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EDITORIAL ADVISORY BOARD G.S. Agarwal Ahmedabad, India G. Agrawal Rochester, USA T. Asakura Sapporo, Japan A. As...

Author: Emil Wolf

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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


[ 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


[ 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


[ 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]


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


[ 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]


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


[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]


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


[ 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]


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


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]


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


[ 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]


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 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


[ 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


[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


[ 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


[ 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]


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


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-


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


[ 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


[ 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]


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


[ 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


1, w 4]

1.50

I

I

'

'

I

51 |

' -.

-

..

..... ~ ~" 9

..

(a) 9

9.

9 all

m ee I

9 .-

~D

....'....

1.40

O

9 ."

9

9 -....

.-'-.-" -;

.. =.. .. 9 .

,...-

..

es

... , . . . ' . . - 9,

.. 9

9

.=-

1.30

m~

,

I

.

I

1.30

,

I

,

I

1.32

,

1.34

Injection current J]Jth

(b) m

..g. =

--

.

~D 9 m

-

9

~, :,

:r ,~

4. 94 .,..

I,,.,. ,='i"

'".

:".':

."

~ '

,~

. .

~.,

"~,"

,~ ,i i

43. , i"

'%" 9 ""

""':

"'~

~.i~

::', : '.... , Jill

:.

~"

,,

,ill t-" ,,tllll< , 4 ,

1.30

ii I

, .,ii| ,,ill

...., l tlllllll

|" 9

9

~...

! ::

~;:

E"

-:-. -,.

J. 9 -;, ,~,,

:." 4 'G :.,.

" G :=.1

":'

" ;

~-

- :'" 9

"!I

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


[ 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


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]


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)


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

..r 9o

li,:'-'"'::

:'t

.:i 9

,,

..

.... ": ..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

9 ".o.,:.

9

:i

0""-

"0: "'i

.... 0,8 :s- "

0.0

I

i

I

i

I

[

(a) Total

":

I

I

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]


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


[ 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]


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


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]


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.


62

t

'

'

'

"'

I

'

'

'

[ 1, w 5

(a)

'

c,,,I q

~

0.5

0 2

i -

'

"

I

'

w"

,

Et

--4

(b)

I

~

1 -

,-,

I I

,~-

0

I0

time

i

i

,

,

,

I

'

'

'

'

-_

20

t (ns)

I

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]


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

'~. ' 1

-

I\|

-

/

-

/

/

14

llil

.:-' rilJ t

""--'"'"

. . . . .

"-"

"

,4J

i ~.t

.........

%%,.'*~ . . . . . 9

,,...~

.........

~ ,,~,1

I

~,~

,

I

F'~ "

-140 ___

lO-a

,

,

~

m-=

f

I

I

Illl[

~o-~

I

~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]


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


[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]


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


[ 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]


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


[ 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]


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


[ 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


[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


1, w 5]

77

14.0 12.0 I0.0 ~_,

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--_

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/

(b)

~

-

--

-0.1

'E 14.0 12.0 ~o.o

~,

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(c)

=, 4.0

:~

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(d)

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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


[ 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


[ 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

Page I nt ro d u c t i o n

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Femtosecond pulses-

w 2.

Observation .

w 3.

Manipulation

w 4.

Control .

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Photonic crystal fibers

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Acknowledgements

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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

88

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]

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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

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[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

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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]).

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[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.

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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


[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]

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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


[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 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]

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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 > 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 ~ 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

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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"

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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> 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~

~

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 > (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)

190

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

192

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> 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.

236

Optical coherence tomography

[4, w3

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.

4, w3]

237

OCT signal generation

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.

238


[4, w3

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


4, w 3]

239

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.

240


[4, w 3

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

4, w 3]


241

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


242

[4, w 3

-\ 10 -I

r

-2

c-- lO :D

-

0-3

"O

10 .4

D i -_.. _

"

-

c)

E lo-s

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.

OCT resolution and sensitivity

4, w5]

273

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


[4, w5

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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275

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


276

[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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277

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

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

Optical coherencetomography

282

[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].

4, w 6]

Light sources and delay lines

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.


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.

4, w6]

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

286


[4, w6

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

4, w 6]

Light sotoves and delay lines

287

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

288


[4, w 6

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

4, w 6]


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

290


[4, w 6

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.

4, w 6]

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


292

[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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293

[ 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


294

[4, w 6

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

303

Contents

Page w 1.

Introduction .

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w 2.

MI in h o m o g e n e o u s media .

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w 3.

MI in periodically i n h o m o g e n e o u s media . . . . . . .

314

w 4.

MI in r a n d o m media

332

w 5.

MI in nonlinear discrete optical systems

w 6.

Conclusions .

Acknowledgements References .

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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

306

MI in inhomogeneous and discrete media

[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


[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]

MI in homogeneous media

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[ (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


[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 0 (/3 < 0) for anomalous

334


[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 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


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


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


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


[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


[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 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 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]


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


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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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


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]


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 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


[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]


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).

358


[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


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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[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 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


[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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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

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