Electromagnetic Wave Propagation

in Nonlinear Kerr Media

Timo A. Laine

Stockholm 2000

Electromagnetic Wave Propagation

in Nonlinear Kerr Media

Timo A. Laine

Doctoral Thesis

Department of Physics

Royal Institute of Technology

Stockholm 2000

TRITA FYS-2201 ISSN 0280-316X

iv

Laine, Timo A. Electromagnetic Wave Propagation in Nonlinear Kerr Media

Royal Institute of Technology (KTH), Department of Physics { Optics

SE{100 44 Stockholm, Sweden.

Dissertation for the degree of Doctor of Technology to be presented with due

permission for public examination and debate in Auditorium D1 at Royal

Institute of Technology on the 9th of June, 2000 at 10:00 a.m.

Abstract

Electromagnetic �elds within a nonlinear medium satisfy Maxwell's equations.

In Kerr-type media the nonlinearity is created through an intensity dependent

refractive index. We examine the Maxwell equations with a Kerr nonlinear-

ity using four di�erent approaches; rigorous vector theory of volume gratings,

Hamilton's canonical perturbation theory of classical mechanics, variational

calculus that approximates the exact nonlinear solution, and analytical meth-

ods that produce, for example, the nonlinear solutions in the form of self-guided

waves. The latter three techniques make use of a mathematical construction

called the action.

We consider models that consist of several in�nite layers of (non)linear

media. Structures of this kind present simple and widely used geometries in

optics, photonics, and laser technology. In the nonlinear wave theory the Kerr-

Maxwell equations simplify signi�cantly due to the layered structure. From the

experimental point of view the fabrication of such samples is highly developed.

Most (non)linear phenomena, such as guided waves, bistability, and solitons,

are readily demonstrated in layered media.

Our results show that the exact electromagnetic theory of volume gratings

predicts similar nonlinear phenomena that are found in experiments. However,

we make evident that in many cases simple analytical approximations for the

nonlinear �elds are still more useful and accurate enough to give a qualitatively

correct nonlinear behavior. We also show that the action integral formalism

is an advanced and e�ective technique when it is applied to layered nonlinear

problems. In particular, our approach gives new insight into optical systems

with Kerr-type media and it elucidates the complex theory of electromagnetic

nonlinear waves.

v

List of Publications

This thesis is based on the following publications by the author:

I. T. A. Laine and A. T. Friberg, \Rigorous volume grating solution to

distortion correction in nonlinear layered media near a phase-conjugate

mirror," Opt. Commun. 159, 93{98 (1999).

II. T. A. Laine and A. T. Friberg, \Optical bistability and waveguide reso-

nances in a nonlinear layered structure with phase conjugation," J. Appl.

Phys. (submitted).

III. T. A. Laine and A. T. Friberg, \Nonlinear thin-layer theory for strati�ed

Kerr medium," Appl. Phys. Lett. 74, 3248{3250 (1999).

IV. T. A. Laine and A. T. Friberg, \Thin-layer theory of nonlinear �elds,"

J. Opt. Soc. Am. B (submitted).

V. T. A. Laine and A. T. Friberg, \Canonical perturbative approach to non-

linear systems with application to optical waves in layered Kerr media,"

Phys. Rev. E (in press).

VI. T. A. Laine and A. T. Friberg, \Variational analysis and angular bista-

bility in layered nonlinear Kerr media with phase conjugation, " J. Opt.

Soc. Am. B (in press).

VII. T. A. Laine and A. T. Friberg, \Self-guided waves and exact solutions

of the nonlinear Helmholtz equation," J. Opt. Soc. Am. B 17, 751{757

(2000).

vi

The author has also contributed to the following reports and publications not

included in the thesis:

1. T. A. Laine and A. T. Friberg, \Phase-conjugate specular-re ection can-

cellation for multilayered nonlinear medium," in Proc. OII '98, Optics

for Information Infrastructure, G. G. Mu, Editor, J. of Optoelectronics

� Laser (JOEL), Vol. 9, Supp., pp. 421{423 (1998).

2. T. A. Laine and A. T. Friberg, \Wave distortion correction and optical

bistability in a nonlinear medium near phase-conjugate mirror," in 3rd

Iberoamerican Optics Meeting and 6th Latin American Meeting on Op-

tics, Lasers, and Their Applications, A. M. Guzman, Editor, Proc. SPIE

3572, 210{217 (1999); invited paper.

3. T. A. Laine and A. T. Friberg, \Nonlinear thin-layer theory and approx-

imative techniques," in 18th Congress of the International Commission

for Optics, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T.

Asakura, Editors, Proc. SPIE 3749, 570{571 (1999).

4. T. A. Laine and A. T. Friberg, \Thin-layer theory and parallel techniques

for nonlinear Kerr medium," in Optics and Optoelectronics; Theory,

Devices and Applications, O. P. Nijhawan, A. K. Gupta, A. K. Musla, K.

Singh, Editors (Narosa Publishing House, New Delhi, India), pp. 152{158

(1999); invited paper. Reprinted in Selected Papers from International

Conference on Optics and Optoelectronics '98, K. Singh, O. P. Nijhawan,

A. K. Gupta, A. K. Musla, Editors, Proc. SPIE 3729, 160{166 (1999).

5. A. T. Friberg and T. A. Laine, \Characteristic matrices in linear and

nonlinear thin-�lm optics," in International Conference on Optical Sci-

ences and Applications for Sustainable Development, Dakar, Senegal,

April 10-14, 2000.

vii

viii

Acknowledgments

I am very much obliged to my adviser Professor Ari T. Friberg for giving

me the opportunity of being part of his research group, and for continuous

support and guidance during my time at Stockholm. It is also a great plea-

sure to acknowledge the warm and inspiring atmosphere at the Department of

Physics{Optics. The opportunity to interact with creative and talented people

from various �elds of optics, physics, and mathematics has much enriched and

stimulated my life and research.

I also thank P. Hautoj�arvi, J. Mickelsson, R. Nieminen, R. Paulsson, H. Ru-

binstein, S. Stenholm, and K.-A. Suominen for o�ering me help and support

when I was entering my studies at KTH.

I thank my parents and brother for continuous encouragement and support.

The work was partly founded by support from the Vilho, Yrj�o and Kalle V�ais�al�a

Foundation, the Magnus Ehrnrooth Foundation, and the Alfred Kordelin Foun-

dation.

Stockholm, June 2000

Timo A. Laine

ix

x

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1 Introduction 1

2 Kerr medium 4

2.1 Nonlinear Maxwell's equations . . . . . . . . . . . . . . . . . . . 4

2.1.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . 7

2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Phase conjugation . . . . . . . . . . . . . . . . . . . . . 8

2.2.2 Bistability . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.3 Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Numerical techniques in layered structures 19

3.1 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Fourier eigenmode method . . . . . . . . . . . . . . . . . . . . . 22

3.3 Local thin-layer theory . . . . . . . . . . . . . . . . . . . . . . . 23

3.3.1 Quadrature forms . . . . . . . . . . . . . . . . . . . . . . 25

4 Action integral formalism 26

4.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3.1 Evaluation of an action . . . . . . . . . . . . . . . . . . . 31

4.3.2 Evanescent solution . . . . . . . . . . . . . . . . . . . . . 32

4.3.3 Gauge �eld theory . . . . . . . . . . . . . . . . . . . . . 34

5 Towards integrability 36

5.1 Liouville's theorem . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2 Spatial solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6 Conclusions 40

Bibliography 41

xi

1 Introduction

Ever increasing ows of information have created a need for optical communi-

cation systems. Optics enhances the capacity of semiconductor based micro-

electronics in the form of broad band optoelectronics, parallel processing and

high-speed optical interconnections. Nonlinear interactions and the optimiza-

tion of associated materials relax the limitations of linear optics. Nonlinear

optical phenomena are the underlying mechanisms in such essential optical sig-

nal processing functions as frequency conversion and modulation, ampli�cation

and emission, directional switching, logical gates based on optical bistability,

noise reduction, and many others [1{3].

The basic, and perhaps the most well-known, nonlinear phenomenon is

the harmonics generation [4{7]. If the incident light is monochromatic, the

nonlinearity may produce electromagnetic waves with higher harmonics. Ap-

plications include compact visible and UV solid-state lasers, that are promising

devices for a high-density optical storage and in biomedical utilization [8]. If,

in addition, several waves are simultaneously involved, the mixing of di�erent

frequencies is possible. In frequency conversion two monochromatic waves cre-

ate a third wave whose frequency is the sum of the frequencies of the original

waves. In parametric ampli�cation two waves are used to amplify a third one.

A feedback to a parametric ampli�er creates a parametric oscillator. Many

optical devices utilize mixing; frequency converters are used to �ll the opti-

cal bandwidth, and ampli�ers magnify weak optical signals that propagate in

optical �bers [9, 10].

A special situation occurs when nonlinearity creates a �eld whose frequency

coincides with the incident wave. The refractive index of the medium then is

linearly proportional to the intensity of the beam. The phenomenon is called

the Kerr e�ect, and together with the electromagnetic theory it characterizes

the nonlinear wave propagation in a Kerr medium [5{7]. Three particularly

noteworthy e�ects are associated with an intensity-dependent refractive index,

namely, phase conjugation, bistability, and solitons. In phase conjugation the

phase and the propagation direction of a light beam are exactly reversed. The

backward wave will retrace the original path of the input wave [11]. This

is contrary to a conventional mirror in which the redirected wave satis�es

the laws of re ection. Optical phase conjugation has many applications, for

example, in real-time holography, information processing, spectroscopy, and

1

image transmission [12{14].

Under suitable conditions a nonlinear material may exhibit bistability; the

re ection has two di�erent values with a single initial intensity [15{17]. By

changing the power of the input one may control the optical state of the out-

put. Bistable devices can be used as switches and memory elements in optical

computing and high-speed parallel processing [18,19]. Closely related to bista-

bility are solitary waves or solitons. These pulse-like objects can travel in a

nonlinear medium without altering their initial pro�le. The waves are called

either optical or spatial solitons depending on whether the pulse compression is

due to temporal or spatial e�ects. Solitons have many important applications

in transmitters and in optical �ber telecommunication [20{22]. In self-focusing

the nonlinearity creates an intense focus at the center of the beam [23{25]. The

e�ective intensity in the middle of the beam may be so large that the material

is damaged.

Nonlinear optical materials can be divided into several di�erent subcate-

gories. Nonlinear glasses [26] are important media in view of their compat-

ibility with existing �bre networks and the sub-picosecond response times of

many glass compositions. However, the nonlinearities of silica are very low.

Particularly attractive media are semiconductors [27, 28] owing to their large

nonlinearities at photon energies close to the band-gap energy. Moreover, the

novel nonlinear crystals, including materials like LiNbO3, KTiOPO4, CdSe,

and Ag3AsS3, have superior optical properties compared to conventional non-

linear media [29, 30]. Together they have played an important role in the

establishment of nonlinear optics as a major area of laser science. Other inter-

esting nonlinear materials are the Langmuir-Blodgett �lms, i.e., exceedingly

thin �lms with a highly organized layer structure, liquid crystals, polymers,

and molecular crystals [31, 32].

In this thesis we concentrate on the theoretical aspects of the intensity

dependent Kerr e�ect. The general formalism, i.e., the electromagnetic wave

theory, is based on Maxwell's equations [33]. In Section 2 the nonlinear Kerr-

Maxwell equations are derived from the �rst principles for both polarizations,

TE and TM. These formulas underlie the nonlinear wave propagation in Kerr

media and they are the basis of all our analytical investigations. Section 3

is devoted to layered structures that are con�gurations commonly used in ex-

periments and in theoretical modeling. Layered structures have many advan-

tages over other geometries; if the refractive indices of the media are properly

2

chosen, a guided wave may be created inside the structure. Guided waves

propagate via internal re ections and therefore they are used, for example, in

signal and energy transmissions [3, 34, 35]. From the theoretical point of view

layered structures simplify the electromagnetic wave theory signi�cantly and

make many systems analytically solvable [36,37]. In Section 3 we present two

numerical methods that are used to solve the �eld equations in a nonlinear

medium. The �rst approach uses the rigorous electromagnetic theory of vol-

ume gratings (scattering theory) [38], while the second method is based on the

exact thin-layer theory.

The action integral formalism is established in Section 4. The method sup-

plies a universal technique to investigate classical systems, their symmetries,

and the conserved quantities [39]. We apply the action principle in two di�erent

forms; the nonlinear �eld solution can be calculated perturbatively by using

the Hamilton-Jacobi theory and the canonical perturbation theory [40, 41].

The second method makes use of the calculus of variations that yields sim-

ple analytical forms for the solutions [42{45]. In Section 4 we further show

some other examples of the use of the action principle. If one �nds suÆciently

many conserved quantities, the system may be integrable, i.e., the motion of

the system can be exactly determined for the speci�c initial values [46{48].

Liouville's theorem of integrability for Hamiltonian systems is presented in

Section 5. The existence of soliton solutions is intimately associated with the

complete solubility of the system. In Section 5 we show exact spatial soliton

solutions for the Kerr-Maxwell equations. The main results and conclusions of

the thesis are summarized in Section 6.

3

2 Kerr medium

Electromagnetic waves in materials propagate with a velocity that is charac-

terized by the refractive index of the medium. If the index is independent

of the transmitting �eld, the system is called linear. In some situations the

nonlinear self-interactions of the �elds change the velocity of propagation. In

Kerr media the index of refraction will depend linearly on the �eld intensity,

satisfying the so-called Kerr law,

n = n0 + n2jEj2; (1)

where n0 is the linear index of refraction, n2 is a small nonlinearity coeÆcient,

and jEj2 is the �eld intensity.

In this section we �rst derive the main equations for the nonlinear wave

optics. We then discuss three separate nonlinear applications.

2.1 Nonlinear Maxwell's equations

The theory of electromagnetic waves is described by Maxwell's equations. In

many cases they are written in the charge rationalized mks units. This system

has the virtue of including the practical electrical units of potential di�erence

(volt), current (ampere), resistance (ohm), etc. Another popular unit system

is known as the cgs Gaussian system. It simpli�es the equations in which the

Coulomb potential plays a central role. Also the velocity always enters in the

dimensionless form prescribed by the Lorentz transformation. Throughout this

thesis we will use the Gaussian units since the �eld theoretical considerations,

presented later in the thesis, favor the system.

The Maxwell equations in the presence of a medium (in cgs units) are [33]

r�H�1

c

@

@tD =

4�

cj; (2)

r�E+1

c

@

@tB = 0; (3)

r �D = 4��; (4)

r �B = 0; (5)

where E and H are the electric and the magnetic �elds, respectively. The

electric displacement is D and vector B is the magnetic induction. The free

4

charge density is � and the current density is denoted by j. The nonlinear

response of the medium is given by D = E+ 4�P and B = H + 4�M, where

P is called the electric polarization and M is the magnetization. Especially,

we are interested in nonmagnetic, charge and current free media, for which

M = 0 and � = j = 0.

First we consider the situation when the incident �eld is TE polarized. By

using Eqs. (2)-(4) one may derive

r2E�

1

c2

@2

@t2E =

4�

c2

@2

@t2P; (6)

where the Coulomb gauge r � E = 0 has been used. For intense beams the

polarization can be expanded in a Taylor series with respect to �eld E. In a

component form P becomes (with summations over repeated indices)

Pi(t) � �(1)ij Ej(t) + �

(2)ijkEj(t)Ek(t) + �

(3)ijklEj(t)Ek(t)El(t); (7)

where �(1)ij , �

(2)ijk, and �

(3)ijkl are the �rst, second, and third order susceptibilities.

The higher order terms are neglected. The nonlinear medium is assumed to

be invariant under operations of inversion, re ection, and rotation. It follows

that �(2)ijk = 0 and the second term on the right hand side of Eq. (7) drops out.

For isotropic materials the linear susceptibility reduces to a constant, P(1) =

�(1)E, and the third order nonlinear susceptibility has only three independent

components.

By making use of the previous assumptions we take the electric �eld as

linearly polarized along the x-axis,

E =1

2x[Ee�i!t + c:c]; (8)

where E is a component along x-direction, and calculate the third order po-

larization,

(P (3)x )!;3! =

�(3)xxxx

8[E3

e�i3!t + 3jEj2Ee�i!t + c:c]: (9)

The �rst term represents the third harmonic generation process. If the phase

is not carefully matched the generation of third harmonics will be weak and

can be neglected. The remaining part is known as a Kerr nonlinearity. We

further assume that there is no interaction between the fundamental and the

third harmonics. By considering the �elds only in x-direction we may adopt

5

the scalar notations. Polarization P (1)x and Eq. (9) are substituted into Eq. (6).

The result is the nonlinear Helmholtz equation,

r2E + k

2E + jEj

2E = 0; (10)

where k = k0n0, = 2k20n0n2, k0 = !=c = 2�=�0, n0 = (1 + 4��(1))1=2, and

n2 = 3��(3)=(4n0).

Result (10) can be also written in the form

r2E + k

20n

2E = 0; (11)

where the e�ective index of refraction is

n = n0 + n2jEj2; (12)

and I = jEj2 = E �E

� is the optical intensity. However, the Kerr coeÆcient n2

must be suÆciently small, n2I � 1, for Eq. (11) to hold. In physical systems

this condition is readily satis�ed since the change of the refractive index due

to nonlinearity is usually much smaller that the linear index.

In the case of TM polarized �eld the material relations are taken in a

multiplicative form, i.e., D = �E and B = �H, where � = 1 + 4�� is the

electric permittivity and � = 1+4�� is the magnetic permeability. As before,

� is the dielectric susceptibility and � is the magnetic susceptibility. When

Maxwell's equations (2) and (3) are taken in a component form, the following

nonlinear equation can be derived,

@

@y

1

k20�

@H

@y

!+

@

@z

1

k20�

@H

@z

!+ �H = 0; (13)

where H = Hx(y; z) and n2 = ��. If n also satis�es the Kerr law (12), Eq. (13)

is called the Kerr nonlinear Maxwell equation for TM polarization. In our

examples we investigate nonmagnetic media, for which � = 1.

6

2.1.1 Boundary conditions

Maxwell's equations are valid in the regions of space where the physical prop-

erties of the medium, characterized by � and �, are continuous. At a discon-

tinuous boundary of two media the boundary conditions take on the form [33]

n12 � (B(2)�B(1)) = 0; (14)

n12 � (D(2)�D(1)) = 4��; (15)

n12 � (E(2)�E(1)) = 0; (16)

n12 � (H(2)�H(1)) =

4�

cj; (17)

where n12 is the unit normal pointing from medium 1 into 2, � is the surface

charge density, and j is the surface current density. Functions � and j are

nonzero only if one of the media is a perfect conductor. It is interesting to

note from Eqs. (14) and (17) that for nonmagnetic media H is continuous

across the boundary.

As an illustration of the derivation of the boundary conditions we show the

rule of Eq. (14). We consider an in�nitesimal thin bill box that lies on the

boundary of media 1 and 2, see Fig. 1. Areas ÆA1 and ÆA2 are assumed to be

small, so that B has constant values in both areas. Hence, one may calculate

B(1)�n1ÆA1+B

(2)�n2ÆA2 = 0. When rules n12 = �n1 = n2 are used, one gets

the boundary condition of Eq. (14).

�

�

� � �

� � �

� � �

� �

� �

Figure 1: Boundary of media 1 and 2.

7

2.2 Applications

We introduce three nonlinear phenomena that are studied in the research pa-

pers of this thesis, namely, phase conjugation, bistability, and optical solitons.

We also give a simple theoretical description in each case by using the electro-

magnetic wave theory.

2.2.1 Phase conjugation

In optical phase conjugation nonlinear optical e�ects are incorporated to pro-

duce wavefront-reversed electromagnetic waves [11, 12]. A beam that is re-

ected by a phase-conjugate mirror (PCM) retraces its original path. This is

contrary to a conventional mirror in which a ray is redirected according to the

law of re ection, see Fig. 2.

� � � �

Figure 2: Comparison of re ection properties from a conventional mirror, (a),

and a PCM, (b).

We denote the incident electric �eld as E(r; t) = A(r) exp(i(k � r � !t)),

where A is a complex vector amplitude and k is a wave vector. On re ection

from a conventional mirror (Em) and a PCM (Ep), the resulting �elds have

the following forms

Em(r; t) = �A(r)ei(km�r�!t); (18)

Ep(r; t) = �A�(r)ei(�k�r�!t); (19)

where km is obtained from k by re ection and the asterisk implies a complex

conjugate of amplitudeA. Re ection coeÆcient � characterizes losses, j�j < 1,

or gains, j�j > 1. If the incident wave is, for example, circularly polarized, the

8

handedness is reversed by an ordinary mirror whereas it remains unaltered

with the PCM.

The retrore ecting property of phase-conjugate waves has many applica-

tions, for instance, in automated pointing and tracking [49], phase aberration

corrections [50], phase-conjugate resonators [13, 51], gyros [52], and in inter-

ferometers [53, 54] (see also Fig. 3). Some other speci�c examples include

polarization selective optical phase conjugation [55] and mode dispersion can-

cellation in �bers [14]. The recent experimental investigations deal with the

phase conjugation of an optical near �eld [56] and scattered light enhancement

near a phase-conjugate mirror [57].

� � � �

� � � � � � � � �

� � � � �

Figure 3: Phase-conjugate interferometer. A plane wave incident on the mirror

produces a re ected and transmitted wave. After interacting with a PCM, the

phase-conjugate wave forms a new pair of re ected and transmitted waves.

In linear media the compensation of wavefront distortions by phase conju-

gation has been explored in detail within the scalar and electromagnetic the-

ories [58{60]. In the case of a nonlinear medium Agarwal [61] proved a more

general theorem that includes full-wave propagation and multiple scattering.

He showed that, in the absence of losses and gains, the distortions generated by

a nonlinear medium with real susceptibility are completely corrected. Agar-

wal's theorem has been tested in layered geometries using a slowly varying

envelope approximation [62]. In our research paper I we apply the rigorous

theory of volume gratings. The layered sample is divided into small shells in

each of which the nonlinear Maxwell equations are exactly solved using an

iterative algorithm. We introduce this method in more detail in Section 3.2.

9

� �

� � � � � � � � � � � � � � �

� �

� �

� �

�

Figure 4: Experimental arrangement for phase conjugation by nonlinear four-

wave mixing.

The electromagnetic theory of optical phase conjugation is based on the

degenerate four-wave mixing [7, 63]. We consider this theory brie y here. We

take a �eld of three waves with angular frequencies !1, !2, and !3,

E = xX

q=�1;�2;�3

1

2E(!q)e

i(kq �r�!qt); (20)

where !�q = �!q, E(�!q) = E

�(!q) = E�

q , and k�q = �kq. The third order

polarization, from Eq. (7), becomes

P(3)x =

�(3)

8

Xi;j;k=�1;�2;�3

E(!i)E(!j)E(!k)ei(ki+kj+kk)�r�i(!i+!j+!k)t: (21)

Waves E1 and E2 are called pump beams. We also assume a situation in which

the frequency and phase matching conditions are ful�lled,

!1 + !2 = !3 + !4; (22)

k1 + k2 = k3 + k4; (23)

and all frequencies are degenerate, i.e., !1 = !2 = !3 = !4 = !. The geometry

of the system is illustrated in Fig. 4, in which k1 + k2 = 0. We also take

k3 = kz. The nonlinear term due to the mixing of waves 1, 2, and 3 is then

P(3);(!=!+!�!)x =

3�(3)

4(E1E2E

�

3ei(�kz�!t) + c:c:): (24)

10

The factor of 3/4 comes from di�erent permutations. The polarization of

Eq. (24) will excite a wave in the form E4 = x[E4 exp(i(�kz � !t)) + c:c:)]=2,

where E4 is a complex amplitude proportional to E�

3 . This new wave will mix

with waves E1 and E2 generating a polarization which has the same frequency

and propagation vector as E3. As a result the latter polarization will interact

strongly with wave E3.

It is further assumed that the intensities of the pump beams are strong

and not a�ected by the power exchange. Hence E1 and E2 can be taken as

constants, and especially so that jE1j = jE2j. Then using Eq. (6) one may

derive

(r2 + k2)E3 = �E

�

4 ; (25)

(r2 + k2)E4 = �E

�

3 ; (26)

where � = 3�(3)E1E2=2 is constant. Results (25) and (26) are the basic equa-

tions that are commonly used in analyses. Waves 3 and 4 are called a probe

wave and a conjugate wave, respectively.

2.2.2 Bistability

A bistable device is a system for which the output can take two distinct values

with one input value [15, 16, 18]. Switching between the states is achieved,

for instance, by a temporal change of the level of the input. In Fig. 5 we

show a typical bistable input-output relation. When the input (from the left)

�

�� � � � � � � � �

� � � � � �

Figure 5: Input-output relation in a bistable device.

11

exceeds a critical threshold value, I2, the output jumps from the lower state

to the higher state. Correspondingly, at I1 a decrease of the input drops the

output value to the lower state. The region between I1 and I2 forms a so-called

hysteresis loop. By detuning the system parameters thresholds I1 and I2 may

also be placed on the top of each other. In this situation a sharp switching

power increase is obtained.

The nonlinear material itself does not necessarily guarantee the bistable

behavior. Another important element is the feedback, i.e., the outgoing �eld

is coupled back to the medium [16], see Fig. 6. The input-output relation

can have the form It = F(It)I0, where F is some bell-shaped function, I0 is

the input, for example, the incident intensity, and It denotes the output �eld,

such as the re ected light or the total �eld. The feedback may be achieved by

inserting the nonlinear medium into a laser cavity or by the use of some other

type of resonator.

� �� �

� � � � �

Figure 6: Both nonlinearity and feedback are usually required to produce a

bistable signal.

The classi�cation of the bistable devices is typically made in two di�erent

ways [15]. A system may be dispersive or absorptive, and it may be intrinsic

[17] or hybrid [64]. In dispersive (absorptive) nonlinear elements the index of

refraction (absorption coeÆcient) is a function of the optical intensity. When

the intensity dependence arises from a direct interaction of the light with

matter, the bistable device is called an intrinsic system. Alternatively, in

hybrid bistable devices a detector monitors the transmitted intensity, and the

feedback is obtained through an electrical signal.

One of the �rst theoretical observations of hysteresis in scattered intensi-

ties near a nonlinear boundary was reported by Kaplan [65]. Later optical

12

�

� �� �

�

Figure 7: Mach-Zehnder interferometer.

bistability has been investigated in the context of surface plasmons [66{68],

nonlinear waveguides [69{71], and optical solitons [72]. On the experimental

side, bistability has been observed, for instance, in a sodium �lled Fabry-Perot

interferometer [73] and in excitations of nonlinear waves [74{76].

Bistable devices are important in digital circuits used in communication,

signal processing and computing. They can be utilized as switches, logic gates,

and memory elements. The Kerr-type media that exhibit optical bistability

are, for example, sodium vapor, disul�de, and nitrobenzene. Nonlinearities are

typically small in these media. However, the nonlinear e�ect can be ampli�ed

by taking a sample with a long interaction region. Semiconductors, such as

GaAs, InSb, InAs, and CdS, have strong optical nonlinearities due to excitonic

e�ects near the bandgap [77, 78].

As an example of dispersive bistable devices we consider a Mach-Zehnder

interferometer. The experimental arrangement is shown in Fig. 7. For two

electric �elds, Ej = I1=2j exp(i�j) where Ij = jEjj

2 and j = 1; 2, the interference

equation is

It = jE1 + E2j2 = I1 + I2 + 2(I1I2)

1=2 cos�; (27)

where the phase factor is � = �1��2. For an interferometer in which the non-

linear Kerr medium is placed in one branch, the electric �elds at the outcoupler

take the forms I1 = I2 = I0, �1 = k0(n0 + n2I0)d + k0n0

d1, and �2 = k0n0

d2

where n0 and n0 are the indices of refraction, d is the length of the active

13

� �� � � � � � � � �

� � � � �

Figure 8: Output intensity of the Mach-Zehnder interferometer.

medium, and d1 and d2 are constants. The interference equation becomes

IKt = 2I0[1 + cos(k0n2dI0 + �0)]; (28)

where �0 = k0nd + k0n0(d1 � d2) is a constant. Ratio I

Kt =I0 is a periodic

function of I0. This bistable output intensity is shown in Fig. 8, i.e., when I0

varies within one period IKt =I0 has two exactly same values.

14

2.2.3 Solitons

The pulse-like stationary solutions of nonlinear equations are called solitary

waves. If the solutions also satisfy certain stability properties they are called

solitons [46]. A soliton is a fairly localized wave, whose amplitude is propor-

tional to its velocity. It travels in a speci�c direction maintaining or regener-

ating its initial pro�le, and for example, in a collision of two solitons the two

components emerge unscathed.

Solitons are usually divided into two categories: optical and spatial soli-

tons [79]. Optical solitons [20,21] are results of the interplay between self-phase

modulation and group-velocity dispersion. In a linear dispersive medium a

pulse of light alters its shape because the constituent frequency components

travel at di�erent velocities. If the medium is nonlinear, the self-phase modula-

tion (for example due to the optical Kerr e�ect) alters the phase and therefore

the frequency. The result in overall pulse spreading or pulse compression then

depends on the magnitudes of these two processes. When the e�ects are fully

compensated optical solitons are created.

Spatial solitons are monochromatic waves that are localized spatially in

the transverse plane [80, 81]. These solitary waves can travel in a nonlinear

medium without altering their spatial distribution. The balance happens now

between di�raction and self-phase modulation, and the beam con�nes to its

self-created waveguide. Spatial solitons are the transverse analogs of longitu-

dinal (temporal) optical solitons.

The theoretical considerations of solitons deal with self-trapping [23, 44]

and self-focusing of optical beams [24,25,45]. Self-focusing was experimentally

observed in CS2 liquid and shown in some cases to create chaotic instabilities

[82, 83]. Solitons are readily detected in nonlinear waveguides made from, for

instance, glass [84] and semiconducting media [85]. Two soliton beams may

interact when placed close to each other. The attraction and repulsion of

the beams is due to the relative phase and spacing between the two solitons

[86{88]. Under the combined e�ect of di�raction, anomalous dispersion, and

nonlinear refraction, an optical pulse can collapse simultaneously in time and

space [89]. Depending on the sign of the nonlinearity coeÆcient solitons are

called bright [90] or dark [91].

Solitons have signi�cant importance in optical �ber communication and in

long-distance transmitters [22, 92]. The performance of such systems is often

15

limited by �ber dispersion that allocates the pulse beyond the bit slot, and by

�ber loss. The problems can be avoided by a) amplifying solitons periodically

to recover their original width and peak power, and b) using the so-called

fundamental solitons as an information bit. Fundamental solitons are the

most stable solutions, since their pulse shape does not change in the course of

propagation.

The soliton theory is obtained from Maxwell's equations. We give a short

derivation for spatial solitons which are our main interest. We start from

Eq. (10) by considering a system with two spatial coordinates. The Laplacian

has the form r2 = @

2=@x

2+@2=@y

2. The wave propagation is taken as a plane

wave in one spatial direction, for example, in y-direction. The electric �eld is

then written as E(x; y) = A(x; y)eiky, where A(x; y) is a complex envelope and

k is a propagation constant. By employing a slowly-varying envelope approx-

imation, jk@A=@yj � j@2A=@y

2j � jk

2Aj, one gets a nonlinear Schr�odinger

equation (NSE),

i@

@�A+

@2

@x2A+ jAj

2A = 0; (29)

where � = 2ky. Equation (29) is the central formula in most soliton theories.

In the case of optical solitons the envelope equation takes the same form as

Eq. (29), but now � is to be identi�ed as the time coordinate.

In the case of bright solitons, > 0, the fundamental solution, N = 1, has

the form

ABN=1(�; x) = A0 sech

�� A

20

2

�1=2(x� 2q�)

�exp

�i(qx� [q2 �

A20

2]�)

�; (30)

where q and A0 are arbitrary real constants (amplitude A0 could also be com-

plex, but for simplicity, we have chosen it as real). The intensity pro�le of

solution (30) with q = 0 is illustrated in Fig. 9a. As one can observe from

the �gure, the soliton does not spread in � . This property makes fundamental

solitons very attractive in �ber communication.

A remarkable property of the NSE is that it has N -soliton solutions. Physi-

cally they describe, for example, soliton excitations. When N = 2, the solution

takes the form

ABN=2(�; x) = 4A0

eiT=2[cosh(3X) + 3ei4T cosh(X)]

cosh(4X) + 4 cosh(2X) + 3 cos(4T ); (31)

16

! � � � �

� � � ��

� �

! � � � � � � � � �

� � �� �

� �

� �

Figure 9: a) Fundamental soliton, N = 1. b) Soliton N = 2.

17

where X = [ A20=2]

1=2x and T = A

20� . In Fig. 9b we show the intensity pro�le

of solution (31) when T goes from zero to T = �=2. The solution oscillates in

time but it will recover its original shape in periods of T = �=4. At � = 0 the

fundamental and N -soliton coincide so that ABN(0; x) = NA

BN=1(0; x).

Nonlinearity coeÆcient can also be negative, < 0. In this situation the

envelope function will not be �nite perpendicularly to the soliton cross-section.

These solutions of the NSE are called dark solitons. When N = 1 the solution

has an analytical form

ADN=1(�; x) = �A0 tanh

��j jA

20

2

�1=2(x� 2q�)

�exp

�i(qx� [q + j jA2

0]�)

�: (32)

The signs (�) refer to two di�erent types of \solitons" (kink and antikink).

They have identical pro�les of intensity but the solutions have a di�erent

topological charge.

The existence of soliton solutions is closely related to the integrability of

a class of nonlinear di�erential equations (see Section 5). Some well known

integrable systems are the Korteweg-de Vries equation (KdV), the nonlinear

Schr�odinger equation (NSE), and the sine-Gordon equation (sG). Together

they furnish universal mathematical models that cover many nonlinear physical

phenomena.

18

3 Numerical techniques in layered structures

Layered structures are simple and widely used geometries in optical model-

ing. The fabrication of systems containing separate media with di�erent layer

thicknesses is highly developed. One important property of layered structures

is that they support guided waves [35, 93, 94]. In linear media these �elds

propagate long distances via internal re ections. The low threshold nonlinear

phenomena like bistability and the enhancement of the nonlinear susceptibil-

ity [95] are also readily observed in waveguide geometries. Layered structures

are therefore useful in device applications.

The linear properties of layered media are well understood [3, 33, 34, 96].

The nonlinear theories have been studied in the context of strati�ed me-

dia [36,37,97], surface polaritons and plasmons [98{100], and the Fabry-Perot

interferometer [101]. In the e�ective-medium approach the layer thickness is

assumed to be much smaller than an optical wavelength [102]. However, in

many instances numerical approaches are the only way to �nd the character-

istics of the nonlinear solutions [62,103]. The refractive indices or the surfaces

of the media may also be periodically corrugated [104, 105].

� � � � � � � �

� � � � � � � � � � � � � �

�

!"

�

# � � � � � � � �

Figure 10: Layered structure.

In Papers I-VI we investigate optical systems in which only one layer con-

tains a nonlinear Kerr medium while the other layers have linear indices of

refraction, see Fig. 10. We apply two separate numerical methods to �elds in

19

a nonlinear medium. In the �rst case we use a theory of volume gratings (also

called a Fourier eigenmode method) to solve the electromagnetic �elds rigor-

ously [38, 106{109]. The numerical procedure is iterative. The applications

include the study of Agarwal's theorem of wavefront distortion correction by

phase conjugation [Paper I] and bistability in nonlinear waveguides [Papers II,

VI]. In the second method we develop a new numerical algorithm to solve the

nonlinear Kerr-Maxwell equations exactly using very thin layers [Papers III,

IV]. This nonlinear thin-layer theory is accurate and easily implemented on a

computer. We employ the latter method to assess the validity of the analytical

approximative results in Paper V.

If the layered structure is placed on a phase-conjugate mirror (PCM), the

directed waves make the nonlinear medium inhomogenous in y-direction (see

Fig. 10) [Papers I, II]. In this situation the conventional guided-wave theory

cannot be used in an exact electromagnetic treatment, but the rigorous nonlin-

ear solution is obtained, for example, by the Fourier eigenmode method. Our

numerical techniques make use of the periodicity of the electromagnetic �elds.

We will introduce this concept in the next subsection. We also brie y describe

the theory of the Fourier eigenmode method and the nonlinear thin-layer the-

ory.

3.1 Periodicity

When the incident light propagates at an angle to the layered structure, the

electromagnetic �elds become periodic along the direction of the layer. In the

case of linear media this is easily deduced from the plane wave expansions of

the �elds. Nonlinearity and the presence of the PCM make the �eld more

complicated within the nonlinear medium, but still the periodicity of the �elds

is maintained.

We use an iterative argument to determine the period in nonlinear layered

systems [see Papers I, II] (one may also apply symmetries in the periodicity

condition, Section 4.2). In the �rst iteration step all media are considered as

linear. The incident plane wave propagates in a medium where the index is

n0, and the angle of incidence is denoted by �. In each layer the exact �eld

solution consists of several plane waves. They all satisfy periodicity condition

r0d = 2�, where d is the period, r0 = k0n0 sin �, and k0 = !=c. Now comes

the second iteration step. By using the exact solutions of the linear �elds one

20

may compute the optical intensities. Then applying the Kerr law, one may

calculate a new index for the layer that represents the nonlinear medium. We

note that all intensities and also the nonlinear index satisfy same period d.

Because the nonlinear index is d-periodic, it follows from the Floquet-Bloch

theorem that in the second iteration step all �eld components in the nonlinear

medium must be pseudoperiodic in y-direction [110],

E(y) = E(y + d)eir0d: (33)

But, in fact, the phase factor in Eq. (33) disappears, since d = 2�=r0, making

the boundary contributions in y-direction unimportant. The �eld equations are

solved in each separate layer, but at this time the nonlinear �eld is computed

using the Fourier eigenmode method applied to the new modi�ed index. After

the solutions have been found, the nonlinear index is modi�ed according to

the Kerr law and the procedure above is repeated. During the next iterations

the incident �eld may become a superposition of plane waves, but they will all

satisfy the same periodicity condition d = 2�=r0 and hence the period remains

unchanged.

As an example of periodic systems we study the e�ect of boundary terms

in Eq. (33). In particular, we consider the system presented in Paper II and

derive Eq. (12) again. By using Fourier transformations, whose de�nitions are

given in Paper II, one may write

@2E

@y2= �

4�2

d2[E �

1

d

dZ0

E dy] + F1 + 2F1 cos�2�y

d

�+ i2F2(1) sin

�2�yd

�; (34)

where F1 = d�1F3(r0)@E(0)=@y, F2(k) = id

�22�kF3(r0)E(0), and F3(r0) =

exp(�ir0d)�1. If the �eld assumes period d = 2�=r0, then F3(r0) = 0, and F1

and F2(1) vanish. The �nal form of Eq. (34) will be the same as Eq. (12) in

Paper II. However, in general the boundary terms will contribute to the exact

electromagnetic solution.

21

3.2 Fourier eigenmode method

The nonlinear medium is generally inhomogenous along the direction of the

layer. Hence the rigorous electromagnetic solution will not be a superposition

of plane waves, but the exact solution must be found di�erently. One approach

is to apply the optical scattering theory [Papers I, II]. In the method developed

by Knop, the linear Maxwell equations are �rst solved in the Fourier space,

after which the solution is transformed into the real space [38]. This so-called

Fourier eigenmode method is fairly general because it can be used in most

optical systems in which the electromagnetic �elds are periodic. The method

is then also applicable in (non)linear layered structures, see Fig. 11.

� � � � �

� � � � � � � $ $

Figure 11: Index of refraction within the nonlinear medium.

In our approach, the nonlinear medium is divided into N sublayers, each of

which is further divided into small cells. Within one cell the �eld is represented

by plane waves to which the boundary conditions are applied. The theory is

local in contrast to the conventional linear theory. In one thin sublayer the

electric �eld (in TE polarization) is written as [38]

Ej(y; z) =Xlk

Blkjeirly[Ckje

�igkjz +Dkjeigkjz]; (35)

22

where rk = (k + 1)r0. Constants gkj are the eigenvalues and Blkj are the

eigenvectors of the Fourier-transformed Maxwell equation,

g2Bl +

Xk

(Ælkr2k � �l�kk

20)Bk = 0: (36)

Here coeÆcients �k are the Fourier transforms of the square of the refractive

index, and Ælk is the Kronecker delta. Amplitudes Ckj and Dkj describe two

eigenmodes travelling in opposite directions.

The rigorous electromagnetic solution is calculated iteratively. The �rst

iteration step corresponds to the linear theory. After each iteration step the

e�ective index of refraction is modi�ed according to Kerr law (12). Because

the index of the nonlinear medium becomes periodic along the direction of the

layer, the Fourier eigenmode method can be used to solve the nonlinear �eld

con�guration in the next iteration steps. When the nonlinear index does not

change during the iteration process, we consider that we have found the exact

electromagnetic solution. A typical nonlinear index pro�le is shown in Fig. 11.

The periodic structure of the �elds is clearly observed.

3.3 Local thin-layer theory

The exact solution to the nonlinear Maxwell equations can also be computed

di�erently. In a local thin-layer theory the nonlinear medium is again divided

into N thin sublayers, see Fig. 12. In each sublayer the Kerr-Maxwell equation

has the form

@2

@y2Ej +

@2

@z2Ej + k

2Ej + jEjj

2Ej = 0; (37)

where subindex j is the layer index, (j = 1 : : : N). The \ansatz" solution for

the electric �eld is taken as

Ej(y; z) = Aj(y) cos(kj;1z) +iBj(y)

kj;2sin(kj;2z); (38)

where Aj(y) and Bj(y) are y-dependent functions, and kj;1 and kj;2 are wave

vectors. The sublayer is considered to be so thin that within one sublayer the

Kerr-Maxwell equation is rigorously solved to the �rst order in z. The higher

order terms are neglected.

23

% � & � �

% � & � �

% � & � '

� � � � � � � �

� � � � � �

�

"

� � � � � � �

Figure 12: Periodic layered structure.

On substituting the �eld of Eq. (38) into Eq. (37), and also using a thin-slab

approximation (kj;i�z)2� 0, one may derive an expression for kj;1,

k2j;1(y) =

1

Aj

(@2

@y2Aj + k

2Aj + jAjj

2Aj): (39)

Operator @2=@y2 re ects the local nature of the theory, and it makes the wave

vector periodic in y-direction.

The boundary conditions between the �elds in two consecutive slabs deter-

mine the characteristic matrix (or transfer matrix) of the nonlinear medium

in a sublayer,

Mj(y) =

0@ 1 ih

ihk2j;1(y) 1

1A ; (40)

where h is the nonlinear slab thickness. MatrixMj(y) is periodic in y-direction

and especially it is independent of k2j;2. If the �eld in the nonlinear medium is

strati�ed in z-direction, wave vector kj;1 reduces to k2j;1 = k

2+ jAjj2 and ma-

trixMj becomes y-independent [Paper III]. If the index of refraction moreover

is constant (linear medium), then = 0, k2j;1 = k2, and the ordinary thin-�lm

theory is recovered [33].

The local thin-layer method can be derived in both polarizations, TE and

TM. Further details of these theories are found in Papers III and IV.

24

3.3.1 Quadrature forms

The accuracy of the nonlinear thin-layer theory can be assessed with the exact

quadrature solutions. These formulas are obtained when the complex valued

nonlinear Maxwell equations are �rst divided into real and imaginary parts,

and the resulting equations are integrated over the space domain.

In TE polarization the electric �eld is written with the help of two real

functions, E(z) and �(z), i.e., E(y; z) = E(z) exp(i�(z)) exp(ikyy), where ky

is a propagation constant in y-direction. The expression is substituted into

Eq. (10) and integrated over z. The result is [Paper II]

E2 = I; (41)

E02 = (k2y � k

20�)E

2 + J �C

21

E2+ C2: (42)

The Kerr law is now in the form � = �0 + �f(jEj2), where �0 = n20, � is a

constant, and f(jEj2) is a function of optical intensity I = I(��) = jEj2.

We also use the notations J = J(��) =R�0

I(��)dz and �� = � � �0. The

prime denotes a di�erentiation with respect to z. CoeÆcients C1 and C2 are

constants that are �xed by the boundary conditions.

When the �elds are TM polarized, the magnetic �eld is taken as H(y; z) =

H(z) exp(i'(z)) exp(ikyy), where H(z) and '(z) are real functions. On sub-

stitution into Eq. (13), the following set of equations is obtained [Paper V],

H2 =

�

2k2y � �[�I � k

�20 J +D1]; (43)

H02 = k

20�

2I � k

2yH

2�D

22�

2

H2 : (44)

Again, coeÆcients D1 and D2 are constants that are determined by the bound-

ary conditions of the �elds.

The numerical results obtained from the thin-layer theory are compared

with the solutions of the exact quadrature forms. In this way one may identify

the numerical error that is included in the thin-layer theory. One advantage of

the technique is that solutions (41) and (42) are directly related to coeÆcients

Aj and Bj of Eq. (38) (when Aj and Bj are y-independent).

25

4 Action integral formalism

It is a most impressive and beautiful fact that all the fundamental laws of

classical mechanics can be described by the mathematical construction called

the action. It yields the classical equations of motion in a simple and elegant

manner. The study of the invariances of the action lead to conserved quantities

which are associated to classical motion. Furthermore, as Dirac and Feynman

showed, the action functional shows its full power in quantum physics. The

Feynman path integral provides an exquisite transition from classical physics

to the world of quanta [39].

In the �eld of optics the action principle has mainly been applied in the

form of variational calculus. For a long time is was thought that the utilization

of the variational technique is just a reformulation of the same optical problem

without any new insight into physics. Recently, the variational approach has

become more popular because of its algebraic simplicity, exibility, and a better

understanding of the formalism.

The main idea in the calculus of variations is that the �rst variation of the

variational functional is made to vanish with trial functions [42, 43, 111]. The

aim is to �nd simple analytical formulas that approximate the rigorous solu-

tion but still keep the essential physics in it. The choice of the trial function

has signi�cant e�ects on the accuracy of the solution, and hence the approx-

imative result agrees usually only qualitatively with the exact one. The �rst

developments in the usage of the averaged variational principle were made by

Whitham [112]. Anderson and his collaborators established the method in the

studies of the nonlinear self-focusing of laser beams [45,44] and in the nonlin-

ear pulse propagation in optical �bers [113]. Later the method has been used

in many applications, for example, in the studies of optical �bers [114{116],

pulse propagation in parametrically ampli�ed optical systems [117], collapse

of optical pulses [118], dark-soliton generation [119], and the Zakharov-Shabat

scattering problem [120{123]. An interesting unusual approach is the utiliza-

tion of canonical theory in electron optics [124].

In this section we introduce the action integral formalism. We emphasize

the importance of symmetries and show the validity of Noether's theorem.

Speci�cally, we present ideas how one could use the action integrals in a more

e�ective way when the technique is applied to optical problems. In Papers V

and VI we make use of these methods.

26

We consider action functionals of the form

S =

x2Zx1

L(�i(x); @��i(x)) dx; (45)

where L is called the Lagrangian density, or Lagrangian for short. Functional

�i(x) is any collection of local �elds, and x1 and x2 are the endpoints of inte-

grations. We also use the notations @� � @=@x� = (@=@t;r). For simplicity,

in the forthcoming sections we suppress all subindices of the �elds and also

the limits of integrations.

We will base our discussion on the principle of least action. It states that

among all trajectories �i(x), the physical one gives the stationary value for

the action.

4.1 Equations of motion

The central issue in the action integrals is to �nd the classical equations of

motion. In the �eld of optics, the (non)linear Maxwell equations should co-

incide with these classical trajectories. Then the corresponding Lagrangian is

considered to describe the optical system.

Under an arbitrary change in �, denoted as �, the change in action S to

the �rst order is

ÆS =

ZÆL dx =

Z �@L

@�� +

@L

@[@��]Æ(@��)

�dx: (46)

Since x does not change in this variation, Æ(@��) = @�Æ�. The use of the chain

rule results in

ÆS =

Z �@L

@�� @�

@L

@[@��]

�� dx+

I �@L

@[@��]

�� d��; (47)

where � is the boundary surface and d�� is the surface element. For all varied

trajectories the surface term must vanish. By requiring S to be stationary, we

get a condition

@L

@�� @�

@L

@[@��]= 0: (48)

Result (48) is the famous Euler-Lagrange equation that determines the classical

equations of motion for L. The second variation of S, Æ2S, gives information

27

about the Jacobi �elds, i.e., whether the classical trajectory is a minimum or

a maximum of the action.

As an example of the action formalism we consider Lagrangian

L =@E

@z

@E�

@z� k

2EE

�

�

2E

2(E�)2: (49)

The Euler-Lagrange equation with respect to E� is

@2E

@z2+ k

2E + jEj

2E = 0; (50)

which is the nonlinear Helmholtz equation. We further write Eq. (50) with

E(z) = E(z) exp(i�(z)), where E(z) and �(z) are z-dependent real functions.

The transformed equations of motion are

P1 = �0

E2 = �

0(0)E2(0) = constant; (51)

P2 = E02 +

P21

E2+ k

2E2 +

2E4 = constant: (52)

Equation (51) implies a conservation of the angular momentum, and Eq. (52)

can be interpreted as a Hamiltonian of a particle that moves in an even poten-

tial. Results (51) and (52) can also be derived di�erently using the canonical

theory [see Paper V].

4.2 Symmetries

The main task in optics is, of course, to solve Maxwell's equations (equa-

tions of motion) with appropriate boundary conditions. General properties of

solutions, such as symmetries, are helpful because they usually simplify the

calculations. Symmetries lead to conserved quantities, for instance, energy,

momentum, and charge, and they enable one to generate a family of solutions

when one such solution is known. Unfortunately, so far the e�ective utilization

of symmetries has not reached any signi�cant attention among the researchers

in optics.

In the action integral formalism symmetries play a crucial role. If the

Lagrangian is invariant for some speci�c symmetry transformation, there is al-

ways a quantity that is classically conserved (Noether's theorem). An anomaly

may break this symmetry, but it usually happens only on a quantum level. The

particularly interesting invariance of the action is a so-called local symmetry,

28

i.e., a transformation that is coordinate dependent. In most cases an addi-

tional gauge �eld is required to create this local symmetry. Gauge symmetries

facilitate the study of complicated systems. In particular, the theory is suÆ-

ciently general to describe solitonic objects. Local gauge theories have many

consequences to physical systems and their e�ects can be detected in experi-

ments. Hence local symmetries are also attractive from the practical point of

view. In Section 4.3.3 we will show an optical system that has a property of a

gauge �eld.

Figure 13: Torus is a manifold that is assembled of two circles, T = S1� S

1.

Another mathematical technique that uses the invariances of a system is

the symmetry reduction; by removing the speci�c symmetry the number of

physical states can be decreased [90]. The most well-known conventional sym-

metries are space-time translations, rotations, boosts, inversion, and Lorentz-,

Galileo-, dilation-, and conformal-invariances. When a point particle moves on

a surface of a torus, the equations of motion are invariant for two independent

operations of rotation, see Fig. 13.

As an example of a conventional symmetry transformation we consider the

response of the action to changes due to translation,

L(x+ a) � L[�(x + a); @��(x + a)]; (53)

where a is a constant. We take an in�nitesimal x-dependent transformation in

the form

Æ�(x) = Æa�(x)@��(x); (54)

Æ@��(x) = Æa�(x)@�@��(x) + @�[Æa

�(x)]@��(x); (55)

29

and use the rules in the action. The change in S becomes

ÆS =

Z �Æa

�@�L+

@L

@�� +

@L

@[@��]Æ(@��)

�dx (56)

=

Z �@�L� @�

�@L

@[@��]

�@��

�Æa

�dx: (57)

The requirement that the �rst variation vanish de�nes a so-called energy mo-

mentum tensor,

T�� =

@L

@[@��]@��� g��L: (58)

Here g�� denotes a metric tensor. In the Euclidean space it has the form

g�� = �� . The conservation of the energy momentum tensor, @�T�� = 0, shows

the validity of Noether's theorem for a classical �eld theory; a conservation

equation is a consequence of the invariance of the action.

We apply this symmetry law to our simple nonlinear problem. First we

note that Eq. (50) is invariant under space translation. Then by using de�ni-

tion (58), one may derive a conserved quantity,

Tzz =

@E

@z

@E�

@z+ k

2jEj

2 +

2jEj

4 = constant: (59)

We note that result (59) is obtained from the classical equations of motion,

Eqs. (51) and (52), when P1 and P2 are combined.

30

4.3 Applications

We present three di�erent applications of the action principle. The �rst two

examples, an evaluation of an action and an evanescent solution, are related

to the methods used in Papers V and VI. The third example makes use of

the gauge �eld theory and it is aimed to present the direction of our work in

future.

4.3.1 Evaluation of an action

In our �rst example we evaluate action S with the Lagrangian of Eq. (49),

where = 0. The same subject has been discussed in Paper V.

The Euler-Lagrange equation (classical equation of motion) with respect

to E� yields an equation of the harmonic oscillator for complex E,

@2E

@z2+ k

2E = 0: (60)

We start the calculation by integrating action S over z,

S =

�@E

@zE�

�z2z1

�

z2Zz1

E�[@2

@z2+ k

2]E dz =

�@E

@zE�

�z2z1

: (61)

The integral expression vanishes with the help of Eq. (60). The evolution of

the system in con�guration space then follows the path about which general

variations produce only endpoint contributions.

Next we eliminate the phase factor from complex E. We write the electric

�eld with two real functions, i.e., E = E(z) exp(i�(z)). Then by applying

Eq. (51), we get

@E

@zE� = (E 0ei� + i�

0

Eei�)Ee�i� = EE

0 + iP1: (62)

We note that P1 is a constant. Action (61) is now written as

S =

�@E

@zE + iP1

�z2z1

= E0(z2)E(z2)� E

0(z1)E(z1): (63)

The value of the action is independent of P1 and only the boundary terms of

the envelope function will have an a�ect on the �nal result.

In Paper V we show that the solution of the envelope has the form

E2(z) = A+B sin(qz + q0); (64)

31

where A, B, q and q0 are some constants. We rewrite this as follows,

E2(z) = A+B[sin[q(z � z1)] cos(qz1 + q0) + sin(qz1 + q0) cos[q(z � z1)]]

= A+ sin[q(z � z1)]@

q@zE2(z1) + [E2(z1)� A] cos[q(z � z1)]; (65)

where a sum law of trigonometric functions and Eq. (64) have been used. For

particular values z = z2 and z = z1 we then �nd

@

@zE2(z1) =

q

sin(qZ)[E2(z2)� A� [E2(z1)� A] cos(qZ)]; (66)

@

@zE2(z2) =

�q

sin(qZ)[E2(z1)� A� [E2(z2)� A] cos(qZ)]; (67)

where Z = z2 � z1. By using Eqs. (66) and (67), we get the �nal expression

for action (63),

S =q

2 sin(qZ)[cos(qZ)� 1][E2(z2) + E

2(z1)� 2A]: (68)

The result resembles the action of the harmonic oscillator with real coeÆcients

[41], but the characteristics of these two solutions are rather di�erent; for

example, action (68) does not have conjugate points (caustics). One could

continue the analysis and, for instance, investigate the quantum mechanical

properties of the classical result (68).

4.3.2 Evanescent solution

An interesting special solution can be derived for evanescent waves that appear

at the boundary of a linear and a nonlinear medium.

We consider the following two-dimensional Lagrangian,

L =@E

@y

@E�

@y+@E

@z

@E�

@z� k

2jEj

2�

2jEj

4; (69)

where E is a complex �eld. We take �eld E as E(y; z) = E(z) exp(ikyy), where

ky is the wave vector in y-direction. Because the solutions that we are looking

for are evanescent, the electric �eld does not alter the phase in z-direction, i.e.,

E(z) is a real function. The Lagrangian becomes

L = E02 + C

2E2�

2E4: (70)

32

� � � � � � � � �

� � �

� � �

� � �

� � �

�

� �

�

� � & � �

� � & � � ( �� �

�

Figure 14: Nonlinear evanescent solutions, C = 1 and �0 = 0.

The prime denotes a di�erentiation with respect to z, and C2 = k

2y � k

2 is a

positive constant. The equation of motion for �eld E is

E00

� C2E + E

3 = 0: (71)

The nonlinear evanescent waves are the solutions of Eq. (71). But as usual,

the nonlinearity makes �nding these solutions complicated, and one has to use

other techniques to establish them.

The problem is solved in a Hamiltonian formalism [40, 41] (discussed also

in Paper V). The Legendre transformation is applied to the Lagrangian of

Eq. (70), after which Hamiltonian H takes the form

H = E0

P � L = E02� C

2E2 +

2E4 = E0; (72)

where P = 2E 0 is the canonical momentum and E0 is a (constant) total en-

ergy. The evanescent waves are zero-energy solutions, E0 = 0. The resulting

Hamiltonian can further be integrated over z and two independent analytical

solutions are found,

E1(z) =4CeC(z+�0)

2 e2Cz + e2C�0; (73)

E2(z) =4CeC(z+�0)

e2Cz + 2 e2C�0: (74)

Here �0 is an integration constant that is �xed by the boundary conditions.

One can verify that both (73) and (74) satisfy Eq. (71) separately but not

33

together. Solution (74) is also illustrated in Fig. 14 with two -values. As one

can note, the nonlinearity lowers the e�ective intensity.

When taking the linear limit, ! 0, Eqs. (73) and (74) reduce to

E1(z) ! 4CeC(z��0); (75)

E2(z) ! 4Ce�C(z��0): (76)

Hence nonlinear formulas (73) and (74) yield the correct evanescent solutions

of the linear system.

4.3.3 Gauge �eld theory

In our last example we present the gauge �eld interpretation of the nonlinear

Helmholtz equation [125].

We take the Lagrangian as follows,

L = ikA�

� @@y� ig�

�A� ikA

� @@y

+ ig�

�A�

�

����@A@y

����2 �����@A@z

����2; (77)

where A = A(y; z) is a complex �eld, g = =(4k) is a (coupling) constant, and

� = jA(y; z)j2 is to be identi�ed as a gauge �eld. The Euler-Lagrange equation

with respect to A� results in

i2k@A

@y+@2A

@y2+@2A

@z2+ jAj

2A = 0: (78)

The classical equation of motion coincides with Eq. (10) in two spatial dimen-

sions.

We perform a local gauge transformation by making a coordinate dependent

transformation, � = �(y; z),

ÆA = i�A; ÆA� = �i�A

�

; � = 0: (79)

The change in the action becomes

ÆS = 2

Z�(y; z)

hk@

@y�+

@

@yj1 +

@

@zj2

idydz; (80)

where \current densities" j1 and j2 are de�ned as

j1 = ImhA�

@

@yA

i; j2 = Im

hA�

@

@zA

i: (81)

34

Using the equation of motion, Eq. (78), one can show that

k@

@y� +

@

@yj1 +

@

@zj2 = 0; (82)

and hence the �rst variation of S vanishes, ÆS = 0. \Continuity equation" (82)

implies a local charge conservation and � represents a \probability density". If

�eld A is rewritten as A(y; z) = A(y; z) exp(i�(y; z)), the conservation equation

takes on the form

k@

@yA

2 +@

@y[@�

@yA

2] +@

@z[@�

@zA

2] = 0: (83)

We derive this same result in Paper VII, but using a di�erent mathematical

approach.

The speci�c form of Lagrangian (77) is interesting because the �rst two

terms can be written in the form of the Dirac operator. The remaining part

is an ordinary Helmholtz operator. Thus it may be possible to calculate the

index for Lagrangian (77). It provides topological information on the manifold

in which the operators are de�ned.

35

5 Towards integrability

In this thesis we have investigated classical systems, their equations of motion,

symmetries of the system, and conserved quantities. Especially, we have tried

to identify constants of integration (or motion) in the course of classical motion.

It turns out that if one �nds suÆciently many constants, the system may be

integrable, i.e., the motion of the system can be exactly determined for the

speci�c initial values. The solubility is intimately associated with the existence

of soliton solutions.

When the slowly varying envelope approximation is applied to the nonlinear

Helmholtz equation (NHE), it reduces to the nonlinear Schr�odinger equation

(NSE). The NSE belongs to a special class of completely integrable equations

that can be exactly solved using the inverse scattering theory [47, 48]. The

study of integrability can be approached from various points of view. We show

�rst Liouville's de�nition of integrability for Hamiltonian systems. We then

discuss the possible applications of integrability in the context of the NHE.

5.1 Liouville's theorem

A mechanical system is typically described with canonical variables; coor-

dinates qi and momenta pi, where i = 1 : : : N . The evolution equation for

arbitrary operator A = A(qi; pi; t) is determined as

_A = fA;Hg+@A

@t; (84)

where the dot denotes d=dt, H is the Hamiltonian, and the fundamental Pois-

son brackets f�; �g are de�ned as

fAi; Bjg =@Ai

@qk

@Bj

@pk�@Ai

@pk

@Bj

@qk: (85)

The evolution of the Hamiltonian system follows the set of Hamilton's equa-

tions,

_qi =@H

@pi; _pi = �

@H

@qi; (86)

that are in accordance with Eqs. (84) and (85). The dynamical variables (qi; pi)

span the 2N dimensional phase space.

36

In a continuum system the partial derivatives are replaced by the functional

derivatives. Equations (84) and (85) transform as

_u(x) = fu(x); Hg =

Zdy f(x; y)

ÆH

Æu(y); (87)

fA[u]; B[u]g =

Zdxdy

ÆA

Æu(x)f(x; y)

ÆB

Æu(y); (88)

where f(x; y) = fu(x); u(y)g. The speci�c form of f(x; y) depends on the

problem and it may be di�erent for di�erent systems. Here we use the standard

de�nition of the functional derivative,

ÆF [u(x)]

Æu(y)= lim

�!0

1

�(F [u(x) + �Æ(x� y)]� F [u(x)]); (89)

where Æ(x� y) is a delta-function.

Given a Hamiltonian system, one may ask as to when it is integrable.

Liouville concluded the following: A Hamiltonian system whose phase space is

2N dimensional is integrable by the method of quadratures if and only if there

exist exactly N functionally independent conserved quantities which are in

involution. In other words, the Poisson brackets of these conserved quantities

with one another vanish. In the case of continuum systems one should �nd

in�nite number of conserved quantities that all are in involution.

There are well-established techniques to determine the conserved quantities.

A point particle problem is typically treated with the action-angle variables,

i.e., the system is expressed with such canonical variables that the classical

trajectory forms an N -dimensional torus in a phase-space [40, 41] (compare

also with Fig. 13). A continuum system can be handled by the method of Lax

and the inverse scattering theory [46{48,126].

As an example of integrable models we consider the NSE as a Hamiltonian

system. One may verify that when the Poisson structure is taken as follows

fA(x; t); A�(y; t)g = �iÆ(x� y); (90)

fA(x; t); A(y; t)g = 0 = fA�(x; t); A�(y; t)g; (91)

H =

Zdx

�����@A@x

����2 �

2jAj

4

�; (92)

the evolution equation for A, i.e., _A = fA(x; t); Hg, gives

37

i@

@tA+

@2

@x2A+ jAj

2A = 0: (93)

This is exactly the NSE, Eq. (29). We also note that Hamiltonian (92) is one

conserved quantity (out of in�nity) of the system.

5.2 Spatial solitons

We have shown that the NSE is closely related to the NHE. Speci�cally, the

equations become equal when the slowly varying envelope approximation is ap-

plied to the NHE. An interesting approach would be to utilize the integrability

of the NSE also for the NHE, and solve the nonlinear evolution of the NHE

exactly with arbitrary initial conditions. We have achieved some progress in

this area in the form of exact special solutions of the NHE.

Integrable models are based on the existence of soliton solutions, i.e., the

soliton solutions are related to conserved quantities of the nonlinear equation.

In fact, in Paper VII we show that the NHE indeed has an exact fundamental

soliton solution. When > 0 (bright soliton), the solution takes the form

EB(x; y) = A0 sech

�� A

20

A20 + 2k2

�1=2��k2� q

2 + A

20

2

�1=2x� q y

��

� exp

�iq x+ i

�k2� q

2 + A

20

2

�1=2y

�; (94)

where k and are de�ned as in Eq. (10). Parameters A0 and q are constants,

q2< k

2. Result (94) describes a self-guided beam that creates its own nonlinear

waveguide. The solution is con�ned onto an in�nite plane.

In a similar way, one can also �nd a dark soliton solution, < 0. Its

analytical form is [Paper VII]

ED(x; y) = �A0 tanh

�� A

20

2(k2 � j jA20)

�1=2��k2� q

2� j jA

20

�1=2x� q y

��

� exp

�iq x + i

�k2� q

2� j jA

20

�1=2y

�: (95)

A typical intensity pro�le of the dark spatial soliton is shown in Fig. 15.

Solitons (94) and (95) are exact solutions of the NHE; no approximations have

been made in their derivation. As was explained in Section 2.2.3, the NSE has

38

! � ��

" � � � �

Figure 15: Dark spatial soliton.

N -soliton solutions, from which the N = 1 solitons resemble results (94) and

(95). However, solutions (94) and (95) are unique in a sense that the NHE

does not have similar type of N > 1 solitons. Nevertheless, the existence of

the exact forms of the NHE imply that one could maybe apply the techniques

of integrable systems to the NHE. This would give more precise information

about the NHE and its nonlinear solutions.

39

6 Conclusions

In this work we have analyzed the consequences of the Kerr e�ect in wave

propagation. Using numerical and analytical techniques we investigated three

nonlinear phenomena, namely, optical scattering in the presence of a phase-

conjugate mirror, bistability, and spatial solitons. The major advances are the

following:

� The rigorous theory of volume gratings is shown to be a exible and

accurate numerical method to �nd exact solutions to the nonlinear

Maxwell equations.

� We derived an entirely new numerical algorithm to �nd rigorous non-

linear electromagnetic solutions. The method is based on a thin-layer

approximation. It uses characteristic matrices and it resembles the

linear thin-�lm theory.

� We applied the action integral formalism to layered nonlinear prob-

lems in the forms of canonical perturbative theory and variational

analysis. We found the technique advanced and e�ective. Speci�-

cally, we showed that the concepts of classical mechanics can be used

in nonlinear optical problems.

� We found exact spatial soliton solutions to the nonlinear Helmholtz

equation. Since the existence of the soliton solutions is closely related

to integrability, we believe that the knowledge of the integrable sys-

tems can also be used with the nonlinear Helmholtz equation to �nd

more precise information about its solutions.

In summary, we used new and modern tools to track the complicated nonlinear

evolution of an optical system. We hope that using our methods and ideas

one may develop further the exact nonlinear theories and �nd additional new

interesting results. Although the nonlinear Helmholtz equation appears simple

and trivial, it still contains a rich physics that de�nitely deserves more detailed

analysis.

40

References

[1] R. W. Munn and C. N. Ironside, ed., Principles and Applications of NonlinearOptical Materials (CRC, Florida, 1993).

[2] H. Kuhn and J. Robillard, eds., Nonlinear Optical Materials (CRC, Florida,1992).

[3] S. Dutta Gupta, \Nonlinear optics of strati�ed media," in Progress in Optics,Vol. 38, E. Wolf, ed. (Elsevier, Amsderdam, 1998), pp. 1{84.

[4] P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, \Generation ofoptical harmonics," Phys. Rev. Lett. 7, 118{119 (1961).

[5] A. Yariv, Quantum Electronics (Wiley, New York, 1989).

[6] Y. R. Shen, The Principles of Nonlinear Optics (Wiley, Chichester, 1984).

[7] B. E. A. Saleh and M. C. Teich, Fundamental of Photonics (Wiley, New York,1991).

[8] E. J. Lim, M. M. Fejer, R. L. Byer, andW. J. Kozlovsky, \Blue light generationby frequency doubling in poled lithium niobate channel waveguide," Electron.Lett. 25, 731{732 (1989).

[9] V. Mizrahi and J. E. Sipe, \The mystery of frequency doubling in optical�bers," Optics and Photonics News 2, 16{20 (1991).

[10] W. Brunner and H. Paul, \Theory of optical parametric ampli�cation andoscillation," in Progress in Optics, Vol. 15, E. Wolf, ed. (Elsevier, Amsterdam,1977), pp. 1{75.

[11] J.-I. Sakai, Phase Conjugate Optics (McGraw-Hill, New York, 1992).

[12] R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).

[13] Special issue on nonlinear phase conjugation, J. Opt. Soc. Am. 73 (May 1983).

[14] G. J. Dunning and R. C. Lind, \Demonstration of image transmission through�bers by optical phase conjugation," Opt. Lett. 7, 558{560 (1982).

[15] H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, NewYork, 1985).

[16] S. A. Collins, Jr. and K. C. Wasmundt, \Optical feedback and bistability: areview," Opt. Eng. 19, 478{487 (1980).

[17] H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, \Optical bistable devices:the basic components of all-optical systems," Opt. Eng. 19, 463{468 (1980).

[18] A. Sz�oke, V. Daneu, J. Goldhar, and N. A. Kurnit, \Bistable optical elementsand its applications," Appl. Phys. Lett. 15, 376{379 (1969).

[19] Special issue on optical computing and nonlinear materials, Proc. SPIE, Vol.881 (1988).

[20] J. R. Taylor, Optical Solitons { Theory and Experiment (Cambridge Univ.Press, UK, 1992).

41

[21] F. Abdullaev, S. Darmanyan, and P. Khabibullaev, Optical Solitons (Springer,Heidelberg, 1993).

[22] H. A. Haus and W. S. Wong, \Solitons in optical communications," Rev. Mod.Phys. 68, 423{444 (1996).

[23] R. Y. Chiao, E. Garmire, and C. H. Townes, \Self-trapping of optical beams,"Phys. Rev. Lett. 13, 479{482 (1964).

[24] P. L. Kelley, \Self-focusing of optical beams," Phys. Rev. Lett. 15, 1005{1008(1965).

[25] V. I. Talanov, \Focusing of light in cubic media," JETP Lett. 11, 199{201(1970).

[26] E. M. Vogel, M. J. Weber, and D. M. Krol, \Nonlinear optical phenomena inglasses," Phys. Chem. Glass 32, 231{254 (1991).

[27] C. H. L. Goodman, \Semiconductor and non-linear optical materials," Semi-cond. Sci. Technol. 6, 725{729 (1991).

[28] I. Broser and J. Gutowski, \Optical nonlinearity of CdS," Appl. Phys. B 46,1{17 (1988).

[29] K. F. Hulme, \Nonlinear optical crystals and their applications," Rep. Prog.Phys. 36, 497{540 (1973).

[30] D. N. Nikogosyan, \Nonlinear optics crystals (review and summary of data),"Sov. J. Quantum Electron. 7, 1{13 (1977).

[31] P. N. Prasad and D. J. Williams, Introduction to Non-Linear Optical E�ects

in Molecules and Polymers (Wiley, New York, 1990).

[32] A. Ulman, Introduction to Ultrathin Organic Films (Academic, San Diego,1991).

[33] M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1970).

[34] P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

[35] G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, \Nonlinear wavesguided by thin �lms," Appl. Phys. Lett. 44, 830{832 (1984).

[36] W. Chen and D. L. Mills, \Optical response of a nonlinear dielectric �lm,"Phys. Rev. B 35, 524{532 (1986).

[37] W. Chen and D. L. Mills, \Optical response of nonlinear multilayer structures:bilayers and superlattices," Phys. Rev. B 36, 6269{6278 (1987).

[38] K. Knop, \Rigorous di�raction theory for transmission phase gratings withdeep rectangular grooves," J. Opt. Soc. Am. 68, 1206{1210 (1978).

[39] P. Ramond, Field Theory: A Modern Primer (Addison, New York, 1990).

[40] H. Goldstein, Classical Mechanics (Addison, New York, 1980).

[41] W. Dittrich and M. Reuter, Classical and Quantum Dynamics from Classical

Paths to Path Integrals (Springer, New York, 1996).

[42] R. Weinstock, Calculus of Variations (McGraw, New York, 1952).

42

[43] K. Rektorys, Variational Methods in Mathematics, Science, and Engineering

(Reidel, Boston, 1977).

[44] D. Anderson, M. Bonnedal, and M. Lisak, \Self-trapped cylindrical laserbeams," Phys. Fluids 22, 1838{1840 (1979).

[45] D. Anderson and M. Bonnedal, \Variational approach to nonlinear self-focusing of Gaussian laser beams," Phys. Fluids 22, 105{109 (1979).

[46] A. Das, Integrable Models (World Scienti�c, Singapore, 1989).

[47] C. S. Gardner, J. M. Greene, M. D. Krushal, and R. M. Miura, \Methodfor solving the Korteweg-de Vries equation," Phys. Rev. Lett. 19, 1095{1097(1967).

[48] V. E. Zakharov and A. B. Shabat, \Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,"Sov. Phys. JETP 34, 62{69 (1972).

[49] A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

[50] B. Ya. Zel'dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of PhaseConjugation (Springer, Berlin, 1985).

[51] A. E. Siegman, P. A. Belanger, and A. Hardy, \Optical resonators usingphase-conjugate mirrors," in Optical Phase Conjugation, R. A. Fisher, ed.(Academic, New York, 1983), pp. 465{535.

[52] J.-C. Diels and I. C. McMichael, \In uence of wave-front-conjugated couplingon the operation of a laser gyro," Opt. Lett. 6, 219{221 (1981).

[53] J. Feinberg, \Self-pumped, continuos-wave phase conjugator using internalre ection," Opt. Lett. 7, 486{488 (1982).

[54] M. Nieto-Vesperinas, \Fields generated by a Fabry-perot interferometer witha phase-conjugate mirror," J. Opt. Soc. 2, 427{436 (1985).

[55] A. E. Neeves and M. H. Birnboim, \Polarization selective optical phase con-jugation in a Kerr-like medium," J. Opt. Soc. Am. B 5, 701{708 (1988).

[56] S. I. Bozhevolnyi, O. Keller, and I. Smolyaninov, \Phase conjugation of anoptical near �eld," Opt. Lett. 19, 1601{1603 (1994).

[57] S. I. Bozhevolnyi, O. Keller, and I. Smolyaninov, \Scattered light enhancementnear a phase conjugating mirror," Opt. Commun. 115, 115{120 (1995).

[58] A. Yariv, \Phase-conjugate optics and real-time holography," IEEE J. Quan-tum Electron. 14, 650{660 (1978).

[59] G. S. Agarwal, A. T. Friberg, and E. Wolf, \Scattering theory of distortioncorrection by phase conjugation," J. Opt. Soc. Am. 73, 529{538 (1983).

[60] P. D. Drummond and A. T. Friberg, \Specular re ection cancellation in aninterferometer with a phase-conjugate mirror," J. Appl. Phys. 54, 5618{5625(1983).

[61] G. S. Agarwal, \Elimination of distortions produced by a non-linear mediumby phase conjugation without losses or gains,"Opt. Commun. 47, 77{78 (1983).

43

[62] J. Jose and S. Dutta Gupta, \Phase conjugation induced distortion correctionand optical multistability in enhanced back scattering in nonlinear layeredmedia," Opt. Commun. 145, 220{226 (1998).

[63] M.~H. Farzad and M. T. Tavassoly, \Degenerate four-wave mixing withoutslowly varying amplitude approximation," J. Opt. Soc. Am. B 14, 1707{1715(1997).

[64] P. W. Smith, \Hybrid bistable optical device," Opt. Eng. 19, 456{468 (1980).

[65] A. E. Kaplan, \Hysteresis re ection and refraction by a nonlinear boundary {a new class of e�ects in nonlinear optics," JETP Lett. 24, 114{119 (1976).

[66] G. M. Wysin, H. J. Simon, and R. T. Deck, \Optical bistability with surfaceplasmons," Opt. Lett. 6, 30{32 (1981).

[67] G. S. Agarwal and S. Dutta Gupta, \Exact result on optical bistability withsurface plasmons in layered media," Phys. Rev. B 34, 5239{5243 (1986).

[68] S. Dutta Gupta and G. S. Agarwal, \Optical bistability with surface plasmonsbeyond plane waves in a nonlinear dielectric," J. Opt. Soc. Am. B 3, 236{238(1986).

[69] R. Reinisch, P. Arlot, G. Vitrant, and E. Pic, \Bistable optical behavior of anonlinear attenuated total re ection device: resonant excitation of nonlinearguided modes," Appl. Phys. Lett. 47, 1248{1250 (1985).

[70] V. J. Montemayor and R. T. Deck, \Optical bistability with the waveguidemode," J. Opt. Soc. Am. B 2, 1010{1013 (1985).

[71] V. J. Montemayor and R. T. Deck, \Optical bistability with the waveguidemode: the case of a �nite-width incident beam," J. Opt. Soc. Am. B 3, 1211{1220 (1986).

[72] A. E. Kaplan, \Bistable optical solitons," J. Physique 49, 69{74 (1988).

[73] H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesen, \Di�erential gain andbistability using a sodium �lled Fabry Perot interferometer," Phys. Rev. Lett.36, 1135{1138 (1976).

[74] P. Martinot, A. Koster, S. Laval, and W. Carvalho, \Optical bistability fromsurface plasmon excitation," Appl. Phys. B 29, 172{173 (1982).

[75] P. Martinot, A. Koster, and S. Laval, \Experimental observation of opticalbistability by excitation of a surface plasmon wave," IEEE J. Quantum Elec-tron. 21, 1140{1143 (1985).

[76] P. Dannberg and E. Broose, \Observation of optical bistability by prism exci-tation of a nonlinear �lm-guided wave," Appl. Opt. 27, 1612{1614 (1988).

[77] H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner,and W. Wiegmann, \Optical bistability in semiconductors," Appl. Phys. Lett.35, 451{453 (1979).

[78] D. A. B. Miller, S. D. Smith, and A. Johnston, \Optical bistability and sig-nal ampli�cation in a semiconductor crystal: applications of new low-powernonlinear e�ects in InSb," Appl. Phys. Lett. 35, 658{660 (1979).

44

[79] G. P. Agrawal and R. W. Boyd, eds., Contemporary Nonlinear Optics, (Aca-demic, New York, 1992).

[80] G. I. Stegeman, \The growing family of spatial solitons," Optica Applicata26, 239{248 (1996).

[81] A. W. Snyder, D. J. Mitchell, and Y. Chen, \Spatial solitons of Maxwell'sequations," Opt. Lett. 19, 524{526 (1994).

[82] A. Barthelemy, S. Maneuf, and C. Froehly, \Propagation soliton et auto-con�nement de faisceaux laser par non linearit�e optique de Kerr," Opt. Com-mun. 55, 201{206 (1985).

[83] S. Maneuf, R. Desailly, and C. Froehly, \Stable self-trapping of laser beams:observation in a nonlinear planar waveguide," Opt. Commun. 65, 193{198(1988).

[84] J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E.Leaird, E. M. Vogel, and P. W. E. Smith, \Observation of spatial opticalsolitons in a nonlinear glass waveguide," Opt. Lett. 15, 471{473 (1990).

[85] P. Dumais, A. Villeneuve, and J. S. Aitchison, \Bright temporal solitonlikepulses in self-focusing AlGaAs waveguides near 800 nm," Opt. Lett. 21, 260{262 (1996).

[86] J. P. Gordon, \Interaction forces among solitons in optical �bers," Opt. Lett.8, 596{598 (1983).

[87] F. Reynaud and A. Barthelemy, \Optically controlled interaction between twofundamental soliton beams," Europhys. Lett. 12, 401{405 (1990).

[88] J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver,J. L. Jackel, and P. W. E. Smith, \Experimental observation of spatial solitoninteractions," Opt. Lett. 16, 15{17 (1991).

[89] Y. Silberberg, \Collapse of optical pulses," Opt. Lett. 15, 1282{1284 (1990).

[90] L. Gagnon, \Exact solutions for optical wave propagation including transversee�ects," J. Opt. Soc. Am. B 7, 1098{1102 (1990).

[91] W. J. Tomlinson, R. J. Hawkins, A. M. Weiner, J. P. Heritage, and R. N.Thurston, \Dark optical solitons with �nite-width background pulses," J. Opt.Soc. Am. B 6, 329{334 (1989).

[92] L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, \Experimental observationof picosecond pulse narrowing and solitons in optical �bers," Phys. Rev. Lett.45, 1095{1098 (1980).

[93] G. I. Stegeman, \Comparison of guided wave approaches to optical bistability,"Appl. Phys. Lett. 41, 214{216 (1982).

[94] P. S. Balourdos, D. J. Frantzeskakis, M. C. Tsilis, and I. G. Tigelis, \Re ec-tivity of a nonlinear discontinuity in optical waveguides," Pure Appl. Opt. 7,1{11 (1998).

45

[95] G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe,and L. A. Weller-Brophy, \Enhanced nonlinear optical response of compositematerials," Phys. Rev. Lett. 74, 1871{1874 (1995).

[96] S. Dutta Gupta and J. Jose, \Enhanced back-scattering with surface andguided modes coupled to a phase conjugate mirror," Opt. Commun. 125,105{112 (1996).

[97] K. M. Leung, \Exact results for the scattering of electromagnetic waves witha nonlinear �lm," Phys. Rev. B 39, 3590{3598 (1988).

[98] K. M. Leung, \Propagation of nonlinear surface polaritons," Phys. Rev. A 31,1189{1192 (1985).

[99] K. M. Leung, \p-polarized nonlinear surface polaritons in materials withintensity-dependent dielectric functions," Phys. Rev. B 32, 5093{5101 (1985).

[100] M. B. Pande and S. Dutta Gupta, \Nonlinearity-induced resonances and opti-cal multistability with coupled surface plasmons in a symmetric layered struc-ture," Opt. Lett. 15, 944{946 (1990).

[101] J. H. Marburger and F. S. Felber, \Theory of lossless nonlinear Fabry-Perotinterferometer," Phys. Rev. A 17, 335{342 (1978).

[102] R. W. Boyd and J. E. Sipe, \Nonlinear optical susceptibilities of layered com-posite materials," J. Opt. Soc. Am. B 11, 297{303 (1994).

[103] V. M. Agranovich, S. A. Kiselev, and D. L. Mills, \Optical multistability innonlinear superlattices with very thin layers," Phys. Rev. B 44, 10917{10920(1991).

[104] A. P. Mayer, A. A. Maradudin, R. Garcia-Molina, and A. D. Boardman, \Non-linear surface electromagnetic waves on a periodically corrugated surface,"Opt. Commun. 72, 244{248 (1989).

[105] G. Li, J. A. Tobin, and D. D. Denton, \Excitation of nonlinear guided wavesin Kerr-type media by use of chirped and tapered gratings," J. Opt. Soc. Am.B 10, 2290{2297 (1993).

[106] C. B. Burckhardt, \Di�raction of a plane wave at a sinusoidally strati�eddielectric grating," J. Opt. Soc. Am. 56, 1502{1509 (1966).

[107] F. G. Kaspar, \Di�raction by thick, periodically strati�ed gratings with com-plex dielectric constant," J. Opt. Soc. Am. 63, 37{45 (1973).

[108] M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, \Stableimplementation of the rigorous couple-wave analysis for surface-relief gratings:enhanced transmittance matrix approach," J. Opt. Soc. Am. A 12, 1077{1086(1995).

[109] F. Montiel and M. Nevi�ere, \Electromagnetic study of a photolithographysetup for periodic masks and applications to nonperiodic masks," J. Opt. Soc.Am. A 13, 1429{1438 (1996).

[110] R. Petit, ed., Electromagnetic Theory of Gratings (Springer, Berlin, 1980).

46

[111] S. Chavez Cerda, Self-Con�ned Beam Propagation and Pattern Formation in

Nonlinear Optics, Ph. D. Thesis (Imperial College, London, 1994).

[112] G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).

[113] D. Anderson, \Variational approach to nonlinear pulse propagation in optical�bers," Phys. Rev. A 27, 3135{3145 (1983).

[114] D. Anderson, M. Lisak, and T. Reichel, \Asymptotic propagation propertiesof pulses in a soliton-based optical-�ber communication system," J. Opt. Soc.Am. B 5, 207{210 (1988).

[115] D. J. Kaup, T. I. Lakoba, and B. A. Malomed, \Asymmetric solitons in mis-matched dual-core optical �bers," J. Opt. Soc. Am. B 14, 1199{1206 (1997).

[116] F. Kh. Abdullaev, \Validation of the variational approach for chirped pulsesin �bers with periodic dispersion," Phys. Rev. E 58, 6637{6648 (1998).

[117] S. Longhi, G. Steinmeyer, and W. S. Wong, \Variational approach to pulsepropagation in parametrically ampli�ed optical systems," J. Opt. Soc. Am. B14, 2167{2173 (1997).

[118] M. Desaix, D. Anderson, and M. Lisak, \Variational approach to collapse ofoptical pulses," J. Opt. Soc. Am. B 8, 2082{2086 (1991).

[119] F. Kh. Abdullaev, N. K. Nurmanov, and E. N. Tsoy, \Variational approach tothe problem of dark-soliton generation," Phys. Rev. E 56, 3638{3644 (1997).

[120] M. Desaix, D. Anderson, andM. Lisak, \Variational approach to the Zakharov-Shabat scattering problem," Phys. Rev. E 50, 2253{2256 (1994).

[121] D. J. Kaup and B. A. Malomed, \Variational principle for the Zakharov-Shabat equations," Physica D 84, 319{328 (1995).

[122] M. Desaix, D. Anderson, M. Lisak, and M. L. Quiroga-Texeiro, \Variationallyobtained approximate eigenvalues of the Zakharov-Shabat scattering problemfor real potentials," Phys. Lett. A 212, 332{338 (1996).

[123] M. Jaworski, \Rayleigh-Ritz procedure for the Zakharov-Shabat system,"Phys. Rev. E 56, 6142{6146 (1997).

[124] J. Ximen, \Canonical theory in electron optics," Adv. in Electronics and Elec-tron Phys. 81, 231{277 (1991).

[125] R. Jackiw, \A nonrelativistic chiral soliton in one dimension," Nonlin. Math.Phys. 4, 261{270 (1997).

[126] Y. S. Kivshar and B. A. Malomed, \Dynamics of solitons in nearly integrablesystems," Rev. Mod. Phys. 61, 763{915 (1989).

47