book for the help for writing thesis by Suzanne M. Sears

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.
5

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
8

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].
Figure 3. Two one dimensional solitons collide and recover.
While Russell observed solitons in nature as far back as 1834, it was not until
1964, after the invention of the laser, that self-focusing behavior was reported in the
laboratory [12]. Narrow wave-packets could propagate undistorted for seemingly
indeterminate 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 3
He [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
2
2
2
A i A
i A A
z
β γ
τ
∂ ∂
= +
∂ ∂
, (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 t
E t z A t z e x
ω−
=
r
(0.2)
18

where we are back in the true coordinate frame, ( ),t z , and linear polarization in the ˆx -
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
2
2 2
1
1
M
j j
o
j j
n
α ω
ω
ω 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 E
r r
, and not
E
r
, since an E
r
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
2
2
( )
2
o
o
ck n
n
n
ω
ω
= + E
r
. (0.4)
Thus ( )2
,k k Eω=
r
, and for frequencies near to the carrier frequency, oω , we may
approximate
( ) ( ) ( )
2
22
22
1
2
o o o
k k k
k k E E
E
ω ω ω ω
ω ω
∂ ∂ ∂
− = − + − + −
∂ ∂ ∂
2
o
r r
r , (0.5)
where all of the derivatives are constants evaluated at ,ok oω , and
2
oE
r
(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− i
z
∂
−
∂
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 2
22 2
1
0
2
o
A k A k A k
i
z t t Aω ω
∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞
+ − + −⎜ ⎟
∂ ∂ ∂ ∂ ∂⎝ ⎠ ∂
A A A = , (0.6)
20

where we have used the fact
2 2
E 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 2
2
2
o
A i A
i A A A
z
β γ
τ
∂ ∂
= + −
∂ ∂
) (0.7)
where we have introduced the notation β and γ for the constants in Eq. (0.6). Note that
2
oA is a constant and thus this term will simply introduce a phase
2
oi A z
e
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 exp
2
o o
i
A 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 andoP 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
2
2
1
0
2
A A
i A
z 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
0
2
A
i A A
z
⊥ 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 22ReE 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 m
m
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 ofincoher ( )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 isr
( ) ( ) ( ) ( )
2 2 2 2
, , , , ,m m
m m
2
A 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 x
r
,
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 x
r
to be ( ) ( ) ( )*
2 1 2 1oR − =x x x xoφ φ
r r r r
autocorrelation function of the source spectrum is
. Now, the
( ) ( ) ( ) ( )* 2 2
2 1 2 1
ˆ ˆ 4o o Gπ δ⊥ ⊥ ⊥ ⊥Φ Φ = −k k k k k
r r
1⊥ , where
r r r
( )G ⊥k
r
is the Fourier transform
30

of ( )R x
r
. Since ( )
2
ˆΦ 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 ⊥k
r
δ -
function implies that there is no correlation between i⊥k
r
and j⊥k
r
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 2
G ⊥k
r
, and each propagates out at an angle
kθ ⊥= k
r r
(we have assumed here that ( )G ⊥k
r
falls off rapidly and that the only
ons are for k⊥k
r
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 x
r 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 x
rr r r
.
ber of point sources, radiating outThus, we can think of the input as being an infinite num
at every position x
r
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 ⊥k
r
( )oI x
r
.
Alternatively, this is equivalent to an infinite number of coherent profiles of shape ( )f x
r
all propagating out at different angles, kθ ⊥= k
r r
, where each component is weighted by
the square root of the power spectral density of the source, ( )1 2
G ⊥k
r
. Since we have
shown above that each is uncorrelated to all of the other transverse wavevectors,i⊥k
r
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 k
r 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 θ
r
:
( )2
g , 0
2 2
o
o
i u u I z
z k n
θ ⊥ ⊥
1 ku∂⎛ ⎞
+ ⋅
r r r
∇ + ∇ + =⎡ ⎤⎜ ⎟ ⎣ ⎦∂⎝ ⎠
x . (0.14)
Here, ( )g ,I z⎡ ⎤⎣ ⎦x
r
is a function of the total intensity of the beam, ( ),I zx
r
, and represents
the non-linear ch ction:ange to the index of refra ( )2 2
g ,on n I z= + ⎡ ⎤⎣ ⎦x
r
. The total intensity
components, ( )
of the beam is given by the integral of the intensities of vidual coherentthe indi
( )
2
,I z
π π
π π
, ,u z dθ θ∫ x
− −
= ∫x
r 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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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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Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
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41

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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:
( ) 2
2
2
0
2
i
i
z T
ψ 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, ( )
0i
β < 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. Equation2 0n > z t
(1) has a fundamental soliton solution of the form
( )
( )
( )
( ){ }
( )
( )
02( )
0
2( ) ( )
0
, sech
exp 2
i
i
i
i i
z T T T
T
iz T
β
ψ
γ
β
=
⎡ ⎤× ⎢ ⎥⎣ ⎦
(0.16)
where is the temporal full width half maximum of( )
01.76274 i
T ( )
2
,z Tψ and
( ) ( )
2( ) ( ) ( )
0 2i i
oz Tπ= i
β is the soliton period for fiber segment i. The N-order soliton (at
) of Eq. (1) can be obtained by multiplying0z = ( ), 0z Tψ = from Eq. (2) by a factor of
. A higher order soliton of a given propagates in a periodic fashion. In the firstN 1N >
48

half of the soliton period ( )( )
0 2i
z , 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 i
β 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 i
z 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 N
T
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 by8N = 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 for8N =
(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
i
T )
β .
52

The same method is used to generate the doublet Cantor set fractals in Fig. 3, where an
soliton is propagated for5N = ( )
00.1623 i
z 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 i
z .
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) 2
127.6 ps kmβ = − (assuming 1
1.62W 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
i
T )
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
i
T )
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) 2
12.2 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) 2
1.2 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 2
ˆ ˆ ˆ
2 2
y yx x
x y
r rr r
r x r y x yˆ
2
++ +
= = + = +
r r
r
and
( ) ( ) ( )1 1 2
ˆ 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
2
0
, , , ,
2 2 2 2
x x y y
y yx x
x y x y
B i
B
z k r r
in
n r r z n r r z B
k 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 2
1
1
12
0
0
1
, , 0, 0 ,
2 2
,
, , 0, 0 ,
2 2
x x y y
yx
x y x y
yx
x y x y
B i
B
z k r r
B r r z
in
B
k c
B r r z
ρ ρ
ρρ
ρ ρ
ω
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= Λy 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 y
x y
x x y y
M
k
M B
c
α α
ρ 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 x
x y x y
i k c
M k k
i
k k
k
B k k B k k
ω
α α
α αα α
⎡ ⎤
⎢ ⎥
= ×⎢ ⎥
⎢ ⎥Ω − +
⎣ ⎦
⎡ ⎤⎛ ⎞ ⎛
+ + − − −⎢ ⎥⎜ ⎟ ⎜
⎝ ⎠ ⎝⎣ ⎦
,
⎞
⎟
⎠
(0.22)
65

where ( ) ( ) ( )
2
ˆ 1 2 i
F d F eπ
∞ ∞
−∞ −∞
= ∫ ∫
k ρ
k 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 x
x y x y
x x y y
dk dk
c
B k k B k k
k k
i
k
y
ωκ
α αα α
α α
∞ ∞
−∞ −∞
= −
⎡ ⎤⎛ ⎞ ⎛
+ + − − −⎢ ⎥⎜ ⎟ ⎜
⎝ ⎠ ⎝⎢ ⎥×
+⎢ ⎥
Ω +⎢ ⎥
⎢ ⎥⎣ ⎦
∫ ∫
⎞
⎟
⎠
.
(0.23)
Here, stands for the Fourier transform of( )0
ˆB 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( )0
ˆB 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 2
2 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 0
ˆ 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 (
( )0
ˆB k
),x yα α , we see that
( )0
0 0
ˆ1
ˆ ˆ
2 2
y y
x x
x
x
dk B k
c
B k B k
dk
k
i
k
ωκ
α α
α
∞
−∞
∞
−∞
= −
⎡ ⎤⎛ ⎞ ⎛
+ − −⎜ ⎟ ⎜
⎞
⎟⎢ ⎥
⎝ ⎠ ⎝ ⎠⎢ ⎥×
⎢ ⎥Ω +
⎢ ⎥⎣ ⎦
∫
∫
. (0.24)
67

Now since is identical with respect to k( )0
ˆB k x and ky and normalized
, integration over k( ) ( )( )0 0
ˆ. ., 0,0i e d B B I
∞ ∞
−∞ −∞
⎡ = = =
⎢⎣ ∫ ∫ k k ρ 0
⎤
⎥⎦
y further reduces this
constraint to
0 0
ˆ ˆ
2 2
⎞
⎟
⎠1
x x
x
x
B k B k
dk
kc i
k
α α
ωκ
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
( )0
ˆB 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
0
0 2 2
0 0 0 0
ˆ , exp
yx
x y
x y x y
kI k
B k k
k 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
( )0
ˆB 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 0
ˆB 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
x
x k k
x x
x
x x
dk dk
c
B
k
k
i
k
α α
x
ωκ
α
α α
α
α α
∞ ∞
−∞ −∞
=
= =
= −
⎡ ⎤
∂⎢ ⎥
⎢ ⎥∂
⎢ ⎥×
⎢ ⎥⎛ ⎞
∂Ω ∂ Ω⎢ ⎥⎜ ⎟Ω + + +
⎢ ⎥⎜ ⎟∂ ∂
⎢ ⎥⎝ ⎠⎣ ⎦
∫ ∫
, (0.27)
which reduces to
'
0
'
ˆ1
1
x x
x y
x x k k
B
dk dk
c 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 ( )0
ˆB k . Choosing the same
double-Gaussian form as above [Eq. (9)], we find that MI will occur in the x direction if
2
0 0
0 2
2
x
x threshold
n k
n I
k
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 0
0 2
2
y
y threshold
n k
n I
k
κ−∆ ≡ ≥ (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, ( )
2
1B r , in the spatial domain, with high intensity represented by
white shading, low by dark. (b) shows the corresponding angular power spectrum
( )
2
1
ˆB 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 ( )0
ˆB 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 2
0 0
ˆ , x y
x y
x x y y
I k k
B 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 ( )0
ˆB k
in Eq. (6), one can then obtain exactly the gain curve ( ),x yα αΩ that is,
( ) 0 0
1/ 22 2
2 2 0
2
0
,
4
yx
x y x y
x y
x y
k k
k k
I
n k
αα
α α
α ακ
α 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

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