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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 39, NO. 1, JANUARY 2003
31
Spatial Optical Solitons in Waveguide Arrays
Andrey A. Sukhorukov, Yuri S. Kivshar, Hagai S. Eisenberg, and Yaron Silberberg
Invited Paper
Abstract—We overview theoretical and experimental results on
spatial optical solitons excited in arrays of nonlinear waveguides.
First, we briefly summarize the basic properties of the discrete
nonlinear Schrödinger (NLS) equation frequently employed to
study spatially localized modes in arrays, the so-called discrete
solitons. Then, we introduce an improved analytical model that
describes a periodic structure of thin-film nonlinear waveguides
embedded into an otherwise linear dielectric medium. Such
a model of waveguide arrays goes beyond the discrete NLS
equation and allows studying many new features of the nonlinear
dynamics in arrays, including the complete bandgap spectrum,
modulational instability of extended modes, different types of gap
solitons, the mode oscillatory instability, the instability-induced
soliton dynamics, etc. Additionally, we summarize the recent
experimental results on the generation and steering of spatial
solitons and diffraction management in waveguide arrays. We also
demonstrate that many effects associated with the dynamics of
discrete gap solitons can be observed in a binary waveguide array.
Finally, we discuss the important concept of two-dimensional (2-D)
networks of nonlinear waveguides, not yet verified experimentally,
which provides a roadmap for the future developments of this
field. In particular, 2-D networks of nonlinear waveguides may
allow a possibility of realizing useful functional operations with
discrete solitons such as blocking, routing, and time gating.
Index Terms—Bloch oscillations, Bloch waves, coupled-mode
theory, diffraction management, discrete diffraction, gap solitons,
modulational instability, optical waveguide networks, spatial
solitons, waveguide arrays.
I. INTRODUCTION
T
HE CONCEPT of discrete spatial optical solitons was
introduced first by Christodoulides and Joseph [1], who
studied theoretically spatially localized modes of periodic
optical structures created by arrays of equivalent optical
waveguides, based on the analogy with the localized modes
of discrete lattices [2], [3]. Since then, the discrete solitons
have been studied extensively in a number of theoretical
papers (see, e.g., [4]–[7], and also review papers [8], [9]), and
they have recently been observed experimentally in arrays of
nonlinear single-mode optical waveguides [10]–[14] (see also
[15]). A standard theoretical approach to study discrete spatial
Manuscript received March 2, 2002; revised August 5, 2002. This work was
supported in part by the Australian Research Council and the U.S. Air Force Far
East (AROFE) Office.
A. A. Sukhorukov and Yu. S. Kivshar are with the Nonlinear Physics Group,
Research School of Physical Sciences and Engineering, Australian National
University, Canberra ACT 0200, Australia.
H. S. Eisenberg and Y. Silberberg are with the Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel.
Digital Object Identifier 10.1109/JQE.2002.806184
Fig. 1. Periodic array of single-mode waveguides: SEM image (top) and
illustration of the layered structure (bottom).
optical solitons is based on the decomposition of the total field
in a sum of weakly coupled fundamental modes excited in
each waveguide of the array; a similar approach is known in
solid-state physics as the tight-binding approximation. Then,
the wave dynamics can be described in the framework of an
effective discrete nonlinear Schrödinger equation [1] that has
stationary localized solutions in the form of discrete localized
modes (see, e.g., [9]). The analogous concepts appear in other
contexts, such as the study of the nonlinear dynamics of the
Bose–Einstein condensates in optical lattices [16].
A phenomenological model suggested for describing the dynamics of a periodic array of nonlinear optical waveguides is
based on a generalization of a two-core optical coupler to the
case of a periodic array of evanescently coupled waveguides,
as shown in Fig. 1. The main assumption of the model is that
the evolution of the slowly varying envelopes of the individual
guided modes of a homogeneous array of single-mode waveguides can be described by a discrete equation that takes into account the nearest neighbor interaction of the weakly overlapping
guided modes. For lossless waveguides with a Kerr nonlinearity,
we can write a straightforward generalization of a two-mode
coupler for the case of waveguides in the form [1]
(1)
is the mode amplitude in the th waveguide (
,
is the number of waveguides, and
), is the
the boundary conditions are
where
0018-9197/03$17.00 © 2003 IEEE
32
linear propagation constant, is the coupling coefficient,
, where
is the optical angular frequency,
is the nonlinear coefficient, and
is the effective area of the
waveguide modes. This equation is referred to as the discrete
nonlinear Schrödinger equation. In a general case of arbitrary
, the system (1) possesses two conserved quantities (see, e.g.,
[3] and [6]): the total power
(2)
Fig. 2. Example of the simplest spatially localized modes of the discrete NLS
equation (1). (a) Stable odd mode. (b) Unstable even mode. Spatial profiles jE j
of the solutions are shown as a function of the lattice index n.
and the Hamiltonian (or the total system energy)
(3)
, the model equation (1) is known to be completely
For
integrable, and it describes a two-core nonlinear optical coupler
[17], [18], [103]. However, the model equation (1) is not inte, and even for
both self-trapping and
grable for
chaotic dynamics can occur (see, e.g., [19]).
Let us now consider the properties of an extended periodic
) in more details. At low powers, when the
system (
nonlinear term in (1) can be neglected, the infinite set of linear
ordinary differential equations is analytically integrable [20, pp.
519–524]. The general solution can be written using the Green’s
function, that defines the field evolution in the case when only
at
.
one waveguide is excited at the input, i.e.,
in
Under such conditions, the solution for the electrical field
the th waveguide is given by the result
(4)
is the Bessel function of the order . As the light propwhere
agates along the waveguides, the energy spreads into two main
lobes with several secondary peaks between them. The solution
under any other initial conditions will be a linear superposition
of the functions (4).
When the light intensity becomes larger, a general analytical solution cannot be obtained, since (1) is not integrable.
However, when the intensity varies slowly over adjacent waveguides (i.e., for the moderately localized solitons [8]), the discrete set of ordinary differential equations (1) can be reduced
to a single continuous NLS equation similar to that describing
conventional spatial solitons in a planar geometry [1].
In the general case, stationary solutions of (1) can be found
numerically in the form of nonlinear modes localized in a few
waveguides [21], [22], and the two simplest localized modes
are shown in Fig. 2. For each power level, there is one solution centered on a single waveguide [Fig. 2(a)] and one centered in between two waveguides [Fig. 2(b)]. Only the first is
stable and represents a local minimum of the Hamiltonian [5],
[23]. If a soliton is forced to move sideways, it has to jump
from waveguide to waveguide, passing from a stable to an unstable configuration. The difference between the Hamiltonians
in the respective cases—the so-called Peierls–Nabarro potential
(PNP)—accounts for the resistance that the soliton has to overcome during transverse propagation [23]. For increasing power
levels, the PNP increases, resulting in a strong localization of
the soliton, mainly in a single waveguide which is effectively
decoupled from the rest of the array. Although the existence of
moving discrete solitons has not been proven in a strict mathematical sense, numerical studies seem to indicate their existence
[21], [24]. Such solutions resemble discrete breathers, whose intensity profile and phase front vary periodically along the propagation direction. However, there have been no comprehensive
studies of their stability, apart from some direct numerical simulations. Nevertheless, moving solitons can be robust enough to
be observable in experiments, as we discuss in Section V below.
Numerical simulations show that the discrete solitons share
a few basic properties with conventional solitons and differ in a
few others. For example, an input beam having a linear-phase
gradient in the transverse plane can excite moving solitons,
which propagate at an angle to the waveguides. However, as
the soliton power is increased, the direction of propagation
may change, since the soliton cannot overcome PNP potential,
as discussed above. These and other intriguing properties of
the discrete solitons have initiated a number of studies of the
soliton-based optical switching and beam steering in waveguide
arrays [5], [25]–[30].
We should also note that there can also exist more complicated solutions, e.g., bound states of discrete solitons [31],
which do not have continuous counterparts [9]. One example
is “twisted” localized modes [32], [104], whose properties are
discussed in Section IV-B.
The discrete NLS equation has been a subject of many theoretical studies as the basic model for describing the dynamics
of the waveguide and fiber arrays. In the context of the fiber arrays, (1) should include the temporal dispersion characterizing
the properties of an individual fiber. The resulting semi-discrete
and semi-continuous (2 1)-dimensional NLS equation represents a system of coupled continuous NLS equations, which
is known to display a complex dynamics, even in the simpler
[33]. However, the system of
coucases such as
pled NLS equations for the fiber array was shown to possess
stable soliton-like pulses localized both in time and in spatial
domain in the direction perpendicular to the propagation direction [34]–[37]. Such multidimensional spatiotemporal solitons
can be formed through a collapse mechanism of the energy localization in an array [36]; this mechanism is associated with
the collapse suppression due to discreteness, the property first
revealed for the (1 1)-dimensional generalized NLS equation
with the power nonlinearity [38], [39].
The phenomenological theory of discrete solitons is often
based on the discrete NLS equation, and it is valid for weakly
coupled waveguides. On the other hand, it has been suggested
that in the case of strong coupling the spatial gap solitons can
SUKHORUKOV et al.: SPATIAL OPTICAL SOLITONS IN WAVEGUIDE ARRAYS
be described by coupled-mode equations [40]. It is interesting
to provide a link between the two approaches, and also consider the intermediate case. In order to do this, in the following
we introduce an improved analytical model that allows studying
spatial solitons in arrays beyond the approximation imposed by
the conventional discrete NLS equation. We also provide and
overview of the basic physics of spatial solitons in arrays of
nonlinear waveguides, as well as summarize the experimental
results and observations obtained to date, including the localized mode stability.
33
Fig. 3. Array of thin-film nonlinear waveguides embedded into a linear slab
waveguide. Gray shading shows a profile of a nonlinear localized mode.
II. MODEL AND DISCRETE DISPERSION
A. Improved Analytical Model
One of the main features of wave propagation in periodic
structures (which follows from the Floquet–Bloch theory) is
the existence of a set of forbidden bandgaps in the transmission spectrum. Therefore, the nonlinearly induced wave localization can become possible in each of these gaps. However, the
effective discrete NLS equation derived in the tight-binding approximation describes only one transmission band surrounded
by two semi-infinite bandgaps and, therefore, a fine structure
of the bandgap spectrum associated with the wave transmission in a periodic medium is lost. On the other hand, the other
well-known approach usually employed to analyze the wave
propagation in periodic structures, the coupled-mode theory for
gap solitons (see [41], [105], [42], [43], [106], [44]), describes
only the modes localized in an isolated narrow gap, and does
not consider simultaneously the gap solitons and conventional
spatial solitons, i.e., nonlinear guided waves localized due to the
total internal reflection (IR). The complete bandgap structure of
the transmission spectrum and simultaneous existence of localized modes of different types are very important issues in the
analysis of stability of nonlinear localized modes [45]. Such an
analysis is especially important for the theory of nonlinear localized modes and nonlinear waveguides in realistic models of
nonlinear photonic crystals (see, e.g., the recent paper [46]).
In this section, we introduce a simple model of waveguide
arrays that describes a periodic structure of thin-film nonlinear
waveguides embedded into an otherwise linear dielectric
medium (see also [47] and [107]). Such a structure can be regarded as a nonlinear analog of the so-called Dirac comb lattice
[48], [49], [108], where the effects of the linear periodicity and
bandgap spectrum are taken into account explicitly, whereas
nonlinear effects enter the corresponding matching conditions
allowing a direct analytical study.
We consider the electromagnetic waves propagating along the
direction of a slab-waveguide structure created by a periodic
array of thin-film nonlinear waveguides (see Fig. 3). Assuming
that the field structure in the direction is defined by the linear
guided mode of the slab waveguide, where this guide can be
weakly inhomogeneous in the transverse direction , we separate the dimensions presenting the electric field as
c.c.
(5)
describes
where is the angular frequency, the function
the fundamental mode (TE) of the slab waveguide with electric
(the effects of inhomogeneities are neglected
permittivity
at this stage), and
is the corresponding linear propagation
describes the scalar
constant. The complex function
wave envelope which accounts for the effects of nonlinear selfaction, diffraction, and linear refraction in the direction. We
assume that the modulation is weak, and that the wave vector
is almost parallel to the propagation coordinate , so that the
spatial evolution of the complex wave envelope can be described
in the frames of the parabolic approximation, rather than the full
set of Maxwell’s equations [50]. Then, we seek solution in the
form of (5), average over the transverse direction , and finally
obtain a (1 1)-D nonlinear Schrödinger equation
(6)
where is the speed of light in vacuum and the effective variation of electric permittivity is found to be
(7)
defines spatial and intensity-dependent
where the function
(due to high-frequency Kerr effect) variations of the permittivity.
In order to reduce the number of parameters, we normalize
is the complex field
(6) as follows:
and
are dimensionenvelope,
and are the characteristic transverse
less coordinates, and
scale and field amplitude, respectively. Then, the normalized
nonlinear Schrödinger equation has the form
(8)
Here,
is the normalized intensity, and the real function
(9)
describes both nonlinear and inhomogeneous properties of an
optical medium.
We note that the system (8) is Hamiltonian and, since we are
considering a lossless medium, for spatially localized solutions,
it conserves the total power as follows:
34
We look for stationary localized solutions of the normalized
equation (8) in the standard form
(10)
where
is the propagation constant, and the amplitude function
satisfies the stationary nonlinear equation
(11)
If there is no energy flow along the transverse direction , then
is real, up to a constant phase which can be
the function
. This is always
removed by a coordinate shift
the case for spatially localized solutions with vanishing asymp.
totics,
To simplify our analysis further, we assume that the linear
periodicity is associated only with the presence of an array of the
thin-film waveguides, as illustrated in Fig. 3. If the width of the
wave envelope is much larger than the thickness of the layers,
then only the lowest-order guided waves, which profiles do not
contain nodes, can be exited in each of the waveguides. Then,
the response function in the model (8) can be approximated as
follows:
cident at the normal angle to the periodic structure. In this configuration, IR cannot occur, and therefore, those results cannot
be directly applied to our case.
B. Diffraction Properties and Discrete Equations
First, we present the stationary modes defined by (11) and
(12) as a sum of the counter-propagating waves in each of the
linear slab waveguides
(15)
. Then, we express the coefficients
where
and
in terms of the wave amplitudes at the nonlinear layers
(16)
. Finally, we substitute (15) and (16) into
where
(11), (12), and (14), and find that the normalized amplitudes
satisfy a stationary form of the discrete NLS
equation
(17)
(12)
is the spacing between the neighboring thin-film
where
is an integer, and the
waveguides (the lattice period),
delta-function amplitudes are found to be
(13)
assuming that the intensity is constant inside the integration
outside the waveguides. It follows that
range, and
if the guiding layer has an effectively higher refractive
if the index is
index than the surrounding structure, and
lower. In the following, we imply that the waveguides possess a
cubic (or Kerr-type) nonlinear response, i.e.,
(14)
where the real parameters and describe both linear and
nonlinear properties of the layer, respectively. Under proper
scaling of the dimensionless wave amplitude, the absolute value
of the nonlinear coefficient can be normalized to unity, so that
corresponds to self-focusing and
to self-defo) defines the
cusing nonlinearity. The linear coefficient (
low-intensity response, and it characterizes the corresponding
coupling strength between the waveguides.
The structure modeled by (12) can be regarded as a nonlinear
analog of the so-called Dirac comb lattice [48], [49], [108],
where the effects of the linear periodicity and bandgap spectrum
are taken into account explicitly, while nonlinear effects enter
the corresponding matching conditions, thus allowing a direct
analytical study. A delta-function approximation has also been
successfully employed in the study of the bandgap structure of
photonic superlattices [51] and in the analysis of gap solitons
in nonlinear multilayered structures [52]. We note that, in [52],
analysis was performed for the case when the input wave is in-
where
, and
(18)
According to the Floquet–Bloch theory [53], linear solutions
of (11) (with vanishing intensity) can be represented as a superposition of Bloch waves (BWs). Such an approach has been earlier used in optics to describe the propagation of light in corrugated planar waveguides both in time [54] and space [55], [56],
and also in a two-dimensional (2-D) optical lattice [57]. The BW
profiles possess the symmetry property
(19)
is the BW number and is an integer. By comparing
where
vanishes) and (19), we immedi(17) (when the term
ately come to the conclusion that the BW number satisfies the
relation
(20)
can exist for
Therefore, extended linear solutions with real
. On the other hand, nonlinear localized modes with
exponentially decaying asymptotics can appear only for
(when
is imaginary), this condition defines the bandgap
structure of the spectrum. Characteristic dependencies of ,
, and
versus the propagation constant are presented in
Fig. 4(a)–(c), where the bands are shown by gray shading. The
first (semi-infinite) bandgap corresponds to the total IR. On the
other hand, at smaller , the spectrum bandgaps appear due
to the resonant Bragg-type reflection (BR) from the periodic
structure.
Following the standard definition, we can introduce the effective diffraction coefficient as follows:
(21)
35
In general, the solutions of this eigenvalue problem fall into one
of the following categories: 1) internal modes with real eigenvalues that describe periodic oscillations (“breathing”) of the
localized state; 2) instability modes that correspond to purely
imaginary eigenvalues with
; and 3) oscillatory unstable modes that appear when the eigenvalues are complex (and
). Additionally, there can exist decaying modes when
. However, from the structure of the eigenvalue equations, it follows that the eigenvalue spectrum is invariant with
, if all the amplitudes
respect to the transformation
are real, where is an arbitrary constant phase. In such a
case exponentially growing and decaying modes always coexist,
and the latter do not significantly affect the wave dynamics.
It is important to note that the grating properties are exand
in both stationary
pressed through the functions
and perturbation equations. Therefore, the functions
and
fully characterize the existence and linear stability
properties of localized and extended stationary solutions of the
original model (8), (12), and (14).
D. Analytical Approximations
D
Fig. 4. Characteristic dependencies of the parameters , , K , and on the
propagation constant . Gray shading marks the transmission bands. The lattice
parameters are h = 0:5 and = 10.
Characteristic dependence of the diffraction coefficient on the
propagation constant is presented in Fig. 4(d). We see that the
sign of the diffraction coefficient is inverted in the outer parts
of the Brillouin zone, and therefore light beams can experience
anomalous diffraction, i.e., of the opposite sign to that experienced in homogeneous structures. Moreover, diffraction com. Recently, a
pletely disappears around the points
similar effect was shown to happen in photonic crystals [58].
The remarkable properties of waves in waveguide arrays make
it possible to manage diffraction in specially engineered structures, as we discuss in Section V-B.
1) Tight-Binding Approximation: When each of the
thin-film waveguides of the periodic structure supports a fundamental mode that weakly overlaps with the similar mode of
the neighboring waveguide, the modes become weakly coupled
via a small change of the refractive index in the waveguides.
Then, the mode properties can be analyzed in the framework of
the so-called tight-binding approximation, well developed for
the problems of solid-state physics [59].
In order to employ this approximation in our case, first we
analyze the properties of a single thin-film waveguide in the
to find the
linear regime and solve (11) with
spatial profile of a linear guided mode
To study linear stability of localized modes, we should consider the evolution of small-amplitude perturbations of the localized state presenting the solution in the form
and the corresponding value of its propagation constant
. Second, we consider the interaction between the waveguides in the array assuming that the total field can be presented
as a superposition of slightly perturbed waves localized at the
isolated waveguides. Specifically, we assume that the propa(i.e.,
gation constant remains close to its unperturbed value
), and neglect small variations in the
spatial profiles of the localized modes. Then, we seek a general
solution of (8) and (12) in the form
(22)
(24)
From (8) and (12), we can obtain the linear eigenvalue problem
and
. Following the procedure outlined
for small
above, we reduce this problem to a set of the discrete equations
for the amplitudes at the layers
In such an approximation, the wave evolution is characterized
only. In order to find the corby the amplitude functions
responding evolutionary equations, we substitute (24) into (8)
,
and (12) and, multiplying the resulting equation by
integrate it over the transverse profile. According to the original
assumption of weakly interacting waveguides (valid for
and
), in the lowest order approximation, we
and neglect the overlap integrals
have
, with
, which
produce the terms of higher orders. Finally, we derive a system
C. Linear Stability
(23)
36
of coupled discrete equations for the field amplitudes at the
nonlinear layers [up to small perturbations, since
]
(25)
Stationary solutions of (25) have the form
are given by (17), with
the amplitudes
, where
(26)
Relations (26) present a series expansion of the original dispersion relations (18) near the edges of the first transmission band,
.
for
2) Coupled-Mode Theory: Now we consider the opposite
when the first BR gap is narrow, i.e.,
limit
, where
are the propagation constants at the
. From (18), it
gap edges, defined by the condition
and
.
follows that
To find solutions close to the BR gap, we present the total field
in the form
(27)
are unknown nonlinear amplitudes and
where
are the linear Bloch functions which satisfy (11) and
. The Bloch functions can be found
(12) at
, and
in an explicit form:
for
. Note that the field amplitudes at the layers
, since
.
are
,
In order to find the equations for the amplitudes
we substitute (27) into the original model equations (8) and
(12). Next, we use the fact that the gap is narrow, and close
to its edges the Bloch functions are weakly modulated, i.e.,
. This assumption allows us to keep only
the lowest order terms. Then, we multiply the resulting equation
and integrate it over one grating period. Finally,
by
the coupled equations for the modulation amplitudes are
(28)
The equations in (28) allow a direct comparison between the
coupled-mode theory and the general results for the stationary
. Addilocalized solutions for which
tionally, since the functions are weakly modulated, the spatial
derivatives can be approximated by finite differences between
the amplitudes at the layers (note that such an approximation is
). After simple algebra, we obtain a disonly valid for
crete equation for the field amplitudes at the layers which have
the form of the discrete NLS equation (17) with
(29)
Similar to the case of the tight-binding approximation, the dispersion relations (29) can be found from the general result (18)
by performing a series expansion near the band-edge value .
3) A Two-Component Discrete Model: The principal limitations of both the tight-binding approximation and the coupled-mode theory are explained by the fact that they are both
valid in local narrow regions of the general bandgap structure
and under special assumptions. Indeed, these two approaches
is either
are applicable when the dimensionless parameter
small or large, and none of those approaches covers the intermediate cases. While a general study should rely on the numerical solutions with the exact dispersion relations (18), it is useful
to consider a simplified model which can (at least qualitatively)
) in the
describe the wave properties close to the BR gap (
as well. To achieve
transitional region, being valid for
this goal, we extend the tight-binding approximation and the
corresponding discrete NLS equation introduced above in Section II-D1.
We note that (25) can be considered as a rough discretization
of the original model (8), with only one node per grating period
. Then, the natural generalization is to include
located at
additional nodes located between the nonlinear layers at the po. Since the refractive indices at the node
sitions
position are now different, we obtain a new system of coupled
discrete equations which correspond to a two-component superlattice
(30)
We find that, similar to the general case, the stationary mode
profiles can be expressed in terms of the amplitudes , which
satisfy the normalized discrete NLS equation (17) with the following parameters:
(31)
One can immediately see that our new model (30) describes
an effective system with a semi-infinite IR gap and a BR gap
of a finite width. We note that, although dispersion relations in
(31) and (29) look similar, the latter relation is only valid for
, so that the coupled-mode theory describes only a
single isolated BR gap. Therefore, the model (30) provides an
important generalization of the discrete NLS theory, which also
has a wider applicability than the coupled-mode theory.
The three different approximate models discussed above
allow a simple analysis of the limiting cases, and they will be
used below in calculating different properties of the extended
and localized modes.
III. MODULATIONAL INSTABILITY
In the previous section, we demonstrated a relation between
the original model and various approximate theories, and discussed their validity. At this point, it is also important to comment on the limits of applicability of the Kronig–Penney model.
This model provides an adequate description if the refractive
index of the medium in between the waveguides remains almost unchanged. This can happen if either: 1) the underlying
medium is linear or 2) if most of the energy is localized in the
waveguides, so that a nonlinear correction to the refractive index
outside the waveguides is minimal. It is because of these assumptions that the second equations in models (28) and (30) are
linear.
In a nonlinear medium, condition 2) is valid for solitons
or nonlinear BWs, which have propagation constants in the
vicinity of the first transmission band. Then, one might wonder
what is the benefit of using a more complicated model instead
of the conventional DNLS equation? As we demonstrate in the
following, nonlinear waves can exhibit oscillatory instabilities,
which appear due to a resonant coupling between different
bands, resulting in excitation of side-band spatial frequencies.
The possibility of such a resonant coupling is determined by
dispersion properties of linear waves in higher order transmission bands, and therefore cannot be studied within the frames of
a conventional DNLS model, which only accounts for a single
transmission band. On the contrary, the Kronig–Penney model
adequately describes the properties of linear waves in multiple
bands, which may be responsible for oscillatory instabilities.
A. General Analysis
First, we analyze the properties of the simplest extended (plane-wave) solutions of the model [(8), (12), and
(14)] which have equal intensities at the nonlinear layers,
and correspond to the first transmission
band. These solutions have the form of BWs which satisfy
, where
is
the symmetry relation (19), i.e.,
the real wave number. Using (17), we find that the nonlinear
dispersion relation has the form of (20), where the function
is replaced with
, which is calculated for a periodic
structure with the layer response modified by nonlinearity, i.e.,
. Since the transmission bands are defined by the
(see Section II-B), the band structure shifts
condition
as the intensity increases. Indeed, by resolving the dispersion
relation, we determine a relation between the propagation
constant and the wave intensity
(32)
, we have
Since in the first transmission band
(see also Fig. 4), and it follows from (18) that the
propagation constant increases at higher intensities in a self), and decreases in a self-defocusing
focusing medium (
).
medium (
One of the main problems associated with the nonlinear
BWs is their instability to periodic modulations of a certain
wavelength, known as modulational instability (see [60], and
references therein). In order to describe the stability properties
of the periodic BW solutions, we analyze the evolution of
weak perturbations described by the eigenvalue problem
, it
(23). Due to periodicity of the background solution
follows from the Bloch theorem that the eigenmodes of (23)
and
should also be periodic, i.e.,
37
. Then, we obtain the following
solvability condition:
(33)
Possible eigenvalues are determined from the condition that
the spatial modulation frequencies , which are found from
(33), are all real. Therefore, the eigenvalue spectrum consists
of bands, and the instability growth rate can only change
continuously from zero to some maximum value. We also note
, and it is
that the spectrum possesses a symmetry
.
sufficient to study only the solutions with
B. Stability of Staggered and Unstaggered Modes
In what follows, we consider two characteristic cases of the
)
stationary nonlinear BWs, when they are unstaggered (
). To describe a transition from purely real
or staggered (
to complex linear eigenvalues, we consider the function
defined from (33), and extend the solution from the real
axis to the complex plane by writing a series expansion:
, where the prime denotes differentiation with respect to the argument. Since the
modulation frequency should remain real, the second term in
the series expansion should vanish. Then, we conclude that
complex eigenvalues only appear at the critical points, where
(34)
Therefore, (in)stability can be predicted by studying the funcon the real axis only, and then extending the solution
tion
to the complex plane at the critical points (if they are present)
determined by (34). The real modulation frequencies are found
in the interval
; therefore, modas
ulational instability only appears at the critical points where
or
(see an example in Fig. 5).
Modulational instability of the nonlinear BWs in a periodic
medium has been earlier studied for the Bose–Einstein condensates in optical lattices [61]–[63] in the mean-field approximation based on the Gross–Pitaevskii equation, which is mathe, where
matically equivalent to (8) with
defines the periodic potential. It was demonthe function
strated that the unstaggered modes are always modulationally
), and
unstable in a self-focusing medium (
). It can be
they are stable in a self-defocusing medium (
shown that the similar results are also valid for our model. Indeed, since the unstaggered waves are the fundamental modes of
the self-induced periodic potential, oscillatory instabilities can
not occur, and modulational instability can only correspond to
purely imaginary eigenvalues . Such an instability should ap, which are
pear at the critical points defined by (34) at
. Therefore, the range of the unstable modfound as
,
,
ulation frequencies is
,
.
and
38
(a)
(b)
Fig. 5. Modulation frequency q versus (a) real and (b) imaginary parts of the
perturbation eigenvalue 0 for the staggered BWs in a self-focusing medium
(K = , = 7, = 3, h = 0:5, = +1). Solid lines: stable (Im 0 = 0).
Dashed lines: unstable (Im 0 =
6 0).
According to (32), for unstaggered modes we have
, and the (in)stability results follow immediately. We note
) the modulational inthat at small intensities (when
stability in a self-focusing medium corresponds to long-wave
modulations.
)
The staggered BWs in a self-defocusing medium (
are also modulationally unstable [61]–[63]. This happens
because the staggered waves experience effectively anomaand,
lous diffraction [12]. Such waves exist for
similar to the case of unstaggered waves in a self-focusing
medium, we identify the range of unstable frequencies cor),
responding to the purely imaginary eigenvalues (
,
, and
,
.
Finally, we analyze stability of staggered BW modes in a
). Note that the stability propself-focusing medium (
erties of such waves have not been fully analyzed, and it was
only noted that such waves may be stable [62, Sec. III-B2].
, the domain
Since such modes exist for
at
does not correspond to physically possible
modulation frequencies. However, the oscillatory instabilities
(i.e., those with complex ) can appear due to resonances between the modes that belong to different bands. Such instabilities appear in a certain region of the wave intensities in the
case of shallow modulations, below a certain threshold value
), as shown in Fig. 6. We find the fol(when
lowing asymptotic expression for the low-intensity instability
threshold:
(35)
while the upper boundary is given by the relation
(36)
These analytical estimates are shown by dashed lines in Fig. 6.
It is interesting to compare our results with those obtained
in the framework of the continuous coupled-mode theory (see
Section II-D2), valid for the case of a narrow bandgap (i.e., for
small and small ). Although the nonlinear coupling coefficients in (28) are different compared to a coupled-mode model
for shallow gratings [64], the key stability result remains the
Fig. 6. Modulationally unstable staggered BW modes (gray shading) in a
self-focusing medium, shown as the intensity I versus the parameter (at
h = 0:5, = +1). Dashed lines are the analytical approximations (35) and
(36) for the low- and high-intensity instability thresholds, respectively.
same, and the oscillatory instability appears above a certain critical intensity proportional to the bandgap width. In our case,
, and we observe good
the bandgap width is
agreement with the results of the coupled-mode theory. However, the coupled-mode model (28) cannot predict the stability
region at high intensities, since in this region the approximation
is no longer valid.
In the limit of large , the BW dynamics can be studied
with the help of the tight-binding approximation (see Section II-D-1). Then, the effective discrete NLS equation (25)
predicts the stability of the staggered modes in a self-focusing
medium [65]–[67]. Numerical and analytical results confirm
that our solutions are, indeed, stable in the corresponding
parameter region.
The two-component discrete model introduced in Section II-D3 predicts the existence of oscillatory instabilities of
the BWs in a certain region of . Importantly, the model (30)
predicts the key pattern of oscillatory modulational instability:
1) instability appears only for a finite range of intensities
when the grating depth ( ) is below a critical value and 2)
modulational instability is completely suppressed for large .
Thus, unlike the discrete NLS equation, the two-component
discrete model (30) predicts qualitatively all major features of
modulational instability in a periodic medium.
We find that, in a self-focusing medium, the staggered waves
) are always stable with respect to low-frequency mod(
ulations. However, at larger intensities, unstable frequencies appear close to the middle and the edge of the Brillouin zone,
) where the inand they are shifted toward the edge (
stability growth rate is the highest. The corresponding modulational instability manifests itself through the development of
the period-doubling modulations, as shown in Fig. 7.
IV. BRIGHT AND DARK SOLITONS
A. Odd and Even Localized Modes
Stationary localized modes in the form of discrete bright solitons can exist with the propagation constant inside the bandgaps,
. Additionally, such solutions can exist only if
when
the nonlinearity and dispersion sign are different, i.e., when
. It follows from (18) that
and
in
the IR gap and the first BR gap (see also Fig. 4), so that the type
Fig. 7. Development of the instability-induced period-doubling modulations.
Initial profile corresponds to a slightly perturbed staggered mode. Parameters
,h
: ,
, and I
: .
are = 3 = 0 5 = +1
' 29 87
of the nonlinear response is fixed by the medium characteris. Therefore, self-focusing nonlinearity
tics, since
),
can support bright solitons in the IR region (where
i.e., in the conventional wave-guiding regime. In the case of the
self-defocusing response, bright solitons can exist in the first BR
gap, owing to the fact that the sign of the effective diffraction is
). In the latter case, the mode localization occurs
inverted (
in the so-called anti-waveguiding regime.
Let us now consider the properties of two basic types of the
localized modes: odd, centered at a nonlinear thin-film waveguide, and even, centered between the neighboring waveguides,
, where
, respectively. For
so that
discrete lattices, such solutions have already been studied in the
literature (see, e.g., [23]), and it has been found that the mode
) if
. On the other
profile is “unstaggered” (i.e.,
hand, (17) possesses a symmetry
(37)
.
which means that the solutions become “staggered” at
Because of this symmetry, it is sufficient to find localized soluand
. The earlier suggested
tions of (17) for
approximate solutions give accurate results only in the case of
) (see [9], [68], and references
highly localized modes (
) (see, e.g., [6]).
therein) and in the continuous limit (
Recently, a new approach was suggested that can be used to accurately describe the mode profile for arbitrary values of [69].
Our linear stability analysis reveals that even modes are always unstable with respect to a translational shift along the
axis. On the other hand, odd modes are always stable in the
self-focusing regime (see Fig. 8, top), but can exhibit oscillatory instabilities in the self-defocusing case when the power exceeds a certain critical value (see Fig. 9, top). At this point, an
eigenmode of the linearized problem resonates with the bandgap
moves inside the band, and a nonzero
edge, the value
imaginary part of the eigenvalue appears. Such an instability
scenario is similar to one earlier identified for gap solitons [70],
and also for the modes localized at a single nonlinear layer in a
linear periodic structure [45].
39
Fig. 8. Top: power versus propagation constant for odd (black) and even
(dashed gray) localized modes in a self-focusing (
) regime. Solid
line: stable. Dashed line: unstable. Gray shading marks the transmission band.
Bottom: profiles of the localized modes corresponding to the marked points
(a) and (b) in the top plot. Black dots: analytical approximation for the node
amplitudes. The lattice parameters are the same as in Fig. 4.
= +1
=
Fig. 9. Top: power versus propagation constant in the self-defocusing (
) regime. Dotted line: oscillatory unstable modes. Other notations are the
same as in Fig. 8.
01
B. Soliton Bound States: “Twisted” Modes
Due to a periodic modulation of the medium refractive index,
solitons can form bound states [31]. In particular, the so-called
“twisted” localized mode, first predicted for the discrete NLS
equation [32], [104], [71], [72], is a combination of two out-ofphase bright solitons. Such solutions do not have their continuous counterparts, and they can only exist when the discreteness
. Properties of the twisted
effects are strong, i.e., for
modes depend on the separation between the modes forming a
bound state. We consider the cases of two lowest order solu) in-between
tions of: 1) “even” type with zero nodes (
) at the
the peaks and 2) “odd” type with one node (
. The corresponding symmetry properties
middle, with
. Additionally, (37) also holds,
are
.
so that we only have to construct solutions for
We determine twisted-mode solution starting with the highly
localized modes earlier described anti-continuous limit (
40
) regime. Notations
(gray) twisted localized waves in a self-focusing (
are the same as in Figs. 8 and 9.
= +1
) [32], [104], and then gradually decrease the absolute value of
,
parameter . We find that a solution exists for
where the critical parameter values are
and
[69]. The characteristic profiles of the
twisted modes are shown in Fig. 10(a) and (b).
In the self-focusing regime, stability properties of the twisted
modes in the IR gap can be similar to those earlier identified in
the framework of the discrete NLS model [32], [104], [71], [72].
In the example shown in Fig. 10, the modes are stable at larger
values of the propagation constant, and they become oscillatory
unstable closer to the boundary of the existence region. Important to note is that the stability region is much wider in the case
of odd twisted modes due to a larger separation between the individual solitons of the bound state. Oscillatory instabilities of
twisted modes are associated with an exponential increase of
periodic amplitude modulations and the emission of radiation
waves.
C. Dark Solitons
Similar to the continuous NLS equation with self-defocusing
nonlinearity [73] or the discrete NLS equation [74]–[76],
the analytical model under consideration supports dark solitons—localized modes on the BW background. However, dark
stationary localized modes in a periodic medium can exist for
both signs of nonlinearity, and this is a direct manifestation
of the anomalous diffraction observed in the systems with
periodically varying parameters.
To be specific, let us consider the case of a background
introduced
corresponding to the BW solutions with
in Section III. Then, dark-mode solutions can appear at the
, since in such a case
bandgap edge where
nonlinear and dispersion terms have the same signs [73].
In analogy with the bright solitons discussed above, two basic
types of dark spatial solitons can be identified; namely, odd
localized modes centered at a nonlinear thin-film waveguide,
and even localized modes centered between the neighboring
thin-film waveguides [74], [75]. All such modes satisfy the sym, where
metry condition
for even and odd modes, respectively. The BW background is
Fig. 11. Top: complementary power versus propagation constant for odd
(black) and even (gray) dark localized solitons in a self-focusing (
)
regime. Notations are the same as in Fig. 8, but (in)stability regions are not
indicated.
= +1
unstaggered if
, and it is staggered for
; the corresponding solutions can be constructed with the help of a symmetry transformation (37). However, the stability properties of
these two types of localized states can be quite different. Indeed,
it has been demonstrated in Section III that in a self-focusing
) the staggered background can become unmedium (
stable. On the contrary, the unstaggered background is always
.
stable if
Two types of dark spatial solitons in our model are shown for
the staggered Bloch-wave backgrounds in Fig. 11. We characterize the family of dark solitons by the complimentary power
defined as
where is integer.
Numerical study of the propagation dynamics demonstrates
that, similar to the case of bright solitons, even dark-soliton
modes are unstable with respect to asymmetric perturbations.
On the other hand, odd modes can propagate in a stable (or
weakly unstable) manner. Dark solitons can exhibit oscillatory
instabilities close to the continuum limit [75] (at small intensities), but a detailed analysis of the discreteness-induced oscillatory instability has been performed only for dark solitons of
the discrete NLS equation [75] and unbounded standing waves
[77].
V. EXPERIMENTAL OBSERVATIONS
A. Self-Focusing and Spatial Solitons
It is difficult to control diffraction in a bulk medium or a
planar waveguide, since distortion of the phase front is a geometrical phenomenon. However, one way to gain control over
diffraction is to let the light propagate in a periodic medium
such as an array of coupled waveguides. In this case, the light
expansion is due to coupling between the neighboring waveguides, and it is obvious that expansion rate (the so-called “discrete diffraction”) is controlled by the strength of the coupling
Fig. 12. Experimental setup for discrete soliton observation. Inset: schematic
drawing of the sample (see also Fig. 1) [10].
and, therefore, light diffracts slowly in a weakly coupled array.
Moreover, even the sign of diffraction can be chosen in a given
array by controlling the input conditions. Those properties are
a direct manifestation of the discreteness of the system under
consideration; many of them were observed experimentally as
is discussed below.
Most of the experiments to date have been performed in waveguide arrays made of the AlGaAs (see, e.g., [10]), which has
been the main material system for spatial soliton research in slab
waveguides (see a recent review [78]). Many similar phenomena
have also been verified for other experimental systems, such as
an array of single-mode polymer waveguides [15]. In the case
of the arrays made of AlGaAs, 4- m-wide single-mode waveguides have been created by means of the reactive-ion etching.
The arrays typically contained 40–60 waveguides. The strength
of coupling between the neighboring waveguides was controlled
through the spacing between the waveguides, that varied between 2 and 7 m (see inset in Fig. 12). The light source in these
experiments was a synchronously pumped optical parametric
oscillator (Spectra-Physics OPAL system) which emits pulses in
the 100–200-fs range, tunable near the wavelength of 1.5 m,
which is below the half-bandgap of the semiconductor, hence
reducing nonlinear absorption. The short pulses were needed in
order to reach the required peak powers, and their maximum
peak power was 1.5 kW.
The first experimental study of the formation of discrete solitons in this system was reported in classical experiments by
Eisenberg et al. [10]. Light was launched into a single waveguide on the input side of a 6-mm-long sample, and the light
distribution at the output end was recorded, as shown schematically in Fig. 12. At low light power, the propagation is linear,
and the light has expanded over some 30 waveguides (top row
in Fig. 13). From this distribution, it is possible to determine the
coupling length, about 1.4 mm. As the input power is increased,
the output distribution shrinks, as shown in Fig. 13. At a power
of 500 W, light is confined to about five waveguides around the
input waveguide.
When light is launched into a single waveguide, the soliton
that is eventually formed propagates along the array. However,
discrete solitons do not have to propagate in the direction of the
waveguides. When the input is into several guides, by imposing
41
Fig. 13. Images of the output facet of a 6-mm-long sample with 4-m
spacing between waveguides for different powers. (a) Peak power 70 W. Linear
propagation—light is diffracting in two main lobes and a few secondary peaks
in between. (b) Peak power 320 W. Intermediate power—the distribution is
narrowing. (c) Peak power is 500 W. A discrete soliton is formed [10].
a linear phase slope at the input side, the transverse momentum
leads to transverse motion of the soliton across the waveguides.
However, while the slab soliton is rotationally and translationally invariant, the discrete soliton is not. It behaves differently
when it is launched in different directions and from translated
locations. A demonstration of these properties was reported by
Morandotti et al. [11], who measured that the output location of
the solitons shifted quite drastically by imposing small changes
on the input.
According to the theory, there are two types of elementary
(simplest) solitons that propagate along the waveguides, one
centered on a single waveguide and one centered in between
two waveguides. In an experiment performed by Morandotti et
al. [11], solitons were steered sideways due to the asymmetry in
the input coupling. The input distribution without any phase tilt
was scanned across the input facet, and the output distribution
was recorded. Whenever the input distribution was centered on
a waveguide, or exactly between two waveguides, the soliton
propagated in the waveguide direction without any transverse
motion.
However, when the input condition was not symmetric, a
transverse motion was excited due to a phase tilt that was induced through the Kerr effect. The periodic motion of the output
location versus the input location is clearly observed in Fig. 14.
Also obvious is the higher sensitivity to the input location of the
unstable soliton state (dashed lines in Fig. 14).
B. Diffraction Management
The beam diffraction in an array can be controlled through
the choice of input parameters, and, under some conditions, the
sign of diffraction can be reversed. This anomalous diffraction
seems a counterintuitive phenomenon: it means that high (spatial) frequency components have slower transverse velocity as
compared with low frequency components. In discrete diffraction, this happens when the input is excited with a tilt so that adjacent waveguides are excited out of phase, and the BW number
is close to the outer edge of Brillouin zone (see the discussion in Section II-B). As far as linear optics is concerned, both
“normal” and “anomalous” diffraction leads to broadening of
a finite width beam. However, while normal diffraction leads
to self-focusing, anomalous diffraction leads to self-defocusing,
42
Fig. 14. Soliton steering by an internally induced velocity. Output field
distribution for fixed input peak power I
1500 W as a function of
the input beam position. (a) Experimentally recorded power distributions for
two input positions relative to the central guide. (b) Output field distributions
for various positions of the initial excitation. (Solid line: beam centered on a
waveguide. Dashed lines: input beam centered in between two waveguides).
(c) Simulation of (b) [11].
that is, even faster broadening at high optical powers. This result
was recently demonstrated by Morandotti et al. [13] in a specially designed sample with 61 waveguides, each 3 m wide,
separated by 3 m.
The experimental results reported in [13] allowed us to compare nonlinear self focusing and self defocusing in the same
array of waveguides, for slightly different initial conditions. For
normal dispersion, i.e., when a light beam is normal to the input
facet, the beam slightly broadens through discrete diffraction.
W), the field shrinks and evolves
At high power (
into a discrete bright soliton, as in Fig. 13. However, the beam
injected at an angle of 2.6 0.3 inside the array, experiences
anomalous dispersion, and it broadens significantly due to self
defocusing. In this latter case, the adjacent waveguides have the
phase difference.
The evolution of the input beam that has a phase flip across its
center, as shown in Fig. 15(a), was also analyzed in the regimes
with effectively positive and negative diffraction [13]. In the
normal diffraction regime, a pair of out-of-phase bright solitons
was excited [see Fig. 15(b) and (c)], and these can form a bound
state, as discussed in Section IV-B. On the other hand, in the
anomalous diffraction regime, the central region resembles the
core of a dark soliton, while the background broadens due to the
self-defocusing effect, as shown in Fig. 15(d) and (e).
By using the diffraction properties of waveguide arrays,
Eisenberg et al. [12] proposed a scheme to produce structures
with designed diffraction. They fabricated arrays with reduced,
Fig. 15. Generation of discrete solitary wave. (a) An input beam with a phase
shift in the center. Normal diffraction regime at (b) low and (c) high powers.
Anomalous diffraction at (d) low and (e) high powers [13].
canceled, and even reversed diffraction. Results of experiments
with such waveguides were presented in [12] and compared
with the predictions made by the discrete NLS equation and the
coupled-mode theory. Instead of a tilted input beam, the beam
is launched normally to the waveguides, but the arrays are tilted
from the normal to the input facet to achieve easy coupling
conditions into a specific range of modes.The angles corre(where
sponded to various values of the parameter
is the distance between the centers of two adjacent waveguides
is the transverse wave number) vary between 0 and ,
and
where in the experiment configuration the maximum value is
.
equivalent to a tilt angle of
Motivated by recent experimental observations of anomalous
diffraction in linear waveguide array, Ablowitz and Musslimani
[79] proposed a novel model governing the propagation of an
optical beam in diffraction-managed nonlinear waveguide arrays. This model supports discrete spatial solitons whose beam
width and peak amplitude evolve periodically. A nonlocal integral equation governing the slow evolution of the soliton amplitude has been derived, and its stationary soliton solutions
have been obtained [79]. Importantly, this new discrete equation governing the evolution of an optical beam in a waveguide
array with varying diffraction has a novel type of discrete spatial
soliton solution which breathes under propagation and, as a result, gains a nonlinear chirp. A full recovery of the soliton initial
power is achieved at the end of each diffraction map, similar to
Fig. 16. Experimentally measured field profiles at the output for an array with
a defect at the central guide. (a) Low power. (b) High power (1-kW coupled
power). (c) Sketch of the waveguides [83].
the well-known properties of the dispersion-managed solitons
(see, e.g., a review [80]). This opens the possibility of fabricating a customized waveguide array which admits specialized
diffraction-managed spatially confined solitary waves.
C. Defect-Induced Localization
Waveguides created in periodically modulated structures
may posses many interesting properties. For example, it is
possible to guide waves through a low-index core with no
radiation losses, which cannot be achieved with conventional
IR waveguides [81], [82]. This type of localization can only
occur in the BR gap, when the sign of diffraction is effectively
inverted, as discussed above. Such a phenomenon was shown to
occur in a nonuniform waveguide array [83], where the width
of a single waveguide was reduced, as illustrated in Fig. 16(c).
The defect appears mainly as a reduction of the effective index
of the corresponding guide. At low powers, this defect layer
supports linear guided waves in the BR regime, so that their
profiles resemble those of gap solitons shown above in Fig. 9.
Indeed, pronounced intensity dips between the neighboring
waveguides were observed at the output of the waveguide array
[see Fig. 16(a)].
It was found in theoretical studies [84], [45] that defect
modes, localized in the BR regime, can exhibit oscillatory
instabilities when the power is increased above some critical
level. Such instabilities appear due to a resonant coupling
with linear waves outside the bandgap, resulting in a strong
emission of radiation away from the defect layer. Such a
threshold dependence of the wave dynamics on the input power
was observed experimentally, cf. Fig. 16(a) and (b). Whereas
43
Fig. 17. Generation of nonlinear defect modes and discrete solitons by a
high-power input beam, which is centered at the defect location (shown with
a dashed line). Experimental (left) and numerical (right) results are shown for
the attractive (a), (b) and repulsive (c), (d) cases [85].
the output beam is localized at the three central waveguides
at low powers, the width increases to seven waveguides at
higher powers. In the latter case, the dips in the field between
the individual guides disappear, indicating that the conditions
for the diffraction inversion and wave localization in the BR
regime are not satisfied.
Attractive or repulsive defects can also be created by decreasing or increasing the spacing between a pair of waveguides, respectively. Such an approach was used in a recent experimental study by Morandotti et al. [85], who have investigated
interaction of discrete solitons with such defects. It was found
that the outcome strongly depends on the inclination angle of
an input beam. At small angles the soliton velocity is slow, and
it is reflected from a defect. On the other hand, transmission is
observed for large angles. Another set of results was obtained
for the case when the input beam was centered at the defect
(see Fig. 17). An attractive defect increases the Peierls–Nabarro
potential, and large inclination angles are required to make the
soliton escape from the defect site [Fig. 17(a) and (b)]. On the
contrary, small perturbations are sufficient to move the soliton
away from a repulsive defect [Fig. 17(c) and (d)]. The latter phenomenon can be potentially useful for switching applications,
since the output position is very sensitive to the sign of the inclination angle. Such sensitivity is a result of instability development, which is similar to the symmetry-breaking instability
of “even” localized modes (cf. Fig. 14).
Soliton tunneling through a wide gap between two uniform
waveguide arrays was also studied experimentally, see Fig. 18.
Such a gap creates an effectively repulsive defect; however,
there are some differences compared to narrow defects. It
was found that moderately localized discrete solitons can get
trapped at the edge of the waveguide boundary [Fig. 18(a)].
44
Fig. 18. Interaction of a broad discrete soliton with a large repulsive defect
(wide gap between two arrays, shown with dashed regions). The beam was
injected about one waveguide away from the defect. (a) Low-power case—the
beam is trapped near the boundary when the inclination angle is not sufficient to
achieve tunneling. (b) In the high-power case, a soliton forms, which is reflected
from the boundary if tunneling does not occur [85].
At larger input powers, highly localized solitons are formed,
and they rather get reflected from the boundary [Fig. 18(b)]. In
both cases, solitons tunnel through the gap when the inclination
angle of the input beam is increased beyond some critical level.
D. Linear and Nonlinear Wannier–Stark States
Discrete systems such as semiconductor superlattices or
waveguide arrays share a lot of interesting features. One of
the most remarkable is the occurrence of Bloch oscillations
[86], [87], when in an ideal crystal, a charge carrier placed
in a uniform electric field exhibits periodic oscillations: The
carrier is accelerated by the electric field until its momentum
satisfies the Bragg condition associated with the periodic
potential and is reflected. Such oscillations are a generic feature
of localized eigenmodes, which are called Wannier–Stark
states, and their eigenvalues are equally spaced. The similar
effect is known in the physics of semiconductor superlattices,
where an oscillating current is generated if a static electric
field is applied, in contrast to the direct current (dc) flow
observed in bulk materials [88]. Because of the fundamental
relevance of discreteness in nature, we expect to find similar
effects in other systems of quite different origin. In fact, Bloch
oscillations were predicted to occur in molecular chains [89],
periodically spaced curved optical waveguides [90], and they
were experimentally observed for cold atoms in optical lattices
[91], [92]. Similar evolution equations in optics and quantum
mechanics indicate the relevance of Bloch oscillations in
Fig. 19. Experimentally measured power distribution as a function of the
propagation length. Low-power Bloch oscillations for (a) a single waveguide
excitation and (b) a broad-beam excitation. High-power field evolution
(around 2000-W peak power) for the same input conditions as in (a) and (b),
respectively. Solid lines: array boundaries. Dashed lines: input waveguide.
Arrow: direction of growing index [14].
optical systems under appropriate conditions. Recently, it was
suggested that waveguide arrays with a varying effective index
of the individual guides are an ideal environment to observe
optical Bloch oscillations in the space domain [93].
Morandotti et al. [14] experimentally demonstrated the
occurrence of optical Bloch oscillations in a waveguide array
with linearly growing effective index of the individual guides,
that played a role of an effective dc electric field in semiconductor superlattices. Operating in a linear limit of weak input
powers, these authors monitored the output profiles for varying
propagation lengths and observed a periodic transverse motion
of the field and a complete recovery of the initial excitation
[see Fig. 19(a) and (b)], in excellent agreement with basic
theory of the particle motion and Bloch oscillations in a tilted
periodic potentials. In contrast, a broad beam moves across the
array without changing its shape considerably. For larger input
powers, Morandotti et al. [14] observed a loss of recovery of
the Bloch oscillations accompanied by symmetry breaking and
power-induced beam spreading, as shown in Fig. 19(c) and
(d). These effects can be explained by the nonlinearity-induced
phase shift that produces an effective perturbation to the linear
phase relations making the recovery incomplete and the field
distribution starts to disperse. In general, the onset of the
nonlinearity tends to randomize the field distribution. The
strong coherence disappears and the light starts to behave
more and more in a conventional way. Similar effects are
observed in quantum mechanical systems for higher densities,
where point-like scattering suppresses coherent recovery and
particles become classical objects. The effect induced by the
nonlinearity as discussed here can be well reproduced by
numerical simulations. These strong power-induced changes
45
Fig. 21. Schematic of a binary array of waveguides with alternating widths d
and d and separation d .
Fig. 20.
Power dependencies for the families of hybrid discrete solitons [96].
of the field distribution might be useful for signal steering and
switching applications. With the onset of nonlinearity, the beam
almost completely scans the array [14]. Similar observations of
optical Bloch oscillations in waveguide arrays were reported
by Pertsch et al. [94], [95]. The required linear variation of the
propagation constant across an array of polymer waveguides
was obtained by applying a temperature gradient.
It was recently predicted theoretically [96] that hybrid discrete solitons can exist in waveguide arrays exhibiting a linear
variation of the effective index. Such solitons can be considered
as stationary bound states of a conventional discrete soliton and
the linear Wannier–Stark state satellite “tail.” There exist different soliton families, which are characterized by the symmetry
(odd or even) and the spatial separation between the two soliton
components in a bound state, as illustrated in Fig. 20. Such a
multiplicity of states can be considered as a nonlinear analog of
the linear Wannier–Stark ladder of eigenvalues.
Linear stability analysis and direct numerical simulations reveal that the hybrid solitons with strong satellite content are unstable. It is quite remarkable that even when the soliton decays
due to the development of instability, the power remain rather
localized. This happens because, as the soliton broadens, the effect of nonlinearity decreases, and the field evolves as a linear
superposition of Wannier–Stark states, which are localized.
The excitation dynamics of hybrid solitons is also unusual.
A realistic input beam cannot match the soliton profile exactly,
and therefore, radiation waves are generated as well. However,
since the linear waves are localized, the radiation cannot escape,
in contrast to homogeneous waveguide arrays where radiation
waves can propagate freely. Therefore, stable hybrid solitons
can be excited only if they can withstand repeating interactions
with radiation waves [96].
VI. DISCRETE-GAP SOLITONS
Based on the Kronig–Penney theoretical model discussed in
Sections II–IV above, we may conclude that the solitons appearing in the BR gaps can possess many unusual properties,
compared to their counterparts localized in the IR regime. Gap
solitons, existing in the vicinity of the first transmission band,
can be studied in the frames of the Kronig–Penney model, but
in this section, we develop an analytical description of discrete
gap solitons existing in the whole (first) BR gap, which can be
engineered by an appropriate design of the waveguide array. We
also address an important issue of the optimal conditions for excitation of such solitons.
In Sections II-D3 and III-B above, we have demonstrated that
the two-component discrete model (30) can describe many of
the properties of discrete gap solitons. Based on these results,
we can suggest a fundamental idea of the array engineering
and consider a binary waveguide array composed of alternating
wide and narrow waveguides [97], as illustrated in Fig. 21. Propagation of waves in such a structure can be approximately described by the DNLS equation (1), where is no longer con, and
. Note
stant, and it takes two values
that such a model is, indeed, very similar to (30).
This structure can be compared to the waveguide arrays with
defects, which we have discussed in Section V-C. The normalized detuning between the modes of the thin (defect) and thick
[83], and we
waveguides, shown in Fig. 16(c), is about
use this value in numerical simulations presented below.
The properties of linear BWs can be calculated using the approach described in Section II-B. We find that the transmission
, appear when
bands, corresponding to real
or
, where
. On the
other hand, an additional gap, also called the Rowland ghost
, and it increases for a
gap [98], appears for
larger difference between the widths of the neighboring waveguides. A characteristic dispersion relation and the corresponding
bandgap structure are presented in Fig. 22(a). We also calcu, and the effective
late the group velocity,
, of the binary array.
diffraction coefficient,
Characteristic dependencies of these parameters on the propagation constant are shown in Fig. 22(b) and (c). In both transmission bands, there exist regions with effective normal and anomalous diffraction separated by a zero-diffraction point.
We find solutions for the discrete gap solitons and determine
their linear stability properties. Some of these results are presented in Fig. 23, where we show the power of “odd”- and
“even”-type localized modes, and plot the characteristic soliton
profiles. Note that the soliton energy is primarily concentrated at
the thin waveguides (which have odd numbers). The even solitons are always unstable, whereas odd ones become unstable
only above some critical power level. This is very similar to
the stability properties of staggered solitons in a self-defocusing
medium discussed in Section IV-A and defect modes described
in Section V-C. We perform the linear stability analysis and reveal that the discrete gap solitons become oscillatory unstable
due to one (or both) of the following mechanisms: 1) internal
resonance within the gap, first discovered for fiber Bragg gratings [70] and 2) inter-band resonances, earlier discussed for
nonlinear defect modes in a layered medium [45].
46
Fig. 22. Characteristic dependencies of the BW number (K ), group velocity
(v ), and the effective diffraction coefficient (D ) on the propagation constant
. Gray shadings mark the transmission bands. Normalized detuning between
the thick and thin waveguides is : [97].
=15
=
Fig. 24. Excitation of discrete gap solitons in a self-focusing medium (
) by input beams with : , n
,
, and normalized peak
intensities: (a) C
: ; (b) 0.75; (c) 0.9; and (d) 5. The waveguide array
parameters correspond to Fig. 22.
+1
j j = 03
= 06
=1 =4
If the beam is incident at a normal angle, then
. In
this case, discrete solitons can be excited in the upper (IR) gap,
and their properties are similar to those in homogeneous arrays.
(dashed gray) discrete gap solitons. Solid lines: stable. Dashed lines: unstable.
of (a)
Dotted lines: oscillatory unstable. Bottom: soliton profiles at even and (b) odd symmetries. Circles and squares mark amplitudes at the narrow
and wide waveguides, respectively. The array parameters are the same as in
Fig. 22.
= 01
We now consider excitation of the discrete gap solitons by an
input Gaussian beam
(38)
is the position of the beam center, is the beam
where
width, and characterizes the inclination angle. Such an input
beam can be presented as a superposition of two modulated
correspond to the
linear eigenmodes, whose eigenvalues
. The two modes always
BW number
have opposite group velocities and diffraction coefficients (see
Fig. 22).
The optimum conditions for generating discrete gap solitons
, and
. It can
can be realized when
be demonstrated that under such conditions the second mode,
which experiences self focusing, is dominant. Moreover, the
BW envelopes have opposite group velocities, so that they move
apart in the opposite directions. In Fig. 24(a), we show that
two beams are indeed generated at the input. The beam which
moves to the right is localized at odd (i.e., thin) waveguides,
and it transforms into a gap soliton. On the contrary, the other
beam moves to the left, and it experiences self defocusing and
broadens.
Stationary gap solitons can be generated at higher input intensities. Under such conditions, two Bloch modes are initially
trapped together, resulting in a periodic beating that can strongly
affect the soliton formation process. If the emerging soliton has
a small velocity, it is not able to overcome the Peierls–Nabarro
potential of the periodic structure, and it becomes trapped [6]. In
this case, the gap soliton can remain in the center, as is shown in
Fig. 24(b). However, if the soliton acquires a higher velocity, it
can still move across the array, as is shown in Fig. 24(c). Somewhat surprisingly, in the latter case, the soliton moves to the left,
i.e., in the direction opposite to the propagation direction of the
input beam. If the intensity is increased even further, then the
gap solitons become oscillatory unstable, as discussed above.
These results indicate that the binary waveguide arrays with
alternating waveguide widths can be used to experimentally
study the intriguing dynamics of discrete gap solitons and
provide a link with earlier theoretical studies.
Fig. 25. (a) A discrete soliton, propagating along a 120 branch. (b) Soliton
intensity at z = 6:5 cm (after traversing the bend). (c) Same as (a) for a 90
bend. (d) Soliton intensity at 6.4 cm when the signal is already on the vertical
branch [99].
VII. WAVEGUIDE ARRAY NETWORKS
Discrete solitons are possible by balancing the effect of discrete diffraction, arising from the coupling between neighboring
waveguides, with that of material nonlinearity. As pointed out
above, optical discrete solitons differ fundamentally from their
bulk counterparts in several ways. For example, a unique property that follows from the discrete nature of waveguide arrays is
that of diffraction management and reverse diffraction.
The experimental observations of the discrete solitons raise
the prospect of using waveguide arrays for data processing applications. However, because of their one-dimensional (1-D)
topology, the functionality of linear waveguide arrays for signal
manipulation purposes is considerably limited. Thus, it would
be highly desirable to study new geometrical arrangements capable of exhibiting superior performance in terms of realizing
intelligent operations.
An important concept of the discrete soliton networks, based
on the propagation of discrete optical solitons in 2-D waveguide
array networks, was introduced by Christodoulides and Eugenieva [99], [100], [109], who demonstrated that such waveguide
networks can provide a rich environment for all-optical data processing applications, effectively acting like soliton wires along
which these self-trapped entities can travel. Moreover, by appropriate design, reflection losses occurring along sharp bends
in 2-D array networks can be almost eliminated.
Fig. 25 demonstrates the basic concept of the network. A 2-D
network of nonlinear waveguide arrays involves identical elements. Each branch is composed of regularly spaced waveguides separated from each other by distance . Every waveguide
is designed to be single moded at the operating wavelength. A
moderately confined discrete soliton (e.g., extending over five to
seven sites) is excited in one of the branches of the network. In
this limit, the soliton is known to be highly mobile with the lattice, and its envelope profile is approximately described by a hy,
perbolic secant function
where describes the phase tilt necessary to make this soliton
move along the array. Calculations in [99], [100], and [109] have
47
been performed for the index
, the index difference 3
, the effective core radius 5.3 m, the waveguide spacing
m, and the wavelength
m.
Imposing a linear phase chirp (tilt) on the beam profile, a discrete soliton can be set in motion along one of the axis of a network (e.g., the horizontal axis of the structure which involves
two array branches intersecting at 120 ), as shown in Fig. 25(a).
The discrete soliton slides along the horizontal branch and after
passing the intersection moves to the upper branch. Fig. 25(b)
shows that this occurred with almost no change in the soliton
intensity profile or speed. Even more importantly, the losses because of the bend are extremely low (less than 0.7%). In other
words, the soliton travels only along the network pathways, i.e.,
the array behaves like a soliton wire. These low levels of losses
at 120 junctions suggest that discrete solitons can be effectively used as information carriers in honeycomb-like networks.
During propagation, the soliton exhibits robustness and it does
not suffer from transverse-modulation instability. This is because the soliton itself is linearly guided by the array in the direction normal to its motion. Similar results were obtained for
a 90 bend, as depicted in Fig. 25(c) and (d). In this case, the
soliton traverses the bend with a loss of 5%. Fig. 25(d) depicts
the soliton intensity after 6.4 cm of propagation (after the 90
bend), indicating that the discrete soliton remained practically
invariant. This behavior is to some extent reminiscent of that of
waves propagating along sharp bends in photonic crystal waveguides [101].
By appropriate design, reflection losses occurring along
very sharp bends (e.g., the bends less than 90 ) in 2-D discrete
soliton networks can be almost eliminated. This can be accomplished by modification of the guiding properties of the corner
waveguide [100], [109]. Indeed, analytical calculations of the
three-site element modeling the waveguide corner suggest that,
for a specific phase speed and the waveguide couplings the bend
reflection losses can be completely eliminated. Effectively, this
is achieved by introduction of a defect site at the bend corner.
In addition, by using vector/incoherent interactions at network junctions, soliton signals can be routed at will on specific
pathways. Therefore, the discrete solitons can be navigated anywhere within a 2-D network of nonlinear waveguides. Useful
functional operations such as blocking, routing, logic functions,
and time gating can also be realized [99]. By appropriately engineering the intersection site, the switching efficiency of the junctions in 2-D discrete-soliton networks can be further improved
allowing to design routing junctions with specified operational
characteristics [102].
VIII. CONCLUSIONS
We have presented a comprehensive overview of the fundamental properties of spatial optical solitons in arrays of nonlinear waveguides based on the recently introduced analytical
model describing a periodic structure of thin-film waveguides.
Such a model allows to study both the conventional spatial solitons in arrays (also called discrete solitons), as well as the localized modes in the gaps of the linear spectrum, known as gap
solitons. Generally speaking, an array of optical waveguides is
an example of a physical system with 1-D periodicity that may
48
be associated with the structure and properties of 1-D photonic
crystals.
We have demonstrated how the known approximate models,
such as the discrete NLS equation and coupled-mode theory,
can be introduced in the framework of our analytical theory. We
have presented a summary of the most remarkable experimental
observations of the soliton generation and dynamics in waveguide arrays. We have also demonstrated that binary waveguide
arrays can be used as a testing ground for the study of discrete
gap solitons. Additionally, we have discussed a novel concept
of the waveguide array networks that, we believe, would allow
further rapid progress in this field.
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ACKNOWLEDGMENT
The authors would like to thank J. S. Aitchison, D.
Christodoulides, and R. Morandotti for useful discussions and
collaboration.
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Andrey A. Sukhorukov graduated from the Physics
Faculty of the Moscow State University, Moscow,
Russia, in 1996 and received the Ph.D. degree
in physical science from the Australian National
University, Canberra, ACT, in 2002.
He has recently accepted a Postdoctoral Research
position at the Research School of Physical Sciences
and Engineering, the Australian National University.
His current research interests are in the areas of nonlinear optics and photonic crystals, including soliton
formation and frequency conversion.
Yuri S. Kivshar received the Ph.D. degree from the
USSR Academy of Science in 1984.
From 1988 to 1993, he worked at different
research centers in the U.S., France, Spain, and
Germany. In 1993, he accepted a position at the Research School of Physical Sciences and Engineering,
Institute of Advanced Studies, Australian National
University, Canberra, ACT, where he founded the
Nonlinear Physics Group in 2000. In 1999, he was
appointed as an Associate Editor (the first Australian
and second non-American) of the Physical Review.
He has published more than 250 research papers. His research interests include
nonlinear guided waves and solitons, photonic crystals and waveguides, and
nonlinear atom optics.
Dr. Kivshar was a recipient of the Medal and Award of the Ukrainian
Academy of Science (1989), the International Pnevmatikos Prize in Nonlinear
Physics (1995), and the Pawsey Medal of the Australian Academy of Science
(1998). In 2002, he was elected to the Australian Academy of Science.
Hagai S. Eisenberg graduated in physics from the
Technion—Israel Institute of Technology, Haifa, in
1994. He received the M.Sc. degree in 1997 from
the Weizmann Institute of Science, Rehovot, Israel,
where he is currently working toward the Ph.D. degree.
Yaron Silberberg received the Ph.D. degree from the
Weizmann Institute of Science, Rehovot, Israel, in
1984.
He joined Bellcore, Red Bank, NJ, where he was a
Member of the Technical Staff. In 1994, he returned
to the Weizmann Institute as a Professor in the Department of Physics of Complex Systems. He is currently serving as a Dean of the Physics Faculty.

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