Progress in Optics, Volume 35

EDITORIAL ADVISORY BOARD G. S. AGARWAL,

Ahmedabad, India

T. ASAKURA,

Sapporo, Japan

C. COHEN-TANNOUDJI, Paris, France

V L. GINZBURG,

Moscow, Russia

F. GORI,

Rome, Itab

A. KUJAWSKI,

Warsaw, Poland

J.

Olomouc, Czech Republic

€"A,

R. M. SILLITTO,

Edinburgh, Scotland

H. WALTHER,

Garching, Germany

PROGRESS IN OPTICS VOLUME XXXV

EDITED BY

E. WOLF University of Rochester. N X , U.S.A.

Contributors E. ARIMONDO, J. BERNARD, R. BROWN, Ts. GANTSOG, K. ITOH, B. LOUNIS, A. MIRANOWICZ, M. ORRIT, D. PAOLETTI, N. N. ROSANOV, G. SCHIRRIPA SPAGNOLO, R. TANAS

1996

ELSEVIER AMSTERDAM. LAUSANNE .NEW YORK . OXFORD. SHANNON. TOKYO

ELSEVIER SCIENCE B.V. SARA BURGERHARTSTRAAT 25 P.O. BOX 21 1 1000 AE AMSTERDAM THE NETHERLANDS

Library of Congress Catalog Card Number: 6 1 - 19297 ISBN Volume XXXV 0 444 82309 3

0 1996

ELSEVIER SCIENCE B.V.

All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means. electronic. mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher: Elsevier Science B. Y , Rights & Permissions Department, PO. Box 521, IOOOAM Amsterdam, The Netherlands. Special regulations for readers in the USA: This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers. M A 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred to the Publisher: unless otherwise specijied. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products. instructions or ideas contained in the material herein.

PRINTED ON ACID-FREE PAPER PRINTED IN THE NETHERLANDS

PREFACE This volume contains six review articles on various topics of modem optics and related subjects. The first article, by N.N. Rosanov, discusses transverse light patterns in nonlinear media, lasers and wide-aperture interferometers. Such patterns may be almost periodic in transverse directions, when they appear as light filaments, or they may be localized, as spatial solitons. They are manifestations of optical “self-organization” and are of interest in connection with information processing. The second article, by M. Orrit, J. Bernard, R. Brown and B. Lounis, deals with the detection and spectroscopic studies of single molecules in transparent solids at low temperature. The isolated spectral line of a single molecule makes it possible to perform basic quantum measurements, and allows probing in unprecedented detail of the surrounding solid matrix. The article also includes some suggestions for future research in this field. The article by K. Itoh which follows, reviews interferometric techniques for retrieving multispectral images with a large number of spectral channels. Special attention is paid to the theory of interferometric multispectral imaging which unifies the theories of coherence-based image retrieval and spectrum recovery. Various techniques are compared, especially in terms of signal-to-noise-ratio. In the fourth article D. Paoletti and G. Schirripa Spagnolo present a review of holographic and electronic speckle interferometric techniques applied to artwork diagnostics. It describes the most important tests performed on models and real artwork. The next article, by E. Arimondo, discusses coherent population trapping in laser spectroscopy and reviews experiments on the detection and utilization of trapping. The coherent superposition of states, which is an essential part of the phenomenon, arises in laser spectroscopy, optical bistability, four-wave mixing, light-induced drift, laser cooling, adiabatic transfer, lasing without inversion, pulse matching, photon statistics and atomic and molecular ionization. Aspects of the theoretical analysis and of experimental observations are described and discussed with the view to some possible future applications. The last article, by R. TanaS, A. Miranowicz and Ts. Gantsog, presents a review of quantum phase properties of optical fields generated in some non-

vi

PREFACE

linear optical processes. Various states of the field, such as coherent states, squeezed states, anharmonic oscillator states and second- and sub-harmonic fields, exhibit different phase properties. Modern formalisms, such as the PeggBarnett Hermitian phase formalism and the formalism based on the so-called s-parametrized quasi-distribution functions for example, are used to elucidate such properties in a systematic way. In view of the wide range of topics discussed in this volume, we hope that most readers will find in this book something that is of interest to them. Emil Wolf Department of Physics and Astronomy University of Rochester Rochester, New York 14627, USA February 1996

E. WOLF, PROGRESS IN OPTICS XXXV 0 1996 ELSEVIER SCIENCE B.V. ALL RIGHTS RESERVED

I

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS BY

NIKOLAY N. ROSANOV Institute of Laser Physics, S.I. Vavilov State Optical Institute, I99034 St. Petersburg, Russian Federation

I

CONTENTS

PAGE

9 1. INTRODUCTION . . . . . . . . . . . . . . . . . . .

3

4 2. FILAMENTATION . . . . . . . . . . . . . . . . . . .

5

SPATIAL SOLITONS . . . . . . . . . . . . . . . . . .

21

tj 3.

9 4. SWITCHING WAVES AND SPATIAL HYSTERESIS. . . . . 8 5. DIFFRACTIVE AUTOSOLITONS IN NONLINEAR

INTERFEROMETERS . . . . . . . . . . . . . . . . .

34 41

9: 6. AUTOSOLITONS IN LASERS AND NONLINEAR WAVEGUIDES

49

CONCLUSION. . . . . . . . . . . . . . . . . . . . . . .

54

REFERENCES . . . . . . . . . . . . . . . . . . . . . . .

57

2

5

1. Introduction

Spontaneous symmetry breaking that results in the formation of light patterns in transversely homogeneous nonlinear optical systems has attracted the attention of investigators for about three decades. These investigations began with radiation self-focusing, including ( 1) large-scale self-focusing in the form of self-trapping, or spatial solitons, when diffractive spread of the propagating beam is compensated by its focusing with a nonlinear medium (Askar’yan [1962], Chiao, Garmire and Townes [1964], Talanov [1964]); and (2) smallscale self-focusing, or filamentation, that is, instability of nonlinear propagation of a plane wave and its breakup into separate filaments (Bespalov and Talanov [ 19661). Special attention was paid to filamentation, which created a real problem for laser investigators, because the filamentation prevented the increase of radiation brightness in high-power laser systems. As a result, several effective ways of filamentation suppression were proposed and realized (Mak, Soms, Fromsel and Yashin [ 19901). Spatial solitons, for example, in a Kerr medium, were found to be unstable. Temporal solitons in nonlinear optical fibers, mathematically equivalent to 1D (one-dimensional) spatial solitons, were shown to have high application potential, however, and they were thoroughly investigated, both theoretically and experimentally (Hasegawa [ 19891). Recently the situation has changed. Stable spatial transversely 1D solitons were demonstrated in planar waveguides with Kerr optical nonlinearity (Barthelemy, Maneuf and Froehly [ 19851). The idea of spatiotemporal solitons (“light bullets”) was suggested for a homogeneous medium with selffocusing nonlinearity and anomalous dispersion (Silberberg [ 19901). Stable or metastable transversely 2D solitons with wavefront dislocations, or vortices, were demonstrated for a Kerr medium and for a medium with saturable nonlinearity (Kruglov, Volkov, Vlasov and Drits [ 19871, Swartzlander and Law [ 19921). New types of transverse patterns with rather striking features were found in the systems with feedback (wide-aperture nonlinear interferometers, lasers, etc.). Recognition of the optical patterns as a manifestation of self-organization phenomena was useful and instructive. Optical instabilities and filamentation 3

4

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I,

9:

1

became popular, and nonlinear investigators successfully tried to destabilize increasing numbers of optical systems. Currently the field of optical transverse patterns is rather broad, and it is impossible to describe all known results in this chapter (see also Abraham and Firth [ 19901, Rosanov, Mak and Grasiuk [ 19921, Lugiato [ 19941, and references therein). I will try to review the main ideas and emphasize some new features of self-organization specific for optics, compared with features typical for other nonlinear physical, chemical, and biological objects (Nicolis and Prigogine [ 19771, Haken [ 19781, Cross and Hohenberg [ 19931). The chapter starts with the classical problem of small-scale self-focusing (filamentation, or modulational instability) for one and two plane waves (9; 2. l), partly because of the simplicity and general character of its theory. To answer the question of whether any patterns will arise in the given wide-aperture system, it is useful to check the possibility of filamentation in the corresponding ideal transversely homogeneous system (plane-wave instability). In the case of filamentation, it is clear that radiation in a wide-aperture system will eventually break up into many filaments. Therefore, some transverse patterns have to arise in the system under conditions derived from the simple filamentation approach. A more realistic theory of wide-beam filamentation is given in 9; 2.2. The chapter then describes surface and guided waves filamentation (§ 2.3). In such systems (i.e., without feedback) instability has a convective character (perturbations grow with a longitudinal coordinate), whereas in the systems with feedback (4 2.4), instability is absolute (perturbations grow with time). In 9; 3 different types of stable filaments, spatial solitons, are described. Their formation and interaction determine the final form of the filamentation in the wide-aperture system. An almost exhaustive description of such phenomena is known for ID geometry and for a medium with Kerr nonlinearity (9; 3.1). Section 3.2 presents computer simulations of interaction of solitons for transversely 2D schemes, which give some insight into a more complicated picture of nonlinear propagation and interaction of high-power radiation beams. Spatiotemporal solitons (“light bullets”) are discussed in 9; 3.3. The second part of the review (9 4-9 6) examines the essentially different types of spatial patterns that are inherent in systems with feedback. Filamentation instability is not needed for their formation, hence they can be generated only by a sufficiently large initial perturbation. The examples are switching waves (9; 4) and diffractive autosolitons (9; 5). The autosolitons are reviewed mainly for the scheme of a nonlinear interferometer; similar structures are also described for the laser with saturable absorption (9 6.1) and for the waveguide with saturable amplification and absorption (9; 6.2). They are particle-like field structures with

1,

9: 21

FILAMENTATION

5

rather striking “quantum” and “mechanical” features. We would like to underline that, contrary to the usual solitons, the diffractive autosolitons have a discrete spectrum of their width. This difference is fundamental, and leads to new physics and possible new applications, which are discussed briefly in the conclusion.

5

2. Filamentation

In this section we consider the simplest case of nonlinear propagation of radiation. The electromagnetic field is taken to be quasimonochromatic, and its polarization state does not vary significantly. Then the field can be characterized by a scalar complex amplitude whose envelope E slowly varies in space (in the scale of light wavelength A)

E

E ( r 1 , t ) = i E ( r 1 )exp[i(kz - wt)] + c.c., I

(2.1)

where z is the longitudinal coordinate and r l =x,y is a vector of the transverse coordinates. The isotropic transparent medium is characterized by nonlinear electrical permittivity & =

&o + bE(lEl2),

8&(O) = 0

For the Kerr medium 6~ = E 2 IEl2, where ~2 is the coefficient of nonlinearity. For resonant nonlinearity (two-level scheme far off the absorption line, I , = \Esl is intensity of saturation)

The envelope E obeys the standard paraxial equation

dE 2ikdZ

a& + ADE + k2-E EO

=

0,

where AD is the transverse Laplacian (D=1 or 2, depending on the scheme geometry). A plane wave serves as a solution of eq. (2.3)

E

= Eo exp(ipz).

Here, Eo = const., p

=

(2.4) ~ o ~ E ( / o ) / ~and E o10 , = I Eo 12.

6

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I, 5 2

2. I . FILAMENTATION OF A PLANE WAVE

We consider here the conditions of instability of a plane wave propagating through a nonlinear medium (Bespalov and Talanov [ 19661). Let us assume that E = Eo exp(ipz)( 1 + SE). For small perturbations SE the linearized equation is valid

dSE 2ikdZ

+ ADSE + B(SE + S E * ) = 0,

where

The solutions of the linear eq. (2.5) can be represented in the form of superposition (Fourier’s integral) of components with different spatial frequencies q=(qx,qy). The perturbations with the opposite frequencies ( q and -4) are interrelated. For them

SE

=

aexp(iqr1 + y z ) + b*exp(-iqrl

+ y*z),

(2.6)

where r l = ( x , y ) . Substituting (2.6) into (2.5), we obtain the following dependence of y on q2:

(2ky)2 = q2(2B - 42).

(2.7)

The exponential increase of the perturbations corresponds to the real increments y, which is possible only if B > 0 ( S E > ~ 0) in the range of spatial frequencies 0 0 filamentation instability takes place for any intensity value. Such media are called self-focusing. The sign of 6 ~ may ; vary for other types of nonlinearity, 2 for example, when SE = ~2 JE) - ~4 \El4. Such a medium is self-focusing for I < ~ 2 / 2 ~and 4 self-defocusing for larger intensities (we let ~ 2 , 4> 0). Note that the filamentation conditions do not include the transverse dimensionality D of the scheme. If we know the increments y and the initial amplitudes of the perturbation spatial harmonics (at z = 0 ) , it is not difficult to determine the subsequent evolution of the transverse structure of radiation at z > 0. At the linear stage of filamentation, the perturbations with spatial frequency qm = (2B)”2

1,

D 21

I

FILAMENTATION

increase more rapidly. The corresponding length of small-scale self-focusing is L, = y;’ = 2k/B. Because of the increase of the initially small perturbations, the radiation transverse profile becomes deeply modulated (filamentation). To determine the final structure, the nonlinear stage of filamentation has to be analyzed. In addition the actual radiation beam is confined within transverse direction, whereas a plane wave has infinite power. These factors will be taken into account in the following sections. Note that this instability is convective; that is, the perturbations increase with the coordinate z (but not with time in points with fixed space coordinates, as in the case with absolute instability). The case of a medium with frequency dispersion can be treated similarly. In this case instead of (2.6), we let

6E = aexp(iqr1 + iQt

+ yz) + b*exp(-iqrl-

iSZt + y*z).

(2.8)

The dependence of the complex increment y on the spatial frequency q and on the modulation frequency SZ can be obtained from linearized non-steady-state paraxial equation. More interesting is the case where the increment has maximum for q # 0 and SZ * 0. Then the initial plane monochromatic wave will break up into separate “light clots” (3D-solitons). According to eq. (2.7), the perturbation with fixed spatial frequency can both increase ( y > 0) and decrease ( y < 0), transferring its power to the main beam (plane wave). The direction of power transfer is determined by phase relations; that is, phase difference @ = @(z) between the perturbation and the unperturbed wave. An arbitrary (small) initial perturbation can be expanded in exponentially increasing and decreasing components. The perturbation complex amplitude can be conveniently represented as a two-component vector with real elements: Re 6E(z) 6E(z)= (Im8E(z))

=

1Wz)I

(

)

cos @(z) sin qZ)‘

The input (z = 0) and output (z = 1) perturbation amplitudes are connected by a linear relation, which can be written in the matrix form

6E,,, = U6Ein. (2.10) The expressions for the matrix elements follow from the general solution of eq. (2.5) (Rosanov and Sniirnov [ 19801): U=

(

cash Y, -r]sinh Y -q-’sinh Y , cosh Y cos Y,

,

q2 < 2B;

(2.11)

-qsin Y

I ”’,

1-

I /2

where Y = ( 1 / 2 k ) q 12B - q2 r] = q / 2 B - q2 . When linear absorption or amplification of radiation is taken into account, the elements of the transfer

8

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I, 5 2

matrix are represented by Bessel functions. In a linear medium (the limit B 4 0 ) U corresponds to the rotation matrix (2.12) If radiation passes through a system consisting of N layers of nonlinear and linear media, the resulting transfer matrix of the system W is a product of transfer matrices of the separate layers:

w = u, x u,-1

x

... x

u2

x

u,.

(2.13)

Lens (telescopic) systems also may be described by similar matrices; however, in the case of magnification coefficients M # 1, changes of perturbation frequency have to be taken into account. Let us define the system transfer coefficient as a ratio of modulation depths of the output and input perturbations with fixed spatial frequency (2.14)

The transfer coefficient K depends significantly on the phase of initial perturbation Yo. Let us introduce the maximum, minimum, and average values of the transfer coefficient

(the angular brackets denote averaging over Do).The coefficients K,, coincide with the matrix W singular values

and Kmin

(2.16)

where S is the Hilbert-Schmidt norm of the matrix W ,

The difference between singular values and eigenvalues is due to the nonHermitian (asymmetrical) form of the matrix W , which is a consequence of

1, § 21

FILAMENTATION

9

K

20

15

10

5

1

0

Fig. 1. Dependence of maximum (solid curves) and average (dashed curves) transfer coefficient on spatial frequency of perturbations; B r = 1 ( I ) , 2 (2), and 3 (3).

power transfer between the unperturbed wave and perturbation in a nonlinear medium. For low frequencies (2.17) where Br is the so-called breakup integral,

and 1, is the length of the nth element. Note that separation of radiation on the main (unperturbed) beam and perturbation are justified if the angle qlk at which the perturbation propagates exceeds the main beam angular divergence. This condition coincides with the requirement of a significant difference between the transverse scales of the main beam and perturbation (see 5 2.2). Figure 1 shows the dependence of the transfer coefficient on perturbation spatial frequency for continuous nonlinear medium ( N = 1). It is remarkable that the maximum transfer coefficient may exceed unity even outside the range of instability (for$ > 2B). In this case, however, the increase of the transfer

10

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I,

82

coefficient with nonlinear medium length 1 is not exponential, but only linear. An important problem facing the designers of high-power, solid-state lasers is the limitation of K,,, . Examples of calculations and comparison with experiments can be found in the literature (e.g., Mak, Soms, Fromsel and Yashin [ 19901). The filamentation of two waves in isotropic or anisotropic Kerr media is described by a set of coupled paraxial equations

(2.18)

The waves may differ by frequencies ( W I # c o ~ ) ,directions of propagation (kl = - k 2 ) , and/or polarization states. For unperturbed plane waves one has E I =El0 exp(ifilz), E2 =E20 exp(ifi2z). The propagation constants f i l , 2 are defined by the wave intensities. Linearized eqs. (2.18) admit solutions of the form (2.6). Increments y are the roots of a biquadratic equation. Contrary to the case of one-wave filamentation, there are now two branches of the dependence y2(q2). Filamentation is possible even for media self-defocusing (with respect to one beam), when a1 I < 0, aZ2< 0. Among problems of this type the filamentation of counterpropagating waves was first studied by Vlasov [1984]. Detailed analysis was performed later by Firth and Penman [1992]. Note that filamentation of counterpropagating electromagnetic waves is possible even in a pure vacuum because of vacuum polarization, but for very large radiation intensities (Rosanov [ 1993a1). 2.2. FILAMENTATION OF A WIDE BEAM

Although the Bespalov-Talanov theory gives some insight into the main features of filamentation, its generalization to the case of transversely confined beams is important. In fact, a beam induces “a nonlinear waveguide” in a medium, which results in the possibility of propagation of transversely localized perturbations corresponding to the discrete spectrum of the waveguide modes. For beams it is precisely the discrete spectrum that corresponds to exponentially increasing perturbations. The theory of filamentation of an axially symmetrical beam in a Kerr medium proposed by Rosanov and Smirnov [1976, 19781 is asymptotic with a large parameter PIP,, where P is beam power and P , is self-focusing critical power. The typical length of small-scale self-focusing is then much less than for largescale self-focusing. Therefore, large-scale beam distortions are of secondary

1,

9 21

I1

FI LAMENTATION

importance. We can neglect them in the lowest-order approximation, which permits us to determine the nonlinear waveguide modes. Some details of this approach are given below. We use the cylindrical coordinates ( r , cp,z). Let us represent the field envelope E as a superposition of the envelopes of the main (cylindrically symmetrical) beam Eo(Y,z) and of small perturbation El ( r , cp, z), whose transverse scales differ significantly: ro >> r l . For the distances z less than the length of large-scale self-focusing 2 1 , these envelopes can be written in the form

EO(r, z) = Eo(r)exp

,

El (Y,cp, z ) = 6E(r, cp, z)exp

(2.19) The quantity B jEo(r)12 now depends on the radial profile of intensity of the main beam. The large parameter of the theory is

-

M

=

BA'2ro

~I

- ro/rI - (P/P,)"2,

where Bo = max B(r). For A4 >> 1 it is justified to use the linearized equation, which is similar to eq. (2.5)

6E + B(r)(GE+ 6 E * )= 0.

(2.20)

The solution of eq. (2.20) has the form 6E(r, cp,z)

=

W(r)exp(yz

m = 0 , 1 , 2, . . . .

+ im cp) + X*(r)exp( y*z - im cp),

(2.21)

For radial eigenfunctions Il/,x and for eigenvalues y we have the set of two coupled equations (Rosanov and Smirnov [ 19761). The perturbation increment is determined by the real part of y. Decomposing the arbitrary (small) initial perturbation 6E(r, cp, 0) in these eigenfunctions, the perturbation at any distance z << zg = r i / k M will be determined using eq. (2.21). The continuous spectrum corresponds to perturbations not increasing with z and to the purely imaginary y. A discrete spectrum of eigen solutions are also present that are finite at r 4 0 and decrease sufficiently rapidly for r + 00. The initial perturbation can be decomposed in the functions of continuous and discrete spectra. The latter are of more interest for us, because they correspond to perturbations exponentially increasing with z. The eigenfunctions are characterized by the azimuthal m and radial n integer indexes. For the discrete spectrum y < Bol2k. To determine the eigenfunctions

12

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I,

8

2

and eigenvalues, the radial profile B ( r ) has to be specified. For a bell-shaped radial profile and for m = 0 the increments y are determined by the condition

n = 0 , 1, 2,....

(2.22)

Here, rb is a branching point radius defined by the condition B(rb)=2ky. The number of discrete radial modes is

Of more interest are the rapidly increasing perturbations, localized in the beam center where radiation intensity is maximum. More precisely, their amplitude is large in the region 0 < r < r b ; for the Gaussian beam with the width w= ro we have r: = rz21/2(1 + 2n)/M << r;. In this region, qi(r) and ~ ( r oscillate ) frequently with the typical period rI = 2 7 ~ B ; ”< ~< r b ; for r > Yb the radial functions decrease exponentially. The oscillation period Y I for the lowest-order modes (with small indexes n ) corresponds in the case of plane waves to the transverse scale of the most rapidly increasing perturbations in the case of a plane wave (9: 2.1). The value rb characterizes the confined beams. The scale rl determines the distance between adjacent filaments, and the value rb may be connected with the width of the region of filamentation. Note that rb < m, the discrete spectrum is absent and exponential increase of perturbations is not possible. For larger N , the increment decrease with m becomes slower, which is due to an approximation to the case of a plane wave ( N =a, dashed line), with increments independent of m. The character of filamentation changes radically in the case where the unperturbed beam contains a vortex with the azimuthal (topological) index ma; that is, A 0 = &(r, cp) = IAo(r)l exp(im0q) (Rosanov [1993b]). Then intensity is maximum at some distance ro from the axis, and in this region the most rapidly increasing perturbations are localized. The azimuthal index m of such perturbations may be estimated from the condition my0 = rl (linear period of the

13

FILAMENTATION

m2/B r 2 a o

Fig. 2. Dependence of increments on azimuthal index of perturbations for Gaussian ( N = 2 ) and super-Gaussian ( N > 2) beams.

most rapidly increasing perturbations in the approximation of plane waves). With the distance z the filaments rotate around the beam axis with the constant angular velocity Q=cmolkr& where c is the light velocity. It follows from the presented simplified linear approach that for cylindrically symmetrical beams with regular wavefront (without dislocations), the cylindrically symmetrical perturbations develop more rapidly. For developed radial modulation of radiation, however, the perturbations with azimuthal field modulation have greater increments; that is, the beam tends to break up into separate filaments (see also 0 3). So far we have neglected the longitudinal variation of the main beam profile assuming z << Z E . With these variations taken into account, interference of large- and small-scale self-focusing and coupling of perturbation modes occur. The equations of coupled modes and their analysis are given by Rosanov and Smirnov [ 19781. Simultaneuos filamentation and fragmentation, or decay, of a quasimonochromatic plane wave both in transverse and longitudinal directions, were studied by Rosanov [ 1994~1for media with saturable gain and absorption. 2.3. STRIPE STATIONARY BEAMS, SURFACE AND GUIDED WAVES

For other types of beams the field envelope does not depend on one of the transverse coordinates, so E = E ( x , z ) (stripe beams). The stationary profile, E(x, z ) = F(x) exp(iPz),

(2.23)

can be realized both in a homogeneous self-focusing medium (due to balance between linear diffraction spreading and nonlinear focusing) and in layered (even

14

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I, 5 2

linear) systems. Since the total power of the beam is infinite, the beams have no chance to be stable in self-focusing media; their breakup in the y-direction into filaments seems to be unavoidable. However, the length of development of the filamentation may be sufficiently large for successful studies of such important structures. Let us consider stationary stripe beams in a homogeneous nonlinear medium. For them, introducing the real amplitude G and phase @, F = G(x) exp[i@(x)], we obtain d2G - dU d@ (2.24) c2= c = const., dx2 dG’ dx C2 &(G2) G dG. where U = -- 2k/3G2+ 2 G2 The integral of motion ~

(f )

1

12 (dd Gx) 2 + u = w = const.

(2.25)

allows us to determine the general form of the transverse distribution of the field. For a Kerr medium C2 2 U = -- 2kfiG2 + c2G4. 2G2 For bright stationary beams C = 0, therefore, the phase @ = const. The second of eqs. (2.24) can be interpreted as the mechanical equation of motion for a particle in the field of force with the potential U ( G ) (Haus [1966]). The case C # 0 describes dark stationary beams, which exist in self-defocusing media where filamentation is absent. Since the dark beams have infinite size also in the y-direction, using this notion in actual situations needs some qualification. In any case, for the experimental realization of dark stationary beams, rather high radiation power is needed, and such structures are unstable (Kuznetsov and Turitsyn [1988], McDonald, Syed and Firth [1992]). For these reasons we will restrict discussion to only bright beams. Nevertheless, the 2D dark solitons and optical vortex solitons have been studied intensively both theoretically and experimentally (see Swartzlander and Law [1992], McDonald, Syed and Firth [ 19921 and references therein). Of special interest are the stationary beams in a Kerr medium, the envelope of which has the following (dimensionless) form (Talanov [ 19641, Chiao, Garmire and Townes [ 19641)

(E)

(2.26) E = gsech[ g(x - xo)] exp(ifiz), fi = 2g Note that the “area” A = J IE(x, 0)l dx does not change with variation of parameter fi (or g), contrary to power P = J J E ( ~0)l2 , dx g (fig. 3a). ’

-

1,

P 21

FILAMENTATION

15

P b

P

0

P

0

i../

P

/".

*.

"/

0

P

~

1

Ln P

10

-

5

.-

*....' , 0

I

e

-,-; ,

0.5

,

,

,

,

I

1.0

I3

Fig. 3 . Dependence of power of spatial solitons with different dimensionality D = 1 (a, d), 2 (b), and 3 (c) on propagation constant /3 for a Kerr medium (a, b), for a medium with saturation of nonlinearity (c,e), and for more complex nonlinearity (d). (e) solid curve: n = O , m=O; dotted curve: n = 0, M = 1 ; dashed curve: n = I , in = 0.

The instability of such a structure was proved by Zakharov and Rubenchik [ 19731. To describe this quantitatively, one must determine the increments for perturbations of a different type. Let us introduce a small perturbation E = [ F ( x )+ GE(x,y,z)] exp(i/?z), and linearize the paraxial equation. Because

16

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I, 5 2

of the uniformity of the problem in the y direction, one can expand an arbitrary small perturbation GE(x,y, z) in the Fourier integral with components 6E = a(x) exp(iqx + yz) + b*(x)exp(-iqx + y*z), where q is the spatial frequency of the perturbation in the y-direction and y is its increment. To find the increments, one needs the solution of a linear eigenvalue problem similar to that discussed in (3 2.2. The result, which is qualitatively similar to the corresponding dependence for a plane wave, is shown in fig. 4 for q = 0 , where y’ and q’ are a dimensionless increment and a spatial frequency (Vyssotina, Rosanov and Smirnov 119871). Let us consider now the waves guided by a plane interface, in which one or both media are nonlinear (nonlinear surface waves, Litvak and Mironov [ 1968]), or the waves propagating in layered systems (nonlinear guided waves in planar waveguides). For small variations of the dielectric permittivity of a medium, we can neglect changes in the state of field polarization, and characterize the field by the electric field component E,. Then the dielectric permittivity E = E ( X , IE,”1’). The stationary waves have the stable amplitude profile E, = F ( x ) exp(ipz) determined by eq. (2.24). For Kerr media the solutions are expressed in elliptical functions. The conditions at infinity (x = *GO) and at the interfaces determine the propagation constant and the amplitude profile F ( x ) of unperturbed surface and guided waves. To analyze the stability of these waves with respect to a transversely 2D modulation, we use the method of Vyssotina, Rosanov and Smirnov [ 19841. We linearize the wave (or paraxial) equation, then we expand small perturbations GE(x,y,z) in the Fourier integral and solve the corresponding eigen problem. The full stability analysis was performed analytically for nonlinear surface waves (Vyssotina, Rosanov and Smirnov [1984, 19871) and for guided waves in different variants of layered system geometry (Akhmediev and Ostrovskaya [1988], Vyssotina, Rosanov and Smirnov [ 1988, 19901). Naturally, all these waves propagating in self-focusing media are found to be unstable. This analytical result differs from the results of numerical calculations (Moloney [1987]) where stability of one type of nonlinear guided waves was stated. However, this was probably due to a lack of accuracy of a direct computer solution of the nonlinear paraxial equation in the case of small increments. An example of increment dependence on the dimensionless spatial frequency q’ and the parameter q connected with surface wave maximum intensity is shown in fig. 4. Filamentation of nonstationary stripe beams in a layered nonlinear medium has more complex kinetics. The reflection of a wide beam from an interface between linear and nonlinear media was considered for the first time by Kolokolov

FILAMENTATION

17

Fig. 4. Dependence of dimensionless increment y’ on dimensionless spatial frequency 9’ of perturbations and on the parameter of nonlinear surface wave 7.

and Sukov [ 19781 and Rosanov [ 19791; filamentation and spatial solitons were discovered for cases of high-power radiation. The important subseqent studies have been reviewed by Stegeman, Wright, Finlayson, Zanoni and Seaton [ 19881 and Moloney, Newell and Aceves [1992]. A primary problem here is the possibility of bistability for the nonlinear reflections of beams. The solution of the full wave equation is needed to solve the theoretical problem, because in this case an approximation of the paraxial equation is insufficient to describe the effects of spatial hysteresis (Rosanov [1979]) (see $4). In this way the spatial bistability for nonlinear reflection of a high-power beam was obtained in simulations (Rosanov and Khodova [ 1986a1). 2.4. FILAMENTATION IN SYSTEMS WITH FEEDBACK

The wide-aperture systems with feedback are nonlinear interferometers, lasers with additional nonlinear elements, nonlinear reflection of beams, and a nonlinear layer with an additional mirror. In such systems the instability is absolute; that is, the perturbations grow with time in points with fixed space coordinates. Due to feedback, perturbation may grow even in the case of self-defocusing nonlinearity. Historically, the first example was the filamentation

18

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I, 5 2

Fig. 5 . The scheme of a nonlinear ring interferometer. M are mirrors. The layer of the nonlinear medium is dashed. In the dashed contour additional linear elements can be present, e.g., a spatial filter.

in a nonlinear interferometer excited by outer radiation (Rosanov and Semenov [1980]) where bistability is also possible (Lugiato [1984], Gibbs [1985]. Of course, bistability is not needed for filamentation phenomena. The latter exist, for example, in lasers and even in a simple system such as a nonlinear layer with an additional mirror (Firth [ 19901, Rosanov, Fedorov and Khodova [1991]). For the detailed numerical simulations and illustrations of field hexagon formation in the layer-mirror scheme, we refer readers to work by Firth [1993]. Let us consider a ring interferometer partly filled with a nonlinear medium and excited by external coherent Ladiation with the envelope E , (fig. 5). If the incident radiation is a plane wave with the intensity Ii = IE;I2 the field inside the interferometer with an infinite aperture also may be assumed to be a plane-wave one with the envelope E. The filamentation analysis given by Rosanov and Semenov [ 19801 is similar to that presented in 9; 2.1. We introduce a small field perturbation, and describe the change of their Fourier components (corresponding to different spatial frequencies) during one round trip across the interferometer by the transfer matrices (2.11) in a nonlinear medium and by (2.12) in a linear medium. If the modulus of the resulting transfer matrix eigenvalues is greater than unity, the perturbation with the corresponding spatial frequency will grow with time (absolute filamentation instability). Depending on the phase detuning of the interferometer, on losses, and on the field intensity I = IEI2, filamentation exists both for self-focusing and self-defocusing medium nonlinearity, Another approach, yielding similar results, is based on simplified

FI LAMENTATION

19

a

d

1 - R

0

-

Br

1 - R b

e

q2

0 C

\

f

\ I Fig. 6. Domains of instability of stationary modes (dashed lines) in an interferometer with selffocusing ( a x ) and defocusing (d-f) Kerr nonlinearity; (a) Aph >-&(I -R); (b) &( 1 - R ) < -Aph < 2(1-R);(C)-dph > 2 ( 1 - R ) ; ( d ) d p h < & ( l - R ) ; ( e ) J S ( I - R ) < d p h <2(1-R);(f)Aph >2(1-R).

equations for the field envelope E averaged in the direction of the interferometer axis 1 dE c dt

--

i 2nk + -ADE + i-6P

2k

EO

+ -1 { [1 - Rexp(iA,h)] 1

E

-

Ei} = 0.

(2.27)

Here, 6P is the medium nonlinear polarization, R is the product of the mirrors’ coefficients of reflection, 1 is the cavity length, Aph is the phase detuning, and E , is the envelope of the incident radiation (Lugiato and Lefever [1987]). The boundaries of stability for different situations for a Kerr medium are shown in fig. 6.

20

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I,

5

2

k 1 0

-W

-w

-u)

W

w

0

0

f

z

u)

Fig. 7. Transverse distribution of the laser intensity Br for Fresnel number N and 4.5 (d).

= 60

(a), 25 (b), 12 (c),

The simple picture of filamentation in the wide-aperture interferometer presented here has to be refined by taking into account the finite aperture of the scheme and of the exciting beam, the existence of additional linear and nonlinear elements inside the interferometer, and other factors. The first publication in 1980 was followed by many theoretical and experimental studies of the peculiar features of filamentation in nonlinear interferometers with different optical schemes (see, e.g., Abraham and Firth [ 19901, Akhmanov, Vorontsov,

1,

5 31

SPATIAL SOLITONS

21

Ivanov, Larichev and Zheleznykh [ 19921, Rosanov, Mak and Grasiuk [ 19921, and references therein). A peculiarity of filamentation in an array of coupled nonlinear cavities was studied by Rosanov [ 1993~1. Strictly speakmg, in the case of lasers we cannot, as previously, use the simplest ideal model of the system with an infinite aperture because the total gain would be infinite in the nearly transverse directions, and the corresponding radiation would be amplified and dominating. This points to the importance of cavity mirror edges or of other aperture limitations for transverse laser patterns. Note that the mirror edges serve as a source of perturbation needed for the development of filamentation. On the other hand, an effective filtering of perturbations occurs in the cavity with a finite aperture. The final pattern depends on the balance of these factors. Also important is the relaxation mechanism and constituent equations. When the adiabatic elimination of the medium variables is possible and the medium inside the cavity is self-focusing, filamentation of laser radiation occurs (Rosanov and Semenov [1982]. Figure 7 shows that filamentation is typical for wideaperture lasers, whereas for small Fresnel numbers the beam profiles are smooth. For many filaments the kinetics of lasing may be nonstationary and even chaotic. Such kinetics are typical of many solid-state lasers. For gas lasers with high average power, however, additional factors are important (radiation heating of the medium, generation of acoustic waves, etc.). 'The filamentation in pulsed COZ lasers was reviewed by Yur'ev [1992]. Interesting results can also be found in the review by Abraham and Firth [1990] and in the references cited therein.

0

3. Spatial Solitons

An important type of radiation structure in a nonlinear medium corresponds to the radiation stationary transverse profile E ( r 1 , z) = F ( r 1 )exp(ipz).

(3.1)

For such beams a diffractive spreading is compensated by their nonlinear compression. The propagation constant p and the envelope F ( r l ) are determined as eigenvalue and eigenfunction of the nonlinear equation resulting from eq. (2.3):

1

A D F + [ ( ; ) ' S E ( I F ~ ~-2kp ) F=O. The stationary stripe beams F = F(x), and formally, even plane wave F =Eo, [eq. (2.4)], are in this class. In this section we are interested in bright spatial solitons

22

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I,

5

3

that, contrary to the plane wave and dark solitons (Kivshar [1993]), have finite power and, correspondingly, an amplitude decrease at periphery: F ( r l ) + 0 for lrl I + 00. From this condition, taking into account the definition 6&(0)= 0, it follows that /3 > 0; the decrease of intensity for lrl I co is then exponential. In addition, /3 < (o/c)~max 6d2k (Vakhitov and Kolokolov [ 19731). Therefore, the bright spatial solitons can exist only in a medium that is self-focusing, at least for small intensity, in which the filamentation of the plane wave and wide beams develops. The radiation structures of the type (3.1), if stable or metastable, are called spatial solitons. The dependence of their features on the type of nonlinearity and the scheme geometry is conveniently illustrated by the case of the model nonlinearity 6~ = lEI2'. The dimensionless form of the paraxial equation is ---f

dE i - + ADE + IE I 2a E

dz

=

0.

(3.3)

The parameter plane (D, a) consists of three regions: subcritical 0 < u < 210, critical u= 210, and supercritical: 210 < u < 00 for D < 2 and 210 < u < 2 4 0 - 2) for D > 2 (Talanov and Vlasov [1989]). The case of a Kerr medium u = 1 corresponds to the subcritical region when D = 1, to the critical one when D = 2, and to the supercritical one when D = 3 (the latter is realized for the anomalous frequency dispersion, see 4 6.2). Spatial structures of the form (3.1) exist in every region, but stable spatial solitons are present only in the subcritical region, where they consist of one-parametric families of spatial solitons, the parameter being the propagation constant p. In critical and supercritical regions such structures are unstable, and for small perturbations they spread with z or collapse at a finite distance (radiation intensity increases infinitely with approach to the point of collapse). The intensity increase is restricted by additional factors, for example, nonlinearity saturation [see eq. (2.2)]. Note that for some materials, such as photorefractive media, optical nonlinearity has a nonlocal character. In such media the spatial solitons can be formed at a rather low radiation intensity; for a strontium barium niobate crystal, it is possible with cw argon-ion laser radiation with intensity -200 mW/cm2 (Duree, Shultz, Salamo, Segev, Yariv, Di Porto, Sharp and Neurgaonkar [1993]). 3.1. ID SPATIAL SOLITONS AND BEAMS

Let us consider the propagation of a stripe beam in a planar waveguide. For a sufficiently thin waveguide and a small nonlinear shift 6~ the amplitude shape in the y-direction is close to the profile of the one-mode linear waveguide (in

1,

D 31

23

SPATIAL SOLITONS

this direction), V ( y ) . It is then possible to suppose E = V ( y )E’(x,z) and to introduce the intensity [El2= p IE’I2 averaged by y , where p = 1-’ I V 2 ( y )dy. As a result, the paraxial equation will be substituted by the simpler transversely one-dimensional one (see § 2.3):

so

2ik-dEf + d2E’

az

~

dx2

+ (:)2S~’(IE’12)E’=0,

(3.4)

where S&’(IE’l*)= S E ( P lEfI2) (the primes will be omitted in the following). Therefore, the stationary waves are determined by “the mechanical analogy”, but the previous statement about their filamentation instability is not longer valid. In the case of Kerr nonlinearity, the dimensionless eq. (3.4) takes the form

.dE 1 d2E 1-+--+1El dz 2dx2

E=O.

(3.5)

This so-called “nonlinear Schrodinger equation” has an exact general solution obtainable by the inverse scattering transform technique (Zakharov and Shabat [1972]). The initial stage of evolution of a radiation transverse profile can be determined more easely by a numerical solution of eq. (3.5). The same equation describes the temporal solitons in the one-mode nonlinear waveguide (see, e.g., Hasegawa [ 1989]), and is mathematically equivalent to the equations for the propagation of pulses of self-induced transparency (Maimistov, Basharov, Elyutin and Sklyarov [1990]). We are also reminded that for a transversely 1D scheme, Kerr nonlinearity corresponds to the subcritical situation (distant from the critical one), and therefore the results depend only slightly on the type of nonlinearity. Factors such as the change of nonlinearity, absorption, and others may be taken into account by different variants of the perturbation theory and the variational approach (see Maimistov [ 19931 and references therein). In this connection, numerous results are appropriate for describing initial and final stages the formation, propagation, and interaction of solitons. In the general case, radiation can be represented as a finite set of (stable) solitons and also as a continuous spectrum that dissipates at z + 00. A onesoliton solution has the form (2.26). The pure multisoliton states (without a continuous spectrum) are a generalization, because for them the relation (3.1) is not valid; periodic oscillations of a transverse profile (bound states of solitons), and the spatial separation of solitons with z are possible in this case. The field in the two-soliton state may be found in an explicit form from the general Zakharov-Shabat theory both for periodic profile variations (Gordon [ 19831) and for separating solitons (Desem and Chu [1987]).

24

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I,

5

3

The number and form of the solitons are determined by the.transverse profiles of the initial radiation amplitude and phase (at z = 0). If phase modulation is absent and the intensity profile is smooth and bell-shaped, the soliton number is completely determined by the “area’’ A . To form one soliton, it is necessary to have A > 1.32. The detailed results for the initial profile are E(x, 0) = ugosech( gox) (Satsuma and Yajima [1974]). If < a < a soliton is generated with z of the form (2.26) with g = (2a - 1)go. The kinetics of soliton generation include its form oscillations and scattering of a part of the initial beam power. The number of solitons N increases with u and is the greatest integer that does not exceed the value ( a + The interaction of soliton-like beams can be conveniently considered for the initial field profile (Desem and Chu [ 19871):

i

i,

i).

E(x, 0) = gl sech[ gl ( x

-

d ) ] + g2ei@sech[ gZ(x + d)] .

(3.6)

For larger d ( d >> g;,;), the field corresponds to two nonoverlapping solitons with different amplitudes g1,2 with the phase difference Qi. The latter determines the behavior of the solitons asymptotical for z 4 m. For @= 0 (the “inphase” launching of solitons) the initial profile will evolve to form a bound system of solitons, that is, formation of the field transverse profile periodically varying with z. For different soliton amplitudes gl # g2, the minimum distance of periodic approaching of the solitons differs from zero. It increases with initial soliton separation 2d and is close to 2d for a large difference among soliton amplitudes, in other words, different solitons interact only weekly. If Qi # 0, an unbound system results, where the two solitons eventually separate. The initial stage, the duration of which increases with the decrease of Qi, may be accompanied by oscillations. For a developed phase modulation of the initial profile, the number of solitons N is determined not only by the beam “area” A but also by modulation parameters. Sufficiently deep modulation decreases the number of solitons and provokes the breakup of multisoliton structures one-soliton beams. When a noise with subcritical intensity is imposed on an initial soliton-like beam, the field is eventually separated into two beams moving farther apart, one of which is close to the soliton and the other has the dominant noise component (Maimistov, Basharov, Elyutin and Sklyarov [ 19901). The results presented lead to the following picture. In the central part of the initially smooth radiation beam with a sufficiently large “area” A >> 1, filamentation instability develops. At its initial stage, the modulation depth grows, and filaments (spatial solitons) are generated. furthermore, they interact

1,

o 31

SPATIAL SOLITONS

25

with each other, and depending on phase relation, this interaction is attractive or repulsive. If the phase modulation of the initial radiation transverse profile is weak, at least during very long distance z the solitons periodically attract and repulse each other (the bound system). A deep phase modulation may result in the separation of the solitons. Experimentally, the fundamental spatial soliton, as well as two and threesoliton states, were demonstrated in the planar waveguide filled with the medium with Kerr nonlinearity: CSz-liquid (Barthelemy, Maneuf and Froehly [ 19851, Maneuf, Barthelemy and Froehly [1986], Maneuf, Desailly and Froehly [ 19881, Maneuf and Reynaud [ 19881) or glass (Aitchison, Weiner, Silberberg, Oliver, Jackel, Leaird, Vogel and Smith [1990]). The weak dependence of the features of 1D spatial solitons on the type of nonlinearity is confirmed by the fact that similar structures were found in a self-focusing semiconductor gain medium (Khitrova, Gibbs, Kawamura, Iwamura, Ikegama, Sipe and Ming [ 19931). Figure 8 illustrates the formation of the fundamental-soliton and twosoliton states in this scheme. The dependence of soliton power on the propagation constant for the Kerr medium is shown in fig. 3a (above). For non-Kerr media, solitons’ strong interaction may result in a change of their number (Gatz and Herrmann 119921). For some types of nonlinearity this dependence is multivalued (Kaplan [ 1985a,b], see fig. 3d). In this case there are two or more nonlinear waves with different transverse profiles (for a fixed number of nodes, e.g., for fundamental modes), different phase velocities, and identical power. Due to some similarity of their dispersion relation to the transfer function of bistable systems (Gibbs [ 19851 and 9; 4. l), such solitons are often called bistable. This term, however, seems inadequate because of the absence of the feedback necessary for the real bistability (for cw-radiation); therefore, for a given input field (for z = 0) the output field (at any distance z ) is unambiguously determined by eq. (2.3). For more details of features of solitons with a multivalued dispersion relation the reader is referred to the excellent review by Enns, Edmundson, Rangnekar and Kaplan [ 19921). It should be noted that, strictly speaking, the nonlinear propagation of radiation in the planar waveguide, which has one linear mode in the “additional” transverse direction, should be described by the transversely 2D paraxial eq. (2.3), which admits collapse for Kerr nonlinearity and high-power radiation. Therefore, if the input beam is sufficiently narrow and powerful, it can collapse at the length of large-scale self-focusing ze less than the distance of mode formation (the length of separation of the discrete mode from the continuous spectrum, L, Ihd - hcl-’, where hd,c are the propagation constants for the discrete mode

-

26

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

-20

0

[I,

53

20

x (Pm) Fig. 8. Transverse intensity distribution in a self-focusing gain medium for different injection currents. (After Khitrova, Gibbs, Kawamura, Iwamura, Ikegama, Sipe and Ming [1993].)

and for the boundary of the continuous spectrum). This statement is valid also for nonlinear optical fibers and places some restrictions on the simplified approach. 3.2. 2D SPATIAL SOLITONS

We will now consider the propagation of radiation through the transversely 2D medium. In cylindrical coordinates (r,q , z ) the field envelope for the stationary spatial structure with the azimuthal index (topological charge) m = O , & l , f 2 , . . . , has the form E,,D

=

F,,o(r) exp(ipz + imcp).

(3.7)

Therefore, for m ; r O the radiation wavefront contains a dislocation of the mth order. The radial index n characterizes the number of zeros of the radial function

1, I 31

SPATIAL SOLITONS

in the interval 0 < r < 00. The real radial function F,,p(r) constant are determined by the following equation:

-+--+ d2F 1 d F dr2

r dr

21

and the propagation

o2

(-6&(F2)---Zk/3 m2 c2 r2

-

with the boundary conditions F(r) rlml for r -+ 0 and F(r) + 0 for r -+ 00. The propagation constant /3 changes continuously in the interval 0 < /3 < (~BE,/~Eo), where 6 ~ ,= max 6&,and the case of a Kerr medium corresponds to the limit fl+ 0. Such structures, if stable or metastable, are spatial solitons. As indicated earlier, the Kerr nonlinearity corresponds to the critical regime for the transversely 2D geometry. In this case the stationary beams with different propagation constants p have the same (critical) power P,,b = P,, where Pnrn/3=

*

JJ

Fimp dx dy =

J

F;,@(r) r dr.

(3.9)

Therefore, aP,,pla/3 = 0. This fact causes instability of stationary structures, because for a small deviation of power from the critical value the beam spreads ( P < P,) or collapses ( P >Pc). For the stability of the fundamental soliton, according to the Vakhitov-Kolokolov criterion (Vakhitov and Kolokolov [ 19731; see also its generalization proposed by Mitchell and Snyder [ 1993]), the inequality dPld/3 > 0 has to be satisfied. This condition is valid for a medium with saturable nonlinearity (2.2) and D = 2. The dependence of spatial soliton power on the propagation constant for such a medium is shown in fig. 3e (above) for n,m = 0, and 1. The positive sign of the derivative dP/d/3 > 0 leads to the stability of spatial solitons with respect to small perturbations, for n,m = 0. Therefore, for transversely 2D spatial bright solitons, the model of saturable nonlinearity of the refraction index is a basic one. A review of the first theoretical and experimental studies of the self-trapping in a medium with saturable nonlinearity was given by Marburger [ 19751. Recent experiments and numerical simulations of patterns for astigmatic laser-beam propagation through a sodium vapor cell demonstrate their high sensitivity to aberrations (Grantham, Gibbs, Khitrova, Valley and Xu Jiajin [ 19911). The distributions of the field with the invariable transverse profile and integer azimuthal index (topological charge) were described by Fetter El9661 and Vlasov, Gaponov, Eremina and Piskunova [1978]. Others analyzed such distributions in the Kerr medium and a medium with saturable nonlinearity (Kruglov and Vlasov [ 19851, Kruglov, Volkov, Vlasov and Drits [ 19871). Higher-order distributions (with n # 0 andor m # 0) were found to be unstable (Kolokolov and Sukov [ 19751, Soto-Crespo, Heatley, Wright

28

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I, 5 3

and Akhmediev [ 19913, Vyssotina, Nesterov, Rosanov and Smirnov [ 19961). This section examines the work of Vyssotina, Rosanov and Smirnov [ 19941 and Rosanov, Smirnov and Vyssotina [ 19941. The paraxial eq. (2.3) for the dimensionless quantities takes the form (3.10)

where A2 is the transverse Laplacian:

82 + _1 _a A2 = dr2 r d r

+--.1 82

r2dq

The dimensionless propagation constant

p

changes continuously in the interval

0
Interactions of two solitons are considered for their initial field distribution in the form

Here, En,p are the stationary field patterns of the form (3.7), A q is the difference in phases of the two solitons, 2d is their initial transverse distance, and a is a quantity determining the angle between the first soliton and the axis z (coincident with the axis of the second soliton). Every soliton induces an inhomogeneity of the refraction index in the vicinity of the other soliton that results in the refraction and deformation of soliton beams. Naturally, when the solitons have a large initial distance and the angle between their axes is chosen so that this distance increases with z, they will interact little. If this angle exceeds the value (wlA)&, where w is a characteristic width of the solitons, their interaction will be weak. Of more interest are the cases, where the solitons approach each other to sufficiently small distance. We will present the intensity distributions in the transverse plane for different z and discuss the transverse motion of solitons. We first consider the interaction of two solitons that are initially cylindrically symmetrical (ml=m2 =0) and propagate parallel to the axis z (a=0, without interaction the solitons are transversely motionless). Figure 9 shows the interaction of two identical fundamental solitons. At a large distance the solitons attract each other. Then for z=25 the solitons begin to merge. Later they separate, but by a smaller distance. The amplitude of oscillations of the distance between the soliton intensity maxima tends to decrease (fig. 10). Finally, the

1,

P 31

29

SPATIAL SOLITONS

Fig. 9 . Intensity distribution I(x,y;z ) for initially coparaliel propagation of two solitons: a =O, in1 =m2=0,/?I =/32=0.5, Acp=O, nl = n 2 = 0 , d = 5 . 9 .

c

DX -

0

60

120

z

Fig. 10. Distance 2DX between intensity maxima of two solitons versus z for the parameters as for fig. 9 ; d = D X ( z = O ) .

solitons merge, forming a new structure with a single intensity maximum. This transverse structure oscillates in the z-direction (stretching alternately in the xand y-directions), the near-steady-state intensity distribution corresponding to the soliton with p M 0.6. The oscillations are slowly damped out with distance z. Solitons with a radial index n > O have a much larger value of power (see fig. 3e, above) and are more unstable than with m > 0 with respect to relatively strong perturbations. Their interaction leads to each soliton breaking up into a

30

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

0

20

40

[I,

03

z

Fig. 1 1. Trajectories of the soliton intensity maxima XC 1, XC2 in the plane x , z and dependence of theirpowerPl,2 onz: a=O,rnl = r n z = O , P l =P~=O.5,Arp=n/2;d=4.8(solidline)and5.9(dashed line).

number of fragments with bell-shaped intensity profiles. These fragments interact and tend to form separate fundamental solitons. Because solitons with radial indices n > 0 are unstable in practice, we will later consider only the solitons with n = 0 and will drop the index n. The dynamics of the transverse movement of the soliton depends on the difference of their phases A p . For A p = q the solitons push off. For sin A p # 0 their transverse movement is accompanied by the exchange of power. The simulations of the propagation of solitons with A p = in are shown in fig. 11. An effective power exchange arises also in the case A p = 0 if p1 # 8 2 . The collision of two solitons ( a= 0.2) with equal individual propagation constants p1=/32 = 0.5 is depicted in fig. 12. Initially the first (left) soliton moves in the plane ( x , y ) along the x-axis toward the second (right) motionless soliton. For z = 10 an intense transfer of power from the first soliton to the second begins. The distance between the soliton intensity maxima is minimal for z = 12. The first soliton reflects from the second, the latter acquiring velocity along the xaxis; therefore, some analogue of the conservation of linear momentum exists. In the simulations just described, the maximum soliton intensity far exceeds the saturation intensity. In the case of a weaker saturation the first soliton transfers its power to the second almost completely. The dynamics presented of collisions of transversely 2D solitons differ radically from the case of transversely 1D solitons in a Kerr medium, where they regain their profiles after the collision (see 9 3.1). Let us consider the interaction of two solitons when one or both are vortices of the first order. The distributions of intensity for two initial solitons with mi = 0, m2 = 1 are shown in fig. 13. At first, the solitons are almost motionless, but

1,

P 31

SPATIAL SOLITONS

31

Fig. 12. Intensity distribution I ( x , y ; z ) for collision of two solitons: a=0.2, ml = m 2 =0, /31 =/32=0.5, Aw=O, d z 5 . 9 .

Fig. 13. Intensity distributions for a=O, ml =0, m2= 1, 81 =/32=0.5, Arp=O, d = 5 . 9 .

their interaction results in the breaking of the second soliton (with index m2 = 1) into two similar fragments, each of which is close to the soliton with m=O and /3=0.55. One of the fragments (“new soliton”) moves away along the xaxis, and the second, coupled with the first (“old”) soliton (mi =O), begins to move in the opposite direction. When two interacting solitons have topological charges with different signs (ml = 1, m2 =-1; see fig. 14), their breakup occurs at a much larger distance. Every soliton decays into two fragments (the new solitons). The fragments of the first soliton with ml = 1 rotate around their center clockwise, whereas the fragments of the second, with m2 =-1,

32

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I,

5

3

rotate anticlockwise. Therefore, some analogue of conservation of the angular momentum also exists. Later the two new solitons with a lesser value of y move away, whereas the pair of new solitons with a larger coordinate y forms a unified structure. As a result of the interaction, two solitons with equal topological indices (ml=m2 =-I) break up similarly (fig. 15), but both pairs of fragments rotate anticlockwise. Of particular interest are the dynamics of a powerful soliton that forms when laser beams propagate in the nonlinear medium. The initial stage of decay of high-power beams was considered in 5 2.2. Numerical simulations of

1,

P 31

SPATIAL SOLITONS

33

powerful Gaussian beam propagation in the medium with saturable self-focusing nonlinearity were presented by Rosanov, Smirnov and Vyssotina [ 19941. The beam power P=30Pc, where P, is the critical power of self-focusing in a Kerr medium, and the maximum intensity of the initial beam I , M 4Zs. In the absence of initial perturbations the beam breaks up into rings whose structure oscillates with z. When the angular modulation is introduced, the beam filamentation occurs. The beam breaks up into spots that exchange the power with a central spot with increasing z. The total power of the spots decreases because of largeangle radiation scattering. As a result, a powerful soliton forms with m=O. The researchers also demonstrated the breakup of the second-order Gaussian beam, which contains two dislocations of its wavefront, and different cases of an interaction of powerful beams in the medium with saturable nonlinearity. Analytic perturbation theory of weak interactions of 2D fundamental solitons and simulations of their strong interaction with changes of solitons’ number are given by Vyssotina, Nesterov, Rosanov and Smirnov [ 19961. Equation (3.10) also describes the propagation of the radiation pulses in a planar waveguide with saturable nonlinearity and anomalous dispersion. Therefore, all results presented here are valid for such a scheme, including those for optical (now spatiotemporal) vortices. 3.3. SPATIOTEMPORAL SOLITONS

If a medium is characterized by self-focusing Kerr nonlinearity and, simultaneously, by anomalous dispersion (as for temporal solitons in fibers), an initial optical pulse can collapse both in time and space (Silberberg [1990]). In this case the time-dependent paraxial equation is (3.12) where ug is the group velocity of light, D = -d2k/dw2 ( D > 0 for the anomalous dispersion). In the moving system of coordinates, t = ( t -zlug)(klD)1’2, and eq. (2.3) takes the form

aE 2ik-

az

6E + A3E + k2-E EO

= 0,

(3.13)

where A3 = d2/ax2+ d2/dy2 + d 2 / d t 2 . For the Kerr nonlinearity, eq. (3.13) describes “weak collapse” (radiation nonlinear focusing not in a point, but in a line) (Talanov and Vlasov [1989]). Due to the nonlinearity saturation, a stable

34

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I,

5

4

spatiotemporal soliton may be formed. Such a “light bullet” has a field structure of the form (Silberberg [ 19901)

An interesting feature of such 3D solitons is the two-valued dependence of their power P on the propagation constant (fig. 3c, above). The branch with the negative slope (8Plap < 0) was found to be unstable (Vakhitov and Kolokolov [ 19731). Therefore, stable solitons correspond to the branch with a larger propagation constant. In the medium with saturable nonlinearity an initial light pulseheam with high energy transforms into a train of light bullets (Akhmediev and Soto-Crespo [ 19931). For the characteristics of light bullets in a medium with more complex nonlinearity (resulting in multivalued dependence of a propagation constant on energy), see the review by Enns, Edmundson, Rangnekar and Kaplan [ 19921 and recent papers by Enns and Edmundson [ 19931 and Edmundson and Enns [ 19931.

(i 4. Switching Waves and Spatial Hysteresis

The second part of this chapter ($4) is devoted to the new types of transverse patterns that differ from the filaments. In this case small-scale selffocusing is absent. These patterns arise in wide-aperture systems with feedback where bistable phenomena take place. Their excitation is hard. Actually, if homogeneous (or smooth) field distributions are stable with respect to small perturbations, a sufficiently large initial perturbation is needed to generate a new pattern. Nonlinear schemes with feedback are characterized by the phenomenon of bistability or multistability. The basic spatiotemporal structures are switching waves that determine the kinetics of switching in spatially distributed nonlinear systems. 4.1. DIFFUSIVE AND DIFFRACTIVE SWITCHING WAVES

The switching wave is an asymptotic notion that corresponds to the excitation of a bistable system with infinite aperture by external radiation in the form of a plane wave. There is no transverse inhomogeneity in the system. One of the two stable transversely uniform field distributions is possible for the intensity of external radiation in the region of bistability (fig. 16a). At the beginning, for the fixed value of intensity of cw holding radiation in this region, all the points of the

1,

I 41

SWITCHING WAVES AND SPATIAL HYSTERESIS

35

I

0

-x

Fig. 16. (a) Transfer function of point-wise bistable system and (b) dependence of velocity of left (c) and right (d) switching waves on intensity of the incident radiation.

infinite transverse section are in the state corresponding to the lower branch of the transfer function (fig. 16a). Let us transform the state of the system on one half of the aperture to the state of the upper branch by a pulse of external radiation, while the lower state is preserved on another half of the aperture. After this operation, when the pulse of additional radiation is moved off, the front between the states of the lower and upper branches moves in the transverse directions x,y with the velocity u = u,, uy (figs. 16bd). The field and system characteristics f, depend on the combination of variables

f,= f , ( r l - ut) = f j ( x - u,t,y

- uyt).

(4.1)

This is a switching wave, or a progressive switching of the transverse section of wide-aperture bistable optical systems. Its front velocity u is determined by the intensity of holding radiation, Zi, and usually turns to 0 for some Maxwellian value Zi = 1 0 . The switching waves are conveniently illustrated by the example of a 1D scheme of increasing absorption bistability (Rosanov [1980, 19811. Let

36

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I, 4 4

, V'

J

1,

n

I

d

f It

V

2

Fig. 17. ID Scheme of increasing absorption bistability.

us consider a thin rod (fig. 17) of a medium whose absorption coefficient increases with the intensity of optical radiation Ii. This is a mechanism of optical

nonlinearity, whereas longitudinal feedback and transverse coupling are provided by the thermoconductivity of the medium. The theoretical description of the transverse patterns is given by the unsteady heat conduction equation averaged in the longitudinal direction z

Here, po is the medium-specific density, cv is the specific heat, A is the thermal conductivity, and the function F( T ) represents the heat balance. A more precise 3D treatment of such schemes was also realized (Rosanov, Fedorov and Shashkin [1991]. The Maxwellian value of intensity 10is given analytically by a simple condition (Rosanov [1980, 19811. To find the dependence u(li), numerical calculations are usually needed. Similar switching waves exist in a wide-aperture nonlinear interferometer excited by external radiation in the form of a plane wave (Rosanov, Semenov and Khodova [1982, 19831); they also can be treated in the frames of eq. (2.27) (Rosanov [1991]). Some of their specific features are important here. For the interferometers a new, diffractive (additional to the diffusive) mechanism of transverse coupling arises. If the characteristic diffractive length [width of the Fresnel zone (hI)'/2]exceeds the length of diffusion [(DT,)~'~,with D the coefficient of diffusion and ,z the relaxation time of the medium], then the intensity profile of a switching wave contains diffractive oscillations decaying with distance from the wavefront. For the case of oblique incidence of the external radiation, the symmetry of the directions x and -x is broken, and an

1 , s 41

SWITCHING WAVES AND SPATIAL HYSTERESIS

31

additional component arises in the switching wave velocity proportional to the (small) angle of incidence 8. For some conditions the velocity of the switching wave does not turn to zero in the whole interval of bistability (Rosanov, Semenov and Khodova [1982, 19831). The number of types of switching waves increases in multistable systems (Grigor’yants, Golik, Rzhanov, Elinson and Balkarei [ 19841, Grigor’yants, Golik, Rzhanov, Balkarei and Elinson [1987]). The notion of switching waves may be generalized to describe the waves of the progressive spatial switching between different (not necessarily stationary) states, including the waves of modulation and the waves of dynamic chaos (Rosanov, Fedorov and Khodova [1988], Rosanov and Khodova [1989]). Concerning the problem of spatial switching (§4.2), it would be reasonable to introduce the notion of wave switching between stable and unstable states. If two switching waves are excited in the system, they propagate independently while their fronts are distant from each other. When the fronts come together up to the distance of about the width of the switching wavefront, their interaction begins, the result of which depends on the dominant mechanism of transverse coupling (diffusive or diffractive). For the diffusive mechanism the colliding waves are mutually annihilated, and the entire system switches into the stable transversely homogeneous state. For diffractive switching waves the result of the collision can be different, see $5.1. 4.2. SPATIAL SWITCHING AND SPATIAL HYSTERESIS

The switching waves progressively switch different parts of the wide-aperture system from one state to the other. Their existence results in fundamental changes of kinetics of switching when compared with the well-known hysteresis in pointwise (lumped) systems. This is best illustrated by the following two examples. 1. We consider spatial switching of a transversely homogeneous system that in point-wise (or plane-wave) description is bistable within some interval of holding intensity I;: /min

< Ii < I,,,.

(4.3)

In other words, two transversely homogeneous states are stable with respect to small perturbations. The problem consists of determining what (large) perturbations will switch the system from one stable state to the other. To begin with, we restrict the possible shape of the perturbation. Let the initial conditions correspond to a (wide) area switched on the upper branch while the

38

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

Rc r

[I,

54

R

Fig. 18. Evolution of large perturbations over the background of the metastable state (the domain of overcritical parameters is dashed; 10 < I , <

remaining part of the system stays in the lower-branch state. Then, because of the propagation of the switching waves, a continuously increasing part of the system switches onto the upper-branch state if the following condition is satisfied I0

< Ij
(4.4)

Therefore, the lower state is metastable in this region: wide initial perturbations are supercritical; that is, they switch the system, and small perturbations dissolve (Rosanov [ 19831). If the intensity I, is within the interval

then the initially wide perturbation will shrink with time. The result depends on the type of the transverse coupling (diffusive or diffiactive). For the diffusive coupling the perturbation collapses with time, and the entire system switches off and turns into the lower-branch state. In this case the system is also stable under small perturbations. It is not easy to solve the problem of spatial switching in the case of an arbitrary shape of perturbation; therefore we need to simplify it. Let us consider the scheme of the increasing absorption bistability (diffusive coupling). We will characterize the temperature perturbation against the background of a

39

SWITCHING WAVES AND SPATIAL HYSTERESIS

1 , s 41 T

b

a

I

I11 1

X

Fig. 19. Scheme of determination of transverse intensity profiles for two cases of the input radiation beam (a, b).

homogeneous stable state by only two parameters: its width R and height T(m) (maximum overheating). The phase plane of these parameters is subdivided into the regions of perturbations, which are either undercritical (dissolving) or supercritical (resulting in switching), as shown in fig. 18 for intensity within the range (4.4) (Rosanov and Khodova [ 1986b1). Figure 18 illustrates the trajectories of temporal changes of these parameters (the curves with arrows that show the direction of time increase). The areas of undercritical and supercritical parameters are divided by a separatrix of the point corresponding to the critical nucleus (C). Near the separatrix we see nonmonotonic temporal changes of the perturbation widths (curves 5,7,8) and height (curves 6,8). 2. Let us consider spatial hysteresis for a wide incident radiation beam with the characteristic width wb, for instance, a Gaussian beam, I,(x) = I,exp(-x2/wi). The width wb substantially exceeds the width of the switching waves’ front. The problem is to find possible steady-state profiles of medium temperature or output intensity (spatial bistability) and to describe spatiotemporal switching between these profiles with slow variation of the intensity I,. When the width of the front of switching waves is negligible compared with the radiation beam width wb, we achieve the scheme of determination of the stationary temperature profile depicted in fig. 19. Quadrant I presents an S-like dependence of the temperature T = 0 on the radiation intensity I;

40

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I,

5

4

for the transversely homogeneous distributions, which coincides with fig. 16a. A bell-like profile of intensity I , ( x ) is shown in quadrant IV. With these two dependencies and the known value of l o , we find the profiles for steady-state temperature T ( x ) (quadrant 11). If the intensity I , = I,(O) maximal over the beam cross-section is lower than 10,that is I , < 10,only one temperature profile T(x) is completely determined by the lower branch of the hysteresis curve. The solution at I , >Imaxwill also be single. In this case, however, the temperature profile is not smooth, but combined. It consists of the peripheral part determined by the lower branch ( I , ( x )< l o ) , with sharp switching between the branches in the vicinity of the Maxwell value of intensity I , FS 1 0 , see quadrant 11, b. Bistability of the temperature profile exists only in the interval of the intensities

Under this condition there are two possible temperature profiles: the smooth profile 1 and the combined profile 2, with switching between the branches (quadrant 11, a) for fixed parameters of the beam with the bell-like profile. It is evident that bistability of the temperature profile is accompanied by bistability of the profile of intensity of transmitted radiation. Now consider the temperature profile variation in a rod with slow temporal variation of maximal intensity of incident radiation: I , =I,(t). We assume that coordinate dependence of intensity of the radiation wide beam is fixed. We can use the notion of switching waves with slowly varying front propagation velocity determined by the local radiation intensity in the beam at the place of the front location. With a slow temporal increase of intensity I , from small values up to intensity of the lower branch edge I,,,, the temperature profile T ( x , t ) at every time moment will be smooth (type 1 in fig. 19), corresponding to the lower branch of the hysteresis curve. At the moment when I , exceeds the value I,=, a narrow and sharp local perturbation of temperature appears in the center of the beam, which, even if I , stabilizes, will gradually widen in the form of two divergent (stable) switching waves. The velocity of propagation of these temperature waves for wide beams is close to the velocity u=u(If,) determined previously for external radiation in the form of a plane wave (see fig. 16b), where If,is the local intensity of the incident radiation in the vicinity of the wavefront. Therefore, for the bell-like beams the propagation of the front will decelerate, because of the decrease of the local radiation intensity If,. Eventually the front will stop at the point where the local intensity is I f , = 10.Thus, the hysteresis transition

1, I 51

DIFFRACTWE AUTOSOLITONS IN NONLINEAR INTERFEROMETERS

41

takes place not simultaneously over the entire beam cross-section, but only in its narrow zone, the propagating front of the switching wave. Accordingly, the duration of hysteresis transition (switching on) is determined by the time of the transverse propagation of the switching wave. If, after that, the maximal value of intensity starts to decrease (switching up), the kinetics of variation of the temperature profile will be as follows: At first, the temperature profile will remain combined, with sharp spatial switching in the domain If,= I 0 (of the type 2 in fig. 19). With the decrease of I,, however, the central domain of the beam switched to the upper state will gradually narrow, and at I , = I 0 this central local perturbation will disappear entirely. Therefore, within the intensity range (4.6) bistability and hysteresis of the temperature profiles and of transmitted radiation intensity will take place. One of the two profiles, which is smooth at I , > l o , is metastable; the other, combined, with the central part switched into the upper state, is the stable profile. Then, in view of the results just presented, the probability of the metastable profile’s fluctuational switching into the stable state is negligible outside a close vicinity of the branch edge ( I , = Imax).Therefore metastability here is almost indistinguishable from stability. If the intensity profile of the incident radiation has several spaced spatial oscillations, at each such oscillation individual spatial hysteresis can be realized. In this case we obtain a multichannel memory on the basis of a single (but wideaperture) bistable element (Rosanov, Semenov and Khodova [ 1983]), which can be of considerable practical interest. This scheme of spatial hysteresis is valid for the difhsive type of transverse coupling. The same kind of kinetics of switching-on is valid for the diffractive mechanism of coupling. In this case the switching-up process is seriously affected by the existence of new types of particle-like structures, the diffractive autosolitons.

9

5. Diffractive Autosolitons in Nonlinear Interferometers

5.1. SINGLE AND COUPLED AUTOSOLITONS

In contrast to the case of diffusive transverse coupling, in diffractive transverse coupling the switching wavefront includes oscillations of the field. We can show that this causes essential changes in the nature of interaction of switching waves, and brings about a new class of particle-like spatial structures, diffractive autosolitons in the wide-aperture nonlinear interferometers (Rosanov and Khodova [1988, 19901). We start our discussion with the transversely ID scheme.

42

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I,

5

5

I a

1.2

0.8 cvv

Fig. 20. Single bright (a, b) and dark (c) diffractive autosolitons

Let two switching waves be excited in the interferometer, and the spacing of their fronts be considerably greater than the front width. Then the interaction of these waves is weak, and their fronts, for example, approach each other with a velocity close to the double velocity of the individual switching wave 2u. With the fronts coming together, however, diffractive oscillations of the field near the wavefront become important. Thus, in addition to transversely homogeneous external radiation, the overlapping wing of the right front of the switching wave with field oscillations affects the left front. Therefore, the left front may stop at such oscillations of the total field if they are not too weak. The right front will stop for the same reason. Thus in the nonlinear interferometer the coupled state of the switching waves can occur, which we call the “diffractive autosoliton”. Note that these diffractive autosolitons differ significantly from more familiar “diffusive autosolitons” (Kerner and Osipov [1991]), which can be formed in interferometers with two competing nonlinearities (Balkarei, Grigor’yants and Rzhanov [ 19871, Grigor’yants, Rzhanov, Balkarei and Elinson [1987], Rzhanov, Grigor’yants, Balkarei and Elinson [ 19871, Balkarei, Grigor’yants, Rzhanov and Elinson [ 19881, Rzhanov, Richardson, Hagberg and Moloney [ 19931). Figure 20 shows different types of single autosolitons in an interferometer with off-resonance nonlinearity; the incidence of external radiation is normal (0 = 0). “Positive”, or “bright”, autosolitons of different widths (with the intensity in the center being above the background) and “negative”, or “dark”, solitons (with the reverse relation of intensities) are shown. The regions of existence of these autosolitons differ: the autosoliton with the greater width (“excited” state of autosoliton) exists in a narrower range of intensities Zi than the narrow autosoliton (in the “ground” state).

1,

P 51

43

DIFFRACTIVE AUTOSOLITONS IN NONLINEAR INTERFEROMETERS

k c

wo

Fig. 21. Scheme of determination of parameters of single autosolitons (b) and their formation (a, I ,< I 0 and c, I ,z l o ) .

The switching waves with low velocities are more easily stopped by the inhomogeneities. This corresponds to the case of the intensity of the incident wave being close to the Maxwell value ( I ,“10). Therefore, autosolitons exist in a narrower range of intensities Ii than the region of bistability, but at a range that includes the Maxwell value Zi = l o . At Ii + l o only a finite number of oscillations is supercritical (significantly large to stop a switching wave), whereas the remainder of oscillations are subcritical. Therefore, at fixed parameters of the nonlinear interferometer and at a fixed intensity of incident radiation Ii # l o it , is natural to expect the existence of a finite number of autosolitons specified by different widths was.When I , approaches l o , more and more oscillations become supercritical, and the number of autosolitons with different widths increases. This is illustrated by fig. 2 1, obtained from a simple analytical consideration (Rosanov [ 19911). Autosolitons can be formed by a collision of two switching waves or by a hard excitation by an initial perturbation of the field additional to the holding plane wave. Since transversely homogeneous states are stable with respect to small perturbations, the local perturbations with small amplitudes up or the widths wp will dissolve with time. At large values of up and wp (and with due relation of intensities Ii and l o ) , the perturbations can transform into two switching waves running apart (see fig. 21c). With time these waves will convert the entire nonlinear interferometer into a transversely homogeneous state (different from the orginal one). Therefore, autosolitons will be excited only in some intermediate region of the perturbation parameters up and wp.

44

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I,

8

5

As the autosolitons are stationary, the relaxation time does not influence their characteristics, but it can affect the stability and kinetics of autosoliton formation. The calculations confirm stability of autosolitons being in the wide range of the ratio z,~/z values (where Z,I is the relaxation time and t is the time of interferometer roundtrip by light), including the cases of fast nonlinearity z,,~ << z and comparable times t,,~ M z. In the case of oblique incidence of the plane wave on the interferometer we obtain autosoliton drift due to geometric drift of rays in the interferometer. The transverse-drift velocity of the autosoliton is proportional to the angle of incidence 8, o = c 8 (02<< 1). This velocity decreases in the presence of spatial filtration of radiation or inertia of response of the medium. Diffractive autosolitons exist also if stationary transversely homogeneous states are unstable (Rosanov, Fedorov and Khodova [ 19881). Modulational instability of the states of the upper branch does not influence the features of the fundamental autosoliton, because there is only a small region in which its amplitude is close to this branch value. Stationary localized stable structures exist even if both the upper and lower branches correspond to unstable states; in this case the autosoliton background is spatially modulated. Such autosolitons were found on the basis of the parabolic equation (2.27) (Fedorov, Khodova and Rosanov [ 19921). The same localized structures were later described by the approximated Swift-Hohenberg equation (Tlidi, Mandel and Lefever [ 19941). The equivalence of these approaches was demonstrated by Scroggie, McDonald, Firth, Tlidi, Lefever and Lugiato [ 19941. Diffractive autosolitons were found experimentally for an interferometer with liquid-crystal medium and spatial filter (Rakhmanov and Shmalhausen [ 19931). 5.2. MECHANICS OF AUTOSOLITONS

In the case of interferometer excitation by a radiation beam, the autosolitons retain their function and basic properties if the characteristic beam width exceeds by a considerable amount the autosoliton width. Here we have the mechanics of an autosoliton in a smoothly inhomogeneous external field, which underlines its particle-like properties. The variant of oblique incidence considered earlier is a special case of inhomogeneity of external radiation phase. The equation of motion of a soliton in a smoothly inhomogeneous field has the form io

=

-vu,

where U=-(a@i+bI,). The coefficients a and b can be found from the analysis of simpler transversely homogeneous distributions. Thus, since the

1,

5 51

DlFFRACTIVE AUTOSOLITONS IN NONLINEAR INTERFEROMETERS

45

phase gradient V@, differs from zero at oblique incidence of a plane wave on the interferometer, for the case of fast nonlinearity a = uo/ko. The term with the coefficient b represents nonzero velocity of the soliton caused by the difference between its left and right front velocities, which is due to the inhomogeneity of the intensity .I;. It can be expressed through the velocity of a single switching wave, u, and the autosoliton characteristic width w,: b- w,duldl;. In this expression the coefficient b depends on the intensity in the place of autosoliton location: b = b(l;(ro)). However, we will assume that in the considered domain of autosoliton motion the intensity of external radiation varies within narrow limits. Then, since in eq. (5.1) b is a factor at the small gradient, one can neglect its variation, assuming b to be constant. In the case of absorptive nonlinearity, b > 0 for a positive autosoliton, and b < 0 for a negative one. Then positive autosolitons propagate in the direction of larger intensities, and negative ones propagate to the domain of smaller intensities. Hence it follows that, with time, a single positive autosoliton will move to the centre of a beam if the nonlinear interferometer is excited by a wide beam with a bell-like intensity profile. One can interpret eq. (5.1) mechanically as the equation for 2D motion (in the xy-plane) of a lightweight particle in a viscous medium under the action of the force F = V U , that is, as the limit of the Newtonian equation

when the particle mass m + 0. If the phase of incident radiation @; is specified unambiguously, the “potential” U(x,y)is determined unambiguously as well. An autosoliton then will move along the lines of the steepest descent of this surface. With time it will approach either the minimum of the potential U (settling of a stationary regime) or the boundary of existence of autosolitons of this type where eq. (5.1) is violated (see below). The behavior of the autosoliton near the boundary of its existence depends on its type and on the relation between the local intensity Ii and the Maxwell value of intensity 10.A positive autosoliton is transformed into switching waves running apart at I ; > I o . At the boundaries with I ;
46

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I,

5

5

a dislocation Ii = 0, and if we follow a closed contour around the dislocation, the phase changes by 2n.In these conditions the potential U ( x , y ) becomes a multivalued function of coordinates, whereas the phase gradient V@i and force F are unambiguous. The trajectories of the autosoliton in the presence of wave front dislocation may be spirals winding along a stable circle (Rosanov [1992a,c]). In preceding sections we discussed the interferometer with an infinite aperture. Near the mirror edges diffractive oscillations of the field prevent the autosoliton from leaving the central region, stopping it near the mirror edge. The existence of the autosoliton greatly changes the kinetics of spatial switching of the bistable interferometer (see 9 4.2). This occurs because, although autosolitons are analogous to the “critical nuclei”, to a certain degree, they are nevertheless stable and have an excitation threshold. They also change the kinetics of spatial hysteresis (its switching-down stage). If the intensity of a (Gaussian) beam incident on the nonlinear interferometer is great enough, so that the width of the region switched to the upper state considerably exceeds the width of the switching wavefront, the fronts of spatial switching serving as the boundaries of this region will be located far apart. At a slow temporary decrease of maximum beam intensity I,, these fronts will draw together and form a positive autosoliton. The autosoliton will still be there during any further decrease of I , up to the boundary of existence of the autosolitons I , . Therefore switching down occurs not at I , = I0 as it did for the case of difhsive coupling, but at I,,, = I1 < Io. The possibility of artificially exciting different types of autosolitons leads to many new variants of spatial hysteresis in the wide-aperture nonlinear interferometer with diffractive transverse coupling. 5.3. INTERACTION OF AUTOSOLITONS

Let us return to the nonlinear interferometer excited by a plane wave (the case of normal incidence, O = O ) . If we form two autosolitons initially separated in the transverse direction by the distance do, they will interact because of their wings overlapping. At greater separation do the field oscillations on the wings are weak (subcritical), and we can assume that each of the autosolitons is under the influence of the mean gradient of radiation intensity corresponding to the wing of the other autosoliton. Therefore with self-focusing nonlinearity two distant autosolitons attract each other. As they draw together, however, the oscillations become essential and, depending on the value do, attraction can be replaced by repulsion. Correspondingly, a limited set of equilibrium distances d,, occurs between these two autosolitons.

I , § 51

DIFFRACTIVE AUTOSOLITONS IN NONLINEAR INTERFEROMETERS

47

Establishing the variant of the distance da, depends on the initial distance do between the autosolitons (the dependence is similar to that in fig. 21a). At sufficiently small initial distances do two autosolitons merge. The distance d,, also depends on the type of interacting autosolitons, each of which can be in the “ground” or “excited” state. With self-defocusing nonlinearity distant autosolitons repulse; however, there is also a set of equilibrium distances between them. Therefore autosolitons can form bound multiparticle structures. Moreover, we can construct an unlimited number of multiparticle autosoliton structures (for an unlimited aperture of the interferometer). Computer simulations of transversely 2D schemes confirm the stability of diffractive switching waves and autosolitons of different types (Rosanov, Fedorov and Khodova [ 19911). Single axially symmetrical autosolitons are shown in fig. 22a, and two- and three-particle structures in figs. 22b, c. The “mechanical” equations for such a set of several interacting autosolitons are a generalization of eq. (5.1):

Here, we approximate the interaction by two-body forces (Rosanov [ 1992a,c]). The form of two-body forces repeats the asymptotic behavior of the field of a single autosoliton on the periphery. Several effects are not described in the present simplest approximation of the weak interaction of autosolitons. For instance, the fact that asymmetrical autosoliton structures move transversely follows from analytical examination (Rosanov [1992b]), and is confirmed by the computer simulation shown in fig. 23 (Rosanov, Fedorov and Khodova [ 19931). Two asymmetrical autosolitons moving with the velocities U I and u2 will travel through each other almost unperturbed if the following inequality is satisfied: (Ul

-

If it is violated, the interaction of the autosolitons can result in their significant distortions, including the formation of their coupled states. For a more detailed description of features of the diffractive autosolitons see (Rosanov [ 1992a1). The field structures discussed earlier can be used for optical processing of information (see the Conclusion). Their particle-like quantum and mechanical properties are also of interest. If we consider an autosoliton as an atom, then such an atom has a restricted set of discrete stationary states, and “atomic” configurations form different “molecules”, including “polymers.”

48

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

Fig. 22a. 2D Intensity distribution for single autosoliton.

Fig. 22b. 2D Intensity distribution for two combined autosolitons

[I, 4 5

AUTOSOLITONS IN LASERS AND NONLINEAR WAVEGUIDES

1, § 61

49

Fig. 22c. 2D Intensity distribution for three combined autosolitons.

Fig. 23. Moving asymmetrical pair of autosolitons

0

6. Autosolitons in Lasers and Nonlinear Waveguides

Patterns in wide-aperture, or multitransverse-mode lasers have many features similar to those in passive nonlinear optical systems (Abraham and Firth [ 19901). Of special interest for many investigators were the optical vortices, described by Coulet, Gil and Rocca [1989], Abraham, Balle and Chen [1992], Arecchi, Giacomelli, Ramazza and Residori [ 19921, Brambilla, Lugiato, Pirovano and Prati [ 19921, Dangoisse, Hennenquin, Lepers, Louvergneux and Glorieux [ 19921, Brambilla, Cattaneo, Lugiato, Pirovano, Prati, Kent, Oppo, Coates, Weiss, Green,

50

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I,

56

D'Angelo and Tredicce [ 19941, Coates, Weiss, Green, D'Angelo, Tredicce, Brambilla, Cattaneo, Lugiato, Pirovano, Prati, Kent and Oppo [ 19941 and references therein. In this section we consider briefly only the patterns in wideaperture bistable lasers because of the prevalence of nontrivial phenomena in such systems. 6.1. SWITCHING WAVES AND AUTOSOLITONS IN BISTABLE LASERS

A laser with a hard excitation of generation, for example, a laser with an additional saturable absorber inside its resonator was one of the first schemes of optical bistability (e.g., Gibbs [1985]). In this section we consider switching waves and autosolitons in a similar wide-aperture laser. The initial equation is the paraxial one for an electric field envelope E averaged in the longitudinal direction (Suchkov [ 19661):

Here, t is dimensionless time; the 2D vector of the transverse coordinates is r l = (x,y); and the function a(/ E l 2 ) represents the combined saturable gain, absorption, and losses for a class A laser and has the form as shown in fig. 24. We assume a(0)= a0 < 0 (then the generationless mode I = 0 is stable with respect to small perturbations), and a' = aa/X(,=,; < 0, which results in stability of cw lasing with intensity I =I : . Therefore, the scheme parameters are chosen in the interval of bistability, where two modes with intensities I = 0 and I = I : are stable (see fig. 24). A general form of the envelope of an electric field for the switching waves, diffractive autosolitons, and their coupled structures is

E

= F ( r l - ut)exp(ivt) = F ( x - u,t,y - uyt) exp(ivt).

(6.2)

Here, u = (ux,uy) is the velocity of transverse motion, and Y is the frequency shift in the running frame of reference Q), moving with the velocity u with respect to the initial one (5: = x - u,t, u=y - u,,t). For the function F we have

(c,

The field phase can be shifted by an arbitrary constant value. Equation (6.3) is supplemented with the conditions at 5, r] -+f c a when, depending on whether the structure is a switching wave or an autosoliton, intensity approaches 0 or I,.

I, § 61

AUTOSOLITONS IN LASERS AND NONLINEAR WAVEGUIDES

51

Fig. 24. Dependence of the medium gain-losses difference on radiation intensity.

It is clear from physical considerations that the transverse velocity u can be arbitrary in some interval, on which the frequency shift Y = Y ( U ) and envelope profile F(E,q) are defined. This corresponds to oblique propagation of the radiation inside the cavity with respect to the cavity axis (Rosanov [1994b]). Computer simulations of the laser switching waves and autosolitons have been performed for transversely 1D and 2D schemes (Rosanov and Fedorov [1992], Fedorov, Khodova and Rosanov [ 19921, Rosanov, Fedorov, Fedorov and Khodova [1993]). Figure 25 shows the dynamics of the formation of two switching waves, their approach to each other, and the formation of a transversely moving autosoliton in the case of asymmetrical initial perturbation. Laser autosolitons exist in a gain interval which is narrower than the interval of bistability. With increasing gain they are transformed into pulsing localized structures. There are also 1D multi-autosoliton structures, both stationary and pulsing (Rosanov, Fedorov and Khodova [1996]). Contrary to the case of spatial solitons in passive media ( Q 3), 2D laser autosolitons are stable not only with in = 0, but also with wavefront screw dislocations with in = 2~1, rn = *2, . . . (Rosanov, Fedorov, Fedorov and Khodova [ 1995a1). Moreover, Rosanov, Fedorov, Fedorov and Khodova [1995b, 19961 have shown the existence of stable laser autosolitons with axially asymmetric intensity distribution rotating with constant angular velocity. The transverse velocity of the autosoliton varies in a laser, where the optical length S of the cavity andor pump or losses depend on the transverse

52

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I,

56

coordinates: S = S ( r l )= S(x,y), a. = ao(x,y). We assume that the deviations of a0 from their axial values are small and smooth on the scale of the autosoliton width. In addition the direction of the autosoliton propagation is supposed to be close to the cavity axis. The deformations of the envelope profile of the autosoliton then may be neglected, and the autosoliton can be characterized only by the coordinates of its “center of gravity” R = ( x , y ) . In this case the

S and

1, § 61

AUTOSOLITONS IN LASERS AND NONLINEAR WAVEGUIDES

53

simplified mechanical approach to the transverse motion of the autosoliton is valid (Rosanov [ 1994133):

R

=

-vu,

where U = - x S ( R ) + cao(R).This is a Newtonian equation for the 2D motion of a material point caused by the force with the potential U.The meaning of the two coefficients n and 5 is similar to that in the case of the interferometer (6 5.2); for example, H = S 0 / t 2 ,where SO is the axial value of the optical path, z is the round-trip time of the cavity, t=So/c, and c is the light velocity. Interaction of laser autosoliton with edge of cavity mirrors was studied by Rosanov, Fedorov, Fedorov and Khodova [ 1995al and Rosanov, Fedorov and Fedorov [1996]. The solution of eq. (6.4) is equivalent to the solution of the classical mechanical problem of the motion of a material point in a central field. Using the mechanical integrals of motion corresponding to the conservation of the angular momentum and energy, a general solution of eq. (6.4) can be obtained for an arbitrary dependence S(R). In a laser with a stable configuration of the cavity, an autosoliton oscillates near the cavity axis, and therefore the mirror edges count very little. Depending on the cavity configuration, the autosoliton trajectories are closed or open (Rosanov [1994b]). The laser autosolitons interaction has some features similar to those in the case of nonlinear interferometers,including the criterion of a weak interaction (6 5.3, see Rosanov, Fedorov, Fedorov and Khodova [1995a] and Rosanov, Fedorov and Fedorov [ 19961). 6.2. TEMPORAL ANALOGS IN NONLINEAR WAVEGUIDES

The propagation of pulses in a single-mode optical waveguide is described by the standard paraxial equation for the electric field envelope E in the frame of reference running with the group velocity ug, dE 2ik-

dz

6E + D-d2E + k2-E ax2 E~

= 0.

Here, k is the wavenumber; z is the longitudinal coordinate along the waveguide axis; D=-d2kldw2 (we take D>O, which corresponds to the region of anomalous dispersion); z = ( t -z/u,), EO is the unperturbed value of dielectric permittivity; and its deviation 6~= 6 ~+’i6d’ = 6 ~+, is a complex value including absorption (a) and gain (g) in the waveguide medium.

54

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

[I

Let us now make use of the spatiotemporal analogy. When both absorption and amplification are present in the medium, for example,

6.8 = 0,

6 ~ =” am +

a0

-

go

1 + I/Z,’ after redefining the variables, eq. (6.5) coincides with the transversely 1D variant of eq. (6.1) for a laser with saturable absorption (Rosanov [1994a]. The quantities am, a0 and go are proportional to coefficients of off-resonant and resonant absorption and amplification in a weak field, whereas I a and I , are saturation intensities of absorption and amplification, or gain. The absence of inertia of the medium nonlinearity assumed by the form of eq. (6.6) means that the pulse duration exceeds the relaxation times of the medium. In this case the results obtained for a laser ( 5 6.1) are also valid for a nonlinear waveguide. In particular, dependence of the steady-state intensity on the system parameters (e.g., on the gain coefficient go), can be two-valued. This resembles the phenomenon of optical bistability (Lugiato [1984], Gibbs 19851). In this case, however, the longitudinal feedback needed for bistability is somewhat peculiar, in that the longitudinal coupling for different parts of the pulse is provided by dispersion. Considering the analogy of the switching waves, we find that the propagating pulse of sufficient initial duration will widen or shrink, depending on the pump. In some interval of parameters it is possible to excite stable bright and dark pulses that are analogs of diffractive laser autosolitons. In the form of bright pulses they represent the islands of lasing, whereas dark pulses show the absence of it. Therefore, the level of maximum intensity of stationary pulses is not arbitrary here (in contrast to the usual solitons, $3); on the contrary, it is fixed and close to the value Is. These autosolitons are stable. A different picture forms in the case of pulses in a waveguide without gain and losses saturation (no bistability, see Grigoryan, Maimistov and Sklyarov [ 19881 and Vanin, Korytin, Sergeev, Anderson, Lisak and Vizquez [1994]). These are unstable because of amplification of perturbations before the pulses’ front. Similar structures occur in schemes of higher dimensionality. For example, in a continuous medium with the same characteristics as for the waveguide just considered (viz. two-level amplifying and absorbing media), it is possible, in contrast to the light bullets, to form 3D localized clots of light, with fixed levels of maximum intensity and sizes. ~

1 + I/I,

~

Conclusion We conclude by comparing the features of particle-like patterns (i.e., bright spatial solitons) in the three transversely uniform nonlinear optical systems

I1

CONCLUSION

55

described earlier: (i) spatial solitons in a transparent medium with a nonlinear index of refraction ($3); (ii) diffractive autosolitons in a nonlinear interferometer excited by external radiation ( $ 5 ) ; and (iii) autosolitons in a laser with hard excitation ($ 6.1). The first system has no feedback; the propagation of a plane wave is accompanied by its instability with respect to small perturbations: in other words, smallscale self-focusing or filamentation takes place here; the radiation frequency and the main direction of radiation propagation are arbitrary; and solitons have a continuous spectrum of characteristics such as maximum intensity. The second and the third systems have feedback, and for cw radiation bistability is possible; filamentation may be absent, and therefore the autosolitons’ excitation is hard, meaning that a sufficiently large initial perturbation is needed to form the autosoliton while initially small perturbations dissolve with time; and the maximum intensity of the autosoliton is close to the intensity of the transversely homogeneous upper state. For the second system, the frequency and transverse velocity of the autosoliton are determined by external, holding radiation; the spectrum of its characteristics is discrete; the mechanics of the transverse motion of the autosoliton is “Aristotelian” which means that “force” (i.e., optical inhomogeneities) determines the velocity. In the third, or laser, system there is no external radiation; the direction and velocity of the transverse motion of the autosoliton are arbitrary and determined by initial conditions; the frequency shift depends on the transverse velocity of the autosoliton; and the transverse motion is described, approximately, by Newtonian mechanics. Therefore, the laser,autosolitons occupy an intermediate position between the spatial solitons in systems (i) and (ii). Our examination of these systems was based on the classical treatment of light, although we admit that its quantum fluctuations could be of considerable significance. It is remarkable, however, that the quantum noise can be greatly reduced in special nonclassical, “squeezed” states of light (Teich and Saleh [1989]). As (Kolobov and Sokolov [1989]) showed it is possible to suppress the quantum noise for the light registration, not only in time but also in space. The applications of such quantum-noise-free light beams could significantly improve accuracy, sensitivity and information capacity in the fields of optical images, Fourier optics, holography, optical computing, and other fields (Sokolov [1992a,b]). In the context of pattern formation the transverse effects in squeezing were studied theoretically for several nonlinear optical systems (Belinskii and Rosanov [1992], Lugiato and Castelli [ 19921, Brambilla, Camesasca, Gatti and Lugiato [1993]). Figure 26 illustrates the scheme of light passing through a nonlinear interferometer. One can see here a considerable reduction of

56

TRANSVERSE PATTERNS IN WIDE-APERTURE NONLINEAR OPTICAL SYSTEMS

. 0

Fig. 26. Spectral density of photocurrent noise; its unit level corresponds to shot noise for coherent radiation detection.

photocurrent shot noise over a definite range of the dimensionless spatial (4’) and temporal (a’) frequencies of the noise. The features of optical spatial solitons outlined earlier indicate their potential for different applications. Thus, on the basis of spatial solitons several fast optical switches were proposed (Wright [ 19921). Contrary to the ordinary Schrodinger solitons, the stability, or absence of fluctuations of maximum intensity of temporal solitons in the single-mode optical waveguide with gain and nonlinear losses makes these solitons promising for information transmission. The threshold of autosoliton excitation results in the suppression of lowintensity noise, because radiation with an intensity lower than the threshold value

I1

REFERENCES

57

decays with propagation distance. These factors are important for information transmission and fast logic schemes. Of particular importance are the opportunities of using autosolitons in parallel optical processing and optical computing. The parameters obtained for bistable optical elements are sufficient for some applications in digital operations, primarily in specialized systems. The use of the spatial distributivity effects in wide-aperture bistable devices ensures the adaptability of architecture inherent in these devices and allows one to combine the advantages of the digital computing technique, such as its preciseness and reliability, with those of an analogous one, namely, the high rate of parallel operations. Such a digital-analogous mode of operation can be illustrated by the examples of a shift register (Rosanov and Fedorov [1990]) and a full adder (summator) (Rosanov 1992b, Rosanov, Fedorov and Khodova [1993]) on the basis of a wide-aperture bistable interferometer.

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Rosanov, N.N., and V.E. Semenov, 1980, Opt. Spectrosc. 48, 59. Rosanov, N.N., and YE. Semenov, 1982, Opt. Spectrosc. 52, 934. Rosanov, N.N., V.E. Semenov and G.V. Khodova, 1982, Sov. J. Quantum Electron. 12, 193. Rosanov, N.N., V.E. Semenov and G.V. Khodova, 1983, Sov. J. Quantum Electron. 13, 1534. Rosanov, N.N., and V.A. Smirnov, 1976, Sov. Phys.-JETP 43, 1075. Rosanov, N.N., and V.A. Smirnov, 1978, Sov. J. Quantum Electron. 8, 1429. Rosanov, N.N., and V.A. Smirnov, 1980, Sov. J. Quantum Electron. 10, 232. Rosanov, N.N., V.A. Smirnov and N.V. Vyssotina, 1994, Chaos, Solitons and Fractals 4, 1767. Rzhanov, Yu.A., A.V. Grigor’yants, Yu.1. Balkarei and M.I. Elinson, 1987, Sov. J. Quantum Electron. 20, 419. Rzhanov, Yu.A., H. Richardson, A.A. Hagberg and J.V. Moloney, 1993, Phys. Rev. A 47, 1480. Satsuma, J., and N. Yajima, 1974, Progr. Theor. Phys. Suppl. 55, 284. Scroggie, A.J., G.S. McDonald, W.J. Firth, M. Tlidi, R. Lefever and L.A. Lugiato, 1994, Chaos, Solitons and Fractals 4, 1323. Silberberg, Y., 1990, Opt. Lett. 15, 1282. Sokolov, I.V., 1992%Proc. SPIE 1840, 282. Sokolov, I.V., 1992b, Opt. Spectrosc. 73, 1158. Soto-Crespo, J.M., D.R. Heatley, E.M. Wright and N.N. Akhmediev, 1991, Phys. Rev. A44, 636. Stegeman, G.I., E.M. Wright, N. Finlayson, R. Zanoni and C.T. Seaton, 1988, J. Lightwave Technol. 6, 953. Suchkov, A.F., 1966, SOV.Phys.-JETP 22, 1026. Swartzlander, G.A., and C.T. Law, 1992, Phys. Rev. Lett. 69, 2503. Talanov, W., 1964, Izv. Vuzov, Radiophysics 7, 564. Talanov, VI., and S.N. Vlasov, 1989, Distributed Wave Collapse in the Nonlinear Schrodinger Equation, in: Nonlinear Waves; Dynamics and Evolution, eds A.V. Gaponov-Grekhov and M.I. Rabinovich (Nauka, Moscow) p. 218. Teich, M.C., and B.E.A. Saleh, 1989, Quantum Opt. 1, 153. Tlidi, M., P. Mandel and R. Lefever, 1994, Phys. Rev. Lett. 73, 640. Vakhitov, N.G., and A.A. Kolokolov, 1973, Izv. Vuzov, Radiophysics 16, 1020. Vanin, E.V., A.1. Korytin, A.M. Sergeev, D. Anderson, M. Lisak and L. Vbquez, 1994, Phys. Rev. A 49, 2806. Vlasov, S.N., 1984, Sov. J. Quantum Electron. 14, 1233. Vlasov, S.N., V.A. Gaponov, 1.V. Eremina and L.V. Piskunova, 1978, Izv. Vuzov, Radiophysics 21, 521. Vyssotina, N.V., L.A. Nesterov, N.N. Rosanov and V.A. Smirnov, 1996, Quantum Electronics, submitted. Vyssotina, N.V., N.N. Rosanov and V.A. Smirnov, 1984, Sov. Phys. Techn. Phys. Lett. 10, 1206. Vyssotina, N.V., N.N. Rosanov and YA. Smirnov, 1987, Sov. Phys. Techn. Phys. 32, 104. Vyssotina, N.V., N.N. Rosanov and VA. Smirnov, 1988, Opt. Spectrosc. 65, 222. Vyssotina, N.V., N.N. Rosanov and V.A. Smirnov, 1990, J. Opt. SOC.Am. B 7, 1281. Vyssotina, N.V., N.N. Rosanov and V.A. Smirnov, 1994, Opt. Spectrosc. 76(5). Wright, E.M., ed., 1992, Special Issue on All-Optical Switching Using Solitons, Optical & Quantum Electron. 24(11). Yur’ev, M.S., 1992, Proc. SPIE 1840, 228. Zakharov, YE., and A.P. Rubenchik, 1973, Sov. Phys.-JETP 65, 997. Zakharov, YE., and A.B. Shabat, 1972, Sov. Phys.-JETP 34, 62.

E. WOLF, PROGRESS IN OPTICS XXXV 0 1996 ELSEVIER SCIENCE B.V. ALL RIGHTS RESERVED

I1 OPTICAL SPECTROSCOPY OF SINGLE MOLECULES IN SOLIDS BY

M. ORRIT,J. BERNARD, R. BROWN AND B. LOUNIS Centre de Physique Moldculaire Optique et Hertzienne, u.a. 283 du C.N.R.S.. Uniuersitd Bordeaux I, 33405 Talence, France

61

CONTENTS

PAGE

5 1.

INTRODUCTION . . . . . . . . . . . . . . . . . . .

5 2. PRINCIPLESANDTHEORETICALBACKGROUND

63

. . . .

66

§ 3 . EXPERIMENTAL METHODS . . . . . . . . . . . . . .

83

5 4. REVIEWOFRESULTS . . . . . . . . . . . . . . . . .

95

5 5 . OUTLOOK . . . . . . . . . . . . . . . . . . . . . .

130

. . . . . . . . . . . . . . . . . . . .

138

§ 6.

CONCLUSION

ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . . .

138

NOTE ADDED IN PROOF . . . . . . . . . . . . . . . . . .

139

REFERENCES . . . . . . . . . . . . . . . . . . . . . . .

139

62

0

1. Introduction

The turn of the last century saw a steady accumulation of indirect proofs of the existence of atoms and molecules. The convenient but somewhat philosophical atomic hypothesis gradually gained credibility with the kinetic theory of gases, the study of thermodynamic fluctuations and of Brownian motion, the observation of discretely charged particles, the diffraction of X-rays by crystals and so on. Yet optical physics, quite a fashionable field at that time, had little to bring in the way of evidence for the atomic hypothesis. Early attempts, for example that by J. Perrin to directly see fluctuations of the numbers of fluorescing molecules in ultra-thin films (Perrin [ 1923]), failed because of the weakness of the signals and the absence of sensitive enough detectors. Optical spectroscopy contributed, of course, to the first steps toward quantum mechanics but always dealt with large populations of atoms or molecules. In the course of this century, atoms and molecules came to be detected in extremely small numbers or even individually in very sensitive experiments, using for example radioactivity or tenuous atomic and molecular beams. The imaging of atoms and molecules in the condensed phase was barely within reach of the best electronic microscopes, at the expense of heavy irradiation by high energy electrons (Spence [ 19811). All such experiments, though improved upon up to the present, were difficult and restricted to atoms and molecules in very special situations. The outlook changed dramatically in the early eighties, with the advent of the scanning electron tunneling microscope (STM; Binnig, Rohrer, Gerber and Weibel [1983]). In this apparatus, a fine metal tip is scanned within atomic distances from the surface of a conducting material. The tunneling current between tip and substrate is kept constant during rastering in the transverse directions and the tip’s altitude control signal provides an image of the topography and of other material properties. The STM breakthrough had two important consequences. First, it introduced a new attitude toward imaging at atomic scales, in which only a very small region is probed by a solid tip, in contrast to the classical methods where particles (e.g., photons, electrons) are directed at the sample. It was very soon realized that besides the tunneling current several other physical quantities 63

64

SINGLE-MOLECULE SPECTROSCOPY

[IL 0 1

like forces or optical waves could be used to probe surfaces. More than ten years after the first STM experiments, these kinds of near-field microscopies have now become widespread in materials science (Meyer, Howald, Overney, Heinzelmann, Frommer, Giintherodt, Wagner, Schier and Roth [ 19911, Griffith and Grigg [ 19931, Sand [1991]). Second, the STM lifted a psychological barrier by showing that the direct observation of atoms and molecules was well within reach of today’s techniques, even in condensed matter. Since then, several new methods have been devised to deal with atoms and molecules individually. The best known examples of the direct observation of single simple systems without any tip are found in atomic physics. In the seventies, single atoms were detected in tenuous atomic beams (Kimble, Dagenais and Mandel [ 19771). Later, radiofrequency or laser traps were used to study a single electron, atom, or ion for several days or even months (Neuhauser, Hohenstatt, Toschek and Dehmelt [1980], Bergquist, Hulet, Itano and Wineland [1986], Diedrich and Walther [1987], Neuhauser, Hohenstatt, Toschek and Dehmelt [ 19801). In these studies, the particle interacts with radiofrequency and electromagnetic fields only, which are able to cool and trap it. The same kind of experiment was extended recently to single molecules in liquid solutions at room temperature. Single molecules started to be detected a few years ago in small volumes of a flowing solution (Shera, Seitzinger, Davis, Keller and Soper [ 19901, Rigler, Widengren and Mets [1992]). The method was later improved by gating the detection of fluorescence photons (Wilkerson, Goodwin, Ambrose, Martin and Keller [1993]), and the emission spectrum and lifetime were used to identify molecules (Soper, Davis and Shera [1992]), including in the infra-red range (Soper, Mattingly and Vegunta [ 19931). Very recently, an inhomogeneous electric field was used to trap a single molecular ion (Eigen and Rigler [1994]). These methods, which allow searching large volumes of liquid for a few molecules of interest, have enormous potential in biological applications. The spectroscopic isolation of molecules which is the subject of the present chapter is another of these new methods for detecting single colored molecules embedded in a solid (Moerner and Basche [ 19931, Orrit, Bernard and Personov [1993], Moerner [1994]). Leaving the discussion of experimental details for 9 3, let us state briefly here how it is done. The method is closely akin to the magnetic resonance imaging well known in biology and medicine. A monochromatic wave addresses selectively those centers resonant with it. For magnetic resonance imaging, the resonance frequencies of the different spins are spread by an inhomogeneous static magnetic field. The incoming radio wave selects a large number of resonant spins in each volume element to form a measurable magnetic resonance signal. In optical experiments, the electronic resonance of every

11, § 11

INTRODUCTION

65

single localized state in a solid is shifted randomly by the particular molecular disorder around it. At low enough temperatures, the resonance frequencies may be considered as constant. The incoming laser wave will again excite selectively the resonant centers lying within the illuminated volume. This method therefore combines spatial selection by the shape of the exciting beam with spectral selection by its frequency. Because the same center can absorb and emit successively a large number of photons each of which can be detected individually, the optical experiment has a much higher sensitivity, down to a single center. Although this scheme has so far been implemented mainly with electronic states of molecules, it was already generalized to quantum dot structures in semiconductors (Brunner, Bockelmann, Abstreiter, Walther, Bohm, Trankle and Wiemann [ 19921, Birotheau, Izrael, Marzin, Azoulay, Thierry-Mieg and Ladan [1992]) and may eventually be extended to other localized excited states of solids in other ranges of the electromagnetic spectrum, provided the detection efficiency is sufficient. The emergence, within one decade or so, of several new methods dealing with individual atoms and molecules is no coincidence. The unprecedented progress of microelectronics towards ever smaller scales has led solid state physicists to contemplate devices with working parts on the scale of a few nanometers, the so-called molecular scale (Aviram and Ratner [ 19741, Aviram [ 19891). However, several features of mechanics and electronics at the molecular scale might prevent the application of macroscopic concepts to such devices. For instance, the electronic wavelength is no longer negligible compared with device size. New phenomena connected with the particle nature of electrons start to rule the workings of nano-electronic devices. For example, in the Coulomb blockade mechanism, a single electron charging a capacitor can block the conduction of other electrons, leading to single electron devices (Fulton and Dolan [ 19871, Devoret, Esteve, Grabert, Ingold, Pothier and Urbina [ 19901). Such concepts are clearly very close to those developed over the years by physical chemists to describe charge transfer between molecules (Delhaes and Yartsev [ 19931). Such considerations, together with the example of the wonderful success of life in processing and storing information at molecular scales, suggest that a possible way to realize nanoscopic devices could be by arranging different molecules. This dream of a molecular electronics calls for a better fundamental understanding of the structure and dynamics of matter at the molecular level. Observation of single molecules in condensed matter is, of course, an important step toward this goal. It gives information from the particular surroundings of the selected probe molecule, which eventually may be related to natural or artificial structures in the neighborhood. Moreover, as this information is free from the

66

SINGLE-MOLECULE SPECTROSCOPY

[II,

5

2

usual averages which obscure phenomena in real disordered samples, it will be easier to compare to theoretical models. Finally, the spectral selection of single molecules opens new perspectives for nanoscopic devices. Despite their tremendous importance for fundamental purposes, tip techniques may be limited eventually to applications in working devices because they can only access the surface of a sample, and because a single tip cannot access several spots simultaneously. Each tip must process information in a serial way, which does not take advantage of the smallness of a nanoscopic device. Optical addressing of single molecules can offer better outlooks on these two points. Use of a suitable superposition of frequencies can address simultaneously several points at different locations in a threedimensional sample. In a less speculative way, it is a safe bet that optical spectroscopy of single molecules has an important role to play in stimulating the birth of new schemes and the combination of existing ones for a better control and understanding of nanoscopic physics. This chapter is organized as follows: $ 2 deals with basic theoretical considerations of the optical lineshapes of an absorbing impurity in a solid matrix at low temperature. Special attention is directed to the timescale of the measurement and to its influence on the absorption and emission spectra. The experimental setups and practical methods used so far for the detection and spectroscopy of single molecules are described in Q 3, and a simple discussion of the expected signal-to-noise ratio is presented. $ 4 is devoted to the results obtained by the different groups working on this theme. Of course, special emphasis is placed on the authors’ results, which are used whenever possible to illustrate the possibilities of the method. $ 5 is a perilous exercise in predicting the future of the fast growing field of single-molecule spectroscopy and its connections with other new methods like near-field optical microscopy.

Q 2. Principles and Theoretical Background 2.1. GENERAL PRINCIPLES

Granted the density of organic solids, molecules/cm3,it is clear that one of the main difficulties in optical detection of a single molecule will be sifting its response from that of the matrix. This has been achieved in the work considered here by combining moderate spatial and severe spectral selection. With suitable filtering, the signal of just one molecule may be quite high compared to the dark

11, § 21

PRINCIPLES AND THEORETICAL BACKGROUND

I

I

61

ZERO PHONON

Fig. 1. Solvation shifts of the transition frequency of the probe molecule arise because of differing interactions with the matrix in the probe’s ground and excited states. Disorder in real solids causes a spread of the intermolecular stabilization of the probe molecule, or static (temperature independent) or “inhomogeneous” spread of the zero-phonon line resonances (average stabilization of a site). “Site selection” spectroscopies, like hole burning and fluorescence-line narrowing, pick out ensembles of resonant molecules by laser excitation of their zero-phonon lines. The zero-phonon lines are broadened by fast modulation of the stabilization, or “dephasing”, for example by the rapid librational fluctuations about the mean position of the probe at its site. This homogeneous broadening increases rapidly with temperature so that site-selection spectroscopy is usually performed at liquid helium temperatures. The broadening of the zero-phonon line is exaggerated, zero-phonon lines at helium temperatures being in practice 103-104 times narrower than the phonon wing.

count rate of a photomultiplier, and may be detected with an appropriate choice of method (see Q 3) and carefully designed and adjusted optics. The electronic states of probe molecules in condensed phases are perturbed by the presence of the host or “matrix” (solid or liquid), but the perturbation is small, as is shown by the strong similarity of their spectra to those in the gas phase. The solvation of the electronic transition can be split into a slowly varying, or even static contribution, reflecting a molecule’s “solvation site”, and fast fluctuations about this average geometry. The former gives rise to “inhomogeneous” broadening. It may amount to full widths at half maximum of rinhom M 102-103cm-’ (3-30 THz) in disordered solids like glassy solutions, or only a fraction of a wavenumber in crystalline matrices where all impurities have very similar environments; see fig. 1 (Personov [1983]). At liquid helium temperatures, the amplitude of thermal agitation (timescale A), so fluctuations are small compared to the average s) is small (< interaction. The resultant “homogeneous broadening”, rhom, can then be as

68

SINGLE-MOLECULESPECTROSCOPY

A

> I0000

1

nolecules

1000

A

xi00 10 ~~

FREQUENCY Fig. 2. Spatial and spectral selection to achieve single-molecule spectroscopy. As the volume or concentration (or both) of the excited sample is reduced from top to bottom, statistical fine structure appears in the spectrum, due to fluctuations in the number of molecules absorbing per frequency interval of the order of the homogeneous linewidth. When the total number of molecules is less than the ratio of the inhomogeneous to the homogeneous widths, chance coincidences are rare and individual lines can be excited with a tunable, monochromatic source.

small as the lifetime broadening of the excited state, say -1C100MHz. At room temperature it may be as large as the inhomogeneous contribution, and in liquids the separation between the two contributions is blurred (Brito Cruz, Fork, Knox and Shank [ 19861, Nibbering, Wiersma and Duppen [ 19921). Spectral selection of a molecule is therefore not possible at room temperature, but in a low temperature solid, narrow band laser excitation will pick out those molecules which are resonant to within z : f r h o m . Their number will be very large in macroscopic samples, so that laser excitation just maps out the smooth density of states of the inhomogeneous band. As the sample is made smaller, “statistical fine structure” (Moerner and Carter [ 19871, Carter, Manavi and Moerner [1988]) appears in the spectrum, owing to fluctuations in the number of absorbing centers, N , of relative amplitude -N‘l2/N = 1/N‘/’, and finally, individual resonances are distinguishable when the probability of chance

11,

5 21

PRINCIPLES AND THEORETICAL BACKGROUND

69

superposition is small. Figure 2 illustrates the relative increase of statistical fine structure as the concentration is reduced, until the single-molecule regime is reached. 2.2. THEORETICAL BACKGROUND

The theory of the optical response of impurity centers in cold matrices was well worked out over the last two decades, to account for the phenomena of fluorescence line narrowing, spectral hole burning and spectral diffusion (Rebane [1970], Osad’ko [ 19831, Sild and Haller [1988]). It applies to single-molecule spectroscopy as a special, simplified case in which no configurational averaging is necessary, making comparison with experiment particularly direct. The theory is thus not new, and we shall present here only an outline, referring the reader to the literature for details. 2.2.1. Optical spectrum of an impurity center in a cold matrix Molecular spectra in cold matrices generally can be understood on the basis of the Bom-oppenheimer adiabatic separability of degrees of freedom with very different characteristic frequencies - here the electronic excitations on the one hand, and intramolecular vibrational modes and intermolecular phonons on the other. This is justified by the close similarity between the spectra in the gas and in the solid states. The weak coupling is also reflected by the general smallness of latent heats of condensed phases compared to molecular heats of formation. In the adiabatic approximation, the total electronic and nuclear Schrodinger equation is solved in two stages. First, the electronic movement is determined for a given configuration of the nuclear coordinates, leading to the definition of electronic energy surfaces dependent on the “slow” nuclear coordinates, R (fig. 3). The electronic energy of these surfaces appears as the potential, driving the motion of the nuclei, which for small amplitudes can be decomposed into harmonic normal vibrational modes. It is usual to make the further simplification of the Condon approximation (Atkins [1983]), in which the dependence of the electronic properties on the nuclear coordinates is neglected, and properties are calculated from the electronic wavefunctions at the corresponding minimum of the energy surface. The motion of the nuclear coordinates, R, follows from analysis of the energy surface, E(R). The energy E(R) is traditionally decomposed into intramolecular parts, such as bond stretching, bending or twisting, and intermolecular contributions such as

70

SINGLE-MOLECULE SPECTROSCOPY

PHONON WING

,’ I/

, .

ZERO PHONON

8

LlNF

I

I

Rw

.

I

I R. 0

F R

Fig. 3. Illustration of the shift and the change of curvature of the Born-Oppenheimer electronic energy surfaces in the ground state and the excited states, because of electronic reorganization of a probe molecule. Nuclear degrees of freedom (phonons and vibrations) are approximated by harmonic modes of the multi-dimensional energy surface. Solid arrows represent transitions contributing to the zero-phonon line, while the dashed arrows contribute to the phonon wing. The zero-phonon line broadens and collapses as the temperature is raised from T = 0 K.

multipolar electrostatic interactions, contact repulsion, attractive van der Waals or dispersion forces and polarization (Claverie [ 19781). Adiabatic separation of intramolecular and intermolecular nuclear coordinates is a less granted approximation, because the lowest intramolecular modes of a large molecule may be of comparable energy to some intermolecular modes of the matrix. Therefore, although there are many cases in which a progression of sharp intramolecular vibronic bands can be identified in a matrix isolated species, we should bear in mind that the separation is not always as clear cut as suggested by the simplified notation below. This being granted, the intermolecular interactions depend, of course, on the intramolecular charge distributions described by the electronic and vibrational - or “vibronic” - state of the molecules. Formally, we may model a host matrix containing a single impurity, with ground state 10) and one vibronic excited state I *) localized on the guest impurity, g (e.g., the lowest vibrational state of the lowest excited singlet state for example), by writing a very simplified model Hamiltonian:

In this equation, T R is the nuclear kinetic energy operator. The total ground state stabilization corresponding to the nuclear (intermolecular) coordinates R is

11,

5 21

PRINCIPLES AND THEORETICAL BACKGROUND

71

En,,

written as Eo(R) = D:\(R), where D\: is the interaction between molecules m and n in their ground states. The solvent shift of the electronic transition is a difference of stabilizations by interaction with the matrix, dependent on the configuration:

This term is commonly negative, because of the increase of the van der Waals forces in the excited state, leading to a red shift of the impurity spectrum relative to the gas phase (Claverie [1978]). Even in crystalline systems, there is some spread of the value of A,(R) - Ao(R), caused by defects. This gives rise to a statistical effect, static or “inhomogeneous broadening” of the spectrum of solid solutions. Inhomogeneous broadening in well-grown crystals may be less than -0.05 cm-’ . In disordered systems like glasses, it may reach -1000cm-’. Having made the adiabatic separation between nuclear and electronic coordinates in eq. (l), one can introduce the phonons or normal modes of vibration, q, by harmonic analysis of the energy surface near its minima. In pure crystals, phonons are Bloch waves, or vibrational states delocalized over all sites with equal amplitude, and q has the meaning of a wave vector. Translations and librations are then well separated in the long wavelength modes, corresponding respectively to acoustic and optical phonons (Ziman [ 19601). In organic systems, the acoustic mode spectrum stretches up to the first optical modes at -50 cm-’. Optical modes may exist up to -200 cm-I, in which case mixing with intramolecular vibrations is possible. In disordered systems like glasses, translational and librational motions are generally less well separated and high-frequency modes may be spatially localized around a specific defect (Dean [1972]). In particular, a guest molecule may create localized modes in its vicinity. There is also evidence of localized low-frequency modes (-10cm-’) in glasses (Buchenau, Galperin, Gurevich, Parshin, Ramos and Schober [ 1992]), which may be related to tunneling two-level systems and free volume. These modes may also affect optical lineshapes (Hizhnyakov and Reineker [ 19891). With these restrictions, we may however expect very long wavelength phonons (energy of the order of 1 cm-’ or less) to be insensitive to atomic disorder, and continue to treat q as a wave vector. In general, the equilibrium intermolecular coordinates of the electronic energy surfaces are shifted, say from Roo = 0 in the ground state to R,o in the excited state, and the curvature of the potential well will change, leading to slightly

12

SINGLE-MOLECULE SPECTROSCOPY

w,§ 2

different normal mode frequencies. The notation of eq. (1) states explicity the change in the normal modes. The alternative is to introduce the difference in the intermolecular couplings in states 10) and I *), as a coupling between the electrons and the phonons. Developing the vertical transition energy as a difference of the Taylor series for A, and A0 around their equilibria, yields to second-order terms linear and quadratic in the nuclear displacements, proportional to the shift of the normal mode coordinates at equilibrium and to the change in the harmonic frequency, respectively:

H = H o + V, V

=

H~=EO~O)(O~+E,I*)(*I+A~(R)+TR,

(A*(R) - Ao(R))

I*) (*I.

(2)

The last line of this form, which is physically equivalent to eq. (l), shows a coupling, V , between the phonons and the electronic excited state. Now the spectrum of the impurity center is given by (Lax [1952], Kubo [1962], Anderson [ 19541):

/"

00

a(w)

=

dte-iw' ( ~ ( t ) p ( O ) ) ~

where p is the electronic transition dipole operator in Heisenberg representation and the average ( . . . ) T is over the thermal distribution of the phonon-bath states in the electronic ground state. PT(a) is the probability of occupation of state a of the bath at thermal equilibrium. In this equation, the spectral response to a short, white light pulse is expressed as the Fourier transform of its temporal response, which is just the autocorrelation function of the dipole operator because according to linear response theory, the sharp pulse merely reveals the characteristic times present in the equilibrium fluctuations of the observable. Interaction with the phonon bath appears here as a modulation, V , of the resonance frequency, or change in the phase of the dipole operator compared to the unperturbed evolution (Hamiltonian Ho) at frequency w , ~ Without . going into details, which may be found in Osad'ko [1983], the nuclear displacement can be expressed in terms of the creation and annihilation operators of groundstate phonons (phonons must be quantified at low temperatures), showing up one and two phonon creation and annihilation processes on excitation of the guest molecule. In a simplified treatment (see below), the full quantum trace is replaced by considering that V [i.e., R in eq. (2)], is a stochastic function, fluctuating because of thermal agitation.

11,

5 21

73

PRINCIPLES AND THEORETICAL BACKGROUND

Evaluation of eq. (3) leads to a zero-phonon line and a phonon wing (Osad’ko [19831),

The “zero-phonon line” corresponds to quasi-elastic scattering of the electronic excitation on the phonon bath. It has a finite width, stemming from the quadratic coupling, because of the thermal average over transitions between states with the same phonon occupation numbers, but slightly different quanta in the ground and excited states (solid arrows in fig. 3), and because of dephasing processes like absorption and emission of quanta with nearly the same energy. Such Raman-like processes amount to a fluctuation of the transition energy, or dephasing, in the semi-classical picture. The “phonon wing” corresponds to optical excitation with inelastic scattering on the phonons; i.e., absorption with creation of one or more quanta of vibrational energy. The phonon wing is analogous to vibronic bands in molecular spectra. Phonon wings at low temperatures commonly stretch to 100cm-’ , from the zero-phonon line, with one or more maxima in the region -3050 cm-’ (Personov [ 19831). The relative integrated intensity of the zero-phonon line and the shape and intensity of the phonon wing depend on the FranckCondon overlap factors between the initial and final phonon states [eq. (l)], or equivalently on the thermal average in eq. (3). A phonon wing to lower frequency may also be present due to absorption from thermally populated ground-state modes, but its intensity is very weak at low temperatures. Figure 3 illustrates these considerations. The zero-phonon line undergoes Lorentzian broadening because of the elastic scattering on phonons. Neglecting a slight shift of the resonance,

-

This “homogeneous broadening” of the zero-phonon line depends on details of the spectrum of active modes, on their coupling to phonons (Hsu and Skinner [ 1985]), and their coupling to the impurity. For interaction with acoustic phonons at low temperatures, one obtains Y ( T )M T7 (Osad’ko [1983]). Interaction with optical phonons or a localized mode of mean energy A leads to activated behavior, f ( T ) M exp(-AlT). The relaxation of this mode is essential for understanding the dephasing.

74

SINGLE-MOLECULE SPECTROSCOPY

The Debye-Waller factor is given by ao(T) = e-f(T), where J

the phonon function is given by f(y,

TI

=

+1 ) f 0 ( ~+ ) f i ( - y ) f ~ ( - ~ ) , and

e$is the mean squared shift of the modes of frequency

~ o ( Y= ) 6NEcp(~).

(scaling like 1/N where there is one impurity molecule in a matrix of N - 1 host molecules), and p(v) is the density of states of the phonon modes. E(Y) is the thermal population of modes of frequency Y . An equivalent form (Rebane [ 19701) is: Y

in whch S, is the “Stokes loss”, or the difference between the vertical absorption and emission energies from the ground and excited-state equilibrium positions due to mode q. Clearly, a( T ) decreases rather quickly with increasing temperature and is smaller in matrices where the electron-phonon coupling is large. For this reason, flexible molecules and molecules undergoing a large charge reorganization in the excited state, which are the causes of equilibrium shifts, usually have very weak zero-phonon lines. The phonon wing contains contributions from 1 , 2 , . . . , m,phonon processes:

x ao(o - ( W * o

+ Y I + . . . + Y m ) , T).

(5) The zero-phonon line decays with increasing temperature, while the phonon wing simultaneously rises and broadens. Combining the natural lifetime T I of the excited state and the matrix induced components of the dephasing rate, commonly called l / T : , the full linewidth at half maximum of the zero-phonon line is: 1 + -. 1 2 n T 1 JET;

Yhom = -

The rapid decay of 1/T; with decreasing temperature means that in crystalline hosts at helium temperatures, where other causes of dephasing are absent (see

11, § 21

PRINCIPLES AND THEORETICAL BACKGROUND

75

below), the width of the zero-phonon line may approach the lifetime-limited value &,m = (2nTl)-'. If = 5 ns, a reasonable value for an allowed transition in an aromatic molecule is Yhom M 30 MHz. This is extremely sharp compared to the phonon wing (-1 THz). Figure 1 (above) summarizes these optical properties of impurity centers. Single-molecule spectroscopy up to now made use of these properties to pick out an individual, sharp zero-phonon line, carrying most of the oscillator strength at liquid helium temperatures, by selective laser excitation. 2.2.2. Spectral diffusion and dephasing It soon became clear from spectral hole burning experiments that zero-phonon line widths in amorphous systems had properties different from those in crystals ( B r e d , Friedrich and Haarer [ 19841, Zschokke [ 19861, Moerner [ 19881). Even at low burning powers and low temperatures, zero-phonon lines were commonly much broader than in crystals. It was also found that the temperature dependence was weak, T ( T )c( T a , with a in the range 1-2, very often 1.3 (Volker [1985], Thijssen and Volker [1986], Volker [1989]), and that the holes broadened slowly with increasing delay between burning and recording. These differences between the optical properties of glasses and crystals are thought to have the same origin as differences in other properties, namely the presence of two-level systems in amorphous materials. The existence of two-level systems was proposed originally to account for the anomalous specific heat and thermal conductivity of glasses compared with crystals below 1 K (Zeller and Pohl [ 19711, Pohl [ 19811). The excess specific heat of glasses varies in proportion to the temperature, whereas the phonon specific heat present in both glasses and crystals varies as T 3 at low temperatures. This shows that there are extra degrees of freedom in glasses, with a roughly constant density of states in the energy range up to -1 cm-'. Saturation of the absorption of ultrasound waves (Hunklinger and v. Schickfus [ 198 l]), or of microwaves (Golding and Graebner [1981], Golding, Graebner and Haemmerle [ 1980]), shows that these excitations behave like very anharmonic oscillators or like two-level systems. The time dependence of the properties of glasses at low temperatures, stretching over many decades of time and confirmed by each new experimental technique, such as hole burning (Kharlamov, Personov and Bykovskaia [ 19741, Gorokhovskii, Kaarli and Rebane [1976], Weber [ 19871, Breinl, Friedrich and Haarer [ 19841, Walsh, Berg, Narasimhan and Fayer [1987], Littau and Fayer [1991], Miiller and Haarer [ 19911, Wannemacher, Koedijk and Volker [1993]), or photon echoes (Meijers and Wjersma [1992]) in the optical domain, is consistent with the assumption that these two-level systems are atoms or groups of atoms tunneling between

76

SINGLE-MOLECULE SPECTROSCOPY

kd

111, § 2

w

Fig. 4. Coupling of a probe molecule to two-level systems in the surrounding matrix. Small changes in the solvation of the zero-phonon line of the probe result from elastic strain and electrostatic coupling when a group of atoms tunnels between metastable configurations in double-well features of the intermolecular potential energy surface. The characteristic times of tunneling of different TLS's are spread out widely because of the spread of the microscopic parameters like height, V , and width, d , of the barrier, and mass of the tunneling entity, m. The asymmetry of the wells AE governs their equilibrium populations.

wells in double-well features of the total potential energy surface of the glass (Anderson, Halperin and Varma [ 19721, Phillips [1972]); cf. fig. 4. The tunneling rate for a particle of mass m going through a barrier of height V and width d between wells where the harmonic frequency is of order W O , is the product of the frequency of approach, W O , by the transmission probability: 1 =k

-o,exp(-2m%).

t

Thus, even a moderate spread of the parameters m, V , and d in a disordered matrix leads to a wide spread of rates via the stretching effect of the exponential function. Relaxation by tunneling between non-degenerate wells requires interaction with other degrees of freedom to make up the energy difference. Just as one can define phonon modes for different electronic states, phonon modes associated with the double-well configurations will differ in general. A simple model of coupling between two-level systems and acoustic phonons, closely analagous to the theory of phonon-assisted intermolecular energy transfer (Holstein, Lyo and Orbach [ 198l]), predicts that the relaxation rate of weakly asymmetrical wells, as defined by AE < kBT (see fig. 4), should vary slowly with the temperature k ( T ) 0:T ' J ,where r] = 1 for inelastic one-phonon coupling, and v = 3 for a Raman-like two-phonon process (Skinner and Trommsdorff [1988], Kassner and Silbey [1989], Kagan [1992]).

11,

5 21

PMNCIPLES AND THEORETICAL BACKGROUND

I1

As shown by hole burning, a set of molecules, all at the same initial frequency, is spread out spectrally on different time scales. This can be explained by the fact that switching (on different time scales) of two-level systems near a molecule leads to changes of its stabilization energy. Two mechanisms by which this may occur are changes in the strain field around the two-level systems, which will change the density of the matrix around the optical probe molecule and hence its van der Waals stabilization, and electrostatic interactions. Because two-level systems have a rather low concentration, it is reasonable to write the stabilization energy as a sum of independent time-dependent perturbations analagous to the semi-classical picture of coupling to phonons (see fig. 4):

H

= Ho

+ V(t),

Ho = h00 10) (01 + hw, ) .1

i= 1

(*I,

i= 1

In this expression, switching of the ith two-level system causes a change 26w(ri) of the stabilization energy, dependent on the electronic state of the probe molecule as explained above. This perturbation is expected to be of dipolar character, both for electrostatic interaction and for modification of the strain field around the two-level system (Joffrin and Levelut [ 19751). The stochastic variable Ei(t) has the values f l in either of the states of the two-level system. If there are N two-level systems, the shift of the zero-phonon line may in general have 2N values between Ci 16m,(rj)- 6wo(ri)l. Switching of the two-level systems with respect to their initial state causes the resonance frequency to wander from its initial value. In this model, the resonance frequency performs a random walk, to which analytical theories may be applied (Zumofen and Klafter [ 19941). This “spectral dzfusion” causes broadening of spectral holes in glassy systems (Breinl, Friedrich and Haarer [ 19841). In single-molecule spectroscopy, the spectraljumps may be followed directly by repeated fast scanning of the zerophonon line (see § 4.3). Figure 5 shows an example of Monte-Carlo simulation of spectral diffusion, with parameters chosen to resemble the experimental results obtained on single terrylene molecules in polyethylene at 1.7 K (Zumbusch, Fleury, Brown, Bernard and Orrit [ 19931, Fleury, Zumbusch, Orrit, Brown and Bernard [1993]). The frequency shift of the zero-phonon line was a sum of dipolar terms of the form

*

78

[II, 5 2

SINGLE-MOLECULESPECTROSCOPY

200

ABS (a.u)

400

600

800

1000

TIME (ms)

Fig. 5. Simulation of spectral diffusion of a probe molecule coupled to a particular configuration of two-level systems drawn from the distribution discussed in 5 2.2. Repeated scanning of the laser over a period longer than all the characteristic times would give the equilibrium spectrum on the left.

for a two-level system at distance ri. The angular factor Qi accounts for the dipolar character of the coupling, with parameters D M 1 GHz and r,in M 50 A. Two-level-system switching rates, l / t = k , where drawn from a flat logarithmic distribution as prescribed by the standard two-level system model (Anderson, Halperin and Varma [1972], Phillips [ 1972]), here covering the range from 1 GHz to lo4 Hz. Two-level system asymmetries were also taken from a flat distribution up to 10 K and the rates toward well i, ki,with k = kl + k2 (= l / t ) were calculated from k assuming Boltzmann equilibrium. Figure 5 shows an example of a frequency trajectory for a particular configuration of two-level systems, and the corresponding equilibrium spectrum. The great sensitivity of zero-phonon lines to small perturbations of the matrix makes them useful probes of matrix dynamics. This property was already exploited widely in spectral hole burning, in which spectral diffusion of an ensemble of molecules makes the hole properties time dependent. Similar spectral measurements, such as determining the distribution of linewidths, are possible for individual molecules. Besides this, correlation plays an important part in single-molecule spectroscopy. For example, the experimental, normalized fluorescence correlation

11, § 21

PRlNClPLES AND THEORETlCAL BACKGROUND

79

function is defined by the rate of detecting pairs of photons separated by interval z (to within dz), relative to the rate if photons were independent (Loudon [ 19731):

where Z ( t ) is the photon counting rate (fluorescence intensity) at time t ; the process is supposed to be stationary and the average is over a period much longer than all the characteristic times of the intensity fluctuations. For N independent sources, with intensities Zi(t):

Z ( t ) I ( t + z)

=

c

Ii(t)Ii(t + z) +

c

I i ( t ) I j ( t + z).

i tj

1

This contains N terms in which correlation may appear and N ( N - 1) uncorrelated terms, so that correlation effects are weak for ensembles. The most favorable case for observation of correlation is N = 1. Deviation of g(2)(z) from unity measures the amplitude and duration of intensity fluctuations of the source. In the present example, if we tune a laser to a single-molecule line and record the fluorescence intensity as a function of time, spectral diffusion will cause fluctuations of the signal by moving the zero-phonon line in and out of resonance with the laser; see fig. 6. Assuming a Lorentzian lineshape of natural width Yhom, the fluorescence autocorrelation function corresponding to the model above can be written as

where I and F are particular initial and final states of the N two-level systems, P T ( I ) is the thermal population of the initial state (dependent on the two-level system’s splitting) and L(Y)is a Lorentzian of width Yhom centered on the laser frequency YL. The conditional probability of reaching state F at time z by any sequence of flips is a product of N independent conditional probabilities corresponding to the initial and final states of the individual two-level systems:

P(l,zll,O)=

-klk+

kzexp(-kz) k ’

with similar expressions for state 2. This shows the richness of the correlation function, which contains components with the time constants of all the twolevel systems coupled to the probe molecule. The exponential growth of the

80

SINGLE-MOLECULE SPECTROSCOPY

t Vl

t

v2

vL

Fig. 6 . Tunneling of a group of atoms between metastable configurations represented as double-well features of the potential, causes modulation of the resonance frequency and a “random telegraph” fluorescence signal, as the probe molecule is moved in and out of resonance with the fixed laser frequency VL .

number of terms in eq. (8) precludes exact evaluation. The correlation function in fig. 7 was therefore calculated by Monte-Carlo sampling. However, the result is particularly simple when only one two-level system is coupled to the molecule (Fleury, Zumbusch, Orrit, Brown and Bernard [ 19931):

with 11 = L ( Y ~ )and 1 2 =L(Y~). The correlation function is a monoexponential from which the tunneling rate of the two-level system can be deduced. If both lines have been identified in the spectrum, then the tunneling system’s asymmetry may be deduced from the ratio of the integrated intensities of the lines as a function of temperature. The frequency autocorrelation function, C V ( r ) is , also important (Reilly and Skinner [ 19931):

A simple calculation for the model in system (6) yields:

11,

9 21

81

PRINCIPLES AND THEORETICAL BACKGROUND

0.3 I

--m

7 0.2

h

+

N

v

0.1

-

c

0.0

-2

0

2

4

-2

0

2

4

0.8

-g

0.6 0.4

0.2

L-2L__-c-I

. 1

3-

m

0

-2

2

0

4

Log, &/W

Fig. 7. Jumping of a molecular resonance frequency because of two-level systems causes bunching of fluorescence photons in bursts whenever the frequency is resonant with the fixed excitation frequency. The fluorescence autocorrelation function g(*)(r) (top) and the frequency autocorrelation h c t i o n C,(t) of the resonance frequency (middle) map out the characteristic times of the two-level systems. Here, simulated correlation functions are compared with the distribution of TLS times (bottom), weighted by the corresponding amplitudes of the spectral jumps.

Spectral diffusion is not the only possible effect of two-level systems. They may also contribute to dephasing of the electronic resonance, since there is no reason to suppose a connection between the switching rate l/z, and the distance from the probe molecule (induced frequency shift 6w).In the present case, the

82

[K§ 2

SINGLE-MOLECULE SPECTROSCOPY

time dependence of the dipole operator, p = pI0 (10) (*I easily from eq. (6), yielding:

1, 00

I ( w ) = pIo

dt exp[i(w - W * O )tl

(exp (-i

1’

+) . 1

(Ol), is calculated

dt’ 6w(t’))) .

(1 1)

This is a single line at frequency w , ~ modulated , by the phase fluctuation due to two-level systems. Consider for simplicity a single two-level system, switching states with a characteristic time z. If the modulation is slow, the spectral diffusion limit is recovered, with two lines of width of order l/z, separated by 6w = 2 16w,(i) - 6wo(i)(.But consider now a fast two-level system such that l/z >> 60. Suppose that the two-level system switches state with probability after a time z. Then the phase in eq. (3) can be approximated by:

1

$(t) =

J’ dt’6m(t’) 0

M

t6w&z),

(12)

j=I

where the time interval [0, t] has been divided into M M t l z parts of length z. Each flip can be seen as a right or left turn along a phase axis, so that $(t) can be viewed as a random walk process, obeying regular diffusion with diffusion constant D = ( 6 w ~ ) ~ / = 2 z6w2z/2. Then (@(t)) = O and (@’(t)) =2Dt = ~ ( 6 w ) ~ t . Performing a cumulant expansion of the average, we have:

which on substitution in the lineshape formula yields a Lorentzian at frequency w,o with width 60(6wz/2) << 6w. This is the motional or exchange narrowing limit of dephasing (Bloembergen, Purcell and Pound [ 19481, Anderson and Weiss [1953], Abragam [1961], Molenkamp and Wiersma [1985]), in which the energy fluctuations are so fast that the phase excursions take much longer than the time 1/60 to build up. Two-level systems in this situation contribute very little to the dephasing. Two-level systems in between the exchange narrowing limit and the spectral diffusion limit contribute to dephasing. A molecule coupled to all kinds of two-level systems could be modelled by the Anderson matrix theory of lineshape (Anderson and Weiss [1953], Abragam [1961]), but we expect the corrections to the simple model of spectral diffusion to be small, because the wide spread of tunneling rates means that the fraction of two-level systems switching at the right rate to cause significant dephasing will not be

11, § 31

EXPERIMENTAL METHODS

83

large. From a practical point of view, the distinction between dephasing and spectral diffusion can be made by performing experiments on different time scales, like hole burning and photon echoes (Walsh, Berg, Narasimhan and Fayer [ 19871, Littau and Fayer [ 19911, Wannemacher, Koedijk and Volker [1993], Meijers and Wiersma [ 19921).

5

3. Experimental Methods

Although thought experiments and physical concepts have been discussed for a long time in terms of “an” atom or “a” molecule, actual experiments were performed until recently on large populations. The isolation of a single atom or molecule was only made possible by new experimental methods, some of which will be described in this section. We deal only with the work on single molecules in solids at low temperature, called single-molecule spectroscopy. 3.1. EARLY DEVELOPMENT AND OVERVIEW

The first step to observe single molecules in condensed matter must be spatial selection. One must focus on a small volume containing the molecules of interest. Here we call molecules any nanometer structure possessing specific localized states enabling their detection. Our molecules could also be ions, traps, or quantum dots in semiconductors, etc. Even in small volumes of the order of one cubic micron (a cube with sides a few wavelengths of light), the number of molecules, about 109, is daunting. The molecules of interest must therefore be very scarce. The method to detect them must be as blind as possible to the much more numerous matrix molecules. The concentration of active molecules must be less than 10-9mole/l if less than one of them is to be present on average in the selected volume. The concentration can be higher if another technique can distinguish between active molecules themselves. For example, if each molecule has a different resonance frequency, a spectrum of the selected part of a sample may allow resolution of the resonances of different molecules. Optical excitation of the electronic states of molecules is very selective and is the only method of interest in the present work, but one could imagine using other techniques in the future, like infrared or X-ray absorption. The idea of spectral selection in order to separate different sub-populations of molecules is not new. It is exploited in magnetic resonance imaging, where an inhomogeneous magnetic field shifts the resonance of nuclear spins in different positions, allowing recording of signals from quite small volume elements.

84

SINGLE-MOLECULE SPECTROSCOPY

[II,

03

Similarly, persistent spectral hole burning (Kharlamov, Personov and Bykovskaia [1974], Gorokhovskii, Kaarli and Rebane [ 19761, Moerner [1988]) selects a subpopulation of molecules which are resonant with a monochromatic excitation laser. These molecules, when excited, may undergo photo-induced processes which remove them from the spectrum, leaving a persistent spectral hole. The number of molecules thus removed can be as small as one million. In another method, site-selective fluorescence (Personov [ 1983]), even smaller populations of molecules of the order of a few thousands or hundreds are detected. What is new however, is the possibility to push this selectivity down to a single molecule. The low-temperature spectra of organic molecules in solids have particular features which were summarized in 8 2.2.1 and have important consequences for the detection and study of single molecules. The most important feature is the existence of a narrow zero-phonon line for each individual molecule. This line’s narrowness makes it extremely sensitive to external perturbations or to subtle processes in the surrounding matrix like rearrangements caused by thermal fluctuations. This sensitivity is the foundation of the many experiments that were performed earlier on persistent spectral holes and are being applied to single molecules at low temperature. Such experiments would be very difficult or impossible to reproduce at room temperature. The other important consequence of the narrowness of the homogeneous line is the resonant enhancement of the absorption cross-section. The whole oscillator strength of each molecule’s zerophonon transition is now concentrated on an intense and narrow line. The crosssection (J at resonance is given by (Jackson [1975])

where 0 is the angle between the exciting light polarization and the transition dipole moment of the molecule; A is the wavelength of the transition; (TT is the Debye-Waller factor; and yrad and Yhom are the radiative and homogeneous widths of the transition. This expression shows that (J can reach values comparable with the square of A, orders of magnitude larger than the physical area of the molecule, or than its room-temperature cross-section, on the order of a fraction of a nm2 (Drexhage [1973]). This obviously facilitates the detection of single molecules against background signals such as Raman scattering of the matrix (an important source of noise in room temperature spectra) or fluorescence by residual impurities. The optimization of (J and of the signal, by choosing the host-guest system, is discussed in 53.2.5. Although low temperature makes detection of single molecules easier in principle, it has certain

11, § 31

EXPERIMENTAL METHODS

85

drawbacks. One must tune a narrow exciting source (a single-mode laser) to the resonance, which can be difficult or lengthy in dilute samples. Further, the molecule's resonance frequency may change suddenly because of spontaneous or photo-induced jumps. In this case, the resonance is lost and a new molecule must be searched for. Dealing with small numbers in physics automatically entails statistical fluctuations of the physical quantities measured. For example, electronic signals are subject to shot noise arising from the quantification of charge. In a similar way, the absorption strength of a small sample, integrated over the whole frequency range of the transition, will fluctuate from sample to sample with the number of absorbers. The same phenomenon arises within the inhomogeneous absorption spectrum of a given sample, when passing from one frequency to another. These fluctuations, called the statistical fine structure of the absorption spectrum (Carter, Manavi and Moerner [1988]), are those of the number N of molecules absorbing at less than their homogeneous width from a given frequency. The statistical fine structure will appear as a random but reproducible noise in the absorption spectrum of small samples (see fig. 1). This structure is a general feature of any inhomogeneously broadened spectrum, scaling like N"2; i.e., becoming negligible in relative value for the very large numbers of molecules involved in usual experiments. The characteristicfrequency interval over which these fluctuations, F ( Y ) ,vary little is the homogeneous width. This quantity appears in the frequency autocorrelation function (F(v)F(v + Y ' ) ) of the fluctuations. The statistical fine structure was first observed in 1987 (Moerner and Carter [1987]) in the absorption spectrum of pentacene in a para-terphenyl crystal. It was detected by the absorption of a frequency-modulated laser beam. The frequency dependence of the statistical fine structure was transformed into an amplitude variation, which was sensitively detected with a lock-in amplifier. As expected, the amplitude of the fluctuations was directly related to the number of absorbing molecules, and their characteristic correlation frequency agreed with the homogeneous width of the optical transition. Although the statistical fine structure is an elegant solution to the problem of determining the homogeneous width of absorbing centers in solids with linear optics (without the need for a complex hole-burning process), it still arises from large numbers of molecules in most experiments and therefore gives only averaged information. The first detection of a single molecule (Moerner and Kador [1989]) was achieved by the same method as for the detection of the statistical fine structure. The change in absorption of a tightly focused beam was sensitively detected by modulating the exciting frequency and selecting the signal with a lockin amplifier. Because this signal was still too noisy, a further modulation-

86

SINGLE-MOLECULE SPECTROSCOPY

[II, § 3

demodulation step was added by modulating the molecular frequency at rate w, either by an electric field (quadratic Stark effect, demodulation at 2 0 ) , or by an ultrasound wave (pressure shift, demodulation at w). Moerner and Kador were thus able to find reproducible structures in the red wing of the 0 1 site of pentacene in a para-terphenyl crystal (Pc/pTP). The signal-to-noise ratio (S/N) of the absorption spectrum was sufficient for detection of some molecules, but far too weak for spectroscopic investigations. Let us compare the absorption signal S obtained with exciting intensity I (in photon/s/cm2) during integration time t , with the background B arising here from the bulk of unabsorbed, but detected photons. The molecular absorption crosssection u(I) depends on the exciting intensity because of saturation:

u(I)=

U

~

1 + I/Is’

where u is given by eq. ( 1 3) and I , is the saturation intensity of the molecule. The cross-section a(I) being much smaller than the beam cross-section A , we may write: S = Itu(I),

B = IAt.

The signal must in fact be compared to the noise N due to the background ( N = Bl/2 = (IA t ) ‘I2):

5N = U ( I ) E . The frequency-modulation technique works best for intense signals, for which photon noise is negligible. However, intense beams saturate the molecule and the cross-section u(1) is very low. Since even for diffraction-limited laser waists u(/)/Ais about 10-3-10-4, the above result shows that very long accumulation times are needed to reach SIN = 1 . Drifts may then occur. The absorption method will work best for smaller laser spots, ideally when the focus spot becomes comparable to or even smaller than the molecular cross-section. This can in principle be the case with the optical near-field tips (see 9 5 . 9 , where spots of less than lo-’ pm2 could be obtained easily. The method could also be improved by detecting the photons scattered resonantly by the molecule. For samples of good optical quality, the background would be orders of magnitude weaker than in the transmission experiment. A third possibility is red-shifted fluorescence, emitted with creation of high-frequency intramolecular vibrations. This can be

11,

5

31

EXPERIMENTAL METHODS

87

filtered efficiently from excitation photons, the signal then appearing on an essentially dark background. This is the principle of the fluorescence-excitation method which we will now discuss. Two of us proposed fluorescence excitation in 1990 to detect single molecules in the PcIpTP system (Orrit and Bernard [ 19901). The integral fluorescence signal is monitored as a function of the exciting frequency. Whenever a resonance of the sample is reached, the fluorescence intensity rises. The SIN of this detection is much better than that of the absorption, because fluorescence photons are discriminated efficiently from the excitation. Following the above notations, we have: S

=

qo(l)It,

r] being the overall detection efficiency. r] is the product of the fluorescence yield of the molecule, the fraction of the solid angle where fluorescence is collected (including the fraction of light emerging from the sample, Lukosz [ 1979]), the transmission of all optical components between the sample and the detector, and the detection yield. The background in this method arises from Raman scattering from the matrix (total cross-section OR), from fluorescence of out-of-focus molecules excited by scattered exciting light, from fluorescence of residual impurities in the sample or in and on the optical components, notably the red-pass glass filter usually inserted in front of the detector (total cross-section for stray fluorescence photons OF), and from the dark counts 6 of the detector. Unlike the signal of the single molecule, Raman and stray fluorescence crosssecticns do not saturate at high intensity. Moreover, UR and OF are proportional to the excited volume. We compare the average fluorescence signal S when the molecule is excited at resonance to the noise N due to the fluctuations of B for off-resonant excitation. The background is:

leading to the following signal-to-noise ratio:

The Raman scattering cross-section for an intense band in an organic compound like benzene is of the order of 10-20pm2 per molecule (Fernandez-Sanchez and Montero [1990]). Even for a large illuminated volume of lo3 pm3, the total Raman cross-section is only about pm2; i.e., completely negligible as

88

SINGLE-MOLECULE SPECTROSCOPY

[IL

9

3

compared with the resonant absorption cross-section u of a single molecule. (This is no longer the case at room temperature where the huge homogeneous width reduces the cross-section u to about pm’.) The residual fluorescence signal is much more difficult to evaluate. It depends on the purity of the matrix and on the cleanliness of all components in the optical path. For example, the background in polymers is much larger than in a molecular crystal like pterphenyl. This fluorescence background arises from broad emissions and can in principle be much reduced if not eliminated by dispersing the fluorescence. However, etKcient transmission of the fluorescence light into the spectrograph demands a very high quality of the optical path. Equations (14) and (1 5) explain why the single-molecule signal i s much better in fluorescence-excitation spectra than in absorption spectra. The background is so weak that its photon noise is much less than the signal of the single molecule. Recent experiments in singlemolecule spectroscopy have all made use of the fluorescence-excitation method. The next section discusses the experimental setups in more detail. 3.2. FLUORESCENCE EXCITATION OF SINGLE MOLECULES

Figure 8 is a block diagram of a fluorescence-excitation setup for single-molecule spectroscopy. The total fluorescence intensity is detected as a function of the frequency of the exciting light, which is eliminated from the detection by a colored filter, a holographic notch filter, or both. The advantage of this filtering is its high efficiency for the fluorescence. The total emitted intensity, apart from the near-resonance bands, is detected. However, the detection does not eliminate fluorescence from other impurities, as would be the case for example with a monochromator. The different components of the setup are discussed with their functions in the following section.

3.2.1. Spectral selection 3.2.1.1. Low-temperature setup. As explained in 9 2.2, narrow homogeneous lines are observed only at sufficiently low temperature. Therefore, an essential part of the setup is the helium cryostat to keep the sample at a temperature usually varying between 1.7K and 4.2K in a pumped bath of helium 4, or between 4.2 K and 10 K or more, in a flow of helium gas. Because some selection andor collection optics must be placed very close to the sample, the cryostat must accept rather bulky optical or mechanical components (see 0 3.2.2). The diameter of the central tube is usually 30 to 50mm.

11, § 31

89

EXPERIMENTAL METHODS

ARGON LASER

I RING DYE

LASER

I I

DETECTION CUT OFF FILTER

FREQUENCY AND POWER STABILIZATION

CRYOSTAT

Fig. 8. Block diagram of an experimental setup for single-molecule spectroscopy. A thin sample, placed on the end of a monomode optical fiber in a liquid helium cryostat, is excited with a monomode dye laser. Fluorescence emitted when the laser is tuned to a single-molecule zero-phonon line is collected efficiently with a parabolic mirror and detected after filtering to remove scattered laser light.

3.2.1.2. Monochromatic light source. The second piece of equipment necessary to exploit narrow lines is a monochromatic light source, usually a singlefrequency tunable laser. In our setup, we use a commercial CR-699-21 ring dye laser pumped by all lines of a 5 W argon-ion laser. The commercial laser, stabilized on an external cavity, has a frequency resolution of a few MHz. The cavity presents long-term drifts of the order of lOOMHz/hour, which must be compensated for high-resolution measurements lasting several minutes. This can be done by building an additional stabilization. For example, we used an actively stabilized helium-neon laser as a frequency standard to stabilize a Fabry-Perot cavity. The dye laser was stabilized in turn on this cavity, and we scanned the exciting frequency by means of an acousto-optical modulator. This gave us 200 MHz scans with a resolution better than 1 MHz, free of drifts (Bernard, Fleury, Talon and Orrit [1993]). Variants of this system can be designed, for instance, by using the dye laser in its normal mode, recording the transmission

90

SINGLE-MOLECULE SPECTROSCOPY

[I13

5

3

of the auxiliary cavity together with the signal and compensating the drift by software (Croci, Miischenborn, Guttler, Renn and Wild [ 19931). 3.2.1.3. Beamprocrssing. Although beam quality and stability are not essential for the detection of single molecules, they are important for delicate experiments such as optically detected magnetic resonance (see 5 4.2.6). We describe here the beam processing done on our setup to improve the overall stability of the signal. Pointing instabilities of the dye laser beam often occur because of pointing jitter of the argon-ion pump beam or turbulence of the dye jet. These fluctuations are eliminated passively by passing the dye-laser beam through a single-mode polarization-preserving optical fiber. We thus obtain the fiber mode at the output. Feedback into the dye laser would be disastrous for the frequency stability. It is avoided by polishing both ends of the fiber at an angle to obtain Brewster incidence (one could also use a tunable optical isolator). There are still power fluctuations due to the laser itself or to geometrical changes. They are damped by an electro-optical modulator (commercial noise eater Lass11 from Conoptics) up to a frequency of about 500 kHz. The intensity of the laser beam is then adjusted at will by inserting neutral density filters. 3.2.2. Spatial selection

The selection of a sufficiently small sample is an essential step for singlemolecule spectroscopy. Most of the experiments done so far have made use of diffraction-limited beams to selectively excite small volumes. The cross-section of such beams was of the order of a few pm2, and the sample also had to be thin to reduce the excited volume. For example, it is easy to grow sublimation flakes of molecular crystals a few pm thick. Polymers can be pressed to thin sheets a few tens of pm thick, or spin-coated on substrates if they are soluble. The general Langmuir-Blodgett method (Kuhn, Mobius and Bucher [ 19721) allows the preparation of ultrathin films of molecular thicknesses, but these have not been used yet for single-molecule spectroscopy. Three different optical designs have been used so far and are described hereafter (see fig. 9). 3.2.2.1. Optical fiber. We obtained most of our results by exciting the sample through a polarization-preserving single-mode fiber. The core diameter is about 4pm, and the acceptance half-angle is 5 degrees. The advantage of the fiber is its ease of operation. No adjustment is required in the cryostat and the stray-light level is very low. Tbe sample holder is also very compact, which

EXPENMENTAL METHODS

91

--+ >

Fig. 9. Examples of three ways of achieving the moderate spatial selection and efficient collection of emission, sufficient for single-molecule spectroscopy at helium temperatures; (a) excitation through a monomode optical fiber (Orrit and Bernard [1990]); (b) excitation with a convergent lens (Ambrose, Basch6 and Moerner [1991]); (c) excitation through a pinhole (Wild, Guttler, Pirotta and Renn [1992]). Collection is achieved with a parabolic mirror of large numerical aperture in the cryostat in (a) and (b), and with a lens outside the cryostat in (c).

facilitates the insertion of additional devices like electrodes for the Stark effect (Orrit, Bernard, Zumbusch and Personov [1992]), or a wire loop for microwaves (Wrachtrup, von Borczyskowski, Bernard, Orrit and Brown [ 19934). There are, however, several drawbacks. Intimate contact between the end of the fiber and the sample is required. This gives rise to defects when the sample is cooled down. The excitation spot cannot be scanned across the sample, and the linear polarization angle cannot be varied continuously. (For an arbitrary polarization at the input, the ellipticity of the output light varies strongly with temperature, stress, and shape of the fiber.)

92

SINGLE-MOLECULE SPECTROSCOPY

PI,

5

3

3.2.2.2. Lens. Moerner’s group uses the same design for spatial selection of single molecules (Ambrose, BaschB and Moerner [1991]) as for the measurement of statistical fine structure (Carter, Manavi and Moerner [ 19881). A lens of focal length 10 rnm focuses the incident light on the sample. Focusing is done within the cryostat by moving the lens with a magnetic force and spring assembly. The excitation spot can be scanned to optimize the excitation of a molecule, or to image the sample. The polarization can also be adjusted. 3.2.2.3. Pinhole. The design used by Wild’s group (Wild, Giittler, Pirotta and Renn [1992]) is based on a pinhole 5 pm in diameter, illuminated on one side by a nearly plane wave. The sample, placed on the other side, is crossed by the diffracted light. In this way, the contact between pinhole and sample is minimal, and the polarization can be varied (Giittler, Sepiol, Plakhotnik, Mitterdorfer, Renn and Wild [ 19931). However, the laser spot cannot be scanned and the beam quality depends critically on the quality of the pinhole. With such a large pinhole, the light beam is still dieaction-limited in most of the illuminated sample. In principle, much smaller holes could be used (for example, holes obtained by evaporating a metal onto a substrate with small latex spheres (Fischer [1985]), but such designs would be very close to the near-field technique, based on tapered optical fibers and other small structures (see 9 5.5). 3.2.3. Fluorescence collection Because the emission signal of a single molecule is weak, it is important to collect as many photons as possible. The overall detection yield is determined by the solid angle of collection (including the losses due to total reflection on emergence of light from the sample), the transmission of the optics, windows and filters, and the quantum yield of the photodetector, usually a photomultiplier tube. The most important factor is the solid angle. A lens with numerical aperture 0.2 will collect 1% of the emission, while a microscope objective with N.A. 1 collects 50%. However, the correction of aberrations and aplanatism of a good objective, are not necessary to collect light. In practice, much simpler solutions can be adopted.

3.2.3.1. Parabolic mirror. A large fraction of the light emerging from the sample can be collected by placing the sample at the focus of a concave parabolic mirror with large numerical aperture (Orrit and Bernard [1990]). The optical quality of the mirror is not very high, but the main aim is to refocus the fluorescence beam onto the photocathode of the multiplier tube (e.g., an RCA

11,

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EXPERIMENTAL METHODS

93

C-3 1034 with AsGa photocathode), about 1 cm2 in area. Therefore, adjustment of the illuminated sample at the focus is not critical and may be done at room temperature. 3.2.3.2. Wide-angle lens. The fluorescence can also be collected by a lens outside the cryostat (Wild, Giittler, Pirotta and Renn [ 19921). The cryostat windows then must be very large (more than 45" viewing angle), but a significant fraction of the fluorescence light is still lost. Finally, it is possible to insert some commercial microscope objectives in the cryostat in order to image the sample. The detector may be a sensitive CCD camera (Giittler, Irngartinger, Plakhotnik, Renn and Wild [1994]). This has the advantage of enabling study of several single molecules at once. 3.2.4. Additional devices Some experiments require perturbing the sample with external fields. In Stark effect measurements, an electric field is applied to the sample. A pair of electrodes can be placed around the end of the fiber (Orrit, Bernard, Zumbusch and Personov [1992]) to achieve moderate fields of the order of 1 kV/cm. Stronger fields of 30 kV/cm were applied between the pinhole and a transparent IT0 electrode (Wild, Guttler, Pirotta and Renn [1992]) to detect the quadratic Stark effect on single molecules. It will be possible to obtain stronger fields yet by applying moderate voltages at the end of a metal-coated fiber tip in near-field experiments. Pressure effects can also be observed by varying the pressure on the helium bath (the temperature also changes, but temperature shifts of resonance frequency are usually negligible in this temperature range), or by building a pressure cell in which the sample is isolated, as done by Croci, Muschenborn, Giittler, Renn and Wild [1993]. The measurement of magnetic resonance on single-molecule lines requires oscillators and amplifiers to generate the microwave field, and means to apply it to a very small sample. This can be done with a wire loop short-circuiting the end of the coaxial cable of the microwave supply. 3.2.5. Choice of the host-guest system The variety of possible organic host-guest systems is almost infinite. To retain the best candidates for single-molecule spectroscopy at low temperature, we have to optimize guest, host, and the coupled system. The optimal conditions, as stated first by Orrit and Bernard [1990], can be deduced from inspection of eqs. (14) and (1 5):

94

[K § 3

SINGLE-MOLECULE SPECTROSCOPY

PVB

TERRYLENE

r

PERYLENE

PS

$H3

1

PMMA

Fig. 10. Chemical structures of the molecules studied in single-molecule spectroscopy up to the time of writing: pentacene, terrylene, and perylene. The host matrices are shown on the right: para-terphenyl, and the polymers polyethylene (PE), polystyrene (PS), polyvinylbutyral (PVB), and polymethylmethacrylate (PMMA).

(i)

(ii) (iii)

(iv)

(v)

The electronic transition involved should be strong, in order to overcome residual fluorescence from the matrix and setup (see 5 3.1). Alternatively, slowly decaying emissions could be separated from an exciting light pulse and from prompt stray fluorescence in the time domain. The fluorescence yield should be high, at least if laser-induced fluorescence is to be used. The absorption of non-fluorescent molecules could be measured directly with the sub-wavelength sources of near-field microscopy. The saturation intensity should be as high as possible, in order for the detection rate of fluorescence photons to be higher than the dark rate of the detector. Metastable states (such as the triplet in most organic molecules) should have a low population rate and a short lifetime. A significant part of the absorption strength should be concentrated in the zero-phonon line (high Debye-Waller factor a T ) , in order to obtain narrow and intense single-molecule structures. This condition holds only at low temperature. In particular, spontaneous changes of the matrix (leading to spectral diffusion) can broaden the zero-phonon line. The host-guest system should be stable versus photo-induced changes (hole-burning processes) for a long accumulation time of the fluorescence signal. Hole burning is very efficient in hydrogen-bonded systems (Fearey, Carter and Small [1983], Kokai, Tanaka, Brauman and Fayer [1988]), so they are not good matrices for single-molecule spectroscopy.

11, 5 41

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95

These conditions are met satisfactorily by several crystalline or polymeric systems in which single molecules have been studied already. The first system used was pentacene in a para-terphenyl crystal (Pc/pTP). Then, perylene and terrylene were included in semi-crystalline polyethylene (Pr/PE: Basche and Moerner [ 19921; Tr/PE: Orrit, Bernard, Zumbusch and Personov [ 19921). Terrylene was also studied in the Shpol’ski matrix hexadecane (Plakhotnik, Moerner, Irngartinger and Wild [ 1994]), and in amorphous polymers (polyisobutylene, PIB: Kettner, Tittel, BaschC and Brauchle [ 19941; polyvinylbutyral, PVB, polymethylmethacrylate, PMMA, and polystyrene, PS: Kozankiewicz, Bernard and Orrit [ 19941). The chemical structures of these molecules and matrices are shown in fig. 10. New fluorescing molecules and new hosts have appeared in the last few months, see note added in proof.

5

4. Review of Results

4.1. DIRECT SPECTROSCOPY OF SINGLE MOLECULES

4. I . 1. Pentacene in para-terphenyl crystals

Pentacene in a crystal of para-terphenyl (Pc/pTP) was the first system in which statistical fine structure (Moerner and Carter [ 1987]), single-molecule detection by absorption (Moerner and Kador [19891, Kador, Xorne and Moerner [1990]), and subsequently by fluorescence excitation (Bernard and Orrit [ 19901, Orrit and Bernard [ 19901) were demonstrated. The wide variety of experiments in the field of single-molecule spectroscopy were done mainly on PcIpTP. Since this system has become a standard for researchers interested in single molecules, we will lescribe its properties in some detail. Para-terphenyl (pTP) crystallizes in the usual monoclinic herringbone structure of many aromatic hydrocarbons, with two molecules in the unit cell, related by a glide plane. However, this structure is stable at high temperature only. At 193 K, the crystal undergoes a phase transition to a triclinic phase, with four inequivalent molecules in its unit cell (Baudour, Delugeard and Cailleau [ 19761, Baudour, Cailleau and Yelon [ 19761, Baudour [1991]). The essential difference between the high-temperature and low-temperature crystal phases is the position of the central ring of pTP for the different molecules in the unit cell. Steric hindrance in the free molecule prevents the central phenyl ring from lying in the plane of the two outer rings. In the high-temperature phase, the two symmetrical positions of the central ring with respect to that plane are populated equally, so that the molecules are planar on average. In the low-temperature triclinic phase,

96

SINGLE-MOLECULE SPECTROSCOPY

(11,

Q4

the symmetry of the two positions is broken, in a different way for the four molecules in the unit cell. The distortion angles between rings for two of the molecules in the cell are larger (23" and 27") than for the other two (1 5" and I 8"). There are two possible ways of breaking the high-temperature symmetry, leading to two kinds of domains at low temperature, related by a glide plane (Baudour, Delugeard and Cailleau [1976]). We may thus surmise the existence of walls separating the domains in single crystals. Pentacene (Pc) molecules have approximately the same size as pTP molecules. They fit rather well in the crystal of pTP. At liquid-helium temperatures, there are four different insertion sites, giving rise to four intense zero-phonon lines in the absorption spectrum of Pc/pTP. These are labeled 0 1 , 0 2 , 0 3 , and 04. and lie at 16883.0, 16886.8, 17006.1, and 17064.6cm-l, respectively (de Vries and Wiersma [1978], Orlowski and Zewail [1979], Patterson, Lee, Wilson and Fayer [1984]). The inhomogeneous width of these lines, measured on mclt-grown samples, is about 1 cm-' . Their photophysical parameters have been reported in detail in the literature. The lifetime of Pc is 23.5ns for 01 and 0 2 and the emission yield is 78%. The lifetime is about Ions for 0 3 and 0 4 . The intersystem crossing (ISC) yield is 0.5% for 01 and 0 2 , but 50% for 0 3 and O4 (de Vries and Wiersma [1978]). The cause of this big difference is not clear. It could be due to distortion of the pentacene molecule (Kryschi, Wagner, Gorgas and Schmid [1992], Kryschi, Fleischmann and Wagner [1992]), or alternatively to the coincidence of a higher triplet state with the excited singlet S I (Patterson, Lee, Wilson and Fayer [ 19841, Astilean, Chitta, Corval, Miller and Trommsdorff [ 19941, Corval, Kryschi, Astilean and Trommsdorff [ 19941). The triplet lifetime is comparable for the four sites, about 5 0 p . The dephasing rate of the singlet-singlet transition, about 20 MHz, has been measured by photon echoes (Hesselink and Wiersma [ 19801, Patterson, Lee, Wilson and Fayer [ 1984]), and corresponds to a homogeneous width of 8 MHz. From these data, we can see that Pc/pTP fulfills satisfactorily the requirements of a good system for single-molecule spectroscopy: allowed transition, high emission yield, relatively inefficient triplet bottleneck, and high Debye-Waller factor. 4.1.2. Distribution of lines, homogeneous lineshape and width

Figure 11 shows how a small section of the inhomogeneous spectrum of a small and dilute sample of Pc/pTP resolves into a set of sharp homogeneous lines at high resolution. Several arguments, notably sudden frequency jumps, confirm that they indeed stem from single molecules (Orrit and Bernard [ 19901). The distribution of these lines reproduces the inhomogeneous fluorescence

11,

P 41

REVIEW OF RESULTS

97

5 GHz

H

FREQUENCY Fig. 1 1. Appearance of a single-molecule fluorescence excitation spectrum at different scales, ranging from a full scan of a monomode dye laser (top, 30 GHz) to a scan of 300 MHz, showing a profile close to the Lorentzian expected from the lifetime broadening of this pentacene molecule in p-terphenyl at 1.8 K.

excitation profile, as shown by fig. 12 for a more concentrated sample. Singlemolecule lines can be found either in the wings of the inhomogeneous line of a concentrated sample or in the center of a diluted one. Because the inhomogeneous broadening depends on defects, crystal quality determines the distribution of single-molecule lines. A thin crystal stuck to the end of an optical fiber is likely to undergo severe strain upon cooling, leading to disordered and mobile environments. Such crystals can show lines as far as a few nanometers (50cm-I) away from the center of 01and 0 2 lines. On the other hand, free-mounted sublimation flakes display inhomogeneous lines as narrow as 0.8 GHz, revealing the weak satellite structure due to isotopic substitution in the Pc molecule by I3C atoms (BaschC, Kummer and Brauchle [1994]).

98

w,5 4

SINGLE-MOLECULE SPECTROSCOPY 16

12 8

t 4 20 15

10

5

0 -3

-2

-

-1 AV, IGHZ I

0

Fig. 12. Example of statistical fine structure of the excitation spectrum of the 01 site of a small flake of p-terphenyl crystal very lightly doped with pentacene (top). Reproducible single-molecule resonances are visible in two spectra of the wing of the inhomogeneous distribution (bottom). (From Ambrose, Basche and Moerner [ I99 1I).

At intensities higher than a few mW/cm2, the lines begin to broaden by saturation (see 5 4.2.1 for a discussion of this effect), but even at low intensity, lines can have different widths (see fig. 13). Some lines have the homogeneous width of 8MHz expected from photon echo measurements in bulk samples (Morsink, Aartsma and Wiersma [1977], de Vries, de Bree and Wiersma [1977]), while others are broader (Talon, Fleury, Bernard and Orrit [1992]). However, the number of broadened lines seems to depend on the quality of the crystals used. The broadening of the lines was attributed to spectral diffusion during the few minutes required to record a spectrum. The smooth curves in fig. 13 are Lorentzian fits. Note that Lorentzian profiles may arise not only from dephasing processes, but also from spectral diffusion. The lines of single Pc molecules may also present sudden frequency jumps, with rate independent of the exciting intensity (Ambrose and Moerner [1991]). Jumping lines occur more frequently away from the center of the inhomogeneous distribution, as expected for molecules in more disordered environments (Ambrose, BaschC and Moerner [1991]). They reveal directly the phenomenon of spectral diffusion and will be discussed in more detail in 5 4.3.

REVIEW OF RESULTS

99

,

7.9 M

4

$-

13.3MHz . :

-100

-50

0

50

100

FREQUENCY (MHz) Fig. 13. Two examples of resonances of single pentacene molecule in p-terphenyl at 1.8 K illustrating (top) lifetime broadening with a full width at half maximum of 7.9MHz, implying that dephasing and spectral diffusion are negligible; (bottom) a different molecule whose linewidth of 13.3 MHz probably may be ascribed to spectral diffusion during the recording of the spectrum (-5 minutes).

4.1.3. Saturation and dependence on temperature The intensity and width of single-molecule lines depend on the exciting intensity. Their behavior shows a clear saturation (Ambrose, Bascht and Moerner [1991]) which can be attributed to the 3-level system formed by the two singlets and a metastable triplet state. This saturation is discussed in 54.2.1. It is also straightforward to study the dependence of single-molecule lines on temperature. Most of the molecules broaden quickly above 4K, as shown by a study between 1.8 and 10K (Ambrose, Bascht and Moerner [1991]; see fig. 14). The broadening is well described by an activation law, the activation temperature being 38 K (26 cm-'), in agreement with earlier measurements in bulk crystals.

100

tII, § 4

SINGLE-MOLECULE SPECTROSCOPY I

I

I

I

I

I

I

I

I

I

I

1

1000

h N

I

I

v

5 U

100

._

ti

._ -

-

2

10

1

0

2

4 6 Temperoture

(K)

8

10

Fig. 14. Temperature-dependent line broadening of a single pentacene molecule in p-terphenyl. The temperature-independent contribution is lifetime broadening, while the activation behavior of dephasing above 4.5 K was explained by coupling of the electronic transition to a localized librational mode of pentacene. (From Ambrose, Bascht and Moerner [1991]).

As in many host-guest systems, this broadening probably arises from dephasing by a low-frequency librational mode of Pc in the host crystal (Ambrose, Basche and Moerner [ 19911). Occasionally, more complex behaviors are found. For example, two cases were found of a Pc molecule presenting two peaks at low temperature. The peaks merge into a single line above 3 K (Talon, Fleury, Bernard and Orrit [1992]). One of the possible explanations involves the motional narrowing model of 9 2.2.2, when the hopping rate becomes larger than the frequency difference between the two lines, 4.1.4. Perturbation by external fields

Narrow spectral structures are much more sensitive to external perturbations than broad inhomogeneous bands. The sensitivity increase is of the order of the ratio of inhomogeneous to homogeneous width; i.e., in the range of 103-105. Narrow spectral structures obtained by persistent spectral hole burning are currently used for spectroscopy under external perturbations such as electric fields (Kador, Haarer and Personov [1987], Meixner, Renn and Wild [1992]), magnetic fields (Personov 119831, Ulitsky, Kharlamov and Personov [1990], van den Berg, van der Laan and Volker [1987]), pressure (Sesselmann, Richter, Haarer and Morawitz [ 19871, Zollfrank and Friedrich [ 1992]), stress (Sesselmann, Richter, Haarer and Morawitz [ 19871, Sesselmann, Kador, Richter and Haarer [ 1988]),

11,

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REVIEW OF RESULTS

101

etc. Similar techniques have been applied to the sharp lines of single molecules, as the examples given here will show. 4.1.4.1. Electricfield. The Stark effect is the shift of electronic levels of atoms or molecules under an applied electric field. In the case of a centrosymmetric system, the Stark effect should be quadratic (the resonance frequency, being a scalar quantity, must be the same for opposite values of an applied vector). The Stark shift can be linear in non-centrosymmetric systems. The environment of Pc molecules in the pTP matrix is normally centrosymmetric, like the crystal structure. The study of single molecules (Wild, Giittler, Pirotta and Renn [ 19921) shows that their lines shift mainly quadratically with the applied field, as was already observed in bulk samples (Meyling and Wiersma [1973]). However, single Pc molecules show small and random terms linear in the electric field (as already observed by Kador, Horne and Moerner [1990]). Wild, Giittler, Pirotta and Renn [ 19921 also found some dispersion of the quadratic terms from molecule to molecule. At least a part of this dispersion, however, could be due to some inhomogeneity of the applied field in the vicinity of the pinhole. At any rate, the dispersion in the quadratic contribution is much less than in the linear one. The linear terms were attributed to slight breaking of the molecular symmetry by defects in the neighborhood. Their effect, corresponding to an electric dipole change of about D, can be compared to that of a (uniform) internal field of about lo3 kV/cm, but such a uniform field is not likely to occur in the non-polar pTP crystal, and slight molecular distortions are a more likely source of the observed linear Stark effect. The inhomogeneous broadening on which the method is based is in part also caused by similar distortions. The Stark effect was also investigated in a disordered polymer sample (polyethylene, PE), with a different probe molecule, terrylene (Tr; see 9 4.3). Tr is also centrosymmetric, but lines of single Tr molecules shift linearly with applied electric field, as shown in fig. 15 (Orrit, Bernard, Zumbusch and Personov [ 19921). The molecular symmetry is thus broken strongly by the local environment in the disordered matrix, as is observed commonly in hole-burning experiments (Agranovich, Ivanov, Personov and Rasumova [ 19861). The linear shift has random magnitude and sign and depends on the orientation of the particular molecule and of the polymer chains around it with respect to the applied field. The measurement of several molecules shows a deviation from the flat histogram expected for an isotropic distribution of dipole-moment changes with constant magnitude (Kador, Haarer and Personov [1987]). The average magnitude of the dipole-moment changes is about 1 D, but some of them are very large (up to 2.5D) and could correspond to highly distorted molecules,

102

SINGLE-MOLECULE SPECTROSCOPY

2.0

1.5

1.o

0.5

0.0

-0.5

-1.o

-1.5

-

I \

\

-2.0

ELECTRIC FIELD (kV/cm)

Fig. 15. Terrylene has no permanent dipole moments, by symmetry, but this symmetry is broken by distortions and polarization induced by the host matrix (here polyethylene). The change of the projection of the dipole moment between the ground state and the excited state can therefore be measured from the linear Stark effect on applying an external electric field, h v = h v o - E d p . The spread of Stark shifts reflects both the random orientation of the molecules with respect to the field and intrinsic spread of the dipole moments.

or to molecules with strong charges in their surroundings. For some singlemolecule lines, it was possible to measure the Stark effect before and after a spectral jump of a few GHz. In all these cases, the dipole-moment change was the same within experimental error before and after the jump, showing that such frequency changes, which are negligible as compared to the inhomogeneous width, correspond to very subtle motion without a strong modification of the host-guest configuration. 4.1.4.2. Pressure eflect. The frequencies of molecular transitions shift (usually to the red) when going from vapor to condensed matter, because of the

11, P 41

103

REVIEW OF RESULTS A pi[liPa]

504

400

320

189

07 0

I

0.0

.

8

200.0

400.0

I

600.0

Frequency / [MHzJ

Fig. 16. Zero-phonon lines are very sensitive to perturbation. Here, pressure applied to a sample of pentacene in p-terphenyl increases the density and therefore the matrix stabilization of the line. The linear pressure shift can be related to the compressibility of the sample around the probe molecule. Small differences (-20%) in the frequency shift per unit pressure for different molecules reflect slightly differing environments in this crystalline host. This amounts to polling a disperse population which would produce pressure broadening of spectral holes in an ensemble measurement (From Croci, Miischenborn, Giittler, Renn and Wild [1993]).

differential effect of microscopic forces on ground and excited states (see 9: 2). Pressure applied externally will modify the microscopic forces and shift singlemolecule lines. The pressure changes applied in high-resolution experiments such as hole burning are negligible as compared to internal pressure in molecular crystals, which is of the order of several kbar. Therefore, the pressure shift of single-molecule lines is linear and reversible. Croci, Muschenborn, Giittler, Renn and Wild [1993] measured the pressure effect on single Pc/pTP lines between lo3 and lo5 Pa using a hydrostatic pressure cell (see fig. 16). Different molecules showed different linear pressure shifts, the slopes ranging from -7.4 to -1OkHdPa (about 1 GHz/atm). The different shifts are attributed to the slightly different environments responsible for the inhomogeneous broadening. The effect of the disorder on the pressure coefficient seems, however, to be much larger than the ratio of the inhomogeneous broadening to the solvent shift. The pressure effect on a spectral hole burned in this sample would thus lead to a broadening and therefore to a fading of the hole. It is a potential advantage of single-molecule spectroscopy to allow high-pressure measurements without fading of the signal, in a range where the shift may have quadratic and higherorder contn butions (Croci, Muschenborn, Guttler, Renn and Wild [ 19931).

104

SINGLE-MOLECULE SPECTROSCOPY

“1,

54

4.1.5. Dispersed fluorescence

In site-selective electronic spectroscopy (Personov [ 1983]), a wealth of information is contained in the emission spectra. They show directly the vibrational modes of the molecule in its ground state. The frequencies and intensities of these modes are a “fingerprint” of the molecule, and may give important information about its surroundings. The integral fluorescence-excitation method used in most experiments on single molecules does not discriminate between the red-shifted fluorescence photons. They are all counted together in order to optimize the signal. However, this has a further disadvantage: stray photons and residual fluorescence contribute as a background added to the signal. The dispersion of the fluorescence in a multichannel detector (spectrograph with a CCD camera) records all collected photons and allows discrimination of singlemolecule emission from broad background structures. The quality of the optical images is very important in these experiments, in order to transmit efficiently a significant part of the emitted signal into the entrance slit of the spectrograph. The first dispersed-fluorescence experiments were done recently on Pc/pTP by Tchknio, Myers and Moerner [ 1993~1.The single-molecule fluorescence bands agreed very well with the bulk spectrum, confirming the attribution of the cxcitation lirres to Pc molecules. The sensitivity of the fluorescence spectrum c o d d also be used to distinguish isotopically wbstituted molecules or dightlj, different isomers of more complex molecules. The same group later studied the dispersed fluorescence spectra of single Tr molecules in PE (Tchknio, Myers and Moerner [1993a,b]). Figure 17 shows the fluorescence spectra in the 12001600 cm-’ range of the bulk sample and of different single Tr molecules. Striking differences in the spectra led the investigators to distinguish between two classes of Tr molecules in the semi-crystalline PE matrix, presenting different vibrational frequencies and different spectra. They attributed the simple spectra of easily burning molecules (type-2) to molecules in the amorphous parts of PE, while thc more stable (type-I) molecules with more complex spectra would lie in contact with the crystalline areas, where their symmetry would be lowered. Some vibrations seem to be very sensitive to the environment; for example, the low-frequency mode at 243cm-‘ of type-1 molecules is shifted down to 2 12 cm-’ for some type-2 molecules. Study of aromatic molecules included in the semi-crystalline PE matrix showed the presence of molecules in the amorphous domains, and of molecules adsorbed on the facettes of polyethylene microcrystals (Phillips [ 19901). This study illustrates the potential of dispersed fluorescence for the detailed study of micro-environments. In combination with the selection of a single

REVIEW OF RESULTS

1200

1400 1500 Wavenumber (cm-’)

1300

105

1600

Fig. 17. Recording of fluorescence spectra of single molecules of terrylene in polyethylene at 1.5 K. Note the small changes in the vibronic frequencies in these spectra of the “fingerprint” region and the large variation in the intensities of some lines. The differences between spectra E and H and the rest were interpreted as an indication of two kinds of environment, possibly amorphous and crystalline, respectively. Such detailed information goes unnoticed in the top spectrum of an ensemble. (From Tchinio, Myers and Moerner [1993a]).

molecule, it opens the door to the study of correlations of vibration frequencies and intensities with many other molecular quantities (like frequency, spectral width of the lines, Stark or pressure shifts, etc.). The absence of any strong correlation in the quantities studied so far (Myers, Tchenio and Moerner [1994]) shows the complexity and the multidimensional character of disordered solids. 4. I . 6. Localization of molecules An important aspect of the work on single molecules, which might become more and more related to their spectroscopy in the future, is the determination of their physical location in the sample. Microscopy of small samples could help identify the effect of perturbing structures or of inhomogeneous applied fields. This is

106

PI,

SINGLE-MOLECULE SPECTROSCOPY

54

especially important for possible applications of single molecules as parts of molecular devices.

4.1.6. I . Polarization of the exciting light. Equation (13) shows the dependence of the molecular cross-section on the angle between its dipole moment and the electric field of the incident laser. In a birefringent crystal like pTP, the state of polarization of the incident light depends on the linear polarization of the incident laser with respect to crystal axes. For an arbitrary incident polarization, the polarization state of the exciting light seen by the molecule depends on its depth in the sample (see fig. 18). Guttler, Sepiol, Plakhotnik, Mitterdorfer, Renn and Wild [ 19931 have used this effect to determine the orientation of molecular dipole moments and the depth of single Pc molecules in a pTP crystal.

0

%L

in

%

-

200

y1

c

.

150

1 .

5c 100 v)

J 8 50 P

bachgrouna

--

1ew

3

'0

90

270 360 450 540 630 orientation 01 incoming laser light I degree

180

720

Fig. 18. The intensity of the fluorescence of a single molecule depends on the angle between the molecular transition dipole and the optical electric field. Here we see recordings of fluorescence intensity as a function of polarization with respect to the crystal axes, for pentacene in p-terphenyl. The phase shift between the curves can be analyzed in terms of a spread of orientations of the guest molecules in the host and their depths from the front face of the sample, since the polarization of the excitation beam changes as it progresses through this birefringent crystal. (From Guttler, Sepiol, Plakhotnik, Mitterdorfer, Renn and Wild [ 19931).

4. I . 6.2. Two-dimensional microscopy. By putting a microscope objective in the cryostat, the same group has been able to produce images of the sample showing single Pc molecules in a pTP crystal (Guttler, Imgartinger, Plakhotnik, Renn and Wild [1994]; see fig. 19). Although the spatial resolution of these images is still well below the diffraction limit reached by the best immersion objectives (working at room temperature), it should be possible to improve them

11, D 41

REVIEW OF RESULTS

107

130pm

Yl 0 Fig. 19. Spatial localization of single molecules is important for several reasons; e.g., correlating spectral properties with visible defects, and sure identification of lines after spectral jumps. This is achieved here in a fluorescence microscope coupled to a micro-channel plate-image intensifier. Recording with a video camera adds temporal resolution to the information. An additional advantage of such multi-channel recording will be improvement of the statistics of single-molecule polling. This grey-scale image of a p-terphenyl crystal doped with pentacene shows an alignment of molecules absorbing close to the laser frequency. (From Guttler, Imgartinger, Plakhotnik, Renn and Wild [ 19941).

by devising special optics. Such images could be exploited to study several single molecules in parallel, thereby improving the acquisition times and statistics in single-molecule studies. Images can also help study the distribution of molecules in the sample, for example the concentration of impurity molecules around defects of the host crystal. This conventional parallel imaging method would complement the new optical near-field serial technique (see 9 5.5). 4.2. INTRAMOLECULAR DYNAMICS

One of the most common uses of spectroscopy is the study of relaxation between the electronic excited states of atoms and molecules. In the condensed phase, microscopic conditions around each molecule vary, giving rise to inhomogeneous broadening of the transition frequencies. Other molecular parameters, like the transition rates, dipole moments, etc., are also influenced by the surrounding matrix, and therefore may be distributed in solid solutions. Measurements on large molecular populations, even by selective spectroscopy (fluorescence-line narrowing, hole burning) and photon echoes (Golding, Graebner and Haemmerle [ 19801) will only give averaged quantities. The measurement of a molecular

108

SINGLE-MOLECULE SPECTROSCOPY

[II, § 4

parameter on single molecules enables one to find the full distribution of this parameter, and therefore how it is influenced by the surrounding solvent. Moreover, it becomes possible to study the correlation between different parameters and to gain insight into the mechanisms of interaction between the impurity and the matrix. Finally, some specific effects can only be observed with single molecules. They disappear in averages or simply have no equivalent when dealing with large populations. 4.2.I . Optical saturation

The postulates of quantum mechanics indicate that physical measurement of a quantum system projects it into one of the eigenstates of the measured observable. This projection is an intrinsically discontinuous and random process, which seems difficult to describe analytically. The density-matrix formalism only gives the probability of the measurement’s outcome. This can be compared to the actual measurement on a very large number of identical systems or, in the case of a single system, to the result of many measurements, repeated over a long period of time (ergodic hypothesis). Since single molecules are measured over long acquisition times, during which many excitation-emission cycles are performed, the density-matrix formalism can be used to discuss results. Molecules in solids at low temperatures may often be described by a simple model including electronic states only. The simplification of neglecting the many vibrational sublevels of these states is valid at low temperature because the population of the excited vibrational levels is very low and because the relaxation to the lowest vibrational sublevel in each electronic manifold takes place in picoseconds; i.e., is much faster than all other characteristic times in the system. The simplest model of a single molecule is the 2-level system of its electronic ground and excited states. The movement of the density matrix in the corresponding Hilbert space under a quasi-resonant laser excitation is described by optical Bloch equations (Abragam [ 19611, Cohen-Tannoudji, Dupont-Roc and Grynberg [1992]). The main phenomenon connected to the 2-level system is saturation. At high laser intensity I, the average populations of the two levels equalize and the absorption line of the transition broadens. The Bloch vector representing the density matrix of the molecule precesses in the rotating frame at the Rabi frequency Q=pElh; p being the transition dipole moment of the molecule, and E c o s o t being the applied laser field. The population of the excited state therefore oscillates around its average value, a phenomenon called Rabi oscillation. The saturation is

11, § 41

REVIEW OF RESULTS

109

Fig. 20. Photon bunching and anti-bunching are hallmarks of emission from single quantum systems. Absorption~fluorescencecycles from the upper singlet Sl are interrupted occasionally by intersystem crossing to the long-lived triplet state T I ,with three sublevels split by dipolar spin-spin interaction between the unpaired electrons. Emission resumes on return to the ground state SO,after the triplet lifetime (ks to s). Photons are therefore bunched in bursts. On a finer scale, photons within a burst are anti-bunched because of the finite time needed to pump the molecule back to the upper singlet state after each emission.

conveniently characterized by the saturation parameter s or the saturation intensity 1 2 :

where k2 is the decay rate of the excited level, and T ( T ) the dephasing rate of the transition (2ntimes the half width at half maximum of the frequency spectrum). Organic molecules usually have a metastable triplet state T I in-between their ground SO and excited SI singlet states (see fig. 20). When this new level is introduced, with the intersystem crossing (ISC) rates from S1 to T I , k23, and from T I to SO,k31, the molecule behaves like a 3-level system. The saturation parameter becomes: S Z -I 1

12

(k31 + ik23) Q2 k 2 r ( T ) k31

The saturation intensity is now reduced by the buildup of a metastable population in the triplet level, which may be very large for a high triplet yield and a long triplet lifetime. The degeneracy of the three triplet sublevels is in fact lifted by magnetic dipole-dipole interaction between the unpaired electrons in the triplet state

110

PI, § 4

SINGLE-MOLECULE SPECTROSCOPY

(zero-field splitting). The molecule must therefore be considered as a 5-level system, and the ISC rates depend on the particular triplet sublevel. Transitions between triplet sublevels can be induced by a resonant microwave; i.e., electronspin magnetic resonance. The evolution of the density matrix under laser and microwave irradiation is also described by a set of Bloch equations (Cohen-Tannoudji, Dupont-Roc and Grynberg [ 19921). More generally, several frequencies may be applied to different transitions in the molecule, offering the possibility of non-linear optics on single molecules.

I

0

200

400

Time Ins]

600

800

lb00

Fig. 2 1. Fluorescence excitation spectroscopy leads to signal-to-noise levels quite sufficient for most experiments on ensembles to be carried over to single molecules. Here the decay of the upper singlet state of an individual pentacene molecule in p-terphenyl (lifetime -23 ns) is determined by time-correlated single photon counting. (From Pirotta, Guttler, Gygax, Renn, Sepiol and Wild [1993]).

4.2.2. Fluorescence lifetime

The first rate to consider in the molecular two-level system is the fluorescence decay time, or the lifetime of the excited singlet 5’1. Fluorescence lifetimes are measured classically by time-correlated single-photon counting. A short light pulse excites the molecule and gives the start signal on a clock. The first fluorescence photon detected stops the clock. One can show that, if the number of detected photons per pulse is much smaller than unity, the histogram of times gives the fluorescence decay. The method was applied to single PcIpTP molecules by Pirotta, Guttler, Gygax, Renn, Sepiol and Wild [1993], the excitation pulse being cut out of a cw laser beam by acousto-optic modulators. The obtained fluorescence decay is shown in fig. 21. The lifetimes obtained for four different molecules were identical within experimental error (23 ?C 1 ns) and agreed very

o

11, 41

REVlEW OF RESULTS

Ill

well with measurement on bulk crystals. This result was expected, since the main contributions to S1 decay in 0, and 0 2 sites of PcJpTP are radiative decay (78%) and internal conversion to vibrations (22%). Both of these intramolecular processes are thought to be only mildly sensitive to the environment, through changes of the transition dipole moment by polarization effects and through variations of the internal conversion channel by molecular distortions. 4.2.3. Correlation functions

In the method of time-correlated single-photon counting, the state of the molecules just before the exciting flash is known, since the period between flashes is much longer than the decay rate to the ground state. Having selected single molecules, it is possible to do related experiments with a cw excitation. the start signal being given by one of the fluorescence photons, the stop signal by another. It is essential in such experiments to deal with a single emitter (or at least with a small number of them). If the number N of emitters is large, the useful information given by N events, where start and stop photons are from the same emitter, will be drowned by N(N - 1) uncorrelated events corresponding to a start by one emitter and to a stop by another (Schaefer and Berne [1972], Rigler, Widengren and Mets [ 19921). The single-emitter case is, of course, free of spurious events. The quantities given by such experiments are a class of second-order correlation functions (here, auto-correlation of the fluorescence intensity). If only consecutive photons are considered, as in the start-stop experiment, the histogram of delay between photons will give the distribution of consecutive photon pairs. This function is related to the normalized autocorrelation function g(2)(t) (Loudon [ 19731, Pecora [ 19851, Cummins and Pike [ 1974]), which includes all possible photon pairs during a long acquisition time, g(2)(z)= ( I ( t ) l ( t + z))/ ( Z ( t ) ) 2 . If there is no memory effect extending beyond the detection of the next photon, a simple relation connects the Laplace transforms of these two functions (Reynaud [ 19831). Practically, the autocorrelation function will be especially useful for slow phenomena, when many photons can be detected during the characteristic variation time of the function. For short times, pairs of consecutive photons hold the essence of the information and the corresponding function is easier to access with a start-stop setup of the Hanbury-Brown-Twiss type (HanburyBrown and Twiss [ 19561). 4.2.4. Anti-bunching

The second-order correlation function g(2)(z) of a classical fluctuating function

112

SINGLE-MOLECULE SPECTROSCOPY

[II, § 4

320

280 240 A

.c n 200 v)

C

40

0

$

a

30

;20 (I:

3

0 0

W

n

10 12

tv

= J 3

4

0

-100

0

100

200

Time (ns) Fig. 22. Emission of a photon by an illuminated molecule starting from the ground state requires a finite time, described fully by a probabilistic density-matrix approach, dependent on the laser power (Rabi frequency) and damping of population and coherence. This leads to anti-bunching of photons on a nanosecond scale. The figure shows the consequent drop in the rate of detection of pairs of consecutive photons at short delays for a pentacene molecule in p-terphenyl at 1.5 K. The Rabi frequencies were, from top to bottom, 11.2, 26.2 and 68.9MHz. (From Basche, Moemer, Orrit and Talon [1992]).

of time / ( t ) must have a larger value at t = O than at z = m , as follows from its definition [eq. (7)]. The light emitted by a single quantum system (atom or molecule) has a quantum nature which allows g(2)(0)< g(2)(m),corresponding to a lesser probability to observe a second photon immediately after the first one was detected (Cohen-Tannoudji [ 19771). This effect, called photon anti-bunching (Teich and Saleh [1988]), is a signature of the quantum nature of light, and was first observed several years ago in tenuous atomic beams (Kimble, Dagenais and Mandel [1977]) where at any time only a few atoms are present in the beam, and more recently on single trapped ions (Diedrich and Walther [ 19871).

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113

The anti-bunching effect was also predicted for dilute molecular solutions at room temperature (Ehrenberg and Rigler [1974]). Its mechanism is connected intimately to the measurement process. The observation of the first photon projects the system’s wave function into the ground state, whence a second photon cannot be emitted immediately. Because the buildup of excited-state population needs time, the correlation function will increase at short times. The same experiment was performed recently on a single Pc/pTP molecule (Basche, Moerner, Orrit and Talon [1992]). The setup was of the HanburyBrown-Twiss type (Hanbury-Brown and Twiss [ 19561). The fluorescence beam is split between two photomultipliers, one for the start and one for the stop signal. In this way, dead-time and afterpulse effects are avoided. The main difficulty of the experiment was to decrease the background sufficiently to observe the correlation signal of a single molecule, even at a high level of saturation. The experimental results are shown in fig. 22. The correlation hole at short times is quite clear, even at high intensity, when background correlation counts add to the signal. The hole becomes shorter for increasing intensity, and Rabi oscillations start to appear for the highest Rabi frequency of 68.9MHz, when the Rabi frequency becomes larger than the spontaneous emission rate. The agreement of the data with fitted theoretical curves is excellent, and gives values of the Rabi frequencies which agree within a few percent with those deduced from the measured intensity of the laser and the transition dipole moment from bulk measurements (de Vries and Wiersma [ 19781). Observation of such subtle quantum-mechanical effects in large organic molecules at low temperature shows that in some respects they are not much more complicated than atoms or atomic ions. 4.2.5. Bunching

While photons tend to avoid each other in the range of tens of ns, they are bunched in packets on longer timescales. This is a consequence of the saturation by the metastable triplet level, which leads to a specific decrease of the correlation function in the timescale of 1&5Ops for Pc/pTP (Orrit and Bernard [1990]). After the laser has been turned on, the molecule absorbs and emits photons, giving intense fluorescence. When a transition to the triplet takes place, the fluorescence stops altogether during the lifetime of the triplet, because there is no resonant absorption from that state. The triplet finally decays to the ground state, and the whole cycle can start again (see fig. 20). The fluorescence photons are therefore grouped in bunches, separated by dark intervals which last for the triplet lifetime on average. The theory of the saturation of this three-level

114

SINGLE-MOLECULE SPECTROSCOPY

111,

94

system predicts an exponential decrease of the correlation function at long times (longer than the characteristic times of antibunching). While the average duration of the dark intervals is always the triplet lifetime, that of the bright periods depends on the exciting intensity. For strong intensity, the bright intervals are short, the contrast of the correlation is high, and the decay rate of g ( 2 ) ( t tends ) to k31 + i k 2 3 , the sum of rates into and out of the triplet (the factor arises because the singlet transition is saturated at high intensity). For weak excitation, the contrast is weak and the decay rate tends to k31, because the population of the triplet becomes negligible. The study of the correlation decay rate as a function of exciting intensity thus gives access to the ISC rates k23 and k31. Moreover, since this information is obtained directly in the time domain, it is more reliable than the combinations of ISC rates obtained from saturation, which might be influenced by other phenomena. Finally, the three-level model predicts a singleexponential decrease of the correlation. If, however, the three sublevels of the triplet state are taken into account, up to three distinct exponentials can appear in the correlation function, depending on the population and decay rates of the sublevels (see 5 4.2.6). Photon bunching in the fluorescence of Pc/pTP single molecules was demonstrated and used as a proof of its mononiolecular origin in the first work on single molecules by fluorescence excitation (Orrit and Bernard [1990]). The correlation decay rate and contrast were later studied as a function of the exciting intensity. The three-level system was sufficient to account €or the data in the available time range, 1-loops (Bernard, Fleury, Talon and Orrit [1993]). Figure 23 shows examples of correlation functions and spectra of a single molecule recorded at different exciting intensities. The dependence of the decay on exciting intensity allows one to determine the ISC rates k23 and k31. The rates obtained are very accurate on the high-intensity side but much less so on the low-intensity side where the fluorescence signal is weak, because the correlation function is proportional to the square of the signal. The determination of the triplet decay rate k31 is therefore much less accurate than that of k23. Comparing the ISC of 8 different molecules gave the main result of this study, a dispersion of triplet population rates k23 over more than 20% of its average value, and a systematic deviation to larger values than in the bulk crystal (de Vries and Wiersma [1978]). The rates of correlation are compared for two molecules in fig. 24. This difference in rate is rather surprising, since the ISC rates are normally determined by intramolecular dynamics. However, the case of PcIpTP is anomalous in this respect, since k23 changes by two orders of magnitude between the 01 and 0 2 sites on one hand and the O3 and 0 4 sites on the other. It is therefore not unreasonable to assume that defects in highly strained

11,

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REVIEW OF RESULTS

115

k0.75 rnW/crn2 lOOr

70

6o0

50

100

150

1=3 mW/cm2

P a

W

U

0

800

2oo0

50

100

150

0

20

40

60

80

s

Lu

3

1=30 mW/cm2

':j".

14000

12000 10000

1200

6000

1000

0 5 - 0

FREQUENCY (MHz)

0

20

40

60

6000 60

DELAY fus)

Fig. 23. Examples of photon bunching from a pentacene molecule in p-terphenyl at 1.8 K. As the laser power is increased from top to bottom, the resonance undergoes power broadening, while the contrast in the correlation function increases and the decay time (approximately mono-exponentlal) decreases. This corresponds to intense bursts of emission interrupted by the intersystem crossing to the "dark" triplet state.

crystals at the end of the fiber can lead to significant variations of k23. Two possible mechanisms can explain the sensitivity of k23 to defects in the molecular environment. One involves a near coincidence of the excited singlet with a higher triplet state, a situation leading to strong ISC variations in substituted anthracenes and other hydrocarbons (de Vries and Wiersma [1978], Orlowski and Zewail [ 19791, Patterson, Lee, Wilson and Fayer [ 19841). Similar variations of ISC were found for different deuterated analogues of Pc in a benzoic acid crystal, as well as

1 I6

[II, § 4

SINGLE-MOLECULE SPECTROSCOPY

t 0.15

-

v)

5 0.10t

0.05 -

0.00

3

2

4

I

-log,Jl)

Fig. 24. Variation of the decay rate of photon correlation from two molecules of pentacene in pterphenyl at 1.8 K, as the laser power is increased. The difference between the curves may reflect the influence on intersystem crossing of small, site-dependent deviations from perfect planar symmetry, for which the population of the triplet would be forbidden.

variations under an applied magnetic field (Astilean, Chitta, Corval, Miller and Trommsdorff [ 19941, Corval, Kryschi, Astilean and Trommsdorff [ 19941). They were attributed to coincidences of vibronic levels. Another explanation involves distortions of the Pc molecule (Kryschi, Wagner, Gorgas and Schmid [ 19921, Kryschi, Fleischmann and Wagner [ 19921). Spin-orbit coupling between S1 and T I is normally forbidden in flat hydrocarbon molecules. Even small distortions of the molecule could therefore cause large enhancement of the ISC rate as compared to the weakly distorted molecule in the 01 site. ISC would thus act as an amplifier for the slight molecular distortions responsible for inhomogeneous broadening and for linear contributions to the Stark effect (54.1.4). 4.2.6. Optically detected magnetic resonance Magnetic resonance spectroscopy has many applications in the characterization of molecular structures and interactions. Unfortunately, the sensitivity of conventional approaches is about 10” electronic spins or 10l6 nuclear spins. This can be improved by applying double-resonance methods to modulate an optical emission by pumping a magnetic resonance transition of the same system. Thus, “optical detection of magnetic resonance”, described below, can detect 1O5 electronic spins (Suter [ 19921) in classical spectroscopy, and was recently combined with single-molecule spectroscopy to detect single electronic spins (Kohler, Disselhorst, Donckers, Groenen, Schmidt and Moerner [ 19931, Wrachtrup, von Borczyskowski, Bernard, Orrit and Brown [ 1993a1).

11, 5 41

REVIEW OF RESULTS

117

Figure 20 (above) illustrates the principle of a fluorescence-detected magnetic resonance experiment. Molecules are pumped repeatedly through absorptionemission cycles between the ground and the excited singlet states, SO and S I . Occasionally molecules cross by spin-orbit coupling to the triplet state, T I .The electrons in the open orbitals in the triplet state have parallel electronic spins, so that the total spin is S = 1 and there are 2s + 1 = 3 sub-levels, whose degeneracy is lifted by the average dipolar spin-spin interaction. The splitting induced is of order 0.1 cm-I. In zero external magnetic field the spin lies in one of the principal symmetry planes of the molecule; e.g., XY for the [Z) sublevel, and so on (McGlynn, Azumi and Kinoshita [1969]). The triplet lifetime may range from microseconds to seconds because the transition to the singlet ground state is spinforbidden to first approximation. The sub-levels in fact have different lifetimes and population rates from Sl . Under steady-state excitation, an equilibrium is reached with a fluorescence signal reflecting the probabilities of branching from SI to the triplet sub-levels IX), I Y) and (Z) and their lifetimes. Application of microwaves (MW) to flip the spins, e.g., from a short-lived, strongly populated level (like IX) in the case in hand) to a long-lived, weakly populated level (IZ)), lengthens the dark intervals. Therefore, resonance of the MW field with the IX)H 12) transition causes a readily observed change (here a drop) in the fluorescence signal. This standard method can be applied to a single molecule, with the added advantage that because a single quantum system is involved, the temporal distribution of fluorescence photons can be exploited; microwave pumping influences the fluorescence autocorrelation function by modulating the duration of the dark periods in the photon bunching of a single molecule (see 0 4.2.5). Figure 25 shows the ODMR signal of a single pentacene molecule in a p-terphenyl host at 1.8 K. Two resonances were observed and attributed to the IX) H lZ) and I Y) H lZ) transitions, by analogy with the transitions observed in bulk samples (van Strien and Schmidt [1980]). The depths of the ODMR resonances depend on the details of the population and decay rates of all the excited states. The complete kinetic scheme can be worked out from an analysis of the steady-state ODMR effect under optical and microwave saturation, the transient ODMR effect after a microwave pulse, and the influence of microwaves on the photon bunching (fluorescence correlation function; (Brown, Wrachtrup, Orrit, Bernard and von Borczyskowski [ 19941). It turns out that the IX)H I Y) transition cannot be detected with our sensitivity because the lifetimes of IX)and I Y) are too close together. Hypefine coupling between the electronic spin and the 14 protons of the pentacene molecule is responsible for the asymmetry of the lines. The broadening is on the high-energy side of

118

SINGLE-MOLECULE SPECTROSCOPY

2400

cn I-

z

tII, § 4

I

2200

3

8 W

0 2000

z w

0

cn W U 0

3

1800

LL

1600

1350

1400

1450

1500

MICROWAVE FREQUENCY (MHz) Fig. 25. Detection of single electronic spins is possible by combining the site selection of singlemolecule spectroscopy with the well-established methods of optical detection of magnetic resonance. Here, fluorescence from a pentacene molecule in p-terphenyl under optical saturation is modulated by using a microwave to move the molecule from the short-lived X and Y triplet sublevels to the longer-lived Z level, resulting in longer dark periods between photon bursts. Resonance of the microwave field with the spin transitions therefore results in a drop of the fluorescence signal. Inset: Recording of the X-Z transition at a lower microwave power to avoid saturation broadening. The asymmetrical broadening of the resonances is due to second-order hyperfine interaction between the electronic spin and the protons of pentacene, which undergo spin difision during the recording.

the IX) H lZ) transition and on the low-energy side of the IY) H lZ) one, because it is by symmetry a second-order perturbation tending to repel levels. The broadening of the resonances shows that during the recording, which lasted some minutes, many of the 214 proton states were sampled. This is most probably due to spin difision of the pentacene protons, which are coupled to those of the matrix, during the periods when the molecule is in the singlet manifold (van Strien and Schmidt [ 19801). Spin-coherence effects on single spins were also observed. By coherence we mean any effect that requires preparation of a quantum superposition of states with definite phase relations, such as may be realized for ensembles by periodic excitation. This is captured by the density-matrix formalism, in whch the ensemble averaging makes populations and coherences smooth functions of time. On the other hand, a single quantum system from such an ensemble undergoes discrete jumps. These aspects are reconciled by the ergodic assumption that a

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PULSE DURATION (ps)

Fig. 26. Spin coherence of a single electronic spin of pentacene in its triplet state. In this experiment a microwave pulse of variable length is applied to the molecule and the fluorescence is recorded for 1 ms. Starting on average from the most populated X sublevel for zero pulse width, the fluorescence is modulated according to the lifetime of the dark state in which the pulse leaves the spin. Because only one center is involved, the usual causes of damping of spin nutation (like inhomogeneity of the microwave field) are absent. In the present case, both modulation of the microwave resonance frequency by spin diffusion and intrinsic dephasing of the spin may contribute to the decay of the nutation pattern. The dashed line is an oscillating function with damping time -5 ps.

temporal average for a single quantum system, over a period much longer than all the relaxation times, is equivalent to an instantaneous ensemble average. This is illustrated by the agreement of a density-matrix model with observations of fluorescence transients and fluorescence correlation (Brown, Wrachtrup, Orrit, Bernard and von Borczyskowski [ 19941). Figure 26 shows the result of a spin-nutation experiment on a single pentacene molecule (Wrachtrup, von Borczyskowski, Bernard, Orrit and Brown [ 1993b1). In this experiment, the fluorescence under steady laser excitation was modulated by applying microwave pulses to the molecule at a repetition rate of 200Hz. The fluorescence was recorded for 1 ms after each pulse. On those occasions when the molecule was in the triplet manifold during the pulse, the microwaves coherently drove the spin between the IX) and 12) levels. The fluorescence intensity recorded just after the pulse depends on the state in which the molecule was left by the periodic microwave pumping. On average it starts in the ( X )state and is left in the long-lived l Z ) state, implying a fluorescence minimum, for a JC pulse (half a Rabi period, lOOns in the example). The decay of the nutation pattern in ensemble measurements generally results from the superposition of nutation patterns with different Rabi frequencies, due to inhomogeneity of the microwave field (Torry [1949]). This does not apply to a single molecule. Here,

120

[It § 4

SINGLE-MOLECULE SPECTROSCOPY

30 25

-

0

I 0

3 20 yI- 15 U 10 Y

LT z

n 0

5

0

100 1000 TIME AFTER TRIGGER (ps) I

0.1

1

I

I......

I

10

m

....I..

I

100

,

. I . I..,

I

1000

,

,..10000

DELAY (ps) Fig. 27. A pulsed ODMR experiment may be synchronized with a single molecule by triggering a n microwave pulse with variable delay on recording of a fluorescence photon (implying the molecule

must be in the ground state just after detection). The pulse takes effect only if it is sufficiently delayed to fall outside the photon burst of the trigger, when the molecule has a high probability of being in the triplet state. The inset shows the average fluorescence transient following a pulse delayed by 2 0 ~ s . The smooth curve was calculated with the optical-microwave Bloch equations.

two causes of decay of the nutation pattern may be at play. One is the spread of the resonance frequency due to the “inhomogeneous” broadening effect of spin diffusion between stays in the triplet state (Breiland, Brenner and Harris [ 19751). This can be eliminated in spin-echo experiments on single pentacene molecules, which show up the other cause, loss of transverse magnetization by dephasing of the triplet spin. Both are in the microsecond range for this system (Wrachtrup, von Borczyskowski, Bernard, Orrit and Brown [ 1993bl). An interesting aspect of single-molecule ODMR is the possibility to synchronize the microwave pulse with the molecule, by triggering the pulse with a variable delay on a fluorescence photon and taking advantage of the bunching of photons in bursts under optical saturation by the population of the triplet state (see 9 4.2.5; Wrachtrup, von Borczyskowski, Bernard, Orrit and Brown [1993bl). Here, the burst duration is of order 10 ps. If the pulse delay after the triggering photon is shorter than this, the molecule has a high probability of still being in the same burst (singlet manifold), so that the pulse has no effect. As the pulse is moved out of the burst, the ODMR effect increases and may even exceed the untriggered value because the synchronization leads to more hits in the triplet state than with periodic triggering, which has no particular phase relation

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with the state of the molecule. Figure 27 shows this effect for a synchronized n pulse. The effect subsides to the untriggered value at very long delays because of loss of correlation. 4.3. DYNAMICS OF THE HOST MATRIX

4.3.1. Spectral difusion

As mentioned in 0 2.2, zero-phonon lines are extremely sensitive to the changes in solvation by the matrix which may accompany even small structural changes. While these changes are usually much more frequent in glasses, notably because of the presence of two-level systems, they were first observed for single molecules in a crystalline host, where as it turns out, spectral diffusion can be explained by assuming the existence of two-level systems at domain boundaries. Sudden drop and return of the fluorescence of single-molecule lines was first seen in p-terphenyl crystals doped with pentacene (Orrit and Bernard [1990]), and soon interpreted as spontaneous spectral diffusion due to switching of unidentified two-level systems (Moerner and Ambrose [1991]), in studies of the random surges of fluorescence power observed after tuning the laser to a particular pentacene molecule. One should pause here to distinguish two effects. The first is spectral diffusion by spontaneous switching of two-level systems, independent of the optical excitation of the probe. The second is structural changes induced by the absorption of light by the probe. They may be distinguished by varying the power of the optical excitation. It seems so far that only spontaneous spectral jumps have been observed in the crystalline system pentacene in p-terphenyl (Ambrose, BaschC and Moerner [19911); whereas perylene and terrylene in polyethylene show both effects (BaschC, Ambrose and Moerner [ 19921, Fleury, Zumbusch, Orrit, Brown and Bernard [1993]). An example of photo-induced jumping will be discussed shortly. By working at sufficiently low powers, one can hope to minimize photophysical effects, so that spectral diffusion measures the spontaneous dynamics of the matrix. Photo-induced effects are discussed in 9 4.3.5. 4.3.2. Frequency trajectories

Two classes of pentacene molecules were distinguished in p-terphenyl (Ambrose, Baschk and Moerner [1991]); those with stable frequencies near the center of the inhomogeneous distribution, and those whose frequency vaned with time,

122

SINGLE-MOLECULESPECTROSCOPY

-200

-100

100

0

AV, [MHz]

200

200

1

100

l'n

[MHz]

0

-100

-200

0

100

I

,

, 200

,

,

I151

,

-

300

,

400

,

1

500

Fig. 28. Spectral diffusion or random wandering of the resonance frequency of a molecule because of relaxation of the surrounding matrix (e.g., two-level systems, structural relaxation) is characteristic of amorphous hosts, but may nonetheless be observed in some crystals. In p-terphenyl, tunneling of the central phenyl ring of molecules in domain walls may cause spectral diffision of nearby pentacene molecules (see text). (Top) Spectra of a single molecule of pentacene in p-terphenyl recorded at 1 s intervals. (Bottom) Position of the peak over a period of 500s. (From Ambrose, Basche and Moemer [1991]).

predominantly found in the wings. Shifts in the resonance frequency were monitored by repeating scans of the laser at 1 Hz repetition rate. Figure 28 shows an example of the behavior found for molecules in the wing of the inhomogeneous distribution, which we may presume to correspond to more disordered environments than those of the stable molecules at the line center. This particular molecule appears to be coupled to many two-level systems. It should be borne in mind that a very wide variety of behaviors can be obtained, ranging from bi-stable behavior, through chaotic multi-frequency jumps, to molecules whose frequency creeps gradually in one direction (Ambrose, Baschk and Moerner [1991]). The latter case gives evidence of structural relaxation of the matrix, not describable by the standard model of coupling to independent two-level systems.

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Frequency trajectories may be discussed more quantitatively by introducing the frequency autocorrelation function (eq. lo), which contains components from all the dynamical variables coupled to the probe molecule. It was found to decay mono-exponentially for a particular pentacene molecule with a wandering zero-phonon line (Reilly and Skinner [19931). This behavior, implying coupling to several two-level systems with approximately the same time constants, was understood by noting that the central phenyl ring of p-terphenyl can occupy two stable positions twisted out of the plane of the outer rings, between which it may tunnel at low temperature. The asymmetry is very large in the bulk of the crystal, except possibly for molecules at walls between domains of different ordering of the phenyl rings in the low-temperature triclinic phase of p-terphenyl. The double wells, supposed to be responsible for the spectral diffusion, were found to be slightly non-degenerate, from analysis of the temperature dependence of the correlation function. Finally, the distribution of frequency drift in a given time was used to conclude that the two-level systems should lie in planes, rather than being either randomly or uniformly distributed in space or along linear faults. These results suggest identifying the two-level systems with p-terphenyl molecules in domain walls. It is to be hoped that more analyses of this kind will be performed in the future to confirm and refine this interesting interpretation. A point worth further exploration is whether neighboring molecules in the domain wall can be considered as independent two-level systems, or whether their movements are correlated. Autocorrelation of the instantaneous frequency (see above) is therefore likely to become an important tool for the analysis of broadening, since it in principle captures all the times and amplitudes of coupling to many two-level systems. A practical drawback of the method is that the temporal resolution is restricted to the order of 1 s by the scan rates of dye lasers and the fluorescence rate of the molecule. It is to be hoped that frequency autocorrelation will be extended to the microsecond timescale by use of diode lasers, which have already been used to study spectral hole broadening in this range (Wannemacher, Koedijk and Volker [1993]). This would also imply carehl choice of the probe molecule to reach a measurable fluorescence signal in one sweep of the zero-phonon line. 4.3.3. Single-molecule lines of terrylene in polyethylene

Spectral diffusion of single-molecule lines has been observed in a number of polymer matrices (Kozankiewicz, Bernard and Orrit [ 19941). Here we focus on the much studied system of terrylene in polyethylene. Terrylene (see fig. 10) is a higher analogue of naphthalene and perylene (Clar [1964]), with a strong

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n 0.wo 100

200

300

400

0

HOMOGENEOUSWIDTH (MHz)

Fig. 29. Distribution of zero-phonon linewidths of terrylene in polyethylefle at 1.8 K. The spread of this “homogeneous” width may be understood in terms of spectral diffusion, or changes in the stabilization energy of the guest molecule following relaxation of two-level systems in the neighborhood during the recording. The lower cutoff corresponds to the lifetime limit. Polling of single-molecule properties has great potential for direct comparison between theory and experiment, without configurational averaging. Readers should be warned, however, about possible bias of data from single molecules; e.g., if experimenters have a tendency to select the stronger and therefore the narrower lines. A very broad single-molecule resonance could go undetected in the background level. The continuous line is derived from a simple analytical model (Fleury, Zumbusch, Orrit, Brown and Bernard [1993]).

absorption band at about 570 nm in polyethylene and a high fluorescence yield of 0.7 (Bohnen, Koch, Liittke and Mullen [1990]). This system yields intense zero-phonon lines with widths in the range 50400 MHz at 1.8 K. Most lines are well represented by a Lorentzian profile, but some are notably asymmetrical, or have secondary maxima or shoulders in their wings. Figure 29 shows the distribution obtained from -180 lines (Fleury, Zumbusch, Orrit, Brown and Bernard [ 19931). The sharp drop at -50 MHz corresponds to the lifetime broadening of the excited state 1/27dT1 =42 MHz, with the measured lifetime found to be 3.78 ns (Plakhotnik, Moerner, Irngartinger and Wild [1994]), implying that some molecular resonances are not broadened by transitions of two-level systems. They may be associated with crystalline parts of the polymer, or with chance holes in the two-level distribution in the amorphous parts. On the other hand, the long tail on the other side of the peak corresponds to molecules with more than the average number of twolevel systems in their neighborhood. Figure 29 underlines the point that zerophonon linewidths may not be “homogeneous” in disordered systems, but vary widely from one molecule to another. Similarly, the linewidths are found to have

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different temperature dependences for different molecules (Fleury, Zumbusch, Orrit, Brown and Bernard [ 19931). The model presented in $2.2 describes the histogram reasonably well. An approximate analytical form follows from equating the half width at half maximum of the molecular resonance in a particular neighborhood of two-level systems with the variance of the instantaneous frequency:

The distribution of ytom may be found by applying the statistical method (Stoneham [1969]) to the sum of pair perturbations, yielding the solid line in fig. 29 (Fleury, Zumbusch, Orrit, Brown and Bernard [1993]). This neglects possible, but most probably, rare dephasing contributions of very fast two-level systems, along the lines of the discussion in 5 2.2.2 and the fact that close twolevel systems will cause splitting rather than broadening of the lines. Indeed, the cutoff in the tail of the distribution comes from setting a lower bound on ri in the summation. 4.3.4. Fluorescence autocorrelation

Up to now we considered the influence of spectral diffusion on the broadening of single-molecule resonances in steady-state spectra, assuming all two-level system time constants are short compared to the recording time. In real systems, line broadening is time dependent because of the spread of characteristic times of spectral diffusion. The autocorrelation of the fluorescence intensity also responds to all the different time components [see eq. (S)]. Compared with correlation of the frequency, it has the advantage of already covering the time scale from -1Ons to -100s. On the other hand, one does not know where the molecular resonance lies during the dark periods. The principle of the method is shown in fig. 6 (above). A jumping two-level system shifts the molecule in and out of resonance with the laser at fixed frequency. A typical “random telegraph” signal, similar to photon bunchmg, is observed (see fig. 28). g(’)(z) does not characterize the interactions with two-level systems in the same way as C,(t). We may expect g(’)(z) to be more sensitive to distant two-level systems, and C,(z) to those nearer when both are present. Most molecules of terrylene in polyethylene showed no correlation, to within noise, between lop6s and 10 s. About 20% showed some decay of the correlation

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--

0.4

7

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0

3u U

o.2

U

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0.1

0.0 0.001

0.01

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TIME (ms)

Fig. 30. Three examples of the fluorescence autocorrelation function of single terrylene molecules in polyethylene at 1.7 K. Curve (a), which is monoexponential over 6 decades of time (see fit), is interpreted as corresponding to a molecule coupled to just one two-level system with a time constant in the range of the recording. Curve (b), which has two exponential components, may arise from a molecule coupled to a pair of two-level systems. Curve (c), showing a continuous spectrum of decay times, may be associated with a molecule coupled to many two-level systems or to a few with varying time constants because of structural relaxation.

in this range. Figure 30 shows examples of fluorescence auto-correlation functions obtained for different molecules of terrylene in polyethylene (Fleury, Zumbusch, Orrit, Brown and Bernard [1993]). According to eq. (9), the monoexponential decay of the fluorescence correlation in curve (a) corresponds to coupling of the molecule to a single two-level system with a time constant in the range covered by the autocorrelator, 1 ps to 10s. The second curve, with two exponential components, means a pair of two-level systems are involved. The continuous decay of curve (c) could mean that many two-level systems are involved. On the other hand, one might suppose that slow structural relaxation of the glass during the recording was modifying the barrier properties of one, or of a small number of two-level systems. Coupling between two-level systems was also observed when the correlation function was quite different before and after a small spectral jump (smaller than the 30 GHz scan range of the laser). As mentioned in 4 2.2, coupling to a single close two-level system should split the zero-phonon line of a nearby probe molecule. Most of the time, such pairs of lines could not be identified since the splitting is unknown. Occasionally, two lines were observed to jump after a few minutes to new positions, with the same splitting. This was interpreted as a molecule coupled to a “fast” two-level system (compared to the scan time) and to a “slow” one. One pair of lines jumped

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0.01

0.1

1

10

100

1000

TIME z (ms) Fig. 3 1. Example of a pair of resonances identified with a single terrylene line split by coupling to one two-level system. In agreement with the model of 5 2.2.2, the correlation functions of the peaks have the same decay time and contrasts in inverse ratio of the intensities. Increasing the temperature accelerates phonon-assisted tunneling and brings the occupations of the two-level system states closer to equality.

back and forth several times over an hour or so. Identification of the pair of lines with the splitting caused by one two-level system was confirmed by the fact that the fluorescence autocorrelation functions of both lines had the same characteristic time, with the stronger contrast on the weaker line, in agreement with the prediction of eq. (9) (see fig. 3 1). Once a pair of lines has been identified, one can also determine the asymmetry of the two-level system by assuming their relative intensities reflect Boltzmann equilibrium. For example, the asymmetry of the two-level system studied in fig. 3 1 was 2.7 K. The temperature dependence of the tunneling rate, k ( T ) = kl + kZ, could also be determined from measurement of the fluorescence correlation function as a function of temperature, for cases where g @ ) ( t )showed a well-defined monoexponential decay in the time range covered by the correlator. k ( T ) generally increased comparatively slowly with temperature; i.e., by a factor of less than 100 between 1.4 K and 4.5 K. Higher temperatures were difficult to study because of large, irreversible (on our timescale) spectral jumps. The slow temperature dependence is in agreement with tunneling rather than activated barrier crossing

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I"

1.5

2

3

4

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TEMPERATURE (K)

Fig. 32. Two-level system tunneling rates can be deduced from the decay time of the fluorescence autocorrelation. The slow power-law dependence of the rate on temperature for molecules (a) and (b), T' and T 3 , respectively, follows from a model of phonon-assisted tunneling between quasidegenerate two-level states, involving one and two acoustic phonons. The activated law (c), exp(-TolT) with To = 15 K, may indicate tunneling between wells stabilized by a polaron effect (see text). Note the variety of laws in a given material (polyethylene) at a given temperature (1.7 K).

which would be described by k(T)= w oexp(-V/kBT), where wo M 1013s d is the expected oscillation frequency in either well. The barrier height has to be large to reduce the effect of the prefactor to the observed timescale (milliseconds),but then the temperature dependence would be much faster than observed. Most of the time, the k(T) followed power laws, k(T)- TO, with r]= 1 or 3 (see fig. 32). These values are in agreement with tunneling between weakly asymmetrical wells (AE< kBT) assisted by coupling to acoustic phonons. The one-phonon absorption or emission process should give r]= 1, while a Raman-like twophonon process, in whch a phonon is emitted and another of nearly equal energy is absorbed, should give r] = 3. While these were not unexpected, being in agreement with a variety of other bulk measurements (e.g., thermal and dielectric properties of glasses; Phillips [1981]), the above results show the possibility of very direct verification of tunneling theory, without the complication of configurational averaging. It should also be noted that in the same sample at the same temperature, several temperature dependences were observed, a point which should therefore be taken into account in configurational averaging for bulk properties. One law appears not to have been observed previously: k(T) M exp(-TOlT), with a prefactor much smaller than expected for activation and TO= 15 K. This behavior cannot be explained by one-phonon-assisted tunneling, because the asymmetry would be too large to observe both wells at

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low temperatures. It is predicted for tunneling when a large polaron effect stabilization of the two-level system by coupling to phonons - does not follow adiabatically the tunneling entity (Kagan [ 19921). Bulk optical measurements on glasses, such as hole burning and photon echoes, have already shown great potential for the investigation of matrix dynamics over many decades. Measurement of individual two-level systems by spectral difision of single-molecule lines and fluoresence autocorrelation seems a promising way of refining our picture of the dynamics of glasses at low temperatures by eliminating configurational averaging. It will be most interesting to study systems by all three methods, combining, so to say, the statistics of a full election with the insight of an opinion poll. I

400 1

1800

1600 1400

1200 1000

200

-

400

tlSl

600

Fig. 33. Reversible photophysical “hole burning” of a single perylene molecule in polyethylene at 1.5 K. The laser was tuned to the molecular resonance at low power (upper trace) and then the power was increased by a factor of six (lower trace). Photon bursts are then much shorter, showing that the spectral jumps out of resonance, which produce the random telegraph signal, are at least in part photo-induced. (From Baschk, Ambrose and Moerner [ 19921.)

4.3.5. Photo-induced spectral diflusion We now turn to photo-induced spectral diffusion in polymer matrices. Here, tunneling may be assisted by excitation of the probe molecule (Basche and Moerner [1992], Basche, Ambrose and Moerner [1992]). Figure 33 shows an example of a photoinduced effect for a perylene molecule in polyethylene at 1.5 K. It is, in fact, an example of behavior akin to photo-physical hole burning. The laser is first tuned at low power to a single-molecule line. Bistable behavior of the fluorescence intensity suggests that only one two-level

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system may be involved (see 0 2.2). At higher power, the molecule remains in resonance for shorter times. This may reflect coupling between the probe and the two-level system, such that the barrier is reduced when the molecule is in its excited state (Hayes, Stout and Small [1981]). Consideration should also be given to the release of heat by the molecule, in completely non-radiative and in radiative vibronic transitions. Fleury, Zumbusch, Orrit, Brown and Bernard [1993] found that some single-molecule lines of terrylene in polyethylene were sensitive to irradiation, while others were not. A simple model was proposed to fit the fluorescence autocorrelation function, in which it was assumed that the rate of photo-induced jumps was proportional to the excitation intensity.

Q 5. Outlook In the few years since study of single molecules began, a wealth of experimental results has been gathered. But what are the advantages of studying single molecules, when most experiments - even selective ones such as hole burning involve large populations of molecules? First, single molecules give direct access to the distributions of molecular parameters and to correlation between different parameters, though at the expense of extensive measurements on large numbers of individuals. Conventional methods usually give only the average value of a parameter or at most the second moment of its distribution. Second, comparison to theoretical models is greatly simplified in single-molecule spectroscopy, because the final averaging over the sampled population is not necessary. The more direct comparison may increase our confidence in the pictures used, or may eventually lead us to refine them. Third, some experiments are possible only with single molecules. For example, the fluorescence-correlationmethod and the triggering of a perturbation by the photon emitted by the molecule itself have no equivalent for ensembles. Fourth, the single-molecule method has the power to address a single localized spot in a solid or on a surface. This feature is interesting for a fundamental understanding of the interaction of the molecule with known structures, and also for potential applications in molecular electronics, where molecules would be associated in nanometer-sized devices. In the following, we examine different areas of solid-state physics and chemical physics in which the introduction of the single-molecule method could bring interesting progress. 5.1. OPTICALLY DETECTED MAGNETIC RESONANCE (ODMR)

ODMR experiments on single pentacene molecules have been demonstrated and are expected to be generalized to other systems as they emerge. The main interest

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of ODMR measurements is to provide information about spin-spin interactions and therefore about the environment of the electron. Magnetic resonance can be observed in continuous-wave measurements, and the fluorescence-correlation method has been shown to be very helpful by providing direct time-domain results. But microwave excitation pulses can also be applied, and the observed transient signals may be selectively sensitive to spin dephasing and other processes, giving a clearer picture of the dynamics of the spin landscape at the nanometer level. Such experiments are likely to develop in the next few years. The ODMR done so far used the zero-field splitting of a triplet state, but the effect of an external magnetic field can be envisaged immediately. The spin levels and intensities as functions of the orientation of the static magnetic field will give information about molecular wave functions (Groenen, Poluetkov, Matsushita, van der Waals, Schmidt and Meijer [ 1992]), and their variations from molecule to molecule. Further, we saw that nuclear spins broaden the electron-spin resonance by hyperfine interactions. Repeating the experiment with deuterated host and guest molecules, under isotopic substitution by 'H and I3Cin controlled positions is a very exciting perspective, where splittings and broadenings of the magnetic resonance lines could be accessed precisely, see note added in proof. Once nuclear spin levels are identified safely, a third step might be to address and polarize a single nuclear spin in the molecule. Such techniques as electronnuclear double resonance (ENDOR) could be applied (although the resonance here would be triple: the laser with the orbital movement of the electron, the microwave with its spin, and the radiofrequency with the nuclear spin). The transfer of order would then go all the way from laser radiation down to polarization of a nuclear spin. In order to become a real spectroscopic tool, the method must first be extended to triplet states of other molecules, and also to other species like the doublet states of free radicals. To achieve magnetic resonance in such systems, the usual degeneracy of the doublet should be lifted by a magnetic field. If the ODMR method is applicable to a wider class of single molecules, it would be very helpful for investigation of biologically interesting systems or of molecules that are difficult to crystallize or to orient. 5.2. SPECTRAL DIFFUSION, TWO-LEVEL SYSTEMS, AND GLASSES

The investigation of spectral diffusion and of two-level systems shows that single-molecule spectroscopy has a huge potential for the study of the structure and dynamics of disordered solids at low temperature. This direction of research will probably develop further. For example, single two-level systems could

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be followed as external perturbations like pressure or static electric fields are applied. These could perturb the asymmetry and the barrier height of the doublewell potential. Double-resonance experiments can be imagined, in which the twolevel system would be driven by an electric or acoustic wave. The correlation functions of single-molecule fluorescence show not only wellcharacterized features attributable to two-level systems, but also other structures which could arise from interacting two-level systems or from more complex defects. Some evidence for interaction between two-level systems is provided by changes in correlation rates corresponding to one system, after a frequency jump corresponding to another. Fundamental knowledge of these elementary processes would be useful to test more sophisticated models of the structural relaxation of glasses. The most interesting step in persistent spectral hole burning is the photophysical modification leading to a jump of the resonance frequency of the molecule. In a hole-burning experiment, a large population of molecules is removed from the initial burning frequency and spread over at least part of the inhomogeneous band. The detection of the broad product band is much more difficult than that of the narrow hole. A single molecule, on the other hand, keeps its narrow resonance line and may be located precisely in the spectrum after jumping. Extensive statistics of these jumps would give insight into the burning process. 5.3. STATISTICAL STUDIES OF MOLECULAR PARAMETERS

True statistical studies of single molecules require larger samples than the few hundreds of molecules that can currently be studied manually. Parallel excitation and acquisition of data on many molecules simultaneously is an essential step toward better statistics and a deeper understanding of microscopic heterogeneity. These studies would provide not only the average values of molecular parameters such as homogeneous width, dipole moments, lifetimes, relaxation rates, and the like, but also their full distributions. Hole-burning measurements can be performed at various wavelengths within the inhomogeneous band. Molecular parameters deduced from the hole can then be correlated with the frequency. The results can be compared with theoretical models, as for example the color effect in the pressure shift (Sesselmann, Richter, Haarer and Morawitz [1987], Laird and Skinner [ 19891). With single-molecule spectroscopy, such studies are not limited to correlations between the absorption frequency and another parameter, but can be extended to correlations between any two parameters that can be measured on the same single molecule. This kind of information is especially

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usehl in the study of very complex systems (as in astronomy, medicine, social sciences, etc.), and it seems promising to apply it to the difficult problem of structure and dynamics of disordered solids. 5.4. NON-LINEAR AND QUANTUM OPTICS

Among single quantum systems, the most studied experimentally are atoms and trapped ions isolated in beams or traps (Kimble, Dagenais and Mandel [1977], Neuhauser, Hohenstatt, Toschek and Dehmelt [ 19801, Diedrich and Walther [1987]). Their simplicity makes them valuable as models in non-linear optics and to test fundamental quantum optics and electrodynamics. Among the many experiments performed on single systems in recent years, several of them could be translated for single molecules, to give new information about quantum behavior in condensed matter. A simple non-linear optical thought experiment consists of exciting the molecule with two frequencies. An example of non-linear behavior is the change in fluorescence signal obtained in ODMR (see 5 4.2.6). Similar double-resonance experiments could be done with two optical excitations. Each frequency could address a particular transition in the molecule. The additional difficulty of finding sharp resonances for two transitions would be compensated by the richness of the three-level system. For example, a strong (say, singlet-singlet) transition could be probed by a blue laser, while a red laser would be scanned through the resonance of a weak (say, singlet-triplet) transition. In ODMR, the resonance signal appears in the steady-state fluorescence and in its correlation function. Similarly, changes in fluorescence intensities under double excitation could be detected with continuous excitation, or as quantum jumps (Nagourney, Sandberg and Dehmelt [ 19861, Cook [ 19901, Cohen-Tannoudji and Dalibard [ 19861). Very interesting spectroscopy of nearly forbidden transitions would then become possible. A molecule could also be used to test subtle phenomena like the Zen0 effect (Misra and Sudarshan [1976], Itano, Heinzen, Bollinger and Wineland [ 19901). Frequent measurement of the population of the ground singlet state by the blue laser delays or prevents buildup of population in the triplet state by the red laser. Excitation of one level of single molecules with two photons is also imaginable. It could have the advantage of producing a sum or difference of frequencies, which could be discriminated easily from the exciting frequencies. Direct observation of the resonant fluorescence would then become possible, while it must be rejected with the exciting light in the usual one-photon excitation method. The resonant fluorescence line holds much information

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about dynamical processes affecting the emitting dipole. The emission line of a molecular dipole subject to dephasing contains a sharp line, resonant with the laser, and a broadened Lorentzian component indicative of dephasing (Hochstrasser and Novak [ 19771). Slow spectral diffusion would affect the intensity and the width of the elastic component. At high exciting intensity, the fluorescence line should exhibit three lines (the Mollow triplet; Mollow [1969]), arising from the modulation of spontaneous emission by Rabi oscillations in the laser field. Analysis of the fluorescence triplet would also give insight into dephasing mechanisms. The ac Stark effect (also called light shift) may also be measured with two frequencies (Cohen-Tannoudji and Kastler [ 19661, Cohen-Tannoudji, DupontRoc and Grynberg [1992]). The light shift arises from a differential shift of ground and excited molecular states by an intense quasi-resonant excitation. A strong laser beam would shift the molecular resonance and a weak resonant beam would probe it, see note added in proof. Interesting phenomena should appear when a single molecule is coupled to a single mode of the electromagnetic field (Meystre [1992], Ng, Whitten, Ramsey and Arnold [1992]). If the mode belongs to a cavity with a very high quality factor, feedback from the cavity is sufficient to alter the spontaneous emission rate, which can be enhanced or inhibited, depending on detuning between molecule and cavity (Haroche and Kleppner [ 19891). Similarly, even a very small number of photons present in the cavity could lead to a measurable light shift. Since the light shift depends on the number of photons, it provides a means to count photons in the cavity mode without destroying them; i.e., a quantum non-demolition measurement. By this method, it would be possible to prepare states of the cavity field with a well-defined number of photons (nonclassical Fock states; Haroche [1992]). Many other experiments, like the preparation of EPR correlated systems, which are proposed for tests of quantum mechanics in atomic physics could perhaps be performed with single molecules. For example, a superposition of different quantum states of a single molecule would resemble more closely the macroscopic Schrodinger cat than does an atom, and could help testing quantum mechanics on mesoscopic systems in condensed matter. 5 . 5 . NEAR-FIELD OPTICS

As mentioned in 1, the techniques of near-field microscopies can be applied with all sorts of signals and in particular with light. This new type of microscopy, called near-field optics (Pohl and Couqon [ 1993]), consist of probing surfaces

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Fig. 34. Near-field optical images of single cyanine dye molecules in PMMA at room temperature. Molecules appear as spots whose various shapes give information about the field at the tip and about molecular orientation. (From Betzig and Chichester [1993].)

with light waves at a sub-wavelength resolution. At such length scales, the optical waves are confined by material structures; e.g., metals or highly refringent materials. In a typical design, a tapered optical fiber is coated with a metal except at its end (Betzig and Chichester [1993]). The small hole thus left at the end is used as a source of an optical near-field to probe substrates. The character of the waves in the vicinity of such obstacles is akin to the well-known evanescent waves of classical optics. The interaction of molecular systems with such waves is more complex than with ordinary propagating waves. For example, the classical concept of cross-section loses its meaning because it assumes the field to be homogeneous over areas of the order of the cross-section itself. For a high quality-factor resonance such as the excited state of a molecule, the crosssection, of the order of h2, is much larger than the hole at the fiber’s end. The interaction of the molecule with the near-field is then determined by the actual value of the electric field at the molecule. Spectacular images of single molecules at room temperature show the potential of this method (fig. 34). The main advantage of near-field optical designs stems from the smallness of the probed area. The waist area A of the exciting beam has been reduced to a sub-wavelength size. The signal-to-noise ratio discussed in tj 3 can therefore

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85

reach huge values. In practice, while the detection of single molecules in roomtemperature liquids is possible with highly efficient, diffraction-limited confocal optics, their detection on scattering and “dirty” surfaces such as polymers has been made possible by near-field optics only. The application of this excitation method to low-temperature spectroscopy of single quantum systems (Hess, Betzig, Harris, Pfeiffer and West [1994]) would clearly improve by orders of magnitude the signal-to-noise ratio. Measurements of many more host-guest pairs, including absorption measurements of non-fluorescent molecules, would then become possible. The near-field method has the unique potential to bridge between spectroscopy at liquid helium temperatures and the high-temperature observations on the same systems, see note added in proof. But a near-field microscope working at helium temperature would also present other specific advantages. The spatial resolution accessible with a near-field probe allows high resolution imaging and the localization of molecules. It would then be possible to obtain a mapping of the spatial coordinates in the sample onto the frequency axis with high resolution. In a second step, one could address specific spots at will by an external parameter, the laser frequency, without moving any tip. In particular, parallel addressing of several molecules at once would become possible. Localization of molecules is also interesting for studying their interaction with other structures in the sample. These can be natural (e.g., surfaces, defects, impurities, two-level systems, charges) or artificial (e.g., wires, electrodes, coatings, absorbers), including the tip itself. The effect of the structures on the molecular resonance could be shifts (Stark effect), dynamic effects, quenching or quantum cavity enhancement, or reduction of radiative rates (Ambrose, Goodwin, Martin and Keller [ 1994]), etc. Because the spatial resolution is very high, the spectral selection of molecules is no longer essential, and time-resolved measurement with broad-spectrum pulses would be possible on single molecules. This would open many possibilities of non-linear optics with short-lived levels, and of studies of photochemical processes at low temperature in well-defined single systems. Spectral diffusion would also be easier to follow throughout the whole inhomogeneous band because the molecule’s coordinates allow a safe identification. Finally, a very exciting perspective is the study of solid solutions at high concentrations. The combination of the very high spatial selectivity of near-field optics with the spectral selection of low temperature allows contemplation of single-molecule spectroscopy with concentrations around 10-2 mole/l or higher. At such concentrations, the distance between molecules is of the order of a few nm and intermolecular interactions become very important. Since the

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resonance of molecules could be tuned by applied fields, it would be possible to study resonant and non-resonant energy transfer, excitonic effects, and the like. Interactions between excited molecules can usually be studied with high excitation densities only. With near-field excitation, two weak resonant beams could excite interacting molecules simultaneously, to create a wealth of new nonlinear effects. 5.6. SINGLE LOCALIZED STATES IN OTHER MATERIALS

Single-molecule spectroscopy has been demonstrated so far on a very small number of host-guest couples. Although several kinds of matrices have been used, from molecular crystals to various polymers (Kozankiewicz, Bernard and Orrit [1994]), the guest so far has been an aromatic molecule. Many other kinds of guest molecules could be tried. Dyes, for example, have good emission properties. However, their excitation involves strong charge transfer, so that their Debye-Waller factor is usually too low to observe zero-phonon lines. Transitionmetal ions or rare-earth ions are well characterized guests of inorganic insulators. There is little doubt that well-chosen ionic systems could allow observation of single ions. Substantial improvements in light collection and background reduction would make single-molecule or ion spectroscopy possible in many systems which are not optimal according to the conditions of § 3.2.5. A very important point in applying single-molecule spectroscopy to devices is the control of the molecular environment. While the control of the quality of the environment is very difficult today, progress in this direction is imaginable. Selfassembly or Langmuir-Blodgett techniques can lead to well-defined molecular structures. Another possibility would be to apply an inhomogeneous field to an otherwise perfect crystal with low impurity concentration. The inhomogeneous shifts of electronic resonances would be directly related to the spatial coordinates of the molecules. Single localized electronic states can also be found in semiconductors, either as natural structures (trap states) or as artificial ones. The spectroscopy of single quantum dots and wires has recently been demonstrated (Brunner, Bockelmann, Abstreiter, Walther, Bohm, Trankle and Wiemann [ 19923, Birotheau, Izrael, Marzin, Azoulay, Thierry-Mieg and Ladan [ 1992]), and has very close kinship to single-molecule spectroscopy. This is not surprising, since the pi-orbital of an aromatic molecule is nothing else than a conveniently self-assembled potential well for an optical electron. Many of the concepts discussed in this chapter are therefore applicable to quantum dots.

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PI

6. Conclusion

In the few years since the first experiments, the field of single-molecule spectroscopy has been enriched gradually with many new results. Among these are classical spectroscopic measurements done on single molecules, like determination of linewidths, lineshapes, and the effect of external fields. More complex phenomena were also demonstrated, for example the pure quantum anti-bunching of fluorescence photons, or the double-resonance experiments of ODMR, where two resonant waves excite the same single molecule. Some of these observations, like the isolation of single two-level systems or the study in real time of spectral diffusion, have no direct equivalent for a large population of independent molecules. It is worth noting that most of these results were obtained with classical spectroscopic tools, available in many laboratories working on optical properties of solids at low temperatures. The next years will probably see the application of more sophisticated techniques to single-molecule studies, in particular those of near-field microscopies or microfabrication. One can then expect that the combination of nanotechnologies with single-molecule spectroscopy will shed new light on solid-state physics and chemistry at nanometer level. The particular advantages of single-molecule spectroscopy will then be useful: access to distributions and correlations of molecular parameters, elimination of averages, selection of single nanoscopic areas within solids, specific quantum effects. The results summarized in this chapter have relation to several other fields of modern physics and physical chemistry, like fundamental quantum mechanics, solid-state science, and others. We think that they also pertain to the emerging field of molecular electronics, spawned by the dream of building devices and machines out of single molecules. Even if this ambitious aim is out of our reach today, we hope that the study and manipulation of single molecules will support its long-term credibility, if only by removing psychological barriers and helping to gather new fundamental knowledge.

Acknowledgement It is our pleasure to acknowledge stimulating collaboration with both our colleagues and visitors, with whom many of the experiments were done. The ODMR work was done in collaboration with Dr. J. Wrachtrup and Prof. C. von Borczyskowski, then at the Freie Universitat Berlin. The Stark effect on terrylene in polyethylene was measured with Prof. R.I. Personov of the

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Institute of Spectroscopy of the Russian Academy of Sciences, Troitsk. We should also like to thank L. Fleury for performing the Monte-Carlo simulations in $2.2. He and A. Zumbusch performed many measurements on two-level systems. Finally, we are grateful to Ph. Tamarat for reading the manuscript.

Note added in proof

Since this review was written (July 1994), several important developments occurred in the field of single molecule spectroscopy at low temperatures. Let us briefly mention the ones that we find most significant: Single molecules have been observed in new systems, such as terrylene in p-terphenyl crystals (Kummer, BaschC and Brauchle [ 1994]), pentacene in naphthalene crystals and other highly fluorescing molecules in various matrices. The proposed ac-Stark effect experiment has been carried out on single terrylene molecules in p-terphenyl. The observed light-shift and Autler-Towneslike structures are in full agreement with theoretical expectations (Tamarat, Lounis, Bernard, Orrit, Kummer, Kettner, Mais and BaschC [1995]). Magnetic resonance experiments on single molecules showed the influence of a single 13C atom in the molecule, whose transitions cause shifts of the ODMR line (Kohler, Brouwer, Groenen and Schmidt [1995]). Shifts and broadening of ODMR lines under magnetic field were measured and interpreted (Gruber, Vogel, Wrachtrup and von Borczyskowski [ 19951). New insight has been gained about spectral diffusion from a comparison of experimental frequency trajectories of single molecules with theory (Reilly and Skinner [ 19951). First experiments have been presented using near-field optical excitation of single pentacene molecules in a p-terphenyl crystal in liquid helium (Moerner, Plakhotnik, Irngartinger, Wild, Pohl and Hecht [ 19941). The lines were shifted by a dc voltage applied to the metal-coated probing tip. Further experiments are in progress.

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E. WOLF, PROGRESS IN OPTICS XXXV 0 1996 ELSEVIER SCIENCE B.V ALL RIGHTS RESERVED

INTERFEROMETRIC MULTISPECTRAL IMAGING BY

KAZUYOSHI ITOH Department of Applied Physics, Graduate School of Engineering, Osaka Wniversiv Suita, Osaka 565, Japan

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. . . . . . . . . . . . . . . . . . .

5 2 . CLASSIFICATION OF THE SPECTRAL IMAGING

TECHNIQUES . . . . . . . . . . . . . . . . . . . .

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9 5. NOISE LIMITATIONS . . . . . . . . . . . . . . . . . 5 6. CONCLUSIONS . . . . . . . . . . . . . . . . . . . .

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LIST OF SYMBOLS AND ABBREVIATIONS . . . . . . . . . .

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0

1. Introduction

Astronomy is one of the important frontiers of optics. Spectrometry and imaging have long been the standard tools for observation in astronomy. For speedy observation, considerable efforts have been made to develop instruments which can simultaneously acquire spatial and spectral information. We will see some of the results of these efforts below. Simultaneous acquisition of spatial and spectral information has also become popular in the field of remote sensing, and is termed multispectral imaging. Recent technological progress in optics and electronics now allows us to obtain and handle easily multispectral images having a number of spectral channels as many as the number of spatial pixels in one dimension. We may call this class of multispectral imaging techniques supermultispectral imaging or simply spectral imaging. Efficient techniques for spectral imaging have potential applications in such diverse fields as global assessment of the environment and natural resources, surveillance for agriculture, and visual inspection in industries and biomedicine. The advent of new types of sources of low-coherence or wide-band radiation, such as ultra-short laser pulses and synchrotron orbital radiation, will also increase the need for spectral imaging techniques. The first elegant idea for simultaneous imaging and spectrometry may be attributed to Lippmann (vide Born and Wolf [1970]; 5 7.5). Suppose that a photographic plate coated with a transparent, fine-grain emulsion is placed in the image plane of an image-forming lens, with the emulsion side away from the incident light and in contact with a reflecting surface of mercury. If an area of interest is exposed to normally incident quasi-monochromatic light of mean wavelength ;lo, the silver at this area in the developed plate forms a system of equidistant layers, parallel to the surface of the emulsion and with separation &/2. This system of partially reflecting layers acts as a selective reflector for light of the wavelength used to expose the area. Recording can be done by using the light of multiple spectral lines, and a colored image can be retrieved by white light. The first practical and scientific instrument dedicated to spectral imaging may be attributed to Harwit [1971, 19731. He used the multiplexing technique to obtain the whole information for spectral imaging. He used a two-dimensional (2D) encoding mask on the image of an object located at 147

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the entrance pupil of a dispersive spectrometer and a one-dimensional(1D) mask on the exit pupil of the spectrometer. These masks modulate the light flux and give substantially the Hadamard transforms of the input signal distributions over the respective pupils. A single detector was placed behind the 1D mask to detect the modulated signal. In the current state of technology, we may not necessarily use such a one-detector system, because two-dimensional arrays of detectors with acceptable uniformity are already available. However, we will have to use partly the multiplexing technique to fill the dimensional gap between the 2D detector array and the 3D data of spectral images. It seems hopeless to build a highresolution 3D array of detectors. The only existing 3D detection systems are the layered structure of photographic color film and the volume of photosensitive media, such as a photographic emulsion or photorefractive crystal. However, these detector systems or elements currently seem to lack in 3D density of information capacity, and are difficult to interconnect tightly with modem signal processors. Examples of succesfd multiplexing techniques for image formation and spectral recovery are radio interferometry and Fourier spectrometry. These seemingly different issues are dealt with thoroughly in the framework of the coherence theory (Bornand Wolf [ 19701). In this review article, we will see that the principles of radio interferometry and Fourier spectrometry can be deduced from a unified theory suggested by Itoh and Ohtsuka [1986a]. This unified theory is based on the theoretical framework developed by Wolf and Carter [1978] for the coherence and radiant intensity associated with primary planar sources. Such a unified understanding is important from the academic and practical viewpoints. The unified treatment suggested a new interferometric technique for spectral imaging (Itoh and Ohtsuka [1986b1). It later provided a theoretical background to a similar technique for astronomical observation suggested independently by Mariotti and Ridgway [ 19881 and developed as an advanced technique by Zhao, LCna, Mariotti and Coude du Forest0 [1994]. Another aspect of spectral imaging is worth calling to the reader’s attention. Spectral imaging is expected to provide a powerful tool for scene analysis, or understanding by machine vision. Understanding of a scene by machine vision is one of the biggest dreams in the scientific and industrial communities.The author believes that to convert this dream into reality, we should increase drastically the amount of usable information to compensate for the lack of computing power of current computers and the absence of efficient algorithms for scene analysis. The most effective way of increasing the amount of usable information may be to add another dimension to the input signal. The use of a time series of a moving picture is one of the simplest solutions, and is suited to the recognition or identification of moving targets. The vision systems of vertebrates

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and insects seem to make good use of this abundance of information in timevarying signals. In contrast, spectral images are suited to the analysis of static but complex scenes full of shadings and occlusions. The fundamental difficulty of image segmentation may hopefully be overcome by identifying the materials that comprise each object. In $2, the spectral imaging techniques are surveyed and categorized in a simple manner. The key to the simple categorization is that the basic techniques for imaging and spectrometry are classified in the same manner. The unified theory of interferometric imaging and spectrometry first suggested by Itoh and Ohtsuka [1986a] and later refined by Itoh, Inoue and Ichioka [1987, 1990al is presented in $ 3. The methodology for coherence detection and applications to 3D Fourier transformation are introduced. The fast techniques for spectral imaging are reviewed in Q 4. Noise limitations associated with specific interferometric spectral-imaging systems are summarized along with those of some conventional spectral-imaging systems in Q 5. Experimental results are also presented. This review is summarized briefly in Q 6.

9

2. Classification of the Spectral Imaging Techniques

To provide a brief survey of various conventional techniques for spectral imaging, let us first categorize separately the basic techniques for imaging and spectrometry. It is interesting that the basic techniques for the optical power analysis in these two different areas of imaging and spectrometry can be categorized in the same manner. They are categorized essentially into three techniques: scanning, multiplexing and multichanneling techniques. In the early 1960s, Mertz [1960a,b, 1961, 19641 suggested the use of Fresnel transform coding to improve the signal-to-noise ratios of imaging and spectrometry. The analogy between imaging and spectrometry implies the unified framework for the spectral imaging techniques discussed in Q 2.2. 2.1. DETECTION OF SPATIAL AND SPECTRAL DENSITIES OF OPTICAL POWER

In real applications, the categorization of techniques for imaging and spectrometry is a little more complex than implied above. The categories are listed in table 1 along with typical examples. Most of the examples are well known, and definitions of the categories may be learned from these examples. The simplest technique for imaging and spectroscopy is to scan the region of interest with a slit or a pinhole in front of a single detector. The typical

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INTERFEROMETRIC MULTISPECTRAL IMAGING

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Table 1 Basic techniques and possible hybrids for photodetection Region

Scanning

Multiplexscanning

Spatial

Film scanning

Michelson stellar interferometer

Spectral

Monochromator None

Technique Multiplexmultichanneling

Multiplexing

Multichanneling

Incoherent holography; Multipleaperture synthesis

Hadamardtransform imaging

Photography; Imaging by a detector array

Holographic spectroscopy

Fouriertransform spectroscopy

Spectrography; Multichannel photometry

example is a film scanner which is used to digitize the images recorded on a photographic film. The monochromator is the counterpart for spectrometry. In contrast to scanning, the multichanneling technique, located at the rightmost column, detects simultaneously the whole distribution of optical signals (e.g., the dispersed spectrum) by using an array of detectors. Multichannel photometers, which have recently been used very widely, are the typical examples. In imaging and spectrometry, the optical power density is sometimes modulated intentionally before detection. The multiplexing techniques make it possible to obtain the desired signal distribution by using a single detector without spatial scanning. For example, Fourier-transform spectrometers give us the time series of multiplexed signals to be detected by a single detector. We can recover the power spectral density from the multiplexed interferograms by making use of the Fourier-transform relationship known as Wiener-Khintchine theorem (Born and Wolf [1970]; 9 10.3). Hadamard-transform spectrometry is also included in this category. Hadamard-transform spectrometry was reviewed extensively by Hanvit and Decker [1974]. We can imagine a Hadamardtransform imager (Decker [1970]), if we neglect the spectrometric section in the Hadamard-transform spectrometric imager mentioned before. The incident optical power is modulated by masking the image of an object with a series of spatial masks which are substantially orthogonal to one another. The modulated time series which are detected by a single detector may be decoded to retrieve the image. The simple scanning and multichannel techniques can be coupled with the multiplexing technique to comprise hybrid techniques. The possible hybrids are listed between the basic techniques in table 1, along with typical examples. The Michelson stellar interferometer measures sequentially the

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complex visibility or coherence function that is multiplexed by the propagation process (Born and Wolf [1970]; 5 10.4). The resultant coherence function is measured as a function of the vector spacing of the two apertures. If the incident wave front is folded (Murty [ 1964]), sheared rotationally (Armitage and Lohmann [1965]) or sheared radially (Hariharan and Sen [1961]), the optical disturbances at different points in the pupil plane are made to interfere simultaneously. The coherence function can be measured simultaneously by using a photographic plate or an array of detectors. This is the multiplexmultichanneling type. Mertz ([1965]; §§ 3,4) discussed thoroughly these wavefront shearing interferometers and incoherent holography in the early days. The wave-front shearing interferometers suggested more recently have been reviewed extensively by Roddier [ 19881. Note that rotational shearing interferometers and radial shearing interferometers transform a point object into a sinusoidal fringe pattern and a Fresnel zone plate, respectively. These patterns associated with different points are orthogonal to one another. Hence, some instruments for measuring the coherence function can be considered as carrying out image coding or coded aperture imaging (Rogers [ 19771). The multiple-aperture synthesis systems measure simultaneously the coherence of optical disturbances at various pairs of apertures (Goodman [1970]). The apertures are usually located such that all of the pairs make different base lines to one another. In holographic spectroscopy, only the temporal information on the optical disturbances is of concern, and the optical disturbances at two different occasions are made to interfere. The original idea of holographic spectroscopy dates back to the pioneering work by Stroke and Funkhouser [1965]. The basic techniques for imaging and spectrometry classified in table 1 are arranged in such a way that the signal-to-noise ratio associated with each category increases roughly from left to right. More thorough discussions of the signal-to-noise ratios of these techniques will be presented in $ 5 . Multichanneling in photo-detection is straightforward and has become very popular since the advent of charge-coupled device (CCD) image sensors. This technique has been proven to be efficient by the results of various applications. The diminishing cost and improving quality of CCDs will increase further widespread applications. Table 1 is made as simple as possible, and is not exhaustive. For example, computerized tomography techniques of image recovery from projections (Barrett [1984]) may not fit into any of the categories. 2.2. CLASSIFICATION OF INTERFEROMETRIC SPECTRAL IMAGING TECHNIQUES

So far, most of the techniques which have been suggested for spectral imaging

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are combinations of the basic techniques for spectrometry and imaging. They can be classified by using a table of possible combinations of the categories listed in the previous table. Table 2 shows the combined categories and provides typical examples. The basic techniques are classified in the same manner as in table 1, yet the categories involving spectral multiplex-scanning are omitted as no examples are available. Some non-interferometric techniques and those for radio waves are included to illustrate the principle of classification. Some spectral imaging techniques based on principles which are not separable into the aforementioned basic techniques may escape from this framework. The top row in table 2 shows examples which use the scanning technique for imaging. The scanning monochromator is the simplest spectral imager which uses the scanning technique for both imaging and spectrometry. Some Raman microprobes adopt this combination (Nakashima and Hangyo [ 1989]), yet some of the earliest versions made use of the multichannel configuration for imaging (Delhaye and Dhamelincourt [ 19751, Dhamelincourt and Bisson [ 19771). If we spatially scan a holographic spectrometer which is dedicated to the measurement of a small area, such a configuration falls in the second category of “spatial scanning” + spectral multiplex-multichanneling”. This combination will not be very meritorious, because the speed advantage of multichanneling is spoiled by the scanning process in the spatial domain. From the viewpoint of signal-to-noise ratio, scanning spatially a pinhole attached to a Fouriertransform spectrometer (FTS) is more advantageous. It will be shown later that multiplexing techniques have equivalent or better signal-to-noise ratios than multiplex-multichannel techniques, depending on the class of noise source. The multi-spectral scanner (MSS) and thematic mapper (TM) are the imaging radiometers with multiple spectral bands which were launched on the Landsat satellite (Goetz, Wellman and Barnes [1985]). They scan a strip of the earth’s surface by taking advantage of its orbital motion. Although the number of channels is small, all spectral channels are detected simultaneously by the dedicated detectors, and spectral signals are processed separately. The multiplex-scanning technique is seldom used for image formation in the visible range. This technique is widely used, however, in radio-frequency regions to achieve high angular resolutions. Two possible types of syntheticaperture radio telescopes with two antenna elements are listed at both ends of the second row of table 2. By changing sequentially the vector spacing of the two antenna elements, synthetic-aperture radio telescopes measure the amplitude correlation of radio waves, called the complex visibility in this field, as a function of the vector spacing. The relative position of one of the apertures is changed mechanically or by using the earth’s rotation (Fomalont [ 19731). Radio

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Table 2 Spectral imaging techniques; combinations of the basic techniques ~

Spatlal\Spectral

Scanning

Multiplexmultic hanneling

Multiplexing

Scanning

Scanning monochromator

(Scanning holographic spectroscopy)

(Scanning F T S ~ ) M S S ~ ,T M ~ (Landsat)

Multiplexscanning

Earth-rotation aperture synthesis (with a tunable filter)

-

Double Fourier spatio-spectral interferometry

Lippmann holography; Holographic spectral imaging

FTSI/P

Multiplexmultichanneling

-

Multiplexing

-

Multichanneling

AOTFh with CCD

-

Lippmann natural-color photography

‘

Multichanneling

Earth-rotation aperture synthesis (with a filter bank); FX d; OASIS VLAg

Hadamardtransform spectrometric imaging

-

WAMDII’ (Spacelab); FTSIIIJ

Color TV camera; AIS k; Medusa spectrograph

Fourier transform spectrometer. Multi-spectral scanner. Thematic mapper. Wide-band digital cross-spectrum analyzer. Optical aperture synthesis in space. Fourier transform spectral imaging in pupil plane. g Very Large Array. Acousto-optic tunable filter. Wide-angle Michelson Doppler imaging interferometer. J Fourier transform spectral imaging in image plane. Airborne imaging spectrometer. a

’

image formation and related topics have been reviewed concisely by Bracewell [ 19791. The simplest radio telescope of this class may be equipped with a tunable filter which scans sequentially a single spectral band. If a radio telescope is equipped with a filter bank and multiple bands are analyzed simultaneously such a radio telescope may be included in the class of “spatial multiplex-scanning + spectral multichanneling”. The wide-band digital cross-spectrum analyzer (FX)

I54

rw § 2

NERFEROMETFX MULTISPECTRAL IMAGING

1

c

h 7

X

(Side View)

Detectlon Plane (Boaom View) '.

.:...

:..

i

Detection Plane

Fig. 1. Schematic diagrams of the OASIS (Optical Aperture Synthesis in Space) concept; (a) the whole system and (b) details of the optical layout.

at Nobeyama Ra&o Observatory (Chikada, Ishiguro, Hirabayashi, Morimoto, Morita, Kanzawa, Iwashita, Nakazima, Ishikawa, Takahashi, Hanada, Kasuga and Okamura [19871) may be classified between spatial multiplex-scanning and spatial multiplex-multichmnneling classes, because FX has multiple (viz., six) antenna elements and yet needs help of the earth's rotation for image recovery. The wide-band (320 MHz) TF signals from the six-element radio interferometer of FX are simultaneously A/D-converted and numerically Fourier transformed in real time. The converted complex data,streams are correlated mutually with regard to all (1024) spectral channels and averaged to produce the cross-spectrum at every spectral channel. The resultant cross-spectra are processed by the host computer for spectral image recovery. This spectral imager is included in this class, since the number of elements is not large enough for image recovery at a snapshot. In this regard, we may categorize the OASIS (optical aperture synthesis in space; Noordam, Greenaway, Bregman and le Poole [1987]) system in the same class. This system suggests the use of a dilute array of telescopes in space for spectral imaging. The optical disturbances collected by the array of telescopes are carried by optical fibers and launched from the ends of fibers which are aligned with nonredundant spacing as shown in fig. la. Each beam from the fiber is first collimated by a normal lens and then focused by a cylinder lens along the direction normal to the line of fiber ends as shown in fig. 1b. The focused beam is dispersed at the same time by a prism. In the detection plane, colored fringes

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Fig. 2. Schematic diagram of the double Fourier spatiotemporal interferometer.

with different spatial frequencies of various amplitudes and phases are formed. This pattern of colored fringes has all of the information required for spectral image recovery. The double Fourier spatio-temporal interferometry suggested by Mariotti and Ridgway [1988] takes up a simple but novel combination of the well-known interferometers; viz., the Michelson interferometer and the Michelson stellar interferometer. The schematic diagram is shown in fig. 2. The principle underlying spectral image recovery is a combination of the principles of Fourier spectroscopy and Michelson stellar interferometry. This principle was first derived in the framework of the coherence theory by Itoh and Ohtsuka [ 1986al. They developed the theory for the single-aperture technique called Fourier transform spectral imaging in pupil plane (FTSIP). This technique is categorized into a different class below. Multiplex-multichanneling for image formation implies shearing interferometry or multiple-aperture synthesis, as discussed previously. Lippmann-Bragg holograms or, alternatively, volume reflection holograms (Collier, Burckhardt and Lin [1971]) are also based on interference of light. This technique uses several separate spectral lines of coherent illuminations.The spectral information is coded as the spatial frequency of the interference fringes, and the spatial information is coded substantially as the direction of the fringes. All of the information is recorded simultaneouslyin the hologram. The multiplexed s p a t i e spectral information is decoded holographically in the reconstruction process by Bragg diffraction. Holographic spectral imaging, an incoherent version of volume reflection holography suggested by Itoh and Ohtsuka [ 1986b], is to be discussed later. Fourier-transform spectral imaging in pupil plane (FTSVP) uses shearing interferometry for spatial information recoding and the

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rm I 2

multiplering technique for spectral information coding. Similar to the double Fourier spatio-spectral interferometry, this technique is a realization of the fundamental principle ,of coherence-based spectral imaging. This technique will be thoroughly discussed later. The Very Large Array (VLA, Napier, Thompson and Ekers [1983]) is a 27-element radio interferometer located in New Mexico, USA. This array digitizes the IF signal as does FX at Nobeyama, but first correlates the digitized signals and then Fourier-analyzes the resultant correlation functions. It should be noted that radio interferometers detect directly the amplitude of radiation in radio-frequency regions. The noise characteristics of these systems seem entirely different from their counterparts in optical frequency regions. Since we focus here on the optical frequency regions, radio interferometers will not be discussed hrther. It is also noted that the newest version of the double Fourier interferometer proposes to use an array of telescopes and combine them with single-mode fiber arms (Zhao, LBna, Mariotti and Coudk du Foresto [19941, Zhao, Mariotti, Gna, Coudi du Foresto and Zhou [1994]). Such a system will fall in the same category as FTSIP. The multiplering technique is seldom used for image coding in spectral imaging. If one must use a single detector, one must use the multiplexing technique for both image coding and spectral coding. The Hadamardtransform spectral imager suggested by Harwit [1971, 19731 is in this class: a practical spectral imager dedicated for use in space. Multichanneling in image detection is the exclusive technique since the advent of CCD image sensors. From the theoretical viewpoint of signal-to-noise ratio, the multichanneling technique is the best, as stated previously. The techniques listed in the bottom row use the multichanneling technique for collecting the two-dimensional image information, and appear to constitute a group of efficient spectral imaging techniques. The combination of multichannel image detection and spectral scanning is a natural and simple choice. The acoustooptic tunable filter (AOTF) suggested by Hams and Wallace [I9691 is an electronically tuned optical filter which operates on the principle of acouste optic difiction in an anisotropic medium. This idea was applied successfully to astronomical spectral imaging by Wattson, Rappaport and Frederick [19761, and to rapid scanning spectroscopy and spectral imaging by Kurtz, Dwelle and Katzka [1987]. The imaging spectrometers with a non-collinear AOTF suggested by Chang [1974] are currently being applied to remote sensing (Cheng, Hamilton, Mahoney and Reyes [19941) and target detection (Cheng, Mahoney, Reyes and Suiter [1994]) by a research group at NASA. Lippmann’s color photography (introduced in Q 1) is an old technique for recording colors of an image formed by natural light. The spectral information is coded in the emulsion

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as a period of the standing waves generated by reflection of light from the surface of mercury. The emulsion acts as a 3D photodetector. Application of this idea to astronomical spectral imaging has been suggested by Lindegren and Dravins [ 19781. The wide-angle Michelson Doppler imaging interferometer (WAMDII) suggested by Shepherd, Gault, Miller, Pasturczyk, Johnston, Kosteniuk, Haslett, Kendall and Wimperis [1985] is a combination of a field-widened Michelson interferometer (Hilliard and Shepherd [1966]) and a CCD imager. This instrument is intended to measure the spatial distribution of upper atmospheric winds and temperatures using Spacelab as a platform. Maps of the velocities and temperatures are recovered from sets of four quarter-wave phase-stepped images obtained by the field-widened Michelson interferometer. The unique feature of this technique is that a Gaussian model of spectral line shape is used for the particular emission lines, and the spectral widths and shifts are deduced from the data set taken at the substantially fixed optical path difference. For the spectral imaging of general objects such as the planets, the optical path difference must be scanned fully. In July of 1989, the author discussed this combination of a Michelson interferometer and a CCD array with S.T. Ridgway at Kitt Peak National Observatory. He told me that he and M.J.S. Belton had already tried this combination for planetary astronomy, and showed me a spectral image of Saturn. Unfortunately, they only briefly described the result in an abstract in the Bulletin of the American Astronomical Society (BAAS 10, 547) in 1978. The result was not satisfactory owing to atmospheric seeing fluctuations and the noise of array detectors available at that time. Simons and Cowie [1990] tied this combination and Simons, Mailard and Cowie [1991] succeeded in reconstructing IR spectral images of nebula NGC 7027. The Fourier-transform spectral imaging in image plane (FTSI/I) suggested by Inoue, Itoh and Ichioka [1991] also makes use of the same configuration. Itoh, Inoue and Ichioka [1990d] suggested the use of a liquid-crystal polarization interferometer in place of the Michelson interferometer. Mailhes, Vermande and CastaniC [ 19901 proposed a slightly different arrangement of this combination of a CCD and a Michelson interferometer for use on a satellite. One of the mirrors in the interferometer is inclined and the motion of the scene on the tilted mirror is used for the scanning of optical path difference. This instrument is equivalent to the previous spectral imagers with normal optical path scanning, if the imagers can track the motion of the scene. T l s group comprises the most efficient practical spectral imagers, because the ideal configuration of an all-multichannel system is difficult to build owing to the lack of a 3D detector array. In this regard, the all-multichannel systems located at the rightmost column of the bottom row in

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INTERFEROMETRICMULT'ISPECTRAL IMAGING

Grating or Prism

2-D Array Detectors

Fig. 3. Schematic diagram of the AIS (airborne imaging spectrometer) concept.

table 2 are imperfect or compromise systems. Professional color TV cameras are equipped with three separate spectral channels for red, green and blue video signals. Such a system is possible because the number of spectral channels is small. The airborne imaging spectrometer (AIS) suggested by Goetz, Wellman and Barnes [1985] has a 2D array of detectors for imaging spectrometry. The AIS concept is illustrated in fig. 3. Light flux from the scene collected by the fore-optics is focused on a slit. After passing through the slit, the light beam is collimated and then dispersed by a grating or prism. A set of spatially resolved spectra is detected simultaneously by the 2D array of detectors. The spatial information is limited to one dimension. This limitation poses no problem, because this instrument is airborne. The Medusa spectrograph, named and built by Hill, Angel, Scott, Lindley and Hintzen [1980], is a very special spectral imager dedicated to astronomical use. The Medusa spectrograph is illustrated schematically in fig. 4. Several tens of stars or galaxies in the focal plane of a telescope are selected. The light from each image is coupled into a fbsed silica fiber. The fiber ends are aligned on the spectrograph slit. The spectra of all stars or galaxies are obtained simultaneously without overlapping. Note that this unique instrument does not give us a real spectral image. It gives only a table of spectra.

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Photographic Plate or CCD Fig. 4. Schematic diagram of the Medusa spectrograph.

0

3. Unified Theory of Coherence Detection and Multispectral Imaging

The coherence of light has usually been discussed in relation to two separate situations: image formation and spectrometry. In image formation, the spatial coherence is of concern. In common and practical situations, the spatial coherence of light is governed by the well-known van Cittert-Zernike theorem (Born and Wolf [1970]; 4 10.4). The coherence function in the far zone of an incoherent light source at a given narrow spectral band is given by the spatial frequency spectrum of the intensity distribution of the incoherent object. This is the basis of radio interferometry and incoherent holography. The coherence conditions of object illumination in the partially coherent optical systems are also discussed in this framework. On the other hand, the temporal coherence is involved in the spectral recovery of Fourier spectrometry, where the spectral density function and the correlation function of the optical disturbances are connected by the Fourier-transform relationship. This relationship is referred to as the Wiener-Khintchine theorem. In general, however, these separate fragments of information about the spatial and spectral coherence are not sufficient for us to identify a general polychromatic object at a distance. The information necessary for identifying a general polychromatic object must be in the form of a nonseparable 3D function. In this section, we will be concerned with the 3D correlation function of optical disturbances at two spatial points in a volume. We may call this correlation function the spatial coherence function because no time delay is

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9: 3

involved. In this framework, we will see that in special cases the spatial and spectral information of the light source can be retrieved simultaneously from this 3D spatial coherence function. As stated previously, building on the formulas of optical propagation of coherence (Wolf and Carter [1978]), Itoh and Ohtsuka [1986a] derived a simple relationship between the spatial and spectral distributions of the light source and the 3D spatial coherence function. Let us follow the essential part of their derivation.

Fig. 5 . Geometry of free-space propagation and detection of optical coherence. Detection of spatial coherence is limited to the small area.

3.1. SPATIAL COHERENCE FUNCTION OF HOMOGENEOUS OPTICAL FIELD

Consider a plane polychromatic incoherent source (S) in the plane of z = O in a Cartesian coordinate system as shown in fig. 5. Let the cross-spectral density function of the source be denoted by

where PIand P2 are two-dimensional position vectors in the source plane, w is the optical angular frequency, and 6(2)(P1-P z ) is a two-dimensional D i m delta function. Let the mutual coherence function without the time delay at two points in the propagation region be denoted by r(Q1,Q2,0),where Ql and Q2 are 3D position vectors. Let the distance between the centers of the observation area and the source plane be Z, and the position vector of t h s center of observation be given by Z. The mutual coherence function without the time delay can be

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161

obtained by integrating the cross-spectral density function with respect to the optical angular frequency. By using the result of Wolf and Carter [1978] in the framework of the second-order coherence theory of scalar wave field [their eq. (2.21)], we get:

where R , = Q , - P for m = 1 and 2, I(O)(P,w) is the spectral density of the primary source at position P, and d2r=dxdy. This relationship involves simultaneously the spatial and spectral contents of the light source, and may be considered as a generalized form of the van Cittert-Zernike theorem. We assume here that we can detect the field correlation function within a limited area of space, and that the area of observation is so small that the incident light within it is approximated by a superposition of statistically independent plane waves. Since the light source is incoherent, the assumption of statistical independence between the plane waves with different wave vectors is reasonable. In this case, we can make an approximation that

where k = k(Z - P)/IZ -PI and r = Rl - R2 = Ql - (92. Note that k is the wave vector (kx,k,, kz), for the elementary monochromatic plane wave with an angular frequency of o = c k , where c stands for the velocity of light. Note also that the elementary plane wave is associated with light emanating from point P in the source plane, and that k is a function of P, or conversely, that P is a function of k . Fortunately, eq. (3.2) conforms to rather common situations where the incident optical dlsturbances can be regarded as an incoherent superposition of plane waves. Such a requirement can largely be met by the usual astronomical observations and optical observations of remote objects through an aperture of a moderate optical instrument. Let the observational points QI and Qz be located inside a sphere of radius a, and let the sphere be centered on point Z as shown in fig. 5. The incident wave fronts within the observation area are well approximated by an ensemble of plane waves if the following condition holds:

where A is the wavelength of light. If the light source is centered on the origin of the coordinates in the source plane, P M 0. This approximation is related closely - but in an unusual fashion - to the Fraunhofer approximation. If the

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5

3

sphere is replaced by a circular aperture of the same diameter and the aperture is illuminated from behind by a plane monochromatic light, the expression above is identical with the condition for observing the Fraunhofer diffraction pattern on the source plane. Note that under this condition the field correlation function is independent of the shift of origin, provided that the two points of concern are within the area of observation. Roughly speaking, or in the practical sense, the field within the observation area is homogeneous. Now that the incident light can be regarded as an ensemble of plane waves, let us recall that the origin of these plane waves associated with the wave vector k is point P on the source plane. Thus, we may put P = P(k). Under the assumption that Z >> a (which follows from inequality 3.3), we may put: IRllM

IR21

Zk

= k,'

(3.4)

Then, from eq. (3.2) we have: T ( r )=

///

G(k)exp(ik . r ) d3k,

(3.5)

where d3k = dk, dk, dk,,

and

G(k)= c(k k,)-' I(O)(P(k),ck).

(3.7)

This is the essence of their results. The spatial and spectral information of the light source is included in G(k), a simple 3 D Fourier transform of the mutual coherence function without the time delay. G(k) represents the 3D power spectral density of the homogeneous optical random field. We have derived the relationship between the power spectral density and the spatial and spectral intensity distribution of the light source. We may call G(k) the spectral image. The correlation function is related inversely to the power spectral density:

G(k)

(A)///

T ( r )exp(-ik . r ) d3r,

where d3r=dxdydz. The pair of equations (3.5) and (3.8) is the equivalent of the Wiener-Khintchine statement for the 3 D homogeneous random optical field (Yaglom [ 19621).

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I63

This theory of spectral image recovery from spatial coherence is applicable only to homogeneous optical fields. It is important to note that a certain class of inhomogeneous fields can be transformed to practically homogeneous ones by using an appropriate lens, as suggested by Itoh and Ohtsuka [1986a]. Suppose that the observation area is so close to a light source of a finite extent that the field within the observation area cannot be considered to be homogeneous. This inhomogeneity is caused mainly by the phase front curvatures of wavelets emanating from the light source, because their phase factors change more rapidly than their amplitudes. If an appropriate lens is placed at a correct position between the source and observation area, the phase fronts of the wavelets from each position over the light source will be transformed from spherical to planar phase fronts. If we can neglect the amplitude inhomogeneity of each wave front, the optical field behind the lens can be considered to be homogeneous. Let us discuss further, but briefly, the homogeneity of the optical random field within the observation area. We may first assume that the amplitude of the homogeneous random field has a 3D spectral representation:

/// F

V ( r )=

F

F

u(k)exp(ik . r ) d3k.

(3.9)

In a homogeneous random field, the spectrum must satisfy the condition (Yaglom ~9621)

where 60,(k) is a 3D Dirac delta function and G(k) is the power spectral density or the spectral image. This condition is satisfied by the fact that the light source is incoherent and the observation area is small. However, eq. (3.9) contradicts the assumption that the field is homogeneous, because the spectral representation of eq. (3.9) implies that the field is absolute integrable. Homogeneous random fields are not absolute integrable. Strictly speaking, we should assume a quasihomogeneous (Carter and Wolf [ 19771) random field which vanishes in the far zone. It is appropriate to point out the relationship between the 3D power spectral density and the radiometric radiance or the generalized radiance. This was suggested by Walther [ 1968, 19731 and further discussed subsequently by Marchand and Wolf [1974] and by Martinez-Herrero and Mejias [1984]. If we locate the center of observation in a propagation region, wave vector k is related uniquely to source point P(k). Then, G ( k ) is related closely to the radiance B(P(k),klk) of a special case. This relationship between the radiance

164

INTERFEROMETRIC MULTISPECTRAL IMAGING

PI, 0 3

of the source and the field correlation function in the propagation region has not been discussed in the theory of coherence and radiometry. It is also noted that eq. (3.7) does not seem faultless. When I(O)(P(k),ck)is given and k, becomes very small, G(k)must become very large. This situation of very slanted observation of an incoherent light source disagrees with common observation. If we need a more realistic description, we should assume a more elaborate model for the light source. Let us realize that eq. (3.5) unifies the principles of Fourier spectroscopy and stellar interferometry. These principles are called the Wiener-Khintchine and van Cittert-Zernike theorems, respectively. Suppose that a light propagates along the z-axis of a Cartesian coordinate system. In Fourier spectrometry, we measure the one-dimensional correlation function along the z-axis. Let a position vector r, denote (0, 0, z). Then, we see from eq. (3.5) that we measure

I]] "

r(r,) =

"

a

(3.1 1)

G(k)exp(-ik,z) d3k.

The Fourier inversion of eq. (3.1 1) with respect to z is: 1 G(l)(kz)= T(r,) exp(ik,z) dz = G(k)dk, dk,.

1

ss

(3.12)

Thus in Fourier spectrometry the spatial structure in G(k) is related to a single value for the given k,. The significance of eq. (3.12) is that we can evaluate the maximum field of view or signal degradation in Fourier spectrometry by use of this equation. Note that even if the light source is uniform spatially, if the object covers a finite field of view, G(k)is dependent on the direction of k or on (k,, ky), and integration in the k,-k, plane smears out the original spectral information in G(k).This is related closely to the instrument profile as discussed by Vanasse and Sakai [ 19671 in Fourier spectrometry. If the spatial resolution of coherence detection is sufficiently high, the spectral recovery based on the unified theory keeps the ultimate spectral resolution for extended objects. In Michelson stellar interferometry or incoherent holography, we measure the two-dimensional spatial coherence function. If we denote (x,y, 0) by rxy we which is given by: measure r(rXy),

r(rxy)=

I//

G(k)exp[-i(k,x

+ k,y)] d3k.

(3.13)

The image recovered by two-dimensional Fourier inversion contains the information as:

111,

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UNIFIED THEORY OF COHERENCE DETECTION AND MULTISPECTRAL IMAGING

165

This is the van Cittert-Zernike theorem (Born and Wolf [ 19701; 4 10.4) in the narrow sense. Radio interferometers operate on this principle. In the earthrotation synthesis, however, the spatial coherence function is not measured in the flat x-y plane because of the curvature of the earth’s surface. The solution to the problem of image estimation from a coherence function measured in a 3D volume still seems to be an open question (Bracewell [1979]). This problem may be approached from the framework of spectral image recovery from the 3D spatial coherence. Finally, let us consider an alternative, common situation where an object with a rough surface is located at the source position in fig. 5 and is illuminated with light of very low coherence. The surface of the object is assumed to be sufficiently rough so that no specular component reaches the observation area. We may be concerned with the local statistical average of reflectivity of the surface of the object. The degree of spatial localization may be determined by the discernible size associated with the size of the observation area. If the coherence area of illumination over the object surface is sufficiently smaller than this discernible size associated with the observation area, we may assume that the object scatters substantially incoherent light. If we reconstruct a spectral image from coherence measurements on the scattered light, the recovered spectral image G(k) reflects the distribution of local radiance of the scattered light in the practical sense. The detailed statistical optical characteristics of the scattered light may be determined by the microscopic structures and materials of the scattering surface and the coherence condition of illumination. However, this is a laborious problem which involves multiple-scattering phenomena (Ishimaru [1978]). 3.2. DETECTION OF 3D SPATIAL COHERENCE

Detection of optical coherence dates back to the proposal of Fizeau and the pioneering work of Michelson (vide Born and Wolf [1970]; fj7.3.6). However, detection of the 3D coherence function is rather new. The large arrays of radio interferometers sample unintentionally the 3D spatial coherence of radio wave fields due to the curvature of the rotating earth’s surface, as mentioned previously. However, so far this effect has not been welcomed (Bracewell [ 19791). The rotational-shear volume interferometer to be discussed here is dedicated to the detection of 3D spatial coherence functions. The rotational shear volume interferometer suggested by Itoh and Ohtsuka [ 1986al is shown schematically in fig. 6. This is an extension of the rotational shearing interferometers.Readers are referred to the extensive review of shearing interferometers by Roddier [19881.

166

INTERFEROMETRlC MULTISPECTRAL IMAGING

rm 5 3

Fig. 6. Schematic of the rotational shear volume interferometer. Rotational shear is introduced by tilting the two right angle prisms around the optical axis. Longitudinal shear is created by shifting one of the prisms along the optical axis.

The incident light beam is split by a beam splitter (BS). The split beams are then reflected by right-angle prisms. Both of the split wave fronts are reversed left to right by the prisms (Pl,P,) and then superposed again on the BS. As two right-angle prisms are rotated slightly around the optical axis, a rotational type of shear is created between the wave fronts. Longitudinal shear is then introduced by moving one of the prisms along the optical axis. We take Cartesian coordinates whose z-axis coincides with the optical axis and whose x-y plane corresponds to the observation plane. Let ex, ey and e, denote the unit vectors taken along the respective axes indicated by the suffixes. Now let the rotation angle of the two prisms be 812 and let the longitudinal path difference be 22. Then, the optical power collected by the detector element placed at a position (x,y, 0) is proportional to:

where rg = 2(-ye,

+ ney)sin 8 + 2ze,.

(3.16)

The three-dimensional Fourier transform of these data is proportional to G(ks), where: ko=-

2

(-sine’ _kx_ _

ky

sine’

k,).

(3.17)

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UNIFIED THEORY OF COHERENCE DETECTION AND MULTISPECTRAL IMAGING

167

Fig. 7. Schematic of the field-folding volume interferometer proposed for complete recording of transient optical phenomena. The bulk storage material is expected to record the three-dimensional interferogram.

Equation (3.17) shows that the spatial scale of the reconstructed spectral image is variable, and a monochromatic object lies no longer on the surface of a sphere but on the surface of a spheroid. Another method for 3D coherence detection is a simple combination of a Michelson interferometer and a Michelson stellar interferometer (Mariotti and Ridgway [ 19881). The method - dubbed double Fourier spatio-spectral interferometry - has already been tried in practice, although spectral images have yet to be reconstructed. The configuration has been shown in fig. 2 (above). In this configuration, one must scan the path difference as well as the 2D base line of the two apertures. The significant advantage of thls method is the capability of aperture synthesis. If this system is mounted on a satellite, it will exhibit its great potential for high-resolution spectral imaging. An appropriate use of a 3D array of detectors speeds up the coherence detection because no scanning is required for obtaining the 3D interferometric data. The idea of a holographic technique which uses a 3D storage material has been suggested by Itoh and Ohtsuka [1986b]. The rotational shearing type of configuration is shown in fig. 7. The incident wave is split by the beam splitter (BS) and the split beams are counter-propagated. The two beams are rotated by Dove prisms (PI, P2) so that the lateral field distribution is sheared rotationally. Figure 7 shows the optical configuration of 180-degree rotational shear. We take a coordinate system centered on the position of zero path difference and denote one of the rotated incident fields by V ( r ) , and the other

168

INTERFEROMETRICMULTISPECTRAL IMAGING

w,8 3

by V(-r). The intensity distribution near the position of zero path difference is then proportional to: l o , ( r ) = r ( 2 r ) + r'(2r)

+ 2r(O).

(3.18)

The critical issues are whether or not a suitable recording material is available and how to read out faithfully and quickly the high-density information stored in the medium. The resolution limits of interferometric spectral imaging are inversely proportional to the dimensions of the observation area (Itoh and Ohtsuka [ 1986a]), and conform to those of Fourier spectrometry and interferometric imaging when separate observations are made.

Fig. 8. Picture of a rotational shear volume interferometer.

3.3. RECOVERY OF SOURCE INFORMATION FROM SPATIAL COHERENCE

A picture of the rotational shear volume interferometerconstructedby Itoh, Inoue and Ichioka [1987, 1990al is shown in fig. 8. The interferometer is composed of a cube beam splitter and a pair of right-angle prisms. The prisms are mounted such that they can be rotated precisely by a small amount. The shear angle (1.2 degree) is so small that the inclination of the prisms is not discernible in this picture. The imaging lens attached to the video camera is focused on the apexes of the right-angle prisms rather than on the object. This interferometer is used for the first spectral recovery from the 3D spatial coherence function.

111,

5

31

UNIFIED THEORY OF COHERENCE DETECTION AND MULTISPECTRAL IMAGING

169

Fig. 9. Picture of an object. This flower is composed of five petals; the central part is red.

A series of 64 interference patterns composed of 64x64 pixels were taken by successively changing the path difference of the interferometer. For the precise control of path difference, a piezoelectric translator was used. The pattern at each path difference was digitized by an eight-bit AID converter and was averaged numerically to improve the signal-to-noise ratio. The object shown in fig. 9 is composed of five petals whose central part is red (gray area in the picture). A portion of the green leaf is illuminated so as to be barely detectable. A series of measured interference patterns are partly displayed in fig. 10. These patterns are the cross-sections of the real part of the 3D mutual coherence function as given by eq. (3.5). The reconstructed spectral image is partly shown in fig. 1 1 . The images displayed are the cross-sections of G(k) perpendicular to the k,-axis. The cross-sections show approximately the image of constant spectral content because G(k) usually have a small angular distribution with respect to k . The approximate wavelengths are indicated in the pictures in nm. The central bright spot that appears in each cross-section was identified with the scintillation noise in the illuminating light. The second object shown (fig. 12) is a postage stamp. This stamp has white rims at the sides, orange surroundings, and two aspects of the earth composed of the five continents finished in green and the seven seas in blue. The reconstructed spectral image is partly shown in fig. 13. In this case, the central pixel which includes the scintillation noise is replaced by a nearby pixel. Referring to the wavelength given to each cross-section, one can realize the color distributions described above. Note that the wavelength dependence of the size of the crosssections is seen clearly. Figure 14 shows the spectra at several locations on the object.

170

INTEFWEROMETRIC MULTISPECTRAL IMAGING

Fig. 10. A part of the series of measured interference patterns.

Fig. 11. Cross-sections of a spectral image. The cross-sections show approximately the images of constant spectral content of the flower presented in fig. 9. The approximate wavelengths are indicated in nm.

We can find a real application of the coherence-based spectral-imaging technique to astronomy. The experimental results of the first trial of double-Fourier

111,

5 31

UNIFIED THEORY OF COHERENCE DETECTION AND MULTISPECTRAL IMAGING

171

Fig. 12. Picture of an object. This stamp has white rims at the sides, orange surroundings and two aspects of the earth composed of the five continents finished in green and the seven s e a in blue.

Fig. 13. Cross-sections of a spectral image. The cross-sections show approximately the images of constant spectral content of the stamp presented in fig. 12.

spatio-spectral interferometry on the 4-meter telescope at Kitt Peak National Observatory were reported by Mariotti and Ridgway [1988]. They obtained interferograms at the 2.2pm band of the unresolved star Betelgeuse. The interferograms were obtained with pupil masks whose separations correspond to

172

INTERFEROMETIUC MULTISPECTRAL IMAGING

wavenumber (an-') [wavelength(nm)l Fig. 14. Spectra at several locations on the object. The object is the stamp shown in fig. 12.

base lines of 0,0.6,1.0 and 1.4m. At long base lines, the fringe jitters increased owing to the atmospheric turbulence, yet the fringe visibility remained high. If this method is assisted by an appropriate phase retrieval algorithm, recovery of a spectral image might be possible even in the presence of the turbulent atmosphere. However, the most attractive application of this method might be its use in space, as stated previously. A refined version of the double Fourier technique can be found in literature (Zhao, Ltna, Mariotti and Coudt du Foresto [1994], Zhao, Mariotti,L h a , Coude du Foresto and Zhou [1994]). 3.4. OPTICAL 3D FOURIER TRANSFORM

Since the coherent-optical information processing systems have attracted remarkable attention because of the capability of two-dimensional Fourier transformation, the 3 D Fourier transform relationship between the spectral image and field correlation function (discussed in Q 3.3) may also attract considerable attention. Itoh and Ichioka [1990] suggested the feasibility of the optical 3D Fourier transformation using an incoherent optical system. Itoh, Inoue and Ichioka [199Oc] demonstrated the experimental results of optical 3D Fourier transformation. The basic equations have been introduced in Q 3.1 as:

T ( r )= and G(k) =

///

G(k)exp(ik . r) d3k,

(A) ///

T ( r )exp(-ik . r ) d3r.

(3.5)

111,

5

31

UNIFIED THEORY OF COHERENCE DETECTION AND MULTISPECTRAL IMAGING

173

Fig. 15. Basic idea of the polychromatic object generator

To construct the optical computing machine for the 3D Fourier transformation, we need two important subsystems; a polychromatic object generator and a 3D spatial coherence detector. A schematic diagram of the basic idea of the object generator is shown in fig. 15. Note that spectral density G(k) of a monochromatic object of wave number kl comprises a sphere (lkl = k ~in) wavevector space. An image whose brightness distribution is proportional to the projection of G(k) (i.e., Ikl = k , = 2z/A,) onto the k,-k, plane is displayed on each CRT, and each beam emanating from the CRT is filtered by the respective wavelength-selective mirror (A,,; n = 1,2,. . . ,N ) . All of the spectral components are superposed and form an object of a given power spectral density of G(k). The CRTs may be replaced by liquid-crystal light valves with an incoherent illuminator or incoherent arrays of laser diodes. From the viewpoint of signalto-noise ratio, an array of multispectral diode lasers is desired. As for the 3D coherence detector, we have the rotational shear volume interferometer introduced in the previous sections. The fundamental limitation of the present computing system is decided by the photon noise. In this limiting case, the noise generated in the detector is governed by the Poisson statistics, and the signal-to-noise ratio is given by:

P = g 7

(3.19)

where N , is the total number of detected photons and N , is the total number of sampling points. If we assume N , = lo9 and p = lo3, we need 1015 photons

174

INTERFEROMETRIC MULTISPECTRAL IMAGING

[III,

54

or a radiation energy of approximately 4x lop5J at 500nm. This limitation is not critical unless very fast calculation ( 5 1 ps) is required. In practice, the speed of calculation will be limited substantially by the data transfer from the detector array to the host computer. The commercial high-resolution CCDs currently allow us to obtain on the order of ten images with 1O6 pixels per second. This class of CCDs will limit the speed of computing the 3D Fourier transform of lo3x lo3x lo3 data points to the order of one minute. It should be noted that 3D Fourier transformation by the conventional electronic computer with a single central processing unit (CPU) demands an extremely long computation time. For example, suppose that we have an electronic computer capable of ten million executions of multiplication followed by addition per second and equipped with a main memory in excess of several gigabytes. This computer will take more than an hour to compute the 3D Fourier transform of lo3x lo3x lo’ data points, when the standard fast Fourier transform algorithm is used. The precision of computing might be a serious factor. The precision will be limited practically by the dynamic range of the detector and is strongly dependent on the choice of detectors. A cooled CCD with a dynamic range of 1: 100000 is readily available and we may expect to see an improvement in this factor.

3

4. Interferometric Multispectral Imaging in the Image Plane

The Fourier transform spectral imaging systems discussed in $ 3 work in the pupil plane of an imaging system. Inoue, Itoh and Ichioka [1991] showed that these pupil-plane techniques have a poor signal-to-noise ratio when used in combination with a single-aperture optical system. A spectral imaging technique that has the highest efficiency or highest signal-to noise ratio might be the allmultichanneling technique. As suggested in § 2, however, the essential difficulty of an all-multichanneling system with uniform high resolution in the spatial and spectral dimensions is that we have no efficient 3D data acquisition system. A practical solution is to deal with one of the three dimensions by the secondbest technique, the multiplexing technique. The spectral imaging system to be discussed here applies this compromise; a combination of an image-sensing array and a Fourier spectrometer. 4.1, FUNDAMENTAL SYSTEM

The basic optical system of Fourier transform spectral imaging suggested by Inoue, Itoh and Ichloka [1991] is shown schematically in fig. 16. This

111,

5 41

INTERFEROMETNC MULTISPECTRAL IMAGING IN THE IMAGE PLANE

175

M.

CCD Fig. 16. Basic optical system of Fourier transform spectral imaging in image plane.

configuration is substantially the same as those used by Belton and Ridgway in 1978 (see 5 2), Gay and Mekarnia [1987], and Simons and Cowie [1990]. A light beam from an object is collimated by a lens (Ll). While passing through the Michelson interferometer, the incident beam is split into two and the split beams are combined with a path difference. The path difference is generated by the mechanical translation of a mirror. Each of the split beams makes each image after passing through the second lens (L2). If these images are superposed upon each other, they interfere. Let us assume that two mirrors in the interferometer are strictly normal to the optical axis and that a light beam incident from a certain point designated by a two-dimensional position vector P on a planar object makes an angle 8(P)with respect to the optical axis. One of these mirrors (M2) is translated along the optical axis by an amount of z/2 from the origin of the zero path difference. The optical path difference between the interfering beams is then given by zcos8. The intensity variation at a point located by two-dimensional position vector S taken in the detection plane is proportional to:

I(S,z) = c

] I(O)(P,ck) { 1 + cos[2kzcos 8(P)]}dk,

(4.1)

where Z(O)(P,ck) is the spectral density of the light source as in 5 3. For simplicity, we may assume an imaging system of unit magnification where S = - P . Equation (4.1) is identical to that for the conventional Fourier spectrometry except that the intensity variation at each sensor element is detected by a separate channel. Note that if one uses a commercially available array sensor with 1000x 1000 pixels, one million independent Fourier spectrometers work in parallel. Fourier cosine inversion of I(S,z) with respect to

176

INTERFEROMETRIC MULTISPECTRAL IMAGING

Fig. 17. Cross-sections of a spectral image. Spectral image of a plastic doll is reconstructed by the technique of Fourier transform spectral imaging in image plane.

z gives substantially I(') {P(k), ck/[2cosO(P)]}. This method inherits Fellgett

and Jaquinot (throughput) advantages from Fourier spectrometry. This multichannel configuration allows a wider field of view than conventional Fourier spectrometers because the angular dependent interferograms are detected by separate detectors in the array. The interferograms which are out of phase are not mixed together. The resolution limit is identical to the conventional Fourier spectrometers except with regard to this angular dependence. Thus, the spectral resolution limit is position dependent. The signal-to-noise ratio will be discussed later. The limiting factor of this spectral imaging technique is currently the data transfer from the sensor array to the mass storage equipment (Simons and Cowie [ 19901). A spectral image reconstructed by Inoue, Itoh and Ichioka [I9911 is shown in fig. 17. The object is a plastic doll which has blue arms and legs, yellow hands and face, and black eyes and mouth. The body has a red stripe in the middle and white and black stripes on both sides. The object was illuminated by a tungsten lamp. Each image is the cross-section of the reconstructed three-dimensional spectral image. The cross-sections are perpendicular to the wave number axis. Each cross-section is composed of

111, D 41

INTERFEROMETRIC MULTISPECTRAL IMAGING IN THE IMAGE PLANE

177

64x64 pixels. The number of recovered spectral channels was 128, and the spectral resolution can be estimated from fig. 17 as approximately 20nm at a wavelength of 600 nm.

4.2. LIQUID CRYSTAL POLARIZATION INTERFEROMETER

Liquid crystals are used in various optical devices owing to their strong birefringence. Among the various liquid crystals, nematic liquid crystals (NLCs) are attractive because their birefringence can be controlled electrically by a small voltage. Adaptive optical elements such as a tunable filter (Wu [ 1989]), a retarder (Wu, Efron and Hess [1984]) and a variable-focus lens (Sato, Sugiyama and Sat0 [ 19851) have been suggested. A polarization-based interferometer which is equivalent to a Michelson interferometer can be constructed by using the NLC. A Fourier-transform spectral imager was suggested by Itoh, Inoue and Ichioka [ 1990dl. The optical configuration is illustrated in fig. 18. The interferometer (called liquid crystal polarization interferometer) consists of a thick NLC layer formed by a pair of glass plates whch are coated with thin layers of transparent conductive material.

P

LC

A

Fig. 18. Liquid crystal polarization interferometer for the Fourier transform spectral imaging in image plane.

Let us take a Cartesian coordinate system with its z-axis perpendicular to the NLC layer. The NLC molecules are aligned homogeneously with their optic axis on the y-axis of the coordinate system. One of the polarizers located at the front side selects linearly polarized light whose electric field vector makes an angle of n/4rad with respect to the y-axis, while the other at the rear side selects the component of 3 d 4 rad. Let us denote the phase difference between the orthogonal components of the optical wave field that is created during the

178

INTERFEROMETRIC MULTISPECTRAL IMAGING

"5 4

propagation in the NLC by q5. The intensity of light passing through the rear polarizer is proportional to: Z(P, V ) = c

] Z(')(P,ck) (1 + cos q5) dk,

(4.2)

where I(')(P,ck) is the intensity of the input beam at position P = ( x , y ) , and angular frequency o = ck. The phase difference may be given by:

where An(T, A) is the birefringence of the NLC at temperature T and wavelength A, d ( P ) is the thickness of the NLC layer at position P, and f ( V ) is the monotone dependence of retardation on voltage which varies nonlinearly from 0 to 1 with respect to voltage V . The birefringence is dependent on the degree of order of the NLC molecules and hence is dependent on its temperature. However, if we assume a temperature-controlled environment, this dependence might not be a serious problem. To calibrate the normalized retardation f ( V ) , one must detect the variation of beam intensity Z(P0, V ) at a particular position PO by using monochromatic laser light. In principle, the Hilbert transform of this interferogram gives the imaginary part of the interferogram, (Zo(P0,o)sin[kAn(T, A) d(Po)f(V)]}/2, and the retardation can then be calculated from the real and imaginary parts. In practice, one may use the Fourier-transform method suggested by Takeda, Ina and Kobayashi [1982]. The effects of the wavelength dependence of An may be corrected readily after the spectra are recovered from the interferograms, provided that the wavelength dependence [An(T,A)] and the voltage dependence If(V)] are separable as in eq. (4.2). The study by Wu [I9891 appears to support the use of this approximation. Strictly speaking, however, this separability does not hold as studied by Mada and Kobayashi [1976]. Interferograms obtained by Itoh, Inoue and Ichioka [1990d] are shown in fig. 19. The interferograms are obtained at a particular position on an image sensor by using a He-Ne laser. The retardation of the liquid crystal (ROTN403; Hoffmann La Roche) is plotted in fig. 20 as a function of the applied ac voltage. The nonlinearity of retardation f(V ) can be evaluated from this curve. The interferogram obtained by using the corrected voltage sequence is plotted in fig. 21. The light source was a tungsten lamp. The recovered spectra are shown in fig. 22. The spectral shape was confirmed by using a conventional grating spectrometer. The resolution limit was estimated at 60 nm from the reconstructed line width of the He-Ne laser light. The discrepancy

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INTERFEROMETRIC MULTISPECTRAL IMAGING IN THE IMAGE PLANE

179

APPLIED VOLTAGE ( VAc)

Fig. 19. Interferogram of a He-Ne laser light plotted as a function of applied ac voltage. The interferogram was detected at a particular position on the image sensor.

0

2

4

6

8

APPLIED VOLTAGE (Volts)

Fig. 20. Retardation of a liquid crystal (ROTN403; Hoffmann La Roche) plotted as a function of the applied ac voltage.

between the theoretical and practical resolution limits was ascribed to the absence of temperature control during the experiment and imperfect correction for the nonlinearity of retardation. It is appropriate to touch on the dependence of the path difference on the angle of incidence of such a polarization interferometer. Assume that a light beam is incident on a layer of NLC as shown in fig. 23, and take a coordinate system as in the figure. The angles of refraction are denoted by O1 and 02, and the angle between the plane of incidence and the y-z plane is I$. The optic axis lies in the y-z plane and forms an angle a with respect to the y-axis. Suppose that the

180

INTERFEROMETRIC MULTISPECTRAL IMAGING

E z

1.0

3

>

a d U

t m a

d

. 0.5.

-I

9 0

u)

-I

4

2n DATA NO.

Fig. 21. Interferogram of light from a tungsten lamp obtained by using the corrected voltage sequence.

F I1.0

3

WAVELENGTH

(nm)

Fig. 22. Recovered spectrum of light from a tungsten lamp.

difference of refractive indices associated with the ordinary and extraordinary rays is small compared with the average, and that the angle of incidence of the light beam is sufficiently small. The phase difference between the ordinary and extraordinary rays is then given (Born and Wolf [1970]; 0 14.4) as:

6 = 2nhsin2y-

ne

- no

ACOS

o, ’

(4.4)

where h is the thickness of the layer, y is the angle between the optic axis and the extraordinary ray, Oa is the average.of 81 and 132, and no and n, are the principal refractive indices of the ordinary and extraordinary rays, respectively. We can reason from eq. (4.4) that, under the condition a = n14 and @ = nJ2, a

111,

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INTERFEROMETRIC MULTISPECTRAL IMAGING IN THE IMAGE PLANE

181

Fig. 23. Geometry of optical paths in a layer of nematic liquid crystal.

P

LC Cells

A

Fig. 24. Three-layered liquid crystal polarization interi’crometer,

large angular dependence of S on y arises when y approaches nl4rad. This angular dependence of phase difference severely limits the angle of view of a polarization-based optical system. The solution to this problem will be discussed in the following. A three-layered NLC polarization interferometer was suggested by Inoue, Ohta, Itoh and Ichioka [1994] to remove the difficulties of the single-layer configuration. The suggested interferometer is shown schematically in fig. 24. The first and second cells are dynamic ones whose glass plates are coated with transparent conductive material. The path difference created in these cells is controlled by an ac voltage. The NLC molecules in these cells tilt symmetrically with respect to the glass plate between the layers. This symmetric arrangement cancels the anti-symmetric angular dependence of path difference created in these cells. The idea of this symmetric arrangement is based on the suggestion for the LC display by Saito [1986]. No voltage

182

INTERFEROMETRlC MLJLTISPECTRAL IMAGING

Fig. 25. Spectral image of Russian dolls reconstructed by using a three-layered liquid crystal polarization interferometer.

is applied to the third cell, and the path difference in this static cell is so arranged as to cancel the bias of the path difference created in the dynamic cells. The optic axis of this static cell is aligned perpendicular to the optic axis of the dynamic cells so that the extraordinary ray in the dynamic cells becomes the ordinary in the static cell and vice versa. This bias of path difference allows us to detect interferograms with respect to positive and negative path differences. The major phase correction algorithms need single-sided interferograms whch include short intervals of signal in the opposite side. A spectral image of polychromatic objects was reconstructed by using this three-layered liquid-crystal polarization interferometer. The result is shown in fig. 25. The objects are well-known Russian “Matryoshka” dolls, which were illuminated by a tungsten lamp. The big one has a red hood and a green body with floral patterns in green, yellow and blue. The small one has a yellow hood and a red body with the patterns in light green, white and blue. The number of spectral channels is 84 and the number of pixels is 64x64. The number in each cross-section indicates the wavelength in nm.

111, P 41

INTERFEROMETRIC MULTISPECTRAL IMAGING IN THE IMAGE PLANE

A

I83

Mirror 1

Object

Fig, 26. Schematic of multiple-image Fourier-transform spectral imaging system. 4.3. MULTIPLE-IMAGE PARALLEL INTERFEROMETER

If an array of lenses is attached to a Michelson interferometer with a tilted mirror, multiple images with different path differences are produced simultaneously. If the mirror is inclined appropriately, these images contain all of the interferometric data necessary for spectral imaging. Such a spectral imaging system is desired for the observation of a fast phenomenon. The principle of multiple-image Fourier transform spectral imaging suggested by Hirai, Inoue, Itoh and Ichioka [1994a,b] is illustrated in fig. 26. The light beam from the object passes through a lens array and is divided into an array of narrow beams. The array of narrow beams is split by the beam splitter. The split multiple beams are reflected by the respective mirrors and are recombined by the beam splitter. Each pair of combined beams produce an image on the surface of the image sensor. Let us take a Cartesian coordinate system whose z-axis coincides with the optical axis. Suppose that one of the mirrors in the interferometer is set at an angle such that its surface is tilted by an angle 8 along the x-axis and by 4 along the y-axis. Then the path difference at position (x,y) on the mirror surface is given by: z(x,y) = 2(xtan 8 +ytan $) + ZO,

(4.5)

where zo is the path difference at the origin (0,O) located at the comer of the mirror in fig. 26. Let us assume that we have N , images along the x-axis with spacing X , and N , images along the y-axis with spacing Y . Each image is located by a pair of numbers (n,,n,,), (n, =0, 1,. . . , N x - 1,

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[IIL

§4

nu = 0,1,. . . ,Ny - 1); here n, means the n,th column and ny the nyth row. These images are ordered by a serial number of m=n,Ny+n,,. We take local 2D coordinates on each image, and a pixel on each image is specified by a position vector (x‘,y’) (0 <x’ <X , 0
NOISE LIMITATIONS

185

Fig. 27. Spectral image of a rapidly rotating picture of a bird illuminated by a single flash of strobe light. This instantaneous spectral image is reconstructed by the multiple-image Fourier-transform spectral imaging system.

to produce the position-dependent optical path difference for multiple-image parallel interferometry. The wedged liquid-crystal interferometer is a compact and robust equivalent to the Michelson interferometer with the tilted mirror. § 5. Noise Limitations

The noise limitations of major spectral imaging techniques will be summarized in terms of the signal-to-noise ratio (SNR). This summary is based on the analysis of a spectrometric technique by Mertz ([1965]; 5 1) and that of interferometric imaging by Ribak, Roddier, Roddier and Breckinridge [1988]. To clarify the discussions involving spectrometry and imaging, simplistic models for the photon and detector noise are used. 5.1. PHOTON AND DETECTOR NOISE IN DETECTION OF OPTICAL ENERGY

DISTRIBUTIONS

Signal-to-noise ratios associated with spectrometry have been discussed by many authors. Winefordner, Avni, Chester, Fitzgerald, Hart, Johnson and Plankey

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[III, § 5

[ 19761 extensively investigated the SNRs associated with three types of schemes for spectral measurements - the scanning, multiplex, and multichannel types. They showed that the SNR of the multichannel type is the best, and that the multiplex method could find considerable use in emission spectroscopy if the density of spectral lines is not great and the background noise is low. The following discussion will summarize first the S N R associated with the five types listed in table 1. For simplicity of representation, the analysis will deal with the SNRs of only spectral data acquisition; the results are readily applicable to those of spatial data acquisition. For simplicity, we deal with a sampled version of a band-limited power spectral density. The spectral density function is assumed to consist of M non-zero spectral elements out of the N (M < N ) total spectral elements in the measurable range. The magnitudes of the non-zero spectral elements are assumed to be uniform. The time loss during the scanning process is neglected, and all schemes are assumed to have the same observation time and aperture size. We may assume a single detector or an array of detectors, depending on the schemes. The detector array consists of N detector elements; no gap is assumed between the elements. Each element in the array and the single detector are assumed to have the same quantum efficiency, say r]. Let the time interval for the detection of one spectral element be TOand the average number of photoelectrons detected within a unit time be ZO. Since we assume Poisson statistics for the number of photoelectrons, the SNR of the photon-noise-limited detection is given by p p = ( Z ~ T ~ ) 1We / 2 . may assume that when the detector noise is predominant, a random number of photoelectrons are superposed on the signal photoelectrons. The noise component at every detector element is assumed to have a constant root mean square (rms) value. If we denote the rms noise component per unit time interval by no, the rms in the signals that are averaged within To is given by nd =no(To)’l2.Then, the SNR of the detectornoise-limited detection is pd =IoTo/nd =lo( ~ o ) ” ~ / n o . The magnitudes of the spectral elements are measured directly in the scanning and multichannel methods. However, the spectral elements are measured indirectly in the multiplex and multiplex-multichannel methods. Mertz ([ 19651; 4 1) gave the relationship between the SNRs of the spectral density and the interf‘erogram in FTS. Ribak, Roddier, Roddier and Breckinridge [1988] modified Mertz’s result in their study of interferometric imaging. According to them, the SNR of the multiplexed signal, px, is related to that of the demultiplexed signal, PO,as:

dN

Po = -px. M

111, § 51

187

NOISE LIMITATIONS

Substitution of the photon-noise-limited and detector-noise-limited SNRs of the interferograms into eq. (5.1) gives the expression for the SNRs of the multiplex and multiplex-multichannel methods. Now let us define the following two general parameters: the total number of photons, P , incident on the aperture within the unit time, and the total observation time, T . The SNRs are given for the scanning method by:

because I0 = qPlM and To = TIN. The SNRs for the multichannel method are given by:

because I0 = qP/M and TO= T . It should be noted that a significant part of the incident photon flux is lost in the encoding process in some of the multiplex and multiplex-multichannel 'techniques. The incident photons are partly reflected by optical elements or stopped by the coding mask. However, for simplicity we may neglect t h s deficiency of light flux. To obtain the SNRs for the multiplexing methods we use eq. (5.1) along with I o = qP and TO=T/N. Then, we have for the genuine multiplexing method:

For the multiplex-multichannel method, we have the SNRs are given by:

I 0 = qP/N

and To = T . Then,

(5.5) The multiplex-scanning technique is different from the multiplex-multichannel technique only in that To = T/N. The results are:

Table 3 summarizes these results, which are normalized by the SNRs of the scanning method in the photon-noise-limited and detector-noise-limited cases,

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INTERFEROMETRIC MULTISPECTRAL IMAGING

“11,

5

5

Table 3 Normalized signal-to-noise ratios associated with various photodetection techniques Noise Scanning

Multiplexscanning

Technique Multiplexmultichannel

Multiplexing

Multichannel

Photon noise

1

4x7

rn

m

Detector noise

1

dT dT

m

1

fi

respectively. The basic techniques listed in table 1 are ordered so that the SNR increases from left to right, except in the case of the multiplex-scanning technique. Apparently the multiplex-scanning technique has the worst SNR. The advantage of the multiplex-scanning technique is that it permits the mode of aperture synthesis pointed out in 5 2.2. 5.2. NOISE LIMITATIONS OF MULTISPECTRAL IMAGING TECHNIQUES

By combining the expressions for the SNRs of the five types of data acquisition methods given in eqs. (5.2)-(5.6), we can readily obtain the expressions for the SNRs of any category of spectral imaging technique tabulated in table 2. Let us derive the SNRs of some of the practical techniques in table 2. The simplest is to collect a series of spatial intensity distributions of successive spectral bands through an imaging system by using a tunable spectral filter. A typical example is the AOTF spectral imager listed in table 2.We may name this technique the multispectral camera (MSC). Another might be the multichannel spectrometer, which has a slit entrance aperture and scans the aperture over the image of an object. The light beam passing through the entrance slit is dispersed on a two-dimensional detector array. An example is AIS in table 2,which is schematically presented in fig. 3. This teclmque may be called the multichannel scanning spectrometer (MCSS). Strictly speaking this MCSS technique is a hybrid; 1D spatial information is acquired by the multichannel method and the other 1D information by the scanning method. We will also derive SNRs for FTSIP and FTSI/I introduced in the previous section. Let us take a Cartesian coordinate system for a spectral image; x- and y-axes for the spatial dimensions and z-axis for the spectral dimension. The maximum numbers of measurable image elements along the respective axes are assumed to be N,, N y and N , . The object is assumed to cover a rectangular solid of M x M y M z elements in the 3D spectral image and to have uniform intensity levels at these elements; M,, M y and M , denote the numbers of non-zero elements

111,

0 51

189

NOISE LIMITATIONS

Table 4 Normalized signal-to-noise ratios associated with various spectral imaging techniques Technique

Noise FTSVIa Photon noise

1

Detector noise

1

a

MCSS

m m

MSCC

FTSI/P~

4imc

Jimq JW

Fourier transform spectral imaging in image plane. Multichannel scanning spectrometer. Multispectral camera. Fourier transform spectral imaging in pupil plane.

along the respective axes. The same notations T and P as in the previous discussion are used for the average number of detected photons per unit time and the total observation time, respectively. In MCSS, one of the two spatial axes taken in the plane of the detector array corresponds to the z-axis of the spectral image and the other to the x-axis, respectively. Thus, the number of elements in the detector array is N,N, in MCSS. The spatial distribution along the y-axis is obtained by the motion (scanning) of the instrument. Except for MCSS, the two spatial coordinates on the detector array correspond to the two spatial dimensions of the spectral image. The number of the cells in the detector array is N,Ny. The results are summarized in table 4. The SNRs are normalized so that the SNR of FTSI/I is unity. The SNR for FTSVI is given by:

for the photon-noise-limited case and the detector-noise-limited case, respectively. From table 4, we see that in the detector-noise-limited case the SNR of FTSI/I is the highest among the four techniques. In the photon-noise-limited case, the S N R of FTSIA is equivalent to those of MCSS and MSC, when N y M M , and when N,M M,, respectively. When the inequality M , > N y is satisfied, MCSS has the highest SNR. This is a special case where a slitlike object with a wide spectral continuum is observed by MCSS without the scanning motion. In both photon- and detector-noise-limited cases, FTSVP has the worst SNR. Only when M,My is a small number and the measurement is under the photon-noise-limited condition, the SNR of FTSI/P is equivalent to the

190

INTERFEROMETRIC MULTISPECTRAL IMAGING

5

5

SNRs of other techniques. This condition will sometimes be satisfied in astronomical applications. If small objects such as stars and/or small galaxies are observed in front of a wide and dark background area, M x M y is small. The only distinction between FTSVI and FTSI/P is the mode of image formation. The former adopts the multichannel imaging mode and the latter the multiplex-multichannel imaging mode. Note that the present investigation involves single-aperture systems. These results do not apply to syntheticaperture systems (Mariotti and Ridgway [1988]). The synthesized large area of aperture provides a significant advantage over the single-aperture systems. A synthetic-aperture spectral imaging system in space (OASIS) and double Fourier spatio-spectral interferometry (introduced in 9 2.2) are attractive from this viewpoint. 5.3. EXPERIMENTAL COMPARISON

The noise characteristics of FTSVI and FTSUP systems were compared experimentally by Itoh, Inoue and Ichioka [1990b]. They first verified that the predominant noise in the video signal from a commercial CCD sensor was the detector noise. They prepared a square sheet of white tracing paper as an object. The object was illuminated by a stabilized tungsten lamp. The object was located at the same distance from FTSVI and FTSIP systems having entrance pupils of equal size. Figure 28 shows two sets of nine cross-sections of the spectral images of the white tracing paper measured by FTSVI and FTSVP. The left set is the result measured by FTSI/P, and the right by FTSI/I. These cross-sections are per-

Fig. 28. Cross-sections of spectral images reconstructed from measurements on a white tracing paper by (a) FTSVP and (b) FTSI/I techniques.

111, § 61

CONCLUSIONS

191

pendicular to the spectral axis of the’3Dspectral image. The white small numbers in each cross-section indicate the wavelength in nm. We can see the difference between the SNRs of these single-aperture techniques. The number of data points in these spectral images is 64 x 64 x 64. The average SNRs of these spectral images were 215 for FTSVI and 5.52 for FTSIP. The ratio of these SNRs is approximately 39: 1 and agrees roughly with the analytical result of (NxNy)”*:1 = 64: 1.

Q 6. Conclusions We have seen the simple categorization of various spectral imaging techniques as summarized in table 2. Although the table is not exhaustive, most of the conventional and many of the new techniques are included. The techniques which involve multichanneling for the acquisition of image information are of great practical importance, because the techniques in this group are expected to have the highest SNRs under the usual situations. Among them, the combination of multichannel-imagingand multiplex-spectrometry will keep the highest SNR, at least until an efficient 3D detector array is brought to realization. The signalto-noise ratios of the FTSVI and WAMDII group have been compared with those of AOTF with CCD, AIS, and FTSVP, and the former was shown to have almost the best SNR. Please note, however, that this comparison involves only single-aperture systems of equal aperture size. If aperture synthesis is included in considerations, the techniques of multiplex-scanning and multiplexmultichanneling for image formation will have the decided advantages of very high resolution in imaging and very high SNRs owing to the large amount of light flux collected by the large apertures. These aperture-synthesis systems will exhibit their full potential in future use in space. We have also seen that the unified theory of interferometric spectral imaging provided the basis for the new techniques of FTSI/F’, holographic spectral imaging, and double-Fourier spatiespectral interferometry, and established the connection between conventional interferometricimaging and spectrometry. This theory also suggested the possibility of computing the 3D Fourier transform by optical means. The author hopes that the present theory will be extended to be applicable to inhomogeneous optical fields. The author anticipates that spectral imaging techniques will be used extensively for robotic vision and surveillance for agriculture as well as for scientific measurements in industrial and biomedical applications and astronomy. In nonscientific applications such as the robotic visions and surveillance, all pixels need not necessarily have the same spectral resolution. The spectral images may have

192

INTERFEROMETRIC MULTISPECTRAL IMAGING

[111

the shift-variant spectral resolution, as does the human vision system. A new concept may be necessary for designing such a spectral imager.

Acknowledgements The research work related to this article has been achieved with the proficient guidance and encouragement of Y. Ohtsuka and Y. Ichioka. Stimulating discussions with M. Takeda, S.T. Ridgway, F. Roddier, C. Roddier and J.B. Breckinridge are gratehlly acknowledged. Comments from Y. Chikada and materials supplied by A.H. Greenaway and L.-J. Cheng are appreciated. The author also thanks T. Inoue, K. Yoshimori, A. Asano and A. Hirai for interesting discussions held during conduct of the research and for support during preparation of the

List of Symbols and Abbreviations one-dimensional two-dimensional three-dimensional radius of the detection area radiance of a light source thickness of the NLC layer at position P monotone dependence of retardation on voltage Fourier transform spectrometry Fourier transform spectral imaging in pupil plane Fourier transform spectral imaging in image plane power spectral density of the optical field or spectral image thickness of the LC layer average number of photoelectrons spectral density of the primary source at position P three-dimensional wave vector (kx,k,,,k,) wave number; = [kI number of non-zero spectral elements multispectral camera multichannel scanning spectrometer total number of spectral channels total number of detected photons

1111

LIST OF SYMBOLS AND ABBREVIATIONS

I93

total number of sampling points principal refractive index for the ordinary ray principal refractive index for the extraordinary ray rms noise component per unit time interval photon noise detector noise birefringence of the NLC at temperature 7' and wavelength 3, total number of incident photons two-dimensional position vectors in the source plane three-dimensional position vectors in the detection area = Q , , - P for m = l , 2 root mean square

z

=Ri - & = e l - Q ~ = ( x , Y , z ) =( X , Y , 0) = (0,0,4 two-dimensional position vector in the detection plane total observation time time interval for detecting one spectral element cross-spectral density function of the source three-dimensional position vector of the center of detection area distance between the centers of the source plane and detection area: = 1 2 1 mutual coherence function without the time delay at two points Q i , Q2 = ~ ( Q,Qz, I 0) two-dimensional Dirac delta function three-dimensional Dirac delta function quantum efficiency of detectors wavelength of light SNR S N R of the demultiplexed signal SNR of the multiplexed signal SNR of detector-noise-limited detection SNR of photon-noise-limited detection angular frequency of light

194

INTERFEROMETRIC MULTISPECTRAL IMAGING

[III

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E. WOLF, PROGRESS IN OPTICS XXXV 0 1996 ELSEVIER SCIENCE B.V ALL RIGHTS RESERVED

IV INTERFEROMETRIC METHODS FOR ARTWORK DIAGNOSTICS BY

DOMENICA PAOLETTI AND GIUSEPPE SCHIRFUPA SPAGNOLO Energetics Department, University of L'Aquila. Localitci Monteluco di Roio. 67040 Roio Poggio (L'Aquila). Italy

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5 1.

INTRODUCTION . . . . . . . . . . . . . . . . . . .

199

5 2.

GENERAL REMARKS ON HOLOGRAPHIC INTERFEROMETRY . . . . . . . . . . . . . . . . . .

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5 3.

ELECTRONIC SPECKLE PATTERN INTERFEROMETRY

. .

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CONCLUSION . . . . . . . . . . . . . . . . . . . . . . .

251

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . .

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REFERENCES . . . . . . . . . . . . . . . . . . . . . . .

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Q 1. Introduction In recent decades, there has been a world-wide growth of interest in cultural heritage protection. The increasing deterioration of our artistic patrimony has accelerated the efforts to search for new methods for diagnosis of its state of conservation, because, apart from the immense costs of restoration, the damages, in particular cases, could lead to the irretrievable loss of the artwork. The study of internal stress (due to variations of environmental parameters or external load) in frescoes, panel paintings or statues, the analysis of the behavior of small deformations or material discontinuities, in wooden or mural supports, and a knowledge of incipient and invisible flaws are a priority task for the preservation of these objects. X-ray fluorescence and radiography, ultraviolet and infrared reflectography, atomic spectroscopy, neutron activation, tomography and sonography are some of the tools commonly used by restoration scientists (Brill [ 19801, Halmshaw [ 19871, Non-destructive Testing Handbook [ 19851). More recently, laser-based methods such as holographic interferometry and electronic speckle pattern interferometry have been proposed for solving some important problems of diagnostics in the art conservation field. However, in spite of obvious advantages, the restoration communities, in general, have been slow to accept the holographic techniques as a tool for nondestructive testing. In fact these methods have been mostly confined to research laboratories. This can be attributed to a variety of factors, ranging from skepticism of new technologies to economic aspects. After the pioneering works in the early 1970s, the use of holographic interferometry has grown from a topic of speculative research into one of more general application only in the last few years, coinciding with the advent of electronic or digital holographic techniques combined with optical fibers and modern methods of image processing. These last techniques, proposed independently in Germany and in Italy, facilitate the inspection task of the artworker, and at the same time allow the development of portable interferometers for diagnosis outside the laboratory. Now nearly all types of artwork may be investigated. It is the purpose of this chapter to provide a survey of the different interferometric methods and their application to nondestructive artwork testing. The subject under review is of interest not only in optics, but also in a number of other spheres of science and engineering. For this reason, 199

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1

we have explicated the description of the basic principles of each technique, thereby facilitating comprehension by users who work in nonoptical fields. We then present the results of the experimental investigations carried out over a large range of artworks with different deterioration problems, and discuss the advantages and the limits of the various methods. 1.1. GENERALITY ON ARTWORK SUBJECTS

Many complex problems are connected with art conservation and restoration. Some are of a rather general nature; they refer, for example, to the study of the behavior of typical materials used in artworks or in restoration, say marble, wood, canvas, under the action of chemical or biological agents as well as mechanical or thermal stress. Very often, the modifications induced in these materials by the applied agent finally lead to morphological changes of the specimen under test. In these cases, holographic and speckle interferometry techniques can be used as a tool for detecting such changes. Natural ageing, the cumulative effects of fatigue, excess loading, modification of the environment, atmospheric pollution and human errors are all factors which lead to deterioration of varying degrees of severity of stone monuments and other works of art. We give a brief description of the types of damage which interests us (Del Monte [1991]). The damages are always the result of an interaction between two groups of factors, intinsic and extrinsic to the artifact; the former concerns the nature and the form of the substance, the latter the environment to which the artwork is exposed in time. Intrinsic factors arise from the chemical, physical, and/or biological characteristics of the material (limestone, marble, sandstone, wood). In stone artifacts, for example, damage depends on the mineralogy, chemistry, structure, and texture as well as compression breaking load, thermal linear expansion coefficient, impact and wear resistance. Mortars and plasters are by definition non-homogeneous; the damage affecting frescoes oRen originates from these primary inhomogeneities. Bronzes also reveal frequent and widespread primary inhomogeneities; e.g., microcracks forming during and after solidification, inclusions, oxidation compounds, and so on. Wood is composed of lignin and structural carbohydrates, cellulose and hemicellulose; their distribution is nonhomogeneous and this can accentuate the effects of variations in environmental parameters giving rise to damage. Temperature and water, among the extrinsic factors, are fundamental to all chemical reactions involved in damage: hydration, carbonation, and sulfation. Many physical causes of damage can also be correlated with climate: creep due to thermal expansion or freezing, corrosion due to wind or acid air pollution. Biological factors are also fundamental to

w § 11

INTRODUCTION

20 1

Fig. 1. A typical layered structure of a wooden panel in use in paintings since the 13th century.

the majority of materials. It is inevitable that stone undergoes colonization by bacteria, lichens etc., while wood is affected by woodworm tunnelling and termites. Also well known is the action of micro-organisms and microflora on a fresco, where they give rise to alteration in color, surface flaking and disintegregation of the underlying plaster. The study of the various types of damage requires an evaluation case by case. The relationship between the type of material, environmental factors and damage entity is in many cases unknown. It is not yet possible to provide a quantitative expression of damage related to various factors; only its quality can be indicated. The initial studies with holographic interferometry on artworks were made on wooden artifacts (Amadesi, Gori, Grella and Guattari [ 19741). Wooden panels are the artifacts particularly suitable for optical testing techniques. In fact, a painting on wood can be considered as a layered structure with a support. Figure 1 shows a typical layered structure on a wooden panel as used in paintings since the 13th century. The wood is coated with some priming layers or plaster, which constitute the basis for the painting. These layers, normally made of a mixture of gesso and glue, are thinner and more fragile than the support. Expansion and contraction of the support due to daily fluctuations of the ambient parameters can produce large strains and eventually cracks in the priming layers, as they become less flexible with age. Furthermore, abrupt changes of humidity and temperature, traffic-induced vibrations and heat exposure may also cause unpredictable stress distributions in the heterogeneous materials of the support, with consequent damage to the painting. All of these mechanisms may lead to the formation of detachments and cracks. It is very important to know how the presence of support cracks and discontinuities alter the movements of the painted surface. In addition, destructive insects can attack wooden artifacts and cause serious damages. During their growth, the larvae make tunnels in the wood; they may become very extensive, reducing the interior of a panel or a statue to a kind of honeycomb without any external signs

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of damage. Wood thus riddled by insects naturally reacts very differently from any sound wood in its vicinity, and panels weakened by “ ~ o r m s ”tend to crack more easily. Defects in painted structures may also be due to intrinsic factors such as irregular wood grain or structural anomalies. A force acting across the grain can cause the wood to collapse by compression or to break by tension. Often some forces develop within the wood itself, and its weakness may become extreme; even without external restraints, a piece of wood may show cracks running in the direction of the grain. The behavior of wood models with or without primer layers has been analyzed with holographic methods in order to obtain some information about the basic components of a painting. The frescoes in ancient churches or in open environments present the same problems as wooden panel paintings but the conditions of conservation are more complex for the frescoes, because they are generally situated in ancient buildings or exposed to atmospheric adversities in open environments. Because the fresco is a layered structure, temperature and humidity variations cause variations of the internal stresses in the fresco material, which lead to microscopic destruction and the formation of microcracks. The strain and the stress of the wall cause deformation or cracks in the painted surface, finally resulting in complete disintegration of the artwork. The study of such cracks or small deformations is a part of the decay process, and their dependence on microclimatic conditions is therefore very important for the preservation of ancient frescoes. Realistic investigations, however, must often be carried out in testing environments, like climatic chambers, or directly in situ.

0

2. General Remarks on Holographic Interferometry

In this part of the review, the holographic methods used so far for artwork diagnos’tics are analyzed, and some optical arrangements are considered. Practically, it is impossible to analyze all the relevant studies in a single review, and a simple enumeration of them seems to be inappropriate. For this reason we will consider only some of the basic arrangements that are directly concerned with the diagnosis of artworks. 2.1. PRINCIPLE OF HOLOGRAPHY

Because the principle of holographic interferometry (established in the mid1960s by Horman [19651, and by Powell and Stetson [ 19651) is directly related to the hologram properties, we first give a brief practical description of the

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Y

X

Hologram

"\\,;.?

/'

LLaser L 3 - - 0'

'ror

Object ~~

Bea m s 1) 11t t e 1-

Fig. 2. Schematic diagram for off-axis holography.

basic principles of off-axis holography. This is at present the most common and versatile holographic method generally used in nondestructive artwork testing. A detailed description of the general principles of holography can be found in the literature (Hariharan [1987]). The original holographic technique was invented by the Nobel laureate Gabor [1948, 1949l.Off-axis holography was developed by Leith and Upatnieks [1962]. It consists of two distinct steps: recording and reconstruction. During recording the coherent light wavefront, scattered by the object, is made to interfere across a photographic plate with a reference beam. The resulting intensity distribution contains all the information about the amplitude and phase of the two waves. Figure 2 shows schematically the off-axis hologram recording setup. The two complex amplitudes of the object and the reference beams at the recording plane xy are respectively:

where IO(x,y)l and IR(x,y)l are the real amplitudes of the light waves, and @(x,y),@(x,y) are the phases. In its simplest form, the reference wave R(x,y) is a plane wave of uniform intensity (IR(x,y)l = A R = const.), which strikes on the recording plane in a different angular direction from the object wave. If we assume that the average direction of the object wave is the z-direction (i.e,,

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normal to the hologram plane), the phase of the reference wavefront can be written as:

where A is the wavelength of the laser illumination, and is the angle between the object and reference wave. Furthermore, the linear progression along y indicates that the incident wave is off-axis in the y-dimension. It is mainly the Qstribution of the phase @(x,y) which carries the desired information. This information is extracted by superposition of O(x,y) and R(x,y). The resultant intensity can be written as:

where * denotes the complex conjugate. For simplicity, the amplitude transmittance T(x,y) of the processed photographic plate can be assumed to be a linear function of the intensity, and can be given by the relation:

where T o is a constant background transmittance, t is the exposure time and

fi is a constant factor which depends on the emulsion characteristics and the development process. The resultant amplitude transmittance of the hologram is then T(x,y)= TO-fi&-fif

1

10(x,y)12 - 2 f i t A R Io(x,Y)l cos y ~ s 1 n E - # ( x , ~ )

[2n-

(2.6) In this way the original information, IO(x,y)l exp[i@(x,y)], amplitude and phase modulation of a monochromatic spatially coherent light wave, has been transferred to a spatial carrier wave cos[y(2n/A)sinE]. For this reason the holographic material must have a very high resolution in order to resolve fringes at the carrier frequency of the hologram. Special photographic plates are made for this purpose (Saxby [1988]). Moreover, the hologram must be made in an environment entirely free of vibration.

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By illuminating the developed hologram with the same reference beam R(x,y), the complex amplitude U ( x , y ) of the transmitted wave can be written as:

where

The first term of these components, U , ( x , y ) ,is a uniformly attenuated version of the incident beam, while the second term U ~ ( yX),yields a halo surrounding it, which is spatially varying. The angular spread of this halo is determined by the angular extent of the object. The third term, U,(x,y), is identical to the original object wave, except for a constant factor, --PtA;, and produces a virtual image of the object in its original position. Similarly, the fourth term, Ud(x,y), corresponds to the conjugate image, which in this case is a real image; it is deflected off axis at an angle approximately equal to 2E. If the offset angle of the reference beam is made large enough, the virtual image can be separated from the directly transmitted beam and the conjugate image. It is to be noted that the shift of U4(x,y), for large E, determines a distortion of the real image. Since in holographic interferometry the virtual image is mainly used, in this context the effect of distortion can be ignored. Figure 3 shows the reconstructed image, obtained by illuminating the hologram with the same reference beam used for recording. If the hologram is placed, during reconstruction, in the same position occupied when making it, and is illuminated once again with the same reference beam used for recording, the image has the same size as the original object and coincides exactly with it. In any other case, the image may exhibit aberration. However, any change of the position or wavelength of the source used for reconstruction results in a change of the position and magnification of the reconstructed image. Generally, the hologram recorded on a photographic film has been treated as equivalent, in a first approximation, to a grating of negligible thickness with a spatially varying transmittance. Under this condition, volume effects can be neglected (Hariharan [1987]). By means of holography, we are able to store the wavefront scattered by an object in a hologram. We can, then, recreate this wavefront by hologram

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v 1I'tu a

iinaye

[IV,

52

I

\ Real

irnage

Fig. 3. Image reconstruction by a hologram recorded in an off-axis configuration.

reconstruction, where and when we choose. This gives us a new opportunity in interferometry. This technique, usually called holographic interferometry, is similar to conventional interferometry and permits comparison of arbitrary waves, originally separated in space and in time. There are three variations of the basic holographic interferometry techruques: real-time, double-exposure, and sandwich holography. Each possesses certain advantages over the others in particular test situations. In the following we present the fundamental procedures and analyses. 2.2. REAL-TIME HOLOGRAPHY

Real-time holography involves the recording of a conventional hologram of the investigated object and the live observation of the formation and evolution of the fringes. A single holographic exposure of the test object in an undeformed state is recorded with the holographic setup of fig. 2. The developed hologram is replaced exactly in the same position at which it was recorded; when it is reconstructed with the identical reference beam used in the recording process, the virtual image is superimposed on the object. If, however, the shape of the object changes (very slightly), two light waves reach the observer, one being the reconstructed wave (corresponding to the object before the change) and the other the wave directly produced by the object in its present state. The actual object

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wave interferes with its holographic replica; the latter serves as a comparison field value and allows one to follow the changes of the sample as a function of time. The particular advantage of this technique is that different types of motions, both dynamic and static, can be studied from a single holographic exposure rather than making a series of holograms for each state. However, there are some disadvantages in its use. The precision with which the hologram must be replaced is of the order of wavelengths. For this reason, a specially designed plate holder must be used. Otherwise, spurious fringes due to imprecise repositioning of the hologram or rigid body movements could mask the deformation under study. Additionally, the fringe contrast is generally lower for real-time than for other types of holographic interferometry. As elsewhere mentioned, during most practical work with real-time holography, the object must be stressed sufficiently to cause observable differential changes in the region of defects or anomalies; it can become very difficult to distinguish the fringes caused by the required faults, from those due to the total object deformation or to rigid-body motion (Erf [1974]). Fringe control techniques can be used to manipulate the fringe pattern so as to vividly bring out the areas of local deformation. The most direct compensation method would be to move the object itself, during its observation in real time, by using a micropositioning device. This method is complicated and extremely insufficient if, at reconstruction, there is a gross anisotropic deformation of the structure (e.g., wooden samples) from thermal stressing. In this case, unwanted fringes could be eliminated (Shakher and Sirohi [ 19801) by positioning some optical elements in the object illumination and reference wave paths. Another simple method of fringe control is obtained by shearing the hologram plate in its plane at reconstruction. A translation of the hologram plate will introduce shearing fringes on the holographic image that are either added to or subtracted from fringes caused by object deformation (Paoletti, Amadesi and D’Altorio [ 19841). 2.3. DOUBLE-EXPOSURE HOLOGRAPHIC INTERFEROMETRY

It is also possible to record two holograms on the same photographic plate, with each one capturing the object in a different state, separated by a fixed time interval. During the reconstruction the two waves, scattered from the object in its two states, will be reconstructed simultaneously and interfere, producing onto the three-dimensional virtual image of the object an interference pattern that in general represents a contour map of the object changes. This double-exposed hologram can be stored and later reconstructed for analysis.

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This technique is less critical than real-time holography, because the two interfering waves are always reconstructed in exact register. Distortions of the emulsion affect both images equally, and no special care needs to be taken in illuminating the hologram during the reconstruction. In addition, since the two diffracted wavefronts are similarly polarized and have almost the same amplitude, the fringes have a very good visibility. Useful characteristics of holographic interferometry include simple, wholefield visual display and applicability to inspection of components with complicated shapes. Limitations of the technique include, in general, stringent requirements of mechanical stability and a restricted range of sensitivity. In particular, for double exposure, the two holographic images cannot be changed during reconstruction. The information on intermediate states of the object is lost; undesired fringes cannot be eliminated. Another disadvantage is that, since the images are on the same holographic plate, one cannot know which image was recorded first. Therefore it is impossible to determine the direction, say forward or backward, of the physical displacement. 2.4. SANDWICH HOLOGRAPHY

A compensation of rigid body motion with elimination of sign ambiguities can be obtained by sandwich holography, a very useful variation of double-exposure holographic interferometry (Abramson [ 1974,19811. This technique gives more versatility to the double-exposure setup; instead of making a double exposure on one holographic plate, the two exposures are made on different plates which are then combined in a plate holder. In the sandwich hologram technique the reconstructed images fiom two holograms, relative to different states of deformation of an object under study, are recorded on different plates. The two hologram plates are simply placed one in front of the other during reconstruction with their emulsions towards the object. To make the plate position optically and mechanically identical during recording and reconstruction, a particular plate holder is used. The image, reconstructed from a sandwich hologram with the reference beam used for recording, gives the same fringes of a conventional double-exposure hologram but, by having the images on two different plates, it is possible to manipulate them and hence the fringe pattern. In the original technique proposed by Abramson [19741, the fringe manipulation is realized by tilting the sandwich with respect to its initial position. Subsequently, Amadesi, D’Altorio and Paoletti [1982a] and Paoletti, Schirripa Spagnolo and D’Altorio [1985] have proposed a new method of fringe manipulation by shearing one

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hologram plate with respect to the other with the help of an accurately adjustable plate holder. There are several advantages in using this new version of the method. For example, a number of holograms can be made, each one recording a single state of the object, in a temporal sequence. Afterwards the plates can be combined in pairs as desired, allowing one to compare any two hologram plates to study interferometrically any changes in object, in order to have a quasi continuous monitoring of the object response to external stress. Probably the most important advantage of sandwich holography is the possibility of manipulating the fringe pattern; it allows us to eliminate unwanted fringes caused by rigid body motion of the object investigated. In fact, as the two reconstructed images can be moved with respect to each other, spurious movements of the object between the exposures can be compensated by a movement of the holographic image during reconstruction. Moreover, the possibility of a posteriori manipulation of the fringe pattern permits an evaluation of local deformation even in presence of much larger displacement of the object under study (about 1000 fringes can be eliminated by a simple sandwich-hologram tilt or shearing). In some instances, for large three-dimensional structures, where the tension stresses are not distributed uniformly, we can effectuate a compensation over local regions. Practically, a continuous scanning of the specimen can be done by shifting the center of maximum sensitivity of the fringe pattern in order to produce a clear display of a flaw, with a configuration that outlines the flaw and gives an approximate indication of its size and shape. In our opinion, this versatility of the sandwich hologram is of particular interest in art diagnostics. Furthermore, since the images are on two plates, the displacement and/or sign of the object can be easily resolved because it is equivalent to the sandwichhologram shift of direction and sign needed for fringe-free object reconstruction. Finally, by manipulating the sandwich pair, a system of linear fringes can be added to or subtracted from the interferogram; these fringes act as a spatial heterodyne carrier. The combination of carrier fringes and deformation fringes gives a resultant pattern suitable for quantitative analysis of deformation (Diubur and VukiEeviC [1984], Paoletti and Schirripa Spagnolo [ 19941). Many problems of double-exposure and real-time holography are so resolved, but there are also some disadvantages: the plates must be repositioned with high accuracy during reconstruction, the freedom of fringe manipulation also contains the seeds of possible errors, and, to achieve the built-in accuracy of double exposure, one must record reference points to guarantee that the fringe pattern being observed reflects actual object changes rather than being a pattern created by moving the images relative to one another. A correct use of the technique

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generally requires the worker to construct a special holographic plate holder which readily allows fringe manipulation. 2.5. INTERPRETATION OF THE INTERFERENCE PATTERN

If the object under observation deformsfdisplaces between the two exposures of the plate (for double-exposure hologram and for sandwich hologram, or between hologram and actual state for real-time hologram), the phase difference between light beams arriving at a specific observation point, from the object and its displaceddeformed copy, is encoded in form of fringe patterns (Vest [1979], Hariharan [1993], Yamaguchi [1985], Ostrovsky and Shchepinov [1992]). These fringes are described by a fringe-locus function A@, constant values of which define fringe-loci on the object's surface. This fringe-locus function can be related to the optical path length variation of the light. The problem is then to find the relation between the fringe-locus function and the deformatioddisplacement. With reference to fig. (4a), the optical path length of the light for the object in the normal state is denoted by L N :

where S = point of light source, ON = point on the surface of the object in its normal state, and H = point in the hologram plane. When the surface of the object is deformed, the point ON changes in OD.With the object in the deformed configuration, the optical path length of the light, denoted by L D , is:

The optical path change related to the object deformation is A1 = LD - LN.

(2.10)

The path variation A1 is related to A@ by 2nfA; therefore, the phase change can be written as: A@=

2n ~ ( LDLN)'

(2.1 1)

This equation, although completely general, is not in a convenient form for numerical use. If the displacement I ONODI is considerably smaller than the distance between the object and the point of divergence S of the object illumination beam and of the distance between the object and the holographic

IY § 21

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3

21 1

H

Fig. 4. Geometry of fringe formation for object deformation: (a) general illumination and viewing geometry; (b) the change in the optical path associated with a displacement.

plate, the illumination and viewing directions for ON and OD can be assumed to be the same. We denote with a unit vector ns the direction of SON, and with nH the direction of O N H .Note that the image used in holographic interferometry is the virtual image. This image cannot be examined directly but can be viewed by an imaging system. The direction O N H depends on the viewing point and on the form of viewing used. If we consider fig. 4b, the optical path change LD - LN can be considered to be made up of two components, Als + A ~ HExplicitly, . we have:

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"5 2

(The assumption that ISON(>> 10~0~1 is equivalent to assuming that SON and

SOD are approximately parallel, i.e., nk M ns. Likewise, the assumption that (ONHI >> IONOD(is equivalent to assuming n& M nH.) Thus, the phase change

due to the displacement I ONODI is given by:

(2.14) When the point source and the point of observation are placed at finite distance from the object, the illumination and observation directions (ns and nH) will vary across the object surface. The sensitivity (displacement per fringe) then varies across the object surface, but for small objects and reasonably long distances to the source point and the observation point, this variation will be quite small. As mentioned earlier, because of the random phase variations across the object wave front, only waves from corresponding points on the two interfering wave fronts contribute effectively to the interference fringes. For a specific viewing direction, the phase difference A$ between the waves from two points (ON, OD) is given by eq. (2.14). This phase difference will therefore vary over the range of viewing directions defined by the aperture of the viewing lens, resulting in a loss of contrast of the fringes. However, it is possible to find a plane in which the variation of A# is minimum over this range of viewing directions; this is the plane of localization of the fringes (Vest [ 19791, Rastogi [ 19941). The position of the plane of localization depends on the type of displacement and on the aperture of the viewing lens. When the fringe localization plane is widely separated from the object surface, a precise identification between a point on the fringe plane and a point on the object cannot be made. For a correct interpretation of the fringe pattern, it must be localized on the object surface. By using a sufficiently small observing aperture, i.e., a sufficiently large fnumber, distinct fringes can be observed in the plane of the object, and can be related to the object surface deformatioddisplacement. For more details see Velzel [1970], Vest [1979] and Gisvik [1987]. 2.6. HOLOGRAPHIC INTERFEROMETRY AS NONDESTRUCTIVE TESTING

Because of its extreme sensitivity to surface deformation, holographic interferometry can be used to gain meaningful information with regard to the structural characteristics of an object, by observing the surface movement produced when

rv,

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it is subjected to a stressing force. As such, it offers the potential for many nondestructive inspections wherein the parameter of interest (e.g., cracks, voids, debonds, delaminations) can be made manifest as discontinuities in surface displacement. The discontinuities appear as an anomaly in an otherwise regular interferometic fringe pattern andhence enable the region of fault to be identified (Vest [1979], Jones and Wykes [1989]) For this reason a stressing technique must be devised in such a way that the anomalies induce detectable perturbations in the surface deformation. Usually the stressing method (a temperature gradient obtained by heating the surface by an infrared lamp or a stream of moderately warm air) is chosen empirically with guidance provided by an analysis of anticipated deformation and by previous results obtained from programmed models. Generally, artworks are analyzed in thermal drift (with a brief thermal irradiation which raises the surface temperature by somejegrees), or in ambient drift (under ambient parameter variations). It seems that the thermal-drift method is considerably better in detecting detached regions than the ambient-drift method: many more detachments are detected, and they are much more defined. Finding the best method and test procedure is partially an empirical process, and it is a function of the component and the expected nature of the fault. On the basis of calibration experiments, it is reasonable to assume that the fringes will generally have a higher spatial density in regions of a structure where strain concentrations are present. Furthermore, one would expect fringe irregularities to exist in regions of uneven load distribution. Co'nversely, a structure which exhibits uniform load-carrying characteristics should generate a deformation fringe pattern of predominantly uniform spacing. In conclusion, the overall appearance of the fringe pattern may be used both as a qualitative guide to the homogeneitiesiof the deformation of a complex structure and as a means for quantitative measurement of strain and stresses. 2.7. HOLOGRAPHIC INTERFEROMETRY APPLICATION IN ARTWORK DIAGNOSTICS

In this section, we present the state of the art of holographic investigations in artworks conservation. A little more than twenty years ago, Munk and Murk [ 19711 proposed a feasibility study to determine whether holographic techniques could be applied in the field (i.e., outside of the laboratory) to actual artistic materials and their problems. Some experimental investigations were conducted in Venice, during 1972, to demonstrate the feasibility of in situ holography of life-sized statuary and wood-carvings (Asmus, Guattari, Lazzarini, Musumeci and Wuerker [ 19731).

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Fig. 5. Double-exposure holographic interferogram showing detached regions of various shapes in a 16x 13x2cm3 panel of primed poplar wood. (From Amadesi, Gori, Grella and Guattari [1974].)

Emphasis was placed on subjects undergoing accelerating disintegration, as a result of environmental processess. Image records of several Venetian art objects to place in archives were produced by pulsed Q-switched ruby laser holography. In addition, interferometric techniques were found to be of potential utility in statue restoration by means of the location of hidden patches, cracks and flaws. To obviate problems of environmental vibration and ambient light, double-exposure holograms with a pulsed Q-switched ruby laser were effected, in particular, on a gold-leafed and painted wooden carving (St. John the Baptist by Donatello). Its wooden base and complex surface were ideal for interferometric nondestructive testing. Some components of the carving (head, leg, etc.) were analyzed with thermally and local-humidity induced deformations. In most of the subsequent investigations carried out so far, conventional double-exposure holographic interferometry with continuous laser was employed, mainly for detecting the presence of surface and invisible subsurface defects. The first detailed and accurate study of the complex behavior of panel paintings was made by Amadesi, Gori, Grella and Guattari [1974]. In this work, preliminary experiments were carried out on some models simulating a wooden painting support, either uncoated or coated with the usual priming layers in order to obtain information about the basic components of a painting. A two-slab structure was chosen in order to reproduce the actual support of an ancient panel painting, on which subsequent tests have been carried out, and its response in thermal drift was examined. A number of detached regions of known

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Fig. 6 . Interferogram of an ancient 38x49x4cm3 panel painting on poplar wood (Suntu Calerina, by Pier Francesco Fiorentino, 15th century). (From Amadesi, Gon, Grella and Guattari [1974].)

shape and extension were provoked inside the layered structure by the insertion of suitable plastic sheets to simulate the presence of defects. An example of an interferogram, where detached regions are visible by islands of fringes, is provided by fig. 5. These fringes have the same structure as a debonded region between the core and skin in a honeycomb panel (Vest [ 19791). Subsequently, a panel painting of the fifteenth century (namely Sunta Caterina by Pier Francesco Fiorentino; circa 1450) was analyzed by double-exposure holography in laboratory. This wooden panel painting consists of a support (two slabs of poplar wood glued together), priming layers of gesso and glue, and the painting. Figure 6 shows a particular double-exposure hologram of this artwork. The interferogram was taken by changing the temperature of the subject between the two exposures by heating the surface with a stream of moderately warm air. It will be observed that a number of detachments (revealed by clusters of distorted

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Fig. 7. Holographic interferogram of Madonna della Misericordia from the Perugina school. The circled areas indicate detachments as evidenced by irregular fringe patterns. (From Westlake, Wuerker and Asmus [ 19761.)

fringes) are concentrated along a vertical line, which corresponds to the junction line between the two wooden slabs constituting the support. Some detachments at an incipient stage located between the layers, that could not be easily detected by the unaided eye, have also been evidenced. An obvious question to ask is whether the fringe anomalies really correspond to detached regions. In order to answer this question, some works of art were restored through injection of suitable adhesives. Some holographic tests were made before and after a panel painting was restored. In all cases, fringe anoma-

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217

Fig. 8 This interferogram shows the Mudonna dellu Misericordia after restoration attempts on four of the detachments. The repaired regions, shown through the ellipses, indicate a successful bond between the painting and the wood backing. (From Westlake, Wuerker and Asmus 119761.)

lies were greatly reduced or disappeared altogether. This was especially true for those works of art where the priming layers had reduced thickness (Gori [1976]). A representation of an interferometric reconstruction of the Madonna della Misericordia from the Perugino school is found in fig. 7; the circled areas indicate detachments as evidenced by fringe distortions. The interferogram of fig. 8, obtained after restoration, shows how four of the detachment zones were repaired and a proper bond between the primer ground and wooden support was reinstated (the ellipses indicate the zone of restoration).

218

INTERFEROMETRIC METHODS FOR ARTWORK DIAGNOSTICS

[IV,

52

Later, holographic tests (double-exposure and real-time holography) were carried out by Bertani, Cetica and Molesini [1982], in the laboratory of the Istituto Nazionale di Ottica (Florence Italy) as part of a research program on the state of conservation and restoration methods of a panel by Lorenzo Ghiberti. The artifact is a bronze cast which belongs to the right portal of the Door of Paradise of the Baptistery of Florence, and shows the Li#e of .Joseph. In this study, the authors have shown the kinetic behavior of the-artifact in response to small thermal stress, simulating the larger deformation ,caused by the daily thermal variation in the natural environment. In this way, it has been demonstrated how temperature variations affect both particular parts of the object and its overall structure, acting in time as an agent of deterioration. In the same years, holographic interferometry was also proposed for fresco diagnostics, but very little work, to our knowledge, has been done on this subject until now. Vlasov, Ginzburg, Novgorodov, Protsenko, Stepanov and Ushkov [ 19761 applied holographic interference methods to determine the optimal temperature-humidity conditions for preservation of frescoes located in closed environments (e.g., churches, museums). Experimental investigations were conducted Ifor genuine samples of construction materials from some cathedrals of Moscow. Rizzi [ 19831 used double-exposure holography to study a fresco by Bramante (Gentihtomo in armi) detached from the wall of a house in Milan (Italy) and now in Accademia di Breru. The studies were made by analyzing the fresco in thermal drift in order to detect the delaminations in the fresco layers. Further applications of double-exposure holographic interferometry were made by Accardo and Micheli [I9831 to study a reversible mechanical system for assembling bronze statues, and by VukiEeviC, von Bally and Sommerfeld [I9931 for nondestructive analysis of anjlancient plate with cuneiform inscriptions. Oftcn the interferograms obtaincd in these proofs do not allow thc visualization of small irregularities masked by the overall deformation. During the early 198Os, sandwich holography (Abramson [1981]) was found to be useful for a better localization of defects. By manipulating the fringe pattern, as already mentioned, it is possible to explore the surface of the subject in order to define the defect contours in the best way. The technique was proved by Amadesi, D' Altorio and Paoletti [ 1982a1 and by Amadesi, D'Altorio, Paoletti and Petraroia [I9833 first on test models, simulating flat panel paintings, andisubsequently on 3D models. These models were made with simulated detachments of various depth and several configurations with the same technique proposed by Amadesi, Gori, Grella and Guattari [1974], and by God [1976]. All test samples were studied in ambient drift and thermal drift. Figure 9a shows the profile of an undulating primed panel, and fig. 9b is a sandwich reconstruction of the same

Iv, 9; 21

GENERAL REMARKS ON HOLOGRAPHIC INTERFEROMETRY

219

Fig. 9. (a) Profile of an undulating primed panel of poplar wood; (b) sandwich hologram reconstruction; (c) manipulated fringe pattern.

panel examined in ambient drift. One can see that the deep flaws located in highly convex areas are more evident than those in less curved regions. Figure 9c shows a manipulated fringe pattern from the sandwich hologram of the same panel in thermal drift; note that, by shearing the sandwich hologram, the location of deep flaws is more defined. This method was used by Amadesi, D’Altorio and Paoletti [1982b] for studying the state of conservation of an Italian wooden panel painting of the fifteenth century (Madonna con Bambino of the Perugina School). A fringe pattern of a sandwich hologram obtained by heating the painting surface using a flow of moderately warm air is shown in fig. 10. The presence of faults is not very evident, being masked by the overall deformation of the object. By manipulating

220

INTERFEROMETRTC METHODS FOR ARTWORK DIAGNOSTICS

Fig. 10. Fringe pattern from a sandwich hologram of the painting Madonna con Bambino; some detachments are detected by the presence of locally closed fringes (as indicated by arrows). (From Amadesi, D'Altorio and Paoletti [1982b].)

the fringe pattern, some hidden faults are easily detected (see fig. 11) in the region of interest. An example of a particular versatility of the sandwich method as applied to statue is illustrated in fig. 12 (head of St. John the Baptist, 16th century). The unwanted fringes are eliminated on the face, in such a way that the fringes at the nose reveal local anomalies (ancient restoration process). Figure 13 shows

rv, § 21

GENERAL REMARKS ON HOLOGRAPHIC INTERFEROMETRY

22 1

Fig. 11. Fringe pattern obtained by shearing the sandwich hologram, the locations of some faults are indicated. (From Amadesi, D’Altorio and Paoletti [1982b].)

the same sandwich hologram of fig. 12 after manipulation, so that the area with anomalies is free of fringes. Real-time holography with fringe manipulation was applied successfully to the detection of incipient microcracks in a particular painting (icon). This technique was used by Amadesi, D’Altorio and Paoletti [1983a] to detect the presence of a crack at the initial phase on a golden painting of the 14th-century Madonna con Bambino of the Byzantine-Venetian school. Figure 14 is a photograph of

222

INTERFEROMETNC METHODS FOR ARTWORK DIAGNOSTICS

Fig. 12. An example of a sandwich hologram of the head of S. Giooanni Battista (16th century). Note the versatility of the technique. The unwanted fringes are eliminated on the face in such a way that the fringes at the nose reveal local anomalies. (From Amadesi, D’Altorio and Paoletti [1983b].)

Fig. 13. The same sandwich hologram as in fig. 12 is here manipulated so that the area with anomalies is free of fringes. (From Amadesi, D’Altorio and Paoletti [1983b].)

Iv, § 21

GENERAL REMARKS ON HOLOGRAPHIC INTERFEROMETRY

223

Fig. 14. A golden icon of the 14th century. A very wide crack is visible. (From Amadesi, D’Altono and Paoletti [ 1983al.)

the panel painting with a very wide surface crack. Figure 15 is a photograph of an interferogram in real-time with a stress obtained in thermal drift (by raising the surface temperature by about 1°C above the ambient temperature). Careful examination of the fringe pattern shows that the regularity of the local mean behavior of the fringes is altered in a number of regions, but the detection is poorly defined. The fringe pattern can be manipulated and its center shifted on the painted surface during temporal evolution in order to improve the detection. Figure 16 shows a photograph of a fringe pattern obtained after the manipulation of the interferogram of fig. 15. The painting shows several abrupt discontinuities along the trend of some fringes; these discontinuities correspond to subsurface cracks formed over the years by the varying tension between the painted layer and the ground layer. Fringes become locally closed along the trace of a deep crack

224

INTERFEROMETRIC METHODS FOR ARTWORK DIAGNOSTICS

Fig. 15. Photograph of an interferngram realized in real-time on the icon of fig. 14 in thermal drift. (From Amadesi, D’Altorio and Paoletti [1983a].)

present on the wooden support. The location and the path of a wide vertical crack, at the initial stage, parallel to that just visible on the surface has been clearly brought out. A combined system, utilizing sandwich and real-time holography, was used by Amadesi, D’Altorio and Paoletti [ 1983bl to yield information about the state of conservation and the dynamic behavior of a large painted 14th-century gessowood statue of the Abruzzi School. De Angelis and Garosi [I9881 and Caponero and De Angelis [1991] have checked the possibility of using double-exposure holographic interferometry to study works of art with large size and/or complicated structure. A bronze equestrian statue of the Musei Capitolini of Rome was analyzed to detect the presence of flaws and to estimate their importance. In these studies a holographic

GENERAL REMARKS ON HOLOGRAPHIC INTERFEROMETRY

225

Fig. 16. Fringe pattern obtained by shearing the hologram plate of fig. 15; the path of a vertical crack is indicated by the presence of locally closed fringes along a vertical line (as indicated by arrows). (From Amadesi, D’Altorio and Paoletti [1983b].)

system employing a double-pulse, high-coherence ruby laser was used. The statue has a linear dimension of about 2 meters and a weight of about 600 kg. The experimental results have evidenced the presence of fractures, discontinuities in the welding, cracks and voids. An interferogram relative to the head and the neck of the artwork is shown in fig. 17. The fringe pattern shows an irregularity on the neck (at the bottom and left). This anomaly is relative to a zone with a fracture. An interferogram relative to the shoulder is presented in fig. 18. A very large crack can be identified in this interferogram. The abrupt discontinuities along the trend of some fringes in correspondence to the shoulder and to the foreleg is related to voids and cracking present in this zone. In precedence, some other diagnostic tests using a particular speckle interferometer based on speckle photography (Accardo, De Santis, Gori,

226

INTERFEROMETRIC METHODS FOR ARTWORK DIAGNOSTICS

[WP 2

Fig. 17. Double-exposure interferograms of the left part of the head and of the neck; the fringe pattern shows an irregularity on the neck. (Photo ccurtesy of Albert0 De Angelis)

Guattari and Webster [1983], De Santis, Gori, Guattari and Webster [1985]) were made to study the behavior under mechanical stress of the bronze horse statue, of roman age, Murco Aurelio [placed by Michelangelo at Campidoglio, Rome, from 15381 under restoration. Some other review papers on holographic nondestructive testing applied to artwork diagnostics have been written by Westlake, Wuerker and Asmus [1976], Paoletti, Schirripa Spagnolo and D’Altorio [ 19891, and Markov [1993]. 2.8. IJMITING FACTORS OF HOLOGRAPHIC INTERFEROMETRY

Any art object which scatters or reflects light can, in principle, he studied with holographic techniques. Indeed, it has been widely demonstrated that the

Iv, § 21

GENERAL REMARKS ON HOLOGRAPHIC INTERFEROMETRY

227

Fig. 18. Interferogram of the left shoulder and of the left foreleg; the fringe pattern shows some irregularities in correspondence of the shoulder and the foreleg (indicated by arrows). (Photo courtesy of Albert0 De Angelis.)

interaction of an artifact with its surroundings can be successfully studied and that the location and size of defects, areas of excessive stress, or material discontinuities can be identified. In spite of these obvious advantages, the actual application of holographic techniques to routine inspection of works of art is rather limited. This can be attributed to a variety of factors, including: timeconsuming wet processing of holograms on silver-halide emulsions together with the requirement of high operator skill, relatively fixed sensitivity of holographic interferometry, and the high cost of the system. A major problem which prevents widespread application of the technique is caused by the need to transport the object to an optical laboratory. In fact, the practical application of holographic interferometry has some disadvantages, which have, in general, hindered its widespread use in situ. In the description of the holographic recording process, we assume that the spatial phase of both the object and reference wave are time independent during the exposure. It is clear, however, that relative

228

INTERFEROMETRIC METHODS FOR ARTWORK DIAGNOSTICS

tIv, § 3

movements between the different optical components (like mirrors, beam splitter, holographic plate, etc.) in the hologram setup will introduce phase changes. Furthermore, to obtain a holographic interference pattern of good quality the optical components, including the surrounding environment, must remain stable to at least A/8; with a continuous laser the required exposure time is usually on the order of seconds or tens of seconds for a hologram. Consequently, the required stability during the exposure must be better than 0.1 pm to ensure a high-quality image. This fact imposes stringent requirements on the stability of the setup. Therefore, the standard methods of holography are normally performed on vibration-isolated heavy tables with the optical components mounted in massive holders. This makes the holographic technique not readily compatible with in-field conditions unless pulsed lasers are used (Tyrer [ 19861). However, these lasers are not very flexible since an external water cooling circuit should be provided, and most systems have a large size, weight and cost. Moreover, their high energy radiation pulses and high voltages dictate that special safety precautions be taken. A measuring instrument, in order to become popular among art workers, should visualize micro deformation and/or displacement, perform real-time analysis, operate under field conditions, and require a minimum of adjustments. These requirements can be fulfilled with electronic speckle pattern interferometry (ESPI), a technique currently used as an industrial inspection tool. In addition, the analogy between ESPI and image plane holography (Lskberg [ 19801) allows some holographic techniques to be translated directly into ESPI. Giilker, Hinsch, Holscher, Kramer and Neunaber [1989] have recently demonstrated the use of ESPI to monitor deformation and deterioration processes in historical buildings and monuments. 3. Electronic Speckle Pattern Interferometry

A diffusely scattering surface illuminated by laser light appears covered by a pattern of bright and dark spots, or speckles, distributed randomly in space. This occurs because neighboring microscopic elements making up the surface produce random differences of the optical path for the scattered light. At any point, therefore, scattered waves arrive from many of these elements simultaneously and, as they are highly correlated, their instantaneous amplitudes add coherently. However, as the phases are random they may provide at any point constructive (bright speckle) as well as destructive (dark speckle) interference (Goodman [1975]). Considered by the majority of holographers as the stain of coherent

IV,

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31

ELECTRONJC SPECKLE PATTERN INTERFEROMETRY

229

illumination, this “annoyance” can still relay information about the surface characteristics as well as its displacement. The combination of the imaged speckle pattern and a reference wavefront to produce a phase-referenced speckle pattern can be used in a way similar to holographic interferometry. The idea of linking this speckle interferogram with a TV camera was the advantage that set this technique apart from holography. The ability to capture images by a video system combined with enhancements achieved by subsequent electronic processing allowed the development of a new, more efficient method, called electronic speckle pattern interferometry. 3.1. BASIC PRINCIPLES OF ELECTRONIC SPECKLE PATTERN INTERFEROMETRY

The electronic speckle pattern interferometry (ESPI; also called TV-holography) was developed in the early 1970s as a method of producing interferometric data without using traditional holographic recording techniques (Butters and Leendertz [ 19711, Macovski, Ramsey and Schaefer [ 19711, Schwomma [ 19721). The ESPI system is usually categorized as a laser speckle method rather than a holographic one, but it is closely related to real-time holographic interferometry (Lprkberg [1980]). Practically, ESPI can be viewed as the combination of holography and speckle interferometry (Ennos [ 1975]), the holographic film being replaced by a TV camera as the recording medium. Obviously, the photosensor of the TV camera is not suitable for optical reconstruction of the hologram; therefore the reconstruction process is performed electronically and visualized on a monitor. The signal picked up by the TV camera is converted into a corresponding video signal by the scanning action of a video camera. This video signal is electronically processed through an intermediate recording medium (commonly a frame grabber) before being displayed on a TV screen, so that the variations in the texture of the speckle are converted into a variation of brightness. This image is entirely equivalent to a holographic reconstructed image and possesses the same interferometric sensitivity. Note that each processed video frame can be considered as a recorded and reconstructed hologram. Consequently, 25 holograms (by the European TV standard; 30 holograms by the American TV standard) are presented on the TV monitor each second. This high sampling rate combined with the relatively short exposure time of 40ms (33ms American standard) is the reason for the technique’s improved immunity to interferometric instabilities that would ruin most conventional hologram interferometric recordings (Lprkberg and Malmo [ 19881).

230

INTERFEROMETRIC METHODS FOR ARTWORK DIAGNOSTICS

[IV, § 3

In this section we give only a brief practical description of the principle of the method; the full theory is well covered in Jones and Wykes [1989]. If the object surface is illuminated by laser light and is imaged by a lens, the intensity of the scattered field has a characteristic granular aspect known as the objective speckle pattern (Ennos [ 19781). When the imaged speckle pattern is combined with a second (reference) beam, a new speckle pattern is produced as a result of the interference between the two beams. The complex amplitudes of the object beam at starting steady conditions and of the reference beam can be expressed, at a given point (x,y) on the image plane, as

respectively, where I Uo(x,y)l and I U R ( X , ~are ) ~the real amplitudes, and @o(x,y) is the phase of the light scattered from the object surface, varying randomly across the image plane, while @R(x,Y) is the phase associated with the reference beam. The latter may vary randomly or may be constant, depending on the form and type of the reference beam. The resulting intensity rl, at the point ( x , ~ )on the image plane, with the test surface in its equilibrium condition, is then given by : rl(x,y) = IuO(x,y)12 + IuR(x,y)12

+ 2 IuO(x,y)l IuR(x,y>lcos[@O(xiy)-@R(X,Y)I.

(3.3)

When the object surface is displaced or deformed, the speckle pattern changes. If we assume that the displacement andor the displacement gradient are not too large, it is possible to consider that all the components scattered from the resolution area to the image plane point change their phase by the same amount, A@(x,y)(Wykes [ 19821). With this approximation, after a small static object deformation andor displacement, the intensity at point (x, y ) will be changed to r2,where:

r 2 ( x , y )=

IuO(X,y)l2 +

IuR(x,y)l2

+ 2 IuO(x,y)l IuR(x,y)l cos[@O(x,r)-@R(X,y)-A@(x,y)l. (3.4) If the phase change A@(x,y)is not significantly large, the scattered object beam amplitude (1 U O ( Xy)l), , the reference beam amplitude (1 U~(x,y)l),and the phase difference between object and reference beam (@o(x,y)- @R(x,Y))are assumed

rv, 9: 31

23 I

ELECTRONIC SPECKLE PATTERN INTERFEROMETRY

to be the same in eqs. (3.3) and (3.4). Moreover, we assume for simplicity that the speckle patterns recorded by the TV camera are fully resolved by its spatial resolution and that the speckle decorrelation (Owner-Petersen [ 199 11) betwcen the two combined exposures is negligible. The correlation between the intensities TI and r2 can be obtained by subtractingiadding eq. (3.4) from!to eq. (3.3). The subtraction process is the one most often used at the present time, because it ensures a significant reduction of the fixed pattern noise (optical noise). Moreover the subtraction fringes have intrinsically better visibility than addition fringes. In the subtraction process the interferometric speckle pattern r l is focused by an imaging lcns onto the photosensor of the TV camera. The video signal corresponding to rl is digitized and sioreti by a frame buffer and a host computer. The live video signal of Tz is thcn subtracted from the stored w v e form. The result is squared arid then displayed on a T V monitor, where live correlation fringes may be observed. The output camera signals C’I and I.’* are proportional to the input image intensities rl anti r2 respectively. The subtracted and squared image is given by:

vs = [( vi

v,)]’

‘K

(r‘~ --

r2):

(3.5)

= n(X, J)) E(X,y),

where 2 . 2

n(x,y) = 8 I(hk?’)12 IU/K(X,Y)I sln and

{ [ @ O ( X , Y )- @K(X,Y)I+ ;A@cx?Y>}

E ( x . ~=) 1 - COS[A@(X,-Y)].

9

(3.6) (3.7)

Here n(x,J)) represents the speckle noisc and E(x,y) describes the fringe pattern. Averaging cq. (3.5) over all speckle phases, we obtain

The angular brackets denote ensemble averaging. Apart from the sign of the cosine function, eq. (3.8) is equivalent to that in classical holographic interferometry and describes perfect correlation fringes without speckles, and with a unit visibility. Since the phase change A @ ( x , y )is a function of the displacement of the surface, information about the relative displacement of different parts of the surface can be obtained. There will be dark fringes where

A @ ( x , y )= 2nn, n = 1,2,3, . . . , whereas other areas will exhibit a maximum mean intensity wherc A@(x,y)= (2n + l)n, n = 1,2,3, . . . .

(3.9) (3.10)

232

PRINTER

1NTERFEROMETRIC METHODS FOR ARTWORK DIAGNOSTICS

",ii 3

1

BEAMSPI.ITTER

[OBJECT

BEAM

1

REFERENCE BEAM

Fig. 19. An optical ESPI arrangement giving fringes representing out-of-plane displacements.

3.2. OPTICAL GEOMETRY IN ESPI SYSTEMS

In the following it will be seen that for an ESPI system, the geometry of illuminating and viewing directions can be arranged so that the phase change A @ ( x , y )is sensitive to displacement in a particular direction. An out-of-plane sensitive interferometer can be made by replacing one mirror in a MachZehnder interferometer with the test object (for a discussion of Mach-Zehnder interferometry, see Born and Wolf [1980]). The arrangement, shown in fig. 19, is commonly used to give fringes which are predominantly related to out-of-plane displacement. Coherent light coming from a laser is coupled to a single-mode fiber and a bi-directional coupler splits the incident light into reference and object beams. Single-mode fibers have a narrow glass core of uniform refractive index profile (-3 to -8 pm diameter) and transmit only a single mode for light of a specific wavelength range and linearly polarized state. They produce a Gaussian spatial intensity distribution at their end. Because the fiber core diameter is very small, the irradiance distribution of the laser light from the fiber ends can be considered to be homogeneous, as for light emerging from a spatial filter. The object beam diverges from the point S and impinges at a small angle y on the surface. The light scattered from the object is collected by a lens and imaged onto the sensitive area of a TV camera where it mixes coherently, by means of a beam splitter cube located between the imaging lens and photosensor plane of the TV camera, with the reference beam which diverges from the point R . The point R would be approximately conjugated with the center C of the viewing lens (e.g., the

IV, § 31

ELECTRONIC SPECKLE PATTERN INTERFEROMETRY

233

Fig. 20. The change in optical path associated with a displacement vector d (for out-of-plane ESP1 interferometer).

beam would appear to diverge from C); this is necessary because the interference between the object and reference beam must not fluctuate at a spatial frequency which is too high to be resolved by the video camera. In order to satisfy the coherence conditions, the optical path difference between the reference and object beams can be changed by coupling (i.e., through a rod GRIN lens) a single-mode fiber of appropriate length to the arm of the reference beam (this problem, present with conventional gas lasers, is removed utilizing solid-state lasers with a long coherence length; Chung and Shay [1988]). Moreover, by controlling the coupling, the intensity ratio of the reference and object waves can be adjusted and adapted to the object reflectivity and to the distance. Now we consider fig. 20, where a point 0, on the object is moved along the displacement vector d to the point O D after a deformation of the object. If the displacement d is considerably smaller than the distance between the object and the point of divergence S of the illuminating source (or if we use collimated illumination), we can consider the object to be illuminated by a plane wave with the propagation direction n,. Besides, if d is smaller than the distance between the object and the viewing lens, it is possible to assume that the object is looking from infinity along the direction nL. In this configuration the angle between n, and n~ is y. The geometrical path length from the light source via the object point to the point of observation will be different before and after the deformation has

234

INTERFEROMETRIC METHODS FOR ARTWORK DIAGNOSTICS

1

\ I I R~

m OMI I

T I ti

[IV, 9: 3

1

Fig 2 I An optical ESP1 arrangement giving fringes representing in-plane displacements

taken place. In our case, this difference is equal to the path length Al= ds therefore the phase change is

+ d,,

which by simple trigonometric relations becomes 2n A#(x,y) = - [d,( 1 + cos y ) + d,sin y] , A

(3.1 1)

where d, is the out-of-plane displacement and dy is the in-plane displacement. The possible displacement in the x direction (with the approximation about the illumination beam and viewing direction mentioned above) does not cause phase change. If an illumination angle y of at most a few degrees is used, one can put 1 + cos y >> sin y and cos y M 1. Thus we see from eqs. (3.9) and (3.1 I) that dark fringes are observed in the subtracted speckle pattern when

A d,=n-, 2

n = l , 2 , 3, . . .

(3.12)

Thus, the fringes represent contours of constant out-of-plane displacement. An in-plane sensitive interferometer can be made by using the apparatus illustrated in fig. 21. Here the coherent light coming from a laser is coupled to a single-mode fiber and then amplitude-divided and spatially filtered into two beams of equal intensity, which illuminate the object at equal and opposite

IV,

I 31

235

ELECTRONIC SPECKLE PATTERN INTERFEROMETRY

----

I

I

A l , =d,,+d, -d, ( 1 I c o s y ) + d , s n y

A l , =d,,+d, = d , ( 1 +cosy)-d, srny

Fig. 22. Phase change related to a general displacement d (for in-plane ESP1 interferometer).

angles y to the normal surface. The surface is viewed in the normal direction by an imaging lens. The phase changes A@, and A42 in the two speckle patterns, due to a general displacement d , are given (with the same approximations about the illumination beam and viewing direction mentioned above) by (see fig. 22): 2n A@I= - [d,(l

A

2n

A@2= - [d,(l

A

+ cos y ) + dysin y ] ,

(3.13)

+ cos y ) - d,sin

(3.14)

y] .

The possible displacement present in the x direction does not cause phase change. Hence, the relative phase change between the two back-scattered beams is given bY

A#

=

2n A@I-A@* = - (2d,sin y ) ,

A

(3.15)

and dark fringes are observed in the subtracted speckle patterns when

A .

dY=n-siny, 2

n = 1,2,3, ...,

(3.16)

so that the fringes represent contours of constant in-plane displacement (independent of the presence of out-of-plane displacements). Ideally, the beams should be collimated in order to have a constant fringe sensitivity across the target surface, but in typical nondestructive testing it is

236

INTERFEROMETRIC METHODS FOR ARTWORK DIAGNOSTICS

[IY§ 3

acceptable and more convenient to approximate plane wavefront by diverging the beam over a large distance relative to the target dimensions. To complete an in-plane study, it is necessary to rotate the illumination beam (or target) through 90" about the viewing axis, to make the system sensitive to x direction in-plane displacements. Several ESPI equipment, in-plane and/or out-of plane displacement sensitive, have been proposed in the literature (Slettemoen [1980], Lerkberg and Krakhella [ 19811, Nakadate and Saito [1985], Creath [1985a], Tatam, Davies, Buckberry and Jones [ 19901, Aswendt, Hofling and Totzauer [ 19901, Virdee, Williams, Banyard and Nassar [ 19901, Davies, Buckberry, Jones and Pannell [ 19871, Ford, Atcha and Tatam [ 19931, Kato, Yamaguchi and Ping [19931); furthermore, TV speckle interferometers are commercially available. 3.3. APPLICATIONS OF ESPI TO ARTWORK NONDESTRUCTIVE TESTING

The first application of ESPI to artwork diagnostics (study of decay mechanisms of historic monuments) was made by Giilker, Hinsch, Holscher, Kramer and Neunaber [ 19901. These authors developed a practical electronic speckle pattern interferometry system that operates directly at the monument, and allows longterm monitoring of deformation processes, even under field conditions. The early systematic studies on the applicability of the ESPI in artwork testing were made by Paoletti, Schirripa Spagnolo, Facchini and Zanetta [1992], Paoletti and Schirripa Spagnolo [19931, Paoletti, Schirripa Spagnolo, Facchini and Zanetta [ 19931 to identify, in situ, the presence of defects (detachments and cracks) in ancient paintings on mural supports. First, the method was applied on test objects. Experimental investigations were conducted on a sandstone sample simulating a block of an ancient wall. The ingredients of the model are a mix of portland cement and sand with sufficient water added to give the required water-cement ratio. The aggregate inserted into the paste during the molding process consists of some stones. The dimensions of the specimen are 16x 13 cm2 with a thickness of 10cm. After artificial ageing, a visible surface crack was investigated under thermal stress realized with an infrared lamp. During the cooling process the out-of-plane deformation was measured. The speckle fringes are interrupted along the crack as shown in fig. 23. Subsequently a real fresco (see fig. 24), kindly provided by the Museo Nazionale d 'Abruzzo L'AquiIa (Italy), was investigated. For restoration purposes the plaster was separated from the support wall. In the optical laboratory a thermal stress was induced on the fresco surface by a flow of moderately warm air. In fig. 25 the ESPI fringes depict the out-of-plane deformation with a subsurface anomaly of the fresco;

IY

D

31

ELECTRONIC SPECKLE PATTERN INTERFEROMETRY

237

Fig. 23. Out-of-plane displacement in the presence of a surface crack in a sandstone sample.

the interrupted fringes suggest the presence of an incipient crack from the lower middle left part to the upper right corner of the inspected area. In addition it should also be noted that, on the left-hand side of the crack, there is a very particular narrow whte fringe between two wider bright lobes; in this case the classification of the defect is not straightforward and would require more detailed analysis or comparison with the results obtained with other techniques. Further tests on an ancient painted surface have been made in situ. To this aim a portable optical-fiber-based ESPI system (Paoletti and Schirripa Spagnolo [1993]) was realized. This equipment was mounted in two closed, dust-free, mechanically and thermally stable portable boxes: one for the laser and fiber optic system and the other for the optical components and TV camera. The optical head was mounted on a vibration-free tripod. Subsequently, Zanetta [ 19941 has also proposed a system utilizing the long coherence length of infrared radiation from a temperature-controlled semiconductor laser to enhance the compactness of the ESPI system. Some in-field specklegrams were effected on mural frescoes; Incoronuzione dellilssuntu (1 5th century) located inside the 5'. Maria di Collemaggio Basilica (13th century) at L'Aquila, Italy (Paoletti, Schirripa Spagnolo, Facchini and Zanetta [1993]). Figure 26 shows the mural fresco as it appears on the wall of the church. Correlation fringes relative to three different zones of the fresco are depicted (see fig. 27); the fresco was analyzed in ambient drift. The presence of a microcrack, identified by the abrupt interruptions of the interferometric fringes, was revealed using an out-of-plane sensitive configuration (see fig. 27a) of the ESPI system. The profile of the crack can readily be deduced by the location of fringe discontinuities in the image.

238

INTERFEROMETRIC METHODS FOR ARTWORK DIAGNOSTICS

[IV,

5

3

Fig. 24. The Sanfo Vescooo, a 15th-century mural fresco. (From Paoletti, Schirripa Spagnolo, Facchini and Zanetta [1993].)

Fig. 25. Fringe pattern showing the presence of a detachment relative to the mural fresco of fig. 24. (From Paoletti, Schirripa Spagnolo, Facchini and Zanetta [1993].)

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Fig. 26. Ancient mural fresco (lncoronazione deN’,4ssunta, 15th century) as it appears on the wall of the Basilica S. Maria di Collemaggio (L‘ Aquila, Italy). (From Paoletti, Schirripa Spagnolo, Facchini and Zanetta [1993].)

The interferogranis also revealed the presence of invisible microdefects. In particular, fig. 27b refers to a region with a subsurface detachment of the plaster and possible local material discontinuities, especially in correspondence to the bright area inside the defect fringes. Another interesting fringe pattern (see fig. 27c) was obtained for a region with some invisible cracks. Fringe discontinuities were detected in apparently faultless areas, indicating that the damage was more extensive than indicated by visual inspection. The

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[IY 5 3

Fig. 27. ESP1 fringes relative to three zones of the fresco shown in fig. 26. (a) Speckle interferogram of a region with crossed microcracks; (b) fringes revealing the presence of a detachment between thc mural support and the priming layers; (c) fringes showing the presence of cracks. (From Paoletti, Schirripa Spagnolo, Facchini and Zanetta [ 19931.)

possibility to operate in an open environment was also demonstrated by Paoletti and Schirripa Spagnolo [1993]. Figure 28 shows the ESP1 system operating outside the laboratory. The interferograms of fig. 29 are relative to the fresco Madonna con Bambino of the thirteenth century situated in the Piazza del Popolo Fontecchio (L'Aquila, Italy). The presence of some cracks is signalled by the appearance of discontinuities in the fringe pattern, corresponding to the in-plane configuration (fig. 29a), while with the out-of-plane sensitive interferometer, an incipient detachment has been revealed (fig. 29b). In this experiment the distance between the test object and the TV-camera was 40 cm, the illuminated area was about 100cm2, and the test was realized in evening hours under ambient drift.

IV,

ii 31

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24 1

Fig. 28 The ESP1 system operating in an open environment on a frcsco (Madonna con Bambino. 13th century) situated in the Piazza del Popolo, Fontecchio (L'Aquila, Italy). (From Paoletti and Schirripa Spagnolo [ 19931.)

Another application has been made in determining whether or not repair work has been effective. In fact, while the aim of most types of treatment is to remedy several defects, none of the methods used can be claimed to remove all flaws. The circumstances in each particular case, the materials used, and the condition of the fresco may have a direct and important bearing on the results. A particular procedure may bring out an improvement in certain circumstances, and the restorer is guided in such matters by his judgement rather than by any rule. After a type of treatment it could be important to control in real-time the state of the artwork in order to take the best decisions. Up to now the examination has been primarily a matter of experience. On the other hand, for the particular field of application it is also important for the restorers, during the restoration process, to know the exact location of the defective area on the fresco as indicated by

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Fig. 29. ESPI interferograms relative to the fresco of fig. 28. (a) lnterferogram showing some cracks recorded .with in-plane configuration; (b) fringe pattern with a hidden detachment recorded with out-uCplane configuration. (From Paoietti and Schirripa Spagnolo [ 19931.)

fringe irregularities. For this reason it can be helpful to capture and to store the image of the illuminated region; it can be subsequently modulated by relative ESPI fringes, so to obtain superimposition on the video monitor of the region of interest with the ESPI fringe. An application of the versatility of the instrument during a restoration process in determining whether or not repair work has been effective was effected on a fresco in the church of St. Maria della Croce of Koio-L'Aquila (Adorazione dei Magi by Farrelli; 1667). Figure 30 shows the mural fresco as it appears on the wall of thc church. The map of some blind cleavages between priming layers and the support is presented in fig. 3 1. Figure 32a shows a deformation map (under thermal stress realized with a brief thermal irradiation) of a selected region (square zone indicated in fig. 31). Abrupt deviations along the trend of some correlation fringes are evident. These discontinuities correspond to the presence

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243

Fig. 30. Mural fresco Adoruzione dei Magi by Farelli (1667), in the church of the S. Maria della Croce, Roio (L'Aquila, Italy).

of cracks. The same region was analyzed some days after the consolidation of the damaged area (with similar stress conditions), and no anomalies were detected (see fig. 32b). Recently some investigations on wooden panel paintings were made. As previously, preliminary studies were made on models realized with the most common defects. Figure 33 shows a specklegram of a model with a simulated defect, recorded with an ESP1 system in an out-of-plane configuration. This model consists of a poplar wood support, coated with some superposed priming layers, with a thin plastic sheet intentionally inserted among the layers to simulate the presence of a detachment (the model test was realized in a way similar to that used by Amadesi, Gori, Grella and Guattari [1974]). By observing the thermal deformation of the surface, the location and extent of the defect was evaluated precisely. Figure 34 shows correlation fringes relative to a typical crack on a wooden

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[IV, 9: 3

Fig. 3 1. Drawing of the fresco, Adorazione dei Magi, with highlighted zones of detachment.

panel, recorded with an ESPI system using an in-plane configuration. The crack is revealed by abrupt interruptions of the interferometric fringes. Subsequently, the portable ESPI instrument has been used for examining ancient wooden panel paintings at the restoration laboratories of Opijicio delle Pietre Dure e Laboratori di Restauro (Florence, Italy) (Lucia, Zanetta, Franchi, Aldrovandi, Cianfarelli, Matteini and Riitano [ 19941). The interferometric measurements were performed during working hours in a building having a large floor. In particular, a wooden panel painting (1 90 x 97 cm2) entitled Madonna in trono con Bambino by Giotto (see fig. 35) was inspected. Figure 36 presents an interferogram relative to an area in the middle of the panel where a nail was driven in from the back and bent down to the front surface. This nail originally served to fix two crossed planks to the panel in order to increase the rigidity of the support. The anomalous density and shape of the fringes at the center of the image clearly indicate the extension of the detachment and the occurrence of a crack along the horizontal direction. The panel was composed of two parts held together by wooden joints. Although the vertical separation between the two parts was very small, the corresponding point under the priming layers could lead to the formation of detachments. One of these detachments has been successfully

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Fig. 32. (a) Specklegram relative to the square zone of fig. 30 showing the presence of a crack; (b) the same zone after restoration.

detected, and the resulting interferogram is presented in fig. 37. The defect is revealed by the elliptical labelled black fringe. Some tests were performed by the same authors on the Incoronazione della Vergine (by Lorenzo Monaco; 1414); this composition comes from Galleria degli Ufizi (Florence, Italy) and consists of a panel (450x350 cm’), a predella and three cuspidi. By inspection of an extended detachment in one of the cuspidi, the effects of the craquelure on the interferometric pattern were observed. As shown in fig. 38, the fringes follow the pattern of surface microcracks and assume a circular shape. Finally, some investigations were made on a wooden panel (42x 125 cm2) Trionjh di David by an imitator of “Pesellino” (15th century), from the Museo Horme (Florence, Italy) before and after repair work on areas of the painting containing detachments. Figure 39a shows the inspected area. From the

246

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rIv. B 3

Fig. 33. Specklegram relative to a poplar wood support with three simulated detachments at several depths.

Fig. 34. Correlation fringes relative to a typical crack on a wooden panel recorded with an ESPI system in-plane configuration.

interferogram of fig. 39b, a large detachment was readily detected. Moreover, the small closed fringes above this defect indicate the presence of another detachment. The same test was performed one day after consolidation of the damaged area. The closed fringes, typical of detachments, were absent (see fig. 39c). To illustrate the possibility of operating in very difficult conditions, the example of application of ESPI on a terracotta “Angel”, located inside the Portinari chapel (15th century) in the church of S. Eustorgio (Milan, Italy) is also reported. The angel is part of the tripudio Angelico, which occupies a vast area of the interior wall of the chapel at a height of approximately 13 meters

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Fig. 35. Madonna in trono con Bambino, by Giotto, under restoration at Opificio delle Pietre Dure e Laboratorio di Restauro (Florence, Italy).

Fig. 36. Interferogram relative to an area in the middle of the panel shown in fig. 35, where a detachment and a crack are visible.

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[IV,

5

3

Fig. 37. A detachment is revealed along the separation between the two parts of the panel.

Fig. 38. ESPI fringes produced by the craquelure of the painting. (Photo courtesy of Paolo Zanetta.)

from the ground. For this survey the ESPI equipment was taken to the sixth floor of the scaffolding (see fig. 40). One foot of the tripod was placed on the scaffolding, while the other rested on a small platform by the chapel wall. The personal computer and the TV-monitor were positioned at the ground floor in order to control the acquisition of the interferogram from a point external to the scaffolding. A 22-meter cable was used for connecting the optical head to the acquisition unit. The object surface had been heated with an infrared lamp for a few seconds between the exposures. The resulting interferogram is presented in fig. 4 1, where the fringes clearly indicate the extension of the detached layers. In ESPI techniques for artwork diagnostics, as previously mentioned, thermal

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249

Fig. 39. (a) Inspected area of the panel Trionfo di Dauide; (b) ESPI interferogram before repair; (c) ESPI interferogram after repair. (Photos courtesy of Pa010 Zanetta.)

Fig. 40. Inspection of the angel of the Dipudio Angelico. (Photo courtesy of Pa010 Zanetta.)

or ambient drift stresses are commonly used. Recently, a different approach (alternative to thermal stress) has been proposed by Gulker, Hinsch, Holscher,

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[I% § 3

Fig. 41. ESPI interferogram showing detached layers. (Photo courtesy of Paolo Zanetta.)

Meinlschmidt and Wolff [ 19931 to map plaster detachments in historical mural paintings. These authors have used an acoustical excitation of vibrations to identify the plaster debonding of the artifact. 3.4. LIMITATIONS OF ESPI MEASUREMENTS ON ARTWORKS

The ESPI method has some obvious advantages over holographic interferometry, with makes it well suited to routine inspection and out-of-lab operation. First of all, the use of a video camera and digital equipment allows conversion of the visual information into an electrical signal, which can be readily processed electronically; the real-time correlation fringes are displayed directly upon a TV monitor without recourse to any form of photographic processing, plate relocation, etc. Moreover, because the images are acquired and stored at the video frame rate, the exposure time is very short and so it is possible to handle difficult measuring problems with regard to object stability and hostile environments. Operating in subtraction mode eliminates the need for darkening to record the interferograms; by using narrow band filters, centered at the laser wavelength, ESPI systems can also operate in daylight conditions. However some limitations are present, in comparison with holographic nondestructive testing. One major limitation in speckle correlation interferometry is that the individual speckles must be resolved by the camera. If the separation of the maxima and minima of cos [ @ ~ ( x , y-)&(x,JJ)] in eq. (3.6) is less than the spatial resolution of the TV camera, the last term in that equation will be averaged to zero and no variation in correlation will be observed. This requires that the imaging system

IVI

CONCLUSION

25 1

numerical aperture be matched to a particular sensor. The mean speckle size must be equal to the size of a single pixel on the camera. This will optimize the performance as well as the fringe visibility. With a typical pixel size of x 10 ym an f/ of 16 is necessary. The use of a relatively large f-number giving large speckles affects the performance of the system in several ways. Firstly, it reduces the intensity of light in the image with the consequence of limiting the possibility of analyzing larger or less reflective objects, unless a higher-power laser and/or camera of greater sensivity are used. A second consequence is that the speckles are seen in the monitor image, and the clarity of the fringes is therefore less than that of good-quality holographic fringes. In the case of complicated fringe patterns, this can make it difficult to identify their positions. For this reason it is necessary to use a despeckle function to analyze this speckle pattern (Crennell [ 19931). The number of fringes which can be seen and analyzed is limited. If we consider that a minimum of about 20 speckles is required for a resolvable fringe, with a typical detector array having 5 12x 5 12 resolution elements, then about 25 equally spaced fringes would be resolvable. Modem camera technology offers devices with. spatial resolutions that are much higher than the standard television resolution (for both tube and CCD camera). These devices can, however, be applied only with non-standard signal readout and transmission, and they demand an image-processing system with a sufficient resolution. The commercial introduction of high-definition television, being near at hand, will offer a new generation of standard video systems with high spatial resolution compared to current television standards. Finally, the biggest limitation of speckle correlation interferometry deals with decorrelation of the speckle patterns (Yamaguchi [1981], Creath [ 1985b, 19931). In fact, in ESP1 theory there is always the assumption that the displacements and/or deformations do not alter significantly the random phase and the amplitude of the speckle pattern. This approximation is valid only if all displacements stay below a maximum value which depends on the optical setup and the type of deformations. The effect of decorrelation will be a decrease in fringe visibility until the limit of total decorrelation, when no fringes will be formed. This determines the maximum displacement and/or deformation that can be observed (Owner-Petersen [1991]).

-

Conclusion The aim of this review has been to provide a valuable and up-to-date source of information in the rapidly expanding field of application of optical methods

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[IV

in nondestructive artwork testing. Ample space has been given both to a comprehensive treatment of the phenomena of holographic and speckle fringe formation and to the practical applications to solve specific problems, being a work addressed to both scientists and conservationists. The development of simple and effective instruments for in situ measurements has been considered. Unfortunately, this chapter cannot report all the contributions that have been made in the field over the years; the authors have only mentioned the works which have come to their attention. From the early investigations, carried out with sophisticated laboratory equipment and great limitations, to the development of portable digital speckle interferometers of the last years, a great impulse has been given to the research in this field, bringing together scientists and artworkers. The results presented here have demonstrated that holographic deformation analysis is well suited to the investigation of initial damage on frescoes and paintings. Moreover, a correspondence between the anomalies of the fringe patterns and the most common types of defects (cracks, detachments) has been demonstrated. ESP1 inspections are valid on artworks located outside the laboratory in difficult situations; the effectiveness of the repair work can also be evaluated a posteriori by comparison of the interferometric fringes obtained before and after restoration. In general, the intensity of the laser beams used for the interferometric measurements is low and there is no risk of damage to the object. In conclusion, we can say that the interferometric techniques complement other methods of nondestructive testing commonly used in artwork diagnostics, but are not yet sufficient to give an optimal response to the many problems of the restoration field. A more effective collaboration with conservation scientists and restorers in the future might play an important role in familiarizationwith various problems of conservation and yield to the development of portable, inexpensive and easy to use instruments.

Acknowledgements We specially wish to thank Prof. Franco Gori for his helpful suggestions. We are also grateful to Paolo Zanetta (Joint Research Centre, Institute for Systems Engineering and Informatics, Ispra, Italy) and Albert0 De Angelis (Enea, Area Inn.,Dip. Sviluppo Tecnologie di Punta, Frascati, Italy) for the use of their photographic material. Furthermore, we wish to thank the Sovrintendenza per i Beni Ambientali Architettonici Artistici e Storici per L'Abruzzo, and the Opificio Delle Pietre Dure e Laboratori di Restauro (Florence, Italy) for the use of their artworks.

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E. WOLF, PROGRESS IN OPTICS XXXV 0 I996 ELSEVIER SCIENCE B.V ALL RIGHTS RESERVED

V

COHERENT POPULATION TRAPPING IN LASER SPECTROSCOPY BY

E. AIUMONDO~ JILA, Uniuersi@ of Colorado at Bouldev. Boulder, C O 80309-0440, USA

* JILA Visiting Fellow 1994-1 995. Permanent address: Dipartimento di Fisica dell’Universita di Pisa, Piazza Torricelli 2, 56126 Pisa, Italy.

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CONTENTS

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§ 1. INTRODUCTION

. . . . . . . . . . . . . . . . . . .

259

§ 2 . ANALYSIS FOR DISCRETE STATES . . . . . . . . . . .

263

9; 3 . SPECTROSCOPY FOR DISCRETE STATES . . . . . . . .

288

9 4 . COHERENT POPULATION TRAPPING IN THE CONTINUUM 309 § 5 . LASER COOLING . . . . . . . . . . . . . . . . . . .

310

§ 6 . ADIABATIC TRANSFER . . . . . . . . . . . . . . . .

325

§ 7 . “LASING WITHOUT INVERSION” . . . . . . . . . . . .

329

.

340

PULSE-MATCHING AND PHOTON STATISTICS . . . . . .

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§ 8. COHERENCES CREATED BY SPONTANEOUS EMISSION § 9.

9 10. CONCLUSIONS . . . . . . . . . . . . . . . . . . . .

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ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . .

346

REFERENCES . . . . . . . . . . . . . . . . . . . . . . .

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This article is dedicated to the memory of A. Gozzini and G.K Series, great teachers in atomic spectroscopy.

9

1. Introduction

Spectroscopic investigations of atoms and molecules are based on the interaction of near-resonant radiation with the species under investigation. The use of monochromatic, intense, and continuously tunable radiation sources, permits very high sensitivity and accuracy to be attained in the determination of the atomic or molecular levels. Although the attention of spectroscopists was restricted to two-level systems for a long time, the possibility of irradiating samples by several electromagnetic fields simultaneously has produced multiphoton transitions and other nonlinear phenomena, whose application has been explored in the continuously expanding field of nonlinear spectroscopy. In comparison to the two-level system, the three-level system, interacting with two monochromatic radiation fields, represents a configuration in which the nonlinear phenomena are greatly enhanced both in the number of possible laser configurations and in the magnitude of the nonlinearities. The development of monochromatic and tunable laser sources has produced a large variety of high-resolution spectroscopic investigations on three-level systems. Among the different nonlinear processes associated with the three-level atomic systems, the application of two continuous wave radiation fields leads to the preparation of the atom in a coherent superposition of states, which is stable against absorption from the radiation field. This phenomenon has been designated as coherent population trapping, to indicate the presence of a coherent superposition of the atomic states and the stability of the population. Coherent population trapping may be also described as the pumping of the atomic system in a particular state, the coherent superposition of the atomic states, which is a nonabsorbing state. The exciting radiation creates an atomic coherence such that the atom’s evolution is prepared exactly out of phase with the incoming radiation and no absorption takes place. This phenomenon was observed for the first time by Alzetta, Gozzini, Moi and Orriols [1976], as a decrease in the fluorescent emission in a laser optical pumping experiment on sodium atoms, involving a three-level system with two ground levels and one excited level. In that experiment, because an inhomogeneous magnetic field was applied along the sodium cell axis, the nonabsorption was produced in only a small region inside the cell, and the phenomenon appeared as a dark line inside 259

260

COHERENT POPULATION TRAPPING

[v, §

1

the bright fluorescent cell. As a consequence, names such as dark resonance or nonabsorption resonance have been used in the literature to describe coherent population trapping. At the same time, and independently, the pumping and trapping originated by two laser fields resonant with two coupled transitions was investigated theoretically for three levels in cascade by Whitley and Stroud [1976], and experimentally in sodium atoms with two ground levels and one excited level by Gray, Whitley and Stroud [1978]. The theoretical analyses by Arimondo and Orriols [I9761 and Gray, Whitley and Stroud [1978], pointed out that the sodium atoms were pumped in a nonabsorbing state because of the presence of interfering processes. Population trapping during laser-induced molecular excitation and dissociation was examined theoretically by Stettler, Bowden, Witriol and Eberly [1979]. The title of a paper by Gray, Whitley and Stroud [I9781 contains, for the first time, the term coherent trapping, and in the conclusion of the paper, the phenomenon is defined as a coherent trapping of a population. The complete designation of coherent population trapping appeared for the first time in the abstract of a paper by Agrawal [ 19813 on the possibilities of using three-level systems for optical bistability. Then the title of a work by Dalton and Knight [1982a] contained the full designation of coherent population trapping with evidence on the main characteristic of the phenomenon. The process remained a sort of amusing scientific curiosity for some time. If one resonant laser beam is switched on inside a sodium cell, some very bright fluorescence is emitted by the cell. If a second laser beam is sent into the cell, this second laser being slightly detuned from the first one but also in near resonance with the sodium atoms, it produces its own bright fluorescence. The simultaneous application of the two lasers produces coherent population trapping and eliminates the sodium bright fluorescence. This interference effect, produced by the presence of atomic coherence, appears at the macroscopic level. Coherences between states of a quantum-mechanical system are generated whenever an interaction or measurement leaves the system in a linear superposition of the energy eigenstates defined in the absence of the field. Interferences produced by the presence of coherences have been known since the development of quantum mechanics, and their creation has been largely exploited in spectroscopy and quantum optics. However, a macroscopic effect such as the total suppression of fluorescence emission by coherent population trapping is quite unusual. In the early 1980s some theoretical attention was given to the process of coherent population trapping, with extensions to the case in which the upper state of the three-level system lies in the continuum, e.g., Knight [ 19841. However, the

Y § 11

INTRODUCTION

26 1

real rise in interest waited for the extensions or applications on the experimental side. The first application, to metrology, is linked to the work by Tench, Peuse, Hemmer, Thomas, Ezekiel, Leiby Jr, Picard and Willis [1981], and Thomas, Hemmer, Ezekiel, Leiby Jr, Picard and Willis [1982] (see also discussion in Knight [ 1982]), who demonstrated how very high-frequency accuracy in the measurement of the sodium ground-state hyperfine splitting could be obtained by using what they called Ramsey fringes in Raman three-level transitions. That now should be defined as a Ramsey fringe investigation of coherent population trapping. The next application was to optical bistability; Walls and Zoller [ 19801 investigated theoretically the optical bistability from three-level atoms contained in an optical cavity and driven into the coherent-trapping superposition. That optical bistability was observed for the first time by Mlynek, Mitschke, Deserno and Lange [ 19821. After those early observations, the phenomenon of coherent population trapping has been exploited in very different applications: in highresolution spectroscopy, laser multiphoton ionization, four-wave mixing, and laser-induced structures in the continuum. Increased attention is due to the work of Aspect, Arimondo, Kaiser, Vansteenkiste and Cohen-Tannoudji [ 19881 on the application of velocity-selective coherent population trapping to laser cooling. Very soon other interesting phenomena such as adiabatic transfer, lasing without inversion, matched pulse propagation, and photon statistics, strictly connected to the trapping properties of the three-level system, were discovered. It should be noted here that even if some theoretical work has considered the extension to molecular systems, the evidence of coherent population trapping in molecules is still very limited and has been associated with the adiabatic transfer experiments of Gaubatz, Rudecki, Schiemann and Bergmann [ 19901 and of Dam, Oudejans and Reuss [ 19901. For the most important steps in the theoretical understanding of coherent population trapping, Hioe and Eberly [1981], and later Hioe [1983, 1984a, 1984b], have shown a relation with the invariants in the density matrix equations: coherent population trapping is related to the SU(3) group symmetries of the Hamiltonian and to some conservation laws satisfied by the density matrix elements of a three-level system during the time evolution. Smirnov, Tumaikin and Yudin [ 19891 and Tumaikin and Yudin [ 19901 have presented generalizations in the construction of the coherent population-trapping atomic superposition. Radmore and Knight [ 19821 and Dalibard, Reynaud and Cohen-Tannoudji [ 19871 have derived the dressed atom description. Coherent trapping as an interference phenomenon is closely related to other interference processes well exploited in spectroscopy, such as the Fano windows in the autoionization profile, the level crossing, or the Hanle effect. A strict

262

COHERENT POPULATION TRAPPING

[v, ii

I

connection also exists between coherent population trapping and the weak interaction decay of the K O and K O mesons. In coherent population trapping, the two linear superpositions of ground atomic or molecular states present very different lifetimes for the interaction of these superpositions with the radiation field. Owing to weak interaction mixing between the KO and mesons, their linear superpositions, K s and K L , should be considered. Those superpositions have different lifetimes, short and long, with respect to the weak interaction decay. However, it should be noted that the reduction of coherent population trapping to an interference feature is a reductive description: the experimental observations are strongly based on the role played by optical pumping in the atomic preparation into that quantum superposition that presents interference in the absorption or radiative decay. Coherent population trapping has been examined in several review papers. An early review was presented by Dalton and Knight [1983]. Yo0 and Eberly [I9851 presented an analysis of the most important theoretical features of the phenomenon, although their attention was devoted to three-level atoms inside an optical cavity. A review by Arimondo [1987] summarized the experimental observations at that time. A more recent review, with more attention toward the theoretical features and the extensions to laser cooling, has been written by Agap’ev, Gornyi and Matisov [1993]. The organization of this chapter is as follows. In 5 2 a theoretical introduction presents the basic properties of an atomic system prepared with the coherent population-trapping superposition of states. 5 3 deals with several experimental observations concerned with the establishment of coherent trapping in different discrete systems. 5 4 very briefly treats the theoretical and experimental aspects of trapping which involves states of the continuum, because a recent review on that subject has been written by Knight, Lauder and Dalton [1990]. The remaining sections are devoted to a review of both the theoretical and experimental features associated with coherent population trapping in laser cooling, adiabatic transfer, lasing without inversion, pulse matching, and photon statistics. The large amount of theoretical work that has been published with respect to the phenomenon and to the constants of motion with relation to the SU(3) symmetry will not be reported here; the book by Shore [1990] and the recent review by Agap’ev, Gornyi and Matisov [1993] deal with most of those features. Nevertheless, 9 8 is devoted to the theoretical aspect of coherent population trapping created by spontaneous emission, which is a possibility considered in some theoretical papers. Even if there is little chance of observing the phenomenon, it is presented because of its fundamental connection with coherent population trapping and lasing without inversion.

v, 5 21

263

ANALYSIS FOR DISCRETE STATES

5

2. Analysis for Discrete States

2.1. DENSITY MATRIX

The atomic (or molecular) three-level systems considered here are shown in fig. 1. The levels, defined in the absence of any radiation field, are labeled as [ I ) , 12), and lo), with their energies E, (j=O, 1, 2), and the energy separation between levels li) and l j ) denoted as hwY=Ei-Ej. The three cases of fig. 1 will be denoted as A, cascade, and V configurations. The Hamiltonian 'Hothat describes the internal energy of the three levels is written as:

The common level 10) is coupled to both levels 11) and 12) through electric dipole transitions induced by two applied classical laser fields,

where ui, WL,, and @i (i= 1, 2 ) are the unit vector, the angular frequency, and the phase of each field, respectively. The three-level configurations of fig. 1 may be treated by a single formalism if proper definitions of the atomic energies and laser frequencies are introduced. The convention to be used here is slightly different from that introduced by most authors (e.g., Feld and Javan [ 19691 or Kelley, Harshman, Blum and Gustafson [ 1994]), because attention is concentrated on the A scheme. In the A scheme of fig. la, the energies are in the order E l < E2 < Eo and the laser frequencies are all positive. In the cascade

I o>

l1>

p I1>

12,

lo>

4 Fig. 1. Three-level systems in (a) A, (b) cascade, and (c) V configurations. Levels 1 1 ) and 12) are connected by dipole transitions only to level 10). See text for signs of the energy splittings and laser frequencies in each configuration.

264

[v, 9: 2

COHERENT POPULATION TRAPPING

< <

scheme of fig. Ib, with E l Eo E 2 , w01 and OLI are positive, whereas 0 0 2 and wL2 are negative. Finally, in the V scheme of fig. lc, with Eo < El < E 2 , all the angular frequencies are negative. In the preliminary analysis the atoms will be supposed to be at rest along the propagation direction of the laser fields and located at the z position of the coordinate. It will be supposed in all of our analyses that, either due to selection rules produced by a proper choice of the polarizations of the two laser fields or because of the properly chosen frequency detunings, each laser field acts only on one dipole transition, and specifically, that laser 1 acts on the 11) -+ 10) transition, and laser 2 on the second one. It is convenient to define the detuning, &I and 6 ~ 2 of , each laser from its resonance: 6 L l = WLI - 0 0 1 ,

6L2 = 0 L 2 - 002,

(2.2a,b)

and also the Raman detuning 8~ from the Raman two photon-resonance:

Introducing the electric dipole moments pol and p 0 2 between the three states, the Rabi frequencies characterize the atom laser interaction:

where, for simplicity, very often the Rabi frequencies are supposed to be real but should be considered complex in the general case. The atom-laser interaction Hamiltonian VAL may be written as:

The standard rotating wave approximation (RWA) has been used to eliminate the nonresonant terms within the interaction Hamiltonian. The two levels 11) and 12), due to the electric dipole selection rules, have the same parity so that electric dipole transitions cannot be induced between them, but magnetic dipole transitions are possible. In order to complete the description of the three-level system, the relaxation terms produced by spontaneous emission, collisions, and any other damping mechanism, have to be considered. All those relaxation terms of the p density

v, I21

ANALYSIS FOR DISCRETE STATES

265

matrix will be introduced through an operator R in the optical Bloch evolution equation: - - [KO + VAL, p(t)] + %p(t). (2.5) h dt The relaxation operator % contains Tri terms describing the dephasing rates of the pij off-diagonal element; Ti/+i terms describing the rates of population transfer from state i’ to state i; and finally r l ,the total rate of pii population decay from state i. The general formulation introduced by Kelley, Harshman, Blum and Gustafson [ 19941 allows the relaxation terms and steady-state solution of the density matrix to be written in the most general case. However, in most cases of coherent population trapping, very simple relaxation processes, such as those produced by the spontaneous emission rates rip(i = 0, 1, 2), must be considered. For instance, for the A configuration only the following rates are different from zero: T o=rip, = rO+, = Tip/2, and rol = Tlo= rO2 = r20 = T:’/2. When experiments in the presence of collisions are to be analyzed, the decay rate rI2 = r21 for the coherences p12 and p21between the two lower levels, the population transfers r 1 - 2 and r 2 - 1 , and the population losses r , and r2, have to be introduced, in addition to spontaneous emission. For the A system, the following density matrix elements in the interaction picture may be introduced to eliminate the fast dependencies in the time evolution equations: 1-.dp(t)

Pol = poiexp[i(oL,t + 41)I

(i

=

1, 21,

(41- 4211).

i512 = pl2exp{-i [(OM- W L Z+ )~

Then the evolution equations are: dpoo dt

(2.6)

.a;;,+ a;;,+ 1-p02 + c.c., 2 2

-= -Topoo i-pol

with pi0 = &. The steady-state solutions of these equations have been reported by Kelley, Harshman, Blum and Gustafson [1994]. Here it is relevant only to

266

COHERENT POPULATION TRAPPING

[v, $ 2

report that the two-photon Raman resonance between the I I ) and 12) levels is described by the two-photon operator ^r:

with the resonant two-photon denominator given by

(2.10a) and the resonant one-photon denominators given by

D. = -is

Li

+ roi (i

=

1, 2).

(2. lob)

All of the above expressions may also be used for the cascade and V schemes (fig. 1) when the appropriate sign changes, as defined above, are applied to the angular frequencies. It will be helpful for later analysis to report the dependence of the coherence term p12 on the laser frequencies in the case of the cascade configuration:

2.2. NUMERICAL RESULTS

As shown by Arimondo and Orriols [1976], Gray, Whitley and Stroud [1978], and Orriols [1979], the simplest way to illustrate the phenomenon of coherent population trapping is to present the numerical solution for the steady-state values of the density equations. In this respect it may be stated that nothing really new was discovered by the numerical analyses referred to above because everything was already contained in the general density matrix general solution, for instance, by Wilcox and Lamb Jr [ 19601, Schlossberg and Javan [ 19661, Hansch and Toschek [ 19701, and Feldman and Feld [ 19721. However, the clear appearance of coherent population trapping requires a proper choice of the density matrix relaxation rates, and that represents an original contribution by Arimondo and Orriols [1976], Gray, Whitley and Stroud [1978], and Orriols [1979].

v, P

21 30

-

25 20

ANALYSIS FOR DISCRETE STATES 1

Po0

261

PNC,NC-0 8

-

- 06 - 04 - 02

I

Yf-

pc,c-

05

00 -0 5

-1 0 -4

-2

0

2

W

4-4

O

-2

0

2

4

M - 0

Fig. 2. (a) Steady-state excited-state population p& for A system as a function of the Raman detuning 6 ~ with . the typical central dip associated with the coherent population-trapping phenomenon; (b) occupation of the noncoupled and coupled states (see $2.3), p&NC and p & , respectively, versus 6 ~ (c) ; imaginary and (d) real parts of the optical coherence on the 11) + 10)transition, Im(pol) and Re( pol), respectively, versus 6 ~The . lineshapes ofthe absorption coefficient, and the index of refraction versus the laser frequency are given by these plots. Parameters: 6~2=0, .QR~ =.Qu=O.2To and rl2=0.001 ro.

Very peculiar narrow features appear for the occupation of the 10) upper level in the A system when the two-photon condition is satisfied; i.e., around the two-photon Raman resonance condition & = S L-~&2 cv 0. Figure 2a shows numerical results for the steady-state solution of the density matrix pto in the closed A system, with values for the Rabi frequencies and relaxation rates as in typical experimental investigations. In fig. 2a, laser 2 is fixed at its resonance value 8 L 2 =0, whereas laser 1 is swept around its optical resonance. When the two laser field frequencies are scanned around the Raman resonance value, the population of the upper 10) level increases following the Lorentzian profile of the absorption line, and a maximum value of pto is expected with both lasers in resonance. However, a strong decrease occurs in a narrow region around the Raman resonance (whose width is given in (I 2.3), where the atoms remain distributed over the lower 11) and 12) levels, without occupation of the upper level; the population of the A system is trapped in the lower states. The coherent nature of the trapping cannot be derived from this numerical analysis and requires further physical investigation of the nature of the process. The fluorescent emission from the upper level, which is proportional to the excited-

268

COHERENT POPULATION TRAPPING

201 15

-3

-2

1

0

1

2

3

GLlro Fig. 3 . p& for A system as a function of the laser 1 detuning 6 ~ 1at 6 ~ 2 = r o parameters: ; Q R ~=O.6To, Qm=0.1 T o , and r 1 2 = 0 . 0 0 1 ro.

state population, presents a strong resonant decrease at the Raman resonance value, which has been denoted as a dark line by Alzetta, Gozzini, Moi and Orriols [1976]. Away from the Raman resonance, the occupation of the three states is determined by the process of optical pumping (Cohen-Tannoudji and Kastler [ 19661). Even if laser 2 is on resonance, when laser 1 is not on resonance, i.e., at large values of 1 6 1,~ the excited-state population remains small. The optical pumping process by laser 2 transfers all the population to the 1) level. Furthermore, because of the optical pumping redistribution among the three levels, the results of fig. 2a do not depend on the initial level populations. Figure 3 reports a similar numerical analysis for the excited-state population in the A system with laser 2 fixed at a value not corresponding to the optical resonance (SL2 # 0), and scanning the frequency of laser 1 around the resonance Raman condition BR = 0, i.e., 6 ~ = 1 6 ~ 2Because . of the detuning of laser 2 from the optical resonance, at the optical resonance for laser 1, 6 ~ 1 = 0i.e., , 6~ =-6~2, some population is transferred to the upper state, with a limit imposed by the optical pumping. However, at the Raman resonance, coherent population trapping is again produced and a drastic decrease in the excited-state occupation takes place. It can be barely noticed from the data of fig. 3, that the minimum of the excited-state occupation does not take place precisely at SR = O but presents a shift from that value. That shift, the ac Stark shift or light shift, appears when the lasers are not in resonance with the optical transitions, and originates from a modification in the energies of the atomic levels through nonresonant transitions (see Cohen-Tannoudji, Dupont-Roc and Grynberg [1992]). That light shift is exactly equal to the term that appears in the denominator of the two-photon operator so that the Raman resonance occurs at:

F,

6R

A

-Slight.

(2.12)

In fig. 3, the occupation of the excited-state population versus the Raman

v, § 21

ANALYSIS FOR DISCRETE STATES

269

detuning presents a large increase on one side of the coherent populationtrapping dip. That anomalous increase of upper state population was detected by Alzetta, Moi and Orriols [ 19791 as an increase in the emitted fluorescence in their first experimental observation of coherent population trapping, and was described as a bright line, in contrast to the dark line in the fluorescence emission corresponding to the trapping process. Radmore and Knight [ 19821 also pointed out the presence of that dispersive feature and investigated its dependence on the laser parameters. A physical explanation for the appearance of that peak in the excited-state population has been derived by Lounis and CohenTannoudji [ 19921 through the quantum interference described in $2.5, and the peak characteristics have been examined in the theoretical and experimental analyses of Janik, Nagourney and Dehmelt [1985], Hemmer, Ontai and Ezekiel [ 19861, and Siemers, Schubert, Blatt, Neuhauser and Toschek [ 19921. Hemmer, Ontai and Ezekiel [ 19861 compared, with good agreement, experimental and theoretical lineshapes. Siemers, Schubert, Blatt, Neuhauser and Toschek [ 19921 investigated the dependence of the bright line peak resonance on the intensities of the lasers. That bright line peak appears also in the cascade scheme, as in experimental and theoretical observations by Kaivola, Bjerre, Poulsen and Javanainen [1984] and Gea-Banacloche, Li, Jin and Xiao [1995]. The narrow resonance produced by the coherent population-trapping phenomenon may also be observed on the absorption coefficient a or on the index of refraction n for each of the two laser fields acting on the three-level system. a and n are derived from the imaginary and real parts of the susceptibility, and X I , respectively, that connect the optical polarization P to the applied electric field (Sargent, Scully and Lamb Jr [1974]). The complex polarization of N atoms or molecules associated with the atomic transitions 11) 10) and 12) ---t 10) is given by:

x”

---f

p

= N(P0lPOl

The Fourier components at frequencies

P (z,O L i )

(2.13)

+ P02Po2) + C.C.

= EO ( X ’ ( O L i )

OL;

(i = 1, 2) are given by:

+ ix”(OLi)) EL;(z).

(2.14)

The above equation leads to:

The dependence of the real and imaginary parts of the optical coherences poi on the laser frequencies determines the lineshapes of the index of refraction

270

COHERENT POPULATION TRAPPING

[v, 5 2

and of the absorption coefficient of the medium under coherent populationtrapping resonance. Figures 2c and 2d show the steady-state values of the imaginary and real parts of pol, respectively. The imaginary part in fig. 2c, which determines the absorption coefficient a, has the same lineshape as the excited-state population poo of fig. 2a, with an increase around the optical resonance whose linewidth is given by spontaneous emission and saturation broadening, and a narrow decrease in absorption in the central region near the Raman resonance condition. The real part of pol, which determines the index of refraction n, presents two dispersion lineshapes, the broad one with linewidth given by spontaneous emission and saturation broadening, and a narrow inverted one, produced by the coherent population trapping, around the Raman resonance, whose width is discussed in 0 2.3. 2.3. COUPLED AND NONCOUPLED STATES

A unitary transformation is very useful for understanding the process of coherent population trapping. Let us analyze first the simple case in which the two lower levels, 11) and 12), are degenerate and the two laser fields have the same frequency wL and phase #L. Let us consider the following two orthogonal linear combinations of the lower states in the A configuration: where G is given by

JzFiZz

(2.16b) G= This defines the noncoupled and coupled states which have the property that, according to the atom-laser interaction Hamiltonian of eq. (2.4), the transition matrix element between INC) and 10) vanishes: (01 VAL INC) = 0, whereas

(2.1 7a) (2.17b)

Consequently, an atom in the noncoupled state /NC) cannot absorb photons and cannot be excited to 10). For an atom prepared in the INC) state, the Schrodinger equation under the Hamiltonian 'Ho+ VAL results in: (2.18) Thus an atom prepared in INC) remains in that state and cannot leave it either by the free evolution (effect of the free Hamiltonian ?-lo), or by absorption of

v, § 21

27 1

ANALYSIS FOR DISCRETE STATES

I o>

r' Fig. 4.Couplings and effective loss rates for the states lo), IC), and INC). IC) is coupled to 10) by the laser interaction, with matrix element R R ~as;a result of this coupling IC) acquires a loss rate 1NC) and IC) are coupled through ground-state relaxation rates, Raman detuning 8 ~ and, , for laser cooling, the kinetic energy operator..

r'.

a laser photon (effect of the atom-laser interaction VAL). Moreover, because INC) is a linear combination of the two ground states, and is radiatively stable, the atom cannot leave INC) by spontaneous emission. The various couplings between INC), IC), and lo), due to XO and VAL, are represented in fig. 4. The IC) and 10) states are coupled by the atom-laser interaction with Rabi frequency G. The excited state 10) has an effective loss rate TO determined by spontaneous emission. For resonant excitation (&I = 6 ~ =24) and weak-intensity limit (G << r),the Rabi coupling between IC) and 10) gives to the state IC) an effective loss rate of d dt

-pc,c

(2.19)

=

The preparation of the three-level system in the couplednoncoupled states implies the production of a coherence between the 11) and 12) states, as follows from the relations for the density matrix elements in the two bases: pC,C =

1 2 5 ( I Q R I I PI1 + I Q R 2 I 2 P 2 2

PNC,NC =

f

5 (I Q R 2 I 2 PI I + I Q R I l2 p 2 2 1

QRIGi2p12

+Qi1QR222l)

-Q R I Q i 2 P 1 2 -

7

(2.20)

Q ~Q RI2 2 2 I ) .

The time evolution of the coupledhoncoupled states should be completed by the evolution associated with the relaxation processes. The relaxation terms for the time evolution for the density matrix elements in the couplednoncoupled basis are derived from the density matrix eqs. (2.7). The spontaneous emission decay leads to the following terms:

(2.21)

272

[v, § 2

COHERENT POPULATION TRAPPING

This equation shows that through spontaneous emission the population occupations of the coupled and noncoupled states increase with a rate T012. However, the coupled state IC) experiences the loss rate r’ because of absorption to the excited state, whereas the noncoupled state is stable against absorption. Then through the process of depopulation pumping of the IC) state and spontaneous emission into the INC) state, all atoms could be accumulated in the noncoupled state of the coherent population trapping: a pure INC) state is formed through the irreversible process of spontaneous emission. It may be surprising to find that the rate of filling the two states IC) and INC) does not depend on the values of the laser Rabi frequencies that determine the two coherent superpositions of eq. (2.16a). The noncoupled state is made up of two particular laser modes, laser 1 and laser 2, so that it is decoupled from those two modes. On the contrary, in spontaneous emission the decay takes place through emission into all the possible modes of the radiation field, so that no modification of the spontaneous emission process is produced by the coherent population trapping, and in an average over all the emission directions the coupled and noncoupled states are equivalent. To be precise, the pumping process of the noncoupled state takes place through a more subtle mechanism: the preparation in that state occurs through a filtering process in the selective light absorption (Aspect and Kaiser [ 19901). Inverting eq. (2.16), the ground states are written:

Within those superpositions JNC) is perfectly stable, whereas IC) is not because of the excitation by the lasers at a rate f ’. After a long time, compared to (r’)-’, the atom will be either in INC) where it will remain trapped, or it will be involved in some absorption and fluorescence cycle. Thus, this selective light absorption process acts as a filter and leaves a fraction of the atoms (determined by the Rabi frequencies) in the noncoupled state, bringing the remaining ones into absorptiodfluorescence cycles. The optical pumping into the noncoupled state arises from the filtering process in laser absorption. The physical mechanism involved in the filtering process is the laser interaction that builds up a coherence between the I 1) and 12) states. From the solution of the Schrodinger equation for the three-level system under the Hamiltonian 3i0+ VAL and the relaxation processes, as shown by Agap’ev, Gornyi and Matisov [1993] and Scully [1994], it is possible to examine the filtering process starting from each atomic state. When the decay rate rI2 of the ground state coherence and the =r 2 = rp population losses, with rl+2= f 2- 1 = 0, are included into the density matrix

r,

v, § 21

ANALYSIS FOR DISCRETE STATES

273

equations, one arrives at the equations reported here for the simple case of QR1 = QR2:

From these equations it appears that the states INC) and (C) are coupled through the ground-state relaxation processes, so that the coherent population trapping in the JNC) state is limited by those relaxation processes.. The last limiting factor in the preparation of the couplednoncoupled states is the laser detuning from the Raman condition. This part of couplednoncoupled evolution is derived by applying an adiabatic elimination of the optical coherence and of the excited-state occupation under the assumption that the Rabi frequencies are small compared to spontaneous emission decay. If the laser frequencies are not in resonance with the optical transitions, but their detunings &I and 6 ~ are 2 small compared to T O ,the evolution of ground state populations and coherence results in: (2.24) Figure 4 shows the ground-state relaxation and the Raman detuning &, as the loss mechanisms between the coupled and noncoupled states. Figure 2b shows the occupation p c , and ~ PNC,NC of the coupled and noncoupled states for the same conditions as fig. 2a. It may be noticed that at the Raman resonance condition the entire population is concentrated in the noncoupled state, and that the occupation of the coupled state, given by the ratio between the ground-state relaxation rates and the spontaneous emission rate, is very small. From eqs. (2.18)-(2.24) the occupations of the following couplednoncoupled states are derived:

PNC,NC - pc,c =

r/ r/+ rI2

1

~

+

i.e., for & I , &2 halfwidth is

(< To, they

s1,2= r/+ r I 2 .

(2.25a)

(&)2'

are described through a Lorentzian lineshape whose (2.25b)

274

COHERENT POPULATION TRAPPING

rv, 9 2

These expressions show that the coherent population-trapping resonance is modified at r' = r12, i.e., when the laser intensity reaches the value I , where: (2.26) The coherence saturation intensity I,, being determined by the ground-state coherence relaxation rate, is much smaller than the saturation intensity I , of the optical transition (see Agrawal [ 1983a1, Kocharovskaya and Khanin [ 19861, Gornyi, Matisov and Rozhdestvensky [1989a]). I , is given by an expression similar to eq. (2.26), replacing r 1 2 by ro2. 2.4. DRESSED STATES

The dressed states are obtained whenever a time-independent matrix is obtained for the Hamiltonian ?lo+ VAL. Even if the dressed state description does not include the influence of dissipative processes, it still provides a good understanding of the physical phenomena. In semiclassical theory, a unitary transformation derives a time-independent Hamiltonian for a three-level system (see Series [1978]), and more recently Narducci, Scully, Oppo, Ru and Tredicce [1990], Manka, Doss, Narducci, Ru and Oppo [1991], Glushko and Kryzhanovsky [ 19921). Quantization of the laser field is another powerful approach, which has been applied to three-level systems by Cohen-Tannoudji and Reynaud [ 19771, Radmore and Knight [ 19821, Swain [ 19821, and by Dalibard, Reynaud and Cohen-Tannoudji [ 19871. In the case of an asymmetric A system with an energy separation between the ground states and with different laser frequencies, the wavefunctions of the three-level system may be written as:

(2.27)

where A, and C,, are determined by imposing the condition that iv,) are the cigenstates of the Hamiltonian ?lo + VAL (Glushko and Kryzhanovsky [ 19921). The phases of the laser fields have been explicitly included into the atomic wavefunctions because of their important role in the noncoupled-state evolution.

v, I 2 1

ANALYSIS FOR DISCRETE STATES

275

The set of states I q;)has an energy hA; with respect to that of the I I ) state. At the Raman resonance, aR= 0, the following eigenvalues are obtained:

A I = 0,

A2,3 = 2

[

1 f (1

+

g)"'1

,

(2.2 8a)

State I V l ) represents the noncoupled superposition of lower levels which satisfies eq. (2.17a) when the lower states do not have the same energy. In contrast to the symmetrical A system, for which the time dependence of the noncoupled state is trivial, here, owing to the time dependence of eq. (2.28b), INC) at different times is a different combination of ground states. The dependence of the noncoupled state on the relative phase of the two lasers evidences the requirement on their stability in any experimental situation. In the quantized field approach, the states are denoted through the atomic quantum number ( i = O , 1, 2) and the number of laser photons, nl for laser 1, and n2 for laser 2. The Hamiltonian is written: (2.29a) with the modes at frequencies OR^ (a&) characterized by the annihilation and creation operators U R I and (aR2 and ui,). For radiation fields with large nl and n2 photon numbers, the constants g R 1 and g R 2 of the interaction Hamiltonian give matrix elements equivalent to those of eq. (2.4):

ail

These matrix elements define a closed family .F(nl,n~) whose states are coupled through the absorption and stimulated emission processes:

Eigenstates of the dressed-state quantized Hamiltonian of eq. (2.29a) are linear combinations of these states. Simple expressions may be derived in

276

COHERENT POPULATION TRAPPING

[Y § 2

relevant cases. At the Raman resonance and laser resonance &I = &,2 = & = 0 with QRl = QR2 = QR, the eigenstates within the family .F(nl,n2) are (CohenTannoudji and Reynaud [ 19771):

(2.3 la)

with eigenenergies (2.31b) The dressed state INC) of eq. (2.31a), corresponding to the semiclassical /NC) state defined in $2.3, presents a similar interference in the absorption process. The IC) state does not have a correspondence with the It) and Is) dressed states, because it has been derived under the assumption of weak laser fields. Equation (2.3 Ib) shows that the noncoupled state is not perturbed in its energy, whereas the other states experience an energy shift. The equidistance of the It) and Is) from INC) is a consequence of the simplifying assumption of one- and two-photon resonance conditions. The spontaneous emission process produces a jump from the states Is) and It) of the F(n1,n2) family to one with a lower energy, F(nl- 1, n2) or F(nl,n2 - I ) (Cohen-Tannoudji, Zambon and Arimondo [1993]). If the INC) state is reached after a spontaneous emission, the time evolution stops because JNC) is stable against laser absorption and spontaneous emission. Cohen-Tannoudji and Reynaud [ 19771 have shown how the relaxation processes may be included into the dressed-state evolution to determine the occupations of the atomic states. In the analysis by Narducci, Doss, Ru, Scully, Zhu and Keitel [I9911 for lasing without inversion in an asymmetric A system, the two lasers acting on the two arms of the A system had the same frequency, nearly halfway between the two optical transitions, i.e., =-& = ~2112,as in fig. 5a. The coherent population trapping in this configuration may be explored either by changing the laser frequency, as in lasing without inversion experiments, or by modifying the frequency separation 021 between the lower levels, for instance by applying a magnetic field to the three-level system. Numerical results of that coherent population trapping are shown in fig. 5b for the occupation p& versus the laser

v, ii

21

ANALYSIS FOR DlSCRETE STATES

271

Fig. 5. (a) Schematic level diagram for coherent population trapping with a single laser of frequency wL acting on the three-level system; (b) pg0 as a function of laser detuning 6 ~ =1- 6 ~ 2 ; parameters: w21 =4To. Q R ~ = Q R ~ = O .and ~ Tr12=0.1 ~, To.

frequency, at fixed 1 3 2 1 separation. The decrease at the center is evidence for the presence of coherent population trapping. 2.5. QUANTUM INTERFERENCES

The dispersion lineshape presented in fig. 3 of the excited-state population versus the laser frequency is very similar to the Fano lineshape produced when a discrete state is coupled to a continuum, as in the autoionizing resonances (Fano [ 19611). In that case the quantum interference between discrete and continuum excitations produces an asymmetric characteristic lineshape. The connection between coherent population trapping and Fano lineshape has been worked out by Lounis and Cohen-Tannoudji [1992]. Their analysis treated a A system, in the case of QR1 << Qw, and determined the absorption of the weak Q R ~ probe at frequency wL1, as perturbed by a strong QR2 field. They examined the different contributions to the wLl absorption cross section produced by the scattering of one wL1 photon from the initial state Il), with decay back to that state emitting a spontaneous emission photon of frequency OIL. Because the excitedstate population is proportional to that scattering cross section, the determination of the physical processes contributing to the cross section provided a physical interpretation for the frequency dependence of the excited-state population. The simplest process for w ~ 1absorption, with transfer of an atom to the excited state 10) and decay back to the 11) state by spontaneous emission of photon wL, is represented by fig. 6a. A similar atomic evolution may take place through a stimulated Raman process, bringing the atom from 11) to (2) with absorption of wL1 and stimulated emission of wL2, followed by

278

COHERENT POPULATION TRAPPING

I z> I o> I1>

Fig. 6. Diagrammatic representation of the scattering processes of an W L photon for an atom in initial state 1 I ) and absorbing one (OLI photon, in the presence of a strong ( U L ~field. The interference behveen the two diagrams produces a dispersion-like Fano profile in the excited-state population.

another spontaneous Raman process from (2) to 11) with oL2absorption and 01, spontaneous emission, as in fig. 6b. When the contribution of the O L ~photons is considered nonperturbatively to all orders, this second process leads to a scattering amplitude comparablc to or even larger than the amplitude of the first path. However, the second path interferes with the first one, and that destructive interference between the two diagrams of fig. 6 produces at i ) =~O a zero value for the scattering cross section and also for the excited-state population, neglecting ground-state relaxation processes. Moreover, that interference creates the asymmetric lineshape for the case dLI,cSLz # 0. A similar destructive interference between diagrams occurs for .QR~ << QR,, in the case of initial occupation of the 12) state; thus, in the general case of comparable Rabi frcquencies, both interferences play an important role in determining the lineshape associated with coherent population trapping. Another key feature of the phenomenon is thc atomic preparation in a coherent superposition of the two ground states, the couplcNnoncouplcd states; additional interferences, constructive or destructive for the two superpositions, respectively, occur between symmetric scattering processes starting from each of the ground states. The difference between the Fano lineshape interference of fig. 6 and the interference of the noncoupled state is that the first one is prcscnt for any atomic preparation in the ground state, whereas the noncoupled-state interference requires an atomic preparation in a ground superposition state, and hence

v 9 1'

ANALYSIS FOR DISCKETE STATES

279

requires a preliminary efficient depopulation pumping or the light filtering process of 4 2.3. 2 6 L W E I - CONFIGURATIONS

i'he simplcst atomic level configuration for the realization of coherent population trapping is the Jg = 1 -J, = 1 configuration shown in fig. 7a, under the application of d,(J- laser beams: with those excitations the

form a closed A system (Aspect, Arimondo. Kaiser, Vansteenkiste and CohenTannoudji [ 1 9881). Furthermore, through optical pumping the entire population af the l.Iy, := I , mJ = 0) level is transferred into the other states, so that finally the complete population can be trapped in the noncoupled state. Also, the .Jg == 1 J, = 0 configuration has a simple A structure for the 0 ' . m excitation; however, because it is not a closed one, incoherent pumping on the == 1 , mJ = 0 ) + I J,= 0) transition is required to avoid population loss. The c' , (1.. excitation for coherent population trapping can be extended to other level schemes (Hioe and Carroll [ 19881, Smirnov, Tumaikin and Yudin [ 19891, rumaikin and Yudin [1990], Papoff, Mauri and Arimondo [1992]). The basic requirement is an odd number of levels ( N 3 3) between ground and excited states, with the ground-level number being larger by one than the excited one, so that a11 excitation channels from the ground levels interfere destmctively. For o+.0.. excitation on the .Ig= 2 J, = I transition, as shown in fig. 7c, the Zeeman-level structure produces two chains of Zeeman levels. The chain marked i n the tigure, called inverted W or M , contains three ground levels interfenng in all transitions to the excited state. In the other chain, not marked in the figure. the ground extrems elements of the chain do not have another channel with which to interfere, so no coherent population trapping can be constructed on that chain. Coherent population-trapping superpositions with a', O' are possible on the ,I .I and J ---,.I -- I transitions. hut not on the .I --t J -t 1 ones. Additional laser cmfiguratiotis. tor instance with elliptical polarized light and Zeeman levels separated by an applied magnetic field, were considered by Tumaikin and Yudin [ 19901, confirming the above requirements on the angular momenta. The linear superpositions of ground-state wavefunctions corresponding to the different configurations have been reported by Smirnov. Tumaikin and Yudin [ 19x91. Use of (J and ~tlight produces additional trapping superpositions: this scheme applied to the JF= 4 -+Jc = transition allows the preparation of a --f

-+

--j

4

280

[v, 5

COHERENT POPULATION TRAPPING

C)

=2

J=1/2

2

O A

d)

z

=2

-2

-2

0 -1

0

-1

.

0

1

+l

2

F=l

2

-2

J=1/2

-1

0

-112

1

112

2

Fig. 7. Optical transitions for coherent population-trapping investigations. (a) and (b) Three-level A schemes in the J , = I + J e = 1 transition induced by d , u - circular polarized, and lin Ilin linearly polarized light, respectively; (c) u', u- excitation of the J , = 2 + J e = 2 transition; (d) o+,x excitation of J , = 1/2 + J e = 1/2 transition; (e) and ( f ) different schemes for coherent population trapping between the hyperfine levels of Na using D I excitation (by Alzetta, Gozzini. Moi and Orriols [1976] and Fry, Li, Nikonov, Padmabandu, Scully, Smith, Tittel, Wang, Wilkinson and Zhu [ 19931, respectively).

trapping superposition involving Zeeman levels with Am., = 1, as in fig. 7d. A combination of (5 and n light is obtained when in the presence of a magnetic field defining the quantization axis, a circularly polarized laser beam makes an angle a different from 0 or ix with the magnetic field direction. That configuration of laser propagation in the presence of an applied magnetic field was used by Alzetta, Gozzini, Moi and Orriols [1976], Alzetta, Moi and Orriols [1979], and Xu [1994] in sodium investigations. Their level scheme is shown in fig. 7e: the ground Zeeman levels of the trapping superposition were (132S~/2, F = 1,mF = 1) , 132S~/2,F = 2, mF = 2 ) ) making a closed A scheme with the excited 13*P1/2,F = 2, mF = 2 ) level under d,n excitation. Another sodium-level configuration for coherent population trapping between ground hyperfine states has been used by Fry, Li, Nikonov, Padmabandu, Scully, Smith, Tittel, Wang, Wilkinson and Zhu [ 19931 in the context of lasing without inversion. The scheme, shown in fig. 7f, is based on two u+ lasers generating a trapping superposition for each couple of levels (132S1/2,F = l , m F ) , 13'S1/2,F = 2 , m ~ )with ) (mF=- 1, 0, I). Several coherent trapping superpositions are created simultaneously in the ground state, and

v, 9: 21

28 1

ANALYSIS FOR DISCRETE STATES

all of them contribute independently to the nonabsorption from ground state. The A schemes of this configuration are open, i.e., with spontaneous emission from the excited state of one A scheme to ground levels not belonging to that scheme. Thus, the ground state coherence created within a given interaction time is reduced. Using the D2 resonance line in alkalis, F -+ F + 1 transitions are excited, so that a completely destructive interference does not take place, and the preparation of coherent population-trapping superposition is not so efficient as for the Dl resonance (Alzetta, Moi and Orriols [1979], de Lignie and Eliel [1989], Eliel and de Lignie [1989,1990], Eliel [1993]). For an angular momentum of J = 1, the wavefunction describing the Zeeman levels has three components and can be treated as a vector. This leads to interesting symmetry properties for transitions involving a J = I state. For those states a linear atomic basis can be defined with respect to the eigenvectors IJ = 1, i) ( i = x , y , z) with zero eigenvalue of the operators J , ( i = x , y , z ) (Mauri and Arimondo [1991, 19921, Arimondo [1992]). The (J = I,x) and IJ = 1,y) states are the symmetric and antisymmetric combinations of the IJ = 1, mJ - 1) and lJ = I, mJ = 1) states, respectively. The convenience of the linear atomic basis results when we consider the selection rules for transitions induced by a laser field E, linearly polarized along t h c j axis, as for Jg=l+Je=l: A

(Je = 1 , iI E,

IJg =

1 , k ) = CE;jk

(i

= x,y, z),

(2.32a)

where C is a constant depending on the oscillator strength and &yk is the thirdrank unit antisymmetric tensor. The transitions induced by linearly polarized electric fields, for the J g = 1 -J,= 1 case, are marked in fig. 7b. It turns out that coherent population-trapping superpositions with A schemes may be formed using orthogonally linearly polarized lasers. For the J, = 1 J, = 0 transition the linear atomic basis leads to the selection rules:

where bjk is the Kronecker delta. The Hanle effect for the ground state (Hanle [ 19231, Corney [ 19771) is closely related to the preparation of the coherent population-trapping superposition, and its lineshape is that of fig. 2a (McLean, Ballagh and Warrington [1985]). The Hanle effect occurs when an optical transition, excited by a single optical frequency radiation, is split into its Zeeman components by an applied magnetic field B. At zero magnetic field the Zeeman levels are degenerate, and the interaction with the radiation is described through a two-level system. For an applied

282

COHERENT POPULATION TRAPPING

[V. 4 2

magnetic field different from zero: the Zeeman structure becomes relevant. and coherent superpositions of the Zeeman levels can be created. A resonance is observed when scanning the magnetic, field around the zero value and monitoring the fluorescence emitted under irradiation by light linearly polarized along a direction perpendicular to the magnetic field. The simplest process to analyze is the transition J, = 1 -+J, = 0 in the linear basis (leu) , lgi) ii = x . y z ) ) , with selection rules given by eq. (2.32b). For light linearly polarized along the x axis: the lJz = 1,x) IJ, = 0 ) transition is excited, and optical pumping out of the !gx)state takes place, bringing atoms to ground states lg,.) and Igz); as a consequence absorption and fluorescence do not take place. Applying the magnetic field, the states /us)and IsjJ)become nonstationary because they are not eigenstates of the magnetic field interaction Hamiltonian. Thus, even in the presence of optical pumping, the state Igx)becomes occupied, and fluorescent emission is observed. The width of the resonance observed around the zero field depends on the ground-state coherence relaxation. For cases where the excited state has an angular momentum larger than zero, the interpretation of the signal becomes more involved because of thc coupling by the ground state coherence to the excited-state evolution (Gorlicki and Dumont [ 19741, Decomps, Dumont and Ducloy [19?6]). The coherent superposition of states created in the Hanle effect for magnetic tields different from zero is closely related to the coherent superposition created in the scheme offig. 5a. A combination of the Hanle effect and excitation by (7, x light, i.e., with the creation o f Zeeman coherences and their destruction through a longitudinal pumping, was studied by Ballagh and Parigger [ 19861 for experiments on the magnetic control of polarization switching by Parigger, Hannaford, Sandle and Ballagh [ 19851, Parigger, Hannaford and Sandle [1986], and Sandle, Parigger and Ballagh [1986], as discussed in 9 3.5. The Hanle effect in the excited state has a similar coherent origin, with a similar resonant lineshape, except that the width of the central dip in the resonance fluorescence is determined by the optical linewidth. The nonlinear Hanle effect, introduced by Feld, Sanchez. Javan and Feldman [1974], again produces a resonance centered at zero magnetic field. Because this process has been described through saturation population only, and the creation of coherence between lower levels apparently does not play a role, it will be not analyzed here. Smirnov, Tumaikin and Yudin [1989] have also considered in their general treatment the case of atomic preparation in a state not interacting with the radiation, but for which the atomic wavefunction is not written as a linear superposition of atomic states. For instance, optical pumping with u' light on a J --J transition pumps atoms in the ground IJ, m.1 = .I) state. This state is nonabsorbing, or dark, and is described by a single eigenstate in the atomic

-

v, a

21

ANALYSIS FOR DISCRETE STATES

283

basis with the z quantization axis. However, if considered in the atomic basis with the quantization axis along the x or y axes, that state is represented through a coherent superposition of atomic states. This transformation through different bases, reducing the wavefunction from a coherent superposition of states to a single one or vice versa, has been used in the above treatment of the Jg= 1-&=O Hanle effect. The question arises whether it is always possible to find an appropriate basis where, instead of coherent superposition of states presenting destructively interfering transitions to the excited state, a dark state not excited by the light is obtained. That basis transformation can be found when the atomic states forming the coherent superposition have the same energy and same total angular momentum. When these conditions do not apply, the coherent population-trapping superposition corresponds to the creation of coherences that cannot be eliminated in any basis. The close connection between coherent population-trapping superpositions and dark states also results from the limiting case of eqs. (2.16) when one Rabi frequency is very small compared to the other one (Cohen-Tannoudji, Zambon and Arimondo [1993]). For Q R I << Q R ~ , the coupled and noncoupled states become: INC)

M

Il),

IC)

M

12).

(2.33)

Thus, the dark state 11) coincides with the noncoupled state. However, for Q R I which are very small, but different from zero, the small value of the absorption on the 11) + 10) transition arises from the quantum interference of 9 2.5. 2.7. DOPPLER BROADENING

Doppler broadening for a three-level laser interaction is treated by means of the ensemblc-averaged density matrix p(u,z, f), which describes an ensemble of thrcc-level systems at coordinates { z , r } moving with velocity u along the z axis of propagation of the laser radiation. For a given z position, the three-level system is describcd through the velocity-averaged density matrix. The simplest case is for A systems moving at constant velocity u, with z=zcJ +of, interacting with two traveling-wave lasers described by &I.I U I {cxp[i(wL.lf - k l z + $I)] + c.c.}i2 and E r . 2 ~ 2{exp[i(co:t - k2z + @*)I -1- c.c.}/2, with wavenumbers kl and k2. The atom-laser interaction Haniiltonian V,, of eq. (2.4) is slightly modified and the unitary transformation of eq. (2.6) to eliminate the fast time dependence of the out diagonal density matrix elements becomes:

284

COHERENT POPULATION TRAPPING

[v, 9: 2

Thus, for each class with velocity u , the laser detunings are modified in order to include the Doppler shift:

where the Doppler shifts on the two transitions have the same sign for copropagating lasers and opposite signs for counterpropagating lasers. By using those velocity-dependent detunings, the steady-state solutions of 5 2.1 may be used to describe the density matrix elements of each velocity class. If the copropagating lasers acting on the A system have k l = k2, the velocity dependence in the Raman detuning & disappears and all the velocity classes present the Raman resonance at the same values of the laser frequencies. Thus, the configuration with copropagating lasers is the most convenient for investigating experimentally the coherent population trapping in the A configuration of a Doppler broadened medium. The copropagating configuration for the A system is related to the laser configuration requirement for realizing two transitions without Doppler effect; a compensation of the Doppler shifts occurs for the copropagating configuration in the A system and for counterpropagating lasers in the cascade system. The influence of Doppler broadening on coherent population trapping has been investigated by several authors through numerical calculations for different sets of parameters (Orriols [1979], Kaivola, Thorsen and Poulsen [ 19851, Manka, Doss, Narducci, Ru and Oppo [1991], Meyer, Rathe, Graf, Zhu, Fry, Scully, Herling and Narducci [1994], Gea-Banacloche, Li, Jin and Xiao [1995]). These results confirm that with Doppler broadening coherent population trapping is preserved, although it is somewhat reduced. The case of two optical transitions driven by a standing wave laser field requires a much more involved theoretical analysis, as presented by Feldman and Feld [1972]. However, from the experimental point of view, there is no particular advantage to using that configuration except for applications in laser cooling, as discussed in 8 5. 2.8. RELAXATION PROCESSES

The relaxation rates r,2, Ti and r,-, ( & j = 1,2) effecting the coherent population trapping have been investigated by McLean, Ballagh and Warrington [ 19851, Parigger, Hannaford and Sandle [ 19861, Eliel [ 19931, Nottelman, Peters

v, a 21

ANALYSIS FOR DISCRETE STATES

285

and Lange [ 19931, and Graf, Arimondo, Fry, Nikonov, Padmabandu, Scully and Zhu [1995]. A detailed analysis of the relaxation processes has never been performed, even though a large amount of results in different experimental conditions has been obtained by Nicolini [1980] and Xu [1994]. For isotropic relaxation, the rates should be expressed through their components in the basis of the irreducible tensorial components of the density matrix (Omont [ 19771). Thus, for coherent population trapping in the J, = 1 ground state as illustrated in fig. 7a, the coherence relaxation rate is equal to tht. alignment decay rate, whereas the population transfer rates depend on the decay rates of both the orientation and the alignment components. For the case of superposition between hyperfine levels, e.g., figs. 7e and 7f, the tensorial component approach is more elaborate. The relaxation rates of the orientation and alignment tensorial components have been measured quite accurately in samarium by Parigger, Hannaford and Sandle [1986]. Because the depopulation pumping between the ground-state coupled and noncoupled states is affected by the relaxation processes, the treatment of relaxation in optical pumping (Cohen-Tannoudji and Kastler [ 19661, Happer [ 19721) should also apply to coherent population trapping. In coherent population-trapping experiments on a Doppler-broadened atomic vapor in a cell, the presence of a buffer gas modifies the creation of the ground-state coherences because (i) the diffusion through the buffer gas increases the interaction time between atoms and radiation, and (ii) the collisions with the buffer gas may destroy the coherence. By using the optical pumping theory, these relaxation processes can be described through the escape rate Tt from the laser beam with radius R and collision decay rcollproduced by the buffer gas: (2.36) where for simplicity the loss rates of populations and coherences are supposed to be equal. In the second equality the escape rate Tt depends on the atomic diffusion coefficient D, in the buffer gas, the mean free path A, with 2.405 the lowest zero of the zeroth-order Bessel function and c, a numerical constant depending on the kind of collisional process (Happer [ 19721). This theory is most appropriate for steady state conditions in experiments in a cell, which should be matched approximately to the zeroth order difhsion mode. In the presence of Doppler broadening of the optical transitions, the effective laser detunings depend on the atomic velocity, as evident from eqs. (2.35). Moreover, through velocity-changing collisions with the buffer gas, the atoms travel back and forth through the velocity space, experiencing different resonance conditions with

286

COHERENT POPULATION TKAPPING

[v, 6

2

the lasers. It is reasonable to suppose that for coherent trapping superpositions between Zeeman or hyperfine levels, the velocity-changing collisions preserve the ground state coherence, because the interatomic potential is insensitive to the nuclear spin orientation involved in the superposition, as pointed out by de Lignie and Eliel [1989]. However, this hypothesis has not been verified. Velocitychanging collisions produce an atomic transfer to velocity classes on the wing of the Doppler profile where an atom is in resonance with only one of the two lasers producing the coherent superposition. Under those conditions the coherent trapping superposition is destroyed by the absorption process (Eliel [ 19931, Graf, Arimondo, Fry, Nikonov, Padmabandu, Scully and Zhu [ 1995]), and this process represents an additional loss for ground-state coherence. The dependence of the coherent population-trapping coherences on the buffer gas collisions has been tested experimentally by Xu [1994] and by Nikonov, Rathe, Scully, Zhu, Fry, Li, Padmabandu and Fleischhauer [ 19941. Figure 8 presents experimental results obtained by Xu for the strength of sodium coherent population-trapping resonance versus helium buffer gas pressures, as observed on the fluorescent emission, using the level configuration of fig. 7e. Figure 8 evidences a slow decrease of the coherent population-trapping efficiency for helium pressures up to several hundred torr. The experimental results of Nikonov, Rathe, Scully, Zhu, Fry, Li, Padmabandu and Fleischhauer [ 19941 were obtained in the context of lasing without inversion using the level configuration of fig. 7f; in that case, the dependence of the ground-state coherence on the helium buffer gas pressure was more drastic, and a helium pressure of IOtorr destroyed completely the coherent population trapping. The main difference between the two investigations is the different hyperfine levels involved in the two coherent superpositions. The buffer gas presence enhances optical pumping which with ot light brings atoms into the Zeeman level with the highest mF number. In the investigations by Alzetta, Gozzini, Moi and Orriols [I9761 and Xu [1994], the optical pumping brings atoms into one of the levels involved in the coherent trapping superposition, thereby favoring coherent population trapping. In the investigation by Nikonov, Rathe, Scully, Zhu, Fry, Li, Padmabandu and Fleischhauer [1994], optical pumping brings atoms into a Zeeman level inaccessible for the coherent population-trapping preparation, and thereby inhibits the phenomenon. A fit of the results of fig. 8 with eqs. (2.25) and (2.36) should determine the effective collisional relaxation rate of the ground-state coherence, which would include the role of velocity-changing collisions. The preceding analysis has been based on two monochromatic radiation fields acting on the three-level scheme: the creation of an atomic coherence in the system arises from the spatial and temporal coherence in the radiation

287

ANALYSIS FOR I)ISC~KIJ'I'F. STATES

0

200

400

600

800

1000

P(He) (torr) Fig. 8. Experimental results for the signal, in arbitrary units, of the coherent population-trapping resonancc in sodium versus the helium buffer gas pressure (from Xu [1994]), with a lascr intensity of a few Watts/cm2 and a laser beam radius R = 0. I7 cm.

fields, as noted explicitly in eqs. (2.1 I ) and (2.28b). Very relevant to the experimental detection of coherent population trapping is the role playcd by the lascr bandwidth (Dalton and Knight [1982a,b, 19831, Kennedy and Swain [ 19841, Dalton, McDuff and Knight [ 19851, Pegg [ 19861, Mazets and Matisov [ 19921, Taichenachev, Tumaikin and Yudin [ 19941). Dalton and Knight [ 1982aI examined the effects of laser fluctuations on coherent population trapping using a phase-diffusion model for the laser fields. The laser spectra were considered to be Lorentzian, and a cross-correlation was introduced to describe the relatiT;c phase diffusions of the two lasers. The laser bandwidths dephase the atoiiiic cohcrencc. leading to a destruction of the coherent population-trapping superposition. The laser bandwidth may be taken into account in the previous analysis by introducing an effective relaxation rate r$ for the ground statc coherence:

where A L I m d Al 2 are the bandwidths of the two lascrs, and A , 1 1 2 is the cioss-correlated bandwidth For completely cioss-correlated laser beams, as for those generated through frequency modulation from a single source, the crosscorrelated bandwidth is equal to the bandwidth of each beam, and the laser bandwidth terms cancel out in eq. (2.37). 2.9. THREE-LEVEL SPECTROSCOPY

In the late 1960s a great eff'ort was devoted to high-resolution spectroscopy in three-level systems owing to their interesting features for saturation Dopplerfree spectroscopy. In these systems, besides the resonances associated with

288

COHERENT POPULATION TRAPPING

[V? ij 3

the saturation of the population distribution, additional Raman resonances are produced (Letokhov and Chebotayev [ 19771). In the A scheme those Raman resonances are obtained when a weak laser field ELI is applied to one arm of the A-scheme and a strong laser field E L 2 is applied to the second arm. These Raman resonances have been investigated theoretically (Notkin, Rautian and Feoktistov [1967], Feld and Javan [1969], Hansch and Toschek [1970], Feldman and Feld [ 19721) and have produced very precise experimental determinations (Beterov and Chebotayev [ 19691, Hansch, Keil, Schabert and Toschek [ 19691). They have the same features as coherent population trapping: they are connected to the creation of a p i 2 coherence in the ground state and their characteristics arc determined by the relaxation rate r 1 2 of that coherence. A density-matrix calculation for the absorption coefficient aL1of a weak probe was performed by Feld, Burns, Kuhl, Pappas and Murnick [ 19801 including the excited state decay into the ground states (see also Murnick, Feld, Burns, Kuhl and Pappas [1979]). That calculation, in the limit of weak saturation (intensity I L I of laser 1 smaller than coherence intensity I c ) , gave:

with 001 the small-signal absorption coefficient of the 11) 4 10) transition. The lineshape of eq. (2.3 8) reproduces the coherent population-trapping resonance with a narrow resonant decrease in absorption having linewidth r l 2 (the Raman-type contribution produced by the p 1 2 coherence), superimposed on a broader resonance structure with width T O representing , the population saturation contribution.

8

3. Spectroscopy for Discrete States

3.1. EARLY EXPERIMENTAL WORK

In the investigation by Bell and Bloom [1961] on sodium optical pumping in a transverse magnetic field, broadband pumping light from a circularly polarized resonance lamp was modulated at a frequency equal to the Zeeman splitting frequency in the ground state of sodium atoms in the MHz range. On the light transmitted through the cell, an optical pumping signal was observed at the modulation frequency. Those observations are evidence of coherent population trapping. The three-level system was composed of two

v, a

31

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Zeenian levels in the ground state and one excited level. The modulation of the light source generated sidebands of the pumping light at the modulation frequency. Two neighboring sidebands produced an optical pumping into the noncoupled state, when the sideband separation matched the Zeeman splitting. The transmitted light monitored the noncoupled state preparation as an incrcase of thc transmission at the modulation frequency. In an experiment by Takagi, Curl and Su [ 19751, sodium atoms were irradiated by Dl resonant laser light, containing a central mode and optical sidebands, and an increase in the transparency of the sodium vapor was observed when thc frequency separation of the optical sidebands from the central mode matched the Zeeman splitting in the sodium ground state. The signal was analyzed on the basis of populations only, and the creation of a coherence was not considercd. This mode-crossing technique, i.e., the simultaneous saturation of two transitions in a three level system, has been used routinely for the determination of molecular structures. However, no evidence related to the creation of groundstate coherence trapping has been reported in those molecular observations. It should be pointed out that in most molecular systems, owing to the high-level dcgeneracy, the coherences are expected to have a short lifetime, and thus the coherent trapping efficiency is expected to be quite low. 3.2. EXPERIMENTAL EVIDENCE

The first experimental observation by Alzetta, Gozzini, Moi and Orriols [ 19761 took place in an optical pumping experiment performed on sodium atoms with radiation from a niultimode dye lascr tuned to the resonant transition between the 3*S1/2 ground state and the 32P1/2 excited state. The frequency separation between the laser modes was around 350MHz, so that the separation between six adjacent modes matched the hyperfine separation of the sodium ground state. The experiment was performed in the presence of a magnetic field B, whose variation allowed matching the lascr frequencies to the level separation (see fig. 9). The experimental set-up presented a nice way to make the optical pumping phenomena clearly visible: a magnetic field gradient was introduced along the laser beam to change the splitting of the Zeeman sublevels in the hyperfine states. In this way the coherent population trapping; ie., the decrease in the excited-state population, appeared as a black line along the fluorescent path of the laser beam across the sodium cell. Figure 10, from a more recent investigation (Xu [1994]), shows an experimental record of the decrease in fluorescent light at the magnetic field position corresponding to the matching between the laser separation and the sodium level

290

-

[Y 9 3

COHERENT POPULATION I'R,\I'!"NG

21

B

a lens photomultiplier

amplifier

recorder

Fig. 9. Sketch of the experimental apparatus used for the observation of coherent population trapping i n sodium atoms in the presence of an inhomogeneous magnetic field (adapted from Alietta, Moi and Orriols [1979]).

splitting. Alzetta, Gozzini, Moi and Orriols [ 19761 reported atomic preparation in coherent trapping superposition for different values of the magnctic field and different Zeeman sublevels of the sodium hyperfine states. Alzetta [ 19781, and Alzelta, Moi and Orriols [I9791 completed that investigation by reporting a deformation of the fluorescent emission lineshape, similar to that shown in fig. 3 , when the laser was tuned within the Doppler profile. In the Xu [I9941 investigation, a dye laser with only three modes was used, and the separation between the two external modes matched the sodium hyperfine splitting. The jitter in the frequency separation between the two laser modes producing coherent population trapping was in the lOkHz range. I n order to realize a three-level schetne and optical pumping at thl: same time, circularly polarized laser light was used. Let us imagine 0- with the laser propagation direction at an angle between 0" and 50" with the magnetic field direction. With respect to the magnetic field, the laser light contained 0' and n components. and also a a- component. The O+ and ;[. components produced the atomic superposition of the three levels reported in fig. 7e; the 0- component did not produce coherent trapping, and, for the angles investigated in the experiment, did not significantly perturb the coherent superposition. Nicolini [ 19801 and Xu [ 19941 investigated the width of the coherent population-trapping resonance as a function of the buffer gas and of the sodium relaxation rates. Very narrow features were obtained when the interaction time of the sodium atoms with the radiation fields was increased by expanding the laser bean1 diameter or adding a buffer gas. Xu [ 19941 used a special cell whose walls were coated by a polydimethylsiloxane (as

V,

4 31

29 1

SPECTROSCOPY FOR DISCRETE STATES

w

B

Laser beam

Fig. 10. CCD image of the fluorescence of sodium vapor irradiated by a Dl resonant multimode laser. A black line in the fluorescence occurs at the magnetic field resonance condition for coherent population trapping (from Xu [1994]). Laser intensity 180mW on three modes and beam radius R = 0.075 cm.

described in Gozzini, Mango, Xu, Alzetta, Maccarrone and Bernheim [ 1993]), so that even at room temperature the sodium density was large enough for detection of the fluorescence. Accurate experiments on sodium atoms were performed by Gray, Whitley and Stroud [1978] who used two independent single-mode dye lasers. The decrease in population of the uppcr level in the coherent trapping conditions was observed as a function of one laser tuning. The experimental observations by that group are reproduced in fig. 1 1 , together with their theoretical analysis, to evidence the very good agreement with the experiment results. The resonant decrease in the excited-state population associated with the coherent population trapping in sodium was examined by Murnick, Feld, Burns, Kuhl and Pappas [1979] and by Feld, Burns, Kiihl, Pappas and Murnick [ 19801 using two independent lasers, with resonance linewidth around 6 MHz, determined by the relative frequency fluctuations of the two lasers, as in all experiments with independent sources. The decrease of the upper-state fluorescence in the three-level system was investigated. both experimentally and theoretically within the context of laser cooling experiments on barium ions in a radiofrequency trap, by Toschek and Neuhauser [1981], Janik, Nagourney and Dehmelt [1985], and Siemers, Schubert, Blatt, Neuhauser and Toschek [ 19921. The coherent populationtrapping features appeared when, besides the cooling laser at 493 nm from the

292

\Ki

COHERENT POPULATION TRAPPING

2-

c )r L

e

0

[Y § 3

0

v

0

Q0 -60-30 0 3 0 6 0

-50-30 0

30 6 0

Fig. 1 1. (a) Theoretical predictions and (b) expenmental results for the steady-state excited population poo monitored in fluorescence by Gray, Whitley and Stroud [1978] on a sodium atomic beam. One fixed frequency laser was at exact resonance, 6 ~ =10 , with Rabi frequency Q R ~x 1.8 ro. The second laser, with Rabi frequency Qm x 2.9 To, was tuned around the resonance value.

6’S1/2 to 62P1,2 states, a second laser at 650nm was simultaneously applied to the ions in order to avoid the loss of the ions into the metastable 52D3/2 state. The second laser exciting from the 5 ’ D ~ z state was required to repump the ions and keep them in interaction with the cooling laser. Janik, Nagourney and Dehmelt [ 19851 performed an analysis of the coherent trapping lineshape and noticed the reduction in coherent population trapping arising with detuned lasers and different Rabi frequencies. They reported coherent population-trapping resonances associated with the different Zeeman levels split by an applied magnetic field. Schubert, Siemers and Blatt [ 19891 performed a detailed analysis for the fluorescence lineshapes produced by the ions in the trap illuminated by the two lasers, taking into account the ion motion, and they evidenced lineshape distortions similar to those shown in fig. 3. The three-level A scheme pumped by two copropagating lasers, with the deformation in the resonant lineshape, was carefully investigated by Kaivola, Thorsen and Poulsen [1985]. A fast beam of calcium atoms excited by two red dye lasers was the medium. Very good agreement was realized between the experimental spectra and the Voigt profile obtained from a numerical integration over the velocity distribution of the steady-state poo density matrix. A similar careful analysis for fluorescence lineshape in the presence of inhomogeneous broadening, with a comparison to the density equation model, was performed by McLean, Ballagh and Warrington [1985] on the neon

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SPECTROSCOPY FOR DISCRETE STATES

293

Is& = I ) + 2p3(J = 0) transition. Their experimental set-up was similar to that used for the Hanle effect in the ground state, as discussed in 5 2.6. Cesium ground-state coherences with excitation based on the different components of the D2 line in a Hanle-type configuration were created by Theobald, Dimarcq, Giordano and Cerez [1989]. A laser diode source with a 30 MHz linewidth was employed. The linewidth of the coherent populationtrapping resonance around zero magnetic field was investigated as a function of several parameters, such as the laser intensity and the interaction time. Thc authors verified that for the measured linewidth good agreement was obtained with the results of a density matrix numerical calculation including all the cesium levels. The decay to a metastable level and the use of a second laser to repump atoms from that metastable level also occurred in the study of laser isotope separation of gadolinium by Adachi, Niki, Izawa, Nakai and Yamanaka [1991]. The threelevel system was formed by the ground state 6s2 9D2,the excited 6s6p9F2, and the metastable 6s' 9D3. The laser sources employed for the excitation, two narrowband pulsed dye lasers (with Rhodamine 6G) injected from two continuous wave (c.w.) lasers, presented excellent spectral qualities. Detection was based on ionization of the upper level through a third broad-band laser at the rate rlon. The authors noticed that the maximum ionized fraction of gadolinium atoms was around 50%, with this upper limit imposed by the gadolinium pumping into the coherent population-trapping state. In order to avoid trapping and produce the maximum ionization, very different Rabi frequencies Q R I and QR2 were used, the two lasers were detuned far from resonance, and finally the Rabi frequencies were equal to the rlon rate from the upper level, controlled by the third laser. Those experimental observations evidenced that coherent population-trapping preparation could also be realized on the nanosecond scale, but with Rabi frequencies and excited-state loss rate on an even faster timescale of -2x lo-'" s. The decrease in fluorescence from the excited-state population was examined by Young, Dinneen and Mansour [1988] on a Sct beam, with a three-level A scheme based on different hyperfine levels of the ground state and generating a sideband through an electro-optic modulation around 140MHz off the main laser beam. Continuing with the A configuration, coherent population trapping has been examined in rubidium atoms by Akul'shin, Celikov and Velichansky [ 199I] using two independent laser diodes tuned to the D I line. They reported a minimum linewidth of the coherent population-trapping resonance (FWHM) of 70 kHz, which led them to estimate the sum of the two laser linewidths to 50kHz, with additional broadening contributions from the transit time of the atoms

294

[v, !$

COHERENT POPULATION TRAPPING

zoov

3

100 ”

u.0

0.2

0.4

0.6

0.8

Fig. 12. Data, from Akul’shin, Celikov and Velichansky [1991], for the linewidth Av, of the *’Rb coherent population trapping versus the laser intensity I , in units of the optical transition saturation intensity I,, and linear fit with the saturation broadening law of eq. (2.25) (from Arimondo [1994]). A coherence saturation intensity I , = 70 pW/cm2 was obtained from the fit.

through the laser beam and from magnctic field inhomogeneity. That result, the narrowest resonance measured in cxperimcnts with cells, represents quitc an achievcrnent for an observation with two independent laser sources, and evidences the advances in the stability of laser diodes. In fig. 12 the dependence of the resonance linewidth on the applied laser intensity has been adapted from the original data in order to derive the coherence saturation intensity IC via eq. (2.26) (Arimondo [1994]). From the fit, I , is 70pWcm’. compared with I , = 1.6mW cm2, the saturation intensity of the rubidium D I optical transition. Akul’shin, Celikov and Velichansky [1991] reported a maximum contrast C of 60%, the contrast being defined as the ratio of the maximum and minimum in the fluorescence emission, or equivalently in the excited-state occupation. More recently, Akul’shin and Ohtsu [ 19941 demonstrated frequency pulling by the coherent population-trapping resonance of a rubidium cell within a laser diode cavity, pulling that could be used for the stabilization of the laser frequency. A A scheme based on the ground hyperfine F = 1 and F = 2 52Sl,2 states and excited .5?P1/2 F = 2 state of 87Rb has been investigated experimentally and compared to theory by Li and Xiao [ 19951. A quite different nonoptical method of studying the coherent population trapping of sodium in an atomic beam under D1 light pumping has been developed by Gieler, Aumayr and Windholz [ 19921. Their detection scheme was based on the electron capture of Na*(’P1,2) atoms by 10 keV He” projectile ions, with production of He+ (n = 5 ) . The charge-cxchanged Het ions were

F

i'

i

0.200 m 2 : ----r---r&

850

860

870

880

890

900

910

920

0 000 930

separated froin the primary ions by a deceleration lens and measured in their translational energy through an cncrgy analyzer. The three-level excitation of sodium atoms was based on laser light tuned at the center of the hyperfine ground structure. with production of the first sidebands through an electrooptic modulator around 886 MHz (one half the hypei-fine splitting). While keeping the dye laser frequency fixed, the microwave frequency was slowly scanned until the separation between the sidebands matched the ground-state hyperfine splitting. Thc experimental results are shown in fig. 13: both the Na D I fluorescence and the charge-oxchange signal were monitored. The lineshapes observed on both signals presented a decrease at the center, which i s evidence of coherent population trapping. The lineshapes produced by the coherent population trapping were quite different in the two detections. The authors claimed that the difference arose because the charge exchange probed only the central part of' the divergent Na beam, whereas the fluorescence detector collected light emitted from sodium atoms present in the Doppler-shifted outer beam re,'"lolls. The cascade configuration of fig. Ib has been investigated in ail experiment by Kaivola, Bjerre, Poulsen and Javanainen [1984], again on a fast beam of '"Ne atoms in the metastable 3s1:3/2I2 state. Laser excitation at 592.Snm to

296

COHERENT POPULATION TRAPPING

[Y ii 3

the 2p'[3/2]2 state, and from there at 248.0nm to the 4d'[5/2]3 state, was used. The cascade configuration requires counterpropagating beams in order to rcalize Doppler-free resonances. Using only one laser, reflected back on itself along the direction of the fast moving 2"Ne atoms, the laser frequency W L of the copropagating or counterpropagating photons appeared blue- or red-shifted in the rest frame of the atom. Thus the two-photon resonance condition was realized by modifying the velocity of the fast atoms. The fluorescence from the intermediate 10) level was monitored as a function of the laser frequency, and at Rabi frequencies comparable to the spontaneous emission decay rates from the intermediate and upper levels, a narrow dispersion-like structure, similar to the bright line feature of the A scheme of fig. 3, was observed on the Doppler-broadened profile of the emitted fluorescence. That dispersionlike structure was not reproduced precisely by the numerical analysis taking into account Doppler line broadening. However, that dip on the fluorescent spectrum from the intermediate level is related to the population trapped into the two terminal states while the intermediate level acquires an anomalously small population. The dispersion lineshape distortion on the cascade configuration has been confirmed by Gea-Banacloche, Li, Jin and Xiao [1995] in experiments on rubidium atoms in a cell using two laser diodes to excite the 52Sl,2 F = 3 to 52P3/2 F = 4 hyperfine component of the D2 line at 780nm and from there the 52D5,2 F = 5 state with 776 nm radiation, using orthogonal polarizations for the two lasers. The Rabi frequency ! 2 ~ 2 for the laser acting on the upper transition was quite large, with the laser intensity around ten times the optical saturation intensity, so that the linewidth of the coherent population-trapping resonance was deterrnincd from that Rabi frequency. The experimental results were fitted through the numerical solution of the density matrix equations, taking the Doppler broadening properly into account. A very good agreement was obtained when the diode laser bandwidth was included in the density matrix equation as an effective relaxation rate ryy [see eq. (2.37)]. 3.3. MODULATED AND PULSED LIGHT

The periodic modulation of the laser source for the creation of ground state coherences has been reinvestigated by some authors in a laser version of the Bell and Bloom [ 196 I ] work. Mlynek, Drake, Kersten, Frolich and Lange [ 198 I] used an electro-optic modulator to generate low-frequency sidebands on laser light resonant with the sodium D, line. The modulation created by the sodium Zeeman coherences was detected on the transmission of a probe laser beam. The periodic modulation of light from a diode laser through an acousto-optic modulator for the

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291

creation of cesium ground-state coherences in the presence of a magnetic field was investigated by Mishina, Fukuda and Hashi [1988]. Their geometry, with light propagating along the z axis of the magnetic field and polarized along the x axis, was similar to a Hanle effect geometry with modulated light. Thus, they detected the production of ground-state coherences either at zero magnetic field or when the modulation frequency matched a Zeeman splitting. Thc detection scheme was based on the detection by a probe beam of the anisotropy produced by the coherence precession in the magnetic field. A different set of experiments has been performed using mode-locked lasers with pulses in the nanosecond or picosecond range, whose repetition rate could bc prccisely controlled. The main idea was to generate and monitor coherences p21 in A- or V-three-level systems. The generation was based on the excitation of atoms by a short pulse of radiation: if the pulse duration T~ is short compared to the precession time 1 / 0 2 1 , or the Fourier spectral width l/z, of the pulse is large compared to the energy splitting 021, the pulse creates a pl2 coherence in the atomic system (Mlynek, Drake, Lange and Brand [1979]). In the presence of an applied magnetic field a precession of the coherence takes place, which is detected by the polarization rotation of a probe laser beam. The apparatus used by Mlynek, Lange, Harde and Burggraf [ 19811 and Harde, Burggraf, Mlynek and Lange [1981] is shown in fig. 14. In those experiments a circularly polarized 10 ps pulse was used to generate hyperfine coherences in the ground and excited states of the sodium DI line. The sodium coherences produced a polarization rotation for a probe pulsed laser beam, linearly polarized, and delayed up to 13 ns by a variable optical line. A typical beat structure could be observed on the detected signal as function of the time delay, with Fourier components corresponding to the hyperfine splittings. A modification of the preparation phase allowed a great increase in resolution (Mlynek, Lange, Harde and Burggraf [ 19811). The exciting light was composed of mode-locked pulses, and each short pulse, which was broad-band, produced an atomic coherence. If the pulse repetition frequency wp matched the evolution frequency 0 2 1 of the atomic coherence, the coherence generation and evolution corresponded to a forced oscillation regime with a large improvement in frequency resolution. Because the ground-state atomic coherences have a long decay time, a driving of the atomic coherence could be realized even when the pulse frequency alp was a very high subharmonic of the 1 3 2 1 frequency: w21 =

qwp,

with q >, 1. At first, q subharmonic orders up to 7 were realized, but Harde and Burggraf [1982, 1983, 19841 obtained q orders up to 2000. As a consequence,

298

COHERENT POPULATION TRAPPING

Fig. 14. Schematic of the experimental set-up for the generation and detection of three-level coherences used by Mlynek, Lange, Harde and Burggraf [1981].

the determination of sodium and rubidium ground-state hyperfine splittings and their pressure shifts were comparable in accuracy to that from standard microwave spectroscopy. In order to tune through the resonance condition, either the pulse repetition frequency cop was fixed and a magnetic field acting on the sodium atoms modified the energy splitting, or, more appropriately, the repetition frequency was varied. Experiments with pulse trains were previously performed in the cascade system of fig. lb, probing the coherence in the optical range generated between the levels 12) and 11) (Teets, Eckstein and Hansch [1977], Eckstein, Ferguson and Hansch [ 19781). The major difference between the A- and cascade schemes is the different sensitivity to the optical phase: in the A scheme the p21 coherence depends on the optical phase difference between the two optical excitation, whereas in the cascade scheme it depends on the sum of the optical phases, as presented explicitly in eq. (2.1 1). Thus, for the cascade transition the requirements on the optical phase stability between successive laser pulses and the larger linewidth of the coherence resonance do not allow the Ascheme accuracy to be attained. The creation of the ground-state coherent population trapping in experiments with pulsed light or a train of ultrashort light pulses was investigated theoretically by Kocharovskaya and Khanin [ 19861 and Koeharovskaya [ 19901. 3.4. METROLOGY

In a search for new atomic frequency standards based on optical transitions but not affected by laser jitter and most line-broadening mechanisms, Ezekiel and his colleagues at MIT and the Rome Laboratory of Hanscom Air Force Base have

Y

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it

Fig. 15. (a) Schematic of the experimental set-up for obtaining Ramsey fringes in the coherent population-trapping preparation and detection; (b) Ramsey fringes for the sodium F = 1, mF = 0 + F = 2, mF = 0 ground hyperfine transition, for a distance L = 30 cm between preparation and interrogation zones (from Thomas, Hemmer, Ezekiel, Leiby Jr, Picard and Willis [1982]).

used the stimulated resonant Raman transitions in a three-level A system based on sodium ground-state hyperfine splitting (see Tench, Peuse, Hemmer, Thomas, Ezekiel, Leiby Jr, Picard and Willis [1981], Thomas, Hemmer, Ezekiel, Leiby Jr, Picard and Willis [1982]). In the resonant Raman transitions, they prepared the ground-state coherent population-trapping superposition using two resonant lasers, and monitored that preparation by light absorption. In order to reduce the transit-time broadening, they used a two-zone excitation for the Raman interaction, as illustrated by fig. 15a. That scheme is analogous to Ramsey’s method of separated-microwave field excitation on an atomic beam (Ramsey [ 19631). In separated-field excitations the atomic coherence created in the two interaction zones interfere and produce fringes whose widths are characteristic of the atomic transit time between the two zones. In the laser apparatus for obtaining Ramsey fringes, laser radiation at frequency wL2 was obtained from laser radiation at frequency wLI through an acousto-optic shifter at the sodium hyperfine splitting with excellent frequency stability. The two circularly polarized laser beams were combined on a beam splitter so that they interacted with the sodium atomic beam in a region of 2 mm diameter at positions A and B separated up to L = 30 cm. Fluorescence from the B region monitored the Ramsey fringes on the F = 1, mF = 0 4 F = 2, mF = 0 Raman transition. Ramsey fringes with a measured width of 650 Hz (HWHM) were obtained, as reported in fig. 15b. That

300

COHERENT POPULATION TRAPPING

[Y 8

3

value represents the narrowest coherent population-trapping resonance measured so far. The laser power required to saturate the Raman process, 40pW, was determined from the interaction time. Hemmer, Ontai and Ezekiel [ 19861 studied the Ramsey fringe detection, and Hemmer, Shahriar, Natoli and Ezekiel [ 19891 examined the ac Stark shift, or light shift, of the Raman resonances, occurring when the laser fields are detuned from the resonance with the upper state, as shown by eq. (2.12), for the application of the resonance as an atomic clock. A previous theoretical analysis of the light shifts was reported by de Clercq and Ctrez [ 19831. Hemmer, Ezekiel and Leiby Jr [1983] and Hemmer, Ontai and Ezekiel [ 19861 stabilized a microwave oscillator using the Ramsey fringes of coherent population trapping in sodium atoms, with a linewidth of 2.6 kHz and a fractional stability of 4 x 10-I' x-''~, where z is the averaging time. Hemmer, Shahriar, Lamela-Rivera, Smith, Bernacki and Ezekiel [ 19931 repeated the experiment on cesium atoms, using a laser diode with microwave sidebands at the cesium ground-state hyperfine splitting. The Ramsey fringes for a 15 cm separation of the interaction zones produced a 1 kHz width, and led the authors to project a 6 x 10-l' frequency stability. Thus, the Ramsey-fringe detection of the coherent trapping evolution has interesting metrological applications. A nice extension of those Ramsey-fringe investigations was performed by Shahriar and Hemmer [ 19901. For a A system, the coherent trapping leads to the preparation of an atomic coherence ~2~ that, as in eq. (2.6), evolves at angular frequency W L I - 0 ~ 2 with , a phase $1 - $2 determined by the relative phase of two laser fields. Shahriar and Hemmer [1990] probed, or even perturbed, the evolution of the atomic coherence through the application of a microwave field at the angular frequency WLI - w L ~ :a microwave field may induce magnetic dipole transitions between the two hyperfine ground levels of sodium. The microwave perturbation was applied in a region intermediate between zones A and B of fig. 15a, with the atomic coherence already formed by the first interaction with the laser fields, before it was probed. When all atoms were prepared in the coherent superposition and the applied microwave field was exactly in phase with their evolution, no deformation of the coherent evolution took place and no modification was detected on the Ramsey fringes in the second zone. In contrast, if the time dependence of the microwave field was out of phase with the coherence time dependence, the atomic evolution was modified by the microwave interaction and detected in the B-zone fringes. In order to have a microwave field with the proper phase with respect to $1 -$2 established by the lasers, the microwave field was generated by detecting and amplifying the beat between the two optical fields generated on a fast avalanche photodiode. This experiment

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confirmed that a pure state of coherent atomic superposition was prepared by the Raman process in the first interaction region. A three-level system, with optical and microwave radiation fields acting on all the transitions, represents a closed loop system, where the net phase of the fields has a critical effect upon population dynamics. Three-level and fourlevel closed loops have been examined theoretically by Buckle, Barnett, Knight, Lauder and Pegg [ 19861, and Kosachiov, Matisov and Rozhdestvensky [ 199 I , 1992a,b]. Buckle, Barnett, Knight, Lauder and Pegg [ 19861 pointed out that those multilevel loops could be applied in more elaborate Ramsey-fringe interference investigations. In order to improve the signal to noise ratio in the cesium-based primary frequency standard and therefore its performance, Lewis and Feldman [ 198 I], Lewis, Feldman and Bergquist [1981], and Lewis [I9841 proposed the use of two polarized laser fields tuned to hyperfine transitions of the cesium D2 line to increase the optical pumping of cesium atoms in a specific ground state, either the F = 4, mF = 0 or the F = 3, mF = 0 state. Different polarizations may be used for the two lasers; however, in some cases, for instance when using two linearly polarized lasers, a three-level A system is formed. In such cases, the cesium preparation in a coherent population-trapping superposition produced results quite different from the intuitive ones. de Clercq, de Labachellerie, Avila, Cerez and Tttu [ 19841 examined theoretically cesium pumping with two linearly polarized lasers and stated that because of coherent population trapping the use of two lasers did not produce the expected increase in cesium pumping. In order to avoid coherent population trapping, these authors explored theoretically some interesting alternatives: to eliminate coherent population trapping and still produce an efficient optical pumping, the two monochromatic laser excitations must be applied alternatively, or broadband and uncorrelated laser sources must be used, so that coherence processes could not take place, leaving the optical pumping to increase the population difference. 3.5. OPTICAL BISTABILITY

A novel mechanism for optical bistability, because of the presence of the lower-level coherence, in the three-level A system was proposed by Walls and Zoller [ 19801, Walls, Zoller and Steyn-Ross [ 19811, and Agrawal [1981]. Walls and Zoller [1980], examining the case of a single laser inducing both optical transitions, mentioned the advantages offered by that system: a lower threshold for the bistability produced by the nonlinear response at laser intensity comparable to I,, as well as a Doppler-free mechanism for the

3 02

[Y I 3

COHERENT POPULATION TRAPPING

Magnetic field (mT)

Magnetic field (mT)

Fig 16 (a) Measured and (b) calculated dispersive optical bistability of sodium atoms, with argon butler gas, contained in a Fabry-Perot resonator, as detected in the output power from the cavity (from Mlynek. Mitschke, Deserno and Lange [1984]). For an accurate list of both experimental and theoretical parameters, see original reference.

copropagating configuration and the relative insensitivity to the laser phase. Experimental observations of optical bistability for sodium atoms contained in an optical cavity and driven to the coherent trapping superposition, were reported by Mlynek, Mitschke, Deserno and Lange [1982, 19841. The resonance condition for coherent population trapping was examined through the groundstate Hanle effect. This was performed on the sodium D1 line with a cell containing a large pressure of argon buffer gas inserted into a Fabry-Perot resonator. Both absorptive and dispersive regimes of optical bistability were examined with the coherent population-trapping atomic response determining both regimes. Experimental results for the dispersive bistability, as observed on the light power transmitted from the sodium filled resonator, are shown in fig. 16a. Figure 16b shows the calculated dispersive bistability. The optical hysteresis in thc resonator transmission is quite evident and is well reproduced by the analysis. A broad triple-peaked profile, observed by Schulz, MacGillivray and Standage [I9831 and MacGillivray [I9831 on the intensity transmitted from a FabryPerot cavity containing sodium vapor, was also interpreted as optical bistability produced by coherent population trapping. A theoretical analysis by Pegg and Schulz [ 19851 cxamined the trapping process in the case o f a standing wave laser field, considering that an atom, moving along the standing wave, experiences an elcctric strength frequency modulated because of the atomic velocity. When the frequency of that modulation becomes comparable to the splitting of the lower lcvel i n the A system, the cohe,rent population-trapping preparation turns out to

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SPECTROSCOPY FOK DISCRETE STATES

303

depend on the laser intensity. Pegg and Schulz claimed that dispersive effects associated with that intensity-dependent resonance account for the experimental results of Schulz, MacGillivray and Standage [1983], and MacGillivray [ 19831. The generation of ground state coherences, and their evolution in the presence of an applied magnetic field, is the origin of another phenomenon of nonlinear dynamics, denoted as magnetically induced polarization switching, which occurs in the same configuration of optical bistability with atoms contained in a Fabry-Perot cavity and illuminated by near resonant laser light. Polarization switching describes the particular atom-cavity responsc in which a significant asymmetry between the output amplitudes of d and o-Dolarizcd waves develops spontaneously, for a given symmetric laser input of thc atom-cavity with equal amplitudes of o+ and o--polarizations, i.e., linc;irly polarized laser light. In polarization switching, fluctuations in the atom and cavity produce laser-output polarization configurations different from thc input oiic. For 110 applied magnetic field, an input linear polarization is conscrvcd at thc outout. The application of a magnetic field to the atoms. paralicl t u tlic light propagation direction, produces a circularly polarized output h n i . %lrlgiictic;illy induced polarization switching has been studied by h i . ILIannrtford, Sandlc and Ballagh [ 19851, Parigger, Hannaford and Saritllc I1 and Snndlc, Parigger and Ballagh [ 19861 on samarium atoms using thc transition from the lower 4f66s27 F ~ state to the upper 4f66s6p 'F: state at 570.7 nm; the Jg -= I +./, = O transition allows a simple physical interpretation. I n the lincnr basis L: linearly polarized input laser light, say E,, has a very high s\.iixiictry, so ihat at zero magnetic field the output laser beam from the Fabry-Perat rcsoiiatot prcscrvcs that linear polarization. I n the presence of a large longitudinal Inagiletic field B,, the l.Jg,i) ( i = x , .v,z ) states are no longer stationary eigenstatcs of thc atomic Hamiltonian. In this condition the linear symmetry of the input electric ficld i s broken and nnalysii of the stcady stntc shows that undcr proper conditions niultistability ma? (ICCUI' with a circularly po1:irixci I;iscr output from thc cavit? contnininy thc : ~ ~ i n i x i u atoms. ni I n that niultistiibilit)., :in iiiiporlant rolc is p l a y i by thc rehuntion rate of thc Sround-state cohcrcnccs. ~

Four-mave mixing is a nonlinear optical phase conjugation technique which allows a n optical field with well detincd optical charactcristics to be generated from an absorbing medium. The tvpical gcometry o f optical beams applied in backward four-wave mixing is shown schematically i n iig. 17a. E l and E., the pump laser waves, determinc the population and cohcrences o f the absorbing

304

E

[v, 5

COHERENT POPULATION TRAPPING

3

MEDIUM w

‘A -0.4

0

0.4

MAONETIC FIELD (mT)

Fig. 17. (a) Four-wave mixing geometry for investigating the coherent population-trapping nonlinearity. E l , E2, the pump laser waves; E3, the probe; Eq, the conjugate generated wave. (b) Phase conjugate reflectivity E:/E: versus transverse magnetic field B for different intensities of the El and E2 pump waves; (A) 0.09 W/cm2; (B) 0.26 W/cm2; (C) 0.94 W/cm2, for an E j probe wave intensity of 0.08 W/cm2 (Koster, Mlynek and Lange [ 19851).

medium. The probe wave E3 generates the conjugate wave Ed through the fourwave mixing process. The pump and probe waves create a grating of population and coherences within the medium; i.e., a periodic variation in position and time of populations and coherences. The phase conjugate wave E4 arises from the scattering of the E3 wave off that grating. The four-wave mixing is denoted as degenerate if the four waves have the same frequency, as for the cases analyzed here. In most cases, four-wave mixing is based on the optical nonlinearities created in a saturated two-level medium. Agrawal [1983a,b] pointed out that in a A three-level system driven from a single laser, the optical nonlinearities linked to the creation of a Zeeman coherence provides a new mechanism for phase conjugation. Furthermore, he noticed that the coherence saturation intensity I , of eq. (2.26) allowed four-wave mixing in a three-level system at laser intensities lower than in a two-level system. In his analysis, based on a geometry with pump waves having opposite circular polarization, the E4 generation efficiency, when plotted versus the lower level Zeeman splitting, presented the characteristic central dip of coherent population trapping. These theoretical predictions were confirmed by the experimental observations of Mlynek, Mitschke, Koster and Lange [1984], Mlynek, Koster, Kolbe and Lange [ 19841, Koster, Mlynek and Lange [ 19851, and Lange, Koster and Mlynek [1986] on four-wave mixing in sodium atoms using a ground-state Hanleeffect geometry. Mlynek, Mitschke, Koster and Lange [ 19841 tested whether a coherent population-trapping resonance could be observed on the sodium

v, 6

31

SPECTROSCOPY FOR DISCRETE STATES

305

absorption coefficient by scanning the magnetic field around the zero valuc. They also measured the phase-conjugate signal on the gcnerated E4 wave, producing results illustrated in fig. 17b. The characteristic lineshape of coherent populationtrapping resonance is evident. The pump intensities applied to the sodium atoms to realize a coherent population-trapping regime wcre large because the experiment operated on sodium atoms in the presence of a large buffer gas pressure (argon at 170 mbar). While the observations by these authors were based on nearly resonant laser excitation, another set of experimental observations was performed, always on sodium atoms, using laser light detuned up to 150GHz from the D I and D2 resonance lines (Bloembergen [1985, 19871, Bloembergen and Zou [1985], Bloembergen, Zou and Rothberg [1985], Zou and Bloembergen [ 19861). Because of the largc detuning, the excitation to the sodium-excited states relied on the presence of collisions with a buffer gas, helium or argon, so that the phenomenon was denoted as collision-enhanced four-wave mixing. Again, narrow resonances were observed, associated with the production of ground-state coherences prccessing in the magnetic field. Careful measurements of the signal intensity versus different experimental parameters such as laser power, laser detuning, and buffer gas pressure were performed. A time-delayed four-wave mixing experiment was performed by Bouchenc, DCbarrc, Keller, Lc Gouet, Tchhnio, Finkelstein and Bcrman [ 1992a,b] on

'

an atomic vapor of strontium atoms using the 5s2 So --t 5s5p P I transition at 689 nm. Driven by two linear cross-polarized laser fields, this experiment behaved as the three-level V scheme of fig. IC. Thc two pump laser pulses produced a coherent population trapping between the upper states whose decay time, the spontaneous emission-time of the strontium-excited state, was longer than the I o n s pulse duration. The four-wave mixing signal was measured as a function of the delay time between the applied pump and probe pulses. A coherent population-trapping resonance was observed on the delayed time dependence of the generated four-wave mixing signal. At weak pump Rabi frequencies the trapping resonance linewidth coincided with the frequency bandwidth of the pump lasers, i.e., the inverse of their coherence time, 120ps. At large pump Rabi frequencies the resonance linewidth was narrower than the pump laser bandwidth. The theoretical analysis by Finkelstein [ 199I] has shown that at large Rabi frequencies the upper-state coherence contains some transient components with long decay times generating the observed narrow linewidth. In a recent theoretical analysis Schmidt-Iglesias [ 19931 has pointed out the possibility of separating within the four-wave mixing signal the contribution

3 06

COHERENT POPULATION TRAPPING

[YI 3

of the coherent population-trapping mechanism from that of other population mechanisms. 3 7. LIGHT-INDUCED DRIFT

I n light-induced drift, a transport of a laser-irradiated species inside a buffer gas takes place owing to the difference between ground and excited-state scattering cross sections with a buffer gas (Werij and Woerdman [198X], Eliel [1993]). The phenomenon depends on the number of atoms present in the excited state. Laser irradiation with two resonant laser sources may be employed to avoid hyperfine pumping or loss of atoms to another state as in other cases of spectroscopic investigations. As a consequence, in a three-level system the coherent populationtrapping mechanism may be operational, with a large decrease in the excited-state population, and also a decrease in the light-induced drift. Modification in the light-induced drift due to coherent population trapping using two-laser irradiation has been reported by de Lignie and Eliel [ I9 891, Eliel and de Lignie [1989, 19901, and by Eliel [1993] in measurements on a sodium vapor contained in a capillary cell in the presence of xenon buffer gas. As is standard in sodium experiments, the two lasers excited the transitions from the two ground hyperfine states to either the 32Pl,2 or 3*P3,2 state. Experimental results for light-induced drift versus the detuning of laser 2, with laser 1 fixed in frequency, are reported in figs. 18a and 1% for the Dl and D 2 excitation, respectively. The two single-mode lasers had orthogonal polarizations. Notice that in both experiments laser 1 was not in resonance, but detuned to the red side of the absorption transition, because light-induced drift is produced for that laser detuning. The D, excitation of fig. 18a, as usual, leads to a clearer evidence of a decrease in the excited-state population, and therefore in the amount of the light-induced drift. For an et'ficient preparation of sodium atoms in the coherent trapping superposition with Dl excitation, the authors satisfied the condition that Rabi frequency amplitudes be small as compared to the excited-state hyperfine splitting. In effect, for Rabi frequencies comparablc to the hyperfine splitting in the excited state, not all the excitation processes experience interference. For the light-induced drift with D2 excitation, some excitcd-state hyperfine levels present absorption from a single hyperfine ground level, whence they do not experience the interference process, as noticed in 5 2.6. Thus, a very small effect o f population trapping could be observed (see fig. I X b j . The linewidth of the observed coherent-trapping resonance was determined mainly by the frequency jitter of the two lasers in the hlHz range. For the measurements on the D1 data good agreement was found with a theoretical model.

v,

31

SPECTROSCOPY FOR DISCRETE STATES

0

-1

0

1

Detuning laser

2

2 (GHz)

307

3

I

I ’ - 8

cE

- 6

-1

0

1

Detuning laser

2

2 (GHz)

3

f i g . 1 X Medsuremetlls of the light-induced drift velocity of sodium in I .5 Torr X e as a function of the detuning of laser 2 for (a) D I excitation and (b) D, excitation. The positions of the resonant frequencies tor the transitions starting from the F = I and F = 2 ground hyperfine levels are marked by bars near the horizontal axes. i n hoth cases, laser I was tuned around 0.6GHz 011 the red side of the F = 2 resciiiiiiicc, its marked. The intensities of hoth lasers were around 3 W/cm2 The solid Iinc in ( a ) rcprchenls the result of a model calculation (fioni dc Lignie and liliel [1989]).

Thc largc variation in the index of refraction associated with the cohercnttrapping resonaiicc, as presented in fig. 2c, has not reccived the same attention ;is the absorption coefficient. However, in thc context of lasing without inversion, it has been suggested by Scully El9911 and Scully and Zhu [I9921 to make use of that large increase in index of refraction in several applications to be discussed in 4 7. Very recently some observations of the index of’ refraction around the coherent population-trapping resonance have been pcrformed. Experinrental results havc bccn obtaincd by Schmidt, Hussein, Wynands and Mcschede [1993. I W S ] on a A cesium system composed of the 1 ; = 3 and

308

:::m L!Y! [Y §

COHERENT POPULATION TRAPPING

0.08 J

d

C

.-

0

0.06

p

0.04

9

0.02

0

3

0.00

-0.2

-500

0

500

-500

0

500

Probe detuning (MHz)

Fig. 19. Measured absorption aL (left) and dispersion nL (right), both in arbitrary units, in a rubidium cell with length L=7.6cm versus the detuning 6 ~ 1of the weak probe laser on the 52S1/2F = 3 -52P3,2F=4 transition in the presence of a strong resonant pump laser on the 52P3/2 F = 4 + ?i2D.j/2 F = 5 transition with an intensity of 250 W/cm2 (from Xiao, Li, Jin and Gea-Banacloche [ 19951).

F = 4 ground states and the F ' = 4 excited state: a strong laser diode excited the F = 3 F' = 4 transition, whereas a weak beam from an independent laser diode probed the F = 4 + F' = 4 transition. Xiao, Li, Jin and Gea-Banacloche [ 19951 have reported measurements of the index of refraction, combined with measurements of the absorption coefficient, on a cascade configuration based on hyperfine levels of the 5S1/2 ground, SP3/2 first-excited, and 5D5/2 secondexcited levels of rubidium, again using independent diode lasers, similar to the lineshape measurements discussed in 4 3.2. A Mach-Zender-type interferometric configuration was used in both experiments to measure the index of refraction. Experimental results are presented in fig. 19 for the measurements of the absorption coefficient and the refraction index by Xiao, Li, Jin and GeaBanacloche [1995] for a probe laser field on the 11) + (0) transition and for a strong pump laser on the 10) + (2) transition of the cascade scheme. These results should be compared to the theoretical ones of figs. 2c and 2d. Even if the theoretical and experimental absorption coefficients and indices of refraction refer to different three-level configurations, the cascade and A scheme, respectively, the lineshape is a general character of the phenomenon. In the experimental results for the cascade configuration the linewidth of the onephoton transition was determined by Doppler broadening, whereas the linewidth of the coherent population-trapping resonance was determined by the Rabi frequency, as in eq. (2.25b). In the observations by Schmidt, Hussein, Wynands and Meschede [1993, 19951 on the A system, the final linewidth was around seventy kHz, limited partly by the transit time broadening and the residual Doppler broadening due to the angle between pump and probe laser beams. ---f

Y 5 41

COHERENT POPULATION TRAPPING IN THE CONTINUUM

tj

3 09

4. Coherent Population Trapping in the Continuum

The phenomena presented in the preceding section for bound states are also produced in the case of a continuum representing the upper state of the A system, as reviewed by Knight [1984] and Knight, Lauder and Dalton [1990]. Furthermore, coherent population-trapping features appear for a level scheme involving either a nonstructured continuum or an energy region in which the continuum is structured, as, for instance, in the presence of autoionization or predissociation resonances in a molecule. These processes involving the continuum are shown in fig. 20. Figure 20a presents the excitation by two lasers at frequencies W L I and O L ~ from the discrete states 11) and 12) to a continuum IC). Figure 20b presents an autoionizing or dissociating state lo), coupled to the continuum IC) and excited by lasers from 11) and 12) levels. The first scheme is associated with induced continuum structures, that originated through the coupling between the continuum and the dressed state corresponding to the absorption of one laser photon from either the 11) or 12) state. Whenever a discrete state and a continuum are in resonance, the mixing of those states leads ta the distorted absorption lineshape known as the Fano profile (Fano [1961]). In the case of coupling between the dressed states and the continuum, a Fano lineshape is also obtained. For the continuum as the upper state, the equivalent of the excited-state occupation for discrete states is the ionization probability, i.e., the probability of being in the continuum IC). That probability determines the number of electrons or ions to

I1>

I1>

4

b)

Fig. 20. Simplified three-level structures involving discrete and continuum states investigated for coherent population trapping. (a) Two transitions from discrete states are coupled to an upper continuum (C) by h+o laser radiations; (b) the upper state (0) of a three-level A scheme is coupled to the continuum IC) by a predissociation or autoionization mechanism.

310

COHERENT POPULATION TRAPPING

[v, § 5

be measured in an experiment. Thus, the ionization probability may reveal the coherent population-trapping features; in applying the laser fields in the scheme of fig. 20a, only a part of the population in the ground states is ionized because the remaining population is trapped coherently in the superposition of states 11) and 12). In the scheme of fig. 20b, two separate three-level A schemes are present, with Coulomb coupling between the upper level 10) and the continuum. Application of a single laser field, either on the 11) 4 10) or 12) 4 10) transition, leads to a Fano lineshape profile due to the coupling between the 10) state and the continuum. The simultaneous presence of two lasers produces a coherent population superposition in the lower states and generates a distortion of the Fano profile. There have not been many experimental observations of continuum phenomena; those up to 1990 have been reviewed by Knight, Lauder and Dalton [ 19901. More recently, coherent population trapping associated with a continuum structure, as in fig. 20a, has been observed in sodium ionization experiments (Shao, Charalambidis, Fotakis, Zhang and Lambropoulos [ 19911, Cavalieri, Pavone and Matera [ 19911). Observations of the third harmonic generation through autoionization states of calcium (Faucher, Charalambidis, Fotakis, Zhang and Lambropoulos [ 19931) and birefringence and dichroism in the autoionization of cesium (Cavalieri, Matera, Pavone, Zhang, Lambropoulos and Nakajima [ 19931) have provided additional evidence of the phenomenon.

5

5. Laser Cooling

In the laser manipulation of atoms, the velocity-selective coherent population trapping (VSCPT) is one method that has permitted a temperature to be reached that is lower than the landmark posed by the single-photon energy recoil (Aspect, Arimondo, Kaiser, Vansteenkiste and Cohen-Tannoudji [ 19881). The other method, by Kasevich and Chu [1992], is always in a A system and uses sequences of stimulated Raman and optical pumping pulses with appropriate shape, the frequency spectrum of the light being tailored so that atoms with nearly zero velocity are not excited. In contrast, the basic idea of VSCPT is to pump atoms into a noncoupled state having a well-defined momentum, where atoms do not interact with the laser radiation. The final requirement for realizing a cooling scheme is to bring all atoms, or a large majority of them, in the noncoupled state with a well-defined momentum. 5.1.

Jg = 1 +Je = 1 ONE-DIMENSIONAL VELOCITY-SELECTIVE COHERENT POPULATION TRAPPING

The simplest system for VSCPT is the A system of fig. 21a composed of the Ig-l), Ig+l), and 10.) ground and excited Zeeman sublevels of the

2313,

0

-

+1

2hk

-

I

I

I

I

I

-1.0

-0.5

0.0

0.5

1.0

Fig. 21. VSCPT on the A system of the 2 3 S -Z3Pl ~ 4He transition. (a) The ground sublevels IJ,=I, m J = - I ) and IJ,=l, m J = l ) are connected to the excited state IJ,=1, m J = O ) by counterpropagating a+ and u- laser beams. The opposite Clebsch-Gordan coefficient on the u' and o- transitions are shown; (b) measured position density profile of the atomic distribution at the detector, as produced by one-dimensional VSCPT in 4He with 0= 0.3 ms and SZR = 0.6 ro (from Bardou, Saubamea, Lawall, Shimizu, Emile, Westbrook, Aspect and Cohen-Tannoudji [ 19941). The distance between the two peaks corresponds to 2 hk; from the width of each peak the temperature was estimated at T = TR/20 x 200 nK.

J , = 1 +J, = 1 transition and excited by o+ and o- circularly polarized electric fields. The atomic Hamiltonian of eq. (2.1) should be modified to include the kinetic energy for an atom moving along the z axis with momentum p z . If the energy of the degenerate Ig-1) and Ig+l)ground states is assumed equal to zero, for an atom of mass M the Hamiltonian is: (5.1)

The kinetic energy describing the motion along the x and y axes does not appear explicitly in the atomic evolution, but a trace over the momenta along those axes should be performed. Coherent population trapping with velocity selection is obtained when the traveling electric fields acting on the two arms of the A system are counterpropagating along the z axis, whereas it was noticed in $2.7 that the configuration of copropagating laser fields generates coherent population trapping in all the velocity classes of a Doppler-broadened medium. If the counterpropagating laser fields of frequency w~and wavenumber k are supposed

312

COHERENT POPULATION TRAPPING

[Y

§ 5

to have the same amplitude EL and phase $L, the interaction Hamiltonian of eq. (2.4) may be written: Q R

VAL = - {-exp[-i(wLt 2

where the Rabi frequency

52R

-

kz + qh)]leo) (g-11

(5.2a)

is given by (5.2b)

With ,u,,~-~ = -peOg+, because of the opposite Clebsch-Gordan coefficients on the two components of the J, = 1 +J, = 1 transition, as shown in fig. 21a. The atom-

laser interaction of the atom with each counterpropagating laser beam modifies by k h k the atomic momentum along the z axis. If the atomic momentum basis la,pz) with a E { leo) , Ig-1) , Ig+l)}is used, in order to describe this change in the atomic momentum taking place along the z axis, the VAL interaction may be written:

(5.3) Equation (5.3) shows that by absorption and stimulated emission, the interaction VAL couples only the following three states:

with q the atomic momentum along the z axis. As long as spontaneous emission is not taken into account, these states form a closed set to be defined as the F(q) family, where the q label denotes the atomic momentum in the excited state belonging to a given family. The basic role of spontaneous emission is to produce a redistribution among the different families. In effect, in a spontaneous emission process, an atom in the excited state leo,q) of the F(q) family emits a fluorescence photon directed arbitrarily in space, so that the atomic momentum q changes by any value between -hk and hk. The main role of spontaneous emission is to produce a diffhion in the momentum space leading the atoms to the specific velocity where accumulation takes place.

v, P

51

LASER COOLING

313

A clear insight into the VSCPT mechanism is obtained by using the basis of couplednoncoupled states. Within the 3 ( q ) family the following couplednoncoupled states are formed:

(5.5)

where the properties of eq. (2.17a) still apply; i.e., no transition element of the interaction Hamiltonian VAL exists between leo(q)) and INC(q)). It may be noticed that in eq. (5.5) the noncoupled state is the symmetric linear combination of ground states, whereas in eq. (2.16) the noncoupled state is associated with the antisymmetric combination. This change arises from the opposite sign of the Rabi frequencies on the two optical transitions of fig. 21a. At this point the p,2/2M kinetic energy term in the 7-lo Hamiltonian plays a key role. Although the Ig*l, q f h k ) are eigenstates of the kinetic energy operator, INC(q)) and IC(q)) are not, and the kinetic energy operator has a matrix element between them:

In the schematic representation of fig. 4 for coupling and effective loss rates of the states, the kinetic energy matrix element of eq. (5.6) becomes the most important coupling between the noncoupled and coupled states if 8~ = 0 and the ground-state relaxation processes are not present. Equation (5.6) shows that the state INC(q)), noncoupled with respect to the absorption of laser radiation, is not stationary for the kinetic energy evolution unless the q value is equal to zero. The state INC(0)) is a perfect trap state because it is stable against the atom-laser interaction and also against kinetic energy coupling. The optical Bloch equations for the density matrix of eqs. (2.5) and (2.7) have to be modified by using the Hamiltonians of eqs. (5.1) and (5.2a). Moreover, for the relaxation produced by the spontaneous emission, which at this stage is supposed to be the only relaxation process, it should be taken into account that the repopulation of the ground states occurs with a diffusion in the momentum space. As a consequence, the generalized optical Bloch equations required to describe the VSCPT process are nonlocal in the momentum space (Aspect, Arimondo, Kaiser, Vansteenkiste and Cohen-Tannoudji [ 19891, Castin, Wallis and Dalibard [ 19891).

314

COHERENT POPULATION TRAPPING

In the limit of q < qo =Aka, with the parameter a given by

and the recoil frequency

WR

defined by

the f ” ( q ) loss rate for the INC(q)) state results in: (5.9) Because f ” ( q ) is the probability per unit time that an atom will leave the state INC(q)), for an interaction time 0, only atoms with r”(q)O < 1 remain trapped in the noncoupled state, and that condition is satisfied by an interval 6q of atomic momenta such that:

(2)’<%G a

(5.10)

The trap state INC(0)) is filled up by the spontaneous emission diffusion in the momentum space discussed above. A significant fraction of the atoms may be trapped in the atomic momentum region defined by eq. (5.10) in a time OUR=a,which defines the VCSPT time scale (Korsunsky, Kosachiov, Matisov, Rozhdestvensky, Windholz and Neureiter [ 19931, Bardou, Bouchaud, Emile, Aspect and Cohen-Tannoudji [ 19941). The trap state INC(0)) superposition has the following expression: INC(0))

1

=

- (Ig-1, -hk)

&

+ k+l,hk))

1

(5.1 1)

and its spatial dependence is given by (zlNC(0)) = exp(+ikz) Ig-1) + exp(-ikz) / g + l ) .

(5.12)

A measurement of the atomic momentum on the INC(0)) states givesp, = H z k . The momentum distribution of the atoms cooled by VSCPT presents two peaks at *Ak with a width given by eq. (5. lo), as in the experiments. VSCPT experiments have been performed on 4He metastable atoms using the transition Z3S, + Z3Pl,

v, I 51

LASER COOLING

315

where the recoil temperature T R ,defined as ~ B T R= ~/ W ~ Ris, 4 pK (Aspect, Arimondo, Kaiser, Vansteenkiste and Cohen-Tannoudji [ 19881, Bardou, Saubamea, Lawall, Shimizu, Emile, Westbrook, Aspect and Cohen-Tannoudji [ 19941, Doery, Widmer, Bellanca, Buell, Bergeman, Metcalf and Vredenbregt [ 19951). Figure 21b illustrates the atomic distribution measured by Bardou, Saubamea, Lawall, Shimizu, Emile, Westbrook, Aspect and Cohen-Tannoudji [ 19941 in an experiment on helium atoms precooled by sub-Doppler laser-cooling techniques, in order to realize a very long interaction time 0 with the laser radiation and a very narrow final distribution of the atomic momenta. Converting the final momentum distribution into an effective temperature by k B TI2 = (6q)212M, the authors associated an effective temperature of 200nK with the data of fig. 21b. The atomic momentum distributions of the J, = 1 +J, = 1 VSCPT cooling are well described theoretically through the solution of the generalized optical Bloch equations (Aspect, Arimondo, Kaiser, Vansteenkiste and Cohen-Tannoudji [ 19891, Arimondo [ 19921). Doery, Widmer, Bellanca, Buell, Bergeman, Metcalf and Vredenbregt [ 19951 have analyzed the atomic momentum distribution through the quantum-mechanical eigenstates of the total ground-state Hamiltonian, using both analytical expressions and numerical calculations. At very long interaction times 0 the two peaks of fig. 21b become more narrow, as indicated by eq. (5.10), but a question arises whether the number of cooled atoms keeps increasing or starts to decrease. Bardou, Bouchaud, Emile, Aspect and Cohen-Tannoudji [ 19941 pointed out the very peculiar nonstationary behavior, with a break of the ergodicity, presented by the VSCPT at interaction times longer than a/wR. In fact, only atoms in the (NC(q = 0)) state, a set of zero measure, are perfectly trapped. In contrast, at any interaction time, however long, the atoms in INC(g = 0)) states have a finite probability to escape from the noncoupled state and to start a diffusion in the momentum space. Because the atomic evolution is dominated by a random sequence of those rare events, the VSCPT is described through a regime, denoted as the Levy flight regime, different from the Brownian motion distribution of other cooling schemes. The authors have examined the VSCPT efficiency through the Levy flight statistical analysis. An alternative approach for the VSCPT at long interaction times is based on quantum Monte Carlo simulations, as introduced by Cohen-Tannoudji, Bardou and Aspect [1992]. Figure 22 shows results from a Monte Carlo simulation for the fraction of trapped atoms (defined as those atoms in the noncoupled states with 1Aq1 < hk/8) versus the interaction time 0, for parameters similar to those of fig. 2 1b. It may be observed that at interaction times much longer than @OR = a, the fraction of trapped atoms decreases. The above analysis can be transferred in toto to another laser configuration

316

[Y P 5

COHERENT POPULATION TRAPPING

0.0

I ' 1

10

100

1000

10000

100000

Fig. 22. Fraction of the trapped atoms, defined as 16ql <0.25Ak, versus the interaction time OWRcalculated from a quantum Monte Carlo analysis for one-dimensional VSCPT 4He with RR= 0.333 To,and a = 1. The vertical bar represents the uncertainties originating in the quantum Monte Carlo analysis.

with a relevant role in laser cooling, the so-called lin Ilin configuration. This is composed of two counterpropagating traveling laser fields, linearly polarized along orthogonal axes, the x and y axes. In fact, the analysis of the Jg= 1 +J, = 1 transition on the linear atomic basis of fig. 7b has shown that the { Ig,) , lez) , Igy)} states give rise to a three-level A system that can be excited by two laser fields linearly polarized along they and x axes, respectively. VSCPT can then be realized with excitation by counterpropagating lin Ilin laser fields, as reported in the 4He experiment by Aspect, Arimondo, Kaiser, Vansteenkiste and Cohen-Tannoudji [1988] and discussed by Kaiser [ 19901. Alekseev and Krylova [1992a,b, 19931 developed an original approach to solving analytically the density matrix equations for the VSCPT with traveling counterpropagating waves. Through a Fourier transformation for the momentum dependence and a Laplace transformation for the time dependence of the density matrix elements, a high-order local differential equation was derived and solved. They obtained a Lorentzian form for the momentum atomic distribution that well interpreted the results of the numerical calculations by Aspect, Arimondo, Kaiser, Vansteenkiste and Cohen-Tannoudji [ 19891 and Arimondo [ 19921 for times 0 6 a/wR. Thus, that analytical form could not reproduce the long-term Levy flight regime. Alekseev and Krylova [1992b,1993], Mauri [1990], and Arimondo [ 19921 have derived, through a simple model with atomic momentum diffusion and a trap state, a rate equation that could analyze the VSCPT at intermediate interaction times. The Jg= 1 -, J, = 1 VSCPT in one dimension has been theoretically extended to the laser configuration with two circularly polarized standing-wave electromagnetic fields acting on the two transitions of the A system (Hemmer,

v, 9; 51

LASER COOLING

317

Prentiss, Shahriar and Bigelow [ 19921, Shahriar, Hemmer, Prentiss, Chu, Katz and Bigelow [ 19931, Shahriar, Hemmer, Prentiss, Marte, Mervis, Katz, Bigelow and Cai [ 19931, Marte, Dum, TaYeb, Zoller, Shahriar and Prentiss [ 19941, Weidemiiller, Esslinger, Ol’shanii, Hemmerich and Hansch [ 19941). The standingwave electric fields are written as EL cos (kz + $2112) a+[exp(iwLlt) + c.c.112 and EL cos (kz - $21/2)a-[exp(iw~lt) + c.c.]/2, where a+ and a- represent the unit vectors for the circular polarizations. For simplicity the amplitudes of the two waves have been assumed equal. An important role in this laser configuration is played by the relative phase difference $21. For instance, at $21 = n / 4 the standing-wave configuration is equivalent to the laser configuration of lin Ilin counterpropagating waves, where VSCPT produces a noncoupled state similar to that of eq. (5.12) but a combination of Jg,) and ( g y )states. For the standingwave configuration the VAL interaction may be written:

For this atom-laser interaction, the following noncoupled state exists: INC(q)) = f [ e x p ( - i ~ d 2 )(Ig-1, q - fik) + lg-1, q + f i k ) )

as can be verified by applying the VAL of eq. (5.13) to this state. For the INC(q)) state the evolution under the kinetic energy Hamiltonian leads to a stationary state only for q = 0. Thus, the standing-wave laser configuration leads to a noncoupled state with atomic momentum components at &hk, similar to those of the counterpropagating traveling-wave configuration. However, in the coordinate space the INC(0)) trap state of the standing-wave configuration results:

(5.15) with a spatial dependence different from eq. (5.12). The analysis of the fraction of trapped atoms for the standing-wave configuration is more difficult than for the traveling-wave configuration, because, due to the form of the atom-laser interaction of eq. (5.13), each of the ground states is connected to two excited states. Therefore, it is not possible to find closed families of states. For the

318

COHERENT POPULATION TRAPPING

[v, I 5

values #21= 0 and $21 = x of the relative phase of the two standing waves, the VSCPT mechanism does not act any more; for those phases the two laser fields produce a standing-wave field linearly polarized along the y and x axes, respectively. The linear atomic basis of fig. 7b shows that a linearly polarized laser, acting on the J, = 1 +Je = 1 transition, produces a depopulation pumping into a dark state without any velocity selection. The real advantage of the standing-wave configuration is the presence of a force produced by the laser cooling polarization gradient mechanism (Shahriar, Hemmer, Prentiss, Marte, Mervis, Katz, Bigelow and Cai [1993], Marte, Dum, TaYeb, Zoller, Shahriar and Prentiss [1994]). The role of the polarization gradient is twofold: to produce a precooling of the initial atomic momentum distribution at short atom-laser interaction times, and to provide a confinement of the atomic velocities in the long interaction time limit. In effect, the traveling-wave VSCPT suffers the drawbacks of relying on the spontaneous emission atomic diffusion only for reaching the trap state and on the occurrence of the Levy flight regime at interaction times longer than a/mR with a decrease in the fraction of trapped atoms. It is expected that for the standing-wave configuration at those long interaction times, the fraction of trapped atoms does not give the decrease shown in fig. 22. Marte, Dum, Tai’eb, Zoller, Shahriar and Prentiss [1994] have pointed out that it is not possible to satisfy simultaneously the requirements on laser detuning and Rabi frequency for efficient use of the polarization gradient force and for trapping the atoms in a narrow momentum distribution. Thus, they proposed alternating the laser configuration between standing-wave and traveling-wave in order to take good advantage of both schemes. Numerical calculations confirmed the advantages obtained through this alternate operation, but in an experiment the preservation of the ground-state phase coherence in the operations could be difficult to realize. The use of a Doppler-cooling force to increase the laser cooling efficiency on A schemes with k l f k2 or To+l f ro+2 has been discussed by Korsunsky, Kosachiov, Matisov, Rozhdestvensky, Windholz and Neureiter [ 19931. Much theoretical work has concentrated on the use of a semiclassical theory, based on the Fokker-Planck equation, to describe the atomic velocity distribution function in the VSCPT (Minogin and Rozhdestvensky [ 19851, Minogin, Ol’shanii and Shulga [1989], Gornyi and Matisov [1989], Minogin, Ol’shanii, Rozhdestvensky and Yakobson [ 19901, Korsunsky, Kosachiov, Matisov, Rozhdestvensky, Windholz and Neureiter [ 19931, Korsunsky, Snegiriov, Gordienko, Matisov and Windholz [ 19941). The Fokker-Planck equation is a powerful approach widely used for describing laser cooling (Minogin and Letokhov [1986]), but it is based on the hypothesis that the atomic velocity

v, 51

LASER COOLING

319

distribution is wider that the photon momentum, a hypothesis not satisfied in the subrecoil regime reached by VSCPT. Thus, the Fokker-Planck equation can be used to describe the three-level A system laser-cooling only in the regime above the recoil limit. However, as pointed out by Korsunsky, Kosachiov, Matisov, Rozhdestvensky, Windholz and Neureiter [ 19931, some general conclusions about the dynamics of laser cooling based on coherent population trapping, which can be drawn from the semiclassical theory, ought to be valid in all the regimes of laser cooling, including the subrecoil VSCPT. The Fokker-Planck equation, in Rozhdestvensky and Yakobson [ 19911, Kosachiov, Matisov and Rozhdestvensky [ 1992d], Korsunsky, Kosachiov, Matisov, Rozhdestvensky, Windholz and Neureiter [ 19931, Korsunsky, Snegiriov, Gordienko, Matisov and Windholz [ 19941, and the Fourier-Laplace transformation of the density matrix equations in Matisov, Korsunsky, Gordienko and Windholz [1994], have been used to investigate the modification in the coherent population-trapping laser cooling produced by the presence of a relaxation mechanism for the ground-state coherence not included in the evolution of the couplednoncoupled states reported above. As in $2.3 for the decrease in coherent population trapping produced by the relaxation of the ground-state coherence, the VSCPT efficiency also decreases because of that relaxation. The VSCPT limit associated with the presence of the r12 relaxation rate has been derived in Arimondo [1994]. The loss rate of the INC(qtmp)) state with the maximum qtrapmomentum should be equal to r 1 2 :

and using eq. (5.9) the minimum VSCPT temperature Tmin produced in the presence of coherence relaxation is 2 qtrap

-

522,r12

kgTmin = - - 4h-. M TO WR

(5.16b)

With the typical values of coherence relaxation rate r12, this limit temperature is smaller than that reached in Doppler or polarization-gradient cooling. Kosachiov, Matisov and Rozhdestvensky [ 1992c, 19931 examined the laser cooling associated with coherent population trapping in the configuration known as the double-A scheme, where both ground states 11) and 12) are connected by dipole transitions to two different excited states, a configuration extensively introduced within the context of lasing without inversion. They found that the efficiency of the laser-cooling process is strongly affected by the relative phase of the ground-state coherences formed by the two separate A schemes. Matisov,

320

COHERENT POPULATION TRAPPING

[v, § 5

Gordienko, Korsunsky and Windholz [ 19951 have investigated through different approaches the noncoupled states formed in the atomic configuration of three levels in cascade, obtaining some results that appear promising for extending VSCPT to highly excited states. Gornyi, Matisov and Rozhdestvensky [ 1989b], Korsunsky, Matisov and Rozhdestvensky [ 19921, and Kosachiov, Matisov and Rozhdestvensky [ 19 9 2 ~ 1considered laser cooling associated with coherent population trapping for A- or cascade schemes where one transition is in the optical region and the other one in the radiofrequency or microwave region, so that the photon momenta for the two transitions are largely different. Thus, VSCPT can be applied to a wider class of atomic systems. Goldstein, Pax, Schernthanner, Taylor and Meystre [19951 considered the influence of the dipole-dipole interaction between ground and excited atoms on the J = 1 +J = 1 VSCPT in one dimension. The main result is that, although the noncoupled state survives the inclusion of the dipole-dipole interactions, the presence of this interaction significantly slows down the cooling process, without affecting the final temperature. 5.2. H I G H J ONE-DIMENSIONAL VELOCITY-SELECTIVE COHERENT POPULATION TRAPPING

A one-dimensional generalization to atomic transitions of the type J +J and J +J - 1 has been reported by Mauri, Papoff and Arimondo [1991] and Papoff, Mauri and Arimondo [ 19921. Under irradiation with u+,u- counterpropagating traveling waves, a superposition of ground states can be formed which experiences interference in all the transitions to excited states. For instance, for the Jg= 2 +J, = 2 transition between hyperfine states of the "Rb resonance line, the M structure of Zeeman sublevels presented in fig. 7c can be extracted, where for each A scheme a destructive interference inhibits the absorption to an excited state. For that case, with equal amplitudes of the two traveling wave lasers, the following noncoupled superposition exists:

where the coefficients of the ground state wavefunctions are determined by the Clebsch-Gordan coefficients for the adjacent d , u - transitions. However, this noncoupled state, at variance with that of eq. (5.11) for the J = 1 4 J = 1 transition, does not present the nice feature of being an eigenstate of the kineticenergy operator for some q value. Even for q=O, the three wavefunctions of

v, I 51

LASER COOLING

32 1

eq. (5.17) present different kinetic energy values. That kinetic energy difference, 4hwR for a heavy mass atom, is a very small quantity, so that the INC(0)) state is a transient or leaky VSCPT state, i.e., not stable but with a long lifetime. This leaky VSCPT is not a very efficient way to realize VSCPT sub-recoil cooling. Ol’shanii [ 19911 has suggested, through application of the Stark effect, a way to transform the noncoupled leaky states into nonleaky ones. If the atomic Hamiltonian part is modified so that the atomic energy difference between the (go) and lgsz) states is exactly equal to 4h0R3,the three wavefunctions of the INC(0)) superposition of eq. (5.17) will have again the same energy by summing up the atomic and kinetic energies. As a consequence the INC(0)) state becomes an eigenstate of the total energy and a nonleaky VSCPT regime can be realized. Whereas the original suggestion was based on the atomic energy shift of NO,through the Stark effect of an applied dc electric field, more recently Ol’shanii [ 19941 and Foot, Wu, Arimondo and Morigi [ 19941 have shown that the same result could be obtained more simply by making use of the dynamic Stark shift produced by a separate nonresonant laser field. Figure 23 shows the theoretical prediction for the measured atomic momentum distribution P(pz) in the case of VSCPT with 0+,6 laser fields interacting with the resonant F = 2 4 F = 2 transition of 87Rb for an interaction time of 4 ms. The peaks at pt = 0, +2hk are produced by a momentum measurement on the I NC(0)) state of eq. (5.17). It may be noticed that when the ac Stark effect is present in order to produce nonleaky noncoupled states, the efficiency in the VSCPT preparation is greatly enhanced. Doery, Gupta, Bergeman, Metcalf and Vredenbregt [ 19951 have proposed another approach for thc preparation of a nonleaky VSCPT trap state, using a configuration similar to that of fig. 7c, but having asymmetrical arms in each A transition. Furthermore, detunings of the two lasers from the Raman resonance condition are essential. For realizing the asymmetric A scheme, the ground-state Zeeman levels m = 0 and m = A 2 should belong to two different hyperfine states, F = l and F = 2 , respectively, and the laser beams should not have the same frequency. For the 87Rb hyperfine states, supposing equal Rabi frequencies for the two circularly polarized counterpropagating beams, the following noncoupled combination shouId be considered: 1 INC(q, t ) ) = - [ IF = 2, mF = -2; q - 2hk) - e-i(BU-bLI)f IF = 1, mF = 0; q )

J5

+ IF=2,rnF=2;q+2hk)],

(5.18) in analogy to eq. (2.28b). The realization of a nonleaky state is based on the fact that, as indicated in fig. 4, the effective loss rate of the noncoupled state

322

COHERENT POPULATION TRAPPING

[v, § 5

Fig. 23. Theoretical evolution of the atomic momentum distribution P ( p , ) for onedimensional VSCPT on the s2s1/2F = 2 + 52P3/2 F = 2 87Rb transition with a nonleaky noncoupled state (continuous line), and with a leaky one (dashed line). Parameters S Z R ~= 9=0.2 ~TO at OWR= 100, i.e., 0 FZ 4 ms. The dotted line represents the initial atomic momentum distribution.

depends on both the kinetic energy and the Raman detuning. For suitable laser frequencies, the Raman detuning may compensate exactly the kinetic energy leakiness, so that a nonleaky state is produced. All these schemes have yet to be tested experimentally. 5.3. TWO- AND THREE-DIMENSIONAL VELOCITY-SELECTIVE COHERENT POPULATION TRAPPING

The generalization from 1D to higher dimensions was started by Aspect, Arimondo, Kaiser, Vansteenkiste and Cohen-Tannoudji [ 19891 with a laser configuration which gave rise to a trapping state but had poor efficiency in the pumping process. Mauri, Papoff and Arimondo [1991], Mauri and Arimondo [1991, 19921, and Arimondo [ 19921 have discussed configurations for VSCPT in two dimensions for the Jg = 1 --+ J, = 1 and Jg = 1 -+ Je = 0 transitions with an efficient filling of the trap states provided by the presence of Doppler- and polarization-gradient forces acting on the atoms, also calculating the VSCPT efficiency. Ol’shanii and Minogin [ 1991a,b, 19921, Taichenachev, Tumaikin, Ol’shanii and Yudin [ 19911 and Taichenachev, Tumaikin, Yudin and Ol’shanii [ 1992a,b] pointed out other schemes for generalizing VSCPT states in 2D and 3D on the basis of elegant formal relations. Their approach is based on a few

v, 5

51

LASER COOLING

323

formal statements: for a J = 1 state, as for a spin one particle, the spatial wavefunction is described by a vector state q ( r ) ; the transition amplitude induced by the VAL interaction with an electric field of amplitude EL(r), for the J, = 1 +J, = 1 case with vector states q,(r) and qe(r),in analogy to eq. (2.32a), is proportional to the following integral:

with a vector product between € ~ ( r and ) q,(r), and a scalar product with qe(r). If the following spatial dependence is imposed on the ground-state wavefunction: (5.20) with a being c-number, then vr c( r ) has a zero transition amplitude for the laser excitation and, thus, describes a noncoupled state. Moreover, the wave equation satisfied by the electric field amplitude E L ( r ) has the same formal structure as the Schrodinger equation with kinetic energy satisfied by the wavefunction qrc(r), with eigenvalue hwR. Thus, the laser spatial configuration determines the spatial and momentum characteristics of the noncoupled state. For the Jg = 1 4 Je = 0 case, the transition amplitude induced by the VAL interaction is proportional to the following integral, in analogy to eq. (2.32b): (5.21) so that the spatial vector of the noncoupled state is given by a vector product operation

q r ( r )= e

x

EL(r)

(5.22)

with e the electric field polarization. Also for this case qFC(r)is an eigenstate of the kinetic energy with eigenvalue hwR. Because from eqs. (5.20) and (5.22) the spatial distribution 1 qFC(r)I2of the VSCPT atomic wavefunction depends on I E L ( r ) 12, nonuniform polarization and nonuniform intensity produce a localization of the trapped atoms at the maxima of the electric field. Laser configurations with d , u - counterpropagating plane waves along two or three axes were examined in detail by Ol’shanii and Minogin [1991a,b, 19921: the momentum representation of the noncoupled state contained momentum components at the positions fZlk along two or three axes. For the

324

COHERENT POPULATION TRAPPING

[v, 0 5

Fig. 24. (a) Schematic diagram of the laser configuration for two-dimensional VSCPT, with counterpropagating lin Ilin laser beams along two directions; (b) solid circles represent the four atomic momentum components, at *-hk along the two axes, contributing to the noncoupled state, with interfering transitions to excited states (open circles). Circles represent families of states; the wavefunctions of the noncoupled state belong to several families; (c) image of the detected atomic position distribution for VSCPT in two dimensions in 4He, with experimental parameters: SZR =0.8 ro; 6~=0.5 ro;interaction time 0 = 0 . 5 m s . The momentum distribution consists of four peaks as in (b); the peak widths are evidence of a subrecoil two-dimensional VSCPT (from Lawall, Bardou, Saubamea, Shimizu, Leduc, Aspect and Cohen-Tannoudji [ 19941).

J, = 1 -+J, = 1 case, the lin Ilin configuration of counterpropagating lasers in two dimensions, examined by Arimondo [ 19921, is represented in fig. 24a. Using the selection rules for the linear polarization basis of fig. 7b, the corresponding noncoupled state is obtained as the superposition of lgi,q,,qy) (i=x, y , z) wavefunctions. This is represented schematically in fig. 24b, together with the interference channels in the excitation to upper states. For both 0',6 and lin Ilin configurations, the wavefunctions composing the' noncoupled state do not belong to closed families, so that the efficiency in the preparation of the noncoupled state has not yet been calculated. The configuration of counterpropagating o+,o- laser fields for VSCPT in two dimensions has been realized by Lawall, Bardou, Saubamea, Shimizu, Leduc, Aspect and Cohen-

V, § 61

ADIABATIC TRANSFER

325

Tannoudji [1994] for 4He atoms precooled in a magneto-optical trap using the same Jg= 1 +J, = 1 transition of VSCPT in one dimension. After precooling, the helium atoms interacted for 0.5 ms with the four counterpropagating laser beams and were prepared in an noncoupled state similar to that represented in fig. 24b. After the VSCPT cooling, the atoms were falling, due to gravity, onto a CCD camera that recorded an image of the atomic position distribution. Figure 24c shows an image from this camera, with the four spots produced by the detection of the wavefunctions composing the two-dimensional noncoupled state. From the measured width of the individual peaks, as on the vertical profile on the peak at the right of fig. 24c, a temperature around sixteen times smaller than the helium recoil limit was estimated. Examining VSCPT as a function of the laser parameters, the authors deduced the presence of nonspecified laser cooling forces that contribute to the efficiency in the filling of the noncoupled state. Combinations of linearly and circularly polarized laser fields were examined by Taichenachev, Tumaikin, Yudin and Ol’shanii [1992a,b] for VSCPT on the Jg = 312 +J, = 112 and Jg= 2 4 J, = 1 transitions. The idea of atomic states decoupled from the laser field because of quantum interferences was extended by Dum, Marte, Pellizzari and Zoller [ 19941 to cases in which the electric field amplitudes have a spatial dependence limited in space similar to that of a laser trap, so that the spatial atomic wavefunction also would present a confinement.

0

6. Adiabatic Transfer

The aim of adiabatic transfer is to transfer an atom or molecule from one lower level of the A scheme to the other one by using properly tailored laser pulses, with as large an efficiency as possible. The adiabatic transfer in threeand multilevel systems was investigated theoretically by Oreg, Hioe and Eberly [1984], and Carroll and Hioe [ 19881. They demonstrated that by using conditions for the time dependence of the Rabi frequencies &(t) and Q,,(t) defined as anti-intuitive, a complete transfer of population from level 11) to level 12) is realized. Adiabatic transfer is a consequence of one of the INC) properties already discussed. From eq. (2.18), INC) is an eigenstate of the K O+ VAL Hamiltonian with zero eigenvalue. If the Hamiltonian is modified adiabatically, so that the system remains in this state, the occupation of the INC) remains constant. Let us consider a sequence of laser pulses applied to the 11) + 10) and 12) + 10) transitions, with the time evolution of the electric field amplitudes

326

COHERENT POPULATION TRAPPING

tv, I 6

described by functions Q ~ l ( t ) and G R 2 ( t ) , so that INC(t)) assumes the following form:

with G given by eq. (2.16b). Adiabatic transfer is realized for a sequence of Rabi frequency time dependencies such as those shown in fig. 25a. If the laser acting on the 12) 10) transition is applied initially, the INC) state coincides with state 11) where the entire atomic or molecular system is supposed to be concentrated. If the laser acting on the 12) 10) transition is progressively switched off while the laser pulse acting on the 11) 10) transition is switched on (see fig. 25a), eq. (6.1) shows that at the end INC(t)) coincides with 12), so that in the adiabatic regime the system occupies state 12). The counterintuitive pulse sequence is based on a laser being applied to the second transition at the beginning, and a laser to the first transition at the end. The requirements on the validity of the adiabaticity condition are (Oreg, Hioe and Eberly [1984], Kuklinski, Gaubatz, Hioe and Bergmann [ 19891, Carroll and Hioe [ 19901, Band and Julienne [1991a], Shore, Bergmann, Oreg and Rosenwaks [ 19911, Marte, Zoller and Hall [ I99 11, Shore, Bergmann and Oreg [ 19921): -+

--f

-+

G R 1 , QR2

>> TO,

QR1,

QR2

1

>> 79

(6.2)

where T represents the time duration of the two laser pulses. If broad-band lasers are used for the excitation, the spontaneous emission damping rate in the first relation of eq. (6.2) should be replaced by the laser bandwidth, which is equivalent to a damping rate of the optical coherences, as in 42.8 (He, Kuhn, Schiemann and Bergmann [ 19901, Kuhn, Coulston, He, Schiemann and Bergmann [ 19921). The first experimental results of adiabatic transfer were obtained on a Naz beam with a transfer from the electronic ground and vibrationally excited level X'C; ( Y = 0 , J = 5 ) to another vibrational one X'C; ( Y = 5, J = 5) using a three-level A system with upper electronic excited level A'C; ( ~ = 7 , J = 6) (Gaubatz, Rudecki, Becker, Schiemann, Kulz and Bergmann [ 19881, Kuklinski, Gaubatz, Hioe and Bergmann [ 19891, Gaubatz, Rudecki, Schiemann and Bergmann [ 19901). The counterintuitive time-dependent pulse sequence was created using the time-dependent interaction of sodium molecules crossing C.W. laser beams with separate excitations by the two lasers at different positions along the beam axis. That separated excitation modified drastically the amount of the population transferred to the final state, such as for the realization of

v, § 61

327

ADIABATIC TRANSFER

-4

-2

0

5!

4

t

0120

-80 -10

DETUNING

0

10

80

4 /2n [MHz]

Fig. 25. (a) Schematic representation of the time dependence for the Rabi frequencies required for adiabatic transfer from level 11) to level (2); (b) expenmental results for the adiabatic transfer efficiency between the XI ( v = 0, J = 5 ) and X' 2; ( v = 5 , J = 5) levels of Naz molecules versus the laser 2 frequency, with laser 1 in resonance, and the temporal pulse sequence which maximizes the transfer; (c) experimental results for the fluorescence from excited state A'Z: ( v = 7, J = 6) involved in the adiabatic transfer showing the resonant decrease associated with coherent population trapping (from Gaubatz, Rudecki, Schiemann and Bergmann [1990]).

the proper pulsed laser sequence. Figure 25b shows experimental results from Gaubatz, Rudeclu, Schiemann and Bergmann [ 19901 for the transfer efficiency at laser fixed positions scanning the frequency of laser 2, with laser in resonance, with a maximum efficiency around 0.8. Figure 25c shows experimental results for the fluorescence from the upper excited state versus the laser 2 frequency. Corresponding to the maximum of the adiabatic transfer, a minimum was obtained in the excited-state population versus laser frequency, as evidence of coherent population trapping. A similar setup for adiabatic transfer was used by Liedenbaum, Stolte and Reuss [1989], and Dam, Oudejans and Reuss [1990] in a molecular beam of ethylene, C2H4, in a three-level cascade configuration. A C02 laser induced the transition gs(4,1,3) ---t v7(5,0,5), and from there a color center laser induced the transition to v7 +v9(5, 1,4). Schiemann, Kuhn, Steuenvald and Bergmann [ 19931 demonstrated a highly efficient and selective population transfer in NO2 molecules in the electronic ground X2II,/2 state from vibrational level v = O to level v = 6 , using pulsed lasers properly delayed to realize the counterintuitive sequence of fig. 25a. Sussman, Neuhauser and Neusser [1994] have realized the adiabatic transfer on a A system of the C6H6 molecule, with pulsed lasers in the counterintuitive time sequence, for preparation in a specific rotational state of a vibronic state of that polyatomic molecule. That transfer was detected through the decrease in the population of the excited state in the A system, which represents the coherent populationtrapping characteristics.

328

COHERENT POPULATION TRAPPING

N§6

The adiabatic transfer with a continuum as the upper level of the A scheme has been examined theoretically by Carroll and Hioe [ 19921. Coulston and Bergmann [ 19921 have examined the modifications in the process produced by the presence of additional levels close to 10)and 12), such as for the vibra-rotational manifolds of molecular states. Band and Julienne [1991b] have considered the adiabatic transfer in a four-level system and demonstrated that a significant population transfer can be achieved, even if a four-level system does not support coherent population-trapping states. The concept of adiabatic transfer has been extended by Marte, Zoller and Hall [ 19911 to the case of noncoupled entangled states involving atomic momentum, such as those involved in the VSCPT processes [see eqs. (5.5) and (5.17)]. For a three-level Jg= 1 +J, = 1 atom interacting with two counterpropagating d/o- laser beams, such that the linear superposition of eq. (5.5) describes the noncoupled state, the counterintuitive application of a d , u - sequence will transfer an atom from the initial lg-1, p - hk) state to the final Ig+l,p + hk) state with a modification of the atomic momentum by two times Ak. For a higher J transition, such as J, = 2 +J, =2 with the noncoupled state given by eq. (5.17), the adiabatic transfer produces a larger coherent and selective modification of the atomic momentum by 4hk. The adiabaticity conditions for the photon momentum transfer are those of eq. (6.2), and are quite easily satisfied. Theoretical analyses for alkali atoms have been performed by Weitz, Young and Chu [1994a], and Foot, Wu, Arimondo and Morigi [1994]. The adiabatic transfer between the entangled states of internal and momentum variables has been realized experimentally by different groups. Pillet, Valentin, Yuan and Yu [ 19931, Goldner, Gerz, Spreeuw, Rolston, Westbrook, Phillips, Marte and Zoller [ 1994a,b], and Valentin, Yu and Pillet [ 19941 have demonstrated, using the D2 excitation of cesium atoms precooled by sub-Doppler techniques, a momentum transfer up to 8hk between the extreme Zeeman sublevels of the F = 4 hyperfine level, with an efficiency up to 0.5. The off-resonant transitions to other excited states of the 32P3/2 manifold limited the efficiency, which should be larger using the D1 excitation to the 62P1/2 manifold where the hyperfine separation is larger. Lawall and Prentiss [19941 realized the adiabatic transfer on 4He metastable atoms in a beam using the 2 3 S 4 ~ 23P~ transition, with an efficiency of transfer up to 0.9 for a change of atomic momentum by 2hk. Using multiple interaction of the lasers with the atomic beam, Lawall and Prentiss [I9941 obtained momentum changes up to 6hk. An immediate application is in atomic interferometry with the adiabatic transfers used as atomic beam splitters. In fact, an atomic interferometerbased on adiabatic transfer between the cesium 6’S1/2 hyperfine states ( F = 3, T ~ = F 0) and IF = 4, m F = 0) using a

v, I 71

“LASING WITHOUT INVERSION”

329

u+,u- polarization configuration has been demonstrated by Weitz, Young and Chu

[1994b]. Using the D1 excitation, a coherent transfer of 140 photon momenta to cesium atoms with an efficiency of 0.95 per exchanged photon was reported.

8

7. “Lasing Without Inversion”

The idea of lasing without inversion was developed independently by two groups in Russia and the United States, Kocharovskaya and Khanin [1988] and Scully, Zhu and Gavrielides [1989], respectively. The aim is to produce a laser system where the population in the excited atomic, or molecular, state is smaller that the population in the lower state. The states without inversion are defined according to the atomic, or molecular, basis, in the absence of applied radiation fields. Much attention has been given to the subject, both theoretically and experimentally, due to the attractive possibility of converting a coherent low-frequency input into a coherent high-frequency output, without any requirement on population inversion between the highfrequency emitting levels. At the present stage, several mechanisms giving rise to the phenomenon of “lasing without inversion” have been identified, as seen in the reviews by Kocharovskaya [I9921 and Scully [1992]. One of those mechanisms has been based on coherent population trapping, and up to now the large majority of the experimental verifications of amplification without inversion is based on this phenomenon. In effect, this mechanism of “lasing without inversion” should be classified as an inversion in the hidden basis of the coupledhoncoupled states. This mechanism of “lasing without inversion” (or more precisely, amplification without inversion (AWI), because a cavity is required to convert an amplifier into a laser), can be easily analyzed when the transformation from the atomic basis { 11) , 12), 10)) to the coherenttrapping basis {INC) , IC) , 10)) is applied, as in fig. 26 (Kocharovskaya, Mauri and Arimondo [ 19911, Kocharovskaya, Mauri, Zambon and Arimondo [ 19921). For the coupledhoncoupled basis, the interaction of the atoms with the externally applied electric field acts only on the IC) and 10) states. In a system at thermal equilibrium, the states 11) and 12) are equally populated, as represented schematically in fig. 26a, and the excited state 10) contains a small population. The population of the IC) state may be transferred to the INC) state through one of the appropriate mechanisms for creating coherent population trapping already described in this review, or one of those to be discussed later in this section. Perfect coherent population trapping is realized with p c , = ~ 0, and an efficient trapping corresponds to ~ N C , N C>> p c , ~as , shown

330

COHERENT POPULATION TRAPPING

Fig. 26. (a) and (b) Energy levels for amplification without inversion for the A scheme, and (c) and (d) for the double-A scheme. In (a) and (c) the bare atomic basis is used, and in (b) and (d) the couplednoncoupled basis is used. The depopulation pumping between coupled and noncoupled levels is schematically represented by the transfer of the black dots.

in fig. 26b. In these conditions, if a population inversion is realized between the levels 10) and IC); i.e.,

an amplification of the radiation on the 10) + IC) transition could be obtained. This amplification is produced by an inversion between states (0) and IC) in the atomic basis of couplednoncoupled states, but in the atomic basis of bare states { / I ) , 12), 10)) no population inversion exists, because p1,l + p2,2 = p c , +~PNC,NC >> p o , ~ !This simple presentation exemplifies the concept of amplification in a hidden basis, the basis of the couplednoncoupled states. The possibility of obtaining gain in a A system on the condition of coherent population trapping is also understood by examining the plot of Im(pol), proportional to the absorption coefficient, as shown in fig. 2c, with the very narrow peak at the center which has been defined as electromagnetic-induced transparency (Kocharovskaya [ 19921, Scully [ 19921). By pumping a small amount of the population into the excited state lo), a contribution with opposite sign is added to the absorption coefficient, which could bring that peak above the horizontal axis and create a condition of amplification. If the condition p0,o >>pc,c corresponds to AWI, the operation of a laser requires three levels inside a cavity with the gain larger than the cavity losses. It should be noted that the amplification on the couplednoncoupled basis corresponds to an amplification of a bichromatic field, i.e., two electromagnetic field waves with frequencies W L and ~ W L ~ equal , to the two transitions (0) -+ 1 I ) and 10) -+ 12) in the bare atomic basis. If the two frequencies have the same cavity loss K , and the two optical transitions of the three-level system have

v, P

71

“LASING WITHOUT INVERSION

331

the same absorption coefficient a, the condition for lasing is a straightforward application of the condition for lasing based on a two-level system: Po0

- pc,c

2K

3 -, aL

where L is the length of the cavity, supposed to be filled uniformly with the three-level medium, presenting amplification without inversion. The condition of amplification, or more precisely of no absorption, is valid only with respect to the INC), IC) states of eqs. (2.16), as determined by the process that has prepared those states. In order to realize amplification of a bichromatic electromagnetic field, composed of two different components ELIexp[-i(oLl ~+QLI)] and &~2exp[-i(u~2t+Q~~)], the INC) state should really be noncoupled for that amplified field. As a consequence, the atomic amplitudes and the amplitudes of the electric field components to be amplified by the noninverted medium, should satisfy a matching condition. If the 11) and 12) levels are degenerate in energy, the amplified field has only one frequency component, and the separation between the two modes of the electric field originates from polarization selection rules. On the contrary, if 11) and 12) are separated in energy, the amplified bichromatic field has components at two different frequencies O L ~and 0 ~ 2 In . this scheme, defined by Fill, Scully and Zhu [1990] as a quantum beat laser, the beat frequency WLI - W L should ~ match the evolution frequency of the ground state coherence. Amplification of a bichromatic field also imposes a phase matching condition: in a laser cavity, neglecting cavity losses and frequency pulling with respect to the interaction with the three-level system, the laser field relative phase should be opposite to that of the groundstate coherence. Fill, Scully and Zhu [1990] have derived, for different schemes of creation of the coherent population trapping, the phase-matching equation to be satisfied by the relative phase Q L -~ 4 ~ 2 of the two lasers. The relation (2.20) between the density matrix elements in the coupled noncoupled basis and in the bare atomic basis shows, that in order to have a small value of p c , ~ the , occupation of the noncoupled state, the atomic coherence Jp12I should be large. The different applications of coherent population trapping for lasing without inversion, both theoretically and experimentally, are really connected to the differences in the preparation of a large atomic coherence p12 with a small occupation of the IC) state and a large occupation of the INC) state. An efficient process for the realization of amplification without inversion is based on the double-A scheme, as shown in fig. 26c, with the two lower levels, 11) and 12) connected by dipole transitions to the upper levels 10) and 10‘) (Fill, Scully and Zhu [1990], Kocharovskaya, Li and

332

COHERENT POPULATION TRAPPING

[v, 9: 7

Mandel [19901, Kocharovskaya and Mandel [ 19901, Khanin and Kocharovskaya [ 19901, Kocharovskaya, Mauri and Arimondo [ 19911, Kocharovskaya, Mauri, Zambon and Arimondo [ 19921). A pump bichromatic laser field, with amplitudes €[,exp[-i&] and €[2exp[-i@!2] resonant with the 11) + 1 0 ' ) and 12) + 10') transitions to the pumping level lo'), prepares the coherent trapping superposition in the ground state, so that an amplification of a bichromatic field from state lo), with amplitudes ELlexp(-iq+l) and E~zexp(-i@~z), takes place. The preparation of the ICp)and INCP)states, shown in fig. 26d, takes place through the depopulation pumping process, as discussed in 5 2.3, with coupled and noncoupled states given by: INC')

l2 +

J/PopIf:l

Ic')

(porzEL2e-ieL2 11) - p o r l ~ ~ l e - i ~12)) El ,

1

=

lP0p2~:212

1

=

J/POYIEL

I2

+ IPOP2E:2I2

(pop,

e-@l 1 I ) + por ~ ~ ~ e p ~12)) e i o. :

(7.3) However, these coupled and noncoupled states should coincide with those on the transitions to the 10) state, so that the following self-consistent condition between the amplitudes of all the fields applied to the double-A system should be satisfied:

Kocharovskaya and Mandel [ 19901 have derived the conditions for the realization of steady-state AWI in the double-A scheme taking into account the simultaneous interaction of the four-level system with the two pairs of bichromatic fields. The important condition to be satisfied for the realization of this AWI was that the population of the lasing level 10) should be larger than that of the pumping level 10'). More precisely, the following condition results:

This relation states than in order to obtain AWI, a population inversion should be realized between the two upper levels, but of course no population inversion is required between these levels and the ground ones. The relation derives from the competition in the creation of coherent population trapping between the two separate A schemes of the double A. In fact, the population in the

v, D 71

“LASING WITHOIJT INVERSION’

333

10’) state contributes through spontaneous emission or one-photon processes to the pumping of the population in the coupled state IC),whose presence decreases the amplification and increases the threshold of amplification without inversion. In the case of generation of short-wavelength radiation by pumping with a longer wavelength laser, level 10) is higher in energy than level lo‘), and in thermodynamic equilibrium the population of the 10) top level is smaller than that of the 10‘) intermediate level. Thus, the above threshold condition (7.5) cannot be satisfied without external pumping. The above discussion points out the close equivalence between the doubleA scheme and a four-level laser. In fact, optical pumping in a four-level system represents an alternative way to realize amplification on the same doubleA scheme without creation of coherences. For instance, on the same level structure of fig. 26c, pumping from the 12) state to the 10’) state followed by spontaneous emission down to the 11) level could produce an inversion between 10) and 12). The threshold condition required for amplification on this optical pumping scheme is exactly equivalent to those for AWI in the double A (Kocharovskaya, Mauri, Zambon and Arimondo [ 19921, Fleischhauer and Scully [1994]). These last authors also pointed out that schemes combining optical pumping with the creation of coherences could produce a further reduction of the threshold. An alternative way to create a lower-state coherence is through the application of a microwave field resonant with the lower-state splitting, as shown schematically in fig. 27a (Scully, Zhu and Gavrielides [1989], Fill, Scully and Zhu [1990], Khanin and Kocharovskaya [ 19901). The generated lower-state coherence can be expressed through couplednoncoupled states. Again, for a population of the 10) level larger than that of the coupled state, amplification takes place, with inversion in the basis of the couplednoncoupled states. The main difference between the double-A scheme and the microwave field is that in the double-A scheme the depopulation pumping of the coupled state leads to the preparation of a pure density matrix state, i.e., with all the atoms prepared in the noncoupled state. For microwave-generated coherence, starting from the nonpure state of the thermal occupation of ground states, the application of a microwave-coherent field cannot produce a pure state. As a consequence, the gain in the microwave case is always smaller than in the double-A scheme. The thermal nonpure occupation of the ground states leads to the following condition for AWI: poo 3 min(pll, p22) (Mandel and Kocharovskaya [1993]). Another scheme for the realization of amplification without inversion, actually the first one proposed by Kocharovskaya and Khanin [ 19881 and examined later by Fleischhauer, Keitel, Scully and Su [1992], is based on the application to the

334

&*;I

COHERENT POPULATION TRAPPING

12 I1

Pulse

Fig. 27. Additional schemes for amplification without inversion based on coherent population trapping: (a) preparation of the coherent trapping superposition through a microwave field and amplification of the bichromatic field; (b) coherent trapping superposition formed by an ultrashort laser pulse with duration tp;(c) four-level scheme with pump laser to the level 10') tuned halfway between the ground levels with amplified laser also tuned halfivay between ground levels.

three-level system of a short laser pulse, as in fig. 27b and as for the experiments described in $3.3. In order for the short pulse with temporal duration zp to interact with both the 0 0 1 and 0 0 2 optical transitions and to probe the lowfrequency coherence p21, the relation llz, >> 0 2 1 must be satisfied. As usual, the gain is based on the preparation of a small population in the upper 10) level, with population in the lower states lying in the INC) state. The INC) occupation should have been realized before the pulse arrival, by applying a microwave field, using the double-A scheme, or by applying a train of pulses as described in 9 3.3. The last scheme involving coherent population trapping, proposed by Narducci, Doss, Ru, Scully, Zhu and Keitel [1991], is based on the combination of the double-A scheme and the dressed-state approach of $2.4 and fig. 5a: a coherent population-trapping preparation is performed on one A system and amplification is achieved on the second A system, as shown in fig. 27c. Only one laser is required for the preparation stage, and only one laser is used for the amplification process, both lasers being tuned at the center frequency between the two groundstate levels. The scheme of fig. 27c was tested in the first experiment of inversionless amplification performed on sodium atoms by Gao, Guo, Guo, Jin, Wang, Zhao, Zhang, Jiang, Wang and Jiang [ 19921, with lower levels being the hyperfine states of 3*S1/2 ground level, 3'P3/2 as the 10) amplification state, and 32P1/2 as the 10') preparation state. In the experiment a discharge through the heliudargon buffer gas prepared the required small occupation in the excited sodium states, and a strong pulsed laser on the { [ I ) , 12)) + 10') transitions produced the

v, P

71

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335

coherent trapping superposition. The amplification of a C.W.dye laser on the 10) + { 11) , 12)) transitions was monitored, through a boxcar detector, during the application of the pulsed laser. These authors have reported AWI, and Gao, Zhang, Cui, Guo, Jiang, Wang, Jin and Li [1994] have measured the excitedstate population through the absorption of a second C.W. dye laser from the excited state in order to verify that no population inversion in the bare states was created by the strong preparation pulse. The positive result of that experiment has generated some discussion in the lasing-without-inversion community: the possibility of a real population inversion between the excited state and ground states produced by the pulsed laser was ruled out in their direct absorption measurements. Because the experiment was performed in a transient regime, a theoretical analysis of the transient AWI was performed by Doss, Narducci, Scully and Gao [ 19931, proving that amplification is also reached in the transient regime. A later analysis by Meyer, Rathe, Graf, Zhu, Fry, Scully, Herling and Narducci [1994] showed that no coherence between the ground state hyperfine levels could have been created in that sodium experiment. Clearer evidence of AWI, based on the scheme of fig. 27c, was obtained by Kleinfeld and Streater [1994] in potassium atoms, using the ground 42S~/2hyperfine levels, 42P3/2 as the preparation state, and 42P1/2 as the amplifying state. Continuous lasers were used for both preparation and amplification, tuned at the center between the two ground hyperfine states separated by 462MHz. The upper-state population in the amplifying state was produced making use of the transfer from 42P3/2 to 42Pl/2 in collisions between potassium and helium buffer gas. The experimental results were very similar to those predicted in the theoretical analysis by Narducci, Doss, Ru, Scully, Zhu and Keitel [1991], with some unexplained features of additional absorption dips just outside the gain peaks. In the experiments by Nottelman, Peters and Lange [1993] and by Lange, Nottelman and Peters [1994], a coherent population trapping in the ground state of a A scheme was created through a train of picosecond pulses, as shown in fig. 27b and as analyzed in $3.3. A second picosecond pulse probed, at different delay times, the amplification without inversion. Samarium atoms on the J, = 1 J, = 0 transition (as in the experiment by Parigger, Hannaford and Sandle [ 19861 discussed in $ 3.5), in the presence of an applied magnetic field along the z axis, were irradiated by a train of 30ps laser pulses, with electric field polarization along the y axis, a ground-state Hanle-effect configuration. When the matching condition of eq. (3.1) was satisfied, the picosecond pulsetrain pumped atoms out of the IJ,,C) = IJ, = 1,y) state and created the IJ,,NC) = IJg = 1,x) coherent superposition of states, as from the selection ----f

336

COHERENT POPULATION TRAPPING

[v, 9 7

rules of eq. (2.32b). However, in the presence of a magnetic field B, that superposition is not an eigenstate and the atomic wavefunction experiences a time evolution. Because of the energy separation w21=2gp.~Blhbetween the IJ,, mJ = 1) and IJ,, mJ = -1) eigenstates of the atomic Hamiltonian, starting from a perfect atomic preparation at time t = 0 in the noncoupled state, the atomic wavefunction Iq,(t)) at time t is: W2I

Iqg(t)) = cos-

2

t

.

INC) -sin-

W2lt

2

IC)

From eq. (7.6) it can be seen that the absorption from the IC) part of lqg(t)) varies with the delay time of the probe pulse. At td = 1612021, the occupation of the coupled state is equal to one half the initial value; at td =.76/021 the occupation of the coupled state is equal to 1, and it is 0 at t d =2n/o21. In order to realize AWI, a third pulse, linearly polarized along the z axis, pumped a few atoms from the ground IJ, = 1, z ) state to the IJ, = 0) state, and the amplification between the excited (J, = 0) state and the IC) = IJ, = 1, y ) state was probed by the delayed pulse. Depending on the delay time, the coupled-state occupation produced different contributions, so that at a proper delay time an inversion between IJ, = 0) and IJg = I , C) could be realized with no population inversion in the Zeeman atomic basis. Actually, the experiment was operated slightly differently from that presented: at a fixed delay time of the probe, the tuning of the occupation of the coupled state was realized by varying the splitting ~ 2 through 1 an applied magnetic field B. Moreover, the maximum value of the generated ground-state coherence was only 0.14, so that the full occupation of the coupled or noncoupled states could not be realized. Finally, while the relatively long decay time of the ground-state coherence (- 15 ns) was beneficial for the experiment, the comparable decay time of the optical coherence (= 9 ns) implied that the atomic dispersion affected the pulse propagation, and the length of the samarium cell could not be increased. Thus, as stated by the authors, the measured amplification of 7% did not seem exciting, but was obtained with an optically thin sample. Another experiment by Fry, Li, Nikonov, Padmabandu, Scully, Smith, Tittel, Wang, Wilkinson and Zhu [1993] was based on the sodium D1 resonance line. A detailed analysis and presentation of the experimental results has been published in a series of four papers: Meyer, Rathe, Graf, Zhu, Fry, Scully, Herling and Narducci [ 19941, Nikonov, Rathe, Scully, Zhu, Fry, Li, Padmabandu and Fleischhauer [1994], Padmabandu, Li, Su, Fry, Nikonov, Zhu, Meyer and Scully [1994], and Graf, Arimondo, Fry, Nikonov, Padmabandu, Scully and Zhu [1995]. The level configuration involved in this experiment was based on two

v, I 71

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337

(5’ circularly polarized lasers exciting hyperfine components of the D I line and has already been presented in fig. 7f. The two laser beams, with linewidth M 30 MHz and frequency difference matching the 1.77 GHz ground hyperfine splitting, were generated through an acousto-optic frequency shifter. The first step in the experiment was to test the production of the coherent populationtrapping superposition by the bichromatic O+ radiation: one of the pumping lasers was switched off through a fast Q-switch and the transient absorption of the sodium atoms on the remaining pumping beam, as a consequence the destruction of the coherent trapping was monitored. The time evolution of the transmitted light was in good agreement with theoretical predictions. AWI was realized by pumping atoms to the excited F = 2 state from the ground F = 2, mF = 2 level not involved in the coherent trapping superposition, through application of a weak excitation up polarized light. As soon as the population inversion was established, an amplification of the bichromatic u+ radiation was observed. The amount of coherence established between the ground levels was not specified; however, in a theoretical analysis, which well reproduced the experimental results, a ground-state coherence around 0.1 G O . 12 was reported. The observed dependence of the coherent trapping superposition on the helium buffer gas has been discussed in 9 2.8. The last experiment in the coherent population-trapping application by van der Veer, van Dienst, Donszelmann and van Linden van den Heuvell [ 19931, operated on a cascade scheme based on the Il2Cd 5s2 ‘So 5s5p3P1-+ 5s6s3S1 levels, with transitions at wavelengths 326 nm and 308 nm. A longitudinal magnetic field B, in the mT range produced an energy splitting of the excited 3Pl state. Nanosecond-pulsed dye lasers, with frequency bandwidths in the GHz range to match the Doppler-broadening of the absorption lines, counterpropagated through a cadmium cell. The two lasers were linearly polarized, and the preparation, as well as the amplification processes are well understood in the level scheme based on the linearly polarized atomic basis of fig. 28a. Laser 1, linearly polarized along the x axis, excited the cadmium atoms from the I’S, J, = 0) state to the I3P1, x) state. Laser 2, linearly polarized along the y axis, transferred atoms to the I3S1, z) state. From there, amplification could be produced with emission towards the I3P1, y ) state. This interpretation of AWI comes out very naturally in the hidden basis, whereas in the Zeeman atomic basis the interpretation of amplification without inversion requires a careful analysis of the atomic coherences created in the intermediate 3PI state. The presence of amplification was tested by monitoring the gain of a seed laser transmitted through the cadmium cell, and a gain of 4.3 was measured. The amplification was monitored in two different regimes. In the first one, laser 2 --f

338

[Y §

COHERENT POPULATION TRAPPING

7

12,>

1

Ioy>

IS 0

o-l

p0,o - pc,c

0.0

0.2 0.1

8

(I

-4

.z

a

2

4

Magnetic field rtrenglh (mT)

8

I

~

-2

-1

0

1

2

Magnetic field strength (10mT)

Fig. 28. (a) Level scheme, in the linear atomic basis (see 5 2.6), for the "'Cd AWI experiment, and (b) measured AWI gain versus applied magnetic field, with a loss of gain at the larger magnetic field due to the absence of population inversion (from van der Veer, van Dienst, Donszelmann and van Linden van den Heuvell [1993]); (c) hidden basis population inversion in a s7Rb double-A scheme versus an applied magnetic field; spontaneous emission rate To = 0.24 r;, pumping rate 0.05 to excited state lo), Rabi frequencies Rf;] =RL=O.l2 r;, and interaction time for preparation of the couplednoncoupled state @ = 83 r; (adapted from Kocharovskaya, Mauri and Anmondo [1991]).

was delayed by 30ns with respect to laser 1, and when the amplification was measured versus the applied magnetic field, the periodic evolution between the coupled and noncoupled states could be monitored. In the second regime the two lasers produced simultaneous excitation to the top level of the cascade scheme, and the amplification was observed as a function of the magnetic field with

v, o 71

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339

experimental results reported in fig. 28b. Figure 28c reports the results of a theoretical analysis for the population inversion in the hidden basis, proportional to the gain, as a function of the applied magnetic field, derived for the doubleA scheme of 87Rb atoms in Kocharovskaya, Mauri and Arimondo [1991]. The strong similarity between the two figures evidences the common features of evolution between couplednoncoupled or lx)/ Iy) states. The use of the index of refraction has been considered in the context of amplification without inversion (Scully [ 1991, 19921, Fleischhauer, Keitel, Scully and Su [1992], Fleischhauer, Keitel, Scully, Su, Ulrich and Zhu [1992], Friedmann and Wilson-Gordon [1993]). It has already been noted (see fig. 2d and Q 3.8) that a large index of refraction can be generated in the conditions of coherent population trapping. The use of that large index of refraction could be inhibited by the absorption coefficient of the material, which is large outside of the Raman resonance (see fig. 2c). However, by preparing the three-level system with a small population in the upper 10) state, a contribution to the absorption with opposite sign could be created, which is really an amplification, so that a regime may be realized with the large index of refraction occurring at a laser frequency where the absorption coefficient is effectively equal to zero. Applications of the enhancement of the index of refraction, considered by Scully [1991], are to the realization of phase-matching in the laser acceleration of electrons, to the increase of the resolving power in a microscope, and to the development of a new class of magnetometers. The experiments by Schmidt, Hussein, Wynands and Meschede [1993, 19951 and Xiao, Li, Jin and GeaBanacloche [ 19951 on the index of refraction, as discussed in 9 3.8, have been performed with the aim of modifying the group velocity for propagation inside a medium pumped so that it gives rise to coherent population trapping. A group velocity ug = d l 2 8 0 has been reported by the first group of authors. Attention here has been concentrated on schemes where no population inversion exists in the basis of the bare atomic states, but a population inversion is found in the basis of couplednoncoupled states, or equivalently in the basis of dressed states. Other schemes of amplification without inversion have been identified where population inversion does not appear to occur in any basis (Kocharovskaya [ 19921). However, the transformation from the bare-atomic basis to the dressed-state basis transforms population differences into coherences, so that the gain can be associated with the creation of coherences (Aganval [1991a], Bhanu Prasad and Aganval [1991]). Even if the amplification cannot be described simply through a population inversion in an appropriate basis, the role of coherence population trapping cannot be excluded. For instance, in an asymmetrical A scheme with level 12) metastable and the Rabi frequency l 2 ~ 2

340

COHERENT POPULATION TRAPPING

[Y § 8

quite large, AWI can be realized on the 10) -+ 11) transition (Imamoglu, Field and Harris [ 199 11). For that process, Cohen-Tannoudji, Zambon and Arimondo [ 19931 pointed out the important role played by coherent population trapping. For 5 2 ~<< 1 Q R ~ , the ground state 11) coincides with the noncoupled state, so that even a small population in the (0) state may be able to produce amplification. Dowling and Bowden [ 19931 have considered AWI in a dense medium, where the near dipoledipole interactions modify the local microscopic electric field through volume polarization, as in the Loreni-Lorentz local field correction. This dipoledipole interaction also would affect the phenomenon of coherent population trapping in a dense medium, with a frequency shift and a distortion of the resonance lineshapes.

0

8. Coherences Created by Spontaneous Emission

The possibility of creating ground-state coherences in a A or V system was first considered by Aganval [ 19741, but has received more attention recently in the context of lasing without inversion (Imamoglu [ 19891, Javanainen [ 19921, Fleischhauer, Keitel, Narducci, Scully, Zhu and Zubairy [1992]). In general the role of spontaneous emission is to erase the coherences through destructive interferences from the vacuum modes contributing to the excited state decay. However, in some particular cases those interferences do not cancel completely and a coherence may even be created by the spontaneous emission process. The most relevant case is for the A system when the two optical transitions 10) + 11) and 10) + 12) are completely equivalent from the point of view of the electric dipole emission, which requires two transitions at the same frequency but also with the same angular momentum quantum numbers. Javanainen [ 19921 considered the decay from a IJe = 1, mJ = 0) level to two degenerate 11) and 12) ground levels, both of them with J, = 0 quantum numbers. It is quite unlikely that such a configuration will be found in atoms, but it should not be completely excluded for molecular levels. In the case of these degenerate levels being found, the spontaneous emission terms in eq. (2.8) should be modified because of constructive interference in the upper level decay to both ground states. A new term should appear in the evolution of the ground state coherence:

Thus, ground-state coherence would be created in the spontaneous emission decay from the 10) state. Javanainen [ 19921has pointed out that the superposition

v, 5 91

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341

principle requires the creation of that coherence. If the linear symmetric and antisymmetric combinations of ground states are considered, i.e., the coupled and noncoupled states, the upper state is dipole-connected to the coupled state only, and by spontaneous emission it will decay only to that coupled state. An atomic preparation of the coupled state through spontaneous emission implies that a coherence between the 11) and 12) ground states is formed. It is quite obvious that any experiment dealing with coherent population trapping will be affected dramatically by the presence of that coherence.

0

9. Pulse-Matching and Photon Statistics

The research on lasing without inversion has shown that the preparation of a coherent population-trapping superposition modifies the interaction with radiation. As a consequence, properly tailored atomic superpositions may produce particular properties of the electromagnetic fields interacting with those superpositions. This section presents the application of coherent population trapping for modifying radiation field properties. In the proposal of matched pulses by Harris [1993], a three-level system, supposed to be prepared in the INC) state, was probed by a bichromatic electromagnetic field, ELI(z, t) u1 cos ( W L I ~+ $1) and &(z, t) u2 cos ( 0 ~ 2 t+ qh), with time-varying envelope, applied to the two arms of a A system. Interacting with the INC) state, the Fourier components of the input electromagnetic field, which are matched in their frequency difference, amplitude, and phase so as to preserve the atomic preparation in the noncoupled state, do not experience any absorption. On the contrary, the nonmatched Fourier components experience an absorption and are attenuated in the propagation. As a consequence, after a characteristic propagation distance, the transmitted field contains only Fourier components matched in amplitude and phase to the noncoupled state, and those components do not experience any further absorption. This concept of matched pulses has a strict connection with the observation by Dalton and Knight [ 1982a,b], reported in $2.8, that critical cross-correlated fields acting on the two arms of the A system may preserve the coherent population-trapping preparation. The idea of pulse matching has been formalized by the introduction of the normal modes by Harris [I9941 and dressed field modes by Eberly, Pons and Haq [1994]. In effect, as the atomic time evolution is greatly simplified by using the INC) state, for which eq. (2.18) expresses a time-independent evolution, a similar relation may be written for a combination of the electric field ELI(z, t) and & L ~ ( t) z , components propagating through the three-level medium. The matched-

342

COHERENT POPULATION TRAPPING

[v, 5 9

or dressed field combinations are linear combinations of the slowly varying envelopes €L,(z, t ) and € L ~ ( zt), , and can be derived from the Maxwell equations for propagation through the three-level system. The dressed fields represent a more general concept. If the atomic wavefunction Iyjg) is written through the amplitudes a1 and a2 for the ground states of the A system:

the dressed-field states are defined through their Rabi frequencies QC and

QNC

where QRI and Q R ~ are the Rabi frequencies for the fields acting on the 11) -+ 10) and 12) -, 10) transitions. This linear transformation produces a coupledlnoncoupled combination of the electric fields such that the QC coupled field component is heavily absorbed during the propagation whereas the QNC noncoupled field component propagates without attenuation. Under conditions of fast evolution of the excited-state population and optical coherences and of slow evolution of the ground-state coherences (fast and slow as compared to the pulse duration) the atomic amplitudes a1 and a2 of eq. (9.1) are determined by the initial conditions of the electric field amplitude. As a consequence, the input electric field determines the occupation amplitudes of the atomic wavefunction and fixes the linear combination QNC of the nonattenuated electric field: the normal modes of Harris El9931 are defined by those atomic amplitudes and the Qc, QNC Rabi frequencies. Normal modes may be realized in the propagation of a bichromatic pulse pair where the occupation of the coupled state is fixed through an initial interaction with the pulse pair. The pulse-pair propagation causes a distortion of the initial pulse edge and an unperturbed propagation of the remaining part of the pulse. Other methods for the preparation of the coherent trapping superposition may be also devised. Cerboneschi and Arimondo [ 19951 pointed out that the double4 scheme of fig. 26c is convenient for realizing pulse matching with great flexibility, because a pulse pair on one A system prepares the noncoupled superposition, whereas the pulse pair on the second A experiences pulse matching without any distortion of the initial edge. Notice that the definition of the dressed-field pulses by Eberly, Pons and Haq [I9941 treats on the same footing the atomic superposition states and those classified as dark states in 9 2.6.

v, I 91

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Aganval [ 19931 has generalized the idea of coherent population trapping by considering a quantized electromagnetic field in an approach similar to that used in the case of dressed states, but considering electromagnetic states with a very low photon number. Starting from the quantized field Hamiltonian of eq. (2.29) describing two field modes interacting with a A system, Aganval searched for a wavefunction corresponding to a generalized atom-field noncoupled state, in the form of an entangled state of the atomic and field variables:

with the C I and c2 coefficients and the N normalization constant to be determined. For the state INCAF) to be an eigenvalue of the quantized Hamiltonian with zero eigenvalue, the radiation field I qfield) wavefunction should satisfy the following equation:

with U R ~and OR2 the annihilation operators of the two field modes. The general solution of eq. (9.4) is obtained in terms of coherent states 1 ~ ~ zR2) 1 , associated with the two modes of the field:

with the C I , C ~coefficients and the function q(zRI) fixed from the initial conditions of the system. The important point of this equation is that it describes two fields matched in their mode mean value: (9.6) but also in their photon statistics, because from eq. (9.5) it results that the coherent states of the two modes are replicas of each other, only scaled by the c1Ic2 factor. The coherent population-trapping mechanism has determined the correlations of the two field modes. Jain [1994] has shown how the correlation in the phase noise of the two modes can be utilized for measurements limited only by the coherent-state shot noise. Aganval, Scully and Walther [ 19941 have described how the ground-state coherence leads to a noise-free transfer of energy between a pump laser and a probe laser acting on the two arms of the A scheme. Fleischhauer [ 19941, investigating the correlations in the phase

344

COHERENT POPULATION TRAPPING

[Y ii

10

fluctuations of the laser beams propagating through a three-level A system, pointed out the importance of the adiabatic response of the atomic coherence with respect to the phase or amplitude fluctuations of the field. In the adiabatic limit the atomic system responds promptly, and any modification in the field drives the atom into a new coherent superposition, again decoupled from the fields. In contrast, in the regime of nonadiabatic response, the atom remains in the initial noncoupled state on the time scale of the phaseiamplitude field fluctuations, and those fluctuations will be damped out. In closely related papers involving the application of coherences and preparation of atoms in ground-state superpositions, the possibility of reduction of the intensity or phase noise in the three-level system has been discussed by several authors (Aganval [ 1991b1, Gheri and Walls [1992], Fleischhauer, Rathe and Scully [1992]). A discussion of those results will not be reported here because a clear definition of the role of coherent population trapping has not been established. A linear superposition of ground states noncoupled to laser radiation has been considered by Cirac, Parkins, Blatt and Zoller [ 19931 for the motion of an ion in a trap. The ground states were associated with different vibrational states of the ionic motion. The noncoupled state formed by that superposition should present properties of squeezing in the atomic motion. Parkins, Marte, Zoller and Kimble [1993], and Parkins, Marte, Zoller, Carnal and Kimble [1995] have proposed a scheme for the preparation of general coherent superpositions of photon-number states. Three-level atoms initially in state 11) should experience an adiabatic transfer process, through proper time dependencies of Rabi frequencies, crossing a cavity where a laser field resonant with the transition 11) + 10) is present, whereas a vacuum field acts on the 10) + 12) transition. As a consequence of the atomic adiabatic transfer process to state (2), a cavity mode with one photon at frequency wL2 is created. The atomic adiabatic transfer allows the generation of photons in the cavity mode starting from the laser field acting on the 11) + (0) transition. Sequences of atoms passing through the cavity can be used to generate Fock states of higher photon number. The adiabatic transfer from properly tailored noncoupled atomic superpositions in h i g h 4 angular momentum states allows the generation of arbitrary superpositions of Fock states.

FJ 10. Conclusions Quantum-mechanical interference effects involving the existence of two different paths for the final process are well known in laser spectroscopy investigations, and they have been fully exploited in order to improve the spectroscopy

Y 5 101

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resolution. Coherent population trapping is an interesting example of that quantum interference, based on the preparation of a quantum-mechanical superposition state and an interference in the interaction with two modes of the electromagnetic field. As described above, the phenomenon has been observed and exploited in a large variety of experimental configurations. Furthermore, references in this review point out that the number of suggested new applications presented in the last few years is quite large. The most exciting and most promising applications for further development are those connected with the manipulations of atoms and molecules and electromagnetic fields. Adiabatic transfer and velocity-selective coherent population trapping allow preparation in a given state of internal and/or external variables, for which elegant theoretical schemes and experimental results have been reported. The manipulation of electromagnetic fields through coherent trapping superposition states represents its counterpart; the coherence created in the quantum-mechanical superposition state should allow the production of electromagnetic fields with specific and interesting correlation properties, with the initial information to be transferred to the optical field stored inside the three-level atomic coherence. Up to now, only several theoretical ideas and few experimental realizations have been reported, if we include within this class of manipulation the experiments on amplification without inversion. In any case, it is interesting to point out that, because coherently prepared systems have so many particular properties, the researchers in this field have been inspired to classify them as a new state of matter called “phaseonium”, short for “phase coherent atomic ensemble” (Scully [ 19921). Devices based on phaseonium, referred to also as “phasers”, are not yet available, but the interest into the subject is growing among quantum optics researchers. Furthermore, the proposals for extensions and applications of coherent population trapping are expanding. Not all of them have been reported here because of space limitations, but it is noteworthy that Tumaikin and Yudin [ 199 11 have proposed to transfer to superconductivity the concept of coherent population trapping, looking for a quantum superposition of electron states decoupled from the electron-photon interaction. Applications to time-domain optical data storage have been presented by Hemmer, Cheng, Kierstead, Shahriar and Kim [1994]. Also, it has been pointed out by Zaretsky and Sazonov [1994] that in an experiment of nuclear decay, the creation of a coherent populationtrapping superposition of atomic states with different hyprfine couplings could modify the angular distribution of the emitted y rays. Moreover, the coherent superposition of states has been extended theoretically by Lindberg and Binder [ 19951 to a three-level A system in a GaAs quantum well, the lower state being associated to the heavy-hole and light-hole excitons. In this system the absorption

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of a bichromatic laser pulse should show up through the characteristic coherent population-trapping decrease. Some features of coherent population trapping have become more or less understood, but they are not yet fully exploited. For instance, the laser cooling application of coherent population trapping is based on the atomic preparation in a quantum-mechanical superposition of two states with different space dependence. That quantum-mechanical superposition may produce an atomic interference, either over a limited region of time and spacc or in consistent way over all of time and space. This interference has been considered by the researchers active in velocity-selective coherent population trapping, and its implications for atomic interferometry are quite important (Aspect, Arimondo, Kaiser, Vansteenkiste and Cohen-Tannoudji [ 19891, Lawall, Bardou, Saubamea, Shimizu, Leduc, Aspect and Cohen-Tannoudji [ 19941). However, those experiments certainly pose a high technical challenge in atomic manipulation. Coherent population trapping represents only an example of the richness of phenomena taking place in the interaction of radiation fields with multilevel atomic or molecular systems. More complicated interference schemes involving a larger number of levels and of radiation fields, as in the closed-loop multilevel structures, may provide new possibilities not yet explored. Similar to adiabatic transfer whose exploitation has taken a considerable amount of time after its original proposition, other interesting phenomena may be buried inside the theoretical analyses of multilevel systems. Coherent population trapping is connected to a conservation law, valid for a three-level system. Some other interesting possibilities may remain to be exploited within the conservation laws related to different observables in experiments involving a multilevel system interacting with several radiation fields. Several of those conservation laws involve excited states, and thus superpositions that are, not stable against spontaneous emission, which may lead to applications in proc,esses involving short pulse sequences. In any case, this field of research is very active and its extensions are too widely spread to be accurately forecast. Acknowledgments This work largely originated from the shared interest in the subject and collaboration throughout the years with G. Alzetta, C. Cohen-Tannoudji, O.A. Kocharovskaya, and M.O. Scully. I am grateful to G. Agarwal for having suggested the preparation of this review, to all those authors who permitted the use of their figures, to J. Cooper and J. Hall for a careful reading of the manuscript, and to C. Menotti for checking the proofs.

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E. WOLF, PROGRESS IN OPTICS XXXV 0 1996 ELSEVIER SCIENCE B.V. ALL RIGHTS RESERVED

VI

QUANTUM PHASE PROPERTIES OF NONLINEAR OPTICAL PHENOMENA BY

R. TANA~ AND A. MIRANOWICZ Nonlinear Optics Division. Institute of Physics, Adam Mickiewicz University, 60-780 Poznari, Poland

AND

Ts. GANTS~G Max-Planck-Institut fur Quantenoptik, 85748 Garching bei Munchen. Germany and Department of Theoretical Physics, National University of Mongolia. 210646 Ulaanbaatar, Mongolia

355

CONTENTS

PAGE

5 1.

INTRODUCTION . . . . . . . . . . . . . . . . . . .

357

5 2.

PHASE FORMALISMS . . . . . . . . . . . . . . . . .

362

5 3. PHASE PROPERTIES OF SINGLE-MODE OPTICAL FIELDS . 5 4 . PHASE PROPERTIES OF TWO-MODE OPTICAL FIELDS . .

378

§ 5 . CONCLUSION

426

. . . . . . . . . . . . . . . . . . . .

399

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . .

426

APPENDIX A . GARRISON-WONG PHASE FORMALISM . . . .

427

APPENDIX B. STATES FOR THE PEGG-BARNETT PHASE FORMALISM . . . . . . . . . . . . . . . . . . . . .

430

REFERENCES . . . . . . . . . . . . . . . . . . . .

434

356

5

1. Introduction

In classical optics, the concepts of intensity and phase of optical fields have a well-defined meaning. The oscillating real electromagnetic field associated with one mode, E = A cos(@), has a well defined amplitude and phase. Apart from a constant factor, the squared real amplitude, A’, is the intensity of the field. In classical electrodynamics, contrary to quantum electrodynamics, there is no real need to use complex numbers to describe the field. However, it is convenient to work with exponentials rather than cosine and sine functions, and complex amplitudes of the field, E = A exp (-@), are commonly used to describe the field. The modulus squared of such an amplitude is the intensity of the field and the argument is the phase. Both the intensity and the phase can be measured simultaneously in classical optics. In quantum optics, it was quite natural to associate the photon number operator with the intensity of the field and somehow construct the phase operator conjugate to the number operator. The latter task, however, turned out not to be easy. The first attempts to construct explicitly a quantum phase operator as a quantity conjugate to the number operator were made by Dirac [1927]. His idea was to perform a polar decomposition of the annihilation operator, similar to the polar decomposition of the complex amplitude performed for classical fields. It turned out later that such a decomposition suffers from serious drawbacks, and the phase operator introduced in this way cannot be considered as a properly defined Hermitian phase operator. Susskind and Glogower [ 19641 exposed the contradictions inherent in Dirac’s polar decomposition and introduced, instead of the phase operator that appeared to be non-Hermitian, the operators cTs @SG and G @s, correFonding to the cosine and sine of the phase. Unfortunately, these CTS@SG and sin@sG operators, although Hermitian, do not commute, so that they cannot represent a single phase angle. Historically, the idea to use cTs @SG and @SG as Hermitian operators describing the phase, was first raised by Louise11 [1963] in his short Letter, but he did not construct the explicit form of these operators. Carruthers and Nieto [1968] in their review paper, gave a detailed record of the problems encountered on the way to constructing the Hermitian phase operator and discussed thoroughly the properties of cTs @SG and G @ S G operators. From their analysis it became clear that it is the boundedness 357

358

QUANTUM PHASE PROPERTIES

[VI, § 1

of the photon number spectrum which does not allow for negative values and which precludes the existence of a properly defined Hermitian phase operator in the infinite-dimensional Hilbert space. The difficulty in finding the form of the Hermitian phase operator led to the widespread belief that no such operator exists, although there were a number of ingenious attempts to construct a suitable operator within the infinite-dimensional Hilbert space (Garrison and Wong [1970], Turski [1972], Popov and Yarunin [1973, 19921, Paul [1974], Damaskinsky and Yarunin [19781, Galindo [ 1984a,b]). It was known (Newton [1980], Barnett and Pegg [1986], LukB and Peiinovi [1991], LukS, Peiinova and Kiepelka [1992a]) that extension of the oscillator energy spectrum to negative values allows for the mathematical construction of the Hermitian phase operator, but this solution was unsatisfactory because of its recourse to unphysical states. Recently, Pegg and Barnett [1988, 19891 (see also Barnett and Pegg [1989, 1990, 1992, 19931, Pegg, Barnett and Vaccaro [ 19891, Barnett, Pegg and Vaccaro 119901, Pegg, Vaccaro and Barnett [1990], Vaccaro and Pegg [1990a,b, 19931, Vaccaro, Barnett and Pegg [ 19921) have found a way out of the difficulties with construction of a Hermitian phase operator. The key idea in the development of the Hermitian optical phase operator was abandonment of the conventional infinite dimensional Hilbert space for the description of quantum states of a single-field mode. They introduced, instead, a state space '/Y(') of formally finite dimension together with a prescription for taking the infinite-dimensional limit only after c-number expectation values and moments have been calculated. This idea reintroduced a symmetry to the photon number spectrum, which became bounded not only from below but also from above, and removed the main obstacle in constructing a Hermitian phase operator. An essential and indispensable ingredient of the Pegg-Barnett construction is the way the infinite dimensional limit is taken, which distinguishes it from the quantummechanical constructions based on finite-dimensional spaces that have been studied before (Levy-Leblond [ 1973, 1976, 19771, Santhanam and Tekumalla [1976], Santhanam [1976, 1977a,b], Santhanam and Sinha 119781, Goldhirsh [1980]), but in which, when the limiting procedure has been applied for the phase operator, the original problems reappeared. The consequences of the limiting procedure in the Pegg-Barnett approach have been discussed by Barnett and Pegg [1992] and Gantsog, Miranowicz and Tanai [1992]. The proposal of the Pegg-Barnett approach has renewed interest in the problem of the proper description of the quantum-optical phase. Almost at the same time, Shapiro, Shepard and Wong [1989] (see also Shapiro and Shepard [1991]) used an alternative approach based on the quantum estimation theory and probability operator measures (Helstrom [ 19761) to

VL

8 11

INTRODUCTION

359

describe the phase properties of optical fields. This approach does not rely on the existence of the Hermitian phase operator but rather on the existence of the eigenstates of the Susskind-Glogower nonunitary exponential phase operator (Susskind and Glogower [ 19641). The eigenstates of the Susskind-Glogower phase operators form a basis for the probability operator measures. The Shapiro, Shepard and Wong [1989] idea has gained some popularity (Hall [1991, 19931, Hall and Fuss [ 19911, Schleich, Dowling and Horowicz [ 199 I], Braunstein, Lane and Caves [1992], Braunstein [ 19921, Hradil [1992a,b], Hradil and Shapiro [ 19921, Lane, Braunstein and Caves [ 19931, Jones [ 19931, Shapiro [ 19931). It turned out, however, that it gives for physical states [i.e., states with a finite mean number of photons (finite energy and its higher moments)], the same results as the Pegg-Barnett approach after the limit transition to the infinite-dimensional space. The eigenkets of the Susskind-Glogower exponential phase operators can be used in a similar fashion as coherent states (eigenkets of the annihilation operator) to define the resolution of the system operators; e.g., the phase operator (LukS and Peiinova [1991, 1993b1, Brif and Ben-Aryeh [1994a,b], Vaccaro and Ben-Aryeh [ 19951. In this case the ordering of the phase exponentials is relevant, and, if the antinormal ordering is taken, the results agree with those obtained from the Pegg-Barnett formalism. Another way to describe the phase properties of the field is to use quasiprobability distribution functions. The idea behind this approach is relatively simple: to integrate the suitable quasiprobability distribution function, such as the GlauberSudarshan P-function, Wigner function, or Husimi @function, over the “radial” variable and getting in this way a corresponding phase distribution, which can then be applied in calculations of the mean values of the phase-dependent quantities. Since the quasiprobability description of the quantum state of the system can be in some cases associated with realistic measurements performed on the system, this approach to the phase problem has focused the attention of many authors (see $2.2). The phase distribution functions obtained by integrating the quasidistributions are different for different quasidistributions, and they are all different from the Pegg-Barnett phase distribution. The situation is even worse, because for some states of the field the P-function and the Wigner function take on negative values, and the corresponding phase distribution can also be negative. T h s means that such phase distributions must be used with some care, but in many cases this approach gives results describing properly the phase properties of the field. Noh, Fougeres and Mandel [ 199 1, 1992a,b, 1993a-el (see also Fougeres, Monken and Mandel [1994], Barnett and Pegg [1993], Hradil [1993a, 19951, Hradil and Bajer [1993]) presented an operational approach to the quantum

360

tVI, 9: 1

QUANTUM PHASE PROPERTIES

phase problem. Their idea is to define phase in terms of what actually is, or can be, measured. They do not search for a phase operator which would satisfy some mathematical criteria, but start their considerations from the experimental schemes usual in classical phase measurements. Examining various measuring schemes, they identify certain operators, and corresponding to the measured cosine and sine of the phase difference between two fields. As a result, Noh, Fougeres and Mandel came to the conclusion that there is no unique phase operator, and that different measuring schemes correspond to different operators. Nevertheless, recent theoretical studies (Riegler and Wodkiewicz [ 19941, Englert and Wodkiewicz [ 19951, Englert, Wbdkiewicz and Riegler [ 19951) show that the intrinsic Hermitian phase operator associated with the Noh, Fougeres and Mandel apparatus can be found. The phase distribution measured in this experiment, under some conditions, is the radius-integrated Q-function (Freyberger and Schleich [ 19931, Leonhardt and Paul [1993a], Freyberger, Vogel and Schleich [1993a,b], Bandilla [1993], Khan and Chaudhry [1994]). The measurements of Noh, Fougeres and Mandel are important since until then only isolated phase measurements were available performed by Gerhardt, Buchler and Litfin [ 19741 (Gerhardt, Welling and Frolich [ 19731, and for theoretical analysis see also Nieto [1977], LCvy-Leblond [1977], Lynch [1987, 1990, 1993, 19951, Gerry and Urbanski [1990], Bandilla [1991], Tsui and Reid [1992]), Matthys and Jaynes [1980], and Walker and Carroll [1984, 19861. Quite recently, another experimental technique, optical homodyne tomography, was invented and applied to measurements of the quantum state of the field (Smithey, Faridani and Raymer [1993], Beck, Smithey and Raymer [1993], Beck, Smithey, Cooper and Raymer [ 19931, Smithey, Beck, Cooper and Raymer [ 19931, Smithey, Beck, Cooper, Raymer and Faridani [ 19931) allowing quantum phase mean values to be calculated from the measured field density matrix. This technique opens new possibilities for quantum measurements. Overviews of various techniques for measuring phase distributions are presented by, e.g., Leonhardt and Paul [ 1993b], Paul and Leonhardt [1994], Peiina, Hradil and JurEo [1994], and Lynch [1995]. Another interesting approach to the phase problem was presented by Bergou and Englert [1991]. They introduced the idea of phasors and phasor bases that can be used for studying possible candidates for the quantum phase operators. Different phasor bases lead to different phase operators, and according to Bergou and Englert [1991] extrapolation of the classical concept of phase to the quantum regime is not unique. Since the absolute phase of the single-mode field is not accessible for measurements, and it is always the difference with respect to a reference phase

ZM,

VL § 11

INTRODUCTION

36 I

that we are forced to deal with in real measurements, it is tempting to define the phase-difference operator as a fundamental quantity describing the optical phase. Luis and Sanchez-Soto [ 1993c, 19941 have defined a Hermitian phase-difference operator, which is in fact a polar decomposition of the Stokes operators for the two-mode field, and it is not the same as the difference of the two PeggBarnett operators. The difference between the two is most pronounced for weak fields. Ban [ 1991a-c, 19921 has introduced yet another phase operator, based on the two-mode description of the field. In recent years, many different aspects of the quantum phase problem have been studied (Barnett, Stenholm and Pegg [ 19891, Nath and Kumar [ 1989, 1990, 1991b], Chaichian and Ellinas [ 19901, Lakshmi and Swain [1990], Summy and Pegg [1990], Hradil [1990, 1993b], Lukl and Pefinova [1990, 1992, 1993a, 19941, Vourdas [1990, 1992, 19931, Adam, Janszky and Vinogradov [ 19911, Cibils, Cuche, Marvulle and Wreszinski [1991], Dowling [1991], Ellinas [1991a,b, 19921, Gantsog and TanaS [1991d], Nath and Kumar [1991a], Orszag and Saavedra [1991a,b], Paul [1991], TanaS [ 19911, Wilson-Gordon, Buiek and Knight [ 199I], Aganval, Chaturvedi, Tara and Srinivasan [ 19921, Bandilla [ 19921, Burak and Wodkiewicz [ 19921, Janszky, Adam, Bertolotti and Sibilia [ 19921, Luki, Pefinovi and Kiepelka [ 1992b, 19941, Ritze [ 19921, Smith, Dubin and Hennings [ 19921, Tsui and Reid [ 19921, Aganval [ 19931, Aganval, Scully and Walther [ 19931, Ban [I 9931, Bialynicki-Birula, Freyberger and Schleich [ 19931, Chizhov, Gantsog and Murzakhmetov [ 19931, Chizhov and Murzakhmetov [ 19931, Daeubler, Miller, Risken and Schoendorff [1993], D’Ariano and Paris [1993, 19941, Jex and Drobny [1993], Luis and Sanchez-Soto [ 1993a1, Stenholm [ 19931, Schieve and McGowan [ 19931, Tu and Gong [1993], Aganval, Graf, Orszag, Scully and Walther [1994], Belavkin and Bendjaballah [ 19941, Vaccaro and Pegg [ 1994a-c], Bialynicka-Birula and Bialynicki-Birula [ 1994, 19951, Das [ 19941, Franson [ 19941, Gennaro, Leonardi, Lillo, Vaglica and Vetri [1994], Opatrny [1994], Sanchez-Soto and Luis [1994], Schaufler, Freyberger and Schleich [ 1994]), and the number of publications on the subject is growing steadily. Various conceptions of the quantum-optical phase have been described by Barnett and Dalton [ 19931 in a special issue of Physica Scripta devoted to “Quantum phase and phase dependent measurements”, and in the same issue one can find very interesting historical facts, given by Nieto [ 19931, concerning the development of our knowledge on quantum phase. Nowadays, although the quantum phase is still a subject of some controversy, significant progress has been achieved in clarifying the status of the quantummechanical phase operator, describing the phase properties of optical fields in terms of various phase distribution functions, and measuring phase-dependent physical quantities. We can now risk the statement that, despite the existence

3 62

QUANTUM PHASE PROPERTIES

[VL

P2

of various different conceptions of phase, we are en route to unified view and understanding of the quantum-optical phase. It is not our aim in this review article to give a detailed account of different descriptions of the quantum phase showing their similarities and differences. We will not focus our attention on the quantum phase formalisms, which are interesting on their own right and deserve separate treatment. We shall instead concentrate on the description of the quantum properties of real-field states which are generated in various nonlinear optical processes. Nonlinear optical phenomena are sources of optical fields, the statistical properties of which have been changed in a nontrivial way as a result of nonlinear transformation. Quantum phase properties are among those statistical properties which undergo nonlinear changes, and fields generated in different nonlinear processes have different phase properties. With the existing phase formalisms, the quantum phase properties of such fields can be studied in a systematic way, and quantitative comparisons between different quantum-field states can be made. Using the Pegg-Barnett phase formalism and the phase formalism based on the sparametrized quasidistribution functions, we will study a number of both singleand two-mode field states from the point of view of their phase properties.

8

2. Phase formalisms

At the very beginning of quantum electrodynamics, Dirac [1927] raised the idea that the optical phase should be described by a Hermitian phase oper,tor canonically conjugate to the number operator; that is, for the one-mode field the two operators should obey the canonical commutation relation:

[5,i ]

= -i.

This commutation relation implies directly the “traditional” Heisenberg uncertainty relation

which appeared to be wrong (Susskind and Glogower [1964], Carruthers and Nieto [1968]). Closer investigation of the commutator (2.1) showed that the

VI, § 21

PHASE FORMALISMS

matrix elements of the phase operator (Louisell [ 19631);

363

5 in a number-state basis are undefined

Since it was suggested that the problems in eq. (2.3) are due to the multivalued nature of $, Louisell [I9631 introduced the operators S s @ and % @ which should, as he suggested, satisfy the commutation relations

[CTS Q, fi]

h

=

i sin @,

[-sin Q, ri1

= -i

CT@. S

(2.4)

However, Louisell [I9631 did not give the explicit form of the cosine and sine operators; thus, his idea did not help much in solving the phase problem. Moreover, it turned out that the problem expressed in eq. (2.3) was not due to the multivalued nature of 5,but rather to the improper application of the correspondence between the Poisson bracket and the commutator. Susskind and Glogower [I9641 returned to the original Dirac idea of polar decomposition of the creation and annihilation operators and introduced the exponential phase operators: M

n=O

and

From eqs. (2.5) and (2.6), one obtains:

which explicitly shows the non-unitarity of the Susskind-Glogower phase operator. The Susskind-Glogower exponential operators (2.5) and (2.6) allow construction of two Hermitian combinations corresponding to cosine and sine of the phase. However, the two combinations do not commute, so they cannot be considered as describing a single phase angle. Despite this deficiency, the Susskind-Glogower phase operators were widely used in description of optical

3 64

QUANTUM PHASE PROPERTIES

[VI,

5

2

fields until recently. The eigenkets of the Susskind-Glogower operator (2.5), given by .

w

generate the resolution of the identity, and, despite their nonorthogonality, they can be used to form the probability operator measure applied to the phase description by Shapiro and Shepard [ 19911. Garrison and Wong [1970] made an attempt to construct a Hermitian phase operator in the infinite-dimensional Hilbert space which, as they demanded, should satisfy the canonical commutation relation (2.1). Their work was almost completely forgotten. Bergou and Englert [1991] have made a comparison of the Garrison-Wong and Pegg-Barnett phase operators, pointing out that if the limit to infinite-dimensional space is performed on the latter operator (but not on the expectation values), the former operator is obtained. In their view the Pegg-Barnett phase formalism is only an approximation to the “correct” phase formalism. Bergou and Englert [1991] introduced their own quantum phase description, which has not gained broader acceptance. Nevertheless, their paper is an essential contribution which clarifies a number of points in the quantum phase problem. The Garrison-Wong approach turned out to lead to phase distributions which are asymmetric and difficult to accept on physical grounds (Gantsog, Miranowicz and Tanai [1992]); e.g., even the vacuum is phaseanisotropic. For reference, we give a sketch of their approach in Appendix A. The renewed interest in quantum phase problems has resulted in a reexamination of some of the earlier approaches and the creation of other, completely new descriptions. From a number of different phase formalisms that are now available, we choose only two, which we shall apply for the description of the phase properties of fields generated in various nonlinear optical processes. These are: the Pegg-Barnett phase formalism, which represents the canonical phase formalism based on the idea of finding a Hermitian phase operator, and another formalism based on the description of the optical phase through sparametrized phase distributions, which for some values of s represents the experimentally measured phase probability distributions. 2.1. PEGG-BARNETT PHASE FORMALISM

Pegg and Barnett [1988, 19891 (and Barnett and Pegg [1989]) introduced the Hetmitian phase formalism, which is based on the observation that in a

VI, § 21

PHASE FORMALISMS

365

finite-dimensional state space, the states with well-defined phase exist (Loudon [ 19731). Thus, they restrict the state space to a finite ( a + 1)-dimensional Hilbert space spanned by the number states lo), I I ) , . . . , la). In this space they define a complete orthonormal set of phase states by:

where the values of Om are given by 2nm (2.10) 8, = 00 + a + 1‘ The value of 00 is arbitrary and defines a particular basis set of (a + 1) mutually orthogonal phase states. The number state In) can be expanded in terms of the I%,) phase-state basis as: U

C7

(2.1 1) From eqs. (2.9) and (2.11) we see that a system in a number state is equally likely to be found in any state \ O m ) , and a system in a phase state is equally likely to be found in any number state In). The Pegg-Barnett (PB) Hermitian phase operator is defined as: U

(2.12) Of course, the phase states (2.9) are eigenstates of the phase operator (2.12) with the eigenvalues 8, restricted to lie within a phase window between 00 and 80 + 2 n a / ( a + 1). The Pegg-Barnett prescription is to evaluate any observable of interest in the finite basis (2.9), and only after that to take the limit a + 03. Since the phase states (2.9) are orthonormal, ( 8 m l ~ m= !6,,t, ) the kth power of the Pegg-Barnett phase operator (2.12) can be written as: (2.13) Substituting eqs. (2.9) and (2.10) into eq. (2.12) and performing summation over m yields explicitly the phase operator in the Fock basis:

,.

@o

= 80

c,

an 2n +a + 1 a+ln s n +

exp [i(n - n’)80] In) (n’l exp[i(n - n’)2n/(a + 111 - 1.

(2.14)

It is readily apparent that the Hermitian phase operator 5 0 has well-defined matrix elements in the number-state basis and does not suffer from such problems

366

QUANTUM PHASE PROPERTIES

tVI, § 2

as the original Dirac phase operator. A detailed analysis of the properties of the Hermitian phase operator was given by Pegg and Barnett [ 19891 and Barnett and Pegg [1989]. The unitary phase operator exp(i58) can be defined as the exponential function of the Hermitian operator $0. This operator acting on the eigenstate 16,) gives the eigenvalue exp(iO,), and can be written as (Pegg and Barnett [1988, 19891): 0-

exp(i5.s) z

1

C in) (n + 11 + exp[i(a + I)O,] la)(01.

(2.15)

n=O

Its Hermitian conjugate is [exp(i+o)]

t

=

exp(-iZ,e),

(2.16)

with the same set of eigenstates lonj)but with eigenvalues exp(-i0,). This is the last term in eq. (2.15) that distinguishes the unitary phase operator from the Susskind-Glogower phase operator (Susskind and Glogower [ 19641). The first sum in eq. (2.15) reproduces the Susskind-Glogower phase operator if the limit a -+ cm is taken. In contrast to the Pegg-Barnett unitary phase operator, the Susskind-Glogower exponential operator (2.5) is defined as a whole and is not unitary, as is apparent from eq. (2.7). The sine and cosine operators in the Pegg-Barnett formalism are the sine and cosine functions of the Hermitian phase operator $0. They are more consistent with the classical notion of phase than their counterparts in the Susskind-Glogower phase formalism. In particular, they satisfy the “natural” relations: cos2 5 8+ sin2 $8 [cos $0, sin $01

= =

1,

0,

(nIcos’$oln)=(nlsin25sln)=l.

(2.17) (2.18) (2.19)

The last relation is also true for the vacuum state, in sharp contrast to the Susskind-Glogower phase operators. This is consistent with the phase of vacuum being random, as well as for any other number state. The Pegg-Barnett phase operator (2.14), expressed in the Fock basis, readily gives the phase-number commutator (Pegg and Barnett [ 19891):

Equation (2.20) looks very different from the famous Dirac postulate of the phase-number commutator (2.1). Santhanam [ 19761 and Pegg, Vaccaro

VI, § 21

PHASE FORMALISMS

361

and Barnett [ 19901 examined canonically conjugate operators in the finitedimensional Hilbert space. According to the generalized definition by Pegg, Vaccaro and Barnett, the photon number and phase are indeed canonically conjugate variables, similar to momentum and position or angular momentum and azimuthal phase angle. As the Hermitian phase operator is defined, one can calculate the expectation value and variance of this operator for a given state of the field I f ) . Moreover, the Pegg-Barnett phase formalism allows the introduction of the continuous phase probability distribution which is a representation of the quantum state of the field and describes the phase properties of the field in a very spectacular fashion. Examples of such phase distributions for particular states of the field will be given later. A general pure state of the field mode with a decomposition 0

(2.21) n=O

can be re-expressed in the phase-state basis, according to eq. (2.1 I), as: (2.22)

We should remark here that the coefficients c, in the decomposition (2.21) in a finite-dimensional space should differ from the coefficients in the infinitedimensional space if the state I f ) is to be normalized. A short discussion of this problem is given in Appendix B. The phase probability distribution is given by (Pegg and Barnett [ 19891); (2.23)

which leads to the expectation value and variance of the phase operator (2.24)

(2.25) If the field state

If)

cn = bnel"q,

is a partial phase state, i.e., if the amplitude has the form (2.26)

368

QUANTUM PHASE PROPERTIES

the phase probability distribution (2.23) becomes:

The mean and variance of $0 will depend on the chosen value of 00. Judge [1964], in his description of the uncertainty relation for angle variables, suggested that the choice of phase window which minimizes the variance can be used to specify uniquely the mean or variance. For the partial phase state, the most convenient and physically transparent way of choosing 80 is to symmetrize the phase window with respect to the phase p. This means the choice

na a +1'

6lO=q--

(2.28)

and after introducing a new phase label,

a

p=m----, 2

(2.29)

the phase probability distribution (2.27) becomes (2.30)

with p, which goes in integer steps from -012 to 012. Since the distribution (2.30) is symmetrical in p, we immediately get, according to eqs. (2.28)(2.301,

(fl&If)= v1.

(2.3 1)

This result means that for a partial phase state with phase q,the choice of 80 as in eq. (2.28) relates directly the expectation value of the phase operator with the phase cp. With this choice of 8 0 , the variance of the phase operator has a particularly simple form: (2.32)

So-called physical states play a significant role in the Pegg-Barnett formalism. Physical states Ip) are defined by Pegg and Barnett [ 19891 as the states of finite

VI, § 21

3 69

PHASE FORMALISMS

energy. Most of the expressions in the Pegg-Barnett formalism take a much simpler form for physical states. For example, the commutator (2.20) reduces to

a form more similar to the standard canonical commutation relation (2.1). On the other hand, when physical states are considered, we can simplify the calculation of the sum in eqs. (2.24) and (2.25) by replacing it by the integral in the limit as a tends to infinity. Since the density of states is (a+1)/2n, we can write the expectation value of the kth power of the phase operator as: (2.34) where the continuous-phase distribution P( 0) is introduced by (2.35) and O,,* has been replaced by the continuous-phase variable 0. If the state I f ) has the number-state decomposition (2.2 1), then the Pegg-Barnett phase distribution is (Pegg and Barnett [ 19891): (2.36) In the case of fields being in mixed states described by the density matrix formula (2.36) generalizes to

216

pmnexp[-i(m

-

n)0]

5,

(2.37)

where pmn= ( m 1 51n) are the density matrix elements in the number-state basis. Formulas (2.36) or (2.37) can be used for calculations of the Pegg-Barnett phase distribution for any state with known amplitudes cn or matrix elements pmn. Despite the fact that they are exact, they can rarely be summed up into a closed form, and usually numerical summation must be performed to find the phase distribution. Such numerical summations have been widely applied in studying the phase properties of optical fields. The Pegg-Barnett phase distribution,

370

QUANTUM PHASE PROPERTIES

[VI, § 2

eqs. (2.36) or (2.37), is obviously 2~-periodic,and for all states with the density matrix diagonal in the number states, the phase distribution is uniform over the 2n-wide phase window. These are nondiagonal elements of the density matrix that lead to the structure of the phase distribution. The Pegg-Barnett distribution is positive definite and normalized. After introducing the continuous-phase distribution P( O), formula (2.32) for the phase variance, if the symmetrization is performed, can be rewritten into the form: ((AZs)’)

=

/“ -n

02P(0)d0.

(2.38)

This means that as the phase distribution function P(O) is known, all quantummechanical phase characteristics can be calculated with this function in a classical-like manner. The result for the variance (2.38) is (Barnett and Pegg [ 19891): (2.39) The value n213 is the variance for the uniformly distributed phase, as in the case of a single-number state. For physical states there are some additional useful relations betwcen the expectation values of the Pegg-Rarnett phase operators and of the SusskiiidGlogower phase operators. For cxample, the following relations hold (Vaccaro and Pegg [ 19891):

(2.42)

where the subscript p refers to a physical state expectation value.

VI, 9: 21

PHASE FORMALISMS

371

2.2. PHASE DISTRIBUTIONS FROM S-PARAMETRIZED QUASIDISTRIBUTIONS

According to Cahill and Glauber [ 1969a,b], the s-parametrized quasidistribution function W ( s ) ( adescribing ) a field state, can be derived from the formula (2.45)

n where the operator ^r(s)(a) is given by: -

n

a*E)E(.')(E> dzE,

(2.46)

(2.47) with E(E) being the displacement operator; p^ is the density matrix of the field, and we have introduced the extra l/n factor with respect to the original definition of Cahill and Glauber [1969b]. The operator ?')(a) can be rewritten in the form (Cahill and Glauber [ 1969a1): (2.48) which gives explicitly its s-dependence. So, the s-parametrized quasidistribution function W ( " ) ( ahas ) the following form in the number-state basis: 1

W'"( a ) = n

c

pmn(nI

F( a ) I m) ,

(2.49)

m,n

where the matrix elements of the operator (2.46) are given by (Cahill and Glauber [ 1969a1):

in terms of the associate Laguerre polynomials L;-"(x). In eq. (2.50) we have also separated explicitly the phase of the complex number a by writing:

a = lc11 el'.

(2.51)

In the following, the phase H will be treated as the quantity representing the field phase.

372

QUANTUM PHASE PROPERTIES

[VI, 9: 2

With the quasiprobability distributions W(’)(a),the expectation values of the s-ordered products of the creation and annihilation operators can be obtained by proper integrations in the complex a plane. In particular, for s = 1,0, - 1, the sordered products are normal, symmetrical, and antinormal ordered products of the creation and annihilation operators, and the corresponding quasiprobability distributions are the Glauber-Sudarshan P-function, Wigner function, and Husimi Q-function. By virtue of the relation inverse to eq. (2.49), given by (Cahill and Glauber [ 1969b1) (2.52) the Pegg-Barnett phase distribution (2.37) can be related to the s-parametrized quasidistribution function (2.45) as follows (Eiselt and Risken [ 199 11):

P(0)=

s

d2a@(a, 0) “(’)(a),

(2.53)

where the kernel is given by (2.54) in terms of the matrix elements (2.50) for (-s). The kernel (2.54) is convergent for s > -1 only. Nevertheless, :he remaining relation between the Husimi Qfunction and the Pegg-Barnett distribution can also be expressed by eq. (2.53), albeit with the following kernel (Miranowicz [ 19941):

(2.55)

On integrating the quasiprobability distribution W(’)(a)over the “radial” variable 1 a 1, we get the “phase distribution” associated with this quasiprobability distribution. The s-parametrized phase distribution is thus given by: (2.56)

VI, § 21

PHASE FORMALISMS

313

or equivalently by (2.57) where integration is performed over the intensity W = IaI2.Inserting eq. (2.49) into eq. (2.56) yields:

If the definition of the Laguerre polynomial is invoked, the integrations in eq. (2.58) can be performed explicitly, and we get for the s-parametrized phase distribution a formula similar to the Pegg-Barnett phase distribution (2.37), which reads: (2.59) The difference between eqs. (2.37) and (2.59) lies in the coefficients @)(rn,n), which are given by:

(2.60)

The s-parametrized coefficients @)(m,n) [eq. (2.60)] can be rewritten in a compact form (Miranowicz [ 19941, Leonhardt, Vaccaro, Bohmer and Paul [ 19951) ( m 3 n):

(L) (m-ny2

G(s)(-,n)=/$(+)' s-1 x

r

m-n

1-s

(2.61)

314

QUANTUM PHASE PROPERTIES

in terms of the Jacobi polynomials P?')(x),

or equivalently as (m-n)/2

G(s)(m,n) = $ n! (m-n)! !! (! sE -L 1 ) 1-s

(1)

x

r(

m-n + 1) 2 F I(-n, 2 + 1 , m - n + 1,-

(2.62) 1+ s

using the hypergeometric (Gauss) function #',(a, b, c, x). For s = 0, we have the coefficients for the Wigner phase distribution P(O)(13); i.e., the phase distribution associated with the Wigner function. In this special case of s=O, eq. (2.60) reduces to the expression obtained by TanaS, Murzakhmetov, Gantsog and Chizhov [1992], whereas eq. (2.62) goes over into the expression of Garraway and Knight [ 1992, 19931:

c

21'n-"yL.\li i

\ \

I

[(n - 1)/21!

,

n odd.

Equations (2.61)-(2.63) are given for m > n . Otherwise the indices m and n should be interchanged. For s=-1, only the term with 1=0 survives in eq. (2.60), and we get the coefficients for the Husimi phase distribution P(-')(I3);i.e., the phase distribution associated with the @function. Now, eq. (2.60) reduces to (Paul [ 19741, TanaS, Gantsog, Miranowicz and Kielich [1991], TanaS and Gantsog [ 1992b1): (2.64) It is apparent from eqs. (2.59)-(2.62) that for the phase distribution associated with the P-function (s = l), the coefficients G(S)(m, n) become infinity, and one can conclude that the phase distribution P(')(@ is indeterminate. However, at least for a special class of states, summation can be performed numerically or even analytically for P(O(I3).For instance, for the states described by the density matrix p^ of the form P m n = JPmnl

exp[-i(m - n)601,

(2.65)

the s-parametrized phase distribution P("( 0) can be rewritten as (Miranowicz [ 19941): (2.66)

VI, § 21

375

PHASE FORMALISMS

with the coefficients 00

u:)

Ipm+n,nI G'"(m

=

+ n,n).

(2.67)

n=O

Equations (2.66) and (2.67), for s = O and &=O, go over into expressions obtained by Bandilla and Ritze [ 19931. Numerical calculation of lim, a$) is usually straightforward. For coherent states, the coefficients are equal to unity. Hence, PiLi(O), given by eq. (2.66), is the Dirac delta function b ( O - 8 0 ) [see 4 3.11. Formulas (2.59)-(2.62) allow calculation of the s-parametrized phase distributions for any state with known pmn,and their comparison with the PeggBarnett phase distribution, for which G@)(m, n) E 1. The phase distributions associated with particular quasiprobability distributions have been used widely in the literature to describe the phase properties of field states. For example, the Husimi phase distribution P(-')(O) was used by Bandilla and Paul [1969], Paul [1974], Freyberger and Schleich [1993], Freyberger, Vogel and Schleich [1993a,b], Leonhardt and Paul [1993a], Bandilla [1993], and Khan and Chaudhry [ 19941, in their schemes for phase measurement. Braunstein and Caves [ 19901 applied P(-')(0) to describe the phase properties of generalized squeezed states. The Wigner phase distribution P(O)(O)was used by Schleich, Horowicz and Varro [1989a,b] in their description of the phase probability distribution for highly squeezed states. Herzog, Paul and Richter [1993] showed in general that the Wigner phase distribution can be interpreted as an approximation of the PeggBarnett distribution. To estimate the difference between the P(o)(0)and P(O), they analyzed the deviation of the Wigner function W(O)(a)for a phase state from Dirac's delta function. Recently, Hillery, Freyberger and Schleich [ 19951 have compared the Pegg-Barnett, Husimi, and Wigner phase distributions for largeamplitude classical states. Eiselt and Risken [ 19911 applied the s-parametrized quasiprobability distributions to study properties of the Jaynes-Cummings model with cavity damping. For some field states the quasiprobability distribution functions W@)(a) can be found in a closed form via direct integrations according to the definitions (2.45)(2.47), and sometimes the next integration leading to the s-parametrized phase distributions can also be performed according to definition (2.56). In the next sections, we shall illustrate the differences between the PeggBarnett phase distribution and the s-parametrized phase distributions obtained by integrating the s-parametrized quasiprobability distribution functions. For any field with known number-state matrix elements pmnof the density matrix, the sparametrized phase distribution can be calculated according to formula (2.59) ~

,

376

QUANTUM PHASE PROPERTIES

[VL

5

2

Fig. 1. The coefficients G@)(m,n) for (a) s = 0, and (b) s = -1.

with the coefficients G(S)(m,n)given by eq. (2.60). The distribution of the coefficients Gb)(m,n), for s = 0, -1, is illustrated in fig. 1. It is apparent that for the Husimi phase distribution the coefficients decrease monotonically as we go further away from the diagonal. This means that all nondiagonal elements pmn are weighted with numbers that are less than unity, and the phase distribution for s=-1 is always broader than the Pegg-Barnett phase distribution (for which G(S)(rn,n) 2 1). For s=O the situation is not so simple, because the coefficients G(O)(m,n) show even-odd oscillations with values that are both smaller and larger than unity. This usually leads to a phase structure sharper than the Pegg-Barnett distribution. Moreover, since the Wigner function (s = 0) can take negative values, the positive definiteness of the Wigner phase distribution is not guaranteed. Also, the oscillatory behavior of the coefficient G(O)(m,n) suggests that, at least for some states, the Wigner phase distribution P(O)(0)can exhibit negative values. This nonclassical feature of P(O)(0) was shown explicitly by Garraway and Knight [ 1992, 19931for the simple example of the number state superposition (only for convenience, we assume that m > n):

I Y )= 2-”2(ln) + 1.2)).

(2.68)

In a straightforward manner, the general expressions for the phase distributions P ( 0 ) [eq. (2.37)] and P(S)(0)[eq. (2.59)] reduce to: 1 ~ ( 0=) -(I 2n

+ cos[(m - n)el),

(2.69)

1 P(”(0) = - (1 + G(S)(m, n) C O S [ ( ~n)0]) , 2n

(2.70)

and

respectively. The Pegg-Barnett, P(@, and Husimi, P(-’)(0),phase distributions are obviously positive definite for any superposition (2.68). As seen in fig. 2,

VI, § 21

377

PHASE FORMALISMS

-8

-0.10

I:,, ,

,

,

.

,

,

I

,

,

0

-0.15

0

2

4

6

8

In> + ln+3> In>+ln+4>

10 12 14 16 18

n

20

Fig. 2. The minima of the Wigner phase distributions l"')(e), eq. (2.70), for the superpositions of two number states (2.68) for various values of n and m - n = 1, 2, 3, 4.

the Wigner phase distribution P(')(O) is positive for superpositions with odd n. However, it takes negative values for even n. In this case, the smaller is n for fixed m - n, or the higher is the value of m - n for a given n, the minimum of the Wigner phase distribution is more strongly negative. Hence, one obtains the greatest negativity for the superposition (10) + 12m))/& in the limit of m -+ 00. As was emphasized by Garraway and Knight [1993] (see also fig. 2), for large values of n the Pegg-Barnett distribution is approached for both even and odd m. It is highly illustrative to consider analytically the special case of eq. (2.68) when m - n = 2 (Garraway and Knight [1992, 19931). These results will be helpful in the analysis presented in Q 3.2 for even and odd coherent states. Now, the coefficients G($)(m,n), given by eqs. (2.60)-(2.62), can be rewritten in a much simpler form: G(')(n + 2, n) = 2J(n

+ + '-' l)(n

2) [ ( s ) n + 2 - l ]

+

(5)"'. (2.71)

For s=O, eq. (2.71) goes over into (Garraway and Knight [1992, 19931): (2.72) and for s = - 1 one obtains

G(-')(n+ 2, n) =

(S) . l/'

(2.73)

Equation (2.72) provides direct proof of the oscillatory behavior of G(O)(n+ 2, n) with increasing n. For even n, the right-hand side of eq. (2.72) is greater than

378

QUANTUM PHASE PROPERTIES

[VI, § 3

unity, whxh implies a negative minimum of the Wigner phase distribution (2.70) [solid circles in fig. 21. However, for odd n, the coefficients G(O)(n+ 2, n) are less than unity and equal to G(-’)(n+ 2, n). Hence the Husimi and Wigner phase distributions for such states (with odd n) are equal and positive definite. From the form of the coefficients G(S)(m, n) it is evident that there is no s such that G(’)(m,n) = 1 for all m, n. This means that there is no “phase ordering” of the field operators; that is, the ordering for which P(S)(13)would be equal to P(I3). However, for a given state of the field one can find s such that the two distributions are “almost identical”. Formula (2.59) is quite general, and it was used in earlier studies of the phase properties of the anharmonic oscillator (Tanah, Gantsog, Miranowicz and Kielich [ 199l]), parametric down conversion (Tanah and Gantsog [ 1992b]), and displaced number states (TanaS, Murzakhmetov, Gantsog and Chizhov [1992]). A disadvantage of formula (2.59) is the fact that the numerical summations can be time consuming and even difficult to perform for field states with slowly converging number-state expansions. This, for example, is the case for highly squeezed states. In some cases, instead of using the number-state expansions, we can find analytical formulas for P(’)‘(13) via direct integrations, as shown in 4 3. In many cases such formulas can be treated as good approximations to the Pegg-Barnett phase distribution.

Q 3. Phase properties of single-mode optical fields Optical fields produced in nonlinear optical processes have specific phase properties which depend on the nonlinear process in which the field is produced and on the state of the field before it undergoes the nonlinear transformation. Since there is a large variety of nonlinear optical processes, there is the possibility to generate fields with different phase properties. Here, some examples of such field states will be given and their phase properties discussed briefly. 3.1. COHERENT STATES

The most common single-mode field in quantum optics is a Glauber coherent state. Its phase properties have probably been analyzed within each known phase formalism. We shall focus our attention on two of them only.

VI, § 31

PHASE PROPERTIES OF SINGLE-MODE OPTICAL FIELDS

379

The s-parametrized quasiprobability distribution function for a coherent state, lao) = &o) lo), can be calculated from eqs. (2.45)42.47) as:

= 1 n2

-

/

(3.1)

-1

exp [(a- ao)E*- (a*- a;))E+ s El2 (0 2

--exp XI-s

{

Ib(E)lO)d2E

2 2 Ia-ao12). 1 -s

--

(3 4 The corresponding s-parametrized phase distribution is (Tanas, Miranowicz and Gantsog [ 19931; for s = 0, see also Garraway and Knight [ 19931 and Bandilla and Ritze [ 19931): P(”(0)

= =

where

lrn

W(”)(a)la1 dial

1 -exp 216

[-(x; -x’)>] { exp(-X2) + &X(I + erf(X))} ,

(3.3)

and X o = X ( y ) ( 6 0 ) 60 ; is the phase of ao. The phase distribution P(l)(8) associated with the P-function can be obtained from eqs. (3.3) and (3.4) in the limit of s 4 1: P(l)(e) = s(8 - 60),

(3.5)

which is the Dirac delta function. This result can also be achieved from eq. (2.66). As was shown by Miranowicz [1994], the coefficients a:) are unity for arbitrary m. Hence, eq. (2.66) reduces to:

which is the desired function (3.5). This example shows that the general expression (2.59) for the s-parametrized phase distributions is also valid in the special case of s = 1.

380

PI,5 3

QUANTUM PHASE PROPERTIES

c a

n

0.5

0.0

-3.14

-1.57

0.00

e

1.57

0.10

-3.14

3.14

-1.57

0.00

e

1.57

3.14

Fig. 3 . Phase distributions for the coherent states with the mean number of photons: (a) laOl2 = 2, and (b) la0I2 = 0.01; the Pegg-Barnett distribution (solid line), the Wigner phase function P(O)(O) (dashed line), and the Husimi phase distribution $-I)(@ (dotteddashed line).

Formula (3.3) is exact, it is 2n-periodic, positive definite and normalized, so it satisfies all requirements for the phase distribution. Moreover, formula (3.3) has a quite simple and transparent structure. For small laoJ,the first term in braces plays an essential role, and for la01 + 0 we get a uniform phase distribution. For large ( a o (the , second term in the braces predominates, and if we replace erf(X) by unity, we obtain the approximate asymptotic formula given by Schleich, Dowling, Horowicz and Varro [ 19901 (for s = 0):

which however, can be applied only for -in < (6 - 60)< in.After linearization of formula (3.7) with respect to 6, the approximate formula for coherent states with large mean number of photons obtained by Barnett and Pegg [ 19891 is recovered. The presence of the error function in eq. (3.3) handles properly the phase behavior in the whole range of phase values -n (0 - 60)< n. The Pegg-Barnett distribution P(6) for the coherent state lao) can be calculated from eq. (2.36) with the superposition coefficients

<

c,

=

b, exp(indo),

b,

= exp

(

-la;l2)

12;

-

The exact formula for the s-parametrized phase distributions P(”( 0) for coherent states is given by eqs. (3.3) and (3.4). Alternatively, the P(’)(O) are given by eq. (2.59) after insertion of c, given by eq. (3.8). In fig. 3 we show the phase distributions P(0), P(o)(6),and P(-’)(6) for a coherent state with the mean number of photons Iao12 = 2 (a), and = 0.01 (b). It is seen that the Pegg-Barnett phase distribution is located somewhere between the Wigner

VI,

I 31

PHASE PROPERTIES OF SINGLE-MODE OPTICAL FIELDS

381

and Husimi phase distributions. It becomes closer to P(O)(O)for (a0l2>> 1, and closer to P(-')(O) for laOl2 << 1. For laol* --t m, the Pegg-Barnett distribution tends to the Wigner phase distribution (Schleich, Horowicz and Varro [1989a], Barnett and Pegg [1989]), and for [aOl2+ 0 all the distributions tend to the uniform distribution, but the Pegg-Barnett distribution in this region tends to the Husimi phase distribution. T h s means that for coherent states with large mean numbers of photons, P(O)(O)is a good approximation to the Pegg-Barnett phase distribution, while for small numbers of photons P(-')(O) becomes a good approximation to the Pegg-Barnett distribution. 3.2. SUPERPOSITIONS OF COHERENT STATES

Superpositions of macroscopically distinguishable coherent states have attracted much interest (see, for example, Buiek and Knight [ 19951 and references therein) due to their property of being prototypes for the Schrodinger cats, and important nonclassical properties, such as sub-Poissonian photon statistics, quadrature squeezing, higher-order squeezing, etc. Their phase properties have also been a subject of interest. Let us consider the normalized superposition Iq) of coherent states defined as:

I+)

N =

ck I exp(@khO).

(3.9)

k= 1

This superposition of two well-separated components is called a Schrodinger cat, whereas for N > 2 the notions Schrodinger cat-like state or kitten states are often used. The phase distributions P(O) [eq. (2.37)] and P(s)(0)[eq. (2.59)] for the state (3.9) can be rewritten in a form showing explicitly the superposition structure (Tanas, Gantsog, Miranowicz and Kielich [19911, Garraway and Knight [1992, 19931, Buiek, Gantsog and Kim [1993], Buiek, Kim and Gantsog [1993], Tara, Aganval and Chaturvedi [ 19931, Hach 111 and Gerry [1993], Buiek [1993], Miranowicz [1994], Buiek and Knight [1995]). The Pegg-Barnett phase distribution P(O) splits into two terms (TanaS, Gantsog, Miranowicz and Kielich [1991]):

P(O) = Po(O) +pidO),

(3.10)

where (3.1 1)

382

QUANTUM PHASE PROPERTIES

[VI, § 3

is the sum of phase distributions (3.12) for the coherent states of the superposition, and the second distribution N

Pint(6)=

C

ckc;pki(O),

(3.13)

k,l=l k t l

represents interference terms emerging from the quantum interference between the component states of the superposition. In fig. 4, the phase distributions (3.10) and (3.1 1) are presented in polar coordinates for the discrete superpositions of coherent states in the anharmonic oscillator model [see $3.5, eq. (3.57)]. It is evident from fig. 4 that as the number of components in the superposition becomes larger, the interference terms play an increasing role and the symmetry of the Pegg-Barnett distribution [eq. (3.10)] is destroyed. These terms are negligible for well-separated components of the superposition only (Tanai, Gantsog, Miranowicz and Kielich [ 19911). Analogously, the s-parametrized quasidistribution W@)( a ) for the superposition state (3.9) is represented as (Miranowicz [ 19941; for s = -1, see Miranowicz, Tanai and Kielich [1990], and for s=O, see also Buiek and Knight [1995]):

W(’)(a)= Wf’(a> + ~ i ’ ; f ( a > ,

(3.15)

where the sum of coherent terms is (3.16) with

W f ’ ( a )= --

nl l -2s

exp

{A --

(3.17)

(3.18)

VI,

:Elo.oi ,El 5 31

383

PHASE PROPERTIES OF SINGLE-MODE OPTICAL FIELDS 0.4

0.0

r = 2 r 1 20.2

-0.2

0.0

r=2x13

-0.6

-0.6

0.0

-0.4 -0.4

0.0

:#

0.6

.0.4

-0.4

0.0

-0.4 -0.4

0.0

0.4

.--

'

-0.4 -0.4

= ';

0.0

2r I 5

0.4

r=2~16 0.4

r=2nlI 0.4

Fig. 4. The Pegg-Bamett phase distributions in polar coordinates for the discrete superpositions of coherent states (3.57) with N = 2-7 components in the anharmonic oscillator model for the initial mean photon number la0I2 = 4; the exact phase distributions P ( @ , eq. (3.10) [solid lines], and the functions Po(@, eq. (3.11) [dashed lines], without interference contribution.

with

(3.19)

In eq. (3.19) the phases are Yk = Arg Ck, 8 = Arg a , 290 = Arg ao,and $k appears in the definition (3.9). On integration, we obtain the following form of the sparametrized phase dlstribution P")( 8) (Miranowicz [ 19941):

pe)((j)= p(s) 0 (6) + P : h

(3.20)

i.e., a simple sum N

(3.21) k=l

384

QUANTUM PHASE PROPERTIES

PI, § 3

of coherent terms pf)(e)=

1

exp [-

( ~ 2 ,- x:)]{ exp (-xi)+ f i x k

where

[I + erf(~k)l}, (3.22)

(3.23) and the sum (3.24)

of the interference terms (3.25) with (3.26)

(3.27) The Schrodinger cat of the form

with the normalization

N ~ [=2 ( i + cosyexp(-21a12))]

-1/2

,

(3.29)

is a special case of the superposition state (3.9). The cat (3.28) consists of two coherent states la) and )-.I , which are $ = n out of phase with respect to each other. The state (3.28) is not only of theoretical interest, since several methods were proposed for generation and measurement of this Schrodinger cat (e.g.,

VI, 5 31

PHASE PROPERTIES OF SINGLE-MODE OPTICAL FIELDS

385

Brune, Haroche, Raimond, Davidovich and Zagury [ 19921, Garraway, Sherman, Moya-Cessa, Knight and Kurizki [1994]). The state la, in) (i.e., for y = in) is called the Yurke-Stoler coherent state (Yurke and Stoler [1986, 19881). This state can be generated in the anharmonic oscillator model (see 9 3.5). For other choices of y, the state (3.28) goes over into the even coherent state la, 0 ) or the odd coherent state I a, X),which have the following Fock representations (Peiina [1991]): (3.30) 00

12n + 1 )

(3.3 1)

n=O

The dissimilar phase properties of the evedodd coherent states were analyzed by Garraway and Knight [1993] (see also Buiek and Knight [1995]). Their phase distributions P(O) and P(s)(0)can be obtained readily from the general expressions (3.10) and (3.20), respectively. Obviously, they consist of the normalized sum of the phase distributions Pl,z(O) [or Pr$O)] for coherent states located at a and (-a) in the phase space plane, together with an additional interference term P12(0) [or Pyi(O)]. As was shown by Garraway and Knight [ 19931, the Wigner phase distribution P(O)(O)for the even coherent state [eq. (3.30)] can exhibit negative values, in contrast to P(O)(O) for the odd coherent state [eq. (3.31)], which never does. The Fock expansion [eq. (3.30)] of the even coherent state contains only even photon numbers, similar to the superposition Ineven)+ Ineven+ 2) discussed in 0 2.1 [see eq. (2.68) and fig. 21. Analogously, the odd coherent state [eq. (3.31)] and Inodd) + Inodd + 2) contain only odd number states. Hence, these dissimilar features of the functions P(O)(O) for I a, 0 ) and I a, n)are well understood for the same reasons as those given in § 2.1 in the analysis of the Wigner phase distribution for the superposition of the two number states and the interpretation of the oscillatory behavior of the coefficients G(O)(rn,n)[fig. la]. 3.3. SQUEEZED STATES

Squeezed states have phase-sensitive noise properties, and it is particularly interesting to study their phase properties. Sanders, Barnett and Knight [ 19861, Yao [1987], Loudon and Knight [1987], and Fan and Zaidi [1988] used the Susskind-Glogower formalism in a description of the phase fluctuations of

386

[VI,

QUANTUM PHASE PROPERTIES

5

3

squeezed states. Lynch [ 19871 applied the measured-phase formalism of Barnett and Pegg [ 19861. Vaccaro and Pegg [ 19891and Vaccaro, Barnett and Pegg [ 19921 investigated phase properties of a single-mode squeezed state from the point of view of the new Pegg-Barnett phase formalism. Grranbech-Jensen, Chnstiansen and Ramanujam [1989] made comparisons of the phase properties of a singlemode squeezed state obtained according to different phase formalisms, including that of Pegg and Barnett. Burak and Wbdkiewicz [1992] introduced a phasespace propensity description of quantum phase fluctuations and analyzed, in particular, squeezed vacuum. The phase properties of the squeezed states have recently been studied by Cohen, Ben-Aryeh and Mann [1992], and by Collett [1993a,b]. Various measures of phase uncertainty and their dependence on the average number of photons were studied by Freyberger and Schleich [ 19941. Squeezed states (ideal squeezed states, two-photon coherent states) are defined by (see Loudon and Knight [1987]):

c) &ao) &) 10) where s^(c)is the squeezing operator lao,

=

(3.32)

9

s(()= exp(ic*ii2 icii2),

A

(3.33)

-

and

c is the complex squeeze parameter c r-e2iq, Y 1 ~ 1 . =

(3.34)

=

The direct integrations lead to the s-parametrized quasiprobability distribution (for r] = 0): 1 W(”(a)= -

2

-=

d(P- W - l

x exp{

2

- s)

[~m(a ao)12-

5 2

-

(3.35) where we have used the notation p=exp(2r). After integration over lal, assuming that a. is real, we arrive at the formula (Tanai, Miranowicz and Gantsog [ 19931):

d(P- W - l - s> P‘S’(8) = 1 2 n ( p - s) cos2 8 + (p-1 - s) sin2 8

(3.36)

VI, § 31

where

x = x(q8)=

PHASE PROPERTIES OF SINGLE-MODE OPTICAL FIELDS

/-+

aoJiu‘scos 8

2

,/(p

- s)

cos2 8 + (p-1 - s) sin2 8

387

(3.37)

Although the variable X is slightly different, the main structure of the phase distribution is preserved. Formula (3.36) is valid for both small and large ao. For a. = 0 we have the result for squeezed vacuum. After appropriate approximations, one easily obtains the formula derived by Schleich, Horowicz and Varro [ 1989al for a highly squeezed state. The exact analytical formula for the s-parametrized phase distribution for squeezed states, as given by eqs. (3.36) and (3.37), for the squeezed vacuum takes the form (3.38) where p = exp (2r). This formula exhibits a two-peak structure with peaks for 8= *in (for Y > 0). It is easy to find that the peak heights are: (3.39) meaning that for s=O, the peak height is proportional to p. One can easily check that the Pegg-Barnett result lies between the s=O and s=-1 results. Qualitatively, all three distributions give the same two-peak phase distributions, but they differ quantitatively: the sharpest peaks are those of P(O)(O), and the broadest those of P(-’)(8). For squeezed states with non-zero displacement ao,an additional factor of a form identical with that for coherent states, except for the different meaning of X(@, appears in the phase distribution P(”(8). Since this extra factor shows a peak at 8=0, a competition arises between the two-peak structure of the squeezed vacuum and the single-peak structure of the coherent component. This competition leads to the bifurcation in the phase distribution discussed by Schleich, Horowicz and Varro [ 1989a,b]. Figure 5 illustrates such a bifurcation for a0 = 1, as exhibited by the Wigner and Husimi phase distributions plotted on the same scale to visualize the differences. Qualitatively,the figures are quite similar, and differ only in the widths of the peaks. The Pegg-Barnett distribution in this case is very close to the Wigner phase distribution, and for this reason

388

QUANTUM PHASE PROPERTIES

Fig. 5. Pictures of the phase bifurcation for the squeezed state with the mean number of photons laoI2= 1. The distributions are: (a) F"O)(e),and (b) &'))@). The Pegg-Barnett distribution is very close to (a).

we omit it here. To calculate the Pegg-Barnett phase distribution one can apply formula (2.36) with cn given by (see Loudon and Knight [ 19871):

(3.40) assuming r]=O (results for r]= in can be obtained on replacing Y by -7). Approximate analytical formulas for the phase variance as well as cosine and sine variances were obtained by Vaccaro and Pegg [1989] for weakly squeezed vacuum. For large squeezing, the squeezed vacuum phase variance asymptotically approaches n214, which corresponds to the phase distribution with two symmetrically placed delta functions:

Ideal squeezed vacuum is generated in the parametric down-conversion process, in which the pump field is treated as a constant classical quantity. Tahng into account the quantum character of the pump, one finds that the signal field is no longer the ideal squeezed vacuum and its phase properties are different (Tanai and Gantsog [1992a]) [see 5 4.51. 3.4. JAYNES-CUMMINGS MODEL

The Jaynes-Cummings model (Jaynes and Cummings [ 19631) (see reviews by Yo0 and Eberly [1985] and Shore and Knight [1993]) is the most popular

VI, § 31

PHASE PROPERTIES OF SINGLE-MODE OPTICAL FIELDS

389

model used to describe the resonant interaction of one two-level atom with one mode of the electromagnetic field. One of the most remarkable effects predicted theoretically (Eberly, Narozhny and Sanchez-Mondragon [ 19801, Narozhny, Sanchez-Mondragon and Eberly [ 198I]) and then observed experimentally (Rempe, Walther and Klein [ 19871) in the Jaynes-Cummings model are collapses and revivals of the atomic inversion. Eiselt and Risken [1989], using the Qfunction, have shown that the collapses and revivals can be understood in terms of interferences in phase space. Phoenix and Knight [1990] mentioned the splitting of the phase probability distribution into two counter-rotating satellite distributions in a model consisting of two degenerate atomic levels, coupled through a virtual level by a Raman-type transition. Dung, Tanai and Shumovsky [ 19901 discussed the collapses and revivals in this model from the point of view of the field-mode phase properties studied in the framework of the Pegg-Barnett formalism. The model is described by the Hamiltonian (at exact resonance):

ii = hw(& + 2)+ hg(&

+kit),

(3.42)

where B t and d are the creation and annihilation operators for the field mode; the two-level atom is described by the raising, gt, and lowering, E, operators and the inversion operator @, and g is the coupling constant. To study the phase properties of the field mode, we must know the state evolution of the system. After dropping the free evolution terms, which change the phase in a trivial way, and assuming that the atom is initially in its ground state and that the field is in a coherent state lao) ,the state of the system is found to be: 00

b,exp(in&) [ c o s ( h g t >In,g) -isin(>)

11)(t))=

In- l,e)] ,

(3.43)

n=O

where 18) and Ie) denote the ground and excited states of the atom, the coefficients b, are given by eq. (3.8), and 60is the coherent state phase. According to the Pegg-Barnett formalism, one gets the phase distribution P ( 0 ) in the form (Dung, Tanai and Shumovsky [1990])

{

P(@= 2 n 1 + 2 1 bnbkcos[(n-k)O] cos[(fi-&)gt]}. n>k

This formula can be rewritten into the form

(3.44)

390

[VI,

QUANTUM PHASE PROPERTIES

0.5

*

T4.4

5

3

(b)

’I,

o.a

-0.5 ~~

-2.0

0.0

2.0

4.0

-1.5

-1.0 -0.5 0.0 0.5

1.0

Fig. 6 . The Pegg-Barnett phase distribution (3.44) of the Jaynes-Cummings model as a function of scaled time T = g f / ( 2 nlaol) for the initial mean photon number lao12 = 20.

where

which shows explicitly that as time elapses, the phase distribution P ( 0 ) splits into two distinct, right and left rotating, distributions in the polar coordinate system. Polar plots of the phase distribution are shown in fig. 6 (the time T = gt/(27c lao/) is scaled in the revival times). So, after a certain interval of time, the two counterrotating distributions “collide”, and at that time the components of the field oscillate in phase and one can expect the revival of the atomic inversion. The numerical calculations corroborate this statement (Dung, TanaS and Shumovsky [1990]). This behavior of the phase distribution resembles the behavior of the Q-function studied by Eiselt and Risken [1991]. The time behavior of the phase variance together with the phase-probability density distribution carries certain information about the collapses and revivals. To show this, we first give the explicit expression for the variance. Using eqs. (2.36) and (3.44), one obtains: (3.47) Variance (3.47) is illustrated graphically in fig. 7 for laof = 20. The variance goes up initially and reaches a maximum at the scaled time T = 1, which is the first revival time. The revival times correspond to the extrema of the phase variance. In this way, the well-known phenomenon of collapses and revivals has obtained clear interpretation in terms of the cavity-mode phase. More details can be found in the paper by Dung, TanaS and Shumovsky [1990]. The dynamical

VI, § 31

391

PHASE PROPERTIES OF SINGLE-MODE OPTICAL FIELDS

21.0

a

I

I

20.8

20.6

V 20.4 20.2 20.0

10.0

.

,

I

I . ,

~

,

I

T Fig. 7. Evolution of (a) the mean photon number (li) and (b) the variance Barnett phase operator for the Jaynes-Cummings model as a function of scaled time T = gt/(2?c Iaol) for la012 = 20.

properties of the field phase in the Jaynes-Cummings model were studied by Dung, TanaS and Shumovsky [1991a], and the effects of cavity damping by Dung and Shumovsky [ 19921. Some generalizations of this simple model were also considered from the point of view of their phase properties (Dung, TanaS and Shumovsky [1991b], Meng and Chai [1991], Meystre, Slosser and Wilkens [1991], Dung, Huyen and Shumovsky [1992], Meng, Chai and Zhang [1992], Peng and Li [1992], Peng, Li and Zhou [1992], Wagner, Brecha, Schenzle and Walther [1992, 19931, Fan [1993], Jex, Matsuoka and Koashi [1993], Drobnf, Gantsog and Jex [1994], Fan and Wang [1994], Meng, Guo and Xing [1994]).

3.5. ANHARMONIC OSCILLATOR MODEL

The anharmonic oscillator model is described by the Hamiltonian

@ = h0JGtd + ; h K G t 2 i 2 ,

(3.48)

where G and Gt are the annihilation and creation operators of the field mode, and K is the coupling constant, whch is real and can be related to the nonlinear susceptibility x(3)of the medium if the anharmonic oscillator is used to describe propagation of laser light (with right or left circular polarization) in a nonlinear

392

QUANTUM PHASE PROPERTIES

[VI, § 3

Kerr medium. If the state of the incoming beam is a coherent state lao), the resulting state of the outgoing beam is given by:

where z = - - ~ t . In the problem of light propagating in a Kerr medium, one can make the replacement t=-z/u to describe the spatial evolution of the field instead of the time evolution. On introducing the notation a. = lao[exp(i&), the state (3.49) can be written as (3.50) where b, is given by eq. (3.8). The appearance of the nonlinear phase factor in the state (3.50) modifies essentially the properties of the field represented by such a state with respect to the initial coherent state I ao). It was shown by TanaS [ 19841 that a high degree of squeezing can be obtained in the anharmonic oscillator model. Squeezing in the same process was later considered by Kitagawa and Yamamoto [1986], who used the name crescent squeezing because of the crescent shape of the quasiprobability distribution contours obtained in the process. The evolution of the quasiprobability distribution Q(a,a') in the anharmonic oscillator model was considered by Milburn [1986], Milburn and Holmes [1986], PeiinovL and Luki [1988, 19901, Daniel and Milburn [1989], and Peiinovi, LukS and Karska [1990]. The states that differ from coherent states by extra phase factors, as in eq. (3.50), are the generalized coherent states introduced by Titulaer and Glauber [ 19661 and discussed by Stoler [ 19711. Bialynicka-Birula [ 19681 has shown that, under appropriate periodic conditions, such states become discrete superpositions of coherent states. Yurke and Stoler [ 19861, and Tombesi and Mecozzi [ 19871 discussed the possibility of generating quantummechanical superpositions of macroscopically distinguishable states in the course of evolution for the anharmonic oscillator. Miranowicz, TanaS and Kielich [ 19901 have shown that superpositions with not only even but also odd numbers of components can be obtained. The Pegg-Barnett Hermitian phase formalism has been applied to the study of the phase properties of the states (3.50) by Gerry [1990], who discussed the limiting cases of very low and very high light intensities, and by Gantsog and TanaS [1991fl, who gave a more systematic discussion of the exact results. Phase

VI,

P

31

393

PHASE PROPERTIES OF SINGLE-MODE OPTICAL FIELDS

fluctuations in the anharmonic oscillator model were also analyzed within former phase formalisms (Gerry [1987], Lynch [1987]). The continuous Pegg-Barnett phase probability distribution (2.36) for the field state (3.50) takes the following form:

and the s-parametrized quasiprobability distribution function is now given by (Miranowicz [ 19941):

4

+ J o ( i1- - s la1 laol)}, (3.52) where Jo(x) is the Bessel function. For t = O , W(.)(a),given by eq. (3.52), is the coherent-state distribution [eq. (3.2)]. In the special case, for @function (s= -l), eq. (3.52) reduces to:

lx

1

Q(a,t)= -exp(- 1aI2- laoI2)

n

O0

(a*aO)n

n!exp

fl=O

The s-parametrized phase distribution resulting from eq. (3.52) is:

2n z x cos (m - n)(e - 6 0 ) - - [m(m- 1) - n(n - 1111 2

[

},

(3.54) where the coefficients G(’)(m,n) are given by eq. (2.60). Symmetrization of the phase window with respect to the phase 60 as done for the Pegg-Barnett phase distribution [eq. (3.5 l)] is equivalent to introduction of the relative phase variable 8 - fro, and the two formulas differ only by the coefficients G(”(m, n),

394

QUANTUM PHASE PROPERTIES

[VI, § 3

as in eq. (2.59). For t = O , eqs. (3.51) and (3.54) describe the phase probability distributions for the initial coherent state lao).When the nonlinear evolution is on (z # 0), the distributions P ( 0 ) and P(’)(e) acquire some new and very interesting features. A systematic discussion of the properties as well as the plots of P ( 0 ) and P(-’)(O) are given by Tanai, Gantsog, Miranowicz and Kielich [1991], and by Gantsog and Tanai [1991fl. The phase distribution P ( 0 ) can be used to calculate the mean and the variance of the phase operator, defined by eqs. (2.24) and (2.25). The results are (Gantsog and TanaS [1991fl):

;n2 =-

3

+4

[n(n - 1) - k(k - 1)]} n>k

(3.56)

[n(n - 1) - k ( k - l)]} For t = 0, we recover the results for a coherent state with the phase 60[eqs. (2.31) and (2.39)]. The nonlinear evolution of the system leads to a nonlinear shift of the mean phase and essentially modifies the variance. An example is illustrated in fig. 8, where the evolution of the mean phase (fig. 8a) and its variance (fig. 8b) are plotted against z for various values of I a0 1’ . We have assumed 60= 0, and the window of the phase values is taken between -nand n.The evolution is periodic with the period 2n, so the initial values are restored for z = 2n. Figure 8a shows the intensity-dependent phase shift. The amplitude of the mean-phase oscillation becomes larger with increasing mean number of photons. The line n2/3 in fig. 8b marks the variance for the state with random distribution of phase. It is seen clearly from fig. 8b that the stronger the initial field, the higher the phase variance. For (ao12= 4, the phase variance increases rapidly and most of the period oscillates around n2/3- the value for uniform phase distribution. This means that the phase is randomized during the evolution, although it periodically reproduces its initial values. This tendency is even more pronounced when the mean number of photons increases. The periodicity of the evolution is destroyed

VI,

I 31

395

PHASE PROPERTIES OF SINGLE-MODE OPTICAL FIELDS

6.0r

I

I

1.5

Q

A

v 0.0

-1.5

I

Fig. 8. Evolution of (a) the mean phase (3.55) and (b) the phase variance (3.56) for the anharmonic oscillator model.

by damping (Gantsog and TanaS [1991b]). The sine and cosine functions of the phase were also calculated and the results compared with other approaches (Gantsog and TanaQ[ 1991fl). The local minima in (AGO) apparent in fig. 8 indicate the points in the evolution in which superpositions of coherent states are formed, and the phase variance decreases at these points. This occurs for z=2nMIN (N =2, 3, 4, . . . , and M , N are mutually prime numbers), for which the P ( 0 ) and P(')(0)plotted in polar coordinates show N-fold symmetry, confirming the generation of discrete superpositions of coherent states with 2, 3, 4, . . . , components:

(

'>

(3.57)

where the phases @k are simply 7d

@k=zk,

k = l , ..., 2N,

(3.58)

and the superposition coefficients Ck, representing the so-called fractional revivals, are given by (Averbukh and Perelman [1989], Tanag, Gantsog, Miranowicz and Kielich [ 199 11) ck =

1 2N

2N

-

eXp (-in=l

n N

[nk - Mn(n - I)])

(3.59)

Such superpositions, created during the evolution of the anharmonic oscillator model, have very specific phase properties discussed in 0 3.2. Plots of the phase

396

QUANTUM PHASE PROPERTIES

[VI,

53

distributions (3.10) and (3.11) (where N should be replaced by 2N) for the superpositions (3.58) with several components are presented in fig. 4. The phase distribution indicates the superpositions in a very spectacular way, as shown by TanaB, Gantsog, Miranowicz and Kielich [ 199I], Gantsog and TanaB [ 1991fl and Sanders [19921for the anharmonic oscillator model, and by Paprzycka and Tanai [ 19921 for the model with higher nonlinearities. 3.6. DISPLACED NUMBER STATES

Other states which are interesting from the point of view of their phase properties are the displaced number states lao,no) generated by the action of the displacement operator @ao) on a Fock state 10.) (see De Oliveira, Kim, Knight and Buiek [ 19901); lao, no) = &o>

Ino) ‘

(3.60)

In a special case, when no =0, the states (3.60) become a coherent state loo). The s-parametrized quasiprobability distribution for the state (3.60) is

whereas the s-parametrized phase distribution becomes (TanaS, Miranowicz and Gantsog [ 19931):

k=O

(3.62)

here,

(3.63)

(3.64)

VI, § 31

PHASE PROPERTIES OF SINGLE-MODE OPTlCAL FIELDS

391

and the normalization constant is equal to

N,

=

1+

The X variable in this case is

and we have assumed a. as real. Despite its more complex structure, formula (3.62) contains phase distributions P,(X) that exhibit the main features of the previous phase distributions P(s)(0);i.e., eq. (3.3) for a coherent state and eq. (3.36) for a squeezed state. Displaced number states have the following Fock expansion

(3.67) n

where the amplitudes b, and phase factors @,, are:

= exp(-i

(fl)

la0l'> n,!

1/2

(-1)-

+ no - n-,

n-

= min{n,

@,,

= (n - no) Arg a. = (n - no) fl0,

no},

n+

=n

laOln+-nL;;-"-(lao12),

(3.68) (3.69) (3.70)

which on insertion into eq. (2.36) give explicitly the Pegg-Barnett distribution P ( 0 ) . Both for coherent states and squeezed states, there was no qualitative difference between various phase distributions. Thus, one could say that at least qualitatively, all the phase distributions carried the same phase information. Here, we give an example of states for which the above statement is no longer true. These are displaced number states. The phase properties of such states were discussed earlier by Tanas, Murzakhmetov, Gantsog and Chizhov [ 19921. It was shown that there is a qualitative difference between the Husimi phase distribution on one side, and the Pegg-Barnett and Wigner phase distributions

398

QUANTUM PHASE PROPERTIES

[VI, § 3

Fig. 9. Phase distributions for the displaced number state with n = 2 and a0 = 3 . Meaning of the lines is the same as in fig. 3.

on the other. There is an essential loss of information in the case of the Husimi phase distribution. The differences can be interpreted easily when the concept of the area of overlap in phase space introduced by Schleich and Wheeler [ 19871 is invoked. Formula (3.62) provides the possibility of deeper insight into the structure of the s-parametrized phase distributions. The phase distribution P(”( 0) is a result of competition between the functions P,(X), which are peaked at 8 = 0, and the functions (Xi- X 2 ) ‘ ,which have peaks at 8 = fn/2. For s = - 1, only the term with n - k = 0 survives, and there is no modulation due to the (-l)”-k factor. This is the reason why the Husimi phase distribution can have at most two peaks, no matter how large is n. Both for the Pegg-Barnett phase distribution and P(’)(O) there are n + 1 peaks. It is also worth noting that despite the fact that the Wigner function W(’) [eq. (3.61)] oscillates between positive and negative values, the Wigner phase distribution P(O)(O)[eq. (3.62)] is positive definite. An illustration of the differences between the phase distributions for the displaced number states 2 with n = 2 and (ao( = 9 is shown in fig. 9. It is seen that the Pegg-Barnett phase distribution is very close to P(’)(O),and that they carry basically the same phase information, while there is an essential loss of phase information carried by P(-’)(8).The Pegg-Barnett and P(’)(8) distributions are very similar for given n, while P(-’)(O)has at most two peaks that become broader as n increases. This example shows the difference between a “pure” canonical phase distribution such as the Pegg-Barnett distribution, whch could be associated with a “pure” phase measurement, and a “measured” phase distribution such as P(-’)( O), which can be associated with the noisy measurement of the phase. The noise introduced by the measurement process reduces the phase information that can be inferred from the measured data.

VI, I 41

PHASE PROPERTIES OF TWO-MODE OPTICAL FIELDS

0

399

4. Phase Properties of Two-Mode Optical Fields

The single-mode version of the Pegg-Barnett phase formalism can be extended easily into the two-mode fields (Barnett and Pegg [ 19901) that are often a subject of consideration in quantum optics. This leads to the joint phase probability distribution for the phases of the two modes, and allows the study of not only the individual mode phase characteristics discussed above but also essentially two-mode phase characteristics such as correlation between the phases of the two modes. The phase properties of a two-mode field are simply constructed from the single-mode phases (see 0 2.1). The two-mode joint phase distribution is given by

This phase distribution can be applied, similar to the one-mode case, for calculations of the mean values of the phase-dependent quantities, such as individual phases, their variances, etc. We are often interested not in the individual phases corresponding to either mode, but rather in the operators or distributions representing the sum and difference of the single-mode phases, which can also be calculated using the joint phase distribution [eq. (4.1)]. However, the phase sum and difference values will cover the 4 n range, and the integrations over the phase sum and difference variable should be performed over the whole range. This approach, although fully justified, is not compatible with the idea that the individual phase should be 2n-periodic, and there should be a way to cast the phase sum and difference into the 2n range. Such a casting procedure was proposed by Barnett and Pegg [1990]. The two approaches, however, give different values for the phase sum and difference variances, for example, and one should be aware of the differences. Sometimes the original calculations based on the joint phase distribution (4.1) have a more transparent interpretation, especially when one considers the intermode phase correlations. We shall adduce here examples of both approaches (the quantities obtained with the use of the casting procedure will be distinguished by the subscript 2 ~ )The . casting procedure is described briefly below. The possible eigenvalues of the phase-sum operator are: 2n om+= 80,+ 60, + -mi, u+ 1 where m, = 0, 1, . . . , 2u, and the eigenvalues of the phase-difference operator are 2n 8,- = 00,- O0, + -m-, (4.3) cT+ 1

400

QUANTUM PHASE PROPERTIES

PI,5 4

where m- = -a, -u + 1 . . . , a . It is seen that the eigenvalue spectra (4.2t(4.3) of the phase sum and difference operators have widths of 4n. Since phases differing by 2 n are physically indistinguishable, the phase sum and difference operators and distributions should be cast into a 2 n range (Barnett and Pegg [1990]). The casting procedure can be applied to the joint continuous-phase distribution, P4,(0+, &), defined as:

(4.4)

As was stressed by Barnett and Pegg [1990], there are many ways to apply the casting procedure. However, if the distribution P4,(0+, &) is sharply peaked, we must avoid splitting the original single peak into two parts, one at each end of the 2n interval. Such a poor choice of the 2n range leads to the same interpretation problems encountered for a poor choice of 80 in the single-mode case (Barnett and Pegg [1989]). The casting procedure can be applied as follows:

where the shifts 61 and

Iv. 61 = 0,

a2 = -216,

62

are dependent on the values of 8- and 8,:

for

{ e+

(eo_ n,eo_+ n), E (eo+ + 3n,4 n - le- - o0_I + eo+).

0- E

-

(4.7) This analysis of four regions in the (O+, &)-plane to be cut and shifted is close to the original idea of Barnett and Pegg [1990], and can be easily understood in a geometrical representation of the variable transformation. Moreover, as a further consequence of the 2n-periodicity of eq. (4.6), one can keep the same

VI,

P 41

PHASE PROPERTIES OF TWO-MODE OPTICAL FIELDS

401

shifts 61 and 6 2 in the whole (8+,8-)-plane without distinguishing any regions. Let us only mention some of the possible simplified castings:

and combinations of the shifts satisfying the condition 1611 + (621= 2 n or 11611 - 16211 = 2n. The resulting joint distribution P2=(0++, 0-) is 2n-periodic in 8, and 8-. Alternatively, one can apply the casting procedure to phase distribution (4.1):

4

The factor occurring in eqs. (4.4) and (4.9) comes from the Jacobian of the transformation (4.5) for the variables. The marginal mod(2n) phase-sum, Pzn( 8+),and phase-difference, P2n(8-), distributions are given by: (4.10) where

In the above approach, the casting was prior to the integration. There is another equivalent manner of obtaining mod(2n) marginal phase sum and difference distributions in which the casting is applied after integration. In this approach (Barnett and Pegg [ 1990]), one starts from eq. (4.4) to calculate the 4n-periodic marginal distributions P4,(8,): (4.12)

(4.13)

402

QUANTUM PHASE PROPERTIES

[VI, § 4

Contrary to the former approach, the casting procedure is now applied to the single-mode distributions P4,(8*) (Barnett and Pegg [ 19901):

p4,(e-) + P,,(& p4,(8-) + P4,(8-

+ 2n) -

2n)

if 80-- n < 8- 6 80-, if 80-6 8- 6 80-+ n.

Again, due to the 2n-periodicity of P2,(8*) recipes (4.14) and (4.15) to one of the forms:

(4.15)

in 8*, one can simplify the

(4.16) in the whole interval 80* < 0* < 80*+ 2n. One can analyze analogously the two-mode s-parametrized phase distributions. Here we give only one expression for the mod(2n) s-parametrized phasedifference distribution for arbitrary density matrix p^ and any s:

(4.17) x exp [i(k - Z)O-] (1, n - 1 lp^l k , n - k ) ,

with the coefficients G(’)(k,1) given by eq. (2.60). Also, by putting G(”)(k,1) + 1, the mod(2n) Pegg-Barnett phase-difference distribution is obtained as derived by Luis, Sinchez-Soto and Tanai [1995]. 4.1. TWO-MODE SQUEEZED VACUUM

Single-mode squeezed states, discussed in 9 3.3, differ essentially from the twomode squeezed states discussed extensively by Caves and Schumaker [ 19851 and Schumaker and Caves [ 19851. The Pegg-Barnett phase formalism was applied by Barnett and Pegg [1990], and by Gantsog and Tanai [1991g] to study the phase properties of the two-mode squeezed vacuum, and some of the results are adduced here.

VI,

5 41

403

PHASE PROPERTIES OF TWO-MODE OPTICAL FIELDS

The two-mode squeezed vacuum state is defined by applying the two-mode squeeze operator &r, cp) on the two mode vacuum, and is given by (Schumaker and Caves [ 19851): 10, O),,,,

= s^(r, cp)

10,O)

= (cosh r)-'exp 00

=

(e2@tanhrciicif) 10,O)

(4.18)

(e2',tanhr)" In,n) ,

(cosh r)-' n=O

where Ci! and 6; are the creation operators for the two modes, r (0 < r < 00) is the strength of squeezing, and v, ( - d 2 < cp < n / 2 ) is the phase (note the shift in phase by 7612 with respect to the usual choice of cp). The state (4.18), when the procedure described earlier is applied to it, leads to the joint probability distribution for the phases 81 and 82 of the two modes in the form (Barnett and Pegg [1990]): P ( O I , 0 2 ) =(4n2cosh2r)-1(l+tanh2r-2tanhrcos(01

+ 02))-'.

(4.19)

One important property of the two-mode squeezed vacuum, which is apparent from eq. (4.19), is that P(Ol,&) depends on the sum of the two phases only. Integrating P(Ol,&) over one of the phases gives the marginal phase distribution P ( & ) or P(O2) for the phase 81or 8 2 : F k

(4.20)

meaning that the phases 81 and 62 of the individual modes are distributed uniformly. This gives: (4.2 1) and (4.22) Thus, the phase-sum operator is related to the phase 2cp defining the two-mode squeezed vacuum state (4.18).

404

QUANTUM PHASE PROPERTIES

Fig. 10. The joint probability distribution P(O1,O2), eq. (4.19), for the two-mode squeezed vacuum with r = 0 . 5 .

The two-mode squeezed vacuum has very specific phase properties: the individual phases as well as the phase difference are random, and the only nonrandom phase is the phase sum. Figure 10 shows an example of the joint phase probability distribution P(8,, 02). The ridge, which is parallel to the diagonal of the phase window square, reflects the dependence of P(81,82) on 81 + 82 only. The phase distribution P(81,82) [eq. (4.19)] is an explicit function of the phase sum, but not of the phase difference. This suggests expression of eq. (4.19) in new variables (8+,&). After applying the casting procedure (see introduction to $4) the joint mod(2n) phase distribution is (Barnett and Pegg [1990]): P2n(8+,I%)

= (4n2 cosh2 r)-l(l

+ tanh2r

-

2 tanh r cos 8+)-',

(4.23)

whereas the marginal phase distributions are

P 2 48,)

= (216 cosh2r)-I (1

P2n(8-)

=

1 2n

-.

+ tanh2r - 2 tanh r cos

(4.24) (4.25)

The uniform shape of function (4.25.) signifies randomness of the phase difference in the field [eq. (4.18)]. If the casting procedure is not applied, the marginal distributions P(8,) = P4n(8*) have more complicated structures

v1, (i 41

PHASE PROPERTIES OF TWO-MODE OPTICAL FIELDS

2

1

0

3

r Fig. 1 1 . Phase variances V12 = [A(@o, + go,)]*),eq. (4.29), and (

405

4

V , = ([A($o, + @ o , ) ] ~ ) ~ ~ , eq. (4.30),and the phase correlation function C12, eq. (4.28), against the squeeze parameter r for the two-mode squeezed vacuum.

(Barnett and Pegg [1990]). In particular, P4n(&) is not uniform because of the integration limits in eq. (4.13). In general, the mod(4n) distribution has no unique shape signifying randomness of the phase sum or difference. There are many distributions in the 4n range leading to a flat mod(2n) function. The two-mode variance of the phase-sum operator can be calculated according to the general formula:

in terms of the individual phase variances function (correlation coefficient)

(

and the phase correlation

(4.27)

(

The variances are simply n2/3 [because of eq. (4.20)], and the phase correlation function C12 is equal to: C12

-2(cosh

1

rYk C (tanh (n - k)2

= -2

dilog( 1 - tanh r).

(4.28)

n>k

This correlation function describes the correlation between the phases of the two modes of the two-mode squeezed vacuum. In fig. 11 the correlation coefficient

406

[VI, § 4

QUANTUM PHASE PROPERTIES

as well as the phase variances are plotted against the squeeze parameter r. The correlation is negative and, as r tends to infinity, approaches -n2/3 asymptotically. Finally, phase variance (4.26) has the form: 4 dilog( 1 - tanh r).

(4.29)

The strong negative correlation between the two phases lowers the variance (4.29) of the phase-sum operator. For r -+ 00, this variance tends asymtotically to zero, which means that for very high squeezing the sum of the two phases becomes well-defined (phase-locking effect). The (“single-mode”) mod(2n) phase-sum variance is (Barnett and Pegg [ 19901):

n3 -_ +4dilog(l -

3

+ tanhr).

As the squeezing parameter r vanes from 0 to

00, the

(4.30)

mod(2n) variance [eq. (4.30)] decreases from n2/3 to zero, whereas the two-mode phase-sum variance [eq. (4.29)] changes from 2n2/3 to zero with increasing r. Hence, both variances (4.29) and (4.30), reveal the fact that the phase sum becomes perfectly locked in the limit of large squeezing (r + 00). The value n2/3 of the variance (4.30) describes random phase sum for zero squeezing. In this case of r=O, the two-mode variance (4.29) is twice as much as the mod(2n) phasesum variance (4.30), since it shows randomization of the two phases, $0, and goz,separately. As was stressed in $4, both the original distributions, given by eqs. (4.19) and (4.20), and the mod(2n) distributions, given by eqs. (4.23)(4.25), are valid and useful. However, some care is required when interpreting the results obtained in both ways. The phase-sum variance has generally different values, as seen from fig. 1 1 , in the two approaches. The original distributions are better for understanding the intermode phase correlation, which can be calculated explicitly from eq. (4.27), while for the mod(2n) distribution the correlation is concealed in the value of the phase variance (4.30) and is not seen explicitly. On the other hand, the mod(2n) results have a clear interpretation for the sum and difference of the individual phases treated as single-phase variables. Generalizing formula (2.15) and taking into account the fact that the twomode squeezed vacuum is a “physical state”, we can calculate the expectation

VI, 9: 41

407

PHASE PROPERTIES OF TWO-MODE OPTICAL FIELDS

values of the phase exponential operators in the following way (Gantsog and Tanai [1991g]):

n,k = 0 m , l = 0 =

(e2i9tanh Y) d,, , m z , ml

where for brevity we denote ((. . .)) = (r,pi)(O, 0 I(. . .)I 0, O),,,,.

(4.3 1)

The operators (4.32)

are the Susskind-Glogower phase operators for the two modes. Formula (4.3 1) is strikingly simple, and shows that only exponentials of the phase sum have nonzero expectation values. Using eq. (4.3 l), the following results for the cosine and sine of the phase-sum operator are obtained (Gantsog and Tanah [ 1991gl): (cos( $0,

+ Go2)) = tanh r cos 2p,

(cos2( (sin2($o,

(

(sin($@, + $0,))

+ $0,)) = + i(tanh r)2 cos 4p, + go2))= i i(tanhr)2 C O S ~ -

2

) (

~ C O S ( $ S+, go2)]

=

bsin($O,

= tanh r

sin 2p,

(4.33)

(4.34) ~ ,

+ $02)]

2

)

=

i(coshr)-*.

(4.35)

For very large squeezing (r + m, tanh r + 1, cosh r --t m), the expectation values (4.33) and (4.34) of the functions of the phase-sum operator become asymptotically corresponding functions of the phase 2p, confirming the relation between the phase sum and 2 p that is already apparent from eq. (4.22). It is interesting that the expectation value of the phase-sum operator is equal to 2 p irrespective of the value of r, whereas for the sine and cosine functions correspondence is obtained only asymptotically. The variances (4.35) then become zero and the sine and cosine of the phase sum are well-defined. It should, however, be emphasized that the expectation values calculated according to the Pegg-Barnett formalism depend on the choice of the particular

408

QUANTUM PHASE PROPERTIES

tVI,

54

window of the phase eigenvalues. If a choice different from that made above were made, the clear picture of the phase properties of the two-mode squeezed vacuum would be disturbed. For example, the value of the correlation coefficient (4.28) would be different, and the phase-sum variance (4.26) would not tend asymtotically to zero. However, formulas (4.3 1)-(4.35), because of the way they have been calculated, do not, in fact, depend on the choice of the phase window. This gives us the opportunity to make a choice which introduces consistency in the behavior of the phase itself and its sine and cosine functions. Another way of making the choice is to minimize the variance (4.26) of the phase-sum operator. 4.2. PAIR COHERENT STATES

Pair coherent states introduced by Aganval [1986, 19881 are quantum states of the two-mode electromagnetic field, which are simultaneous eigenstates of the pair annihilation operator and the difference in the number operators of the two modes of the field. Aganval [ 19881 has discussed the nonclassical properties of such states, showing that they exhibit remarkable quantum features such as sub-Poissonian statistics, correlations in the number fluctuations, squeezing, and violations of the Cauchy-Schwarz inequalities. He has also presented results for fluctuations in the phase of the field using the Susskind-Glogower phase formalism. The phase properties of such states on the basis of the Pegg-Barnett formalism were studied by Gantsog and TanaB [ 199le], and by Gou [ 19931. Phase distributions for squeezed pair coherent states were analyzed by Gerry [ 19951. The pair coherent states are defined by Aganval [I9881 as eigenstates of the pair-annihilation operator: (4.36)

c

where is a complex eigenvalue and q is the degeneracy parameter, which can be fixed by the requirement that lc, q ) is an eigenstate of the difference of the number operators for the two modes -

6th) lc, 4 ) = 4 It,4 ) .

(4.37)

The solution to the above eigenvalue problem, assuming q to be positive, is given by (Aganval [ 19881): (4.38)

VL

41

PHASE PROPERTIES OF TWO-MODE OPTICAL FIELDS

409

where N, is the normalization constant (4.39)

The state In + q, n) is a Fock state with n + q photons in mode a and n photons in mode b. If the complex number is written in the form

c

the state (4.38) can be written as 00

(4.41) n=O

where lrln

(4.42)

Now, the phase properties of the state (4.41) can be studied easily using the Pegg-Barnett formalism in a standard way as described above. The resulting joint probability distribution for the phases 8u and 8 b of the two modes is given by (Gantsog and Tanas [ 199 1 el) (4.43)

where b, is given by eq. (4.42). For q=07 formula (4.43) can be written in the following simple form:

As in the case of the two-mode squeezed vacuum, the joint phase probability distribution depends on the sum of the two phases only, which means strong correlation between the two phases. Again, the only non-uniformly distributed phase quantity is the phase sum 8,+8b. This suggests re-expression of the phase distribution (4.43) in new variables of the phase sum, 8+=8,+8b, and

410

[VL

QUANTUM PHASE PROPERTIES

5

4

phase difference, 8- = 8, - 0 b . AAer applying the casting procedure, the mod(2n) Pegg-Barnett distribution P 2 4 8+,&) takes the form w

(4.45)

and the marginal distributions are (4.46) 1 -.

(4.47) 2n For completeness of our discussion and by analogy with our presentation of the singlemode models, we now give expressions for various s-parametrized phase distributions. Thus, the mod(2n) two-mode s-parametrized phase distribution is equal to

P248-)

=

{

1+2

C bnbkG(S)(n,k ) G‘”(n + 4, k + 4 ) cos[(n 00

n>k

-

I

k)8+] ,

(4.48) where the coefficients G(S)(n,k)are given by eqs. (2.60)-(2.62). The mod(2n) marginal s-parametrized phase-sum distribution is

(4.49) The mod(2n) s-parametrized phase-difference distribution P f i ( O-) and the single-mode ones, P(’)( 8,) and PCS)( 8 b ) , are uniform: 1 (4.50) P g ( 8 - ) = P(S’(0,) = P ( q 8 b ) = -. 2n The distributions (4.48)-(4.50), similar to the distributions (4.45)-(4.47), reveal the fundamental phase properties of pair coherent states. The correlation coefficient C a b , eq. (4.27), [subscripts 1,2 should be replaced by a and b, respectively] is given in this case by the formula (4.5 1) where bn is given by eq. (4.42). This correlation is negative and lowers the variance of the phase-sum operator. For ----t m, this coefficient approaches -n2/3,

VL § 41

PHASE PROPERTIES OF TWO-MODE OPTICAL FIELDS

41 I

the phase-sum variance becomes zero, and we have the classical situation of perfectly defined phase sum (the phase-locking effect). This phase-correlation coefficient can be contrasted with the photon-number correlation coefficient, considered by Aganval [1988], which increases as II;I increases. The sine and cosine functions of the phase-sum operator were also obtained by Gantsog and Tanah [ 199le] and compared to their counterparts obtained by Aganval [ 19881, who used the Susskind-Glogower approach. 4.3. ELLIPTICALLY POLARIZED LIGHT PROPAGATING IN A NONLINEAR

KERR MEDIUM

To describe propagation of elliptically polarized light in a nonlinear Kerr medium, a two-mode description of the field is needed. The quantum nature of the field results in the appearance of such quantum effects as photon antibunching (Ritze and Bandilla [ 19791, TanaS and Kielich [19791, Ritze [ 19801) and squeezing (Tanai and Kielich [1983, 19841). Tana8 and Kielich have shown that as much as 98 percent of squeezing can be obtained when intense light propagates in a nonlinear Kerr medium. They referred to this effect as selfsqueezing. Aganval and Pun [ 19891 re-examined the problem of propagation of elliptically polarized light through a Kerr medium, considering not only the Heisenberg equations of motion for the field operators, but also the evolution of the states themselves. Quantum fluctuations in the Stokes parameters of light propagating in a Kerr medium were discussed by Tanai and Kielich [ 19901, and by Tana8 and Gantsog [1992b]. The following effective interaction Hamiltonian can be used to describe the propagation of elliptically polarized light in a Kerr medium (Tana8 and Kielich [1983, 19841)

where cil and ci2 are the annihilation operators for the circularly right- (“1”) and left- (“2”) polarized modes of the field, K is the coupling constant, which is real and related to the nonlinear Susceptibility tensor of the medium, and d is the asymmetry parameter describing the coupling between the two modes. For a hlly symmetrical susceptibility tensor, d = 1. Otherwise, d # 0 and describes the asymmetry of the nonlinear properties of the medium (Ritze [ 19801, Tanai and Kielich [1983, 19841). Using the Hamiltonian (4.52), one can obtain the evolution operator G(z), and assuming that the initial state of the field is a coherent state of the elliptically

412

QUANTUM PHASE PROPERTIES

[VI, § 4

polarized light, one gets for the resulting state of the field (Aganval and Puri [ 19891): INz))

at)

1a1, a2)

=c =

b n l b n 2 e x ~ i ( n+in~2i ~ 2 )

(4.53)

n1,m

+ i i t [ n l ( n l - 1) + m(n2 - 1) + 4dnln21) InI,n2),

where z = n ( w ) k z / c (with n(w) the refractive index), and the coefficients b,,,, are given by eq. (3.8) with [all2and la2I2as the mean numbers of photons for the circularly right- and left-polarized modes, respectively, whereas ql,2 are the phases of the coherent states of the two modes. The state (4.53) is the two-mode state of the field, and the two-mode generalization of the Pegg-Barnett formalism used by Gantsog and TanaS [ 1991~1leads to the following joint probability distribution for the continuousphase variables, 81 and 82, of the two modes:

+ i-2z [nl(nl - 1) + n2(n2 - 1) + 4dnlnzl

(4.54)

The phase distribution function P(81, 6,) describes the phase properties of elliptically polarized light propagating through a Kerr medium, which were discussed in detail by Gantsog and Tanas [ 1991c]. Figure 12 shows an example of the evolution of P(81,82). It is seen that the peak is shifted and broadened during the evolution. Since the numbers of photons in the two modes are different, one can see that the shift of the peak and its broadening is asymmetric. The intesitydependent phase shift is bigger for the mode with higher number of photons. This corresponds to the classical effect of self-phase modulation in a nonlinear Kerr medium. The quantum description shows not only the shift but also the boadening of the phase distribution (phase diffusion). Integration of the distribution function P ( & , 82) over one of the phases 81or 132 leads to the marginal distribution P(&) or P(8l) for the individual phases. All single-mode phase characteristics of the field can be calculated using these distributions, and the corresponding formulas were given by Gantsog and TanaS [ 1991c]. In addition to the phase properties of the individual modes, it is interesting, in the two-mode case, to study the behavior of the phase difference between the two

VL

P 41

PHASE PROPERTIES OF TWO-MODE OPTICAL FIELDS

413

Fig. 12. Evolution of the joint probability distribution P(B1, &), eq. (4.54), of light propagating in a Kerr medium: la1I2= 0.25, la2I2 = 4 and d = 1; (a) t = O , (b) t=0.1, ( c ) 2=0.2, (d) t = 0 . 3 .

modes. In the Pegg-Barnett formalism, the phase-difference operator is simply the difference of the phase operators for the two modes, so the mean value of the phase-difference operator is the difference of the mean values of the single-mode phase operators. To calculate the variance of the phase-difference operator, we can use the relation

( ~ ( C O&d] , ’) -

=

(

+ ((A&,2)2) - 2 c I 2 ,

(4.55)

where the last term is the correlation coefficient between the phases of the two modes and can be calculated by integration of P(81,@) according to eq. (4.27). Thus, the resulting formula is (Gantsog and Tanah [1991c]): C12(T) =

cc

nl

> n i nz > n ;

f i 2 h -

(4.56)

414

[VL

QUANTUM PHASE PROPERTIES

P4

1.5 1 .o 0.5

u" 0.0 -0.5 ' 0.1-

0.00

1.57

3.14

4.71

6.28

2:

-0.1

0.00

1.57

3.14

4.71

6.28

2:

Fig. 13. Evolution of the internode phase correlation function C l 2 ( t ) , eq. (4.56), and the phaseeq. (4.55), of light propagating in a Kerr medium. Thin difference variance ([A(G,, - 002)]2), solid line: la1 l2 = 0.25, la2I2 = 4 and d = I ; bold solid line: la1 l2 = 0.25, 1 ~ x 2 = 1 ~4 and d = $; thin dashed line: la1 l2 = 0.25, Ja2I2= 0.25 and d = 1; bold dashed line: la, l2 = 0.25, la2I2 = 0.25 and d=i.

where J j = 2b,

b,,I I

(- 1p -4 ~

-

ni - ni

}

n:) [ni + ni - 1 + 2d(nj + nj)] .

(4.57)

A graphic illustration of the correlation fhction (4.56) is shown in the left-hand panel of fig. 13. The strength of the correlation depends crucially on the value of the asymmetry parameter d . The highest values of the correlation are obtained for d = This means that the minimum of the phase-difference variance, in view of eq. (4.55), is obtained for d = The phase-difference variance is shown in the right-hand panel of fig. 13. It was shown (TanaS and Gantsog [1991]) that, similar to the single-mode case, dissipation destroys the periodicity of the evolution and broadens the phase distribution. Recently, the phase properties of light propagating in a Kerr medium have been reconsidered (Luis, Sanchez-Soto and TanaS [ 19951) from the point of view of the Hermitian phase-difference operator introduced by Luis and SanchezSoto [1993b, 19941, which is based on the polar decomposition of the Stokes operators. This example shows clearly the difference between the Pegg-Barnett and Luis-Sanchez-Soto phase-difference formalism, which is most visible for weak fields. The Luis-Sanchez-Soto phase-difference operator differs from the Pegg-Barnett phase-difference operator, which is simply the difference of the phase operators of the two modes. For strong fields both formalisms give the same results. The nonlinear Kerr medium appears to be a good testing ground for different phase approaches.

i.

i.

VI,

P 41

PHASE PROPERTIES OF TWO-MODE OPTICAL FIELDS

415

Fig. 14. The joint probability distribution P ( @ , &), , eq. (4.54), of light propagating in a Kerr medium. lal12=1a212=4, r = 2 ~ / 2 , a n d ( a ) d = O ; ( b ) d = ; .

As shown by Gantsog and Tanai [1991a], superpositions with any number of components can be obtained in the process of light propagation in the Kerr medium (similar to the anharmonic oscillator model described in 9 3.5) if the evolution time z is taken as a fraction MIN of the period, where M and N are mutually prime integers. Exact analytical formulas for finding the superposition coefficients were given for any N . The joint phase probability distribution P(81,&) splits into separate peaks if the state of the field becomes a discrete superposition of coherent states, and this is a very spectacular way of presenting such superpositions. Some examples are shown in fig. 14.

4.4. SECOND-HARMONIC GENERATION

Second-harmonic generation is probably the best known nonlinear optical process. In the quantum picture we deal with a nonlinear process in which two photons are annihilated and one photon with doubled frequency is created. The quantum states of the field generated in the process exhibit a number of unique quantum features such as photon antibunching (Kozierowski and Tanai [ 19771) and squeezing (Mandel [1982], Wu, Kimble, Hall and Wu [1986]) for both the fundamental and second-harmonic modes (for a review and literature see Kielich, Kozierowski and Tanai [1985]). Nikitin and Masalov [I9911 discussed the properties of the quantum state of the fundamental mode, calculating numerically the quasiprobability distribution function Q(a, a*)for it. They suggested that the quantum state of the fundamental mode evolves, in the course of the secondharmonic generation, into a superposition of two macroscopically distinguishable states, similar to the superpositions obtained for the anharmonic oscillator model (Yurke and Stoler [1986], Tombesi and Mecozzi [1987], Miranowicz, Tanai and

416

QUANTUM PHASE PROPERTIES

[VI, 8 4

Kielich [1990], Gantsog and Tanai [1991fl), or a Kerr medium (Aganval and Pun [1989], Gantsog and Tanai [1991a]). Gantsog, Tanai and Zawodny [1991a] discussed the phase properties of the field produced in the second-harmonic generation process. To describe second-harmonic generation, the following model Hamiltonian is used:

ii = f i 0 + fi[= hWcitci + 2ho6t6 + Ag(6tLi2 + 6cit2),

(4.58)

where Li (Lit) and 6 (St) are the annihilation (creation) operators of the fundamental mode of frequency w and the second-harmonic mode at frequency 20, respectively. The coupling constant g, which is real, describes the coupling between the two modes. Since fi0 and fi, commute, there are two constants of motion: I?,and f i 1 , 30determines the total energy stored in both modes, which is conserved by the interaction fi,.The free evolution can be thus factored out, and the resulting state of the field can be written as: (4.59) I ~ ( t )=)exp(-ijiIt/h) I ~ ( 0 ), ) where I Y ( 0 ) )is the initial state of the field. Since the interaction Hamiltonian

is not diagonal in the number-state basis, the numerical method of diagonalization of GI may be applied to find the state evolution (Walls and Barakat [ 19701). Let us assume that initially there are n photons in the fundamental mode and no photons in the second-harmonic mode; i.e., the initial state of the field is In,O) = In) 10). Since f i 0 is a constant of motion, we have the relation:

fi1

(citLi)

->

+ 2 (” btb

= constant = n,

(4.60)

which implies that the creation of k photons of the second-harmonic mode requires annihilation of 2k photons of the fundamental mode. Thus, for given n, we can introduce the states

I$))=

In-2k,k),

k = 0 , 1 ) . . . )[n/2],

(4.61)

where [n/2]denotes the integer part of n/2, which form a complete basis of states of the field for given n. We have

(v;’) I vt)) = bnn’bkk’

7

(4.62)

meaning that the constant of motion 20splits the field space into orthogonal subspaces, which for given n have the number of components equal to [n/2] + 1.

VI, 9: 41

PHASE PROPERTIES OF TWO-MODE OPTICAL FIELDS

417

In such a basis, the interaction Hamiltonian has the following nonzero matrix elements:

which form a symmetric matrix of the dimension ([n/2] + 1) x ([n/2] + 1) with real nonzero elements (we have assumed g real) located on the two diagonals immediately above and below the principal diagonal. Such a matrix can be easily diagonalized numerically (Walls and Barakat [ 19701). To find the state evolution, we need the matrix elements of the evolution operator: (4.64) If the matrix 5 is the unitary matrix that diagonalizes the interaction Hamiltonian matrix given by eq. (4.63), i.e.,

G-‘@‘)fi

=

fig x diag(A0, A,,.

. . ,A-[n/21),

(4.65)

then the coefficients dn,k(t) can be written as [421

dn,k(t) =

exp(-igtAi)

ukiu;)i,

(4.66)

i=O

where Ai are the eigenvalues of the interaction Hamiltonian in units of fig. Of course, the matrix fi as well as the eigenvalues Ai are defined for given n and should have the additional index n, which we have omitted to shorten the notation. Moreover, for real g the interaction Hamiltonian matrix is real, and the transformation matrix fi is a real orthogonal matrix, so the asterisk can also be dropped. The numerical diagonalization procedure gives the eigenvalues Ai as well as the elements of the matrix 6, and thus the coefficients dn,k(t) can be calculated according to eq. (4.66). It is worthwhile to note, however, that due to the symmetry of the Hamiltonian the eigenvalues Ai are distributed symmetrically with respect to zero, with one eigenvalue equal to zero if there is an odd number of them. When the eigenvalues are numbered from the lowest to the highest value, there is an additional symmetry relation: ukiuoi = (-1) k

Uk,[n/~l-iU~,[n/~]-i,

(4.67)

which makes the coefficients dn,k(t) either real ( k even) or imaginary ( k odd). This property of the coefficients dn,k(t) is very important, and in some cases allows exact analytical results to be obtained.

418

QUANTUM PHASE PROPERTIES

[VI, § 4

With the coefficients dn,k(t)available, the resulting state of the field (4.59) can be written, for the initial state In,O), as: (4.68) The typical initial conditions for the second-harmonic generation are: a coherent state of the fundamental mode and the vacuum of the second-harmonic mode. The initial state of the field can thus be written as: m

(4.69) n=O

where c, = b,e'"% is the Poissonian weighting factor (3.8) of the coherent state Iao) with the phase q, = Arg ao. With these initial conditions, the resulting state (4.59) is given by m

(4.70) n=O

n=O

k=O

Equation (4.70), describing the evolution of the system, is the starting point for a further discussion of second-harmonic generation. If the initial state of the fundamental mode is not a coherent state, but has a decomposition into number states of the form (4.69) with different amplitudes c, eq. (4.70) is still valid if appropriate c,'s are taken. This is true, for example, for an initially squeezed state of the fundamental mode. The coefficients dn,k(t) have been calculated numerically to find the evolution of the field state (4.59), and consequently, its phase properties (TanaS, Gantsog and Zawodny [ 1991a,b], Gantsog, TanaS and Zawodny [ 1991a]). Repeating the standard procedure of the Pegg-Barnett formalism with the field state (4.59), the joint phase probability distribution is obtained in the form

X

lg 2 bn

k=O

dn,k(t)

/2

exp {-i [(n - 2k) 0, + k e b - k(2Va - qb)]} ,

(4.71) where 8, and 6 b are continuous-phase variables for the fundamental and second-harmonic modes, and the phases q, and V b are the initial phases

VL

5 41

PHASE PROPERTIES OF TWO-MODE OPTICAL FIELDS

419

with respect to which the distribution is symmetrized. It is interesting that formula (4.71) depends, in fact, on the difference 2 q , -V)b, which reproduces the classical phase relation for second-harmonic generation. Classically, for the initial conditions chosen here, this phase difference takes the value in,which turns out to be a good choice to fix the phase windows in the quantum description as well. The evolution of the joint probability distribution P(OU,Oh), given by eq. (4.71), is illustrated graphically in fig. 15. At the initial stage of the evolution the phase distribution in the Oa direction (fundamental mode) is broadened, while a peak of the second-harmonic mode phase starts to grow. The emergence of the peak at 6 b = 0 confirms the classical relation 2cpu- V)h = which has been applied to fix the phase window. The phase distribution in the %b direction narrows at the beginning of the evolution, meaning less uncertainty in the phase of the second harmonic. However, for later times the distribution P(eU,6,) splits into two peaks, which resembles the splitting of the Q(a,a*) function found by Nikitin and Masalov [1991]. For still later times, more and more peaks appear in the distribution P(B,, O b ) , and this distribution becomes more and more uniform, which means randomization of the phase. The route to the random phase distribution, however, goes through a sequence of increasing numbers of peaks. The splitting of the joint phase distribution can be understood if one realizes that the mean number of photons of the second harmonic oscillates and after reaching the maximum the secondharmonic generation becomes, as a matter of fact, the down-conversion process which exhibits a two-peak structure of the phase distribution in the direction of the fundamental mode (see $4.5). The appearance of new peaks may thus be interpreted as a transition of the process from the second-harmonic to the downconversion regime, and vice versa. The phase variances for both modes tend asymptotically to the value n2/3of the randomly distributed phase (Gantsog, Tanah and Zawodny [1991a]); however, it has turned out that partial revivals of the phase structure can be observed during the evolution (Drobnj, and Jex [1992]). It is also interesting to study the phase distribution of the field produced by second-harmonic generation with other than coherent initial states of the fundamental mode. Such studies were performed by Tanai, Gantsog and Zawodny [1991b], showing for example that even for a second harmonic generated by an initial number state the joint phase probability distribution has a modulation structure owing to the intermode correlation that develops in the process of the evolution.

420

QUANTUM PHASE PROPERTIES

[VI, § 4

Fig. 15. Evolution of the joint phase probability distribution P(B,, Bb), eq. (4.71), in the secondharmonic generation. The initial mean number of photons of the fundamental beam is la012 = 4, and gt is the dimensionless scaled time.

v1, 5 41

PHASE PROPERTIES OF TWO-MODE OPTICAL FIELDS

42 1

4.5.PARAMETRIC DOWN CONVERSION WITH QUANTUM PUMP

The parametric down-conversion process with quantum pump, which is a subharmonic-generation process, can be described by the same model Hamiltonian [eq. (4.58)] as the second-harmonic generation. The initial conditions distinguishing the two processes are the following: For the subharmonic generation process, mode b is initially populated while mode a is in the vacuum state. The distinction between the two processes is far from trivial, and the states generated in the two processes have quite different properties (Gantsog, Tanas and Zawodny [1991b], Jex, Drobnjr and Matsuoka [1992], Tanas and Gantsog [ 1992a,b], Gantsog, Tana.4 and Zawodny [1993]). Let us assume, in analogy to our analysis of second-harmonic generation, that initially there are n photons in the pump mode (6) and no photons in the signal mode (a); i.e., the initial state of the field is I0,n) = lo), In)b. Since f i 0 is a constant of motion, we have the relation: (6th) + 2(6+6>= constant

= 2n,

(4.72)

which implies that the annihilation of k photons of the pump mode requires creation of 2k photons of the signal mode. Thus, for given n, we can introduce the states

1q::i)

=

12k,n-k),

k = 0 , 1 , . . . , n,

(4.73)

which should be compared to the corresponding expression (4.61) for the secondharmonic generation. Proceeding along the same lines as in second-harmonic generation, the resulting state of the field can be written as (4.74) where the coefficients d2,,k(t) are given by (4.75) whereas now the c, = b, exp(incpb) are the Poissonian weighting factors (3.8) for the initially coherent state [PO= [Pol exp(icpb)) of the mode b. Again, the method of numerical diagonalization is used to calculate the coefficients d2n,k(t) and,

422

QUANTUM PHASE PROPERTIES

[VI,

54

in effect, the phase properties of the state (4.74). The joint phase probability distribution in this case is given by

lz

bn

d2n,k(t) k:O

exp 1-i [2kea + ( n - k) e b + k(2qa - qb)]}

li

,

(4.76) As for the second-harmonic generation, we similarly take 2q, - q b = to fix the phase windows. The evolution of the joint probability distribution P(8,, 8,) for parametric down conversion with the mean number of photons = 4 is shown in fig. 16. Comparison of figs. 15 and 16 shows immediately a striking difference between the phase properties of the fields obtained in the two processes. The state produced in the down-conversion process acquires from the very beginning the two-peak structure in the Oa direction, which suggests the appearance of a superposition of two states in the resulting field. The two peaks which appear at the beginning of the evolution correspond, in fact, to the two-peak phase distribution of the squeezed states (see 5 3.3). At later stages of the evolution randomization of the two phases takes place, similarly as for the second harmonic. However, the symmetry with respect to 8, is preserved. The two-peak structure of the phase distribution has already appeared, although not in its pure form, in the phase distribution for second-harmonic generation (fig. 15). Its appearance can be ascribed to the down-conversion process that has overcome second-harmonic generation at this stage of the evolution. The transition from the one-peak phase distribution to the two-peak distribution makes a qualitative difference between the two field states, and is a sort of “phase transition”. Once the joint phase distribution P(oa, 6,) is known, all quantum-mechanical phase expectation values can be calculated. In particular, the phase variance for the signal mode can be calculated according to the formula

VI,

P 41

PHASE PROPERTIES OF TWO-MODE OPTICAL FIELDS

423

Fig. 16. Evolution of the joint phase probability distribution P(0,. 0*), eq. (4.76), in the process of parametric down conversion with quantum pump. The initial mean number of photons in the mode b is I /3012 = 4.

424

**'I ,.I- =

1

QUANTUM PHASE PROPERTIES

i'.-,

..

2

SIONAL. MODE

- - - bz%&EDVACUUM

VACUUM STATB VALUE

6.0 4.0

W

v

.

2.0

0.0 -* 0.0

---------

i i , ,

,

0

...........................

.........I .........

-- - --

\

,

1.0

,

,

I

,

2..=7/<--

,

,

,

3.0

2.0

[VI, § 4

,

,

,

,

4.0

I

5.0

gt Fig. 17. Evolution of the phase variances ((A$0J2),

eq. (4.77), and

eq. (4.78), in the

parametric down conversion. The initial mean number of photons in the mode b is I pol2 = 4.

and for the pump mode we have

where we have used eq. (4.76), and we take 2g?, - g?b = in.The time evolution of the phase variances can be calculated numerically using these expressions for given initial field states. The dynamical behavior of the phase variances calculated from eqs. (4.77) and (4.78) is illustrated graphically in fig. 17 for \pol2 = 4. The dashed line n2/3marks the variance for the state with random distribution of phase. It is apparent that the phase variance of the signal mode starts from the value n2/3,dips into the minimum, and after a few oscillations again becomes close to n2/3.For comparison, the phase variance for the ideal squeezed state is also shown. The two variances are initially indistinguishable, but the phase variance for the squeezed state approaches monotonically its asymptotical value n2/4,while for the quantum pump case the phase variance of the signal mode begins to oscillate at later times. This confirms the statement that there a limit is imposed by the quantum fluctuations of the pump on the applicability of the ideal down-converter model. The phase variance of the pump mode increases rapidly from its initial value for the coherent state, and also shows oscillatory behavior approaching the value n2/3 at the long-time

VL

B 41

PHASE PROPERTIES OF TWO-MODE OPTICAL FIELDS

425

@a Fig. 18. Phase distribution P(Ou), eq. (4.79), for gt=0.3, in the parametric down conversion. ..

..

-

.

.

"

.

..

,...

iimit. I nus, tne long-time errect or tne quantum nuctuations or the pump mode is the randomization of the phase distribution for both signal and pump modes. This randomization process is not monotonous, and it turned out that at least partial revivals of the phase structure are possible during the evolution (Gantsog [1992], Gantsog, TanaS and Zawodny [ 19931). Integrating P(8,, 0,) over one of the phases leads to the marginal phase distributions P(8,) and P(8b) for the phases 8, and 8 b of the individual modes. We have: 1 .

rnl

1

C . l

(4.79)

. ~ , I x expL-i(n - n')&J .

(4.80)

1

The phase distribution P(8,) for the signal mode is shown in fig. 18 for gt = 0.3; ie., for the time at whch the squeezing in the signal mode has its maximum value. For comparison we show the phase distribution for the squeezed vacuum for Y = 2 gt = 1.2. The effect of quantum fluctuations of the pump is seen as the broadening of the phase distribution with respect to that for the ideal squeezed state.

426

QUANTUM PHASE PROPERTIES

0

[VI

5. Conclusion

In this article we have reviewed some recent results concerning the quantum phase description of optical fields. We have focused our attention on the real fields that can be generated in practice in various nonlinear optical processes. So, we rather avoided discussions of the phase formalisms as such and tried to exploit their practical applicability in the description of optical fields. In the description of the phase properties we used two different, though related formalisms: the Pegg-Barnett Hermitian phase formalism and the formalism based on sparametrized phase distributions. The Pegg-Barnett Hermitian phase formalism is a good example of the concept of the phase as a physical property of a single field mode represented by a Hermitian phase operator canonically conjugate one to the number operator. It allows one to obtain the phase distributions for the fields, mean values and variances of the phase, and other phase characteristics of the field in a reasonably simple way, both from the conceptual as well as the calculational point of view. The phase distributions obtained from this formalism are 2n-periodic, positive definite and normalized. They can be treated as a good representation of the quantum state of the field and can be referred to as canonical phase distributions. Another description of the optical phase used by us is that based on the sparametrized quasiprobability distributions, which can give phase distributions that can be both narrower and broader (depending on s) than the Pegg-Barnett phase distribution, but these distributions with s < 1 can be associated with some noisy, real measurements of the phase probability distribution and can be referred to as measured phase distributions. Using the examples of real field states presented here, we tried to show the similarities and differences that one encounters when various phase distributions are applied to describe a particular field state. Our choice of the field states is, of course, a bit arbitrary, and we relied to a large extent on our own results. We believe, however, that our review covers a number of field states important for quantum optics, and that the results presented here may prove interesting. We have also attempted to give a more or less complete review of the literature on the subject, but the subject of quantum phase is still a “hot” one and the literature is growing rapidly.

Acknowledgments This work was partially supported by the Polish Committee for Scientific Research (KBN) under the grants No. 2 P03B1288 and 2 P03B1888.

VII

427

APPENDIX A. GARRIS0N;WONG PHASE FORMALISM

A.M. is particularly indebted to Dr. Stephen M. Barnett for his hospitality and scientific guidance at Oxford University, and he is grateful to the Foundation for Polish Science for the Fellowship. Ts.G. would like to thank Professor Herbert Walther for his hospitality at the Max-Planck-Institut fiir Quantenoptik, and the Alexander von Humboldt Foundation for the Research Fellowship. Appendix A. Garrison-Wong Phase Formalism

Garrison and Wong [1970] constructed the phase operator relation:

$GW

using the

for any g, f E X2, where Id2 is the Hilbert space in the unit disk of the complex plane, and 0 0 is arbitrary. Here, we have changed the sign with respect to the original Garrison-Wong paper and introduced arbitrary 0 0 . The inner product in X2 is defined by (glf) =

eo

d0g*(e-i6)f(e-io)

The boundary value off is given by a convergent Fourier series,

which does not contain coefficients c, with negative n. Subsequently, Popov and Yarunin [ 19731 established the connection of this operator to the Susskind and Glogower [ 19641 exponential phase operators g* of the form $GW

=

eo+ n + i [ln(1-

eisOE+)- ln(1- e+

The operators g- and g+ = @)t operators B and Bt of the mode A

E-

=

(GtG

K ) ].

(A.4)

are defined by the annihilation and creation

+ l)-1'2G, g+= Gt(BtG + 1)-1'2,

[-E-,E+ -1

=

10) (01,

(A.5)

where 10) is the vacuum state [g-is another notation for the Susskind-Glogower exponential operator (2.5)].

428

QUANTUM PHASE PROPERTlES

tVI

Let us consider the "phase states"

which are the right and left eigenstates of the operators k and E+:

E- 10)= exp(i8) lo),

(elE+= exp(-iO) ( 0 1 .

(A.7)

The states (A.6) are not orthogonal, but allow for the resolution of the identity operator

lo

0"+2Z

d 0 18) (01 = i.

(A4

With the aid of eqs. (A.7) and (A.8) to the operator, eq. (A.4) can be rewritten in the form (Bergou and Englert [1991])

Since the states (A.6) are not orthogonal, they are not eigenstates of the Garrison-Wong phase operator. From eq. (A.9), we have: (A.lO) Taking the field states

If)

in the form (A.11)

n=O

we then have: (A. 12)

which has the same form as eq. (A.3), and we can consider the phase operators (A.l) and (A.4) as equivalent. However, we should keep in mind that the

VII

429

APPENDIX A. GARRIS0N;WONG PHASE FORMALISM

Garrison-Wong phase operator is defined on a dense set of state vectors, which for mathematical consistency and the requirement that the number-phase commutator should be -i, imply f(-1) = 0. Unfortunately, when approximating even simple physical states on this dense set, one finds rather undesirable properties (Bergou and Englert [1991]). Since the states (A.6) are not orthogonal, we have (A. 13) and for the expectation values (A. 14) This means that the quantity I(Olf)12 cannot be interpreted as a phase distribution function. To find the Garrison-Wong phase distribution function, we must calculate the quantity (A. 15) I

where the vector I e)GW is the eigenvector of the Garrison-Wong phase operator. The function (el.) has a quite complex structure (Garrison and Wong [ 19701, Popov and Yarunin [ 1973, 1992]), but it can be found from the recursive formulas given by Garrison and Wong [1970], which are (A.16) where, for n 3 1, (A.17)

and 1 ro(e)= --21 + [ ( 2 +~ eo - e) ln(2n + O0 - 0) + (0 - 0,) ln(8 4n

-

0011.

(A.20) The formulas (A.16t(A.20) were used by Gantsog, Miranowicz and Tanai [ 19921 to calculate the Garrison-Wong phase distribution for some real states

430

[VI

QUANTUM PHASE PROPERTIES

of the field, showing that their symmetry is incompatible with the symmetry of the phase distributions obtained from the Pegg-Barnett as well as the sparametrized phase approaches. In the Garrison-Wong approach, even vacuum has a preferred phase, which is hardly acceptable on physical grounds. The recursive relation (A. 17) has the following solution (Miranowicz [ 19941): (A.21)

x i =nimi ,

where the sum over {ni, mi} is taken under the condition k after integration the functions y n ( 0 ) [eq. (A.l8)] take the form

{

1 yn(e> = - eineo [In(

2nin

~

2n % - 80

-

1)

-

=

n, and

in]

+cine (Ei[in(2n + 80- O)] - Ei[-in(O

(A.22) -

OO)]) €lo)])

in terms of the exponential integral Ei(x). Equations (A.21)-(A.22) are more convenient for numerical calculations than eqs. (A. 17)
@GW =

Oo+n+

c

nzn'

exp[i(n

n') 801 In) (n'l , i(n - n') -

(A.23)

leading to the number-phase commutator [$GW,ii+ii]

= - i ( ~-2xl0oj ( ~ o l ) ,

(A.24)

and for the states for which ( & I f ) = 0, the second term on the right-hand side vanishes, giving the value demanded by Garrison and Wong [1970]. A detailed comparison of the Garrison-Wong and Pegg-Barnett formalisms was given by Barnett and Pegg [I9921 and by Gantsog, Miranowicz and TanaS [1992]. The difference between the two formalisms is, in mathematical sense, the difference between the weak and strong limits for the phase operators that is taken when a + co (Vaccaro and Pegg [1993]). Appendix B. States for the Pegg-Barnett Phase Formalism The Pegg-Barnett optical phase operator (2.12) is constructed in a finite (a+1)dimensional Hilbert space 'H(') spanned by the number states 10) , \ I ) ,. . . , I a).

VII

APPENDlX B. STATES FOR THE PEGG;BARNETT PHASE FORMALISM

43 I

Hence, all other quantities, such as states, operators or probability distributions, analyzed within the Pegg-Barnett formalism, should also be defined in the same ( a+ 1)-dimensional state space .H('). Buiek, Wilson-Gordon, Knight and Lai [1992] emphasized that it is not strictly correct to apply the definition (2.23) of the finite-dimensional phase distribution P(&)

=

I

(wrong),

(U)(@nIf)l2

(B. 1)

for the infinite-dimensional state M

n=O

The problem of the precise definition of states in .H(') can be overcome by assuming that a is large enough so that the differences between the states in the finite-dimensional, .H("),and infinite-dimensional, X,spaces can be arbitrarily small in the sense of the Cauchy condition (Pegg and Barnett [1989]): (7

E (7

n=O

The precise finite-dimensional phase distribution reads as follows (Buiek, Wilson-Gordon, Knight and Lai [ 19921):

P(6m) = I ( U ) ( M f ) ( U ) I

2

(B.4)

for the ( a + 1)-dimensional state

c (7

=

c?)

n=O

1.)

9

03.5)

which is properly normalized,

n=O

for arbitrary a. The main problem resides in the construction of the normalized ( a+ 1)-dimensional states We restrict our attention to finitedimensional coherent states only. However, other finite-dimensional states of the electromagnetic field can be defined in a similar manner; e.g., squeezed states

432

QUANTUM PHASE PROPERTIES

[VI

(Buiek, Wilson-Gordon, Knight and Lai [ 1992]), even and odd coherent states (Zhu and Kuang [1994]), phase coherent states (Kuang and Chen [1994a,b], Gangopadhyay [ 19941) and displaced phase states (Gangopadhyay [ 19941). There exist several generalizations of coherent states comprising the finitedimensional case (see Zhang, Feng and Gilmore [ 19901 and references therein). It is possible to define coherent states using the concept of Lie group representations (see, e.g., Peiina, Hradil and JurEo [1994]), or to postulate the validity of some properties of the infinite-dimensional Hilbert-space coherent states for the finite-dimensional coherent states. We present two definitions of the latter case. Firstly, the coherent states I a)(,) in (a+ 1)-dimensional Hilbert space of a harmonic oscillator can be defined in the Glauber sense by the action of an analogue of the Glauber displacement operator 6(")(a)on the vacuum state 10) (Buiek, Wilson-Gordon, Knight and Lai [ 19921):

The operator E(")(a)is given in terms of the modified annihilation operator

and modified creation operator 6'. The coherent states la)(,) are close analogues of Glauber's (i.e., infinite-dimensional) coherent states I a).They were introduced and discussed by Buiek, Wilson-Gordon, Knight and Lai [1992], and their analytical Fock expansion was found by Miranowicz, Piatek and Tanai [1994] in the form [eq. (BS)] = la), with the superposition coefficients

If)

Here, xi =xj"+') are the roots of the modified Hermite polynomial of order (a+l), He,+,(x,) = 0, He&) 3 2-"/'H,,(x/fi), and a = la1 exp(i8). Kuang, Wang and Zhou [ 1993,19941 defined the normalized finite-dimensional coherent states in another manner by truncating the Fock-basis expansion of the Glauber infinite-dimensional coherent states or, equivalently, by the action of the formally designed "displacement" operator exp(ticit) exp(-tici) on the vacuum state. This approach is close to that of Vaccaro and Pegg [199Ob] in

VII

433

APPENDIX B. STATES FOR THE PEGQBARNETT PHASE FORMALISM

the construction of a finite-dimensional Wigner function for coherent states. The states I&)(,,) can be defined as follows (Kuang, Wang and Zhou [1993]):

1cp) In), LJ

= N(LJ)exp(6rit)10) =

(B. 10)

n=O

where

(B. 11) and the normalization constant is (Opatmy, Miranowicz and Bajer [ 19951) (B. 12) in terms of generalized Laguerre polynomials L",x). The differences between the finite-dimensional coherent states (B.7) and (B. 10) were discussed in detail by Opatmy, Miranowicz and Bajer [ 19951 using the finite-dimensional Wigner function (Wootters [19871, Vaccaro and Pegg [1990b]) and in terms of the Stokes parameters. go over into ( a = E): In the limit 0 -+ 00, the coherent states la)(,) and (B. 13) as was shown analytically by Opatrny, Miranowicz and Bajer [1995]. Howare essentially different, particularly for ever, the states l a)(,) and l~?)(, ) l a / ,lC?l 3 a'/*,from the ordinary (infinite-dimensional) Glauber coherent states la) as revealed by their photon-number, squeezing and phase properties (Buiek, Wilson-Gordon, Knight and Lai [1992], Kuang, Wang and Zhou [1993, 19941, Miranowicz, Piatek and TanaS [1994]). Let us only mention that the well-known property of the ordinary coherent state la) for the mean photon number is not fulfilled in the case of the finite-dimensional coherent states:

(B. 14) The finite-dimensional states discussed here are not only mathematical structures. A framework for their physical interpretation is provided by cavity quantum electrodynamics and atomic physics. Moreover, they can be generated,

434

QUANTUM PHASE PROPERTIES

[VI

e.g., in a single-mode resonator. Several methods have been proposed for the preparation of an arbitrary field state (e.g., Vogel, Akulin and Schleich [1993], Garraway, Sherman, Moya-Cessa, Knight and Kurizki [ 19941 and references therein), which can be readily applied for the preparation of these finitedimensional states. Recently, Leonski and TanaS [ 19941 have presented a scheme of field generation in a cavity containing a nonlinear Kerr medium, kicked periodically with classical pulses. The field generated in this process is actually the finite-dimensional coherent state la)(,) in Hilbert space of arbitrary dimension o + 1.

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AUTHOR INDEX FOR VOLUME XXXV

A Aartsrna, T.J. 98 Abragam, A. 82, 108 Abraham, N.B. 4, 20, 21, 49 Abramson, N. 208, 218 Abstreiter, G. 65, 137 Accardo, G. 218, 225 Aceves, A.B. 17 Adachi, S. 293 Adam, P. 361 Agap’ev, B.D. 262, 272 Aganval, G.S. 339, 340, 343, 344, 361, 381, 408, 41 1, 412, 416 Agranovich, VM. 101 Agrawal, G.P. 260, 274, 301, 304 Aitchison, J.S. 25 Akhmanov, S.A. 20 Akhmediev, N.N. 16, 27, 34 Akulin, VM. 434 Akul’shin, A.M. 293, 294 Aldrovandi, A. 244 Alekseev, VZ. 3 16 Alzetta, G. 259, 268, 269, 280, 281, 286, 289-29 1 Arnadesi, S. 201, 207, 208, 214, 215, 218225, 243 Ambrose, W.P. 64, 91, 92, 98-100, 121, 122, 129, 136 Anderson, D. 54 Anderson, P.W. 72, 76, 78, 82 Angel, J.R.P. 158 Arecchi, F.T. 49 Arimondo, E. 260-262, 266, 276, 279, 281, 283,285,286,294,310,313,315,316,319322, 324, 328, 329, 332, 333, 336, 338-340, 342, 346 Armitage Jr, J.D. 151 Arnold, S. 134 Askar’yan, G.A. 3

Asrnus, J.F. 213, 216, 217, 226 Aspect, A. 261,272,279,310,311,313-316, 322, 324, 346 Astilean, S. 96, 116 Aswendt, P. 236 Atcha, H. 236 Atkins, PW. 69 Aurnayr, F. 294, 295 Averbukh, J.S. 395 Avila, G. 301 Aviram, A. 65 Avni, R. 185 Azoulay, R. 65, 137 Azurni, T. 117

B

Bajer, J. 359, 433 Balkarei, Yu.1. 37, 42 Ballagh, R.J. 281, 282, 284, 292, 303 Balle, S. 49 Ban,M. 361 Band, Y.B. 326, 328 Bandilla, A. 360, 361, 375, 379, 41 1 Banyard, J.E. 236 Barakat, R. 416, 417 Bardou, F. 3 1 1, 3 14, 3 15, 324, 346 Barnes, W.L. 152, 158 Barnett, S.M. 301, 358, 359, 361, 364, 36& 370, 380, 381, 385, 386, 399406, 430, 43 1 Barrett, H.H. 151 Barthelerny, A. 3, 25 Baschk, T. 64, 91,92, 98-100, 121, 122, 139 BaschC, Th. 95, 97, 112, 113, 121, 129 Basharov, A.M. 23, 24 Baudour, J.-L. 95, 96 Beck, M. 360 Becker, M. 326 Belavkin, VP. 361 447

448

AUTHOR INDEX FOR VOLUME XXXV

Belinskii, A.V. 55 Bell, W.E. 288, 296 Bellanca, M.J. 315 Ben-Aryeh, Y. 359, 386 Bendjaballah, C. 361 Berg, M. 75, 83 Bergeman, T.H. 315, 321 Bergmann, K. 261, 326-328 Bergou, J. 360, 364, 428, 429 Bergquist, J.C. 64, 301 Berman, P.R. 305 Bernacki, B.E. 300 Bernard, J. 64, 77, 80, 87, 89, 91-93, 95, 96, 98, 100, 101, 113, 114, 116, 117, 119-121, 123-126, 130, 137, 139 Berne, B.J. 1 1 1 Bernheim, R.A. 291 Bertani, D. 218 Bertolotti, M. 361 Bespalov, VI. 3, 6 Beterov, 1.M. 288 Betzig, E. 135, 136 Bhanu Prasad, G. 339 Biatynicka-Birula, Z. 361, 392 Bialynicki-Birula, I. 361 Bigelow, N.P. 3 16-3 18 Binder, R. 345 Binnig, G. 63 Birotheau, L. 65, 137 Bisson, P. 152 Bjerre, N. 269, 295 Blatt, R. 269, 291, 292, 344 Bloembergen, N. 82, 305 Bloom, H.L. 288, 296 Blum, 0. 263, 265 Bockelmann, U. 65, 137 Bohm, G. 65, 137 Bohmer, B. 373 Bohnen, A. 124 Bollinger, J.J. 133 Born,M. 147, 148, 150, 151, 159, 165, 180, 232 Bouchaud, J.P. 314, 315 Bouchene, M.A. 305 Bowden, C.M. 260, 340 Bracewell, R.N. 153, 165 Brambilla, M. 49, 50, 55 Brand, H. 297 Brauchle, C. 95, 97, 139 Braurnan, J.I. 94 Braunstein, S.L. 359, 375

Brecha, R.J. 391 Breckinridge, J.B. 185, 186 Bregman, J.D. 154 Breiland, W.G. 120 Breinl, W. 75, 77 Brenner, H.C. 120 Brif, C. 359 Brill, T.B. 199 Brito Cruz, C.H. 68 Brouwer, A.C.J. 139 Brown, R. 77, 80, 91, 116, 117, 119-121, 124-126, 130 Brune, M. 385 Brunner, K. 65, 137 Buchenau, U. 71 Bucher, H. 90 Biichler, U. 360 Buckberry, C.H. 236 Buckle, S.J. 301 Buell, W.F. 315 Burak, D. 361, 386 Burckhardt, C.B. 155 Burggraf, H. 297, 298 Burns, M.M. 288, 291 Butters, J.N. 229 Buiek, V. 361, 381, 382, 385, 396, 4 3 1 4 3 3 Bykovskaia, L.A. 75, 84

C Cahill, K.E. 371, 372 Cai, T. 317, 318 Cailleau, H. 95, 96 Camesasca, D. 55 Caponero, M.A. 224 Carnal, 0. 344 Carroll, C.E. 279, 325, 326, 328 Carroll, J.E. 360 Carruthers, P. 357, 362 Carter, T.P. 68, 85, 92, 94, 95 Carter, W.H. 148, 160, 161, 163 Castanit, F. 157 Castelli, F. 55 Castin, Y. 313 Cattaneo, M. 49, 50 Cavalieri, S. 310 Caves, C.M. 359, 375, 402, 403 Celikov, A.A. 293, 294 Cerboneschi, E. 342 CBrez, P. 293, 300, 301 Cetica, M. 218 Chai, C.L. 391

AUTHOR INDEX FOR VOLUME XXXV

Chaichian, M. 361 Chang, I.C. 156 Charalambidis, D. 3 10 Chaturvedi, S. 361, 381 Chaudhry, M.A. 360, 375 Chebotayev, V.P. 288 Chen, X. 432 Chen, Z. 49 Cheng, K.Z. 345 Cheng, L.J. 156 Chester, T.L. 185 Chiao, R.Y. 3, 14 Chichester, R.J. 135 Chikada, Y. 154 Chitta, V.A. 96, 116 Chizhov, A.V. 361, 374, 378, 397 Christiansen, P.L. 386 Chu,A. 317 Chu, P.L. 23, 24 Chu, S. 310, 328, 329 Chung, Y.C. 233 Cianfarelli, T. 244 Cibils, M.B. 361 Cirac, J.I. 344 Clar, E. 123 Claverie, P. 70, 71 Coates, A.B. 49, 50 Cohen, D. 386 Cohen-Tannoudji, C. 108, 110, 112, 133, 134, 261, 268, 269, 274, 276, 277, 279, 283, 285, 310, 311, 313-316, 322, 324, 340, 346 Collett, M.J. 386 Collier, R.J. 155 Cook, R.J. 133 Cooper, J. 360 Corney, A. 281 Corval, A. 96, 116 Coude du Foresto, V. 148, 156, 172 Coulet, P. 49 Coulston, G.W. 326, 328 Courjon, D. 134 Cowie, L. 157, 175, 176 Creath, K. 236, 251 Crennell, K. 251 Croci, M. 90, 93, 103 Cross, M.C. 4 Cuche, Y. 361 Cui, H.F. 335 Cummings, F.W. 388 Cummins, H.Z. 1 1 1 Curl, R.F. 289

D

449

Daeubler, B. 361 Dagenais, M. 64, 112, 133 Dalibard, J. 133, 261, 274, 3 13 Dalton, B.J. 260, 262, 287, 309, 310, 341, 361 D’Altorio, A. 207, 208, 21 8-226 Dam, N. 261, 327 Damaskinsky, E.V. 358 D’Angelo, E.J. 49, 50 Dangoisse, D. 49 Daniel, D.J. 392 D’Ariano, G.M. 361 Das, H.K. 361 Davidovich, L. 385 Davies, J.C. 236 Davis, L.M. 64 De Angelis, A. 224 de Bree, P. 98 de Clercq, E. 300, 301 de Labachellerie, M. 301 de Lignie, M.C. 281, 286, 306, 307 De Oliveira, F.A.M. 396 De Santis, P. 225, 226 de Vries, H. 96, 98, 113-1 15 Dean, P. 71 DBbarre, A. 305 Decker Jr, J.A. I50 Decomps, B. 282 Dehmelt, H. 64, 133, 269, 291, 292 Del Monte, M. 200 Delhaes, P. 65 Delhaye, M. 152 Delugeard, Y. 95, 96 Desailly, R. 25 Desem, C. 23, 24 Deserno, R. 261, 302 Devoret, M.H. 65 Dhamelincourt, P. 152 Di Porto, B. 22 Diedrich, F. 64, 112, 133 Dimarcq, N. 293 Dinneen, T. 293 Dirac, P.A.M. 357, 362 Disselhorst, J.A.J.M. 116 Doery, M.R. 315, 321 Dolan, G.J. 65 Donckers, M.C.J.M. 116 Donszelmann, A. 337, 338 Doss, H.M. 274, 276, 284, 334, 335 Dowling, J.P. 340, 359, 361, 380

450

AUTHOR INDEX FOR VOLUME XXXV

Drake, K.H. 296, 297 Dravins, D. 157 Drexhage, K.H. 84 Drits, V.V. 3, 27 DrobnL, G. 361, 391, 419, 421 Dubin, D.A. 361 Ducloy, M. 282 Duguay, M.A. 184 Dum, R. 317, 318, 325 Dumont, M. 282 Dung, H.T. 389-391 Dupont-Roc, J. 108, 1 10, 134, 268 Duppen, K. 68 Duree Jr, G.C. 22 Dwelle, R. 156 Diubur, A. 209

E Eberly, J.H. 260-262, 325, 326, 341, 342, 388, 389 Eckstein, J.N. 298 Edmundson, D.E. 25, 34 Efron, U. 177 Ehrenberg, M. 113 Eigen, M. 64 Eiselt, J. 372, 375, 389, 390 Ekers, R.D. 156 Eliel, E.R. 281, 284, 286, 306, 307 Elinson, M.I. 37, 42 Ellinas, D. 361 Elyutin, S . 0 . 23, 24 Emile, 0. 311, 314, 315 Englert, B.G. 360, 364, 428, 429 Ennos, A.E. 229, 230 Ems, R.H. 25, 34 Eremina, I.V. 27 Erf, R.K. 207 Esslinger, T. 3 17 Esteve, D. 65 Ezekiel, S. 261, 269, 299, 300

F

Facchini, M. 236-240 Fan, A.F. 391 Fan, H.Y. 385 Fano, U. 277, 309 Faridani, M.B.A. 360 Faucher, 0. 310 Fayer, M.D. 75, 83, 94, 96, 1 I5 Fearey, B.L. 94 Fedorov, A.V. 18, 36, 37, 44, 47, 51, 53, 57

Fedorov, S.V. 44, 51, 53 Feld, M.S. 263, 266, 282, 284, 288, 291 Feldman, B.J. 266, 282, 284, 288 Feldman, M. 301 Feng, D.H. 432 Feoktistov, A.A. 288 Ferguson, A.I. 298 Fernandez-Sanchez, J.M. 87 Fetter, A.L. 27 Field, J.E. 340 Fill, E.E. 331, 333 Finkelstein, V. 305 Finlayson, N. 17 Firth, W.J. 4, 10, 14, 18, 20, 21, 44, 49 Fischer, UCh. 92 Fitzgerald, J.J. 185 Fleischhauer, M. 286, 333, 336, 339, 340, 343,344 Fleischmann, H.-C. 96, 116 Fleury, L. 77, 80, 89, 98, 100, 114, 121, 124126, 130 Fomalont, E.B. 152 Foot, C.J. 321, 328 Ford, H.D. 236 Fork, R.L. 68 Fotakis, C. 310 Fougkres, A. 359 Franchi, M. 244 Franson, J.D. 361 Frederick, E.E. 156 Freyberger, M. 360, 361, 375, 386 Friedmann, H. 339 Friedrich, J. 75, 77, 100 Froehly, C. 3, 25 Frolich, D. 296, 360 Frommer, J. 64 Fromsel, V.A. 3, 10 Fry, E.S. 280, 284-286, 335, 336 Fukuda, Y. 297 Fulton, T.A. 65 Funkhomer, A.T. 151 Fuss, I.G. 359

G Gabor, D. 203 Galindo, A. 358 Galperin, Y.M. 71 Gangopadhyay, G. 432 Gantsog, Ts. 358, 361, 364, 374, 378, 379, 381, 382, 386, 388, 391, 392, 394-397,402,

AUTHOR INDEX FOR VOLUME XXXV

407409, 41 1 4 1 6 , 418, 419, 421, 425, 429, 430 Gao, J.Y. 334, 335 Gaponov, VA. 27 Garmire, E. 3, 14 Garosi, F. 224 Garraway, B.M. 374,376, 377, 379,381, 385, 434 Garrison, J.C. 358, 364, 427, 429, 430 Ghvik, K.J. 212 Gatti, A. 55 Gatz, S. 25 Gaubatz, U. 261, 326, 327 Gault, W.A. 157 Gavrielides, A. 329, 333 Gay, J. 175 Gea-Banacloche, J. 269, 284, 296, 308, 339 Gennaro, G. 361 Gerber, Ch. 63 Gerhardt, H. 360 Gerry, C.C. 360, 381, 392, 393, 408 Gerz, C. 328 Gheri, K.M. 344 Giacomelli, G. 49 Gibbs, H.M. 18, 25-27, 50, 54 Gieler, M. 294, 295 Gil, L. 49 Gilmore, R. 432 Ginzburg, VM. 218 Giordano, V 293 Glauber, R.J. 371, 372, 392 Glogower, J. 357, 359, 362, 363, 366, 421 Glorieux, P. 49 Glushko, B. 274 Goetz, A.F.H. 152, 158 Goldhirsh, I. 358 Golding, B. 75, 107 Goldner, L.S. 328 Goldstein, E. 320 Golik, L.L. 37 Gong, C.D. 361 Goodman, J.W. 151, 228 Goodwin, PM. 64, 136 Gordienko, V 318, 319 Gordon, J.P. 23 Gorgas, W. 96, 116 Gori, F. 201, 214, 215, 217, 218, 225, 226, 243 Gorlicki, M. 282 Gornyi, M.B. 262, 272, 274, 318, 320 Gorokhovskii, A.A. 75, 84

45 1

Gou, S.C. 408 Gozzini, A. 259, 268, 280, 286, 289-291 Grabert, H. 65 Graebner, J.E. 75, 107 Graf, M. 284-286, 335, 336, 361 Grantham, J.W. 27 Grasiuk, A.Z. 4, 21 Gray, H.R. 260, 266, 291, 292 Green, C. 49, 50 Greenaway, A.H. 154 Grella, R. 201, 214, 215, 218, 243 Griffith, J.E. 64 Grigg, D.A. 64 Grigoryan, VS. 54 Grigor'yants, A.V '37, 42 Groenen, E.J.J. 116, 131, 139 Grnnbech-Jensen, N. 386 Gruber, A. 139 Grynberg, G. 108, 110, 134, 268 Guattari, G. 201, 213-215, 218, 225, 226, 243 Giilker, G. 228, 236, 249 Guntherodt, H.-J. 64 Guo, A.Q. 391 Guo, C. 334 Guo, X.Z. 334, 335 Gupta, R. 321 Gurevich, VL. 71 Gustafson, T.K. 263, 265 Giittler, F. 90-93, 101, 103, 106, 107, I 10 Gygax, H. 110

H Haarer, D. 75, 77, 100, 101, 132 Hach 111, E.E. 381 Haemmerle, W.H. 75, 107 Hagberg, A.A. 42 Haken, H. 4 Hall, J.L. 326, 328, 415 Hall, M.J.W. 359 Haller, K. 69 Halmshaw, R. 199 Halperin, B.I. 76, 78 Hamilton, M. 156 Hanada, K. 154 Hanbury-Brown, R. 11 1, 1 13 Hangyo, M. 152 Hanle, W. 281 Hannaford, P. 282, 284, 285, 303, 335 Hiinsch, T. 266, 288 Hiinsch, T.W. 298, 317

452

AUTHOR INDEX FOR VOLUME XXXV

Happer, W. 285 Haq, H.R. 341, 342 Harde, H. 297, 298 Hariharan, P. 151, 203, 205, 210 Haroche, S. 134, 385 Harris, C.B. 120 Harris, S.E. 156, 340-342 Harris, T.D. 136 Harshman, P.J. 263, 265 Hart, L.P. 185 Hanvit, M. 147, 150, 156 Hasegawa, A. 3, 23 Hashi, T. 297 Haslett, J.W. 157 Haus, H.A. 14 Hayes, J.M. 130 He, G.Z. 326 Heatley, D.R. 27 Hecht, B. 139 Heinzelmann, H. 64 Heinzen, D.J. 133 Helstrom, C.W. 358 Hemmer, P.R. 261, 269, 299, 300, 316-318, 345 Hemmerich, A. 3 17 Hennenquin, D. 49 Hennings, M.A. 361 Herling, G.H. 284, 335, 336 Herrmann, J. 25 Herzog, U. 375 Hess, H.F. 136 Hess, L.D. 177 Hesselink, W.H. 96 Hill, J.M. 158 Hillery, M. 375 Hilliard, R.L. 157 Hinsch, K. 228, 236, 249 Hintzen, P 158 Hioe, F.T. 261, 279, 325, 326, 328 Hirabayashi, H. 154 Hirai, A. 183, 184 Hizhnyakov, VV 71 Hochstrasser, R.M. 134 Hofling, R. 236 Hohenberg, P.C. 4 Hohenstatt, M. 64, 133 Holmes, C.A. 392 Holscher, C. 228, 236, 249 Holstein, T. 76 Horman, M.H. 202 Home, D.E. 95, 101

Horowicz, R.J. 359, 375, 380, 381, 387 Howald, L. 64 Hradil, Z. 359-361, 432 Hsu, D. 73 Hulet, R.G. 64 Hunklinger, S. 75 Hussein, Z. 307, 308, 339 Huyen, N.D. 391

I Ichioka, Y. 149, 157, 168, 172, 174, 176-178, 181, 183, 184, 190 Ikegama, T. 25, 26 Imamoglu, A. 340 h a , H. 178 Ingold, G.L. 65 Inoue, T. 149, 157, 168, 172, 174, 176-178, 181, 183, 184, 190 Imgartinger, T. 93, 95, 106, 107, 124, 139 Ishiguro, M. 154 Ishikawa, S. 154 Ishimaru, A. 165 Itano, W.M. 64, 133 Itoh, K. 148, 149, 155, 157, 160, 163, 165, 167, 168, 172, 174, 176-178, 181, 183, 184, 190 Ivanov, V.K. 101 Ivanov, VYu. 20 Iwamura, H. 25, 26 Iwashita, H. 154 Izawa, Y. 293 Izrael, A. 65, 137

J Jackel, J.L. 25 Jackson, J.D. 84 Jain, M. 343 Janik, G. 269, 291, 292 Janszky, J. 361 Javan, A. 263, 266, 282, 288 Javanainen, J. 269, 295, 340 Jaynes, E.T. 360, 388 Jex, I. 361, 391, 419, 421 Jiang, D. 334 Jiang, Y. 334, 335 Jin, G.X. 334, 335 Jin, S. 269, 284, 296, 308, 339 Joffrin, J. 77 Johnson, D.J. 185 Johnston, S.F. 157 Jones, J.D.C. 236 Jones, K.R.W. 359

AUTHOR INDEX FOR VOLUME XXXV

Jones, R. 213,230 Judge, D. 368 Julienne, P.S. 326, 328 JurEo, B. 360, 432

K

Kaarli, R.K. 75, 84 Kador, L. 85, 95, 100, 101 Kagan, Yu. 76, 129 Kaiser, R. 261, 272, 279, 310, 313, 315, 316, 322, 346 Kaivola, M. 269, 284, 292, 295 Kanzawa, T. 154 Kaplan, A.E. 25, 34 Kkski, M. 392 Kasevich, M. 310 Kassner, K. 76 Kastler, A. 134, 268, 285 Kasuga, T. 154 Kato, J. 236 Katz, D.P. 3 17, 3 18 Katzka, P. 156 Kawamura, Y. 25, 26 Keil, R. 288 Keitel, C.H. 276, 333-335, 339, 340 Keller, J.-C. 305 Keller, R.A. 64, 136 Kelley, P.L. 263, 265 Kendall, D.J.W. 157 Kennedy, T.A.B. 287 Kent, A.J. 49, 50 Kemer, B.S. 42 Kersten, G. 296 Kettner, R. 95, 139 Khan, M.A. 360, 375 Khanin, Y.I. 274, 298, 329, 332, 333 Kharlamov, B.M. 75, 84, 100 Khitrova, G .

25-27

Khodova, G.V. 17, 18, 36, 37, 39, 41, 44,47, 51, 53, 57 Kielich, S. 374,378, 381, 382, 392,394-396, 411, 415 Kierstead, J. 345 Kim, M.K. 345 Kim. M . S .

3 8 1 . 396

Kimble, H.J. 64, 112, Kinoshita, M. 117 Kitagawa, H. 392 Kivshar, YuS. 22 Klafter, J. 77 Klein, N. 389

133, 344,415

453

Kleinfeld, J.A. 335 Kleppner, D. 134 Knight, P.L. 260-262, 269, 274, 287, 301, 309, 310, 341, 361, 374, 376, 377, 379, 381, 382, 385, 386, 388, 389, 396,431434 Knox, WH. 68 Koashi, M. 391 Kobayashi, S. 178 Koch, K.-H. 124 Kocharovskaya, O.A. 274, 298, 329-333, 338, 339 Koedijk, J.M.A. 75, 83, 123 Kohler, J. 1 16, 139 Kokai, F. 94 Kolbe, J. 304 Kolobov, M.I. 55 Kolokolov, A.A. 16, 22, 27, 34 Korsunsky, E.A. 3 14, 3 18-320 Korytin, A.I. 54 Kosachiov, D.V. 301, 314, 318-320 Kosteniuk, PR. 157 Koster, E. 304 Kozankiewicz, B. 95, 123, 137 Kozierowski, M. 415 Krakhella, K. 236 Kramer, A. 228, 236 Kiepelka, J. 358, 361 Kruglov, V.I. 3, 27 Krylova, D.D. 3 16 Kryschi, C. 96, 116 Kryzhanovsky, B. 274 Kuang, L.M. 432,433 Kubo, R. 72 Kiihl, T.U. 288, 291 Kuhn, A. 326, 327 Kuhn, H. 90 Kuklinski, J.R. 326 Kiilz, M. 326

Kumar, P 361 Kummer, S. 97, 139 Kurizki, G. 385,434 Kurtz, I. 156 Kuznetsov, E.A. 14 L

Ladan, F.R.

65, 1 3 7

Lai, W.K. 4 3 1 4 3 3 Laird, B.B. 132 Lakshmi, PA. 361 Lamb Jr, W.E. 266, 269 Lambropoulos, P 310

454

AUTHOR INDEX FOR VOLUME XXXV

Lamela-Rivera, H. 300 Lane, A.S. 359 Lange, W. 261, 284, 296-298, 302, 304, 335 Larichev, A.V. 20 Lauder, M.A. 262, 301, 309, 310 Law, C.T. 3, 14 Lawall, J. 3 11, 3 15, 324, 328, 346 Lax,M. 72 Lazzarini, L. 2 13 Le Gouet, J.L. 305 le Poole, R.S. 154 Leaird, D.E. 25 Leduc, M. 324, 346 Lee, H.W.H. 96, 115 Leendertz, J.A. 229 Lefever, R. 19, 44 Leiby Jr, C.C. 261, 299, 300 Leith, E.N. 203 Lena, P. 148, 156, 172 Leonardi, C. 361 Leonhardt, U. 360, 373, 375 Leonski, W. 434 Lepers, C. 49 Letokhov, V.S. 288, 318 Levelut, A. 77 Levy-Leblond, J.M. 358, 360 Lewis, L. 301 Lewis, L.L. 301 Li, G.X. 391 Li, J.S. 335 Li, R.D. 331 Li, X. 280, 286, 336 Li, Y. 269, 284, 294, 296, 308, 339 Liedenbaum, C. 327 Lillo, E 361 Lin, L.H. 155 Lindberg, M. 345 Lindegren, L. 157 Lindley, D. 158 Lisak, M. 54 Litfin, G. 360 Littau, K.A. 75, 83 Litvak, A.G. 16 Lohmann, A. 151 Lskberg, O.J. 228, 229, 236 Loudon, R. 79, 111, 365, 385, 386, 388 Louisell, W.H. 357, 363 Lounis, B. 139, 269, 277 Louvergneux, E. 49 Lucia, A.C. 244 Lugiato, L.A. 4, 18, 19, 44, 49, 50, 54, 55

Luis, A. 361, 402, 414 Lukosz, W. 87 LukS, A. 358, 359, 361, 392 Luttke, W. 124 Lynch, R. 360, 386, 393 Lyo, S.K. 76

M Maccarrone, E 291 MacGillivray, W.R. 302, 303 Macovski, A. 229 Mada, H. 178 Mahoney, J.C. 156 Mailard, J.P. 157 Mailhes, C. 157 Mairnistov, A.I. 23, 24, 54 Mais, S. 139 Mak, A.A. 3, 4, 10, 21 Malmo, J.T. 229 Manavi, M. 68, 85, 92 Mandel, L. 64, 112, 133, 359, 415 Mandel, P. 44, 331-333 Maneuf, S. 3, 25 Mango, E 291 Manka, A S . 274, 284 Mann,A. 386 Mansour, N.B. 293 Marburger, J.H. 27 Marchand, E.W. 163 Mariotti, J.-M. 148, 155, 156, 167, 171, 172, 190 Markov, VB. 226 Marte, P. 317, 318, 325, 326, 328, 344 Martin, J.C. 64, 136 Martinez-Herrero, R. 163 Marvulle, V 361 Marzin, J.Y. 65, 137 Masalov, A.V. 415, 419 Matera, M. 310 Matisov, B.G. 262, 272, 274, 287, 301, 314, 3 18-320 Matsuoka, M. 391, 421 Matsushita, M. 131 Matteini, M. 244 Matthys, D.R. 360 Mattick, A.T. 184 Mattingly, Q.L. 64 Mauri, F. 279, 281, 316, 320, 322, 329, 332, 333, 338, 339 Mazets, I.E. 287 McDonald, G.S. 14, 44

AUTHOR INDEX FOR VOLUME XXXV

McDuff, R. 287 McGlynn, S.P. 117 McGowan, R.R. 361 McLean, R.J. 281, 284, 292 Mecozzi, A. 392, 415 Meijer, G. 13 1 Meijers, H.C. 75, 83 Meinlschmidt, P. 249 Meixner, A.J. 100 Mejias, PM. 163 Mekamia, D. 175 Meng, H.X. 391 Mertz, L. 149, 151, 185, 186 Mervis, J. 317, 318 Meschede, D. 307, 308, 339 Metcalf, H. 315, 321 Mets, U. 64, 11 1 Meyer, E. 64 Meyer, G.M. 284, 335, 336 Meyling, J.H. 101 Meystre, P. 134, 320, 391 Micheli, M. 218 Milburn, G.J. 392 Miller, C. 361 Miller, D.W. 157 Miller, R.D.J. 96, 116 Ming, L. 25, 26 Minogin, V.G. 318, 322, 323 Miranowicz, A. 358,364, 372-374,378,379, 381-383, 386, 392-396,415, 429,430, 432, 433 Mironov, V.A. 16 Mishina, T. 297 Misra, B. 133 Mitchell, D.F. 27 Mitschke, F. 261, 302, 304 Mitterdorfer, A. 92, 106 Mlynek, J. 261, 296-298, 302, 304 Mobius, D. 90 Moerner, W.E. 64, 68, 75, 84, 85, 91, 92, 95, 98-101, 104, 105, 112, 113, 116, 121, 122, 124, 129, 139 Moi, L. 259, 268, 269, 280, 281, 286, 289, 290 Molenkamp, L.M. 82 Molesini, G. 218 Mollow, B.R. 134 Moloney, J.V. 16, 17, 42 Monken, C.H. 359 Montero, S. 87 Morawitz, H. 100, 132

455

Morigi, G. 321, 328 Morimoto, M. 154 Morita, K. 154 Morsink, B.W. 98 Moya-Cessa, H. 385, 434 Miillen, K. 124 Miiller, K.P. 75 Munk, J. 213 Munk, W. 213 Murnick, D.E. 288, 291 Murty, M.VR.K. 151 Murzakhmetov, B.K. 361, 374, 378, 397 Miischenborn, H.J. 90, 93, 103 Musumeci, G. 213 Myers, A.B. 104, 105

N

Nagourney, W. 133, 269, 291, 292 Nakadate, S. 236 Nakai, S. 293 Nakajima, T. 3 10 Nakashima, S. 152 Nakazima, K. 154 Napier, P.J. 156 Narasimhan, L.R. 75, 83 Narducci, L.M. 274, 276, 284, 334-336, 340 Narozhny, N.B. 389 Nassar, N.S. 236 Nath, R. 361 Natoli, V.D. 300 Nesterov, L.A. 28, 33 Neuhauser, R. 327 Neuhauser, W. 64,133, 269, 291 Neunaber, H. 228, 236 Neureiter, C. 314, 318, 319 Neurgaonkar, R.R. 22 Neusser, H.J. 327 Newell, A.C. 17 Newton, R.G. 358 Ng, K.C. 134 Nibbering, E.T.J. 68 Nicolini, C. 285, 290 Nicolis, G. 4 Nieto, M. 357, 362 Nieto, M.M. 360, 361 Niki, H. 293 Nikitin, S.P. 415, 419 Nikonov, D.E. 280, 285, 286, 336 Noh, J.W. 359 Non-destructive Testing Handbook 199 Noordam, J.E. 154

456

AUTHOR INDEX FOR VOLUME XXXV

Notkin, G.E. 288 Nottelman, A. 284, 335 Novak, F.A. 134 Novgorodov, VG. 218 0

Ohta,T. 181 Ohtsu, M. 294 Ohtsuka, Y. 148, 149, 155, 160, 163, 165, 167, 168 Okamura, S. 154 Oliver, M.K. 25 Ol’shanii, M.A. 317, 318, 321-323, 325 Omont, A. 285 Ontai, G. 269, 300 Opatmy, T. 361, 433 OPPO,G.-L. 49, 50, 274, 284 Orbach, R. 76 Oreg, J. 325, 326 Orlowski, T.E. 96, 115 Orriols, G. 259, 260, 266, 268, 269, 280, 281, 284, 286, 289, 290 Orrit, M. 64, 77, 80, 87, 89, 91-93, 95, 96, 98, 100, 101, 112-114, 116, 117, 119-121, 123-126, 130, 137, 139 Orszag, M. 361 Osad’ko, L.S. 69, 72, 73 Osipov, VV 42 Ostrovskaya, N.V 16 Ostrovsky, I. 2 10 Oudejans, L. 261, 327 Ovemey, R.M. 64 Owner-Petersen, M. 231, 251

P

Padmabandu, G.G. 280, 285, 286, 336 Pannell, C.N. 236 Paoletti, D. 207-209, 218-226, 23C242 Papoff, F. 279, 320, 322 Pappas, P.G. 288, 291 Paprzycka, M. 396 Parigger, C. 282, 284, 285, 303, 335 Paris, M.G.A. 361 Parkins, A.S. 344 Parshin, D.A. 71 Pasturczyk, 2. 157 Patterson, F.G. 96, 115 Paul, H. 358, 360, 361, 373-375 Pavone, F.S. 310 Pax, P. 320 Pecora, R. 111

Pegg, D.T. 287, 301, 302, 358, 359, 361, 364, 366-370, 380, 381, 386, 388, 399406, 430-433 Pellizzari, T. 325 Peng, J.S. 391 Penman, C. 10 Perelman, N.F. 395 Peiina, J. 360, 385, 432 Pesnovi, V 358, 359, 361, 392 Perrin, J. 63 Personov, R.I. 64, 67, 73, 75, 84, 91, 93, 95, 100, 101, 104 Peters, C. 284, 335 Petraroia, I? 2 18 Peuse, B.W. 261, 299 Pfeiffer, L.N. 136 Phillips, P.J. 104 Phillips, W.A. 76, 78, 128 Phillips, W.D. 328 Phoenix, S.J.D. 389 Pi@ek, K. 432, 433 Picard, R.H. 261, 299 Pike, E.R. 11 1 Pillet, I? 328 Ping, Q. 236 Pirotta, M. 91-93, 101, 110 Pirovano, R. 49, 50 Piskunova, L.V. 27 Plakhotnik, T. 92, 93, 95, 106, 107, 124, 139 Plankey, F.W. 185 Pohl, D.W. 134, 139 Pohl, R.O. 75 Poluetkov, O.G. 131 Pons, M.L. 341, 342 Popov, VN. 358, 427, 429 Pothier, H. 65 Poulsen, 0. 269, 284, 292, 295 Pound, R.V 82 Powell, R.L. 202 Prati, F. 49, 50 Prentiss, M. 328 Prentiss, M.G. 3 16-3 18 Prigogine, I. 4 Protsenko, VN. 2 18 Purcell, E.M. 82 Puri, R.R. 41 1, 412, 416

R Radmore, P.M. 261, 269, 274 Raimond, J.M. 385 Rakhmanov, A.N. 44

AUTHOR INDEX FOR VOLUME XXXV

Ramanujam, P.S. 386 Ramazza, P.L. 49 Ramos, M.A. 71 Ramsey, J.M. 134 Ramsey, N. 299 Ramsey, S.D. 229 Rangnekar, S.S. 25, 34 Rappaport, S.A. 156 Rastogi, P.K. 212 Rasumova, N,V 101 Rathe, U. 344 Rathe, U.W. 284, 286, 335, 336 Ratner, R.A. 65 Rautian, S.G. 288 Raymer, M.G. 360 Rebane, K.K. 69, 74 Rebane, L.A. 75, 84 Reid, M.F. 360, 361 Reilly, PD. 80, 123, 139 Reineker, F! 71 Rempe, G. 389 Renn, A. 90-93, 100, 101, 103, 106, 107, 110 Residori, S. 49 Reuss, J. 261, 327 Reyes, G.F. 156 Reynaud, F. 25 Reynaud, S. 11 1, 261, 274, 276 Ribak, E. 185, 186 Richardson, H. 42 Richter, T. 375 Richter, W. 100, 132 Ridgway, S.T. 148, 155, 167, 171, 190 Riegler, P. 360 Rigler, R. 64, 1 11, 1 I3 Riitano, P. 244 Risken, H. 361, 372, 375, 389, 390 Ritze, H.H. 361, 375, 379, 41 1 Rizzi, M.L. 218 Rocca, F. 49 Roddier, C. 185, 186 Roddier, F. 151, 165, 185, 186 Rogers, G.L. 151 Rohrer, H. 63 Rolston, S.L. 328 Rosanov, N.N. 4, 7, 10-13, 16-18, 21, 28, 33, 35-39, 41, 43, 44, 46, 47, 51, 53-55, 57 Rosenwaks, S. 326 Roth, S. 64 Rothberg, L.J. 305

457

Rozhdestvensky, Yu.V 274, 301, 314, 318320 Ru, P. 274, 276, 284, 334, 335 Rubenchik, A.F! 15 Rudecki, F! 261, 326, 327 Rzhanov, Yu.A. 37, 42 S Saavedra, C. 361 Saito, H. 236 Saito, S. 181 Sakai, H. 164 Salamo, G.J. 22 Saleh, B.E.A. 55, 112 Sanchez, A. 282 Sanchez-Mondragon, J.J. 389 Shchez-Soto, L.L. 361, 402, 414 Sandberg, J. 133 Sanders, B.C. 385, 396 Sandle, W.J. 282, 284, 285, 303, 335 Santhanam, T.S. 358, 366 Sargent 111, M. 269 Sarid, D. 64 Sato, R. 177 Sato, S. 177 Satsuma, J. 24 Saubamea, B. 31 1, 315, 324, 346 Saxby, G. 204 Sazonov, S.B. 345 Schabert, A. 288 Schaefer, D.W. 11 I Schaefer, L.F. 229 Schaufler, S. 361 Schenzle, A. 391 Schernthanner, K.J. 320 Schiemann, S. 261, 326, 327 Schier, H. 64 Schieve, W.C. 361 Schirripa Spagnolo, G. 208, 209, 226, 2 3 6 242 Schleich, W. 360, 361, 375, 381, 386, 387, 398 Schleich, W.P. 359, 361, 380, 434 Schlossberg, H.R. 266 Schmid, D. 96, 116 Schmidt, J. 116118, 131, 139 Schmidt, 0. 307, 308, 339 Schmidt-Iglesias, C. 305 Schober, H.R. 71 Schoendofl, L. 361 Schubert, M. 269, 291, 292

458

AUTHOR INDEX FOR VOLUME XXXV

Schulz, W.E. 302, 303 Schumaker, B.L. 402, 403 Schwomma, 0. 229 Scott, J.S. 158 Scroggie, A.J. 44 Scully, M.O. 269, 272, 274, 276, 280, 284286, 307, 329-331, 333-336, 339, 340, 343-345, 361 Seaton, C.T. 17 Segev, M. 22 Seitzinger, N.K. 64 Semenov, YE. 18, 21, 36, 37, 41 Sen, D. 151 Sepiol, J. 92, 106, 110 Sergeev, A.M. 54 Series, G.W. 274 Sesselmann, Th. 100, 132 Shabat, A.B. 23 Shahriar, M.S. 300, 316-318, 345 Shakher, C. 207 Shank, C.V. 68 Shao, Y.L. 310 Shapiro, J.H. 358, 359, 364 Sharp, E.J. 22 Shashkin, V.V. 36 Shay, T.M. 233 Shchepinov, V.P. 210 Shepard, S.R. 358, 359, 364 Shepherd, G.G. 157 Shera, E.B. 64 Sherman, B. 385, 434 Shimizu, K. 3 I 1, 3 15, 324, 346 Shmalhausen, V.I. 44 Shore, B.W. 262, 326, 388 Shulga, S.U. 3 18 Shultz, J.L. 22 Shumovsky, A.S. 389-391 Sibilia, C. 361 Siemers, I. 269, 291, 292 Silberberg, Y. 3, 25, 33, 34 Silbey, R. 76 Sild, 0. 69 Simons, D. 157, 175, 176 Sinha, K.B. 358 Sipe, J.E. 25, 26 Sirohi, R.S. 207 Skinner, J.L. 73, 76, 80, 123, 132, 139 Sklyarov, Yu.M. 23, 24, 54 Slettemoen, G.A. 236 Slosser, J. 391 Small, G.J. 94, 130

Smirnov, V.A. 7, 10, 11, 13, 16, 28, 33 Smirnov, V.S. 261, 279, 282 Smith, A.V. 280, 336 Smith, P.W.E. 25 Smith, S.P. 300 Smith, T.B. 361 Smithey, D.T. 360 Snegiriov, A. 318, 319 Snyder, A.W. 21 Sokolov, I.V. 55 Sommerfeld, W. 218 Soms, L.N. 3, 10 Soper, S.A. 64 Soto-Crespo, J.M. 27, 34 Spence, J.C.H. 63 Spreeuw, R.J.C. 328 Srinivasan, V. 361 Standage, M.C. 302, 303 Stegeman, G.I. 17 Stenholm, S. 361 Stepanov, B.M. 218 Stetson, K.A. 202 Stettler, J.D. 260 Steuenvald, S. 327 Steyn-Ross, M.L. 301 Stoler, D. 385, 392, 415 Stoke, S. 327 Stoneham, A.M. 125 Stout, R.P. 130 Streater, A.D. 335 Stroke, G.W. 151 Stroud Jr, C.R. 260, 266, 291, 292 Su, C. 333, 336, 339 Su, R.T.M. 289 Suchkov, A.F. 50 Sudarshan, E.C.G. 133 Sugiyama, A. 177 Suiter, H.R. 156 Sukov, A.I. 16, 27 Summy, G.S. 361 Susskind, L. 357, 359, 362, 363, 366, 427 Sussman, R. 327 Suter, D. 116 Swain, S. 274, 287, 361 Swartzlander, G.A. 3, 14 Syed, K.S. 14

T

Taichenachev, A.V. 287, 322, 325 Tai’eb, R. 317, 318 Takagi, K. 289

AUTHOR INDEX FOR VOLUME XXXV

Takahashi, T. 154 Takeda, M. 178 Talanov, VI. 3, 6, 14, 22, 33 Talon, H. 89, 98, 100, 112-114 Tamarat, Ph. 139 Tanaka, H. 94 Tan&, R. 358, 361, 364, 374, 378, 379, 381, 382, 386, 388-392, 394-397, 402, 407409, 411416, 418, 419, 421, 425, 429, 430, 432434 Tara, K. 361, 381 Tatam, R.P. 236 Taylor, B. 320 Tchenio, P. 104, 105, 305 Teets, R. 298 Teich, M.C. 55, 112 Tekumalla, A . R . 358 Tench, R.E. 261, 299 Tltu, M. 301 Thiobald, G . 293 Thierry-Mieg, V. 65, 137 Thijssen, H.P.H. 75 Thomas, J.E. 261, 299 Thompson, A . R . 156 Thorsen, P. 284, 292 Tittel, F.K. 280, 336 Tittel, J. 95 Titulaer, U.M. 392 Tlidi, M. 44 Tombesi, P. 392, 415 Torry, H.C. 119 Toschek, P. 64, 133, 266, 288 Toschek, P.E. 269, 291 Totzauer, W. 236 Townes, C.H. 3, 14 Trankle, G. 65, 137 Tredicce, J.R. 49, 50, 274 Trommsdorff, H.P. 76, 96, 116 Tsui, Y.K. 360, 361 Tu, H.T. 361 Tumaikin, A.M. 261, 279, 282, 287, 322, 325, 345 Turitsyn, S.K. 14 Turski, L.A. 358 Twiss, R.Q. 1 11, 1 13 Tyrer, J.R. 228

U

Ulitsky, N.I. 100 Ulrich, B.T. 339

459

Upatnieks, J. 203 Urbanski, K.E. 360 Urbina, C. 65 Ushkov, F.V. 218 V v. Schickfus, M. 75 Vaccaro, J.A. 358, 359, 361, 366, 370, 373, 386, 388, 430, 432, 433 Vaglica, A. 361 Vakhitov, N.G. 22, 27, 34 Valentin, C. 328 Valley, J.F. 27 van den Berg, R. 100 van der Laan, H. 100 van der Veer, W.E. 337, 338 van der Waals, J.H. 131 van Dienst, R.J.J. 337, 338 van Linden van den Heuvell, H.B. 337, 338 van Strien, A.J. 117, 118 Vanasse, G.A. 164 Vanin, E.V 54 Vansteenkiste, N. 261, 279, 3 10, 3 13, 3 15, 316, 322, 346 Varma, C.M. 76, 78 Varro, S. 375, 380, 381, 387 Vkquez, L. 54 Vegunta, P. 64 Velichansky, VL. 293, 294 Velzel, C.H.F. 212 Vermande, P. 157 Vest, C.M. 210, 212, 213, 215 Vetri, G. 361 Vinogradov, A.V. 361 Virdee, M.S. 236 Vlasov, N.G. 218 Vlasov, R.A. 3, 27 Vlasov, S.N. 10, 22,27, 33 Vogel, E.M. 25 Vogel, K. 360, 375, 434 Vogel, M. 139 Volker, S. 75, 83, 100, 123 Volkov, V.M. 3, 27 von Bally, G. 2 18 von Borczyskowski, C. 91, 116, 117, 119, 120, 139 Vorontsov, M.A. 20 Vourdas, A. 361 Vredenbregt, E.J.D. 315, 321 VukiEeviC, D. 209, 218 Vyssotina, N.V 16, 28, 33

460

AUTHOR INDEX FOR VOLUME XXXV

W Wagner, B. 96, 1 16 Wagner, C. 391 Wagner, T. 64 Walker, N.G. 360 Wallace, R.W. 156 Wallis, H. 313 Walls, D.F. 261, 301, 344, 416, 417 Walsh, C.A. 75, 83 Walther, A. 163 Walther, H. 64, 112, 133, 343, 361, 389, 391 Walther, M. 65, 137 Wang, C. 280, 336 Wang, D.H. 334 Wang, F.B. 432, 433 Wang, p. 334 Wang, Q.W. 335 Wang, Z.W. 391 Wannemacher, R. 75, 83, 123 Wamngton, D.M. 281, 284, 292 Wattson, R.B. 156 Weber, M.J. 75 Webster, J.M. 225, 226 Weibel, E. 63 Weidemuller, M. 3 17 Weiner, A.M. 25 Weiss, C.O. 49, 50 Weiss, P.R. 82 Weitz, M. 328, 329 Welling, H. 360 Wellman, J.B. 152, 158 Werij, H.G.C. 306 West, K.N. 136 Westbrook, C. 3 I 1, 3 1 5 Westbrook, C.I. 328 Westlake, D. 216, 217, 226 Wheeler, J.A. 398 Whitley, R.M. 260, 266, 291, 292 Whitten, W.B. 134 Widengren, J. 64, 11 1 Widmer, M.T. 315 Wiemann, G. 65, 137 Wiersma, D.A. 68, 75, 82, 83, 96, 98, 101, 1 13-1 15 Wilcox, L.R. 266 Wild, U.P. 90-93, 95, 100, 101, 103, 106, 107, 110, 124, 139 Wilkens, M. 391 Wilkerson, C.W. 64 Wilkinson, S.R. 280, 336 Williams, D.C. 236

Willis, C.R. 261, 299 Wilson, W.L. 96, 115 Wilson-Gordon, A.D. 339, 361, 4 3 1 4 3 3 Wimperis, J.R. 157 Windholz, L. 294, 295, 3 14, 3 18, 3 19 Winefordner, J.D. 185 Wineland, D.J. 64, 133 Witriol, N.M. 260 Wbdkiewicz, K. 360, 361, 386 Woerdman, J.P. 306 Wolf, E. 147, 148, 150, 151, 159-161, 163, 165, 180, 232 Wolff, K. 249 Wong, J. 358, 364, 427, 429,430 Wong, W.C. 358, 359 Wootters, W.K. 433 Wrachtrup, J. 91, 116, 117, 119, 120, 139 Wreszinski, W.F. 361 Wright, E.M. 17, 27, 56 Wu, H. 321, 328, 415 Wu, L. 415 WU, S.-T. 177, 178 Wuerker, R.F. 213, 216, 217, 226 Wykes, C. 213, 230 Wynands, R. 307, 308, 339

X

Xiao, M. 269, 284, 294, 296, 308, 339 Xing, C.Z. 391 XU, J.H. 280, 285-287, 289-291 Xu Jiajin 27

Y Yaglom, A.M. 162, 163 Yajima, N. 24 Yakobson, N.N. 3 18, 3 19 Yamaguchi, I. 210, 236, 251 Yamamoto, Y. 392 Yamanaka, C. 293 Yao, D.M. 385 Yariv, A. 22 Yartsev, VM. 65 Yarunin, VS. 358, 427, 429 Yashin, VE. 3, 10 Yelon, W.B. 95 Yoo, H.I. 262, 388 Young, B.C. 328, 329 Young, L. 293 Yu, J. 328

AUTHOR INDEX FOR VOLUME XXXV

Yuan, R.-L. 328 Yudin, V.1. 261, 279, 282, 287, 322, 325, 345 Yur’ev, M.S. 21 Yurke, B. 385, 392, 415 Z

Zagury, N. 385 Zaidi, H.R. 385 Zakharov, YE. 15, 23 Zambon, B. 276, 283, 329, 332, 333, 340 Zanetta, P. 236-240, 244 Zanoni, R. 17 Zaretsky, D.F. 345 Zawodny, R. 416, 418, 419,421, 425 Zeller, R.C. 75 Zewail, A.H. 96, 115 Zhang, H.Z. 334, 335 Zhang, J. 310 Zhang, W.M. 432

46 1

Zhang, Z.M. 391 Zhao, J. 334 Zhao, P. 148, 156, 172 Zheleznykh, N.I. 20 Zhou, B. 156, 172 Zhou, P. 391 Zhou, Y.G. 432, 433 Zhu, J.Y. 432 Zhu, S.Y. 276, 280, 284-286, 307, 329, 331, 333-336, 339, 340 Ziman, J.M. 71 Zoller, P. 261, 301, 317, 318, 325, 326, 328, 344 Zollfrank, J. 100 Zou, Y.H. 305 Zschokke, I. 75 Zubairy, M.S. 340 Zumbusch, A. 77, 80, 91, 93, 95, 101, 121, 124-126, 130 Zumofen, G. 77

SUBJECT INDEX FOR VOLUME XXXV

A ac Stark effect 134, 268, 321 acousto-optic diffraction 156 - modulator 296 - tunable filter (AOTF) 156 adiabatic transfer 325-329 airborne imaging spectrometer (AIS) 158 anharmonic oscillator model 391-396 antibunching 111-113, 411, 415 atomic interferometry 328 - spectroscopy 199 autosoliton, in bistable lasers 50 -, - lasers and nonlinear waveguides 49-54 -, interaction of 46, 47 -, mechanism of 44-46 -, single and coupled 4 1 4 4

superpositions of 38 1, 395 two-photon 386 - -, Yurke-Stoler 385 Condon approximation 69 correlation function 1 1 1, 159, 164 - -, second-order 11 1 - -, - -,

D

Debye-Waller factor 84, 94, 96, 137 displaced number state 396-398 - phase state 432 Doppler broadening 283, 284, 296 dressed state 274-277

E electro-optic modulation 293

F

Fabry-Perot cavity 303 filamentation 3-21, 55 -, in systems with feedback 17-21 -, instability 6 -, of a plane wave 6-10 -, - - wide beam 10-1 3 fluorescence autocorrelation 125-1 29 - line narrowing 69 Fokker-Planck equation 3 18 four-wave mixing 261, 303-306 Fourier optics 55 fractional revivals 395 Fresnel transform coding 149

B Bespalov-Talanov theory 10 birefringence 177, 178 bistability 18, 34-36, 38, 40, 41, 43 -, optical 50, 54, 301-303 Bragg diffraction 155 Brownian motion 63 bunching 1 13-1 16

C

Cauchy-Schwan inequalities 408 Clebsch-Gordan coefficients 3 12 coherence function, mutual 160 - -, spatial 159, 160, 164 coherent population trapping 257-346 - - -, velocity-selective (VSCPT) 261, 310, 320-325 - state 359, 378, 380 - -, even 385, 432 - -, odd 385, 432 -, pair 4 0 8 4 1 1 - -, phase 432

G Garrison-Wong phase formalism 427-430 Glauber-Sudarshan P-function 359, 372 H Hadamard transform 148 Hadamard-transform imager - spectrometry I50

-

463

150

464

SUBJECT INDEX FOR VOLUME XXXV

Hanbury-Brown-Twiss setup 1 13 Hanle effect 261, 282, 283, 297, 302, 335 Heisenberg uncertainty relation 362 Hilbert space 358 holographic interferometry 20 1-228 - -, application in artwork diagnostics 2 13226 - -, as nondestructive testing 212, 213 - -, double-exposure 207, 208, 224 - spectroscopy 15 1 holography 55 -, double-exposure 21 8 -, incoherent 164 -, principle of 202-206 -, real-time 206, 207, 218, 221, 224 -, sandwich 208-210, 218, 224 Husimi phase distribution 374-376, 378, 381, 387 - Q-function 359, 372 hysteresis, spatial 34, 37, 41

I interferometer, bistable 46, 57 -, liquid crystal polarization 177-1 82, 184 -, multiple-image parallel 183-1 85 -, nonlinear 3, 4, 20, 4 1 4 3 , 46, 53, 55 J Jaynes-Cummings model

375, 388-391

K

Kerr medium 3, 10, 14-16, 25, 27, 30, 392, 41 1 4 1 5 - nonlinearity 4, 5, 23, 25, 27, 33 kinetic theory of gases 63

L Langmuir-Blodgett method 90 laser cooling 262, 310, 315, 316, 318, 319 lasing without inversion 261, 262, 276, 307, 329-341 level crossing 261 linear response theory 72 M Mach-Zehnder interferometer 232 Medusa spectrograph I58 metrology 298-301 Michelson Doppler imaging interferometer (WAMDII) 157 -interferometer 155, 157, 167, 175, 177 - stellar interferometer 150, 155, 164, 167

multichannel scanning spectrometer (MCSS) 188, 189 multichanneling 156, 174 multiphoton ionization 261 multiplex-multichanneling 151, 152, 155 multiplex-scanning 152-1 54 multiplexing technique 147, 148, 150, 156, 174

N

near-field optics 134-1 37 neutron activation 199 number operator 357 0 optical Bloch equations 108, 3 13 - computing 55, 173 -fiber 90 - homodyne tomography 360 - phase conjugation 303 - spectroscopy 63 optically detected magnetic resonance (ODMR)

130, 131, 138

P

parametric down-conversion 388,421425 paraxial equation 15, 17, 23 - -, nonlinear 16 Pegg-Barnett approach 358 - - phase distribution 359, 369, 372, 373, 375-377, 380-383, 387, 388, 390, 393, 398, 410 - _ - formalism 362, 364370, 386, 389, 392, 399, 402, 407410,413, 418, 426, 430 pentacene in para-terphenyl crystal 95, 96, 121 phase operator 357, 361, 366 - -, Hermitian 358, 360 - -, Pegg-Barnett 364-366, 370 - -, Susskind-Glogower 359, 363, 364, 366, 370, 407, 408 - probability distribution 367 phaseonium 345 photorefractive crystal 148 Poisson bracket 363 polymer 90

Q

quantum beat laser 33 1 -dot 137 - interferences 277-279, 283 - non-demolition measurement

134

465

SUBJECT INDEX FOR VOLUME XXXV

R

Rabi frequency 108, 1 13,264,267,272,278, 283,292,293, 296, 305, 306, 325, 326, 342 - oscillation 108, 113 radiography 199 radiometry I64 Raman scattering 87 Ramsey fringes 261, 299, 300 reflectography 199 S

s-parametrized quasidistribution 37 1, 374, 375, 393 scanning electron tunneling microscope (STM) 63 Schrodinger cat 134, 381, 384 - equation 69, 323 - -, nonlinear 23 -kitten state 381 second-harmonic generation 415-4 19, 422 self-focusing 3, 4, 6, 7, 10, 11, 13, 16, 33, 34, 46, 55 self-induced transparency 23 self-trapping 3 soliton, I D spatial 22-26 -, 2D spatial 26-33 -, dark 14 -, spatial 3, 17, 21-34, 54-56 sonography 199 speckle interferometer 225 - pattern interferometry, electronic 228-250 - photography 225

spectral diffusion 69, 75, 78, 8 1, 121, 131 photo-induced 129, 130 - hole burning 69, 75, 84, 100, 132 -imaging 147, 151, 170, 171, 189, 191 - multichanneling 153 spectrometry 147, 152 -, spatiotemporal 33, 34 -, spectroscopy 259 spectroscopy, Fourier 164 spontaneous symmetry breaking 3 squeezed state 55, 378, 385-388, 424, 425, 43 1 - -, generalized 375 - vacuum 387, 388 - -, two-mode 402408 Stark effect 101, 102 switching wave 34, 36-38, 50, 51 - -,

T

terrylene in polyethylene 123 tomography 151, 199

V

van Cittert-Zernike theorem 165 - der Waals force 70

159, 161, 164,

W Wiener-Khintchine theorem 150, 159, 164 Wigner function 359, 372, 374 - phase distribution 374-378, 380, 385, 387

CONTENTS OF PREVIOUS VOLUMES

VOLUME I(1961)

I I1 111

IV V VI VII VIII

1- 29 The Modern Development of Hamiltonian Optics, R.J. PEGIS 31- 66 Wave Optics and Geometrical Optics in Optical Design, K. MIYAMOTO The Intensity Distribution and Total Illumination of Aberration-Free Diffraction 67-108 Images, R. BARAKAT 109-1 53 Light and Information, D. GABOR On Basic Analogies and Principal Differences between Optical and Electronic 155-210 Information, H. WOLTER 211-251 Interference Color, H. KUBOTA 253-288 Dynamic Characteristics of Visual Processes, A. FIORENTINI 289-329 Modem Alignment Devices, A.C.S. VANHEEL

VOLUME I1 (1963) Ruling, Testing and Use of Optical Gratings for High-Resolution Spectroscopy, 1- 72 G.W. STROKE 73-108 I1 The Metrological Applications of Diffraction Gratings, J.M. BURCH 09-129 111 Diffusion Through Non-Uniform Media, R.G. GIOVANELLI IV Correction of Optical Images by Compensation of Aberrations and by Spatial 31-180 Frequency Filtering, J. TSUJIUCHI 181-248 V Fluctuations of Light Beams, L. MANDEL 249-288 VI Methods for Determining Optical Parameters of Thin Films, F. ABELBS I

VOLUME I11 (1964)

I I1

The Elements of Radiative Transfer, F. KOTTLER Apodisation, P. JACQUINOT, B. ROIZEN-DOSSIER 111 Matrix Treatment of Partial Coherence, H. GAMO

1- 28 29-1 86 187-332

VOLUME IV (1965)

I

Higher Order Aberration Theory, J. FOCKE

IV V VI VII

Optical Constants of Thin Films, F! ROUARD,P. BOUSQUET The Miyamoto-Wolf Diffraction Wave, A. RUBINOWICZ Aberration Theory of Gratings and Grating Mountings, W.T. WELFORD Diffraction at a Black Screen, Part I: Kirchhoff’s Theory, F. KOTTLER

II Applications of Shearing Interferometry, 0. BRYNGDAHL I11 Surface Deterioration of Optical Glasses, K. KINOSITA

467

1- 36 37- 83 85-143 145-197 199-240 24 1-28 0 281-314

468

CONTENTS OF PREVIOUS VOLUMES

VOLUME V (1966)

I Optical Pumping, C. COHEN-TANNOUDII, A. KASTLER 1- 81 I1 Non-Linear Optics, P.S. PERSHAN 83-144 I11 Two-Beam Interferometry, W.H. STEEL 145-1 97 199-245 IV Instruments for the Measuring of Optical Transfer Functions, K. MURATA V Light Reflection from Films of Continuously Varying Refractive Index, R. JACOBSSON247-286 VI X-Ray Crystal-Structure Determination as a Branch of Physical Optics, H. LIPSON, C.A. TAYLOR 287-350 VII The Wave of a Moving Classical Electron, J. PICHT 351-370 VOLUME VI (1967)

I I1 111 IV V VI

VII VIII

Recent Advances in Holography, E.N. LEITH,J. UPATNIEKS 1- 52 53- 69 Scattering of Light by Rough Surfaces, P. BECKMANN S. MALLICK 71-104 Measurement of the Second Order Degree of Coherence, M. FRANCON, 105-170 Design of Zoom Lenses, K. YAMAJI 171-209 Some Applications of Lasers to Interferometry, D.R. HERRIOT Experimental Studies of Intensity Fluctuations in Lasers, J.A. ARMSTRONG, 21 1-257 A.W. SMITH 259-330 Fourier Spectroscopy, G.A. VANASSE, H. SAKAI Diffraction at a Black Screen, Part 11: Electromagnetic Theory, F. KOITLER 331-377 VOLUME VII (1969)

I 11 111 IV V VI VII

Multiple-Beam Interference and Natural Modes in Open Resonators, G. KOPPELMAN I- 66 Methods of Synthesis for Dielectric Multilayer Filters, E. DELANO, R.J. PEGIS 67-137 Echoes at Optical Frequencies, I.D. ABELLA 139-168 Image Formation with Partially Coherent Light, B.J. THOMPSON 169-230 Quasi-Classical Theory of Laser Radiation, A.L. MIKAELIAN, M.L. TER-MIKAELIAN 23 1-297 The Photographic Image, S. OOUE 299-358 Interaction of Very Intense Light with Free Electrons, J.H. EBERLY 359415 VOLUME VIII (1970)

1 Synthetic-Aperture Optics, J.W. GOODMAN I1 The Optical Performance of the Human Eye, G.A. FRY 111 Light Beating Spectroscopy, H.Z. CUMMINS, H.L. SWINNEY

1- 50 51-131 133-200 201-237 239-294

IV Multilayer Antireflection Coatings, A. MUSSET,A. THELEN V Statistical Properties of Laser Light, H. RISKEN VI Coherence Theory of Source-Size Compensation in Interference Microscopy, T. YAMAMOTO 295-34 1 VII Vision in Communication, L. LEVI 343-372 VIII Theory of Photoelectron Counting, C.L. MEHTA 373440 VOLUME IX (1971)

I

Gas Lasers and their Application to Precise Length Measurements, A.L. BLOOM 11 Picosecond Laser Pulses, A.J. DEMARIA 111 Optical Propagation Through the Turbulent Atmosphere, J.W. STROHBEHN IV Synthesis of Optical Birefringent Networks, E.O. AMMANN

1- 30 31- 71 73-122 123-177

CONTENTS OF PREVIOUS VOLUMES

469

179-234 V Mode Locking in Gas Lasers, L. ALLEN,D.G.C. JONES VI Crystal Optics with Spatial Dispersion, V.M. AGRANOVICH, V.L. GINZBURG 235-280 VII Applications of Optical Methods in the Diffraction Theory of Elastic Waves, J. PETYKIEWICZ 281-310 K. GNIADEK, VIII Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions, B.R. FRIEDEN 3 1 1-407

VOLUME X (1972) I 11 111

IV V VI VII

Bandwidth Compression of Optical Images, T.S. HUANG The Use of Image Tubes as Shutters, R.W. SMITH Tools of Theoretical Quantum Optics, M.O. SCULLY, K.G. WHITNEY Field Correctors for Astronomical Telescopes, C.G. WYNNE Optical Absorption Strength of Defects in Insulators, D.Y. SMITH,D.L. DEXTER Elastooptic Light Modulation and Deflection, E.K. SITTIG Quantum Detection Theory, C.W. HELSTROM

1- 44 45- 87 89-135 137-1 64 165-228 229-288 2 89-3 69

VOLUME XI (1973) I I1 111

IV V VI VII

Master Equation Methods in Quantum Optics, G.S. AGARWAL Recent Developments in Far Infrared Spectroscopic Techniques, H. YOSHINAGA Interaction of Light and Acoustic Surface Waves, E.G. LEAN Evanescent Waves in Optical Imaging, 0. BRYNGDAHL Production of Electron Probes Using a Field Emission Source, A.V. CREWE Hamiltonian Theory of Beam Mode Propagation, J.A. ARNAUD Gradient Index Lenses, E.W. MARCHAND

I- 76 77-122 123-1 66 167-22 1 223-246 247-304 305-337

VOLUME XI1 (1974) I I1 111 IV V VI

Self-Focusing, Self-Trapping, and Self-phase Modulation of Laser Beams, 0.SVELTO I- 51 Self-Induced Transparency, R.E. SLUSHER 53-100 101-1 62 Modulation Techniques in Spectrometry, M. HARWIT, J.A. DECKERJR Interaction of Light with Monomolecular Dye Layers, K.H. DREXHAGE 163-232 The Phase Transition Concept and Coherence in Atomic Emission, R. GRAHAM 233-286 Beam-Foil Spectroscopy, S. BASHKIN 287-344

VOLUME XI11 (1976) On the Validity of Kirchhoff’s Law of Heat Radiation for a Body in a Nonequilibrium 1- 25 Environment, H.P. BALTES 27- 68 I1 The Case For and Against Semiclassical Radiation Theory, L. MANDEL 111 Objective and Subjective Spherical Aberration Measurements of the Human Eye, 69- 91 W.M. ROSENBLLM, J.L. CHRISTENSEN IV Interferometric Testing of Smooth Surfaces, G. SCHULZ, J. SCHWIDER 93-1 67 V Self-Focusing of Laser Beams in Plasmas and Semiconductors, M.S. SOOHA, A.K. GHATAK, V.K. TRIPATHI 169-265 VI Aplanatism and Isoplanatism, W.T. WELFORD 267-292

1

470

CONTENTS OF PREVIOUS VOLUMES

VOLUME XIV (1 976) I I1 I11 IV V VI VII

The Statistics of Speckle Patterns, J.C. DAINTY High-Resolution Techniques in Optical Astronomy, A. LABEYRIE Relaxation Phenomena in Rare-Earth Luminescence, L.A. RISEBERG, M.J. WEBER The Ultrafast Optical K e n Shutter, M.A. DUGUAY D. RUDOLPH Holographic Diffraction Gratings, G. SCHMAHL, Photoemission, P.J. VERNIER Optical Fibre Waveguides - A Review, P.J.B. CLARRICOATS

1- 46 47- 87 89-1 59 161-193 195-244 245-325 327-402

VOLUME XV (1977)

I I1 I11 IV V

Theory of Optical Parametric Amplification and Oscillation, W. BRUNNER, H. PAUL Optical Properties of Thin Metal Films, P. ROUARD, A. MEESSEN Projection-Type Holography, T. OKosw Quasi-Optical Techniques of Radio Astronomy, T.W. COLE Foundations of the Macroscopic Electromagnetic Theory of Dielectric Media, J. VAN KRANENDONK. J.E. S l P E

1- 75 77-137 139-1 85 187-244

245-350

VOLUME XVI (1978)

I I1 I11 IV V

1- 69 Laser Selective Photophysics and Photochemistry, V.S. LETOKHOV Recent Advances in Phase Profiles Generation, J.J. CLAIR,C.I. ABITBOL 71-1 17 119-232 Computer-Generated Holograms: Techniques and Applications, W.-H. LEE 233-288 Speckle Interferometry, A.E. ENNOS Deformation Invariant, Space-Variant Optical Pattern Recognition, D. CASASENT, 2 89-3 5 6 D. PSALTIS 35741 1 v1 Light Emission From High-Current Surface-Spark Discharges, R.E. BEVERLY I11 VII Semiclassical Radiation Theory Within a Quantum-Mechanical Framework, 4 13-448 I.R. SENITZKY

VOLUME XVII (1980) 1- 84 I Heterodyne Holographic Interferometry, R. DANDLIKER 85-161 I1 Doppler-Free Multiphoton Spectroscopy, E. GIACOBINO, B. CACNAC 111 The Mutual Dependence Between Coherence Properties of Light and Nonlinear 163-238 Optical Processes, M. SCHUBERT, B. WILHELMI 239-217 IV Michelson Stellar Interferometry, W.J. TANGO,R.Q. Twrss 279-345 V Self-Focusing Media with Variable Index of Refraction, A.L. MIKAELIAN

VOLUME XVIII (1 980)

1-126 Graded Index Optical Waveguides: A Review, A. GHATAK, K. THYACARAJAN Photocount Statistics of Radiation Propagating Through Random and Nonlinear Media, J. PEAINA 127-203 I11 Strong Fluctuations in Light Propagation in a Randomly Inhomogeneous Medium, VI. TATARSKII, V u . ZAVOROTNVI 204-256 IV Catastrophe Optics: Morphologies of Caustics and their Diffraction Patterns, M.V BERRY.C. UPSTILL 257-346 I I1

CONTENTS OF PREVIOUS VOLUMES

47 1

VOLUME XIX (1981) Theory of Intensity Dependent Resonance Light Scattering and Resonance 1- 43 Fluorescence, B.R. MOLLOW II Surface and Size Effects on the Light Scattering Spectra of Solids, D.L. MILLS, K.R. SUBBASWAMY 45-137 I11 Light Scattering Spectroscopy of Surface Electromagnetic Waves in Solids, 1 39-2 1 0 s. USHIODA IV Principles of Optical Data-Processing, H.J. BUTTERWECK 21 1-280 V The Effects of Atmospheric Turbulence in Optical Astronomy, F. RODDIER 281-376 I

VOLUME XX (1983)

I

Some New Optical Designs for Ultra-Violet Bidimensional Detection of Astronom1- 61 M. SA%E ical Objects, G. COURTES,F! CRuvELLiEq M. DETAILLE, I1 Shaping and Analysis of Picosecond Light Pulses, C. FROEHLY, B. COLOMBEAU, M. VAMPOUILLE 63-153 III Multi-Photon Scattering Molecular Spectroscopy, S. KIELICH 155-261 263-324 IV Colour Holography, P. HARIHARAN V Generation of Tunable Coherent Vacuum-Ultraviolet Radiation, W. JAMROZ, 325-380 B.P. STOICHEFF

VOLUME XXI (1984)

I

1- 67 Rigorous Vector Theories of Diffraction Gratings, D. MAYSTRE 11 Theory of Optical Bistability, L.A. LuGlATo 69-2 16 2 17-286 111 The Radon Transform and its Applications, H.H. BAR RE^ D.W. SWEENEY287-354 IV Zone Plate Coded Imaging: Theory and Applications, N.M. CEGLIO, V Fluctuations, Instabilities and Chaos in the Laser-Driven Nonlinear Ring Cavity, 355428 J.C. ENGLIJND, R.R. SNAPP,W.C. SCHIEVE

VOLUME XXII (1985) 1- 76 Optical and Electronic Processing of Medical Images, D. MALACARA Quantum Fluctuations in Vision, M.A. BOUMAN, W.A. VANDE GRIND,P. ZUIDEMA 77-144 Spectral and Temporal Fluctuations of Broad-Band Laser Radiation, A.V. MASALOV145-196 Holographic Methods of Plasma Diagnostics, G.V. OSTROVSKAYA, Yu.1. OSTROVSKY197-270 Fringe Formations in Deformation and Vibration Measurements using Laser Light, I . YAMAGUCHI 271-340 341-398 V1 Wave Propagation in Random Media: A Systems Approach, R.L. FANTE

I I1 III IV V

VOLUME XXIII (1986) I

Analytical Techniques for Multiple Scattering from Rough Surfaces, J.A. DESANTO, G.S. BROWN 1- 62 II Paraxial Theory in Optical Design in Terms of Gaussian Brackets, K. TANAKA 63-1 I 1 III Optical Films Produced by Ion-Based Techniques, P.J. MARTIN, R.P. NETTERFIELD 113-182 IV Electron Holography, A. TONOMURA 183-220 V Principles of Optical Processing with Partially Coherent Light, F.T.S. Yu 221-275

472

CONTENTS OF PREVIOUS VOLUMES

VOLUME XXIV (1 987) Micro Fresnel Lenses, H. NISHIHARA, T. SUHARA Dephasing-Induced Coherent Phenomena, L. ROTHBERG 111 Interferometry with Lasers, F! HARIHARAN IV Unstable Resonator Modes, K.E. OUCHSTUN V Information Processing with Spatially Incoherent Light, I. GLASER 1

I1

1- 37 39-101 103-1 64 165-387 389-509

VOLUME XXV (1988) I

Dynamical Instabilities and Pulsations in Lasers, N.B. ABRAHAM, P. MANDEL, 1-190 L.M. NARDUCCI 191-278 I1 Coherence in Semiconductor Lasers, M. OHTSU,T. TAKO L. RONCHI Ill Principles and Design of Optical Arrays, WANGSHAOMIN, 279-348 349415 IV Aspheric Surfaces, G. SCHULZ

VOLUME XXVI (1988)

I

I1 111 IV V

Photon Bunching and Antibunching, M.C. TEICH,B.E.A. SALEH Nonlinear Optics of Liquid Crystals, I.C. KHOO Single-Longitudinal-Mode Semiconductor Lasers, G.F! AGRAWAL Rays and Caustics as Physical Objects, Yu.A. KRAVTSOV Phase-Measurement Interferometry Techniques, K. CREATH

1-104 105-161 163-225 227-348 349-393

VOLUME XXVll (1989)

I The Self-Imaging Phenomenon and Its Applications, K. PATORSKI 1-108 I1 Axicons and Meso-Optical Imaging Devices, L.M. SOROKO 109-1 60 I11 Nonimaging Optics for Flux Concentration, I.M. BASSETT, W.T. WELFORD, R. WINSTON 161-226 IV Nonlinear Wave Propagation in Planar Structures, D. MIHALACHE, M. BERTOLOITI, C. SIBILIA 227-3 13 V Generalized Holography with Application to Inverse Scattering and Inverse Source Problems, R.P. PORTER 3 15-397

VOLUME XXVIlI (1990) Digital Holography - Computer-Generated Holograms, 0. BRYNGDAHL, F. WYROWSKI 1- 86 Quantum Mechanical Limit in Optical Precision Measurement and Communication, S. MACHIDA, S. SAITO,N. IMOTO,T. YANAGAWA, M. KITAGAWA, Y. YAMAMOTO, G. BJORK 87-179 111 The Quantum Coherence Properties of Stimulated Raman Scattering, M.G. RAYMER, I.A. WALMSLEY 181-270 IV Advanced Evaluation Techniques in Interferometry, J. SCHWIDER 271-359 V Quantum Jumps, R.J. COOK 361416 1

I1

CONTENTS OF PREVIOUS VOLUMES

473

VOLUME XXIX (1 991 )

I

Optical Waveguide Diffraction Gratings: Coupling between Guided Modes, I- 63 D.G. HALL I1 Enhanced Backscattering in Optics, Yu.N. BARABANENKOV, Yu.A. KRAVTSOV, V.D. OZRIN,A.I. SAICHEV 65-197 199-29 1 III Generation and Propagation of Ultrashort Optical Pulses, I.P. CHRISTOV 293-3 19 IV Triple-Correlation Imaging in Optical Astronomy, G. WEICELT V Nonlinear Optics in Composite Materials. 1. Semiconductor and Metal Crystallites in Dielectrics, C. FLYTZANIS, E HACHE,M.C. KLEIN,D. RICARD, PH. ROIJSSIGNOL 32141 1

VOLUME XXX (1992) Quantum Fluctuations in Optical Systems, S. REYNAUD, A. HEIDMANN, E. GIACOBINO, I- 85 C. FABRE II Correlation Holographic and Speckle Interferometry, Yu.1. OSTROVSKY, V.P. SHCHEPINOV 87-135 111 Localization of Waves in Media with One-Dimensional Disorder, VD. FREILIKHER, S.A. GREDESKUL 137-203 IV Theoretical Foundation of Optical-Soliton Concept in Fibers, Y. KODAMA, A. HASEGAWA 205-259 26 1-3 55 V Cavity Quantum Optics and the Quantum Measurement Process., I? MEYSTRE I

VOLUME XXXI ( 1993)

I II

Atoms in Strong Fields: Photoionization and Chaos, P.W. MILONNI, B. SIJNDARAM 1-137 Light Diffraction by Relief Gratings: A Macroscopic and Microscopic View, E. POPOV 139-187 111 Optical Amplifiers, N.K. DUTTA,J.R. SIMPSON 189-226 227-26 1 Y. QIAO IV Adaptive Multilayer Optical Networks, D. PSALTIS, V Optical Atoms, R.J.C. SPREEUW, J.P. WOERDMAN 263-3 19 A. RENIERI, VI Theory of Compton Free Electron Lasers, G. DAITOLI,L. GIANNESSI, 321412 A. TORRE

VOLUME XXXII (1993) 1- 59 Guided-Wave Optics on Silicon: Physics, Technology and Status, B.P. PAL 61-144 Optical Neural Networks: Architecture, Design and Models, F.T.S. YIJ The Theory of Optimal Methods for Localization of Objects in Pictures, 145-20 1 L.P. YAROSLAVSKY IV Wave Propagation Theories in Random Media Based on the Path-Integral Approach, 203-266 M.I. CHARNOTSKII, I. GOZANI, V.I. TATARSKII, V.U. ZAVOROTNY V Radiation by Uniformly Moving Sources. Vavilov-Cherenkov effect, Doppler effect 267-3 12 in a medium, transition radiation and associated phenomena, V.L. GINZBURG VI Nonlinear Processes in Atoms and in Weakly Relativistic Plasmas, G. MAINFRAY, C. MANUS 313-361 1 I1 Ill

474

CONTENTS OF PREVIOUS VOLUMES

VOLUME XXXIII (1994)

I

The Imbedding Method in Statistical Boundary-Value Wave Problems, V.1.

fiy-

1-127 129-202 I1 Quantum Statistics of Dissipative Nonlinear Oscillators, V. PEEINOVA, A. LUKS 203-260 111 Gap Solitons, C.M. DE STERKE,J.E. SIPE IV Direct Spatial Reconstruction of Optical Phase from Phase-Modulated Images, 261-3 17 V.I. VLAD,D. MALACARA 319-388 J. OZ-VCGT V Imaging through Turbulence in the Atmosphere, M.J. BERAN, T. SCHEERMESSER, VI Digital Halftoning: Synthesis of Binary Images, 0. BRYNGDAHL, F. WYROWSKI 389463 ATSKlN

VOLUME XXXIV (1 995)

I I1

Quantum Interference, Superposition States of Light, and Nonclassical Effects, V BUBEK,P.L. KNIGHT Wave Propagation in Inhomogeneous Media: Phase-Shift Approach, L.P. P u s -

1-158

NYAKOV 159-1 8 1 183-248 T. ASAKURA 111 The Statistics of Dynamic Speckles, T. OKAMOTO, IV Scattering of Light from Multilayer Systems with Rough Boundaries, 1. OHLIDAL, 249-33 1 K. N A V R ~ T M. I L ,OHL~DAL V Random Walk and Diffusion-Like Models of Photon Migration in Turbid Media, G.H. WENS 333402 A.H. GANDJBAKHCHE,

CUMULATIVE INDEX - VOLUMES I-XXXV

ABELES,F., Methods for Determining Optical Parameters of Thin Films 11, 249 ABELLA,I.D., Echoes at Optical Frequencies VII, 139 ABITBOL, C.I., see Clair, J.J. XVI, 71 ABRAHAM, N.B., P. MANDEL, L.M. NARDUCCI, Dynamical Instabilities and Pulsations in Lasers xxv, 1 AGARWAL, G.S., Master Equation Methods in Quantum Optics XI, 1 IX, 235 AGRANOVICH, V.M., V.L. GiNzBuRc, Crystal Optics with Spatial Dispersion XXVI, 163 G.P., Single-Longitudinal-Mode Semiconductor Lasers AGRAWAL, IX, 179 ALLEN,L., D.G.C. JONES,Mode Locking in Gas Lasers IX, 123 AMMANN, E.O., Synthesis of Optical Birefringent Networks XXXV, 257 ARIMONDO, E., Coherent Population Trapping in Laser Spectroscopy ARMSTRONG, J.A., A.W. SMITH,Experimental Studies of Intensity Fluctuations in Lasers v1, 211 XI, 247 ARNAUD, J.A., Hamiltonian Theory of Beam Mode Propagation XXXIV, 183 ASAKLIRA, T., see Okamoto, T. BALTES,H.P., On the Validity of Kirchhoff’s Law of Heat Radiation for a Body in a Nonequilibrium Environment VD. OZRIN,A.I. SAICHEV,Enhanced BARABANENKOV, Yu.N., Yu.A. KRAVTSOV, Backscattering in Optics BARAKAT, R., The Intensity Distribution and Total Illumination of Aberration-Free Diffraction Images BARRETT, H.H., The Radon Transform and its Applications BASHKIN, S . , Beam-Foil Spectroscopy BASSETT, I.M., W.T. WELFORD, R. WINSTON, Nonimaging Optics for Flux Concentration BECKMANN, P., Scattering of Light by Rough Surfaces BERAN,M.J., J. OZ-VOGT,Imaging through Turbulence in the Atmosphere BERNARD, J., see Orrit, M. BERRY,M.V., C. UPSTILL,Catastrophe Optics: Morphologies of Caustics and their Diffraction Patterns BERTOLOTTI, M., see Mihalache, D. BEVERLY 111, R.E., Light Emission From High-Current Surface-Spark Discharges BJORK,G., see Yamamoto, Y. BLOOM, A.L., Gas Lasers and their Application to Precise Length Measurements Quantum Fluctuations in Vision BOUMAN, M.A., W.A. VANDE GRIND,P. ZUIDEMA, BOUSQUET, P., see Rouard, P. BROWN,G.S., see DeSanto, J.A. BROWN, R., see Orrit, M. BRUNNER, W., H. PAUL,Theory of Optical Parametric Amplification and Oscillation

475

XIII.

1

XXIX, 65 I, XXI, XU, XXVII, VI, XXXIII, XXXV,

67 217 287 161 53 319 61

XVIII, XXVII, XVI, XXVIII, IX, XXII, IV, XXIII, XXXV, xv,

257 227 357 87 I 77 145 1 61 1

476

CUMULATIVE INDEX - VOLUMES I-XXXV

BRYNGDAHL, O., Applications of Shearing Interferometry IV, 37 BRYNGDAHL, O., Evanescent Waves in Optical Imaging XI, 167 O., F. WYROWSKI, Digital Holography - Computer-Generated Holograms XXVIII, BRYNGDAHL, 1 BRYNGDAHL, O., T. SCHEERMESSER, F. WYROWSKI, Digital Halftoning: Synthesis of Binary Images XXXIII, 389 BURCH,J.M., The Metrological Applications of Diffraction Gratings 11, 73 BUTTERWECK, H.J., Principles of Optical Data-Processing XIX, 211 B U ~ E KV.,, PL. KNIGHT,Quantum Interference, Superposition States of Light, and Nonclassical Effects XXXIV, 1 CAGNAC, B., see Giacobino, E. XVII, 85 CASASENT, D., D. PSALTIS,Deformation Invariant, Space-Variant Optical Pattern Recognition XVI, 289 CEGLIO, N.M., D.W. SWEENEY, Zone Plate Coded Imaging: Theory and Applications XXI, 287 CHARNOTSKII, M.I., J. GOZANI, V.1. TATARSKII, WJ.ZAVOROTNY, Wave Propagation Theories in Random Media Based on the Path-Integral Approach XXXII, 203 CHRISTENSEN, J.L., see Rosenblum, W.M. XIII, 69 CHRISTOV, I.P., Generation and Propagation of Ultrashort Optical Pulses XXIX, 199 CLAIR,J.J., C.I. ABITBOL, Recent Advances in Phase Profiles Generation XVI, 71 XIV, 327 CLARRICOATS, P.J.B., Optical Fibre Waveguides - A Review COHEN-TANNOUDJI, C., A. KASTLER, Optical Pumping v, 1 XV, 187 COLE,T.W., Quasi-Optical Techniques of Radio Astronomy COLOMBEAU, B., see Froehly, C. XX, 63 COOK,R.J., Quantum Jumps XXVIII, 361 COIJRTBS,G., P. CRUVELLIER, M. DETAILLE, M. SA~SSE, Some New Optical Designs for Ultra-Violet Bidimensional Detection of Astronomical Objects xx, 1 XXVI, 349 CREATH, K., Phase-Measurement Interferometry Techniques XI, 223 CREWE, A,V, Production of Electron Probes Using a Field Emission Source CRWELLIER, P., see Court&, G. xx, 1 VIII, 133 CUMMINS, H.Z., H.L. SWINNEY, Light Beating Spectroscopy 1 DAINTY, J.C., The Statistics of Speckle Patterns XIV, DANDLIKER, R., Heterodyne Holographic Interferometry XVII, 1 DATTOLI, G., L. GIANNESSI, A. RENIERI, A. TORRE,Theory of Compton Free Electron Lasers XXXI, 321 DE STEFKE,C.M., J.E. SIPE,Gap Solitons XXXIII, 203 DECKER JR, J.A., see Hanvit, M. XII, 101 DELANO, E., R.J. PEGIS,Methods of Synthesis for Dielectric Multilayer Filters VII, 67 DEMARIA, A.J., Picosecond Laser Pulses IX, 31 DESANTO,J.A., G.S. BROWN,Analytical Techniques for Multiple Scattering from 1 Rough Surfaces XXIII, DETAILLE, M., see CourtBs, G. xx, 1 X, 165 DEXTER, D.L., see Smith, D.Y. DREXHAGE, K.H., Interaction of Light with Monomolecular Dye Layers XU, 163 DUGUAY, M.A., The Ultrafast Optical Kerr Shutter XIV, 161 XXXI, 189 DUTTA,N.K., J.R. SIMPSON, Optical Amplifiers

EBERLY, J.H., Interaction of Very Intense Light with Free Electrons ENGLUND, J.C., R.R. SNAPP, W.C. SCHIEVE, Fluctuations, Instabilities and Chaos in the Laser-Driven Nonlinear Ring Cavity ENNOS,A.E., Speckle Interferometry

VII, 359 XXI, 355 XVI, 233

CUMULATIVE INDEX

- VOLUMES I-XXXV

FABRE,C., see Reynaud, S. FANTE,R.L., Wave Propagation in Random Media: A Systems Approach FIORENTINI, A,, Dynamic Characteristics of Visual Processes Nonlinear Optics FLYTZANIS, c.,F. HACHE,M.C. KLEIN, D. RICARD,PH. ROUSSIGNOL, in Composite Materials. 1. Semiconductor and Metal Crystallites in Dielectrics FOCKE,J., Higher Order Aberration Theory FRANCON, M., S. MALLICK, Measurement of the Second Order Degree of Coherence FREILIKHER, VD., S.A. GREDESKUL, Localization of Waves in Media with OneDimensional Disorder FRIEDEN, B.R., Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions FROEHLY, C., B. COLOMBEAU, M. VAMPOUILLE, Shaping and Analysis of Picosecond Light Pulses FRY,G.A., The Optical Performance of the Human Eye GABOR, D., Light and Information GAMO,H., Matrix Treatment of Partial Coherence

GANDJBAKHCHE, A.H., G.H. WEISS,Random Walk and Diffusion-Like Models of Photon Migration in Turbid Media GANTSOG, Ts., see Tanai, R. GHATAK, A,, K. THYAGARAJAN, Graded Index Optical Waveguides: A Review GHATAK, A.K., see Sodha, M.S. GIACOBINO, E., B. CAGNAC, Doppler-Free Multiphoton Spectroscopy GIACOBINO, E., see Reynaud, S. GIANNESSI, L., see Dattoli, G. GlNzBURG, VL., see Agranovich, VM. GINZBURG, VL., Radiation by Uniformly Moving Sources. Vavilov
ITOH,K., Interferometric Multispectral lmaging

477

xxx,

I XXII, 341 I, 253

XXIX, 321 IV, 1 VI, 71 XXX, 137 IX, 311 XX, 63 VIII, 51 I, 109 111, 187 XXXIV, xxxv, XVIII, XIII, XVII, xxx, XXXI, IX,

333 355 1

169 85 1 321 235

XXXII, 267 11, 109 XXIV, 389 IX, 281 VIII, 1 XXXII, 203 XII, 233 XXX, 137 XXIX, 321 XXIX, XX, XXIV, XII, XXX,

I 263 103 101 205 xxx, 1 X, 289 VI, 171 x, 1

XXVIII, 87 XXXV, 145

478

CUMULATIVE INDEX - VOLUMES I-XXXV

JACOBSSON, R., Light Reflection from Films of Continuously Varying Refractive Index P, B. ROIZEN-DOSSIER, Apodisation JACQUINOT, JAMROZ,W., B.P. STOICHEFF, Generation of Tunable Coherent Vacuum-Ultraviolet Radiation JONES,D.G.C., see Allen, L.

V, 247 111, 29 XX, 325 IX, 179

KASTLER,A,, see Cohen-Tannoudji, C. v, 1 KHOO,I.C., Nonlinear Optics of Liquid Crystals XXVI, 105 KIELICH, S., Multi-Photon Scattering Molecular Spectroscopy XX, 155 KINOSITA, K., Surface Deterioration of Optical Glasses IV, 85 XXVIII, 87 KITAGAWA, M., see Yamamoto, Y. XXIX, 321 KLEIN,M.C., see Flytzanis, C. V.I., The Imbedding Method in Statistical Boundary-Value Wave Problems XXXIII, 1 KLYATSKIN, KNIGHT, P.L., see Buiek, V. XXXIV, 1 KODAMA, Y., A. HASEGAWA, Theoretical Foundation of Optical-Soliton Concept in XXX, 205 Fibers KOPPELMAN, G., Multiple-Beam Interference and Natural Modes in Open Resonators 1 VII, 111, 1 KOTTLER,F., The Elements of Radiative Transfer IV, 281 KOTTLER, F., Diffraction at a Black Screen, Part I: Kirchhoff’s Theory VI, 331 KOTTLER, F., Diffraction at a Black Screen, Part 11: Electromagnetic Theory KRAVTSOV, Yu.A., Rays and Caustics as Physical Objects XXVI, 227 KRAVTSOV, Yu.A., see Barabanenkov, Yu.N. XXIX, 65 H., Interference Color I, 211 KUBOTA, LABEYRIE, A,, High-Resolution Techniques in Optical Astronomy XIV, 41 XI, 123 LEAN,E.G., Interaction of Light and Acoustic Surface Waves XVI, 119 LEE,W.-H., Computer-Generated Holograms: Techniques and Applications VI, 1 Recent Advances in Holography LEITH,E.N., J. UPATNIEKS, V.S., Laser Selective Photophysics and Photochemistry 1 LETOKHOV, XVI, VIII, 343 LEVI,L., Vision in Communication LIPSON,H., C.A. TAYLOR,X-Ray Crystal-Structure Determination as a Branch of V, 287 Physical Optics XXXV, 61 LOUNIS, B., see Orrit, M. LUGIATO, L.A., Theory of Optical Bistability XXI, 69 XXXIII, 129 LuKS, A,, see Peiinovi, V. MACHIDA, S., see Yamamoto, Y. G., C. MANUS,Nonlinear Processes in Atoms and in Weakly Relativistic MAINFRAY, Plasmas MALACARA, D., Optical and Electronic Processing of Medical Images MALACARA, D., see Vlad, V.1. S., see Franpon, M. MALLICK, MANDEL, L., Fluctuations of Light Beams MANDEL, L., The Case For and Against Semiclassical Radiation Theory MANDEL, P., see Abraham, N.B. MANUS, C.,see Mainfray, G. MARCHAND, E.W., Gradient Index Lenses MARTIN,P.J., R.P. NETTERFIELD, Optical Films Produced by Ion-Based Techniques MASALOV, A.V., Spectral and Temporal Fluctuations of Broad-Band Laser Radiation MAYSTRE, D., Rigorous Vector Theories of Diffraction Gratings MEESSEN, A., see Rouard, P.

XXVIII, 87 XXXII, XXII, XXXIII, VI, 11, XIII, xxv, XXXII, XI, XXIII, XXII, XXI, xv,

313

I

261 71 181 27 1 313 305 113 145

I

77

479

CUMULATIVE INDEX - VOLUMES I-XXXV

MEHTA,C.L., Theory of Photoelectron Counting VIII, 373 MEYSTRE, I?, Cavity Quantum Optics and the Quantum Measurement Process. XXX, 261 MIHALACHE, D., M. BERToLom, C. SIBILIA, Nonlinear Wave Propagation in Planar Structures XXVII, 227 MIKAELIAN, A.L., M.L. TER-MIKAELIAN, Quasi-Classical Theory of Laser Radiation VII, 231 MIKAELIAN, A.L., Self-Focusing Media with Variable Index of Refraction XVII, 279 MILLS,D.L., K.R. SUBBASWAMY, Surface and Size Effects on the Light Scattering Spectra of Solids XIX, 45 MILONNI, P.W., B. SUNDAM, Atoms in Strong Fields: Photoionization and Chaos 1 XXXI, MIRANOWICZ, A., see TanaS, R. xxxv, 355 MIYAMOTO, K., Wave Optics and Geometrical Optics in Optical Design I, 31 MOLLOW,B.R., Theory of Intensity Dependent Resonance Light Scattering and Resonance Fluorescence XIX, 1 MURATA, K., Instruments for the Measuring of Optical Transfer Functions V, 199 MUSSET,A., A. THELEN, Multilayer Antireflection Coatings VIII, 201 NARDUCCI, L.M., see Abraham, N.B. NAVR~TIL, K., see Ohlidal, I. NETTERFIELD, R.P., see Martin, PJ. NISHIHARA, H., T. SUHARA, Micro Fresnel Lenses OHL~DAL, I., K. NAVR~TIL, M. OHLIDAL, Scattering of Light from Multilayer Systems with Rough Boundaries OHLIDAL, M., see Ohlidal, 1. OHTSU,M., T. TAKO,Coherence in Semiconductor Lasers OKAMOTO, T., T. ASAKURA, The Statistics of Dynamic Speckles OKOSHI, T., Projection-Type Holography OOUE,S., The Photographic Image ORRiT, M., J. BERNARD, R. BROWN,B. LOUNIS,Optical Spectroscopy of Single Molecules in Solids OSTROVSKAYA, G.V, Yu.1. OSTROVSKY, Holographic Methods of Plasma Diagnostics OSTROVSKY, Yu.I., see Ostrovskaya, G.V OSTROVSKY, Yu.I., VP. SHCHEPINOV, Correlation Holographic and Speckle Interferometry OUGHSTUN, K.E., Unstable Resonator Modes OZ-VOGT,J., see Beran, M.J. OZRIN,VD., see Barabanenkov, Yu.N. PAL,B.P., Guided-Wave Optics on Silicon: Physics, Technology and Status PAOLETTI,D., G. SCHIRRIPA SPAGNOLO, Interferometric Methods for Artwork Diagnostics PATORSKI, K., The Self-Imaging Phenomenon and Its Applications PAUL,H., see Brunner, W. PEGIS,R.J., The Modern Development of Hamiltonian Optics PEGIS,R.J., see Delano, E. PEWINA,J., Photocount Statistics of Radiation Propagating Through Random and Nonlinear Media PEWINOVA, V, A. L u K ~Quantum , Statistics of Dissipative Nonlinear Oscillators PERSHAN, P.S., Non-Linear Optics PETYKIEWICZ, J., see Gniadek, K. PICHT,J., The Wave of a Moving Classical Electron

xxv, I XXXIV, 249 XXIII, 113 1 XXIV, XXXIV, XXXIV, XXV, XXXIV, XV, VII,

249 249 191 183 139 299

XXXV, 61 XXII, I97 XXII, 197 XXX, XXIV, XXXIII, XXIX,

87 165 319 65

XXXII,

1

XXXV, 197 XXVII, 1

xv,

1

1,

1

XVIII, XXXIII, V, IX, V,

127 129 83 281 351

VII, 67

480

CUMULATIVE INDEX - VOLUMES I-XXXV

Porov, E., Light Diffraction by Relief Gratings: A Macroscopic and Microscopic View XXXI, 139 PORTER,R.P., Generalized Holography with Application to Inverse Scattering and Inverse Source Problems XXVII, 315 PRESNYAKOV, L.P., Wave Propagation in Inhomogeneous Media: Phase-Shift Approach XXXIV, 159 PSALTIS, D., see Casasent, D. XVI, 289 PSALTIS, D., Y. QIAO,Adaptive Multilayer Optical Networks XXXI, 227 QIAo, Y., see Psaltis, D.

XXXI, 227

RAYMER, M.G., LA. WALMSLEY, The Quantum Coherence Properties of Stimulated Raman Scattering XXVIII, 181 RENIERI, A,, see Dattoli, G. XXXI, 321 REYNALID, S., A. HEIDMANN, E. GIACOBINO, C. FABRE, Quantum Fluctuations in Optical Systems xxx, 1 RICARD, D., see Flytzanis, C. XXIX, 321 RISEBERG, L.A., M.J. WEBER,Relaxation Phenomena in Rare-Earth Luminescence XIV, 89 RISKEN, H., Statistical Properties of Laser Light VIII, 239 RODDIER, F., The Effects of Atmospheric Turbulence in Optical Astronomy XIX, 281 ROIZEN-DOSSIER, B., see Jacquinot, P. 111, 29 L., see Wang Shaomin RONCHI, XXV, 279 ROSANOV, N.N., Transverse Patterns in Wide-Aperture Nonlinear Optical Systems xxxv, 1 ROSENBLLJM, W.M., J.L. CHRISTENSEN, Objective and Subjective Spherical Aberration Measurements of the Human Eye XIII, 69 ROTHBERG, L., Dephasing-Induced Coherent Phenomena XXIV, 39 ROUARD, P., I? BOUSQUET, Optical Constants of Thin Films IV, 145 ROUARD, P., A. MEESSEN, Optical Properties of Thin Metal Films x v , 77 ROUSSIGNOL, PH.,see Flytzanis, C. XXIX, 321 RUBINOWICZ, A,, The Miyamoto-Wolf Diffraction Wave IV, 199 RUDOLPH, D., see Schmahl, G. XIV, 195 SAICHEV, A.I., see Barabanenkov, Yu.N.

SAYSSE,M., see Courtks, G.

XXIX, 65 xx, 1 XXVIII, 87 VI, 259 XXVI, 1 XXXIII, 389 XXI, 355 XXXV, 197 XIV, 195

SAITO,S., see Yamamoto, Y. SAKAI, H., see Vanasse, G.A. SALEH, B.E.A., see Teich, M.C. SCHEERMESSER, T., see Bryngdahl, 0. SCHIEVE, W.C., see Englund, J.C. SCHIRRIPA SPAGNOLO, G., see Paoletti, D. G., D. RUDOLPH, Holographic Diffraction Gratings SCHMAHL, SCHUBERT, M., B. WiLHELMi, The Mutual Dependence Between Coherence Properties of Light and Nonlinear Optical Processes XVII, 163 SCHULZ, G., J. ScHwiDER, Interferometric Testing of Smooth Surfaces XIII, 93 G., Aspheric Surfaces SCHULZ, x x v , 349 SCHWIDER, J., see Schulz, G. XIII, 93 SCHWIDER, J., Advanced Evaluation Techniques in Interferometry XXVIII, 271 Tools of Theoretical Quantum Optics SCULLY, M.O., K.G. WHITNEY, X, 89 SENITZKY, I.R., Semiclassical Radiation Theory Within a Quantum-Mechanical Framework XVI, 413 SHCHEPINOV, VI?, see Ostrovsky, Yu.1. XXX, 87 SiBiLiA, C., see Mihalache, D. XXVII, 227

CUMULATIVE INDEX

- VOLUMES I-XXXV

48 1

SIMPSON, J.R., see Dutta, N.K. XXXI, 189 SIPE,J.E., see Van Kranendonk, J. XV, 245 SIPE,J.E., see De Sterke, C.M. XXXIII, 203 SITTIG,E.K., Elastooptic Light Modulation and Deflection X, 229 XII, 53 SLUSHER, R.E., Self-Induced Transparency v1, 211 SMITH,A.W., see Armstrong, J.A. X, 165 SMITH,D.Y., D.L. DEXTER,Optical Absorption Strength of Defects in Insulators x, 45 SMITH,R.W., The Use of Image Tubes as Shutters XXI, 355 SNAPP,R.R., see Englund, J.C. SODHA, M.S., A.K. GHATAK, V.K. TRIPATHI, Self-Focusing of Laser Beams in Plasmas XIII, 169 and Semiconductors SOROKO, L.M., Axicons and Meso-Optical Imaging Devices XXVII, 109 XXXI, 263 SPREEUW, R.J.C., J.P. WOERDMAN, Optical Atoms V, 145 STEEL,W.H., Two-Beam Interferometry XX, 325 STOICHEFF, B.P., see Jamroz, W. IX, 73 STROHBEHN, J. W., Optical Propagation Through the Turbulent Atmosphere STROKE,G.W., Ruling, Testing and Use of Optical Gratings for High-Resolution 11, I Spectroscopy XIX, 45 SUBBASWAMY, K.R., see Mills, D.L. 1 XXIV, SUHARA, T., see Nishihara, H. XXXI, 1 SUNDARAM, B., see Milonni, P.W. SVELTO, O., Self-Focusing, Self-Trapping, and Self-phase Modulation of Laser Beams XII, 1 XXI, 287 SWEENEY, D.W., see Ceglio, N.M. VIII, 133 SWINNEY, H.L., see Cummins, H.Z. XXV, 191 TAKO,T., see Ohtsu, M. XXIII. 63 TANAKA, K., Paraxial Theory in Optical Design in Terms of Gaussian Brackets TANAB,R., A. MIRANOWICZ, Ts. GANTSOG,Quantum Phase Properties of Nonlinear xxxv, 355 Optical Phenomena XVII, 239 TANGO,W.J., R.Q. Twiss, Michelson Stellar Interferometry V.I., V.U. ZAVOROTWI, Strong Fluctuations in Light Propagation in a TATARSKII, XVIII, 204 Randomly Inhomogeneous Medium XXXII, 203 V.I., see Charnotskii, M.I. TATARSKII, V, 287 TAYLOR, C.A., see Lipson, H. XXVI, 1 TEICH,M.C., B.E.A. SALEH,Photon Bunching and Antibunching VII, 231 TER-MIKAELIAN, M.L., see Mikaelian, A.L. VIII, 201 THELEN, A., see Musset, A. VII, 169 THOMPSON, B.J., Image Formation with Partially Coherent Light XVIII, 1 THYAGARAJAN, K., see Ghatak, A. XXIII, 183 TONOMURA, A,, Electron Holography XXXI, 321 TORRE,A., see Dattoli, G. XIII, 169 TRIPATHI, V.K., see Sodha, M.S. TSUIIUCHI, J., Correction of Optical Images by Compensation of Aberrations and by Spatial Frequency Filtering 11, 131 XVII, 239 Twiss, R.Q., see Tango, W.J. UPATNIEKS, J., see Leith, E.N. UPSTILL,C., see Berry, M.V. USHIODA, S., Light Scattering Spectroscopy of Surface Electromagnetic Waves in Solids

VI, 1 XVIII, 257 XIX, 139

482

CUMULATIVE INDEX - VOLUMES I-XXXV

VAMPOUILLE, M., see Froehly, C. XX, VANDE GRIND, W.A., see Bouman, M.A. XXII, VANHEEL,A.C.S., Modern Alignment Devices I, VANKRANENDONK,J., J.E. SIPE,Foundations of the Macroscopic Electromagnetic Theory of Dielectric Media XV, VANASSE, G.A., H. SAKAI, Fourier Spectroscopy VI, VERNIER, P.J., Photoemission XIV, VLAD,VI., D. MALACARA, Direct Spatial Reconstruction of Optical Phase from PhaseModulated Images XXXIII,

63 77 289 245 259 245

261

WALMSLEY, I.A., see Raymer, M.G. XXVIII, 181 WANGSHAOMIN, L. RONCHI, Principles and Design of Optical Arrays XXV, 279 XIV, 89 WEBER,M.J., see Riseberg, L.A. G., Triple-Correlation Imaging in Optical Astronomy WEIGELT, XXIX, 293 WEISS,G.H., see Gandjbakhche, A.H. XXXIV, 333 WELFORD, W.T., Aberration Theory of Gratings and Grating Mountings IV, 241 WELFORD, W.T., Aplanatism and Isoplanatism XIII, 267 WELFORD, W.T., see Bassett, I.M. XXVII, 161 WHITNEY, K.G., see Scully, M.O. X, 89 WILHELMI, B., see Schubert, M. XVII, 163 R., see Bassett, I.M. XXVII, 161 WINSTON, WOERDMAN, J.P., see Spreeuw, R.J.C. XXXI, 263 WOLTER,H., On Basic Analogies and Principal Differences between Optical and Electronic Information I, 155 WYNNE,C.G., Field Correctors for Astronomical Telescopes X, 137 WYROWSKI, F., see Bryngdahl, 0. XXVIII, I WYROWSKI, F., see Bryngdahl, 0. XXXIII, 389 YAMAGUCHI, I., Fringe Formations in Deformation and Vibration Measurements using Laser Light YAMAJi, K., Design of Zoom Lenses YAMAMOTO, T., Coherence Theory of Source-Size Compensation in Interference Microscopy s. SAITO,N. IMOTO, T. YANAGAWA, M. KITAGAWA, YAMAMOTO, Y., s. MACHIDA, G. BJORK,Quantum Mechanical Limit in Optical Precision Measurement and Communication YANAGAWA, T.,see Yamamoto, Y. YAROSLAVSKY, L.P., The Theory of Optimal Methods for Localization of Objects in Pictures YOSHINAGA, H., Recent Developments in Far Infrared Spectroscopic Techniques Yu, F.T.S., Principles of Optical Processing with Partially Coherent Light Yu, F.T.S., Optical Neural Networks: Architecture, Design and Models VU., see Charnotskii, M.1 ZAVOROTNY, VU., see Tatarskii, V1. ZAVOROTNYI, ZUIDEMA, P.,see Bouman, M.A.

XXII, 271 VI, 105 VIII, 295 XXVIII, 87 XXVIII, 87 XXXII, XI, XXIII, XXXII,

145 77 221 61

XXXII, 203 XVIII, 204 XXII, 77