SOLITON INTERACTIONS AND THE

FORMATION OF SOLITONIC PATTERNS

Suzanne M. Sears

A DISSERTATION

PRESENTED TO THE FACULTY

1

OF PRINCETON UNIVERSITY

IN CANDIDACY FOR THE DEGREE

OF DOCTOR OF PHILOSOPHY

RECOMMENDED FOR ACCEPTANCE

BY THE DEPARTMENT OF PHYSICS

June 2004

2

© Copyright by Suzanne Marie Sears, 2004. All rights reserved.

3

Acknowledgements

Without the help of colleagues, friends, and family, the work in this thesis would not

have been possible. First, I would like to thank Moti Segev, for introducing me to the

intriguing science of solitons, and for his support as my advisor. His great love for

solitons has taught me that a passion for one’s work is truly the greatest asset any

scientist can bring to their endeavors. Many thanks are in order to Demetri

Christodoulides as well, for his guidance and interesting discussions.

To the others whom I shared a lab with over the years, thanks for many fun memories. I

will always remember Beate Mikulla, Ricky Leng, Evgenia Evgenieva, Alexandra

Grandpierre, and Claude Pigier in particular. Special thanks to Marin Soljacic, for all of

the conversations and collaborations over the years. Thanks to Judith Castellino and Mike

Nolta for entertainment in Jadwin. And much love to Elena Peteva, for many happy times

under the sun and stars.

Mom and Dad, many were the times when your faith in me and loving support made all

the difference. I love you both.

To Marc, with all my love.

4

Abstract

From the stripes of a zebra, to the spirals of cream in a hot cup of coffee, we are

surrounded by patterns in the natural world. But why are there patterns? Why drives their

formation? In this thesis we study some of the diverse ways patterns can arise due to the

interactions between solitary waves in nonlinear systems, sometimes starting from

nothing more than random noise.

What follows is a set of three studies. In the first, we show how a nonlinear

system that supports solitons can be driven to generate exact (regular) Cantor set fractals.

As an example, we use numerical simulations to demonstrate the formation of Cantor set

fractals by temporal optical solitons. This fractal formation occurs in a cascade of

nonlinear optical fibers through the dynamical evolution of a single input soliton.

In the second study, we investigate pattern formation initiated by modulation

instability in nonlinear partially coherent wave fronts and show that anisotropic noise

and/or anisotropic correlation statistics can lead to ordered patterns such as grids and

stripes.

For the final study, we demonstrate the spontaneous clustering of solitons in

partially coherent wavefronts during the final stages of pattern formation initiated by

modulation instability and noise. Experimental observations are in agreement with

theoretical predictions and are confirmed using numerical simulations.

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Acknowledgements

Without the help of colleagues, friends, and family, the work in this thesis would not

have been possible. First, I would like to thank Moti Segev, for introducing me to the

intriguing science of solitons, and for his support as my advisor. His great love for

solitons has taught me that a passion for one’s work is truly the greatest asset any

scientist can bring to their endeavors. Many thanks are in order to Demetri

Christodoulides as well, for his guidance and interesting discussions.

To the others whom I shared a lab with over the years, thanks for many fun memories. I

will always remember Beate Mikulla, Ricky Leng, Evgenia Evgenieva, Alexandra

Grandpierre, and Claude Pigier in particular. Special thanks to Marin Soljacic, for all of

the conversations and collaborations over the years. Outside of the graduate college,

Princeton simply would not have been the same without Elena Peteva.

Mom and Dad, many were the times when your faith in me and loving support made all

the difference. I love you both.

And for Marc, with all my love.

6

Table of Contents

1 Introduction ............................................................................................................... 9

1.1 Solitons and the dynamics of pattern formation ......................................... 9

1.2 A brief history of solitons............................................................................... 11

1.3 Optical solitons................................................................................................ 17

1.3.1 Optical temporal solitons ....................................................................... 17

1.3.2 Optical spatial solitons ........................................................................... 23

1.4 Incoherent solitons......................................................................................... 26

1.5 Modulation instability ..................................................................................... 33

1.6 References ....................................................................................................... 36

2 Cantor Set Fractals from Solitons......................................................................... 43

2.1 About fractals .................................................................................................. 43

2.2 The generation of Cantor set fractals.......................................................... 44

2.3 Optical fibers provide a possible environment for fractals....................... 46

2.4 Numerical simulations confirm theoretical predictions ............................. 51

2.5 References ....................................................................................................... 58

3 Pattern formation via symmetry breaking in nonlinear weakly correlated

systems ............................................................................................................................ 59

3.1 Spontaneous pattern formation ................................................................... 59

3.2 Modulation Instability..................................................................................... 59

3.3 Stripes and lattices from two-transverse dimensional MI ........................ 61

3.4 Modulation instability with anisotropic correlation function..................... 73

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3.5 Conclusion........................................................................................................ 86

3.6 References ....................................................................................................... 89

4 Clustering of Solitons in Weakly Correlated Wavefronts .................................. 92

4.1 Universality of clustering phenomena ......................................................... 92

4.2 Clustering of optical spatial solitons ............................................................ 93

4.3 Solitons............................................................................................................. 94

4.3.1 A review of some basics ........................................................................... 94

4.3.2 Incoherent solitons................................................................................. 95

4.3.3 Modulation Instability............................................................................. 96

4.4 Clustering – theory and simulations ............................................................ 97

4.5 Clustering - experiment ............................................................................... 105

4.6 Conclusion...................................................................................................... 110

4.7 References ..................................................................................................... 111

5 Conclusion and future directions........................................................................ 117

5.1 References ..................................................................................................... 123

6 Publications............................................................................................................ 124

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1 Introduction

1.1 Solitons and the dynamics of pattern formation

Everyone in America knows what a fractal is. Take a stroll around any college

campus and you will pass by young computer scientists wearing T-shirts emblazoned

with brightly colored spiraling patterns. In the corporate world, fractals thrive in the after

hours as screen-savers come to life on cubicle workstations. Similarly, pattern formation

has also captured the imagination of the media. Coffee-table picture books and websites

abound showing images of zebras next to striped tropical fish, or brains compared to

coral. But while these familiar images surround us in nature and on computer screens,

something of a gap remains. Although truly a breakthrough in many respects, the ability

to generate a fractal picture of a leaf on a computer screen does not necessarily enhance

our understanding of the physical mechanisms that actually caused the leaf to form in the

manner that it did. Too often the dynamics of pattern formation are no less mysterious

than they ever were.

We have found that non-linear systems supporting solitons provide a rich

theoretical and physical environment in which to study the dynamics of pattern

formation. There are two main reasons. First, soliton interactions and properties are

complex, and the range of behavior to explore is vast and very interesting. Second,

solitons exist in a wide range of non-linear media, and are not an isolated phenomena.

Despite the diversity of physical systems capable of supporting solitons, they are

universal and manifestations in different systems share many common features. Results

in any one particular field are often broadly applicable.

9

In this thesis, we present several mechanisms leading to pattern formation in

soliton-supporting media. In Chapter 2, we propagate optical temporal solitons in a multi-

stage fiber optic system to generate exact Cantor Set fractals [52]. The fractal is

generated from a single input soliton. This soliton separates into several self-similar

“daughter” solitons as it propagates; when then next stage of the setup is reached, the

breakup of each of these “daughter” solitons is triggered. The process is repeated again

and again, exhibiting self-similarity at every stage. At the output a train of pulses with

temporal spacing corresponding to an exact Cantor Set fractal is produced.

In Chapter 3, we explore the formation of grid and stripe patterns from initially

featureless white noise [53]. A broad beam is the input to the system; as it propagates

small perturbations cause the beam to fragment into narrow beamlets due to an imbalance

of non-linear and linear forces. Some of the resulting beamlets will be stable, and the

frequencies these beamlets are composed of become amplified by the system, leading

eventually to stripes, and grids at those frequencies.

In Chapter 4, clustering of solitons in partially coherent wavefronts is observed.

Solitons in such systems experience only attractive forces, and each soliton moves

towards its nearest neighbor [54]. Clustering is observed.

The remainder of the introductory chapter discusses relevant background material

concerning the history and variety of optical solitons and their theoretical underpinnings.

10

1.2 A brief history of solitons

It was in 1834 that the first officially documented observation of a soliton

occurred. John S. Russell, a Scottish scientist, was riding his horse along a shallow canal,

when he noticed in it a “well defined heap of water” elevated above the smooth water

around it travelling “without change of form or diminution of speed” [1]. He was able to

follow it on horseback for some distance until it finally disappeared. Today, science

recognizes what Russell saw as a soliton, a phenomena related to tsunamis and tidal

waves. Solitons are by no means restricted to water waves; the mechanim is universal,

appearing in numerous nonlinear systems capable of supporting waves. Loosely

speaking, a soliton may refer to any solitary, localized wave packet that remains

unchanged as it propagates.

Soliton formation results from the interplay between the linear and non-linear

responses of the propagation medium. In linear systems, dispersion or diffraction

generally will cause wave-packets to spread as they propagate. Any wave-packet can be

decomposed into a linear superposition of plane-waves of different frequencies using

Fourier methods; broadening of a pulse will occur if these plane-waves of different

frequencies travel at different velocities (chromatic dispersion) or at different angles

(diffraction). Although the spectral contents of the pulse will remain unchanged, the

dispersion (or diffraction) will introduce a frequency dependent phase-shift to each of the

plane wave components, causing the overall intensity profile that is their superposition to

grow wider. In non-linear materials, these broadening tendencies can be countered by

focusing of the wave-packet caused by intensity dependent properties of the

11

12

Figure 1. Modern day re-creation of the soliton observed by Russell in 1834. [Union Canal near

Edinburgh, Scotland, July 1995, at a conference on nonlinear waves at Heriot-Watt University.]

propagation medium. In optics, for example, the refractive index of the material may be

affected by the presence of light; in self-focusing materials the refractive index will

increase with the intensity of the beam. This can in turn lead to the effective creation of

an induced “lens” which “focuses” the beam. To think about this in another way, both the

linear and nonlinear responses introduce phase differences among different plane wave

components of the beam. These changes can offset one another, and the nonlinear effect

may cause a beam widened by dispersion (diffraction) to narrow again. If the

characteristics of the wave-packet and the properties of the material are such that the

linear spreading and non-linear self-focusing effects exactly counter one another, a

soliton will be created.

13

Figure 2. A. Diffraction (or dispersion) of a one-transverse dimensional beam

propagating in linear media. B. Propagation of a similar beam in non-linear media: the

properties of the material and beam are such that the linearity and nonlinearity exactly

balance, resulting in a soliton.

14

While soliton formation is in itself a very interesting phenomena, interactions between

solitons are one of their most fascinating aspects. Intriguing parallels can be drawn

between soliton interaction “forces” and those of particles. In some respects, solitons

behave like “quasi-particles”. A single soliton travels as a unique, well formed,

unchanging entity. These defining properties are indifferent to close-range interactions

(or even collisions) with other solitons. For the class of integrable systems, soliton

collisions have been proven to be fully elastic [9,11]; not only is the number of solitons

conserved, but also each soliton retains its respective power and velocity. Furthermore,

soliton collisions are not just the result of two solitons blindly crossing paths; rather

effective “forces” exist between solitons and the particle-like wave-packets may either

attract or repel one another, depending on their phase properties. Unique and quite varied

dynamics, such as spiraling, fusion, and fission may be observed [10].

W

1964, af

laborator

indeterm

Figure 3. Two one dimensional solitons collide and recover.

hile Russell observed solitons in nature as far back as 1834, it was not until

ter the invention of the laser, that self-focusing behavior was reported in the

y [12]. Narrow wave-packets could propagate undistorted for seemingly

inate distances. Many fundamental results in soliton science followed within a

15

few years. In 1965, Kruskal showed mathematically that, like particles, the beams could

intersect with one another and continue to propagate undisturbed. This behavior was

likened to “collisions” and the new “particles” were christened “solitons” [11]. After

more pioneering work such as the superposition of soliton solutions and Lax-pairs,

inverse-scattering methods were used in 1972 to find exact solutions to the (1+1)D

Nonlinear Schroedinger Equation (NLS) with Kerr non-linearity [9]. (The Kerr-type non-

linearity is a real quantity, linear in the local intensity I. To first order, the

non-linearity in almost any system can be modeled this way, provided the frequency is

far from any resonances so that the anharmonicity is relatively weak. Typical values of

∆n giving rise to optical spatial solitons are on the order 10

2NLn n∆ = I

-4.)

In the years since then, solitons have been found in many other systems,

illustrating their universality. The solitons first discovered in 1964 were optical spatial

solitons. That is, these solitons were optical and had constant spatial profiles. In 1973,

another sort of optical soliton, the optical temporal soliton, was theoretically shown to be

possible by Hasegawa and Tappert [14]. These are one dimensional solitons consisting of

a beam of light trapped in its transverse spatial dimensions by a waveguide, while pulsed

in the direction of propagation; it is this temporal profile which is solitonic and remains

unchanged during propagation over huge distances. The first temporal solitons were

observed experimentally in optical fibers by Mollenauer, Stolen, and Gordon in 1980 [13]

and have since then been much studied for potential use in long-haul communication

systems [14-16]. Although optical solitons are probably the easiest to study nowadays,

and the most commonly researched, solitons are universal and have been discovered in

16

many non-linear media allowing the propagation of waves. Plasma waves [2], sound

waves in 3He [3], and waves in CS2 [5], glass [6], semiconductor [7], and polymer

waveguides [8] have all been shown to support solitons. An incredible variety of solitons

have been classified since the early days, exhibiting a remarkable range of forms:

photorefractive solitons [39,40], quadratic solitons [41,42], multicomponent vector

solitons [43], incoherent solitons [44-46], discrete solitons [47,48], optical “bullets” [49],

and cavity solitons [50,51] are just a few examples.

1.3 Optical solitons

1.3.1 Optical temporal solitons

In optics, we speak of two generic kinds of solitons: temporal and spatial.

Temporal solitons can be seen in optical fibers, where the propagation of light is

goverened by the Non-Linear Shroedinger equation (NLS),

2

222

A i A i A Az

β γτ

∂ ∂= +

∂ ∂, (0.1)

where A refers to the slowly varying electric field envelope of a short pulse of light with

carrier frequency oω ; β and γ are real constants reflecting, respectively, the strength of

the linear and non-linear responses. The coordinate, z, corresponds to the distance the

light pulse has propagated along the fiber, and τ is the time coordinate in the reference

frame of the pulse (the time variable has been shifted linearly as a function of z so that the

17

coordinate frame moves at the group velocity of the pulse). Although, of course, there are

three spatial dimensions, only one appears in the equation; this is because the light is

assumed to be an unchanging mode of the optical fiber waveguide in the transverse x and

y directions, which cancels out of the equations. Such a system is referred to as (1 + 1) D,

meaning 1 transverse (or trapping) dimension and 1 propagation dimension.

Examining Eq (0.1), we can see both dispersion and non-linear focusing, or self-

phase modulation, at work. The first term on the right hand side represents linear

chromatic dispersion, and the second, the nonlinear response of the medium resulting

from the dipole movements of the electrons in the material in response to the electric

field waves passing through it. Eq. (0.1) is known as the Non-Linear Schroedinger

equation (NLS) due to its resemblance to the Schroedinger equation in quantum

mechanics. Along these lines, we can intuitively think of the non-linear term as creating a

“potential well”. In this case, a soliton can be thought of as being a “bound state” of the

potential which it itself induces (the so-called “self-consistency principle”) [20].

To better understand this important equation, it is instructive to consider its origin

[17]. As stated above, ( , )A zτ represents the slowly varying envelope of the electric field

at a carrier frequency oω :

( ) ( ) ( ) ˆ, , o oi k z tE t z A t z e xω−=r

(0.2)

18

where we are back in the true coordinate frame, ( ),t z , and linear polarization in the x -

direction is assumed; the wavevector of the carrier in vacuum is 2o VACk π λ= , where

VACλ is the wavelength.

The response of the medium to the light, (both the dispersion and the non-

linearity), are embodied in the form of its index of refraction:

( ) ( )2 22

2, on E n n Eω ω= +2r r. (0.3)

where ( )on ω represents chromatic dispersion, and (far from the resonances of the

material) may be well approximated by the Sellmeier equation ( )2

22 2

1

1M

j jo

j j

nα ω

ωω ω=

= +−∑ ,

where the sum, , is over each of the M resonances of the material [18]. The non-linear

response of the medium is assumed to be linear in the intensity, proportional to the

constant, . This results by assuming that the electric field is sufficiently weak enough

for the response to be approximated as a Taylor’s series with only the lowest non-zero

term retained; for centro-symmetric materials this must be proportional to

j

2n

*E Er r

, and not

Er

, since an Er

term would indicate a directional preference in the material.

The wave-vector, , is related to the index of refraction (keeping first order terms

only):

k

19

2 22( )

2oo

ck nnn

ωω

= + Er

. (0.4)

Thus ( )2,k k Eω=r

, and for frequencies near to the carrier frequency, oω , we may

approximate

( ) ( ) ( )2 2222

12o o o

k k kk k E EE

ω ω ω ωω ω

∂ ∂ ∂− = − + − + −

∂ ∂ ∂

2

o

r rr , (0.5)

where all of the derivatives are constants evaluated at , ok oω , and 2

oEr

(the average

amplitude). Knowing that the electric field may be represented in the Fourier domain as

well as in time and space, and that, at infinity, , we can use integration by parts to

replace with the spatial operator

0E →

ok k− iz

∂−

∂ and oω ω− with the temporal operator i

t∂∂

.

Making these replacements in the equation above, and operating on the field envelope,

( ),A t z , we get:

( )2 2

2 222 2

1 02 o

A k A k A kiz t t Aω ω

∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞+ − + −⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ∂A A A = , (0.6)

20

where we have used the fact 2 2E A=

r. Remembering that the derivatives with respect to

are constants, and moving to a frame of reference, k ( )/ ,gt z zτ υ= − , that moves with

the group velocity of the pulse ( )g kυ ω= ∂ ∂ , we have

(2

2 222 o

A i A i A A Az

β γτ

∂ ∂= + −

∂ ∂ ) (0.7)

where we have introduced the notation β and γ for the constants in Eq. (0.6). Note that

2oA is a constant and thus this term will simply introduce a phase

2oi A ze γ that is constant

across the profile of the pulse, introduces no new physics, and may be renormalized out,

reducing Eq. (0.7) to the NLS as desired. If the dispersion constant, 0β > , then the

material is said to have anomalous dispersion, and the equation can be solved exactly

using the inverse-scattering method developed by Zahkarov and Shabat [9] for bright1

solitons of the form

( ) ( ) 2, sech exp2o oiA z P z oτ τ τ β τ⎛= ⎜

⎝ ⎠⎞⎟

(0.8)

1 It is also possible to have dark solitons; such beams are “negative images” of bright solitons and are of

high intensity everywhere except in the center, where the absence of light can create a dark soliton which is

as stable as its counterpart of the inverse shape.

21

where is the peak power of the pulse and oP oτ is the temporal width of the pulse. The

intensity profile of the pulse, ( ) ( )2,A z fτ τ= , has no dependence and thus the pulse

is truly stationary and a soliton.

z

Since the 1980s, most of the research on temporal solitons has focused on

applications to long-distance fiber optic communications [14-16]. However, temporal

solitons are also intrinsically interesting from a scientific point of view and much about

the general behavior of self-trapped waves in non-linear systems can be learned by

examining their behavior. As discussed in the introduction, two solitons in close

proximity to one another will interact. If two solitons of the form of Eq. (0.8) are near to

one another with no relative phase difference between them, then the two pulses will

attract one another, and eventually pass right through one another, “colliding”. The

solitons have momentum and will continue to separate after the collision, but the

attraction will act as a restoring force, eventually drawing the two back together. The pair

of solitons will continue to pass through one another, again and again, with perfect

periodicity. On the other hand, if the solitons are initially π out of phase with respect to

one another, then they will repel.

A phenomenon related to solitons is that of “higher-order” solitons. If N solitons,

all in phase, are initially exactly overlapping in both time and space, then the initial pulse

profile will look like

( ) ( ), 0 sechN oA z N P oτ τ τ= = . (0.9)

22

As the solitons propagate, the interactive forces between the solitons will cause them to

oscillate, and various patterns will form as the pulse eventually breaks into peaks.

The behavior is periodic, and the pulse shape will continually return to the same profile

as in Eq. (0.9). The behavior of higher-order solitons is explored further in Chapter 5,

where we show how fractals can be formed by triggering each of the peaks of an N

1N −

1N −

th-order soliton to break up into 1N − peaks. If the process is performed recursively,

exact Cantor set fractals result.

1.3.2 Optical spatial solitons

The temporal solitons in Section 1.3.1 are able to exist because temporal changes

in the intensity of the pulse create a temporal gradient in the index of refraction of the

material, causing it to act as a time-dependent waveguide for the pulse. Since the electric

field in Eq. (0.2) is essentially uniform in space (the fluctuations in space and time due to

the envelope’s carrier wave are very rapid and average out) only the derivative of the

slowly varying electric field envelope with respect to time matters. However, time is a

coordinate like any other, and in fact, variations in the intensity of a beam in space can

also give rise to an altered index of refraction and an optically-induced waveguide. If the

characteristics of the incident beam coincide with those of a mode of the waveguide

which it induces, then the light will propagate, (“self-trapped” by its own waveguide), as

a soliton.

23

One-dimensional CW optical beams with spatial intensity structures propagating

in a Kerr non-linear self-focusing media obey the following normalized equation:

2

22

1 02

A Ai Az x

∂ ∂ A+ + =∂ ∂

, (0.10)

which is identical to Eq. (0.1). A (1+1)D (one transverse (or trapping) dimension, one

propagation dimension) spatial soliton can occur in dielectric planar waveguides, or by

using beams which are very broad and uniform in one transverse dimension, and narrow

in the other (such beams are unstable and will break up due to “transverse instability”,

discussed further in section 1.5).

Immediately, an important difference between spatial and temporal solitons

becomes apparent: (as far as we know) only one time dimension exists, therefore

temporal solitons are inherently limited to be one-dimensional! Work over recent years

has shown a rich variety of possibilities for spatial solitons, and solitons trapped in two

transverse spatial dimensions ((2+1)D) have been shown to exist as well as solitons

trapped in both transverse spatial dimensions and the time dimension ((2+1+1)D solitons,

or “light bullets”). In two spatial dimensions, the NLS (with Kerr-type non-linearity)

looks like:

221 02

Ai A Az ⊥ A∂

+ ∇ + =∂

(0.11)

24

The non-linearity in Eq. (0.11) is only one possibility; many other forms exist, for

example the saturable non-linearity ( )2~ 1n A A∆ + 2

is commonly found.

In addition to providing an extra dimension for solitons to propagate in, moving to

the spatial domain also allows an extra dimension for solitons to interact in, and for the

definition of inherently high-dimensional quantities such as angular momentum. Now,

intriguing behaviors such as soliton spiralling and vortex solitons are possible. Overall,

the spatial domain provides a very rich environment for studying the fundamental

properties of solitons.

One simple way to the understand the existence of spatial solitons is to view them

as a balance between spreading due to linear diffraction, and focusing caused by a non-

linearly induced “lens”. An alternative, and very illustrative, picture of spatial soliton

phenomena was presented by Askar’yan in 1962 and expanded upon by Snyder et al in

1991 [20]. Consider a material of the self-focusing type - for bright beams, the refractive

index will be highest at the center of the beam where the intensity of the beam is greatest.

The structure is identical to a graded-index waveguide: a higher index core is surrounded

by material with a lower index of refraction, causing waves to reflect internally. Such

waveguides may have guided modes for which these reflections interfere constructively,

allowing these modes to propagate in the waveguide with their intensity profiles

unchanged. Our spatial soliton example is no different: the higher index of refraction in

the center sets up a waveguide which may allow the propagation of certain modes. If the

profile of the incident beam is the same as one of the modes of the waveguide, then the

25

incident beam can propagate unchanged. In such a case, the incident beam induces a

waveguide in the material, and then proceeds to propagate in it as a guided mode! The

soliton is said to be “self-trapped”.

1.4 Incoherent solitons

All of the solitons discussed in Sec. 1.3 above are coherent solitons; that is to say,

if the phase of the electric field is known at one particular time (place) then the phase of

the electric field at any other time (place) can also be predicted. For example, consider

the temporal soliton solution of Eq. (0.1) given in Eq. (0.8); at the input, we know the

amplitude and phase of the electric field at every point and time:

( ) ( ) ˆ, 0 secho oE z Pτ = =r

xτ τ (the phase is simply uniform everywhere). The solution,

Eq. (0.8), also dictates the amplitude and phase of the electric field at every point in time

and space. Furthermore, for any input electric field amplitude whatsoever, Eqs. (0.1) and

(0.11) can be used to calculate the phase at any later point, provided the phase of the

initial condition is specified. This is what is meant by coherent.

While coherence is certainly not a property of light in general, it is a reasonably

good characterization of the light produced by the lasers used in many experiments. Since

lasers produce light by stimulated emission, their beams are indeed highly coherent. On

the other hand, light from Light Emitting Diodes (LEDs) and from natural sources, such

as the sun or light bulbs, is incoherent, and the phase varies randomly with time and

space across the beam. Some light is partially incoherent, and for distances smaller than

26

the coherence length, lc, (or times shorter than the coherence time), the phase is correlated

(for coherent light ). cl → ∞

The double slit experiment illustrates the meaning of coherence well. Consider a

board with two very small slits, spaced apart on order of a wavelength at positions x1 and

x2, placed before a beam which has a coherence length lc. If lc is much greater than x2 - x1,

then the situation is the same as if two point sources radiating in unison (with a constant

phase difference between them) were placed on the slits. The total light passing through

the slits will be the time averaged sum of the intensity from each “source” plus the

interference between them: ( )2 2 2

1 2 1 22 ReE E E E E ∗= + +r r r r r

, where r is the response

time of the detector. If lc is much smaller than x2 - x1, then it will seem as if each of the

slits were an independent point source (as long as the fluctuations in the phase difference

between E1 and E2 are rapid compared to the response time of the detector), and the

resulting light will be of an intensity: 2 2

1

2

2E E E= +r r r

. If the board were taken away

altogether, what one would see (if our eyes worked much faster and on a much finer

scale!) would be a beam with random speckles, constantly changing their positions in

time and space. These speckles would be of average diameter lc, and correspond to

regions of the beam where the phases were correlated and constructively interfered. Some

highly monochromatic laser beams are partially incoherent in space, but strongly

correlated in time; if the speckles are of a large enough size, the human eye will be able

to see them (when projected onto a flat surface), as they can last for hours, or even

longer.

27

For many years, only coherent optical solitons were known to exist, and it was

assumed that this property was a necessity. It was thought that the instantaneous speckles

inherent in incoherent beams would each be individually self-focused by the non-

linearity, resulting in filamentation and the breakup of the wavefront. This all changed,

when in 1995, Mitchell, Chen, Shih, and Segev from Princeton University experimentally

demonstrated self-trapping of incoherent light, (with randomly varying phase both in time

and in space), using an SBN photorefractive crystal with a slow non-linearity [26]. Key to

the success of the experiment was the use of a medium with a response time long

compared to the characteristic phase fluctuation time across the beam. In this way, the

non-linearity could respond only to the smooth and steady time-averaged intensity

profile, and was not affected by the momentary speckles. Since then, much research has

been done both experimentally and theoretically in nonlinear media in general, greatly

increasing understanding of this new type of soliton and propagation of incoherent optical

beams.

This

waveguide may have many modes, and the soliton may be decomposed into a sum

Perhaps the simplest way to explain incoherent solitons is the multi-modal theory.

Whether the wave is incoherent or not, in a self-focusing medium, the refractive index

will be highest where the intensity of the incident beam is highest. In crystals with a slow

non-linearity, the refractive index of the material will increase where the time-averaged

intensity of the beam increases and, for example, for a Gaussian beam with highest

intensity in the center, this will lead to the creation of an induced wave-guide.

28

( ) ( ) ( ) ( ), , , , expm m mm

A x y z t c t U x y i zβ= ∑ (0.12)

where ( ),x y e mode profile of the mmU is th th mode, mβ is the propagation constant of

mode mU , and ( )mc t is its instantaneous relative weight. Due to the random nature of

ent beams, the amplitude and phase of incoher ( )mc t will also randomly fluctuate, and

( )mc t will nction. Thus, no correlations can exist betwbe tic fu een different

and

a stochas

modes ( ) ( )m n mc t c t nδ∗ = . The time-averaged p ofile of the soliton is

r

( ) ( ) ( ) ( )2 2 2 2, , , , ,m mm m

2A x y z t c t U x y d U x y= =∑ ∑ , (0.13)

where ( )m md c t= is the time-averaged population of mode m. In this way, the time-

averaged intensity of the soliton can be decomposed into a sum of the modes of the

induced waveguide. Of course, the time-averaged population of each of the modes will

remain stationary as it propagates in the waveguide, so the sum of their time-averaged

populations must also remain stationary. Since the waveguide was induced by the

intensity profile in the first place, what we have is a genuine soliton. This explanation

implies three requirements for the existence of incoherent solitons: (1) the response time

of the non-linearity must be slower than the characteristic time of phase fluctuations, (2)

the incoherent beam must be able to induce a multi-mode waveguide, and (3) the slowly

29

varying envelope of the partially incoherent beam must be an appropriate superposition

of these

is the Fourier transform of the correlation

function. Since the coherent density method will be used extensively in Chapters 2 and 3,

it is of

modes of the waveguide, so that it is commensurate with the modal weights.

Although the modal perspective of incoherent soliton formation is informative

and useful for finding stationary soliton solutions, it offers no insight into the dynamic

properties of incoherent solitons and cannot say anything at all about incoherent non-

solitonic beams. A quite different approach, the coherent-density method [27], is

excellently suited to studying these problems. In this model, infinitely many “coherent

components” propagate at all possible angles (i.e. values of the wave-vector (kx, ky)) and

interact with one another only through the non-linearity, which is a function of the time-

averaged total intensity. The shapes of the initial intensity profile for each of these

coherent components are the same, but the relative weights are given by the angular

power spectrum of the source beam, which

much use to thoroughly detail it now.

First, consider an incoherent field of uniform time-averaged intensity, ( ),o tφ xr ,

representing only the statistical fluctuations of 0 . Since our

concern here is the degree of incoherence of the source, let us define the spatial statistical

our source at the input,

autocorrelation function of

z =

( ),o tφ xr to be ( ) ( ) ( )*2 1 2 1oR − =x x x xoφ φr r r r

autocorrelation function of the source spectrum is

. Now, the

( ) ( ) ( ) ( )* 2 22 1 2 1

ˆ ˆ 4o o Gπ δ⊥ ⊥ ⊥ ⊥Φ Φ = −k k k k kr r

1⊥ , where r r r ( )G ⊥k

r is the Fourier transform

30

of ( )R xr . Since ( ) 2Φ k

r is the intensity density in the spectral domain, physical , o ⊥ ly

must be the angular power spectrum density of the source. Examining the

autocorrelation function of the source spectrum, we see that the presence of the

( )G ⊥kr

δ -

function implies that there is no correlation between i⊥kr

and j⊥kr

for any . Thus, we

may think of our source as a set of plane waves, all statistically uncorre where the

en by

i j≠

lated,

amplitude of each wave is giv ( )1 2G ⊥kr

, and each propagates out at an angle

kθ ⊥= kr r

(we have assumed here that ( )G ⊥kr

falls off rapidly and that the only

ons are for k⊥kr

significant contributi ).

Now consider our total input; the source is spatially modulated by some spatial

function, such that ( ) ( ) ( )ˆ, 0, ,oE z t x f tφ= =x x xr r r r . Taking the average intensity of the

background statistical source to be unity, we have ( ) ( ) ( )2 2

, 0 ,o oI z E t f= = =x x xrr r r .

ber of point sources, radiating out Thus, we can think of the input as being an infinite num

at every position xr in all directions (with the power going at each angle weighted

according to ) and with the total power density at each point given by( )G ⊥kr

( )oI xr .

Alternatively, this is equivalent to an infinite number of coherent profiles of shape ( )f xr

all propagating out at different angles, kθ ⊥= kr r

, where each component is weighted by

the square root of the power spectral density of the source, ( )1 2G ⊥kr

. Since we have

shown above that each is uncorrelated to all of the other transverse wavevectors, i⊥kr

31

there is no statistical correlation between any of the so-called coherent-components,

( ) ( ) ( )1 2, 0,u z k f G kθ θ⊥ ⊥= = = ⋅ =x k x kr rr rr r nt will propagate

unaffected by the others, except for the non-linear changes caused in the common

refractive in

he propagation of a single coherent component is governed simply by the NLS

with on to account for the angle of propagation,

. Each compone

dex of the material by the presence of their intensities.

T

e additional term θr

:

( )2 g , 02 2

o

o

i u u I zz k n

θ ⊥ ⊥1 ku∂⎛ ⎞+ ⋅

r r r∇ + ∇ + =⎡ ⎤⎜ ⎟ ⎣ ⎦∂⎝ ⎠

x . (0.14)

Here, ( )g ,I z⎡ ⎤⎣ ⎦xr is a function of the total intensity of the beam, ( ),I zxr , and represents

the non-linear ch ction: ange to the index of refra ( )2 2 g ,on n I z= + ⎡ ⎤⎣ ⎦xr . The total intensity

components, ( )

of the beam is given by the integral of the intensities of vidual coherent the indi

( )2

,I zπ π

π π

, ,u z dθ θ∫ x− −

= ∫xr rr r m

d above, they are statistically uncorrelated. The

wavevector,

, with no interference ter s between the

components, since, as discusse

2o VACk π λ= , is that of the carrier wave in vacuum, and is the index of

refraction in the absence of light.

on

The coherent-density approach can easily be adapted for computer; all that is

required is to supply the initial conditions and to approximate the infinite number of

coherent components by a discrete, finite number (replacing integrals by summations). In

32

practice, a large number of components are required to simulate beams with even a small

partial incoherence; in two spatial dimensions, the number can exceed 100 x 100. For

problems with a fair amount of spatial variation, each of the 100 x 100 components may

require on the order of 2048 x 2048 spatial grid points as well. Thus, a modest problem

might require 41,943,040,000 points just for the grid, and due to the sensitive nature of

non-linear dynamics, these usually are required to be 32-bit double precision (that’s more

than 156 Giga-Bytes just to store the incoherent wave profile!). The amount of

computational power required for the problem quickly escalates! In fact, without access

to a supercomputer, most problems can not reasonably be attempted. Fortunately, the

nature of the problem is highly parallel and naturally suited to massively parallel

machines. In Chapters 2 and 3, research was made possible thanks to the use of the

Pittsburg Supercomputing facility; computations were performed using parallel

programming techniques on up to as many as 512 processors.

1.5 Modulation instability

Closely related to the formation of solitons is the process of modulation instability

(MI). In the regime of soliton formation, a very broad, flat beam (a beam much wider

than the corresponding soliton of equivalent peak intensity) propagating under the

influences of linear and non-linear influences will be unstable, since the linear diffraction

effect is quite small compared to the non-linear effects. Interestingly, due to random

background noise, the wavefront may have small amplitude perturbations of width

similar to little “quasi-solitons” and each may individually start to “self-focus”. These

initially infinitesimal fluctuations may grow in amplitude, causing the beam to fragment

and breakup into narrow filaments [28-31] that often are almost ideal solitons [32,33]. In

33

the context of certain pursuits, the behavior is undesirable; it is well known in fiber optic

communications that signals containing long, broad pulses may disintegrate into random

trains of short pulses. This mechanism is known as modulation instability (MI) and is

observed with both temporal and spatial optical wavefronts, in both one and two

dimensions2. MI is not exclusive to optics, but is a universal phenomena, occurring in

many non-linear environments including waves in fluids [36], plasmas [37], and

dielectr

han they are being

washed-out by linear diffusion. Above this point, MI will occur. Such a “threshold” is

unique to incoherent MI and has no counterpart in coherent systems.

ic materials [38].

While the existence of MI in coherent non-linear systems has been well known

for many years, MI in incoherent systems remained largely unexplored for a long time.

Approaching the topic naively, it might at first appear that incoherence would eliminate

modulation instability. The less coherent a wavepacket is, the more rapidly it will diffuse,

and so any growth of “filaments” due to MI tends to be “washed-out” by this linear

diffusion. However, recent theoretical and experimental work has established the

presence of MI in partially-coherent systems [34,35]. As the strength of the non-linear

response of the material is increased, the strength of the MI mechanism also “increases”:

filaments will form more and more quickly. It has been shown that by continuing to

increase the non-linearity, eventually the filaments will form faster t

2 (1+1)D solitons formed by propagating an two-transverse-dimensional optical beam which is broad in one

transverse dimension and narrow and solitonic in the other are subject to breakup in the broad dimension

due to the related “transverse instability”.

34

Not only is the onset of MI different in incoherent and coherent systems, but

interesting differences in the dynamics of the resulting filaments can be seen between the

two systems as well. In coherent systems, solitonic filaments of random phases are

created and forces between any pair of solitons can be either attractive or repulsive

depending upon the phase difference between the solitons. In incoherent system, as I

show in this thesis, on scales greater than the correlation length only attractive forces will

be of significance since incoherent solitons can never be out of phase with one another

and the effect of increased intensity nearby always deepens the effective potential well.

As a result, the solitons group together and soliton “clusters” are created. We show this

behavior experimentally and theoretically, using numerical simulations, in Chapter 4. We

believe this phenomena important not only in that it offers an opportunity to observe rich

non-linear dynamics, but also in that clustering behavior is common to many non-linear

systems. Natural systems are often impossible to control or explore in different parameter

regimes or initial conditions and optical spatial solitons provide a well-controlled and

very flexible environment to study clustering.

35

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37

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38

I. Stegeman, Phys. Rev. A 53, 1138 (1996); G. I. Stegeman, D. J. Hagan, and L. Torner,

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39

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41

[51] V. P. Taranenko, I. Ganne, R. J. Kuszelewicz, and C. O. Weiss, Phys. Rev. A., to be

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42

Cantor Set Fractals from Solitons

We show how a nonlinear system that supports solitons can be driven to generate exact (regular)

Cantor set fractals [7]. As an example, we use numerical simulations to demonstrate the formation of

Cantor set fractals by temporal optical solitons. This fractal formation occurs in a cascade of nonlinear

optical fibers through the dynamical evolution from a single input soliton.

1.7 About fractals

A fractal, as defined by Mandelbrot, “is a shape made of parts similar to the whole in

some way” [1]. Fractals can be classified in numerous manners, of which one stands out

rather distinctly: exact (regular) fractals versus statistical (random) fractals. An exact

fractal is an “object which appears self-similar under varying degrees of magnification...

in effect, possessing symmetry across scale, with each small part replicating the structure

of the whole” [1]. Taken literally, when the same object replicates itself on successively

smaller scales, even though the number of scales in the physical world is never infinite,

we call this object an “exact fractal.” When, on the other hand, the object replicates itself

in its statistical properties only, it is defined as a “statistical fractal.” Statistical fractals

have been observed in many physical systems, ranging from material structures

(polymers, aggregation, interfaces, etc.), to biology, medicine, electric circuits, computer

interconnects, galactic clusters, and many other surprising areas, including stock market

price fluctuations [1]. In optics, fractals were identified in conjunction with the Talbot

effect and diffraction from a binary grating [2] and with unstable cavity modes [3]. Exact

fractals, on the other hand, such as the Cantor set, occur rarely in nature except as

43

mathematical constructs. In this chapter we describe how a Cantor set of exact fractals

can be constructed, under proper nonadiabatic conditions, in systems described by the

(1+1)D cubic self-focusing nonlinear Schrödinger equation (NLSE). We demonstrate

exact Cantor set fractals of temporal light pulses in a sequence of nonlinear optical fibers.

We calculate their fractal similarity dimensions and explain how these results can be

produced experimentally (see Sears et al. Cantor set fractals from solitons, Phys. Rev.

Lett. 84, 1902 (2002) [7]).

A Cantor set is best characterized by describing its generation [1]. Starting with a single

line segment, the middle third is removed to leave behind two segments, each with length

one-third of the original. From each of these segments, the middle third is again removed,

and so on, ad infinitum. At every stage of the process, the result is self-similar to the

previous stage, i.e., identical upon rescaling. This “triplet set” is not the only possible

Cantor set: any arbitrary cascaded removal of portions of the line segment may form the

repetitive structure.

1.8 The generation of Cantor set fractals

This experiment is based on a recent idea [4] that nonlinear soliton-supporting systems

can evolve under nonadiabatic conditions to give rise to self-similarity and fractals. Such

fractals should be observable in many systems, and their existence depends on two

requirements: (i) the system does not possess a natural length scale; i.e., the physics is the

same on all scales (or, any natural scale is invisible in the parameter range of interest) and

(ii) the system undergoes abrupt, nonadiabatic changes in at least one of its properties [4].

44

To illustrate generating fractals from solitons, Ref. [4] showed optical fractals evolving

dynamically from a single input pulse or beam. The idea is to repetitively induce the

breakup of the pulse (beam) into smaller pulses by abruptly modifying the balance

between dispersion (diffraction) and nonlinearity. Consider a broad pulse launched into a

nonlinear dispersive medium. The pulse is broad in the sense that its width is much larger

than that of the characteristic fundamental soliton, given the peak intensity. This

fundamental soliton width is determined by properties of the medium such as the

dispersion and nonlinearity coefficients as well as by the soliton peak power. A broad

pulse will always break up, either due to modulation instability [5] when random noise

dominates or by soliton dynamics-induced breakup [4] when the noise is weak. The result

of the breakup is a number of smaller pulses or “daughter solitons,” which propagate

stably in the medium in which the “mother pulse” broke up. The daughter solitons are

self-similar to one another in the sense that they can be mapped (by change of scale only)

onto one another, because they all have the same shape (hyperbolic secant for the Kerr-

type nonlinearity). Now, if an adequately abrupt change is made to a property of the

medium (e.g., the dispersion or the nonlinear coefficient [6]), then each of the daughter

solitons seems broad and therefore unstable in the “modified” medium. The daughter

solitons undergo the same instability-induced breakup experienced by the initial mother

pulse and generate even smaller “granddaughter solitons.” Successive changes to the

medium properties thus create successive generations of solitons on successively smaller

scales. The resultant structure after every breakup is self-similar with the products of the

first breakup. The successive generations of breakups of each soliton into many daughter

45

solitons leads to a structure which is self-similar on widely varying scales, and each part

breaks up again in a structure replicating the whole. The entire structure is therefore a

fractal.

In the general case, this method of generating fractals from solitons gives rise to

statistical fractals. In the fractal which results from each breakup, the amplitudes of the

individual solitons, the distances between them, and their relation to the solitons of a

different “layer” are random. Thus, the self-similarity between the structures at different

scales is only in their statistical properties. Here we show that the principle of “fractals

from solitons” can be applied to create exact (regular) fractals, in the form of an exact

Cantor set. The requirement is that after every breakup stage, all of the “daughter pulses”

must be identical to one another. In this case, all the daughter pulses can be rescaled from

one breakup stage to the next by the same constant, and the entire propagation dynamics

repeats itself in an exact rescaled fashion. The resulting scaling on all length scales

constitutes an exact Cantor set. In this manner, one can obtain exact Cantor set fractals

from solitons. This represents one of the rare examples of a physical system that supports

exact (regular), as opposed to statistical (random), fractals [1].

1.9 Optical fibers provide a possible environment for fractals

To illustrate the idea of generating Cantor set fractals from solitons, we analyze the

propagation of a temporal optical pulse in a sequence of nonlinear fiber stages with

dispersion coefficients and lengths specifically chosen to impose a constant rescaling

factor between consecutive breakup products. We solve the (1+1)D cubic self-focusing

NLSE, vary the dispersion coefficient in a manner designed to generate doublet- and

46

quadruplet-Cantor set fractals, and show the formation of temporal optical soliton Cantor

set fractals (Fig. 1).

Figure 1. Illustration of a sequence of nonlinear optical fiber segments with their

disperson constants and lengths specifically chosen to generate exact Cantor set fractals.

47

The nonlinear propagation and breakup process in fiber segment “i” is described by the

(1+1)D cubic NLSE:

( ) 22

2 02

i

iz Tψ β ψ γ ψ ψ∂ ∂

− +∂ ∂

= , (0.15)

where ( ),z Tψ is the slowly varying envelope of the pulse, gT t z υ= − is the time in the

propagation frame, gυ is the group velocity, ( ) 0iβ < is the (anomalous) group velocity

dispersion coefficient of segment i , and 0γ > is proportional to the nonlinearity

( ); is the spatial variable in the direction of propagation and is time. Equation 2 0n > z t

(1) has a fundamental soliton solution of the form

( )

( )( )

( )

( )( )

02( )0

2( ) ( )0

, sech

exp 2

ii

i

i i

z T T TT

iz T

βψ

γ

β

=

⎡ ⎤× ⎢ ⎥⎣ ⎦

(0.16)

where is the temporal full width half maximum of ( )01.76274 iT ( ) 2

,z Tψ and

( ) ( )2( ) ( ) ( )0 2i i

oz Tπ= iβ is the soliton period for fiber segment i. The N-order soliton (at

) of Eq. (1) can be obtained by multiplying 0z = ( ), 0z Tψ = from Eq. (2) by a factor of

. A higher order soliton of a given propagates in a periodic fashion. In the first N 1N >

48

half of the soliton period ( )( )0 2iz , the pulse splits into two pulses, then into three, then

into four, etc., up to pulses [5]. In the second half of the period the process reverses

itself until all the pulses have recombined into a single pulse identical to the original one.

While attempting to generate Cantor set fractals from solitons, we observed that, if we

start with an -order soliton, it splits into

1N −

N M N< pulses, each of which reaches an

approximately hyperbolic secant shape. Furthermore, there is always a region in the

evolution where all the M daughter pulses are almost fully identical and possess the same

height. The breakup can be reproduced if we cut the fiber at this point and couple the

pulses into a new fiber with a dispersion coefficient chosen such that each of the pulses

launched into the second fiber is an -order soliton. Each of the daughter pulses

generated in the first fiber exactly replicates the breakup of the “mother soliton,” on a

smaller scale. Because Eq. (1) is the same on all scales, the entire second breakup process

of each daughter pulse is a rescaled replica of the initial mother-pulse breakup. In fact,

we can redefine the coordinates in the second fiber by simple rescaling, so that in the new

coordinates the equation is identical to the equation (including all coefficients) describing

the pulse dynamics in the first fiber. In this manner, we can continue the process

recursively many times, resulting in an exact fractal structure that reproduces, on

successively smaller scales, not only the final “product” (the pulses emerging from each

fiber segment), but also the entire breakup evolution.

N

What remains to be specified is how we choose the sequence of fibers and the relations

between their dispersion coefficients and lengths. Consider a sequence in which the ratio

between the dispersion coefficients of every pair of consecutive segments is fixed

49

( 1) ( )i iβ β+ =η , where 1η < . This implies that the periods of the fundamental solitons in

consecutive segments are related through ( ) ( ) [ ]2 2( 1) ( ) ( 1) ( )0 0 0 0 1i i i iz z T T η+ +⎡ ⎤= ⎢ ⎥⎣ ⎦

.

Numerically, we launch an -order soliton into the first fiber segment and let it

propagate until it breaks into

N

M hyperbolic-secant-like pulses of almost identical heights

and widths. At this location we terminate the first fiber and label the distance propagated

in it . From the simulations we find the peak power (1)L (1)MP and the temporal width (1)

MT

of the M almost-identical pulses emerging from the first segment. The M pulses are then

launched into the second segment. Our goal is to have, in the second segment, a rescaled

replica of the evolution in the first segment. To achieve this, we require that each of these

M pulses will become an -order soliton in the second segment. Thus we equate the

peak power in each of the

N

M pulses in the first fiber to the peak power of an -order

soliton in the second fiber:

N

( )

(2)(1) (2) 2

2(2)0

M NP P NT

β

γ= = , (0.17)

where since it is the width of the input pulse to the second fiber. From Eq. (3)

we find the dispersion coefficient in the second fiber,

(1) (2)0MT T=

(2)β . The ratio η between the

dispersion coefficients in consecutive fibers determines the scaling of the similarity

transformation. Using η and we calculate the period (2)0T (2)

0z . Requiring that the

evolution in the second fiber is a rescaled replica of that in the first fiber, we get

(2) (2) (1) (1)0L z L z= 0 . Each of the M pulses in the second fiber exactly reproduces the

dynamics of the original soliton in the first fiber but on a smaller scale. At the end of the

50

second stage, each of the M pulses transforms into M pulses, resulting in M sets of

M pulses. The logic used to calculate the second stage parameters is used repeatedly to

create many successive stages, each producing a factor of M pulses more than the

previous stage.

1.10 Numerical simulations confirm theoretical predictions

We provide examples of Cantor set fractals from solitons by numerically solving Eq. (1).

The order of the soliton used and the fraction of a soliton period propagated vary

depending on the desired number of pulses, M . Figure 2 shows a quadruplet Cantor set

fractal. We launch an soliton into the first fiber characterized by 8N = 1γ = and

and let it propagate for . At this point the pulse has separated into

four nearly identical hyperbolic secant shaped pulses. We launch the emerging four

pulses into the next fiber, characterized by and

(1) 1β = − (1)00.1261 z

(2) 0.01285β = − 1γ = . Each of the four

solitons is an soliton in the second fiber. We let the four soliton set propagate for 8N =

(2)00.1261 z , which is identical to (1)

00.03290 z . The scaling factor η is 0.012 85. We

repeat this procedure with the third fiber and let the four sets of four solitons propagate

for (3)00.1261 z , so there are three stages total. The output consists of four sets of four sets

of four solitons. This evolution is shown in Fig. 2(a), where the degree of darkness is

proportional to ( ) 2,z Tψ . In Fig. 2(b), we show a magnified version of the lowermost

branch of the fractal of Fig. 2(a). Figure 2(c) shows a magnified version of the lower

branch of Fig. 2(b).

51

Figure 2. Evolution of pulse envelope during the generation of a quadruplet Cantor set.

The darkness is proportional to the pulse intensity. (a) shows the entire process. An N=8

soliton is propagated for (1)00.3112 z and then propagated in a rescaled environment so

that the input to that stage is four N=8 solitons. The procedure is repeated for one more

stage. (b) shows the magnification of the second stage; (c) shows the third stage. Units

are normalized: , peak power =1, and 1 unit of distance = 0 1T = 2 (0

iT )β .

52

The same method is used to generate the doublet Cantor set fractals in Fig. 3, where an

soliton is propagated for 5N = ( )00.1623 iz in each segment. Figure 3(a) shows the two

output pulses emerging from the first segment. The two pulses are then fed into the

rescaled environment, where they mimic the original 5N = soliton, each breaking up

into two more pulses [Fig. 3(b)]. Figure 3(c) shows the output after the third segment. At

this stage we have two sets of two sets of two pulses, which is a Cantor set prefractal. If

one could construct an infinite number of fiber segments, then it would be an exact

regular Cantor set fractal in the mathematical sense. In physical systems, limitations such

as high order dispersion, dissipation, and Raman scattering place a bound on the number

of stages.

As with any physical fractal, the breakups are prefractals rather than fractals; yet, we

expect at least three stages in a real fiber sequence. To prove the generation of an exact

Cantor set fractal, we choose random selections from each of the three panels of Fig. 3

and plot them on the same scale in Fig. 4: They fully coincide with one another. The

exactness of the overlap in Fig. 4 indicates that this indeed is an exact Cantor set fractal.

Similarly, we verify that the quadruplet fractals from Fig. 2 are exact. We have also

generated a triplet Cantor set fractal from an 6N = soliton, propagated for ( )00.1649 iz .

One can design an experiment of Cantor set fractals in a fiber optic system. For example,

a doublet Cantor set can be generated from the breakup of an 3N = soliton. In the first

stage a 50 ps FWHM pulse of 0.88 W power is launched into a 6 km long fiber with

(1) 2127.6 ps kmβ = − (assuming 11.62W kmγ 1− −= for all fibers). At the end of this fiber,

53

Figure 3. Temporal pulse envelope after each of the three stages for doublet Cantor set.

(a) shows the output from the first stage, (b) shows the result from the second, and (c)

shows those from the third. The inset in (c) shows a magnification of one of the four sets

of two. Units are normalized so that 0 1T = , peak power = 1, and 1 unit of distance =

2 (0

iT )β .

54

Figure 4. Illustration of exact self-similarity of pulse envelopes after each of the stages of

the doublet Cantor set. The three panels shown in Fig. 3 have been appropriately

rescaled, shifted, and overlapped. Units are 0 1T = , peak power = 1, and 1 unit of distance

= 2 (0

iT )β .

55

which corresponds to the midpoint of the soliton period, the input pulse has broken into

two pulses of peak power 1.2 W and width 13.2 ps spaced 42 ps apart. These pulses are

then coupled into a second 4.1 km long fiber characterized by a dispersion parameter of

(2) 212.2 ps kmβ = − . The pulses exiting this second stage are each 3.3 ps in duration and

peak power 1.9 W. They are grouped in pairs separated by 9.9 ps. Finally, the pulses are

propagated in a third 2.7 km long stage with (3) 21.2 ps kmβ = − .This results in two sets

of two sets of two pulses, each of width 816 fs and peak power 3 W, grouped in pairs

separated by 2.4 ps. These results have been confirmed through simulations including

third order dispersion, fiber loss, and Raman scattering. The inclusion of these additional

terms in Eq. (1) limits the number of stages which may be realistically obtained

experimentally. The example system given above is consistent with readily available

fibers. One may use specialty fibers (dispersion flattened or dispersion decreasing fibers)

to combat effects of third order dispersion and loss to expand the number of

experimentally realizable stages.

The Cantor set fractals in the fiber optic system are robust to a variety of perturbations in

the fiber parameters and variations in the initial pulse conditions. We simulated the

evolution of the Cantor set fractals under 5% deviations in the pulse peak power, pulse

width, fiber length, and dispersion. We also added 2% (of the power) of excess Gaussian

white noise and launched a Gaussian initial pulse shape. Under all these variations, the

resulting Cantor set fractals exhibit excellent similarity to the ideal case.

Although we generate only prefractals, we can calculate the fractal dimension for an

56

equivalent infinite number of stages. There are various definitions of fractal dimensions;

here we calculate the similarity dimension . In the construction of a fractal an original

object is replicated into many rescaled copies. If the length of the original object is unity,

SD

ε is the length of each new copy, and is the number of copies. The similarity

dimension is [1]:

N

( ) ( )log log 1SD N ε= . For the doublet Cantor set fractal from Fig. 3,

and for the quadruplet Cantor set fractal from Fig. 2, 0.2702SD = 0.4318SD = .

In conclusion, we have shown how a nonlinear soliton supporting system can be driven to

generate exact (regular) Cantor set fractals and have demonstrated theoretically optical

temporal Cantor set fractals in nonlinear fibers (see Sears et al. Cantor set fractals from

solitons, Phys. Rev. Lett. 84, 1902 (2002) [7]). The next challenge is to observe Cantor

set fractals experimentally.

57

1.11 References

[1] P. S. Addison, “Fractals and Chaos” (Institute of Physics, Bristol, 1997).

[2] M.V. Berry and S. Klein, J. Mod. Opt. 43, 2139 (1996).

[3] G. P. Karman and J. P. Woerdman, Opt. Lett. 23, 1909 (1998).

[4] M. Soljacic, M. Segev, and C. R. Menyuk, Phys. Rev. E 61, 1048 (2000).

[5] G. P. Agrawal, “Nonlinear Fiber Optics” (Academic Press, San Diego, 1995).

[6] The change in the conditions must be abrupt; an adiabatic change does not cause a

breakup, but instead the pulse adapts and evolves smoothly into a narrower soliton.

[7] S. Sears, M. Soljacic, M. Segev, D. Krylov and K. Bergman, Cantor set fractals from

solitons, Physical Review Letters 84, 1902 (2000).

58

2 Pattern formation via symmetry breaking in

nonlinear weakly correlated systems

We study pattern formation initiated by modulation instability in nonlinear partially coherent wave fronts

and show that anisotropic noise and/or anisotropic correlation statistics can lead to ordered patterns [18].

2.1 Spontaneous pattern formation

The decay of signals and the growth of disorder are everyday occurrences in physical

systems. Naively speaking, this is just a manifestation of the law of increase of entropy or

second law of thermodynamics. Interestingly, however, in some circumstances order may

appear spontaneously out of noise. Starting from an initially featureless background,

random fluctuations may generate structures that naturally balance the various forces in

the system and are stable. These may grow, as further fluctuations lead the system

towards even more stable states. Such processes of ordered structures emerging from

noise, or spontaneous pattern formation, are typically associated with phase-transition

phenomena. In optics, spontaneous pattern formation has been demonstrated in many

systems [1], in some cases arising from feedback, and in other occurring in the absence of

feedback, i.e., during one-way propagation.

2.2 Modulation Instability

Perhaps the best known example of pattern formation during unidirectional propagation

is the process of modulation instability (MI), manifested as the breakup of a uniform

59

‘‘plane wave’’ [2] or of a very long pulse in time [3]. Such an MI process can lead to the

spontaneous creation of stable localized wave packets with particlelike features, namely,

solitons, in nonlinear self-focusing media. Depending upon the nonlinear properties of

the medium, perturbations of certain frequencies are naturally favored; these frequencies

emerge out of white noise and gain in strength. These sinusoidal oscillations grow,

becoming more and more peaky, until eventually the wave fragments into localized

soliton-like wave packets. Until recently, MI was considered to be strictly a coherent

process. But during the last two years, a series of theoretical and experimental studies [4-

8] has demonstrated that modulation instability can also occur in random-phase (or

weakly correlated) wave fronts, in both the spatial domain [4–8] and the temporal domain

[9]. The main difference between MI in such partially coherent systems and the

‘‘traditional’’ MI experienced by coherent waves, is the existence of a threshold. In other

words, in incoherent systems MI appears only if the ‘‘strength’’ of the nonlinearity

exceeds a well-defined threshold that depends on the coherence properties (correlation

distance) of the wave front. Thus far, incoherent MI has been demonstrated

experimentally in both (1+1)D (one transverse dimension) [5,6,8] and (2+1)D (two

transverse dimensions) [5,7] systems. Yet theoretically, analytic studies of incoherent MI

were reported only for the (1+1)D case [4,8,9] and so far, the only theoretical work

carried out in (2+1)D systems has addressed a very different problem [7]. Furthermore,

the experiments with (2+1)D incoherent MI [5,7,8] have left many open questions. For

example, is there a threshold for (2+1)D incoherent MI? And if such a threshold exists,

how does it relate to the threshold in (1+1)D systems? But beyond all other questions, the

ability to explore (2+1)D incoherent MI adds another degree of freedom to the problem:

60

anisotropy between the transverse dimensions that may lead to symmetry breaking and to

the formation of asymmetric patterns. The anisotropy can arise from the nonlinearity,

from the two-dimensional coherence function (that is, the correlation statistics of the

random wave front), and interestingly enough, from the noise that serves as a ‘‘seed’’ for

MI.

2.3 Stripes and lattices from two-transverse dimensional MI

In this chapter, and in the paper we have published on the subject (S. M. Sears et al,

Pattern formation via symmetry breaking in nonlinear weakly correlated systems,

Physical Review E 65, 36620 (2002) [18]), we formulate the theory of two-transverse-

dimensional modulation instability in partially incoherent nonlinear systems, and study

specific intriguing cases of broken symmetry between the two transverse dimensions. We

show that quasi-ordered stripes, rolls, lattices, and grid-like patterns can form

spontaneously from random noise in partially incoherent wave fronts in self-focusing

non-instantaneous media. We show that the cases of broken symmetries (e.g., stripes and

grids) can be generated by manipulating the correlation statistics of the incident wave

front and/or by having anisotropic noise. We emphasize that, in fully coherent systems,

the existence of features associated with broken symmetries is not surprising and has

been demonstrated before [10]. But in partially incoherent (that is, random-phase and

weakly correlated) systems, the very fact that anisotropy in the correlation statistics or in

the statistics of the noise causes symmetry breaking and determines the evolving patterns

is a new, exciting, and unique feature in the area of nonlinear dynamics and solitons.

61

We begin by considering a partially spatially incoherent optical beam propagating in the z

direction that has a spatial correlation distance much smaller than its temporal coherence

length; i.e., the beam is partially spatially incoherent and quasi-monochromatic, and the

wavelength of light λ is much smaller than either of these coherence lengths. The

nonlinear material has a non-instantaneous response; the nonlinear index change is a

function of the optical intensity, time averaged over the response time of the medium τ

that is much longer than the coherence time tc. Assuming the light is linearly polarized

and that its field is given by E(r,z,t) [r = (x,y) being the transverse Cartesian coordinate

vector], we can define the associated mutual coherence function

( ) ( ) ( )*1 2 2 1, , , , , ,B z E z t E z t=r r r r . The brackets denote the time average over time

period τ. By setting

( ) 1 21 2 1 2ˆ ˆ ˆ2 2

y yx xx y

r rr rr x r y x y2++ +

= = + = +r r

r

and

( ) ( ) ( )1 1 2ˆ ˆ ˆx y x x y y1 2 ˆx y r r x r rρ ρ= − = + = − + −2ρ r r y

as the midpoint and difference coordinates B(r,ρ,z) becomes the spatial correlation

function in the new system. Note that B(r, ρ=0,z) = I(r,z) = ( ) 2, ,E z tr where I(r,z) is

the time-averaged intensity. We emphasize that in this model only time-independent

perturbations can lead to MI; any rapid fluctuations will average out over the response

62

time of the material τ and have no significant bearing on the final result. From the

paraxial wave equation [4,11,12], we derive in (2+1)D an equation governing the

evolution of the correlation function, B(r,ρ,z),

2 2

20 , , , ,

2 2 2 2

x x y y

y yx xx y x y

B i Bz k r r

in n r r z n r r z Bk c

ρ ρ

ρ ρρ ρω

⎧ ⎫∂ ∂ ∂⎪ ⎪− +⎨ ⎬∂ ∂ ∂ ∂ ∂⎪ ⎪⎩ ⎭⎧ ⎫⎛ ⎞ ⎛⎪ ⎪⎛ ⎞= ∆ + + − ∆ − −⎨ ⎬⎜ ⎟ ⎜⎜ ⎟

⎝ ⎠ ⎪ ⎪⎝ ⎠ ⎝⎩ ⎭,

⎞⎟⎠

(0.18)

where ω is the carrier frequency of the light, k is the carrier wave vector, n0 is the index

of refraction of the material without illumination, and ∆n is the intensity-dependent

nonlinear addition to the index of refraction ( 0n n∆ ).

MI is manifested in the development of a small intensity perturbation on top of an

otherwise uniform beam. This can be expressed mathematically by taking

( ) ( ) (0 1, , , , )B z B B z= +r ρ ρ r ρ , where B0 is the uniform beam B1 is the perturbation to be

affected by MI, and 1 0B B . Substituting this latter form of B in Eq. (1) we obtain

63

( )

( )( )

2 21

1

120

0

1

, , 0, 0 ,2 2

,

, , 0, 0 ,2 2

x x y y

yxx y x y

yxx y x y

B i Bz k r r

B r r zin Bk c

B r r z

ρ ρ

ρρ ρ ρω κ ρ

ρρ ρ ρ

⎧ ⎫∂ ∂ ∂⎪ ⎪− +⎨ ⎬∂ ∂ ∂ ∂ ∂⎪ ⎪⎩ ⎭⎧ ⎫⎡ ⎤⎛ ⎞

+ + = =⎪ ⎪⎢ ⎥⎜ ⎟⎝ ⎠⎪ ⎪⎣ ⎦⎛ ⎞= ⎨ ⎬⎜ ⎟

⎝ ⎠ ⎡ ⎤⎛ ⎞⎪ ⎪− − − = =⎢ ⎥⎜ ⎟⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

(0.19)

where we have defined the marginal nonlinear index change evaluated at intensity I0, to

be ( )0I

d n I dIκ = ∆⎡ ⎤⎣ ⎦ . Equation (2) is linear in B1 and has translational invariance with

respect to r. Thus B1 can be investigated in terms of its plane-wave (Fourier) constituents,

i.e., B1 can be taken as proportional to ( )exp x x y yi r rα α⎡ ⎤+⎣ ⎦ , where 2x xα π= Λ and

2yα π= Λ y are the wave vectors of the oscillations, and are taken to be real. From the

structure of Eq. (2), we expect that perturbations will grow exponentially with

propagation distance z and so we assume B1 to be proportional to exp(Ωz), where Ω is the

growth rate of the MI at a particular set of spatial wave vectors ( αx , αy). In fact, B1 has

to be exponential in z because of the translational invariance of Eq. (2) in z. Note that B1

has no time dependence: any rapid perturbations will average out over the response time

of the material τ. Thus, we can write the eigenmodes of Eq. (2) as

( ) ( ) ( ) ( )

( ) ( )

*1

*

exp exp exp

exp ,

B z i L

i L

φ

φ

= Ω + + Ω⎡ ⎤⎣ ⎦

× − + −⎡ ⎤⎣ ⎦

α r ρ

α r ρ

z (0.20)

64

where φ is an arbitrary real phase, and L(ρ) are a set of modes that contain all the

dependence on ρ, and can be obtained for each ( αx , αy) [4]. These eigenmodes satisfy

B1(r,ρ,z) = B1*(r,-ρ,z), which is required from the definition of B(r,ρ,z) given above. By

introducing M(ρ) = L(ρ)/L(ρ=(0,0)) into Eq. (2) and integrating over z, we arrive at

( )

( ) ( )0

1

2 sin 0.2

x yx y

x x y y

Mk

M Bc

α αρ ρ

α ρ α ρωκ

⎧ ⎫∂ ∂⎪ ⎪Ω + +⎨ ⎬∂ ∂⎪ ⎪⎩ ⎭+⎛ ⎞

× + =⎜ ⎟⎝ ⎠

ρ

ρ ρ

(0.21)

Since growth can only occur for this form of the ansatz for 1B if Ω has a real component

greater than zero, we look for particular and homogeneous solutions to Eq. (4) For which

this is the case. Physically, for growing modes, the homogeneous solution must be zero

as must be bounded for large ( )M ρ ρ . By taking the Fourier transform of Eq. (4) we

find that

( )

( )

0 0

ˆ ,

ˆ ˆ, ,2 2 2 2

x y

x x y y

y yx xx y x y

i k cM k k i k kk

B k k B k k

ω

α α

α αα α

⎡ ⎤⎢ ⎥

= ×⎢ ⎥⎢ ⎥Ω − +⎣ ⎦

⎡ ⎤⎛ ⎞ ⎛+ + − − −⎢ ⎥⎜ ⎟ ⎜

⎝ ⎠ ⎝⎣ ⎦,

⎞⎟⎠

(0.22)

65

where ( ) ( ) ( )2ˆ 1 2 iF d F eπ∞ ∞

−∞ −∞= ∫ ∫ k ρk ρ ρ denotes the Fourier transform of . From

the definition of

( )F ρ

( )M ρ above, it can be seen that

( )( ) ( )( ) ( )( )0,0 0,0 0,0 1M L L= = = =ρ ρ ρ = . Hence we arrive at the constraint,

0 0

1

ˆ ˆ, ,2 2 2 2

x y

yx xx y x y

x x y y

dk dkc

B k k B k k

k ki

k

y

ωκ

α αα α

α α

∞ ∞

−∞ −∞= −

⎡ ⎤⎛ ⎞ ⎛+ + − − −⎢ ⎥⎜ ⎟ ⎜

⎝ ⎠ ⎝⎢ ⎥×+⎢ ⎥

Ω +⎢ ⎥⎢ ⎥⎣ ⎦

∫ ∫⎞⎟⎠

.

(0.23)

Here, stands for the Fourier transform of ( )0B k ( )0B ρ as expected, but note that this

function also physically represents the angular power distribution of the beam. This can

be seen by keeping in mind that ( ),x yk k k k=θ also represents the angle of

propagation, as long as kx and ky are small compared to k. Once a form is chosen for

, Eq. (6) uniquely determines the growth rate ( )0B k Ω as a function of the wave vector

( ,x y )α α and contains all the information about how quickly the MI will grow and which

spatial frequencies of perturbations will dominate.

We show now that if the radial symmetry in the transverse (x-y) plane is not broken,

either by the medium or by the beam itself, many parallels can be drawn between the

behaviors of the one- and two-transverse-dimensional systems. More specifically, the

relation between the one- and two-dimensional growth rates,

66

( ) ( ) ( )2 22 2,D x y D x y D1α α α αΩ = Ω + = Ω α , can be shown to be true for any case in

which the intensity of the beam is uniform and its correlation function is radially

symmetric and separable: ( ) ( ) ( ) ( )0 0 0 0ˆ ˆ ˆ ˆ,x y x yB k k B k B k B= = k . This separation is not

just for mathematical convenience, but in fact separable correlation functions do exist in

numerous physical settings. For example, transverse modulation instabilities of (1+1)D

solitons in a 3D bulk medium can be eliminated by making use of a separable correlation

function (although in that case the correlation function is also not radially symmetric)

[11]. This implies that both the magnitude of the spatial frequencies of maximum growth

and their corresponding growth rates must be identical in one-and two-transverse-

dimensional systems. This important conclusion can be proven by the following

argument. Since both the beam and the medium possess radial symmetry, the gain curve

can have no dependence on angular orientation and thus must be a function only of the

magnitude of α. Therefore, we may pick 0,y xα α α= = , and solve for the case 0α ≥

without loss of generality. Rewriting the constraint Eq. (6) using this form for and

these values for (

( )0B k

),x yα α , we see that

( )0

0 0

ˆ1

ˆ ˆ2 2

y y

x x

xx

dk B kc

B k B kdk ki

k

ωκ

α α

α

∞

−∞

∞

−∞

= −

⎡ ⎤⎛ ⎞ ⎛+ − −⎜ ⎟ ⎜⎞⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎢ ⎥×

⎢ ⎥Ω +⎢ ⎥⎣ ⎦

∫

∫. (0.24)

67

Now since is identical with respect to k( )0B k x and ky and normalized

, integration over k( ) ( )( )0 0ˆ. ., 0,0i e d B B I

∞ ∞

−∞ −∞

⎡ = = =⎢⎣ ∫ ∫ k k ρ 0⎤⎥⎦ y further reduces this

constraint to

0 0ˆ ˆ

2 2⎞⎟⎠1

x x

xx

B k B kdk kc i

k

α αωκ

α∞

−∞

⎡ ⎤⎛ ⎞ ⎛+ − −⎜ ⎟ ⎜⎢ ⎥⎝ ⎠ ⎝⎢ ⎥= −⎢ ⎥Ω +⎢ ⎥⎣ ⎦

∫ (0.25)

where is now the one-dimensional normalized angular power spectrum. This is

identical to that obtained in the (1+1)D case [4]. Therefore, since this equation gives the

gain curve

( )0B k

( )Ω α , the curve itself, and all quantities derived from it, the wave vector of

maximum growth αMAX must be the same in both the (1+1)D and the (2+1)D cases.

To better understand the behavior of two-dimensional incoherent MI, we now consider a

particular form of angular power spectrum, the double-Gaussian distribution,

( )22

00 2 2

0 0 0 0

ˆ , exp yxx y

x y x y

kI kB k kk k k kπ

⎡ ⎤⎛ ⎞= − +⎢ ⎥⎜⎜ ⎟⎟⎢ ⎥⎝ ⎠⎣ ⎦

, (0.26)

which is realizable experimentally. By numerically solving Eq. (6) for ( ),x yα αΩ , we

find that the results are exactly identical to those obtained in the (1+1)D case using one-

dimensional Gaussian statistics; i.e., the magnitude of the frequency of maximum growth

68

and the growth rate as a function of frequency are the same in both one and two

dimensions. These computations were performed using the coherent density approach

[13,14] that describes the propagation of incoherent light in media with a

noninstantaneous nonlinearity. In this model, infinitely many ‘‘coherent components’’

propagate at all possible angles [i.e., values of the wave vector (kx ,ky)] and interact with

one another through the nonlinearity that is a function of the time-averaged intensity. The

shapes of the initial intensity profile for each of these coherent components are the same,

but the relative weights are given by the angular power spectrum of the source beam,

which is , the Fourier transform of the correlation function. The nonlinear change

in the refractive index is taken to be saturable and of the form

( )0B k

( )1MAX N Nn n I I∆ = ∆ +⎡ ⎤⎣ ⎦ ,

where ∆nMAX is the maximum nonlinear index change possible and N SATI I I= , ISAT is

the saturation intensity of the material.

Our numerical simulations (Fig. 1) confirm the analytic conclusion: the spatial frequency

of maximum growth and its rate of growth are the same in (1+1)D and (2+1)D systems,

provided that the nonlinearity, seed noise and the spatial correlation function are all fully

isotropic. The (1+1)D case, [Fig. 1(a)] reveals strong peaks (the spatial frequency of

maximum growth) occurring at 0.0350k =α , in accordance with the analytic theory.

The (2+1)D case contains a ring of wave vectors [a side slice of which is shown in Fig.

1(b)] at 0.0350k =α , exactly the same magnitude as in the (1+1)D case. The

parameters chosen were n0 = 2.3, λ = 0.5 mm, k = 28.903 µm-1, ∆nMAX = 5 x 10-3, and

( )0 0 0 13.85x x yk k mradθ θ≡ = = , which are representative of typical values in biased

69

photorefractives. The input wave front was taken to be a very broad (~500 µm), flat beam

of height 1 in normalized units [with radial symmetry in the (2+1)D case], seeded with

random Gaussian white noise [15] at a level of 10-5. In both cases, the beams were

allowed to propagate for 1.2 mm, and the intensity of the background beam was 1 in

normalized units. As predicted by the theory, numerics confirm that the one- and two-

dimensional cases grow at the same rates and at the same spatial frequencies. If the

system is fully isotropic, that is, if the nonlinearity, input beam (both in its input intensity

distribution and in its correlation function), and the noise, are all fully isotropic, then the

(1+1)D case is fully equivalent to the (2+1)D case.

To conclude the section dealing with incoherent MI of input beams with isotropic

properties (correlation function and seed noise), in fully isotropic nonlinear media, we

emphasize that, because (2+1)D incoherent MI has no preference whatsoever with respect

to any directionality in the transverse plane [as manifested by Eqs. (5)–(8)], the resultant

patterns such as 1D stripes, 2D square lattices, and 2D triangular lattices, etc., all have the

same growth rate and MI threshold. In other words, the system as it is does not

differentiate between such patterns. This could lead to a naive conclusion that all possible

states of this system are equally likely to occur. But this conclusion is wrong: our

simulations clearly indicate that, in spite of the fact that all possible 2D patterns in a fully

isotropic system have the same threshold for incoherent MI, some patterns are more

likely to emerge than others. The reason for that is statistical: the likelihood for the

emergence of filaments of a random distribution in space (for which the distribution in

Fourier space is isotropic) is much greater than the likelihood of stripes (for which the

70

peaks in Fourier space are lined up in some direction). Equally important, we note that

our analytic calculation relies on a linearized stability analyis. After a long enough

propagation distance, when the perturbations gain sufficiently high amplitudes, we expect

that they will compete with one another, and some patterns will prevail over others, even

Figure 1. Comparison between the angular power spectra of the features resulting from

incoherent modulation instability in the (1+1)D (a) and in the (2+1)D (b) cases, for a

beam with the input power spectrum of Eq. (9) with 0 13.85x mradθ = . The beam was

propagated for 1.2 mm in a material with a saturable nonlinearity ( )1MAXn n I I∆ = ∆ +⎡ ⎤⎣ ⎦ ,

where ∆nMAX = 5.0 x 10-3. The parameters used in both cases were identical, and only the

71

number of spatial dimensions was varied. The figure shows the power spectrum (in

arbitrary units) as a function of the transverse wave vector α normalized to the wave

vector of the light k. The uniform incoherent background intensity has been subtracted

out so that the statistics of the perturbations alone is shown. In (b) the results are radially

symmetric and we show a representative slice through the plane αy = 0.

72

if both have initially the same gain. In fact, our simulations reveal just that: some 2D

structures emerge and others do not, even though they initially have the same gain.

2.4 Modulation instability with anisotropic correlation function

Next we consider a case where the correlation function 0B is anisotropic, that is, the

radial symmetry in the correlation statistics is broken: 0x 0 yθ θ≠ with the noise remaining

fully isotropic. We will show that the extra spatial dimension allows for complex

behaviors with no counterpart whatsoever in a one-dimensional system. In one

dimension, it has been established that for sufficiently incoherent wave fronts, MI is

totally suppressed [4]. In a 2D system with 0x 0 yθ θ≠ one may ask, what kind of features

will emerge if the gain of the spatial frequencies in one direction is above the MI

threshold, while the gain for those spatial frequencies in the other transverse direction are

below threshold. To answer such questions, we must first derive constraints governing

the onset of MI. Although the difference in behaviors above and below the threshold is

very marked (MI either occurs or it does not) the transition between the two regimes is

continuous, and so it must be that at this threshold both the gain Ω and its derivative

dΩ/d|α| are zero when |α| = 0 [4]. Let us first consider the threshold for MI to occur in

the x direction, and set αy = 0. For small values of αx, Eq. (6) becomes (to first order in

αx)

73

'

' '

0'

2 2

' '2

0 0

1

ˆ

2

x x

x x

x y

xx k k

x xx

x x

dk dkc

Bk

kik

α α

x

ωκ

α

α ααα α

∞ ∞

−∞ −∞

=

= =

= −

⎡ ⎤∂⎢ ⎥

⎢ ⎥∂⎢ ⎥×⎢ ⎥⎛ ⎞∂Ω ∂ Ω⎢ ⎥⎜ ⎟Ω + + +⎢ ⎥⎜ ⎟∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦

∫ ∫

, (0.27)

which reduces to

'

0'

ˆ11x x

x yx x k k

Bdk dkc k

ωκ ∞ ∞

−∞ −∞ k=

∂= −

∂∫ ∫ . (0.28)

Equation (11) can be solved exactly for kx0 , the threshold width of the angular power

spectrum, for any form of the angular power spectrum ( )0B k . Choosing the same

double-Gaussian form as above [Eq. (9)], we find that MI will occur in the x direction if

2

0 00 22

xx threshold

n kn Ik

κ−∆ ≡ ≥ ; (0.29)

thus, if the nonlinearly induced index change ∆n exceeds the threshold value on the right-

hand side of Eq. (12), then MI will form stripes with periodicities (spatial frequencies)

along the x direction. Since the initial constraint Eq. (6) is unchanged by interchanging kx

and ky, it follows that y-direction MI must also be subject to a similar inequality,

74

2

0 00 22

yy threshold

n kn I

kκ−∆ ≡ ≥ (0.30)

Although Eqs. (13) and (14) are identical functions with respect to kx0 and ky0 , there is no

reason that the actual threshold values must be the same. It is, therefore, possible that if,

for example, the beam is more coherent along the y direction than along the x direction,

only MI with y directionality will occur. To test this analytic prediction, we use the

coherent density approach [13,14] to simulate the propagation of a beam with

‘‘elliptical’’ double-Gaussian statistics, as in Eq. (9). The initial beam is more coherent in

the y direction, with 0 0 2.2y yk k mradθ = = , but much more widely distributed in the x

direction ( 0 0 9.6x xk k mradθ = = ). In the simulation, the input beam is a very wide

(~500

µm), flat, and radially symmetric wave front of intensity 1 in normalized units, with

random Gaussian white noise added at a level of 10-5. The beam is propagated for 1 mm

in a material with a Kerr-type nonlinearity of the form 0 NL Nn n n I= + ∆ , where n0 = 2.3

and ∆nNL = 5 x 10-4. We find that the extra incoherence in the x direction inhibits the MI,

as expected, and that the formation of stripes occurs preferentially in the more coherent y-

direction. These results are presented in Fig. 2, where the emergence of MI in y and not in

x is manifested in both the development of the spatial intensity fluctuations [Fig. 2(a)]

and in the corresponding Fourier spectra [Fig. 2(b)]. Figure 2(b) shows that a narrow

band of wave vectors dominates the pattern formation process with significant MI

occurring only for a very limited range of values for αy /k (~0.03).

75

Figure 2. Features resulting from incoherent modulation instability for an input beam of

an elliptical double-Gaussian angular power spectrum [Eq. (9)] with 0 9.6x mradθ = and

. The beam was propagated for 1 mm in material with a refractive index

of the form , where n

0 2.2y mradθ =

0 NLn n n I= + ∆ 0 = 2.3 and ∆nNL = 5 x 10-4. (a) shows the intensity

of the perturbations, ( ) 21B r , in the spatial domain, with high intensity represented by

white shading, low by dark. (b) shows the corresponding angular power spectrum

( )2

1B α , where the uniform background intensity has been subtracted out.

76

While the example of elliptical double-Gaussian correlation statistics begins to illustrate

some of the variety that an extra spatial dimension can introduce, other forms for ( )0B k

can lead to even more complex and completely different patterns. One interesting case

that happens to be exactly solvable analytically is that of a partially incoherent optical

beam with an angular power spectrum in the form of a double Lorentzian distribution

( ) ( )( )0 0 0

2 2 2 2 20 0

ˆ , x yx y

x x y y

I k kB k k

k k k kπ=

+ + , (0.31)

which, while identical along the x and y directions, lacks radial symmetry and is narrower

along the +45° directions than along the 0° and 90° directions in the transverse plane.

From this insight, one may naively expect that MI will appear first along the +45°

directions. But, in this case intuition is misleading. Using this particular form for ( )0B k

in Eq. (6), one can then obtain exactly the gain curve ( ),x yα αΩ that is,

( ) 0 0

1/ 22 22 2 0

20

,

4

yxx y x y

x yx y

k kk k

In k

ααα α

α ακα α

Ω = − −

⎛ ⎞++ + −⎜ ⎟⎜ ⎟

⎝ ⎠

. (0.32)

Equation (0.32) predicts that the strongest gain will occur along the 0° and 90° directions,

and not along the 45° axis (as might be naively expected). Solving for the thresholds

along the 0° (αy= 0) and 45° directions and provided that (αx = αy = k0), we find,

77

20 0

0 245

20 0

0 20 ,90

2 ando

o o

threshold

threshold

n kn Ik

n kn Ik

κ

κ

−

−

∆ ≡ ≥

∆ ≡ ≥

(0.33)

Thus, the threshold value for the nonlinear index change ∆nthreshold is indeed lower along

the 0° and 90° directions than along those tilted by 45°, even though the angular power

spectrum is wider along the 0° and 90° directions than along the 45° tilted directions. In

other words, in this intriguing example the MI grows fastest along the directions with the

widest angular distribution of power, since it has the lowest threshold, and the winner

takes it all. Such a phenomenon has no analog in (1+1)D, where widening the angular

power spectrum always decreases MI growth [4]. Just why MI grows first along the

directions with a wider distribution in k space (despite the intuition drawn from the 1D

case) can be understood by considering the Fourier transform of the angular power

spectrum, that is, the correlation function, ( )0 , ,B zr ρ . In general, in a 1D transform, a

wider distribution in k space has a narrower distribution in r space, but in 2D this is not

always the case and the actual geometry must be considered. In fact, the Fourier

transform of a 2D double Lorentzian spectrum is broadest in r space (real space) along

the same directions it is broadest in k space. So in fact, the beam is most strongly

correlated along the 0° and 90° directions, even though these are the directions along

which the angular power spectrum is the widest. From this example, it is apparent that MI

grows preferentially in the most strongly correlated direction, and that this may or may

not correspond to the direction with the most angularly concentrated distribution of the

power.

78

We confirm these results using numerical simulations in the same material and beam

parameters described above but with a saturable nonlinearity, ( 0 , ,n kλ )

( )1MAXn n I I∆ = ∆ +⎡⎣ ⎤⎦

θ= = .

, with ∆nMAX = 1.8 x 10-3. The input is a very broad (800 µm), flat

wave front in the spatial domain, seeded with random Gaussian white noise at a level of

10-5, with the degree of incoherence set by 0 0θ The results after 6 mm of

propagation are shown in Figs. 3(a) and 3(b); the axis has been tilted by 20° to isolate any

boundary artifacts of using a square grid to store data points. The result is just as

predicted: MI occurs only on the 0° and 90° degree directions, as is evident in Fig. 3(b)

that shows the power Fourier spectrum of the intensity perturbations,

12x y mrad

( )2

1B α .

79

Figure 3. Features resulting from incoherent modulation instability for an input beam of a

double-Lorentzian angular power spectrum [Eq. (14)]. The beam was propagated for 6

mm in a material with a saturable nonlinearity of the form ( )1MAXn n I I∆ = ∆ +⎡ ⎤⎣ ⎦ , with

∆nMAX = 1.8 x 10-3. (a) shows the intensity of the perturbations, ( ) 21B r for

0 0 12x y mradθ θ= = . The magnitude of the intensity is represented by the shading of the

figure, with white representing maxima and black minima. (b) shows the corresponding

angular power spectrum ( )2

1B α . (c) shows ( ) 21B r for 0 12x mradθ = and

. (d) shows 0 6y mradθ = ( )2

1B α .

80

The perturbations that experience highest gain occur near |α|/k ~ 0.0125, which compares

well with 0.01, the value predicted by Eq. (15). We attribute this slightly lower value of

the wave vector to neglecting higher-order terms in Eq. (2), where the intensity-

dependent change in ∆n, was approximated as ( )0

0 0II d n I dI Iκ = ∆⎡ ⎤⎣ ⎦ × . This result is

exact in the Kerr case (∆n = ∆nNLI), but for saturable nonlinearities, which is what we use

in our simulation (and which is also encountered in experiments, otherwise the patterns

emerging from the MI are unstable), the approximation introduces a small error in finding

the spatial frequency that grows fastest. Going back to the spatial domain, the emerging

MI pattern is manifested as a grid of localized wave packets; overlapping the stripes in

the x direction with those in the y direction results in increased intensity at the

intersections of the grid.

To further explore the emergence of incoherent MI when the correlation statistics are

anisotropic, we again use the form of the angular power spectrum used in Fig. 3, but

distort it so that the correlation function of the beam is stretched in one direction with

respect to the other. In this particular example, the angular widths are 0 12x mradθ = and

, while all other parameters are kept the same as in the previous example (of

Fig. 3(a) and (b), where . In this case, Eq. (15) predicts the formation

of strong peaks in the Fourier domain on the y axis at the 90° and 270° marks at |α|/k =

0.016. As Figs. 3(c) and 3(d) show, the numerical simulations confirm the analytic

prediction: significant MI forms only in the y direction near |α|/k ~ 0.017 (again, as

above, the axes have been tilted by 20° to isolate artifacts of using a square grid to store

0 6y mradθ =

0 0 12x y mradθ θ= =

81

data points). Note, that in the spatial domain the patterns appear similar to those produced

using elliptical statistics, but the Fourier analysis, depicted in Fig. 3(d), reveals that the

range of frequencies present is actually much narrower, with little spread in either the x or

y directions, and thus stronger striping is seen overall. The strong intensity stripes seen

here are very similar to patterns observed experimentally in photorefractive

crystals [5,8].

Until this point, we have only used input beams that are inherently asymmetric in their

correlation statistics (coherence properties) to produce symmetry breaking. But, it is

legitimate to ask: Can a beam that is radially symmetric and of perfectly isotropic

coherence properties give rise to anisotropic MI, that is, to spontaneous formation of

patterns that lack radial symmetry? For example, can such a fully radially symmetric

beam transform into stripes or another geometrically ordered grid-type state? The answer

lies in the propagation dynamics. Obviously, asymmetry or anisotropy in the nonlinear

medium can give rise to such phenomena, as is the case for the photorefractive

nonlinearity and for nonlinearities in liquid crystals. But there exists another alternative

that is actually much more interesting: asymmetry can exist in the noise that seeds the MI

process. For example, inorganic photorefractive crystals have striations that appear in the

form of planes of index inhomogeneities. As as result, random variations in the index of

refraction (noise) are much greater along the direction normal to these planes. We

investigate this phenomenon of pattern formation from incoherent MI in the presence of

anisotropic noise by propagating a perfectly isotropic wave front with a radially

symmetric angular power spectrum in a medium with broken symmetry. To model these

82

kind of irregularities, we seed our initial input to the numerical simulations with

predominantly one-dimensional noise that fluctuates strongly in the y direction, while

remaining almost constant across the x direction [16]. The medium had a saturable

nonlinearity of the form ( )1MAXn n I I∆ = ∆ +⎡ ⎤⎣ ⎦ , where ∆nMAX = 3.3 x 10-3; other

parameters were the same as in the simulation shown in Fig. 1. The angular power

spectrum of the beam was a double Gaussian, as in Eq. (9), with 0 0 2.2x y mradθ θ= = .

The results are shown in Figs. 4(a) and 4(b), after 2 mm of propagation and we see that

indeed the random yet anisotropic noise breaks the symmetry and gives rise to stripes in

the preferential direction. The angular power spectrum of the perturbation ( ) 21B α

shown in Fig. 4(b), reveals that the mechanism behind the symmetry breaking process

that leads to striping is different from both the methods to produce striping studied above.

It is apparent that there is an overall background of fluctuations as in the radially

symmetric case, but the MI is dominated by a very strong preferential growth of stripes at

x = 0, with

spreading in the y direction.

Before closing, we wish to note two generic results that links 2D to 1D systems. (I)

Whenever the input beam has correlation statistics that are separable and radially

symmetric as ( ) ( ) ( ) ( )0 0 0 0ˆ ˆ ˆ ˆ,x y x yB k k B k B k B= = k , and the nonlinearity and the noise are

fully isotropic, the features of 2D incoherent MI exactly reproduce those of 1D

incoherent MI. The 2D system relates to the 1D system in a straightforward manner: the

83

MI threshold and the growth rates are identical. (II) Whenever the input beam has

correlation statistics that are separable but are not radially symmetric, such as

84

Figure 4. Features resulting from incoherent modulation instability for an input beam

with a radially symmetric Gaussian angular power spectrum (with ),

but with preferential (white) noise in the y direction that is 10

0 0 2.2x y mradθ θ= =

2 times stronger than the

noise in the x direction. (a) shows the spatial distribution of the perturbation, ( ) 21B r ,

after 2 mm of propagation in a material with a saturable nonlinearity of the form

, where ∆n( )/ 1MAXn n I I∆ = ∆ +⎡⎣ ⎤⎦ MAX = 3.3 x 10-3, and (b) shows the corresponding

angular power spectrum, ( )2

1B α .

85

( ) ( ) ( ) ( )0 0 0 0ˆ ˆ ˆ ˆ,x y x yB k k B k B k B= ≠ k , and the nonlinearity and the noise are fully

isotropic, the features of 2D incoherent MI can be mapped onto two independent 1D

systems, corresponding to the two transverse dimensions, each of which having its own

properties, such as MI threshold, growth rates, the spatial frequency off maximum

growth, etc. We note, however, that these types of beams account for only a subset of all

possible physical cases and the set of experiments that can be performed with two-

dimensional partially incoherent beams. In fact, in many cases, either the correlation

statistics are not separable, or anisotropic noise introduces directional preferences.

2.5 Conclusion

We have provided in this chapter and in the paper we have published on the subject, (S.

M. Sears et al, (2002) [18]), an analytical framework for studying (2+1)D incoherent MI,

and using numerical simulations, we have shown that the predictions of this theory are

accurate. However, we may ask what will happen as the MI continues to grow. Are the

patterns that evolve from incoherent MI stable or will they develop into something

different? Or will they break apart? Initially the perturbations are only a very small

sinusoidal wave on top of an otherwise uniform background. But perturbations that grow

exponentially as the beam propagates must eventually reach the same order of magnitude

as the background intensity ( )0B ρ and the linearization assumption 1 0B B can no

longer hold. Earlier works [5,7,8] have shown that as the MI grows large, the character of

the dynamics changes, resulting in a transition from sinusoids on top of a constant

background to individual, localized wave packets of increasing height. The onset of this

behavior can be seen in Fig. 4 above; the long thin ripples gradually become punctuated

86

by small round peaks arranged in a gridlike structure; such grids have been observed

experimentally [5,8]. The subsequent evolution of the system now depends on the nature

of the nonlinearity and on the correlation statistics of the beam. In nonlinear Kerr media,

the localized isolated peaks continue to grow in height and become narrower until a

‘‘collapse’’ occurs [17]. The system’s long-range evolution is completely different in

saturable nonlinearities, where the isolated intensity peaks stabilize and remain mostly

unchanged in shape by further propagation. The fact that incoherent MI in 2D saturable

systems leads to a grid of isolated intensity peaks might by naively mistaken to be

thought as a grid of localized islands of coherent, that is, possibly each isolated wave

packet is an individual fully coherent (or fully correlated) entity. However, this is not the

case: each of these isolated wave packets is still partially incoherent, albeit being slightly

more coherent than the uniform beam that initiated them. Furthermore, the separation

between two adjacent isolated wave packets is several times larger than the correlation

distance. In the limit where this distance is not too large, long-range attraction forces

between these localized wave packets lead to clustering of solitons, as was recently

demonstrated experimentally and theoretically [7]. This means that the correlation

statistics play a crucial role not only in determining the MI threshold and the dominating

spatial frequencies, but also in determining the long-range evolution of the emerging

patterns. This subject is described elsewhere [7], but for completeness, we briefly discuss

the main ideas. When the initial beam is fully coherent and the nonlinearity is saturable, a

stable grid of localized wave packets emerges. These wave packets propagate without

further change in their width, i.e., they behave like quasisolitons. The interactions among

these solitons are coherent, therefore, the interaction forces between adjacent localized

87

wave packets can be either attractive or repelling, depending upon the phase between

them. Coherent MI, however, always produces features (quasisolitons) that are π out of

phase with one another, thus, the dominating force between adjacent solitons is always

repulsive. This leads to a grid of evenly spaced localized wave packets [5]. However, if

the initial beam is sufficiently incoherent leading to incoherent MI, the phase-dependent

interactions between and among the ‘‘MI products’’ (the localized isolated wave packets)

that result from interference terms average out and only a net attractive force among these

solitons survives. As a result, the solitons begin to draw nearer to their neighbors and

cluster in aggregates of fine-scale structures: clusters of solitons [7].

In summary, we have studied theoretically modulation instability in (2+1)D partially

spatially incoherent systems. Our study reveals different and interesting dynamics that do

not exist in (1+1)D incoherent systems. In particular, we observe the ordering of the MI

perturbations into stripes and grid-like features, which occurs if the symmetry of the

system is broken is some manner. Some of these interesting dynamics of pattern

formation from incoherent modulation instability have already been demonstrated

experimentally in Refs. [5], [7], [8], but many other features are yet to be observed.

Furthermore, such behavior should be observable in other natural systems, since solitons,

MI, and incoherence are phenomena universal to many nonlinear systems. The discovery

of incoherent MI has implications for many other nonlinear systems beyond optics. It

implies that patterns can form spontaneously (from noise) in nonlinear many-body

systems involving weakly correlated particles, such as, atomic gases at (or slightly above)

the Bose-Einstein-Condensation temperatures.

88

2.6 References

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(1980); K. Tai, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1981). For a

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89

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90

[15] White noise was added in the frequency domain by adding a random number chosen

from a Gaussian distribution separately to both the real and imaginary parts of each

Fourier component. The width of the Gaussian distribution was chosen so that the

average power added by the noise would be some small fraction of the total power, in our

case 10-5.

[16] Unlike the previous simulations, here we also added noise in the spatial domain. A

random number chosen from a Gaussian distribution was added separately to the real and

imaginary parts of each component of the spatial profile. The noise was added first being

held constant in the y direction; then another layer was added being held constant in the x

direction. The ratio of the two layers of noise was 10-2. An equal amount of noise was

then added in the Fourier domain, consistent with the method used in Ref. [15]. The total

power in the noise was 10-5 of the total.

[17] N. N. Akhmediev, Opt. Quantum Electron 30, 535 (1998).

[18] S. M. Sears, M. Soljacic, D. N. Christodoulides and M. Segev, Pattern formation via

symmetry breaking in nonlinear weakly correlated systems, Physical Review E 65, 36620

(2002).

91

3 Clustering of Solitons in Weakly Correlated

Wavefronts

We demonstrate the spontaneous clustering of solitons in partially coherent wavefronts during the final

stages of pattern formation initiated by modulation instability and noise [37]. Experimental observations

are in agreement with theoretical predictions and are confirmed using numerical simulations.

3.1 Universality of clustering phenomena

Clustering, or the gross-scale aggregation of fine-scale structures, has been observed

in many diverse physical systems: from galactic clusters [1] to molecular aggregates [2],

from self-assembled quantum dots [3] to biological systems [4], just to name a few.

Despite the great variety of physical systems in which these clustering phenomena occur,

the underlying processes are fundamentally similar in several ways. This similarity can

be better understood by considering the following arguments: (1) the fine structure results

from the equilibrium of opposing effects or forces, and (2) the cluster forms because of

attraction between these individual "fine scale elements". For example, a protein

molecule may be made up by more than one polypeptide chain. In the case of

hemoglobin, four separate polypeptide chains, or subunits, are clustered together (held

together by van der Waals and ionic forces). In astrophysics, gravitational attraction is

known to lead to the formation of galaxies and galactic clusters. Given the universality of

these processes, one may be able to study clustering of fine scale elements in a number of

completely different physical systems. In fact, propositions were recently made to use

Bose-Einstein condensates to simulate galactic environments [5]. It would be very

92

interesting if similar dynamics, namely, the clustering of fine-scale elements, could be

observed in optical settings where the ensemble interaction forces can be varied at will

and the underlying theory is well understood.

3.2 Clustering of optical spatial solitons

Here and in our paper [37], we report the first experimental observation of clustering of

optical solitons. The clustering of solitons occurs spontaneously, when a partially-

incoherent optical wavefront disintegrates in a non-instantaneous nonlinear medium with

a large enough self-focusing nonlinearity. This process is initiated by noise-driven

modulation instability (MI), which in turn leads to the formation of soliton-like self-

trapped filaments. These solitonic filaments, tend to attract one another, eventually

leading to the formation of clusters of solitons. The incoherence of the wavefront (which

can be varied in a controlled manner), along with the non-instantaneous nature of the

nonlinearity, give rise to attractive forces between the solitonic filaments involved. The

experimental results are in agreement with theoretical predictions and are confirmed

using numerical simulations.

To further elaborate on these ideas, we introduce some aspects of solitons, and in

particular, the ideas underlying incoherent or random-phase solitons. Other relevant

topics, such as those pertaining to the recent discoveries of modulation instability and

pattern formation in incoherent (or weakly-correlated) nonlinear wave systems, will also

be discussed.

93

3.3 Solitons

3.3.1 A review of some basics

Solitons are stationary localized wave-packets that travel "without change of shape

or diminution of speed" in dispersive nonlinear wave systems [6]. Here we use the term

"soliton" to denote any solitary wave-packet, i.e. in the broader definition of the word

that includes self-trapped solutions of non-integrable systems [7]. Solitons share many

features with real particles: for example, their total energy and momentum are conserved

even when they interact with one another. In addition, solitons retain their shape and

identity after a collision event. Thus far, solitons have been predicted and their existence

has been demonstrated in many physical systems. Such examples include surface solitary

waves in shallow water [6], plasma solitons [8] and sound waves in 3He [9], short

temporal soliton pulses in fibers [10], and optical spatial solitons [11,12]. In spite of this

diversity, the main principles behind soliton formation and soliton interactions are the

same. Intuitively, solitons form when the broadening tendency of diffraction (or

dispersion) is balanced by nonlinear self-focusing. Until 1990, most research on optical

solitons concentrated on trapping in a single dimension. Examples of such one-

dimensional self-trapped wave-packets are temporal fiber solitons [10] and spatial

solitons [13] in slab waveguides. In general two-dimensional bright solitons in Kerr

media are known to be highly unstable and undergo catastrophic collapse [14]. However,

in the past decade, major progress has been made with solitons in saturable nonlinear

media where stable solitons of higher dimensionality can be generated. These include

94

two-dimensional spatial solitons in bulk media [12], spatio-temporal solitons that can be

self-trapped in one dimension in space and in time [15], and even “optical bullets” which

are self-trapped in both transverse spatial dimensions and in time simultaneously [16].

This general view of solitons being the result of a balance between diffraction/dispersion

and self-focusing also holds in all of these cases of a higher dimensionality provided that

the wavepacket exhibits stable self-trapped propagation. Another way to understand

soliton formation comes from the so-called self-consistency principle: this idea implies

that a soliton forms when a localized wave-packet induces (via the nonlinearity) a

waveguide and in turn is "captured" in it, thus becoming a bound state in its own induced

potential [17]. In the spatial domain of optics, a soliton results when a very narrow

optical beam induces, via self-focusing, a waveguide structure and guides itself in it.

Thus, interactions (collisions) between solitons can be viewed as interactions between

bound states of a jointly-induced potential well, or between bound states of different

wells located at close proximity [12]. In non-integrable systems (such as those with

saturable nonlinearities), interactions between solitons exhibit very rich behavior

compared to those in integrable [12].

3.3.2 Incoherent solitons

Relevant to our discussion is the class of incoherent solitons. For decades, solitons

were believed to be solely coherent entities. This perception changed, however, just a few

years ago, when partially-spatially-incoherent solitons were first observed in 1996 [18].

Observations of temporally and spatially incoherent ("white") light solitons [19] followed

soon thereafter. These experiments proved that indeed solitons made of random-phase (or

95

incoherent) wavepackets can exist. As a result, entirely new directions in soliton science

have opened up. Shortly thereafter, the theory of incoherent solitons was developed [20-

23] and dark incoherent solitons were observed [24]. Further studies considered their

interactions [25], their stability properties [26], and their relation to multimode composite

solitons [27]. Crucial to the existence of incoherent solitons is the non-instantaneous

nature of the nonlinearity, which responds only to the time-averaged intensity structure of

the beam, rather than to the instantaneous, highly speckled and fragmented, wave-front.

In other words, the response time of the nonlinear medium must be much longer than the

average time of phase fluctuations across the beam. Through the nonlinearity, the time-

averaged intensity induces a multimode waveguide structure (a potential well that can

bind many states), whose guided modes are populated by the optical field with its

instantaneous speckled structure.

3.3.3 Modulation Instability

Central to our discussion is the concept of modulation instability (MI) and its

occurrence in random-phase (or incoherent) systems. MI is a universal process that

appears in most nonlinear wave systems in nature. MI causes small amplitude and phase

perturbations (from noise) to grow rapidly under the combined effects of nonlinearity and

dispersion/diffraction. As a result, uniform excitations (such as broad optical beams in the

spatial domain in optics or quasi-CW pulses in the temporal domain) tend to disintegrate

during propagation [28,29], leading to filamentation or break-up into pulse trains. The

relation between MI and solitons is best illustrated by the fact that the filaments (or the

pulse trains) that emerge from the MI process are actually trains of almost ideal solitons.

96

MI can therefore be considered as a precursor to soliton formation [30]. Over the years,

MI has been systematically investigated in connection with numerous nonlinear

processes; yet it was always believed that MI is inherently a coherent process and can

only appear in nonlinear systems with a perfect degree of spatial/temporal coherence.

Recent theoretical and experimental studies [31,32] have shown that MI can also occur in

partially-incoherent (or random phase) wavefronts, and have demonstrated that, even in

such a system of weakly-correlated “particles”, patterns can form spontaneously.

However, such incoherent MI appears only if the ‘strength’ of the nonlinearity exceeds a

well-defined threshold that depends on the coherence properties (correlation distance) of

the wavefront. The discovery of incoherent MI has implications for many other nonlinear

systems beyond optics. It implies that patterns can form spontaneously (from noise) in

nonlinear many-body systems involving weakly-correlated particles, such as, for

example, atomic gases at (or slightly above) the Bose-Einstein-Condensation (BEC)

temperatures [32].

3.4 Clustering – theory and simulations

In light of the above, one may wonder how the solitonic filaments emerging from

the MI and breakup of a partially coherent yet uniform wave-front will ultimately behave.

In fully coherent systems with saturable nonlinearities, such solitary filaments are stable

and interact in the same manner as solitons: they may either attract or repel one another,

depending on their relative phase. As a result, the filaments arising from MI in coherent

(saturable) systems do not cluster together; instead, the presence of repulsive forces leads

to almost evenly-spaced solitons in a quasi-ordered lattice structure [33]. On the other

97

hand, in incoherent self-focusing systems soliton interactions over scales larger than the

correlation length are always attractive [34]. This is because the relative phase between

adjacent solitons varies much faster than the response time of the nonlinear medium

(recall that the non-instantaneous nature of the nonlinearity is a prerequisite for the

formation of incoherent solitons and incoherent MI). Thus, when two incoherent solitons

are brought to close proximity, their intensities add in the center region between them,

leading to an increase in the refractive index. This, in turn, attracts more light to the

center, moving the centroid of each soliton towards it and hence the solitons appear to

attract one another [12].

To analyze this process theoretically we employ the coherent density approach [20]

that describes the propagation dynamics of partially spatially-incoherent (quasi-

monochromatic) optical beams in non-instantaneous nonlinear media. For the

propagation medium, we choose a saturable nonlinearity of the type ( ) ( )1o S

S

n I In II I

∆∆ =

+

where n is the nonlinear change in the refractive index as a function of the total intensity

I, n0 is the maximum change in the refractive index, and IS is the saturation intensity (a

constant factor indicating the degree of saturation). This specific form of nonlinearity

represents the true nonlinear response of a homogeneously-broadened 2-level system at

the vicinity of an electronic transition, and, to a reasonable approximation, it also

represents the photorefractive screening nonlinearity [35]. We study clustering in a (2+1)

D system, in which the optical beam propagates along the z-direction and undergoes

diffraction or self-trapping in two transverse (x and y) dimensions. In such a system it is

essential to employ a saturable nonlinearity (e.g., the nonlinearity employed in our

98

experiment). Otherwise, all self-focusing effects that start from a beam with power

exceeding a particular value (the critical power), lead to catastrophic collapse, in which

the self-focusing processes never stabilize into a 2D filament [14].

In the coherent density approach, the propagation of incoherent light in slow

responding nonlinear media (such as biased photorefractive crystals) is described by

superimposing infinitely many “coherent components” or quasi-particles all interacting

via the nonlinearity. In this picture the initial relative weights of these components are

given by the angular power spectrum of the source beam, which is physically the Fourier

transform of the correlation function. Mathematically this is modeled using the coherent

density function, f , from which one can obtain both the intensity and the correlation

function of a partially coherent beam during propagation. The coherent density function

is governed by the following integro-differential equation [20]:

( )( )2 2

02 2

1 , , 02x y N

f f f f fi kz x y k x y

θ θ⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂

+ + + + + =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠g I x y z f ,

where

( ) 2, , ,N x y x yI f x y d dθ θ θ= ∫∫ θ

and, at z = 0,

( ) ( ) ( )1/ 200, , , , , ,x y x yNf z x y x yGθ θ θ θ φ= = .

99

In the equation above, xθ and yθ are angles (in radians) with respect to the z axis, k =

kono, and 0 2k 0π λ= . The function ( ),, , x yf x y θ θ is a band-limited function, and is of

negligible amplitude outside of the narrow paraxial angular range. Thus, even though the

integration is formally over all transverse momentum space, i.e., k-space, the only

contributing range is an angular range of ≈ ±0.1 radians. ( ),N x yG θ θ is the normalized

angular power spectrum of the incoherent source, and ( )0 ,x yφ is the wavefront’s input

spatial modulation function. IN = I/IS and g(IN(x,y,z)) represents the intensity dependence

of the nonlinearity given by ( )2 2 2o o Nn n n g I= + . As previously mentioned, here the

nonlinearity is taken to be of the form ( ) ( )1N o N Ng I n I I= ∆ + , (saturable non-linearity).

In our simulations the linear index is no = 2.3, and the maximum nonlinear index change

is taken to be (which is roughly the maximum attainable index change in

inorganic photorefractive crystals). The wavelength of the light source is λo = 0.488 µm,

and thus k = 29.613 µm

32.5 10n −∆ = ×

-1, and ko = 12.875 µm-1. The angular power spectrum is assumed

to be of the Gaussian type ( ) ( ) ( )( )2 2 2, 1 expN x y o x y oG 2θ θ πθ θ θ= − + θ and ( ),o x yφ is

taken to be a very broad, yet finite, flat wavefront ( ( ) ( )( )2 2 2, exp 2m m

o ox y x y Wφ = − + ,

where m = 4 and WO = 500 µm) . The pictures in Figure 1 show the results of numerical

simulations carried out at an intensity ratio IN = 1 in normalized units, seeded with

random Gaussian white noise at a level of 10-5 .The figures on the left side depict the

100

Figure 1. Numerical simulation of propagation of partially coherent wavepacket in

saturable non-linear media. (A) shows the growth of perturbations after 1 mm of

propagation, and (B) its Fourier transform. (C) shows the development of individual

solitons at 2 mm, and (D) its Fourier transform. After 3 mm, clustering develops (E), and

(F) shows its Fourier transform.

101

intensity of the partially coherent wavefront whereas those on the right side show the

two-dimensional Fourier transform of the intensity pattern. The input to the system was a

partially coherent, spatially uniform broad beam. The width of the angular power

spectrum θΟ is assumed to be 13.85 millirads, which corresponds to an initial correlation

length of 6.3 µm. In Figure 1A, we see as expected, that perturbations of certain spatial

frequencies are favored by the MI process and have begun to grow on top of the input

[31,32]; Figure 1B shows that these frequencies are contained within a rather narrow

ring. As the propagation continues, the ripples grow stronger until the beam disintegrates

into solitary filaments (Figure 1C), indicating a balance between the effects of diffraction

and nonlinearity. Interestingly enough, little has changed in the frequency domain (Figure

1D); there is still a single thin ring of a well-defined radius. This radius or spatial

frequency is in fact related to average distance between solitary filaments (or "particles"),

which is fairly uniform. Now however, the particle-like nature of the solitons starts to

affect the overall dynamics of the partially coherent system, signaling the onset of a

qualitatively new stage of behavior. As discussed above, in incoherent systems, only

attractive forces between the solitary filaments ("particles") need be considered for

separation distances longer than the correlation length. Thus, as small random

movements accidentally bring two solitons closer together, the mutual potential well

caused by their joint overlapping intensities will further strengthen the attractive force

between them. As a result, the peaks will be drawn towards one another. What ensues is

best described as “clustering”; the interplay of the forces among particles eventually leads

to the grouping of quasi-particles with their nearest neighbors (see Figure 1E). The

overall size of each of the clusters continues to shrink as the particles move inwards and

102

in general this motion is rather complex. For example, the particles may spiral around one

another along seemingly chaotic orbits. Spectral analysis of the clusters of Fig. 1E reveals

a quite different picture in which a new, lower spatial frequency has begun to dominate;

this is the frequency of the inter-cluster spacing. The spatial intensity pattern is now

characterized by sparseness as the clusters compact and the distances between their edges

grow.

For comparison, the simulations were redone using a fully coherent wavefront as

input. In Figure 2, the spatial intensity pattern at the output is displayed (the simulation

parameters correspond to Figure 1E). As can be seen, the results contrast starkly with

those of Figure 1. Initially, the development is similar to the incoherent case. On top of

the featureless beam used as input, modulation instability seeded by noise causes ripples

to grow, developing into solitary filaments as the diffractive and nonlinear forces counter

each other. After this stage, the two cases are no longer comparable. In the coherent

regime, both attractive and repelling forces between the solitary filaments ("particles")

are present, and we find that the "particles" will be subject to too many conflicting

interactions for any definite course to evolve. Depending on the initial conditions,

ordered grid-like patterns may form, or the particles may simply remain well spaced

apart. In other words, the solitary filaments developing from MI in coherent wavefronts

do not cluster. Only when the spatial coherence of the beam is low enough for the long

range repelling forces to disappear can the attractive forces (that survive even when the

beam is totally incoherent, that is, the correlation distance is zero) dominate and cause the

solitary filaments produced by MI to cluster together.

103

Figure 2. Simulation of coherent wavefronts in non-linear media after 3 mm of

propagation. Shown is the (output) intensity pattern displaying multiple evenly spaced

solitonic filaments.

104

3.5 Clustering - experiment

Our experiments on soliton clustering were performed in a photorefractive

nonlinear optical system. A partially spatially incoherent beam was generated by passing

an argon ion laser beam (λ = 488 nm) through a rotating diffuser. The spatial coherence

of the scattered light from the diffuser was varied by changing the width of the laser

beam incident upon the diffuser. The degree of spatial coherence, (namely the transverse

correlation distance), was monitored by imaging the speckles on the front face of our

nonlinear crystal while the diffuser was held stationary. The average speckle size was

roughly equal to the transverse correlation distance lC, representing the longest distance

between two points on the transverse plane within which the points are still phase-

correlated. A biased Strontium Barium Niobate (SBN) photorefractive crystal was used

as a slow saturable nonlinear medium, with a response time on the order of 10 seconds.

As pointed out in the introduction, this response time must be much longer than the

characteristic random phase fluctuation time created by the rotating diffuser (1

microsecond in our experiments). The experimental setup was similar to that used in

earlier experiments on incoherent MI [32, 36]. In our experiments, a broad and uniform

extraordinarily polarized optical beam with a controllable degree of spatial coherence

was launched into the biased crystal. (The strength of the self-focusing nonlinearity of the

crystal was controlled by varying the external bias field and the intensity of the beam

[35]). The intensity patterns of the incoherent beam at the crystal output face were

monitored using an imaging lens and a CCD camera.

Typical experimental results are presented in Fig. 3. These were obtained by using

105

an SBN:60 crystal (5x10x5 mm3, r33=280 pm/V). When the nonlinearity was set to zero

(zero bias field), the output beam remained essentially the same uniform broad beam that

entered the crystal. As the magnitude of the nonlinearity was increased (by increasing the

dc field applied to our nonlinear crystal), the output beam remained uniform until the

nonlinearity reached the threshold value for incoherent MI to occur [31,32]. After the

threshold nonlinearity, a rather sharp transition in pattern dynamics was observed: the

incoherent wavefront disintegrated into 1D stripes at the output [32]. Further increases in

the nonlinearity led to the appearance of 2D solitary filaments, or "particles", with a

characteristic width of about 12 micrometers (similar to the structures observed in Ref.

[32]). To appreciate these solitary filaments, we note that, if filaments of this

characteristic width are launched in a linear medium, they diffract and broaden to at least

6 times wider after 10-mm of linear propagation (which is the propagation length in our

crystal). The self-focusing nonlinearity in our crystal keeps them as nondiffracting

solitary light spots, even though the length of our crystal corresponds to roughly 5

diffraction lengths. Finally, increasing the nonlinearity to even higher levels caused these

2D filaments to cluster together in lumps of fine-scale structures, opening empty voids in

other regions upon the beam. The intensity outside the clusters did not drop to zero

completely, but clearly more energy was concentrated in the cluster region. This pattern

of behavior is in good agreement with our numerical results presented in Figure 1.

In addition, we have carried out a series of experiments in different regimes of

parameters by varying the nonlinearity saturation and the degree of spatial coherence. In

principle, as long as the spatial coherence was below a certain level, repulsion forces

106

Figure 3. Experimental results showing pattern development in a biased photorefractive

crystal. The coherence length of the beam at the input is 10 µm. Shown are the intensity

patterns taken at crystal output face (after 10 mm of propagation) for a bias field of 1

kV/cm (left), 1.8 kV/cm (middle), and 2.6 kV/cm (right).

107

between the 2D filaments were practically eliminated on length-scales comparable to the

correlation length. As a result, forces of an attractive nature dominate the dynamics and

clusters of MI filaments form. It seems that the smallest distance between two adjacent

fine-scale elements (solitons) in a cluster is determined by the correlation distance: when

two filaments become so close that they start to be phase coherent, the repulsive forces

will push them apart and prevent them from getting any closer to each other.

The results shown in Fig. 3 depict typical intensity patterns taken from the output

face of the crystal at various values of the bias field (keeping all other experimental

parameters constant). In this particular experiment, the spatial correlation distance across

the beam was roughly 10 µm, and the average intensity of the beam at crystal input face

was 0.75 W/cm2. Similar experiments were performed with different SBN crystals under

various conditions and correlation distances. Figure 4 shows experimental results of

pattern formation for different degrees of spatial coherence obtained with an SBN:75

crystal (6-mm cube, r33 = 870 pm/V). When the input beam is a spatially-coherent

wavefront (taken directly from the argon laser without the diffuser), the uniform beam

disintegrates as a result of MI, and the resultant filaments tend to form individual well-

separated solitons (Fig. 4a). Even as we increase the nonlinearity further, these soliton

filaments still stand on their own and do not merge together. On the other hand, when the

beam is made sufficiently incoherent, the correlation between individual filaments

becomes insignificant, and any slight overlapping of their intensity profiles will drag

them closer together due to incoherent interaction [34]. As a result, a broad uniform

incoherent beam experiences a global weakly attractive force and tends to form patterns

108

Figure 4. Experimental results showing pattern development as the coherence of the

beam is reduced. The bias field across the crystal is 0.9 kV/cm. Shown are the intensity

patterns taken at the output face of the crystal (after 6 mm of propagation) for a

coherence length of 6 mm (left), 30 µm (middle), and 12 µm (right). (The intensity of the

last photograph has been enhanced for better visualization).

109

of dense groups of solitary filaments (clusters of 2D solitons). Specifically, in Fig. 4,

when the correlation distance is sufficiently reduced to lc < 30 µm, the onset of soliton

clustering occurs (Fig. 4b and 4c). Observing the clustering as a function of decreasing

coherence reveals that the peak intensity of individual solitons decreases, and the overall

size of soliton clusters increases as more neighboring solitons group together. The size

and the shape of each individual cluster as well as the dynamics inside clusters appear to

be random and driven by noise.

3.6 Conclusion

In summary, we have demonstrated both experimentally and theoretically the

spontaneous clustering of solitons in partially coherent wavefronts initiated by random

noise (see our paper, [37]). Soliton clustering is an intriguing outcome of the interplay

between random noise, weak correlation, and high nonlinearity. Together, these processes

lead to incoherent modulation instability, formation of 2D solitary filaments, and

eventually to clustering of 2D solitons. Yet all of these fascinating features are not unique

to optics. Nonlinear systems involving weakly correlated particles are abundant in nature

and so our results may prove relevant to other areas and fields.

110

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modulation instability in the temporal domain, see Agrawal, G. P. in Nonlinear Fiber

Optics, Second Ed. (Academic Press, San Diego, 1995), Chap. 5.

[29] Dianov, E. M. et al. (1989) Opt. Lett. 14, 1008; Mamyshev, P. V. et al. (1994) J.Opt.

Soc. Am. B 11, 1254; Iturbe-Castillo, M. D. et al. (1995) Opt. Lett. 20, 1853; Carvalho,

M. I., Singh, S. R., & Christodoulides, D. N. (1996) Opt. Comm. 126, 167.

[30] Interestingly, this view holds also for the breakup of one-dimensional solitons in

bulk media, the so-called transverse instability, which leads to the breakup of a one-

dimensional beam into an array of 2D filaments. In Kerr media, 2D beams are unstable,

so the entire structure is unstable and the whole beam quickly disintegrates (see

Zakharov, V.E. & Rubenchik, A. M. (1974) Sov. Phys. JETP 38, 494). However, in

saturable nonlinear media, such an array of 2D filaments is stable and is in fact an array

of 2D solitons (see, e.g., Anastassiou, C., Soljacic, M., Segev, M., Kip, D., Eugenieva, E.,

Christodoulides, D. N. & Musslimani, Z. H. (2000) Phys. Rev. Lett. 85, 4888.

[31] Soljacic, M., Segev, M., Coskun, T. H., Christodoulides, D. N., & Vishwanath, A.

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(2000) Phys. Rev. Lett. 84, 467.

[32] Kip, D., Soljacic, M., Segev, M., Eugenieva, E., & Christodoulides, D. N. (2000)

Science 290, 495.

[33] If the underlying nonlinearity is of the Kerr-type, then the products of transverse

instability are 2D filaments which are highly unstable, and tend to disintegrate, thereby

cannot form such a structure.

[34] See review on soliton interactions in Ref. 11. The interaction forces between solitons

in such systems were first studied theoretically by Anderson, D. & Lisak, M. (1985)

Phys. Rev. A 32, 2270. The first experimental demonstration of incoherent interaction

between solitons was reported by Shih, M. & Segev, M. (1996) Opt. Lett. 21, 1538; Shih,

M., Chen, Z., Segev, M., Coskun, T., & Christodoulides, D. N. (1996) Appl. Phys. Lett.

69, 4151.

[35] Segev, M., Valley, G. C., Crosignani, B., DiPorto, P., & Yariv, A. (1994) Phys. Rev.

Lett. 73, 3211; Christodoulides, D. N. & Carvalho, M. I. (1995) J. Opt. Soc. Am. B 12,

1628; Segev, M., Shih, M., & Valley, G. C. (1996) J. Opt. Soc. Am. B 13, 706.

[36] Klinger, J., Martin, H., & Chen, Z. (2001) Opt. Lett. 26, 271.

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[37] Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides and M. Segev, Clustering of

solitons in weakly correlated systems, Proceedings of the US National Academy of

Science (PNAS), 99, 5223 (2002).

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4 Conclusion and future directions

In this dissertation, we have explored some of the diverse patterns which can arise

when solitary waves interact in non-linear systems. Although optical systems have been

the stage for our experiments, it is important to keep in mind that perhaps the most

intriguing aspect of this work is its universality. Non-linear systems are everywhere in the

natural world and the opportunities to spot connections are frequent and often startling. In

this conclusion, I revisit some of our work and discuss a few systems which would be

interesting to examine more closely in the future for similarities with the dynamics

studied in this thesis.

In Chapter 2, we showed how exact Cantor Set fractals (an at first seemingly

mathematical and artifical construct) could be generated in a physical laboratory (see also

our paper on the topic, S. Sears, M. Soljacic, M. Segev, D. Krylov and K. Bergman,

Cantor set fractals from solitons, Physical Review Letters 84, 1902 (2000).[1]). A high-

order soliton is launched into a cascaded series of optical fibers. The soliton separates

into its first order component “daughter” solitons as it propagates, creating self-similar

branches. When a new fiber stage is reached, the process is triggered and begins again,

each “daughter” solitons branching out into new self-similar “grand-daughter” solitons.

After an infinite number of stages, an exact Cantor Set fractal is generated.

Now consider the Tent Venus Clam (see Figure 1). The pattern on the shell is

immediately recognizable as a Cantor Set fractal. The formation of these designs is not

well understood, although similar patterns have been generated using cellular automatons

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[2]. The Tent Venus Clam illustrates perfectly how successive stages, such as used in our

fiber optic experimental setup, can occur naturally. The shell grows in layers, creating

ridges as new material piles on top of old (clearly seen in Figure 1). Each of the

branching patterns on the Tent Venus Clam begins at one of these ridges, just as the

branches in our Cantor Set fractals were triggered by reaching a new stage. In our optical

setup, the characteristics of each new stage were scaled in relationship to the previous.

This is also the case for the Tent Venus Clam ridges: the curvature of the ridges grows

sharper as the hinge of the shell is approached.

Figure 1. The shell of the Tent Venus Clam exhibits a Cantor Set-like pattern.

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In Chapter 3, we showed how modulation instability can create ordered, symmetry

breaking patterns of stripes and grids from completely random initial noise (see our

paper, S. M. Sears, M. Soljacic, D. N. Christodoulides and M. Segev, Pattern formation

via symmetry breaking in nonlinear weakly correlated systems, Physical Review E 65,

36620 (2002) [3]). As an initially broad optical beam propagates, the linear diffractive

effect is not strong enough to counter the non-linear self-focusing, leading to

disintegration. The beam continues to focus until the linear tendency is balanced by the

self-focusing, and a striped pattern emerges at a stable frequency.

Other examples of rippling occurring as a broad expanse collapses can be found

in nature. There are remarkable stripes in desert sand dune system [4] which can exist

even in such exotic environments as Mars and Venus, sometimes stretching out for

hundreds of kilometers.

Some unusual parallels can be drawn between sand dunes and solitons. Dunes

also exist as a balance between two forces: wind and gravity. As sand begins to pile up,

winds blowing horizontally across the desert encounter the proto-dune and are forced

upward by the elevated sand. The wind becomes compressed and the shear velocity along

the side of the elevation increases, causing more sand to be transported upward. The force

of gravity eventually counters the upward force of the wind, and a stable dune is created.

Like solitons, “Aeolian” dunes have been shown to migrate without changing shape at a

speed inversely proportional to their height. Phenomena reminiscent of MI occurs in dune

systems as well. Broad raised planes of sand can become unstable as over time wind

directions and velocities may change. Small perturbations will occur, and the broad plane

will collapse. As was mentioned above, stable ripples can appear (see Figure 2).

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Rippling situations are also known to occur in other geophysical environments,

such as artic vegetation on hillsides (see Figure 3).

Figure 2. Sand dune ripples, Death Valley, California.

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Figure 3. Hillside ripples (with vegetation), Myvatn, Iceland.

In Chapter 4 we theoretically and experimentally illustrated clustering, or the gross scale

aggregation of fine scale elements, in optical systems (see our paper Z. Chen, S. M.

Sears, H. Martin, D. N. Christodoulides and M. Segev, Clustering of solitons in weakly

correlated systems, Proceedings of the US National Academy of Science (PNAS), 99,

5223 (2002) [5]). We noted a number of systems outside of optics exhibiting clustering

behavior, such as galactic clusters, molecular aggregates, self-assembled quantum dots,

and biological systems. The fundamental similarity between these processes can be better

understood by considering the following arguments: (1) the fine structure results from the

equilibrium of opposing effects or forces, and (2) the cluster forms because of attraction

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between these individual "fine scale elements". For example, a protein molecule may be

made up by more than one polypeptide chain. In the case of hemoglobin, four separate

polypeptide chains, or subunits, are clustered together (held together by van der Waals

and ionic forces). In astrophysics, gravitational attraction is known to lead to the

formation of galaxies and galactic clusters. Given the universality of these processes, one

may be able to study clustering of fine scale elements in a number of completely different

physical systems.

We have examined in this thesis only a small fraction of the many possibilities for

pattern formation in non-linear soliton supporting systems. The possibilities for future

research are rich and many interesting phenomena remain to explore.

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4.1 References

[1] S. Sears, M. Soljacic, M. Segev, D. Krylov and K. Bergman, Cantor set fractals from

solitons, Physical Review Letters 84, 1902 (2000).

[2] S. Wolfram in A New Kind of Science (Wolfram Media, Champaign, IL, 2002), Chap.

8.

[3] S. M. Sears, M. Soljacic, D. N. Christodoulides and M. Segev, Pattern formation via

symmetry breaking in nonlinear weakly correlated systems, Physical Review E 65, 36620

(2002).

[4] H. Momiji in Mathematical Modelling of the Dynamics and Morphology of Aeolian

Dunes and Dune Fields (PhD Thesis, University College London).

[5] Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides and M. Segev, Clustering of

solitons in weakly correlated systems, Proceedings of the US National Academy of

Science (PNAS), 99, 5223 (2002).

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5 Publications

[1] N. K. Efremidis, S. M. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev,

Discrete solitons in photorefractive optically-induced photonic lattices, Phys. Rev. E 66,

46602 (2002).

[2] Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides and M. Segev, Clustering of

solitons in weakly correlated systems, Proceedings of the US National Academy of

Science (PNAS), 99, 5223 (2002).

[3] D. Kip, M. Soljacic, M. Segev, S. M. Sears, and D. N. Christodoulides, (1+1)

Dimensional modulation instability of spatially-incoherent light, Journal of Optical

Society of America B 19, 502 (2002).

[4] S. M. Sears, M. Soljacic, D. N. Christodoulides and M. Segev, Pattern formation via

symmetry breaking in nonlinear weakly correlated systems, Physical Review E 65, 36620

(2002).

[5] S. Sears, M. Soljacic, M. Segev, D. Krylov and K. Bergman, Cantor set fractals from

solitons, Physical Review Letters 84, 1902 (2000).

[6] M. Soljacic, S. Sears, M. Segev, D. Krylov and K. Bergman, Self-similarity and

fractals driven by soliton dynamics, Invited Paper, Special Issue on Solitons, Photonics

124

Science News 5(1), 3-12 (1999).

[7] M. Soljacic, S. Sears and M. Segev, Self-trapping of necklace beams in self-focusing

Kerr media, Physical Review Letters 81, 4851 (1998).

[8] M. Soljacic, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier,

Collisions of two solitons in an arbitrary number of coupled nonlinear Schrodinger

equations, Phys. Rev. Lett. 90, 254102 (2003).

125