Download Soliton Interactions and the Formation of Solitonic Patterns

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
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© Copyright by Suzanne Marie Sears, 2004. All rights reserved.
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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. 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.
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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
2
3
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
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
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
7
4
5
3.5
Conclusion........................................................................................................ 86
3.6
References ....................................................................................................... 89
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
Conclusion and future directions........................................................................ 117
5.1
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References ..................................................................................................... 123
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.
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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 multistage 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.
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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
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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.]
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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.
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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.
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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
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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 nonlinearity ∆nNL = n2 I 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 10-4.)
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
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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),
∂A i ∂ 2 A
2
= β 2 + iγ A A ,
∂z 2 ∂τ
(0.1)
where A refers to the slowly varying electric field envelope of a short pulse of light with
carrier frequency ω o ; 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
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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 selfphase 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 (τ , z ) represents the slowly varying envelope of the electric field
at a carrier frequency ω o :
r
i k z −ω t
E ( t , z ) = A ( t , z ) e ( o o ) xˆ
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(0.2)
where we are back in the true coordinate frame, ( t , z ) , and linear polarization in the x̂ direction is assumed; the wavevector of the carrier in vacuum is ko = 2π λVAC , where
λVAC is the wavelength.
The response of the medium to the light, (both the dispersion and the nonlinearity), are embodied in the form of its index of refraction:
(
)
r2
r2
2
n 2 ω , E = no (ω ) + n2 E .
(0.3)
where no (ω ) represents chromatic dispersion, and (far from the resonances of the
M
α jω j 2
j =1
ω j2 −ω 2
material) may be well approximated by the Sellmeier equation no (ω ) = 1 + ∑
2
,
where the sum, j , 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, n2 . 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
r r
term retained; for centro-symmetric materials this must be proportional to E E * , and not
r
r
E , since an E term would indicate a directional preference in the material.
The wave-vector, k , is related to the index of refraction (keeping first order terms
only):
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ck
ω
= no (ω ) +
n22 r 2
E .
2no
(0.4)
)
(
r2
Thus k = k ω , E , and for frequencies near to the carrier frequency, ω o , we may
approximate
k − ko =
(
r2 r
∂k
1 ∂2k
∂k
2
−
+
E
− Eo
ω
ω
(ω − ω o ) +
(
)
r2
o
∂ω
2 ∂ω 2
∂E
2
),
(0.5)
r 2
where all of the derivatives are constants evaluated at ko , ω o , and Eo (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, E → 0 , we can use integration by parts to
replace k − ko with the spatial operator −i
∂
∂
and ω − ω o with the temporal operator i .
∂z
∂t
Making these replacements in the equation above, and operating on the field envelope,
A ( t , z ) , we get:
(
2
2
∂k
2
⎛ ∂A ∂k ∂A ⎞ 1 ∂ k ∂ A
+
−
+
i⎜
A − Ao
⎟
2
2
2
⎝ ∂z ∂ω ∂t ⎠ 2 ∂ω ∂t
∂ A
20
2
) A = 0,
(0.6)
r2
2
where we have used the fact E = A . Remembering that the derivatives with respect to
k are constants, and moving to a frame of reference, (τ = t − z / υ g , z ) , that moves with
the group velocity of the pulse (υ g = ∂ω ∂k ) , we have
(
∂A i ∂ 2 A
2
= β 2 + iγ A − Ao
∂z 2 ∂τ
2
)A
(0.7)
where we have introduced the notation β and γ for the constants in Eq. (0.6). Note that
Ao
2
is a constant and thus this term will simply introduce a phase e
iγ Ao 2 z
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, β > 0 , 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
⎛i
2⎞
A (τ , z ) = Po sech (τ τ o ) exp ⎜ z β τ o ⎟
⎝2
⎠
1
(0.8)
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.
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where Po is the peak power of the pulse and τ o is the temporal width of the pulse. The
intensity profile of the pulse, A (τ , z ) = f (τ ) , has no z dependence and thus the pulse
2
is truly stationary and a soliton.
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
AN (τ , z = 0 ) = N Po sech (τ τ o ) .
22
(0.9)
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 N − 1 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 N − 1 peaks of an N
th
-order soliton to break up into N − 1 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.
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One-dimensional CW optical beams with spatial intensity structures propagating
in a Kerr non-linear self-focusing media obey the following normalized equation:
i
∂A 1 ∂ 2 A
2
+
+ A A=0,
2
∂z 2 ∂x
(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:
i
∂A 1 2
2
+ ∇⊥ A + A A = 0
∂z 2
24
(0.11)
The non-linearity in Eq. (0.11) is only one possibility; many other forms exist, for
example the saturable non-linearity ∆n ~ A
2
(1 + A ) is commonly found.
2
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 nonlinearly 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:
r
E (τ , z = 0 ) = Po sech (τ τ o ) xˆ (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 lc → ∞ ).
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
r2
r 2 r 2
r r∗
interference between them: E = E1 + E2 + 2 Re E1 E2 , 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
r2 r 2 r 2
resulting light will be of an intensity: E = E1 + E2 . 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 nonlinearity, 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.
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. This
waveguide may have many modes, and the soliton may be decomposed into a sum
28
A ( x, y, z, t ) = ∑ cm ( t ) U m ( x, y ) exp ( i β m z )
(0.12)
m
where U m ( x, y ) is the mode profile of the mth mode, β m is the propagation constant of
mode U m , and cm ( t ) is its instantaneous relative weight. Due to the random nature of
incoherent beams, the amplitude and phase of cm ( t ) will also randomly fluctuate, and
cm ( t ) will be a stochastic function. Thus, no correlations can exist between different
modes and cm ( t ) cn ( t ) = δ mn . The time-averaged profile of the soliton is
∗
A ( x, y , z , t )
2
= ∑ cm ( t )
m
where d m = cm ( t )
2
U ( x, y ) = ∑ d m U ( x, y ) ,
2
2
2
(0.13)
m
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 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 nonsolitonic 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 timeaveraged 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 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 much use to thoroughly detail it now.
r
First, consider an incoherent field of uniform time-averaged intensity, φo ( x, t ) ,
representing only the statistical fluctuations of our source at the input, z = 0 . Since our
concern here is the degree of incoherence of the source, let us define the spatial statistical
r r
r
r
r
autocorrelation function of φo ( x, t ) to be R ( x 2 − x1 ) = φo ( x 2 ) φo* ( x1 ) . Now, the
autocorrelation
r
r
ˆ k Φ
ˆ* k
Φ
2⊥
1⊥
o
o
( ) ( )
function
of
the
source
r
r
r
r
= 4π 2δ 2 k 2⊥ − k 1⊥ G k 1⊥ , where G k ⊥
(
) ( )
30
( )
spectrum
is
is the Fourier transform
r
of R ( x ) . Since
r
G k⊥
( )
r
Φ̂ o k ⊥
( )
2
is the intensity density in the spectral domain, physically,
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 δ -
r
r
function implies that there is no correlation between k i ⊥ and k j ⊥ for any i ≠ j . Thus, we
may think of our source as a set of plane waves, all statistically uncorrelated, where the
r
amplitude of each wave is given by G 1 2 k ⊥ , and each propagates out at an angle
( )
r
r
r
θ = k ⊥ k (we have assumed here that G ( k ⊥ ) falls off rapidly and that the only
r
significant contributions are for k ⊥
k ).
Now consider our total input; the source is spatially modulated by some spatial
r r
r
r
function, such that E ( x, z = 0, t ) = xˆ f ( x ) φo ( x, t ) . Taking the average intensity of the
r r 2
r
r 2
background statistical source to be unity, we have I o ( x, z = 0 ) = Eo ( x, t ) = f ( x ) .
Thus, we can think of the input as being an infinite number of point sources, radiating out
r
at every position x in all directions (with the power going at each angle weighted
r
r
according to G k ⊥ ) and with the total power density at each point given by I o ( x ) .
( )
r
Alternatively, this is equivalent to an infinite number of coherent profiles of shape f ( x )
r r
all propagating out at different angles, θ = k ⊥ k , where each component is weighted by
r
the square root of the power spectral density of the source, G 1 2 k ⊥ . Since we have
( )
r
shown above that each k i ⊥ is uncorrelated to all of the other transverse wavevectors,
31
there is no statistical correlation between any of the so-called coherent-components,
r r
r r
r
r
u x, z = 0,θ = k ⊥ k = f ( x ) ⋅ G 1 2 θ = k ⊥ k .
(
)
(
)
Each
component
will
propagate
unaffected by the others, except for the non-linear changes caused in the common
refractive index of the material by the presence of their intensities.
The propagation of a single coherent component is governed simply by the NLS
r
with one additional term to account for the angle of propagation, θ :
k
r
⎛ ∂u r r ⎞ 1
i ⎜ + θ ⋅ ∇ ⊥u ⎟ + ∇ ⊥2 u + o g ⎡⎣ I ( x, z ) ⎤⎦ = 0 .
2no
⎝ ∂z
⎠ 2k
(0.14)
r
r
Here, g ⎡⎣ I ( x, z ) ⎤⎦ is a function of the total intensity of the beam, I ( x, z ) , and represents
r
the non-linear change to the index of refraction: n 2 = no2 + g ⎡⎣ I ( x, z ) ⎤⎦ . The total intensity
of the beam is given by the integral of the intensities of the individual coherent
r
components, I ( x, z ) =
π π
r r
u x, z ,θ
∫∫ (
−π −π
)
2
r
dθ , with no interference terms between the
components, since, as discussed above, they are statistically uncorrelated.
The
wavevector, ko = 2π λVAC , is that of the carrier wave in vacuum, and no is the index of
refraction in the absence of light.
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
dielectric 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 than 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.
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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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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Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
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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 Kerrtype 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:
i
∂ψ
∂z
−
β ( i ) ∂ 2ψ
2 ∂T
2
+ γψ ψ
2
= 0,
(0.15)
where ψ ( z , T ) is the slowly varying envelope of the pulse, T = t − z υ g is the time in the
propagation frame, υ g is the group velocity, β (i ) < 0 is the (anomalous) group velocity
dispersion coefficient of segment i , and γ > 0 is proportional to the nonlinearity
( n2 > 0 ); z is the spatial variable in the direction of propagation and t is time. Equation
(1) has a fundamental soliton solution of the form
ψ ( z, T ) =
β (i )
γ (T
)
(i ) 2
0
sech (T T0( i ) )
{
}
(0.16)
2
× exp iz β (i ) ⎡⎢ 2 (T0(i ) ) ⎤⎥
⎣
⎦
where 1.76274 T0(i ) is the temporal full width half maximum of ψ ( z , T )
zo( i ) = π (T0(i ) )
2
2
and
( 2 β ) is the soliton period for fiber segment i. The N-order soliton (at
(i )
z = 0 ) of Eq. (1) can be obtained by multiplying ψ ( z , T = 0 ) from Eq. (2) by a factor of
N . A higher order soliton of a given N > 1 propagates in a periodic fashion. In the first
48
half of the soliton period ( z0(i ) 2 ) , the pulse splits into two pulses, then into three, then
into four, etc., up to N − 1 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 N -order soliton, it splits into 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 N -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.
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
β (i +1) β ( i ) = η , where η < 1 . This implies that the periods of the fundamental solitons in
2
consecutive segments are related through z0(i +1) z0(i ) = ⎡⎢(T0(i +1) )
⎣
(T ) ⎤⎥⎦ [1 η ] .
(i ) 2
0
Numerically, we launch an N -order soliton into the first fiber segment and let it
propagate until it breaks into 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 L(1) . From the simulations we find the peak power PM(1) and the temporal width TM(1)
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 N -order soliton in the second segment. Thus we equate the
peak power in each of the M pulses in the first fiber to the peak power of an N -order
soliton in the second fiber:
(1)
M
P
=P
(2)
N
=
β (2)
γ (T
)
(2) 2
0
N2 ,
(0.17)
where TM(1) = T0(2) 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, β (2) . The ratio η between the
dispersion coefficients in consecutive fibers determines the scaling of the similarity
transformation. Using η and T0(2) we calculate the period z0(2) . Requiring that the
evolution in the second fiber is a rescaled replica of that in the first fiber, we get
L(2) z0(2) = L(1) z0(1) . 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 N = 8 soliton into the first fiber characterized by γ = 1 and
β (1) = −1 and let it propagate for 0.1261 z0(1) . 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 β (2) = − 0.01285 and γ = 1 . Each of the four
solitons is an N = 8 soliton in the second fiber. We let the four soliton set propagate for
0.1261 z0(2) , which is identical to 0.03290 z0(1) . 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 0.1261 z0(3) , 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 ψ ( z , T ) . In Fig. 2(b), we show a magnified version of the lowermost
2
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 0.3112 z0(1) 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: T0 = 1 , peak power =1, and 1 unit of distance = T02 β ( i ) .
52
The same method is used to generate the doublet Cantor set fractals in Fig. 3, where an
N = 5 soliton is propagated for 0.1623 z0(i ) 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 N = 5 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 N = 6 soliton, propagated for 0.1649 z0(i ) .
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 N = 3 soliton. In the first
stage a 50 ps FWHM pulse of 0.88 W power is launched into a 6 km long fiber with
β (1) = −127.6 ps 2 km (assuming γ = 1.62 W −1 km −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 T0 = 1 , peak power = 1, and 1 unit of distance =
T02 β ( i ) .
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 T0 = 1 , peak power = 1, and 1 unit of distance
= T02 β ( i ) .
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
β (2) = −12.2 ps 2 km . 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 β (3) = −1.2 ps 2 km .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 DS . In the construction of a fractal an original
object is replicated into many rescaled copies. If the length of the original object is unity,
ε is the length of each new copy, and N is the number of copies. The similarity
dimension is [1]: DS = log ( N ) log (1 ε ) . For the doublet Cantor set fractal from Fig. 3,
DS = 0.2702 and for the quadruplet Cantor set fractal from Fig. 2, DS = 0.4318 .
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 [48] 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-transversedimensional 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
B ( r1 , r2 , z ) = E * ( r2 , z , t ) E ( r1 , z , t ) . The brackets denote the time average over time
period τ. By setting
r =
( r1 + r2 )
2
= rx xˆ + ry yˆ =
r1 y + r2 y
r1x + r2 x
xˆ +
ŷ
2
2
and
ρ = ( r1 − r2 ) = ρ x xˆ + ρ y yˆ = ( r1x − r2 x ) xˆ + ( r1 y − r2 y ) yˆ
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) =
E ( r, z, t )
2
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),
∂B i ⎧⎪ ∂ 2
∂ 2 ⎫⎪
− ⎨
+
⎬B
∂z k ⎪⎩ ∂rx ∂ρ x ∂ry ∂ρ y ⎭⎪
in ⎛ ω ⎞
= 0⎜ ⎟
k ⎝c⎠
2
ρ
ρ
⎞
⎛
⎞ ⎪⎫
ρx
ρ
⎪⎧ ⎛
, ry + y , z ⎟ − ∆n ⎜ rx − x , ry − y , z ⎟ ⎬ B,
⎨∆n ⎜ rx +
2
2 ⎠
2
2 ⎠ ⎪⎭
⎪⎩ ⎝
⎝
(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 ( ∆n
n0 ).
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
B ( r, ρ, z ) = B0 ( ρ ) + B1 ( r, ρ, z ) , where B0 is the uniform beam B1 is the perturbation to be
affected by MI, and B1
B0 . Substituting this latter form of B in Eq. (1) we obtain
63
∂B1 i ⎧⎪ ∂ 2
∂ 2 ⎫⎪
− ⎨
+
⎬ B1
∂z k ⎪⎩ ∂rx ∂ρ x ∂ry ∂ρ y ⎪⎭
⎧ ⎡⎛
⎫
⎤
ρy ⎞
ρx
(0.19)
+
+
=
=
,
,
0,
0
,
ρ
ρ
B
r
r
z
(
)
⎪
⎪
⎢
⎥
⎟
y
x
y
1 ⎜ x
2
2
2 ⎠
in0 ⎛ ω ⎞ ⎪ ⎣⎝
⎪
⎦
=
⎬ B0 ( ρ ) ,
⎜ ⎟ κ⎨
k ⎝c⎠ ⎪
⎡⎛
⎤⎪
ρy ⎞
ρx
− B1 ⎢⎜ rx −
, ry −
⎟ , ( ρ x = 0, ρ y = 0 ) , z ⎥ ⎪
⎪
2
2
⎝
⎠
⎣
⎦⎭
⎩
where we have defined the marginal nonlinear index change evaluated at intensity I0, to
be κ = d ⎡⎣ ∆n ( I ) ⎤⎦ dI I . Equation (2) is linear in B1 and has translational invariance with
0
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 ⎡⎣i (α x rx + α y ry ) ⎤⎦ , where α x = 2π Λ x and
α y = 2π 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
B1 = exp ( Ωz ) exp ⎡⎣i ( α r + φ ) ⎤⎦ L ( ρ ) + exp ( Ω* z )
× exp ⎡⎣ −i ( α r + φ ) ⎤⎦ L* ( −ρ ) ,
64
(0.20)
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
Ω M (ρ) +
∂
∂ ⎫⎪
1 ⎧⎪
+ αy
⎨α x
⎬
∂ρ y ⎭⎪
k ⎩⎪ ∂ρ x
⎛ α x ρx + α y ρ y
2ωκ
× M (ρ) +
sin ⎜
2
c
⎝
⎞
⎟ B0 ( ρ ) = 0.
⎠
(0.21)
Since growth can only occur for this form of the ansatz for B1 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 M ( ρ ) must be bounded for large ρ . By taking the Fourier transform of Eq. (4) we
find that
Mˆ ( k x , k y )
⎡
⎤
⎢
⎥
iω k c
= ⎢
⎥ ×
i
⎢ Ω − (α x k x + α y k y ) ⎥
k
⎣
⎦
⎡ˆ ⎛
αy ⎞
α y ⎞⎤
αx
αx
ˆ ⎛
, ky +
, ky −
⎢ B0 ⎜ k x +
⎟ − B0 ⎜ k x −
⎟⎥ ,
2
2 ⎠
2
2 ⎠⎦
⎝
⎣ ⎝
65
(0.22)
2 ∞
where Fˆ ( k ) = (1 2π ) ∫
∫
∞
−∞ −∞
the
definition
dρF ( ρ ) eik ρ denotes the Fourier transform of F ( ρ ) . From
M (ρ)
of
above,
it
can
be
seen
that
M ( ρ = ( 0, 0 ) ) = L ( ρ = ( 0, 0 ) ) L ( ρ = ( 0, 0 ) ) = 1 . Hence we arrive at the constraint,
1= −
ωκ
∫
∞
−∞
∞
dk x ∫ dk y
−∞
c
⎡ ⎛
α ⎞
α
⎛
αx
α
, k y + y ⎟ − Bˆ0 ⎜ k x − x , k y − y
⎢ Bˆ0 ⎜ k x +
2
2 ⎠
2
2
⎝
×⎢ ⎝
α k + α yky
⎢
iΩ + x x
⎢
k
⎢⎣
⎞⎤
⎟⎥
⎠⎥
⎥
⎥
⎥⎦ .
(0.23)
Here, B̂0 ( k ) stands for the Fourier transform of B0 ( 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 θ = ( k x k , k y k ) also represents the angle of
propagation, as long as kx and ky are small compared to k. Once a form is chosen for
B̂0 ( k ) , Eq. (6) uniquely determines the growth rate Ω as a function of the wave vector
(α
x
, α y ) 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
66
two-dimensional
growth
rates,
Ω 2 D (α x , α y ) = Ω 2 D
(
)
α x 2 + α y 2 = Ω1D (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: Bˆ0 ( k x , k y ) = Bˆ0 ( k x ) Bˆ0 ( k y ) = Bˆ0 ( 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-transversedimensional 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 α y = 0, α x = α , and solve for the case α ≥ 0
without loss of generality. Rewriting the constraint Eq. (6) using this form for B̂0 ( k ) and
these values for (α x , α y ) , we see that
1= −
ωκ
c
∫
∞
−∞
dk y Bˆ0 ( k y )
⎡ˆ ⎛
α⎞
α ⎞⎤
⎛
B0 ⎜ k x + ⎟ − Bˆ0 ⎜ k x − ⎟ ⎥ .
⎢
∞
2⎠
2⎠
⎝
⎥
× ∫ dk x ⎢ ⎝
−∞
α kx
⎢
⎥
iΩ +
⎢⎣
⎥⎦
k
67
(0.24)
Now since
B̂0 ( k )
is identical with respect to kx and ky and normalized
⎡i.e., ∞ ∞ dk Bˆ ( k ) = B ( ρ = ( 0, 0 ) ) = I ⎤ , integration over ky further reduces this
0
0⎥
∫−∞ ∫−∞ 0
⎢⎣
⎦
constraint to
⎡ˆ ⎛
α⎞
α ⎞⎤
⎛
B0 ⎜ k x + ⎟ − Bˆ0 ⎜ k x − ⎟ ⎥
⎢
ωκ ∞
2⎠
2⎠
⎝
⎥
dk x ⎢ ⎝
1= −
∫
α kx
c −∞
⎢
⎥
iΩ +
⎢⎣
⎥⎦
k
(0.25)
where B̂0 ( k ) 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 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,
Bˆ0 ( k x , k y ) =
⎡ ⎛ k2
k y2 ⎞ ⎤
I0
x
exp ⎢ − ⎜ 2 + 2 ⎟ ⎥ ,
π kx0k y 0
⎢⎣ ⎜⎝ k x 0 k y 0 ⎟⎠ ⎥⎦
(0.26)
which is realizable experimentally. By numerically solving Eq. (6) for Ω (α x , α y ) , we
find that the results are exactly identical to those obtained in the (1+1)D case using onedimensional 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 B̂0 ( k ) , the Fourier transform of the correlation function. The nonlinear change
in the refractive index is taken to be saturable and of the form ∆n = ∆nMAX ⎡⎣ I N (1 + I N ) ⎤⎦ ,
where ∆nMAX is the maximum nonlinear index change possible and I N = I I SAT , 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 α k = 0.0350 , 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 α k = 0.0350 , 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
θ 0 x ≡ ( k x 0 k ) = θ 0 y = 13.85 mrad , 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 twodimensional 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 θ 0 x = 13.85 mrad . The beam was
propagated for 1.2 mm in a material with a saturable nonlinearity ∆n = ∆nMAX ⎡⎣ I (1 + 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 B̂0 is anisotropic, that is, the
radial symmetry in the correlation statistics is broken: θ 0 x ≠ θ 0 y 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 θ 0 x ≠ θ 0 y 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
1= −
ωκ
∫
∞
∞
dk x ∫ dk y
−∞
c −∞
⎡
⎤
∂Bˆ0
⎢
⎥
αx '
⎢
⎥ ,
∂ kx '
kx =kx
⎢
⎥
×
⎢ ⎛
⎥
⎞
2
2
αx ∂ Ω ⎟
α x kx ⎥
∂Ω
⎢i ⎜ Ω + α
+
+
x
⎢ ⎜
2 ∂ α x'2 ' ⎟
k ⎥
∂ α x' '
α x =0
α x =0 ⎠
⎥⎦
⎣⎢ ⎝
(0.27)
which reduces to
1= −
ωκ
c
∫
∞
−∞
∞
dk x ∫ dk y
−∞
1 ∂Bˆ0
k x ∂k x'
.
(0.28)
k x' = k x
Equation (11) can be solved exactly for kx0 , the threshold width of the angular power
spectrum, for any form of the angular power spectrum B̂0 ( k ) . Choosing the same
double-Gaussian form as above [Eq. (9)], we find that MI will occur in the x direction if
∆nx −threshold ≡ κ I 0 ≥
n0 k x20
2k 2
;
(0.29)
thus, if the nonlinearly induced index change ∆n exceeds the threshold value on the righthand 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
∆n y −threshold ≡ κ I 0 ≥
n0 k y20
(0.30)
2k 2
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 θ 0 y = k y 0 k = 2.2 mrad , but much more widely distributed in the x
direction ( θ 0 x = k x 0 k = 9.6 mrad ). 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 n = n0 + ∆nNL I N , 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 ydirection. 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
75
of
values
for
αy
/k
(~0.03).
Figure 2. Features resulting from incoherent modulation instability for an input beam of
an elliptical double-Gaussian angular power spectrum [Eq. (9)] with θ 0 x = 9.6 mrad and
θ0 y = 2.2 mrad . The beam was propagated for 1 mm in material with a refractive index
of the form n = n0 + ∆nNL I , where n0 = 2.3 and ∆nNL = 5 x 10-4. (a) shows the intensity
of the perturbations, B1 ( r ) , in the spatial domain, with high intensity represented by
2
white shading, low by dark. (b) shows the corresponding angular power spectrum
B̂1 ( α ) , where the uniform background intensity has been subtracted out.
2
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 B̂0 ( 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
Bˆ ( k x , k y ) =
π
2
(k
I0kx0k y0
2
x0
,
+ k x2 )( k y20 + k y2 )
(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 B̂0 ( k )
in Eq. (6), one can then obtain exactly the gain curve Ω (α x , α y ) that is,
Ω (α x , α y ) = − k x 0
αx
k
− k y0
αy
k
1/ 2
⎛ κ I 0 α x2 + α y2 ⎞
2
2
+ αx +α y ⎜
−
⎟⎟
2
⎜ n
k
4
0
⎝
⎠
.
(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
∆n45o − threshold ≡ κ I 0 ≥
2n0 k02
k2
and
(0.33)
∆n0o ,90o − threshold ≡ κ I 0 ≥
n0 k02
k2
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, B0 ( r, ρ, z ) . 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
( n0 , λ , k )
described
above
but
with
a
saturable
nonlinearity,
∆n = ∆nMAX ⎡⎣ I (1 + I ) ⎤⎦ , 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 θ 0 x = θ 0 y = 12 mrad . 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, B̂1 ( α ) .
2
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 ∆n = ∆nMAX ⎡⎣ I (1 + I ) ⎤⎦ , with
∆nMAX = 1.8 x 10-3. (a) shows the intensity of the perturbations,
B1 ( r )
2
for
θ 0 x = θ 0 y = 12 mrad . 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
B̂1 ( α ) . (c) shows
2
θ0 y = 6 mrad . (d) shows B̂1 ( α ) .
2
80
B1 ( r )
2
for
θ 0 x = 12 mrad and
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 intensitydependent change in ∆n, was approximated as κ I 0 = d ⎡⎣ ∆n ( I ) ⎤⎦ dI
I0
× I 0 . 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 θ 0 x = 12 mrad and
θ0 y = 6 mrad , while all other parameters are kept the same as in the previous example (of
Fig. 3(a) and (b), where θ 0 x = θ 0 y = 12 mrad . 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
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 ∆n = ∆nMAX ⎡⎣ I (1 + 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 θ 0 x = θ 0 y = 2.2 mrad .
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 B1 ( α )
2
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 Bˆ0 ( k x , k y ) = Bˆ 0 ( k x ) Bˆ 0 ( k y ) = Bˆ 0 ( 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 θ 0 x = θ 0 y = 2.2 mrad ),
but with preferential (white) noise in the y direction that is 102 times stronger than the
noise in the x direction. (a) shows the spatial distribution of the perturbation, B1 ( r ) ,
2
after 2 mm of propagation in a material with a saturable nonlinearity of the form
∆n = ∆nMAX ⎡⎣ I / (1 + I ) ⎤⎦ , where ∆nMAX = 3.3 x 10-3, and (b) shows the corresponding
angular power spectrum, B̂1 ( α ) .
2
85
Bˆ0 ( k x , k y ) = Bˆ0 ( k x ) Bˆ0 ( k y ) ≠ Bˆ0 ( 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 twodimensional 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 B0 ( ρ ) and the linearization assumption B1
B0 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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[2] E. M. Dianov et al., Opt. Lett. 14, 1008 (1989); P. V. Mamyshev et al., J. Opt. Soc.
Am. B 11, 1254 (1994); M. D. Iturbe-Castillo et al., Opt. Lett. 20, 1853 (1995); M. I.
Carvalho, S. R. Singh, and D. N. Christodoulides, Opt. Commun. 126, 167 (1996).
[3] V. I. Bespalov and V. I. Talanov, JETP Lett. 3, 307 (1966); V. I. Karpman, ibid. 6,
277 (1967); G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987); S. Wabnitz, Phys. Rev. A 38,
2018 (1988). A. Hasegawa and W. F. Brinkman, IEEE J. Quantum Electron. 16, 694
(1980); K. Tai, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1981). For a
review on modulation instability in the temporal domain, see G. P. Agrawal, “Nonlinear
Fiber Optics”, 2nd ed. (Academic, San Diego, 1995), Chap. 5.
[4] M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys.
Rev. Lett. 84, 467 (2000).
[5] D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, Science 290,
495 (2000).
[6] J. Klinger, H. Martin, and Z. Chen, Opt. Lett. 26, 271 (2000).
89
[7] Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, Proc. Natl.
Acad. Sci. (to be published).
[8] D. Kip, M. Soljacic, M. Segev, S. M. Sears, and D. N. Christodoulides, J. Opt. Soc.
Am. B 19, 502 (2002).
[9] B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Modulational
Instability and Photon Landau Damping of Incoherent Light Wave Packets, OSA
Nonlinear Guided Waves Topical Meeting, Clearwater, Fl, March 2001.
[10] A. V. Mamaev, M. Saffman, A. A. Zazulya, Phys. Rev. A 54, 870 (1996).
[11] C. Anastassiou, M. Soljacic, M. Segev, E. Eugenieva, D. N. Christodoulides, D. Kip,
Z. H. Musslimani, and J. P. Torres, Phys. Rev. Lett. 85, 4888 (2000).
[12] V. V. Shkunov and D. Z. Anderson, Phys. Rev. Lett. 81, 2683 (1998).
[13] D. N. Christodoulides, E. Eugenieva, T. Coskun, M. Segev, and M. Mitchell, Phys.
Rev. E 63, R35601 (2001).
[14] D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett.
78, 646 (1997); 80, 2310 (1998).
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 partiallyincoherent 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 selftrapped 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 onedimensional 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 [2023] 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 timeaveraged 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 (quasimonochromatic) optical beams in non-instantaneous nonlinear media. For the
propagation medium, we choose a saturable nonlinearity of the type ∆n ( I ) =
∆no I I S
(1 + I I S )
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]:
⎛ ∂f
∂f
∂f ⎞
+θy ⎟ +
i ⎜ + θx
∂x
∂y ⎠
⎝ ∂z
1 ⎛ ∂2 f ∂2 f ⎞
+
⎜
⎟ + k 0 g ( I N ( x, y , z ) ) f
2k ⎝ ∂x 2 ∂y 2 ⎠
where
IN
=
∫∫ f ( x, y,θ
,θ y ) dθ x dθ y
2
x
and, at z = 0,
f ( z = 0, x, y,θ x ,θ y ) =
99
G (θ
1/ 2
N
x
, θ y ) φ 0 ( x, y ) .
= 0,
In the equation above, θ x and θ y are angles (in radians) with respect to the z axis, k =
kono, and k0 = 2π λ0 . The function f ( x, y,θ x ,θ y ) 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. GN (θ x ,θ y ) is the normalized
angular power spectrum of the incoherent source, and φ0 ( x, y ) 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 n 2 = no2 + 2no g ( I N ) . As previously mentioned, here the
nonlinearity is taken to be of the form g ( I N ) = ∆no I N (1 + I N ) , (saturable non-linearity).
In our simulations the linear index is no = 2.3, and the maximum nonlinear index change
is taken to be ∆n = 2.5 ×10−3 (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-1, and ko = 12.875 µm-1. The angular power spectrum is assumed
(
to be of the Gaussian type GN (θ x ,θ y ) = 1 (πθ o2 ) exp − (θ x2 + θ y2 ) θ o2
(
)
and φo ( x, y ) is
taken to be a very broad, yet finite, flat wavefront ( φo ( x, y ) = exp − ( x 2 + y 2 )
m
)
2Wo 2 m ,
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 phasecorrelated. 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 wellseparated 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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[15] Liu, X., Qian, L. J. & Wise, F. W. (1999) Phys. Rev. Lett. 82, 4631.
[16] P. Di Trapani, University of Como, Italy, presented at the Workshop on Optical
Solitons, Orlando, Florida, March 2001.
[17] Snyder, A. W., Mitchell, D. J., Polodian, L. & Ladouceur, F. (1991) Opt. Lett. 16,
112
21.
[18] Mitchell, M., Chen, Z., Shih, M., & Segev, M. (1996) Phys. Rev. Lett. 77, 490.
[19] Mitchell, M. & Segev, M. (1997) Nature 387, 880.
[20] Christodoulides, D. N., Coskun, T. H., Mitchell, M., & Segev, M. (1997) Phys. Rev.
Lett. 78, 646; ibid (1998) 80, 2310.
[21] Mitchell, M., Segev, M., Coskun, T. H., & Christodoulides, D. N. (1997) Phys. Rev.
Lett. 79, 4990; ibid (1998) 80, 5113.
[22] Snyder, A. W. & Mitchell, D. J. (1998) Phys. Rev. Lett. 80, 1422.
[23] Shkunov, V. V. & Anderson, D. Z. (1998) Phys. Rev. Lett. 81, 2683.
[24] Chen, Z., Mitchell, M., Segev, M., Coskun, T. H., & Christodoulides, D. N. (1998)
Science 280, 889.
[25] Akhmediev, N., Krolikowski, W., & Snyder, A. W. (1998) Phys. Rev. Lett. 81, 4632.
[26] Bang, O., Edmundson, D., & Krolikowski, W. (1999) Phys. Rev. Lett. 83, 4740.
113
[27] Mitchell, M., Segev, M., & Christodoulides, D. N. (1998) Phys. Rev. Lett. 80, 4657.
[28] Bespalov, V. I. & Talanov, V. I. (1966) JETP Lett. 3, 307; Karpman, V. I. (1967)
JETP Lett. 6, 277; Agrawal, G. P. (1987) Phys. Rev. Lett. 59, 880; Wabnitz, S. (1988)
Phys. Rev. A 38, 2018; Hasegawa, A. & Brinkman, W. F. (1980) J. Quant. Elect. 16, 694;
Tai, K., Hasegawa, A., & Tomita, A. (1981) Phys. Rev. Lett. 56, 135. For a review on
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 onedimensional 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.
114
(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.
115
[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).
116
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 highorder 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
117
[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.
118
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).
119
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.
120
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
121
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).
123
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