev. A42, 4(177-4084 (199()). [(i.(i] F. (Ji()c(:i, (l. l)att()li, C. Mari ml(l A. Torte, "Mea.slu'illg tll(; Wigll('r rotatiol~ with (;le(:tr()~ I)(;m~s", l)t~ys. Rev. A 46, 514.()-5153 (1.().()2). [6.7] I). ttm~, Y.S. Kiln m~(l M.E. Noz, "Wig~mr rotatio~s m~(l lwasawa (leco~l)ositio~s i~ l)()larizati()~ ()l)ti('s", l)l~ys, l/ev. E 60, 1()36-1()41 (1.().().()). [6.8] .l..l. Mo~z6~ m~(l I~.I~. Shnchez-Soto, "M~fltilayer optics as m~ m~al()g (:()~np~ter for t,est,i~g Sl)(~'ial relativity", l)l~ys. I~ett,. A262, 18-26 (1.().().()). [6.9] T. Yo~te, .1..1. Mo~zdn, L.L. S~h~(:t~ez-Soto, .].F. Carifie~m mM C. l~dl)ez-Lacasta, "U~(l(;rsta~(li~g ~n~ltilay(~'rs fi'()~ a ge()~etri('al viewp()i~t", .1. ()pt. S()(:. A~. 19, 6()3-6().() ( 2()()2). [7]

F()r a (lis(:~ssi(n~ ()[ I1~(; lwasawa (l('(:()~l)()sti()~ t l~(;()r('J~ wi~ l~i~ t l~(' g('~mral group tl~(;()l'y (:(nd.(;xt w(; ~ a y S~lgg(;st

[7.1] K. Iwasawa, "Ol~ the represe~tations of Lie algel)r;~s", .1I)~..1. Math. 19, 513-523 (1.948). [7.2] K. Iwasawa, "Oil soll~le types of topological grolll)S", Alul. Math. 50, 507-558 (1.948). [7.:3] S. Helga,s()~, Diffc~:ntial Gcomt:try and Syrrml,t:t~c b'pact;.s (A(:a(texnic Press, New York, 1.962), pp. 21.9-225. [7.4] S. Lm~g, "SL2(1~,)" (h(ltlis()~ Wesley, New York, 1993), Pl). [i59-68.0. [8] Ai)I)li(:ati()~s ()f the Iwasawa (h'~('()nq)osition t()()t)ti('al t)r()t)le~s are ill~strated in [:3.(i], [(i.7] [(i.:~], [:3.7], ('l~. 9, l)l). 173-177, and i~ [8.1] l-l,. Simo~ and N. Muk~n~(la, "Iwasawa deco~nposition for SU(1, l) and the Giioy effect for squeeze(t states", Opt. Commun. 95, 39-45 (1993). [8.2] R. Simo~ and N. Muk~(la, "Iwasawa deco~nposition in first-or(let optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams", J. Opt. Soc. A~n. A 15, 2146-2155 (1998). [.9] E.L. Ince, O~'dinary DifJ'e,'rential Equations (Dover Publ., New York, 1956). [10] For a detailed ray tra,(',ing in a t)arabolic-index m e d i u m we suggest [10.1] [10.2] [10.3] [10.4]

D. Marcuse, Light Transmission Optics (Van Nostrand-Reinhold, Princeton, 1972). E.W. Marchand, Gradient Index Optics (Academic Press, New York, 1978). S. Cornbleet, Microwave and Geometrical Optics (Academic Press, London, 1994). V. Lakshminarayanan, A.K. Ghatak and K. Thyagarajan, Lagrangian Optics (Kluwer Academic Publ., Boston, 2002).

4 Wave-Optical Picture of First-Order Optical Systems

4.1

Introduction

We elaborate the wave optical picture of light propagation through first-order optical systems within the context of a.n operator treatment, which parallels the matrix treatment of linear ray optics, as presented in the previous chapters. In accord with the scalar wave model of light, by which all the components of the electric or magnetic fieht associated with tile optical wave can be treated independently, we deal with a single scala, r flmction of position a,nd time. Further we suppose that such a, scalar function be the comt)lex-valued amplitude of a single monochromatic component of the wave field at a certain temporal frequency co, and so only its dependence on the spatial coordinates will be object of interest. Accordingly, the scalar Helmholtz equation and the parabolic wave equa,tion, resulting from the former in the para,xial approximation, come to be the basic equations in our discussion. As earlier emphasized, geometrical optics and classical mechanics display a structural similarity, both developing through the Hamiltonian formalism from analogous principles; the optics of light rays develops from the extrelnal Fermat principle as the inechalfiCS of ma,terial particles from the minimal Hainilton principle [1]. Likewise, the structural similarity between wave optics and quantum mechanics manifests into the forma,1 identity of the Hehnholtz wave equation for the scalar light field to the Klein-Gordon equation for the quantum mechanical wave function of a spinless free particle. Thus the parabolic wave equation for slowly varying envelope wave fields looks like the SchrSdinger equation for non relativistic quantum particles [2]. Guided by these similarities, we will translate the matrix formalism of ray optics into an operator formalism, plainly retracing the line from classical to quantum mechanics. Since linear ray optics is in a sense like the particle dynamics under quadratic Hamiltonia, ns, we will operate a kind of "waviza-

168

Linear Ray and Wave Optics in Phase Space

Geometrica[OpticsI ~

~

ClassicalMechanics

Waviza tion

Quantization

WaveOptics

QuantumMechanics

F I G U I I E 4.1. F r o ~ g(;o~etrical optics to wave optics by a ki~(1 of a w a v i z a t i o n proce(h~re.

ti()l~" [:).7] ()f t,lm tia.,,~ilt()l~ia.,~ (ly,m,~fi(:s llll(t(;r (tl~ra,t,i(: tx)lyl~(),~ials (Fig. 4.1). Ttnls we will elal)()rate t,lm fi)rnm,lization of t)a.ra.xial wave ()l)l,i(~s fl'()lll tha,t~ of I)ara,xia,1 ra,y ()I)tics, l)y 1)r()lx;rly r(;i)la,(:illg ray v(;('t()l'S t)y wa,v(; fllll(:l,i()xlS ml(t a(:(:(w(tizlgly ray-trmlsfi;r ()lx;ra,t()I'S I)y wa,w;-trallsfi;r ()I)(;ra,t()i's. M()r(: l)recis(;ly, nfirr(willg the ray ()l)I,ics l)i(~l,~r(' ()f light t)r()l)agati()~ in l,('r~s ()f syl~l)le(:t,i(: tra~sfiwnm,tio~s a,('t('(t ])y ot)tical syst('~s ()1~ t)()i~ts i~ t,lm 4D (()r, 2D) l)l~a,se st)a,(:(;, w(; will (l(;s(:rit)(; tl~(' ligl~t t)i'()t)a,ga,ti()~ i~ l,(~,rlllS ()f ~ i t a r y tra~sflw~m,l,i()l~S a(:l,('~(| l)y ()l)l,i('al ,";ysl,t;lllS ()11 (:()nq)h;x-val~('~(l f~ll~(:l,i()l~S 1)(;h)l~gil~g 1,() tl~e Itillx'rt sI)a,(:c ()f Stl~m,rt;-il~l,(;gra,l)l(: flu~('.l,it)l~s ()~ 1,1~(;rea.1 libra. We will nm,i~l,a,i~ l,l~(; t)i(:tl~re ()f l,hc oI)tit'al system a,s a, s()rt ()f "t)la,t:k l)ox t)(;tw(;t;n tw() rt;f(;r(;~(:(; s(:i't;(;ns" a,Iltt that ()f light I)r()t)a,ga,tit)n as a, "tra,nsf()rlna,ti()~" ()f the ligl~t (list~n'l)m~(:(; l)etw(;('~ll the tw() e~(1-t)la,l~eS. TI~(; ill(X)lllillg light <list~u'|)a,~ce, wt~t)se (listl'il)~ti()~ i~ tt~e i~t)~l, i)lmm is s~i)t)t)s(;(l I,() I)(; known, (:onstit~tes the "excitati()~" t() whit'h the syst(;~ rest)ontts t)rt)(ll~t:i~g a.I~other light disturbant:e, wh()se (tistril)~tit)n in the outt)~t plane is t()i)(; (tetermined. Ttn~s the rau t)i('t~r('~ ()f ()I)l,i(:a,1 systems a,s "ot)erators" ~mI)I)ing phase spa,co points into t)ha,sc st)a,ce points, is t)a,ra,llclctl by tim wave-ot)tit:s picture of optical systems a,s "()t)(;ra.l,()rs" m a p p i n g squa.re-integra,t)le (:omt)lc, x-va,lued flmctions int, o sq~m,re-intt'grat)lc (:()mt)lex-va,hle(t flmctioi~s. Lincarity is a, ha,sic systcn~ property. It permits one to ttt'compose any propagation problem into "t)ieces", which can easily be "a,na,lyzed" or plainly "solved"; then one must t)e able to compose the individual solved "pieces" back together 1~o indica, tc the solution of the original problem. We are accordingly allowed to represent or to look for representing optica,1 signals as weighted sums (or, integrals) of a, set of pre-selected waveforms, which are so chosen that the responses of the system to any one of t h e m be easily obtained a,nd easily composed back to reconstruct the response of the system to the original signal. Delta flmctions a,nd complex exponentials naturally arise as convenient m a t h e m a t i c a l waveforms for such a signal and system performance piecewise decomposition. Evoked as eigenstates of the position and momentum

Wave-Optical Picture of First-Order Optical Systems

169

operators, delta, functions and complex exponentials are tile inathenmtical description of the intuitive view of idealized point-like and unidirectional disturbances. They will be used to elaborate two equivalent signal representations, as delta, functions are functional to the spatial representation of the signal and complex exponentials are functional to the spectral representation of the signal. Notably, delta functions and complex exponentials are found to be the interchangeable object-image pair of a physically realizable optical s y s t e m two identical free-space sections intertwined by a proper thin lens, i.e. the Fourier transformer. In fact, whilst in general systems Fourier transforms do not represent physical signals, the optical Fourier transformer produces on the output plane the detectable (scaled) Fourier transform of the input waveform. As a. conclusion, we will see that a wave-optical phase space can be conceived as formed by the Fourier conjugate variables. Such variables span the precise domains where the two equivalent signal descriptions, spatial and spectral, may be elaborated, i.e., the domain of the spatial coordinates, over which the signal extends, and the domain of the propagation angles of the plane waves components contained in the signal. However, analysis, computation and visualization of the light propagation, and hence of the optical system behaviour, in this wave-optical phase space need a joint space-angle description of the signal and, of course, a knowledge of the action of tile various elements of paraxial optics on such a joint signal representation. It is assumed that the reader is familiar with the basics of wave optics, which therefore is only briefly reviewed in Sect. 4.2. In Sect. 4.3 the wavetransfer operator and the associated point-spread flmction are introduced as basic tools for the wave-optical description of optical systems. The formalization of paraxial wave optics by a kind of a quantization procedure of paraxial ray optics is elaborated in Sect. 4.4. The basic waveforms, i.e., unit impulses and spatial harmonics, for signal decompositions are discussed in Sect. 4.5. Finally, in Sect. 4.6 the spatial Fourier representation of the signal is described, and the conceptual introduction of the wave-optical phase space as a counterpart of the symplectic geometric-optical phase space is commented on.

4.2

Essentials of the scalar wave model of light. The paraxial wave equation in a quadratic medium

In accordance with the scalar wave model of light, the optical wave is described by a single scalar function of position and time ~(x, y, z; t), called the optical wave function. It may be related to any one of the rectangular components of the electric or magnetic field associated with the optical wave [3]. In a linear homogeneous and isotropic medium of refractive index n, the

170

Linear Ray and Wave Optics in Phase Space

opti(:al wave f u n c t i o n o b e y s tile wave equation 02

02

02

02

~2 ot2 '(' - 0, ,,~

(o-~ + 0_~ + ~z~)~

(4.2.1)

w h e r e (" is tile st)eed of ligtlt in the va,(:luun. E(tlm,tioIl (4.2.1) is a,t)t)li(:able also t() illllonlOgCnCous nlcdia, as lollg a.s tllc rcla, tivc refractive index 'n(x, y, z) clmllgCS |)llt, sliglltly over (tistallt:(;s ('Ollll)ar('~(t wittl t,ll(; Wa,v(;l(;llgtll ()f tim c,lc(:l,I'()Illa,~II(;I,i(: ti('l(t, as it, is tll(; ('as(; wll(:Ii ()I)ti(:al fi(;l(ls arc iiw()lv(;(l, hi fil(:t, the a,l)I)l'()xilllal,t;ly s(:alar ('lmrat:tt;r ()f lll()sl, ()l)tical sysl,t;lllS Sl,t;~llS fl'()lll I,}l(~ fa,(:t, l,lmt llslmlly t,ll('~ (:harat:t('risti(' t)llysi(:al (]illlCllSi()llS ()f the systcln (:()lllI)()ncnts gr(;a,tly (;x(:(',(;(l 1,11(; WaV('~l(;llgtll ()f ligllt. M()r(;<~v(;r, it, is (:()llsist(;Id, with tile l)a,raxial al)l)rt)xillm,ti()ll tlm, t mllt)lnd,S I,() sau l,lm,l, l,ll(: wa,v(; ti(;l(l is t)urcly trmlsvcrs(;. Tiffs is tllc inlmrcnt wa.vt;-()i)ti(',al (:ha ra(:tcrizati()n ()f ligld, t)rot)a,ga,l,i()ll l,llI'()llgll tirst-()r(h;r ()l)t,i(',a,1 sysl,(;IllS, wlli('ll art; ()f l)rillmry ild,t;I't;st h('xc. Ii1 g(;Ii(;I'al, all illIillil,(; lnnlfl)(;r ()f s()llll,i()lls t() 1,1~(; wav(; (;(t~al,i()]~ (4.2.1)

(;xis(,s. I~ fit(:(,, all (,l~(; wa,vt;ft)rl~lS ()f (,l~(: tyt)(;

,/,(.r, .q, z; t) -~/,(t..~.r + t.,~q +

t..z

- ,~t).

(4.2.2)

fi)r a~y (:l~()i('(; ()f l,l~(; lmra~('~t(;rs L':,:, L'u, L'z a~(l ~ sat,isfyi~g t,l~(: r(:lat,i()~

/,:2 _/~:;., +/~:u~ + 1,.2 -- 'n2 ,~2 ~

(:2

(4 2.3) '

art; s()l~ti()~ls ()f t,ll(', wav(; (;(t~mti()~l. Tll(; ~u~i(l~W s()l~l,i(m is i~(tivi(l~mlizt;(t t)y l,ll(: Sl)(:(~iti(' ])()~l(Iary (~()~(lil,i()llS fi)r tit(' I)r()I)a.~ati()~l I)r()t)l(;~ll a.t lm~l(]. T h i s ]~l(;ans tlm,t at a. (:(;rt, ai~ t,i~l(; tll(; wav(; fi(;l(l ()v(;r a (:(;rta,i~l r(;gi()~l i~ t,]l(; st)a,cc is giv(,~, a.l~(I I,l~(; t)r()l)a,ga,I,i()~ i~l sire,(:(; a,ll(t t,illlC is l,ll(;ll (:a.h:~tla,t(;(l 1)y (4.2.1). It, is a (:()~n~l()~l I)ra.('ti(:(; t,() i~wcst, iga,t('~ th(; ])chavi()~u" ()f tll('~ sinai(; ~()no(:hronmti(: ('()~i)()~l(;nts ()f th(; ligllt wave, s(;1)a,r a t i n g tim rat)i(Ily tinlc-w~,rying factor (,-i~t a,l, l,ll(; l,(;~lli)()ral fl'(;(t~c~l('y ~v fl'()~ll t,ll(; sta.tio~mry a.nlt)lit~(tc as fi)llows

~(,(x, y, z; t) - r

y, z)c -i"~t.

(4.2.4)

Thc~l tllc t)osition (tct)cxl(tcnt complex amplitude r y, z) a,ssumcs the role of the dcscrit)tor of tim wave; it, o b e y s the reduced or time-free wave equation (a,lso kn()wn a,s tile scalar Hclmholtz equation) ( 02

0~

02

+ 0 7 + ~z~)r + ~ r

0,

(4.~.5)

the (tist)crsion rcla,tion (4.2.3) linking tile w~ve mmfl)er k in the me(liuin to the tcnlt)oral f r e q u e n c y ~ of the wave. Since the refractive index is allowed to v a r y in space, t h e wave n u m b e r m a y a c c o r d i n g l y be r e g a r d e d as a p o s i t i o n d e p e n d e n t function: k(x, y, z) - kon(X , y, z), w i t h a space scale large c o m p a r e d w i t h the m e a n w a v e l e n g t h A of t h e wave; k 0 _ _ 0-~2 is t h e v ~ c u u m wave n u m b e r .

Wave-Optical Picture of First-Order Optical Systems

171

As we solely consider monochromatic waves, the terms wave function and complex amplitude are interchangeably used to denote a complex function of position only, obeying the time-free wave equation (4.2.5). Clearly the wave flmction and the complex amplitude are just mathematical entities; it is the real part of the wave function, i.e., of the complex amplitude times the harmonic exponential e - i w t , that may represent a physical quantity like, as noted, the amplitude of any component of the electric or Inagnetic field vectors. The plane waves and the spherical waves represent two sets of elementary solutions to the Helmholtz equation (4.2.5) in a homogeneous medium. Described respectively by the complex amplitudes (~p(X, y , Z) - - e i ( k x x + k u y + k z z )

y, z) - -7

,

- - Ci k ' r

(4.2.6)

r>A

the plane and the spherical waves can pictorially be visualized by the relative sets of constant phase surfaces, separated by/~ - -~ and extending through the whole 3D space with no spatial confinement. Thus the plane waves are depicted as an unlimited sequence of parallel planes orthogonal to the positionindependent wave vector k = (k.~,ky, kz), which individualizes the direction of propagation of the wave, whilst the spherical waves a.re viewed as a set of far to infinite reaching concentric spheres. The plane waves are unidirectional, with the intensity of the wave (and so the energy flow), uniformly distributed over the infinite consta.nt-phase planes, being constant a,long the direction of propagation, i.e. along k. In contrast, the spherical waves have multidirectional features, with the intensity of the wave d e c r e e i n g with the distance r fl'om the central radiating source according to the inverse square law. The hypothesis of paraxiality implies that the propagation occurs mostly in one direction, the z direction, for instance [4]. Then we can reasonably set (4.2.7)

kz ~ k,

and accordingly write the complex amplitude for the general solution to (4.2.5) in the nearly plane-wave form y,

-

kz,

k-

(4.2.8)

where the modulating amplitude ~(x, y, z) varies slowly with z in comparison with the leading plane w a v e e i k z . More in general, when propagation occurs in a weakly inhomogeneous medium the carrier phase factor is replaced as ~ik o n o z -----+

Giko

z no ( z' )dz' f zi ,

(4.2.9)

thus accounting for the proper phase shift i k o fzz n o ( z ' ) d z ' experienced by the wave field along the z axis from the assigned reference plane at zi; no(Z) -

Linear Ray and Wave Optics in Phase Space

172

n(0, 0, z) d e n o t e s indeed the on-axis va.lue of the refractive index. T h e slowly v a r y i n g envelope f ( x , y, z) t h e n obeys the parabolic wave equation i 0 //~ --

~:;, o:

-

1

-

( 0 2 q_ 0 2

1

2

no(Z)

~ ) ' ( ' + 2,,,o(~) [no(Z) - 'n2 - i ~. ](,,

2kg,,,o(~ ) , 0 , ~.

(4.2.10)

1 , 2 2,,,o(:)[n o(z) - 'n 2 - i "o(:) % ](, (tisa,t)t)ca,ring w h e n

w i t h the t)()tcntia.l-likc t c r ~

l)rt)Imga,titnl takes t)la,(:(', ill a ll(nll()gC~c()~S ~ ( ' . ( l i ~ : 'tt,(X, y, Z) ---- '~t,o. Tt~('. t)ri~c (h;n()tt;s (hu'iw~tivc wi(,ll resl)e(:t t() z. ()t~ g()i~g fi'(n~ (4.2.5) 1,() (4.2.11)) l,l~r()~gl~ (4.2.8) tll(; (litf'~siw: (,(:r~ ~i ) z'~ lms t)('~(;1~ ~(:gh'~(:(,(;(l witl~ resl)(;t:t 1,() :o: (:(n~fiw~fity witl~ tl~(~ a s s ~ l ) t i ( n ~ tl~a,t sig~ifi(:a,tiw; wu'iati(n~s ()f (, ()(:(:~n" ()w;r (tistm~(:cs (:(n~si(h;ralfly ex(:e(;(lil~g (,l~c wavch;~g(,l~ ()f ligld,. ~I'1~(; hyt)()l,l~(;sis ()f tmraxia.lity (4.2.7) a,~()~nd, s 1,() l,~ln~i~g l,l~(; sl)l~('r(; ()f wave w;(:l,()rs ( 4 . 2 . 3 ) i ~ i . ( ) a Imral)(fl()i(l. I~ flu:t, (4.2.7) ix als()('(l~iwd(u~i, l() k.,.,/s

<< k:

,~ k,

()r

k•

<
(1.2.11)

wll(;l"C /~:_t_ rCl)l'(',Sel~i,s (,l~e l,l'a.llSV(WSt; r ()f (,l~(; w a v e vc(:l,()r {:()l~l)l'isi~g th(; .r a n(l y ('()llll)()ll(qltS a.S 1~'~_ /~:i~ q- /~:;q" 2 Tln~s (,ll(; (lisl)(;rsi(n~ s~trfa,:(;, wt~i(:h -

-

t)y (4.2.3) ill(livi(l~m,lizcs a, sl)ln;rc il~ (,l~('~3 D sl)a(:('~ ()f l,ln; (Su't,(;sia.~ (:(n~l)(n~(',l~ts /v.,,,/,':q ml(l/% ()f k, ()r a (:in'h; i~ (,lie 21) Slm(:(; ()f (,ll(; (nu~SW;l'S(; m~(1 h)l@t~uli~ml way(: v(:(:l,()r ('()l~q)()~nq~l,s/,'t a~l(l /,':, (,~r1~s i~(,()

-

-

" 2/,'

-

--

~'

-

-2-/a, ' '

(,1.2.12)

whi(:tl (:a,ll 1)(; vi(;w(;(l as a Imrat)()h)i(l ill (,ll(; 3 D sI)a(:(; Sl)allZl(;(l ])y (k.,,,/~::q,/,:z) ()r a, t)ara,t)(fla i~l tl~(; 21) Sl)a.(:('. sI)a,~('~(t |)y (k• L::) (Fig. 4.2).

ki

4.2. Tile paraxial a p p r o x i m a t i o n a m o u n t s to replacing the circle k~ + k2z - k 2 by the p a r a b o l a kz - k - k,.2 / 2 k . (Inspired by [3.3]) FIGURE

Evidently, tile s a m e occurs in g e o m e t r i c optics w h e r e tile spher e of ray toomenta, p2 + p2 + Pv2 -- 'n2 (Eq. (1.2.12)) is a,p p r o x i n m t e d by the pa,raboloid 2

2

Px+PY2nw h e n ,

p a r a x i a l p r o p a g a t i o n is a s s u m e d .

Wave-Optical Picture of First-Order Optical Systems

173

We can further shape the parabolic equation (4.2.10) for the wave amplitude ~(x, y, z), assuming that ~b changes smoothly through the transverse plane and is mostly concentrated near the axis so that the second-order approximation of the refractive index with respect to the transverse coordinates is allowed. Replacing n2(x, y, z) by the second-order expansion in x and y,

n~(x, y, ~) ~ ~o~(~) - ~o (~)~.~(~)~ ~ - ~o ( ~ ) ~ ( ~ ) y

~,

(4.2.13)

one obtains the parabolic wave equation appropriate to describe the propagation of a wave field in a quadratic-like medium; namely, i

0

k 00z ~/)

1

2k2n 0

02 (~.~2 +

02

~

)~ +

1

-2

[n~x 2

@ Tt2y

y2]~ _ i % ~. 2kon 0

(4.2.14)

with the z-dependence of tile refractive-index related functions no, n2x, n2y and n to being implied. The term --in~ V(x,v,z) above can be absorbed into a convenient redefinition 2k 0 n o

rv/n0(z ) . The resulting equation

of the field amplitude from ~/~(x, y, z) to ~/~

for (I) formally resembles the SchrSdingcr equation for a quantum oscillator with two independent degrees of freedom and "time"-dependent "inass" and "frequencies". In fact, by a simple separation of the variables x and y as

r

y, ~) = ~x(*~, ~ ) ~ , ( y , ~),

(4.2.15)

the 2D parabolic equation (4.2.14) turns into two 1D SchrSdinger-likc equations of the form

~ihO~(q.t)

__ ~1[__!_1 m(,~)~2

+.,~(t)~(t)~]~(q. t).

(4.2.~6)

with the mechanics-to-optics correspondences:

h ~ -~,

t --~ z

r e ( t ) ~ no(Z ) '

~2(t)---,

(4.2.17) ~o(~)'

and the usual quantum mechanical definition of the position and momentum operators: ~ - q and ~ - - ~ h9 ~o (see w 4.4.1) q symbolizing any one of the profile coordinates x or y. On deriving the parabolic wave equation (4.2.10) from the time-free wave equation (4.2.5) on the basis of the paraxial approximation, the questions concerning the compatibility of the inherent assumptions with the exact Maxwell's equations as well as the description of nonparaxial propagation have been ignored. The paraxial description of light propagation may safely be applied when the variations in the transverse profile of the optical field per wavelength are small and the propagation distance is not too long compared with

174

Linear Ray and Wave Optics in Phase Space

the d e p t h of focus of the optical field. Naturally, there are situations where the paraxial description is no longer a d e q u a t e a,s it is the case of tightly focused light beams, semi(:onductor laser b(;ams with large divergence angh;s a,nd nont)araxial eigenmo(tes of stable resonators. A grea,t variety of appr()a,(:hes has t)een (tevise(t t()exten(t the treatnleilt ()f s(:alar wave t)r()t)agation t)ey()n(1 the t)araxial a,t)t)r()ximati(m as well as t() extend the tI'(;atnl(;nt of s(:a,lar waves t() ve(:t()r wa.v(:s, lm.ill(:ly t()el(;(:tI'()Ilm.gll(:ti(: waves satisfyi~lg Maxw(;ll's (;(ilia,ti()lls. In [5.1], f()r instan(:(:, a,ll ('xl)a,nsi()n s(:lmnl(; ()f th(; f()rllm,l s()lllti()n t() th(; (;xa(:t Maxw(;ll's e(tlm,ti()xls fi)r tll(: el(:(:tri(" ti(:l(1 v(:('t()r ill l)()w(:rs ()f a, sllita,1)ly (t('Iill(;(1 l)armll('t(:r is t)r(;s(:llt(:(t, ml(l a(:(:()r(tixlgly ttl(' r(;(:ln'r(;ll(:(; (;(tllati()ns for the (:()rr(;(:ti()ll t(:nlls ()f (lilh,r(,m, ()r(l(,rs ar(: (l(;rive(1. Th(, (:xI)ansi()n l)a,ra,nl('J:,(;r, wtfi(:tl (:()1Ill)ares t,ll(; st)a,(:(' s(:a,l(:s ()f tll(', trmlsv(:rs('~ ml(t l()llgil, ll(lillal wu'iations ()f 1,11(; wav(: mlq)lil,lut(;, is lnl(lerst()()(l as a nl(:a.sllr(, ()f 1,11(: 1)(:an1 (liv(:rg(:n(:e angle ml(l s() ()f l,tl(: (l(;I)artllr(: fl'(nll l,ll(: ll(,a,rly t)a,ralM t)r()I)agati()ll ()f l,ll(: t)ea,m t() 1,11(; axis. Fllrtlmr illv(:st,igati(nls Ira,v(; 1)(:(;11 I)(;rfi)nll(;(l ill ()r(l(:r t()()l)ta.in ('xa('t rattl(;r tliall I)(',rtllrl)ativ(; s()l~d,i()~s t() l,l~('~ It(;l~d~()ltz way(: e(l~m.ti()n, hi [.5.~:~] tw() fm~ili(:s ()f (:xa(:t s()hlti()~s ()f tl~(: Ih:lll~l~()ltz (:(l~m,ti()~ a,r(: l)r(:s(:~d,(;(t: ()n(; in 1,11(; ()l)lat(', st)l~('r()i(lal ('()()r(li~mt(: sysl,(:~ll m~(l tile ()tiler i~l tim l)r()la,te st)h('r()i(lal (:()()r(li~mt('~ syst(:~l~, tll(, flnl(lal~l(:llta.1 lll()(t(:S f()r ill(' tw() fm~ili('s t)(;i~lg r(;sl)(:(:l,iv('ly 1,1~('l)r()lmgal,i~g (lmlssia~ l)(:m~ls m~(l 1,1~('~sl)l~(;ri(:al wa,v(;s. T h e tra~liti()~ml (Im~ssim~ t)('m~ is l,l~(,~ r(:(:()v(:r('~(l witl~i~ a x~()~l)araxial (i('s(:rit)ti()n ()f wav(;-t)r()I)a,gal,i()~l, r(;-ild,(:rl)r(:l,ixlg 1,11(: l)(:an~ way(: fr()~d, as a s(:(:l,i()~l ()f an ()t)lat(' ('llil)s()i(l ra,tl~('r tl~m~ ()f a sI)l~('r('. Also, tll(' p()t('~ltial ()f tl~(: (',()nq)lcx I)()int s()~r(:('~ g(:~l(:ra,t(;(t tiel(ls [s.u)], whi(:l~ are rig()r()~s s()h~ti()ns ()f l~la,xw(;ll's (;(tuati()~ls, t() (l('s(:ril)(, t)llysi(:a.1 II()~lt)araxia,1 t)('mn-lik(; waves tins (te(;l)ly been inv(~stigat(,(l, tln~s S~lgg(,sti~lg a, (:(:rta,in ~n~ld)(;r ()f m()(Ms t() a,v()i(t the ii()iit)hysi(:a,1 si~g~larities i~fl~(:r(:l~tly (:a,rri(:(l l)y tiffs kind ()f s()l~ti()~ls Is.12]. For a, (t(;taile(1 ixlv(;stiga, ti()n, the r(:a,(h;r xnay (:()~s~flt [5].

4.3

Ray and wave optics

Geometri(: a,n(t wave ot)tics are intimat(;ly (:()nnected, with tile results of geometrical ot)ti('s being recov(;rable as a linfiting case of wave opti(-s, but not the other way a,rolm(t. In tile short wavelength limit, in fact, (or, more precisely, in tile asymt)toti(" a p p r o x i m a t i o n for large wave numbers) the solution of the time-iIl(teI)en(hmt wave e q u a t i o n yiehts the eikonal equation of geometric optics [6]. In contrast, wave optics cannot be recovered from geometrical optics by purely algebraic transformations, but by a sort of q u a n t i z a t i o n procedure. Wave optics can be regarded as a quantized geometrical optics. A quantum t h e o r y of light rays has been e l a b o r a t e d from both a differential and integral equation apt)roach, a p p l y i n g the same rules for transition from classical to

Wave-Optical Picture of First-Order Optical Systems

175

quantum mechanics, according to which variables are replaced by linear operators [7]. Notably, quantization of the equations of geometric optics leads to the scalar Helmholtz equation, and accordingly the paraxial limit of wave optics corresponds to the non relativistic limit of quantum mechanics. Thus, the relation between wave optics and ray optics is similar to the relation between quantum mechanics and classical mechanics, the light wavelength in the vacuum being like the Planck constant. We recall also that the axial coordinate of optics is like the time coordinate of mechanics. Geometrical optics grounds on the concept of ray of light, whose trajectories are uniquely determined by the ray-equation, or equivalently by the optical Hatniltonian equations, supplemented with the proper initial values for tile ray position and direction. Correspondingly, wave optics is based on the concept of optical wave function, whose behaviour is uniquely determined by the reduced wave equation, supplemented with the appropriate boundary conditions for the optical field. In both cases, the optical properties of the medium, in which the wave is propagating, are accounted for by the refractive index, assigned (in tile case of a nondispersive medium of concern here) as a scalar function of space position only. ()n the other hand, instead with the properties and the tracing of the ray paths, geometrical optics may alternatively be concerned with the properties and the construction of the surfaces to which the ray trajectories are normal, the so called wave-surfaces or wave fronts. The eikonal is representative of such a "wave front" at)t)roa(:h ()f geometrical optics to light propagation; it is a scalar potential that ('an be interpreted as aIl optical path length from some reference wavefront surface. As Fermat's principle is the building block for the ray-tracing procedure, the eikonal equation is the basic tool for the wavefront-evolving investigations, since it determines the transformations experienced through propagation by the cquilcvcl surfaces of the eikonal. The eikonal is recovered in wave optics as the phase of the quasi-plane wave expression of the light field as it results from the zero-order term of the asymptotic representation of the solution to the Helmholtz equation in powers of the inverse of the wave number. In wave optics the concept of ray is not elementary at all; its place is taken by the simplest solutions to the Hehnholtz equation in a homogeneous medium: the spatial harmonic plane waves and the spherical waves, which, originating respectively from a harmonic disturbance distributed over a plane and from a light point, retain in the respective sources the basic attributes of the ray, the direction and the location. The concept of light ray contrasts with the uncertainty relation, as it inherently implies the precise specification of both the location and the direction of the ray, which is precluded by the classical wave properties. In wave optics, light rays are individualized as the trajectories perpendicular to the wave fronts of the propagating optical disturbance. Figure 4.3 offers a naive illustration of the ray-to-wave relation in the case we are considering of plane and spherical

176

Linear Ray and Wave Optics in Phase Space Rays

~ .

\ ~

Way

FIGURE 4.3. TI~c ray of gco~ctrical optics are oI'tl~ogo~ml to tim wave fr()~fl,s of wave optics. Tt~c wave fl'oI~ts are (,l~c s~rfa(:cs al()~g wlfi(:l~ (,t~c cik()~ml rc~nai~s (:ox~stm~t.

wa.v(:s, rI'll(: (:()llsl,md, l)lm,s(: sllrfil,('(:S ar(: t ) a r a l M l)la,n(:s ml(l ('()ll(:(:ld,ri(: st)ll(:r(:s r(:sl)(:(:tiv(:ly, a.ll([ a.(:(:()r(lillgly tim r a y (,l'a,j(:(:l,()ri(:s I,ra.(:(: sl,raiglfl, lill(:s, I'(:('ii)r()('a.lly l)a.I'a,lh:l ill (.ll(; (lir(:('ti()ll ()f l)r()I)agaI,i()ll ()r (:llm.~la(,illg ()ll(,war(l i~l all (lir(:(:l,i()Ii fi'()111 l, ll(: (:(:Ire'a,1 s(nlr('(:. th)w(:v(:r, as t l l ( : " r a y " , t)()t, ll tll(: t)lml(: ml(l st)h(:ri(:a.1 wav(:s ar(: i(l(:aliz(:(t (:ll(,i(,i(:s, tim(, (:all lmr(lly 1)(: r(:aliz(:(l ill l)ra.(~(,i(:(: (l('llm,ll(lillg fi)r all ildilli(,(' (:x(,(:nt t)la,nar s()llr('.(: ()r ill (:()nl, ra,st a l)()ild,-lik(: s()llr(:(:, r(:st)(:(:tiv(:ly. N(:v(:rl, l~(:l(:ss, th(:y a.r(: ~ts(:fl~l l~mt,l~(',~mI,i('al ~()(l(:ls fi)r wu'i()~s sit,~a,t,i()ns. Pla~(: wa,v(:s at(: th(: 1)asi(: t()()ls ()f l"(n~ri(:r al~alysis, wl~i(:l~ I'(:l)l'(:s(:~ts tl~(: wa,v(: fi(:l(l as a, s m ~ of tntr(:ly ha,r~()ni(: wa,v(:s, a.n(l s(), r(:sti~g ()n tim ~tni(lir(:(:ti()na,1 fi:al,~r(: ()f s~(:l~ wa,v(:s, (l(:t(:r~i~(:s tl~(: (lir(:(:ti()~al (:()nt(:~t ()f tl~(: wav(: fi(:l(1. ()~ tl~(: ()tl~(:r han(t, (,I~(: sl)l~(:ri(:al w~v(:s ~l,I'(: i ~ s ( , r ~ ( : ~ ( , a l t() ll~lyg(:~s' (:()ns(,r~u:(,i()~ ()f a(lwu~(:ing th(: wa,v(: fi'(nd, s ()f (,lm ()l)(,i('al (tis(,~rl)m~(:(:. W(: als()()t)s(:rv(: t l m t far awa,y fr()i~ tl~(: s()~r(:(: snm,ll s(:g~(:~d,s ()f a sI)tmI'i('a,1 wav(: x~my l)(: (:(msi(l(:r(:(1 a.s a,t)I)r()xiIna,t(:ly t)lm~(: wa.v(:s. Tl~(: ()l)l,i('a,1 wa.v(: ti(:l(ls (,lm,(, ~ i ~ i ~ i z ( : tl~(: ~n~(:(:l'(,ai~(,y t)r()(l~(:t, t h u s ~m,x~if(:sti~g a ray-lik(: fi:a,(,~n'(:, m~(l ~-i,i'(~,t)()(,t~ matt~(:n~ati(:a.lly r(:I)r(:s(;I~ta})l(: a,n(t t)l~ysi(:ally r(:aliza,t)h: a,r(: (,l~(: r(:al (~a,~ssim~ wa,v(:f()rn~s. Tlmr(:fi)r(: th(:y (:a,n b e a,ssimila,t(:(t ()~ th(: t)rot)(:r q-t,: s(:a,l(: of th(: wave-optical phase space to a, pro(lu(:t of d e l t a fl~i~(:tions like b(q - q0)~(t,: - t,:,)), j u s t a,s t h e ligl~t r a y c a n be r e p r e s c n t c ( t ()n t h e t)r()t)(:r q-p s(:a,lc of t h e g(:()nmtric-ot)tical t)ha,s(: st)a,(:e by a single p o i n t , a n d hcn(:c b y t h e t ) r o d u c t b ( q - q o ) b ( p - Po) (a). As we will see in C h a , p t e r 6, t h e W i g n e r r e p r e s e n t a t i o n el a,borat es a p i c t u r e of t h e light d i s t u r b a n ( : e in b o t h spa,ce a,nd f r e q u e n c y (i.e. d i r e c t i o n ) , a,s a,n a Tile symbol ~ denotes the spatial frequency, we will introduce later as the Fourier conjugate variable to the spatial coordinate q; it relates to the direction of propagation of the wave. Thus, % and ~o localize the "centres" of the distributions of the beam in space and direction of propagation respectively, whose relative spreads multiply to the minimum value 1 5, allowed by the uncertainty relation of Fourier analysis. It is in the light of the uncertainty product based "granular texture" of the wave-optical phase space that the product 5 ( q qo)5( ~ - ~o) must be interpreted as signifying the "granule" in that space.

Wave-Optical Picture of First-Order Optical Systems

177

alternative to the separate space and frequency descriptions of wave optics, yielded respectively by the wave function and its Fourier transform. Thus the ray-like model of geometrical optics, and so the feasibility to precisely detail an "optical entity" by both position and direction attributes, is recovered to a certain extent within the Wigner space, composing information concerning both the space and angular distributions of the signal into a single entity, the Wigner distribution function. However, due to its intrinsic nonpositiveness, the Wigner distribution function cannot be interpreted as the wave energy distribution in space and frequency. Notably, Gaussian signals are the only waveforms, for which such an interpretation of the relevant Wigner distribution function is feasible, being it everywhere positive through the phase space. 4.3.1

Reviewing the operator model of ray optics

From the viewpoint of ray optics the optical system is seen to operate a mapping of the ray position and momentum coordinates from the values, pertaining to the reference screen assigned at the entrance to the system, to the values, pertaining to the reference screen assigned at the exit fl'om the system (Fig. 4.4.a)). Evidently such a view is not concerned with the internal workings of the system, but only with the relation between the system inputs and outputs. It finds the prot)e,r formulation in the operator relatioil (~ 1.4.1) A

Uo - 9)l (Zo, zi) ui,

(4.3.1)

where the cohmm vector u - (qx, qy,Px,py) T comprises the four ray coordihates. The ray-transfer operator 9)l (zo, zi) is the mathematical descriptor to account for the "finite-step" transformations of the ray-variables due to propagation through the system from the intmt plane to the output plane, located along the optical axis at the axial positions conventionally denoted as zi and Zo. In fact, ui and Uo localize the representative points of the incoming and emerging ray in the 4D geometrical optics phase space, where the bilinear antisymmetric metric = (u, v) = ~jdj~v~ (4.3.2) A

is defined (~ 1.6.4). For notational convenience the ray vector components are indexed by the integers 1,.., 4, summation over repeated indices being implied. As noted in w 1.3.1, the numbers Jjl above, that can only be 0, 1, and - 1 according to the rules @l = ~l,j+2, j = 1, 2 and Jjl = -~l,j-2, j = 3, 4, form the symplectic unit matrix J. When dealing with a 2D optical phase space as in the case of stigmatic and separable astigmatic optical systems, whose behaviour can effectively be depicted on the phase plane spanned by only one pair of conjugate ray variables, the product (4.3.2) is related to the oriented area of the parallelogram spanned by the two phase-plane vectors u and v.

178

Linear Ray and Wave Optics in Phase Space

qV

q).

qY

qV

(qx,,.qv,, 7 Optical device

Optical device

9Ti

M

(q.,,.q,',).,,~

(I) :qo,zo) ' -o

q.,.

q,.

(W

q.,.~/

q,.

(h)

FIGIIIIE 4.,1. (a) Ray v(:('(,()rs m~(! ray t,rm~sfcr ()l)crators l)('rtai~ t() tlw ray-()l)ti('s f()r~na,lisin as (b) way(: fiu~(:ti()~s m~(l wave trax~si'(:r ()l)(:rat()rs l)erl, ai~ t() 1,1~(:wave-()l)ti(:s fl)n~mlis~n. Tlx(: ~ml)i)i~g (
H ( z ) M ( z zi) ' '

M ( z i , zi) - I,

(4.3.4)

where the matrix H(z) obtains from the paraxial form of the Hamiltonian of the system under Poisson bracket with the conjugate variables q, p (w 1.7.3). The matrix M belongs to the Lie group Sp(4, IR), or Sp(2, IR) in the case of 1D systems, of 4 x 4 (or, 2 x 2) symplcctic matrices, and so M q - J M - J. Accordingly the generator matrix H can be placed in the symplectic Lie a 1gebra, sp(4, IR), or sp(2, R), M a,nd H being related by a,n exponentia,1 Ina,t)ping (w 2.3) [8]. In fact, by (4.3.4) the ray matrix is locally expressible in the exponential form M ( z i + dz, zi) - c H(z~)dz for any elemental distance dz from zi. The prot)aga,tion of the ray through the optical system progresses by multiplying the infinitcsima,1 ray-inatrices c H(~)az, thus obta,ining for any desired propaga,tion distance Z o - zi the infinite-term product M(zo, Zi) cH ( z ~ cH(z~+&)dzc H(~)az. The one-exponential form over the entire propagation distance Zo - zi is allowed when H does not depend explicitly on the axial parameter z. In which event, the exponents of the infinite-exponential form plainly sum up to yield the global representa, tion of M ( z o , zi) a,s the single exponential M ( z o , zi) - cU (Zo-Z~). (4.3.5) The exponential form (4.3.5) has been basic to the considerations developed in Chapters 2 and 3. It will be recovered here, formulated within the operator representation we will elaborate to describe the light wave propagation. 4.3.2

I n t r o d u c i n g the operator m o d e l o f wave optics. The w a v e - o p t i c a l t r a n s f e r operator

From the viewpoint of wave optics, the optical system is seen to operate a mapping of a set of functions given in the input plane into a set of functions in the output plane. The "objects" of such a mapping are not the position a,nd

180

Linear Ray and Wave Optics in Phase Space

m o m e n t u m coordina,tes of the incoming and o u t g o i n g rays a,s in (4.3.1), b u t the light d i s t r i b u t i o n s before a n d after the passage t h r o u g h the system, a,nd h e n c e the wave fimctions in the a,ssigned profile pla,ncs (Fig2. 4.4.b)). Accordingly, tllc tool of tile ma.t)t)ing is tile wave transfe:r ope','ator M(zo, zi), which a('ts Oil the wa.v(; flmction 'Ih(qi, zi) ()f ttl(; incoming ligtlt t() yicl(t tll(; wave flmcti()n Oo(qo, zo) of the ()~,tg()ing light, a.s f()rma,lly cxt)r(;ss('~(t t)y ttl('~ ()t)(;ra.t()r rcla,tion

,I,,,(q,,, z,,) .... M(z,, zi)'I)i(qi, zi).

(4.3.6)

~I'll(; l)(,siti(,~l w;(:t,,ws q.i = (q.,.,, q:,/,) ml(l q , = (q.,.,, %,) i(h;~tify l)(,illI,S WSt)C('tiv(;ly ill 1,11(; i~ll)~l, m~(l ()~tl,l)~l, l)la.~(; })y l)airs ()f (~a.rt('siall ('()()r(li~m.t('~s with r('~si)(;(:l, 1,~) l,llc a.ssigl~(;(l r(,fi;rc~('(; fl'm~l(;s ill t.ll(; r(;st)(;(:tiw; I)lm~(;s. ~I'1~; (wigins ()f I,ll(;st; l'(~f(~r(;ll(:(; fl'il,lllO, S 1,1,1"(~, (:()~v('~i('~ltly I,ak(;ll ()ll tll(; ()l)l,i(:al z axis, ml
The poi'nt-sp'1"ead .fitucti~m Evi(h;llt, ly the, "tillit,('~ stcl;' Ifi(~t,lu'(; fiw ligllt, tn'(JImga, ti()21 I)ertill('~llt t()tllc ot)('ra,t,(n" r(:lat, i(nl (,1.3.6) is (;(llliwd(;llt t,() t,ll(; "(;l(;lll(;lltal st(:l)" vi(;w l)(:rt, ill(;llt, t() tll(; wa,v('~ (;(tlta.ti(nl. ']'ll(;ll, as tll(~ wave (~(llm,ti(nl is lilmar, tlw, llla.l)l)illg (4.3.6) is lill(;ar a,s w(~ll. "l'll('~ r(;sl)()llS(: ()[ I,]l(; sys|,(~,lll |,() a, lill(;ar (:()lll])illaI,i()ll ()f va,ri()llS ill(l(;l)('~ll(l(;llt illl)11ts is Sillll)ly t,ll(; Salll(; lili(;ar (:()llfl)ilm,ti()ll ()f t, ll(; r('sl)(nls(;s ()f tlw~ sysI,(;lll t() l,ll(; ill(livi(l~ml i~I)~ts. I~ f()r~m,1 t,('m~s, l,l~(; wa.v('~-t.rm~sf(;r ()I)(;rat()r M(z,, zi) (:(m~(;s I,()1>(; li~(:a.r, m~(l l~(;n(:c l,hc~ rcla, ti(m i [a(I)i(qi, zi) ] b ~ ( q ~ , zi)] t^ aM'I~i(q.i, zi) + bMtPi(q/, zi) = a'I~o(q,,, z.) + btP,,(q,,, z,,),

(4.3.7)

l~()l(ts fl)r all (:()n~t)l('x (:()llStl, l,lltS (I i-I,ll(t D a.ll(1 wa.v(; flm(:ti(n~s 'hi m~(l ~iTI~(; t,I'a,l~sf(;r I'(;la,ti()n (,/1.3.6) (',~1,11t,l~c~ t)c i'cca,st as t,h(; 2 D i~lt(;gral t,ra,nsf()rn~ O.(q.,

z.) -

/ ~5(qo, q,~; Zo, z,~),~(q,~, z~)dq~

(4.:~.S)

which is in(tcc(t tile continlu~lS w;I'sion of (4.3.7) expressing the wa.vc function a,t position qo in the o u t p u t plane as the weighted s u p e r p o s i t i o n of the contrib u t i o n s a,t different positions qi in the input pla,ne. Naturally, a. 2 D integration is implied by (4.3.s), the expression f ...dq~ r e p r e s e n t i n g f f ...dqx~dq.v~. As in the Chat)ter all integrals w i t h o u t limits of integrations are to be t a k e n from - o c to + o c , i n t e g r a t i o n in (4.3.8) is i n t e n d e d to e x t e n d over the entire plane. T h e weighting function O(qo, qi;Zo, Zi) is referred to a,s the point-spread (or impulse 'response) function of tile systeIn. It is tile response of tile s y s t e m to a trim point object, mathema, tically r e p r e s e n t e d by the delta function as ~ ( q - qi, zi) = (5(qx - q.~., zi)(5(q.v - q.w, zi). As m a t h e m a t i c a l r e p r e s e n t a t i o n s of point sources or unit impulsive signals, the functions ~(q, z) p r o v i d e , in fact,

Wave-Optical Picture of First-Order Optical Systems

181

a convenient set of elementary waveforms, into which any optical wave can be decoinposed. Exploiting tile sifting property of tile ~ function, we can write the wave function (I)i(qi, zi) in the input plane as the weighted superposition - /

-

(4.3.9)

the weighting factors being given by the numbers Oi(q~, zi). Inserting the integral representation (4.3.9) into the input-output relation (4.3.6) and invoking the linearity of M, which then can separately operate on the individual unit impulses, we end up with the integral transform (4.3.8). Also, the formal expression for the spread function O(qo, qi; Zo, zi) obtains as O(qo, qi; Zo, zi) -- M(zo, zi)5(q~ - qi, zi),

(4.3.10)

thus confirming that it represents the image distribution generated by the system responding to the point object at (q/, zi). An ideal optical system is expected to transfer a point object into a point image between two planes determined by the system; thereby, the response function of an ideal system with respect to suitably assigned profile planes may be a delta function. An optical system is then completely characterized if its responses to unit impulses placed at all possible points in the input plane are known. The response to any optical wave is obtained by composing such responses with suitable weighting factors as just provided by the incoming wave. Of course, the spre~d function depends on points in both the object and image planes; hence tile forln of tile rest)onse in tile image plane generally varies with tile position of the unit impulse over the object plane. In this respect, it is straightforward to recognize the practical relevance of the space-invariant systems, for which the shape of the image for a given object does not depend on the specific location of the object in the input plane; only the location of the image over the output plane changes as the point object explores the input screen. Space invariant systems are discussed in w 4.6.3. Notably, they have complex exponentials as their eigenfunctions, and hence the Fourier-transform description is valuable for their analysis. It will be proved in Chapter 5 that, in the frame of situations allowing for the first-order approximation and hence for the matrix representation of tile optical systems, the impulse response O(qo, qi; Zo, zi) is expressible in terms of the entries of the optical matrix M of linear ray optics. This establishes a relation between the geometric- and wave-optics pictures (4.a.3) and (4.3.8) of light propagation for the case of first-order optical systems. Evidently such a relation has both a conceptual charm, connecting different mathematical structures, and practical advantages, enabling calculations with operators be replaced by calculations with 4 x 4 matrices, or even more simply with 2 x 2 matrices if the single transverse coordinate depiction is allowed.

182

Linear Ray and Wave Optics in Phase Space

One-dimensional model: the line-spread function In order to ma,intadn the correspondence with the 1D picture of tile previous cha.ptcrs, we limit the (ict)enden(:e of the opti(:al wave fimction to a single profile (:oor(tinatc. E x t e n s i o n to two tra,nsvcrse dimensions is straightforward, a p a r t fronl the unavoidatfle in(:rea.sc in fornm,1 (:omt)lexity. Natura.lly the dest:rit)ti(nl in t(;rms ()f a singl~; transverse Ca,rtesia, n (:()()r(lilm,te is (:onsistent with a. sinq)lc astignmti(: ()t)ti('al systcln, f()r whi(:ll, in fat:t, t,ll(~' 2D tra.nsfcr ()t)('xator ('.i-i,ll ])C st)lit i11t,() I,w() 1D ()i)(;i'a.t,ors: M - M.rM:,j, ('a.(:ll ()t)cra.t, iIlg (hi a, single tra.Ilsv(;rs(; (liIll(;nsi(m. Tlnls, sllt)I)()siIlg ttl(; iIll)llt wav(; fllIl(:ti(nl t)C scI)ara.|)lc a.s w(;ll, i.('.., ~F~(q.r, , q:~j,, zi) -- ~., (q.r, , zi)~,.,/, (q~j, , zi), ()i1(; (:a.~l mmlyz(; t,ll(; t,cllaviour ()f tllc systtull ()n(: (tim(;nsi()Ii at a. tillm (so(; ,~ 4.2). ()f ('.()llrs('~, ill(; 2D (l('~s(:rit)t,i(m ()f t,l~(; sysl,(u~, r(;s~lti~g |)y l)l'()l)(;rly (:(n~l)()si~g t|~(; ()|)tai~m(I 1D (I('s(:riI)ti()~s, is wdi(1 a.ls() f()r ~()~-sct)a.ra,I)h; tyI)(; i~I)~ts. Axis-sy~u~(;t,l'i(: syst,(u~s, I)(;i~g cvi(lc~d, ly SCl)aralfl(;, ~1,i'(~(l(;s(:ril)a|)h; |)y a si~glc tra.~sv(;rsc (~art('sim~ (:o()r(Iinate as w(;ll; l~()w(;vcr, i~ tlm.t (;as(: tim (:ir(:~la.r syllogist;try is n()t f()l'~mlly a.t~(',o~mte(] f()r si~('(; the i~tmr(u~t Ol)(;rat()rs Ira.v(; a, (tift't;rt:~t r(;l)r(;s('~t,a,ti()~ i~ t(WIIIS ()f the i)r()t)cr ratlial (:(~()l'(li~mtc i~ t,l~c l)r(~tilt; s(:rt:tu~. TII(;II w(; (l('al i~ g(m(:ral wiI, t~ ('()~q)l(:x-val~e(l fluwt,i()~s ~(q, z), r(;ga,r(t(;(t as flm(:ti()l~S ()f tim, t)r()fih; t:()()r(li~mtc q, tl~(; axial ('()()r(li~atc z I)(;ing like a, Im.rm~ctcr. Nat~rally, ~(q, z) is S~l)l)Osc(l t,(~ I)ch)l~g t() tl~(: llill)(;rt Sl)a,(:(;/~2(I~) ()f the s(tlla,r(; i~t(,gra,t)lc fl~(:ti()~s (tcfi~c(l ()~ tl~c r('M li~r F()r a.~y tw() fllll(:tiolls

/.

(.~/,,~)(z) -

.~/,*(q,z)~(q,z)dq,

(4.3.11)

~

the ii()rlll of a.~y fimctio~ t)ci~g a,(:cor(ti~gly giv(;~ a.s ~

(z) - (~, ~ ) ( z ) ~

/

I~(q, z)l 2 dq.

(4.3.12)

T h e asterisk as ~s~m,1 (te~otes (:omt)lex (:o~tj~ga,t, ion. As is well k~own, two functions having vanishing inner t)ro(tu(:t, ('~/,, p) = 0, are ret)orted as orthogona], and flm(:ti(ms with unit n()nn, (~, p) = 1, arc said to be ~ormalizcd to unity. In (:onfon~fity with the single transverse coordinate picture, the inputo u t p u t relation (4.3.8) t u r n s into the 1D integral t r a n s f o r m

po(qo,

Zo) - / g(qo, qi; Zo, zi)pi(qi, zi)dqi,

(4.3.13)

and correspondingly the inherent spread flmction, forma,lly given as

g(qo, qi; Zo, zi)

-

M(zo, zi)~(q~ - qi, zi),

(4.3.14)

comes to represent the response of the system to a line source in the input plane at q~ represented by the ~-distribution ~ ( q ~ - q~,z~). Hence it should

Wave-Optical Picture of First-Order Optical Systems

183

properly be called the line spread function. As previously mentioned, as far as propagation through first-order optical systems is concerned, an explicit relation between tile spread function and the ray matrix of geoInetrical optics can be established. In fact, in the next chapter we will be able to relate the line spread function g(qo, qi; Zo, Zi) to the ray-matrix entries A, B, D by a quadratic phase function. For ease of writing, we may simply denote the spread flmction &s g(qo, qi), omitting to display the axial coordinates zi and Zo referring to the location of the input and the output planes, unless of specific relevance to the topic under discussion. As earlier noted, the complex amplitude is related to any one component of the electric or magnetic field of the wave. Therefore, disregarding the precise dimensions, the magnitude squared of the complex amplitude can be interpreted as the optical intensity or tile optical energy density, tile two quantities differing by the light speed in the medium. Then, ignoring the further limits arising from the single profile-coordinate description of the light disturbance, we are dealing with, we may consider the quantity I~(q, z)l 2 as the optical intensity, or the optical energy density. Accordingly, the non negative integral in (4.3.12) relates to the optical power or energy of the wave field. We will consider lossless and gainless optical systems, which may reshape the amplitude and phase of the input wave function without affecting however the energy content. The conservation of the optical power or energy through propagatioi~ is mathematically expressed by the identity of the n o r m s o f the incoming and outgoing wave functions, namely

I]poll (Zo) = Ilp,;ll (zi).

(4.3.15)

this identity signifies the property of the transfer operator M(z, zi) to preserve the norm of the wave function, which is reported as unitarity. Clearly, relation (4.3.15) does not apply to optical systems having limiting apertures, like optical resonators, for instance, or media with gain or loss profiles, which are in fact described by non unitary transfer operators. Needless to say, "physical" signals arc represented by wave functions which are smooth and negligible outside a definite region of space, and so the energyintegral (4.3.12) converges to a finite value. We will consider as well "idealized" signals described by wave functions that are not normalizable in the common sense of the energy-like integral (4.3.12), as, for instance, the impulse and the monofrequency signal, respectively represented by the delta and the linear phase function. Such idealized signals can be handled mathematically as limiting processes of normalizable signals or by resorting to the generalized function method. We shall not dwell on the mathematical details of such procedures, and use the delta and linear-phase functions as powerful tools in a kind of interplay between physical intuition and mathematical rigor. As

~o

-

M)gi,

Linear Ray and Wave Optics in Phase Space

184 r-i i-- l-i 0

l--ll

1-12

FIN_ I

FIN_ 2

r l N -- 1-I o I

i

i Emerging signal

Incoming signal

I

A

q~o

MN-1

(ON

I

i i I

Input plane

q)~

~

qN-:

qN-~

Output plane

F I G I I I / E 4.5. Tim wave trm~sfer o p e r a t o r of a co~nposite(2ptical s ystexn is the pro(luct of the trm~sfer ot)erators ()f tim wtri(n~s ol)tical ele~m~d,s: M - M~ .... M.,M~.

Com, po.sition 'J'uh' fir~" th, c t'm,',,.sfi:'~" opcra, to'~" a,nd the .s'p'rca,d .functio'~,

In ()rde, r t() elal)orate the C()llll)r rllle fi)r the wa,w;-l, ransfi;r ()tmral, or, wr r(;s()rl, 1,() tll(; syst(;zll (h's(:ri|)e(l ill !} 3.2, r ()f N ()l)tical eh'.llwnts (:(mne(:t,(;(t t,()gel,ll(:r i~ s(;(l~('~r fl'()~ zo I,() z~, fr(n~ z~ t,() z~ a.~(l s()()~ ~t) t() z N (Fig. 4.5). I)e~()l,i~g I)y M,(z,, z, ,) t,l~e trm~sfer {)l)eral,()r a,<:('(n~l,i~g fin" tt~(; t)r()l)a,gal, i(ni fi'(n~ z., ~ I,() zj l,l~r()~gl~ l,lie .j-tl~ ()l)l,i(:al ele~le~t,, j - 1, .., N , we ~my fi)ll()w t,l~(; i~l)~t, sig~m.l ~,)(q., z~)) wlfilst, t)r()t)a,gal,i~g l,ln'(n~gl~ l,l~e wu'ious (:()~t)()~ent, s. Tlm~, we write ; , (q,, z, ) -

M, (z,, z,,)~,,(q,,, z,,),

r

M.~(z.~,z,)r

,

(4.3.16)

(I~N(qN' ZN ) -- MN (ZN' ZN-I )~N-I (qN-l' ZN-I )' which s(mIlr as t,lle wave opt, ical w;rsioils of tlle ra.y-ve(:tor relati()Ils (3.2.1). The ()t)ti(:a.1 Im)Imgat(w M(z N, z0) for the ()w;rall system fronl z o to z N results from nnfltit)lyillg tlu; tra,xlsfi;r ()I)(;ra,t()t's ()f tll('~ va,ri(nls ()ptical (;h;ments, and so

M ( z N , z , , ) - MN(ZN,ZN_I) .... M.,(z2,z~)Ml(Z~,Zo),

(4.3.17)

in the same way as the ray ma, trix for the con~posed system is obtained multiplying the ray matrices of the various components (Eq. (3.2.a)). As the matrices, the transfer opera,tors must be a,i'ra,llged in the p r o d u c t in the reverse order to t h a t the wa,ve hits the corresponding elements, and so from right to left,, being the p r o p a g a t i o n conventionally assumed to occur from left to right. It is easy to turn the above composition rule for M into the composition rule for the impulse response of the system, which writes as g(qN, qo)

-- [ ' ' " [ dqN-I ...dql NN(qN, qN--l)"'gl (ql' q0)" J

J

(4.3.18)

Wave-Optical Picture of First-Order Optical Systems

185

The spread flmction of a cascade of N systems is then calculated as an ( N - 1)fold integral of the product of the spread functions of the various components, the integration being performed over all the interInediate planes from 1 to N - 1, if one numbers the input and output planes respectively by 0 and N. We learned iil the foregoing chapters that in general optical systems are "noncommuting" objects, and so a change of the order of the single optical elements in a cascade may change the properties of the overall system. This manifests into the in general non commuta,tivity of the transfer operators corresponding to different optical systems.

Unitarity and inverse operator The unitarity of the transfer operator, which as noted before expresses the condition that the system be lossless, turns into a specific relation obeyed by the corresponding spread function. In fact, let p~(q~, z,) and Po(qo, Zo) = M(zo, z~)~(q,, z~) designate as usual the complex amplitudes of the input and output signals, respectively. The conservation of the norm (4.3.19)

=

re(tlfire.s that the eqlmlity

/'/./" ~t(t~(tq~' (tqog * (qo, q~' ) g(qo, q~) ~ * (q~, ' z~ ) ~ (q~, z~) - / dq~ ~ * (q~, z~ ) ~ (q~, z~ ) (4.3.20) be sa.tisfied for any input signal p~ (q~, z~), thus yielding for g the relation

./dqog*(qo, ql) g(qo, q,) - 5(q~ - q,).

(4.3.21)

In this connection, it is interesting to think, from a purely conceptual viewpoint, of a. cascade of two optical systems whose output distribution be exactly equal to the input distribution. Namely, given an optical system described by the transfer opera.tor M, we question ~bout the existence of an associated optical system, the relevant transfer operator being denoted as Minv, such that cascaded with M produces ali optical systeln that has no effect on the input signal whatever it be, apart from a trivial translation along the optical axis. This is mathematically expressed by the relation A

A

A

U(z1,2Ji)Minv(Zo, Z1) - Minv(Zi, z2)M(zo, Z2) --i,

(4.3.22)

z 1 ail(t z 2 denoting some intermediate points between zi and z ~ deterinined for each sequence, i.e., MMinv and MinvM, according to the axial extension of the system and its inverse. Relation (4.3.22) allows us to identify Mi~v as the inverse of M" Minv = [~--1, and the unitarity of IVl assures the existence of [~]--1, in the salne way A

t".

A

A

A

A

-~

186

Linear Ray and Wave Optics in Phase Space

a,s tile symplecticity of the ray transfer m a t r i x guara,ntees tile existence of the (unique) inverse m a t r i x M -1. This means that the inverse of ally optical system is conceptually conceivable, its physica,1 realiza,bility being however a. m a t t e r ()f system synthesis. The dis(:ussion, we t)resented in w 3.3.1, a,1)()ut tile realizability of the inverse of tile free-medium se(:tion makes (:lea,r the (tifference t)etween the ma,thema, tical existence a,n(t the t)hysi(:a,1 fea,sil)ility ()f a, system.

4.4

From the ray-optical matrix to the wave-optical operator

Wt; l,ra,llslal,e ]lt)w l,llt; llm,l,rix fl)rxlmlis~ll, l)rt;vit)llsly ilhlstratt;tl witllill 1,t1(',(:ox> text ()f tirst-t)r(h:r ray-t)I)tics, illt,t) a,~l ()I)erat, tw fiWIlmlis~ll, st)et:ilic:a,lly a,tle(tlm,te t() t,lle wa.w; ()l)tit:a.l (h:st:ril)titnl ()f ligllt l)rt)l)agati{)ll, tlnls s(;tl, illg a "t)ritlgc" t)etweell l,]le lilmar ray-()I)ti(:s f{)rlIla.lislll, review(;(l ill ~ 4.3.1, ml(l tlm li~mar waw:-()l)tics f~w~a,lis~, i~l,r()(l~u'(;(l i~ .~ 4.3.2. ~I~)l,]~is (;l~(l, w(: will (:xl)hfit the r iza,ti(n~ l)r()('e(l~x'(;, l,~tr]~i]~g a,ll l,l~e w~.ria,l)les ()f ray ()t)l,i(:s il~l,() lil~ea,r ot)era,t()rs [7]. As a natured })y-l)rt)(l~u:l,, l,}~e ray t,ra.~sfi',r ~m,l,ri(:es will })e t,~n'ne(1 ix~t() (h;ti~il,{; waw: l,ra~sf~;r ()l)t:ra,l,()rs, a(:(:()~n~l,i~g f()r l,l~e tra.~sf(w~a,l,i()~s ()f the waw; ('()~q)h:x-a.]~q)lil,~l(h; wlml~ g()i~g l,ln'(n~g]~ l,l~e ()l)l,i(:a.1 sysl,(:~ ()f r [9]. V~r~; sl~a.ll ~()1, (l(;l,~dl 1,1~(' (t~m~l.izati()~ {)f ray {)i)l,i(:s, ]nil, ~t;r(',ly ()~ll,li~(; th{', t)a,si(: stel)S h;a{lil~g t,() tl~(; {,wfl~l,i(n~ e(l~la.l,i(n~ fin" l,l~e waw;-tra,~sfi;r ()l)(;ra.t,or in tl~(' I)a.raxia.1 rdgix~e. Tl~is will ~m.t~ra,lly s~ggesl, ~ts t,t~e strategy 1,(){)t)t,ain t,he ()t)(:ra.l,()r rel)r(:st:nl,a.l,i(n~ ()f tirst-()r(h;r ()l)l,i(:a,1 sysl, en~s, l,ha.l, will l)e ~s(;(t in the next t:lmI)t(;r t,()(h;tl~('(; l,ll(,, fin~(:ti(ma.1 f()rl~ ()f tile Sl)l'(;a,(l fln~(:ti(m of any tm,raxia.1 ()I)ti(:a,l sysl,(;~ in l,(',rn~s ()f tl~e entries ()f the relatiw; ray ~mtrix. As sai(1 earlier, I)ara,xial ray ()t)ti(:s is generated l)y (t~m.(lrati(: l)()lynomials ~(1(;1" P(fiss(n~ I)ra,(:kel,s a,n(l l~latrix exl)(mel~tiati()n. A(:(:()r(li~gly we may (;xt)e(:t that t)araxial wa,ve ()t)ti(:s 1)(; g(;nera.te(l t)y quantized (t~m(lra,ti(: poly~()~fia,ls ~(l(;r l,ie t)ra,('kets m~(1 ()t)erat()r exl)()~e~tia.tion. In fa.(:t, as we will see, tlm r(:lati()~ t~et,w(',('a~ t)a,raxia,1 ray ()t)ti(:s a,l~(1 t)ara,xia,1 wave ()t)ti(:s (;a,n be intert)r(;te(t i~ tl~(: light ()f th(; relatiox~ t)etwee~ tim synq)h;(:ti(: gr()~q) ()f the ray tI'aa~sfer ~na,tri(:es aa~(t the nmtat)h;(:ti(: gr()~t)()f wave tra,nsf(;r ()t)era,tors. The t)resenta, tio~ t)elow is kept (m a nonte(:tnfi(:al level, refining later representa,tions firstly ~lse(t invoking intuition from, and analogy with, classical and q~a,ntmn me('hanics.

4.4.1

Wave-optical position and momentum operators. Hamiltonian operator for parazial wave optics

E x t r a p o l a t i n g from geometrical to wave optics, the canonical va,riables q and p of ray-optics are replaced by tile canonical Hermitian operators ~ and ~ [10]. In the spatial coordinate representation, the position operator ~ acting on the

Wave-Optical Picture o f First-Order Optical Systems

187

optical wave function has the effect of multiplying by the coordinate value q: - q,

(4.4.1)

whereas the momentum operator ~ acts by the derivative with respect to q: __i~0 0 _ 2~ Oq -

~ 0 ko O q

(4.4.2)

We emphasize again that the vacuum wavelength Ao 2~ corresponds to k0 Planck's constant h of the quantum theory. As the Poisson bracket of the classical functions q and p is { q , p } - - 1,

(4.4.3)

the Lie bracket of the corresponding operators ~ and ~ is A

A

i

2- - k0 ,

(4.4.4)

I denoting the identity operator. The operators { ~ , ~ , Z } , with the commutation brackets [fi,~] - i2-, [~,Z ] - 0,

[~,Z] - 0,

(4.4.5)

form the Heisenberg-Weyl algebra [11], which parallels tile algebra spanned by the linear monomials {q,p, 1} under Poisson bracket operation, on which we have briefly commented in w 1.3.2 and w 1.5.1. The quantization procedure proceeds then oil turning the scalar equation of geometrical optics (Eq. (1.2.12)): H(q,p,z)

-

-

-Pz,

(4.4.6)

into the proper operator form A

A

-

-pz,

(4.4.7)

substituting for the Hamiltonian flmction H ( q , p , z) and the z-component of the optical moment Pz the respective operator realizations, ~ and Pz. Since, of course, "

Pz --

i O

ko OZ '

(4.4.8)

squaring both sides of (4.4.7) yields the operator form of the exact (i.e., circular) dispersion relation of the optical momenta (see w 4.2), which then, once applied to the optical wave function, results into tile 2D Helmholtz equation.

188

Linear Ray and Wave Optics in Phase Space

Similarly, taking on the left, side of (4.4.7) the paraxial form of the H a m i l t o n i a n functioIl yields the operator form of the approxilnatc (i.e., parabolic) dispersion relation of the optical m o m e n t a (see w 4.2), which then, once applied to the ot)ti(:a,1 wave finlction, reslllts into the 2D parabolic wa,ve equation. Nt;(;(ll(;ss t() say, the tta,l~filtonian ot)(;rator 7-t(~, ~, z) is ot)tain(;(t from the Ha.nfilt(mian flm(:tion H(q, p, z) ()f ge(ml(;tI'i('al ()I)ti(:s })y m(;a.ns of the slfita.blc (:(n'r(':Sl)(nl(l(;11(:(; I'lll(;. W(; sllall n()t (lw(;ll her(; on tll(; illtl'insi(: (litfi(:lll(,y ()f going fl'(nll a ('lassi(:al fllll(:ti(nl ()f tlle (:a,n(nli('al t)()siti(m mitt lll(nll(:ntllnl varia|)lcs q ml(l p 1,() a,ll ()I)('ra.l,()r finl('l,i(nl ()f 1,11(;(tlla.~ltlml ()t)s(:I'val)les ~ a.Ii(l ~, ('()ns(;(tll(;nt 1,() tll(: ll()ll ('()lllllnlta, t,iw; r(;la,ti(nl (4.4.4), wlii('ti a,lh)ws i~l(l(;(;(l fi)r (lifti;i'(;xit non (:(l~iwd(:~(, (:()rr(:sl)(n~(l(:~('(:s. 'l'l~(: i~t(:r(:s(,(:(l r(:a,(l(:r is (lir(:('(,(:(l 1,() tl~(: r(:f(:r(:n(:es (l~u)(,(:(l i~ [13]. lh:r(: w(: sil~I)ly say (,l~at (:a.(:l~ (:lassi(:al I)Im,s(:-Sl)a.(:(: fin~(:tion f(q, p) is (,ra.,,s('rit)(:(l i,,(,()a. Ih:r,,,itia.,, ()I)(:ra,t()r .f(~, ~), (),,(:(: (,t,(: sI)(:(:ifi(" r,,l(: ()f a.ss()(:ia,(,i()~ l~as l)(:(:~ (:t~()s(:~. In flu:(,, (,l~(:r(: is ~()~l~i(l~U: (l~md, iza.ti()l~ s(:h(:n~(:, (,ha,t (:()~1(1 !)(: (l(:(l~u'(:(1 fr(n~ (,1~(: t)asi(' t)ril~(:iI)l(:s, fi)r assig~i~g a H(:r~fitia,n ()t)(:ra.(,()r fiul('(,i()l~ ()f (,1~(; ~()~('(n~mn~(,ing ~ ml(1 ~ (,() a, giv(:~ ~'-ln~lnl)(;r fin~(:ti()n ()f q m~(l p. As m~ (:xm~q)l(:, w(: ~my ('(n~si(l(:r (,l~(: si~q)l(: ~ix(:(l ~()~()~fial qp. In t)ri~(:il)l(; it (:(n~l(l (:(l~a, lly well l)(: t~n~(:(l i~d,() tim ~()l~(:(l~fival(:~(, ()l)(:rator f()r~s ~ , ~ ()I" ( q p + p q ) / 2 . Si,~(:(: (,I~(, lirs(, (,w() f()r1,~s (I()~()(, l(:a
is n()(, a, s~fli(:i(:~(, (:I'i(,(:ri()~ f()r ~mi(l~u:ly (l(:(,(:rndning (,I~(: ()l)(:ra(,()r fim(:ti()n t() wl~i(:l~ tl~(: (:lassi(:al 1~()~()~ia.l q2p2 ~igl~t l)(: (:a.I'ri(:(l t(). U~i(l~U:~(:ss ()f th(: (:()rr(;sl)(),,(l(:,,('(: f(q. p) ---" .[('~, "~) is a.ss,n'(:(1 (),~ly wi(,l,i,~ a l)arti(',~lar r,ll(: ()f a,sso(:ia,(,i()~, tit:r(: w(: ai)t)ly t}~(: Weyl r~l(: |)e(:m~s(: ()f i(,s i~d,il~m,t(: r(:la.tio~ to (,lit: Wig~(:r r(;i)i'(:s(:~d,a,(,i()l~, (,lm.t is th(: ~a,i~ t()I)i(" ()f (,1~(: f()r(,l~('()nfix~g (~hat)t(:rs (see in I)arti(:~la, r w 8.2.3, a,n(l the refi:r(:nc(:s th(:r(: a
A

A

Wave-Optical Picture of First-Order Optical Systems

189

nomials, except for the special classes of fimctions which are linear in one of the two canonical variables, and so are like qf(p) + g(P) or pf(q) + g(q). We remind the reader that the geometrical optical Hamiltonian H(q, p, z) pertinent to the propagation through first-order optical systems displays separate quadratic contributions from the ray variables as (w167 1.6-1.7)

1 p2

H(q, p, z) - 2no(,)

+

n2(z)

2 q2,

(4.4.9)

the meaning of the medium parameters n o and n~ being elucidated in w 1.7.5. The quantized form of the "classical" Hamiltonian (4.4.9) is then uniquely obtained according to the correspondences

p2___~ ~2,

q2____~ ~2,

qP-

1 5(qP+P'q),

(4.4.10)

which hence yield the Hamiltonian operator appropriate to first-order wave optics in the Hermitian form A

A

"]"~p . . . . ial

4.4.2

(q, P, z)

-

H(~,

p, z)

1 ~2 + n~z)~2. 2no(z)

(4.4.11)

Evolution equation for the wave transfer operator in the paraxial regime

Tile evolution equation we will deal with takes therefore tile form i 0

ko

~-

Oz,(q, z) - H(~, ~, z)p(q, z),

p(q, zi) - Pi(q),

(4.4.12)

where ~i(q) specifics the light wave distribution in the input reference screen. On account of the explicit functional form (4.4.11) of H(~, ~, z), the reader may easily verify the correspondence of the above equation with tile 2D parabolic wave equation (4.2.16) in quadratic media. Accordingly, by (4.3.6), Eq. (4.4.12) yields the equation for the wave transfer operator M(z, zi) in the paraxial regime, i.e., A

A

/'---0 M(z, zi)- ~t(~,~,z)M(z, zi), k o Oz

M(zi,z i ) - [

(4.4.13)

which represents the wave-optical version of the ray-matrix equation (4.3.4). As already noted, matrices and operators are strictly similar entities. Within the context of the correspondence between (4.3.4) and (4.4.13), matrices are intended to act on the geometrical phase space of the ray vectors, as operators are intended to act on the Hilbert space s (R) of the square integrable optical wave functions. Therefore, the analysis elaborated in w167 1.4.1 and 1.7.2-3 in relation to the solving techniques of the ray-matrix equation (4.3.4) is adequate

Linear Ray and Wave Optics in Phase Space

190

to the wave-operator equation (4.4.13) as well. In particula,r, if the Hamiltonia,n operator H(~, ~, z) does not explicitly depend on the axial coordinate z: H(~, ~, z) - H(~, ~), exponentiation of H over the finite propa,gation distance z - z i can loosely be perf(wnmd, thus giving the sollltion to (4.4.13) in the forth M ( z , zi) - ~e-i~:,,a(~-z~).

(4.4.14)

In fact, rel,ra,(:ing the {tist:llssi()n t)rc,scnt(~(t ill ~ 1.4.1 a,n(1 1.7.2-3 (re('alh~(t in 4.3.1) ()~ tl~('~{~v()l~ti(,~ (~(l~m,ti()l~ (4.3.4)a,~(t tt~('~(~xI)()~e~tial ~m,t)t)i~g (4.3.5)a,s rela,tiw~ s()lvi~g f~)r~, w(: ~m,y say ttm,t 1)y (4.4.13) 1,1~(;l,ra,~sfi~r ()t)era.t(w M(z, z~) fi'{m~ zi 1,() z a,riscs fl'(m~ a,~ intinil,(~ sl,ring ()f ~l~il,a,ry ilfli~il,(;sinml {)I)(;ra,t()rs {)f 1,1~{; typ(~ (,-i~.H(:),tz fi)r s~u:(:(~ssive (~l(u~(~ta,1 i~(:rm~(~l,s dz ()f l,t~e (~v{)h~tion I)a,ra,~eter fl'{)~ zi ~l) I,() z. IIe,~(:e if H {l{)(~s ~()1, (lel)e~(l ()~ z, 1,1~(~{w(lere(t tm)(t~(:t ()f all l,l~()sr, i~ti~itesi~ml ~fita.ry ()l)era.l,(ws (:()~l)i~(~s I,() tim ~n~it,a,ry si~@e-exI)(n~{~tial fiW~l~ (4.4.14). Evi(t(n~tly, ('~xI)r(~ssi()~ (4.4.14) (:(n~l)lctcs the (:()I'r(;sl)t)~(t(;~t't;s witl~ ray ,)l)tit:s, t)(;iI~g it, tl~(; 'wave-opti(:al w;rsi()~ ()f (4.3.5). ()ira nectls ()~ly s~|)stit,~m;-ik,,H fl)r H i~ the exl)ressi()~ fl)r tlm ray-tra.nsfcr ~m.trix, wl~el,l~(;r we (:()~si(l(;r I)r( )l )a ga l,i(n~ ()v(;x" i~ti~it(:si~al ()r Iil~it(; (listm~(:c. Tt~e (,~(;-1,,,-(,~(; (',,rresl),)~(le.~,(:(;s (4.4.10)m,(l tt~(; (:xI)()~e~tia.1 la,w (4.4.14) rct)rescnl, l,lm w;ry l)a,si(' (li(:ti(nmry t,o tral~slal,e l,lm ray ()l)l,i(:s (h;st:rit)tit)n of first-tml(;r ()l)tica,l syst(:~s i~t,() tlw. la,~g~agc a,l~(l fiw~m,lis~ ()f waw'~ ()l)tics. As a. ti~a.1 (:()l~n~m~d,, we ~()t,(; t,tm.t, a. ra.y-t)t)t,it:s I)ase(l (l(;sc.ril)ti()~ ()f light f()I'ma,tioI~ |)~lI, la(:ks t|m t)lm,se inf()i'~na,ti(m. I1~ fi~,(:l,, l,l~(~ l~(n~()g(n~(~()~s t(~rn~ 'n0(z ) a.risi~g ill t,]l('. (tlla~lra,ti(: a,t)I)roxil~m,ti()~ ()f l,]~('~ ()t)tit:al IIm~ilt()nia~ is Inerely (tr()t)t)(;(l ()t~t, ill ray ()t)t,i(:s, in a.(:(:or(l with t,lie flu't, I,ha, t a,s a l()(:al the()ry geonietri(:al ()I)t,i('s ('a.~ slfift t,tm origili of t,lm (ql(:rgy. hl(l(',(;(l, tliis term is uninfl~m~ltial t() ttm ray I)a,tti cvol~tioIl, si~lce its IAc ()I)era,t()r is z(:ro. If retained in tim "cla,ssi(:a,l" tla, miltonia,n function a,n(1 then t)asse(t into t,hc "qua, nt,,ms Hm,,ilt,(,~,im, ()I)(:ra.t()r, it, wo,fld yi(;l(t t,h(~ a(tr tern, 89 in th(; 1D (;w)l~titn~ (;(t~m,tit)~ (4.4.12). ()f (:()~rs(;, this tt;n~ (:()~1(t t)(; ~d)sorbcd into t,hc wave a,nq)lit~ute t,l~ro~gh the phase factor c i ~ '[~ ,,o(~)d~, which, once properly co~nt)oscd a,ccortli~g to the decomposition scheme (4.2.15) with the identical phase factor associated with the other profile coordina,te, would yield the pa, raxia,1 representation (4.2.8) of the wave field. The axial phase shift, is not a,cco~lntcd for by the ray optics description of light propagation, and hence is not recovera,blc from the ray m a t r i x formalism. In situations where phase information are rcq~fired iIl addition to amplitude information, as, for instan(:c, when studying holography or optical resonators, the on-axis phase i~o "f~i no(z)dz shift c must be separately calculated and added to the complex amplitude ~(qx, qy, z) to obtain a complete wave optical representation of the paraxia, lly p r o p a g a t i n g disturbance. i:

Wave-Optical Picture of First-Order Optical Systems

191

The wave transfer operator from the perspective of quantum mechanics

4.4.3

The correspondence of the transfer operator M(z, zi) to the time evolution operator U(t, ti) of quantum mechaifics is naturally suggested by the transfer relation (4.3.6) as well as by the evolution equation (4.4.13). As the propagation operator M(z, zi) transfers the optical wave flmction in the plane zi to the optical wave function in the plane z on account of the optical properties of the medium, likewise the time evolution operator U(t, ti) transfers the system wave function at time ti to the system wave function at time t on account of the dynamical features of the problem: A

t) -

(4.4.15)

u(t,

where x shortly signifies the appropriate set of system coordinates It is well known that O(t, ti) obeys the SchrSdinger equation

ih-avO(t, ti) - H(x, t)U(t, ti),

U(ti, ti) - I

[10]. (4.4.16)

o~ A

where H(x, t) is the Hamiltonian operator of the system. As said earlier, Eq. (4.4.13) for M(z,z~) is in a, sense the "SchrSdinger equation" of wave optics. The similarity between M(z, zi) a,nd U(t, ti) involves other mathematical aspects; indeed, it reflects the similarity of the physica.1 problems the two operators describe: propagation of light waves and dynamical evolution of quantum states. In this connection, we recall that for any linear operator 0 acting on the Hilbert space s of the squa,re integrable functions p(q) defined on the real line, the hermitian conjugate (or, adjoint) operator 6t obeys the relation A

(~b, 6 t p ) - (6~, p),

(4.4.17)

involving the inner product (4.3.11) between flmctions. The relation between the operator O and its hermitian conjugate 6t turns into specific properties of utmost relevance to the physical interpretation of 0 itself, and hence to its role within the physical context of concern. It is well known that Hermitian (or, self-adjoint) operators, for which the formal identity 6 - 0*, (4.4.18) holds, may correspond to physical observables, a,s their eigenvalues being real can be interpreted as possible values of dynamical variables. Strictly speaking, self-adjoint operators are characterized by the further condition that the domain of O* is the same as that of O; throughout the book we will consider the terms Hermitian and self-adjoint as equivalent.

Linear Ray and Wave Optics in Phase Space

192

On the other hand, u n i t a r y operators, characterized by the relation ~)-1_

(~,,

(4.4.19)

have the distinglfished p r o p e r t y of preserving the inner produ(:t t)etween fllnCtions. Hr the n o r m of the transfornmd flmction O p is the same as the n o r m ()f p. I~ synfl)t)ls we may writ,(; (or',,

-

(,1,,

-

(4.4.20) A

As rellm.rke(l <'arlier, tlle wave trmlsfi,r ot)era.t()r M(z, zi) fin" gaillless and lossh;ss systelllS is llllil, ary. Ttw, time r162 i()II ()I)erat()r U(t,.. ti)f()r I1()I1 (lispersive (tlla, lltlllll sta, tes is llllitttry as well. T h e lmil, axity ()f M a,n(t U is l,lle nm.thenm,ti(:al (;xl)rt;ssi()ll ()f l,ll(: ('()zlserwttit)n ()f l,he t)I)t,it'al t)r ttllx all(l l,lle t)r()|)al)ility I,() tilul tim (llm.~d,lllll sysl,e111 ill rely r ()f its t)<)ssil)le (lylm, llfit:al states. It (lire(:tly fiflh)ws fr()~ the relewmt ev(fl~ti(n~ er i()~s (4.4.13) aJ~r (4.4.16), a.(:(:or(ti~g t,) wl~i('l~ |)()tl~ M a.~(l U res~lt fl'()~ l,tw, eXl)(nm~d, ial of a IIernfitian ot)era.tt)r |)y a InlI'(;ly i~agi~a.ry t)aranmter, h~ fi~,:t, if i~ g('amra.1 6 - (.irH, with w real aJ~(l H llen~fitia~, it, folh)ws t,lmt (5 t ( '-ir~t -'irH l Evi(le~d,l.y, ~ i l , arity l)lays wil, hi~ t,l~r wa.vr fra,lllewt)rk l,l~(; stone role a.s syl~l)le(:l,i(:ity l)lays wil, l~i~ tim g(;()l~mtI'i(:al-()l)ti(:al (:()~d,exl,. I~ l)arl, i(:~lar, (:()n(:en~i~g l)rol)a,gati()~ t,l~rougt~ first,-orr ot)ti(:al syst, c~ns, tim tra(:ele, ss nmti'ix H (',r t() tt~(', Ih'n~fitian ()t)('.rat()r H, rest)('.('tively (:hara(:terize(t by the t)erl, i~m~d, rela.l,i(ms --

JH+H

TJ - 0

~

-:~

H -

Hr.

~"

--

O* " - -

9

(4.4.21)

Also, as ext)~melltiatioll ~)f H l)y a real t)arallmter generates the ray-t)r{)l)agating matrix, likewise ext)olmld;ia.tion of H t)y a purely inm.ginary t)arameter generates the wa.w;-t)rot)agating ot)erator. As an example, for an optically homogeneous ine(timll ahmg tim z axis, we have. the correspon(h;nce c H(*-z~)

~

c -i%H(z-~)

(4.4.22)

between the ray-optics and wave-optics transfer o p e r a t o r s from zi to z. 4.4.4

Further exploring the correspondence between ray and wave optics pictures. I

According to the matrix relation (4.3.5), the evolution of the ray variables is ruled t)y the Lie o p e r a t o r k H - {., H } , with which the m a t r i x representative H is associated via Poisson bracket operation. In contrast, as displayed by (4.4.13), the evolution of the wave function is ruled by the Hamiltonian

Wave-Optical Picture of First-Order Optical Systems

193

A

operator H(~, ~, z). This seeming incongruence between the ray and waveoptical description of light propagation originates from the fact that ray optics is concerned with the evolution of the ray variables (q, p) whereas wave optics is concerned with the evolution of the wave function ~(q,z), i.e., with the Schrhdinger-like representation of the light disturbance. The correspondence between the ray and wave optics pictures looks straightforward once investigated from tile viewpoint of tile Heisenberg operators ~(z) and ~(z), whose evolution is ruled by i d k0 dzq( ,:

) -- [q, a(q,p, -

k o dz

~

z)],

(4.4.23)

~

which stand out as the wave-optical form of Hamilton's equations of ray optics. Hence, in the light of the optical Schrhdinger equation (4.4.12), the (paraxial) "propagation" of ~(z) and ~(z) through an optical system is formally interpreted in the sense of tile similarity transformations (by the inverse operator ~(z) - M-l(z,.. ~(z)

-- M-I(z,

zi)'~(zi)M(z,.., zi), zi)'p(zi)M(z, zi),

(4.4.24)

of the SchrSdinger operators (4.4.1) and (4.4.2), accordingly regarded as the "input" operators "~(zi) and "~(zi), i.e. "~(zi) -'q and "~(zi)-'p [10]. We have learned in w 1.3.2 that operators (;an be associated with flmctions by Poisson bracket. In an exactly similar manner, operators can as well be associated with operators by Lie product. Indeed, the operator

adO - [-, 0].

(4.4.25)

generated by an operator 0, which acts as the right commutator bracket with 0, is called the adjoint operator of O [14]. Introducing the operator-valued vector ~ r ( z ) - (~(z), ~(z))T, Heisenberg's equations are then recast in a form similar to (1.4.3) for the ray vector u; namely, i d ~ko dz @(z) -- adH(z) @(z), @(zi) - wi, (4.4.26) where, of course, "wi-(~(zi), ~(zi)) q-. Therefore, the evolution relations (4.4.24) for ~ and ~ can coinpactly be expressed through the solving form of (4.4.26). In particular, if H is z-independent, one has ~r(Z)

-- e -ik~

VCi.

(4.4.27)

The reader may prove that (4.4.27) is equivalent to the similarity transformations (4.4.24) with the transfer operator H being given the pertinent functional A

Linear Ray and Wave Optics in Phase Space

194

A

form (4.4.14) (Problem 2). N a t u r a l l y the exponential operator corresponds to the power series

C- i k ~

A

(,-- iko (z-- z~ ) adH

- ~

(-~ ~

( z - z~)J(adH) :~,

(4.4.28)

j=O 1--

A

A

A

th(: I)()wer (adH)J - [[...[[-, HI, HI, ...], H] (lenm,n(ling f()r th(: j-f()l(l (',()x~m~1~tators wi(,l~ H. Tl~(:r(:f()r(: (,he (:v()h~ti()n r(:lati()n (4.4.27) exl)li(:itly n~(:ans

T h e a,lmh)gy witll (,h(: (:v()lll(,i()ll r(:la.ti()ll (1.4.4) wli(:r(: (,If(: lfi(', tra.nsfl)rma,(,i(nl (1.4.5) a(:ts (nl (,If(: ray varia.l)l(:s ()f g(:()]ll(:(,l'i(:al ()l)ti('s is (:vi(l(:ll(,. Tll(: (lir(:(:t a.lm.l()gy witll (,ll(: f()nlm,lisIil ()f ray ()l)(,i(:s r(:Iim,ills ill s()i11(: s(:~is(: lfi(l(l(:n, sit1('(: wa,v(: ()I)(,i('s (h:als witll th(: wav(: finl(:ti()ll ~(q, Z) ratll(:r (,Im.n with th(: l)()si(,i()ll a.n(l lll()In(:nt, ltlll ()l)s(:rwd)h:s ~ aal(l ~. Th(: wav(: fiUl(:(,i()ll (l()(:s not ]my(: a, (liI'(:('t (:()l~t(:I'l)a,rt wit, l~i~ ray ()l)ti('s, tl~(: l()(:al ('l~a,ra.(:t(:r i~l~(:i'(:~t in tl~(: ra,y I)i(:t~Ir(: ('()~tI'a,sti~g witl~ t]~e ~()~ I()('.al (:]lara.(:t(:r ()f t]l(: wa.v(: fin~(:ti()n l)i(:t,u'(:. Locality is 1)ar(,ia.lly I'(~,,:,,v(;r(;,| is, (,1,(: ,,,(:a.,, wd,,(:s ( ~ ) a l,(l {~) ()f the ()l)(:rat()rs ~ aa~(l ~ i~ (,11(:sl,a,(,(: r z), wl~i(:l~, as w(; will s(;(: i~ (',lml)(,(:r 5, evolve (,lu'()1~gh (,1~(: laws ()f ray ()l)(,i(:s i~ (,l~(: (:()~si(h:r(:(l (:as(: ()f (l~m,(lrati(: lla,nfil(,()nia,n, a.~(l s() r(:lativ(:ly (,() (,I~(: I)aI'axial I)I'()I)a,ga(,i()~. h~ atI(li(,i(n~, i~ w 5.5. I i(, is sh()wn (,ha(, f()r (l~m(Ira(,i(' llal~fil(,()~fia,l~ ()l)(:ra,(,()rs like (4.4.11) ~m(,rix r(:l)r(:s(:~d,a.(,iv(:s (:a,n l)e ass()t:ia,(,(:(l witl~ (,l~(: a(l.j()i~d, ()I)(:rat()rs (4.4.25).j~s(, al)l)lyi~g a. r~le sinfilar t() (1.7.7) fin" Li(: ()l)(:ra,(,()rs, witl~ fiu~(:(,i()~is l)(:i~tg r(:l)la(:(:(l l)y ()l)era.tors a,n(l P()iss()l~ ])ra('k(:(,s l)y Li('. ])ra,(:k(:(,s.

4.5

Eigenfunctions of (:t and if point-like and spatial harmonic waveforms

We give a, (:loser look at tim eigenflmctions of the position a,n(l n l o m e n t u m ot)erators ~ mid ~. This is a, genuine qllalltlllll niechanica,1 problem, tha, t we review wittiiIl tim ot)ti(:a,1 (:()xltext. So, tile t)osition eigenstates ~(q), as we said (;a,rlier, are i11tert)retcd as representing unit impulsive signals, generated by s()llrces con(:e~tra.ted into a t)()int. Correspondingly, the m o m e n t u m eigenstates c ik~ are. interpreted as ret)resenting plane waves, the relative m o m e n t u m eigenvalues st)ecifying the direction of p r o p a g a t i o n of the wave. B o t h the delta and linear phase functions can be regarded as "elementary" signals, into which any optical d i s t u r b a n c e can be decomposed; a p r o p e r t y already exploited in w 4.3.2 to express the wave-optical i n p u t - o u t p u t relationships into the integral form (4.3.13). Thus, we are naturally led to introduce the Fourier integral as

Wave-Optical Picture of First-Order Optical Systems

195

an alternative representation of the optical disturbance. Both ~ and ~ have a continuous spectrum of eigenvalues [10]. 4.5.1

The impulse function" 5(q)

The eigenfunction )gq,(q) of ~ belonging to the coordinate eigenvalue q~, for which q~q,(q) - q'~q,(q), (4.5.1) is represented by the delta function: ~q,(q) -- ~ ( q - q'). We will not detail the mathematical aspects of the theory of the distributions, or generalized functions, of which tile &function is an exatnple [15]. We will sitnt)ly use the defining properties of the 6"-function, whose intuitive understanding rests on the rough description as a "function" having an infinite height and zero width such that its area is equal to unity, as formalized by

8(q - q') -- O,

q # q';

f (5(q)dq - 1,

(4.5.2)

which evidently cannot describe an "ordinary" function. In conformity with the view of the ~-filnction being highly concentrated in one point, it is frequently used as mathematical representation of unphysical idealized quantities exhibiting "point" or "line" characteristics, as point- or line-like sour(:es, or any other quantities highly localized in some coordinate systeln. The ~-fimction is as well defined by the remarkable property of sifting out the value of any suita,bly continuous function p(q) in a single desired point, according to the integral relation ~(qo) - / ~(q)~(q - qo)dq,

(4.~ 3)

which suggests the conceptually and practically pleasing possibility of representing •(q) as a superposition of a large number of suitably positioned and weighted point-like functions. Tile derivatives of the delta, flmction exhibit tile same sifting property. In fact, being / ~(q)~(-~)(q _ qo)dq _ (--)m~(m)(q0) '

(4.5.4)

the m-th derivative ~(m)(q) sifts out the value ~('~)(q0) of the ruth derivative of in tile desired point q0, provided of course this value exists. The delta runetion is an even function: ~(-q) = ~(q), whilst the derivatives are alternatively odd or ~v~n: ~(~)(-q) - ( _ ) m ~ ( m ) ( q ) . Also worth mention is the relation

/

6(q - 4 ) 6 ( q - q")dq - 6(q' - q"),

(4.5.~)

Linear Ray and Wave Optics in Phase Space

196

whi(:h along w i t h the s u p e r p o s i t i o n integra,1 (4.5.3) allows llS to consider the ensemble of the eigenfunctions { Fq, (q), q' E I~ } a,s a, c o m p l e t e o r t h o n o r m a l set. I n t e r e s t i n g l y the ~-flm(:ti(m (:an |)e visualize(t, and even forma, lly defined, as the limit of a, sequence of a,t)t)rot)ria,te flmctions of (te(:rea.si~g width, in(:rea,sing height a,n(t unit a,rea,. In t)a,rti(:~flar, we re(:a,ll the (h'finition ()f the (h;lta, flm(:tion a,s limit of a, s(;q~('~n(:e of rea,1 or (:(m~t)h'x Gaalssitm flm(:ti()~s a,(:(:()r(tiI~g to 2

~(q) - l i ~

~

-. - '~-

the [)a,ra,n~eter a 1)ein~; a,llowe(l t,~) |)(' real ~)r [)~r(:ly in~a,gi~a,ry. Fina,lly, it is ~l,(:w~rl,l~y l,l~(: i~d,(:gra,1 r(:t)res(:~t,a,l,i()I~ ~(q)-

,/

~'+i"qd', I,

~.

(4.5.7)

('stal)lishi~g a rt;lal,i~)~ ~)f F~)~ri~;r (x)~Lj~lga.l,i~x~ l)(,t,w~;~,~ 1,1~; ~h;lta. f ~ ( ' l , i ~ a n(t tl~(, lmrl~,,~fi(' fl~('ti,,~ (.• (see ]';(IS. (4.6.1)a,~(l (4.6.2)). E(t~ati,,~ (4.5.7) (:a,l~ l)(, (t(:(l~l(:(:(l fl'()lii i,]l(: r(:l)r(:.s(:lita.l,i()li ()f ,,',(q) a.s liniit, (~f a, s(:(t~l(:ll(:(: ()f sill(: fllll(:l,i()llS: b ( q ) - Kli~l~ - ~ ~i,,(Kq) ,rq ( l)r()l)h:lll 3) As a, l)la,il~ (:()llS(~(tll('.ll(~O. ()f (4.5.7), "

a(''') (q) - (~)'"

/.

2~r .

4.5.2

q,,,

~'+i'tqd'q"

(4 5.8)

7'he spatial harTnonic ~l,ave~foTwt: c i'q

Tlle r162

In'Critic;In fi~," l,llr lll~lllr

~q~r162

~ -

o is r ~:,~Oq

t)y t,h(; (xpla, titm i 0

Oq ,, (q) -

which is solv('(| |W the 1D ha,nn(mi(: wa,vr of ~ axe t,he~l c~)ilv~:nient, ly w r i t t e n a,s _

(q),

(/ikopq.

1 ,,~,,q ,

(4.5.0)

T h e eigr

~ . (q)

(4.5 10)

a b s o r b i n g t,h('~ eigenva, lue p into the t)rodu(:t L:0p -= t~. Clea,rly the eigenvalue p, and so a:, is real, o t h e r w i s e p~(q) will |)low up either as q ~ - c ~ or as q ---+ +cx~ 9T h e fa.(:tor ~ 1 ensures tha,t the functions {p~(q), t~, E R} fulfil the orthogona,lity rela.tion

f"

(q)dq - ~(t~ - tJ).

(4.5.11)

C o m p a r i s o n of (4.5.10) w i t h (4.2.6) suggests to i n t e r p r e t the m o m e n t u m eigenf u n c t i o n p~ (q) for any ~ as the profile of a plane wave along a chosen transversal c o o r d i n a t e axis, let us say qx, in a designa~ted reference plane at some z = zi.

Wave-Optical Picture of First-Order Optical Systems

197

Propagation direction of the w a v e f r o n t

= no kosin =k$ kz = nokocos~9

=k

z

F I G U R E 4.6. T h e s p a t i a l f r e q u e n c y as t r a n s v e r s e wave n u m b e r .

T h e harmonic profile (4.5.10) can be continued into a 3D plane wave, which m a y be supposed to p r o p a g a t e t h r o u g h some m e d i u m of uniform index n o in a direction uniquely identified by the eigenvalue ~ for a given v a c u u m wave n u m b e r k 0. T h e a m p l i t u d e of the wave can in fact be w r i t t e n as (Eq. (4.2.6)) ~2,~(qx, qy, z) -- ~1 ei[,~q~+k~(z-z~)] ,

(4.5.12)

with then (Eq. (4.2.3)) ~2 + k~ _ n o2 k ..2o -

k2 .

(4.5.13)

T h e eigenvalue ~ identifies the transverse c o m p o n e n t of the wave vector, so t h a t , if ~ is the angle of the wave vector relative to the z axis, h: is (Fig. 4.6) = k s i n t9 = kon o sin tg,

or

p = n o sin tg,

(4.5.14)

thus yielding a direct relation of tile eigenvalues ~c to tile directions of propagation in any chosen meridian plane t h r o u g h the z-axis in conformity with the 2D picture we are considering. Also, as the direction of the wave vector is the direction of the light ray in the geometrical optics picture, we are enabled to establish a direct correspondence between the eigenvalue p of the m o m e n t u m o p e r a t o r ~ and the ray m o m e n t u m p - n o sin t9 of geometrical optics. Interestingly we see from (4.5.13) t h a t kz - •

o - ~2.

(4.5.15)

In order t h a t kz be real the range of a t u r n s out to be b o u n d e d by n0ko, 2 T h e s p e c t r u m of ~ cannot e x t e n d b e y o n d the ini.e., n 2 _< no2 k o2 or p2 _< n 0. terval [ - n o , no] , fixed by the refractive index n o of the medium, just as the range of the "classical" variable. However, in conformity with first-order ray optics, paraxial wave optics assigns small values to m o m e n t observable ~. T h e eigenvalues of ~ are supposed to r e m a i n well confined within the n a t u r a l range [ - n o , no] , so t h a t one can consider the s p e c t r u m as it e x t e n d e d over all the real line, thus avoiding the difficulties inherent in a limited s p e c t r u m . In fact,

Linear Ray and Wave Optics in Phase Space

198

ac(:or(ting to tile Stone-yon Neuma,nn theorem, tile SchrSdinger realizations (4.4.1) a,nd (4.4.2) in s the ca,noni('a, lly self-a,(tjoint ot)erators ~ and ~, fl~lfilling the Heiscnl)crg-Wcyl c o m m u t a t i o n rclati()n (4.4.4), hohts only if b o t h ot)crators arc unt)()undcd. A('(:()unt of the li~llitc(t st)c(:trunl ()f the optical m()nle~lt~nl, a,n(t hen(:e of the gl()|)a,1 range of tile ra,y (tirecti()ns over the flfll st)her(~ t)(W()n(t t)()th tim I)araxial an(1 nletaxial regi~lles, ha,s le(t t() the fornl~fla,t,i()n ()f the E~l('li(lea,n ii~()(lel ()f ()l)ti(:s, t,tl('~ ~ul(lcrlyi~lg algct)ra, t)cing the E~l(:li(t(~a~l a lg(~l)i'a, i,,~o(3) (i.so(2)) iilstea,(l ()f th(~ Heis(~lt)crg-W(,yl a.lg(,t)ra, [16]. In th(~ t)araxia,l a,I)I)r()xi~m,ti()~l tile ext)ressi()~l ()f ~', ()r p i~l tenlis ()f the~ (tire(:ti()~l ()f tt~(~ ray si~q)lifies as ~: -

k()p _~ ],',,n,,'d,

()r

p _~ '~(,d,

(4.5.16)

t)y tlle a,ll()we(l a,l)l)r()xil~m,ti()~ si~ 'd _~ 'd f()r 'd << 1 (Fig. 4.6).

4.6

Spatial Fourier representation of optical wave fields

Pa.ra.llelillg r(~l)r('s(:lltati()ll (4.5.3) ()f tll(~ wav(~ mlq)litll(te ill tenllS ()f the &flnl(:ti()llS, W(' ('ml (le('()~lll)()S(: ~ ( q , z ) i~ |,(WlllS ()f |,]l(~ fl'e,(l~m~('y ('ig('~wa,ves ~ . (q) f()r a,~W [ix('.(! va,l~a('~()f tl~(' axial I)a,ra~(:t(:r z. W(' writ(; I,l~(:~ t,l~(~ i~t(~t4ral ,

r

z) - ./" ~(,~-, z)~, (q)d~:

1

,/~.

~(h:, z)~ ,iKq dh:,

(4.6.1)

whi(:ll, ill tlw, ligllt ()f tile t)llysi(:al siglliti(:all(:('~ ()f ~. (q), lla,s tile lllea,llillg of a, (te(:(mq)()siti()ll ()f tll(; 2D ()t)ti(:al wa,vc flm(:ti(m ~(q, z) a,(:rt)ss a.lly a.ssigne(| z-plane i]lto 2D i)lmm wave, s tra vellillg a.lollg sliglltly (tiffere, llt (tire(:tioils, (lefinitely si)e, cifie,(l |)y l,ll(: va.llms ()f h" for a give,]l wave, ]umfl)er ];'= ];'o'no. The, wcightillg fa(:t()rs ~(h', z), I'(',ga,I'(t(',(t a,s a flnl(:ti()ll ()f t~:, f()rlll ttm, Fourier tran,s'form of ~(q, z), wlli(:|l t)y (4.5.11) sha,t)es ill turn a,s the illtegra,1

~(~-,~) - /"9 ~* .(q)r

,)~tq - v ~ .

/" P(q' z)c-i~qdq"

(4.6.2)

T h e =sig~l over a flnlt:tion sym|)ol will be llse(1 t h r o u g h o u t to identify the Fourier tra,nsfornl of the flm(:tion. In the (:onun(m language of Fourier a,na, lysis, intcgx'a,1 (4.6.1) is referred to a,s the i'nv(~'r,s(: Fourier tra,nsform. Note tile s y m m e t r y of ext)rcssions (4.6.1) a,n(t (4.6.2) with respect to the variables q, ~. T h u s the cigcnvahm h; acquires the signifi('a,nce as the Fourier conjugate variable to q; indeed, it is reported as spatial f, eq'uency. In addition, the Fourier t r a n s f o r m (4.6.2) is also referred to as the angular spectrum, since it describes the directional content of the signal a m p l i t u d e across any transverse plane in terms of plane wave components, ca,ch of which, according to relations (4.5.14) or (4.5.16), can equivalently be labelled by either the spatial frequency ~ or tile angle 0 of the direction of p r o p a g a t i o n of tile wave relative to the z axis.

Wave-Optical Picture of First-Order Optical Systems

199

Remarkably both ~(q, z) and ~5(~, z) provide a complete representation of the light disturbance separately in the spatial coordinate and the spatial frequency domains, respectively. Paralleling the interpretation of I~(q,z)l 2 as indicative of the energy density in space, we can relate the squared modulus of ~5(~,z) to the energy density per unit frequency, thus understanding I~(~,z)] 2 as indicative ofthe optical energy density spectrum. Evidently, [~5(~, z)] 2 tells us which spatial frequencies (i.e. directions) are present through the total extent of the waveform at every z plane. Of course, no indication can be drawn from ~5(~, z) about precisely where in space such frequencies occur. The physical ideas, that have suggested to, and the mathematical machinery, that has been developed in order to elaborate a joint space-frequency description of the wave field will be the subject of Chapters 6 to 9. Finally, we like to note that the angular spectrum ~5(~, z) mirrors the momentum wave function of quantum mechanics, the Fourier transform operation being as well the intertwining tool between the position and momentum representations of the quantum system state. It is in fact obvious that we can equally well describe the optical wave field in the momentum space, with the releva,nt rela,tions colmecting the q- and p-space being, of course,

@(p, z)eikoPqdq, ~(q,

Z)C-ikoPqdp.

(4.6.3)

To fa,cilitate the compa.rison with the quantum mechanical relations we recall 1 corresponds to h. once again that Fo Integrals (4.5.3) and (4.6.1) realize the decomposition of the optical disturbance into impulsive and monofrequency signals. The choice of one with respect to the other is essentially suggested by the symmetry properties of tile system, as reflected t)y tile relative point spread function, as we will see in ,~w5.2.1 and 5.2.2 when describing the propagation through a free-medium section and a thin lens. 4.6.1

Basics of the Fourier transform

As the Fourier representation of the optical disturbance finds wide use in the later discussion, we list without proof a few of the basic mathematical properties of the Fourier transform, addressing the interested reader to [17] for detailed proofs as well as wider comments. For notational convenience, we regard @(~, z) as resulting from the action of the Fourier transform operator 9c on the complex amplitude ~(q, z): A

(4.6.4)

200

Linear Ray and Wave Optics in Phase Space

and inversely, 7: results from the action of the inverse operator ) c - l on @: (~'-l@)(q,

Z) -- 7:(q, Z)

1

_

j

(4.6.5)

We sha,ll n()t dwell on the "existence (:(mditions" t h a t (:()nft;r a, ma,ttwnmtical sense to the. at)ove integrals, directing the rea,(ter to the q u o t e d references. In(tee(t, a,s we are illtereste(t i~l the Follrier transfl)rm fr(nll a t)llysi(:a,1 ra,t,lwx tha,n ma,tlwllm,tica,1 vi(;wt)()illt, w('. lllay jllstifia,1)ly (:(nlsi(ler tlm t)llysicaJ existence of a wa,v('forIil as a. sulIit'ient ('()ll(titioIl f()r (lcfiIfillg its F()lH'i(;r tra,nsf()r~ll. E m m m r a , t i n g tlm I)r()I)(;rt, i(;s ()f th(; Ft)llri(;r ()t)erat()r, w('. ()~lli(, t() ('.xI)li(:itly (list)lay I,lm ll()w irrelewud, (lel)e~(ten(:(; ()f 7:a,n(1 ~? (n~ the axial (:()()r(lilm,te z. TI~(, linearity property a,ll()ws to (:a,lc~la,t('~ the Ftnn'i('r trm~sflm~ ()f a, linear (:()~flfi~al, io~ ()f fl~t:l,i()~s a,s tlw. sa,~lw li~wa,r co~flfi~m,ti(n~ of tim Fourier tra,nsfln'~s ()f ttm in(livi(l~m,1 (:(mq)(ments, i.e.,

Y{aT:, +b,i:.~}-aYT:,(q)+bY~.,(q)-a~,(~,)+t@.,(h'),

(4.6.6)

tl~e a.l~a.lysis (ff w 3.4.2, we will s(~e tha,t tim t'r tra.~sfr (:m~ tm lw.rfr t)y si~lfle. (n~(; ()r t,w()-h:~s ()l)ti(:a.1 syst(n~s. Tl~e FCn~rier ()tmra.t()r ~my tl~en be r(:gar(l(:(l as a, wa,v(:-l,ra,~sf(:r ()l)(:ra,t()r, m~(l a.s s~u:l~ ('(n~(:s 1,() I)(: li~(:ar. Tim si'mila,'ity prop(:'rty Sl)(:(:ifi(:s (,I~(: (:ff(:(:t ()Ii tlm F(nn'i(:r trm~sf()rl~ of a, s(:a,li~g ()f (,Ira i~I)~(, wu'ia,l)h:. S(:a,li~g (,I~(: q-wtria,l)h: l)y (,Ira r(:a,l I)arm~(:tcr a ca,us(:s tl~(; h:-va,ria,l)l(; t() s(:a,h: a,(:c()r(li~gly l)y tl~(: i~v(:rs(: fa,(:t()r a m~(l I)r()(luces a,ls() a, (:Im,~g(: i~ (,I~(: ()v(:rall ai~q>lit~ul(: ()f (,I~(: F()~ri(:r (,rm~sf()r~ a,s 1 ~; Y^ 7:(aq) - ~75(-~). (4.6.7) As s(:a,lc (,ra,~sfi)r~m,(,i()~s a.rc t)('.rforme(1 ()t)ti(:a,lly t)y i(h'.a,1 mag~fifyi~g systems, relatit)x~ (4.6.7) (l(:s(:ril)(:s tlm l)erfornm,n('(: ()f a F(mri(:r (,i'a~sft)ri~(;r-i(h:a,l ma, gnifier co~q)()sit(: s y s t e ~ . Tim shift property is (:on(:erned with the effe(:t on the st)e(:t, r m ~ ()f a tra,nslation ()f tim ()rigin ()f the (:oordiim, te axis, whi(:h in fa,(:t t)ro(tu(:es a linear modula,tio~ of the Fourier tra, I~sform, t)eing A

- qo) -

(4.6.8)

T h e modulation property in t u r n a,c(:ounts for tile shift of tile Fourier transform in response to a, linear frequency modula,tion of the w~tveform; natmely, .~'[(,inoq 7:(q)] __ @(N __ NO)"

(4.6.9)

T h e Parseval relahlon establishes the c o n s e r v a t i o n of the signal energy, t h a t can intuitively .justified considering the p e r f o r m a n c e of a lossless optical Fourier t r a n s f o r m e r . Thus, we can write

[

J

- f

J

dq.

(4.6.10)

Wave-Optical Picture of First-Order Optical Systems

201

The convolution property expresses the particularly interesting and useful result that the Fourier transform of the convolution of two functions equals the product of the Fourier transforms of the convolved functions. In formal terms, given the wave function ~(q) as the convolution integral (~(q)

-

-

J (/:)1(q -- q')~2 (q') dq'

--

(~1

*F2,

(4.6.11)

the Fourier transform is proved to be simply the product

(/~(~) -- ~/~(1~1 (~) (~2 (/~)"

(4.6.12)

The multiplication property mirrors the convolution property in the spatial frequency domain, stating that the Fourier transform of the product of two functions equals the convolution of the respective Fourier transforms. Thus, the spectrum of the product =

(q),

(4.6.13)

comes to be the convolution integral (4.6.14) Remarkably, nmltipli(:ation of F(q) times q is turned by Fourier transformation into the diflhrentiation of @(h;) with respect to ~:, as A d 9r [q F ( q ) ] - i~--s @(h;).

(4.6.15)

Conversely, differentiation of F(q) with respect to q produces under Fourier transform the multiplication of @(~) times i~: t- d )c [-77 P] -i~c~(~).

aq

(4.6.16)

Relations (4.6.15) and (4.6.16) show that the position and momentum operators ~ and ~ are Fourier imaged respectively into (i times) differentia,tion and (~0 times) multiplication in the frequency domain: A q p

)r 1...

o

>

>

ion,

(4.6.17)

t~

The relations above are used in w 4.6.6 to identify the momentum representation of the operators ~ and ~, and accordingly the frequency representation of the position and frequency operators ~, ~ (there introduced).

202

Linear Ray and Wave Optics in Phase Space

Ailticipa,ting the results of later sections, we mention that tile convolution (4.6.11) and the product (4.6.13) m a y be regarded as the tra, nsformation laws for light signa,ls t)ropaga.ting respectively through free-spa,(:e se(:tions a,nd thin lenses. One of the functions in (4.6.11) and (4.6.13) (:an t)(: interpreted as the int)ut wavefonn a,nd the other as the impulse rest)oxlse of tlm free-sI)acc section or (:orrest)oll(lingly tlle t)ha.se-mo(tulation fllll(:tion of the h:lls. A(:cor(tingly, tile l)i'()(lli(:(, (4.6.12)a.1~(l tl~(: (:()I~v()h~ti()n (4.6.14) r(:l)resel~t th(: (,ra,nsfer la,ws f()I" the a l)ove (tll()(,(:(t syst(ullS ill tll(: sI)atia.l-fre(llU:llt:y (t()lim,i11. As a (:()llvt)llltivc-like r(:la,tioIl ill the q-dollm.i1~ (,~n~s i~t,() a, ~n~ltiI)li('ativ(~lik(: r(:lati(n~ i~ tl~(: F()~rier (:onj~ga,t(: (t()~mil~, i.(:., (,l~(: h'-(h)l~m,i~, sil~ila,I'ly a l~nfl(,iI)li(:a,tiv(:-like r(:lati()n in the q-(t(n~m,i~ t,~n~s i~t() a (',()l~v()lutiv(:-lik(: r(:lati(n~ i~ t,l~(: h;-(l()~m,i~. C()nvol~lti(n~ m~(I ~nll(,il)li(:a.(,i()~ a.r(: (:()~ti,~ga.(,(: ()I)(:ra.ti()~s wi(,]~ r(:sl)(:(:(, (,() F()~ri(:r trm~sf()r~m.(,i()~. Fr(:(: l)r()l)agati()~ m~(l l(:~s (,ra~sf(:r (m~(l, Iii()r(: i~ general, sl)a(:e-invaria.~t m~(l sl)r(:a(lh'ss sys(,(n~s) at(' .i~s( (,I~(' ('(n~('r(:t(: r(:a.liza(,i()n ()f s~(:tl a, ~lm,tll(:~la,t,i(:al r(:la(,i()~l. Ill tiffs (:()llll(:(:(,i()ll, WO. ll()tO. (,ha,I, (h:l(,a a,~(l Imn~(n~i(" fln~('t,i()~s ar(: (:()~tj~ga,te witl~ resl)(:(:t (,() l'(n~ri(,r (,ra~isf()r~im(,i()~ a.s l)r()v(:(l ])y (,If(: r(:lat,i()~is _

q)

-

.1

-

1

.1",

(5(h ' t - ~) -- -if-4,["

,ie,'(q'-q)dH,, -

-

.1

-

-")dq,

(4.6.18)

,,:),t,,-',

(:xl)r(:ssing (,ll(: fl)r~m,l li~k l)(:(,w(:(:~l (,l~(:s(: (,w()i(h:a,liz(:(l ()l)ti(:a,l (list,~Ii'l)ml(:es, wll()s(: |)ella vi()~H's i~l t,ll(: q m~(l h" (l(nlmi~is at(: (:()~l)h:~ll(:~It, ary t()()~l(: a,~()tl~er. ll,(:lati()l~S (,I.6.18) fi~rl~isll (,I~(: ~ImI,ll(:llm,(,i(:a,l gr()lll~(I f()r (,If(: l)(:rf()nl~a,~l(:(: ()f (,l~e ()l)(,i(:a.l F()~ri(:r (,i'a.llSf()rlll(:r, wl~i(:l~ i~mg(:s a ln~(ll(: ()f l)arall(:l rays i~l(,() a I)()i~(, (i.e., sl)atia, l ]la,i'lll()lli(:s ill(,() b-finl(:ti()~s) ill (,l~(: F()IIri(:r l)lalm a~(l (:()xlversely a l)oint i~l (,I)(' i~l)~t l)la~(: i~(() a l)~l(ll(: ()f l)aralh;l rays (i.('., b-fi~l('(,i()~is into spatial harl~()~fi(:s). 4.6.2

Input-output 7"elations in the spatial .frequency representation: the. wave-,~'trre.ad function

The integral tra, nsf~)rm (4.3.13) is the ma,thematica,1 model for the light wave propa, gation through oI)tical systems, exenlt)lified in ternls of a. relation between the opti('a] (tisturt)a,ll(:es entering to and emerging from the system. In it both the input a.nd olltt)llt signals are (h:scribed in the space domain and hence are ret)i'esentc(t t)y th(: (:oinplex amt)litudes ~o:,(q, z;) an(t Po (q, Zo) a,t the a,ssigned int)ut and o u t t m t reference planes. Equiva,lent input-outI)ut relations can be elabora,ted, in which the input and o u t p u t signals are (tescribe(t in (tiffereIlt domains. One can relate, for instance, the frequency spectra of the input and o u t p u t signals @; (t~,, z ) and @o(~c, z o), or the field distribution at one plane and the Fourier transform at the other plane, namely p,:(q, z,:) and @o(~,Zo), or inversely ~5~(~, zg) and Po(q, Zo). In any case

Wave-Optical Picture of First-Order Optical Systems

203

the transfer relation takes the form of ail integral transform similar to (4.3.13), with the inherent spread functions being specific of the involved domains. Evidently, the existence of four equivalent forms of input-output relations with the respective spread functions is a direct consequence of the existence of four pairs of input-output domains in which the signals can be represented, i.e., spacespace, frequency-frequency, space-frequency and frequency-space. This can be brought into relation with the existence of exactly four Hamilton's cha,ra,cteristics, i.e., the point characteristic, the angle characteristic and the two mixed characteristics. In fact, a definite relation can be shown to hold between tile four system functions of wave optics and tile corresponding Hamilton's characteristics of geometrical optics. As it is beyond the scope of this book to present a, more deta,iled ana, lysis of the subject, the interested reader is referred to [18]. Here we deduce the relation between the Fourier transforms of the object and image distributions ~5~ and @o- Fourier transforming both sides of (4.3.13) with respect to qo and then expressing the input amplitude p~ (q, zi) in terms of the relevant frequency spectrum @~(~,z~) by (4.6.1), we end up with the integral relation

@o(h;o,Zo) - f dh;~h(h;o,h;~;Zo,Z~)@~(~;~,z~),

(4.6.19)

formally identical to (4.3.13). The weighting function h is explicitly given by

h(~;o, ~;~,9Zo, z~) _ ~1 / / d q ~ d q o (~-i(~o qo-~q~) g(qo, q~; Zo, z~),

(4.6.20)

with the reciprocal expression of g versus h,

g(qo,qg;Zo,Z~) __ ~I / / d l ~ i d t i , ~ ci(~o qo-~iqi)

h(/~ ~ , h;i;

Zo , zi).

(4.6.21)

It is evident that h and g are related by a Fourier transform-like relation; however, due to the minus sign in the exponent (~o qo - t~q~), they do not constitute a Fourier-integral pair (~). For ease of writing, as in the case of g(qo, q~) we simply write h(rco,rC~) instead of h(~o, ~ ; Zo, z~). Interestingly, if the input to the system is a spatial harmonic ~ (q~, z~) eit~,i qi

, the system responds producing the output

~o(qo Z o ) -

1 f dqig(qo qi)c i~q~

c The Fourier transform of a 2D function /~'~ ( ~ 1 , t%) -- ~1

(4.6.22)

f(ql, q2) is defined by the 2D integral

f f d%dq~ e - - i ( ~ 1

ql -t- ~2 q2 )

f(%, q2).

W h e n the inner product (n~ql + ~2q2) in the exponent is replaced by the skew-product (~1 q~ - ~2q2), the resulting transform is also reported as syrnplectic Fourier transform.

204

Linear Ray and Wave Optics in Phase Space

which first by (4.6.21) a,nd t h e n by (4.6.18) is seen to be ~o(qo, Zo) -- ~ .1

/

dh; o h(h;o, t;,i )(,it% qo . .

(4.6.23)

T h e flmction h yiehls therefore tlm st)e(:tra,1 ret)rcscIlta.tiol~ of tllc systenl rcst)()Ilse to a st)atia,1 lmn~l()ni(: wav(;fornl, whit:h is r(;t)I'Cs(mt(;(1 1)y a &fim(:ti(m in h',-st)a.(:('~" ~ (h,,, z, ) -- b(h:-h;,), th;n(:,;, h ix r(;t)ort,;(l as ,,he wa't,,'.-,,q,'t','.ad ((,r 'wa'~,~: 'r(:spo'n.s~:) .flt'n(:lio'n (~f tll(; syst(',lll. TtI(' il~t)~t-()~tl)~t r('lati(n~ (,1.6.19) lm.s ttl(; sm~l(' sigIliti(:ml('(' as (4.3.13), si~l(:(' ttl(; r('.Sl)()~ls(' ()f tll(, sysl,(;lll ill the h;&st, il~nlli b ( h ; - h',), (:()rr(;sl)()~l(li~lg l,() fl'(;(t~l(;ll(:y ('~ig('llWa,W;S ill l,ll('~ q-(l()l~m,in. As a (:()Ii(:l~lsi(~l ~f |,llis a.('('()~l|, a|)()~l, l,lw, wav(', st)r('a.(t flnl(:l,i~)~l, W(; say tlmt t~(;iIlg h a.Ii(t g li~lk('(l ])y (4.6.2{)) ml(l (4.6.21), all rela.ti()IlS f i w g (:()rr(;sl)(m(t t() a,Im,l()g()~s r('lati()~s fin" h, a,~l(l vi(:(', versa.. Ill 1)arti('~flar, relal, i(~l (4.3.21) (;xt)r(;ssing 1,11(; h)ssl(:ssl~:ss ()f |,l~(; ~)l)l,i(:al sysl,(;l~l is I11irr()r('~(l ill|,~

/&.

~

o

*

t

o

*

t

ti,

) i

"

Tll(' r(;a(l('r ('m~ ~:asily (l~;(l~u'~' t,l~' ('~n~lI~(~sit.i(n~ r~l(; fiw h, wlli('ll Imrall('ls, ()f ('()~rs(;, r('la,|,i~)~ (.1.3.18) fl)r g. 4.6.3

S"pac(:-inva'ria'nt a,n d sprcadh~,s',s ,s'ystcm,s

Spa, cc-i'n.'~,aria,'nt syst~'lIls, wlli('ll i,l.r(, ('llara('t~'riz~'(l t~y l ll(' r(,lllarkalfi~, l)I'()l)(;l'l,y l,lmt tll(; il~m,g~; (listl'it)lll,i()ll (I~J~;S ll()l, (1('I)('11(1 ()ll t,tl~; St)e(:iti(' h)(:a.ti()tl ()f tim ()t)je(:t in the illl),lt, t)lml~; [17]. 'l'lnls, wu'ia,ti()llS in t,lxe lo(:a,ti~)~l ()f the illI)11t, (listri])lltion ()w'r l,lm ()l)j(;(:t l)lml(' (1() n()t a,lf'e('t 1,11(; st, rlu:t~r~: ()f t,lm ()~t,t)~t, (listriln~ti(m, t)~l, ()I~ly its l()('ati(nl ill l,ll~; iina,g(, I)la.ne, t)(;ing ill(h:e(l sllift(;(t t)y the sa,n~c i-l,lll()llIll, ~1,"; 1,1~(' ()])j~',(:|, (lis|,ril)~ti(m ('t). Shifting the il~l)~|, ()v('r |,h~' ()t)j(;(:t I)la,ne as gp~(q~ - Q , z~) , t,l~(; (n~tt)~lt is (:orrest)(m(tingly sllift('(l |,hr(~gll 1,t1(' ilna.g('~ l)lml(; ill|,() r ( q o - Q , z ) fl~r a.ny shift a,n~ount Q. Ttlis il~lI)lies ttla,t ttl(' st)r(;a.(t finmti(nl (lo(;s not (1(;I)(;11(1 s(;t)aratcly on its a,rg~m~t;nl, s t)~t, ()lily t,hr()~gh th(; (liffer(;n('(; qo - q~" g ( q . , q~ ) -- ~(qo - q~ ).

(4.G.25)

T h e iIlt)ut-outt)llt relation for stfift-invariant s y s t e m s writes t h e n in the form ~o(qo, Zo) - f d q ~ ( q o - q,~)~(q~, z~) - ~ 9 ~,~, J

(4.6.26)

d In practice, optical systems are seldom space-invariant over all tile object plane, but it is usually possible to divide the input plane into small regions through which the system is approximately invariant. Thus, within the linear approximation of our concern, the condition for space-invariance may be checked over a small region of tile object plane near tile axis.

Wave-Optical Picture of First-Order Optical Systems

205

thus showing that the image distribution is the convolution of the object distribution with the point-spread function of the system. Taking advantage of the convolution property of the Fourier transforin, the above relation takes the simple multiplicative form

JPo(t~, Zo) -- b(t~)~, (t~, z,). in the t~-space. The function b(n) is, apart from a ~ 1 transform of the point-spread function g(q) according to

(4.6.27) factor, the Fourier

I~(~) -- / d q c -i~q g(q) - v"2~(t~).

(4.6.28)

It is reported as the optical transfer function. The "local" character of the input-output relation (4.6.27) is evident" no mixing of the spatial frequencies is produced by the system, each spectral component of the input being transmitted into the output signal though roodulated by the transfer function b- In particular, spatial tmrmonics are simply transferred to the output plane. Naturally, by an appropriate design of the optical system, i.e., of the relative b-function, one may remove or preferentially pass certain desired spatial frequency component of the signal; in other terms, one may operate a spatial frequency filtering of the input signal. Equation (4.6.27) relates the Fourier transform of the output signal @o to the Fourier transform of the input ~?~ through a multiplication by the system function b. This suggests an alternative procedure to that indicated by the space-domain convolution (4.6.26) to find the system output, requiring the often simpler sequence with a detour through the frequency domain:

space-invariant systems: g(qo, q~) -- g(qo - q ~ ) q-space

~-space

~

Fourier transformation

convolution by ~

multiplication by [} 0r

1 ~o

-'~

1 inverse Fourier transformation

<

--"

~o

In contrast with the non local character of the transfer relation (4.6.26) for space-invariant systems, spreadless systems are characterized by a local relation connecting the output to the input signal as

~o(q, Zo) --~(q)~(q,z,).

(4.6.29)

The input signal is simply modulated by the function ~(q). Then, in full analogy with (4.6.27), one may suitably reshape the input disturbance as desired,

Linear Ray and Wave Optics in Phase Space

206

r e m o v i n g or preferentially passing certain desired p o r t i o n s of the distribution; in o t h e r words, o n e m a y p e r f o r m a filtering opera,tion in the space domain. In t e r m s of the spread fimction, spreadless systems are identifie(t t)y g(qo, q~) = [(qo)b(q, - qo).

(4.6.30)

It is (:vi(l(:~d, t h a t ()~ a.(:(:()unt of tt~(: ~mdtipli(:ati()x~ I)r()t)erty (4.6.14)()f the F()~n'i(:r (,rm~sfi)rn~, l()(:ality is l()st ix~ th(: fl'(:(t~m~l(:y (l(n~m,in, wh(:r(: (,l~(: int)~t()~t,l)~(, r(:la,(,i()l~ tak(:s (,l~(: (:(nw()11~(,iv(: S t I ' l l ( : t l l l ' ( ~. ~'I,S (4.6.2(i), ~alnely, -

)

-

1

.

/

-

-

-

z , ) , t + , . '

-

' ~,~,

-

(4.6.31)

.

Tl~(:r(:f()r(:, tl~(: l)i(:t~r(: a,1)()v(: sk(:t(:l~(:(t r(:la,tiv(:ly t() tl~(: I)(:rf()r~m~(:(: ()f a Sl)a(:(:-i~waria~d, sys(,(:~ ('m~ I)(: r(:l)r()(l~(:(:(l r(:la,(,iv(:ly (,() (,I~(: l)(:Im,vi(n~r of a sl)r(:a(lh:ss sys(,(:~ as well wi(,l~ (,I~(: ilatllr(:, a,ll(l I~(:~('(: tl~(: ~m,(,l~(:~tm,(,i(:a,l r(:l)r(:s(:nt, a,(,i()~, ()f (,l~(: r(:sl)()~s(:s ()f tim (,w() sys(,(:~s l)(:i~g i~d,(:r(:lm,~g(:(l l)(:tw(:(:~ (,I~(: sl)a,(,ia,l m~(l fl'(:(111(:~(:y (l()~m,il~s: spreadless systems:

g(q,,, q, ) = [(q,,)l~(q, - % )

q-space

~,--space l'o~rier

tra~mfornmtion

-

n:t:lt, ii)licat i,)::

c(mv,)luti,m

',y t

',y t / v ~

I{(:lati()~ls like (4.6.29) a,ll(l (4.6.27), wlmre tl~(: i~l)l~(, a,n(1 the (,~i,l)l~(, ()f a li~lear sys,,clll are r(:la.te(l ill tlle st)a(,ial ()r stmtia,1 fi'e(tlu:ll(:y (1Ollm,ill l)y a, siuq)le ~lufltitfli(:a.(,ive rela(,i(nl (,llrr t,lle sys(,(nll fllllC[,i()ll, a,l'(: r(:t)resexd,a,(,iv(: ()f w h a t a,re rel)(wte(l as 'modulato'r:~ (Jr spatial .fr~quency filte'ns, a,ccr (,() w h e t h e r the mllltiI)licative rela,tioll ll()l(ts in tim sl)a,(:e ()r fl'e(tue~my (tonm, i21. Evidently, any st)a(:e iiwa,riant systenl o p e r a t e s a.s a, st)a,tia,1 freq~mn(:y filter. A region of holnog(:~mous st)a,(:e flm~ishes a, rea,lizat)le exm~q)le ()f a, st)a,(:e-inva,I'ia,~t system, whilst a, tt~i~ lens is a.~ e x a m p l e of a, m()(l~lator, i.e., a, filter i~ the space doma,in. Linear filters axe basic to the (opti(:al a,n(t not) signa,1 t)rocessing, which comprises, for instance, spectral analysis, p a t t e r n recognition as well as signal re(:(wery and (tetection [15.3, 17]. It is easily seen t h a t (:ascading a,ppropriate mo(tula, ting and filtering devices one o b t a i n s a, Fourier transformer. In fact, given a waveform pi(q, zi), we s u p p o s e to firstly multiply it by the linear chirp c -'~ ~ (~), i.e., ~, e Here, by linear chirps we signify the pure imaginary Gaussians e ir

(4.6.32)

~ real.

207

Wave-Optical Picture of First-Order Optical Systems

2

~01;q, zi )

~,eq-e

l e":'

e

ir162 2 ~ ( / 3 q , zi) la

2

2

(a)

(• )

FIGURE 4.7. (a) Electrical and (b) optical systems implementation of the sequence of chirp multiplication and chirp convolution, producing the Fourier transform of the input signal.

e~~q22 g~

which is then input into a filter with a chirp-like impulse response g(q) and hence a chirp-like transfer function b(~) -

~o(q, Zo) --

1 f

c --

d

-i

V . The output,

ci~q2 2 @i(/3q, zi),

(4.6.33)

is a chirp-modulated signal whose envelope is the Fourier transform of the input. Since in the linear approximation thin lenses act as modulators by chirps and free-space sections as frequency filters by chirps, the above sequence of processes can be implemented by a lens followed by a free-space section, as sketched in Fig. 4.7.b), which of course supports the basic scheme of a Fourier transformer drawn in Fig. 3.7. In Fig. 4.7.a) the analogous electrical cascade is depicted in order to evidence the optical-electrical devices analogy [17.2]. 4.6.4

T h e optical u n c e r t a i n t y relation. I

It is generally observed that spatially "narrow" signals have "wide" frequency spectra,, and conversely. Tile scaling property (4.6.7) is a manifestation of this reciprocal relation between the spreads of any Fourier transform pair (~, q~). The uncertainty product gives a quantitative account of this empiric observation, relating the "width" of the signal to the "width" of its Fourier transform through the inequality 1 CrqCr~ > ~.

(4.6.34)

Here the variance of [~l 2 is used as a measure of the width of ~ and similarly the variance of I~l 2 is used as measure of the width of @. Accordingly, the

Linear Ray and Wave Optics in Phase Space

208

symbols C~q and cr~ above m e a n the square roots of (Yq2(z)

f(q _ wz(~))~ I~(q, z)l 2 dq,

1

I1~112 1 f(K, - N,(z)) 2 I@(h;, z)[ 2 dh',.

2(Z) CYtr

(4.6.35)

I1~11 ~ ,

()~l a, g(;(n~wtri(:a.1 a.(:(:()~d,, c~q a.~(l c&- axe tile ra,(lii ()f gyrat, i()i~ ()f the areas ,l,,,t,,r tit,, ,:,lrv,;s I~(q,z)l 2 a,,~,l I@(~;,z)l 2 a,|,(,,lt a,,, axis tmssi,,g t,l,i'r tlie reh'wml, ~ea,n va,l~u's g/(z) al~r g,(z), resl)e(:tively given a,s

~(z)

/

i1~11~ ' q I~(q,z)l

2

dq,

~(z)

' i1~11~,

/

~1@(~,

~)1"

d~-.

(4.~.3~)

Fr fr r162162 tim wave finl(:l, iCnl is Sllt)I)r162 I,~ I~r llr lize(t to lnlity; a,lsr tllr ~t~'I)r162162 r ,c aJ~r ~, a,s wr as r ~/, g, crq a.~r c&., r the axia,1 r162162162 te z will ~r ~r162 I)~, r162 A fi~rtlmr si~iflitica.ticn~ tff exI)rcssir (4.(i.35) is a,ttai~m{t |w taking a.s zero tl~c t.~;~m.~fi~l ~f [~[2 mul [~512, i.~;., r / - 1 ) a , ~ r ~, - ( ) . Tl~is t:~uliticm t:m~ easily |)~; realizetl t)y slfifti~g 4~ !W - g / m u l ~ |)y - g . Tlm rt;a,tlr ~ a y easily verify l,lm.t c,q a,x~l c&. are i~s,;~sil, iw; I,,~ s~u'l~ Slm.(:r a.~l~l frt;ttllt;Ilt'y sl~ifts ~f ,~ a,ml ~3. ~l'l~t; witttl~ l)ri~cil)h; (4.6.3,1) grtnn~tls t)l~ tim varim~r ~u;a.s~m; t~f "width" a.t:r t,t) (4.6.::15) m~r tn~ t,l~t; Ft)~ricr l,ra.llsforlll rela.titn~ llt)ltti~g })(;l,weell a~(l ~}. I~ tim t, it a,1)l)lies tt~ a~y l"tn~rier i~d,egra,1 l)air witl~ tleti~il,itn~s (4.6.35) I~r ~s r162162 tlu' fl'r162 ,,,o,l,fl,,s I~?(~')l2 - ~ * ( a ) ~ ( a ) w c

wu'ia.~t:r c,~.. 2 As t~; is rr ,:a,,, write

2

/

c< -

2

I~:~(~')1 d ~ : -

2

I1~',~>11

~n~fr162

t,l~r s(t~m,re

(4.6.37)

~

whi(:h then (:a,n be iIltcrt)reted a.s the norm of h:#?. We ha.w, learned in w 4.6.1 t h a t tim fre(tlmncy nnfltit)lica, tion of tim tra.nsfornl a.rises frolll the transform of the (lerivative of the signal; na,nMy,

~:~(~) -

^

d

f [-~ aq -7

~].

(4.6.38)

As Parseva,l's t h e o r e m says tha.t the nornl of the Fourier t r a n s f o r m equals the n o r m of the fimction, one can identify c~2 as the n o r m of tile deriw~tive of ~. (The i m a g i n a r y unit i disat)pears, of course, when multiplying -idp/dq by the ('~omplex conjugate ida*/dq). Therefore, we can write

2

/d

(4.6.39)

Wave-Optical Picture of First-Order Optical Systems

209

Likewise, the spa,tiM va,riance O'q2 is seen to be the norm of qF, i.e.,

2

Crq

flq~(q)12dq_

--

iqFl12

(4.6.40)

J

2 Then, upon multiplying O'q2 a,nd Cry" -

J'

J'

Iq (q)l dq-

d

aq

(4.6.41)

dq,

we resort to Schwarz's inequality (I) to compose the product of the two integrals into a single integral, thus finally obtaining 2> O'q2 O'~

d

qp*-~qFdq

_

12

1 -2>-"

(4.6.42)

The conclusive inequa, lity above follows from the estimation of the intermedia,te quantity through the steps: (4.6.43)

tile la,st equa,lity resulting from a,n integration by parts of the two integrals embraced by the modulus symbol under the assumption that the ('onlt)lex amplitude •(q) goes to zero a,t infinity more ra.pidly than v~" 1 Rela,tion (4.6.34) with definitions (4.6.35) for the signal variances in the space a,nd frequency doma,ins is strongly reminiscent of Heisenberg's uncerta,inty relation of qua,ntum mechanics. Of course, the analogy is formal only, although the use of the same word "uncertainty" may be misleading. In this respect, we further remark tha,t (4.6.34) expresses a, definite rela, tion between the energy density of the signal in space and in spatial frequency, respectively, represented by I~12 and I@ 12. Such a relation takes the ultimate form (4.6.34) as a consequence of the mathematical definitions (4.6.35) being adopted as mea,sures of the spa, ce and frequency extents of the signa,1, a,nd above all a,s consequence of the Fourier relation holding between p and @. The plain f We recall that the Schwarz inequality states that b

b

f(x)g* (x)dx a

<

/ a

b

f(x) 2.

J

g(x)l 2 d x -

fl[ 2

912 ,

a

for any pair of coinplex-vahled functions f(x) and 9(x), which are supposed to be integrable on the interval [a, b]. The equality holds if and only if f and g are linearly dependent functions.

Linear Ray and Wave Optics in Phase Space

210

c o n t e n t ()f (4.6.34) is t h a t tile energy densities in space an(t frequency of any signal c a n n o t b o t h be m a d e narrow. It is c u s t o m a r y to take 2c~q as r e p r e s e n t a t i v e of the linear extent of the signal" S -- 2Crq, a,nd correst)on(tingly ~ : a,s ret)resentative ()f the signaJ 1)an(twidth (i.e., the interval of frequ(nl('y I=1 _< over wllich the fre(t~len(:y st)ectrun~ ()f the signal (lifters signifi(:antly from zer()). A(:(:()r(tingly, ine(tlm,lity (4.6.34), wlfi(:ll s() t,llnlS illt() H / ] >_ 1 (4.6.44) is a,ls() l"et)()rte(l a,s l,ll(~ .,q.t('~'-ba'n,dwidth,l)l"()(l~l(:t l"(qa,ti()]l. Alt(u'~lativ(~, m~(l ~l()r(' ('()~v(:~i(;~t,, (l('fil~iti(n~s fl)r t,l~('~ wa,v(;fi)rl~ (~,xt(ult a,n(1 t)ml(lwi(ltll lm,ve ])eel~ a( l( )l )te( l witl~i~l tll(; sig~m,1 ml(l illf()rllla,ti()ll I,l~('~()ry, with wlli(',ll (lefi~liti(nls tl~e a,lml()gy witl~ (l~m,l~t~ l~(:(:lml~i(:s wu~isl~(~s. S(:hwarz's ille(l~mlity tlll'llS i~ll,()an (;(l~mlity wl~(;~l l,ll(~ tw() fl~l(rti(nlS .f a,n(l .q a,r(~ t)r()t)()rti()~al t,() (:a~'l~ ()tl~(u'. ll~(l(;r tl~is Iwl)()tll(~sis, i~ tl~(: (:as(; ()f t,h(~ flm(:ti()ns (',llteri~lg the illtegrals ill (,l.(i.,il), ()lie sll()~fl(l llave:

d

dq~.(q) -,'q~(q).

(4.(i.45)

wl~i(:ll yiehls l,lie ~()l'~mliz(~(l (:(n~q)l(,x m~l)lit~l(h:, ( ,.,. + s,.~ ) ,2?_

-

lr

a.u(l (:i i(l(ultifyillg tile r('a.l ml(l illm.gi~m.ry l)arts ()f tile ( : ( ) l l s l , a , l l | , ('" ('-- (~'r +ici. Inserting (4.6.46) i~lt()tl~(~ l)r()(l~(:t ()f i~t(~grals in (,1.6.,11) ()r ('.(l~iwdently ill th(~ singl(', illt(~gra,1 ill (4.(i.42), ()ll(~ ()l)taills Cr

/"

d, Iq~(q)l2dq" /" ]~q "Pl2dq-

9

1 (:r

5~-

((,~ -Jr-(,2) (:~ 7r - ~(1 1 ,"? + ~),

(4.6.47)

,

which t u r n s illto ttw~ vahle ~1 if ml(t ()nly if (:i -- (). Accor(ti~lgly, tim waveforms, which minimize the un(:erta,inty t)r()(tllct t() ~, ' slmt)e as rea.l Ga,llssia.ns , i.e. , (r

: 1/2 (q) -- ( __lTra2)~ (,-2,2 ,

a rea.1.

(4.6.48)

An examt)le of a. m i n i n u u n un(:ertainty t)ro(tuct waveform, accomlting ~lso f()r t)ossible non zero centr()i(ls of p and ~3, is provided by tim Gabor function (pG(q)

__ (

1 ( q--qo ) 2 + iK'o (q-- q-~2) ~ 1 ) -31 C__ "~a'2

(4.6.49)

which is of great interest w i t h i n b o t h opticaJ and not strictly optical contexts. T h e p a . r a m e t e r a is a positive quantity. As the reader m a y easily verify, ~ - q0, /g-

--

/%0, O'q2 -_ _-

a 2 2 --~--, O'~; -_ _-

2 a1 2

a n d so CrqCr~ -- 1

Waveform

(4.6.49) consists of

Wave-Optical Picture of First-Order Optical Systems

211

sinusoid modulated by a Gaussian form; its intensity is concentrated around the centroid at q0 through a Gaussian profile, whose width is measured by a. Real-valued functions of space and time, formally similar to (4.6.49), have originally been introduced and studied by Gabor in his investigation of the essence of the information, conveyed by channels of communications [20]. Gabor elaborated a "quantum theory" of information, dominated by uncerta, inty relations analogous to those formulated by Heisenberg in qua, ntum mecha, nics; accordingly, rephrasing (4.6.34) in the time-frequency context, the simultaneous specificity of a signal in time and temporal frequency is limited by a lower bound of the signal bandwidth and time duration. In Gabor's theory, Gaussian modulated sinusoids like (4.6.49) represent the elementa, ry signals, i.e., the quanta of information, which minimize the indeterminacy product. As the original Gabor functions of time, complex-valued space functions of the type (4.6.49) retain the attractive feature of being compact in both space and spatial frequency as they Ininimize the product of their variances in space aim spatial frequency. Since this product specifics roughly the ~tmount of information conveyed by a, signal, it is evident its relevance for resea,rch on information encoding and tra,nstnission in va,rious fields of physical interest. In this respect, we mention that Gabor's signal expansion in terms of Gaussian-wcighted COlnplex exponentials like PG (q) represents a, ba.sic example of a joint space-frequency signal representa,tion (see w 7.2.2). In optical terms, wa,veforms like (4.6.49) may be interprete(t a,s the q-profiles a,t the waist of Gaussian beams, a,s those t)rodlwc(t, for instance, by stable resonator laser sources. The frequency spectrum ~?G(h',)has the s~mc a,na]ytica.1 form as PG (q)' with the Fourier conjuga,te variables q and t~ interchanged; explicitly we find @G(~

)

(a 2 -

1

a 2

--~-)~ c- -5- (~-~0)2-- 7"q0 ( ~ - ~2 )

(4.6.50)

which displays ~ Gaussian profile centered around the mean frequency ~0A comparison between PG (q) and @G(~C) manifests the general property of Ga,ussian forms being inva,ria,nt under Fourier tra,nsform, a,s the reader may prove by using suitable normalized coordinates" q ~ qa ~ ~ ~ ~ca. Specifically, focussing on the strictly Gaussian factors in (4.6.49) m~d (4.6.50), we see that x 2

9c[c--7 -] -- c

x 2

2.

(4.6.51)

In Chapter 9 we will obtain ~t more precise form of the optical uncertainty 1 relation (4.6.34) showing indeed an additional term besides ~.

4.6.5

The wave-optical phase space

From the discussion above it clearly emerges that a wave-optical phase space is conceivable with coordina,tes natura,lly suggested by the space a,nd spa,tiM

212

Linear Ray and Wave Optics in Phase Space

frequency variables q, ~. Tile geometric-optical phase space, which has been the a m b i e n c e of our description of first-order opti(:al s y s t e m s in the foregoing chapters, ix identified by the tx)sition a,nd angle ray-va.riabl('s. T h e "t)oint" in the g('()~netrica,l-ot)tics t)hase-st)ace ix the geonletri(: ot)je('.t sllit, al)le to the ray t)ase(t, a,n(t hen(:e lo('a,1, (test:rit)tion ()f light cla,l)()rat, e(t t)y g(;(ml(;t, ri('al ()t)t,it:s. T h e t)()siti(m anti lll(nll('ntlllll (:()()r(lina,l,('s ar(' ill fa,(:t ild,Cll(l(;(1 as Sillllllt;a,ll(;l)llsly a,ll(1 t)r('(:is('ly lll(;a,SlU'a,])l(; al,l, ril)lll,(;s ()f l,ll(', ray. As sll()Wll ill Clm,t)l,('rs 2 a,ll(1 3, ()11('~(:~1,11l"et)r('.s('~lll, a, ray ill l)llysi(:al st)at:(; |)y a, I)()illt, ill I)lm,s(; st)a,(:(;, ml(l a,(:('()r(tingly (tcs(:ril)c t, ll(, ligllt, l)l'()t)a.ga,ti()ll |)y (:llrv('s ill t)lm.s(' Sl)a.(:(;, sill(:(' l)()illt, s a,r('~ t,l'a.llsfornl('(1 ild,() l)()ild,s a.s l,ll(; ray l)r()l)a,ga,l,(;s ill r(;a.l st)a.(:(;. Tile "l)()ild, '' ill t)llas('~ st)a(:e st, ill rellla,ills ('.(;ld,ral t,() t,ll(: g(;()l~w.t,ri(>()I)l,i(:s (l(;scI'iI)t,i()ll ()f ligtd, t)r()l)a,gat, ion ('v('n wll(:ll a ray (l(;llsil,y (lisl,ril)l~l,i(n~ ix ilw()lv(;(t, sin(:(; 1,1~('~ray (l(;ilsil,y (:m~ |)(; ev()lv(;(l l,l~r(n~gl~ I,lw tra,j(;('l,()ri('s ()f 1,1~(' si~gl(; rays i~ 1,1~(; I)~(ll('. Evi(l(;~d, ly a, wavc-()t)l,i(:al t)lm,s(;-Sl)a,('(; r(;t)rcsent, a t,i()l~ ()f ligld, t)r()l)a,ga.l,i(n~ ix still ~()i, f(,asil)h; il~ l,('~n~s ()f 1,1~(;l~il,l~('rt,()('la,1)()ra,l,e(l (t('s(:ril)l,i()~s ()f ()I)l,i(:al wa,v(; fi(;hts as l,lwy i~v()lv(; S(:l)aral,(:ly 1,1~('~sl)a.l,ial wa.v(:fl)n~, sig~iti('~(l l)y w(q), m~(l I,l~(; fr(;(pu;~(:y st)(;('.l,r~t~, sigl~ifi(;(l I)y @(h'). Tll(;r(;fl)r(;, a .i()i~ll, r('t)r(:s(;~d,a,l,i(n~ ()f ()t)l,i(:a.1 wav(; ti(;hls l,()g(,l,lll,r i~ Slm(:(' a,ll(l fl'(;(l~t(;~(:y ~l~sl, 1)(; ('la,l)()l'al,('~(l, a,ll(l a,(> (:()r(li~@y tl~(' rt'l('wug t,rm~sfl,r laws i(l('~d, ifi('(l, i~ ()l'(l('r t,()('~al)h' a I)l~a,s('-st)a(:(' (l(;s(:rit)ti(n~ ()f ()l)l,i(:al sig~m,ls m~(l l,l~('ir l)rt)l)aga,l,i(m t,l~l'()~gl~ ()l)l,i(:al sysl,(;lllS. H()wt'vt;r, w(' l~my g~lt;ss tim.l, (;v('~ wl~(;~ |,1~(' (l(;sir(;(l jtfild, Sl)a(:t;-fl'(;tl~U;nt:y rcl)rest;~d, at, i()~ ()f oI)ti(:a,l wa.v('~t'(n'~s will 1)(' (;la.|)oI'a.t,(:(l, "I)(fi~d,s" i~ wave()I)l,i('al I)l~as(; Sl)a('(; will still l'(;l~m.ill t)t~ysit'ally ~(;a~i~glt'ss. Si~(',(: l,l~t, sillllllt, a~(;(n~s a.n(l l)r('(:is(' sI)('(:itit:ity ()f m~y sig~m,l i~ sI)a,('(; a,ll(l fl'l;tllll;llt:y ix i~l~(;r('~d, ly li~fit('(t t)y tl~(' ~ ( : ( ' r t a i ~ t y r('lati(~, w(; ~my ('Xl)('ct tl~at ally jt)illt, s i ~ a l rt't)r(;s(~'~d,a,t,i(n~ will retle(:I, s~u'l~ a. lil~fita.ti(n~, lla,lllely, Sll(:ll il.ll ilm('.(:(;ssil)ility ()f "t)()i~t,s" i~ l,l~(; wav(;-()l)ti(:a,1 l)l~a,s(; st)a('(;. Tln~s, whilst a. 1)]~as('~ sire.(:(; (list, titration lik(' a ( q - qo)(~(p- p,,) ix l(,git, i~m,t('ly (:()n(:('iva,|)l(' i~ g('()~('t, ri(: ()t)t,i(:s, ret)res(;~d,i~g a, si~gle ray at, % i~ tt~(; (lir(;(',ti()~ P0, tlw, a,~a,l()g()~s I)lm.se-st)a,(:e dist, ri|)~t,i(m (5(q-q0)(~(h;-h',,) ix n()t ('(n~(:eiva,|)le in wave ()t)t,i(:s. Ii~ this respect, the m i n i ~ n ~ llllcert~aillt, y t)r()(t~ct, wa,v('forlllS, i.e., t;h(; Gaussia,~ wave packets like (4.6.49), m a y be int, e~(led as the signals closest to the ra.y-concept of geometri(: ()t)t,i(:s, having the l~m,ximmn a,llowed space and K e q u e n c y localization. Finally, w(; note t h a t t,h(; g e o m e t r y of a space is of course d e t e r m i n e d by the invarian('e laws u n d e r the inherent t r a n s f o r m a t i o n s of t h a t space inl;o itself. T h e geon~etI'ic-optical p h a s e space is indeed symplectic since areas are preserved t)y the p r o p a g a t i n g ray t,ransforma,t, ions in accord with the m~(terlying H a m i l t o n i a n dynamics. Likewise the g e o m e t r y of the wave-optical phase space will be d e t e r m i n e d by the invariance laws of the individualized space-frequency d e s c r i p t o r w h e n evolved by propagatioi~ t h r o u g h optical systems. Specifically it has been proved t h a t in the W i g n e r r e p r e s e n t a t i o n the wave optical phase

213

Wave-Optical Picture of First-Order Optical Systems

space (also called Mock space) is an affine space [21]. As a conclusion, it is interesting to reInark that, oil account of the definite relation between the optical m o m e n t u m p and the spatial frequency ~, the matrix formalism of ray optics may also be considered to yield relations between the spatial coordinates and the spatial frequencies of the light waveform from the input to the output plane. In fact, taking account of (4.5.14), the inputoutput relations of ray optics, involving the ray position and slope (q,p) and the A B C D matrix, can be turned into analogous relations, involving the space and frequency variables q and t~ and a slightly different transfer matrix, where 1 the off-diagonal elements are scaled by ~0 and lco with respect to those in the ray-optics matrix. In fact, it is easy to verify the correspondence: canonically conjugate variables (q,p)

Fourier conjugate variables (q,t~)

(qo)

Of course, the matrix relation on the left is visualizable in phase space identifying the meaningful trajectory of the ray under paraxial propagation. In contrast, the matrix relation on the right has only a mathematical valence linking the conjugate variables q, n from the input to the output plane. As earlier said, it may acquire a physical content only if it is brought into relation with the transfer laws for some signal representative in the wave-optical phase space. Indeed, we will see in w 5.5 that the centroids ~ and g of any waveform p(q) and its spectrum @(t~) evolve under paraxial propagation through the matrix relation on the left of (4.6.52), i.e., just as the ray-coordinates of geometrical optics. Also, such a relation comes to be crucially functional to the paraxial propagation law for the Wigner distribution function, whose arguments, in fact, transform just through that relation (see w 8.3). 4.6.6

Coordinate and frequency operators in the q and ec representations

Expressions (4.4.1) and (4.4.2) define the operators ~ and ~ in the coordinate representation, i.e., through their action on the spatial waveform p(q); namely, 0 ~(q) P~(q) -- - k o O---q "

qF(q) -- qF(q)'

(4.6.53)

The corresponding expressions in the m o m e n t u m representation identify the operators ~ and ~ through their action on the m o m e n t u m wave function ~5(p). We recall that if the operator A defines a mapping of s into itself, every operator 6 is changed to O ' - ~,OA -1, thus bearing the same relation to the transformed wavefunctions Ap as the original operator 0 to the original wavefunctions ~, i.e., A

if

p

6,

~

then

A

A (~s

5'=ASA-I ' A~.

(4.6.54)

Linear Ray and Wave Optics in Phase Space

214

Then, confirming rela,tions (4.6.17), one can give the ot)erators ~, ~ in the p-reI)resentation a,s *"

q -

i

Oq

"-

% Op'

P -

(4.6.55)

p'

It is (:onvenient in the i)reseId, (:o~text t() intr()(hwe the st)a.tia,1 fre(l~mncy ()t)eI'a,t()r ~,, just ti'a,nsla, ting into ot)era,t()r f()l'ni the s(:a.la,r relation h: - / % p , i.e.,

W(', will I,~()stly (l(;al i,~ tlm f()ll()wi,~g witl~ tl~e I,air (~,~) i,~stea.(l ()f (~,~). Evi(l(;ntly, l,l~(; ()I)(;ra.l,()r ~ (:a.~ 1)(; i(l(;~tili(;(! as 1,1~(;(lifli;r('~d, ia.ti(n~ ()l)(;l'a,t()r

i) Oq

a- - - i - - .

(4.6.57)

when a(:tillg (hi ~(q), a,n(t as l,h(; llnfltiI)li(:ati()ll ()l)(;ra,t()r

awhen acting ()n ~(h'). Likewise, t,ll('~(:()()r(lillal,(; ()t)erat()r ~ ill tll(; h'-r(;l)r(;s('~nta.ti()n writes a,s i)

--io-7,

ml(t ll(;ll(:(; l,lle (:()l~lllnll,a,l,i()zl l)ra.(:k(;t l)(:l,w(:(:zl ~ ml(t a, (:(nll(;S l,() !)(:

a ] - ii. The Hermitia.n ()t)era,tors {~, k:,,i}, wil,ll ill,: reh:w,,,lt Lie In',,(l,u:ts [~. ~ ] - i]', A

A

[~, I] - (), [~,, I] - (), SI)a,ll tl,(; Ileise, fl)(,rg-W(,yl a,lg(;l)ra is, tl,(; sa,,m way as the ot)erator set {~, ~ , I } , lm.vi,,g il,(le,xl l,l,e sa.,,,e (:(,,,m,,,tal,(,rs (4.4.5).

4.7

Summary

We ha,ve presented a fornmliza,tion of wa,ve ()t)tics, simila,r to that of quantum mechanics 5, la Schr6dinger-Dirac, namely in terms of wave fllnctions ~(q) t,elonging to s and unitary tra,nsforma,tions M acting on s As g)(q) is the mathema, tical representation of light (tistributions, which are subject to an evolution under propagation through optical systeins, M is the Ina,thelnatical representa,tion of the "agents" of this evol~ltion, i.e., the optical systems. The representation of optical systems in terms of unitary operators M has been obtained by means of a quantization-like procedure applied to ray optics, by turning variables into appropriately defined operators. As a first step, the ray varia,bles q and p have been replaced by the position and momentum A

A

Wave-Optical Picture of First-Order Optical Systems

2 15

i o, whose Lie bracket [~, ~ ] - ~i mirrors the Poisson operators ~ and b k00q bracket of the classical variables {q, p} - 1. The complete sets of eigensta,tes of ~ and b have been seen to be basic to two complementary representations of optical wavefields, carrying information about the relevant energy distributions in space and spatial frequency (i.e., direction). Owing to the Fourier relation linking these two signal representations, one can establish an inequality for the product of the space and frequency variances of the signal, only formally similar to Heisenberg's inequality. Problems 1. Consider the set of up-to-second degree polynomials in q and p: ~<2 - {q, P, q2, p2, qp} and the set of the corresponding Hermitian operators, having the uni-que transcription as P <2 -- {q, P, ~2, ~ , ~ }. Verify t h a t the Poisson bracket of any pair of polynomials in P<2 i is taken to the Lie bracket of the corresponding operators in P_<2 by the factor k--~0. It is not so for higher-order polynomials. Verify, for instance, t h a t the Poisson bracket of the quartic polynomial q2p2, which by the Weyl rule is carried into (q2p2)w - ~ - 2 ~ _ #__~oq p2i~ _~_2--~21 ' 1 with any one of the polynomials in P<2 does not equal the c o m m u t a t o r bracket of (q2p2)w with the corresponding operator in P~2" A

A

A

A

A

2. Prove the identity" c -A B CA--C adA B, for arbitrary operators A, B. (Hint: Introduce A

A

A

A

parameter-depending operators such as F(T) - - e - t A B e r A , with ~- real. Write down the difA ferential equation obeyed by F(T). T h e n take the solution for 7 - 1.) 3. Consider a sequence of frequency spectra being flat over increasing frequency intervals:

{1 o

I~l > n~,,

,

with n integer and ~o arbitrary. Denoting by ~,. (q) the corresponding waveforms, prove t h a t lira ~ ( q ) - l i mK - - , o ~ ~i,,(gq) ~q ' with K - - n e % . n----+ oo

4. Ientify the realness and s y m m e t r y properties of the Fourier transform in relation with properties of the signal like (a) realness: ~(q) - g)* (q), (b) imaginariness: g)(q) - ~* (q), (c) evencss: ~(q) -- ~ ( - q ) , (d) oddness: g)(q) - - g ) ( - q ) , (c) hermiticity: ~(q) -- 99"(-q), and (f) antihermiticity: ~(q) - - ~ * ( - q ) . 5. Write down the hybrid transfer relations with the object and image waveforms being represented the q and r~ space, and vice versa. Deduce the relations between the response functions relevant to the four forms of the i n p u t - o u t p u t relations. 6. Show t h a t (a) for a system, in which the o u t p u t is a replica of the input, the transfer function is b (~) - 1, whereas (b) for a system, in which the o u t p u t is a replica of the input shifted by Q, the transfer function is b(n) - e ~Q. 7. Prove that the signal g)(q), which is supposed to be centered at ~ in space and at in frequency, and its space and frequency shifted version ~(q) - - e -i-y(q+-~) ~ ( q nt- -q) have the same spatial and frequency variances. 8. Expressing the complex amplitude as ~(q) - f(q)e -r with f and 4) real and f f ~ d q - 1, prove t h a t

If q~*aq dq[ 2 - - I - - ~1+ i f q f

2-~qdq de 2,

yielding the general form of the width principle as

~

2

2

1

2

>- -s + I f qf ~dql

2

Linear Ray and Wave Optics in Phase Space

216

References [1]

T h e (:()lle(:tions of H a m i l t ( m ' s pat)ers specifi(:ally (:on(:erning light ()t)ti(:s a,n(t I)artitle dyimmics are quoted rest)ectively in [1] and [2] of oh. 1.

[2]

F()r the mechanical-t)t)tical analogy sec [6] iu (:h. 1. A dctaih;r investigation of tlw (:orrcsl)()ll(h;ll('c ])CI,W(;CII (tllalltlllll lll(;cllalli(:s a lid classit'al ()l)tics ill its varie(t a,st)(',t:l,s, as ill t)arti(:lllar t lint rt',lat(;(l t() tilt; t)lmst',-st)a(:(; vi(;w, is 1)r(;s(;llt('(1 ill [2.1] D. Dr~4goinan, "P]las(; sire.co, c()rrc'st)OlldCllCc 1)ctwccll classical ()pries ml(l qllmd, llIn IlleCllaIfiCs", ill l'rog'rc.s.s in Optic.s, E. Wr (c(l.) (Elsevier, AIIlStel'(laIll, 2002), Vol. 43, c|~. 5, 433-49(i. [2.2] D. Drag()iimll ml(l M. l)rag<)IimIl, Qaa'rtt'ttm-(Tla.s.sical A'n,al~qlie.s (Sl)riIlgc'r, Berlin,

2004). [3]

Tllc wav(; I ll(;()ry ()f liglll is ilhlslral('(1 ill llmlly l(;xtl)()()ks as, fi)r itlslall('('~,

M. l/on~ m~(l E. Wolf, I)riuciph',s of Optics (1)ergm**o,~ Press, ()xfi)r(l, (i '~' e(l., 1980). D. Marc~se, Light Transmission Optic.s (Vm~ Nostrax~(l-Iiei~i~ohl, l'ri~(:(;l,r 1!)72). .I.A. An~mlr lIcam and Fiber Optics (Acmlenlic Press, New Yr 1976). A.K. (ll~at, ak m~(l K. 'l'l~yagm'ajm~, Conlcmporar!l Optics (l)lt;~nu~ l'ress, New York, 1978). [3.5] B.E.A. Sal('l~ m~,! M.(',. 'l'ci('l~, l,;undamcntaLs of l'h,otonic.s (.l()t~ Wiley & S()~s, New York, 1!)!)1). [3.6] S. S()li~m~(), 1~. (',r()sig~m~i m~(l l ). l)i l)()rl,(), Guiding, D~[fraction and (;o,~:fincmcnt of Optical Radiation (A('a(h;~fi(: l'rcss, Orlm~(lo, 1985).

[3.1] [3.2] [3.:~] [3.,1]

[4]

Tl~c I)r()(:('(l~H'(' t()r('I)la('(' ttw fi~ll wave ('(l~mli()~ l)y llw t)aral)()li(' way(' ('(l~mti()~, ()~ tl~(, li~(, ()f t l~(; ()rigi~ml s~gg(,,st, i()~ 1)y F()(:k m~(l L(,()I~t()vi(r is ill~strat(,(1 in th(; t)asi(' t(,xtt)()()ks i~ [3]; i~ l)arti('~llar, w(' a(l(lr(',ss tlm r('a(h;r t()[:~.:~], (:l~. 3, I)t). 173-177. I~ F()('k's l)al)('r [4.:~] t l~(; s i ~ i l a r i l y I)(,lw(;(,~ t lm t)aral)()li(', way(' (;(tuati()n and tlw t i~(;-(l('~l)('~(t(;~d S('t~rii(ling(;r ('(l~mti()~ was ()l)serve(l fi)r tim first ti~(;. [4.1] M.A. Leo~tovicl~, "A ~mt,l~o(1 of solutioI~ of prot)le~ns of clectro~nag~mtic wave propagatio~ along the earth's surface", Bullcti~ (to l'Acad6Inic (los Sciex~ccs (te I'URSS, s6r. phys 8,16-22 (1944) (in Russian). [4.2] M.A. Leontovict~ m~(l V.A. Fock, "Solutio~ of the problein of propagation of electroinagnetic waves alo~g the earth's surface I)y the method of parabolic equations", Sov. J. Phys. 10, 13-20 (1946); see also V.A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon Press, Oxford, 1965), ch. 11, pp. 213-234. [4.3] V.A. Fock, "Theory of radiowave propagation in an inhomogeneous (stratified) atmosphere for a raised source", in Electromagnetic Diffraction and Propagation Problems, (Pergamon Press, Oxford, 1965), oh. 14, pp. 276-307.

[5]

T h e description of n o n p a r a x i a l p r o p a g a t i o n of beam-like wave fields has been investigated in several papers by using rather different approaches, ranging from the p e r t u b a t i o n a p p r o a c h to the Maxwell's equation in a b e a m divergence-related

Wave-Optical Picture of First-Order Optical Systems

217

p a r a m e t e r to the angular s p e c t r u m representation of the electric field and to the spheroidal coordinate system representation of the ttelmholtz wave equation as well a.s to the complex point-source model. Below only a few of t h e m are listed. [,5.1] M. Lax, W.H. Louisell and W.B. McKnight, "Prom Maxwell to paraxial wave optics", Phys. Rev. A 11, 1365-1370 (1975). [5.2] L.W. Davis, "Theory of electromagnetic fieds", Phys. Rev. A 19, 1177-1179 (1979). [5.3] G.P. Agrawal and D.N. Pattanayak, "Gaussian beam propagation beyond the paraxial approximation", J. Opt. Soc. Am. 69, 575-578 (1979). [5.4] W.H. Louisell, "From Maxwell to optical resonators", in Laser Induced Fusion and X-ray Laser Studies, S.F. Jacobs, M.O. Scully, M. Sargent III and C.D. Cantrell III (eds.) (Addison-Wesley, Reading, 1976), pp. 369-485. [5.5] S.R. Seshadri, "Electromagnetic Gaussian beam", J. Opt. Soc. Am. A15, 2712-2719 (1998). [5.6] H.-C. Kim and Y.H. Lee, "Higher-order corrections to the electric field vector of a Gaussian beam", J. Opt. Soc. Am. A 16, 2232-2238 (1999). [5.7] Q. Cao, "Corrections to the paraxial approximation solutions in transversely nonuniform refractive-index media", J. Opt. Soc. Am. A 16, 2494-2499 (1999). [5.8] A. Ciattoni, B. Crosignani and P. Di Porto, "Vectorial free-space optical propagation: a simple approach for generating all-order non paraxial corrections", Opt. Comm. 177, 9-13 (2000). [5.9] G.A. Deschamps, "Gaussian beam as a bundle of complex rays", Electron. Lett. 7, 684-685 (1971). [5.10] S.Y. Shin and L.B. Felsen, "Gaussian beam modes by multipoles with complex source points", J. Opt. Soc. Am. 67, 699-700 (1977). [5.11] E. Heyman and L.B. Felsen, "Complex-source pulsed-beam fields", J. Opt. Soc. Am. A 6, 806-817 (1989). [5.12] C.J.R. Sheppard and S. Saghafi, "Electromagnetic Gaussian beams beyond the paraxial approxiInation", J. Opt. Soc. Am. A 16, 1381-1386 (1999). [5.13] B.T. Landesman and H.H. Barrett, "Gaussian amplitude flmctions that are exact solutions to the scalar Hehnholtz equation", J. Opt. Soc. Am. A 5, 1610-1619 (1988). [5.14] Z. Ulanowski and I.K. Ludlow, "Scalar field of nonparaxial Gaussian beams", Opt. Lett. 25, 1792-1794 (2000). [5.15] G. Rodrfguez-Morales and S. Ch~vez-Cerda, "Exact nonparaxial beams of the scalar Helmholtz equation", Opt. Lctt. 29, 430-432 (2004). [6]

An exhaustive analysis of the relation of geometrical to wave optics, and in general to Maxwell's electromagnetic theory, is presented in [3.1], [3.6] as well as in the refs. below. Keller's paper [6.5], introducing the "geometrical theory of diffraction", had signified a great increase of the domain of application of ray methods. [6.1] R.K. Luneburg, Mathematical Theory of Optics (California Univ. Press, Berkeley, 1964). [6.2] M. Kline and I.W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965). [6.3] G.A. Deschamps, "Ray techniques in electromagnetics", Proc. IEEE 60, 1022-1035 (1972). [6.4] S.W. McDonald, "Phase-space representations of wave equations with applications to the eikonal approximation for short-wavelength waves", Phys. Rep. 158, 337-416 (1988).

218

Linear Ray and Wave Optics in Phase Space

[6.5] J.B. Keller, "Geometrical theory of diffraction", J. Opt. Soc. Am. 52, 116-119 (1962).

[7]

T h e formalization of wave ot)ti(:s ~ la (tuantmn Inechani(:s is illustrated in [3.2], t)t). 103-111, and [3.4], pp. 29-30, on the basis of tile original elaboration by Gloge a ml Marcuse [7.2]. Prelinfinary suggestiolls a n(t wi(ter investigati(ms are in the a(t(tit, iolml tith;s t)eh)w. We als()sllgg(;st, [a.3], i)p. 44-46, 51-52. [7.1] N. Clmko, "Slu- le t)a,ssagc tie l'Ol)l, iqlm gdt)lIldt,riqlm h l'Ot)t, ittlm Oll(llflatoirc", Co~11t)t. l{ell(l. 251, 852-853 (l.()(i()). [7.2] D. Gh)gc, ml(l D. NIarcllse, "Ieorllml (tllmltlllll tlmory of ligtlt rays", J. Of)l,. Soc. AiIl. 59, 1(i2.()-1631 (1!)69). [7.3] G. Eictllnmnl, "Prol)agatioll of ligld, ])emns ill Colltilul(nlsly focllsillg lens-like lnedia", Proc. IRE 58, 837-839 (197()). [7.4] G. EicluIuulzl, "Qlmsi-ge(nlmtric optics ()f ilm(lia wit,tl iIlll()lI~oge~eO~m i~(lex of rekactioxF', .1. Opt. Soc. Am. 61, 161-168 (1.()71). [7.5] M. FehlI~a~, "Mo(les i~ gestural gm~ssim~ optical sysl,(;IllS IlSillp._s q~m~d,~un-~eclmifics forxnalis~n", .1. Opt. Soc. A~n. 61,446-44.9 (1971). [7.(i] T. Pra(lt~m~, "Maxwell's e(l~at, i()~s fl'()~ ge()~mt,rica, l ()l)tics '', l)l~ys, l,et,t,. A 122, 397 398 (1987). F()r a gt'~mral | l'(~,a|,lll(;ll|. ()f l l~(; sy~I)h;(:li(' gr()ll 1) wil,l~i~ tim (:()ll|,(;xl ()f l l~(; grout) tlm()ry ,see [9] ()f ('l~. 1, wl~ilst fi)r a Sl)(;('iti(' tr(:all~t;ld s(;(; [11] ()f (:1~. 2. S()~(; t,ith;s ('()l~('('l'l~('(l will~ i~w('sligali()~s r('lal('(I l()ll~(; ()l)(;ral()r alg('l)ra str~mturc ()f I)araxial wave ()l)t,i(:s as arisil~g fr(n~ a (pm]gizati()]~-lik(; l)r()(:css ()f I)araxial ray ()i)ti(:s, I)ase(l ()~ 11~(; lt(;ist;~l)(;rg-W(;yl alg(;l)ra, itl'( ~, (pu)t,(;(l ]~(:I'(;. St'(' als() [16]. [9.1] A. Vm~(h'r I,~gl,, "Ol)crati(nml ~()l,al, i(n~ fi)r tim m~alysis m~r sy~d,lmsis r ()t)ti(:al (lata-proressi~g syste~ns", l)roc. IEEE, 54, 1()55-1()63 (1.()6(i). [9.2] H.J. B~ttc'rwe('k, "General tl~eory ()f linear, (:oherc~t,, optical (lata-pro('essi~g systeins", .1. ()pt. So('. A~. 67, 60-70 (1.()77). [9.3] M. Nazaratl~y m~(l ,I. Sluunir, "Fo~rier optics (lescribed by ol)erat()r algebra", J. Opt. So(:. A~n. 70, 150-15(.) (1.()80). [.9.4] H.J. Butterweck, "I)riI~cit)les of optical data-processing", in Prog~vss in Optics, XIX, E. Wolf (e(l.) (Nort|l-HollaIl(I, Ainst(;r(laIn, 1.()81), oh. 4, pp. 211-28(). [9.5] M. Nazaratl~y and J. Shmnir, "First-order optics- a canonical operator representations: lossless systClilS", J. Opt. Soc. Aln. 72, 356-364 (1982). "First,-or(ler optics: operator representation for systems with loss or gain", ibid., 1398-1408. [9.6] H. Raszillier m~(l W. Sche~npp, "Fourier optics from the perspective of the Heisenberg gro~p", in Lie Methods in Optics, J. Ss Mondragdn and K.B. Wolf (eds.) (Springer-Verlag, Berlin, 1986), oh. 2, pp. 29-43. [9.7] O. Castafios, E. L6pez-Moreno and K.B. Wolf, "Canoifical transforms for paraxial wave optics", in Lie Methods in Optics, J. Ss Mondragdn and K.B. Wolf (eds.) (Springer-Vcrlag, Berlin, 1986), ch. 5, pp. 159-182. [9.8] K. Wittig, A. Giesen and H.Hiigel, "Algebraic approach to characterizing paraxial optical systems", Appl. Opt. 33, 3837-3848 (1994). [9.9] R. Simon and N. Mukunda, "Optical phase space, Wigner representation, and invariant quality parameters", J. Opt. Soc. Am. A 17, 2440-2463 (2000). [9.10] V. Guillemin and S. Sternberg, Symplectic Techniques in Physics (Cambridge Univ. Press, New York, 1990)

Wave-Optical Picture of First-Order Optical Systems

219

[9.11] J. Shamir, Optical Systems and Processes (SPIE, Bellingham, 1999). [10] The reading list on nonrelativistic and/or relativistic quantum mechanics should be extremely long. Among the rich variety of titles, only a few are listed below. In particular, we signalize [10.5] in relation to the theory of linear operators in the Hilbert space of the square integrable functions. [10.1] [10.2] [10.3] [10.4]

A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1961). A.S. Davydov, Quantum Mechanics (Pergamon Press, Oxford, 1976). L.E. Ballantine, Quantum Mechanics (Prentice Hall, Englewood Cliffs, 1990). W. Greiner, Relativistic Quantum Mechanics- Wave Equations (Springer-Verlag, Berlin, 1990). [10.5] T.F. Jordan, Linear Operators for Quantum Mechanics (John Wiley ~ Sons, New York, 1969). [11] A presentation of the Heisenberg-Weyl group from the perspective of quantum mechanics, to which it is intimately related, is given in [11.1] H. Wcyl, The Theory of Groups and Quantum Mechanics (Dover, New York, 1931). [11.2] K.B. Wolf, "The Heisenberg-Weyl ring in quantum mechanics", in Group Theory and Its Applications, Vol. 3, E.M. Loebl (ed.) (Academic Press, New York, 1975), pp. 189-247. [13] The methods for obtaining the operators that correspond to particular dynamical variables are illustrated in [11], and in [13.1] J.R. Shewell, "Oi1 the formation of quantum-mechanical operators", Am. J. Phys. 27, 16-21 (1959). [13.2] H. Daughaday and B.P. Nigam, "Function in Quantum Mechanics which corresponds to a given flmction in Classical Mechanics", Phys. Rev. B 139, 1436-1442 (1964). [13.3] G.S. Agarwal and E. Wolf, "Calculus for functions of non-commuting operators and general phase-space methods in quantum mechanics. I. Mapping theorems and ordering of flmtions of non commuting operators", Phys. Rev. D 2, 2161-2186 (1970). [14] S. Steinberg, "Lie serie, Lie transformations, and their applications", in Lie Methods in Optics, J. SSnchez Mondragdn and K.B. Wolf (eds.) (Springer-Verlag, Berlin, 1986), ch. 3, pp. 45-104. [15] For an enlightening physics-oriented introduction to the theory of the distributions we suggest [17.3], pp. 50-66; a rigorous mathematical treatment is in [15.1] R. Bracewell, The Fourier TransfoT~n and Its Applications (McGraw-Hill, New York, 2"d ed., 1986), ch. 5, pp. 69-97. [15.2] M. J. Lighthill, Introduction to Fourier Analysis and Generalised Functions (Cambridge Univ. Press, London, 1958). [15.3] A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1962), pp. 269-282. [15.4] K.B. Wolf Integral Transforms in Science and Engineering (Plenum Press, New York, 1979). [15.5] R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. II, pp. 766-798.

220

Linear Ray and Wave Optics in Phase Space

[16] The Euclidean model of optics, as arising from a quantization-like process of paraxial ray optics, based oil the Euclidean algebra, is presented in the titles below; p r d i m i n a r y suggestions can be found in [9.7]. [16.1] K.B. Wolf, "Elements of Elmli(lcan optics", in Lie Methods in Optics II, K.B. Wolf (cd.) (Springer-Verlag, Bcrlii~, 1989), oh. 6, pp. 115-162. [1~i.2] K.B. Wolf, Geo'met'r~ic Optics on Phase Space (Springer, Bcrli~, 2(}{}4), pp.135-151. [17] Tlle lIlatllelllati(:s ()f tile F()llrier tra llsf<wl~l is (tes(:ril)e(1 ill tlm (I(;v()I,(',(I I extl)()()ks [15.1]- [15.4] , wl~ilsl ~tll ()t)lics-tlawmrcd il~v(',stigalitnJ is ()t[cr(,(l i~ [17.1] .I.W. G()()(l~m~, Introd'uction to t,buricr Optics (McGraw-ttill, New York, 1968). [17.2] A. l'apo~flis, Systems and 7;ran.@n'ms with, Applications in Optics (McGraw-Hill, New Y()rk, 1.()(i8). I lore, tim clc'ctrical-optical (lcvi(:cs mlalogy is cxprc'sscly cvi(lcilcc(1. [17.3] .l.l). (]askill, Linear Systems, t,'o'uricr 7}'an.@nwts and Optics (Wih,y & Soils, New Y()rk, 1.978). [18] Tll(~ t)asi(:s ()f tim tlalllilt()ll ('tlara(:t(~risti(:s l~mtl~()(l a r(~ I)r(~se~tc(l i~ tim t(~xtl)()()ks [3.1] all(l [3.6]; i-I lll()l'(~ exl~a~stivc l)res(~l~tati()~ is ()ffere(l i~ tim t)()()ks I)y Sy~ge [18.1], l~x~ct)urg [6.1] all(t lh~(:l~(lal~l [is.2]. Tim rclati()~ ()f tta~ill(n~'s ('lmract(wist i(:s ()f g('~()~ctri('al ()t)ti('s ~() tl~(', sl)rca(l fi~(:li()~ts ()f wav(~ ()l)ti('s is i~v(:stigat(~(l in [~)..5], a~(1 i~ tim l)al)cr I)y Basliaa~s [~s.:~], wlmre ttm Wig~mr (lislril)~ti()~ fi~(:ti()~ is I)r()ve(l t() i)ri(lg('~ ray m~(l wav(~ ()l)li('s. [18.1] .l.l,. Sy~ge, (h'omct'uical Optics. An introduction to tlamilton '.s 'm~:th,od ( ( ',mnl)ri(lgc Ill,iv. l'rcss, I,()~(1()~, 1.()37). [18.2] II.A. II~wt~(latfl, An Int~vduction to llamiltonian Optics (Cm~fl)ri(lg('. Ih~iv. Press, New York, 1970). [IS.3] M.,I. I~stiam~s, "Th(; Wign(;r (tistrib~tio~ finwtio~ a~(l tlamilto~'s characteristics of a gco~netric-optical systcI~", Opt. C()I~H~. 30, 321-32(i (197.()). [19] F()r~al i~vcstigati()~s ()f vm'i(n~s i~m(t~mlitics l)('rtai~fi~g t() t lm F(n~ri(u" a xmlysis ('al~ t)e f(n~(l i~ [15.1]- [15.3] . See als()[171. F()r a (tis('.ussi()~ ()f tim ~('.crtainty I)ri~(:il)lc witl~ix~ t t~e (lua~IuI~ i~m(:lmi~i('al fi'a~(~w()I'k we (lirc(:t l lm r(~a(lcr t() [10]. Ad(litional rcfs. arc given in (:11.9. [20] Wc (tu()tc t)ch)w the seminal Gat)()r's t)at)(ws [20.1] ()n the tlm()ry ()f information and the later t)aI)cr [20.2] ()n "()t)ti(:s fr()n~ tim vi(~wt)()int ()f inf()rn~ati()n theory": [20.1] D. Gabor, "Theory of coinmunication. I: The analysis of inforination", J. of the IEE 93, 429-441 (1946); II: The analysis of hearing", ibid., 442-445; III: Frequency compression and expansion", ibid., 445-457. [20.2] D. Gabor, "Light and hfformation", in Progress in Optics, E. Wolf (ed.) (NorthHolland, Amsterdam, 1961), Vol. I , ch. 4, pp. 109-153. [21] For an excmplificative account of affine space geometry we suggest [21.1] H.J. Wolters, "Affine and projective geometries. A tutorial.", http://www.hpl.hp.com/rcscarch/mmsl/prcscntations/3d/GcomctryTutorial.pdf

in

5 1D First-Order Optical Systems: The Huygens-Fresnel Integral

5.1

Introduction

It has been emphasized ma,ny times that the quadra, tic Hamiltonia, n model for paraxial propagation is central to several fields of physics in its "classical" as well as "quantum" form, as pictorially sketched in Fig. 3.1. The presenta,tion elaborated in Chapters 2 and 3 was based on the classicallike Hamiltonian model, which had provided the basic algebra for the formaliza,tion of pa,raxial ray-optics. The presentation started in Chapter 4 is based on the quantum-like Hamiltonian model, which in turns provides the basic a,lgebra suitable to the forma,liza,tion of paraxial wave-optics. Such a, formalization emerges from a, quantization procedure, by which the description of the (monochromatic) optica,1 disturbance in terms of rays of light is replaced by the description in terms of optical wave functions, and correspondingly the phase-space variables of geometrical optics are replaced by linear Hermitian operators acting on the Hilbert space L;2(IR) of the square-integrable wave functions F(q): light ray --~ optical wavefunction phase-space variables ~ linear H e r m i t i a n operators on

E/22(R)

/22(I~)

(5.1.1)

Accordingly, the position and m o m e n t u m variables are promoted into the operators ~ and ~, which act on L;2(R) according to the SchrSdinger realization (strictly admissible only under paraxial approximation)" i 0

"fig~(q)

~g~(q) -- qg~(q), Their Lie bracket"

ko Oqg~(q).

(5.1.2)

A A

[~, ~] - i2-,

A

2-

I -k0 ,

(5.1.3)

222

Linear Ray and Wave Optics in Phase Space

is in a, sense the "quantized" form of the Poisson bracket of the classical variables: {q,p} = 1. Also, the ray-transfer matrix M(zo, zi), which characterizes first-order optical systems identifying the releva,nt input-output relations in terms of the ray-coordinates at the en(t-t)la,nes, is ret)laced t)y tile wave-tra,nsfer operator M(zo, zi), which (:orre, spondi~gly identifies the int)~t-outt)~t rela,tions in terms of tl~(; "excitation" p,~(q) il~ tl~(; int)~t pla,nc m~d tl~(; systcn~ "rest)()l~SC" g~o((t) i~ tt~c ()~tt)~lt t)la~('~, i.e. i

r

-- M(zo, zi)~i(q).

As a ~m,t~tral |)y-i)r()(t~u:t <)f tl~(', (l~a,nl, izal,i()~ I)r()(:(;ss, 1,1~(;(',(l~ta,ti()i~ ()f ~n()ti()n for M, a,l~(t tl~c fil~it(;-st(;l) fll~(:ti(nm,1 ext)ressio~ ()f M a,s a, si~gh; (;xt)()~mntia,1 fi~(t tlw, ir wavc-()I)ti(:a,1 tra,~s(:I'il)ti<)~s r('st)('(:tivcly i~ tl~(; ()l)ti(:al S('l~ri~(tingcr (;(l~a,l,i()n f()r M:

i 0 ~A(z zi) - A(~ ~, z)~A(z zi) ko Oz

M(zi, zi) - i ,

(5.1.5)

a~
[V](zo, zi)

A

-- (,-ik,,H(z,,-z,),

(5.1 .6)

a,risi~lg, ~)f (:out'so, a.s a S~411t,i~nl t~ (5.1.5) ix1 tile ~:a.sc ~)f a z-iIl
p2 ___~ .~2,

q2 ___, -~2,

qp ~ ~1 ( ~ + p ' ~ ) ,

(5.1.7)

i

the Ha,miltonia,n ot)era.tor H(O, b, z)entering (5.1.5)a,n(t (5.1.6)is ,miquely defined a,s a plain transcription of the paraxial form of the ray-optics Hamiltonian function (see w167 4.4.1-2). Accordingly, relations (5.1.7) and (5.1.6) are central to the present analysis. They will be used, in fact, to elaborate the operator description of first-order optical systetns, which eventually will converge into the well known HuygensFresnel integral in a form that, establishing the link with ray optics in the linear approximation, allows us to benefit from the effectiveness of the ray matrix forma, lism [1]. Specifically, paralleling the analysis in Chapter 2, in Sect. 5.2 we 1A2 consider the quadratic operators ~1 2 , ~q and ~1 (q'P+P"q), whose exponentiated forms provide the operator representations of a free-medium section, a thin lens, and a positive scaler. Besides, each of these operator representations is

1D First-Order Optical Systems: The Huygens-FresnelIntegral Symplectic algebra

Lie algebra of homogeneous

of ray optics

quadratic polynomials

223

Metaplectic algebra of wave optics

F I G U R E 5.1. The quadratic homogeneous polynomials provide the basic algebraic structure of optics, yielding the Lie algebra of the symplectic ray matrices of ray optics as well as the Lie algebra of the metaplectic transfer operators of wave optics.

given an explicit functional form, individualizing for each tile definite expression of the relevant line-spread function g(q, q'), which represents the kernel of the integral transform expression of the transfer relations (5.1.4), i.e.

~o(q) -- /

g(q, q')qpi(q')dq',

(5.1.8)

holding in general as an obvious by-product of the linearity of M. The operator representations for optical systems, we will deduce in the light of the quantization procedure, can as well be inferred from the parabolic wave equation for qua(Ira,tic media, once the underlying operator algebra, has been recognized, in full analogy with the group theoretical approach to the quantmn mechanical SchrSdinger equation [1]. The procedure, we will follow, reveals more clearly the link of pa,raxia,1 wave optics to paraxial ray optics, and hence, within the group theoretical framework, of the metaplectic group Mp(2, R) to tile sylnplectic group Sp(2, IR), both having a Comlnon root ill tile algebra of quadratic polylmlnials (Fig. 5.1). In Sect. 5.3 the obta,ined transfer relations for free-propagation and lens transformation are exploited to infer the input-output relationships for a,n arbitrary ABCD system. This natura.lly conveys the link between the HuygensFresnel integral of the diffraction theory and the ray-matrix picture of geometrica,1 optics. In fact, the line-spread function is found to be a purely imaginary exponential of a quadratic polynomial in the profile coordinates at the endplanes, the relevant coefficients being defined in terms of the entries of the ray-matrix. The position-momentum identification of light rays in geoInetrical optics is mirrored in wave-optics by the space and spatial frequency representations of optical fields, respectively conveyed by the wave function ~(q) and the Fourier transform ~5(ec). Remarkably, tile Fourier transform of an optical signal is not an abstract entity; in fact, it turns out to be physically detectable by use of plain bulk optics tools, and hence its properties have a definite physical interpretation and specific utilization in practical applications, as, for instance, in the design of devices for signal analysis and synthesis. However, whilst the position-momentum pair in ray-optics yields a "joint" (i.e., local) representa-

224

Linear Ray and Wave Optics in Phase Space

tion of the optical disturbance, the Fourier t r a n s f o r m pair (F, ~) yield separate (global) information a b o u t tile signal. In the forthcoming cha,pters we will address the issue to ela, b o r a t e a, st)a(:e-frequency ret)resenta,tion of optica,1 signals t)eing a,s close a,s t)ossit)le to the inherently Iota,1 ret)resexlta,tion of ray ot)tics. Se(:tion 5.4 t)resellts a, rather (h;taile(t (tiscllssi()n ()n the ()t)ti(:a,1 Fourier l,ra,~lsf()rIil a.11(1 the ot)ti('al fi'a('tional F(nlrier tra,nsfi)rlll. In S('('t. 5.5 th(: z-(:v()lld, i()~l ()f the ('xI)e(:ta,ti()n wdlms ()f 1,11(; ()I)(;rat()rs a.n(1 ~ is (:()llsi(h;re(l. Tll(;y are s(;('~l t()(,v()lv(' in sillfila,r faslli()ll as 1,11(; ray (:()()r(ti~mt(;s q a,ll(t p il~ i)a,raxial ray-()t)ti(:s. Fi~la,lly i~1 Se(:t. 5.6 it is sl~()w~l tlmt the i~ltegra,1 r('t)r(;s('~ltati()~l ()f ttl(; ~n(;ta,t)h;(:ti(: ()t)(;ra.l,()rs (i.(;., 1,11(;ABCD i~m;gra,1 f()rnl ()f 1,11(;wav(;-l,ral~Sfi;r ()t)(;rat()rs IVI) a,ris(;s as a. lmita, ry r(;l)r(:s('~d,a,l,i()~l ()n 1,11(; Hill)(;rl, st)a,('e s linea,r (:a.xlo~fi(:a,1 l,ra,~sfi)r~m, ti()~ls ()f ~ ml(t ~, r(;t)res(;llt('(l I)y tim ray ~m.l,ri(:es M. Tile r(;sulti~lg (:()rr('~Sl)(nl(h;l~(:('~ ()f M t() M is s('(;~ t() 1)(; (h'fi~('(1 ~t) t() a sig~. Ih;n(:e, wlfilst a,~l is()~l()rt)lfisl~ (:a,~ l)e esta,1)listle(t t)etw(;(;~l ttm ~(l(;rlyi~g a,lg(;t)rai(: str~(:t~wes ()f l)a.ra,xia.l ray a.~(l wa,v(; ()l)ti(:s, i.(;., ttl(; syx~q)l(;(:l,i(' a,n(l ~ml, a,t)h;(:tic alg(;1)ras sp(2, !R) m~(l 'ttq~(2, !R), a,l~ is()~ll()rI)lfimll ~l) t()a. sigma l~()l(ls I)(;tw(;en the (',()rrest)()~(li~lg gr(n~t)s Sp(2, !t~) m~(t ]lip(2, !R).

5.2

Quadratic Hamiltonians and metaplectic Lie algebra

As .jllsl, reviewe(t, l,ll(' (pm(lra.ti(' Ill()II()IIlials ()f ray ()I)ti('s i-I,I'(~,llIli(llu:ly l,akeIl to 1,t1(; (12m,(lrati('..th;rIllil, ia,II ()t)era,t()rs ()f wave ot)t,i('s, we resI)e('l, ively (lell()t,e t)y K , K~, a,n(l K:~, i.e.,

lp2

,

fi

_ IF2

1 q2 2

,

K+ ^

-~q

_

89

,

~

(5.2.1)

1..2

K,- ~(~+N),

which a,re indeed the wave-optical trans(:ription of (2.3.6). T h e (tinmnsiona, lity of the above operators (tirectly folh)ws fronl tha,t of ~, having the, (timension of length, and ~, being dimensionless. The, Lie bra,cket of any two operators on the right of (5.2.1) corresponds to tile Poisson bracket of the (:orresponding c - n u m b e r functions on the left (see P r o b l e m 4.1), being in fact [ K~"+ , K _~"] - F o

'(2fi~),

[ E ~ , E ]+ - ' Fo ( - K +~") ,

[ K~"3 , K ~" _]

- ' ~ o ( K~"_ ) .

(5.2.2)

In w 1.3.2 it has been proved t h a t tile homogeneous q u a d r a t i c polynomials span a Lie algebra with respect to the Poisson bracket operation. T h e n in w 2.3 the i s o m o r p h i s m of this algebra with the Lie algebra sp(2, R) of the real traceless 2 x 2 matrices { K _ , K + , K 3 }, which provide the underlying algebraic

1D First-Order Optical Systems: The Huygens-Fresnel Integral

225

structure of first-order ray optics, has been evidenced. The correspondences 1 2 (5.2.1) identify an operator realization of the Lie algebra of the monomials ~p , lq2 and lqp. The commutation relations (5.2 2) prove that such an operator 2 realization spans a Lie algebra as well, isomorphic to the former, since the commutator between operators just retraces the Poisson bracket between the corresponding polynomials; the factor i marks the passage from the rayto the wave-optical context, as evidenced by the correspondence of the Lie brackets of the q-numbers ~, ~ to the Poisson brackets of the c-numbers q, p. Moreover, the exponential rule (5.1.6) mirrors the similar rule (4.3.5) that takes the traceless generating matrices in the symplectic algebra sp(2, R) to the ray-optical matrices in the symplectic group Sp(2, IR), whose correspondence to physically realizable optical systems has been elucidated in the previous chapters. Accordingly, by (5.1.6) the Hermitian oAperatAors in the metaplectic Lie algebra mp(2, R) spanned by the set {K , K+, Ka} are carried to the unitary wave-optical operators in the metaplectic group Mp(2, N), which gives a two-valued representation of the symplectic group Sp(2, R) [2]. We can therefore rephrase within the wave-optical framework the path (3 91.2) froln the quadratic monomials _~p 1 2 , _~q2 1 , 89 of first-order ray-optics to A

A

A

A

the exponential mappings e - i k o d K - , C- i k o p K+ and C-ikosKa of first-order wave optics, which then can justifiably be recoglfized as the paraxial propagators respectively for a free-medium section, a thin lens and a, positive magnifier: g e n e r a t i n g function

generating operator

A

p2

free-medium section

1 2

thin lens

1 2 ~q

positive magnifier

1

optica I p r o p a g a t o r

~

~,

--~

A K+

-

__ 1 ~ 2

___>

c-iko d if_

-- ~1

--,

c-i%p K+

A

"

2

(5.2.3)

A

1

Needless to say, inA tile q-domain representation (5.1.2) of ~ and ~, tile generating operators {K_, K+, Ka} turn out to signify A

~-

1

K_ --

02

2~2oOq2,

A

--

1

!~+ - - E

q2

" '

i

K3

0

-4k---~(1 + 2q~q).

(5.2.4)

Accordingly, the meaning of the real parameters d, 7) and s is the same as in the ray matrices of Chapters 2 and 3. We shall not discuss further the algebra and group related aspects implied by the sketched path (5.2.3), we are going to follow. As we go on, insight will gradually be gained. So, we will inspect the action of the wave-transfer operators e -ikodK- , e-ikoT)K+ and e-ikosK3 on a given signal in order to infer the explicit forms of the input-output relation (5.1.8), and hence the functional expressions of the spread functions g(q, q~) of the systems they represent. A

A

A

Linear Ray and Wave Optics in Phase 5pace

226

5.2.1

Prvpagation

thrvugh

a fr'ee-mediurn

section

Firstly, w(; coi~si(ter the ot)crator --

T(zo,

z,) -

-i~(<,-z~)'~

~' -o

-

= c

-i--~-(<,-z~)'~'~ ~,,0 .

(5.2.5)

wlfich, as ca.rlicr noted, describes the propaga.tion tl~rougt~ a hon~ogcncous ~(',(|i~m~ ()f rcfra.('tivc in(tex 'tt,, cxtcn(ti~g frt)m z~ t() z,,. 'l'lu'~ ()|)j(;(:t m~(t into,go a.~l~t)lil,~l(h's a.rc l,l~c~ I'('~lal,(;(1 l)y . . . . - T(z,,,z,)g,(U)

r

i .,-~-(:,,-- ) ~ ~",, "' r

-,'

7;l'a'nsfc.'l" 'l'c.lation in th, c. q-'l'r162

(5.2.6)

fl'ce-'nu'diwm, li'ne-sp'l't:ad fu'nction

gd

Ill <)r(t(;r 1,() g i v e a,~l exI)li(:il, fllll(:l,i()Im,1 f()rlll t,() l, ll(; a.|)(w(: ()l)(;I'a.|,()r r ( ' . l a t i ( m , it, is (:(mvcni(ml, 1,() cxt)rcss tile ('(nllt)h;x alllI)lil, lute ~ (q) in re.rills ()f l,lle eigenflln(:ti,,~s (4.5.10),,f ~, i.,:. 4L(q) - ~ 1 (,it,-q , h " - k,,p. I'l,is is |,,:,.m~sc ,,f a ~,,,ta.t,lc t)r~)t)crty ~f tll~', eig~;llfilIlc:ti~nlS ~f all ~)I)erat~w O t)ch~Ilging t~ s~)nle cigc~wd~u; / t, wl~i('li rr ~;ig~;~fi~:l,i~s ~f m~y ~l)crat~w fi~r f ( O ) i~f O witl~ cigcnva.l~u', f(It,), tl~e fin~(:t,i~al relati~n~ Cxl)resse(l t~y .f I~ci~g tra~sfi',rre~t ~ the i;igcnwd~u; It. 'I'ln~s, a I)l)lyi~g l,l~l; ~)l)l;ral,I)r T ( z , z,) 1,r l,l~t, r wl', 14~ta,in *"

i ( z~

T(z,,, zi),y~ (q) - ,'-

z i ) t';~2

2,,,,~.,,

r (q),

(5.2.7)

sh()wing tlmt the t)r()t)aga.ti(~ thr~n~gl~ a. ~mifi)rn~ refi'a.(:tiw: in(lex n~er (toes ~ot n~o(|ify tt~r t)r()t)aga.ti~ ( l i r e c t i ~ ~f a. I)lm~e waw;. Si~filarly, tl~c ray slope is not a.ltcre(1 t)y tl~c ray ,,,at, fix T(d) (E,t. (2.4.3)). th)w,;v(;I', l,|~,; l)r,,tmga,tion iml)arts 1,o l,hc sl)a,tial lm.n~()~i(: ( 'it''q a, I)lm.se sl~ifl, l)rr I,() ~.2. Then, ext)ressing 1,1~(; il~t)~l, a.~l)lil,~lth; r (q) in l,crn~s ()f l,l~c F
-

Turning ba,(:k to ~ ( q ) by the inverse Fourier transform and carrying out the intcgra.tion over ~:, the cxt)li(:it form (~f th(; imag{;-to-()b.jc(:t relation obtains --

2d ( q -

p~ (q') d q ~,

d -

%

where d denotes the section length scaled by the refractive index, in conformity to the notation used for the ray-matrix T(d) (a). By comparison with the a We recall that e - ~ ~' - ~ X d x - c-"-z sgn(;)

2~ e" 2~, ~

real

1D First-Order Optical Systems: The Huygens-Fresnel Integral

227

general form (5.1.8), we may individualize the line-spread function gd(q, q') for free-propagation over tile reduced distance d as the quadratic phase function gd(q' q')

-

-

~2k~deikO(q-q')22d.

(5.2.10)

71" The argument of the square root is taken in the interval ( - 571", ~]; this means that the phase of v~, with d being allowed to be both positive and negative (see w 3.3.1), is either 0 or 2" Like in the matrix relation (2.4.4) for the ray variables (q, p), the positions of the input and output planes enter (5.2.9) through the distance Zo - z~. The system is then invariant for translations along the optical axis, provided the input signal be equally translated. So, the operator T ( z o , z~) may equally be denoted as T(zo - z~), or T(d). Finally, resorting to tile general considerations in w 4.4.2, we note that the transfer relation (5.2.9) does not account for the phase accumulation as the light propagates froln tile plane at z~ to z o, tile spread function gd modeling only the transverse profile of the propagating wave. Therefore, the axial phaseshift c i % % ( Z o - q ) / 2 should be added to the propagated signal amplitude Po(q) in (5.2.9) for a complete account of the propagation effects.

T h e l i n e - s p r e a d f u n c t i o n gd as the r e s p o n s e o f the system, to a line-like signal

The line-spread function represents the response of the system to the unit impulse 6 ( q - q'), as formally expressed by Eq. (4.3.14), which in terms of T and gd writes gd(q, q') -- T ( z o , z~)6"(q - q'), (5.2.11) from which expression (5.2.10) for g~(q, q') is easily obtained, on account of the Fourier expansion (4.5.7) of the 6 function. Within the original 2D model introduced in w 4.3.2, the (5 function yields the mathematical representation of the point source of a spherical wave, and hence gq. (5.2.11) should describe the propagation over the distance z o - z ~ of a spherical wave originated at point q' in the input plane. In a homogeneous (isotropic) medium a spherical wave radiates uniformly through concentric spherical wave fronts with increasing radii. Within the planar model of optical propagation of our concern here, we must properly consider a cross section of the emanated spherical wave in a plane through the center, i.e. a circular wave. In fact, the picture suited to a planar geometry is that of a line source and an emanated cylindrical wave. Either case, however, the cross section of the wave in any meridional plane through the point source is represented by conforming to the choice of the argument of the square root in the interval (-~, ~]. The symbol sgn(-) means the sign flmction, which gives 4-1 according to the sign of the argument.

Linear Ray and Wave Optics in Phase Space

228

,

;

i

S = tan '('~-ql)

.,.\

z,,\- z,

\ n

~",

\

';q

"",,

',,i

q -- q'

9 !

o !

o

I

o

i

o

Zi

I:I(I[JII,E 5.2. Tim li~w sprca~l fl~t~cl,iot~ ge (q, q') cm~ I)c ~n~lcrst,~M as tim (:r m~plit~r at l)()i~t, q i~ t}~(.' ()t~l,l)~l, l)lm~c ()f a cili~(lrical wave ct~m~at, i~g f r o ~ l)()i~d, q' i~ tim ii~p~t plmm, with both q m~l q' rm~gi~g t,l~ro~gl~ object a ~ l ixnagc rcgio~m re:ross tim c'~l-plm~es wlficl~ arc s~pposc~l to ])e ~mu'l~ mnaller tt~m~ tim p l a ~ t o - p l a s w . (tistmmc.

V/

ik (,i.n()k op

v~

(5.2.12)

witll p (h~ll~)l,illg ttlc (listml(:~; fl'~nll l,ll~'~s(nlr(:(; i~l tile q-z I)la,xl(; (Fig. 5.2). If the t)()i~t s,)~r(:(; is l()r a,t (q', z~), a,s s~q)t),~sc(t i~ (5.2.11), tim sig~m,1 a.t (q,z,) stl()lfl(1 tlav(; tll(; fi)rIIl as (5.2.12), witll p = v / ( q - q ' ) 2 + ( z , , - ~ ) 2. Ull(l(;r th(; llyl)()tlmsis tlmt ()nly snm,ll r(;gi()ns a,(:r()ss th(; ()l)ti(:a,1 axis i~ t)()th th(; int)~t m~(l ()~t,t)~t, I)la,~(;s ar(; r int(;rest, their ~m,xi~mn~ lal,(;ra, l (;xtensions l)(;ing ~n~(:h s~m.ll(;r ~f tim ~)v(;rall axia.1 (lista.~(:e z , , - z~, w(; r (',xt)m~(l the i,,mgr162 (list, anr p with rest)(;(:t t,()thr snmll a,ngh; I q - ' / I << 1, and

[) ~" Z ~ -- Z i Jr- "~ (q~--q--z2.--~ 9

Zo

--

~" i

Wc r('placc p by ttw~ zcro-or(tcr term Zo - z; in the nmgnitudc of (5.2.12) and by the sc(:(m(t-()r(tcr extm,nsi(m (5.2.13) in the phase, being the t)hase much sensitiw'~ t() errors. Then the (:ircular wa,vefront (tisturt)an(:e (5.2.12) turns into the t)a,ra|)oli(" wavefroilt (tistllrbance i

~ ( q , Zo ) ~

nOkO

ci 2(% nOkO -zi ) (q_qt)2 cinoko (Zo-Zi)

,~(Zo-Z~ ) -

(5.2.14)

which, a.t)art fl'om the t)la,ne wave factor, has tile very flmctionM form of tile line spread fllnction gd- Tile object-image relation (5.2.9) relevant to freem e d i u m sections acquires therefore tile meaning of the composition on the

1D First-Order Optical Systems: The Huygens-Fresnel Integral

229

output plane of the parabolic waves emanated from each point on the input plane with strengths fixed by the input signal, the linear sizes of the involved regions over both the input and output planes being much smaller than the axial (tistance through which the waves are propagated. This is the Fresnel approximation to the Huygens integral, which in its general form involves the composition of the propagated spherical (or circular) waves with appropriate weighting factors [3, 6]. The Fresnel (or paraxial) approximation to the general diffraction theory, which allows us to replace spherical profiles by parabolic profiles, implies roughly that (wi--F-Wo) 4 4()~o/no)(Zo_Z i )x % C,

09~+~9~176 4(~0/~0)

or

< c

(5.2 . 15)

if c is the maximum angle error, in units of 7r, tolerated in the phase in (5.2.12). Here, wi and Wo denote the largest radial extensions of the regions in the two end-planes within which the input and output signals may be confined. Correspondingly, tgi and 0o are the angles subtended by wi and Wo at Zo and zi. The above reasoning shows that the Fresnel condition (5.2.15) of paraxial wave optics corresponds, and give a quantitative account, to the small angles and off-axis distances condition of paraxial ray optics. Transfer 'relation in the t~-rcpresentation: free-medium transfer function b~ It is evident that the impulse response g~ of a homogeneous-medium section depends on the coinbination q-q~. With the notation of ~ 4.6.3, we may write ga(q, q') -- ~la(q - q'),

(5.2.16)

where, according to (5.2.10), gd is

~d(q)- ~

e

i~dq2

.

(5.2.17)

The system is shift-invariant under transverse translations, as expected due to the uniformity of tile refractive index. Translations of the object over the input plane turn into translations of the image over the output plane by the same amount. The same occurs in the simple picture of ray optics, where a shift of the ray height turns into an equal shift in the height of the propagated ray without any alteration of the ray inclination. Accordingly, the transfer relation in the spatial frequency domain writes in the form (see Eq. (4.6.27)) @o(t~) - b~ (t~) ~5~(~).

(5.2.18)

The transfer function b~(n), relating to the Fourier transform of gd, writes as b~(t~) -- c

- i d t~2 ~ ,

(5.2.19)

230

Linear Ray and Wave Optics in Phase Space

i.e., just as the q u a d r a t i c pha,se flmction entering (5.2.7), whi(:h is in fact a ma,nifesta,tion of relation (5.2.18) for the specifi(: (:a,se of plane waves. N o t a b l y b o t h the spread and tra,nsfer flmctio~l g~, b~ have the functiona.1 form of a quadrati(:-phase fllnction, whi(:h a,s note(t in w 4.6.3 is typically reported in the signal ttleory a.s a (lilma,r) chirp. Tlle sil~lila,rity ()f the flmctional forn~s ()f g~ a n(l I}~ is (t~(', t() the tmsi(: t)rot)erty ()f ttl(; (:lfirI), wlfi(:h being a,n inm gilm,ry Gallssia,~ f()r~ is t)asit'ally t~lnie(t i~d,() its (:()~t)lex (:()~l.j~gate t)y Ft)llri(;r |,l'a,llSf()r111a, l,i()11. Ill fi~,('t, (me has

wl~i(:l~ ImralMs r(:lati()~ (,1.6.51) fi)r real (lm~ssia,~ f()rlllN. S() w(; see l,lm.l, 1.11(; l,rallsfer n'lati()11 fin" fl'(:(, I)r()l)aga.l,i()11 11m.l,ll(;1na,ti(:ally sl~aI)(;s a,s a (:t~irl) ('()~v()l~(i()~ i~ I,ll(' sI)a,('(' (l()~ai~ al~(l as a, ('l~irl) l~nlltil)li(:a,tion ill tile fl'(~r (l()~a,il~:

Propagation through a homogeneous medium spatial domain c()~w()l~l, i o ~ I W t h e c l l i r p

frequency domain

t,.

~" ~~,/q2

.

l m f l l , i l f l i c a l , i o ~ I W t i m cl~irl) ~'

.__xL ~.2

' 2t,.()

C()nw)lllti()n a,ll(l llnlltil)li(:a,ti(nl I)y a (:lfirt) a,r(; F(nn'ier (:()~l.jllgat('. ()l)(;ra,tions. 'l'llc r t) (:()lw()lllti()ll is kll()w11 llla,tlw.11mti(:ally a,s l,ll(; l,'r(;s~l('.l trmlsfornl. 'l'll(;r(;f()r('., t,tl(' fl'(~(' l)r()l)a.gal.i()ll })(;tween tw() 1)lmles S(;l)arate(l })y a. (:ertain (tista,ll(:(; d t)I'()vi(h;s a, l)ra.(:ti(:al)h: t()()l t,() t)llysi(:ally i1111)l(;111eld, the Fresucl t,ra,nsf()rlll ()f a 11m.tll(;llm.ti(',al fi~ll(:ti()11, tlmt is t)r()i)erly t,~rn(;(l illto a,n optica,1 signal 1,() t)e t)r()t)a,ga,te(1. Tyl)i(:a,lly, (,11c fim(:l,i()11 is nm,t)t)e(l t() the (:omplex a,xlq)lit~ute tra, nslllitl, a,ll(:(; ()f a, ti'a,llSi)a.r(;ll(:y , whi(:ll is t)la,(:e(1 i~ 1,11(;()|)ject plane a,n(t l,l~ell illunfi~m.te(1 l)y a, ~l]lit-mlq)litu(le t)lml(; wave. 111 a,(:(:()l"(t with the t,;-donm,iI~ transfer rela, tioll (5.2.18), the Frem~el I,ra,11sfor111 is e(tuivMent to a sI)a, tial frequency filter, witll 1)hase-(mly filtering flm(:tion t}d. All the fl'equency coInt)()nents of the int)~lt are l)r()I)a,gate(t without a,l,tem~atioi~, lint with a phase dist)ersion quadra,tic in the fl'equellcy. 5.2.2

PTvpagation thTvugh a

thin lens

As earlier remarke(t, the operat()r k(f)-('.

/

- c -*2/q

(5.2.21)

describes the effect of a thin lens with (reduced) focal length f (and power

1D First-Order Optical Systems: The Huygens-FresnelIntegral

I / l/

\ !

!"

i

/

"i

231

///

-i

"---" f ---~,

(b) F I G U R E 5.3. A thin lens transforms a normally incident plane wave into a spherical wave (a) converging to the real secondary focal point F' (if f > 0) or (b) diverging from the virtual secondary focal point F' (if f < 0).

Transfer relation in the q-representation: lens line-spread function gf As in the space-coordinate representation the operator ~ has the effect of multiplying the wave function by the coordinate value q, the thin lens operator k(f) acts as a phase-only transformer, affecting the incident wave simply by tile quadratic-phase factor c-i% q2/2f. The complex amplitudes p~ (q) and Po (q) of the optical waves incident on and emerging from the lens arc then linked by

_i #'Oq2 (q)

-

(q),

tile reference planes being a~ uslml locate(] immediately in front of and t)ehind tile lens. Unlike a section of homogeneous inediuin, a thin lens a.cts as a modulator with the phase-only modulating function c-i% q2/2f. Since, as previously noted, the chirp factor in (5.2.22) may be interpreted as the paraxial approximation to the circular profile of a spherical wave, the above relation means that a thin lens transforms a normally incident unit amplitude t)lane wave (p~ (q) - 1) into a spherical wave centered about a point on tile lens axis. In particular, as illustrated in Fig. 5.3, if the lens is convergent (f > 0), the emerging wave is convergent toward the real secondary focal point F'; correspondingly, if the lens is divergent ( f < 0), the emerging wave is divergent about the virtual secondary focal point F' (w 1.8.3). Accordingly a thin lens maps linear profiles into circular profiles (and hence planar wave fronts into spherical wave fronts, within a proper 2D picture of light propagation) in the proximity of the optical axis. Under non paraxial conditions the emerging wave fronts exhibit deviations from perfect sphericity, as an effect of the aberrations. The input-output relation (5.2.22) can easily be cast in the integral form

Po (q) -- f dq' gf (q, q')F~ (q'), J

(5.2.23)

232

Linear Ray and Wave Optics in Phase Space

ret)ro(tucing the genera,1 form (15.1.8). The imt)ulse response g~(q, q') m'ites a,s .._~q~ g~ (q, q') -- k ( f ) ~ ( q - q') -- c-~*~/ b(q - q'). T h e a,t)()ve rela.tion (:an a.ls() t)e rega,r(h;(t a,s tim eig(;x~va.l~('~ e(t~a.ti()~ fi)r the thin lens ()t)(;rat()r k ( f ) , with the b flm(:t,i(n~s t)eing tt~(; (;ig(;~fim('ti()~s ()f ~. The (h's:(mq)()siti(m ()f tlx(; int)~t sit4na,1 i~ tel'IllS ()f ~ fllll(:t,i()llS l)r()vi(h's tl~eref()r(' 1,1~(; syste~n. I~ (,l~is r('sI)c(:(,, w(; r('(:all thai,, a(:(:()r(li~t4 t() 1,1~('~nkv ()I)tit:s I)i(:t,~r('~, a (,lfin lens (l()(;s 11()|, afl'e(:t 1,1~(;ray I)()sit, i(n~, l)(,ix~g q,, = q,. C~l('arly tll(', i~l)~l, ()~lt,l)~ll, relati()x~ (5.2.22) (l()(;s 1~()1, a.(',(:()llld, ft)r l,l~(; axial t)hase slfifl, a.s t,l~(; wa.ve t)r()I)aga.tes l,l~r()~tZ]~ l,l~(; lens, (t~u' f()r i~si,a,~(:(' t() tl~e fi~i(,(:, a.Ii,l~()~gl~ ~(:gligil)h:, (,l~i('kn(:ss ()f (,l~(: h:ns. TI~(:~, if ~(:(:(h:(l, (,h(: axial I>has(: (May ~'it'()"'~'~ I~n~s(, l)(: a
l("u,s u~a,'~('-sp'l'('a,d flm,('tion

hf

A('(:()r(ti~g t()tl~(' g('~(;ral r~l('s (4.(i.19) ma(l (4.6.2()), t,l~(; l('~s t,ra~Lsfi'r i~ t,l~(' Sl)a,l,ia,1 fl'(;(llU;ll(:y (l()l~laill is (l(;s(:rit)('(l ])y l,ll(; SllI)('rl)()sil,i()ll ild,('gral

~o(h:) -- [ d/-,:' hf(h',/',:')~,,i(h:'),

(5.2.25)

!

wll(;r(; th(; wa,v(;-sl)rca,(1 flnl('l,i()ll hf is

hf(~:, h") -- ~/ f ( 'i-/-(~: - ~.,)2 2rrik() 2/,.,,

(5.2.26)

the t)ha,s(; ()f tl~(' s(t~m,re r()()t t)eing yet again tak(;n in th(; i~m;rwd ( - ~ , ~]. Evi(tently, in th(; h:-(h)main the lens t)(;ha,v(;s a,s a shift-inwwia,nt syst('m, and so the tra,nsf('r relation in th(' q-domain is si~t)ly nmltipli(:a,tiv(,. Relation (5.2.25) with th(' relevant kernel (5.2.26) resenfl)h;s the q-doma,in transfer rela,ti()Ii (5.2.9) for the propagati()n t h r o u g h a, hom()geneous mediuIn, the wa,ve-st)r('a,(1 flmction hf, as a flm(:tion ()f h:, shaping like the line-spread flmction gd, a,s a, flmction of q. The focusing power P -- ~ is placed in (5.2.26) as the re(tu(:e(t (listance d in (5.2.10). Correst)ondingly, the h;-(tomain transfer relation (5.2.18) for propaga,tion t h r o u g h a homogeneous m e d i u m has the same multiplicative s t r u c t u r e of the q - d o m m n transfer relation (5.2.22) for lens focusing. In b o t h cases, the input is simply affected t h r o u g h the chirps c - i a ~ / 2 } ~ and c -ik~ the correspondence of P to d being yet again evident. This is a m a n i f e s t a t i o n of the duality relationship holding between free propagation

1D First-Order Optical Systems: The Huygens-Fresnel Integral

233

and lens transfer. Chirp multiplication and chirp convolution are in fact the m a t h e m a t i c a l operation acted by a thin lens respectively in the space and frequency domain through an inverted scheme with respect to that pertaining to the propagatioll through a holnogeneous medium:

Propagation through a thin lens spatial domain multiplication

frequency domain

by the chirp

9

e

9.f

~i

convolution

'

~o

by the chirp

e

,2k

0

~i

'

~o

PTvpagation through a positive pure magnifier

5.2.3

We describe now the wave-optical representation of a positive pure magnifier, whose wave transfer operator writes as A

A

.k0s

s(,,,,.) - ,:-~kosK~ _ ,:-~ 4 ~ + ~ q ~ ,

(5.2.27)

where the real parameters m and s are linked by m - e S/2. Taking the explicitexpressions of Ka in the coordinate representation, the effect of the operator S(m) on the object wave function ~ (q) follows from A

t"-

s

s

0

~o (q) - s ( ~ , , ) ~ , (q) - ~-~ ~ - ~

(5.9.28)

~ (q).

It is convenient to de(:onlpose the input wave p~(q) into 6-like components. Then, after analysing the action of S(m) on each component 6(q) by inspection of the spread function A

s

s

0

gm (q, q') -- ~ - ~ ~ - ~ T q

t(q' -- q),

(5.2.29)

we recompose the output wave according to the superposition integral

~o (q) -- f dq' gm(q, q')~9, (q'). Changing from q to q with q s

o cq, so t h a t q oq

(5.2.30) 0 the scaling operator Oq'

0

s 0

e - s q N with respect to q is carried to the shift operator e 2 o, with respect to tile new variable. Thus, being ~o C

2q~q(~(q'

-- q)

--

C

2 0q ~ ( q '

, -- cq)

--

(~(q

~q_~ --

_ ) --

(~(q'

) q m

~

(5.2.31)

the spread function gm (q, q') takes the explicit expression

gm(q' q') - v/~(q '- ~),

(s.2.32)

Linear Ray and Wave Optics in Phase Space

234

whi(,h conforms to the ray-optics descrit)tion of the ideal Inagnifier transferring point objects at qi into point images at q o = mqi (~ 2.4.3). A('cor(tingly, tile integral transform (5.2.30) yields the inl)ut-outtmt relation for tile positive ideal magnifier in the simt)le form

r (q) - ~ , , r (~,,,) ,

(~.2.aa)

sh()wiIlg tllal, t,ll(; (~llt,gt~illg sigllal fl't~Ill l,llt' sysl,('~]ll S(ttt) is a s('al(;(l rt:t)li(:a ()f //I tlnas I)reservi]~g l,l~'~ (:ll(WK~. R e l a l , i ~ (5.2.33) is as w~;ll ~t~l,ai~al)h', fi'~nll (5.2.28) l~y tll~'~ (:lm,~lge (~f w~,riat)le q---+ q m~l 1,1~'~~:~n~s~',(t~u;~t trm~sfiw~al,i~ q ~ --+ No a.s i~ (5.2.31). Tlle al, enlt)lmsizing tll~; mla.l~)gy witl~ the; ray-~l)ti('s r nl ~f l,ll~'~syst~,nl. Finally, w~: ~:,~si~ler tl~'~ ~',[ti'~(:t (~f tlw, ~l)~;rat~w S(t~) i~ 1,1~' h'-(l(nlm.i~l. The fi'e(l~U'~r Sl)~:cl,ra ~f tl~: ~l~.j~;cl, m~l i~mg~, ~lisl,ril~l,i~s are r~'lal~;~t l~y A

~,, (h') .....

/

(5.2.34)

dr;' h,, (h', h")~i(t;'),

wl~:r~, a~'c~)r~lil~g 1~) (4.(i.2()) 1,1~, wa,w, rt,sl)~)]~se h m is

h,, (h', h") = ~)-~b(h" --t,th'),

(5.2.35)

which t)a,ralMs l,ll(: ray ~)l)l,ir (les(:rit)ti~nl ~f the sysl,(',lll, a~:c~r~ling l,~ whi(:h l,]l(; ray lll()lll(:lll,lllll Pi is trallSfi)r]lle(l i]d,<) p,, - ~I t l , (~ 2.4.3). Tll(' i~It('gral trmlsf()r~ (5.2.34) t,i~r]~s I,I~(;~ lid,() t]~(: s('ali]ig r(:lati()]i Z,(~') = ~ , ~ ,

5.3

(5.2.36)

(',,,,~).

Wave-optical transfer relations for an ABCD system

We at)t)r~m,(:h tll~', i11Ind,-~)lltI)ltt I)rol)lenl f()I' a 1 0 first-~)rr ~)I)ti(:a.1 system, genera, lly (les(',ril)e(1 1)y t,lm ray-transfer nmtrix M - - ( c A , g ) . We resort to the e(tllivalell(:es establislled ill Chat)ter 3 of any optical system to apt)roI)riate seqlmll('es of a. finite num|)er of free-medium sections and thin lenses. I~l I)arti(:ldar, we consider the sixllt)le configllration (:onsisting of two free-me(tillnl se(:tions and one thin lens, (lescribed by Eqs. (3.4.1-2). W i t h i n the t)I'esent forma,lism, the equivalence in (3.4.1) can be restated as k

M(M)

/

T ( d 2 ) L ( f ) T ( d , ),

-

(5.3.1)

with the relevant parameters being specified by dl

A-1

c

>0,

d2 _

-

D-1

c

>0,

f

c1,

B

-- d 1 + d 2 -

dl / d2 > 0 . ( 5 . 3 . 2 )

1D First-Order Optical Systems: The Huygens-Fresnel Integral 5.3.1

235

Transfer relation in the q-representation: line-spread f u n c t i o n of an A B C D s y s t e m

Using relations (5.2.9) and (5.2.22) to account for the effects respectively of the operators T(dl) , T(d2), and L(f), we easily obtain the complex amplitude ~o(q) of the emerging signal from the single lens setup (5.3.1) in the form

~o(q)

. k__O_ q, 2 C?~2d2 . k__~_(q _ q, ) 2 Ci 2d1 k__o__( q, _ 2 7 ko r i ~ fj dql jf dqtte-/'2f

qt,)2

~i(qtt)" (5.3.3)

We perform the integration over the d u m m y variable q~ and exploit relations (5.3.2), thus fina,lly obtaining the integral tra,nsform

~o(q)

-

M(M)p~ (q) - f dq'gABcD(q,q')p~(q' ),

(5.3.4)

where the relevant line spread gABCD(q, q~) explicitly writes as

gASCD(q, qt)

__

ko

"ko

~/27riBC't2B(Aq'

2

+ Dq 2 - 2qq')

involving the entries of the ray-matrix of the overall TLT configuration. Although the TLT configuration (5.3.1) ('omprises a, limited set of ABCD systems, if one regards d~ and d 2 a,s positive section lengths, we a,ssmnc rcla,tion (5.3.5) as a general prescription to construct the representing kernel for the intcgra,1 transfornl (5.3.4), which defines the wave-optica,1 operator M associa.ble to the ray-optica,1 matrix M, with the alrca,dy mentioned convention for the pha,sc of the squa.re root being taken in the interval ( - 2 , ~]" We will address in ,~ 5.6 the question of the uniqueness of the operator M a,ssocia.ble to M through (5.3.4), and ~ccordingly of ~o(q) resulting from F~(q) through M. The spread function gABCDof a,ny first-order optica,1 system is evidently a,n imaginary Gaussian exponential of the profile coordinates at the end planes, with coefficients determined by the ray-matrix entries. The factor in front of the Ga.ussian exponentiM guarantees the losslessness of the sytem. For ease of writing, the subscript ABCD to g will be dropped; our discussion is in fa,ct concerned exclusively with first-order, i.e., " A B C D " , optical systems. Remarkably the integral transform in (5.3.4) is the Huygens-Fresnel integral of diffraction theory. In fact, it can also be inferred from the Huygens principle through the Fresnel approximation to the Huygens wavelets, with the relevant eikonal function being expressed in terms of the entries of the overall A B C D m~trix of the system [4]. This ca,lculation automa, ticaJly includes the phase shift factor, that on the contrary, as noted m~ny times and further stressed below, is not accounted for by a, ray-matrix related a,pproach. Expression (5.3.4) of the input-output relation for an arbitrary opticM system represents the cardinal result of the chapter. It states that, within the

236

Linear Ray and Wave Optics in Phase Space

p a r a x i a l a,pproximation, aIl a r b i t r a r y optical wave (:an be p r o p a g a t e d t h r o u g h an opti(:al device, on a c c o u n t as well of all the diffraction efibcts, 11sing knowlc(tge only of the ovcra.ll ra.y n m t r i x of the system, wlli(:tl synthesizes the (tctailed stI'uctltr(~ of the device. Also, it csta,l)lishcs the fllll (:()rrcst)(m(tcn(:(~ t)ctween tllc ray-()l)tical (lcs(:rit)tio~l ()f ot)tical systcnls in tcrllls ()f ray nmtri(:cs and the wa,v(;-()I)tic'al (lcsc:riI)ti()ll ill tcrllls ()f wa,vc tra~lsfr ()l)eratc)rs, tllc flnl(:ticma,1 fi)r111 ()f 1,11(; latter l)(;illg Cxl)r(;ssc(l ill l,('r~lls ()f tll(; (;l~tri('s ()f tllc fi)rlllcr ()xlcs. Th~ls, a ll ()t)ti(:al sys(,('lli (:all 1)(' a~la,lyz('~(l t)y ~lsi~g (,l~('~ ray ~ai,rix fln'~mlis~11 ()r 1,11('~waw' ()l)('ra,l,(w f i ) r ~ a l i s ~ , wl~i(:l~('~v(;r is ~l()r(; (:()~w'~li(;~d,. ~I'l~(; ~r()~ t) ()f first,-(w(h;r ()l)ti(:a,1 syst(;~lis is 1,11(' l)llysi(:al rea,liza,ti()~l ()f 1,11('~sy~111)l(;(:(,i(' gr()~l t) ()f

c~11t)l~asiz('~ 1,1~(;lmsi(' c()~('('l)t~ml ('()~1,(:~1, ()f r('la,ti()~s (5.3.,1) a~(l (5.3.5), wlli(:ll I)r()vi(l(; t,l~(' li~k ])(;tw(:(:~ (,l~(; i~l~('r('~tt, ly l()('a,1 l)ict,~r('~ ()f ligld, i)r()l)aga,ti()~ i~1 ray ()l)(,i(:s a,~(l 1,I~(: i~l~(:r(:Id, ly ~()~ h)(:al l)i(:(,~r(: i~ way(: ()l)(,i(:s. 'I'() fi~r(,11('r al)I)r(:('ia((' (,I~(: sig~ili('a~('(: ()f |';(is. (5.3.,I) m~(] (5.3.5), w(', m a y (:()~ll)a.r(' 1,11(:si~1~l)l(' fi~(:l,i()~al f()rIil tl~(:y s~gg(',s(, f()r (,l~(' sl)r(:a,(l fi~(:(,i()~ g witll

sys(,(,~. ~I'll(' flw~lml si~l)li('ity ()f r~th' (,1.3.17), (l(;~m~(li~g t'(w (,l~(' l)r()(l,t(:t ()f I,h(~ si~('(, I,l~(; l)r()(l~('t ()f r ~my yi(;l(l ral,l~(~r i~tw)lv(;(l fi~('l,i()~al f,)r~s, a,s l)r()v('~(l, fiw i~si,a.~(:(', 1)y l,l~(; (l()~tl)l(' i~t('t4ra,1 f(w~ i~ (5.3.3) a('(:()~t~li~t4 f()I" the ()f ~lal, I'i(:t:s (:a,~1 (:a,sily I)(: l)(:rf()r~(:(l. 'FIH~s, 1,I~(' ra.v-(,()-wav(: (:()rr(:sl)()~i(l(:~(:(: (:sta,l)lisl~(:(l l)y r(:la,(,i()~is (5.3.4) a,~(l (5.3.5) a lh)ws (,()r(:I)la,(:(' (,lw (:()~q)li(:a,(,(: (:a,l(:~fla,(,i()~ i~fl~(:r(:~1(, ii~ (4.3.18) l)y si~II)l(: ~11~fl(,iI)li(:a(,i()~Is ()f 2 x 2 ~m(,ri(:(:s. Fr()~11 tit(: gI'()ll I) t,l~(;()l'(:ti(:a,l vi(:wl)()i~It, r(:la,ti()~is (5.3.4)-(5.3.5) sig~lify a, corrcsl)o~1(h:x~(:(: ()f ~mix11()(l~flar ~m(,ri(:(:s in (,h(: synq)l(:('(,i(: Lie gr()~l) Sp(2,]~) to unita,ry ()l)(:ra,t()rs i~l 1,11(: ~11(:(,al)l(:(:ti(' l~i(: gr()~II) Alp(2, I~)" A

M c Sz,(2,R)

> M e MZ,(2,R),

(5.3.6)

the latter ()llOS acting (m the IIill)ert st)ace s of wa,ve fllncti(ms t h r o u g h the integral t r a n s f o r m (5.3.4) with kernel d e t e r m i n e d by M actor(ling to (5.3.5). It is ill the virtlm of sll(:h a r t h a t all the consi(terations (tcvelopc(t in C h a p t e r 3 rela,tivcly t() t)ropagation proi)lcms expressed in t e r m s of ray matrices a p p l y to prot)agation p r o b l e m s expressed in t e r m s of waveoptical t)roI)agators. This is to sa,y, we can solve the o p e r a t o r eqlm, tion (5.1.5) ruling the p a r a x i a l p r o p a g a t i o n t h r o u g h a q u a d r a t i c m e d i u m by solving the c o r r e s p o n d i n g m a t r i x e q u a t i o n (4.3.4), with rules (4.4.10) and (5.3.5) providing the key to t u r n from the o p e r a t o r to the m a t r i x formalism and vice versa.

1D First-Order Optical Systems: The Huygens-Fresnel Integral

237

However, the correspondence between symplectic matrices and metaplectic opera, tots is not one-to-one, a,s stra, ightforwa,rdly will emerge in w 5.6, where we will frame the wave-optical relation between ray-matrices and wavepropaga,tors within the general context of unita,ry representations of linear canonical tra,nsforma.tions. We will see that the unitary operator M associable to M is specified to within an arbitrary phase. Even though the choice of the phase can be narrowed by requiring that the operators reproduce the multiplication rule of the matrices, a sign ambiguity still remains. In fa,ct, the mapping (5.3.6) is
M~( M ~ ) M 2 ( M 2 ) A

.~.

A

- +M(MIM2)

,

(5.3.7)

and accordingly, [[~(M)] -1

-

-

-t-M(M -1),

M -1

-

(_~ AB).

-

(5.3.8)

Clearly this sign ambiguity implies a sign ambiguity on the propagated wave function as well. So, whilst the ray-variable evolution is uniquely determined through the optical matrix M, the wave flmction evolution is determined through M ut) to a sign. If one is interested in the energy distribution of the prot)agated wave, su(:h a, sign, which is indeed ot)scrvable [10], ca,n be ignored. Fina,lly, we stress yet a ga,in that the input-output relation (5.3.4), being derived from the ray inatrix picture of optical systems, does not include the on-axis phase shift fa,ctor c ik~ ext)erienced by the optica,1 wave as it travels through the system. This factor must be separately evaluated and added to the outgoing amplitude, if it is of releva, nce for the problem under study. Naturally the optical path length 12 along the a,xis of the system is the sum of the individua,1 on-axis optical path lengths s for all optical elements in the system: s = ~ j s each s being corrected by the factor 1/2 when required. 5.3.2

Transfer relation relation in the ~-representation: wave-spread function of an ABCD system

The relation between the spatial frequency spectra of the incoming and outgoing field distributions ~i(e~), ~o(~) for an ABCD system writes (see w 4.6.2) ~o(t~) -- ff dn'h(n, n')~i(n').

(5.3.9)

The wave spread function h(t~, n'), resulting from the double Fourier transform of the impulse response (5.3.5), has the explicit expression h(n, n')

_

/V 2~k~

-i

~

+ An2

-

"

(5.3.10)

Linear Ray and Wave Optics in Phase Space

238 Hi

Flo

Optical system

(fli (q )

Hi

~oi(h )

~o,,(q )

/k

Ho

Dual Optical system

)

/k

M

M

2t

Zi

-

(a)

(h)

t,.

l"l(~[ll{l'; 5.4. lh;lat, io~ I~et,wee~ (a) a~ Ol)tical syst,(;~n m~(l (b) its (l~ml. As M acts i~ the real A

spa('(: ()** tim ligt~l, (list,ri]n~l,i(),~ ~, (q), M acts i,~ tim h-(lo,nai,~ ~)~ l,t,(; b'~)~n'ier l.rm~sf()r,n @(h;).

wil,1, l,l,(; llslml (:(nwel,l,i(ni fi~r l,l,~' a,l'gt~,(;l,l, (~f t,l,(; s(t~ta.I'(, r()()t,. N()l,al~ly, g(q, q') ml(l h(n. n') lia,w' si~ilar i~mgi~ary (lm~ssia~ fin'~s wit, l, (;n(;rA~C-l)r~;s~;rving ('(n,,I)h;x fa(:l,()rs, 1,lint, (:a.1, |)(; sy~,l.l,(;siz(;(1 i1,1,~) _

IA," +

_

2,,'1

(5.3.11)

rl'll(; l)a.ra.lll(;|,(;l'S .A, /~ all(l D ~;llt~;rillg t,ll(; a.l)()v(; (:a.1, 1)(; ln,(l~;rst<)()(l as t,he l,,a,l,rix (~ ~ ) , wl,i('l, is al1,,;r,,at, iv,:ly i,l,:l,l,ili,;(t wit, l,

,;1,l,ri,;s ,,f 1,1,,; ,,,,i,,,,,,l,,la,r

%

t,l,(; ra,y-()t)l,i(:s matrix M -

/

( At.,,(: ,J/h,,,) f(w the Im,ir (q, -) wl,e,1, (:(,l,si(l(;ring V

t,h('~ i11,l)llls(; r(;st)()l,s(; g(q, q'),,I" witll tl,(; llmtrix 1 ~ -

( - 11D/t.,, -/,',,(:).4 f()r the

t)a,ir (h',-q) wl,('.l, (:()llsich;,'i~lg l ll,; waw, r,;sI)oI~s(; h(n, n') (t,). As w(; kn,,w fron~ w 3.6.1, l,h,; I,air (q,n) (:a,n 1,(; ta,k(n~ 1,,, l,l~,; Imir (h:,-q) (apart fl'(~ii, scaling fa,('l,(n's l,tm,l, (:a,], t)(; a,|)sort)(;d |)y slfitat)l(; 1,(~rllm.liza,tions) |)y a F(nu'i(;r tra,nsforiIm.tioIl, I)ro(tlu:iI,g a (:lo(:kwis(; r(~tation t)y ge in the phase pla,ne. It is (;vi(tent fi'(m~ (5.3.11) that if ~;i(q) is the int)~t signal t(~ the optical systenl, (tescribed |)y t,h(; ray nm.t,rix M, a.nd @i(a;) is the int)ut siglml to the system, (h;s(:ribed |)y t,ll(; ray ma,trix M, then the olltt)ut signa,1 from M is just ~o(h:), that is the F(n~ri('.r transform ()f th(; output signa.1 ~o(q) fr()n, M (Fig. 5.4). In that sense, the system M is referred to as the dual system of M [5]. It ha,s been earlier r(;Ilm.rk(;(t that the single spa,(:e-angle dc's(:ription of wave fields, (:()nveyed through th(; ray variables (q,p) ()f geometri(:al ()t)tics, is ret)la,ce(t |)y two (Fouri(;r-transform linked) descriptions in wave optics. The hAs noted in w 4.6.5, tile same syinbol M will be used to signify transformations of both the ray-variables (q, p) and the Fourier conjugate pair (q, ec), in the latter case being implied that tile off-diagonal elements of M be scaled to B / k o and k oC.

1D First-Order Optical Systems: The Huygens-FresnelIntegral

239

q-domain representation, conveyed through the complex amplitude F(q), is naturally associated with the (q, ~) phase space, as the ~c-domain representation conveyed through the frequency spectrum ~5(n), is naturally associated with the ( - ~ , q) phase space, in which the role of p(q) and ~5(n) are symmetrically reversed. Accordingly, the optical matrix M of ray optics is involved into the wave-optical transfer relation in the q-domain, whilst the dual matrix

~ I - (O 1)M(0 01)

(5.3.12)

is involved into the wave-optical transfer relations in the t~-domain. The similarity transformation by tile pure ~-rotation matrix in (5.3.12) reflects the fact that, tile phase-plane axes (q, t~) and tile dual phase-plane axes ( - ~ , q) are geometrically linked by a ~ counterclockwise rotation. In accord with the prescriptions of w4.6.6, it is evident that in the frequency domain the operators K_, K+, and Ka have the meaning as in the scheme below, where, for purpose of comparison, the corresponding q-domain realizations (5.2.4) are also reported:

AK_=I~2

q-domain 1 02

g-domain 1 h;2

2k20 0 q 2 1 2

2k20 1 02

1_~'2 K+ = ~ q A

-~q

1

i

2

'~'rl+2qO'

(5.3.13)

o,~2

i

"

The reader will then find intuitive tile following correspondences of the basic optical systelns to their dual systems:

Optical system

Dual System

free-space section, d

thin lens,

thin lens, f

free-space section, d= 1 / f

positive scaler, m

positive scaler, 1 / m

f=I/d

(5.3.14)

which of course are another way to express the frequently remarked dual relation between free-space propagation (i.e., chirp convolution) and lens transfer (i.e., chirp modulation), and the transfer relations (5.2.33) and (5.2.36), relevant to the positive scaler. In relation with the correspondences above, note that in the space domain one deals with a length-unit variable whilst in the frequency domain one deals with a length-l-unit variable. The correspondences in (5.3.14) are intended in the light of the structure of the involved ray-matrices, being lower-triangular, upper-triangular or diagonal, which structure ultimately determines the functional form of the system response, whichever be the input, p or ~5.

Linear Ray and Wave Optics in Phase Space

240

E:~poncntialrepresentation of the optical p~vpagator. Operator equivalent of the Wei-Norman matrix synthesis

5.3.3

Ext)r(',ssi(m (5.3.11), ta.il()r('d t() rct)rcs(mt the rcst)()lls(; ()f an arbitra,ry first()r(ter ()I)ti(:al systenl ill tlw, st)atial ()r fl'(~(tlmn(:y (]()nmin a('~(:()r(ting t() the st)e(:ifi(: sctt, int4s ()f the ()t)tit:a.1 I)a,ra.lllcters A, B a,n(t D, (urn.tiles 1is t() wI'ite (t()wn the trmlsf(~r rt~la,ti(nls f()r a,ll ()I)l,i(:a.1 systtull ill the q ()r t~ (t(nlm.ill ill ()11(~ f()I'lll

)

-

/.

'

0;(.4).

(5.3.15)

v

It, is tl~(;i'c,f~)r(' (,a,sy t() r('(:()t4~iz(' t,t~(, l)()ssil)ilit, y ()f i~t(',rI)r(;t,i~ t,l~(' t,ransf('~r I'(~la,ti(n~s ft)r a,~l ()I)ti('a,l syst(u~i ill t)()tl~ 1,1~(~st)aria.1 a.~(l fr(~(t~('~l(:Y (l()~m,i~ a.s

~/,i(..ij)

chirp convolution

scale transformation

chirp modulation

t'/,,,(~//)

~I'l~(~(~Xl)li(:it(>[>t,rat~>rf,)r~ ()f tl~(, tra~sf()r~mti(n~s, ~,~t,~,ri~gt]~(' general

t, ra,llSf(;r s(:ll(;lll(' (5.3.16), i~ t,(;r~s ()f t,l~(' t4r()~l) t4(;~(;rat,()rs {K , K,, K:, } ttct)(ul(ls ()ll wll(;tll(;r it, (:(n~(~('~nls t,l~('~ waw; flul(:t,i()~l ~i(q) ()r its fl'('~(ll~(;~(:Y st)cct,r~uil ~;(~,). l,()()ki~lt4 ill lm.rti('~llar at. t,l~(' Sl)a.(:(', (l()l~mi~l, it" t,ll(: ra.y-llm.l.rix cnt, ry A > (), wlli(:l~ a,1]()ws ~s t,() lil~k t,ll(; s(:a,h; t,ra,llsf()r~m,ti()ll s()h;ly t,() K:~, w(; ()l)t,a,in f()r t,l~(' i~l~('i'(;~t, wa.w;-()l)ti('al ()l)Cra,t()r t,l~(', ()i'(tcr(:(l I)r()(l~(',t f()r~ [~..~]-I~.7] ~-

M(M)

-

()~""-~ K+ (' ;2~,.,,~"(A)K:'~ '--~;/'''~K- , I.t

(5.3.17)

wt~i(:tl ('~vi(lellt, ly tI'a,llsla,t(;s ill t,('rl~lS (~f lll(~ta,1)l('~('ti(' (>l)(~I'a,t()l'S 1,11(' W~',i-N()rnm,n (l(~(:o~lI)()sit,i(n~ ()f si~lll)l(',(:t,i(: llla, l,l'it:cs ill |,tWillS ()f a, ~1"(~(~ t)r()l)a.ga.l,i(nl, a. t)ositivc s(:a,lil~g mitt a, lens 1,1"l'l,llSftw, slll).j(~('l, l,()th(~ (:(m(liti()l~ t h a t A > () (!i 3.5)"

M

-

C B L(-~)S(A)T(~).

(5.3.18)

It has t)c(u~ n o t e d i~l w 3.5 tlmt (|(;(:()~q)()siti()n (5.3.18) is a,lh)w('~(| i~l a neighborhoo(t of the i n p u t t)lan(; in any t)r()t)agation problem. This m e a n s t h a t any prot)a,gation t)roblem can 1)(; ~m,thcnm,ti(:a,lly t h o u g h t of as a, suita|)ly parameterizc(| sequence of a Fr(,s~l(,1 tra.nsf()nn, a scale t r a n s f o r m and a ('hirp modulation, a,nd optically a,s a, s u i t a b l y t)aramcterized cascade of a, free-medium section, a positive scaler and a thin lens. T h e e x p o n e n t i a l opera.tor tra,ns('ription of the transfer scheme (5.3.16)in

the frequency domain needs, of course, to represent the basis operators K , K+, K 3 in the Fourier space. Using indeed the appropriate expressions given in (5.3.13), the reader may easily obtain the dual form of (5.3.17) and verify the transfer relation in the ~-domain having the form (5.3.9) with (5.3.10). A

A

1D First-Order Optical Systems: The Huygens-Fresnel Integral

5.3.4

241

Imaging systems ( B ----O)

The case B = 0 deserves some comments since expression (5.3.5) of the line spead function seems to be meaningful only for B ~: 0. We recall from w 1.8.4 that the vanishing B-element in the optical matrix characterizes imaging systems, with the matrix entry A specifying the scaling factor by which the object is expanded or contra,cted in the image plane. A thin lens is the basic example of an imaging system, with unit magnification. In terms of wave optics, imaging means tha,t the amplitude distribution of the input signal is reproduced on the output plane, apart from possible scaling, folding about the origin and phase shift effects. The image is perfect when the corresponding amplitude distribution has not phase factors other than those present in the amplitude distribution of the input signal. A thin lens always produces an imperfect image, since, according to (5.2.22), the image amplitude has a quadratic phase term that is not present in the object amplitude. It is convenient to recast the spread function (5.3.5) in the form g(q, q') -- i 2~iB ko eis2, (q'-~ )2ci s2A ,i_2,

(5.3.19)

which, on a,ccount of the limit relation (4.5.6), immediately yields

V)

-/--2-" "_~k~ 2

-

v/

c '

(5.3.20)

Imaging systems are therefore ('hara(:terized |)y intmt-output relations like -

2.

:,(

) -

(.

- o),

(5.3.21)

the back foca,1 length f2 A being introduce(| in the last expression. c Equation (5.3.21) genera.lizes the imaging relation (5.2.22) of a, thin lens to arbitrary imaging systems. The input signal is scaled into the output signal by the matrix entry A; the factor ~/~ preserves norma,lization. As for a, thin lens, the image is not perfect due to the presence of the quadratic phase term e-ik~ It represents (in the parabolic approximation) a circular wa,ve converging towa, rd (if f2 > 0) or diverging from (if f2 < 0) the secondary focal point F' (Fig. 5.5), whose wavefront profile So with curva,ture radius 7~ = f~ is then ta, ngent to the output plane IIo, where the emerging signal is observed. The qua,dra.tic pha,se term in (5.3.19) is compensated if the output signaJ is observed on So, which thereby represents the profile where a perfect image ma,y be detected. If C -- 0 (i.e., 7~ : f2 --~ cxD), the pha, se factor disappears and the perfect image is observed on the straight profile of the end-plane [i o. In that case, (5.3.21) yields the transfer relation for the pure magnifier, /

:o

(q)

-

/5.3.22)

242

Linear Ray and Wave Optics in Phase Space Hi

[-]o

i

!~

So

(' ol !I

..t

i

|:,

,

./: FIGURE 5.5. Ilnagixlg syst, czl~ (// .... ()). Tlu:

pcTfi:cl image ()f tim i~l>~(, sigzml lll~ty t)e observed

oi~ tim ('irc~lar I)r()tih' $,,. wlfi(:l~ Ira,s, ()f (:()~rs(:, tl~(: sm~(' f()nl~ as (5.2.33), In~(, ~ a y a.('(:(n~t as w(',ll fi)r a, f()l(lil~g ()f (,li(: s i g l m l a l)()~(, 1,I~(: ()rigil~.

5.4

The optical Fourier transform

11, Ires l)(:(:ll (:a,l'li(:r n:llm,rk(;(l l,lla,(, (,ll(: l:(nu'i(;r (,ra,llsl'~)rlll ()f ~1,11()l)(,i('al siglm,1 is ll()t, a,ll al)st, ra,(:t (:~d,ity. Ill fa.(:l,, it, is ()|)s(:rwd)l(:, f()r illsl,ml(:(:, ()ll (,If(: ])a(:k f()(:a,l l)Im)(: ()f a, (,llill l(:llS. 'l'll(: l"()llri(:r (,ra.llsf()nll is (,If(: llla,(,ll(:~m,(,i(:al (,()()I (,() (,urn fr()I~ (,I~(: Sl)a,(:(:-(l()l~mi~ r(:l)r(:s(:~l,a(,i()~ (,() (,l~(: Sl)a,l,ial-fr(:(1)U:l~(:y (l()~m,ii~ r(:l)r(:s(:~(,a,(,i<)~, (lisl)la,vi~g (,l~(: (lir(:(:(,i()~al (:()~1,(:~(, ()f (,l~(: sig~al. 'l'l~(: ()l)s(:rwd)ility ()f t,]~(: F()~I'i(:r (,rm~sf()n~i i~l)li(:s (,1~(: ()l)s(:rwd)ilil,y ()f (,l~(: ~:-(l()~ai~ r(:l)r(:se~ta(,i()~ ()f a, sig~m,], (,]~(: f()(:al I)]m)(:s ()f a, l(:~s I)r()vi(li~g (,]~(: (:l(:~(:~(,ary I)]~ysi(:a,l l)la,n(:s wh(:r(: tl~(: tw() F()~n'i(:r r(:('il)r()(:al (l(:s(:ril)ti()~s ()f (,h(: siglml l~my l)(: obscrv(:(I, l{(;nm,rka,l)ly, (,l~(: "ga,l)" l)(:(,w(:(:~ (,l~(;s(: l)lm~(:s is f(n~l~(I (,()l)(: filled by a, (:o~t,i~nnm~ ()f l)Im~(:s, fracl, io'n, ally F(mri(:r (:o~kj~gat(: (,() (:a,(:h ()(,l~(:r, where a,n int, en~(:(lia,te rel)r(:s(:~ta,(,i()~ is ()l)s(:rwd)h:, disl)layil~g l)()(,l~ (,l~(: sl)a,(:(: and frequen(:y f(:at~n'(:s ()f (,I~(: signal. Tiffs is th(: ()l).j(:(:t, ()f (,I~(: f()r(;h(:on~ing (liscussion. 5.4.1

Four/or

tT"ansform

Tile geamra,1 c o n d i t i o n for t,ral~sft;rring the i n p u t sig~ml to (,he, fo(:al plane of an optical s y s t e m is t h a t (,he ray-~lmtrix e n t r y A be zero (,~ 3.4.2). W i t h A = 0, tile ti'a,nsfer rela,t,i(m (5.3.4) })(',(:()m(',s 27rif

-

f

qq'~ (q') --

w h e r e 11se h a s b e e n d o n e of t h e r e l a t i o n B -

-~

~c* -

1

-

f.

2f Dq2

1D

First-Order Optical Systems: The Huygens-Fresnel Integral

243

The amplitude of the output signal, detected at the point q on the secondary focal plane, is directly linked to the Fourier transform of the input signal at the spatial frequency t~ = koq/f. However, the Fourier transform relation between the object and the image is not an exact one, due to the presence of the quadratic phase factor ~ikoDqg/2f, which specifies the circular profile, of radius - Z, D on which the exact Fourier transform of the signal can be observed. If D = 0, the quadratic phase term in (5.4.1) disappears; thus, the amplitude distribution at the secondary focal plane is the exact Fourier transform of the input amplitude at the primary focal plane. As mentioned in w3.4.2, the system is said to respectively perform the imperfect or perfect Fourier transform of the input illumination. Either case, the intensity distribution oil the focal plane reproduces the spatial frequency spectrum of the object signal:

[~~

2 - %S

9

(5.4.2)

Remarkably, relation (5.4.1) gives a physical content to the rather formal treatnlent of the spatial Fourier transform presented in w 4.6. On the back focal plane of any optical system, one may "see" the Fourier transfornl of tile input signal, apart from a. possible quadratic phase factor (whi('h can be avoided looking on an output circular t)rofile instead of a plana,r one). However, tile frequency domain of the "mathematical" Fourier transform is not directly mapped into the output space domain of tile opti('al system, the s('aling constant -~ [length -2] being indeed required: -

q.

Also, the additional constant complex factor ~

(5.4.3) appears. Hence, the relation A

between the "mathematical" Fourier transform 5 (w 4.6) and the "optical" Fourier transform F[F(f)] - Ff writes in the form A

A

-

(5.4.4)

As an elementary example of the tranfer relation (5.4.1), we may think of a thin lens, whose back focal plane is connected by a Fourier transforming relation like (5.4.1) to any plane, placed at a certain distance d against the lens (see Fig. 3.7.b) and the scheme of Fig. 4.7). Indeed, the lens produces on its back focal plane the imperfect Fourier transform of the object, the quadratic phase factor in (5.4.1) specifically being eik~162 )q2/2f. It disappears, of course, if the object is placed at the front focal plane (d = f) of the lens, which so produces the perfect Fourier transform of the object (see Fig. 3.8).

Linear Ray and Wave Optics in Phase Space

244

Either case, the lens images pla,ne waves with different spatial frequencies, i.e. different directions of propaga, tion, into different, points of the fo(:a,1 t)lane, and hence separa,tes the c o n t r i b u t i o n s of the various pla, ne waves which compose the int),lt signal. In particular, the t)la,nc wave t)rot)aga,t,i~g a.t a,n a,~gl(', '0 with r(;st)(;(:t t() th(', ()t)ti(:al axis is f()(:usc(t int() the t)()int q = '~,of'O (s~q)t)()si~g t h a t the mc(tia, oil t,l~e lens si(les hay(; the sa,n~(; refl'a,(:tiv(; in(lex n,,).

5.4.2 In

{}

F~actior~o,1 I, b u r i c r

l:ran,@rrm,

3.6.1 w(; lmvc ('(msi(|(;l"(;(| t,l~(; (l~a,(ll"al,i(: l~()n~()g(ql(:()lls l)()ly~(n~ial l 2 + ~q l 2 II,~.,,. - -~I)

(5.4.5)

whcrr l,lm slfl)sr a.(). siglla,lizcs l,lle mmh)gy wil,ll l,lle IIalllill,(nlia.ll r l a tl, ra.(',t,iw' ()s(:illal,(w, llavillg ill l)art, iclllar llllit 11ross a ll(l fi'('~CtlU'llcy is. ~I'llr r

llmtrix, r162 F"(.f s) -

tll('~ ray l)r()l)agatir

_ ~l sill ~/~

(:()sO

axis m~(1 tl~(', fi)(:al l(u~gtl~ t)aral~('t('r .fs" 4)

,

'~'.

an

rllh'r ]).V (5.4.5), is -Tr < 0 <_ ft.

(5.4.(5)

I~ tl~(', ligl~t ()f t,l~('~ a.l~al()gy

wiI,l~ l,l~(; ha.r~()~i(: ()s(:illa,l,()r, il, ~my l)(', s(;(;~ I,() (:()rr(;sl)(n~(l 1,() l,l~(; l)l~a.s(; , c ( t t,,) ()f tl~(; ()s(:illa,t()r. Als(), 4) is ~s~ally giiv('~ as a, "fi'a.('ti()~" ()f g_, tl~r(n~gh l,h(~ 1)a,ra,m(;t(;r rt s(;l,l,ing (/5 - - (t-~. -~ sigl~ili(',s 1,1~(; ()r(lilm,ry F(),n'i(;r tra,nsf()n~: F ~(.fs) - F ( f s ) , wl~i(:l~ l,l~(u~ (:a~ t)(; ,~(l(:rsl,()()(l as a st)(;(:ifi(: (t(;t(;rl~fi~m.ti()x~ ()f t,t~(; (:()x~l,i~nun~s (,,v()l~l,i()~, g()v(;r~(:(l ])y (5..~1.(i), at (l(;ti~ite 2 + 4't,~ :2, t)r()fih; s(:r(;(;x~s i(l(u~tifi(;(t l)y 4),, -~ " tt~ - (), 1,2... (,(nw('rs(;ly, |)y va,hu;s () < () < ~ ()~('~ l~my il~st)c(:t l,l~(; ray-('v()l~lti()~ at, i~m:i'~(:(lial,(; t)lanes within t,h(; 1)a,si(: t:()~rier tra.~sf()rn~ ~mit,, wlfi(:l~ is s() fi'ag~(;~t,(;(1 i~ fluctional ~mits [7], a.n(t, a(:(:()r(ting t() th(; t)(;ri()(ti(:ity ()f t,h(; harm()ni(' f~m(:t,i(ms, at, any other t)la.ne "s(',t)a.ra.tc(t" fr()l~ t,h(; f()n~(;r t)y A ~ - 2~,7r,,t~,- 1, 2... ()pposite vahlcs of () signify t,hc inv(;rs(; syst(;n~, t)eing in(te(xt IF(*(.5)] -1 - F-(~(A); also, we recall the renm,rka,l)h'~ relation: F"(.fs ) - [ F ( , f s ) ] ~. In w 3.6.1 we ha,ve indivi(hla,lized tim ot)tica,1 implemcntati()n ()f t,lm system F a ( f s ) in t e r m s of b o t h rcfl'a,ctivc l(mscs and gra,dcd-in(t(;x m(xtia,. In fa,ct, mirroring the single- a,nd doul)le-lens geometries, which t)('xforn~ tim ordinary Fourier tra,nsforin, the basic })ulk optics realizations of the syst('m (5.4.6) are obta,incd interleaving free-medimn sections a.nd thin lenses a,c('or(ti~g to the schenlcs" 71"

koh mann's

type

I"

[ohmann's

type II:

F a( f s ) i

-

T(.fs t'an(~)) L(fssinr

F ctII( L ) -- L( tan(~b/2)fs) T ( L sin r

t,an(~ ))

tan(~)/2) )f8 .

(5.4.7)

1D First-Order Optical Systems: The Huygens-Fresnel Integral

245

Also, a quadratic graded-index focusing mediuin is appropriate to implement the fractional Fourier transforming operation with a continuously ranging order a(z). The bulk optics realizations (5.4.7) are in fact paralleled by the graded-index medium realization synthesized into the scheme: quadratic-index medium of length L (n(q,z)=no-niq2/2, n2>0 )

FC~ ~:

)"

2 .//-~

1

(fs) (a(L)=-TrVn0-n-2"g, f S - v / n o n

2 )"

(5.4.8) In particular, with c~ = 1, the resulting optical systems perform the ordinary Fourier transform of the object. Specifically, it amounts to choosing all the lengths involved in (5.4.7) equal to fs, and correspondingly the length of the quadratic medium in (5.4.8) as (or, as quartic multiples of) LF -- ~ ~ . Finally, we recall from w 3.6.1 the single exponential representation of the 1 matrix (8.4.6). It writes in terms of the matrix K 1(fs) - ~(fs K - + ~1K + ) !2 ( 0. - -1 / f s /s)which is nothing but one half the Fourier transform matrix with 0.. ' focal length fs- Explicitly, one finds F~ (fs) -- ~2r

(fs).

(5.4.9)

All the above reviewed results can be read from the wave-optical perspective just replacing the inherent scalar functions of q and p by operator fimctions of ~ and ~, and accordingly the inherent matrices by integral transforms. We can therefore say that the qua,ntum harmonic oscilla,tor-like Hamiltonian "" 1"2 + ~ $ 2 ~ 2 , H~.o.~p

(5.4.10)

genera, tes the optical propagator ~c~ -i2k~ Is -- C

-

+ , A -~SK+].

(5.4.11)

As F~(fs) ' F~fs is intended to connect two planes separated by a finite or infinitesima,1 distance according to whether the order ct is given a, definite specification within the proper interval ct E (-2,2] or is allowed to range continuously in conformity with the propagation variable z. Of course, IF;s] - 1 -

F-~ fs ' The formal input-to-output relation

~o(q)-

F;s -- [Ffs] ~

(5.4.12)

F~is [p~(q)]

(5.4.13)

for the fractional Fourier transforming systems (5.4.7) finds its own integral representation through rule (5.3.5), which indeed yields )90 (q) --

i

ko

2TOilssin r

sin r [q' 2 cos r + q2 cos r - 2qq'] / c i 2fs ko

~i ( q t ) d q t,

~ -- c[2"

(5.4.14)

Linear Ray and Wave Optics in Phase Space

246

Of (:ourse, the above equa,tion Inky describe as well the paraxia,1 propagation of a, light disturbance through a, quadratic index medium, if a a,nd fs are given a,s in (5.4.8), an(t so directly linked to the t)ropaga,tion length z - zi and the n,e(tiunl t)a,ra.meters ~t(), ~t~. Therefore, it t)rovides the ext)li(:it s()hlti(),, to the t)a,ra|)oli(" wa,ve e(l~mti()~ f()r a (tua,(h'a,ti(" f()('~lsing nle(ti~u1~, wh()se ray ()t)ti(:s a,ccount ha,s t)ee~ (letaile(1 i~l ~ 3.9. Equa,ti~)n (5.4.14) is .jllst tlw, wa.w',-ot)tical I,ra,l~s(:rit)l,i()l~ ()f I,l~(', ra,y t,ra,~sfiw rela,t,ir (3.9.1(i). It rel)r(;sents a.1~exa,nq)le of l,]w, i~d,eI'l)la:v l)(;l,w(',(',ll i~m.l,rix a.]l(l ()l)(Wa,l,t)r ]~l(;I,]~()(ls l,]~a,l, (:a,]~ (',[t'(',(:l,ivt',ly 1)een ('xt)l()it(',(l t() s()lv(; (;v()l~lti()ua,ry (liI[er(mtia,1 (Xt~la.l,i()llS. Tl~e, a:-(l()~m.il~ r(;t)r(',s(;~l, al,i()11 ()f t,l~e i~l)~ll,-()~lt,t)]ll, r(~la,l,i()]l (5.4.1d) is easily ()l)ta.ine(l l,l~I'()]ltKl~ l,l~(; r~h~ (5.3.1()), t)y ~()l,i~t4 tlm.t tl~e (l,lal ~m.trix F'~(.fs) writes j,lst a,s (5.4.(i) witl~ fa,l,fily l)ara,ll,t,t.(;r f-~s' 1,m,,t;ly, (.fs)(.f~)Th(~ si~gh;-(;xI)()~,;~tial f()rlll (5./1 11) ()f t,l~(; ()I)(wa,t,()r F '~ is t)ara,lh;h;(l t)y the 9 t~ liutll,i-ext)()~|('~fl,ial f(n'l~s, (lir(,('l,ly ()])tail~(;(l fl'()]l~ (5.,1.7) })y a, l~(;r(' r(;l)la(:ement ()f ma,l, rir l)y ()l)('ra,l,()rs. I~1 fad',t, l,l~e wa,v(',-()t)ti(:al l,ra.]~s(',ril)ti()~s ()f (5.4.7) yiel(1 1,11(;f()ll(~wi~g <)r
I type I

type

F,,

~C.i(q )

f,.;

,,C.i(q)

r

A

fs. ta,,( ~' ) K

i~,.,,

chirp convolution

F'* fs

II

_.

..... r

r

i ~-2/si,, 'h K fs

chirp modulation

- i ~-2tt ta,,(-~) K -ilq, Is t (.

chirp modulation>

...

i/,',, fs. ta,,( ~ ) K

Ir

69

chirp convolution

f.,.;si,,,/,K .

-i~'~ r

:,,(,t)

tin'(7)

Is

chirp convolution> (,> chirp modulation,

,~(,(q)

(5.4.15) Tlle rea,~ter is ilwit~'(1 t~) writ~,, (t()wll the, t)111k ~)I)ti~:s r~'alizati~)ll ~)f tlw, (hla,1 ,)t,era.tor Ffs (,:,)rr,',st,,)l,,li,~g t , ) t h e ,hml l,la,trix F " ( f ~ ) ) , wl~,)s,; tyI)e I m~d type II i1111)]elll(;11ta,ti()]IS 1,1]nl o11t to l)e .illst the (llla]s resl)ective]y ~)f tile type II and t,yI)t', I rea,liza,tit)lls t)f F '*fs (and so F '* (.fs))" We (:()11(:111(t(', t,llis se(:l,i(nl resorting to the fre(tlwafl, ly outlill('(l a,~m,logy of the ray- a,n(1 wa,ve-ot)ti(:s spacc-propaga,tion un(ler the (tuadrati(: Hamiltonia,ns (5.4.5)a,~l(t (5.4.10) with the t i m c - e w ) h l t i o n of a, mechanica,1 (respectively, cla, ssica,1 aal(t (t~m~fl,~lm) lm,r~noific oscilla,tor. In the, light of this analogy, realizations (5.4.7) and (5.4.8)(:a,n t)e interpreted as opti(:al a,nalogs of the harmonic oscilla,tor (tyna,mi(:s, whi(:t~ is indeed optically repro(hl(:e(t at "sampled times" a,nd at "continuous times". The ana, logy with the qua,nt~m~ harmonic oscillator dynamics is further evidenced by the eigenvalue equa, tion for the fractionM Fourier tra, nsform opera,tor F~ , s

A

Fa

(q) -

(q)

(5.4.16)

1D First-Order Optical Systems: The Huygens-Fresnel Integral

247

Physically, it identifies light distributions Fn(q), whose transverse profile is not altered (apart from the complex factor I~, I nl 2 - 1) when propagating through a fractional Fourier transforming system. Since F~ fs can equally be interpreted as the time-evolution operator of the quantum harmonic oscillator described by the Hamiltonian (5.4.10), it is natural to look for ~ ' s in the form of the oscillator energy eigenstates, and hence satisfying the equation Ha.o. Fn(q) - En~n(q).

(5.4.17)

Resorting to the well known oscillator energy eigenstates, we then write (q) --

1

1/2

(5.4.18)

where Hn denotes the Hermite polynomial of degree n. Also, on account of tile already mentioned correspondences: oscillator

mass

-----+

oscillator frequency Planck's constant

1

--+ &1 ---+ ~o

(5.4.19)

tile parameter w ill (5.4.18) and tile "energy" eigenvaluc cc,~ specialize as w-

@So

'

g~

1 ~ (n + ~).

kof s

(5.4.20)

A

Then, applying F~ to ~n we obtain the explicit expression of the propagation ' fs factor A,,~; namely,

F~spn(q)- c-i~2("+ 89

(-i)~(n+ 89

(5.4.21)

The fractional Fourier transform operator F~ fs can equally be identified through the eigenvMue equation (5.4.21) being required to be solved by the HermiteGauss wave functions (5.4.18) with eigenvalues An - ( - i ) ~(n+l). In particular, with c~ = 1 one recovers the well-known property of the Hermite-Gauss functions being eigenfunctions of the ordinary Fourier transform operator: t"-

1

Ffs %On(q) -- ( - - i ) n + - ~ n ( q ) .

(5.4.22)

Relation (5.4.22) for n = 0 is in accord with (4.6.51) on account of (5.4.4). It is worth noting that in the original formulation by Namias [7] an eigenvalue equation similar to (5.4.21) (see Eq. (5.4.42) below) was proposed as defining equation of the fractional Fourier transform operator, which is then

248

Linear Ray and Wave Optics in Phase Space

conceive(t to have the same eigenfllnctions of the o r d i n a r y Fourier t r a n s f o r m but with eigenva,lues sepa,rated by a, fra,ction of the, inmgina, ry unit. E q u a t i o n (5.4.21) merely ret)roduces the time evohltion of the, os(:illator energy eigenstate, s, whi('tl in fa,(:t ()(:curs t h r o u g h the, t)lmse fa,(:tor (~-is 1 (-i)'~("'~+~). It howev('~r states s()nlethillg new, sllggesting t() i~ltert)rel, tim timeevohlti()ll ()t)era,tor ()f tlle (tlla.lltllnl ()s(:illa,tor a,s a, fra,(:ti()lml F(mrier t,ra,l~sform ()t)era,t()r. Tl~is is a rat, l~(;r g('al(;l'a,1 r(;s~flt rega,r(ling il~(h:e(l ally (l~a.(lrati(: tta,nfilt,()liiali a.ll(1 ll(;ll('('~ rely l)a.raxia,1 wav('~ l)l'()I)aga,ti()li, sill(:(;, as w(: will s(:('~ ill t,he ~(;xl, 1)aragra,I)h, a,lly AH(7I)ild,(;gra,1, i.('~, m~y l)araxial ()t)ti(:al l)r()l)~,g~tl,()r, is r('la,t,(:(l t() t,l~(; fl'a(:ti()~al F()~ri(;r 1,ra,llSf()rlll. A(l(liti()l~ally, sil~(:(', t,lm, q-l)r()tih; ()f tl~(; (;ig(',~u~()(l(:s ()f a (l~m,(lI'a,l,i(" f()('~sing nn',(ti~n~ ~()(l(:l .i~sl, as (5.4.18), E(t. (5.4.21) (',a,l~ a.s well 1)(: i~m:rt)ret(;(t ('n+ 7 ) eigel~l~()(l(;s, t,l~(; (:()rresl)(nnli~g t)r()l)aga,ti()~ (:()~st, a,~ts t)(;i~g s() s - ~ . ) ~ . Tl~is, ()f (:(n~rs(;, (:(n~tir~s tin: (;arli(;r ~eld, i()~(;(l view ()f t,l~(; fl'a.(:t,i()lml F()~riex l,rmisf()r~ ()l)('rat()r as tin; ()l)ti('al l)r()l)agat,()r il~ a (l~a.(lI'a,ti(' f()(:~si~g ~(',(li~l~. lh;lati(n~ (5.4.21) ('(n~l(l (:(t~ally I)(; ()l)tail~(:(l fl'()~ (5.4.22) ()ll H(:(:()llllt ()f l,l~e s(',(:()l~(l ()f (5.4.12)', l,lm (lis('~ssi(n~ a,l)()ve el~q)l~asizes l,lw r()le ()f F '~Is a,s a,n l

A

5.4.3

AHCI)

i'ntq]'ml and fl'actional t, bv,'~'icr t'ra'n,@~'lv.,. Opt'tarot

('quivah':nt of the: Jwa,sau,a matri:r ,s'ynth.c,s'i,s A(:(:()r(li~lg t() tin; Iwasawa (l(;(:(nlll)()sit,i()ll ()f syllll)l(x:t,i(', llla,l,l'i(:(;S (!i 2{.7), every tirst-()r(ler ()l)ti('a.1 sysl,(;lll (:ml 1)(~ sylgll(;size(l 1)y a S(:(llWll('(: ()f a i)lmse-t)la,ne r()tat()r, a I)()sitive s('ah;r a ll(l a t,llill l('~ls, as (;xI)li('itly ('xl)r('ss(:(l |)y

M = L(7))S('~,~)F '' (.f.~), the real I)a,ra,Ineters 'I1~,, 7) ml(t 4) |)eiIlg B2 tan r ~tt~'2 - A2 + 7-~s'

m "f s '

T) -

~ m--2

( A C + D~.~s, ) "

(5.4.24)

Here an in genera,1 non 11nit vahm of fs has been (:onsidered with respect to the parameterizatioIl re, p o r t e d in Eq.(3.7.3). Thus, simply tra,nslating i~ opera, tor form the above ma,trix p r o d u c t , we see t h a t cve, ry optical t)r()t)agator ca,n be represented a,s the ordered product of a lens ot)erator, a scale o p e r a t o r and a fractional Fourier t r a n s f o r m operator: A

A

A

A

M(M) - L('P)S(m)F~fs'

(5.4.25)

with relations (5.4.24) for the involved parameters. The exponential representations of the three components of the product above in terms of the algebra basis {K_, K+, Ka} are respectively given by (5.2.21), (5.2.27) and (5.4.11).

1D First-Order Optical Systems: The Huygens-Fresnel Integral A

A

249 A

Clearly. while F~fs generates the diffraction integral (5.4.14), S(m) and L(7)) manifest on the incoming signal Pi(q), respectively, through the scaling operation (5.2.33) and the chirp-modulation (5.2.22). Any ABCD system can therefore be represented by the transfer relation 2

o(q) -

1

[Fs

ikoT)~-2 ~"c~

m

(5.4.26)

which expresses the amplitude distribution ~Po(q) over the output plane as the scaled fractional Fourier transform of the amplitude distribution pi(q) over the input plane with a further quadratic phase term, and hence as the imperfect scaled fractionM Fourier transform of the input (~). The above offers an equivalent representation of the ABCD integral, which then ca,n alterimtively be characterized by the scale factor m, the curvature parameter 7) and the angle r and hence the order ~ of the fractional Fourier transform. Evidently, 7) specifies the curvature radius T~ - - ~ of the circular profile on which the perfect (scaled) fractional Fourier transform can be observed. When 7) - 0, the quadratic phase factor disappears and the perfect (scaled) fractional Fourier transform is observed on a planar profile ( T ~ - c~). The pure fra,ctional Fourier transform corresponds to m - 1 and 7) - 0 . The ABCD integra,1 represents the explicit fim('tional solution to any paraxial propagation problem ruled by a polynomial Hamiltonian quadratic in q and p. Ae('.ordingly, the LST and LSF (~ decompositions (5.3.17) and (5.4.25) yield a valuable solving tool permitting, in the same way as their matrix counterparts (5.3.18) and (5.4.23) do for the relevant matrix equation (4.3.4), to decompose the overall problem into three subprobleIns, whose dynamics might be more easily investigated. This may simplify the solving procedure and/or suggest interesting and enlightening analogies. It is therefore evident the physical and mathematical relevance of decomposition rules in relation with evolution problems ruled by second-order differential equations. In particular, relation (5.4.26) will be used in w 5.4.5 to relate the freemedium propagator to the quadratic medium propagator, i.e. the Fresnel transform to the fractional Fourier transform.

5.4.4

Fresncl and Fourier transform

It is interesting to investigate the link between the Fourier transform of a wave field and the relevant far-field pattern. To this end, we transform back to the q-domain the frequency spectrum @o(~, Zo) of the free-propagated signal c One m a y talk of imperfect fractional Fourier t r a n s f o r m in the same sense as one talks of imperfect Fourier transform. An imperfect fractional Fourier tr a n s f o r m displays a residual quadratic phase factor in the o u t p u t variable with respect to the basic performance (5.4.14).

Linear Ray and Wave

250

Optics

in Phase Space

amplitu(te ~o(q, Zo) over the distance z o - %. T h e n , using (5.2.18), we o b t a i n

1 / ,/5-;

/ cigq~ ~o(q, Zo)_ ,~-if-;. l dh; J 92o(fl;,Zo)

(/h;

citW -i(z~ c

2"()to

~i(fl', z i )

'"

(5.4.27) W r i t i n g z o - %-5 f a,n(1 r(;lmn(tling the a,rgmllent ()f the (;xt)()ll(;ld, ia,1 flm(:ti()n, 1,11(; a l)()ve i n t e g r a l ('a,n t)e r(;(:a.st in tlx; form ~,,(q,z

,

+(')

/'/ d,~' - , i ~""~"' ~' (,_,,,~k(),., ' )--

- ~l , ' i ~ , F

-

wlfi(',h, ()11 a.(:(:()lnfl, ()f tim r(,lal,i()11 (4.5.(i), in tll(: lilllii, t'---~ OC a ll(l fillit,(; extent

()f the, signa,1, tllrlls i11t() 2,

f ---,oo

V

if

~ i

(

f

, z,) 9

(5.4.29)

TII(; a,l)()v(; r(;lati()ll sillll)lY says tlmt 1,11(; far-Ii(:l(l l)att(:r11, i.(;., tll(: fi'(;ely t)r()t)a,ga,t(;(t fiel(t (listril)llt, i()ll at larg(; (listml('es fl'()lll tim illl)l~t, 1)la,n(; (f--* oc), is (;(t~iwd(;Ilt, a I)a.r(, fl'()~ a. (l~a(lra,ti(: l)l~as('~ fa,('t()r, t() tim sI)al, ial F()~n'ier I,ra.nsf()n~ ()f tl~(~ i~lln~t sig~ml (tl~(', ~,~ar-fi(',l(l (listritn~ti(n~), (wal~m.t('(l at fi'('.(l~l('a~(:ies "o%q h', f . A(:(:()r(lil~gly, tl~('~ ()l)s(~rv(;(1 sigxm.1 a(:r()ss tl~(; 1)a(:k fi)(:al l)lm~(; ()f a. tllin l('~s (:an t)e ~(l('rsl()()(! as l,l~(' far ti(;l(l I)al,l,('r~ ()f 1,1~(' i~l),~l, sig~al, if fr(:(;ly t)r()t)a,gat(;(l ()vex la,rg(; (listm~(:(;s. R,(;la,t,i()~l ( 5 . 4 . 2 9 ) (',xl)r(;sses 1,1~(; w(;ll-k~x)w~ r(;s~ll, ()f (lilh'a.(:l,i()~ tl~(:()ry a(:(:or(ling to whi(:ll fi)r large (lisl,m~(:(;s a,ll(1 fillit,(~ ('.xl,(;lll, ()f l,l~('~sig;~m,1, t,l~(; Fr(;snel (tiffra(:tion t)ass('~s i~d,() tile l"ra,~u~h()fi;r (lifli'a.(:ti()l~ [(i]. W(~ will see i~1 tlx; next subse(:tion tha.t 1,11(; "gal)" ])(;l,w(;e~l t,h(; Fres~M a.ll(l Frm~l~l~()fi;r trmlsf()r~ns is filled t)y a ('ont, ilnn~x~ ()f fl'm:ti()~m.1 F()~u'i(;r traa~sfi)r~s, tln~s r(~l)r()(l~(:i~lg f()r a, h o m o g e n e o u s n l e , ( l i ~ 1,11(; v i e w ()f t)r()t)a.ga.ti()~ tl~l'(),~gl~ fl'a.(:ti()~m.1 trmlsform t)la,nes s~ggest(;(t f()r a, (l~m(lra,ti(: f()(:~lsing me(li~n~l.

5.4.5

Fvcsncl and fi'actional kbuvicv transform

Interestingly, tlle ext)ressi()II (5.2.9) of the Fremml illtegral (:()llll(',(:tillg two pla: na, r surfa(:es can t)e a(tai)te(| t() the ext)ressi(m (5.4.14) ()f the fra(:ti(mal Fourier tra,nsform, thus pa, r a l M i n g the relation of the Fralmh()fer integral to tlle sta, nd a r d Fourier tra,nsfonn, previously discussed. In fact, it is ea.sy to see t h a t the Fresnel t r a n s f o r m ('a,n |)e i n t e r p r e t e d a,s a ~lm,gnifie(| fra(:tional Fourier t r a n s f o r m with a, residlml t)hase-(:urvature. Ext)li(:itly, by (5.4.26) one m a y (|ecomt)ose the f r e e - m e d i u m t)r()t)a,ga, t()r T(d) into T(d) - L(P)S(m)F~fs '

(5.4.30)

w i t h the parameters m, 7) , (b specifically given as m2

_

1.5

d2 --fs2 '

7)

_ ~2fs2 d ,

tanr

~d.

(5.4.31)

1D First-Order Optical Systems: The Huygens-Fresnel Integral Wavefront profiles

q

t~--...,..~._

I

~

i

: w z)/ I

9

I

251

~

1

',. I

I

.

,

..,

l

i

Z

FIGURE 5.6. Schematic of a Gaussian beam, showing the parabolic profiles of the wave fronts and the hyperbolic profiles of the beam width. Here, the scale p a r a m e t e r can be conveniently set to fs = k0T h e Fresnel diffraction p a t t e r n can therefore be i n t e r p r e t e d as the scaled fractional Fourier t r a n s f o r m of the signal diffracting from the e n t r a n c e plane Hi. T h e t r a n s f o r m is observed on the spherical surface Eo with c u r v a t u r e radius n o - d(1 + ~ ) ,

(5.4.32)

a,t dista,nce d from Hi. Tile order of the t r a n s f o r m increases monotonically with the (tistan(:e d. As d --~ oc, tile angle 4) apt)roaches ~ and hence tile fractional t r a n s f o r m turns into the ordina,ry Fourier transform. Correspondingly, since d and ~ o ~ d, one recovers the far-field diffraction p a t t e r n , which according to (5.4.29) is the Fourier tra.nsform of the incoming signal. A unifying m a t h e m a t i c a l description of the diffraction in a free m e d i u m is therefore provided by the fractional Fourier t r a n s f o r m formalism, which embraces the ordinary Fourier formulation as limiting case. Notably, tile a m p l i t u d e distributions on the t)lane IIi and on the spherical surface Eo of curva,ture radius ~ o are directly connected by a fractionM Fourier t r a n s f o r m a t i o n . T h e expression (5.4.32) of T~o closely resembles t h a t for the c u r v a t u r e radius R of the wavefront at d of a Gaussian b e a m having its wa,ist at z2

the plane IIi, i.e. R ( d ) - d ( l + - ~ ) ([8], see also w 9.5.1). T h e scale p a r a m e t e r fs takes t h e n the m e a n i n g as the Rayleigh length z R of the beam: z R _ ~lkw2o , Wo denoting the waist radius (a). Besides, the magnification factor m has the form of the normalized b e a m width ~ at distance d from the waist, which, according z2 to the law ~ 2 ( z ) - ~2(z) 1 + ~- , yields rn 2 ~s in (5.4.31) with z - d and 2 w0

R

zR = fs. Finally, the order c~ of the fractional t r a n s f o r m is p r o p o r t i o n a l to the d We recall that when dealing with the propagation laws of Gaussian beam parameters, the z coordinate is commonly measured with respect to the beam waist.

Linear Ray and Wave Optics in Phase 5pace

252

G o u y I)hase ( of the bea,m, which writing

((z) -- ta,n-l(--~z ),

(5.4.33)

turns int() the a,ngh~ O ()f tl~e tra,nsfl)i'u~ by t)roperly setting z = d a,~(t z u = fsTlu~s, we (:a.n say t h a t t,l~e t)r()t)aga,ti(m ()f a, Gm~ssia,n t)ea.m i~ fl'ec st)a,(:e lm,t~u'a,lly ()(:(:~lrs tl~r()~gl~ s(:al('~(t fra(:ti()na,1 F()~rier tra.~sfl)r~mti()~s, wl~()se or(t(:r is (lir('('tly li~k(,(l t() l,l~(, (,~()~y I)l~a,s(; sl~ift wl~(~r(,as (,1~(' s('a,l(' fa('~.()r a('('()~u~ts fi)r tt~(~ variati()~ ()f tl~(~ 1)(,a~a wi(ltl~ witl~ l)r()t)agati()~l. 5.4.6

7'he A H C I ) integral as a scaled fi'actional b'ov,'l'ic'l' t'l'a'nsform of a scaled i'npu, t

Un~h,r (h;tilfite, (:(m(lit, i()lls, the ra.y lim.trix M ca,ll 1)(' (le(:()llq)()se(l as M-

(5.4.34)

S( ,,,.2)F'~(.f.~)S( ,,,, ).

(:()llv(~yillg l,l~(~ vi('w ()f t,ll(', ()l)ti(:al syl,(~lll (les(:ril)(',(l I)y M a,~ l)(',rfi)rllfillg tim fi'a,(:ti()lml F()ln'i(',r trmlsfi)rlll ()f tim i111)llt, sigxml witll tw() S(:a.l(~ fa(~t()rs, ()ira fi)r S()lvi~g (5.4.34) fl)r t,l~(' r(d(wmd, Imral~(q,('~I'S, l)r()vi(l(~s ill(' g(~('ral (~()~(liI,i()x~ ~u~(l('a' whi('l~ (le('()~l)()siti()~ (5.4.34) is fl'~a,sit)l('~. W(~ ()l)tai~, i~ fa(:t,

(~()s" 0 - A D ,

(5.4.35)

() <_ A D <_ 1,

(5.4.36)

whi(:l~ yiehts the i~(~(t~a,lity a,s a, ~e(:essa,ry a,n(t s~dfi(:ie~t (:()~(titi()~ t() interl)r(',t tl~c ()l)ti(:al tra~sfi)rn~ by M ()f the int)~t i~t() tim ()~t,t)~t t)lmm as a fl'a,cti()~ml F()~n'i(',r tI'aX~sfi)r~ ()f real

or(l(',r. C()rI'(~st)oI~(tingly, tim s(:a,l(~ fa,(:tors '~tt~ and ~n~ sI)e(:ializ(', a.s B '~

-4 .

,~ni

_

D

B2

~ _

_

f.,2 A(1 - A D ) '

m~

A

_

,

f~2 D(1 - A D )

(5.4.37)

In the general ca,se of (:omt)lex or(ter transforlns, the (:oI~(titioi~ (5.4.36) can be relaxe(t; the scaJe t)ara,n~et('.rs 'ttt,~ and m 2 axe then aJh)we.(t to |)e (:onq)lex. Relation (5.4.34) has suggested a generalized form of the fractiona.1 Fourier transform, as depending on the order and on two s(,ale factors, which can as well be complex to a,ccount for possible field curvatures [7.6]. In a. sense, the decomposition (5.4.34) generalizes to the Huygens-Fresnel integral the three factor decomposition (3.6.28), with the scale factors for the input and the o u t p u t being allowed to be complex and not necessarily reciprocal to each other.

1D First-Order Optical Systems: The Huygens-Fresnel Integral

5.4.7

253

Basic mathematics of the fractional Fourier transform

Although introduced within a closely optical context, the fractional Fourier transform has its pure m a t h e m a t i c a l countepart in the same way as the optical Fourier transform F/s is paralleled by the "mathematical" operator $c, to which is linked by relation (5.4.4). A similar relation is naturally suggested between the "optical" fractional transform F~ and the "mathematical" fractionM t--

-~.

A

transform ~ ~. Explicilty, one may write" AvT

-

-

(5.4.3s)

which can be recast in a further "abstract" form changing from the dimensional variable q to the dimensionless x - -~q, as we will do below. In fact, conforming to the common mathematical practice, we use here the same symbol to denote the arguments of both the function p(x) and its transform ~5~(x). The pertinent physical context gives, of course, specific meaning, role and dimension to the arguments of the tilded and untilded functions. Accordingly, ; r ~ is defined to signify the following transformation . J

t.

/

f

2

x2

27r

(5.4.3v) which correst)onds to the exponential operator representation )~ ~ - c.i~ c - i ~ ,

(5.4.40)

being tile "mathenmtical" generator reproducing tile harmonic-oscillator Hamiltonian: ~" 02 21 o-~ + 89 z2 -- 1 ~ 2 + 1 ~ 2 (5.4.41) The argument of the square root in (5.4.39) is as usual taken in the interval ( - 5 , 5 ] " Of course, r ctS, with ct c ( - 2 , 2]. The position- and m o m e n t u m like operators X, P in (5.4.41) signify 2 - x and P - - i dax " The mathelnatical definition (5.4.39) of the fractional Fourier transform is properly obtained, within the purely formal context of linear canonical transforms, as unitary representation of the canonical transformation, which produces a rotation, governed by the order p a r a m e t e r c~, in the plane of the Fourier conjugate variables (see also w 5.6). It is required to have the basic properties: i. 9~ is continuous with respect to the order p a r a m e t e r c~, i.e. ~ tends to ~ ~ as/3 tends to c~, ii. U s obeys the semigroup property, so t h a t composing two fractional transforms of order (~1 and c~2 yields the fractional transform of order (~1 -~71"

A

O~21

71"

71"

Linear Ray and Wave Optics in Phase Space

254

A

A

.7"~1 ? %

-

A

A

.7"'~.7"~

A

-

.7"'~ §

A

iii. ? " re(h]ces to the i(teI~tity ()t)era, t,()I" with (~, - 0 an(t t,() the ()r(lina, ry Fourier t r a n s f o r m with ( , - 1" .~o _ ~-a,n(t . ~ ' - .~. T h e r(;a,(ter ca,n easily re(:()gnize t)rot)(:rties i.-iii, a,n]()]]g th()s(; (tis(:~sse(t in rela,tion witt~ the ()t)t,i(:al t r a n s f o r m within b o t h the ray-ot)ti(:s (w 3.6.1) a,nd wave-ot)ti('s (,~ 5.4.2) fra,n](:w()rks. W(: will (lis('~ss in th(: next s(;(:(,i()n s()n~e ()ther l)r()l)er(,ies ()f tlm fl'ac:(,i()~m,l b'()~n'i(:r (,I'~tIISf()I'III .j~s(, l)a,ra,ll(:li~g (,l~()s(: (h:scril)e(l in w 4.6.1 in r(:ga,r(Is (,() (,h(: ()r(lina,ry F()~ri(:r (,ra,nsf()r~. Firstly, i(, ~my b(: int(:r(:stil~g t() Io()k a(, tlm (:ff(:('t ()f (,h(: ()I)(:ra,t()r ,~'" on s()x~(: l)asi(: ()I)(,i('a,l siglm]S. I~ (,l~is (:o~n~(:(:(,i(n~, w(: ~my i~(:(lia.(,(:ly ~()(,(: tha.t the eige~wal~u: (:(l~a.(,i()]~ (5.4.21) says tlmt tl~(: H(:n~it(:-Ga.~ssia.~ fi~(:tions (5.4.18) a,r(: r(:I)r()(I~(:(:(l ~n~(I(:r (,h(: a,('(,i()l~ ()f ~ ' " . h~ (,Ira I)r(:s(:~(, (:(nm:xt, Eq. (5.4.21) ('a,]~ i]~ fact t)(: r(:writt,en a,s

?A" u , , ( . r ) -

(-i)"')u,,(.r),

u,,(x) - (

~

~ )1

1/2

tt,,(v~:,,),,

x2

(5.4.42)

v

Als(), i(, is ins(,r~u:(,iv(: (,()('()~si(l(;r (,l~(: fi'a.(:(,i()~ml F()~n'i(:r (,ra,]~sfi)r~ ()f t]~(: lix~(:-

like, mo~()fi'(:(t~mI~(:y a,x~(l (:l~irI) sigxm,ls, r(;st)(:(:(,iv(:ly r(:t)r(:s(:~(,(;(l |)y a,/~ fi~(:tion, a.n(t |)y a li~(:m" m~(1 a (l~m.(lra.(,i(: l)l~as(: fi~('ti()~. 'I'll(: r(:a(h:r is i~vi(,(:(l (,() ca.try ()n the r(:l(:wn~(, (:al(:~lal,i()x~s, yi(:h|i~g a,~ ti~ml (:xl)r(:ssi()]~s (,1~(: fi)lh)wi~g" Function

Fractional Fourier transform

V/ r (,iflla: "2 9

27r

("

V I + i tall 0 ( '

V/

i;r2 2 ,.,-)s4, (:r 2 sill CA-'~2 sill O--2"y.r)

(5.4.43)

1 -t- i t a n (D (, ~ia.2( Ifl )/~t;m,/, t..... 4, ) 1 +fl tan 0

Wc see tha, t t]~(: chirt) fll~(:ti()~ <:i/~ (:a~ t)(: fl'a,ctiona,lly trm~sfi)n~m{1 into a delta fim(:ti()n if the or(t(:r ()f the transf()nll is (:ll()sc:n t() r~ -- _2~ (,aal-~ (~)., this p r o p e r t y is ~ls(:(t to rem()v(: (:hirp-tyt)e ~()is(:s from signa.ls. We fina,lly ~l~ention tlm.t fi'a(:tiona] F()llri(:r tra,nsforn~s of (:()~t)l(:x ()r(h:r have been intro(h~(:(:(t as well. In the light ()f the interpreta,ti(m in terms ()f rotations in the F(mrier plan(:, complex-or(let transforms are seen t() (:orrespond to complex-a,ngh: rota,tions, from which the a,ssociated integral transforms emerge in the base form (5.4.39) a,nd with the b~sic properties i. and ii., but with substantial differen(-es with respect to the real-order t r a n s f o r m s a,s far a,s variable (tomains, fllnction spaces and o p e r a t o r properties, like b o u n d n e s s an(t unitarity, are concerned. C o m p l e x orders n a t u r a l l y arise when dealing with complex optical systems; gaussian lenses, characterized by complex focal lengths, and ga, ussian apertures, modelled by i m a g i n a r y focal lengths, are examples of complex optical systems, described by complex A B C D matrices [4.2].

1D First-Order Optical Systems: The Huygens-Fresnel Integral

255

Some properties of the fractional Fourier transform We will discuss in particular the similarity, tile multiplication and tile derivative properties of the fractional Fourier transform, that can interestingly be brought into relation with the corresponding properties of the ordinary Fourier transform, expressed through (4.6.7) and (4.6.17). We will mostly make use of the exponential representation (5.4.40) and the LSF ~ decomposition (5.4.25) of ~ , whose "mathematical" content will then be emphasized. Let us start deducing the similarity property of the fractional Fourier transform 9c a , recalling that for the Fourier transform it writes as A

A

A

1

x

9c {p(ax)} - ]-a-f~5(~).

(5.4.44)

with a being an arbitrary real parameter. The function p(ax) can be thought of as resulting from a scale transformation A

1

~ ( X ~ x -~- 1

of F(x) acted by the operator S(~) - c-~ to the optical analogy (5.4.38) we can write

"~

~), with a -

.F {p(ax)} - ~

c ~. So, resorting

~)~(x).

(5.4.45)

On account of the Iwasawa decomposition (5.4.25) with the associated relaA A 1 tions (5.4.24), the optical propagator F~ S(~) can alternatively be realized by the ordered product

p, g ( 1 )

_

with the parameters being specifically identified through tan tan rr

{:I2

,

~)

-

-

-

cot r

-

c~ r162r

-

m-

a ~sin , 4)

~t

O/t -

-

~.

(5.4.46)

Then turning back to the "mathematical" context, we end up with ~'~ {~(glX)} -- ~-~ 1 V/1-i ~ot..~.. ~2 (1- co2 ~' x sine' 1_icot4),r z 2 ~o~2 ; )cot@ @c~'(~ sine )"

~"

(5.4

.47)

The c~-th order fractional Fourier transform of ~(ax) is not the scaled version of @,(x), as in (5.4.44); indeed, it is the scaled and chirp-modulated version of @~,(x) for a different order c~~, determined by the order c~ and the scale factor a as prescribed by the first of (5.4.46). The multiplication property is concerned with the fractional Fourier transform of the product x~(x) that comes to be a rotational mixture of products by x in the normal and Fourier sense. In fact, we write 9c ~ {xp(x)} - [9C~x9c - ~ ] iF ~ {p(x)} - X(c~)~5~(x).

(5.4.48)

The operator X(c~), defined through the similarity transformation A

A

A

A

A

A

X(c~) -- ~ C ~ x ~ - ~ - .7~X.7 "-~,

(5.4.49)

256

Linear Ray and Wave Optics in Phase Space A

A

is evidently the operator into which X is transformed under ? ~ . Using the exponential representation (5.4.40) of ?(~, we ca,n give X ( a ) the explicit form A

A

A

A

A

A

X((t) - c - i * ~ Ni (:ie~ _ c,,s 4) X - sin (/5P,

(5.4.~o)

{.r:(.r)}the ext,ressi,,n

A

th,,s Iwovi(ling for ? "

.T" {:,':(:r)} -

,:,)sO:,'+

isi,,(P~,:r

@,,(:r).

(5.4.51)

d tiona,1 N)l~'i(;r tra.l~SfiWl~ ()f (,l~c (teriwtt, iw~ ,T,, ~(.r) i1~ te.i'~s ()f t,l~e fl'a,(:tio~ml F()~u'i(:r (,ra.nsf()rm ()f ~. As l)(;fi)r(:, w(,. wri(,(:

p,,{d

:(:")

}-

d ,,1 p , , [?,,~.~

A

wh(:r(: (,I~(: ()i)(:ra.(,()r I)(()) (l(;li~(:(l ^

{:(.,.)) -

if>(,~)~,,(.,.),

(,~.4.~)

l)y d

.--

i;(,,) - y " ( - ~ ) P - "

^

^

(,~.4.,~3)

- y" ~,y-",

A

A

is l,tl(; ()I)era,l,()r illl,() wlli(:ll I ) is l,ra,llsflw~ll('(1 11~l(h'r if'*. I1, Sl)('(:ia,liz('s illt,r A

IS((~) _

A

A

,-i,/,~ ~ (i,I,~ _

A

sill qSX ) (:()s~/5I',

(5.4.54)

t,lnls h;a.(ti~lg (,()

A(,, }Eisin96.r+cosO(T"1

~"

~(.r)

-

r

@,,(:v).

which looks like a, r()ta,ti()xm,l mixtllre of deriva,tives with re.st)e(:t t,() :r in the norma,1 a,nd F(nlrier seIlse. As rela, tions (4.6.17) s
A

A

z-domain

A

Fourier domain

A

X P A

A

--~ ~

-P X A

z-domain

Fractional Fourier domain

A

X P A

A

--~ ---,

A

cos ~ X - sin ~ P sin 0 X + cos 0 P .~

t..

The explicit calculation of the fractional-Fourier domain representations (5.4.50) and (5.4.54) of the operators X and P follow from rule (4.4.25) (see

1D First-Order Optical Systems: The Huygens-Fresnel Integral A

257 A

A

also Problem 4.2) on account of the commutator brackets of 7-/with X and P being [~, X ] - - i P and [~, P ] - iX. We may guess from the expressions in the above table, showing that X and P are transformed by the fractional Fourier transform operator into a mixture of X and P, the potential of the fractional Fourier transform to yield a representation of the signal not purely in space as p(q), or in frequency as ~5(n), but in both space aim frequency with a contiimous, and adjustable, degree of emphasis on space or frequency features. In later chapters, in fact, it will emerge the intimate link of the fractional Fourier transform to the Wigner distribution function (and the ambiguity function), the exemplary joint representation of optical signals.

5.5

Recovering the ray-optical description

As expected, we may recover from the A B C D integral (5.3.4) the transfer law of ray optics, according to which the ray variables from one plane to another plane are linked by a linear relation formally expressed through tile ray matrix pertaining to the involved planes. To this end, we consider the expectation values of tile operators ~ and in tile state qp(q, z), which are defined by the inner products

(~)(z) -- f dq~*(q, z)~ ~(q, z), J

(~}(z)- f dq~*(q,z)~v~(q,z),

(5.5.1)

with the complex amplitude being assumed to be normalized to one: II ll- 1. Using the input-output relation for the optical system to express the signal amplitude at different planes along the optical axis, we can relate the expectation values at the plane z, (~)(z), (~ }(z), to the expectation values at the input plane zi, (~}(zi), (~)(zi). It is evident that the expectation values (5.5.1) equal the mean values ~ and -g/k o introduced in Eq. (4.6.36), i.e.,

The first identity is obvious. As to the second, using the inverse Fourier transform, one obtains

1 j/

.t

/

Accordingly, the z-evolution of the expectation values (~)(z), ( ~ ) ( z ) y i e l d s information on the "motion" (position and direction) of the centroid of the wave field.

258

Linear Ray and Wave Optics in Phase Space

In t)a,rti(:~flar, for the propaga.tion through a homogeneous medium over the redu(:e(t (tista,nce d we ha,ve

( q )o = 2red. dq

dq'.

dq"q

C - z 2(t (q--q')2ci 2d

JPi ((tl)~gi(qll),

(5.5.4)

wl~i(:l~, |)y ~Se c)f relaticn~s (4.5.8) a n(t (4.5.4) (:o~me,r~i~g tl~e first (terivative of ( ~ ),, = ( ~ ).~ ~ ,t( ~ )i.

('p),, - .e,~,,.~

~ ,~,/

,~,/,q,

2,

(5.5.5)

r (q)(-7.o,, ~)" ~'' ("-"" ~i(q"),

,

(~.~.~) whi(:h llI~nl {:alc:ltla.ti~nl giw;s (~),, = ( ~ ).~.

(5.5.7)

l/{;lal,icnls (5.5.5) ml{l (5.5.7) llav{; l,ll{; Salll{; f{n'lll as t.ll{: lllal,rix r{'lati{}ll t'~w tll(; ray varia,lJh's (q, p) I~{'l,w{;{;ll l,ll{; {;llr {;llr l,ll{; lll{'{lilllll. l,ik{:wis{', fl'{nll l.ll{' l.rallsfi,r r{:la.l,ir (5.2.22) fin" a l,llill h;llS, il {'a.sily f{}lh~ws fr 1,11r {;Xl)r wdlu:s l,]l{' sa,llle fr rr lil~ki~g tl~{; ray wu'iat)les i~ ~m,l,rix r162 i.{;.

(~)"

-

(~)~'

(n.a.s)

In get,era.I, ~si~g t,l~{; i~In~t,-{n~ttmt r{'~laticn~ (5.3.4) fin" a~ a rlfitrary A B C D systen~, a.fl,er sr a,lg{;]}ra w{; ~;~(t ut) wit,]~

thus prc,vi,,g tlm.t tl~e ,,~{'a.,~ vahu;s ( ~ ) ( z ) a.,~cl ( ~ ) ( z ) tra,,~sf{,r,,~ a.s the ra,y va,ria,bles c)f ray r Eq~m.ti(ms (5.5.9) ~,my t,,, regarded a.s the wa,ve-c)t)tic'a.1 version (in the linear apprc~xi~m,tic)~) of l,}lc Elirenh;st theoren~ of q~la,Iltllli~ n~echanics, which is well known to t)rovide the law of motion of the mem~ values of the conjugate ot)servat)h;s ~ an(t ~ [:}]. The Ehrenfest theorem states that the equations of ,,~otion ot)eye(t by (~) a,,(1 (~) are forma,lly i(tenti(:a,1 to Hamilton's equa,tions of the (:la,ssi(:al theory, ex(:et)t that the quantities whi(:h o(:cur on both sides of the cla,ssica,1 equations must be replaced by their expe(:tation values. This does not mean in general that the expectation values follow the classical motion. In fa,ct, relations (5.5.9), which establish indeed that (~) and (~) follow the ray motion of geometrica,1 optics, hold only in the first-order approximation, and hen('e only for Hamiltonians, which are quadratic flmctions of ~ and ~.

1D First-Order Optical Systems: The Huygens-Fresnel Integral

259

To see this, consider that the z-evolution of the expectation value of any wave-optical operator O is ruled by the equation

i d ~ ~ ~ ~ 06> dz - <[O, H I ) + ~-0 <~zz

,

(5.5~0)

the second termA in the right hand side accounting for the possible explicit dependence of O on the propagation variable z. When the canonical operators ~ and ~ are concerned, Eq. (5.5.10) gives i

d

~?0dz -- ([q, HI}, i

d

(5.5.11)

<[~, HI>

~-o ~z<~>-

As the commutators on the right hand side yield the derivatives of H with respect to ~ and ~, we can write A i

d

~0~z<~> i

-

d

k-0dz

--

H -O ~ -, < > OH --< -Nq >'

(5.5.12)

which represent the definite form of Ehrenfest's theorem. Evidently, although Eqs. (5.5.12) reselnble Hamilton's equations of geometrical optics, one can not say that the expectation values (~} and (~) follow the laws of ray optics, unless the following identities (5.5.13) can be stated. It is easy to verify that the above identities hold true only if the Hamiltonian operator is a quadratic function of ~ and ~, and hence only in the paraxial approximation. In that case only, the expectation values of the position and momentum operators follow tilt "classical" motion.

Further czploring the correspondence between the ray and wave optics pictures. II

5.5.1

Continuing the line of the presentation in ~ 4.4.4, we reconsider again the evolution equations for the Heisenberg operators ~(z) and ~(z), we rewrite in the slightly different form dz

d D(z) -- [D, 7~], dz A

A

with 7 - I - - i k o H .

(5.5.14)

Linear Ray and Wave Optics in Phase Space

260

R e p h r a s i n g the presentation of w 1.7.1, concerning the q u a d r a t i c polynomials and the associated m a t r i x representatives under Poisson brackets, we (:onsider a.s a g(;~eral H a m i l t o n i a n o p e r a t o r the t)ilinear form in ~ and ~: (5.5.15) A

A

T h e Lie t)ra,('k(;ts [~, ~] a,,~(1 [~, ~] ('()m(; t,() t)(', li,~(;a,r in ~ a,~(l ~, t,('~ing, in fa.(:t,

[~, 7~] - -a~ - (' ~,

(5.5.16)

wlfi(:h a,ll()ws ~s I,()i(l(u~tify 1,1~('~,xm.t,rix r(',l)r('s(',~l,ativ(, ()f ~ a,s

H-(

b) ' - a(: -(:

(5.5.17)

(?l('arly t,ll('~ tra,(:(;h:ss llm,tI'ix H 1)(',l()~s t() ~,1~('~sy~q)l(',:ti(: alg(,~t)ra .,q)(2, R). TI~(,~ tt(:is(:~fi)(:rg (:(l~al,i(n~s (5.5.14) ('a.~ l,l~(~ 1)('~r('~(:ast i~l,()a ~m,l,rix r('la,l,i(n~ si~ila,r t()!';(I. (1.7.4) fl)r tl~('~ ray v(;(:t()r u; l~m~(:ly, d dz~(Z)

-

H-N(z),

w(zi)--~r+

(5.5.1S)

i,,v()lvi,~g tt,,: ,,l,,,ra, t()r-val,,(:(l v(','.t()r ~ ( z ) - (~(z), ~(z)) T (s,:(; !i 4.4.4). It, ix illll(',r(;llt i~l tll('~ a,1)()V(; ('*tlm,ti()ll a, lill('ar trmlsf()r~lla,ti()ll f()r~ a,n(t ~, wlfi('ll ret)r()(llu:('s tlm.t f()r t,ll('~ c-znlnfi)(;r ray va,rial)l(:s q ml(! p. Ill fa,('t, siil(',e the (:hang(; ()ftll(', v(','t()r ~ ( z ) ()ver a,n infilfit('si~lm.l (tistml('('~ d z fr()Ill ttl(; initia,1 l)lan(; zi ix (;t[(~('t(',(l t)y t,ll(; 2 x 2 symt)l(;('ti(" Illatrix ~'H(:')'l:a,('(:()r(liilg I,()

(5.5.19)

'~r(zi -Jr- dz) -- ('H(zi)dz'vr

the evoluti(nl ()f @(z) ()ver tll(; finite distml(:(; z - zi ix (:ffi;(:t(;(l t)y a , l l infinite ordered string ()f 2 x 2 nm,tri('(;s like tha,t, a,t)()v(;, whi(:ll ev(mtlm.lly y M d s for the opera,t()rs (~, ~) the, sa,1~(,, finite linear symt)le(:ti(" tra,nsf()rmati()n ruling the evolution of the ray-va,ria,|)les (q, p). Interestingly, if w(; ('()nv(',niently define the ot)erators A

A

/C - -ik0K

A

,

A

A

/C+ - - i k 0 K + ,

A

/Ca - -'ikoK:~

(5.5.20)

we can rewrite the comInuta.tion relations (5.2.2) a,s [/C+,/C_] - 2/Ca,

[/C3,/C•

- :F/C+,

(5.5.21)

which, of course, p e r t a i n to tile metaplectic Lie algebra mp(2, R), and exactly mirror the c o m m u t a t i o n relations (2.3.8) of the sp(2, R) m a t r i x basis

261

1D First-Order Optical Systems: The Huygens-Fresnel Integral

{K_, K+, K s }. Also, by rule (5.5.16) the expected correspondences between the rap(2, IR) operator basis, and the sp(2, N) matrix basis are recovered, being

]~_--K_- (0 1), J~+---~K+- (~176

]~3 ----~Ka-- 1(01 ~

The inverse a,rrows, from matrices to operators, work through the operator form of (1.7.11), which writes ~(~, ~) _ 1 ( H e ,

J@)

- I@THTJ@.

(5.5.23)

An isomorphism can evidently be established between the symplectic and metaplectic a,lgebras sp(2, R) and rap(2, R). We will see later that the respective groups are isomorphic up to a sign.

5.6

Wave-optical propagators as unitary representations of linear canonical transformations i

It is noteworthy that the wave-optical propagators M(z, zi) in their integral transform representation (5.3.5) arise as unitary operators acting oil tile Hilbert space s of wave functions, which correspond to unita,ry tra,nsformations mapping the Heisenberg-Weyl algebra operators ~ and ~ into real linear combinations of themselves [10]. As said in w4.4.4, the propagation of a wave fieht through an optical system described by the propagator M reflects into the the similarity transformations (by M-l) of the Heisenberg operators ~(z) and ~(z), which then evolve from tile SchrSdinger operators (5.1.2) according to

q(z)-

M-l(z, zi)q(zi)M(z, zi),

p(z) - M-l(z, zi)P(zi)M(z, zi),

(5.6.24)

where, of course, ~(zi)--~ and ~ ( z i ) - ~. On the other hand, guided by the considerations developed in the previous section, showing the formal identity of the evolution laws for the q-number pair (~(z), ~(z)) to those of the c-number variables (q(z),p(z)), we may relate the propagated operators ~(z) and ~(z) to the input operators ~(zi) and ~(zi) through the optical matrix M of ray-optics. So, we explicitly write

"~(z) - A(z, zi)'~(zi) + B ( z , ~(~) - C(~, ~ ) ~ ( ~ ) + D(~,

zi)~(zi), ~)~(~).

(5.6.25)

The unimodularity of M guarantees that the evolved operators ~(z) and b(z), as ~(zi) and b(zi), obey the Lie bracket (5.1.3)" [~(z),~(z)] - [~(z~), ~(z~)] i ~ .

Linear Ray and Wave Optics in Phase 5pace

262

Th(~ question arises about the st)e('ific flmctionM form of the tra,nsfer op-

erator M(z, zi), which through (5.6.24), leads to the linear ca,nonical transforina, tion (5.6.25) for q a,n(t p, nam(;ly

M-I(~,

~)~(,~)M(,, ~)

~r

zi)p(z,i)~C~(Z, Zi)

_ A(~, ,~)~(,~) + -

-

~(z, ,~)~(~),

C(Z, Zi)q(Zi) J- D(z, z,i)~(z,i) .

(5.G.2G)

Ill ()tlmr words, wc ar(" tryillg t()(t('till(' a ~mt)l)illg ()f sylllt)le('ti(', illatri('~es M into llnitary ()t)(~ra,t()rs M" t..

M E Hp(2, I~)

~ M C ]Ilp(2, IR),

(5.6.27)

t)a,se(l ()~ the r(',(lllil'(~llleld, t|la.t t,ll(', ()l)(~I'al,()rs q(z), p(z) f()lh)w tll(~ S~-t,lll( ~, ~m,trix law ()f tim ray varial)h~s q(z), p(z), wlml~ tim wave li(;l(l l)r()I)aga,t(~s tl~i'()~gt~ tim systelll, a u(1 h(ul(:e tlm ('()rresl)(nl(li~lg wave fiUl(:ti()ll is a,(:t(~(l |)y M. Evi(h;ntly, siu(:e 1)()th M al~(l s (ud,(:r (5.6.26), ally ()t)(u'at()r I~l, satisfying (5.6.26), is (tetilm,I)h~ llI) t()all a,rl)itra,ry t)lm,se fa,(%()r. ()llr alla,lysis t)eh)w is ai~lm(t at (~xt)h)I'il~g tt~(; l)()ssil)ility t() ~u~i(t~mly (l(~fi~(' m~('l~ a I)lm.s(~ i~ a,(:(:()r(t witt~ tlm (tesir(;(l I)r(:s(:rw~ti(n~ ()f tl~(~ gr()~ t) str~(:t~u'(' ()ftlm s(:t ()f M's. We will see that the t)lm,se il~(t(~t(:n~i~a~'y (:m~ 1)e r(~(h~(:c(l ~l) t()a sig~. As s~lggeste(t t)y (5.1.8), w(' a s s ~ ( ' tl~at M(z. z~) ('()~('r(,tiz('s i~t,() tim integral tra~sf()I'~

M(z, zi)r

zi) -

/. dq'G(q, q', z, zi)r

zi),

(5.6.28)

s() tlm,t the prolJ(u~ ()f (l('.t(~r~lli~i~g M is t~u'~(',(l i~t,() tl~a.t ()f (l(~t,er~i~fi~g the functioIml form ()f th(~ k(~'~(~l G(q, q', z, z~) a,ss~ri~g tl~at r(da.ti(n~s (5.6.25) hoht. We rewrite rela.ti()~s (5.6.26) ill tile e(t~fiwde~d, f~n'~

~ ^M - M(A~~.+ B~), ~M - M(C~ + D~),

(5.~.~9)

the dependence of the vari()~s fim(:tions and ot)erat()rs Oil tim axial (:()ordinate z being implie(t. Acting (n~ tim wave flm(:tion ~(q) t)y tim ()t)(;I'a,tors on both sides of (5.6.29), we ot)ta,in the i(l(u~tities

f dq'G(q, q')(A~ + B~)~(q') - ~ f dq'(E(q, q')~(q'), y J

(5.6.30)

On account of (5.1.2), we may integrate by parts the left-ha,nd sides of both

equations. Then, from the resulting identities, that must hold for any 7:(q), we obtain the following set of linear partial differential equations for G(q, q')" i B o )G(q, q') - q G ( q , q'), (Aq'+ ~ ~ o ~(q, q,)

(cq' + ~oD ~ )C(q, 4)

~o~

(5.6.31)

1D First-Order Optical Systems: The Huygens-FresnelIntegral

263

We may propose a solution in the form

G(q, q')

(5.6.32)

-- 7 e i(#q2+~'q'2+rqq'),

with the constant parameters (i.e. independent from q and q~) 7, P, L,, 7 being to be appropriately defined. Indeed, inserting (5.6.32) into (5.6.31), we obtain for p, u and 7 the definite expressions / ] - - ]go a

P - - k02B

7-

-

(5.6.33)

--k o 1

-

In order to specialize ~,, we resort to relation (4.3.21) (adapted to G), which reflects the unitarity of M. With (5.6.32) and (5.6.33), Eq. (4.3.21) yields 712-

k~

(5.6.34)

2 IBI' by which 7 comes to be defined up to a phase factor: 7 - ei~

27r[BI 9

Accordingly, we can establish for G(q, q~) the flmctional form k0B C~" -2" ~ (Aq'2-bDq2-2qq') G(q, q')=c i( i 2~ri

(5.6.35)

with an undefined phase ~. It, couht seem (:onvenient, in order to form a one-to-one mat)t)ing of symplectic matrices M into operators M, to envisage a rule to assign a unique phase 0(M) to every matrix M. With such a choice, however, the resulting set, of operators M(M) (to not close under multit)lication, an(t hence it, is not a B1) and M 2 _ (A2 group , a s it would be desirable 9 In fact, let M , - (A1 c 1 D1 C2 D2 h

\

\

/

B2) \

2"

be two syinplectic matrices, which are imaged respectively into i 1 aIl(] i 2 , with r e p r e s e n t i I l g kernels (~1 a n d C__g2 . Let M - (A B) denote the product matrix M = M 1M~; of course, ]

\

J

( A1A2+ M

-

B1C2

A1B2-~-

B1D2)

A 2 C 1 -Jr- C 2 D 1 B 2 C 1 -Jl- D 1 D 2

(5.636) "

Finally, let M12 and M respectively denote the operator resulting from the product M~2 - MIM 2 and that corresponding to the product matrix M through (5.6.35). From (5.6.28) it follows that the representative kernels G 1 and G 2 of i I &rid M 2 compose as (see also Eq. (4.3.18)) 1-.

A

G12 (q, qt)

_ / dqttG1 (q, qtt)(~2 (qtt qt).

(5.6.37)

In order to verify that the product of operators exactly reproduces the multiplication of matrices according to A

M(M)

t".

A

- M 1 (M1)M2 (M2) ,

(5.6.38)

264

Linear Ray and Wave Optics in Phase Space A

A

for a.ny choice of M 1 a.n(t M.,, we will work out the explicit expression of G~2(q, q') in terms of the entries of the product m a t r i x (5.6.36), a.n(t thenA we will comI)are it with the exI)ression of the representing kernel G(q, q') ()f M. By (5.6.35), the integral in (5.6.37) yiehts G~2( q qt) __ r162

_

_

I"0

,'t~ 2 ('42

~.

2

/__~.q2)

u2q' + "~

k~ (,i~l (,i~'2 r ~sg~(B,HB:t ) r 2n"/%13 (' i 21~

is t)()sitiv(: ()r ~(:gal,iv(' (s(:(; fi)()l,~()t(; (n~ I). 226). ()n tl~(', ()tl~(:r lm.~(t, tl~(; r(',t)r(',s(u~t,i~g k(;r~(~,l G(q, r / G(q q') - (,i~ ~/ l,.o

dq"('

ikO[ !~ qtt22(,___g__F,_51~__)] '2 " ~ ' 2

"~

"'2

+l)q 2 2qq'),

()f M is

i kO ( Aq~'2 + D q 2 _ 2 q q ~)

Evi(h',~tly, fl)r t,l~(', i(l(',~tity (5.6.38) t() l~()ht, (n~(', sl~()~l(l lmv( ~

7r u ) ((M) - ( ( M , ) + ( ( M ~ ) + - ~ s g , ~ ( u, n~ 9

71C(M) - - 7~sg~(B),

(5.6.41)

(5.6.42)

1)y wlli(:ll We evi(h',lltly fi)rlll a (nl(',-t,(~-(nlC a,ssr 1)etweell syllll)h;cti(: rim,trices zul(l zllt,'tat)h;t:ti(: ()t)erat(n's. Bllt witll tiffs ct~oi(:c tl~c t)llase ()f tim l)rottuct era, tors 1)y the, ternz

)] (5.6.43) which in(h;t;(1, as tilt; r(;a(tt;r may v(;rify t)y direct cht'.t:k, tlu'xls i~lto 0 or +re according t() the signs ()f tllc involve(t B-entries. This mea.ns tha,t the set of opera.tors M ( M ) we obtain with this choice do not form a grout), since on multiplying two opera,t(ws, M~(M~) a.n(t M,,(M~), it (:ould hat)t)en that, depending on M l and M2, the result be - M ( M ) , which is not in the set. Therefore in or(ler to obtMn a. group, wel.mllst let t h e c o r r e s p o n d e n c e (5.6.27) be not one-to-one, allowing tha,t both M ( M ) and - M ( M ) be a,ssociable to each symple('tic m a t r i x M . The resulting set of operators closes under multiplication, a.nd yields a two-valued representation (r of Sp(2, IR), imaging linear e Note that this is not strictly a representation, which is intended to admit only one "image" corresponding to possibly more than one "object".

1D First-Order Optical Systems: The Huygens-Fresnel Integral

265

canonical transformations of the Heisenberg-Weyl algebra operators ~, ~ into transformations of the nilbcrt space s of wave functions ~(q). The integral kernel G(q, q') for the operator M corresponding to M is then, k0 5~ G(q, q') -- -t-~/ 27riBr (Aq'2+Dq2-2qq'),

(5.6.44)

thus evidencing that M is uifiquely identified by M along with the choice of the sign. The phase of the square root is taken within the interval ( - ~ , ~]. It is a rather common convention to write the kernel C, without the sign, letting implied the possibility that two unitary operators, M(M) and -M(M), may correspond to every sytnplectic Inatrix M. Evidently, the line spread function gABCD(q, q') in (5.3.5) has been written in conformity with this convention. In the next paragraph, we illustrate an example of this sign ambiguity. As a conclusion, we may say that, whilst an isomorphism holds between the symplectic and metaplectic algebras sp(2, R) and rap(2, R), the respective groups Sp(2, R) and Mp(2, R) are globally homomorphic (precisely, isomorphic up to a sign).

An example Let us consider the optical Fourier transform, which is represented in ray optics by the symplectic matrix F(f)-

-1/f 0 '

and in wave optics by the unitary operator FI, discussed in w 5.4.1. We have seen in w 3.6.1 that tile Fourier transform matrix is cyclic by 4. A cascade of four identical Fourier transformers compose to the identity; namely,

F(f)F(f)F(f)F(f)- I

(5.6.46)

In contrast, the operator Ff is not cyclic by 4 but by 8, thus differing from the "mathematical" operator 9c, which is indeed cyclic by 4. Evidently the different periodicity of the "optical" operator Ff with respect to .7- arises from the phase factor c - i 4 , which basically distinguishes between tile optical and the mathematical Fourier transforms. In fact, propagating an optical signal ~(q) through the cascade (5.6.46), one obtains at the intermediate planes, firstly (q)

_

_

y

and then (q)

-

l(q)

-

(5.6.47)

266

Linear Ray and Wave Optics in Phase Space

aga,in leads to

So, ot)tically F o u r i e r t r a n s f o r m i n g

~.~ (q)

F f ~2 (q) - -

-

(5.6.49)

i/@(-~q)"

F i n a l l y after a, fi~rther tra,nsfiwm wc (',u(t ~t) a,t the. exit t)la,lw, witll ~., (q) - F / ~ : , (q) - - r

~Fhis nl(',mls |,lm.t

#} -

(5.6.50)

-i,

a,n(111(;~1(:(;, wil,l~i~ tl~(' f()rl~mlisl~ ()f 1,11(; a,l)()v(' l)aragral)l~, tim t,

IVI(F)IVI(F)IVl(F)M(F) - -IVI(F '1),

(5.6.52)

1,11(; nlim~s sigl~ sig~lifyi~lg a. l)l~ase sllift 1)y 7r ()f 1,11(; ()l)l,i(:al wav(; g()i~lg l,ln'()~@l 1,11(' 4-F()~ri('r l,ra.~sfi)r~l(;rs (:as(:a(l(; [,().~()]. Tl~('r(;fi)r(', ()l~ ()~(; si(h' r(;la, ti()~ (5.(i.4(i) sig~ifi('s l,lm,t a, 271- r()|,ati()~ i~ l)tlas(; l)lml(; s~lfi('('s t() l,~n~ l,l~(' ray v(;(:t()r i~m) its(:lf. ()~ t,ll(' ()tlwr si(h:, r('la,ti()n (5.(i.51) sig~ili('s l,lm,l, a, 47r r()l,a,l,i()~ is ~(;('(h'(l 1,() r(;(:()v(:r tl~(: i(l('~tity ()l)(:ra,t()r I, anl()~ml,i~lg 1,() a~l (;igl,l~ t)()w(:r ()f 1,11(;~mta,l)h;('ti(: ( )l )(:ra t( )r F / . Tll(; ~mtal)l(:(:ti(: gr()~ 1) 3Ip(2. IR) r(;aliz(:s a (l()~l)l(; (:()v('r ()f tl~(, sy~q)h;('l,i(' gr(),~ l) Np(2, ll~) [2].

5.7

Summary

W(' lmv(; ('la.1)()ra,l,(,(i tll(' wav(;-()t)ti(:a,l r('l)r(:s(:lltati()ll ()f tll(' 1)asi(' ()l)ti(:al sys1 2, t(;lllS, i(h;lltili(:(l wil, llill tll(; ray-()t)ti(:s (:()lll,('Xl, I)y l,ll('~ (tlm,(lra,l,i(: lll()ll()lllia,ls -~l) _1q22 a,n(l lqp, ill t,(;nlls ()f llllit,ary. ()t)('ra.t()rs 1)(;l()l~gil~g 1.() l,ll(; lll(;l,a,I)h;('l,i(' gr()ut) AIp(2, N). Tllis is syl~th(;siz('(l ill th(; tal)h, l)('h)w, wlli(:ll (:()lllI)l(;t('S Fig. 5.1 t ) a r a l M i n g tll(; ray-()l)ti(:s r(;l)r(;s(;llta,ti()n ()f [irst-()r(l('r ()l)ti(:a,l sysl,(;llls in t(;rms of mfim()(llfla.r 2 x 2 llm,tri(:(,s M , with tll(; wav(; ()t)ti(:s r(;t)r(;s(;lltati(nl iu t(;rms ()f u n i t a r y ()t)('rat()rs M, |)()tl~ (h't)a,rting fi'()x~ th(; (pu)t(;(1 ~()x~()~ia,ls" Ray Optics

4:=

@(2,~) T(d)-

(1 r

Quadratic Monomials as generators of

Wave

Optics

1~lp(2,lk~)

=~

0

free-mediuml propagation

T (d ) - ~

lenstransferl 2

"L(f ) -- c-i~q2

positive scaling

A

p2

L(f)-

( _~/f )01

S(IYI, ) - -

((;n

lO/.,n

5q

-2qPl

S('m) __ ~1c

(tql (',ia'2-~d(q-q')2

- ln('m) q~q

T h e opera, tor representation of an a,rbitra,ry A B C D s y s t e m has then been obtained in the form of an integral transform w h o s e kernel is d e t e r m i n e d by

1D First-Order Optical Systems: The Huygens-Fresnel Integral

267

the entries of the ray-matrix of the system, thus yielding the operator t" __ k0 / "._~0__[Aq'2+Dq2 2qq'] M(M) V/2~iB dq'e ~2B

.____+

M-

A

A

with an implicit possibility that both M(M) a,nd - M ( M ) are associable to each matrix M. This means that the symplcctic and the metaplcctic groups, Sp(2, R) and Mp(2, R), are isomporphic up to a sign. In the forthcoining chapters we will describe the Wigner representa, tion, which composes both the ray and the wave optics approaches to first-order optical systems, retaining the wave optical description in its kinematic structure and displaying the ray optical features in its dynamics. Problems 1. Prove t h a t the fractional Fourier t r a n s f o r m of

(a) the space-shifted flmction ~ ( x -

x,, ) is X0) } -- e - i x t , ( x - : ~ 2

.~c~ { ( r

and of

cosqS) s i n r (~c~ ( X -

X 0 COS r

(b) the frequency-shifted function e-~*,, ~ ( x ) is

b o t h manifesting an c~-dependent

shift variance.

2. Write down the evolution law ml(ler ~ ~ of the Heisenberg-Weyl operators c ~p and c ~ax, a and b being real parameters, which are known to produce respectively space and frequency shifts of the wave fimction. 3. Prove t h a t the fractional Fourier transform of the convolution

~(~) - / ~1 (x')~2 (~ - ~')dx', A

.)L-'c~ {(r

=

e--igt

..... /,

/

r 12

( i ~ z _ - tanc~[~_c~(~2](X/)(~?1

I,".... /,I

(x--x t )dx,

.

It obtains chirp-modulating the fractional Fourier transform of one fllnction, convolving with the scaled version of tile other function and finally imlltiplying by a chirp-fimction and a A scale factor. (Hint: use the result of P r o b l e m 2 to specialize the effect of .7- ~ on ~2 (x - x'). 4. T h e complex amplitude of a Gaussian b e a m is typically written as ~ (q, z) - -

2(~(z) e - < ( z ) , A(z)e ~k'-/-L-

where r denotes the Gouy phase and Q(z) is the complex b e a m p a r a m e t e r , depending on the wave front radius curvature and the b e a m width; k is the light wavenumber in the medium. Verify t h a t on propagating t h r o u g h an A B C D system the Gaussian b e a m remains exactly Gaussian, and t h a t specifically the complex p a r a m e t e r changes to

AQ(z~) + B Q(zo) -

CQ(z~) + D '

provided t h a t the same m e d i u m is in the object and image spaces.

Linear Ray and Wave Optics in Phase 5pace

268

References [1]

S o m e basic titles c o n c e r n e d w i t h t h e flwmalization of p a r a x i a l wave t)I)t, ics by a (tuantizati()n-likc t)ri)ccss r tmraxial ray optics are listc(t in [9] ()f (:ll. 4. As an altcI'llativc to tim (lllaldizati()ll l)r()cc(lurc, tlm direct gr(mt) tlm(w(~tit:al a t)t)r()ach t() tim t)ara])()li(: wave e(tllati()ll is l)l'(~s(~lg(~(t ill [1.1]

l). St()lcr, "OI)crat()r lImt, ll()(l ill I)llysi(:M ()I)ti(:s", ,]. OI)t. S()('. Atli. 71, 3:t4-33.()

(t.(~sl).

[2]

[1.2]

II. l/acry ml(l El. (:a(lillm(', "hlctaI)hwti(: gr()lq) all(l F()lu'icr ()I)ti('s", Pllys. I{cv. A 23, 253:~-253(i ( 1.981 ).

[1.3]

II. Ba('ry, "(;r()llp tllt',()ry ml(I paraxial ()pries", ill Pro(;. XIII tj' IllI,r C()II. ()n Gr()lq) 'l'ltr N,lctli, ill l)ltys., W.W. Za,clm.ry (c(l.) (Worl(l So. l)lil)l., Si|lga, l)()re, 1,()84), t)P. 215-224.

[1.4]

.I. Ojc(la-(',~L~tafic(la mltl A. N()yr "Ditt'urcld, ial ()pcrat()r for s('alar wave cqlm.ti()lF, .]. ()l)t. Soc. Azll. A 5, 16()5-16().() (1.988).

[1.5]

(l. l)at, t,()li, S. S()lilll('ll() ml(l A. T()rr(;, "Algcl)raic view ()f tim ()l)tical l)r()l)aga, ti()ll il~

[1.6]

(]. I)attr

[1.7]

(I. I)attoli ;tl~(] A. 'l'()rr(', "A g('~(;ral view t() I~i(' alg(;1)raic ~('l,]~()(ls i~ al)l)li('(1 ~atl~e~mti(:s, ()I)tics m~(l trm~sl)()rt ,syst(,~s f()r ('l~argc(l I)em~ a('('clcrat()rs", i~ l)y'namir Sy'm,mct'~c.s and (:h.aoti~ lh'h.a~riour in l)h.ysical Systems, (;. lklai~(), I~. l'u m~(1 M. P,.~ti,i~fi ((:(Is.) (W()rl(l So. l'~fl)l., Si~gap()r(', l!)!)l), pp. 21-8,!).

.l. (',. (lallm'r m~r A. 'l'(wrc, "A~ algol)talc' view tr tim r orttcri~g m~r it,s applicati()~s to optics", Riv. N~mvo (',i~. 11, 11, 1-7!) (1.()88).

F()ra l)asi(' |l'('allll('lli ()f ilm sy~l)l(~(:li(' gr()~q) ill llm get,oral (:()lll(;xt ()f |ll(~ gl'()ll t) tlm()ry scc [9] ()f ('1~. 1; sI)e('iti(' i~v(~sligali()~s arc i~ [l l] ()f ('l~. 2. Sl~a(li(,s ()~ /tw, x~ctal)l(~(:ti(: r(~l)r('sc~lali()~, firslly i~lr()(l~w(;(l I)y Weil [2.1], all(l ()ll ils r(4ati()n t() the sy~ll)le('ti(' g(;()~(',try ar(, l)l'(',sc~t(',(1 i~ tim titles 1)cl()w. We ,sig~mlize the (mlightcning l)i'(',s(',l~lal,i()l~S i~ [2.6], an(l, withix~ tim ()t)ti(:al (:()~t(',xt, i~ [2.4] an(t [2.1]

A. Wcil, "S~lr ccrtai~s gro~q)cs (l'opdratc~rs ~fitaircs", Acta Math. 111, 143-211 (1964); also ill Collected l'apcrs, Vol. 3 (SpriIlgcr-Vcrlag, Hci(h;lbcrg, 1.()8()), pp. 1-69.

[2.2]

S. Stcrnberg, "Sodom record, rcs~llts o~ the ~netaplectic rcprcs(;~d, atio~", in Group Theoretical Methods in Physics, P. KraIncr m~(l A. Rieckcrs (c(ls.) (Springcr-Verlag, Berlin, 1978), pp. 117-143.

[2.3]

V. Guillcinin and S. Stenfl)erg, "The mctaplcctic representation, Weyl operators, and spectral theory", J. F~H~ctioiml Anal. 42, 129-225 (1981).

[2.4]

V. Guillemin and S. Sternberg, Symplcctic Techniques in Physics (Cambridge Un. Press, New York, 1984), oh. 1, pp. 1-104.

[2.5]

M. Garc/a-Bull6, W. Lassner and K.B. Wolf, " T h e metaplectic group within the Hcisenberg-Weyl ring", J. Math. Phys. 27, 29-36 (1986).

1D First-Order Optical Systems: The Huygens-Fresnel Integral

269

[2.6] R. G. Littlejohn, "Tile semiclassical evolution of wave packets", Phys. Rep. 138, 198-291 (1986). [2.7] R. Simon and N. Mukunda, "The two-dimensional symplectic and metaplectic groups and their universal cover", in Symmetries in Science VI, B. Gruber (ed.) (Plenum Press, New York, 1993), pp. 659-689. [2.8] R. Simon and N. Mukunda, "Optical phase space, Wigner representation, and invariant quality parameters", J. Opt. Soc. Am. A 17, 2440-2463 (2000). [2.9] H. ter Morsche and P.J. Oonincx, "On the integral representations for metaplectic operators", J. Fourier Anal. & Appl. 8, 245-257 (2002). [3]

For the diffraction theory the reader is directed to [3] of ch. 4, and to [6] below.

[4]

The link between the ray-matrix and wave-operator formulations of first-order optics through the Huygens-Fresnel integral, firstly recognized by Collins [4.1], is clearly elucidated in the textbook by Siegman [4.2]. [4.1] S.A. Collins, "Diffraction-integral written in terms of matrix-optics", J. Opt. Soc. Am. 60, 1168-1177 (1970). [4.2] A.E. Siegman, Lasers (Univ. Science Book, Mill Valley, 1986), ch. 20, pp. 777-782.

[5]

For a discussion of the concept of duality in relation with optical systems, see [5.1] A. Papoulis, "Dual optical systems", J. Opt. Soc. Am. 58, 653-654 (1968). [5.2] H.J. Butterweck, "General theory of linear, coherent, optical data-processing systerns", J. Opt. Soc. Am. 67, 60-70 (1977). [5.3] H.J. Butterweck, "Principles of optical data-processing',' in Progress in Optics, Vol. XIX, E. Wolf (ed.) (North-Holland, Amsterdam, 1981), oh. 4, pp. 211-280. [5.4] A. Lohmann, "Ein neues Dualit/itsprinzip in der Optik',' Optik 11, 478-488 (1954). For the English translation, see: A. Lohmann, "Duality in optics',' Optik 89, 93-97 (1992).

[6]

For a mathematics- and/or physics-oriented treatment of the Fourier transform the reader may consult the textbooks in [15] and [16] of oh. 4. In particular, for a discussion of the Fresnel and Fraunhofer diffraction regimes see: [6.1] J.W. Goodman, Introduction to Fo'ur~er" Optics (McGraw-Hill, New York, 1968), ch. 4, pp. 57-76. [6.2] A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), ch. 9, pp. 315-345. [6.3] J.D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley & Sons, New York, 1978), pp. 361-385.

[7]

An essential bibliography about the fractional Fourier transform is in [8] of ch. 3. We quote below tile paper by Nalnias, in which the fractional Fourier transform is introduced through an eigenvalue equation like (5.4.42), and those by Agarwal and Simon, and Dattoli, Torre and Mazzacurati, the former dealing with an interpretation of the fractional Fourier transform in terms of the time evolution operator of the quantum oscillator, and the latter with an extension of such an interpretation to an in general quadratic Hamiltonian. Papers concerned with fractional Fourier transforms of complex order and generalized forms of the fractional Fourier transform are also suggested. For a discussion of the fractional Fourier transform within the general context of real and complex linear canonical transforms, see [10].

270

Linear Ray and Wave Optics in Phase 5pace

[7.1] S.V. Namias, " T h e fractional order Fourier transforin and its application to quantum mechanics", J. Inst. Maths. Applics. 25, 241-265 (1980). [7.2] G.S. Agarwal and R. Simon, "A simple realization of fractional Fourier transformation and relation to harInonic oscillator Green's fllnction',' Opt. Coimn. 110, 23-26 (1994). [7.3] G. Dattoli, A. Torre all(1 G. Mazzacllrati, "All altcnlative point of view to the theory of fractional Follricr trailsfonn", IMA ,l. Appl. Mattl. 60, 215-224 (1998). [7.4] C.-C. Stfitl, "Ot)tical illt,crl)rctatioll of a colIlI)lcx-or(lcr Foln'icr trmlsfiml?', Opt. Lctt. 20, 1178-1180 ( 1.().()5). [7.5] L.M. Bcnmr(lo a.ii(l O.O.I). Soarcs, "()ptical fl'a(:ti()llal t)r(h'r l"olu'icr trmlsforllls wittl complex or(lcrs", AppI. Opt. 35, 31(i3-31(i(i (1.9.()(i). [7.6] ,]. Hua, L. I~ill mlr (I. IA, "Exl, cll(lcd fl'actioilal Follricr trmlsforlll", ,I. Opt. Soc. Ain. A 14, 331(i-3322 (19!}7). [s]

A (;a~ssim~ l)(:a~ relale(l I)il)li()grat)l~y (:al~ ])c f(nl~(l i~ [13] ()f (:l~. 2.

scc, tl~c tcxtl)()()ks i~ [l()] ()f r ,1. 'l'l~(; I)rcsc.~dali()~ i~ !i 5.5 is i~sl)irc(l l)y t h a t in [!).l] D. MarctlSC, Light 7}'ansmis.~ion Optics (Vm~ Nostrm~(l-ll.ci~l~ohl, l'ri~ceto~, 1972), cir. 3, pp. 1()7-1()9. [1()] F()llowi~g tt~(, s~gg(;sli()~ i~ [~().l], l l~(; l)r(;s(;~flali()~ i~ !i 5.6 r(;l)l~ras(;s witl~i~a tl~(; ()t)ti(:al (:()nt('xt t l~(; g(,~(,ral ('()~si(l(;l'ati()l~S (:()ll(:('rllillg lll('~ illlcgral lrallSf()rln as1)cots ()f tl~(; ('(nlstn~('ti()~ ()f ~l~litary r(,l)res(;~ltati()~s ()f li~(,ar ('a~()~li('al trmisf()rl~mti()ns ()f ll~(, H(,is(,~d)(,r~-Wcyl algel)ra ()t)(,l'ai()rs q, p ()~ il~(' tIill)rl'l si)a(:e t)cl()w. Sccals() [2.4]. A~ (;~ligl~lc~i~g (;xl)laI~ali()~ ()f I l~(', 7r-l)lmse sl~ifl, s~ff'crc(1 t)y a light wave I)a,ssing tl~r(~gl~ a ('()~j~gatc I)()i~d is given i~ II().l()]. [10.1] M. Nazara.i,l~y m~(l ,]. Slm~ir, "First-or(ler optics- a (:mm~fi('al ()l)erat()r reprcsci~tat,i()~: lossless syst, c~s", .l. Opt. S()(:. A~n. 72, 35(i-3(i4 (1.982). [10.2] L. IIffcl(l m~(l ,]. I)lcl)m~ski, "()~ it certain class of ~fit, ary trm~sform~tio~s",Acta Phys. Polon. 14, 41-75 (1.q54). [10.3] V. Bargmmu~, "O~ a Hill)err Sl)aCe of m~alyti(: fl~('ti()~s m~(! m~ ~mso('iatc(l integral transform, r', ( ; o x ~ . lhm,~ & AI)t)I. Matt~. 14, 187-214 (1.()61); II. A fiunily of related fllnction spaces aPl)li(:ati()~ t()(listril)~d, ion theory, i|)i(l., 20, 1-101 (1967). [10.4] M. Moshi~sky m~(l C. Q~leS~c, "Li~ear (:anonical transf()r~mtiox~s mul tlmir unitary representatio~s", J. Matl~. Pt~ys. 12, 1772-1780 (1971); C. Q~ms~c an(l M. Moshinsky, "Canonical tral~sformat, i()~s and matrix elements", ibid., 1780-1783. [10.5] M. Moshinsky, "Cm~o~dcal transformations an(l quantllln ~nechanics", SlAM J. Appl. Math. :15, 1772-1780 (1973). [10.6] K.B. Wolf, "Canonical transforlns. I. Complex linear transfortns", ,J. Math. Phys. 15, 1295-1301 (1974); II. Complex radial transforms", ibid., 2102-2111. [10.7] P. Kramer, M. Moshinsky and T.H. Seligman, "Complex extensions of canonical transformations and qllalltlllIl inechanics", in Group Theory and Its Applications, Vol. 3, E.M. Loebl (ed.) (Academic Press, New York, 1975), pp. 249-332. [10.8] K.B. Wolf, Integral Transforms in Science and Engineerin 9 (Plenum Press, New York, 1979), ch. 9, pp. 381-416. [10.9] K.B. Wolf, Geometric Optics On Phase Space (Springer, Berlin, 2004), pp. 277-282. [10.10] V. Guillemin and S. Sternberg, Geometric Asymptotics (Am. Math. Sot., Providence, 1977), ch. 1, pp. 1-19.

6 The VVigner Distribution Function: Analytical Evaluation

6.1

Introduction

The Wigner distribution function was originally introduced by Wigner [1] as the simplest quantum analogue of the classical phase space distribution function permitting to find probabilities and expectation values of quantum operators using as much classical language and methods as allowed. The Wig:mr method establishes a rule to associate a c-imln|)er function in phase-space with every operator being a fimction of position and momentum operators. Thus, the expectation vahms of quantum mechanical observables can be calculated in the same mathematical form as the averages of tile classical statistical nmchanics rather than through tile operator fornmlism of quantum inechanics. As an illustrative example, we consider a particle moving along a line. In classical mechanics the dynamical state of the particle at any instant of time is fully described by the position q(t) and the momentum p(t) of the particle [2]. The motion of the particle is generated by the cla,ssica,1 Hamiltonia,n p2

H(q, p) -- 2m + V(q),

(6.1.1)

according to which the classical equations of motion obtain as

dq dt

p m

,

dp dt

-

-V'(q)

(6.1.2)

m denoting the mass of the particle and V(q) the potential it experiences (V'(q) - dV/dq). If the initial conditions q0 - q(O) and Po - p(O) arc known, the evolution of the state of the particle is uniquely determined by the solution (q(t), p(t)) of (6.1.2), which individualizes as well a definite trajectory in the particle phase-space. It may happen however that the initial state of the particle be not specified through the definite values q0 and Po of the position and momentum of the

272

Linear Ray and Wave Optics in Phase Space

particle at t = 0, but r a t h e r t h r o u g h the .joint probability distribution of q and p at t = 0. In t h a t case, the d y n a m i c s of the particle will not be determined by Ha nfilton's equations (6.1.2) for q(t) and p(t), t)ut by Liouville's equa,tion for the t)ha,se-spa(:e. (tistributi(m flm(:ti(m Pc(q, P; t),

0 p 0 V' 0 otp(;( q, p; t) - {p(,, H } . . . 'mOq . P(" + (q) ~ p(,.

(6.1.3)

Clea.rly, p(,(q, p; t) lll(~.tl.ll,S 1,11{;l)r()lm.1)ility (lellsit,y fin" tilt; tmrt, it:le I,() 1)e a,t the t)]m.sr l)r (q, p) at t,i,,l(: I. It,:,,(',;, p(,(q, p; l) >_ ()m,(l Ji/'P,,(q, P; t)dqdp 1 a,t a.lLy t, tlu; ilm;gra,ti(nL l~r (~w;r tlLr ilL1Lr l)lLas(; st)a(:r r tlLr I)article. If tlle t)rr r ir r t,hr (lylm,nfica,1 wu'ia,lfles q a,llr p a,t the initial t,i~l~; is k~lr p,,(q, p; ()) = p,,(q, p), t11r lw(~tm,l~ility r p(,(q, p; t) a.t a.lly slll~sr162162 t,ilLLr is lllLir (lr 1)y t, lLr sr i(~]L r Er (6.1.3). hi ()tiLer W()l'(ls, ill (:la,ssi(:al ~Le(:lm~Li(:s t,lle illf()rllla,t,i()li ('(nitaille(l Lilt,() the (:ml(nli(',ally (:~)ll.j~lg~at,(; wu'ialfl~;s q mL(l p, r(;l)r(;s(;~lt, i~Lg: t,lL(; lmrt,ir st,a,te as a, t)(~i~lt, i~1 lfllase-Slm.('e, r t,ra.~lsf~;rre~l int,()t, ll(; (list,riln~t,itnl fllll(:l,i()ll p(,(q, p; t), ret)res(;nt, ing the t)r()lm.I)ility fl)r every t)()int in t)hase st)a(:(; t,() ln'~ a. l)(~ssil)l(; st, a.l,c fin" 111(; l)arti('l(;. Ar 1,11(;(;v()l~lt,i()11 ()f 1,11(; Imrti(:l(' st.at(, ('i.lll ~zli(l~z(;ly 1)e a.(:('(nnm;(l f()l" ])y t,ll~; l~l()l,i~)ll ()f t,l,; l)lm,se l)la.ll(; I,~,i~lt, (q(t),p(t)), ,)r 1)y the ew)l~t,i(nL ()f t.lL~' (list,ril)~t,i~n~ fl~L('t,i~nL p(;(q, p; t). W~; 1L()t,(; t,lLa,t I,i()~vill~:'s (;(l~m,t,i,)~ ILas t)(;(;~L (l(:riw;(l ill !i 2.2.1 wit,lli~L t,ll(; ()f ligllt, rays, t,ll(: relewud, (ly~lmlli(:al flnl(:t,i(~l p(q,p; z) l~('~i~lg i~d,erl)r(;t(;(t a,s the l)lmse-sl)a,(:e ray ~lensit,y a,t every ref(;re~l(:e z t)lmu;, hi efIi;(:l,, p(q, p; z)dqdp yiehls 1,11e ~n~td)(~r ()f rays a,t z izl l,ll(; (;le~lexd,a,l v()l~l(; dqdp ar, nn11(l 1,11(;t)ha,sest)a,(:e t)()i~lt (q,p); l,m,(:e a,t ea,(:tl z l)la,ne .]i['vp(q,p;z)dqdp = N" (1,11(; total lnn~fl)(;r ()f rays i~1 l,lle l)ea~tl), i,t1(; izltegration inv()Ivi~lg tlm i~lw~,rimlt t)ha,se st)a,(:e ret)rese~d, ative v()h~11e 12 of 1,11(; |n~1(:11 ()f rays. It is (;vi(l(;~lt tlmt, within the me(:ha,~fi(:al-()t)ti(:a,l t)a,si(: (:()rr(;st)(nl(t(;n(:(; ()f the temt)()ra.1 (:()(w(lina.t,e t to the axia.1 (:o()rdina, te z, the (listrit)~d,i(m fim(:tions p(:(q,p; t) a,n(l p(q,p; z) are equiwdent, a,t)art fi'()I~1 the relative n()rIna.liza, tion settings. T h e I)r()l)a,bilistic picture inllerent in p(;(q, p; t) i~lt)li(:itly involves, in N,('t,, the e~lsmlfl)le picture of a hug(; nm~d)(;r ()f i(tenti(:a,1 systenls, i.e. t)artMcs with mass 7n, (tistributed at ea,ch tinle t ()vet" all the possible (lyna,mi(:a,1 states (q,p) a,(:(:()r(ling to the releva,nt weighting function Pc (q, P; t). E(t~m,tion (6.1.3)(:orrest)on(ts to (6.1.2) when a statistical ensemble picture of the system d y n a m i c s is concerned. Accordingly, the average value of any dynami(:al filnction f(q, p) is calculated by the integral ( / ) c (t) - . f / p(: (q, p; t)f(q,p)dqdp,

(6.1.4)

the integration involving the entire phase space d o m a i n p e r t a i n i n g to the syst e m [2].

The Wigner Distribution Function: Analytical Evaluation

273

Integral (6.1.4) formally expresses the nature of the statistical description of classical mechanics, where the mea,n values of dynamical variables are obtained by averaging over uniquely determined processes. This clea,rly mirrors tile fact that in classical mechanics the position and IIlOilleil[,lliil of tile particle a,s well as any function of q and p, a,nd hence the probability distribution Pc(q, P; t), can exactly be known at any time, the relevant time evolution being uniquely determined by deterministic laws of evolution; the whole uncertainty in classical mechanics is indeed contained in the form of the initial probability distribution of the dytmmical variables specifying the given system. In contrast, in quantum mechanics the simultaneous specification of the position and monlentunl of the particle with arbitrary precision is impossible. The dynamical state of the particle is described by specifying at each time t the wa,ve function ~(q,t), which determines the probability density of the position coordinate according to [3] OQ (q, t) - I~(q, t) 2,

(6.1.5)

and separately the probability density of the momentum coordinate through the Fourier tra,nsform ~/,(p, t) according to q3q (p, t) -- I ~(p, t)12.

(6.1.6)

In addition, the knowledge of '(.,(q,t) enables us to find the expectation value of any observable .f(~, ~) in the state ~/~ by the scalar product

If)q(t)

- (~/~(q,t)]f(~,~)lt/~(q,t))

- ]~/~*(q,t)f(~,~)~/~(q,t)dq.

(6.1.7)

In particular, if f depends only on the coordinate operator ~, i.e., f(~, ~) h.(~), we have

( f )q (t) -

~Q (q, t)h(q)dq,

(6.1.8)

whereas, if f depends only on the momentum operator ~, namely: f(~, ~) ~(~), one obtains

( f }q (t) --

j 9i3q (p, t)9(p)dp,

(6.1.9)

where h ( q ) a n d g(P) arc the classical functions corresponding to the quantum operators h(~) and ~(~). The evolution of the dynamical state of the particle is determined by the SchrSdinger equation for the wave function ~h(q, t),

ih Ot O__~h(q, t) - H(~, ~- ~)~h(q, t),

(6.1.10)

Linear Ray and Wave Optics in Phase 5pace

274

with H(~, ~) - ~ 2 / 2 m + 0 ( ~ ) being the q u a n t u m Hamiltonian operator corrcspon(ting to the classical function (6.1.1) [3]. As II()t(;(t above, a, classi(:al joint probability distriblltion p(,(q, p; t) can be (teviser whose tra,nsfornmtion with tinlc is I'111c(t by Liollvillc's equa,ti()n (6.1.3), tllllS a.lh)wi~lg for a, flflly t)r()|)a,1)ilisti(:-likc a.t)t)roa('ll to ('la.ssi('al dynmlli(:al t)robh;nls witll()ld, cxpli('itly res()rting to tll(; (t(;s(:rit)tir in torlllS of the; t)osition all(| lllt)lllt;lltlllll c()()r(lilm,t(;s, a ll(l h(;ll(:(; t() IImllilt()ll'S (;(tlla,ti()liS. It, is llal.llral t,() illv(,stiga,tt, t,lw. t)()ssil)ility ()f cla.l)()ratillg a (:lassi('al-lik(; (l(;s(Til)l,i(nl ()f l,ll(' r sta,l,('. ()f tll(; (lllalltlllll systr (t,h(; I)()illl, I)ai'ti(:l(; in l,ll(; t)rt;st;ld, (list:llssit)ll) a ll(l its ('~vr i()ll wil,ll()lll, (;xl)li(:itly rt;s()rl,ill~ t() tll(; wav<; filll('l,i()ll, a ll(t ll(u~('<; I,()tl~(; $(:l~rii(li~g(;r (;
tit(; sil~nlll,a.l~(:()llS all(| l)r(;(:is(; val~t(;s of

1 , ~ t)l~ast; Sl)at'e, wl~el'(; a "l)rr (listril)~tl,i()~ fi)r m~y lmir ()f q a n(l p vahw, s (~(~hl ]~(~
//' pq (q, p; t).f(q, p)dqdp.

( 6.1.11 )

where .f(q, p) sl~()~ll(l rclm;Sc~t the (:lassica,1 finl(:ti(~ (:~x'r<;st)()~(li~g t<~ tl~c qua,nl,~m~ ,,|,s,,rval,h, .f(~, ~), .i,tst a,s h(q) a.~,l g(p) in (6.1.8) a,n,t (6.1.9) in relation with h,(~) a ~ l .0(P). l{~u~m,rkal)ly, there is ~()a,nfl)ig~ity in tl~'. (:()rrcsI)()n(tcncc |)ctwecn tl~e ,t,m,nt,,n~ ,,t),;rat,)rs h,(~) a.n,t .0(P) an,1 the ,:la,ssi(:al f,m,:tions h(q) anr g(p), a,s they rcsl~e(:tiw'Jy depth(1 only on ~ an(t ~. In (:(mtrast, the corrcspon(lcncc t)ctwccn the (pm.ntum fim(:tion f(~, ~) of the n o n - c o m m u t i n g opcra,tors ~ a,n,t ~ a.n,t the classical function f(q, p) of the ('.onmmting variables q and p is ~ot ~mi(t~(;. Classi(:M mecha.nics opcra.tes with a,lgebraic fi~nctions, whcrea,s (t~m,~tmn n~ccha~i(:s works witt~ operators, and there is no unique rule to tra,nslatc from one nm,thema, ticM context to the other. If the operators and ~ are in(tis('riminatcly rcpla,ced by the sca,lars q and p, different operator fm~(:ti()~s .f(~, ~ ) c o m e to correspond to the s~m~c c-function f(q,p). The ambiguity is of the order of h (see P r o b l e m 1 and w 8.2.3). We may therefore guess that phase space distributions in q u a n t u m mechanics arc not unique and that their definition is closely connected with the correspondence rule set up A

A

The Wigner Distribution Function: Analytical Evaluation Classical

275

Mechanics

f Inform a t i o n c o n v e y e d by the c o n j u g a t e v a r i a b l e s q ( t ) and p ( t )

I n f o r m a t i o n c o n v e y e d by the w a v e f u n c t i o n q/ ( q , t )

are transferred

are expected transferred

into

to be into

%

the p h a s e - s p a c e d i s t r i b u t i o n f u n c t i o n Pc" ( q , p ; t )

the p h a s e - s p a c e d i s t r i b u t i o n function p s (q, p ;t) J

Quantum

Mechanics

FIGURE 6.1. Paralleling the mapping of the Hamiltonian view into the Liouvillian view of classical mechanics, which transfers the information about the system state from the canonically conjugate variables q(t), p(t) into the phase-space distribution function Pc (q, P; t), the phase-space formulation of quantum mechanics is expected to map the SchrSdinger view into the Wigner view, transferring the information about the system state conveyed by the wave function ~p(q, t) into a suitably defined phase-space distribution function pq (q, p; t). between q u a n t u I n observables a,nd classical functions. It is of course desirable t h a t pq (q, p; t) be in the classical limit the pheusespace d i s t r i b u t i o n flmction Pc: (q, P; t). Also, in view of its role as a .joint probability density, it should be real and n o n n e g a t i v e for all values of q and p: pq (q, p; t) _> 0,

(6.1.12)

and give the t)rot)er probabilities for the c o o r d i n a t e q and m o m e n t u m p, if integrate(t respectively over p and q: P,a (q, P; t)dp - I ~/~(q, t)I z ,

pq (q, p; t)dq - I~h(p, t)12.

(6.1.13)

T h e integrals above are k n o w n in the p r o b a b i l i t y t h e o r y as the marginal dist r i b u t i o n s of the given (q, p) d i s t r i b u t i o n p~ (q, p; t). F u r t h e r conditions are usually required to be satisfied by pq(q, p; t), which however do not lead to a unique d e t e r m i n a t i o n of pQ. T h e W i g n e r d i s t r i b u t i o n function is one of the possible d i s t r i b u t i o n s pQ (q, p; t), t h a t can be e l a b o r a t e d to sa,tis .fy the required conditions at a la,rge extent. It benefits from the greatest m a t h e m a t i c a l simplicity and elegance. It is typically used as a joint p r o b a b i l i t y density, a l t h o u g h a strict i n t e r p r e t a t i o n in this sense is not possible as it m a y in general take on negative values in some regions of the q u a n t u m phase space, thus violating the usual axioms of p r o b a b i l i t y t h e o r y (a) . a The reader interested in a discussion of phase-space distribution functions in the light of the instance for extended probabilities, which are not necessarily positive, may consult the stimulating review: W. Muckenheim, "A review of extended probabilities", Phys. Rep. 133, 337-401 (1986).

276

Linear Ray and Wave Optics in Phase Space

As a l r e a d y noted, tile W i g n e r d i s t r i b u t i o n flmction was first i n t r o d u c e d by W i g n e r for a, s y s t e m of m a n y particles in o r der to s t u d y the qua,ntunl corrections to ttmrmodynanfi(:s flm(:tions a,n(t t h e n a,t)t)lie(t a(tva.ntage, ollsly to a large va,riety of qua,nt, mll 11m('hani(',a,1 t)r()l)lenls [4]. It, was re, intr()(tll(',e(1 ])y Ville in the, signal a.na,lysis [5] a.n(1 IlI()I'C ret:ently t)y Bast, iambs [(i] in the, ()t)ti(',a,1 (:(mtext w h e r e it, fr wi(le a,l)I)li(',a,tions wit, lfi~l t)()t, ll t,lw, st)a,(:e-fl'e(tlm, n(:y all(| timefi'c,(tlll~,ll(:y (hnlm,ills. Na,tllrally th(', st)a,('(',-fl'(~,(tllt',llt',y (a,s well t,illl(',-fr(',(tllC, ll(;y ) Wigllt~l' (lisl, l'il)lll, i()ll fllll(:ti()ll ill ()l)ti(:s (as w(;ll ill g(~ll(~ral ill sig~lal tlll',()l'y) is lllatll(nllati(:ally i(l(:lll,i(:al 1,() l,tl(', (Ilia.lit,11111 Wigll(:r (lisl,l'il)lll, i()ll filll(:ti(nl ()f t)()siti(nl a.ll(l lll()lllt~ll[,lllll. 'l'll(~ (llm,ld, lllll lll(~(:lm,l~it:al res~fll, s l~a,ve s~ita,l)ly t)ccn rt;I)l~ra.s(:(l wil,l~i~ tl~(' ()l)l,i(:al (:()lll,(~xl,, wil,l~ l,l~e sig~m,1 al~l)lil,~(le ml(l r(4e, va.nt ff(;(l~m~(:y st)(~(:l,r~l~l r(~l)la,(:illg l,l~(~ I)()siti(n~ a.ll(l lll()lll(',llI,11111 (t~m.nl,~l~ wave, tim(:ti(~l~s. 'I'll(; illt(;l'l)l'l;ta,ti()ll ()f tll(', ()t)ti(:al Wigll(;r (listril)~ti(nl fllll(:ti()ll ill t(',I'nls ()f tim siglml (~ll(',l'~y (Iistril)~tti()n lm,s als() 1)(~(n~ ~tl,t,(u~q~t(~(l t>aralh'li~lg t h a t ()f l,lle l)()sil,i()ll-lll()lll('~d,~l~l (listril)~l,i(nl ,)f l,lle (l~mld,~ll~ l,]lt~tIl'y ill t,('r~llS ()t"a. l)r()l)al)ility (lisl,ri])~ll,i()~l. As ill 1,11(: (:as(~ ()f l,ll(~ (l~lalll,~ll~ l,ll(;()ry, ll()w(w(u', t,ll(~ stri(:t ild,(U'l)r(,l,al,i()ll ()t" l,ll(~ W i g l m r fllll(:l,i(nl as all (ul('rgy (listril)~lti(n~ is l~()t, l)()ssit)le as it, i~ g(u~('ral l,ak(;s ()~ ~('gal,ive va,lll('S ill l,ll(' l)l~as(' Sl)a,('r Tll(' Wigll('r <listril)~ll.i~nl fl~l~(:l,i,)ll l'(~l)l'(~.St~lll,,S ~l lm,rl,i<'~flarly ~ls(:fl~l I()()1 in (11111.11t11111 ()l)ti('s ;~s w('ll. 'I'll(' Wigll/;l'-Sl)~t(:/'~ l)i(:l,~r(' yi(;l(ls ;u~ il~l,~iliv(~ vis~a,1izal,i()l~ ()f l,ll(' (ly~m~li(:s ()f (:()ll(~l'(~lll, a ll(l ,S(l~e(~z('(l Sl,a.l,(',S, wlli(:l~ ('m~ 1)e ~(;ne, l'i-I.t(',(l fI'()lll 1,11(: Va.(:lllllll ,S[,t,l.[,(', l)y I)lm.s('-Sl)a('(: l'()ta.l,il)ll, l,ra.l~sla,t,i(nl a~l(l s(t~e, cze, ()l)(;ra.t()rs, all lm,villg tll(:ir g(;(n~(',tI'i('a,1 l'a,y-()I)ti(:s a.~m,h)gs [7]. It Ires t)(:(u~ alr(~a.(ly ('~q)lm,siz(;(l 1,1~(; anah)gy 1)('l,we(;n light m~(l lnagneti(: ()l)t,i(:s, wlli('ll I)r()vi(l('s tll(; t'()~u~(la,ti()lls ti)r slm,rillg t)()t, ll ('(nl(:(',t)t,s ml(l l~le,th()cts. At:t:()r(lillgly l,ll('~ a,l)I)li(:a,ti()l~ ()f l,ll(', Wigll(;r l)i('l,~lr(: 1,() llla,~ll(',l,i(: ()i)ti(:s lms a ls() 1)e,(;n s~gg('sl,(',(1, as, fin" i~sl,a,~(:(',, i~ rt;la,l,i()ll 1,() 1,1~(;(h;s(:rit)l,i(n~ a,lltl mmlysis ()f th(', ()t)ti('al ('lm,ra~'t(;risti(:s ()f syllf'llrf)tl'()ll ra.(liati()l~ s()~r(:(;s Is]. S(;(:ti(nl 6.2 i~ll,r()(l~t:(~s l,tl(', f()rnm,1 (t(;tinil, i()ll ()f 1,1~(~ op/ical Wigll('r (listI'it)~ti()ll flnl(:ti()n, wll()s(~ t)r()t)eI'tie, s are, (h',sc:rit)c(1 il~ (l(',ta,il i~l S(;(:t. 6.3. Seve, ral exa,~I)les a.i'c ill~stl'a,te(t ill Se(:t. 6.4 ill ()r(ter t() (:rea,te the, feeling fl)r tlle, hybrid na, t~re ()f l,ll(~ Wigller ful~cti(n~, whicll i~fl~erently ill(:()rI)ora,t(',s , a,n(t hence ca, n dist)lay, tl~e, |)a,si(" fe,a,t~res of the ray-optics descrit)tion of light signa,ls within a n d thr()~gh the se,enfingly irreconcilable c o n t e x t a n d m e t h o d s of wa,ve optics. T h e a,t)()ve discussi(m has h a d a, t)urc illustra,tive t)urpose. Wc considered in fa,ct the sin~t)le ('ase ()f a s y s t e m w i t h one degree ()f f r e e d o m in a, p u r e state. M o r e in gei,eral a, s y s t e m w i t h m a n y degrees of free(tom a n d in a, m i x e d s t a t e m a y t)e (:()nsi(terc(t, tluls (tema,nding for a certa,in nunfl)er of pa,irs of canonically conj~,ga,t(~(t va,ria,blcs as well a,s for the d e n s i t y m a t r i x formalism, for which the intcreste(t r e a d e r is a d d r e s s e d to [1.1, 4, 7]. Fina.lly, we note that, a l t h o u g h our discussion ha,s c o n c e r n e d w i t h t h e position a n d m o m e n t u m o p e r a t o r s , the W i g n e r reprcsenta,tion can as well be e l a b o r a t e d for any pair of c o m p l e t e sets

The Wigner Distribution Function: Analytical Evaluation

277

of mutually incompatible Hermitian operators [9].

6.2

The optical Wigner distribution function basic concepts

We consider a time-harmonic 1D optical signal, represented in the space and spatial frequency domains, respectively, by the complex amplitude ~(q, z) and the respective Fourier transform @(ec,z). Linked by the reciprocal Fourier relations (w 4.a.1.2), ~ ( q , Z) -- ~ ~/~(/g,Z) __

@(/%' 1

z)~i~qdtg'

(6.2.1)

g~(q, z)c-z'~qdq,

p(q, z) and ~(ec, z) are understood as square-integrable complex-valued functions of the Fourier conjugate variables q and ec for any value of the real parameter z. We will also consider signals, which are not normalizable in the common sense of the energy-like integral, i.e., by the requirement that f IP(q, z)l 2 dq < oo. As noted in w 4.3.2, such signals may be handled as limiting processes of normalizable signals or by resorting to the generalized function method. The impulse and the pure harmonic signal, respectively represented by the delta. and the complex exponential, are prime examples of such idealized signals. The Wigner distribution function 1/V(q, ec) of the 1D optica.1 signal ~(q) is defined as a function of the Fourier conjugate va.riables q and t~ by the integral in the space domain [6] ) & ( q , ~) -- ~1

/ j)* (q

-- (tl)~(q 2

+ q'2 )c-i~q' dq'.

(6.2.2)

A basically identical integral in the frequency domain, which then involves the signal spectrum @(t~), is also given as an alternative definition, i.e. lA?(q, t~) - ~1

/ @*(l'C-- -ff)~(n ~' t~' ciK,'qdl,c,-. + -if)

(6.2.3)

As usual all the integrM in the chapter without limits of integration are intended from - e c to +oc. The equivalence of the integrals (6.2.2) and (6.2.3) (:an easily be proved. In fact expressing the amplitudes ~* ( q - q') 2 and ~(q + q') 2 in (6.2.2) in terms of the signal spectrum according to (6.2.1), we obtain

lff ,

W(q, ~) - ~

1f

(~')~5(~") ei(t~"--t~')q [ ~

~ -'~)dq']dt~'dn".

(6.2.4)

As the bracketed expression above turns to be ~[~'+~" t~] the double inte2 gral over frequency simplifies in the form (6.2.3). Inversely, using the Fourier integral for @ in (6.2.3), we regain the q-domain definition (6.2.2).

278

Linear Ray and Wave Optics in Phase Space

T h e d e p e n d e n c e of tile signa,1 a m p l i t u d e ~s and tile Fourier t r a n s f o r m ~3 on the axial c o o r d i n a t e z is implied in the integrals (6.2.2) and (6.2.3), which are u n d e r s t o o d as t)erfornmd a,t z = const, planes. Hen(:e the. W i g n e r distribution flln(:tion d e p e n d s (m z as well, but this det)e.nden(:e will in general not be (tist)layed. Als(), a slfita,t)le slfl)s('rit)t, referri~lg to tim. sI)e(:ifi(: signa.1 a,t hand, will t)e a,tta,(:he(t to tlle ba,si(" symbol W(q, h:), wtlexl ~lee.(le(l for ('la,rity's sake. T h e r e is a,II evi(le~lt forxlml sixifila.rity t)etwee~l t,lle Slm(:e a ll(t fre(llU:n(:y do~mi~ (t(:fi~il, i()~s (6.2.2) a~(l (6.2.3), so tlmt (n~e (:a~ 1)(: (lel, en~i~e(1 fr(m~ the ()(,lmr t)y ilfl,cr(:lmllgillg l,ll(: r(flcs ()f q a.ll(1 h'. Tiffs fi:a.tln'(: is ill (:(mf()rmil,y witll tll(: (lcsire(l l~yl)ri(l l~al,~r(: ()f tl~e Wig~(:r (listl'il)~ti(n~ fi~l~(:ti()l~, whi(:h is (;xt)e(:t('~(1 (,()(h;s(:ril)('~ l,l~(~ sig~m,1 i~ st>a(:e aa~(l fl'(;(t~(;~(:Y si~n~li,mm()~My. I(, lllil,y t)t', i~sl,r~('l,iv(; (,() give a (:l()s('~r l()()k a,t 1,t~(; s y ~ ( ; l , r i ( ' i)r()(t~(:(, ~* ( q q' )~(q + ~ ) wl~i(:l~ is (X:lfl,ral (,()1,1~(: (:wd~al,i(n~ ()f 1,1~(; Wigi~(:r i~t(:gra,1. T()hell) 2 vis~mlizati(n~ w(: (:()~si(l(;r a real a l~(1 ~()~n~(;ga,i,iv(; sig~a,l. 11, is 1,1~(;~ al)tm,reI~t (,l~a,l, (,l~c Wigl~(;r (lisl,l'il)lt(,i()~ fl~n('(,i()~ a l, l)()ilfl, q is ()l)(,ail~(;(l l)y a(l(ling ~t) |,tWillS (:(n~sisl, i~g ()f 1,1~(; l)r()(l~(:l, ()f 1,1~(; siglml m~t)lil,~ul('.s a,t l)()ilfl, s syl~mmtri(:ally (tist)()s(;(l ()~ 1,1~(' h;f(, m~(l ()11 l,h(', rigid, ()f q. Wi(,l~ (,l~is vi('w ill ~i~(l, after tlm ~m~nmr ()f [~().2, r,.s, .~.7], w(' (:tl,ii ga,il~ a, vis~ml a.(:t:(n~t ()f s()~(: g('~(:ra,l l)rOl)(:r(,i(:s ()f the Wig~(:r (listriln~ti()l~ fin~(:l,i()~ i~sl)(:('(,il~g (,l~(: sil~q)h: wav(:f()rn>, (l(:l)i('t(:(l i~ Fig. 6.2. l)()tl) sig;~Is (,xt(:~(l ()v(;r (.l~(, fi~i(,(' i~(,(,rval (Q,,, Q,,), ()~(: sig~m,l sl~()wil~g a gal) as 1,I~(' ()(,l~(:r a si~n~s()i(la.l l)(:rl,~rl)m~(:(: f()r a sl~()r(, ild, erva,l ()v(:r a,~i ()th(:rwis(: ~ i f ( ) I ' ~ a,~I)lit,~ul(:. 'l']~i~ki~g i~ ()I)ti('a,l),(:n~s, tl~(: wa.vef()n~s (lel)i(:t(:(l llia,y 1)(: l,l~()~glfl, ()f a,s ()l)(,ah~(:(l l)y a, ~n~if()r~ ill~n~ilm,l, ion of (,w() (:()nsta,nt m~l)li(,~ul(: (,l'a.llSiiiil, l,tl.Ii(:(: slits, Sl)a,(,ia.lly S(:l)ara,(,(:(l l)y a,~ ()l)a.(lUe s(:('t,i()~ in Fig. (i.2.a,) aa~(l I)y a siln~s()i(tal a,l~t)lil,~ul(: (,ra~s~il,l,m~(:(: gl'a.t,ing in Fig. 6.2.t)). Evi(h:ld, ly, (,1~(: Wign(:r (tisl,ri|)~d,i(n~ fim(:l,i()~s ()f |)()(,1~ sig~m,ls wmish (n~tsi(lc the il~(,(:rwd (Q,,, (~),,). I~ fa.(:(,, a,t a l l y t)()ilfl, q <_ (~),, tim sig~m.1 O l l the righ(, ()vcrlat)s ()v(:r (,h(: zer() siglml on the left, a,n(1 si~filarly a,(, any t)()int q > Q)~, tim signal ()~ tl~(: h:ft ()v(:rlat)s over the zero signal ()1~ tim rigl~t, tlms yielding in b o t h (:a,ses n() (:()n(,ril)~tion to the integral (6.2.2). ()n th(: other hand, in regard to the wa,v(:f()n~ ()f Fig. 6.2.a), we (:a,~ see t h a t tt~(: Wig~(:r (listribution flln(:tion is n()(, z(:ro tt~r()~gl~ the whole ga,p, i.(;., f()r tl~(: i~(,(:rva.1 (Q,., Q~). At point q~, for instance, th(: a.n~t)litudes outside the interval (Qc, Q~) c o n t r i b u t e a non zero overlap to the W i g n e r integral (6.2.2). In fact, as a, general property, the Wign(;r disti'ibuti(m flmction is limited to the sa,Ine interval of the signal, but is not ncccssa,rily zero at points where the signa,1 vanishes. Similarly, the h a r m o n i c p e r t u r b a n c e a p p e a r i n g in the waveform of Fig. 6.2.b) is seen to yield no c o n t r i b u t i o n to the W i g n e r integral at the s a m p l e d points ql and qa, where the sinusoid overlaps over the vanishing, respectively, left and right part of the signal. In contrast, it contributes to the Wigner integral at point q2, having here a nonzero overlap with the signal amplitude at the left.

The Wigner Distribution Function: Analytical Evaluation

279

cp(q )

I

Q.

Qc

.

q, (a)

I

Qh

Qu

q~(q )

%/k/k/V ~ Q,

ql

q2

q3 Qt,

(b)

F I G U R E 6.2. (a) The Wigner distribution function of the depicted signal does not vanish throughout tile gap extending from Q,. to Qd. Tile left and right part of tile signal about the point ql, for instance, have a non zero overlap, thus yielding a non zero value of 1A2. (b) T h e sinusoidal p e r t u r b a n c e extending through the short interval (Q,., Qa) does not contribute to tile Wigner function at the points q~ and %, whilst it does at the point q2. (Inspired by [5.7])

Of course, the same (:onsi(terations hold for the frequency (tomain, due to the symmetry between the space and frequency domain definitions. Interestingly, by an easy manipulation, expressions (6.2.2) and (6.2.3) are recast in the equivalent forms o

_ qt

~ 2 ( q , h;) -- ~ .

-2i~(q'-q)

(6.2.5)

W(q, ,,:) - ~ f ~* (2,~ - ,~')~(,~')~('~'-'~)d,~', clearly evidencing the convolution nature of the Wigner integrals. In particular,

w(q, o) - 1 J ~, (2q - q')~(q')dq' 7"f

w(o, ~) - 1 ]" ~* (2~ - ,~')~(,~')a~',

(6.2.6)

saying that the profiles of the Wigner distribution function on the q and ~ axes equal (apart from i ) the autoconvolutions of the signal and its spectrum. The Wigner distribution function can equally be defined as a function of tile canonically conjugate variables q and p. In fact, being ~c = kop (as usual, 71"

k0 denotes the v~cuum w~ve number), the integrals in (6.2.2) ~nd (6.2.3) b~ r ~ t in t~rm~ of p ~,d ~ ( p ) ~ o r d i ~ g to ( ~ ~1~o Eq. (4.6.3))

+ q'2 )e-ikopq' dq' }V(q, p) - ~1 ~ ~, (q - 9J_)~(q 2 27r

2

2

can

(6.2.7)

Linear Ray and Wave Optics in Phase Space

280

where F(q) and @(p) are linked by the Fourier integrals (4.6.3). We win mainly u~o d~i~ition~ (~.e.e) ~,~d (~.~.3) i~ torm~ of tho p, ir (q, ,~). 6.2.1

The W i g n c r distribution of light signals: simple examples.

We give a I)I'i~mry ilh~stration of tim va.h;n(:e ()f the Wigiler distrilmtit)n tim(:t,i()~ t,r (:()~vt'~y a,~ i~d,r162 signal (h;s(:rit)ti
wllr

Wig~r

r162

~,,,, (q) - a(q _ q,,).

(o.~.s)

~')('r

t,() t)r a b-fl~(:ti,)~ a.s well:

fl~(:t,ir

1 4 ; (q,

>,.. (q..)

_ 5~(,~

_ q..).

(6.2.9)

iIl(l('t)r162 r t," a,llr flllly r162162 ahnlg I llr lillr lmralh;11,r162 h-axis at q = q,,; all tl~(, frr art; I)res(;ld,
~,,

(q) - ~ l ~''ih;~ ,

(6.2.10)

is descri}~r in ray (q)tit's a,s a (:<)lle(:ti(nl ()f t)ara.lh;1 rays a,ll tm)t)a,gating in the dii'e('ti()Ii Po = t%/]%, whi(:hever })e the st)atial (:()()r(tina.te. T h e W i g n e r distributioli fun(:l, itm lN~.~ (q, re) is easily ()bta.ine(t as _

~(~:

),

(6.2.11)

which r e p r o d u c e s the ray picture of tile optical d i s t u r b a n c e (6.2.10), and ac(:ordingly individualizes on the (q, ~:) plane a straight line parallel to the q-axis t h r o u g h the point t~:- ~:o- T h e frequency s p e c t r u m ~ o (~') - ~ ( h : - ~ 0 ) reveals t h a t only the frequency t~,o is present in the signal, whilst tile flmction W% (q, ~) provides the f u r t h e r i n f o r m a t i o n t h a t ~o is present at all points. It is also interesting to consider a spherical wave, p r o p a g a t i n g in a uniform m e d i u m . As n o t e d in w 5.2.1, a spherical wave is r e p r e s e n t e d in the linear

The Wigner Distribution Function: Analytical Evaluation

no

FI

281

tc=k q R

/I

q

J

(a)

(b)

FIGURE 6.3. Tile Wigner distribution flmction of (a) a spherical wave paraxially crossing the reference plane II at q into the direction 0 _~ q/R is (b) a line delta, whose trace in the Wigner plane (q, n) is the straight line n = kq/R. p l a n a r model by the q u a d r a t i c - p h a s e waveform ~s(q)-

c i ~ q2 ,

(6.2.12)

where R d e n o t e s the c u r v a t u r e radius of the para,bolic wa,vefront a,t the z-plane, which p~(q) is referred to, and k = n0k 0 is the wave n u m b e r in the m e d i u m . C a r r y i n g on the calculations prescribed by (6.2.2), the W i g n e r d i s t r i b u t i o n flmction of the signal (6.2.12) takes the form l/~2s(q, ~,) - (5(h; - / , : ~ ) ,

(6.2.13)

yiehting a, definite rela.tion of the st)ace ('ooI'(tina, te q to the fi'equency ~; (i.e. the direction of prot)agation) of the wave. At the point q in the reference plane one finds the only frequency t~ - k_q R, in accord with the ray optics r e p r e s e n t a t i o n of the spherical wave, as the ray crossing the reference plane at q p r o p a g a t e s in the pa,ra.xial direction '0 ~ ~. We reca, ll tha,t in the wa,ve-optica,1 picture the rays a,re u n d e r s t o o d as the norlnals to the wavefronts. T h e sut)port (b) of }Vs in the W i g n e r plane (q h:) is evidently the straight line ~; - kq (Fig. 6.3). R As a conclusive example, due to its relevance a,s b o t h a, mathema, tical model and a,n analysing ~nd synthesizing tool for optical as well a,s not strictly optical signals, we consider the G a b o r fimction _ ~ )2 %) ~G(q) - ( 7 ~2)1/4e 2-X-~(q-q~ +i%(q- ~ ,

(6.2.14)

where a is a positive quantity. As e m p h a s i z e d in w 4.6.4, the above fimction describes an optical signal with an intensity c o n c e n t r a t e d a r o u n d the point q0 b We recall that the support of a function is defined as tile region where tile function takes on non negligible values.

282

Linear Ray and Wave Optics in Phase Space

t h r o u g h a Gaussian profile, whose width is m e a s u r e d by a. T h e fl'equency spect r u m @~ ha, s the same a.nalytical form as ~ , the Fourier conjugate va,ria,bles q and ~ being interchai~ged. In fact, we obtain the Ga,ussia.n amt)litude

~G (h;)

--

( ~a2

)1/4C.-"~-~(t''-t%)~-iq~

2

(6.2.15)

(:e~tere(t aro~m(t t,l~e fl'(:(lll(:l~(:y h;o, wl~()se wi(tth is ~easl~re(l t)y 1/(~,. Tl~e W i g n e r (lisl, ril)~ti()l~ fllll(:t,i()ll )/~G(q' h;) t'(:a,(ls a,S (q..)

-

;1 (,-.-~(q--qo

I~t,(',r~;st,i~gly it, (:(~d)i~;s l,ll(' f(m,t,llI'~;s (~f r

(6.2.16)

--~

m~(l ~ ;

i~ flu't,, we' s~;~; t,lmt,

N()tal)ly, it, is l)(~sit,iv(; (;v(,rywl~(;r(; ix~ l,l~e (q, ~:) I)lm~(;. Sl)(;('iti('ally, it, is (~aJxssia~ ixx t)()l,l~ q mx(l ~" a,r()lx]~(l l,l~(; Sl)a(:(',-fr(;(px(n~(:y I)()ilg (q,~,~',~) t,l~r()~xgl~ wi(tt, hs al)l)r()xi~m,t,(4y ()f Ct, m~(l l/a,, t,l~s (:(n~v(;yil~g t,l~(; vi(;w ()f t,l~(; (laid(n" sigl~al a,s a, kil~(l ()f a "wav(; 1)at'kag(:" l()(:at,('(l al, l)()sit,i()]~ q,~ m~(l Sl)al,ial fl'('(l~l(:]~(:y t~',).

6.3

The Wigner distribution function" basic properties

Ill this se(:ti()n we (h's(:ril)r ill S()lll(; (teta,il a, mmflmr (~f l)r(~l)ertir (ff the Wigner (lisl, rit~l~ti(m flu~t:t,i(~ [1o1. Wr will s(;r that W(q,p,) is a rr162 fl~n(:tit)n, inva,rianl, t() t,ra,uslat,i(nls ml(l reversil)le. T h e siglml ('aal 1)r re(:~)llst,rll(',t,(;(t fi'()m it, ut) t,(~ a (:OllStmlt, l)tm,s~; fa,(:t,(n'. 'l'll(; Wigller (tist,rit~llt,i(nl fllll(:t,i~ll yiehts the ('(wre(:t llm,rgiilal (tistritnlti(~lls fin" mW a rbitra,ry siglm.1 a.ll(l llas fillit(: slq)t)ort t)rot)erty. However, ill gelleral it (l()(;s ll()t b(;lm,v(; a,s a, 1)r()l)er (;~(;rgy density fi~n(:ti(m, |)e(:a,l~se it, is ~()t, everywl~ere t)()sitive t,l~r(~gl~ 1,1~(; Sl)a('e-fi'eq~mncy doma,in. Also, t)(;ix~g a. fiu~(:t,i(m (~f two w~,riabh;s, (lerive(t fi'(m~ a. fiu~(:t,i()n of one varia, t)h;, i.e. the signal a,~q)lit,~ule or the (:orrest)o~(li~g fi'C(lUe~(:y spectrum, it ImlSt satisfy terra,in (:(mr a n(t thereby ~()t, ev(~'t'y t)ha.sr st)ace function config~res as the W i g n e r (tistri|mtion flm('tion of s()me signal. As r e m a r k e d in the Intr()(lucti(m, the formalism rela.ted to the Wigner distribution function is widely use(1 within the qua,ntui~ n~echa~fi(:aJ context a,s well a,s within the general signal theory. Therefl)re the (:tm.ra(:terizations, we report below to identify tim t)rol)erties of the Wigner distrit)ution, resort to the la,nguage of b o t h prol)at)ilisti(: and signal the()ry. Firstly, we note tha,t t)y a,pt)li(:ation of Schwarz's inequality to the Wigner integraJs (6.2.2) and (6.2.3), a,n estimation of the u p p e r limit for the absolute value of the W i g n e r distribution function obtains as

IW(q,~)F ~ I~ ~.

(6.3.1)

The Wigner Distribution Function: Analytical Evaluation

283

Reality and s y m m e t r y p r o p e r t y

6.3.1

The Wigner distribution function is always real irrespective of that the function p is real or complex:

w ~ (q, ~) - w ~ (q, ~)

v ~.

(6.3.2)

It is also invariant with respect to space and frequency reflection of ~, i.e.

if

if

~(q) --~ F ( - q )

then

W~(q,t~) ~

-~

then

W~(q,n)

~(K,)

~(-n)

-~

W~(q,-t~),

(6.3.3)

W~(-q,n).

Besides, the Wigner distribution of a real function, ~(q) = ~*(q), is a,n even function of n, as the Wigner distribution of a function having a real frequency spectrum, ~5(t~) = @*(t~), is a,n even function of q. Since a real function possesses a spectrum, which turns into the complex conjugate under frequency reversal: ~ ( - ~ ) = ~*(t~,), and similarly a real s p e c t r u m corresponds to a function with the reflection property: F ( - q ) - F.(q)(C), we can synthesize the s y m m e t r y properties of the Wigner distribution function in relation to the s y m m e t r y properties of the signal and its frequency spectrum as

if if 6.3.2

~ r~l" ~ ( - ~ ) - ~*(~) ~ ro.]" ~ ( - q ) - ~*(q)

th~o then

W ~ ( q , - ~ ) - W~ (q, ~), W~ ( - q , ~:) - W~ (q, ~;)

(6.3 4)

Shift and m o d u l a t i o n p r o p e r t y

Tile Wigner distribution flmction is shifted in st)a,ce and frequency just as are shifted the signal and its frequency spectrum. The reader can verify that

if

~(q) -~ ~(q - q0)

th~o

W~(q, ~:) ~

W~ (q - qo, ~:);

if

@(n) --~

@(t~-h;0)

then

W~(q,~)

W~(q,t~-t%).

-~

(6.3.~)

As illustrated in w 4.6.1, a space shift of the signal corresponds to a linear modulation of the spectrum, and conversely a frequency shift of the s p e c t r u m corresponds to a linear modulation of the signal; namely

if if

~(q)-+

~(q-qo)

~(~) -+ ~ ( ~ - ~ 0 )

then then

@(n) ~ @(t~)ci~q~ cp(q) --+ cp(q)ci~o q.

(6.3.6)

Combining space and frequency shifts, we may see t h a t the Wigner distribution of the sp~ce and frequency shifted function

r

- ~(q - qo)ei~o q

(6.3.7)

c Functions with the reflection property: f ( - x ) = f* (x) are referred to as Hcrmitian, or also symmetric, oil account of the consequent symmetry of the modulus: If(-x)l = If(x)l.

Linear Ray and Wave Optics in Phase Space

284

is just the Wigner distribution of the original flmction ~(q) equally translated t,hro~gh the (q, ~:) plane to the point (q0, t%) as YVv, ( q, ~: ) = YV~ ( q - q,, , ~: - t% ) .

(6.3.8)

The 1)ehavi<mr <)f tl~e Wiglmr
6.3.3

Firtitc

mtppo't't p'~YW:r'tfl

The Wigner <listrilntti()~ fl~(:ti(m refl(;(:ts t,l~(; Sl)a(:(; a,~(l fl'e(l~lt;~(:y rang(; limit,ati<)ns <)f tl~c, sit4~ml. h~ fi~<',t, if t,l~(; flu~('t,i(n~ is lil~fit,c(1 t,()a ('(;rtai~ Slm.(x; i~t,('rval m~ t, ll(; sa~l(; i~lt, tTval, l,ik(;wis(~, if t, li(' fI'(;(lll(~ll('y Sl)(;(~t,I'lllll is ~l<)lizcI'O ()~ly ()ver a. ti~il,(; fl't;tt~tt;~t'y i~tt;rval, tl~(; Wig~t;r i~t,t'gral (6.2.3) is (lifl'(;rent fr<)~ zer()<)~ly witl~i~ t,l~(~ s;u~(; fI'(;(t~(~(:y i~t(;rwd. I~ sy~d)()ls, we (:a~ writ(;

if if

r -- (), q it (q,,, q,,) ~(h~)- (), h it (h ,h,,)

t h e n ]/V+(q,h) - (), q i~ (q,,, q,,); t h e n l/V~(q.h.)- (). h r ,h,,,).

(6.3.9)

~l'll(; a t)<)ve (;xl>r(;ss a (listill(:l,iv(; l)r<)t><;rl,y (~f l,ll(' Wigll('r (listril)lll,i(nl, wld<'ll is r(;l)()rte
Non

linearity

The Wigner
(/5~(q) -- ~1 (q) Jr- (/5)2(q)"

(6.3.10)

The Wigner Distribution Function: Analytical Evaluation

285

From definition (6.2.2) it follows that the Wigner distribution of p reads as

/

)&ha(q, ~) -- Wha, (q, ~) + lA2ha2 (q, ~) 1 . (r , ( q --~--2~{~--~

q')~2(q + q')e-i~q'dq '} , 2 2

(6.3.11)

whose non linearity in relation to superposition (6.3.10) is clearly manifested by the term enclosed between brackets (the symbol ~ signifying the real part of the argument), which adds to the two signal-components arising t e r m s ~Vha1 and W % . This term is referred to as the cross-(or, rnized-)Wigner distribution function of the functions (r 1 and ~2- Denoted by Whal, % (q, ~), it generalizes integral (6.2.2) as displayed in (6.3.11), i.e by the expression [4.4, .9] Whal 'ha2 ( q' ec) - ~1

, qt ~1 (q -- q'2 )~2 (q @ -'2 ) C-i~q'

dq I

(6.a.12)

which, of course, with (~1 --- (~2 = (~ turns into the expression (6.2.2), often referred to a,s the auto-(or, self-)Wigner distribution function of ~ (see g 9.3.2). Needless to say, l/YhaI ,% (q, h;) writes in terms of the pertitmnt frequency spectra, @1(~;) and ~52(K,) in the plain form . 1 -* ~;' ~' ) gin' + (6.3.13) The integrals in (6.3.12) and (6.3.13) (:an be given the same visual interpretation a,s the auto-Wigner integrals (6.2.2) and (6.2.3). Clearly, the folding of the signal at position q (or frequency h;) involves one signal (or frequency spectrum) component on the left and the other component on the right. Thus, we ca,n express the Wigner distribution function of the composed signal (6.3.10) in terlns of the auto and c'ross-Wigner distributioil flmctions of the two components Pl and ~2 according to ~e~a (q, ~;) -- ~'?~aI (q, 15;) + Wqa 2 (q, Pi~) -11-~]~31' qa2 (q, ~i;) + ~V~a2, hal (q, K,),

(6.3.14)

which turns into expression (6.3.11) on account of the easily provable property )/~2"qa2, ~31 of the cross Wigner distribution function t h a t W~al, ~32 As we will see in relation with Moyal's formula, the cross-Wigner distribution function is used within the q u a n t u m mechanical context as a tool to investigate the interference between q u a n t u m mechanical wave functions, and accordingly the occurrence of transitions between different q u a n t u m states. The q u a n t u m issue is in a sense paralleled by the wave-optical issue of determining the efficiency of light launching into waveguides, for the purpose of optimizing the transfer of energy from the incident beam into a definite mode of the guiding structure.

Linear Ray and Wave Optics in Phase Space

286

A brief comment on the cwss-term behaviour Here we detail the interplay between the self and cross-terms in relation to the overall distribution, after the maturer of [10.3, 10.4]. Relation (6.3.14) can ca,sily be generalized to the (:as(: of a signal (:omposed |)y a, ccrta.in mmfl)cr, s~W N, of (:omt)()nc~ts lo(:a,tc(t a,t (litii:r(:~t region,s in the st)a,(:(:-fl'(:(l~(:n(:y t)la.nc, h~ fat:t, if N -

j=l N

N

j=l

N

j - I /=1 j>~

Ea.(:h siglml (:()llll)()llt:llt ~, (:(nltrilnlt~:s t(~ 142~ witll its (~wll siglml te.rxll ]/V~ , 9

j

a n(t (;a,(:ll Imir (~f siglml (:~nlq)~nlel~l,s (,~ , ~ ) l)r,~(l~l(:es a (:(n'r(~sl)~(ii~g (:r()ss t(;rn~, giv(;l~ |)y tl~(; r(;a,1 l)a.rt (~f tt~(: (:r()ss ~Vig~(;r (listri|nlti(~ fi~(:ti(~ W%, ~ ()f siglml (6.3.15) will (lisl)lay. N sigl~a.1 t(,.l'lllS a,ll(l (N2) (:r()ss tt;l'~s ((:~lst()~m,i'ily. l,('n~(;(l i~ l,l~; sig~a,1 tl~(;()ry (:()x~t('xt as i'nte'tfi:'rence I,(WIlIS).

the s(t~a,re(l l~l(~(t~l~s ()f 1,1~; (:r(~ss Wig~mr (tistril)~ti()l~ fiu~(:ti(n~ ]/Y~ ,~ (q, h;) is cxt)ressil~l~; i~ l,~'~rl~S()f 1,1~; m~l,(~ Wig~mr (listrit~ti(n~ fi~(:ti(n~s ~f tl~(', i~wolve(t signals, i.('~., 142~j a,n(t 142~. Ext)li(:itly, after s(~nl(; a,lg(;ln'a (~n(~ fi~l(ls II/V~,., ~, (q,h;)l 2 -

'

W~ ( q - ~

2--~

.

hb/

~

h:--ff)l/V~(q+~

/~/

~

h'+-ff)dq'dh;'

~

(6.3 17) 9

giving W ~ , ~t a,s the st)a,(:~; a,n(t fl'c(tlmn(:y (:(mvohlti(m ()f the Wign(;r (tistribulions of tim in(tividua,1 signals. Tile 2D integral on the right rcpro(hl(:es with respect to tim space and frcqucn(:y wl,riablcs the structure of the cross-Wigncr intcgra,1 (6.3.12) at zero frequcn(:y. Hcn('c, resorting to tile hinge-like picture of the self-Wigner integral (6.2.2) (:la,borate(t in w 6.2, it (:an bc visualized a,s arising at any point (q, ~) in the Wigner plane by summing up the contributions of )/V~j and )/V~ respe(:tively at points lying symmetrically to tile chosen point (q, ~;). Thus, if )d;~j and 142~ sharp around the points (ql,/~1 ) and (q2, ~2) ii1 tile (q, ~) plane, the cross contribution 142~j, ~t (q' t~) develops mainly in the surroundings of the midway point (qm Nm) with qm '

~

~

ql +% and /~m 2

~1-+-~2 2 "

The

WignerDistribution Function: Analytical Evaluation

287

F I G U R E 6.4. G e o m e t r y in the W i g n e r plane of the self and cross t e r m s in the W i g n e r distribution flmction of the stun of two identical signals centered on (ql, ~,) and (%, t%).

Specifically, let us consider a signal ~h(q) composed by two identical copies of a model signa,1 p(q). We suppose that p(q) is mainly concentrated around the origin of the space-frequency plane, and that the two signal replicas are shifted in space and frequency to the points (q,, ~,) and (%, ~ ) . So we write --

-

ql ) c

-Jr-

-

q2 ) C'iK2q ,

(6.3.18)

a,nd accordingly,

)/Vr (q, h;) -- ~;~,(q--ql,~ -- /"~'1) "31-)"V~(q--q2,~-- ~2) -11-2 COS[/'~d (q -- qm) -- qd (h; -- h;m) + h;dqn,]~V ~ (q -- qn,, t~ -- h;m).

(6.3.19)

Here, 142 is the Wigner distribution flxnction of the model signal ~; qm and /~m are as defined before, whilst qd' /fi~dare tile difference coordinates: qd --- ql --q2 a n d / ~ d = /5~1-1~2" The cross-term above has been obtained from an appropriate manipula,tion of the cross-Wigner integral (6.3.12) explicitly involving the two spatially and spatial frequency shifted versions of the model signal. The geometry of the terms entering (6.3.19) is illustrated in Fig. 6.4. As expected, the two signa,1 contributions in W e are located a,t the regions in the space-frequency plane where the relative signals are located, whilst the cross contribution lies midway between the signal terms. Moreover, the envelope of the cross term shapes just as the Wigner distribution W~ of the model signal with a double amplitude, but is modulated by an oscillation occurring transversely to the line through the centres of the self terms, with a frequency governed by the separation between those centres. Although established in the specific case of a signal like (6.3.18), i.e., composed by two identical waveforms, the behaviour exemplified in Fig. 6.4 is quite generally displayed by the self-

Linear Ray and Wave Optics in Phase Space

288

and (:ross-terms of the W i g n e r distribution function of a composed signal, even when the signal terms have not the same form. Specifically, the g r a p h in Fig. 6.4 refers to a G a u s s i a n - s h a p e d model signal (see w 6.4.5). It is evi(tent fl'onl (6.3.19) t h a t even t h o u g h the siglla,1 ternls are t)ositive everywhere thr(mgh th(,~ st)a.(:e-fl'e(tlu'~n(:y t)la.ne, as it, ()(:(:lu's in the (rely (:ase ()f a Gmlssiml-stm.t)e(t sigilal, the ()s(:illa,ti~lg |)ehavioln" ()f tlle (:ross ternl unav
~1

, w

....... ,, ....,(q, ~-1 -

/.7'

W(q',

~-')C(q - q', ~ - ~ ' ) , t q ' , t ~ ' ,

(~.:~.2())

wl~i('l~, ()f (:(n~rs('~, lllllSt ()l)ti~mlly t)e (l(:sig~(;(l a(:t:()r(li~g t() t,l~(: l'ett~ir('l~('a~ts of l,lm t)r()l)l(;~ ,)f illI,(;rt;s|, ill ()l'tler I,() I'(;(lll('(; |,ll(; (tist~rl)i~g ('r()ss 1,(;r~s witl~()~lt atfe(:ti~g 1,()() ~nt(:h i,l~(: sig~ml l,(;n~s, m~(l s() i~ gen(;ral wil,l~()~t alt(;ring too Inth;t;(l, altl~()~@~ n~,)stly t)ften ~mtlesire(1, tl~e (:rt)ss l,(;r~s (:arry l~()wever sig~lifi(',ative i~lfoI'~m,tio~ I'ega,r(lixig the I'elat, i()~s })etweel~ t,lic sig~ml ('()~l~l)onents, as sh()w~ l)y (6.3.17), wl~i(:l~ tra,(:es tl~e ~<)(hfl~s ()f l,lm (:r()ss l,(;r~ I)a,(:k to the (:()rrest)(m(lil~g signal temps, lV()r instal~(:e, they all()w ()x~(; I,()(lis|,i~g~fisl~ 1)etween a, meaxfix~gfl~l aa~(t a. fi(:titi()~s (te(:()~x~t)()sition ()f a, sig~al i~d,() (:()~l)()~ents. In fact, if ()~(; (;xt)r(;ss(;s a sig~ml a,s the s ~ of tw() i)arts a,s "(,(q) = a ~ ( q ) + (1 - a ) ~ ( q ) ,

the relevald, Wigner (listrit)llti()n fimcti(nl turns to |)e

w~ (q. ~-.) - I..12w~ (q. ~)+11-a.12w~ (q. h~) +2!s

(1-a)]W~ (q. h:) - W~ (q. t~).

(6.3.21) and no oscillations arise in the (fictiti(ms) cross distribution as a modulation of the signal generate(t envelope W~. Thus, inspe(:tion of the s t r u c t u r e of the W i g n e r distribution fimction can be an effective tool to individualize multicon~t)onent signals, i.e., signals which in the spa(:e-frequency planes are well delineated into definite (and possibly intersecting) regions [10.5, 5.5, 5.7]. Indeed, reliable suppression or reduction of the cross-terlns must wisely be designed in order to preserve the information a b o u t the various independent space-frequency "entities", which m a y compose the observed signal.

The WignerDistribution Function:Analytical Evaluation 6.3.5

289

Marginal distributions and inversion formulas

The Wigner integrals (6.2.2) and (6.2.3) manifestly display a Fourier transform structure. Expression (6.2.2) in fact conveys the Wigner distribution function as the Fourier transform of the product p(q + q')~*(q - q') regarded as a 2 2 function of q' with q taken as a parameter. So, inverse transforming W(q, n) to the q'-domain, we obtain ~(q + q'2)~* (q - q')2 -- / W(q, n)ei'~q'dn,

(6.3.22)

which, redefining the arguments of ~; and ~*, is recast in the form

~9(q)~* (q') -- f ~)( +9@2t, I~) Cig(q-qt)dl~.

(6.3.23)

In a like manner, by (6.2.3) the Wigner distribution function relates also to th~ i~v~r~ Fourier tr~I,~form of th~ product ~(,~ + ~) ('~ - 7 ) with r ~ p ~ t to ~', thus enabling the Fourier integra,] of t/V(q, ~c) to yield

~, ~ ( ~ -~- -if)

@*(t~; - -if) ~'

-- /

1/V(q,h~)c-i~'qdq ,

(6.3.24)

which by ('hanging the variables writes in a form similar to (6.3.23), i.e.,

#(~)~*(~') - / W(q, ~--~)c-~(~-~')~q~ .

(6.~.25)

Evidently, setting q' = q and ~ ' = ~ respectively in (6.3.23) and (6.3.25), wc obtain on one side the square moduli of ~(q) and @(t~;), and on the other side the integra,ls of W(q, ~) over the ~- and q-va.ria,ble, namely

I~(q)l 2 - f 142(q, re)dt~;, [@(~;)l2 - f )/V(q, t~;)dq.

(6.3.26)

The integral of 142 over the frequency variable at position q yields the energy density of the signal at q. Correspondingly, integrating t42 over the space variable at the spatial frequency rc gives the energy density spectrum of the signal at t~. The projections of the Wigner distribution onto the q- and ~-axis, mathematically represented by the integrals in (6.3.26), are respectively named the space and frequency marginal distributions (or simply, marginals) of 142(q, ~). Indeed, they yield the desired quantities in conformity with the basic properties (6.1.13), one should expect in general for a phase-space distribution having the definite role of an energy density distribution. In accord with the marginal relations, the signal energy contained within a certain space interval [q~,qo] is given by the integral of W(q, ~) over the vertical strip - o c < ~c < +oo, q~ _< q _< qo in the q-t~ plane, as correspondingly

290

Linear Ray and Wave Optics in Phase Space

the content of energy within the frequency interva,1 [t~, t~o] is obtained by integra,ting the Wigner distribution flmction over the horizontal strip - c ~ < q < +cx~, n~ _< h; _< ~o. E v i d e n t l y the integration of W(q, ~;) over the whole q-~; t)lane t)rovides th(: t()ta.l energy a.(:(:or(ting t()

E(tlml,i()ns ((i.3.26) an(l (G.3.27) (:sta,l)lish a. link ()f (,1~(:ra, th(:r a,l)stra(:(, Wigncr fl~l~(:ti()~ wi(,l~ tl~(: ~(:as)n'a|)l(: i~d,(:~sil,y a.~(l I)Ow(:r Sl)(:(:l,r)~ ()f tl~(: siglml, and s() wil,l~ 1,1~(: l,()(,a.l (:~(:rgy ('a,rri(:(l l)y tl~(: ()I)(,i(:a,1 l)(;rl,)~rl)m~('(;. As w(; will see in Cl~a,l)l,(;r 9, (,l~(:y ar(: (:r~u:ial l,()(;sl,a,l)lisl~ (,lint (,l~(: ~()~(:~(,s ()f (,lu, Wigner dis(,ril)~(,i()~ f~(:(,i()~ ar(: (,l)('~ ~()~(:~(,s ()f (,l~(: sig~ml a,~(l i(,s fr(:(l~(u~(:y Sl)(:(:t,r~; ~()n~(:~d,s ()ll'(:r a l~ all,(:r~mtiv(: aa~(l (:ll'(:(:tiv(: (,()()I (,() (:()~v(:y as w(:ll as t()(:x(,ra('t inf()rnm,(,i()~ a,l)()~(, (,h(: sig~a,l, as f()r i~s(,an(:(: in l)a(,(,(:n~ r(:(:()gni(,i()~ l)r()(:(:ss(:s. N()ta,l)ly i(, is I)()ssil)h: (,() r(:('()~s(,r~(:(, (,I~(: Sl)a,(:(: m~(l fr(:(l~U:l~(:y r(:I)r(:s(:Id, ati()ns ,;(q) a~(l .~(t,)()f (,l~(; sig, m.l fr()~ (,l~(: Wig,~(:r f,,,~(:(,i(),~. I,~ fa('(,, wi(,l~ a,ssign(:(l va.hu:s ()f q' - q() m~(l +;' - t,'(), ((i.:).23) m,(l (6.3.25) g/iv(: (,l~(: i,~v(:,'si(),~ f()r~nlla,s .

W('~+%` +.)<.i,,.(q-q,)) dr,',

/'

-

,;* (q())

Q(,,-)

....

.

! ,,.. (,,())

2

"

+

'% ) ,

-iq(,,.-

)dq,

,

fr()~n whi('l~ ,;(q) a.l~(l ,~(h') ~m.y ])(: r(:(:()v(:r(:(! ~II)(,()a. (:()l~s(,a.x~(, [)lm.s(: fa,(:(,or, si11('e (nlly ill(: IIla.gllitluI(:S ()f <;* (q()) a,ll(l V}*(+,'()) (:a,ll))(: kll()Wll fr()lll tll(: squared mo(I1di in (6.3.26). ()n(: iim.y (:()nv(:ni(ultly (:h()()s(: % = () a,ll(| t,'() = (), if of (:(),,t's(: ,;*(()) a,~(l Q* (())a,r(: ,,()(, Z(;l'(). A('.(:()r(lingly, the siglml a.,~q)li(,,u|(: ~2(q) is r(~'tri(:v(:(l fl'()lll ill(' i11v(:rs(: I"()in'i(:r (,rmlsf()rnl with r(:Sl)(:(:t t() ~ ()f ill(: Wigner (listril)llti()ll a,t I)()siti()n ~2" Likewise, th(: fr(:qlu:nt:y Sl)(:t:(,rlllll Q(h') r(:sldts from the Ft)11ri(:r (,ra,1)sf()nll ()f I/V(q, -~) wit]~ r(:sl)e(:t t() q. TI~(: i~v(:rsi()~ f()rm~la.(: (G.3.28) offt:r (:vitlt:nc(: (,hat, (,I~(' Wig~(:r (listribution (:ontaix~s a,s ~nu:t~ i~f()r~m, ti()n a,s t,tm sigx~a,1 itself, |)(:i~g (,t~(: sigiml ~miquely r(:coveral)h: fr(nn it, ~1t) t() a, (:()nsta,nt t)ha,s(:. In ()ther t(:n~s, w(: ~m,y say that the ext)e(:ta,(,i()ns t)i(:torially (:xpressed i~ Fig. 6.1 are sa(,isfa.(:torily fidfilled by the Wig~mr (tistrilmtion fln~(:tion. T h e information a,bou(, the system state are equiva,lently (tist)la,yed by the wa,ve fim(:tion and the W i g n e r distribution, on (l~m a(:(:o~m(, of (:()~rse of tl~(: intrinsi(: lindt, ations of the W i g n e r view, inherent in the inhi|)ited intert)retatioi~ of W(q, ~:) as a true energy density distribution. From the applica,tive viewpoint, relations (6.3.28) a,re the m a t h e m a t i c a l expressi(ms for the inverse problems of signal re(:onstr~u'tion and synthesis t h r o u g h the W i g n e r distribution function, and possibly suitable synthesis functions. Signal reconstruction and synthesis are basically concerned with an est i m a t e of the signal waveform ~(q) from the knowledge of some other signal h',

The Wigner Distribution Function: Analytical Evaluation

291

representation related to F, with the possible aid of auxiliary tools. In fact, the reader may easily prove that synthesis of a signal is also possible by use of a "key" function r (as it is coIiliilonly said in the context of radar/soimr applications), in accord with the relation 7)(q) -- J / r

-- q)lA?4),~(q', t~')ei2E(q-q')dq'd~; ',

(6.3.29)

holding if the key function is normalized to unity. )/Vr is the cross-Wigner distribution function defined by (6.3.12). Thus, according to Eqs. (6.3.28) and (6.3.29), it, is possible to synthesize a signal having the desired space-frequency characteristics. Constructing a Wigner distribution function, or a cross-Wigner distribution function in conjunction with a suitable key function, with the desired attributes, one can obtain tile signal which produces that auto-, or crossWigner distribution, and hence has the desired space-frequency properties. The inference of the system wave function from the Wigner distribution function is of a great relevance in quantum mechanics/optics. From l/V~(q, t~), which is typically reconstructed by quantum tomography (see w 8.4.3), we can numerically construct not only the squared modulus of the wave function, but the very complex wave function of the system under investigation (perhaps a particle). The phase indeterminacy of the inferred wave function is, of course, irrelevant, since, as we know, any wave flmction (:a,n be nmltit)lie(t by an arbitrary t)hase factor without altering the physical state.

A brief comment on partially coherent signals As remarked in ~ 6.3.4 when illustrating the effects of tile cross-terms, the bilinear (h;pendence of YV(q, re) oil tile signal amplitude may be a (tisadvantage in some applications eus, for instance, in pattern recognition, since tile nonlinearity generated terIns may reduce tile readability of the signal pattern. In contrast, it can reveal advantageous in other applications where the field enters the theory quadratically, as in the theory of partial coherence. In fact, the concept of Wigner distribution function with the relevant formalism has been extended from deterministic to stochastic signals, without increasing the dimensionality of the formulae. Roughly speaking, it seems natural, retracing the lines of the definition of the Wigner distribution in quantum mechanics, to replace in the defining integral (6.2.2) the product ~*(ql)7:(%), appropriate to completely coherent signals, comparable to the pure states of quantum statistics, by the ensemble a.verage (7:* (q~)P(q~)}, suitable to describe partially coherent signals, comparable to the quantum statistical mixtures. In effect, as the complex amplitude p(q) is the proper descriptor of coherent signals, the appropriate descriptors for randomly fluctuating perturbances are known, from the optical coherence theory, to be the coherence function F(ql, q2,7") and the cross-spectral density r(ql, q~, ~) [11]. The former describes the spatial correla-

Linear Ray and Wave Optics in Phase Space

292

tion of the tempora,lly sta,tiona,ry stocha.stic signal g~(q, t) a,t pairs of positions ql and q,, as a function of time r t h r o u g h the ensemble average (d) V ( q l , q~,

r)

-

(~/,* (q,, t)'('(q2, t + 7-)},

(6.3.30)

whilst the latter descrit)es the st)atiaJ (:orrelatiou of (,(q, t) as a flnl(:tion of frequency w, arising fl'()~ t,l~(; t,enq)oraJ Fo~u'ier tra,llSfOI'lll of F(q~, q2,7-), i.e.

l'(q,,q.~,~) --

/. I'(q,,q.~,

7-)('--/~rd7 -.

(6.3.31)

D('tiniti()~ (6.2.2) ('.()~d,a,i~s 1)y its(;lf 1,1~(: l)r()(l~(:t ()f 1,1~r siglml wHv(;f()l'lllS in a, ft)rlll rt;sr tlla,t ill (6.3.3()). II('11(:(;, within th(; sl)irit r (6.2.2), it, is na,t~u'al 1,()(l(;ti~(' tl~(; Wig~(;r (lisl,ril)~ti()~ flu~('ti()l~ f()r t)artially (:()Ii(;l'(;lll, light t)y th(; F(n~ri(;r-lik(' im(;gra, l ()f l,l~(; (:r()ss Sl)(;(:tra,1 (l('~sil,y as [12]

l/V(q, ,,) - ~

/. f'(q - ~, q + ~, w)('-i~'q'dq'.

(6.3.32)

Th(; Wig~(;r r ()f t)artially ('()l~crr light is t)('y(n~(l tl~(' l)~rI)~s(:s ()f the 1)()()k; l,l~t; illi,(;l't;st(;(1 r(;a.tlt;r is (lir('(:tc(1 1,() [12]. F~rl,l~(;r (lisr ix i~ .~ 7.5. D(;fi~fil,i()t~ ( ( i . 3 . 3 2 ) is s i ~ i l a r I,()l,l~at s~t~gt;sl,(;(l 1)y Wall,l~,'r as g~'n,c't'alized la.tr 6.3.6

ilwr162162

m~(l rr162162 lw Wr

[~a.e].

Nonpositivity

It ha,s 1)r162 r162162 ixl t,hr ild,rr162 nt)tr tlla.t i~l ()i'(l(;r f()r 3, St)it.COfl't'~ttll(;ll(:y (Iist,ril)llti(ni filll(:tir t() t)e t)rr162 ild,crt)rr162 a,s l,llc siglla,1 (;ll(;rgy (lcnsity ill Sl)a(:(; ml(1 fl'(;(llU'~zl(:y, it, lllllSl, ])(; t)()sitiv(' (l('till('(l ()v(w all tlic st)a,(:cK('(tu('xl(:y (l()llm.ill ml(l, ()21(:(; ild,('gl'a,t(;(l ()v('r (hi(' ()f t,ll(' varial)l(,s, it, llulst yield tllc energy r with r(;st)e(:t to the ()tll(;r varial)le. A(:(:r162 tr (6.3.26), the Wigner (listril)ld, ion flmct, ir sa,t,isfi(;s |,lit; ma,rginaJ requirenl(;Id,s a,s its spa,(:(; a,II(t fr(;quen(:y llm.rgina,1 (tistril)lltioi~s .j~st (;qua,1 the space m,
(/(t, ..)/-

f(t, ..)dr. r

The Wigner Distribution Function: Analytical Evaluation

293

energy distribution of ~ in space and frequency, actually a strict interpretation of W(q, a) as an energy density distribution is not possible. Thus information about how the energy of the signal is distributed in q and ~ can be gained from the marginal distributions of W(q, ~;), but this does not enable the Wigner distribution to be truly interpreted as an energy density distribution, although it is commonly treated and manipulated as an energy density distribution. As Wigner showed, bilinear distributions, which have the correct Inarginal distributions and are at the same time nonnegative defined for arbitrary state functions, do not exist [1.2]. The only waveforms for which the Wigner distribution function is everywhere nonnegative are represented by exponentials of quadratic polynomials, i.e., Gaussian functions [10.6]. Expression (6.2.16), we obtained for the Wigner distribution flmction 142G(q, K,) of the Gabor amplitude (6.2.14), is in agreement with this assertion (see also w 6.4.5). The existence of space-frequency distributions that satisfy the marginal conditions and are as well positive for arbitrary signals has been demonstrated. They are not bilinear in the signal amplitude or frequency spectrum; also, a method to generate them within the general class of mixed space-frequency distributions has been elaborated [10.r]. Notably, among all the bilinear distributions satisfying the Inarginal properties, the Wigner distribution flmction is the closest to being positive, in the sense that it requires the least amount of Gaussian smoothing to become nonnegative everywhere in the space-frequency plane [10.12].

Smoothing as a tool to obtain a positive phase-space distribution function We will not enter the details of the positivity question, addressing the reader to w 8.2.2 for a more accurate account. However, we can gain an immediate feeling for the nonpositivity of the Wigner distribution flmction, examining its value at the origin of the (q, a) plane, which explicitly writes as 14;(0,0)- 1 71-

l

9

/0

~ * ( - q ) ~ ( q ) d q - 2_~[ OO

71"

~*(-q)qp(q)dq].

(6.3.33)

For illustrative purposes, we consider the real function ~(q) = q. By (6.a.a3), the corresponding Wigner distribution attains at (0, 0) a, nega,tive value, being

t4;~ (0, O) --

q2dq 2 -~ j~0~176

< 0.

(6.3.34)

In general, for any signal such that q~(q) > 0 for all q or q~(q) < 0 for all q it happens that 142~(0, 0) < 0 in both cases (Problem 3). In w 6.4 we will calculate the Wigner distribution function of a certain number of optical waveforms. As we will see, for all of them, except that for the Gaussian waveform, the Wigner distribution function takes on negative values over some regions of the space-frequency plane.

Linear Ray and Wave Optics in Phase Space

294

T h e s m o o t h i n g of 142 by an a p p r o p r i a t e kernel W(q, ~)

~ W ...... ,,,~,,, (q, t~:) -- . f f W ( q ' , ~;')~(q - q', t~; - t~')dq'dn',

(6.3.35)

ha,s thor()~ghly t)(;(;n i~v(;stigatc(t a.s a, t)ossi|)l(; tool to yi(;l(1 a, l~O~m('gativc (tistri|)~ti(m fln~(:ti()~. Tl~(; s~()()thing t)r()(:(;(t~rc (6.3.35) t)a,ra,lh;ls, a,n(t a,(',tually (:a,n (:()nq)l('a~wad,, (6.3.2()), ('ad,a,il('~(t t()r('~(l~(:c the (:r()ss-t,(~n~s (',()ntrit)~ti(m. Th(: (]a,llsSiall sinai)(; fi)r tl~('~ sm()()tlfing k('rn('l ~(q, ~:) is a ~ a t ~ r a l (:h()i(:c: -

- 89

wh('xc, ()f (:()~rs(;, 7 m~(l/t at(', SllI)I)()s(;(l 1,() ])('~ l)()sitiv('~. F()r (:()llv(;ni('~ll(X','s sake, tim ~nlll,il)lyi~g fa(:t()r al)()v(; is set t()l~a,v(; .[J" {~(q, t~')dqdh" = 1. It l~as 1)C(;l~ 1)r()ve(l tlmt if tl~('~ (la,~ssim~ k(;n~('~l ((i.3.3(i) is as wi(l(; as ()i" wi(tcr l,lla.II tll('. ~ i ~ i ~ n n ~ ~('(~rt,ai~ty way(; t)a(:k(;t, i.(;., if /3~ _< ,1,

((i.3.37)

the resltltillg (Iistri|)lll,i()ll (6.3.35) is ll()lllmga,t, iv(; cverywlw, r(; ill t,lm t)ha,sci)lml(; [l().l,l]. As w(; will lll(;llti()ll in .~ (i.3.1(), a, 2 D Slm,('(;-fl'(:(tllCll(:y (lmlssiml (listri])~ti()~ Ilk(', ((i.3.:16), satisfyi~g tl~(; (;(l~a.lity i~ ((i.3.37), I,~rl~s t() 1)(; itself a Wigll(;r (list, ril)llt, i()ll fllllr t)rct:is(;ly t,ll(; Wiglmr (list, ril)llt, i()~l flnwt, i(m ()f a, I'r (Imlssiml sigIlal, wll()s(; st)a,tia,1 ml(l Sl)a,tial fr(;(llmll(:y (;Xl,(;llsiollS arc faithflflly r('l)r()(lllr i~ll,() ttl(; r(',st)c(:tiv(; (;Xt(',Ilsi()~lS ()f tll(~ Wigll(',r r (s(;c als() w167 6.2.1-6.4.5). 'l'lnls, tim s11~()()l,l~i~g |)y a ~ i ~ i ~ n ~ ~u~(:t;rtai~d,y 2D

2

_'3__

Then, t)y (6.;3.;36) the
~/~ 2rr

1A2(q', h;')c-~ (q-q')2-~ (~-#)2dq'dh;' .

i',[.i".

~,~2 .I,I. I, . , So* (q, _ q,,)~(q, + q,,)c-~(q-q')~-~(,~,-,,;') 2 ~'- 2i,,;'q" dq" dq ~d~'. '2

(6.3.39) Firstly wc intcgra,tc t)y t~:~ a,s ~/~

t

~ - ~ (~-~')2e-2i*~'q' d~' - v / ~ c - ~

2

tt 2

q -2i,~q",

(6.3.40)

and then rc(tefinc the integration variable to finally obtain ~smoothed (q, /~) _ 2~/~e 4~r2

-~//

f(x)f.(y)e~Ydxdy,

(6.3.41)

The Wigner Distribution Function: Analytical Evaluation

295

where we have set x2

f(x)-

~(x)e-(;~'r+4)~ +('rq-2i~) ~,

~- ~(4-/3~).

(6.3.42)

Expanding the cross exponential in (6.3.41) as (3<)

1 ~nxnyn '

(6.3.43)

n--0

~smoothed(q' 153)can

be recast into (x~

Wsmoothed

(q, ,c) - ~ , o2- ~. q

~

2

~1 ~, f xn f (x)dx

n=0

d

.

(6.3.44)

Evidently, the sign of ~smoothed depends on that of ~; indeed, if ~ is nonnegative tile sxnoothed Wigner distribution turns to be positive. It is evident then positive ~'s amount to relation (6.3.37). GenerMization of the above result to obliquely oriented Gaussians, forinally described by 6(q, ~) - (/31'-432)1/2 1 27r C--2 (')'q2--2~ (6.3.45) the degree of obliqueness of the Gaussian "bell" with respect to the phaseplane axes being measured by a, has been proved under the condition that ~ , _< a 2 + 4.

(6.3.46)

Needless to say, this inequality correspoilds to (6.3.37) for vanishing vMues of the obliquity parameter a. Of course, in order that (6.3.45) be a 2D Gaussian, ~, ~ > 0 and ~ 432 > 0. Also, note that again f f G(q, ,c)dqd~ = 1. The reader may verify the above assertion retracing the steps of the proof previously elaborated in regard to the kernel (6.3.36). In particular, if the parameters in (6.3.45) are chosen to satisfy the equality in (6.3.46), the resulting 2D Gaussian function is interpretable as a Wigner distribution function; specifically, the Wigner (tistribution fimction of the comt)lex Gaussian amt)litude _q2 (~ pg(q)

_

(~_~)1/4e -~(1

-i-~).

(6.3.47)

Evidently, one may regard (6.3.38) as the amplitude profile of a Gaussian beam at the waist with beam radius equal to v ~ , and accordingly (6.3.47) as the amplitude profile of a Gaussian beam at a plane different froin the waist, where specifically the wave front curvature is ~ and the beam radius is V/-~. The smoothing by a minimum uncertainty Gaussian has been originally introduced by Husimi in his extensive investigation of the formal properties of the density matrix [10.13]. Accordingly, the phase-space distribution resulting

296

Linear Ray and Wave Optics in Phase Space

from the convohltion process (6.3.35) by a, m i n i m u n i u n c e r t a i n t y Ga,ussia,n is frequently r e p o r t e d as the H~simi distribution flmction. It is one of the basic t)hase-st)a,(:e~ (tistril)llti(ms of (tua.ntunl opti(:s, (:orr(;st)()n(ting to the n o r m a l ordering of the })oson (:rea,/,ion and a,nnihila,ti(m ot)erat, ors. T h e s m o o t h i n g t)r()(:(;(ture imt)lie(t by (6.3.35) t,ra,nslat,(;s in an (;l(;ga,nt, and a,(tat)t,a})l(; fornl I,}l(; naive i(t(;a, of (:()a,rsely r(;t)la,(:ing t,ll(; wdll(: ()f the W i g n e r (lisl, ril)lll, i(nl fllll(:l,i()ll a,I, (;a,('}| l)()illI, ill t,}l<; I)}la,s(; l)]ml(' wil,}l l,}l(' va,111(' ()f t,h(; im('gral ()f W ()V(~,I"il. l)r()l)('rly (:}l()S(;~l r(;gi(nl a r()llll(l l,}m,l, t)<)ild,. Ill a s(;lls(; l,his is illst)ir('(1 })y l,ll(' llll(:(:rl, a,illl,y l'(;la,I,i()ll ()f F()lu'i('r mmlysis, wllit:ll is (:(nlnllonly int('II(ted t() lira,k(' tll(: "I)()ild," iIl I)}m,sc t)lmle a, t)}lysi('a,lly lll(;mli~lgl('sS ('~ltity. Th(' single t)()ild,s a,r(; ll()l, I)rm:l,i(:a.lly a,('('(;ssi})l(;, l,]l(; l~fil~i~nu~ siz(; ()f l,ll(; t)has(;t)la,n(' r(;gii(m l,l~a,l, (:a~ 1)(; r<;s()lv(,(l l)(,~i~g ('('l, (:l()s('ly tt~(; Wign(;r <list, r i l ~ t i o n flln(:l,i()li |)elm,vi()~u" in Ke(p~en('y, re(l~(:ing in 1,11rli t,l~(; rcsolvibilil,y ill st)a,('e. ~I'1~(; ~ i ~ i ~ ~ llll(:(;l'(,a,illI, y (]a,llSSia,ll (lisl, r i l ~ t i ( ~ I'('~])I'(;S(;IlI, S t,he ()t)ti~n~ni (:()l~lI)l'()lliis(; 1,()<:<)~r tlic a,(:t'llra,t:y rt~r s witl~ tile limita,ti(n~s imt~os<;
The

WignerDistribution Function: Analytical Evaluation

297

performed on the signal is invariably affected by the measuring apparatus. 6.3.7

Moyal formula

The Moyal formula relates the overlap of two functions to the overlap of the corresponding Wigner distribution functions. We consider two functions ~1 and ~2 [4.2, 12.1, 4.4, 5.3]. The inner product of their respective Wigner distribution f u n c t i o n s WqpI and 142~2 writes as (Wcp I

,

Wcp 2 ) -,

/]

W*

qo1

(q, /~)W~p 2 9

(q ec) dqdn ~

(6.3.4S) ,

the asterisk being actually uninfluential because of the realness of t42~1. Inserting the explicit forms of "~1 and 142~2 according to (6.2.2), after an easy algebra we obtain the expression * q2' ) ~ 2 ( q - q2' ) ~ ( q + q')dqdq', (6.3.49) (~2~1,~2~ 2) -- ~1 J ] ~l(q -+- q'2)~l(q-2 which, by a suitable change of variables, turns into the form as the square modulus of the inner product of 7:1 and 7:2- Explicitly, we have y.

(W~ , W~2 ) -- ~1 /, / dq~,*(q)~2(q)

2

1

2

- ~ l(~,,~2)l ,

(6.3.50)

thus linking the overlap of the filnctions 7:~ and (P2 to the overlap of the respective Wigner distribution fimctions. Result (6.3.50) says that the average of one Wigner distribution fim(:tion with another Wigner distribution function is always nonnegative. In particular, if (7:1,7:2)- 0, the overlap integral of W ~ a.nd lN~2 vanishes:

J'W* (q, /~)~ 2 (q,t~)dqdt~ - 0 if ((Pl, (~2) -- 0.

(6.3.51)

As the equality above cannot be satisfied if both W~I and W~2 are positivevalued everywhere in the Wigner plane (q, a), it represents a further proof of the fact that the Wigner distribution function can attain negative values. By use of Schwarz's inequality: 1(7:1,7:2)1 -< I]Flll" liP21[, and relation (6.3.27), Eq. (6.3.50) gives

Jf

)/V~I (q, n)W~2

(q, n)dqdn < ~1 ff ~ 1 (q' .c)dqdn jj" W.2 (q, n)dqdn,

(6.3.52) which can be regarded as a kind of Schwarz's inequality for the Wigner distribution. The equality holds if and only if 7:1 = 7:2 = P, or more in general if the two functions differ by a complex constant; in that case, we obtain

IIw l 2 - J / W (q,

)dqd - 1 I[ ll 4 -

1 (

j

/ (q) t 2)dqdt

(6.3.53)

298

Linear Ray and Wave Optics in Phase Space

which relates the n o r m of the flmction to t h a t of the c o r r e s p o n d i n g W i g n e r d i s t r i t m t i o n fi~nction. In a, sense, (6.3.53) m a y be t h o u g h t of a,s a, kind of Pa,rseval's relation for the W i g n c r transforln, in the light of its intert)retation as tim Fo~rier tra,nsfonn of the signal intensity or the t)ower st)e(:tr~n~l. W(; emt)hasizc t h a t M()yal's rela.ti(m, in l,ll(; ft)rnl rr in (6.3.5()), is a. I)a,I't,i(:~fla,r (:as(; of tim ~l()r(; g(;~l(;I'a.1 f()r~nfla. _

I

/

*(q)~c,e(q)dq

/

O, (q)c/52(q)dq

l

).

wlli(:ll illw)lv(;s fi)llr siglla,ls a.ll~l ll~;ll('~; I,W~ (:r~)ss-Wigll(:r ~lisl,rilnll,i~l filll(:t,i()ns. T h e rr is i11vit,c(l I,() l)r(Jve l,lle al)()vc f()rllnlla, ltsillg (l(;tillil,i(nl (6.3.12) f()r tt1r cr{~ss-Wigllcr flnl(:t,i(nlS W a.11(l W r !

"4' 1

~c2 ' 'h2

"

M(~yal's r(;lal,i~nl (lire(:l,ly illl,r~(lll(:(;s llS i111,()a, n(;w (lira,hi,lUll lll(;(:lmld(',al inl,crt)r(;l,al, i~nl ~)f 1,11c s['alar l)r()(lll('l, ()f l,w~) wav(; flu~('l,i~n~s i~ llill)(:rl, Sl)a,(:{;, as l)r()t){)s(;(l ill [10.22, 1().23]. F()r ill~tsl,ral, iv(; In~rI)()s(;s, w(; l~my (:(nlsi{h;r l,w~ (t~mn1,~1 I)m'l,i(:l~;s, wl~()s(; sl, a,l,(; l)ei~g reSl)(;(:l, iv(;ly (h;s(:rilm(l l)y l,l~e wav(: fl~l~(:l,i~nls, v/-- 1

()f ()11(; sl,a,l,(; ()1~ 1,11(; ()l,ll(;l', lkh~yal's f ~ w ~ d a s~l)sl,il,~l,(:s l,l~(: s('a.la.r l)n)(l~r {)f , r a~l~l ,f2, i.(', 1,11(; ()w;rla I) i~ 1,11(; ('{)l~tig~ral,i()11 Sl)a(:('~, l)y. t,l~(; ~)v(;rlal) ~f l,hc (:~wr('sl)~n~(lil~g Wigl~:r (lisl, riln~l,i~n~ fi~(:l,i~n~s ill t)l~as(' Sl)a,:(;. S~, (n~ ~n~c si(tc it, (:r l,l~(; (;(l~iwd(;~(:(; (~f l,l~; r ~l(;(:lm~i(:s f(~n~t~lal,i(~ ix~ l,(;I'nls {)f S(:hri~{ling~;r way(; fi~(:l,i(nl aal(1 Wigl~;r (tisl, ri|)~l,i()~l fi~n{:l,i(nl. ()~ l,hc ~)l,hcr si~te, si~:e i~ (t~ta~ll,~u~ l~e~'l~a,~i~:s 1,11(; Wign~;r (lisl,ri|)~ll,i(n~ fi~l('l,i~)l~ ]/V~ (:a,n t)(; inl,('q)rcl,(:(l as 1,11(: l)I'()lml)ilil,y l,lml, 1,11(; t)arl, i(:l(; i~1 sl, a.l,(; ~2 lla.s a st)(;(:ifi(: (:o()r(lilml,(', ml(l a st)(;('iti(: 111()I11(~,111,11111,I,}IC t)r()(l~l(:l, )TV~, ] A 2 , (;~ll,(;ri~g (6.3.50) is 1,11(; t)r()lm,l)ility l,lm.l, 1,1~(; l,w() t)a,rl, i('h;s, ill 1,1m (:()l~si(l(;r(;(l sl,a,l,(;s, }~a,vc tim sa,nl(; st)(;(:iti(: l)()sil,i()~ ml(l 1,11(; sa,nl('~ Sl)(;(:iti(" nl()m(;~l,~l. Tll(;r(;fin'(;, 1,11(;integral ()f t h a t I)I'()(tuct ()v('r t|m w|l()l(; I)]las(; st)a(:(; ix a, lll(~,it.SllI'( ~, ()f t,ll('~ tm)ximity of 1,t1(; l,w() sl, a,t(;s, g~ a.~l(l g)2' il~ t)lm,s(', st)a,(:e , a,n(t s()r l,t~(', "i)r()i)(;~sity''. of the t,w() tmrti(:lcs "to have the sanle t)()sit, ions and the stun(; nl()~l(;~lta" [10.22]. Also, sllifl, i~lg one ()f l,lle states, say ~2 in bot, h t)osil, ion m~(t ~l(n~l('ntum by st)ecifi(: a,mo~mt, s, t,he r(;lcva,nt W i g n e r (tistri|)~t, ion flin(:tion is eq~m, lly shifted in t)ositi(m and mom(;ntunl, an(t h(;n(:e t,h(; scalar t)ro(hlct in I)tmse space is at ~1masure of the " p r o p e n s i t y " of the two particle, in t,he consi(tcre(t sta,tes, "to lmve positions and mo~ne~ll, a differing" just by those specific a,IllOllnt, s [10.231. This int, ert)retation in t e r m s of " p r o x i m i t y of states in phase pla,ne" gives further insight into the mea,surenmnt process, since one of the particles, p r e p a r e d in t,he stat, e ~1, can be u n d e r s t o o d a,s a, part of the a.ppara,tus for mea,suring certain p r o p e r t i e s of the o t h e r particle in the s t a t e ~ . Finally, as the reader ca, n verify, the inner p r o d u c t of two functions equals the phase space integral of the cross W i g n e r d i s t r i b u t i o n flmction of the two

The Wigner Distribution Function: Analytical Evaluation

299

functions (see w 6.3.4 a,nd w 9.3.2):

./ (Pl(q)(P2(q)dq -- /I

W~O1 ''2 (q' ~'~)dqdl'~.

(6.3.55)

Therefore, Moyal's formula can be forInulated in the alternative forin

2 1

2~

1

-- (Wy~l , W~a2 ) -- 2--~

// ~)~al'~2 (q' t~)dqdt~

9 (6.3.56)

In the light of the discussion above, (6.3.56) relates the "proximity" of the two states p,, ~2 to the phase space integral of the cross-Wigner distribution function of the two states. We have learned in w 6.3.4 that the cross Wigner distribution function expresses the cross-term contribution to the Wigner distribution function of the (possible) superposition of the two states; in a sense it is a, measure of the "amount" of interference [10.24, 7]. The analogue in optics is found in the launching of a,n optical signal into a mode-supporting structure like a, waveguide [10.25]-[~0.28]. Evidently, the scalar product f ~l(q)F2(q)dq rela,tes to the coupling coefficient between the incoming signal, say ~ , and a, given mode, F2, of the wa,veguide. By Moyal's formula., such a coupling coefficient comes to be interpretable in the light of the overlap of the Wigner distrit)ution flmctions of the signa,1 an(t the waveguide ino(te, and so in the light of the "proximity" of the signal a,nd the mode to occupy the same region in the spa,ce-angle pla,ne. In a sense, it is a measure of the signal-mode interfering a,rea in phase plane. This view has as well practical advanta, ges since the pha,se pla,ne representa,tion may straightforwardly incort)orates the effects of the finite angular aperture of the receiving structure. Also, it may be easier to obtain information about the Wigner distribution fllnction rather than about the spatial distribution of the signal at the plane of interest. Besides, the Wigner distribution function, unlike the signal amplitude, benefits from very simple transfer laws (see w 8.3). Hence, it can be measured at any plane and then propagated up to the desired pla,ne. Finally, as far as paraxial propagation is concerned, phase plane ellipses, pertaining to the signal and the waveguide modes, can be individualized in terms of the second-order moments of the respective Wigner distribution functions. Accordingly, the optimiza,tion of the coupling efficiency of the light launching into the wa,veguide acquires the intuitive visualization in terms of the matching of the phase-plane emittance of the incident signal to the phase-plane acceptance of the receiver's mode.

6.3.8

Scalingproperty

Under linear scaling the Wigner distribution function reproduces in its space and frequency dependence the scaling laws for the signal and its spectrum.

300

Linear Ray and Wave Optics in Phase Space

Sut)t)ose in fact to scale tile signal vP(q) to O(q) -- ~

(6.3.57)

p(aq),

witll a t)ositive real t)aranl(',ti;r. Tile square root fa,(:t()r is r(',(l~fir(;(l t() k(',e,t) the l~()rnmlizati()n ()f t,h(', s(:a.l(;(t sig~m,1 O(q) th(; saa~l(; a,s for ttl(; ()rigi~m.1 sig~m,1 ~(q). Tll(; F()~n'i(;r st)(;(:t,r~u~ s('al(;s i~v('.rs(;ly a,s tl~(; siglm.l a.(:(:()r(lillg t,() (.~ 4.6.1)

0;(-)

1

-

h"

).

l,ik(;wis(;, l)y (6.2.2) w(: ()l)l,ai,~ f()r W,/,(q, ~') l.l,(; s('ali~,t4 ,'(;la.li(),, w,,

(,1,

-

(au,

(s.s.s9)

wl~i('l~ (;xI)r(;ss(:s 1,1~(' s('al(' i'n'~,a't'ia'n,('(' ()f 1,1~(' Wigll('l" (lisl,ril)~i.i()~ fllll('|,i()ll s(:a,1i~g i~ si)a('(; a~(l fr('(l~(:~('y tl~(' sal~(, way as t l~(' s i ~ a l al~(I its Sl)(,('l.r~u~. 6.3.9

Convolution

and 'multiplication

p'ropc'l'ty

l,i()lls is giw;ll r(',Sl)(;('tiw',ly ! W ill(; Slm,(:(', (l()llmill (~r fr(',(lll(~llcy (l()~mi~ r

c2(q) --

()f th(; fi~n(:ti()~s ~, (q) aa~(l ~ (q), t)('. (;a.sily (;va,hm.t(;(l l~m,ki~g ~s(; fi'()lll ,~ 4.6.1 tlmt tl~(', fr(;(l*~(;~('y ~2 ()f tt~(; (:()~v()lv(;(l fim(:t,i()~s ~

/.

r (q -- q')r

]/V~(q,/~;) -- 2

/.

((J.3.(i())

1,1~(;r(;l('.Va.lfl, Wigl~(;r (listril)~l,i(~ fi~l~(:l,i()n (:a,n ()f tl~(; h;-(l()~m,i~ (l(;ti~fiti()l~ ((i.2.3). l{,('.(:alling Sl)(;(:trlllll ~9(h') r(;la,t(;s t() tl~(; Sl)(;('tra @, a,n(t al~(t ~2 t)y tl~(; t)r()(t~(:t

~:(t,) - ' v / ~ , we ()t)tai~

dq'

(t,:) ~ (t,),

(6.3.61)

~1(/~;_ /~;t)~ 1 (/~; _}_ /~;t)~2(/~; _ /~;t)v~2 (/~; _qL_i~;t)(,2i~':'q dl~;t.

(6.3.62)

Then, applying relation (6.3.25) to tile t)r()(hl(:ts Vpl ( t ~ - ~:')V~(t~ + /,~I ) and ~ (t~-t~:')~ 2 (~:+~:'), we en(l '~t) with the (:onv()hltiou of the Wiguer (tistributions W~I and W~2 in the spa,(:(; w~,riable; namely 9

~*

(q, ~) - 2Tr f "~Vp 1 (q - ql, ~:)W~2 ( q t t;,)dq t .

(6.3.63)

Simila,rly, given 0(q) as the product of the functions 01 (q) and 02 (q), i.e.

0(q) -- 01 (q)02 (q),

(6"3"6~x)

The Wigner Distribution Function: Analyt&al Evaluation

301

it is convenient to use definition (6.2.2) and rela,tion (6.3.23) to deterinine the relevant Wigner distribution function. The resulting expression,

(q,

- f

(q,

-

(6.3.65)

(q,

is just the convolution of the Wigner distributions W(b1 and W,2 along the frequency dimension. We note that the Fourier spectrum of (6.3.64) is the convolution of the spectra 01 and r of the functions 01 and r (w 4.6.1)

(~(N) _-

1 / (~ (~ _ t~t)~2 (h;t) dt~t

(6.3.66)

A slightly more general result accounts for a, possible scaling factor in (6.3.64). If the function ~(q)is obtained as (6.3.67)

qS(q) = O~ (q) O2(aq),

with a > 0, the Wigner distribution 142| (q, ~) still comes to be the frequency domain convolution of W,1 and I/F,2 , the arguments of the latter being appropriately scaled according to (6.3.59), thus giving .

W

(q,

-

~/

(6.3.6s)

(~1(t~ - t~')~ 2(-j) dt~'

(6.3.69)

1

Note that the Fourier spectrunl of (6.3.67) reads 0(/.@)

1

v~l

Expressions (6.3.63) a.nd (6.3.65) of the Wigner distributions W, (q, a~) and We (q, ~) relative to tile convolutions (6.3.60) a,nd (6.3.66) make evident that the convolution is transmitted from the functions to the corresponding Wigner distributions, and involves the space or frequency dimension according to whether the functions or the respective Fourier transforms are convolved. Interestingly, the convolutions (6.3.63) and (6.3.65) may be interpreted as the space-frequency representations of the transformations described by products (6.3.61) and (6.3.64). Thinking in physical terms, the latters may describe filtering operations respectively in the frequency and space domains, with one function representing the signal and the other the filter function. Thus, both the frequency and spatial filtering manifest in the q-~ plane through the convolution of the Wigner distributions of the signal and the filter function along the space and frequency dimension, respectively. Within the optical context, relations (6.3.60) and (6.3.67) may be seen as describing the transfer properties of space-invariant and diffraction-free optical systems, which, as noted in w 4.6.3, act as frequency and spatial filters for optical signals. Convolution (6.3.60) and

Linear Ray and Wave Optics in Phase Space

302

p r o d u c t (6.3.64) express therefore the transforma, tion la,ws for optical signals un(ter t)rot)agation t h r o u g h a, h o m o g e n e o u s m e d i u m and a, thin lens, one of the flm(:titms representing the int)ut wa,veform, the other one the line spread functioI~ of the me(tim~ or ('orrest)ondingly the t)ha,sc n~o(lulation fl~n(:tion of the lens. Ac(:or(til~gly, (:(revolutions (6.3.63) an(1 (6.3.65) nmy t)(; rega,r(ted as the transfer la,ws ()f the Wig~(;r (listributioi~ fl~I~('ti()~ f()r fl'ee-t)r()t)a,ga.tion and lens transfer, a,s they relate the Wig~mr (tisl,ritnd,i()~ fiu~(:(,i()~ ()f the int)~t a n(t t,h('~ ()~l(,t)lll, siglia,ls. W(; will r('(:()vt',r l,h('lii ill !i 8.3.1 wll(;n illv('stigatint4 the t,rm~sfer laws f()r t,l~(' Wig]~(,r (listri|)~ti()~ flll~(:(,i()l~ llI~(l(;r l)r()i)a,gati()~ t,hrougt~ a fl'(','=n~e(li~m~ S(;(:(,i()ll a,ll(l a, l,llill l(',llS.

l~,ep'l'esen,tability

6.3.10

As a. na,t,lira.l (:()ll(:llisi()li t() tliis s(;('.ti(nl, w(; (:()liilil(;llt ()li ill(; 'l'~'p'l'~'m"ntability ()f i,lle Wigll('r (lisi,rii)ilt,i(nl filll('li()ll. Evi(t(;lltly, ll()t ('~v(',ry filll('t,i()ll ()f Sl)a,(:(; a.ll(1 fi'e(tlU;ll(:y (:ml 1)(; 1111(Iersto()d as a. Wiglmr (list, rilnlti()ll filll('t,i()ll, l)e('a,llse tllere lllay ~()l, (:xist, a sig~al, fl'()~ wl~i(:l~ it, ~igl~t, l)e ge~(;rate(I. 'l'l~s, a 21) fllll(%i()ll ()f Sl)m'(' m~(l fI'(;(lll(;l~(:y is sa.i(t t,() t)e Wig~wr 'rt'p't't',s~'n.tabh' ()r 't'calizabh', if it is ~(;~erat(;(l I)y s()~(' signal. A~ exl)lit:it (:()~(lili()~ ('m~ l)('~ ('stal)lisl~('(l as a l~('a~ I() as('t;rtai~ wl~ether ()r ~()t, [4.13, (i.7, 1o.21]. Basit:a,lly, it grt)~na(ls ()n tl~e (:()~t)a, tilfility ()f rela, tions (6.3.23), (6.3.25) a,~(l (6.3.28); ilk(lee(I, l()()kil~g at (6.3.23) m~(l (6.3.25), wc sec t,l~at a, fiu~(:ti()~ ()f q m~(1 t," is a W i g n e r (listril)~ti()~ if m~(l ()i~ly if l,t~e i~d,egrals on the right-lm,~(l si(les res~lt i~d,() the Sel)aral)le fl)n~s al)l)eari~g in the leftlmn(1 si(l('s. St)('(:iti(:ally, w('~ (:a~ I)hrase t,l~(' ll('~(:(;sSa.l'y a l~(t s~ffti('ie~d, (:()l~(titi(m for a flmt:ti()~ ()f the F()~rier-t'.onj~gate variM)les 142(q, ~') l,() |)(, a r(;t)resentat)le~ W i g n e r (tistri|)~d,i(ni ill t(',l'lllS ()f the fl~n(:ti(n~ itself l, hr()~gl~ th(; r('Ja.ti()~ [4.13, 6.7]

W(q,, _

_

-

-

1 27r.

/

2

+ u'

+

W ( ( ] o -Jr- ~ ' I'," o -Jr- - ~ ) W ( q

-

'2,1 t,;~} -

-~)

o -- ~ ' H,o -- -~)('.-i(ml'-qtc~)d(]tdh;t

9

which m~st h()l(t fi)r any % and t% (see P r o b l e m 5). In pra.ctica,1 terms, in order to establish w h e t h e r a fimction is representable or not, we a,ss~m~e the flmcti(m, we are dealing with, be a W i g n e r distribution fl~nction of sonm signal, wtfi(:h can be found from the inverse formula (6.3.28). Then, we calculate the W i g n e r d i s t r i b u t i o n of the derived signM and compare it with the originM flmction. Evidently, if we recow;r the same function, the d i s t r i b u t i o n at h a n d is a, representable W i g n e r distribution function. In w 6.4.6 we will illustrate an example of a flmction of q and t~:, which is not representable W i g n e r d i s t r i b u t i o n function. As we will see, the features of the p o s s i b l e g e n e r a s signal, derived from the inverse f o r m u l a (6.3.28), are

The Wigner Distribution Function: Analytical Evaluation

303

in contrast with relations (6.3.23) and (6.3.26). Also, we know that the Wigner distribution function attains always negative values somewhere in the q-ec plane, except when it is generated by a Gaussian signal. Thus, any nonnega,tive 2D function can immediately be recognized as a non representable Wigner distribution function, unless it is in the Gaussian form (6.3.45) with the relevant parameters satisfying the equality in (6.3.46).

6.4

The Wigner distribution function of light signals further examples

We will further illustrate the potential of the Wigner distribution function to act as a local frequency spectrum, simultaneously displaying the space and frequency features of the signal. We will consider typical waveforms (e) and derive the analytical expression of the corresponding Wigner distribution functions and/or show on graphs the 3D images of it [5, 10]. We will start by considering signals limited within a, finite-length interval in space or frequency. The mathematical representation for such waveforms is the rectangular function, whose dual representation is the sine-function. Then, resorting to the prime Fourier-transform pair, i.e., the monofrequency and &like signals, discussed in w 6.2.1, we will firstly consider a cosine signal, arising as the sum of two st)atial lmrlnoni('s of equal an(t ()t)posite frequencies. The obvious generalization is evidently the infinite (numera, ble) sum of pure spatia,1 harlnoni(:s, whi(:h for suitable coeffi(:ients (:a.n t)e rcgar(ted as tile Fourier series of spa,tially periodic wavefields. As a basic example of periodic signal, we will consider the comb function. It is of great relevance in connection with the analysis of optical effects, like, for instance, the Talbot and Lau effects, produced by proper illumination of suitably arranged periodic structures. Frequency-modulated and amplitude-modulated signals will then be COilsidered. In the first case, we will start describing the pure quadratic phase signals, which are the basic waveforms of first-order wave optics, na,turally arising in fact in the Fresnel approximation as model function for the response function of optical systems as well as for the phase front of the incoming disturbance. As a natural continuation, the pure polynomial-phase signals are briefly coinmented on, being seen as a tool for investigations of the optical propagation properties beyond the Fresnel approximation. As a primary example of amplitude-modulated signal, we will consider the real Gaussian amplitude which provides, for instance, the model function for the transverse profile of the fundamental mode of the field oscillations in a stable resonator laser as well as of the quantum mechanical harmonic oscillator. c A detailed description of many typical waveforms of relevance in optics is [14].

Linear Ray and Wave Optics in Phase 5pace

304

T h e H e r m i t e - G a u s s i a n b e a m s will also be discussed a,s well as the more recently i n t r o d u c e d Hcrmitc-cosh-Gaussia, n modes. Wc will give a numerical a,ccount of the supcrGa, ussian envelope wa,vcforms, whi(:h m a y rnodcl the la,tcral profile of b e a m s froi~l high-ga,in unsta,ble resonator lasers slfital)ly e(01ipped with sut)crGaussia,~l transnlitta.ncc or refle('tivity mirrors. It is worth noting t h a t , sill(:(; the W i g n c r (tistrilmtion flln(:tion is (tcfincd 1)y a,n iIlfinite integral of a sllt)t)ose(lly (:ontimlolls flul(:tion, its cxa(:t cvalllal,i()ll is ill g(;n(;ral imt)()ssil)h'. In(l(;(;(l, f()r a,llll()Sl, all ()f l,ll(; wav('~f()rnls, w(; will (:()llsi(l(;l" ill wlmt f()ll()ws, a,ll (;xa,(:l, (;xt)r(;ssi(nl ()f l,ll(; l"(;st)(;(:l,iv(; Wigll(;r (tistrit)lll, i(nl fllll('ti()ll (:il,ll l)(; ()t)l,a.ill(:(l. Tllis is s(), (;xt:eI)t ill l,ll(; (:as(; ()f t,llc (tuarti(: ml(1 (llfixlti(: t)()lyll()lllia.l I)llas('~ siglmls, mi(l ill l,llc (:a.s(; ()f l, ll(', slq)(;rGaalssian a.lllt)lil,~dc, ft)r wlli(:l~ a, ~nn~t'rit:al a,t:(:t)~t is givtnl. ()f (:()lll'S(~., tiffs inlt)lies a, (tis('rt;l,iza,titnl t)f tilt: sig~ml m~tl st) a (lis(:rt;l,iza, titnl ()f tilt; rt:s~ll,i~g (tistril)~ti()l~. W(; slmll ~l()l, tlwt;ll (hi tilt; (l~U;sl,itms rclatt'~tt t,() tilt; tlt:til~ititnl ()f a (tiscrctc v(~,rsi()~l ()f l,l~(; Wigll(;r (lisl,ril)~l,i()~l fi~(:l,i()~ a~(1 (hi t]~(; (:()llS(;(tll(~,ll(:(; ()f this (lis(:rel,iza,ti()ll, fi)r wl~i<:l~ l,llc r(mtl(;r is (lire(:l,(;(l 1,() [,5.a, .5.,1]. I~ fi~,(:l,, il, is ()~r l)~rt)()sc ()~l sll()Wil~g 1,11(;31) gral)l~s a~l(1 l,ll(; i'eh;vmd, l)lm.s(;-l)lml(; (:()~ll,()~r I)l()ts ()f l,hc a.f()r(;n~c~d,i()~(;(l sig~m.ls, ()~ly t() giv(; a fi;(;li~g fi>r 1,l~('~ir ]>(;l~avi<)~r; ~l()exa(:tit~ule is ll(;(:(l(;
7'h('~

Wig'ne:rphase-plane

It, lms t)ccn rr ill !i ,1.6.5 t,lmt as t,ll(; IImllilt<)llia,ll ('ot)ti(:al I)lmse sire(:(;. It llas also |)ceil not,(;(1 that the nmtrix rclal,i(m (:(mncct, ing the ray wl,rial)lcs (q, p) a.t different t)lanes along the i)rot)agati()ll axis, (:a,n eqllally 1)(; ('xt, en(l(;(t 1,() tll(; wave-optica,1 pair (q, ~:), of (:(nlrs(;, ill a, t)llrely formal fa,shi(nl. T h e W i g n e r <listrilnlt, i()~l fllll(:tion (:omposes 1)ottl the st)atial mid frequency features of the signal into a, single representa,tioll. So it, is ta,ilored to be that "entity" in ternls of which we can afford a t)hasc-t)la, nc visua,lization of the signal p r o p a g a t i o n I)rot)crtics, as we will do indeed in the examples below. However, such a picture in the phase plane (that we will also refer to as the W{gner phase-plane) suflbrs from an inherent limitation arising from the u n c e r t a i n t y relation of Fourier analysis, which, as it is c o m m o n l y interpreted, ma, kes "points" in the pha,se pla,ne ina,ccessible a,nd hence va,lues of the joint r e p r e s e n t a t i o n at points in phase plane not measurable, and so meaningless. A critical c o m m e n t on this c o m m o n l y held notion will be addressed in w 9.4.4. W e know that the g e o m e t r i c a l optical p h a s e - s p a c e has a s y m p l e c t i c struc-

The Wigner Distribution Function: Analytical Evaluation

305

ture. The Wigner phase-space has its OWil structure as well, that a n accurate investigation has revealed to be confor:n to an affine metric [4.6, 4.10], being invariant under unimodular linear transformations (a subgroup, indeed, of symplectic transformations, under which the geometricM-opticM phase space is invariant). Unimodular linea, r transforma, tions preserve stra,ightness, mapping straight lines into straight lines (see w 6.4.4). Within the paraxial approximation, amounting to symplectic transformations of the ray variables which actually are linear (represented,as they are, by the ray matrices), the difference between the geometrical-opticM phase-space and the Wigner phase-space becomes imperceptible (apart from, of course, that arising from the indeterminacy relation). Also, it is beyond the purposes of our studies here and a,ccordingly we shall not dwell on it any further. Let us finally remark that, as also pointed out when dealing with geometricaloptical phase-plane, in order that the worked relations be meailit:gful we must think of the Wigner plane axes q and t~ as suitably scaled (possibly by some characteristic length of the problem at hand), so that they have the same dimensiona,lity. It is in the light of this implicit normalization that the mathematical relations, that will subsequently be derived, must be understood. Accordingly, no mention will be given to the milts in relation to the va,lues a,ssigned to the pa,rameters in the gra,t)hs (which will appear just a,s numbers), the specifi(: units being, of course, determined in accord with the (:hosen normaliza,tion settings. 6.4.2

Finite-extent waw~fo'rms in space or frequency Rc.ctangular signal

A unit-magnitude uniform signal, limited to the spa.ce interval of length 2Q, :nay be described as Pa(q) - rect(sqOQ) -

1 0

[q[-< Q' otherwise,

(6.4 1)

where Q is a positive number. The frequency spectrum ~?(n) behaves as a sine-function, being I

~5R (g) - V/L~sinc(t~Q).

(6.4.2)

Using definition (6.2.2), it is easy to work out for the Wigner distribution function W R (q, t~) of ~a the explicit expression )d?a (q, K.) --

--~-(1 - --)sinc[2Q~;(1 - --)]

ql -< Q,

0

ql>Q.

(6.4.3)

As expected by virtue of the finite support property (w 6.3.3), being the function (6.4.1) concentrated within the space interval - Q _< q _< Q, the

306

Linear Ray and Wave Optics in Phase Space

I " I G U I { E 6.5. (a) 31) ph)t of tlm Wig~wr (lis(,ril)~ti()~ fi~('(,i()~ ()f a r(wtm~gh.,-sig~ud ()f wi(Ith 2 Q = 5, (b) reh:wu~t profile vs. t," at q - 0 m~(l (c) Fmu'ier ( . r a ~ s f o r ~ ()f t,l~(, r(;ctm~gh:-sig~ml.

rt;levalit Wigxl(;r (listl'il)lll,i(nl flnl('l,i(nl (:(hires 1,() t)(; (X)ll('(:lll,ra(,(;(l ill (,lie t)lmse l)lml(; in(,() l,lle verti('al stri 1) (,llr()llgll (,ll(; Stall(; Sl)a(:e ilfl,(;rwd - Q < q <_ Q . ll(:r(;, I/V. (q, t,,) (lisl)lays

a, sill('-lik(:

slla,l)(: a,l()llg (,If(: fl'( :( l l l( 'l l( '.y (lilll(:llsi()ll.

F()r

(:a,(:ll q E ( - Q , Q), (,l~e ~mi~ I()I)(: is (:(:~(,(:r(:(l a,(, h" = (), m~(l its wi(l(,l~ i~(:r(:as(:s witl~ in('rea.si~g wd~u:s ()f ]ql, wl~ilst (,I~(: l~(:igl~(, (h:('r(:as(:s (,() z(:r(). I~ l)a.rti(:~dar, W,,(q,t,;) a t t a i n s its ma.xi~un~ a.(, (,I~(; Sl)a,(:(:-fr(:(l~U;n(:y I,,)i~d, ((), ()) with th(: wd~u:: W,,((), ()) = 2Q/rr, ix~ a.(:(:()r(l wi(,l~ (,I~(: lix~i(, (:xl)r(:ss(:(l i~ (6.3.1). Nota.l)ly, I42, (q, h:) is ~u:ga.tiv(: in (:(:r(,a.i~ r(:gi(n~s ()f (,I~(: q-t<, I)Im~(:. Fi~ml]y, tlm z(:r() wd~u: h)(:i ()f the (listril)~(,i(n~ (6.4.3) at(: th(: sy~l~(:(,ri(:a,l hyl)(:rl)()las

(:orresl)onding to integer va,]ues ()f n = -l-l, i 2 , ... Figure 6.5.a) shows th(: 3D I)h)t of l/V. (q, ~,:) (:()rrest)m~(li~g t() a given va,hle of Q. The sim:-like behavio~n" v(:rs~s h; is evi(h:nt:e(1 in Fig. 6.5.t)), sh()wing the t)rofile of the distributi(m vers~ls h; a,t q = 0, which ca,i~ t)(: (x)mI)a,r(:(t with the frequency s p e c t r u m 73(h:) tra,(:ed in Fig. 6.5.(:).

Si'n,c signal Tile n a t u r a l counteI'pa,rt of tile rectangle signa,l is sinc-anq)litude signa,1,

which in t u r n has the finite-extent frequency s p e c t r u m , ~,

~s (~) - rect()-~),

K -

1

Q.

(6.4.6)

The WiEner Distribution Function: Analytical Evaluation

307

FIGURE 6.6. Contour plots of the Wigner distribution functions of (a) the rectagle-waveform (6.4.1) (2Q = 5) and (b) the sinc-wavcform (6.4.5) (2K = 5). Using (6.2.3) the rdeva,nt Wigner distribution flmction )/Vs (q, t;) obta,ins as

w~(q,~:)

-

2K -7-( 1 o

JS[)sinc[2Kq(1 -- IN)] K K

It~l < K,

(6.4.7)

I~,:1 > K,

which evidently has tile same behaviour with rest)ect to the space va,riable a.s (6.4.3) with rest)cot to the frequency va,riable. Figure 6.6 shows the (q, ~:)contour images of the distributions (6.4.3) and (6.4.7) for the same numericaJ va,lue of (~) and K; we :nay see the characteristic structure of nested symmetric hyperbolas, which axe rota,ted by 571" one to the other, being confined respectively within a, vertical and a horizontaJ strip. Rectangle and sinc signaJs relate to each other by the Fourier integral, and accordingly the respective Wigner charts in the q-t~ plane relate to each other by a,~-rotation. Resorting to the results of w 6.2.1, we note that the same ~-rota,tion in the q-~ plane rela,tes the Wigner distribution functions of the impulse and the spatia,1 ha,rmonic, which axe a, Fourier-transform pair a,s well.

Modulated and shifted rectangular signal As noted in w 6.3.2, a modulation of the recta,ngula,r signal (6.4.1) by the harmonic factor ci~0q according to F(q) -- rect(2-~)c i~0q

(6.4.8)

produces a, frequency shift of the corresponding Wigner distribution (6.4.3) as

w ~ (q, ~) = w ~ (q, ~ - ~0).

(6.4.9)

Linear Ray and Wave Optics in Phase Space

308

Tit(; ab()ve b e h a v e s a,s (6.4.3), a p a r t fl'om a, shift of the center of g r a v i t y fl'om the t)lm.sc-t)la.ne t)oint ((), 0) to the point (0, ~:o)Trm~slating signal (6.4.8) a,hmg the q line t)y qo as h;a(ls t()

~(q) - r e t : t ( ~ ) ,

'i%q,

W~(q, ~:) - W.,(q - q,,, ~: - t,:,,),

(6.4.10) (6.4.11)

wl~()se ])('~l~avi()llr agai]~ r(;l)li('at('s tl~al ()f (tisl,l'i|)~]ti()]l (6.4.3) witl~ 11~' ('e]d,r()i(t t)(;illg sl~it'l,('~(l l,() l,ll(; l)()il~l, (q~,, t~,~) i~l 1,11(; Sl)a,(:t;-fl'c(t~l(;~l(:y l)lall(',. Sig~lals (6.4.8)a~(l (6.d.l()) l~av('~ l,l~(' sa~('~ sill(:2-lik(; 1)()wt;r Sl)(;(:l,l'lllll. Ill fa(:l,, l,ll(;ir fl'(;(lll(;ll(:y Sl)(;(:l,ra,, r(;sl)(;t:l,iv(;ly giv('ll ])y sill(:[(2(t~"- t~',,)]r '--i(~- ~"")q", (6.4.12) si~t)ly (litt'('r fin" l,ll(; l)l~as(', fl~l(:l,i(n~ ~,-i(,,.-,,.~,)%. ~l'l~(' (:(n'r(;sl)(n~(lil~g s(pm,rc ~mg~lit~t~h:s slml)~' l,ll~:r~;fi)r~: fi)r 1)~)1,1~ sig~mls as l,ll~; si~('2-fi~('ti~n~ w(t;) --

sill(:[(2(t~"- t~',,,)],

~(t~') --

I~(u)l 2 -- 2(a2sil,(:2[Q)(t; -t;,,,)],

(6.d.13)

71"

whit:ll evi~h'~lltly t:()ll|,a.illS lit) illf~)l'llla,|,i~)ll al)t~ltt l,llt; h~t:atitnl ill Sl~a~'t: ~f tlle (;ll(;rgy (h;llsily. I~l (~
Monofrcquency and impulse related wavcform,s' Cosine ,signal

T h e W i g n c r d i s t r i b u t i o n of a, cosine signal, d e s c r i b e d by the amplitu(te

:(q) - ~,, cos(ecoq ) - ~~ 2

+ %c 2 -~oq ~

(6 94.14)

Po being in general a, c o m p l e x c o n s t a n t , is easily obta,incd m a k i n g use of property (6.3.11) a n d result (6.2.11). In fact, as the cosine wave can be r e p r e s e n t e d

The Wigner Distribution Function: Analytical Evaluation

309

a.s the sum of two plane waves of frequencies n o and - n o , the W i g n e r distribution fimction obtains as

~2+, (q, h;)

2 ~o4 {(5(n -- t~;o)-+- (5(h; -J-/%)-4- 2(5(h;) co8(2t-,;oq)}.

(6.4.15)

It displays two q-independent self-terms concentrated along the lines ~ = h~o and ec = -ec 0 in the q-n plane, and a q-dependent cross-term concentra,ted on the line ec = 0. In accord with the general considerations of w 6.3.4, the cross t e r m exhibits an oscillatory behaviour at the frequency 2eCo; it accounts in fact for the sinusoidal oscillations of the signal power with q occurring at the frequency 2n 0 a r o u n d the stable level the only value of frequency, where the which s h a r p e n s at t~ = t% and ~ = overlap of the left over the right part,

%12 2 . Also, it is sharp at n - 0; this is s p e c t r u m @(t~;) c< 8(t~; - ~o) + 8(t~; + eCo), - ~ o , can be folded to yield a nonzero and so a nonzero cross-term.

S u m of two spatial harmonics. Fourier series As a more general case, we m a y consider the s u m of two spatial harmonics of frequencies K,1 and n2, weighted by the complex coefficients F01 and P02:

(r

-- (/2)01Cit~lq -Jr- (/~02 Gi~2q.

(6.4.16)

In t h a t case, the Wigner distribution flmction comes to be

W~ (q, ,~) -- I~0, 12~'(~' - ", ) + I % I ~ ( ~: - ~:2)+

(6.4.17)

q- 219~019~021cos[(K, 1 - ~:=)q -t- r]gS(t;, - "1 +-'22 ) where r denotes the t)hase differen(:e t)etween tile two waves a,t q = 0, i.e.

Io1%1 Distribution (6.4.17) is concentrated along the a u t o - t e r m s lines at the two c o m p o n e n t frequencies t~:1 and t%, and along the cross-term line at the m e a n frequency ~+~2 2 " Like the signal power, the cross t e r m manifests an oscillating behaviour at the difference frequency n I - K 2 with a c o n s t a n t phase fixed by the initial phase difference r between the two c o m p o n e n t waves. The graphs in Fig. 6.7 show the W i g n e r distribution function of a cosine a m p l i t u d e and the relevant space-frequency contour image. Specifically, Fig. 6.7.a) pertains to an unlimited cosine signal. T h e contributions from the selfand the cross-terms, respectively located at the c o m p o n e n t frequencies and at the m e a n frequency, are evident. In turn, Fig. 6.7.b) refers to a finite-length cosine envelope. The delta spikes of Fig. 6.7.a), typical of unlimited sinusoids, widen into sinc-like profiles, proper of finite extent sinusoids.

Linear Ray and Wave Optics in Phase Space

310

Fl(lIJlll'; 6.7.31) view witl~ tim rch, vm~t (q, h)-l)lmm (:o~to~r l)h)ts ()f tl~(' Wiggler (listril)utio~ fil~(:ti(n~s ()f (a) l,lm (:()si~c sig~utl (:()s(2q), m~(l ()f (b) t,l~c sa,~c ('()si~c sig~al li~fitc(l to the ti~fit,c-lc~gtl~ i~d,t'rval - 2 <_ q _< 2.

W(; i~vit,(' tl~(; r(;a.(h;r t,() g(:l~(;raliz(: (;xl)r(;ssi(n~ ((i.4.17) r(:l(:va~t t,() a sig~al o~

wll(:r(; ~s,'s ml(l h,'s at(: a,,~sigxl(;(l (:()llS(,~-l,ll[,S [1().2.()]. N()tal)ly, wav(:f()rlll (6.4. l.()) (:all 1)(: ilm:rl)r(:(,(:(l as a F()llri(:r s(:ri(;s (h;s('rit)ing f()r i~st, a~(:(; a I)(:ri()(li(: sig~a.l: ~(q) -- ~(q + d) (Pr()t)l(:~ 8). As is w(:ll k~()wn, ill tlmt ('as(;, ill(; fl'(:(tluul('i(;s t~;,, ()ftll(: (:()lllt)()ll(;llt, lm,nll()l~i('s i~ tl~(; s(;ri(;s ca, n |)e (;xi)rcss('(l a,s i~t(;g('x ~n~ltit)l('s ()f tl~(; .fu'ndamr tal fl'(:(l~W~(:y h~: a',, = 'nh:0, t~;(, t)('~i~g (l(;t(:n~ix~(',l t)y t,l~(' sI)atial I)(;ri()(l as h;(, - ~'. d Ih;si(h;s, tl~(' w('ights ~,, ()f th(; wtri()~s lm.rn~()ni(:s tak(' th(' ~ ( ' a n i ~ g as th(' F()~n'i(,r (:()(;tfi(:i(n~ts, linked to the sig~m,1 wav('f()rn~ r t)y the i~t(;grals

~"

__ 1 /.d/2 d .I-d~2 ~(q)('-i'~% qdq"

(G.4.2o)

Sum of two impulses Tile W i g n e r (tistribution flm(:tion of a, signal, tha,t arises a,s the sut)crt)osition ()f two imt)llls(;s, loca,tcd a.t q - q~ a.nd q - q2, ~(q)

-- (~(q -- ql ) -[- ~'(q -- q2),

is forma, lly similar to tile W i g n e r distribution (6.4.17) of tim sum of two spatial harmonics, with the variables q and h; being exchanged. Explicitly we find 1

)/~2 (q, ~) -- -ff-~{ ( 5 ( q - ql ) -I-(~(q--q2 ) -t- 2 ~ ( q - -

q~+%)c~

-- q2)~]},

(6.4.22)

The Wigner Distribution Function: Analytical Evaluation

311

F I G U R E 6.8. 3 D view of the W i g n e r d i s t r i b u t i o n f u n c t i o n of t h e suin of an i m p u l s e a n d a s p a t i a l h a r m o n i c , h a v i n g ~1 -- qD2 = 1, q~ - - 0 . 5 a n d t% = 1 (see P r o b l e m 9).

which shows two contributions at the lines q = ql and q = q~, arising from the self-terms, and the cross-terms contribution, placing values along the intermediate lines q ql+q~ in the space-frequency plane. 2 The reader may deduce the expression of the Wigner distribution function of the slim of a b-fllnction and a. spatia,1 harmonic, whose 3D plot is shown in Fig. 6.8 for given va.lues of the releva,nt pa,rameters (see Problem 9). Train of ~5-like w a v c f o r m s

The sum of two (5-like waveforms in (6.4.21) can be generalized to (:omprise the infinite sequence of impulses

oo

(6.4.23)

-

m z

--oo

which is one of the various representation reported in the literature for the comb function [14]; d denotes the spacing between any two consecutive ~ functions. Graphically tile series (6.4.23) is usually depicted as an infinite array of unit-height spikes located a,t positions q - rod, m - 0, +1, -t-2,... (Fig. 6.9.a)). Clearly, p(q) is periodic and hence admits the Fourier series representation

oo

jp(q) _ ~ ~

r

no =

271"

d"

(6.4.24)

nz--CX)

The Wigner distribution writes therefore as (DO

Wcomb(q, 153): ~2 ~ 'gt,~ -- O0

(X)

~ ,T't,/--- - - O 0

(5(Jki;-- ~ 2 ~o)e -i(~-~')~~

(6.4.25)

Linear Ray and Wave Optics in Phase Space

312

F I G [ I I I E 6..9. (a) Scl~e~uttic of tim c(~fl) fl~mt, i(,~ wit, l~ d - rr, m~(l WigI~er <list, riln~ti~ni flt~wl,io~ wil, li tim c<)rresp()~(ti~g l a t t i c e ()f l)r

1

*

oo

oo

Z

2 i2,,'~,,,q E

I~ I - - - - ( ~

'2

I~l----

CxD

l>(:ri,)
~

~ ,,~'2,,'~-,,q _

--i,,,~,',,q

<)f t, ll<; <'
(X~

'

of t,lw relative (hi t,lie (q, h:)-plmm.

~

wit, ll st>a.tial

d

~

b ( q - 'n=,). '1'1~;~, ('xl,l()iti~g t,t~,; w<,ll k:~own

t>i'()t)(;ri,y <)f tl~e (:r flm<'tir t<> (;Xt,l'a,<:l, t,l~r wd~ws r a fl~<'tir rcg~lal'ly sl)a<'r162l)y l,ll<; I)<;ri<)
at t)<)sit,i(ms

oo

d

'II--

((i.'1 '20)

(X)

'Fll<', first s<'ri<;s a,1)<)vr is.jllst t,ll<; l"<)ln'i<;r sr d.

(b) 31) view

(-i

-- (X)

'~ b ( q - ,,~).

(6.4.27)

'rl,--- -- (X)

we (;11(t 11t) witll ttl(; ext)ressioll cx)

w,, ....,,(q,~:)- ~

~ n---

cx)

( - ) ' " " ~ ( q - , , , ~d) ~ ( ~ ; - , , , , ~ ) .

~ -- (X)

~tl,--

(s.4.28)

-- (X)

We :lot(; tha,t, tile p r o d u c t of tile two delta flmctions (:a,n a.lso be e m b e d d e d into the two-dimensional 5-flmction [14]

a(q _ ~ ) ~ ( ~ _ , , ~ )

_ ~(q _ ~

~ _ ~,~o) 2

"

(6.4.29)

Distribut, ion (6.4.28) identifies a lattice of delta flmctions with q- and ecspacings respectively of d and -%2. On graphs the ~5 functions are visualized a,s spikes of height ~d located at the points ( rid, m~2 ); so, each of t h e m has the

The Wigner Distribution Function: Analytical Evaluation

313

volume ~ . However, l/Vcomb(q, t~) c~nnot be interpreted as a 2D comb function, as tile sign of the various (5~ flmctions (tepends on the lattice points an(t hence on the integers n and m. The ~ functions located at points identified by a,n even n or m are positive, whilst those placed at points identified by an odd n have alternate signs with m and vice versa (Fig. 6.9.b)). The comb function reveals a useful mathematical tool when dealing, for instance, with the sampling of a signal at regular intervals. As an example, we may consider an opaque screen containing a certain number of very narrow identical slits, regularly spaced by d. The screen is uniformly illuminated on one side by a collimated light beam, whose optical axis, for formal convenience, may be supposed to pass through some one of the slits. Then, a,s the amplitudes contributed by the individual slits are of the same magnitudes, the signal amplitude ~(q) on the other side of the screen can be represented by N

- A

Z m,---

(6.4.3o) N

where the (:onstant A relates to the illuminating amplitude through the transmittance of the slits and the sum extends over all the 2N + 1 slits comprised within the t)eanl width 2w0, i.e., Nd <_ w o. Due to the sa,nq)liilg t)ropei'ty of the (:onfl) flln(:tion, we can also write CX~

b ( q - rod),

e,(q) - A

(6.4.31)

' H t -~- - - 0 0

thus conveniently expressing tile tra,nsmitted amplitude as tile product of the comb flm(:tion and the rectangle fllnction. Exploiting the windowing property (6.3.65) and the proper expressions (6.4.3) and (6.4.28) for 142a and Wcomb, tile reader may deduce the explicit expression of the Wigner distribution function of the finite ~-tra.in (6.4.31). As earlier mentioned, periodic structure are of relevance in connection with investigations of self-imaging phenoxnena, in which the lateral periodicity of the wave field is mirrored into a, longitudinal periodicity as well. The Wigner distribution function-based description yields a clear, a,hnost visualizable, account of the properties of tile phenomenon [10.30, 10.29]. As we will see in w 8.3.4, the correspondence of the lateral to the longitudinal periodicity finds a plain interpretation in the light of the propagation properties of the Wigner distribution function. 6.4.4

Frequency-modulated wavcforms Quadratic phase signal

The quadratic-phase signal is described by the purely imaginary Gaussian

Linear Ray and Wave Optics in Phase Space

314

1

I

~" o

g

-I

-2

1

o (a)

2

q

l:l(~tIl:~,l'] 6.10. (a) l:~('al l)art all(l

-2

(b)

2

0

q

(b)

illlagillary l)art of (,lie cllirl)

r

2 ,#(q) _ , i/~.

(6.4.32)

witl~/4 1)(:i~g a, r('al lnllll])(:I'. It, sl~()~hl 1)(: ~(ufl, i()l~e(l, l)(~rlm.l)S, tim t, i~ (,l~(: l;u~g~mg(' ()f sig~ml (,l~('ory, wa.vef()n~ (G.,1.32) is ref(u'r(~(l 1,() as a chi't7~ sig'n,al (()r (:l~irl), fi)r sl~()r(,). I{()~ghly Sl)(:aki~g, (:l~irl)s ar(: i~ g(:~(:ral i~d,(u~(l(:(I as wav(:f()r~s ()f (,l~(: tyl)(: :(q)

-

,,(q), ~'/'(',>.

((i.4.:)a)

wll(:r(, (,If(; a,llll)lii,lul(' filll('l,i()II~s(q) is Sl ll)l)()S('(l l,() l)r()(llw(' a. Slll()()l,ll lll()(l-

llla,(,i()ll ()f l,ll(: ral)i(l ()s('illati()lls l)r()(llu'(:(l t)y (.ll(: l)lm.s(: filll('l,i()~l r The (pm,~lra,ti(:-lflms(; sigllal ((i.,1.32) is a Im,rti~:llla,r (:as(: ()f l,ll(: ling'm" chirp, whi(:h l~a,vi~g (:()~s(,a~l, a,~I)lil,~ul(', a.~(l (t~m,(lral,i(: I)()ly~()~ial I)l~as(: (h:sig~a,l,(:s i~ g(:~(:ral a, li~(:a,rly fl't:tl~Uq~(:y ~()(l~lal,t:(l sig~a.1. N(:t:(llt:ss t,()say, waw:f()r~ (6.4.32) is ()f great r(:lewm(:(: i~ ()l)t,i(:s. I~ fi~<:t,, we l~aw: f()~(t i~ S(:('t. 5.2 t,lmt it is rcla,t(:s t,() t,t~c lilac sl)r(:a,(l ()f t,l~c first,-()r
tra,Ilsforlll

Wchirp (q~ 1~) -- b'(K, -- /4q) -- b ' ( h ~ - ~ ( q ) ) ,

(6.4.34)

for the W i g n e r distribution flmction of (6.4.32). It is totally concentrated along the local frequency curve and accordingly it identifies on the q-~ plane

The Wigner Distribution Function: Analytkal Evaluation

315

straight line through the origin making the angle 0 = tan -1/3 with the q-axis. In w 6.2.1, the Wigner distribution function of the spherical wave (6.2.12) has been found to be confined along a ~-line, making the angle 0 = tan-l(ko/R) (R is now intended to be scaJed by the refractive index medium no) with the q-axis in the space-frequency plane. Indeed, pertinently with the linear approximation (i.e. small spatial and angular extensions across the reference pla.ne), the amplitude (6.2.12), representative of the spherical wave with curvature radius R at the observation point q, can be assimilated to a chirp signal with coefficient ~ = ko/R. In the same approximation, a thin lens is known to produce a modulation of the signal amplitude by the chirp e-iko q2/2f. More in general, a linear chirp accounts for both space and frequency shifts with respect to the envelope (6.4.32) according to ~Ochir p (q) -- ~ i~(q-qO)2-4-it~Oq.

(6.4.35)

the local frequency r = t% + ~ ( q - qo) being still linear in q. The representative line of the distribution (6.4.34) in the phase plane is then translated to pass through the point (%, ~o)- We obtain in fact ~chirp(q'

/E) = ~(h; -- /~;0 -- /~(q -- qo)) = (~(h;

J~(q)).

(6.4.36)

As noted in w 6.2.1, the Wigner distribution of both an iInt)ulse and a nlonofrequency signal configures a,s a (5-line. We have found here that a b-line form as well (:haracterizes the Wigner distribution function of a chirp. Due to the relevance that these signals have in optics, we summarize the results concerning the respective Wigner distributions into the table below with the visual aid of Fig. 6.11, showing the pertinent supports in the q-h; plane.

~--iqo~ -

q0)

1 27r

_

v~

qo)

1

ci~2(q--qo)2+it%q

(~'(/~- /40 -- f l ( q -

q0))

/3

C

i q o ( t~ -- t% )

(6.4.37) In all cases the support of the Wigner distribution in the q-~, plane is a straight lille, respectively disposed vertical through the point q = q0, horizontal through the point ~ = ~o, and inclinated by the angle 0 = tan -1 c~ to the qaxis through the point (qo,~o). As already remarked, the impulse and the harmonic wave are Fourier conjugate to each other, and tile supports of the respective Wigner distribution functions relate to each other by a 7r/2-rotation in the q-~ plane. It is worth noting that from chirp (6.4.35) one can recover the

Linear Ray and Wave Optics in Phase Space

316

K"

spatialharmonic signal k'o y ~"~quadratic-phase. I V s,gnal q

I ~.~ike signal FI(;(IIiE (i. 11. S~il)l)()rl,s ()t' tlw Wigl~(;r (listrilnil,i()i~ fliil('ti()lis ()f a b-like, a Sl)atial tmrll~()ific an(l a q~m(lrati('-l)l~as('~ sig~al. inq)lllse a,ll(l ill(; slmtial lm, rlll()lli(:, sen(lillg resl)e(:tiw;ly ~ ---+ oo ml(l
The Wigner Distribution Function: Analytical Evaluation

317

FIGURE 6.12. 3D image and relevant (q, ec)-contour plots of the Wigner distribution flmctions of (a) the sum of the chirps e ~(q2+nq) and e ~(q2-4q), and of (b) the chirp e~q21imited to the finite extent interval - 4 <_ q < 4. Here, it is also shown the 6-function wall, with the relevant straight trace on the (q, r~)-plane, representative of the Wigner distribution function of the unlimited chirp e ~q2.

Cubic phase signal Tile cubic phase signal is rcprcs(~nte(t by the (:omt)lex amt)litude ~Ocubic(q)

-

(~iflq3 3,

(6.4.38)

which displays a local frequen('y quadratically varying with q as A(q) -/~q2. Inserting waveform (6.4.38) into the Wigner integral (6.2.2), we obtain (q, t~:)

-

1 J ci[2-/3q'a+2(~t(q)--~)] -~ 3 dq',

(6.4.39)

which, on due account of the integral representation of the Airy function [18]

Ai( (3a)1/3 x ) -- (3a)1/ 3 f ei(ax'3+xx')dx t 27r

(6.4.40)

is recast in the closed form ~)cubic(q~ N ) -

(~)l/3Ai((~)l/3(j~(q)

t~)).

(6.4.41)

As shown in Fig. 6.13.a), the Airy function oscillates for negative arguments whilst shows a, fast dec~y for positive arguments. Therefore, distribution (6.4.41) is expected to have a fast decay at the convex side of the local frequency curve (q, A(q)) in the q-t~ plane and in contrast an oscillatory trend at

318

Linear Ray and Wave Optics in Phase Space

F I G U R E (i. 13. (a) Airy f,~('ti()~ Ai(:r), ~u~(t (b) 3D ilnag(: ()f 1,1~r Wig~(:r (listri|),~(,i()~ f)mcti()i~

]/V(.,,i,i(" (q, t,) ()f th(' ('~t)i(" l)l~a.se sig~tl ~,' :) witl~ tim rel(:wu~t q-t," c()~(,()~r i~tg(:. (,tl(: coll(:aw' si(h: ()f (,ll(: cllrw;. Tllis is l,lle ])(:lm.vi()ln" we llm.y (4)s(;rw: illsI)(:(:tiIlg; the sln'fit(:(; gral)ll all(l l,ll(: (:()rr(:sI)()ll(lillg (:()lll,()~r l)l()ts ()f VV ,,,., sll()w~ in Fig. 6.13.t)) fiw a, giw:~ va.l~(: ()f/4. Evi(lenl, ly, ~ l i k ( : ~.,...,(q, t,'), l,l~(:Wig~(:r (listrit)~d,i(m fi~l('l,i()~ ()f a c~lfiC-l)lmse sig~lal (h)('s ~()l, (h'w'hH) excl~siw;ly a r()~m(l the l()(:al fi'(:(t~(:~(:Y li~(' t~" A(q), alt, l~()~gt~ it, l'(;~m.i~s ~a,i~ly (:()~(:(:~trat('(l a l)()~lt tha,t li~(:. I~ (',lml)l,(:r !) w(: will s(:(' (,l~a,(, (,l~(: l()(ml fr(:(l~U:~('y (i.(:. (,l~(: (h:riva, tiv(: of the siglml l)l~as(:) is (,l~(: first-()r(|(:r h)(:al ~()~(:~1, ()f flu: Wig~(:r (listribution function i~ (,lie freque~cy variable, and l~c~cc in a, se~s(: it, rcpresc~ts the local (:enter ()f gravi(,y ()f 1,1~(:(lisl, ril)~d,i()n with r(:st)(:(:l, 1,() l,h(: fl'(:(l~(:n(:y (lin~(:nsion. It is th(:n higl~ly (h;sira,l)h: (,]m.l, (,h(: Wign(:r flm(:l,i()n a,s a, s()rl, ()f st)a,(:(:-fr(;(tlmn(:y energy (l(:~sil,y (lisl, ril),d,i()~ 1)(: nm,inly (:()n(:(:ld, ra,(,(:(l ~(:a,r (,l~(: h)(:a,l fl'(:(lucncy line i~ tl~(: sl)a,(:(.,-fl'(:(lll(:l~(:y l)la~(:, ar()~n(l whi(:l~ t,l~(: sigi~ml (:~(:rgy is (:Xl)(;(:t(:(l to be lo(:at(:(l. "l'l~is is i~(l(:(:(l the tyl)i(:a,l l)(:Im.vi()~n" ()f (,l~(: Wig~(:r (listribution function as w(: will see i~ r(:la, ti(m to o(,h(:r signals a,s w(:ll [10.4, 5.7].

Polynomial phase signals Figure 6.14 shows the treIl(t of the W i g n e r distributi()ll fim('tions of the pure phase signal Tt

p , . ( q ) - c':;3~,

n _> 2

(6.4.42)

with in t)a,rti(:ular n = 4 an(t n = 5, whose h)cal frequ(m('y lines are respectively given by tim cubic curve ~q(q) = / 3 q 3 an(t the quartic (:11rve t~(q) = / 3 q 4. Note t h a t an (;xact expression of the W i g n e r d i s t r i b u t i o n flm('ti(m for the polynomial phase signal (6.4.42) is not obtainable. We see that, like }/Vubic, )/Vquartic(q, t~) a n d ]/Vquintic(q, t~) model essentially in conformity to the local frequency line in the q-ec plane, displaying a rapid

The Wigner Distribution Function: Analytical Evaluation

319

FIGURE 6.14. 3D image with the relevant q-t~ contour image of the Wigner distribution functions of (a) the quartic phase signal ei4 and (b) the quintic phase signal ei@. decay at the convex side of that curve and an oscillatory trend at the concave side. Note also that for even powers the Wigncr distribution function at the origin of the q-~; t)lane (:omes to be a if-point, being 1A;~.(0, h;) = ?i(~;). The n-th power t)hase signals of the tyt)c (6.4.42) arc a, parti(:ular ('asc of the more general t)olynomial phase signals ret)r(;scnte(t by (~pol(q)

--

Ci ~jN~I ~jqj ,

(6.~.43)

Due to the Weierstrass theoreIn, allowing for a unifornl polynomial approximation, with properly chosen coefficients, of any continuous function over a closed interval, waveforms of the type (6.4.43) may model the response function of different kinds of systems. From the optical viewpoint, higher-order powers than the second in the phase function in (6.4.43) may signify the corrections to the parabolic approximation of the distance of the image to the object, or also to the exact phase of the response function of a pure phasemodulation system. The Wigner distribution function may accordingly offer an effective tool for testing the departure of the response function of a thin lens from ideality, and thereby for studying the relative wave aberrations. In fact, an ideal lens should affect the signal by a pure quadratic-phase modulation, imaging for instance a unit-amplitude normally incident plane wave into the spherical wave front signal e-iko q2/2f (apart from a, constant phase factor accounting for the real thickness of the lens). If we could produce and display the Wigner distribution function of the transmitted wave, we might check the presence of aberrations produced by the lens, inspecting the deviations of the Wigner chart from the rectilinear pattern, expected for a pure quadratic-phase

Linear Ray and Wave Optics in Phase Space

320

modulation

(see w 7.8)[12.3, 10.32].

Sin'usoidal phase signal As a, [-iilal e x a m p l e of fr(~,(tU(;ll(:y n~()(hllated signals, we (:()nsi(ter the simis()i(tally fr(;(t~ml~(:y mo(tulate(t envel()t)(; ~(q) _ (,i[~-,)q+~ ,~,(~:rq)].

(6.4.44)

T h e l()(:a,l frc(t~u~,~(:y ()f tl~(; sig~m,l l~a,s a u ()s(:illa,t()ry w~,riati(m witl~ q a,1)()~t the ('arrit;r frt:(tllOll('y a,s .f~(q) --- h'()+ K,r"/(:(,s(h;fq)/'2. I[(;ll(:(; K,r is I,]|(; lll()(lllla,ti()ll fl'(;(t~(;~('y an(t ~/ st)(;(:iti(;s tl~(; ~m,xiuuu~l (;x(:~rsi()l~ ()f th(; l)lm.sr va,ria,ti(m. l~y (6.2.2), w(; ()})l,a,i~ fi)r l,l~(; Wig~(;r (listril)~ti(ni flu~(:ti(n~ 1,1~(; iut(;gral

1 [ (,-2i(h'--a'.o)q' ( i~co.~(.rq).~i,,(.'rq')dq~ o

(6.4.45)

!

whi(:l~ (:m~ t)c filrtl~;r l~m~(lh;(l ])y r~;s()rtil~g t() tl~e well k~()w~ i~h'.~tity O0

<.""+.s,,(.,.).

'l I --~- - - 0 ( 3

wlmr(; ,L, is tlie llr W , (q,

fllIl(:tir

r tiI'st ki~(t ()f ()r(lr '

'n [IN]. 'l'lu~s, w(; (:aJ~ write _,Lr

lt:~-- O0 O0

=

7t~

E

-

((5.4.47)

,, ),

wlli('ll slI()WS a, lill(' str~(:t~rr wit, ll ll()riz(nltal lille's tlll'(~llgtl t,ll(; l)r h'~ = h;o + 'lt,h:f/2, equally st)a(:r t)y lmlf r th(; m()(tula.ti()ll fl'C(tlU;Zl(:y. ~l'll(; c()lltrit)ution (hi ea,(:ll of t,hes(; lill(;s va,ri(;s 1)(;ri()(li(:ally witll q I,llt'()llgli Z~m,gllii,lul(;s gr 1)y t,lm Bcssel flnl(:ti(ms. Simlsoi(tal t)has(; sigImls lik(; (6.4.44) are at)I)r()t)riate t()(tr the light (tistri|)lltion tra,nsl~itt,(;(l |W a si~uls()idal I)has(; gratil~g, wl~()s(; l,ra,~s~fittance fl~nction tph(q) nlay in(lee(1 |~(; w r i t t e n a,s tph(q)-

c i~ Siil(~fq).

(6.4.48)

If the g r a t i n g is illlmfinat(;(1 (m one side by a unit-anlt)litlut(;, n()nllally incident plane wave, the light (tistrit)lltion on the o t h e r si(te is given t)y (6.4.48). We also mention, as countert)art ()f (6.4.48), the sinusoi(tal a.mt)litll(te grating, whose t r a n s m i t t a n c e t~m(q) is nm,tllenmtically r e p r e s e n t e d by the fim(:tion t~m(q) = 1 + cos(t~fq).

(6.4.49)

hlterestingly, w h e r e a s tile t r a n s m i t t a n c e function (6.4.48) t r a n s f o r m s an incident plane wave into the infinite s u p e r p o s i t i o n of diffracted waves J~(22 )cin~f q, the tra,nsmittance (6.4.49) splits the incident wave into three waves: the undiffracted wave a n d the + l s t order diffracted waves. T h e r e a d e r m a y calculate the W i g n e r d i s t r i b u t i o n function of the t r a n s m i t t a n c e (6.4.49) (see also w 8.3.4)

The WignerDistribution Function: Analytical Evaluation

321

FIGURE 6.15. (a) 3D plot and (b) relevant q-n elliptical contours of the Wigner distribution 1 1/4e-q2 function I/V~(q, ~) of the Gaussian waveform (5-g~) /4 6.4.5

Amplitude-modulated wavcforms Gaussian signal

As a prinm,ry ('~xample of amplitude modulated wa,vef()rms, we consider the real Gaussia,n signal 1 ~G (q) -- (~2)1/4C-~-;-2q27rw2, (6.4.50) where w is a, positive quantity, representing the e -e intensity extension of the signal. The corresponding Wigner distribution fimction W~ (q, t~) is easily obtained in the form 2 q2 w 2 t~2 W~(q,t~;)- xe---;-~~ ---{- , (6.4.51) which exhibits a Ga, ussia,n profile in both space and frequency dimensions in the (q, t~) plane. It is seen that )/V~ is the same as the Wigner distribution function W G of the Gabor signal (6.2.14), on due account of the response of the Wigner distribution function to space and frequency shifts of the signal. The Gabor amplitude (6.2.14) results in fact froin the plain Gaussian amplitude (6.4.50) through a suitable sequence of space and frequency shifts. Figure 6.15.a) shows the familiar "bell" shape of W~ (q, ~), with the relevant space-frequency contours being pictured in 6.15.b). Described by the equation ~2 ~2 w22 q2 + --U

-- C,

(6.4.52)

for arbitrary positive numbers c, the level curves are identified as ellipses, centered on the phase plane origin with axes determined by the signal p a r a m e t e r w and the assigned number c. The ellipse corresponding to the value G = 2, and

Linear Ray and Wave Optics in Phase Space

322

hen(:e t() the vahm ~: e -2 ~ 0.04 of ]A2~ is m a r k e d in the figure. It relates to the concet)ts of pha, se st)ace e m i t t a n c e a n d acceptance. We have ea,rlier touched on these (:()n(:et)ts ill w 2.5, w h e n illustra,ting the geometric-ot)ti(:s t)hase-t)lane view of the ev()lllti()n ()f the Gmlssian ray-(tensity (2.2.83), wt:()s(; spa,(:e-fre(tuency (:()nt()lu" illmg(~ ('(nlsists a,s well ()f Ii(;ste(1 ellit)ses (Fig 2.13). It ha,s t)(,(;n m(;ntione(t ttlere tha,t the (:oxmet)ts ()f emitta,n(:e a,n(l a,(:(:el)ta,n(:e t)r()vi(te a valid t()()lill rega,r(l, f()r illsta,ll(:(:,t() ill(;(lesigll ()f ()l)ti(:al(:()llnlnini(:a,ti()ll(:Im,nnels, ])(:illg il,st, rl:l**(:~md l() (t(:(,('l'llfil,(; 1.1,(: (:fli(:i(:ll(:y ()ftll(: (:(n:l)li*lg ()f 1,11(: ligld, fr()lll l.h(: s()llr('(: 1() file r(;('(:iv(:r ()r lil()r(: gt:ll(:rally fr()ill ()11(: ('()llll)()ll(:ll(, (,() a ll()l,h(;r ill

1,11(: (:lla.lll,(:l [1().25]-[1().28]. V~r(: a.ls()(;111I)l,i.,.SiZe(|

tl,~l.{, {,l,(; (t()ll(:(:I){,S ()f I)l,~.,.,.;(;-st)i-,.ce

c'm, itta'u,(:r ml(l ru'('t:pla'n(:t: at(: fmlliliar as w(:ll t() tll(: ()l)ti('s ()f (:lm.rg(:(l t)artich: 1)(:mils, 1)(:illg a.('('()r(lillgly Ils(:(l ill l,ll(', (l(:sigll ()f l,ra~lSl)()r(. (:lm llll('ls f()r I)a.rl, i('1(: a(:(:('h:ral.()rs. Ill l.llis ('(nlll(:('li():l, W(: 11()I.(; l.lml. 1,11(' l)lms( ' l)lall(: ()f l~lagll(:l,i(: ()l)l.i('S is Sl)m~('(! I)y l,l~(' ('()~j~gal(' wwial)l('s (:r,.r'), r('l)r('s('~li~g r(,sl)(;(:tiv(;ly (,11(: Sllmll l,I'a.llSV(:rs(: (h:via(,i()~ a~1(l 1,11(: r(:l(:wu~(, sh)l)(: ()f (,l~(' "r(:a,l" l)al,]~ ()f tli(: (:Imrg(:(l l)arti(:l(: witl~ r(:sl)(:(:t t() tl~(: de.,~ign, l,ra:je.~'l,ory ()f (,l~(: trm~sl)()rt (:lm.ln1(:l. Evi(l(:mly, (.r,.r') I)ara.lh:l (I,(: l)air (q, p) ()f g(:()***(:(ri('al ()I)(i('s. (7o'mph':r (la'u.,,'.,~ia,'n, a'mplil'ud~'

TI~(; hi(w(; g(;~(;ra.l c(nlq)h;x (lmlssia~ ml~l)lit~uh; fi~('l,i(n~

,)2 ((1) .

( '2 )1/4,, ~Tll? 2

,,,77--~(q-q() )~+i;~-~(q-q() )~ )-i,r

(C).4.5S)

is ()l)ta,il1(:(l fr(),,, a (l~m(Ira,ti(: l)l,as(: 111()(l~,la,(,i()ll ()f (,I~('. r(:al (;m:ssia.** a,1,1)litu(h: (6.4.50), fin'(,11(:r shift(:(l a,l()l,g (,li(: Sl)a,(:(: (linl(:,isi()n ])y %. I,l ()I)(,i(:al tern>, wav(:f()rnl (6..I.5()) ,imy a(:('()~nlts f()r t,11(, q-I)r()fil(: ()f tl1(, ml,I)lit~ul(: (listril)~,ti()n ()f a Gm:ssia** I)(:a,l, at, its wais(,; w(' **my (,lfink, f()r inslal,('(:, ()f (,1,(' (,ransverse (:nvel()l)e ()f a, TEI~I()() 111()(l(: fl'()nl a sial)h: ~m(:()llfil1(:(l lal'g(: l"r(:sl~(:l-mn,l|)er ()I)ti(:al res()11a(,()r a.(, tlle I)()siti()n ()f its 111ilfinmlll lat(:ral (:x(,(:,isi(),l [15]. CorreSl)()n(lingly, (6.4.53) nmy (l(:s(:ril)(: the q-l)r()fih: ()f th(: a,nll)lit, lul(: (listril)~:tion of a Ga,ussiml 1)(:mll a.t th(, t)()sit, i()n a,l()ilg its ()wn ()t)ti(:a.1 axis wller(: it, exhibits w as l)eanl ra,(li~ls a,l(l/~ a s wa,v(~,fr()ll(, (:~lrvat,~,re, s(:ale(t by (,ll(~ light, wa,v(~number; t)()ssit)ly, tlm ()t)(,i(:al axis ()f the |)eanl l~my not |)e a,liglm(t with tlle (:hosen z axis t,hlls a(:(',()~mting for (,lm space a,n(t frequency shifts (tist)laye(t in (6.4.53). C()nv()lvillg ill the fre(t~mn(:y va,ria,l)le the W i g n e r (tistri|)uti()ll flmctions (6.4.51) a,n(l (6.4.34) of the rea,1 Ga,ussia,n a,nlt)litude a,n(t the (:hirt), we o b t a i n f()r the W i g n e r (tistributi()n flmt:tion of the envelot)e (6.4.53) the expression

W~(q,~:)- -}~-~ (q-q`,>~-~-~I'~-~o-/~(q-qo)J= - %(q)12[@~(~-,)l2.

(6.4.54)

T h e elliptical contours of W~ (q, r~) axe traced in Fig. ??, the relevant equation being ~22 ( q - - q o )

2 +--ifw~ [/~; --/~o -- fl(q -- qo)] 2 - - : .

(6.4.55)

The Wigner Distribution Function: Analytical Evaluation

323

FIGURE 6.16. The elliptical contours of the Wigner distribution function of the chirped and shifted Gaussian signal (6.4.53) may be seen as resulting from a translation and a ~c-shear of the elliptical contours of the Wigner distribution function of the basic Gaussian amplitude (6.4.50) (see Fig. 6.15). In the graph, w = 2, q~, = 2, ~, = 4. A compa.rison with tile ellipses relevant to the Wigner distribution flmction (6.4.51), also traced in the figure, Mlows one to recognize a translation of the ellipse center from tile origin to the t)hase-pla,ne point (qo, ~',0), a,nd a h',-shcar, (tctcrinined by the curvature t)a,ra,Inetcr/~. W i t h increasing the value of w the (:hirp (tomiim, tes over the Ga,ussian amt)litu(te, which widens a.lmost uniformly over the q-axis. On graphs, with w ~ + o c the phasc-t)la,IlC ellipse (:ollapscs into the ret)resentative line of the chirp signal, i.e. the line delta, 5(~;-~; o - [ J ( q - q o ) ) Likewise, for small va,hles of w the Ga,ussian is almost like a, (5-flm('tion, and so the representative pha,se-plane ellipse collapses into the vertical line ~ ( q - %) proper of a point source. Notably, distributions (6.4.51) and (6.4.54) are positive t h r o u g h the whole phase plane, in accord with the Hudson theorem [10.6], which, a.s mentioned in w 6.3.6, identifies the Gaussia,n signals as the only wavcforms whose W i g n c r distribution function is positive everywhere in the space-frequency plane. S u m o f two G a u s s i a n s i g n a l s

As a filrther example of the non linearity of the W i g n e r distribution fimction a,nd accordingly of the beha,viour of the cross-terms entering the expression of W(q, ~c) when the superposition of signals is involved, we consider a signa,1 composed by two identical Gaussian signals, which are however shifted one to the other in b o t h space and frequency. So, we consider the a m p l i t u d e 1 1 q2(q) -- ( ~ 2 ) l / 4 e - - s ) 2+i~lq + (~_~w2 (6.4.56)

)l/4~--~--~(q-q~)9+i~2q.

Linear Ray and Wave Optics in Phase Space

324

l;l(;tIl{l,] 6.17.31) i~mgc m~(l rclcvm~t Sl)aC(.,-fr(:q~c~wy c(nmnu" ph)ts ()f tl~c Wiggler (listrib~l(,i()~ fi~c(,i()~ ()f (,l~(' sig~al (6.4.5(i), c()s~sp()s(,(l ()f (,w() (;m~ssias~ aSS~l)lit,~t(l(:s wi(,l~ (a)')v - 1.5, q~ - q., 1.5, +,') ~'., - 1.5, m~(! ( b ) q, - q., - 2.5, t<~ - K,, - 2.5. Afl,(:r s()IIl( ~. l(:~gl,l~y a.lg(:l)ra w(: (:l~(I ~II) wi(,l~ 1,I~(' (:xl)li(:i(, (:Xl)I'(:ssi()l~ ()f (,li(: Wig~(:r (lis(;ril)~(,i()~ fln~('(,i()~ wl~i('l~ wri(,(:s

II('r(', W~; is (,ll(' Wigll('r (tisl,ril)llti()ll filll(:I,i()ll (6.4.51) ()f 1,11(; r('al ( ; m l s s i a n mllI)li(,ll(l(; (6./1.5()); q,,,, h,,, a ll(l q , i , h',! (1(:1~()(,(: (,l~(: 111(~.~1,11~i,11(1 (lift'(:l"(:~(:(: St)a,(:(: a.l~(I fI'(;(l~lt;l~(:y (:()()r(li~a(,(:s, wl~ils(, ~ is a (:()llSl,a.ll(. I)l~as(' as q,,,

-

q'+% 2

, h',,,,

~'+~~ ., 2

q,l

.

.

q ~ -

.

q2,

h:,l

h'l

h'~", ~ - q,,h'~

9

(6.4.58)

Ill a,(:(:or(t wil,ll (;lit: (:t)llsith:ra,(;itnls ()f ~ 6.3.,1, ttis(;rit)llI, i()ll (6.4.57) (list)lays l,wo sclf-t,(:rnls, nm, illly l()('a,(;c(1 a,rt)lmtl (;h(: st)a,(:t'-Kt:ttllt:ll(:y (:(;nl,rt)itls ()f t,hc (;wo COnlt)OIlcnt, si~lmls, i.(:. (q,,~,,) anti (q2, h,2). Tilt: crt)ss (;crm n~ainly develops aroun(1 (;lie (:(;llI,l'al I)()ild, (q .... ~,; ) w i t h (;lit: Ga,llssia,n t)rt)filc |)(:izlg sinllsoidally mo(tlfla,(,(:(t; t,lit; ()st:illa,(;it)ns ()(:('11r or(;llogOlla,lly (;() t,h(: li~l(: (:(nnlt:(:(;i~lg (;he two signal ccn(;crs (q,, t,;,) a,,,d (q2, ~';2)- As it can |)(; sccn c o m p a r i n g (;he surface plots of Figs. 6.17.a,) a n d 6.17.1)), corrcst)on(ting (;o (;he sa.mc va,luc of w b u t difl'crcn(; va,hlcs of t,hc st)a,('c a,nd fr(;qucncy shift, s, t,hc visil)ili(;y of t, hc c r o s s - t e r m oscilla,t, ions t h r o u g h tll(: o v e r l a p of (;he sclf-tcrnls ix g o v e r n e d by t h e rcla(;ivc value of t,he t)ha,s(;-t)la, nc (tista,n(:c be(;ween (;lie t,wo self-(;erms (:enters to (;he r(:spectiv(; spa,c(: a,n(t fr(;quency ex(;en(;s, fixed |)y t h e signal t)a,ra,meter w. Intcr(:stingly, gra,phs like t h o s e in Fig. 6.17 are e n c o u n t e r e d in q u a n t u m o p t i c s w h e n d e a l i n g w i t h t h e superposi(;ion of sta,(;es. W e m a y m e n t i o n , for insta, n(:(;, t h e q u a n t u m s t a t e o b t a i n e d s u p e r p o s i n g two c o h e r e n t s t a t e s of (;he s a m e q u a n t u m n u m b e r a n d a given p h a s e difference ((;he S c h r 6 d i n g e r cat

The Wigner Distribution Function: Analytical Evaluation

325

state). Of course, the occurrence of the cross term in the Wigner distribution function of such a state is a consequence of the superposition and the bilinearity of the Wigner integral. Accordingly, the cross-term shape is a manifestation of the separation of the two states in the pertinent phase-plane (i.e., the q-p oscillator phase-plane), ultimately determined by the relative value of the phase difference to the quantum number. Thus, as noted in w 6.3.7, the cross-term behaviour in the Wigner surface, yields a key-tool to individualize the occurrence of overlap, and hence of interference and transitions, between the individual states in the superposition state. Also, it is closely associated with the features of the relevant photon statistics [10.24].

Hermite-Gaussian beams As is well known, the Gaussian signal (6.4.53) plays a significant role in optics as both an effective signal and a convenient model signal. Gaussian beams are introduced in optics in several different ways; for instance, as solutions to the paraxial Helmholtz equation in uniform and quadratic index media, or as fui~dainei~tal modes of optical stable resonators. The basic features of signal (6.4.53) are the paraboloidal profile of the wavefronts and the Gaussian shape of the intensity distribution; the former is accounted for by the complex quadratic exponential, the parameter/~ identifying the curvature of the wavefront, and the latter is accounted for by the real qlmdratic exponential, the parameter w fixing the beam radius. The Gaussian envelope is not the only solution to the aforementioned optical problems. The other solutions, known as Hermite-Gaussian beams (or modes), have the paraboloidal wavefronts as (6.4.53), but do not exhibit a plain Gaussian distribution of the intensity across the axis as the Gaussian envelope is modulated by Hermite polynomials [15]. We do not dwell on the mathematical and physical properties of the Hermite-Gaussian modes, but simply consider the basic real Hermite-Gaussian amplitude, Fn~n (q) -- (2~n!V'~)

1/2

Hn

(~)~

-

-~22q2

,

(6.4.59)

and calculate the relevant Wigner distribution function. Here Hn(-) denotes the Hermite polynomial of degree n and the constant factor is fixed to satisfy the condition: 12 - 1 [18]. With n - 0 we recover the fundamental Gaussian waveform (6.4.50), as Ho(x ) - 1. In Fig. 6.18.a) the profiles (6.4.59) are graphed vs. q/w for some values of n; note that the Hermite polynomials are even or odd functions according to whether n is even or odd. It should be mentioned, perhaps, that (6.4.59) may be interpreted as the position representation of the energy eigenstates of the quantum inechanical harmonic oscillator, and so of the Fock states of quantum optics (see also w q2 5.4.2). Indeed, within the quantum mechanical picture, the argument ~2/2 +

Linear Ray and Wave Optics in Phase Space

326

1

y,

? G

.'~ ()

n~l 9.......n 2

0.4

~

I

I

I

I

q

|

,o

/'". "",'~-" n 4/-'x

,,! X

(

0.2

X

nl

q

n-2

,,"/ .~_.'?_._~_.L~"~"~tt~~x"-'-'~"~ "

9-' Y'7"-()'2 -! (I

I I

1 2

i 3

-0.4

(.t/14" 4

()

(a)

FIGIIRE

I

i

I

I

2

4

(,

8

I

E

i0

(~)

6.18. ( a ) l l c r ~ i t ( ; - ( l a ~ m s i m ~ 1)(;m~ pr()tih;s vs. q / w fi)r s()~m I)(,m~ ()r(h,rs '~,, m~(t (b)

(:()rrcst)()~(li~g W . ~

,,

~u, 2 + "' 2~'~

-l)r()fih;s vs. E -

"

.u~2/.2 p2 2

()f tim (;Xl)()~m~d,ia.l fa.('t()r, (;~(,(;ri~g tim Wi~lll;r (lisl,rit)~(,i()~ fim(:ti()~ W~ a,s w(;ll a,s W.~,, t)(:l()w, ('all ])(; ~ll~(l(;rs(,()()(! as (,1~(; (;~(;rgy fi~('l,i()n ()f tim t~a,r~()~i(' ()s(:illat()r, wl~i(:l~ fi)r giv(;~ i~l)~l, (lata i~(livi(l~m.liz(; i~ tim q-p ()s(:illa,t()r l)lms('-l)la,~(; tl~(' fa,~filia,r (;lliI)ti(:al (:(n~t()~rs lik('~ tl~()s(; i~ Fig. 6.15.|)). Tl~(; r('a,(h;r is i~vit(;(l t()('()~In~t(; tl~(; fr(;(t~U;l~(:y Sl)(;(:tr~ll~ ~,~;,, (h'). h~ this ('()~n~('(:ti()~, w(; ~()t(; tl~at tt~(; II('r~fit(;-(;a,~ssim~ fi~(:ti()~s ((i.4.59) a,re (;ig(;nfi~n(:ti()ns ()f tl~(; l'()~u'i('r trm~sf()r~ ()t)(;rat()r ,~, f()rn~i~g i~(l(;(;(l a~ ()rtl~()n()rma,1 1)()th th(; sI)a(:(; m~(1 fr(;(l~u;~('y (l(n~mi~s, i.(;., s(;tti~g !1)

ki,

"It)

ill th(; tw()(tim,1 (l()llm,ills. ~l'll(;ll, in the .~' (:()()r(tina,t(; ()ll(; fill(ls

since in tile (liln(;nsi()nh;ss va,ria,1)le :r -

K, 11)

~/~ th(', F()llri('a" trmlsf()rnl @~a, 9(x)

takes th(; sa,lll(; f()rlll as tll('~ wa,v(; fim(:ti()ll ~u.~,, (;r) ill tll(; (:()rr(;st)()n(ling va,riable x - v ~ . As m(;nti(m(;(t in w 5.4.1 an(t 5.4.5, th(; cigcnva,hm equation (6.4.60) lm,s t)(;(;n th(; k(;y-I)()int for intr()(hming the fra(:ti()na,1 F()lH'ier t r a n s f o r m o p e r a t o r ,T"~, (tefinc(t in fa,(:t t h r o u g h a, g(;neraliza, ti()n ()f (6.4.60) to eigenvalues evenly set)a,rate(t t)y a, fraction of the i m a g i n a r y unit; ~m,xnely, i -

(6.4.61)

We use (6.2.2), so tha,t a,fter rea,rranging tile variables we o b t a i n 2nn! v/~ C

w2

'2

(6.4.6e)

The Wigner Distribution Function: Analytical Evaluation

327

FIGURE 6.19. Surface plots and q-t~ contour plots of the Wigner distribution fimction of the Hermite-Gaussian signal (6.4.59) with w = 1.5 and order (a) n = 4 and (b) n = 9. Taking advantage of the parity p r o p e r t y of the Hermite polynomials: H~ ( - x ) = ( - ) ~ H ~ ( x ) and the integral [19] +

+

(6.4.63)

-

where L~ (-) denotes the Laguerre polynomial of degree n [18], we end up with the nice expression n (q,

-

2(12

w2 2 ~2 -

L,~[2(2~-~~ +

w

2N2

2

)]"

(6.4.64)

Expression (6.4.51) for the Wigner distribution function of the f u n d a m e n t a l Gaussian inode (6.4.50) follows froln (13.4.64) with n = 0, being L 0 (z) = 1. In Fig. 6.18.a) the amplitudes (6.4.59) are graphed vs. q / w for some values of n. Notably, on graphs the contour plots of W u ~ reproduce the elliptical structures (6.4.52) pertinent to W~, with the basic difference being that, unlike W~, the W ~ , ' s take on negative values in some regions of the space-frequency plane in accord with the order n and the value of the a r g u m e n t of the Laguerre polynomial. In Fig. 6.18.b) we have plotted W ~ , as a function of the q u a n t i t y g - 2q2 -~ ~ w 2

2t~2 2 "

We see t h a t at the phase-plane origin

W~ n (0, O)

(-)~, 7l-

(6.4.65)

turning to be positive or negative according to whether the mode order is even or odd. Also, we see t h a t Wu~ n oscillates versus g roughly up to g ~ 2n + 1, and then for greater values decays rather quickly. Accordingly, as it can be seen

328

Linear Ray and Wave Optics in Phase Space i n

I

1

0

I

I|--

I

rl 2.

!

rl 0.5

I

I

1

I

G-:3.5

, 5

g" o ~,~,o.~-,~

/k

.

-',

nos

"

"

,'.

9-

t ,' / X ' ,

<,

n 4

/

-4

-2

-O.5 0

?

I -4

q/n'4

(a)

1.5 I -2

1 0

2

1 q/w 4

(b)

F I ( ; I J I I E 6.20. l l e r ~ i t ~ c ~ s l ~ - ( l a , ~ s s i m ~ lw.m~ Ira)files vs. q / w fear Sl~e<'ilie
i~ l,ll(; e x e ~ l ) l a r y l)h)ts ()f Fig. 6.19, i't;h;wtld, l,t)l,l~(', (~r(h,,rs t~,-~ 4 a.~(1 ',, = 9, 14~,uo,, (q, h') ix ~m,i~ly (:(n~(:(;~d,ral,(;(l ()w;r 11~(: r(,gi()l~ ()f 1,1~('SlmC(;-fl'(:r l)la~e w m~(l V / 2 n,,+ 1 ~-l,(:l'()SS wl~(;r(; t,l~(; sig~ml its(;lf ix t:()llt't;~d, ra.l,(;(l, i.~',, l'()~lgl~ly V/2~ + 1 ~tim ()rigil~ resl)e(:tiv~;ly a hn~g tl~e q a.~l(t h" ax~;s. I~ a.~l(lit,i ~ , w~' ~ a y ~(~t(; tliat l/V.u,:, ' (q,h) ()s('illah's i~si(h; l,l~(; (;llil)ti(:al (:()lll.()llr i(l(:~d,iti(:(l as w(;ll t)y E(t. (6.4.52) fin 1,1~: wtl~l~; ~f e ~ 2(2~ + 1), wliilst (l(;('ayil~g rai,l~;r fast ~l,si(h;.

He'rmitr

(ht'u.s.~ia'n .sig',,a,l,s'

A l~(;w set ()f t)(,,~,l.Ill s()llll, i(~I~s ()f the t)a,ralJ(fli(: waw; (:(llm,l,i(ni lias l)een re(:(uit, ly (;la,l)()rat,(;~l, g(;ll(;rally r(;t)()rl,e(l a.s si'n'usoidal lh'.'t'mit~'-Ga'ussia',, m()(les I16]. S l l ( : h s()llll,i()lis at(; ill l, llt; fl)rni ()f silllls()i(la.1 fllll(:tit)llS ()f a. t)()ssil)ly (:oniph;x a.rgllliient tiliies a, r i()lia.1 th;rliiite-C,a,llssiali t)ealii fa,r W e (:ox~si(l(;r ll(;rc ()ll(; ()f tile; si~llI)h;st f(n'llis ()f tl~(;s(; ll(;w s()l~lti(~ls, lia,incly,

the Hernfite-(:()sll-Gmlssiali t)eani, which is represelite(1 t)y tlie rea,1 a.liiplitude 2

~n.,..g,, (q) -- (2,,.,,!v,5,,:&,(t,))l/2Hr',(v~q)

_

1

"'-7~"2(:('sl'(~q),

(6.4.66)

w h e r e t~ is a, rea,1 t)a,ra,Iimter, and the factor ~,,, (~) Colifoi'nis to the n o r m a t i o n condition for lmity, a,n(t llell(:e ~,,,(t~)- 1 + e

f~2w2

2 L,~,(_f~2w2).

(6.4.67)

T h e H e r m i t e - c o s h - G a u s s i a n modes can optically be pro(time(t, for instance, by passing an H e r m i t e - G a u s s i a n m o d e from a stable r e s o n a t o r laser t h r o u g h an a p e r t u r e h a v i n g a c o s h - a n i p l i t u d e t r a n s m i s s i o n fllnction. N o t i n g t h a t cosh(flq) o( e aq + e -aq, we m a y i n t e r p r e t the b e a m s (6.4.66) as resulting from the s u p e r p o s i t i o n of two conventional H e r m i t e - G a u s s i a n modes

The Wigner Distribution Function: Analytical Evaluation

329

F I G U R E 6.21. Surface plots and relevant q-~ contour plots of the Wigner distribution flmctions of the cosh-Gaussian signal (6.4.66) with w = 1.5 for (a) r / = 2 and (b) rl = 5.

weighted by the factors c ~q and C-f]q. It becomes evident therefore that it is the value of Ft in relation to that of the Gaussian beam radius w to determine the overall shape of tile resulting beam. In fact, in Fig. 6.20 we have plotted tile amt)litu(te profiles versus q / w ()f tile flm(tamenta,1 mo(te ('n = 0) an(t of an higher order mode (n = 4) for increa,sing vaJues of the t)arameter r / = ~w. Referring to the flmdamental mode, we see that the profile firstly becomes fla.t, closely resenfl)ling the shape of super-Gaussiail fieht distributions, that a,re greatly desired in high-t)ower la,sers due to their potentiaJ of efficiently extracting energy fi'om the amplifying medium (see next pa.ragraph). With increasing r/, a, center deep appears which becomes deeper and larger with increasing values of ~1, thus fina,lly modelling the overall distribution into a double-lobe envelope. This is basically the effect a,cted by the cosh-factor in (6.4.66) for higher order modes; the near-a,xis oscillations of the conventional Hermite-Gaussian modes are more or less strongly reduced in favour of a magnification of the lateral lobes, in which then the beam energy comes to be concentrated. Hence we may guess that the Wigner distribution function will come to resemble that of the sum of two Ga,ussia,n-like signa, ls. It is an easy task to verify that (6.4.66) can be recast as v ~7-t.c.gn (q) -- (2"n!v/-~wen(~))1

{C -(-q-

-Jr- C (~w-~2) },

T] z ~ W .

(6.4.68) Accordingly, carrying on the calculations, we explicitly find W~.~.~ (q, ~)

1 {~],~ ~n (q, ~)cosh(2f~q) + R[142~ n (q, ~ - it))]} ~(a)

(6.4.69)

Figure 6.21 shows the 3D image with the relevant contour plots in the q-~

Linear Ray and Wave Optics in Phase Space

330

'

d~

'

0.5

"~"

0.5

0

-2

o

qln'

0

2

(a)

-5

0

K )t'

5

(h)

I,'IG[)I{I,~ (;.22. l>r()files ()f (a) tim S,~l)er(;m~ssia~ (list,rit)~ti()~s (6.4.7()) vs. q/u, fi)r 'tt,- 2, 5 a~(l 2(), ;u~(t ()f (b) tim ('()l'r(:sl)()~(li~g fi'(:(l~(;~('Y st)(;('tra vs. ~',u,. t)laa~(: ()f (,1~(: Wig~(:r (lisl,rii)~ti()~ fit~(:(,i()n ((i.4.(i.()) f()r (,1~(: fi~(la,~(:ntal m()(l(: i~ (:()rr(:sl)()~(l(:~,:(: wi(,l~ (w() val~(:s ()f (,l~(: l)ara.l~(:(,(:r q.

SupcrGaussia'n signal W~: (~()~lsi(ter (,ll(, Slll)('l'(;allssia,ll sig;llal, wlli('ll Call lflaillly I)(' r('tw(;s(ud,e(t t)y (,If(: a11q)li(,ll(l(: filll(:(,i()ll :(q)

-- :,)~'--(~" )",

(0.4.7())

'l~ t)(:ing a, t)()si(,iv(; ill(,(;g(:r; wi(,ll 'n = 2 w(: r(:ga.in (,h(: (~a,llssia,ll signal (6.4.50). T() a,i(l(: ild, lli(,i()ll, Fig. 6.22 sll()ws (,ll(: t)r()til(:s ()f (,ll(: sigllal (6.4.7()) a,n(1 the ('()rr(:st)()n(lillg l"()llri(:r (,l'll,llSf()l'lll ill (,ll(: (:as(:s l~, = 2, 5, 2(). 'I'll(: Sl)a,(,ial I)r()fil(: is l)a,si(:a,lly Ila,t (,llr()llgl~ a (:(:rl,ai~ in(,(:rval a,n(l Sl~()()(,l~ly g()(:s t() z(:r(), wi(,h the t)r()fil(: wi(ltl~ a,~(l (,1~(; (l(:gr('(: ()f s~()()tl~(:ss I)(:i~g g()v(:n~(:(l l)y (,1~(: l)ara,m(:ter u~ a,n(l th(: ()r(l(:r ~. (',()rr(:sl)()n(lingly, in (,h(: fi'(:(0~(u~(:y (l()~m,i~ (,I~(: (lis(,ril)~ti()n (lisl)lays a, hyl)ri(l l)(:l~avi()~r l)(:(,w(:(:~ a, Ga~ssia,x~ aa~(l a si~(: fi~(:ti()~. S~q)erGa,~ssia,~ a,~I)]i(,~(l(: (listril)~ti()ns lik(: (6.4.70) a,r(: I)r()(l~(:(:(l l)y a, wi(le (:lass ()f la,s(:r syst,(:n~s, as, f()r i~sta,n(:(:, la,s(:rs with higl~-ga.ii~, sa, t~ra.(,(:(t active ~n(:(lia, as well a,s la,s(:rs (:(t~iI)I)(:(t with va,ria))l(: r(:fl(:(:tivity ~irr()r (:avities [17]. T h e Wig~(:r (tistril)~ti()~ fln~(:ti()n of the signal (6.4.70) f()r n > 2 (:annot be analyti(:a, lly (:a,l(:ula,t(:(1. Fig~r(: 6.23 shows the trend of th(: Wign(:r distribution fimction of th(: signal (6.4.70) for two values of n. As (:xt)(:('t(:(t, the Wigner distribution flm(:tion ten(Is to t)e mainly con(:entrate(t wittfin a. region of the q-t~ plane, whose extensiol~s rela,tc to the widths of th(: energy distribution of the signa,1 in the q and t~: (tomains, exhibiting os(:illa,tions inside and fast decay outside.. For a given w with increasing the order n the. st)atial profile of the signal (6.4.70) approaches t h a t of a recta,ngular signal a,nd, accordingly, the W i g n e r distribution function comes to display a sin(:-like behavior with respect to the frequency variable.

The Wigner Distribution Function: Analytical Evaluation

331

FIGURE 6.23.3D images and q-s contour plots of the Wigner distribution fimctions of the superGaussian amplitude (6.4.70) with w = 2, and (a) n = 5 and (b) n = 13. 6.4.6

An example of Wigncr nonreprcscntability

As an examt)le of Wigner nonrepresentability, we consider a fimction W(q, ~;) a t t a i n i n g a constant va,hle over a rectangular region of the (q, ~;) pla,ne, i.e.

W(q, ~;) -

1

0

Iql ~ Q ;

()therwise.

I~:1 ~ / ~

(6.4.71)

We a.ssume W(q, h;) be a, Wigner (tistrilmtion fllnction arising from a, signal, whose amplitude, by the inversion foi'nmla (6.3.28), would obtain in the form

7:(q) _

2K sinc(Kq) ~*(0) 0

[ql < 2Q Iql > 2Q.

(6.4.72)

The absolute value of 7:* (0) could be inferred from (6.3.26), which, indeed, on account of (6.4.71), gives

i~(q)l 2 - ~ 2 K [ 0 ~ . a h~nr

Iql-~ Q Iql >Q,

(6.4.73)

t~*(0)1 ~ - 2K.

We see t h a t results (6.4.72) and (6.4.73) arc in contrast to one another. In fact, (6.4.72) yields for the signal m a g n i t u d e 17:(q)l a q-dependent expression vanishing outside tile interval - 2 Q <_ q < 2Q. In contrast, (6.4.73) gives a constant value for ]7:(q)l, which also would differ from zero within the smaller interval - Q _< q < Q. Thus, we m a y argue t h a t the 2D function (6.4.71) is

332

Linear Ray and Wave Optics in Phase Space

FI(iI.~RE 6.24. (,,) 3I) vi(,w a~(l (b) r(,h,va~t, ('(,~t()~ar l)i()ts ()f tl~(, Wi~(,r (listril)==t.i(~==t'~=~('ti()~ ()f t,l~(, sig~al (6.4.74) witl~ (2 = l a=~(l K = 2. ~()t, a. r('])r('s('~ld,a])l(' W i g ~ ( ' r fl~('ti()l~. I~ fa('t, t,l~(: r('~a,(l('r l~my (:a,,-;ily v('rif.v t,l~at,

~('I)

-

q ~/2Ksi,~('( Kq),'(;(~t (-~ )

r162 liot~ r('I)I'r162162 I,tl('~ r t'llll('I,i()li (6.4.71). It, r ])(, ('Xlm'SS('~r a,s t,llc (:()llvr ir r ~ ()f t,w() sill(: ['llll(:I,i()llS, 1)r(;(:iscly a sill(: t'llll(:I,i()ll lik(~ (6.4.3) (:()rrcst)(m(Iin ~ I,() i,ll(; r('~r a n(I a, sillr fllll(:/,i()ll like (6.4.7) r t() t,he sin('~-si~lml, lll(I(:ctt, t,ll(; l'(;a(i(;r IlI,I:V (;a,sily W()l'k ()llt, t,ll('~ mmlyti(:a.1 cxI)rcssion of 147~(q, h',) illVOIvill~ a r162 illt,(;~4ral ()v('.r I,ll(; t'r(:(lll('~llr varia])le; a l,~o, it, t~n'll,~ ()~11, l,()])(; (:()llfill('~(l wit,]lill I,ll(; vcrtic',al sl, ril)Jq] < 2Q. As a,n (;xamt)le , Fig. 6.24.a,) sll()ws l,ll(; 3 D illlag(; of l,lle Wigll('~l' (lisl, ril)lll,i()zl fllll(:t, ion of t h e Wa,V(~foI'Ill (6.4.74) witll Q - 1 mlcl K -- 2. Ev(~II t, ll()l~gli it, l,('~Iltls t() |)e m a i n l y co~l(:(mtra,t,('~(1 withi~l a, r('x:t,a,~lg~fla,r i'('gi(m i~l t h e (q, ~') I)lml('~, a,s cl(~a,rly e v i d e n c e d |)y t,h(~ (:()rI'cSt)on(ling (:()ilI,()~u" t)l()t,s ()f Fig. 6.24.])), it, cxl,(m(ls indefinitely t h r o u g h t,he ~:-diwc'tion; also, it, (loes not a,tta,in a, c()nsta,nt va,lue over t h e region of ma,xim~lm localiza.tioll. W e will se(m in Cha, t)tcr 9 t,ha,t th(~ st)a.c:e a,ll(] st)a, tia,1 ext(msi()ns of t~he signal reflect into t h e (:orrest)oncIing ext, cmsiolls of t h e W i g n e r (tist, ri|)~ltion fl~nction. So, (6.4.71) w o u l d corI'espon(] to a, signal t h a t s h o u l d be linfit('~(] to a, finite interva,1 in b o t h t h e spa,ce a,nd frcqllency d o m a i n in c o n t r a s t to the w i d t h principle of F o u r i e r ana,lysis. T h i s w o u l d ilnlnedia,tely s u g g e s t the n o n r e p r e s e l i t a b i l i t y of 2 D f u n c t i o n like (6.4.71) a,s W i g n e r d i s t r i b u t i o n fulIctions.

The Wigner Distribution Function: Analytical Evaluation

6.5

333

Summary

After i n t r o d u c i n g the forma,1 definition of the optical W i g n e r d i s t r i b u t i o n function, which indeed draws its r o o t in the issue of expressing q u a n t u m mechanics in a, form as similar as possible to classical mechanics, we have e v a l u a t e d a n d displayed on g r a p h s the W i g n e r d i s t r i b u t i o n flmctions, a n d the relevant W i g n e r charts in the (q, K,) plane, of a large variety of signals, which are of interest in optics as well as in q u a n t u m a n d signal theory. We have gained a feeling for t h e p o t e n t i a l of the W i g n e r d i s t r i b u t i o n function to convey i n f o r m a t i o n a b o u t b o t h the space a n d frequency features of t h e signal in accord as well w i t h the inherent limitations, d i c t a t e d by the u n c e r t a i n t y relation, concerning the entire space a n d the entire f r e q u e n c y ext e n t s of the W i g n e r d i s t r i b u t i o n flmction. R e m a r k a b l y , it has been a p p a r e n t t h a t in general the W i g n e r d i s t r i b u t i o n f u n c t i o n takes on negative values over some regions in the phase plane. This feature prevents us from i n t e r p r e t i n g the W i g n e r d i s t r i b u t i o n function as a t r u e space-frequency e n e r g y distribution even t h o u g h its m a r g i n a l s correctly convey the energy d i s t r i b u t i o n s of the signal s e p a r a t e l y in space a n d in frequency. In the next (:haptcr we will rc(:onsider the W i g n c r ret)rescnta,tion, a,nd the rela,ted issue of c o m p o s i n g into a single " e n t i t y " the descriptioits s e p a r a t e l y (:onveyed by the signal a m p l i t u d e a n d the Fourier s p e c t r u m , w i t h i n the (:ontext of signal theory, t h u s para,lleling the quantum-me(:hani(:s oriented p r e s e n t a t i o n given here.

Problems

1. (a) Prove that the quantum observables f~ (~, ~) = (q'P+ "P'q)/2 and f2 (q, "P) - q'P differ by -ih/2. However, they would turn into tile same classical flmction f -- qp if we trivially replaced the operators q and p with the classical variable q and p. (b) Verify that the same occurs with tile observables f'~ ( ~ , ~ ) - ~--~2~and f2 ( q , P ) - (~2~2 +~2~2)/2, which differ by h 2. Similarly both would turn into the same classical function if one simply replaced ~ -+ q and p --+ p. 2. Show that (a) if two functions 9~, (q) and qP2(q) vanish outside the space intervals (Qa, Q6) and (Qc, Qd), respectively, then the corresponding cross-Wigner distribution fllnction turns to be zero outside the vertical strip Q"+Q~: 5 q <_ Q~,+Qa in the phase-plane, and likewise (b) if the frequency spectra of two flmctions vanish outside the frequency intervals (ha, n6) and (no, ha), respectively, the relevant cross-Wigner distribution function is nonzero only within the horizontal strip ~~+~'~ _< t~ <_ ~++~,a2 in the phase-plane. 3. Verify that the W~ (0, 0) < 0 for functions ~(q), such that q~(q) > 0 or qqa(q) < 0 for all q over the real line. Check this property for the power functions: qa(q) - q~, distinguishing between odd and even exponents. 4. Prove that smoothing the Wigner distribution function of a given signal qp(q) by the Wigner distribution function of any normalized signal, say ga(q), the resulting distribution I/V~.......~ho,,(q, h) still satisfies the energy relation, i.e,

//

l/V~..... thea (q, n)dqdh - ] / Y V ( q ,

h)dqdt~,

Linear Ray and Wave Optics in Phase Space

334

whilst t h e m a r g i n a l s are lost, since pro.jecti~g W.~....... ,~....~(q, n) on the q a n d t~ axis one o b t a i n s t h e a v e r a g e of t h e power s p e c t r u m a n d t h e e n e r g y d e n s i t y of # t h r o u g h the power s p e c t r u m a n d t h e e n e r g y d e n s i t y of ~, a c c o r d i n g to

/' /

w ......... , .... , (q,,~,)d,~

-

142 ........ ,....,(q,~,)d~,, -

,

/' /

,.,5(,.,:')12 .g(,,: -

,.,-,')l:~a,,: ',

~(q,)12 ,t/~(q - q') 2dq'.

5. l)r(we rclatir ((i.3.7()) as ttw, ~ecessary mul Slltii(:icilt ('r162162 fl)r a give~ flnmt,io~ W ( q , t ~ ) t,() be a rel)resexfl, at)le Wig~mr (list,ritn~l,i()~ fl~wl,i<)~. (Iliad,: relal,i(~ ((i.3.7()) is straiglfl, forwar
for wlfict~/:~, 7 > () mlr /#? m~(l "), satisfy l,lie er

-

4(~ 2 > (), is Wigi~cr relm'se~mdfle ()Ifly if l.lie l)arm~leters (,, /~ /~'~ = (,2 + 4.

7. W r i t e ~h)w~l tlw e x p r e s s i ~ of l,lw Wig~wr ~lisl,riln~l, io~ ftnwti()~l ~)f tim ~hnflfle rectm~gle ~1 { , - , , ~ t ( ~~q,, ),.

~(q)

~,"

," + feet( ~'~ '~'' )~"~'-'}. ',

Sll(~w l,]mt il, is Zel'~ f~l" q <.... % ()tcq" 2 - 0,,~ m~l q > , q t ~ 2tq,,- ~(~.,- , alibi l,]lal, ill tim li~nit (2, -- (2._, --+ cyo t~r~s i~t,<~ tim exl)ressi(~l (6.4.17) relewu~t to tim S~lmrp~sitiCn~ of two 2"

8. (',alc~tlal, c t,lw \Vig~er ~lisl,ril~tt, ir

~(q) -

l~l~('l, it~l~ r

~

~'

tim rt;ctm~gle-wavt; fln~cl,ioli

,.,.,,:t,(~)

forlne(t t)y all ixflilfit, e seqlu;xme of rectaligle wavefonli, 2(2 wi(le mltl space(l I)y d. (llixit: express g)(q) as a Fol~rier series r a c c t n ~ t of its perio(licity IW d.) Verify tl~at i~ l,l~c.' lit~fit Q --~ 0, where the r e c t a n g l e wave fluwti(~ll tllrns iilto the colntr-flllmt, ioli, the obl.aiIle{l expression for W~ (q, h;) t,lrI~s ii~to qi.4.2S). 9. C a l c u l a t e the Wigller (list, ributioll flumtio~l for tlle ct~llq~osed siglmls ~(q)

-

~ , ~ ( q - q, ) +

~p(q)

=

~-~-c~, ~2 + / ~ ' ' 1 q 2 "

#(q)

-

-

rect(--q-)c ' : " Z2 2Q

~.~ c,~.~ ,, 4-

+iK, o q

In p a r t i c u l a r , show t h a t w i t h a 1 a p p r o a c h i n g a 2 9(.l{1 for a single chirp.

~-a(,i'~ '-~+2'~"2q :z

-----+

(132, we regain tile expression (6.4.34)

The Wigner Distribution Function: Analytical Evaluation

335

References [1]

We quote below the fundamental papers by Wigner concerned with tile Wigner method. In [1.1] the Wigner distribution function is introduced in an almost heuristic manner in connection with the problem of determining the quantum correction to thermodynamic flmctions. In the later paper [1.2] the formal procedure to elaborate an in general quantum phase-space distribution on account of the specific requirements and inherent limitations is described. [1.1] E.P. Wigner, "On the quantum correction for thermodynamic equilibrium", Phys. Rev. 40, 749-759 (1932). [1.2] E.P. Wigner, "Quantum-mechanical distribution function revisited", in Perspectives in Quantum Theory, W. Yourgrau and A. van der Merwe (eds.) (MIT Press, Cambridge, 1971), pp. 25-36.

[2]

A few titles on classical statistical mechanics are given in [4] of ch. 2. We signalize the textbook by Balescu, wherein the Wigner method is described and used. [2.1] R. Balescu, Equilibrium and Nonequilibrinm Statistical Mechanics (John Wiley Soils, New York, 1975), ch. 3, pp. 89-104.

[3]

Titles on quantum mechanics are listed in [10] of oh. 4. The textbook by Blum is specifically concerned with the density matrix formalism. [3.1] K. Blum, Density Matrix Thco W and Applications (Plenum Press, New York, 1981).

[4]

The literature concerned with the Wigner representation in the quantum mechanical franlework is considerably rich. Only a restricted list of exemplary t)at)ers and reviews is given here. A comprehensive aim lysis of the Wigner distribution function in quantuin mechanics was firstly presented in the book by KiIn and Noz [4.12]. See also the review paper by Dragonmn [4.11], offering an overall investigation of the correspondence between the phase-space description of classical optics and quantum mechanics, and the fascinating presentations in [2.1], and, within the context of quantum electrodynamics, in [4.13]. Additional references are suggested in [1] of ch. 8. For investigations about the structure of the (quantuIn) Wigner phase space we address the reader to the enlightening analysis in [4.6]. [4.1] H.J. Groenewold, "On the principles of elementary quantum mechanics", Physica 12, 405-460 (1946). [4.2] J.E. Moyal, "Quantum mechanics as a statistical theory", Proc. Cambridge Phil. Soc. 45, 99-124 (1949). [4.3] H. Mori, I. Oppenheim and J. Ross, "Some topics in quantum statistics. The Wigner function and transport theory", in Studies in Statistical Mechanics, J. de Boer and G.E. Uhlenbeck (eds.) (North-Holland, Amsterdam, 1962), Vol. 1, pp. 212-298. [4.4] N.G. de Bruijn, "A theory of generalized flmctions, with applications to Wigner distribution and Weyl correspondence", Nieuw Archief voor Wiskunde (3), Vol XXI, 205-280 (1973).

Linear Ray and Wave Optics in Phase Space

336

[4.5] M.V. Berry, "Semi-classical mechanics in phase-space: a study ofWigner's flmctioif', Phil. Trans. Roy. Soc. 287, 237-271 (1977). [4.6] N.L. Balazs, "Weyl's association, Wigner's flmction and atIine geometry", Physica 102A, 236-254 (1980). [4.7] J.P. Dahl, "Tile Wigll(:r fllnctiolf', Pllysica l 1 4 A , 439-444 (1982). [4.8] R,.F. O'Co~mell, "Ttm Wigl~er distrib~tio~ fluwt, io~ - 5()tl~ birtt~(lay", Fo~md. Phys. 13, 83-92 (1.()83). [,1.9] V.I. Tatarskif, "'I'll(: Wig~(:r repr(:se~d, atio~ of q~m.~t~n~ ~(:(:}m~ics", Soy. Pl~.ys. Usp. 26, 311-327 ( l (.)8,3). [4.1()] N.I~. l~alazs m~(l II.K..l(;~u~i~gs, "Wig~er's fin~ctio~ m~(l otl~er (listril)~d.io~ fl~ct,io~s i~ tl~e M()(:k l)lms(: sl)a('es", l>l~ys. Rel). 104, :~47-3.()1 (1.()~;4). [4.1()] M. Hillery, ll..l". ()'(',()~mll, M.(). S(:~flly m~(l E.! ). Wig~mr, "l)istril)~ti()~ fln~ctio~s i~ I)t~ysi(:s: flu~(lm~m~tals", Plws. ReI). 106, 12 l- 167 (1984). [4.11] D. Drag()~m~, "I>lms(: Sl)a('(' ('()rr('.st)()~(le~(:e 1)etwee~ ('lassi('al ()l)ti('s m~(l q~ml~t~H~ ~eclm~fi(:s", i~ l)r~jrc.,~.,~ in Optic.~, V()l. 43, E. W()lf (e(I.) (Els(,vier, A~stcr(lmn, 2()()2), (:t~. 5, I)I). 433-4.()(i. [4.12] Y.S. K i ~ m~(! NI.E. Noz, l'ha.,~r b'pacc l'icturc r Q'u,.'n,tum Mr (Worl(l Sci('~gifi(', Si~gal)()r(;, 1.9.(tl). [.1.13] S. l/,. l)c (~r()()t m~(l l~.(]. S~t,t.()rl), bb'wn,datio'n..s of l'Jh,ct.'rodyna'm.ics (N()rtl~-H()llm~(l, A~ster(lm~, 1!172), ('1~. VI-VII, l)l ). 317-4()(i. 'l'i~(: l)r()l)(:rti(:s ()f tim W(:yl l,rm~sfor~ m~(l tl~e Wiggler (listril)~ti(,~ fl~('ti()~ are sp(,('ifi('ally (l(,s('ril)(,(I ie~ l~(, al)l)e~(lix ()f (:l~. VI, pp. 341-3(i4. [5]

A I'(:w (i(.h:s (:()~('('r~('(l wi(l~ (I)(' \u r(:l)r('s('~(a(i()~ i~ r('lali()~ (() (l~(: sig~ml (.l~(:()ry at(' list(:(l l)(:r('. S('e als()[.l]-[(i] ()f (:I~. 7 i~ r(,la(i())) (()(l)(, W~l'i()us a Sl)(:(:ts ()f th(,. j()in( sig~ml l'('l)r(:S('l|l al i()ll (l~(:()ry. [5.1] ,l. Ville, "'l'lld()rie et al)l)li('ati(,ls (h: la. llol, i()ll (h: siglml mlalil, i(llle '', (~al)l(:s el, TrmlsIifissiolls 2A, (il-74 (1948). [5.2] A.W.R.itmczek, "Siglmi elmrgy (list,ril)lltioll ill l,iIIle ml(l freqllellcy", IEEE Trans. Inf. Theory, I T - 14, 3(i,()-374 (1.(t68). [5.a] T.C.A.M. (;lmlseil ml(l W.F.G. Mecklellbr/iilker, "Tim Wiglmr (lisl, riblltiozl- a tool for tilne-freqlltuwy sipglml mlalysis. I: contiimolls-tilne sigluds", Plfilips ,J. R,es. 35, 217-250 (1980); II: r iiile signals", ibid., 276-30(); Ill: relatiollS wittl other time-frequelmy sigzml trazlsfonnations", ibid., 372-389. [5.4] F. PeyriIl ml(1 Ii,. l)r()st, "A ~nified defilfitim~ for the (liscrel, e-l,i~ne, discretefrequency, m~(l (liscrete-tiIne/frequency Wigner distril)~tio~ls", IEEE Trans. on Aco~st., Speech an(t Sig~ml Process. 34, 858-867 (1986). [5.5] L. Cohen, "Tilne-freq~mncy distributions-A Review", Proc. IEEE 77, 941-981

(1989).

[5.6] F. Hlawatsch an(1 G.F. Bou(treaux-Bartels, "Linear an(t quadratic Lime-frequency signal representations," IEEE Signal Process. Mag., 21-27, April 1992. [5.7] L. Cohen, Time-Frequency Analysis (Prentice-Hall, Englewoo(1 Cliffs, 1995). [5.8] The Wigner Distrv;bution: Theory and Applications in, Signal Processing, W. Mecklenbrguker and F. Hlawatsch (eds.) (Elsevier, Amsterdam, 1997).

[6]

T h e pioneering p a p e r s by B a s t i a a n s i n t r o d u c i n g the W i g n e r r e p r e s e n t a t i o n in optics and investigating m a n y related theoretical and applicative issues are q u o t e d below. We also address the reader to the review p a p e r s [6.8] and [6.9], and to the J O S A issue [6.10].

The Wigner Distribution Function: Analytical Evaluation

337

[6.1]

M.J. Bastiaans, "Wigner distribution flmction applied to optical signals and systems", Opt. Comm. 25, 26-30 (1978). [6.2] M.J. Bastiaans, "The Wigner distribution flmction and Hamilton's characteristics of a geometrical-optical system", Opt. Comm. 30, 321-326 (1979). [6.a] M.J. Bastiaans, "Wigner distribution flmction and its application to first-order optics", J. Opt. Soc. Am. 69, 1710-1716 (1979). [6.4] M.J. Bastiaans, "The Wigner distribution flmction and its applications to optics", in @tics in Four Dimensions-1980, AIP Conf. Proc. 65, M.A. Machado and L.M. Narducci (eds.) (AIP, New York, 1981), ch. 9, pp. 292-312. [6.5] M.J. Bastiaans, "Signal description by means of a local frequency spectrum", ill Transformations in Optical Signal Processing, W.T. Rhodes, J.R. Fienup and B.E.A. Saleh (eds.) SPIE 373, 49-62 (1981). [6.6] M.J. Bastiaans, "Use of tile Wigner distribution function ill optical problems", in 198~ European Conference in Optics, Optical Systems and Applications, SPIE 492, 251-262 (1984). [G.7] M.J. Bastiaans, "Application of the Wigner distribution function in optics", in The WiDner Distribution: Theory and Applications in Signal Processing, W. Mecklenbrguker and F. Hlawatsch (eds.) (Elsevier, Amsterdam, 1997), pp. 375-426. [~.s] G. Cristdbal, C. Gonzalo and J. Bescds, "hnage filtering an(t analysis through the Wigner distribution", in Advances in Electronics and Electron Physics, Vol. 80, P.W. Hawkes (ed.) (Academic Press, San Diego, 1991), pp. 309-397. [6.9] D. Dragoman, "The Wigner distribution flmction in Optics and Optoelectroidcs", in ProDress in Optics, Vol. XXXVII, E. Wolf (e(t.) (Elsevier, Amsterdam, 1997), oh. 1, pp. 1-56. [6.10] Wigner Distribution and Phase Space in Optics, G.W. Forbes, V.I. Man'ko, H.M. Ozaktas, R. Simon and K.B. Wolf (e(ts.), J. Opt. Soc. Am. A 17 (12), 2273-2542

(2000). [7]

The W i g n e r distribution flmction is one of the basic t()ol of q u a n t u m optics, from which the whole class of s-parameterized distribution functions can be obtained. We signalize the seminal papers by Cahill and Glauber [7.1] and some fimdamcntal textbooks on q u a n t u m optics, dealing as well with the Wigner representation, specifically addressed in the quoted pages. [7.1] K.E. Cahill and R.J. Glaubcr, "Ordcred expansions in boson amplitude operators", Phys. Rev. 177, 1857-1881 (1969); "Density operators and quasiprobability distributions", ibid., 1882-1902. [7.2] W.H. Louisell, Quantum Statistical Properties of Radiation (John Wiley & Sons, New York, 1973), ch. 3, pp. 168-176. [7.a] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), pp. 541-542. [r.4] U. Leonardt, Measuring the Quantum State of Light (Cambridge U. Press, New York, 1997), ch. 3, pp. 37-54. [7.5] W.P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, Berlin, 2001), ch. 3, pp. 67-98.

[8]

Presentations concerned with the application of the Wigner representation to electron optics in b o t h synchrotron radiation and microscope transfer function related issues are in [s.1] K.-J. Kim, "Brightness, coherence and propagation characteristics of synchrotron radiation", Nucl. Instr. & Meth. in Phys. Res. A 246, 71-76 (1986).

338

Linear Ray and Wave Optics in Phase Space

[S.2] V.M. Castafio, "A generalized display of tile contrast transfer fimction in transmission electron microscopy", Optik 81, 35-37 (1988). [s.a] G. Dattoli, C. Marl ail(t A. Torre, "A note oil Wigilcr-phase space and electron beam transport", ENEA R T / I N N / 9 1 / 3 0 . [S.4] V.M. C~stafio, P. Smltiago, A. G&Ilez, "linage (tisplacciileilts ill trmlslldssion dectron lliicroscopy: a pliasc space approacll", Optik 95, :~1-33 (1993). Is.5]

F. Ciocci, (]. Dattoli, A. Torrc ml~l A. Rcliicri, hlsertioll I)eviccs for SylmhrotroIl Ra~liat, ic)ll a,ll~! lq'ee Eh;ctr~,l I.~Lser (W~wl~l S('., Sixlgal)~We, 2()()()), oil. 5, t)I). 167-178.

[!)] (',ellm'alizal,i~)ll ~)f 1]1~' l)llasC-Slm('c fiWllnllali~ll ~f (lllallllllll lll~;~'lla~i~'s 1~ a~y set t)f llt)ll r C~lljllgalt' t)l)Cl'nl~l'S al'~' ~'la])t)l'al~'{l ill lilt lml)Crs l)cl~)w. Sr l il h;s ~:~('~;r~er wi111 s i ~ i l a r ~'lal~wal i~n~s will~il~ 11~; ('r r sigl~al I lm(~ry (:a~

I~; f i ~ { l i~ [5] ~f ~'1~.7. [!~.1] kl.(). S(uflly ~tll~l l,. (:()}ICll, "Q~si-ln'~flm,lfility {lisl,riln~l,it~s fin" arl)il,l';try ~l)erators '', i~ 7'h.c l'h.ysics of l'h.a.~c b'pa~:~', Y.S. K i ~ m~l W.W. Z~u'l~al'y (e~ls.) (Spri~ger-Vcrlag, New Ycwk, 1.q87), 1)!). 253-26(). [9.2] A. ]~i~,~l)ale a~r lkl. ]{azavy, "Q~lalltlllll-lllc('llallical l)ll;~S~:space: a gt~'~cralizal,io~ of Wig~mr 1)lmsc-spacc fi~r~n~lal,i{~ I,~ al'l)itl'al'y c()r162 ~ systclllS", l~l~ys. Rev. 38,

A 41, 515(i-6()54 (1.q.gt)). crlics ~f 1l~t; W i g ~ ' r (lisl rii)~ll i(~ fit,~('l i ~ , f i ' ~ 1)~11~ l l~'()r~,l i~'al a~ul al)t)li~:alivc vi,;wl)(,i~ls, (:a,, i),, f i , ~ , l i~ [1], [4], [8] a,~r il~ l l~t; lml)e,'s 1,,;l,,w. A,l,liti,,~,al tith;s (:an l)e fr i~ l l~' f()l'ill('()~i~g ('lml)i(;rs. co~(liti~s f(~r ~iq~u,~ess", l~t~ys, l~ett. A 83, 145-14~, (1.981). [10.2] I~. Co|m~i, "()1~ a fi~(la~w~tal l)r~)pcrty ~f tl~c Wig~wr (listriln~ti{)~C, IEEE Trans. o~ Ac(~st., Sl)cec}~, mul Sig~ml Pro('cssi~g, ASSP-35, 55.%561 (1987). [10.a] F. Hlawatsch, "l~tc'rfcre~mc l,e r ~ s i~ tim Wig~mr ~lisl.ri])~l,i~)~", i~ Digital Signal l'rocessing-84, V. (',appellilli alibi A.G. Collstalltillir (e~ls.) (Elsevier, Alllster(laln, 1.984), pp. 363-367. [10.4] A.,I.E.M. Jansscn, "O,~ thr loc~s a~{l sprea(l of pse~ulo-r time-frequency I)la~c", PlfiliI)s ,1. Res. 37, 79-110 (1982).

fin~ctio~is in the

[10.5] L. Cohen, " W h a t is a ~mdticompo~e]~t signal?", Proc. I E E E , I(',ASSP .92 5, 113-116 (1992). [10.6] Hudson, R.L., "Wlmn is tl~e Wigner q~msi-probability (lensity ~o~>~mgative?", Rep. Math. Phys. 6, 249-252 (1974). [10.7] L. Cohen and Y.I. Zaparovmmy, "Positive quantum joint distributions", J. Math. Phys. 21, 794-796 (1980). [10.8] R.F. O'Com~ell an(t E. P. Wigimr, "Some properties of non-x~egative quantmnmechanical distribution fimction ", Phys. Lett. A 85, 121-126 (1981). [10.9] F. Soto and P. Claveric, "When is the Wigner function of multi-dimensional systems non-negative?", J. Math. Phys 24, 97-100 (1983). [10.10] A.J.E.M. Janssen, "A note on Hudson's theorem about fimctions with non-negative Wigner distributions", SIAM J. Math. Anal. 15, 170-176 (1984). [10.11] A.J.E.M. Janssen, "Positivity properties of phase-plane distribution functions", J. Math. Phys. 25, 2240-2252 (1984).

The Wigner Distribution Function: Analytical Evaluation

339

[10.12] A.J.E.M. Jansscn and T.A.C.M. Claasen, "On positivity of time-frequency distributions", IEEE Trans. Acoust., Speech, Signal Processing, ASSP-32, 81-94 (1985). [10.13] K. Husimi, "Some s properties of the density matrix", Proc. Phys. Math. Soc. Japan 22, 264-314 (1940). [10.14] N.D. Cartwright, "A non-negative Wigncr-type distribution", Physica 83A, 210212 (1976). [10.15] P. Bertrand, J.P. Doremus, B. Izrar, V.T. Nguyen and M.R. Feix, "Obtaining non-negative quantum mechanical distribution function", Phys. Lett. A 94, 415417 (1983). [10.16] T.B. Smith, "A comment on the smoothed Wigner function", Phys. Lett. A 127, 75-78 (1988). [10.17] D. Dragoman, "Tile origin of negative values of tile Wigner distribution function", Optik 111, 179-183 (2000). [10.18] R. Jagannathan, R. Simon, E.C.G. Sudarshan and R. Vasudevan, "Dynamical maps and nonnegative phase-space distribution flmctions in quantum mechanics", Phys. Lett. A 120, 161-164 (1987). [10.19] F.J. Narcowich, "Conditions for the convolution of two Wigner distributions to be itself a Wigner distribution", J. Math. Phys. 29, 2036-2041 (1988). [10.20] H.H. Szu and J.A. Blodgett, "Wigner distribution and mnbiguity flmction", in Optics in Four Dimensions-1980, AlP Conf. Proc. 65, M.A. Machado and L.M. Narducci (eds.) (ALP, New York, 1981), pp. a s s - a s 1 [10.21] F.J. Narcowich and R.F. O'Connell, "Necessary and sufficient conditions for a phase-space flmction to be a Wigner distribution", Phys. Rev. A 34, 1-6 (1986). [10.22] Y. Aharonov, D.Z. Albert and C.K. Au, "New interpretation of the scalar product in Hilbert space", Phys. Rev. Lett. 47, 1029-1031 (1981). [10.22] R. F. O'Connell and A.K. Rajagopal, "New interpretation of the scalar product in Hilbert space", Phys. Rev. Lett. 48, 525-526 (1981). [10.24] V. Buzek and P.L. Knight, "Quantum interference, superpositioll states of light, and nonclassical effects", in Progress in Optics, Vol. XXXIV, E. Wolf (ed.) (Elsevier, Amsterdam, 1995), ch. 1, pp. 1-158. [10.25] D. Onciul, "Eflqciency of light launching into waveguides", Optik 96, 20-24 (1994). [10.26] D. Onciul, "Waveguide launching efficiency for multimode and partially coherent light sources", Optik 97, 75-77 (1994). [10.27] D. Dragoman, "Phase space representation of modes in optical waveguides", J. Mod. Opt. 42, 1815-1823 (1995). [10.28] D. Dragoman, "Wigner distribution function representation of the coupling coefficient", Appl. Opt. 35, 6758-6763 (1995). [10.29] M. Testorf and J. O.jeda-Castafieda, "Fractional Talbot effect: analysis in phase space", J. Opt. Soc. Am. A 13, 119-125 (1996). [10.30] J. Ojeda-Castafieda and E.E. Sicre, "Quasi ray-optical approach to longitudinal periodicities of free and bounded wavefields", Opt. Acta 32, 17-26 (1985). [10.31] K.-H. Brenner and A.W. Lohmann, "Wigner distribution function display of complex 1D signals", Opt. Comm. 42, 310-314 (1982). [10.32] A.W. Lohmann, J. Ojeda-Castafieda and N. Streibl, "The influence of wave aberrations on the Wigner distribution", Opt. Appl. 13,465-471 (1983). [11] The concept of the m u t u a l coherence function was firstly introduced in the seminal paper by Wolf [11.1]. Some representative t e x t b o o k s are listed here. See also [7.2] and [7.3].

340

Linear Ray and Wave Optics in Phase Space

[11.1] E. Wolf, "A macroscopic theory of interference and diffraction of light from finite sources. II. Fiehts with a spectral range of arbitrary width", Proc. R,oy. Soc. (Londo,~) 230, 246-265 (1955). [11.2] M. Born and E. Wolf, Pri'n,ciple,s of Optics (Pergamon, Oxford, 6th,1980), ch. 10, pp. 491-555. [11.3] M.,J. Berall ml(l (].B. l'arrent, ,Jr., 7'hi'or!/ of l'artial Cohere~,ce (l'reiltice-Hall, Ellglewwo(1 CIiIN, 1964). [11.4] .l.W. (loor Statistical Optics (.lo}~ Wiley & So~s, New York, 1985). [1 1.5] B.E.A. Salctl ml~l lkl.(:. Tci('ll, Ftm,da'ntt"n,/als of Ptm/,o'n,i~:s (.It~lHl Wiley & Sculls, New York, 1991), r 1(1, 1)1). :H2-383. [12] F'~r tll~' Wigll~'r-n'l)rt's~'lltatit~ll ~f l)artially <'~ll~:l'~'llt ~l)tical fi<'l
~ay

(:()"SI'II ['i.'|l, [(i.(i I ~111(1 [12.1] M..I. l~astiam~s, "'Flu; Wiggler r riln~tir finmtio~ of partially cot~crr ligtd,", Optica Acta 28, 1215-1224 ( 1!)81 ). [12.2] r~l..l, l~astiam~s, "Al~plicati~n~s ~f tt~c Wig~mr r finmi,i~ I.<~Imrtially colmre~fl, ligi~t", .1. ()pt. S~m. A~. A 3, 1227-1238 (1986). In~ti~mhaikh, "Propagatioi~ properties of Herinite-cosl~-Gmlssian laser heaths", Opt. (Jo~n~. 186, 269-276 (2000). [17] For tl~(: l)asi(: tI'cat~l(:~t ()f mlt)erGaussian 1)eanis w(: s~gg(:st [17.1] S. De Silvestri, P. Laporta, V. Magni and O. Svelto, "Solid-state laser unstable resonators with tapere(t reflectivity mirrors: the s~pergm~ssian approach", IEEE J. Qum~t. ElectroI~. 24, 1172-1177 (1988). [17.2] C. Pahna an(t V. Bagini, "Propagation of super-Gaussim~ beains", Opt. Comm. 111, 6-10 (1994). [18] W. Magmts and F. ()t)crhcttinger, For'mulas arm Th, eorems fi)'r" the Special Functio~,s of Mathematical Physics (Chelsea, New York, 1949). [19] I.S. G r a d s h t e y n and I.M. Ryzhik, Table of h~tegr'als, Se'~'ies arid P'rvd~ucts (Academi(: Press, New York, 1965), p. 838, # 7.377.

7 The Wigner Distribution Function: Optical Production

7.1

Introduction

We have let~rned in the previous chapters tha,t an optical signa,1 can be fully and globally represented by the complex amplitude ~(q) ~s well a,s by the releva,nt Fourier tra,nsform ~(h;). The former conveys information about the spatial distribution of tile signal and tile latter a|)ollt the frequency content. However, on pcrfornfing the Fourier a.na.lysis wc loose a,ll information about the st)a,(:c loca,liza,tion of a. given fl'e(tuen(:y (:omt)oncnt. in contrast, we may wish to know the very t)osition at whi('h a given frcqucn(:y comt)oncnt occurs. In a,(hlition, Fourier analysis is rather "uneconomical" and highly unstable with respect to pcrtl:rbation. In fact, one needs to add an infinite series of a,lternating contributions even to reproduce a, uniform signa,1 over a finite space interval. Also, if one adds a sma,ll-amplitude term to a, linea,r superposition of sinusoidal waveforms, the spa,tial distribution of the overall signa,1 changes very little, whilst the frequency spectrum is strongly modified. This is due to the global character of the Fourier tra,nsform. Wha.t we wish indeed is to construct a local frequency representation of a signal. The instance is then how to infer a, local informa,tion from knowledge of p and @, how to join the two parallel information conveyed by p and in order to obtain an "intersection" of information, enabling us to know the position at which given frequencies ~rc present. This is the same question as was approached in the Introduction of Chapter 6 within the qua,ntum-mecha, nical context, even though in a slightly different formulation. In that context, the Wigner distribution flmction emerged as one of the possible (and most convenient) solutions to the posed problem. Here the question is posed within the context (and accordingly with the inherent terminology) of signal theory, where indeed the Wigner distribution function was reintroduced (by Ville) some 15 years after its originM introduction (by Wigner). The most popular example of a joint signal representation in the

Linear Ray and Wave Optics in Phase Space

342

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M",.,.,

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J

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q

. ~..

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~F

-

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I q ( ; U I i E 7.1. A ~n~si('al s('(~rc is l)~si('~tlly a .j()i~t rct)rcsc~l,al, i(~ t)f a sig/~ml ( f r ~ Pi6c(;s Ig'()i(h's-l)m~s(;s (h' 'lYav(,rs, ~(). 3).

Sa, l,ic's

illll(;r(;lll, l,illle-fr(;clll(;llt:y (l(,~,~ai,~ is 1,1~(; ,~n~si(:a.1 s(:(,,'(: [1] (Fig. 7.1). Tllell we llmy say tirol, We are l()()killg f~)r a IXl(:l,ll(J(l I,(~ (:()nsl,rll(:l, l,lle "llnlsica.1 s(:()re" ()f a.ll ()l)t,i{:al siglm,1. As l)r()v{:(l 1)y l,ll{; wl.ri(nls ('xmlq)les ilhlsl, ral.(:(t i~l tim I)revi()lts (:lm.l)la'.r, l, ll(: Wig~(:r (lisl,ril)~l,i(n~ fin~(:l,i()~ (:m~ l)(; (:()~si(h;r(;(l a,s a, kin(t ()f s~t(:l~ a (l(:sir('(l ~n~si(:al s(:()r(;. W(: giv(; l~(',r(; a, ra,l,l~(;r ~mTfv(: (l(;~()~sl,ral, i()l~ ()f l,l~is ()~ l,l~c 1)asis ()f ()l)('ral,iv(' (:()~si(h'ral,i()~s. It, is, i~ fa('l,, ('h'ar tl~a,l, i~ ()r(h;r 1,(~ i~(livi(l~mliz(; l,l~(' l()(:al f(;al,~r(;s ()f a signal ()11(; lllay i~sl)(;('l, il, l)i(;('~' l)y l)i(;(:(: will~ all a~alyzi~g fit~(:l,i()li. 'l'l~is l(;a, ls 1,() l,h(; f()r~mliza,t,i()~ ()f t,l~(~ ('(~(:(;l)l,s ()f sli(li~tg-wi~(l()w F()~ri(:r t,rm~sf()n~ m~(l st)(;(:trogra,n~, as (l(',s(:ril)(;(l ill ~q(;('l,. 7.2. As a,~ (;xl,(;rI~a,1 l)r()l~(; ~um.v()i(la.1)ly i~dt~mn(:(;s t,h(; r(;s~fll,, ()~(; ~my 1)r(;f(;r 1,() ~s(; l,l~(; siglml itself ()r its ~irr()r(;(l v(;rsi()n a,s a,na,lyzil~g fin~(:l,i()~, l,lnls r(;sl)(;(:l, iv(;ly ()])l,a,iili~g l,t~(; a~l)ig~il, y fl~(:l,i()~ a,n(1 t,h(; W i g n e r (lisl, l'il)~l,i()~ fl~(:l,i()~, ll()t, ti at(; l,tl(',~l i~d,I'i~si(: l,l'a,llsf()l'lllS, a(:l,~m,ting t,he f()rn~cr a, (:()rrclativ(;-tyl)(; i~sl)(;(:tiol~ ()f t,h(; sig~al as t,l~(; la,/,t,(;r a, (:()~v()lutivct,yt)c insl)(;(:l,i()~. A I)ri(;f a,(:(:()~nd, ()f 1,1~(; t)I'()t)(;rl, i(;s ()f l,l~(; a.l~fl)ig~lil,y filn(:t, ion a.n(I its r(;la,l,i()~sl~il) I,() l,h(', Wign(;r (tisl,ril)~l,i()n flu~(:l,i()l~ is giv(;~ i~ S(;(:I,. 7.4. St)(;(:t,r()gra,I~, W i g n e r (list, ri|)ut, i()n fl~n(:ti()~l a~(l m~t)ig~fil,y fiu~(:t,i()~ are all r(;t)res(',~d,a,t, iv(;s ()f a. (:()~I)r(',l~(;nsiv(; a,~(l wi(tcr (:lass ()f .joint (1)ili~(;a.r a,n(t not) signa,1 r(;t)I'(;s(;~d,a.t,i()~s, ()~ wlfi('h we (:()~xmx(mt, ra,t,l~(;r 1)riefly in S(;(:t,. 7.3. Th(; ild, ri~si(: n()n lin(;a,rity of tim Wign(;r (tist, rit)~d,i(m a,n(t a,mt)ig~lit, y functions nat,~ra,lly fra,n~(', t,l~(;~n w i t h i n the context, of t,hc t)art, ia,1 (:()hcr(;~(:e theory, a,s the t)r()(l~u:t, fim(:ti(m p * ( q - q2 ' ) ~ ( q + q') e n t e r i n g b o t h the W i g n e r and 2 a m b i g u i t y flu~(:t,i()~ i~tegra,ls, is i n t e r p r e t a b l e in tlm light, of the se(:ond-order statistics ()f l,lm wa,vc field, as discussed in Sect. 7.5. Fina,lly, t,h(', ot)t,i(:a,1 t)roduction of the W i g n e r d i s t r i b u t i o n function is discusse(t i~ a, (:ert, a,in (tet, a.il, firstly f r a m i n g it wit, hin t,he f a s c i n a t i n g issue of optical c o m p u t i n g (Se(:t. 7.6) a n d t h e n detailing the optical a r c h i t e c t u r e s t h a t have t)e(m devised to display the W i g n e r integral for b o t h rea,1 (Sect. 7.7) and c o m p l e x (Sect. 7.8) 1D signals. I l l u s t r a t i o n of the optical i m p l e m e n t a t i o n of the Ga,ussian s m o o t h e d W i g n e r d i s t r i b u t i o n (Sect. 7.9) closes the chapter.

The Wigner Distribution Function: Optical Production

7.2

343

The sliding-window Fourier transform

Tile simplest way one can devise to answer tile question "which frequencies pertain to which position" is that of sectioning tile entire spatial distribution into a certain number of separate portions of some length and then Fourier analyse each selected portion to ascertain the frequencies present in that portion. Tile collection of all tile obtained local spectra provides us with a kind of a joint space-frequency description of the signal. The reliability and the sharphess of the resulting joint representation is determined by the details of the sectioning procedure. However, they are inherently limited since the "identity" of the original signal is at a certain extent lost when sectioning is operated; also, the position and frequency resolutions of the obtained representation are intimately related to tile extent of tile various signal sections, that we have created, through the uncertainty relation of Fourier analysis. More specifically, such an intermediate description can operatively be realized by windowing the signal p(q) by a suitable function w(q) and then Fourier transforming the windowed signal for running locations of w(q). Namely, for a given position q we multiply the signal amplitude p(q') by tile conjugate and shifted version of the window fllnction w * ( q ' - q), and then take the Fourier transform of the t)roduct p ( q ' ) w * ( q ' - q). Shifting the window function w(q) (:ontinuously over the signal, we create a fimction of q and ec, which allows us to estimate the frequency (:ontcnt of the signal at different positions within tile relevant space domain. The obtained sliding-window Fourier transform [2] ~(q,

h;; W) --

1 l

v~ d

~(qt)w* (q' -- q)c -inq'dq'

(7.2.1)

can in fact be seen as a local frequency spectrum of the signal (Fig. 7.2). The window function w may be chosen rather arbitrarily; however, being the procedure implied by (7.2.1) aimed at gaining information about the space localization of the frequency components of the signal, it is convenient that w(q)

~ q ) ( q ) w

(q)

FIGURE 7.2. Sliding-window Fourier spectrum as a local frequency spectrum: moving the window function w(q) over the waveform qa(q) to be analysed, definite sections of ~(q) are selected whose frequency content can then be estimated upon Fourier transformation.

344

Linear Ray and Wave Optics in Phase Space

be relatively "narrow". Thus, the integration limits in (7.2.1) are in practice set a(:(:()r(ting to the (:hos(ul t)osition and the. wi(tth of tile window flmction. Evi(tently, ~ ( q , ~,; w) is linear a,n(t in genera,1 (:()mtflcx va,llm(1. It t)I'eserves fl'e(tllen(:y shifts i~l tl~(: siglm,1, and st)a.(:c shifts ut) to a, n~o(hllatioll, thus all()willg for a, lnfifornl mm.lysis ()f t,lle siglml featln'es over tlle wll()le (q, a;) t)la,ne (s(:(: Ih'()tflenl 7.1). Ill a(t(lil, iim, I,h(: signal (:ml lufi(tllely t)(: rt:i:tnlsl,r11(:I,(:(t from tht: l'(',l(:vallt will(lt)w(:(l sl)(:t:tl'lllll (7.2.1). h~ fa(:t, t,rm~sf()r~i~g 1)a(:k (7.2.1) t,() (,ll(' Sl)a('(: (t()llmill rt:sllli,s Jill,() l,ll(: ()t)vi()lls ilwt:rsi()ll ['()rlllllla.: -

/

1

q) -

w),

ih"

All,(:rlla.l,iv(:ly, ()If(: IIlay (,rallsfl)r~ l)a(:k (7.2.1) (,()(,I~(: Sl)a.(:(' (l()~a.i~, ~utll,il)ly l)y tl~(: wil~(l()w m~(l (,I~(:~ i~(,(:gra,(,(: ()v(:r (,I~(: Sl)a(:(: w~.rial)h:, (,In~s lilmlly ()l)tail~il~g

//

'V~2~.

~(q'

; w)[w(q --

q'),'i""~],tq',th",

(7.2.3)

(,l~(: wi~(l()w fl~(.(,i(n~ l)(:i~g S~ll)l)()s(:(l (,() lmv(, a ~ , i ( (,~(,rgy: j' lwl2(q)dq = I. I~(,(:r(:s(,i~gly, i~ (7.2.3) (,1~(: sig~ml a(, l)()si(,i(n~ q is (:xl)r('ss(:(! as a w(:igl~(,('~(l S~l)(:l'l)()sit, i()~ ()f wl~a,t (:a~ 1)(: r(:gar(l(:(l a,s t)asis siglm.ls .s,~,,,, (q) w(q--q~)( 'i~'q, (:(:~(,(:r(:(l ()~ 1,I)(' l)()i~t (q~, h"~) i~ tl~(: l)l~as(: l)Im~(:, if a,s ~s~ml tl~(' wi~(l()w flu~(:(,i(n~ is (:(:l~(,(:I'(:(l a.l'()lllI(l (,If(: ()rigid) i~) l)()(,l~ (,li(: Sl)a.(:(; m)(l fl'(:(lll(:II('y (l()~ai~. A Iii()I'O, ~;(:~(:ra,l f()n~i ()f tl~(: .,,'!]'n,lh,(',,~i,,~ f()rI~n~la (7.2.3) is (h:s('ril)(:(l i~ l)r()l)l(:~ 7.1. ~I'I~(: ~m,g~i(,~(l(: S(l~mr(:(l ()f (,I~(: sli(li~g-wi~(l()w Sl)(:(:(,r~, ~m~(:ly

s (q. , ;

w) -

w)l 2

(7.2.4)

yi(:l(Is tl~e (l(:sil'(:(l Sl)a('(:-fl'(:(l~(u~(:y l"(:I)r(:s(:~d,a.ti()l~ ()f (I~(: s i ~ m l (:(nw(:ying inf()rnm(,i()n a.l)()~( (,l~(: (;II(:I'g'V (h:~si(,y (listril)~ti()l~ i~ (,I~(: Sl)atial l)()siti()n a,n(l fr(:q~(:~(:y. I~ tl~(: sig~a,l tl~(:()ry, $~ (q, +,;;w) is ~m,~(:(l (,I~(: sp('~:Irogra,'n~,, lh)wever, a. certah~ wtri(:ty ()f ~m,l~(:s exists. Tlu~s, f()r il~stal~(:(:, N~ is as w(:II caJled the comph::r ,spectrogram, a~(t $~ the physical ,spe('.tr'u'm, ()f ~(q). It, is wortl~ ()l)s(:rvi~g (,l~at (,l~(: sli(ling-wi~(l()w Sl)(:t:(,r~n~ (:~d,a.ngl(:s the signa, l and the win(low, m~(l 1~(:~(:(: u~m.voi(tably the t)rot)erti(:s ()f the. siglml are admixed with th()se ()f the wi~(t()w flm(:tion. As a,n examt)h:, n()te that integra,ting the st)ectrogram (7.2.4) over the entire. (q, t,:) t)la,ne on(' ()t)ta.ins . / / ' $~((t, h;; w)dqdh; -- I

(q)llall (q)ll l,

(7.~.5)

i.e., the produ(:t of the total energies of the signa,1 and the window function. AnaJogously, a,s the reader may easily verify, the margina.1 distributions of the s p e c t r o g r a m do not relate to the "pure" energy densities of the signaJ, the energy density of the window being as well involved in the relevant expressions. Relation (7.2.5) makes evident the interchangeability of p(q) and w(q) as "signal" and "window function". It is in fact evident that the signal and the

The

~o(q )

WignerDistribution Function: 4~rc q)(q)e

Optical Production

-itcq

345

q~(p ( q , ~c "w )

w (q)

FIGURE 7.3. The windowed Fourier spectrum can be seen as a filtering operation of the 1 linearly frequency modulated amplitude -~7)(q)e -~q by the conjugate spectrum of the window function. window play essentially syinmetric roles; it is in a certain measure aInbiguolls whether a window w(q) is used to compute the spectrogram of a signal ~(q) or inversely a window 7:(q) is used to compute the spectrograln of a signal w(q). Indeed, one has that ~ ( q , t~; ~) - - r _~; w). O n e may remedy the inherent ambiguity in the roles of the signal and the window, using the signal itself, or its mirrored version, as window. This leads to the ambiguity function, or the Wigner distribution function, as we will see in w 7.2.3. Finally we note that the short-space Fourier transform N~(q, h;; w) can well t)e interpreted as the "output" from a, space-invariant system, whose spread flm(:tion is basically w*(q), with the int)llt being the linearly fl'e(tuency mo(tulated amt)litu(le ~ ~: ( q ) c -i'~q . The modulation of ~(q) serves to shift the fre(tueilcy spectrum of ~(q) at frequen(:y h; to 0 fre(tuen(:y. Accordingly, the windowed-Fourier tra.nsforming operation imt)lied |)y ~ ( q , ~; w) is ml(tel'stoo(t as a filtering opexation of the shifted spectrum of ~(q) by the conjugate spect r u m of the window flmction (Fig. 7.3). 7.2.1

The frequency-domain

view o f ~ ( q , t~; w)

The windowed Fourier transform (7.2.1) is amenable to a windowing-like interpretation in the frequency domain as well. In fact, being the Fourier transform of the product Oq(q') = ~ ( q ' ) w * ( q ' - q ) , regarded as a function of the rmming position q' for a. fixed q, 8~(q, n; w) comes to be the convolution of the spectra, of 7~ and the shifted window. Hence, the alternative expression P

~(q

re;w) -- _A_:~-i~q / @(t~;')W*(h;' /

~i;)e.in'qdh;'

(7.2.6)

immediately follows. The integral over t~~ signifies evidently a windowing operation in the frequency domain with tile window flmction being tile very Fourier transform of w(q). Then it can be interpreted as a sliding-window space transform, and accordingly denoted as

/

-

(7.2.7)

346

Linear Ray and Wave Optics in Phase 5pace

Expression (7.2.6), we m a y suita.bly rewrite a,s itcq-~(tt

h;; ~/),

(7.2.8)

shows t h a t win(h)wing the signal ~(q) in sI)a,(:e t)y a, fimction w(q) a,l~toma,titally illq)lics wi~l(h)wil~g tim st)(;(:trmn ~(h',) t)y 1,11(; F~n~ricr tra.nsf(,n~ w(t,:). It, signifies t,lm e(t~fiwd(;~(:(; ()f the win(towi~g ()t)(;ra.ti(m in sI)a(:e a.n(1 fre(l~t(;n('y if, ()f (;()llrSO, tlm will(l()w fllll('ti()lis ill tlm t,w() (h)Iliailis a r(; jllst a. F()~lri(q" transfi)rn~ trait. Eit,l~(;r (:a,s(;, w(; ()t)l,ain 1,11(; sa~lm st)a,t:c-fl'(;(lll(;lit'y (listril)~l,i(m $~, wld(:]~ a,'c(w, li~gly (:a~ l)(; ~s(;(l 1,() i1~st)(;(:l, t,l~(; fI'(',(lll(;ll('y l)l'()t)('.rti('.s ()f the signal at giv(u~ l)()siti()l~S al~(l i~v('rs('ly l,l~('~ Sl)a~:(' l)r()l)Cl'li(;s a,l, giv(:ll fl'(;(l~(q~(:i('s. TI~(: ()l)s(;I'W~l,i()l~ a,l)()v(: x~mk(;s ('.vi(h;~t a~ i~d,ri~sir lil~fit ()f l,l~(; Sl)a(:(:-fl'(;(t~mn('y r(;I)r(;s(:ld,ati(n~ lms('~(l ()ll a. wil~(h)wi~Z-l)l'()('(;(l~rc. II~ (w(h;r I,()a.(:l~i(:v('~ g()()(l st)a('.(; h)(:a,lizati(n~ ()f tl~('~ fl'(;(t~un~cy lm)l)(;I'l,i(;s a ~mrr()w m~alysis wi~(h)w w(q) il~ st)a.c(; is r('~(t~fir(;(l, l~ik('~wi~;, fiw a, g()()(l fl'(;(l~(u~(:y h)(:alizati(n~ ()f 1,1~('. st)a,(:(; t)r()tmrti(;s a ~mrr()w wil~(h)w ~t(h',) i~ fl'(;(tl~(u~(:y w(n~l(l lm ~m('~(h;(l. ()f (:(n~rs('~, th(; (l(~t)(;l~(li~z (n~ l,l~(' al)l)li('al,i(n~s. ~,u kl~()w l,l~at l l~(' ('xl,('l~si()~s ()f a fl~('l,i()l~ m~(l its F()~n'i(:r t,l'a.llSf()l'lll (:l,l.ll ll()|, I)(~ arl)itrarily (:l~()s(:~, I)ei~g lil~k('(l I)y tim ~(:(~rta,i~d,y r(4ati()l~. W(' ('m~ ~()t l~av(' lli4rr()w wi~(l()ws i~ l)()l,l~ t,l~(' Sl)a('(' m~(1 tlm l)y a I)a,si(' r(;s()l~lti()li tra(l(' ()tf" a,s illll)r()vilig tll('~ Sl)a(:('~ r(:s()l~t,i()li I)y lmrr()wina tll(~ wi~(l()w r('s~lts il~ a l()ss ()f t'I'(;(l~l('~(:y r('s()l~ti(n~ m~(l vi('(: V(~I',SiI.. (]rit(u'ia f()r fixi~g t,l~(', ()l)til~ml wi~(h)w finl(:ti(n~ (:a,~ t)(~ s(~t ~I) a.('(:()r(li~g 1,() tl~(: I)r()|)h;m ~m(l(u" i~v(:stigati()~.

7.2.2

Some e.:ram,ple.s

Tlm Sl)(;(~tr()grm~l ([(;1)(nl(ls (;vi(hnd, ly ()ll tlm ('ll(fi(~(~. ()f t,lm will(l()w flmction. For illllstrati()ll l)llrl)oSeS w(; calculat(' tim Sl)(',ctr()grallls ()f S()lll(; (:h;llmnta,ry signals, 11siilg r('~(:taIlglllar ml(t/or Galmsiml wixl(t()ws; ar162 t,tm w~,rious tm,ram(.'t(',rs in th(; gral)lls a r(; st)et:ifie(t sinq)ly ])y lnlllll)(;rs, llllits ])(Jng implicit. As n()l,e(l ill w 4.6.4, (]a,ussia,ll fllll(:tiolls llave tlm llota,t)le t)r()t)erty of self-ret)r()(tlw.illg llll(l(;I" Fourier tra,nsfor~ll wit, t1 tlm I'(J(~.va,nt stm,ce-|)andwidth pro(hl(:t ta,ki~lg tim 11dnillnml va,lue. As we will s(;(; in w 7.2.4, the s p e c t r o g r a m of a signal |)y a Gaussia.n win(tow is also intert)reta])h', a,s a, G m l s s i a n - s m o o t h e d W i g n c r (tistI'il)ld, i(nl flln(:tion (see also ,~ 6.3.6). Firstly w(', (:onq)are tlm st)ectrograms of the qua(tratic t)ha,se signal

(1Ochirp (q) -- ci~ q2 ,

(7.2.9)

obta,ine(t using a re(:ta,ngula,r and a, Ga,llssian window. A ~ n i t - e n e r g y recta,ngular fimction is, of course, des(:ribed by

{ 1

wa (q) _

2x/rff~,

0

Iq] -< Q~,

Iql > Q .

(7.2.10)

The

WignerDistribution Function: Optical Production

347

FIGURE 7.4. Spectrograms of the chirp (}~)(:hii.t)(q) -- e~;4~q2using (a) the rectangular window (7.2.10) and (b) the Gaussian window (7.2.13) with the respective optimal window widths. A direct calculation leads to the expression ~chirp (q, t~; Wa ) --

v/

i

C--*V{erf( ~-~q +

Qw) - erf(/~~-/3q __ '/--~-Qw)}V 2/ '

.-

y

'

(7.2.11) for the windowed spectrum of (7.2.9) using (7.2.10). Here, eft(x) is the error flmction defined by [13] eft(x)-

~2

L

C -t2

dt.

(7.2.12)

It can be guessed from (7.2.11) that the spectrogram 8,.hirp(q~h;;WR) of the (:hirt) (7.2.9) develops around the local frequency line t ~ - / 3 q (recall the linedelta, ~i(h;-/3q) obtained for the Wigner distribution function). The frequency resolution is optimal for a window width roughly estima, ted as Q~ ~ v~//3 (see Problein 7.2). In Fig. 7.4.a) the 3D graph o f ~chirp(q, h;; WR) is shown in the case fl - 0.5 for the corresponding optimal window width Q~ - 2. Correspondingly, a, unit-energy Ga,ussia,n window shapes a,s q2

_

1

o(q)

-

2~2

1 )~ (? w

W

.

(7.2 13)

W i t h this a,nalysis window the spectrogram of (7.2.9) explicitly writes as ~chirp ( q ' I'{;; ~ / G )

___J__lC -

- - V/-~cr

~2

(7.2.14)

with cr being 0"2

1-~-/32cr4 0.2

w

w

9

(7.2 15)

It develops as well around the line ~ - / 3 q in the (q, ~) plane with a frequency localization depending on /3 and crw It is in fact Gaussian in the variable

348

Linear Ray and Wave Optics in Phase Space

h:-/~q, and would turn into a line-delta, (5(t,:- ~q) with c7 ~ 0, a value however tha,t is not reacha,ble. The best we can do is to choose crW in order to minimize or, whi(:ll reqlfire, nlent is flllfilled with cr - 1 A 3D view of ~h~,.p (q a,; w(;) W

--

V/~"

9

for tlle clfirt) 14 = 0.5 is given in Fig. 7.4.t)) ixl (:orrc, st)ondcn(:e, witll the ot)timal va,hw, c~W - 1.41. Evidently the Ga.ussia~l wi~l(l()w a.ccommoda,tes fi)r a bette, r fl'e(lll(;ll(:y l~('a.liza,ti~nl witll rest)(;ct l,~ 1,11r r~;('ta.~lgh; win(l()w. All a cclmil,e a,(:(:~lud, (~f sew:ra,1 will(l~w filll(:l,i~)llS tyt)i(:ally llS~;~l ill st)e(:l,r~)is giv,;l~ i~ [2.2], [2.3] a.l~,l [,i.a]. hi Im,rti(:,~la.r, il~ [2.:,] l,l~c rcla.l, i, )l, l),;l,wc(m I,l~c sI)~r sig~al syld,l~esis m~(l l,l~e (la,l~)r sig~al ('Xl)m~si~n~ is elu(:i(la,t(;(l. ~l'h(~ I'(',a,(ll;r lllay t 4 1 1 l ~ S S 1,1~(~lm,l,~re ()f Sll('ll a, r(~la,l,i<m fl'()~ (7.2.3), which a,s l~()t(;(l (:xl)r(',ss(:s tl~(; sig~m.l as a w(;iglm:(l Sl ~l)(;rl )( )sit, i(n~ ()f I)asis sig~mls in 1,1~(; Sl)a,(:(;/sI)a,tial fr(;(lluu~('y I)lmm like ,~/,~, (q) - w(q - qt)~,i~,'q. Witl~ a. Ga,~ssim~ al~l)lil,~(l(; tak(:~ as a sy~l,l~(;sis wilx(t()w i~ 1,1~(: t)a,sis sig~mls .~,,,, (q), we 1)a,si(:ally ()])ta,il~ l,lm (lal)()r (;h;l~Xelfl,a,ry sigl~als Ca (q), r(;l)()rte,(l i~ w 4.6.4 i~ the, f()rlll .....

I

I/.1

,-

~

(q-- q()

+ i~'.o ( q -

2

,

witl~ a 1)(:it~g tl~(: t-it~t(:t~sity,, wi(ltl~. As sai(l it~ !i .1.6.4, ,y~ (q) Ires tl~(; s~m.llesi, sl)a,('('-lm,~lwi~ltl~ l)r~)~l~(:l, l)(U'~il,t(;(l l)y i,l~'~ Ill|('~u'l,ailli,y r~:lati~)~ ~)f F(nu'ier analysis. Tl~e sig~ml t;xtm,nsi()l~ s~ggcsl,c(1 ill (7.2.3) is l~()wever il~ l,(;rnxs ()f 1)asis sig~als tl~a.t a r(; (:(n~t,i~lu)~sly I)la,('(;(l i~ tl~(; (q, t,') l)lm~(;, wl~(;r(;as (la,l)()r's signal eXl)m~si~n~ is ix~ t,~'r~s ~f lmsis sig~als l,lml ax'~' ~lis~'r~'l,ely l)la~:~:~l i~ tim (q, t~:) I)lan('~. hx(l(;e(t, i~l l)ra(:ti(:al l)r()l)l(u~s tim "(:(nxl,ixut~txxf' ()f l,lie st)a('(;-fl'(;(l~len(:y t)la,~(; is r( ;l )la,(:(;(l l)y a. la.tti(:(; ()f I)()i~xt,s, nm,(l(~ l)y r(',('ta,nglflar (:('lls ('(n~veniently The 1)asis signals - (w lm.lOnS - was ()rigi~mlly ('lx()sr 1)y (la.l),)r as a, set of Ga~ssia.~-w(;ight(;(l ('(n~lfl(;x ('xt)()IX(;~tials lik(~ (7.2.16) [:~]. Ea.r l()g()l~ is l()ca,te(t in only o~m (:ell ()f tim lattice and a,t)t)rot)ria,tely size(l to t)e higl~ly (:()~xcentra,ted in its ()w~ (:ell, tln~s ~fininfizi~g the ()verlat) with the other h)g(ms, an(1 accordingly yir a "(l~a.~t~m~ ()f iifforlnatio~f' iI~(h',t)e~(lcnt ()f th(~sc t)r()vided by the, ott~er t)asis sig~a,ls. Tlnls the xmmber of log(ms for an eifb,ctive signal representa,tion tlm~s to t)e eq~m,1 to the (finite) nuint)cr of degrees ()f freedom of the signal. Ilxtcrcstingly, in the light of the, ray-like, picture of the Gaussia, nweighte(l c(n~t)h;x (;xt)onential signals, the relewmt rins widths in space and frequen(:y x~u~ltil)lyi~g to the, m i n i m u m allowed value, Ga,bor's signa,1 expansion is amena,t)le to the ni(:e interpreta,tion a,s representing the ot)ti(:a,1 signal in the form ()f a, s~t)ert)osition of a certa, in mmfl)er of "rays" appearing in discrete positions in the spa,ce and with discrete dire(-tions, uniquely determined by the definite G a b o r la,ttice in the pertinent space-frequency domain [3.3]. T h e accurate investigations of the G a b o r signal expansion variedly developed in t h e literature have g e n e r a l i z e d t h e original Gabor's expansion to rather arbitra,rily shaped elementary signals and to partially coherent light as well as to

The

WignerDistribution Function: Optical Production

349

discrete-time signals and oversampling [3]. Relation (7.2.3) signifies the link of the windowed Fourier transform-based signa,1 representa,tion with the Gabor signa,1 expansion, the coefficients of the latter being obtained as sampling values of the former at the definite spacefrequency lattice points. Gabor's signa,1 expansion represents indeed a primary example in a discrete view of a joint space-frequency signal representation. As an example of the resolution trade-off affecting the spectrogram, it may be instructive to consider the sum of a monofrequency signal a,nd a.n impulse [4.7]. We take as analysis window the unit-energy Gaussian (7.2.13). Firstly, we note that for the spatia,1 ha.rmonic ~ 0 (q) - ~ c ~~ one obtains 1

o"w

-(~-~o)2o-2

w,

9

(7.2.17)

and correspondingly for the point-like signal ~q0 (q) - ~ ( q - q0), one has _ (q--q0) 2 c,2

1

The reader may compare the above expressions with those for the corresponding Wigncr (tistribution flmctions being two delta-lin(~s at a - t% and q - %, respe('tiv(fly. The spe('trogran~ (7.2.17) turns into a. delta, line: &o (q, h;; w(~) 1 ~ ( ~ _ ~(,) for c~W ~ oc, whilst in turn Sqo(q ~;w(~) ---+ ~1( q %)with c~ ~ 0. Thus the st)('~(:trogram Sh+~(q, t;,"w(;) of the signal -

qo)+

1

" q,

(7.2.19)

displays a, good frequency resolution when tile signal is windowed by a "wide" Ga,ussian at the expense of the spa,ce resolution (Fig. 7.5.a,)). Conversely a, good spatial resolution is attained by using a, "narrow" Gaussia,n, thus reducing the frequency loca,liza,tion (Fig. 7.5.b)). An intermediate situation is reached by an intermedia,te va,lue of c~w (Fig. 7.5.c)). It is interesting to compare the distribution shown in Fig. 7.5.c) with that representing the Wigner distribution function as plotted in Fig. 6.8. Note that Sh+~(q, n;wc.) presents as well a cross term, which is oscillatory in n with a period governed by c~W and q0 and modulated by a double Ga,ussia,n. Hence it is significa,tive mostly a,round the point (q0, n0) in the phase plane inside an elliptical contour approximately of width 2v/2/cr w along the n direction and 2x/~crw along the q direction. We close this section briefly commenting on the behaviour of the cross terms that unavoidably enter the spectrogram when a multicomponent signal is considered. In fact, although the short-space Fourier transform is a linear transform, the spectrogram is evidently a bilinear transform, and hence, like the Wigner distribution function, has cross-terms. In particular, for a twocomponent signal like ~(q) = ~ (q) + ~ (q), (7.2.20)

Linear Ray and Wave Optics in Phase 5pace

350

FIGUII.I,; 7.5. SI)Ccl, r()grm~s of l,l~t: sig~al ~p(q) - b ( q - 1) + ~ (7.2.13) witll (liffcrcIlt wi(lths CYw'S.

1

r

~si~g t,l~e (/a, msim~ window

we haw~

S~(q, ~-; w) - l;~, (q, ~; w)l 2 ~ liL~ (q, ~; w)l 2 + 2!~{;~, (q, ,,:; ~)iL.~ (q, ~:; w)}, (7.2.2a) whi(:h ('h;a.rly sh()ws tlm.t tile (:ross-terl~ (:ontribllti~)1~ a~l~ts to t,tlc a.uto terms [~'~ (q, t~:;w)[2 a.n(t [ ~ 2 (q, ~; w)12 thr(,,lgh an intcrfcrcn(:c-likc nlc~:hanism. Then, h;t 11s consi(h;r a sI)atial harln()ni(: starting from q = (); namely,

{ (:'i~,oq q >__O, 9~+ (q) --

0

elsewhere.

(7.2.22)

Taking a,s a window fimction the lmit-energy rectangh: waveform (7.2.10), a straightforward calculation leads to

sin2[(tc--~~ Sqo + ((],

--(~w < q < Q~

~Q~(~-~o)2

h;; Wit ) --

,

_

sin2[(tc_~c0)Qw] ~rQw(~--~,o)2

,

q >

,

(7.2.23)

Qw.

It is evident t h a t ,S'~+ is non zero even for values q < 0, specifically over the intervM - Q w < q -< 0, where the signal does not exist. As expected, the frequency and space localization depends on Qw t h r o u g h an inverse relation.

The Wigner Distribution Function: Optical Production

351

FIGURE 7.6. Spectrograms of the signal (7.2.25) with Q1 = Q2 = 3. (a) ~,,~ = --~,,2 = 3; qol = --qo2 = 3. (b) ~o~ = -~o2 = 1; qo~ = -qo2 = 1. The subtle lines represent the spectrograms of the individual signal components. Tile reader is suggested to ca,lcula,te tile spectrogram of a, complex sinusoida] signal limited to a segment of width 2Q, i.e.,

{

d~~ 0

~e (q) -

Iql < Q, elsewhere,

(7.2.24)

using again the rectangular window (7.2.10). It turns out to wmish outside the vertical strip - Q ~ - Q < q _< Q~ + Q, and be mostly concentrated within the frequency interval centered on ~c0 of width 2~ Qw" The graphs shown in Fig. 7.6 arc relevant to the spectrogram of the twocomponent signa,1 ~(q) - (IOQ1 (q)@ ~Q2 (q)' (7.2.25) ea,ch component being a shifted version of the finite-length ha,rmonic wa,veform (7.2.24), namely i~~ (qc

P% (q) -

0

J

) qoj

Iq-

q% I < - Qj,

j-

1, 2.

(7.2.26)

elsewhere.

Two sets of values of the involved parameters have been considered; again, a rectangular window has been used (Q~ = 1.5). The displayed graphs are aimed at providing a certain feeling for the mechanism of the cross-terms in the spectrogram, also in comparison with that of the Wigner distribution function (see w6.3.4). In fact, by inspecting the various contour plots in connection with the corresponding set of p a r a m e t e r values, we may infer that in general the cross-terms between the signal components have a significant effect on the overall shape of the spectrogram of a multicomponent signal only if there is an effective overlap of the self-terms, which

Linear Ray and Wave Optics in Phase 5pace

352

ultimately det)ends on the signal structure a,nd the window used [4.6, 2.5]. Then, if the, a,uto-terms axe sufficiently far apart in the st)a(:e-fre(tuen(:y t)la,ne, the (:ross-terms axe significantly a,ttenuate(t a,n(t maske(t by the s(;lf-teri~s. In contrast, as n()te(l in w 6.3.4, th(; ('ross-t(;rms ()f the Wign(;r (tistrit)~ti()u fim(:ti()n a.lways ()(:(:ur ~um.tt('mmte(l llfi(tway 1)(;twe(;x~ (;a.(:l~ t)a.ir ()f signal (:()uq)()nents wl~(;tl~(;r t,l~(; (x)I'r('st)()1~(ling a.~xt()-t,(;r~x~s ()v(;I'la t) ()r ~()t,. A gr(;a.t vari(;ty ()f fllrt,l~(;r (;xa~I)l(;s ()f Sl)(~('(,r()~rm~s ('m~ t)(; f()~l~(t ix~ [/1.7] ((:l~. 7, t)I)- 93-112), 1)y wl~i(:l~ tlxis s~d)s(;(:ti()l~ is t)a.rtly i~sl)ir(;(l. 7.2.3

F'ivm the sh, o'r't-spacc spcct'l"um to the W'ignc,'l' dist'r'ibutio'n and ambiguily function

A (:h)s(;r h)()k a.t (,1~(; (;xl)r(;ssi()~ (7.2.1) ()f ~ ( q , h:; w) r(:v(;als a (:(;rtai~ si~ila.rity witll tll(: ('r()ss-Wig~l(;r (listril)llti()ll fllll('(,i()ll ((i.3.12) ml(l 1)('(,l.('r y(:(, witll l,l~(~ (:r()ss-m~fi)ig~ity fitl~(:t,i()~ (7.4.1()) [~2..~, 4..~]. TI~(; s i ~ i l a r i t y 1)(;(:()~(;s l~()r(~ (;vi(h;nt if w(; r('~writ(; l,ll(; (~xl~r(;ssi()ll (7.2.1) izi a, syllllX~(;tri(' fi)r~ I~y (,l~(; sinq)l(~ (:lm~g(; ()f varial)h' ~{ ~ q", I)(;i~g q' ..... q" t q/2. ~I'1~(, i~t('gral i~ (7.2.1) t~ln~S iut() ~(,/~-..

w ) -- -~..;, ,, -i,,.q/'.,

/.

. ;(q'

~ '~) w* (q'

- ',,-1)~,- i"q'dq'

(7.2.27)

(:h;arly I'(;s(;~fi)li~g tl~(; (;xl)r(;ssi()~ (7.4.1 ()) ()f tl~(; (:I'()ss-a~d)ig~ity flu~(:ti()n ()f t,t~('~ itself, i.e., w(q) = ~(q), tt~(; i~t(;gTa.1 at)()v(; j~st r(;t)r()(h~('(;s tlmt (l(:fi~i~g the a.~t()-a,~fl)itz~ity flu~(:ti()~ (7.4.1). 'I'll(; s('~lf-wi~(l()w(;(l st)(;(:tr~u~ t)(;(:()~(;s i~(tee(t

~,(q, +,';~(q)) - ff-~'-~"q/2A~,(q, t,').

(7.2.28)

()n the ()(,l~(;r lm~(1, wi(,l~ (,1~(: (:h()i(:(; w(q) = ~ ( - q ) , w(; ()|)(,aix~

/.

~ ( q , ~-,,9~ ( - q ) ) - , /l___L_(,-ih'q/2 ~ . ~(~ + q') ~ * (~ - q')~, -ih'q ~dq',

(7.~.~9)

the integral 1)(,iug imme(tia,tely re(:ogifize(t as 7r14;~(q/2, ~:/2). Hen(:e,

~ ( q , ~; ~ ( - q ) ) - V ~ ~"

142~(~2, -~). 'r

(7.2.30)

Thus, we (:a,n synthesize the windowing-ot)era, tion pi(:ture of the Wigner distribution an(t ambiguity fimction in reJation with the short-space Fourier transform through the dia,gram w(q) = ~(q) ~/

A~(q, ~)

\ ~ ( q ) = ~(-q)

w ~ ( q, ~) (7.2.31)

The Wigner Distribution Function: Optical Production

353

which outlines the basic correspondences, disregarding the precise m a t h e m a t ical relations demanding for the cross quadratic phase-term e -i~q/2 and some constant factors. It appears evident that a direct correspondence holds between the spacefrequency variables of ~ and ,A~, representing in either case the position and frequency diflhrence-coordinates of the signal and its spectrum to the corresponding sliding replicas. In contrast, the space-frequency variables of the Wigner distribution function represent the position and frequency centrecoordinates of the signal and its spectrum to their mirrored sliding replicas. The considerations above support the interpretation of the Wigner distribution function suggested in w 6.2, and anticipate a possible interpretation of the ambiguity function, on which we will comment in more detail in Sect. 7.4. 7.2.4

The spectrogram as a smoothed Wigner distribution function

As previously noted, the sliding-window Fourier transform can also be related to the smoothed Wigner distribution function when a gaussian-shaped window is involved in. To see this, wc resort to the Moyal formula, (6.3.50) which expresses a definite link between the overlap integral of two signals and that of the respective Wigner distribution functions, i.e.,

1 / ~1*(q)~2 (q)dq 2---Y

2 -

/ / lA?+l (q, t;)lN~u (q, t~)dqdh;.

(7.2.32)

The defining integral (7.2.1) of the short-space spectrunl is interpretable as the inner product of the signal p(q'), whose Wigner distribution function is W+(q', h;'), an
S (q,

w) -- f / l / Y w ( q ' - q, ~'

~c))4~(q', ~')dq'dn'.

(7.2.33)

The double integral above is evidently the convolution of the Wigner distribution function of the signal with that of the window. Therefore, due to the arbitrariness of the window function, we may conclude that the convolution of any two Wigner distribution functions can be interpreted as a spectrogram. The reader may deduce (7.2.30) from (7.2.33) with w(q) = ~ ( - q ) . Interestingly, the convolution in (7.2.33) can also be viewed as resulting from a windowing operation in the phase plane of the Wigner distribution function of the signal, whose value at the phase-plane point (q', ~') could be picked up by a rather narrow distribution 142w. Thus, the windowing operation

354

Linear Ray and Wave Optics in Phase Space

in the space domain of a, given signal, or in the frequency domain of the releva.nt Fourier spectrum, corresponds to a, windowing ot)era,tion in the spacefrequency t)la,ne of the Wigner distribution flxnction of the signal. In the light of this view, tlle resohlti()n tra,(te-off ()f the st)e(:trogranl is (:learly evi(tence(t, a,s we know tha,t tlle Sllt)l)()rt ()f t,lle Wigner (tistril)utioxl fim(:ti()ll exa,(:tly r('~t)ro(tuces tha.t ()f th(; signal in sl)a.(:e a.n(t ()f the releva,nt st)e(:trllnl in fre(tlu;n(:y. Also, we s(;(', fl'()nl (7.2.33) tlmt tll(; will(l()willg ()t)(;I'ati()ll ill ttl(; Wigll(;r t)lml(; t)y any Wiggler (listril)llti()l~ fi~(:ti()~ I)r()(l~(:(;s a sI)a('e-fr(~(l~l('~(:y (listri|)~d,i(ni wlfi(:h is

(~v(;rywl~('t'(; l)()sitiv(', t,lir()~gli tl~(; wl~()l(; (q, ~',) I)lm~(',. In t)a,rti(:~fia,r, taki~g a, (lm~ssim~ wi~(l()w, wl~()s('~ Wig~(',r (listril~ti(~ tim(:l,i()~ is as well (la,~lssim~ i~ I)()l,t~ sI)a,('e al~(l fi'(;(t~(;~(:Y, (hi(; ()l)ta,i~ls ()l~ tt~('~ right si(le ()f (7.2.33) tl~(' Ga,,u,,s,sia,'n,-,s'm,ooth,(,d Wigl~er (listl'ii)~ti()~ fi~(:ti(n~ ()f ~(q), wl~i(:l~, as (les('ril)(:(l i~l w (i.3.(i, is ()t)tai~l(',(l t)y l,l~(: ('()~v()l~l,i()~ ()f l,l~(; Wig~mr (tistrit)~ti()~ fi~(:t,i()~ ()f tim sig~ml witl~ a sI)a,('(~-fi'(;(l~U',~l('y (la~ssia.~l (listri|)~lti(n~, lmvi~lg at)l)r()t)riat(;ly r(:lat(;(l Sl)a,(:(; m~(l fi'(;(l~(:~l('Y r ~ s wi(lt]ls. Evi(l(;ntly, r(;la,ti(n~ (7.2.;~3) (;sta.I)lislms tlm,t l,l~(; (lm~ssia,l~-Sl~()()tl~(,,(l Wiglmr (listril)~ti()n fiu~(:ti()~ is (',v(~ryw]~(',r(' I)()sitiv(; l,l~r()~gl~ l,ll(; wl~()l(, (q. t;) l)lm~(;, i~ a('('()r(t witl~ tl~(; (lis('~ssi(n~ ()f ~ (i.3.6. I~l !i 7.9 w(; will (l(;s(:ril)(; tll(; ()l)ti(:al i~lll)l(;~(;l~tati()n

()f tl~(; Gm~ssia~l-s~l()()tll('(l

Wig~l(;r (listril)~t.i()~.

A(:(:()l'(li~lg t,() (7.2.33) t,l~(; (/a.~ssia.li-Sl~l()()tll(;(l Wig~l(;r (listril)~ti()~l fini(:ti()n (:a,n t)(; ilit(;rt)r(;t(;(1 a,s l,ll(', Sl)(',(:tr()grmn ()f l,ll(; sig~ml I)y a Gm~ssiali wi~l(l()w, an(t tll('ai |)(; r(;lat(;(1 t() tl~(; l)r()j(;(:ti()~l ()f tll(; sig~ml ~l)()~l a (lal)()r-lik(; wav(;f()rm, i.(;., a (la~lssim~-w('iglll('(l ('(n~ll)h'x ('xt)()~l('~ltial.

7.3

The Wigner distribution function and the general class of

space-frequency signal representations

Tile st)ectr()grmll, tlw, Wiglm, r (listrit)utioll fimctioll aal(l tile a,lllt)igllity flmction are ret)resexitatives ()f a, wi(le (:lass of bilinea,r st)a,ce-fl'er r fuI1ctions, whi(:tl are rela, ter t,()ea,(:h other through linear transf()rmations. More in general, they are r ilm) a c o m m o n a,na,lytica,1 fi'a,nlework, in(:luding as well non|)ilinea,r signal reI)resentations, which has t)een r by Cohen [4.1] within the (:ontext ()f (tlm,Id, lml mechanics and then a t)t)lie(t to time-frequency as well as stm,ce-frequency signal analysis [4]. It is w()rth noting tha, t t)resently the "strategies" mlr the joint spacefrequency (as well tinw~-fi'equency) signa,1 analysis dew,lop through two channels. On one. hand, the linear at)proa, ch is inspire(t by the Fourier-like philosophy, and so it a,ttemt)ts to directly decompose the signal into components, which are. simultaneously localized in space and frequency. Formally it can be regarded a.s a linear m a p p i n g between the set of the 1D complex functions of spatial position or frequency, p(q) or ~5(~;), to the set of 2D in general

The Wigner Distribution Function: Optical Production

355

complex-valued functions of position and frequency: s

~(q) E 122(R)

> 12~(q, t~;;~).

(7.3.1)

As earlier said, Gabor's signal expansion is the primary example of such a linear approach, the analysing components being the highly space- and frequencylocalized Gaussian-weighted complex sinusoids. The windowed short-space Fourier transform is conceptually based on the same decomposition strategy, performing the Fourier analysis of the signal as it appears when seen through an ensemble of identical appropriately chosen windows shifted in position with respect to each other. The symbol 2E generally signifies the auxiliary tools, as, for instance, the analysing or the synthesizing window function, which pertain to the specific transform and/or the signal under examination. On the other hand, the bilinear approach attempts to combine the instantaneous power I~(q)l 2 and the power spectral density I@()12 of the signal in a single 2D distribution such that it could indicate the energy content of the signal silnultaneously in position and frequency. Formally, it realizes a. inapping between the pairs of 1D complex functions of spatial position or frequency to the set of 2D in general complex-valued functions of t)osition and frequency:

c (R) • c (R) Tile Wigncr distribution flmction is tile primary cxa.nq)le of such a quadrati(: approach, and as note(t in w 7.2.3 it can be seen as arising from a windowing-like pro(:c(ture with the ensemble of analysing windows being provided by replicas of the signal itself, mirrored and shifted in position. Indeed, the scheme (7.2.31) exemplifies tile general relation holding between the linem" t2~ and the bilinear ~3~ joint representations, with the former turning into the latter when the waveform to be analysed is itself used as the analysing tool. It is shown that any bilinear signal representation is obtainable as a linear transformation, and also under inore specific conditions as a linearly filtered version, of the Wigner distribution flmction of the signal to be analysed (Problem 6). Again, 2E quite generally denotes any auxiliary tool that can be used to make tile mapping (7.3.2) adjustable and selectable in accord with the specific requirements inherent in the problem under examination. More in general, may depend on the signal itself, thus allowing for higher-order representations. Evidently, this provides a further degree of freedom in the light of the optimal "design" of the mapping (7.3.2) in relatioi: with the signal and tile specific processing/analysis requirements. In order to gain a feeling for the "position" of the Wigner distribution within the general class of the joint signal representations, we report below the unifying formal expression from which all the space-frequency distributions can be generated. Only a few comments arc given, addressing the reader to [4] and [5] for a detailed treatment of the subject and the several related topics.

Linear Ray and Wave Optics in Phase Space

356

7.3.1

The general class of space-frequency signal representations

All tile st)a.ce-frequency ret)resenta,tions of tile 1D signal p(q) can be obtained from the triple integraJ [4]

C~(q, ~;; ~) - ~-~ . ~ /././' ~* ( q ' - ")~(q'2 + ")~ (",2

"l)~'-~(""'+""-"~')~Zq'~t'ld', (7.~.3)

wl~('xe tl~(; a uxilim'y fin~('l,i()~ ~P(u, q), r(;ferr(;(l t()as tl~(; k~','~"n,~',l[4.:t], is 1)asi(: t() i~l(livi(llm,lize tll(; (lifIi;r(;ld, r(;l)l'(;S(;ld,a.ti()llS a,ss()(:ia])l(; witll l,ll('~ sig;lm,1 ~(q). Tlle Wi~n(;r ild,(;gra,1 (:()rr(;sl)()ll(ls t() tll(; i(l(;ld, ity k(;rll(;l ~ w (",

"t) -- 1"

(7.3.4)

Tll(; Sl)(;('tr()gra,lll als() ])(;l()llgs t() tll(; g(;ll(;ra,liz(;(l (:lass ()f siglml r(;I)r(;s(;uta,tions (7.3.3). It, is ill(livi(llm.liz('(l ])y t,ll(; k(;r~l(;1

/. ~ ( q -

'#:,(,,.,i)--

.

~)~*(q + ~),,'"',dq,

(7.:3.~)

(t{;fi~l(;(1 1)y tile; wi~(l()w fil~l(:t,i()~l w(q) l,lir()~lgli a,~l a x~fl)ig~lity-lik(' i~lt(;gra,1 f()rni. ~(~) l~()l(ls a,s w(;ll"

C~(q, ~;; ~) - ~ l.

/l'/'

~*

(H.I

" t~.l t/ , ~)~( + ~)~(,,, ,~)('i(tltCt-Fljq-qtr

__ lJ

(tt~;t(hJ(t'q

(7.s.~)

a,s it, ~my easily I)(; v(;rifi(:(l 1)y (;xt)r(;ssi~g tl~(' sig~ml a~l)lil,~(h' l,l~r()~glt th(; i~v(;rs(; F()~n'i(;r i~d,(;gra,l" ~ ( q ) - ~ 1 .[' ~(~:),"~qd~:. Th(, t)r()t)erti(;s ()f a, {lisl, ril)~l,i()~ C~ a,r(; l'(;th;(:t(;(1 il~l,() st)(;(:ifi(: (:()~stra,ints ()u the (:orrest)()u(li~g k(;n~(;l ~. It(;u(:(;, it, is t)()ssit)l(; t() a,~a,lys(; tl~(; I)rot)erties ()f a t)arti(:~flar r(;l)r(;s(;~d,a,ti()~ i~ a, systemati(' wa,y ()~(:t: tl~(; kt;nml is given; aJso, ()~(; can ('l~()()s(', tl~()s(; k(',~'~(',ls tha,t t)ro(l~(:(', st)a.('(;-fl'(;(t~(;~('y (listributions sa,tisfying the (lcsir(;(1 t)r()I)('xti(;s for th(; st)e(:ifi(: I)r()t)l(;~ a,t lm.~(1. F()r examt)le, the distributions sa,tisfyi~g the ma,rgina,1 req~ir(;~nents a,r(; g(;~mxa,te(t by kernels having the simt)h,' t)I'()t)(',rty: ~(t,, 0) - ~(0, 'q) - 1 (Prot)h;~ 3.a,). In that case, the one-dimensional (i.e., in st)a,(:(; a,nd frequen('y) e,nergy (tensities of the signal directly relate to the freque~(:y and space integraJs of C~(q, t,:, ~) according to

./'C~(,~,~ 9

j c~(q, As a, consequence, the signa,1 t r i b u t i o n over the p e r t i n e n t in w 6.3.5 when c o m m e n t i n g bution function, properties

~:, ~ ) d ~

I~(q) l2,

(7.3.7)

~)dq - I~(~-.)1=.

energy ca, n be derived by integrating the joint disspace-frequency domain. However, as also noted on the marginal properties of the Wigner distri(7.3.7) do not w a r r a n t the interpretation of the

The Wigner Distribution Function: Optical Production

357

relative distribution as a space-frequency energy density at each point in tile space-frequency plane, being also such a pointwise interpretation commonly intended as precluded by the fundamental resolution limitation imposed by the uncertainty product. Reality is as well another desirable property. Real distributions arise from kernels satisfying the relation: ~(u, r/) = ~*(-~,,-~/). Evidently, both ~ w and ~ s generate real distributions, but whilst ~ w generates a distribution with tile proper marginal properties, a,s confirmed by Eqs. (6.3.26), ~ s does not. The spectrogram is an example of a signal representation that, as opposed to W, is positive but does not yield the correct marginal distributions, its marginals being not exactly equal to the energy densities of the signal. Although not shown in tile notation, tile kernel ~(p, rl) may depend explicitly on the space and frequency variables; it may also be a functional of the signal itself. Independence of the kernel of space and frequency assures the invariance of the corresponding distribution under shifts of the signal in space and frequency (Problem 3). Hence, the analysis characteristics of the signal representation do not change with position and frequency, the signal being then uniformly analyzed with the same space resolution and the same frequency resolution at various space-frequency points through the pertinent dolnain. Also, we have already noted that the st)a.(:e an(t frequen(:y-shift invariance is essential to guarantee the corresponden('e of the space and frequency variables of the distribution to those of the signal and its spectrum respe(:tively. In(tependence of the kernel of the signal gives the associated distribution a quadra,tic dependence on both the signal and the spectrmn; in that case, the distribution is said to be bilinear. As seen in w 6.4, the Wigner distribution function is both bilinear and space and frequency-shift invariant, in accord with the mathematical form (7.3.4) of the relative kernel ~w- It has also been noted in w 6.3.6 in relation with the positiveness of the Wigner distribution function that, as a general property, bilinear joint distributions cannot be positive and simultaneously satisfy the marginal requirements (see also w 8.2.2). Invertibility is of course a filrther important issue to be assessed. For signalindependent kernels the following inversion formula is readily deduced: P(q)-

1

2~*(o).

/jr/ C~(q',g';~) cit~,q_i(q,_~2)~,,dqtdt~d~,~ q,(~,',q)

(7.3.8)

It is therefore possible to recover tile signal from the relevant distribution C~ (up to a constant phase-factor) only if the division by the kernel is allowed for all values of u t, q. The reader may easily regain the inversion formulas for the spectrogram and the Wigner distribution function from this general relation. A strictly related topic is concerned with tile representability of a 2D function, i.e. with the individualization of the condition under which a 2D runetion could be considered as the space-frequency representation of a physical signal. This is the mathematicM formulation of tile practical problem of signal

358

Linear Ray and Wave Optics in Phase Space

synthesis from a space-frequency distribution, i.e., a surfa, ce over the spacefrequency pla,ne. In a. typica.1 spa, ce-frequency representatioi~-base(t scheme for signal synthesis, the sl)a,ce-freq~w, ncy ret)resentation is modifie(t (e.g., 1)y masking or smoothii~g) in order l,() fit, sr (tesirer t)()sit, ional a,n(t st)e(:t, ra,1 t)rot)crties. This nmy result into a "~onwdi(t" st)a(:e-freque~cy (listl'i|)~ti()~ which then is ~r l(mger t,lw~ st)a,(:(;-fl'(;(l~l(;~(:y r(;t)res(;lfl, a,t,i()I~ ()f a.ny siglml. We have a,(t(; (h;vise(1. TI~(: i~l,el'eSter l'(;a,(ter llltl;'y" (',o~s~fll, the relal,er lil,eral,~re [5].

7.4

The ambiguity function

rl'lle ambiguity flmction A({l, [,') is (t<'filler llnu:ll tile Smile way as tlw Wigner ~listri|nlti(ni fllllc't,i~ll. It lllirr<ws t,ll~' Wigll~'r ~iistritnlti~ll fllllr ill t,lm wi11(h~we(l F(nlri~;r trmlsfin'lll arisillg sr (7.2.31). It lla,S finlll~l wi~le a.l)t)lica,ti~nls ill ra,(la,r m~l s~m,r as ~sefi~l l~ma,s~m', ~f Sillnlltm~t',t)llS rm~g~' a,l~(] velo(:ity target i~ffor~m,l,i~)~ [~i]. Its a,l)I)li~:ati~)~ i~ tl~,', ,)l)ti(:a.1 (:~)~m;xt t~as tirstly t)ee~l ,lisr162 t)y Pal),~lis [7.~], a~,l later i~vestigale, t 1)y ~ a ~ y m~tl~,n's [7]. The t)rot)eI'ties of tl~e m~t)ig~fit,y f l l l l ( : I , i ( ) l l il,I'C V(~,I'y well ~(h;rst,()()(l; als(), several ()t)t,i(:a,1 sel, ut)s lla,v(; ])(;(;11 i~q)h;n~(;~lt(;(t t() w(~rk a.s ()I)l,i(:a,1 Im)r for (list)laying the a.nfl)ig~it,y fim(:l,i()~ fin" t)()l,h real aalr (:()l~q)lex siglmls [8]. Due to the formal similarity ])(;t,ween W a,~(t ,,4, oI)tir a,r(:hitect~res have l)e(;n (levise(1 to ()t)erate a.s l)r()t:ess()rs fi)r |)r the Wig~mr (tistri|)~ti(n~ aa~r a.m|)ig~fity flm(:t, ions with l~i~()r ()r no (:ha,nges in the ha,rr re. In the t)a,ra,gra,I)hs below we will evi(ten(:e the analogies a,n(] differen(:es t)etween t,he Wigner and ambiguity integrals, e~t)hasizing t,t~eir intert)reta.ticm a,s F()~lrier t,ra.ilsf()rlns of the /K t h; t s y m m e t r i c pro(h,cts ~*(q @)~(q + @) and @ * ( n - -ff)@(h: + -if). We will see t h a t the direction of the transform, q/q~ or ~:/h:~, is central to determine the meaning, the interpreta, tion a,n(t the use of 14; and ,4 as signal (tescriptors over the space-freq~u;ncy plane.

7.4.1

Basic definitions

The ambiguity function A(O, ~) is defined as a c o m p l e x cross-correlation of the signal with a version of itself which is both spatially and spatial-frequency shifted (a). It arises therefore as a function of the Fourier conjugate variables The (complex) cross-correlation function 7f,g(x) of two deterministic signals represented by the functions f(x) and g(x) is defined by the integral [10.15, 10.16] a

The Wigner Distribution Function: Optical Production

359

and ~ through the space-domain integral

A(q, ~) - 1

~* (q'

~2)~(q' + ~2)e-inq'dq ',

(7.4.1)

as well as through tile frequency-domain integral

A(q, ~)

__

~1

@* (t~' __ -~)@(t,;' ~; s -Jr--~)e in'

qdt~' .

(7.4.2)

Unlike the Wigner distribution flmction, which is always real, the ambiguity function is in general complex. It is worth mentioning some differences in terminology concerning the ambiguity function. We may report, for instance, the alternative definition

x(q, ~) -- ~1

/

~*

(q' - @7:(q')e- i ~q' dq',

(7.4.3)

reproducing tile integral pertinent to tile windowed Fourier transform (7.2.1), and hence being equivalent to (7.4.1) within a linear phase term. Also, the complex conjugates of (7.4.1) or (7.4.3) as well as the respective absolute values, or squared absolute values, are often referred to as tile ambiguity function. As said earlier, the alnbiguity flmction has primarily been used in radar applications as a tool for predicting the ability of a radar signal to measure the range and the velocity of a, target via, Doppler shift [6]. In this context, it is properly defined a,s a, function of the dual variables (T, L,) (the "time" equivalents of the "space" pa,ir ((~,~), perta.ining to the optical context); 7represents tile time-delay of an echo produced by the target as ~, the Doppler frequency shift due to the target inotion. 7.4.2

Relating the Wigner distribution and ambiguity functions

Notably the Wigner distribution and the ambiguity function are related by a double Fourier-like transform relation, being

1

jj A(q~ ~)e-i(~q-q~)dqd~,

A(q, ~) - -2-~

(7.4.4)

W(q,~)ei(nq-qn)dqdt~

W(x) -- / f(~')g*(~' - x)dx' - / f(x' + x)g*(x')dx'. In general, f(x) and g(x) may signify b o t h the signal waveforms and their spectra; correspondingly, 7f,g(x) may represent the crosscorrelation of the signals or their spectra. In particular, if g(x) = f(x) one obtains the (complex) autocorrelation function 7f of f. At the origin the modulus [Tf(x)l attains its m a x i m u m value, which is just the norm of f:

[Tf(X)[ ~ "yf(O) --I[f[[ 2

Linear Ray and Wave Optics in Phase Space

360

Resorting to the n o t a t i o n of w 4.6.1 for the Fourier transform, we can write (7.4.4) in the elegant and enlightening form W(q,

-

'

^

-

(,,,,)

o

(q..)

9

Tll(; slfl)s('rit>ts (>11 l,lle F()ln'i(:r (>t)(:ra,l,(w syllfl)()Is sigllify 1,1)(: l)air <)f l:()llrier <'(mjllgat,(: varia])h:s ilJw)lw:
" Y(,,,~) h(.r, ',]) - ~ .~

/. d.~"< - i.,.;~

(7.4.6)

] d'[/~'i":'l~'l('[I '1~) - h(.r q)

(7.4.7)

'

h(.r, '1) - .q(')). q).

Evi
I

f

(.,',,~) (:stalflisl~(:s a F()~ri('r-lik(: r(:lal,i{)~ ]>('l,w(:e~ l.l~(' fi~('l,i()~s h(.r,q) a~(! .f(.r,.?), 1,I~(: wu'ial)h:s i~w()Iv(:(l i~ tl~(: F()~ri(,r (,ra.~sf()rx~ti()~ l)(:i~g, ()f <'()~rs(:, :? a,n(l 'q. A(:<:()r(li~gly, tl~(: Wig~(:r <listril)~ti()l~ fi~<:ti()~ l~my I)(: tl~()~gil~t as I'(:s~llting fl'()~ tlm m~fl)ig~ity fi~('(,i<)~ l)y a,~ i~w(:rs(: F<)~ri(:r (,ra,~sf()r~ wi(,]~ l'(:Sl)(;(:t to (,l~(: fr(:(:i~g t~I'~(:f (,I~(: tw<) tra,~sf()r~m,ti<)~s i~ E
q

<

.A=st)ace variable

q

<

'

.A-frequency

variable

er

]A?-ffequency

variable

K,

(7.4.s)

showing that in the a,nfl)iguity function space-fr('qlwamy t)lane, the "horizontal" axis <:<)rrest)on
IliA(q.

~)12 d~d~ -//pV2

(q. t~)dqdt~ -

1~ll4 27r

"

(7.4.9)

The

WignerDistribution Function: Optical Production

361

For future use, we write below the expression of the cross-aInbiguity function A~I,~ 2 ((t, K,) relevant to the pair of functions (/91(q) aIld (/92 (q)" A~I ,~2 ((t' ~';) __ ~1

(~19(ql -- ~)(~2 (ql

+ ~2)C-it~qldql,

(7.4.10)

which evidently parallels the cross-Wigner integral (6.3.12). Interestingly, whereas W~I ,% (q, n) is commutative in the signals, being W~2,~ ' (q, n) - W~, ,% (q, n), the cross-ambiguity function is not, since

Aq02,~l ((t, ~v) -- A;l,q02 (--q,--~)

(7.4.11)

Equation (7.4.10) supports the consideration in w 7.2.3 concerning the relation between the windowed Fourier transform and the cross-ambiguity fimction. 7.4.3

Energetic and correlative classes of the bilinear space-frequency signal representations

Within the general class of space-frequency (tistributions represented by (7.3.3) with signal-independent kernels, the ambiguity integrals (7.4.1) and (7.4.2) are obtained in correspondence with the kernel ~ ( . , ~) = 2 ~ 5 ( ~ - ~ ) 5 ( .

- ~,).

(7.4.12)

Sin(:e qJA (tct)(;nds on both the spa(:(; and fr(;qucncy variables, the ambigllity flmction is not invariant under shifts of the signal in space and frequency. In fact, translating the signal or the spectrum affects the flmctiotl A(q, ~,;) by a linear phase factor (Problem 4.b). In addition, unlike the Wigner distribution, the amt)iguity flmction does not satisfy the marginal properties. It satisfies the so called correlative marginal properties expressed by the relations A(q, ~)d~ -- ~(~)~*(-~)'_ _ A(0, ~)d0

We also see that

- ~(~

(7.4.13)

)~* (-~).

.

~(q') ~* (q' - q-)dq',

A(q, O) __ ~1

(7.4.14) A(0,~)

-

~1

f

J

~(~')

@* ( H t

- ~)d~'

showing that the profiles of A along the 0" and ~ axes just equal (apart from the factor ~1) the autocorrelation functions of the signal and its spectrum (see footnote on p. 358). Then, the energy densities of the signal obtain by Fourier transforming the zero-space and zero-frequency profiles of A; namely,

I~(q)l 2 -

A(O,~)d~qd~c, (7.4.15)

362

Linear Ray and Wave Optics in Phase 5pace

Moreover, it follows from (7.4.14) t h a t

A ( 0 0, ) -

II:II 2. ~

9

(7.4.16)

T h e anfi)ig~fity fl~xmti()n a,t tll('~ ()rigin is t)r()I)()rti()na.1 t() tl~('~ t()ta,1 (;~mrgy ()f the signal, a.n(t ll('~l(:('~ A((), ()) > 0. hi (:()~tra.st, tll('~ Wign(;r (tistril)~ti()n fiu~(:tion a,t the ()rigin ~lmy a.t,taixl a n('gat, iv('~ vahm, as cvi(tcn(:(,~(1 i~l !i 6.3.6. M()rc stri('.tly, t)y S(:l~warz's i~('~(tl~ality w(" ti~(l

[A(0, ~)l <_ A((), ()),

(7.4.17)

a,n(1 hone:(; 1,11(;lll()(llll~ts ()f ,,4 is always grea, t('~sl, at t,h('~ ()rigin ()f 1,11e (~, k) t)la,nc. Si~l(.(,~ (hi(; l~s~m.lly (:(n~si(l(;rs ~lliil,-('~l(~I'gy sig~mls, r('~lal,i(n~s (7.4.!)) a,~(l (7.4.16) ta.kc tile ~-il|([(:l)('lRlcld, f()n~s

a~(l

//IA<,

<7

1 I A(O, ~-)] _< A((), ()) - ,,_~.

(7.4.1~)

lh;lati()ll (7.4.18) stat(;s tllat t,llc w)llllll(; Illl(lcr t,ll(' a l~llfigllit,y sllrfiu:(; is ill(l(;l)(;ll(h;zlt ()f l,ll(; siglml wa.v(;fi)nll. Als(), it, sll()ws t,lm.t, t,lm a,llfi)igllity finl(:t, ion lillE";|, (tistri})lll,(~ i,llr()ltgll()lll, l,ll(; (q, t~') l)lml(~; il, (:alnl()t 1)~' (',()ll(:(;~ll,ratc(t a,r|)itrarily (:h)s('~ 1,()1,1~(; ()rigi~. A(:(:()r(lil~g l,() (7..1.19) a (l('~Ii~ii,(; "I)<wti(n~" ()f A(~, ~,;) ~nlst rc~lai~l i~l tll(' l)r()xi~lil,y ()f t,t~(; ()rigi~. Tlicrcfi)rc w(; ~ a y arg~m tliat if t,hc region ()v(;r wl~i('l~ t,lle a~fifig~ity fi~l(:l,i()ll is siglfifi(:a,~d,ly
The Wigner Distribution Function: Optical Production

363

we are not allowed to interpret the space and frequency variables of the ambiguity function ~, ~ as corresponding to those of the signal, as we loosely do with the space and frequency variables of the Wigner distribution. As already emphasized in w 7.2.3 an(t here evidenced by (7.4.14), the ambiguity flmction variables (~, ~) are properly interpreted as the space lag and the fl'equency lag; we will further discuss this point w 7.5.1. Therefore, unlike the Wigner plane, the (~, ~) plane can not be identified with the signal phase plane. The space and frequency variables proper to 147 and A are linked by the duality relation implied by the Fourier integrals (7.4.4), and displayed in (7.4.8). As a conclusion, a few words on the invertibility of the mapping (7.4.1), and the representability of a. space-frequency distribution as the ambiguity function of some signal. The reader may easily verify that the signal amplitude can be recovered from the ambiguity function up to a constant phase factor as 1 ~*(0)

~(q)

r

~,)c~qn d~

(7.4.20)

a dual formula holding for the signal spectrum. Finally, carrying on the various steps suggested in w 6.3.10 in connection with the representability condition (6.3.70) for l/V, it can be shown that the necessary and sufficient condition for a f~mction of the F(mrier-conjllgate variables (q,;:) to be interpretable as the ambiguity fllnction of sonic signal ix expressit)le through the relation A(O+

1 f/A((~t

27r,

qo 2

2~

2

2

(q -

%

2 ~

2~

2

(7.4.21)

2

which must hold for any q0 and ~07.4.4

Comparing the ambiguity fltnction and the Wigner distribution function through some ezamples

In order to gain an intuitive insight into the differences between the Wigner distribution and the ambiguity function, we may calculate A(O,~) for the simple signals in the table (6.4.39), i.e., the ~-like, monofrequency and chirplike signals. We easily obtain -

qo)

1 Gi~oq i~(q-qo)2+inoq

27r~(q)c -iqO ,

,

~,

A~ 0 (~, ~)

1 ~(~)Ci~oq 27r e

(7.4.22) -

tiff).

In all cases the ambiguity function is confined along a straight line in the (if, ~) plane, respectively the ~-axis, the if-axis and the line ~ = c@ The absolute

Linear Ray and Wave Optics in Phase Space

364

quadratic-phase signal --

, ~

~

spatial harmonic ~signal

f

J

q -~r

Ilk e s i g n a l

FI(IIJI{E 7.7. S~pp~rts <~ftim a~flfig~ail,y finwt,ir (~f t,l~e h-like, ~{~{~f~'r m~l q~m~lratic I)l~asc sig~mls. 'l'l~' tl;~sl~:tl li~e al~lwarm~cc is ai~m~l at ev~ki~g/ l,lu'~ si~stfitlal l~clm,viCnn" of tlm m~l)ig~fity fi]~r twer tim i~dmre~d, S~l)l)orl,s. wfllu;s ~)f A,,,~, A,,,, ml(l .A~h~rp ar~, illsellsil)h: 1,~) fl'l:(llll:llCy lll~)~llllal,i~nlS mill st)ar sllifl,s ~f l,ll~; siglla.l, wlli~:ll sillll)ly llm,llifl;st l,llr~ntgll lillCm" l)llas~; t~:rlllS. Ill Fig. 7.7 tll~; slq)t)(Jrts ~f A,,,, A,~, ml~l A~h~,p ill l,llt: 1)1"(11)1',1" (0' h') l)lmH; ~1.1"1'~h;t)it:tc(l, , Imralh;lillg Fig. 6.11 tlmt ill~stral,t's tllt: S~tl)l)~)rt.s ~)f tilt; c~wr~'sl~nl~li~lg Wiggler (listl"ilntl,i~ms I/V,,,,, W ~ . a l l ( l ~flchirp ill tll~'~ r(;sl)(;('tiv(, (q h) l~lml~:. l"~n" t'~lt~lr~', l'(~,f(q'lql(:(~,, W(~ giw' l ll~; ~'Xlm;ssi~nl r t ll~; a llll)ig~lil,y f~nl(:t,i(m A,, (q, k) ~f a r~'ct, a~lgh~sig~lal ~f wi~ltll 2(2:

A,, (,/.

-

{ ~(l-~)si,lr

~)1

~

[q < 2 Q ,

2q

r

-

101 >

(7.4.23)

wlfi('tl the rca(lr ~lay i~ll,r162 r162162 witll tllr r162162 ~;xl)rr (6.4.3) r the Wig~lr r f~nl(:l,ir ~2r (q. h'). V~), s(;(' l,lml, .A.r ((1, ~;) is (:r162 withi~ tlle w'rti('al stril) - 2 ( 2 < 0 _< 2Q, witll l,ll~' z~;rr wd~u; loci ])r the synm~ctrir hyl)crl~r (Fig. 7.8.a)) 7171"

-

I, 1 -< 2(2,

) -

',, -

1,2,..

(7.4.24)

It is also inst, I'~l(:l,iv(; 1,() (:a,h:~fla,t(; the a,~nl)ig~il,y flln(:ti(m of tllr real Gmlssian a mplit~uh; 1 P~ (q) -- (---~2) 1/4('-aq2~-w2 , (7.4.25) fiw whi(:h we obtaill __ 1

- 5 ( ,,2

)

(7.4.26)

Like Wo(q, ~;) (see Eq. (6.4.51)), it, displays a, Gal,ssia,n profile a.hmg both the and ~,: axis, but with variances cr2 - w 2 and or-2 - 4 (b) R o u g h l y speakq

t~

~

"

b The variances for a 2D function are defined by double integrals arising as an obvious generalization of (4.6.35) appropriate to a 1D function (see w 9.2.2).

The Wigner Distribution Function: Optical Production

365

FIGURE 7.8. Contour plots of the ambiguity flmction of (a) the rectangle signal

1)1/4 e _A4 Pn (q) - rect(~4) and (b) the Gaussian waveform ~ (q) - (7-~ ing, we may say that ld2~ extends over the phase plane significa, ntly within the w w x [ - ~, thus reproducing the (at)t)roxinmt(~) extensions of rc(:ta, ngle [--if,-if] 1 V,,], 1 ~ (q) in the spa,(:e and frequen(:y domains. In (:ontrast, A~ extends in the pertincnt (/~, ?~:) plan(; significantly over the la,rgcr rc(:tangh; [ - w w] x [ _ 2 s the '

~

,tt)2 ~, 2

"ll~ ~ "It)

~

relevant level (:llrves t)cing in fact the ellit)scs ~-~ + 4 2~, which evidently w 2 for any given value of a have (toul)h~ axes with r(~st)e(:t t() tile (:orrest)onding elliptical contours of 1/Fo(q, h;)(Eq. (6.4.52), Fig. 7.8.b)). To holt) intuition, in Fig. 7.8 wc show the contour plots of A~ ((}, ~,). and Ac,(q, ~,). In this conne(:tion, note that the ambiguity flmction is in general complex; therefore the modulus IAI or tile squared modulus IAI 2 is ordinarily plotted. In particular, the ambiguity function of the rectangle and Gaussia,n signals as well a,s of all the wavcforms we consider in this paragra,ph are real. Hence they can directly be shown in a, 3D plot or in a, phase-plane plot. Examples (7.4.23) and (7.4.26) illustra, te the genera] property of the ambiguity flmction, whose support in the pertinent space-frequency plane h~s (~ and t2 extensions double than those of the signal and its spectrum, respectively (Problem 6). We may g~in an intuitive justification for this property, by resorting to the naive picture, elaborated in w 6.2.1 in relation with the Wigncr distribution function, and further supported by the considerations developed in ~ 7.2.3. As in w 6.2.1, we consider a reM and nonnegative signal. Thus, on computing the Wigner integral at a point we may think of folding the part of the signal to the left over to the right, the resulting overlap contributing to the Wigner integration. Correspondingly, on computing the ambiguity integral at a point we may think of shifting the signal to the chosen point, the overlap with the ~moved replica of the signal contributing to the ambiguity function

366

Linear Ray and Wave Optics in Phase 5pace

FIG[IRE 7.9. 3i) gral)l~s a~l(l rch'wu~(, l)lm.sC-l)lm~(' l)h)t,s ()f t,ll(' a~l~l)ig~it,y fl~('(,i()~ of the II('l'~fi(,(:-Ga,~ssia,~ waw'f()r~ (7.4.27) ()f ()r(h,r 'n - 4 m~(! '~,- 9. int('grati()l~. S(), wlfilst, th(; fi)l(li~lg giv(;s z('r().j~st, at tlio (,xtr(;~l(;s ()f l,ll(, signal i~lt(;I'Wd, t,ll(; ()v(;I'lal) wuiisll(;s a,t (l()~l)h: ()f l,ll(; (;xt,r(;~l~(;s. "1'1~(; a,~ll)igt~ity flnlcI,i()Ii is in fa(:t a ~l(;a,s~r(; ()f l,ll(; (I(;gr('('~ ()f si~lila,l'il,y ])(;l,w(~('~l~ l,ll(; sig~lal mid its r(;t)lit:a t)(;i~lg sliift,(;(l i~ sI)a,(x; a,~l(l fl'(;tl~(;l~(:y ()~l('~ I,() l,l~(; ()l,]~(;r.

~.,,,~,, (q) - (2.,,,,.!yrS.,)

(7.4.27)

II,,(., )(

F()ll()wi~lg t,ll(; li~l(;s ()f l,l~(; (:al(:~llati()~l l(;a,(li~lg l,(~ ((i.4.6(i), w(; (~t)tai~

.A..:;.,

k)-

,,,2

(7.4.2s/

+

whet(; L.(-)(t(;~(,l,('s 1,1~(, Img~(;rr(; I,(,ly~(,~ial (,f (l(;~I'(~(~ II [l:]]. T h e rms wi(lths ()f '#m;,, aIl(l its F~)llri('~l" tra,llSfiWlll ~n.~,, are ro.sl)e(:tiw'~ly inca-

s,u'e(t t)y t,h(, va,'ia,,(:(;s (,f ]r

12 a,,(t [O~a,, [2 '

})(;iIlg

o 2, , , , , ---

"''~ 4 (2',, + 1) and

~72~;,, ,,,12(2n + 1) . I{(~ll(;C, f()r ilicr(~a,sing, va, hles of tile ()r(l(;r 'H, t,ll(; wi(lt, hs of the st)a,ce and Koqll('al(:y (h)xlia, ilis ()f W~,, incrcas(;, a,ll(] s() (]() as well t,hc silt)ports of 142~,, a,n(1 A ~ , . T h e rea,(lcr m a y guess t h a t t,tle sut)I)ort of A,~r in tile relevant (~, Y;) t)la,ne involves a, region with rnls ~ a,nd ~,; ('xt, exlsions just given by 4aq .... and 4or ....... . TheI'os ill a,ccord with tile consideratioIls suggested by (7.4.18) a,nd (7.4.19), with increa, sing n the a,nlbiguity fui~cti()n becomes Inore s h a r p l y p e a k e d a b o u t the origin ill the (~, &) plane whilst a,cconmlodating its invariant v o l u m e over a, larger region. F i g u r e 7.9 shows the 3 D g r a p h s with tile phase-pla,ne plots of the a m b i g u i t y function of tile H c r m i t e Ga, ussia,n amplitudes (7.4.27) of order 'r~ - 4 and n - 9, t h a t Call be colnpared with the c o r r e s p o n d i n g plots of the W i g n e r d i s t r i b u t i o n function ill Fig. 6.19. E x a m p l e s (7.4.23), (7.4.26) and (7.4.28) are in accord with the relation: W(q,

- 2A(2q,

(7.4.29)

The Wigner Distribution Function: Optical Production

FIGURE

7.10. 3 D g r a p h s

and relevant contour

component signal (7.4.30) with w = ql = - % = 2.5, ec1 = -~c 2 = 2.5.

1.5 a n d

plots of the ambiguity

(a)

q~ =

-q2

=

367

flmction of the two

1.5, n~ =

-~%

=

1.5;

(b)

holding between the Wigner distribution and ambiguity fimctions in the case of an even signal, ~ ( - q ) = ~(q), in which case in fact the ambiguity function turns to be real. Finally, to close this tracking shot on the characteristics of the ambiguity function we illustrate the behaviour of the cross-terms in the ~tmbiguity full('tion, considering a, two component signal. Such a behaviour is usefully compared with tha, t of the (:ross-terms in the Wigncr distribution fimction. Due to its analytica,1 computability, we consider the sum of two identical Gaussian amplitudes being both spatially and spa,tial-freqllency shifted. Namely, =

l(q) +

(7.4.30)

(q)

where specifically

~)j(q) .

(zrw22

1

)2

)l/4c--~-g(q-q j +i~jq

j = 1, 2

(7.4.31)

In general the ambiguity function of a two component waveform is composed by four terms, i.e.,

A~(q, ~) -- A~I (q, ~) + M~2 (q, ~ ) + A~I,~ 2 (q, ~) + A~2,~ 1 (q, ~),

(7.4.32)

where according to (7.4.11) A~2,~ 1 (q, ~) - ~A.;1,~2 ( - q ~ - ~ ) . The reader is invited to work out the explicit expression of A~ in the general case of arbitrary spatial and frequency shifts: ql, %, I'i~l and t~2. Here we report, and comment on, the expression of A~ relevant to the case of identical and opposite shifts, i.e., ql = - % = q0 and pi~1 = --/%2 = ~i~0"The pictorial view of A~ in that case is offered by Fig. 7.10 for specific values of q0, n0 and w. The self-terms A W l and A~2 shape both as A~, apart from a linear phase factor accounting for the space shift and tile frequency modulation of

Linear Ray and Wave Optics in Phase .Space

368

FIGI;III'3 7.11. Cr~ss- a~r self-l,(,r~,s ~ct)~lCl, l'y t)f t,l~c ;u~fl)ig~il,y f~('l,i()~.

[A~, ! A~.2](,i,k)

_l,. ~(~,~.."~ ~-)(~,,,s(~,,,~-q,,k) 71"

- 2A,,(,~ #)(',,,s(~,,,~-q,,~) I

~

(7.~.:~:~) wl~i(:l~ (',vi(l(',~l,ly (lisl)lays a~ ()s(:illa,t()ry 1)('l~avi()~l', g()v(',r~(;(l I)y q,, a,~(1%,, a,n(t ~(~(h'lh'(t l~y l l~(' (la~ssia~ ('~w('h)l)(' as (7.,1.26). I~ (:()~tra,st, tt~r (:r~,s,s-t,('r~,s A:~ ,:~ a~(I A : 2 , : ~ yi(;l(l i~ gr tw() sct)a,ra.tc (q, -q~, ~, - ~ ~ ) i~ 1,1~(;(q, k) t)la.l~(;. II~ tlm~ t)art,i(~flar (:a.s(; ()f (:(l~al a~(l ()t)t)osite slfifts w(: at(: (t(,ali~g witl~, ()~(' ()l)tains, in fit(:l,, I,

A~,,~

-

)2

,,,~

2]

)~

,,2

~]

(0, ~) - ~1, ' -~[,,-~(q+2q o +-r(;,+2h:o)

A~,~, (~,~,-) - ~r

_l

I

~[,,,-3('~-2% +~-(;:-2~,,/

:

A:~(O+

2q,,. ~- + 2h-,,),

(7.4.34)

: A , : ( 4 - 2q,,,Y,: - 2t%).

Both tez'nis lia,ve tlie sa,liie Ga,ussia.n envelot)e as Aa((~, ~:), a,nr are sylnmetri(:a,lly h)('alize(t ill t,tie st)a.(:(~-fre(tu(mcy plane a,r()lmd the t)()i~lts ( - 2 q 0, - 2 n 0 ) and (2qo , 2h:o). Tim ('ontz'i|)lttion fi'om ca,oh ('xoss-tcrm is tla.lf ttmt of the selfterms arising COllti'ibution (7.4.33). Figure 7.11 schcma, tically synthesizes the. above (tescrit)(;(I bo.haviour of the ambiguity flmction, which is a.s noted ra,ther general. Indeed, Figs. 7.10 and 7.11 pa,ra,llcl Figs. 6.17 a,nd 6.4, which arc relevant to the Wigncr distribution function. Specifically, Figs. 7.11 and 6.4 evidence the c o m p l e m e n t a r y geometry of the ambiguity and Wigner distribution flmction cross-terms. In fact, in the case of the ambiguity function the signal terms are located around the origin in the (~, ~) plane, whilst the cross terms are separately centered at

The

Distribution Function: Optical Production

Wigner

369

(-q12,-/5];12) and (q12, /~12)' q12 and ~12 being the space and frequency difference coordinates between the central coordinates of the component signal. On the contrary, as illustrated in w 6.3.4, the self-terms of the Wigner distribution function are concentrated at the same space-frequency points around which the signal components occur, whereas the cross-terms contribution is oscillatory around a point located midway between the self-terms centres. Notably such a complementarity may effectively been used to filter out the undesired cross-terms contributions in the Wigner distribution function, since from the viewpoint of the ambiguity function an appropriate weighting function sharply concentrated around the origin of the space-frequency plane may allow the signal terms to pass whilst reducing or suppressing the (:ross terms contributions. 7.5

Understanding the Wigner and ambiguity functions from the viewpoint of the mutual intensity function

Here we further emphasize the mathematical interpretation of the Wigner distribution and ambiguity function as 1D Fourier transfortns of 2D functions, formed by the product of shifted versions of the signal or its spectrum, which ultimately can be interpreted a,s the mutual intensity and spectrum. Both 1/V a,nd .4 arc seen to arise from a, t)roper sequence of a, multipli(:ation a,nd a Follrier tra,nsfornl t)r()ce~ss, thus sllggesting a, t)()ssit)le ()t)tical imt)lementati(m for the Wigner (tistriblltion and ambiguity fim(:tions display. Also, the forth(:oming discussion conveys the signifi('an('e of the Wigner distribution and ambiguity fim(:tions within the partial coherence theory (see also w 6.3.5). 7.5.1

Signal and spectrum generated functions Try(q, ~) and 7v/(n, ?~)

Consider two signMs p~ (q) a,nd ~2(q), with the respective Fourier spectra ~1 (~) and ~2 (~). We regard ~1 (q) and ~2(q) as two independent, entities and accordingly associate to each its own space and frequency coordinates. Thus, we write ~ 1 ( q l ) ~ n d ~1(~1), and correspondingly ~ . ( q ~ ) a n d ~2(t%). On describing the relations between the two signals in both the space and frequency domains, it is rather natural to introduce the (:entre and difference coordinates: q

ql + q2 2

q --

q2 -

tr -k- tr 2

2

]%

ql,

~

1 --

/~2

9 --

]~1"

Evidently, the original coordinates are recovered by inverting (7.5.1) as

q2--q+~,

t%--~+~.

Also, in order to individualize the interrelations between the two signals, it is natural to consider the space-domain product ~1 (ql)~2(q~) as well as the

Linear Ray and Wave Optics in Phase Space

370

frequency-domain p r o d u c t @, (h;~) ~2(~,2). On a,ccount of (7.5.2), such spaceand frequency-domain p r o d u c t s can equiva, lently be interpreted as fllnctions of the (:entre a,nd difference variable pairs (q, q) a,nd (~:, ~), respectively. Of (:ours(;, we m a y take as in(le.pe.ndent signals any wa.vefl)rm p(q) a,n(t its (:()mt)h:x (:onjuga,te ~*(q). th:n(:(;, the a,for(:menti()n(;(t st)a,(:(:-(l()nm,in t)roduct t,~r~s to t>e

[4.3, 12.5,

7.4, 4.8]

~ , (9q . O) - r

- ~)r 2

(7.s.a)

+ ~) 2 "

~m,nifi:stly (:Xl)l"(:ss(:(l i~ (,(:l'n~s ()f (,I~(: (:(:II|,l'(: m~(l (lifli:r(:l~(:(: Sl)a(,ial (:()()r(lilmt(:s q a,~(l ~.

W(: (:H,II t,l~(:~ r(:writ(: tl~(: Sl)a.('(:-(l()~mi~ i~(:(;gral (l(:fi~)i~g (,l~(' Wig~(:r (listril)~(,i()~ i~ (,I~(: f()l'~ o

.~

~r,(q, 0), -;" "dO - --~,~ ' ?(,~.,)I ~ , ( q , 0)1,

w ( q , ~-) - '.,~. -

(7.~.~)

r(:lat, i~g (,I~(: Wig~H:r (lisl,ril)~(,i()ll fl~(:(,i()~ (,() (,l~(: F(nn'i(:r (,ra,~sf()r~ ()f (,l~(: l>r()(l~u:(, fl~(:ti
(7.a.a)

w ( ~ : , ~:) - ,> (" - ,~):(~: + : ) ,

t)eillg a flm(:ti(nl ()f tll('. (:('.Ill,r(' ml(l (lifli'rell('(, (st)a,tial) fl'(:(llU'll(:y variat)h,.s h: a.n(1 ~;;. In ternis ()f (7.5.5) (,ll(; fi'(;(l~U:ll(:y-(l(nim.i~l ild,(;gral (:(nl(:(:r~lilig (,lie Wigner (listI'it)~lti()n flln(:l,i()n wril,(',s a,s ~V(q, ~;) - ~ .

w(~,

x)

",/~: - ~

(,,,,)

,

(7.5.6)

i.e., a,pa.rt fronl the lmixlflll('alt,ial fa.(:t,()r-~, a.s (,l~c illw'.rs(; F~)ln'io.r tra.nsform of 7r/(~:, Tc) with rest)e(:t t() the (liff(;ren(:e fi'e(tlu;n(:y wtria,t)l(; k. Simila.r relations can t)e esta.t)lislle(t t)etweell the a,lnt)iguity fllnction A((}, K) and the 2D flmctions 7c~ a ll(t 7rf. hi fa,('t, rewriting the a,llfl)iguity integrals (7.4.1) and (7.4.2) in terms ()f ~T~ an(t 7Tf, i.e., p

1 /

A(V,~~) - ~1

(,-i-

~ 7l-f(fl;,~;)(,.i~q~dfl;

1

A

_

A-1 "" ~ 1 f(~,~) ~s(~:, ~:).

(7.5.7)

we see that, the a m b i g u i t y function (:an be interpreted (apart from the factot" ~ )1 a,s the Fourier transforln of the space-domain p r o d u c t %(q, ~ with respect to the centre space-coordinate q and likewise as the inverse Fourier t r a n s f o r m of the frequency-domain p r o d u c t 7c7 (t~, ~) with respect to the centre

The Wigner Distribution Function: Optical Production

371

~ " V ( q. u ) A

j,,r ~.~

( q . i(; )

zr ~. t q . gj )

A_ I

FIGURE 7.12. Schematic of the Fourier transform relation between the product functions Try(q, ~ , 7rf(~, ~) and the Wigner distribution and ambiguity functions (inspired by [12.5]). The various space, frequency and hybrid 2D domains are then recognized.

frequency-coordinate ~. The diagram in Fig. 7.12 synthesizes the Fourier-like rela.tions between 7c~, 7ri, and 142 a.nd A [12.5, 7.4, 4.8]. Relations (7.5.4)-(7.5.7) are in evident accord with the established correspondences between the two sets of variables (q, t~) and (q, ~) perta.ining to the Wigner distribution fllnction and the ambiguity function, sketched in (7.4.8). With these ('orrespon(tences in mind, we ('an write down the formal ext)ression of the 2D Fourier transform of 7rs(q, (}), namely -

d d

7)c

which is oasily proved to equal the frequoncy-donlain product

"rcf(#<,,, ~;)" (7.5.9)

The Fourier transform relation between tile signal and spectrum generated functions 7cs and 7cI is also evidenced in the diagram of Fig. 7.12. As a conclusion, we synthesize the relations between the space and frequency variables (q, (~,t~, s involved in the Wigner and ambiguity function representations of the signal. Tile space variables (q, @ span tile 2D domain of the signal-generated function 7r,(q, q) as the frequency variables (t~, ~) span the 2D domain of the spectrum-generated function 7of (t~, s The two subspaces are duals in accord with the Fourier transform relation (7.5.8). On the other hand, the space-frequency pairs (q, t~.) and (~, ~) span the space-frequency domains of the Wigner distribution and the ambiguity function, respectively. These hybrid domains as well are duals according to (7.4.4). Although both (q, ~) and (~, s are pairs of dual variables through the Fourier transform relation holding between the signal and its spectrum, when framed within the Wigner and ambiguity function representation they acquire different roles, (q, ~c) corresponding directly to the space and frequency variables of the signal, whilst (q, ~) being associable with the space and frequency lags.

Linear Ray and Wave Optics in Phase Space

372

7.5.2

Relating the Wigner distribution and ambiguity flmctions to the mutual intensity function

T h e t)r()(tu(:t s t r u c t u r e of the fmlct, ion 7r.~(q, ~) suggests to r(4a.tc tim Wigner (tistrit)uti()n a.n(1 ambigllity flmctions t() the ~lnltua,1 i~lt(;llsity flm(:ti()n. We re(:a,ll from tlm th(',()ry of t)a.rtia,1 (:oh(;r(;n(:(~ [9] tha,t the nllltlm,l int, ellsit,y flm(:tion (t(;s(:ril)es tim st)a.tial (:()rr(4a.t,i()ns ()f t,h(~ st)a.(:('.- a n(l t,inl(;-va.ryillg way(', flm(:ti()n ~/,(r, t), r('I)r(;s('ld,illg t,ll(; s(:alm" ()t)I,i('a,1 (listllrt)ml('.(; at t)()sil,i(nl r ml(l |,il~l('~ t, t llr()llgll tll('. ('llS(~llfi)l(~ a v('rag(; (()r (~(llfival(~ld, ly, t,ll(' illfilfit('-tilll(' av('rag('.)

J(r,, r~) = (,/,* (r,,/)~/,(r~, l) )

(7.5.1())

fl)r any l)air ()f l)()siti()~s r, = (q.,:,, %1, z~ ) a.n(l r~ - (q.,.:,, %.,, z~) a,(:r()ss (,he t,l~e ti~(~-a,v('ra.g(:, wl~i('l~ ar(~ i~(l(~('.(l i~d,(~r(:l~m~g(;al)l(; (l~('. t() tl~(' l~yl)()tl~(;siz(',(| (;rg()(|i('ity ()f tl~(' ()l)ti(:al wav(; ii(;l(l; m~(l l~('~('('

l)~/,(r.~.l)) -

(~/,*(r '

li~ @ 7'--*~ -

.

T

~/,*(r,.l)~/,(r.~.t)dt.

Evi(l(nd, ly, J(r~,r~) .vi('l(ls tl~(~ ~n~t~ml (:()l~(:r(;~(:(, fi~(:ti()~ l ' ( r , , r ~ ; r ) (l('lay r, m~(l tl~(; li(;l(l il~t(;~sity f()r r, = r~:

I(r) .... (I '/'(r,l.)l 2 ).

(7.5.11) al ~ll

(7.5.12)

C(nff()n~i~g will~ I1~(; f()rlllalislll ()f (11(' ])()()k, w(' ('()~si(l('r (lisl ril)~l i(n~s ()f l,h(~ ()t)l,i(:al ti(4(l a(:r()ss l)r()til(~ l)la~ms a l, tix(;(l l()~@l,~(li~ml z (:()()r(li~ml,(~s, s() tha, t, with th(; fln'tl~(~r si~q)lili(:ati()~ i~d,r()(l~(:(',(l 1)y tim singl(~ t,rm~sv(:rs('-(:()()r(lina,t(; a.(:(:()~nd,, l,l~(', g(',~(;ra.1 (l(',ti~iti()l~ (7.5.1()) l,~r~s i~l,() l,l~(~ si~l)l(~r

J(q, ,q.~) -- ( ,/,*(q, ,t),/,(q.~, t)),

(7.5.13)

the (t(;I)c~(t(;~(:(; ()~ z as a, i)ara,l~l(;t('.r l)(;i~g i~t)li('.(l. T h e nn~l,~a,1 i~t(~nsity fin~(:ti()n yM(ls a, (:()~q)l(~t(~ (t(~s(:riI)l,i(n~ ()f l,l~(; st)aria,1 (:orrela,ti()~ ()f th(~ ()t)l,i(:a,1 (tisl, urba,~('(; for (l~msi-~()~()('tm)~a,t,i(: ti(;l(ls, whos(~ temt)ora,1 fr(;(l~mn(:y st)r(;a,(t Ac~ is in(t(~e(t sma,ll (:()mt)a,r('~(l t() the (:()rr(~st)onding mca,n fr(;(t~mn(:y a~: Aa~/~ << 1. In tha,t (:as(', m~(l(;r th(; a.(t(liti()na,1 restriction t h a t all tim ()t)ti(:al t)a,th (tifferen(:es A s inh(;rent in the ot)ti(:a,1 s y s t e m are much shorter t h a n th(; t(m~t)ora,1 (:()her(~n(:(; length ()f tim wavefi('l(t: A s << 27re~Ace, the light nmy t)e (:onsidere(t to effectively t)()sses (:omplet(~ t(~n~t)oral (:oherence. Hence the m u t u a l cohercn(:(~ flmction F(q~,%; 7-) takes the form of a simple modula,tion of J(q~ ,q~) a,t tt~e m e a n temt)oral frequency ~: F(q~,q~; ~-) ~ J(ql ,q2) c i ~ .

(7.5.14)

Besides, u n d e r the above specified restrictions, the time-varying wave function ~b(q, t), representing a nonmonochroma,tic field is shown to obey the same

The Wigner Distribution Function: Optical Production

373

laws of propagation as the time-independent amplitude p(q) representing a monochromatic disturbance at temporal frequency ~. Evidently, this is crucial in order to establish the transformation relationship for the mutual intensity fun(:tion under propaga,tion through opti(:al systems. In particular, for quasi-monochromatic spa,tia,lly coherent fields, the infinitetime average in (7.5.10) yields the simple product of the time-independent amplitude ~(q) mea,sured at the points of coIIcerIl, q~ and q2; namely, J(ql ~q2) ---- (~* (ql)(~(q2)"

(7.5.15)

In contrast, in the opposite (and idealized) limit of spatially incoherent fields, in which case light from each point a,cross the plane of interest is assumed to be statistically independent of light from every other point, J(ql ~q2) writes as J(qt'q2) -- <] :~[)(ql't)l 2 }(~(ql -- q2) -- I ( q l ) ( ~ ( q l -- q2)' (7.5.16) where the intensity I(q) = ([ ~/:(q,t)] 2 } turns into the square modulus of the time-independent amplitude for quasi-monochromatic fields. As noted in w 4.3.2, due to the restriction to 1D description, the physica,1 quantities involved into our a,nalysis may (:owe not to have the right dimensions. Aga,in, wc invite the reader to disregard this accident, focussing a,ttcntion ra,ther on the mathcma,ti(:a,1 relations t)etween the involve(I qua,ntities, the right (timcnsions being rc(:()vera|)le in a (:orre(:t 2D a,imlysis. Relations (7.5.15) a,nd (7.5.16), rclcwmt to the ot)t)ositc limits of spatially (:ohcrcnt and incoherent wavcficlds, are ba,si(: to demonstrate tha,t ot)tica,1 pro(:cssing under (:ohcrcnt illumina,tion can be dcscI'ibc(t as a, linca,r ma,pping of complex amplitudes, whilst optical processing under incoherent illumina,tion amounts to a linear ma,t)ping of rea,1 intensities. In the intermediate case of pa,rtia,lly coherent light, it is necessary to establish the tra, nsformation rcla,tionships for the mutual intensity function under propaga, tion from the object to the receiving plane of the optica.1 set-up under exainina,tion. Indeed, it can easily be argued that the mutual intensity is in general expressible in the form

J(ql,q2;Zo) --

//'

dq'ldq;J(q'l,q;;zi)g*(ql,qI1)g(q2,q;),

(7.5.17)

where g(q, q') denotes the line-spread function of the optical system, and zi and Zo individualize as usual the location of the input (i.e., illuminating plane and the output (i.e., observing) plane. In this connection, we mention that in [7.2] it has been shown that the paraxial propagation into the Fraunhofer region and the thin-lens transfer to the back focal plane affect the mutual intensity function by a, double Fourier-like transform. In fact, recalling that the far-field distribution of a freely propagatcd object amplitude looks likes the imperfect Fourier transform of the object itself (see w 5.4.3), one finds

Po(q, zi + ~) ~ ~ c i ~ q 2 ~ i ( ~ ,

zi)

(7.5.18)

374

Linear Ray and Wave Optics in Phase Space ~W 2

where, a,s required by the Fra,unhofer condition, g >> - - y , 2w being the maximuln w i d t h of the object distribution; k - nok o is the light w a v e l m m b e r in the me
( , o ( q , z~ + ;; t) ~

2 -

g

v ~e

e,

z ;t - -). ,.

(7.5.19)

Insertil~g 1,1~<;a.t)()ve ext)ressio~ ii~to tl~e il~fi~fite-ti~<; average i~ (7.5.10) with (,,,, m~(l a<:<'(w(li~gly ('i, l)<'~il~gl,aken at l,w()(1it[r162 I)()ixd,s q~ an(l q.,, w(; ()})t,ain "~* ( ~ t ' Zi; [ , - {'/")'/'i( kq2 g J(ql,q2;zi Jr [') -- k it"'-Yi(q~-q'~ . )( t/'i t ' Zi; [ , - -[.))

_ -

-

//

'

~. i~(,t~-,t~)

2rrg ~'

dq'dq"J(q',q";

Zi)('-i'[(q2qtt-qlqt) k

when; l,ll<; s<'~(:()ll(l st(q) lms Iw.eu {~l~taillcr l~y {;xl)r<;ssillg l,ll<'~ l)r{~{lll<'l, 1)etween ln'a<:kcts lm,<:k I,~ l,lle Slmtia,1 ~l~nlm.ill t l l n n @ l the l"~nlri<:r r<'~la.l,i~nl. Then, tile llnltlla,1 ild,{'.llsil,y flUl<:l,i{nls at l)lmles zi ml
a tlfi~ l~,~s, witl~ { ta.ki~g tl~<:~ tl~c ~oa.~i~g ~f tl~<: h:~s fi~<:al h'.~gtl~. Afl,<;r l,l~is In'<:li~|i~m.ry a.~l
J(q, q ) -

{'('*(q- ~2 ' t)(,(q + ~2 ' t))

'

(7.5.21)

thus yiel(lillg just the signal generate(t fim(:ti<)n 7r.~(q, q) in the (:ase of strictly (:oherent wave ficl(ls:

J(q, (1) -- ~ * ( q -

for coherent signals.

q)~(q -+- ~) -- 7rs(q q) 2

T h e W i g n e r (tistril)llt, i()Ii and a m b i g u i t y flmctions a,re a,c(:ordingly interp r e t a b l e a,s the 1D Fourier t r a n s f o r m s of the m u t u a l intensity function J(q, q~ with respect to the (tifferen(:e and centre spatial coor(tinates ~ and q"

J(q, (7.5.22) w(q,

A(O,

The Wigner Distribution Function: Optical Production

375

This naturally introduces tile Wigner distribution and ambiguity functions within tile context of the partial coherence theory. Formally, one has 142(q, n) -- ~1

/

J(q,~)c -

znqd~

-~

] @*(q-~,t)~/;(q+~,t)}c-i~qdO,

(7.5.23)

and correspondingly A ( q , ~) -

1 ~

J(q,

q~)e-i~qdq - ~1

(~/J* (q - ~,t)~p(q +

~, t ) ) e - ~ q""~d q .

(7.5.24)

We note that, referring to the pertinent discussion in w 6.3.5, one can recover expression (7.5.23) for W in terms of the mutual intensity J(q.q-) from expression (6.3.33) giving W in terms of the cross spectral density F(q 1,q2; aJ), since under the restrictions inherent in (7.5.14) s turns into

s

,q~; ~) - J(ql ,q2)(5(c6

-

-

~).

(7.5.25)

Remarkably, relating the Wigner distribution and ambiguity function to the mutual intensity function has the further valence of offering a tool to de(tu('e the transformation laws for J under paraxial propagation through optical systems from those for the Wigner (tistribution or tile ambiguity function. In turn, from the tra,nsfer relationships for J under paraxia,1 t)ropaga,tion into tile Frmmhofer region or thin-lens tra,nsfer to tile back focal plane (for strictly (:oherent light), it is possible to deduce the corresponding transfer laws for tile Wigner distribution and ambiguity fimctions. As a (:onclusion, we rewrite (7.5.20) in terms of the (:entre and difference coor(tinates q and (~, thus obtaining

J(q, ~; z~ + e) -

k c ~kq~ f / dq'd~ ,

,j ( q, ,~ , ;z~) e_ik(~q,+q~,)

~

(7.5

9

26)

which shows that the transformation law of J(q, @ for tile Fraunhofer diffraction is a pure 2D Fourier tra,nsform with respect to q and ~. Also, the integral above can directly be expressed in terms of the Wigner distribution and ambiguity function as J(q, q; zi

.k

+ g) - ~e i7 dq'W(q', e, k-ci-~q~ d~A(~' , k_k__~; zi)c-i-}q~, ,

(7.5.27)

thus showing that W and A may naturally enter optical propagation problems under both coherent and partially coherent illumination, with W or A coming into play according to whether it is the centre or the difference coordinate to have relevance for the specific problem under examination. In Sect. 8.3 when illustrating the paraxial propagation properties of the Wigner distribution function, we will see that ld?(q, ~), defined in terms of wave

376

Linear Ray and Wave Optics in Phase Space

ot)ti(:s, is closely related to the ray-ot)tical (tes(:rit)tioI~ of light t)rot)agation, being il~ fa,(:t intert)reta,t)le as the "a,mt)litu(te" ()f g(;()metri(:a,1 rays seen from the, vi(;wt)()i~t ()f wave ()t)ti(:s. Hcn(:e, rela,ti()ns like (7.5.27) signify a, vMua,ble link 1)(',tw(;(;~ tl~(; wav(;- m~(1 ray-ot)ti(:s tools a n(l r(;t)r(;s(;ntati()~s. 7.5.3

"1t~.cambiguity .fwnction and the optical t'ran,@;'rfunctio'n

Tll(; ~nlalysis l)r(;s(:llt,('(l ill t,ll(' l)r('~(:(~(lillt4 l)aragral)ll llas ('vi(l('ll('(;(l l ll(; signifi(:ml(:('~ ()f t.ll(; Wigll(;r (lisl,ril)lll,i(nl a,ll(l mlfi)i~;llil,y filll('t,i()ll witllill t,ll(; (:(nm;xt (,f ('(,ll(;r(;lit [7.1] a.s w(;ll as tmrtially (:(,ln;r(;lll, [7.2]- [7..1] ilim.~(; I'(,rlll~,ti(nl, a,s (:()nvey(;(l I)y t,ll(; Im.si(' trmlsfi;r law (7.5.2(i), wlli(:ll lllay illv()lv(; W a,~l(l ,A a,(:(:()r(li~lg 1,() 1,1~(; f~;a,t,~res ()f tin; ()l)l,i(',a.1 s(;l,~l) al, lm~(l. Ih;r(;, fiflh)wi~lg [7.r,1 w(; will ill~tsl,ra.l,(; t,ll(; li~lk I)(;i,w(;(;~l tin; a,~flfig~tit,y tim(> l,i()~ a~(l l,ln', ()l)l,i('al l,rm~sf(;r f~t~(:t,i(n~, wl~i(:l~ is I)a,si(: t() 1,1~(', Sl)a.('(:-invaria,nt i~lag(; f()r~llal,i(n~ ml(l (wa.l~m,t,i()~l l,ll(;()ry ~11(1(',~"sl,ri(:tly i~('()ln;r(;~t ill~l~i~m,ti(n~, r(',t)r(',s(;~d,i~l~ t,ll(', Sl)atial fl'(;(t~(;~l('y r(;sl)(n~s(~ ()f tin' ()l)ti('al syste,~11 t() il,ll ill(:()I~(W(',d, ()l)j('('t ([~().~a], ('1,. (i, I)I ). 113 125; [~().~r,], (:11. 11, I)I)..183-5()4). Fi~;,,r(; 7.13 sln)ws l ln' tyl)i('al ()l)I,i('al ~(;(n~l(,l,ry f()r il~m,g(' f(n'~ml,i(ni, witll l,ll(', i~t)~d, ()r ()l)j(;('l, 1)lain' m~(l tin, ()~ltlnll, ()r il~mg( ~ 1)lal~(' 1)('illg r(qal,(;(l 1)y t,l~(' lens law: 1

d,)

1

d,

1

.f

Wil, ln)~l, (;~ll,('ri~lg 1,11(; (h;tails ()f 1,1~(; l,l~(;()ry ()f tin; ()l)ti(:al l,rm~sfi;r fim(:ti()~, w(', will si~11I)ly r(;I)()rl, t,ll(; r(q(wm~t (l(;ti~lil,i(nl as ~.11(' ~n)n~m,liz(~(l mal,()-(:()rr(~la,ti()~ ()f l,ln; g(',~(',ralizc,(l in~l)il fi~(:l,i()~ P(~). Ex1)li(:itly, ()In' l~as II(u) - .]' P• (~-,~)P(~+~ ),l.r ./

'

(7.5.29)

wher(; ( mnl u (leIl()l,(', ~()n~m,lize,(t Sl)al,ia,1 fl'(',(tllel~(:y varia.l)l(;s t() l,ll(; (:()tmrent ('ut-()ff fr(;(t~l(~(:y t%,.,,,,,. _- t:, ,r 2a t)eing the lens a l)crt~r(~ (lianw, t,er m~(l k - nok o the ligld, wa.v(',mu~fi)(w in l,lm me(li~un. Tile, g(',~(walize,(l Inq)il f~ll~(:ti()n P(~) is (tefi~l('(l as P ( ( ) - P ( ( ) c -i~'[W(~)+~2] - P , , ( ( ) , ' - i ~ ' ~

(7.5.30)

T h e (:(n~q)h;x tmt)il fin~('tion P0(() = P(()cxt)[-ikW(()] a,c(:om~ts for the effective exte,~t of the lens a,t)erture a n d the a b e r r a t i o n s . In fa(:t, W ( ( ) is t h e a,berra,ti(m fimction, whilst the simt)lc pupil fi~n(:ti(m p ( c ) is defined to be u n i t y (m the ot)ening a n d zero otherwise; in t e r m s of the n o r m a l i z e d frequency varia,t)le ~ one has 1 _< 1, (7.5.31) P ( ~ ) -0 otherwise. T h e q u a d r a t i c p h a s e term, which is of m a i n concern to us, originates from an out-of-focus of t h e i m a g e p l a n e location w i t h respect to the focus condition

The Wigner Distribution Function: Optical Production

Vii

FI o

i

I

i

f

i

I i ! i i I I

I i I I i t I

i t

FIGURE

377

i do

i

dl

7.13. T y p i c a l g e o m e t r y for i m a g e f o r m a t i o n by a t h i n lens.

expressed by (7.5.28). Indeed, the parameter c yields the maximum pathlength error due to a departure of the relative positions of the object and image planes from the imaging condition (7.5.28), being a2 1 1 c - - 7 - ( ) . f dl

1 (7.5.32)

do

Evi(tently, since Ip(<)l 2 = p2(<), it follows that

/ IP(~C)l?d~-F

P 2 ( < ) d < - 2.

(7.5.33)

Then, inserting (7.5.30) a,nd (7.5.33) into (7.5.29) a,n(t evi(tencing the (lependen('e of the optical transfer fimction on the defocusing error c, we can write .

H(u; b) - 51

u u P o* (~ - 5)Po({ + 5)c. -~23~'~d{,

h - ke.

(7.5.34)

The integral a.bove has the very form of the ambiguity integral for tile "signal" P0({), with u and 2hu being placed as the spatiaJ a.nd frequency variables of the ambiguity flmction. Accordingly, we can relate the monochromatic optical transfer function to the ambiguity function of the complex pupil P0 as (7.5.35)

H0,; ~) - ~ A p o (~,, 2~,,).

Resorting then to the expression (7.4.23) for the ambiguity function of a slit of width 2Q, the ambiguity function Ap ~(u, v) of the complex pupil P0(~c), in the absence of aberrations (i.e., P0(4) - P(g)) and in the normalized set of variables we are working with, readily obtains as A p ~ (u, v) -

~1( 1 0

_[_u__[)sinc[v(1 2

- ~2) ]

lur _< 2, f~l > 2.

(7.5.36)

378

Linear Ray and Wave Optics in Phase Space

FI(~IJI1E 7.14. (a) l)lmutr l)lot of tl~c finwti()~ ~r.Ap,,('u,,~,) rcl)rcsc~ti~g tl~c ol)tical traI~sfcr fi~(:ti()~ ()f tt~e syste~ witl~ w~ryi,~g (lef()('~s parm~eter ~. (b) I)r()tiles ()f ~r~4p,,(u, v) alo~g tl~e straigld, lix~c v -- 2bu for (leli~fitc wd~tes ()f b - L'r (l~sl)irc~! i)y [7.5].) Cl(~a,rly, (hi l'(;la,t,illN .,4p,~ t() 1t ill(, r('.a(till~ ()f Mp,, ill ill(' l)(~l'till(;llt (tt, t,) I)lane llnlSt 1)c l)erfi)r~;(l a.l(~g t,l~e straigld, lilac t , - 2~tt. '1'1~' wd~w, ()f .Ap ~ a.t tl~c

(5 = L:c, als() ()l)tai~(;(l gral)l~i(:ally as tll(; tmig(;~lt ()f 1,11(; 1)()lar a.llgl(; /;/ ()f th(; line t , - 2(~u t() tl~(' u axis: 2b - l,m~/~, h~ t)arti('l~lar i~ tl~(' ('()l~(liti(m ()f f()(:lts (7.5.28), a~()~l~d,i~g t() b - (), tl~(' vahws ()f Ap,, a l()~g l,l~(, tt axis yiel(t the va.l~w,s ()f tt~(; il~-f()('~s ()t)ti(:a,1 tra~sfer fit~(:ti()~ [7.3]. Als(), tl~e i~t(,rs(~,(:ti()ns ()f the straigtd, line ~ , - 2(5u witli tl~(', s y ~ c t I ' i ( " t~yl)(;rt)()la.s ,'-

2-1-I .

I-I

.

< 2, ' n . . .

1 2,

(7.5.37)

whi(:tl as notc(t ill w 7.4.4 are tile zero wdlu; lo(:i ()f flnl(:ti(nl (7.5.36), (tetermine the points a,t which the ()I)ti(:a,1 transfer fimctioxl II(u; ~) va,xlishcs. In Fig. 7.14.a,) the t)la.lm,r plot of 7rApo (u, ~,) is rct)ortc(t, with a straight line '~ = 2~'u, for a (:h()scll vallw, of (5, t)eing showll a.l()~lg whi(:tl the reading of the ()t)ti(:a.1 tra.nsfcr flln(:ti(m for t h a t va.lue of the (tcf()(:lls nlllst t)c t)crformed. Fina.lly, the profiles of 7rAp ~ (u, ~,) a.long the lines ~, - 2(~u for different values of the dcfocusing t)a.ramct(;r b, i.e., c, are shown in Fig. 7.14.b). Ilm;rcstingly, if P0(~c) is mo(tellc(1 by a symmctri(:al flm(:ti(m, a,n equivalent (tcs(:rit)tion in t e r m s of the W i g n c r distribution flmction can be elaborated in virtue of relation (7.4.29). Finally, it is w o r t h m e n t i o n i n g t h a t the a m b i g u i t y flmction-based account of the optical transfer function can effectively be used to obtain the polychro-

The Wigner Distribution Function: Optical Production

379

matic optical transfer function. In fact, since longitudinal chromatic aberrations, which affect imaging with polychromatic light, amount to a wavelengthdependent defocus parameter c(A), planar plots like that in Fig. 7.14.a) can be interpreted as polar displays of all the monochromatic optical transfer functions of the system, and hence used to calculated the relevant polychromatic optical transfer function.

7.6

Optical production of the Wigner distribution function general considerations

The optical production of the Wigner distribution function is a particular case of the more general issue concerning the realization of coherent optical computing systems. Noncoherent optical techniques for signal processing have been developed as well. As noted before, coherent illuminated optical processors operate on typically complex amplitudes, whereas incoherent illuminated optical processors operate on intensities, and so on inherently real signals. Conforming with the limited purpose of the present analysis, we consider here coherently illuminated optical processors. Optical processors may be thought of as analog computers, which by use of optical architectures are capable of performing a certain class of operations [10]. Opti('al systems can inherently handle 2D data, displayed on the surface of the object or image plane. They can therefore operate simultaneously and independently oil two separate channels as opposed to serially as in digita.1 processing, whi(:h employs electronic devices having only time as an independent variable. The Fresnel and Fourier transforms are prime examples of matheInatical operations that can be implemented by very simple optical systems. As noted in w 5.2, the propagatioil of coherent light between two planes in a homogeneous and isotropic medium provides a means to perform the 2D Fresnel transform. Similarly, the propagation between the focal planes (front to back) of a positive sphericM lens provides a means to perform the 2D Fourier transform. In both cases, obviously, the respective optical setups must satisfy the Fresnel approximation, demanding for small extensions of the light distribution in both the object and observation regions with respect to the propagation distance. The Fresnel and Fourier transformations are specific realizations of a general integral transform-like relation, expressing through the mathematical form of a superposition integral the relation between the input and output signals for linear systems. We recall, in fact, from the introductory considerations of w 4.3.2 that for a 2D linear system the input ~i(ql, q2) and the output (I)o(ql, q~) are linked through the 2D superposition integral

~o(ql,q2) -- //e(ql,q2;qtl,qt~)~i(qtl,qt2)dqtldqt2, JJ

(7.6.1)

Linear Ray and Wave Optics in Phase Space

380

.

!

the t)oint-sprea,d function O(q~, %, q'~, q2) b e i n g tile mathema,ti('a,1 model for the t)hysi(:al t r a n s m i s s i o n t)r()t)erties of the s y s t e m . As in w 4.3.2, q~ and q2 (ten()te Ca,rtesia, n (:oor(tinates in the referen('e t)la,nes, tra,usverse t() the optical z axis, the sa,nlc synlt)()ls t)(;ing ~se(t for b()th the int)~t ml(t (n~tt)~t (:()()r(tinates. Th(: t)()int sI)r(:a,(l fim('(,i()n O(q,, (12;q'~, q;) signifies th(: r(:st)()ns(: ()f th(: s y s t e m a,(, t)()sil,i(m (q~, q2) i~ (,t~(: ()~(,t)~(, t)lm~(: t ( ) a (l(:l(,a sig~m,1 gS(q"- q'~)b(q~' - (1~), aI)I)li(:(1 at I)()sil,i(nl (q',, q ~ ) i ~ (,l~(: inI),~(, t)lal~(:. A sI)(:('ial ('as(' is wl~(:~ t,l~(' r(:sI)()~s(: fiu~(:t,i()~ O(q,, q.~; q',q~) fa('t()rs as .

I

l

l

O(q,, '12, '1,, %) - g~ (ql' q, )g2('12'

I

'12

)'

(7.6.2)

(list)layi~g i~l(l(:l)(ul(l('.~t l'(:Sl)()llS(:s ()f tl~(: syst(ul~ i~ ()I'I,l~()g()~al (lir(:('ti()~s. Tiffs ha.s 1)(:(ul 1,11(: ('(:~I,ral l~yl)()tl~(:sis ()f 1,11(: (:()usi(h:rati()l~S (l(:v(:l()l)(:(l i~ l,ll(: t)r(:vi()~s (:l~al)t(:rs, l~a,vi~g (:~ml)h:(l ~s t()I,~rl~ fl'(n~ a 21) I,()a l I) (h:s(:l'il)I,i()ll (i.(:., i~l I,(:r~s ()f a si~gl(: l)r()til(' (:()()r(li~mt(:) ()f I.l~(' ligl~I, l)I'()t)a,gaI.i()~l. As u()t('(t a, ~U~l~d)(:r ()f I,i~(:s, fa('i,()riza(,i()~s lik(: (7.6.2) al)l)ly I,() sl, ig~m, ti(: m~(t six~q)l(: a,sl,ig~mti(: sysl,(:IIlS. (',h:a.rly, (,l~(: a rra.~g(:l~(:l~tS fi)r (,I~(: ()l)ti('al i~l~l)l(:~(:~tati()~ ()f (,I~(: l:r(:Sl~(:l (,rm~sf()rl~, ~m~(:ly, a s(,ra,igl~(, s(:(~l,i(n~ ()f a. l~()~()g(u~(:()~s ~(:(li~l~, a.s well as ()f (,I~(: l"()~ri(,r (,rm~sf()r~, i.(:., a sl)l~('ri('al l('~s ()r a s('(, ()f ('yli~(Irical l(:~s(:s, alh)w f()r (I~(' (l(,('()~q)()si(i(n~ ()f (,l)(: I)()i~( sl)r('a(l fi~('(,i()~ O(q,, q.~; q',, q~) i~(,() tw() li~(: sl)r(:a(l fi~('(,i(n~s g(q, q') a,s ix~ (7.6.2). A('('()r(li~gly, f()r (:a(:l)()f (,I~(: (,w() (',a,rt(:sia,~l (lir(:(:l,i()l~S w(: l~my wri(,(:, iu (,l~(: lh'(:s~(:I a,l)l)r()xil~m,(,i()n, gF ..... I,( q, q~,)

V/ k

( i -fft, 9t,.t ( q - - q t )2

II(:l'(:, ]~: (l(ul()(,(:s (,ll(: Wa,V(:llllllfi)(:r ill (,ll(: lll(:(lilllll, a,S d ml(l .f Sl)(:(:ify l'(:st)(:(:(,iv(:ly (,ll(: s(:(:ti()n l(ulg(,ll a.ll(l (,lm l(:lls f()(:a,1 l(:ngth ('). Als() int(:r(:stillg, a,u(l l)(:r(,ill(Ult t() th(: f()rth(:(nlfing (lis(:llssi(nl, is (,ll(: imaging (,rmlsf()rnm,ti()ll, wlli(:tl is (l(;s(:ril)(:(l t)y a, set)ara.l)h: illlI)llls(' r(:sl)()lls(: i~) (,ll(: fornl a.s (7.6.2), ea,(:h (lir(:(:t,i()n l)(:i~lg a(:t(:(1 a,(:(:or(tixlg (,()

(q, r

-

Mq),

(7.G.4)

w h e r e M d e n o t e s the ilmgllifica.tion fa,ctoI'. Tim ot)tical inq)lellmnta,tion of a,n ilna,ging t r a n s f o r m m a y (:(nlsist of a simple two-lens system; a sequence of two single-lens FOUl'i(',r tra,ilsforxners (a, pure ina,gxlifier), for iilsta,ii(:e, performs an i m a g i n g t r a n s f o r m a t i ( m , with m a g n i f i c a t i o n I)rot)ortional t,() tt~e ra,tio of the lens f(,(:a,1 lengths: A4 = -f,2/.fl (see w 3.4.3 and Figs. 3.9-10). T h e st)rea(t fi~n('ti(ms gFour~er(q' q') a n d g~m~g~.g(q,q') a|)()ve t)(;rta,in to ot)tical setlH)s t h a t realize a,n (;xa,(:t Fourier t r a n s f o r m i n g or i m a g i n g relation b e t w e e n ~ We evidence that the expressions of the line spread flmctions for tile Fresnel and Fourier transforinations given in w167 5.2.1-2 contain the wavenumber k() in the vacuum space, and accordingly the reduced section and focal lengths. Of course, they are the same as those in (7.6.3), where the refractive index is absorbed into the wavenumber k in the medium, and hence d and f signify the physical section and focal lengths.

The Wigner Distribution Function: Optical Production

381

the input and output planes. In general a nonexact Fourier transforming or imaging relation between the input a,nd output planes gives rise to a spatially varying quadratic phase factor in the output signM that is not present in the input signal. If, for instance, the signal is out of the front focal plane, the lens produces the Fourier tra,nsform of the input with a, quadratic pha,se factor depending on the signal misloca, tion. Similarly, if in the double-lens arramgcmcnt of an imaging system the back focM plane of the first lens is not perfectly superimposed on the front focal plane of the second, the imaged distribution presents a, quadratic pha,se fa,ctor depending on the misloca,tion of the two planes relatively one to the other. 7.6.1

The Wigncr transform as an optical 2D linear transformation

It has been already empha, sized that the Wigner integrals represent a, bilinear transformation of 1D signals. The general expression for a bilinear [11.1] transformation of the signal ~(q) to ~/~(q) has the form of the double integra,1 ~* (q:)~(q;)L~(q; q'~, q; )dq'1dq;,

~/~(q) -- ~ !

(7.6.5)

!

where the kernel Q(q; ql'q2) may t)e interprete(t as tile response of the system to two impulses applie(t a,t positions q'l and q~. Although not cxpli('itly cvi(tcncc(t in the notation, the kernel ~(q; qtl' (t~) may (tet)en(t (m s()m(; other pa,ramcters a,nd hence also tlle flmction 't/,(q) will. Evidently the Wigncr distriblltion and ambiguity flm(:tion arc in conformity with the scheme (7.6.5), t)oth converting a 1D flmction of spa,ce or fl'equency into a, 2D flmction of spa.ce and frequency together. It is a.lso evident tha,t self-(:orrela,tion and self-convolution arc fllrthcr examples of bilincar transformations. Moreover, we mention that a bilincar transformation like (7.6.5) relates tile optica,1 intensity ~h(q) in the image plaile to the optica,1 transmitta,nce p(q) of a tra,nsparency illuminated by partia, lly coherent light, the kernel accounting then for both the response of the system and the coherence degree of the illuminating light. Relation (7.5.20) with ql = q2 illustrates the possible bilinea,rity of the response of a,n optical system to pa,rtially coherent illumina,tion. Indeed, it has been evidenced in w167 6.3.5 and 7.5.2 that the Wigncr distribution naturally finds an eminent place within the partial coherence theory, also due to its intrinsic bilincarity. As observed iI1 [11.1], Eq. (7.6.5) describes more optica,1 systems a,nd processes than the linear transformation equation (7.6.1), as a consequence of the quadratic relation between the optical intensity and the optical field amplitude. The reader may readily verify that the Wigner transform is recovered from the general expression (7.6.5) with ~vv(q;

qll~ q12) -- l7r~ ( q l

-~- q2 --

2q) ~i~(ql-q2)

(7.G.G)

382

Linear Ray and Wave Optics in Phase Space

wherea.s the ambiguity function is obtained setting t

t

1

,i--

L~A (q; ql' q2) -- 2-~a(ql -- q2 + (]) c-Tn(q' +q2).

(7.6.7)

The spatial fre(t~mncies h: and ;: enter the above kernels a,s t)araniete, rs on which the Wigner (tistriblltion a.n(1 the ambiguity flmction (:onie then to depend. The flul(:tiolial fornis of L~w ml(t ~A Ilia,nifestly stl()w the relati()li (ff the st)a,ce va.ria,t)l(;s q a.ll(] (l ()f)42 a,li(t A t,() t|le (:(;lit(;r (:o(ir(tiliate" q '#~+q~ a.n(t t() i,li(: (lilI'(:r(;li(:(: (:()()r(lilla,(,(:: O = q~ - q l , r(:sl)(:(:i'iv(:lY" W(; ilivi(,(: (,lle r(:adcr to (h;(lll(:(; i, ll(: ali,(;rlia.i, iv(; (;xl)r(;ssi()li ()f (,]l(; (,ra,lisfi)rliia,(,i()li (7.6.5) ill (,(;rills ()f (,lie siglla,l sI)(:(:trlllll ~(/;,), a,ll(l a,(:(',()r( l i l lgly t,]l(: fr(:(lll(:ll(:y-(l()lllaili (:Xl)l'(:ssi()ns ()f i,li(; k(;rn(;ls (7.6.6)a,li(l (7.6.7). It is easy (,() r(:(:()gniz(: (,]i(: I)()ssil)ility ()f r(:la.tillg (,lle l)ilili(:ar (,rallsf()rma,tion (7.0.5) (,()a. 2D lil~(:a,r ()l)(:ra.i,i()~, wl~()s(: ~m.(,l~(:~m.ti(:a.l f()r~ is giv(:,~ l)y (7.6.1). Ill fa.(:(,, w(: lilay a.rra.lig(: a. 21) sys(,(;lii wh()s(,, sl)r(;a.(l fllll(:(,i()li O(qi, q~, qi' q~) t)e 9

+(q, q; q',, q;) Tii(;li,

~(q)

,s,, #

#

#

(7.s.s)

#

l(;t,t, ilig t, ll(; 21) ()t),i(:('t, siglia,l Oi(qi, q.2) !)(; g(;li(;l'at(;(l tlu'()llgli (,lie l)I'()(t~l(:(,

I)y t, ll(; 1 l) wa.vcf()rni

~i(q,, q2) -- ~o*(q,),,c(q.~),

(7.6.9)

wc nia.y ()l)serv(; tile finl(:ti(ni (,(q) a.l()lig tll(; lille q~ = q~ ill tll(; t)lltt)llt t)la.nc ()f th(; sysl,(;lli, lla,lll(;ly tl,(q) = r q). (7.6.10)

Evi~tei~t, ly, ()tll('x (:ll(fices fin" tile s.vst('aIl r(;sl)(nlse a.ll(1 llell(:(: fin" ill(; geometry of the olltt)llt (h;te(:tioli are a,lh)we(1. Th~ls, fin" iIlsi, a,il('e,, witli #

#

#

(); q,, %) we nla.y observe ()(q) t41Olig the line q2

-

#

q,, %), -

(7.6.11)

0 in t,tie ollt, t)llt t)la.ll(;, t)eiiig

('(q) -- (Po(q, ()).

(7.6.12)

In w 7.5.1 we proved tha,t tile Wigner distribution and tile a,nfl)iguity function can both t)e interpreted a,s 1D Fourier tra, nsforms of the 2D product function Try(q, q) ~ ( q - ~)~o(q + ~), (7.6 13) the former a,s Fourier tra.nsfi)rm with respect to the difference, space coordinate q and the latter as Fourier transform with respect to the mean spa,ce coordinate q (see Fig. 7.12). Here we conveniently rewrite Eqs. (7.5.4) and (7.5.7) in the slightly different but equivalent forms

W(q, t~)

1

x/~ i i l_l_

t

~t

--it~q

-r-r ~"(q ' q )c t ~t

JJ

~ "

--i~q ~

/

5(q ""t

q)dq'dq',

(7.6.14)

The Wigner Distribution Function: Optical Production

383

Accordingly, by comparison with (7.6.1), it becomes evident that 142(q, ~) or A(~, ~) (:an be computed optically by a 2D linear transformation where the product function Try(q, ~) be assigned as the 2D object signal and the optical processor be designed to ha.ve the response function

r

(q, i~; q' , i]')

-- ~ 1 c - i ~ q ' ~ ( q ' - q ) ,

(7.6.15)

to compute the Wigner distribution, or (~A (q, q; q' , q')

--

- ~1c

-i

t~ q'~(Ol

~),

(7.6.16)

to compute the ambiguity function. Clearly, tile space variables q and ~ come to acquire the role as the orthogonal coordinates of the Cartesian frames in the input and output planes. It is evident that ~5w describes an optical system capable to perform the Fourier transform in the q direction and the imaging transformation in the ~ direction, and similarly for ~A with the role of the two directions interchanged. The optical implementation of the Wigner and/or ambiguity integrals requires therefore the following two processes: i) a multiplica.tion process to form the 2D product flmction 7r,(q, ~) from the 1D signal ~(q), and then ii) a. para.llel imaging/Fourier transforming process involving the two orthogonal directions of the object 7r.~(q,q); specifically, the q-axis direction is imaged whilst q is Fourier tra.nsformed in order to display 1A2 and vice versa,, q is Fourier tra,nsformed whilst O is imaged to display A. It is worth emphasizing that since the ambiguity function is in general complex whilst the Wigner distribution function is always real, when the detection scheme of the display setup makes use of devices responding only to light intensity (like, for instance, a photographic film), information regarding the phase of the ambiguity function atll(t correst)otl(tiilgly the negative regions of the Wigner distribution function are got lost. The optical realization of the processes i) and ii) is illustrated in the following section. Within this respect, we mention that, due to its relevance to radar applications, the generation aond display of the ambiguity function have received a great attention [8]. Since the Wigner and ambiguity integrals sireply differ in the direction along which the signal-generated 2D product function 7rs(q, @ is Fourier transformed, the optical implementation of the Wigner transform has greatly benefited from the optical arrangements for the ambiguity function display [ 1 2 . 1 ] ; also, optical setups operating as both Wigner distribution and ambiguity function processors have been arranged. In the considerations to follow, we shall refer specifically to the Wigner transform.

384

7.7

Linear Ray and Wave Optics in Phase Space

Wigner processor for I D real signals: basic configurations

Since ()in' x~m,in c o n c e r n is to ilhistrate how the "ma,thenm,ti(:s" of the W i g n e r integral ('oul(t t)e iInt)h;l~e~te(t t)y ()t)ti('a,1 ~ma,~s, we (les(:ri])c ttw, tm,si(: s(:he~nes ()f ttw, (list)la,y sct~q)s witl~ (u~q)lm,sis ()x~ tl~e (:(m(:et)ts i'a,tl~(;r tila.ll ()11 tlw, t,e(:h~i(:al (l(;tails. 'I't~(' illl,(:rt's|,('.(] I'('.a,(lt',r is (lir(;(:l,(;(l t() l,l~(: lil,(',l'a,l, llI'(; f()r a. ~()re ('()]~q)l(',l,(' a]~([ ~l)([a,l,(' a('t:()~]]d, ()f l,l~(; s~]l)j(;('l,, ('()]~('('n~(',(1 a ls() wit,]~ l t~(; (ligil,al a.l~(l l~yl)ri(l i~t)l(:~(;~d,a.l,i(n~s ()f l,l~(~ Wig~(;r (lisl,ril)~ll,i(n~ a,s well as with l,]~c ()l)l,it:al vislm,lizal,i()~s l,]~al, ]~av(; ])('~(q~ l)l'()l)()s('(l 1,()(lisl)lay 1,1~(',,1/) Wigl~(;r (tisl, ril)~l,i()~ fl~(:l,i()~ ()f 21) sig~m,ls [12]. W(; I)(;gi~ (:()l~si(l(',ri]~g ~(n~('ga.l,iv(; r('a,1 sig~als, s() l,]m,l, 1,1~('~]~n~ll,il)li(:al, iv(; t)r()(:css I,() fl)n~ 1,1~(; 21) l)r()(l~u:t f~u~(:l,i(n~ 7r.~(q,~) i]~v()lv(,,s 1,1~('~sy~l~(;l,ri(:a,lly shifl,e(l a,~(! r(;a.l v(;rsi(n~s, ~ ( q - '}) a,~(l ,c(q-t-~) a l'(; t)ri('tly (:
7.7.1

,.,

~

()f 1,1~(',sig~al (',()l~l)l(;x sig~mls ~

rr.~-p't'oduct flrnction modelling setup .fo't" 1 I) 'real .~ig'nals: t'~t~o obj(:(:t l'ra,'nspa't'c'ncic.~

()t)ti(:ally ~nl~' llall~ll~'s l,w~ (lillwllsi(nlS, l,ll~'~21) ~l~.}~;(:l, (lisl,ril~llti~ll (~11 l,h(', int)ld, t)lml(; Iii l)~;illg l,rallsf(n'~ll~',(l Jill,() 2 D illlag~; (lisl,ril)lll, i~ll {~ll l,ll~; ()111,])111, t)lmlc 11o. Mal,l~
(7.7.1)

which is t)ra(:ti(:ally rea,lizc(1 by a, t)la,na,r tra,nst)aren(:y tha, t va,ries in the qdirection a,(:(:or(tiilg to the signal a m p l i t u d e and is consta,nt in the (}-direction. As an exa,mt)h;, in the simt)le case of a, rectangle wa,vefornl, illustrated in Fig. 7.15.1)), the tra,nsparency m a y be designed to ha,ve a, non zero uniform t r a n s m i t t a , n c e t h r o u g h a vertical strip with the s a m e w i d t h a,s the signal. Altema,tively, the d i s t r i b u t i o n in the transparency plane m a y be concentrated along a, single line, which is m a t h e m a t i c a l l y represented by the &function as

p(q) --+ r

~ = ~(q)h(q-'),

(7.7.2)

The Wigner Distribution Function: Optical Production

385

Transparencyplane

q~(q)

I

-(2

(2

-Q

q

Q q

(a)

~

-Q

Q q

(c)

FIGURE 7.15.2D representations of (a) a rectangle signal, obtained (b) by a nonzero uniform amplitude transmittance through the vertical strip q _
single signal 7r.~ is ~(q-

q) and ~(q + ~) Opti(:ally 7r~ can be produced by multiplying two planar 2

"

'

tra.nsmittances, one representing ~ ( q - ~) a.s the other ~(q + ~). Typi(:ally the imfltipli(:ation of two real sigIm,ls can be a.ccomt)lished opti(:ally by placing the two transparencies, representing the signals, adjacent to each other thus yielding as an equivalent transmittmme the product of the two signals. Alternatively one of the signal, more precisely its 2D representation, can be transmitted on the tra.i~sparency representing the other, thus again effecting the desired multiplicative operation of the two signals. How ca,n one form the planar transmittances of the shifted signals ~ ( q - ~2) and 7:(q + ~)? Note tha,t, if we have a transmittance ~(q, ~) of the signM ~(q), the signal 7:(c~q + / ~ ) , where c~ and/3 are arbitrary real numbers, can be obtained by a coordinate transformation of the transmittance plane with respect to the fixed (q, ~) coordinate frame of the working plane, that is the input plane for the optical processor we intend to build. Rotating, for instance, the transparency ~(q, ~) counterclockwise by the angle 0 about the origin, we produce in the reference plane the transmittance ~0 (q, @ - ~(q cos 0 + qsin 0).

(7.7.3)

In particular, a rotation by 0 - 45 ~ generates the scaled and shifted signal

(q,

-

(q+q

(7.7.4)

Linear Ray and Wave Optics in Phase Space

386

Transparency plane

I

Input plane ]l

....' i

"".

/

m) Transparency plane

i

~~o( 71&45 ' ~/)

Input plane I I ,

ks

",,,.r

..d

q~ ( '1 ' '1

4:

--W,

0,)

FI(;UFIE 7.16. lles, fll.m~t tra,~s,~itt.m~co (~f tim ~ l l , i l ) i i c a l , i ~ ,)f l.w~ i,h'~tical tra.~spm'c~cics r()l,a,tc(1 l)y .9(1" r(,lat.ivoly l.,, em']~ ,~tlwr i~ tlw ca.,.;c ()f (a) a si~gh'--recl,m~gh, sig~al m~(l (b) a ,l{,,,|,le-re('ta,,glc sig,,al (i,,sl,ir,',! I,y [s.2]).

whcrca,s a, r()l,a,l,i()ll ])y 0 = --.15 ~ l'('~sllll,s ild,() 'I' 4,~,,,(q, 0) - ~ ; ( ~ ) .

(7.7.5)

Evi(t(;ntly, at)art fl'~)lli a sr fa,cl,()r, t,ra, llSlllit, l,all(:t;S (7.7.4) aal~l (7.7.5)(lisplay l,hc sam(; am()lmt c~f slfifl, aal(1 in l,hc ()t)t)osit(; (lir(x:l,i(nls, as re(tltir(;(l l,o form the t)ro(tll(:t fllll(:l,i()II rr.~(q, (}). Tlmn, w(: lllay t)la(:(: i~l l,tl(: l)lmu: Ili two identi(:al t,raIlsI)ar(:]l('i(:s, (hi wlli(:}l l,ll(: 2 D I'(:I)r(:s(;]d,a,l,i()~l O(q, ~1) ()f l,ll(; signal is r(;(-oi'(t(;(t, trot; r()l,a,l,t:(l ]W 45~ an(t l,h(; ()l,h(;r t)y - 4 5 ~ Ull(h:r lnlif()rnl (:()h(;rellt illuminatitm, l,hc l,ransIlfitl,t:r signal r(:t)r()thu:cs l,hc (lesir(;r t)r()(tllt:t flm(:tion 7r~(q,O) (or, so,,,c s('a.h:(l version of it), fro,,, wt,i(:tl l,llt: Wigncr (tistribution flmction (:a,ii t)c gcilcra.tc(l. Figure 7.16 illustrates tim rcslfltmlt t r a n s m i t t a n c e of the m,,ltipli(:at, ion of t,l,(; two rcpresci, ta.tions ~(1~-2/) m,(t qp(t~2 ) for b o t h a single and a (toublc r(;(:ta,l~glc waveform. Likewise, supposing to r()tatc by 0 an(t dilate by ~t the axes, we arc able to generate the tra,nsmittan(:(;

~,,.0 (q, q) - ~ ( m q cos O + m ~ s i n (4),

(7.7.6)

which (:an be made to fit the required shifts q + ~ by at)propriately arranging 1 'm and O. Indeed, requiring t h a t r n c o s 0 - 1 an(t r n s i n 0 - +-~, one obtains 1 .6 o 'm~ 1 and 0 - a r c c o s ( ~ ) ~ 26 .

The Wigner Distribution Function: Optical Production

7.7.2

387

Optical processing of the product function Try(q, ~) to display the Wigner distribution function

Once tile 2D product flmction 7r~(q, ~), or soine scaled version of it, is available, it remains to arrange a,n optical system capable to perform the 1D Fourier transform of % with respect to the ~ direction to yield the Wigner distribution function. As we know, a.n optical Fourier transformer produces an output signal proportional to the Fourier transform of the input signal by a constant factor or a quadratic-phase term in the space coordinate. Here, the input signal is the field transmitted through the two transparencies arrangement under plane-wa,ve illumination a,nd hence is proportional to the product ~ ) -- 7 r , ( ~ , x/~q~. Then supposing that the 1D Fourier transform of tile input signal with respect to ~ has been t)erformed by an optical setut) with effective focal length f, the resultant signal Oo(q, ~ on the output plane should be proportional to the Fourier integral

Oo(q, q) oc ff P( q-~' )~( q+~' v/~ ) c - ~ ' d

.

(7.7.7)

As a general property of optical Fourier transformers, the frequency variable h; of the transf()rmation is related to the out, trot (:oordinat, e ~ as (see w 5.4))

~, - -~q. By changing the variable to y -

apo(q,O) c~

/ ~(~

(7.7.8)

v/2~~, the integral above turns into --i

t~.

- ~)~(-~2 + ~)c2 "~Ydy,

(7.7.9)

thus clearly reveaJing the relation of the output signal to the Wigner distribution of ~(q), being in fact (I)o(q, ~) o( ~ V ( ~ , 722)"

(7.7.10)

Thus we may genera.te and display the scaled version of the Wigner distribution function of the given signal ~(q) in the output plane of a 1D Fourier transforming optical processor. Notably, since the frequency variable of the displayed Wigner distribution comes to correspond directly to the output coordinate ~, the entire output plane of the optical processor is utilized, corresponding directly to the Wigner pla,ne (q, n).

Parallel imaging/Fourier transforming operations of the 2D real signal 7c~ Central to this is the possibility of performing the Fourier transform in only one direction. As said earlier, the required functional form (7.6.15) of the impulse response reveals that the optical processor should ~ct as a Fourier transformer

388

Linear Ray and Wave Optics in Phase 5pace

in one (lirection and as a, unity magnification imaging system in the orthogonal (tire(:ti()n. Therefore, we must a,rra, nge an a,stignmti(: optica,1 system, capa,ble ()f t)erf()rming t)ara,llel ot)era,tions in orthogona,1 (lire('tions; t)re(:isely, imaging i,1 (),re (ti111(;nsi(),l a,,l(t F(nlri(;r trmlsfornl in the ()th(;r. We (t(',s(:ril)e th(', ()t)ti(:a.1 a.t'(:hit(:(:tllr(~,s tha.t have t)(~(~n r(~a.lize(t t() t)(',I'f()rIll sll(:h illmging/F()llrier transf()rnfing I)aralM ()t)erati()ns. Evi(lently (:ylin(lri(:al l(:ns(;s at(; tll(; ha, tin'a.1 (:ml(li(la,t,(;s a,s (:(nllt)()ll(;Id,s ()f an ()t)ti(:al s(;t,~l) wh(;r(; t,h(; sigl~al is a.iti,('t('(l (liff(:i'(:~ltly i~ tll(' t,w()()rttl()g()~lal (lii'(:(:ti()~ls. Va.ri()~s (:()~lfig~rati()~ls, wl~i(:l~ (:~l)l()y (:yli~(lri(:al l(;~ls(;s i~ (:()~fi)i~a.ti()~ as w(ql witl~ sI)tl(;ri('al l(;~ls(:s, lmv(: 1)(,(,~ s~gg(,sl,(:(l [1().(;]. 7',u~o-/~"n,,~' ~o'nJi~.lu,'~'a,tio'n,.,,'. W(: 1)(:gil~ ()~r (lis(:~lssi()ll (:()l~si(l(:ril~g 1,11(; (:()niig~ra.ti()~ (l(;I)i(:l,(;(l i~ Fig. 7.17.a) [s.2, s.a, s.v, l().(i], wl~(:r(; w(; l~av(' a,ssig~l(;(l th(': varial)l(;s q m~(l ~ r(;sl)(:('l,iv(;ly I,() l,l~(; l~()riz()~l,al m~(l v('rti('al axis i~ 1,11(;r('f(;r(;~(:(; l)lm~(:s. A ('yli~(lri(:al al~(l a sl)l~(;ri(:a.1 l(;~s, I)()ll~ l~avi~g fi)('al l(;~gtll f, a.r(; t)la(:(;(l at (lis|,m~(:(; .f r(;sl)(;('l,iv(;ly fr()~l 1,11(:illl)lll al~(l 1,1~(;()~ll,l)~ll, l)lm~(;; tl~(; t,w() l(;ns(;s ar(: S(:l)arat(:(l 1)y 2 f t()(;a~:l~ ()tll(:r, ml(! tl~(; (:yli~(lri(:al l('~s is ()ri(;nt(;(t witll its (:yli~l(lri('al axis l)a,rall(4 I,()l,ll(~ r)-axis. W(: ~ a y m~alys(' tl~(; trm~sf()r~ati()~s a,(:t(;(l ()~ tl~(' i~l)~t si~a,l s('~t)aI'a,t(;ly ill th(', t,w<)()rtll()g(n~al (lir(:('ti(n~s. l';vi(l(;~l,ly, i~ ~,l~(, l~()riz()l~,al (lir(;('ti(nl th(; si~m,1 "s(~(~s" a l)(~rf('('l, i~mgi~/4 sysl,('lll, f()rlll('.(l I)y l,w()(:a.s('a~l('(l l:()~l'i('r l,ra.llSf()rlll('.l'S, wlli(:ll th(;n 1)r()(l~t(:(:s a ~('ga,tiv(' ~ll~il,y nla.g~liti(:a,ti()~l. I~ (:()~ltra.st, in t,h(; ()(tir(:(',ti()~l ()~lly tl~(: sl)l~(:ri(:a,l l(;~s is a,(:tiv(;, 1)(;i~g at (lisl,m~(:(;s 3 f m~(l .f fi'()n~ 1,11(;i~1t)~1~,ml(11,1~(; ()~l,l)~t l)lm~(:, r(;sl)(;(:tiv(;ly. ~1'1~(;(l()~fi)l(:-l(;~s s(:t~l) is tl~(;r(;for(; (;(t~iva,l(':nt t() a,~l i~ll)(;rfi;(:l, F()~ri(;r l,ra,~lsf()rlll(;r, wl~()s(; ray ~mtrix lm,s D = - 2 . Th(; inq)~ls(, r(;sl)()l~S(; ()f l,l~(, syst(;lll is (;X(;l~l)lifi('(t t)y ~,11(' fial~('l,i()~ e ( q . O; q . q ) .... V 2 . /

,,

7 q'~(q' + q).

an(1 a,(:(:()r(tinglv9 t,ll(; i,~l),lt (bi(q , gl)- ~ ( q ~- ~ i ' ) ~ s ( ~ )

(7.7.11)

is |,r~l,llSf(,l'lll(;(| |,(, (7.7.12)

where, a,s already note(t, tim Slm,tia,] frequency a; is re a,(l al )l e, a,lo~lg tile o u t p u t axis (~ t h r o u g h tlle scaling fa,(:t()r ~" h; kq iThe system is seen to perforill the, desired para,llel inmging a,n(t Fourier transforming operations, thus generating in the plane IIo the s('ale(1 a,nd q-inverted version of the W i g n e r (tistritmtion fllnction of the assigne(t 1D signal ~(q) with a, q u a d r a t i c phase factor in the, w',rtical direction. This factor can be removed by placing a negative cylindrical lens i m m e d i a t e l y in front of the o u t p u t plane. Alternatively, in the astigmatic system of Fig. 7.17.b) the spherical and cylindrica,1 lenses are in conta, ct [10.a, 12.3]. T h e lenses have the same focal length f and are at distance f from b o t h the input and o u t p u t planes; as before, the cylindrical lens has its axis parallel to the ~ axis. Since b o t h lenses

The Wigner Distribution Function: Optical Production

389

Kc~q

Ho

.f

(a)

f

t

l-lo[

(b)

.

q

q

FIGURE 7.17. Schematic of astigmatic optical systems performing (a) parallel perfect imaging/imperfect Fourier transforming operations; (b), (c) parallel imperfect imaging/perfect Fourier transforming operations, an(t (d) parallel pcrfcct imaging/perfect Fouricr transforming operations.

are active in tile horizonta,1 dire(:tion, their combinatioIl is oquivalent to a single h;ns with fo(:al h;ngth fT given by 1

2

fT : 7 + 7

1

1

7'

(7.7.13)

Therefore the input signal dPi(q, q) experiences a,n imperfect imaging operation with a negative unity magnification and a curvature factor Cikq2/f. In contrast, in the vertical direction we have a perfect Fourier transforming arrangement. The impulse response of the system is seen to be

e(q, O; q', g) - V ~~/-~kei}q~ e(q' + q)~ - ~ ~~

(7.7.14)

It produces on the output plane a scaled and q-inverted version of the Wigner distribution function according to .kq2

9 o(q, q3 -

~'7

q tc ), w ( - - ,/~' ,/~

(7.7.15)

with a quadratic phase factor in tile horizontal direction, which can be removed by placing a positive cylindrical lens immediately in front of the plane 11o. Finally, a totally equivalent configuration can be realized by placing two cylindrical lenses, L 1 and L2, of focal lengths f and f back to back as sketched

390

Linear Ray and Wave Optics in Phase Space

t"IG[II/E 7.18. A~fl)ig~ity fit~wt,i,)~ (~n(),l~l~s sq~mr,,,l)(lisl)lay ,)f (b) a single sqt~are t)~flse ;tll(l (c) ~ (loul)le square ln~lse, ol)tailm(l l)y (a) a st)her()-(:yli~(lri(' (l()~l)let set~l). (ll,et)l'Oduce(l witl~ l)ernfissi()~ fi'()~ [s.2].) ill Fig. 7.17.(:). 'File l(~llSeS ar(~ at, (list,au(:(~ f fr()lll 1)()t,ll tll(~ illl)llt, ml(t ()llt,t)llt, t)lml(~ ml(1 t,ll(qr a.x(~s a,r(' r('(:il)r()(:ally I)(~rl)('ll(li('lllar, witll tllat, ()f L l t)ei~lg t)a,ralM t,() tll(', q axis. Th're.e-lens ('o'nJig'u, ralio'n,. Ill ()r(ler t() av()i(l tll(' l)r(',s(',ll(:(', ()f a, Sl)atially varyi~lg (pm,(trati(' I)has(' ill ill(' ()lll,I)llt., as ()('('llrs ill 111(' I)r('vi(nMy (l('s('ri|)(;(1 t)r()(:(;ss()rs 11sing a. sI)lmr()(:ylill(lri(: (t()ll|)h't, w(; llmy ~s(: tier(;(; (:yli~(tri(:a,1 lenses a.rrange(t a,s illl~stra,t(,(l i~ Fig. 7.17.(t), ttu~s (:r('a,ti~g (;xa,(:t i~a,gi~g m~(l Fourier t r a n s f o r m i n g i'ela,ti()~s I)(~t,we(n~ the input m~(t ()~1,t)~I, t)lmms [s.~, ~().a, s.2, 10.6, s.5]. Lenses L~ m~(1 L:~ ()f fi)(:al le~gth .f a,re v(~rtically ()ri(~te(t, s(',t)a,ra.t(',(t |)y 2 f a,n(t (tista,n(:e(11)y .f r(~sl)(~(:tiv('Jy fl'()u~ t)la,ne Iii (n~ tl~(~ rigl~t m~(l l)lmm Iio on the left. Lens L~ ha,s a fo(:aJ le~gth double of ttm,t ()f L1 m~(t La; it is h()rizontMly oriented an(t t)la.(:e(1 ~i(lwa.y t)(',t,w(~e,n L~ m~(l L:~. Evidently, lens L~ g(',~(~ra,t,(~s a,ii exa,(:t F()~rie, r traJ~sf()r~ r(~lati()n for the (~ t r a n s f o r m a t i o n , whilst the t,w()-lei~s g e o m e t r y in t,he h()riz()ntaJ (tire(Mon produces a, perfe(:t imaging ()I)eration with negative unit ma,gnifi('a,tion. T h e syst e m response writes as (7.7.16) which t h e n yields the outt)ut (tistribution 9 o(q,

- V

(---%

(7.7.17)

kq tile frequency variable rc being rela,ted to the o u t p u t coordina,te ~ as 2f" Figures 7.18.b)-c) show the a m b i g u i t y function display of a single- and a, d o u b l e - r e c t a n g l e signal, as o b t a i n e d by the two-lens a s t i g m a t i c processor,

The Wigner Distribution Function" Optical Production

391

sketched in part (a) of the figure [8.2]. The optica,1 setup is practically the same as that pictured in Fig. 7.17.a), but rotated by 90 ~ around the optical axis in order to perform the Fourier transform in tile horizonta.1 q-direction thus displaying the ambiguity function instead of the Wigncr distribution function. The notation in Fig. 7.18 is in conformity with the time-frequency domain pertaining to the quoted paper. Thus, the pair (t, r) corresponds to the pair (q, 0) relevant to the space-frequency domain of our analysis. Also, the function s(t r) correspond to the product function rr~(q~q--) as u corresponds to 27r" ~ Finally, the function u(u, 7), which is the outcome of the displayed processor, is related to the scaled version of the asymmetric ambiguity function defined through tile integral (7.4.3) without the factor 2"n'" 1 7.7.3

Try-productfunction modelling setups for 1D real signals: a single object transparency

The technique, described in w 7.7.1, to generate the shifted product Try(q, ~) requires two identical transparencies that are placed close together in the plane IIi and suitably rotated relatively to each other. [li

Optical setup producing vertical coordinate

transparency plane

Ilo

plitter

q + ii

q)( ---7--- ) 'V:

parallel imaging/Fourier transforming processor

inversion of the forth and back propagating field

Wigner plane

e(~) plane wave illumination

F I G U R E 7.19. Schematic of the optical setup for displaying the Wigner distribution function of a real 1D signal using a single input transparency.

Alternative schemes have been devised demanding for a single input transpa,rency. Tile ba,sic idea, is that of placing in the plane IIi a single signal tra,nsparency properly rotated to produce any one of the shifted waveforms needed to produce the product rr~(q @ ~ let us say ~(q+q) ~ ~ The transparency is then illuminated from the right by a plane wave reflected by a beam splitter (Fig. 7.19). Tile transmitted field is allowed to propagate forth and back through an optical setup suitably designed so that the 0-inverted field distribution is returned to IIi, i.e., ~(t~2 ). Thus, after transmission through the ~ ( W ) transparency, the d ired product ~ ( ~ ) ~ ( W ) i s produced on the right of IIi, which then can be input in any one of the previously described optical stages for the Wigner distribution function display. A three-lens setup, as that

392

Linear Ray and Wave Optics in Phase 5pace

IrI(Itllll'; 7.2(). (~,,) A ('<)lwre~t, l)r(wess<w fi~r m~ll)igt~it,y fi~cti()~l (lisl)lay tisi~g a si~lgl(, 1D i~in~t,. (b) Tim m~l)ig~ity f~(:t,i(~ (~()(lt~l~s s(itmro(l ) (lisI)lay ()f a si~tgh' S(l~mre InCise. (l{el)r()(l~co(! wit, l~ l)er~nissi()n fr()tn [~.5].)

tfictln'e(l ill Fig. 7.16.(1), ml(l a, t,()t,ally rotl(,(:l,ing lllirr(w t)lace(l at 11 yM(1, fin' ()f t,ll~' ()}).j~'(:l, (listril)lll, i~ll fi'
l,ll(; l)r()l)a,ga,l,i()~ l,lir(n~gli a, ( ',a,s(:a,(l( ', ()f fi)~n" F()~n'i(,r l,ra,~sfi)r~li~lg sysl,(n~s, which l)r()(l~(:(;s a, ~ i l , I~la,g~lilica,t,i()~ i~la,gi~g (~t)(;ra,t,i()~l; ]~(;~lc(;, i~ t,]~(; q (lir(,('t,i(m the l(,l~S 1.2 g(;~era,l, es a (:a,s(:a,(l(; (~f l,w() F()~ri(;r l,ra,nsfiw~('rs, wl~i(:l~ l)r(~(l~(:(;s t,hc (l(;sir(;(l (:()(w(tinal,(' i~w;rsi(n~. Fig~r(; 7.2() r(;1)I'()(l~u:(;s fr()~ [~.s] tl~(; si~lgl(;-il~t)~t s(;t~l) fi)r m~l)ig~fity tim(:single s(t~mr(; t)~fls(;, tt(;r(;, \(u, 7-) (l(;~()t,(;s t,h(; asy~l~l(;t, ri(: a~l~t)ig~il,y fin~(:t,ion " and t,t~(; fa(:t()r ~~ t)(;i~g ()mit, t(;d (i(;fi~(;(1 t)y (7.4.::I) wit, l~ 1,, (:()rr(;st)on(ling t,() )-~, Th(; (t~m,ntiti(;s (v all(l /~ st)(;(:ify t,t~(; r()ta,t,i()n angle ()f t,l~(; t,ra.nst)ar(;~(:y with rest)(;(:t, t,() t,h(; i'(;f(;r(',~(:(; t)lm~e: (t = (:()sO a,n(t/~ =-Sill 0. Evi(lently, t,h(; t)lanes P1, P,, and P:~ ('orr(;st)on(t t() t,h(; t)la,n(;s, Hi, II a,n(t IIo ()f Fig. 7.19. An()th(;r s(:hen~(; is sh()w~ in Fig. 7.21, ret)ro(iu('ing fI'(n~ [ai.:~] t,he singleint)~lt, s(;t,~t) and t,h(; r(;s~lt, i~g Wign(;r (tist, ril)~t, ion fim(:t, ion (tist)la,y ()f a, slit. T h e left a r m of t,he t)ro(:(;ssor (:omt)ris(;s a, 2 D Fourier t,ra.nsfornmr a,n(t a, roof 1;op t)rism, wh()se refle(:t, ion a,n(1 ()ne-(:()or(tilm,te inversion t)r()t)(wt, ies are sket, ched in l~he t)a,rt, (c) of the figure. As a,n ov(;ra,ll effect of the t)rot)aga,tioI~ t h r o u g h the lens-prism arrangemeI~l;

the rotated t r a n s p a r e n c y ~(q+q) is imaged into the v~ oppositely rota,ted transmitta,nce F ( ~ 2 )" T h e pair (x,y)in [12.3] corresponds to the pair (q, q-) of our ana,lysis as u to ~tg; also, the W i g n e r distribution function W(x, u) is defined by the expression (6.2.3) w i t h o u t the factor ~1. '

The Wigner Distribution Function: Optical Production

393

We observe that use of a single transparency representing the signal eliminates the problem of producing two identical object transparencies and of ma.intainiilg them aligned with respect to one another. However, part of the input power is wasted in passing through the beam splitter twice, ~md also the number of optical components in the processor is increased.

FIGURE 7.21. (a) Optical setup for displaying the Wigner distribution flmction of a real 1D signal using a single input transparency (BS: beam splitter, R: roof top prism), by which (b) the Wigner distribution function of a slit has been obtained. (Reprinted with permission from [12.3].) (c) Schematic of a roof top prism. As suggested in [12.10], a,n approt)riatc polarization-sensitive setup ('a,n be devised to avoid tile loss of power, n()t'ma,lly consequent to tile use of beamsplitting devices. The optical scheme, described in [12.10], makes use of a cubic polarizing beam splitter, which in combination with two retardation plates is a,ble to convey the incident beam firstly into tile 7cs-product flmction modelling processor and then into the parallel imaging/Fourier transforming processor with an ahnost intact power. A polarizing beam-splitter cube normally separates the incident beam into two linearly polarized beams propagating at right angle to each other and with complementary states of polarization. The beam which straightly passes through the cube emerges a,s a p-polarized beam, the electric field oscillating in a plane parallel to tile plane of incidence. Tile beaIn transmitted at the right angle to the incident beam emerges as an s-polarized beam with the plane of the electric vector being orthogonal to the incidence plane. The state of polarization of the incident light governs the distribution of the optical power among the two exiting beams. In the experimental setup of [12.10], whose sensitive beam splitting section is reproduced in Fig. 7.22, the collimated linearly polarized coherent beam from a He-Ne laser is made to emerge from a properly oriented ~ plate as a pure ppolarized beam. Thus, it can be totally transmitted straightly by the polarizing cube into the 7c~-function modelling section, which is designed in conformity

Linear Ray and Wave Optics in Phase Space

394

cubic polarizing

2 plate r___~

p-polarized t

beam-splitter

beam

transparency

l~~ne

ongoi ng b e a m

rt-function forming optical setup

s- polarized returning beam

linearly polarized collimated laser beam

plate p- polarized

s- polarized emerging beam

circularly 3olari/ed beams

Wigner processor I

FIGUIIE 7.22. Ol)tical scl~c~m for flu: gc~cral,io~ of tim Wig~mr (listriln~tio~ fi~mtio~ of a 1D ol).jcct ~si~g a si~gle trm~spm'c~my. It expl<)its a s~fitatfle arrm~ge~u'.~t of p(flarization s(;~sitiv(; (tevices to ('o~vey almost tim wl~()h; ()ptical power of l,l~(; i~('i(l('~l, I)em~ i~to tim Wig~mr t),'occssor (i~,sl)iretl I)y [12.10]). or a 21)

witll ttl(; a,t)(,v,'~ illllstral,t;(l ,,l,tit:al sr A ~ l)late is illsert,;r 1,r th(', l)()la.rizing 1)emil Slflil, l,er ('lll,e a ll(l tim ,)l).j,:ct l)lall,: lli wil,ll its ,)t)l,i(: axis a,ligller al, d5 ~ 1,~ l,ll~: w;rl, i(:al axis. Tints l,ll~: illil,ial ill(:isl)lil,l,(;r (:lfl)e as a, t)lu'c s-t){fla.rize~l l)em~. Ih'.~(:~; il, ~u~('.rg~s fr~)~ 1,1~(~l)(',a,ll~-sl)littcr at 90 ~ with respect 1,~) tim ret, m'~d~g tma,l~ a,~(l witl~ a,h~r the sa,n~e i~l)~t, t)ower a,s tlm r ])ea,~. ()f (:(~rs~;, l,l~e rigl~l, a,~gh: se(:l,i~n~ r l,l~: ~l)l,i(:al s(:lmn~e is (tesigne(1 t~ l)~;rf{n'~l~ 1,1~; Im.ra.ll{'J it~m.gi~g/b'r l,ra,~sf~r~l~i~g ~imra, t,ion, wtfi(:h serves t{~ g(umra.t,c tim Wigx~er r icn~ fil~r ~f tim 1D ()t).ject ln~(tcr (',xa,nfi~a,ti~n~ a,s I)revi(n~sly (tcs(:ril)e~l. F i n a l l y we ~1,~; tl~a,t tl~e si~@e-i~tnlt s~;l,~l) is h;ss tl~;xit~l~; l,lm,~ 1,1~('~(toubh;i~t)ut a,rra,l~g(u~m~d, slime its ~lsc is lil~fil, er I,~ the r ~f tim self-Wigner (tistri|mtion fln~(:ti(n~ (and self-m~fl)ig~fity flu~(:t,i(n~). Cross-Wig~u" (listribution a n d (:ross-am|)ig~fity fl~m:ti~n~ r re(l~fire, {)f (:()~rs(;, l,w{) tra,~st)a,relmies ret)r(;s(;nting the two multit)lying signals (;nl,(;ring th(; t)(;rtin(;~t integrals.

7.8

Wigner processor for 1 D complex signals basic configurations

In m a n y practical s i t u a t i o n s 1D (or 2D) comph',x signals are encountered. E v i d e n t l y w h e n dealing w i t h c o m p l e x signa,ls one needs to p r o d u c e the complex c o n j u g a t e version of the signal in order to m o d e l the p r o d u c t function

7r,(q, ~) o< ~*(q- ~)~(q + ~2)- We describe here the basics of s o m e of t h e proposed m e t h o d s to effect process i); of course, process ii) can be effected by the

The Wigner Distribution Function: Optical Production

395

sa,ine optical arra, ngements, described in the foregoing paragraphs. 7.8.1

rc~-productfunction modelling setups for 1D complex signals

A certain nutnber of %-function modelling a,rrangenmnts for coinplex signals have been proposed and implemented [12]. A proposed method makes use of a (possibly computer generated) hologram of the signal, replacing the tra,nsparency at the object plane (OBJ) in the optical setup of Fig. 7.21.a) [12.3]. The hologram conta,ins informa,tion a,bout both the signal and its complex conjugate. Thus, by use of suitable masks, it is possible to select the informa, tion about p or p*, a,s desired. We may suppose, for insta,nce, tha,t under coherent illumination in the first pa,ssage through the hologram (a,t)prot)riately rotated according to the t)revious (tiscussion) the reconstructed image of ~ ( ~ ) is transmitted, which image is returned back as p(q+q) whilst in the second passage the complex conjugate p * ( ~ ) is v~' delivered. The obtained product function p * ( ~ ) ~ ( ~ ) i s then input into the Wigner distribution flmction processor on the right of the object plane. Alterna, tively, an interference ba.sed method ca,n be used to effect the wanted phase conjugation of the complex signal. In fact, the interfcren(:e pattern of the sum of two signals when yiehted by an intensity recording ma,teria,1 like, for instance, a holographic fihn, always contains, at its + / - first order, the t)roduct of the amplitude of one signa,1 a.nd the complex conjuga, te amplitude of the other and vice versa,. Using therefore a non linear ma.terial, one can produce the required product function to be used for the Wigner distribution display of the original signal. Also, illuminating the complex object at a certain a.ngle may improve the visibility of the final Wigner distribution function display. Referring to Fig. 7.23.a), inspired by [12.3], we may suppose to illuminate the complex object representing the rota,ted signa,1 ~(~-q2) by a monochromatic plane wave at a certain angle Oi with respect to the axis, thus producing the signal ~(qq-q --~)C ikq s i n Oi behind the object plane. The signal is then separated in "

two parts at 90 ~ apart by a bea, m splitter. One part is reflected back to the beam-splitter without change by a mirror. The other is reflected back a.s well, but inverted with respect to the vertical direction by a roof top prism; it is then formally represented as (~ ( q--q ~ )e -ikq sin0i The two parts are again combined and added on some recording material yielding the modulus squared, '.

ikqsin Oi -q-~9(~22 )G--ikqsinOi12

(7.8.18)

Evidently it contains the self-terms ~ ~(q+q ' v~)l 2 and - 2 yielding two ~mirrored contributions proportional to the Fourier transform of the instantaneous power ~(q)l 2 at frequency ! + - v/2k}, modulated by a q-depending

Linear Ray and Wave Optics in Phase Space

396

complex object

mirror

q

at angle (h

~

order

I1

transparency ( p ( - ~ )

n-.- r -

x oc ~

[[

q+q "~

I t beam-splitter

roof top prism

~tl---

D

.

q.,)(q + q )el]~~i .s'JnO,lO, [[ I

/

2k, s ~

,~)

2k. ~ m ~

/

~

0~ = 0

,,,,, ' ~~ ,

~,

q~'( q- q )~, -i~q .sin(),

~ / 0 +

I

I I recording material

aO

1 order

(h)

l,'l(;lllll,; 7.23. (a) Sclu,~mtic fin-tlu, Wiggler (iisplay ()f a 11) c()~ph,x sig~a.l. (h)Tl~e (lelt,a lilies t)r()(l~u'e(l ()~ tim ()~tl)~l. l)lmm I)y l.l~e + / ()r(ler l,en~s wril, te~ ()~ tim re('()r(li~lg til~ii i~l l,l~(, Wig~wr l)r()c('ss()r sclu,~w of l)arl (a,)i~ tl~(, cas(,()f a h'~s-lik(' (l~a(lralic l)lu~s(' wav('fi)r~l.

sil'e(l ('(n~ll)h;x ln'i)il~u'l, fi~l~l'l,i()li l~n~ll,il)lil;(l 1)y all I'Xl)i)~li'~l,ial. 'l'l~s, (;a,(:li l,(;l'lll ~ll(h'r (~xa.l~ilia.l,i(ni. l)ri;i:is(;ly, l,li(; { / -

,,* ~ a,il(1

),t2 ( ~

til'sl, (n'(hu" l,(~r~ls ri;sl)(;(:l,iw'ly yi('~l(l

)(,2il,'q si,, Oi

~( ~,/5, );* ( '~,/5, )"-'-';*"; ..i,, o,

2k

, W.(_%

(7.8,.19)

_ , +'_,,...,, o,

As ll()l,e(l, l,lle Sl)a('illg t)etweell tile t,vr (tistrit)llti(ms is (:(ml,r()lh;(l t)y the illlmfixmti()ll a llgh' Oi. In Fig. 7.23.1)) the ext)('(:te(l vi('w ()f the two) ild,(;rfere~n('ca,risi~g tra('('s i~ tll(' (nltI)~lt l)lmie is sll()w~l ill tll~' ca s(, ()f a 1D (tlm(lrati(:-I)hase wa,vef()nl~ a,s l,lm,l, l)r()(l~t:t;tl t)y a, r162 a,1)(;rra,ti()l~-fre(; t,l~i~ h;~s. It can be ct)~q)a,rt;tl witl~ tl~e Wigncr dist)lay of a, (:ylil~tll'i(:al lens, sl~t)w~ in Fig. 7.24.t)), wl~ir has t)ee~ ot)tained l~sing the setlq) sketr in Fig. 7.24.a), b o t h rel)r()(hu:e(t fi'()m [12.3]; here, a, holograt)hi(: film ha,s |)(;(;n enq)l()yed ~ a recording device. In t)a,rt ((:) of the figure the Wigner distribution fi~n(:tion of a third orr t)hase ot).jc(:t t)ro(tuced with the sa,nw~ setup is shown. Tlw~ notation in the figure is a,s described in relation to Fig. 7.21. Moreover, as r in [12.11], optical phase r via, a nonlinear four-wave mixing process can be used to model the p r o d u c t of the signal and its complex conjugate. To this end, a, collimated monochronlatic plane wave is s e p a r a t e d into three beams, which are then suitably redirected to a t h i r d - o r d e r optical nonlinear material. As a result of the interaction of the three beams inside the m~terial, ~ fourth b e a m is generated which carries the

The Wigner Distribution Function: Optical Production

397

FIGURE 7.24. Wigner dispay for truly complex signals. M: mirror, BS: beam splitter, R: roof top prism. Wigner distribution flmction of (b) a cylindrical lens had (c) a third order phase function, both produced with the setup in (a). (Reprinted with permission from [12.3].) product of the three ongoing t)ealns, one of them entering the t)rodu('t through its comt)lex conjuga.t,e version. Referring to the s(:henl(~ of Fig. 7.25, a,da,t)te(t from [12.11] to the t)resent, discussion, we see that, the (',omt)lex object is inserted b(;twccn t,hc |)cam st)litt,('~rs B 1 and B 2 . It, nla,y |)C sut)t)()scd to be a,t)prot)ria,tcly oriented in the input, plane of the processor in or(ter t() produce the 7rs-flmction factor ~ ( ~ ) upon illumination from the left by the undeflected t)ortion of the ongoing monochromat, ic plane wave. The reflected portion of the incoming wave at B 1 is made to follow an appropriate path up to the nonlinear material. On the other hand, the signal ~ ( ~ ) is separated into two identical copies by the beam-splitter B 2. One copy is imaged without change to the nonlinear material along the path B2-B3-NLM. The other copy is conveyed through the path B2-M~-NI,M, along which the required coordinate inversion is effected (by a Dove prism), thus reaching the non linear material in counterpropagating with respect to the unaltered illumination signal. The signal s(q, ~ produced by the nonlinear material carries just the product

q-) travelling opposedly to the signal copy ~ ( ~ ) . It can then be input into a strictly Wigner distribution function processor. The details of the optical setup, described in [12.11], have been disregarded in the basic scheme of Fig. 7.25. The reader is directed to the quoted paper for a view of the complete scheme of the optical setup used to display the Wigner

Linear Ray and Wave Optics in Phase Space

398

/

M~ / ~e W~gner

display

1:/\/

colkmated

__

monochromatic

p~aRowave

(

~

/

B1

....

: t

tI ~ t I

%

XXXXJX /

B2

(P( q q )

M2

Input object

FI(~tIF/t,; 7.25. Scl~t,~,uttic ~f tim Wig~,,r l~roct~'ssor sectim~ (~lescrilu,~! i,~ [12.11]) for~fi~g the rr.~-i)ro(l~ct fl~ct, io~ for a co~ph,x sig~ml. (lisl,ril)lll,i{)n fllllr

7.9

a,~l wil,l~ sligld, ~{lilir

l,l~: a l~lfig~il,y f]lll(:ti()n.

The smoothed Wigner distribution function and the cross-ambiguity

function

optical p r o d u c t i o n

Following the s~lgg(',sl,i()ll ()f [12.2], we (h;s(:ril)e tll(', ()t)ti(:a,1 illll)lt;lll(;lll,~tI, i()ll of loll(', smoothed Wigxlcr (list, l'il)llti()zl filll(:t,i(nl ill t)a.rl,i(:lflar t)y l,ll(', Wigller (list, ribul,ion function of a Ga,llssiml siglml. The rclater (lisr will filrther (:onfirm tlle link, ba,sica,lly roote(l ill tile Moyal I'ela, ti()ll, a,lllollg a,ll tim (:(nmet)ts intro(tuced and illllstrate(t iIl tlu', I)revious se(:tiolls, i.e. tim st)(;(:trogram, the (smoothed) Wig,mr (tist, riln~ti()I~ f, umtion and tim (cross) m,flfig,fity function. We resort to the C, al)()r elellmnta,ry signal (7.2.16), whose Wigner (tistribution fllnction, a,s showil in w 6.2.1, sha,pes as a, 2D rea,1 Gmlssia,n centered at the point (q0, n0) in the spa,(:e-frequeimy plane with rela,tiw; n n s w i d t h s ~ and 1 v% in t,he q and ~: directions, respectively. Nanmly,

1 W~ (q, ~:) - 7e

(q--qo )2 a2

- a 2 (K,-~,0)2

.

(7.9.1)

The overlap of any Wigner distribution flmction W~ with ]/VG, i.e.,

(

ff dqd~W~(q,~)WG(q,

h;),

(7.9.2)

is evidently equivalent to the convolution of W~ with the at the origin centered

The Wigner Distribution Function: Optical Production q2 _a2~ 2

2D Gaussian 1 (~-- a2

. In

fact, S/

(

(q,

qo,

-

399

1

dqdtd42~(q, ~)c --

(q--q0)2

~2

- a2 (a;--a;0)2 ,

(7.9.3)

where the dependence on the parameters of the Gabor signal q0,~0 and a is explicitly displayed. Typically tile convolution of ldd~ with another appropriately chosen Wigncr distribution function is reported as the smoothed Wigncr distribution function. The smoothing has been suggested to overcome the problem of the in general non positiveness of the Wigner distribution function (see 6.3.6) as well as that of the cross-terms contributions (see w 6.3.4). In accord with the Moyal formula, the integral in Eq. (7.9.3) is related to the inner product of the signal p(q) with the Gabor function PG" Namely,

1/

( )N~ )~G (q' t~; qo, t%, a) -- ~

2

~(q)Fa (q)dq

.

(7.9.4)

Using the explicit expression of ~G(q), a trivial manipulation of the integral above yields the enlightening relation, (W~}~G ( q , h ; ; q o , t , ; o , a ) -

27rlA~G,~(qo, h:o)l 2,

(7.9.5)

between the smoothed Wigner distribution flmction and the modulus squared of tile cross-ambiguity function A~G,~ of the Gabor signal and the signal under investigation, nleasured at the space and frequency mean values, qo and h:o, of ]~G(q)] 2 and its st)ectrum 193G(t~;)l2. This result conforms with the analysis of .~ 7.2.3. Also, it is a particular case of the general result linking the convolution of two cross-Wigner distribution functions to the product of two cross-ambiguity flmctions, the involved signals in the former being crossly coupled in the latter (see [6.4] and Problem 10). As noted in w 7.2.3, the smoothed Wigner distribution function (7.9.2) can also be interpreted as a windowed Wigner distribution function. Therefore, the phase-space concentration of (I/F~}~G is controlled by the windowing distribution 142G,and hence the position resolution is determined by the rms width of FG and similarly the frequency resolution is deterinined by tile corresponding diffraction angle. On account of relation (7.9.5), the implementation of the smoothed Wigner distribution function can be realized by an ambiguity function processor, and so by any one of the previously described optical setups with the roles of the q and ff coordinates being interchanged. Also, two different signals are written on the two transparencies, which are rotated by 90 ~ one to the other, as previously described. The signal under study is indeed written on one transparency, whilst for the Gaussian amplitude an appropriately rotated Gaussian transmission mask can be used or as well the Gaussian profile output from a laser. Figure 7.26 reproduces from [12.2] the smoothed Wigner distribution runetion of a monofrequency signal of finite duration and a linearly increasing frequency signal. The symbols in these figures are as in Fig. 7.21.

Linear Ray and Wave Optics in Phase Space

400

FI(;I;III'; 7.26. S~<~<~tl~e~lWig~<,r <listril~ti(~ f~('l,i(~ (~f (a) a ('~si~(, sig~ml ~f ti~dte h,~gtl~ m~l (b) a li~mm'ly i~wresi~g fl'<,q~e~('y sig~ml. (llcl)ri~d,e~l wil,l~ l,en~issi<,~ fl'~,~ [12.2].)

7.10 Summary rl'llt; illStall<'e ~f tim ill~livi~lllalizal.i~ll ~f a lllt;t.ll~l t~ C'~nlslrllc'l t ll~' l<)('al fl'e(lll(:ll(:y S])(',(:l,l'llllI ()f a, sigllal llas ])('~(;ll ild,r()i(:<'('-I)Y-l)i('~<'(' illSl)('('l,i()ll ()f l,]l(; siglml wil,]l all ('xl(:nml t )r<)l )(',. ki~(l ()f wi~(l<)w(:(l l"()~u'ie,r tra~sf()l'~, wl~(;~ 1,1~(',sig~al il,s(:lf ()r its ~firr()re(t versi()~ are ~ls(;(l as ~Imlyzi~g wi~(l()w. I~ sl)il,(; ()f t]~('~ si]~il~ril,y ()f 1,11(' ~ml, he~na, l,i(:al (l(;ti~il,i()l~S, 1,1~(; illf()rlllal,i<)ll is (;~('()(1(;<1 i~ 1,1~(; tw() sig~al r(:l)r(;s(:~tati<)ns

T h e Wigx~er (listril)~ll, i~n~ a.~(l m~d)ig~lil, y fl~c:l,i~)~s lmv~; l)~;~,~ s~',~'~ l,~) t)e rela,ted t)y a Fom'ier l,ra.nsfor~ to t,l~e ~n~tua,1 intensity f~l~(:t,i()]~, l,lnls i~d,ro(hlcing their (',o~cet)t,~al a,n(1 a,i)i)li(:ativ(; I)()tentia.1 int() l,l~e i)arl, ial (:()t~(;r(;~(:e theory. Finally, the ()t)ti(:al a,i'(:t~ite(:t~lres i'ea.lized t() i~t)l(;I~(;~t tl~(; "~m,thematics" of the W i g n e r integral (an(l, l)y slight modificatio~s, of tl~e anfl)iguity function integra,1 as well) have t)ee~ (tescril)ed, restrictedly to 1D signals. In the next chat)ter we will res()rt to the original qua,ntllln n~e(:hanica,1 context, giving a,ccomit of th(; fornm,1 con(:et)t of the W i g n e r (tistriln]tion function on the basis of definite r e q u i r e m e n t s formula,ted in the prot)er t e r m i n o l o g y of q u a n t u m mechanics. This comt)letes the q u a n t u m mechanica,1 view of the W i g n e r r e p r e s e n t a t i o n an(t evidences the peculiar role of (classica,1) optics comprising the instances of b o t h q u a n t u m mechanics and signa,1 theory. Problems 1. (a) Verify that the sliding-window Fourier transform is invariant with respect to

The Wigner Distribution Function: Optical Production

401

shifts of the signal in space (up to a modulation) and frequency; nmnely,

qp(q) ~ qo(q)

if if

qD(q--qo), ~(q)e i~~

--+

then then

~(q,n;w) ~(q,n;w)

,

a~ (q - q,, , n; w) e ..... %

> ~r.,(q,~-n,,;w).

C o m p a r e with the corresponding relations for the Wigner distribution and the ambiguity function. (b) Show that the following representation of the signal,

~(q)_

1

~

~ q' ~v~(,

n' ;w)[8(q

-- qt)c iK''q]dq'dn',

in terms of the relevant windowed s p e c t r u m is actuable for any function s(q) such t h a t f s(q)w* (q)dq = 1. This representation generalizes (7.2.3), assigning coinplementary roles to the function w(q) as "analysis" window, and to s(q) as "synthesis" window. As a n a t u r a l choice, s(q) is taken as w(q), thus leading to (7.2.3). (c) Prove t h a t the condition for the representability of any function defined over the (q, n) plane as a sliding-window s p e c t r u m of a signal qp(q) writes as a~(q, n; w) -- 1 / / d q ' d n ' a ~ ( q ' ,

dq"w* ( q " - q)w(q" - q')e -iq''('~-'~'')

n'; w) /

for any functioi, w(q) such t h a t j ' w 2(q)dq- 1. 2. Prove t h a t tile sliding-window Fourier transforin a,.h~,.,,(q, n; W~) of the chirp (7.2.9) with a unit-energy rectangular window can also be written as (x) achi,.l, (q, /~;; VVI{) .

4

2/ ~----~w C

~(/~q2--2qK)~-~ (2 ....1q-l)

//r~(~f~ )2m,+ 1 ( V -~Ww H 2 r n ( ~:--(~q )

m=O

where H2 .... is tile Herinite polynomial of degree 2rn [13]. 3. Prove t h a t (a) if the kernel qJ(q,u) does not depend on the space and freq~mncy variables, shifts of tile signal in space or frequency result in corresponding shifts of the associated distribution, i.e., ~(q)

, ~ ( q - %)

yields

C ~ ( q n; g2) '

, C ~ ( q - % n; ~), '

qp(q)

> ~ ( q ) e i'q'q

yields

Cv~(q, n; kI/)

, C ~ ( q , n -- /%o "' II/) '

if ~ ( r / , v ) i s

independent of q, if qJ(rl,v ) is of K,, independent

and (b) if the kernel ~(q, u) is signal-independent and depends on its arguments t h r o u g h tile product" ~(q, u) -- vq, tile corresponding signal representation is scale invariant, and hence with ~(q) 4. Verify that

,

v/-aqp(aq), a > 0

one has

C~(q, n; ~)

>C~(aq, ~; q2).

(a) the Moyal relation for the ambiguity function writes in the form / g)*~(q)qP2(q)dq

and (b) shifts of the signal in space and frequency result in linear phase factors in the ambiguity flmction, i.e., if

~b(q) - qp(q - q0),

then

Me (q, ~) -- e -'~q~ A~ (q, ~);

if

r

then

Ar

- ~(n-

no),

-

e~q'~~

Linear Ray and Wave Optics in Phase Space

402

5. Prove t h a t tile a m b i g u i t y flmction of (a) tile sum of two rectangle-signals of tile same wi(tth 2Q centere(t on the positions q~ and q2 writes as

A(~, ~.) - ~-'~'~,,, {~A. (~, ~,) ~ o ~ ( . ~ ) where q,,, = (q, + % ) / 2 m~(l q,, = q~

-

+ A . (~ + q,,, ;;) + ~4,~ ( ~ - q,,, ~)},

q2; all(l (b) a (livergi~g spherical wave isstfi~g from a :t

2

slit of wi(lth 2Q, i.e., w(q) - r('ct(~)c/;=- ' ' , is

A(,~, ;4 - A,, (,~, ~- -/~,~). 6. (@ Sl~()w t,lmt l,l~(; rtq)r(,st,~t,at,iv(, C ; ( q , ~ ; q~) ()f tim .i()i,~t (list, rii)~ti()~s (7.3.3) ('al~ t)e ('xt)r('ss(:(l iza l,('r~s ()f tit(' Wig~('r (iistril~ti(~a fiu~('ti()~ I)y a 21) ('(~v(~lttti(~t, ilk(' C:(q, h'; ~P) -- ~

dq'dh"'.l)(q

witl~

4,(,1, ,,:) "l']lis lllCallS tlla(,

il, l l y

i

/y

-

q',h"- h")I4;:(q',~'),

d'qduOd(u,

lll(:llll)(,r ()f (~()]IU.ll'S ('l,k~s

(7.3.3)

q)c '('''t

,,,)

Call l)('. ()l)taille(l l)y a lillear t r m l s f o r -

liial.i()li ()f tlic Wiglier clist.ril)tltioli filiwt, ioll ()f t.lie sig~al at lm,~l, witli ,tit "iilli)tllse rest)ollse" ,I,(q, ~) t,lmt is rclat.('(l t,(, tim kt~'rtml q~(u, q) i,y a, (i(,ttl)h'. Fr t.ra;~sfi,rtt;. (b) Pr(wc also t.lic alt,crliativc cxl)r(:ssi()l| C ; ( q , h . qJ)

-

~

,hfl,,q,(,.,, ,i), ,'~'''' "").A;(~l,,:),

l i l l k i l l g l, lle gcllcric C,p(q, h'; * ) l,t) tlm m l d ) i g l l i t y fllll(:l,i()ll .,4~ l, llr()llgll it l)rt)(lll('t all(l a (lol]l)le F o l u ' i c r - t r a x l s f o n l l relatiozl. (c) As ml ()l)vi()lls collscqlmlw, e, w ; r i f y tllat, (l('fillillg tile d u a l of

C:(q, h:; qJ) by the s a m e (lo~d)lt~ Fo,~rier-likr relatioii liiiki~lg W ml(l ,4, i.e.,

// if the kenml ~P(~/,*I) is Imsiti(n~, fr('qll('.llC.V all(l siglia] ill(lept'.ll(l('llt, wo (4)t,ai~l fiw 79; t]l(~ nndtit)li(:ative fonl~

v:(,i, ~; , ) - ,(;~, ,~)A:(,i, ~-). This is a key relation for ol)ti~lizi~lg tile s~l()othing of tim Wig~mr (listril)t~tiozl fi~zwtion aime(t at sl~pt)ressi~g tile ~H~(tesire(l (:ross-teri~ Colitributiolis. 7. Prove that, if tim sig~ml is r(,strict(;(l to the space i~d,(;rwd ( - Q , Q), tim St~l)l)ort of tile ambiguity fimction extell(ls tlir()~tgti the ~1 (limension from - 2 Q to 2Q, all(l s() is (louble of t h a t of the sig~ml itself. A si~nilar resl~lt tiohts in the freq~m~l(:y (tomaiii, tile h'-extension of r ~,) being double of t h a t of the sig~ml spectrmn. 8. Deduce the transfer laws for the Wiglier distributioll all(l the mnl)igtfity fi~wtion u n d e r free-propagation into the Frm~diofer regio~l exploiting relatioxis (7.5.27), lixiki~lg YV, A and J, and the transformatio~l relation (7.5.26) of J for Fraunhofer diffraction. 9. As alternatives to the i n t e r p r e t a t i o n s (7.5.4),(7.5.6) and (7.5.7)), verify by a simple lnanipldation, t h a t the W i g n e r and ambiguity integrals can as well be considered as resulting froln a process of multiplicatio~l, and a process of convolution in the case of tile Wigner (listribution and correlation in the case of the a m b i g u i t y fimction. 10. Prove t h a t the cross-correlation of two cross-Wigner distribution fimctions equals the pro(hlct of two cross-ambiguity fimctions, the involved signals in the former being correspondingly coupled in the latter as aq ag V V . l , . 2 (q' - q,

- t~)l/V~,l,:2 (q, ec') -- 27rA.~,:z (q, ec)A
The Wigner Distribution Function: Optical Production

403

References [1]

N.G. De Bruijn, "Uncertainty principles in Fourier analysis", in Inequalities, O.Shisha (ed.) (Academic Press, New York, 1967), pp. 57-71.

[2]

Specific considerations oil the windowed Fourier spectrum and the spectrogram within the general context of the joint (space-frequency or time-frequency) signal representations can be found in [4.3]-[4.8]. In particular, in [4.7], ch. 7, pp. 93-112, and in [4.8], ch. 3, pp. 63-65, the windowed Fourier transform is discussed from the viewpoint of the time-frequency and space-frequency signal representations, respectively. The space-frequency view of the sliding-window Fourier transform being interpreted as a local frequency spectrum is elaborated in detail in [2.3]. Several examples of windows extensively used in Fourier spectrum estimation related problems are illustrated in detail in [2.2], [2.3] and in [6.3], ch. 7, pp. 234243. We also suggest reading [10.16], oh. 4, with a. comparative description of four representative window functions being specifically addressed on pp. 160-163.

[2. 1] [2.2] [2.3] [2.4] [2.5] [3]

J.B. Allen and L.R. Rabiner, "A unified approach to short-time Fourier analysis and synthesis", Proc. IEEE 65, 1558-1564 (1977). F.J. Harris, "On the use of windows for harmonic analysis with the discrete Fourier transform", Proc. IEEE 66, 51-83 (1978). M.J. Bastiaans, "Signal description by means of a local freqllency spectrum", in Transformations in Optical Signal Processing, W.T. Rhodes, J.R. Ficnup and B.E.A. Salch (eds.) SPIE 373, 49-62 (1981). S.H. Nawab and T.F. Quaticri, "Short-time Fourier transform", in Advanccd Topics in Signal Processing, J.S. Lira and A.V. Oppenheim (cds.) (Prentice Hall, Englewood Clifl~, N J, 1988), pp. 289-337. J. Jeong and W.J. Williams, "Mechanism of the cross-terms in spectrograms", IEEE Trans. Acoust., Speech, Signal Process. 40, 2608-2613 (1992).

The original papers by Gabor are quoted in [19] of ch. 4. For accurate investigations of Gabor's signal expansion in relation with the determination of the a lmlysis function for a given synthesis function as well a.s the application of Gabor's concepts to discrete time signals and to the case of oversampling we suggest the papers below. Discussions about Gabor's signal expansion are also in [2.3], [4.5], [4.7], oh. 13, pp. 186-187, and in [4.8], ch. 3, pp. 65-67.

[3. 1] [3.21

[3.3] [a.4]

M.J. Bastiaans, "Gabor's expansion of a signal into Gaussian elementary signals", Proc. IEEE 68, 538-539 (1980). M.J. Bastiaans, "A sampling theorem for the complex spectrogram, and Gabor's expansion of a signal in Gaussian elementary signals", Opt. Eng. 20, 594-598 (1981). M.J. Bastiaans, "Gabor's signal expansion and degrees of freedom of a signal", Optica Acta 29, 1223-1229 (1982). M.J. Bastiaans, "Gabor's signal expansion applied to partially coherent light", Opt. Comm. 86, 14-18 (1991).

Linear Ray and Wave Optics in Phase Space

404

[3.5] M.J. Bastiaans, "Gabor's signal expansion and Zak transform", Appl. Opt. 33, 5241-5255 (1994). [3.6] M.J. Bastiaans and M.C.W. Geilen, "Oil the discrete Gabor transform aIl(t the discrete Zak transfornf', Signal Process. 49, 151-166 (1996). [4]

Tl~e general ext)rcssi(m (7.3.3) f()r t l~e.]t)i~t, sigiml rel)rese~dati()lLS was firsly given ill [4.1] wittlill a (lllaldlllil llm(:lmlli(:al (:()lltext. Tile l'elati()lls l)(;tw(;('ll S()lllO st)o(:iti(: rt;l)r('scldativ(;,s ()f (7.3.3) wiIll sigllal-ill(l('l)('~ll(h;lli k(;rll(;ls arc ilhLstrate(1 at (liv('rs(' (h'l)llls ill lllally l)al)('rs all(l l('xll)()()ks; a fi;w ()f lll(~lll at(' (pl()l(;(1 l)(;l()w. Ill l)arti('lllar w(' sig~aliz(; Ill(; ('()lllI)rt;ll(;llsiv<; issll<; [4.!)1. [,1.1] L. Collcll, "(lellcralize(l I)ll~se space (listril)~t, iol~ fi~('tio~s", ,I. Nlatl~. l)l~ys. 7, 781-

7s(~ (~)(~(~). [4.2]

L. (~()lm~, "Q~m~tizati()~ l)r()l)lc~ ;u~(l vm'iati()~ml I)ri~cil)le i~ tl~e 1)lm.,~(~'-st)a,('c fori~nilal, i()~ ()f (l~m~t,~ ~wclm~fi('s", .I. Nlatl~. l)l~ys. 17, 1863-1866 (l!}7(i). [,l.:l] T.C.A.NI. ('laas(,~ a,~(l W.F.G. r~l(,ckle~d)r~h~k(,r, "'I'll(, Wig~(,r (iist.riln,.i()~- a tool fi)r ti~e-fl'('q~m~('y sig~a] mmlysis. Ill: rclati()~s witl~ ()tl~er ti~(~fl'('(l~W~cy sig~al trm~sf()r~mti()~s", l)l~ilii)s .l. lies. 35, 372-389 (1.()8()). [.|.4] L. C()lm~, "Ti~(;-fr('q,m~wy (listril),~ti(),~s - A r(,vi(,w", I)r()( ". IEEE 77, .()41-.()81

(~.()s.()).

[.1.5] I~.I)..la.('()l)s()~ ;u~(i |I. W('('l~sl(w, ".I()i~t Sl)atial/sl)atia, l-fr(',l~('~wy r('l)r('se~(,ati()~", Sig~m.l l)r()('ess. 14, 37-68 (1.()88). [1.(;] F. Hlawats('l~ m~(i (;.F. i~()~(lr(,a.~x-l~a.rtt,ls, "l.i~(,a.r a~(l (i~a(lrati(" l.i~('-fl'eq~w~('y sig~al rel)r(,s(,,fl.al.i()~s", IEI'~E Sig~ml l)r()('(,ss. Nlag., 21-27, April 1.()!}2. [4.7] I~. (',()l~(,~, 7'imc-],}'cqucn.cy Analy.,~i.,~ (l)r('~ti('e-llall, l'h~glew()()(l (',lifts, 1!)95). [4.S] ll.M. ()zaktas, Z. Zal('vsky m~(l NI. Alper K~tay, 7'tu' l"rm'tio'n, al l"o'urier T~unsfozvn with, Application, s in Optics and Signal l)ro('e.ssi'n,g (.l()l,~ Wil( W & So~s Ltd., ('~]~ic]~csl,cr, 2()()1 ). [,1.9] 7~im,c-t,'~quc'n, cy Signal A'n,alysis and l'roc~'s.si'n,g. A Co'mp'n'tu'nsi'~c l{~'fcrc/ncc. I3. [5]

Tl~e literat~r(' ()~ 11~(' .i()i,~ sig~al r('I)r('se~tati()~s is ('xtr(,~('ly ri('l,, ('()~I)risi~g as well Sl)(~('iti(' i~v(~stigali()~s regar(li~g, f()r i~sta~('(', (t~mlity relali()~-I)ase(t (:lassifi(:ati()~, ()l)ti~izati()n l)r()(:e(lure f()r sig~ml sy~tl~csis ()r kcr~el (l(;sig,~, ai~e(t at re(tu(:i~g tim (:r()ss-ter~s ('()~Iril)~ti(n~s a~(l at (:()~stI'~(:til~ I)()sitivc (listrit)~tions. Also, elat)()rati()~ ()f new k(~r~els t() fillfil st)c(:itic rc(t~fircn~(~ds, i~vesligati()ns of the rclati()n I)(~twc(m (liff(~r(~nt siglml r(~t)r(~s(~ntati()ns, in(tivi(l~mlizati()l~ ()f general ot)crator t)r()(:e(turc t() g(merat(~ .j()il~t signal reI)reseI~tations ()f a r b i t r a r y variables, an(t so (m. Here we give a sh()rt list ()f titles (',)nc(~rn(~d with the a l)()ve instances. A comt)rch(msiv(~ a(:(:om~t is ()ffcre(l l)y [4.9]. [5.1] H. Choi and W.J. Williams, "hnproved tiine-frequency represeI~tation of multicomponent sigi~al ~lsing exponential kernels", IEEE Trans. on Signal Process. 37, 862-871 (1989). [5.2] Y. Zhao, L.E. Atlas ai~(l R.J. Marks, II, "The use of cone-shape kernels for generMize(t time-frequency representation of nonstationary signals", IEEE Trans. on Signal Process. 38, 1084-1091 (1990). [5.3] F. Hlawatsch, "Duality and classification of bilinear time-frequency signal representations", IEEE Trans. on Signal Process. 39, 1564-1574 (1991). [5.4] F. Hlawatsch and W. Krattenthaler, "Bilinear signal synthesis", IEEE Trans. on Signal Process. 40, 352-363 (1992).

The Wigner Distribution Function: Optical Production

405

[5.5] J. Jeong and W.J. Williams, "Kernel design for reduced interference distributions", IEEE Trans. on Signal Process. 40, 401-412 (1992). [5.6] A. Kumar, D.R. Fuhrmann, M. Frazier and B.D. Jawerth, "A new transform for time-frequency analysis", IEEE Trans. on Signal Process. 40, 1697-1707 (1992). [5.8] P.J. Loughlin, J. W. Pitton and L.E. Atlas, "Bilinear time-frequency representations: new insights and properties", IEEE Trans. on Signal Process. 41,750-766 (1993). [5.9] R.G. Baraniuk and D.L. Jones, "A signal-dependent time-frequency representation: optimal kernel design", IEEE Trans. on Signal Process. 41, 1589-1602 (1993). [5.10] F. Hlawatsch and R.L. Urbanke, "Bilinear time-frequency representations of signals: the shift-scale invariant class", IEEE Trans. on Signal Process. 42, 357-366 (1994). [5.11] P.J. Loughlin, J. W. Pitton and L.E. Atlas, "Construction of positive time-frequency distributions", IEEE Trans. on Signal Process. 42, 2697-2705 (1994). [5.12] R.G. Baraniuk and L. Cohen, "On joint distribution of arbitrary variables", IEEE Signal Lett. 2, 10-12 (1995). [5.13] R.G. Baraniuk and D.L. Jones, "Unitary equivalence: a new twist on signal processing", IEEE Trans. Acoust., Speech, Signal Proccss. 43, 2269-2282 (1995). [5.15] F. Hlawatsch, T.G. Manickam, R.L. Urbanke and W. Jones, "Smoothed pseudoWigner distribution, Choi-Williains distribution and cone-kernel representation: ambiguity-domain analysis and experimental comparison", Signal Process. 43, 149168 (1995). [5.16] R.G. Shenoy and T.W. Parks, "Wide-band ambiguity flmctions and affine Wigner distributions", Signal Process. 41,339-363 (1995). [5.17] A.H. Costa and G.F. Boudreaux-Bartels, "Design of time-frequency representations using a multiform, tiltable exponential kernel", IEEE Trans. on Signal Process. 43, 2283-2301 (1995). [5.18] L. Cohen, "A general approach for obtaining joint representations in signal analysisPart I: characteristic flmction operator method", IEEE Trans. on Signal Process. 44, 1080-1090 (1996); Part II: general class, mean and local values, and bandwidth", ibid., 1091-1098. [5.20] R.G. Baraniuk, "Joint distributions of arbitrary variables made easy", Multdim. Sys. and Signal Process. 9, 341-348 (1998).

[6]

Many titles offer careful and thorough descriptions of the properties and the applicative potential of the ambiguity fimction within the context of the signal analysis. A few of t h e m are quoted here. T h e reader is also addressed to [4.3]-[4.6], as well as to [4.7], where tile ambiguity function is considered ill relation with tile characteristic function of the Wigner distribution function, and to [4.8], ch. 3, pp. 73-76. The Fourier transform duality of the Wigner distribution and ambiguity function is discussed in detail in [6.4]. Various aspects of the ambiguity function and its generalizations are as well investigated in some of refs. in [5]. [6.1] P . M . Woodward, Probability and Information Theory with Applications to Radar (Pergamon, New York, 1953). [6.2] J.R. Klauder, "The design of radar signals having both high range resolution and high velocity resolution", Bell Syst. Tech. J. 39, 809-820 (1960). [6.3] A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), ch. 8, pp. 284-295. [6.4] H.II. Szu and J.A. Blodgett, "Wigner distribution and ambiguity function", in Optics in Four Dimensions-1980, AIP Conf. Proc. 65, M.A. Machado and L.M. Narducci (eds.) (AIP, New York, 1981), pp. 355-381. [6.5] H.H. Szu and H.J. Caulfield, "The mutual time-frequency content of two signals", Proc. IEEE 72, 902-908 (1984).

Linear Ray and Wave Optics in Phase 5pace

406 [7]

Many a pt)lications of tile ambiguity flmctions to optics have been devised starting fl'oIn the seminal pat)er by Papoulis [7.1]. Below only tile titles specifically (:oIl(:eI'~m(1 with tim presentation in ~w 7.5.2-7.5.3 are ret)()rte(t. [7.1] A. I)apoulis, "Ambiguity flmction in Follricr optics", ,l. Opt. Soc. Ain. 64, 779-788 (1.()74). [7.2] K. l)lltta aIl(l .I.W. Goo(lInan, "RcconstructioIls of ilnagcs of partially coherent objects frolIl smlqflcs ()f lIllltlml illt,cllsity", 3. Opt. Soc. AIn. 67, 7.q(i-8():~ (1977). [7.3] .].-]). (llligay, "Tim a,l~llfiglfity flnlcl, i()ll ill (iiI[racl,i()li ml(l is()I)lalmtic iIim,giilg by partially ('()tmr(ult I)(;mlls", Opt. (',()lIllll. 26, 13(;-1:~8 (1978). [7.4] K.-ll. l~r('~mr a~,! .I. ().jt,(la,-(~a.,~ta.i,t,(la., "Ati~l,igt,ity ftt~,('t.i,~t, a.,~(l Wigt,t,r (listribl~t,i(n~ al)pli(~'(l t() partially ('()lmre]d, i~mgcry," Opti('a A('ta 31, 213-223 (1.()84). [7.5] K.-II. l~r('.~,mr, A.W. l,()ln~m~ m~(l J. ().j('(la-(',a.,4.afit~'(la, "Tlw a~fi,ig~fit,y flH~cl,io~ as a l)()lar (lisl)lay ()f l,l~(' OTF", Opt. C ( ) ~ . 44, 323-32(i (1.()83).

[~,] A~()~g l l~(' wi(h' lil('ra! ~r(' ('()~c(;r~('(l wil l~ l l~(' ()1)1i('al i)r()(l~('! i()~ ()f l l~(' ax~fi)ig~ily fi~('li()~ w(, r('l)()rl a fi'w r(;I)r(;s('~tativ(; l)al)eI'S, a(l(lr(;ssi~ag l l~(; r('a(h;r a ls() t() [(~.4] a~(l [12.4] fi)r a ('()~i)aralivc a~alysis ()f tim a~l)ig~ily fil~('ti(n~ i~,~l)l(;~(;Id, at i()~ i~ r(;lati()~ r('.Sl)(;('~iv(;ly will~ l l~(; Wig~(;r (lislril)~li()~ fl~('li()~ a~(l I1~(; C()I~('~ g(:~(,i'aliz(,(l Sl)a('(,-fi'(,(l~(,~('y (list ril)~! i()~ fi~('! i(n~s. [s. l]

[s.2]

[s.:~] -[s.4] [8.5] [8.6] [8.7] [9]

R.A.K. Sai(! m~(! I).(;. C()()p('r, "Cr<)ssl)at]~ r(;al-ti~(; ()pti('al <'()rr(:iat<)r m~(I a~nbig~lity fi~wl, i()~ l)r()('css()r", ])r()('. IEE 120, 423-428 (l!]73). l{..I, lklarks [l, .I.l". Walkout> a~(l 'l'.l". Krih,, "A~fl)ig~ity fi~('l,i()~a (lisl)lay: ai~ i~l)r()v('(l ('()l~cr(;,~t l)r(w('ss()r", AI)t)l. ()l)t. 16, 746-75() (1(.)77). "A(l(h',~(l~", 16, 1777 (1.()77). l). (~a.sa.sc~t m~(l I~.V.K. Vi.jaya K ~ m r , "Ol)ti(:al i~age l)lmm (:()rrclal,()r for anti)ig~fity s~rfa(:c ('()~l)~tati(ns Al)t)l. ()l)t. 18, 1(;73-1{i7~1 (1.979). 1). l)salt.is ml(l I). (',asas('lit, "Sl)a('(;-varialit alill)ig~fil.y fl~ll('t,i()li t)r()('(.'ss()r", AI)pl. ()pt. 18, 18,6.()-1874 (1.()7.()). l{. ,J. Marks II ml(l M.W. l|all, "Aiifl)ig~ity f~nicti()li (lisplay l~siiig a siiiglc ID input", Appl. ()pt. 18, 2539-254() (1979). T.C. Lcc, ,J. l~(4)liolz, I).N. 'I,t~m~ra ;ul~l .I. Li~),iq,fist, "Alnt)ig~fity pro('(,ssiiig by joint F(),~ri(:r tra~lsfi)r~n ]lolograpliy", Appl. Opt. 19, 8,.()5-8.().9 (1.98()). I).N. Tmmira, .J..J. I{ct)holz and T.C. Lee, "Axnbig~fity-fi~xiction gc~lcration using I)a.ssivc optical r-stfift tcctmiqlm", Opt. Lctt. 5, 401-403 (198()).

Basi(: titles ()~ tl~(; l)aI'tial ('()her(m(:(; tl~e()I'y are s~ggeste(1 i~ [11] ()f (:l~. 6.

[10] The sul).je(:t ()f ()I)tical pr()cessing corot)rises a t)lethora of techniques, differing from 1)()th the ot)erating t)rinciples amt the processing t)rot)lems. Her(; a selection of titles is given i~ I'elati()i~ to b o t h coherent [?]- [~0.10] and incoherent [10.8]-[10.12] illuini~mti()n, (tisregar(ling the specificity of the described applications. Some basic textt)()()ks are also quoted; the application of the fractional Fourier transformrelate(1 n~eth()ds t()()ptical signal processing is accurately e x a m i n a t e d in [4.8]. For an a(:(:ount of "digital optics" being the n a t u r a l b y - p r o d u c t of the theoretical and experimental work on optical c o m p u t i n g see [10.19]-[10.22]. [10.1] K.G. Birch, "Spatial filtering in optical-data processing", Rep. Prog. Phys. 35, 1265-1314 (1972). [10.2] A. VanderLugt, "Coherent Optical processing", Proc. IEEE 62, 1300-1319 (1974).

The Wigner Distribution Function: Optical Production

407

[10.3] W.T. Rhodes and J. M. Florence, "Frequency-variant optical signal analysis", Appl. Opt. 15, 3073-3079 (1976). [10.4] J.W.Goodman, "Operations achievable with coherent optical information processing systems", Proc. IEEE 65, 29-38 (1977). [10.5] R.J. Marks II, J.F. Walkup, M.O. Hagler and T.F. Krile, "Space-variant processing of 1D signals", Appl.Opt. 16, 739-745 (1977). [10.6] R.J. Marks II, and S.V. Bell, "Astigmatic coherent processor analysis", Opt. Eng. 17, 167-169 (1978). [10.7] J.F. Walkup, "Space-variant coherent optical processing", Opt. Eng. 19, 339-346 (1980). [10.8] W.T. Rhodes, "Incoherent spatial filtering", Opt. Eng. 19, 323-330 (1980). [10.9] W.T. Rhodes and A.A. Sawchuk, "Incoherent optical processing", in Optical Information Processing Fundamentals, S.H. Lee (ed.) (Springer, New York, 1981), pp. 69-110. [10.10] H. Bartelt and A.W. Lohmann, "Optical processing of one-dimensional signals", Opt. Comm. 42, 87-91 (1982). [10.11] T.E. Bell, "Optical computing: a field in flux", IEEE Spectrum, 23, no. 8, 34-57 (1986). [10.12] I. Glaser, "Information processing with spatially incoherent light", in Progress in Optics, Vol. XXV, E. Wolf (ed.) (North Holland, Amsterdam, 1987), ch. 5, pp. 389-509. [10.13] J.W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968). [10.14] G.L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977). [10.15] J.D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley & Sons, New York, 1978). [10.16] A. VanderLugt, Optical Signal Processing (John Wiley & Sons, New York, 1992). [10.17] J. Shamir, Optical Systems and Processes (SPIE, Bellingham, USA, 1999). [10.18] Digital Optical Computing, special issue in Opt. Eng. 25, 1-116 (1986). [10.19] A.W. Lohmann, "What classical optics can do for the digital optical computer", Appl. Opt. 25, 1543-1549 (1986). [i0.20] K.-H. Brenner, "Digital optical computing", Appl. Phys. B 46, 111-120 (1988). [10.21] N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A.W. Lohmann, D.A.B. Miller, M. Murdocca, M.E. Prise and T. Sizer, "Digital optics", Proc. IEEE 77, 1954-1968 (1989). [10.22] H.M. Ozaktas and D.A.B. Miller, "Digital Fourier optics", Appl. Opt. 35, 12121219 (1996). [11] For an exhaustive analysis of the bilinear transformations in the light of their formal and applicative properties in the optical context as well as their relation to the Wigner distribution function and the ambiguity function, we suggest [11.1] B.E.A. Saleh, "Bilinear processing of 1D signals by use of linear 2D coherent optical processors", Appl. Opt. 17, 3408-3411 (1978). [11.2] B.E.A. Saleh, "Optical bilinear transformations: general properties", Optica Acta 26, 777-799 (1979). [11.3] B.E.A. Saleh, "Linearity of quasi-homogeneous partially coherent imagery", J. Opt. Soc. Am. 72, 1095-1096 (1982). [11.4] B.E.A. Saleh, "Bilinear transformations in optical signal-processing", Proc. Soc. Photo-Opt. Inst. 373, 125-134 (1984).

408

Linear Ray and Wave Optics in Phase Space

[11.5] J. Ojeda-Castaiieda and E.E. Sicre, "Bilinear optical systems. The Wigner distritmtion flmction and ambiguity flmction representations", Optica Acta 31, 255-260 (1984). [12] T h e oI)tical generation of the Wigner distrit)ution fllncti(m is doclunentc(t in a very large varie.ty of pat)ers in relatioil with truly ~)t)ti(:al as well as not t)rot)erly (~t)ti(:al Im~l)lelns. Also (llloted are SOIllC reI)resexdaliw'~ l)at)ers (:()ll(:(;rn(;(1 with tlle Wigller ~lisI)lay ~)f 2D siglials, wlii('li t~)I)i(: l~eilig ll~t t~ll(:tl(;~l ~11 ill tile text. S~,~, als~ tile revi~;w 1)al)ers [12.15], wliere a synthetic: ~l~'.s~:ril)li{)li {)f l llc' setlq)s f()I"

[12.1] M..I. 13astiam~s, "Wiggler ~listril)~fl,io~ fi~l~ction {lisplay: a m~l)ple~eld, 1,(~aml)ig~fity fi~cl.i~ {lisplay ~si~g a si~@e 1D i~I)~l.", Appl. ()l)l,. 19, 192 (198(I). [12.2] l].(), llm'l,elt, K.-II. llrm~er m~l A.W. l,~)l~m~l~, "'l'lm Wiggler <listriln~tio~ fin~ctio~ mtr its optical l)ro-
8 1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function

8.1

Introduction

In Chapter 6 we ha,ve introduced the Wigner distribution function within a, purely quantum nm(:tlanical context as an effe(:tive and appealing tool allowing for a, formulation of (luantunl mechanics in terms of s()me kind of phase spa.co fornmla,tion similar to tha,t of classical mccha, ni('s. Using the Wigncr method, it is possible to cal(:ulatc a class of quantun~ mcchani('a] averages in the sa,mc ma,nner a,s the (:la,ssi(:a,1 t)hase st)a,(:c distribution is use(t to ca,lculatc (:lassi(:al averages. As one merely multiplies the distribution flm(:tion and the classical phase spa,ce flm(:tion, whose average is desired, and then integrates over the entire phase spa, ce, likewise one merely nmltiplics the Wigner distribution function and the pha,se space flnlction ('orrest)onding (by a precisely defined corresponden(:c rule) to the qua, ntum mechanical observable whose expectation value is desired, and then integrates over the entire pha, sc space [1]. We have not dwelled in detail on the difference between the classical (symplectic) phase-space and the Wigncr phase-space, which, as noted in w 6.4.1, has its own structure according to the inherent invariance la,ws. We have simply dealt with the Wigner phase-space as the mere Cartesian product of the eigenvalues q and p (or t~) of the coordinate and momentum (or spatial frequency) operators. In a sense, therefore, we have considered the Wigner representation like an extension of quantum mechanics into the "classical" phase space, focussing on the operative advantages of such a representation rather than on the structure of the manifold where it is defined (which is indeed object of interesting investigations). Naturally, whilst a point representation in phase space is allowed by the classical formalism, and so the motion can be viewed as a simple substitution of one point for another, the concept of moving point in phase space is replaced within the quantum mechanical formalism by a "quasi-probability" distribution which changes with time through non local

410

Linear Ray and Wave Optics in Phase Space

tra,nsfornmtions. After i m p o r t i n g the fl)rmal concept of the W i g n e r distribllti(m flmction into optics [2], our a t t e n t i o n has accordingly been h)('usse(t on the Wigner representation of optical signa,ls, which yields a. sigim.1 (tes('rit)tioll displaying t)otll the spatial and frcqlu;ncy fca,tln'cs. Finally, in Chat)tot 7 wc haw; fl'a,mcd the Wigncr (tistrilnlti(m fllIl(:ti()n withill the g(ul(:ral (:(nlt(;xt ()f siglml theory in r(;lati(m to the isslu'~ ()f in(livi(tlm,lizil~g l,he "nnlsi(:al s(:(n'e" ()f a signal. hi ()I'(h;r t() (:()lllt)l(;t,(; trat:illg tll{'~ (:()ll(:r lm,tll t() tll(; (h;filliti(nl ()f tllc Wiggler intcgra.1, wc I'(',1,111"11 l,() tllc (tlla,lll, lllll lll(;(:tmllit:al (:()lll,(',Xl, il.ll(l SllOW ill S(;(',t. 8.2 that t,hc Wigllt;r (lisl,ril)ltl,i(nl fl~n(:t,i()~ ~ml,~u'a.lly rcv(:rl,s i~l,()l, hc t)asi(: forln wh(m re(luir(;(l l,() sal,isfy a (',(;rl,a,in set ()f well (h:finc(l (X)l~(lil,it)l~S sllgg(~st, e(I |)y tll(; (h'~sir(;(1 l'()h: it, st~()~l(l l)lay as a .j()i~t (listritn~ti(n~ ()f t,l~(: (:()~.j~lgat(; Val'ia,1)lcs q a,~(t p as w(;ll as l)y 1,1~(: (l(;sil'al)h; rt:lal,i()l~ il, ~igl~l, l~av(; t,() th(; (:la,ssi(:a,1 t)lm,se-stm(:e (listril)~ti(n~. '1'1~(; rea,(h;r will easily rt;t'()g~iz('~ s(n~(; ()f tim t)r()t)('rti(;s (h's(:ril)('(l i~ !i 6.3 f()r 1,1~(' ()t)l,i(:a,1 Wig~(:r (lisl,ril),~l,i()~ fin~(:l,i()~ in l,h()s(' (:(n~si(h'r(,(l i~ !i 8.2 ('v(u~ l,l~()~gil~ tl~(, ~()tiwtti(n~s fi)r, a,~ul tlw r(,h;vm~(:('~ ()f, l~('~(:lm,~i(:al i~st('.a,(1 ()f way(; ()l)ti(:al) wl~('r('~ l,l~(; (lis(:~,~si(n~ ()f !i 11.2 is l)la,(:('(1. Tl~(:~, t)a,ra,lh'li~g l,l~(, "slal,i("' vi('wl)()i~t ()f l,l~(; (h's('ril)l,i()~ giv(u~ in Chal)t(;r (i, wt~(;re tl~(; Wig~('r (lisl,ril)~ti()~ fllll(:l,i()ll lla.S ])(;(:11 (:al(:~lat('(l fi)r a, large vari(;ty ()f signals, i~ !i 8.3 w(' a l)l)l'()a('l~ tlw t)r()l)l(:~ ,)f tl~(: "(ly~m~i(:s" ()f th(' Wigl~(:I" (listrit)~l,i()~ fl~(:l,i()~, m~(l s() w(; ilw(;sl,iga.l,('~ l~()w l,l~('~ Wig~(;r (tistrit)~l|,i()~ flu~(:ti(n~ ()f a l,rav(;lli~g sigl~al is ~()(tifi('(l t)y I)r()l)a,ga.l,i()~. Tl~(' ray-()t)ti(:s (h;s(:rit)l,i()n ()f t)a,raxial 1)r()t)agal,i()l~ i~ t,(;n~s ()f sy~ll)h;('l,i(: l~m,l,ri(:(:s is r(;(:()v(;rc(t withi~ l,h(~ Wign(;r t)l~as(;-Sl)a,(:(', f()n~m.lis~, tln~s (;stal)lisl~il~g a,~ i~l,er(;sting similarity with the (;v()l~ti(n~ ()f tlm l)lm,s(;-sI)a(:(', ray-(l(u~sity (listri|)~t,i()~ p(q, p; z) (w 2.2). T h e (:()rr(;sl)()~(hu~(:(; |)(;l,w('e~ l,t~c fieht (lisl,ri|)~ll, i()~, ()r its fl'(~(t~mn(:y st)e(:tr~m~, a,n(t tl~(; Wigll(;r (tistrit)~ti(n~ flll~(:ti()ll is tl~(',~ r(',i~f()r('(;(1. Not only we (:a,n s u b s t i t u t e th(' l)a,rallcl a,n(t int(;rr(;la, te(t sl)a,(:(' a,n(t fl'(;(tuc~(:y signal representa,tions with tlm i~(tir(;(:t tnlt ~u~i(t,m Wign(;r r('t)r('s(u~tati()~, t)~t also we know t~()w to "t)r()t)a,ga,t(" t,l~('~Wign(;r (tistributi(n~ fln~(:ti(n~ thr()~gt~ first-order optical systems, as it rc~m,rka,bly tra,nsforms by a,~ a,flinc tra,~sf()rl~m,tion of its coordinates involving .j~st the tra,nsfer matrices ()f ra,y-ot)ti(:s (Fig. 8.1). As a completion of su(:l~ a "dynami('a,l" investigation, in w 8.5 we derive the transt)ort equation for tl~c Wigner (tistribution flm(:tion, the wavc-ot)tica,1 a,nalog ()f th(; quantum Li()~vilh, (;(tua,ti(m. As ext)ectc(t, we fin(t t h a t f()r quadratic polynoinia,1 g e n e r a t e d systems (and in particular for free-prot)agation systems), the qua,ntum Liouville cqua,tion turns into the classical Liouvillc equation. M()r(;ovcr, some spa, co (w 8.4) is devoted to illustrate the relation between the Wigncr distribution function and the fractional Fourier transform, which is of b o t h conceptual and applicative interest, being basic to the phase-space t o m o g r a p h i c investigation of complex wave fields, be they classical optical

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 411

Tools o f w a v e

optics

9 (p ( q ) , # ( K ) 9

Fresnel

<

Tools o f W i g n e r

:>

9

Integral

9

W

optics

(q,u)

ra y -tra n s fe r m a trix

FIGURE 8.1. The identification of the propagation properties of the Wigner distribution fimction allows for a complete phase-space description of propagating signals.

disturba,nces or q u a n t u m matter-waves.

8.2

From the wave function to the phase space representation

We consider tile one-dimensional motion of a single particle of mass m moving in a potential V(q). Accordingly the relevant phase space is that of the particle position and m o m e n t u m q and p, where both are treated as ordinary real variables instead of Hermitian operators. Our concern is to identify a mapping ~b(q, t) C/Z 2 (R) --~ q3(q, p, t) E s (N2), allowing for a unique (:orresponden(:e of the tile fllnct, ion 9(3(q,p, t) ef the t)ositional and (and time t), we would like to interpret as parti('h', ('oordinate and m o m e n t m n have the 8.2.1

(8.2.1)

particle wave flmction ~/;(q,t) to m o m e n t u m coordinates q and p giving the t)robability that the values q and p at tinw, t.

Identifying q3(q,p, t) as the Wigncr distribution function

The desired probabilistic interpretation of q3(q, p, t) naturally formalizes into a certain set of requirements. First of all, it is desirable that q3 be real. This suggests relating q3(q, p, t) to the wave function ~h(q, t), it represents, through the expectation value in the state ~h of some Hermitian operator depending on ~ and ~. It is therefore reasonable to write down the relation connecting the phase space distribution function q3(q, p, t) to the system wave function '(~(q, t) in tile forin

q3(q, p) - g with the kernel function

K(q, p; x, x')~h* (x)'(J(x')dxdx',

(8.2.2)

K(q, p; x, x') turning to have the basic property /~(q, v;

~, ~')

- K * (q, v;

~', *),

(8.2.3)

in order that q3(q, p) be real. For formal convenience, the time t is not explicitly denoted in (8.2.2).

Linear Ray and Wave Optics in Phase Space

412

Further, it is n a t u r a l to iInt)ose tha, t the phrase space representa,tive g3(q, p) has the same invariances of the wa,ve flm('tion, it corresponds to. Basically we req~fire tha,t g3 be Galilei invariant, i.e. invariant for (tist)lacements in both the t)ositi(maJ an(1 ~n()nmnt~m~ (:oor(tina,t(;s. The inva,ria,n(:e ()f q3 fin' t)()siti(mal (list)la(:cments ()f the wave flmction is (;xI)r(;ss(;(t l)y t,lm (:(n~(titio~ l,lm,t if

,/,(q)

---,

+ Q),

t/,(q

(8.2.2), wr ~,,t,,: tl~a,t

(,(q-+ (2) lms tl~; r

//

,/'( q + (2) ~

|,11(:11 ~ ( q . ])) ---+ q[l(q @ ('~. p).

r,;Im;S,;~m~,tiCn~

(s.2.5)

It(q. p; .,., .,.'),/,* (.,.+ ~2),/,(.,.' + (2 )d.,'d.,".

F,(l~m,ti~g tl~r al,t,v,, i~m;gra.l t,, q3(q + Q, p), wl~ir V(q

(s.2.~)

+ (~.p) - ././"

~,(q

writes

+ (~,, ~,;.,...,.'),,'," (.,),;',(.,.'),~.,.d.,.'.

(s.2.~)

yi,;l(ls t,l~r fifll,,wi~g r,'la,l,i,,~ fi,r 1,1~r kr I , (q. p; .,.,

.,.')

i~la.1, rclat, i ~ l (8.2.7) is s~;,:~l l,,, 1~,: ~;(t~liwd~;~ll, l,,~ t,lic (lilf'r 0

i)

(-~q + ~

(8.2.7)

I , (q - (2, p; ., - (2..," - (2).

,:r

0

+ O.r--7)Ig(q,p;.,',.,")-I}.

(8.2.8)

This lllr tlm.t K, ill ar tr p, (lr ls ()11 tllr (tlmlltiti,:s q, .r aJl(1 :r ~ r thrr tllr (:r nls q 2 anti . r - :/, (~r er through q - :~: anti q - .~". Sillfilarly, l,llt~ illwu'iml(:e t~f g3 witll rest)et:t 1,() a. ~llonlenl, lnl~ (lisl)la.(:e~nent is ext)resse(t t)y tim retluirenm~d, tt~at if .t/,(p) ---+

.t/;(p + P),

then

q3(q, p) ---+ q3(q, p -t- P ) ,

(8.2.9)

wha,tever P t)e. Sin(:(; a, shift ()f the monmntunl ret)re, sentative 't}(p) is equiva,lent to a linear mo(tula, tion of the wave flmction t/'(q) according to

~(p) --+ ~(I) nL P) ~ o n e easily obtains for

l/~(q) --+ "(~(q)e-iPq/h,

(8.2.10)

K(q,p; x , x ~) the relation

K ( q , p; ~, . ' ) - K ( q , p + p; . , / ) ~ - ~ . ( x - x , ) / h .

(8.2.11)

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 413

Since P is arbitrary, setting P = - p we are able to further define the dependence of the kernel on the m o m e n t u m variable in the form

K(q, p; x, z') -- K(q, 0; z, z')c ip(z-~')/h.

(8.2.12)

Also, according to the above considerations, the functional form of the kernel at p = 0 can be written as

K(q, O; z , z') -- K, (q

x+x' 2

, z -

z , ),

(8.2.13)

or K(q, O; z, z') = K 2 (q - z, q - z').

(8.2.14)

To fix mind, in the following we will use the first form so t h a t

K(q, p; x, x')

- - K 1 (q

- ~+'~'2 , x - x')eiP(X-x')/h.

(8.2.15)

Evidently the Galilei invariance of 043(q,p), we have inquired above, corresponds to the space and frequency shift invariance of the optical Wigner distribution function described in ~ 6.3.2. As emphasized in Chapter 6, it is desirable that q3(q,p), when integrated over q and p, yields the correct marginal probability distributions. Thus, integrating over p, we should obtain tile t)rot)er probability distribution for the space coordinate, given by the square modulus of tile wave fllnction:

I,(,(q)l 2

./'q3(q, p)dp -

(8.2.16)

Likewise, integrating over q we should obtain tile t)ro|)a|)ility distribution fi)r the m o m e n t u m variable, given by tile square modulus of the Fourier transform of ~/5: --2-~,

-

.

(8.2.17)

On account of relation (8.2.12), integrating (8.2.2) over p gives

043(q p)dp - 27rh / d z K 1(q - z, 0)I~(,~)

12

(8.2.18)

so that in order to meet condition (8.2.16) the kernel must satisfy the relation K 1

,

0) --

(8.2.19)

1

Similarly, requirement (8.2.17), which explicitly writes as

J" dzdx'

f

dqKl

(q 1 27r h

2 ,x x+.~, dxdxtc

-

x t ) e *p

(x)~(x t) (x-x')/hg,. 9

h (z--z')if),

(X)~)(xt)

(s.2.2o)

Linear Ray and Wave Optics in Phase Space

414

a,lh)ws l~s to establish yet a,n o t h c r rcla, tion for the kernel as ;/?/) /9 d~K . 1 (~., .T --

1

27rh "

(8 "2.21)

However, K(q,p; :r,:r') is still not uniqlu;ly define(t. As a,~ a~hliti()~ml si~q)h: i'eql~(;st ()~ tt~e t)hase sI)a,ce (tistritn~tio~ fln~(:tio~ q3(q, p), l,lmt l~igld, I)(;r~il, a, ~n~i(t~u; (:()rr(;st)(m(h;l~(:(; ()f q3 t()(,, w(; nmy (lema,n(1 t,t~at i~ t,lm~ ('a,se ~f a fl'~;~; lmrti(:le tl~; ('~(t~la,ti()l~ ~f I~()ti()I~ ~f q3 t)~'~~'~(l~al t(~ tl~(; (:la.ssi(:al (n~r fin" tim I)l~a,s(', sl)a,(:(; (tistril)~tir p(q, p; t), i.e. l,i(n~vilh;'s (;(t~m,tio~ fi)r tl~e (:lassi~:al fl'ee I)artir

i) i)-~.p(q, p; t) --

p i) ',,,, Oql,(q, p; t).

(8.2.22)

Let ~s l,l~(;I'(;f~)r(: (litfi;r('.~tia.t(; q3(q, p) witli r(;st)(:(:t I,() t"

o i)t~(q,p)-

//"

, .,0 d.,',t.r'K(q p;.r, )/-~[(,*(.,')(,(.r')].

(s.2.~a)

o

o [,/,.(.,.),/,(.,.,)] _ it, o~ o~ o~ 2-;7,,,['/'* (') o:,.' ~ '/'(:'") - '/'("")~5.~ '/'* (*)l,

(s.2.24)

wllcr(; 1is(: llas 1)(:(:ll ilia(h: ()f l,ll(: S(dlrihlillger (:(lllal,i()ll fi)r l ll(: fl'~:e l)arti(d(', i)

h 2 02

i)-t.'('tql' '

2'1,,. Oq 2

(S.2.25)

Inserting (8.2.24) inl,~) (8.2.23), after i n t e g r a t i n g 1)y t)arts we ellr IIt) with

0 Otq3(q,p)_

ff 02 it, / . / da'd.rt(,*(.r)'(,(:rt)[O:r, 2

02 O:r2]l<(q,p;.r .r')

(s.2.26)

Equa,ting the t-(h;riwl,tive of q3 to the q-derivative ttlr(nlgh the fimtor _ 2in we finally o|)tain

ih 02 2 [Oa"'2

0z 0 ff Oa.2 ]K(q, p; :r, x') - -p-~qK(q, p; :r, ).

(8.2.27)

It is ca, sy to express the (tcrivativc of K w i t h respect to q, z amd x' in t e r m s of tim dcriw~,tiv(,s of K~ (~, q) with respect to the relative a,rgumcnts ~ - q X - ~2- X t a n d q - : r - .f,~. Thus, after a little calculation, Eq. (8.2.27) t u r n s into 02

--KI 0~0~

(4, ~) - 0,

(S.2.2S)

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 415

which suggests /~1 (.~, 77) to depend separately on ~ and r/; namely (8.2.29)

/~1 (~' ~]) --(1{(~) -4- /~(~]).

Since according to Eq. (8.2.21) the integral of/s over c takes on a constant value,/3(r/) is a constant function, and hence can be absorbed into a(~). Being therefore Kt (~, ~/)= a(~) = K, (~, 0), by (8.2.19) we immediately obtain that

(8.2.3o) The kernel K(q, p; z, z') comes to be unambiguously defined as

I~w(q,p;x ' X ' ) -

~

1 ~(q

._V__-LT'_)eip(.T-x')/h 2 '

(8.2.31)

and hence the phase space distribution function takes the explicit expression W(q, p) __ ~1 f ~/; (q -- q' )~/(q + q'

(8.2.32)

as firstly given by Wigner in his seminal paper (Eq. (5) in [3]). Expression (8.2.32) establishes a unique correspondence of tile state fimction ~(q) to the function W(q, p), defined in a, classical-like manner in the c-number space (q, p), where q a,nd p arc real parameters turning into the classical coordinate and momentum in the limit h, ~ O. The link with the optica,1 Wigner distribution fimction W(q, t,;) defined through (6.2.2) is immedia, tely recognized by noting that in the optica,1 context the reduced wavelength ~0 27r corresponds to h and acc()r(tingly the spatial frequeilcy t~; t)lays the role as p Naturally, in the presence of forces, W(q, p) does not satisfy the same equation as Liouville's equation for the classica,1 phase spa, ce distribution p(q, p; t), 9

9

Op Ot

p Op V' Op rn Oq F (q) Op'

h"

(8.2.33)

except in the case when the potential V(q) is linear or quadratic. We shall consider this aspect of the problem within the optical context. In fact, in 8.5 we will show that in the Fresnel approximation (corresponding to the nonrelativistic regime of quantum mechanics, to which all the above considerations apply) the transport equation for the optical Wigner distribution function W(q, F;) shapes in the stone form as the classical Liouville equation for p(q, p; z), only if the refractive index function n(q, z) depends quadratically on the off-axis distance. Within the mechanical-optical analogy the refractive index corresponds to the mechanical potential. It is worth noting that the (q, p) representation of the quantum system as yielded by (8.2.32) is consistent with the assumption that the system state is completely described by the single wave function ~(q), i.e., that the system is

Linear Ray and Wave Optics in Phase Space

416

in the pure s t a t e ,~b(q). T h e fea,sibility of such a univocal description by ~/~(q) is s(mmtimes t)revented by a lack of a, pre(:ise knowle(tge of the s y s t e m state. In m a n y case, in fact, we m a y know only the pro|)a|)ility t h a t the s y s t e m be in eacll of its own t)ossible (tylm,nfical states. Ac(:or(tingly tile (tescrit)tion of the systenl is suitably t)erfornle(t t)y nma,lls of the (tensity n m t r i x p(q~, q), which writes in the ('()()r(linat,e r(;t)res(nd, al, i()n as(")

P(q'" q) - E u,j ~/,.~(q')(,j(q), J

(8.2.3./1)

tlm t)()ssi|)le tilll(~ (lel)(~ll(l(~,l(:(: I)('illg agaill i*llt)li(:(l. ~l'll(~ Slllll illw)lv('s ()f (:()llrse all tlle t)()ssil)h'. (lyllallli(:al stat(:s ()f tllr syst(ull, t)re('is(4y (h:fill~:(l 1)y tlle rcst)e(:t, ive wave fllll(:ti~nlS 'l/'j(q); |,lie W~',igld, 'U'j Sl)e(:iti~;s l,ll~; lm4m,lfilil,y l,ha,t l,]lO sysI,Olll 1)0 ill l,]l(', s t a t e ~/,.j. As tl.ll ()l)vi()lls (:(nlSer ill(; 1)llase st)at:e (tisl, I'il)lll, i(m fllll(:l,i()ll (:()lll(~,s l,() 1)1~,exl)rr t)y l,lm sa,~m ilfl,(;gra,1 [()Fill tin in (8.2.32) wit, l~ l,l~e si~,gh,, l)r,,,l,~('t ,/'* (q - ,t'2 ),/,(q + ~) I,,,il,g r,'l)la,'e,l I,y 1,1,(, s,u,~ ,,f w(,iglll,r162 1)r,)(l~u:ts W(q,

p)

....

~/'.i(q -

' f

)~/'J(q + ,1'2), ~m,~(;ly .,/,j ( q -

J

,j *

+

.

.

Ill l,ll(~ ()l)ti(:al (:()l~l,(~xl, I,l~('~(h'~l~sity ~lm.l,rix relaI,(:(l Sllllllllal, i()ll SI~()~I(I 1)(~ret)la(:e(t

1)y |,1~(~(:()herell(:(: fl~(:|,i(n~, |,lnls yi(~l(li~lg tlm (~xl)r(:ssi(n~ fi)r |,1~(: ()t)|,i(:al Wiggler (lisl,ril)~ti()n fl~(:l,i()~ s~il,alfle |,()(les(:I'it)e t)arlially (:()l~('a'(',~t ligl~|,, as earlier ii()te(t in ~ 6.3.5. N()tal)ly i~l |,1~(: li|,('~rat~r(; the Wig~(',r (lis|,ri|)~|,i()~ fl~ll(:tion is ~l()st, ly ext)ress(',(I i~ |,1~(: fi)rl~ (8.2.35), i~lv()lvi~lg |,1~(: (h:~si|,y llla|,rix ()r ()I)erat~()r, wlii(:ti is ~miv(;rsally (x)~lsi(h;re(1 as |,tie fl~n(la.~l('~tal (lesr ()f a (l~m~lt,~lnl system. Ext)ressi()~l (8.2.32)is (h~r ffo~n (8.2.35)i~ |,ll,~ ('a,s(', ()f a, I)~re st, at, e. Tim (leriwtti(n~ ()f exl)ressi()~ (8.2.32), t)rese~te(t lwre, is ])ase(l ()~ a (tefinite set, of requireme~d,s, tlmt hav(~ a.ll()we(t for a ~lni(l~m (leter~film, t,i(nl ()f the ker~lel ()f the tra,nsfi)r~m,ti()~l (8.2.2) relating the wave flul(:tio~ |,~) a (tistrilmtion fl~n(:ti()n in the (q,p) I)lmm. In gen(',ral nm,ny ()t,ll(;r r(;aS()llal)h~ re(t~irements ma,y |)e (mvisa,ge(t t,hat (:ml |)e restri(:tive for tile kennel ill (8.2.2); they can be variously com|)ine(t t() f()rm seve, ral sets of con(titi(ms, m i n i m a l in mmfl)er, each uniquely yielding the s a m e W i g n e r kernel (8.2.31). T h e re(t~firements we have e x a m i n e d her(;, i.e. G a l i M i~lwMa,nce, free partit:le limit a,nd correct nlarginals, arc effective for providing a unique definition of the kernel of (8.2.2) in the form (8.2.31) only in the one dimensional ca,se. T h e r e a d e r interested in t h e general instance of the necessary conditions for yielding a unique definition of t h e p h a s e s p a c e d i s t r i b u t i o n t i m e | i o n m a y c o n s u l t t h e refs. in [4], a m o n g w h i c h we signalize [4.3] offering an exhm~stive investigation of the q~mstion. a We refer to the density matrix by the customarily used symbol p(ql, q), hoping that this do not originate any confusion with the classical phase space distribution function, p(q,p; t).

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 417

8.2.2

The non positiveness of the Wigncr distribution function

The properties of the Wigner distribution function W(q,p), discussed above, are shared by any classicaJ pha,se-spa, ce distribution. However, a, classical phase space density is everywhere non negative, whilst the Wigner distribution function c~n in generM attains negative vMues over some regions of the (q, p) space for some states. The non-positive property of the optical Wigner distribution function ~Y(q, ~) has briefly been commented on in w6.3.6, and verified in a certain number of examples in w 6.4. There we evidenced that only the quadratic phase signM (i.e., the chirp), which comprises ~s limiting case the impulse and the inonofrequency signal, and the Gaussian signal have noimegative Wigner distributions, respectively represented by a line delta and a Gaussian. The non positiveness is not a peculiar property of the Wigner distribution function, but is a distinguishing feature of any bilincar real functional of the state vector ~(J(q) satisfying as well the marginal conditions. In fact, as proved ill [4.1], t h e r e is no distribution function gl(q, p) of the type (8.2.2), i.e. bilinear in ~/~(q) ,which satisfies the marginal integrals,

q3(q, p)dp - I~(q) 2, 2

(8.2.36)

for all wave fimctions ~(q) a,nd is at the same time nonnegative

~(q, p) > O,

(8.2.37)

for all ~(~'s and all values ()f q an(t p. Following [4.11, we firstly consider a, wave function ~b(q), which is supposed to va,nish at all points outside a limited interval 1I of the real line: ~(q) =/- 0

only for q E lI C R.

Let q3w(q, p) be the distribution function corresponding to t/J(q) through the bilinear functional (8.2.2). We suppose that q3w is nonnegative definite. Also, we consider an observable ~(~), whose associated classical function is of course .q(q). We may suppose that .q(q) is zero inside the interval 1I, and is nowhere negative in its own domain. The expectation value of g(~t) in the state ~ is evidently zero, g(q) and I~/~(q)l 2 yielding a null overlap integral; namely <~) - f ~* (q)9(q)g~(q)dq - O. On the other hand, in accord with the interpretation of glr tation value ~ can as well be calculated by the (q, p) integral (~} - f f

JJ

9t3~(q,p)9(q)dqdp,

(8.2.38) p), the expec-

(8.2.39)

Linear Ray and Wave Optics in Phase Space

418

which then m u s t vanish. T h e integrand flmction with respect to p above is evi(tently i~onnega,tive, being the integra,1 of a n()nnega.tive fim(:tion, i.e. the t)I'o(tuct, of the nom~ega, tive functions g3r p) and .q(q). T h e n it must be zero: f q3.0(q, p)g(q)dq = 0, and hence, ~va(q, P).q(q) = 0 a.s well. As a ('ons('quence, the (listrit)~ti()n fimctio~ q3v,(q, p) vanishes (exceI)t for a, set of meas~r(' zero) for all va.hw, s ()f q ()~tsi(h' the (toma.in II ()f (,. W('~ (:()llSi(l(;i" ll()W t w ( ) WI't,VC fllll(:ti()IlS, 't/' l m~(l ('2, wl~i(:l~ tt.i'(~, s~I)l)()s('(1 to Wtl~isl~ ()~l,si(l(', l,w()(tisj()ilfl, t)()~u~(h;(t i~m',rvals, w(; (h;l~()l,(; r('st)('t:tiv(;ly as T[~ a,ll(l II2. I~ sy~fl)()ls, w(' lmv(' (,, (q) ~/() (n~ly f()rq ~ II, C R

aa~(l

(,.~(q) ~/() (),,ly f()rq ~ I[~C ~.

II('a~(:e ~/,, (q)~/,~ (q) == (). L(',t q3,/,~ (q, p) m~(l q3r (q, p) t)(, l,lw (listrit)~lti(n~ fil1~('ti()~s a,ss()(:iat(',t t()(,, an(l ('~ 1)y (8.2.2). W('~ S~ll)l)()se tl~at q3~j, m~(l q3,/,~ are ~n)~ln;gative fi)r all wfl~w~s ()f q aa~(l p as well as l,lml, l,l~(',y give l,lm (x)rr(;(:t ~mrgilml (listril)~ti()~ls when i~t(;gl'a,l,(',(l wil,l~ r(;sl)e('t l,() q m~(l p. 'l'l~er(,fl)r(,, a(x:()r(li~g t() 1,1~('l)revi()~s (tis(:ussi(n~, o43,,/,' m~(l q3,;,~ vm~isl~ (n~l,si(l(, t,l~(' v(;rti(:al stril)s i~ tin' (q. p) l)lmn' lil~fit(',l })y l,l~(', (t()~ai~s ()f tin' way(' f~n~('l,i(n~s tl~(;y r(;l)r(;se~l,, m~(l ln'~(:(' q3,r wmishes ()~l,si(h' 1,1~(' sl,ril) II, x IR a,s 9(3,/,2wu~is}~('s (nll,si(l(, tin, sl,ri I) II.2 x IR. We sll()w t,liat tll('r(' (;xists a~l a l)~ll~tlall(:(; ()f way(' fi~l('ti()~s, w(' limy (:()llstruct as li~('ar (:Ol~fl)ilmtio~s of (,~ a,~(1 '('2, wln)se l)lm.se-l)lal~e (listribution fi~(:I,i()~s fail I.()satisfy at t,}~(; sm~(; I,i~(' i'(',(l~ir('l~('l~tS (8.2.37)~u~(l (8.2.3(J). \V('~ ~my (:(n~si(l(;r, f()r il~sl,a.~('(', 1,1~(; wavt; fiu~('l,itn~ ('(q) si~ll)ly ~)l)l,ailn;(l as ,/,(q) -

(q) +

(q),

(s.?,.4o)

tll(,~ r('h'~wult (listl'il)llti()ll filll(:t,i()ll 1)(;illg (l(;ll()t(;(1 as q3,,/,(q, p). ~I'ln; assllmt)ti()ll tt~a,t 9(3,,/,(q,p) sal,isfi(;s I)()tl~ r(;(lllirt;ln(',llI, s (8.2.37) all(l (8.2.36) h;a(ls I,()a (:()ntra(li(:ti()ll. Ill fa('t, a(;(:()r(ling t() the g(,n(;ral integral f()rm (tefiifiti()n (8.2.2), ()n(; fin(Is t h a t

whi(:h is t,l~(, t)a,si(' nmnif(',sta,tion of the bilinea, rity ()f q3r (see w 6.3.4); of course,

q3"e~,V'2(q, P) --

/y dxdx'K(q, p;

x, x')~/,~ (x)'(,2 (:r') - q3,;2,V,, (q, p).

(8.2.42)

Evidently, a.s q3,el and q3,e,2 are nonzero only within their respective domains lIl x R and IT2 x R, in order that ft3~(q,p) >_ 0 everywhere t h r o u g h the (q,p) plane the (:ross distri|)lltion q3r m u s t vanish, thus ensuring for q3~ the definitely n o n n e g a t i v e expression: q3, (q, p) - q3% + q3%.

(8.2.43)

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 419

It is easy to verify that the above relation is in contrast with the assumption that g3~ yields the proper marginal distributions when either one of the variables is integrated over. In fact, on the ground of such an hypothesis, integrating g3.e(q,p) over q we should have

(.)I'+

the last expression being in accord with (8.2.40) and_ the linearityof the Fourier

transform operation, which implies that ~h(p) - ~/)1( P ) + @2(P)- On the other hand, according to (8.2.43) the integration of g3~(q, p) with respect to q gives

,/9t3~(q,p)dq

--1'~1(p)12 _~_[,f2(p)[2

(8.2.45)

since both q3~1 and g1% are supposed to yield the correct marginals. In order that (8.2.44) and (8.2.45) be equal, the product ~/,~(p)~,(p)should vanish. Such a condition could be satisfied only if the Fourier transforms ~/)1(P) and ~2(P) were_ nonzero over two nonoverlapping bounded intervals. But, being ~/)1 and ~/h the Fourier transforms of functions defined over bounded intervals, they can not be limited to finite intervals as well. As a conclusion, we can state that phase spa,(:(; distribution functions involving the state bilinearly and having correct marginals must locally take on negative values for (:ertain states. This raises the question about the interpretation of tile g3.e,"s, whi(:h clearly. cannot strictly represent the probability of fin(ling the t)article at q with motnentum p. The non positiveness of the (bilinear) quantum phase-spa(:e distribution flmctions is a plain inanifestation of the departure of the joint position momentum description of quantum inechanics frotil the classical behaviour [5]. It is evident that not every phase space distribution flmction is realizable in the quantum mechanical case. As an example, we may consider a phase-plane point, described by tile product of delta functions ~(q - qo)~(p- Po). Such a distribution is classically realizable, representing a single particle at qo with momentum P0, which moves according to Hamilton's equations of motion, and thereby individualizes a definite trajectory in phase plane. Delta functions are evolved into delta functions by the classical Liouville equation (8.2.33). In contrast, the realizability of the point-like distribution function is inhibited within the quantum mechanical context by the uncertainty principle. Rephrasing, for example, properties (6.3.1) and (6.3.53) for the quantum distribution (8.2.32), we see that for any normalized pure state function @ the corresponding Wigner distribution W(q, p) must satisfy the quantum constraints [W(q,p)[ < --

/[W(q,

1 71"h

~

-

1 27rh.

(8.2.46)

Linear Ray and Wave Optics in Phase 5pace

420

which evidently prevent W(q, p) from being peaked on one single point of the (q, p) plane. Indeed, they yield a, value of the order of It, for the ndnimal area in the (q, p) t)la,nc over whi('h the W i g n e r (tistribution flm(:tion st)r(;a,(ts out. 8.2.3

Wigne'r distribution function and Weyl t'ran,s'fo'r'm

As a.h'('a(ly ll()t(;(l, i,ll(; .j()iill, I)()siti()II-lll()lll(;ld, lUil (tr ()f a, (llm,ld,llln systClll is (Ii(:ta.t('.([ 1)y tilt; (Icsir(; ()f tr(;a,tillt4 tile (tlla,lltlllll tll(;()ry witllill a, fra.lllt;w()I'k silililar as tllat ()ftll(: (:lassi(:al statisti(:al Illr tllllS ~Imkillg f(;a,sit)lc, f()r illsl, ml('(:, l,ll(; ('alt'ltlal,i()ll ()f l,ll(' (;Xl)t't'l,al,i()ll valltt;s ()f (llla,lll,lllll

()l)s('~I'Va,1)lt'~s

t)y a (q, p)-illlt;gra, titni illv()lvillg lunllt;rit:al-valll(;(l fllllt:l,it)llS ()f q ml(l p ra,tht'~r tlla.~l t)y a q-illt,(;gral,i()ll illv()lvilitZ ()I)(;ra.ttw-valll(;
/

A

()t)(;rat()r .f(~. ~)i,, t,l,(; st, a,t(: ,/,,


/' (,*(q)f(i~,fi)~/,(q)dq, ^

q

(8.2.47)

.

a,ll(l l,h('~ (4assit'al-lik(' (;xl)rt'ssi()l~ fi)r 1,1~(;"av(;rag(;" ()f 1,1~('s('alar filll('li()ll .f(q, p)

(.f) -

/./"q3(q, p) f (q, p)dqdp,

(8.2.48)

q3(q, p) l)(;i~g tr(~at(',(l as a. (:lassi(:a.1 t)r()l)M)ility (l(;~sity, yi(;l(l tl~(~ smn('~ vahw~. W(; rc-(;~l)lmsiz(: tlm,t i~d,(;gra.1 (8.2.47) i~v()lv(;s ()l)(;rat()rs wl~cr(;as (8.2.48) s(:alar fl~(:l,i()~s; s(:alar fl~(:l,i()~s nm,y ~()r(; easily l)(; lm~(ll(;(l a.~(l i~v(;stigatcd l)y t)oth a lg(;l)rai(' ~al~il)~flati()ns an(t int~litivc t)(;rt:('.l)l,i()~s. As i~ l,l~('~I)rcvious t)ara,graI)l~s, w(; <:<)~si(h;r a system ()f one (:()ntim~t)~s (h;gr(;(; ()f fl'(;(;(h)m, so tha,t tl~e pair ()f (~()~ti~gat(; w~.rial)lcs (q, p) suffi(:cs t() st)m~ tl~('~ i~l~(;r(;~t I)lmsc st)a,cc. The (t(;siI'(;(1 i(l(;lltity ()f(8.2.47) a n(t (8.2.48), a,ll(t tilt; i(lcntifit:a,ti()n of tlle centra.1 (t~md,ity q3(q,p) is t)r()|)lcmatic (i~c t() the la,('k ()f a, ~mi(pw~ rule for associating a (:la.ssi('al flm('ticm of the co~tjugatc va,ria|)l(;s q an(t p to a given qua, n t m ~ f~(:ti()~ ()f the (:(mjuga,tc opcra,tors ~ a,n(t ~, a.n(t vi(:e versa. One (:mmot trivially rct)la(:('~ the ()t)erators ~ and ~ in the cxt)rcssion of .f(~,~) by the scala.I'S q and p, a,s so doing different o p e r a t o r flmctions would come to correst)ond to the same cla,ssical function. Therefore, the identity of the two expressions (8.2.47) an(1 (8.2.48) would fail to occur, thus preventing the individua,lization of the probability distribution. A cout)lc of examples illustrating this aspect of the problem has been given in P r o b l e m 6.1. As a further illustration, we ma,y consider the exponential o p e r a t o r c ~q+~p. The simple s u b s t i t u t i o n of ~ and ~ by q and p would yield the scalar exponential e i~q+~P,

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 421

which according to the algebra of c-numbers can equally be written as ei~q+irlP

1 (ci~qcirlp Jr- cirlpci~q ). C i~q C irlp -- GirlpC i~q __ -~

In contrast, due to the non commutativity of ~ and ~: [~, ~] - ih, the operators e i~q+irlp i~q and 51 (gigq~irlp + c i r ~ g i ~ q ) are clearly not equal. By .. A, ei~qe . it/p, . . eirlpe . tile BCH forlnula (E( t. (2.2.60)) one has

It is then apparent that tile functional forIn of tile phase space distribution function is closely related to the choice of the correspondence rule between operators and c-number functions. Conversely, each phase space distribution function, defined through suitable physical reasonings possibly suggested by the specific class of problems at hand, individualizes a well defined rule of association between classical and quantum observables. Thus, tile t)hase space ret)resentation of quantum mechanics, defining tile correspondence (8.2.1) between quantum states and phase space distribution flmctions, establishes as well a one-to-one mapping between self-adjoint operators f(~, ~) on the Hilbert space of the square-integrable complex-valued functions of the space coordinate and scalar functions f(q, p) on the Cartesian product of the t)osition and monmntum ('oordinates: f(q, p) ~

.f(fi, ~).

(8.2.49)

As the joint q-p (tistribution q3(q, p) reI)la(:es the state function '(,(q), similarly tile c-1111111ber fullct, ioll f ( q , p ) replaces the q-nunl|)er function f(~, p). Note that in w 4.4.1 we have been concerned with the direction f --~ f in the association (8.2.49). Here we are concerned with the direction f + f . Considering in particular the Wigner distribution function W(q,p), let fw(q,P) denote the Wither phase plane function corresponding to the quanturn operator f(~, ~), which allows for the desired identity:
.//J./W(q, P P

1 r_t2~ 2~ f f dqdp f d r ~ * ( q - Tr h,) ~ ( q + --)c

ipr

fw(q,P)

.

(8.2.5o)

Using the Fourier representation of fw(q, P), namely

fw (q, P) -- ~1

p P

]]

JJ

d~drlf~w (~, rl) r

,

(8.2.51)

fw(~c, r]) being the Fourier transform of fw(q, P), one can recast (8.2.50) into

Linear Ray and Wave Optics in Phase Space

422

after noting t h a t integration with respect to p gives 2 : r h ( r - r/). It is convenient to at)t)r()t)ria, tely redefine the involved va,riabh~s in the integral ()n the right of (8.2.52) and then to rewrite the rcslfiting integrand flm(:ti(m in the ()t)era,tor form, thus ()t)ta,ining

.1:1:1dqdC~dq"/'* ( q)<~/'(q -t- ',~h,)fw (<, ,~)(,~eq,,~O,',/2 .l:lJ @,t< ,/.,~,/,* (q) ./~w(<, ,~), ~e,,,,7~, ,ea~ ;,,~ ,/,(q).

(.f} (w> -

(8.2.53)

TI~(: last i(l(:~d,i(,y is a.~ ()l)vi()~s (:()i~s(:(l~l(:l~(:(: ()f (,I~(: bt('(,, (,lint (,I~(: sl~ift(:(l way(: fi~(:(,i()~ )/,(qA+qh,) (:a~ I)(: s(:(n~ a,s r(:s~I(,i~g; fr()~ (,I~(: a,('ti()l~ ()f (,I~(: (lisl)la,('(:n~(:nt ()l)(:ra,l,()r ('i'qP a,(:(:()l"(li~g (,()

,~"'Q,(q) -,"~'g,/,(q)

- ,/,(q + uh).

Fi~mlly, a,l)l)lyi~g (,l~(: I}(',II f()l'IIllIla (,() I'(~,(:~'l,,";(,(,If( ~, l)r()(l~u'(, ()f tln: t,w()(:Xl)()~e~tia,l ()I)(:rat()rs al)()v(, i~(,() a si~gI(: (:Xl)()~i('l~(,ial ()I)(:ra,(,()r l)y (,Ira well k~()w r(:(:il)(:: f,i~q(,~tlp

t,-ih(q/'2 (,i(q+iqp

_

(s.~.~)

w(: (:~(t ~t) wi(,l~ .

<.f>(w>

.

dq ~/'* ( q )

E(l~la,ti~g tl~(: al)()v(: (:xl)r(:ssi(n~ witl~ tl~(: (llIa,ll(,llIIl (:Xl)(:(:t,a,ti()~ wd~u: (8.2.47) all,)ws ,,s t()(l(:ti,,(: (,1,(: ,'(:lati(),, t)(:tw(:(:n .f(~, ~) a,,,l .fw (q, P) as

.f (.~, r

_ ~

/'

~

A

A

d&tq.f w (~ , ,q)r ieq+i,,p

(8.2.57)

eqlliva.le~d, ly, (hi a(:(:()lnd, ()f (8.2.51),

f"(Cl P) --

'

-~-~2~2 .

/ / / ' [ d~d'qdq,Ofw (q, p)(,i<(~-q)+ i,,('~ ,,) 1.1.1.1

(s.2.ss)

To invert (8.2.57) ()I" (8.2.58) we eva,lua, t(: the m a t r i x (:1(:~(:~(, of the opera.tot .f(~,~) between two cigc~sta,tcs of the position operator (:()rresponding, for instance, to the eigenwdlu:s q' and q. Using the Dira.(' llr162 we write (8.2.59) where the inner p r o d u c t


-- J" dq"~5(q" - q')eihr162

~5(q" -4- hrl - q) --cihr

' + hrl - q).

(8.2.60)

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 423

Therefore, (8.2.59) turns into __

d~f w(~ ~ )

1

c~-

(8.2.61)

more symmetrically written as

It is immediate to obtain fw ({, r/) through

and accordingly, by an inverse Fourier transforma, tion, fw (q, P) as

fw(q,v) In the literature the function fw(q, P) is re.ported as the Weyl ..(~ WignerWeyl) transform of the quantum operator f(~,~), fw(q,P)and f(~,~)being related through (8.2.57)and (8.2.64). Remarkably, expression (8.2.57) mirrors the Weyl quantization procedure for associating a quantunl operator with a classical filtration. The Weyl rule establishes in fa.ct to decompose the classical function by harmoni(: analysis ill exponential flmctions as in (8.2.51) a.nd then to replace the exponential functions by exponential ot)erators in which the (tua,ntmn operators ~ and take the place of the sca,lars q and p, thus leading to a,n expression of the form (8.2.57). In a, way the "classical" orthogonal a,nd complete set of harmonic c-number exponentials ci~q+i,w is replaced by the '~ua,ntum" orthogonal a,nd complete set of harmonic exponential opera.tors ci~q+i~lp [6]. Therefore, Weyl transforms arc related to quantum operators by a, transformation which reproduces the Weyl rule. However, they do not obey the commutative a,lgebra, of classical functions. As a,n example, the Weyl tra, nsibrm F w (q, p) of the quantum operator F(~, ~) - f(~, ~) ~(~, ~) is markedly different from the product of the Weyl transforms of f(~, ~) and ~(~, ~); namely, Fw (q, P) r fw (q, P)gw (q, P). Moreover, F w (q, p) #- G w (q, p), with G w (q, p) denoting the Weyl transform of the inverse order operator product G(~, ~) ~(~, ~) f(~, ~) (Problem 1). Consequently, one cannot state in general that if A

A

A

f

fw A

the Weyl image of any function H(f) is then the function H(fw) , i.e., that I..

H(f) ~ H ( f w). This feature of the Wigner phase space functions along with the non positiveness of the Wigner distribution unequivocally evidence the difference of the quantum phase space formulation to the classical stochastic theory [~.2].

424

8.3

Linear Ray and Wave Optics in Phase Space

First-order optical systems" propagation law for the Wigner distribution function

T h e Wigner distribut,i(n~ flu~(:tion is "I)r()I)agated" tl~r()~gl~ li~ea,r ()t)t,i(:a,1 syst,t;~s a,('(',()rr to t,l~e ray ~m,t,rix fornm.lisln of geo~et, ri('al ot)t,i(:s [2]. This is a, t)la,il~ (:()llS(;(tllell(:(~ ()f l,l~(; trm~sf(;r rela, t,i(n~s, whi<:l~ fin" li~ear ()t)tit:al systems writ(; ix~ 1,1~(; f()r~ ()f S~l)tU'l)()siti()l~ ilfl,t;grals wlml,(;v(;r 1)('~ l,l~(: r(:l)l'(:s(:nta,t,i()n a,tt()t)t,(;(l t<)
71"an,@:'~' relations fro' th,~ basic optical s y s t e m s

Ill ~n'~l~;r I,~ gaill all illlll.'.~lia, t~ illsiglll, illl,c~ l,ll{'~ Im~lmgal,i~nl l)n~l)~'rties ()f the

l)r~)l)a,ga,ti~n~ l,hr~)~gl~ 1,1~', Im,si~: ~)l)l,ir syst~u~s, w~' ~my ~lir~x:tly ass~)~:ia,te with tlm se(:{n~r (legree ~ n ~ n ~ i a l s i~ tl~e ray-{)t)tica,1 l)~)siti~)l~ a,ll(l lll()lll(:lll, lllll va,rias (tis('~sse(t in Cl~aI)t~'.rs 2-5. 'I'l~e (tis{:~ssi(m is r

~n~ I1~' ]msis ~)f the

sente(1 i~ tl~e next s~])se(:l,i{n~; a,(:c~)r{lil~gly t,lm res~lts r tl~is l)aragra,t)h are re(x)w'.re,(t within ttw. (:r ~f the ray rest)onse ~f tl~e syst~;~. As ca.rlier sta,te(t ill relati()n to I)lmse-tfla,nc gwa,t)l~s, the reh;wuit axes axe to be t,ho~ght of a,s b(;i~g s~it, a,t)ly s(:ale(t in ()r(ter t,o have the Sallle (lin~ensionality. Section of h,omogeneo'u,s medium T h e prot)agation thi'ougt~ a, liomogeneous m e d i u m is described ill t,he q-domain by the i n p u t - o u t p u t i'elati~m (w 5.2.1)

~o(q) -- ~

2~ ~(q')dq', f ei~(q-q')~

(8.3.1)

where d (tenotes the length of the section through which the input signal ~, (q) 1 of the refractive index. is freely propagated, factored by the reciprocal no The transfer relation (8.3.1) involves the convolution of the i n p u t a l n p l i m d e i

~, (q) and the chirp g(q) - V/k--~e~k~

, which reproduces in proximity of the

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 425 optical axis the circular profile of a spherical wave with curvature -~. Hence, by result (6.3.62), the Wigner distribution function 142o(q, ~) of tile output signal ~o(q) turns out to be the convolution in the space variable q of the Wigner distribution functions 142~(q, ~) and 142~(q, ~c)of ~ ( q ) and l~(q); n~mely,

]A2o (q, ~) -- 27r f )/F~ (q - q', t~)l/12~(q', t~) dq'.

(8.3.2)

Thus, on account of result (6.4.34), by which W 0 (q, t~) takes the expression (q,

k _ 2_~d6(h;_ _~q)

_

1

-

d /~,)

F0

,

(8.3.3)

we obtain the input-output rela,tion for a, free-medium section in tile form

a

(8.3.4)

Let us resort to the ray-optical picture of free propagation, according to which the ray-direction p remains unchanged whilst the ray-position changes in rela.tion with the actuaJ direction. The ray transfer matrix T(d) for free propagation along the reduced distance d reads, in fact, (Eq. (3.3.1)) T(d)-

1 d) 0 1 "

(8.3.5)

As emi)hasize(t in .~ 4.6.5, tile (q,K,)-ret)resentation of light rays allows for the ray-matrix picture as well, the off-diagonal elenlents of the ray-transfer matrix of the geometrical optical (q, p)-ret)resentation being suitably scale(t by 1/k 0 and k0, respectively. It is therefore evident that the arguments of IA2~ in (8.3.4) relate to those of 1420 through the inverse matrix of (8.3.5) in the (q, t~)description, namely T - l ( d )

-

(0-d/k~ ). Gathering the Fourier conjugate

variables (q, t~) into the column vector u - (q, t~) T, we may recast the transfer relation (8.3.4) for a free-medium section into the compact form (b) 142o (u) -- 142~(T

-1

u),

(8.3.6)

the specification of the section length d being omitted for simplicity's sake. b A remark is in order. When denoting the arguments of a function by a multicomponent column vector, the correct notation would require the use of the transpose vector, as for instance in Eq. (2.2.67). Thus, Eq. (8.3.6) should more precisely be written as ~2o (U -V) -- ")/Vi ( [ T - i n ] T) -- ~2i, (u-I- [T-1]T),

u y __ (q, ~).

For sake of formal convenience, we adopt in the following the shorthand notation of Eq. (8.3.6), thus implying transposition in the vector arguments on both sides.

426

Linear Ray and Wave Optics in Phase Space

l'I(][Jl/l'~ 8.2. l)lmse-Sl)~We m~r l)l~ysical-sl)aCe views r t,l~e t,ra~sfr162 t~r 1)y a (]m~ssim~ sig~al ~r162 fl'ee t)rt)l)agatir tl~e latter l)ei~g exe~l)lilier l)y tim free propagat, io~ r a (;m~ssim~ i~em~ fl'o~n tim waist. (a.) q-slmar of tim Wig~mr clmrt of tim sig~ml 1 1 /,1 ~; (q) - (.~) ~. q 2 / , i ~ fr()Id, S ()f a freely pr()l)agati~g (,a~ssim~ l)(,;u~ are like spl~eri('al wav(,fr()~l,s witl~ ('~rvat,~re r a ( l i ~ s 17 ~ z.

Ill !i 2.5 wr 11r162162ill,It tile' frr162 l)rr162 lms tile: gral)llir vir in t,hr gr162162162 l~llasr l~lmlr (q, p) a,s 1,11r tlcn'izcnli,al sllr r 1,11r ray (tistri|)llI, i()ll /)(q,p) (Fig. 2.13). Si~ila,rly I,l~e |,l'~-i.ll,,-;f(~r l'(;la,|,i()ll ( 8 . 3 . 6 ) ( ' m l ])(; visllalizr162 ill 1,11(', Wigllr l)lmlr (q,t,') as a, ll()l'iZ()lll, a,1 S]l(~,il,19 r t,]l(; Wigzlr (:hart,, as sll()wll t)y Fig. 8.2.a), W]l(~,l'(', t,]l(~ Wigll('~r r s ()f tll(; (;mlssimi signal ~:~ (q) _ (1)1/4~,-,/2/4, I)('f()r(, m~(l af(,(;r (,l~c I)r()t)aga(,i()~ (,l~r()~lgl~ t,lu; se(:t, ion, arc (t(:t)i(:te(l. Tile ellit)l,i('al ('()ld,()llrs r(;l(;vmd, t,() t,ll(; \Vigll(;r (Iis(,ril)ll(,i(nl fun(:t,i(m ]/V~ ()f t,hc |lit)lit, sigl~al at(; sl~(;ar(',(l al()ng the q axis wi(,]~i~ (,]|e l~()rizont, al st, riI) - v ~'U) -<- ~,: -<- v'11}~ . I~ fi~'(,, sin('e (,lu'~ fl'(;e I)roI)aga(,i()~ (l()(~s ~()1, affect t h e ray r tl~e ~: (i.r a~g~dar) exl,('a~sir of tim sig~m,1 is ext)(;r162 t,() r e m a i n iln(:hanger Also, t h e int,ersr162 icms wil,h t,he q axis are ~m~r162 I)ei~g represcnta,tivcs ()f r a y s (i.r thr ~r t() tl~c t)cam wave fl'r s) t, lm, t r the se(:tio~ I)ara,llel to t,l~e r axis a,~(l so t)rot)agat, e ~II(l('fle(:t(',(t. Clea,rly, w i t h in(:reasi~g the, sectio~ l(n~gt, t~ d --~ oc, the ellipses telM to tl~i~ a l)()~t, the seginent ma,king t h e a,ngle t,a,n/~ = t:o/d w i t h t h e q axis. As i~ot,e(t in w 6.4.5, real G m l s s i a n w a v e f o r m s m a y ret)t'esen~ t h e q-profiles a.t tlm waist of the a m p l i t u d e of G a u s s | a n bea,ms e n t e r i n g the section w i t h waist ra.dius w. At far d i s t a n c e d f r o m the, waist: d >> 'n,ot:ow 2 a n d wit, bin tile b e a m I'a,di~ts, t, he wa,vefronts of a Gaa~ssian b e a m c a n |)e assilnilat, ed to t)ort, ions of wave fi'tmts of a spherical wa,ve ema.na,t, ing f r o m a, p o i n t s o u r c e a,t t h e waist, a,nd so ha,ving a (reduced) c u r v a t u r e r a d i u s R - d (Fig. 8.2.b)) [7]. A c c o r d i n g l y , t h e W i g n e r c h a r t c o m e s to be a s t r a i g h t line m a k i n g t h e angle t a n / 3 = ko/d w i t h t h e q axis.

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 427 T h i n lens

The t)rot)agation through a thin lens t)ro(hlces a quadratic t)hase modulation of the input amplitude p~(q) according to (w 5.2.2) ~o ( q )

- c-- i k'-~fq'27:, (q),

(8.3.7)

with f being the lens focal dista,nce. Having proved in w 6.3.9 that the Wigner distribution function of the product of two signals is the convolution in the frequency variable of the Wigner distributions of the two signals, we may write

(8.3.8)

Wo(q...) - f wf(q.~:- ~:')W~(q. ~') u . where Wf(q, ~) is the Wigner distribution function of the chirp f(q) &nd

--

e-ikoq2/2f,

8o

Wf(q, ~) - e(~ + ~q).

(8.3.9)

Wo (q. ~:) - ~ (q, ~ + ~ q ) .

(8.3.10)

It immediately follows that

Re,sorting aga,in to the ray-optica,1 picture, we recall tha, t a thin lens is represented by the ray matrix (Eq. 3.3.2) L(f)-

1

0)

-1/f

1

'

(8.3

11)

with the off-diagonal entry being sea,led into - ~ when deaJing with the spatial frequency ~ instead of the momentum p. Thus, the transfer relation (8.3.10) can be recast in a. form simila.r to (8.3.6), i.e., d

}/Vo(u) = I/V,:(L

-1

u),

(8.3.12)

involving the Wigner plane vector u = (q, r~)T and the inverse lens matrix in (1 0 1). The specification of the lens focal the (q, ~)-representation L - l(f) - .%/f distance f is omitted. As seen in ~ 2.5, under the L-transformation the phase-plane ray-distribution is sheared along the p axis. Likewise, the transformation (8.3.12) produces a vertical shear of the Wigner chart. As an example, Fig. 8.3.a) shows the shear induced by (8.3.12) on the Wigner chart of the Gaussian signal ~ (q) = /

2

( ~ ) 1 / 4 e - ~ . The canonical ellipses of the initial distribution are sheared in the ~ direction within the vertical s t r i p - w ~ v/ ~" _< q _< w~//-~. As the passage V "

428

Linear Ray and Wave Optics in Phase Space

l~'l(l[Jl/l'; 8.3. l~lm.se-Sl)a('c m~! lfl~ysi(:al Sl)a,cc views ()f t,l~c trm~sfin'~ati~n~ ~n~lcrg(n~c 1)y a (lm~ssim~ sig~ml ~n~h,r l,l,i~ h,~s trm~sfi,r, tim latter Iwi~g exe~lflifie~l !W l,l~e wav(;fl'(n~t c~n'val,~re trm:sfin'~mti~n~ cf[c('tc~l I)y a hu~s (n~ a (lm~ssim~ I)cm,~. (a) h',-sl~cm" ()f l,]~(,. Wigl~cr ctmrt of tl~e sigl~al v~;(q) ()-k;)~/~c '~/~ ~n~ler l,l~i~ h,~s trm~sfi,r. (b) S('l~'~al, i(" of t,l~c -

',llr(lllt~ll a t,llill h'llS lm.s ll~l ~;tii'<:t, ~lll t,ll~; ray II~lsit,i(nl, 1,11~;q ~'Xl.~'llsi~nl ~lf 1,11r siglml (h)(~s 11()[, <'llallg~'. Als<~, t l ~ ilm~rs(~(:l,i~nls witl~ t,ll~' h" axis r~llmi~ ~l~(:lm,nge(1, l)~:ing r~;1)r~:s~'~l,al,iw~s ~f rays l,lmt 1)assil~g l,l~r()~gl~ l l~' ('('III,(W ~)f 1,1~(;h;ns t)r()t)l,l~e clliI)s~s 1,~(! l,~l t,l~i~ a.l~(l~t t,l~e s(~g~(~t, ~mki~g l,l~' a~@~ ~ tm~/4 - - k o / f with tt~(~ h axis. W~ r~('a,ll tlm,t a. hu~s Ifla.(:~(l a,t 1,1~' waist ~lf a, (Im~ssia,~ Imam tra,nsf(~r~s l,l~ t)lax~e wavefi'(mt ()f th(~ i~(:(mfing l~ean~ i~l,~ a st)l~ri(:al (more t)re(:isely, l)ara,l,(,li(:)(,~e lm,vi~g tl~c (re(l,~(:c(l)c,~rva,t,n'c ra(li,,s 17 = - f (Fig. 8.3.1))) [7]. W(~ kl~(iw l,lm.t tl~t ~.Wig~(~r (:ha,rl, ()f th(~ ~(~a,r-axis l)r~lih~ (if a. st)hcri(:al wa,vc is a, stra,igl~t li~c ~m,ki~g the a,ngh; tan/4 = -1,',~/I? witl~ t,l~e h" a,xis. Note tl~a.t r(~la,ti(n~s (8.3.2) a,t)t)lics t,() (8.3.8) a,r~ r~l)r~se~ta.l,iw~ r~sl)e(:tivcly of shift-i~wu'ia~t a.~(t st)r(~a(lh~ss systems, a,s a,nti(:it)a,t(~.(l ill !i 6.3.,9.

Ideal 'mag'n'lifie.'r T h e int)llt,-()lltt)llt rela,ti()ll f(w the idea] ~lmgnificr writes a,s (w 5.3.3) ~o(q)

--

~ p1 ~ ( q ) , m

(8.3.13)

where ~/t is the magnifi(:ation factor. Tlmrefi)r(~, on account of result (6.3.59), we easily o[)ta,in the i n p u t - o u t p u t relationshit) for the W i g n e r distribution flmction as

Wo(q, ~) - W~(~, mn) .

(8.3.14)

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 429

F I G U R E 8.4. Phase-space and physical-space views of the transformation undergone by a Gaussian signal under scale transform, the latter being exemplified by the passage of a Gaussian b e a m through a telescopic system. (a) t~-shear of the Wigner chart of the sig1 1 nal ~%(q) -- ()-7~) / 4 e - q2/4 under propagation through a positive scaler with v.~ > 1. (a) Schematic of the waist radius magnifying effect of a Galilean telescope on a Gaussian beam.

Evidently, being tile system represented in ray ot)ti('s by the transfer matrix

(Eq. (3.4.2(i)) 0

o)

1/~n

'

(8.3.15)

for both tile (q, p) a,nd (q, h;) des(:riptions of the light ray, the a,bove relation is equivalent to the following -1

Wo(U) = I/F~(S u ) .

(8.3.16)

As shown in Fig. 8.4.a), tile transformation (8.3.16) is depicted in the phase plane (q, ec) as a squeeze of the Wigner chart in tile ec or q direction according to whether m > 1 or m < 1. In Fig. 8.4.b) the magnifying effect of a, Galileanlike telescope on the beam radius of a Gaussian bealn is sketched. Evidently, with the limiting values m --+ oc and m ~ O, we recover the monofrequency and impulsive signals respectively, and accordingly the ellipses in Fig. 8.4.a) should crush over the horizontal and vertical axis, respectively. Pheose-space and physical space views of the transformation undergone by a Gaussian signal under scale transform, the latter being exemplified by passage of a Gaussian beam through a telescopic system. 8.3.2

The ray spread f u n c t i o n

We deduce now the transformation law for the Wigner distribution function under propagation through a linear optical system within a general context.

Linear Ray and Wave Optics in Phase Space

430

Let }/V(q,t~:,Zo) = VVo(q, t,:) and }/V(q, t,:, z,) = }iVy(q, t,.)denote the Wigner distribution fimctions of the signa,ls ~ (q) and ~o (q) respectively entering and emerging fr(m~ the ()t)tica,1 system. As ~.~ (q) a.nd 7:~,(q) are linked by the space (tonm,in transfer relation, 0

~,,(q) --

g(q,q')~.~(q')dq',

(8.3.17)

il, is t)~ssilflr t,r r162 1,1~; Wig~:r (lisl,rit~l,i~ fi~(:l,ir ~V,, (q, ~.) r l,l~r (mttmt sig~m,1 i~ t,~'.r~s ~)t"l,l~e li~e-Sl)I'ea(l flu~(:t,io~ g(q, q') ~)f l,l~{; sysl,r m~(l t,l~e int)~d, sig~ml a.n~I)lil,~ul{' r (q). I~ fa.~:t,, a.(:(t(w(|i~g I,{~ the q-(l~m.i~ ~l~;ti~il,i~m (6.2.2) of l,l~c Wig~wr r fin~{:l,i~m, we; write

w,,(q,

1

l'/.'/",

-iml'g*

-

(u

,

4')g(q

,

,/")< (q")r (,/")d,/d,/'dq'" *

.

[Jsi~g r~;lal,i~ ((i.3.23) I,~ r~:(:~w~'r l,l~: Wig~(:r ~lisl,ril~l,i~ fll~('l,i~ r l,l~; i~i)~t sig~,~,l fl"(llll l, ll{ ' [)l'{){lllf'(,., "* (q")r (q'"), a ~ l s~italfiy a,rrm~gi~g l,l~; va.rir i~l,egrals a.~{I wn'ia])h:s cqderi~g l.l~, rr i~g exl)rr w~: r ~I) wil,l~ a (lcti~il, e rela.ti~)n ~)f }/V(q. h') I,~ }IV (q, ~') i~ 1,1~: f ~ ' ~ ~)f 1,1~: S~ll)~;rl)~sil.i~ i~d,r

W,,(q. ~') -

//

G(q, ~, q', ~")I/V, (q'. t;')dq'd~".

(8.3.19)

~I'1~(; kenl(,1 G(q, ~,', q', ~,") (:xl)li(:itly writ(:s a.s G(q,~,;,q',t;')--

~l

g* ( q - ~2

q'" 2 )g(q t q'' 2 " q'

, q'-

t

u"' 2 )~-i(,q"-,~'q'")dq"dq'"

,

(s.a.20) a.ll(l is SeCll t,r lJ~: tile' syzlll)lr162

D)ln'ir

t,ra.llsfr

r l,llr t)rr162

g*(q-- ~2 ,q'--

2 )g(q + ~ 2 ' q ' + q"')2 rr162 as a t'~r162 eft q" m~r q'" Naturally the l,rm~sfer relation fi)r t,l~e Wigner (listrit)~ti(m fl~n(:tion can also })e w()rk~;~l ~n~t wil, l~i~ l,t~e fl'(;(tll(;II(:y ~t~m.in rel)res~;~d,a,l,i~)~ ~f th(; system, ~sing the t)crtix~r i~t)~t-r relations,

j h(~:, h;')~ (h:') d~;'. 0

~o (h;) --

(8.3.21)

In fact, on a,:co,l,~t of deft,fit, ion (6.2.3) of Wo (q, ~c) in tern,s of the Fourier spectrum ~o(~:) a,n(t relatio~ (6.3.25) for the product of two spectra, a,t different frequencies, we obtain the superposition integral (8.3.19) with the kernel G(q, n, q', ~,:') being expressed in terms of the wave-st,tea(1 function h(t~, n') in the same form as (8.3.20), namely

// h * ( K - Z -~" , n' ~ c

G(q, n,q', n')

__

~1

,.i,.!

__

/~// ~ " ' ) h ( n + Z-, -T-

nl

/~ttt + -g)c i(q~''- q/ /~/t/)dn"dn"'

(8.3.22)

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 431

The equMity of the two expressions on the right of (8.3.20) and (8.3.22), implied by the use of the same symbol G(q, ~, q~, ~ ) , can rigorously be proved making use of the relations (4.6.20) of g(q, q') to h(~;, t~') or (4.6.21) of h(t~, ~c') to g(q, q'). The dual correspondence of the two expressions in (8.3.20) and (8.3.22) is evident. Accordingly, G(q, ~, q', ~') can also be interpreted as the inverse symp]ectic Fourier tra,nsform of the product h*(~- -5-, - -~-)h(~ + /~/// ~"- ~ + ) with respect to ~:" and ~"~. The function G(q, t~, q~, h;~) is completely determined by the optical system through the point-spread function g(q, q') or the wave-spread function h(n, n'); it provides a characterization of the system appropriate to the space-frequency represent.atio~ of light signals conveyed by the Wigner representation. As the reader may verify, the hybrid input-output relations for the optical system, where the input and output signals arc described in different domains, yield forlnM expressions for the kernel G(q,t~,q~,~c~), which involve the hybrid response functions in the same way a.s g(q, q') or h(~;, ~;') in (8.3.20) and (8.3.22), tile integrations being obviously perforined over both the q and ~; variables, and not exclusively in the space or frequency domain as in (8.3.20) and (8.3.22). Notably, G(q, h',, q~, h;') can be understood as the response of the system to a signal represented by the Wigner distribution fimction 142~(q, ~) = (5(qq~)~(t~- t~;~), for which the transfer relation (8.3.19) would give 1A2o(q,t~,) G(q, h;, q~, h',~). ()f (:ours(;, this interpretation has only a formal significance, since there does not exist any real optical signal, whose Wigner distribution function is a point delta in the (q, t~) plane, i.e., 142~(q, t~) = (~(q- q~)~(h;- t~ ). However, taking a,c(:omlt of the geometrical optics representation of light rays, we inay consider the nonret)resentable distribution lNi ((t, ~;) = ~(q-tL )(~(~;- ~;~) as the mathematical representation of a single ray, entering the system at the position q~ with the optical momentum (i.e., direction) p~, and hence with the frequency ~,~ - k0p;. The kernel G(q, ~c, q~, ~ ) is accordingly reported in the literature as the ray-spread fltnction or the ray response of the system [2]. The ray spread function G(q, ~, q~, ~ ) represents a sort of trait d'union between the ray and wave optical description of light propagation in the inherent paraxial and Frcsnel approximations. In fact, on one side it marks the transformation law of the Wigner distribution function, which is defined in terms of the signal w~veform p(q), and on the other side it reproduces the ray-opticM picture of propagation, which is based on tile transfer matrix formalism.

A little more about the ray-spread function It is interesting to note that expressions (8.3.20) and (8.3.22) of G(q, ~c, q', ec') have the structure of a "double" Wigner distribution function; hence, the function G(q, t~, q~, n ~) has all the properties of a Wigner function.

Linear Ray and Wave Optics in Phase Space

432

In fiwt, G is real. Also, it is e a s y to d e d u c e t h e r e l a t i o n s

g(q,q')g*(q",q

ttt ) -- ~1 ./z./.Gz.

t.." q' +q'' ( ~ 2 ,'~', 2 , t~') ci~(q-q'')c-i~;'(q'-q''')dmtt~;',

" --~t~'")(.-iq(t~-h;")(~iq'(t~'-t.,-'")dqdq, ' h(t,;, ~',')h*(~:", h,,,, ) -= ~1 / 7._. ._. G(q, ~;-~-t~" 2 ,q', ~'2

(s.:~.~:~) wt~i(:l~ h~t~k very si~fila,r t(~ (6.3.24) a.~(t (0.3.25). I~(h'e.(t, we als()~t)taiu

/*

]g(q, q,) 1'2 - 2~'

G(q. ~. , q' .,-')d~-,Z~-' " Ih( ~;,

')

~") I-

-- 2--~ ~

//

G(q , ~", q' , t~"' )dqdq', (s.:~.~)

~qll(l

g(q, q')

/7'

-

,.,

~

~ ~;,)(,i(t~q t
G(2. h-,

2~g)(l~J~) ' " ' G(q, ~, q', ~- )~'-i(~q h(t;, t;' ) - .~Ty-~hli,, j, ) . / /

(s.:~.25)

~"u')dqdq'

wl~i('l~ fiw~mlly ~'~:s~;~l)lr r~:lal,i~n~s (6.3.26)m~cl (6.3.28), r~:sl)r It is i~fl,er~;sl, illg 1,<)r l,ln; ray sl)r~',a.r~';ul fill~:l,i~s ~f 1,1~r v~ri~)lts c~lX~lXnnufl,s, l{efi'rri~g l,o the s{;~l~U;l~:~: ~)f N ~)l)ti{:al syst{'lliS, ~'~nmi{h:readr <)f r~;fi'r~;l~r162l)l~ln;s l,ln'~ il~l)~ll.-~nltl~tl, r<',l~.ti(n~

w , (q. ,,.) -

//

G, (q. ,,. ,/. ,,")w

,

(,/. ,,.'),~,/,m'.

.j

~.... x

(s.:~.~)

wl~;rc ),V i ~ m~l }'V, ~lcln)l,e, l,ln; W i g ~ ; r ~listril)~ll,i l,l~(; W i g n e r (listI'it~tion W,,(q. h)
/7

G(q,t;,,q'.t;/)lA?,,(q,h')dq'dt;',

(8.3.27)

w h e r e G(q, t,;, q'. M) ha,s tln~ ~ea,~fing as t h e ray r e s l x m s e ~)f the <werall s y s t e m , al~(t writes i~ t,e r ~ s ~f tl~e ray reslX~ses of tl~e s y s t e ~ ( : ~ l > ~ ; ~ t s a,s the ( N - 1)-(timensional integral G(q, ~;, q', 1~;1) -- ,1'''" ,1' dq N 1(t~.N 1dqN 2 dt3",N_2 "'(tql (t~: 1 G N (qN ' t~'N ' qN l ' ~;X--, ) 9 t,; 'qs ,t~ ) .G (q,,t,; q',h:')

(8.3.28) The, sin~ilarity w i t h rela, tio~ (4.3.18) for the i m p u l s e r e s p o n s e of a c o m p o u n d s y s t e m is a p p a r e n t . Finally, it is a.n e a s y ta s k to verify t h a t for a lossless ot)tical s y s t e m the ray spread flmction obeys the relation

ff G(q,

q', ,~')dqd,~ -

1.

(8.3.29)

1D First-Order Optkal Systems: Transfer Laws for the Wigner Distribution Function 433

8.3.3

The ray response of an arbitrary A B C D system. Transformation law for" the Wigner distribution function

We consider now an arbitrary linear optical system, characterized by the raytransfer matrix M - (~ ~ and accordingly by the point-spread flmction V-"

g(q, q') - r

k0

~ [Aq'2+Dq2-2qq ']

ci 2~

(8.3.30)

Inserting the expression of g(q, q') into (8.3.20), after a simple algebra we obtain the following expression for the ray response

G(q, t~, q', h;')

koB ]/dq,,dq,,,ci_%_q,,[Dq_q, " 4rr2 R - ; B~]c~-%-q"'[Aq'-q+ z ;~']

(8.3.31)

which turns into the product of delta functions according to G(q, t~:,q', t~:')- 6(Aq' + EB /~/ -q)b(kCq' + Dr,;'- t~:). Resorting to the formalism of the Wigner plane vector u = (q, t,:)qfilrther recast the above expression into the vectoriaJ fornl

G(u, u') = b ( M u ' - u).

(8.3.32) we

call

(8.3.33)

Therefore, rewriting the rela,tion (8.3.19) as

Wo(U) -

G(u, u')W (u')du',

(S.3.34)

it is imme(tiate to establish tile tra,nsforma,tion law of tile Wigner (listriblltion flmction under propagation through a first-order optical system in the form 1/~o (U) -- )/~i (M-111)'~

in accord with relations (8.3.6), (8.3.12) a,nd (8.3.16). In formal terms the above relation states that the Wigner distribution function at a given phase plane point relevant to a tra,nsverse plane at the axial coordinate z is the same as tile Wigner distribution function at the corresponding phase plane point releva,nt to a,nother transverse plane, the correspondence being determined by the optical transformation, i.e. by the ray transfer matrix M connecting the chosen reference planes. Hence, the Wigner distribution function turns out to be invariant along the "pa,raxial geometrical ray-path". In particular, as the origin of the phase plane transforms into itself for any optical matrix, the Wigner distribution function at the origin of the phase plane is invariant:

We (o, o) =

(o, o).

(8.3.36)

434

Linear Ray and Wave Optics in Phase 5pace

It is w o r t h empha,sizing the formal similarity b e t w e e n tile transfer relation (8.3.35) for the W i g n e r d i s t r i b u t i o n function and the transfer relation (2.2.67) for the ray (tensity d i s t r i b u t i o n p(q, p, z), approt)riate to the geometrical opti('a,1 r of ray prot)a,gation throllgh first-or(let ()t)ti(:a,1 syst, enls. Thus, t,ll(nlgh t,he W i g n c r (tist, ri|)lltion filll(:t,i()ll (:mm()t 1)r int,(;rt)ret,c(t a.s a.n (;ncrgy (l(;llsit, y (tist, ril)utioll, tll(; r(;s(;nlt)la.n(:('~ r (8.3.35) t,o (2.2.67) (:()lltirillS t,ll('~ role of th( ~, Wign(;r (tist,rit)llti(nl filll(:ti()ll as a llm, tll(;llmti('a,1 nl(;mis t'()r a, .]()im, st)ac(;Sl)al, ial fl'(;tlllC]l(:y ch'st'ril)l,i(nl ()f l,ll(; liglll, l)crl, ltrl)a]l(:(: ('v()llll,i()~l, wlfi(:ll, yet, r162 as fa.r as tirsl-()r(h:r ()I)l,i('s is ('()lW(;nl(;(l, ('a.]l t)(; (l(:alt wit, ll t)y ('~(tlm,t,i(ms fi)rl~m,lly sil~filar 1,() tll()s(; 1)(;rtill('alI, t() t,ll(; g(;,)lll(;trir ()l)ti(:s (:ollnI,(;l'])a,l'|, D(q, ]), z), h'giti]~at(;ly ill|,r as all (;ll(;l'~y (l(;~sity (listril)~t, io~. ~1'1~r Wigl~('~l" (list,ril)~l,i(n~ flll~(:I,i()l~ a,l)t)(;ars I,l~(;rt;f()l'(; a.s a l~ ctfir a,l~(] i~d,~itiv(;ly a,1)l)(;a.li~g l,()()l t,() li~k l,l~(; l:()~ri('r ()l)l,i('s a l~(l l,l~(; til'si,-()l'(h:r ray ()t)t,i(:s, ()fl'(;ri~g it,s('lf t,() I)(; i~t,('rl)r(;t(;(l as tl~(; "a~l)lit,~(h"' ()f a Im.raxial ray I)assi~g l,]~r()~gli tl~(, I)()i]lt (q, z) i~ l.l~(: ('l()s(, vi('i~ity ()f t,l~(, ,)I)I.i('al axis wii, l~ i~('.li~mt, i()~ (lir(;(:l, ly (h;(l~u'il)h; fi'tn~ t,l~(; st)atial fl'(;r ~. Firsl,-()r(h:r g('~()l~cl,rir ()l)l,i(:s is ~mt~rally ass()('iat(,(i wil,l~ (lylm.l~i(:al ()l)servat)h;s (pm.(lrati(:ally ([el)(:l~(li~g (n~ tim l)()sil,i()]~ (:()()r(li~mi,(; q al~(I its Ca.ll()lli('al (',()nj~gaI,r i.(:., l,l~(: ~ll()~l(:~d,~t~l~ p. (',(wr(:sl)(n~r , ill t,llt; wave ()t)tit'al ('.()ll|,(;xl, t)('rt,i]l(qd, ly l,() t,]~(' t:r(;s~el a,l)l)r()xi~a,l,i()l~ i~l~(l I,]~(' a ss~l~l)i,i(n~ ()f t,l~i]~ h;~s('s a,l~(] t,ra.l~Sl~issiv(; (;h;l~C~t,s t,l~(; ()])s(;rwd)h's (t~la(lrati(:Mly (1('~1)(;~(1 ()n tl~r I)()siti()n ()t)(;ra.t()r ~ an(t its ('a,~(~i(:al (:()lljllga,I,(;, i.(;., 1,1~(;Ill()lll(;ll[,lllll ()])(;ra,|,()I" p or t,lw~ sI)atial fr(;(t~u'~(:y ()l)('rat()r K. II('~('e it, is ~mt~n'al t ( ) l ) r c s ~ ( ' tlmt ~m(h'x t,l~(;s(; aSSlllllt)t,i()llS l,ll(; Wig~('x" r fllll(:l,i()ll f()r~ally ])(,l~a,v(; a,s t,l~(~ray (tensit,y (listril)~ti()l~. In flu't, i~ w 8.5 w(; will s(~(; tl~a,t i~ g('~(;ra,1 tl~(' tra,nst)()rt, (~(t~a,ti<m fi)r t,l~r Wigl~(~r tlistril)~lti()~ fin~('ti()l~
1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 435

Linear Canonical Transformations of the c o n j u g a t e o p e r a t o r s ~ and 15

described by 2x2 symplectic matrices

Integral t r a n s f o r m s for the w a v e f u n c t i o n s

T r a n s f e r laws for the W i g n e r distribution function

by quadratic phase kernels

by affine transformations of its coordinates

FIGURE 8.5. The transfer laws for the optical wavefunction and the Wigner distribution function are parallel representations of linear canonical transformations, in the Hilbert space /22(IR) and in the Wigner phase-plane (q, t~). tion of ~ and ~. Thus, a definite link between the light signal representations as conveyed by ray optics and wave optics, with the respective mathematical machineries, is established in the Fresnel approximation. An alternative form of the (paraxial) transfer law for the Wigner distribution function in terms of the relevant second-order moments will be described in Chapter 9. 8.3.4

Ray, wave and Wigner optics. I

Relation (8.3.35) recovers to pa,raxia,1 wave optics the practical and visually appealing potential of the matrix formalism of paraxial ray optics. It a,ppea,rs evident the utility of relation (s.a.as) for analyzing and synthesizing optical systems. In fa,ct, whilst the propagation law for the wave field distribution or its spatial-frequency spectrum demands for the calculation of chirp-like integrals (that may reveal rather odd oil a computational footing), the transfer law for the Wigner distribution function is based on the simple matrix-vector multiplication of ray-optics, thus offering a modular computational scheme, highly desirable for computer calculation. Hence, all the equivalence relations and decomposition theorems, described in Chapter 3, apply to Wigner optics as well. Also, the field distribution and its frequency spectrum can easily be obtained (by a 1D Fourier transform) from the Wigner distribution function. Moreover, as noted in w 7.5.2, the Wigner representation is still valid when dealing with light of arbitrary coherence, the two-point field amplitude product in the Wigner integral being replaced by the mutual intensity, which is in fact the proper descriptor for partially coherent light (see also Problem 3). As a final remark we emphasize that the matrices involved in Wigner optics calculation explicitly contains the wavelength of the light, by which, in fact, the off-diagonal entries of the ray-optics matrices are scaled. Accordingly, design or performance investigations of optical systems naturally display the wavelength dependence of the problem at hand [8]. This wavelength dependence can, of

436

Linear Ray and Wave Optics in Phase Space

course, suitably be exploited when working, for instance, with polychromatic light to individualize the system p a r a m e t e r s t h a t can be adjusted in view of a,n a,chromatization of the system [s.a, 8.4].

"Ray"-optical description of longitudinal periodicities of optical 'wave fields As a plain examt)h ~,of the t)ra,cti(:a,1 ma,nagea,bility an(t (:()n(:(;t)tlm.1 transt)arency ()f the Wigxler ret)reseld,a, ti()~l 1)ase(1 at)t)r()a(:ll t() wa,ve-()t)t,i(:al t)r()t)a,gati()n t)r()t)l(~llls, we ilhlstrat(', t,tle a.t)t)li(:a,ti()ll ()f t,lm t)araxia,1 l)r()t)agati()ll rifle for the Wigner (list, ril)llt, i(nl finl('ti()~l t ( ) t l u ~.iIlvestigat, i()xl ()f l()Ilgit,luli~ml l)(~,ri()(li('i ties ()f ()I)ti('a.1 wa,vefi(~hls, wlli('h Ilm,llifl~St illt,() tlle Ta,lt)()t, (~[f'e(:t lnl(l(~r (:()lmrellt ilhlnli~mti(nl, ml(1 int() tlm Laal ('fl'e(:t lnl(ler t)a,rtially ('()tl('.rellt illlnlliIlati(nl [,()]. Using tim Wiglmr r(~.t)res(~.lll,ati()ll, it is t)()ssit)le 1,() a.(:(:()~d, fi)r tl~('s(~ i~l~ere~d,ly (litf'ra.(:tiv(~ eff(~('ts 1)y a, si~I)le m~(| vis~mlly a.l)l)(m.li~g fi)r~m.lis~, wlfich i~(|ee(l, (l~m t,() t,t~(' i~t,('rl)r('.tat,i()~ ()f tl~e Wig~mr (list,riln~ti()~ fi~mt, i()~ a,s the "a~I)lit~ule" ()f g(~()~(~l,ri(:al rays, res()rts t() th(~ ray-t)i(:t~re m~(l fi)n~mlisn~ of I)a,raxia,l g(~(n~mtri('a,l ()t)ti('s. Tim l)rese~tati()~ 1)eh)w fi)lh)ws tlmt ela,I)()ra, te(1 in [!).,~], wl~()s(' title is i~ul(~e(l rel)lm~s(~(1 i~ t,t~(~ se('l,i()~ titl(~ al)()v(~; 1,1~(~r(~a,(ter is (lir(~('l,e(l t() l,l~e (l~u)te(l r(~fi~.re~('(~ fi)r a wi(ler a.~mlysis. We sI)e('ili('ally fi)('~s ()~ t,l~e trm~sfl,r law fl)r tl~e Wiggler (lisl,l'il)~ti()~ fln~('ti(n~, wl~i(:l~, as (t(~()~st, ra,t(~(l, writes i~ t,lm fi)r~

w ( q ~-; z

-

z~)

-

w(q

-

(:-:')" ~-; z~)

(s.a.a7)

fi)r t)a.ra,xial I)r(~t)agati()~ fr()~ tt~e t)lm~(~ a,t zi t() tt~e t)lm~e a,l, z. I~ tl~e f()lh)wing, zi is (:()l~Ve~ie~d,ly set t() zi --(). Tlm~, l(~t ~s (:()~si(h~r a ('()l~(~r(~t,ly illm~fi~a,tx~(t si~n~s()i(lal-al~t)lit~uh~ grating i~ or(h~r 1,() ha,v(~ ()n t,l~e grating l)lal~(~ (tl~(; z -- () t)lm~(~) a wave a~l)lit~ute like r

1 + (:(,s(h;oq),

(8.3.38)

which exhit)its, ()f ('ours(;, the spa, tia,1 I)eriodicity ()f the grating, d - 9~. The h- 0 relevant Wig~mr (tistrit)l~tio~ flm(-tio~ is easily cal('ula, t(;(l t()

W(q, h:; ()) - (5(~)[1 + 89(:()s(2~-,0q)] + ~ [~5(~-,- ~;o) + ~(~; + h:,))] +[~(~:- ~2) + ~(~-~ + -~)] co~(~oq) , 2

(8.3.39)

whose support in the (q, ~;) is pictorially sketched in Fig. 8.6. At plane z > 0, the Wigner distribution function changes to

W ( q K , ; z ) - W(q, ec; O) + b(~c - -ffa ) cos[~:o (q - Ea2)] + b(r~ + -~2 ) cos[~;o (q + ~2 k 0 )] 2 2k 0 (8.3.40) w h i c h , a,s explicitly evidenced, differs from the distribution at t h e input plane gV(q, ~:;0) only along the two cross-terms lines ~ - -Fe%/2, displaying t h e r e cos-like amplitudes with opposite z-depending initial phases.

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 437

K

o uniform

amplitude

K0/2

un,form

f

background

a cosinusoidal

plus

amplitude

-Ko/2

cosinusoidal

amplitude

maxima

minima

and

spaced

-K 0

with

equally

by x/~o

FIGURE 8.6. Phase-plane contours of the Wigner distribution function of the signal (8.3.38) showing the self- and cross-terms generated lines.

It is then evident that at planes z such that y-~ 2ko - 2mTr which means d2

z -- 2m A---~'

(8.3.41)

where A0 is the vacuum wavelength of the incident light, the initial distribution is fully reproduced, and the field intensity as well. Recalling that according to our convention the propagation variable z is scaled by the refractive index no, it turns out that actually the physical length scales with the reciprocal of the wavelength in the medium, A = Ao/n o. On the other hand, at planes d2 z - (2m + 1)A---~,

(8.3.42)

the amplitudes along the lines ~c = -4-~o/2 have a contrast reversion. Finally, at planes z - - ~l(2m + 1)a0' a2 (8.3.43) the field intensity exhibits the lowest possible contrast. Notably the three formulae above just correspond to the locations of the positive-, negative- and zero-contrast Talbot planes; the axial distance z T = d2/Ao is in fact reported as the Talbot distance. In contrast, if the grating (8.3.38) is incoherently illuminated, the description of the wave field must resort to the mutual intensity J(q, ~; z), which as noted in w 7.5.2 is expressible in terms of the Wigner distribution function as

14?(q, t~; z) -- ~1 / J(q,

""~ "~; z)e-'~qd~,

(8.3.44)

where for the grating transmission function, we are considering, the mutual intensity turns out to specifically be J(q, ~; O) - [1 + cos(~coq)]2~(q-).

(8.3.45)

Linear Ray and Wave Optics in Phase Space

438

FIGURE 8.7. The (werlap r t,t~<;Wiggler ct~arl,s fi)r l,tw, sig~ml m~(l t,lu.' ~oise r if pro.jecte(t on axes ~tt, so,he m~gh' to tim Fo~rier axes.

(:lea rly l)c t)(;rfr

8.4

I)e Ininiinized

i~ l,l~e sal~; way as ])eft)re, as s~ggesl,e~l i~ Pr~)l)leln 2.

The Wigner distribution function and the optical Fourier transform" linking Fourier optics to Wigner optics

T h e fra,cl, i<ma,1 Fr l,rmlsfc)rlll }n'i~lges l,}lt; ga, I) })el,w(;(ul Fcnn'ier a,ll~l Wigner ()t)ti('s, I,}mt is, |)el, weeIl ()I)l,ir r ill l,(~,I'lIl,q ()f t,lle (:()Illl)lellmlll,a,ry st)a,ce and spa l,ial fl'eqllml(T siglml rel)reseld, a t.i()llS, t,]l(' lalter t)('illg ('(nlveye(I t)y t,he F()llrier t,ransforln, a,11(I r r in I,(;rlllS ()f i,ll(; jr Slm(:e a,nd fre(tlmIl(:y sigIla,1 l'ct)res(',Id,a,l,i(nl, a,s (:()Ilveye,(l |)y l,ll(; Wigiler (Iist, ril)llI, i()ll flln(:tion. In fa,(:t, t)()tll t,lle Wiggler (tistri|mtioll fllll(:t,i()ll a,n(t tll(; fra~'ti()lml Fourier t,ransfonn yield tw|)ri(l signal rcI)resentations, (list)layiIlg st)a(:e ml(l spa,tia,1 frequen(T fca.tau'es t,ogetllei'. Ill a,(:(:or(1 witll tl~(; aa~alysis t)l'evi()~sly t)erformed in connection witt~ the, fractional Fourier t,ra,~sform in t)ot, h ray (,~ 3.6.1) and wa,ve (,~ 5.4.1) oI)t,i('s, it is n()t s~rt)rising that, as we will see, t)(;rf()rlning the fra,ctiona,1 Fo~lrier t,ra,i~sfornl of a, signal ma,nifesl, s into a, clo(:kwise rotation of tile corresponding Wigner cha,rt in tile Wigner- t)la,ne by a,n a,ngle ~2" This correst)on(tence t)etween the Wigner dist, rilmtion flmction and the fractional Fourier transform is forma, lly expressed through tile R a d o n t,ra, nsform. The. relation of tile fractional Fourier transform to the Wigner distribution function was firstly asserted by L o h m m m [10.2]. As a supposedly natural extension of the beha,viour of the ordinary Fourier tra,nsform, the fractional Fourier transform of order a was defined as the transformation displayed by the function when the relevant Wigner distribution flmction is rotated by the angle a g in the inherent Wigner phase space. This relation has evidently applicative advantages, suggesting, for instance, optical designs for efficient signal

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 439

multiplexing or for optimal filtering of signals from degradating noise terms [10.4, 10.7]. As sketched in Fig. 8.7, it may happen that the Wigner distribution functions of the signal and the noise overlap, when projected on both the space and spatial frequency axes, but have little or no overlap on some other direction, rotated by some angle with respect to the Fourier axes. Removal of the undesired signal components can then be actuated by using suitable amplitude masks after performing the appropriate fractional Fourier transform of the signal, and hence the appropriate rotation in the Wigner plane. We describe in the forthcoming paragraphs this rotation based relation between the Wigner distribution function and the fractional Fourier transform. Firstly we examine the plain case of the ordinary Fourier transform. 8.4.1

Fourier transform and Wigncr distribution function: optical analog of the ~-rotation in the Wigner phase plane

In order to focus on the pure rotation part of the overall transformation induced by the optical Fourier transform in the Wigncr phase plane, we explicitly ./'7"'--

sca,le the q and g variables to the unitless q v / ~ and a ~ / ~ , although the familiar q and ~ are still used to refer as well to the unitless coordinates. Note that u

U

i - - 2 - _

the re(:iprocal scaling of q and h; by the same factor V/-~ does not affect the formal definition of the Wigncr distribution fimction, being tile wave fimction scaled as well according to the normation condition. Accordingly, the Fourier transform matrix F comes to correspond to the rotation matrix with angle ~"

F

(21 1),

, R(2)_

unitless variables

(8.4.1) A

thus implying a direct relation between the optical Fourier transform Ff and the mathematical Fourier transform 5 (see Eq. (5.4.4)), apart from the factor e-i'~/4, which however has no effect on the Wigncr distribution function. Hence, the transformation of the Wigner distribution function from lA;i(q, ec) to A

Wo(q,

-

q)

(8.4.2)

manifests into a clockwise rotation by ~ about the phase plane origin. This ~-rotation based relation in the Wigner plane between the Fourier transform and the Wigner distribution function has already been remarked when comparing the Wigner charts of some primary Fourier transform pairs, like the impulse and monofrequency signals (w 6.4.4), or the rectangle and sinc-like signals (w 6.4.2). In more detail, consider the point-like signal ~i(q%) with the relative Wigner distribution function (q) - 6(q - q0)

,

w % (q,

-

16(q

_ q0),

(8.4.3)

Linear Ray and Wave Optics in Phase Space

440

K" !

8-1ik e s i g n a l

i

%.

!

/

Fourier transformed -q.

~,~'~ ~ ~i~

0,,

'(i/,b').~"

=-tan

"% '%

'~'

0 = tan '(fl)

"~

ignal

chirp

Fourier transformed

"~hirp

(h)

(a)

F I G [ I R E 8.8. ~-,'otati(,,~ of t,l~e Wig,~er clmrts of (a) a h-like sig,~al a,~(l Fo~ri(,r t,rm~sfor~nati()~.

(b)

a chirp, ,,I~(ter

wll()s(; s~lI)t)(n't ill l,ll(; l)lm,s(; l)lmm is tim verti('a,l lilw t,lir(nlgli tim i)()ilit, q -- %.

_

-" [F%,,,l(q)

; ~ (q) -

~

l__l__(,iqo q

,

~" --- q , .

(8.d.d)

wll()s(; tra,(:(; ()~l l,tl(', (q, h') I)lmm is a, ll()rizozll,a,l lill(; l,hr(nlgll l,lw, l)()ild, h: = --q0, ,'(;s,llting ('vi(h',ltly fr(),,l a, g2 (:h)(:kwis(, r()ta,ti(m ()f the W,,,-lin(', (Fig. 8.8.a.)). Needless t() sa,y, th(; (:()llq)h;lll(;nl,a,ry view ()(:(:llrs ill l,h(; ('ase ()f th(; st)a,tia,1 harnloni(:, the ll()riz(mta,1 line ()f l,ll('~ corr('st)(m(tillg Wig~l(,r (:ha,rt |)eing rotated ild,() a vertical lilw inl(h;r Dmri(,r tra.Ilsfor~lm.ti()l~. As a f~rth(;r examI)h;, we (:(msi(h;r the In,r(; (:hirt) ( 'iflq'2/2 with the tortest)on(ling Wiggler (list, ril)~ti(m flm(:tion (q) - (#.4

,

(q, .-.) -

(8.4.6)

the relevant sllt)I)ort in the Wig~mr plane t)eing the stra,igld, line h: - flq (w 6.4.4). T h e opti(:a,1 Fourier transform of ~h~r, (q) is ~" i [F~gd, irp](q ) -- ~ ( ' .

which, a,s expected, is still a chirp (see flnlcti()n is then

- t 2/~

,

Eq. (5.2.2)).

(8.4.7)

The Wigner distribution

F~o

which clearly individualizes on the (q, ~) plane the straight line, ~ - - q / 3 , orthogonal to the line s - / 3 q (Fig. 8.8.b)).

1D First-OrderOpticalSystems: TransferLaws for the WignerDistribution Function 441 Then, one can synthesize the (optical) Fourier transform/Wigner distribution fllnction relation through the scheme A F

71" ,

1 unitless

8.4.2

' axes

~qO[ (0_ 1 1 ) ( q ) ]

Fractional Fourier transform and Wigner distribution function: optical analog of the C-rotation in the Wigner phase plane

The fractional Fourier transforming operation produces a clockwise rotation by the angle r of the Wigner distribution fllnctioil in the (q, ec) plane. In fact, the optical fra,ctiona,1 Fourier tra,nsform matrix F ~ corresponds to the rotation matrix by the angle r = (~rc/2 in the (q, ~) plane with unitless axes: F ~

unitless variables R ( @ ) -

( -sinq5 cosr cosr sine) "

(8.4.9)

Accordingly, the optical and mathema,tica,1 fra,ctiona.1 Fourier transforms F~Y~, 2 ~ come to differ simply by the phase factor c -~'~/4 (see Eq. (5.4.38)). So, as the signal g)i(q) is acted by the optical fractional Fourier transform operator: '

--,

o(q)

the corresponding Wigner distribution function changes from Wi(q,/~;) to qsinr + ncosr (8.4.10) which is obtained rotating Wi(q, a~) clockwise by the angle r ~. Evidently this mirrors the behaviour of the ray density distribution p(q, p; z) (see Fig. 3.16). Also, it suggests an alternative and equivalent definition of the fractional Fourier tra, nsform as originally proposed in Ira.2]; accordingly, the fractional Fourier transform of order c~ of the signal ~i(q) can implicitly be defined as the transformation experienced by ~i(q), when the associated Wigner distribution function is rotated in the (q, t~) plane by the c~-th fraction of ~. As learned in w 3.6.1, it turns out that such a Wigner rotation in phase plane can be accomplished by a simple bulk optics implementation consisting of free-medium lengths and thin lenses, which realize the proper three-stage transformation, i.e., the consecutive q, K, and q shears or ec, q, and K, shears, yielding eventually an overall rotation of the Wigner distribution function. It was just by this reasoning that the fractional Fourier transform setups, Lohmann's type I and II, described in w 3.6.1, have been conceived. Lohmann's definition parallels the definition given by Mendlovic and Ozaktas [10.1], modeled on the variation of the signal amplitude when propagating along a quadratic graded-index medium by a length proportional to the order W(OZ)(q, / ~ ) -

Wi([R(r

W/(qcosr

nsinr

Linear Ray and Wave Optics in Phase Space

442

!

6-like s i g n a l

i

;

o

r

.........................

I

q" i.,........"'-"q',lc~

..............i 9

q,,'sin(

~"

.

n" + 0 - ~b

~*~

~'~

~'~.

q

,

.Z. "~

.

"6

.

.~.~"~-~"~

chirp

O : ,a,, ' , , , -J

q fractional Fourier

! fractional Fourier transformed signal

~'" ~ r a n s f o r m e d

(a)

chirp

(h)

FIGURE 8.9. 4)-r()tatio~ of tt~c Wiggler clmrts of fra(:tioiml F()~rier trm~sfi)n~mti~.

(a) a

h-like sig~al m~(l

(b) a

clfirp, ~m(tcr

~)f the trmlsfiwIll, as ~tis~alsse~l iIl .~ ;3.5. Tile eCtllivah;ll~:e ~)f tile tWO)~h'finitiaras was t)r~we~l ill [10.a]. Ba,sir it, relies ~nl tile fiu:t tim, t, as r~;21m,rkc~t in .~ 8.3.3, lillca.r ~:ml~nlir tra,llsflmlm.t, iolls, wllicll are I'Cl)rcsclltc~l 1)y syllq)h'.ctic llla.tI'i(:e.s (ill l)arti(:lllar, 2 • 2 fi)r ()Xl(;-(lillwllsi(~lm.1 syst
F '~

1

ci@

2

sin4~cosr

cotr

(8.4.11) T h e W i g n e r (listrit~d, ion fimction is therefore a b fimction, namely -

1

~5(/~-qcot~+qosill-X(~)),

(8.4.12)

whose s u p p o r t results evi(lently from a rotation by ~b of the input Wigner line q = q0 (Fig. 8.9.a). Finally, let us c o m p a r e the Wigncr distribution flmctioI~ O(t~- [~q) of the chirp (~?chirp (q) -- ei/3q2/2 with that of the fractionally Fourier transformed signal, which writes (see Table (5.4.39))

[F

~a

i

(/9chirp] (q)

1 e~ q 4cos q%{_~sin q~

2 /O--tanr 1 -{-/3 t a n ~b ,

(8.4.13)

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function

443

The Wigner distribution function is then )/W~_~

~chirp

(q,/%) _

COS r

1

sin r

~(/% _ q /3--tan ) 1+/3 tan q5 '

(8.4.14)

which identifies a line through the phase plane origin making the angle 0~ fl--tan r tan-l[l+fltanr -- 7r q- 0 -- r with the q axis, thus confirining the expected clockwise C-rotation of the input Wigner line (Fig. 8.9.b)); 0 = tan-l(/3). Then, one can synthesize the relation between the (optical) fractional Fourier transform and the Wigner distribution function through the scheme 7r

F~'~ ~ e - ia i ~ o~( q ) "~---~ }/1]9~( q ,1~)

8.4.3

-

R(r

)

/'cos r r

-sin r cosr

Fractional Fourier transform and Radon-Wigner transform

As a simple restatement of the Wigner rotation property, the connection between the Radon-Wigner transform and the fractional Fourier transform can be demonstrated [11.8], making it possible to transpose the signal processing capabilities of the Radon-Wigner transform into the fractional Fourier field, which benefits from easy digital compllta,tion and optical realization. The 2D Radon transform is a set of 1D functions obtained by integrating a 2D flmction over all possible lines in the pertinent 2D domain. It is convcni{;nt to think of tile Radon transform of a 2D function v(x', y') as a collection of projections of v(x', y') onto lines, each being identifiable as a coordinate axis into new frames (x, y)'s suitable rotated with respect to the original frame (x', y') (Fig. 8.10). The Radon transform consists then of two steps: rotation followed by projection, as expressed in the mathematical definition [11]

y')] (x, r -

v([R(r

--1

X

(~))dy,

(8.4.15)

where R(r denotes as usual the clockwise rotation matrix and (x, y) represent a coordinate system counterclockwise rotated by r with respect to the original set (x', y'). For a fixed angle r the above yields integrals of v(x', y') along lines parallel to the new y axis; it is evidently a 1D function, we denote as [Tgv]r When applied to the Wigner distribution flmction the above definition straightforwardly leads to the squared modulus of the fractional Fourier transform. In fact, applying the definition (8.4.15) to the Wigner distribution runetion of a given signal ~(q), i.e., A

- fR W([R(r

~(2))d~,

(8.4.16)

we obtain the Radon-Wigner transform [11.6, 11.7]. Then, on account of the above established Wigner rotation property, the direct correspondence between the Radon-Wigner transform (8.4.16) and the squared modulus of the

444

Linear Ray and Wave Optics in Phase Space

X ,

/ line of integration

F I G U R E 8.10. Ge()~netry f()r the 2D R,a(l()~ trm~sf()r~.

fra,(:ti()~m,1 F()~ri('r tra,~sf()r,~ ()f tt~(, sig~ml ~(q) i~n~m(lia,t(,ly fi)lh)ws a,~ [ll.S, 11.7]

[9~r

2

(q)-- ]~,~(q)l 2,

(8.4.17)

wh(;r('. ()l)vi()llsly. 0 - (~-~ " (an(l, w(: r('('a,ll, we at(: still w()rkillg witll unitless a.xes ill ttl(; Wigll(;r l)lmm). Evi(l(,Iltly th(, a,|)()v(, r(;la,ti()Ii lms a, (;()Ilsi(l('rat)l(' a,I)I)li(',ativ(', r('l('van(:(, a,s it, r('la,t(;s tim fra,('ti()lml l"()llri('r tra,llsf()rlll, wtli(:ll is illll)l('almllta.I)l(; I)y simt)le ()t)ti(~al llWa.llS, t() tll(' Ra(l()ll-Wigll('r t,ra,llsf()r~ll, wlli('ll is a I)asi(~ t()()l for r('('()~st,r~(:t,i()n ()f th(; Wig~(',r (listril)~t,i(m fln~('t,i()l~ m~(l a,(:(:()r(li~gly ()f t,l~(; ('.()rrest)()~(li~g "sig~m.l" ~()t ()~ly i~ ()t)ti(~s |)~t als(), f()i" i~staa~(:(;, i~ sig~m.l l)r()cessing a.n(1 i~ m('as~r('m('nts (:()~(:(,rn(,(1 witl~ th(' (t~lant~ll~ n~('(:tmni(:al wave fun(:tion. I~l ()tll('r w()r(ts, r('lati()~ (8.4.17) s~gg('sts a,n easily t)ra,(:ti(:at)l(' ta.(:ti(:s f()r to'm, og~up/t,i(~ re(:onstr~(:ti()l~ ()f a.n "()t)je('t". h~ I)arti(~flar, if w(" tak(' (, = () ()r (, = 1, w(" ()t)tai~ ()~ tim left side of (8.4.17) the integral ()f the Wig~mr (tistrit)~tion fln~(:ti()n al()~g a li~m in the a.n(t 7r-(tire(:ti()n, r(;st)e(:tiv('.ly, a,n(1 (:()rrest)on(lingly ()~ tim right th(; a,mt)litu(te s(t~m.re(t ()f the sig~m,1 a.n(t its F()~ri('x st)(;ctr~m~. In ()ther w()r(ls, we recover rest)e(:tiv('~ly the sI)a,(:e aa~(t the fr(;(t~('.n(:y margina, ls (E(ts. (6.3.26)), which indeed, as t)r()je(',ti()ns ()f th(; Wigner (tistribution flm(:ti()n ()nt() th(, q- a,nd t~-~xis, are t)arti(:ula,r realizations ()f the general relation (8.4.17) |)etween the RadonW i g n e r an(t fractional Fourier transforms. This also confirms the hybrid character ()f the fracti()na,1 Fourier transf()rm, which in a sense interpolates between the pure st)ace and fr('qu('n(:y signal representations; relation (8.4.17) conveys a geon~('tri(:al view for such a hybrid character. In this connection, we mention t h a t a space-order, i.e. (q, (~), signal representation has been suggested as an alternative way to the space-frequency representation, with the intent of displaying signal features t h a t may remain unrevealed by the space-frequency (as well as other) signal r e p r e s e n t a t i o n [10.5]. Also, an optical system to generate the (q, a) chart of a signal has been implemented, as described in [~0.~].

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 445

Relation (8.4.17) can therefore be viewed as a "generalized marginal" involving projections of the Wigner distribution flmction onto arbitrary lines in the (q, n) plane. Interestingly, as the space and frequency margina]s (6.3.26) find their generalization in (8.4.17), likewise the inversion formulas (6.3.28) for the space and frequency signal representations find their generalization in [11.9]

~ ( q ) ~ ; ( 0 ) - - / ~ I~V(~2cos r - nsin r ~ s i n e +

~cosr

(8.4.18)

that the reader may prove along the same line yielding (6.3.28) from (6.2.2). Reconstruction of the Wigner distribution function from the Radon transform is made in principle possible by the projection-slice theorem [11.2]-[11.4]. This theorem states that the 1D Fourier transform of a projection of a 2D function yields one line through the 2D Fourier transform of the flmction itself, whose Fourier frequency space can therefore be sampled on a set of lines through the origin by transforming projections at different angles r To see this, we firstly note that a slice of a 2D function f(x', y') is defined by slice[f(x', y')](.~) = f(~ cosr ~ sin r

(8.4.19)

which evidently for fixed angle r and variable { E IR selects a slice of the function along the line ( x ' = { cos r y ' = { sin r in the inherent plane (x', y'). Take now the 1D Fourier transform of the Radon transform (8.4.15),

.-. A

1 /

--i~x s

which, turning back to the old axes " " {,)t"[~PvV]qh}(~ )

(x', y')

v([n(r -1 x

(s.4.20)

in the double integral, yields

-- ~ 1 f dx' f dy'v(x', y,)g--i[x'{cosr = v/~[9~v] (~ cos Oh,~ sin r

qh]

(8.4.21)

i.e., the slice of the 2D Fourier transform of the original function v(x', y') through the line (u = { cos r # = { sin r in the Fourier plane (u, #). Accordingly, reconstruction of the Wigner distribution function, and so of the corresponding signal, by tomographic measurements could in principle be achieved through the steps: 9 take 1D projections of the 2D Wigner distribution function at successive angles r covering at least 180 ~ or equivalently take the fractional Fourier transform of the signal for orders corresponding to the sampled C-range (this provides the so called (q, c~) or Radon-Wigner display, of the signal); 9 Fourier transform all projections, thus obtaining slices of the 2D Fourier transform of the Wigner distribution function at the same sampling angles r in the relevant Fourier space;

Linear Ray and Wave Optics in Phase Space

446

.

\

.\

W(q',E)

,k.,

I

, '~ \ ".11\. 5Z~'\ \

,.

/. I:rVVl(~,,:,

"~ projection data in the Wigner plane

4, Wl(q)

I----.. [

radially sampled data

Fourier Transform

in the Fourier plane

FIGURE 8.11. Schematic of the projection-slice theoreln, accordi~g 1,o wlficl~ tim collection of project, ion data in the (q', ~') plmm gmmrate by Fourier transform a collc~:tio~ of data in the Vo~rier plane alo~g radial li~es at tim stone angles as tim pro.jectio~ axes i~ the (q', ~:') do~nai,~ (i,~sI)iretl I)y [11.5]).

9 ewdlm.te tim 2D illw;rse F~llri~'.r l,ra.llSfiWlll ~f all ~l~t,ailm~l ~la.ta. I,~ ~tist)la.y the 2D Wig21er r rit)ut, i~)ll fllnr fr~)nl wllir l,tle sigxml aIIll)lil, lule. can be reconstrlu:tr162 t,~) wit, tlill a. t)lmse r162 a,llt. Figllrc 8.1 1 illllstra, t~'.s ttw. F~nM~;r r~'.la,ti~ll t~'.twe~;zl tll~'. ~:~lh'.~:ti~ll ~f t)roject,ion ~ta.ta, ~)f W, sa.~q)h;~t a.t ~tis~:rete i~l,ervals a.l~n~g t,l~'~ q-li~es a.t l,l~e chosen angles (/5, a.nr the r r162 of Fr res~ll, ing ~la.l,a, representing the 2D F~)~rier tra.x~sforx~ of W a.t, ~liscret, e intervals alo~g li~es ~naking in the relevant Folu'ier t)lmm tl~e saline m~gles as the i'est)er iw; l)r().j(',r axes in t,he Wigner t)lane. We nmnti~m that, in t)ra,ctice, the above sketche{t iI~versioI~ s(:he~Im is rarely used because firstly all (tata xnust be ()t)tained before reconstr~u:tioI~ can begin and secon{tly 2D D)~ri~;~" tra.nsfiwm are quite ext)ensive in (:~)n~t)~ting time. An alternative widely applied method is the filtered back project.ion, by which the coInplete 2D reconstruction of the Wigner distribution fimction of the invest, igate(t "object" is obtained by adding modified versions of each of the projections taken over at least 180 ~ The modification a m o u n t s t,o convoluting the original projection with a spatial filter function, called the R,adon or ramp filter, which is the same for all projections and compensate for the decreasing deal of information when moving radially far from the origin in the Fourier plane. Yet alternative m e t h o d s have been devised for q u a n t u m - s t a t e tomography, since the filtering procedure used in the filtered back projection algorithm introduces a certain smearing of the data, which can hide some important fea-

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 447

tures of the quantum states. The reader interested in this data-processing related issue may consult the relative literature [12]. Before closing this section, we wish to evidence a relation between the Fourier transform of the Radon-Wigner transform and the ambiguity function. On account of the Fourier-like relation of the Wigner distribution function W(q, n) to the ambiguity function A(~, ~), it becomes evident that the slice of the 2D Fourier transform of the Wigner distribution function entering (8.4.21) is as well a. slice of the ambiguity function A ( - ~ , ~) . Explicitly, we may write {9~[7~W]r (~) - v/-2~A(-4 cos r ~ sin r

(8.4.22)

Of course, the slice is oriented at the angle q5 with respect to the horizontal axis, which in the ambiguity function plane turns out to be the g-axis. Finally, note that by (8.4.17) the above relation is equivalent to

[~l@~(q)12](~) - v / ~ A ( - ~ cos r ~ sin r

(8.4.23)

which remarkably signifies a direct Fourier transform relation between measurable fractional Fourier transform intensities and ambiguity function slices in the relative (~, ~) plane. Evidently, it suggests an alternative ambiguityfunction based approach to phase-space tomographic reconstruction of "signals" to the Wigner distribution function based approach, represented by (8.4.17); in fact, we have learned in w 7. that the ambiguity function is linked through a 1D Fourier transform to the mutual intensity for both coherent and partially coherent optical wave fields (as well as to the density matrix in the quantum mechanica.1 context).

A bit of mathematics about the Radon transform Different and equivalent expressions can be given for the Radon transform of a 2D function. Let be v(x', y') the function to be transformed. The Radon transform is the set of lille integrals of v(x', y') along all possible lines ill the (x', y') plane:

{v(x', y') } - .~ v(x', y')de,

(8.4.24)

where df is the elemental Euclidean arc length along the line g. Of course, the above is a 2D function, since a line in the (x r, yr) plane is specified by two parameters. Specifically, g can be given in the slope intercept form or, equivalently, in the normal form; namely as y ~ = r / x I+~-

or x ~ c o s r 1 6 2

(s.4.25)

where r is the polar angle and r the perpendicular distance from the origin to the line (see Fig. 8.10). Accordingly, the integral in (8.4.24) takes the form -

[[ JJ

-

+

(8.4.26)

Linear Ray and Wave Optics in Phase Space

448

or, [7~v] (r, r

//

-

y')(5(r - x' cos r + y' sin r

dx'dy'v(x',

(8.4.27)

Of course, if we define new axes (x, y) rotated by r counterclockwise with respect to the original (x', y') so t h a t ~i - [ R ( r -I (.~j), we end up with the expression previously used to introdu(:e the Ra.don-Wigner tra.nsform. Then, w(; may say tha,t th(; Ra,(t()n t,ra, nsform is a ma, t)t)ing from th(; (.r', y') (toma,in to the 21) (t()i~m,i~ i~(livi(l~mlize(1 1)y 1,1~(;(:hos(;l~ li~(; t)araa~(;t(;r s(;ts. N(;(;(lless t() say, t)()th (8.4.26) a,ll(t (8.4.27) yM(t a I D filll(:ti()Ii if ()lie ()f tll(; line t)aI'aI~eters are fixe(t, like, fl)r iI~sl,a,~(:(;, tim sl()t)(; 'q in (8.4.26) aa~(t tl~(; I)()lar a,l@e (p in (8.4.27). Als(), if w(; (t(;ti~(; new a,x(;s (.r, y) r()ta,t(;(l t)y 05 ('(nnm;r(:l()(:kwise with rest)e('t t ( ) t h e (,hl (:r',y') s(, tlm.t ,, - [R(r (y), we e~(1 , q ) w i t h the ext)ressi()n t)revi(n~sly ~se(t to i~d,r()(t~(:e tt~e lla.(l()~-Wig~mr tI'a.~sfi)r~. The Ra.(t(m tra.nsf(w~ is linear; lining

()

A

[n(,,, + .,,.~)1(,, ~)

A

A

[n.,,,](,, ~) + [re,,~](,,

-

~),

(s.4.2s)

all(1 s(:a,les wittl ttm filIl(:l,i()ll a,:(:()r(lixlg t() "" 1 [.-~?,](b [n,'(,~.,", b:,/)](,, ~') -- N a'' bT).

(8.4.29)

hlt(;r(;sti~@y, t,lm IIa,(l()ll tra, llsf()rlll ()f the 1D (:(nlv()lllti()ll h(:r', fl') ()f a, 2D fi~mti()n v(x', y') witl~ a, I O fi~(:ti(n~ t ( y ' ) , i.(;., ,

.,(.'. :,/- ,)t(,)d,,

h(.,.', :,/) -

(8.4.30)

tllrns out to t)e the 1D (:()llv()lllti(nl ()f the I{a~l()ll l,raalsfl)rIll of '~(x', y') with t(y') with rest)e(:t t() the varia|)le r; ext)lMtly, we fin(t

A

[ n h ] ( , , ~) -

/A

[n,,](~, ~ - ~-')t(~-')d~-'.

(8.4.31)

The reader may (;a,sily verify t)r(,t)erties (8.4.28)-(8.4.31). ()tlmr t)r()t)(;rties can be found in the related literature in [11]. As an illustrative exanq)le, we calculate the R a d o n trm~sform of the 2D symmetrical Ga.ussian xt2+y t2

G(.r', y ' ) -

2-~2e-

2~2

(8.4.32)

Using definition (8.4.26), we obtain A

//"

[J-2v~](77, 7-) --

1

27ra

2

T2 f 1 2e 2fro.

2-g2 J

xt2q-y t2

dx'dytc __ (1-+-r]2)Xt2xt2

d x te

2~,2

232 ~_X t

- ~,2

5(y'-

(,.' +

~)) r 2

1 2E2(a,~) Z~7~ (~,,) e

(s.n.aa)

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 449

FIGURE 8.12. (a) 3D plot of the Radon transform of the Gaussian {7(x',y')- ~-~e-(='2+y'2)/u, and (b) corresponding projections at different line slopes. whose 3D graph is shown in Fig. 8.12.a). For a fixed slope r/, it turns into a 1D Gaussian in the intercept r (Fig. 8.12.b)), whose width E(cr, r/) is determined by the slope r/and the width c~ of the original function according to E(cr, q) - 1 1 + r/2cr.

(s.4.a4)

Fractional Fourier transform based phase-space tomography Relation (8.4.17) has a powerflll applicative content. It tell 118 that it is possible to determine a, complex wave field (i.e., its amplitude and phase a,s functions of position) by measurements of intensity only and, yet remarkably, using refractive optics only (so that less power fieht is lost). Also, although we have derived relation (8.4.17) within a purely (classical) optical context, it holds as well within a quantum Inechanical context oil due account, of course, of the frequently remarked analogy with the quantum-mechanical formulation of the phase-space representation. Thus, ~(q) on one side may refer to a scalar coherent monochromatic classical electromagnetic wave or to a quantum-mechanical matter wave. Correspondingly, the line integral on the other side may signify the marginal distribution of the Wigner distribution function for the totated phase-plane variable defined with respect to the rotation angle r or the quasiprobability distribution for the rotated quadrature defined with respect the reference phase ~b [12.1, 12.2]. Moreover, as briefly noted many times, also partial coherence as well as statistical mixture of quantum states can be accounted for by (8.4.17), with the Wigner distribution function being in that case expressed in terms of the mutual intensity or the density matrix. As an example of the application potential of the fractional Fourier transform based approach to phase-space tomography, we synthesize below the detailed report of [12.4], where indeed that potential has been demonstrated (see also [12.s]). Figure 8.13.b) reproduces the scheme of the experimental setup de-

450

Linear Ray and Wave Optics in Phase Space

FIGURE 8.13. (a) Mea,s~m~'(l s('ale(l i~te~sity (listrit)~ti()~ for tl~(,' flflly (;ot~cr(,~t t)cain o|)t,ai~(,(l t).7 tl~(, (;xi)(;ri~n(,~tal s(;t~q) s|~()w~ i~ (b). (c) Tlw i~t(,~sity l)r()fih' (soli(! c~rv(;) m~(t tl~e phase stm~(:t~rc ((laslm(l (:~rve,s) ()f the rccoI~str~mt(~(l (:o~nplcx wave tiel(l obtaix~e(1 in tt~c fl~lly coherent cw~c. Wl~(;~ ttm i~t,('~sit,y is ~mar zcr() th(; plum(, is ~(l('Ii~('(l m~(! is ~()t plottc(1 (a(lapte(1 witl~ pcrx~lissio~ fr()~ [12..'1]).

l()('a.t(;(t ill t,llc z --~ (I l)lall(; (.r is ~'~w~lillat~; ill tll(; ll()riz()llt,al trmlsw;rse (|ircct,i(nl). T h e o~xt,t)~t, ()f a I h : - N e lase, a,fl(;r t)eing slmtially tilter(;(l il~ a. s i l @ e - m o d e i,d,() tw() sI)a,t, ia,lly s('I)a.ral,('(l, I,(;arly I)a,ra.ll(;1 t)(;m,~s. " A ('( )l lS|,i.l.ll|, r('la.tiv(' I)has(; is **m,i,,ta,in('~(t |)('t,w(;('** t,l,('s(; ])('m**s s() tl~a,t t,()g(,tl~(;r tl,(;y r('~I)r(',s(',**t, a. fi('l(t w i t h fl,ll st)atial (:()h('x(,**('('~." B y I)assi**g tl,r()~,gh th(; tw()(:ylill(tri(:a,1 l('~l,S(;S (~L1 an(t CL2, th(' two |)(;a.***s a.r(, ('xI)m~(l('(l il, t,l,c vcrti('al (lir(;(:t,i(),~, tln,s yi('J(ting a fi('J(t (t(~t)(;n(ting ()nly ()i1 i-i, si**gl(; l,raIiSV('Xs(; (:o()r(ti,m,t(;. T h e field t h e n I)r()i)aga,t(;s tl~r()~igl, t,l~(; (:ylin(lri(:a,l l(;,~s CI~3, l()(:a,l,('~(l a,t, z - d a,n(t o r i e n t e d 90 ~ w i t h rcsI)(;('t t() l)()t,h CL1 a.n(t C L 2 s() a.s t,() aff(;(:t the field o n l y a l o n g the horizold, a.1 (lir(',(:ti()x~. As notc(t i~ w 5.4.5 , a.x~ A B C D optical syst('m ('an bc rcgar(t(',(t a,s t)(',rfi)n~fing a s(:a,l(',(t fra,(:ti(mal F(n~ri('x t r a n s f o r m of s(',a,le(t i n p u t w i t h st)(',(:ifi(: r(;stri(:tiol~s b e i n g imt)ose(1 on th(', m a t r i x (',l(',n~ents if a real o r d e r t r a n s f o r m is re(luir(',(t a.s in t h e case w(', a,re (tea, ling with. T h e r e f o r e , v a r y i n g d a,nd D for fixed f a m o u n t s to v a r y i n g tim or(t(~,r of t h e fractional tra,nsform a n d so t h e rotati()n a,ngl(', in t h e pha,s(', t)la,n(',, whi(:h in [12.4] is d e n o t e d by 0. " F o r each c o m b i n a t i o n ()f d an(t D, t h e i n t e n s i t y (tistributi()n I(x, D, d, f) was digitized, n o r m a l i z e d , a n d scaled to give t h e m a r g i n a l d i s t r i b u t i o n s h o w n in" Fig. 8.13.a). T h e v a r i a b l e X0 d e n o t e s t h e r o t a t e d (a, nd scaled to s o m e c h a r a c t e r i s t i c l e n g t h ) p h a s e - p l a n e variable, a n d X~ is obta, ined f r o m ;V0 by a furloher convenienloly O - d e p e n d i n g scaling. A filtered b a c k p r o j e c t i o n a l g o r i t h m was used to r e c o n s t r u c t t h e W i g n e r

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 451

distribution function, form which the complex wave field can be determined to within a constant phase (Fig. 8.13.c)). For more details the reader is directly addressed to the quoted paper, where the reconstruction of the mutual intensity for a partially coherent wave field is also reported. In fact, the dotted line in Fig. 8.13.b) represents the path of third beam, that was used to produce, by reflections by mirrors M1 and M2, phase randomization by a, piezo-electric translator (PZ) and lower reflected beam block by a beam block (BB), "a field similar in intensity to the field reconstructed above, except that it is pa,rtially coherent." Finally we mention that an alternative approach to the phase-space tomography based on the relation between the fractional Fourier transform and the ambiguity function (Eq. (8.4.23)) has been proposed in [12.5, 12.6].

8.5

Transport equation for the Wigner distribution function

Here we derive tile transt)ort equation for tile Wigner distribution flmction, which in a sense parallels Liouville's equation for the ray-density distribution p(q, p; z). It can be regarded as the continuous version of the input-to-output relations, discussed in w 8.3, which of course are recovered as solution to tile transport equation for pa,ra,xial propagation. Evidently, as the propagation of light rays involves a, corresponding propagation of the a,ssocia,ted ray density distribution, the propagation of light waves involves a corresponding propagation of the associated distribution flm(:tion. In both cases, the equations for the phase space distributions are derived by the equations modelling respectively the ray propagation and the wave propagation, i.e. the Hamilton equations for the ray position and momentum variables, and the Hehnholtz equation for the wave complex amplitude. In other words, we establish a relation between the evolution equa,tion for the system and that for the associated phase space distribution function. Firstly we devote a short paragraph to elaborate a bit of operator forxnalism, which will revea,1 its conceptual and practical convenience later when deriving the transport equation for the Wigner distribution function.

8.5.1

"Wigner representation" of the position and momentum operators

Let us start with the question concerned with the possibility of m~king the following identities -

W~,_i~r

~)

-

KW~,~(q, n),

(8.5.1)

Linear Ray and Wave Optics in Phase 5pace

452

to be true, individualizing definite expressions for the operators Q and K, one may guess should depend on the position and frequency operators ~, ?~. Said in a, different manner, within an operator picture one may consider the cross Wigner distrit)ution flmction ldd~,v, (q, t~:) as the result of the operator W~ a,('ting ()n the Hilt)ert st)a,(:e of squa,re-integra,ble fi,n(:ti()ns for a, (:hosen in the st)a,('e, i.e., W ~ "1/' e ~2(I~)

(s.s.~)

---+ V ~ , , ( , ( q . t.;,) ~ /~2(I~2).

Ther(;fi,r(;, E(ts. (8.5.1)caJ~ 1~r ret)lira,se(l as A

A

wr

A

~-) - Q[W~,/,](q, ~:),

W~ol~,(,](q, t,:) - K[Wr

(s.5.3)

h,).

'File a,t)()w; arc rc111illisr ()f relati(nl (4.6.54), wlii(:ll is Imsir t() tile i~l(tividlm,liza,t,ir r ~tiff'~;relit r~3)rese~d, a l,ir r a,xi ~)I)erat~)r. I~l~h;er l,lie ~)l)ex'at,()rs Q a,n(t K ~liglit, t)e (:(nisich;red a,s the Wiggler ret)reselitaticm ()f t,lie I)osition a,n(t fre(t~le~l('y r162162 7q aJl(l k [14]. Atm,rt fi'r tll~;ir i~t~;rl)rr Cni wtii(:ti we slm.ll ~1r r we slm.ll regarr t,lie r Q a,~lr K as inliere~d, ly elegant a,xl~l ~la,~mgeal)le ('a,h'~flal,i~nml l,~ls. rla,ki~g 1,1~r ext)lir ext)ressi~n~s (~f 1,1~; left l~a.nr sidles ~f (8.5.3), ~me fin(ls

~1 ./

(,1 - ~)(q + ~),/'(q + ~),-~"'@' - (q + ~)w~,~,(q, ~:) - Q[W~,/'](u, ~:), __

~, i

(s.s.4)

it

a,llr

= 4-,~ ./ ~ (q- 2 1{ ~'/'(q + 2 ) + 2~.,/,(q + '21:)},, i O = (,,-- ~)w~,~(q,,,:)= ~,[~/,](q,,,:). i

,

q

O

q'

,

O

- i^-q'

@'

(8.5.5)

Accor(tillgly, the. ot)erators Q and K st)ecia, lize as A i 0 Q - q -4 2 0 t ~ ' K-t,: io 20q ~

(8.5.6)

which at)t)ear as a m i x t u r e of the q- and g-representations respectively of the position a,n(t frequency ot)erators ~, ~ (see w 4.6). Interestingly, [Q,K] - i .

(8.5.7)

The above operators have been originally introduced by Bopp, and recovered in more recent investigations relatively to the formulation of q u a n t u m mechanics

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function

453

in phase-space [14.3] or to the elaboration of phase-space operators aimed at simplifying the technical aspects of the Weyl transforms [14.2]. It is evident that for any power of ~ and s we find A

A

A

A

w ~ [ f i ~ l ( q , ~) - o n [ w ~ ] ( q , n) - onlA?~,,(q, ~c),

~)

W#[?~nr

~)

-- K~[W#r

-- K~142~,r

t~).

(s.5.s)

A

Hence, for any operator function O(~, s of ~ and s which is supposed to be expressible as a series expansion of ~ and s the following relation is easily obtained [W~O(~, ~)~](q, ~) - O(O, K)ld2#,~(q, t~). (8.5.9) A

A

A

A

A

Of course, the function O(x, y) is the same of its arguments on both sides. Notably, a pair of complex conjugate operators naturally correspond to the pair (Q, K). In fact, the initial question concerning the feasibility of identities (8.5.1) can legitimately be concerned with the feasibility of the alternative identities: }'Vq,,cp(q,l'C)

- - Q * }32r

(q,/~;),

W_i~q,,~ (q, ~) - K*W,,~(q, n),

(8.5.10)

whose operator formulation is evidently A

A

A

W.,~[~b](q, n) - Q* [W.,~b] (q, t{), W.,~[~](q, ~) - K*[W. ~ ] ( q , ~).

(8.5.11)

The operator W.,~ parallels W~ (which should more properly written as W~,.), on account of the two possible ways by which a function can enter the Wigner integral. Hence by a direct calculation along the lines leading to (8.5.6), we obtain A i 0 Q*-q

2o ,

A

i 0

K* --I%@ 20q'

which, as announced by the use of the asterisk, are the very complex conjugates of (8.5.6). Naturally relation (8.5.9) has its complex conjugate counterpart A

A

A

A

A

[W.,~O(~, ~)~](q, t~) -- O(Q*, K* )ld2,,~ (q, ~). 8.5.2

(8.5.13)

The transport equation for the Wigner distribution function

We recall from w 4.2 the Helmholtz equation for the 1D picture of scalar wave propagation 02 02 (-a----s + _-~--~)r z) + k2(q, z)r z) - O, (8.5.14)

oq"

Linear Ray and Wave Optics in Phase Space

454

where the possible weak inhomogeneity of the m e d i u m is accounted for by the position d e p e n d e n t w a v e n u m b e r (8.5.15)

=

Since only t)rot)agation in the positive z direction is consir Eq. (8.5.14) r sa.fcly bc "solvc(t" f(~r tim z-(tcriwt,tivc (ff the wave a,mt)litlI(lc, ttnls yielding(C)

0

i-:~z(/)(q, z) -- -

i /~:2(~,z)

- ;.2 (/)(q, z) -- ~ ( ~ , ~,)(/5(q, z).

(8.5.16)

The q-(t()ina,in (tcfinition ()f tim Wigner (tistritnlti(m flm(:ti(nl i111I)li(~s the integral ld2(~(q, h;; z) - ~ 4'* (q - ~ )r + ~ )('-i~q'dq ', (8.5.17)

/

the z-dct)en(lcn('c ()f tx)th q5 a,n(l 1424, t)cing ext)li(:itly r Ditti~rentiating ]42e,(q, h;; z) with respect to tile t~r(~tmga,ti(m wu'iat)lc z, cmc (fl~ta,ins (J) 0

i~zl/V4,(q, h;; z) - 1A2_i~,o(q,,,,,

h'; z) + 142(~,i~4~(q,_

h'; z),

(8.5.18)

wtli(:tl, ()ll a,(:(:()lnd, ()f 1,11e waw~ cr (8.5.16) a.ll(t tll(~ ()l)t~l'at()r ~()tatio~l t)revio~My intrr162 (:a~ I)e writtcli as

i~zlA24,(q, h;; z) - [ W 0 ~ ( ~ , ~ : ) 0 ] ( q , h : ; z ) - [W. ,/, ~ ( ~ , ~)e](q, h'; z).

(S.5.19)

Th(~rcfi)rc, t)y r~tles (8.5.9) a,~(l (8.5.13), w(' i~n~e(liatcly ()t~tai~ ((tr()t)t)ing out the now ~mcSSCld,ia,1 sul)s(:ril~I,)

i-~zlg)(q,~,; z) - [~(Q, K) - ~ ( Q * , K*)]IA2(q, h'; z),

(8.5.20)

which rot)resents the tra,l~Sl)()rt e(l~m.ti()li fi)r t,lm Wigner r fl~n(:tion. It para,lMs the Hclmh()ltz c(tlm, ti(m fi)r tt~c wave (:(m~t)lcx a,~l)lit~utc. It is rct)ortcd a,s the quantum L io'uvilh' ('q'uation. Explicitly, we write

O W ( q , h;;z )_{V//~:2 (q - ~i ~-~-s o z) - ( h ; -~- ~i qo) 2 i(-~z _

]~2(q _~_~__~,

~_~q)2

W(q, h;; Z),

(8.5.21)

which is useflfl to identify the bchaviour of the Wigner distribution function under specific conditions. In t)a.rticular, we are. interested in determining the paraxial forn~ of the t r a n s p o r t equation for the W i g n c r distribution function. c Note that this is nothing but Eq. (4.4.7) with the flfll expression of Hamiltonian operator (in terms of ~ instead of~) being used. d Naturally, it is assumed that the conditions under which the derivative of the integral can be turned into the integral of the derivative of the integrand function are satisfied [see, for instance, E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, (Cambridge Univ. Press, 4 th ed., 1965), ch. 4, p. 67].

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 455

Recovering the classical Liouville equation for quadratic refractive index

8.5.3

Let us expand the operators (a • ~i )" 2 under the square roots in (8.5.21) above, thus obtaining i

" Z)k2(q ~ -~i 0"--~0,

I

i" (t~ -~- ~qq)2 -- ~ ]~2 __ t~ 2 :~: i l~ ~q" Jr- 41 002 q2 ,

(S.~.22)

where, for formal convenience, the dependence of the wave number k on the opA i erators Q, Q*, and z is not explicitly shown on the right side. Then, according to the usual paraxial approximation expansion, we may write

~/

1 0__2_ 2 ~

k~-~Ti~+~Oq~

i 1 1 o2 k0~(qT~,~)+~o~o(Z)[~~Ti~ ~] , (s .5.23)

which once inserted into (8.5.21) yields 09 lA?(q, ~;; z) --

Oz

1

(9

- kono(z ) n-~qI,V(q, t~; z) o ~)]W(q, ~~ o~,

+iko [,~(q + ~ ~ , ~) - ~(q

(s.5.24)

~; ~)

where evidently tile first term accounts for the kinetic contribution whilst the second for the "potential" contribution. Also, expanding the refractive index flmction with respect to tile transverse coordinate q and grouping tile terms with equal powers, we end up with OG

()Z(9W ( q ,

i,~; Z) --

no(z)p__~ql

() W ( q ,

,l'~"Z) +~-~

(_)j+l

1

n (2j+l)

(2j-I-i)! (2ko)J

oq2j+l

Op2j--I-1

FV(q ec; z) , ,

j=O __

tO2j-F1

where rt (2j+l) - Oq2j+ln(q,z). Note that we have turned from the Fourier conjugate pair (q, ec) to tlle Hamilton conjugate pair (q,p) to evidence the dependence on the wave number. In fact, the potential term turns into a series expansion in terms of the reduced wavelength ~1, the optical analog of h. Naturally in the geometrical optics limit: ~1 ~ 0, we recover the classical Liouville equation (as it should be appropriate to paraxiM propagation, which

p2

amounts to the optical Hamiltonian 2% (z) - n(q, z)), since only the j - 0 term survives in the series above. In general the refractive-index depending term is not expressible in the paraxial regime classical-like form, i.e, ~n(q, z), except that in the case of free propagation: n(q, z) = no(Z), or quadratic refractive 1 q2 9In the latter case, in fact, the resulting index: n(q,z) - no(Z ) --~n2(z) equation can be written in the "classicM" form 0

o--;W(q, ,~; ~) - - { w ,

H},

W(q, ,~; ~) - W, (q, ,~),

(S.5.26)

Linear Ray and Wave Optics in Phase Space

456

where H is the pa,raxial Hamiltonia,n of geometrical optics. It is therefore evident tha, t the Wigner distribution comes to have a, constant vaJue along the geometrical optics ray-path-

d )/~(q, h;; z) - O, dz

along any "ray trajectory"

(8.5.27)

ml(t a(:(:()r(lingly th(~ t,railsfer relati()ns t)etween two r(;f(w(~Ii(:e, i)la,nes, we ha,ve ()t)t,a,i~l(~(t llsillg tim t"reslml ixltegral, follow illllll(~(lia,t,ely, i.e.,

142o(u) -- ] 4 ; i ( M - l u ) ,

(8.5.28)

M t)ei~lg, ()f (:()llrs(;, tlm. ()l)ti('a.1 ray-Ilmtrix witll k,)-s(:a,h'.(1 ()f[-(tiag()im,1 entries. E(tua, ti()]l (8.5.28) lll(~a,llS tim,t, .i~st as ill l,lle ('ase of tll(~ ray-(l(nlsity (list,rit)llti()n p(q, p; z), we (:an rlul ill(livi(llm,1 traj(~(:t()ri(~s t() (teterllfilm tlm z-ev()lutioil ()f the Wigner (listrit)llti()n fim(:ti()u a,n(1 h(ul(:e ()f tim rehwa,nt signa,1. Blxt this is tr~e f()r tim Wiglmr (listril)l~t,i()n f~('t,i()~ ()Idy f()r l)araxia, l l)r()t)aga,ti()n m~(ler (t~m,(traJ,i(' r(~fi'a,(:tiv(~ i~(lex. I~ genera,l, as tim (lll;mtllln Li()~ville ()t)era,tor i~w()lves (leriwd, ives ()f tl~ir(l an(t hi~h(;r ()r(ler, h)('ality is h)st. As fa,r a,s the li~ea,r a,I)I)r()xi~m,t,i()~ is ('()l~('(u']~e(t, we ~my (']~()()se, l)()ssil)ly a('('()r(ti~g t() sym~ e t r y I)r()l)(wti('s ()f tlm syst,(:~, a I)ha,s(:-l)la,~(' ('l()s(:(l (l()~ai~ m~(l f()ll()w the (;v()l~t,i()~ ()f s~u'l~ (l(n~m.i~ a,s r(q)rese~ta.tiv(~ ()f tim, (w()l~t,i(n~ ()f tim Wigner (lisl, rit)~ti()~ fluml, i()~ (m~(l s()()f the (',()rresl)()n(ling sig~a,l) ~m(ter I)r()t)agation tl~r()ugh tt~e syst(u~ a,t hm~(l. ()n a,('~(:ount ()f tt~(~ (:()~si(l(wati(n~s ()f w 2.5, we nmy g~mss tha,t l)lm,se-l)la.l~(~ (l()l~m,ins t)()~m(t(,'(l 1)y stra, ight li~(~s ()r l)y (',llit)ses a,re very (:()nv(nfient t() w()rk witl~. In(le(xt, in tlm ~ext (:ha,l)t(~r we will investiga,te tim transf()r~m,ti()~s (',tf(~('t,(~(l ()~ I)tm,s(;-t)lm~(~ (l()l~mil~s, I)(nl~(t(~(l I)y ellit)ses, t)y t)r()t)a,ga,ti()~ tl~r()~gh l)araxial ()t)ti(:aJ systemics. We will re(:()v(~r th(~ t)i(:t()rial visua,lizatio~s ()f tim Wigl~er (:harts eh~(:i(ta,t(~(t i~ w 8.3 i~ temps ()f the se(',on(torder mom(u~ts of the Wig~mr (tistributio~ fim(:ti()n.

8.6

Summary

We have investiga,te(t the t)araxiaJ propagation properties of tile Wigner distribution function. Unlike the four possible i n p u t - o u t p u t relations for optical systems when the signals a,re represented through the conq)lex amplitude or the respective Fourier spe(:trum, there is only one transfer relation when the signals are represented through the Wigner distribution functions. It basically involves an aitine transformation of the variables of the Wigner distribution function in the same form as that of the ray variables of geometrical optics. Thus, on one side the Wigner distribution function, being defined in terms of the optical wave function, captures all the wave aspects of light. On the other side, due to its transfer law, it retains in its dynamics the local features of

1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 457

r a y p r o p a g a t i o n , local values of the o u t p u t W i g n c r d i s t r i b u t i o n f u n c t i o n b e i n g d e t e r m i n e d by local values of t h e i n p u t W i g n e r d i s t r i b u t i o n function. T h e n , p a r t i c u l a r a t t e n t i o n has b e e n d e v o t e d to t h e r e l a t i o n b e t w e e n t h e W i g n c r d i s t r i b u t i o n f u n c t i o n a n d t h e f r a c t i o n a l Fourier t r a n s f o r m , w h i c h has b e e n recognized to p r o d u c e a r o t a t i o n of t h e W i g n e r c h a r t . Such a r o t a t i o n b a s e d r e l a t i o n has b e e n seen to be basic to t h e p h a s e - s p a c e t o m o g r a p h i c s t r a t egy for r e c o n s t r u c t i o n of c o m p l e x wave-fields, be t h e y classical o p t i c a l d i s t u r b a n c e s or q u a n t u m m a t t e r - w a v e s . Finally, t h e g e n e r a l form of t h e t r a n s p o r t e q u a t i o n for t h e W i g n e r distrib u t i o n f u n c t i o n has b e e n derived. It b e c o m e s t h e sarnc as Liouvillc's e q u a t i o n for t h e classical p h a s e space d i s t r i b u t i o n f u n c t i o n in t h e case of p a r a x i a l propa g a t i o n t h r o u g h q u a d r a t i c index profile media. Problems

1. (a) Prove that tile Weyl transform of tile operator F(~, ~) -- f(q', ~)~(~, ~) is Fw

(q,p)

P P P P

--

~ ////l J,l

, - , - , , - ,, ~

aq ap aq ap Jw(q'

,

,, p,,) -~t(.'-."

, P ) 9 w (q ,

e

)p-(p'

-~"

)q+p'

~"-~"~'

]/h

,l i

where fw(q,P) and 9w(q,P) are the Weyl transforms of f(~,~) and ~(~,~) respectively. (b) Verify that Fw(q,p) r Gw(q,p), where Gw is the Weyl transform of G(~,~) - g(q,p)f(q,p). 2. Prove that the the paraxial propagation law for the ambiguity fimction .A(q, ~) writes as

A,,(~,~)=/]C~(

~,~,~,~').a,(~,~')|

JJ

with a definite relation between GA (q, ~, q', ~') and G(q, K,,q', g'), i.e. between the system responses to the incoming signal in the Wigner and ambiguity function representations; or equivalently as A,, (u) -- A~ (M -1 u), where M is the ray-matrix entering as well (8.3.35) and u-- (q, ~)T denotes vectors in the ambiguity flmction phase plane. 3. Using the propagation properties of the Wigner distribution function, prove that the propagation of an incoherent wave field, described by the mutual intensity flmction (see Eq. (7.5.16)) J(ql,q2;zi)---- I(ql;z~)5(ql --q2), through an A B C D optical system, is like that of a coherent wave field described by the coinplex amplitude ~(q~ ) -- I(q 1; z~)g* (0, q~ ), g* being of course the spread function of the system. The above is known as the Van CittertZernike theorem. 4. Find out the positive, negative and zero-contrast planes for a freely propagating wave field, described at the plane z -- 0 by the mutual intensity (8.3.45), using the paraxial propagation properties of the Wigner distribution function. Take, in particular, ~ -- ~0-

458

Linear Ray and Wave Optics in Phase 5pace

References [1]

A list of titles concerned with the W i g n e r distribution flln(:ti()n a xl(t related issues within the (tlmntlun me(:hani(:al ('ontext, enri(:hing [4] ()f oh.6, is l)el()w. For tile (tuantunl ot)ti(:s view c)f the Wigner rel)I'eSeld,at~ioil see [12.9] ml(l [12.10]. [1.1] T. Takabayasi, "Tile forlmflatioil of qlmntlml mechanics ill tenlls of enseinble in phrase spa, c(,", Pr()gr. Tll('(w. Pllys. 11, :H1-373 (1954). [1.2] G.A. Baker, "ForlIl, llal, ioll of (llmIlt,ml xnecllmlics based oil l,lle qlm.si-prot)al)ility tlistritmtit)ll illtlll('c;(l ()11 phase space", Ptlys. Rev. 109, 2198-2206 (1.q58). [1.3] R. Klfi)o, "Wiglmr rcpreselltatioll of qlmld, lu~ operators a~tl its applicatioi~s to electro~s iI~ a ~mg~ml,ic tiel(l", .I. l)l~ys. Soc..lap~ 19, 2127-2139 (1!)64). [1.4] K. h~m;, E. (~)ziz~ir M. l{(me~fim~ m~(l P.F. Zweifl;1, "Wig, mr ~mtl~()(l i~ q~mi~t~m~ statistical ~mclm~fics", .l. Matin. l~l~ys. 8, 10.97-11(18, (1967). [1.5] G.,I. Ih~ggcri, "()~ lfl~sc,-space (h;scriptio~ ()f (l~mI~t~u~ ~m('l~a~ics", l)r()gr. Theor. l't~ys. 46, 17():g1712 (1.971). [1.6] E..I. tlc'lh.'r, "Wig~mr l)lmse Sl)aCe ~mtl~o(1. A~alysis for sc~,~icb~ssical al)l)licatiol~s", .]. Clm~n Phys. 65, 128.()-12(.)8 (1.()76). [1.7] I). I I o t ~ a~t(! l$..I, llih'y, "()~t a (lli~-tlltliIIl algelmdc al)l)r()aclt to a gt'~mralize
[2]

As e,Int)hasiz('~(l i~ (:l~. 6, t}~(', i~lI'()(tu(:tion ()ftlm Wign(;r ()t)ti(:s was sugg(;ste(l t)y Bastiaans, whose seIninal t)at)ers gations, concerning as well the I)ropagation properties of flm(:tion in relati()n t() first-()r(ler optical systems, are in

(listril)~lti()~l fllnction in an(1 su])s(;(lu(;nt investithe Wiglmr distribution [6] ()f (:h. 6.

[3]

E.P. Wigncr, " O n the qua ntux~l correction for thermodynami(: e(t~filibrium", Phys. Rev. 40, 74%759 (1932).

[4]

An accurate analysis of th(; (:(mditions t h a t uniquely lead to tile formulation of tile q u a n t u m phase st)ace ret)resentative in the form of tile Wigner distribution function can be found in

1D First-Order Optical Systems: TransFer Laws for the Wigner Distribution Function 459

[4.1] E.P. Wigner, "Quantum-mechanical distribution functions-Revisited", in Perspectives in Quantum Theory, W. Yourgrau and A. van der Merwe (eds.) (MIT Press, Cambridge, 1971), pp. 25-36. [4.2] R.F. O'Connell and E.P. Wigner, "Quantum-mechanical distribution functions: conditions for uniqueness", Phys. Lett. 83A, 145-148 (1981). [4.3] J.G. Kr/iger and A. Poffyn, "Quantum mechanics in phase space I. Unicity of the Wigncr distribution function", Physica 85A, 84-100 (1976). [5]

In relation with the negativeness of the Wigner distribution function see [5.1] L. Cohen, "Generalized phase-space distribution functions", J. Math. Phys. 7, 781786 (1966). [5.2] L. Cohen, "Can quantum mechanics be formulated as a classical probabilistic theory?", Philos. Sc. 33, 317-322 (1966). [5.3] M.D. Srinivas and E. Wolf, "Some nonclassical features of phase-space representations of quantum mechanics", Phys. Rev. D 11, 1477-1485 (1975). [5.4] R. Groblicki, "A generalization of Cohcn's theorem", J. Math Phys. 24, 841-843 (1983). [5.5] D. Dragoman, "The origin of negative values of the Wigner distribution flmction", Optik 111, 179-183 (2000).

[6]

As to the Weyl formalism and its relation to the Wigncr method see also [5.], 5.3] and [1.4]. The paper by Shenoy and Parks [6.6] presents an elegant and interesting investigation of the relation between the Weyl correspondence (with the related Heisenberg group formalism) and the time-frequency analysis. [6.1] C.L. Metha, "Phase-space formulation of the dynamics of canonical variables," J. Math. Phys. 5, 677-686 (1964). [6.2] B. Leaf, "Wcyl transformation and the classical limit of quantum mechanics", J. Math. Phys. 9, 65-72 (1968). "Weyl transform in nonrelativistic quantum mechanics", ibid., 769-781. [6.3] L. Cohen, "Quantization problem and variational principle in the phase-space formulation of quantum mechanics", J. Math. Phys. 17, 1863-1866 (1976). [6.4] M. Springborg, "Phase space functions and correspondence rules", J. Phys. A: Math. Gen. 16, 535-542 (1983). [6.5] A.M. Ozorio De Almcida, "The Wcyl representation in classical and quantum mechanics", Phys. Rep. 295, 265-342 (1998). [6.6] R.G. Shenoy and T.W. Parks, "The Weyl correspondence and time-frequency analysis", IEEE Trans. Signal Process. 42, 318-331 (1994).

[7]

Basic references concerned with Gaussian beams are in [13] of ch. 2.

[8]

As an illustration of the Wigner representation based approach to analysis and synthesis of optical systems on account as well of the wavelength dependence of the Wigner optics matrices, we suggest [8.1] W.D. Furlan, G. Saavedra and J. Lancis, "Phase-space representations as a tool for the evaluation of the polychromatic OTF", Opt. Comm. 96, 208-213 (1993). [8.2] W.D. Furlan, G. Saavedra, E. Silvestre, P. Andr6s and M. J. Yzuel, "Polychromatic axial behavior of aberrated optical systems: Wigner distribution function approach", Appl. Opt. 36, 9146-9151 (1997).

Linear Ray and Wave Optics in Phase 5pace

460

[9]

[8.3]

J. Lancis, E.E. Sicre, A. Pons and G. Saavedra, "Achromatic white self-imaging phenomenon: an appproach using the Wigner distribution function", J. Mod. Opt. 42, 425-434 (1995).

[8.4]

D. Wang, A. Pe'er, A.W. LohmanI~ and A.A. FricscIn, "Wigner algebra as a tool for the design of achromatic optical processing systems", Opt. Eng. 39, 3014-3()24 (2000).

An (;xhaustiv(; t)it)li()grat)hy (m the Tall)ot-Lau (;ffe(:t, ut)(lat(;d t() 1997, is in [9.6]. Bcl()w arc (tu()tc(1 t,t~c s(;~fi~ml t)at)(;I'S 1)y Tal|)()t [:).l] a~(t La~ [:).2], 11~(; |)asi(: rcvi(;w 1)y Pat()rski [!).:~] a~(l l l~c r(',(:c~d survey [!).7], wl~(;r(; al()~g witl~ tfistori(:al ~(~l(;s I1~(; l)()l~',~lial ()f I1~(; Talt)()l ('fl'e('t f()r at)l)li('ati()1~ t()()l)ti('al ('()~l)~ting an(t ()l)ti('M lIl(~a,'-;llr(;Ill(;llt, is (;~l)t~asiz(;(l. Pal)(',rs [:).4, .~).s] ar(; Sl)(;('ifi('ally r('lal(;(1 t() 1,11(; [9.1]

tt. F. q,tll)()l,, "F;tcts relati~g 1,() ()I)ticM Scie~me", l)l~il, lklag. 9, 401-407 (1836).

[9.2]

E. l~m~, "lh;~g~gsersclmilnu~ge~ a~ l)()l)l)elraster~", A ~ . I'l~ys. 6, 417-4237 (1.948).

[9.3]

K. Pat, orski, "'l'l~e self i~,tgi~g l)Im~
[9.4] A.W. l~olu~m~, .l. ()je(la-(;astafie(la m~(l N. Strcil)l, "Sl)al, ial l)erio(licities ix~ coherr tu~(l partially (:()here~t fiehls", Oi)t. Acta 30, 125.()-1266 (1.()83). [!).5] .l.O.je(la-(~a,stafier m~
l.lw

'l'MI)()l,-l,a~

<'tFe(:t"

ix~

[.9.7] C. Ztlr W. Wmig, E. I)ai alibi L. Li~l, "Si~lll)h; l)ri~l(:il)les of tll<' 'l,dl)()t effect", Optics & l'l~ot. News 15 (11), 46-50 (2()()4), a~(l refere~(:es tl~erei~. [10] A Illis(:cllazw, a ()f tit l(;s (:()ll(:crll(',(t wit ll t,lw, al)l)li('ati(nh'~ r tll(', fi'a('li
1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function 461 [10.8] A.W. Lohmann, D. Mendlovic and Z. Zalevsky, "Fractional transformations in optics," in Progress in Optics, Vol. XXXVIII, E. Wolf (ed.) (Elsevier, Amsterdam, 1997), ch. 4, pp. 263-342. [10.9] A. Torre, "The fractional Fourier transform and some of its applications to optics," in Progress in Optics, Vol. 43, E. Wolf (ed.) (Elsevier, Amsterdam, 2002), ch. 7, pp. 521-586. [10.10] T. Alieva and M.J. Bastiaans, "Wigner distribution and fractional Fourier transform" in Time-Frequency Signal Analysis and Processing. A comprehensive Reference, B. Boashash (ed.) (Elsevier, Oxford, 2003), sect. 4.8, pp.145-152. [10.11] H.M. Ozaktas, Z. Zalevsky and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley & Sons Ltd, Chichester, 2001). [11] Basic titles on tile Radon transform, whose original application was mainly for medical computed tomography, are [11.1]-[11.5]. The classic paper by Radon, where he showed how a reconstruction from projections was possible, is reproduced in facsimile in [11.2], and in English translation in [11.3]. Formalization and applicative suggestions regarding the Radon-Wigner transform and its relation to the fractional Fourier tra.nsforin are respectively in [11.6], [11.7] and [11.8]-[11.10]. [11.1] J. Radon, "Uber die Bestimmung von Funktionen durch ihre Integralwerte lgngs gewisser Mannigfaltigkeiten", Ber. Saechs. Akad. Wiss. (Leipzig), Math-Phys. K 69, 262-277 (1917). [11.2] S. Helgason, The Radon Transform, Progress in Mathematics, Vol. 5 (Birkhafiser, Basel, 1980). [11.3] S.R. Deans, The Radon Transform and some of its Applications (Wiley, New York,

~983).

[11.4]

H. H. Barrett, "Tile Radon transform and its applications", in Progress in Optics, Vol. XXI, E. Wolf (ed.) (Elsevier, Amsterdam, 1984), ch. III, pp. 217-286. [11.5] C.N. Guy, "Tile second revolution in medical imaging", Contcmp. Phys. 37, 15-45 (1996). [11.6] J. C. Wood and D. T. Berry, "Radon transform of the Wigner spectrum", in Advanced Signal Processing Algorithms, Architectures, and Implementations III, F.T. Luk (ed.), Proc. SPIE 1770, 358-375 (1992). [11.71 J. C. Wood and D. T. Berry, "Linear signal synthesis using the Radon-Wigner transform", IEEE Trans. Signal Process 42, 2105-2111 (1994); "Radon transformation of tilne-frequency distributions for analysis of multicomponent signals", ibid., 3166-3177. [11.8] A.W. Lohmann and B.H. Softer, "Relationships between the Radon-Wigner and fractional Fourier transforms", J. Opt. Soc. Am. A 11, 1798-1801 (1994). [11.9] D. Mustard, "The fractional Fourier transformation and the Wigner distribution", Austral. Math. Soc. Ser B 38, 209-219 (1996). [11.10] T. Alieva and M.J. Bastiaans, "Radon-Wigner transform for optical field analysis", Proc. Int. Conf. on Opt. and Optoelectron., Dec. 9-12, 1998, Derha, India, Vol. 1, pp. 132-135; http://www.sps.ele.tue.nl/members/M.J.Bastiaans/pdfs/ico198_ 5.pdf [12] Since the pioneering paper by [12.8], where the tomographic strategy for the construction of the Wigner function was firstly suggested, the q u a n t u m t o m o g r a p h y has found a great deal of applications, having also gained new insight from the

462

Linear Ray and Wave Optics in Phase Space recognized R a d o n - W i g n e r / f r a c t i o n a l Fourier transfornl relation. A few titles concerned with the phase-space t o m o g r a p h i c r e c o n s t r u c t i o n of complex wave-field are listed here. We direct the reader to [12.9] an(l [12.10] for all a('curate a(:count of q u a n t u i n t()Inograt)hy. An enlightening survey of the statlls of art of tim q u a n t u m states re('(mstru('.tion 1)ased oi1 the Ra(l(m-Wigncr traIlsform is in [12.s].

[12.1] K. Vogel and H. Riskcll, "I)ctcrmination of qua.siprobability distributioIls in terms of probal)ility (listril)llti()Ils for tim rotatc(l qimllys, l~.cV, l~c(,t. 70, 1244-1247 (1.().()3). [12.3] M.(;. [~;tyIIl(;r, M. lh'.(:k all(i I).F. M(:Alistcr, "C()~q)h:x wav(~ticl(! rc(:(),,str~ctio~ ~si,~g i)lms(~st);wc (,()~ogral)l~y" , F'l~ys. I{cv. I,(;(,t. 72, I 137-1140 (1.9.94). [12.4] I).F. McAlistcr, M. Bc('k, I~. Clark(;, A. Mayer a~(] M.G. IbW~m'r, "OI)ti(:al phase r(;tri(;val |)y phase sl)a('c t()~()grai)t W m~(l fi'a('ti()~ml-()r(lcr F()~ricr trm~sfi)r~s", Opt. Lctt. 20, 1 lt~l-1183 (1.().()5). [12.5] .J. "I'll all(I S. Tmmtra, "Wav(; ticl(l (lctcr~fi~at, i()~ ,~si~,g t()~()gral)l,y ()f tim a~nl)ig~fity f~('ti(),,", P|Ws. R,:v. E 55, 1.()46-194.() ( 1.().()7). [12.6] .J. T,I ;LII(I S. "l'a~n~ra, " i , m l y t i c rclati()~a fi)r r(;(:()vcri~g tim ~H~t~ml i~t.(;nsity by ~(;a~s ()f i~tc~sity i~fi)r~mti()~", J. Opt. S()c. A~. A 15, 202-20(i (1.().()8). l)l~;m(' rc('()vcry ()f vcrti('al-('avity s~rfa('(;-(,~itti,~g las(;r I)y llSC ()f tt~(; a~d)ig~it, y fi~ct,i()~", Ol)t. l,(;tt. 27, 151.()-IS21 (2()()2). [12.8] .J. l~('xl,ra~(! m~(! l ). I~(;rt,ra~s(l, "A t,()~()gral)l~i(: al)l)r()a(:l~ t() Wig~mr's fi~(:tion", l"()~n~(l, l)lws. 17, 397-4()5 (1987). [12.8] M.(~. l{ay~ncr, "Me~-,~ri~g the q~m~t~n ~ncctm~i(:al wave fi~nction", (~ontcmp. l)t~ys. 38, 343-355 (1.().(17). [12.9] U. Leo~dmr(tt, Measuring the q'uantwm stat~ of light ((,'a~fi)ri(lgc [h~iv. l)rcss, Camt)ri(lgc, 1.().()7), ('l~. 5, pp..()t~- 143. [12.10] W.I ). S('t~l(;i(:]~, Quantum Optics in I'hasc b'pa('c (Wilcy-VC]t, Bcrli~, 2()()1), pp. 143-146. [13] F()r a~ a('(:~r;~t(; a~alysis ()f tim tra~sl)()rt ('q~ati()~ fi)r tim Wig~mr (tistri|)~tion fml('ti()li wc sligg(;st tim t)at)er 1)eh)w. R(;a(leI'S ilit.(;r('st,(',(l ii~ the (tllalitlllll inechanical a sI)(;('ts ()f tll(' ('v()lllti()xi (;(tuati()n ar(; (lir(;('.t(;(1 t() [1..()]. [13.1] It. Brcmmcr, "General remarks concerning theories dealing with scattering and (liffractioii ixl ra~l(lo~n inc(lia", Radio So. 8, 511-534 (1973). [13.2] M.J. Bastiaans, "Transport equations for the Wigncr distribution fimction", Opt. Acta 26, 1265-1272 (1979); "Transport equations for the Wigner distribution function in an inhomogcncous and dispersive medium", ibid., 1333-1344. [14] For the W i g n e r r e p r e s e n t a t i o n of n u m b e r flmctions see [1.3], [1.9] and [14.1] F. Bopp, "Statistischc Mcchanik bci StSrung des Zustands tines physikalischcn Systems durch die Beobachtung", in Werncr Hcisenbcrg und die Physik unserer Zcit (F. Vicweg, Braunschweig, 1961), 128-149. [14.2] N.L. Balazs and H.C. Pauli, "Quantum-mechanical operators and phase-space oporators", Z. Physik A 282, 221-225 (1977). [14.3] Go. Torres-Vega and J.H. Frederick," Quantum mechanics in phase space: new approaches to the correspondence principle", J. Chem. Phys. 93, 8862-8874 (1990).

9 1D First-Order Optical Systems: Moments of the Wigner Distribution Function

9.1

Introduction

Although the Wigner distribution function of a signal ~ cannot be interpreted as the energy distribution of p in space and frequency, it provides an effective characterization of the signal. For a specific signal, indeed, we can evaluate (at least in principle) the Wigner distribution function and from the analysis of this function we can get an idea of how the energy of the signal is distributed in space and frequency. As all example, tile energy of a Gaussian signal comes to spread out over all the (q, ~) plane, though remaining mainly concentrated within a region with definite rms extensions in both q and ~. Alternatively, it is possible to characterize the signal without giving the complete Wigner representation, i.e. the values of the Wigner distribution function throughout the whole (q, ~) plane. Also, due to the uncertainty relation, such a point-wise description is commonly intended as meaningless. In effect, an equally effective characterization of the signal can be pursued by means of the moments of the Wigner distribution function, which, as we will see, are directly linked to the moments of the signal. Moment methods for treating general problems in image manipulation and analysis have been pioneered by Hu [1.1], and later investigated and refined by many authors [1]. Object reconstruction is one of the basic instances addressed by the moment methods. The moments of a 2D function f (x, y), as, for instance, the image irradiance distribution, are defined through the integrals

#~h (X f f ~'~h(X,y)f(x, y)dxdy.

(9.1.1)

The subindices r and h are positive integers. The sum r t h defines the order of the moment. The functions {L'~h(X,y)}~,h are known as the basis set. The determination of the integration region 9l, the choice of the basis set {U~h}~,h

464

Linear Ray and Wave Optics in Phase Space

and accordingly the integration variables may be addressed in relation with tile specific problem of concern. As d e m o n s t r a t e d by Teague [1.3], for instance, 2D image inoinents with respect to Zernike polynomials have very convenient transforInation t)rot)crties in rest)onsc to image transforInations like scaling, translations or rotations. Also, an image has a uni(tuc apt)roximatc reconstruction in either real ()r Follricr space in terms of a fiifitc set of slmh moments. The sinai)lest f()I'iil of (9.1.1) takes as a t)a.sis set tim ~ o n o i n i a l s {atrflh }r,h,, l,lnls yicl(lillg tllc~ g~'o'm,(:tri(: 'm,o'm,('.'nt,s m,~h, t h a t will t)e (:(msi(lere(t in what foll()ws. Gc(nlmtri(' lll(ml(ud,s lmve ml ild, llit, ivc, ra,tllcr visi|)lc ild,crprctatio~l. In flu:t, the tirst-(n'(lcr st)atial m()nlcnts Sl)c(:ify the l()(:ation of tlie (:cater of tllc t-M(I (tistritnlti(nl, tile sc('(nl(t-()r(lcr stm.tia.1 nl()Illcllts (:tm.ra.(:tcrizc tile size a,ii(1 tile oric~tatio~ ()f tlm sig~m,1, m~(l l~iglmI'-or(te.r ~ o ~ e ~ t s (lcteI'~i~m fi~rthcr (t(;tails ()f the tichl (listrit)~ti(n~, lik(; tl~c (t(~gr('~('. ()f syllnll(;t,I'y l)()ssil)ly with rest)e(:t t,() tlm (:e~t,roi(t ()r tl~e (tegree ()f I)eake(l~ess. Her(;, we slmll })c (:()l~('erl~(;(l witl~ 1,11(; g(~(n~(;tri(: (K(n~ ll()W 1)11 sin~t)ly refcrr(;(l t() a,s ~()~(u~t,s) ()f tl~(~ Wig~(;r (listl'il)~ti(n~ fllllt:t,i()ll. We lllay (listinguish |)(;tw(;en local m~(l global ~()~(mt, s. Tl~(; l()(:al ~()~(ud, s arc (l(;t(u'~d~(;(t t)y treati~g tl~(; Wigt~(;r (listril)~ti(n~ as a fln~('ti()~ ()f tim fr(;(l~UU~(:y h" f()r a giv(u~ st)a(:(; wn'ia.1)h', q ()r as a fi~l~(:t,i(n~ ()f the Sl)a(:e, w~,rial)le q f()r a. tix(;(l fr(',(llUU~(:y w~,riat)le ~:; tl~(;rcf()r(;, t,l~(;y (l(;l)(u~(l ()~ q ()r h', m~(l giv(; i~(li(~a,ti(n~ a.1)()~d, t,l~(: l()(:a.1 1)clmvi()~r ()f t,l~(; Wig~(;r (tistrit)~d,i(n~ flln(:t,i()ll. Iil (:()lltrast, I,l~e gh)l)al moments arc (l(;tcrx~i~(;(l |)y (:(n~si(l(;ri~g th(; Wign(;r (list, ri|)~ti()~ as a fl~('t,i(n~ ()f t)()th q ai~(l h:; a,(:('()r(li~gly, tt~(w are i~(l(',t)(u~(l(ud, ()f t)()tt~ va,riat)h',s aI~(l 1)r()vi(lc gl()|)al iI~f()rnm,t,i()n al)(n~t tl~c Wig~(;r (tist,ri|)~d,i()n fi~('ti(n~. I~ (:()nf()r~ity witl~ tl~e linear ai)t)r()xi~m.ti()~, wlfi('l~ lm,s ~n~(terli(;(t the t)res(ud,ati()ns l,l~r()l~gl~()~tl, tt~(; I)()()k, we li~fit (n~rs(;1v(;s t() (:()l~si(lcr (n~ly first- an(t s(;(:()l~(l-()r(l(;r ~(n~(ud, s. W(; will s(;(; tlm, t (:()~t)l(;x (~,aussian sigl~als arc fully in(livi(iualizc(1 t)y tl~c s(;(:()~(t-or(tcr n~()ments, tim r(',hwa,nt Wigncr (tistribution fun(:ti()n being m~i(l~u;ly ext)ressi|)le i~ terms of tl~en~. Wl~(',~ (:(n~si(lering art)it, ra, ry non Ga, ussian signals, (:onsi(l(;rati()n of the s(',(:on(t-or(ter ~noments is enough t() give a gl()t)al charactcrizati(m ()f the sig~m,1 an(t ()f tim transformations it ext)(',ri(mc(',s ()n t)assing through A B C D ot)tical systems. Also, they enable a pictorial visualization of the signal t)ropagation in the Wigner plane, which propagation being mirrored into the phase-t)lane transformations of the variance-inatrix ellit)se. Notably, the t)ropagation laws for the first-order and the second-order moments directly involve the transfer m a t r i x of linear ray-optics. Thus, we will see that, just rephrasing the results of w 5.5, the first-order moments, which represent the centroid of the Wigner distribution function, evolve in the same way as the ray-variables. Correspondingly, the second-order moments transform t h r o u g h a congruence transformation by the ray-transfer matrix, once properly embedded into the variance matrix. This enables a unique determi-

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

465

Tools of Wigner optics (paraxial optical systems)

9 kV (q, to) 9 transfer laws for kV(q,h")

and/or

9 v a r i a n c e m a t r i x , Y. 9 propagation

l a w s for 7_.

F I G U R E 9.1. T h e second-order moments of the Wigner distribution function yield an effective description of the light propagation properties t h r o u g h paraxial optical systems.

nation of the second moments of the signal at the observation plane in terms of the second moments of the signal at the object plane. Therefore, we may add to the "tools of Wigner optics" for the description of light propagation through paraxial optical systems, that comprise the Wigner distribution function a,nd the relevant transfer laws (see Fig. 8.1), the second-order moments and the pertinent propagation laws (Fig. 9.1). This reinforces the link of Wigner optics to ray optics. It results from the presentation given in Chapter 8 that the Wigner distribution function retains in its kinematic structure the wave-optical description of signals. In contrast, in its dynamics it resorts to the transformation laws of ray-optics, its arguments being transformed as the geometrical ray variables. So, the results here will prove that such a ray-optical like characterization of the Wigner distribution function dynamics, in relation with paraxial optical systems, reflects a,s well in the propagation laws of its second-order moments through such systems. Basic notions concerning the definition of the moments of a 1D function, not necessarily interpretable as a density function, and of the local and global inoments of a 2D fimction, not necessarily interpretable as a (tensity fimction, are given in Sect. 9.2. Section 9.3 develops some preliminary results, useful for the subsequent calculations. Section 9.4 addresses the specific calculation of the first-order and second-order, both local and global, moments of the Wigner distribution function. A variety of expressions are worked out, each having its counterpart in the signM analysis context. The results of Sect. 9.4 are then applied to Gaussian Wigner distribution functions (Sect. 9.5), which, as already remarked, are fully characterized by the relevant second-order moments. The variance matrix and the associated ellipse in the Wigner plane are introduced. The transfer laws for the variance matrix are finally deduced. Section 9.6 extends the results of Sect. 9.4 to arbitrary non Gaussian Wigner distribution functions. The propagation laws of the second-moments in arbitrary A B C D systems are explicitly written down and visualized in terms of the transformations of the variance ellipse in the Wigner plane, specifically in relation to the propagation through a free-medium section, a thin lens, a beam expander and

Linear Ray and Wave Optics in Phase Space

466

a Fourier t r a n s f o r m e r . Finally, a brief a c c o u n t of t h e h i g h e r - o r d e r m o m e n t s

a,nd the i n h e r e n t p r o p a g a t i o n laws is p r e s e n t e d in Sect. 9.7.

9.2

Basic notions on moments

Wc give 11()w tll(' tm.si(: (tefilfititms ()f the lll()nl(;~lts ()f t)()tll ()11~'~a,II(t two dilnensi~nml fln~:ti~n~s. Ewn~ t l ~ g l ~ tl~(; (t(;fi~i~g relati~n~s fiw~ally relm~(l~(:e tt~ose c ~ n n ( n d y ild,r()(l~(:c(l wl~',l~ ~lcali~lg witll ~h'l~sil,y fitl~(:l,i(nls, 1,11c inl,(;rl)rcl, al,i()~ ()f 1,11(; r(;s~ll, illg q~aid, il,i(;s ~my (:(n~l(; 1,() l)(; ~1()1, 1,11(: sm~l(', si~(:(; we (:(msi(l(;r fl~n(:ti()ns, 1,trot i1~ gcn(',ral (:ml~l()t t)(; ~ulth:rst()()(l as (h;llsil,y fll~l('l,i()ns [2]. T h e r~'a(h;r will r(;(:()g~liz(', l,l~at s(m~(', (~f 1,11(;(h'tixlil,i(nls ]M(~w llaw; lw,t;n llSe(1 ill I)revi(n~s mmlys(;s wl~<;~ (l('ali~g witl~ flu; sl)a(:(: a.l~(l/()r fr(:(pu;~l('y (;xl,(;~lsi()~ ()f a, sig~m,1 ()r a 1)lms(;-I)la,~(; (lisl,ri|)~ti,)~, as, f,w il~sl, a,~(:(:, i~ w 4.6.4 i~ (:()~(;('ti()~ with the ()I)ti(:a,1 ~(:erl,ai~1,y rclati(m, wld('l~ i~l(h;e(l will I~e ref()rnnflate(1 in l,(;r~s ()f 1,1~(; s(;(:~)~l-~)r~l(;r ~ ) ~ ; ~ t , s

9.2.1

()f 1,11(; Wigl~(:r (lisl,i'il)~lt, i~)~l fl1~1(:1,i()~1.

Moments of a one-dimen,s'ional.flm, ctio'n

C()llsi(l(;r a real finl('ti()ll D(q) (l(;tin(;(l ()v(:r t,ll(; r('al li~l(:: q E IR, whi(:h is slq)t)()s(;(t t() llav(; a 1)()sitiv(' average, lm,lll(:ly m,,-

/ 7)(q)dq

> (),

(9.2.2)

als() (:alh;(t 1,11(; Ze'l'o-m'&"l" 'mo'm~:'nt ()f D. As llmm.l, llllless ()tll(;rwis(,, st)(;(:iti(;(t, im(;gra.ls arc ilm,n(h;(1 ()v(;r the wh()h; (l(nlm,ill ()f tlu', illt(:grml(l finl(:ti()n. hi g('al(',ra,1, D ( q ) is ll()t a,ssl1111(',(t t() |)(', l)()sitiv(; f()r all tlu', vallu;s ()f q in its (l()nmin, a,n(t so it ('aal 11()t l)(; ilm;rt)r(;l,e(t as a (icllsity flnl(:ti()ll, tt()w(;v(;r, res()rting t() t,ll(; st, rat('gy (~()Ilnll()nly a,t)t)lie(t wll(;ll (l(:alillg witll (l(:11sity fllnctions, ()ira m a y (tcfinc t,lm first-oT"dcr moment ()f 7) t)y t,llc 11sire,1 ild, egra,l: __!_1

Ill I - - m o ,/qV(q)dq.

(9.2.3)

Evidently, m~ lo(:atcs tim center of gravity of D, i.e., tim vahm of q for which, t h i n k i n g of 79 as a mass (listribution along a s t r a i g h t line and allowing for imgative nmsscs where D(q) < 0, the d i s t r i b u t i o n is pcrfe(:tly t)alan(:ed. Likewise the secon(t-(~i'(tcr ordinary and central nlomcnts, 1l 2 and m2, write

a,S 112__L1 m0.

/q2~)(q)dq,

m2_

1 ~0

f

2 (q -- ml ) 2 D(q)dq - n 2 - m~.

(9.2.4)

If D(q) is n o n n e g a t i v e , the above can be considered as the variances of D respectively a b o u t the origin q = 0 and t h e centroid m 1. In fact, within t h e

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

467

intuitive analogy of 79 with a mass distribution we can say that n 2 and m 2 give indication about the spread of the masses along the wire with respect to the origin and the center of gravity m l , respectively. If 7:) goes negative somewhere, this interpretation must be handled with care, since then, as we will see, both n 2 and m 2 inay become negative. Notably, regarding m 2 as a fimction of m~, the expression (9.2.3) just furnishes the value of m~ for which m 2 is minimized. The higher-order moments n~ and m., r > 2, arc then

n~

__

1 -~o

/

qr

7P(q)dq,

m~

__

~1

/

(q-ml)

r

T)(q)dq.

(9.2.5)

In particular, the central moment of order r relate to tile ordinary moInents of order zero up to 7". In fact, using tile binonfial expansion ( q - m l ) r E~=0 (~)(--)~-hqhm[ -h in the expression for m~ and integrating term by term, we obtain m r --

L

(rh)(--)r-hIlhm r-h I ,

(9.2.6)

h=0

where specifically n o = 1 and n 1 = m 1. W i t h r = 2 we regain the relation of m 2 to n 2 and ml, as shown in (9.2.4). Also, we note t h a t by an appropriate translation of the coordinate origin to m I the central moments (:an be made to coincide with the ordinary moments. The symbol q used here to denote the independent variable does not strictly relate to the space coordinate of optics, signifying in general a real quantity. Thus, the definitions above can be applied to the complex amplitude and the frequency spectrum of an optical signal as well. In that case, the pertinent 7)function identifies as the absolute square IP(q)12 or I@(t~)l2, which actually have both the conceptual valence and the m a t h e m a t i c a l properties to be considered as density functions. Besides, as we will see in the next section, the moments of Ip(q)l 2 and I@(n)l 2 can be interpreted in terms of the expectation values of the position aim frequency operators, and their powers, in the "state" p.

Averages in the space and frequency domains as expectation values of the operators ~ and ~ and their powers We show that the moments of Ip(q)l 2 and I~(n)l 2 are interpretable as expectation values of the position and spatial-frequency operators ~ and ~, and their powers in the "state" p. This conforms to the parallelism between the geometric/wave and classical/quantum correspondences. Also, the seemingly abstract operator picture, we have used in connection with the wave-optical description of the light propagation properties of paraxial optical systems, gains both calculational and physical merit revealing a profitable tool for the characterization of optical signals and systems.

468

Linear Ray and Wave Optics in Phase Space

According2 to the coiilIilOn definition, tile ext)ectation, or average, value of an o p e r a t o r O in a given "state" is i n t e r p r e t e d in the sense of tim inner product

(0)--(-,0.).

(9.2.7)

T h e dots can inter(:ha.ngea,t)ly t)e ret)laced t)y p(q)r 75(h') witt~ tlie involved ()t)erat()r, i.e. ~ anr ~:,, t)r ac(:(mlingly giv('.ll ill tllr q r rr fi)rlils. Ttle illtr162 illll~lir1621)y (9.2.7) will l.llr 1~' r t llr q- r h,-(h)llmill. It is evi,hn~t tl~a,t tl~,', zc'r,, ,,rch~r ~,,u~r ,,f I,;(q)l 2 a~,l I,~(~-)l ~ i(lentify the 11()I'111 II~l[ ~- I[~ I'2, wliir lllay, c,,,lveifielitly 1,,; set ,',tilal t,, ,,lie. Theli h't its (:~lill~lit~; tllr lil~:ali wfi~u; ~ (~f tlie Slm tia.1 fr~:(llU:ll(W. Tlic (letinilig relati(m (9.2.3), wlii(:li sl)~;~:i|i(:ally writes a.s ~, -

is ~'xtm'ssilfl~: a,s

/

~: I@(,,-)1.2 d~:,

(9.2.s)

~, - ( ~ , ~ ) .

(9.2.9)

The; ('.ah'.~fla,ti~,~ ,,f (!).2.!t) r~,~l~irr t,l~ k~,w'h,~lg;~, ,,f t l~'~ 1;~n~ri~,r Sl,~,~:tr~m~ ~(~). Ih~wew~r, it, ('a.ii a.lsr t~e exl)ressr162 ill teniis r tllr siglm.1 a.liltflitllr ~(q) in a very enligl~i.~:~i~g fl)rlli. II~ far ~mfr i,lie S~l~mrr ~)~l~h~s [~?(h~)l2 = i,~tr we: r rr162 (9 2.8) as Wa*(h;)v?(h:) re,el ~akil~g ,,sr ~,f tl~e l"r

/

F,-

~?*(h)h'~.(h')d~" - 77

///

dh;dqdq'2*(q)~(q' ), i'~qh','-i'~"/dq'.

(9.2.10)

9

d idT/c-

Ile(:~gnizing t,lmt ~.~.-i~,/

--

h; -9

/7'

r

ih'q ~

, w~'~ ~:a.li ilitx,gra.t~, t~y Imrt witli r~'~st)e(:t

d dT/,~(q')]i~(q' -- q)dqdq' -

/'

~ * ( q ) [ - i ~ 2 d( q ) ] d

q.

(9.2.11)

()n a,('(:olmt of tlie rel~reselml.ti(m (4.6.57) ()f ~., the a.t)~w~ ilitegral i(hmtifies the inner t)r(~(hl(:t ~, - ( ~ , ~,~). (9.2.12) T h e first ortter mon~ei~t ()f [~(h:)[2 is then interpretat)le a.s the ext)e(:tation value of the spatial-frequency ot)erator a,, i.e., --@).

T h e explicit calculation t r u m @(~) according to takes the form - i ~ or Likewise the average

~

(9.2.13)

involves the wave flmction p(q) or the frequency specour convenience, and correspondingly the operator h;. value h;~ of the n - t h power h;" can be calculated as -

]

~

I@(~)12 d,~ -

(@, ~@),

(9.2.14)

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

469

and alternatively as

t~n _ f ~ , ( q ) ( _ i ~ ) r ~ ( q ) d q _ (~, ~n~).

(9.2.15)

Therefore, the n-th order moment of 1~5(~)[2 can be brought into relation with the operator 9~ through the average value of ?~, namely ~ = (~),

(9.2.16)

the choice for the wave flmction and operator representation being a matter of convenience in accord with the problem at hand. Finally, one can straightforwardly infer that the average f(~) of a frequency function f ( a ) i n the state ~(q), defined by the integral f(n) - f f(n)I@(~)12

- (@, f(?~)@),

(9.2.17)

whose inherent inner-product interpretation has been displayed, is also obtainable a,s the integra,1

f(t~) - J ~*(q)f(s

- (~, f(?~)~),

(9.2.18)

and hence interpretable as the expectation value of the operator function f(~) in the state ~(q): f(a) = (f(~)). (9.2.19) It has implicitly been assumed that the function f(h;) be expressible through the Taylor series (x)

f(g) _ ~

f~gn _ f0 + fi n + f2 h;2 -+-"",

(9.2.20)

n--O

with fn = fn(O)/n!. Then, applying result (9.2.16) to the series above, which amounts to substituting the operator ?~ for ~c in each occurrence, we end up with the operator function f(~), oo

f(~) -- ~

fn ~ ,

(9.2.21)

n--0

in both (9.2.17) and (9.2.18), the appropriate operator forms being ~ and - i ~ , respectively. The computational relevance of relations (9.2.16) and (9.2.19) is apparent. The calculation of the average ~ , or, more in general, of the average f(~) of a frequency function, does not necessarily require the knowledge of the spectrum ~5(~). It can alternatively be calculated using the signal amplitude ~(q), which

Linear Ray and Wave Optics in Phase Space

470

is indeed simply ma, nipulated t h r o u g h a p p r o p r i a t e differentiations and then inserted into the integraJs implied by (9.2.16), or (9.2.19). Clea,rly, simila,r results hold a,s well for the average of the spa, tia,1 variable q and, in general, of a.ny spatial function 9(q). In fa,ct, sta,rting from the definiti()~s q" q" I~(q)l 2 dq, .q(q) 9(q) I~(q)l 2 dq, (9.2.22) ,

/

/

t,ll(; a lt('rlmtiv(; (:xl)r(;ssi()llS (:a,ll t)('~ worke(l ()~l(, -,

d

)'n.

__

"

q " - . / r (~)(i= ~(~)d,~ (~,~"r :~(q) -./ r

(9.2.2:~)

- (~, .q(~)~),

1,() 1,11(: ( : X l ) I ' ( : s s i ( ) l l ()f ~ i l l (,ll(; h', r(:l)r(:s(:lll,al,i(n1" ~ - i ? ~ .o. Ttl(;r(;f~)r(; r(;lati()~is ((,).2.16) m~(l (9.2.19) f()r tl1(: fr(:(l~U:~1(:y av(:ra,g(:s a.r(: l)aralh:h:(l |)y

a,(:(:()I'(liIlg

q,-V_ (~,,),

:/(q)- (:/(~)),

(9.2.24)

fin" (,11('. Sl)a.l,ial a.v('rag(;s. Tluls, a.s r('gar(ls 1.11(: ('a.l('lllat.i()ll ()f sl)a(,ial a.vera,g(;s 1,11(:frt:(llU:11(:y Sl)(:('l,nIII1 ,~(h;) l)lays 1,11(:sa.111(: r()h: as 1,11('Sl)a,l,ial m~11)lil,~1(le r wl1(:ll (l(;alillg wi(,ll (,If(: ('ah'llla(,i()ll ()f fr(:(lll('11('y av(;rag('s. Fi~mlly, 11()1,(' flint, if (:v(ul l)()w(:rs a r(: illv()Iv(:(l i~l (!).2.15), 1,11(' fillal (:xl)r(:ssi(nl is fiu'(,11(:r si1111)lili(:(l. Ill flu:l,, wi(,ll l ~ - 21)~, (!).2.15) yi('l(Is ' :* ( q ) ( - i ~ a ) 2,, :(q),tq.

~"~'" -

(9.2.25)

whi(:h l)y a,~l '~-fl)l(l ill(,(:gra(,i()n l)y l)ar(,s r(:(:asl,s i~11,() h;2.m, __

9

__

'

d

m

2

,~

.

In t)arti(:ula,r, th(; avera,g(', sr

frequem:y (:an 1)(; (:al('~fla,l,('sl t)y (,ll(; integral h;2 - - . / I ~ q ~ ( q ) 12 dq ,

(9.2.27)

simply inv(flving the first-()r(ter deriva, tive of the spatial (tist, ritn~tir The correspon(ting exlm:ssion for q~" is, of course, '

"'

d ~ ; - IIq''@ll ~.

(9.2.28)

In w 4.6.4 we have discussed the well known uncerta,inty inequality 1

CrqCL > ~

(9.2.29)

for the p r o d u c t of the space and frequency widths of the optical wave function C~q, c~, which, according to the accepted variance measures (4.6.35), are just

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

471

the second-order moments of I~(q)l 2 and ~5(n)l 2, and hence the average values (p, ~2~) and (~, ~2qp). In effect, (9.2.29) is a consequence of the relation holding between the operators ~ and ~ as expressed by the Lie bracket" [~, ~] - i. It is a particular case of a more general theorem holding for any pair of variables, whose associated operators do not commute, as it is for ~ and s The proof of the uncertainty relation on the ground of the operator representation is elegant and simple; also it evidences the generality of this principle which applies to any pair of non commuting observables (see, for instance, [3.10], pp. 216-218).

9.2.2

Moments of a two-dimensional function

Moment notions apply as well to a 2D real function 13(q, n); of course, q and n denote now in general real independent quantities, without any close reference to the Fourier pair of optics. Evidently, the variety of possibilities is greatly enriched as both local and global moments can be defined. The formers depend on q or n, as the latter, being constant, characterize the function 13(q, t~) in a global sense. It is not required that B(q, ~) be nonnegative over the relevant domain, but only that the local and global averages be positive. The global average is obtained by integrating B(q, t~) over the whole q-~ plane to the consta,nt value

//s(q, t~;)dqdh; > O.

moo

(9.2.30)

In turn, the local averages are determined by partial integration of B(q, ~) along the ~ or q line, thus yielding the q and ~ depending quantities o(e) -

f

./B(q,~)dq > O.

> 0,

It is apparent t h a t

(9.2.31)

/,

s

moo -- I [~o(q)dq - I qo (t~)dt~. d

(9.2.32)

d

Local moments In the light of the averages (9.2.31), one defines the local moments of B(q, ~) with respect to n and q simply reproducing the previously introduced formulae. Thus, the local first-order moments are given by the integrals

1~1(q)

-

l/~B(q,~)dn tt0(q)

'

ql(~ ) -

i f ql(q,

q0(~ )

(9.2.33)

and respectively interpreted as the ~-average value of B at q and the q-average value of B at ~. In general, the local higher-order moments with respect to ~ and q write as 1

t~r (q) -- t~0 (q)

][/~--{~l(q)]rB(q,

~)d~

'

qr(~)

1 q o~,~)

/

[q

--ql ( ~ ) ] r ~ ( q ' ~)dq.

(9.2.34)

Linear Ray and Wave Optics in Phase Space

472

Global m o m e n t s Similarly, following (9.2.30), the global first-order moments are obtained by integration over the whole q-h: plane as mo

i

_~

1

m oo

jj 'h:B(q, h:)dqdt~:,

mr0 ~_

1

ITI O0

/'/'

q]3(q, h:)dqd~,.

(9 2.35)

Evi(teId, ly, tll(;y st)c(:ify t,lle l()(:atioIl of the ceid, r()i(t ()f B(q, h'), i.e. ()f the point (re,o, m,,, ) ()11 t,l~(,,st)a.(:e-fl'cq~le]~(:y t)la,Im (q, h',) a,t whicl~ B, t,l~t)~lgld, of as a, ma.ss (tistril)ld, i()ll ()vet tim (q, h:) lfla.lm, is ill lm,la.ll(:e. Th(:ll, lillfil,iIlg ()~n's(:lvt:s (,()(:(msi(h:r tile (:md,ral lll()lll(:ll(,S, it is ()l)vi(nlS tha,t the se(:(n~(l-()r(ter m()lllellts write too2

__ 1 mi)(}

m II

--

~2o

--- m o O

(h"-

' .

(q

1 1

(q

WI{)()

mot

~

)2~(q,

H,)dq(tH,,

m,,))(t," - m,),)/~(q,' t;)dqdh"

(9.2.36)

m,())2~(q, t~,)dqdt~,,

as a lm,(,~ral (:xt(:nsi()~ of m~ i~ (9.2.4), whi(:l~ in(|e(:(| is f()l(h:(] i~t() tl~ree quantiti(:s i~ ()r(h:r t()(:l~ara(:t(:riz(: (,l~(: (:xt(:~si(n~ ()f/3 wi(,l~ r(:sl)(:(:(, t() q a~d t,: a,s well as its ()ri(:~d,a.(,i()~ iI~ tlm (q, ~,) I)la.~e. hi g(umral, tim gl()l)a,1 x~()~(:~(,s a,r(; ~m.rk(:(t t)y tw() i~t,(:g(:I" ~n~x~fi)(:rs, being lT~rl)

---

' m()()

//(q

~

m,() )'(t;

--

too, )hB(q , t;)dqdt; .

(9.2.37)

I.I

As th(: (:()()r(linat(: ()rigi~ <:a,~ always 1)(: (:hoscn ()r nm,(l(:, l)y a.I)I)r()I)riate t,ranslations, t() c()incide with th(: (:entroi(I of B, the first-or(|er ~nom(:nts m~0 and m()~ (:a.n |)(: (Ir()I)ped ()~t fr()~ (9.2.37), tln~s yi(:Iding th(: si~I)lifi(:(I (:xl)ressions ITIr h

=

mool

/'f

qr t,:hB(q, ~)dqdh:.

(9 2.38)

Id

9.3

Preliminaries to the calculation of the moments of the Wigner distribution function

It is evident tha,t the eva,llmtion of tile moments of the Wigner distribution function requires the integra,tion over q o r / a n d t~ of products of ~Y(q, ~) by monomials like q r t~h and q"t~ "~. Thus, for purposes of future convenience, we firstly consider in some detail products of the type

q~W(q, ~),

n h W ( q , t~),

qn~"~W(q, ~),

(9.3.1)

and express t h e m in a form, which will be usefully exploited when integrating (9.3.1) with respect to q o r / a n d t~.

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

473

It turns out that the integration of the products (9.3.1) comes to involve the cross-Wigner distribution of functions, which relate to the signal amplitude and frequency spectrum through the coordinate and frequency operators ~ and ~. Therefore, we devote a later section to a brief description of the properties of the cross-Wigner distribution function. Although the presentation in the chapter is limited to the second-order moments of the Wigner distribution function, the discussion below offers a description of a useful operatorial machinery for possible advanced uses. The reader may also recognize the link with the forlnMism illustrated in ~ 8.5.1. 9.3.1

About the products q~W(q, t~), ~hW(q, t~) and q~tC~W(q, t~)

Let us start with the product qW(q, tj. Using for W(q, t~) the definition (6.2.3) involving the signal amplitude p(q), we conveniently arrange the factors under the integral as

q')F(q + ql)c-iec q' dq' ql&(q, t~) -- -51 ~( 1~ [ (q - q')F,(q_ 2 2 2 J -Jc--ff-~][ ( q _[_ q~) ~* ( -- q~) ~ ( -~- q~) c- i~q' dqt ) "

(9.3.2)

Since (q-

2 (q + 2

_

2 + r

2

-

2 , 2 '

+

(9.3.3)

(9.3.2) can compactly be expressed in terms of the mixed Wigner distribution function, as defined in (6.3.12), thus yielding

ql/V(q, t~) - ~1 [~V~,~ (q , ~) + ~V~ , ~ (q, t~)]

-

-

~[W~ , ~ (q, n)].

(9.3.4)

The final expression above is due to the property of the cross-Wigner distribution: I/V~I,~2 (q, t~) - 1/V~2,~1 (q, n), already mentioned in w 6.3.4 (see also below). Notably, tile expression on the right-hand side is reminiscent of tile rule for the derivative of a product. It is an easy matter to obtain for qrl/V(q, ~) the expression

qr)&(q, n) -- (-~1 ) r L (

A. ~) W~_jv),q~v)(q , n),

(9.3.5)

j=0 where, of course, according to the coordinate representation of the operator ~, simply implying ~ h ~ _ qh~, the term )/V~-r-~,~j~(q, n) explicitly reads 142~>_jv),~-r

1/

, ~c) -- ~

(q- 2

(q- 2

2

2

dq'. (9.3.6)

Linear Ray and Wave Optics in Phase Space

474

Correspondingly, tile product ~W(q, K) can be ha,ndled as follows: glA;(q, t,;) - 51- ~ .f/ ~ * ~

1 .f~.

+ q'2 )c-i,~q'dq, (q - 9~_)p(q 2

,t -i,./ )dq', (q - ~ )~(q + q')(i ~ ~,,~

(9.3.7)

whi(:h, aft(;r integrating t)y t)a,rts, yi(;l(ts

.W(q, h-) - ~ [w~,~(q, ~-) +

w~, ~ (q, ~;)1- !~[w~, ~ (q, .)1 .

(9~.s)

Agaili, IlS(; lm.s t)(;exl (l()ll(; ()f t,tm ()t)(;rat()rial Ii()tat, i()Ii ()ll St)e('ifyillg l,h(; tim(:t,i()us (~lfl,(~l'ing t,he (:r()ss-Wigller (list, ril)Ilt,i()ll fim(:ti()li. ()f (:()In's(,', h(,'r('~ ~ _ i d,dq~Y a(:('()r(ling t() tim ('()()r(lina.t(', ret)r('~s(;nta,ti()n (4.6.58) ()f tim fr(;(llmn(:y ()t)(;rat()r ~,. Tlm fi)n~ml si~fila.rity ()f (:xI)ressi()l~S (9.3.4) a n(l (9.3.8) fi)r ql/V(q,h;) a i~(t t,;1A;(q, h'), with the ()l)erat()rs ~ a,n(l ~ ilm~r(:ha, nge(l, s~lgge,sts t() write, th(: t)r()(1u(:t h;hW(q, h-,) i~ tl~e f()r~ re~inis(:(,nt ()f (9.3.5), i.e. -

-

h.

1 h

j =0

wl~(;r(;

~,,.

"k'",y -- (-i)'" ,tq'' ~'

(9.3.10)

an(t l~('~x~(:(:142~h J~,~.J~o(q, h') is (;xl)li(:itly given t)y W;,, ~r

-

i ''-'2~ [

dh-~

d~

2, .lldq h - j r

~)[d-~/j~l(q+ ~)''

-i,,q'

dq'. (9.3.11)

Tlm rea llless of tile rigllt-lm,i~(t si(les of both (9.3.5) a.l~{l (9.3.9) follows again fron~ the co~.j~gat, io~ prolmrty of the (:ross-Wigner (lisl,ril)~l,i(n~ fimctioi~ in rest)(mse t,() the inter(:ha.nge of the fimctions. The convenience {)f the operatorial notation a,(t()t)t,('(t in t,t~e previous relat, i()ns, inv()lving t,h(; nfixe(t Wigner (tistribut, ion fin~(:t,i()n, nmnif(;st, s, among other things, in the fa.(:t tha, t the ()t)ta,ine(t ext)ressi()~s (9.3.5) a.x~(t (9.3.9) for the pr()(tu(:ts q~14;(q, h;) a,n(t h;hlA;(q, t~;) ('a,n be recast in terms ()f the signal frequen(:y st)(;ctrum @(h;) a,(:(:or(ting to

qrl4;(q, h;) - (71 )r s

j=0

(~.) ~],)~,q~-.~, ~ ~'9~(h;,

q),

h,

(9.3.12)

1 h ~, ())W2h-~ 2jyc(t~ q) nhW(q, ~) -- (-~) j=O

~'

'

'

where, taking into account expressions (4.6.60) and (4.6.59) for the operators and ~ in the frequency representation, we have d~ d ~ - p,

~'~# -

#.

(9.3.13)

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

475

Also, the cross Wigner distribution functions appearing on the right hand sides are understood as A

~

'

--

27r

*

t~! -2

@,

t~'

,]Ld~r'-3

1 /

~'

dJ

tr ) c iq ~1 -2 er )j

t~' )ciqe;'

../

(9.3.14) Accordingly, it easy to see that the product qnn'~W(q, ~) comes to be

1

m (~)qnW2m_ ~ ~j (q, t~)

q n l ~ m l / V ( q , I~) -- (_~)m ~

j=0

~'

(9.3.15)

(1) W'~ .... j'~ ..... ~.~,'~j'~.~(q,t~).

(9.3.16)

n

1 nj~ 0 which can further be written as Tt

q n n m w ( q ' n ) -- (5)

~}q,

~ E j=0/=0

In conclusion, we see that the products of the Wigncr distribution function with the monomials q~, ~;h and qn~m can be expressed in the form of sums of cross-Wigncr distribution functions, involving the signal amplitude or the frequency spectrum, t)rot)crly acted by powers of the operators ~ and ~. The advantages of such a rci)resentation result from that, as we will see below, when integrated with respect to the variables q or t~, the cross-Wigncr distribution function behaves ii~ the sixnilar way as the self-Wigner (tistrib~tion fimction. As a useful exercise, we invite the reader to deduce the basic relations of this section, summarized by (9.3.16) within the Bopp operators formalism, t)resented in w 8.5.1. It is evident, in fact, that

q-

1

Q, ),

1

~,

(9.3.17)

)

Since the operators Q and Q* commute with each other as well as K and K*, powers of q or t~ simply follow the binomial rule.

9.3.2

Basics of the cross- Wigner distribution f u n c t i o n

The cross-Wigner distribution function W~I ,~2 (q, n) of the two functions (Pl (q) and ~2 (q) has been introduced in w 6.3.4 through the q-domain convolution

W991,~92 (q, ~)

1 ./

- ~

(JOe* (q -

q'

q' e--i~q'

2 )~2 (q + 2 )

dq/,

(9.3.18)

and, correspondingly, through the ~c-domain convolution W ~ I , ~ 2 (/5;, q) - ~1 fj

~-, (1~ - -2-)~2 ~' t~' eiqt~'dfj - WqP1,99 2 (q, ~) , (/~1 (/~ -31- --2-)

(9.3.19)

Linear Ray and Wave Optics in Phase Space

476

involving the Fourier transforms ~51(n) and @2(h;). Needless to say, both Pl (q) a,nd P2(q) are supposed to be Lebesque integrable. Also, as a,lrea, dy note(t, it results that W991,992(q, h;) - W;2,qOl (q, J~;),

(9.3.20)

ix~t)lyi~g tlm rea.hmss ()f tlm (a.l~to-)Wig~er (listriln~ti~m fiuwti~)~ ( ~ = ~2). Basi(:s t~/tll~; fi~rtll(:~nllillg (:ollsi(tera,ti(nls, are tlw (:r
,,(q-

~)~.~(q + ~) -

~;(~:-

N. ! i~, ! =_,)~(~. + =,) -

/ w~,,~(,~,~.),~,/a~. / W,~,,,,.~(q.,,.)~.-,~',,d,l,

(9.a.21)

whir like (6.3.22) m~l (6.3.24), rela.te tlm IIr~l~w(,s ~ll~ (,Ira left l~a.mt sir r~:ga,r(le~l resl)e~:l,ively as fitl~l:l,i~ms (if q' alt~l h", l.~) l,l~e F~)~u'ier l.ransfi/rms of l,lm r ~listri|ntti~n~ fint(:tir )/V~,,% (q, h:) witl~ resl)ect t~l t,- a,~r q. F~rtlmr relati~n~s caat lie i~fi:rre(1 fi'~n~ (.(1.3.21). Sl:tt,it~g fin" it~sl,a,l~(:~: q' = () a t~(l h" --{1, w~: t~my ga.it~ the IIr~r ~ (q)z.~(q) I~y it~l,egral, i~g }/V~,~ 2 over tl~: fr~:r variallh:, t~att~t:ly "r', (q)~.e(q) --

}/Vso ' , ~ (q, h:)dh',

(9.3.22)

w, (h:)~ (h') fl'()~,~ t,t~(: i~m:gra,ti(/x~ of W~,~o2 (iv(',r tlu; (:(l(ir(li~mtc: ~, (h')~5~ (h;) -

,

W~o, ,~.~ (q, h')dq.

(9.3.23)

Consequ(ud, ly, t,t~r i ~ e r tm~t~u:t of the tw~ ful~t:l,itms ~, (q) a,ntl ~ (q), as well as of the (:(IrrestItn~r Ft,,lrit;r tra.xmforn~s @~ (h')m~tl ~2(h'), si~l,ly etl,m,ls the integral of ]/~o1,~o2 (q, h;)(}vP, r the wh()l( ~, t)la.ll( ~, (q, h;), ~m.nmly

(~,, ~ )

- (~>l, (~2) 9

/7 w+,,+.~ (q, ~:)
~:)dqd~:,

~

(9.3.24) which gives (6.3.26) with ~1 = P~. Moreover, the inversion formulas for both the flmctions and the Fourier transforms can t)e inferre(1 from the rela,tions (9.3.21) a,s well. In particular, for the amt)litlutes ~ (q) a,n(t ~2(q) we find (r ( q ) ~2 (q)

.1

/

*

~2(q(,) '

1

~ l (qo) .

(q+qo h;)c-i~(q-qo)dh;, 2'

W991 'q02" .

}/~q~, (q+qo h;)ei~(q-qo)dh;, '% "

~

(9.3.25)

'

indicating that both ~ (q) and ~ ( q ) can be reconstructed from }/Y~l,~ ~ up to a constant phase.

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

9.4

477

W i g n e r distribution function" local and global m o m e n t s .

We apply the notion of moments, as described in w 9.2.2, to the Wigner distribution function. Specifically, we consider the local moments of W(q, n) with respect to the spatial frequency ,c and the coordinate q in w 9.4.1 and w 9.4.2. The global moments and related issues are considered in w167 9.4.3-9.4.5. 9.4.1

Local moments in the angular variable

As earlier remarked, the moments with respect to one variable depend on the other and therefore yield informations ~bout the local (i.e., on a chosen plane parallel to one or the other direction) properties of the Wigner distribution function, as, for instance, the relative center and extension. On applying the scheme of w 9.2.2, we are led to consider the average t~0 (q) of 1/V with respect to the frequency, i.e. the marginal distribution (9.4.1)

e o (q) - f W ( q , ~ ) d ~ - I~(q)l 2 .

which is, of course, nonnegative. Accordingly, we can define the local higherorder moments of the Wigner distribution flmction with respect to n only at positions for which 7:(q) =/= 0.

The first-orclcr moment: ~1(q) The first-order monlent t~1(q) is then given as 1

~1 (q) -- ~o(q)

/

~,~(q, ~:)d~:,

(9.4.2)

and interpreted as the average frequency of W at position q. Taking into a,ccount relation (9.3.8), we can write t~1 (q)

2t~0(q)l f [w ~r ) (q , ~;) + ~]7 ) , ~qP (q, K,)]dn.

(9.4.3)

Then, exploiting the property (9.3.22) of the cross-Wigner distribution function, after some algebra we end up with i 1 d _ _1_ d p , ) _ , ~ [ ~1 ( q ) -- -- -2 ( --~ -~q (t~ (i~?* d q

]

~ (q ) '

(9.4.4)

where 7/(q) - d~p/dq and the symbol .~ denotes the imaginary part. In order to gain a deeper insight into the above expression, it is convenient to represent the signal in the form

~(q) - f (q)c ir

(9.4.5)

Linear Ray and Wave Optics in Phase Space

478

b o t h the envelope and the phase, f(q) and r being rea, l-valued fimctions. T h e formal expression (9.4.4) yields t h e n the p h a s e derivative of the signal: d 1~,(q) -- -~qr

(9.4.6)

It, sllgg(;sts to int(;rt)r('~t [~1(q) as tim lo('a,1 (tir('~(:t,i()Ii of t)rot)agati()ll ()f th(; signal. This is in (:()iff()nllity witll th(; way(; ()I)ti(:s vi(',w ()f ligllt rays as tll(: ll()r~llals t() th(; wa,v(;fr()llt ()f tll(; siglm,1, wlli(:ll h('n(:(; a,r(', llm.tll(;llm,l,i(:ally r(;1)r(;s(;nt(;(l |)y tim gra(ti(;nt (m~(t s() t)y tl~(', (t(:rivative i~ a. 1D ~()(1(;1) ()f t.l~(', 1)lm,s(;. S~u:l~ m~ int(;rt)rctati()~ is a,ls() s~gg(;st('d t)y tt~('~ (t('fi~i~g ('xI)r('ssi()~ (9.4.2), whi(:t~ wit, tfi~ tim t)i(:t~r(', ()f l,lm sig~m,l a.s a. 1)(;a.~ ()f rays, l~my 1)(,, ~l~(l(',l'sl,()()(l as tt~(; l()('al ~(;m~ va,h~(' ()f l,l~(' ray (lir(;(:ti(n~s h: - k'0 i~ tl~(; i)(;m~, 1)(;i~g ()l)l,a,i~(',(1 1)y a,v(;i'a,ging over all tim l)()ssil)h; ray (lir(',(:ti(n~s, ('~a~:l~wcigl~t(',(l l)y t,l~(; r(;l('~va,~t ra,y "m~q)lit~ul(;" t~V(q, t~). Wit, l~i~ tlm t,i~(',-fl'(',(t~('~(:y (l()~m.i~ ()f tlm sig~ml th(;()ry tt~(; ti~(; (h;riva,tiv(; ()f tim I)has(; is r(;I)()rt('~(t a,s ttu; i'n,sla'nlanco'u,,~ frcqu~:ncy [a]. As a,~ cxa,~l)l(;, w(; ~my (:()~si(l(;r tl~(; I)lmm wa.v(',

(,a.4.r) wl~()s(; l()(:a,1 fl'(;(lll('~ll('y (:()lll(;S I,() 1)(' 1,11(; q-in(t(;1)(;~(l(;l~t vallu; (s(;(' }i 6.2.1)

(u) - ,,:,,. Correspon(tillgly, ill l.ll(' ('as(' ()f tile 1)llr(; ('llirl)

(,a.4.,9) we find (9.4.10)

(,1)

in accord with tim considcrati()ns of w 6.4.4 r e g a r d i n g tim W i g n e r (tistribution fllnction of a plll't ~,(:hirt). Likewise, for tlm slnll of two chirps, writtcIl a,s

~(q)

_

~~ Ci~l g2_

2

2 + -Eaei~22

q2 2 ,

(9.4.11)

wc find ~', (q) --

1(ZIG -~-Z2q),

(0.4.12)

which gives indeed the angle within the cone b e t w e e n '01 -/4~ q a,nd ~2 - / 3 2 q . Interestingly, the average frequency {~ (q) of the W i g n c r d i s t r i b u t i o n function of a real signal, i.e., r - O, is zero"

[~1( q ) - 0

for a real signal,

(9.4.13)

which, in effect, emerges also from (9.4.2) on account of t h a t the W i g n e r function of real signals is an even flmction of t~.

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

479

In addition, if ~(q) is the product of two signals (9.4.14)

(~(q) ---- (~1 (q)(J~2 (q)'

the average frequency of the Wigner distribution function turns out to be

~1 (q) -- t~l,qOl(q) -t-- ~l,qo2 (q),

(9.4.15)

i.e. the sum of the average frequencies of the Wigner distribution functions "~1 and 1/Vr of the two signals in the product. Thus, for the frequency-modulated chirp

~(q) -- ~o er

F.it%q,

(9.4.16)

we have ~1(q) =/3q + ~0.

(9.4.17)

The second-order moment: ~2(q) According to the general expressions in (9.2.34), the second-order moment of 142 with respect to t~ is given by

t2(q)

-

1 /[g--t~ 1 (q)]2142(q, a)dec --

to(q )

1

i~a(q) 2

f g2 }IV(q, ec)dec-t~(q). (9.4.18)

Using relation (9.3.9) for the product n2W(q, ec), namely

/%21/~2(q,/~) __ 41{~2"~2~,~ (q, ec) + 2I/V~,y~(q, n) + 1/V~,22~(q, ec)},

(9.4.19)

and property (9.3.22) of the cross-Wigner distribution function, it is easy to calculate 1~2(q) in the form

1 d ~'

d ~'*]_

~2(q) - --~ [--~q~ + ~ ~,

1

d ~'(q)

--2 ~[ dq ~(q) ]"

(9.4.20)

The above is interestingly compared with the expression (9.4.4) for ~1 (q)" Notably, since l/V(q, ec) is not always positive, the local second-order moment t~2(q) is not necessarily positive. Thereby, as already remarked, it cannot be interpreted as the variance of 142 in the t~ direction at a given position q. Using for v)(q) the representation (9.4.5) in terms of the envelope and phase, we readily obtain

~2(q)

1 d f'(q) - ~ dq [ f(q) ],

(9.4.21)

thus showing that the local second-order moment is determined by the signal envelope f(q). As before, we have set f'(q) = df/dq. The second-order moment t~2(q) for the chirp (9.4.9), for instance, is zero.

Linear Ray and Wave Optics in Phase Space

480

In general, t~ (q) -- 0 whenever the signal envelope boos the form f(q) - fo e/~q with f0 a n d / 3 arbitrary constants. The a.dditivity property, obeyed by the average frequency t~1(q) when a product of signals is conccrnc(t, holds for the second-order m o m e n t as well. In fact, for tt~(; signal (9.4.14) we have -

(q) +

(q).

A s a.~ ~;xm~l)l('~, h;t ~s c(n~si(h:r tt~c fr(;(t~w,~('y-xno(l~la,t(;(t (;a.~ssia~ signal 1

1

.

(9.4.23)

7rw2

for whi(:l~ wc til~(l t~ (q) - ~:,,,

1 ~ ( q ) - ,u,2,

(9.4.24)

i~ a(:(:()r(l witli tll(; (:la,i~le(t r()h; ()f tll(; Im.ra.xil(;t(;r 'u, rt~li~lg tim sI)atia,1 a,~l(t fre(t~m~(:y extmd, s ()f t|m relevm~t Wigx~er (tistriln~ti()~ fi~x~('l,i()~ (s~',e, w 6.4.5). 9.4.2

Local 'moments in tit(; space variable

Tlle h~('a,1 lll(nll(;llt,S ill l,llc mlglflar varial)h~ a r~', l)a,ralleh',(l 1)y l,ll~ h)(:al lll()nwnts in the st)atia,l variat)h;. 'File a,v(;rag(~ (ff W(q, h:) witll rcst)(;('t t(~ q is given by the s(t~mrc a,mt)lit~uh'. (~f tim fr(;(t~mn(:y Sl)(;(:tr~m~ ~(~:) (E(t. 6.3.26) .

q,,(~:) -

W ( q , h,)dq -

I@(h-,)] 2 .

(9.4.25)

,

Th(;ii, wc llla,.v (tetiile the lo(:al m(m~cnts iI~ the st)a.cc va,ria,lfl(; (nfly f(~r fr(;quen(:its h', at w|,i('l, ~(~) # 0. T h e ]irst-o'rde'l" m o m e n t :

q, (h;)

Tile lo(:al first-order in(relent q~ (h:) is ttmreh)re

1/

ql (t;,) -- qo(,~)

qlN(q, ~)dq,

(9.4.26)

which, on account of the picture of the signal as a collection of rays and of the Wigner distribution function l/V(q, n) as the "ray amplitude", may be interpreted as the average ray position at the frequency (i.e. direction) t~. It is convenient to use the expression (9.3.12) for the product qW(q, t~) in terms of the frequency s p e c t r u m ~(t~), so that, due to the relation (9.3.23) obeyed by the cross-Wigner distribution function, we obtain i 1 d ~ ql (t~) -- ~ ( ~ ~--~g~

1 dff.)__@[ ~* dn

]

~(~) '

(9.4.27)

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

481

where, of course, 0'(t~) -- d~/dn. Expression (9.4.27) can be regarded as the dual of (9.4.4) for t~1(q). The correspondence between q~ (~) and ~1(q) is better evidenced, writing the spectrum in a form similar to (9.4.5), i.e. ~ 5 ( ~ ) - h(t~)e/r

(9.4.28)

where h(~c) and ~p(~c) are real functions. Thus, q~ (~) equals the derivative of the phase of the spectrum d ql (~;) -- --~-s162

(9.4.29)

As an example, we consider the chirp (9.4.9), whose Fourier transform is

~.~

~

p ( ~ c ) - Po

.~2 e-r

(j3 > 0)

(9.4.30)

and hence q~ (a) - ~,

(9.4.31)

with an evident correspondence with (9.4.10). In order to clarify the meaning of ql (g) we may consider a, space-invariant system (see w 4.6.3). The relevant input-output relation in the a domain writes in the simple multiplicative form

~o(,~)- 0(,~)~,(,~),

(9.4.32)

involving the spectra of the input and output signals, ~o(n) and ~,(~), and th~ tr~.~f~r f.n~tio. 0(") of the ~y~t~m. W~ writ~ 0(") a~

0(~)- h(~)d~(-),

(9.4.33)

and suppose that both h and ~ vary sufficiently slow about some frequency t~o to make reasonable the approximations h(~c) ~ h(~o) and ~(t~) _~ ~ ( ~ o ) + ~'(t%)~c o. Accordingly, on reconstructing the amplitude Po(q) of the output signal, we obtain Vo(q)

-

h(~o)dIr162

+ ~'(~o)).

(9.4.34)

Also, we assume the input signal to have a spatial envelope fi(q) linearly modulated at the frequency ~Co, namely

~,(q)- f~(q)~.~o~.

(9.4.35)

Then, the output amplitude (9.4.34) becomes ~o(q)

-- h ( l ' ~ o ) f i ( q --

ql ( ~ o ) ) Cit~O[q-qp(t~~

(9.4.36)

Linear Ray and Wave Optics in Phase Space

482

(Pi(q)

.---l~ (P o ( q )

F I G U I / E 9.2. Typical (liagrmn ill~mtrati~g tt~e i~mmfi~g of q~ ai~(l qv a,,~e~welope m~(1 phase slfifi,s.

wtmre we tmvr (:xI)l(fite(l exl)ressi(n~ ((,}.4.2(,}) fi)r q, (h',) m~r r cr tl~e ratio ~'(~~ ~'("~ l';vi(lcl~tly, q,(h,,,) i(tc~t, ifics tl~e stm,(:e-sl~ift l)ro(luced h',{} as q,,(h',) h',() l)y tlm systeIll (hi t,lm eIlw~l~l~' fi(q) ()f ill (~. i~ln~t sig~ml. ()~ tlm ()tl~er han(t, thr t)hase a.lsr is a lf'(~r162 I)y a sl~ift, sI)r162162 l)y qp(h',,). 'l'l~e fm~iliar view ilh~stra.tix~g tl~(; r162 aJ~(l l)l~a.sr sl~ifts, q, (h',,,) a,l~(I qp(h'o) , is r in Fig. 9.2. Tl~esc |,wr sl~ifts are cr r wl~r thr pha,sc rr162 '(,(h;) of tim systen~ va.rics li~ma,rly with fr(;(t~m~wy. W(; re(:a.ll tl~a.t f()r ti~a(;-(hq)('.n(tent signals the (t~aJ~tities (:()rrest)()~(li~g t() ql aI~(l qp, i.e. --d~/,(t) a,~(1 - - ~ , are rest)e(:tively (:alh;(l tlm g'~v'up delay a.~(t tl~(; phase, delay [3].

The second-o'l'dc'l" moment: q2 (h') Tile sec()n(t-()r(l(;r lll(nIlcnt q2 (h:), (lefiIle(t t)y q=(';') -

,

qo(~)

/

[q - q~ (")1

2

W(q,,;,)dq

-

1

/

i~(~)1 = '

q2

W( ,h;)dq q

_ q

12(h;), (9.4.37)

takes the expli('it expression lid

@'

d ~5'*

1

d @'(a) ~(-~-) ]'

q2(h;)- - 4 dt~-~ + (-t-~K,-~] -- - 2 ~ [ d ~

(9.4.38)

after using rela,tions (9.3.12) and (9.3.23). The correspondence with t~(q)is evident. Also, note that with the representation (9.4.28) of ~(~c) one finds

1 d h'(h;) q2(~) -

2 d~ [ h(n) ]"

(9.4.39)

Hence, tile local second order moment q~(n) depends only on tile amplitude of the spectrum of the signal.

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

483

As an example, we consider the Gaussian signal (9.4.23), for which, being ~.~

W 2

1

w 2

p ~ ( e c ) - (57)ae--a-(~-%)~,

Eqs. (9.4.29) and (9.4.39)yield ql

9.4.3

q2(t~) - w2 4

"

(9.4.40) (9.4.41)

Global moments

The global moments of the Wigner distribution function are obtained by integration over the whole space-frequency plane (q, ~) and hence they are constants, which characterize the Wigner distribution function in a global sense. We shall limit to consider tile zero-, first- and second-order moments. A brief comment on the higher-order moments is given in w 9.7; a more accurate discussion is beyond the scope of tile book. It is noteworthy the variety of expressions and relations, that, as we will see, can be deduced. This evidently conveys the effectiveness of the moments of the Wigner distribution function as inspection tools of tile nature and tile properties of a signal. Thus, the average of 142 is given by moo

-//147(q, ~)dqd~,

(9.4.42)

and, on account of (6.3.27), identified as the total power of the signal, being moo

f I~(q)[ 2 dq

-

- f

1~5(~)12 dt~.

(9.4.43)

Being the signal power always positive for all p r 0, the global higher-order moments of the Wigner distribution function can safely be introduced.

The global first-or'der" moments

toOl

and ml0

The general definition (9.2.35) yields for the global average frequency 11101 the formal expression

1 //t~J(q, moo

tool

t~)dqdt~

(9.4.44)

Integrating with respect to q, one obtains m01

_

1

ii l

./"

2

_

--

(9.4.45)

which clearly shows that tool can be interpreted as the average frequency of the signal frequency spectrum (see w 9.2.1). Alternatively, the integration with respect to ~, which by (9.4.2) gives

f~14i(q, ~)d~-

]~(q)12

(q), yields

toOl

as the average of the local frequency

484

Linear Ray and Wave Optics in Phase Space

[~, (q), with the local power I~(q)] 2 acting a,s the pertinent density function. Explicitly, we have m01 -

' f ~ ~( q ) I ~ ( , , ) l ~ ,,,, - ,, (q) I1~112

(9.4.46)

Finally, a,(:(:ordiI~g to the. relation (9.4.6) at)t)ropria,te to the representation (9.4.5) of the signal, the. nleaal spatial ffeqlw.n(:y mo~ a,ccpfires the fllrther nmanillg; as the aw~ra,ge ()f the t)ha.se (teriva.tive ()f tim. siglml: m')l

_

/

1

[I-i-~.

~t

(q)f 2 (q)dq- (qS'(q)}.

(9.4.47)

Rela,ti(ms (9.4.45) a,n(1 (9.4.46), implying the e(t~mlity m01 - ~ - {~, (q),

(9.4.48)

enabh', the eqllivaleIl(:e of the weighte(t averages in the h', and q r a,ll(l t~ (q), a(:r t() wlli(:h the (:a,rrier freqlu,,llc:y ()f the siglm,1 e(llla,ls the a,w',ra,ge of l,ln; local fi'er162 of the signal. 'Finis li~k was firstly re(x)gnize(t t)y Franks wi(,l~i~ the (:o~m'xt of the sig~m,l the(wy [a.s]. The (li~:~ssion a,t)(we evi(teimes the fl)nnal an(l concet)tual ea,siness tt~rough which that equivalence (:a,~ t)e i~fi:rr(;(l wil, l~i~ tin; fi)r~mlis~ inln;re~t i~ the Wiggler rel)rese~tati()n. Using exI)ressi()~ (9.4.4) fin" t~, (q), we (:a,~ fi~rtlw,r re('ast (9.4.46) as i

m,,, = '21~ '~

/

d ~* _ ~* d

N~)dq.

(~@

(9.4.49)

Similarly, 1,1~(;a,v(;ra,g(' Sl)m:e (:()or(tina,l,e m~,,, ()l)l,a.in(',(l l,l~r()~gl~ 1~I0

-~

~ f f q142(q, t~:)dqda;,

(9.4.50)

rno() '

turns out t() l)(: tln: (:(:~tr()i(l ()f the loom l)ower, sin(:(: l)(:rf()rl~fix~g tln: il~t(:gration with resl)(:(:t t() h" w(: ()l)(,a,i~ m,,,-

~

1

q I~(q)l 2

dq-q-

Also, as the integra.ti()~ with respect to q yM(ts

(q).

/ qW(q,

(9.4.51) =)dq -

I@(K,)I 2 ql (K,),

the alternative interpretation of ml0 as the mean value of the local first order moment ql (t~) is in accord with the resulting expression -

m,,) -

'

i1 112

/ l ~ ( , ~ ) l = q, (t,:)dn - q, (t,:) .

(9.4.52)

Then, on account of (9.4.29), the first order moment ml0 is the weighted average of the phase derivative of the signal spectrum, i.e. m~0 - ( - ~ ' ( ~ ) ) . Evidently, relations (9.4.51) and (9.4.52) imply the equality ml0

--

q

--

ql

(N),

and so the equivalence of the two averages 0 and ql (K)-

(9.4.53)

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

485

The global second-order moments rno2, rn2o and 11111 We start considering the global second-order m o m e n t in the frequency variable.

The global second-order moment rno2 As a direct transcription of the general definition in (9.2.36), the global secondorder moment rno2 with respect to the frequency is formally given by r~o2 -- mo1 o

/

(/~ _ m o I

)2142(q, ~)dqdn.

(9.4.54)

Integrating with respect to q and using (9.4.45), we find

i1111 f

(9.4.55)

It is evident that m02 is always non-negative and so can be interpreted as the spread of the Wigner distribution function in the n-direction with respect to the centroid (rnlo,rn01). It is also evident that rn02 is just the variance cr~2 of the Fourier transformed signal ~5(t~):

2 lll02 -- O-t~"

(9.4.56)

Besides, in conformity with the formalism illustrated in w 9.2.1, we may relate m02 to the expectation value of the squared operator ~2. mo2 _._,

(~2},

(9.4.57) A

and so to the matrix element in the state F of the group generator K_ (w 5.2). Interestingly, tile global moment rno2 relates to tile local moments 1~1 (q) and t~(q). In fact, on account of (9.4.2) and (9.4.18), it is easy to recast (9.4.54) as mo 2 m

1

IIv)ll2

t~2(q)]~(q)12 dq +

1

/[t~ l(q) _ ~]2 iv~(q) 2 dq

(9.4.58)

1~2 (q) -]- ((1~1 (q) -- ~)2}.

We see that m02 comprises two components: the first term represents the average ~2 (q) of the local second-order moment ~2 (q), and the other is the variance of the local frequency ~1 (q)- Within the representation (9.4.5) of the signal, it becomes evident that the magnit~lde and the phase of the signM yield separate contributions to rn02, on due account of expressions (9.4.21) and (9.4.6) relating ~2 (q) and t~1(q) respectively to the envelope and the phase of the signal. We have seen that the global moment m01 is just equal to the average of the corresponding local moment 1~1 (q)" In contrast, rn02 can not be identified with the average of t~2(q), unless the second average in (9.4.58) vanishes, as it happens, for instance, for signals displaying a phase linearly depending on t~.

Linear Ray and Wave Opticsin Phase Space

486

Further expressions for the second-order moment

too2

Tile second-order moinent m02 is aInenable to alternative ext)ressi()ns, which involve integrals over b o t h the space and frequency domains. Besides being conceptua.lly interesting and offering a, challenge for their interpretation a,s well as a deeper insight into the intert)lay of the st)a,ce a,n(t fre(tlu;n(:y variables, they t)rovide useful t()()ls f()r t)ra~:ti(:al (:a,lculations. Let 1is (:onsi(ler l,lie fl'e(tlmll(:y (hnlm,ill fonlnlla (9.4.55), yi~;l(lillg too2 as tlle variance of ]~(t~:)l 2. By th(, Pa,rscwfl th,;orem, it ('an t,,; r,;wu'itt,,,, a,s

mr,=

1 /

ii~ii =

I("~- ~ ) ~ ( " )

12d,,: -

l / I.~- I [(h;- ~,,)@](q) 12dq

~,~i=

('~4,~9)

H(,I'(, .~'-l[(h;- N,)~]((I)(l(;I,()t,(;S t,|,(; i,,v(;rs(' V(),,ri(;r tra,,sf()r,,, ()f tl,(' f,l,,(:ti(),, (t~:- ~,)@(h;), fl)r wl,i(:h a,, easy (:a,l(:,,la,ti(,,, gives $__l[(h;

__ ~)@](q)

__ __i(3~q ~d

[~(q),:- i-~.q] .

(,~.4.~0)

It, is t,herefl)re illlllm(liat, e t,~ write (lowll t,lle space (t(nlla,ill f(~rIlulla f(n" too2 as

-im"ql ' 1~ dq"

m,,~ = l l -1~ . /l'l ~ [ r

(9.4.61)

whi(:h rela,t, es too2 t,(J tim ll(n'lll of t,tl(; (t(;riwl,tiv(; (ff t,h(; sig~ml (~ll(:('~ tilt,(;r(;(t of the harn~()i~ic (:()~l)()~(;~t r a,t tl~(; (:a,i'ri(;r fl'(;(tll(;I~(:y nl,)~ - ~,. Mor(;ovor, ext)a,,l(li,~g t,l~(' (l('riva, tiv(' a,t)()v(, w(' fil~(t l

m,,~ _

11~-~-~.

_

/

I

r

(q)l

2

,tq-

2 __

m,,,

-

I1:'11 ~ 2 i1~11~ - m,,,.

(9.4.62)

Another interesti~g cxt)ressio~ for mo~ is obtaina, bh, fr(m~ (9.4.58) t)y use of expressions (9.4.6) and (9.4.21) for ~1 (q) and 1~2(q) in t('r,,,s of tl,(' I)tmse and the envelope of the signal. In fa,(:t, we end up with mo ~ = IlIII ~1

/ fie(q)@ + ~ ./

[r

-

g]212(q)dq.

(9.4.63)

It explMtly shows that, as noted before, the two averages in (9.4.58) depend one only on the envelope an(t the other only on the t)hase of the signal. It is useful to c o m m e n t on the two components of too., in (9.4.63). We consider a satmple signal O(q), which is supposed to retain the envelope of p(q) w h i l e h a v i n g _zer~ namely r f(q). Being 9 real, the relevant s p e c t r u m q~(n) - I ( n ) is symmetric, i.e., O ( - n ) - O*(n). Thus, the mean of z~ro ~nd hence the second t e r m in (9.4.59) vanishes. Moreover, the corresponding variance turns out to be simply

~ (~)

1

II~ I =

f~2 i~(,~)l 2

d~

1

Ilfl ~

/ft2(q)dq,

(9.4.64)

1 D First-Order O p t i c a l S y s t e m s : M o m e n t s o f t h e W i g n e r D i s t r i b u t i o n F u n c t i o n

487

the last equality being in accord with the Parseval theorem. Thus, the first term in (9.4.63) acquires the meaning as the variance ~2(~) of the spectrum of an envelope-only equivalent signal of ~(q). The other term is the variance of the phase derivative r and so relates to a certain type of smoothness of 0(q). As said earlier, for a linearly varying phase, this term vanishes. A further interesting relation is obtained by use of (9.4.6) and (9.4.21) in (9.4.58); namely, __ 1 too2 -- 2[~a

1 de d e ~p, 5[~*~-~q2~ + 7)~q2 ]}dq - -

--

2

toOl.

(9.4.65)

Interestingly, the integrand function can be seen as resulting from the action of the Kaiser-Teager energy operator s of signal analysis [4.6, 5], defined by A

..

s

__ [

~, )[l~tfq'12 -

d2 d2 1 ~[P*~q2 P + P~q2 ~*]"

(9.4.66)

The global second-order moment meo Tile global second InoInent m2o with respect to tile space coordinate is m2o -- mloo

/j

(q - l l l l O ) 2 W ( q ,

n)dqdn,

(9.4.67)

2 dq,

(9.4.68)

and can easily be recast into m~o

__

-

l f(q

I1~1 ~

-

~)2

]~(q)l

showing that me0 is a non-negative quantity, and accordingly interpretable as the spread of the Wigner distribution function along the q-direction with 2 of tile respect to the centroid (mlo,mol). In addition, it is just the variance Crq local power I~(q)l 22 (9.4.69) m2o -- Oq. The relation of m2o to the expectation value of the operator ~2 is apparent" me~ ~

(~2),

(9.4.70)

which then, paralleling (9.4.57), relates me0 to the matrix element in the state of the group generator K+ (see ~ 5.2). Again, the relation turns into an identity if one takes, as it is usually done, I1~112- 1 and ~ - 0. Like m02 , also meo can be expressed in terms of the local moments ql (/~) and q2 (~) and thereby through integrals over the frequency domain. An easy manipulation of (9.4.67), on account of (9.4.26) and (9.4.37), leads in fact to m~0

-

11~11~

q~(~) I@(~)l d ~ + ii~ll~

-- qe(t~) - [ - ( ( q l ( / ~ ) - ~ ) 2 } ,

(9.4.71)

488

Linear Ray and Wave Optics in Phase Space

which pa, ra,llels the corresponding expression (9.4.58) for m02. Thus, m20 can be sepa,ra,ted into two components, one being the average q2(~) of the local second-order m o m e n t q~(~) and the other the va,riance of the local centroid q~ (~,). As we will see below, these two terms are related to the m a g n i t u d e and pha.se c o m p o n e n t s of the Fourier trm~sforme(1 signal ~(~;). The se(:on(t-or(tcr m()In(:~lt m2o equa.ls th(: a,v(;ra,g(: q~ (h:) ()I11y wtmn the t)ha,se (l(:I)ending t,(:r~ i~1 (9.4.71) va,~lish(:s. Tiffs lla,t)t)(:~ls, f()r i~lst,a~l(:(:, i~1 th(: (:asc wh(:re t,tl(: l)lm,s(: ()f tim st)(:(:I,rllnl @(h,) is a, linear flnl(:(,i()~l ()f h;. FuT'th, eT" e:rFre,ssio'n.~ fin" the second-o~'de'r 'moment m~()

The exl)ressi()l~S f()r m~o giv(:~ l)elow a,re l)a,si(:a.IIy tl~(: (l~a.l f()n~is ()f tl1()se (:sta,blishe(I f()r mo~. Tl1(:n, w(: (h:s('ril)(: v(:ry litth: tlm deta.ils ()f th(: ('a,h:~da,ti()~is, tha,t as l)efore gr(n~1(l ()~ tlm Pa,rs(:wd (,l~(:()r(:~11, (:()~mect, i~g (,lm Sl)a('(: a.~(l fr(:(l~U:n(:y W(: a,I)I)ly (,11(: Pars(:wd tlm()r(:~1 (,() tl1(: q-(]()~m,i~ f()n~n~la, (9.4.68) f()r m2o. Tlu~s, ta,ki~ig i~i(,())u:(:()~(, (,l~at (,h(: Fo~lri(:r (,ra.~isf()r~11 ()f ( q - Y/)~(q) is [(q

w(: ()l)ta,i~1

-

-

i(

d

m~,)- 11~112. ~[@(h')r

~]

.-

(9.4.72)

dh',,

(9.4.73)

whi(:h (:(mn(:(:(,s (,h(: s(:(:()n(l-()r(h:r m()ment m~,) t() (,I~(: (h:riw~,tiv(: ()f (,h(: Sl)(:(:trum, whose I)ha,s(: |)(:illg 11()rllm.liz(:(l to (,11(: ha.rIIl()lli(' (:()lIII)()II(:ll(, ('iml()~': . hi terms ()f (,1~(; a,~,,l)li(,~uh: a,,l(t t)tlas(: ()f @(h:) a.s giv(;ll i~ (9.4.28), w(: fin(t

l /

m2o .... ilhll2

2

h t (h)dh: ! ~

1 /

22

[(,'(h,) t7/] h, (h',)dh:,

(9.4.74)

,

which (:lea,rly s(;t)ara.t(;s tile (:ontritmtions fl'(nll t,lm lnagllitlul(', h(h',) a,n(t the phase ~(/(h;) of th(: sI)(;(:t,rlml. The intert)reta,tion ()f these tw() (:ontributions parallels tha,t illllstrate(t fi)r the compou:11ts of m()2 ii1 (9.4_63). Consider a, sample signal tp(q), wt,ose sI)ectrum be ~(h',) - h ( t , ; ) . B(:i~lg ~(h',) rea,1, tp(q)is symmetric. Hence, the ~l(:an of I~(q)l 2 is zero a,nd the t)hase depending term in (9.4.74) vanishes. As to the first term, since h'(~) - ~'(~), we see that, applying again the Pa,rseva,1 formula, it can be understood a,s the variance of the local I)ower I~(q)l 2. Ext)licitly, we have

Or2(lI/)

IlqJl 21 / q2 IlI/(q) 12 dq

]hi21 / h / 2(g)d/4.

(9.4.75)

Therefore, the second-order moment m20 can be separated into the sum of two variances" the variance a 2 ( ~ ) of an equivalent signal of ~(q), which retains the m o d u l u s of the spectrum, and the variance of the phase derivative ~p'(~) of the Fourier transformed signal ~5(~), which vanishes if ~p(t~) is linear.

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

489

Finally, inserting the expressions (9.4.27) and (9.4.38) of ql(~) and q2(~) in (9.4.71), we obtain the dual form of (9.4.65), i.e., 1 ./ 11 11 .

-

2

1

d2

~ d2

- ~[~*~5

2

+ 7)7fi~2qS*]}dt~ - mxo ,

(9.4.76)

the function under the integral Aarising from the frequency spectrum ~5(~) when acted by the energy operator g according to the rule (9.4.66) [4.6, 5].

The global second-order moment m~ The mixed second-order moment ml1 means the double integral mll

mOO1/ l ( q

--

_ mlo)(t ~ _

mol)}/V(q,n)dqd~"

(9.4.77)

Espanding the product ( q - m l o ) ( N - m o l ) , a n d using (9.4.44)and (9.4.50), we end up with the equivalent expression 1 1Tll 1 --_ mo o

i / q t d / V ( q , ~;)dqdt~ - m~omo~ ,

(9.4 78)

which can be further elaborated recasting the integrand in terms of the first order moments t~ 1 (q) or qx (/g)" In fact, it is easy to obtain the q-domain formula

]

m~ ---- i~1121 (q

--

1T[10)(~l

(q)

--

mol ) I~(q) 12dq,

(9.4.79)

and correspondingly the ~-domain formula

1/

mll

[[~[[2

(ql (N) -- m l o ) ( N -- t o O l ) I @ ( ~ ) l 2 d~.

(9.4.80)

Further expressions are obtained by explicitly calculating the integral over ec or q. In fact, noting that m l l __ - mo1 o / d q q [ i t ~ V ( q , t ~ ) d N ] _ m l o m o l

'

(9.4.81)

oil account of (9.3.8) and tile first of (9.3.21), we obtain m11

mlomo

(9.4.82)

j" d ~ [ j " ql/V(q, ~)dql - mlomol ,

(9.4.83)

dqq[v:*'(q)g~(q)-

2moo

f

v:*(q) ~' (q)]-

1"

Similarly, since 1 ml

I

_- -

mo o

using (9.3.12) with r = 1 and the second of (9.3.21), we end up with IT'll

i

211100

dntr

'(n)0(ec) - qS*(n)~5'(ec)] - ltllOmOl.

(9.4.84)

490

Linear Ray and Wave Optics in Phase Space

The apex means, of course, derivation with respect to the specified argument. Both (9.4.82) and (9.4.84) convey the relation of m~x to the expectation va,lue of the opera,tor ~1 ( ~ , + ~,~)" m,,

+ A

a n(t s() t() tim ina,trix elei~mnt in tt~e state ~ ()f th(; gr()l~t) g(;nera,t()r Ka (w 5.2). (J()llf()rllling t() tile ll()ta,ti()ll (9.4.69) a,n(t (9.4.56) llSe(1 ill I'ela,ti()n with the st)a,(:e an(t fr(;q~mn(:y varia, Ii(:es, the mixe(1 Ill()llle, l l t is a~x:()r(li~gly (tenotc(t as ml~

9.4.4

=

aq~..

(9.4.86)

Local valence of the, uncertainty relation

AI~ intcr(',st, ing set ()f iIl(;(llla,li|,i(;s (:ml l)c (t(;(t11(:(;(1 ill rela, ti()ll witll tlm secondorder 211()lIl(;llts ()f tim Wigxmr (list, rilnlti()ll finmtiozl. The i~ltercstc(l rca(ter llm.y (:onsult the relate(t lit,eratlm; [6.1, ~.2]. Ih',re we c(mmmnt ()n the glot)a,l or local va.leIwc llm,y be attrilnltc(t t,() tim I)asic illeqllality, illwflviIlg tim ~'~(:()lld-order lnOlllelltS m20 a.ll(l m.2, (lire(:tly illfi'xre(l fronl tim wi(ltl~ l)ri~('il)le of Fo~rier m~alysis (,~ 4.6.4) m~(l ttm i(h:l~tifi('a, ti()~ ()f m~,, a.~(i m,,~ wil, l~ t,lm wtria,n('es of t,lm sig~m,1 a,n(t its Sl)er A(:r usi~g tim ('()~()l~ ~()l,ation a,s in (9.4.69) a,n(t (9.4.56), wc r(',write (m(:e aga,in th(; stm~(lar(t fi)r~ ()f the uncerl,a,i~ty r(;la,tion, i.(:. 1

aqa~ >_ -~.

(9.4.87)

()f r tim r a,t)()w; refi',r to tim wl~()h; Sl)a,tial rm~ge and the whole frequency rm~ge of tim signal. Notwithstm~(ti~g. i~m,(l~mlity (9.4.87) is comn~only given a "local" va,h;~t:e, t)eing inten(tc(t as setti~g a, li~it to what can be done locally. It is thercfi)re im,tura.1 to ix~(t~lire al)ol~t tim correctness and the general validity of i~terI)reting (9.4.87) i~ ten~s of local t)roperties of the signal. In other worr it is natural to incpfire al)()ut the possibility of defining "local" qua,ntities ana, logous to the glo[)a,1 o~ms cnteri~g (9.4.87) and accordingly about the possibility of individualizing bouI~(ts on their product, t h a t so might be interpreted as a local uncertainty product. This issue is beautifully addressed in [6.3], where a local uncertainty product is suggested, which product is then shown to be upper bounded, under definite conditions, by the global uncertatinty product (9.4.87). Specific investigations are then presented within the time-frequency domain pertaining to the context of signal theory. Following the suggestion of [6.3], we illustrate the question specifically in relation with the Wigner distribution function. Accordingly, let us go back to Eqs. (9.4.71) and (9.4.58) relating the global second-order moments to the local second-order moments. In view of such relations, one may ask whether

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

491

inequality (9.4.87) may place a limit to some analogous product involving tile local variances. Indeed, multiplying both sides of these equations we obtain

2 2

CrqOt~

q2 (N;)t~2(q) @ other terms,

--

(9.4.88)

the "other terms" specifically contributing as other terms - q2(u) ((~1 (q) - g)2} ~)2} nL_ ((ql (/~) -- ~)2} <([~1(q)

+t~2 (q) <(ql (/~)

-

-

(9.4.89) -

g)2)

Evidently, the variances of the local centroids ql (n) and t~1(q) are positive quantities. As to the averages q2(~) and t~2(q) of the local variances, we may recognize inspecting the first terms in (9.4.74) and (9.4.63), that, although nothing can be said a priori on the sign of q2 (n) and t~2 (q), their averages turn out to be unequivocally positive quantities. Therefore, equality (9.4.88) implies the inequality q2 (~)t~ (q) _< aq2cry2,

(9.490)

between the product of the averages of the local variances on the left and the product of the global variances on the right. It seems natural to consider the former product as a local uncertainty product:

q2 (~c)[~2(q)

local uncertainty product,

(9.4.91)

for which inequality (9.4.90) fixes an upper limit through (9.4.87). Thus, we see that, although the global uncertainty product cannot be sinaller than the lower limit set by (9.4.87), the local uncertainty product is allowed to be smaller than the global uncertainty product. Can it be arbitrarily small or does it have its own nonzero lower limit? Also, may inequality (9.4.90) stilnulate a critical revision of the comlnonly held belief that the global uncertainty relation somehow limits the local quantities? Leaving these fascinating questions open, as they are indeed, we address the reader to the related literature [6]. 9.4.5

Moments of the Wigner distribution function and ambiguity function

It does exist an interesting relation between the moments of the Wigner distribution function and the derivatives of the ambiguity function, which is indeed implicit in the Fourier transform relation linking 142 to A [4.9]. In fact, just rewriting relation (7.4.16) we see that moo = 27rA(O, 0).

(9.4.92)

Linear Ray and Wave Optics in Phase Space

492

Also, taking account of the Fourier transform W - A relation and of the obvious derivative relations --

Or _

-ciqtr

t~h c - i ~ q

.

_ ih

Oh

(,-in, q

(9.4.93)

it is easy to t)r()ve for the (:(;ntroi(t of tile Wigner distributi()xl flm(:tion the i~lteresti~lg relati()ns m~() = 2~ri

07r

m(, - -27ri 0.A((~, 00

q, ~=()

k~)l

(9.4.94)

~,~=0 "

Ttw.n, translati~lg 1,11(' origin1 ()f tt~e anfi)iguity fun(:ti()n t)ha.se t)lan(; (tl, h;) to th(; (:entr()i(t (7/, ~) ()f the Wig~l('r (tistrit)~lti()~l fl~n(:ti(m, w(; l~my ext)r('ss tile se('(m(t nl()meJlts ()f W(q, ~;) i~1 t(;r~ls ()f the se(:()x~(t (t('riw~.tiv(;s ()f A(~, ;c) at the ()rigi~l; na.~mJy, m~oEl 1

-2~r

o~ OqO~

]

~,~_()

'

m,)~ -

-2~r ~

o,'i '~

I~, ;r

(9.4.95)

lq,~_()

Evi
9.5

Gaussian Wigner distribution functions the variance matrix and its evolution

In order to gain a (:ertain insight into the definitions given M)ovc, we consider Gaussian Wigner distributions. We have already considered Gaussian distributions in w 6.3.6 when illustrating the smoothing procedure acted on a Wigner distribution function in order to generate distributions being positive everywhere through the space-frequency plane. There, it was noted that a Gaussian distribution is Wigner representable under definite conditions for the relative parameters, the generating signal being in general complex Gaussian. In w 6.4.5 we have calculated the Wigner distribution functions of both real and complex Gaussian amplitudes which have been noted to be interpretable as the amplitudes of a Gaussian beam respectively at the waist and at a plane

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

493

far from the waist. We consider here the expression of a propagating Gaussian beam as it is usually represented, and focus our attention on the relevant Wigner distribution function. On calculating the moments, we will be able to put them in relation with the beam parameters and identify the relevant evolution law, which will be generalized to arbitrary non Gaussian Wigner distribution functions, and so to arbitrary non Gaussian (coherent) signals. 9.5.1

Gaussian beams: review of the relevant parameters

A Gaussian beam propagating in the z-direction, as obtained, for instance, from the lowest-order low-divergence solution of the parabolic wave equation in a free medium or in a quadratic index medium, is typically represented by the complex amplitude [7]

k q2

_ ( i~Q2(z))-~e kQ0 /[2Q(Z)

kz] .

(9.5.1)

For complctcncss's sake, also the on-axis phase factor C- i ~ ( z ) + i k z h a s been included in the expression above even though it is not functional to the calculations of the Wigner distribution fimction and the relevant moments, for which only the q-dependent part of the waveform has relevance. 1 Central to the representation (9.5.1) is the complex curvature Q--~, which contains information about both the spatial width and the angular divergence of the beam, being 1

1

=

+ i

2

(9.5.2)

As is well known, R(z) is the curvature radius of the beam wavefront at z, which in accord with the paraxial approximation inherent in the waveform (9.5.1) has a parabolic profile; it, of course, determines the transverse deviation of the beam phase from the plane-wave phase. Correspondingly, w(z) gives indication of the radial extent of the beam, being the off-axis distance where the beam intensity reduces to e -2 ~ 0.135 of its value on the axis. As the beam propagates from zi to Zo, the complex parameter Q transforms simply as

e( o)

=

+

-

(9.5.3)

It can therefore be understood as a kind of a complex curvature radius since it propagates just as the real curvature r a d i u s / ~ of a spherical wave emanating from a point source at z = 0, for which, in fact, one has

Rs(zo) = Rs(Zi) + Zo - zi.

(9.5.4)

The analogy Q-Rs is better yet confirmed by the fact that in general, when propagating through an A B C D system, the complex beam parameter changes

Linear Ray and Wave Optics in Phase Space

494

according to the hyperbolic MSbius t r a n s f o r m a t i o n (see also P r o b l e m 2.10)

AQ(zi)/no, + B Q ( z o ) - no~ C Q ( z ~ ) / n o ,

(9.5 5)

+ D'

whictl is the same suffered by the c u r v a t u r e radius It~ of a st)herical wave t)r()t)agating thr(nlgh the s y s t e m fr()nl a t)()int s()llr(:e a,t z - 0. As 11ram,l, n 0 9 l and n,),, (t(m()t(~ th(; refl'a,(:tive ill(li('(;S resI)e(:tiv(4y i~l ttl(; (fl).je(:t ml(t tlle image st)a,(:e. Evi(t(mtly relati(n~ (9.5.3) is a Sl)(;('ia,1 (:as(; (ff tl~(; g(;u(;ra,1 law (9.5.5). W i t h the t)lane zi = () I)la(:e.(t at the t)(;am waist, wh(;re l)y (h;finiti(m the b e a m wave front is I)lan(;: R(()) = co, an(t the t)(;a,~ ra(ti~s is nfinimun~: w(()) = 'u~0, the (:()nq)l('x 1)(;m~ I)ara,n~(~ter Q at tl~is t)lmm is I)~rcly i~mgi~mry; nmnely, Q(())

-

--Tff- = -iz,r

(9.5.6)

Ih;n,'.(;, i,~ a(:,:()r,l witl~ (9.5.3), tl,,; l)(,m,~ t)arm,,,:t,,rs R(z), w ( z ) a r e (w()lve a.l()lig tll(; l)r()l)agati()li axis as

seen t,)

z2

R(z)-

z(1 + :-J~2) :>>:,,

Z~

.,,2

()

~

(9.5.7)

4z 2

u' 2(z) .... u,2(l + " ) z ~,;> Zl/

~;w 2

Th(; Rayb:igh 'ru'n,9c z,, 2 '~ (th(; wa,ist a,r(;a (livi(lc(l |)y A) ('ntering the al)()ve fi)rnmlae, r(;t)r(;s('~llts tlle axial (listml(:(~, fI'()lll ttl(; waist at wlli(:ll the beain ra,dius in,:reases t)y a, fa(:t()r v~: "t,'(z, t) - V/2w,,, a,,(t s,)t,l,e t)(,a,,, area ,tout)les: 7rw2(za) - 7 c w 2. Tim regi()n fi'()nl -z~r t() z,t can t,tmr(;f()r(; |)e i(telltilie(t as the fo(:al zone, or the region over which tlle |)cam is a.t)I)r()xiIlmtely (:()lliIImted. The fllll dista,n(:e bctwe(;n the two w(z~t) b(;a,nl radii is r(;fi'xre(t t ( ) a s tll(; confocal paTu'meter of the beam. We n()te also t h a t at z - z R the (:urvatllre ra(tius takes the m i n i m u m wdue R(z~t ) - 2z,t. Then, in the rcgi(m z >> zxt th(; b e a m is similar t() a spheri(:al wave creating from a t)oint s(nlr(:e at z - 0, t)eing in(teed 1 The two regioils z << z a and z >> z,t are generally known Q(1z ) z>> zr~ n(1z ) ~" -;. as near-field region and far-field region respectively. A n o t h e r interesting p a r a m e t e r related to the shape and the p r o p a g a t i o n features of the b e a m (9.5.1) is the b e a m divergence angle O(z)(tefiim(t by

O(z)-

w(z) Z

z2 w~ V/ 1 + z2 Z

T h e far-field divergence angle 0~ of the b e a m radius dependence; p r o p a g a t i n g towards infinity. In are reciprocal p a r a m e t e r s , which

R

z>>zs

O~ = ~ ./~

(9.5.8)

71-Wo

represents the slope of the oblique a s y m p t o t e it describes the spreading of the b e a m when accord with the width principle, 0~ and w 0 m u l t i p l y to the z - i n d e p e n d e n t value A/Tr.

1D First-Order Optkal Systems: Moments of the Wigner Distribution Function

495

FIGURE 9.3. (a) Intensity of a Gaussian beam, (b) corresponding beam width w(z) all(| wave front parabolic profiles, and (c) Guoy phase shift, as functions of the axial variable z.

F i n a l l y we recall t h a t tile G u o y p h a s e z ), ( ( z ) -- t a n - l ( z--~

(9.5.9)

which adds to the plane-wave p h a s e kz to form the l o n g i t u d i n a l p h a s e of the b e a m , a c c o u n t s for the excess delay of the b e a m wave front in c o m p a r i s o n w i t h a plane or a spherical wave. It ranges from - ~71" at z - - o c to 571" at z - oc, t h u s yielding a t o t a l excess delay of 7r as the b e a m travels from - o c to + o c . Figure 9.3 depicts the g e o m e t r y of a p r o p a g a t i n g G a u s s i a n b e a m . 9.5.2

Gaussian beams: the matrix of second m o m e n t s

T h e W i g n e r d i s t r i b u t i o n function of (9.5.1) is r e a d i l y c a l c u l a t e d to 2q2 _ ~ 2 ( z ) [ n _ R-~)] 2

_

w (q,

-

_

~.2(z)

7I"

2

.

(9.5.1o)

T h e zero- a n d first-order global m o m e n t s are of course moo(Z)

-

1,

rI~lo(Z ) --tirol (z) - - 0 ,

(9.5.11)

Linear Ray and Wave Optics in Phase Space

496

in accord with the n o r m a t i o n condition and the on-axis p r o p a g a t i o n of (9.5.1). Interestingly, for the global second-order m o m e n t s we find

w~(z) 4 '

m~o(Z )

m~l (z)

--

4R(z)

1

--

2z R

m,,~ (z) - ,,,,(:) + 4n~(:)

As (~xI)e(:tr

t,llr sl)a,('e sr162

ilitI'ilisi(: t)em~l t)r()I)r i.r wave inlllfl)ei', wlfi(:li is i]l(ler

(9.5.12)

'

~:2~1~2( Z)

/,:2 0o ~

1

"w()2 --

4

~ll()~ll(~]lt, m~o(z ) r(;lat(:s t,()t,l~(: h)(:a,l I)r

ra,(ti~s

l,llr fa.r tir (tiw~rgr 0 , l,lirr l,llr light t,lie 1)l'()l)(~I" "(:()listmit" t,()l,~mi fl'(nxi a l@es t()

st)a,tial fr(~(t~u',n('ies. Fi~lally, wr see; tlm.t a.t t,tie i~ll)tfl, t)la,llr z -- {1, m2~~({)) . 4 whi(:li t,ll('ai linflt, il)lies wit, li m.2(()) t,(~ tlic lililii]lnl]~l alh)wr 1)y tile; ()t)ti(:al 1 ~m('(;rl,ail,ty r(;lati(,,l (9.2.29)" m2,,(l))m.~ (1)) a" It is (:()liv(;ni(;lll, l,() a,rra,llgr tim a])(we, lil()llient, s i~ll,() a, ~latrix s(:lm~lie (h;filfing tilt; 'l~a'l'iit'lt(:(', (()r, .,~c('oltd "nto'nl~"ltt ) "ntall"i:r E a s

ml, (z)

mo2(z )

kw2 (z)

1

L.2.w2(z )

.

(9.5.13)

A(:(:(n'(tingly tim s(~(:()ll(l Ill(nlielll,s will a,lterIm,tivr t)~; (l~llr a,ls() a,s m2o ~-~qq, 11102 -- }-~t,:t,; ~l.II(l m,~ - Eq~=. N(~r t(~ say, E is (h'~tillr a.s a, rc;a,1 symlllr (all(l llr llr llm, t,rix, wll()se (~tf-(liagr r162162 S are (:lea,rly ,~ith~ss wliilst t,l~r (lia,gonal (rues have (limensio~s rcslm(:tiw:ly (~f h;l~gt, h ~ a,n(1 h;x~gtli -2. T h e (teternfinaJlt r IE, wt~i(:t~ is easily seen to l)c 1 (let ]E(z) - ~,

(9.5.14)

comes to t)e an iIlvaria,nt ()f the nlotion. T h e inverse of E is ttmil 1 ~-']--l(z) -- ( t e t ~

(mo2(Z) --roll(z)

--m~(z) ) m2o(Z )

(9.5.15) "

Also, it is w o r t h m e n t i o n i n g t h a t ]E is positive definite. Note t h a t a, H e r m i t i a n m a t r i x A is said to be positive definite if for &Ily vector u it results t h a t [8] (u, A u ) > 0.

(9.5.16)

It is a, well e s t a b l i s h e d result of m a t r i x t h e o r y t h a t the necessary and sutficient condition for a H e r m i t i a n m a t r i x A to be positive is t h a t all leading principal

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

497

minors of A are positive. Since rn0~(z ) > 0 (and as well m~0(z ) > 0) and det E(z) > 0, the variance matrix E is evidently positive definite. Notably, relation (9.5.14) implies the optical uncertainty relation (9.2.29). In fact, writing the explicit expression of the determinant of E, i.e., 2 Crq2Cr~

1

-

2 _ ~, Crq~

-

(9.5.17)

we obtain for the space-bandwidth product the relation O'qO'~ - - ~1V/ 1

+ 4cr2~ > - 2 '-1

(9.5.18)

which at a primary level confirms inequality (9.2.29). At a deeper level, it yields a more precise relation for the space and frequency signal widths product being linked to the beam-depending mixed moment Crq~r (see also Problem 4.8). 9.5.3

Gaussian beams: propagation law of the variance rnatriz

The utility of introducing the variance matrix is conveyed by the evident possibility of expressing the Wigner distribution function in a compact and elegant form in terms of E. In fact, one may note that the exponent of (9.5.10) directly involves the entries of E, i.e., tile second moments, being 2q2 q_w22(z)[N

~(z)

kq 2

1 T[~-'](Z)]-I

R(z)] - ~u

u,

(9.5.19)

where as usual u = (q, h;)T. Therefore, according to the normalization of Ga,ussian functions (a) we can finally write

}&g(q,n;z)

-

-

1

2rrv/det ~

e-2

l u T [Y](Z)]-I

u

(9.5.20)

thus relating the Wigner distribution function of the Gaussian beam (9.5.1) to the relevant variance matrix E. Remarkably, this allows us to obtain tile propagation law for E and hence for the second moments. In fact, the paraxial transfer law for the Wigner distribution function signifies that 1426(u; Zo) -- ] / V G ( M - I u ; zi),

(9.5.21)

where M is the ray-matrix connecting the planes at zi and Zo between which the beam (9.5.1) is propagating. Recall that when dealing with the Wignerplane pair (q, ~) instead of the geometrical-optics phase-plane pair (q, p), the off-diagonal entries of the optical matrix are scaled respectively by ~0 and k 0. Recall that for an n-dimensional Gaussian function G(u) - e - l u T Au, with A an n (2~)n/2 positive definite matrix, one has f ~ ( u ) d u - d,/~A a

x

n

Linear Ray and Wave Optics in Phase Space

498

In terms of the variaIme inatrices ~(zi) a,nd X(Zo) at the end-planes, relation (9.5.21) mea,ns 1 ( ; - - ~ l u T [Y](Zo)] - 1 U __ _1 C - 1 [ M 7r 71"

1 u ] T [~--](Zo)] - 1 [ M - 1 u]

whi(:h imt)lies of (:ourse t,l~(; i(te~tity ()f tl~(; ext)()nent. H(;~(:(;, w(; (;n(1 ~t) with

~(zo) -- M ~ ( z i ) M T,

(9.5.22)

whi(:h is t,tl(; (',entral r(;la.ti(ni ()f ()llr mlalysis. Alth(nlgtl it, lm,s t)(:(;ii inf(;rrc(t ill (:()nXl(;(:(,i()ll with tll(; ('~asily llm,lm,g(;a,1)h; Wigll(;r (lisi,ril)ll(,i()ll fiU~(:ti()n ()f the signal (9.5.1), it h()l(ls tr~u~ fi)r m~y Wiglmr (listrit)~ti(n~ fi~(:ti()~, m~(l s()()t)ti(:a.1 (tist~lrl)m~(:(; (i~ 1,t~(; 1)a,raxia,1 a,l)l)r()xi~m,ti()~). ()~(:e ~fi)l(l(:(l, it, yicl(ls the (;xt)li(:it (;v()l~ti(n~ laws fi)r (;a(:l~ ()f tl~(; s(:(:(n~(l ()r(l(;r ~()~(;~ts, tln~s (:stat)lislfix~g tt~(; r~l(,s, l)y wl~i('l~ 1,t~(' s(:('()~(l-~()~m~t t)a,s('(l I)l~as(~l)lm~(: r(:l)r(:s('~tativ(; regi(n~ a,ss()(:iat('~(l witt~ tt~(', I)(;a,~ (:lm,~g(;s (l~ri~g I)r()l)a,gati()~, witl~ (,1~(',rel(;va,i~t a.r(:a r(:~m,i~i~g (:()IIS(,a.II(,, ~IS S(,a.[,(:(I l)y tlm i~va.ria,~(:(: ()f (h:t ]E. Wc st~a.ll (lis(:~lSS r('la(,i()~ (9.5.22) lal,(:r i~ a ~()re g('~(;ral (,(rim;x(,. th're we ex~q)ha,size (,l~a,(, it (l()es yM(l tl~e AI:1(71) law (9.5.5) fi)r (,Ira (',()~l)l(;x 1)eax~ l)ara,~(:t(:r Q(z). N()t(:, i~ fa('(,, (,Ira(, (2 vn'it,(:s i~ [,(~,l'IIIS ()f (,l~(: E-(,~t,ri(:s a.s

•

z,,. + iv/,l(,t E

(9.5.23)

or, c(tlfivah;lltly

c2 - L,(z,,. - i v / , l , ; t

(,9.5.24)

()f collrs(;, v/(let E - 89 ill (,lie (:a,s(; ()f (,ll(; filll(la,lllClltal (la,llssia,ll 1)cam we are (tealing with, howew;r l(;tl,ing v/(let E lm(lcfinite i~ (,h(,~ a.I)()w; exl)ressions confer thein a certain gelmrality, that will revea,1 usefill f(w filtllre sllggestions. Inferring f()rm (9.5.22) the r(4ati(m ()f the m a t r i x (',h'mcxlts Eq~ and E ~ between the en(t-pla,nes, i.e., Y]q~(Zo)

--

ACkoEqq(Zi) + (AD + BC)Eq,~(zi) + ~. BDE,r 2 C 2 k)Eqq(Zi ) + 2koCDEq,~(zi ) + D~E,~,,~(zi),

and noting from (9.5.24) t h a t 2/,:Eqr Q, E~ = Q +

-2ikv/det E E~ = Q-

k2Eqq Q*'

E~

, QQ ,

(9.5.26)

after some algebra we recover the A B C D transfer law (9.5.5) for the complex beam parameter under propagation from the plane at zi to the plane at Zo being connected by the ray m a t r i x M - (A ~). \v /

\

1D First-OrderOptical Systems: 9.5.4

Moments of the

WignerDistribution Function

499

Gaussian beams: the phase-plane ellipse of second moments

Sections of the Wigner distribution function (9.5.20) at planes parallel to the (q, ~) plane individualize elliptical contours defined by the equation (b) luT

[E(z)] - 1 u -

c.

(9.5.27)

[E(z)] - 1 u -

1,

(9.5.28)

In particular, the ellipse (c = 2) 1u T

occurs at, an height which is 0.135 of the maximum height of the 3 9 plot of the Wigner distribution function. Of course, any one of tile ellipses (9.5.27) can be taken as a phase-plane representative region of the corresponding Wigner distribution function (and Ga,ussian beam) and hence used to inspect the modifica,tion suffered by the beam under propagation through the systen under investigation. Mantaining the correspondence with the analysis in previous chapters, we specifically consider the phase-plane ellipse identified by (9.5.28), which explicitly writes }--~.#~t~(z)q2

-

-

2Y]q~(Z)qh; Jr- Y]qq(Z)#~2 -- 1.

(9.5.29)

For a, descriptiom of the geometrical properties of (9.5.29), the reader is diretted to w 9.6.4, where the topic is illustrated in relation with the general case 1 of an a,rbitra,ry non-Gaussian Wigner distribution flmction (v/det E -r ~).

9.6

Propagation laws for the moments of the Wigner distribution function in first-order optical systems

Here we reformulate tile propagation laws for tile secon(] nnoinents of the Wigner distribution function in first-order optical systems just described in connection with Gaussian beams, within the general context pertaining to arbitrary non Gaussian optical signals [4.1, 4.2, 4.4]. Firstly we consider the first-order moments ~, g. In conformity to the vector notation u - (q,ec)T, we may concisely write (9.4.44) and (9.4.50) into one form U(z) -- N

u W ( u ; z)du,

(9.6.1)

where, of course, f f ..du = f f ..dqd~. As we know, the propagation of the signal through an ABCD system implies a transformation of the first-moment vector (9.6.1), explicitly expressed by

U(Zo)

-

-

[l JJ

U}/~2(U;

zo)du

-

-

if JJ

u~2(M-Xu;zi)

du.

b N a t u r a l l y the conic g1 u T[r~(z)]-i u is recognized as an ellipse since det ~ > 0.

(9.6.2)

Linear Ray and Wave Optics in Phase Space

500

C h a n g i n g the variables from u to v -- M - l u , h)r which d u due to the u n i m o d u l a r i t y of M , we end up w i t h

~(Zo)- M f /" vl/V(v; zi)dv-

det M d v - d v

(9.6.3)

M~(zi).

1,!

As ext)e(:t(',(t, the, first-()r(ter m()me, nts ()f the, W i g n e r (tistrit)llti()n fim(:ti()n evolve j u s t as the ra,y-va,ria,t)les ()f t)a,raxial ray-()t)ti(:s, in a,(:(:()r(l evi(lently with tim reslllt, s (~stal)lish(~(t in ~ 5.2 regar(lillg l,lle l)r()l)a,ga,ti()ll law ()f l,h('~ m e a n va,hms, ~ ml(t ~ ()ftll(; l)()sil, i()ll ml(t fl'e(lll(;ll('y ()l)(~I'a,l,()rs. In ()r(ler t()(:()n(:isely write tim se('()ll(t-()r(ler lll()lllents, We f()rlll tlm Kr()lm('ker I)r()(llu:t ()f tlle Wigll(:r-I)lmle ve('t()r u witll l,ll(~ trmlsl)()s(~ ve('t()r u T, tluls ()t)taining tim 2 x 2 syIiulmtri(: llmtrix(") qh- ,~.2

9

(9.(i.4)

A('(:()r(lil@y, t)y (9.6.1) l,ll(, l)r()(lll('t ii C4 ii T l)r()(llu'('s l,tm lilatrix ,,

.

'l'lle (l(~i)eli(l(~1i(:(' ()ll tile l)r()l)a,gati()ll wu'ia,1)h~ z is ill~l)li(~(l il~ (9.6.4) all(t (9.6.5). Tlm wu'im~('(~ ~m.trix E , i~l,r()(l~l(:(~(l i~ ((.).5.13) i~ ('.()~m(:l,i()~ witl~ Gm~ssia,n })ea,~s, is tl~er(~fi)re se(u~ a.s a, l)arti(:~la,r (:ase ()f l,l~(', g(~era,1 (l(~fi~il, i()~ E(z) 1.!

TyI)i('ally it is a s s ~ ( ; ( t tt~at ~ - (), tlu~s si~al)lifying tl~e w()rki~g f()r~mlae. h~ or(ter t() i~fer tt~e trm~sfi,r law f()r 1,1~(, varia.~(:(' l~mtrix E ( z ) i~ first-order ()t)ti(:aJ systen~s, we a,t)t)ly tlm t)rot)agati()~ law for tlm W i g n e r (tistrit)ution function, c h a n g e tim integrati()n varia.t)h~s m~(l tt~ex~ ext)loit tim, sy~q)le(:ti(:ity of t h e ray-ma,trix M in t h e s a m e way as ()n (leriving (9.6.3). We fi~m.lly fin(t

~(Zo)

-- / / U 9

-

M

(~ UT]&(U; zo)du - / / u j/.

~ uT]A;(M-lu;

zi)du (9.6.7)

I,!

v | v T W ( v ; z i ) d v M T,

~The Kroneckcr (or direct) product of an n x m matrix E and an n' x rn' matrix F produces tile nn' x ram' matrix E | F by tile rule [8] el: F

999 el,,, F '~

. . . . .

E|

e,,i F

.~i

) ~

e ...... F

Evidently, the Kronecker product is not commutative. It satisfies, among the others, the properties" (a) E | --(E|174 E|174 (b) E | -- E | 1 7 4 (c) (E | F ) ( H | G) -- E H | FG, provided that the ordinary products E H and F G may be formed; (d) ( E Q F ) T -- E T | (f) ( E Q F ) - 1 - E -1 | -1, i f E and F are non-singular.

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

501

which, after identifying the double integral as E ( z i ) , yields E(Zo) - M E ( z i ) M T,

(9.6.8)

reproducing, as expected, relation (9.5.22). Relation (9.6.8) represents a congruence transformation (a). It implies however a sitnilarity transformation for the matrix E ( z ) J , for which, in fact, we firstly obtain E(Zo)J - M E ( z i ) M T J , (9.6.9) and then, being M T J -

J M -1, ]E(Zo)J - M ] E ( z i ) J M -1

(9.6.10)

Similarity transformations have the notable property of preserving eigenvalues and traces. Therefore, Eq. (9.6.10) may provide a useful supply of invariants. The interested reader may consult the related literature [4.2, 4.4, 4.7, 4.5]; here, we only detail the basic invariant yielded by the determinat of E. It is, in fact, evident that, due to the unimodularity of the optical matrix M, det E(Zo) - det E(zi),

(9.6.11)

as in tile prevously described case of Gaussian Wigner distribution functions. 9.6.1

The optical uncertainty

relation. H

The explicit form of (9.6.11) gives EqqEnn -- E2q~ -- const.

(9.6.12)

In particular, choosing a transfer matrix M such that E comes to be diagonal: Eq~ = 0, we have d e t E ---Y]qq}-]t~n, (9.6.13) which according to the width principle yields 1 d e t E _ > ~.

(9.6

14)

Since, whichever be the optical matrix M, the variance matrix resulting from (9.6.8) and obeyng (9.6.14) corresponds to a physical signal, i.e., the signal transformed by some other signal by propagation through the system represented by M , relation (9.6.14) should hold for all variance matrices. This d Given a matrix A and a matrix B, the product BAB T realizes the congruence transformation of the matrix A by B, whilst, if B is non-singular, the product BAB -1 realizes the similarity transformation of the matrix A by B.

Linear Ray and Wave Optics in Phase Space

502

establishes a b o u n d for the invariai~t value in (9.6.12), which then can be written as 1 2 1 E q q E ~ _> ~(1 + 4Eq~) -> -'4 (9.6.15) It ret)rcsents the con~t)h'te form of the ot)tical ~ln(:(;rl,a,ild,y relal, i()i~, il~volving a t)re(:isc signa.l-(h;t)cn(h;nt t(;n~ (relate(t to tim mixe(1 ~(nn(;~d, Eq~,) t)csi(h;s the 1 signa,l-i~(let)e~(le~t valll(; 4" 9.6.2

l~vpagation la,~s for the seco'nd-orde.'r mommy'his

Unfr tt~(', trm~sfi.'r law (9.6.8) fi)r 1,1~('~w~,ria,~(:(; ~ a t r i x E ( z ) , (n~(: gets the exI)li(:il, i~l)~l,-()~l,l)~t, r(;la.ti()l~S fin' 1,1~(;se('()l~r ~()l~(;~d,s, l,ha.l, (:a,n as well l)e a,rra.l~ge(1 i~d,() a ~m.l,rix st:l~(;l~e. I~ fa,(:l,, il, is sl, ra.igl~l,fi)rwa,r(l 1,~)wI'il, e

Eq,,.

(z,,)

Es:Sl

k,,A(7

AD + BC

/,:2(.-,2

2/% (71)

--HI) 1) 2

Eq,~

(zi).

We will st)c(:iti(:a.lly exm~i~w, tl~e eflb,('l, r s{n~m r162 tra.~sfr l~l(n~r la,l,~;r i~ r~:lal,ir witl~ 1,1~r lflm,s~;-l)lm~e s~'.('~r162162 9.6.3

(9.6.16)

i{)~s ()~ tl~e ~:llil)Sr

Positive definiteness of E

Firstly, let 1is ~l~t,~' tlm,t r~:lati~m (9.6.16) ('ml }~e lls~'~t t~ l)r~v~; tlla.t l,lle va,ria,n(:e m a t r i x E is t)~sitiw; ~h;tiIdt~:. Tim tirst r~w, fin" izlstml(:e, writes

2

+ v.

+

1 /3 2

-

> o.

(,9.6.17)

Evidently ttl<' left-Ira,fill si
u. Eu,

(9.6.18)

with the ve(:tor u t){;i~g givex~ by the ~natrix entries u - (A, B) q-. S~u:h a scalar p r o d u c t turns to |)e equal to a positive quantity, i.e., Eqq(Zo). Then, being u arbitrary, one (:m~ state the positive definiteness of the m o m e n t matrix, a,ccording to the t)ertincnt definition reca,lled in w 9.5.1a (Eq. (9.5.16)).

9.6.4

The ellipse of second-order moments

According to tile results of w 8.5.3, as far as tile linear a p p r o x i m a t i o n is concerncd, the t)hase-plane d y n a m i c s of the W i g n e r chart mirrors t h a t of the raydensity distribution in the geometrical optical phase-plane. This means t h a t one m a y choose any closed d o m a i n in the W i g n c r plane to inspect the t r a n s f o f m a t i o n s experienced by tile W i g n e r distribution flmction under p r o p a g a t i o n

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

503

through first-order optical systems. As it can be inferred from the pictorial views presented in the preceding chapters (in full analogy with those pertaining to the "classical" ray-density distribution p(q, p; z); see, for instance, w 2.5), it is convenient to work with regions delimited by straight lines or ellipses. In fact, &line distributions transform into &line distributions and Gaussian distributions transform into Gaussian distributions (w 6.4.4-5), straight lines and ellipses respectively being the phase-plane equilevel contours pertaining to such distributions. It is a common practice to adopt as "model" distribution the Gaussian distribution, and accordingly to focus on any one of the phase-plane elliptical regions individualized by the second-order moments according to (9.5.27), i.e. luT [Y](Z)]-lu =c,

(9.6.19)

for real values of c. In a sense we consider the Gaussian Wigner distribution function that best fits the Wigner distribution function under study, and hence the Gaussian signal that best fits the signal under examination. To maintain the correspondence with the presentation in w 9.5.4, we will definitely examine the ellipse corresponding to the value c = 2; namely, ~ g a ( z ) q 2 -- 2~qg(Z)qt~ + ~qq(Z)h; 2 -- 4( 2,

(9.6.20)

where we have denoted by e2 tile invari~mt va,lue of tile determinant of E, i.e. (let E _----e2.

(9.6.21)

The basic geometri(:al properties of (9.6.20) are enumerated below; in this respect, it ix worth recalling once more that in order that tile worked formulae be meaningful one must think of properly scaling the axes so that they have the same dimensionality. The reader can easily verify that (Fig. 9.4) 9 (9.6.20) is a tilted ellipse centered on the origin of the phase plane. 9 The points at which it intersects the q and h; axes are identified by the coordinates qint

-

-

-[- ~ -2E -~ ,

/~int

_

_

2~ ~//-~qq"

(9.6.22)

The semiaxes a+ of (9.6.20) are given by fl-F -- lY]qq-JF }-]nn -t-- 2~--~- V/}-]qq -t- Y]nn -- 2~ ,

(9.6.23)

and disposed in conformity with the two solutions, spaced by ~, of tile equation tan(20)

2r~q~ y]qq__y]n

,

(9.6.24)

504

Linear Ray and Wave Optics in Phase Space

9

t

//

K" /

~ ' m a ~ ~ K'in t

V",,,,

/

,'IN 2 I -

k.,"

.,i"/

X,,.._... ,~

I

8

I qim

2~" =

'

FIGURE !).4. S(:l~e,~at,i(' ()f t,l~e se(:()~(l-~()x~m~d, l)t~tsr

ellipse fi)r a~ arl)it,rary sig~al.

f()r tt~(; ()ri(;~ta, ti(n~ ()f tl~(; (;llil)s(; axes. . T h e ~xm.xixxnn~ (;x(:~rsi()~s a,l(n~g 1,1~(; (:()()r(li~m,l,e ax(;s a,I'(; Sl)e(:iti(;(1 1)y 1,1~(', sI)a,(:(; ml(I fr(;(t~(;n(:y va.riall('(;s a(:(:()r(ti~g 1,() ,, ......

-

. ......

-

9 TI~(', a,r(:a, (~f (9.6.2()) (:m~ t)(; t)r()~gld, i~d,() r('lal,i(n~ wil, l~ (1(;I, ~ , 1)(;i~g

A = 7ra+a_ = drrr

(9.6.26)

Tile inva.ria.xl(:e ()f t,lle (l(:t,enllixmxlt, ()f ~ t,heil relates t,o l~iollville's illva.riance ()f t)ha,se-t)la.n(: a.r(:as (set: !i 2.2.1 a,n(l .~ 8.5.3). Tile ixwa,ria,xl('e ()f l,lu; a,rea of the Se(:on(t-lll(nlU;llt (;llil)Se lla.s l)ra.(:l,i(:a,1 (:{)IlSe(tllell('(;s ()11 tim (:(nll)lillg efficiency when l,ra,lisfi;rri~lg ()I)l,i(:a,1 l)()wer fi'()m ()I1(; (:onlt)cnl(;ld, 1,() a.xl()tlier ill a, given optica.1 r i(m. At:c(~r(ling 1,o Li(nlvilh;'s l,lm()rel~, l,h(;re is nt)()ptica,1 systen~ l,l~a,t, r re(l~u:e 1,1~(; area, ()f the ret)res(',~d,a,tiw; l)l~a,s(;-l)la~('~ (l(n~m,in of a given light, twa,m, whi(:l~, in fact, can be defon~e(t tn~t n()t trm~sfi)rn~ed into a, sma,lh;r a,r(;a, (l(mm,i~. Ttu;refi)re, if the a,rea of t,he ret)resenta.tive ellipse, of a, given opt, i(:a,1 s y s t e ~ (tl~e accepta~,ce, Inent, ione(t i~ w 2.5) is sx~m,ller t,han the ellipse area (the e'm,ittance, still mentione(t in w 2.5) of tl~e ra,(tiation that must be couple(1 to it, some of the optica,1 power will be lost (Fig. 9.5). T h e design of optical (tevi(:es, where light is coupled to some receiver, greatly benefits from the eva, h~a.ti(m ()f the maxi~mmi tout)ling efficiency a.(:hieva.ble. We see t h a t |)a,sica, lly it turns out to be a problem of m a t c h i n g of t)ha,se-pla, ne areas. As said ea, rlier, the concepts of e m i t t a n c e and ac(:epta,nce are familiar within the context of cha,rged-particle optics. Indeed, the b e a m ellipse in Fig. 9.4 m a y equally ret)resent the phase-t)laIm ellipse of a charged b e a m travelling along a t r a n s p o r t channel. In this respect, the b e a m ellipse is p a r a m e t e r i z e d in terms of the second-order m o m e n t s of the phase-plane b e a m distribution in full analogy wit, h optical b e a m s (typically this is done in beamlines), or in terms of

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

505

FIGURE 9.5. The optimization of the energy launching of a beam into a receiver requires the optimization of the matching of the phase-plane ellipse of the incoming beam to the phase-plane acceptance domain of the receiving structure (conveniently arranged through an elliptical shape as well). A beam ellipse both smaller and greater than the receiver ellipse produces uneconomical situations, implying in one case unneeded large sizes of the receiver and in the other case losses of the incident beam energy. the Twiss parameters (typically in rings), linked to the second-order m o m e n t s of the beam distribution by a,n emittance depending factor. Accordingly, the phase plane views of Fig. 9.6 may equally depict the transformations effected by definite magnetic components on the travelling p a r t M c beam in terms of the transformations of the corresponding phase-plane ellipse. The free-medium section corresponds in magnetic optics to a field-free section, and a thin lens to a thin quadrupole lens. Also, the Fourier transform-like matrix may accounts for the flight of the beam between the two focal planes of a magnetic quadrupole lens. As in light optics, the beam expander is a, composite system. 9.6.5

The second-moment ellipse and optical transformations

We briefly investigate the effects of the elementary optical systems considered many times through the book on the second-order moments of the Wigner distribution function. The detailed investigation of such effects has been presented in Chapter 8. Here we shortly rephrase t h a t analysis in terms of the second order moments, and so in terms of the transformations of the beam ellipse in the Wigner phase-plane (Fig. 9.6). We will see t h a t each of the optical transformations t h a t have been central to previous analyses, are characterized by the invariance or the skew invariance of one of the second order moments. A diffraction region of length d leaves unchanged E ~ :

E~,c(Zo) =

E~(zi)

free-medium section,

(9.6.27)

and hence leaves unchanged the m a x i m u m excursions ~max along the ~-axis

506

Linear Ray and Wave Optics in Phase Space

FIG URE 9.6. I)tmse-lflmm view of tile trallsfl)rIimtioIls I)y wlficll tile S(;(',)llr162 ellipse is (b) a tt~i~ le,~s, (c) a positive scaler, and (d) a Fo,~rier tra~mfor~ner.

effecte(l ~H~(ler prol)agal, i()~ tl~r(),~gl~ (a) a fre('.-~m(li~ secl, io~,

a,n(t tim intersections q~,,, {)ll the q-axis (Fig. 9.6.a)). Tlmll, tim G~::inwMa,nce tmCOlneS a, distinr r {)f the lflmSe-lflaxie area l,ra.lisf{)rmation a,c:ted by a free-t)rot)agati()li that has t)reviollsly |)(;(;ll i(leld, ili(:(t a,s a, q-shea,r. Correspon(tingly, the I)r()Imgal, i(m ttlrollgh a thin lells leaves lulr the stmtiaJ va,ria,nce of l,lle siglm,1, i.e.

Eqq(Zo) = Eqq(Zi)

thin lens.

(9.6.28)

The m a x i m u m ex(:ursions q...... a,long the q-axis a,n(t the iIlterse(:tions %nt with the t~-axis do not change, ttuls inr of course, a p-shear (Fig. 9.6.b)). As expected, a positive scaler is chara,cterized by the inva,ria,Ime of the mixed moment

P.q~(Zo) = Eq,c,(zi),

positive scaler

(9.6.29)

whilst b o t h qm~x and qint a r e s(:a,led by the magnification fa,ctor m, a,nd, reciproca,lly, ~m~x and /~int are scaled by the inverse 1l r t (Fig. 9.6.c)). Finally, a Fourier transformer basically carries the space and frequency m o m e n t s one into the other, producing, as we know, a scaled ~7t" rotation in the W i g n c r plane. Accordingly, the mixed m o m e n t is changed as (Fig. 9.6.d)). ~-]~q~(Zo) = --~-]~qt~(Zi) ,

Fourier transformer.

(9.6.30)

1DFirst-Order Optical Systems: Moments of the Wigner Distribution Function 9.6.6

507

G e n e r a l i z e d c u r v a t u r e o f a signal. B e a m quality p a r a m e t e r

Following the suggestion put forward in w 9.5.3 when dealing with Gaussian beams, whose parameters have been redefined in terms of the variance matrix entries, we introduce the generalized curvature parameter ~ of an arbitrary non Gaussian beam through an obvious restatement of (9.5.23), i.e. 1

Eq,~ + ie

Q

kEqq

=

Y]qt~

kEqq

+ i~.

e

kEqq

(9.6.31)

Interpreting the real part of ~ as the real curvature -~ of the beam wavefront, and relating the b e a m radius w to the second order m o m e n t }-]qq in the same way as suggested by (9.5.12) for a Gaussian beam [7.4, 7.s], i.e. w 2 --

4Eqq,

(9.6.32)

we obtain for ~ an expression formally similar to (9.5.2) relevant to a Gaussia,n b ea, Iil: 1 1 4e 1 2M 2 --x -R + i kw----~ - -~ + i k w---T . (9.6.33) The factor M 2 introduced above (following a common practice) marks the difference with the beam radius w of a, Gaussia,n beam, signifying, in fact, that w 2 -- M 2 w 2.

(9.6.34)

In tile literature related to laser beam characterizations, it is known a,s tile M 2 factor. It is adopted as a measure of the (inverse) beam quality, with M 2 = 1 corresponding to a perfect Gaussian beam, and M 2 > 1 signifying a "poorer" quality [7.4, 7.6, 7.5]. Evidently, it is nothing but (2re) -1 the area of tile phase-plane bealn ellipse. Remarkably, the generalized beam curvature p a r a m e t e r ~ obeys the same A B C D law obeyed by the relevant p a r a m e t e r ~ of a Gaussian beam, as the reader can guess on account of the presentation given in g 9.5.3. As an alternative to tile M 2 factor, tile b e a m quality is also defined as the diameter-divergence product, i.e. the product of the b e a m diameter at the focus with the far-field divergence angle (recall the space-bandwidth product mentioned in w 4.6.4, and the geometrical extent of a beam considered in w 2.2.4). It is evident that the concept of " b e a m quality" cannot be unique or universal, since the "quality" of a laser b e a m is evaluated in accord with the applications for which the laser is intended. Yet, the adopted definitions of b e a m quality yield useful criteria to characterize a laser beam, being basically a measure of how tightly it can be focused under certain conditions (as, for instance, with a limited b e a m divergence). A great variety of methods and instruments have been envisaged and implemented to yield a quantitative measure of the

Linear Ray and Wave Optics in Phase Space

508

laser b e a m properties[7]. In the next section, we will describe a, simple optical device, first i n t r o d u c e d by Nemes [7.2] which is able to directly yield a, measure of the b e a m d i a m e t e r - d i v e r g e n c e t)roduct.

9.6.7

Phase-space beam analyzer

B(;fl)r('. (h,s(:ril)ing th(' t)has('.-st)a(:(; i)(;a,m a,na,lys('r [7.2, 7.:), 7.~3], w('. giv(; a fi'.w ~()t(,s ()~ tl~(, 4 x 4 ray-~mtri(:('s ()f ()t)ti(:al syst('~s, tl~at (:()~(' i~t() l)lay wl~('~ ~()~ s('t)a,ra|)h' ()t)ti('.a.1 ('()~q)()n(;i~(,s a.r(' t)r('s('nt, as i~ tim ('a.s(' t)(;l()w . W(; list h(;r(; witl~()~t t)r()()f th(; ra,y-trm~sfi;r ~m,tri(:(;s fi)r fie(' t)r()t)a,ga,t,i()~ a.~(l tl~i~ l('~s ()t)('ra,ti()x~ (tt~r()~gl~ (tiff('r('~t g('()~('tri('s) s~ita|)h' t() a, 2 D t)a,raxia,1 ~()(l('l ()f light t)r()t)a,ga,ti()~, in wl~i(:l~ i~(t('('(1 tl~(' ray is I)a,ra,n~('t(;riz('(1 t)y th(; 4-(:()mI)()n('nt v(;('t()r II - -

qy, ]':,', Py)T.

(q:r,,

(().(J.:,}5)

Th(; r(;a.(t('r nmy fin(l a.ll illtllitiv(: jllstifi(:a, ti()ll t() tim fa,('t tllat, fie(; l)r()t)a,ga,ti()n tllI'()llgll a S('(:ti()ll ()f l ( ' l l g t l l d is a(:(:()llllt(',(| })y tlw Ilm,trix 1 0 d 0) T(d)-

()( ) l ()

()I

(),t

() ()

()

1

'

(9.6.36)

wtlilst (:ylill(lri('a,l l(;llS(;S witll tlwir axis r('sI)('(:tiv(;ly al()xlg tll(' q~. ml(l q:/ (tire('ti()ll (wlli(:ll t,tmll at(; a.('tiv(; ill tll(; l)lmms q,-z ml(l q.,;-z) a,r(; (t(;s(:ril)(;(1 t)y('")

L(72:/)

--

o()

ol

ol

() -'P:/

o

0

'

L(/2r)_

() 01 o1 o0 () () 0 l

-1o~

() 1

"

(96.37) 9

A(x:()r(tillgly, a st)lwri(:al lens a,Ill()llnts t() ttl(' ()t)ti(:a,1 llmtrix L(~D)_

1

0

0 () /

o

1

o

o

1

o

0

-1 ~

0

1

-p

o

9

(9.6.38)

Finally, a, qua.(trut)ole le~ls, whi(:h consists of a, imgative and a positive cylinder lens, - f and f , the respective axes being rota.ted one to the other by 90 ~ and by 45 ~ with respect to the q,-axis [7.3], is described by the 4 x 4 m a t r i x Q(~))_

1

0

0 0 -P

1 -p 0

o o/

0 0 1 0 0 1

1

,

7)_

(9.6.39)

f

We refer to Fig. 9.7 for the description of the pha, se-space b e a m analyzer, which as we will see is an effective device "to d e t e r m i n e the one-dimensional c According to our n o t a t i o n , we refer to lens matrices, in relation to b o t h cylindrical a n d spherical lenses, in t e r m s of the focal length f or the focusing power 7) -- 71' and so by L ( f ) and L(7)), the possible directional valence of b o t h p a r a m e t e r s being implied.

1D First-Order Optical Systems: Moments of the Wigner Distribution Function qy,

FIi[

509

qy,,

slit1

slit2

lens f,

quadrupolelensf

Flo l

qx,

i

dl

I=

:1

d

fo

-I-

"U.o -,

FIGURE 9.7. Schematic of the phase space beam analyzer. (Inspired by [7.9]). beam p a r a m e t e r product (beam quality) by a single measurement." [r.9] A slit 1 is located at the input plane, having an aperture in the qy direction sufficiently small to eliminate any wave field structure. It serves to reduce the incoming 2D optical wave to a line-like distribution, which then can be considered as a 1D input beam. A second slit acts as a collimator, selecting only rays with zero inclination in the % direction. In order that the subsequent spherical lens f0 be reached by a, collima,ted beam, the distance between the slit 2 and the spherical lens should be as small as possible (i.e., d ~ 0). Finally a quadrupole lens f is placed in the back focal plane of the spherical lens. The o u t p u t plane is at distance d o from the spherical lens, which in turn is at distance d, from the input pla,ne. Distances ~t~ a.nd d o fulfil the lens law for imaging: 1

1

1

d~ + d o = fo"

(9.6.40)

Therefore, in the absence of the quadrupole lens we would obtain the image of the slit on the observation plane. We present here an account of the phase-space beam analyser by means of geometrical matrix optics, which is enough to gain a first undertanding. A deeper investigation on account as well of diffractive effects is given in [7.9]. The overall matrix of the system is, of course, obtained by multiplying the matrices of the various components, and so Mbeam analyzer

=

T(do - fo)q(f)T(fo)L(fo)T(di),

(9.6.41)

which explicitly reads

Mbe,m o , zer--

A

0

0

A

-1/Yo -1/f o

0

0 B

1/A

0-fo/f

B 0

-fo/f 1/A

'

(9.6.42)

510

Linear Ray and Wave Optics in Phase Space

where A -

fo - d o fo , B

fo(fo

- do) f

.

(9.6.43)

We can n()w insI)e(:t, the effe(:t that the system in Fig. 9.7 t)ro(tu(:e,s on the rays ff()m the int)~t plane. By (:onstr~(:ti(m, t,he rays t)a.ssing t,hr()~gh the slit 1 a,r(', (tes(:ri})(;(t |)y ra.y ve(:t()rs like u.~ -

(q:,:,, (), t,.,.,, o ) r ,

(,9.~.44)

ml(1 p.,. ()v(;r l,l~(: 4- l,l~c ~liw'rge~(:e a,~gl(; i~ l,l~e q.,. (lir(;(:l,i~n~. Tim % a,n(t py (:r ~)f 1,11(; ray v(:('l,{)r al)()ve lla,v(: |)(',e~ s(:l, 1,() z(;r~), as a tirsl, a.t)t)r()xi~na,l,i()n, sil~{:(; q:q, is r ])y l,hc a,1)(;rl,~tre, S~l)t)()se(l s~lm.ll, ()f l,he slit 1, say a,~, i.(',. /kq:,j, ~ 2a~, a.~l(l si~ldlarly Pu, is (lel,(;rl~li~le([ I)y tim a,l)('rl,~m,, , st, ill s~t)I)~s(;~l s~a.ll, ~f 1,1~(:slit 2, say %, m~(t s~ Ap:,** ~,, 2a.~/d~. Li~nil,i~g ~rs~'~lw;s i,~ ('~n~si{ler (ndy l,l~e Sl)al,ial (:{)lllt){)lll;lll,S ~)f the I'a,yv(;(:t,~)rs r(;1)r(:s(nd,i~lg 1,11(; rays r(;a,(:l~i~g 1,t1(; ~)~l,ln~l, l)laal(;, w(; s(;(; l,tlal, q.,. - Aq.,.,, qu,, = I J p . , .

(9.6.45)

T h e n the real a.l)l)eara~me ~)f l,lle lighl, r ~)~1 1,11~;()111,i)111, Sf:l'een is a (lire(:l, (a,tmrl, f i ' ~ l 1,11~;s(:ali~lg fa,(:l,~)rs A a.~l 13) l)llas~:-l)lall(; r(;l)r(;s~;~d,a,l,ion of 1,1~; 1 I ) ~lisl,rit)~l,i~)~ i~ 1,1~; i~l)~l, I)laa~;. ]';vi~t~;~l,ly, ~)~: a,ls~)~t)l,a.i~s l,l~'~ |)ea,~ (lia,nlel, er-r162 I)r~)(l~u'l,. 9.6.8

R,a:q, wave and Wigner optics. II

T h e t)resent, a,t,ir given ill (',lla.t)ter 8 tlas reveale~(l l,lla.l, l,ll~; Wigll(:r r fllnction is wa,w;-~)t)t,i(:al ill its st, rll(:tllre, I'esortillg ill its ~telilfil,i(~ll I,() l,lle wave{)t)tical (les{:rit)t,{ws {~f ligllt wave fi{',l(ls. Ii1 c{nltra,st, tll~; {lylm,llli(:s ~f tll(; Wigner (tistributi(m fllnction 1111r tm)pa,ga.tion ()f the, sig~la,1 tllr(nlgll first-(wr opti(:al systems is (tet, ermine(l ]~y the tra,nsformation rifles ~f the ray-variables in gc()metrica,1 n m t r i x {~t)ti{:s. Tlnls, in fllll mml()gy witll tim {:la,ssi(:a.1 ray-density, we can run the "indivi~tlm,1 I)lm,se-pla,ne ra:y-tra,je(:tories" to follow tile evolution of the W i g n e r distri|)~tion fllnction under protmgation t h r o u g h A B C D optical systems. In fact, we have also seen in w 8.5.3 that, a,s far a,s the linear approxiilnm, tion (i.e. silm,ll i~mlination angles to a,n(t s~Im,ll distances oF/' the axis) is concerne(t, the tra,xist)~)rt equation for the W i g n e r ~tistribution function turns into the cla,ssical Lio~ville's equation for the ray-{tensity distribution. Here we have seen that the Wigner distribution flmction of a Gaussia,n b e a m is completely cha, racterized t)y its second-order m o m e n t s t h r o u g h a, quadra,tic form. It identifies in the W i g n e r plane an elliptical shape, whose tr~msformations mirror those of the signal under p r o p a g a t i o n t h r o u g h the system. Rema,rkably, the paraxial propaga,tion laws for the second-order moments involve

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

511

FIGURE 9.8. The perfect beauty of Borromini's dome of Sant'Ivo alla Sapienza in Rome superbly conveys the aestethical and emotional valence of a spiral. (Picture taken by the author.) the ray transfer matrix of geonletrica,1 optics in a form conforming to Liouville's invariance of phase-plane areas, whose formulation in terms of second-order moments yields indeed the optical uncerta, inty relation. Indeed, this is not surprising since Gaussia,n beams, evolving from the rainimum uncertainty amplitude, i.e. the real Gaussian signal, may be regarded as "ray-like beams". So it might be expected that the Wigner representation of a Gaussian beam operates a synthesis between ray and way optics, recovering the simple transformation rules and phase-plane depictions of the former but enriched by the intrinsic wave-optical nature of light distriblltion conveyed by the latter. It is noticeable that the same holds true for arbitrary non Gaussian beams, by virtue of Liouville's theorem, holding true even for Wigner optics as far as the linear approximation is concerned (w 8.5.3). The Wigner representation amalgamates ray and wave optics, embedding the wave-optical features of light propagation into the ray-optics formalism. Resorting to the beginning of our presentation started in Chapter 1 with the illustration of the geometrical matrix tools, we may say that, in a sense, we have moved along a spiral. So let us conclude our "path" with the view given in Fig. 9.8 of a charming architectural example of a spiral.

512

9.7

Linear Ray and Wave Optics in Phase Space

Higher-order moments of the Wigner distribution function

For (:()mt)leteness'sake, we tra,ce the, line of the investiga,ti()n ()f higher-order m o m e n t s ()f the, W i g n e r (tistrit)uti(m flm(:ti()n a.ii(t their t)r()t)agati()n la,ws, f()llowing [4.s]. As sllggc, st(',(t t)y ttle (tire,(:t t)r()(t~u:t s(:lm,~m, (9.6.4) yiel(ting tim se(:on(t~n()~l(',~d, ~la,trix E , we ~ a y (',nil)e,(1 tim tllir(l-()r(ter ~l()nl(;~lts i~lt,() a ~mtrix as weJ1 1)y fl~l'tll(',r a,t)t)lyiilg tim Kr()~e,(:k(',r l)r()(l~(:t ()t)era.ti()~l t() tlm ray vc,(:t()rs. Tlu~s, w(; ()l)taiu t,ll(', l,llir(t-~l~()n~eld, ~lm,trix E:~ 1)y I,lm t,l11"(;(',-v(~,(:t()r l)r()(l~(:t s

-- (u r u T O0 u>,

(9.7.1)

wher(; ill(; slm.rt) 1)ra.(:kets lll(;~l,ll ( ) f (:()llrS(', a.ll a,V(;l'~l,g(; l)r()(:(;(lltre, ()ver the, t)ha.sc t)lml(; t,llr()llgll tlm Wiglle, r (tistrit)llti()n flnl(:ti()ll a,s ill (9.6.6), i.e.

(...) - ].'/...W(u;

z)du.

(9.7.2)

Exl)li('itly, 1,11(: tllir(l l)r()(lll('t al)()v(, yi(;l(ls 1,11(; 4 x 2 lllatrix !1 (;':) HT

@:) U

q2h-, qh. 2

(!.2~. ,!.~

-

(.().7.3)

,

q~2 K3

(:()llta, illillg tim, I)()w(;rs ()f q a,ll(l h;, qa, q2h; ' qH2 ml(l h;:~, t)(',rta,inillg t() tim f()rIlm,ti()n ()f I,tl(; (tesir(;(l tlfir(l-()r(ter I~()ll~e,nts. Tim, I)r()t)a,gati(nl la,w f()l" tile 11m,trix Ea (',ml I)(; ()l)ta,ilm,(l ill tim sanle wa,y as t)(;fi)rc, hi fa(:t, (:llmlgillg tim ild,(;grat, i()11 wu'ia,I)h; f I ' ( ) l l l U t ( ) V - - M - i u in the, integral (9.7.2), i~q)lie(1 1)y (9.7.1), we s(,,e tlmt the thr(',e-v(;('t()r t)r()(tu(:t U (;x) U T ~ U t, l l t ' I l S int() U ( ~ U T (>q U

v=M

lu

,

Mv ~ vTM T ~ Mv.

(9.7.4)

I{e,t)(',a,te,(t a.t)l)li(:ati()lls ()f the (lire(:t I)ro(lu(:t t)r()t)erty (see f()()tll()te, ()Il t). 500)

(E r F ) ( H r G) - E H r FG,

(9.7.5)

M v ~ v T M T ~ M v - ( M | M ) v ~ v n- | v M T.

(9.7.6)

leads to the nice form

It is therefl)re evi(tc, llt tha,t the prot)a,ga, tion la,w for E 3 writes as E3(Zo ) - ( M @ M ) E a ( z i ) M

T.

(9.7.7)

Tile t)rocedure can be iterated, thus yielding for the n - t h order m o i n e n t m a t r i x E~ the average of the n-vector direct p r o d u c t

E. --(u|

T|174 n - t l rues

T|

(97.8)

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

513

with u and u T being alternatingly involved into the sequence of products. It is straightforwardly, although a little lengthy, using the properties of the Kronecker product, to verify that the matrix E n propagates according to En(Zo ) --(M|174

(9.7.9)

Here, [~s yields tile closest integer to ~, and (M|

=M | M |174

M

(9.7.10)

h-times

denotes the h-th direct product of the ray matrix M. Needless to say, with n - 2 we regain the propagation law (9.6.8) for the variance matrix E, which indeed within the formalism of this section could be identified as E - E 2. Let us separately consider the cases of n even and odd. For n even" n - 2j, relation (9.7.9) yields the symmetric equation

E2j(Zo) --(M@j)E2j(zi)(MT@j).

(9.7.11)

Note that E2j is a 2j • 2 j matrix. It (:an also bc seen as arising from ~2j --(]E2|

(9.7.12)

Multiplying both sides by the 2J x 2J matrix (J| using the relation M n- = J M - 1 J -1 and repeatedly applying (9.7.5), we obtain on the right-hand side

[(M~2(zi)JM-lj-1)@J](J@J) - [ ( M E 2 ( z i ) J M - 1 j - 1 j ) @ J ] [(ME~(zi)JM-1)| - (M|174174

(9.7.13)

Finally, on due account of the further property of the direct product

(E @ F) -1 - E -1 | F -1,

(9.7.14)

( M - I @ j ) - (M@j) -1,

(9.7.15)

by which,

the propagation law (9.7.11) is seen to be equivalent to the similarity transformation

E~j(Zo)(J|

- (M|174174

-~,

(9.7.16)

of the 2J x 2J symmetric matrix E2j (zi)(J| by the 2J x 2J matrix ( M | ). A similar result, even though in a less direct form, can be worked out for the case of n odd. Indeed, with n - 2j + 1 in (9.7.9) we obtain

~--~2j+l( Z O )

--

for the 2 j+l • 2j matrix E~j+ 1.

(M@j-t-1)Y]2j+I(Zi)(gT @j),

(9.7.17)

Linear Ray and Wave Optics in Phase Space

514

Multiply b o t h sides by ( J |

~-]~2j+~(Zo)(J(/~)j)

--

a,nd r e p e a t e d l y use (9.7.5), to fina,lly write

(M@j+I)~2j+I ( z i ) ( J @ j ) ( M - l @ j ) ,

(9.7.18)

t)()th si(tes mea,ning a 2 j+l • 2 j matrix. C()rrest)on(tingly, taking the tra,nspose of (9.7.17) a,n(t multit)lyi**g t)y ( J | we hav(;

Er.,j,l (z,,) (Jc4j+ l ) - (Mc~j)E,T +, (zi)(Jc~j+,)(M -1 c~j+l ),

(9.7.19)

t)()tl, si(l(;s i**q)lyi**g ()f (:(n,rs(; a, 2 j x 2j+ l**m,trix. Finally, 1,1,(; l)r()(l~u:l, ()f (9.7.19) l,i***(;s (9.7.18) r(;s~,lts i**l,()

NT2 j

! 1

(z,,)(Jc4j+ l)N

2j { 1

(z,,)(Jc4j)

= (Mc+j)[IE~,, (zi)(J~j+l)~.~j+, (zi)(a(/oj)l(McAj) -1 ,

(9.7.20)

wl,i('l, (:a.,~ I)(: r(:('()g,~iz(:(l as tl,(; si**filarity tra.**sf()nlm.ti(),~ ()f ttm 2 j+l • 2 j+l **m.trix ]E 2.i1 q- 1 (z,,)(Jc4j+~)lE.~j - + 1 (z,,)(Jc4j) l)y t,l,(; (tire('t l)r()(l~u:t l,m.trix (Me+j) As (;arli('.r n()l,e(l, si**filarity tra,**sf()r,lm,l,i()**s are very ('()nV(:ld(;nt 1,() w()rk wil,l,, si**(:(', l,l,('.y l)r('.s('rv(: 1)()I,1~ (:ig(:~wal~u:s m~(l tra,('(:s. 'l'l,(:r(;f()r(;, 1,1,(, i(l(;**tifi(:a.ti()n ()f si~ila,rity trm~sfi)n~m,ti()l~s ill r(;lati()~ witl~ s(;(:(n~(l-()r(l(;r as well as l~igl~(;r-()r(h'.r ~()~(:~l,s ()f ligl~l, (lisl,rit)l~t,i()~s t)r()vi(l(; m~ (;tli'.(:tiv('~ t()()l t() in(tivi(t~la,liz(; i~wa.rim~t (l~mlity l)a.ra,~(:t('rs i~ t(;r~ ()f wl~i(:l~ ()t)ti(:al sig~a,ls (:a,~ s~fita,l)ly t)(; (:lmra.('t(;riz(;(l a.l~(l a.(:(:()r(ti~gly ()l)ti(:al sysl,(;~s ('.tfi;('tiv('ly (l(;sign(;(t.

9.8

Summary

After illtr()(llu'illg t,ll(' l)asi(: II()ti()llS ()f lll()lll(;Ilts ()f 1D ml(1 2D flnl(:ti()ns, w(; have fo(:lls(;(t (nlr a,l,teIlti()ll ()ll the first- a n(t s('r lll()21u'nts of the Wigner (tistrit)llti(nl fllll(:ti()21, fi)r wlfi(:tl a variety ()f siNlitir relati(ms have been esta,blishe(1 ill gmlera,1 aald then specia, lize(t to tlu; case of tim, Wigner distribution fllncti()Ii ()f a Gm~ssian signal, which is il~(tee(t 2D Gaussian. 2D Gaussia,n (tistributio~ flumtion have relevance a,s well within the context of pa, rtial coherence tt~eory. In particular, a, suita, bh; m a t r i x scheme has |)een envisaged ena,bling a compact a,n(t elegant f()rmula,tion of the t)r()paga,tioI, laws for the second-order m o m e n t s ~,n(ter t)rot)agati()n ()f the signal t h r o u g h A B C D ()t)tical systeins. A similarity tra,nsformati()n ha,s been individualize(t, whi(:h involves the secondm o m e n t ma,trix a,nd the ray transfer ma,trix of geometri(:a,1 optics. From it a va,riety ()f interesting a.n(t 1,s(;fl,1 relations can t)e (t(',(t~,('(;(t. Of p a r t i c u l a r releva,n(:e is the inva,riance of the determina,nt of the variance matrix, interpretable as the area of the s e c o n d - m o m e n t phase-plane ellipse. This has suggested the identification of suitable p a r a m e t e r s for a meaningful definition (and operative p r o c e d u r e for the measure) of the " b e a m quality", as well as possible generalizations to partially coherent beams.

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

515

Problems

1. (a) Prove the general relation rn,.~ -- 27ri- ~+s 0 ~+ "~-A(q',~) 9

0~0~

~

q, v ; = 0

'

holding between the central moments of the Wigner distribution function and the derivatives of the ambiguity function at the origin. It is implied above t h a t the ambiguity plane origin has been translated at the mean values point of the Wigner distribution function. (b) Prove t h a t the ambiguity function can be constructed by knowledge of the Wigner distribution function moments as i l-m, A(~, ~) - ~ ~ m ....., q ~ , t,rn=0

thus also implying the reconstruction of the Wigner distribution function by knowledge (possibly gained by experimental observations) of its moments. n 2. Verify that, the application of the energy operator g to a signal of the form (9.4.5), under the assumption t h a t b o t h the a m p l i t u d e f(q) and the local frequency t~1 (q) - d_~ dq do not vary too rapidly, results into ~f~l, i.e. the product of the squared amplitude and the squared (local) frequency of the signal. 3. Prove t h a t for the Gaussian b e a m (9.5.1), the local uncertainty product, defined in (9.4.91), takcs the explicit z-depending expression 2 ZR

q~(~)~

(q) -

4 ( z~ + z ~ )

C o m m e n t oil it in the light of the considerations presented in w (9.4.4). 4. Prove that in order that, under propagation through an A B C D system, the secondorder moments of the light distribution do not change with respect to the respective input values, it must be

B ~qq ( Zi ) --

]%

~

E~,,~-z, )

Eq,~(z,)-sin/~'

sin/3'

e sin/~

A-D ~

'

where cos/3 = A+D It is implied, of course, that A + D < 2 (stability condition) 2 " 5. Consider a light b e a m propagating t h r o u g h a quadratic index mediuln (see w 3.9). (a) Prove that the b e a m radius, defined by (9.6.32) evolves according to the Pinney equation d2w

dz 2

16e 2 n2 _2~2,.,3 + w - 0. t ~0 r~o vv

(Hint: use Eq. (9.6.8) for 2E and Eq. (1.7.18) for M to deduce the paraxial equations of motion for the second-order moments in a general form. Specialize the result to the case of propagation in a quadratic index medium.) (b) In particular, verify t h a t with n 2 = 0 (free propagation) an evolution law for w(z) is obtainable in the similar form as for Gaussian beams, i.e. Z2

w2(z) --Wo2(1 -+- -22-)' ZR

demanding for Z R --

in accord with relation (9.6.34).

ZR M2

'

516

Linear Ray and Wave Optics in Phase 5pace

References [1] Only a h~w titles on the nlonlent m c t hods in relation wittl inmge characterization, evahmti()II an(1 nlallitnllal, i()II, (:OmpI'isillg tlm t)ioneeriIlg t)at)er 1)y tlll and the subsequent investigatioils by Teaguc, are listed here. The. reader m a y find an cxhallstive t)resentati(m in the text,1)t)t)k l)y Mukml(lan an(l liamakrishlmn. [1.1] M.K. Hll, "Vislml pattenl recogidtioll |W inolneilt iilvarimlts", IIi E 'IYmls. I~ffo. Theory, IT-8, 179-187 (1962). [1.2] M.R. Teague, "Optical ('al('lllatioll of irrar InoInmlts", AppI. OI)t. 19, 1353-1356

(l:)s0). [1.3] M.IL 'IL'aglm, "linage mlalysis via tim gmleral tlmory of IllOIlmllts", .1. Opt. Soc. Am. 70, 92()-93() (198()). [1.4] l{. rVl,lk,nl{lml ml{l K.I{. llmlmkrislllmll, Moment ],;wn,ctions in Image Analy.~is: Theory and Appli~ntions (W()rl(l So., Siilgap()rc, 1.().()8). [2] Tlu: l)r(:s(;~m~ti()~ i~ w 9.2 is ])(,a~ltif~lly (:()vet(;(1 i~ tim (l~l()t(,(I l)ag(;s ()f [2.1]. S()lll(; r(;l)r('sc~m~tiv(', titles ('()n(:cr~m(l witl~ tim ttm()ry ()f g('~mral slalisti('s ar(; s~gg(;st, e(t.

[2.1] I,. Mm~(lel m~(l E. Wolf, Optical (7ohcrcncc

and Quantum Optics ((:mn|)ri(lge

UIliv.,

N(',w Y()rk, l .().()5), oh. l, pp. 1-40.

[2.2] II. (;rm~6r, Matlwmatical Mcthod,s ~g Statistics (1)ri~a('(;l,()~ I,J~fiv. l)rcss, l)ri~meto~,

1:)4(~). [2.3] M. Kc~(tall ai~(l A. St~mrt, Tk,c Advancc, d Thco'rv of Stati.~tics (Griffins, l~()~(l()~, 1958). [2.4] B.IL Martin, Stati.~t,ic.~ for l)hy.sicist.s (Aca(lc~nic Press, I~()~(l()~, l.()Tl). [3] (~()Im(;I)tS m~(t (lefi~iti()~s (:()~(:(;r~(;(l witt~ tl~e i~()it~(,,~l,s (l()('al m~(l gl()t)al)()f a ligl~(, (lisl, ril)uti()~ ti~ul (,lmir (:()~(,(;rl)ar(, il~ tim (',()ll(,Cxl, ()f sig~al th(;()ry withil~ the t)cr(i~mld, tiI~C-fl'(~,(l~U;~(:y (l()~mi~. S()~(; I)('~rtil~(;nt (i(h;s arc sugg(:s(e(l I)clow. In t)ar(i(:ular wc signaliz(; th(; l)ar( I ()f (h(; thr(;(;-t)ar( Imt)(;r l)y ('laascn a~(1 3l(;(:klcnl)riiukcr [3.1], wlmr(; tim ~()~(;~(,s of (,t~c Wigner (lisl, ril)~ti()~ f~('ti()~ m~(1 t)crtin(',xd, in(;(tualities at(; (:()l~si(l(',re(1. W(;ll-k~()wn tcxt|)()()ks (:()~m(;i'~m(l witl~ l,l~(', signal th(;ory and the r(;levaI~t tin~(;-fre(tucn(:y rct)r(;seI~tati()~ issue(; at(; als() listed. [3.1] T.C.A.M. Claasci~ ai~(t W.F.G. Mccklenbr/4uker, "The Wigner (listribution- a tool for time-frequcncy signal m~alysis. I: continuous-tiIne sig~mls", Philit)s ,]. Res. 35, 217-250 (1980). [3.2] S.C. Pohlig, "Signal duration and the Fourier transforin", Proc. IEEE 68, 629 (1980). [3.3] L. Cohen, "InstaI~taImO~m mean qum~titics in tiine-freq~mncy analysis", Proc. IEEE ICASSP-88, 2188-2191 (1988). [3.4] L. Cohen and C. Lee, "InstaI~taimous frequency, its standard deviation and multicomponent signals", in Advanced Algorithms and Architectures for Signal Processing. III SPIE Vol. 975, 191-208 (1988). [3.6] P.,]. Loughlin and B. Tacer, "Instantaneous frequency and the conditional mean frequeimy of a signal", Signal Process. 60, 153-162 (1997).

1D First-Order Optical Systems: Moments of the Wigner Distribution Function

517

[3.7] B. Boashash, "Estimating and interpreting the instantaneous frequency of a signal. I: Pundamentals", Proc. IEEE 80, 520-538 (1992); "II: Algorithms and applications", ibid. 540-568. [3.8] L.E. Franks, Signal Theory (Prentice Hall, London, 1971). [3.9] A. Papoulis, Signal Analysis (Mc-Graw Hill, New York, 1977). [3.10] L. Cohen, Time-Frequency Analysis (Prentice-Hall, Englewood Cliffs, 1995). [4] The papers by Bastiaans [4.1, 4.2], concerned with the definition of the secondorder moments of the Wigner distribution function, relative propagation laws and related invariants, have been basic to subsequent investigations and generalizations [4.4, 4.5], also in relation with partia,lly coherent beams [4.3]. Tile paper by Sheppard and Larkin [4.6] presents a variety of interesting relations involving the secondmoments, some of which reproduced in w 9.4.3. The fascinating review by Simon and Mukunda [4.8] elucidates the relation between the ray, wave and Wigner optics formalisms, in relation as well to the partial coherence theory. The link of the second-moments of the Wigner distribution function to the second derivatives of the ambiguity function has been evidenced by Papoulis [4.9]. [4.1] M.J. Bastiaans, "Propagation laws for the second-order moments of the Wigner distribution fimction in first-order optical systems", Optik 82, 173-181 (1989). [4.2] M.J. Bastiaans, "Second-order moments of the Wigner distribution function in firstorder optical systems", Optik 88, 163-168 (1991). [4.3] M.J. Bastiaans, "ABCD law for partially coherent Gaussian light, propagating through first-order optical systems", Opt. Quantum Electron. 24, $1011-$1019 (1992). [4.4] D. Onciul, "Invariance properties of general astigmatic beams through first-order optical systems", J. Opt. Soc. Am. A 10, 295-299 (1993). [4.5] D. Dragoman, "Higher-order moments of the Wigner distribution fimction in firstorder optical systems", J. Opt. Soc. Am. A 11, 2643-2646 (1994). [4.6] C.J.R. Sheppard and K. G. Larkin, "Focal shift, optical transfer function, and phase space representation", J. Opt. Soc. Am. A 17, 772-779 (2000). [4.7] V.V. Dodonov and O.V. Man'ko, "Universal invariants of quantum-mechanical and optical systems", J. Opt. Soc. Am. A 17, 2403-2410 (2000). [4.8] R. Simon and N. Mukunda, "Optical phase space, Wigner representation, and invariant quality parameters", J. Opt. Soc. Am. A 17, 2440-2463 (2000). [4.9] A. Papoulis, "Ambiguity flmction in Fourier optics", J. Opt. Soc. Am. 64, 779-788, (1974). [5] Tile non linear energy operator g was originally developed by Teager and then systematically used by Kaiser. In the context of signal analysis, it has revealed an efficient tool for extracting amplitude and frequency modulation information in signals of the form s(t) = a(t)cos(r As representative papers, we suggest [5.1] H.M. Teager, "Some observations on oral air flow during phonation", IEEE Trans. Acoust., Speech, Signal Process., ASSP-28,599-601 (1980). [5.2] H.M. Teager and S.M. Teager, "Evidence for nonlinear sound production mechanisms in the vocal tract", in Speech Production and Speech Modelling, J. Hardcastle and A. Marchal (eds.) (Kluwer, Boston, 1990), pp. 241-261. [5.3] J.F. Kaiser, "Some useful properties of Teager's energy operator", IEEE ICASSP-93, Vol. 3, 149-152 (1993).

Linear Ray and Wave Optics in Phase Space

518

[5.4] P. Maragos, J.F. Kaiser and T. F. Quatieri, "On amplitude and frequency demodulation using energy operators", IEEE Trails. Signal Process. 41, 1532-1550 (1993). [6] T h e richness of inequalities (:on(:erne, d with the nlonmnts of the W i g n e r distributi(nl flUl(',ti(nl is well c(nlve, ye(t |)y (I(; Bi'lfi.j~l's t)apers [6.1, 6.2], the se('()n(1 offering one ()f tim first, malh(;mat, i(:al f()rmalizalion ()f the W i g n e r ret)resentation. T h e t)res(;Idati()ll ixl !i 9.,1.,1 is illst)ire(t ])y I.lm l)aI)er ])y Ix)llglllill a Il(l (',()lmll [6.3]. [6.1] N.G. (le Brlfi.jli, "UI|(:erl,~dIlty t)rilwil)le ill Follrier mmlysis", ill ['ncq'ualilic,s, O. Shisha (e(l.) (A(:~(h;lIfi(" l)ress, New York, 1.()(i7), pp. 57-71. [6.2] N.G. (te Bruijll, "A tlmory of gelmralize(l flulcti(nlS, wittl applicatiolls to Wigner (listribld, ion an(l Weyl c()rresl)ollth;llCe," Niellw Arclfief vt)or Wisklultle 21(3), 2()52s0 (1:}73). [6.3] I)..I. I~ollg|llill ml, l I~. (;()lleIl, "Tim Iillcer|,a.illty prillciple: glol)al, local, or bottf, z'', IEEE 'IYmls. SigI~al lh'(wess. 52, 1218-1227 (2()()4).

[7] A r('a(li~g list (,~ (;a~ssia~ I)(:a~s is i~ [131 ()f (:1~. 2. Iler,, we lis~ a f('w t i t l e s reh'va~! t ( ) t i m I)rese~tati()~ i~ !i!i 9.6.6-9.6.7. a~(l 1~)(l,'ti~ili, n~s. al)l)licali(n~s a~r ~ e a s ~ r e ~ m ~ t s ()f relew~('e t,) tim i~(livi(l~mlizati()~ (~f s~ital)le i)ara~mters f()r laser 1)ea~s/(levi(:(~s (:lmra<'lerizati(n~ a~
[7.6] NI.W. Sa~sIlel, t,, "l)rol)agatic)~ of ~m~lti~no(l('~ b~sc'r |)('runs. Tim M 2 fi~ctor", ill fltc l'h,y.,4c.s and 7Y.ctm,ol~oy of La.~cr Re.~o'n,ator.s, I).R. llall m~(l i~.E..]a('ks(n~ (e(ls.) (A(lmn Hilger, Bristol, 1989). [7.7] P.-A. BSlanger, "Bem~ propagati()~l m~(I l,l~e ABCD ray ~natrices", ()pt. Lett. 16, 1 :)(~-~:)s (1991). [7.8] A. E. Sieginm~, "l)efi~fi~g tim effectiw; ra
A Lie Algebras and Lie Groups" Basic Notions

A Lie algebra A is a linear space over some field F, say, for instance, the field of real numbers R or of complex numbers C, together with a binary operation [.,-], which maps pair of elelnents in A into all elelnent in A:

[., .]: (h,.9) c A • A

, [h,.q] E .4.

(A.1)

It is called the Lie bracket and satisfies the following properties: 9 bilinearity:

[~h + Vk, ~] = ~[h, ~] + V[k, ~],

[h, ~g + vk] = ~[h, g] + bib, k],

(A.2)

for all a, b C F and h, k,g E A; 9 anticommutativity:

[h,.q] = -[.q, hi,

(A.3)

[h, [k, g]] + [k, [9, h]l + [~, [h, k] = o.

(A.4)

for all h, g r A; 9 Jacobi identity:

for all h, k, g E A. It is evident that in general the Lie bracket is not associative, i.e.

[h, [k, g]] r [[h, k], g],

(A.5)

the associative law being replaced by the Jacobi identity. The dimension of A is, of course, determined by the cardinality of the basis set {hj }j, which spans the underlying linear space. As a concrete example of a Lie algebra, we may mention that the Euclidean space R 3 becomes a three-dimensional Lie algebra with the Lie bracket taken as the cross product of vectors. Also, any associative algebra, with associative product f g (here simply denoted by justaposition), becomes a Lie algebra under the definition of the Lie bracket as the commutator, also called Lie product, of elements in the algebra: {h, g] = h~ - gh.

(A.6)

Linear Ray and Wave Optics in Phase 5pace

520

In a sense the c o m n m t a t o r is a measure of how n o n - c o m m u t a t i v e the algebra is. The algebra of invertible real square matrices is m a d e into a Lie algebra under the. Lie t)rodu('t (A.6), involving the familiar row-by-col~mnl multiplication of ma.triccs. Besi(tes, in the text it is seen that the algebra of the 2D illhomogelu;O~lS linear t)olynonfials (witll rest)cot to a,(tr a.cquircs a Lie a.lget)ra s t n u : t u r c lul(ter tilt; Poissoll bra,(:ket ot)eratio~l, as sinlilarly (toes the algel)ra ~f 21) l~on~ogelm~n~s (t~a,r l)~)lyll~)~fials. A h,omomo't7~h, ism a 1)(:t,w(:(:ll l,w() l.ie a,lg;el~ra,s A m u l / 3 ()v(:r t,l~(: sanle ti(:l(t /~': a:A

, B

(A.7)

(letixu~s a, li~('a,r ~lm,l)t>i~g ~f A i~t() B, s~u:t~ tlm,t tl~e I.ie Iwa,t'ket i~ B ~f illm,ge(t elellltUltS ()f ~4 is ta,ktu~ I,() tlw illm,g('. ()f tile l.ie lwa~:ket ill A ()f 1,11e ()rigina,1 (~lelliell|,S; l~m~l('.ly:

[~(h), ~(g)l = ~([h, g])

v h, g e ,4.

(A.S)

If slu:ll a ll()l,,()l,l(~I't)llisll, is ()ll('~-t()-()ll(; (()r, .fitit/r s() tlmt tll(' ilw(;rs(; l~m.t)t)illg exist,s, it, is <:alh;(l a,n isomo't'phism. Ft)r all t)ra,t:tir In~rl)t)ses, is(ml()l'lflfi(: l,i(; algt;twas il,l'(~ i(l(;lltir A s'u,ba,ltl~'bra ,5 i~ .,4 is a, li~ear s~fl)Sl)a,(:e ()f A, wl~i(J~ is (:h)s(:(l ~ ( h ; r Lie lwa~:k(;t ()f ('~l(;~(;~l,s ()f ,5 witl~ (;l(;l~(;~t,s ()f $:

IS, Sl c s.

(A.0)

All ideal ill A is a lill(;a.r slfl)sl)ar J ()f A, wtfit:h is (:l()s(;(l llll(l(;r Lie |)ra(:ket ~f ~;h;lllellts ~)f J witll elellmIlt.s ~)f A:

[j, .4] c J .

(A.IO)

The dcT"ived series is reI)resente(t t)y the se(tuen(:e of sulm.lgebras ill A rec11rsively ot)ta,ilm(t Ulltler Lie 1)ra,(:ket a,t tim same level ()f (:ollnlnltal,i(m: A D [A,A] D [[A, A], [A, A]I D IlIA, A], [A, A]], [[A, A], [A, A]]] D ....

(A.11)

Corr('~st)(nl(tingly, th(; lowe'l" cent'ml series is intended as the sequence of sllba.lgcl)ras in A obtained under rct)cate(t (right) Lie bracket with A itself:

A ~ [A, A] D [A, [A, A]] ~ [A, [A,, [A, A]]] D ....

(A.12)

A Lie algebra A is said abclian if [A, A] = 0; is said solvable if at a certain level of c o n u n u t a t i o n tile derived series terminates; and is said nilpotent if the lower central series b e c o m e s zero eventually. The Heisenberg-Weyl algebra {~, ~, I} yields a significative example of a nilpotent algebra. A

Appendix

521

Also, a Lie algebra A is simple if it contains no nonzero ideals, whilst it is semi-simple if it contains no nonzero abelian ideals. A simple Lie algebra is semi-simple. It turns out that semi-simple Lie algebras are the direct sum of simple (non abelian) Lie algebras. A Lie group ~ combines the continuous structure of a differentiable manifold with the basic properties of groups. A group is a set of eleinents in which it is defined a binary operation (the group multiplication), which maps any two elements in G into an clement in G as well, and satisfies the properties: 9 associativity: H.(G.K)=(H.G).K (A. 13) for all H, G, K E {7; 9 existence of identity: 3 IEG

9 I .H-H

(A.14)

for a l l H EG;

9existence of unique inverse: 3 H -1 E ~} 9 H - 1 - H -

H - H -1 - I.

(A.15)

fl)r every H C G. A familiar example is the group 7),. of all p e r m u t a t i o n s of n objects, which consists of n! operations. Also, the collection of rotations of the circle in a plane through multiples of a given angle tg, say t9 - 2~ with respect to a fixed n ~ point, constitutes a group Inade of n distinct operations. In order that G be a Lie group, its group operation must be consistent with the continuous structure of the underlying manifold, and so it must diffcrcntiable as well. This na'/vely means t h a t if H and G in G multiply to W - H - G in ~, then the group operation takes any elements J and K , respectively drawn from a neighborhood of H and a neighborhood of G, to the group element Y in a neighborhood of W. As a simple example of Lie groups, we may mention that the set of real numbers, excluding 07 forms a Lie group with respect to the familiar multiplication of numbers. The set of translations of a straight line with respect to a reference point also constitutes a Lie group, each element of which being individualizable (i.e. parameterizable) by a real number (which specifies the shift from the identity). Also, the ensemble of rotations of the circle with respect to a fixed point by an angle ~9, allowed to range continuously through the interval 0 <_ g _< 27r, generates a Lie group, whose elements can be parameterized just by the continuous angle g. It is seen in the text t h a t the group of the unimodular 2 x 2 real matrices (under the ordinary multiplication of matrices) forms a Lie group, and each matrix in it is parameterized by three real numbers, which

522

Linear Ray and Wave Optics in Phase Space

can be (tifferently interprete(1 a(:cording to the a d o p t e d (tec()mt)osition. Ano t h e r e x a m p l e is the g r o u p GL(~t) of n x n invertible m a t r i c e s , or tl~e relevant sut)gro~H) O(n) of o r t h o g ( m a l matrices. 1R.()~ghly st)caking, Lie a.lget)ras ext)(mentiate to Lie gr(nq)s. M()re t)re(:isely, the t a n g e ~ t st)a<'e at the i(hmtity of a Lie gr(n~t) is always a Lie alget)ra. This Lie algel)ra (h;tei'nfines the h)(:al str~(:t~r(; ()f the Lie gro~lt) thr(n~gh the ext)()nential nm.I). (~,()~si(t('.ri~g, f()r ill~strative tn~rI)()s(;s, a. ()n(;-t)aI'a.~u;ter I,i(; gr(nq) {7, S)~l )l )( )s(: tlm.l, (;(t) r(;l)r(:selll,s a sni()()l,h (:lit'v(: in ~; l)assil~g (,l~r()~gh (,h(: g r o u p i(le~Itity, (:.g. I -: (;(()). F()r vahu:s bt ()f (,l~(: r(:al l)ara~(:(,(:r t il~li~dl,(:simally (:h)s(; t() (), (,lle (:()rr(:sl)()~uli~g (:l(:~n(;~t (;(bt) iI~ ~ (:()~(:s t() l)(: i~fi~fi(,(:simally (:]()se (,() tl~(: i(](:~tity I = (1(()), i.(:. (A.16) tllllS (;St,alflisllillg a ~llal)l)illg ()f (;l(;111(;lll,S (;'(()) ill tll(' l,i(', alg('.l)ra t() tt~(; i~ti~fitesi~m.l ge~mrat()rs G((St) ()f (,l~(; Lie gr()~q). It ix (;vi(h;~t tlm.t l,l~(; sl, r~u:t,~r(; ()f tt~(', a lg(;1)ra ix r(;fi(;('l.(:(! l()('ally i~(,() tlm.(, ()f l,l~(; gr()~ l) i~ a ~(;iglfl)()rh()()(t ()f tl~(; i(le~tity. S~q)t)()s(; ~l()w (,() l,ak(; a. fi~ii,e ~nn~d)(;r t ~ IR. ()f ('.()~n's('., ()~(; ('a~ g() i~finitesi~m,lly (:l()s(; 1,()(), 1)y l)r()l)(,rly fra('l,i()~i~g t,; s(;(, r()~gl~ly, bl - •I I I . a~(l let 1,1~(; in(,(:g(:r ')))g() (,() infilfi(,y: '))) --+ c~. Sill(:(:, 1)y vir(,~i(: ()f (,lle ('l()s~n'(: l)r()l)erty ()f tl~(: gr()~q), f()r a~y i~(,(:g(:r '))),, [(7'(bt)]'" l)(:l()~gs (,o (,l~(: gr()~l), w(: fi~m.lly lm.ve t

[(;(~t)]'" ,.~ [I + - - _ (7,' (())] '" ~ (' to,' (()) ,

(1.17)

Ill,

/,lnls esl, at)lisllillg all (:xt)()Imlltial nmt)t)i21g l,llr(mgll (,ll(; real lm.rmllel,er t ()f eh:nl(:nl,s G ' ( 0 ) in (,he Li(: alg(:l)ra (,()" finil,(:" eh:nlenl, s ()f l,h(: I,i(: gr(nq). Histori(:ally, lSe algel)ras lm.ve enlerge(t |)y (:oIlsi(h:rillg (:l(:IIl(:lll, s in Lie gTout)s infinitesimally (:h)se t() the gr()lq) i(tentity. D u e to this i n t i m a t e link l)etween Lic algc|)ras a.n(1 I, ie g r o u p s (,lu: (:haracteriza.tion ()f Lic algebras ~llent, ione(t a,t)()ve reflects in an analog(ms (:hara,cterization (or, nlore precisely, classification) of Lie groups. A r e p r e s e i l t a t i o n of a Lie g r o u p {~ on a vector spa,(:(; V is a h o m o m o r p h i s m of {~ into the g r o u p of a u t o m o r p h i s m of V (i.e. of transforma, tion of V into itself). If a tin.sis set is chosen for the vector space V, the r e p r e s e n t a t i o n can equally t)e intende(t as a. h o m o m o r p h i s m of {7 to the g r o u p GL(n), a n d a c c o r d i n g l y it t)rovides a m a t r i x r e p r e s e n t a t i o n of t h e group. In a sense, a g, oup r e t ) r e s e n t a t i o n Inay yieht a d e s c r i p t i o n of a g r o u p in t e r m s of m o r e "visible" ot)jects.

Index A

Aberrations, 32 Acceptance, 105, 504 Ambiguity flmction, 358-369 cross terms, 368 marginals (correlative), 361 optical transfer flmction, and, 376 representability, 363 Wigner distribution function, and, 359 Axisymmetric systems, 32, 34 B

Baker Campbell Hausdorff formula, 75, 77 Beam quality, 105, 507 Bopp operators, 452, 475-476 C Coherence fllnction, 291 Collimating systems, synthesis of, 123 Composite optical system, 113 ray-transfer matrix, 114 wave-transfer operator, 184 Cross-spectral density, 291 Cross-Wigner distribution flmction, 285, 475 D

5-fllnction, 195 Dual systems, 238 E

Ehrenfest theorem, 258 Eikonal, 175 Elementary beam of rays, 86 Emittance, 105,504 F

Fermat's principle, 3, 5 First-order optical systems (1D) free-medium section, 94 ideal magnifier (positive), 98 input-output relation, ~-domain, 203 input-output relation, q-domain, 182 quadratic monomials, and, 93 two-media interface, 96 Focal length back, 47 front, 48

primary, 48 secondary, 48 Focal plane back, 47 front, 48 Focusing power, 49 Focusing systems, synthesis of, 123 Fourier transform, 198 linearity property, 200 modulation property, 200 multiplication property, 201 Parseval relation, 200 shift property, 200 similarity property, 200 Fourier transformer Fourier tubes, 125 imperfect, 123 input-output relation, 242 perfect, 124 2f system, 124 Fractional Fourier transform derivative property, 256 eigenfunctions, 254 multiplication property, 255 similarity property, 255 Fractional Fourier transformer, 137 ABCD integral, and, 248, 252 Fresnel transform, and, 250 harmonic oscillator, and, 135 input-output relation, 245 Lohmann's type I geometry, 139 Lohmann's type II geometry, 140 phase space rotator, as, 143 ray-transfer matrix, 137,244 standard focal length, 141 subgroup of ray matrices for, 138 wave-transfer operator, 245 Free propagation, 94 input-output relation, t~-domain, 229 input-output relation, q-domain, 226 line spread function, 227 negative distance, 118 optical transfer function, 230 ray matrix, 115 subgroup of matrices for, 116 wave transfer operator, 226 Frequency operator coordinate representation, 214

Linear Ray and Wave Optics in Phase Space

524

frequency representation, 214 Wigner representation, 452 Fresnel approximation, 229 G

Gabor flmctim~, 210, 348 Gauss' (livergmlce theoreIn, (i6 Gmmsian sigllal, 176, 493 ~l~wertai~fl,y r('lal,io~l, m~(t, 21()

(;(;~mraliz(:(! Cllrvatlir(; paranmt('r, 5()7 H

Ha~nilto~l's (;(l~mtions light ray, for tile, 7, 19, 5.9 ligl~t ray, ()f the, paraxial fi)r~l ()f, 37 Itainiltoi~'s prin(:iple, 3 t ta~filto~fiml fluwti(n~, paraxial fi)nn, 31 ray, of tim, (i ttamiltoxfian operator, 18,8 I)araxial fi)r~l, 189 lleiseld)erg-Weyl algebra, 187 th;l~nholtz eq~at.i()x~ (scalar), 170 lh'rt~lit(;-('()sh-(;a~lssia~ t)(;m~s, 328 lh,rmite-(;a~tssian I)ea~ns, 325 I

hleal magnifier F(nlrier traIlsf()r~ner, relati(n~ witll, 126 iIlt)ut-()~tI)~d, relati()ll f()r l)ositive, ~(lo~nai~, 234 input-output relation for positive, q(lolnai~l, 234 line spread fimction for positive, 233 subgroup of, 128 s~bgro~p of positive, 129 synthesis of, 125 wave spread fimction for positive, 234 wave trmlsfer operator for positive, 233 hnaging systems input-output relation for, 241 line spread fimction for, 241 magnifier, or, 51 Irradiance, 85 Iwasava (tecompostition metaplectic operators, 248 symplectic matrices, 152 L

Lagrange invariant, 43

Lie algebra Inetaplectic rap(2, R), 225, 260 symplectic, sp(2, R), 91,179, 224,260 Lie group Lie transformations, of, 73 metaplectic Mp(2, R), 225,264 symplectic, Sp(2, R), 91, 179, 225, 264 Lie operator, 18, 5.(), 6.9 alge.l)ra of, 19 IJe trailsformatioll, 21, 69 lixiear l)()lytl()llfials, getmrate(l l)y, 25 P()issol>t)racket preservation property, 72 pro(hlct I)reservatioll property, 7() qlm(h'atic l)olyIloznials, gelmrat('.(l I)y, 26 l~ille-st)rea(1 flilwti()ll, 183 l,i(nlville tlmoreln, 67 Liollville's eqlmti(nl, 62, 65, 67, 455 M

M 2 factor, 507 MagIlilic'at,i()li, 125 ray ('()(n'(litmt(', 46 ray lIl()lIleltttull, 46 Meri(li(nml l)lmm (or zneri(liml), 35 Mometlts, 463 1D fllIl('t,i()II, 466 2D fllIlcti(nl, 471 2D filxmtioIl, global, 472 2D fimcti(nl, local, 471 Moment~Hn operator coor(linate rel)reseIltati(nl , 187 eigenfllxlctiolis of, 1.()6 lnomentllln representation, 214 Moyal fi)rmllla, 2.(}7 Mlltlml illtensity finlct, ioIl, 372 ambigllity fimction, and, 374 Wigner distriblltion flnlction, and, 374 N

Nodal planes, 50 O Optical computing Fourier transform, 380 Fresnel transform, 380 Wigner transform, 384-399 Optical SchrSdinger equation, 189 Optical transfer flmction, 205 Orthogonal optical systems, 32, 33

Subject Index P

Paraxial approximation Helmholtz equation, in the, 171 Path length geometrical, 4 optical, 4 Phase space beam analyzer, 509-510 ellipse of 2rid-order moment, 503 geometrical optical, 2, 7, 99 mechanical, 2 ray density distribution, 63, 78 trajectory, 62 volume, 62, 65, 69, 504 wave optical, 8, 176, 211 Phase space picture fractional Fourier transformation, 136 free propagation, 99 refraction, 101 scale transformation (positive), 102 Phase space rotations, 136 generating matrices of, 150 Phase space squeezes, 147 generating matrices of, 150 Point-spread function, 180 Poisson brackets, 15-18 Position operator coordinate representation, 186 eigenflmctions of, 195 frequency representation, 214 moinentum representation, 214 Wigner representation, 452 Principal plane primary, 48 secondary, 48

Q Quadratic graded-index medium, 43, 158 defocusing, ray-matrix for, 161 focusing, ray matrix for, 159 Fourier transformer, and, 160 fractional Fourier transformer, and, 160 Quadrupole lens, 508 R

Radiance (or, brightness), 80 definition, of, 81 phase space function, as a, 83 phase-space ray density, and, 84 Radiant intensity, 85

525

Radon transform, 443, 447 projection-slice theorem, 445 Radon-Wigner transform, 443 fractional Fourier transform, and, 443 Ray (light), 1, 175 axial, 41 coordinates, 9 equation, 6 field, 41 meridional, 35 momenta, 6, 9 paraxial, 30 paraxial, coordinates of, 36 skew, 35 Ray-spread flmction, 431 Ray-transfer matrix, 40, 60, 179 single-exponential form, of the, 42, 179 Ray-transfer operator, 22,177 single-exponential form, of the, 22 Refracting power, 44, 96 S Shift-(or, space-)invariant systems, 181,204 Simple astigmatic systems, 33, 61 Skewness, 35 Sliding-window Fourier transform, 343 ambiguity function, and, 352 inversion formula, 344 Wigner distribution flmction, and, 352 Space-bandwidth product, 210 Space-frequency distribution flmctions, 354 correlative classes of, 361 energetic classes of, 361 inversion formula, 357 kernels, of the, 356 marginals, 356 Spatial harmonic, 196 Spectrogram, 344 cross-terms, 350 smoothed Wigner distribution function, as, 353 Spreadless systems, 205 Squeezer, ray matrix, 145 Stigmatic systems, 34, 61 Symplectic condition, 12 transformation, 11 unit matrix, 11 Symplectic matrices canonical representation, 154

526

Linear Ray and Wave Optics in Phase Space group of real, 12 LSF '~ synthesis, 152 LST synthesis, 132 LTL factorization, 121 non,canonical representations, 154 optical ~natrices, as, 130 TLT factorization, 121

T 'l'(q(,s('()pi(' systemics, sy~ti~esis ()f, 122 Tlai(:k le~s, 114 'l'l~i~ le~s, 114 (:()~vei'gi~g, ()r p()sitive, 115 (livergi~g, ()r ~egative, 115 i~t)~t-ol~t,l)~d, relati()~, ~-(l()~mi~, 232 i~I)~t,-()~lt,I)~ll. relati()~, q-(h)~nai~, 231 liI~e spr(:a(l fluwt.io~, 231 ray i~mtrix, 115 md)gr()~q) of ~mtri('es fi)r, l l[i wave sprea(l fiu~(:ti()~, 232 ~lbI~ograplw (ol)tical), 4,1.1 U Uiwertai~ty relations, 2()7, 4.()(), 5()1 gh)l)al, 490 lo('al, 4.90 V Varim~ce matrix, 5()() Gm~ssiaI~ sig~ml, 496 transfer laws, 50()

W Wave trm~sfer operator, 222 unitary represex~tatiox~ of lixmar cax~o~ical transformatioim, as, 261-265 Wei-Norman (tecompositio~ metaplectic operators, 240 symplectic Inatrices, 132 Wcyl transform, 420 Window function, 343 Wigner distribution flmction, 271 chirp, 314 315 comb-signal, 312 condition for uifiqueness, 411 convolutio~ property, 300 cosine-sigIml, 308 cross-terms, 286 cubic-phase signal, 317 cylin(lrical wave, 281 :i-fimctioi~, 280

finite support property, 284 l"'-or(ler global InoIneIlts, 483-484 l"t-or(ter local moment, h;-wtrial)le, 477 l'~t-or(ler local lIlomellt, q-wtriable, 480 Follrier trallsfi)rnl, all(l, 43.() fractioiml Follrier trmmfi)nn, ml(l, 441 (lallssiml siglml, 321:122 (;mmsiml SlIl()ot.llillg, 2.94 l l(;riifite-c()sll-(Imlssiall siglml, 32!) lt(,rlifit(,-(lallssiml siglml, :127 lligller-()r(l(;r lll()llleIlts, 512-514 illversioll forlmd~Ls, 2.()() llmrgilml (list,ril)llti()llS, 28!) IIudtil)licatioll l)rop(wty, 301 ll()ll lillearity, 284 ll()lll)()sitiv(:lmSS, 2.92, 417 ()l)ti('s, il~, 277 l)artially ('()lmr(;~t sig~mls, 2.91 r ~ne('ha~fi('s, i~, 275 r('alit.y l)r()l)(,rty , 283 r(:('tm~gh,-sig~al, 305 r('l)resentability, 3()2,332 s('ali~g l)r()perty, 2!)!) 2"'L~)r(ler h)('al ~()~e~t, h-w~rial)le, 479 2"d-()r(h'r l()cal ~o~nei~l., q-w~riable, 482 self-ter~ns, 286 sii~(:-sig~al, 3()7 si~u~soi(lal plume sig~ml, 320 spatial harmo~fic, 280 s~nn of two (~mmsim~ sig~mls, 324 s~un ()f two i~nl)~lses, :11() symmetry property, 283 trm~sfer laws, ABCD syste~, 433 transfer laws, free ~ne(ti~m~, 425 transfer laws, positive ~nagnifier, 429 trm~sfer laws, tl~ix~ lex~s, 427 transport eq~mtionr, 454 Talbot effect, and, 436 Wigner phase plane, 304 Wig~mr processor Gaussiax~ s~noothed WigImr (listrib~tion fiH~ctio~, 398 1D real signals, 384 1D coinplex sigImls, 394