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Rochester, USA
T. Asakura
Sapporo, Japan
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EDITORIAL ADVISORY BOARD
G.S. Agarwal
Ahmedabad, India
G. Agrawal
Rochester, USA
T. Asakura
Sapporo, Japan
A. Aspect
Orsay, France
M.V. Berry
Bristol, England
A.T. Friberg
Stockholm, Sweden
V.L. Ginzburg
Moscow, Russia
E Gori
Rome, Italy
A. Kuj awski
Warsaw, Poland
L.M. Narducci
Philadelphia, USA
J. Pe~ina
Olomouc, Czech Republic
R.M. Sillitto
Edinburgh, Scotland
H. Walther
Garching, Germany
Preface This volume contains five articles presenting reviews of several topics of current research which are likely to be of interest to many optical scientists and optical engineers. The first article, by J. Ohtsubo, deals with the dynamics of feedback-induced instability and chaos. The characteristics of semiconductor lasers based on the rate equations, including various laser structures, are reviewed and the effects of optical feedback in semiconductor lasers are then discussed. In the second article, by F.G. Omenetto, the progress made in recent years in the general area of the nonlinear interaction of ultrafast pulses with optical and photonic crystal fibers are discussed. In particular, ultrafast pulse measurements, pulse shaping and pulse control are discussed. The next article, by A.B. Shvartsburg and G. Petite, presents a review of transient optical phenomena that take place in the spatial-temporal dynamics of ultrashort pulses. The interplay of diffractive and dispersive phenomena is examined. They include coupled processes of amplitude and phase reshaping, spectral variations and polarity reversal for different types of light pulses. Reflection and refraction effects that take place at the interface between media with time-dependent dielectric susceptibilities are also discussed. The fourth article, by A.E Fercher and C.K. Hitzenberger, outlines the principles of optical coherence tomography (OCT). This is a relatively new discipline with important potential applications in macroscopic, microscopic and endoscopic imaging. The article begins with a brief summary of the field and then describes various OCT interferometer configurations and discusses basic sample signal extraction techniques. The article also covers subjects such as contrast generation techniques, resolution, signal processing techniques for image display, image enhancement, speckle suppression and OCT detection sensitivity. A description of optical delay lines used in OCT is also presented. The concluding article by F.Kh. Abdullaev, S.A. Darmanyan and J. Garnier is concerned with modulation instability (MI) of electromagnetic waves in inhomogeneous and in discrete media. The article pays special attention to the MI of electromagnetic waves in nonlinear optical fibers with periodic amplification, dispersion and birefringence. The MI in random media is also covered. Other topics discussed in this article are discrete nonlinear systems with
vi
Preface
cubic, quadratic and vectorial interactions and nonlinear optical systems such as tunnel-coupled filters. Some of the readers may note that authors from six different countries have contributed to this volume, thus helping to maintain the international character of this series. Emil Wolf Department of Physics and Astronomy and The Institute of Optics University of Rochester Rochester, NY 14627, USA October 2002
E. Wolf, Progress in Optics 44 9 2002 Elsevier Science B. V. All rights reserved
Chapter 1
Chaotic dynamics in semiconductor lasers with optical feedback by
Junji Ohtsubo Faculty of Engineering, Shizuoka University, 3-5-1 Johoku, Hamamatsu, 432-8561, Japan
Contents
Page w 1.
Introduction . . . . . . . . . . . . . . . . . . . . . .
3
w 2.
Semiconductor lasers as class-B lasers . . . . . . . . . . .
6
w 3.
Theory of semiconductor lasers with optical feedback . . . . .
25
w 4.
Chaotic dynamics in semiconductor lasers with optical feedback
40
w 5.
Some applications of semiconductor lasers with optical feedback
59
w 6.
Concluding remarks
. . . . . . . . . . . . . . . .
79
Acknowledgments . . . . . . . . . . . . . . . . . . . . .
81
References . . . . . . . . . . . . . . . . . . . . . . .
81
w 1. Introduction It was in 1963 that Lorenz [1963], investigating the behavior of convective fluid flow as a model for the atmospheric flow, showed that nonlinear systems described by three variables could exhibit chaotic dynamics. Chaos is the phenomenon of irregular variations of system output derived from models described by deterministic equations. In spite of the models being deterministic, we cannot foresee the future of the output since chaos is very sensitive to the initial conditions: each system behaves completely different every time, even if the difference in the initial state is very small. Chaos can be observed in various fields of engineering, physics, chemistry, economics, and biology. Although the fields are different, some of the chaotic systems can be characterized by similar differential equations. Since lasers are nonlinear systems and are typically characterized by three variables: the field, the polarization of matter, and the population inversion, they are candidates for chaotic systems. Indeed, it was proved in the mid-1970s by Haken [1975] that lasers are nonlinear systems similar to the Lorenz model and that they show chaotic behavior in their output power. Haken assumed a ring laser model and considered two-level atoms in a laser medium. Although lasers are not always described by his model, the approximations are reasonable for most lasers. Therefore, nonlinear laser rate equations with three variables and involving chaotic dynamics are called Lorenz-Haken equations (Haken [1985]). However, ordinary lasers do not exhibit chaotic behavior, and only few of the lasers with bad cavity conditions show chaotic dynamics. In the meantime, chaotic behavior were theoretically demonstrated in a ring laser system (Ikeda [1979]). Weiss and Brock [1986] were the first to observe Lorenz-Haken chaos in infrared NH3 lasers. Contrary to Haken's prediction, ordinary lasers are stable systems and only a few infrared lasers systems show chaotic behavior in their output power. Arecchi, Lippi, Puccioni and Tredicce [ 1984] investigated the laser systems from the viewpoint of the characteristic relaxation times of the three variables, and categorized lasers in three classes. According to their classifications, one or two of the relaxation times are very fast compared with the other time scales and most lasers are described by rate equations with one or two variables.
4
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 1
These are, therefore, stable systems, categorized into class-A and class-B lasers. Only class-C lasers are described by the full set of rate equations with three variables and can show chaotic dynamics. However, class-A and class-B lasers can show chaotic dynamics when one or more degrees of freedom are introduced to the laser systems. Class-B lasers are characterized by rate equations for the field and the population inversion, and they are easily destabilized by an additional degree of freedom applied as an external perturbation. For example, solid-state lasers, fiber lasers, and CO2 lasers that are categorized as class-B lasers show unstable oscillation upon external optical injection or modulation of accessible laser parameters. Semiconductor lasers, the main topic of this review, are also classified as class-B lasers; they are very sensitive to optical injection, self-induced optical feedback, optical injection from other lasers, opto-electronic feedback, and injection current modulation. They show chaotic dynamics in the presence of external perturbations. An overview of chaotic instabilities in lasers has been given by Abraham, Mandel and Narducci [ 1988]. Since the early 1980s (Lang and Kobayashi [1980]), feedback-induced instabilities and chaos in semiconductor lasers have been closely examined. In a semiconductor laser, when the light reflected from an external reflector couples with the original field in the laser cavity, the laser oscillation is affected considerably. A variety of dynamical behaviors can be observed in semiconductor lasers with optical feedback and they have been investigated by many researchers for two decades. One of the main differences between semiconductor lasers and other lasers is the low reflectivity of the internal mirrors in the semiconductor laser cavity. It ranges typically only from 10 to 30% of the intensity in Fabry-Perot semiconductor lasers. This makes the feedback effects significant in semiconductor lasers. In the case of the Vertical-Cavity Surface-Emitting Lasers (VCSELs), the reflectivity of the internal mirror is very high at more than 99%, however they are also sensitive to external optical feedback because of the small number of photons in the internal cavity. Therefore, semiconductor lasers of all types are essentially very sensitive to external optical feedback. Another difference is a large absolute value of the linewidth enhancement factor a of the laser media. Values for the linewidth enhancement factor a of 2-7 have been reported in semiconductor lasers depending on the laser materials, while this value is almost zero for other lasers. Then, the coupling between the phase and the cartier density (equivalent to population inversion) is encountered in the laser dynamics. Interestingly, these factors lead to a variety of dynamics quite different from any other lasers. At weak to moderate optical feedback reflectivity, the laser output power shows interesting dynamical
1, w 1]
Introduction
5
behaviors such as a stable state, periodic and quasi-periodic oscillations, and chaos with changes of the system parameters. These ranges of the external reflectivity are not only interesting from a viewpoint of fundamental physics, but also very important in practical applications of semiconductor lasers such as in optical data storage systems and optical communications. Extensive lists of recent literature for the dynamic characteristics in semiconductor lasers with optical feedback can be found in articles by van Tartwijk and Agrawal [1998] and Ohtsubo [ 1999]. Until now, many semiconductor laser devices with different structures have been proposed and fabricated. In spite of the different device structures, the dynamics of semiconductor lasers are the same as long as the laser rate equations are written in the same or similar forms. The dynamics of edge-emitting singlemode semiconductor lasers have been studied extensively for a long time, and many fruitful results have been obtained. However, there are still important issues on the fundamental physics of optical chaos and practical applications to be discussed. On the other hand, little investigation on the dynamics of other laser structures has been done, for example VCSELs, self-pulsating lasers, broad-area lasers. In the meantime, important breakthroughs in the applications of chaos were achieved in the early 1990s. The ideas of chaos control (Ott, Grebogi and Yorke [1990]) and chaos synchronization (Pecora and Carroll [1990]) have been proposed and developed in that decade. The idea of chaos control has been applied immediately to the stabilization of chaotic lasers (Roy, Murphy, Maier, Gills and Hunt [ 1992]). The possibility of chaos communications has been discussed based on chaos synchronization in systems with two chaotic lasers (Colet and Roy [1994]). In this article, we focus on the dynamics and applications in semiconductor lasers with optical feedback. In w2, we first introduce general laser rate equations which reduce to the Lorenz equations, and the classifications of the lasers are given. The instabilities intrinsically involved in the rate equations are studied. Next, semiconductor lasers as class-B lasers are described and the possibility for unstable oscillations of lasers by the introduction of external perturbations is discussed. A solitary semiconductor laser is characterized by two equations, for field and cartier density (population inversion). We then derive the forms of the rate equations for edge-emitting semiconductor lasers. After that, the rate equations of the semiconductor lasers for various laser structures are introduced. In w3, we present the theory of semiconductor lasers with optical feedback and discuss the various effects of feedback-induced characteristics and instabilities. The effects of incoherent optical feedback and phase-conjugate feedback are also discussed. We assume a single-mode oscillation for a semiconductor
6
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 2
laser, however the effects of multimode oscillations in semiconductor lasers are considered in this section. In w4, substantial feedback effects and chaotic dynamics in semiconductor lasers are discussed and both theoretical and experimental results are given for variations of the system parameters. Dynamic properties are presented not only for edge-emitting lasers, but also for various laser structures such as self-pulsating lasers, VCSELs, and broad-area lasers. w5 is devoted to the applications of chaos in laser systems, and we discuss chaos control and chaos synchronization in laser systems. As an application of chaos control in laser systems, the reduction of feedback-induced Relative Intensity Noise (RIN) is demonstrated. Next, we present the possibility for chaos communications based on chaos synchronization in semiconductor lasers with optical feedback. Finally, w6 concludes.
w 2. S e m i c o n d u c t o r lasers as class-B lasers
2.1. Laser model and Bloch equations In this section, we derive the Maxwell-Bloch equations for a two-level laser model and show that lasers are described by the Lorenz-Haken equations. Although theoretical treatment of lasers should be based on quantum field theory, we also use the semi-classical treatment followed by Haken [1985] and van Tartwijk and Agrawal [ 1998]. We consider for simplicity the model of twolevel atoms and their emission and absorption of light, though the model is not always applicable to real lasers. Actual lasers are composed of a FabryPerot resonator, and a few contain an unidirectional ring resonator. Although the description for a unidirectional travelling-wave ring resonator is very simple, it has merits for theoretical treatments, and the theory can be extended to ordinary Fabry-Perot lasers. The model is shown in fig. 2.1. The laser is assumed to operate in a single mode with linear polarization. For field E and polarization of matter P, the equation for the propagation of light in the z direction at time t is described by the Maxwell equation
02E
/12 02E
022
C2 0 t 2
-
02P 110 Ot 2 ,
(2.1)
where r/is the reflectivity of the laser medium,/t0 is the magnetic permeability in the vacuum, and c is the velocity of light in the vacuum. Using the angular
Semiconductor lasers as class-B lasers
1, w2]
7
Fig. 2.1. Unidirectional ring cavity model of laser. frequency tOo of the laser oscillation and the wavenumber k = rltOo/c, the field and the polarization are written as
t) exp[i(kz - tOot)] + c.c.,
(2.2)
P(z, t) = 1B(z, t) exp[i(kz - tOot)] + c.c.,
(2.3)
E(z, t) = 89
where A(z, t) and B(z, t) are assumed to vary slowly compared with the optical frequency (slowly-varying-envelope approximation, SVEA), and c.c. represents the complex conjugate of the preceding terms. Then, by substituting eqs. (2.2) and (2.3) into eq. (2.1) and neglecting the second small infinites, we obtain
OA ~IOA k 0z + -r ~ ="1260//2 B,
(2.4)
where e0 is the electric permeability in the vacuum. The Hamiltonian for the field operator E in two-level atoms is written as the sum of the Hamiltonian H0 without the perturbation and a perturbation term - / t . E , where g = er (e is the elemental charge and r is the position vector) is the moment of the transition between the two levels. For the eigenstates q~. (j = 1,2) of the two levels and the energy of each level h ~ , the quantum state I~p) of the two-level atoms is written by the linear addition of the two states as I~P) = Cl (t) exp(-itol t) l~l ) + c2(t) exp(-itozt) 1992).
(2.5)
8
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 2
From the Schr6dinger equation, the coefficients c l and c2 are given by the coupled Bloch equations as follows: de1 _ ic2 exp(--irOAt) (q~ll,U" E 1(/92) dt h
(2.6)
dc2 _ icl exp(irOAt) (C#21/l. E 199,) (2.7) dt h where rOA -- 0)2 -- rol is the angular frequency corresponding to the photon energy required for the absorption and emission in the two levels. Denoting the density of atoms by Nat, the macroscopic polarization of the matter is calculated as P = Nat =
Nat{p(t) ltl2 +p*(t)/t21 }.
(2.8)
Here, the microscopic polarization p(t) and the moment of the transition ~0 (i,j = 1,2) are given by
p(t) = c~ (t) c2(t) exp(--iroAt), /t0 = (%1 ~ IcPi) 9
(2.9) (2.10)
Substituting these equations into eqs. (2.6) and (2.7), the equations for the polarization and the population inversion w = Ic2(t)]2 - ]cl(t)l 2 are given by = -iroAp + ~Eg21 w, dw
dt
--
2
ihE(
p*
#21 -p/t12),
(2.11) (2.12)
respectively. In the following, we derive the equations describing the laser oscillations that result in chaotic nonlinear equations. By differentiating the polarization equation in eq. (2.3) and using eqs. (2.8) and (2.11), we obtain the macroscopic equation for the polarization as d B _ --i(rOA-- (D0) B + i/t2 W [A + A* exp{-Zi(kz- rOot)}] dt--~ where W = N~tw is the macroscopic population inversion and # = population inversion is calculated from eq. (2.12) as
dW _ 1 [AB* - A B e x p { 2 i ( k z - root)} -c.c.]
(2.13)
1~,21. The
(2.14) dt ih In the above equations, the time variations of the variables are considered to be very slow compared with the optical frequency, and the oscillation terms of the angular frequency of 2090 (rotating-wave approximation) are neglected.
1, w 2]
Semiconductor lasers as class-B lasers
9
In actual lasers, we require pumping for the laser oscillations, so that we add a corresponding term to the right-hand side of eq. (2.14). Further, the phenomenological terms for the damping oscillations are added to each term. Then the Maxwell-Bloch equations for field A, polarization B, and population inversion W are given by (Haken [1985], van Tartwijk and Agrawal [1998]) OA 770+t k ~1 0z + =i B - ~A, c -~2e0~/~ 2TphC OB
-
0t dW dt
i(tOA -- 0~0) B + -
1 (AB* ih
- AB)
+
AW Wo-W T~
(2.15) B T2'
(2.16)
(2.17)
Here, W0 is the population inversion due to the pumping at the threshold, and Tph, T2 and T1 are the relaxation times of the photons (photon lifetime), the polarization (transverse relaxation) and the population inversion (longitudinal relaxation), respectively. Actual lasers exhibit spontaneous emission, so that statistical Langevin noise terms are sometimes added to each MaxwellBloch equation to explain the noise effects in lasers. However, chaos is a phenomenon described by deterministic equations, so such terms are excluded in investigations of the pure laser dynamics. They are introduced to account for noise effects of laser oscillations when necessary.
2.2. Lorenz-Haken equations
By normalizing the field, the polarization, and the population inversion as = v/eocrl/2A, B = k/eorl2v/eocrl/2B and W = asW (with as = la2CooT2/2eohcrl), respectively, and neglecting the term OA/Oz as a small mean field that propagates in the z direction, the Maxwell-Bloch equations are rewritten as follows: dA_i~n~_ dt __
1 2-~phA'
d~
(2.18)
T2--d-~- = - ( 1 - i 6 ) / ~ - iAW,
(2.19)
dW Im[~]*/)] T, --~- - Wo- I~ + Isa~'
(2.20)
where 6 - (o~0- ~OA)T2 is the scaled atomic detuning and/sat = hZcrlfo/2[a 2 T1T2 is the saturation intensity.
10
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 2
In 1963, Lorenz, a meteorologist, proposed a set of equations to model convective fluid flow. For three variables X, Y and Z related to the normalized atmospheric flow and the temperature, Lorenz derived differential equations and proved that the system shows chaotic behavior. Using chaos parameters Z, R and fi, the Lorenz equations are written as dX dt
-
,Z(X
-
Y),
(2.21)
dY - RX- Y-XZ, dt
(2.22)
dZ dt
(2.23)
fiZ+XE.
On the other hand, the Maxwell-Bloch equations are written for the normalized variables x = x/b/Isat~4, y = (iCTph/rl)x/b/Isat[~, and z = (W0- 17V)CTph/rl as dx dt
o(x-y),
(2.24)
(1 - i 6 ) y + (r - z ) x ,
(2.25)
bz + Re[x'y].
(2.26)
@ dt
-
dz dt
where a = Tz/2Tph, r = ITVoCTph/rl and b = T2/TI are the normalized chaos parameters. It is easily proved that eqs. (2.24)-(2.26) are the same as those of the Lorenz model, and thus that the laser described by two-level atoms is a chaotic system. Equations (2.24)-(2.26) are called the Lorenz-Haken equations (Haken [ 1985], van Tartwijk and Agrawal [ 1998]). A laser oscillation starts when the population inversion exceeds a certain value that corresponds to the pumping threshold. The threshold value is theoretically calculated from the linear stability analysis for the rate equations by applying small perturbations on the steady states of the laser variables (van Tartwijk and Agrawal [ 1998]). An example of the linear stability analysis will be presented in the following chapter, but here we show the result of the analysis for the LorenzHaken model. Applying the linear stability analysis for eqs. (2.24)-(2.26), the value r of the laser threshold is calculated as ~2
rth" (1) = 1 + (a +
1)2"
(2.27)
At the laser oscillation, there exists an accompanying frequency that corresponds to the imaginary part of the solutions of the characteristic equation for the
1, w 2]
Semiconductor lasers as class-B lasers
11
stability analysis. This is known as the relaxation oscillation frequency, given by
~"~-- V/o'(rt(~ ) - |).
(2.28)
Without detuning (6 = 0), the threshold is r th Ill = 1 However, the threshold value increases so as to compensate the losses in the laser cavity induced by detuning. r(1) th is known as the first threshold of the laser oscillations since there is another threshold in laser operations. After pumping over the laser threshold, the laser reaches steady-state oscillation and its output should increase proportionally with increasing pumping. However, the laser power does not increase linearly with a further increase of pumping. Of course, there are effects of intensity saturation and other nonlinearities originating from the laser medium, however our concern here is not those effects: we focus on the nonlinear effects intrinsically involved in a nonlinear system described by the Lorenz-Haken equations. Next, we consider unstable oscillations well above the first laser threshold. For a steady laser oscillation above the threshold, we again apply the linear stability analysis. We simply assume that the laser has no detuning between the laser mode and the cavity resonance frequency (which is not often the case for real lasers) and the phase is constant for the time evolution. Then the second threshold for the pumping r is given by /2) = 0"(0" + b + 3) th (y-b- 1 "
(2.29)
When the pumping r exceeds the second threshold r t {2) h , the laser shows unstable behavior and Hopf bifurcations resulting in chaos. There are many routes to chaos in laser systems, such as period-doubling bifurcations (Hopf bifurcations), quasi-periodic bifurcations, intermittent bifurcations (pulsations), and others. However, actual lasers do not always show chaotic behavior. This point will be discussed in the next section. As an example, if the conditions T2 >> TI, b ~ 0, and 0 = 2(T2 = 4Tph) are /2) = 10, a very large value relative satisfied, the second laser threshold becomes rth ~1) = 1. In actuality, the second laser threshold is to the first laser threshold of rth ten or more times the first threshold value. The counterpart frequency of the second threshold is given by
_(2),). f2 = V/ b(o"+ rth
(2.30)
12
Chaotic dynamics in semiconductor lasers with optical feedback
[1, w 2
From eq. (2.29), the necessary condition for the existence of the second threshold in real lasers is given by ~r > b + 1, known as the bad cavity condition. The bad cavity condition is also written by using the time constants as 1
1
1
> -- + --. 2Tph 7"2 T!
(2.31)
Namely, a low quality laser cavity (low Q) and a dissipative system for photons are the conditions for a bad laser resonator.
2.3. Classification of lasers Lasers do not always show instabilities and chaotic behavior with increased pumping, and most lasers are actually stable. Above, we mentioned three variables for laser operation: the field, the polarization of matter, and the population inversion; however we do not always have to take all three variables into account for laser oscillations. When one or two of the time constants of the relaxation oscillations are considered to be very small compared with the other time constants, the rate equations can be written as one or two differential equations. For such cases, lasers are described by differential equations that do not show chaotic behavior. Depending on the scales of the time constants, lasers are categorized into three classes: When the relaxations of the polarization and the population inversion are much faster than that of the photon (i.e., Tph >> Tl, T2) the rate equations for the polarization and the population inversion are adiabatically eliminated and the laser is described only by the field equation. Such a laser is called class-A according to Arecchi, Lippi, Puccioni and Tredicce [1984]. Class-A lasers are the most stable lasers with a high Q factor. Class-A lasers may be destabilized and show chaotic instabilities when two or more degrees of freedom of the lasers are introduced externally. Examples of class-A lasers are visible He-Ne lasers, dye lasers, and Ar-ion lasers. If the time constant of the polarization of matter is much smaller than the other time constants and the latter are of the same order, i.e., Tph, T1 >> T2, the polarization equation is adiabatically eliminated and the rate equations consist of the field and population inversion equations. These lasers are called class-B. Class-A and class-B lasers have no second threshold, and class-B lasers also do not show chaotic behavior. The electric field is complex, so that the complex field equation can be split into two differential equations: the field amplitude and phase equations. However, the phase equation has no coupling
1, w2]
Semiconductor lasers as class-B lasers
13
with other variables, so that these systems can still be characterized by two differential equations. Class-B lasers are intrinsically stable, however they are easily destabilized by the introduction of external perturbations, like the addition of an extra degree of freedom. If the equations for the field amplitude and the phase become coupled through the perturbation, the laser must be described by the rate equations coupled with three variables. The lasers then become chaotic systems and show instabilities. Possible perturbations are modulation of the laser parameters, external optical injections, and optical self-feedback from external optical components. In actuality, semiconductor lasers, categorized into class B, are greatly affected by external optical feedback and show chaotic instabilities as we will discuss in w4. One of the typical features of class-B lasers is a relaxation oscillation that is observed in a step time response when the population inversion does not follow the photon decay rate, i.e., Tl > Tph. Other examples of class-B lasers are CO2 lasers, solid-state lasers, and fiber lasers. When all the time constants of the relaxation oscillations in the rate equations are compatible, we must take into account all three differential equations. In this case, the lasers do have the second threshold and they show instabilities at pumping rates high over the threshold. These are called class-C lasers. A few lasers show such instabilities; typical examples of class-C lasers are infrared lasers, which tend to have the three time constants in the same order, e.g., infrared NH3 lasers, Ne-Xe lasers (3.51 ~tm), and infrared He-Ne lasers (3.39 ~tm). These class-C lasers usually do not have commercial application.
2.4. S e m i c o n d u c t o r lasers a n d rate equations
In this section, we derive the rate equations for semiconductor lasers. Here we treat a laser consisting of a Fabry-Perot resonator with a single active layer of a double heterostructure, however any type of laser that has a narrowstripe edge-emitting structure, such as Multi-Quantum Well (MQW) laser or a Distributed Feedback (DFB) laser can be described by the same rate equations. Lasers described by the same form of rate equations show similar instabilities from the viewpoint of nonlinear dynamics, although the parameter regions for characteristic nonlinear dynamics phenomena depend on the specific laser structure. On the contrary, lasers having structures different from edgeemitting lasers with narrow stripe width, for example VCSELs and broad-area semiconductor lasers, have different forms of rate equations and show fairly different dynamics.
14
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 2
Laser Medium
El,(z)
y
es(z)
z=O
z=l
Z
Fig. 2.2. Model of Fabry-Perot semiconductor laser.
Actual semiconductor lasers are not simply described by the model of twolevel atoms. However, the intra-band relaxation time, ~10 -13 s, in the semiconductor medium is much smaller than the carrier recombination time, ~10-9s (Agrawal and Dutta [1993]). Therefore, we can apply the model of two-level atoms to the operation of semiconductor lasers as a good approximation. As discussed in the previous section, the polarization relaxation time (T2 < 0.1 ps) is much smaller than the other relaxation times (Tph ~ several ps and T1 ,~ several ns), and semiconductor lasers are classified as class-B lasers. Therefore, the polarization equation is adiabatically eliminated and the laser is described by the equations for field and carrier density (equivalent to population inversion). In the dynamics of semiconductor lasers, the laser amplitude (laser intensity or photon number) and the phase play important roles, therefore we must consider three differential equations, for the field amplitude, the phase and the carrier density, as the laser equations. Below, we derive the rate equations of semiconductor lasers following Petermann [1988]. We examine a laser model of a Fabry-Perot resonator with plane mirrors (fig. 2.2). Assuming a resonator distance l, and reflectivities of the front and back facets rl and r2, respectively, the fields Ef and Eb propagating in forward and backward directions are written by Ef(z) = Eof exp{ikz + 51 ( g - a ) z } ,
(2.32)
1 Eb(z) = E0b exp{ik(l - z) + 5(g - z ) ( l - z)},
(2.33)
where g is the gain in the laser medium and a is the total loss due to absorption and scattering in the laser medium. The parameters are all defined for the laser intensity, so that a factor of I is introduced in the above equations. From
1, w 2]
Semiconductor lasers as class-B lasers
15
the boundary conditions at the facets E f ( O ) = r l E b ( O ) and E b ( l ) = r 2 E f ( l ) , the condition for laser oscillation is rlr2 e x p { 2 i k l - ( g - a ) l }
(2.34)
= 1.
From the real part of eq. (2.34), the amplitude condition for laser oscillation is given by
,1 (1)
gth = a + ~- In ~
rl r2
(2.35)
.
Also, the imaginary part of eq. (2.34) gives the phase condition as (2.36)
kl = myt,
where m is an integer. Equation (2.35) is the condition for the laser threshold and is interpreted as the balancing of the gain with the losses of the internal absorption and reflection in the laser medium. However, the actual gain is slightly less than gth due to the existence of spontaneous emission of light. Before introducing the rate equation for the field, we consider the gain in the laser medium. Following the definition of eq. (2.34), the gain G for the round trip in the laser cavity is given by G
= r l r2
exp{2ik/+ (g - a) l}.
(2.37)
After the round trip, the steady-state solution for the traveling wave for the positive direction El(t) must coincide with the previous field, thus the condition is El(t) = GEf(t-
Tin),
(2.38)
where tin is the round-trip time of light in the internal laser cavity. The wavenumber k depends on the refractive index of the laser medium, and is a function of the frequency and also the cartier density n. The wavenumber can be expanded by those parameters as k = r/-
C
-
-
C
r/0 +
~-
( n - - nth ) -k- ~ ( 0 )
(Dth
- - O)th )
,
(2.39)
where nth and O)th are the cartier density and the angular frequency at the threshold, respectively, r/0 is the refractive index below the laser oscillation, and
16
[1, w 2
Chaotic dynamics in semiconductor lasers with optical feedback
r/e is the effective refractive index. Using the fact that im is equivalent to the operator d/dt and applying the same procedure for the field of the backwardtraveling wave in addition to some lengthy calculations, we obtain the field equation as the time evolution
{ -i(oa0 --
dE(t)dt -
O,)th) + ~1
( g/]e c____Tph 1 )}E(t) '
(2.40)
where O9o is the angular frequency of the laser oscillation and Tph is the photon life time that accounts for the scattering and dissipation of light in the laser medium. The photon lifetime is defined by the following relation:
, Tph
-
c{ a+ 1In (1)} --
?~e
7
rlr2
,
(2.41)
with c/rle =Og = 2//tin the group velocity of light in the laser cavity. Before introducing the full expressions for the rate equations, we discuss the relation between the complex susceptibility and the linewidth enhancement factor at laser oscillations, which is an important parameter in semiconductor lasers. We define the susceptibility of the medium before the laser oscillation as X0 = X0' + ix0". Above laser threshold, the change of the susceptibility Z; due to the laser oscillation is added to the original susceptibility. The total susceptibility is a function of the laser frequency and is given by (2.42)
X(m) = Xo(m) + Xt(m) = Xg + Xl" + i(x~' + Xff).
Writing the imaginary part of the refractive index as r/' at laser oscillation, the macroscopic complex refractive index is defined by r/c = ~/-ir/'.
(2.43)
The relation between the imaginary part of the refractive index and the gain is given by
1
r/' = -a-;-, g. z/c0
(2.44)
From eq. (2.44), we obtain the relation between the oscillation frequency and the carrier density as ((.0-- (.Oth) --
(_Oth
Or/(n
tie On
-/'/th) --
(_Oth O~ O~] z
--(n-
tie O rl' On
nth)
,
=
og
~ Ott)g~ n ( n - nth). (2.45)
1, w 2]
Semiconductor lasers as class-B lasers
17
The parameter a introduced in the above equation is very important and called a linewidth enhancement factor. From the definition of susceptibility, the a parameter is defined by a -
Re[x] Im[z]
2
co Or/IOn
(2.46)
c Og/On"
The parameter a is almost zero for most lasers, while it has a positive value of 2-7 for semiconductor lasers, which sets them off from other lasers. The a parameter plays an important role in the determination of the linewidth of the laser oscillation, which is broadened by 1 + a 2 times in semiconductor lasers compared with other lasers. When a laser oscillates very close to the threshold, the gain g is expanded for the carrier density: g = gth + Og (n - nth),
(2.47)
where gth is the gain for the carrier density at transparency no, related as
Og
gth = ~ ( n t h -- no).
(2.48)
At transparency, the loss in the laser medium balances the gain. The gain must exceed this value for laser oscillations. At a laser oscillation well above the threshold, the effect of gain saturation must be taken into account. In that case, we use the coefficient of gain saturation es and obtain the relation g =
gth
(2.49)
1 + es El 2"
Using eqs. (2.45) and (2.47), and the fact that the cartier density is also a function of time, the field equation for the laser oscillation is finally given by dE(t) - 2' [(1 - i a ) G , { n ( t ) dt
nth}] E(t),
(2.50)
where we define the linear gain G,, as G,, = tggOg/On. In actuality, a term fisn/rs (with fis the spontaneous emission coefficient) for spontaneous light emissions must be added to the field equation, however the effect of this term is very small
18
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 2
(fis ~ 10-5). For the complex amplitude E(t) = E o ( t ) e x p { - i q ~ ( t ) } , eq. (2.50) is split into the field amplitude and phase equations as dE(t) ~ [(1 dt _ ~-
ia)Gn{n(t) -
-
nth }] E(t),
dq~(t) _ 21a G n { n ( t ) - n t h } , dt
(2.51) (2.52)
The carrier density corresponds to the population inversion for the approximation of the model of two-level atoms in semiconductor lasers. The equation for the cartier density is dn(t) dt
-
J
n(t)
ed
rs
G , { n ( t ) - no} E 2,
(2.53)
where the first term on the right-hand side of the equation represents the pumping by the injection current density J, with d the thickness of the active layer. The second term is the cartier recombination due to spontaneous emission, where rs denotes the cartier lifetime. The third term represents the carrier recombination induced by the laser oscillation. The photon lifetime and the cartier densities at threshold and transparency have the relation Og On ( n t h
1 --
no)
-
rP h .
(2.54)
In the derivation of eq. (2.50) we assumed that the field is normalized to the square root of the photon number. In actuality, the gain terms in eqs. (2.50) and (2.53) must include a confinement factor that has a value less than unity. Furthermore, Langevin terms accounting for statistical noise effects are added to each equation when necessary. Since semiconductor lasers are class-B lasers, the two equations (2.50) and (2.53) are sufficient to describe the laser dynamics, while the polarization of matter is implicitly included as P = eoZE in eq. (2.50). For a Fabry-Perot type edge-emitting laser with a narrow stripe we ignored the effect of carrier diffusion, but carrier diffusion plays an important role for lasers with other structures, such as VCSELs and broad-area lasers. For a step input, a semiconductor laser shows a typical behavior known as relaxation oscillation. Relaxation oscillation in a semiconductor laser occurs because the population inversion does not follow the photon decay rate as already discussed. The damping rate and its frequency for the relaxation oscillation are
1, w 2]
Semiconductor lasers as class-B lasers
19
easily calculated from a linear stability analysis as discussed in w2.2. Using the rate equations in eqs. (2.51)-(2.53), small perturbations for the steady states are applied and the real and imaginary parts of the solutions for the characteristic equation give the damping rate FR and the damping oscillation angular frequency C0R as (Agrawal and Dutta [1993])
1( G"Es2+ ~,)
FR = - 5
/ GnE~ ~ O R = v 1.ph - r 2.
'
(2.551
(2.56)
Since FR is negative, the oscillation damps out for the time evolution, so that the semiconductor laser shows no instability by the relaxation oscillation. For low 2 output power of the laser oscillation, i.e. F 2 <
S = -~(J--Jth),
(2.57)
where S is the photon number, and Jth = ednth/rs is the threshold current density. Using eq. (2.57), the relaxation oscillation frequency is written as
fR = ~1v/an ~-(J - flth).
(2.58)
The existence of the relaxation oscillation in semiconductor lasers induces various instabilities when external light is injected into a laser or externally reflected light is fed back into a laser. In those cases, the relaxation oscillation is not damped out any more and the laser shows various unstable behaviors. The field defined by eq. (2.51) is a slowly varying field derived from the SVEA approximation. Replacing the field by the photon number as S(t) = ]E(t)] 2 = E2(t) and including the spontaneous emission and the Langevin
20
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 2
noises, the rate equations for the photon number, the phase, and the carrier density are written as (Agrawal and Dutta [1993]) dS(t) -- 21 [G n { n ( t ) dt
nth }] S(t) + Rsp(t)
+ Fs(t),
(2.59)
dqg(t)- 1 a [Gn{n(t)- nth}] + F~(t), dt 2
(2.60)
dn(t) dt
(2.61)
J(t) ed
n(t) rs
Gn{n(t) - no} S(t) + F,,(t),
where Rsp(t) is the contribution from the spontaneous emission and the final terms represent the Langevin noise. The injection current may also be a function of time. The carrier recombination also originates from other processes such as the Auger recombination, however they are not included as small fractions. The terms can be neglected when we treat the pure dynamics of chaos in semiconductor lasers. At a glance, a semiconductor laser seems to be described by three rate equations, however the phase does not affect the other equations. Therefore, the two rate equations for the photon number and the carrier density are essential for the laser dynamics and the laser does not show chaotic dynamics by itself.
2.5. Various semiconductor lasers and rate equations We have derived above the rate equations for the edge-emitting single-mode semiconductor lasers. However, there are various kinds of laser structures in semiconductor lasers, and each has its own characteristic properties. Here, we do not discuss the details of each device and its characteristics, but we introduce the rate equations for such lasers. Before discussing other structures than edgeemitting semiconductor lasers, we present the rate equation for a multi-mode operating edge-emitting laser. The separation between the longitudinal modes of a semiconductor laser is more than 100 GHz (corresponding wavelength is ~1 nm) due to the short internal cavity length. The gain profile is as large as 20 nm or more and it has the possibility of multi-modal oscillations with several oscillation lines. Even a single-mode laser sometimes oscillates in multi-mode due to noise and external perturbations. When a semiconductor laser oscillates in multi-mode with M oscillation lines, the rate equations for the complex
1, w 2]
S e m i c o n d u c t o r lasers as class-B lasers
21
field and the carrier density are written as (Ryan, Agrawal, Gray and Gage [1994])
dEj(t)
1
dt
- ~(1 - i a ) G n { n ( t ) -
tlth } E j ( t )
(2.62)
2
ej IEj(t)l z + ~
0.,~ Era(t) 2 Ej (t),
m=I
dn(t)
dt
-
J ed
n(t) rs
M
Z
[AmGn{n(t)- no}] Em(t)l 2 ,
(2.63)
m= 1
where Ej(t) is the field of the jth mode, w and 0,,!i (m ~ j) are the self- and cross-saturation coefficients, respectively, and Am is the mode gain coefficient. The saturation coefficients do not have large values and they are in the order of 103 s-1 . In the equation for the carrier density, we ignore mode interferences for carrier recombination because these are small effects. A semiconductor laser with a multi-mode oscillation itself is unstable, and it is easily affected by external perturbations. A multi-mode semiconductor laser shows mode competition and mode switching induced by the nonlinear interactions among the modes. Mode partition noise is one of the dominant effects in multi-mode oscillating lasers and is non-negligible. Each oscillation mode includes a very large RIN and this sometimes causes problems in actual use. But in total, a multi-mode semiconductor laser with partition noise has the same order of RIN as a single-mode laser since the partition noise is averaged out (Petermann [1988]). Self-pulsating semiconductor lasers are currently used for DVD (Digital Versatile Disk) systems in optical data storage, since noise induced by the optical feedback is greatly reduced by self-pulsations due to the laser structure. The pulsation frequency depends on device and bias injection current, and ranges typically from several hundreds of MHz to the order of GHz. A self-pulsating semiconductor laser is itself unstable, and it sometimes shows instability for a certain region of bias injection current without external perturbations. The structure of self-pulsating lasers is almost the same as that of edge-emitting lasers, except for saturable absorbing regions adjacent to the active layer. Outside the active area, saturable absorbing areas for carriers are installed in the direction of the layer. The width of the active region is usually the same size as that of the edge-emitting laser. Due to the presence of this region, carriers in the active area rapidly decay toward the absorbing regions and pulsations occur in the laser output. In such a structure, we must take into account the cartier
22
Chaotic dynamics in semiconductor lasers with optical feedback
[1, w 2
density equations in the absorbing regions. The rate equations for the complex amplitude E(t) and the carrier densities nl(t) and n2(t) for the active and absorbing regions describing self-pulsating semiconductor lasers are written as (Yamada [ 1993, 1994]) dE(t) dt dnl(t)
dt dn2(t) dt
_
1
2(1 -- in) [Gnl {t/l ( / ) - rtthl } + G n 2 { n 2 ( t ) - rtth2 }] E(t) + Rspl, (2.64) J ed
nl(t) Tsl
n2(t) rs2
n l ( t ) - n2(t) - G,,1 {nl ( t ) - no1 } [g(t)[ 2 , rl2
n2(t)- nl(t) - G n 2 { n 2 ( t ) - no2} [E(t)[ 2 r21
(2.65)
(2.66)
The subscripts 1 and 2 denote the quantities for the active and absorbing regions, respectively, Rspl is the spontaneous emission in the active region, and rl2 and /'21 are the carrier diffusions from the regions 1 to 2 and vice versa. In actuality the relation between the carrier density and the gain is not linear, but we can use the linear relation within a certain region. Self-pulsating semiconductor lasers were originally aimed at reducing the effect of optical feedback, however the RIN is sometimes enhanced under certain feedback conditions. Vertical-Cavity Surface-Emitting Lasers (VCSELs) are promising future light sources for optical information processing and communications. Currently, VCSELs from visible wavelengths to near infrared (1.5 ~tm) are fabricated and their output powers reach as high as 10 mW. Also, a device with a modulation bandwidth over 10GHz and a RIN of less t h a n - 1 4 0 dB has been fabricated. A VCSEL has a disk structure with light coming out from the top or bottom of the surface. VCSELs have many merits for practical purposes. Since a VCSEL has a symmetric space structure, we can obtain a circular beam as its output, while the beam profile of an edge-emitting laser has astigmatism. Owing to the small cavity length compatible with the wavelength of light and very high reflectivity of light in the internal cavity, the laser is considered to have a very low threshold, as low as ~~tA. From this same reason, we can produce stabilized oscillation with a single mode that has a large mode separation (~40 THz). Another merit of VCSELs is the easiness of manufacturing laser arrays because of the surface-emitting structure. In addition to the time-dependent phenomena, the space structures and the polarization modes give rise to the spatial dynamics in VCSELs. Spatial hole burning and multi-transverse mode oscillations are often observed in the laser output because the spatial structure causes instabilities such as mode and polarization switching. Therefore, the VCSEL in itself is
1, w2]
Semiconductor lasers as class-B lasers
23
an unstable laser depending on the device parameters. The effects originate intrinsically from the instabilities involved in the laser rate equations. The rate equations for VCSELs are similar to those for the edge-emitting laser except for the spatial terms. For a certain polarization mode, the field equation is given by (Valle, Sarma and Shore [1995a,b], Law and Agrawal [1997])
dEj(t) - l(1-ia)Gnj{n(r,(p,z,t)-nth.j} Ej(t), dt
(2.67)
where n(r, c#,z,t) (using radial coordinates (r, c#,z)) is the space-dependent carrier density and nth,j is the threshold carrier density for the jth mode. Ej is the field amplitude for the laser oscillation of the jth spatial component, and the total complex amplitude from a VCSEL is written as
M Etotal(r, q~,z, t) = ~ Z ejEj(t) ~ ( r , gp)Ao sin(fi:z) exp(--itOth,jt) + C.C., j=l 1
(2.68) where M is the total number of spatial modes, kj is the polarization vector for the jth mode, lpj is the eigenfunction for the jth mode, fi- is the propagation constant for the z direction, and A0 is the normalization coefficient. Since the carrier diffusion in the radial direction must be taken into account for the VCSEL oscillation, the rate equation for the carrier density is written as
d J(r,~) dtn(r, dp, t) = DXTZn(r, r t) + ed
rd
n(r,~,t) rs
M
(2.69)
d ~ G.~{.(r. ~. 0 - ~0} IEj(t)~(r. ~) ~. j=l
where D is the coefficient for the carrier diffusion, the subscript T denotes the operation for the transverse coordinates, and Fd is the confinement factor for the longitudinal direction in the active layer, given by
I'd = ~0"d A sin(fiz)[ 2 dz.
(2.70)
In reality, interferences occur among the spatial and polarization modes and this effect must be taken into account in eq. (2.69), however we neglect this effect because the separation of successive modes is as large as --100 GHz. The high-power semiconductor laser is a promising laser device, since the power conversion efficiency from electricity to light in these semiconductor
Chaotic dynamics in semiconductor lasers with optical feedback
24
[1, w 2
lasers is much higher than in other lasers (more than 50%). Such high-power high-efficiency lasers can be used in light sources for laser welding, pumps for solid-state lasers, and laser fusion. Currently, a high-power semiconductor laser with an output >1 kW is devised by stacking lasers as arrays. One of the technologies for high-power semiconductor lasers is a broad-area laser that has a broad stripe width (-100 btm, about twenty times larger than for ordinary semiconductor lasers). The longitudinal dimension is typically 1 mm, however the thickness of the active layer is comparable to that of ordinary semiconductor lasers and the laser oscillates in TE mode. The output power of a broad-area laser is more than 100 mW. Therefore, the effects of cartier diffusion and light diffraction are essential for such a structure. The broad-area semiconductor laser in itself is an unstable device due to the spatial dependence of the laser oscillations. Although broad-area lasers sometimes oscillate with multimodes, we assume single-mode operation here. Starting from the Helmholtz equation for the complex laser field E(x,t) (with x the coordinate perpendicular to the laser thickness in the active layer, i.e. the direction along the laser stripe), the rate equation is given as (Rahman and Winful [1994], Merbach, Hess, Herzel and Scholl [1995], Levy and Hardy [1997])
OE(x, t) 0t
02E(x,
= i D e ~ +0X2
t)
2(1-ia)G,,{n(x,t)-nth}E(x,t)+Rsp,
(2.71)
where De=c/2krl 2 is the diffraction coefficient of light. The first term on the right-hand side of the equation is the diffraction effect due to the broad active area. The diffusion effect must also be included in the rate equation for the carrier density:
On(x,t) O2n(x,t) J Ot -D~ O x ~ ~ ed
n(x,t)
G,,{n(x,t)-no} IE(x,t) 2
(2.72)
rs
where Dn is the diffusion coefficient of the carrier and it is defined by Dn = lZdlrs, with ld the diffusion length). In reality, the injection current is not only a function of time but also of x. Semiconductor laser arrays are also important devices for high-power radiation. The laser may be composed of arrays of broad-area lasers to make an extremely high-power laser device. However, here we assume arrays of ordinary edge-emitting lasers. When the separation between the array lasers is very small, the lasers interfere and instability sometimes occurs in the total laser output. In a strict sense, we must consider all the effects of diffraction and carrier diffusion as we have discussed above (Mtinkel, Kaiser and Hess [1996]). However, here
Theory of semiconductor lasers with optical feedback
1, w 3]
25
we consider the situation that the coupling of light among arrays is dominant over diffraction and carrier diffusion. We also assume that the coupling is strong only between neighboring lasers, as is often the case. Thus, the rate equations for the field amplitude and the carrier density of the jth element are given by (Winful and Rahman [1990], Winful [1992], Nichols and Winful [1993]) dEj(t)
dt
_ 2( 1 1 - i a ) G n { n j ( t ) - n t h } E j ( t ) - i - ~ i tean {Ej +l (t) + Ej - l(t)},
dnj(t) _ J dt ed
nj(t) rs
G, {nj(t) - no} Ej(t)[ 2
(2.73) (2.74)
where tea is the coupling strength between adjacent laser elements. The spontaneous emission term is neglected in above equation. Moreover, the phase %. is assumed to be the same for each laser element. For the numerical calculation of the rate equations, the number of laser arrays is N + 1 and the boundary condition is E0 = EN = 0. The rate equation for the field amplitude has the form of the well-known equation of the Coupled Map Lattice (CML). The CML shows typical spatio-temporal instabilities and chaos. Therefore, semiconductor laser arrays are essentially chaotic systems.
w 3. Theory of semiconductor lasers with optical feedback 3.1. Rate equations f o r semiconductor lasers with optical feedback
Optical feedback effects in semiconductor lasers are often used for mode selection, reduction of frequency chirping, and linewidth narrowing of laser oscillations. However, optical feedback generally induces unwanted phenomena such as instabilities and chaos in laser output. Due to the low reflectivities of the laser facets in semiconductor lasers, they are largely affected by external optical feedback. Other lasers usually have mirror reflectivities in excess of 99% and the light in the cavity is reflected many times by the mirrors, while in semiconductor lasers the light leaves the cavity after only a few reflections. In spite of the low reflectivities of the laser facets, semiconductor materials can be used as laser media because of the high efficiency of the laser oscillations. The situation is fairly different for VCSEL structures, since the reflectivity of VCSELs is almost 100%. However, a VCSEL has a short cavity length and operates with much fewer photons compared to ordinary edge-emitting lasers. As a result, an external perturbation from very few photons can affect the laser oscillation. Therefore, all semiconductor lasers are very sensitive to external
Chaotic dynamics in semiconductor lasers with optical feedback
26
[ 1, w 3
Fig. 3.1. Model of semiconductor laser with optical feedback.
feedback and they show unstable behavior due to the feedback. Semiconductor lasers with optical feedback have been extensively studied for the last two decades (van Tartwijk and Agrawal [1998], Ohtsubo [1999]), since they are excellent models for physical chaotic systems and they are also important from the viewpoint of applications. As already discussed, we must deal with time-dependent rate equations for the laser dynamics, and they must be described not by the photon number but by the complex field amplitude, since the dynamics originate from the coherent coupling between the internal and external fields. Figure 3.1 shows the model of a semiconductor laser with optical feedback. The light emitted from a semiconductor laser is reflected by an external mirror at a distance L from the front facet of the laser and the reflected light is fed back into the laser cavity. The external mirror could be a diffraction grating or a phase-conjugate mirror, but here we assume an ordinary reflection mirror that is located within the coherence length of the laser. The laser sometimes oscillates in multimodes by external feedback, however here we assume single-mode operation. The rate equation for the complex field is then given by (Lang and Kobayashi [1980]) dE(t) dt
_
1
K"
~(1-iot)G,,{n(t)-nth} E(t) + ~inE(t - r) exp(io)or),
(3.1)
where tr is the feedback rate of light from the external mirror, Tin--2tlel/c is the round-trip time of light in the internal cavity, r = 2L/c is the round-trip time of light in the external cavity, and too is the angular frequency of the laser oscillation. The final term is the feedback effect. In eq. (3.1) we have neglected the effect of gain saturation, but it can simply be added as a term proportional to the square of the field amplitude when necessary. The feedback rate tr is derived by considering multiple light reflections between the front facet and the external mirror. However, here we assume that the reflectivity of the external mirror is weak and only a single reflection has significance. This case is very important for
1, w 3]
Theory of semiconductor lasers with optical feedback
27
practical applications and gives rise to a variety of dynamics in the laser output. We also assume that the amplitude reflectivities for the front and back facets of the laser cavity are the same at r0. This assumption is not always true for actual lasers, but the feedback rate for different reflectivities can be calculated straightforwardly. Thus the feedback rate is calculated to be ~c = (1 - r g) rex---~t, ro
(3.2)
where rext is the reflectivity of the external mirror. For a small external reflectivity of less than a few percents of the amplitude, the laser easily destabilized and shows unstable oscillations. The rate equation for the carrier density can be described by the same as that of the solitary laser in eq. (2.53). Writing the complex field as E(t)exp{-iq~(t)}, we obtain a set of rate equations for the amplitude, the phase, and the cartier density as (Petermann [1988], Agrawal and Dutta [1993]) dE(t) dt
_
1
K"
Gn{n(t) - nth } E(t) + -~inE(t - r)cos O(t),
~c E ( t - r) d~(t)dt - ~laGn{n(t)- nth} - - - - - t i nE(t) sin O(t), dn(t) dt
J ed
n(t) rs
G, {n(t) - no } [E(t) 2,
O(t) - O)o r + q ) ( t ) - q ) ( t - r).
(3.3) (3.4)
(3.5) (3.6)
The semiconductor laser with optical feedback is described by coupled delaydifferential equations with three variables, although the system is different from the Lorenz model. Thus, the laser exhibits chaotic behavior in its output. Since the equations include delay terms, the induced chaos is high-dimensional due to the continuous delay-differential nature of the system. We can numerically solve the rate equations (3.3)-(3.5) and we can also investigate the laser dynamics. For numerical calculations, the fourth-order Runge-Kutta algorithm is frequently used for the sake of accuracy.
3.2. Feedback regimes
As already discussed in w3.1, a semiconductor laser is very sensitive to external feedback light, and shows chaotic dynamics. There are many parameters to characterize such instabilities. One important parameter is the external
28
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 3
reflectivity. According to Tkach and Chraplyvy [1986], the laser instabilities may be categorized into the following five regimes depending on the feedback fraction: Regime I. Very small feedback (the amplitude feedback fraction is less than 0.01%) and small effects. The linewidth of the laser oscillation becomes broad or narrow depending on the feedback fraction. Regime II. Small but not negligible effects (less than ~0.1% and the case for C > 1, where the C parameter is a measure of instability discussed later). The generation of the external modes gives rise to mode hopping among the internal and external modes. - Regime III. This is a very narrow region around ~ 0.1% feedback. The mode hopping noise is suppressed and the laser oscillates with a narrow linewidth. - R e g i m e IV. Moderate feedback (around 1%). The relaxation oscillation becomes undamped and the laser linewidth is broadened. The laser shows chaotic behavior and sometimes evolves into unstable oscillations in a coherence collapse state. The noise level is enhanced greatly under this condition. - Regime V Strong feedback regime (higher than 10% feedback). The internal and external cavities conform like a single cavity and the laser oscillates with a single mode. The linewidth of the laser is narrowed greatly. The investigated dynamics were for DFB lasers with a wavelength of 1.55 gm, so the feedback fractions quoted above are not always true for other lasers. On the other hand, the dynamics for other lasers show similar trends for variation of the feedback fraction. We are very interested in regime IV that shows chaotic dynamics, though it has a very small feedback fraction (the fraction of intensity feedback is only 0.01%). In actual applications of semiconductor lasers, this regime is also very important, for example, the amount of feedback into the laser cavity in a CD (Compact Disk) system corresponds to regime IV. A semiconductor laser with optical feedback for regime IV is modeled after the Lang-Kobayashi equations that include the optical feedback effects as already derived in eqs. (3.3)-(3.6). The lasers show the same or similar dynamics as far as the rate equations are written by the same forms. Therefore, edge-emitting semiconductor lasers such as Fabry-Perot, MQW, or DFB lasers exhibit similar chaotic dynamics, though the parameter ranges may be different. -
-
3.3. Gain reduction and linewidth broadening due to optical feedback The optical feedback effects in semiconductor lasers are dynamic in nature for
Theory of semiconductor lasers with optical feedback
1, w 3]
29
time evolution, however the steady-state solutions are important to study in laser oscillations. Here we discuss the condition for the steady-state solution in the presence of optical feedback. We assume that the front facet of the laser and the external mirror are combined and behave like a single mirror. In steady-state operation, the effective reflectivity of the external mirror together with the front facet of the laser is given by (Kakiuchida and Ohtsubo [1994]) reff =
ro +
text
1 + ror
exp(io)o r) exp(icoo r)
We also assume that the external reflectivity is very small, effective reflectivity is written as reff = reffl exp(i0r) = ro + (1 - ro2) r exp(io)o r),
(3.7) rex t
~ 1. Thus, the
(3.8)
where q~r is the phase of the effective mirror. When the feedback coefficient defined in eq. (3.2) is small enough, the amplitude and phase of the effective reflectivity are calculated as
Ireffl = ro{ 1 + x" cos(o)o r)}, Or =
K"sin(~O0r).
(3.9) (3.10)
In the meantime, from eq. (2.34), the condition of the laser oscillation is given by roreff
exp{2ik/+ (g - a) l} = 1.
(3.11)
The gain of the laser oscillation in the presence of optical feedback is written as
1 (1)
gc = a + ~ In
(3.12)
ro [reff [
For a small feedback rate, the difference between the gains in the presence of optical feedback and in the solitary oscillation is given by K"
gc - gth = - - 7 COS(O)0/').
(3.13)
The gain in the presence of optical feedback varies periodically depending on the external cavity length; the maximum gain is attained at to0r = 2mJr. Since the gain changes depending on the external cavity length, we can easily select
30
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 3
the oscillation mode and suppress the side modes by appropriately positioning the external cavity. For example, the gain difference between the successive modes is as small as 0.1 cm -1 in a Fabry-Perot semiconductor laser and the stability condition is calculated as tr < 0.1. Finally, the expected reflectivity for the external cavity is rext < 2 • 10-3 for r0 = 0.32 and l = 300 ~tm. This value corresponds to regime III or regime IV of the previous section and it is a measure for the onset of instability. A linear stability analysis is also applied for solving the steady-state oscillation in a semiconductor laser with optical feedback. The spectral linewidth of the laser is derived from a perturbation for the steady state of the phase in the linear stability analysis. Here we show the result of the calculation and the spectral linewidth given by (Tromborg, Olesen, Pan and Saito [1987]) Afe,,- Af F2,
(3 14)
where Af = Rsp(1 + a2)/4:rS (with S the photon number at the laser oscillation) is the linewidth without optical feedback, and the linewidth reduction (or increment) factor F = dCOth/dCOs(with COsthe steady-state solution for the angular frequency of the laser oscillation) is written by F
-
dCOth
_
1 + C cos(COst + tan -l a).
(3.15)
dCOs C is an important parameter for characterizing the feedback effect in the system; it is defined by C = tc-~rv/1 + ct2.
(3.16)
"gin
The minimum spectral linewidth is attained at COsr = tan -1 or, and is given by Afe• =
6f
(3.17)
(1 + C ) 2"
On the other hand, the linewidth at maximum gain is given by Afex =
Af I+K'~
2"
(3 18)
From eqs. (3.17) and (3.18), the linewidth reduces relative to that of the solitary laser for steady-state conditions. However, with increasing feedback strength the
Theory of semiconductor lasers with optical feedback
1, w 3]
31
laser shows instabilities and the linewidth of the laser oscillation grows much larger than that of the solitary laser. Above a certain feedback level, the laser oscillates among several possible oscillation modes and feedback-induced noise increases. With further increasing feedback, the laser even loses its coherence: this is known as the coherence collapse state. At the coherence collapse state, the spectrum spreads over the range of GHz. On the contrary, with a stronger optical feedback over this range, the laser is stabilized to a single-mode oscillation by the feedback and the linewidth of the laser oscillation becomes very narrow.
3.4. Linear stability analysis and linear modes Linear stability analysis is a very useful technique for investigating the dynamic characteristics around the steady states in nonlinear systems. Here we apply the linear stability analysis for the steady states of field Es, phase (ms- COth)t, and carrier density ns in the presence of the optical feedback. Taking into account the steady-state conditions in the rate equations in eqs. (3.3)-(3.6), we obtain the solutions (Agrawal and Dutta [ 1993], Tromborg, Olesen, Pan and Saito [1987], Murakami, Ohtsubo and Liu [1997]) E2 =
J-ns/rs
(3.19)
G,(nrms - no)' K" (_.0s -- 09th = - - - - { a
ns
2K" = nth --
Tin Gn
COS(O)s T ) -4- sin(rest)},
COS(O)sr).
(3.20) (3.21)
Equation (3.20) is rewritten as O)th T -- (Ds T + C
sin(tOsr + tan -l a),
(3.22)
and the laser oscillation condition is calculated from this equation. The C parameter is already defined by eq. (3.16). The number of solutions for the laser oscillation modes in eq. (3.22) depends on the value of the C parameter and the solutions are graphically calculated. When C < 1 (for small optical feedback and a short external cavity), eq. (3.22) has only a single solution and the laser shows stable oscillation. If C > 1, many possible modes for the laser oscillations (external modes and anti-modes) are generated with the relation among the internal laser mode and the excited external modes, and the laser becomes unstable.
32
Chaotic dynamics in semiconductor lasers with optical feedback
[1, w3
Fig. 3.2. Mode distribution in the phase space of the carrier density and the oscillation angular frequency. Crossing points of the solid and broken sinusoidal waves are the locations of the modes. Modes are on an ellipsoid. Solid dot at center is the solitary oscillation mode. Another expression of possible solutions for the laser oscillation is that for the mode distributions in phase space of the carrier density and the angular frequency of the laser. Figure 3.2 shows the mode distributions in phase space given by (Aoosr- i1a r G n A n ) 2 + (89ra,,An): =
~cr
,
(3.23)
where Ans and A~os are the deviations from the steady state of the carrier density and the oscillation angular frequency, respectively. The dashed sinusoidal curve in fig. 3.2 represents A~os, and the solid one denotes Ans. The crossing points of the two curves are on an ellipsoid (thick line) and these are the solutions for the possible laser oscillations. Those in the lower half of the ellipsoid are stable solutions (external modes) and those in the upper half are unstable solutions (anti-modes). The laser oscillates at one of the external modes, and the maximum gain mode (the minimum cartier density mode) is usually the most probable mode for the laser oscillation. However, the mode of the laser oscillation hops around among the external modes and the anti-modes when the oscillation is unstable. One typical instability is the phenomenon known as lowfrequency fluctuations (LFFs). The solid circle at the center of the ellipsoid in fig. 3.2 is the solution for the laser oscillation in the solitary mode. The stability and instability of laser oscillations are theoretically studied by the linear stability analysis for the stationary solutions of the laser variables, and many papers have been published on the study of the dynamics of semiconductor lasers with optical feedback based on such analysis. For the steady states in
Theory of semiconductor lasers with opticalfeedback
1, w 3]
33
eqs. (3.19)-(3.21), we obtain differential equations for the perturbations from the assumption of a small perturbation for each variable as 6E, 6qJ and 6n. In the derivation of the differential equations, we neglect the higher infinite terms and we linearize the equations, which is the origin of the name "linear stability analysis". Then by assuming the forms of the solutions for the perturbations as 6x(t) = 6x0 exp(yt) (x = E, q~,n), we finally obtain the characteristic equation as follows (Tromborg, Osmundsen and Olesen [ 1984], Murakami, Ohtsubo and Liu [ 1997]): D()*) =
]/3 _1_2
-FR + ~ A cos(cost)
),2
Tin
+ a~
41r
cos(cosr) +
Tin
-
2 x'A2/-'R Tin
+
1r A
y
(3.24)
~in
x'A co2 tin
{cos(cOs r) - a sin(cOsr) } = 0,
where A = I - e x p ( - y r ) , a n d / ' R and cOR are the previously defined parameters for the relaxation oscillation. The above equation cannot be analytically solved since it involves exponential terms for y. The imaginary part of the solution denotes the oscillation frequency of the mode and the real part is related to its stability. The solution is stable if the real part is negative, and unstable when it is positive. For a short external cavity or a small external reflection, tOT/Tin << 1, we can assume that 7r << 1 for the solution of y. Thus we obtain analytical solutions for eqs. (3.24), and the real and imaginary parts are calculated as (Agrawal and Dutta [ 1993]) r~ = FR,
(3.25) ~/
a~ = a,~
1 -1- (K"c -- 0~K's) r/Tin (1 + Ir r/nn)2 + (tr r/tin)2'
(3.26)
where tCc = tccos(cOsr) and tCs=tcsin(cOsr). From eq. (3.26) it is apparent that the relaxation oscillation frequency is greatly affected by the external optical feedback. For a stable oscillation condition, the sign of the expression inside the square root in eq. (3.26) must be positive and we obtain the stability condition 1 + C cos(cOsr + tan -1 a) > 0.
(3.27)
Thus, the laser is stable as long as C < 1. However, the laser shows instabilities for C > 1 as recognized from the above equation. The solutions obtained from the characteristic equation (3.24) are called linear modes.
34
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 3
Fig. 3.3. L F F waveform and chaotic itinerary at LFFs. (a) Experimentally obtained L F F waveform. (b) Calculated chaotic itinerary at J = 1.01Jth , L = 30 cm, and rex t = 10%.
3.5. Low-frequency fluctuations in semiconductor lasers with optical feedback One of the typical routes to chaos in semiconductor lasers with optical feedback is a Hopf bifurcation for variations of the external reflectivity, the injection current modulation, and other parameters. Other evolutions of instabilities exist such as intermittent routes to chaos. Low-frequency fluctuation (LFF) is known as intermittent chaos of saddle node instability in semiconductor lasers with optical feedback (Risch and Voumard [1977], Fujiwara, Kubota and Lang [1981 ], Sano [1994]). A typical feature of LFFs is a sudden power dropout with a following gradual power recovery. LFFs occur irregularly in time depending on the system parameters. Figure 3.3a shows an example of LFFs obtained by experiment. Power dropout in LFFs induces considerable noise in the laser, so the study of this mechanism and the suppression of noise are important issues in semiconductor lasers. Since the frequency of LFFs ranges from one MHz to a hundred MHz, and it is smaller than the relaxation oscillation frequency, the phenomena are called low-frequency fluctuations. On the other hand, LFF includes very small finite time structures within the LFF waveform and it
1, w3]
Theory of semiconductor lasers with opticalfeedback
35
consists of a series of fast pulses on the order of a pico-second. Indeed, this fast pulsation has experimentally been observed by using a streak camera (Fischer, van Tartwijk, Levine, Els~il3er, G6bel and Lenstra [1996]). The chaotic itinerary around external modes can be visualized in phase space by numerical calculation (Sano [1994]). In fig. 3.3b, the laser at first oscillates around the neighborhood of the maximum gain due to a small perturbation originating in the nonlinearity. When the state gets close enough to an anti-mode that may be the counterpart of the maximum gain mode, the laser is trapped in the anti-mode. As a result, it becomes unstable and the cartier density suddenly increases to the threshold of the solitary laser, while the phase is unchanged. The sudden increase of the carrier density induces a sudden increase of the phase, and the oscillation behaves like that of the solitary laser. This is the process of the sudden power dropout in LFFs. After that the laser is trapped in one of the external modes near the solitary oscillation. Then the laser stays near the area of the mode for a few cycles and slips to a lower mode toward the maximum gain mode. The mode slipping continues until the laser reaches the maximum gain mode. The dwelling time in each mode coincides with the time corresponding to the external mode frequency. These parts compose the time recovery process after the power dropout. In actuality, very fast pulsations occur when the laser goes around one of the modes. The power-dropout/power-recovery cycle repeats again and again irregularly in time, then we observe LFFs in the laser output. As a whole, LFF has three time scales; a fast pulsation of the order of picoseconds, a time scale corresponding to the external cavity length, of the order of nanoseconds for an external cavity length of the order of centimeters, and the LFF frequency of the order of microseconds.
3.6. Grating feedback effects A grating is frequently used as an optical external reflector in semiconductor lasers. A grating mirror is used to select the oscillation mode or to stabilize a laser. By the grating feedback, the targeted mode is selected and the linewidth of the laser is narrowed greatly. However, this does not always work for linewidth narrowing and it sometimes gives rise to linewidth broadening and instabilities. In this section, we discuss the effects of grating feedback in semiconductor lasers. In the presence of grating optical feedback, the laser sometimes shows instabilities, but here we consider the effect of linewidth narrowing based on the steady-state analysis and we derive the linewidth. The effective reflectivity of the laser facet and the grating mirror is derived in the same manner as that
36
Chaotic dynamics in semiconductor lasers with optical feedback
[1, w 3
in eq. (3.7) and calculated as (Tromborg, Olesen, Pan and Saito [1987], Genty, Grohn, Talviti, Kaivola and Ludvigsen [2000]) ro + r(tOo) exp(itOo r) reff = [reff[ exp(iq~,.) = 1 + ror(tOo)exp(itOor)"
(3.28)
In the grating feedback, the external reflectivity is a function of the laser angular frequency too. When feedback exists in the laser, the phase change Aq~ becomes a function of the refractive index. From the relation A(r/tog) = tothAr/+ (tOg- tOth)77, the phase change is given by Aq~ = 2/{tothAr/q- (tog - toth) 77} -t-q~,., C
(3.29)
where we assume too = tog. A r/is expanded by the carrier density and the angular frequency as Or/
A r / = -~-(
n
Or/
-- nth) + ~ ~ ( t o g -- toth).
(3.30)
Using the refractive index r/c = 77-it/' together with the equalities
Orl On
Orl' -
a - -
On
ac Og -
2toth On'
(3.31)
the relation between the carrier density and the gain is written as 0 r / ( n _ nth) _ On
aC Og
2toth Or/(gg - gth),
(3.32)
where gg is the gain in the presence of grating feedback. Substituting eqs. (3.30)(3.32) into eq. (3.29), the phase change reads
Aq)=-Ot(g-gth)l+
2r/el (O)g -- (_Oth) --b (~r. c
(3.33)
Putting Ar for a possible solution for the laser oscillation and using Tin = 2 r/el/C, we obtain 1
tog - (Oth = nri -{a(gg
--gth) l-- q~,-}.
(3.34)
1, w3]
37
Theory of semiconductor lasers with opticalfeedback
Then the reduction of gain in the presence of grating feedback is given by
1( ro )
(3.35)
g g - - g t h = l In ]retr(O)g)l ' and the linewidth reduction factor is calculated as dgOth 1 dO,. F g - do)g - 14 rindo)g
a d { ( Tindo)g In [reff(O)g)]
"
The linewidth of the semiconductor laser with grating optical feedback is finally written by Afg - Af F2.
(3.37)
where Af is again the linewidth of the solitary laser. When the laser beam has a Gaussian profile and a certain diffraction order is selected by the grating as a feedback light, the reflectivity is explicitly given by r(O)g) = rg exp{ (O')gA--o)2~GG)2} ,
(3.38)
where m6 is the angular frequency selected by the grating, rs is its reflectivity, and AraB is the width of the grating resolution at that frequency defined by Am6 =ctan O/wo (with 0 the angle of incidence of light onto the grating and 2w0 the diameter of the Gaussian beam). The linewidth of a semiconductor laser is narrowed by a grating feedback under stable oscillation. However, it is again noted that even with a grating feedback the laser becomes unstable for a certain region of feedback strength.
3.7. Incoherent feedback effects
Coherent optical feedback effects are important in applications of semiconductor lasers. Even for a long external cavity where the feedback light has an incoherent coupling with the original light within the laser cavity, the rate equations in eqs. (3.3)-(3.5) are still applicable for investigating the laser dynamics. In incoherent optical feedback, the laser becomes unstable and shows instability and chaos in its output. Besides incoherent feedback from a distant reflector, other
38
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 3
Fig. 3.4. Schematic diagram of incoherent optical feedback system. The polarization direction of the light returned to the laser cavity is perpendicular to that of the emitted light.
incoherent systems can be considered. Figure 3.4 shows a laser with a single longitudinal mode, and injection of feedback light from an external reflector. But in this case the polarization of the returned light is orthogonal to that of the original oscillation due to polarization optics, such as a quarter-wave plate, placed in the path of the external cavity. The returned laser field does not interfere with the inner oscillation, but couples to the carriers. This interaction causes instabilities in the laser. The model is described by the following rate equations (Otsuka [ 1999]): dS(t) - G n { n ( t ) - nth } S(/), dt dn(t)
J
n(t)
dt
ed
rs
{n(t)- no}{S(t) + r ' s ( t - r)},
(3.39)
(3.40)
where to' is the feedback coefficient coupled with the carrier density and the photon number, and r, as before, is the round-trip time of light in the external cavity. We need not consider the phase since the phenomena are from incoherent origin. The rate equations consist of only two differential equations, however they are coupled by the delay-differential term. Thus, we can observe instabilities and chaos in a semiconductor laser with incoherent optical feedback.
3.8. Phase-conjugate feedback
Another form of optical feedback in semiconductor lasers is phase-conjugate feedback. A semiconductor laser is frequently used as a light source in phase-conjugate optics. The light reflected from a phase-conjugate mirror is automatically fed back into the laser cavity without any additional optical components in the external optical path. Therefore, a phase-conjugate mirror is
1, w 3]
Theory of semiconductor lasers with optical feedback
39
sometimes used as an effective optical feedback reflector. The phase is inverted by the conjugate mirror and the reflected light interferes with the internal field of the semiconductor laser. The phase-conjugate feedback induces instabilities in the laser oscillation, and the dynamics of the laser are not always the same as those of an ordinary feedback mirror. The typical time scale in semiconductor lasers with optical feedback is on the order of a nanosecond, defined by the laser relaxation oscillation frequency. Therefore, typical effects of phase-conjugate feedback occur when the phase-conjugate mirrors respond as fast as this time scale of the order of a nanosecond. Such phase-conjugate mirrors are realized in quick-response Kerr media with large third-order susceptibility Z t3) (Agrawal and Klaus [ 1991 ], Agrawal and Gray [1992], Langley and Shore [1994], Bochove [1997], Murakami and Ohtsubo [1998]) and also quick-response photorefractive mirrors of semiconductor materials. On the other hand, the dynamics for slowresponse photorefractive mirrors, where the response is much slower than the time variations of the laser dynamics, are the same as those for ordinary reflection mirrors. For a slow-response photorefractive crystal, for example TiBaO3, the laser light automatically returns into the laser cavity, however the mirror shows the same dynamics of optical feedback as an ordinary reflection mirror. Only the spatial phase-conjugate characteristic is effective in such optical feedback. Here, we assume that the mirror responds immediately, i.e. much faster than the scale of the laser dynamics (typically over a nanosecond), and the generation of a phase-conjugate wave is a four-wave mixing scheme. The angular frequencies of the signal and pump beams at the phase-conjugate mirror are set to be tOo and tOp, respectively, and the generated phase-conjugate wave has a frequency tOc = 2tOp-tOo. Therefore, we consider a frequency detuning 26=2(tOp-tO0) between the laser frequency and that of the feedback light. Thus, the complex field equation for semiconductor lasers with phase-conjugate feedback is given by (van Tartwijk, van der Linden and Lenstra [1992], Gray, Huang and Agrawal [ 1994])
dE(t) dt
_
1
~.(1 - ia) Gn {n(t) - nth } E ( t ) (3.41) + -x" -E(t
"t'in
- r) exp [i26(t - ]i t ) + iq~pCM]
where ~PCMis the phase shift induced by the reflection at the phase-conjugate mirror. The last term in the above equation is the effect of phase-conjugate feed-
40
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 4
back. The rate equations for the field amplitude, the phase, and the carrier density are written as dE(t) dt
_
1
K"
~Gn{n(t)- nth} E(t) + -~inE(t - r)cos O(t),
dcp(t) 1 dt - ~aGn{n(t)- nth} dn(t) dt
J ed
n(t) rs
x" E ( t - r) Tin E(t) sin O(t),
(3.42)
(3.43)
2 ,
(3.44)
O(t) = 2 6 ( t - ~1 T) nt- qg(t) + Cp(t -- r) + 0PCM
(3.45)
Gn {n(t) - no } I E ( t ) [
Equations (3.42)-(3.44) have the same form as eqs. (3.3)-(3.5), however eq. (3.45) is different from eq. (3.6) even for zero detuning (6=0). This makes the laser dynamics of phase-conjugate feedback different from those of ordinary optical feedback. The typical feature of the dynamics is phase locking. Indeed, the solution of the phase under steady-state conditions at zero detuning is q~s = ~1 tan -1 (-a).
(3.46)
Namely, in phase-conjugate feedback the phase is locked, while in ordinary optical feedback it is time-dependent and has multiple solutions of the laser oscillations as shown in eq. (3.22). A laser with ordinary optical feedback is very sensitive to short variations of the external mirror compatible with the optical wavelength. However, the laser with phase-conjugate feedback does not show any change for such a small variation of the external mirror. Here we have discussed the case when the phase-conjugate mirror responds immediately after the arrival of the signal beam. The laser dynamics of semiconductor lasers with a finite response time in a phase-conjugate mirror have been discussed by DeTienne, Gray, Agrawal and Lenstra [1997] and van der Graaf, Pesquera and Lenstra [1998].
w 4. Chaotic dynamics in semiconductor lasers with optical feedback 4.1. Route to chaos and chaotic bifurcations Semiconductor lasers with optical feedback show a rich variety of chaotic dynamics. The dynamics are investigated by numerical calculations for the rate equations in eqs. (3.3)-(3.5). There are many parameters for studying the
1, w 4]
Chaotic dynamics in semiconductor lasers with optical feedback
41
Table 4.1 List of semiconductor laser parameters for edge-emitting laser (GaAs-GaA1As 780 nm wavelength) Parameter
Value
Gain coefficient G n
7.0x 103
Linewidth enhancement factor a
3.0
Facet reflectivity r 0
0.566
Carrier number at threshold nth
2.02 x 1024 m -3
Carrier number at transparency no
1.40 x 1024 m -3
S- 1
Carrier lifetime r s
2.04 ns
Photon lifetime rph
1.93 ps
Round trip time in laser cavity tin
8.00ps
Volume of active region V
1.2 x 10-16 m 3
Gain saturation coefficient e
8.4 x 103 s- l
Wavelength
780 nm
instabilities, however here we focus on the reflectivity of the external mirror, since the external reflectivity is one of the most important measures for the chaotic bifurcations discussed in w3.2. At first, we investigate the route to chaos for an increase of the external reflectivity by numerical simulations. The semiconductor laser we will model is an A1GaAs Channeled Substrate Planar (CSP) laser with the device characteristics listed in table 4.1 (Liu, Kikuchi and Ohtsubo [1995]). The Langevin noise terms are neglected in the simulations in order to see the pure dynamics of the chaotic behavior. The laser shows a stable oscillation at a constant output power when it has either no external feedback or negligibly small feedback. For a small but not negligible optical feedback, the laser shows a periodic oscillation (period-1) as shown in fig. 4.1a. The fundamental oscillation frequency is 2.53 GHz, very close to the relaxation oscillation frequency, 2.50 GHz, of the solitary oscillation. Next, for an increase in the external reflectivity the laser evolves to a period-2 oscillation (fig. 4.1b). With a further increase in the reflectivity, the laser shows chaotic oscillation (fig. 4.1c). Figure 4.2 shows the attractors in phase space of the laser output power and the cartier density corresponding to the time series in fig. 4.1. The periodic oscillation is a closed loop in fig. 4.2a and the period-2 oscillation is a double-closed loop in fig. 4.2b. Finally, the chaotic attractor shows a multiple loop called a strange attractor. In actuality, the attractor is a projection of the trajectory of the multi-dimensional parameters in two-
42
Chaotic dynamics in semiconductor lasers with optical feedback :5 ,--,
(a) i
i
~
0
,
I
,
,
0
,
,
1
10
20
Time[nsec]
"-7.
(b')
t
o~ -i
[ 1, w 4
flllll
II1,I
iff~ltlill!lt't!~lIl I ~f] tjIft"1~!lFt'j: 14il!~II'tiIkJt ~'i IJItJII~I 1'I~~i~t I~lf'JI!|"
O 0
10
Time[nsec]
-~
20
(ci
0
, 0
10 Time[nsec]
20
Fig. 4.1. Chaotic time series of semiconductor laser with optical feedback at J = 1.3Jth and L = 3 cm. The external reflectivities are (a) 0.5, (b) 1.0, and (c) 2.0% in amplitude, respectively. (a) Period-l, (b) period-2, and (c) chaotic oscillations.
dimensional space. Also, the state of the laser oscillation never visits the same point twice in the multi-dimensional space, which is a typical feature of the chaotic itinerary (Mork, Mark and Tromborg [1990], Ye, Li and Mclnerney [1993], Li, Ye and Mclnerney [1993]). The same chaotic evolutions have also been observed experimentally. Another tool for investigating chaotic behaviors is a bifurcation diagram: a plot of the local maxima and minima of a chaotic time series for changes in one of the chaos parameters. Figure 4.3 shows such a plot for a semiconductor laser with optical feedback (Ohtsubo [1999]). The parameter for the chaotic bifurcation is the external reflectivity. When the external reflectivity is less than 0.35%, the laser is in a stable oscillation state (fixed state) like in fig. 4.3a. Above this value, we observe a period-doubling bifurcation. In those regions, period-1
1, w 4]
Chaotic dynamics in semiconductor lasers with optical feedback I
1
I
F
1
i
]
[
(a) ,--7,. :3
..Q
E :3 Z ~)
0
L
I
t
I
l
I
1
I
Output Power[a.u.]
i
i
i
i
i
1
i
!
r
i
I
p
I
(b) ,--..,
5 E Z
I,..
0
I
I
i
1
I
I
1
I
i
i
i
1
Output Power[a.u.]
i
i
i
~
i
1
i
]
i
.
(c)
ai
E z
I
I
I
1
I
1
i
i
Output Power[a.u.]
Fig. 4.2. Chaotic attractors corresponding to fig. 4.1.
43
Chaotic dynamics in semiconductor lasers with optical feedback
44
[ 1, w 4
12
lO ~
(a)
8E
-
~D
,6,-
9 ..
.
o
~
"
_ 9 .
9
"
.:
*o.
-,
"
."
....
..".'..,.
..-
,~.
9
.i"
,....:
~~----
2 -_ 0r
9
"..'....; ."" :.,.-L.~'.,::.',,": d,:,"::" :.-":~ ::' ".-:
4-
,
,
0
,
t
I
0.5
,
I
,
,
,
1
9
-7-
.-:,.-,,-.-,~__
1---
1.5
-
--'~-"-~
2.5
2
rext(%) 12 (b)
10-
~- 8-
:.
E
"~"
6
~'
4
.
9
....-9 ".99 ..,,..""-." .: . . .
.., .:..-.
2 0 0
1
1.5
..
" :-;'. :-...:. ":I ." ;-"" :" ":" ."'.- 9 " . . , ,,~ r. . .,. ~ 9 ~.s'. . . .,. . . ,,,..,..'.... ,~,,..r..... .. -"
0.5
.
9
r
2
~-
.
2.5
rex,(%) Fig. 4.3. Chaotic bifurcations. (a) Periodic bifurcations at J = 1.3Jth and L = 9cm. (b) Quasiperiodic bifurcations at J = 1.3Jth and L = 15 cm.
oscillations appear at first, then period-2, period-4, and finally chaotic states via quasi-periodic oscillations for an increase o f the external reflectivity. With period-1 oscillations, the frequency closest to the relaxation oscillation is excited first. T h e n the excitation o f the external m o d e follows for higher-periodic states, and their m i x e d frequencies, their beats, and their higher h a r m o n i c s appear in the laser output. This typical chaotic route is called a H o p f bifurcation. However, various chaotic routes exist in nonlinear systems. One such route in s e m i c o n d u c t o r lasers with optical feedback is a quasi-periodic bifurcation as shown in fig. 4.3b. I m m e d i a t e l y after period-1 oscillations, the laser has quasi-
Chaotic dynamics in semiconductor lasers with optical feedback
1, w 4]
45
periodic oscillations without showing a period-doubling route. The routes of bifurcations highly depend on the chaos parameters in the nonlinear systems. Another route is an intermittent route to chaos, which is considered LFE We ignored Langevin noise in the numerical simulations. In the presence of noise, the chaotic route and chaotic characteristics do not change, while the maxima and minima of the output in fixed and periodic states in the bifurcation have finite widths.
4.2. Linear modes
The frequency excited at unstable oscillations in a semiconductor laser with optical feedback is numerically calculated from the solutions of eq. (3.24). Figure 4.4 shows an example of the calculated mode distributions for changes in the external reflectivity (Murakami and Ohtsubo [ 1998]). The real and imaginary parts represent the damping rate and oscillation angular frequency of linear modes, respectively. As long as the damping rate of the solution is less than zero, the mode is stable and an oscillation induced by such noise rapidly damps out. But if the real part of the solution exceeds zero, the mode becomes unstable. With increasing reflectivity, the highest mode at a frequency of 2.5 GHz, with the highest value of the real part among the linear modes, is the first to become unstable due to a positive value of the real part. With further increasing reflectivity, the values of the real parts for all the modes increase, and every mode tends
-
9/ "
X :r=O.0%~'~ ~ :r=0.2% I :r=0.4% I :r=0.6% ~:r=O.8%.j
. _Highest mode 9
\\
[]
Second
k.,
/ ,
-2 -
9
El
x
o
"~
9
o o
o
o
-113 )
!
2
3
4
5
if2 / 2Jt(G Hz) Fig. 4.4. Linear mode distributions at J = 1.3Jth and L = 10 cm. The highest mode corresponds to the relaxation oscillation and the second mode to the external cavity mode. With increasing external feedback, the real part of each mode increases and the laser becomes less stabilized.
46
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 4
to be less stable. Then the laser shows an unstable oscillation with an undamped relaxation oscillation. When the highest mode is well over the zero point of the real part, the second mode is excited next, and the instability is extremely enhanced. The second mode is closely related to the fundamental external cavity mode. The value of the C parameter is calculated to be C = 2.8 for the critical unstable point of the reflectivity of 0.4%. Excitations of the unstable modes can be experimentally observed and analyzed by a Fabry-Perot spectrum analyzer (Ye and Ohtsubo [ 1998]). It is noted that the number of linear modes for the solitary laser is only one and its real part is always less than zero, i.e. stable oscillation. In the linear stability analysis done in w3.4, we obtained the characteristic function (3.24). From this equation, we can find the boundary condition for stability in the laser oscillations. For the solution of D ( 7 ) = 0 with 7=i09, the condition for stability is given by (Tromborg, Osmundsen and Olesen [ 1984]) 0) 2 -- (d~ =
-2rR ~o c o t ( 1 09/').
(4.1)
From eq. (4.1), the boundary for the stability is a periodic function of the external cavity length (equivalently the delay r). The delay time at this boundary is given by
2E { 1
} J
tO cot -~ -2FRtO (092 _ 092) + mar . r = --
(4.2)
At the relaxation oscillation 09= 09R, periodic solutions with the time delay r (the equivalent length for the periodic solutions is given by Lres = mc/2fR, with m an integer) have been obtained from the above equation as the stability condition. Namely, the stability of the laser for the external reflectivity is periodically enhanced with a period equal to that calculated from the relaxation oscillation frequency. This periodic enhancement of the stability is explained by the competition of two existing linear modes close to the relaxation oscillation frequency (Murakami, Ohtsubo and Liu [1 997]). Equation (4.1) can be solved graphically. The cross points for 09= 09R in the graph represent the positions where the stability of the laser oscillation is locally enhanced. Such a phenomenon is indeed observed, as will be discussed in the following section.
4.3. Dynamic properties for variations of external cavity length In this section we consider the effects of the external cavity length on the dynamics. The external cavity length plays an important role for the chaotic
1, w 4]
Chaotic dynamics in semiconductor lasers with optical feedback 12
.
8 t(a) . |
==
".'.
.
.
.
I
9 9
o,,
~ 9
4 i ~ " * * * "* ~ .,
O
.
o
.
.
" ."
..
! .
~
". 9
~ ~ ~.n, . . ...
4.20000 0.5
.
47
~
]
~ : , * . " .-.
9 8~ ' . . . . . .
4.20005
4.20010
4.20005
4.20010
0 -0.5 4.20000
External Cavity Length (cm) Fig. 4.5. Chaotic bifurcations and maximum Lyapunov exponent for small change of the external mirror position. (a) Chaotic bifurcations. (b) Maximum Lyapunov exponent. J = 1.3Jth, rext = 7%, and ~. = 780 nm.
dynamics of semiconductor lasers. There are several important scales for the change of the external mirror position in the dynamics. Chaotic dynamics occur for very small changes of the external cavity length, compatible with the optical wavelength ~ (Ikuma and Ohtsubo [1998]): the laser output shows periodic undulation (period 89 and it contains a chaotic bifurcation within the period. In actuality, there is also a hysteresis either for an increase or decrease of the external cavity length. Figure 4.5 shows a bifurcation diagram and the maximum Lyapunov exponent for small variations of the offset mirror position of 4.2 cm. The maximum Lyapunov exponents are almost always positive, and the laser shows chaotic bifurcations with the change of the external cavity length. Since the offset position of the external mirror is within the distance corresponding to the relaxation oscillation frequency, the laser tends to destabilize at the rather higher external mirror reflectivity of 7.0%. For large external feedback reflectivity, a semiconductor laser sometimes oscillates in a multimode. In that case, we observe periodic undulations with periods not only of 2 ~ but also ~1~, l~, and so on, for a small change of the external cavity length. Which period occurs in the laser output power depends on the absolute external mirror position relative to the laser facet. The effects of this small change of the external cavity length occur in every offset of the mirror position as far as the feedback field is coherent. The second case is still for an external mirror positioned within the distance
48
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 4
equivalent to the relaxation oscillation frequency (on the order of several centimeters), but with mirror displacements on the order of millimeters. In this region, the coupling of the internal field with that of the external mirror is strong (the C parameter is small and the number of modes excited is small) and the laser shows stable oscillation. The internal and external cavities couple strongly with each other and we require much stronger optical feedback to destabilize the laser in this region. This sometimes leads to characteristic features in the chaotic dynamics. For example, LFFs occur irregularly in time for large C, while periodic LFFs are observed under strong optical feedback at a high injection current for small C (short external cavity length) (Heil, Fischer, Els~il3er and Gavrielides [2001]). This region of external mirror position is very important from the viewpoint of practical applications in semiconductor lasers, such as optical data-storage systems and optical communications. When the external mirror distance is small enough and less than the internal cavity length, the behavior of the laser oscillation is usually governed by the external mirror, and the laser behaves as if its cavity length is extended with the mirror distance. When the external mirror is positioned outside the distance equivalent to the relaxation oscillation frequency, but within the coherence length of the laser (on the order of several centimeters to several meters), the laser is greatly affected by the external optical feedback. In this region, the number of modes related to the C parameter is large and the laser shows various types of dynamical behavior at a moderate feedback rate. This region is also important for the study of fundamental dynamics and their applications, since the feedback distance and rate in most practical optical systems are of the same order. Thus, we can easily make chaos devices for various applications using these parameter conditions. Figure 4.6 presents a numerical result for the phase diagram of the external mirror position L and the external reflectivity r, showing the boundaries between different states in the bifurcations (Murakami, Ohtsubo and Liu [1997]). For example, we can see a chaotic bifurcation for an increase in the external reflectivity at a fixed external mirror position. The notable feature of this graph is that the critical reflectivity of the fixed stable state shows a periodic structure with variation of the external cavity length, and the stable area is extremely enhanced at the peak positions. The period of the stability enhancement is exactly equal to the length calculated from the relaxation oscillation frequency. The result coincides with the theoretical prediction in eq. (4.2). If the external mirror distance exceeds the coherence length of the laser (more than several meters), the laser still shows chaotic oscillations with external feedback, but these effects are induced by incoherent phenomena (Takiguchi, Liu and Ohtsubo [ 1999]).
1, w 4]
Chaotic dynamics in semiconductor lasers with optical feedback
49
Fig. 4.6. Boundaries between different states for change of the external cavity length and the reflectivity at J = 1.3Jth. Fixed: fixed point, P l: period-l, P2: period-2, and QP: quasi-periodic state.
Instabilities and chaos in semiconductor lasers can be induced not only by incoherent feedback from the laser itself but by from optical injection from another laser source (Rogister, Locquet, Pieroux, Sciamanna, Deparis, Megret and Blondel [2001 ]).
4.4. Dynamic properties for injection current changes The output of a semiconductor laser increases linearly with increasing bias injection current as long as the bias point is not far from the threshold. One of the notable features of optical feedback effects in semiconductor lasers is
50
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 4
threshold reduction (Pan, Shi and Gray [ 1997]). The laser threshold is lowered by the external optical feedback by as much as 10% and the reduction rate depends on the feedback fraction. From the reduction rate, we can estimate the amount of external reflectivity. The other effect in the presence of optical feedback is mode hopping in the L - I characteristics for changes of bias injection current (Fukuchi, Ye and Ohtsubo [1999]). The relation between the injection current and the oscillation frequency is given by
~o0 = ~Oc
Ooooj OJ '
(4.3)
where r is the angular frequency of the laser oscillation at the reference bias injection current and O~oo/OJ is the conversion efficiency from the injection current to the laser oscillation frequency. According to eq. (4.3), successive external modes are selected for increasing injection current. When the mode hops from one mode to another, the laser power jumps to another state. Figure 4.7a shows the numerical calculation of the L - I characteristic in the presence of optical feedback. Each output power jump corresponds to the mode jump calculated from the external mode frequency through the relation (4.3). The conversion efficiency from the injection current to the frequency in ordinary Fabry-Perot lasers is on the order of GHz/mA, so that a mode hop occurs for about every 1 mA of injection current at an external cavity length of ~15cm. Inbetween mode hops, changes of bias injection current lead to chaotic bifurcation depending on the external mirror condition. Figure 4.7b shows bifurcations of the output power. Periodic bifurcations of the output power between two successive jumps are visible in the figure. In reality, the L - I characteristic in the presence of external optical feedback has a hysteresis. In numerical simulations, the L - I characteristics are for a step response and the calculation is for the increase of the injection current. The chaotic scenarios were also investigated experimentally, and the chaotic bifurcations were observed. It is a well-known fact that the laser oscillation is stable for a higher bias injection current. Therefore, a larger optical feedback strength may be required to destabilize the laser at a higher bias injection current.
4.5. Low-frequency fluctuations
Low-frequency fluctuation (LFF) is one of the laser instabilities in the presence of optical feedback as discussed in w3.5. The typical features of LFFs are sudden power drops and gradual power recovery processes. Each LFF has almost the
Chaotic dynamics in semiconductor lasers with optical feedback
1, w 4]
1.50
I
I
'
'
I
51 |
' -.
-
..
..... ~ ~" 9
..
(a) 9
9.
9 all
m ee I
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~D
....'....
1.40
O
9 ."
9
9 -....
.-'-.-" -;
.. =.. .. 9 .
,...-
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.=-
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,
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,
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Injection current J]Jth
(b) m
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.
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-
9
~, :,
:r ,~
4. 94 .,..
I,,.,. ,='i"
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ii I
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...., l tlllllll
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9
~...
! ::
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:." 4 'G :.,.
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":'
" ;
~-
- :'" 9
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9
.' "
..
::.
.~]
llt i llt)' i.,dllt i ,,.Iti" 9 ,
'hill, ll',li" I'".it. l'l',.J , "ill" '1
.-
.;
"Jill
1.32
il . " i"
,,
",i,"l 411 ; -,1( i I,.
,l.,,
,,
1.34
Injection current J/Jtk Fig. 4.7. L-I characteristics and chaotic bifurcations in the presence o f optical feedback at L = 15 c m and r = 1.0%. (a) L-I characteristics. External m o d e s are successively selected for increasing injection current. (b) Chaotic bifurcation within each external mode.
52
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 4
same structure, and the external cavity length plays an important role in the power recovery process after the power dropout. In the power recovery process, the dwelling time of each step is equal to the time calculated from the external cavity frequency. LFFs we usually observe are in time series obtained from a low-pass filtered waveform, however LFFs are composed of irregular pulse trains much faster than a nanosecond as discussed in w3.5. The pulsation occurs due to a fast transition or excursion between one of the external modes and the accompanying anti-mode. Up to now, several physical models for LFFs have been proposed: the noise-driven model (Tromborg, Mork and Valichansky [1997], Eguia, Mindlin and Giudici [1998], Mork, Sabbatier, Sorensen and Tromborg [1999]), the deterministic model of saddle node instability (Sacher, Els~il3er and G6bel [1989], Sacher, Baums, Panknin, Elsfil3er and G6bel [1992]), multimode competition in laser oscillation (Huyet, Balle, Giudici, Green, Giacomelli and Tredicce [1998]), and the crisis between two attractors (van Tartwijk, Levine and Lenstra [1995]). However, LFFs have been experimentally observed for a single-mode laser and they have been simulated for the theoretical model of a single-mode laser without noise. Therefore, LFFs originate from a saddle node instability of the system and one of the intermittent routes to chaos. The frequency of occurrence of LFFs depends on various parameters in the feedback system (Takiguchi, Liu and Ohtsubo [1999]). For example, the frequency of LFFs decreases with increasing C parameter. The C parameter is proportional to the external cavity length and the feedback coefficient, so the frequency decreases with increases both in the external cavity length and in the feedback reflectivity. Namely, the frequency of LFFs decreases with increasing number of linear modes excited by the feedback. The relation is easily verified by numerical simulations and is also observed experimentally. The frequency is also dependent on the bias injection current, increasing with increasing injection current. LFFs are, at first, thought to occur only near the solitary laser injection current. It has been reported by Pan, Shi and Gray [ 1997] that LFFs occur everywhere along the boundary separating feedback regimes IV and V introduced in w3.2. Figure 4.8 shows the possible area for the occurrence of LFFs. Due to the low internal reflectivity of the laser cavity, LFFs occur not only for low injection current close to the laser threshold, but for higher injection current as well. For certain device parameters of semiconductor lasers, the laser threshold is reduced by the external feedback, but the slope efficiency becomes small and the laser power is low compared with that of the solitary laser at a higher injection current. In such cases, LFFs occur at a high injection current and the laser power shows jumps and no dropouts by the occurrence of LFFs. Therefore, LFFs are universal
Chaotic dynamics in semiconductor lasers with optical feedback
1, w 4]
0
9
,
,
53
,
Regime V
133 v
-10
n" -15 ~,/"-..,.,, TFd ~;;alm e -20
.0
1.5
Regime IV
2.0
2.5
3.0
I/Ith Fig. 4.8. LFF regions between regimes IV and V at L = 60 cm. The horizontal axis is the injection current and the vertical axis is the ratio of the feedback light. The external cavity length is 60 cm. Symbols denote experimental results, solid lines theoretical results. (After Pan, Shi and Gray [ 1997]; 9 1997 OSA.)
phenomena observed in semiconductor lasers with optical feedback as saddle node instabilities involved in the nonlinear systems.
4.6. Dynamics in phase-conjugate feedback Due to experimental difficulties, only few papers have reported experimental investigations of the dynamics in semiconductor lasers with phase-conjugate optical feedback (Ktirz and Mukai [1996]). However, phase-conjugate feedback has some merits. For example, an externally reflected light is automatically fed back into the semiconductor laser without any additional optical components in the optical path, so that an optical feedback device in semiconductor lasers is easily constructed. Another merit is the phenomenon of phase locking by phase-conjugate feedback. Owing to the phase locking, the laser becomes insensitive to small variations of the external mirror position compatible with the optical wavelength. The dynamics of phase-conjugate feedback are theoretically described by eqs. (3.42)-(3.45). The effects of phase-conjugate feedback are quite different from those for conventional optical feedback. One difference is the stability enhancement in the phase space of the external reflectivity and the external mirror position. The stability condition for phase-conjugate feedback is also calculated from the linear stability analysis and is given by (Murakami, Ohtsubo and Liu [1997]) -
= -
Tin
+ 2 a ) - -
COS2q~s tan(89~or).
(4.4)
54
Chaotic dynamics in senliconductor lasers with optical feedback
[1, w 4
3.5
i 2.5
II I" I, I, I" ' I
1.5 1 0.5 .
0
.
.
.
I
5
.
.
.
.
I
.
.
10
.
.
I
15
.
.
.
.
I
20
L [cm] Fig. 4.9. Calculated boundary between stable and period-1 oscillations at J = 1.5Jth. The solid and dashed curves indicate the boundary for a phase-conjugate mirror and a conventional mirror, respectively.
This stability condition corresponds to that in eq. (4.1) for conventional optical feedback. We also obtain a periodic condition for the stability enhancement along the external mirror position, however the stability locations are different from those for conventional optical feedback, as easily understood from the comparison between eqs. (4.1) and (4.4). Figure 4.9 shows numerical results for the boundary between stable-state and period-1 oscillations. The solid line shows the result for phase-conjugate feedback, while the dashed line is for conventional feedback as already shown in fig. 4.6. Periodic stability enhancement is observed with a period equal to the length corresponding to the relaxation oscillation frequency of the solitary laser, but stability peaks for phase-conjugate and conventional mirrors are located alternately with each other. Actually, the excited mode frequency in the laser output power in conventional feedback is linearly proportional to the theoretically calculated external mode frequency with a slope of unity. However, in the phaseconjugate case, the excited mode frequency is less than that for the conventional feedback, even if it shows a periodic undulation for a change of the external cavity length. The period is again equal to the relaxation oscillation frequency of the solitary laser (Murakami, Ohtsubo and Liu [1997]). There are several subjects for the study of the dynamics in semiconductor lasers with phase-conjugate optical feedback. One of them is the dynamics
1, w 4]
Chaotic dynamics in semiconductor lasers with optical feedback
55
for phase-conjugate feedback generated from non-degenerate four-wave mixing. In that case, one more parameter, i.e. the phase mismatch, is introduced and the system shows somewhat different dynamics compared with the case of degenerate four-wave mixing (van Tartwijk, van der Linden and Lenstra [ 1992], Gray, Huang and Agrawal [1994]). Another issue in phase-conjugate feedback is the time response of a phase-conjugate mirror. As the time scale of the dynamics in a semiconductor laser with optical feedback is usually a nanosecond or faster, the effect of finite time must be considered in the dynamics for a phase-conjugate mirror with a slow time response (DeTienne, Gray, Agrawal and Lenstra [ 1997], van der Graaf, Pesquera and Lenstra [ 1998]). For much slower time response of the phase-conjugate mirror, the time-dependent features in the dynamics, such as phase locking, are lost. For example, the time response of a photorefractive phase-conjugate medium, such as a photorefractive crystal, is so slow compared with the time fluctuations of a semiconductor laser that a grating formed in a photorefractive crystal can be considered a static grating (Murakami and Ohtsubo [1999]). Once the grating is formed in a photorefractive crystal, the dynamics are only governed by the total feedback loop of the pump beam as an external cavity length. As a result, the dynamics are completely the same as those for a conventional mirror feedback except for the generation of a spatial phaseconjugate wave. The dynamics in such photorefractive phase-conjugate feedback are quite different from those for a fast-response phase-conjugate mirror.
4. 7. D y n a m i c s in m u l t i - m o d e oscillations
Even a semiconductor laser operating in a single mode sometimes oscillates at multimodes in the presence of optical feedback. Also, some semiconductor lasers are intrinsically multimode in steady state, since semiconductor materials have broad gain profiles. Here, we consider the effects of optical feedback for multimode semiconductor lasers. In the presence of optical feedback, the rate equation for the complex field is written from eq. (2.62) as (Ryan, Agrawal, Gray and Gage [1994]) dEj(t) _
dt
89 - i a ) G . { n ( t ) - nth } Ej(t) 1
2
(
~
2
M
Ej(t)t + ~ 0,,,/IE,,,(t)l 111 =
~j
+ --Ej(t Tin
]
- r) exp(ie)j r),
2
)
Ej(t)
(4.5)
Chaotic dynamics in semiconductor lasers with optical feedback
56
6.0
. . . .
,
,
0 . . . o l o eeol o , " .,:.::.,;.:.::.. :',
,
9 ..... : . . . . . 9
3.0
o-"
..................
9 oooo~
..... ,,,,,..:.,~,.~.ttllljll I 9
.or. ~ l . %'4
Ooo."
"e
e~
9 l, O9 0 o
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:'t
.:i 9
,,
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.... ": ..r"-I;: :,:
oeooO
I
9 9
O
IJ
~,...:.l.o _.." OOlooUl,....l:O.~:
o
[ 1, w 4
:
IJ el
.i:.
,,I .-.
~ .:..'..,:q.
i.OOl-ooO ~ . . . . . . .
9
_O_'~ I I d . ' ~ l l . o J l O l . ]
'
........ :...... :,,,-......:..,,' ...... I.IIi,l!'Hgitl,,. 15 ...... ::t~,~;i~:i'::'l" "~:.';::l=!~lt ,':
9
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i
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(a) Total
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l 6.0
3.0 9 (b) Mode 0 0.0
. . . . . . . . . . .
,. . . . . . . . . . . . . . . . . . . . . . .
. _ , . ,
1 6.0
.......~':':-'---:
,
el:o ~
0
9 9
:
e~
~~ 1 7 6~ 9 8 Ooo ..Io
%:1 " 9
.o ""
|o
I..
ili,ll.' ..!..!::.-;ii . . . . :: "".:;'"":.',,'~'t' :"l .... "'::::1;I l.I;;I, -, ~,:! "
3.0
9
,,,=a
9
il"iltl'"ii
o.
o
:~ o,..o. Oeo oo eo 8 ~ i e o eo
0.0
i
I
oe
(c) Mode 1
coo
I.
,
1 6.0
'
E
'
'
................. 9......
3.0
'
I
..... ,mi,'Si.'llllllli:l.,
!.
"'":~!~ .'+'.~.-',~';
-- . . . . . . . . ..... ..:..
"I,;I.,:'.,' . : , , , . . . 9 ::..! ! s
..... :;... 9 :-:
9
2
"i:,
9 -~,l~,i,::::::l :....
0.0
J
0
i
(d) Mode 2 t. . . . . . . . . . . . . . .
1 Reflectivity (%)
9 .......
i
.....
2
Fig. 4.10. Chaotic bifurcations for multi-mode semiconductor laser with optical feedback at J = 1.4Jth and L = 3 cm. Mode 1 is the main mode and mode 0 corresponds to the oscillation for the lowest wavelength.
1, w 4]
Chaotic dynamics in semiconductor lasers with optical feedback
57
where toy and o)j are the feedback coefficient and angular frequency for the jth mode, respectively. The rate equation for the carrier density is the same as eq. (2.63). We show an example of the effects of multimode oscillations in fig. 4.10, resulting from numerical calculations of bifurcations as a function of the external mirror reflectivity. We assume that the laser oscillates in three modes, with mode 1 the main mode. The powers of the laser oscillations for the side modes are assumed to be the same as the solitary oscillations. With the optical feedback, the total power of the laser shows a chaotic bifurcation similar to that for a single-mode laser. However, mode switching occurs with an increase of the external reflectivity, while one of the modes is usually always active at a certain reflectivity. In this simulation, mode 0 corresponds to a lower wavelength of the laser oscillation. Since the gain of semiconductor materials has an asymmetric profile, mode switching (as shown in this figure) is always observed in semiconductor lasers. Here we have assumed three modes for simplicity, however similar trends can be seen for multi-mode oscillations with more than three modes. Also, we have demonstrated chaotic bifurcation only for changes in the external reflectivity, but mode switching is also observed with variations in the other parameters, and it is the main characteristic in multi-mode oscillations of semiconductor lasers with optical feedback.
4.8. Dynamics of semiconductor lasers in various structures
The rate equations for other than edge-emitting semiconductor lasers have been introduced in w2.5. These lasers are intrinsically unstable in solitary oscillation because of their additional degrees of freedom compared with edge-emitting semiconductor lasers, e.g. spatial modes or polarization modes. Optical feedback effects in these lasers are also important, both because of their physical dynamics and for applications. However, these lasers have been developed only recently, and their dynamic behavior has not been thoroughly studied up to now. In the presence of optical feedback, the same feedback term as in eq. (3.1) is added to the rate equations, and the dynamics are numerically studied by the rate equations. Here we briefly review some of the effects. Self-pulsating semiconductor lasers are used as sources for optical data storage systems as discussed in w2.5. By pulsating oscillations in the laser, feedback noise is greatly reduced under certain optical configurations owing to the large compulsive pulsation amplitude. However, for some other conditions the laser loses its stability and shows chaotic behavior. In chaotic oscillations in selfpulsating lasers, the laser still shows periodic pulsations but with irregular
58
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 4
amplitudes or occasional time jitters together with pulsations with irregular amplitudes. The RIN of self-pulsating lasers in the presence of optical feedback has been studied by Yamada [1998]. Without optical feedback, the RIN was very low in the lower frequency regions (below 1 GHz). In the presence of optical feedback, the RIN decreased below the allowed level at a certain external cavity length. However, the RIN increased to as high as about -90 dB/Hz for other external cavity lengths, making the laser unsuitable as a source for optical data storage systems. VCSELs also show instabilities even without external optical feedback, since the polarization and spatial modes are excited simultaneously. In spite of the high reflectivity of the internal cavity in a VCSEL, further instabilities appear in the output by the introduction of optical feedback. The optical feedback effects were numerically investigated by Law and Agrawal [1998] and Spencer, Mirasso, Colet and Shore [1998], and chaotic bifurcations and spatial mode switching were also discussed. In chaotic oscillations, each oscillation mode has a large RIN, but the RIN of the total intensity is greatly reduced relative to those for individual modes, which is always the case for multimode semiconductor lasers. Broad-area semiconductor lasers also show spatial dynamics as their original characteristics. Filamentation of a near-field intensity pattern is a typical feature of the spatio-temporal dynamics of broad-area semiconductor lasers in the absence of optical feedback (Fischer, Hess, Els~iBer and G6bel [ 1996]). Another feature of a broad-area laser is the oscillation intensity with twin or multiple peaks in the far-field pattern. Even for a single-mode model, broad-area lasers show various dynamics depending on the laser parameters such as the bias injection current. The study of the effects of external feedback in broad-area semiconductor lasers is now on its way, however little is known about their dynamics since broad-area semiconductor lasers are relatively new devices and both numerical simulations and experiments are not easy to conduct. The characteristics of the beam emitted from a broad-area laser, for example the beam profile, the coherence, and the longitudinal mode, are not always suited for certain applications, such as a source for the excitation of solid-state lasers. Optical feedback or external optical injection from another laser is expected to permit beam shaping of the laser for the development of high-quality broad-area lasers. A constant stripe width along the cavity is not the only possible structure for broad-area lasers: various structures are being developed, for example flared lasers that have a tapered structure (Levy and Hardy [1997]) and stadium lasers that have ellipsoid shapes (Fukushima, Biellak, Sun and Siegman [ 1998]); optical feedback effects are very important issues in these lasers for the future. Coupled semiconductor laser arrays are different from broad-area semiconduc-
1, w 5]
Some applications of semiconductor lasers with optical feedback
59
tor lasers. We introduced the rate equations for these lasers in w2.5; however, when a strong interaction between the neighborhood lasers exists and the separation of the lasers is small, the rate equations used for a broad-area laser can be applied. In this case, we introduce a laser model with periodic active areas and an infinitely small stripe width instead of an extended broad active area. The dynamics of laser arrays in the presence of optical feedback have been investigated, while dynamic spatio-temporal patterns have been observed in the system as well (Mtinkel, Kaiser and Hess [ 1996], Martin-Regaldo, Balle and Abraham [1996]). Filamentation is also a typical feature of the dynamics in semiconductor laser arrays with and without optical feedback. Coupled laser arrays with only two elements can induce space- and time-dependent complex structures in the laser output power.
w 5. Some applications of semiconductor lasers with optical feedback
5.1. Control of chaos in semiconductor lasers with optical feedback Physical models encountered as real systems are more or less nonlinear; nevertheless, linearization techniques are frequently applied to these systems ,and only the linear parts are considered for convenience. Therefore, chaos induced by nonlinear effects is an unfavorable phenomenon and we avoid it in practical applications. However, chaos is controllable, and the first paper on chaos control was published by Ott, Grebogi and Yorke in 1990. They proposed an algorithm (OGY method) which applies appropriately estimated minute perturbations to an accessible system parameter to select and stabilize a certain periodic orbit (unstable periodic orbit). This idea indicates that a chaotic system can be turned into a system with multi-purpose flexibility, meaning that one can obtain various desired orbits in a simple system without dramatically modifying the configuration of the system. This method is called chaos control. We will not discuss the details of the OGY algorithm, since it is difficult to apply it to real experimental systems. The method comes is rather mathematically based and is applicable only to experimental situations where one knows explicitly the exact parameter values in the dynamical systems, since the parameter values are essential for calculating unstable periodic orbits in the systems. A fundamental problem existing in the OGY algorithm is the applicability to the control of high-dimensional chaotic systems. Although some attempts have been made to adapt this technique to the experimental control of highdimensional dynamics, the requirements for the knowledge about the attractors
Chaotic dynamics in semiconductor lasers with optical feedback
60
[ 1, w 5
P(t) h.._ v
Chaotic system u(t)
P(t)
Delay circuit
/3o Gain P(t- r) Fig. 5.1. Chaos control system based on the Pyragas method. P(t) is the chaotic output and u(t) is the feedback signal, r is the time delay for the signal P(t) and fi0 is the feedback gain.
and their calculations obstruct the application of the algorithm to real-time control of high-dimensional chaos. As alternative methods of chaos control for the applications to real systems, several chaos control techniques have been proposed. One of them is continuous control, proposed by Pyragas [1992]; fig. 5.1 shows a schematic diagram. In the Pyragas method, part of the output in a nonlinear system is detected with a delay that matches the intrinsic period of the chaotic attractor. The difference between the present and delayed outputs is fed back into the original system. This characteristic time can be calculated not only from the theoretical model but also from the experimental estimation. For example, choosing the injection current as a control parameter in a semiconductor laser with optical feedback, the control signal is given by (Naumenko, Loiko, Turovets, Spencer and Shore [1998]) J=Jb
1+
tiP(t- Po P(t)} r) -
(5.1)
where Jb is the bias current, fi is the feedback gain, and P and P0 are the instantaneous and averaged output powers, respectively. As mentioned above, r is the delay time and is chosen to be near or equal to the response time of the system. When the system is controlled, the amount of feedback reduces to zero and the output is stabilized to a periodic or fixed state that corresponds to one of the unstable periodic orbits involved in the nonlinear system. Another powerful method for application in experimental systems is Occasional Proportional Feedback (OPF) (Hunt [1991], Roy, Murphy, Maier, Gills and Hunt [1992], Liu and Ohtsubo [1994]). The OPF method modifies the
1, w 5]
Some applications of semiconductor lasers with optical feedback
61
OGY algorithm in such a way that extensive calculation for the control is unnecessary. The OPF method also perturbs a system control parameter by carefully feeding back part of the output signal. It creates only small alterations in the attractor and gives rise to the stabilization into periodic orbits. Since OPF is accomplished with an analog technique, it can be performed very fast and is applicable to a variety of nonlinear systems. It is also pointed out that the OPF method is essentially a limited case of the OGY algorithm, when the contracting direction of the chaos attractor is infinite in strength. In laser systems, control of chaotic oscillations has been successfully performed based on the OPF method (Gavrielides, Kovanis and Alsing [1993]). Chaotic oscillation in a semiconductor laser has a typical time scale below the nanosecond range, closely related to the several GHz range of relaxation oscillation of the solitary semiconductor laser. The actual time control of chaos for such fast signals is not an easy task, since very high-speed electronic circuits are usually required for processing. Here we discuss another method of chaos control that is very simple and suitable for practical use. In w3.4 we discussed the linear stability analysis for semiconductor lasers with optical feedback. In that analysis, we examined linear modes in laser operation. These modes are the candidates for unstable periodic orbits in the nonlinear system. Therefore, one may stabilize a chaotic oscillation to a periodic state by applying a small perturbation to a chaos parameter. The perturbation may contain a frequency corresponding to one of the mode frequencies as far as the original state is not too far from the periodic state (Liu, Kikuchi and Ohtsubo [1995]). The easiest way to apply this method is sinusoidal modulation of the injection current. Choosing an appropriate frequency J0 from the mode analysis, the frequency modulation with a small modulation coefficient m for the injection current is given by J = Jb { 1 + m s i n ( 2 ~ t ) }.
(5.2)
However, the modulation does not always work for all mode frequencies. Whether or not control is achieved depends on the extent of attractor basin for the control frequency and the modulation depth. Figure 5.2 shows an example of chaos control in a semiconductor laser with optical feedback. The chaotic waveform in fig. 5.2a, due to the presence of optical feedback, is controlled to a period-1 oscillation in fig. 5.2b by modulation of one of the mode frequencies. The modulation depth is only 2.1% of the bias injection current. The method is robust and a finite modulation range of the parameter value exists for successful control; for example, control is achieved within a range of several tens to a hundred MHz around the chosen frequency.
Chaotic dynamics in semiconductor lasers with optical feedback
62
t
'
'
'
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'
'
[ 1, w 5
(a)
'
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~
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,
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--4
(b)
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,-,
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time
i
i
,
,
,
I
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t (ns)
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t_,
.~
O.5
Z
0
0.5
IE o (t)l 2
1
0
0.5
i
1
IE o (t)l 2
Fig. 5.2. Numerical simulations of chaos control at J = 1.1Jth, L=25.5cm, and text = 1.5%. (a) Original chaotic laser output. (b) Output upon controlling the injection current modulation. The control frequency is 1.25 GHz and the modulation depth is 2.1% of the bias injection current. (c) and (d) are the attractors corresponding to (a) and (b), respectively.
1, w 5]
Some applications of semiconductor lasers with optical feedback
63
However, the extent of the attractor (the amplitude of the oscillation) after the control changes slightly depending on the modulation frequency and amplitude. Low-Frequency Fluctuation (LFF) is one typical instability of semiconductor lasers with optical feedback. Typical time scales exist in the optical feedback system and the external mode is usually very close to one of the linear modes derived from the linear stability analysis. Here we discuss the control of LFFs by injection current modulation with a frequency close to the external cavity mode. For modulation detuned from the external mode frequency, the laser output under certain parameter conditions shows power dropouts due to LFFs. However, unstable oscillations with LFFs were experimentally stabilized to a synchronous oscillation with an external mode frequency (Takiguchi, Liu and Ohtsubo [1998, 1999]). The modulation frequencies were not exactly equal to the external mode frequency estimated from the external cavity length. The range of synchronous oscillation was very small at +3 MHz for a modulation frequency of 450 MHz in that experiment. The frequency of LFFs by the modulation increased with increasing detuning from the synchronous oscillation frequency of the external mode. The modulation depth in that case was o n l y - 5 dBm and the modulation was considered the same technique as the chaos control for modulation of one of the linear modes involved in the nonlinear system.
5.2. Noise suppression based on chaos control
The main noise source in free-running semiconductor lasers is the spontaneous emission of photons in laser media. Noise in semiconductor lasers is greatly enhanced by optical feedback. A detailed definition and description of noise characteristics in semiconductor lasers can be found in the book by Petermann [1988]. It is useful to introduce the Relative Intensity Noise (RIN) relating the noise in the optical power 6S to the mean power (S) according to R I N - (6S2) (s)
(5.3) '
where the optical output power from a laser is defined by S(t) = (S) + 6S(t). In actuality, the feedback-induced irregular intensity fluctuation is not noise but a chaotic fluctuation. However, the effects of this phenomenon are similar to those of noise in free-running semiconductor lasers. Therefore, we use the same notation for the feedback-induced irregular fluctuation. Without optical feedback, the RIN for ordinary edge-emitting semiconductor lasers in solitary oscillation is less t h a n - 140 dB/Hz. However, in the presence of optical feedback,
64
[1, w 5
Chaotic dynamics in semiconductor lasers with optical feedback ........
-100
I
........
J
........
I
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-
I\|
-
/
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. . . . .
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lO-a
,
,
~
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Illl[
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~oo
f r e q u e n c y (GHz) Fig. 5.3. Numerical result for noise suppression by chaos control at J = 1.3Jth, L = 15 cm, and rex t =2.5%. The modulation frequency is taken to be 2.38 GHz and the modulation depth is 3.46%. Solid line: solitary laser; dashed line: with optical feedback; and dotted line: upon control.
the RIN increases to much higher than - 120 dB/Hz in regions III-V (chaotic and coherence collapse regimes) as discussed in w3.2. A laser with the RIN around the above value cannot be used as a light source for optical data storage systems because of the increase of bit-rate errors. The noise characteristics are dependent not only on the reflectivity of the external mirror, but also on other system parameters such as the bias injection current and the position of the external mirror. Optically induced instability was stabilized by the introduction of a sinusoidal modulation to the injection current as shown in w5.1. Figure 5.3 shows a numerical result for noise suppression by the proposed method (Kikuchi, Liu and Ohtsubo [1997]). The solid line in fig. 5.3 is the RIN of a semiconductor laser in solitary oscillation. The RIN is about -140 dB/Hz in the lower frequency region. The peak at about 3 GHz is the relaxation oscillation component. In the presence of optical feedback, the noise is extremely enhanced to about -120 dB/Hz in the lower frequency region as shown by the dashed line in the figure. The dotted line shows the result of control. One of the mode frequencies (2.38 GHz in this case) is chosen as the control frequency and the laser is
1, w 5]
Some applications of semiconductor lasers with optical feedback
65
modulated by this frequency through modulation of the injection current. By the modulation, the laser output shows synchronous oscillation (period-I) and the RIN in the lower frequency region is reduced to - 130 dB/Hz. The modulation amplitude is m = 0.15. In reality, high-frequency injection current modulation has been employed to suppress feedback-induced noise in semiconductor lasers in optical data storage systems on an empirical basis (Gray, Ryan, Agrawal and Gage [1993]). However, the modulation depth is much larger than that of the chaos control and the laser is sometimes brought below the threshold by the modulation. The modulation frequency is also determined empirically in those systems. The RIN induced by optical feedback was reduced by the method of chaos control and the noise level was lowered as much as 10dB/Hz by the experiment (Ye and Ohtsubo [1998]). The problem here is how to determine the modulation frequency appropriately in actual experimental systems, since the exact device parameters for the semiconductor laser in question must be specified in advance. Recently, a self-pulsating laser was used as a light source for a DVD in optical data storage systems. The chaos control algorithm introduced here may give us important information for the design of such a device. The essence of chaos control is that the control does not change the original dynamics of the nonlinear system. However, the original dynamics may be changed due to a very small but a non-negligible modulation amplitude. In that case, the idea of chaos control is still effective for the control of an existing unstable periodic orbit as long as the modulation is small.
5.3. Chaos synchronization in semiconductor lasers with optical feedback Since the first prediction of chaos synchronization by Pecora and Carroll [ 1990], synchronization of chaotic oscillations between two nonlinear systems has been reported in various fields of engineering. In chaos synchronization, two similar nonlinear systems showing chaotic dynamics are prepared: the transmitter and receiver systems. The transmitter system is divided into two sub-systems (driving and response systems), but the receiver only consists of one of the sub-systems (response system). Without signal transmission from the transmitter to the receiver, the Outputs from the two systems never show the same waveform since chaos is very sensitive to the initial conditions of a system. On the other hand, when a chaotic output from the driving system in the transmitter is sent to the receiver, the output from the receiver synchronizes with the transmitted signal under certain conditions of the system parameters, thus chaos synchronization is performed. The transmitter must be a chaotic system, however the receiver may
66
Chaotic dynamics in semiconductor lasers with optical feedback
[1, w 5
or may not be an intrinsically chaotic system (Pecora and Carroll [1991]). As long as the conditional Lyapunov exponents between the transmitter and receiver systems have negative values, we can achieve chaos synchronization. Following this technique, chaos synchronization has been successfully demonstrated in various systems. However, the method is not directly applicable to laser systems, since we cannot divide the dynamics of the laser variables into subsystems. As an alternative technique in such a case, the difference between the variables in the transmitter and receiver systems can be used as a control parameter for the synchronization (Pyragas [1992], Annovazzi-Lodi, Donati and Scir+ [1996]). In the field of optics, experimental synchronization between two chaotic laser systems has also been demonstrated in solid-state lasers (Roy and Thornburg [1994]) and CO2 lasers (Sugawara, Tachikawa, Ysukamoto and Shimizu [1994]). In addition, many theoretical studies on the synchronization of chaotic oscillations in semiconductor laser systems have been published. In delay-differential systems, there are solutions for the condition of complete chaos synchronization in which the dynamics of the transmitter and receiver systems can be described by equivalent rate equations (Ahlers, Parlit and Lauterborn [1998]). There is also another possibility for synchronization in laser systems. It is well known that a laser shows synchronous oscillations by a technique based on optical injection locking phenomena under a master-slave configuration (Fujino and Ohtsubo [2000]). Here the synchronization scheme is completely different from that for complete chaos synchronization. The two synchronization schemes can be easily distinguished for delay-differential systems. Recently, experimental chaos synchronization in semiconductor lasers has been reported (Fujino and Ohtsubo [2000], Fischer, Liu and Davis [2000], Sivaprakasam, Shahverdiev and Shore [2000]), however, most experimental results of chaos synchronization were based on optical injection locking phenomena under a master-slave configuration of transmitter and receiver lasers, the so-called generalized synchronization of chaotic oscillations (Rulkov, Sushchik and Tsimring [ 1995]). A few experiments were for complete chaos synchronization (Sivaprakasam, Shahverdiev, Spencer and Shore [2001 ], Tang and Liu [2001 a], Liu, Takiguchi, Davis, Aida, Saito and Liu [2002]). Here we consider chaos synchronization in semiconductor laser systems with optical feedback. The models under consideration are shown in fig. 5.4. We prepare two semiconductor lasers having very similar device characteristics as light sources. Three types of synchronization systems, with unidirectional and mutual coupling, are shown. In fig. 5.4a, the transmitter is a chaotic system consisting of a semiconductor laser with optical feedback. The system is chaotic under certain parameter conditions. On the other hand, the receiver
1, w 5]
Some applications of semiconductor lasers with optical feedback
67
Fig. 5.4. Schematic diagram of chaos synchronization systems in semiconductor lasers with optical feedback. (a) Unidirectional optical injection system. (b) Symmetric system with unidirectional optical injection. (c) Mutual coupling system.
is an intrinsically stable solitary laser. With a chaotic signal injection from the transmitter to the receiver, the receiver synchronizes with the transmitter under appropriate conditions. In fig. 5.4b, the transmitter and the receiver are identical semiconductor lasers with optical feedback. Again, a chaotic signal from the transmitter is injected into the receiver laser and then the receiver laser synchronizes with the transmitter laser. The system in fig. 5.4a can be considered to be a special case of the system in fig. 5.4b as described in the following. Figure 5.4c shows a mutual injection system, where each laser is coupled with the other laser. The system in fig. 5.4c is the same as that in fig. 5.4b, but with the isolator removed. As a result, the injection directions are mutual. Chaotic synchronization is also expected in this system under appropriate parameter conditions. A mutual system behaves differently than unidirectional
68
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 5
optical injection systems (Fujino and Ohtsubo [2001 ]); it has interesting features similar to chaotic systems, however in the following we treat unidirectional injection systems as examples of chaos synchronization. The systems are described by the following set of rate equations for the laser fields E t , r, phases (/gt, r and carrier densities r/t, r (subscripts t and r stand for the transmitter and receiver lasers): Transmitter: dEt(t) 1 dt - ~G, t{nt(t)-nth, t}Et(t)+ '
}
dqgt(t)-latGn't{nt(t)-nth"
dt
dnt(t) dt
2
_
gt
ed
nt(t) rs, t
tCt E t ( t - rt)cos0t(t), Tin, t
tot E t ( t - rt) sin 0t(t), Et(t--------~
rin, t
t - ~
G,,,t {nt(t) - n0,t } Et(t)l 2,
0t(t) = 600, t l" + 0 t ( t ) - q~t(t - gt),
Receiver: dEr(t) 2l G n , r { t / r ( / ) - t'/th, r } E r ( t ) dt tCr + Er(t - irr) COS 0r(t) + tfinjEt(t - re) cos ~(t),
(5.4) (5.5) (5.6) (5.7)
(5.8)
/'i rl, r
dq~r(t) dt _ 21 a r G , , ,r { n r ( t ) -
nth, r }
ICr E r ( t - Tr) - ~ ~ sin Or(t)- tCinjE t ( t - r,,) sin ~(t), rin, r Er(t) Er(t)
dnr(t)
dt
--
Jr
nr(t)
ed
rs, r
G,,,r{rlr(t)- no, r}lEr(t)l 2,
(5.9)
(5.1o)
0 r ( t ) = fO0, r r + 0 r ( t ) - Or(t - /'r),
(5.11)
~ ( t ) = ~o0,t rc + 0 r ( t ) - 0t(t -- re) + A~ot,
(5.12)
where the parameters are the same as in the rate equations for semiconductor lasers with optical feedback in {}3, except for the optical injection terms in the receiver rate equations, tCinj is the injection coefficient from the transmitter to the receiver and rc is the transmission time of light from the transmitter to the receiver. A m = co0,t- m0, r is the detuning between the two lasers. The third terms on the fight-hand sides of equations (5.8) and (5.9) represent the effects of the optical injection from the transmitter to the receiver. These equations were derived for the model in fig. 5.4b. If we put tcr = 0 in eqs. (5.8) and (5.9), the systems are reduced to the model in fig. 5.4a.
1, w5]
Some applications of semiconductor lasers with optical feedback
69
We investigate the possible solutions for chaos synchronization based on the rate equations. To synchronize chaotic waveforms in the two nonlinear systems, the mismatches between the corresponding parameters in the two systems that characterize the nonlinear systems must be very small. Therefore, we at first assume that all parameters in the transmitter and receiver lasers have the same values except for the feedback coefficients tot and tCr. Next, we can easily obtain the conditions for the two sets of equations to be described by equivalent delaydifferential equations (Ahlers, Parlit and Lauterborn [1998]): At),
(5.13)
q~r(t) = Ct(t- A t ) - m0At(mod2~),
(5.14)
nr(t) = nt(t - At),
(5.15)
tCr = tot + tqnj,
(5.16)
At = r c - r,
(5.17)
E~(t) = + E t ( t -
where r = r t = r r . Under the above conditions, the rate equations of the transmitter and receiver lasers are mathematically described by the same equations, and the receiver laser can synchronize with the transmitter laser. This is so-called complete chaos synchronization. In this case, the receiver laser anticipates the chaotic output of the transmitter and it outputs the chaotic signal in advance, with time r as understood from eq. (5.13), thus the scheme is also called anticipating synchronization (Masoller [2001]). Anticipating chaos synchronization is not only observed in higher-dimensional chaotic systems described by delay-differential equations, but it is also observed in lowdimensional continuous systems described by differential equations with more than three variables (Voss [2001 ]). There is yet another possibility of synchronization of chaotic oscillations in semiconductor lasers. An optically injected laser in the receiver system will synchronize with the transmitter laser based on the effects of optical injection locking or amplification due to optical injection. The optical injection locking phenomenon in a semiconductor laser, of course, depends on the detuning of the frequencies between the master and slave lasers, but it is usually observed for a rather large optical injection fraction of several tens of percent of its amplitude fluctuations (corresponding to an intensity injection fraction of several percents or more) for a wide range of the detuning. Under these conditions, the output of the receiver laser is synchronous with the transmitter. The relation between
70
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 5
the two laser fields at this synchronization is written as (Fujino and Ohtsubo [2000]) Er(t) cx Et(t- rc).
(5.18)
Namely, the receiver laser responds immediately after it receives the chaotic signal from the transmitter, since the time lag from the transmitter signal is always behind the transmission time rc of light from the transmitter to the receiver. The scheme is sometimes called generalized synchronization of chaotic oscillations to distinguish it from complete chaos synchronization. Most experimental results in laser systems including semiconductor lasers reported up to now were based on this type of chaos synchronization. Of course, the chaotic attractors for the transmitter and receiver lasers are the same for the case of complete chaos synchronization. On the other hand, a distinct difference of generalized chaos synchronization relative to complete chaos synchronization is the reduction, due to the optical injection, of the carrier density in the receiver laser and of the gain of the receiver laser (Murakami and Ohtsubo [2002]). Therefore, the shapes of the chaotic attractors in the transmitter and receiver lasers in the phase space of the laser output power and the carrier density differ slightly. Thus, the physical origins of complete and generalized chaos synchronizations are completely different. Figure 5.5 shows an example of the experimental results of generalized synchronization of chaotic oscillations. The experimental system is that of fig. 5.4b. Without coupling, the transmitter and receiver lasers show different chaotic oscillations, and the correlation plot between the two laser outputs spreads over the correlation plane. However, the receiver laser synchronizes with the transmitter when a small fraction of the transmitter output is injected into the receiver laser. This scheme is for generalized synchronization of chaotic oscillations. Synchronization of chaotic oscillations is observed for both a system with symmetric configuration and a system with asymmetric configuration, figs. 5.4a or 5.4b, however it seems that the allowance for parameter mismatch in synchronization is much larger in the symmetric system than in the asymmetric system. Synchronization of chaotic oscillations was also observed in systems with low-frequency fluctuation regimes (Takiguchi, Fujino and Ohtsubo [ 1999]). Possible regions for complete and generalized chaos synchronizations are investigated in the phase space of the frequency detuning and the optical injection ratio. Figure 5.6 shows the results. It is noted that the system we are discussing in the following corresponds to that in fig. 5.4a, but similar discussions can be applied to the system in fig. 5.4b. The solid curves show
1, w 5]
Some applications of semiconductor lasers with optical feedback FI,,
I
I
i
71
I
~
9
I 0
20
40 Time [nsl
i
=
-
~
.
i
u
|
I 60
i I
(b)
9 .,gl~ 9
;
o
,?. ,
9
~D 9~ ~D r r
~
,, 9
9
,
Transmitter Output Power [a.u] Fig. 5.5. Experimental result of generalized chaos synchronization for the system of fig. 5.4b. (a) Transmitter and receiver outputs. (b) Correlation plot. The external reflectivities for the master
and slave lasers are 0.93 and 0.48% in intensity, respectively. The bias injection currents for the transmitter and receiver lasers are 1.22Jth and 1.17Jth, respectively, The optical injection to the receiver laser is 4 . 5 6 % . r = 2 ns.
the boundaries between stable and unstable locking areas for ordinary optical injection locking. The dark areas show the regions of chaos synchronization. The area of complete synchronization is situated in a region of unstable injection locking with almost zero detuning with a very small optical injection ratio. On the other hand, synchronization of chaotic oscillations is recognized in a broad area in the phase space. The synchronization area is situated within region of ordinary injection locking, however synchronization of chaotic oscillations does not always occur in that region, but requires appropriate conditions for the injection ratio. The robustness of chaos synchronization for parameter mismatches has been investigated (Murakami and Ohtsubo [2002]). For complete chaos synchronization, the mismatch tolerance for each parameter including
72
Chaotic dynamics in semiconductor lasers with opticalfeedback
[ 1, w 5
Fig. 5.6. Calculated areas for complete and generalized chaos synchronization in the phase space of the frequency detuning 6f and the optical injection rate Rinj (intensity). Quality of synchronization is represented by shades of gray with error rate. The external mirror reflectivity is 1.2% and the optical injection to the receiver is also 1.2%. r = 1 ns, r c = 0 ns, and J = 1.3Jth. device characteristics is very severe, excellent synchronization is performed for zero parameter mismatch, and the accuracy of the synchronization has a symmetric form centered at zero mismatch. On the other hand, synchronization is rather inaccurate for the case o f generalized chaos synchronization. The accuracy is worse than that for the complete case, and the devation is not always symmetric.
5.4. Application f o r chaotic secure communications The study of synchronization o f chaotic lasers is very important in practical applications for optical secure communications. Either digital or analogue methods can be applied for chaotic communications (Kennedy, Rovatti and Setti [2000]). However, the system o f a semiconductor laser with optical feedback is best suited for signal-level communications. Chaos is essentially analogue in nature and is best suited for analogue modulation. Therefore, we restrict
1, w 5]
Some applications of semiconductor lasers with optical feedback
73
Fig. 5.7. Schematic diagram of chaos communications.
the discussion of chaos communications to signal level. Here, one can embed small messages into chaotic carriers. Chaos is very dependent on hardware and it is not easy to guess or analyze chaotic signals without knowing chaos keys. Therefore, we can construct very secure communicating systems. In secure communications based on chaos, a message together with a chaotic carrier is sent to a receiver as schematically shown in fig. 5.7. In the receiver system, only the chaotic carrier from the transmitter is duplicated by chaos synchronization and, thus, the message can easily be decoded. For message encoding and decoding, three typical types of synchronization schemes have been proposed in laser systems: Chaos Modulation (CMO) (VanWiggeren and Roy [1998], Goedgebuer, Larger and Porte [1998], Tang and Liu [2001b]), Chaos Masking (CMA) (Mirasso, Colet and Garcia-Fernfindez [1996], SfinchesDiaz, Mirasso, Colet and Garcia-Fernfindez [1999], Sivaprakasam and Shore [2000a,b]), and Chaos Shift Keying (CSK) (Annovazzi-Lodi, Donati and Scirb [1997], Juang, Hwang, Juang and Lin [2000], Cuenot, Larger, Goedgebuer and Rhodes [2001]). In the CMO scheme, the carrier is simply modulated by a
74
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 5
message, while it is just added to the chaotic carrier in the CMA scheme. On the other hand, in the CSK scheme, two separated states for bit sequences of a message are sent to the receiver and the corresponding states are detected based on chaos synchronization with two receivers. The robustness for communication and the mismatch allowance for system parameters are important issues in chaos communications based on synchronization of chaos (Johnson, Mar, Carroll and Pecora [ 1998]). In semiconductor laser systems, chaotic secure communications have also been investigated theoretically in the early days (Annovazzi-Lodi, Donati and Scir6 [1996]), and experimental verifications have recently been reported (Sivaprakasam and Shore [2000a,b]). Semiconductor lasers are very suitable devices for this purpose. The techniques of CMO, CMA and CSK can be applied for chaotic secure communications using semiconductor lasers under various system configurations. We can use either a complete or generalized synchronization scheme for chaos communications, though the robustness of the communication is different. As light sources, not only ordinary MQW or DFB lasers, but also other various types of lasers such as self-pulsating lasers and VCSELs are used for chaos synchronization and communications (Spencer, Mirasso, Colet and Shore [1998], Jones, Rees, Spencer and Shore [2001]). Several schemes for data transmission with chaos carriers in semiconductor lasers have been proposed for secure communications. Abarbanel, Kennel, Illing, Tang, Chen and Liu [2001] reported a chaotic modulation (CMO) scheme in a system of a semiconductor laser with opto-electronic feedback. On the other hand, the systems of semiconductor lasers with optical feedback are suitable for sources of Chaos Masking (CMA) in chaotic communications. The system considered here is the same as that shown in fig. 5.4a or 5.4b. In the chaotic masking system, a message to be transmitted is directly imposed by modulating the injection current or by external modulation using an electrooptic modulator. The message signal together with a chaotic carrier from the transmitter is fed into the receiver laser, and the chaotic output from the receiver laser is compared with the transmitted signal. The amplitude of the message must be much smaller than the average amplitude of the chaotic carrier for secure communications. It is usually less than a few percent of the amplitude fluctuations. In the receiver system, only the chaotic carrier is synchronized under certain conditions. Then the message is obtained by simply subtracting the receiver output from the transmitter output at synchronization. In the presence of a message in chaotic carriers, this selective synchronization or amplification of a chaotic attractor in the receiver system is not self-evident, and a reasonable explanation has not yet been given. The security of data transmission based on
1, w 5]
Some applications of semiconductor lasers with optical feedback 0 ,~,
75
(a)
-20
~D
o
-40 -60
9
-80 Frequency [GHz]
-20 ~D 9
-40 -60
9
-80
Frequency [GHz] Fig. 5.8. Experimental RF spectra in chaos communications. Spectra of (a) transmitter and (b) receiver outputs. The message is a sinusoidal wave of a frequency of 1.5 GHz. The bias injection currents for the transmitter and receiver lasers are J = 1.50Jth and 1.56Jth, respectively. The feedback fraction in the transmitter system is 3.75% (intensity) and the optical injection is 6.54% (intensity). r is 2.3 ns.
chaos synchronization depends heavily on the chaotic dimension of the system. Since a delay-differential system gives rise to high-dimensional chaos due to its continuous nature, a semiconductor laser with optical feedback is an excellent chaos device for such purpose. In CMA, the receiver laser only outputs the chaotic carrier which is the same as the transmitter chaos. Figure 5.8 shows an experimental example of RF spectra at chaos synchronization with a message. The result is for generalized synchronization. Figure 5.8a is the spectrum for the transmitter output. Beside the broad spectral peaks of the external cavity mode and its higher harmonics, a sharp spectral peak for the message of 1.5 GHz is clearly visible in the spectrum. On the other hand, the spectrum of the receiver output does not show any distinct spectral peak corresponding to the message in fig. 5.8a. However, the overall structure of the spectrum of the receiver output highly resembles that of the transmitter, except for the message component. Thus, only the chaotic carrier was copied in the receiver laser and the message component involved in the
76
Chaotic dynamics in semiconductor lasers with optical feedback
[1, w 5
transmitter signal was extremely suppressed. This fact is true as long as the message has a small amplitude, less than a certain level, however the physical origin of this selective amplification or synchronization has not yet been clarified and it is still an important issue in chaos synchronization and communications. A similar trend was also reported in systems for complete chaos synchronization. By subtracting the receiver chaos from the transmitted signal, we can obtain the message. It seems that the message may be extracted from the transmitter output by simply filtering the waveform with a narrow band-pass, since one can observe the spectral peak in the RF spectrum in fig. 5.8. However, the signal through a narrow band-pass filter is usually different from the exact message waveform. Of course, we obtain a sinusoidal-like signal from the band-pass filtered waveform in this case, however the phase of the signal is not always the same as the message and the filtered waveform is an imperfect sinusoidal signal. Thus, the security of the signal transmission is guaranteed. Figure 5.9 shows an example of the numerical results for Gbit/s data communications based on chaos synchronization (Sfinches-Diaz, Mirasso, Colet and Garcia-Fernfindez [1999]). The system is the same as in fig. 5.4b, and the scheme is for the complete chaos synchronization. Figures 5.9a and 5.9b show the original chaotic transmitter waveform without message and the digital signal that is sent with a maximum rate of 4Gbit/s; the signal sent to the receiver (chaotic cartier with message) is shown in fig. 5.9c. In data transmission, the propagation along the fiber connecting the transmitter and the receiver is considered. The data transmission in the optical fiber is described by the following nonlinear Schrtdinger equation (Agrawal [1989]): i _~zOE= - 89i a f E
+
02E l [~2 _~f_~ -- y E I2E ,
(5.19)
where E(z, T) is the slowly varying complex field, z is the propagation distance, T is the time measured in the reference frame moving at the group velocity, ~' is a nonlinear parameter that takes into account the optical Kerr effect, af is the fiber loss, and/32 is the second-order dispersion parameter. The transmission distance in the optical fiber is 50 km in the simulation. Figure 5.9d shows the synchronized chaotic output from the receiver. The decoded message is shown in fig. 5.9e, and that after an additional low-pass filter is shown in fig. 5.9f. After the transmission through a nonlinear optical fiber, the message can still be recovered with small distortions. In the above discussion, we assumed that both the optical feedback and the optical injection are coherently coupled with the internal laser cavities. The detuning of the frequencies between the two lasers plays an essential role for the
Some applications of semiconductor lasers with optical feedback
1, w 5]
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Fig. 5.9. Numerical example of chaos communications based on the chaotic masking method through optical fiber transmission of 50 km. (a) Chaotic output from the transmitter. (b) Encoded message sequence at a rate of 4Gbit/s. (c) Transmitter output including the message. (d) Output of the receiver. (e) Decoded message. (f) Decoded message after filtering. (After Sfinches-Diaz, Mirasso, Colet and Garcia-Fernfindez [1999]; 9 1999 IEEE.)
78
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 5
synchronization, however it is not easy to tune the frequencies for real lasers, and a frequency drift due to external conditions such as temperature drift, may destroy the synchronization. Therefore, a system based on incoherent optical feedback and optical injection has been proposed in semiconductor lasers with optical feedback (Rogister, Locquet, Pieroux, Sciamanna, Deparis, Megret and Blondel [2001]). The system is almost the same as in fig. 5.4b except for some polarization optics in the optical paths. In that system, chaos synchronization and message transmission were successfully demonstrated without closely paying attention to frequency detuning. A semiconductor laser with optical feedback can also be used as a chaotic generator for the system of Chaos Shift Keying (CSK). In CSK, two chaotic states are generated, for two values for a certain system parameter, and they are transmitted to the receiver system. Chaotic states that are generated in time according to binary message sequences are transmitted to the receiver. For example, the bias injection current is selected as a parameter and two chaotic states corresponding to two different bias injection currents are used. In the receiver system, two chaotic systems are prepared and each system responds and synchronizes with the corresponding chaotic state of the transmitter signals. Then, the message is decoded by the comparison of signals between the outputs from the two systems. Usually, chaos synchronization has a transient time to giving a waveform synchronized exactly with that of the transmitter, so that several periods of the typical cycle of the chaotic carrier are required for synchronization, and the transmission rate for messages becomes much lower than the characteristic time of the chaotic carrier. Two chaotic systems for the receiver are usually required in CSK, however the receiver may be replaced by a single chaotic system. In that case, the receiver system synchronizes only to one of the states of the transmitted signal. If a transmitted message is binary, the message can be decoded by a single chaotic system depending on the signal of the receiver output being synchronized or non-synchronized. Chaos shift keying and chaos masking are essentially different techniques, but the distinction is not clear. In fact, Mirasso, Mulet and Masoller [2002] proposed a theoretical model of CSK using exactly the same system as in fig. 5.4b. In their method, a sequence of binary codes as the injection current modulation was transmitted and the receiver laser was synchronized by either states of the binary signals. Therefore, classification of a system as CMA or CSK is based on the method of decoding purely for convenience. The upper limit of the message frequency in chaos communications is posed by the chaos carrier frequency, which in turn is characterized by the relaxation oscillation of the transmitter and receiver lasers. Enhancement of
1, w 6]
Concluding remarks
79
the modulation bandwidth in semiconductor lasers has been proposed based on strong optical injection (Meng, Chau and Mu [1998], Chen, Liu and Simpson [2000]). This enhancement is recognized in the ordinary optical injection locking region in semiconductor lasers. For a small modulation of the bias injection current, the semiconductor laser typically shows a resonant oscillation at a frequency of several GHz (the relaxation oscillation frequency). Above the relaxation oscillation frequency, the modulation efficiency rapidly decreases with an increasing frequency. Therefore, the relaxation oscillation frequency is one of the measures for the modulation performance of semiconductor lasers. Although the relaxation oscillation frequency depends on the bias injection current and the temperature, it is a characteristic parameter for a solitary laser. However, the relaxation oscillation frequency can be raised greatly by strong optical injection from an external laser source. For example, the modulation bandwidth of about 3 GHz for the solitary laser is expanded to 11 GHz with a strong optical injection ratio of 45% (field amplitude). In that case, all parameter ranges in the phase space of the frequency detuning and the optical injection are within the ordinary optical injection locking region. Such lasers can be used as sources in chaos synchronization and communications with an enhanced modulation bandwidth.
w 6. Concluding remarks In this chapter we have discussed the phenomena of feedback-induced instability and chaos in semiconductor lasers, and their recent applications based on chaos control and synchronization. Semiconductor lasers are categorized as class-B lasers which are very sensitive to external perturbations such as optical injection and optical feedback. The dynamics are important not only for basic studies in nonlinear optical systems, but also in terms of applications. The study of noise effects is still an important issue in semiconductor lasers, and the irregularity of the output power has been clarified by studying chaotic behavior. The nature of stability and instability of the output power in semiconductor lasers with optical feedback is important for system design when using them as a light source. A variety of dynamics are observed in the laser output power as shown in w167 3 and 4. These characteristics must be taken into account for practical use of semiconductor lasers as coherent light sources. Control and suppression of noise induced by optical feedback is very important in semiconductor lasers since practical optical systems more or less include light reflections from optical components. Chaotic secure communications are
80
Chaotic dynamics in semiconductor lasers with optical feedback
[ 1, w 6
another important issue in chaos applications. Chaotic code scrambling based on digital data transmission is one of the promising techniques today, and many studies are now underway (Kennedy, Rovatti and Setti [2000]). These lasers may also be good candidates for optical Code Division Multiple Access (CDMA) systems. On the other hand, the analogue modulation methods discussed here have the shortcoming of a high bit-error rate in data transmission. However, they can be used for restricted applications. The subject of improving the performance of data transmission is an important issue. We did not introduce this in our text, however internal mode selection by chaotic search algorithm (Mork and Tromborg [1990], Mork, Semkow and Tromborg [1990]) and linewidth controlling (Goldberg, Taylor, Dandridge, Weller and Miles [1982], Kfirz and Mukai [1996], Liby and Statman [1996]) are also important applications in semiconductor lasers with optical feedback. In this chapter, we have restricted the discussion to the dynamics of chaotic oscillations in semiconductor lasers with optical feedback. Chaotic effects in semiconductor lasers are not only induced by optical feedback, but also by other possible perturbations such as optical injection and injection current modulation. These are also important issues and interesting results are obtained in such systems. Recently, a variety of device structures for semiconductor lasers (selfpulsating lasers, VCSELs, broad-area lasers, laser arrays, and others) have been proposed; their dynamic behavior has been investigated as discussed in w167 2 and 4. These lasers are intrinsically unstable,and they also show chaotic dynamics for optical feedback. However, very little is known on the dynamics of these newly developed lasers in the presence of feedback, while fruitful results of the chaotic dynamics are expected. Control of the spatial and polarization modes is very important in such lasers, and also coherence control and the shaping of the beam profile are fundamental issues for these lasers. In applications of these lasers, the results obtained in w 5 provide important information. Chaos inevitably arises in practical systems since they are more or less nonlinear systems and chaos is inherent in such systems. Here, the idea of chaos control may be the best way to cope with intrinsically unstable systems as discussed in w5. Studies of the applications of chaos have started only recently, and we have shown some examples of recent research. Useful results are expected in this area since we can construct very fast chaotic systems based on the parallelism of light. The discussion here will also provide important information for other studies on instability and chaos in the field of optics, such as in microcavity systems and other nonlinear optical devices. As discussed in w 1, chaos has been studied not only in semiconductor lasers but also in other lasers, mainly in class-B lasers, and interesting results have
1]
References
81
been obtained. However, semiconductor lasers are very sensitive to external perturbations due to their device structures and the non-zero finite value of the linewidth enhancement factor a. They are also very attractive as light sources in practical applications. The chaotic dynamics of semiconductor lasers have been extensively studied for the past two decades and the studies are still progressing. The time scales of the dynamics of semiconductor lasers are usually very fast, on the order of a nanosecond, while those of other class-B lasers, such as solid state lasers and fiber lasers, are less than a microsecond. Therefore, experimental investigations of the dynamics of semiconductor lasers, especially dynamic properties of time-dependent phenomena over a nanosecond, have been performed only very recently, and many issues are left to be solved. Comparing with other existing chaotic systems, for example, analogue electronic nonlinear circuits, nonlinear solid-state lasers and other nonlinear optical devices, compact and high-speed chaos sources or chaos devices are expected using external cavity semiconductor lasers. The reason for the importance and the development of chaotic dynamics and applications in semiconductor lasers is not only for the fundamental studies of nonlinear optical systems, but also for compact chaos sources or chaos devices with easy control. We expect to see future development of the studies and applications of chaos in semiconductor laser systems. In such studies, semiconductor lasers with optical feedback will play an important role.
Acknowledgments The author would like to thank R Davis, I. Fischer, J.-M. Liu, Y. Liu, A. Murakami, R. Roy, K.A. Shore, and Y. Takiguchi for their valuable discussions. Also, thanks are extended to T. Asakura for encouragement and providing the opportunity of writing this manuscript.
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E. Wolf, Progress in Optics 44 9 2002 Elsevier Science B. V. All rights reserved
Chapter 2 Femtosecond pulses in optical fibers by
Fiorenzo G. Omenetto Materials Science and Technology Division, Condensed Matter and Thermal Physics Group, MST-1 O, MS K764, Los Alamos National LaboratoJy, Los Alamos, NM 87545, USA
85
Contents
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Femtosecond pulses-
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Observation .
w 3.
Manipulation
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Introduction
The generation of optical pulses that last tens to hundreds of femtoseconds across a wavelength spectrum reaching from the ultraviolet to the far infrared has become widespread. These laser pulses are readily available and constitute an invaluable tool to peer into the first instances of motion of fundamental processes. Furthermore, constraining the energy of a light pulse in such a short time interval allows access to unprecedented peak powers in which nonlinear optical interactions between light and matter become dominant and, at times, extreme. The transmission and use of femtosecond pulses in optical fibers holds the promise of being of the utmost utility. In the telecommunications realm, temporally shortening individual pulses from the presently and more traditionally employed picosecond time scale to the femtosecond time scale would lead to a dramatic increase in single channel capacity. The high peak power associated with ultrafast pulses also enables applications and the design of devices based on nonlinear interactions. Furthermore, the ability to deliver ultrafast, high-peakpower pulses through optical fibers has important implications for diagnostic techniques that rely crucially on pulse duration at the sample site. The task, however, is not without challenge: confining such pulses into small cores of material makes the interaction of the light with the fiber constituents become nonlinear even at very low pulse energies, affecting the pulse temporally and spectrally and introducing effects which, depending on the operational context, can be undesirable or catastrophic. Principally, when dealing with ultrafast pulses, it is essential to be able to characterize them in detail, to alter their shape to suit an experimental need and to ultimately use the information gathered to guide the experiment towards a desired solution. In short it is essential to observe, manipulate and control femtosecond pulses. In this review, an attempt is made to address the approaches which have dealt with these three issues in the context of ultrafast (femtosecond) pulse propagation in optical fibers. An overview of ultrafast pulse propagation in photonic crystal fibers will be also included, given the extreme interest in these new structures and their promise of being a crucial component of fiber-based systems and devices at the time of writing. Characterization of femtosecond pulses using phase-sensitive detection techniques is central to the structure of 87
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Femtosecond pulses in optical fibers
[2, w 1
the chapter, and the widely used frequency-resolved optical gating techniques will be illustrated in the context of diagnosis, shaping and control of ultrafast pulses in optical fibers. The chapter is structured to follow the three topics of observation (w 2), manipulation (w 3) and control (w4) of femtosecond pulses. A brief introduction gives a basic overview on nomenclature and quantities in use. Section 2 illustrates the diagnostic techniques for ultrafast pulses and the application of some of these to ultrafast pulse propagation in optical fibers. Experimental results that reveal the behavior of the electric field through the temporal phase function of the pulse are illustrated, and a variety of nonlinear effects ranging from soliton formation, dark soliton formation, pulse breakup, intrapulse Raman scattering to soliton self-frequency shifts are presented. Section 3 treats the ability to shape femtosecond pulses introducing various approaches that have become popular in the past decade. Fourier pulse-shaping is introduced along with a variety of programmable modulator techniques, and some examples of shaped pulses are shown. The application of pulse shaping to propagation in optical fibers is illustrated. The addition of feedback to the pulse-shaping capacity is described in w4 both from a theoretical and an experimental point of view, illustrating the adaptive control methods based on genetic algorithms showing that undistorted pulse propagation can be obtained by suitable preshaping of the input pulse launched in the fiber. Finally, w5 treats the propagation of femtosecond pulses in photonic crystal fibers, illustrating some new (at the time of writing) results on the enhanced nonlinear effects, self-frequency shifting, high-order soliton formation and other aspects such as selective harmonic generation.
w 1. F e m t o s e c o n d pulses - a b r i e f o v e r v i e w
1.1. Pulse description Characterization and understanding of ultrashort pulses is based on the relationship between the mathematical description of the electromagnetic field of the ultrashort pulse and the experimentally observable quantities that are accessible by means of measurement techniques. A detailed overview of femtosecond pulse formalism can be found in a number of excellent textbooks (Akhmanov, Vysloukh and Chirkin [1992], Diels and Rudolph [1996], Rulli~re [1998], Trebino [2002]). Generally, an ultrashort pulse can be described completely in either the time or frequency domain.
2, w 1]
Femtosecond pulses - a brief overview
89
In spite of the fact that the quantities detected experimentally are real functions, it is common practice to represent the pulse amplitude (i.e. its electric field) as a complex function. This is principally due to the added simplicity when modeling pulse propagation. Whenever the temporal description is the framework of major interest for ultrashort pulse propagation, the spatial dependence of the electric field is neglected. The complex electric field in the time domain is usually represented as the product of an amplitude function A(t) and a phase term cO(t)
E(t) = A(t) e i~c~t)
(1)
More specifically, in the description given in eq. (1), the field E(t) contains a component which represents the field envelope A(t)l, and the phase term q~(t) is composed of a central frequency o90 around which the spectral amplitude of the pulse is centered and a time-dependent phase function q~(t), i.e E(t)=A(t)exp{i[c/)(t)+o90t]}. Such a description has its counterpart in the frequency domain through a Fourier relationship which describes the pulse's features in the spectral realm E(o9)=A(o9) e i0t'''). For the scope of this introductory overview, we will make reference to this basic description. Nevertheless, even in this elementary framework lies an important quantity that is responsible for much of the pulse dynamical behavior during propagation (both in the temporal and spectral domains): the temporal modulation on the carrier commonly referred to as the chirp of the pulse. With reference to eq. (1), the phase factor q~(t) used in the description of the pulse has a carrier frequency component and a time-dependent component q~(t)=o90t +q~(t). The carrier frequency component is commonly chosen to be the frequency at the pulse peak (a more accurate choice being the value found for the pulse's intensity weighted average frequency). The time-varying term - or temporal phase function r - holds the information on the chirp of the pulse. Generally, the classification of the electric field modulation is based on the first derivative of q~(t) which yields the instantaneous frequency (i.e. a time-dependent cartier frequency o9(t) = O9o +dcp(t)/dt). When d~(t)/dt is a constant, it simply represents a correction on the carrier frequency. If dq~(t)/dt =f(t), then the pulse is said to be chirped, and the carrier
I In pulse description nomenclature, a distinction is made between the field envelope A(t) and the complex field envelope A(t)=A(t)e ioll) (which in turn allows to rewrite the pulse as
E(t)=,4(t) e"~,t).
Femtosecond pulses in opticalfibers
90
[2, w 1
frequency varies in time, accordingly. Specifically, iff(t) = kt, with k > 0 (k < 0), the pulse is said to positively (negatively) chirped or upchirped (downchirped) and the constant k is the linear chirp coefficient. This factor is also referred to as the chirp factor C in fiber-optics parlance. Higher-order terms in q~(t) (i.e. f ( t ) ~ k2t 2, k3t 3, etc.) often indicate nonlinear perturbation on the pulse during propagation and can be representative of and associated to a variety of effects. As will be discussed later, detection techniques that allow experimental observation of the temporal phase function O(t) constitute a very important analysis tool for detailed investigations of ultrashort pulse features and behavior.
1.2. Pulse propagation The propagation of an ultrafast pulse through an optical fiber is described in the framework of an electromagnetic wavepacket traveling through a dispersive nonlinear medium. The behavior of femtosecond pulses propagating in optical fibers is affected principally by the large bandwidth and high peak powers that are characteristic of these pulses. These features impose careful consideration of the combined effects which act on the optical pulses that originate from the spectral dependence of the index of refraction, and from the nonlinear response of the polarization term of the fiber medium. The large bandwidth of a femtosecond pulse affects the propagation in the linear regime since a different value for the index of refraction is associated to different spectral components. A pulse propagating along the fiber will have, as a consequence, a spectrally dependent propagation constant k(to), which in turn affects the pulse features through the material dispersion. This effect is usually taken into account by expanding the propagation constant so that: k(co) = ~ i=0
1 d/k(o~)
i!
~7
(~o- ~Oo);. ( 0 ---- (0 0
In the fiber-optics literature, the propagation constant k is often referred to as fi, and the coefficients of the Taylor series are defined as tim, with tim = dmfi/dtOm]~o=~o,,. The definition of practical useful parameters as a function of these coefficients is common. Among these, important quantities are the group velocity (Vg-1~ill) which represents the speed at which the pulse envelope moves, and the group-velocity dispersion (GVD) parameter dfil/dto, which affects the broadening of the pulse because of the different velocities at which different spectral components travel.
2, w 1]
Femtosecond pulses - a brief overview
91
An equally used, if not more practical, denomination for the GVD parameter in fibers is the dispersion parameter D, which is the derivative of fil with respect to wavelength. The material is said to be normally dispersive when D < 0 and anomalously dispersive if D > 0 (D =-(2r 2) fi2). The high peak power that is typical of ultrashort pulses causes the fiber material constituents to be excited beyond the linear regime, affecting dramatically the dynamics of pulse propagation. Simply described, the polarization term P describes the influence of the medium on the pulse as well as the medium's response to the pulse. The polarization term is commonly written as the sum of linear and nonlinear contributions, i.e. p = p L +pNL. The linear component of the polarization, p L = e o Z E , is associated with linear optical effects such as dispersion, diffraction, absorption or gain, and is affected by the large bandwidth of the ultrafast pulse. The optical nonlinearity can be described, once again, by a series expansion of the nonlinear component of the polarization vector pNL, i.e. pNL
=
X(2)EE
+
Z (3) E E E
+
Z (4) E E E E
+ ",
where X (i) represent the nonlinear susceptibility tensors associated to the material. Typical z(Z)-dependent effects are second-harmonic frequency generation or parametric processes. In optical fibers whose constituent is silica, the material lacks non-centrosymmetric properties so, in general, there is no second-harmonic generation. Under special conditions, however, an electromagnetically induced non-zero Z (2) can be obtained even in centrosymmetric materials (Orsterberg and Margulis [ 1987], Kayshap [ 1991 ]). The majority of nonlinear effects in optical fibers are due to the Z(3)-dependent contributions. Effects such as third-harmonic generation, four-wave mixing, selfphase modulation, cross-phase modulation and Raman and Brillouin scattering can be ascribed to this component. In this regime, the index of refraction seen by the pulse becomes intensity dependent, giving rise to the so called "self-action" effects. This means that the index of refraction is a function of pulse intensity and therefore affects the pulse's phase velocities and absorption. Self-action effects exist in both the spatial and temporal domains. In the latter, specifically, they give rise to modulation instabilities and cause important envelope effects such as self-phase modulation, pulse chirping, and self-compression (Kelley [2000]). One of the most prominent self-action effects arises from the balance of linear and nonlinear contributions to pulse distortion during propagation. When the group-velocity
92
Femtosecond pulses in optical fibers
[2, w 2
dispersion and the self-phase modulation balance each other, a stable pulse that propagates undistorted (to first order), called a soliton, is generated. In the early 1970s, Hasegawa and Tappert [ 1971 ] predicted that optical fibers could support the formation and propagation of optical solitons thanks to the balance of the nonlinear refractive index and the group-velocity dispersion. This is possible provided that the two effects are of opposite sign, which happens, in regular silica fibers, for wavelengths/l > 1.3 mm (i.e. in the anomalous dispersion region of the fiber). The topic of solitons in optical fibers has been, and continues to be, actively investigated and a number of excellent references are available on the subject (e.g., Hasegawa 1989], Abdullaev, Darmanyan and Khabibullaev [1993], Remoissenet [ 1999], Hasegawa and Kodama [1995]). Solitons in optical fibers bear fundamental interest for telecommunications applications as data carriers since they propagate undistorted and collide with one another elastically. Experimental observation of bright soliton formation in optical fibers has been independently obtained by Mollenauer, Stolen and Gordon [1980] and Golovchenko, Dianov, Pilipetskii, Prokhorov and Serkin [1987] in the early 1980s. Towards the end of that decade, the first experimental observation of a dark soliton (which consists of a sharp dip in the intensity of a broad pulse) in optical fibers was reported by Weiner, Heritage, Hawkins, Thurston, Kirschner, Leaird and Tomlison [1988]. The X (3) effects can be either elastic in nature, when there is no exchange in energy between the electromagnetic field and the medium, or inelastic, in which energy is transferred to the medium. The latter are typically related to the vibrational excitation modes of the fiber material and include, for instance, the optical-phonon-assisted stimulated Raman scattering and the acoustic-phononassisted stimulated Brillouin scattering. These effects become easily observable when propagating ultrafast pulses in optical fibers, and very marked in highly nonlinear structures such as photonic crystal fibers, as will be discussed in w5. A detailed analysis of these effects have been published, and complete models for the treatment of these problems have been developed. Reference to the most significant works in this realm is found in monographs on these topics (Agrawal [1989]) and in the remainder of the present chapter.
w 2. Observation 2.1. Pulse measurement
The measurement and interpretation of femtosecond pulses has provided, in the past decades, a challenge to experimentalists and theorists alike. Part of this challenge lies in having to characterize the fastest man-made events without the
2, w2]
ObseJvation
93
availability of an equally fast (or faster) reference 2. The most commonly used measurement techniques rely upon the nonlinear interaction of the pulse with a replica of itself. Analysis of ultrafast pulses dates back to over thirty years ago, when Maier, Kaiser and Giordmaine [1966] generated the second-order intensity autocorrelation of the pulse. The popular intensity autocorrelation function AC(r) = f [E(t)] 2 Ig(t- r) 2 dt gives an estimate for the pulse duration but does not contain direct information on the shape A(t) of the pulse envelope nor on its phase function r While this method is convenient and effective, it requires assumptions based on the expected mathematical description of the laser pulse as well as its Fourier relationship to the spectral bandwidth of that assumed pulse (i.e. the time-bandwidth product of the pulse). To overcome this problem, variants of the second-order intensity autocorrelation were developed, eventually leading to the first complete characterization of femtosecond pulses, performed with iterative reconstruction from an interferometric autocorrelation signal and pulse spectrum (Diels, Fontaine, McMichael and Simoni [1985], Naganuma, Mogi and Yamada [1989]). In the early 1990s a major breakthrough in the complete understanding of ultrafast pulses was achieved thanks to the work of Kane and Trebino [1993] with frequency-resolved optical gating. The foundations of this work can be traced to Treacy [ 1971 ] and Chilla and Martinez [ 1991 ]. Frequency-Resolved Optical Gating (FROG) creates a two-dimensional image of the pulse called a spectrogram. This trace can be iteratively reconstructed to obtain the pulse envelope and phase information. Mathematically, the FROG trace can be written as
IFRoG(Og,r) = f Esig(t, r) ei~~ dt[ 2 A nonlinear process is needed to generate the temporal component of the trace E~ig(t,r), and this can take different forms depending on the process used (Trebino, DeLong, Fittinghoff, Sweetser, Krumbi.igel and Kane [1997]). An interesting variation of FROG based on a cross-correlation measurement has been introduced by Linden, Giessen and Kuhl [ 1998]. In this version called X-FROG two spectrally distinct pulses form the ensemble needed to generate the signal trace, thereby permitting an unbalance in pulse powers and enhanced sensitivity, on top of an extended spectral range. A FROG apparatus that uses
2
A very detailed book has been dedicated to this topic (Trebino [2002]).
94
Femtosecond pulses in optical fibers
[2, w 2
two-photon absorption (TPA) in an InP crystal as the nonlinear process has been used to characterize pulses at 1.5 ~tm (Ogawa and Pelusi [2000]). GRENOUILLE is a simplified gratingless arrangement of frequency resolved optical gating which uses the spectral dispersion provided by a thick nonlinear crystal used to generate the temporal component of the trace (Trebino, O'Shea, Kimmel and Gu [2001 ]). This greatly simplifies the experimental arrangement and provides the advantage of being able to use thicker nonlinear optical elements thereby increasing the sensitivity of the measurement. A number of diagnostic techniques for complete characterization of ultrafast pulses, with an equal number of inventive acronyms, such as DOSPM (Chu, Heritage, Grant, Liu, Dienes, White and Sullivan [ 1995]), ENSTA (Franco, Lange, Ripoche, Prade and Mysyrowicz [1997]), TROG (Koumans and Yariv [2000]) STRUT (Yang, Fetterman, Davis and Warren [2000]), MI-FROG (Siders, Taylor and Downer [1997]), TIVI (Peatross and Rundquist [1998]) have been proposed. An adaptation of spectral interferometry, introduced by Iaconis and Walmsley [1998], is among the techniques which have generated more interest. The technique called Spectral Phase Interferometry for Direct Electric-field Reconstruction (SPIDER) is based on the non-iterative inversion of a spectrally sheared trace. The non-iterative nature of SPIDER data processing makes it quite suitable for real-time pulse recovery as well, making these two techniques very amenable for direct phase optimization of experiments. Thanks to great improvements in the iterative reconstruction algorithms introduced by Kane [ 1998] and based on the principal-component generalized projections (PCGP) FROG can operate in "real time" 3. Finally, a relatively recent approach which merits attention, especially in an optical fiber context due to its potentially very high sensitivity, is named Phase and Intensity from Correlation and Spectrum Only (PICASO) (Nicholson, Jasapara, Rudolph, Omenetto and Taylor [1999], Nicholson and Rudolph [2002]). The technique is based on traditional correlation and spectral measurement, for instance, on two-photon photocurrent generation in diodes instead of nonlinear crystals. Along these lines, Temporal Information Via Intensity (TIVI) uses traditional autocorrelations and spectra to extract pulse shapes imposing constraints on the non-negative nature of the intensity. The results are informative for fairly simple pulse shapes, and in more complex situations the method can serve as a good starting point for iterative reconstruction methods to a more general technique such as FROG.
3 FROG and SPIDER have been demonstrated to operate at data acquisition rates in excess of 10 Hz.
2, w2]
Observation
95
In face of the contention to determine which one among the many techniques is the most effective, it must be emphasized that it is crucial, above all, to understand the possible ambiguities and limits inherent in the specific technique applied to the specific experiment. Once this necessary step is accomplished, the choice is a matter of experimental convenience and personal taste. In this spirit, throughout this review, frequent references will be made to results obtained using FROG or its variations, which so far has been the phasesensitive technique of choice employed to fully characterize ultrafast pulses in optical fibers.
2.2. Experimental results
The discovery of practical methods for the complete experimental characterization of ultrashort wavepackets has given new impetus to the observation of ultrafast dynamics in optical fibers driven by the never before available degree of precision warranted by phase-sensitive detection techniques. It could become feasible, for instance, to directly observe the progressive balancing-out occurring between the group-velocity dispersion and nonlinear Kerr effect leading to the formation of a bright soliton, or to detect the abrupt 3: phase shift that is characteristic of a dark soliton. Over the last few years, phase-sensitive techniques, and frequency-resolved optical gating (FROG) in particular, have become a popular method to describe pulse propagation in optical fibers. The ability to completely characterize the electric field evolution provides a unique point of view for the investigation of the linear and nonlinear dynamics that take place during propagation of ultrashort pulses in fibers. Such investigations could guide and impact practical realizations ultimately allowing insight into the physical limitations of transmission through optical waveguides. In order to employ phase-sensitive detection techniques to measure the features of ultrafast pulses in optical fibers operating at Jl= 1550 nm, a (relatively) sensitive pulse measurement scheme must be employed. The sensitivity limit for FROG has been previously reported to be extremely high (Fittinghoff, Bowie, Sweetser, Jennings, Krumbiigel, DeLong, Trebino and Walmsley [ 1996]). In this wavelength range a (multi-shot average) single-shot geometry arrangement based on second-harmonic generation in a BBO crystal (fig. 1) has been demonstrated to offer sensitivity to pulses with energies of less than 20 pJ. An obvious strategy to enhance sensitivity is to employ a scanning geometry instead of a single-shot geometry (Taira and Kikuchi [2001]). A further added advantage of the latter
96
Femtosecond pulses in optical fibers
[2, w 2
Fig. 1. Single-shot second-harmonic frequency-resolved gating (FROG) geometry. A traditional single-shot intensity autocorrelation geometry generates a signal which is then spectrally dispersed generating a two-dimensional image of the ultrafast pulse (spectrogram or FROG trace).
geometry is the greater temporal range that can be analyzed. The use of crossphase modulation in a fiber as the nonlinear process to generate Esig has also been employed (Thomson, Dudley, Barry and Harvey [1988]) and X-FROG has been used to study femtosecond pulse evolution in fibers at 1550 nm (Nishizawa and Goto [2001]). Finally, TPA-FROG has recently been proposed for this wavelength, with a reported sensitivity to pulse energies as low as 3.8 pJ (Ogawa and Pelusi [2000]). Frequency-resolved optical gating was first used in an optical fiber realm to characterize fiber based sources with pulses of the order of a few ps (Barry, Dudley, Bollond, Harvey and Leonhardt [ 1996], Dudley, Barry, Bollond, Harvey, Leonhardt and Drummond [1997]). Typically, pulse evolution in an optical fiber is evaluated either by propagating a pulse at fixed energy and varying the length of the fiber, or by keeping the fiber length fixed and varying the pulse energy so as to induce intensity-dependent nonlinear effects that alter the pulse propagation. A full
2, w2]
Observation
97
characterization of the source is extremely important for understanding the pulse dynamics. Retrieving the amplitude and the phase of the input pulse provides the correct initial conditions for numerical calculations performed to simulate the pulse propagation itself. Analysis of the ultrafast pulses even over short distances of optical fiber reveals the richness of information that a single phase-sensitive measurement technique can provide. Such observation techniques have yielded details on nonlinear propagation in bulk media (Diddams, Eaton, Zozulya and Clement [1998], Zozulya, Diddams, Van Engen and Clement [ 1999]) and provide a valuable probe in the case of novel materials, such as the microstructured optical fibers described in w5 (Omenetto, Taylor, Moores, Knight, Russell and Arriaga [2001], Gu, Xu, Kimmel, Zeek, O'Shea, Shreenath, Trebino and Windeler [2002]). A detailed phase study of femtosecond pulse evolution in conventional telecom fiber is illustrative of the amount of information accessible through these measurement techniques. Results of one of such experiment are shown in fig. 2. The experiment illustrates the propagation of t ~ 170 fs chirped pulses in a 10 meter link of standard single-mode fiber. As the input pulse energy is increased, the intensity-dependent nonlinear effects increase correspondingly, and the recovered temporal phase function q~(t) undergoes variations which ultimately lead to its flattening. Such a flat phase across the pulse indicates temporal coherence within the pulse itself and is the signature of a temporal soliton having been achieved. The ability to detect this feature is one of the main traits of using phase-sensitive techniques to measure ultrafast pulses as opposed to more conventional autocorrelation techniques. Second-order intensity correlation relies upon a (physically motivated) conjecture on the pulse's mathematical profile and on the combination of temporal and spectral data to verify the closeness of the experimentally evaluated time-bandwidth product to that corresponding to the mathematical profile selected. If the two are equal (or close) then the experimental pulse is assumed to be transform-limited, which in fiber propagation can lead to the conclusion that an optical soliton has formed. This can be established unambiguously if one is able to experimentally observe the phase function q~(t). When the initial pulse coupled in a fiber is chirped, the dispersive energy associated with the initial chirp gradually leaves the pulse in a series of periodic bursts accompanied by characteristic oscillations in the peak amplitude of the pulse (Desem and Chu [1986], Kuznetsov, Mikhailov and Shimokhin [1995], Kaup, El-Reedy and Malomed [ 1994]). Analysis of pulse propagation indicates that these periodic bursts or oscillations occur because the phase near the center of the pulse increases linearly with Z (in soliton units) whereas it decreases
98
Femtosecond pulses in optical fibers
[2, w 2
Fig. 2. FROG traces and corresponding reconstructed intensity profiles (on a logarithmic scale) and phases of output pulses resulting from the propagation of a r = 170 fs pulse at/~ = 1550 nm through a Z = 10 m length of single-mode optical fibers, for different pulse energies.
with Z in the far wings of the pulse due to the chirp. This causes the envelope to periodically develop two simple zeros (i.e. two Jr phase slips) on either side of the pulse with a frequency of occurrence approximately equal to the soliton's frequency. The phase slips are illustrated in fig. 3 which shows a log-linear plot of the amplitude profiles and the temporal phase function of a propagating femtosecond pulse at the peak of its amplitude oscillations. The (near)-zeros of the amplitude associated with the Jr phase jump are clearly seen, and coincide precisely with the phase slips. Furthermore, the power spectrum of the pulse is in
2, w 2]
99
Observation
Experiment
Simulation
104
104
102
-~102
~
~10 ~ o
10 .2 - 1000
0 TIME (fs)
10 .2 -1000
1000
1500 1550 1600 WAVELENGTH (nm)
1650
10 o z Ill a U <( r'r 10 .2 FO LU [3_
1 0 .2
r.,o 10 .4
1 0 .4
1450
1000
>i--
10 ~
(_..5 iii cl ~
0 TIME (fs)
102
102 >1z LU r~ .d <
10 ~
n
v ._. 1500 1550 1600 WAVELENGTH (rim)
1650
1450
Fig. 3. Illustration of the intensity and spectral profiles for the flat-phased output pulse resulting from the propagation in 10 meters of optical fiber, showing experimental data (left) and corresponding profiles from numerical simulation (right). The dashed curves in the upper panels and the circles in the lower panels show the intensity and spectral profiles of pure solitons of the same peak intensity, for comparison. The (near)-zeros (upper part of the figure) of the amplitude associated with the zr phase jump are clearly seen, and coincide precisely with the phase slips. Furthermore (lower part of the figure), the power spectra of the pulses compared to the calculated spectrum of a pure soliton of the same pulse width with identical fiber parameters are in excellent agreement over multiple orders of magnitude in the intensity. excellent agreement with that expected for a pure soliton o f the same pulse width for several orders o f magnitude in the intensity. This is a notable result which stems from the dynamic range of the FROG measurement used (Omenetto, Luce, Yarotski and Taylor [1999]). A more detailed analysis of the amplitude oscillations can be qualitatively and quantitatively carried out. These results are shown in fig. 4. By comparing the asymptotic soliton profiles to those experimentally detected at 10 meters, it can be seen that for 228 pJ and 255 pJ, the peak amplitudes o f the output pulses appear to lie below that o f the asymptotic soliton. Plots o f the calculated evolution o f the peak amplitude suggest that this occurs for these energies because, for the 10-m fiber used, the pulses are measured near the
1O0 1200
Asymptotic soliton 1
800
228 pJ
0 -1000
I/ / .
~
Asymptoticsoliton P level .............................................!
0
1000
0
1200
5
10
15
1 r
0.8~ , ~ t e d
255 pJ
400 O -1000
pulse power oscillations
0.6 .._'k,___.,,:,,~..,_
g 0
0 4 .............
o
1000
S
,.-
1200 800
294 pJ
o ~
,io
is
...... i
20
I
,
0.8 0.6
400 0 -1000
20
............
1
800
13.
I
L ~ 0.5 .........~
400
0 C)..
[2, w 2
Femtosecond pulses in optical fibers
13.. 0.4 0
0
1000
5
10
1 ,
1200
15
20
~ .......................
i
800
318 pJ
0.8
400 0 - 1000
0.6 0
time (fs)
1000
0
5
10
15
20
Z (meters)
Fig. 4. Chirped pulse dynamics obtained from the deconvolution of the FROG traces. Oscillations in the pulse amplitude occur in dependence of the initial chirp of the pulse. The pulse oscillations are calculated by using the experimentally obtained initial pulse peak power and phase profile (to account exactly for the chirp). The results are shown in the fight-hand panel for the propagation of the pulse through Z=20m of standard single-mode fiber, along with the asymptotic soliton power level for each case. The left-hand panel illustrates the experimental results obtained from the reconstruction of the spectrograms (Omenetto, Luce, Yarotski and Taylor [1999]; shown also in fig. 7). The experimentally observed peak power values at Z = 10 m are illustrated by the crosses in the right-hand panel and match the calculated oscillatory dynamics caused by the input chirp.
minimum o f the first oscillation. The phase profiles have not yet flattened out in the vicinity o f the pulse, because the low amplitude has not yet allowed the Kerr effect to fully balance the dispersion and produce the flat phase characteristic o f solitons. In the case o f 294 pJ the output pulses have almost exactly the same amplitude and pulse width as the asymptotic soliton, which indicates that these pulses are measured exactly at the midpoint o f the first oscillation, consistent with the simulation. This suggestion is strongly reinforced by the fact that :r phase slips are seen to be half-formed in the phase profiles, a feature that theory and simulation predict should coincide precisely with the oscillation midpoint. Finally, for 318 pJ, the amplitude o f the output pulse is seen to exceed
2, w 2]
Obseruation
(a) t N
:1:
101
100
I
'
I
'
I
..................
I~
iit ~
5
~
'EIO.Z
u... -5
.
.
.
.
.
i
I
__ !
I_
-4
-2
0
2
I
i "
4
.
~ c~
O-a-
5 1
LI_
.
0
~
-0.4
0.0 Time (ps)
"~
~ 0.5 -5
~ 0.0 -0.4 TimO(~(~ps) 0.4
0.4
Fig. 5. Terahertz pulse train characterization illustrating (a) the experimental FROG trace, (b) the retrieved FROG trace and (c) the retrieved spectrum. The reconstructed intensity and phase profile are shown in (d) along with the calculated values obtained with a NLSE-based simulation (open circles). The retrieved phase function illustrates a near-Jr (0.86Jr) phase shift across the central portion of the pulses which is indicative of grey-soliton formation. (Courtesy J.M. Dudley.)
that of the asymptotic soliton, and the ~ phase slips are now seen to be fully formed. These facts are consistent with the measurement coinciding with the maximum of the first oscillation, confirming both theory and simulation. In this case, the phase is now almost perfectly linear across the pulse, revealing that features of the asymptotic soliton have become strongly evident. Similar propagation dynamics have been treated by Dudley, Barry, Bollond, Harvey and Leonhardt [1998], studying the evolution of the high-order soliton over half a soliton period. The formation of the previously described r phase slips can be noticed in the set of data reported in their paper as well. Multiple pulse patterns in fibers can also be analyzed with phase-sensitive techniques (Dudley, Gutty, Pitois and Millot [2001 ]). In these cases, reconstruction of FROG traces for pulse trains requires additional care compared to the singlepulse case. The data are deconvolved assuming a circular computation window, which calls for the assumption of trains of pulses of infinite temporal extent. In the experimental demonstration, a 2.5 THz pulse train is generated by a sinusoidally beating signal in a high-birefringence optical fiber (Fibercore HB600) of length Z = 1.8 m. The experimental results are shown in fig. 5.
102
Femtosecond pulses in optical fibers
-=-. J~ =.
z (m)
[2, w2
~amplitude
10
.m if} c _=
-1500-1000
/ ~. x3
-5 0 0
0 500 time (fs)
. z (m)
1000
1500
phase
0
~
~
O. t~ r
-2 0-'
-5()o
'
o
' 5(;o
time (fs)
Fig. 6. Experimental results obtained with the cutback method showing the phase features of a soliton propagating through Z=90m of single-mode conventional fiber (Coming SMF-28). The phase profile is found to be consistently flat throughout propagation, indicating extreme proximity to soliton formation. A slight oscillation is detectable in the temporal values of the pulse, indicating the existence of some residual imbalance between the linear and nonlinear components.
A train of gray solitons can be detected and a characteristic signature is offered by the experimentally detected profile of the phase function q~(t). A phase shift across the center of the pulse equal to ~0.86r appears, revealing a reduced modulation depth compared to the case of an ideal dark soliton (which would exhibit a zr phase jump). For longer fiber distances, by appropriately tuning the input pulse energy, a femtosecond soliton can be followed in its path through the fiber using the cutback method. The energies of the pulses are carefully varied before they are launched into a fixed-length segment of fiber, and FROG traces are correspondingly detected. The fiber is then cut and the process repeated. FROG traces of the output pulses from the cutback experiment (corresponding to a total propagation distance of Z = 90 m), for input pulse energies of ~280 pJ are reconstructed and the temporal intensity and phase profiles are plotted. The phase is found to be noticeably flat across the pulse throughout propagation, indicating temporal phase coherence, which has been noted previously to be a signature of a soliton traveling through the fiber (fig. 6). When the pulse power is increased so that the balance between the linear and nonlinear contributions is perturbed, the pulse undergoes a series of
2, w2]
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Fig. 7. X-FROG images of ultrafast pulse breakup in optical fibers at/~ = 1550nm. The data were taken after propagation in 4 meters of polarization maintaining dispersion-shifted fiber. The figure illustrates the X-FROG trace which gives a depiction of the pulse breakup showing the generated Stokes and anti-Stokes components and their temporal relationship. The original (probe) pulse is shown in the inset (after Nishizawa and Goto [2001]). transformations dependent on the fiber material and on the pulse chirp which results in various effects such as pulse splitting and breakup, self-frequency shifting, self-steepening, to name a few. The most evident of these effects, perhaps, is the Raman-assisted soliton self-frequency shift (Mitschke and Mollenauer [1986], Gordon [1986]) which will be described later. Experimentally, femtosecond pulse breakup at ~, = 1550 nm has been analyzed with X-FROG in polarization-maintaining dispersion-shifted fibers (PM-DSF) (Nishizawa and Goto [2001]). These measurements use the spectrograms to visualize the co-propagating soliton pulse and the generated anti-Stokes pulse. The experimental arrangement consists of a 48-MHz repetition rate, 110fs pulsed laser coupled into a link of PM-DSE The X-FROG trace corresponding to pulse breakup is shown in fig. 7. The temporal relation between the various spectral components can be identified thanks to the spectrogram which helps establish the dynamics of pulse components generated through nonlinear effects, providing a graphic representation of the phenomena occurring. Power-dependent phenomena in different fiber types are shown in fig. 8 (Piccinin, Arioli, Fer6 and Martinelli [2002]). Two different dispersion-decreasing fibers and a PANDA fiber are examined with FROG. The results for the
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Fig. 8. FROG traces of femtosecond pulses propagating through single-mode optical fiber for the cases of (a,b) dispersion-shifted fiber, with different diameters, and (c) PANDA fiber. The traces give (in a,b) a graphical indication of soliton formation, pulse breakup with Stokes and anti-Stokes shifts for the dispersion shifted cases and (in c) soliton self-frequency shift for the PANDA fiber. (Figure courtesy of D. Piccinini, M. Arioli, E Fer~ and M. Martinelli.) dispersion-decreasing fiber show pulse breakup, leading to a Stokes shift in one case, but an anti-Stokes shift in the other, principally due to the different core sizes and compositions. In the PANDA fiber, there is no detectable pulse breakup leading to a clean soliton self-frequency shift. An example of experimentally detected spectrograms of self-frequency shifted pulses are shown in fig. 9. The deconvolution of the experimental traces and the recovery of the phase function O(t) reveals that only under certain propagation conditions does the self-shifted pulse actually assume the features of a soliton, undergoing amplitude and temporal oscillations reminiscent of chirped pulse propagation (Desem and Chu [1986]). Typically, as linearly chirped femtosecond pulses are coupled into a segment of single-mode fiber and the input pulse power is increased, oscillations in the pulse width are detected at the output, in combination with the Raman-induced self-frequency shift which causes the carrier wavelength to vary as it travels through the optical fiber.
Fig. 9. Intrapulse Raman scattering leading to the self-frequency shift of the pulse during its propagation through 90 meters of single-mode conventional fiber. By using phase-sensitive detection techniques it is noticeable that the flat phase, signature of the temporal coherence typical of a soliton, is obtained only for certain conditions (an appropriate combination of pulse power, fiber length and initial pulse characteristics). For fixed initial pulse chirp and varying power, the soliton phase feature is detected in cases (a) and (d).
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While simplistic dispersive wave propagation can be argued to be the dominant cause of the oscillations, a more complex interplay exists mainly because of the frequency dependence of a number of parameters (such as the nonlinear coefficient, the effective area/mode size, and certainly the dipersion curve characteristics) compounded by the chirp-induced oscillations. This was noted by Nishizawa, Okamura and Goto [ 1999] in the characterization of a large Raman shift in polarization maintaining fibers. In the experimental data shown in fig. 9, the maximum shift detected for the Z = 4 0 m link of SMF-28 fiber, with no polarization control, is of the order of 100 nm. An analysis of the interaction of a soliton with dispersive waves was carried out theoretically by Kuznetsov, Mikhailov and Shimokhin [1995], taking into account the oscillatory nature of pulse propagation due to the interplay of the soliton with the dispersive waves. They treat the picosecond pulse regime which offers an integrable case, as opposed to the femtosecond regime which presents a non-integrable case for the NLSE and must rely on suitable approximation and corrections. In spite of this, the modified NLSE is capable of tracking the phase variation while the Raman effect is causing the pulse to self-frequency shift (Omenetto, Yarotski and Taylor [2000]). The phase functions indicate that at a particular instant a near-flat phase is observed across the center of the pulse, thus confirming the formation of a temporal soliton. Modifications of the canonical formulation of the NLSE in order to address femtosecond pulse propagation have been analyzed extensively (Christodoulides and Joseph [ 1985], Mamyshev and Chernikov [ 1990] and Chernikov and Mamyshev [1991 ], among others) to include frequency-dependent effects of the parameters and to account in detail for the dynamics of the material response. Through this small sampling of results, it is apparent that the phase details obtained through reconstruction of the FROG traces and, more generally, obtainable with phase-sensitive techniques allow a new vantage point for the study of femtosecond phenomena in optical fibers.
w 3. Manipulation 3.1. Ultrafast pulse shaping Additional versatility in ultrafast optics has been provided by the capacity to shape femtosecond pulses, allowing the generation of complex ultrafast waveforms which can be designed to suit a specific experimental need.
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Fourier-based pulse-shaping techniques have been first introduced by Froehly, Colombeau and Vampouille [1983] in the picosecond time domain. It was the adaptation of grating-based pulse stretchers by Weiner and co-workers that started and popularized femtosecond pulse shaping, and we refer the reader to very complete reviews on the topic: Weiner [1995, 2000]. Shaping techniques of femtosecond optical pulses have become widespread with the increased availability (and reliability) of ultrafast laser sources and programmable spatial light modulators. Femtosecond pulse shaping is based on a linear filtering operation performed either in the time or in the frequency domain. The most widely adopted and successful approaches operate in the frequency domain, by masking the spatially dispersed optical spectrum of the pulse. The apparatus centers around a so-called "zero-dispersion stretcher". The first grating and lens (G l, fl) spatially disperse the individual optical frequency components contained in the incident pulse, while the second lens and grating (f2, G2) are placed symmetrically opposite the first two and recollimate the beam. Filtering is performed at the point where the frequency components experience the maximal spatial separation (the masking plane or Fourier plane), located midway between the two lenses. The lenses are set up as a unit magnification telescope (fl =f2). The first lens performs a spatial Fourier transform between the plane of the first grating and the masking plane, and the second lens performs a second Fourier transform from the masking plane to the plane of the second grating. If no filter is present in the masking plane (in an optical system without aberrations) the effect of these two Fourier transforms is to leave the input pulse unchanged. Due to the symmetry of the arrangement, the setup can be "folded", employing one grating/lens pair and simplifying alignment. Also, an all-reflection geometry with concave mirrors instead of the lenses can be used to maximize the throughput of the shaper. A schematic of this pulse shaping apparatus is shown in fig. 10. Filtering in the Fourier plane can be performed in a variety of ways. Fixed masks such as slits or lithographically etched phase masks (Weiner [2000], Omenetto, Nicholson and Taylor [ 1999]) can be used. However, programmable devices offer the maximum versatility and are interfaceable with feedback loops to control ultrafast experiments, as will be described later. One of these programmable devices is an acousto-optic modulator (AOM) (Hillegas, Tull, Goswami, Strickland and Warren [1994], Dugan, Tull and Warren [1997]). The AOM crystal, typically TeO2, is driven by a RF voltage signal which is converted to an acoustic wave. The latter travels through the crystal and generates, via the photoelastic effect, a refractive index grating by which the spatially dispersed spectrum of the ultrafast pulse is diffracted
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Fig. 10. Femtosecond pulse shaper, based on a (all-reflective) zero-dispersion stretcher. The first grating and curved mirror (G1,Ml) spatially disperse the individual optical frequency components contained in the incident pulse, while the second mirror and grating (M2,G2) are placed symmetrically opposite the first two and recollimate the beam. Filtering is performed at the point where the frequency components experience the maximal spatial separation (the masking or Fourier plane), located midway between the two mirrors. and modified. Ideally, by modulating the RF drive waveform, the grating can be phase-, amplitude- or frequency-modulated. An important improvement in acousto-optic pulse shapers has been introduced in the last few years (Tournois [1997], Verluise, Laude, Cheng, Spielmann and Tournois [2000]). An in-line shaper, which does not require a zero-dispersion stretcher, based on a long crystal provides the modulation on the pulse. Such a shaper can, in principle, independently and simultaneously control both amplitude and phase. Another method of programmable shaping employs, in a folded geometry, a deformable mirror (DM) (Vdovin, Sarro and Middelhoek [1999]). In a DM, small actuators are placed at the back of a reflecting surface. A deviation of the mirror surface causes a phase change in the spectral components in the area of the deformation. A considerable advantage of the DM over other shaping techniques is its extended wavelength operating range. Deformable mirrors can provide femtosecond pulse shaping over short (blue to ultraviolet) wavelengths at which other devices do not perform well or do not work at all, and they can, in principle, withstand higher power densities (depending on the substrate properties). One of the most popular devices for ultrafast pulse shaping is the liquid crystal spatial light modulator (SLM). In an SLM, a thin layer of nematic liquid crystal is sandwiched between two pieces of glass. The nematic liquid crystal consists of rod-like molecules which, in the absence of an electric field, are aligned along a certain direction.
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Fig. 11. The Fourier transform of a rectangular waveform in the spectral domain yields a sin(t)/t function in time (and the corresponding sinZ(t)/t 2 intensity profile). The calculated SHG-FROG trace corresponding to this particular pulse shape is illustrated. Upon application of an electric field, the liquid-crystal molecules tilt in the direction of the applied field, causing a refractive-index change, thus impressing a phase change on the light traversing them. The inside surface of each piece of glass is coated with a transparent, electrically conducting film of indium-tin oxide. By patterning separate electrodes (also referred to as "pixels") with corresponding electrical connections it is possible to independently control every element, enabling phase or amplitude modulation over the different spectral components. All of the above devices are still rather bulky, and considerable effort is being expended in trying to integrate pulse-shaping devices. Some results have been reported using arrayed waveguide grating routers (Kurokawa, Tsuda, Okamoto, Naganuma, Takenouchi, Inoue and Ishii [ 1997], Takenouchi, Tsuda, Naganuma, Kurokawa, Inoue and Okamoto [1998]) or integrated electro-optic modulators (Diels and Rudolph [1996]). Pulse shaping with fixed masks is illustrated in figs. 11 and 12. In this example, the shaped pulse is altered in a folded zero-dispersion stretcher with a slit transmitting the central portion of the spectrum. Such amplitude modulation in the spectral domain comes close to transmitting a square wave in the frequency domain, which corresponds to a sinc function in time. A FROG trace of the
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Fig. 12. Experimental SHG-FROG trace showing (a) the observed spectrogram, (b) the experimental spectrum after masking in the Fourier plane, and the reconstructed temporal profile on (c) a linear and (d) a logaritmic scale with the corresponding temporal phase profile. The phase profile has near-Jr phase jumps corresponding to the zeros of the temporal profile, in correspondence with the zero-crossings of the electric field.
shaped pulse is taken and reconstructed. Plotting the retrieved intensity and phase on a semilog scale shows the characteristic sidelobes of the (near) sinc function. Furthermore, the retrieved phase exhibits near-:r phase shifts in correspondence of the zero crossings of the electric field. A very nice example which uses a (near) sinc function in the frequency domain to synthesize a square pulse in time was presented by Weiner [2000]; the spectral shaping was performed using a microlithographically patterned amplitude mask which generates the specific spectral filtering function to obtain the desired shape in the time domain. Feedback control methods (described in more detail in w4) have been applied to allow the design of arbitrary femtosecond waveforms (Meshulach and Silberberg [1998]) (fig. 13).
3.2. Ultrafastpulse shaping in optical fibers The ability to alter the temporal shape of ultrafast pulses, their spectral (and temporal) phase and their chirp affords a variety of degrees of freedom introducing new control parameters and optimization variables for the control of femtosecond pulse behavior in optical fibers. The extended use of pulse-
Manipulation
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shaping techniques is likely to extend the spectrum of fiber-based applications, and various solutions and approaches have been explored. Preshaping a femtosecond pulse can optimize nonlinear coupling and optical switching by devising the optical pulse shape that will induce the most efficient switching performance (Weiner, Silberberg, Fouckhardt, Leaird, Saifi, Andrejco and Smith [ 1989]). Proper shaping of the input pulse chirp can alter and control the periodic evolution of solitons (Omenetto, Luce, Yarotski and Taylor [ 1999]).
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Fig. 14. Example of near-distortionless transmission of sub-500 fs pulses in a 3-km link of standard-mode/dispersion-compensated/dispersion-shifted fiber. While the dispersion-compensating fiber included in the link corrects for the majority of the second- (and third-) order components, the residual higher-order components can be corrected with the programmable pulse shaper by appropriately selecting a spectral phase correction function illustrated in the bottom fight corner. The original pulse (upper left) in that case is delivered at the output practically undistorted (bottom left), whereas without compensation it is broadened and significantly distorted (upper right). (Courtesy A.M. Weiner.)
Dark solitons were also generated by means of launching a predefined shaped pulse (Weiner, Heritage, Hawkins, Thurston, Kirschner, Leaird and Tomlison [1988]). Femtosecond pulse encoding can also take place thanks to shaping and code-division multiplexed access (CDMA) schemes have been designed and demonstrated (Sardesai, Chang and Weiner [1998]). In these examples, it is possible to detect the properly encoded signals through an equally properly programmed spectral phase. Pulse-shaping techniques have also been used to filter multiple-wavelength signals in wavelength-division multiplexing applications (Nuss, Knox and Koren [ 1996]). In this case, the shaping is induced passively by using the dispersion of a fiber and exploiting the large bandwidth of the ultrafast pulse, generating in excess of 1000 channels (Collings, Mitchell, Boivin and Knox [2000]). Novel sources can be achieved based on the compensation of nonlinear fiber effects by suitable pre- or post-compensation. The transmission of undistorted short pulses over long distances of optical fiber is especially appealing.
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With the use of a liquid crystal SLM, higher-order dispersion terms were defeated, in the linear regime, in several kilometers of fiber, enabling the undistorted transmission of sub-500fs pulses (Chang, Sardesai and Weiner [ 1998], Shen and Weiner [ 1999]). A combination of standard single-mode and dispersion-compensating fiber link is followed by a liquid-crystal spatial light modulator. The bulk of the quadratic and cubic dispersion compensation is performed in the dispersion-compensating segment of the overall fiber link. The residual dispersion is compensated by programming a suitable spectral phase profile in the SLM. With this approach, the original pulse duration is almost perfectly preserved, indicating that an exact phase correction is possible. Results of this experiment are shown in fig. 14 for transmission of 490 fs through a 3-km fiber link.
w 4. Control 4.1. Coherent control - adaptive pulse shaping
An exciting complement to the capacity to generate user-defined ultrafast pulses lies in the use of feedback from the experiment. The feedback variable acts as a control parameter guiding the ultrafast waveforms until the desired experimental outcome is obtained. The adaptive approach, first proposed by Rabitz (Judson and Rabitz [1992]), holds exciting implications in the basic and applied sciences. With this method, light fields are designed, through a feedback control process, with femtosecond temporal features, guided by a parameter of interest generated by the experiment at hand. The first demonstrations of this method have shown its efficiency and power (e.g., Assion, Baumert, Bergt, Brixner, Kiefer, Seyfried, Strehle and Gerber [ 1998], Bardeen, Yakovlev, Wilson, Carpenter, Weber and Warren [ 1997], Bartels, Backus, Zeek, Misoguti, Vdovin, Christov, Murnane and Kapteyn [2000], Efimov, Moores, Beach, Krause and Reitze [1998], Meshulach and Silberberg [1998], Levis, Menkir and Rabitz [2001]). The power of this method lies in the capacity of evolving towards an optimal optical field in situations where theoretical prediction is difficult or when experimental conditions are either not consistent or complex. The problem of ultrafast pulses propagating in optical fibers in the nonlinear regime is of practical interest and of non-trivial implementation. Understanding and optimizing the propagation of ultrashort pulses in optical fibers finds part of its motivation in the projected need for increased capacity
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Fig. 15. Schematic of the experimental arrangement for femtosecond adaptive pulse shaping. Femtosecond pulses are sent to a pulse shaper implemented as a folded zero-dispersion stretcher with a phase-only spatial light modulator (SLM) located in the Fourier plane, and then sent to an experiment (10-meter link of conventional SMF-28 single-mode optical fiber for the experiment described in w4). The output from the fiber is then sent to a detector which provides the control variable for the genetic algorithm (for the fiber experiment this control signal is the second-harmonic signal revealed by a photodiode).
in telecommunication systems. The application of adaptive control in fibers has been discussed both theoretically (Omenetto, Luce and Taylor [1999]) and experimentally (Omenetto, Reitze, Moores and Taylor [2001]). A schematic of feedback control is shown in fig. 15. The essential components of an ultrafast adaptive control experiment are (besides the shaper!) an efficient detection scheme which samples an observable quantity that measures the performance of the experiment, and an efficient control algorithm that ties the observable to the pulse shaper. The control algorithms need to be able to search a large solution space efficiently to identify a desired pulse shape. Simulated annealing (SA) has been applied to adaptive pulse-shaping problems (Meshulach and Silberberg [1998]). The SA algorithm is a Monte-Carlo based approach to minimize multivariate functions. To apply simulated annealing, an initial configuration is sent to the pulse shaper. An optimizing criterion, in the form of a "cost function", is used to determine the degree of convergence obtained. A random variation is imposed to generate a new configuration. The latter is accepted unconditionally if the imposed criteria are satisfied, otherwise it is accepted probabilistically 4. The process is iterated until the selected criteria are met. Other search strategies, such as neural-network algorithms, can also be employed for this purpose (Haykin [1999]). Genetic algorithms (GA) are widely employed for these problems because of their successful application to a variety of optimization problems that involve a large number of variable parameters (Mitchell [ 1999]), their ease of programming, and their relative strength in converging to a global solution.
4 This procedure is also known as the Metropolis step. This allows the SA algorithm to proceed consistently towards the solution yet allows "jumps" so as to avoid stalling on local maxima/minima.
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4.2. Adaptive pulse shaping- Numerical evaluation A numerical analysis of ultrafast adaptive feedback, based on genetic algorithm optimization, which treats the case of transmission of ultrashort pulses in fibers has been explored. In the model, the pulse-shaping function is represented by a spectral amplitude filter. The latter is updated according to a fitness function which is programmed with the goal of obtaining the shortest pulse possible at the output and is evaluated on the output pulse from the fiber. A decision criterion selects the best filters (that is, the shaped input pulses which yield the shortest output pulses) and the pulse shaper is updated. The process is iterated until pulse propagation has been optimized. The feedback loop employs a Genetic Algorithm (GA). The search space for this simple case, assuming an on/off control of 128 different spectral components, yields 2128 possible filter configurations. In a typical experiment the search space becomes even more extensive since the control parameters that act on the individual spectral components have more degrees of freedom. The basic operation of the GA is outlined in fig. 16. The GA operator employs basic evolutionary paradigms
Fig. 16. Example of simple genetic algorithm routine illustrating the genetic operators, based on the goal of realizing an array of 8 bits of values x = 1.
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(survival of the fittest) to vary the filter configurations (hence the pulse shapes) according to a fitness criterion. The crossover operator swaps part of the content of a filter (i.e. a chromosome) to another and vice versa, exchanging information and generating new filter configurations. The mutation operator, on the other hand, enacts random variations on one component of the filter (i.e. an allele), which, generally speaking, keeps the GA from stalling on local minima. Through the action of the latter two operators a new collection of filters (i.e. population) is generated, and among these the filters that produce the shortest pulse at the output of the fiber (i.e. the fittest) are selected; the process is then iterated. Again, the choice of a suitable fitness function is a crucial factor, especially in view of experimental application, where this choice is also subject to practical feasibility. In our case, the evaluation is performed on the peak intensity of the output pulse, as will be described later. Ultrafast pulse propagation is modeled using a nonlinear Schr6dinger equation (NLSE) with appropriate correction terms to account for operation on the femtosecond time scale (Agrawal [1989]):
OA _ i 02A 2 16A a 0 ~AI 2A - ifiA 0 ]A]2 + y -1~ 03A , OZ 20r 2 +iA I A-5 Or Or 60r 3 where A(Z,r) represents the pulse amplitude, Z and r represent the distance along the fiber and time (in dimensionless units), while 6 takes into account fiber loss and a,fi,), give the self-steepening, stimulated Raman scattering, and third-order dispersive corrections, respectively. The relationship of these coefficients to the physical parameters of the fiber is a = 1/(moto), fi=tR/tO, )'=-fi3/(tofl2), where m0 is the cartier angular frequency, tR is the delayed nonlinear response time, to is the non-dimensionalization unit, fi2 is the groupvelocity dispersion parameter, and fi3 is the third-order dispersion parameter. The calculations are performed to simulate the propagation of a r - 1 7 6 fs pulse at a wavelength of ~.-- 1550 nm transmitted through a 100-meter segment of dispersion-shifted fiber. The parameters used for the fiber are fi2 = - 1 ps2/km, fi3-0.1 ps3/km, tR--3 fs. The initial pulse has the form A(t)= sech(t)exp(-iCt2), where the parameter C represents the linear chirp, with C = 1. Filtering of the input pulse is performed in the spectral domain by transmitting or blocking spectral components. In this model, filtering was achieved by performing a FFT of the input pulse, followed by multiplication of the achieved pulse representation in the spectral domain with a string of ones and zeros (which mimic the transmission or blockage of the spectral components and form the chromosomes for the genetic algorithm). An inverse transformation back to the time domain provides the temporal input to be propagated through the fiber. The
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Fig. 17. Optimally filtered pulse shape obtained by an adaptive pulse-shaping simulation. The temporal pulse shape (shaded pulse) is shown along with the original unshaped pulse (dashed line). Representation of the best amplitude filter superscribed on the original input pulse spectrum is also shown in the inset. Solid lines represent complete transmission while lack of lines represent no transmission of the spectral component.
Fig. 18. Propagation of (top) unfiltered and (bottom) optimally filtered pulse through 100 meters of fiber. The unfiltered pulse is severely dispersed and broadened temporally, while the optimally filtered pulse shape recovers its original peak intensity. filter is composed of 128 individual "alleles" to mimic commercially available filtering devices. When the algorithm has converged, it gives an optimal filter configuration. The corresponding temporal shape is then propagated through the fiber link and the result is shown in figs. 17 and 18. The output resulting from the propagation of the optimally shaped pulse has a duration of ~ 100 fs, and higher peak intensity
118
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than the original pulse. The output of the unfiltered pulse, in contrast, has completely dispersed. According to these results, the maximum peak intensity at the output of the fiber is achieved by propagating a suitably shaped pulse of considerably lower peak intensity. The simulation illustrates the ability of adaptive control to overcome an initial chirp and to produce a pulse that propagates approximately as a soliton while simultaneously prefiguring the pulse so that it produces enhanced intensity at the output (which is in fact greater than a pure soliton of the same energy). The robustness of the solution is also evaluated and reveals that the effect of a 10% fluctuation in the energy of the input pulse employing the same filter configuration found earlier is minimal, and it can be seen that the output pulse characteristics do not vary appreciably. The method is also applied to longer lengths showing excellent performance over 2 km of fiber with the same input conditions (at lower energies). The results suggest that the method can provide significant improvement over conventional soliton transmission. It is interesting to note that even with the crude control provided by discrete on/off filtering, it is possible to successfully propagate a sequence of ultrashort pulses. The results for this simulation are shown in fig. 19, which presents the transmission of a 5-bit word (176 fs pulses with a peak-to-peak separation of 750fs) over 100 meters of fiber. The same optimal filter configuration found for the single-pulse propagation case is used. The optimized pulse stream is recovered successfully at the output of the 100-meter link, further confirming the utility and robustness of the approach.
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It is important to note that adaptive shaping is essentially an optimal precompensation technique, albeit non-trivial. The GA finds the optimal filter at the input of the fiber that produces the desired output pulse after a specific propagation length. The evolution of the pulse envelope prior to and beyond this specific length will in general not be optimized nor will the duration be minimized. It is not possible for adaptive pulse shaping to produce a pulse envelope that does not evolve during propagation unless the conditions for soliton formation are met (and maintained during propagation) (Omenetto, Reitze, Luce, Moores and Taylor [2002]).
4.3. Femtosecond pulse shaping - experiment
An experimental realization of ultrashort pulse delivery through optical fibers not only impacts telecommunications by providing an avenue for high data rates in a single channel, but enables applications in other fields where delivery of an energetic short pulse is essential (for instance, in nonlinear imaging techniques such as multi-photon microscopy (Maiti, Shear, Williams, Zipfel and Webb [1997], Squier and Muller [2001]), in materials processing (Schmidt, Husinsky and Betz [2000]), and in medical diagnostics (Morgner, Drexler, K~irtner, Li, Pitris, Ippen and Fujimoto [2000], Drexler, Morgner, K~irtner, Pitris, Boppart, Li, Ippen and Fujimoto [1999]) to name a few. Efforts to optimize energetic femtosecond pulse delivery in optical fibers have been undertaken at the operational wavelength of Ti:sapphire (~ ~ 800 nm). The effects of prechirping the pulse were analyzed (Myaing, Urayama, Braun and Norris [2000]) and successful delivery of 0.4 nJ pulses of duration r = 82 fs has been reported. This result was obtained by using a sequence of elements (a pulse compressor, a negative GVD arrangement and a spectral compressor) with the purpose of creating an input pulse being the phase conjugate of the pulse detected at the output (Clark, Ilday and Wise [2001]). Adaptive pulse shaping has been applied to ultrashort pulse propagation in optical fibers at an operational wavelength of ~.= 1550nm (Omenetto, Reitze, Moores and Taylor [2001 ]). During propagation in conventional singlemode fibers, in the traditional low-power picosecond regime, the dominant detrimental effect in single-mode fibers is chromatic dispersion (or groupvelocity dispersion) which causes the optical pulse to broaden in time as it propagates through the fiber. In the anomalous dispersion region (~ > 1.3 ~tm) optical solitons are formed by the balance of nonlinearity and chromatic dispersion. With the pulse duration becoming shorter, however, nonlinearities
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start to dominate and eventually can lead to distortion in the pulse shape at the output of the fiber as seen in w2. Applying adaptive feedback techniques can defeat the combined linear and nonlinear effects by controlling the spectral phase of the femtosecond pulses launched in the fiber. In this case, the control loop finds an optimized ultrashort pulse shape which, once launched, will compensate for the nonlinear effects which distort the pulses traveling through the optical fiber. Experimentally, ultrashort pulses are shaped by a computer-controlled Spatial Light Modulator (SLM) and are then launched into an optical fiber link. The output pulse after propagation in the fiber is then sent to a second-harmonic (SH) crystal, which generates the control parameter. As shorter pulses exit the fiber, their peak intensity will increase, leading to the generation of stronger SH signal. The algorithm modifies the spectral phase in the pulse shaper until a maximum value for the SH signal is obtained, indicating that the pulse propagation has been optimized. Operation of the feedback loop relies on a Genetic Algorithm (GA) search that, after acquiring data from the output of the fiber, updates the pulse shaper, controlling the phase parameters in order to maximize the SH signal. Smooth profiles are sent to the spatial light modulator to minimize diffractive effects caused by sharp phase jumps from one pixel to the next. The phase profiles are specified by the third (quadratic in to-to0) and fourth (cubic in to-to0) terms of a Taylor series about too, the center frequency of the source. The coefficients for these terms constitute the chromosomes of the GA. Each chromosome's phase profile is sent to the SLM, and the resulting magnitude of the measured second-harmonic signal from the detector represents the fitness of that individual. The fitnesses are then ranked, and the next generation of individuals is created. This new population is then subjected to the genetic operators of cross-over and mutation. The process is iterated until a satisfying fitness level has been reached. The extent of the search space of the experiment can be estimated by considering that each pixel of the spatial light modulator- i.e., the fs shapercan be addressed with a variable voltage level between 0 and 5 V encoded with a precision of 12 bits. This leads to a total search space of (2~2) 12s. In this case, a reduced set of values for the SLM was used restricting the search space to 64128 More practically, the computer-controlled algorithm generates an initial "population" of spectral phase filters. The corresponding output pulses are evaluated according to a "fitness" function, i.e. the magnitude of the SH signal. The pulses which have the highest peak intensity (i.e. shortest duration) at the fiber output will generate the highest SH signal and are considered the
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w
0
-
;
!
500
1000
-2
9
1500
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Fig. 20. Output after propagation through 10 meters of single-mode fiber of an unshaped t = 204 fs with P = 2 5 m W . The pulse is broadened to r = 3 5 2 fs and asymmetrically distorted. The same unshaped pulse at lower powers (P = 5 mW) is illustrated for comparison; that pulse is broadened, mostly because of linear dispersion, to r = 702 fs.
"fittest"; these are selected, thereby extracting from the initial filter "population" a new group of filters. This selected group is used as a seed to generate a new "population" of filters through cross-over and mutation operators, and the process is iterated until the pulse propagation has been optimized. In the experimental realization, an initially unshaped pulse of average power P = 25 mW (repetition rate of 80 MHz) and duration of 204 fs (i.e. peak power in excess of 1 kW) is coupled into a 10-meter single-mode optical fiber segment (Corning SMF-28). In this case, the dispersion length LD is equal to 2m, whereas the nonlinear length LNL is --20 cm. These length parameters provide the scales over which dispersive and nonlinear effects become important (for pulse propagation in fibers). The ratio LD/LyL= 10 insures that the pulse distortion is governed mostly by nonlinear effects. The experimentally recovered output pulse is illustrated in fig. 20. The asymmetric broadening of the pulse and the structure present in its wings indicate the effect of nonlinearities on the pulse propagation whereas the same input pulse at lower powers is linearly broadened to -1 ps. When the feedback loop is turned on, the SLM applies a phase-only modulation on the pulse to be launched in the fiber. Depending on the level of the SH signal, the SLM is updated, and the process is iterated so as to maximize the SH signal. Once the loop has converged, the iterations are halted. The best filter function is applied to the spatial light modulator, and the resulting shaped pulse is sent through the fiber. Comparisons of the spectrograms for the optimized and unoptimized cases and of the recovered pulses are shown in figs. 21 and 22, respectively. The output pulse resulting from the launch of
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Fig. 21. Spectrograms illustrating the adaptive pulse-shaping experiment in 10 meters of SMF-28 fiber: (a) input unshaped pulse; (b) the output pulse after propagation; (c) input optimally shaped pulse; (d) the output pulse after its propagation.
Fig. 22. Reconstruction of the experimental data obtained at the output of the fiber due to (a) the propagation of the unshaped pulse and of (b) the optimally shaped pulse. In the first case, the output pulse is broadened to t=352 fs, is asymmetrically distorted, and has wing features extending to nearly 1ps. For the optimized case, the output pulse has a duration of 214 fs and is very close to the original input pulse, with much of the asymmetry eliminated. The phase function is smooth despite the nonlinear effects. the optimized shaped input pulse has a time duration of 213 fs, with a phase structure that is noticeably smoother. Much of the nonlinear pulse distortion has been defeated, particularly in the wings. It is illustrative to compare the output resulting from the unshaped and shaped pulse propagation on a logarithmic scale (shown in fig. 23). In the former case, the structure of the pulse is substantial throughout its temporal profile. The optimized output pulse reveals, in contrast, no detectable wing structure over the same range, and a pulse shape approaching that of the original unshaped input pulse, especially on the leading edge. These results demonstrate the feasibility of fiber delivery of energetic
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Control 1
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Optimized out
~
0.1
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1E-3 500
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.
I
~"?~
-500
.
,
0
.
~-,
500
~
,
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time (fs)
Fig. 23. Comparisonof the initial unshaped pulse, the output pulse resulting from the unshaped pulse propagation, and the output pulse resulting from the optimally shaped pulse propagation plotted on a logarithmic scale to the extent of the dynamic range of the measurement. The effect of optimization is remarkable, especially in the wings of the pulse.
ultrashort pulses defeating the nonlinear propagation effects by appropriate preshaping of the input pulse through linear filtering combined with adaptive control. This result is particularly significant because (a) it allows delivery of femtosecond pulses of higher intensity through optical fibers, (b) it defeats the effect of higher order nonlinear components due to fiber properties and (c) it provides a way to compensate for nonlinear effects through linear filtering. From an applied point of view, the concept of precompensation promises to be a powerful approach for future incarnations of photonic networks. The technique provides a solution that advantageously harnesses the same nonlinearities that are presently regarded as a serious limiting factor in the overall network capacity, enabling the use of much shorter and more powerful pulses as carriers of data. From a general standpoint, the possibility of delivering ultrashort pulses through optical fibers provides many advantages. Extensions of this approach to various fiber types and lengths in conjunction with operation at different wavelengths can be foreseen as an important development for the delivery of ultrashort pulses for various applications in the sciences where remote delivery of reasonably powerful ultrashort pulses is needed. The choice of more refined decision criteria that are not focused on the delivery of the shortest pulse possible, can provide the delivery of shaped ultrafast pulses, further extending the ability to control physical processes on the ultrafast time scale to a, at times more convenient, fiber-optics-based realm.
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w 5. Photonic crystal fibers 5.1. Introduction
Research and development on optical fibers has been steady and has dramatically enhanced their properties and performance. For instance, fiber with improved clarity approaching the theoretical limit for SiO2 has been developed recently. Thanks to the use of dry heat sources, this fiber does not exhibit the OH absorption peak between the second and third window of transparency (at about 1.4gm) caused by water diffusion during the draw process, and this greatly extends the usable bandwidth available to communication systems (Thomas, Shraiman, Glodis and Stephen [2000]). An exhaustive overview of the technological advances in the field, however, is beyond the scope of this chapter. Perhaps one of the most disruptive recent designs, holding the promise of the greatest innovation, comprises the waveguide structures broadly classified as photonic crystal fibers (PCF). 5 These fibers have enhanced nonlinear guiding properties and novel dispersion features. The cross section of a PCF is fundamentally different from conventional optical fibers and typically exhibits a pattern of air holes around the core which can be either solid or hollow (fig. 24). In the former case, light is guided by total internal reflection, since the air-filling in the cladding lowers its effective index of refraction, whereas in the latter case a photonic bandgap defines the wavelength guiding range. Such a structure confines the light in the cladding and guides it as a single mode (Knight, Birks, Russell and Atkin [1996], Knight, Broeng, Birks and Russell [1998], Cregan, Mangan, Knight, Birks, Russell, Roberts and Allan [1999]). Progress in the design of these fibers is moving at a fast pace and the constituents are changing. While the majority of the PCF structures have been made out of undoped silica, variations in structure and composition of these fibers are continuously being explored. Gratings have been successfully written in these fibers (Eggleton, Kerbage, Westbrook, Windeler and Hale [2001]). Recently, erbium-doped and ytterbium-doped microstructured fibers have been reported (Wadsworth, Knight, Reeves, Russell and Arriaga [2000]) and photonic crystal fibers have been developed using polymers instead of silica as the constituent material (Van Eijkelenborg, Large, Argyros, Zagari, Manos, Issa, Bassett, Fleming, McPhedran, De Sterke and Nicorovici [2001]). Geometries with cylindrical, dielectric, multilayer omnidirectional reflecting
5 Alternatively, these fibers are called holey fibers, because of their morphology, or microstructured fibers.
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Fig. 24. Scanning electron microscope images of microstructured fibers: (a,b) solid-core fibers that typically guide by total internal reflection; (c,d) hollow-core fibers guiding light through a photonic band gap mechanism. [Images (b)-(d) courtesy of the University of Bath].
mirror boundaries for fibers with photonic bandgap guiding have also been proposed and are currently under development (Ibanescu, Fink, Fan, Thomas and Joannopoulos [2000]). Only a few results will be illustrated below; they mostly concern the propagation of femtosecond pulses in microstructured fibers that guide by total internal reflection. This field is in rapid evolution and the reader is referred to the exponentially increasing multitude of publications on the subject.
5.2. Experimental results." femtosecond pulses and PCFs The enhanced nonlinear effects of photonic crystal fibers come primarily from two physical properties: (1) the light is confined in a smaller-diameter core compared to single-mode fibers and (2) the dispersion properties of PCFs differ greatly from those of conventional fibers because of the difference in waveguide characteristics. The first spectacular demonstration (fig. 25) of the enhanced nonlinear effects in PCFs was achieved with the observation of broad supercontinuum radiation extending from ~400nm to 1600nm by coupling 0.8nJ, 100fs pulses from a Ti:sapphire laser into a 75cm segment of holey fiber (Ranka, Windeler and Stentz [2000]); the fiber was found to have anomalous dispersion in the visible region with a zero-dispersion wavelength estimated around 750 nm (depending on the structure of the fiber), a result later confirmed by other groups (Knight, Arriaga, Birks, Ortigosa-Blanch, Wadsworth and Russell [2000]). This dispersion feature extends the wavelength range over which optical solitons can
Femtosecond pulses in optical fibers
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[2, w 5
,,, I
I
I
I
I
800
1000
1200
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I
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I0 ~
.~ 10
-2
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i:
~ 10.4
10.6
,
400
,
600
..,
,
Wavelength (nm) Fig. 25. Optical spectrum of the supercontinuum radiation generated from coupling ~. = 790nm, 100 fs pulses of energy of 800 pJ in a 75 cm segment of photonic crystal fiber (after Ranka, Windeler and Stentz [2000]).
be formed to the visible. Such effects have been reported over a few meters of fiber at an operating wavelength of 850nm, albeit with conventional pulsedetection schemes (Wadsworth, Knight, Ortigosa-Blanch, Arriaga, Silvestre and Russell [2000]). Under these circumstances, phase-sensitive techniques can provide additional details on the nature and effects of nonlinearities. Frequencyresolved optical gating, with nonlinear crystal dithering to accommodate the wide bandwidth of the PCF output, has been applied to deconvolve the phase structure of the generated supercontinuum (Gu, Xu, Kimmel, Zeek, O'Shea, Shreenath, Trebino and Windeler [2002]), an important endeavor to try to recompress the broad spectrum to generate single-cycle optical pulses in the visible and to understand the mechanisms of supercontinuum generation in PCFs. The results for this case are presented in fig. 26. Experimental results from ultrashort pulse propagation at /~= 1550nm and their phase analysis have recently been performed in our group applying FROG techniques to observe soliton formation and pulse propagation properties in these novel fibers in the same vein as the results obtained for conventional fibers. These structures, given the much lower input pulse energies required to induce a nonlinear response in the medium, provide a convenient setting for the study of nonlinear phenomena, such as soliton-soliton interaction or frequencygeneration processes to name a few. With the experimental arrangement previously described in w3, observation of the N = 1 and N = 2 soliton was obtained thanks to the lowered threshold conditions for higher-order soliton generation. The fiber used is a microstructured fiber with a silica core 2.5 btm in diameter,
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Fig. 26. X-FROG measurement of supercontinuum radiation generated by coupling femtosecond pulses at 800nm in a microstructured fiber: (a) experimental data; (b) reconstructed FROG trace; (c) temporal intensity and phase profile; (d) retrieved spectral intensity. (Courtesy R. Trebino.)
suspended in air by a web of sub-micron silica strands with a cladding diameter of 90 ~tm (the actual fiber used in the experiment is shown in fig. 25a). The data are taken for average powers ranging between 3 and 25 mW, by varying the input power coupled to the PCE Oscillations (breathing) are noticeable in both the temporal and spectral domain as a function of power. The observation of a soliton at the output of the fiber link requires the appropriate combination of fiber length and pulse power while the pulse undergoes oscillations deriving from its initial chirp (further compounded at higher powers by the pulse self-frequency shifting). The FROG data show a pulse
Femtosecond pulses in optical fibers
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[2, w 5
1650
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= 1600 >
1550
9
0
,
10
9
f
9
,
,
,
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20 30 40 50 avg. power (mW)
9
i
60
Fig. 27. Intrapulse Raman scattering spectral shift as a function of input average pulse power for a segment of 25 centimeters of microstructured optical fiber, with a 2.5 ~tm diameter solid core compared to the self-frequency shift in a Z - - 9 0 meter single-mode fiber (Corning SMF-28) link. The maximum shift detected for the PCF is ~ 110 nm for imput pulses of 25 mW.
with a flat temporal phase for an average power of 6 mW at the fundamental wavelength. As the power is increased, the temporal phase varies and the pulse self-frequency shifts towards longer wavelengths. These shifts are particularly enhanced in microstructured and tapered fibers, and large shifts have been reported recently (Liu, Xu, Knox, Chandalia, Eggleton, Kosinski and Windeler [2001]). A comparison of the shift occurring in conventional fibers and in PCFs for the same input pulse conditions is shown in fig. 27. The enhancement of the nonlinear effects in PCFs can be appreciated by comparing these results to the self-frequency shift that occurs in conventional single-mode silica fiber. For similar input conditions (same initial pulsewidth and average power), the amount of shift observed for 95 centimeters of PCF is many times larger than the shift we previously detected in a 90-meter segment of conventional (Corning SMF-28) silica fiber. A flat temporal phase feature appears again for a pulse average power of 25 mW. The frequency shift in this short piece of PCF over this power range is in excess of 100nm, and after the N - 1 soliton is formed, the pulse widths throughout the entire shifted wavelength range stay in the hundreds of femtoseconds range. The two reconstructed FROG traces that represent the solitons are illustrated in fig. 28, revealing pulse durations of r x - i = 220 fs and T N = 2 - 130 fs. Expected values for soliton formation in this fiber are calculated using the published values for the dispersion equation for silica, a nonlinear index of n z - 2 . 6 10-z~ and an initial pulse duration of 170fs. Experimental evaluation of the GVD of this fiber yielded a [32 value b e t w e e n - 1 5 0 and -200 ps/nm kin.
Photonic co,stalfibers
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129
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Fig. 28. Reconstructed pulses from the experimentally detected FROG traces on a logarithmic scale for 170fs pulses at ~ = 1550nm propagating through 25 centimeters of photonic crystal fiber. The pulse widths are rX-- 1 = 220 fs and rx -- 2 = 130 fs, corresponding to pulse energies of 75 pJ (N = 1) and 340 pJ for the N = 2 soliton. Using these recovered pulse-width values, these results are compared to the theoretical sech 2 pulse shape (dashed line) on a log scale with good agreement. Both traces exhibit a flat temporal phase indicating temporal coherence, signature of an achieved soliton state. The experimentally detected evolution of the pulse width as a function of input average pulse power is shown. The dashed lines in the graph indicate the detected N = 1 and N = 2 soliton pulse widths.
Establishing the value for the effective area Aerr occupied by the mode is necessary for evaluating the soliton power. If the core were fully filled, the effective area would amount to 4.9 Ftm2. Assigning this value to calculate the predicted average powers for the soliton formation for the detected pulsewidths gives PN=! = 10mW for an operating wavelength of A= 1550nm, and PN--2= 9 8 m W at ~ = 1660nm (taking into account the variation of the
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Femtosecond pulses in optical fibers
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group-velocity dispersion for the PCF) which is inconsistent with experiment. The relationship between the effective area and the geometric area of the core, however, depends on the normalized frequency parameter of the fiber V, with the ratio Aeff/Acore decreasing with increasing V (Okamoto [2000]). The microstructured fibers used here have a large index mismatch between cladding and core. In this case, effectively, the core's index of refraction at this wavelength (calculated for the fundamental mode) is ncore= 1.5, whereas the cladding can be approximated to be constituted mostly of air, yielding a V parameter of "-6, which could suggest that the effective area Aeff is smaller than the actual core area. The effective mode area that is derived using the experimentally obtained parameters for the fundamental soliton gives a value of Ae~-- 1.6 ~tm2, significantly smaller than expected. Using this value to calculate the average power levels for the fundamental and second soliton gives PN= I = 5.87 mW at /l= 1550nm, and PN_-2=27mW at /l = 1660nm. These results are in excellent agreement with the values observed experimentally. A comparison of the sech 2 soliton waveform with a pulse width equal to the experimentally derived values and the reconstructed intensity profile obtained from the FROG data yields excellent agreement on a logarithmic scale. The comparison is shown also in fig. 28. The results from the NLSE, in these conditions, predict a N - 1 soliton forming at 6mW with a pulse width of 218.1 fs (energy 75pJ) and at 26.33 mW for a pulse width of 111.3 fs (energy 334 pJ), consistent with experiment. This also indicates that the previously employed modified NLSE can be used successfully in simulating pulse propagation in microstructured fibers, with appropriate parameter changes related to the geometrical structure of the fiber.
5.3. Frequency conversion As t h e / l = 1550 nm pulse propagates in PCFs, nonlinear conversion takes place and the generation of visible light is detected at the output. This frequency conversion process is found to differ substantially from the previously reported supercontinuum generation. During the experiments performed, it is observed that the /lvuyD = 1550 nm light propagates in a single mode through the fiber. At low input-pulse average powers, only the IR light corresponding to /lVUND is detected at the output. As the average power is increased above 10 mW (corresponding to a peak power per pulse of---735 W), green-blue light is visible at the output of the fiber exhibiting a six-lobed far-field modal pattern. Increasing the input pulse powers to values
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Fig. 29. FROG trace resulting from the propagation of a P=45 mW, /l = 1550nm, r = 150 fs pulse in 95 cm of PCE The shifted pulse has a center wavelength of 1714 nm and has a pulse duration of r = 129 fs, whereas the unshifted portion of the fundamental has r = 177 fs. The detected spectrum is also shown on the side of the trace with matching scale length. above 22 m W yields yellow radiation as well as green radiation. By filtering out the green light, it is observed that the yellow spatial mode exhibits the same farfield spatial characteristics as the green-blue light, i.e. it has a six-lobed pattern which appears slightly rotated when compared to the modal far-field pattern of the blue-green mode. Spectral analysis of the visible components reveals two distinct features peaking around ~ll = 517 nm for the green-blue component and ~12- 571 nm for the yellow radiation. A FROG trace of the resulting fundamental pulse at the output of the PCF at P - 4 5 m W is shown in fig. 29. For these conditions, both visible radiation components are generated at the output of the PCF, as described earlier. The FROG trace reveals two spectrally distinct pulses separated by 164 nm. Reconstruction of these data gives a pulse duration of rVUyD ----177 fs for the fundamental pulse at ilFUyD = 1550nm whereas the self-frequency shifted pulse, which is centered at Jtv,Shift-- 1714 nm, has a pulse width of rv, Shift 129 fs. The unshifted pulse is soliton-like as demonstrated by the relatively flat phase across the pulse; -
-
the shifted pulse is not a soliton due to its strongly cubic (and higher-order) phase function (Omenetto, Taylor, Moores, Knight, Russell and Arriaga [2001]).
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By comparing the spectral content of the split fundamental pulses with those obtained by spectral analysis of the visible output from the PCF, it can be seen that '~'FUND ~ 3~.1 and ).F,Shift ~ 3~2, or 30)FUND~ 0)l and 3o91:,Shift ~ 0)2. This reveals that a possible mechanism for the generation of visible-frequency components is temporal splitting followed by Raman self-frequency shift of the split pulse, with subsequent third-harmonic frequency conversion of both pulses. The conversion efficiency from the IR to the visible components is estimated to be 0.1%. It must also be pointed out that throughout the range of powers examined no second-harmonic radiation is detected and supercontinuum radiation is not generated. Modal calculations of the PCF have been performed by describing the fiber using a full vector model based on the supercell plane wave method (Ferrando, Silvestre, Miret, P. Andres and M.V. Andres [1999]). The input refractive index profile was a bitmap picture taken from an electron micrograph of the actual structure employed in these experiments. The material dispersion of silica was included by applying the published Sellmeier coefficients. The model provides the modal indices and the transverse modal intensities of all of the guided modes of the structure. The calculated modal index for the fundamental mode propagating in the PCF at 1550nm is n= 1.38, and that at 510nm is n = 1.45. This clearly does not provide the phase matching required for third-harmonic generation. Higherorder modes propagating at 510 nm form a near continuum of decreasing modal indices, and there is a group of modes with a six-lobed near-field pattern whose index of refraction is in the range n = 1.40-1.42. A comparison of the calculated mode field pattern with the experimentally observed far-field mode pattern shows a mode of the correct shape whose modal index is close to the calculated index of the fundamental mode at 1550 nm to allow for phase matching (Omenetto, Taylor, Moores, Knight, Russell and Arriaga [2001 ]). The modal indices of highorder modes are expected to be particularly sensitive to the exact shape and size of the fiber core, so it is quite possible that the true index matching in the fiber is better than calculated.
5.4. Polarization control o f nonlinear conversion in PCF
More interestingly, experimental measurement on the cobweb microstructured fiber have revealed the existence of a control parameter which governs the generation of specific visible components in the microstructured fibers. The propagation of pulses of fixed energy, yet polarized along different directions, yields different visible components at the output, suggesting polarization-
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Fig. 30. Plot of the calculated modal indices (N = 5-25) for ~.= 510nm (squares), ~, = 520nm (circles) and 2. = 530 (triangles), and modal index for the fundamental as a function of wavelength. The inset shows the calculated far-field profile of the N = 21 mode which bears similarity to what is observed experimentally, and for which phase-matching conditions can be identified. For further details see the text.
dependent selectivity for phase-matching according to the input polarization state. These results have been obtained for lower average power values (i.e. up to 25 mW). In this range, the fundamental wavelength ~,VUND= 1550 nm appears to propagate in the fundamental mode through the PCF 6. The input pulse power is controlled by a ~,/2 waveplate and a polarizing cube combination, placed in front of the fiber input. An additional waveplate controls the polarization state of the input to the PCE An intensity transmission analysis as a function of input polarization reveals two principal polarization states ct and 13 orthogonal to one another. When the input pulses are polarized along one of these two directions, one of two visible modes of different colors is detected at the output. These generated visible frequencies are higher-order modes exhibiting an eight-lobed structure and centered at Al = 5 1 4 n m and ~,2 =533 nm (fig. 32, below). It has been noted in w5.3 that the observed components ~,l and ~,z bear a relation to the fundamental such that /~VUND~3/~I and /~F,Shift,~3/~2, (or 3COVtmD~ COl and 3~ov,shift ~ 092, with ~,~.shift the Raman self-frequency shifted portion of the fundamental). The mode structure detected experimentally is, once again, compared to the calculated pattern evaluated by modal calculations of the PCF, and is illustrated in fig. 30.
6 It must be observed that the fiber is not single-mode at the fundamental wavelength.
Femtosecond pulses in optical fibers
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(a) ~ 25 E"
__
1500
---
1550
J
1600
1650
1700
25
o~ 1500
1550 1600 1650 wavelength (nm)
1700
Fig. 31. Spectrum of the fundamental radiation at the output of the photonic crystal fiber as a function of power. The shift at P = 25 mW (indicated by the arrow) is the same for both polarization states.
The modal index of the fundamental mode of the PCF at 1550nm is n = 1.38, whereas higher-order modes propagating around 510nm form a near continuum of decreasing modal indices. Eight-lobed pattern modes whose index of refraction is in the range n= 1.38-1.40 indicate the existence of a correctly shaped mode whose index is close to the calculated fundamental mode (with caution imposed by the model sensitivity to the actual PCF morphology). This is reflected in the graph by adding error bars (0.2%) to the calculated indices. The calculated mode field pattern is also shown in the inset of fig. 30 and closely resembles the experimentally observed one. A power-dependent spectral analysis is carried out on the fundamental and on the visible harmonics for the two different polarizations. The detected spectra for the fundamental pulse are illustrated in figs. 31 a,b, while the results for the generated visible harmonics are shown in figs. 32a,b along with the experimentally observed far-field modes. A comparison between the spectral features of the fundamental pulse polarized along the directions (x (fig. 31a) and [~ (fig. 31b) shows different spectral dynamics as a function of power but a fundamentally similar behavior: in
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Fig. 32. Detected spectra of the visible radiation components which appear at the output of the photonic crystal fiber as a function of power for both polarization states a and [~. The component at 533 nm appears only in one polarization state. The visible spectral features are acquired with a 16-bit CCD camera and the images are integrated over variable time windows to insure maximum sensitivity. Also shown are the far-field modes detected at the output of the microstructured fiber for the two principal polarization states. The input to the fiber are pulses at/~ = 1550nm, r ~ 170 fs, with average power of 25mW. The two outputs are centered at (a) ~,=514nm and (b)/~=533 nm. both cases the spectrum shifts as expected toward longer wavelengths and the magnitude o f the shift (--'50 nm at P - 2 5
m W ) appears to be the same for both
input polarization orientations (independent o f the initial polarization). Since the SRS process is an intensity- (i.e. pulsewidth-) dependent process, this suggests that the dispersion properties are similar for both polarization states. The results shows a clear difference in dependence o f the input polarization state. In the direction a (fig. 32a) a spectral feature centered around ~, = 514 nm is detected for pulses o f average power o f 4 m W at 1550 nm. As the power o f the fundamental is increased, more light is converted to the visible (with an estimated conversion efficiency o f 0.2% for a 25 m W pump) but only a feature at /~l = 5 1 4 n m is observed. The relationship o f this c o m p o n e n t to the fundamental wavelength is consistent with the generation o f a third harmonic o f the fundamental, yet no third harmonic from the self-shifted c o m p o n e n t is observed. W h e n the input pulse polarization is rotated along the direction [3 (fig. 32b), however,
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a sharp feature centered at ~ 2 - - 5 3 4 nm appears for pulses of average powers of 16 mW in addition to the /~l component that subsequently vanishes as the power of the fundamental is further increased. Whereas for both polarizations the fundamental shifts to 1600nm (and beyond for higher input powers), it is only for input pulses polarized along the 13direction that the shifted fundamental is converted to third-harmonic radiation. It must be stated that a combination of the two states (i.e. an input pulse polarized at 45 degrees with respect to the axes a, 13) generates the two visible components simultaneously, a situation that we most likely encountered previously. It is also interesting to note that no frequency components attributable to mixing processes (Sharping, Fiorentino, Coker, Kumar and Windeler [2001]) are detected for the wavelength interval covered by the fundamental and its shifted components. These results provide evidence for the possibility of using the pulse polarization state as the control parameter for the nonlinear processes which occur in a photonic crystal fiber using femtosecond pulses at a wavelength of 1550 nm. Polarization selectivity combined with Raman self-frequency shifting and thirdharmonic generation allows control of the generation of specific harmonics. This approach can provide an interesting path for all-fiber signal control and ultrafast optical switching based on nonlinear phenomena in microstructured fibers.
w 6. Final remarks The evolution of the next generation of optical building blocks that can rival today's electronics, arguably, requires new structures that are to photons what semiconductors are to electrons (Yablonovitch [2001 ]), operating at high speed in excess of present electronic means. Original architectures which operate in this context have to contend with some of the issues that are recognized limits existing in current optical systems. It is inescapable to associate fibers with telecommunications given the central role they occupy in the modern information age. From this point of view, some of these limits are identified, for instance, with physical capacity of a single channel, data rates, coding and quantum limitations. While most present commercial systems operate in the linear regime, the drive towards higher capacity and faster speeds is naturally pushing the timescales into the femtosecond realm, and as a consequence, in a nonlinear operation regime. Ultrafast pulses show promising potential for telecommunication applications (Knox [2000]). The alternative to avoiding nonlinear effects is to be able to control them and integrate their use. The most obvious example of useful nonlinearity at work in a telecommunications context is the optical soliton,
2]
Acknowledgements
137
which has been of central interest for over two decades. With the ability to design femtosecond pulses, a correspondingly designed coherent nonlinear process can be exploited: such an approach can be of key importance to perfect applications which require ultrafast speed in support of all optical interconnects, switches and logics. Pulse shaping has other implications that go beyond the control of the light-matter interaction. In an optical soliton, for instance, photons within the same pulse envelope can be phase-locked to create entangled states (Sizmann and Leuchs [1999]), which provides a liason between quantum information processing and ultrafast operation in optical fibers. Novel structures such as photonic crystal fibers are undergoing a very rapid growth, and their appeal certainly centers on their enhanced nonlinear properties, but also on the selective photonic bandgap guidance of light. Improvement in the design of these structures will provide a new way to think about how to propagate and process light, a statement which is equally compelling in its dual form, in which the need to process light in certain ways will guide the design of new structures. The implications of mastering the nonlinear interaction of new materials with ultrafast light are far-reaching: bringing together coherent control of processes which take place on the femtosecond scale and novel photonic structures and photonic crystal fibers provides a rich context for applications spanning from new compact laser sources to novel ultrahigh capacity communication approaches, from all-optical signal processing to all-optical computing, and from optical probes to optical sensors. With fundamental limits approaching, ultrafast controlled nonlinear photonics seems to be a very promising avenue to find solutions.
Acknowledgements The experimental work presented here was carried out at Los Alamos National Laboratory in Antoinette Taylor's Laboratory for Ultrafast Material and Optical Science (LUMOS). Many many thanks to Toni for letting me pursue this research and to the Los Alamos Office of the Director for the support through the J. Robert Oppenheimer Fellowship. The author wishes to gratefully and enthusiastically acknowledge the people who have shared in this w o r k - Anatoly Efimov, Dzmitry Yarotski, Jeff Nicholson, Mark Moores, Dave Reitze, Ben Luce, Yeojin Chung, and the collaborators at the University of Bath (Jonathan Knight, William Wadsworth, Philip Russell, and Jesus Arriaga) for their help, comments and critiques.
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Femtosecond pulses in optical fibers
[2
References Abdullaev, EKh., S.A. Darmanyan and P.K. Khabibullaev, 1993, Optical Solitons (Springer, Berlin). Agrawal, G.P., 1989, Nonlinear Fiber Optics (Academic Press, Orlando, FL). Akhmanov, S.A., V.A. Vysloukh and A.S. Chirkin, 1992, Optics of Femtosecond Laser Pulses (Springer, Berlin). Assion, A., T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle and G. Gerber, 1998, Science 282, 919. Bardeen, C., V.V. Yakovlev, K.R. Wilson, S.D. Carpenter, PM. Weber and W.S. Warren, 1997, Chem. Phys. Lett. 280, 151. Barry, L.P., J.M. Dudley, P.G. Bollond, J.D. Harvey and R. Leonhardt, 1996, Electron. Lett. 32(25), 2339. Bartels, R., S. Backus, E. Zeek, L. Misoguti, G. Vdovin, I.P. Christov, M.M. Murnane and H.C. Kapteyn, 2000, Nature 406, 164. Chang, C.C., H.P. Sardesai and A.M. Weiner, 1998, Opt. Lett. 23, 283. Chernikov, S.V., and P.V. Mamyshev, 1991, J. Opt. Soc. Am. B 8(8), 1633. Chilla, J.L., and O.E. Martinez, 1991, IEEE J. Quantum. Electron. 27, 1228. Christodoulides, D.N., and R.I. Joseph, 1985, Appl. Phys. Lett. 47(2), 76. Chu, K.C., J.P Heritage, R.S. Grant, K.X. Liu, A. Dienes, W.E. White and A. Sullivan, 1995, Opt. Lett. 20, 904. Clark, S.W., EO. Ilday and EW. Wise, 2001, Opt. Lett. 26(17), 1320. Collings, B., M.L. Mitchell, L. Boivin and W.H. Knox, 2000, IEEE Photonics Technol. Lett. 12,906. Cregan, R.E, B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, P.J. Roberts and D.C. Allan, 1999, Science 285, 1537. Desem, C., and P.L. Chu, 1986, Opt. Lett 11(4), 248. Diddams, S.A., H.K. Eaton, A.A. Zozulya and T.S. Clement, 1998, Opt. Lett. 23(5), 379. Diels, J.C., J.J. Fontaine, I.C. McMichael and E Simoni, 1985, Appl. Opt. 24, 1270. Diels, J.C., and W. Rudolph, 1996, Ultrafast Laser Pulse Phenomena (Academic Press, New York). Drexler, W, U. Morgner, EX. Khrtner, C. Pitris, S.A. Boppart, X.D. Li, E.P. Ippen and J.G. Fujimoto, 1999, Opt. Lett. 24, 1221. Dudley, J.M., L.P. Barry, P.G. Bollond, J.D. Harvey and R. Leonhardt, 1998, Opt. Fib. Technol. 4, 237. Dudley, J.M., L.P. Barry, PG. Bollond, J.D. Harvey, R. Leonhardt and PD. Drummond, 1997, Opt. Lett. 22, 457. Dudley, J.M., E Gutty, S. Pitois and G. Millot, 2001, IEEE J. Quantum Electron. 37(4), 587. Dugan, M.A., J.X. Tull and W.S. Warren, 1997, J. Opt. Soc. Am. B 14, 2348. Efimov, A., M.D. Moores, N.M. Beach, J.L. Krause and D.H. Reitze, 1998, Opt. Lett. 23, 1915. Eggleton, B.J., C. Kerbage, P.S. Westbrook, R.S. Windeler and A. Hale, 2001, Opt. Express 9(13), 698. Ferrando, A., E. Silvestre, J.J. Miret, P. Andres and M.V. Andres, 1999, Opt. Lett. 24, 276. Fittinghoff, D.N., J.L. Bowie, J.N. Sweetser, R.T. Jennings, M.A. Krumbtigel, K.W DeLong, R. Trebino and I.A. Walmsley, 1996, Opt. Lett. 21,884. Franco, M.A., H.R. Lange, J.-E Ripoche, B.S. Prade and A. Mysyrowicz, 1997, Opt. Commun. 140, 331. Froehly, C., B. Colombeau and M. Vampouille, 1983, in: Progress in Optics, Vol. 20, ed. E. Wolf (North-Holland, Amsterdam) ch. 2. Golovchenko, E.A., E.M. Dianov, A.N. Pilipetskii, A.M. Prokhorov and V.N. Serkin, 1987, JETP Lett. 45(2), 91. Gordon, J.P., 1986, Opt. Lett. 11(10), 663.
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Omenetto, EG., B.P. Luce, D. Yarotski and A.J. Taylor, 1999, Opt. Lett. 24, 1392. Omenetto, EG., J.W. Nicholson and A.J. Taylor, 1999, Opt. Lett. 24(23), 1780. Omenetto, EG., D.H. Reitze, B.P. Luce, M.D. Moores and A.J. Taylor, 2002, IEEE J. Sel. Topics Quantum Electron. 8(3), 690. Omenetto, EG., D.H. Reitze, M.D. Moores and A.J. Taylor, 2001, Opt. Lett. 26(12), 938. Omenetto, EG., A.J. Taylor, M.D. Moores, J.C. Knight, P.St.J. Russell and J. Arriaga, 2001, Opt. Lett. 26(15), 1558. Omenetto, EG., D. Yarotski and A.J. Taylor, 2000, in: Ultrafast Phenomena XII, Proc. Int. Conf., Charleston, SC, USA, July 9-13, 2000, ed. T. Elsasser (Springer, New York) p. 186. Orsterberg, U., and W. Margulis, 1987, Opt. Lett. 12, 57. Peatross, J., and A. Rundquist, 1998, J. Opt. Soc. Am. B 15(1), 216. Piccinin, D., M. Arioli, M. Fer6 and M. Martinelli, 2002, unpublished data, presented at CORECOM, Milano, Italy. Ranka, J.K., R.S. Windeler and A.J. Stentz, 2000, Opt. Lett. 25, 25. Remoissenet, M., 1999, Waves Called Solitons, 3rd rev. Ed. (Springer, Berlin). Rulli~re, C., 1998, Femtosecond Laser Pulses (Springer, Berlin). Sardesai, H.P., C.C. Chang and A.M. Weiner, 1998, J. Lightwave Technol. 8, 478. Schmidt, V., W. Husinsky and G. Betz, 2000, Phys. Rev. Lett. 85, 3516. Sharping, J., M. Fiorentino, A. Coker, P. Kumar and R.S. Windeler, 2001, Opt. Lett. 26(14), 1048. Shen, S., and A.M. Weiner, 1999, IEEE Photonics Technol. Lett. 11, 827. Siders, C.W., A.J. Taylor and M.C. Downer, 1997, Opt. Lett. 22(9), 624. Sizmann, A., and G. Leuchs, 1999, in: Progress in Optics, Vol. 39, ed. E. Wolf (North-Holland, Amsterdam) ch. 5. Squier, J., and M. Muller, 2001, Rev. Sci. Instrum. 72(7), 2855. Taira, K., and K. Kikuchi, 2001, Electron. Lett. 37(5), 311. Takenouchi, H., H. Tsuda, K. Naganuma, T. Kurokawa, Y. Inoue and K. Okamoto, 1998, Electron. Lett. 34, 1245. Thomas, G.A., B.I. Shraiman, P.E Glodis and M.J. Stephen, 2000, Nature 404, 262. Thomson, M.D., J.M. Dudley, L.P. Barry and J.D. Harvey, 1988, Opt. Lett. 23(20), 1582. Tournois, P., 1997, Opt. Commun. 140, 245. Treacy, E.B., 1971, J. Appl. Phys. 42, 3848. Trebino, R., 2002, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic Publishers, Boston). Trebino, R., K.W. DeLong, D.N. Fittinghoff, J.N. Sweetser, M.A. Krumbfigel and D.J. Kane, 1997, Rev. Sci. Instrum. 68, 3277. Trebino, R., P. O'Shea, M. Kimmel and X. Gu, 2001, Opt. Photonics News 12(6), 22. Van Eijkelenborg, M.A., M.C.J. Large, A. Argyros, J. Zagari, S. Manos, N.A. Issa, I. Bassett, S. Fleming, R.C. McPhedran, C.M. De Sterke and N.A.P. Nicorovici, 2001, Opt. Express 9(7), 319. Vdovin, G., P.M. Sarro and S. Middelhoek, 1999, J. Micromech. Microeng. 9(2), 8. Verluise, E, V. Laude, Z. Cheng, C. Spielmann and P. Tournois, 2000, Opt. Lett. 25(8), 575. Wadsworth, W.J., J.C. Knight, A. Ortigosa-Blanch, J. Arriaga, E. Silvestre and P.St.J. Russell, 2000, Electron. Lett. 36(1), 53. Wadsworth, W.J., J.C. Knight, W.H. Reeves, P.St.J. Russell and J. Arriaga, 2000, Electron. Lett. 36(17), 1452. Weiner, A.M., 1995, Progr. Quantum Electron. 19, 161. Weiner, A.M., 2000, Rev. Sci. Instrum. 71, 1929. Weiner, A.M., J.P. Heritage, R.J. Hawkins, R.N. Thurston, E.M. Kirschner, D.E. Leaird and W.J. Tomlison, 1988, Phys. Rev. Lett. 61(21), 2445.
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E. Wolf, Progress in Optics 44 9 2002 Elsevier Science B. V. All rights reserved
Chapter 3 Instantaneous optics of ultrashort broadband pulses and rapidly varying media by
Alexander B. Shvartsburg Central Design Bureau for Unique Instrumentation of the RAS, Butlerov Str. 15, Moscow, Russia
and
Guillaume Petite Laboratoire des Solides IrradiOs, CEA/DSM/DRECAM, CNRS (UMR 7642) and Ecole Polytechnique, 91128, Palaiseau, France
143
Contents
Page Introduction
. . . . . . . . . . . . . . . . . . . . . .
145
w 1.
Ultrashort EM pulses: how they are modeled, produced, measured
145
w 2.
Spatiotemporal reshaping of ultrashort pulses in stationary media
163
w 3.
Optics of instantaneous media
. . . . . . . . . . . .
196
w 4.
Conclusion
.
211
Acknowledgements
.
.
.
.
.
.
.
. . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . .
144
212 212
Introduction
This chapter is devoted to the physical fundamentals and mathematical basis of the optics of waveforms whose parameters vary in the course of propagation. The dynamics of instantaneous optical fields, travelling in free space and continuous media, opens many new opportunities for controlled spatiotemporal reshaping of these fields. The ongoing interest in such problems is fueled by several research goals: - to optimize the processes of light pulse transfer through optical systems; this is particularly important in view of the applications to optical communication: the use of ultrashort pulses at high repetition rates is one of the approaches to increase of the transfer rate. - to develop methods for fast non-destructive testing of materials and targets, and for atmospheric sensing, using ultrashort broadband electromagnetic (EM) pulses; - t o reach a comprehensive understanding of such ultrafast processes as amplitude or phase modulation of EM waves interacting with non-stationary media. Moreover, an important task is to elaborate an analytical approach to these topics, which until recently were considered an exclusive field of computer simulations. The investigation of coupled spatial and temporal deformations of localized waveforms is preceded here by a brief description of waveforms widely used in modeling such processes. Both frequency- and time-domain models are presented in w 1.1. We then briefly recall how such waveforms are produced (w 1.2), and measured experimentally (w 1.3). Two opposite statements of the problems of instantaneous optics will then be treated. Section 2 focuses on the spatiotemporal dynamics of localized EM pulses interacting with stationary dielectric media. On the contrary, the reshaping of harmonic CW trains interacting with non-stationary media is discussed in w3. In this latter case, special attention will be given to some exactly solvable models, providing a better physical insight into these problems. w 1. Ultrashort EM pulses: how they are modeled, produced, measured
It is first necessary to agree upon the definition of an ultrashort electromagnetic 145
146
Instantaneous optics of ultrashort broadband pulses and rapidly valying media
[3, w 1
pulse. In the following we will adhere to the following definition: a pulse whose duration is of the order of a few, at most a few tens of, periods of the EM field. It is of course almost equivalent to consider pulses whose spectral width is a substantial fraction of their frequency, except that in this case one rather addresses the coherence length of the pulse than its duration, so that this definition would be strictly valid for radiation with complete temporal coherence only, a criterion which is not even satisfied by all lasers. In this section, we will first recall some mathematical models for such pulses (w 1.1), and then briefly recall how such pulses are produced (w 1.2) and measured (w 1.3).
1.1. Non-sinusoidal waveforms of electromagnetic waves Let us begin this analysis from the traditional spectral approach; to optimize a waveform E(t) = EoF(t) with respect to a bandwidth-limited communication system or to the width of an absorption line, one works with its Fourier transform (FT). Some properties having an essential influence on the spectral bandwidth ofwaveforms F(t), such as their duration and rise time, are considered below.
1.1.1. Square-shaped truncated train of monochromatic waves A square-shaped truncated train of monochromatic waves is represented by the function
F(t)
= ~f cos(~oot);
Itl ~ Itol,
( 0;
It
(1.1)
> Itol.
The FT of this waveform is
F,o(co) =
sin[t0(co0 - co)] coo
--
(1.2)
co
The spectral density of EM energy E(A) in a pulse (1.1), localized in some finite spectral range (co + 89 can be written as 1
E(a) = E012 W(a);
1 f~'~" + ffA
W(a) = ~-[~,,,,, 89 IF,,, 12 dco.
(1.3)
The spectral density E calculated for an infinite spectral range, A ---+c~, is given by the function W(cxD). The spectral bandwidth of a pulse is defined as the range
Ultrashort EM pulses: how they are modeled, produced, measured
3, w 1]
147
of frequencies ~o0 + k containing some given fraction 6 (e.g., 6 = 90% or 99%) of the pulse energy: (1.4)
W(A) = 6 x W(oo).
Substituting the Fourier amplitude (1.2) into (1.3) and introducing a variable x = toA we obtain to[ W(A) = ~ Si(2x)
sin2x 1 x
(1.5)
where Si(x) is the integral sine function, Si(x)
=f 0
sin y dy. Y
(1.6)
Using the value S i ( ~ ) = ~1 , one can present eq. (1.4), governing the value of spectral-time product Kc = to&,, in the form Si(2K~.)
sin 2 Kc Kc
-
1~6.
(1.7)
Considering the central part of the distribution F,,, (1.2), located between the zeros of F,,, (Kc = -+-~), one can see that 90% of the pulse energy is contained in the spectral range ~Oo+;r/to. Any further significant growth of this fraction 6 requires a substantial increase of spectral range: for 6=0.99, the spectral range Kc is increasing by more than a factor 10. 1.1.2. Influence o f increasing rise and fall time
It is worthwhile to compare the broadband square-shaped pulses, characterized by a short rise time, tending formally towards zero, with other waveforms possessing the same characteristic duration, but a finite rise time. To illustrate the influence of increasing rise and fall times, let us consider the Gaussian waveform
(,2)
F(t) = exp -~202 .
(1.S)
the FT of a Gaussian waveform is known to be Gaussian as well: 1 F.) = tox/2--~ exp [-~(COto):] .
(1.9)
148
Instantaneous optics of ultrashort broadband pulses and rapidly valying media
[3, w 1
It is often convenient to characterize the pulse by its full width at halfmaximum (FWHM). The Gaussian waveform (1.8) and its FT (1.9) are defined by the FWHM values tc and co,.: tc = 2.35t0;
coc-
2.35
;
t, co,.- 5.55.
(1.10)
to
Using the first of relations (1.10), one can present the Gaussian profile (1.8) in the form, expressed via the FWHM, (1.11) Calculating the function W ( A ) (1.3), related to the FT (1.9) we obtain the equation governing the bandwidth A~.: erf(t0Ac) = 6,
(1.12)
with erf the error function. Taking, e.g., 6=0.9, we have K~. = toAc. = 1.17, and thus the bandwidth Ac of the Gaussian pulse (1.8) is almost three times smaller than that of the square-shaped pulse (Kc = Jr for 6=0.9). This example shows that an increase of rise time, at fixed pulse duration, induces a strong spectral narrowing. Some pulses with extremely steep leading edges have been proposed as prospective carriers for directed energy transmission in free space. Such a pulse, coined by Wu [ 1985] "electromagnetic missile", yields a diffracted wave decaying slower than z -2 in intensity. Considering a broadband pulse passing through a plane circular aperture with radius a, one can see that the boundaries of the first Fresnel zone zv = coa2c -~ for high frequencies co become extremely distant. Wu [1985] showed that when the pulse spectrum is damping slowly in the high-frequency limit co ~ ~ , the energy collected on a detector of finite dimensions, even though still tending to zero when z ~ ~ , can be made to do so in an arbitrarily slow manner. An example of such pulses, expressed as a function of time via the modified Bessel function K,, of vth order, is
Taking the cosine FT from eq. (1.13), we obtain 1
F~o = v / ~ 2 v - l t ~ + -~) 2 2 (1 + co0t0) ''+ 1/2
(1.14)
Writing v in the form v = -~i + y; 7, > 0, one finds the asymptotic spectral behavior of F,,,],,~-~ ~: ~ co-2:,; thus, following Wu [ 1985], an appropriate choice
3, w 1]
UltrashortEM pulses." how they are modeled, produced, measured
149
of parameter )' yields an arbitrarily slow decrease of the high-frequency limit of eq. (1.14), providing a weakened angular divergence of the pulse at distances
z ~ o)a2c -1
1.1.3. Statistical fluctuations Until now we have discussed ways to broaden or to narrow the spectral bandwidths for waveforms of a given duration 2t0. Steepening of the pulse fronts was shown to cause an increase of the spectral bandwidth; their softening can give rise to a narrowing of the spectrum. However, it should be mentioned that there is an implicit simplification in all the above developments: we have not allowed statistical fluctuations between the different spectral components of the EM field we studied. Any classical source will present such fluctuations, as well as do most lasers. All the above developments thus deal with temporally coherent sources such as single-longitudinal-mode or mode-locked lasers. One often refers to such pulses as "Fourier-Transform limited pulse" (in the sense that the bandwith of a partially coherent pulse will be larger than strictly required by the FT). It should be mentioned that there is a stricter definition of a FT-limited pulse, as that having the narrowest pulse width possible for a given spectral amplitude distribution. The term "amplitude" may be misleading here since the spectral amplitudes are in reality complex, and the actual temporal shape of the pulse will depend on the shape of the "spectral phase" q~(~o). Figure 1 illustrates a)
b)
1.5
(5)
-3 -1
/~ ~r
(3)
~
!
a",.,.at
x
0 U
Fig. 1. (a) Asymmetrical pulse 1 and symmetrical transform-limited (TL) pulse 2, plotted vs. the normalized time u = t/to characterized by the same normalized spectral envelope. (b) Spectral envelope (curve 3) y = F(x)/F(O), m - r x = t0(~o- cot)). Unlike the spectral phase q~ of pulse 1, shown by curve 4, the frequency dependence of the phase of the TL pulse (curve 5) is linear.
150
Instantaneousoptics of ultrashort broadband pulses and rapidly ualying media
[3, w 1
this point by comparing two pulses having the same power spectrum (If(~o)[2), but different spectral phase behaviors. It shows that the pulse with the minimum width (and in this sense, strictly speaking FT-limited) is the one presenting a linear spectral phase frequency dependence. The physical and computational problems that resulted from efforts to adjust the Fourier "oJ-k" language to the dynamics of ultrashort waveforms in dispersive media stimulated the development of time-domain models, i.e. models directly using the temporal dependence of electric and magnetic field strength for such waveforms. 1.1.4. Models o f pulses with unequal rise and fall times
So far we have considered models of pulses with equal rise and fall times. However, many real optical pulses do not have such symmetry. Qian and Yamashita [ 1991 ] discussed the example of a strongly asymmetrical waveform described by a Gaussian rising edge followed by an exponentially falling tail:
E(t)
0exE
=
\ '~
E 0 e x p ( - •" , , , ,
]
;
t <<.O,
(1.15)
t~>0,
This so-called single-cycle exponential waveform has a discontinuity of the derivative at the maximum t=0. Ma and C iric [1992] analyzed transient scattering on small targets using a double-exponential waveform possessing a continuous derivative at the maximum:
For t l >> t: the profile (1.16) resembles a single-sided waveform, but differing from eq. (1.15) by the smooth maximum. Flexible models, describing continuous waveforms with an arbitrary amount of different extrema and unequal distances between zero-crossing points, were used in time-domain optics by Shvartsburg [1999]. These waveforms, characterized by well-expressed leading fronts with finite slope, an arbitrary number of anharmonic oscillations and exponentially damping tails, are defined by the series of Laguerre functions L,, in the time interval 0 ~
a,,L,,
t
~
.
(1.17)
3, w 1]
Ultrashort EM pulses." how the), are modeled, produced, measured 1 , 0 ] ~ " '
i
'
I
'
I
'
J
'
I
'
i
'
I
'
I
'
I
151 '
0,5
g 0,0
-0,5 0
1,0
I
l
1
2
,
3
I
,
4
I
~
5
I
,
6
I
,
7
I
~
8
I
9
'
I
'
i
'
i
'
i
'
0,8
,
!
,
10
b)
0,6
~
F1
---o--
F2
0,4 i ~ 0,2 0,0 -0,2 J
I
-0,4 0
,
2
I
4
,
I
6
,
I
8
,
10
Fig. 2. (a) Laguerre envelopes, Lo(x ) (squares), Li(x) (circles), L2(x) (diamonds); (b) unipolar, Fl(x)-- L o ( x ) - L l ( x ) (squares), and bipolar, Fz(x)= Lo(x)- L2(x) (circles), waveforms, x - t/to.
The Laguerre functions (fig. 2)
L,(x) = exp(Ix) d" [x" n!
dx"
exp(-x)]
(1.18)
are known to be orthonormal in the interval 0 ~< x ~< oo, that is" oc
/ L,(x) Lm(x) dx = 6,,,,,. o
(1.19)
152
Instantaneous optics of ultrashort broadband pulses and rapidly ualying media
[3, w 1
These waveforms, localized in time, are suitable for description of plane wave pulses. However, to consider the dynamics of both spatially and temporally localized fields, other families of waveforms are needed. 1.1.5.
Poisson-spectrum
pulses
The spatiotemporal structure of few-cycle three-dimensional pulsed wave beams can be described by means of so-called Poisson-spectrum pulses" it0 F(t) = Ref(t);
f (t) =
/ '~
t' + ito
'
(1.20)
with R e f the real part of the functionf(t); to > 0 and m ~> 1 are free parameters, t' is the retarded time for points located on the beam axis t ' = t - z c - l . This model, discussed by Porras [ 1998], is suitable for waveforms of any duration and with an arbitrary number of oscillations (fig. 3). The FT of waveform (1.20) gives the Poisson spectrum, also called the power spectrum F~,, = ~t~'
exp(- I~o t0).
r(m)
(1.21)
Unlike the waveforms mentioned above, the function (1.20) can be used for modeling the spatiotemporal evolution of narrow directed pulsed beams I
'
I
'
I
'
'
I
'
I
'
I
0,5-
-
0,0
-0,5
-
9 -3
'
" -2
'
' -1
'
' 0
' 1
-
'
' 2
'
' 3
X
Fig. 3. Poisson-spectrum pulses (1.20) for the values o f parameter m = 1 (curve 171 - 2 (curve 2) are plotted vs. the normalized time x = t/to.
l) and
UltrashortEM pulses: how they are modeled, produced, measured
3, w 1]
153
with curvilinear wavefronts. To describe the non-stationary three-dimensional structure of such beams one has to replace the retarded time t' by the shifted time t' - r 2 / 2 c R , where R and r are the radius of wavefront curvature and the distance between the beam axis and the observation point on the wavefront. This nonseparable waveform, containing both temporal and spatial variables, provides a useful analytical tool for investigation of coupled diffraction and dispersioninduced distortions of localized fields. The spatiotemporal dynamics of all the waveforms mentioned above must be investigated by means of relevant solutions of Maxwell equations. Below we recall, on the contrary, some localized waveforms which are themselves packetlike solutions of Maxwell equations in a free space. 1.1.6.
Packet-like solutions
Search of packet-like solutions of the wave equation in free space 02U OZu 02U Ox---T + - - ~ + Oz 2
1 02U c20t 2 - 0
(1.22)
led to the following family of solutions, presented by Bateman [1955] in the form (1.23)
U ( r , t) = g ( r , t ) f (O),
where r stands for the spatial variables, 0 = O(r, t) is a solution of the eikonal equation
(00)2 (oo) (oO) l(OO) ~-
+
-~-
+
-~z
-Y
-07
=0,
(1.24)
and f is an arbitrary function. Two simple examples of waveform-preserving solutions (1.23) are (a) plane waves: 0 - z - ct, and (b) spherical waves: 0 = R - ct. A third class of solutions (1.23), reported by Hillion [1983], is based on r2
0 = z - ct
+
z + ct - i b '
(1.25)
with b an arbitrary positive constant, having the dimension of a length. Making use of eq. (1.25) one can express the solution U of wave equation (1.23) via an arbitrary function f(0): U =
f(O) z + ct-
(1.26) ib"
154
Instantaneous optics of ultrashort broadband pulses and rapidly varying media
[3, w 1
The free choice of function f ( 0 ) provides a remarkable flexibility in the modeling of localized fields. For instance, Brittingham [1983] chose
f(O) -
exp(iK0),
(1.27)
where K is a free real parameter. Separation of real and imaginary parts in (1.27) permits separation of the amplitude and phase factors in the field presentation (1.26):
exp(-r2/l~) { b z+ctr21} U1 = V/(z + ct)2 + b2 exp -i K(z - ct) + z~+ct ~ b l~
(1.28)
This field is localized around the direction of propagation (z-axis) as a Gaussianlike pulse with transversal width
l•
r (z + ct)2 + b2 Kb
(1.29)
Although the energy of the pulse (1.28) happens to be infinite, this approach gave rise to further improvements of such packet-like solutions, providing them a finite energy. Indeed, taking the function f(O) in the form f(0)=exp
I ( r i0)] 2Kb
1-
1-~
,
(1.30)
Kiselev and Perel [2000] demonstrated the packet-like behavior of solution (1.26) near the point r = 0, z = ct, 0 = 0. This point is viewed as the center of the packet, moving along the z-axis with velocity c. Expansion of the function f (1.30) in the vicinity of this point in the case Kb >> 1 yields the approximation
U2=U'exp[ (z-ct)2 ll~
;
/ll = CbK
(1.31)
Substitution of eq. (1.31) into (1.26) yields the representation of the field near its peak in a form differing from solution U1 (1.28) only by an exponential factor, providing longitudinal localization:
U2=Ulexp[ - (z-ct)21 l~
"
(1.32)
This describes a wave packet, filled with non-sinusoidal oscillations; its envelope decreases in a Gaussian-like manner, both in longitudinal and transversal directions. Unlike the pulse U l, the energy of waveform Ue is finite.
3, w 1]
Ultrashort EM pulses." how the), are modeled, produced, measured
155
1.1.7. New types of waveforms New types of waveforms presenting non-separable solutions to the propagation of a pulse of EM field in a collisionless plasma, characterized by its "plasma frequency" s such that s = 4;rNe2/m (N, e and m are the electron density, charge and mass, respectively) have been suggested by Shvartsburg [1999]. We start with the vector potential of the field A(Ax, 0, 0) such that
OAx (1.33) c Ot' Oz" Substitution of these expressions into the Maxwell equations yields the propagation equation governing the vector potential: 10Ax
E,. -
02Ax OZ2
9
H,,-
1 02Ax C20t 2
f22 C2 A,-.
(1.34)
The traditional solution of eq. (1.34) takes the form of harmonic wave trains: Ax = A0 exp[i(kz- 0)0]. In this case the wave number k and the frequency 0) are linked by the dispersion equation, derived from eq. (1.34): k 2C 2
= 0 ) 2 -- ~-2 2.
(1.35)
However, side by side with these sinusoidal wave trains, there is a huge family of exact non-sinusoidal solutions of eq. (1.34). To analyze the spatiotemporal structure of such anharmonic EM fields in plasma, it is convenient to introduce the normalized variables r and r/and the dimensionless function f :
r = g2t;
rl -
s
;
f = AxAo I.
(1.36)
C
Using these variables one can rewrite eq. (1.34) in a dimensionless form known as the Klein-Gordon (KG) equation:
O2f Orl2
O2f Or 2 - f ;
(1.37)
Solutions of this equation suitable for time-domain optics were presented by Shvartsburg [ 1999]"
f=
Z
dqfq(r, rl),
(1.38)
q = qo
L ( r , r/) = ~
,
( )+2 r-r/
~,(r, ~):
r + ~
4
v / r : - ~2
,
)
(1.39) ,
(1.40)
where Jq is the Bessel function of order q; the coefficients dq and the values q will be determined from the continuity conditions on the boundary plane r]. The
156
Instantaneous optics of ultrashort broadband pulses and rapidly uwying media 0,3
'
I
'
I
'
I
'
[3, w 1
I
0,2
0,1
c--
0,0
O2 -0,1
-0,2
-0,3
'
I 5
0
,
I 10
,
I 15
,
I 20
Fig. 4. Temporal waveforms of electric, e 3 (solid line), and magnetic, h 3 (dashed line), components of non-sinusoidal EM field on the interface of plasma-like medium (2.68) plotted vs. normalized time x = t/to.
non-separable functions ~pq cannot be written in the usual form of a product of time-dependent and coordinate-dependent factors. Let us point out some of their salient features" (i) They have both spatial and temporal derivatives of arbitrary orders, which may be calculated by means of recursive formulae: Ol])q OT
__ 1
Oqll3q 0/7
_
2"(l])q- 1 -- l])q + ! )'
(1.41)
1 2"( l])q_ 1 q- l/)q + 1)-
(1.42)
(ii) The causal condition r i> r/, which is fulfilled for each observation point r//> 0, results in restriction of the magnitudes of harmonics ~q for q/> 0. The function qaq on a plane r/= 0 reduces to lpq I r/= 0 = J q ( r ) .
(1.43)
(iii) The electric and magnetic components of the EM field are also presented by non-separable harmonics. Using eq. (1.47), we obtain O<3
E,-( r, rl) -
Aog2 Z C
OC
dqeq( r, r/); H,.( r, rl) -
Aog2 Z r
q = q0
dqhq( r, r/) (1.44)
q = q0
1 eq(r,/7) = ~(lpq_2 - 21pq + lpq+2),
(1.45)
hq(r, rl)
(1.46)
2-
2).
3, w 1]
Ultrashort E M pulses." how the)' are modeled, produced, measured
157
Examples of electric and magnetic harmonics with q= 3 are shown in fig. 4. One can see that these harmonics are non-sinusoidal, non-stationary and that their spatiotemporal structures are quite different. The models of localized pulses discussed above are far from exhausting the huge variety of non-sinusoidal waveforms, but they were chosen since they will be exploited in w2.
1.2. Production of ultrashort EM pulses Production of ultrashort EM pulses started long before lasers were even thought of, with the advent a long time ago of short-pulse radars. We will not address this point here, and rather concentrate on the production of short optical pulses, and some of their derivatives. Techniques consisting in optically gating a CW laser have been developed but they do not, so far, give access to pulse durations below 1 ps, and thus do not really satisfy the above definition. As can be deduced immediately from the statement above that short pulses have broad spectra, production of short laser pulses requires either the use of materials that can support a large gain bandwidth, or the development of techniques to increase, in the course of propagation, the spectrum of the pulse, which therefore pertain to nonlinear optics. Obviously, the same materials used for broadband tunability can be used in principle for producing ultrashort laser pulses. Indeed, dye lasers were the first to allow, more than twenty years ago, production of pulses with a duration significantly below 10-13 s. However, it is fair to say that in the past decade, all-solid-state systems have definitely outrun dye-based systems in the race for production of high-intensity ultrashort pulses. Several materials have been considered for such applications, including alexandrite, LiSAF crystals, and titanium-doped sapphire (Ti:Sa), definitely the most commonly used nowadays because of its excellent thermal and spectral qualities as well as particular nonlinear properties offering the opportunity of a simple mode-locking mechanism. With a central wavelength in the near IR and a spectral bandwidth in the 100 nm range, one can expect pulse durations in the 10 fs range. Three essential functions have to be realized in such a laser: - Broadband amplification, which is provided by the amplifying material. - M o d e locking, which is based in such lasers on the "Kerr-lens mode locking" mechanism. Because of the high intensities reached at the focus of the Z-shaped subcavity where the amplifying crystal is located, nonlinear contributions to the material's refractive index (Kerr effect) cause the appearance of a "Kerr lens" instead of the parallel slab used, which in turn
158
Instantaneous optics of ulo'ashort broadband pulses and rapidly varying media
[3, w 1
perturbs the cavity stability. This effect can be corrected by a readjustment of the Z-shaped subcavity mirrors, with the consequence that the total cavity is now optimized for the high-intensity (pulsed) regime. Besides, it is possible to select the spatial modes corresponding to a pulsed operation using a slit conveniently placed in the oscillator cavity. It is worth noting that, if one considers a pulse travelling back and forth in the cavity, the gain perturbation caused by the appearance of the Kerr effect occurs every time the pulse passes in the amplifying crystal, i.e. perturbation of the gain occurs at the intermode frequency, a known condition for obtaining mode-locked operation of a laser. This self-mode-locking effect (once known as "magic" modelocking!) was essential in the success of such lasers. -Compensation of the Group-Velocity Dispersion (GVD) induced both in the amplifying crystal and also, to some extent, in the coatings used for the different mirrors included in the cavity. This function is provided by a "negative dispersion line", usually consisting of two identical isoscele prisms placed in one of the arms of the cavity. Such oscillators now permit pulses with durations in the 10 to 20 fs range and energies of 1 nJ or more, i.e. peak powers in excess of a gigawatt, making nonlinear optics experiments accessible in quite comfortable conditions. It should be mentioned that the world record for pulse duration in such systems (5.8 fs, Mamschek, Gallmann, Sutter, Steinmeyer and Keller [2000]) was obtained using a different GVD compensation principle, based on the use of "chirped mirrors" proposed by Szip6cs, Ferencz, Spielmann and Krausz [ 1994]. This solution, now commercially available, also offers excellent compactness and stability. Note that for such short durations the natural bandwidth of the amplifying material is not sufficient. Self Phase Modulation in the amplifying crystal (a temporal counterpart of the Kerr effect) provides the extra bandwith needed. Amplification of such laser pulses in solid-state amplifiers was the occasion of another revolution, with the appearance of the "Chirped Pulse Amplification" technique (Maine, Strickland, Bado, Pessot and Mourou [1988])- first applied to Table-Top Terawatt (T 3) neodymium lasers- which will not be detailed here. Let us just mention that it relies on a three-stage manipulation of the pulse: stretching of the ultrashort pulse to nanosecond durations, amplification (which under these conditions can efficiently extract the energy stored in the amplifier's material without reaching the material breakdown threshold) and recompression of the pulse almost to its original duration. Pulses of typically 25 fs/25 J can be produced this way in the most advanced Ti:Sa systems, allowing to reach intensities in excess of 102~ cm -2. At much lower intensities (typically 10 ~4W cm-2), interaction with dense
Ultrashort EM pulses." how they are modeled, produced, measured
3, w 1]
'
I
'
I
'
I
17
15 13
14
'
I
16
12
I
Argon
f
0_1
'
159
o,,~
>.. 10.2
3: 10 .3
I
70
I
6O
"
50
I
'
40
I
"
I
30
Wavelength (~tm) Fig. 5. High-order harmonic spectrum generated by focusing a high-intensity (1014W cm -2) 50 fs pulse of a titanium-doped sapphire laser in a dense argon jet, together with its second harmonic so that both odd and even harmonics are generated.
gaseous targets allows one to produce with a rather high efficiency a large number of odd harmonics of the incident frequency (Sali~res, UHuillier, Antoine and Lewenstein [1999]). A typical harmonic spectrum generated in such interactions is shown in fig. 5, where both the fundamental beam and its second harmonic have been focused simultaneously to generate a complete spectrum (odd and even harmonics: the fundamental alone generates only odd harmonics). It has a very typical shape, consisting of a rather fast decrease of efficiency for the lowest-order harmonics, followed by a "plateau" whose width and height depends on the gas used (rare gases, most often), and finally a cut-off region. The emitted harmonics have excellent spatial and temporal coherence properties, thanks to the coherent nature of the process producing them, and pulse duration smaller than that of the exciting laser. The number of photons per harmonic pulses is quite high (typically 108 in a common case where one does not seek to produce the shortest possible wavelength), and such sources, in some applications requiring short UV pulses, are a serious competitor to synchrotron radiation (which is still leading the race, though, in terms of average power). Concerning the shortest wavelength that can be generated with such techniques, the latest results showed evidence of generation of the 255th harmonic of the Ti:Sa laser, i.e. a wavelength close to 3 nm! The particular shape of the spectrum in fig. 5 has suggested a possible way of reducing the pulse duration of such harmonics far below 1 fs. If one
160
Instantaneous optics of ultrashort broadband pulses and rapidly varying media
[3, w 1
could lock the phases between the different harmonics in the plateau region, modelisation predicts that pulse durations in the range of a few attoseconds (one attosecond equals 10-18 s) could be obtained. Very recently, the relative phases of the different harmonics were measured using a two-photon IR-VUV ionization experiment (Paul, Toma, Breger, Mullot, Aug6, Balcou, Muller and Agostini [2001 ]), which allowed to reconstruct the harmonic pulse train, arriving to the conclusion that one individual harmonic pulse had a maximum duration of 250 as, the shortest EM pulse produced to date. The search for ultrashort pulses has also been successful in the IR range. It has been known for some times that IR free-electron lasers produce, in the leading edge of the "macropulse" (a train of 100 or more micropulses with picosecond or less duration) characteristic of such machines, pulses with durations of typically one picosecond, which, given the wavelength range considered (10 ~tm or more) satisfies the definition given above. Such lasers now almost routinely operate between 5 and 50 ~tm. In the communication domain, extensive efforts have also been made to develop very high repetition rate-ultrashort optical sources. Usually starting from semiconductor lasers for compactness and cost-efficiency reasons, such sources are based on compression techniques using propagation in different optical fibers. Limited for some time to the ps duration regime, recent progress has allowed compression levels down into the 20 fs range, that is, equivalent to that of the Ti:Sa lasers described above. In particular, starting from a 7.5-ps pulse generated from a gain-switched semiconductor laser at ~l=1.55 ~tm and repetition rate 2 GHz, Matsui, Pelusi and Suzuki [1999] achieved a compression down to 20fs using a four-stage fiber soliton pulse compressor consisting of standard single-mode transmission, Er-doped, dispersion-decreasing, and dispersion-flattened fibers, respectively. They confirmed experimentally that the soliton self-frequency shift plays an important role in obtaining such high compression in very short fibers, and also in minimizing the inherent undesirable pedestal component. Finally, let us mention that the ultrashort laser pulses discussed above have been used to generate single- or half-cycle Terahertz pulses. The principle of the experiment is the following: a piece of semiconductor is irradiated during a short time using a subpicosecond laser pulse (You, Jones, Bucksbaum and Dykaar [ 1993]). The carriers injected into the semiconductor thus allow a current to circulate in it, biased under a high dc voltage, as long as it is maintained in the conducting state by the laser illumination. The field radiated by the moving electrons has a temporal shape close to a single arch, and can be modeled as a half-cycle pulse of a radiation whose frequency is determined by
3, w 1]
Ultrashort EM pulses: how they are modeled, produced, measured
161
the duration of the illumination laser pulse, and falls in the Terahertz range. More recently this technique has been refined to obtain THz emission from unbiased GaAs (Cote, van Driel, DeCamp and Bucksbaum [1999]), originating from ballistic photocurrents generated via quantum interference of one- and twophoton absorption in low-temperature-grown and semi-insulating GaAs. At a 250 kHz repetition rate, up to 3 nW of THz power have been measured. The examples given above show that it is possible, using the different techniques briefly summarized here, to obtain ultrashort EM pulses at almost any wavelength between millimeters (at least hundreds of ~tm) and nanometers.
1.3. Measurement o f ultrashort E M pulses
Another problem is to measure such ultrashort EM pulses. If one excepts the case of ultrashort RF pulses, which can be measured using standard electronic techniques, no electronic equipment possesses a sufficient bandwidth to allow direct measurement of any, e.g., subpicosecond optical pulses. Therefore a number of optical techniques were proposed, and some of them are routinely in use, for measurement not only of the pulse duration, but also of various characteristics of ultrashort laser pulses. The reader is referred to the comprehensive review of such techniques by Dorrer and Joffre [2001 ] for a detailed description of the many different possibilities demonstrated so far; here we will concentrate on the methods that are most commonly applied, and those offering the most complete and detailed information on the pulse characteristics and therefore appearing as the most promising. The first of them is second-harmonic generation (SHG), which was proposed very early as a means of measuring short pulses (Weber, Mathieu and Meyer [ 1966]). One uses two replicas of the pulse to be measured, delayed by a time interval r, to deduce the autocorrelation function of the pulse intensity with the measurement, for different delays, of the quantity oc
IsH~(r) = f
I(t) I(t - r) dt,
(1.47)
--s
where I represents the intensity profile of the pulse to be measured. Autocorrelators based on this technique (which can be exploited in two variants: the intensimetric mode, and the interferometric mode, in which the autocorrelator is simply a Michelson interferometer associated with a frequency-doubling crystal) are part of basically all subpicosecond laser systems. They allow to obtain
162
Instantaneous optics of ulO'ashort broadband pulses and rapidly oalying media
[3, w 1
not only the pulse duration (FWHM), but also some information on the pulse shape, for instance the fact that the pulse shapes generated in femtosecond Ti:Sa oscillators are generally closer to the "squared-cosech" profile than to the exponential one described by eq. (1.11), the latter still being in general use owing to its analytic simplicity. However, due to the symmetrical nature of the autocorrelation function, such a method is helpless in the case of asymmetrical pulses. It should also be mentioned that such methods are limited to measurements in the visible part of the spectrum essentially because the GVD of most nonlinear materials (which would produce an artificial lengthening of the pulse) is large in the UV range, thus limiting their thickness to values that preclude collection of a usable signal. So, for measuring pulse durations in the UV or VUV range, one has to use instead of SHG another nonlinear process allowing to couple the pulse to be measured to a well-characterized optical pulse. Two-photon absorption or ionization has often been employed to this aim. Note that the electronic nature of the nonlinear process is essential since the fast response time of electrons warrants the accuracy of the measurement. For instance, the pulse duration of X-ray pulses generated by intense irradiation of a metallic target could be measured by monitoring the sidebands induced in the Auger electron spectrum of atomic Ar, subject to the combined irradiation of the X-ray pulse and a delayed IR subpicosecond pulse (Schins, Breger, Agostini, Constantinescu, Muller, Grillon, Antonetti and Mysyrowicz [1994]). Needless to say, such methods are difficult to employ and generally do not give birth to apparatus that could be considered as "measurement equipment". It is sometimes not enough to know the pulse duration. This was the case, for instance, in the above example of the measurement of the respective phases of the different high-order harmonics. A common instance of such problems is the measurement of so-called "chirped" pulses, in which the frequency of light varies along the pulse. One then first has to locate the energy density in time-frequency space, which is generally done using spectrographic techniques, for instance one based on the Frequency-Resolved Optical Gating (FROG) principle (Kane and Trebino [1993]), where the spectrum of the pulse is measured after gating through a correlation process. However, retrieval of the field, which requires the use of quite heavy algorithmic techniques, is a slow and uneasy task. At present, techniques based on interferometry appear to be the most powerful, in particular the Spectral-Phase Interferometry for Direct Electric field Reconstruction technique ( S P I D E R - Iaconis and Walmsley [1998]): a frequency-domain counterpart of shearing interferometry, which allows singleshot measurements of the spectral phase of ultrashort pulses (Dorrer, de Beau-
3, w2]
Spatiotemporal reshaping of ultrashort pulses in stationalT media
163
Fig. 6. Principle of the "SPIDER" time-resolved measurement of the amplitude and phase of an ultrashort EM pulse. voir, Le Blanc, Ranc, Rousseau, Rousseau and Chambaret [1999]). The experimental principle of this technique is sketched in fig. 6: two time-delayed frequency-upshifled replicas of the ultrashort pulse to be analyzed are generated by frequency mixing with a chirped pulse. In such a chirped pulse, the instantaneous frequency depends linearly on time ( t o = t a +fi), and will be supposed constant throughout the duration on the ultrashort pulse. If r is the time delay between the two replicas, a frequency shear s = r a is introduced between both replicas. One then measures the spectrum of the pair of upshifted replicas which consists of a series of fringes whose structure is determined principally by the frequency shear f2, but also bears information on the spectral phase of the original pulse as a continuous function of frequency (Fittinghoff, Bowie, Sweetser, Jennings, Krumbtigel, DeLong, Yrebino and Walmsley [ 1996], Lepetit, Cheriaux and Joffre [1995]). This technique thus makes it possible, in a single laser shot, to obtain complete information on both the amplitude and the phase of the EM pulse.
w 2. Spatiotemporal reshaping of ultrashort pulses in stationary media The variety of applications of few-cycle optical pulses in different branches of physics is growing continuously. These applications range from time-domain spectroscopy of dielectrics (Smith, Auston and Nuss [1988]) to impulse photoionization of molecules (Jones, You and Bucksbaum [1993]), new principles of imaging, suggested by Hu and Nuss [ 1995], and shape-dependent absorption of broadband pulses in the space plasma, examined by Akimoto [ 1996], to name just a few.
164
Instantaneousoptics of ultrashort broadbandpulses and rapidly uarying media
[3, w2
In any such study, one has to know the spatiotemporal parameters of the pulse at the target location. However, these parameters may change on the path between source and target. It was argued by Wolf [ 1986, 1987] that freespace propagation of a polychromatic radiation beam produces variations of the beam's spectrum, unless the radiation source possesses some particular coherence properties (which fortunately happens to be the case for many classical sources). Nevertheless, the general rule is that the spectrum does not have to be conserved upon propagation, and this applies particularly when broadband coherent radiation is considered. During the last two decades, great effort has been put into the understanding of the optics of broadband waveforms localized in time and space. Spatial and temporal dynamics of these waveforms cannot be analyzed separately, as in the quasimonochromatic case, but become coupled even in free space. The spatial frequencies that arise due to the finite transverse size of a real wave beam provide free-space dispersion, inducing changes in the pulse shape during propagation. The interplay of these effects in the dynamics of different waveforms, travelling both in dispersionless and in dispersive media, is discussed below.
2.1. Dynamics of ultrashort waveforms in dispersionless optical systems
Propagation of ultrashort few-cycle pulsed beams in linear homogeneous lossless media is accompanied by a coupled evolution of their spatial and temporal parameters. To understand the fundamental role of the coupling between spatial and temporal reshaping of the pulse, it is worthwhile to show first how these processes are developing during propagation in free space (w 2.1.1); w2.1.2 is devoted to spatial and temporal variations of waveforms passing through an optical system, and waveform-preserving reflection on curvilinear mirrors is discussed in w2.1.3.
2.1.1. Coupling between spatial and temporal reshaping in free space
The interplay between transversal, longitudinal and temporal distortions of localized pulses is described by the paraxial equation for the electric field E(r, z, t) (Einziger and Raz [ 1987]):
A•
-
2 02E c OzOt'
(2.1)
Spatiotemporal reshaping of ultrashort pulses in stational y media
3, w 2]
with A z = 0 2 + 0 ,2. .... t' = t eq. (2.1),
E ( r , z , t ) = i L R qF ( t
165
z c - .l We will use a non-separable solution of
~_ ~r2 ) ,
(2.2)
where r 2 = x 2 +y2, q = z + iLR, LR is the diffraction length (Rayleigh range), and F is an arbitrary function. The pulsed beam diffraction, arising due to its finite transversal size, induces, through the factor iLR/q in eq. (2.2), propagation changes in the on-axis waveform. Writing this factor in the form iLR= exp(ic#) . q V~ 1 + (z/LR) 2 ,
q~ = arctan
(z) Lr~'--
(2.3)
one can link the factor [ 1 + (Z/LR)2] -1/2 to the pulse amplitude attenuation, while the phase c# is responsible for the evolution of the pulse shape: eq. (2.2) is real at z - 0 , and purely imaginary for large z >> LR. The parameter cg in eq. (2.3) is known as the Gouy phase shift, and takes values f r o m - 31 ~ to ~1 Jr. Independently of the choice of the function F the coupling of its spatial and temporal variations comes from the complex space-dependent time shift -r2/2cq. Its real part, zr 2
ts = - ~ 2clq 2,
(2.4)
is the actual time of arrival of the pulse at each plane. This shift is connected with the paraxial spherical phasefront of radius
R(z) - Iq12 - z 1 +
~
.
(2.5)
z
The imaginary part of the time shift, iLRr2/2c]q], determines the spatial distribution of pulse attenuation. Let us illustrate the spatiotemporal coupling phenomena, choosing the function F to have the form (1.20), corresponding to a Poisson-spectrum pulse. The 1/e width of the real pulse F is
T = to
( exp
m
1
(2.6)
which can represent a large variety of pulses, the value m = 1 relating to a single maximum of F, whereas large values of parameter m correspond to a growing
166
Instantaneousoptics of ultrashort blvadbandpulses and rapidly •arying media
[3, w 2
number of oscillations with almost constant frequency ~o,,, = mto I in the central part of the pulse. To examine the forming of a spatiotemporal structure in the course ofparaxial propagation of the pulse (2.2) one can use eq. (1.20), replacing the time t' by a complex time t ' - r e / 2 c q and multiplying the function (1.20) by the Gouy factor: E ( r , z , t) = iLR q
I
it0 t~ - ,.2 + it0
I Ill
(2.7)
Using the definition of q, and with the help of eq. (2.5), one can rewrite eq. (2.7) in the form
E(r,z,t) =
iZ R q
I
I !11
to to+
r2LR i
to + 2c ]q[2
i
m (2.8)
r2LR
r2LR
-2-~
+i
t0+
2c Iq]2 )
of which, according to eq. (1.20), one should take the real part. One can see that the solution (2.8) derives from (2.7) through a time shift, connected with the spherical pulse fronts of radius R (2.5) and replacement of parameter to by to + r2LR/2C q]2. These changes result in an increase of the pulse duration T, as compared with (2.6):
T:to(l+ r2R
2cto ]ql2
exp
- 1.
(2.9)
The frequency of oscillations is red-shifted:
r2LR ) o),,, = m
tO + 2c lql 2
(2.10)
As shown by Porras [1999a], the off-axis spatial structure of the pulse (2.8) is characterized by the factor (1 + reLR/2cto ]ql2) -''', indicating a decrease of the pulse amplitude when moving away from the axis. The Gouy factor iLR/q describes the pulse temporal reshaping, including its polarity reversal during the pulse travel from z <<-LR to z >> LR (fig. 7). It should be pointed out that the pulse spectrum has an essential influence on the dynamics of its spatiotemporal evolution. The results mentioned above
3, w 2]
Spatiotemporal reshaping of ultrashort pulses in stationaly media 0,4
E ~
a)
167
.
0,2
9 " o,o
-0,2
-3
-2
-1
0 x
1
2
1,0
3
4
z = 0
0,5
0,0
-0,5 -4
-3
-2
-1
0
1
2
3
4
3
4
x 0,4
z=0.2
0,2
~- o,o
-0,2
-3
-2
-1
0 x
1
2
Fig. 7. Reshaping of Poisson-spectrum pulse (2.8) with 171= 4 and reversal of its polarity on the beam axis in the course of propagation from z = -2LR (fig. 7a) through z = 0 (fig. 7b) and further to z = 2LR (fig. 7c).
relate to Poisson-spectrum pulses (1.21). This spectrum, determined by two parameters o n l y - m and to - proves to be flexible enough for modeling such phenomena as pulse lengthening, frequency red-shifting and polarity reversal. Such transformations were observed experimentally by Feng, Winful and Hellwarth [1998] in the diffraction of ultrashort pulses. The above analysis treated pulse propagation in an homogeneous medium. We now turn to the study of the effects arising from transportation of such pulses through an optical system.
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Instantaneousoptics of ulo'ashort broadband pulses and rapidly ualying media
[3, w2
2.1.2. Spatiotemporal transformation o f an ultrashort Poisson pulse passing through a thin lens
Let us consider the spatiotemporal transformation of an ultrashort pulse with Poisson spectrum (1.21), passing through a thin lens. The main features of such a transformation can be revealed in a simple case, that of the so-called isodiffracting pulses. These pulses, characterized by a frequency-independent Rayleigh range, were examined by Melamed and Felsen [1998]. Feng, Winful and Hellwarth [1999] showed that such pulses can be generated, e.g., in a cavity resonator consisting of two curved mirrors of equal radii of curvature R, separated by a distance L. The confocal parameter for this geometry is given by d -
2yfW 2 ~ - v/L(2R - L).
(2.11)
Here W is the beam waist, W ~ x/~, and the Rayleigh range is LR =
(2.12)
v/L(2R - L).
This fixed confocal parameter provides the same values of wavefront radius of curvature R(z) for all wavelengths and thus for the entire pulse. Making a onesided inverse FT of the isodiffracting pulse with Poisson spectrum (1.21) and m = 3, Feng, Winful and Hellwarth [ 1998] obtained the following expression for the field: iA0 exp[-3i arctan(T)] E (2.13) q (1 + T=) 3/= 1 + ~1.2
[]3,
where the parameter q = z + iLR represents the Gouy phase shift and the z-dependent amplitude attenuation factor; the normalized local time scale 7' is
1 [Z + ~ ,.2]
t- 7 T=
(2.14) to 1 + ~
Here the radius of curvature R(z) is defined as in eq. (2.5), and the beam width a(z) is defined by a 2(z)=2ctoLR
{ /z 1+
~
.
(2.15)
The q-parameter of the pulse is linked to the values of R(z) and a(z) by 1 q
-
1 R(z)
2ict0 a2(z)
(2.16)
Let us consider such a pulse, passing through a thin non-dispersive lens with focal length f , placed in the vacuum at plane z=0. Upon propagation through
3, w 2]
Spatiotemporal reshaping of ultrashort pulses in stationa#y media
169
this lens the entire pulse can be characterized by a new value of parameter q = q2, which is related to the parameter qi of the incoming pulse by the so-called ABCD transformation, developed by Dijaili, Dienes and Smith [ 1990] q2 =
Aq~ + B
(2.17)
Cq~ +D"
Here q i =di +iLn, with d l the distance from the beam waist of the incoming pulse to the lens. For the pulse passing through the aforesaid thin lens to any point z2, the values A, B, C and D were found by Feng, Winful and Hellwarth [1999] to be A = I
Z2 D
f,
B
9
m
z2,
C = - l/f ;
9
-
-
D = 1.
(2.18)
Equations (2.16)-(2.18) give the transformed q-parameter at z2 as q2 =
(1 - z 2 f - 1 ) ( d l + iLR) +z2 1 - ( d i + i L a ) f -I
(2.19)
Making use of (2.16) with the value q=q2 (2.19) yields the new Rayleigh range L~ and new beam waist a2(d2), located at the distance d2 from the lens: L~R _ LR K'
a 2 (d2) - 2cto
a2=~ [d l _f-I 1
(2.20)
K ' ( d 2 _p. L2R)],
K = (1 - d t f -I )2 + ( L R f - i ) 2 .
(2.21)
The total phase shift between the input and output beam waist is 99 = arctan
(i LR ) -d~
"
(2.22)
These results illustrate the spatiotemporal evolution of the pulse (2.14) caused by its passage through a thin dispersionless lens in vacuum. The salient features of such evolution can be summed up as follows: (i) The phase shift, introduced by the lens, can drastically change the pulse's temporal profile at the focus. For instance, for di = f , it follows from 1 . eq. (2.22) that 99 = ~;r. a unipolar (bipolar) pulse is transformed into a bipolar (unipolar) one. (ii) Considering the planes z = d l and z = d 2 as the object and image planes with respect to the beam waist, one can see that the image pulses are different both in space (beam size) and in time (pulse width) at different locations.
170
Instantaneous optics of ultrashort broadband pulses and rapidly varying media
[3, w 2
(iii) The variations of the spatial and temporal structures of the off-axis field at the beam waist (z= d2) can be found by means of substitution of parameters (2.20)-(2.21) into expression (2.14); the on-axis pulse width is invariant. Feng, Winful and Hellwarth [ 1999] showed that the spatiotemporal reshaping of Poisson-spectrum pulses, reflected from a concave spherical mirror with radius R, is described by formulae (2.19)-(2.22) after replacing the lens focal length f by the mirror radius R. Another geometry, with a curvilinear mirror providing invariance of the reflected pulse shape, is considered below.
2.1.3. Curvilinear reflectors for controlling transient fields For the design of curvilinear reflectors to control the properties of transient fields, Bateman [1955] developed an elegant time-domain approach to the analysis of EM fields in free space. Following this approach, let us introduce a vector M which is a linear combination of electric (E) and magnetic (H) components of this field (assumed to be linearly polarized): M = E + ill.
(2.23)
The equation governing M, can be derived from the Maxwell equations: V' x M -
i OM
(2.24)
c Ot"
Further, representing M by scalar fields U and ~p in the form M = V U x V~p
(2.25)
and substituting eq. (2.25) into eq. (2.24), we obtain an equation describing these scalar fields: V U x V~p = - c
--~-V~p---0-TVU
.
(2.26)
It should be noted that, as well as the vector M defined by eq. (2.25), any other vector M1 = F ( U , ~p) V U x V~p
(2.27)
containing any arbitrary function F(U,~p) also obeys eq. (2.24). The determination of M for the scattered field is appreciably easier than the traditional solution
3, w 2]
Spatiotemporal reshaping of ultrashort pulses in stationaJy media
171
of the scattering problem. When the components of this vector are known, the separation of their real and imaginary parts yields the fields E and H (2.24). This notation proves to be important for the analysis of some scattered pulsed fields. A linearly polarized plane wave with components E, and H,,, travelling in the z-direction, may be described by means of scalar functions Ul = x + iy;
(2.28)
~Pl = z + ct.
Substitution of eq. (2.28) into eq. (2.25) yields M,-=i, M , . = - I ; comparing this result with the definition (2.24), we obtain the dimensionless values of the components of this plane wave: Ex = 1, H,. = - 1 . An arbitrary waveform of this field may be written, according to eq. (2.27), as (2.29)
Ml = Fl (~Pl) V U1 x V lPl.
Proceeding in a similar fashion, one can find the representation of a diverging spherical wave, using the coordinates p, 0, tp. Introducing the scalar functions U2 = e-'9~ tan(~~0),
(2.30)
~ = p - ct,
one can find the components of the vector M 2 for this field:
M2p = O,
M20 =
ieiq p(1 + cos 0)'
M2q = -iM2o.
(2.31)
By analogy with eq. (2.29), M2 can be presented in a form containing an arbitrary scalar function: M2 =
F2(q,2) vu2
x
vqJ2.
(2.32)
We can now examine the scattering of an incident plane wave M 2 on some curvilinear surface R. Since eq. (2.24), governing the vector M2, is linear, the linear combination M2 = Fl(lPl) VU1 • V~Pl +KF2(~P2) VU2 • V~P2,
(2.33)
representing the total field of the incident and scattered waves, must also be a solution of this equation; the constant K in eq. (2.33) has to be defined from the boundary conditions at the scattering surface R. Let us consider, using this approach, the scattering of an arbitrarily shaped waveform with a plane wavefront, incident on the convex side of a perfectly
Instantaneous optics of ultrashort broadband pulses and rapidly tmrying media
172
[3, w 2
F
R
0
P0
........................................................................
Fig. 8. Geometry of waveform-preserving reflection of a plane wave P to wave S on a parabolic reflector R: O, P0 and 0 are the pole, radius vector and polar angle of the reflecting mirror R. conducting axisymmetrical surface R along its axis of symmetry z (fig. 8). Assuming this wave to be linearly polarized, one can rewrite the scalar functions UI and *Pl (2.28) in spherical coordinates: q'l = - P cos 0 + ct,
Ul = p sin 0eir
(2.34)
and substituting eq. (2.34) into eq. (2.33), we find the vector M : Mp
= Fl sin 0 e 'Cr,
M0
= Fl cos 0 e i~r +
(2.35)
KF2e-'q
(2.36)
p(1 + cos 0)' M~
-- i F 1 e
iq -
KF2e-iq
p(1 + cos 0)"
(2.37)
It is of interest to examine the case of an arbitrary waveform which remains invariant in the course of scattering, i.e. F~ =Fe=F. Since the surface R is supposed to be perfectly conducting, the tangential component of the electric field E~c = ImM~ c should vanish on this surface, which determines the value of the constant K: K = p0(1 + cos 0),
(2.38)
where P0 and 0 are the polar coordinates of the curve forming the axisymmetric surface by rotation around the z-axis. The curve (2.38) is a parabola; its pole is
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Spatiotemporal reshaping of uln'ashort pulses in stationaly media
173
located at the center of the coordinate system (fig. 8). Hence the surface R is a paraboloid. The components of the EM field scattered by this paraboloid are determined directly from eqs. (2.36)-(2.37): Ho = - E r = Po - - cos q~F ( p - ct), P G =Hp = 0.
H~ = E0 = - -Po - sin q~F ( p - ct), P
(2.39)
Using these fields, one can write the radial component of the energy flow, carried by the scattered waveform, as Pp = - ~c
F2 ( p - ct).
(2.40)
Thus we have obtained the time-domain description of the scattered wave. The main features of this reflector are: (i) the Poynting vector of the scattered field is directed radially: Pp ~ 0; P~ = Pp = 0; (ii) this result is valid for an arbitrary waveform; (iii) the preservation of the scattered waveform is probably a unique property of a paraboloid reflector. This example shows the possibilities of the elegant time-domain method developed by Bateman [1955] for the problem of diffraction and scattering of transients, treated traditionally by computer simulations. It is worth noticing that, more than forty years later, the use of parabolic mirrors is considered as the only way of focusing efficiently (i.e. without lengthening the pulse) ultrashort laser pulses.
2.2. Evolution o f transients in dispersive media
The traditional analysis of wave-dispersion phenomena is based upon the supposition of a slow accumulation of pulse-envelope distortions in the course of propagation in dispersive media. The characteristic length for such accumulation, Ld - 2to/Ig,,,,ol, depends on the second-order dispersion of the medium, described by the factor K,,,,,, = 02k/O0) 2, where k=k(og) is the dispersion relation. Some types of pulse distortions typical, in particular, of fiber optics, are determined by third-order dispersion; the analysis of these effects was given by Agrawal [ 1995]. A further improvement of this approach, taking into account the fourth-order dispersion, was considered by Karlsson and Hook [1994]. Brabec and Krausz [ 1997] derived the generalized equation, describing the influence of a dispersion of arbitrary order on the evolution of a pulse envelope.
174
Instantaneousoptics of ultrashort broadbandpulses and rapidly ualying media
[3, w2
Alternatively, the effect of non-stationary propagation of ultrashort single- or few-cycle waveforms in dispersive media can be investigated by means of the temporal variations of electric strength itself. This approach is presented in this subsection: simple analytical expressions for time-dependent electric strength of a flexible waveform in a medium with second-order dispersion are given in w2.2.1. Sections 2.2.2 and 2.2.3 show the rapid deformation of transient electric and magnetic components in plasma-like media and waveguides by means of exact time-domain non-sinusoidal solutions of Maxwell equations without using the concept of dispersion. 2.2.1. Spatiotemporal evolution of localized waueform in lossless medium with arbitrary dispersion Let us consider first the spatiotemporal evolution of localized waveform in a lossless medium with an arbitrary dispersion o)= o)(k). Following Nodland [1997], we use the flexible representation of an initial waveform as a superposition of spatially modulated Gaussian waveforms: tl
Eo(z) = Z f , ,
exp(-a,,,z z) cos(b,,,z).
(2.41)
1
m:
Here fro, am and bm are real constants, with a,,, > 0. The FT of eq. (2.41) can be written as
1/
~X3
A(k) = ~
Eo(z) exp(-ikz) dz
-~ _ 1 " f,, 4Jr Z m= 1 ~
(2.42) {
[ (k-bin) 2] exp
4a,,,
[ (k+bm)2]} + exp
4am
.
The dominant wavenumbers in eq. (2.42) are k = +b,,,; expanding the dispersive relation up to second order around each point +b,,,, we obtain co(k + b,,,) = so(+b,,,)+ ksl(+bm)+ k2s2(+b,,,),
(2.43)
where the coefficients s l and s2 depend on the derivatives Oo)/Ok and 02 oo/Ok2. Since the expression (2.42) for A(k) is real, we can write the electric field E(z,t) in the form i'*
E(z, t) = / A(k) cos [kz - to(k)] dk. --~X2
(2.44)
Spatiotemporal reshaping of ulo'ashort pulses in stationary media
3, w 2]
175
Substitution of (2.42) and (2.43) into (2.44) brings the expression for electric field as a sum of 2n terms, all of them centered on the major wave numbers
~bm with l ~<m ~
=
1
O(3
. Jdkexp[ 4an,b'2] --O<3
cos{-lso(bm) + k[z - tsl(b,,,)] - k2ts2(bm) + (bm ----+-b,,,) }. (2.45) The integration in eq. (2.45) can be performed analytically, yielding •
E(z,t)
-
lYm
4v/-
m= 1
x { ~ m exp [FmA -:----,,,cos(B,,) ]larctan[4tams2(bm)] -3 4am
+ (b,,, ~ -bin) ,I
(2.46) with 2 2 Am = tbms2(bm)[Z-tsl(b,..)]- ~l[z - ts 1(b,..)]2 -- t 2b,..s2(b..).
Bm= Fm ~ bm[Z -- tSI ( b m ) ]
(
t[s~
-
(2.47)
nt- b2's2(bm)]
16a~,
(2.48)
+ l ts2(bm)[Z - tsl (bm)] - t3 so(bm) s2(b,,,) } ,
Fm = [(4am) -2 + t2s~(bm).] -'
(2.49)
Considering each mth pair in eq. (2.46), one observes that the amplitude of this mth pair, maximal at the points +bm, decays exponentially when one moves away from these points; the amplitudes are 1/e of their maximal value when k is at the distance -+-2v/~m from the point +b,,,. Thus, each pair makes a significant contribution to the field only in two areas of the w(k) line. The advantages of this approximate description can be summarized as follows: (i) the dispersion law is arbitrary, (ii) the presentation of an initial waveform is flexible, each sinusoidal wave packet in eq. (2.41) being characterized by three free parameters, and (iii) the localized field is presented as a sum of elementary functions. At the
176
Instantaneous optics of ultrashort broadband pulses and rapidly varying media
[3, w 2
same time one has to mention some shortcomings of this approach: (i) an initial waveform E(z) (eq. 2.41) is supposed to be known inside the medium. This seems to be justified when considering a waveform near a source located inside the medium. But this waveform remains to be found when the pulse has to travel a large distance z ~ ~okt from the source; (ii) expressions (2.41) are restricted to symmetrical waveforms, and (iii) the result (2.46) is restricted to second-order dispersion, and it is not easy to generalize this result, e.g., to include third-order dispersion. In view of the above, it is worthwhile to discuss another statement of the pulse dynamics problem, considering the pulses generated by a source, located outside the dispersive medium. Using the continuity conditions for EM fields on the interface of this medium, we will find the non-stationary waveforms inside the medium. Herein the impulse reflection and refraction problems have to be solved simultaneously. It is shown below how exact non-stationary time-domain solutions of the Maxwell equations can describe the waveforms dynamics without any suppositions about smallness or slowness of dispersive distortions of fields in the medium; thus, in particular, the expansions (2.43) are not needed in this approach.
2.2.2. Realistic models of ultrashort EM transients Real ultrashort EM transients, generated by existing optical systems, often differ from idealized cos-Gaussian waveforms. They can exhibit uneven distances between zero-crossing points and asymmetric waveforms, including a welldefined leading front and slowly damping tail, and can be, as well as the reflected ones, represented in the time range 0 to ~ by linear superpositions of Laguerre functions L,, (1.18); the same superposition for the reflected pulses can be written replacing coefficients a,, in (1.17) by b,,. Since both electric and magnetic components of these pulses are zero at the pulse's front edge and L,(0)= 1 (fig. 2), the coefficients a,, and b,, must obey the conditions OO
~x3
Zan--0, n = 0
(2.50) n= 0
The coefficients a,, characterizing the incident pulse, are supposed to be known, the coefficients b, have to be found simultaneously with the coefficients dq defining the field components in the plasma-like medium (eq. 1.44). For simplicity let us consider the normal incidence of Laguerre waveforms from vacuum on the half-space r//> 0, filled by a plasma. The continuity conditions
Spatiotenworal reshaping o f ultrashort pulses in stationa W media
3, w 2]
177
for incident Ei,Hi, reflected Er, Hr and transmitted Et, Ht fields at a plane r/= 0 imply Ei +Er = Et,
Hi +Hr = Ht.
(2.51)
Presenting the incident and reflected waveforms on a plane r/=0 by means of Laguerre functions (1.18) and transmitted fields by means of non-separable solutions of Maxwell equations (1.44), one can rewrite the continuity conditions (2.51) as 0(3
Z(a.
+ b,,) L.(t) -
Aof2 Z C
n =0 Cx~
Z(a.
dqeq( r),
(2.52)
dqhq(r)"
(2.53)
qo 9(2
- b,,)L.(t) -
AoQ Z C
n=0
qo
Multiplying eqs. (2.52)-(2.53) by the Laguerre functions L,,(t), integrating both sides of these equations with respect to t and using the orthonormality of Laguerre functions (1.19) we obtain an infinite set of equations: A0t2
a,, + b,, a,-b,,
A0g2
-
Tl,,(a),
(2.54)
T2,,(a),
(2.55)
where a = (2t0 and oC
2<2
r,. = Z dqP..,(.),
qo ~X2
oc
Pnq(O 0 :
(2.56)
v=.. = Z d.,o,..,(.).
qo
f dxL.(x) eq(aX), 0
Q,,q(a) = /
dxL,,(x) hq(aX).
(2.57)
0
The dimensionless quantities P,,q and Q,,q can be considered as matrix elements, describing the excitation of the qth harmonic of the non-sinusoidal field by the nth Laguerre waveform. The system (2.54)-(2.57) allows the simultaneous calculation of coefficients b,, and dq, characterizing the reflected and transmitted EM fields. Let us outline some properties of this system:
178
Instantaneous optics of ultrashort broadband pulses and rapidly ualying media
'
[3, w 2
a)
0,5 "EL.. x
0,0
i1
-0,5
J o
'
~
'
1'o
'
X
b)
-1
,
0
i
2
,
i
4
,
i
6
,
i
,
8
Fig. 9. (a) Lengthening and polarity reversal due to reflection from plasma with a = 0.5: squares, incident waveform El(X ) = Lo(x ) -Ll(x); circles, reflected waveform F2(x), x = t/to. (b) Reflection of Laguerre waveform L 0 from plasma-like dielectric (normal incidence); reflection coefficient R (2.79) is plotted vs. the parameter ct = ~2t0 (2.73).
(i) The coefficients P,,q and Q,,q given in eq. (2.57) depend only on the parameter a, which may be considered as the ratio of two characteristic t i m e s - pulse duration to and plasma period ~2-1 . The values of to and f2 are not restricted in the model discussed here. Thus, speaking about the propagation of "ultrashort" pulse in plasma-like media, one has to keep in mind that the same pulse can be considered as long or short, not depending on the absolute value of its duration but on the comparison of this value with the dispersive time scale of the medium of propagation. (ii) Due to the non-stationary regime of pulse reflection, the reflected transient can contain some Laguerre waveforms that are absent in the incident pulse, so that in the relevant pair of eqs. (2.52)-(2.53) for the nth Laguerre waveform one has a,, = 0, but b,, ~ 0. For instance, considering an
Spatiotemporal reshapingof ultrashortpulses in stationaiy media
3, w2]
179
incident waveform of the type Fl(X)--Lo(x)-Ll(X) (with a0 = 1, al =-1), the reflected pulse contains, along with transients L0 and L1, the series of transients L2 to L7 with significant amplitudes although a2 to a7 are all zero. The generation of these transients, resulting in reshaping of the reflected signal (fig. 9a), can be viewed as the result of shock excitation of the dispersive medium by the ultrashort waveform. Solving the system (2.54)-(2.57) we will find n coefficients b,, for the reflected pulse and n coefficients dq for the refracted field. All these coefficients are real. Substituting these coefficients into eq. (1.44), we can present both electric and magnetic components of the EM field inside the medium. It is to be noted that the coefficients b,, describing the reflected Laguerre waveform L0 can be found without solving the entire system (2.54)-(2.57). Doing this, Shvartsburg [ 1999] showed that the reflection coefficient of Laguerre waveform L0 can be expressed simply as a function of coefficient a only: Ro-
bo a0
-
(aD) 2 -
4a 2
(2.58)
(1 +v/1 + 4 a 2 ) 2"
The graph Ro(a) is shown in fig. 9b. As could be expected, in the case of a rarefied plasma (a --~ 0) the coefficient R0 is small (Ro = - a 2) and proportional to the plasma density; in the opposite case (a >> 1, dense plasma) the reflection is almost total (R0 --~ - 1).
2.2.3. Quasi-optical methods of spatiotemporal pulse shaping The appearance of optoelectronic sources and receivers of subpicosecond pulses of terahertz (THz) EM radiation has generated much interest in quasi-optical methods of spatiotemporal shaping of these pulses. Many methods from the microwave domain were scaled down to the THz domain. The continuous refinement of THz generation techniques during the last decade has made several important applications possible, including time-domain far IR spectroscopy (Keiding [1994]), THz imaging (Hu and Nuss [1995]) and generation of tunable narrow-band THz radiation (Weling, Hu, Froberg and Auston [ 1994]). These applications require a specific pulse form such as, e.g., the half-cycle pulses mentioned in w 1.2. Rapid spatiotemporal shaping of almost half-cycle THz radiation, based on diffraction of waves on a conductive aperture with finite thickness, was observed by Bromage, Radic, Agrawal, Stroud, Fauchet and Sobolewski [ 1998]. An effective method of THz pulse shaping, based on dispersive passage through a segment of a metallic waveguide, was demonstrated by McGowan,
180
Instantaneousoptics of ultrashort broadbandpulses and rapidly varying media [3, w2 400
200
"
-
IL
300
1000
200
-100
,
2O0
100
r~
~
<
-100
;I
-200 5
10
115 Time (ps)
20
25
30
Fig. 10. Pulse streching in a waveguide" electric strength of the pulse, observed by McGowan, Gallot and Grischkowsky [1999] at the output of a span of circular waveguide (radius R = 280 ~tm, length L = 0.4cm), is plotted vs. time (2.85).
Gallot and Grischkowsky [1999]. In their experiment the pulses were injected into the circular metal waveguide through a silicon lens and extracted with a second silicon lens. The ~'1 ps input pulse was stretched to 70ps while propagating through the waveguide segment, equal to 80 initial pulse spatial lengths. The waveguide propagation is characterized by a multimode behavior and late-time oscillations; fig. 10 shows the typical envelope of these oscillations in the emerging pulse based on the measurements made by McGowan, Gallot and Grischkowsky [ 1999]. It should be noted that a similar waveform can be obtained by means of direct time-domain solutions of Maxwell equations for a hollow circular waveguide with perfectly conducting walls. Let us consider the simplest axisymmetrical vortical TE0~ mode in this waveguide, with Eq, Hp and H: components. The component Er is known to obey the wave equation
02Er 10Eq + Op2 p Op
Eq 02Er ~ p2 072
1 02Eq - 0. r 2
(2.59)
Setting E~ to
Eq~ - F( p ) f (z, t)
(2.60)
Spatiotemporal reshaping of ultrashort pulses in stationaly media
3, w 2]
181
and substituting eq. (2.60) into eq. (2.59), we obtain the factorized equations governing the functions F and f
02F 1 OF Op--T + P - ~ + k~F = O, 02f
102f
OZ2
r
(2.61)
- k 2 f = 0.
2
(2.62)
Solutions of eq. (2.61) restricted to positive values of the "separation constant" k 2 are given by the Bessel function
F = J1 (k•
(2.63)
The values of the constant k• determined from the boundary condition E~ =0 on the waveguide inner surface are discrete" Xtl
k•
--
R
~
n = 1 2, 3, ~
9
.
(2.64)
.
where xn are the roots of Jl(x,,)=O. The smallest value of x,, is known to be xl = 3.84. Considering eq. (2.64) and introducing the normalized variables r, = ctk• and r/, = zk• one can rewrite eq. (2.62) in the form of the KleinGordon equation (1.37). Substitution of solutions of this equation, expressed via the non-separable harmonics (1.44)-(1.45), and solutions F (eq. 2.63) into formula (2.60) yields the electric strength Ev expressed in the time domain:
Er = Ao ~ Z Jl(k• n=l
d,,q [~pq_2(r,,, r/,,) - 2~pq(r,,, r/,,) + qJq+ 2(r,,, 17n)].
Z q=3
(2.65) The field components Hp and H- can be found from the Maxwell equations:
Ho -
Ao 4 ZvcJ ' ( k • n=l
-
2 Z n=l
Z
d,,q [IPq_2(r,,, r/,,) - lPq+2(r,,, r/,,)] ,
(2.66)
d,,q I~pq_,(r,,, rh,) - ~Pq+, (r,,, r/,,)].
(2.67)
q=3
Jo(k•
Z q=3
Instantaneous optics of ultrashort broadband pulses and rapidly varying media
182
[3, w 2
These solutions, which are both non-stationary and non-sinusoidal, are totally different from the traditional representation of a monochromatic CW train in the same axisymmetrical modes in the frequency domain:
E~c = Z
Jl(k• ,,p) exp[i([3,,z- tot)],
n=l ~L
Hp
= _c Z
A,,[3,Jl(k•
exp[i(fl,,z- cot)],
/092
n=l oo
17: = ic Z
A,,k• ,,Jo(k• ,,p) exp[i(/3,,z - ~ot)]
h ~
o)
it =
I
(2.68) The excitation efficiency of the modes in a waveguide by an external source depends upon the geometry of the problem, spatial scales, and polarization structure of the exciting pulse. To evaluate the excited modes in the circular waveguide one has to perform a numerical overlap integral between the incoming field and the field of the individual mode over the waveguide cross-section. After this one can calculate the reflected field and the field launched into the waveguide, by means of the time-domain sets of eqs. (2.54)-(2.55), written for each mode. Usually, the main contributions to these fields originate from a few modes only. The spatiotemporal structures of non-separable harmonics in (2.65)-(2.68) are qualitatively similar. The whole pulse envelope contains the weighted sum (2.65) of such harmonics. Moreover, the asymptotics of Bessel functions Jl(u,,), un = V/r,,2 -r/~, for u,, >>1 yields the late-time asymptotics of harmonics eq in the form of damping sinusoids, e.g. e3(r,, rl,) =
2 sin (u,, _ 35sr).
(2.69)
The frequencies of these sinusoids are given by the mode cut-off frequencies (2.64):/2,, = x,,cR -l . Thus, the tail of the pulse will contain the waveguide's eigenfrequencies. This "ringing" with a period of about 1 ps was observed by McGowan, Gallot and Grischkowsky [1999] in a pulse passed through a circular waveguide. Another manifestation of the same effect - "ringing" of reflected radiation- will be considered in the next subsection.
2.3. Broadband reflectivity in transient optics Electromagnetic fields of solitary ultrashort waveforms, reflected from targets, possess some characteristics distinguishing them from CW trains reflected from
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Spatiotemporal reshaping of ultrashort pulses in stationary media
183
the same targets. The response of a target often contains some useful information about the nature of reflection. The discovery of reliable sources of picosecond single-cycle waveforms and femtosecond laser pulses has opened a new field of research, connected with the pulsed excitation of EM eigenoscillations in resonant media by some spectral components of the incident broadband pulse. The re-emission of the excited eigenfrequencies can stimulate a dramatic reshaping of the reflected pulse and formation of a long ringing tail of this pulse. The shorter the pulse and the broader its spectral bandwidth, the more likely such resonant effect will arise. The physical fundamentals of identification of complex targets due to their ringing in the microwave range, presented by Baum, Rothwell, Chen and Nyquist [1991], stimulated the development of similar concepts in impulse optics. Such a ringing is a general wave pulse effect, common to the different scatterers and spectral ranges. For instance, the scattering of broadband IR pulses on a metallic surface containing microtips can lead to excitation of IR eigenmodes in these microtips, and to reshaping of the scattered waveforms. A method of fast non-destructive control of surface mechanical imperfections, based on these resonant deformations of scattered pulses, was suggested by Maradudin [ 1999]. Examples of resonant ringing of continuous dispersive media will be discussed in w2.3.1. On the contrary, the non-resonant phenomena in broadband reflectivity of frequency-selective plane interface of a thin heterogeneous film are considered in w2.3.2. Strong inhomogeneity-induced dispersion of such a film provides drastic distortions of both the reflected and transmitted waveforms. A generalization of Fresnel's reflectivity laws that takes into account the usual geometric dispersion of curvilinear reflecting interfaces of transparent dielectrics is illustrated in w2.3.3.
2.3.1. Resonant ringing o f continuous dispersive media
Reflection spectra of crystals reveal pronounced structures near the material resonances. The reflection should show a transient behavior in the case of pulsed excitation as a consequence of the frequency dependence of the reflection coefficient R(~o). The reflected pulse Er(t) is related to the initial pulse spectrum Ei(o)) through the Fourier integral: 2X~
Er(t) = / R(co) Ei(co) do). --0<3
(2.70)
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Instantaneous optics of ultrashort broadband pulses and rapidly ualying media
[3, w 2
To consider the possibilities of transient reflection one can use the model of a 'non-local' medium, where the dielectric function depends on the wave vector k: e(~o) = t~s +
~O2o - ~o e - i o g F + t,,,,,, k 2
(2.71)
Here ~o0 and T' are the eigenfrequency and damping constant of the oscillator, /3 is the polarizability, es is the background dielectric constant, and M is an effective exciton/polariton mass. In the limiting case M ~ ~ , the dielectric function e(~o) in eq. (2.71) describes a Lorentz-resonant medium. The non-zero spectral bandwidth of the incident pulse and the dispersion of the reflection coefficient R(~o) result in the reshaping of the reflected waveform. This approach was shown by Agrawal, Birman, Pattanayak and Puri [1982] to be suitable for the interpretation of different types of transients, formed by interaction of picosecond pulses in the infrared range with GaAs and CdS crystals near the excitonic resonances. To illustrate the temporal reshaping of the reflected pulses, induced by excitonic resonances, Aaviksoo and Kuhl [1989] used the reflection of P-polarized pulses from the medium (2.71) close to the Brewster angle YB. They showed that the general shape of the reflected pulse consists of two parts: a fast nonresonant contribution, which is due to the background dielectric constant and gives rise to a quasi-steady-state reflection of the incident pulse, and which vanishes at Brewster angle, and the resonant contribution to reflection which is responsible for the delayed tail (fig. 11). One can see that the duration of the ringing tail excited by the pulse may be much longer than the characteristic pulse halfwidth. The tail's damping is related to the decay of induced polarization in the reflecting surface layer of the crystal. The Brewster angle of incidence is chosen since near this angle the phase and amplitude of the reflection coefficient change rapidly with the angle. The shorter the pulse halfwidth, the wider is the angular interval Ay = 1~'- yB[ associated with the non-zero reflection close to the Brewster angle. Thus, Campbell and Fauchet [1988] showed that the reflection of a 100 fs P-polarized IR pulse from a GaAs surface, characterized by a value of Ay < 0.01 rad, results in a strong reshaping of the reflected pulse, while for a 10 ps pulse the reshaping becomes negligible. This transient behavior is expected to arise when the exciting pulse possesses sharp leading or tailing edges, which are shorter than the characteristic relaxation time of the system. A truncated square-shaped pulse is preferable for such "shock" excitation. For instance, Mokhtari and Chesnoy [ 1989] reported the observation of terahertz fluorescence transients, containing only a few oscillations with picosecond period, due to excitement of a dye solution by a femtosecond laser pulse.
3, w 2]
Spatiotemporal reshaping of ultrashort pulses in stationary media
10~
185
//~\
10-2 //
\
~
10.4 I 0
I
\.1 20
I
I 40
Time (ps) Fig. 11. Formation of a tail of reflected P-polarized sech pulse at the Brewster angle of incidence in the vicinity of the excitonic resonance; calculation of the cross-correlation traces of the reflected pulse with the incident ultrashort pulse (dashed curve). The values of parameters in eq. (2.71), used in calculation of these graphs, are es = 12.6; fi = 4:r10-3; g2 = 55THz; M = nine; m e is the free electron mass. Curve 1 corresponds to an idealized model ( n ~ , iv = 0). Weakening of the time-resolved reflectivity due to a finite damping factor (n~ec, iv = 0.8 cm-I ) or a finite electron mass (n = 0.3, iv = 0) are shown by curves (2) and (3), respectively. Adapted from Aaviksoo and Kuhl [ 1989]. 2.3.2. Impulse reflection f r o m a m e d i u m with non-local inhomogeneity-induced dispersion Side by side with the reflectivity of Lorentz media, discussed above, one can consider the impulse reflection from a m e d i u m with non-local inhomogeneityinduced dispersion. An example of such m e d i u m consists in the now widely used "index-gradient" layers. In such a case, Shvartsburg, Petite and Hecquet [2000] showed that a "waveguide-like" dispersion arises in the inhomogeneous dielectric layer with some definite distributions of normalized dielectric susceptibility e(z) = n~U2(z). This strong inhomogeneity-induced dispersion can become a decisive effect in preserving or reshaping broadband ultrashort pulses reflected from inhomogeneous film. Shvartsburg, Petite and Hecquet [2000] examined the broadband anti-reflection properties of such film in the case of normal dispersion. They showed that for some parameters of the film the energy reflection coefficient ]RI 2 did not exceed 5% in the frequency range g21 < to < 3(21, where g21 is the characteristic cutoff frequency of the film (fig. 12). Let us consider, e.g., the reflection of square-shaped pulse with carrier frequency too = 1.77 x 1015 rad s -l (~ = 1060 nm) and a total duration 2t0 = 20 fs on the film corresponding to fig. 12. According to expression (1.7), 90% of this pulse energy is localized in the spectral range ( t o 0 - A , to0+A), with A=0.35(2~ here; thus this pulse occupies a spectral range (1.65-2.35)g21, characterized by a value of the reflection
186
Instantaneous optics of ultrashort broadband pulses and rapidly ualying media
[3, w 2
0,07 0,06 0,05
O4
0,04 m
0,03
0,02 0,01 0,00 1,0
,
I
i
I
i
I
2,0
1,5
2,5
30
x = ~o/f21
Fig. 12. Typical variation of the reflection of thin heterogeneous film with positive dispersion. The intensity reflection coefficient R 2 is plotted vs a normalized frequency x = ~o/g21 . 0,8
'
i
,
!
'
i
'
0,6
m
0,4
0,2
0,0
i
j
1
i
2 x = ~o/..Q
i
3
4
2
Fig. 13. Reflectivity o f thin heterogeneous film with negative dispersion (2.97). The intensity reflection coefficient R 2 is plotted vs a normalized frequency x = ~o/s .
coefficient R[ 2 ~ 3%. The propagation of this broadband pulse through the discussed anti-reflection film will be almost distortionless, though some substantial reshaping may occur in the weak reflected field. On the contrary, an example of strong distortion of a square-shaped pulse would arise from the passage of a broadband pulse through a thin heterogeneous film with abnormal dispersion (fig. 13). This type of film is characterized by another characteristic frequency, here equal to -Q2-- 1.6 rad s -1 . The pulse under discussion differs from the previous one by its shorter wavelength (A = 800 nm)
Spatiotemporalreshaping of ultrashort pulses in stationaly media
3, w 2]
187
and a shorter (but still realistic) duration 2t0 = 10 fs; so, A=0.38f22 . Since the carrier frequency is oo0=2.2Q2 here, the pulse occupies the spectral range (1.82-2.58)Q2. One can see from fig. 13 that the reflection coefficient RI 2 diminishes in this spectral range from ]R]2 = 0.7 to ]R]2 = 0.35. Thus, the film acts as a high-pass filter, and both the reflected and transmitted pulses will be drastically reshaped.
2.3.3.
Geometric dispersion
The dispersive phenomena in the reflectivity of plane dielectric interfaces were discussed above with the aid of Fresnel formulae. However, these formulae have to be modified in order to describe the geometric dispersion that arises in the reflectivity of curvilinear surfaces confining a dispersionless dielectric medium. Owing to this effect, some rays, predicted by Fresnel's laws to be totally reflected from this interface, are reflected only partially. The influence of such "geometric" dispersion is displayed in the reflectionrefraction phenomena near the angle of total internal reflection (TIR). Considering light incident from the optically denser medium onto the curved interface between two dielectric media with slightly different refractive indices n l and n2, so that n l > n2, one can define the complement 0c of the TIR angle by
02 = 1 -
n2
(2.72)
It follows from eq. (2.72) that 0c << 1. Snyder and Mitchell [1973] have derived an expression for the power-transmission coefficient T in this geometry:
1(2),J3 ~
T = ~
l
=
()2J3
2 12
[Ai[!exp(3Jr)] ,
(0c2 - 0 t ) .
(2.73)
(2.74)
Here 0 t is the complement of an angle of incidence ~' (0t = ~1 - ) , ) , p is the radius of curvature in the plane of incidence, K is the wave vector in the denser medium, K = m n l c -l , and Ai is the Airy function. Let us examine the grazing incidence of rays onto this interface, when the angle 0t is small, but not zero:
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Instantaneous optics of ultrashort broadband pulses and rapidly vaJying media
[3, w 2
1 >> 0t >> (K/3) -2/3. When the ray is incident with 0 t > 0c, geometric optics predicts the Fresnel value of the power-transmission coefficient T: T=4
~
- 1
(2.75)
for rays with 0t ~ 0c. However, the asymptotic forms of the Airy function, given by Abramowitz and Stegun [1968] as (2.76)
lim ]Ai(x) x<0, Ixl > l __ - ~1 XI-I/4 lim ]Ai(x)l,. >>1
_
2 3/2 ~I X 1/4 exp(-3x ),
(2.77)
show that eq. (2.73) can be reduced to eq. (2.75) only in the case of large negative values of x in eq. (2.76), when 0 2 -- 0 2 ~
(2.78)
( K p ) -2/3.
When the ray is incident with 0t < 0c, Fresnel's standard value T is zero, while the value given by eq. (2.73) is finite. This effect indicates some radiation energy losses due to leaky rays, penetrating to the medium n2. A simple expression for these losses can be obtained for leaky rays which propagate not too close to the TIR angle: 0c2 - 0 2 >> (Kp) -2/3. The asymptotics (2.77) in this case yield
r176 1 ,,
exp[-
ot)3J]
These values for coefficient T are small. Thus, the efficiency of energy transfer through a curved dielectric interface by means of broadband pulses depends on the dispersive losses. These losses, arising due to formation of leaky rays in the vicinity of the TIR angle, can become important for the energy balance in bent optical fibres. Similar dispersive effects were found by Whitten, Barnes and Ramsey [1997] in the propagation of pulses in the whispering-gallery modes in a narrow region around the equator of a transparent homogeneous dielectric microsphere with a diameter about tens of microns. The discrete spectra of these surface modes were shown to provide coupled phenomena of modal dispersion, modal radiation losses and high quality factors Q,~ 107 of spherical optical microcavities.
2.4. Diffraction-induced transformations of ultrashort pulses in a free space There are problems, involving applications of ultrashort broadband pulses, where the spatiotemporal structure of the localized radiation must be carefully
3, w 2]
Spatiotemporal reshaping of ultrashort pulses in stational T media
189
monitored. Any diffraction of these pulses from slits, apertures or gratings in optical systems results in changes of the pulse structure, since each frequency component has its own diffraction pattern. The superposition of all these diffraction patterns tends to smooth out the intensity distribution. A series of classical results in the theory of diffraction of CW trains was reconsidered recently in pulse optics. One can mention, e.g., the non-stationary generalizations of the problems of Fresnel diffraction from circular and rectangular apertures and from a circular opaque disk, performed receptively by Anderson and Roychoudhuri [1998] and Gu and Gan [1996a]. The pulse diffraction from half plane and grating was examined by Rottbrand [1998] and Ichikawa [1999]. Contrary to w2.1, devoted to the transformations of Poisson-spectrum pulses, the present section is centered mainly on dynamics of pulses which are initially Gaussian both in time and space. On-axis time-derivative behavior of pulses and their diffraction-induced spectral variations in the far field are discussed in w167 2.4.1 and 2.4.3; w2.4.2 is focused on the off-axis diffraction phenomena.
2.4.1. On-axis time-derivative behavior of pulses The time-dependent effects in on-axis propagation of diffracted pulse can be examined starting from the statement of this problem presented by Landau and Lifshitz [1970] (an approximation of the Helmholtz-Kirshhoff theorem using Green's functions, applicable when the source irradiating the aperture is either pointlike or spatially totally incoherent, which can be viewed as a foundation of the Huygens principle- on this point see also Sommerfeld [1954]). The monochromatic field E(x,y,z), diffracted on an opening in a flat screen, may be written in the form
E(x,y,z) = 2 f / Eo(x ,,, , 0 ) OG ~ z~]_---0dx' dr.
(2.80)
Here G = (4;rR) -I exp(ikR) is the Green function, with R the distance between the observation point (x,y,z) and the point of integration (xl,y I, 0) in the pupil plane z'=0: R = V/(x-x') 2 + (y_y,)2 + (z_z,)Z; the integration in eq. (2.80) is performed over the opening. Summing over all frequencies o), we obtain from eq. (2.80) the expression for the polychromatic field:
E(x, y,z, t) =
f/
OG [---0 dx' dy' d~o. Eo(x , , y , , O, oo) exp(-iogt)~z~
(2.81)
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Instantaneousoptics of ultrashort broadbandpulses and rapidly varying media
[3, w 2
Considering the far field (kR >> 1), one can find the derivative of the Green function, Oz oa.~ ikz exp(ikR). 4zrR2
OG [ _ Oz~ 1-, = 0
(2.82)
Substitution of eq. (2.82) into eq. (2.81) yields the generalized time-dependent diffraction integral
E ( x , y , z ) = lcf J O ( ~Eo x',y', O, t -
R ) c~R Y dx' dy',
(2.83)
where cos 7 = zR-~ is the angle between the direction of observation and the normal to the screen plane. Equation (2.83), presented by Wang, Xu and Zhang [ 1997], contains all the information about the spatiotemporal structure of arbitrarily short diffracted pulses in the far field. To illustrate some features of this structure let us consider first the on-axis field of a pulse with duration to, incident normally on an opaque screen with a circular opening of radius a. If the electric field E of an incident pulse is independent of the transversal coordinates x' and y', then the field on the axis of the opening Eon(Z,t) can be easily calculated from (2.83) by substitution: dx' dy ' = 2~R dR,
R
= V/a 2 + Z2 ,
v/~a2 + z 2
E~
1-c / o
O[Eo(tR ~-~ -c)
(2.84) 1 cos y dR.
0 Setting z >> a, cos y ~ 1, and using ~7 = - c 30 , we obtain from eq. (2.84):
Eon(Z, t) = Eo t - c
- E~ t -
c
.
(2.85)
Thus, the pulse diffraction results in the appearance of two heteropolar pulses on the axis of the opening, separated by the time interval At - a2/2zc. The origin of this effect is the difference in propagation time to the on-axis observation point for waves radiated by different points of the aperture. These heteropolar pulses do not overlap as long as the time difference At exceeds the pulse duration to. This condition determines the distance of pulse propagation without overlapping: a2
z < z0 -
2cto
.
(2.86)
Thus, when a=0.1 cm, to = 10fs, the distance z0 is sufficient (33cm). In the far field (z >> z0 or to < At) the increasing overlap of the pulses results in a
3, w2]
Spatiotemporalreshaping of ultrashortpulses in stationaiy media
191
weakening of the diffracted field. In this case the field Eo. of eq. (2.85) can be written as
a2 0 l E o ( t - z ) ]
Eon(Z, t) - 2cz 0t
c
(2.87)
which presents a general physical result: the shape of the diffracted waveform in the far zone is determined by the temporal derivative of the initial waveform. This result remains valid for an arbitrary initial pulse; in particular, the homopolar Gaussian pulse (1.8) transforms in the far field into a single-cycle bipolar pulse Eon-
EOa2 ( t - z/C ) exp [ (t - z/c)21 czto to - 2to .
(2.88)
The time-derivative diffraction-induced transformation was considered above in one of the simplest cases, when the field in the opening is coordinateindependent. However, this time-differential aspect of Fresnel pulse diffraction has been discussed by a series of authors, including Ziolkowsky and Judkins [1992], Gu and Gan [1996b] and Kaplan [1998], for a pulse having an initial Gaussian distribution both in space and time:
E(p, t) = Eo exp
t2 2t 2
p2 2a 2 - i~ot).
(2.89)
These researches revealed the influence of the dimensionless parameter B = At~to on the diffracted pulse dynamics. The diffraction was found to greatly affect the shape of the laser pulse when this parameter B was large. For instance, Jiang, Jacquemin and Eberhardt [1997] showed that for B = 40 the diffracted pulse was double-splitted; its width became more than twice that of the initial waveform, and peak power was less than half of the initial waveform peak power. This complex of non-stationary effects may have to be taken into account whenever the critical thresholds are encountered as, e.g., in nonlinear optics or laser fusion.
2.4.2. Frequency effects The off-axis transformation of diffracted pulses with Poisson spectra, discussed above ({}2.1.1), was characterized by an increase in pulse width (eq. 2.9) and a frequency red-shift (eq. 2.10). However, regardless of the shape of the broadband pulse, its lower-frequency part diffracts more strongly and hence is found mostly off-axis, where the higher frequencies are weaker. Similar effects were revealed in the diffraction of a Gaussian pulse: Anderson and
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Instantaneous optics of ultrashort broadband pulses and rapidly valying media
[3, w 2
Roychoudhuri [1998] showed an essential growth of the low-pass filter function of a Gaussian pulse far from the aperture's axis. An eight-times decrease of intensity W(p) at the periphery of the pulsed beam ( p = a), as compared with the axial intensity follows directly from eq. (2.12) for the Poisson-spectrum pulse; the same intensity decrease, characterized by a factor 8, was computed by Aleshkevich and Peterson [1997] for a pulse consisting of a single-period harmonic EM wave. A peculiar property of off-axis radiation was shown by Bertolotti, Ferrari and Sereda [1995]: the red-shift of the spectrum of the diffraction pattern for off-axis observation points is the same for stationary polychromatic light sources and non-stationary light sources with the same spectral composition. Moreover, generalizing the well-known formula describing the intensity W of a monochromatic CW train (W = W0) diffracted from a slit (-a, a) in the far field (R >> a) under an observation angle ),,
(sin0) o
w=w~
0 = ao)0 sin y,
(2.90)
C
Bertolotti, Ferrari and Sereda [ 1995] obtained the expression for spatiotemporal distribution of intensity W~ of a cos-Gaussian pulse, diffracted from the same slit (-a, a): m~ = K
(a)2 ~
IWI2
(2.91)
with o~
~P- x /to~ /
exp [-~J(o)- o)0)2t02 - io)0t'] sin0 0 do),
(2.92)
--O<3
where t ~ is the retarded time t-R/c and K is a normalization constant. The diffracted pulse, depicted in fig. 14, proves to be split into two partially overlapping pulses, the observation angle being large enough. The shorter is the pulse, the deeper is the minimum resulting from its splitting. The unusual properties of diffracted off-axis fields attract special attention for potential applications of diffractive optical elements to pulsed light. Some properties of such elements can be illustrated by means of a diffraction grating, whose spatial period is comparable with the wavelength- so-called grating in the resonance domain. Comparison of diffraction of 100fs pulsed light with central wavelength ~ = 800nm and CW light with the same wavelength, performed both experimentally and numerically by Ichikawa and Minoshima
3, w2]
Spatiotemporal reshapingof ultrashortpulses in stationaly media 100
193
I
(center) 10-I
o
10-
10"
10-
-3
-2
-1
0 t/T
1
2
3
Fig. 14. Time dependence of the intensity of the diffraction pattern at different diffraction angles for a pulsed source with )' = 1/~o0t0 = 0.32. The intensity is normalized to its maximum value at the center./3 = 2Jr (a//~0)sin 0. (Bertolotti, Ferrari and Sereda [1995].) [1999], revealed a significant divergence of non-zeroth diffraction sidelobes of diffracted pulses, caused by their broad spectra. The difference of diffraction angles, related to different spectral components of the pulse, could be used, as was suggested by Ito [1996], for high-speed signal detection by multipleangle spectral interferometry. Agrawal [1998] showed that a pulsed chirped diffracted beam, initially Gaussian in the transversal coordinates, did not remain Gaussian in the course of propagation, and had its spot size enhanced. Analysis of diffraction of the waveform
E(t) = Eo exp I-( l + iC) t21 2t 2
(2.93)
where C is the chirp parameter, revealed that for C = 5 the enhancement of the RMS spot size was as large as 20%. Changes in the diffracted intensity were shown to be noticeable in the visible range for pulse widths below 5 fs and were increasing for large values of chirp parameter C and for pulses whose spectrum had long tails. This example of pulse-spectrum-induced changes in the diffraction pattern leads us to discuss a symmetrical problem: diffraction-induced changes in the pulse spectrum.
2.4.3. Diffraction-induced spectral uariations in the far field To study the spectral shifts occurring in diffraction of broadband ultrashort
Instantaneous optics of ultrashort broadband pulses and rapidly varying media
194
[3, w 2
pulses, let us analyze the spectral intensity of an axisymmetrical beam Lo(p,z) at a point with cylindrical coordinates p and z:
I , o ( p , z ) - [E,,,(p,z)l 2
(2.94)
To find the spectral amplitude E,,, let us consider a spatially Gaussian field, incident normally on a circular aperture with radius a, located in the plane z - 0: El_= 0 = Eo(t) exp
(p2) -~aSa2
(2.95)
.
A solution of the paraxial approximation (2.1) obeying the condition (2.95) was presented by Kaplan [1998] in the form of a Fourier transform
, ),
Eo(og) exp
E,,(p, z) - 1 - iz/L~~--7
2a 2 1 - iZ/LR
(2.96)
(0
E,, (o9)
= ]Eo(t)exp(-iogt)dt.
(2.97)
--(t)
Here LR = oga2c -1 is the Rayleigh range, and 1 - i z / L R is the Gouy factor. Substitution of eq. (2.97) into eq. (2.94) yields the explicit dependence of spectral intensity I,o on the coordinates: o92Eo[o9112
Lo
-
o92 + g22
exp
(p2 O92 ) a2 0)2 + Q2 ,
g2o = zca -2.
(2.98) (2.99)
To find the shift of the spectral maximum of the diffracted pulse let us consider the Gaussian modulated waveform Eo(t) (eq. 2.89) and its Fourier transform
E,, = Eotov/89 exp[-89
2 - 092) t02].
(2.100)
Substituting E,,, from eq. (2.100) into eq. (2.98), we consider the on-axis spectral intensity L,,(O,z) in the far field (g202 >> o92): 2 o92t 2 exp[-(o9 I+o(O,z)- or5 E--6U-o
o90)2 to2].
(2.101 )
The condition s >> oo2 implies, e.g., that for radiation with wavelength ~, = 1060 nm, passing through an opening with a - 0.1 cm, expression (2.101) is
3, w 2]
Spatiotemporal reshaping of ultrashort pulses in stationalT media
195
valid for distances over 5 cm. The maximum of intensity I,,, corresponds to the frequency O)max _ o90 2 [l + V/I + 4(~o0t0)2]
(2.102)
Inspection of formula (2.102) shows that the on-axis spectral maximum is blue-shifted; however, its largest value does not exceed the spectral width to ~. Proceeding in a similar way, one can find the more complicated expressions for radius-dependent off-axis spectral shifts. 2.4.4. Interplay o f diffraction and dispersion-induced reshaping
Unlike the free-space effects treated above, the propagation of spatially and temporally localized waveforms in dispersive media is accompanied by an interplay of diffraction and dispersion-induced reshaping tendencies. The simultaneous action of these tendencies can lead to drastic changes of the waveform's spatiotemporal parameters in the bulk of dispersive media as well as to stabilization of some of these parameters. Some samples of such phenomena are listed below: (a) Considering the propagation of a beam with a transversal spatial frequency spectrum gK • , EKI ~o = gK•
exp [i(k•177 + k_-(~o)z - ~ot)].
(2.103)
Porras [2001 ] showed that compensation of diffraction-induced and secondorder material dispersion effects could provide diffraction-free and dispersion-free propagation of one fixed transversal wavevector component p of a broadband paraxial beam: k 2 =p2 =
b3btt
'~o'~o
(2.104)
(k~)2 + k0k~" where k0 = ~oon(~oo)c -l is the wavenumber corresponding to the central frequency of the beam ~o0, and k~ and kg~ are the coefficients of the firstand second-order dispersion. On the other hand, the CW monochromatic beam with the amplitude profile, given by Bessel function Jo(u), 0 <~u <~2.4, was shown by Durnin, Micely and Eberly [1987] to propagate in free space without diffractive spreading. Unlike this, a pulsed Bessel beam, possessing some finite spectral bandwidth, is characterized by dispersive distortions. However, as noticed by Porras [2001], choosing the transversal profile of the pulsed beam in the form Jo(pr), where p is given by eq. (2.104), one can cancel the second-order dispersion.
196
Instantaneous optics of ultrashort broadband pulses and rapidly valying media
[3, w 3
(b) The dynamics of reshaping processes in pulsed beams is governed by the competition between diffraction, increasing the pulse front curvature, and dispersion, connected with the frequency red-shift at the off-axis part of the pulse. The GVD being normal (abnormal), the red components, diffracted further from the axis, are travelling faster (more slowly), than the blue components, which are nearer to the axis. Thus, GVD results in an additional pulse-front bending; the curvature radius R of the pulse front, arising from the superposition of diffraction and dispersion-induced curvatures was obtained by Porras [1999b] for Gaussian pulsed beams as
R-
1T -~D R(z) '
(2.105)
where LR is the Rayleigh length and LD is some dispersion characteristic length. One can see from eq. (2.105) that the normal dispersion can weaken the convexity of the wavefront, and can even transform it to concave. In the special case LD = LR the wavefront remains plane. (c) The analysis of the focusing of femtosecond pulses in lenses, performed by Bor [1988], revealed a temporal stretching of these pulses, caused by the dispersion of lens materials. A real lens with normal dispersion delays the central part of the beam as compared with the peripheral parts. Later Kempe, Stamm, Wilhelmi and Rudolph [1992] considered the possibility to effectively cancel this defect by means of an achromatic lens doublet. Unlike this, Ibragimov [ 1995] showed that the pulse stretching could be eliminated using a Fresnel zone plate instead of one of the refractive elements in the achromatic doublet. Due to opposite dispersion of the lens and zone plate the chromatic aberration will be compensated in one focus of this combined system.
w 3. Optics of instantaneous media This section is devoted to the amplitude, phase and frequency modulation of light interacting with media with rapidly varying optical characteristics. Such situations can be encountered in a series of hot problems in astrophysics, energy transfer through non-stationary media and optical diagnostics of ultrafast processes. The temporal variations of dielectric susceptibility e(t) under discussion are characterized by finite relaxation times, which can become comparable with the period of the probing light wave. Hence, the use of such well-known approximations of e(t) as the 6(t)-like model elaborated by Felsen and Whitham
Optics of instantaneous media
3, w3]
197
[1970], adiabatic theory developed by Askaryan and Pogosyan [1974], as well as the perturbative approach presented by Masoliver and Weiss [ 1994], cannot be used here. Some on the tendencies in ultrafast pulse reshaping, caused by finite relaxation times of e(t) for plasma and solid dielectrics, were determined by Koretsky, Kuo and Kim [1998] and Ogusu [2000] by means of numerical simulations, carried out beyond such approximations. However, the optics of instantaneous (e = e(t)) media has been less developed up to now than that of heterogeneous (e = e(z)) materials. To provide some physical insight into the electromagnetics of media with time-dependent dielectric parameters, one needs exact analytical solutions of the Maxwell equations, free of any WKB-like assumptions, for such media. To examine the crucial role of non-stationary variations of e(t), one can consider the simplest case when the relaxation dynamics of the medium is governed not by the field of the travelling wave, but by external sources, e.g. by heating, ionization or phase transitions. In particular, this approach can be related to the optics of the probing wave in the so-called "pump-probe" experiments, using ultrashort pump pulses. To tackle this problem we will model a non-stationary, homogeneous, non-magnetic and lossless medium, writing the dielectric displacement D(t), produced by a variable electric field E(t), as a scalar instantaneous dielectric function
D(t) = e(t)E(t).
(3.1)
The function e(t) can be written as
e(t) = n2 UZ(t),
(3.2)
where no is the time-independent value of the refractive index, and the dimensionless function U(t) accounts for the temporal dependence of the dielectric susceptibility. Below we study the following problems: (a) search for models of e(t) that are analytically solvable, inducing a "non-stationaryinduced dispersion", and related closed-form propagation equations (w 3.1). (b) Dynamical effects in the reflectivity of non-stationary media and timedependent generalization of Fresnel formulae (w 3.2). To avoid the massive mathematics these problems will be examined for the simple case of normal incidence of a plane wave on the interface of a non-stationary medium. (c) Unlike these, some polarization phenomena, pertaining to oblique incidence, are treated in w
198
Instantaneous optics of ultrashort broadband pulses and rapidly varying media
[3, w 3
3.1. Non-stationary-induced dispersion in dielectrics
Let us consider a linearly polarized EM wave, incident normally on the boundary of half-space z ~>0, filled by the non-stationary dielectric. The electric and magnetic components of this wave are linked by the Maxwell equations: OE~
1 OH,, -
Oz
(3.3)
c Ot'
OH,.
10D,~ -
Oz
(3.4)
c Ot"
We have implicitly assumed that the medium is non-magnetic. The electric displacement D is determined in eq. (3.1). Two models for e(t) and two related families of exact analytical solutions of the system (3.3)-(3.4) will be examined now.
3.1.1. Analytical solution
The first way to solve the system (3.3)-(3.4) involves three steps: (a) Let us present the field components E,- and H,. via some generating function ~Pl" 1
E~.
O~Pl
10~Pl
n2U21 O z '
(3.5)
H , . - c Ot "
Substitution of expressions (3.5) into the system (3.3)-(3.4) transforms eq. (3.3) to an identity, while the function ~Pl is governed by eq. (3.6)" 021])1
Oz 2
n2U2(t) 021])1 -
c2
Ot 2
- 0.
(3.6)
Introducing the new variable r, having the dimension of time, and the new dimensionless function F~, ! lPl
Fl = v/U, (t~'
dt~ Ul (tl)'
T=
(3.7)
0
we can rewrite eq. (3.6) in the form 02Fs Oz 2
ng O2F, _ c 2 Or 2
n2F, c2
2 Ot 2
4
Ot-
"
(3.8)
The temporal dependence of the dielectric susceptibility still remains unknown. However, with the new representation (3.7), the time-dependent
3, w 3]
Optics of instantaneous media
199
coefficient is eliminated from the left-hand side of eq. (3.8). The nonstationarity is now accounted for by the expression in brackets on the righthand side of eq. (3.8). Though it is not necessary, a particularly interesting class of Ul(t) relates to the simplest case, when the expression in brackets in eq. (3.8) is equal to some real constant T(2; the quantity T1 has the dimension of time: (0Uj) 2 _
102U11 20t 2
4
~
1
(3.9)
T 2"
The function F1 is then governed by an equation with constant coefficients: 02F' OZ2
(310)
nZO2Fl - n2 F, C2 Or 2 C2 T21 9
Equations (3.9) and (3.10) determine the temporal dependence of dielectric susceptibility e(t) and the solution of wave equations, corresponding to this model Ul(t). Let us find first the model Ul(t). (b) The normalized dielectric susceptibility, described by the solution of eq. (3.9) obeying the condition U1 (0)= 1, is sit
Ul(t) = 1 + - +
tl
s2t 2
t2 '
Sl-0,+l,
s2=0,+l.
(3.11)
Here t l and t2 are positive free parameters. The constant T~-2 can be expressed in these parameters:
Tl2
$2
SI
t2
4t~"
(3.12)
Depending on the values of tl, t2 and sl, s2, the function Ul(t) provides flexible representations of both ascending and descending temporal dependencies of e(t). For instance, in a case where s~ - -1, s2 = +1, t2 < 2tl, the function Ul(t) reaches its minimum (fig. 15) at the time train:
(U1)min = U ] t = t,,,in - - 1 - y2 ,
t2
y -- ~
ztl
~ 1,
train = t2 y.
(3.13)
At time tc = 2tmin this dielectric function returns to its initial value U1 (0) = 1.
200
Instantaneous optics of ultrashort broadband pulses and rapidly vaJying media 1,0
'
I
'
I
'
I
'
I
'
I
'
I
'
I
[3, w 3
'
7
0,8
~
III II
0,6
%
Z~) 0,4
0,2
0,0 0,00
,
I
0,25
,
I
0,50
,
I
0,75
,
I
1,00
,
I
,
1,25
I
1,50
,
I
,
1,75
2,00
t/trn Fig. 15. Exactly solvable models of normalized instantaneous dielectric susceptibility U2(t) for the equal values of Umi 2 n = 0.25 and equal durations tc = 2tm. Solid curve: U2(t) (eq. 3.26); dashed curve: U2(t) (eq. 3.11).
(c) The solution of eq. (3.10) related to the model (3.11) can be written as a travelling wave: F1 = A1 exp[i(qlz- or)], ton0 q l = c Nl, N 2= 1
1 (mTi)2"
(3.14)
Substitution of eq. (3.14) into eq. (3.7) yields the generating function ~l = AI v/UI(t) exp[i(qlz- mr)],
(3.~5)
with A I a normalization constant. Finally, substitution of eq. (3.15) into the definitions (3.5) yields the expressions for the field components inside the non-stationary medium (3.11):
Ex - iAltONI [Ul(t)]_3/2 exp[i(qtz- tor)],
(3.16)
cno
H,, - iAlo) [Ul(t)]_l/2 cno
i OU~) 1 + 2o90t exp[i(qlz-cor)].
(3.17)
Before discussing the properties of this field, let us consider another family of exactly solvable models for the Maxwell equations (3.3)-(3.4).
3, w 3]
201
Optics of instantaneous media
3.1.2. Alternative analytical solution
The second method of solution of the system (3.3)-(3.4) follows the same scheme as in w 3.1.1; however, another generating function is chosen. Expressing the EM field components via the generating function E~ -
1
0/])2
c
Ot'
01~
H,,-
Oz
,
(3.18)
we will transform eq. (3.3) to an identity; eq. (3.4), which determines the function ~P2, will be written as
02v,2 OZ 2
2u t)o2v, C2
Ot 2
-
or,2 C2 0 t
(3.19)
Ot"
Using the variable r (3.7) and introducing the new function F2, F2 = ~P2v/U2(t),
(3.20)
one can eliminate the unknown dependence U2(t) from the left-hand side of eq. (3.19):
OZ2
C2 01-2 --
C2
2
Ot 2 + -~
---07-
.
(3.21)
Equating the expression in brackets in eq. (3.21) to some real constant,
will reduce eq. (3.21) to a form similar to eq. (3.10): 02F2 Oz 2
n2 0 2 F 2 _ c2 0r 2
n 2 /:2. c 2 T22
(3.23)
To find the model U2(t) from eq. (3.22) let us introduce the new function Q = x / ~ ; using the variable r of eq. (3.7), we find from eq. (3.22) 02Q Q = 0. r---0 S +
(3.24)
Subject to the sign of the parameter T2 the function U2(r) can be expressed via the solution of eq. (3.24) in two different forms: U2(r) = [cos(r/T2)+ Ml sin(r/T2)] 2 ,
T} > 0,
(3.25)
U2(r) = [cosh(r/T) + M sinh(r/T)] 2 ,
T 2 = - T 2 > 0.
(3.26)
The special case of the right-hand side of eq. (3.22) being equal to zero is not examined here.
Instantaneousoptics of ultrashortbroadbandpulses and rapidlyuaIyingmedia [3, w3
202
The temporal dependence of U2(r) is represented in eqs. (3.25)-(3.26) by the variable r = r(t). To present the dependence of the family of models Uz(t) on physical time, one has to solve these equations with respect to r and to substitute the dependence r = r(U2) into the definition in eq. (3.7). Thus, for eq. (3.26) one finds, supposing M 2 < 1:
-arccosh
(V/ 1 -u2 M 2 ) -arccosh ( v / 1 -1M 2 ) "
(3.27)
Combining formulae (3.27) and (3.7) we obtain an equation governing the temporal dependence U2(t), containing two free parameters T and M: dU2 2 V/ dt - qzT 1
1-M 2 U
(3.28)
T h e - and + signs in eq. (3.28) relate to the descending and ascending branches of the function. The descending solution of eq. (3.28) describes the decrease of U2(t) from an initial value of 1 towards its minimum Umin = 1 - M during a time train:
t l{ j U2( U2 + M 21-1)
-f = -~ M -
u2 2 )1}(i .29) + ( I - M 2) I arccosh ( v / I -1M 2 ) -arccosh (V/ I - M
{
[
tminT - 21 M + (1 - M 2) arccosh
( v/1 _1M 2 )1 }
,330,
Dividing eq. (3.29) by eq. (3.30) one can eliminate the free parameter T, replacing it by the characteristic evolution time train: t
tmin
arccosh 1-
_M2 )
M + (l _ M2)arccosh (1/v/l _ M 2)
(3.31) Unlike (3.11), this model U2(t) is more easily expressed via an inverse function t = t(U2). The ascending branch, related to the growth of U2(t) from its minimum g r o i n --- 1 - M up to the value U = 1, can also be found starting from eq. (3.28). Both branches are shown in fig. 15.
Optics of instantaneousmedia
3, w 3]
203
Substitution of the solution of eq. (3.23), written in the form of a travelling wave, similar to (3.14)-(3.15), into expression (3.20) yields the generating function A2 exp[i(q2z- COT)] ore0 o)n0 lP2 = , q2 V/1 - ( c o T ) -2 = N2. (3.32) CU2(t)
c
c
The field components E,- and H,. can be found from eq. (3.18):
Ex - iA2r176 [U2(t)]-l/2 (1 -
cno
i OU2) 2oo Ot exp[i(q2z- ~or)],
[U2(t)]_3/2exp[i(q2z-
My = iA20)N2
cno
~or)].
(3.33) (3.34)
The derivative OUz/Ot is given in eq. (3.28). Thus we have obtained, with Ul(t) and U2(t), two models for the nonstationary dielectric susceptibility allowing to obtain exact solutions of the propagation equation in a non-stationary dielectric, in the sense that no "slowly varying envelope" assumption has to be made. The application of these models can be exemplified by the problem of probing wave reflection from a dielectric, in which the carrier density grows steeply due to ionization induced by a high-power laser beam. It is convenient to present the instantaneous dielectric susceptibility by means of functions e = eL 1 -
c~2
= eL 1 - ~oXj
(t),
(3.35)
where eL is the lattice-dependent part of the dielectric susceptibility, f202= e2No(meL)-1, and U(t) will be approximated by a function of the form (3.11). No is the unperturbed electron density, and the function K(t) satisfies the condition K(O)= 1. Let us consider a "symmetrical" regime, where the ionization reaches its peak Nmax = NoKmax in a time tmax and returns thereafter to the initial value in the same time interval. This regime relates to the concave profile of U] depicted in fig. 15. The time scales t l and t2 in U12 can be found from the condition (3.13) at time t=tmax 9
g m a x = g l l t=t = ( 1 - y 2 ) 2 = ( 1 - ~ 2 2 K m a~ x ) ( ....
~22) -1
1 - ~-~
(3.36)
Using the value Kmax one can calculate from eq. (3.36) the parameter y and express the time scales t~, t2 and To via the time interval tmax" tl-
tmax 2y 2 ,
tmax
t2-
Y ,
tmax
To = YV/1 _ y2
(3.37)
This approach will be used below for the analysis of instantaneous plasma reflectivity.
204
Instantaneous optics of ultrashort broadband pulses and rapidly valying media
[3, w 3
3.1.3. Amplitude-phase structure of field components The expressions for EM fields (3.16)-(3.17) and (3.33)-(3.34), propagating in the non-stationary media, described by both models U~ and U2, contain the phase factor exp[i(qz- mr)], expressed using the variable r: in this "r-space", the spatiotemporal structure of the discussed EM fields resembles that of travelling harmonic waves with time-dependent amplitudes. Let us analyze the amplitude-phase structure of field components (3.16)-(3.17), emphasizing some peculiarities of this structure, originating from the strong non-stationarity of the medium they travel in: (a) The wave propagation is characterized by non-stationarity-induced dispersion, described by the factor N in eq. (3.14). The "concave" temporal profile of dielectric susceptibility, depicted in fig. 15, gives rise to normal dispersion (N - V/1-(roT)-2); this waveguide-like formula shows the appearance a non-stationarity-induced cut-off frequency Y2 = T -~. On the contrary, a convex profile of U(t) gives rise to abnormal dispersion (N -- V/1 + (roT)-2), and the cut-off effect does not arise in this case. (b) The electric component (3.16) can be viewed as a non-stationary wave, travelling in (z, r) space with a constant phase velocity. However, in the physical space (z, t) the phase velocity varies in time. These variations give rise to a drastic reshaping of waveforms. To determine this reshaping, one has to find an explicit expression for the parameter r in eq. (3.7). For simplicity, let us consider the case where a wavefront is incident on the boundary at the beginning of temporal variation of dielectric susceptibility t=0. The result of the integration in eq. (3.7) in this case is r(t) (t/To) To-arctan l_ty2/tmax
(3.38) .
The duration of perturbation of dielectric susceptibility is tc = 2tma• At any time tf ~< tc the position z0 of the front of the electric waveform (3.16) is determined by the equation
qzo - mr(tf) = 0.
(3.39)
Let us characterize by its phase difference r with the front edge a waveplane crossing position z0 at a later time td (tf < td ~< tc). Given the constant phase velocity of the wave in the perturbed medium in "r-space", one has: r ( t d ) - r ( t f ) - q~, m
(3.40)
which can be used to determine the unknown to. One can then find the amplitude modulating factor with expression (3.16). Propagation through
3, w 3]
Optics of instantaneous media
205
Fig. 16. Reshaping of the leading part of harmonic cw train (~o = 1.77 rad s- I ) after passing through a non-stationary dielectric described by the U2(t) temporal dependence (3.11 ), shown in fig. 15 by the dashed curve. The normalized electric strength (3.16) e = [ Ui (t)]- 3//2 sin( q l z - ~or) is plotted vs the normalized delay (t~t - t/)/tc in a plane z = ~or(tc)q-(I; tc = 4 fs, U~ain = 0.25.
the area with decreased dielectric susceptibility results in an increase of the electric field amplitudes. Since the temporal perturbation stops at time tc, the parts of the waveform incident on the plane z = 0 at times t > tc traverse the medium without reshaping. Thus, only the leading front of the waveform proves to be distorted (fig. 16). (c) The non-stationarity-induced dispersion is determined by the finite relaxation time tc of the instantaneous dielectric susceptibility. It decreases towards zero with increasing tc, and the quantities N, U and r tend to the values N = 1, U - 1 and t. The fields Ex and H,. approach those of a sinusoidal wave in a stationary dielectric. (d) Finally, it should be mentioned that the results presented for normal incidence here can be generalized to an arbitrary angle of incidence. In that case, the model (3.11) applies only to a P-polarized incident field, and the model (3.26) applies to a S-polarized one. In the case of normal incidence, both polarizations coincide, and it is natural that both models apply. In the general case, the only time dependence which simultaneously applies to both polarizations [with the requirement of a constant right-hand term in eqs. (3.8) and (3.21), which is not strictly necessary] should obey the condition 0 2 U / O t 2 = 0, and thus is of the type U ( t ) = 1 + t/to. Note that this model is obviously contained in expression (3.26), but also in (3.11), with M = 1.
206
Instantaneousoptics of ultrashort broadbandpulses and rapidly varying media [3, w3
3.2. Dynamical regime in reflectivity of non-stationary dielectrics Electromagnetic fields in non-stationary dielectrics were shown to depend not only on the current values of time-dependent dielectric susceptibilities e(t), but also on the temporal derivatives of U ( t ) - see, e.g., eqs. (3.16)-(3.17) and (3.33)(3.34). These derivatives determine the non-stationary phases of reflected and refracted waves. To find the generalized Fresnel formulae describing these dynamical phase effects, let us examine the problem of reflection of a linearly polarized EM wave with frequency to on the non-stationary dielectric, filling the half-space z > 0. Using the continuity conditions for the fields at the interface, one can obtain the reflection coefficients (for the general case, for both S- and P-polarizations, with an arbitrary incidence angle y):
Rs =
c~
sin z
,
ns = noU(t)Ns
(
1
i OU)-' 20o Ot '
(3.41)
cos )' + x/ n~ V
Rp =
-
ngUZ(t)c~ ngU2(t)cosy+
~/
;
(
np= noU(t)Np
iUr)-' 1 + ~-~ , (3.42)
n 2 - s i n 2),
where we used a complex angle of refraction/3, sin ), = ns, p sin/3.
(3.43)
Note that these results are general and apply under the only condition that eq. (3.8) or (3.21) have a constant right-hand side. One important remark concerns the fact that the laws of reflection (non-stationary counterparts of the Descartes/Snell and Fresnel laws) depend on a "reflection" index (ns, p) which differs from the refractive index (Ns, p, determining the propagation inside the dielectric, and accounting for the non-stationarity-induced dispersion described above), and also differs from the "instantaneous refractive index" n(t) = noU(t). Let us now return to the case of normal incidence, in which S- and P-polarizations coincide, and use a model with a very simple dielectric constant, encompassing both cases developed above, i.e.
U(t) = 1
t t0
(3.44)
This allows simple calculations, but nevertheless yields results which can be generalized to the two models for U discussed above, at the expense of some
3, w 3]
Optics of instantaneous media
207
increased algebraic complexity only. Here, one has N 2 = N 2 = 1 + (2cot0) -2= 1 + 1/cp2. The real and imaginary parts of the reflection coefficient are then N(1 - n2(t))
ReR =
N(1 + n2(t))+ 2n(t)' -2in(t)
ImR =
cp0 [N(1 + nZ(t))+ 2n(t)] "
(3.45)
(3.46)
This yields the intensity reflection coefficient [R[ 2 and the phase q~: ]R 2
=
N2[ 1 - n2(t)] 2 + 4n2(t)/cg 2
(3.47)
(N[1 + n2(t)] + 2n(t)) 2 [
q~ = arctan
-2n(t) ] cP0N[1 - n2(t)] "
(3.48)
In the limiting case when the non-stationarity vanishes (t0,r ~ oc), N ---+ 1, ImR vanishes while Re R reduces to the usual Fresnel formula. It is interesting to compare this to a "quasistationary" approximation in which one would simply use the regular Fresnel formulae with the instantaneous value of the "normal" refractive index (we consider the case of a positive dielectric constant). The Fresnel reflection coefficient in that case is R=
1 - n(t) 1 +n(t)
(3.49)
which, firstly, has no imaginary part and, secondly, has a real part that is quite different from that in eq. (3.45). Figure 17 allows one to evaluate the errors that result from the use of the quasistationary approximation. The intensity reflection coefficient (3.47) and the phase (3.48) are compared to that obtained from the quasistationary value (3.49). There are very significant differences up to values of cp0 of a few units. For cP0 = 5 the intensity reflection coefficient is almost equal to the quasistationary value, but some significant effects are still visible on the phase up to cp0 = 10. The phase of the reflected light therefore seems to be a much more sensitive test of such effects than the intensity reflection coefficient. It is also interesting to consider the case of a negative dielectric constant, arising in the area t > to. e(t) can be represented in this area by using the function U(t)=i
(') T0-1
(3.50)
which provides continuity of the dielectric susceptibility and its temporal derivatives at time t = to. With such a U(t), the modulus of the reflection is equal
208
Instantaneous optics of ultrashort broadband pulses and rapidly valying media 1,0
1
0,9
a)
0,8
i
i
i
[3, w 3
,
~ to=0.5 fo=l fo=2 9
0,7
9
0,6
- - Fresnel
0,5
:. ~ ,
I~ 0,4 0,3 0,2 0,1
o , o ~ 0,0
0,2
........
~ ,
0,4
I
0,6
0,8
1,0
t/to 0,0
!
b)
-0,5
D
0
-1,0
.-0
~
O-
V
V
"
9
.0"
o o~::-, .~""
-1,5 -2,0 /
fo=2 fo=5
- - - ~ - fo = 10 -- Fresnel
-2,5 -3,0~
fo = 0.5 fo = 1
', o
: ~ ~ i
0,0
0,2
I
i
0,4
I
0,6
,
I
0,8
i
1
t/t0 Fig. 17. Reflection coefficient of a non-stationary dielectric corresponding to e variations of eq. (3.2), for different values of the parameter q~0, compared to the result of a "quasistatic" approximation using the instantaneous value of the standard Fresnel coefficient: (a) intensity reflection coefficient; (b) phase. The beam is at normal incidence.
to 1 if (and only if) c;0 > 1. In that case, the phase of the reflection coefficient is found to be q~ = arctan [ 2 q ~ o ( t / t o -
L q)o
1)(cgo - 1) 1/2 + 1 - ( t/to - 1)2((.flo - 1)
(3.51)
However, when cr < 1, one finds a real reflection coefficient, equal to R = q)g - (t/to - 1)(1 - q?02) 1/2 cr + (t/to - 1)( 1 - Cr
'
(3.52)
smaller than unity, as it should, which brings in the conclusion that an ultrafastvarying medium of negative dielectric constant may not be totally reflective! Quite normally, when q)0 ---' 1(-), the reflection coefficient (3.52) tends to unity,
Optics of instantaneous media
3, w3]
209
whereas when cp0 ~ 1(+), the phase (3.51) tends to zero, so that continuity between the two regimes is perfect. These phenomena, which cannot be described by the "quasistationary" expression (3.49), illustrate the importance of dynamical effects in the reflectivity of instantaneous dielectrics. It should be noted that the dielectric susceptibility in the model (3.44) tends to zero when t --, to; in this case, which is experimentally realizable, one finds that the fields in the medium - (3.33) and (3.35) - take arbitrarily high values, which questions the validity of the linear approximation used in the definition of the displacement current (3.1). It should also be noted, however, that the strong effects in reflectivity, shown in fig. 17, do not particularly occur in the region where e vanishes, so that this limitation of the model does not affect the above conclusions. Moreover, one would obviously have to take into account in such a case the dissipation processes that occur in any real physical situation. Whether they would be strong enough to reduce the field to values acceptable for a linear theory cannot be ascertained at present, but it is worth showing that dissipation can be included in the above theory relatively simply by using complex time scales, describing the non-stationary complex dielectric susceptibility. Considering the simple generalization of the model (3.44), describing both time-dependent refraction and attenuation,
t
U(t) = 1 - ~ to + itl'
(3.53)
one can present the variable r (eq. 3.7) in complex form: r = Re r + i lmr, Rer = t o D - tiM,
1
D-~ln
I/
M=arctan
Im r = t l D + toM,
tto
1
2
t~) +t~
) ( +
ttl
t2o +t~
-
'
(3.54)
tt~ ) t~ + t~ - tto "
Substitution of eqs. (3.53) and (3.54) into the field representation (3.16)-(3.17) provides finite values for all field components at any time. The non-stationary reflectivity of a medium (3.54) is determined again by eqs. (3.45)-(3.46), with parameter to replaced by to + it l; the expressions for of0 and N become complex: ~0 = 2m(t0 + itl),
N = V/1 + [2to(t0 + itl)] -2.
(3.55)
210
Instantaneous optics of ultrashort broadband pulses and rapidly uarying media
[3, w 3
These results illustrate the d y n a m i c a l regimes o f reflectivity o f non-stationary dielectrics, which d e p e n d essentially on the relaxation times o f their dielectric susceptibility, for n o r m a l incidence [except for the general expressions ( 3 . 4 5 ) (3.48) which include the case o f oblique incidence]. Some d y n a m i c a l effects for oblique incidence are considered below.
3.3. S p e c t r a l d i s t o r t i o n s o f w a u e s r e f l e c t e d f r o m an i n s t a n t a n e o u s d i e l e c t r i c
The t i m e - d e p e n d e n t phase shift q~(t) o f a reflected wave was p r e s e n t e d in eq. (3.51) for n o r m a l incidence. These t e m p o r a l variations o f q~(t) can be
0.35 I o
0.25
-
~176I0.15
O.lO~ 0.05 0.00 0.0
a) ,
012
,
0'.4
,
0'.6
018
'
t/t m
0.0
,
!
:70o:
'
-0.2
-0.4
s -0.6 -0.8
b) -1.0
0.0
,
0'.2
'
0'.4
'
01.6
'
018
'
.0
t/t m
Fig. 18. (a) Modulus of reflection coefficient Rp and (b) relative frequency shift u = Ao/~o for a P-polarized wave reflected from the non-stationary dielectric e(t)= no U2(t), plotted vs the normalized time x = t/tm; the model U(t) is given in eq. (3.11 ), no = 3.5, tm = t~/2t I . The parameters of model (3.11)are: tl = 120fs; t2 = 155 fs.
3, w4]
Conclusion
211
considered as spectral distortions of a refected wave: Aw(t) -
Ot "
(3.56)
Unlike the well-known Doppler frequency shifts, pertaining to the reflectivity of moving media, the spectral shifts (3.56) originate from the temporal variations of the dielectric susceptibility of a motionless medium. So they should rather be viewed as a phase modulation, not self-induced in this particular case where the modifications of the dielectric susceptibility are not due to the EM field we study. These spectral shifts can be quite significant for oblique incidence of the EM wave. If we consider the case of a P-polarized incident beam, the reflection coefficient is given in eq. (3.47). Again, we emphasize the dependence of the reflectivity on both the current value of dielectric susceptibility and its temporal derivative. Figure 18 illustrates two particular features of the time-dependent reflection and the spectral distortions of the reflected wave, corresponding to the descending branch of the model (3.11): (i) the amplitude IRe[ of the reflection coefficient has no zero-crossing point for any angle of incidence y. Thus, the situation ]Rp I - 0, well known for the reflection of P-polarized waves from stationary lossless dielectrics under some angles of incidence (Brewster effect), does not appear for non-stationary dielectrics (fig. 18a). (ii) The waves reflected from the medium with decreasing dielectric susceptibility are red-shifted (fig. 18b).
w 4. Conclusion The analysis of ultrafast transient optical problems in this chapter centered on a number of phenomena of the spatiotemporal dynamics of ultrashort pulses travelling in stationary dielectric media, and interactions of CW trains with nonstationary dielectrics. Paraxial propagation of ultrashort pulses in free space was shown to provide a series of coupled instantaneous effects- spectral distortions, waveform reshaping and polarization reversal. Localization of pulses in a plane orthogonal to the direction of propagation results in a complicated interplay of diffractive and dispersive effects. Several analytical models for the interaction of ultrashort pulses with media characterized by material- and inhomogeneityinduced dispersion were considered both in the frequency domain and in the time domain. A generalization of the Fresnel formulae for the reflectivity of instantaneous dielectrics has been presented based on exact analytical solutions of the Maxwell
212
Instantaneous optics of ulo'ashort bi'oadband pulses and rapidly valying media
[3
equations for these media. The dynamical regimes of instantaneous reflectivity, illustrating the decisive influence of non-stationarity-induced dispersion on amplitude, phase and spectral distortions of reflected waves, were considered. Optics of transient waveforms is nowadays a hot and rapidly expanding branch of electromagnetics, and it is useful to identify some problems that are still open to this date: (i) Optics of nonstationary inhomogeneous media, in which the dielectric susceptibility of the medium is varying in time and space simultaneously. The simplest model, in which the temporal and spatial perturbations of e(t) are supposed to be independent (Stenflo, Shvartsburg and Weiland [ 1997]), reveals a complicated interplay between inhomogeneity- and nonstationarity-induced dispersive phenomena. However, an analysis of coupled spatial and temporal variations of e(t) is lacking. (ii) The search for optimized regimes for the propagation of directed pulsed beams of anharmonic waveforms through non-stationary media. (iii) In all problems mentioned above, the media had well-determined distributions of dielectric parameters. However, the non-stationary optics of random media is still awaiting research. This problem is particularly relevant to the use of ultrashort pulses in the field of atmospheric optics.
Acknowledgements The authors are much obliged to Professors G.P. Agrawal, J. Eberly, S. Haroche, A. Migus, L. Vazquez and E. Wolf for their continuing interest in this work and many valuable discussions.
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E. Wolf, Progress in Optics 44 9 2002 Elsevier Science B. V. All rights reserved
Chapter 4 Optical coherence tomography
A.E Fercher and C.K. Hitzenberger Institute of Medical Physics, University of Vienna, Waehringer Strasse 13, 1090 Wien, Austria
215
Contents
Page w 1.
Introduction .
w 2.
Principles o f O C T
w 3.
O C T signal generation
w 4.
O C T contrast generation . . . . . . . . . . . . . . .
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w 5.
O C T resolution and sensitivity
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w 6.
Light sources and delay lines .
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Conclusion
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w I. Introduction "Tomos" means section. A tomogram is a slice image of the internal structure of a three-dimensional object. Optical coherence tomography (OCT) uses backscattered optical radiation to synthesize a slice image. The standard OCT technique uses a series of adjacent linear depth scans. The latter are similar to the more familiar ultrasound A-scans but use coherence properties of light to obtain superior depth resolution. Optical coherence tomography can exploit any physical property that affects the amplitude, phase, frequency, polarization, or coherence of the probe beam to extract information about the object. The presence is dominated by imaging in the medical field. Several optical tomography techniques have been developed since the early 1980s (Kak and Slaney [1988]). There are three basic approaches: (1) diffraction tomography (Wolf [1996], Born and Wolf [1999]), reconstructing the object structure from the corresponding diffraction pattern; (2) diffuse optical tomography (Yodh and Chance [1995]), deriving spatial maps of absorption and scattering coefficients from the characteristics of multiple scattered light transmitted through the body; and (3) OCT, based on low-time-coherence interferometry (LCI), also called "partial-coherence interferometry". Reflectometric OCT was pioneered by Fujimoto and co-workers (Huang, Swanson, Lin, Schuman, Stinson, Chang, Hee, Flotte, Gregory, Puliafito and Fujimoto [1991]). A first OCT image of the structure of the retinal pigment epithelium of a human eye obtained in t)ic~o by dual-beam OCT was presented in 1990 at the ICO-15 SAT conference by Fercher [1993] and published by Hitzenberger [1991]. The first in viuo OCT images of the human retina were published by Fercher, Hitzenberger, Drexler, Kamp and Sattmann [1993], and Swanson, Izatt, Hee, Huang, Lin, Schuman, Puliafito and Fujimoto [1993]. Later, wavelength tuning interferometry was used to synthesize OCT images by Chinn, Swanson and Fujimoto [1997], whereas H~iusler and Lindner [1998] described the generation of OCT images by spectral interferometry. OCT synthesizes cross-sectional images from a series of adjacent LCI depth scans. In contrast to classical interferometry, based on incremental interference fringes, LCI can measure absolute distances. Whereas in classical interferometry the process to be evaluated changes the order of fringes observed at the 217
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Optical cohelence tomography
[4, w 1
interferometer exit, LCI is based on the occurrence of fringes, if the optical path lengths of reference and object beam coincide within the "coherence gate", which is of the size of the coherence length. OCT has, owing to its LCI basis, some outstanding properties: First of all, depth resolution is decoupled from transverse resolution. High depth resolution is possible even at sites not accessible by high-numerical-aperture beams. Second, depth resolution is of interferometric quality. Subsurface imaging with nearhistological resolution in the 1 gm range is possible. Third, the interferometric technique provides high dynamic range and sensitivity (>ll0dB). Imaging of weakly scattering structures even in a scattering environment is possible, enabling "in situ optical biopsy". There are three basic LCI techniques: one working in the time domain, one in the wavelength domain and one in Fourier space. The time-domain technique uses the correlation performance of white-light two-beam interferometry to obtain the depth distribution of light remitting sites in an object. First applications of the time-domain white-light two-beam interferometric technique to in vivo medical depth ranging have been described by Fercher and Roth [1986], and to optical fiber testing by Danielson and Whittenberg [1987] and by Takada, Yokohama, Chida and Noda [1987]. The fiber-optic reflectometer white-light two-beam interferometric technique was first used to measure intraocular distances in vitro by Huang, Wang, Lin, Puliafito and Fujimoto [1991]. Later, first medical applications of Fourier-domain spectral interferometry were described by Fercher, Hitzenberger, Kamp and E1-Zaiat [1995], and wavelength-tuning LCI was used for in vivo measurement of intraocular distances by Lexer, Hitzenberger, Fercher and Kulhavy [1997]. OCT image generation using wavelength tuning has been described by Chinn, Swanson and Fujimoto [1997] and by Haberland, Blazek and Schmitt [1998]. The first Fourier-domain OCT images were presented by H~iusler and Lindner [1998]. Ophthalmology is, due to the transparent ocular structures, still the main field of OCT application. The first commercial instrument (Humphrey Systems, Dublin, USA) was introduced for ophthalmic diagnostics. Advances in using near-infrared light, however, opened the path for OCT imaging in strongly scattering tissues. Optical in vivo biopsy (Fujimoto, Brezinski, Tearney, Boppart, Bouma, Hee, Southern and Swanson [1995]) and endoscopic OCT (Tearney, Brezinski, Bouma, Boppart, Pitris, Southern and Fujimoto [1997]) are promising fields of OCT in medicine. For the time being, however, technical applications remain an exception. Important subjects of present OCT research are new light sources to increase
4, w2]
Principles of OCT
219
depth resolution, to enable spectrometric OCT, and to get new access to tissue properties. Another open question at the technologic side is the development of parallel ac detectors. On the more basic side there are still open questions concerning the type of information that OCT yields, and how fluorescence and low space-coherence light can be used in OCT imaging. Several reviews on OCT have appeared so far: Fujimoto, Brezinski, Tearney, Boppart, Bouma, Hee, Southern and Swanson [1995], Fercher [1996], Izatt, Kulkarni, Kobayashi, Sivak, Barton and Welch [1997], Fercher and Hitzenberger [1999] and Fercher [2000]. A selection of important original papers on OCT has been published by Masters [2000]. Technical details and an overview on applications of the OCT technique can be found in the Handbook of Optical Coherence Tomography (Bouma and Tearney [2002]). In this report we address basic issues of OCT. After a general discussion of OCT configurations and basic sample signal extraction in w2 we discuss OCT signal generation in w3. OCT contrast generation techniques will be presented in w4. OCT resolution, signal enhancement and sensitivity is discussed in w5. Finally, in w6, we present some OCT specific light sources and delay lines and conclude this report in w7.
w 2. Principles of OCT 2.1. Basic OCT configurations Figure 2.1 presents general components used in OCT systems. OCT-specific light sources and delay lines are discussed in w6. The central part of an OCT system is the low-coherence interferometer, usually of Michelson configuration. From an optical point of view it is a "white-light" interferometer (Hariharan [ 1985]). Several modifications of white-light interferometers have been used in the fields of LCI and OCT. The first LCI interferometer used in an OCT setup for biomedical application was a dual-beam Fabry-Perot interferometer (Hitzenberger [1991]). Later, the dual-beam technique has been used in Michelson configuration for ophthalmologic studies (Drexler, Findl, Menapace, Kruger, Wedrich, Rainer, Baumgartner, Hitzenberger and Fercher [1998]). Today, the majority of OCT interferometers are Michelson configurations implemented in fiber-reflectometer technology as described by Huang, Swanson, Lin, Schuman, Stinson, Chang, Hee, Flotte, Gregory, Puliafito and Fujimoto [1991] or in bulk optics (Hitzenberger [1992]). A discussion of optimal interferometer designs using state-ofthe-art fiber technology has been presented by Rollins and Izatt [1999]. Figure 2.2 depicts basic OCT interferometer schemes. In the reflectometer scheme the sample is positioned in one arm of the interferometer. In the dualbeam scheme the sample is positioned outside the interferometer and illuminated
220
Optical cohelence tomography
[4, w2
Fig. 2.1. Basic components of an OCT system and some of its functions and variations. ASE, amplifiedspontaneousemissionfiber light source; CCD,CCD detectorarray; MML,multimode laser; PC, PC/monitor; PCE photonic crystal fiber; PIN, PIN photodiode; SLD, superluminescent diode; SPDA, smart pixel detector array. by both beams exiting the interferometer. In this scheme the LCI depth scan records light remitting sites in the sample if the interferometer path difference matches sample path differences. This configuration is, therefore, not sensitive to relative movements between sample and interferometer. The unique stability of this technique in depth direction facilitates high-precision depth measurements. It is used in ophthalmology for intraocular distance measurement, and has found a wide range of applications in physiologic studies and in diagnostic biometry (Fercher and Hitzenberger [1999]). En-face OCT generates transversal images, i.e. images in planes parallel to the sample surface, whereas standard OCT generates slice images in depth direction. En-face OCT images can be obtained by fixing the reference mirror and transversely scanning either the sample beam of the interferometer or the object (Izatt, Hee, Owen, Swanson and Fujimoto [ 1994], Podoleanu, Dobre and Jackson [1998]). Often separate phase modulation is used to generate a carrier frequency. OCT synthesizes cross-sectional images from a series of adjacent scans. Motion artifacts can be removed by cross-correlating adjacent scan signals (Swanson, Izatt, Hee, Huang, Lin, Schuman, Puliafito and Fujimoto [1993]). All
4, w 2]
221
Principles of OCT
T
~T
Reference
D
Source
Sample Detector
I
Reference
" s: Je Source
{a} Reflectometer
Detector
{b} Dual beam
Reference
Oe, ,e ce
Modulator Source
__
le Detector
{c} En face
Source
0 Sample I Detector Array
{d} Parallel
Fig. 2.2. Basic OCT interferometer schemes. The open double arrow indicates the rapid (or "priority") scan. (a) Reflectometer: based on Michelson LCI; this is the dominating optical scheme. (b) Dual beam: this configuration is not sensitive to longitudinal movements between sample and interferometer. (c) En face: the fast scan is performed transversally; a separate modulator can be used to generate the carrier frequency. (d) Parallel OCT: The sample is illuminated with an extended beam and imaged on an array of photodetectors.
OCT schemes, except the parallel one, use standard lateral scanning. Most of these lateral scanning schemes are also used in confocal microscopy and related techniques. OCT-specific scanner techniques have been described by Feldchtein, Gelikonov and Gelikonov [2002]. Parallel OCT eliminates the need for lateral scanning and can, therefore, dramatically increase the frame rate. This is important, for example, in in situ optical OCT biopsy which must provide the clinician with near real-time imaging. The sample is illuminated by an extended beam and imaged on an array of photodetectors. A parallel OCT device based on a Linnik interference microscope has achieved image acquisition rates up to 50 Hz (Vabre, Dubois, Beaurepaire and Boccara [2000]). CCD cameras are used as detectors in parallel OCT. Their performance is limited by their integration times and their dynamic range is strongly reduced by dc light. Furthermore, it is difficult to implement heterodyne
222
Optical coherence tomography
[4, w 2
signal detection even if a parallel lock-in technique enables shot-noise limited detection (Beaurepaire, Boccara, Lebec, Blanchot and Saint-Jalmes [1998]). These drawbacks might be overcome by novel ("smart") silicon ac-detector arrays where each single photodetector has an attached electronic circuitry for OCT signal processing (Bourquin, Monterosso, Seitz and Salath~ [2000], Bourquin, Seitz and Salath~ [2001]). Such new detector technology m i g h t besides new light sources, see w6 - open the door for new applications of OCT.
2.2. OCT macroscopy, microscopy and endoscopy At present there are three promising fields of optical imaging for OCT (figs. 2.3 and fig. 2.4): macroscopic imaging of structures which can be seen by the naked eye or using weak magnifications, microscopic imaging using magnifications up to the classical limit of microscopic resolution, and endoscopic imaging, using low and medium magnification. Imaging of macroscopic structures is performed by standard OCT with low numerical aperture focusing to obtain a large depth of field. This choice of parameters is typical in some ophthalmologic applications where the high depth resolution offered by OCT is of significance. However, high-numerical-aperture OCT can provide transversal resolution comparable to conventional histology, but in real time. Therefore, it has a potential to be used for real-time "optical biopsy" (Fujimoto, Brezinski, Tearney, Boppart, Bouma, Hee, Southern and Swanson [1995]). Optical coherence microscopy (OCM) aims at high-resolution tomographic imaging. High-numerical-aperture lenses are used to focus the probe beam at the sample (Izatt, Hee, Owen, Swanson and Fujimoto [1994]). Compared to confocal scanning microscopy (CSM), OCM provides improved rejection of light scattered at defocused planes and, therefore, improved optical sectioning (or depth discrimination) in highly scattering tissues. To achieve depth resolution in the 1 ~tm range like standard optical microscopy, however, extremely broadband light sources, like Kerr-lens mode-locked Ti:sapphire lasers, PCF's or thermal light sources have to be used. An important class of OCM techniques is based on interference microscopy. Beaurepaire, Boccara, Lebec, Blanchot and Saint-Jalmes [1998] describe a Michelson interferometer scheme, whereas Dubois, Boccara and Lebec [1999] present a Linnik interferometer scheme. Owing to their microscopic instrumentation these techniques can yield extremely high transversal resolution. Using a thermal lamp permitted OCT resolution close to 1 ~tm in three dimensions (Vabre, Dubois and Boccara [2002]).
4~
Fig. 2.3. Left: Macroscopic OCT technique implemented in fiber optics. Right: Optical coherence microscopy implemented as parallel OCT in bulk optics. PC, PC monitor.
224
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Fig. 2.4. Endoscopic OCT.Distal end of probe arm. PC, PC monitor. Adapted from Tearney,Boppart, Bouma, Brezinski, Weissman, Southern and Fujimoto [1996]. Endoscopic OCT provides medical imaging of internal organ systems (Tearney, Brezinski, Bouma, Boppart, Pitris, Southern and Fujimoto [1997]) and of precancer and cancer states of human mucosa (Sergeev, V.M. Gelikonov, G.V. Gelikonov, Feldchtein, Kuranov, Gladkova, Shakhova, Snopova, Shakhov, Kuznetzova, Denisenko, Pochinko, Chumakov and Streltzova [ 1997]). The basic components are depicted in fig. 2.4. A device for interstitial OCT imaging using a so-called imaging needle with a diameter as small as 410 ~tm has been developed by Li, Ko and Fujimoto [2001].
2.3. Sample signal extraction Single scattered photons are the dominating source of all recent OCT techniques. It is straightforward to decipher information about the object structure carried by these photons. Multiple scattered photons collect information about several object sites. In scattering media also multiple scattered photons are used, but are treated like single scattered photons. In fact, to rigorously derive the information carried by multiple scattered photons is substantially more complex. Single backscattering is limited to a narrow cone around the illumination direction. Thus the photon flux available for the tomographic depth scan is rather limited. Hence, there is a high demand for additional photons. One such attempt with the aim to exploit not only backscattered but also side-scattered photons was by Lazar, Brunner, Seshek, Meier and Steiner [1998], but it remained ineffective.
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The main problem is that multiple scattered photons travel (within the coherence length) through the sample along different trajectories. This causes averaging of the object structure and reduces contrast. Hence, a delicate balance has to be found between the need of collecting as many photons as possible and the need to reduce averaging of the object structure (see w3.3). Most OCT techniques use a single low-coherent light source. Dual-wavelength OCT techniques, however, have also been investigated. Schmitt, Xiang and Yung [ 1998] used two light-emitting LEDs in a differential absorption technique; Pan and Farkas [ 1998] generated high-resolution two- and three-dimensional singleand dual-wavelength images of microscopic skin structures. Time-coherence properties are invariably used in all OCT techniques at present. One disadvantage of these bandwidth-based techniques is that wavelength-dependent absorption will change the impulse response of the OCT interferometer. Therefore, alternative techniques are of interest. Space-coherence OCT is one such alternative investigated by Rosen and Takeda [2000]. These authors suggested the use of the longitudinal part of spatial coherence for surface profilometry. In this technique the spatial spectrum of the beam illuminating the object is varied by spatial masks. In a first demonstration a He-Ne laser was used as a light source and Fresnel zone plate structures were used to move the coherence gate. The disadvantage of this technique is that depth resolution becomes dependent on the numerical aperture like in classical imaging. Other alternatives to obtain tomographic imaging are based on computational correlation techniques. Either the correlation of the light patterns in the aperture plane or that in the image plane can be used. For example, Wilson, Juskaitis, Neil and Kozubek [1996] described a microscopic technique to obtain realtime confocal images by using an aperture correlation technique. Later, Walker [2001] presented a study showing that correlating full field speckle illuminated image and reference fields can yield depth discrimination and enhanced lateral resolution.
w 3. OCT signal generation 3.1. Time-domain LCI 3.1.1. LCI signal
The key to the internal structure of transparent and scattering objects by LCI is the correlator performance of two-beam interferometry (Born and Wolf [1999]).
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Fig. 3.1. Basic low-time-coherence interferometer (LCI). The interferometer output contains the sample response h(r) convolved with the source coherence function FSource(r). The impact of the object on the illuminating light beam is decoded by correlation of the scattered light with a reference wave, see fig. 3.1. The wave V(t) emitted by the light source LS is convolved with a b-function in the reference beam and with the sample response h(t) in the probe beam. The interferometer output contains- besides the intensities of sample beam (Is) and reference beam (IR) a convolution of the sample response h(t) with the auto-correlation Fsource(r) of the probe light. Low-time-coherence interferometry has already been used in the past. For example, Flournoy, McClure and Wyntjes [1972] have used white-light fringes in an interferometric thickness gauge. Patten [ 1971] describes the use of white light for the interferometric measurement of distances. Fochs [ 1950] uses white light for thickness measurement, and white-light fringes have been used in interference microscopes (Linnik [1933]) for measuring phase steps over sharp discontinuities. These older uses of LCI were limited by the lack of space coherence of the available light sources. At present LCI and OCT are based on low-coherence two-beam interferometry. If sample beam and reference beam of intensities Is and IR interfere, the resulting intensity IpD at the photodetector is given by the two-beam interference law: IPD = IS + IR + GSR.
(3.1)
The variation GSR superimposed on the sum of the intensities of the two beams is called interferogram, sometimes "interference term". This quantity is of central importance in the subsequent presentation. In fact it is twice the real part of the coherence function of sample and reference beam, GsR(r) = 2Re[FsR(r)], see eq. (3.7). The variation GSR appears only if the optical distance from the beam splitter to the reference mirror matches the corresponding optical distance to light
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remitting sites in the sample arm within the coherence length. The corresponding range in the sample arm is called "coherence gate". LCI and OCT use heterodyne techniques to separate the interferogram from the beam intensities. OCT images are based on the photoelectric ac-signal of the low-coherence interferogram - GSR in eq. (3.1) - quantifying their picture function. This signal can be interpreted as the ac part of the photocurrent i(t) or the output voltage U(t) of a transimpedance amplifier following the photodetector. Band-pass filtering yields the ac component U(t)- Um which we shall call "LCI signal" (see also w5.2.1):
U ( t ) - Um O( i ( t ) - im --
qe rl hv
IpD(r) d2r - im,
(3.2)
Ar(r)
where Um is the mean of U(t), im is the mean of i(t), qe is the electronic charge, r/is the quantum efficiency of the detector, h is Planck's constant, v is the optical frequency, and Ar(r) is the sensitive detector area. Frequently, the envelope of the LCI signal is generated by full-wave rectification of the photoelectric ac signal followed by low-pass filtering. Alternatively, amplitude and phase (or the corresponding quadrature components) of the photoelectric ac signal are determined by mixing using a lock-in amplifier. In this paragraph we ignore field quantization and polarization and represent the light waves as scalar, stationary, ergodic, random analytic signals (Mandel and Wolf [1995]). The Fourier integral representation of the electric field E(t) of a light beam is: oo
E(t) = / E(v)exp(-2:rivt)dv = FT -1 {E(v)}, --
(3.3)
CX}
and, therefore, the corresponding analytic signal can be defined as O0
V(t) = 2 r E ( v ) exp(-2~ivt)dv = A(t) exp[iCI)(t)-2Jrivot],
(3.4)
o
where the formulation with amplitude A(t) and phase q)(t) can be used in the quasi-monochromatic case within times short compared to the coherence time; A(t)e i4'(t) is the complex envelope of V(t), A(t) = ]V(t)[ is the real envelope,
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and v0 is the mean frequency of the power spectrum of V(t). The instantaneous intensity is
I(t) = V* (t) V(t).
(3.5)
3.1.2. Autocorrelation interferometry If both the sample arm and the reference arm are terminated by plane mirrors, the interferogram is the real part of the autocorrelation of the source waves V(t), see fig. 3.2. We assume perfect spatial coherence of the light illuminating the interferometer. The depth scan relates the path difference z - u in the interferometer to the time delay r = 2 ( z - u)/c, with c the speed of light. Then the averaged intensity IpD(r) at the photodetector of the interferometer is
I,D(r) = <&(t)) +
(3.6)
where ( . . . ) means time or ensemble average. The interferogram GsR(r) is twice
RM,
,$
RB
D
BS [
,-,,. z
v
LS
I
~l SB
SM
PB
PD 1"2h SP
I
U
9 z-u
I I
~. Z - U
Fig. 3.2. Time-domain low-coherence Michelson interferometerterminated by plane mirrors (RM and SM). U(t)- Um (x GSR(t) equals the real part of the auto-correlation of the source wave; A is the real envelope. RM performs the depth scan (double arrow). BS, beam splitter; LS, low-timecoherence light source; PB, probe beam; PD, photodetector; RB, reference beam; RM, reference mirror; SB, sample beam; SM, sample mirror; SP, signal processor.
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the real part of the time cross-correlation of the analytic signals of the two interfering beams" GSR(r) = 2Re [(V~(t) VR(t + r))] = 2Re [FsR(r)]
(3.7)
= 2V/(Is(t)) (/R(t))lYSR(r)I cos [aSR -- OSR(r)], where YSR(r) is the complex degree of coherence, 6SR(r)= 2~V0 r is the phase delay between sample and reference beam, and aSR is a constant phase depending on the reflectivities of the mirrors and the beam splitter. With balanced interferometer beam intensities and both beams terminated by ideal mirrors, however, the cross-correlation simplifies to an autocorrelation or (self-) coherence function of the source beam: FSR(I")= (Vffource(t) Vsource(t + r ) ) = FSource(r). The coherence function can be obtained from its real part Gsource(r)= 2Re [(V~ou~ce(t) gsource(/+ T))] by analytic continuation: FSource(r) : ~1Gsource(r) + I i HT{Gsource(r)},
(3.8)
where HT means Hilbert transform and Gsource(r) is obtained from the LCI signal. Furthermore, the interferogram Gsource(r) of the "empty" interferometer terminated by plane mirrors can be considered as the impulse response of the interferometer, and A t ( r ) = 2 ]FsR(r)] as its real envelope. Figure 3.3 shows examples obtained from light sources used in LCI and OCT. Corresponding spectral relations are obtained by the Wiener-Khintchine theorem. First of all we note that the power spectrum is obtained from the source self-coherence: Ssource(V) = FT{Fsource(r)}.
(3.9)
Furthermore, with the cross-spectral density function of the radiation exiting the interferometer, WSR(V) = FT{FsR(r)},
(3.10)
the spectral interference law is obtained as
S(v) = Ss(v) + SR(V) + 2Re[ WsR(v)] cos(2~vr).
(3.11)
Many light sources used in LCI and OCT have an approximately Gaussian power spectrum, 0- ) 2 ] Ssource(V) o( exp [-4 ln2 v( v Av 2 [ L
(3 12)
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(1)
o O
e_.0 o
l,=6 gm
O r
(9 -1, -40
0
(a)
40
Time-delay T in fsec
(1)
o
0
8
rj-)
e_.0 Z
r
o m
0 (ao
(9_ 1 -20 (b)
0
-1, -20
20
Time-delay T in fsec
(c)
0
20
Time-delay ~ in fsec
Fig. 3.3. Normalized plot of LCI point spread function Gsource(r)= 2 Re [/-'Source(r)] (see eq. (3.7); also called "signature") and coherence length Ic obtained with (a) a broad-band superluminescent diode, (b) a Kerr-lens mode-locked Ti:sapphire laser, and (c) a halogen lamp as light source.
with the full width at half maximum (FWHM) A v. This spectrum yields a Gaussian self-coherence F(r) o( exp ( -at241n2 Av2 r2 - 2ariv~
(3.13)
with the envelope
I r ( r ) l o( exp -
2 l~]-n2r
.
(3.14)
Finally, we consider the FWHM of the coherence function F(r), as this parameter plays an important role in LCI and OCT. It can be defined as coherence time tc; then c . tc is the corresponding coherence length. For the Gaussian spectrum of eq. (3.12) the FWHM duration of F(r) is tc -
4 ln2 ~A v
(3 15)
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Because of the backscattering configuration of OCT, the so-called round-trip coherence length lc -
ctc 2
-
2 In 2 A2 Jr
A~
(3.16)
is most common in the OCT literature. /1,0 is the center wavelength of the Gaussian spectrum.
3.1.3. Cross-correlation interferometry If we introduce a sample in the probe arm the interferogram is a crosscorrelation, see fig. 3.4. Within linear optics the interferometer can be treated as a linear shift-invariant system. A sample generates a beam oo
Vs(t) = /
V(/) h(t - t') dt' = V(t) 9 h(t),
(3.17)
--0(3
where the asterisk stands for convolution; V(t) is the probe beam signal, h(t) is the response function of the sample, and the Fourier transform
H(v) = FT{h(t)}
(3.18)
is its transfer function. Obviously, h(t) is the local amplitude reflectivity (related to z by the speed of light). The interferogram can readily be shown to be the cross-correlation: GSR(r) = 2Re [(V~(t) 9 VR(t + r))] = 2Re[FsR(r)] (3.19)
From eqs. (3.17) and (3.19) we have: rsR (r) = rSou~ce(r) 9 h(r)
(3.20)
GSR(r) = 2Re [Psource(r) * h(r)] : 2Re [FT-' {Ssource(V)" H ( v ) } ] .
(3.21)
and
Hence, in time-domain LCI the coherence function of the interferometer beams with a sample in the probe arm equals the source coherence function FSource(r)
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RMI RB
Vl, L$
r---~Z
Bs I V
q
I
SB
PB
U PD H~I
U,~
U SP I
~ Z-U 9
ATA A I
I, Z - U
Fig. 3.4. Time-domain LCI with a sample SA. U - U,,, v< GsR(r) is the LCI signal. A is the real envelope. Only two light remitting sites (anterior corneal surface and Bruch's membrane in the fundus of the eye) are indicated. BS, beam splitter; LS, low-time-coherence light source; PB, probe beam; PD, photodetector; RB, reference beam; RM, reference mirror; SB, sample beam; SP, signal processor.
convolved with the sample response function or backscattering profile h(r) (Fuji, Miyata, Kawato, Hattori and Nakatsuka [1997]). Obviously, FSource(r) is the complex impulse response function and Re[Fsource(Z)] plays the role of a LCI depth point-spread function. For the Gaussian spectrum we obtain Re[Fsource(Z)] = exp -4 In 2
~c
cos -~0 z
.
Note that the Fresnel-reflecting interfaces model (see {}3.3.1) associates amplitude reflectivities R V / ~ with the backscattering profile h(r); within that model, the LCI signal (3.2) represents the depth distribution of sample amplitude reflectivities convolved with the LCI depth point-spread function. In the frequency domain we have from eqs. (3.9), (3.10), (3.20) and (3.21): WSR(I/) = Ssource(g)" m(g),
(3.22)
where WSR(V) is the cross-spectral density of sample and reference beam, Ssource(V) is the spectral density of the source light, and H(v) is the sample transfer function.
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3.1.4. Heterodyne interferometry LCI is based on the interference of wave groups. The propagation speed of wave groups is the group velocity v~: dk(~o) do)
UG =
_ c ")o
(3.23)
-- -?lG - '
k(~o) = 2:r/J1 is the wavenumber, ~o0 is the mean frequency, nG = n - / l a~ dn is the group index, and n is the refractive index. The carrier wave moves with phase velocity up: vp -
C n
.
(3.24)
The cross-correlation term is separated from the dc component of the interferometer signal by optical heterodyne detection. One of the interferometer beams is used with a slightly different frequency to produce a carrier frequency of the interferogram signal. Thus various noise contributions like 1/f noise and speckle noise in the interferometer signal can be minimized by band-pass filtering. There are several techniques to implement heterodyne detection, see w6.3. Most common in LCI and OCT is a delay line with a reference mirror moving with speed VMirror. This device simultaneously performs the depth scan. Let time t and reference mirror position z be related by z(t)= VMirrort and assume a 6(z)-like sample interface at z0. Since the reference mirror moves in air we have r(t) = 2
Z0-
VMirrort
(3.25)
and
GSR(t) = 2V/(Is(t))(IR(t))]YSR(r)[ COS(2 ~o___q0VMirror/_//~SR) C
= AG(t) COS(2~OO t -VMirror
(3.26)
-/~R),
C
where fiSR = 2ZoOJo/C.Note: A mirror shift Az is related to a shift d of the coherence degree y(r) and of the real envelope AG(r) in the sample by the group index: d -
Az nG
(3.27)
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The interferogram at the interferometer exit is U ( t ) - U,, cx GSR(t) = 2Re[Fsource(t) * h(t)],
(3.28)
and the real envelope A6(t) of the interferogram can be obtained from AG(t) = v/(Gsa(t)) 2 + (HT{GsR(t)}) 2,
(3.29)
where HT{...} stands for the Hilbert transform; the phase is obtained from
q~(t) = arctan [HT_{GsR!t)} L
GsR(t)
] "
(3.30)
3.2. Fourier-domain LCI In Fourier-domain LCI, the object is again positioned in the probe arm of an interferometer illuminated by low-time-coherence light. But no depth scan occurs. The light exiting the interferometer is dispersed by a spectrometer to display its spectral distribution on a detector array. The depth structure is obtained from the electronic readout spectrum by a Fourier transform. This technique can be based on Wolf's treatment of scattering by first-order Born approximation (Wolf [ 1969]).
3.2.1. First-order Born approximation scattering We consider a monochromatic (wavenumber k) component of a Gaussian probe beam. Furthermore, we assume the sample to be centered at the waist of the beam and to have a depth extension of the order of magnitude of the Rayleigh length of the probe beam. Hence, we can assume the object illuminated by plane wavefronts v(i)(r, k r t): vIi)(r, k ti), t) = A (i). exp(i 9k ti). r - i- w. t),
(3.31)
where k (i) is the wavevector of the illuminating wave and lk(i) I = k = 2:r/X is its wavenumber.
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Wolf [1969] showed that in case of weakly scattering objects the scattered field is obtained by the first Born approximation as an integral extended over the illuminated sample volume Vol(r'):
l/
V~S)(r,k<S),t) : - ~
Fs(r',k) . v(i)(rt, kIi),t) 9 G(Ir- FI)d3r '
(3.32)
Vol(r')
with the Green function
G (Ir- r'[) = exp (i. k is). I r - r'l)
(3.33)
Equation (3.32) is Huygens' principle with the Green function representing the secondary wavelets which combine to form the scattered light (we have underlined the Green function G to avoid confusion with the interferogram G). The scattering potential of the sample Fs(r, k ) = k 2- [mZ(r, k ) - 1]
(3.34)
determines the relative amplitudes of these wavelets; m is the complex refractive index distribution of the sample structure: (3.35)
m(r) = n(r)[1 + ix(r)],
with n(r) the refractive index, and to(r) the attenuation index. In Fourier-domain LCI (as well as in time-domain LCI) we detect backscattered light at a distance d = ] r - r'] >> lc in the far field. Then the scattered wave VtS)(r,k ~s),t) at d can be written as (Fercher, Hitzenberger, Kamp and E1-Zaiat [1995])
A(i)exp(iktS)r - i o g t ) f
V(s)(r, k (s), t) = ~
Fs(r') exp(-iKr') d 3r',
(3.36)
Vol(r')
where the amplitude Ali) of the illuminating wave has been assumed constant within the coherent probe volume. K = k tSt- k (i) is the scattering vector. Hence, in the far-field approximation the amplitude of the scattered wave equals the three-dimensional inverse Fourier transform of the scattering potential of the sample. In backscattering Fs equals the local amplitude reflectivity.
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3.2.2. Fourier-domain LCI Provided the scattering potential Fs at depth z within the illuminated sample volume does not depend on lateral coordinates, we can replace the integrations over x / and f by a constant factor C, chosen proportional to the transversal cross-section of the beam. Then the backscattered light wave at z is proportional to the one-dimensional inverse Fourier transform of the scattering potential: AIi)c V(S)(z,k,t) = 4Jrz e x p ( - i k z - io)t)FT -l {Fs(z)}
(3.37)
= A(S)(K)exp(-ikz- icot). Usually the z-extension of the sample is much smaller than its distance z to the detector; therefore z in the denominator can be assumed constant. Hence, Fs(z) can be obtained by a Fourier transform of the wavelength-dependent scattered field A(S)(K): Fs(z) o( FT{A(S)(K)}.
(3.38)
This is the physical basis of Fourier-domain LCI and OCT. Note that in this description the LCI signal depicts the depth distribution of the scattering potential. There are, however, restrictions: First, AtS)(K) is the complex amplitude of the scattered light field. Therefore, no direct detection of A~S)(K) is possible; this problem is addressed below. Second, only a rather limited range of Fourier data is accessible; this problem is addressed in w3.3. Two techniques can be used to record the spectral amplitudes and phases of A(S)(K). Spectral measurement of the real amplitude is straightforward and can be performed with a spectrometer. To measure the spectral phases r Wojtkowski, Kowalczyk, Leitgeb and Fercher [2001] for example, used phase-shifting spectral interferometry (fig. 3.5). To minimize phase calibration errors, they adopted a 5-frame method introduced by Schwider, Burow, Elssner, Grzanna, Spolaczyk and Merkel [1983], and calculated the phase by the following formula:
r
= arctan
_i[1 ] + 41121 _4i[41 +1151 ) ill] + 1121 _ 61131 + 21141 + 1151 ,
(3.39)
where I["] = I(O(K)+ nzr/2), n = 1, 2 . . . . , 5 are the spectral intensities at wavelengths 2. - 4zr/K.
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PS RM BS
OB
E] LS
PDlllll]lll
I[K],~ AI [FT) I A
J,F,A
i
A
;.
Z
Fig. 3.5. Phase shifting based Fourier-domain LCI. FT, signal processor performing the Fourier transform. I, spectrum; F, scattering potential; A, real envelope of F. BS, beam splitter; LS, light source; OB, object; PD, photodetector array; PS, phase shifter; RM, reference mirror. An alternative approach is based on a correlation technique by Fourier transforming the spectrally resolved intensity I ( K ) at the interferometer exit (Fercher, Hitzenberger and Juchem [ 1991 ]): FT{I(K)} = C ( F * ( z ) . F ( z + Z)) = C . ACFF(Z),
(3.40)
where F ( z ) is the scattering potential of the sum of sample and reference beam. With a &like reflectivity (magnitude R) of the reference mirror at ZR the complete backscattering potential is F ( z ) = F s ( z ) + v ~ 6 ( z - ZR) and I ( K ) = ] V(S)(K)[ 2= C IFT-' {F(z)}] 2. Then the autocorrelation yields four terms: ACFF(Z) = ACFFs (Z) + v/R 9F~ (zR - Z) + v ~ 9Fs(zR + Z) + R . 6(Z). (3.41) Only the third term yields - besides the constant factor v / R - a true representation of the object structure, centered at Z =-zR. The first term yields the autocorrelation of the object structure, the second term yields a complex conjugate version of the object, and the last term yields a strong peak at the origin of the reconstructed object space. To avoid overlapping of the reconstructed terms the reference mirror must be put at a position twice the object depth apart from the next object interface.
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There have been many approaches to suppress the undesired terms in eq. (3.41). The problem is also encountered in digital holography. Cuche, Marquet and Depeursinge [2000] have shown that proper masks can be used in Fourier space to eliminate these terms. Recently, Wojtkowski, Kowalczyk, Leitgeb and Fercher [2001] have presented an autocorrelation technique which detects two scattering spectra AtS~(K), one with a phase shift of Jr introduced into the reference arm. After subtraction the disturbing autocorrelation terms disappear. The advantage of Fourier-domain LCI is that the object structure along the complete depth is obtained by one or a few readouts of the photodetector array. The short acquisition time, the elimination of moving parts integral to the measurement, and the inherently direct access to spectral information, are advantages of the spectral technique in comparison to time-domain LCI and OCT. Disadvantages are the limited resolution and dynamic range of photodetector arrays.
3.3. Single and multiple scattered light In the single-scattering regime depth resolution is determined by the source coherence length, and lateral resolution depends on the numerical aperture of the objective lens. At larger probing depths multiple scattering becomes important (see fig. 3.7, below). Multiple scattering causes image degradation by averaging of the object structure and by intensity loss, generation of a noisy background, and loss of coherence. There are two basic questions in the application of OCT: Which type of information do we obtain, and what is the penetration depth of this technique. A first answer to the first question will be given in w3.3.1. A first answer to the second question can be found in w167 3.3.2-3.3.5.
3.3.1. Single scattering Of course, a model is helpful to understand which type of information we obtain from OCT imaging. Many treatments describe the sample as a collection of Fresnel-reflecting interfaces with amplitude reflectance x/R. This approach is obvious in the ophthalmologic field. In other biomedical fields, however, the microscopic structure of tissue becomes important. Biological tissue is constituted of cells maintained in the extracellular matrix. This matrix is composed of bundles of structure proteins such as collagen and elastin, and is partially filled with hydrated gel. Embedded in tissue are blood vessels, nerve
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fibers, and other structures. Models which account for these structures are the path-length-resolved diffuse reflectance model introduced by Pan, Birngruber, Rosperich and Engelhardt [ 1995], the continuum model of index variations used by Fercher [1996], and the discrete-particle model introduced by Schmitt and Kumar [ 1998]. Phase-contrast microscopic measurements of the refractive index variations of various tissues performed by Schmitt and Kumar [1996] showed that the spectrum of index variations exhibits a power-law behavior for spatial frequencies spanning at least a decade (0.5-5 ~tm-l) and has an outer scale in the range of 4 - 1 0 g m , above which correlations are no longer seen. Results obtained by the discrete-particle model suggest that the refractive-index variation of soft biological tissue has a skewed log-normal distribution function, with a shape specified by a limiting fractal dimension of 3.7. (The calculation of fractal dimensions, however, is highly dependent on the chosen scattering model; Wang [2000], for example, obtained a fractal dimension between 4 and 5). In the wavelength range from 600nm to 1400nm the diameters of the scatterers that contribute most to backscattering were found to be significantly smaller (~-~ than the diameters of the scatterers that cause the greatest extinction of forwardscattered light (3-4X). Inspired by the Born approximation treatment of scattering used by Wolf [1969] we consider the Fourier spectrum of the sample scattering potential. This approach is based on a continuum model of the refractive index and is not limited to a certain tissue model. Due to the small size of the coherent probe volume, backscattered light in LCI and OCT is detected in the far field. The coherent probe volume (we use this term to avoid confusion with the term "coherence volume") equals the depth of the coherence gate multiplied by the corresponding probe beam cross-section. The amplitude of the backscattered wave e q u a l s - besides a constant f a c t o r - the inverse Fourier transform of the scattering potential in the illuminated probe volume, see eq. (3.36):
V(K) = /" Fs(r') exp(-iK, r') d 3r'.
(3.42)
tJ
Vol(r')
The scattering vector K = k <S)- k (i) defines a sphere in K-space (known as "Ewald sphere" in solid-state physics; see also w 13.1 in Born and Wolf [ 1999]). For a fixed wavevector k li) of the probe beam only Fourier data on the surface of that sphere are accessible, see fig. 3.6 (D/indliker and Weiss [ 1970]). The consequence is selective detection of structural properties of the scattering potential.
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Fig. 3.6. Spatial Fourier components accessible in LCI and OCT. Two examples are indicated in the illuminated object volume in front and behind the beam waist (diameter 2w0). E1 and E2 are the Ewald spheres corresponding to )-I and X~ /,l i) is the wavevector of the illuminating beam at -" "'l wavelength Zl" kl"Is) is the wavevector of the scattered beam at wavelength Zl 9K1 is the scattering vector at wavelength ~-l. SA, sample; OL, objective lens. First of all, LCI and O C T signals depend on the refractive index distribution, see eq. (3.34). It is the scattering potential, however, which is Fourier-related to the scattered field. If monochromatic light is used, only Fourier data of the scattering potential located on the surface of the Ewald sphere are accessible (ML in fig. 3.6). Since these data are located on a surface they do not represent true three-dimensional information. Access to a three-dimensional range of Fourier data, however, is obtained with low-time-coherence light (data volume LCL in fig. 3.6). Second, the use of low-time-coherence light of a limited wavelength range [~1,~,2] does only give access to Fourier components of F s ( r ) in the range
[
~,-
or, equivalently, in a range 4st Z0
+
4 ln2 lc
'
(3.43)
where /1,0 is the mean wavelength. O f course, we are imaging only a limited spectral window of the object structure and we do not see structures which absorb or scatter other wavelengths. There is an additional limitation concerning the spatial structure imaged by backscattered light. The accessible Fourier components have spatial period lengths d around d = l~0 if their normals are oriented along the z-axis. If their normals are tilted by 0 with respect to the z-axis, the corresponding spatial period lengths are defined by Bragg's condition for three-dimensional
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gratings: d = 2/(2cos0). Object scattering potential structures not having Fourier components with spatial frequencies in this order of magnitude will not backscatter light. Furthermore, the aperture angle a also limits the orientation of accessible Fourier components, to those whose back-reflected light enters the focusing lens, see fig. 3.6. Fourier components with a larger tilt (0 > l a) will remit the back-reflected light out of the aperture of the objective lens. Hence, LCI and OCT are band-pass limited. There is an additional selectivity depending on the coherence length of the probe light and on the diameter 2w0 of the probe beam. At very small focal diameters 2w0 < d the tilt of the Fourier components will not be resolved within the coherent probe volume and, therefore, this tilt will not be effective. Finally, since the low-time-coherent Fourier data volume ("LCE' in fig. 3.6) is proportional to the square of a mean scattering vector K2 ~< /s ~< K~, the mean amplitude of backscattered light will be proportional to K2m. Hence, from the Fourier derivative theorem the mean signal is proportional to the second derivative of the scattering potential: d2
FT -1 {KZFs(K)} = ~zzFs(z).
(3.44)
Substituting eq. (3.34) into eq. (3.44) yields an expression containing firstand second-order derivatives of the real and the imaginary refractive index. Neglecting attenuation, for example, yields
LCI and OCT signals, therefore, increase with increasing first and second derivative of the object refractive index; the second derivative is weighted by the refractive index itself. Both time-domain and Fourier-domain techniques are subject to the limitations mentioned above. Furthermore, at small coherence lengths lc < d, the grating will not be efficient but the corresponding depth modulation of the scattering potential will be detected by time-domain LCI, whereas at large coherence lengths lc > d, reinforced backscattering will occur (provided the Bragg condition is fulfilled).
3.3.2. Transitionfrom single to multiple scattering An experimental study reported by Bizheva, Siegel and Boas [ 1998] investigated the transition of backscattering from singly scattered light to light diffusion
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-\ 10 -I
r
-2
c-- lO :D
-
0-3
"O
10 .4
D i -_.. _
"
-
c)
E lo-s
<
~
multiplescattered light
-
0
1
0
0
0
0
0
0
0
O0
10 .6
single scattered light~ \ 10
-7
-
,,,,
!
0
500
1000 1500 Path Length [pm]
2000
2500
Fig. 3.7. Lorentzian amplitude as a function of photon path length through the medium. ~tS = 9.43mm -1 " g = 0.2; N A = 0 . 2 . Solid line: single scattered light. Dots: experimental data, indicating the onset of multiple scattering at a path length about 1.3 mm. Redrawn with permission from Bizheva, Siegel and Boas [1998].
using Dynamic Low Coherence Interferometry (DLCI). Power spectra were measured at various depths as a function of the photon path length in highly scattering, monodispersed suspensions. Figure 3.7 shows the amplitude of the Lorentzian photocurrent power spectrum of the backscattered light. The straight solid line represents the exponential decay predicted by single scattering theory. The deviation of the experimental data from the straight line is attributed to detection of multiple scattered light. Experiments performed with highly scattering polystyrene microsphere suspensions showed that the transition from single to multiple scattering depends on the scattering anisotropy g (defined as the mean cosine of the scattering angle). For small scattering anisotropy (g = 0.2) the transition from single to multiple scattering starts at about 6 photon random walks and reaches to about 20 random walks, whereas for large scattering anisotropy (g = 0.75) multiple scattering starts immediately at about 2 photon random walks (one each back and forth). Further experiments performed with different numerical apertures of the objective lens and with a variety of scattering anisotropy factors g revealed that in media with low scattering anisotropy (g ~ 0.2), increasing the NA causes the transition to occur at shorter path lengths. In suspensions with a larger scattering anisotropy (g ~ 0.75) no dependence of the transition
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path length on NA was detectable. Obviously, in media with low scattering anisotropy, coherence gating, small NA and any confocal filtering rejects multiple scattered light and, therefore, leads to persistence of single scattering for greater imaging depths. It must, however, be kept in mind that smaller NA leads to poor transversal resolution and a poor signal-to-noise ratio. In media with high scattering anisotropy, the transition from singly scattered light to diffuse scattered light starts at a depth equal to 1 photon random walk 1/[Its ( 1 - g)]. At this depth the linewidth of the diffuse regime equals the linewidth caused by single scattering.
3.3.3. Coherence of multiple scattered light The coherence properties of multiple scattered light are of great interest not only in LCI and OCT but in a much wider context, for example, in basic phenomena like coherent backscattering and dynamic light scattering, in lighttissue interaction, and in general imaging through scattering media. Of course, multiple scattered light will lose its time coherence if the photon random walk length exceeds the coherence length. The impact of scattering on space coherence seems more difficult to estimate. For example, as stated by the van CittertZernike theorem (Born and Wolf [ 1999]) space coherence is generated simply by propagation. Similarly, an analysis by Sotskii and Anisimov [1973] showed that for small angles between the scattered waves the degree of their coherence is not affected by that of the incident light. Furthermore, Kinsinger and Mallick [1974] showed that the scattered field conserves the high spatial coherence of incident light. In a more recent study Yang, An, Perelman, Dasari and Feld [ 1999] showed that spatial coherence in forward-scattered light diminishes rapidly with the number of elastic scattering events. However, a small but distinct coherent component persists, even for a large number of scattering events.
3.3.4. Spatial distribution of the probing beam intensity Feizulin and Kravtsov [1967] and later Lutomirski and Yura [1971] extended the validity of the Huygens-Fresnel principle to propagation through scattering media. The impact of a random distribution of scatterers between object and detection plane for small-angle forward scattering is represented by a multiplicative factor ~p(p, r) added to the corresponding Green function. ~p(p, r) describes the perturbation of the Green function along the propagation path.
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Fig. 3.8. Geometry at the objective lens. The time-coherence matching point is at the focus E z is the probing depth. Using this principle Yura and Hanson [1996] and Thrane, Yura and Andersen [2000] have shown that the mean intensity distribution at the object plane, see fig. 3.8, can be expressed as (/(r)) c< exp(-ysz)
exp(-r2/w 2)
+
[1 - exp(-/~tsz)]
exp(-r2/w 2) ,
(3.46)
where WH and ws are the 1/e intensity radii at the object plane in the absence (WH) and presence (ws) of scattering, respectively, and w0 is the 1/e intensity radius of these beams at the objective lens with focal length f :
f WH -- kwo'
ws =
w~ +
~0
'
(3.47)
with po(z) the lateral space-coherence length for forward scattering; an analytic form has been given by Lutomirski [ 1978] under the assumption of independent multiple small-angle scattering and uniformly distributed scatterers in object space (d = 0 in fig. 3.8): !
po(z)
= ./ 3
V
~..
(3.48)
~tsz :r0~m~ '
~s is the scattering coefficient, 0rms is the root mean square scattering angle. Assuming typical parameters of skin tissue at X = 800nm, (gs = 2 0 m m -1, 0 r m s --- 0.3 rad), optical parameters f = 10 mm, w0 = 2.6 mm, and a numerical aperture NA = 0.5, eq. (3.46) yields the intensity distribution depicted in fig. 3.9. It is clear from this figure that diffraction-limited focusing in the usual sense can
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1012..
~>.1 0 9
~ r--~
z = 0,015 m m
10 6 L_ z = 0.7 m m - z = 1,2mm 3
10
-
~
z
= 1,72 m m ~ ~ " - ~ " ~
"
(I) = . . . .
10 ~ -~ -
10gm
lj 0
QIm,,-
-5gm
0
m
5~m
m
~
m
10Ftm
Lateral Distance
Fig. 3.9. Mean intensity of a Gaussian probe beam focus in a scattering medium like human skin (semi-log scale), z is the probing depth; A = 800nm; f = 10mm; NA=0.5. The dashed line (z = 0 mm) is the single scattering intensity distribution.
be achieved only in the very superficial layers of a highly scattering tissue. In the depth, however, multiple scattered light will contribute an increasing background with a lateral width of 2ws. At a probing depth of z = 0.02 mm, for example, we have 2WH ~ 1 ~tm and 2ws = 4.6 ~m, whereas at a depth of z = 0.7 mm we have 2ws = 916 ~m. A dramatically reduced transverse resolution will result, though light scattered to positions more distant from the probe beam focus will have reduced time-coherence and will, therefore, be suppressed. Shorter coherence length, therefore, will not only improve depth resolution but will also improve lateral resolution to a certain degree.
3.3.5. OCT probing depth Yura [1979] and later Schmitt and Knfittel [1997] showed that with the heterodyne signal current expressed as
i(z) = 2~-7 1y(r) I Re
VR(p,z) V~(p,z)d2p
(3.49)
its mean square can be written in terms of the mutual space-coherence functions Fs of the sample beam and FR of the reference beam (the latter, of course, is deterministic):
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with a = qerl/(hv) the photodetector radiant sensitivity, and y(r) the timecoherence degree. Here, Vs and consequently i are random variables (in contrast to VR). Thrane, Yura and Andersen [2000] showed that the mean square heterodyne signal current at probing depth z can be expressed as (i2(z)) =
o2pR Ps Ob yg2W2
exp(-2gsz) +
2 exp(-/~sz) [1 - exp(-/~sz)] 2 H 2 1 + Ws/W
[ 1 - exp(-gsz)] 21
+
(3.51)
wl/w
a2Pe.Ps Ob
zcZw~
9 q'(z),
where PR and Ps are the powers of the reference and the probe beams. Ob is the effective backscattering cross-section. The term in the large brackets is the heterodyne efficiency factor tP(z). ~(z) is the reduction in the heterodyne signal-to-noise ratio due to tissue scattering. Furthermore, we have 2w0
po(z)
2}
(3.52)
For a backscattering object structure behind a scattering layer of thickness z, and a distance d(z) between the objective lens and the surface of that layer, see fig. 3.8, the lateral coherence length is p0(z)=@ 3
X
/~sZJr0rms
(nd(z)) 1+
Z
(3.53)
with d(z) = f - z/n. Increasing po(z) decreases ws and, therefore, increases the signal current. The term in parentheses in eq. (3.53) indicates that the lateral coherence length increases with increasing distance d(z) between the sample surface and the pupil. The larger the lateral coherence length, the sharper will the object structure be defined. A multiple scattering layer more close to the object structure causes less wavefront distortion on the waves arriving at the objective lens aperture than a layer immediately at the lens. This phenomenon has been called shower curtain effect and has already been observed by Dror, Sandrov and Kopeika [1998] in light propagation through the atmosphere. It is not present for specularly reflecting object structures; in this case we have the transverse space coherence as
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10o
b,l 10 -2 L_
0 0 10-4 Cl
4--
0>- 10-6 r-~ .
_
k
0-8 I
_
3
:4
10 -lo 0.0mm
0.5mm
1.0mm
1,5mm
2.0mm
Depth z Fig. 3.10. Heterodyne efficiency factor tP(z) as a function of probing depth z (see fig. 3.8). ~, = 814nm, IxS = 20mm -1, 0rms = 0.3rad, n - 1.4, f = 5mm, w0 - 0.5mm. Graph 1: diffuse backscattering with shower curtain effect taken into account; graph 2: diffuse backscattering with shower curtain effect not taken into account; graph 3: specular reflection; graph 4: single scattering. Redrawn from Thrane, Yura and Andersen [2000] by permission of the Optical Society of America.
given by eq. (3.48). Figure 3.10 displays the efficiency factor tp(z) as a function of depth z and demonstrates the shower curtain effect. As can be seen from fig. 3.10 the shower curtain effect has a decisive impact on the magnitude of the LCI signal and, therefore, on the OCT probing depth. Finally, it is clear that the above considerations are valid only if bulk tissue absorption is neglected and, more important, for the case of diffuse backscattering at the optical discontinuity characterized by a corresponding Fresnel reflection coefficient, as it occurs, for example, at skin-layer interfaces. There is no shower curtain effect in case of a specularly reflecting object (graph 3 in fig. 3.10). OCT probing depth can be estimated from the heterodyne efficiency factor tp(z). For example, an OCT system with a power sensitivity of 100dB can detect specularly reflecting discontinuities in depths down to 0.35 mm, however, strong diffusely reflecting discontinuities can be detected in depths down to 2 mm. OCT probing depth is also an important issue in OCM. A single-backscatter model has been used by Izatt, Hee, Owen, Swanson and Fujimoto [1994] to show that OCM has depth penetration superior to confocal microscopy when the light impinging on the photodetector is dominated by light scattered from other depths. This analysis predicts that in a range of approximately 5-15 scattering mean free path lengths, OCM can measure signals that would be obscured in confocal microscopy.
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3.3.6. Measurement of physical sample properties
Depth-scan signals also have the potential to provide depth-localized physical data on the scattering media. The first use of LCI to measure parameters of diffusing multiple-scattering tissue was by Clivaz, Marquis-Weible, Salath6, Novak and Gilgen [1992]. The structure of the wall of postmortem bovine superior mesenteric arteries was analyzed using a LED at a mean wavelength o f / ~ - 1300 nm. From the LCI signal the refractive index of the intima surface, the thickness of the arterial wall and the total transmission coefficient were evaluated. Schmitt, Kntittel and Bonner [1993] have used a commercially available reflectometer operating at ~ = 1300 nm to perform in oivo measurements of attenuation and backscattering coefficients of human skin by fitting the backscattered power curves. Tearney, Brezinski, Southern, Bouma, Hee and Fujimoto [1995] report on two techniques to measure the refractive index of tissue. In an in vitro technique the refractive index is obtained from the measured optical thickness of a sample of known geometrical thickness. Furthermore, a focus tracking technique has been reported, which uses the mismatch of focus position and coherence matching point to derive the refractive index. Popescu and Dogariu [1999] describe an optical path-length spectroscopic technique, which can provide precise data on the boundary reflection properties of unknown scattering systems. Hammer, Schweitzer, Thamm and Kolb [2000] have shown that optical properties of ocular tissues can be obtained from LCI signals. For example, the ratio of the refractive indices at intraocular interfaces can easily be derived from the height of the corresponding signal peaks. Furthermore, scattering coefficient and scattering anisotropy could be derived from the slope of the signal behind the reflection peaks. Thurber, Burgess, Brodsky and Shelley [2000] showed that a nonperturbative solution of Maxwell's equations can be used to determine the correlation length of a suspension and in particular to obtain information about nanoparticle suspensions. The LCI measurement and OCT imaging of the speed of scattering fluids, of birefringence properties and of spectral refraction and absorption of tissues are described in w4.
w 4. OCT contrast generation Biomedical imaging techniques use various contrast generation techniques which make nonabsorbing structures visible and/or give access to functional tissue parameters. The interest in functional parameters in medicine is based on the fact that functional disturbances usually precede anatomic changes. Therefore, these parameters are of great value in early diagnosis of diseases.
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4.1. Doppler Optical Coherence Tomography
Doppler Optical Coherence Tomography (DOCT) provides unprecedented velocity sensitivity, localized velocity data, and at the same time high-resolution structural images. It might find applications in the field of microvasculature research and diagnosis. Potential applications include ophthalmologic perfusion research, burn-depth determination, transplant tissue perfusion control, photodynamic therapy monitoring, and optimization of the laser treatment of port-wine stain. When light scattered by moving particles interferes with a reference beam, beating occurs at the Doppler angular frequency, ~OD= K" Vs,
(4.1)
where K is the scattering vector and vs is the velocity vector of the moving particles. Doppler OCT measures ~OD in highly localized volumina and at the same time provides high-resolution structural information. Structural resolution is subject to OCT resolution limits determined by coherence length and numerical aperture of the objective lens. Frequency resolution is subject to the well-known Fourier uncertainty relation (Bracewell [2000]): 1 At. Aoo >~ ~.
(4.2)
The first use of coherence gating in heterodyne laser Doppler velocimetry (LDV) was by Gusmeroli and Martinelli [1991 ]. They measured the velocity profile of seeded particles in water flowing in a duct by Fourier-transforming the fringe data. A superluminescent diode was used as low-time-coherence light source in a LCI configuration. Wang, Milner and Nelson [1995] were the first to simultaneously obtain structural data and velocity data in a LCI experiment. They used the power spectrum of the interference fringe intensity measured with a spectrum analyzer. The velocity profile of microspheres suspended in the fluid was measured using the centroid of the Doppler-shifted spectrum at each position in the conduit. The velocity profile in a turbid cylindrical collagen conduit was measured with an uncertainty of 7%. 4.1.1. Fourier transformation of fringe data
The Fourier transform of the time-dependent fringe intensity data seems the most straightforward approach to Doppler OCT. Fringe data are defined by the
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corresponding mutual coherence function F(r), see eq. (3.7). With the analytic signal of the Doppler-shifted sample beam Vs(t) = A(t) exp[iq)(t) - i(co0 - coo) t]
(4.3)
we have FSR(r) = FSource(r) exp(--icoDt),
(4.4)
where coD is the Doppler frequency, r = 2vst/c + r0 has been used setting the time offset r0 = 0, and vs is the sample velocity component parallel to the scattering vector, with us/c << 1. Therefore, the cross-spectral density or power spectrum of the interferogram is W(co) = Ssource(co ) * 0 ( c o - coD),
(4.5)
and the Doppler frequency can easily be estimated, provided cod is larger than the width of the source power spectrum Ssource(co), or, in case of a Gaussian spectrum, col) > 8 In 2/tc, where tc is the coherence time. Chen, Milner, Dave and Nelson [1997] use a fiber optic Michelson interferometer and generate Doppler images by sequential lateral scans followed by an incremental sample movement in vertical direction. The carrier frequency is generated by electrically driven piezoelectric cylinders. The interference fringe intensity at each pixel is transformed by a FFT algorithm to obtain the Doppler spectrum. Flow velocity at each pixel is obtained as the difference between the cartier frequency and the centroid frequency of the measured power spectrum. Structural information is obtained from the same signal by calculating the magnitude of the power spectrum at the carrier frequency. In this technique, as in classical OCT, lateral resolution is limited by the diameter of the probe beam spot and depth resolution is limited by the coherence length. Figure 4.1 shows some examples. Velocity resolution is connected with data acquisition time at each pixel by the Fourier uncertainty principle. Here a velocity resolution of ~100 ~tm/s has been obtained (at a data acquisition time of ~3 min for an OCT image of 1 mm• 1 mm). Izatt, Kulkarni, Yazdanfar, Barton and Welch [1997] used conventional OCT depth scans to obtain structural as well as flow information. Phase-sensitive demodulation at the carrier frequency coo - coR provides the complex interferogram modulated by the Doppler frequency. Short-time Fourier transformation (STFT) of the complex interferogram is used to obtain localized Doppler spectra along
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Fig. 4.1. Doppler OCT image of in vivo blood flow in a chicken chorioallantoic vein. Mean wavelength ~ = 850nm, FWHM spectral width A~. - 25 nm. A, color-coded structural image. B, color-coded velocity image. C, velocity profile across the center of the vein; the open circles are experimental data, the solid curve is a parabolic fit. Reproduced from Chen, Milner, Srinivas, Wang, Malekafzali, van Gemert and Nelson [ 1997] by permission of the Optical Society of America. the depth scan. The local m e a n velocity us o f the scatterers is obtained from the centroid f r e q u e n c y COD o f the localized D o p p l e r spectra:
us = 4 ~ n cos 0 ' w h e r e n is the m e a n tissue refractive index, ~
(4.6) is the m e a n wavelength, and
0 is the angle b e t w e e n the incident b e a m and the velocity. Here, spatial resolution
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of the velocity estimate is connected to the STFT window length by the Fourier uncertainty relation. The spectral width of the measured Doppler spectrum increases with increasing coherent probe volume, with increasing velocity gradient and turbulence, and with increasing aperture of the probe beam. Ren, Brecke, Ding, Zhao, Nelson and Chen [2002] have shown that under certain conditions the two extreme Doppler shifts of the two outer boundaries of the probe beam can be used to determine flow velocity. If the flow angle is within + 15~ perpendicular to the probe beam the Doppler bandwidth permits accurate measurement of the flow velocity without precise measurement of the flow direction. A technique which extends the physiological relevance of DOCT to higher velocities, such as those occurring in blood flow through atherosclerotic lesions, has been described by van Leeuwen, Kulkarni, Yazdanfar, Rollins and Izatt [1999]. Using multiple frequency demodulation yielded fringes of constant velocities and shear-rate distributions.
4.1.2. Sequential scan processing
The most severe limitation of the fringe data technique is its limited velocity sensitivity. Using a frame rate of one per second for an image of (only) 100xl00 pixels, for example, eqs. (4.1) and (4.2) yield a minimum resolvable Doppler frequency of 10 kHz corresponding to a velocity sensitivity of approximately 3 mm/s (for A = 1300 nm at an angle of 0 = 80 ~ between probing beam and flow direction). In microvasculature, however, flow velocities occur in the 100 ptm/s range. One technique to overcome this limitation is phase-resolved DOCT using sequential depth scans. It also decouples velocity resolution from spatial resolution. This technique has been developed by Zhao, Chen, Saxer, Xiang, de Boer and Nelson [2000], and was used by Zhao, Chen, Saxer, Shen, Xiang, de Boer and Nelson [2000] and Yazdanfar, Rollins and Izatt [2001 ] to determine the Doppler shift from the phase change of the sample beam between sequential depth scans at the same location or even between sequential frame scans. The corresponding minimal Doppler shift is ~ ar
rOD-- T '
(4.7)
where Ar is the measured phase change, and T is the time interval between successive depth scans or frames at the same location, respectively. The phase
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difference Aq~ between two sequential depth scans (j) and (j + 1) is obtained from the complex coherence functions (4.8)
Aq~ = arg [Fj-(r)] - arg [Fj-+I(T)] .
As sequential depth-scan signals at the same location are compared, speckle modulations cancel each other and speckle noise is reduced in the velocity image. The above procedure can be performed on-line by phase-sensitive demodulation of the photodetector current using a lock-in amplifier (Izatt, Kulkarni, Yazdanfar, Barton and Welch [1997]). Alternatively, the complex coherence function can be computed off-line by analytic continuation of the photodetector LCI signal as described by Zhao, Chen, Saxer, Xiang, de Boer and Nelson [2000]. Recently, an optical implementation of the Hilbert transform has been described, enabling real-time images of in vivo blood flow in human skin (Zhao, Chen, Ding, Ren and Nelson [2002]). Another technique to increase velocity sensitivity is correlation processing. It has been suggested by Yazdanfar, Rollins and Izatt [2001 ] to use correlation processing of sequential depth scans m and m + 1:
(^ira(t) ^ira+l(t + r) ) ,
(4.9)
where ira(t) is the complex signal obtained by phase-sensitive demodulation of the detector current. The local Doppler frequency is obtained from the change in the phase of the cross-correlation:
1 ~ f~
2Jr
ira(t)ira+l(t +
Z')
r=0
(4.10)
(~;,(t)~m+l(t + r))r=o
Phase-sensitive sequential scan processing yields sensitivities in the range of 10 ~tm/s.
4.1.3. Fourier-domain DOCT In Fourier-domain DOCT too, data acquisition occurs from depth scans. However, instead of scanning the depth with the help of a delay line, the light scattered back by the whole sample depth, together with the reference beam is dispersed by a spectrometer on a detector array. The scattering potential of
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the sample is obtained by a Fourier transform of the scattered field (eq. 3.38). The Fourier shift theorem states that the shifted sample scattering potential F(z + Az) has the Fourier transform A(S)(K) exp(iKAz). Hence, a shift Az of the sample may be determined from the phase change KAz of the scattered wave. Depth information is spread over the whole spectrometer signal; the local sample scattering potential F(z). 6 ( z - zo), for example, is encoded in a corresponding harmonic component F(zo). exp(iKz0) of the K-spectrum. A local shift Az of the object structure at z0 generates a phase shift KAz of the corresponding harmonic component of the K-spectrum. Using sequential scans, this shift can be obtained from the phase difference of the Fourier transforms of two sequential scattering spectra A(S)(K) evaluated at z0 (or, equivalently, from the corresponding complex scattering potentials):
FW{AZI X)}]
Z=ZO
= VW{A ' X)}I
z0 .exp[iAq)(z0)] Z=
'
(4.11)
where m is the scan frame number. Hence, the local Doppler frequency is ~oD(z) -
Aq,(z) At
,
(4.12)
with At the scan frame time-delay. An important advantage of Fourier-domain DOCT is the high phase stability and speed of Fourier-domain setups. Furthermore, Leitgeb, Schmetterer, Wojtkowski, Hitzenberger, Sticker and Fercher (2002) have shown that longitudinal velocity components from ~10 gm/s to 2 mm/s might be measured at a rate of 104 depth scans per second without readjusting the experimental parameters.
4.1.4. Hardware solutions Signal-to-noise ratio is an important issue in DOCT. In LCI and OCT the optimal detection electronics bandwidth is approximately twice the bandwidth of the photocurrent signal due to the heterodyne modulation (Swanson, Huang, Hee, Fujimoto, Lin and Puliafito [ 1992]). In DOCT, however, the detection bandwidth must be chosen to match the much wider width corresponding to the range of velocities. Hence, to improve the signal-to-noise ratio in DOCT, a frequencytracking band-pass filter based on a modified phase-lock loop has been suggested (Zvyagin, Fitzgerald, Silva and Sampson [2000]). Another important issue is the speed of data processing. The digital Hilbert transform, being the most time-consuming process of phase-resolved DOCT, can be replaced by an optical Hilbert transform using polarization optics
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(Zhao, Chen, Ding, Ren and Nelson [2002]). Using hardware-implemented autocorrelation a frame rate of 8 per second with 480 depth scans per image and 800 samples per depth scan have been achieved (Westphal, Yazdanfar, Rollins and Izatt [2002]).
4.2. Wavelength-dependentOCT Fourier transform spectroscopy is an important technology for the spectral measurement of frequency-dependent refractive index and absorption of substances. Any sample in OCT introduces a frequency-dependent phase ~Disp(V) to the penetrating beam; its amplitude is determined by a frequency-dependent reflectivity x / ~ v ) . These coefficients determine the cross-spectral density of the interferogram WSR(V) = Ssource(V)" H(v) by the factor H(v), see eq. (3.22). Ssource(V) is the spectral density of the source light, and H(v) is the sample transfer function: (4.13)
H(v) = V ~ v ) exp[iqlSDisp(V)]. The interferogram is: G(r) = 2Re [FT -1 {Ssource(V). H(v)}] = 2Re [r(r) 9 h(r)],
(4.14)
where h(r) is the sample response function. In general the amplitude and phase change introduced to the interferogram will be depth-dependent: H(v;z) = v/R(v;z) exp[iq~isp(V;z)],
(4.15)
with an amplitude spectrum (Schmitt, Xiang and Yung [1998]): z2
~a(V;z')dz'
exp -2
(4.16)
0
where ou(v; z) is the spectral backscatter cross-section of the sample and ga(V; z) is the mean spectral attenuation coefficient between sample surface (assumed at z = 0) and the depth in the sample z. The phase spectrum is
qbDisp(V;Z) =
2~ /[n(v;z')-1] 0
where
dz~=2
J~[[k(v)- ko] dz', 0
n(v;z) is the sample refractive index, and k0 = 2:r//l.0.
(4.17)
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Both amplitude spectrum v/R(v;z) and phase spectrum 0~)isp(V;Z ) of the sample response function can be recovered from the Fourier transform of the interferogram (Fuji, Miyata, Kawato, Hattori and Nakatsuka [1997], Hellmuth and Welle [1998]). The measurement and imaging of the spectral sample reflectance R(v;z) is the subject of spectrometric OCT techniques, whereas measurement and imaging of the phase spectrum 0~)isp(V;Z ) is the subject of spectral refractometry OCT techniques.
4.2.1. Broadband spectroscopic OCT Optical tissue spectrometry can be an important basis for functional diagnostics and detection of tissue pathologies. Tissue oxygenation, hemoglobin oxygen saturation, and exogenous and endogenous chromophores are important features to indicate the functional state and pathologic changes in tissue. Using Beer's law the concentration of the various tissue dyes can be determined if the corresponding path length is known. Most present in uivo tissue spectrometric techniques, however, do neither locate spectroscopic variations with depth nor determine the path length of the measuring beam. Spectrometric OCT offers a solution to these problems. Spectrometric OCT (SOCT) is based on Fourier transform spectroscopy (FTS). In FTS the power spectrum of a light beam is obtained from a Fourier transforrfi of the temporal auto-correlation of the electric field, S(v) = FT{F(r)}, see eq. (3.9). FTS instruments can provide extremely high resolution and high ~tendue and, therefore, have already been used in tissue diagnostics (Ozaki and Kaneuchi [1989]). The similarity between FTS and LCI suggests a Fourier transform of the analytic depth-scan interferogram signal /'SR(I') = (V~(t)" VR(t + r)) to obtain spectroscopic information about the sample. However, the Fourier transform of FSR(r) is a cross-power spectrum WSR(V) and not the power spectrum of the light remitted by the sample. Hence, for example, WSR(V) ~< v/Ss(V)SR(V), where Ss(v) is the spectral intensity of the light remitted by the sample, and SR(V) is the spectral intensity of the reference light. Furthermore it should be noted that the spectral intensity of the light at the interferometer exit is not just the sum of the spectral intensities of the two interfering beams but depends on the spectral degree of coherence of the light remitted by the sample and reflected by the reference mirror (Mandel and Wolf [ 1995]). In a preliminary study, Kulkarni and Izatt [1996] used the sample transfer function H(v) = WSR(v)/Ssource(V) to calculate the spectral characteristics of the
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interface Fresnel reflections of a sample. They observed that, for Mie scatterers, the average of ]H(v)[ 2 correlates with the backscatter spectrum. A more serious problem, however, is encountered in depth resolution with a spectrally distorted LCI beam. Since the sample spectrum is determined by Fourier-transforming the (depth) interferogram signal, the Fourier uncertainty relation connects spectral resolution A v with depth resolution Az: c
Az. Av/> 4 ~
(4.18)
If, for example, a spectral resolution of A v/N is needed to identify a chemical component of the sample, where A v is the spectral width of the source light, the corresponding depth resolution Az is degraded by the same factor N, and becomes NAz. Nevertheless, there have been several promising steps towards SOCT. Morgner, Drexler, Kfirtner, Li, Pitris, Ippen and Fujimoto [2000] presented the first broadband time-domain SOCT technique. Leitgeb, Wojtkowski, Hitzenberger, Sticker, Kowalczyk and Fercher [2000] described the adaptation of Fourier-domain OCT spectroscopic measurements to SOCT. Schmitt, Xiang and Yung [1998] have presented a differential-absorption SOCT technique. In Fourier-domain OCT no depth scan occurs. The light exiting the interferometer is dispersed by a spectrometer to display its spectral distribution on a detector array. The detector signal represents the wavelength-dependent scattered field A(S)(K), from which the object structure is obtained by an inverse Fourier transform, see eq. (3.38). In spectral Fourier-domain OCT a frequency window w(AK; K~) of width AK centered at wavenumber Kn is shifted along the spectrum A(S)(K). The spectral data within each frequency window are Fourier transformed: FT -1 {A(S)(K) 9w(Kn; AK)} = F s ( K , ; z ) * #(AK;z) exp(iK~z).
(4.19)
The object structure is thus reconstructed using a windowed Fourier transform (WFT). #(AK; z) is the Fourier-transformed window function. The central wavenumber Kn of each window represents a monochromatic wavelength; its width AK determines the spectral resolution. The SOCT image is constructed step by step from monochromatic components. Each component can be encoded by a (false) color and thus a spectroscopic false-color image results. Depth resolution is determined by Az - 8 In 2/AK, see w5.1. Experimental SOCT images of a glass filter plate have been obtained by Leitgeb, Wojtkowski, Hitzenberger, Sticker, Kowalczyk and Fercher [2000].
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The use of the WFT presents a basic question: The Fourier uncertainty relation shows that narrow depth windows give good depth resolution, but poor spectral resolution, whereas wide windows give good spectral resolution, but poor depth resolution; what then is the optimal window function? The answer, of course, depends on the application. For example, if the spectral components are well separated from each other in the signal, we may sacrifice some frequency resolution and choose good depth resolution. However, very often there is no such a priori information. The Wavelet Transform (WT) solves this dilemma, at least to a certain extent. An entire spectrum is calculated at each point in the OCT image. Therefore, specific information at a single frequency or bandwidth can also be extracted. Compared to WFT with its constant resolution at all depths and frequencies, the WT yields good depth and poor spectral resolution at high frequencies, and good spectral but poor depth resolution at low frequencies. Morgner, Drexler, K~irtner, Li, Pitris, Ippen and Fujimoto [2000] used stateof-the-art femtosecond Ti:AI203 lasers to obtain spectroscopic information in tissue over a bandwidth from 650 nm to 1000 nm. Spectroscopic detection was performed with a Morlet WT of the interferogram to reduce windowing effects and yielding an entire spectrum at each image point: /.
W(s
r ) = J G(t + r) exp[-(t/to) 2] exp(ig2t)dt
2
.
(4.20)
To display this four-dimensional information (transverse and depth coordinates x and r, backscattering intensity, and spectral data) in false colors the spectral center of mass has been mapped into hue, while keeping the luminance constant, and mapping the backscattered intensity into saturation. Thus intensity and spectral shift of the backscattered light are visualized. A hue-saturation spectroscopic OCT image obtained from an African frog tadpole is depicted in fig. 4.2. It must, however, be kept in mind that OCT does mainly record Fourier components of the object structure with spatial frequencies in the range of 4;r/~, see w3.3.1. We see band-pass components of the object structure illuminated by light which has been attenuated by wavelength-dependent absorption and scattering in the tissue in front of it. Hence, absorption and scattering of deep structures is convolved with properties of the intervening structures.
4.2.2. Differential absorption OCT Schmitt, Xiang and Yung [1998] have presented a technique to generate OCT images of local concentrations of absorbing substances. One aim of their work was a technique to determine the concentration of water in tissue using
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Fig. 4.2. Hue-saturation false-color in uiuo image of tadpole mesenchymal cells. Wavelengthbandwidth 230 nm, mean wavelength 800nm. Geometrical resolution 1 ~tm(depth) x 5 ~tm (transverse). A green hue indicates a short-wavelengthshift of the center of gravity of the spectrum, and a red hue a long-wavelength shift. Arrows indicate melanocytes suspected of increased absorption at shorter wavelengths. Reproduced from Morgner, Drexler, K/irtner, Li, Pitris, Ippen and Fujimoto [2000] (courtesy J.G. Fujimoto, M.I.T.) by permission of the Optical Society of America.
the difference in absorption at two wavelengths. This technique, however, can be used to generate differential absorption OCT images of any substances with sufficiently different absorption coefficients. A pair of light-emitting diodes (1) and (2) were used to illuminate the OCT interferometer. Ideally, one diode emits in the vibrational band of the chemical compound of interest and the other just outside this band. Thus the operating principle is similar to differential-absorption lidar techniques (Sasano [1988]): The integrated differential attenuation within a layer of the sample is obtained in terms of the
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logarithmic ratio of the intensities ll and 12 of the two beams measured at the top (z0) and bottom (z) of the layer:
2 [tAaZ(Zt) -
[Ii(z) I2(zo)] "
(4.21)
IAal (Z')] dz t = In I1 (z0) I2(z)
z0
Since the interferometer is illuminated by two mutually incoherent, broadband light sources, the total power spectrum in the interferometer equals the sum of the two power spectra. From eq. (3.22) the power spectrum of the interferogram is the source power spectrum multiplied by the sample transfer function H(v): SS(V) = [Ssource 1(~') + Ssource 2(v)]" H ( v ) .
(4.22)
Hence the power spectra can be used to estimate the depth dependent intensities. This is eased by the fact that the two wavelengths generate different cartier frequencies. Schmitt, Xiang and Yung [1998] reported on two experiments using LEDs with mean wavelengths of ~.~ = 1.3 [am and ,~2 - 1.46 [am. In the first experiment they used a cuvette first filled with normal water, then with heavy water; in the second experiment an arterial-wall phantom was used. The measured differential absorption coefficients of water showed excellent agreement with those determined by conventional spectrophotometry. The thinnest layer of water that could be measured was approximately 50 [am.
4.2.3. RefractometricOCT Interference refractometry is a standard analytical laboratory technique. Dispersion data are important in environmental science and in diagnostic medicine. Quantitative dispersion data, for example, are important in predicting the propagation of light through ocular media, in photorefractive surgery and in tissue diagnostics. Refractometric OCT, however, is still at an initial stage (Sticker, Pircher, G6tzinger, Sattmann, Fercher and Hitzenberger [2002]). A dispersive sample introduces a frequency- and length-dependent phase ~lDisp(~t; L) to the analytic signal of the sample beam. Hence, the sample transfer function is given by:
H(v; L) -- [H(v; L)[ exp[iq~isp(V; L)], with the amplitude spectrum ]H(v;L)] and the phase spectrum 2:r qbisp(V; L) - - ~ L [n(v)- 1] = L [k(v)- k0]; L is the length of the dispersing path in the sample arm; index of the sample.
(4.23)
(4.24)
n(v) is the refractive
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Recently, a highly sensitive phase-dispersion tomographic technique has been described by Yang, Wax, Dasari and Feld [2001]. A pair of harmonically frequency-related low-coherence light sources was utilized to remove motional artifacts in the interferometer, and local phase differences could be measured with a sensitivity of better than 10-1 rad. This technique is outlined in more detail in w4.4.
4.3. Polarization sensitive O C T
So far we have treated the electric field of the light beams as scalar field and ignored any vector properties. While this simplification allowed us to derive several important properties of intensity-based LCI and OCT signals, it fails if any polarization effects are to be analyzed. However, as is well known from microscopy, many samples yield only poor image contrast if imaged on an intensity basis. Other properties of light, e.g., its polarization state, can be used to improve the contrast or to generate new types of image contrast. In this subsection, treating the electric field as a vector field, we describe how additional information on a sample can be obtained by polarization-sensitive recording of the interferometric signals. Different approaches to polarization-sensitive OCT (PS-OCT) have been reported by Hee, Huang, Swanson and Fujimoto [1992], by de Boer, Milner, van Gemert and Nelson [1997], by Schoenenberger, Colston Jr, Maitland, Da Silva and Everett [1998], by de Boer, Milner and Nelson [1999], by Yao and Wang [1999], by Saxer, de Boer, Park, Zhao, Chen and Nelson [2000], by Roth, Kozak, Yazdanfar, Rollins and Izatt [2001], and by Hitzenberger, GStzinger, Sticker, Pircher and Fercher [2001]. Since OCT detects the coherent component ofbackscattered light the Jones formalism is a means to characterize polarization properties of the sample (Jiao and Wang [2002]). Most of the experiments reported so far were based on the setup initially described by Hee, Huang, Swanson and Fujimoto [1992]. Since this setup is widely used and rather versatile, we base our description of PS-OCT on it. The polarization states of the light beams, and the changes of the polarization states as the beams propagate through the polarizing components of the instrument and the sample, are described by the Jones formalism. Figure 4.3 shows the basic configuration of the PS-OCT setup. It consists of a low-coherent light source, a Michelson interferometer, a two-channel polarization-sensitive detection unit, and additional polarizing components. The light emitted by the low-coherent
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Fig. 4.3. Optical scheme of polarization-sensitive OCT device. LS, low-coherent light source; NPBS, non-polarizing beam splitter; PD1,2, photodetector; PBS, polarizing beam splitter; Pol, polarizer; QWP1,2, quarter-waveplate; RM, reference mirror; SA, sample; SR signal processing unit. light source transmits a polarizer that renders the beam in a linear vertical polarization state. Its Jones vector can be written as
E=Eo (01 ) "
(4.25)
E is the electric field vector, with E0 = A0 exp(icot) the scalar electric field, and A0 the field amplitude. The upper and lower components of the vector in eq. (4.25) correspond to the horizontal and vertical components of the electricfield vector, respectively. The Jones vector of a beam traversing an optical element can be found by multiplying the Jones vector of the incident beam by the Jones matrix corresponding to the optical element. The Jones matrix is a 2 x 2 matrix consisting o f - u s u a l l y c o m p l e x - elements. If more than one optical element is traversed, the input Jones vector has to be multiplied by the Jones matrices of all elements, in the order they are transmitted by the beam. The beam described by eq. (4.25) enters the Michelson interferometer where it is split by a nonpolarizing beam splitter into a reference and a sample beam of equal field amplitudes. The reference beam passes both the beam splitter and the quarter wave plate QWP 1 twice. The effect of the beam splitter is simply to reduce the intensity of the reference beam by a factor of 2 at every pass, leading to a total
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intensity reduction of a factor of 4, equivalent to a reduction in field amplitude by a factor of 2. The Jones matrix of a general retarder of retardation 6 and fast-axis orientation 0 is
M(6, O) =
[ cos2(0) + sin2(0) 9exp(-i6) cos(0), sin(0). [1 - exp(-i6)] ] cos(0), sin(0). [1 - exp(-i6)] cos2(0) 9exp(-i6) + sin2(0) ~" (4.26) 1 For QWP1 6 = ~7t and 0 -_- ~ : t1 . The Jones vector of the reference beam, after double passing QWP1 (and the beam splitter) is
gR(zr)--
L
1MQwp 1X MQWPI 9E0"
E0 / 11) / 01) = ~-~
9exp(-i2kzR),
(4.27) where zR is the (optical) length of the reference arm. This is a linear polarized beam with its polarization axis oriented at 45 ~. This beam provides equal reference intensity for both the horizontal and the vertical polarization component, which are separated by the polarizing beam splitter PBS of the detection unit. Furthermore, no phase shift occurs between the two polarization components (this would cause at least one of the vector components to have an imaginary part). Therefore, the reference beam affects neither the intensity ratio nor the phase of the interference signals recorded in the two detection channels. The sample beam passes both the beam splitter, QWP2, and the sample (amplitude reflectivity: v/-R) twice. Again, the effect of the beam splitter is simply to reduce the field amplitude by a factor of 2. QWP2 has retardance and axis values 1 1 of 6 = ~;r and 0 = ~;r, respectively. Assuming that the polarizing properties of the sample volume can be described by a homogenous retarder (Jones Matrix msample, retardation 6s(z), (constant) fast-axis orientation 0s), the Jones vector of the sample beam, after exiting the interferometer, is (Schoenenberger, Colston Jr, Maitland, Da Silva and Everett [ 1998], Hitzenberger, G6tzinger, Sticker, Pircher and Fercher [2001 ]):
gS(z ) = 1MQwP2 x Msample((~s(Z),0s) " vIM(z)
"Msample(OS(z)'Os)•176 _ 1x/-R~ "Eo. ( 2
( 0)1
cos(6s(z)) exp(-i6s(z)) sin(6s(z)) exp(i(sr - 6 s ( z ) - 20s))
) 9exp(-i2kzh),
(4.28) where z is the geometrical distance from the interferometer beam splitter to a sample reflection site (for simplification, no air gap has been assumed between beam splitter and sample surface); h is the mean refractive index of ordinary and
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extra-ordinary beam, and 6s(z) = An. z. k0 (An is the refractive index difference between ordinary and extra-ordinary beam). After recombination at the interferometer beam splitter, reference and sample beams interfere. The recombined beams travel towards the polarization-sensitive detection unit where they are split by a polarizing beam splitter into horizontal and vertical polarization components. The interference terms IH, V at the detectors, corresponding to the horizontal and vertical polarization channels, respectively, are:
IH, v(Z, AZ) = IH0,v0(Z, 6Z). COS(q)H,v(AZ))
(4.29)
with I.o(z, ~ )
=
RV/ 2v~
9cos(6s(z)). I y(zXz)l (4.30)
Iv0(z, az) =
2~
9sin(Os(z)).
I y(az)l
and q)H(Az) = 2k0Az,
q~v(Az)= 2k0Az + :r - 20s,
(4.31)
where In0,v0 describe the envelopes of the interferometric signals caused by a single reflective site in the sample, q~, v are the corresponding phase functions of the interferometric fringes, Az = ZR- z. h is the optical path-length difference between reference and sample arms, and I )'(Az)] is the modulus of the complex degree of coherence of the interfering beams which describes the width of an interferometric signal. In eq. (4.31) an unimportant common phase term has been omitted. The terms cos(6s(z)) and sin(6s(z)) in eq. (4.30) oscillate slowly with depth z, if a sample with constant birefringence An is assumed, giving rise to intensity oscillations in the horizontal and vertical polarization channels which are g1 Jr out of phase. The cosine term in eq. (4.29) oscillates rapidly with path difference Az and describes the fringes of the interferometric signals. The fringes of the horizontal and vertical components have a phase difference of : r - 20s. To calculate sample reflectivity R(z) and retardance 6s(z), we only need the envelope peaks of the interferometric signals Ino(z, Az = 0) and Iv0(z, Az = 0): R(z)
CK IH0(Z, 0) 2 q- Iv0(Z, 0) 2,
(4.32)
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6s(z) = arctan(Iv~
"
265
(4.33)
These envelopes can be obtained, for instance, by demodulating the signals IH, V at the carrier frequency 2 k -d(Az) - ~ , or by analytic continuation of the recorded signals by use of the Hilbert transform HT (w 3.1.2):
Az))
i,o, vo(z, Az) =
+
(z, Az)}) 2
(4.34)
A closer look at the phase terms of the interferometric signals reveals that the information on the optic axis orientation is encoded entirely in the phase difference A q~ = q ~ - q54 of the two signals (de Boer, Milner, van Gemert and Nelson [ 1997], Schoenenberger, Colston Jr, Maitland, Da Silva and Everett [1998], Hitzenberger, G6tzinger, Sticker, Pircher and Fercher [2001]): 0s = l ( s r - Aq~)
(4.35)
is the angle between the fast axis and the x-axis. To measure 0s, we need a phase-sensitive recording of the full interferometric signals. From these signals, q~ and q~v can be derived by q)H,V = arctan
HT {IH,v(Z, Az) } ] IH, v(Z, Az) "
(4.36)
In biological tissues, birefringence can be caused by two different mechanisms: (a) intrinsic birefringence similar to that observed in anisotropic crystals, caused by anisotropic electrical properties of the molecules; and (b) form birefringence caused by structural anisotropy of otherwise optically isotropic material. An example of intrinsic birefringence in biological tissue is in dental enamel which consists of birefringent hydroxyapatite crystals. Form birefringence is found in several fibrous tissues, e.g., in muscle, tendon, nerve fiber tissue, etc., where oriented fibers are surrounded by a matrix of different refractive index. PS-OCT can be used to quantify birefringence in terms of retardation and also optical axis orientation. Changes in these parameters might be used to detect tissue degradation, and therefore PS-OCT has a potential application in medical diagnostics. Among the suggested applications are burndepth estimation (de Boer, Srinivas, Malekafzali, Chen and Nelson [1998]), caries diagnosis (Baumgartner, Dichtl, Hitzenberger, Sattmann, Robl, Moritz, Fercher and Sperr [2000]), and glaucoma diagnostics (Ducros, de Boer, Huai, Chao, Chen, Nelson, Milner and Rylander III [1999]).
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Fig. 4.4. OCT images recorded in a chicken myocardium in vitro parallel to the heart longitudinal axis. Dimensions are indicated in mm (the ordinate shows optical distance). Mean wavelength = 1310nm, FWHM spectral width A~ = 65nm. (a) Intensity image (color scale: logarithmic intensity); (b) phase retardation image (color scale: retardation in degrees); (c) image of fast-axis orientation distribution equals the muscle fiber orientation distribution (color scale: axis orientation in degrees). As an example of the potential of PS-OCT for imaging birefringence properties in tissue, fig. 4.4 shows PS-OCT tomograms recorded in a chicken myocardium in vitro. The heart was sectioned perpendicular to its longitudinal axis into two halves, OCT images were recorded perpendicular to the section plane (i.e., parallel to the longitudinal axis), approximately across the center of the myocardium, covering ventricular muscle tissue on the left- and righthand side of the tomograms, and tissue corresponding probably to the septum interventriculare near the central image part. Figure 4.4a shows the intensity
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image, fig. 4.4b the phase retardation, and fig. 4.4c the fast-axis distribution in the sample. The intensity image shows very little structure and contains the least information. The retardation image shows rather constant color with depth near the (transversal) central third of the image, and more than one full oscillation, indicating a retardation of > 180~ in the right and left thirds of the image. This is probably due to birefringence caused by the fibrous structure of the muscle tissue. The low birefringence in the central area of the image is probably caused by a steeper orientation of the muscle fibers in this area. The map showing the optic axis orientation (fig. 4.4c) shows information complementary to that of the retardation image. Different orientations of fast optic axis can be observed in the right- and left-hand parts of the tomogram, respectively, indicating different orientations of muscle fibers (the 90 ~ color changes in axial direction at depth positions where the retardation (fig. 4.4b) passes the 90 ~ and 180 ~ values are caused by the data processing algorithm and do not indicate layers of different orientation). Note that eq. (4.28) is based on constant axis orientation in depth. Therefore, only the transverse distribution of axis orientation of the first layer is correct. Recently, new PS-OCT schemes have been reported. De Boer, Milner and Nelson [1999] measured depth-resolved Stokes vectors of the backscattered sample light. Thereby different polarization properties of the sample can be derived such as, for example, birefringence and optic axis orientation. Another method addresses the special problems associated with a fiber-optic implementation of PS-OCT (Saxer, de Boer, Park, Zhao, Chen and Nelson [2000]). Other schemes measure the full Mtiller matrix of the sample which contains the complete information on the sample's polarizing properties (Yao and Wang [1999], Jiao and Wang [2002]).
4.4. P h a s e - c o n t r a s t O C T
In microscopy, phase-contrast techniques have found widespread applications to generate image contrast in transparent samples whose structural compartments are non-absorbing and differ only in refractive index. Especially cell biology has taken advantage from this imaging technology since many cells are transparent and their organelles were almost impossible to image without staining before the invention of the phase-contrast microscope. Only few attempts towards the application of phase contrast to OCT have yet been reported. These are: differential phase-contrast OCT (DPC-OCT) and phase-dispersion optical tomography (PDOT).
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Fig. 4.5. Optical scheme of differential phase contrast OCT device. L, collimating lens; LS, lowcoherent light source; NPBS, non-polarizing beam splitter; PD1,2, photodetector; PBS, polarizing beam splitter; Pol, polarizer; QWP1,2, quarter-wave plate; RM, reference mirror; SA, sample; SP, signal processing unit. DPC-OCT was first reported by Hitzenberger and Fercher [1999]. This technique transferred concepts of differential interference contrast microscopy to OCT. Figure 4.5 shows the optical scheme. It is based on a PS-OCT setup (cf. fig. 4.3) with a modified sample arm: the sample beam, after propagating through QWP2, is in a circular polarization state. This beam traverses a Wollaston prism that splits the beam into two components with mutually perpendicular linear polarization states. The Wollaston prism is located in the focal plane of focusing lens L that converts the two diverging sample-beam components into parallel beams and focuses them at the object. They are reflected at different object interfaces and then travel back through L and the Wollaston prism, where they are recombined into a single beam. After backpropagating through QWP2, this beam generally has an elliptical polarization state, depending on the phase difference A q~ between the two components of the sample beam. In other words, A q~ is encoded in the polarization state and can be retained in the same way as the retardation caused by birefringence in PS-OCT, i.e., with a twochannel polarization-sensitive detection unit (cf. w4.3). Hitzenberger and Fercher [ 1999] demonstrated the imaging of schlieren (local variations of refractive index in otherwise uniform samples) by this technique. With an improved version of this method, Sticker, Hitzenberger, Leitgeb and Fercher [2001] demonstrated phase measurements corresponding to a path-difference resolution on the order
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of 2-3 nm. They also demonstrated that phase measurements and imaging by this technique are possible through scattering layers of thickness up to eight mean free path lengths. A phase-dispersion optical tomography technique has been developed by Yang, Wax, Dasari and Feld [2001]. The key idea of this technique is to use a pair of harmonically related low-coherence sources to record two sample phases of each point simultaneously. Since the wavenumber of the second-harmonic k2 is exactly twice k~ any influence of axial sample jitter during measurement can be eliminated completely by subtracting twice the phase of the kl-data from the phase of the k2-data. Corresponding phase data can be used to characterize interfaces and, if these phase shifts are known, the intrinsic dispersive phase shift of a layer expressed as AO5 =mod2jr [2kl(n2 - nl)L],
(4.37)
where n l and n2 are the refractive indices at the fundamental and second harmonic wavelength, and L is the layer thickness. First experiments demonstrated that phase difference measurements with a resolution on the order of a few degrees are possible.
w 5. OCT resolution and sensitivity 5.1. Resolution and point spread function
Depth resolution in LCI and OCT depends on the coherence length, while lateral resolution depends on the numerical aperture (NA) of the focusing optics. In scattering media the transition from single to multiple scattering degrades image contrast as well as lateral and depth resolution. Since the quantities used in LCI and OCT resolution studies are subject to Fourier relations we define resolution as the corresponding FWHM uncertainty. In time-domain LCI and OCT the coherence function plays the role of a depth point-spread function. Hence, depth resolution is determined by the coherence length. In Fourier-domain OCT, depth resolution 6z can be estimated with the help of the Fourier uncertainty relation. For a Gaussian frequency spectrum the FWHM values 6z and AK of the Fourier variables z and K are related by 6z. AK = 8 In 2, from which 6z - 2 ln2/l~ (5.1) :~ A/I' the same depth resolution as in time-domain OCT is obtained. /10 is the mean wavelength of the Gaussian frequency spectrum. At present, for example,
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superluminescent diodes at a mean wavelength 2,0 = 830 nm have spectral widths of about A~, = 25 nm. A coherence length and depth resolution of 12gm is obtained. Transversal resolution in OCT is determined by the transverse sampling rate and the transverse width of the probe beam. Provided the transversal sampling rate satisfies the sampling theorem, transversal resolution is determined by the transversal width of the probe beam. In a Gaussian beam the minimum spot size occurs at the beam waist. The radius r at which the beam intensity falls to 1/e 2 of its central value is w0. If we define transversal resolution in analogy to depth resolution as FWHM value 6r of the probe beam amplitude at the beam waist (positioned at the probe beam focus) we have: 6r = 2 v / ~ 2 . w0 = 2 lx/ln-2. 2,0 :tO'
(5.2)
where 2,0 is the mean wavelength, and 0 is the angular spread of the Gaussian beam; sin 0 is the numerical aperture of the beam. Of course, a small beam waist radius demands for a large numerical aperture which in turn results in a small confocal beam parameter z0. Twice the confocal beam parameter can be used to define the depth of focus DOF: DOF = 2zo = 2 XY " . jr0 2
(5.3)
Therefore, a compromise has to be found between the desired depth of focus DOF, and the desired transversal resolution 6r. For example, a transversal resolution of 6r = 20 ~tm at a mean wavelength of 2,0 = 830 nm leads to a depth of focus of DOF = 3 mm, whereas a transversal resolution of 6r = 2 ~tm reduces the depth of focus to DOF = 30 gm. Schmitt [1998] defined an OCT point-spread function as the functional dependence of the interferogram signal of a point scatterer scanning throughout the focal space of the objective lens. If the optical path matchpoint lies in the focal plane of a Gaussian probe beam and scattering in the intervening space is neglected, the point spread function has an envelope
H(Ar, A z ) = e x p -
41n1 exp[
(5.4)
where Az is the displacement of the scatterer in the axial direction and Ar is the displacement in the radial direction, a - v / ( L / k R ) e + [(Az/f)R] 2 is a radial
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parameter, R is the 1/e radius of the Gaussian beam, and L = f - Az is the axial distance of the scatterer from the lens with focal length f .
5.2. OCT signal processing 5.2.1. Image display The standard time-domain OCT depth-scan signal is obtained by amplifying, band-pass filtering, rectification and low-pass filtering of the photodetector signal to obtain the envelope AG(t) of the LCI signal. Standard noise-reduction techniques are used in these steps. Finally, quantization is performed which maps the raw data to a series of fixed graylevels or pseudocolors. The LCI signal is proportional to the square root of the sample beam power and thus to sample amplitude reflectivity or sample scattering potential Fs, see eqs. (3.2), (3.7) and (3.36). Hence, the signal-to-noise ratio easily exceeds 50 dB. Therefore, the logarithm of the signal U(t) or i(t) is usually quantized and used as OCT image function value and encoded in graylevels or false colors. From the many false-color maps that have already been applied to OCT, the most widely used is the blue-green-yellow-red-white scale first used in the work of Huang, Swanson, Lin, Schuman, Stinson, Chang, Hee, Flotte, Gregory, Puliafito and Fujimoto [1991 ]. However, it has been noticed by Ishiwaka, Giirses-Ozden, Hoh, Dou, Liebermann and Ritch [2000] that grayscale OCT images can be used to demonstrate additional information not present in false-color images, and, furthermore, that false-color mapping of OCT images may lead to erroneous qualitative image interpretation. Quantization is used for image compression, for adaptation of images to the properties of the human eye, and for image processing. As mentioned, logarithmbased quantization has commonly been used in OCT. It has been shown by Yu, Ji, Wang and Xue [2001], however, that logarithm-based quantization gives relatively inferior contrast, low contrast-to-noise ratio and low signal-tonoise ratio. Since all quantization methods have probability density functions different from that of the sample, there is no single optimal quantization technique. For example, the minimum-distortion method improves contrast and reduces quantization errors, whereas the information-expansion method reveals additional detail information.
5.2.2. Image enhancement by deconvolution The depth-scan signal is the convolution of the sample response function with
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[4, w 5
the self-coherence function, see eq. (3.21). Since the standard deviation of a convolution equals the sum of the standard deviations of its components, the spread of the depth-scan signal is increased, and resolution is degraded. Furthermore, most light sources that can potentially be used in OCT do not have a Gaussian spectrum but a less suitable spectral structure. Therefore, the corresponding depth point-spread function has sidelobes. This leads to spurious image structures. Deconvolution of the depth-scan signal and spectral filtering of the source light can be used to improve the resulting cross-correlation function. Deconvolution can be performed digitally in Fourier space. From eq. (3.22) the cross-spectral density of the depth-scan signal is WsR(V)= Ssource(V)"H(v), where WsR(V) is the cross-spectral density of sample and reference beam, Ssource(V) is the spectral density of the source light, and H(v) is the sample transfer function. Deconvolution in Fourier space is simply performed by a division:
H(v) =
wsR(v)
(5.5)
Ssource(V) + E"
However, deconvolution is fraught with problems. First of all, Ssource(V) must be known and must not contain zeros. Therefore, most deconvolution techniques add a non-zero term [e in eq. (5.5)] to the denominator. Furthermore, deconvolution can be ruined in particular by structured noise. Bashkansky, Duncan, Reintjes and Battle [1998] discuss the problems generated by external incidental reflections common to both reference and signal beams or one of the beams. Using eq. (5.5) they demonstrate a depth-resolution improvement by a factor o f - 2 . Kulkarni, Thomas and Izatt [ 1997] have used a constrained iterative restoration algorithm for deconvolution of an OCT image of an onion specimen. Depth resolution was improved by a factor of >2.5, whereas dynamic range was decreased by only 2 dB. Schmitt [1998] used the CLEAN algorithm, an iterative point-deconvolution algorithm, introduced by Hrgbom [1974] to find positions and strengths of light sources in radio astronomy, to deconvolve OCT images. Some improvement of OCT images of human skin has been obtained. A straightforward application of eq. (5.5) to LCI signals has been described by Wang [1999]. Depth resolution could be improved by a factor of 2-3, but a substantial enhancement of noise occurred. A digital spectral shaping technique was presented by Tripathi, Nassif, Nelson, Park and de Boer [2002]; its efficiency was demonstrated on a composite source of two superluminescent diodes. The Fourier components of the interferometric signal were corrected such that the spectrum became Gaussian. The spectrally shaped image showed significant reduction of spurious image structures whereas the noise floor increased only by 0.9 dB.
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5.2.3. Image enhancement by dispersion compensation The transfer function of a dispersive sample equals the exponential function of the dispersion phase coefficient of eq. (4.17). Hence, the impulse response function of a dispersing sample of length L is FT-' {exp [iqJDisp(co;L)] }.
(5.6)
The main consequence of dispersion in PCI and OCT is loss of depth resolution. Material dispersion k = k(oJ) -- n(oo)r can be developed into various orders: dk (r k - k0 = d--~ ~o0
51 ~d2k ~,,,,(r
r
1 d3k 2 + g h-g5
(r
r
3+...
,oo
"
(5.7) (1) First-order dispersion dk/d~o changes the coherence length to Ic/nG. Since in most media the group index nG is only slightly different from the phase index n, first-order dispersion has only a negligible effect on resolution. (2) Second-order dispersion d2k/d~o 2 causes group-velocity dispersion and, therefore, degrades depth resolution. Hitzenberger, Baumgartner, Drexler and Fercher [1999] have shown that resolution in OCT is degraded by a factor of 1 + (L.
GD. Ar 2,
(5.8)
where L is the length of the dispersive path, and GD = dno/dZ is the group dispersion. (3) Group-velocity dispersion furthermore leads to the phenomenon of chirping, i.e. the instantaneous frequency of light pulses changes. Chirped pulses are not Fourier-transform limited. (4) Because of energy conservation, second- and higher-order dispersion reduce the signal-to-noise ratio in OCT by the factor (5.8). (5) Furthermore, Hitzenberger, Baumgartner and Fercher [1998] have shown that dispersion gives rise to artifacts in LCI and OCT. For example, a beat effect occurs if a thin object behind a dispersive medium is to be measured by LCI or imaged by OCT. (6) Third-order dispersion does also contribute a wavelength-dependent phase to the spectrum but can be neglected in standard OCT. It distorts the shape of the coherence function. High-resolution LCI and OCT in transparent media requires dispersion compensation. Usually a dispersive plate is introduced into the reference arm
274
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of the interferometer with the same dispersion as the sample in the probe arm. Variable object depths can be matched with a depth-dependent dynamic dispersion balancing system. A plate tilting synchronously with the movement of the depth-scanning reference mirror or synchronously shifted dispersive prisms can be used. This, however, requires mechanically moving components and, therefore, is slow, and difficult to implement. Furthermore, if the object and/or its dispersion changes, dispersion-balancing components might have to be replaced. Fercher, Hitzenberger, Sticker, Zawadzki, Karamata and Lasser [2001] have shown that a digital correlation technique can be used to compensate for dispersion-induced resolution loss. This technique is based on the fact that the auto-correlation Q(r) of a quadratic phase term yields a b-function: Q(r) | Q(r) = 6(r).
(5.9)
Since a dispersive sample with second-order dispersion adds a quadratic phase factor Q(to; L) to the spectrum, } d ~2 Q(o);L) = exp{ [i (o) - 0)0)2] d2k(~
,
(5.10)
with L the length of the dispersive sample, the correlation of the response function of the interferometer with the corresponding dispersive impulse response function yields a dispersion-compensated impulse response function GDc(Z';L): GDc(r;L)=[G(r).Q(r;L)]Q[G(r).Q(r;L)]=G(r)|
(5.11)
where | means correlation. Hence, the compensated impulse response function GDc(r;L) equals the autocorrelation of the dispersion-free impulse response function G(r). Advantages of the digital correlation technique are that it is not sensitive to zeros in the response function and that correlation of the experimental signal with a mathematically defined smooth function reduces noise. Another advantage of numerical compensation of dispersion in LCI and OCT depth-scan signals is that it can be performed a posteriori and can meet different dispersion requirements. Of course, object dispersion must be known in the numerical compensation technique as well as in the physical technique. De Boer, Saxer and Nelson [2001] have numerically corrected the quadratic phase shift in the Fourier domain, eliminating the broadening effect of groupvelocity dispersion on the coherence function in the Fourier domain of the interference signal. A similar technique was used by Brinkmeyer and Ulrich [ 1990] in testing integrated-optical waveguides.
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5.3. Speckle in OCT 5.3. I. Speckle properties Experience shows that in a picture of good television standards the signalto-noise ratio is 40dB. Hence a sufficiently noiseless pixel must contain 104 resolved elements. Therefore, laser speckle degrades the linear resolution in television standard pictures obtained with coherent light by about two orders of magnitude (Gabor [ 1970]). On the other hand, speckle does contain information about the corresponding scatterer as evidenced by a wealth of speckle-related techniques (Dainty [1984]). Speckle is generated by interference of waves with random amplitudes and phases. In LCI and OCT speckle is generated in the sample beam. The sample wave is the sum of many wavelets arising at backscattering sites within the coherent volume. These waves can have random phases due to the random depth distribution of scattering sites in the sample and due to the fluctuating refractive index in the sample. In the ideal case the amplitudes and phases of these individually scattered contributions are statistically independent, are equally distributed, and have phases uniformly distributed over (-:r, :r), and the waves are perfectly polarized. Then the resultant phasor is a circular complex Gaussian random variable and its real and imaginary parts are Gaussian random variables (Goodman [ 1976]). The corresponding statistics of the sample intensity are negative exponential, and the probability density function is
Pi~(Is) =
/1 ~
0
exp -~--~
if Is ~> 0,
(5.12)
otherwise,
with moments
(I n) = n! (I)".
(5.133
The contrast of a speckle pattern is defined as the standard deviation related to the mean, C-
aIs (I~)'
(5.14)
leading to the well-known high speckle contrast C = 1 of so-called "fully-developed" polarized speckle (George, Christensen, Bennett and Guenther [1976]). The characteristic depth extension of these speckle can be obtained as the inverse width of the corresponding power spectrum. Here we use, however, a
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[4, w5
qualitative argument to estimate the depth extension of fully developed speckle. In fully developed speckle the intensity correlations can be obtained from the amplitude correlation by (Loudon [1985]):
(5.15)
(I(t)I(t + r ) ) = (l(t)) 2 [1 + IG(r)12].
Hence, the intensity fluctuations of fully developed speckle have a correlation length of the order of the corresponding coherence length lc. The real situation in OCT, however, is more complex: firstly, because of the reference beam we do not have fully developed speckle in LCI and OCT; secondly, light backscattered at specularly reflecting interfaces does not generate speckle.
5.3.2. Speckle in LCI and OCT The field at the photodetector of a LCI system is the sum of backscattered sample waves plus a reference wave (of intensity IR). If the scattered waves are perfectly polarized and the reference wave is co-polarized the statistics of the total intensity I at the photodetector is a modified Rician probability density function (Goodman [ 1984]):
p,(I) =
I 1 (I+IR)(v/I.IR)if ~
exp
0
(Is)
I0
()s)
I)0, (5.16)
otherwise.
I0 is a modified Bessel function of the first kind and second order. The variance of the total intensity is 02 = (Is) 2 (1 + 2r),
(5.17)
and the speckle contrast is C = v/1 + 2r/(1 + r), with the beam ratio r - IR/(Is). By the same argument as above the correlation length of these intensity fluctuations too, is of the order of the coherence length lc. The intensity at the photodetector, therefore, contains the constant reference-beam intensity IR, the fluctuating sample intensity Is, and the cartier-frequency-modulated fluctuating interferogram intensity GSR. The cartier frequency is fc - 2 . VMirror//~0 (with A0 the mean wavelength) and the mean frequency caused by speckle is fSpeckle - 2. VMirror/lc, with VMirror the speed of the reference mirror. Hence, the heterodyne technique can very efficiently suppress speckle fluctuations induced by Is if lc >>/1,0. In high-resolution OCT, however, electronic band-pass filtering
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must be performed very carefully to eliminate this noise. In any case, the speckle noise of the interferogram remains. In standard OCT we have lc >> &0 and the depth-scan signal contains only the interferogram. Hence, the statistics of the depth-scan signal equals the statistics of the interferogram. The interferogram phasor FSR equals the sample wave phasor multiplied by the deterministic reference phasor, see eq. (3.7). The interferogram is twice the real part of the interferogram phasor; therefore, its statistics for polarized light is that of a Gaussian random variable. If the OCT image function S(t) = U ( t ) - Um (eq. 3.2) is obtained by full-wave rectification of the interferogram GSR, the probability density function of S(t) is Gaussian and provides a signal-to-noise ratio of the depth-scan signal of v / 2 / ( a r - 2) = 1.32 (Bashkansky and Reintjes [2000]). However, standard OCT techniques use the real envelope Ao of the interferogram as picture function S. Ac equals the random magnitude of the sample beam amplitude As multiplied by twice the magnitude of the reference wave amplitude AR and the degree of coherence lYSRI. Hence, the standard OCT picture function S will be Rayleigh-distributed: arS ps(S)
=
2(S) 2
9exp
7/'S2 ) 4(S) 2
0
S/> 0,
(5.18)
S<0,
with mean (S) = 2ARITSRI(As), second moment (S 2) =
y4 ( S ) 2 , c o n t r a s t
os _ q _ Cs = (s) _ 4-~
1 = 0.52, and signal-to-noise ratio 1/Cs = 1.91 It must be noted that these fluctuations are caused by the distribution of the scatterers within the coherent probe volume. The depth positions of these scatterers are not resolved. Therefore, the corresponding intensity fluctuations of the sample signal occur on the scale of the coherence length and, therefore, will severely degrade depth resolution. Note, however, that in most OCT techniques the logarithm of the LCI signal is used as OCT picture function. 5.3.3. Suppression o f speckle in O C T
From a fundamental law of probability theory it follows that the sum of M identically distributed, real-valued, uncorrelated random variables has a relative standard deviation of M -1/2 times the standard deviation of one component. This points to possible techniques for speckle suppression. Adding up several speckle patterns reduces speckle contrast in the compound signal. The resulting
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probability density has been shown by Goodman [1965] to be a Gammadistribution with speckle contrast C = M -1/2. Therefore, the corresponding gain in the signal-to-noise ratio is M 1/2. Three conditions, however, must be fulfilled: The summation must occur on an intensity basis; the single speckle patterns must be uncorrelated; and the underlying single object structures must be identical. Goodman [1976] has already discussed conditions for uncorrelated optical speckle, like time, space, frequency, or polarization diversity. Further possibilities are depth-scan signal processing (Forsberg, Healey, Leeman and Jensen [1991]) and image post-processing (Schmitt [1997]). In the space-diversity method the compound signal is obtained from adjacent detectors filling the aperture of the objective lens. If the signals measured by these detectors are uncorrelated, light is detected from outside the diffractionlimited probe beam focus and transverse resolution is degraded. Since the subdetectors record uncorrelated signals, incoherent processing of the single detector signals does not further degrade transverse resolution. In fact the situation is more involved since transverse resolution also depends on the transverse sampling rate. Depth resolution is not affected by a reduction of the effective aperture since the coherence length remains unchanged. Using this technique Schmitt [1997] obtained a signal-to-noise gain close to the theoretical attainable factor (two; using four detectors) without significant loss in resolution. Some other techniques have been tested so far. A practical speckle-reduction technique for en face OCT imaging has been described by Bashkansky and Reintjes [2000]. A wavelet filter with non-linear thresholds was used by Xiang, Zhou and Schmitt [1998] for speckle reduction. Further experimental specklereduction methods are discussed by Schmitt, Xiang and Yung [1999]. The zero-adjustment procedure (Yung, Lee and Schmitt [1999]) reduces speckle contrast in regions with high scatterer densities but tends also to blur sharp boundaries between image features. Rogowska and Brezinski [2000] used an adaptive speckle suppression filter to enhance image features while suppressing a noisy background. 5.4. System sensitivity
An important issue in OCT imaging is detection sensitivity. Sensitivity can be defined as the smallest signal just distinguishable from noise. OCT signals are generated by backscattered light. The amount of backscattered light depends on the incident light intensity and the sample reflectivity Rs. Since we are not interested in the absolute amount of backscattered light but in the sample parameter Rs, a suitable definition for the sensitivity S in the context of OCT is
4, w 5]
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279
the ratio between the reflectivity of a perfectly reflecting mirror (R = 1) and the smallest sample reflection coefficient, Rs, min, that yields a signal power equal to noise (or signal-to-noise ratio SNR = 1): S-
1
(5.19) Rs, rain" To derive the sensitivity of an OCT system, we calculate the signal and noise currents in the photodetector of the OCT interferometer according to eq. (3.2). With the optical powers Ps and PR of signal and reference beam respectively, incident on the photodetectors, the peak effective signal photocurrent is is:
is = -~rlqex/~ v/Psp R = a x/~ v/PspR.
(5.20)
Noise sources are expressed in terms of mean square photocurrent fluctuations (Aft). The main noise sources are shot noise (Ai~h), excess intensity noise (Ai2x), and receiver noise (Air2e). Receiver noise includes both the thermal and the amplifier noise; for a commercial photoreceiver it can be calculated from the manufacturer's specifications, otherwise it can be modeled by thermal noise of a resistance-limited receiver. The random arrival of photons from a broad-band light source, such as a superluminescent diode (SLD) or a solid-state femtosecond pulse laser, is described by Bose-Einstein statistics (Hodara [1965], Morkel, Laming and Payne [1990]). The resulting mean square photocurrent fluctuation can be written as (Podoleanu and Jackson [1999], Rollins and Izatt [1999]) B (Aip2) = (Ais2h)+ (A/e2x) = 2qeB (i)+ (1 +/-/2) (i)2 Aveff'
(5.21)
where B is the electrical bandwidth, H is the source polarization degree, (Ai) is the mean detector photocurrent, and A vef~ is the effective optical linewidth of the source. The first term on the right-hand side of eq. (5.21) represents the shot noise (Ais2h) which would also be present in case of a monochromatic source (a laser), whose photon statistics is described by a Poisson process. The last term on the fight-hand side of eq. (5.21)is the excess photon noise (Aie2x) caused by beating of the various Fourier components within the effective spectrum Avert (Morkel, Laming and Payne [1990]). The SNR can now be defined as the ratio of the mean square signal photocurrent and the total mean square photocurrent fluctuation: SNR =
2a2PspR (Ai2h) + (Ai2x> + (Ai2e)
(5.22)
For an OCT device using a Michelson interferometer configuration, we have
PR =PoRR/4 and Ps =PoRs/4, where P0 is the source optical output power,
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[4, w 5
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RR is the reflectivity of the reference mirror, and Rs is the sample backscattering coefficient. Furthermore, we model the receiver noise as thermal noise of a resistance-limited receiver: /Air2e) = 4kBTB/pL, where kB is Boltzmann's constant, T is the absolute temperature, and PL is the load resistor (amplifier noise can be incorporated into the thermal noise current by replacing PL by an effective load resistor). From the definition of sensitivity S, and assuming that the mean detector photocurrent (i) is determined by the reference intensity (i.e., we neglect intensity contributions backscattered from the sample or backreflected from optical components in the sample arm), we have:
a2 S-8'~ e + l -
p2RR "~otPoRRlA+ H]/eft2 )
4
1
(5.23)
4kBT-B'pL
-t- ~
Equation (5.23) has three different regimes" Receiver noise dominates if 4kBT
PL
(1
1+n2)
>> ~ otPoRR 2qe + ~ aPoRR Aveff
"
(5.24)
This is the case if the incident light power at the detector is low, either because of low source output power or because of low reflectivity of the reference mirror, RR, or attenuation of the reference beam. With the use of low-noise receivers and sufficient source power, this regime can usually be avoided. In case of very high light power at the detector (i.e., the factor PoRR is high), excess photon noise dominates and the sensitivity will be S=
2
RR(1 +/-/2)
B
.
(5.25)
Avef~
In this case, the sensitivity cannot be improved by increasing the source power. Instead, an attenuation of the reference beam can be helpful. In the intermediate regime, shot noise dominates and the sensitivity will be S -
a Po 1 77 Po . 4 qe B 4hv B
(5.26)
Within this shot-noise dominated range, S depends linearly on source output power. This is the regime usually strived for in OCT system design and can be regarded as optimum, since at lower power receiver noise would limit sensitivity,
Light sources and delay lines
4, w 6] 130 -
I
125 -
~ --
Stotal --
--
Sreceive
r
120 -". "" . 9 .~ . . . . . . . Sshot 115 "" ,,. - " - " Sexcess 110
281
""-,.
"
9
~
i
1 1
-" i
,,
, ,,-'"" 9
i
105
._>
lOO
.....................
~..--~.:.
........................
r-
@
95 90 .
'
/
',
85 8O _
lO
I
-....
I 3 ........ i~ ........ i, ........ '
1o
lOR e f e r e n c e reflectivity
10-
Fig. 5.1. Sensitivity of an OCT system as a function of reference arm reflectivity. The total sensitivity, Stota I as given by eq. (5.23), and the sensitivities corresponding to the individual noise sources: Sreceiver (receiver noise), Sshot (shot noise), Sexcess (excess noise), are indicated. Parameters typical for OCT imaging of the retina were assumed: P0 = 1.5 mW; ~. = 0.83 gm; AA = 25 nm; 77 = 0.8; H = 1; receiver noise current 0.5 pA/v/-H-z; B = 100 kHz.
and at higher power no additional sensitivity can be gained because of excess noise. In the shot-noise limited regime, sensitivities of S = 1011 and higher have been reported. Figure 5.1 illustrates the sensitivity of an OCT system as a function of reference reflectivity (eq. 5.23) for typical parameters of retinal OCT. A similar plot, employing parameters for OCT of scattering tissues, can be found in the paper by Rollins and Izatt [1999]. We have restricted the discussion on sensitivity and signal-to-noise ratio in this paragraph to Michelson interferometer configuration. More detailed discussions on other setups, including Mach-Zehnder schemes, dual balanced detection, optical circulators, and special considerations of problems related to internal reflections in fiber-optic setups have been presented by Podoleanu and Jackson [ 1999], Bouma and Tearney [1999], Rollins and Izatt [1999] and Podoleanu [2000]. These papers demonstrate that, at least to a certain extent, sensitivity can be improved by optimal exploitation of the light power backscattered by the sample, and how excess photon noise can be reduced by dual balanced detection. Note that the final OCT image of a scattering object will contain speckle noise too.
w 6. Light sources and delay lines There are some basic criteria for light sources in OCT, like wavelength, space coherence, time coherence, spectral shape, power, and stability. The ideal OCT light source would yield radiation with perfect space coherence but with time coherence approaching zero. This radiation must only contain longitudinal
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[4, w6
radiation modes. This requirement is not generally met by low-coherence lasers. Thermal light sources yield extremely low-time-coherence radiation but suffer from low space-coherence and a corresponding low power (a few ~tW) per coherence volume. At present the most popular light sources in OCT are superluminescent semiconductor diodes (SLD) and amplified spontaneous emission (ASE) sources. New light sources with potential application in LCI and OCT are currently under development. This paragraph presents a short survey.
6.1. Wavelength, spectral width, and shape of the power spectrum 6.1.1. Wavelength The optimal choice of wavelength depends on the application. For example, biological tissues are strongly scattering and absorbing, and the penetration depth of optical radiation is small. Only at the spectral region from about 600 nm to 1300 nm are scattering and absorption small enough to form the socalled "therapeutic window" (fig. 6.1), which can also be used for LCI and OCT. With increasing wavelength, the scattering of light in tissue decreases more or less monotonically in the visible and near-infrared range. Therefore, OCT probing depth in tissue tends to increase with increasing wavelength (see w3.3 for additional details). Figure 6.2 demonstrates the different penetration depths of two different wavelengths (830 nm and 1280 nm) in human dental tissue in vitro. Note the two-fold larger penetration depth at the larger wavelength. 106
~ o r 104. v
=o 103 -
~
102.1
10. 1
~
I,_
|
i-
p
N. o*
,
10-1. 10 -2 0.1
.2
.3
.5 .7 1 2 3 5 Wavelength (#m)
7 10
Fig. 6.1. Spectral absorption of some tissue molecules, and of aortic tissue (dashed line; postmortem). H20 & Aorta: linear absorption coefficient (1 cm-1). HbO2& Melanin: molar extinction coefficient (103 mol-] cm-1). Adapted from Boulnois [1986] and Berlien and Mtiller [1989].
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283
Fig. 6.2. OCT scattering potential images of human dental tissue at the dento-enamel junction: (a) SLD: ~, = 830nm, A~, = 20nm; (b) SLD: ~. = 1280nm, AA, = 34 nm. The magnitude of the scattering potential is logarithmically encoded in false colours. Depth dimensions are in terms of optical path length. Refractive indices are approximately 1.6 for enamel and 1.5 for dentin. Note the extension of the visibility of the dento--enamel junction. (Reproduced from Fercher and Hitzenberger [1999] with permission from Springer Verlag).
6.1.2. Spectral width The spectral w i d t h o f the O C T light source d e t e r m i n e s s p e c t r o m e t r i c r e s o l u t i o n as well as d e p t h resolution, see eqs. (4.18) and (5.1). H i g h r e s o l u t i o n is a v e r y c h a l l e n g i n g objective in all fields o f optical i m a g i n g . D e p t h r e s o l u t i o n in O C T is defined by the s o u r c e p o w e r s p e c t r u m . Figure 6.3 p r e s e n t s a c o m p a r i s o n o f O C T d e p t h r e s o l u t i o n u s i n g a s u p e r l u m i n e s c e n t d i o d e and a state-of-the-art Kerr-
Fig. 6.3. Topographical mapping of retinal layers along ~8.5 mm of the papillomacular axis. The logarithm of the LCI signal is represented on a false-color scale. (a) SLD: mean wavelength ~, = 843 nm, A~, = 30nm, depth resolution 10gm. (b) Ti:AI20 3 laser: mean wavelength ~, = 800nm, AA = 260 nm, depth resolution 3 ~tm. Reproduced from Drexler, Morgner, Ghanta, K~irtner, Schuman and Fujimoto [2001] (courtesy J.G. Fujimoto, M.I.T.) by permission of the Optical Society of America.
Optical coherence tomography
284
[4, w 6
0.8-
c:3
i
0.8
~
,nil .dJu,!
_~ 0,6 _c 0.4 ~ 0.2 t5
~o
o
o
778
[a)
r =
780 Wavel~'~;~h
-3
782 nm
I
1
0.8
0.8
i
0.6 ~ 0.4
0
0.6
18 n m
0 798
808
[b]
818 828 Wave~mgth
838
-o.3 -~.2
848 n m
0.8 0.8
,..
0,6
o.6
L
"6 0.4
260 nm
0,4
t5 0.2 60O
~
~.;
~.2
I
!.s ~___,
1 ,
\
I
0
-b.]
3
Opflcol path difference (In ram]
I
i
-2 -I 0 I 2 Optical path difference[Inmm]
!
680
[c}.
76O
840
920
-15
1000 nm
wc~,~,~
-10 -5 0 5 10 Optical path difference [In lxm)
1 -
1
/
0.8
0.6
\
9 0.8 8
~o
11am
'~ 0 , 4 -
510 nm
~ 0.4
0.6-
~ 0.2o.
0 300
460
620
780
940
1100 n m
-15
--J
~--
-~0 -5 0 5 I'0 Optical path dlfference [In ixm]
Fig. 6.4. (left) Optical output spectrum and (fight) LCI signal envelope of (a) a multimode laser diode (S/--satellites), (b) a superluminescent diode, (c) a Kerr-lens mode-locked Ti:sapphire laser (redrawn from Drexler, Morgner, K~irtner, Pitris, Boppart, Li, Ippen and Fujimoto [1999]), and (d) a tungsten halogen lamp (note: effective spectral width of 320 nm due to limited photodetector spectral sensitivity). Note different abscissa scales.
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285
Light sources and dela)' lines
lens mode-locked Ti:sapphire laser. Note, for example, the clear boundary of the nerve fiber layer in fig. 6.3b. The thickness of this layer is an important indicator of early glaucomatous changes in the eye.
6.1.3. Shape o f the p o w e r spectrum
Another critical spectral parameter besides wavelength and spectral width is the shape of the source power spectrum. As shown in w3, FSource(r) is the impulse response of the LCI interferometer. Its real part plays the role of a depth point-spread function, see eq. (3.21). This function, therefore, must quickly die away with increasing distance from its center and must not have satellites. Since FSource(r) equals the inverse Fourier transform of the power spectrum the latter must have a smooth shape. Figure 6.4 presents a comparison of spectra and corresponding depth point-spread functions of a multimode diode laser, a superluminescent diode, a state-of-the-art Kerr-lens mode-locked Ti:sapphire laser, and a thermal lamp. As can be seen from fig. 6.4, the multimode laser LCI signal envelope suffers from strong satellites (Si) in the coherence function which will generate multiple image structures. These parasitic images will not only disturb the general appearance of OCT images but can hide weak object structures causing blindness. Such spurious images are absent when using smooth spectra like those of SLDs. On the other hand, as can be seen from eq. (5.23) power is an important quantity in OCT (see table 6.1). At present, both requirements, high power and large spectral width, can best be fulfilled by mode-locked solid-state lasers, even at the expense of some satellites, see fig. 6.4c. Table 6.1 Examples of low-time-coherence light sources Light Source
~. (~tm)
A/~ (nm)
SLD
0.8
60
Kerr-lens mode-locked Ti:sapphire laser
0.81
LED
0.840
IC
(~tm)
Power (mW)
4.7
10
260
1.1
400
20
15.6
40-80 ~15
0.1
Amplified spontaneous emission fiber
1.300, 1.550
Photonic crystal fiber
0.400-1.4
370
1
50 6
Thermal tungsten halogen
0.880
320
1.1
0.2 ~tW
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6.2. Low-time-coherence light sources
Except wavelength tuning OCT, all OCT techniques use low-time-coherence light sources. Most techniques depend on the space coherence being high too. Usually, low-time-coherence light sources exhibit low space coherence. Therefore, OCT needs unusual light sources. A short survey is given here.
6.2.1. Single sources
The first light sources used in LCI were multimode laser diodes utilized by Fercher and Roth [1986] for measuring the optical eye length. The performance of these sources is limited by satellites, see fig. 6.4a. Attempts to obtain highspace-coherence but low-time-coherence light with a smooth spectral distribution were based on emission at forward bias current below threshold. This technique has recently been revived by Zvyagin, Garcia-Webb and Sampson [2001], who reported on a high-power semiconductor laser operated below threshold as a high-brightness source of low-time-coherence light, achieving an output power of 1.3 mW and a coherence length of 16 ~tm. Since the time cheap superluminescent diodes at 0.8 ~tm wavelength became available around 1991, these light sources have dominated LCI and OCT. At present SLDs span the wavelength range from about 675 nm to 1600nm, have output powers up to 50mW and spectral widths up to 50 nm. Basing on a tandem structure, Semenov, Batovrin, Garmash, Shidlovsky, Shramenko and Yakubovich [1995] have developed broad-bandwidth SLDs with a spectral width of A~ = 98nm at a mean wavelength of = 820 nm. Another light source that has been used in LCI and OCT at a very early stage is the LED. LED sources of high-space-coherence but low-time-coherence light have been developed for LCI applications in fiber technology. Derickson, Beck, Bagwell, Braun, Fouquet, Forrest, Ludowise, Perez, Ranaganath, Trott and Sloan [1995] reported on a LED with reduced internal reflections operating at 1300 nm and 1550 nm, allowing high-sensitivity LCI without spurious responses. These light sources provide beam powers in the range of some tens of ~tW. Clivaz, Marquis-Weible, Salath~, Novak and Gilgen [1992] used an edgeemitting LED at ~ = 1300 nm with a bandwidth of A~ = 60 nm in reflectometry, and Schmitt, Lee and Yung [1997] used edge-emitting LEDs operating at 1240nm and 1300nm in an optical coherence microscope. Spatially coherent fluorescent light sources have been investigated by several groups. Clivaz, Marquis-Weible and Salath~ [1993] used fluorescence light of a Ti-sapphire crystal pumped by an Ar laser. An extremely broad spectrum of
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A/I = 144 nm was obtained at ~. = 780 nm, yielding a depth resolution of 1.5 ~tm in tissue; unfortunately the power was limited to 2 ~tW. Recently, Kowalevicz, Ko, Hartl, Fujimoto, Pollnau and Salath6 [2002] reported the generation of single-spatial-mode fluorescence with 40gW power and a bandwidth of 138 nm at a central wavelength of 761 nm using a thin superluminescent Ti:A1203 crystal pumped by a 5 W cw diode laser. Liu, Cheng and Wang [1993] report on a point fluorescence source of 1 ~tm diameter in a Rhodamine 590 jet pumped by an argon-ion laser. 9mW power at A2 = 44 nm were achieved; the degree of spatial coherence was 0.97. Presently one of the most promising light sources for high-resolution and high-dynamic-range OCT is the Kerr-lens mode-locked Ti:sapphire laser (see table 6.1). Based on this technology, Bouma, Tearney, Boppart, Hee, Brezinski and Fujimoto [1995] have already demonstrated a high-resolution fiber-optic OCT system with lc < 2 gm. Recently, Drexler, Morgner, Ghanta, K/irtner, Schuman and Fujimoto [2001] have used a similar system to generate ultrahighresolution ophthalmic OCT images (see fig. 6.3). ASE (Amplified Spontaneous Emission) sources are fibers doped with different elements which can yield transverse single-mode fluorescent light several orders of magnitude stronger than a typical thermal light source. These light sources are pumped by semiconductor diode lasers and are available for different wavelengths and large spectral widths. ASE sources based on rare-earth-doped fibers, for example, can give a continuous output spectrum from 1250 nm to 1625 nm. Tm-doped fluoride fibers produce output spectra from 1440 nm to 1500 nm. A superfluorescent Yb-doped high-power fiber ASE source was developed by Bashkansky, Duncan, Goldberg, Koplow and Reintjes [1998]. This source operates at ~. = 1.064 gm, A/~ ~ 30 nm, and yields beam powers of some tens of mW. Photonic crystal fibers, in which light is guided by a bandgap created by a periodic arrangement of air holes surrounding a solid silica core (Ranka, Windeler and Stentz [2000]), and tapered fibers (Birks, Wadsworth and Russell [2000]), can provide supercontinuum light of high space coherence. Hartl, Li, Chudoba, Ghanta, Ko, Fujimoto, Ranka and Windeler [2001] have used a 1 m crystal fiber source operating with 100 fs input pulses generated by a Kerrlens mode-locked Ti:sapphire laser pumped by a frequency-doubled Nd:vanate laser, and obtained 6 mW output power. Ultrahigh resolution OCT has been demonstrated with a depth resolution of approximately 2 ~tm in tissue at 1.3 ~tm mean wavelength and 370nm bandwidth. Finally, there are OCT techniques which not only tolerate but even benefit from low space coherence. One of these techniques is the adaptation of OCT to
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interference microscopy, as described by Beaurepaire, Boccara, Lebec, Blanchot and Saint-Jalmes [1998] and Dubois, Boccara and Lebec [1999]. To reduce speckle noise a spatially low coherent light source is useful, and to enable multiplexed lock-in detection high-frequency modulation must be possible. Therefore driving-voltage modulated LEDs (~ = 840rim, A~ = 20nm) have been used in these techniques. Thermal light sources suffer from low coherent power. The light energy flux q~c emitted into the solid coherence angle can be shown to be q~c = 0.307L~ 2, where L is the radiance and ~ is the mean wavelength of the source. For example, with a halogen lamp a power of approximately 0. l~tW has been obtained per radiation mode. Nevertheless, there are two situations which benefit from thermal light sources. If transverse resolution can be relaxed, several coherent channels can be combined to a probing beam with reasonable power and extremely high depth resolution. A corresponding technique has been described by Fercher, Hitzenberger, Sticker, Moreno-Barriuso, Leitgeb, Drexler and Sattmann [2000]. Furthermore, if high-numerical-aperture beams can be used, as in microscopy, reasonable power is available. Vabre, Dubois and Boccara [2002] have built a high-resolution optical coherence microscope based on parallel OCT using a CCD camera and a tungsten halogen lamp. Table 6.1 shows some striking examples of wavelengths, spectral widths, coherence lengths, and coherent powers of light sources used in OCT.
6.2.2. Synthesized sources and coherence function shaping The basic idea of synthesized sources is to cover a larger bandwidth by combining several light sources with adjacent spectral bands. The width of the composite autocorrelation function of a synthesized source can be narrower than the single autocorrelation functions. An early work in this field was by Rao, Ning and Jackson [1993], who combined two multimode laser diodes with a wavelength difference of 108 nm. Since the two spectra (A~ ~_ 8 nm) in this work were not adjacent to each other, the autocorrelation function was channeled; this improves central fringe identification but cannot be used as point-spread function in OCT. Schmitt, Lee and Yung [1997] combined two edge-emitting LEDs with peak wavelengths of 1240nm and 1300nm to a composite light source, and obtained an improvement of the measured autocorrelation widths from 10.8 ~tm with only one LED to 7.2 ~tm with both LEDs in a tissue phantom. Baumgartner, Hitzenberger, Sattmann, Drexler and Fercher [1998] used two superluminescent diodes spectrally displaced by 25 nm with an effective bandwidth of 50nm, and obtained a depth resolution
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289
of 6 to 7~tm at the retina of a human eye in vivo. Recently, Yan, Sato and Tanno [2001a,b] presented an optimization algorithm to minimize the coherence length of a synthesized light source. Using three LEDs with a mean coherence length of 13.4~tm yielded a synthesized autocorrelation function with a coherence length of 8gm, and sidelobes below 9% of the main peak. Synthesized light sources and many broad-bandwidth sources suffer from spectral ripple. Unfortunately, most new sources too provide non-Gaussian spectra which will generate sidelobes in the envelope of the coherence function and, therefore, generate spurious structures in OCT images. Some problems with non-Gaussian spectral structures of synthesized light sources, and ways to solve these, have been discussed by Bashkansky, Duncan, Reintjes and Battle [1998]. Spectral shaping or coherence-function shaping is used to approximate a Gaussian spectrum. There are two basic possibilities to generate Gaussian-shaped source spectra. Time-domain spectral shaping, for example, uses large amounts of groupvelocity dispersion to temporally stretch a short pulse. Chou, Haus and Brennan III [2000] describe a related technique where a fiber Bragg grating is used to stretch the pulse. A circulator routes the stretched pulses to a LiNbO3 amplitude modulator driven by a 1 GHz arbitrary waveform generator. Deep spectral ripples, due to birefringence in the fiber Bragg grating and the polarizing waveguide in the amplitude modulator have been removed from the output spectrum. Frequency-domain spectral shaping can be performed by spatially displaying the optical spectrum and modulating the amplitudes and/or phases of the frequency components. Techniques like spectral-hole burning and liquid-crystal spatial light modulation can be used. Weiner, Heritage and Kirschner [ 1988] used a grating apparatus and modulated the spectrum by spatially patterned amplitude and phase masks. Hillegas, Tull, Goswami, Strickland and Warren [ 1994] used an acousto-optic modulator to obtain flexible pulse shaping. Weiner, Leaird, Patel and Wullert [1990] used a programmable multi-element liquid-crystal phase modulator; Teramura, K. Suzuki, M. Suzuki and Kannari (1999] too describe the synthesis of a smooth coherence function by modulating the phase of the spectral light components with a programmable liquid-crystal phase modulator. They furthermore demonstrate linear shifting of the coherence function and image detection of selected planes of a multilayer object without mechanical scanning. He and Hotate [1999] describe time-sequential autocorrelation-function synthesis using a broadband tunable laser diode. By stepwise tuning the frequency and synchronous phase modulation a synthesized coherence function is scanned
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through the sample depth. Depth resolution was 475 ~tm, the detection range was 12.2 mm, both measured in air.
6.2.3. Tunable lasers Indium-gallium based semiconductor lasers generate tunable (transversal and longitudinal) single-mode laser light with reasonable power (some tens of mW) from the visible range to the mid-infrared. There are two approaches to tunability: external cavity lasers and distributed Bragg reflection systems. In external cavity lasers the cavity does the tuning, for example by a rotating grating. Tuning range is about 80 nm, tuning speed around 10 nm/s. Distributed feedback (DFB) lasers achieve tuning by a Bragg reflector distributed within the active layer; in distributed Bragg reflector (DBR) lasers tuning is done by a distributed Bragg reflector beyond the active layer. Tuning is performed by the input current; in comparison with external cavity lasers, tuning speed can be higher, for example 1000nm/s, but tuning range is smaller, for example 2 nm. Lexer, Hitzenberger, Fercher and Kulhavy [1997] have used wavelength-tuning interferometry to measure intraocular distances. Chinn, Swanson and Fujimoto [1997] generated OCT images of a glass microslip sandwich structure using tunable lasers. Haberland, Blazek and Schmitt [1998] used tunable lasers to generate OCT images of layered scattering structures.
6.3. Optical delay lines In contrast to Fourier-domain LCI and OCT, time-domain LCI and OCT need a depth scan that is performed with the help of optical delay lines. Optical delay lines shift the coherence gate throughout the object depth. Most delay lines also provide the heterodyne carrier frequency. Delay lines are a deciding factor in terms of signal-to-noise ratio and data acquisition rate. The signal-to-noise ratio in the band-pass filter was shown to be inversely proportional to the bandwidth B of the photoelectric signal in eq. (5.23). The lower limit of B is determined by the stability of the carrier frequency, the bandwidth of the sample beam, and the speed of the delay line. Therefore, there is a high demand for good delay-line linearity. Another important figure determining the performance of an OCT system is the duty cycle of the delay line, defined as the duration of available scan time related to the time period of subsequent depth scans. Low duty cycles increase the data acquisition time.
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291
Light soulves and delay lines
6.3.1. Propagation of the coherence function A shift of the coherence gate can be achieved either by free propagation or by modulating the phases of the monochromatic components of the beam. This can be seen from the Wiener-Khintchine theorem, which shows that a time shift of the mutual coherence function F ( r - A r ) = (V(t + r - A t )
V ,(t)) =
fi( )
exp [-itor + iAq~(to)] do)
,y
(6.1) with AqS(6o) = o a r
(6.2)
can either be performed by directly introducing a time delay A r between the two interferometer beams or by a linear phase ramp Aq~(o)) in one of the beams. A time delay A r between the interferometer beams can, of course, be generated simply by propagating one of the beams in free space and/or in a refractive medium. Free propagation of a beam along a distance Az in space introduces a phase shift q~(co) to its monochromatic components: Aq)(co) = kAz = ~Az. c
(6.3)
Alternatively, a corresponding phase shift can also be introduced by a dispersing device. In the past, optical delay lines have already been used in interferometric spectroscopy, in optical low-coherence reflectometry (OLCR), in heterodyne interferometry, and as diagnostic tools for mode-locked laser pulse-width monitoring. Besides devices modulating the optical path length of one of the interferometer beams several techniques based on modifying the spectral components of this beam have been developed in the field of femtosecond pulse shaping (Weiner, Heritage and Kirschner [1988]).
6.3.2. Optical delay by path length modulation An additional optical path Az can be introduced by either increasing the geometrical path of a beam or/and by increasing its optical path length. Figure 6.5 shows examples for the first case, fig. 6.6 represents examples of the second case. Linear scanning stages are most often used in experimental work because of their flexibility. Different modes of movement and different acceleration phases can be realized with scanning stages operated by stepper motors. These
Optical coherence tomography
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[4, w 6
% lI> (a)
(b) +
Fig. 6.5. Geometrical path modulation. (a) Motorized linear translation stage. (b) Rotating or oscillating retro-reflector with two rooftop prisms 90 ~ twisted against each other.
(a) Fig. 6.6. Optical (and geometrical) path modulation. (a) Shifting prism. (b) Rotating cube. Both delay lines generate variable second-order dispersion.
devices can also achieve high duty cycles. High-speed linear translation stages have been implemented with loudspeaker cones (Sala, Kenney-Wallace and Hall [1980]) and reel-driven stages (Edelstein, Romney and Scheuermann [1991]). The oscillating retro-reflector has been used in the first commercial OCT instrument. The devices shown in figs. 6.5b and 6.6a,b can realize linearities in the + 1% range with duty cycles of some 10%. However, the scan ranges can be quite different. Whereas the devices depicted in fig. 6.6 achieve scan ranges in the order of some mm, the device depicted in fig. 6.5b c a n - with one additional retroreflector- achieve scan ranges in the order of some 100 mm. The linear scanning stage seems the most straightforward implementation of a path-length delay line, see fig. 6.5a. Standard motorized translation stages provide tens of mm/s of speed and tens of mm linear delay. There are many solutions for faster delay scanning; to achieve tens of mm linear delay and a high duty cycle is more difficult. Fork and Beisser [1978] describe the use of a corner cube mounted on an oscillating armature of a commercial shaker assembly. This device was driven by a signal generator at 30 Hz with a maximum displacement of 5 mm. A delay line with an extended range of 2 m for OLCR based on two retroreflectors that multiply bounce an incident light beam has been described by Takada, Yamada, Hibino and Mitachi [ 1995]. As mentioned above, delay lines based on circulating or oscillating rooftop prisms have been used in OCT (Swanson, Izatt, Hee, Huang, Lin, Schuman, Puliafito and Fujimoto
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[ 1993]). Harde and Burggraf [ 1981 ] describe a device based on a rooftop rotating with 30 Hz, providing a path-length modulation amplitude of 7.5 cm. A slightly modified version presented by Guan, Lambsdorff, Kuhl and Wu [ 1988] provided 42 cm at a repetition rate of 50 Hz (since these devices have been used for visual observation of the autocorrelation function of mode-locked pulses there was no need to further increase the repetition rate). A rotating-mirror delay line has been investigated for interferometric high-speed Fourier transform spectroscopy (Campbell, Krug, Falconer, Robinson and Tait [1981]). Two parallel mirrors are arranged on a rotating platform. The exiting beam oscillating in transversal direction is reflected back by a stationary mirror. Repetition rates of 100 Hz have been used. An advantage of the parallel mirror configuration is its insensitivity to vibration (Yasa and Amer [ 1981 ]). Delay lines based on rotating cubes have achieved a scanning speed of 21 m/s in air at repetition rates of 384 Hz. Using a circular array of tilted mirrors a 2 mm scanning range at a repetition rate of 2400 Hz and 94% duty cycle has been demonstrated by Chen and Zhu [2002]. An air-turbine driven rotating cube has achieved a 28.6kHz repetition rate over a range of 2 mm but suffered from extreme noise and wobbling (Szydlo, Delachenal, Gianotti, W~ilti, Bleuler and Salath6 [1998]). The main disadvantage of these delay lines is that masses have to be moved with high speed. All high-speed rotating devices suffer from wobbling of the axis of rotation due to free play in bearings and imbalance of the whole rotating assembly, leading to mechanical vibrations and reducing the signal-to-noise ratio. Therefore, there have been several attempts to avoid the moving of masses. One attempt uses piezo-actuated fiber stretchers (Tearney, Bouma, Boppart, Golubovic, Swanson and Fujimoto [1996]). These devices have achieved scan rates up to 600 Hz, but are not temperature stable, require matching of the fiber dispersion and extra compensation of bending-induced birefringence. Another non-moving delay line has been proposed on the basis of an acousto-optic Bragg cell device (Riza and Yaqoob [2000]). There are also delay lines using a discrete series of increasing delays (Ishida, Yajima and Tanaka [ 1980]) and delay lines which enable the selection of arbitrary delays from a set of discrete delays (Yamaguchi and Hirabayashi [1995], Riza [1998]).
6.3.3. Optical delay by phase modulation Here a linear phase ramp A q~(o9) is introduced to the monochromatic components using a grating, as depicted in fig. 6.7. A corresponding high-speed delay line has been developed by Kwong, Yankelevich, Chu, Heritage and Dienes
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Fig. 6.7. Rapid scanning optical delay line. The mean frequency ~o0(Q = 0) is assumed to propagate along the optical axis. The linear phase ramp is generated by the path length change caused by the lateral offset of the beam components when reflected at the grating.
[1993] and introduced to OCT by Tearney, Bouma and Fujimoto [1997]. A related technique has been described by Zeylikovich, Gilerson and Alfano [1998]. The instrument depicted in fig. 6.7 uses a grating at one focal plane of a lens and a tilting mirror at the other focal plane. The grating spreads the spectrum of the source across the galvanometer-mounted mirror. Tilting the mirror introduces an optical path delay that varies linearly with the frequency. This approach is known as rapid-scanning optical delay (RSOD). The grating diffracts the spectral component g2 = to-tOo into the angle A0 = A)~/(p cos 00), where p is the grating period and A0 = 0 - 00 is the difference between the diffraction angle of the frequency components tO and tOo. The group time delay is:
rg -
O0(f2)
4~f o
c9f2
coop cos 00'
(6.4)
which can be considered as the fundamental equation for RSOD design (Kwong, Yankelevich, Chu, Heritage and Dienes [1993]). With a focal length o f f = 100mm, a mirror tilt of o = 10-2rad, and a grating constant of p = 1/1800mm a modulation depth of approximately 4 m m can be obtained at a wavelength of ~ = 800 nm. The RSOD has the advantage that minute mechanical movements of the mirror result in large modulation depths. Rollins, Kulkarni, Yazdanfar, Ungarunyawee and Izatt [1998] have shown that an offset of the mirror pivot by a distance x normal from the optical axis can be used to change the mean frequency without significantly affecting the group path length. These authors have also demonstrated the use of the RSOD to achieve in uiuo video-rate OCT. Images of 500• pixels of a beating Xenopus embryo heart have been obtained at 30 frames per second. A similar technique has been used by Zeylikovich, Gilerson and Alfano [1998] in their depth-lateral
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microscopy technique. A programmable pulse-shaping device using a multielement liquid-crystal phase modulator, which might be used to implement complex depth scan modes, has been described by Weiner, Leaird, Patel and Wullert [ 1990].
6.3.4. Focus tracking Usually, in OCT the coherence gate is depth-shifted while the objective lens focus remains stationary. Hence, lateral resolution is depth dependent. Furthermore, the Rayleigh length is chosen to match the thickness of the object to be imaged. This provides a more or less constant transversal resolution, but requires a large beam waist radius and results in low transversal resolution. Both a large depth of focus and high lateral resolution cannot be achieved simultaneously using standard lenses. Therefore, different focustracking approaches have been developed. For example, the object itself can be moved through the stationary focus of a high-numerical-aperture lens to perform the coherence scan, or a Fresnel lens can be used. Alternatively, different focusing lenses have been used to shift the focus for imaging object structures at different depths (Izatt, Hee, Swanson, Lin, Huang, Schuman, Puliafito and Fujimoto [1994]). It has also been proposed to dynamically move the focusing lens and the reference mirror by properly synchronized drives (Huang, Fujimoto, Puliafito, Lin and Schuman [1992]). Schmitt, Lee and Yung [1997] have used an ingenious scanning technique to obtain a dynamically moving focus. Reference mirror and focusing lens are mounted together on a common translation stage to provide path-length matching. However, rather large masses have to be moved. Lexer, Hitzenberger, Drexler, Molebny, Sattmann, Sticker and Fercher [ 1999] describe an optical setup which dynamically shifts the focus through the object without changing the optical path length in the measurement arm. Therefore, the coherence gate remains at the beam focus without readjustment of the reference arm. An oscillating mirror forms a probing beam focus oscillating along the depth scan and generating the carrier frequency. Since no moving lens is needed, the system is suitable for high-speed imaging. A digital alternative to a moving focus has been presented by Drexler, Morgner, K~irtner, Pitris, Boppart, Li, Ippen and Fujimoto [1999] who used a zone-focusing and image-fusion technique. Separate images recorded with different focal depths of the probing beam were manually segmented and fusioned. Ding, Ren, Zhao, Nelson and Chen [2002] used an axicon lens to focus the probe beam: 10 ~tm resolution over a focusing depth of at least 6 mm has been obtained.
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w 7. Conclusion Optical coherence tomography is a fascinating imaging technology with a huge future potential. N e w developments comprise all components o f OCT technology: N e w light sources, new detectors, new delay lines, new signal generation techniques, new contrasting techniques, new applications. We have outlined basic principles of these technologies. Ophthalmology is still the main field o f OCT application. However, new light sources and detectors will not only widen the field of medical applications but also increase the not yet fully understood potential for technical applications.
Acknowledgements Our own contributions to this work are based on projects (P7300-MED, P9781MED, P10316-MED, and P 1 4 1 0 3 - M E D ) financed by the Austrian Fonds zur F6rderung der wissenschaftlichen Forschung (FWF) and project No. 7428 supported by the Austrian National Bank. We want to thank K.K. Bizheva, W. Drexler, M. Pircher, and R. Leitgeb for communicating their current research results.
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E. Wolf Progress in Optics 44 9 2002 Elsevier Science B. V. All rights reserved
Chapter 5 Modulational instability of electromagnetic waves in inhomogeneous and in discrete media by
Fatkhulla Kh. Abdullaev Physical- Technical Institute, Uzbek A cadeno' of Sciences, G. Mavlyanov so:, 2-b, 700084, Tashkent-84, Uzbekistan
Sergey A. Darmanyan Institute of Spectroscopy, Russian Academy of Sciences, 142190, Troitsk, Moscow Region, Russia
and
Josselin Garnier Laboratoire de Statistique et ProbabilitO, UniuersitO Patti Sabatie~; 118 Route de Narbonne, Toulouse Cedex, France
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Contents
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w 1. Introduction Modulational instability (MI) represents a fundamental subject in the theory of nonlinear waves. This phenomenon was predicted by Benjamin and Feir [ 1967] for waves on deep water and by Bespalov and Talanov [ 1966] for electromagnetic waves in nonlinear media with cubic nonlinearity. The modulational instability phenomenon consists in the instability of nonlinear plane waves against weak long-scale modulations with wavenumbers (frequencies) lower than some critical value. Long time evolution leads to the growth of sidebands and a periodic exchange of energy between a pump and sidebands during the wave propagation. At present, MI is observed in nonlinear optics, plasma physics, condensed matter physics (fibers, magnetics, Bose-Einstein condensates, long Josephson junctions, etc.). This phenomenon is of great interest both for the general theory of nonlinear waves and for applications. MI exists due to the interplay between the nonlinearity and dispersion/ diffraction effects. Important models for investigating MI of electromagnetic waves in nonlinear media represent the scalar and vectorial nonlinear Schr6dinger (NLS) equations, the system describing evolution of the envelopes of fundamental and second harmonics waves in quadratically nonlinear media, sine-Gordon equation and others. In nonlinear optics the MI is of fundamental importance for a formation of both temporal and spatial solitons. MI of the nonlinear continuous-wave (cw) solution of the scalar and vector nonlinear Schr6dinger equations has attracted a considerable deal of interest in past years (for overviews see, e.g., books by Agrawal [ 1995] and Abdullaev, Darmanyan and Khabibullaev [1993]). It has been demonstrated that MI is among the major factors limiting the transmission capacity of long-haul optical communication systems. On the other hand, MI can be used to generate chains of short optical pulses for high-bit-rate data transmission (Hasegawa [1984], Millot, Seve, Wabnitz and Haelterman [1995]). Much attention has been devoted to investigations of MI in the framework of the NLS equation. The initial stage of the instability can be explored by linear stability analysis, which shows the exponential growth of sidebands. The longtime behavior was investigated by numerical simulations; this approach involves 305
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[5, w 1
truncation of a finite number of modes and finding exact periodic solutions of the NLS equation. Analysis of the long time evolution show that the initial exponential growth of the amplitude of modulations is followed by a decrease, and this process is periodic in time. This phenomenon is analogous to the well-known Fermi-PastaUlam (FPU) recurrence, observed in a chain of coupled nonlinear oscillators. The exact solution of the MI problem expressed via elementary functions has been given by Akhmediev and Korneev [1986] and Its, Rybin and Sail [1988]; an exact solution in terms of elliptic functions using finite zone potentials theory is by Tracy and Chen [1984]. Recently the FPU recurrence has been observed experimentally in optical fibers by Van Simaeys, Emplit and Haelterman [2001 ]. According to the spatio-temporal analogy, these results can be also applied to beam propagation in bulk nonlinear media and waveguides. However, all of the above investigations were concerned with MI in homogeneous media. Real systems have different kinds of inhomogeneities, which can strongly affect the process of MI, introducing new phenomena (Malomed [ 1993]). For example, in optical fibers, in addition to a constant dispersion and nonlinearity, there are effects such as a periodic amplification of pulse and dispersion profiling (or a dispersion compensation) as well as stochastic fluctuations of the dispersion and nonlinearity (for reviews see, e.g., Hasegawa and Kodama [1995], Wabnitz, Kodama and Aceves [1995] and Abdullaev [1999]). In nonlinear photonic crystals the periodic inhomogeneity of linear and nonlinear medium parameters is an intrinsic property. Finally, nonlinear discrete optical systems are also examples of inhomogeneous media. All of these observations motivate the investigation of MI processes in inhomogeneous nonlinear media. In nonlinear fiber optics the influence of the periodic and random amplification on MI has been studied by Matera, Mecozzi, Romagnoli and Settembre [1993] and Abdullaev [1994]. It was shown that new domains of MI appear in this case; parametric resonance of the unstable mode with the modulation period turned out to be the underlying mechanism. Optical fibers with variable dispersion represent another example of fiber inhomogeneities. Existing longhaul transmission systems frequently consist of different fiber pieces, where the dispersion varies randomly around an averaged value from piece to piece. It has been suggested to use fibers with a controlled dispersion profile (called dispersion-managed fibers) to improve the transmission characteristics of solitons (Smith, Knox, Doran, Blow and Bennion [1996], Gabitov and Turitsyn [1997]). Besides the reduction of soliton jitter and deteriorating effects which can be caused by a periodic amplification, dispersion management may also lead to a decrease of the MI domain as well as MI gain in this domain.
5, w 1]
Introduction
307
Analogous phenomena occur in fibers with periodic birefringence. This case is described by a vector NLSE with periodic coefficients. The process of MI in random media represents a separate interest. This problem has been considered recently by Karlsson [1998], Abdullaev [1994], Abdullaev, Darmanyan, Kobyakov and Lederer [1996], Abdullaev, Darmanyan, Bischoff and Sorensen [1997] and Abdullaev and Garnier [1999] for fibers with random amplification, dispersion and birefringence. It should be noted that the problems mentioned above involve inhomogeneities along the evolutional variable - i.e. the coordinate. Another type of problem occurs in nonlinear optical media with periodic modulations of parameters along a spatial variable which is not evolutional. A typical example is the dynamics of an electromagnetic wave in a nonlinear optical medium that exhibits periodic variation of linear parameters at a scale comparable to the wavelength of the propagating wave. In this case the inhomogeneities induce the appearance of a reflected wave and we have the dynamics of two interacting waves moving in opposite directions, which is described by the so-called coupled-mode theory. For the investigation of MI it is often useful to apply coupled-mode theory in combination with linear stability analysis (see recent works by De Sterke [ 1998] and Litchinister, McKinstrie, De Sterke and Agrawal [2001]). This set of problems also includes the phenomena in nonlinear discrete optical systems, such as arrays of planar waveguides and fibers (see, e.g., the review by Lederer, Darmanyan and Kobyakov [2001]). In such systems, the spreading of the initial excitation due to linear coupling, which can be viewed as effective discrete diffraction, can be compensated by a nonlinearity-induced localization. The array of planar waveguides is described by the discrete NLS equation or by a set of discrete Z t2) equations, while the array of fibers allowing for the temporal dispersion is represented by a 2D-continuous-discrete NLS equation. Modulation instability in these systems has many new features not observed in homogeneous and continuous media, in particular the critical dependence of both the MI gain and the MI domain on the wavenumber related to the discrete variable. Particular interest which motivates the MI research in these systems is connected with the possibility of generating discrete optical solitons using MI. A similar class of phenomena has been predicted to appear in arrays of BoseEinstein condensates (BEC) by Abdullaev, Baizakov, Darmanyan, Konotop and Salerno [2001 ], and in BEC in an optical lattice by Konotop and Salerno [2002]. The structure of this review is as follows. In w2 we give a description of modulational instability in homogeneous nonlinear media which can be used as a basis for reading of the subsequent sections. This description involves mainly the linear stability analysis which is valid for studying the initial stage of the
308
MI in inhomogeneous and discrete media
[5, w 2
evolution of modulations. Methods which admit the description of the long time evolution, such as periodic solutions of the NLS equation and the coupledmode theory with three modes, are discussed in brief. In w3 we discuss MI of electromagnetic waves in optical media with periodic inhomogeneities. We start in w3.1 with MI in optical fibers with periodic amplification, i.e. where the power in the fiber is periodically varied. The structures of sidebands are found and the relation with the sidebands generated by a soliton in such a system is discussed. The analogous problem for a fiber with periodically varying dispersion is considered in w3.2, and application to dispersion-management optical communication systems is discussed. An important case of MI in periodic nonlinear media, occurring in a Bragg-grating optical fiber, is considered in w3.3. Coupled-mode theory is employed in order to describe MI in this case. In w3.4 we consider MI in nonlinear media with periodic potential, while MI in birefringent fibers with periodic dispersion and birefringence is investigated in w167 3.5 and 3.6. MI in periodic quadratic media is the subject of w3.7. In the second part of the review we explore the modulational instability of electromagnetic waves in nonlinear media with random parameters. In w4.1 we discuss the origin of the random fluctuations of parameters in optical fibers and other nonlinear optical media. MI in fibers with random amplification and dispersion is investigated in w167 4.2-4.6. MI in randomly birefringent fibers is discussed in w4.7. The final part of the review (w 5) is devoted to the MI of electromagnetic waves in nonlinear discrete optical systems such as an array of planar waveguides and fibers. Particular cases of MI in discrete media with cubic nonlinearity as well as quadratic nonlinearity are investigated.
w 2. MI in homogeneous media The phenomenon of instability of long-wavelength modulations of nonlinear plane waves in homogeneous nonlinear media has been predicted in the 1960s for nonlinear optics by Bespalov and Talanov [1966], for water waves by Benjamin and Feir [ 1967], and for waves in plasma by Vedenov and Rudakov [1964] and Ostrovski [ 1966]. Later this phenomenon has been observed, among others, for spin waves in magnetics and for waves in nonlinear lattices. Two important cases for future consideration are MI in Kerr-like media and MI in media with quadratic nonlinearities [2 't2t media].
5, w 2]
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309
2.1. M e d i a with c u b i c n o n l i n e a r i t y
Let us consider MI in homogeneous optical media with a cubic nonlinearity. Typical example is a nonlinear optical fiber. The evolution of the wave envelope in the fiber is described by the nonlinear Schr6dinger equation, l 2q; i~p_ + ~fi2~t, + ylv, l = o,
(2.1)
where z is the propagation distance, t is the time in the moving reference frame, ~p is the envelope of the electromagnetic wave,/~ is the second-order dispersion coefficient, and y is the coefficient of the cubic nonlinearity. By introducing dimensionless variables, this equation can be written in the form iu:
+
(2.2)
l du,, + lu[2u - O.
Here d - +1 corresponds to the cases of anomalous and normal dispersion respectively. This equation has a nonlinear plane-wave solution ~Pc of the form lPc = A exp(iA2z),
(2.3)
where without loss of generality A is taken real. Let us investigate the stability of this solution against initial small modulations. If we are interested in the initial stage of the evolution we can apply linear stability analysis. Then we can look for a solution in the form ~p = [A + ~Pl(z, t)] exp(iA2z),
[q~l[ << A.
(2.4)
Substituting this equation into eq. (2.2) and linearizing, we get the equation for
i ~21- + -~1dlpl,t + A 2 ( V2, + ~P~{) = O.
(2.5)
Looking for a solution in the form ~Pl " exp(ig2t + Kz) we get the relation K(g2) = -t-g2 dA 2
4I
Q2
.
(2.6)
Thus modulations with frequencies 0 < s < 2A are unstable. This is the modulational instability phenomenon that exists in the anomalous dispersion region d > 0, but is absent from the normal dispersion region d < 0. The MI gain of MI is IK((2)I.
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MI in inhomogeneous and discrete media
[5, w 2
1.0
0.5 I I
\ \ ' \
0.0 -2
I
-I
/ "/ ],/
~\
, I
,
I
I/ u
0
,
,
i
I
2
Fig. 1. MI gain versus modulation frequency for the NLS equation in the anomalous dispersion region. The dashed line corresponds to A = 1, the solid line to A = 2.
In Figure 1 we present the dependence of MI gain on the modulation frequency. In physical units, the maximal gain is at g2m = 4-V/27AX/]fi2] and Km = )'A 2. MI in an optical fiber has been observed experimentally by Tai, Hasegawa and Tomita [1986]. The solution (2.6) describes MI in conservative media. For nonconservative media the linearized equation can be solved too. For example, the problem of MI in fibers with two-photon absorption can be solved exactly (see the work by Tsoy, De Sterke and Abdullaev [2001]). More information can be found in books by Abdullaev, Darmanyan and Khabibullaev [ 1993], Agrawal [ 1995] and Hasegawa and Kodama [ 1995]). The long time evolution of the modulated wave requires to find a periodic solution of the NLS equation. In general the solution requires the application of the finite zone potentials theory. Here we analyze, following Akhmediev and Korneev [1986] and Akhmediev, Eleonsky and Kulagin [1990], the simple periodic solution of the NLS equation describing the evolution of a nonlinear plane wave on which a small periodic solution is superimposed. Introducing the new field as u ~ u e i- and representing the solution as u = v + iw we can rewrite eq. (2.2) as a system of equations for two real functions. Assume that these functions are coupled by the l i n e a r relation v = ~(z)(w
+
1),
where r / = - tanhz, we obtain the Bernoulli equation for the function w: w- + tanh(z) w z + tanh(z) w - 0.
(2.7)
5, w 2]
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311
Thus the solution of eq. (2.2) is - x / 2 sinh(z) + i cos(x/2(t - to)) u(z, t) = V/~ cosh(z) - cos(x/~(t - t0))
(2.8)
This time-periodic solution of the NLS equation can also be obtained using the Darboux dressing method for the nonlinear plane-wave solution (2.2) (see the article by Its, Rybin and Sail [1988]). The solution (2.8) coincides with the solution obtained from the linear stability analysis at z --+ -oo: I ( 26)r u ~ 1 + a 1 + i~5
+b
(
2,3) ] 1 - i~--s e -'~-- c o s ( ~ ( t - t o ) ) ,
(2.9)
with b = 0. Here 6 = E2x/1-/-22/4 is the gain, and a,b are small independent parameters defined by the initial modulation. At z --+ cc the solutions (2.8) and (2.9) coincide when a = 0, Q = x/2, and there is an additional multiplier e i+7. Thus, during the instability the amplitude grows to the maximal value at z = 0 and then evolves at z ---+ cxz to the nonlinear plane-wave solution with the initial amplitude and opposite phase. The recurrence phenomenon can be described considering a finite number of modes. As shown by Infeld [1981], Infeld and Rowlands [ 1990] and Trillo and Wabnitz [ 1991 ], a three-mode approximation for the periodic solution of the NLS equation is effective. These modes are the carrier and the first two sidebands at frequencies ~o + g2. Thus in this approximation the field can be written as
u(z, t) = uo(z) +//-1 (Z) e iat + u l(z) e -i'~2'.
(2.1 O)
Here u0 is the carrier amplitude, and u-l, b/l are the amplitudes of the sidebands at ~o- g2 and oo + g2. Using eqs. (2.10) and (2.2) we find the system for the amplitudes of the modes: iuo- + (2lu_l 12 + 2lull 2 + luo 2)U-I + 2u~u_,ui = O, i u _ l - - ~ 1~,-~2u-l+(21uo I2 + 2 Ul [2 + /d_l [2)/d_ 1 + //2/,/i* = 0 ,
(2.11)
v , 1 =0. iul- 1 "~Q2Ul +(21U0 12 +2]U-II 2 + Ul 2 )Ul +U~U_
Writing the modes as uk = ]uk]e ~ok, introducing variables r/ = luo12/(]U_l12+ ]Ul 12+ ]u012), r = q~_~+ q~, - 2r and using two conserved quantities ofeq. (2.11), we obtain the equations for r/and q~: r/z = -2rl(1 - r/) sin(q~),
q~- = -2(1 - 2r/) cos(q~) + 3r/+ to- 1.
(2.12)
M I in inhomogeneous and discrete media
312
[5, w 2
The corresponding Hamiltonian is H = 2r/(1 - r/) cos(q~) - (to - 1) r/-
:3//2.
(2.13)
The numerical simulation of the full NLS equation and eqs. (2.12) with the same initial modulation shows good agreement. In particular, the reduced system describes the recurrence phenomena very well (Trillo and Wabnitz [ 1991 ]).
2.2. M e d i a w i t h q u a d r a t i c n o n l i n e a r i t y
The second example of importance for the following subsections is MI in media with a quadratic nonlinearity (Trillo and Ferro [1995]). The governing equations, describing the propagation of the envelopes of the fundamental wave (FW) and the second harmonics (SH) in dimensionless variables are 1 u . + U* v e x p ( i 6 k z ) iu: + :[31
= O,
(2.14) iv- + it)gOt + -~1fl2 Utt + -2i u 2 exp(-i6kz) = O.
Here z = X/Xd, Xd = t2/[3~] is the dispersion length, t is the time in the moving reference frame, and 6k is the dimensionless wave-vector mismatch. Looking for solutions of the form u = u0 exp(iq~l),
v = v0 exp(iq~2),
q~l.2 = ~ll,2Z,
we obtain the stationary solution u0 = 4- v/2/t(21t - 6 k ) ,
v0 = It,
(2.15)
where/tl = It. Here we have taken into account that It2 = 21t~- 6k. The total 1 intensity is normalized to 1, :]u I-"~ + Iv]2 - 1, thus It e represents the fraction of SH mode in the total intensity. The relation for the phases is u2
~I
=
VOZ~2 --
"*0 Z
2v0 "
Dispersion and nonlinearity can lead to MI of cw solutions (2.15) against small perturbations. To study MI let us consider the ansatz u = (uo+ut(z,t))exp(iltz),
v = (vo+vl(z,t))exp(21t-Ok).
(2.16)
5, w 2]
MI in homogeneous media
313
Inserting these expressions into eqs. (2.14) and retaining only terms of first order in perturbations we get the set 1 l t u 1 + V o lgl* + U o V 1 = 0 , iul- + -~filblltt-
(2.17) l - 2t.tvt + uou 1 = O. ivy: + iVgvlt + -Sfi2Ultt
Looking for solutions of this linear system of partial differential equations in the form Ul = A ( z ) exp(iI2t) + B ( z ) exp(-if2t),
(2.18) vl - C ( z ) exp(if2t) + D ( z ) exp(-i,Qt),
we arrive at the linear problem dA -MA, dz where A = (A,B, C , D ) r and the complex matrix M is
M = i
-g2~ vo
uo
0
-vo
.Q~
0
-uo
u0
0
-f22 - Vgg2
0
0
-uo
0
.Q~ - vg..Q
(2.19)
with
.-Q~ = +ill ..Q2 + ~,,
~2 = l _
:fi2
~2
+
It2.
MI occurs if an eigenvalue with a positive real part exists. For a group-velocity matched process, i.e. for Vg = 0, the eigenvalues are 2K r
(2.20)
C f 4- v / f 2 - j i ,
where f = -2( ~22 + 4/t2/t) - 2( fi2~t2 + fil ]l) ~,-~2 _ :(fi~l
+fi22 ) ~,~4,
A = Q2 [fi2fi2E26 + 4filfi2(fi2r + fil~'2) ~24 + 4fi~l tzE22 - 16/~t(fi,/t22 + 2fi2/~t2/~t)].
314
[5, w3
MI in inhomogeneous and discrete media
Analysis of this equation shows that MI develops for both normal and anomalous dispersion regions. In the normal dispersion regime, where/~!.2 < 0, MI exists in the region of frequencies
1 -
~ + 8
2 -
+
2
lFt).
For the case 6k = 2 the peak of the gain is achieved at the frequency f2 = v/2. This value can be obtained also from the phase-matching condition kp - ka + ks. This is the condition for the decay of an SH photon with wavevector kp into Stokes (ks) and anti-Stokes (ka) photons. With decreasing 6k the gain band narrows and the gain decreases. In the anomalous dispersion region, ill.2 > 0, MI occurs for 0 < f22 < g2~.
(2.21)
For 6k = 2 the eigenmode is stable, and in the case of a pump with q~ : :r for 6k--2 it is unstable. More details can be found in the articles by Trillo and Ferro [1995] and He, Drummond and Malomed [1996].
w 3. MI in periodically inhomogeneous media 3.1. M I in optical fibers with periodic nonlinearity
The governing wave equation for the envelope of the electric field in a Kerr-like nonlinear medium with a periodic nonlinearity has the form iu: + dou,, + 2f(z) u 2u = 0,
(3.1)
where do = ~: 1 for anomalous and normal group dispersion, respectively, and f ( z ) describes the periodic modulations of the fiber's nonlinear parameters. These modulations are induced either by periodic variations of the effective core area
5, w 3]
MI in periodically inhomogeneous media
315
Aeff of the fiber along z, or by periodically positioned amplifiers. This equation has the steady-state cw solution v = A exp(ir
q~ = 2A 2
J0"7f ( z ' )
dz'.
(3.2)
Without loss of generality we can take A real. Let us consider the stability of the wave solution (3.2) with respect to small modulations tJ = [A + ~p(z, t)] exp(iq~),
Iq) << A.
(3.3)
Substituting eq. (3.3) into eq. (3.2) and linearizing with respect to the correction ~p, we obtain the equation for lp(x, t): i~p: + doq~,, + 2 A 2 f (z) (lP + lp*) = O.
(3.4)
By expressing lp as ~p = C(z) exp(if2t) + B* (z) exp(-ig2t)
(3.5)
and inserting eq. (3.5) into eq. (3.4), we obtain the system of equations = 0,
(3.6)
ic- - dof22b + 4 f (z)A2b = O,
(3.7)
ib- - dos
where c(z) = C ( z ) - B(z) and b(z) = C(z) + B(z). This system can be reduced to
b:- -I- .Q2[.Q2
_4doA2f(z)]b =
O.
(3.8)
Choosing for the variation the particular form f ( z ) = 1 - j ~ cos(az), j~ << 1, we obtain the Mathieu equation instead of eq. (3.8) (Abdullaev [1994], Matera, Mecozzi, Romagnoli and Settembre [1993])" b= + o9211 + h cos(az)] b = 0,
(3.9)
where 4A2fodo h = ~,~2_4doA 2'
~ = (22 o9o (Q2-4A2d~
(3.10)
By applying the standard method (Landau and Lifshitz [1973]) for the case 0 < h << 1, we find that the wave is unstable in the region of parametric
316
MI in inhomogeneous and discrete media
[5, w 3
1 resonance, which occurs when o30 = -~ma, with m = 1,2, 3 , . . . . In the region of the first parametric resonance we have a - 20)o; i.e. the wave is unstable for modulations with frequency
~,-~2 = 2AZd0 + V/4 A 4 + ~la2"
(3 11)
The width 6 of the MI region can be found from the instability condition for the solution of eq. (3.9): 2J~A2 s
16[ < V/s
(3.12)
-4d0A 2"
The corresponding width As AE2~
is found from eqs. (3.11) and (3.12)"
dofoa 2
~ , .Q (V/g24 + a 2 - f2o) v/-Q4 + a 2
(3.13)
where f2g = 4A2d0. The appearance of a new MI region at higher modulation frequencies indicates that it is possible to generate a chain of ultrashort pulses with a high repetition rate. It can be shown that the maximum gain gmax for parametric resonance in the region of anomalous group-velocity dispersion when a >> 4A 2, is equal to gmax ~ J ~ A2.
(3.14)
For small a << 4A 2 we have 8J~A4 gmax ~ ~ . a
(3.15)
For the normal dispersion domain a >> A = 4A 2 we can use the estimate f2 ~ V/2 a - 2A 2.
(3.16)
For small a << 6, E2 is approximated by g2 ~
a
8A
.
(3.17)
Let us now estimate the value of the effect for the case of periodic variation of amplification in fibers (Matera, Mecozzi, Romagnoli and Settembre [ 1993]). The governing equation is iv: + dv,, + 210120 - - i F v ,
where F is the loss coefficient. In each amplifier point the loss is exactly compensated. Let us perform in the interval [0, l], with l the distance between
5, w 3]
MI in periodically inhomogeneous media
317
amplifiers, the transformation u = v exp(-Fz). Then we have eq. (3.1) with f(z) = exp(-2Fz) in the interval 0 < z < l and periodic with period I. If we take the first terms of the Fourier expansion for f(z), ( 1 - e -2rl) ( .2~n '~ (2FI)[1 - i n J r / ( F l ) ] exp -1--[-zj
f(z) = Z ;l ~
for the case
--00
Fl << 1, we obtain the model for f (z) as
2Fl (2:rz) f(z) ~ 1 + ~ sin Jr
--f-
"
In the dimensional units of eq. (2.1) with d0 = - 2 ps 2 k m -1 , y -- 2 W -1 km -1 , F = 0.03 km -1, l = 10 km, P = 0.1 W and j~ = 0.2, the critical value o f / 2 is approximately 0.67 THz, and the increment gmax ~ 0.1 km -~ For Fl/> 1 the position of sidebands is g2p = i V / ?
2A2c~ d'
p=O, 1,2,...,
(3.18)
where co = [ 1 - exp(-2F1)]/(2Fl). The positions of sidebands for ~[ = 1 and d = -0.Sps2km -l, P = 1.5mW, ~' = 3• F = 0.03km -1 and l - 100 km are 40 GHz from the central frequency. Numerical simulation of the amplitude gain in the amplifier chain after 10000 km gives the value -'10 -3 km -l. Experimental observation of sideband generation upon periodic variation of power in a fiber has been reported by Kikuchi, Lorattanasane, Futami and Kaneko [ 1995]. It should be noted that this type of instability is different from the sideband instability occurring for optical soliton propagation in a fiber with periodic variation of power (Matera, Mecozzi, Romagnoli and Settembre [ 1993], Gordon [1992], Kelley [1992]). When a soliton propagates in such a fiber, the periodic variation of the nonlinear coefficient leads to the generation of radiation corresponding to the sidebands in the spectrum of soliton, with frequencies 1
+
4nn
where r is the initial duration of the pulse. MI is a parametric resonance and exists both for even and for odd values of n, but the emission generated by the soliton is due to the resonant process between the soliton with wavevector
318
MI in inhomogeneous and discrete media
[5, w3
]/32[/(2r 2) and a dispersive wave with wavevector 115~2]Q2/2 under periodic modulation with wavevector 2:r/l, and exists only for even values of n.
3.2. MI in fibers with periodic dispersion It has been observed that a new type of soliton can exist in fibers with a periodic variation of the dispersion along the fiber. Remarkably, these solitons exist also in the region of normal averaged dispersion. These solitons, called dispersion-managed (DM) solitons, have enhanced power in comparison with standard solitons in a fiber with a dispersion equal to the average dispersion for the DM case (see Smith, Knox, Doran, Blow and Bennion [1996] and Gabitov and Turitsyn [1997]). It was also observed that four-wave mixing effects are significantly reduced (Kurtzke [1993]). As is well known, the appearance of solitons and MI are closely related problems. It is thus interesting to check the possibility of MI in fibers with periodic modulation of dispersion, for both weak and strong dispersion management. The governing equation for this problem is iu- + d ( z ) u , + ]u 2u = 0,
(3.19)
where d(z) is a periodic function. We start with the case of weak dispersion management, d = do + dl(z), dl << do, where do is the dispersion coefficient averaged over time. The stationary solution of eq. (3.19) is u = A exp(iA2z). To perform the stability analysis we add a small spatio-temporal perturbation ~p(z,t) to this solution, as we did in eq. (3.3). Substituting ~p(z,t) in the form q, = a exp(ig2t)+ b* exp(-ig2t) we obtain the set of equations ia: = -(22d(z)a - A2(a + b),
(3.20)
ib: = (22d(z)b+A2(a + b),
(3.21)
which can easily be reduced to an equation for the amplitude a(z)" a:: +q2 [1 + &,d,(z) + &2d~(z)- i~,d,-(z)] a : 0,
(3.22)
with the coefficients
q 2 _ /-22 2~"
&l = 4(/22do +A 2) ~',
&2 = 2Q2~',
~'
-
-
1.('~2
1 2 + 4A
(3.23)
Because dl(z) is an arbitrary function it is not possible in general to solve eq. (3.22) analytically. Results can be obtained for particular choices of &(z):
5, w 3]
MI in periodically inhomogeneous media
319
periodical modulations (Abdullaev, Darmanyan, Kobyakov and Lederer [1996], Smith and Doran [1996]) and stepwise dispersion-management (Bronski and Kutz [ 1996]). In this section we assume the dispersion is periodically modulated, viz. d l ( z ) - dp cos(kpz). Then we can rewrite eq. (3.22) as a= + Q2 [1 + al cos(kpz) + a2 cos(kpz) + iy sin(kpz)] a - 0,
(3.24)
with the new coefficients Q2 = q2x,
oq --
a l dp
,
012 --
X
~1,2d/2~
2X
,
Y = kp4~Y,
,-2
Z = 1 + a2G 2
(3.25) Equation (3.24) is the generalized Mathieu equation. For small @ << 1 the standard perturbation method can be applied (Landau and Lifshitz [1973]). In doing so we find that the domain of MI is reduced relative to a fiber with constant anomalous dispersion. In addition to this change, parametric resonance gives rise to new domains defined by 1ink, = Q, m = 1,2, 3,.... The first parametric resonance corresponds to
(3.26)
Q : -~kp + 6l ,
where el denotes the demning from the resonance. The increment of the instability Xl of this first resonance is then given by )~ = ~q2(Ot~ _ y2)_ ~ ,
(3.27)
where we have neglected the terms proportional to a2, which yield higher-order corrections. From eq. (3.27) we can easily see that the maximum increment of instability and the width of the resonance e~,,, coincide, amounting to max =
le, mI =
l d p q )'A2-
(3.28)
The center frequency g2~e~l of this resonance is obtained upon substitution of q = q ( Q ) from eq. (3.23) into eq. (3.26) as ~"~r2es 1 -- V / e A 4 + k2 -2d0A 2,
(3.29)
and the width Ag-21 is determined by dp A2 ~'~res ! A Q 1 = 2(Qr2es 1 + 2d0A2)"
(3.30)
The first conclusion to be drawn from eq. (3.29) is that the oscillations of the dispersion entail MI for the case of normal dispersion for any g2 > 0. Secondly,
320
MI in inhomogeneous and discrete media
[5, w 3
for the anomalous dispersion a new domain of MI (parametric instability region) occurs for .(2 > s If the period of the dispersion variations is small (kt~ >> 2A2), we obtain -C2res, - ~ for the location of the resonance. The second parametric resonance appears at Q - k, + c2, with the corresponding resonance frequency
nr2es
-
2(v/A 2 +
-
d0A
).
(3.31)
It can be shown that the instability domain is asymmetric with respect to the central frequency (Abdullaev, Darmanyan, Kobyakov and Lederer [1996]). The instability of soliton sidebands during soliton propagation in a fiber with periodic dispersion has the same origin as the MI resonances (Abdullaev, Caputo and Flytzanis [ 1994]). The analysis of MI in fibers with strong dispersion management (Smith and Doran [1996]) shows that the fundamental MI region is changed. This differs from the case of periodic variations of the amplification considered in the previous section. Also, sideband formation is suppressed under strong dispersion management. Note that in case of periodic variation of amplification the spectral component couples with one spectral sideband, while in DM even one harmonic modulation of d(z) generates a complete set of sidebands. The MI induced by the periodic variation of dispersion has been experimentally observed in a longdistance transmission system by Shiraki, Omae and Horiguchi [1998].
3.3. MI in a fiber Bragg grating In this subsection we consider the MI of electromagnetic waves in a cubic nonlinear dielectric medium with periodically modulated dielectric constant, with the period of modulation close to the optical wavelength. A typical example of such a medium is an optical fiber with Bragg grating. The governing equation is
E--- ~ E t ,
= ltoP,~k,
(3.32)
where c(z) = h 2 + 2~ cos(2k0z), k0 = 2~/do, and do is the period of the modulations of the medium. The electric field has the form
E(r, t) = ~(r) exp(-icot) + c.c. The nonlinear response is pNk = e0(A(~. ~*)~ exp(-icot)+ ~B(~. ~)
exp(-i~ot) + c.c.
Periodic variations of both a linear and a nonlinear index of refraction can be produced in fibers, e.g., by corrugating the core-cladding structure. When
MI in periodically inhomogeneous media
5, w 3]
321
the period of the variations is of the order of the optical wavelength the Bragg reflection is large. As a result, the amplitude of the reflected wave is of the same order as that of the forward-going wave. One of the possible approaches to the study of MI is to use coupled-mode theory, i.e. to consider counter-propagating waves. According to this approach we can look for a solution of the form E(z, t) = E+(z, t) exp[-i(o0t - k0z)] + E_(z, t) exp[-i(co0t + k0z)] + c.c.,
(3.33) where E+ are the envelopes of the forward- and back-propagating waves, respectively. The coupled-mode equations are (Winful and Cooperman [1982], De Sterke and Sipe [1990], Aceves [2000])
i
0:+-0,
E++tcE_+F(IE+2+2E_2)
E+=O,
(3.34)
E_=0,
(3.35)
C
i
- 0 : + - 0 , 1 E _ +l r E + + F ( I E _ c
2+2E+e)
7
where B 2hc (A + -~ ), 0) 2
F
e)0~ tr - 2hc
The cw solution is
E+
a exp[i(Qz- g2t)], v/1 + / 2
E_ = ~ a f exp[i(Qz- g2t)], v/1 + / 2 (3.36)
where
g2=~l tc(f-I + f ) - ~3Fa 2,
l
l
Q = - 3 to(f- + f ) -
l 1 _f2 ~ Fa2 1 - ~ .
It follows that the parameter a is related to the power of the grating through a 2 = IE+I 2 + IE_I2. The meaning of the parameter f can be established by considering the linear limit (a ~ 0). We see that the case f < 0 corresponds to the upper (positive) branch of the dispersive curve and f > 0 corresponds to the lower (negative) branch.
322
MI in inhomogeneous and discrete media
[5, w 3
Let us consider the stability of these waves against small perturbations, following the work by De Sterke [1998]: (3.37)
E+ = (a+ + e+(z, t)) exp[i(Qz - f2t)].
Substituting eq. (3.37) into eq. (3.34) and neglecting terms of order (~2, we get the equations for the small perturbations e+:
i(& + 0:)e+ + re_ - t c f e + + G [(e+ + e+) + 2f(e_ + e*)] = 0, (3.38) i(0, - 0-) e_ + re+ - tcf -l e_ + G [2f(e+ + e+) + f 2(e_ + e_*)] = 0, where G = F a 2 / ( 1 +f2). One can look for a solution of the form (3.39)
e+ = A+ cos(qz- oot) + iB+ sin(qz- ~ot). The solvability condition gives the characteristic equation (0)2 _ q2)2 _ 2;(.2(0)2 _ q2) _ g . 2 f 2 ( o ) + q)2 _ K.2f-2(O) _ q)2
(3.40) + 4 t c G f ( 3 q 2 - (02) = O.
The coefficients of this equation are real. Thus the complex conjugate roots correspond to the MI of the cw solution. For the particular case f = + l , where f = 1 and f - -1 correspond to the top and the bottom of the photonic band gap, respectively, eq. (3.40) can be solved analytically. (a) Top of the photonic band gap, f = 1. This case corresponds to anomalous dispersion and so we can expect the existence of MI. Indeed, the dispersion relation is 602 = q2 + 2to(to- G) -t- V/tC2(tr - G ) 2 + / r 1 6 2
+ 2G).
(3.41)
We have MI of cw for the wavenumbers between -x/'12tcG and +v/12tcG. The maximal growth rate is obtained at the wavenumber
/ qm = ~/ 3tcG
V
2tr + G tc + 2 G
(3.42)
5, w 3]
323
MI in periodically inhomogeneous media
(b) Bottom of the photonic band gap, f eq. (3.40) transforms to
= -1 (normal dispersion). Then
o)2 = q2 + 2to(to + G) 4- 2 V/tC2(tr + G) 2 + tcq2(tr - 2G).
(3.43)
Analysis of this equation shows that the instability of cw exists only for
~/ q >(to+G)
K" 2G-to
For details we refer to the work by De Sterke [ 1998].
3.4. MI in nonlinear media with periodic potential
In this section we study the problem of MI in the NLS equation with a periodic potential V(x + a) = V(x): iu, + du,:,. + V(x)u + Xlul2 u = O.
(3.44)
This problem appears in the investigation of pulse propagation with a periodic phase modulation in fibers, using lumped modulators in the time domain (see the book by Hasegawa and Kodama [1995]). Another example of a physical system described by eq. (3.44) is the Bose-Einstein condensate (BEC) in an optical lattice. We remind that in fibers x and t are interchanged, so the potential V is periodic in time. One of the ways to study the MI in such a system is the reduction of this equation to an effective NLS equation with constant coefficients (P6tting, Meystre and Wright [2000], Konotop and Salerno [2002]). Let us consider a periodic potential of the form V ( x ) = Vo cos2(k0x). Such a form of the potential applies for the BEC condensate in an optical lattice. The derivation procedure consists in using the linear eigenfunctions O,,.k of eq. (3.44):
d2q~n,k
d dx2 4- V0 cos 2(k0x)g),,,k = E,,.~. r
(3.45)
Let us consider the wave packet for a given band index n and wavenumbers distributed in an interval of length Ak around the carrier k0. The parameters of
324
MI in inhomogeneous and discrete media
[5, w 3
the wave packet are assumed to vary slowly on the scale of modulations 2r Then the field function can be written in the form (3.46)
u(x, t) = U(x, t) O,,.k,, exp(-iE,,.k,,t).
Substituting this approximation into eq. (3.44) we obtain the equation for the slowly varying envelope: 1
i(Ut +ugU,-)+ 2mefl U,-,-+ z l g l 2 g
- 0,
(3.47)
where
meff
02E"k (k = ko), Ok 2
u,, - OF.. "k (k = ko). "~ Ok
Here meff is the effective mass of the Bloch state and ug is the group velocity. Depending on the choice of the sign of meff which is the curvature of the energy band (i.e. taking parameters near the bottom or the top of the Bloch zone) we can have stable or unstable cw solutions. This approach explains, for example, why dark (bright) solitons can be created in periodic media with a focusing (defocusing) Kerr nonlinearity.
3.5. M I in birefringent fibers with periodic dispersion Another problem worth studying here is the propagation of two polarized electromagnetic waves in a periodic transmission line with variable dispersion d(z). The governing equation is a system of two coupled modified NLS equations"
iu- + d(z)ut, + y( u 2 + alvlZ)u = 0,
(3.48)
iu= + d ( z ) u , + y(alul" + IuI2) u : o,
(3.49)
where a = 1 for orthogonally polarized waves. For a birefringent fiber, a ~ 23 (Agrawal [1995], Georges [ 1998]). The system has the nonlinear plane-wave solutions u0 = A exp[i?'(A 2 + aB2)z],
u0 = B exp[iy(B 2 + aA2)z].
(3.50)
5, w 3]
MI in periodically inhomogeneous media
325
We perform the linear stability analysis, as described in previous sections, following the work by Abdullaev and Gamier [1999]. Separating corrections ~Pl = a + ib, ~P2 = e + i f we get the system a: = d(z) f22b,
(3.51)
b: = - d ( z ) (22a + 2 y A ( A a + aBe),
(3.52)
e: = d(z) C2Pf ,
(3.53) (3.54)
= - d ( z ) f 2 2 e + 2 y B ( B e + aAa).
Throughout this section we shall consider the particular case A = B which simplifies the algebra. The MI gain is defined as the maximal exponential growth of a 2, b 2, e 2 and f 2 . Let us consider the particular case where d(z) is stepwise constant and takes two different values dl and d2 at regularly spaced intervals with lengths L1 and L2. The system is periodic with period L = L l + L2, i.e. : f dl d(x)
I d2
if x E [nL, nL + L l),
n C IN,
if x C [nL + Ll, (n + 1)L),
n E N.
Equations (3.51)-(3.54) can be solved analytically for this form of d(z). The eigenvalues have the following forms: AIi : a + + v/(a + ) 2 _ 1,
Af = a + - v/(a +)2 _ 1,
(3.55)
where a+ +
r
: : cos(kfL2)cos(kilL,) - 51 ( r + _
k,
+ r +-1
) s i n ( k f L 2 ) sin(k(Ll),
d2
(3.56) (3.57)
k2id I
with k. =1: = V/I~j.Q2(I~j.Q 2 - l+-l ) J
.
'
l• =
2yA2(1 + a)"
Note that a + are always real-valued, as can be checked by simple algebra using the fact that k.]i is either real-valued or purely imaginary. Two cases are possible:
326
[5, w 3
MI in inhomogeneous and discrete media .
.
.
.
I
.
.
.
.
I
.
I
,
.
.
.
I
.
.
.
I:~i
. . . .
.
I
. . . .
0.03
9~
0.02
0.01
i!, i
0
i
!
i
i ~.L
i:
0.1
0
0.02,
,1
,
'
'
'
I
.
~
,
0.2 frequency .
.
.
.
1
.
.
.
.
,.
I
0.3 (THz) I
"
,
,
.
.
.
.
I
(b)
,
0.5
0.4
.
.
.
.
F ............
0.016
G
H
= 0.012 r
0.008 0.004
,A,
fti Ii:
0
0.1
0.2 frequency
0.3 (THz)
0.4
0.5
Fig. 2. (a) MI gain per unit length (in km) versus modulation frequency f2 (in THz) for Y = 2 W -1 km -!, Ll = L2 = 20km, Po = A2 = 5mW, and a - 3" Curve A corresponds to the standard anomalous dispersion /31 = /32 = l ps 2 km -l. Curves B - D all have the same average anomalous dispersion /3 (LI/~ I + L2132)/(L! + L2) = l ps2km - l , but increasing dispersion management; B corresponds to /31 = 4, /32_ = - 2 ps 2 km -l , C corresponds to/31 = 8, /32 = - 6 p s 2 km -] , and D corresponds to/31 = 16,/32 = - 1 5 p s 2 km -1 . (b) Same as (a), for average normal dispersion/3 - -1 ps 2 km -I . F corresponds to/31 = -3,/%__ = 1 ps 2 km -1 , G corresponds to fll = -4,/32 = 2 ps 2 km -I , and H corresponds to fil = -8,/32_ = 6 ps 2 km -I . =
(i)
I f a +2
>
different.
1, t h e n t h e e i g e n v a l u e s Since
the
product
eigenvalues has a modulus
A~ a n d ~.f a r e r e a l - v a l u e d
3.fJ.f
is e q u a l
l a r g e r t h a n 1.
to
1, at
least
and strictly one
of the
5, w 3]
MI in periodically inhomogeneous media
327
(ii) If a +2 ~< 1, then, writing b • = v / 1 - a +2, the eigenvalues are given by 2 ~ = a + + i b + and 22i = a + - i b +. Thus there is stability if and only if a - 2 - 1 and a +2- 1 are non-positive-valued; otherwise the exponential gain is G = ~2 max(lln[[a+l + V/a+ 2 - 1 ] I , Iln[la- I + v/a-2 - 111).
(3.58)
This equation reads as a closed-form expression which allows us to plot G((2) as a function of Y2 for a given set of parameters (fil,~,Li,L2, a,A). For the case L1, L2 << l +, l- we get that the resonant frequencies are g-2p~ 2 -
~p
I/3L
t
1
(3.59)
2/31+
The resulting theoretical resonant peaks (fig. 2a) should be around f2~ ~ 0.31, g2i- ~ 0.29, f2~- ~ 0.42, (22 ~ 0.40, 82~ ~ 0.50, Y23 ~ 0.49 (in YHz). These values agree well with the exact plot of the MI gain in fig. 2a. For fig. 2b the theoretical resonant peaks are predicted as 82~- ~ 0.25, Y2i- ~ 0.275, g2~- ~ 0.375, Y22 ~ 0.39, f2~-~ 0.47, 823 ~ 0.48 (in THz). Figure 2a corresponds to the case of birefringent fibers with an average 2 We have considered the case when anomalous dispersion, when a = 3" y = 2 W -l klT1-1, Ll = L2 = 20km, P0 = A2 = 5 mW. In these conditions the characteristic wavelengths are l + = 30 km and l- = 150 km. It appears that the central peak gain is progressively reduced as the strength of the dispersion management increases, and so are the resonant sidebands. For more details see Abdullaev and Gamier [1999]. Figure 2b corresponds to the case ofbirefringent fibers with an average normal 2 Although there is no MI gain in the uniform case/3 = -1, dispersion, when a = 3" some new sidebands appear when the dispersion management is not zero, but these sidebands tend to disappear for strong dispersion management.
3.6. MI in fibers with periodic birefringence In polarization-preserving birefringent optical fibers, two distinct forms of modulational instability occur: cross-phase modulational instability (Berkhoer and Zakharov [1970], Agrawal [1987]) and polarization modulational instability (PMI, Wabnitz [1988]). The periodic variations of birefringence change the phase-matching condition for the latter case, and as a result, sideband instability can occur for PMI.
MI in inhomogeneous and discrete media
328
[5, w 3
Following the work by Murdoch, Leonhardt, Harvey and Kennedy [1997], we consider a weakly birefringent fiber with periodically varying birefringence. The variation can be achieved, for example, by wrapping the fiber between two spools. The resulting periodic variation is of the form 0, 63(z) =
63,
m(lt + h ) < z < m(ll + 12) + lt,
(3.60)
m(ll + 12)+ Ii < z < (m + 1)(ll + 12),
where 6/3 = k , - - k,., and k, and k,. are the wavenumbers of the principal polarizations. The electric field is E ( r , t) = U ( r ) { x A x ( z , t) exp[i(kz - ~o0t)] + yA).(z, t) exp[i(kz - toot)]} + c.c.
Here U ( r ) is the transverse-mode function. Using a standard procedure for averaging over transverse modes, we obtain the system of equations for the envelopes of waves in different polarizations: iAx._-- [a(z) - 89
A,- - 89
+ R(IA,-I 2 + 3~IA,.I 2)A,. + 5..~.,...,. ' RA*~2 = 0, (3.61)
iA.r,:- [ a ( z ) - 89
A,.- 89
' RA *~2 = 0, (3.62) + RtlA,.I 2 + 32]A,-]2) A,. + 3.~..,...,.
where a ( z ) = k - 5l(k,-+ k,.) in the birefringent section and a ( z ) = 0 in the nonbirefringent section, A,.,,. are slowly varying envelopes along the birefringent axis, R = n2coo/(cAeff), with Aetv the effective fiber core area, and n2 is the nonlinear part of the refractive index. Below we study the stability of the steady-state solutions A.,. = v/-fi exp(i0),
A,. = 0,
0 = RPz-
f0 :
1 ( a - ~6/3) dz /
(3.63)
To perform the linear stability analysis, let us look for the solutions of the form Ax = A.~ + h A ( z , t),
A,. = b(z, t) e '0,
6A, b << IA.,-I.
(3.64)
From eqs. (3.61), (3.63) and (3.64) we get the equation for the small perturbation b(z,t):
ib,. - -~' [?ab tt + 89
- b]
-
6fi(z) b : 0.
(3.65)
The regions of MI can be determined explicitly, since the variations of the birefringence are piecewise (see eq. 3.60). The Floquet discriminant
5, w 3]
MI in periodically inhomogeneous media
329
defining K(g2) can be calculated explicitly. The gain of MI is calculated as g(g2) = 2Im[K(g2)], namely A(g2) = 2 cos(K(.Q) L). The stability region corresponds to - 2 < A(g2) < 2. The calculation of the discriminant gives +
A(g2) = 2 cos(kl It ) cos(k212) -
-
klk2
2
sin(kill)sin(k212)
(3.66)
= 2 cos[k2(ll + 12)], where 1
1 2 2 ~/32s~ - 3RP),
(3.67)
The values of the frequency shift of PMI sidebands can be approximated by the formula 2n~ ) ll +12 "
(3.68)
The analysis of the expression for the PMI gain spectrum (3.66) shows that a series of sidebands is generated. The typical experimental parameter values are: An = n x - n,. = 6.8x 10-6, 12 = 2.53 cm, R = 0.025 m -l W -l,/32_ = 50ps 2 km -1, P = 700 W. We obtain for the sideband with n = 0 and l~ = 0 that the PMI gain is 11.5 m -l and the frequency shift is ~7 THz; with ll = 10 cm the gain is 8.5 m -t . Experimental results obtained by Murdoch, Leonhardt, Harvey and Kennedy [1997] confirm these predictions for the PMI sidebands.
3.7. MI in periodic quadratic nonlinear media The next important example of a nonlinear physical system with periodically modulated parameters comprises quadratic nonlinear media with periodic inhomogeneities. In quadratic nonlinear media the main parameter defining wave interaction is the phase mismatch. Particular values of this parameter give the strongest interaction and mutual transformation of the FW and SH waves. Stable optical solitons can exist in these media at all dimensions, starkly different from the case of Kerr nonlinearity. On the other hand, the existence of the strong cascading limit permits the observation of many phenomena known from Kerr nonlinearity in the quadratic case. Periodic variation of the linear and nonlinear parameters of such a medium lead to compensation of the large phase mismatch
MI in inhomogeneous and discrete media
330
[5, w 3
and as a consequence to the existence of new types of solitons. Experimentally, such modulations were realized using periodically poled LieNbO3 (Fejer, Magel, Jundt and Byer [1992]) and GaAs/GaA1As structures (Petrov [1996]). The influence of the periodic inhomogeneities on the MI in Z t2) media has been studied by Corney and Bang [2001 ]. Below we follow this work. Let us consider beam propagation in a quadratic medium having modulations of linear and nonlinear refraction indices with a period much longer than the optical period. In this case Bragg reflection can be neglected and we can write the system of equations for the envelopes of FW and SH propagating waves. The governing equations for FW and SH waves propagating in periodic quadratic nonlinear 1D media are
iEl,z + -~lEl ~vx + oriel + xEbEc_ exp(iflZ) = 0, (3.69) iE2,z + 1E2~vx + 2a2E2 + zE~ exp(-iflZ) = 0.
Here E i ( X , Z ) are the envelopes of the FW and SH waves. The coordinate X is measured in the input beam width X0, the coordinate Z in the diffraction length Lo = klX2; [3 - AkLo, where Ak = k 2 - 2131, kj = i~onj is the phase mismatch. The normalized linear refractive index grating is a i(Z) - Lo~oAnj(Z), and the normalized nonlinear grating is Ldoodeff(Z)/(nic), where deff - ~1X(2) . The gratings aj and ~. are assumed to be periodic functions with period 2L0 = 2;r/lkg], with kg the spatial period of the grating. It is useful to study rescaled equations for new variables, 1 Xtw, iw_ + ~W,;,- rw + vexp(ibz) = 0, (3.70) 1 ~t XtW2 iv- + ~vx.,-- av + + exp(-ibz) = 0, where w(x,z) = E1 exp(i0), v(x,z) = E2(X,Z)exp(2iO + i/JZ), x = v/-~(X + (2Z), z = rlZ, r / - [A + O2/21, and A and s are the longitudinal wavenumber offset and the transverse wavenumber respectively. The residual phase mismatch is = / 3 - kg and 0 = A Z - f or,(z)dz. Here also r = sgn(A + ~102 ), a = 2 r - /3'. To study this system it is useful to expand the rescaled gratings and fields in Fourier series, i.e.
a'-
a'~-~' g, exp(intcz),
Z ' = d~ + d ' ~ - ~ g,, exp(intcz)
n
(3.71)
!1
and
w =~
w,(z,x)exp(intcz), tl
v- Z
v,, exp(intcz),
(3.72)
!1
where r = kg/rl, a' = 2(a2 - a l ) / r l , and g,, = 2s/(i~n), with s = sgn x" for n odd, and g, = 0 for n even.
5, w 3]
331
M I in p e r i o d i c a l l y i n h o m o g e n e o u s m e d i a
The scales of the problems considered here are the diffraction length Ld = 77, the coherence length Lc = a~/]fi'], and the grating domain length Lo = ~r/]kg]. Typical values for the parameters are k - 100, a' = d~ = 0 in LiNbO3, and d6/d' = 4.6, a ' / k - lO in a GaAs/A1As superstructure. The natural small parameter here is e = Lo/Ld harmonics wn .o, vn .o are of Then the residual mismatch first-order terms in e we can
<< 1, which is satisfied for typical gratings. The order e. We will assume effective phase matching. is small, i.e. ]fi'] << ]kg]. Then by collecting the find the expressions for the harmonics as
I
,
( d ' g , _ l + do6,,,l) WoVo WHr 0
nK
(3.73)
a'g,,oo + (d'g,,+ 1 + d ~ b , , _ l ) w ~ On neq 0 =
rtK"
Using this result we can derive the averaged equations for W, V = wo, vo" iW~+~l
m xr ' _
r W + p W * V + Y(I Vl 2 - I ml 2) m -- o,
(3.74) iV- + a1 Vx..r
_ oV
+
P W2 + 2 y I w I 2 V = O.
For a square grating we have p = 2 i s ( d ' - d ~ a ' / t r
y = [d~2 + d'2(1 - 8 / ~ 2 ) ] / K .
Thus, from the analysis of the averaged equations we can conclude that the two principal effects of fast periodic modulations are the reduction of the effective quadratic interaction and the appearance of new (cubic) nonlinearities. The sign of the induced Kerr nonlinearity determines whether the MI gain is reduced or enhanced. Indeed, the defocusing effective Kerr nonlinearity stabilizes the background. Applying the linear stability analysis as in w 2.2, we obtain the expression for the complex matrix M: a M = i
b
c
d
-b -a -d -c 2c 2d g 0 -2d-2c 0-g
(3.75) '
where a = _ 1 ~/ 2 2 - r + ~'(Io~ 2 - 2w2), b = p o ~ - yw 2, c -- pws + yO~Ws, d = yO~Ws, and g _ _ ~1 Q : - a + 2 yW2s . For r - 0 and r = 1 all nondegenerate and nontrivial solutions are unstable. The result of numerical analysis for other regions can be found in the article by Corney and Bang [2001 ]. These results have been obtained in the framework of the averaged theory. The exact stability analysis involves the analysis of eigenvalues of the matrix of
332
MI in inhomogeneous and discrete media
[5, w 4
the linear stability problem M with periodic coefficients. Application of Floquet theory shows that the averaged theory accurately predicts the properties of the lower part of the MI spectrum. The analysis of the higher part of the spectrum necessitates the application of exact Floquet analysis.
w 4. MI in random media
4.1. Origins of random fluctuations All the results discussed in the previous sections were obtained in media where the characteristic parameters are either constants or periodic functions. In realistic fiber transmission links, the nonlinearity and the chromatic dispersion are not constant but can fluctuate stochastically around their mean values. The inhomogeneity of the medium may be induced by other propagating waves (Agrawal [1987]) or may be intrinsic to the medium (Kodama, Maruta and Hasegawa [1994]). As there are various physical reasons for the fluctuations of the fiber parameters, we may encounter fluctuations whose spectrum has components with short length scales (around meters) and long length scales (around kilometers). Usually the short-scale perturbations originate from variations of the fiber p a r a m e t e r s - such as core radius, fiber geometry or index of r e f r a c t i o n generated during the drawing process. They may also be induced by mechanical distortions of fibers in practical use, such as point-like pressure or twists. The correlation length of such fluctuations is less than 1 meter. Direct measurements of the chromatic dispersion at these scales are usually not feasible, but it can be calculated from measurements of the fiber parameters such as the fiber radius or the index difference. For instance, the standard deviation of the fiber radius is typically about 0.1 ~trad (~1% error). As a result, the standard deviation in the chromatic dispersion is about 0.1 ps 2 km -! . Similarly, the standard deviation of the relative index difference is about 0.02%, and the induced standard deviation in the chromatic dispersion is about 0.4 ps 2 km -l (Kuwaki and Ohashi [1990]). Fluctuations in the nonlinear coefficient are also caused by variations of the core radius and geometry which slightly affect the transverse profile of the mode and the effective nonlinear coefficient. Long-scale fluctuations can also arise in the drawing process, as a result of a slow drift of the operating parameters. They may also be induced by long-scale fluctuations of the environment of the fiber in practical use, such as temperature variations for fibers in the ground. The literature contains
5, w4]
MI in random media
333
many results of measurements for the longitudinal variations of the chromatic dispersion (see for instance Mollenauer, Mamyshev and Neubelt [1996]), since they involve fluctuations of the zero-dispersion wavelength which in turn leads to the degradation of the optical system performance (Karlsson [ 1998]). Typical standard deviations for the chromatic dispersion and the nonlinear coefficient are of the same order as those cited for the short-scale fluctuations, while the typical length scales are of the order of a kilometer or longer (Nakajima, Ohashi and Tateda [ 1997]). The chromatic dispersion and the nonlinear coefficient are not the only parameters that can suffer from random modulations in real optical fibers. Singlemode fibers are actually bimodal because of birefringence (Kaminow [1981]). The details of the birefringence evolution along the fiber are not known, but it is usually assumed that the birefringence is locally linear and that the strength and the orientation of the birefringence vary randomly along the fiber with a typical length scale whose spectrum may have components from a few centimeters to hundreds of meters (Simon and Ulrich [ 1977], Rashleigh [ 1983]). More exactly, the birefringence is small in absolute values in communication fibers, with values of the order of An/n ~ 10-7. The corresponding beat length is only about 10 meters, which is far smaller than the typical nonlinear and dispersive lengths of the order of a few hundreds of kilometers. This would involve dramatic distortion in communication systems, but fortunately the orientation of the birefringence is also randomly varying on a length scale of about 100 meters, which averages out the effect ofbirefringence to zero. The residual effect leads to pulse spreading, referred to as Polarization Mode Dispersion (PMD). The effects due to PMD accumulate along the length of an optical fiber, as do the effects due to chromatic dispersion and nonlinearity.
4.2. The random scalar case 4.2.1. Linear stability analysis
The evolution of the field in random fibers is governed by the NLS equation with random coefficients (Agrawal [1995]): iu- + [3urr + ylul2u = 0,
(4.1)
where we used the standard dimensionless variables, y is a nonlinear coefficient. The Group Velocity Dispersion (GVD) coefficient is/3 > 0 (/3 < 0) for anomalous
334
MI in inhomogeneous and discrete media
[5, w4
(normal) dispersion. Both coefficients fluctuate around their respective mean values Y0 and fi0 so that they can be described as: "/(z) = Yo (1 + m,,,(z)),
fi(z) = rio (1 + m/~(z)) ,
(4.2)
where m/~ and mz, are stationary, zero-mean and random processes. A usual model for a random fiber is the random concatenation of different fiber sections whose coefficients have constant values (Mollenauer, Smith, Gordon and Menyuk [ 1989], Wai, Menyuk and Chen [ 1991 ]). Typically the lengths of these segments are of the order of 10-100 km, which is usually less than the dispersion distance Ld = t2/[ fiol" with the pulse duration to of the order of 10ps and the mean GVD coefficient of the order of 0.1 ps 2 km -l for most glass fibers, Ld ~ 1000 km. A white-noise model is justified for such a configuration. However some fibers have values of GVD in the range 1-10ps2km -l, so that Ld ~ 10-100km becomes of the same order as the correlation length. A more elaborate model (colored noise) is then necessary. Equation (4.1) has continuous-wave solutions
Their linear stability is determined by considering a perturbed solution of the form
(/0
)
By substituting eq. (4.3) into eq. (4.1), and retaining only the first-order terms, one obtains a linear equation for u~(z,t): iul~ + f i u l , + 2?'PoRe(ul) = 0.
(4.4)
Performing the Fourier transform hi = f ul exp(-ioat)dt, and using the complex representation h~ = h~ + it)! i, one obtains a system for the Fourier components of the perturbation term:
dz
uli
Y(z)=
(
hli 0 2y(z)Po-[3(z)oa 2
' /~(z)co2 ) 0
(4.6) "
The basic linearized equations remain unchanged with respect to the homogeneous case, but/3 and ~, are now random processes having the consequence
MI in random media
5, w 4]
335
that Ul is random as well. Because the explicit form of hi cannot be found, stochastic analysis has to be applied for analyzing the stochastic differential equation (SDE) satisfied by ftl. There are basically two ways to analyze the solutions of a SDE. Either we can determine the moments up to any order to describe the statistical properties of the sideband amplitude hI, or (which is more general) we can examine the probability density function of hi. The probability density function satisfies a partial differential equation known as the Fokker-Planck equation, and any expectation value can be determined from it. However the solution of a Fokker-Planck equation is usually intricate and cannot be written in closed form. Furthermore we are not interested in the complete determination of the statistical distribution of the sideband amplitude, but only in the statistical distribution of the growth rate of the modulation. This growth rate is characterized by the Lyapunov exponents of the linear system (4.5). The sample MI gain is defined as the Lyapunov exponent which governs the exponential growth of the modulation: G(co)
:=
lim 1 In ,z ----~oc Z
(z, o)12.
(4.7)
Note that G(to) could be random since/3 and y are. So it should be relevant to study the mean and fluctuations of the MI gain. For this purpose we shall analyze the 2nth mean MI gain defined as the normalized Lyapunov exponent which governs the exponential growth of the nth moment of the intensity of the modulation:
Gz,(O9) := lim l ln,
(4.8)
where the angle brackets stand for the expectation with respect to the distribution of the process (m:,,m/~). Note that this is a generalization of the standard mean MI gain G2 which characterizes the exponential growth of the mean intensity of the modulation. If the MI process were deterministic, then we would have G2n(O)) = G2(o)) for every n, since In [hi(z, to)[ 2'' = n In [hi(z, to)[ 2. But due to randomness this does not hold true since we cannot invert the nonlinear power function "]-]"" and the linear statistical averaging "(.)". Actually, Jensen's inequality establishes that, if n ~> 1, then ([hi(z, to)[ 2''} ~> ([hi(z, ~0)12}n, and consequently G2n(~o) /> G2(~o). If G2,, = G2 = G we can claim that the exponential growth is deterministic, but if G2,, increases with n, it means that the MI process is fluctuating. Note that random matrix products theory applies to the problem (4.5), which gives qualitative information on the Lyapunov exponents (see, for instance,
336
M1 in inhomogeneous and discrete media
[5, w 4
Theorem 4 of Baxendale and Khaminskii [1998]). In the following we present closed-form expressions for the Lyapunov exponents G(to) and G2,,(to) in the natural framework corresponding to telecommunication applications, where the noise level is low.
4.2.2. The moment equations We now present the standard approach applied when the random processes mf~ and m,, can be considered as Gaussian white noise: i
(ml~(x)ml~(y)) = 2o/~6(x-y),
(m;,(x) m;,(y)) = 2o~6(x-y).
This approach is valid as long as the correlation lengths of the random perturbations are shorter than the dispersion distance. The first and most natural step consists in considering the evolution of the expectations of the modulation amplitudes. The solutions coincide with the deterministic and homogeneous case, apart from the inclusion of an exponentially decaying damping term, which in turn involves a reduction of the MI gain. From this we may conclude that random fluctuations always give rise to a reduction of the MI gain for anomalous dispersion where high-frequency modulations are more effectively damped, and do not introduce any MI for normal dispersion. Actually the stochastic resonance responsible for the MI induced by the random fluctuations is not captured by the analysis of the first moments. The first moments of the modulations actually decay because of the uncertainty in phase. Consider a toy model where ht satisfies the equation ~_' = ihtq(z), where q is a Gaussian white noise. A closed-form expression for the solution is hi(z) = hl(O)exp(iW(z)), where W(z) := fo q(z')dz' is a Brownian motion ((W(z) 2) = 2o:z). The expectation thus decays exponentially with z: (hi(z)) = hl(0)exp(-o2z), while the modulus of h~ is constant. Therefore it is misleading to consider only the expectation of the modulation amplitude, and it is relevant and necessary to compute the growth of the modulation intensity given by second-order moments. The problem at hand is actually analogous to a harmonic oscillator with a randomly perturbed frequency •_-: + to~(1 + q(z))u = 0, where q is a random process. The equations for the first moments obey a trivial dynamics. The stochastic parametric resonance is only observed from the second-order moments equations as shown by Klyatskin [1980].
4.2.3. The second-moment MI gain We now analyze the behavior of the second moments (h~r), (h~i) and (h, rh, i) (the crossed moment (hi rhl i) is added so as to close the equations for the second
MI in random media
5, w 4]
337
moments). Applying It6's formula (Revuz and Yor [1991], p. 145) the evolution of the column vector of the second moments X '2' = ((h~r) , (hi rhl i), (u 2 i)) T reads as a closed-form linear system: d X (2)
- M(2)X(2),
dz -232094 2)'0P0 -
M (2) =
Ol~
fioOa2
2300) 2
230094 0 "2fi0 -2
riO (-02
-4fi0r
2 4 Ol3 2 + 8y6P 2 o2{y2 2(2yoPo 23000
-
9
3 0 0 2) _2fi0(_040[~
Instability arises if an eigenvalue of M t21 has a positive real part, and the mean MI gain G2 defined by (4.8) is the largest value of the real parts of the eigenvalues of M (2). The matrix M t2) has three eigenvalues whose complete expressions have been given by Garnier and Abdullaev [2000]. These general expressions coincide with the particular cases studied by Abdullaev, Darmanyan, Bischoff and Sorensen [1997] and Abdullaev, Darmanyan, Kobyakov and Lederer [1996]. We now present and discuss these formulas, where the cut-off frequency e)c = V/2yoPo/lfio] plays an important role. In case of normal dispersion, fi0 < O, two eigenvalues are negative while the last one is positive for every frequency co, which proves that there is instability for all frequencies: 40)2 02((-0)
=
fi2
0)2 + oo2 + 4fi o0)60 -4
The optimal frequency and corresponding MI peak are given by: 0)2
4
2 4-1/3
figO.)c4
= 8fiOO)c
,
_.op,- 1 + 3( fiolO)20"2 fi)2/3
Let us now consider the anomalous dispersion regime, fi0 > 0. In the standard stable region o) > ~Oc, the gain is positive for every frequency: 092 (-Oc4
v
G2((o) = 3 2 0)2 _ (0 2 (O'fi + 0;7).
For the study of the standard unstable region co < COc, we introduce the dimensionless parameter 6 = 02/o 2 Two cases can be distinguished. 7" fi"
338
[5, w 4
MI in inhomogeneous and discrete media
If 6 ~< 1, which means that randomness essentially originates from GVD fluctuations, then the gain is
G 2 ( c o ) = 2fiocor
2 - 092 + 4fioco2 (0)2 _ 0)2-, ~)(co2 _ 0)2+, ~)
2.
where the two characteristic frequencies co_ ,~ and co+,,~ are defined by
co2 2-r -"~ :=
4
2 co~'
co2 2+V/2(1-6) 2 +"~ := 4 co~"
This shows that the random dispersion increases instability for co E (0, co-,,5) and co E (co+,,~,coc), and decreases it for co E (co_,~, co+,,~). The optimal frequency oy2,opt is lower than the homogeneous optimal frequency coc/V~, and the corresponding MI peak is reduced: 2, opt
~O) c -- ~/~00)c (O'/~ -- O'i,),
O2,opt =/~0602 - ~
(4.9) If 6 > 1, which means that randomness essentially originates from fluctuations of the nonlinear coefficient, then the gain is:
G2(~) = 2floco
r
'(60-)2 - 602 +/32co 2 (2092 - c02)2 + -~ ~Oc ~ - ~0~
1)co4
2 0~,
which shows that the random nonlinearity makes instability increase for all frequencies co E (0, coc). In particular the MI peak obtained at co2,opt is enhanced and given by eq. (4.9).
4.2.4. Higher-order moments
The results derived in the previous subsection give accurate expressions for the second-moment MI gain G2, but the relationship with the sample MI gain G is not obvious. Further it may seem arbitrary to characterize the stability of the continuous wave through the analysis of the second moment of the modulation ~1. In this subsection we present a study of the exponential growth of higherorder moments of ftl so as to get a picture of the fluctuations of the exponential growth of hl. Let n be an integer and X ~2''~ be the (2n + 1)-dimensional row
5, w 4]
339
MI in random media
vector of the 2nth moments of the modulation: X/2''I 2,,-iL,,j j := ( rUlr Uli , j = 0 , . . . \
,2n.
/
Applying It6's formula establishes that the vector X/2'') satisfies dX-(2n)
-- M(2n)x(2n) '
dz where M r
is a (2n+ 1) x (2n+ 1) matrix
M t2n) = ( flora 2 - 2 YoPo) A t2'') - rio 092B/2") + /~4(-D40-/~C(2n) -k-4 y2p2 0-2Dr The matrices A (2n), B (2n), C (2n) and I) (2'') are null except for the following (2n) elements: Aj,j +, = 2n - j , B(2n) j,j_, = j, ~ ,(2n) j = -2n - 4 n j + 2j 2, C~,2~')2 = j ( j - 1), C)2~) 1), and Dj.jt2'') 2 = J(J - 1). In this framework nG2,(~o) ,~ + 2 = ( 2 n - j ) ( 2 n - j is equal to the maximum of the real parts of the roots of the characteristic polynomial of the matrix M 12n) whose degree is 2n + 1. This implies that the derivation of a closed-form expression for G2,,(m) for any n >/2 is quite intricate, but it can be carried out numerically. Figures 3-5 display the MI gains Gzn for different values of n, off and 0:,, and compare them with full numerical simulations. As a model for the random processes ml~ and m:, in the simulations we have chosen step-wise constant functions which take independent and random values in { - 0 , o} over elementary intervals with lengths I. This configuration can be approximated by a white-noise model with o/,2, = 0-2 = ~I 0.21 . Comments on the figures, as well as further theoretical investigations, will be given in the next subsection. 4.2.5.
The s a m p l e a n d m e a n growth e x p o n e n t s
Figures 3-5 show that the moments of the modulation Ul grow with different exponential rates. This provides evidence that the behavior of the modulation is not deterministic but exhibits strong random fluctuations. It should thus be relevant to study the complete distribution of the MI gain, and not only the first moments. Furthermore, only the case of white noise has been addressed in the previous subsection, but it should be relevant also to consider general colored noise and to study the influence of the power spectrum of the noise. The answers to these questions can be found by applying elaborate tools of stochastic analysis that compute the expansion of the Lyapunov exponents of randomly perturbed linear systems. Garnier and Abdullaev [2000] have applied this method to compute the statistical distributions of the Lyapunov exponents of the system (4.5). A striking point is that the statistical distribution of the modulation intensity can be expressed in terms of log-normal statistics. The
MI in inhomogeneous and discrete media
340
,
1
3
, ,o.,.0-~. ~ :~, o . ~, j
[5, w 4
.
.
.
.
///f.../- .... "\ ~ Gdet, theo ---. "11............ G2, theo "', II . . . . . G4, theo
(a)
"" U . . . . . . om
theo
~ \,, "X,X" ~ .
9
P'q
G6,
1
0 I
0
,
,
,
I
1
~
,
~
2
3
0)
, ,
,
,
,
.
.
.
, t~o=l, ~ : o . 1
.
(b)
,,..---..\ / , ,/"" " . .......... .
/,...Y
,,
\
,
,
,
,
I
1
~
J
,
,
,
Gdet , h u m
II . . . . .
G4, num
,
-
,, X ", [ ~ ' \ -..
I
.
--...,,'~, ..
0.5
.
~
-.....
. . . .
.
'"" "~" I . . . . . . G6, hum
,,~
~
.
II ............ Gz, num
','\, '\
/;" ..........
"/'!~
t ,
~
~
I
1.5
~
~
J
_ ~. ~ .
............... ~.-~
~
I
2
~
~
,
,
I
2.5
. . . .
3
O) Fig. 3. MI gain curves for/40 - 1, YoPo = 1, deterministic G V D and r a n d o m nonlinear coefficient with o 2 = 0.1 (l = 0.2 and o = 1). The solid curves correspond to the h o m o g e n e o u s case, while the dashed curves correspond to the m e a n Lyapunov exponents G2,. (a) Theoretical formulas obtained with white-noise approximations. (b) N u m e r i c a l simulations o f stochastic N L S E ' s averaged over 100 runs.
log-normal distribution has a such heavy tail that the moments of the intensity have very different behaviors. More exactly, the growth of the intensity of the modulation is governed by an expression of the kind exp(aW~ + bz), where W. is a standard Brownian motion. Accordingly, there are different behaviors for the
5, w 4]
341
MI in random media
-
(a)
~
'
~
.....
/
-
G4, the~
II. . . . . .
/
1.5 -
Gaet,theo G2 ' t h e ~
II ............ II . . . . .
G6'
the~
,m 9.. ~'\,\ ".. i~\ "\ ... ", "\ '.. ,. ,~!\" ~ "...... " ,~ "~
p.q
0.5
_
0 . . . .
0
I
I
. . . .
0.5
,
,
,
I
1
,
,
,
,
1.5
I
,
,
2
,
,
I
,
,
2.5
,
,
3
03
- (b)
Gde , , n u m ............ G2, n u m - . . . . ._G~,n u m -. . . . . G 6 , n u m . . . .
/ /r
.-=
'"~",,, .. ,,,~,\
'l I-.-. "-2"-., "~
1
p-q
-
"'..... 9... "L ....
~'~ ",. ~ ..... \ . ..................... ~i~. i~.ii .!:
. . . .
I
0.5
,
,
,
,
I
1
~
,
,
,
I
1.5
J
,
,
,
I
2
. . . .
I
2.5
. . . .
3
O) Fig. 4. Same as fig. 3, but now the nonlinear coefficient is homogeneous and the GVD is random, 02# =0.1 ( / = 0.2 and o = 1).
mean case and for a typical case, because W_- - v ~ with high probability, but ( e x p a W : ) = exP(2a2z). We shall assume for simplicity that the processes m# and m:, are independent; the dependent case gives rise to crossed terms that do not qualitatively alter the forthcoming results. We assume here that either/30 < 0 or/3o > 0 and that/30~02 > 2y0P0, i.e. the regions where there is no MI in the h o m o g e n e o u s framework. The sample
MI in inhomogeneous and discrete media
342
....
0.4
, ....
Ii [ 3 0 = - 1 ,
, ....
(a) .
."
I;I .Jl
9"
0.2 0.1
0
O +
O
+
, ....
o
G 4 , hum
............ c '-.
:~... .
0..""
....
Crz, theo G 4 , theo G2 , hum
..
o
"
0.3
I
~=0.161,,,,
='
9
o
O
2
[5, w 4
%'-.
d-
-
.~ '..
... +
-
,~ '-..
~
.." .."+
:
".s
_ -
"-....~
-r:
............
-
+ +
. . . .
I , ~ , , I , , ~ , I , , , , [ , , , , I
0.5
....
1
. . . .
1.5
, ....
2
I
2.5
. . . .
I
3
, J [30=-1, cy2=0.161 , . . . .
+
. . . .
3.5
4
, ....
- (b) :-
...............................
.............
~"
............... ...'"'"
......." O C
O
_.
C F
2-
"
.
_
%. -.~ O
9
OO
+
+
@
+
_
o~
P,N
0.5 .~,
,
9
* 9 ~."s / ."
[I ............ G4 ; theo II ~ G2, h u m ~] ~ G4 , h u m 1
2
3
4
5
I I }6
0,)
Fig. 5. MI gain spectra for/3 o = - 1 , 70P0 = 1. (a) Homogeneous nonlinear coefficient and random GVD, with ol] = 0.16 (l = 0.02 and o = 4). (b) Homogeneous G V D and random nonlinear coefficient, with o:2 = 0.16 (l = 0.02 and o = 4). The curves correspond to the theoretical formulas, while the crosses and circles correspond to the numerical results averaged over 1000 simulations.
and m o m e n t Lyapunov exponents have expansions at order 2 with respect to the noise level of the random process:
I(z) = I0 exp ( v / 2 H ( ~ o )
IV= +
G(~o)z),
(4.10)
5, w 4]
343
MI in random media
where W: is a standard Brownian motion, and G is the sample MI gain: =
/3 Oc4092 2(o)2 + (_D2) (a;,(O)) q- OCfi(O.))) ,
(4.11)
with a#(og) =
cos 21/301~o o92 + o92 z
(mi(z) mi(O)) dz,
j = fi, y,
and H(r = G(og). Note that aj is non-negative and proportional to the power spectral density of the process m/by the Wiener-Khintchine theorem (Middleton [ 1960], p. 141). Accordingly, the 2nth-moment MI gain is G2,(co) = (n + 1) G(co). Remember that the sample MI gain is the gain which is actually observed for a typical realization. This provides an affirmative reply to the question whether randomness enhances the MI process in the normal regime. We now assume that/3o > 0 and fi0~o2 < 2y0P0, i.e. the region where the homogeneous MI gain is positive. The sample and moment Lyapunov exponents have expansions at order 2 with respect to the noise level of the random process (4.10), where the sample MI gain G is G(r
/ = 2fioCOv/09c2 - ~02-
&j(w) =
exp -2fi009
fi2 O c4(.02
2(~0c2 -r 2) C02c- 002 z
(&:,(~o) + &fi(o9))
(m/(z)mi(O)> dz,
j = fi, y,
and H is given by fi 2(-O2 2 20)2) 2 4 H((-~ = 0)2_(.02 [( (-0c afi(O)q- O)ca],(O)] . Accordingly, the moment MI gains are:
G2n(O)) = G((.o)+ nH(oo). Analysis of these formulas shows that randomness reduces sample MI peak and optimal frequency:
(d)o2t = ,
2
l
4 (~;,
((_Dc/V~) _+_
344
MI in inhomogeneous and discrete media
[5, w 4
but random fluctuations of the nonlinear coefficient involve a highly fluctuating reduction of the MI peak and optimal frequency, so that we actually observe an enhancement of the mean MI peak: G2,1, opt 0922n, opt
while random fluctuations of GVD involve a deterministic reduction of the MI peak and optimal frequency. These surprising effects can also be observed in figs. 3b and 4b.
4.2.6. Conclusions Stochastic parametric resonances between the wavevectors of the perturbations and those of the modes of the linearized systems lead to MI for normal dispersion. The results represent a straightforward generalization of those obtained in the previous section for periodic variations of the fiber parameters. Because a random variation contains all frequencies, the parametric resonance arises for any frequency of the modulation. For anomalous dispersion the domain of MI increases, accompanied by a reduction of MI gain. More precisely the MI gain peak for almost every realization is described by a non-random quantity in case of GVD fluctuations, where only the Laplace transform of the autocorrelation function of the random process at the optimal frequency appears. In case of fluctuations of the nonlinear coefficient, the results show that the MI peak is reduced in probability, but is enhanced in mean, because there exist some rare events (i.e. realizations of the fluctuations of the nonlinear coefficient) for which the MI peak is drastically increased, and these rare events impose the mean value.
4.3. M1 in fibers with random birefringence We have seen that scalar MI in homogeneous media only occurs when the GVD is negative (anomalous regime). Cross-phase modulation (XPM) between two modes may extend the instability domain to the normal dispersion regime (Agrawal [1987]). This XPM-induced MI is also called Vector Modulational Instability (VMI).
5, w 4]
MI in random media
345
4.3.1. VMI induced by random nonlinearity and GVD
The MI of nonlinear continuous waves in birefringent fibers with random dispersion has been analyzed by Abdullaev and Garnier[ 1999]. The effects of random fluctuations of the chromatic dispersion are very similar to the scalar case. More exactly it is found that in the normal dispersion region all frequencies are modulationally unstable. In the anomalous case the MI spectrum is broadened and the MI gain is reduced. Although analyses have never been performed, the conjecture is that random fluctuations of the nonlinear coefficient give rise to effects that are similar to those observed in the scalar case. The analysis of vector MI induced by random birefringence is more interesting in that birefringence is the fiber parameter that exhibits the most important fluctuations, and it gives rise to the most interesting effects. 4.3.2. VMI induced by random birefringence and PMD
The evolution of polarized fields in randomly birefringent fibers is governed by the coupled NLS equations (Wai and Menyuk [1996]): . + .KA .+ iAAt . + fiA, + 9yATl=0, iA~
(4.12)
where ,4 is the column vector (A1,A2) t that denotes the envelopes of the electric field in the two eigenmodes (polarizations); we use standard dimensionless variables. The matrices K and A describe random fiber birefringence. The GVD coefficient is ft. The N1 term stands for the nonlinear terms: --,
Nl =
(
( Al 2 q._ o]A212)AI
,_,,t2,/,) + 2"'2"'1
(IA2 12 + O~A l] 2) A2 + E2 "d'21d" ' 2*
(4.13) ,
where the cross-phase modulation is a = 3 for linearly birefringent fiber. As shown by Wai and Menyuk [1996], one can eliminate the fast random birefringence variations that appear in eq. (4.12) by means of a change of variables leading to the new vector equation iU: + i ~ G + fiG, + yN2 = 0,
(4.14)
where /.~ - M-~d and U = (u,u) T represents the slow evolution of the field envelopes in the reference frame of the local polarization eigenmodes, and the matrix M obeys the equation iM- + KM = 0. The nonlinear term ~ reads /V2= ( (lu'2 + (~lU 2)u ) (OCIu 2 + IU[2)U
(4.15) '
346
MI in inhomogeneous and discrete media
[5, w 4
where the cross-phase modulation a = 1 after averaging over fiber birefringence (Evangelides, Mollenauer, Gordon and Bergano [1992]). ~ is a z-dependent matrix associated with the coupling between the modes due to perturbations:
~(z) :
( ) o 1
10
S~(z) +
(0i) i 0
S2(z) +
(,0) 0-
&(z),
(4.16)
where ~. are white Gaussian-distributed noises: (Sj(z)) = 0,
(Sj(z) ~.(z')) = 202 ~5(z - z'),
j = 1,2,3.
The presence of the term flU~ is associated with linear coupling between the modes, as well as an accumulation of a mismatch between their phases. In spite of this extension the model remains analytically solvable and it predicts general new features associated with the random nature of polarization MI. The nonlinear continuous-wave solutions of the system (4.14) read as uo(z) = A exp(iy(A 2 + B2)z) and vo(z) = B exp(iy(A 2 + B2)z). Linear stability is evaluated by substituting u(z,t)
= (A + u l ( z , t ) ) e x p [ i y ( A 2 + B2)z],
o(z,t)
= (B + v l ( z , t ) ) e x p [ i y ( B 2 +A2)z],
into eq. (4.14). By retaining only the first-order terms one obtains a linear system of equations for ul and ol"
iu,: + iS, v,, + S2ult q- iS3ult + [~Ultt + 2 y [A 2 t e ( u l ) -+-A B Re(o,)] = 0, iv,-
+ iSlult
-
S2Ult
-
iS3ol, + fivl,, + 2 y [B 2 Re(Vl) + A B Re(u,)] = 0.
The MI gains are defined as the Lyapunov exponents that govern the exponential growths of the Fourier components of the modulations Ul and o,. In the anomalous dispersion regime it has been found by Garnier, Abdullaev, Seve and Wabnitz [2001 ] that the MI region is increased by random birefringence so that all frequencies are unstable as soon as 02 > 0 while the MI peak is reduced. Denoting by P0 = A 2 + B 2 the total power, the MI peak is equal to 2yP0 when a = 0, and it decays as a increases. The first terms of the asymptotic
5, w 4]
347
MI in random media
expansion of the MI gain in the limit of small noise fi-~ 0-2 << 1 can be computed. For 0) < 0)c '= V/27Po/fi, the MI gain is reduced: G2(0)) = 2fi0)r
2 - 0)2 _ 40)20-2.
(4.17)
The MI peak G2,opt = fl0)c2 - - 2 a 2 0 ) c 2 is obtained at 0)22,opt = 2' 0)c2 _ 02 0)2c/fi. For 0) > 0)c the MI gain is positive, while it is zero for a = 0: "~
2
-2
G2(0)) = 40) 22KoKI(0)) - 2K2(0)) + 57-Pofi 02, 3KoKI (0)) + 5K2(0))- 5]/2P02f1-2
(4.18)
where K0(0)) = 0) 2 - 7Pofi -l , Kl(0)) = 0)V/0) 2 - 27Pofi -i, In the normal dispersion regime all frequencies are made unstable by random birefringence. As in the case fi > 0, the closed-form expression for the MI gain is too complicated to be written down explicitly. Nevertheless the first terms of the asymptotic expansion of the MI gain for small PMD fluctuations Ifi1-102 << 1 can be computed. It is found that for any 0) > 0 the MI gain is positive and given by (4.18). The MI gain spectrum is maximal for o~_,opt = ( v ~ - 1) 1/2 CYPo/Il~1 and the corresponding MI peak is 2(x/2 - 1)(2 vr2 + 1) yPo 02 G2, opt --
5 + 3v/2
I/~1
4.4. The case of a random-in-time potential We shall now say few words about the case of a potential that is random in time. The model is the nonlinear Schr6dinger equation:
iu: + flu, +
ylul2u =
V(t)u,
(4.19)
where V(t) is a random time-dependent potential. We could also consider temporal fluctuations of the nonlinear coefficient y(t) or the GVD coefficient fi(t). First of all we would like to point out that the problem at hand is different from the homogeneous or spatially random problems in that there is no continuous-wave solution of eq. (4.19). Therefore the question of the stability of such a stationary wave is meaningless. Actually it is well-known that in linear and random media all eigenstates of the Schr6dinger operator are localized (Anderson [1958]). Previous work on the stationary NLS equation has shown strong evidence that there exist delocalized transmission states (Doucot and Rammal [1987]). However,
348
MI in inhonlogeneous and discrete media
[5, w 5
since only the time-harmonic problem has been addressed, not all of these states are physical, so that a complete study with the time-dependent model (4.19) is required to understand this issue. Numerical experiments have been performed by Caputo, Newell and Shelley [1990]. An input pulse that decomposes into lumps without randomness (standard MI) has been shown to generate new pulses that are NLS solitons. Indeed, solitons are normal propagating modes for the homogeneous NLS equation. A lot of work was then devoted to the propagation of solitons across random media. Theoretical (Kivshar, Gredeskul, Sfinchez and Vfizquez [1990]) and numerical studies (Knapp [1995]) have demonstrated that, for a NLS soliton propagating in a random medium, there exist two distinct regimes of behavior which depend on the soliton parameters. One of these regimes has been shown to be very different from the localization regime in that the soliton retains its mass although it loses velocity.
w 5. MI in nonlinear discrete optical systems To date, quickly developing technologies like epitaxial growth, ion exchange in solids, electrical poling, etc., permit the fabrication of new kinds of thin films, multilayered systems, and different photonic band-gap structures for advanced photonics applications. The discrete nature of such structures gives rise to qualitatively new types of excitations and effects connected with them. Arrays of coupled waveguides represent a prominent example of discrete optical systems, where interplay between inherent discreteness and nonlinearity qualitatively alter the dynamical behavior compared with their continuous counterparts. In particular, spreading of the initial excitation due to linear coupling, which can be viewed as effective discrete diffraction, can be compensated by nonlinearityinduced localization. As a result, localized modes, frequently referred to as discrete solitons (DSs) are formed. Since the original investigations (Dolgov [1986], Sievers and Takeno [1988], Page [1990]) a considerable and steadily growing amount of interest has focused on the study of these localized modes in nonlinear discrete systems because of their relevance in various fields such as solid-state physics, optics, and biology (Scott [1992], Flach and Willis [1998], Hennig and Tsironis [1999], Lederer, Darmanyan and Kobyakov [2001]). Different kinds of bright and dark DSs may exist in this environment (Cristodoulides and Joseph [1988], Cai, Bishop and GronbechJensen [ 1994], Konotop and Salerno [ 1997], Darmanyan, Kobyakov and Lederer [ 1998b], Hennig and Tsironis [1999]), where MI of the nonlinear plane-wave solution is a necessary condition for bright soliton formation. Dark DSs, on the
5, w 5]
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349
contrary, need a modulationally stable background. Experimental demonstrations of these phenomena in nonlinear mechanical and electrical lattices have been reported by Denardo, Galvin, Greenfield, Larraza, Putterman and Wright [ 1992] and Marquie, Bilbaut and Remoissenet [1995]. Specific features of the MI and recurrence phenomena as well as pattern formation in nonlinear lattices have been discussed by Kivshar and Peyrard [1992], Kivshar, Haelterman and Sheppard [1994], Burlakov, Darmanyan and Pyrkov [1995], Burlakov, Darmanyan and Pyrkov [1996], Daumont, Dauxois and Peyrard [1997], Leon and Manna [1999], Burlakov [1998] and Vanossi, Rasmussen, Bishop, Malomed and Bortolani [2000]. In particular, it has turned out that a peculiar feature of many discrete systems consists in the critical dependence of both the MI gain and the MI domain on the wavenumber related to the discrete variable. Nonlinear optical waveguide arrays have been proposed as the basis for different schemes of alloptical signal processing and switching (Bang and Miller [1996], Krolikowski and Kivshar [ 1996], Aceves, De Angelis, Peschel, Muschall, Lederer, Trillo and Wabnitz [ 1996], Peschel, Muschall and Lederer [ 1997], Darmanyan, Kobyakov and Lederer [ 1998b]). Experimental observations of ultrafast switching as well as existence and dynamics of DSs were reported for arrays fabricated on the basis of A1GaAs (Millar, Aitchison, Kang, Stegeman, Villeneuve, Kennedy and Sibbett [1997], Eisenberg, Silberberg, Morandotti, Boyd and Aitchison [1998], Morandotti, Peschel, Aitchison, Eisenberg and Silberberg [1999]). As mentioned above, the stability of nonlinear solutions is an important property which drastically affects the entire dynamics. This is of particular importance for a nonintegrable discrete system where many solutions can be found only approximately. In the following we consider stability of discrete plane-wave solitons in an array of n lossless, identical and equidistant channel waveguides or fibers. The dynamics of this system, as well as of many others, is described by the discrete nonlinear Schrrdinger equation (DNLSE) or by its modifications depending on the kind of nonlinearity, the scalar or vector nature of the system, etc. In w5.1 we will be dealing with the MI of discrete plane-wave solitons of scalar and vectorial DNLSE with Kerr-like nonlinearity; we consider the case of quadratic nonlinearity in w5.2. Finally, w5.3 is devoted to the study of the influence of temporal effects on the MI in an array with cubic nonlinearity.
5.1. M1 in discrete cubic media
For generality we start with the vectorial case where two components with different frequencies or polarization states copropagate in a nonlinear waveguide.
350
[5, w 5
MI in inhomogeneous and discrete media
In this case the cubic nonlinearity provides a self-phase modulation (SPM) of each field component as well as a nonlinear coupling between the components resulting in cross-phase modulation (XPM) and energy exchange. The latter effect can be neglected provided that the wavevector mismatch between both field components is large. In continuous systems the field dynamics can then be described by two incoherently coupled NLSEs, the properties and solutions of which have been studied extensively. In turn, in the discrete case the twocomponent field dynamics is described by two coupled DNLSEs constituting the following set of difference-differential equations" dAn
i---d~- + Cl(An-1 + An+l) + (2,11 An 2 + 2,12[BnlZ)An = 0, dBn
i-~
(5.1)
+ C2(Bn-1 + Bn+l) + (/~21 Anl 2 +/~22 Bn[2)Bn = 0,
(5.2)
where An and Bn represent the field envelopes of both components in the nth channel, the evolution variable Z denotes the spatial coordinate along the waveguide, and C~,2 - :r/2L~,2 are the respective coupling coefficients, with L~,z the half beat lengths of the corresponding two-core coupler. The effective nonlinear coefficients Ai,j = ooini,jr (i,j - 1,2) with Oi,j = f dx dy [Ri[ 2 [Rj 2 / f dx dy IRi['( include the dimensionless functions Ri(x,y) describing the transverse mode profile and the cubic nonlinear coefficients nij. For weakly guided modes in optical fibers we have cri,j "~ 0.5 (Agrawal [1995], Abdullaev, Darmanyan and Khabibullaev [1993]). The existence and stability of bright and dark vectorial discrete solitons as well as MI of plane waves in the system described by eqs. (5.1) has been studied by Kobyakov, Darmanyan, Lederer and Schmidt [1998] and Darmanyan, Kobyakov, Schmidt and Lederer [1998]. Equations (5.1) can be recast in a more convenient dimensionless form: dan
i---~z + Ca(an-1 + an+l) + (~.ala,I 2 + IbnlZ)a,, - 0,
(5.3)
dbn
i--d-fz + cb(b,-1 + bn+l ) + (Jan]2 + ~b[bnlz) bn : 0, where an = ]~21/~,2l~/2AJOm,x, b,
IOmaxl2, Ca,b = CI,2LNL, /~a = ~11/~12 > 0, ~b = ~22/~12 > 0, and Bmax is the peak amplitude of the second field component. Provided that the field envelopes vary slowly with n, eqs. (5.3) can be transformed into a continuous system which describes likewise pulse propagation = On/Bmax,
z
= Z/LNL,
LN 1 -- /~12
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MI in nonlinear discrete optical systems
351
in a dispersive medium and beam evolution in a planar waveguide with SPM and XPM. For the specific case of equal SPM and XPM coefficients the corresponding continuous system has been proven to be integrable. A discussion of various peculiarities of the continuous system, including the formation of bright and dark solitons as well as modulational instability of plane wave solutions, can be found for arbitrary ratios between SPM and XPM coefficients in the article by Agrawal [1995]. As in the continuous model the system (5.3) describes coupling between modes which either oscillate at different frequencies (~,a ~ ~h ~ 0.5) or are orthogonally polarized in a highly birefringent waveguide. In the case of an elliptically birefringent fiber ~,~ = ~+ = / l , where/l varies between 0.5 and 1.5 depending on the angle of ellipticity. Let us consider the MI of the stationary plane-wave solution to eqs. (5.3) that has the form an = a exp [i (qan + kaz)] ,
b, = b exp [i (qhn + kbz)] ,
(5.4)
where ka = 2Ca COSqa + ~,aa2 + b 2,
kb = 2ch cos qh + ~.bb2 + a2.
(5.5)
To investigate the stability of this solution we substitute into eqs. (5.3) a slightly perturbed eq. (5.4), namely, hn = (a + ~n(z))exp[i (qan + kaz)], b~ = (b + ~,(z))exp [i (qbn + kbz)], where ~n(Z) = Ul exp [i (Qn + Kz)] + u2 exp [-i (Qn + K'z)] , (5.6) ~n(z) = v, exp [i (Qn + Kz)] + 02 exp [-i (Qn + K*z)]. Performing linearization with respect to small perturbation we end up with the following eigenvalue problem for the perturbation wave vector K: - K + f + ~a a2 ~a a2
det
K + f_
a2 b 2
a 2b 2
a2 b 2
a2b 2
a 2b 2
a 2b 2 K + g+
aZ b 2
aZ b 2
/thb 2
=0,
(5.7)
~+b2 - K + g _
where f+ = 2Ca[COS(qa -+- Q ) - cos qu] + ~,aa2, g+ = 2cb[cos(qh -+- Q ) - cos qb] + ~bb 2. Thus, according to eq. (5.6), G = IImKI provides us with the MI gain. To resolve the eigenvalue problem (5.7) one needs to apply numerical methods. However in some particular cases it can be treated analytically. In what follows
MI in inhomogeneous and discrete media
352
[5, w 5
we consider two of these cases important for applications, viz., (i) scalar DNLSE and (ii) vectorial DNLSE at q = 0, Jr.
5.1.1. Scalar D N L S E For the scalar D N L S E , dbn i--d-zz + c(bn_l + b~+l)+ Z b,,12b,, = 0,
(5.8)
i.e. for a = Ca =/1.a = 0, the eigenvalue problem (5.7) reduces to that considered by Kivshar and Peyrard [ 1992]. The solution in this case can be found easily: K =-2csinQsinq+
2 qsin2(sQ) - Zb 2 cosq). V / 8csin2(-~Q)(2ccos , 1
(5.9)
Thus for Z cosq > 0 the plane-wave solution of the scalar DNLSE is modulationally unstable with the gain
G = 2 ]sin(sQ) , ] v / 2ccosq(Xb 2 - 2 c c o s q s i n 2 ( Q / 2 ) ) .
(5.10)
It is important to note that for cos q < 0, MI occurs in defocusing media where X < 0. This is in contrast with MI in the continuous case, which can be reproduced from eq. (5.10) by inserting cos q = 1, sin Q ~ Q.
5.1.2. Vector D N L S E at q = O,jr It can be inferred from eqs. (5.7) and (5.10) that the stability of the plane-wave solution (5.4) strongly depends on the carrier wavevectors. This, as mentioned above, is an intrinsic feature of discrete systems which distinguishes them from the continuous analogues. We note that for the strongly localized soliton-like solutions, amplitudes in two adjacent channels are either in phase (so-called unstaggered solution q = 0) or out of phase (staggered solution q = Jr). In what follows, for simplicity, we restrict ourselves to these two cases. This allows a simplification of eq. (5.7), i.e. for q = 0, Jr it can be reduced to the biquadratic equation K4 + p l K 2 +P0 = 0
(5.11)
with the coefficients
Po = f 2 g2 _ ~2[a4 g2 + b4f 2 + a4b4(4_/~2)] + 2a2b2(l~ga 2 + ~j,b 2 _ f g ) ,
(5.12)
Pl
= ~,2(a4 + b 4) _ f 2 _ g2,
(5.13)
f
= ~a 2 - 2Ca(1
- cos Q),
g = Xbe - 2ch(1 - cos Q).
(5.14)
Here we have assumed ~,, = 2h = /~ (different polarizations) and sgn(ca,b) = sgn(Z0 cOSqa,b). Thus, provided that C~.e > 0, which is the only case that is
5, w 5]
MI in nonlinear discrete optical systems
353
physically meaningful, and ~'0 > 0 (focusing nonlinearity) hold, the normalized linear coupling Ca,b is positive (negative) for qa,b = 0 (Jr), respectively. Note that the vector DNLSE can support solutions where an unstaggered component is coupled with a staggered one (e.g. qa = 0, qb = Jr), viz. Ca =--Cb. The MI gain can be calculated straightforwardly from eq. (5.11) as
G =
Imp2 (-p,+
V/p~-4p0)
(5.15)
1 2 It equals zero if 0 < P0 < ~Pl and pl < 0 hold. For equal moduli of the coupling coefficients, ]ca] = Icbl, eq. (5.11) exhibits zero gain for 0 < Q < Jr only provided that
Ca = Cb < 0,
~, >~ 1,
(5.16)
or
Ca =--Cb > O,
2a/(a2+ 1) ~ /l <~ 1,
(5.17)
where without loss of generality we assumed a < b - 1. The solution (5.4) is modulationally unstable in those regions of parameter space where eqs. (5.16) and (5.17) are not satisfied. The respective gain is plotted in fig. 6 as a function of the amplitude of the first component a and the nonlinearity/l. As predicted by eqs. (5.15) and (5.16), the vectorial unstaggered (qa,b - 0 ) plane-wave solution is always unstable in a medium with focusing nonlinearity (Aij > 0) (see fig. 6a), which coincides with the behavior of the corresponding scalar mode. However, if both components are staggered the mode is stable only for /l >/ 1 (see eq. 5.16 and fig. 6b). A vectorial solution which consists of one unstaggered and one staggered wave can be stable only if the weaker component is unstaggered (unstable in the scalar limit) and, in addition, eq. (5.17) is satisfied (see fig. 6c). This condition defines a boundary between the stable and unstable regions: a small periodic perturbation corresponding, e.g., to point S(~ - 0.8) in fig. 6c does not grow and the plane wave propagation is stable. However, if the amplitude of the A-component is slightly increased and exceeds a critical value (point T in fig. 6c), the same small perturbation (1% of the wave amplitude) leads to the formation of a nonstationary cnoidal-like wave (fig. 7). If the stronger component is unstable in the scalar limit, i.e. Ca =--Cb < 0, this vectorial solution is always unstable (fig. 6d). The nonmonotonic behavior of the
354
M I in i n h o m o g e n e o u s a n d discrete media
[5, w 5
Fig. 6. Maximum MI gain as a function of the amplitude of the first component a(b = 1) and the nonlinear coupling A; (a) ca = cb = O.1, (b) Ca = ch - -0.1, (c) ca - -cb = O.1, (d) -Ca = Cb = O. 1.
gain in fig. 6d arises from the fact that the maximum gain is shifted from the edge of the Brillouin zone, Q = :r, to its center. This shift can also be recognized in fig. 8, where the maximum gain is plotted as a function of linear coupling (equal for both components) together with the respective value of Q. For larger Ca,b the maximum gain tends to saturate and the respective value of Q decreases. Thus, no qualitative difference in stability behavior arises upon a change in the linear coupling. The existence of stable regions (see figs. 6b,c) is a prerequisite for the formation of vector dark solitons. An interesting feature of the system studied here consists in the instability of the vectorial plane-wave solution for a certain strength of the nonlinear coupling where either component is stable in the scalar limit. The analysis has shown that the case of equal self- and cross-phase modulation coefficients represents an interesting situation in the discrete scenario as well. As a matter of fact, it represents a stability boundary for solutions consisting of a staggered and an unstaggered component.
5, § 5]
355
M I in n o n l i n e a r discrete optical s y s t e m s
Fig. 7. Modulational instability o f the periodically perturbed plane wave solution corresponding to point T in fig. 6c; the a - c o m p o n e n t is shown, the b - c o m p o n e n t has a similar structure. The waveguides are labelled by positive numbers. Parameters: ca = - c h - 0.1,/~a =/~h = 0.8, a = 0.55,
b - 1, Q-- ½~.
2.0 .=_ 1.5
,• .
Z~ E 1.0
,
1.0 0.8
0.6
.~
0.4
~:~
._ e~
0.5
0.2 0.0
0.0 -0.4
-0.2
0.0 Ca=C
0.2
0.4
b 1.0
1.4 -=e~
",,
1.2
",
eat~
,, E = E ._ E
L
0.8
,
1.0 0.8
,
0.6
'-
0.4 0.2
(b)
,
0.6
",
0.4
~
0.2 0.0
0.0 -0.4
-0.2
0.0 0.2 Ca=-C b
0.4
Fig. 8. M a x i m u m instability gain and the respective wavevector o f modulation Q as a function o f linear coupling for the two components; (a) c a = cb, (b) c~ = - c h. Parameters: a = 0.5, b - 1" solid curves ~'a = Ah = 0.5, dashed curves Aa = ~.h = 1.5.
356
MI in inhomogeneous and discrete media
[5, w 5
5.2. M I in discrete quadratic media
It is a common belief that, compared to the cubic scenario, a quadratic nonlinearity provides a greater variety of effects and, more importantly, that they are obtainable for lower optical power. The particular form of the nonlinearity leads to energy exchange between the field components and additionally brings another crucial parameter, the phase mismatch, into play. Although the equations describing the field dynamics in such media are not integrable, stable mutually locked solitary waves may exist, as was experimentally confirmed in continuous bulk media (Torruelas, Wang, Hagan, Van Stryland, Stegeman, Torner and Menyuk [1995]) and in film waveguides (Schiek, Baek and Stegeman [1996]). As far as discrete media with quadratic nonlinearity is concerned the existence of different families of bright and dark localized two-field states has been demonstrated theoretically, and their fundamental properties have been studied (Peschel, Peschel and Lederer [1998], Darmanyan, Kobyakov and Lederer [1998a], Miller and Bang [1998], Kobyakov, Darmanyan, Pertsch and Lederer [ 1999]). In this subsection we consider the MI of a two-component plane wave in such an environment. The evolution of the two-component field in a waveguide array with quadratic nonlinearity may be described by the following proper normalized set of difference-differential equations (Kobyakov, Darmanyan, Pertsch and Lederer [1999]): dan i--~z + Ca(an-I + an+l) + 2ya,*b,, = O,
(5.18)
db,
i--dTz + cb(bn-1 + bn+l) + fibn + ya2n = O,
where z denotes the propagation direction, an and bn are the amplitudes of the fundamental (FW) and second-harmonic (SH) guided waves, Ca,b and }' are linear and quadratic nonlinear coupling coefficients, respectively, and fi is the wavevector mismatch. Equations (5.18) remind of the well-known system describing the evolution of spatial solitons in a quadratically nonlinear film waveguide where diffraction is replaced by linear coupling. It is evident that in the long-wavelength case (slow variation with n) the dynamic equations (5.18) transform into the continuum limit. As mentioned above, several types of localized solutions to this system have been found both in continuous and in discrete cases, while the discrete case provides more variety of DS solutions. To find the stationary plane-wave solutions to the set (5.18) we insert the ansatz an = a exp [i (qn - kz)] ,
bn = b exp [2i (qn - kz)] .
(5.19)
5, w 5]
MI in nonlinear discrete optical systems
357
This gives us a relation between the FW and SH amplitudes a and b as a 2 = 4b 2 +
b(4ca cos q - 2cb cos 2q - fi)
> O,
(5.20)
and the dispersion law which relates the wavevector in the propagation direction (k) with the transverse wavevector (q) and the SH amplitude b:
k = --2(Ca COSq + yb).
(5.21)
As a matter of fact, a certain SH amplitude b applies to a FW amplitude a of either sign. To check the stationary plane-wave solution (5.19)-(5.21) against MI we repeat all steps of the linear stability analysis of two-component fields described above for the array with cubic nonlinearity. Considering, for the same reason as in the previous subsection, the cases of q = 0, Jr we arrive at the conclusion that MI gain G is described by eq. (5.15) where, instead of eqs. (5.12)-(5.14), parameters P0 and pl are given by the following expressions:
Po = f 2 g 2 + 16y4a4 _ 4 7 2 b 2 g 2 _ 8y2a2fg,
(5.22)
Pl
= 4yZ(b 2 - 2a2+) _ f 2 _ g2,
(5.23)
f
= 2yb + 2Ca(1 -- COSQ),
(5.24)
g = 4yb • 4c,, - 2ch cos Q -/3,
where the upper (lower) sign applies to q = 0 (q = Jr). In the long-wavelength limit (small Q) the MI gain approaches that of the continuum model (Trillo and Ferro [1995]). The maximum MI gain is plotted in fig. 9 as a function of the stationary SH amplitude b and the mismatch fi for in-phase (q = 0) and out-of-phase (q = Jr) solutions. With regard to the dispersion relation (5.21) these two cases correspond to opposite signs of dispersion of the linear waves. Evidently the change of the character of dispersion critically affects the stability behavior. Although the stability ranges for the two regimes differ considerably, a common stable region with (b < 0 and fi < 0) can be identified. To illustrate the effect of the MI of the plane-wave solution it is appropriate to consider evolution of dark solitons. The consequences of the background stability for the dynamics of dark DSs is shown in fig. 10, where the propagation of two dark kink-like DSs which are in close proximity in the b-fi plane is displayed. The obviously stable DS corresponds to the domain where G = 0 holds (fig. 10a). A slight change of the SH amplitude b causes the solution to move to the unstable region. As a result the dark DS gets unstable and decays after some distance (fig. 10b).
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[5, w 5
Fig. 9. Maximum MI gain as a function of the normalized SH amplitude and the wavevector mismatch. No plane-wave solution exists in the shaded region. Bright regions correspond to stable solutions (G = 0). Parameters: c, = ch = 0.1; (a) q = 0, (b) q = Jr. 5.3. Temporal effects in arrays with cubic nonlinearity To study the propagation of short pulses in waveguide arrays one needs to take into account the dispersion that acts in each channel o f the array. In this case an additional variable, time, appears in the governing equation. The equation becomes 1+2 type, where along with the evolution variable there are two others, one of which is discrete and the second is continuous. Such a discrete-continuous combination introduces new interesting features into the system dynamics. In particular, the discreteness stops the collapse that takes place in fully continuous 1+2 systems, and leads to the formation o f stable 2D-localized structures. The existence and stability o f solitary waves localized both along (temporally) and across the array have been considered by Aceves, De Angelis, Luther
MI in nonlinear discrete optical systems
5, w 5]
359
1.0
0.5
-:-
----- -
+,=,,~-.
.,_.
,&>~ . ...~
--~,~-~
' ! r,!,!l,i~ 0.5 ' " "
(b)
':
/ ,/
"~
l/,,.,
',
,
q.-.
-.
,_, :... . . . . . . ,.. .
Fig. 10. P r o p a g a t i o n o f a dark, k i n k - l i k e i n - p h a s e D S (the n o r m a l i z e d F W intensity is shown). P a r a m e t e r s : ca = ch = 0.1, fi =
2; (a) b =
0.76 ---+ Gma • = 0; (b) b -
0.72 --+ Gma x = 0.24.
and Rubenchik [1994], Aceves, De Angelis, Luther, Rubenchik and Turitsyn [1995], Aceves, Luther, De Angelis, Rubenchik and Turitsyn [1995], Buryak and Akhmediev [ 1995], Laedke, Spatschek, Turitsyn and Mezentsev [ 1996] and Darmanyan, Relke and Lederer [1997]). The case of arrays with periodically modulated linear coupling was studied by Relke [1998]. The normalized slowly varying envelope b,, of the optical field in the nth channel of the waveguide array allowing for temporal dispersion is described by the dimensionless discrete-continuous nonlinear Schr6dinger equation
i db,
d 2b,,
-d-Zz +
+
+
:o,
(5.25)
where t is the normalized time in the moving frame, and s = 1 or s = -1 stands for anomalous or normal group velocity dispersion, respectively. In this subsection we study the stability of a moving plane-wave solution to eq. (5.25) that may be written as b , ( z , t) = b
exp[i(kz +
qn -
where k = Z b 2 + 2c cos q - so92.
wt)],
(5.26)
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MI in inhomogeneous and discrete media
[5, w 5
Here q is an arbitrary transverse wavevector and to is the deviation from the carrier frequency. We again perform the familiar linear stability analysis by modulating the unperturbed amplitude b ~ b + ~p,(z, t) in eq. (5.26), where l p , ( t , z ) = u exp[i(Kz + Q n - g2t)] + v exp[-i(K*z + Q n - g2t)].
(5.27)
Inserting eq. (5.26) into (5.25) we arrive at the dispersion relation for the perturbations: (K + 2stog2 + 2c sin Q sin q)2 = F ( F - 2/lb2),
(5.28)
where F ( g 2 , q, Q ) = sf22 + 4 c c o s q s i n 2 ( ~ Q )l .
(5.29)
Equation (5.28) straightforwardly gives the following expression for the MI gain: I m [ K ] - G = v/F(2~b 2 - F) > 0,
(5.30)
which varies with the wavenumber q of the stationary solution as well as with both the frequency Y2 and wavenumber Q of the perturbation. In contrast, the real part of K additionally depends on the frequency to of the carrier wave. Two familiar limiting cases can be read off from eqs. (5.29) and (5.30) as MI of (i) the nonlinear Schr6dinger equation (see, e.g., Abdullaev, Darmanyan and Khabibullaev [1993], Agrawal [1995]) for c cos q = 0 (diffractionless case) and (ii) the discrete nonlinear Schr6dinger equation for s - 0 (dispersionless case), see eq. (5.10). It is obvious that the existence criterion, the domain and the gain of MI are essentially determined by sgn(~) and by F(Y2,q, Q) which, in turn, critically depends on the signs of both s and cosq, see eq. (5.29). For example, in the case of defocusing, ~ = -1, the MI occurs only for - 2 b 2 < F < 0, which is possible at (i) s - 1 (anomalous dispersion) and cos q < 0 (negative "diffraction") for the boundaries of the MI domain given by g22 - 2 b 2 < Y22 < g22, or at (ii) s = - 1 (normal dispersion) and cos q ~> 0 (positive/zero "diffraction"), in which case the MI domain is located in the frequency region g22 + 2b 2 > .(22 > g22, or, lastly, at (iii)s - -1 (normal dispersion) and cos q <~ 0 (negative/zero "diffraction") l for 2b 2 - g2c2 > g22, where Y22 -- 4c Icosq sin2(~Q). Detailed analyses of other cases as well as an analysis of the stability of a soliton array propagating in a waveguide array have been published by Darmanyan, Relke and Lederer [ 1997].
5, w6]
Conclusions
361
w 6. Conclusions In this review we have considered many aspects of the MI in inhomogeneous nonlinear media. The MI of electromagnetic waves in nonlinear optical fibers with periodic variations of amplification, dispersion or birefringence has been investigated. The MI in random media shows new properties such as the extension of fundamental MI domain to all frequencies of modulation and the existence of MI in the region of normal dispersion. Similar features can be observed in the MI in discrete nonlinear media. It is desirable to perform experiments on the observation of MI in random nonlinear media. At present it seems that the best possibilities for realizing these experiments are in optical fibers with random parameters and in arrays of nonlinear polymer waveguides. Some related problems have not been touched on in the review. First this concerns the MI of incoherent beams in nonlinear media. The investigation of this problem is important for understanding the conditions of existence of incoherent solitons, a phenomenon demonstrated in recent experiments (Mitchell and Segev [ 1997]). A second problem is the MI in fiber-ring soliton lasers with periodic dispersion. Recently, modulational instability with sideband generation in a passively mode-locked fiber-ring soliton laser has been observed by Tang, Man, Tam and Demokan [2000] and Tang, Fleming, Man, Tam and Demokan [2001]). The periodic modulation of dispersion was caused by the pulse circulating in the laser cavity. A promising perspective is the study of MI in periodically modulated quadratic media and in media with more complicated nonlinearities. Some preliminary results have been obtained by Corney and Bang [2001]. Another important direction is the MI in periodic media with higher dimensions. This is important for investigations of localized waves in nonlinear photonic crystals. The MI in 2D and 3D Bose-Einstein condensates in an optical lattice and the generation of localized states seems to be another significant problem for future studies. It should be of great interest to extend the approaches outlined in this review to the problem of MI in nonconservative modulated continuous media and discrete nonlinear optical systems with linear and nonlinear amplification or damping. The analysis of the regions of MI can be useful for the search of autosolitons both in continuous and in discrete media. Experimental observations of MI in a system possessing autosolitons, namely in a transmission system with periodic semiconductor amplifiers and in-line filters, have been reported by Goles, Darmanyan, Onishchukov, Shipulin, Bakonyi, Lokhnigin and Lederer [2000]. It is also interesting to investigate the long-time behavior of MI in modulated media. For this purpose it is necessary to go beyond
362
MI in inhomogeneous and discrete media
[5
the linear stability analysis. One of the ways to do this analytically is to utilize the three-mode approximation. It has been shown that this approximation provides a good description of the complicated dynamics of MI under periodic perturbations of the medium. The theory of perturbations involving the dynamics of nonlinear periodic waves in periodic media needs to be developed. Recently, the existence of a new family of stationary nonlinear periodic solutions in NLSE with periodic potential has been reported by Bronski, Carr, Deconinck and Kutz [2001 ].
Acknowledgements EA. and S.D. appreciate valuable collaborations with E Lederer, S. Bishoff, A. Kobyakov and M.P. Soerensen, our coauthors of original papers. S.D. and EA. are grateful to US CRDF (Award ZM2-2095) for the partial financial support.
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Author index for Volume 44
A Aaviksoo, J. 184, 185 Abarbanel, H.D.I. 74 Abdullaev, EKh. 92, 305-307, 310, 315, 319, 320, 325,327, 337, 339, 345,346, 350, 360 Abraham, N.B. 4, 59 Abramowitz, M. 188 Aceves, A.B. 306, 321,349, 358, 359 Agostini, P. 160, 162 Agrawal, G.P. 5, 6, 9, 10, 14, 19-21, 23, 26, 27, 31, 33, 39, 40, 55, 58, 65, 76, 92, 116, 173, 179, 184, 193,305, 307, 310, 324, 327, 332, 333, 344, 350, 351,360 Ahlers, V. 66, 69 Aida, T. 66 Aitchison, J.S. 349 Akhmanov, S.A. 88 Akhmediev, N.N. 306, 310, 359 Akimoto, K. 163 Aleshkevich, V. 192 Alfano, R.R. 294 Allan, D.C. 124 Alsing, P.M. 61 Amer, N.M. 293 An, K. 243 Andersen, P.E. 244, 246, 247 Anderson, J. 189, 191 Anderson, P.W. 347 Andrejco, M.J. 111 Andres, M.V. 132 Andres, P. 132 Anisimov, V.Ya. 243 Annovazzi-Lodi, V. 66, 73, 74 Antoine, P. 159 Antonetti, A. 162 Arecchi, ET. 3, 12 Argyros, A. 124 Arioli, M. 103
Arriaga, J. 97, 124-126, 131, 132 Askaryan, G.A. 197 Assion, A. 113 Atkin, D.M. 124 Aug6, E 160 Auston, D.H. 163, 179 B
Backus, S. 113 Bado, P. 158 Baek, Y. 356 Bagwell, T.L. 286 Baizakov, B.B. 307 Bakonyi, Z. 361 Balcou, P. 160 Balle, S. 52, 59 Bang, O. 330, 331,349, 356, 361 Bardeen, C. 113 Barnes, J. 188 Barry, L.P. 96, 101 Bartels, R. 113 Barton, J.K. 219, 250, 253 Bashkansky, M. 272, 277, 278, 287, 289 Bassett, I. 124 Bateman, H. 153, 170, 173 Batovrin, V.K. 286 Battle, P.R. 272, 289 Baum, C. 183 Baumert, T. 113 Baumgartner, A. 219, 265, 273, 288 Baums, D. 52 Baxendale, P.H. 336 Beach, N.M. 113 Beaurepaire, E. 221,222, 288 Beck, P.A. 286 Beisser, EA. 292 Benjamin, T.B. 305, 308 Bennett, J.S. 275 367
368
Author index for Volume 44
Bennion, I. 306, 318 Bergano, N.S. 346 Bergt, M. 113 Berkhoer, A.I. 327 Berlien, H.P. 282 Bertolotti, M. 192, 193 Bespalov, V.I. 305, 308 Betz, G. 119 Biellak, S.A. 58 Bilbaut, J.M. 349 Birks, T.A. 124, 125, 287 Birman, J. 184 Birngruber, R. 239 Bischoff, S. 307, 337 Bishop, A.R. 348, 349 Bizheva, K.K. 241,242 Blanchot, L. 222, 288 Blazek, V. 218, 290 Bleuler, H. 293 Blondel, M. 49, 78 Blow, K.J. 306, 318 Boas, D.A. 241,242 Boccara, A.C. 221,222, 288 Bochove, E. 39 Boivin, L. 112 Bollond, P.G. 96, 101 Bonner, R.E 248 Boppart, S.A. 119, 218, 219, 222, 224, 284, 287, 293, 295 Bor, Z. 196 Born, M. 217, 225, 239, 243 Bortolani, V. 349 Boulnois, J.-L. 282 Bouma, B.E. 218, 219, 222, 224, 248, 281, 287, 293,294 Bourquin, S. 222 Bowie, J.L. 95, 163 Boyd, A.R. 349 Brabec, T. 173 Bracewell, R.N. 249 Braun, A. 119 Braun, D.M. 286 Brecke, K.M. 252 Breger, P. 160, 162 Brennan III, J.E 289 Brezinski, M.E. 218, 219, 222, 224, 248,278, 287 Brinkmeyer, E. 274 Brittingham, J. 154 Brixner, T. 113
Brock, J. 3 Brodsky, A. 248 Broeng, J. 124 Bromage, J. 179 Bronski, J.C. 319, 362 Brunner, H. 224 Bucksbaum, E 163 Bucksbaum, P.H. 160, 161 Burgess, L.W. 248 Burggraf, H. 293 Burlakov, V.M. 349 Burow, K.E. 236 Buryak, A. 359 Byer, R.L. 330 C Cai, D.L. 348 Campbell, D.J. 293 Campbell, I. 184 Caputo, J.G. 320, 348 Carpenter, S.D. 113 Carr, L.D. 362 Carroll, T.L. 5, 65, 66, 74 Chambaret, J.P. 162 Chance, B. 217 Chandalia, J.K. 128 Chang, C.C. 112, 113 Chang, W. 217, 219, 271 Chao, L.C. 265 Chau, T. 79 Chen, H.E 74, 79 Chen, H.H. 306, 334 Chen, K. 183 Chen, N.G. 293 Chen, Z. 250-253, 255,261,265,267, 295 Cheng, P.H. 287 Cheng, Z. 108 Cheriaux, G. 163 Chemikov, S.V. 106 Chesnoy, J. 184 Chida, K. 218 Chilla, J.L. 93 Chinn, S.R. 217, 218, 290 Chirkin, A.S. 88 Chou, P.C. 289 Chraplyvy, A.R. 28 Christensen, C.R. 275 Christodoulides, D.N. 106 Christov, I.P. 113 Chu, K.C. 94, 293, 294
Author index for Volume 44
Chu, P.L. 97, 104 Chudoba, C. 287 Chumakov, Yu.P. 224 Ciric, I. 150 Clark, S.W. 119 Clement, T.S. 97 Clivaz, X. 248, 286 Coker, A. 136 Colet, P. 5, 58, 73, 74, 76, 77 Collings, B. 112 Colombeau, B. 107 Colston Jr, B.W. 261,263, 265 Constantinescu, R.C. 162 Cooperman, G.D. 321 Corney, J.E 330, 331,361 Cote, D. 161 Cregan, R.E 124 Cristodoulides, D.N. 348 Cuche, E. 238 Cuenot, J.-B. 73 D
Da Silva, L.B. 261,263,265 Dainty, J.C. 275 Diindliker, R. 239 Dandridge, A. 80 Danielson, B.L. 218 Darmanyan, S.A. 92, 305, 307, 310, 319, 320, 337, 348-350, 356, 359-361 Dasari, R.R. 243, 261,269 Daumont, I. 349 Dauxois, T. 349 Dave, D. 250 Davis, J.C. 94 Davis, P. 66 De Angelis, C. 349, 358, 359 de Beauvoir, E. 162 de Boer, J.E 252, 253, 261, 265, 267, 272, 274 De Sterke, C.M. 124, 307, 310, 321-323 DeCamp, M. 161 Deconinck, B. 362 Delachenal, N. 293 DeLong, K.W. 93, 95, 163 Demokan, M.S. 361 Denardo, B. 349 Denisenko, A.N. 224 Deparis, O. 49, 78 Depeursinge, C. 238 Derickson, D.J. 286
369
Desem, C. 97, 104 DeTienne, D.H. 40, 55 Dianov, E.M. 92 Dichtl, S. 265 Diddams, S.A. 97 Diels, J.C. 88, 93, 109 Dienes, A. 94, 169, 293, 294 Dijaili, S. 169 Ding, Z. 252, 253, 255,295 Dobre, G.M. 220 Dogariu, A. 248 Dolgov, A.S. 348 Donati, S. 66, 73, 74 Doran, N.J. 306, 318-320 Dorrer, C. 161, 162 Dou, H.-L. 271 Doucot, B. 347 Downer, M.C. 94 Drexler, W. 119, 217, 219, 257-259, 273, 283, 284, 287, 288, 295 Dror, I. 246 Drummond, P.D. 96, 314 Dubois, A. 221,222, 288 Ducros, M.G. 265 Dudley, J.M. 96, 101 Dugan, M.A. 107 Duncan, M.D. 272, 287, 289 Durnin, J. 195 Dutta, N.K. 14, 19, 20, 27, 31, 33 Dykaar, D.R. 160 E
Eaton, H.K. 97 Eberhardt, W. 191 Eberly, J. 195 Edelstein, D. 292 Efimov, A. 113 Eggleton, B.J. 124, 128 Eguia, M.C. 52 Einziger, P. 164 Eisenberg, H.S. 349 El-Reedy, J. 97 E1-Zaiat, S.Y. 218, 235 Eleonsky, V.M. 310 Elsiil3er, W. 35, 48, 52, 58 Elssner, J. 236 Emplit, Ph. 306 Engelhardt, R. 239 Evangelides, S.G. 346 Everett, M.J. 261,263, 265
370
Author index for Volume 44
F
Futami, E
Falconer, I.S. 293 Fan, S. 125 Farkas, D.L. 225 Fauchet, P. 179, 184 Feir, J.E. 305, 308 Feizulin, Z.I. 243 Fejer, M.M. 330 Feld, M.S. 243,261,269 Feldchtein, EI. 221,224 Felsen, L. 168, 196 Feng, S. 167-170 Fercher, A.E 217-220, 235-239, 254, 257, 260, 261,263, 265, 268, 273,274, 283,286, 288, 290, 295 Fer6, M. 103 Ferencz, K. 158 Ferrando, A. 132 Ferrari, A. 192, 193 Ferro, P. 312, 314, 357 Fetterman, M. 94 Findl, O. 219 Fink, Y. 125 Fiorentino, M. 136 Fischer, I. 35, 48, 58, 66 Fittinghoff, D.N. 93, 95, 163 Fitzgerald, J.B. 254 Flach, S. 348 Fleming, S. 124, 361 Flotte, T. 217, 219, 271 Flournoy, P.A. 226 Flytzanis, N. 320 Fochs, P.D. 226 Fontaine, J.J. 93 Fork, R.L. 292 Forrest, G.K. 286 Forsberg, E 278 Fouckhardt, H. 111 Fouquet, EE. 286 Franco, M.A. 94 Froberg, N.M. 179 Froehly, C. 107 Fuji, T. 232, 256 Fujimoto, J.G. 119, 217-220, 222, 224, 247, 248, 254, 257-259, 261,271,283,284, 287, 290, 292-295 Fujino, H. 66, 68, 70 Fujiwara, M. 34 Fukuchi, S. 50 Fukushima, T. 58
317
G Gabitov, I. 306, 318 Gabor, D. 275 Gage, E.C. 21, 55, 65 Gallmann, L. 158 Gallot, G. 179, 180, 182 Galvin, B. 349 Gan, X. 189, 191 Garcia-Fem~indez, P. 73, 76, 77 Garcia-Webb, M.G. 286 Garmash, I.A. 286 Garnier, J. 307, 325, 327, 337, 339, 345, 346 Gavrielides, A. 48, 61 Gelikonov, G.V. 221,224 Gelikonov, VM. 221,224 Genty, G. 36 George, N. 275 Georges, T. 324 Gerber, G. 113 Ghanta, R.K. 283, 287 Giacomelli, G. 52 Gianotti, R. 293 Giessen, J. 93 Gilerson, A. 294 Gilgen, H.H. 248, 286 Gills, Z. 5, 60 Giordmaine, J.A. 93 Giudici, M. 52 Gladkova, N.D. 224 Glodis, RE 124 G6bel, E.O. 35, 52, 58 Goedgebuer, J.-P. 73 Goldberg, L. 80, 287 Goles, M. 361 Golovchenko, E.A. 92 Golubovic, B. 293 Goodman, J.W. 275, 276, 278 Gordon, J.P. 92, 103, 317, 334, 346 Goswami, D. 107, 289 Goto, T. 96, 103, 106 G6tzinger, E. 260, 261,263, 265 Grant, R.S. 94 Gray, G.R. 21, 39, 40, 50, 52, 53, 55, 65 Grebogi, C. 5, 59 Gredeskul, S.A. 348 Green, C. 52 Greenfield, A. 349
Author index for Volume 44
Gregory, K. 217, 219, 271 Grillon, G. 162 Grischkowsky, D. 179, 180, 182 Grohn, A. 36 Gronbech-Jensen, N. 348 Grzanna, R. 236 Gu, M. 189, 191 Gu, X. 94, 97, 126 Guan, X. 293 Guenther, B.D. 275 Gfirses-0zden, R. 271 Gusmeroli, V. 249 Gutty, E 101 H
Haberland, U.H.P. 218, 290 Haelterman, M. 305, 306, 349 Hagan, D.J. 356 Haken, H. 3, 6, 9, 10 Hale, A. 124 Hall, G.E. 292 Hammer, M. 248 Hanson, S.G. 244 Harde, H. 293 Hardy, A.A. 24, 58 Hariharan, P. 219 Hartl, I. 287 Harvey, J.D. 96, 101,328, 329 Hasegawa, A. 92, 305, 306, 310, 323, 332 Hattori, T. 232, 256 Haus, H.A. 289 H~iusler, G. 217, 218 Hawkins, R.J. 92, 112 Haykin, S.S. 114 He, H. 314 He, Z. 289 Healey, A.J. 278 Hecquet, P. 185 Hee, M.R. 217-220, 222, 247, 248, 254, 261, 271,287, 292, 295 Heil, T. 48 Hellmuth, T. 256 Hellwarth, R. 167-170 Hennig, D. 348 Heritage, J.P. 92, 94, 112, 289, 291, 293, 294 Herzel, H. 24 Hess, O. 24, 58, 59 Hibino, Y. 292 Hillegas, C.W. 107, 289
371
Hillion, P. 153 Hirabayashi, K. 293 Hitzenberger, C.K. 217-220, 235, 237, 254, 257, 260, 261,263, 265, 268, 273,274, 283, 288, 290, 295 Hodara, H. 279 H6gbom, J. 272 Hoh, S.-T. 271 Hook, A. 173 Horiguchi, T. 320 Hotate, K. 289 Hu, B.B. 163, 179 Huai, E.-H. 265 Huang, D. 39, 55, 217-220, 254, 261, 271, 292, 295 Hunt, E.R. 5, 60 Husinsky, W. 119 Huyet, G. 52 Hwang, T.M. 73 I
Iaconis, C. 94, 162 Ibanescu, M. 125 Ibragimov, E. 196 Ichikawa, H. 189, 192 Ikeda, K. 3 Ikuma, Y. 47 Ilday, EO. 119 Illing, L. 74 Infeld, E. 311 Inoue, Y. 109 Ippen, E.P. 119, 257-259, 284, 295 Ishida, Y. 293 Ishii, M. 109 Ishiwaka, H.R. 271 Issa, N.A. 124 Ito, E 193 Its, A.R. 306,311 Izatt, J.A. 217, 219, 220, 222, 247, 250, 252, 253,255,256, 261,272, 279, 281,292, 294, 295 J
Jackson, D.A. 220, 279, 281,288 Jacquemin, R. 191 Jasapara, J. 94 Jennings, R.T. 95, 163 Jensen, J.A. 278 Ji, L. 271 Jiang, Z. 191
372
Author index for Volume 44
Jiao, S. 261,267 Joannopoulos, J.D. 125 Joffre, M. 161, 163 Johnson, G.A. 74 Jones, R.J. 74 Jones, R.R. 160, 163 Joseph, R.I. 106, 348 Juang, C. 73 Juang, J. 73 Juchem, M. 237 Judkins, J. 191 Judson, R.S. 113 Jundt, D.H. 330 Juskaitis, R. 225 K
Kaiser, E 24, 59 Kaiser, W. 93 Kaivola, M. 36 Kak, A.C. 217 Kakiuchida, H. 29 Kaminow, I.P. 333 Kamp, G. 217, 218, 235 Kane, D.J. 93, 94, 162 Kaneko, S. 317 Kaneuchi, E 256 Kang, J.U. 349 Kannari, M. 289 Kaplan, A. 191, 194 Kapteyn, H.C. 113 Karamata, B. 274 Karlsson, M. 173, 307, 333 K/irtner, EX. 119, 257-259, 283, 284, 287, 295 Kaup, D.J. 97 Kawato, S. 232, 256 Kayshap, R. 91 Keiding, S. 179 Keller, U. 158 Kelley, R.L. 91 Kelley, S.M. 317 Kempe, M. 196 Kennedy, G.T. 349 Kennedy, M.P. 72, 80 Kennedy, T.A.B. 328, 329 Kennel, M.B. 74 Kenney-Wallace, G.A. 292 Kerbage, C. 124 Khabibullaev, RK. 92, 305, 310, 350, 360 Khaminskii, R.Z. 336
Kiefer, B. 113 Kikuchi, K. 95, 317 Kikuchi, N. 41, 61, 64 Kim, J. 197 Kimmel, M. 94, 97, 126 Kinsinger, J.B. 243 Kirschner, E.M. 92, 112, 289, 291 Kiselev, A. 154 Kivshar, Yu.S. 348, 349, 352 Klaus, J.T. 39 Klyatskin, V.I. 336 Knapp, R. 348 Knight, J.C. 97, 124-126, 131, 132 Knox, EM. 306, 318 Knox, W.H. 112, 128, 136 Knfittel, A. 245, 248 Ko, T. 224, 287 Kobayashi, K. 4, 26, 219 Kobyakov, A. 307, 319, 320, 337, 348-350, 356 Kodama, Y. 92, 306, 310, 323, 332 Kolb, A. 248 Konotop, VV 307, 323, 348 Kopeika, N.S. 246 Koplow, J.P. 287 Koren, U. 112 Koretsky, E. 197 Korneev, VI. 306, 310 Kosinski, S.G. 128 Koumans, R.G.M.P. 94 Kovanis, V 61 Kowalczyk, A. 236, 238, 257 Kowalevicz, A.M. 287 Kozak, J.A. 261 Kozubek, M. 225 Krause, J.L. 113 Krausz, E 158, 173 Kravtsov, Y.A. 243 Krolikowski, W. 349 Krug, P.A. 293 Kruger, A. 219 Krumbfigel, M.A. 93, 95, 163 Kubota, K. 34 Kuhl, J. 93, 184, 185, 293 Kulagin, N.E. 310 Kulhavy, M. 218, 290 Kulkarni, M.D. 219, 250, 252, 253,256, 272, 294 Kumar, G. 239 Kumar, P. 136
Author index for Volume 44
Kuo, S. 197 Kuranov, R.V 224 Kurokawa, T. 109 Kurtzke, C. 318 Kiirz, R 53, 80 Kutz, J.N. 319, 362 Kuwaki, N. 332 Kuznetsov, E. 97, 106 Kuznetzova, I.A. 224 Kwong, K.E 293,294 L Laedke, E.W. 359 Lambsdorff, M. 293 Laming, R.I. 279 Landau, L.D. 189, 315, 319 Lang, R. 4, 26, 34 Lange, H.R. 94 Langley, L.N. 39 Large, M.C.J. 124 Larger, L. 73 Larraza, A. 349 Lasser, T. 274 Laude, V. 108 Lauterbom, W. 66, 69 Law, J.Y. 23, 58 Lazar, R. 224 Le Blanc, C. 162 Leaird, D.E. 92, 111, 112, 289, 295 Lebec, M. 222, 288 Lederer, F. 307, 319, 320, 337, 348-350, 356, 359-361 Lee, S.L. 278, 286, 288, 295 Leeman, S. 278 Leitgeb, R. 236, 238, 254, 257, 268, 288 Lenstra, D. 35, 39, 40, 52, 55 Leon, J. 349 Leonhardt, R. 96, 101,328, 329 Lepetit, L. 163 Leuchs, G. 137 Levine, A.M. 35, 52 Levis, R.J. 113 Levy, G. 24, 58 Lewenstein, M. 159 Lexer, E 218, 290, 295 UHuillier, A. 159 Li, H. 42 Li, X. 224 Li, X.D. 119, 257-259, 284, 287, 295 Liby, B.W. 80
373
Liebermann, J.M. 271 Lifshitz, E.M. 189, 315, 319 Lin, C.P. 217-220, 254, 271,292, 295 Lin, W.-W. 73 Linden, S. 93 Lindner, M.W. 217, 218 Linnik, W. 226 Lippi, G.L. 3, 12 Litchinister, N. 307 Liu, H.H. 287 Liu, J.M. 66, 73, 74, 79 Liu, K.X. 94 Liu, X. 128 Liu, Y. 31, 33, 41, 46, 48, 52-54, 60, 61, 63, 64, 66 Locquet, A. 49, 78 Loiko, N.A. 60 Lokhnigin, V 361 Lorattanasane, C. 317 Lorenz, E.N. 3, 10 Loudon, R. 276 Luce, B.P. 99, 100, 111, 114, 119 Ludowise, M.J. 286 Ludvigsen, H. 36 Luther, G.G. 358, 359 Lutomirski, R.E 243, 244 M
Ma, J. 150 Magel, G.A. 330 Maier, M. 93 Maier, T.D. 5, 60 Maine, P. 158 Maiti, S. 119 Maitland, D.J. 261,263, 265 Malekafzali, A. 251,265 Mallick, S. 243 Malomed, B.A. 97, 306, 314, 349 Mamyshev, P.V 106, 333 Man, W.S. 361 Mandel, L. 227, 256 Mandel, P. 4 Mangan, B.J. 124 Manna, M. 349 Manos, S. 124 Mar, D.J. 74 Maradudin, A. 183 Margulis, W. 91 Mark, J. 42 Marquet, P. 238
374
Author index for Volume 44
Marquie, P. 349 Marquis-Weible, E 248, 286 Martin-Regaldo, J. 59 Martinelli, M. 103, 249 Martinez, O.E. 93 Maruta, A. 332 Masoliver, J. 197 Masoller, C. 69, 78 Masters, B.R. 219 Matera, E 306, 315-317 Mathieu, E. 161 Matsui, Y. 160 Matuschek, N. 158 McClure, R.W. 226 McGowan, R. 179, 180, 182 McInerney, J.G. 42 McKinstrie, C.J. 307 McMichael, I.C. 93 McPhedran, R.C. 124 Mecozzi, EA. 306, 315-317 Megret, P. 49, 78 Meier, T. 224 Melamed, B. 168 Menapace, R. 219 Meng, X.J. 79 Menkir, G.M. 113 Menyuk, C.R. 334, 345, 356 Merbach, D. 24 Merkel, K. 236 Meshulach, D. 110, 111, 113, 114 Meyer, K.P. 161 Meystre, P. 323 Mezentsev, V.K. 359 Micely, J. 195 Middelhoek, S. 108 Middleton, D. 343 Mikhailov, A.V 97, 106 Miles, R.O. 80 Millar, P. 349 Miller, P.D. 349, 356 Millot, G. 101,305 Milner, T.E. 249-251,261,265, 267 Mindlin, G.B. 52 Minoshima, K. 192 Mirasso, C.R. 58, 73, 74, 76-78 Miret, J.J. 132 Misoguti, L. 113 Mitachi, S. 292 Mitchell, D. 187 Mitchell, M. 114, 361
Mitchell, M.L. 112 Mitschke, EM. 103 Miyata, M. 232, 256 Mogi, H. 93 Mokhtari, A. 184 Molebny, S. 295 Mollenauer, L.E 92, 103, 333,334, 346 Monterosso, V. 222 Moores, M.D. 97, 113, 114, 119, 131, 132 Morandotti, R. 349 Moreno-Barriuso, E. 288 Morgner, U. 119, 257-259, 283, 284, 287, 295 Moritz, A. 265 Mork, J. 42, 52, 80 Morkel, P.R. 279 Mourou, G. 158 Mu, M.C. 79 Mukai, T. 53, 80 Mulet, J. 78 Miiller, G. 282 Muller, H.G. 160, 162 Muller, M. 119 Mullot, G. 160 Mtinkel, M. 24, 59 Murakami, A. 31, 33, 39, 45, 46, 48, 53-55, 70, 71 Murdoch, S.G. 328, 329 Murnane, M.M. 113 Murphy, T.W. 5, 60 Muschall, R. 349 Myaing, M.T. 119 Mysyrowicz, A. 94, 162 N Naganuma, K. 93, 109 Nakajima, K. 333 Nakatsuka, H. 232, 256 Narducci, L.M. 4 Nassif, N. 272 Naumenko, A.V. 60 Neil, M.A.A. 225 Nelson, J.S. 249-253, 255, 261, 265, 267, 272, 274, 295 Neubelt, M.J. 333 Newell, A.C. 348 Nichols, D.T. 25 Nicholson, J.W. 94, 107 Nicorovici, N.A.P. 124 Ning, Y.N. 288
Author index for Volume 44
Nishizawa, N. 96, 103, 106 Noda, J. 218 Nodland, B. 174 Norris, T.B. 119 Novak, R.P. 248, 286 Nuss, M.C. 112, 163, 179 Nyquist, D. 183 O Ogawa, K. 94, 96 Ogusu, K. 197 Ohashi, M. 332, 333 Ohtsubo, J. 5, 26, 29, 31, 33, 39, 41, 42, 45-48, 50, 52-55, 60, 61, 63-66, 68, 70, 71 Okamoto, K. 109, 130 Okamura, R. 106 Olesen, H. 30, 31, 33, 36, 46 Omae, T. 320 Omenetto, EG. 94, 97, 99, 100, 106, 107, 111, 114, 119, 131, 132 Onishchukov, G. 361 Orsterberg, U. 91 Ortigosa-Blanch, A. 125, 126 O'Shea, P. 94, 97, 126 Osmundsen, J.H. 33, 46 Ostrovski, L.A. 308 Otsuka, K. 38 Ott, E. 5, 59 Owen, G.M. 220, 222, 247 Ozaki, Y. 256 P
Page, J.B. 348 Pan, M.-W. 50, 52, 53 Pan, X. 30,31,36 Pan, Y. 225, 239 Panknin, P. 52 Park, B.H. 261,267, 272 Parlit, U. 66, 69 Patel, J.S. 289, 295 Pattanayak, D. 184 Patten, R.A. 226 Paul, P.M. 160 Payne, D.N. 279 Peatross, J. 94 Pecora, L.M. 5, 65, 66, 74 Pelusi, M.D. 94, 96, 160 Perel, M. 154 Perelman, L.T. 243
375
Perez, W.H. 286 Pertsch, T. 356 Peschel, T. 349, 356 Peschel, U. 349, 356 Pesquera, L. 40, 55 Pessot, M. 158 Petermann, K. 14, 21, 27, 63 Peterson, V 192 Petite, G. 185 Petrov, D.V 330 Peyrard, M. 349, 352 Piccinin, D. 103 Pieroux, D. 49, 78 Pilipetskii, A.N. 92 Pircher, M. 260, 261,263, 265 Pitois, S. 101 Pitris, C. 119, 218, 224, 257-259, 284, 295 Pochinko, VV 224 Podoleanu, A.G. 220, 279, 281 Pogosyan, VA. 197 Pollnau, M. 287 Popescu, G. 248 Porras, M. 152, 166, 195, 196 Porte, H. 73 P6tting, S. 323 Prade, B.S. 94 Prokhorov, A.M. 92 Puccioni, G.P. 3, 12 Puliafito, C.A. 217-220, 254, 271,292, 295 Puri, A. 184 Putterman, S. 349 Pyragas, K. 60, 66 Pyrkov, VN. 349
Q Qian, Y.
150
R
Rabitz, H. 113 Radic, S. 179 Rahman, L. 24, 25 Rainer, G. 219 Rammal, R. 347 Ramsey, J. 188 Ranaganath, T.R. 286 Ranc, S. 162 Ranka, J.K. 125, 126, 287 Rao, Y.J. 288 Rashleigh, S.C. 333 Rasmussen, K.O. 349
376
Author index for Volume 44
Raz, S. 164 Rees, P. 74 Reeves, W.H. 124 Reintjes, J. 272, 277, 278, 287, 289 Reitze, D.H. 113, 114, 119 Relke, I. 359, 360 Remoissenet, M. 92, 349 Ren, H. 252, 253,255, 295 Revuz, D. 337 Rhodes, W.T. 73 Ripoche, J.-E 94 Risch, Ch. 34 Ritch, R. 271 Riza, N.A. 293 Roberts, P.J. 124 Robinson, L.C. 293 Robl, B. 265 Rogister, E 49, 78 Rogowska, J. 278 Rollins, A.M. 219, 252, 253, 255, 261,279, 281,294 Romagnoli, M. 306, 315-317 Romney, C.R.B. 292 Rosen, J. 225 Rosperich, J. 239 Roth, E. 218, 286 Roth, J.E. 261 Rothwell, E. 183 Rottbrand, K. 189 Rousseau, J.P. 162 Rousseau, P. 162 Rovatti, R. 72, 80 Rowlands, G. 311 Roy, R. 5, 60, 66, 73 Roychoudhuri, C. 189, 191 Rubenchik, A.M. 358, 359 Rudakov, L.I. 308 Rudolph, W. 88, 94, 109, 196 Rulkov, N.E 66 Rulli+re, C. 88 Rundquist, A. 94 Russell, P.St.J. 97, 124-126, 131, 132, 287 Ryan, A.T. 21, 55, 65 Rybin, A.V. 306, 311 Rylander III, H.G. 265 S
Sabbatier, H. 52 Sacher, J. 52 Saifi, M.A. l l l
Saint-Jalmes, H. 222, 288 Saito, S. 30, 31, 36, 66 Sala, K.L. 292 Salath6, R.P. 222, 248, 286, 287, 293 Salerno, M. 307, 323, 348 Sali~res, P. 159 Sall, M.A. 306, 311 Sampson, D.D. 254, 286 Sfinches-Diaz, A. 73, 76, 77 S~nchez, A. 348 Sandrov, A. 246 Sano, 1". 34, 35 Sardesai, H.P. 112, 113 Sarma, J. 23 Sarro, P.M. 108 Sasano, Y. 259 Sato, M. 289 Sattmann, H. 217, 260, 265, 288, 295 Saxer, C.E. 252, 253, 261,267, 274 Scheuermann, M. 292 Schiek, R. 356 Schins, J.M. 162 Schmetterer, L. 254 Schmidt, E. 350 Schmidt, V. 119 Schmitt, H.J. 218, 290 Schmitt, J.M. 225, 239, 245, 248, 255, 257, 258, 260, 270, 272, 278, 286, 288, 295 Schoenenberger, K. 261,263, 265 Scholl, E. 24 Schuman, J.S. 217, 219, 220, 271,283,287, 292, 295 Schweitzer, D. 248 Schwider, J.R. 236 Sciamanna, M. 49, 78 Scir~, A. 66, 73, 74 Scott, A.C. 348 Segev, M. 361 Seitz, P. 222 Semenov, A.T. 286 Semkow, M. 80 Sereda, L. 192, 193 Sergeev, A.M. 224 Serkin, V.N. 92 Seshek, R. 224 Settembre, M. 306, 315-317 Setti, G. 72, 80 Seve, E. 305, 346 Seyfried, V. 113 Shahverdiev, E.M. 66
Author index for Volume 44
Shakhov, A.V. 224 Shakhova, N.M. 224 Sharping, J. 136 Shear, J.B. 119 Shelley, M. 348 Shelley, P.H. 248 Shen, Q. 252 Shen, S. 113 Sheppard, A.P. 349 Shi, B.-P. 50, 52, 53 Shidlovsky, V.R. 286 Shimizu, T. 66 Shimokhin, I.A. 97, 106 Shipulin, A. 361 Shiraki, K. 320 Shore, K.A. 23, 39, 58, 60, 66, 73, 74 Shraiman, B.I. 124 Shramenko, M.V. 286 Shreenath, A.P. 97, 126 Shvartsburg, A. 150, 155, 179, 185, 212 Sibbett, W. 349 Siders, C.W. 94 Siegel, A.M. 241,242 Siegman, A.E. 58 Sievers, A.J. 348 Silberberg, Y. 110, 111, 113, 114, 349 Silva, K.K.M.B.D. 254 Silvestre, E. 126, 132 Simon, A. 333 Simoni, E 93 Simpson, T.B. 79 Sipe, J.E. 321 Sivak, M.V. 219 Sivaprakasam, S. 66, 73, 74 Sizmann, A. 137 Slaney, M. 217 Sloan, S.R. 286 Smith, J. 169 Smith, K. 334 Smith, M.J. 319, 320 Smith, N.J. 306, 318 Smith, P.R. 163 Smith, P.W. 111 Snopova, L.B. 224 Snyder, A. 187 Sobolewski, R. 179 Sommerfeld, A. 189 Sorensen, M.P. 52, 307, 337 Sotskii, B.A. 243 Southern, J.E 218, 219, 222, 224, 248
377
Spatschek, K.H. 359 Spencer, P.S. 58, 60, 66, 74 Sperr, W. 265 Spielmann, C. 108, 158 Spolaczyk, R. 236 Squier, J. 119 Srinivas, S.M. 251,265 Stamm, U. 196 Statman, D. 80 Stegeman, G.I. 349, 356 Stegun, I. 188 Steiner, R. 224 Steinmeyer, G. 158 Stenflo, L. 212 Stentz, A.J. 125, 126, 287 Stephen, M.J. 124 Sticker, M. 254, 257, 260, 261, 263, 265, 268, 274, 288, 295 Stinson, W.G. 217, 219, 271 Stolen, R.H. 92 Strehle, M. 113 Streltzova, O.S. 224 Strickland, D. 107, 158, 289 Stroud Jr, C. 179 Sugawara, T. 66 Sullivan, A. 94 Sun, Y. 58 Sushchik, M.M. 66 Sutter, D.H. 158 Suzuki, A. 160 Suzuki, K. 289 Suzuki, M. 289 Swanson, E.A. 217-220, 222, 247, 254, 261, 271,290, 292, 293, 295 Sweetser, J.N. 93, 95, 163 Szip6cs, R. 158 Szydlo, J. 293 T Tachikawa, M. 66 Tai, K. 310 Taira, K. 95 Tait, G.D. 293 Takada, K. 218, 292 Takeda, M. 225 Takeno, S. 348 Takenouchi, H. 109 Takiguchi, Y. 48, 52, 63, 66, 70 Talanov, V.I. 305, 308 Talviti, H. 36
378
Author index for Volume 44
Tam, H.Y. 361 Tanaka, Y. 293 Tang, D.Y. 361 Tang, S. 66, 73, 74 Tanno, N. 289 Tappert, E 92 Tateda, M. 333 Taylor, A.J. 94, 97, 99, 100, 106, 107, 111, 114, 119, 131, 132 Taylor, H.E 80 Tearney, G.J. 218, 219, 222, 224, 248, 281, 287, 293, 294 Teramura, Y. 289 Thamm, E. 248 Thomas, C.W. 272 Thomas, E.L. 125 Thomas, G.A. 124 Thomson, M.D. 96 Thornburg, K.S. 66 Thrane, L. 244, 246, 247 Thurber, S.R. 248 Thurston, R.N. 92, 112 Tkach, R.W. 28 Toma, E.S. 160 Tomita, A. 310 Tomlison, W.J. 92, 112 Torner, L. 356 Torruelas, W.E. 356 Tournois, P. 108 Tracy, E.R. 306 Treacy, E.B. 93 Trebino, R. 88, 93-95, 97, 126, 162, 163 Tredicce, J.R. 3, 12, 52 Trillo, S. 311,312, 314, 349, 357 Tripathi, R. 272 Tromborg, B. 30, 31, 33, 36, 42, 46, 52, 80 Trott, G.R. 286 Tsimring, L.S. 66 Tsironis, G.P. 348 Tsoy, E.N. 310 Tsuda, H. 109 Tsukamoto, T. 66 Tull, J.X. 107, 289 Turitsyn, S.K. 306, 318, 359 Turovets, ST 60 U Ulrich, R. 274, 333 Ung-arunyawee, R. 294 Urayama, J. 119
V Vabre, L. 221,222, 288 Valichansky, V 52 Valle, A. 23 Vampouille, M. 107 van der Graaf, WA. 40, 55 van der Linden, H.J.C. 39, 55 van Driel, H. 161 Van Eijkelenborg, M.A. 124 Van Engen, A.G. 97 van Gemert, M.J.C. 251,261,265 van Leeuwen, T.G. 252 Van Simaeys, G. 306 Van Stryland, E.W 356 van Tartwijk, G.H.M. 5, 6, 9, 10, 26, 35, 39, 52, 55 Vanossi, A. 349 VanWiggeren, G.D. 73 Vfizquez, L. 348 Vdovin, G. 108, 113 Vedenov, A.A. 308 Verluise, E 108 Villeneuve, A. 349 Voss, H.U. 69 Voumard, C. 34 Vysloukh, VA. 88
W Wabnitz, S. 305, 306, 311, 312, 327, 346, 349 Wadsworth, W.J. 124-126, 287 Wai, P.K.A. 334, 345 Walker, J. 225 Walmsley, I.A. 94, 95, 162, 163 W/ilti, R. 293 Wang, J. 218 Wang, J.P. 287 Wang, L. 271 Wang, L.V 261,267 Wang, R.K. 239, 272 Wang, X.J. 249, 251 Wang, Z. 190, 356 Warren, W.S. 94, 107, 113, 289 Wax, A. 261,269 Webb, W.W 119 Weber, H.P. 161 Weber, P.M. 113 Wedrich, A. 219 Weiland, J. 212
379 Weiner, A.M. 92, 107, 110-113, 289, 291, 295 Weiss, C.O. 3 Weiss, G. 197 Weiss, K. 239 Weissman, N.J. 224 Welch, A.J. 219, 250, 253 Weling, A.S. 179 Welle, M. 256 Weller, J.E 80 Westbrook, P.S. 124 Westphal, V. 255 White, WE. 94 Whitham, J. 196 Whitten, W. 188 Whittenberg, C.D. 218 Wilhelmi, B. 196 Williams, R.M. 119 Willis, C.R. 348 Wilson, K.R. 113 Wilson, T. 225 Windeler, R.S. 97, 124-126, 128, 136, 287 Winful, H.G. 24, 25, 167-170, 321 Wise, EW. 119 Wojtkowski, M. 236, 238, 254, 257 Wolf, E. 164, 217, 225, 227, 234, 235, 239, 243, 256 Wright, E.M. 323 Wright, W. 349 Wu, B. 293 Wu, T. 148 Wullert, J.R. 289, 295 Wyntjes, G. 226 X Xiang, S. 252, 253 Xiang, S.H. 225, 255, 257, 258, 260, 278 Xu, C. 128 Xu, L. 97, 126 Xu, Z. 190 Xue, P. 271 Y Yablonovitch, E.
136
Yajima, T. 293 Yakovlev, V.V. 113 Yakubovich, S.D. 286 Yamada, H. 93, 292 Yamada, M. 22, 58 Yamaguchi, M. 293 Yamashita, E. 150 Yan, Z. 289 Yang, C. 243, 261,269 Yang, W. 94 Yankelevich, D. 293, 294 Yao, G. 261,267 Yaqoob, Z. 293 Yariv, A. 94 Yarotski, D. 99, 100, 106, 111 Yasa, Z.A. 293 Yazdanfar, S. 250, 252, 253, 255, 261,294 Ye, J. 42 Ye, S.Y. 46, 50, 65 Yelin, D. 111 Yodh, A. 217 Yokohama, I. 218 Yor, M. 337 Yorke, J.A. 5, 59 You, D. 160, 163 Yu, K. 271 Yung, K.M. 225, 255, 257, 258, 260, 278, 286, 288, 295 Yura, H.T. 243-247 Z Zagari, J. 124 Zakharov, V.E. 327 Zawadzki, R. 274 Zeek, E. 97, 113, 126 Zeylikovich, I. 294 Zhang, Z. 190 Zhao, Y. 252, 253, 255, 261,267, 295 Zhou, L. 278 Zhu, Q. 293 Ziolkowsky, R. 191 Zipfel, W.R. 119 Zozulya, A.A. 97 Zvyagin, A.V. 254, 286
Subject index for Volume 44
A adaptive pulse shaping 115 Airy function 187 amplified spontaneous emission Auger recombination 20 autocorrelation 226, 228 -, intensity 93
- - resonator 13 femtosecond pulse 88-93, l l l , 125 - -, chirped 104 - -, pulse shaping 107, 119 Fokker-Planck equation 335 four-wave mixing 55 Fresnel formula 206, 207, 211 - laws 187, 206 - zone 225
282, 287
B
Bernoulli equation 310 Bessel beam 195 - function 155, 182, 195 bifurcations, chaotic 40, 42, 48 - , H o p f 11,44 -, intermittent 11 -, period-doubling 11, 42 -, quasi-periodic 11 Bloch equations 6, 8 - state 324 Born approximation 235, 239 Bose-Einstein condensate 305, 307, 323 - - statistics 279 Bragg condition 240 - grating 320 - reflector 290 Brewster effect 211 Brillouin scattering 91 , stimulated 92 C chaos, intermittent 34 - modulation 73 - synchronization 66, 67, 74 cross-spectral density function
G Gouy phase shift 165, 168 group-velocity dispersion 90, 95, 273 H
Helmholtz equation 24 Hilbert transform 234, 253 I
interferometry, autocorrelation 228 -, cross-correlation 231 -, heterodyne 233 , , low-time-coherence 217, 226 -, partial-coherence 217 -, two-beam 225 J
Jones matrix 263 Josephson junction K
Kerr effect 95, 157, 158 - lens 157, 285 - nonlinearity 331 Klein-Gordon equation 181
229
L Laguerre function 150, 151, 176, 177 Lang-Kobayashi equations 28 laser, classification of 12 -, distributed feedback 13, 28, 290
D
Doppler velocimetry
249
F Fabry-Perot interferometer
305
219 381
382
M I in inhomogeneous and discrete media
-, Fabry-Perot 6, 30 -, multi-quantum well 13 -, self-pulsating 5, 6, 21, 22, 65, 74 -, semiconductor, see semiconductor laser -, vertical-cavity surface-emitting 4, 22 liquid crystal 108, 109 Lorentz-resonant medium 184 Lorenz equations 10 - model 3, 27 Lorenz-Haken chaos 3, 10, 11 - equations 3, 6, 9 Lyapunov exponent 47, 66, 335, 336, 339, 342 M
Mathieu equation 315 Maxwell equations 153, 155, 170, 180, 198, 200, 211 Maxwell-Bloch equations 6, 9, 10 Michelson interferometer 161,222, 250, 261, 262 modulational instability 305 - - in homogeneous media 308-314 - - - nonlinear discrete optical systems 348362 --periodically inhomogeneous medium 314-332 - - - random media 332-348 O optical coherence tomography, contrast generation 249-269 , Doppler 249-255 , phase contrast 267-269 , polarization sensitive 261-267 , refractometric 260 , signal generation 225-248 , processing 271-274 - - - , speckle in 275-278 - - - , wavelength dependent 255-261 - fiber 101, 123 --, femtosecond pulses in 87, 90 --, pulse propagation in 88, 96, 119 , shaping 110 - gating 93 - - , frequency-resolved 162
photorefractive crystal 39, 55 Poisson process 279 - spectrum 168 --pulses 152, 167 polarization mode dispersion 333 population inversion 8 power spectrum 229 R
Raman scattering 91, 105 , stimulated 92, 116 Rayleigh length 196, 234 - r a n g e 165, 168, 169, 194 rotating-wave approximation 8 Runge-Kutta algorithm 27 S Schr6dinger equation 8 - - , nonlinear 76, 116, 305, 307, 310, 311, 323, 333, 349, 360 second-harmonic generation 161 semiconductor laser 5, 6, 13, 14, 18-22, 25, 30, 35, 39, 41, 55, 57, 58, 79 --,chaosin 4, 49 , chaotic dynamics 40, 46, 61, 69 , instabilities in 4, 49 , linewidth of 37 - with optical feedback 34, 59, 65, 68, 72, 74, 78, 80, 81 sine-Gordon equation 305 slowly-varying-envelope approximation 7 soliton 101, 127 -, dark 95, 102-104 -, optical 119, 136, 317 spatial hole burning 22 - light modulator 108, 120, 121 speckle 275-277 spectral density 146 - intensity 256 - phase interferometry 94, 162 T tomography, diffraction 217 - , see also optical coherence tomography V van Cittert-Zernike theorem
243
W wavelet transform 258 Wiener-Khintchine theorem
229, 291,343
P
phase-conjugate feedback 53 - - mirror 38-40, 55 photonic crystal fiber 124-136, 287
Contents of previous volumes*
VOLUME 1 ( 1961 ) 1 The modern development of Hamiltonian optics, R.J. Pegis 2 Wave optics and geometrical optics in optical design, K. Miyamoto 3 The intensity distribution and total illumination of aberration-free diffraction images, R. Barakat 4 Light and information, D. Gabor 5 On basic analogies and principal differences between optical and electronic information, H. Wolter 6 Interference color, H. Kubota 7 Dynamic characteristics of visual processes, A. Fiorentini 8 Modern alignment devices, A.C.S. Van Heel
1- 29 31- 66 67-108 109-153 155-210 211-251 253-288 289-329
VOLUME 2 (1963) 1 Ruling, testing and use of optical gratings for high-resolution spectroscopy, G. W. Stroke 2 The metrological applications of diffraction gratings, JM. Burch 3 Diffusion through non-uniform media, R.G. Giooanelli 4 Correction of optical images by compensation of aberrations and by spatial frequency filtering, J. Tsujiuchi 5 Fluctuations of light beams, L. Mandel 6 Methods for determining optical parameters of thin films, E Abelks
1- 72 73-108 109-129 131-180 181-248 249-288
VOLUME 3 (1964) 1 The elements of radiative transfer, E Kottler 2 Apodisation, P Jacquinot, B. Roizen-Dossier 3 Matrix treatment of partial coherence, H. Gamo
1- 28 29-186 187-332
VOLUME 4 (1965) 1 Higher order aberration theory, J Focke 2 Applications of shearing interferometry, O. Blyngdahl 3 Surface deterioration of optical glasses, K. Kinosita 4 Optical constants of thin films, P Rouard, P Bousquet
* Volumes I-XL were previously distinguished by roman rather than by arabic numerals. 383
1- 36 37- 83 85-143 145-197
384 5 6 7
Contents of previous volumes
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, E Kottler
199-240 241-280 281-314
VOLUME 5 (1966) 1 2 3 4 5 6 7
Optical pumping, C. Cotlen-Tannoudji, A. Kastler Non-linear optics, PS. Persllan Two-beam interferometry, W.H. Steel Instruments for the measuring of optical transfer functions, K. Murata Light reflection from films of continuously varying refractive index, R. Jacobsson X-ray crystal-structure determination as a branch of physical optics, H. Lipson, C.A. Taylor The wave of a moving classical electron, J. Picllt
1-81 83-144 145-197 199-245 247-286 287-350 351-370
VOLUME 6 (1967) 1 2 3 4 5 6 7 8
Recent advances in holography, E.N. Leith, J Upatnieks Scattering of light by rough surfaces, P Beclonann Measurement of the second order degree of coherence, M. Franqon, S. Mallick Design of zoom lenses, K. Yantaji Some applications of lasers to interferometry, D.R. Herriot Experimental studies of intensity fluctuations in lasers, JA. Armstrong, A. W. Smith Fourier spectroscopy, G.A. Vanasse, H. Sakai Diffraction at a black screen, Part II: electromagnetic theory, E Kottler
1
Multiple-beam interference and natural modes in open resonators, G. Koppelman Methods of synthesis for dielectric multilayer filters, E. Delano, R.J. Pegis Echoes at optical frequencies, I.D. Abella Image formation with partially coherent light, B.J. Ttlontpson Quasi-classical theory of laser radiation, A.L. Mikaelian, M.L. Ter-Mikaelian The photographic image, S. Ooue Interaction of very intense light with free electrons, J.H. Eberly
1- 52 53- 69 71-104 105-170 171-209 211-257 259-330 331-377
VOLUME 7 (1969) 2 3 4 5 6 7
1- 66 67-137 139-168 169-230 231-297 299-358 359-415
VOLUME 8 (1970) 1 2 3 4 5 6 7 8
Synthetic-aperture optics, J. W. Goodnlan The optical performance of the human eye, G.A. F~y Light beating spectroscopy, H.Z Cummins, H.L. Swinney Multilayer antireflection coatings, A. Musset, A. Thelen Statistical properties of laser light, H. Risken Coherence theory of source-size compensation in interference microscopy, T. Yamamoto Vision in communication, L. Levi Theory of photoelectron counting, C.L. Mehta
1- 50 51-131 133-200 201-237 239-294 295-341 343-372 373-440
Contents of preuious t)olumes
385
VOLUME 9 (1971) 1 2 3 4 5 6 7
Gas lasers and their application to precise length measurements, A.L. Bloom Picosecond laser pulses, A.J. Demaria Optical propagation through the turbulent atmosphere, J. W. Strohbehn Synthesis of optical birefringent networks, E.O. Ammann Mode locking in gas lasers, L. Allen, D.G.C. Jones Crystal optics with spatial dispersion, V.M. Agranot)ich, V.L. Ginzburg Applications of optical methods in the diffraction theory of elastic waves, K. Gniadek, J. Petylaewicz 8 Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions, B.R. Frieden
1- 30 31- 71 73-122 123-177 179-234 235-280 281-310 311-407
VOLUME 10 (1972) 1 2 3 4 5 6 7
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 89-135 137-164 165-228 229-288 289-369
VOLUME 11 (1973) 1 2 3 4 5 6 7
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, O. 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
1- 76 77-122 123-166 167-221 223-246 247-304 305-337
VOLUME 12 (1974) 1
2 3 4 5 6
Self-focusing, self-trapping, and self-phase modulation of laser beams, O. Svelto Self-induced transparency, R.E. Slusher Modulation techniques in spectrometry, M. HaJ~,it, J.A. Decker Jr Interaction of light with monomolecular dye layers, K.H. Drexhage The phase transition concept and coherence in atomic emission, R. Graham Beam-foil spectroscopy, S. Bashkin
1- 51 53-100 101-162 163-232 233-286 287-344
VOLUME 13 (1976) 1 2 3 4
On the validity of Kirchhoff's law of heat radiation for a body in a nonequilibrium environment, H.P Baltes The case for and against semiclassical radiation theory, L. Mandel Objective and subjective spherical aberration measurements of the human eye, W.M. Rosenblum, J.L. Christensen Interferometric testing of smooth surfaces, G. Schulz, J. Schwider
1- 25 27- 68 69- 91 93-167
386
Contents of previous volumes
5
Self-focusing of laser beams in plasmas and semiconductors, M.S. Sodha, A.K. Ghatak, V.K. Tripathi 6 Aplanatism and isoplanatism, W.T. Welford
169-265 267-292
VOLUME 14 (1976) 1 2 3 4 5 6 7
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 Kerr shutter, M.A. Duguay Holographic diffraction gratings, G. Schmahl, D. Rudolph Photoemission, PJ. Vernier Optical fibre waveguides - a review, PJB. Clarricoats
1 2 3 4 5
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. Okoshi Quasi-optical techniques of radio astronomy, T.W. Cole Foundations of the macroscopic electromagnetic theory of dielectric media, J Van Kranendonk, J.E. Sipe
1- 46 47- 87 89-159 161-193 195-244 245-325 327-402
VOLUME 15 (1977) 1-75 77-137 139-185 187-244 245-350
VOLUME 16 (1978) 1 2 3 4 5
Laser selective photophysics and photochemistry, VS. Letokhov Recent advances in phase profiles generation, J J Clail; C.I. Abitbol Computer-generated holograms: techniques and applications, W.-H. Lee Speckle interferometry, A.E. Ennos Deformation invariant, space-variant optical pattern recognition, D. Casasent, D. Psaltis 6 Light emission from high-current surface-spark discharges, R.E. Beverly III 7 Semiclassical radiation theory within a quantum-mechanical framework, I.R. Senitzky
1- 69 71-117 119-232 233-288 289-356 357-411 413-448
VOLUME 17 (1980) 1 Heterodyne holographic interferometry, R. Ddndliker 2 Doppler-free multiphoton spectroscopy, E. Giacobino, B. Cagnac 3 The mutual dependence between coherence properties of light and nonlinear optical processes, M. Schubert, B. Wl'lhelmi 4 Michelson stellar interferometry, W.J Tango, R.Q. Twiss 5 Self-focusing media with variable index of refraction, A.L. Mikaelian
1- 84 85-161 163-238 239-277 279-345
VOLUME 18 (1980) 1 Graded index optical waveguides: a review, A. Ghatak, K. Thyagarajan 2 Photocount statistics of radiation propagating through random and nonlinear media, J Pe?ina
1-126 127-203
Contents of previous volumes
3 4
Strong fluctuations in light propagation in a randomly inhomogeneous medium, V.I. Tatarskii, V.U Zavorotnyi Catastrophe optics: morphologies of caustics and their diffraction patterns, M. V. Berry, C. Upstill
387
204-256 257-346
VOLUME 19 (1981) 1 Theory of intensity dependent resonance light scattering and resonance fluorescence, B.R. Mollow 2 Surface and size effects on the light scattering spectra of solids, D.L. Mills, K.R. Subbaswamy 3 Light scattering spectroscopy of surface electromagnetic waves in solids, S. Ushioda 4 Principles of optical data-processing, H.J. Butterweck 5 The effects of atmospheric turbulence in optical astronomy, E Roddier
1- 43 45-137 139-210 211-280 281-376
VOLUME 20 (1983) 1 Some new optical designs for ultra-violet bidimensional detection of astronomical objects, G. Courtbs, P Cruvellier, M. Detaille, M. Sai'sse 2 Shaping and analysis of picosecond light pulses, C. Froehly, B. Colombeau, M. Vampouille 3 Multi-photon scattering molecular spectroscopy, S. Kielich 4 Colour holography, P Hariharan 5 Generation of tunable coherent vacuum-ultraviolet radiation, W.Jamroz, B.P Stoicheff
1- 61
63-153 155-261 263-324 325-380
VOLUME 21 (1984) 1 2 3 4 5
Rigorous vector theories of diffraction gratings, D. Maystre Theory of optical bistability, L.A. Lugiato The Radon transform and its applications, H.H. Barrett Zone plate coded imaging: theory and applications, N.M. Ceglio, D.W. Sweeney Fluctuations, instabilities and chaos in the laser-driven nonlinear ring cavity, J C. Englund, R.R. Snapp, W.C. Schieve
1- 67 69-216 217-286 287-354 355-428
VOLUME 22 (1985) 1 2 3 4 5 6
Optical and electronic processing of medical images, D. Malacara Quantum fluctuations in vision, M.A. Bouman, W.A. Van De Grind, P Zuidema Spectral and temporal fluctuations of broad-band laser radiation, A. V. Masalov Holographic methods of plasma diagnostics, G.V. Ostrovskaya, Yu.I. Ostrovsky Fringe formations in deformation and vibration measurements using laser light, I. Yamaguchi Wave propagation in random media: a systems approach, R.L. Fante
1- 76 77-144 145-196 197-270 271-340 341-398
VOLUME 23 (1986) 1 Analytical techniques for multiple scattering from rough surfaces, JA. DeSanto, G.S. Brown 2 Paraxial theory in optical design in terms of Gaussian brackets, K. Tanaka 3 Optical films produced by ion-based techniques, PJ Martin, R.P Nette~field
1- 62 63-111 113-182
388 4 5
Contents of preuious uolumes
Electron holography, A. Tonomura Principles of optical processing with partially coherent light, ET.S. Yu
183-220 221-275
VOLUME 24 (1987) 1 2 3 4 5
Micro Fresnel lenses, H. Nishihara, T. Suhara Dephasing-induced coherent phenomena, L. Rothberg Interferometry with lasers, P Hariharan Unstable resonator modes, K.E. Oughstun Information processing with spatially incoherent light, I. Glaser
1- 37 39-101 103-164 165-387 389-509
VOLUME 25 (1988) 1 Dynamical instabilities and pulsations in lasers, N.B. Abraham, P Mandel, L.M. Narducci 2 Coherence in semiconductor lasers, M. Ohtsu, T. Tako 3 Principles and design of optical arrays, Wang Shaomin, L. Ronchi 4 Aspheric surfaces, G. Schulz
1-190 191-278 279-348 349-4 15
VOLUME 26 (1988) 1 2 3 4 5
Photon bunching and antibunching, M.C. Teich, B.E.A. Saleh Nonlinear optics of liquid crystals,/.C. Khoo Single-longitudinal-mode semiconductor lasers, G.P Agrawal Rays and caustics as physical objects, Yu.A. Krautsou Phase-measurement interferometry techniques, K. Creath
1-104 105-161 163-225 227-348 349-393
VOLUME 27 (1989) 1 2 3 4
The self-imaging phenomenon and its applications, K. Patorski Axicons and meso-optical imaging devices, L.M. Soroko Nonimaging optics for flux concentration, I.M. Bassett, WT. Welford, R. Winston Nonlinear wave propagation in planar structures, D. Mihalache, M. Bertolotti, C. Sibilia 5 Generalized holography with application to inverse scattering and inverse source problems, R.P. Porter
1-108 109-160 161-226 227-313 315-397
VOLUME 28 (1990) 1 Digital holography- computer-generated holograms, O. Bryngdahl, E Wyrowski 2 Quantum mechanical limit in optical precision measurement and communication, Y. Yamamoto, S. Machida, S. Saito, N. Imoto, T. Yanagawa, M. Kitagawa, G. Bjdrk 3 The quantum coherence properties of stimulated Raman scattering, M.G. Raymet, I.A. Walmsley 4 Advanced evaluation techniques in interferometry, J. Schwider 5 Quantum jumps, R.J. Cook
1- 86 87-179 181-270 271-359 361-416
Contents of previous volumes
389
VOLUME 29 ( 1991 ) 1 Optical waveguide diffraction gratings: coupling between guided modes, D.G. Hall 2 Enhanced backscattering in optics, Yu.N. Barabanenkov, Yu.A. Kravtsov, V.D. Ozrin, A.I. Saichev 3 Generation and propagation of ultrashort optical pulses, I.P Christov 4 Triple-correlation imaging in optical astronomy, G. Weigelt 5 Nonlinear optics in composite materials. 1. Semiconductor and metal crystallites in dielectrics, C. Flytzanis, E Hache, M.C. Klein, D. Ricard, Ph. Roussignol
1- 63 65-197 199-291 293-319 321-411
VOLUME 30 (1992) 1 Quantum fluctuations in optical systems, S. Reynaud, A. Heidmann, E. Giacobino, C. Fabre 2 Correlation holographic and speckle interferometry, fil.L Ostrovsky, V.P Shchepinov 3 Localization of waves in media with one-dimensional disorder, V.D. Freilikhet; S.A. Gredeskul 4 Theoretical foundation of optical-soliton concept in fibers, Y. Kodama, A. Hasegawa 5 Cavity quantum optics and the quantum measurement process, P Meystre
1- 85 87-135 137-203 205-259 261-355
VOLUME 31 (1993) 1 2 3 4 5 6
Atoms in strong fields: photoionization and chaos, P W. Milonni, B. Sundaram Light diffraction by relief gratings: a macroscopic and microscopic view, E. Popov Optical amplifiers, N.K. Dutta, JR. Simpson Adaptive multilayer optical networks, D. Psaltis, Y. Qiao Optical atoms, R.JC. Spreeuw; J P Woerdman Theory of Compton free electron lasers, G. Dattoli, L. Giannessi, A. Renieri, A. Torre
1-137 139-187 189-226 227-261 263-319 321-412
VOLUME 32 (1993) Guided-wave optics on silicon: physics, technology and status, B.P Pal Optical neural networks: architecture, design and models, ETS. Yu The theory of optimal methods for localization of objects in pictures, L.P Yaroslavsky Wave propagation theories in random media based on the path-integral approach, M.I. Charnotsla'i, J. Gozani, V.I. Tatarskii, V.U. Zavorotny 5 Radiation by uniformly moving sources. Vavilov-Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena, V.L. Ginzburg 6 Nonlinear processes in atoms and in weakly relativistic plasmas, G. Mainfray, C. Manus 1
2 3 4
1- 59 61-144 145-201 203-266 267-312 313-361
VOLUME 33 (1994) 1 2 3 4
The imbedding method in statistical boundary-value wave problems, V.I. Klyatskin Quantum statistics of dissipative nonlinear oscillators, V. Pe~inov6, A. Lukg Gap solitons, C.M. De Sterke, J.E. Sipe Direct spatial reconstruction of optical phase from phase-modulated images, V.I. Vlad, D. Malacara 5 Imaging through turbulence in the atmosphere, M.J. Beran, J. Oz-Vogt 6 Digital halftoning: synthesis of binary images, O. BJyngdahl, T. Scheermessel; F. Wyrowski
1-127 129-202 203-260 261-317 319-388 389-463
390
Contents of preuious uolumes
VOLUME 34 (1995) 1 Quantum interference, superposition states of light, and nonclassical effects, V. Bu~ek, PL. Knight 2 Wave propagation in inhomogeneous media: phase-shift approach, L.P Presnyakou 3 The statistics of dynamic speckles, T. Okamoto, T. Asakura 4 Scattering of light from multilayer systems with rough boundaries, I. Ohlidal, K. Naorhtil, M. Ohlidal 5 Random walk and diffusion-like models of photon migration in turbid media, A.H. Gandjbakhche, G.H. Weiss
1-158 159-181 183-248 249-331 333-402
VOLUME 35 (1996) 1 Transverse patterns in wide-aperture nonlinear optical systems, N.N. Rosanou 2 Optical spectroscopy of single molecules in solids, M. Orrit, J. Bernard, R. Brown, B. Lounis 3 Interferometric multispectral imaging, K. Itoh 4 Interferometric methods for artwork diagnostics, D. PaolettL G. Schirripa Spagnolo 5 Coherent population trapping in laser spectroscopy, E. Arimondo 6 Quantum phase properties of nonlinear optical phenomena, R. Tana~, A. Miranowicz, Ts. Gantsog
1- 60 61-144 145-196 197-255 257-354 355-446
VOLUME 36 (1996) 1 Nonlinear propagation of strong laser pulses in chalcogenide glass films, V. Chumash, I. Cojocaru, E. Fazio, E Michelotti, M. Bertolotti 2 Quantum phenomena in optical interferometry, P. Hariharan, B.C. Sanders 3 Super-resolution by data inversion, M. Bertero, C. De Mol 4 Radiative transfer: new aspects of the old theory, fit.A. Krautsou, L.A. Apresyan 5 Photon wave function, I. Bialynicki-Birula
1-47 49-128 129-178 179-244 245-294
VOLUME 37 (1997) 1 The Wigner distribution function in optics and optoelectronics, D. Dragoman 2 Dispersion relations and phase retrieval in optical spectroscopy, K.-E. Peiponen, E.M. Vartiainen, T. Asakura 3 Spectra of molecular scattering of light, I.L. Fabelinskii 4 Soliton communication systems, R.-J Essiambre, G.P Agrawal 5 Local fields in linear and nonlinear optics of mesoscopic systems, O. Keller 6 Tunneling times and superluminality, R.Y. Chiao, A.M. Steinberg
1- 56 57- 94 95-184 185-256 257-343 345-405
VOLUME 38 (1998) 1 Nonlinear optics of stratified media, S. Dutta Gupta 2 Optical aspects of interferometric gravitational-wave detectors, P. Hello 3 Thermal properties of vertical-cavity surface-emitting semiconductor lasers, W Nakwaski, M. Osihski 4 Fractional transformations in optics, A. W Lohmann, D. Mendlovic, Z. Zalevsky 5 Pattern recognition with nonlinear techniques in the Fourier domain, B. Javidi, J.L. Homer 6 Free-space optical digital computing and interconnection, J Jahns
1- 84 85-164 165-262 263-342 343-418 419-513
Contents of preuious uolumes
391
VOLUME 39 (1999) 1 Theory and applications of complex rays, Yu.A. Kravtsov, G. W. Forbes, A.A. Asaoyan 2 Homodyne detection and quantum-state reconstruction, D.-G. Welsch, W. Vogel, T. Opatrn~ 3 Scattering of light in the eikonal approximation, S.K. Shalwla, D.J. Somerford 4 The orbital angular momentum of light, L. Allen, M.J Padgett, M. Babiker 5 The optical Kerr effect and quantum optics in fibers, A. Sizmann, G. Leuchs
1- 62 63-211 213-290 291-372 373-469
VOLUME 40 (2000) 1 2 3 4
Polarimetric optical fibers and sensors, T.R. Wolihski Digital optical computing, J Tanida, Y. Ichioka Continuous measurements in quantum optics, V.Pe~'qnou6, A. Luk~ Optical systems with improved resolving power, Z. Zaleusky, D. Mendlouic, A. W. Lohmann 5 Diffractive optics: electromagnetic approach, J. Turunen, M. Kuittinen, E Wyrowski 6 Spectroscopy in polychromatic fields, Z. Ficek and H.S. Freedhoff
1-75 77-114 115-269 271-341 343-388 389-441
VOLUME 41 (2000) 1 Nonlinear optics in microspheres, M.H. Fields, J. Popp, R.K. Chang 2 Principles of optical disk data storage, J. Carriere, R. Narayan, W.-H. Yeh, C. Peng, P. Khulbe, L. Li, R. Anderson, J. Choi, M. Mansuripur 3 Ellipsometry of thin film systems, L Ohlidal, D. Franta 4 Optical true-time delay control systems for wideband phased array antennas, R.T. Chen, Z. Fu 5 Quantum statistics of nonlinear optical couplers, J Pe~ina Jr, J. Pet~ina 6 Quantum phase difference, phase measurements and Stokes operators, A. Luis, L.L. S6nchez-Soto 7 Optical solitons in media with a quadratic nonlinearity, C. Etrich, E Lederer, B.A. Malomed, T. Peschel, U Peschel
1- 95 97-179 181-282 283-358 359-417 419-479 483-567
VOLUME 42 (2001 ) 1 Quanta and information, S. Ya. Kilin 2 Optical solitons in periodic media with resonant and off-resonant nonlinearities, G. Kurizki, A.E. Kozhekin, T. Opatrn~, B.A. Malomed 3 Quantum Zeno and inverse quantum Zeno effects, P Facchi, S. Pascazio 4 Singular optics, M.S. Soskln, M.V. Vasnetsou 5 Multi-photon quantum interferometry, G. Jaeger, A. V. Sergienko 6 Transverse mode shaping and selection in laser resonators, R. Oron, N. Dauidson, A.A. Friesem, E. Hasman
1-91 93-146 147-217 219-276 277-324 325-386
VOLUME 43 (2002) 1 2 3 4
Active optics in modem large optical telescopes, L. Noethe Variational methods in nonlinear fiber optics and related fields, B.A. Malomed Optical works of L.V. Lorenz, O. Keller Canonical quantum description of light propagation in dielectric media, A. Lukg and V. Pe~inov6
1- 69 71-193 195-294
295-431
392
Contents of previous volumes
5
Phase space correspondence between classical optics and quantum mechanics, D. Dragoman 6 "Slow" and "fast" light, R. W. Boyd and D.J. Gauthier 7 The fractional Fourier transform and some of its applications to optics, A. Torre
433-496 497-530 531-596
Cumulative i n d e x - Volumes 1-44"
Abdullaev, EKh., S.A. Darmanyan, J. Gamier: Modulational instability of electromagnetic waves in inhomogeneous and in discrete media Abel6s, E: Methods for determining optical parameters of thin films Abella, I.D.: Echoes at optical frequencies Abitbol, C.I., s e e Clair, J.J. Abraham, N.B., P. Mandel, L.M. Narducci: Dynamical instabilities and pulsations in lasers Agarwal, G.S.: Master equation methods in quantum optics Agranovich, V.M., V.L. Ginzburg: Crystal optics with spatial dispersion Agrawal, G.P.: Single-longitudinal-mode semiconductor lasers Agrawal, G.P., s e e Essiambre, R.-J. Allen, L., D.G.C. Jones: Mode locking in gas lasers Allen, L., M.J. Padgett, M. Babiker: The orbital angular momentum of light Ammann, E.O.: Synthesis of optical birefringent networks Anderson, R., s e e Carriere, J. Apresyan, L.A., s e e Kravtsov, Yu.A. Arimondo, E.: Coherent population trapping in laser spectroscopy Armstrong, J.A., A.W. Smith: Experimental studies of intensity fluctuations in lasers Amaud, J.A.: Hamiltonian theory of beam mode propagation Asakura, T., s e e Okamoto, T. Asakura, T., s e e Peiponen, K.-E. Asatryan, A.A., s e e Kravtsov, Yu.A. Babiker, M., s e e Allen, L. Baltes, H.P.: On the validity of Kirchhoff's law of heat radiation for a body in a nonequilibrium environment Barabanenkov, Yu.N., Yu.A. Kravtsov, V.D. Ozrin, A.I. Saichev: Enhanced 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
* Volumes I-XL were previously distinguished by roman rather than by arabic numerals. 393
44, 303 2, 249 7, 139 16, 71 25, 1 11, 1 9, 235 26, 163 37, 185 9, 179 39, 291 9, 123 41, 97 36, 179 35, 257 6,211 11, 247 34, 183 37, 57 39, 1 39, 291 13,
1
29,
65
1, 21, 12, 27, 6,
67 217 287 161 53
394
Cumulative
index-
Volumes 1-44
Beran, M.J., J. Oz-Vogt: Imaging through turbulence in the atmosphere Bernard, J., s e e Orrit, M. Berry, M.V., C. Upstill: Catastrophe optics: morphologies of caustics and their diffraction patterns Bertero, M., C. De Mol: Super-resolution by data inversion Bertolotti, M., s e e Mihalache, D. Bertolotti, M., s e e Chumash, V. Beverly III, R.E.: Light emission from high-current surface-spark discharges Bialynicki-Birula, I.: Photon wave function Bj6rk, G., s e e Yamamoto, Y. Bloom, A.L.: Gas lasers and their application to precise length measurements Bouman, M.A., W.A. Van De Grind, P. Zuidema: Quantum fluctuations in vision Bousquet, P., s e e Rouard, P. Boyd, R.W. and D.J. Gauthier: "Slow" and "fast" light Brown, G.S., s e e DeSanto, J.A. Brown, R., s e e Orrit, M. Brunner, W., H. Paul: Theory of optical parametric amplification and oscillation Bryngdahl, O.: Applications of shearing interferometry Bryngdahl, O.: Evanescent waves in optical imaging Bryngdahl, O., E Wyrowski: Digital holography - computer-generated holograms Bryngdahl, O., T. Scheermesser, E Wyrowski: Digital halfloning: synthesis of binary images Burch, J.M.: The metrological applications of diffraction gratings Butterweck, H.J.: Principles of optical data-processing Bu2ek, V., EL. Knight: Quantum interference, superposition states of light, and nonclassical effects
33, 319 35, 61
Cagnac, B., s e e Giacobino, E. Carriere, J., R. Narayan, W.-H. Yeh, C. Peng, P. Khulbe, L. Li, R. Anderson, J. Choi, M. Mansuripur: Principles of optical disk data storage Casasent, D., D. Psaltis: Deformation invariant, space-variant optical pattern recognition Ceglio, N.M., D.W. Sweeney: Zone plate coded imaging: theory and applications Chang, R.K., s e e Fields, M.H. Charnotskii, M.I., J. Gozani, V.I. Tatarskii, V.U. Zavorotny: Wave propagation theories in random media based on the path-integral approach Chen, R.T., Z. Fu: Optical true-time delay control systems for wideband phased array antennas Chiao, R.Y., A.M. Steinberg: Tunneling times and superluminality Choi, J., s e e Carriere, J. Christensen, J.L., s e e Rosenblum, W.M. Christov, I.P.: Generation and propagation of ultrashort optical pulses Chumash, V., I. Cojocaru, E. Fazio, E Michelotti, M. Bertolotti: Nonlinear propagation of strong laser pulses in chalcogenide glass films Clair, J.J., C.I. Abitbol: Recent advances in phase profiles generation Clarricoats, P.J.B.: Optical fibre waveguides- a review Cohen-Tannoudji, C., A. Kastler: Optical pumping Cojocaru, I., s e e Chumash, V. Cole, T.W.: Quasi-optical techniques of radio astronomy Colombeau, B., s e e Froehly, C. Cook, R.J.: Quantum jumps
17, 85
18, 36, 27, 36, 16, 36, 28, 9, 22, 4, 43, 23, 35, 15, 4, 11, 28,
257 129 227 1 357 245 87 1 77 145 497 1 61 1 37 167 1
33, 389 2, 73 19, 211 34,
1
41, 97 16, 289 21, 287 41, 1 32, 203 41, 37, 41, 13, 29,
283 345 97 69 199
36, 16, 14, 5, 36, 15, 20, 28,
1 71 327 1 1 187 63 361
Cumulative
index-
395
Volumes 1-44
Court6s, G., P. Cruvellier, M. Detaille, M. Saisse: Some new optical designs for ultra-violet bidimensional detection of astronomical objects Creath, K.: Phase-measurement interferometry techniques Crewe, A.V.: Production of electron probes using a field emission source Cruvellier, P., s e e Courtbs, G. Cummins, H.Z., H.L. Swinney: Light beating spectroscopy Dainty, J.C.: The statistics of speckle patterns D~indliker, R.: Heterodyne holographic interferometry Darmanyan, S.A., s e e Abdullaev, EKh. Dattoli, G., L. Giannessi, A. Renieri, A. Torre: Theory of Compton free electron lasers Davidson, N., s e e Oron, R. De Mol, C., s e e Bertero, M. De Sterke, C.M., J.E. Sipe: Gap solitons Decker Jr, J.A., s e e Harwit, M. Delano, E., R.J. Pegis: Methods of synthesis for dielectric multilayer filters Demaria, A.J.: Picosecond laser pulses DeSanto, J.A., G.S. Brown: Analytical techniques for multiple scattering from rough surfaces Detaille, M., s e e Court~s, G. Dexter, D.L., s e e Smith, D.Y. Dragoman, D.: The Wigner distribution function in optics and optoelectronics Dragoman, D.: Phase space correspondence between classical optics and quantum mechanics Drexhage, K.H.: Interaction of light with monomolecular dye layers Duguay, M.A.: The ultrafast optical Kerr shutter Dutta, N.K., J.R. Simpson: Optical amplifiers Dutta Gupta, S.: Nonlinear optics of stratified media 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 Essiambre, R.-J., G.P. Agrawal: Soliton communication systems Etrich, C., E Lederer, B.A. Malomed, T. Peschel, U. Peschel: Optical solitons in media with a quadratic nonlinearity Fabelinskii, I.L.: Spectra of molecular scattering of light Fabre, C., s e e Reynaud, S. Facchi, P., S. Pascazio: Quantum Zeno and inverse quantum Zeno effects Fante, R.L.: Wave propagation in random media: a systems approach Fazio, E., s e e Chumash, V. Fercher, A.E, C.K. Hitzenberger: Optical coherence tomography Ficek, Z. and H.S. Freedhoff: Spectroscopy in polychromatic fields Fields, M.H., J. Popp, R.K. Chang: Nonlinear optics in microspheres Fiorentini, A.: Dynamic characteristics of visual processes Flytzanis, C., E Hache, M.C. Klein, D. Ricard, Ph. Roussignol: Nonlinear optics in composite materials. 1. Semiconductor and metal crystallites in dielectrics Focke, J.: Higher order aberration theory Forbes, G.W., s e e Kravtsov, Yu.A. Franqon, M., S. Mallick: Measurement of the second order degree of coherence
20, 1 26, 349 11, 223 20, 1 8, 133
14, 17, 44, 31, 42, 36, 33, 12, 7, 9,
1 1 303 321 325 129 203 101 67 31
23, 1 20, 1 10, 165 37, 1 43, 12, 14, 31, 38,
433 163 161 189 1
7, 359 21, 355 16, 233 37, 185 41, 483 37, 30, 42, 22, 36, 44, 40, 41, 1,
95 1 147 341 1 215 389 1 253
29, 321 4, 1 39, 1 6, 71
396
Cumulatiue
index-
Volumes
1-44
Franta, D., s e e Ohlidal, I. Freedhoff, H.S., s e e Ficek, Z. Freilikher, V.D., S.A. Gredeskul: Localization of waves in media with one-dimensional disorder Frieden, B.R.: Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions Friesem, A.A., s e e Oron, R. Froehly, C., B. Colombeau, M. Vampouille: Shaping and analysis of picosecond light pulses Fry, G.A.: The optical performance of the human eye Fu, Z., s e e Chen, R.T.
41, 181 40, 389
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., s e e Tana~, R. Gamier, J., s e e Abdullaev, EKh. Gauthier, D.J., s e e Boyd, R.W. Ghatak, A., K. Thyagarajan: Graded index optical waveguides: a review Ghatak, A.K., s e e Sodha, M.S. Giacobino, E., B. Cagnac: Doppler-free multiphoton spectroscopy Giacobino, E., s e e Reynaud, S. Giannessi, L., s e e Dattoli, G. Ginzburg, V.L., s e e Agranovich, V.M. Ginzburg, V.L.: Radiation by uniformly moving sources. Vavilov-Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena Giovanelli, R.G.: Diffusion through non-uniform media Glaser, I.: Information processing with spatially incoherent light Gniadek, K., J. Petykiewicz: Applications of optical methods in the diffraction theory of elastic waves Goodman, J.W.: Synthetic-aperture optics Gozani, J., s e e Charnotskii, M.I. Graham, R.: The phase transition concept and coherence in atomic emission Gredeskul, S.A., s e e Freilikher, V.D.
1, 109 3, 187
Hache, E, s e e Flytzanis, C. Hall, D.G.: Optical waveguide diffraction gratings: coupling between guided modes Hariharan, P.: Colour holography Hariharan, P.: Interferometry with lasers Hariharan, P., B.C. Sanders: Quantum phenomena in optical interferometry Harwit, M., J.A. Decker Jr: Modulation techniques in spectrometry Hasegawa, A., s e e Kodama, Y. Hasman, E., s e e Oron, R. Heidmann, A., s e e Reynaud, S. Hello, P.: Optical aspects of interferometric gravitational-wave detectors Helstrom, C.W.: Quantum detection theory Herriot, D.R.: Some applications of lasers to interferometry Hitzenberger, C.K., s e e Fercher, A.E Homer, J.L., s e e Javidi, B. Huang, T.S.: Bandwidth compression of optical images
30, 137 9, 311 42, 325 20, 63 8, 51 41, 283
34, 35, 44, 43, 18, 13, 17, 30, 31, 9,
333 355 303 497 1 169 85 1 321 235
32, 267 2, 109 24, 389 9, 8, 32, 12, 30,
281 1 203 233 137
29, 29, 20, 24, 36, 12, 30, 42, 30, 38, 10, 6, 44, 38, 10,
321 1 263 103 49 101 205 325 1 85 289 171 215 343 1
Cumulative
index
-
Volumes
397
1-44
Ichioka, Y., s e e Tanida, J. Imoto, N., s e e Yamamoto, Y. Itoh, K.: Interferometric multispectral imaging
40, 77 28, 87 35, 145
Jacobsson, R.: Light reflection from films of continuously varying refractive index Jacquinot, R, B. Roizen-Dossier: Apodisation Jaeger, G., A.V Sergienko: Multi-photon quantum interferometry Jahns, J.: Free-space optical digital computing and interconnection Jamroz, W., B.P. Stoicheff: Generation of tunable coherent vacuum-ultraviolet radiation Javidi, B., J.L. Homer: Pattern recognition with nonlinear techniques in the Fourier domain Jones, D.G.C., s e e Allen, L.
5, 3, 42, 38, 20,
Kastler, A., s e e Cohen-Tannoudji, C. Keller, O.: Local fields in linear and nonlinear optics of mesoscopic systems Keller, O.: Optical works of L.V Lorenz Khoo, I.C.: Nonlinear optics of liquid crystals Khulbe, P., s e e Carriere, J. Kielich, S.: Multi-photon scattering molecular spectroscopy Kilin, S.Ya.: Quanta and information Kinosita, K.: Surface deterioration of optical glasses Kitagawa, M., s e e Yamamoto, Y. Klein, M.C., s e e Flytzanis, C. Klyatskin, VI.: The imbedding method in statistical boundary-value wave problems Knight, EL., s e e Bu~ek, V Kodama, Y., A. Hasegawa: Theoretical foundation of optical-soliton concept in fibers Koppelman, G.: Multiple-beam interference and natural modes in open resonators Kottler, E: The elements of radiative transfer Kottler, E: Diffraction at a black screen, Part I: Kirchhoff's theory Kottler, E: Diffraction at a black screen, Part II: electromagnetic theory Kozhekin, A.E., s e e Kurizki, G. Kravtsov, Yu.A.: Rays and caustics as physical objects Kravtsov, Yu.A., s e e Barabanenkov, Yu.N. Kravtsov, Yu.A., L.A. Apresyan: Radiative transfer: new aspects of the old theory Kravtsov, Yu.A., G.W. Forbes, A.A. Asatryan: Theory and applications of complex rays Kubota, H.: Interference color Kuittinen, M., s e e Turunen, J. Kurizki, G., A.E. Kozhekin, T. Opatrn~,, B.A. Malomed: Optical solitons in periodic media with resonant and off-resonant nonlinearities
5, 37, 43, 26, 41, 20, 42, 4, 28, 29, 33, 34, 30, 7, 3, 4, 6, 42, 26, 29, 36, 39, 1, 40,
1 257 195 105 97 155 1 85 87 321 1 1 205 1 1 281 331 93 227 65 179 1 211 343
42,
93
Labeyrie, A.: High-resolution techniques in optical astronomy Lean, E.G.: Interaction of light and acoustic surface waves Lederer, E, s e e Etrich, C. Lee, W.-H.: Computer-generated holograms: techniques and applications Leith, E.N., J. Upatnieks: Recent advances in holography Letokhov, V.S.: Laser selective photophysics and photochemistry Leuchs, G., s e e Sizmann, A. Levi, L.: Vision in communication Li, L., s e e Carriere, J. Lipson, H., C.A. Taylor: X-ray crystal-structure determination as a branch of physical optics
14, 11, 41, 16, 6, 16, 39, 8, 41,
47 123 483 119 1 1 373 343 97
247 29 277 419 325
38, 343 9, 179
5, 287
398
Cumulatiue
index-
Volumes
1-44
Lohmann, A.W., D. Mendlovic, Z. Zalevsky: Fractional transformations in optics Lohmann, A.W., s e e Zalevsky, Z. Lounis, B., s e e Orrit, M. Lugiato, L.A.: Theory of optical bistability Luis, A., L.L. Sfinchez-Soto: Quantum phase difference, phase measurements and Stokes operators Luke, A., s e e Pefinovfi, V. Luke, A., s e e Pefinovfi, V. Luke, A. and V. Pefinovfi: Canonical quantum description of light propagation in dielectric media
38, 263 40, 271 35, 61 21, 69
Machida, S., s e e Yamamoto, Y. Mainfray, G., C. Manus: Nonlinear processes in atoms and in weakly relativistic plasmas Malacara, D.: Optical and electronic processing of medical images Malacara, D., s e e Vlad, V.I. Mallick, S., s e e Franqon, M. Malomed, B.A., s e e Etrich, C. Malomed, B.A., s e e Kurizki, G. Malomed, B.A.: Variational methods in nonlinear fiber optics and related fields Mandel, L.: Fluctuations of light beams Mandel, L.: The case for and against semiclassical radiation theory Mandel, P., s e e Abraham, N.B. Mansuripur, M., s e e Carriere, J. Manus, C., s e e 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., s e e Rouard, P. Mehta, C.L.: Theory of photoelectron counting Mendlovic, D., s e e Lohmann, A.W. Mendlovic, D., s e e Zalevsky, Z. Meystre, P.: Cavity quantum optics and the quantum measurement process Michelotti, E, s e e Chumash, V. Mihalache, D., M. Bertolotti, C. Sibilia: Nonlinear wave propagation in planar structures Mikaelian, A.L., M.L. Ter-Mikaelian: Quasi-classical theory of laser radiation Mikaelian, A.L.: Self-focusing media with variable index of refraction Mills, D.L., K.R. Subbaswamy: Surface and size effects on the light scattering spectra of solids Milonni, P.W., B. Sundaram: Atoms in strong fields: photoionization and chaos Miranowicz, A., s e e Tana~, R. Miyamoto, K.: Wave optics and geometrical optics in optical design Mollow, B.R.: Theory of intensity dependent resonance light scattering and resonance fluorescence Murata, K.: Instruments for the measuring of optical transfer functions Musset, A., A. Thelen: Multilayer antireflection coatings
28, 32, 22, 33, 6, 41, 42, 43, 2, 13, 25, 41, 32, 11, 23, 22, 21, 15, 8, 38, 40, 30, 36, 27, 7, 17,
Nakwaski, W., M. Osifiski: Thermal properties of vertical-cavity surface-emitting semiconductor lasers Narayan, R., s e e Carriere, J. Narducci, L.M., s e e Abraham, N.B.
41, 419 33, 129 40, 115 43, 295 87 313 1 261 71 483 93 71 181 27 1 97 313 305 113 145 1 77 373 263 271 261 1 227 231 279
19, 45 31, 1 35, 355 1, 31 19, 1 5, 199 8, 201
38, 165 41, 97 25, 1
Cumulative
index-
Volumes
399
1-44
Navr~til, K., s e e Ohlidal, I. Netterfield, R.E, s e e Martin, EJ. Nishihara, H., T. Suhara: Micro Fresnel lenses Noethe, L.: Active optics in modern large optical telescopes Ohlidal, I., K. Navrfitil, M. Ohlidal: Scattering of light from multilayer systems with rough boundaries Ohlidal, I., D. Franta: Ellipsometry of thin film systems Ohlidal, M., s e e Ohlidal, I. Ohtsu, M., T. Tako: Coherence in semiconductor lasers Ohtsubo, J.: Chaotic dynamics in semiconductor lasers with optical feedback Okamoto, T., T. Asakura: The statistics of dynamic speckles Okoshi, T.: Projection-type holography Omenetto, EG.: Femtosecond pulses in optical fibers Ooue, S.: The photographic image Opatrn~,, T., s e e Welsch, D.-G. Opatrn~,, T., s e e Kurizki, G. Oron, R., N. Davidson, A.A. Friesem, E. Hasman: Transverse mode shaping and selection in laser resonators Orrit, M., J. Bernard, R. Brown, B. Lounis: Optical spectroscopy of single molecules in solids Osiflski, M., s e e Nakwaski, W. Ostrovskaya, G.V., Yu.I. Ostrovsky: Holographic methods of plasma diagnostics Ostrovsky, Yu.I., s e e Ostrovskaya, G.V. Ostrovsky, Yu.I., V.P. Shchepinov: Correlation holographic and speckle interferometry Oughstun, K.E.: Unstable resonator modes Oz-Vogt, J., s e e Beran, M.J. Ozrin, V.D., s e e Barabanenkov, Yu.N. Padgett, M.J., s e e Allen, L. Pal, B.P.: Guided-wave optics on silicon: physics, technology and status Paoletti, D., G. Schirripa Spagnolo: Interferometric methods for artwork diagnostics Pascazio, S., s e e Facchi, P. Patorski, K.: The self-imaging phenomenon and its applications Paul, H., s e e Brunner, W. Pegis, R.J.: The modern development of Hamiltonian optics Pegis, R.J., s e e Delano, E. Peiponen, K.-E., E.M. Vartiainen, T. Asakura: Dispersion relations and phase retrieval in optical spectroscopy Peng, C., s e e Carriere, J. Pefina, J.: Photocount statistics of radiation propagating through random and nonlinear media Pefina, J., s e e Pefina Jr, J. Pefina Jr, J., J. Pefina: Quantum statistics of nonlinear optical couplers Pe~inovfi, V., A. Luke: Quantum statistics of dissipative nonlinear oscillators Pefinovfi, V., A. Luke: Continuous measurements in quantum optics Pe~inovfi, V., s e e Luke, A. Pershan, P.S.: Non-linear optics Peschel, T., s e e Etrich, C. Peschel, U., s e e Etrich, C. Petite, G., s e e Shvartsburg, A.B.
34, 249 23, 113 24, 1 43, 1
34, 41, 34, 25, 44, 34, 15, 44, 7, 39, 42,
249 181 249 191 1 183 139 85 299 63 93
42, 325 35, 38, 22, 22, 30, 24, 33, 29,
61 165 197 197 87 165 319 65
39, 291 32, 1 35, 197 42, 147 27, 1 15, 1 1, 1 7, 67 37, 41,
57 97
18, 41, 41, 33, 40, 43, 5, 41, 41, 44,
127 359 359 129 115 295 83 483 483 143
400
Cumulatiue
index-
Volumes
1-44
Petykiewicz, J., s e e Gniadek, K. Picht, J.: The wave of a moving classical electron Popov, E.: Light diffraction by relief gratings: a macroscopic and microscopic view Popp, J., s e e Fields, M.H. Porter, R.P.: Generalized holography with application to inverse scattering and inverse source problems Presnyakov, L.P.: Wave propagation in inhomogeneous media: phase-shift approach Psaltis, D., s e e Casasent, D. Psaltis, D., Y. Qiao: Adaptive multilayer optical networks
9, 281 5, 351 31, 139 41, 1
Qiao, Y.,
31, 227
see
Psaltis, D.
Raymer, M.G., I.A. Walmsley: The quantum coherence properties of stimulated Raman scattering Renieri, A., s e e Dattoli, G. Reynaud, S., A. Heidmann, E. Giacobino, C. Fabre: Quantum fluctuations in optical systems Ricard, D., s e e Flytzanis, C. Riseberg, L.A., M.J. Weber: Relaxation phenomena in rare-earth luminescence Risken, H.: Statistical properties of laser light Roddier, E: The effects of atmospheric turbulence in optical astronomy Roizen-Dossier, B., s e e Jacquinot, P. Ronchi, L., s e e Wang Shaomin Rosanov, N.N.: Transverse patterns in wide-aperture nonlinear optical systems Rosenblum, W.M., J.L. Christensen: Objective and subjective spherical aberration measurements of the human eye Rothberg, L.: Dephasing-induced coherent phenomena Rouard, P., P. Bousquet: Optical constants of thin films Rouard, P., A. Meessen: Optical properties of thin metal films Roussignol, Ph., s e e Flytzanis, C. Rubinowicz, A.: The Miyamoto-Wolf diffraction wave Rudolph, D., s e e Schmahl, G. Saichev, A.I., s e e Barabanenkov, Yu.N. Saisse, M., s e e Court~s, G. Saito, S., s e e Yamamoto, Y. Sakai, H., s e e Vanasse, G.A. Saleh, B.E.A., s e e Teich, M.C. Sfinchez-Soto, L.L., s e e Luis, A. Sanders, B.C., s e e Hariharan, P. Scheermesser, T., s e e Bryngdahl, O. Schieve, W.C., s e e Englund, J.C. Schirripa Spagnolo, G., s e e Paoletti, D. Schmahl, G., D. Rudolph: Holographic diffraction gratings Schubert, M., B. Wilhelmi: The mutual dependence between coherence properties of light and nonlinear optical processes Schulz, G., J. Schwider: Interferometric testing of smooth surfaces Schulz, G.: Aspheric surfaces Schwider, J., s e e Schulz, G. Schwider, J.: Advanced evaluation techniques in interferometry Scully, M.O., K.G. Whitney: Tools of theoretical quantum optics
27, 34, 16, 31,
315 159 289 227
28, 181 31, 321 30, 29, 14, 8, 19, 3, 25, 35,
1 321 89 239 281 29 279 1
13, 24, 4, 15, 29, 4, 14,
69 39 145 77 321 199 195
29, 20, 28, 6, 26, 41, 36, 33, 21, 35, 14,
65 1 87 259 1 419 49 389 355 197 195
17, 13, 25, 13, 28, 10,
163 93 349 93 271 89
Cumulative
index-
Volumes
401
1-44
Senitzky, I.R.: Semiclassical radiation theory within a quantum-mechanical framework Sergienko, A.V., s e e Jaeger, G. Sharma, S.K., D.J. Somerford: Scattering of light in the eikonal approximation Shchepinov, V.P., s e e Ostrovsky, Yu.I. Shvartsburg, A.B., G. Petite: Instantaneous optics of ultrashort broadband pulses and rapidly varying media Sibilia, C., s e e Mihalache, D. Simpson, J.R., s e e Dutta, N.K. Sipe, J.E., s e e Van Kranendonk, J. Sipe, J.E., s e e De Sterke, C.M. Sittig, E.K.: Elastooptic light modulation and deflection Sizmann, A., G. Leuchs: The optical Kerr effect and quantum optics in fibers Slusher, R.E.: Self-induced transparency Smith, A.W., s e e Armstrong, J.A. Smith, D.Y., D.L. Dexter: Optical absorption strength of defects in insulators Smith, R.W.: The use of image tubes as shutters Snapp, R.R., s e e Englund, J.C. Sodha, M.S., A.K. Ghatak, V.K. Tripathi: Self-focusing of laser beams in plasmas and semiconductors Somerford, D.J., s e e Sharma, S.K. Soroko, L.M.: Axicons and meso-optical imaging devices Soskin, M.S., M.V. Vasnetsov: Singular optics Spreeuw, R.J.C., J.P. Woerdman: Optical atoms Steel, W.H.: Two-beam interferometry Steinberg, A.M., s e e Chiao, R.Y. Stoicheff, B.P., s e e Jamroz, W. Strohbehn, J.W.: Optical propagation through the turbulent atmosphere Stroke, G.W.: Ruling, testing and use of optical gratings for high-resolution spectroscopy Subbaswamy, K.R., s e e Mills, D.L. Suhara, T., s e e Nishihara, H. Sundaram, B., s e e Milonni, P.W. Svelto, O.: Self-focusing, self-trapping, and self-phase modulation of laser beams Sweeney, D.W., s e e Ceglio, N.M. Swinney, H.L., s e e Cummins, H.Z.
16, 413 42, 277 39, 213 30, 87
Tako, T., s e e Ohtsu, M. Tanaka, K.: Paraxial theory in optical design in terms of Gaussian brackets Tana~, R., A. Miranowicz, Ts. Gantsog: Quantum phase properties of nonlinear optical phenomena Tango, W.J., R.Q. Twiss: Michelson stellar interferometry Tanida, J., Y. Ichioka: Digital optical computing Tatarskii, V.I., V.U. Zavorotnyi: Strong fluctuations in light propagation in a randomly inhomogeneous medium Tatarskii, V.I., s e e Charnotskii, M.I. Taylor, C.A., s e e Lipson, H. Teich, M.C., B.E.A. Saleh: Photon bunching and antibunching Ter-Mikaelian, M.L., s e e Mikaelian, A.L. Thelen, A., s e e Musset, A. Thompson, B.J.: Image formation with partially coherent light Thyagarajan, K., s e e Ghatak, A.
25, 191 23, 63
44, 27, 31, 15, 33, 10, 39, 12, 6, 10, 10, 21,
143 227 189 245 203 229 373 53 211 165 45 355
13, 39, 27, 42, 31, 5, 37, 20, 9,
169 213 109 219 263 145 345 325 73
2, 1 19, 45 24, 1 31, 1 12, 1 21, 287 8, 133
35, 355 17, 239 40, 77 18, 32, 5, 26, 7, 8, 7, 18,
204 203 287 1 231 201 169 1
402
Cumulative
index-
Volumes
1-44
Tonomura, A.: Electron holography Torre, A., s e e Dattoli, G. Torre, A.: The fractional Fourier transform and some of its applications to optics Tripathi, V.K., s e e Sodha, M.S. Tsujiuchi, J.: Correction of optical images by compensation of aberrations and by spatial frequency filtering Turunen, J., M. Kuittinen, E Wyrowski: Diffractive optics: electromagnetic approach Twiss, R.Q., s e e Tango, W.J.
23, 31, 43, 13,
Upatnieks, J., s e e Leith, E.N. Upstill, C., s e e Berry, M.V. Ushioda, S.: Light scattering spectroscopy of surface electromagnetic waves in solids
6, 1 18, 257 19, 139
Vampouille, M., s e e Froehly, C. Van De Grind, W.A., s e e Bouman, M.A. Van Heel, A.C.S.: Modern alignment devices Van Kranendonk, J., J.E. Sipe: Foundations of the macroscopic electromagnetic theory of dielectric media Vanasse, G.A., H. Sakai: Fourier spectroscopy Vartiainen, E.M., s e e Peiponen, K.-E. Vasnetsov, M.V., s e e Soskin, M.S. Vernier, P.J.: Photoemission Vlad, V.I., D. Malacara: Direct spatial reconstruction of optical phase from phase-modulated images Vogel, W., s e e Welsch, D.-G.
20, 63 22, 77 1, 289
Walmsley, I.A., s e e Raymer, M.G. Wang Shaomin, L. Ronchi: Principles and design of optical arrays Weber, M.J., s e e Riseberg, L.A. Weigelt, G.: Triple-correlation imaging in optical astronomy Weiss, G.H., s e e Gandjbakhche, A.H. Welford, W.T.: Aberration theory of gratings and grating mountings Welford, W.T.: Aplanatism and isoplanatism Welford, W.T., s e e Bassett, I.M. Welsch, D.-G., W. Vogel, T. Opatrn~,: Homodyne detection and quantum-state reconstruction Whitney, K.G., s e e Scully, M.O. Wilhelmi, B., s e e Schubert, M. Winston, R., s e e Bassett, I.M. Woerdman, J.P., s e e Spreeuw, R.J.C. Wolifiski, T.R." Polarimetric optical fibers and sensors Wolter, H.: On basic analogies and principal differences between optical and electronic information Wynne, C.G.: Field correctors for astronomical telescopes Wyrowski, F., s e e Bryngdahl, O. Wyrowski, F., s e e Bryngdahl, O. Wyrowski, F., s e e Turunen, J.
28, 25, 14, 29, 34, 4, 13, 27,
181 279 89 293 333 241 267 161
39, 10, 17, 27, 31, 40,
63 89 163 161 263 1
1, 10, 28, 33, 40,
155 137 1 389 343
Yamaguchi, I.: Fringe formations in deformation and vibration measurements using laser light Yamaji, K.: Design of zoom lenses
183 321 531 169
2, 131 40, 343 17, 239
15, 6, 37, 42, 14,
245 259 57 219 245
33, 261 39, 63
22, 271 6, 105
Cumulative
index
-
Volumes
403
1-44
Yamamoto, T.: Coherence theory of source-size compensation in interference microscopy Yamamoto, Y., S. Machida, S. Saito, N. Imoto, T. Yanagawa, M. Kitagawa, G. Bjrrk: Quantum mechanical limit in optical precision measurement and communication Yanagawa, T., s e e Yamamoto, Y. Yaroslavsky, L.P.: The theory of optimal methods for localization of objects in pictures Yeh, W.-H., s e e Carriere, J. Yoshinaga, H.: Recent developments in far infrared spectroscopic techniques Yu, ET.S.: Principles of optical processing with partially coherent light Yu, ET.S.: Optical neural networks: architecture, design and models Zalevsky, Z., s e e Lohmann, A.W. Zalevsky, Z., D. Mendlovic, A.W. Lohmann: Optical systems with improved resolving power Zavorotny, V.U., s e e Chamotskii, M.I. Zavorotnyi, V.U., s e e Tatarskii, V.I. Zuidema, P., s e e Bouman, M.A.
8, 295 28, 28, 32, 41, 11, 23, 32,
87 87 145 97 77 221 61
38, 263 40, 32, 18, 22,
271 203 204 77
