Солитоны в оптике

CHAPTER
2
Soliton Shape and Mobility
Control in Optical Lattices
Yaroslav V. Kartashov a , Victor A. Vysloukh b
and Lluis Torner a
a ICFO-Institut de Ciencies Fotoniques, and Universitat
Politecnica de Catalunya, Mediterranean Technology Park, 08860
Castelldefels (Barcelona), Spain
b Departamento de Fisica y Matematicas, Universidad de las
Americas – Puebla, Santa Catarina Martir, 72820, Puebla,
Mexico
Contents
1.
2.
3.
4.
5.
Introduction
Nonlinear Materials
Diffraction Control in Optical Lattices
3.1 Bloch Waves
3.2 One- and Two-dimensional Waveguide Arrays
3.3 AM and FM Transversely Modulated Lattices
and Arrays
3.4 Longitudinally Modulated Lattices and Waveguide Arrays
Optically-induced Lattices
4.1 Optical Lattices Induced by Interference Patterns
4.2 Nondiffracting Linear Beams
4.3 Mathematical Models of Wave Propagation
One-dimensional Lattice Solitons
5.1 Fundamental Solitons
5.2 Beam Shaping and Mobility Control
5.3 Multipole Solitons and Soliton Trains
5.4 Gap Solitons
5.5 Vector Solitons
5.6 Soliton Steering in Dynamical Lattices
5.7 Lattice Solitons in Nonlocal Nonlinear Media
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E-mail address: Yaroslav.Kartashov@icfo.es (Yaroslav V. Kartashov).
Progress in Optics, Volume 52 c 2009 Elsevier B.V.
ISSN 0079-6638, DOI 10.1016/S0079-6638(08)00004-8 All rights reserved.
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Soliton Shape and Mobility Control in Optical Lattices
6.
Two-dimensional Lattice Solitons
6.1 Fundamental Solitons
6.2 Multipole Solitons
6.3 Gap Solitons
6.4 Vector Solitons
6.5 Vortex Solitons
6.6 Topological Soliton Dragging
7. Solitons in Bessel Lattices
7.1
Rotary and Vortex Solitons in Radially Symmetric Bessel Lattices
Multipole and Vortex Solitons in Azimuthally
7.2
Modulated Bessel Lattices
7.3
Soliton Wires and Networks
Mathieu and Parabolic Optical Lattices
7.4
8. Three-dimensional Lattice Solitons
9. Nonlinear Lattices and Soliton Arrays
9.1 Soliton Arrays and Pixels
9.2 Nonlinear Periodic Lattices
10. Defect Modes and Random Lattices
10.1 Defect Modes in Waveguide Arrays and
Optically-induced Lattices
10.2 Anderson Localization
10.3 Soliton Percolation
11. Concluding Remarks
Acknowledgements
References
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1. INTRODUCTION
Nonlinear wave excitations are ubiquitous in Nature. They play an
important role in our description of a vast variety of natural phenomena
and their applications in many branches of science and technology,
from optics to fluid dynamics and oceanography, solid-state physics,
Bose–Einstein condensates (BECs), cosmology, etc. Examples are widespread (see, e.g., Bloembergen (1965), Boyd (1992), Infeld and Rowlands
(1990), Whitham (1999), Moloney and Newell (2004), Hasegawa and
Matsumoto (1989), Akhmediev and Ankiewicz (1997), Kivshar and
Agrawal (2003) and Stegeman and Segev (1999)).
Until recently, research in this area was commonly divided into
two distinct, separated categories: Excitations in systems which can be
described by continuous mathematical models and those in systems
that can be effectively described by discrete equations. The difference
between them is not merely quantitative, but qualitative, and often
drastic. The paradigmatic example is the case of systems described by the
nonlinear Schrödinger equation (NLSE). This is one of the few canonical
Introduction
65
equations that are encountered in many branches of nonlinear science
when phenomena are studied at the proper scale length. The continuous
version of NLSE belongs to that very special class of equations that in onedimensional geometries are referred to as completely-integrable and thus
have rigorous single and multiple soliton solutions, which describe the
stable and robust propagation of wave packets with an infinite number
of conserved quantities. In contrast, systems modeled by the discrete
NLSE may not conserve some of the fundamental quantities, such us
the paraxial wave linear momentum, and thus feature a correspondingly
restricted mobility. From a practical point of view, such qualitative
differences are not necessarily advantageous or disadvantageous. For
example, in the case of propagating light beams, the restricted mobility
inherent in the discrete NLSE suggests new strategies for the routing
or switching of light. By and large, the point is that the key differences
existing between continuous and strictly-discrete mathematical models
manifest themselves in the generation of correspondingly different
physical phenomena.
However, there is a whole world in between systems modeled by
totally continuous and totally discrete evolution equations. This world
has been made accessible to experimental exploration in its full scope
with the advent of optically-induced lattices in optics (see Fleischer, Segev,
Efremidis and Christodoulides (2003)) and in Bose–Einstein condensates
(see Morsch and Oberthaler (2006)). Tuning the strength of the lattices
causes the system behavior to vary between a predominantly continuous
model and a predominantly discrete one. The corresponding powerful
concept which may be termed “tunable discreteness” has direct important
applications, e.g., for all-optical routing and shaping of light in optics, or
for the generation and manipulation of quantum correlated matter waves
in Bose–Einstein condensates. In addition to such applications, from a
broad perspective, the ability to tune the strength of the genuinely discrete
features of nonlinear systems affords important applications in other
areas of nonlinear science where the concept can be implemented. Optical
lattices provide a unique laboratory to undertake such exploration. Here
we present a progress overview focused on soliton control. We focus
our attention on the salient possibilities afforded by the concept of
tunable discreteness for soliton manipulation at large, and we refer the
readers to other reviews for in-depth experimental advances (Lederer,
Stegeman, Christodoulides, Assanto, Segev and Silberberg, 2008). Thus,
we start by addressing the stationary properties of the different types
of lattice solitons, and focus our attention on the dynamical properties
by highlighting the main advances achieved to date. We address optical
solitons, but most of the results hold as well for BECs.
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Soliton Shape and Mobility Control in Optical Lattices
The paper is organized as follows. In Sections 2 and 3 we discuss briefly
the available nonlinear materials and structures that are currently used in
most experiments with optical lattice solitons. In Section 4, we describe
the experimental techniques available for optical lattice induction and
the mathematical models used to describe nonlinear light excitations in
lattices. In Section 5 we address the salient properties of scalar and vector
solitons supported by one-dimensional lattices. We discuss the impact
on the soliton shape and soliton mobility caused by tunable static and
dynamic optical lattices. We also address the potential stabilization of
complex lattice soliton structures. In Section 6, the properties of solitons
supported by two-dimensional optical lattices are considered. Addition of
the second transverse coordinate remarkably affects properties of steadystate nonlinear waves. Optical lattices may have a strong stabilizing action
for two-dimensional solitons and allow the existence of new types of
solitons such as vortex solitons, possibilities that we discuss explicitly.
In Section 7 lattices imprinted with different types of nondiffracting
beams (including Bessel, Mathieu and parabolic ones) are discussed. Such
lattices offer new opportunities for soliton managing and switching, and
constitute one of the key points to be highlighted in this area. In the
absence of refractive index modulation, three-dimensional solitons in selffocusing Kerr-type nonlinear media experience rapid collapse, but under
appropriate conditions optical lattices may stabilize not only groundstate solitons but also higher-order three-dimensional solitons, even if
the dimensionality of the lattice is lower than that of the solitons, as
briefly reviewed in Section 8. The properties of nonlinear optical lattices
(including soliton arrays), together with the techniques that can be used
to cause their stabilization, are addressed in Section 9. In Section 10 we
review briefly the specific properties introduced by defect modes and by
random lattices. Finally, in Section 11 the salient conclusions are given.
2. NONLINEAR MATERIALS
The advent and availability of suitable materials and fabrication
techniques for the generation of optical lattices has been a key ingredient
for the advancement of the field. In this section we briefly introduce the
properties of only a few optical materials which are routinely used for the
experimental excitation of lattice solitons.
Fused silica is known to be a weakly nonlinear material (n 2 '
2.7 × 10−16 cm2 /W). It has found applications in experiments with
lattice solitons due to its unique ability to write two-dimensional optical
waveguides by tightly focusing femtosecond laser pulses generated by
amplified Ti:Sa lasers. Permanent refractive index changes up to 1.3 ×
10−3 have been reached, with a typical spacing between 20 mm long
Nonlinear Materials
67
waveguides of the order of ∼20 µm (Pertsch, Peschel, Lederer, Burghoff,
Will, Nolte and Tünnermann, 2004; Szameit, Blomer, Burghoff, Schreiber,
Pertsch, Nolte and Tünnermann, 2005). Formation of two-dimensional
solitons in such waveguiding arrays was observed by Szameit, Burghoff,
Pertsch, Nolte, Tünnermann and Lederer (2006). Evolution of linear light
beams in arbitrary waveguide arrays, including laser-written ones, was
addressed by Szameit, Pertsch, Dreisow, Nolte, Tünnermann, Peschel and
Lederer (2007).
Nonresonant nonlinearities in semiconductor materials may be orders
of magnitude stronger than in silica. For instance, in AlGaAs at the
wavelength 1.53 µm the nonlinearity coefficient amounts to n 2 ∼
10−13 cm2 /W, and self-action effects are observable already for
light intensities ∼10 GW/cm2 . Such light intensities can be readily
achieved with tightly focused pulsed radiation. Owing to highly
advanced semiconductor processing technologies it is possible to fabricate
individual semiconductor waveguides as well as waveguide arrays
with the effective core areas ∼20 µm2 . Such arrays were used to
demonstrate discrete spatial solitons at power levels ∼500 W (Eisenberg,
Silberberg, Morandotti, Boyd and Aitchison, 1998). Waveguiding arrays
fabricated from polymer inorganic-organic materials also have found
important applications in the field of diffraction management (Pertsch,
Zentgraf, Peschel, Brauer and Lederer, 2002a,b), and in the experimental
observation of the optical analog of Zener tunneling (Trompeter, Pertsch,
Lederer, Michaelis, Streppel and Brauer, 2006).
Metal vapors may feature resonant enhanced nonlinearities. Under
proper conditions, e.g., in the absence of saturation, the nonlinear
coefficient of sodium vapors for large frequency detuning is given by
n 2 = 8π 2 N µ4 /[n 20 c h̄ 3 (ω − ω R )3 ], where N is the atomic concentration, µ
is the dipole moment, and ω R is the resonance frequency (Grischkowsky,
1970). In rubidium vapor, resonant enhancement is achieved close to the
D2 line at λ = 780 nm. The accessible values of the nonlinear coefficient
range from n 2 = 10−10 cm2 /W to 10−9 cm2 /W for typical concentrations
N ∼ 1013 cm−3 . Self-trapping of a CW laser beam in sodium vapor was
observed as early as 1974 (Bjorkholm and Ashkin, 1974). Saturation of
the nonlinearity is typical for rubidium and sodium vapors. Then, the
nonlinear contribution to the refractive index can be approximated as δn =
n 2 I /(1 + I /I S ), where I S is the saturation intensity. The nonlinearity sign,
its strength, and saturation degree may be varied by changing the vapor
concentration and by changing the detuning from resonance. Solitons
were observed in metal vapors at power levels ∼100 mW (Tikhonenko,
Kivshar, Steblina and Zozulya, 1998).
Nematic liquid crystals have emerged as suitable materials for experimentation with lattice solitons, thanks to their strong reorientational non-
68
Soliton Shape and Mobility Control in Optical Lattices
linearity that may exceed the nonlinearity of standard semiconductors by
several orders of magnitude (Simoni, 1997). Such crystals might be used
for fabrication of periodic voltage-controlled waveguide arrays (Fratalocchi, Assanto, Brzdakiewicz and Karpierz, 2004). In such arrays the top
and bottom cell interfaces provide planar anchoring of the liquid crystal molecules along the direction of light propagation. A set of periodically spaced electrodes (typical spacing ∼6 µm) allows a bias to be applied
across a ∼5 µm thick crystal cell, thereby modulating the refractive index
distribution through molecular reorientation. The typical power required
for spatial soliton formation in such arrays is ∼35 mW at biasing voltage
1.2 V (see, e.g., reviews on solitons in nematic liquid crystal arrays by Brzdakiewicz, Karpierz, Fratalocchi, Assanto and Nowinowski-Kruszelnick
(2005); Assanto, Fratalocchi and Peccianti (2007), and Section 5.7 of the
present review).
The photovoltaic nonlinearity of photorefractive LiNbO crystals (see Bian,
Frejlich and Ringhofer (1997)), sets a rich playground for experiments
with spatial solitons. For instance, self-trapping of an optical vortex in
such crystals was reported by Chen, Segev, Wilson, Muller and Maker
(1997). Two-dimensional bright photovoltaic spatial solitons have been
observed in Cu:KNSBN crystal featuring focusing nonlinearity (She, Lee
and Lee, 1999). Photorefractive materials also can be used for fabrication
of waveguide arrays. Such arrays might be created by titanium indiffusion in a copper-doped LiNbO crystal where the optical nonlinearity
(defocusing and saturable) arises from the bulk photovoltaic effect. Such
waveguides combine high nonlinearity with adjustable contrast of linear
refractive index. A typical LiNbO waveguide array consists of 4 µm-wide
titanium doped stripes separated by the same distance. Each channel
forms a single-mode waveguide at 514.5 nm wavelength for refractive
index modulation depth ∼ 3 × 10−3 , while the coupling constant ∼
1 mm−1 . The experimental observation of spatial gap solitons in such
arrays at the power levels ∼µW was reported by Chen, Stepic, Rüter,
Runde, Kip, Shandarov, Manela and Segev (2005).
An excellent material for experimentation with different types of nonlinear lattice waves is a doped photorefractive SBN crystal biased externally with a DC electric field. Due to a strong anisotropy of the electrooptic effect, the ordinary-polarized lattice-forming beams, propagating almost linearly along the crystal, are capable of inducing relatively strong
refractive index modulation for the extraordinary polarized probe beam,
as was suggested by Efremidis, Sears, Christodoulides, Fleischer and
Segev (2002). It was demonstrated that nonlinear lattice waves in photorefractive SBN crystals might be observed at extremely low power levels of a hundred nanowatts (Träger, Fisher, Neshev, Sukhorukov, Denz,
Krolikowski and Kivshar, 2006).
Diffraction Control in Optical Lattices
69
Summarizing, there are a variety of materials suitable for the
exploration of nonlinear wave propagation in periodic media. However,
optical induction in suitable materials constitutes a real landmark advance
for the purpose that we explore in this review, as it affords control of
the shape, strength and properties of the lattice with an unprecedented
flexibility (see Section 4 below for a detailed discussion).
3. DIFFRACTION CONTROL IN OPTICAL LATTICES
The ability to engineer and control diffraction opens broad prospects
for light beam manipulation. In this section we discuss briefly a few
milestones achieved in this area in lattices and waveguide arrays. In
a linear bulk medium the spatial Fourier modes (plane waves) form
an adequate set of eigenfunctions for the analysis of diffraction and
refraction phenomena. Thus, diffraction of a collimated wave beam might
be interpreted as dephasing of spatial harmonics that leads to a convex
wave front. Inside an optical lattice standard Fourier analysis becomes
inefficient because of the energy exchange between an incident and Braggscattered waves. The adequate functional basis for description of wave
propagation in such a periodic environment is provided by the so-called
Floquet–Bloch set of basis functions.
Figure 1 illustrates the dependencies of the propagation constants of
Bloch waves from the different bands on transverse wavenumber and
the corresponding profiles of Bloch waves. In regions of convex band
curvature the central mode propagates faster than its neighbors and the
beam acquires a convex wave front during propagation. There are regions
of normal diffraction where the wave behavior is fully analogous to that in
the homogeneous media. By contrast, a group of modes in concave regions
of band curvature evolve anomalously, producing a concave wave front.
Thus, modification of the input angle of the beam entering the periodic
structure results in different rates and even signs of diffraction. Therefore,
the effect of a periodic lattice on the propagation of a laser beam depends,
to a great extent, on the overall size of the input beam, relative to the lattice
period, and on the depth of refractive index modulation.
3.1 Bloch Waves
The linear interference of optical Bloch waves in periodically stratified
materials was studied both theoretically and experimentally almost
two decades ago (see Russell (1986) and references therein). Eisenberg,
Silberberg, Morandotti and Aitchison (2000) proposed a scheme to design
structures with controllable diffraction properties based on arrays of
evanescently coupled waveguides. The scheme proposed is suitable for
exploiting the negative curvature of the diffraction curve of light in
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Soliton Shape and Mobility Control in Optical Lattices
FIGURE 1 (a) Dependencies of propagation constants of Bloch waves from different
bands on transverse wavenumber k at  = 4 and p = 3. (b) Band-gap lattice structure
at  = 4. Bands are marked with gray color, gaps are shown white. Profiles of Bloch
waves from first (c),(d) and second (e),(f) bands, corresponding to k = 0 (panels (c)
and (e)) and k = 2 (panels (d) and (f)) at  = 4 and p = 3
waveguide arrays in order to control diffraction. Arrays with reduced,
canceled, and even reversed diffraction were technologically fabricated.
Excitation of linear and nonlinear modes belonging to high-order bands
of the Floquet–Bloch spectrum of periodic array was demonstrated by
Mandelik, Eisenberg, Silberberg, Morandotti and Aitchison (2003a). To
excite a single Bloch mode in a waveguide array the light beam was
coupled at a grazing angle to the array from a region of a planar
waveguide. Changing the tilt angle of the input beam allows selective
excitation of periodic waves from different bands. A prism-coupling
method for excitation of Floquet–Bloch modes and direct measurement
of band structure of one-dimensional waveguide arrays was introduced
by Rüter, Wisniewski and Kip (2006). Solitary waves bifurcating from
Bloch-band edges in two-dimensional periodic media have been studied
theoretically by Shi and Yang (2007).
In optically-induced photonic lattices, the Bragg scattering of light
was studied experimentally in a biased photorefractive SBN:60 crystal
(Sukhorukov, Neshev, Krolikowski and Kivshar, 2004). This experiment
showed the splitting of an input slant laser beam into several Bloch modes
due to Bragg scattering. A novel, powerful experimental technique was
presented for linear and nonlinear Brillouin zone spectroscopy of optical
Diffraction Control in Optical Lattices
71
lattices (Bartal, Cohen, Buljan, Fleischer, Manela and Segev, 2005). This
method relies on excitation of different Bloch modes with random-phase
input beams and far-field visualization of the emerging spatial spectrum.
This technique facilitates mapping the borders of extended Brillouin zones
and the areas of normal and anomalous dispersion. The possibility of
acquiring the full bandgap spectrum of a photonic lattice of arbitrary
profile was discussed by Fratalocchi and Assanto (2006c). Nonlinear
adiabatic evolution and emission of coherent Bloch waves in optical
lattices was analyzed by Fratalocchi and Assanto (2007a). Modulational
instability of Bloch waves in one-dimensional saturable waveguide arrays
was studied both theoretically and experimentally (Stepic, Rüter, Kip,
Maluckov and Hadzievski, 2006; Rüter, Wisniewski, Stepic and Kip, 2007).
Various types of two-dimensional Bloch waves were generated in a
square photonic lattice by employing the phase imprinting technique
(Träger, Fisher, Neshev, Sukhorukov, Denz, Krolikowski and Kivshar,
2006). The unique anisotropic properties of lattice dispersion resulting
from the different curvatures of the dispersion surfaces of the first and
second spectral bands were demonstrated experimentally.
Nonlinear interactions between extended waves in optical lattices
may lead also to new phenomena. For example, two Bloch modes
launched into a nonlinear photonic lattice evolve into a comb or a
supercontinuum of spatial frequencies, exhibiting a sensitive dependence
on the difference between the quasi-momenta of the two initially excited
modes (see, Manela, Bartal, Segev and Buljan (2006)). Finally, it is also
worth mentioning that spatial four-wave mixing with Bloch waves was
considered by Bartal, Manela and Segev (2006).
3.2 One- and Two-dimensional Waveguide Arrays
The fabrication of simple GaAs waveguide arrays was achieved and
analyzed more than three decades ago (see Somekh, Garmire, Yariv,
Garvin and Hunsperger (1973)). Channel waveguides were formed by
proton bombardment through a gold mask that allowed creation of
channels with width 2.5 µm separated by 3.9 µm and featuring relatively
deep refractive index modulation ∼0.006. Optical coupling accompanied
by complete light transfer from the incident channel into adjacent ones
was observed.
Since diffraction properties of light beams in lattices strongly depend
on the propagation angle, tandems of different short segments of slant
waveguide arrays (zigzag structure) might be used in order to achieve
a desired average diffraction. This approach for diminishing the power
requirement for lattice soliton formation was suggested by Eisenberg,
Silberberg, Morandotti and Aitchison (2000), and it is closely linked to the
idea of dispersion management from fiber optics technology. It was also
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Soliton Shape and Mobility Control in Optical Lattices
shown that at higher powers, normal diffraction may lead to self-focusing
and formation of bright solitons, but slight modification of the input tilt,
causing change of diffraction sign, results in defocusing. Self-focusing
and self-defocusing were achieved with the same medium, structure,
and wavelength (Morandotti, Eisenberg, Silberberg, Sorel and Aitchison,
2001).
Detailed experimental studies of anomalous refraction and diffraction
of cw radiation in a one-dimensional array imprinted in inorganic-organic
polymer was reported by Pertsch, Zentgraf, Peschel, Brauer and Lederer
(2002b). Discrete Talbot effects in a one-dimensional waveguide array was
observed by Iwanow, May-Arrioja, Christodoulides, Stegeman, Min and
Sohler (2005). It was shown that in discrete configurations a recurrence
process is only possible for a specific set of periodicities of the input
patterns. Experimental demonstration of the discrete Talbot effect was
carried out in an array consisting of 101 waveguides imprinted in a 70 mm
long Z-cut LiNbO3 wafer with lithography and titanium in-diffusion
techniques.
Linear diffraction as well as nonlinear dynamics in two-dimensional
waveguide arrays can have more complicated character and can be much
richer than those in one-dimensional arrays (Hudock, Efremidis and
Christodoulides, 2004; Pertsch, Peschel, Lederer, Burghoff, Will, Nolte
and Tünnermann, 2004). For instance, diffraction in these arrays can
be effectively altered by changing the beam’s transverse Bloch vector
orientation in the first Brillouin zone. In general, diffraction in twodimensional arrays can be made highly anisotropic and therefore permits
the existence of elliptic solitons when nonlinearity comes into play.
Under appropriate conditions not only the strength but also the sign of
diffraction can differ for different directions (e.g., a beam can experience
normal diffraction in one direction and anomalous diffraction in the other
one).
3.3 AM and FM Transversely Modulated Lattices and Arrays
In this subsection we discuss the new physical effects afforded by
rather smooth (at the beam width scale) transverse modulation of the
optical lattice parameters. As was shown in the context of Bose–Einstein
condensates, matter-wave soliton motion can be effectively managed
by means of smooth variations of the parameters of the optical lattice
(Brazhnyi, Konotop and Kuzmiak, 2004). Thus, linear, parabolic, and
spatially-localized modulations of lattice amplitude and frequency were
considered. For instance solitons could be accelerated, decelerated, or
undergo reflection depending on the modulation function profile. The
energy, effective mass, and soliton width can also be effectively controlled
in such lattices. In this case the effective particle approximation provides
Diffraction Control in Optical Lattices
73
a qualitative explanation of the main features of soliton dynamics, if
the soliton width substantially exceeds the lattice period and the lattice
modulation is smooth enough.
Grating-mediated waveguiding was proposed by Cohen, Freedman,
Fleischer, Segev and Christodoulides (2004). Such waveguiding is driven
by a shallow one-dimensional lattice with bell- or trough-shaped
amplitude that slowly varies in the direction normal to the lattice wavevector (lattice with amplitude modulation). Optical lattices with linear
amplitude or frequency modulation were considered specifically for
soliton control (Kartashov, Vysloukh and Torner, 2005d). It was revealed,
that the soliton trajectory can be controlled by varying the lattice depth
and amplitude or frequency modulation rate. Also, it was discovered
that the effective diffraction of a light beam launched into the central
channel of a lattice with a quadratic frequency modulation can be turned
in strength and sign (Kartashov, Torner and Vysloukh, 2005). Finally,
complete suppression of linear diffraction in the broad band of spatial
frequencies is accessible in FM lattices.
A direct visual observation of Bloch oscillations and Zener tunneling
was achieved in two-dimensional lattices photoinduced by four interfering plane waves in biased photorefractive crystals (Trompeter, Krolikowski, Neshev, Desyatnikov, Sukhorukov, Kivshar, Pertsch, Peschel and
Lederer, 2006). In this experiment, a coordinate-dependent background illumination was used to create a refractive index distribution in the form
of a periodic modulation superimposed on to the linearly increasing background.
Previously, it had been shown theoretically that Bloch oscillations can
also emerge in discrete waveguide arrays with propagation constants
linearly varying across the array (Peschel, Pertsch and Lederer, 1998). The
occurrence of Bloch oscillations was demonstrated experimentally in an
array of 25 AlGaAs waveguides with transversally varied refractive index
and spacing (Morandotti, Peschel, Aitchison, Eisenberg and Silberberg,
1999b), as well as in a thermo-optic polymer waveguide array with
applied temperature gradient (Pertsch, Dannberg, Elflein, Brauer and
Lederer, 1999). Photonic Zener tunneling between the bands of the
polymer waveguide array was observed and investigated experimentally
by Trompeter, Pertsch, Lederer, Michaelis, Streppel and Brauer (2006).
Such a one-dimensional polymer waveguide array was fabricated
by ultra-violet lithography from an inorganic-organic polymer, while
creating a temperature gradient in the sample caused a linear refractive
index increase due to the thermo-optic effect. Zener tunneling in onedimensional liquid crystal arrays was demonstrated by Fratallocchi,
Assanto, Brzdakiewich and Karpierz (2006) and Fratalocchi and Assanto
(2006a). A model allowing the description of one-dimensional and more
74
Soliton Shape and Mobility Control in Optical Lattices
general two-dimensional Zener tunneling in two-dimensional periodic
photonic structures and calculation of the corresponding tunneling
probabilities was derived by Shchesnovich, Cavalcanti, Hickmann and
Kivshar (2006). Fratalocchi and Assanto (2007b) investigated nonlinear
energy propagation in an optical lattice with slowly varying transverse
perturbations. Symmetry breaking in such a system can provide a wealth
of new phenomena, including nonreciprocal oscillations between Bloch
bands and macroscopic self-trapping effects.
A binary waveguide array composed of alternating thick and thin
waveguides was introduced by Sukhorukov and Kivshar (2002a). In
such a structure the effective refractive index experiences an additional
transverse modulation. As a result, the existence of discrete gap
solitons that possess the properties of both conventional discrete and
Bragg grating solitons becomes possible. Such solitons and the effect
of interband momentum exchange on soliton steering were observed
experimentally in binary arrays fabricated in AlGaAs (Morandotti,
Mandelik, Silberberg, Aitchison, Sorel, Christodoulides, Sukhorukov and
Kivshar, 2004).
The possibility of controlling the magnitude of dispersion experienced
by a BEC wave-packet at the edges of spectral bands by modifying the
shape of a double-periodic optical super-lattice was explored by Louis,
Ostrovskaya and Kivshar (2005). It was shown that an extra periodicity
opens up additional narrow stopgaps in the band-gap spectrum, while the
effective dispersion at the edges of these mini-gaps can be varied within a
much greater range than that accessible with a single-period lattice.
Nonlinear light propagation and linear diffraction in disordered
fiber arrays (or arrays with random spacing between waveguides)
was investigated experimentally by Pertsch, Peschel, Kobelke, Schuster,
Bartlet, Nolte, Tünnermann and Lederer (2004). It was shown that for
high excitation power, diffusive spreading of a light beam in such arrays is
arrested by the focusing nonlinearity, and formation of a two-dimensional
discrete soliton is possible. The linear transmission and transport of
discrete solitons in quasiperiodic waveguide arrays was studied by
Sukhorukov (2006b) and it was shown that, under proper conditions,
dramatic enhancement of soliton mobility in such arrays is possible.
3.4 Longitudinally Modulated Lattices and Waveguide Arrays
Longitudinal modulation of the diffractive/nonlinear properties of the
transmitting medium is a powerful tool for controlling the light beam
parameters. This technique can be extended to the case of optical lattices
and waveguide arrays. A model governing the propagation of an optical
beam in a diffraction managed nonlinear waveguide array with a steplike diffraction map was introduced by Ablowitz and Musslimani (2001).
Diffraction Control in Optical Lattices
75
This model allows the existence of discrete solitons whose width and peak
amplitude evolve periodically. Its generalization to the case of vectorial
interactions of two polarization modes propagating in a diffraction
managed array was developed later by Ablowitz and Musslimani (2002).
Interactions of diffraction managed solitons have been studied
in detail. Pertsch, Peschel and Lederer (2003) considered soliton
properties in inhomogeneous one-dimensional waveguide arrays in the
framework of a discrete model. It was shown that longitudinal periodic
modulations of the coupling strength may lead to soliton oscillations
and decay. Different types of optically induced dynamical lattices were
considered, including lattices with the simplest longitudinal amplitude
modulation (Kartashov, Vysloukh and Torner, 2004), where amplification
of soliton centre swinging is possible, or more complex three-wave
lattices (Kartashov, Torner and Christodoulides, 2005) that are capable
of dragging stationary solitons along predetermined paths (see also
Section 6.6). Two-dimensional lattices evolving in a longitudinal direction
can be produced with several nondiffracting Bessel beams and it has
been shown that two-dimensional solitons may undergo spiraling motion
with a controllable rotation rate in such lattices (Kartashov, Vysloukh and
Torner, 2005c).
Linear and nonlinear light propagation in a one-dimensional
waveguide array with a periodically bent axis was studied by Longhi
(2005). In the linear regime the analog of the Talbot self-imagining
effect was predicted. The dynamical localization of light in linear
periodically curved waveguide arrays was observed experimentally
(Longhi, Marangoni, Lobino, Ramponi, Laporta, Cianci and Foglietti,
2006; Iyer, Aitchison, Wan, Dignam and Sterke, 2007). A suitable periodic
waveguide bending is capable of suppressing discrete modulational
instability of nonlinear Bloch waves. The dynamics of light in the presence
of longitudinal defects of arbitrary extent in nonlinear waveguide arrays
and uniform nonlinear media was studied by Fratalocchi and Assanto
(2006b,d). Dynamical super-lattices also may be used for manipulation
with solitons, as was suggested by Porter, Kevrekidis, Carretero-Gonzales
and Frantzeskakis (2006), for the case of one-dimensional BECs. Some
soliton modes supported by optical lattices with localized defects can be
driven across the lattice by means of the transverse lattice shift, provided
that this shift is performed with small enough steps (Brazhnyi, Konotop
and Perez-Garcia, 2006a). It was predicted that Rabi-like oscillations and
stimulated mode transitions occur with linear and nonlinear wave states
in properly modulated waveguides and lattices (Kartashov, Vysloukh
and Torner, 2007a). In particular, the possibilities of cascade stimulated
transitions in systems supporting up to three modes of the same parity
were shown. Such phenomena occur also in the nonlinear regime and
76
Soliton Shape and Mobility Control in Optical Lattices
in multimode systems, where the mode oscillation might be viewed as
analogous to interband transitions.
4. OPTICALLY-INDUCED LATTICES
The method of optical lattice induction is especially attractive because
it allows creation of reconfigurable refractive index landscapes that can
be fine-tuned by lattice-creating waves and easily erased, in contrast to
permanent, technologically fabricated waveguiding structures. Periodic,
optically induced lattices may operate on both weakly and strongly
coupled regimes between neighboring lattice guides, depending on
the intensities of waves inducing the lattice. This affords tunability
of soliton behavior in such lattices from predominantly continuous to
predominantly discrete, which is the core idea that we address in this
review. The idea of optical lattice induction was put forward by Efremidis,
Sears, Christodoulides, Fleischer and Segev (2002). For completeness, in
this section we discuss typical parameters of optical lattices that can be
made with this method.
One or two-dimensional photoinduced lattices have to remain
invariable along the propagation distance (typically, up to several
centimeters). Experiments with a photorefractive SBN crystal take
advantage of its strong electro-optic anisotropy. The lattice-writing
beams are polarized in the ordinary direction, while the probe was
polarized extraordinarily along the crystalline c-axis. By applying a static
electric field across this axis, the probe would experience photorefractive
screening nonlinearity, while the lattice beams propagate in linear
regime (Fleischer, Carmon, Segev, Efremidis and Christodoulides, 2003),
which enables an invariable lattice profile in the longitudinal direction
to be obtained. A further benefit of this system is that the sign of
the nonlinearity (focusing/defocusing) can be altered by changing the
polarity of the externally applied voltage.
4.1 Optical Lattices Induced by Interference Patterns
In the pioneering experiments, the periodic refractive index profile was
photoinduced by interfering two ordinary polarized plane waves in a
biased photorefractive SBN crystal (Fleischer, Carmon, Segev, Efremidis
and Christodoulides, 2003). An external static electric field applied to
the crystal creates periodic changes of the refractive index through the
electro-optic effect. In SBN crystals orthogonally polarized waves feature
dramatically different electro-optic coefficients (r33 ' 1340 pm/V, r13 '
67 pm/V), so that ordinary-polarized interfering plane waves propagate
almost linearly and create the stable one-dimensional periodic lattice,
whereas the extraordinary polarized probe beam experiences strong
Optically-induced Lattices
77
nonlinear self-action, which can be described through the nonlinear
change of the refractive index δn ∼ −n 3e r33 E 0 /(Idark + I0 cos2 (π x/d) + I ),
where Idark characterizes dark irradiance level, I0 is the peak intensity
of the lattice with the spatial period d, I is the intensity of the probe
beam, and E 0 is the static electric field. The intersection angle between
plane waves determines the lattice periodicity (d ∼ 10 µm is a
representative value for experiments with optically induced lattices),
while experimentally achievable refractive index modulation depth in
such a lattice is ∼10−3 .
The propagation direction of the lattice-creating waves and probe
beam may not necessarily coincide. Laser beams launched into tilted
lattices may experience anomalous refraction and experience transverse
displacement, as demonstrated by Rosberg, Neshev, Sukhorukov, Kivshar
and Krolikowski (2005). In such an experiment a periodic lattice was
induced in a SBN:60 photorefractive crystal by interfering two ordinary
polarized beams from a frequency-doubled Nd:YVO4 laser at 532 nm.
Variation in bias voltage alters refractive index modulation depth
and bandgap lattice structure, which, in turn, modifies the diffraction
properties of Bloch waves and allows us to observe both positive or
negative refraction of probe beams that selectively excite first or second
spectral bands. Optical lattices might be also made partially incoherent
and created by amplitude modulation rather than by coherent interference
of multiple plane waves (Chen, Martin, Eugenieva, Xu and Bezryadina,
2004). Such lattices feature enhanced stability, and might propagate
even in weakly nonlinear regimes, due to suppression of incoherent
modulation instability.
Optical induction also allows us to produce square (Fleischer, Segev,
Efremidis and Christodoulides, 2003), hexagonal (Efremidis, Sears,
Christodoulides, Fleischer and Segev, 2002), or triangular (Rosberg,
Neshev, Sukhorukov, Krolikowski and Kivshar, 2007) two-dimensional
photonic lattices by interfering four or three plane waves. A representative
example of a lattice generated by four waves is shown in Figure 2.
A second dimension brings fundamentally new features into light
propagation dynamics in periodic environments and allows a wealth of
new lattice geometries.
Spatial light modulators allow us to generate a rich variety of optical
potentials, including nondiffracting ones, that might be used, for example,
for guiding or trapping of atoms, as was suggested by McGloin, Spalding,
Melville, Sibbett and Dholakia (2003). Such modulators can be used as
reconfigurable, dynamically controllable holograms and under proper
conditions replace prefabricated micro-optical devices. For instance,
current spatial light modulators operating in a phase-modulation regime
offer 1024 × 1024 pixels with high diffraction efficiencies, which are
78
Soliton Shape and Mobility Control in Optical Lattices
FIGURE 2 Typical profile of a two-dimensional photonic lattice induced by the
interference of four plane waves in highly anisotropic SBN:75 crystal. Each waveguide
is approximately 7 µm in diameter, with an 11 µm spacing between nearest neighbors
(Fleischer, Segev, Efremidis and Christodoulides, 2003)
comparable with etched holograms produced lithographically. The
technique can be extended for optical lattice induction in photorefractive
materials. A programmable phase modulator was used for engineering
of specific Bloch states from the second band of a two-dimensional
optical lattice (Träger, Fisher, Neshev, Sukhorukov, Denz, Krolikowski and
Kivshar, 2006).
4.2 Nondiffracting Linear Beams
Nondiffractive linear light beams offer very attractive opportunities for
photoinduction of steady-state photonic lattices of diverse topologies.
The properties of solitons supported by photonic lattices imprinted by
complex nondiffracting Bessel and Mathieu beams will be discussed in
Section 7. Here, we briefly describe the most representative features of
nondiffracting beams in linear homogeneous and periodic media.
Propagation of linear beams in uniform media is described by the
three-dimensional Helmholtz equation that admits separation into the
transverse and longitudinal parts only in Cartesian, circular cylindrical,
elliptic cylindrical, and parabolic cylindrical coordinates. Each of these
coordinate systems then gives rise to a certain type of nondiffracting
beam associated with a fundamental solution of the Helmholtz equation
in these coordinates. Details of experimental observation of parabolic,
Bessel, and Mathieu beams can be found in papers by Durnin, Miceli
and Eberly (1987), Gutierrez-Vega, Iturbe-Castillo, Ramirez, Tepichin,
Rodriguez-Dagnino, Chavez-Cerda and New (2001) and Lopez-Mariscal,
Bandres, Gutierrez-Vega and Chavez-Cerda (2005).
Optically-induced Lattices
79
A powerful and universal representation of the amplitude of any
nondiffracting beam is provided by the reduced Whittaker integral:
Q(ζ, η) =
Z
π
G(ϕ) exp [−ik (ζ cos ϕ + η sin ϕ)] dϕ,
(1)
−π
where G(ϕ) is the angular spectrum defined on the ring of radius
k in frequency space. In the simplest case of Cartesian coordinates,
PM
superposition of M plane waves G(ϕ) =
m=1 δ(ϕ − ϕm ) produces
kaleidoscopic patterns (rectangular, honeycomb, etc). The angular
spectrum G(ϕ) = exp(imϕ) produces m th order Bessel beams that
represent fundamental nondiffracting solutions in a circular cylindrical
coordinate system, while G(ϕ) = cem (ϕ; a)+isem (ϕ; a), with cem (ϕ; a) and
sem (ϕ; a), being angular Mathieu functions, produces m th order Mathieu
beams in an elliptic coordinate system. Parabolic nondiffracting optical
fields were discussed by Bandres, Gutierrez-Vega and Chavez-Cerda
(2004). Some representative examples of transverse intensity distributions
in most known types of nondiffracting beams are shown in Figure 3.
Using such beams in the technique of optical induction opens broad
prospects for creation of the refractive index landscapes with novel
types of symmetry. Besides this, nondiffracting beams find applications
in diverse areas of physics, such as optical manipulation of small
particles (McGloin and Dholakia, 2005), frequency doubling, or atom
trapping.
Nondiffracting linear beams may also form in periodic media. Such
beams were considered in two-dimensional periodic lattices (Manela,
Segev and Christodoulides, 2005). The links between the symmetry
properties, phase structure of such beams and the number of spectral
bands that gives rise to the beam have been established. The possibility
of diffraction-free propagation of localized light beams in materials
with both transverse and longitudinal modulation of refractive index
was predicted (Staliunas and Herrero, 2006; Staliunas, Herrero and de
Valcarcel, 2006; Staliunas and Masoller, 2006). In such materials the
dominating (second) order of diffraction may vanish and thus the overall
diffraction broadening of the beam is determined by the fourth-order
dephasing of spatial Fourier modes.
4.3 Mathematical Models of Wave Propagation
Several mathematical models are commonly accepted for description
of soliton evolution in optical lattices. In the case of optical solitons
all of them are based on the parabolic propagation equation for the
slowly varying amplitude of light fields coupled to the material equation
describing the corrections to the refractive index. Thus, in the simplest
80
Soliton Shape and Mobility Control in Optical Lattices
FIGURE 3 Lattices induced by the nondiffracting beams of different type: (a)
zero-order radially symmetric Bessel lattice; (b) first-order radially symmetric Bessel
lattice; (c) lattice produced by the azimuthally modulated Bessel beam of third order;
odd Mathieu lattices with small (d) and large (e) interfocal parameters that closely
resemble azimuthally modulated Bessel and quasi-one-dimensional periodic lattices,
correspondingly; (f) lattice produced by the combination of odd and even parabolic
beams
case of cubic nonlinear media their combination results in the canonical
nonlinear Schrödinger equation describing the evolution of optical wave
packets in the presence of the lattice, which in the case of one transverse
dimension reads:
i
1 ∂ 2q
∂q
=−
+ σ |q|2 q − p R(η)q.
∂ξ
2 ∂η2
−1/2
(2)
Here q = (L dif /L nl )1/2 AI0
is the dimensionless complex amplitude
of the light field; A is the slowly varying envelope; I0 is the input intensity;
the transverse η and longitudinal ξ coordinates are normalized to the
beam width r0 and the diffraction length L dif = n 0 ωr02 /c, respectively; ω
is the carrying frequency; n 0 is the unperturbed refractive index; L nl =
2c/ωn 2 I0 ; σ = −1 for focusing nonlinearity and σ = +1 in the case of
defocusing nonlinearity; p = L dif /L ref is the guiding parameter; L ref =
c/δnω; δn is the refractive index modulation depth, while the function
Optically-induced Lattices
81
R(η) describes the transverse refractive index profile. In the particular
case of optically induced or technologically fabricated harmonic refractive
modulation one can set R(η) = cos(η), where  is the lattice frequency.
The generalization of the evolution equation (2) to the case of two
transverse dimensions requires replacing ∂ 2 /∂η2 by the two-dimensional
transverse Laplacian ∂ 2 /∂η2 + ∂ 2 /∂ζ 2 .
Note the analogy between the equations describing propagation of
optical wave-packets and matter waves. Thus, Equation (2) also describes
dynamics of one-dimensional Bose–Einstein condensate confined in an
optical lattice generated by a standing light wave of wavelength λ. In
this case, q stands for the dimensionless mean-field wave function, the
variable ξ stands for time in units of τ = 2mλ2 /π h, with m being the mass
of the atoms and h the Planck’s constant, η is the coordinate along the axis
of the quasi-one-dimensional condensate expressed in units of λπ −1 . The
parameter p is proportional to the lattice depth E 0 expressed in units of
recoil energy E rec = h 2 /2mλ2 . In quasi-one-dimensional condensates one
has σ = 2λas Na /π `2 , where as is the s-wave scattering length, Na is the
number of atoms, and ` is the harmonic oscillator length.
The evolution equations typically admit of several conserved
quantities. Thus, Equation (2) conserves the total energy flow U and the
Hamiltonian H :
∞
Z
|q|2 dη,
U=
−∞
1
H=
2
Z
(3)
∞
(|∂q/∂η| − 2 p R |q| + σ |q| )dη.
2
2
4
−∞
The method of creation of periodic lattices (especially two-dimensional
ones) in nonlinear crystals generally relies on an optical induction
technique (see Section 3). With this technique an optical lattice can be
imprinted in a photosensitive crystal, for example, by interfering two
or more plane waves. The interference pattern of several plane waves
is a propagation invariant pattern and should not be affected by the
nonlinearity of the crystal. At the same time the soliton beam released
into the lattice should experience strong nonlinearity. Such conditions can
be achieved in materials with a strong anisotropy of nonlinear response,
e.g., in suitable photorefractive crystals. In this case, the lattice-creating
waves and the soliton beam are polarized in orthogonal directions, so that
nonlinearity affects only the soliton beam. In this configuration the lattice
itself is not affected by the soliton beam, while the soliton beam does feel
the periodic potential.
82
Soliton Shape and Mobility Control in Optical Lattices
Since nonlinearity saturation is inherent for the photorefractive
materials, the model equation
i
∂q
1 ∂ 2q
Eq
=−
−
[S |q|2 + R(η)]
2
∂ξ
2 ∂η
1 + S |q|2 + R(η)
(4)
is also frequently used for description of solitons in photorefractive optical
−1/2
lattices. Here q = AI0
, I0 is the input intensity; the normalization for
transverse and longitudinal coordinate is similar to that for Equation (2);
S = I0 /(Idark + Ibg ) is the saturation parameter; Idark and Ibg are dark
and background radiation intensities; R = Ilatt /(Idark + Ibg ), where Ilatt
represents the intensity distribution in the lattice-creating wave; E =
(1/2)(ωr0 n 20 /c)2reff E 0 is the dimensionless static biasing field applied to
the crystal; reff is the effective electro-optic coefficient corresponding to
polarization of the soliton beam. Under the assumptions S |q|2 , R(η, ξ ) 1, E 1, the Equation (4) can be transformed into the cubic nonlinear
Schrödinger equation (2). It is important to stress that the depth of
optically induced lattices can be tuned by changing the intensities of
the lattice-creating waves. The model (4), generalized to the case of two
transverse dimensions by replacing the transverse Laplacian with its
two-dimensional counterpart ∂ 2 /∂η2 + ∂ 2 /∂ζ 2 , is also frequently used in
the literature, though it describes thestrongly anisotropic and nonlocal
response of such crystals only approximately (see Section 9 where fully
anisotropic model of photorefractive response is discussed).
Trivial-phase stationary solutions of Equation (2) or (4) can be obtained
in the form q(η, ξ ) = w(η) exp(ibξ ), where w(η) is a real function
describing the transverse profile, and b is a real propagation constant.
Thus, mathematically, families of lattice solitons are defined by the
propagation constant b, lattice depth p, and particular lattice shape given
by the function R(η). Various families of soliton solutions can be obtained
from a known family by using scaling transformations, which, in the case
of Equation (2), are q(η, ξ, p) → χq(χ η, χ 2 ξ, χ 2 p), where χ is the arbitrary
scaling factor. The equation for soliton profiles obtained upon substitution
of the light field in such form into the evolution equation can be solved
numerically with standard relaxation or spectral methods. Analogously,
a split-step fast-Fourier method may be used to solve the very evolution
equations with a variety of input conditions.
Stability of solitons can be tested by searching for perturbed solutions
in the form q = (w + u + iv) exp(ibξ ), where u(η, ξ ) and v(η, ξ ) are real
and imaginary parts of perturbation, that can grow with a rate δ upon
propagation. Substitution of the light field in such form into Equation (2),
linearization around a stationary solution w and derivation of real and
One-dimensional Lattice Solitons
83
imaginary parts yields the following linear eigenvalue problem
1 ∂ 2v
+ bv + σ w2 v − p Rv,
2 ∂η2
1 ∂ 2u
−δv = −
+ bu + 3σ w 2 u − p Ru,
2 ∂η2
δu = −
(5)
that can be solved numerically with standard linear eigenvalue solvers.
The presence of perturbations corresponding to Re δ 6= 0 indicates an
instability of the soliton solutions, while Re δ ≡ 0 indicates linear stability.
The results of linear stability analysis may additionally be tested by direct
numerical propagation of the soliton solutions perturbed with broadband
input noise. The generalization of these techniques to the case of two
transverse dimensions is straightforward.
5. ONE-DIMENSIONAL LATTICE SOLITONS
In this section we describe the properties of scalar and vector solitons
supported by one-dimensional lattices and briefly discuss the possibilities
for control and manipulation of the soliton shape and width, its internal
structure, and transverse mobility afforded by static and dynamical
optical lattices with tunable parameters, imprinted in the materials with
local and nonlocal nonlinear responses. We discuss the role of the main
factors that can lead to stabilization of complex lattice soliton structures.
Notice that the state of art in the field of spatial optical solitons
in uniform nonlinear materials has been summarized in several books
(Akhmediev and Ankiewicz, 1997; Trillo and Torruellas, 2001; Kivshar
and Agrawal, 2003) and reviews (Stegeman and Segev, 1999; Kivshar
and Pelinovsky, 2000; Torner, 1998; Etrich, Lederer, Malomed, Peschel
and Peschel, 2000; Buryak, Di Trapani, Skryabin and Trillo, 2002; Kivshar
and Luther-Davies, 1998; Malomed, Mihalache, Wise and Torner, 2005;
Conti and Assanto, 2004). Lattice solitons are continuous counterparts of
discrete solitons existing in waveguide arrays (for a recent reviews on
discrete solitons see Aubry (1997), Flach and Willis (1998), Kevrekidis,
Rasmussen and Bishop (2001), Christodoulides, Lederer and Silberberg
(2003) and Aubry (2006)). The discrete NLSE that is used to describe
the evolution of nonlinear excitations in such systems has a rather
universal character and can be used to describe light propagation even in
tensorial systems and in the presence of nonparaxial and vectorial effects
(Fratalocchi and Assanto, 2007c). As mentioned above, mathematically,
the equations describing propagation of laser radiation in periodic media
are analogous to those describing evolution of Bose–Einstein condensates
84
Soliton Shape and Mobility Control in Optical Lattices
in optical lattices, hence, many soliton phenomena predicted in nonlinear
optics were encountered in BECs and vice versa (Dalfovo, Giorgini,
Pitaevskii and Stringari, 1999; Pitaevskii and Stringari, 2003; Abdullaev,
Gammal, Kamchatnov and Tomio, 2005; Morsch and Oberthaler, 2006).
5.1 Fundamental Solitons
Lattice solitons form due to the balance of diffraction, refraction in
the periodic lattice, and nonlinear self-phase-modulation. The existence
of such solitons is closely linked to the band-gap structure of a onedimensional periodic lattice spectrum as depicted in Figure 1. Since the
spectrum is composed of bands where only Bloch waves can propagate,
and gaps where propagation of Bloch waves is forbidden, lattice solitons
emerge as defect modes in the gaps of the lattice spectrum. The band-gap
structure depends crucially on the lattice depth p, but in one-dimensional
it always includes a single semi-infinite gap and an unlimited number of
finite gaps. The internal soliton structure is determined by the position
of the soliton propagation constant inside the corresponding gap. In
focusing media, the simplest fundamental (odd) solitons are formed
in a semi-infinite gap (see Figure 4(a) for an illustrative example of a
soliton profile in the lattice R(η) = cos(η) with  = 4 and focusing
Kerr nonlinearity); the position of the intensity maximum for such a
soliton coincides with one of local lattice maximums. The properties of
odd and other types of lattice solitons have been analyzed in different
physical settings, including photorefractive optical lattices and BECs
(Efremidis, Sears, Christodoulides, Fleischer and Segev, 2002; Louis,
Ostrovskaya, Savage and Kivshar, 2003; Efremidis and Christodoulides,
2003; Kartashov, Vysloukh and Torner, 2004a).
The monotonic increase of energy flow with propagation constant
indicates stability of odd lattice solitons in accordance with the VakhitovKolokolov stability criterion (Vakhitov and Kolokolov, 1973). Solitons
existing near the gap edges transform into Bloch waves with the same
symmetry. Besides odd solitons, optical lattices also support even solitons
centred between the neighboring maxima of R(η) (Figure 4(b)). In Kerr
nonlinear media even solitons are unstable, while strong nonlinearity
saturation may result in stabilization of even solitons accompanied by
destabilization of odd ones (Kartashov, Vysloukh and Torner, 2004a).
Experimentally odd solitons in photorefractive lattices were observed by
Fleischer, Carmon, Segev, Efremidis and Christodoulides (2003) and by
Neshev, Ostrovskaya, Kivshar and Krolikowski (2003). Figure 5 illustrates
the experimental generation of odd and even lattice solitons.
Increasing the optical lattice depth inhibits localization of light in
the vicinity of local lattice maxima, a process accompanied by the
development of strong modulation of soliton profiles. Under such
One-dimensional Lattice Solitons
85
FIGURE 4 Odd (a), even (b), and twisted (c) solitons originating from semi-infinite
gaps of optical lattice with focusing nonlinearity at b = 1.4. Odd (d), even (e), and
twisted (f) gap solitons originating from the first finite gap of an optical lattice with
defocusing nonlinearity at b = −0.5. In all cases lattice depth p = 4
conditions, one can employ the so-called tight-binding approximation to
reduce Equation (2) to a discrete NLSE obtained by Christodoulides
and Joseph (1988) for arrays of evanescently coupled waveguides, where
discrete solitons were observed by Eisenberg, Silberberg, Morandotti,
Boyd and Aitchison (1998). Discrete matter-wave solitons were introduced
by Trombettoni and Smerzi (2001).
Dark discrete solitons in one-dimensional arrays with defocusing
nonlinearity have been also investigated (Smirnov, Rüter, Stepic, Kip and
Shandarov, 2006). Localized nonlinear dark modes displaying a phase
jump in the centre located either on-channel or in-between channels were
observed, and the ability of the induced refractive index structures to
guide light of a low-power probe beam was demonstrated. The interaction
of a probe beam with dark and bright blocker solitons was studied
in counter-propagating geometry by Smirnov, Rüter, Stepic, Shandarov
and Kip (2006). Hadzievski, Maluckov and Stepic (2007) presented a
numerical analysis of the dynamics of dark breathers in lattices with
saturable nonlinearity. Comparison of discrete dark solitons properties in
lattices with saturable and Kerr nonlinearities was performed by Fitrakis,
Kevrekidis, Susanto and Frantzeskakis (2007). Dark matter-wave solitons
in optical lattices were discussed by Louis, Ostrovskaya and Kivshar
(2004). The properties of dark solitons in dynamical lattices with cubic-
86
Soliton Shape and Mobility Control in Optical Lattices
FIGURE 5 Experimental demonstration of odd and even spatial solitons. (a) Input
beam and optical lattice (power 23 µW). (b) Output probe beam at low power
(2 × 10−3 µW). (c) Localized states (87 × 10−3 µW). Left, propagation in the absence of
the grating; middle, even excitation; right, odd excitation. Biasing electric field
3600 V/cm (Neshev, Ostrovskaya, Kivshar and Krolikowski, 2003)
quintic nonlinearity have been addressed by Maluckov, Hadzievski and
Malomed (2007).
5.2 Beam Shaping and Mobility Control
Optical lattices offer rich opportunities for soliton control, by varying
the lattice depth and period. In a lattice, at a given energy flow,
the field amplitude necessary for soliton-like propagation amounts to
lower values than that in homogeneous media. The width of the lattice
soliton also changes with increase of the lattice depth. More importantly,
One-dimensional Lattice Solitons
87
the transverse refractive index modulation profoundly affects soliton
mobility.
The simple effective particle approach (Scharf and Bishop, 1993;
Kartashov, Zelenina, Torner and Vysloukh, 2004), based on the equation
d2
p
hηi =
U
dξ 2
Z
∞
−∞
|q|2
dR
dη
dη
(6)
R∞
for the integral soliton centre hηi = U −1 −∞ |q|2 ηdη, might be used
to identify regimes of soliton propagation in the lattice. This approach
requires substitution of a trial function for the light field, that can be
chosen in the form q = q0 sech[χ (η −hηi)] exp[iα(η −hηi)], with q0 , χ, and α
being amplitude, inverse width, and incident angle, respectively. At small
incident angles Equation (6) predicts periodic oscillations of the soliton
centre inside the input channel. When the input angle exceeds a critical
value αcr = 2[ p(π/2χ ) sinh−1 (π/2χ )]1/2 (associated with the height of
the Peierls–Nabarro barrier created by the periodic potential (Kivshar and
Campbell, 1993)) the soliton leaves the input channel and starts traveling
across the lattice, a process accompanied by radiative losses (Yulin,
Skryabin and Russell, 2003). The radiation may cause soliton trapping
in one of the lattice channels. The controllable trapping in different
lattice locations may find practical applications and has been studied in a
variety of periodic systems, including discrete waveguide arrays (Aceves,
De Angelis, Trillo and Wabnitz, 1994; Krolikowski, Trutschel, CroninGolomb and Schmidt-Hattenberger, 1994; Aceves, De Angelis, Peschel,
Muschall, Lederer, Trillo and Wabnitz, 1996; Krolikowski and Kivshar,
1996; Bang and Miller, 1996; Morandotti, Peschel, Aitchison, Eisenberg
and Silberberg, 1999a; Pertsch, Zentgraf, Peschel, Brauer and Lederer,
2002a; Vicencio, Molina and Kivshar, 2003).
The transverse mobility of lattice solitons can be substantially enhanced
in saturable media in the regime of strong saturation (Hadzievski,
Maluckov, Stepic and Kip, 2004; Stepic, Kip, Hadzievski and Maluckov,
2004; Melvin, Champneys, Kevrekidis and Cuevas, 2006; Oxtoby and
Barashenkov, 2007), an effect related to the stability exchange between
even and odd solitons. Due to radiation emission by the moving
solitons, soliton collisions in such systems are strongly inelastic (Aceves,
De Angelis, Peschel, Muschall, Lederer, Trillo and Wabnitz, 1996;
Meier, Stegeman, Silberberg, Morandotti and Aitchison, 2004; Meier,
Stegeman, Christodoulides, Silberberg, Morandotti, Yang, Salamo, Sorel
and Aitchison, 2005; Cuevas and Eilbeck, 2006).
88
Soliton Shape and Mobility Control in Optical Lattices
5.3 Multipole Solitons and Soliton Trains
Optical lattices can support scalar multipole solitons composed of several
out-of-phase bright spots, when repulsive “forces” acting between spots
in a bulk medium are compensated by the immobilizing action of the
lattice (Figure 4(c)). In discrete waveguide arrays such solitons are
known as twisted strongly localized modes (Darmanyan, Kobyakov and
Lederer, 1998; Kevrekidis, Bishop and Rasmussen, 2001). The important
feature of multipole solitons in continuous lattices is that they become
completely stable when their energy flow exceeds a certain threshold
(Efremidis, Sears, Christodoulides, Fleischer and Segev, 2002; Louis,
Ostrovskaya, Savage and Kivshar, 2003; Efremidis and Christodoulides,
2003; Kartashov, Vysloukh and Torner, 2004a). Such stabilization takes
place for multipole solitons of arbitrary higher order (soliton trains),
containing multiple spots, provided that the phase alternates by π (in
the case of focusing medium) between each two neighboring spots.
This immediately suggests the important possibility of constructing and
manipulating multi-peaked “soliton packets” beyond single “soliton
bits”, a feature that might open a new door in all optical-switching
schemes. Notice that individual solitons can then be extracted or added
into such trains.
The simplest dipole solitons in optical lattices were observed by
Neshev, Ostrovskaya, Kivshar and Krolikowski (2003). He and Wang
(2006) addressed specific dipole solitons residing completely in a single
channel of a low-frequency optical lattice. Individual solitons and soliton
trains have been also studied in lattices with competing cubic-quintic
nonlinearities, where bistability was encountered for several soliton
families (Merhasin, Gisin, Driben and Malomed, 2005; Wang, Ye, Dong,
Cai and Li, 2005), as well as in the pure quintic case (Alfimov, Konotop and
Pacciani, 2007). Periodic lattices also support truly infinite periodic waves
that were studied in the context of BEC (for an overview, see Deconinck,
Frigyik and Kutz (2002)).
5.4 Gap Solitons
The finite gaps of the Floquet–Bloch spectrum of the periodic lattice may
also give rise to solitons that are usually termed gap solitons. In contrast
to solitons emerging from the semi-infinite gap (or total internal reflection
gap), solitons from finite gaps are possible due to Bragg scattering from
the periodic structure and exist in conditions of strong coupling between
modes having opposite transverse wavevector components. Gap solitons
exhibit multiple amplitude oscillations (Figure 4(d)), while their internal
symmetry depends on the particular gap that supports them.
Gap-type excitations in discrete waveguide arrays were analyzed by
Kivshar (1993). One-dimensional gap solitons in continuous optically
One-dimensional Lattice Solitons
89
induced lattices were observed in photorefractive crystals by Fleischer,
Carmon, Segev, Efremidis and Christodoulides (2003). Controlled
generation and steering of gap solitons in optically induced lattices was
reported by Neshev, Sukhorukov, Hanna, Krolikowski and Kivshar (2004).
Matter-wave gap solitons in Bose–Einstein condensates with repulsive
inter-atomic interactions were observed by Eiermann, Anker, Albiez,
Taglieber, Treutlein, Marzlin and Oberthaler (2004). Gap solitons may
also form in arrays of evanescently coupled waveguides fabricated in
AlGaAs (Mandelik, Morandotti, Aitchison and Silberberg, 2004) and
LiNbO (Chen, Stepic, Rüter, Runde, Kip, Shandarov, Manela and Segev,
2005; Matuszewski, Rosberg, Neshev, Sukhorukov, Mitchell, Trippenbach,
Austin, Krolikowski and Kivshar, 2005).
The steering properties of gap solitons strongly depend on the
sign and magnitude of the spatial group-velocity dispersion near the
corresponding gap edge and may be anomalous, i.e. solitons may move
in the direction opposite to the input tilt. Similarly to solitons forming in a
semi-infinite gap, solitons from finite gaps may form trains. In defocusing
media, trains of in-phase gap solitons may be stable (Figure 4(e)), while
twisted modes (Figure 4(f)) undergo strong exponential instabilities.
Trains of gap solitons have been predicted in one-, two-, and threedimensional optical lattices (Kartashov, Vysloukh and Torner, 2004a;
Alexander, Ostrovskaya and Kivshar, 2006). Experimental observation
of higher-order one-dimensional gap solitons in defocusing waveguide
arrays was presented by Smirnov, Rüter, Kip, Kartashov and Torner
(2007).
Gap soliton collisions are determined mostly by soliton localization.
While broad low-energy solitons emerging close to the upper gap edge
may interact almost elastically, the collisions of high-power gap solitons
is inelastic and may lead to soliton fusion (Dabrowska, Osrovskaya and
Kivshar, 2004; Malomed, Mayteevarunyoo, Ostrovskaya and Kivshar,
2005). The interaction between two well-localized parallel solitons in
one-dimensional discrete saturable systems has been investigated using
defocusing lithium niobate nonlinear waveguide arrays by Stepic,
Smirnov, Rüter, Prönneke, Kip and Shandarov (2006). Gap solitons have
been studied in more complicated but still periodic systems such as binary
waveguide arrays (Sukhorukov and Kivshar, 2002a) or quasiperiodic
optical lattices (Sakaguchi and Malomed, 2006).
Gap solitons are typically stable in the middle of the gap, but
near the band edge they feature specific oscillatory instabilities similar
to those observed in fiber Bragg gratings (Barashenkov, Pelinovsky
and Zemlyanaya, 1998). Weak instabilities of gap solitons arise due
to resonant energy redistributions between different gaps, while
90
Soliton Shape and Mobility Control in Optical Lattices
perturbation eigenmodes associated with such instability are poorly
localized (Pelinovsky, Sukhorukov and Kivshar, 2004).
Motzek, Sukhorukov and Kivshar (2006) studied the dynamical
reshaping of polychromatic beams in periodic and semi-infinite photonic
lattices and the formation of polychromatic gap solitons. The propagation
of polychromatic light in nonlinear photonic lattices has been reviewed by
Sukhorukov, Neshev and Kivshar (2007). In particular, the observation of
localization of supercontinuum radiation in waveguide arrays imprinted
in LiNbO3 crystals and the possibilities for dynamical control of the output
spectrum were reported by Neshev, Sukhorukov, Dreischuh, Fischer, Ha,
Bolger, Bui, Krolikowski, Eggleton, Mitchell, Austin and Kivshar (2007).
5.5 Vector Solitons
Vectorial interactions (cross-phase modulation) between several light
fields may significantly enrich the dynamics of soliton propagation in the
periodic structure. The propagation of two mutually incoherent beams
in the lattice with focusing (σ = −1) or defocusing (σ = +1) Kerr
nonlinearity can be described by the system of equations:
i
∂q1,2
1 ∂ 2 q1,2
=−
+ σ q1,2 (|q1 |2 + |q2 |2 ) − p R(η)q1,2 .
∂ξ
2 ∂η2
(7)
The model (7) can be enlarged to take into account coherent interaction
between orthogonally polarized beams in birefringent media, when fourwave-mixing is taken into account. Vectorial coupling between strongly
localized modes in discrete waveguide arrays generates new soliton
families having no counterparts in the scalar case (Darmanyan, Kobyakov,
Schmidt and Lederer, 1998; Ablowitz and Musslimani, 2002). The simplest
vector solitons were observed in AlGaAs arrays in the presence of fourwave-mixing (Meier, Hudock, Christodoulides, Stegeman, Silberberg,
Morandotti and Aitchison, 2003).
In lattices, vectorial interaction between the solitons emerging from
the same gap stabilizes beams that are unstable when propagating alone.
Thus, even solitons in focusing media are stabilized when coupled
with twisted mode, and vice-versa, in defocusing media in-phase trains
stabilize their twisted counterparts (Kartashov, Zelenina, Vysloukh and
Torner, 2004). Localization of two-component Bose–Einstein condensates
and formation of bright-bright and dark-bright matter-wave solitons in
optical lattices was considered by Ostrovskaya and Kivshar (2004a).
Coupling between solitons emerging from the neighboring gaps
of a lattice spectrum may also lead to formation of composite
vector states as predicted by Cohen, Schwartz, Fleischer, Segev and
Christodoulides (2003) and Sukhorukov and Kivshar (2003). Importantly,
One-dimensional Lattice Solitons
91
one of the components in such vector states is always unstable alone
due to its symmetry dictated by the gap number. Transient interband
mutual focusing and defocusing of waves exhibiting diffraction of
different magnitude and sign was observed by Rosberg, Hanna, Neshev,
Sukhorukov, Krolikowski and Kivshar (2005) in optical lattices. The
coherent interactions between two beams closely resembling profiles of
Bloch waves from two neighboring bands also results in formation of
nonlinear localized excitations or breathers that experience beating upon
propagation in the array (Mandelik, Eisenberg, Silberberg, Morandotti
and Aitchison, 2003b). The impact of nonlinearity saturation on
the properties of vector solitons in discrete waveguide arrays was
studied by Fitrakis, Kevrekidis, Malomed and Frantzeskakis (2006),
while experimental observation of vector solitons in saturable discrete
waveguide arrays was performed by Vicencio, Smirnov, Rüter, Kip and
Stepic (2007).
5.6 Soliton Steering in Dynamical Lattices
All lattices considered so far were invariant along the direction of
propagation. Variation of the lattice shape in the longitudinal direction
opens a wealth of new opportunities for soliton control. Several types
of dynamical lattices were considered in the literature, including lattices
with monotonic or periodic variation of depth and width of guiding
channels as well as lattices with periodically curved channels. In
particular, a longitudinal variation of coupling strength in a modulated
waveguide array can resonantly excite internal modes of discrete solitons,
which may lead to soliton breathing, splitting, or transverse motion,
depending on the symmetry of the excited mode (Peschel and Lederer,
2002; Pertsch, Peschel and Lederer, 2003).
Self-imaging at periodic planes may occur in arrays of periodically
curved waveguides, while modulational instability is inhibited under
conditions of self-imaging (Longhi, 2005). Periodic waveguide arrays built
of several tilted segments support breathing discrete diffraction managed
scalar and vector solitons, reproducing their shapes on each period of
the structure (Ablowitz and Musslimani, 2001, 2002). A longitudinal
modulation causes parametric amplification of soliton swinging inside
the lattice channel that may be used to detect submicron soliton
displacements (Kartashov, Vysloukh and Torner, 2004). Modulated lattices
may be used for stopping and trapping moving solitons, while initially
stationary solitons can be transferred to any prescribed position by a
moving lattice (Kevrekidis, Frantzeskakis, Carretero-Gonzalez, Malomed,
Herring and Bishop, 2005; Dabrowska, Osrovskaya and Kivshar, 2006).
Gap solitons undergo abrupt delocalization and compression in an optical
lattice with varying strength (Baizakov and Salerno, 2004).
Soliton Shape and Mobility Control in Optical Lattices
92
An interesting opportunity to control output soliton position was
encountered in dynamical optical lattices that exhibit a finite momentum
in the transverse plane. Such lattices can be induced by three plane waves
A exp(±iαη) exp(−iα 2 ξ/2), B exp(iβη) exp(−iβ 2 ξ/2), where a, b are wave
amplitudes and ±α, β are propagation angles, so that the lattice profile is
given by R(η, ξ ) = 4A2 cos2 (αη) + B 2 + 4AB cos(αη) cos[βη + (α 2 − β 2 )ξ/2].
In this case the transverse momentum can be transferred from the lattice
to the soliton beam, thus leading to a controllable drift (Kartashov, Torner
and Christodoulides, 2005; Garanovich, Sukhorukov and Kivshar, 2005).
Perturbation theory, based on the inverse scattering transform, might be
used to calculate the tilt φ (or instantaneous propagation angle) of the
soliton beam q(η, ξ ) = χ sech[χ (η −φξ )] exp[iφη −i(χ 2 −φ 2 )ξ/2] in a threewave lattice. One gets
φ=
β +α
β −α
−
(β + α) sinh[π(β − α)/2χ] (β − α) sinh[π(β + α)/2χ]
!
β 2 − α2
1 − cos
ξ ,
(8)
2
2π AB
χ
×
Thus, the drift grows linearly with the amplitude B of the third plane
wave. All-optical beam steering in modulated photonic lattices induced
by three-wave interference was observed experimentally by Rosberg,
Garanovich, Sukhorukov, Neshev, Krolikowski and Kivshar (2006) and is
illustrated in Figure 6. Soliton steering and fission in optical lattices that
fade away exponentially along the propagation direction was considered
by Kartashov, Vysloukh and Torner (2006b). Garanovich, Sukhorukov and
Kivshar (2007) discussed the phenomenon of nonlinear light diffusion in
periodically curved arrays of optical waveguides. Soliton steering by a
single continuous wave that dynamically induces a photonic lattice, as
well as interactions of several solitons in the presence of such waves were
addressed by Kominis and Hizanidis (2004) and Tsopelas, Kominis and
Hizanidis (2006).
5.7 Lattice Solitons in Nonlocal Nonlinear Media
Under appropriate conditions the nonlinear response of materials can be
highly nonlocal, a phenomenon that drastically affects the propagation
of light. New effects attributed to nonlocality were encountered, e.g.,
in photorefractive media, liquid crystals and thermo-optical media (see
Krolikowski, Bang, Nikolov, Neshev, Wyller, Rasmussen and Edmundson
(2004) for a review). Nonlocality of the nonlinear response also strongly
affects properties of lattice solitons. Propagation of a laser beam in
nonlocal media with an imprinted transverse refractive index modulation
One-dimensional Lattice Solitons
93
FIGURE 6 (a) Experimental and theoretical (inset) linear output in a straight lattice.
(b) Shift of the nonlinear probe beam output versus the modulating beam power for
k3x = 1.18k12x and ϕ/2π = 0.22, where k3x and k12x stand for the transverse
wavenumbers of the third and first two waves, ϕ is the phase difference between the
third wave and the other two waves. (c),(d) Experimental and theoretical (inset)
nonlinear output in straight and modulated lattice for intensity of the third wave
I3 = 0 and I3 = 4I12 , respectively, and k3x = 1.18k12x . (e),(f) Numerical simulations of
the longitudinal propagation (Rosberg, Garanovich, Sukhorukov, Neshev, Krolikowski
and Kivshar, 2006)
can be described by the equation:
i
∂q
1 ∂ 2q
=−
−q
∂ξ
2 ∂η2
Z
∞
G(η − λ) |q(λ)|2 dλ − p R(η)q,
(9)
−∞
where G(η) is the response function of the nonlocal medium. In the
limit G(η) → δ(η) one recovers the case of local response, while a
94
Soliton Shape and Mobility Control in Optical Lattices
strongly nonlocal medium is described by slowly decaying kernels G(η).
Symmetric strongly nonlocal response reduces symmetry-breaking
instabilities of lattice solitons, including even solitons in focusing and
twisted gap modes in defocusing media. The Peierls–Nabarro potential
barrier for a soliton moving across the lattice is reduced, due to
nonlocality, an effect that results in the corresponding enhancement of
the transverse soliton mobility (Xu, Kartashov and Torner, 2005a). By
tuning the lattice depth one can control the mobility of lattice solitons in
materials with asymmetric nonlocal response (Kartashov, Vysloukh and
Torner, 2004b; Xu, Kartashov and Torner, 2006).
Liquid crystals belong to the rare class of nonlinear materials,
simultaneously featuring intrinsically nonlocal nonlinearity and allowing
for formation of tunable refractive index landscapes (Peccianti, Conti,
Assanto, De Luca and Umeton, 2004; Fratalocchi, Assanto, Brzdakiewicz
and Karpierz, 2004; Fratalocchi and Assanto, 2005). Figure 7 shows
the setup and experimentally observed lattice solitons in a periodically
biased nematic liquid crystal. The lattice depth is tuned by varying the
voltage applied to periodically spaced electrodes on the liquid crystal
cell. A similar technique was used for demonstration of multiband vector
breathers (Fratalocchi, Assanto, Brzdakiewicz and Karpierz, 2005a). The
experimental observation of power-dependent beam steering over angles
of several degrees in a voltage-controlled array of channel waveguides
defined in a nematic liquid crystal via electrooptic reorientational
response was observed by Fratalocchi, Assanto, Brzdakiewicz, and
Karpierz (2005b). Such steering was demonstrated for light beams at
mW power levels. An overview of the latest experimental and theoretical
advances in investigation of discrete light propagation and self-trapping
in nematic liquid crystals is given in Fratalocchi, Brzdakiewicz, Karpierz,
and Assanto (2005).
6. TWO-DIMENSIONAL LATTICE SOLITONS
In this section we describe the properties of solitons supported by twodimensional optical lattices. By and large, adding the second transverse
coordinate profoundly affects properties of self-sustained nonlinear
excitations. In particular, two dimensional solitons in uniform Kerr
nonlinear media may experience catastrophic collapse in contrast to their
one-dimensional counterparts (Berge, 1998). Optical lattices typically play
a strong stabilizing action on two-dimensional solitons and allow for
the existence of new types of solitons with unusual internal structures.
Here, we describe the conditions required for existence of stable twodimensional lattice solitons. Two-dimensional settings also give rise to
vortex solitons that are impossible in lower-dimensional settings. The
Two-dimensional Lattice Solitons
95
FIGURE 7 (Top) Cell for liquid-crystal waveguide array: 3, array period, d, cell
thickness, V , applied voltage, ITO, indium tin oxide electrodes. (Bottom) Experimental
results with light propagation at 1064 nm wavelength: (a) one-dimensional diffraction
at V = 0 V, (b) low-power discrete diffraction at V = 1.2 V and 20 mW input power, (c)
discrete nematicon at V = 1.2 V and 35 mW input power. The residual coupling and
transverse-longitudinal power fluctuations shown are due partly to the limits of the
camera and partly to nonuniformities in the sample (Fratalocchi, Assanto,
Brzdakiewicz and Karpierz, 2004)
properties of optical and matter-wave vortices are described in detail in
a review by Desyatnikov, Torner and Kivshar (2005). In this section we
address the existence and specific features of vortex solitons in periodic
media. Various types of two-dimensional soliton have been studied in
discrete nonlinear systems (for reviews see Aubry (1997), Flach and Willis
(1998), Hennig and Tsironis (1999), Kevrekidis, Rasmussen and Bishop
(2001), Christodoulides, Lederer and Silberberg (2003), Aubry (2006); as
well as the focus issues Physica D 119, 1 (1998) and Physica D 216, 1 (2006)
devoted to discrete solitons in nonlinear lattices). Recent developments in
the field of two-dimensional matter-wave lattice solitons are summarized
by Kevrekidis, Carretero-Gonzalez, Frantzeskakis and Kevrekidis (2004),
Morsch and Oberthaler (2006).
6.1 Fundamental Solitons
As in the case of one-dimensional solitons addressed in Section 5, the
properties of two-dimensional solitons in optical lattices are dictated
96
Soliton Shape and Mobility Control in Optical Lattices
by the band-gap structure of the lattice spectrum. However, in clear
contrast with the one-dimensional case, bands in the spectrum of a twodimensional lattice can strongly overlap, thereby considerably restricting
the number of gaps in the spectrum or even completely eliminating all
finite gaps. This is the main property that distinguishes two-dimensional
lattices from their one-dimensional counterparts possessing an infinite
number of finite gaps. The number of finite gaps in the spectrum of a twodimensional lattice is dictated by the particular lattice shape, depth, and
period.
The simplest two-dimensional lattice solitons in focusing media
emerge from the semi-infinite gap (Efremidis, Sears, Christodoulides,
Fleischer and Segev, 2002; Yang and Musslimani, 2003; Efremidis, Hudock,
Christodoulides, Fleischer, Cohen and Segev, 2003; Baizakov, Malomed
and Salerno, 2003). The absolute intensity maximum for such solitons
coincides with one of the local lattice maxima. In Kerr media, an increase
of the soliton amplitude at a fixed lattice depth is accompanied by a
gradual energy flow concentration within the single lattice channel, while
the soliton profile approaches that of an unstable Townes soliton. When b
approaches the lower edge of the semi-infinite gap, the soliton amplitude
decreases, a process that is accompanied by soliton expansion over many
lattice periods (transition into the corresponding Bloch wave).
In Kerr nonlinear media periodic refractive index modulation results
in collapse suppression and stabilization of the fundamental soliton,
almost in the entire domain of its existence, except for the narrow
region near the lower edge of a semi-infinite gap, where the energy
flow of two-dimensional solitons increases with decrease of b, in contrast
to the energy flow of their one-dimensional counterparts (Musslimani
and Yang, 2004). Fundamental two-dimensional lattice solitons were
observed in a landmark experiment conducted in an optically induced
lattice in a photorefractive SBN crystal (Fleischer, Segev, Efremidis and
Christodoulides, 2003). Photorefractive crystals possess strong electrooptic anisotropy, so that optical lattices produced by four ordinarily
polarized interfering plane waves propagate in the linear regime, while
extraordinarily polarized probe beams experience a significant screening
nonlinearity (see Section 3). Since the nonlinear correction to the refractive
index is determined by the total intensity distribution, the lattice creates
the periodic environment, invariable in propagation direction, where the
probe beam may experience discrete diffraction or form a lattice soliton
when nonlinearity is strong enough (see Figure 8).
Photonic lattices might be induced with partially incoherent light
(Martin, Eugenieva, Chen and Christodoulides, 2004; Chen, Martin,
Eugenieva, Xu and Bezryadina, 2004). In this case the lattice is created
by amplitude modulation rather than by coherent interference of multiple
Two-dimensional Lattice Solitons
97
FIGURE 8 Experimental results presenting the propagation of a probe beam
launched into a single waveguide at normal incidence. (a) Intensity structure of the
probe beam at the exit face of the crystal, displaying discrete diffraction at low
nonlinearity (200 V). (b) Intensity structure of a probe beam at the exit face of the
crystal, displaying an on-axis lattice soliton at high nonlinearity (800 V). (c)
Interferogram, showing constructive interference of the peak and lobes between the
soliton and a plane wave. (d) Reduction of probe intensity by a factor of 8, at the same
voltage as (b) and (c), results in recovery of discrete diffraction pattern (Fleischer,
Segev, Efremidis and Christodoulides, 2003)
waves and might be operated in either linear or nonlinear regimes (the
latter regime is achieved when the lattice is extraordinarily polarized)
due to suppression of incoherent modulational instabilities and allow for
observation of two-dimensional fundamental solitons and soliton trains.
Quasi-one-dimensional (i.e., uniform in one transverse direction and
periodic in the other direction) optical lattices created in a bulk medium
with a pair of interfering plane waves also support quasi-one-dimensional
or two-dimensional solitons.
In focusing media, quasi-one-dimensional lattice solitons are transversally unstable, developing a snake-like shape at low powers or experiencing neck-like break-up at high powers, as observed by Neshev, Sukhorukov, Kivshar and Krolikowski (2004). However, the periodic refractive
index modulation inhibits the development of transverse instability. Note
that the modulational instability of discrete solitons in coupled waveguides with group velocity dispersion may lead to formation of mixed
spatio-temporal quasi-solitons (Yulin, Skryabin and Vladimirov, 2006).
Fully localized two-dimensional solitons in quasi-one-dimensional lattices can be completely stable in most of their existence domain in both
Kerr and saturable media. Such solitons can be set into radiationless
98
Soliton Shape and Mobility Control in Optical Lattices
motion in the direction where the lattice is uniform, which opens a wealth
of opportunities for studying tangential soliton collisions (Baizakov, Malomed and Salerno, 2004; Mayteevarunyoo and Malomed, 2006). Weakly
transversally modulated quasi-one-dimensional lattices support localized
2D solitons even in defocusing media (Ablowitz, Julien, Musslimani and
Weinstein, 2005). Two-dimensional solitons have also been thoroughly
studied in various discrete systems (see for example Pouget, Remoissenet
and Tamga (1993), Mezentsev, Musher, Ryzhenkova and Turitsyn (1994),
Kevrekidis, Rasmussen and Bishop (2000) and reviews mentioned above).
Mobility of two-dimensional solitons in both continuous and discrete
systems can be strongly enhanced in the regime of nonlinearity saturation
(Vicencio and Johansson, 2006). Specific optical discrete X-waves are
possible in normally dispersive nonlinear waveguide arrays with focusing
nonlinearity. Such waves can be excited for a wide range of input
conditions while their properties strongly depart from the properties of
X-waves in bulk or waveguide configurations (Droulias, Hizanidis, Meier
and Christodoulides, 2005). Discrete X-waves were observed in AlGaAs
arrays by Lahini, Frumker, Silberberg, Droulias, Hizanidis, Morandotti
and Christodoulides (2007).
6.2 Multipole Solitons
Multipole solitons in two-dimensional optical lattices may have much
richer shapes than their one-dimensional counterparts. The basic
properties of such solitons were investigated by Musslimani and
Yang (2004) and Kartashov, Egorov, Torner and Christodoulides (2004).
Multipole solitons in focusing media are characterized by a π phase jump
between neighboring spots (two-dimensional lattices also support even
solitons comprising two in-phase spots, but such solitons are typically
unstable). Multipole solitons exist above some minimal energy flow
determined by the lattice parameters. In contrast to fundamental or even
solitons, multipole solitons do not transform into delocalized Bloch waves
at the cutoff for existence and their existence domain does not occupy the
whole semi-infinite gap.
Lattices stabilize multipole solitons even in Kerr nonlinear media when
the energy flow (or b) exceeds a certain critical value. In principle, the
number of solitons that can be incorporated into the multipoles is not
limited, but the energy threshold for stabilization grows monotonically
with the number of poles. This suggests the possibility of packing a
number of individual solitons with properly engineered phases into one
stable matrix to encode complex soliton images and letters. Experimental
observation of dipole solitons was conducted by Yang, Makasyuk,
Bezryadina and Chen (2004a,b) in an optical lattice induced with partially
coherent light beams in photorefractive crystals.
Two-dimensional Lattice Solitons
99
FIGURE 9 Experimental results of charge-4 vortex propagating without (top) and
with (bottom) the two-dimensional lattice for different applied electric fields. (a) input,
(b) linear diffraction, (c) output at a bias field of 90 V/mm, (d) output at 240 V/mm, (e)
discrete diffraction at 90 V/mm, (f)–(h) discrete trapping at 240 V/mm. From (f) to (h),
the vortex was launched into the lattice from off-site, to on-site, and then back to
off-site configurations. Distance of propagation through the crystal is 20 mm; average
distance between adjacent spots in the necklace is 48 µm (Yang, Makasyuk, Kevrekidis,
Martin, Malomed, Frantzeskakis and Chen, 2005)
Higher-order stationary necklace solitons were experimentally generated by launching vortex beams into a partially coherent lattice (Yang,
Makasyuk, Kevrekidis, Martin, Malomed, Frantzeskakis and Chen, 2005).
This effect results in the generation of octagonal necklace solitons featuring π phase difference between adjacent spots (see Figure 9). Notice
that for the observation of formation of necklace-like structures in optically induced lattices, Yang, Makasyuk, Kevrekidis, Martin, Malomed,
Frantzeskakis and Chen (2005) fixed the intensity of the input beam and
increased the voltage applied to the crystal, something that results in a simultaneous increase of the lattice and nonlinearity strengths, in contrast
to conventional method of observation of transition between linear discrete diffraction and nonlinear self-trapping upon increase of the intensity of the input beam at fixed applied voltage (see, e.g. Fleischer, Segev,
Efremidis and Christodoulides, 2003).
It should be pointed out that even more complex soliton trains can
be generated when a relatively broad stripe beam is launched into a
two-dimensional lattice (Chen, Martin, Eugenieva, Xu and Bezryadina,
2004). In this case, due to the highly anisotropic character of the nonlinear
response of the crystal, increasing the nonlinearity facilitates soliton train
formation or, vice versa, enhances discrete diffraction for orthogonal
orientations of the input stripe.
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Soliton Shape and Mobility Control in Optical Lattices
An experimental study of the dynamics of off-site excitations
conducted by Lou, Xu, Tang, Chen and Kevrekidis (2006) has shown
that a single beam launched between two sites of a periodic photonic
lattice excites an asymmetric lattice soliton when nonlinearity reaches a
threshold. This is an indication that the branch corresponding to stable
asymmetric lattice solitons may bifurcate from the unstable even soliton
branch at high powers. Multipole solitons in two-dimensional discrete
systems have been also thoroughly studied (see, for example, paper by
Kevrekidis, Malomed and Bishop (2001) and references therein). Extended
structures in nonlinear discrete systems occupying multiple channels
were addressed by Kevrekidis, Gagnon, Frantzeskakis and Malomed
(2007).
6.3 Gap Solitons
If the parameters of a two-dimensional lattice are chosen in such a
way that the lattice spectrum possesses at least a single finite gap,
conditions are set to form two-dimensional gap solitons. Such solitons
were observed in photorefractive optically induced lattices by Fleischer,
Segev, Efremidis and Christodoulides (2003). A probe beam was launched
at an angle with respect to the lattice plane to ensure that the beam
experiences anomalous diffraction, while the polarity of the biasing field
was chosen to produce defocusing nonlinearity. As a result, a self-trapped
wave packet was observed with a staggered phase structure, where the
phase changed by π between neighboring lattice channels. A theoretical
analysis of the properties of two-dimensional gap solitons shows that
they always exhibit power thresholds for their existence, and they
might be stable provided that the propagation constant is not too close
to the gap edges (Ostrovskaya and Kivshar, 2003; Efremidis, Hudock,
Christodoulides, Fleischer, Cohen and Segev, 2003). Such solitons can form
bound states, whose stability properties are defined by the nonlinearity
sign. For example, in defocusing media the in-phase combinations of
two-dimensional gap solitons are stable, while out-of-phase combinations
(or twisted solitons) may be prone to symmetry-breaking instabilities.
Stability of gap solitons was addressed numerically (Richter, Motzek and
Kaiser, 2007). Experimental observation of gap soliton trains in a twodimensional defocusing photonic lattices was presented by Lou, Wang,
Xu, Chen and Yang (2007). Peleg, Bartal, Freedman, Manela, Segev and
Christodoulides (2007) reported on specific conical diffraction and gap
soliton formation in honeycomb photorefractive lattices. Dipole-like twodimensional gap solitons in defocusing optically induced lattices were
observed by Tang, Lou, Wang, Song, Chen, Xu, Chen, Susanto, Law and
Kevrekidis (2007).
Two-dimensional Lattice Solitons
101
Arrays of two-dimensional gap solitons might be generated via Bloch
wave modulational instabilities (Baizakov, Konotop and Salerno, 2002;
Brazhnyi, Konotop and Kuzmiak, 2006). Novel types of localized beams
supported by the combined effects of total internal and Bragg reflection
in two-dimensional lattices were observed by Fischer, Träger, Neshev,
Sukhorukov, Krolikowski, Denz and Kivshar (2006). Such gap states
originate from the X-symmetry point of the lattice spectrum and possess
a reduced symmetry and highly anisotropic diffraction and mobility
properties. Hybrid semi-gap spatio-temporal solitons may exist in quasione-dimensional lattices, where localization in space is achieved due
to interplay of lattice, diffraction, and defocusing nonlinearity, while
localization in time is due to defocusing nonlinearity and normal group
velocity dispersion (Baizakov, Malomed and Salerno, 2006a). Higherorder solitons in two-dimensional discrete lattices with defocusing
nonlinearity were analyzed by Kevrekidis, Susanto and Chen (2006). Inband (or embedded) solitons that appear in trains and that bifurcate from
Bloch modes at the interior high-symmetry X points within the first band
of lattice spectrum were observed in two-dimensional photonic lattices
with defocusing nonlinearity (Wang, Chen, Wang and Yang, 2007).
6.4 Vector Solitons
In contrast to their one-dimensional counterparts, two-dimensional
vector solitons in periodic lattices have been studied only for the case
of incoherent interactions between soliton components. Experimental
observation of the simplest two-dimensional vector soliton in an optically
induced partially coherent photonic lattice was performed by Chen,
Bezryadina, Makasyuk and Yang (2004), Chen, Martin, Eugenieva, Xu and
Yang (2005). It was demonstrated that two mutually incoherent beams
can lock into a fundamental vector soliton while propagating along the
same lattice site, although each beam alone would experience discrete
diffraction under similar conditions. The components w1,2 of such solitons
feature similar functional shapes and can be considered as φ-projections
of the profile of scalar lattice solitons w(η, ζ ), i.e., w1 = w cos φ and
w2 = w sin φ.
When two mutually incoherent beams are launched into neighboring
lattice sites, they form dipole-like vector solitons featuring strong mutual
coupling between components in both poles. Lattices also support vector
solitons formed by two dipole field components in the case of orthogonal
orientation of dipoles in different components. The total intensity
distribution in such solitons is reminiscent of that for quadrupole scalar
solitons, but stability of vector states is enhanced in comparison with their
scalar counterparts (Rodas-Verde, Michinel and Kivshar, 2006). Lattices
can also support two-dimensional vector solitons whose components
102
Soliton Shape and Mobility Control in Optical Lattices
emerge from the same or different finite gaps of the lattice spectrum.
Such solitons were studied by Gubeskys, Malomed and Merhasin (2006)
for the case of repulsive interspecies and zero intraspecies interactions
in the context of binary BECs. It was found that intragap solitons, that
are typically bound states of tightly and loosely bound components
originating from first and second gaps, can be found in lattices that are
deep enough and are stable in a wide region of their existence.
Vectorial interactions in photonic lattices, optically induced in
photorefractive crystals, result in the formation of counter-propagating
mutually incoherent self-trapped beams analyzed by Belic, Jovic,
Prvanovich, Arsenovic and Petrovic (2006). Optical lattices may play
a strong stabilizing role for such counter-propagating solitons (Koke,
Träger, Jander, Chen, Neshev, Krolikowski Kivshar and Denz, 2007).
The time-dependent rotation of counter-propagating mutually incoherent
self-trapped beams in optically induced lattices was discussed by
Jovic, Prvanovic, Jovanovic and Petrovic (2007). Interactions of counterpropagating discrete solitons in a photorefractive waveguide array were
studied experimentally by Smirnov, Stepic, Rüter, Shandarov and Kip
(2007). Finally, the existence and stability of strongly localized twodimensional vectorial modes in discrete arrays were analyzed by Hudock,
Kevrekidis, Malomed and Christodoulides (2003).
6.5 Vortex Solitons
Vortex solitons are characterized by a special beam shape, in amplitude
and in phase, that carries a nonzero orbital angular momentum, which is
related to energy circulation inside the vortex. Thus, vortex solitons realize
higher-order, excited states of the corresponding nonlinear systems.
Vortex solitons carry screw phase dislocations located at the points where
the intensity vanishes. In uniform focusing media, vortex solitons feature
localized ring-like intensity distribution. However, they are highly prone
to azimuthal modulational instabilities that result in their spontaneous
self-destruction into ground-state solitons (for a review on vortex solitons
see Desyatnikov, Torner and Kivshar (2005)).
The properties of vortex solitons in periodic media differ dramatically
from the properties of their radially symmetric counterparts in uniform
materials. The profiles of lattice vortex solitons have the form q(η, ζ, ξ ) =
[wr (η, ζ )+iwi (η, ζ )] exp(ibξ ), where wr and wi are real and imaginary parts
of the complex field q. The topological winding number m (or dislocation
strength) of such complex scalar field can be defined by the circulation of
the field phase arctan(wi /wr ) around the phase singularity. The result is an
integer. Substitution of the vortex field in such a form into the nonlinear
Two-dimensional Lattice Solitons
103
Schrödinger equation in the simplest case of Kerr nonlinearity yields
1
−
2
∂2
∂2
+ 2
2
∂η
∂ζ
!
1
−
2
∂2
∂2
+
∂η2
∂ζ 2
!
wr + bwr + σ (wr2 + wi2 )wr − p R(η, ζ )wr = 0,
(10)
wi + bwi + σ (wr2
+ wi2 )wi
− p R(η, ζ )wi = 0,
where σ = −1 corresponds to focusing nonlinearity and σ = 1
corresponds to defocusing nonlinearity, while R(η, ζ ) stands for the
profile of the lattice. Linear stability analysis for lattice vortex solitons can
be conducted by considering perturbed solutions in the form q = (wr +U +
iwi + iV ) exp(ibξ ), where U = u(η, ζ ) exp(δξ ), V = v(η, ζ ) exp(δξ ) are real
and imaginary parts of perturbation that can grow with a complex rate δ
upon propagation. The linearized equations for perturbation components
then become
1
δu = −
2
1
−δv = −
2
∂2
∂2
+
∂η2
∂ζ 2
!
∂2
∂2
+
∂η2
∂ζ 2
v + bv + 2σ wi wr u + σ (3wi2 + wr2 )v − p Rv,
(11)
!
u + bu
+ σ (3wr2
+ wi2 )u
+ 2σ wi wr v − p Ru.
Absence of perturbations satisfying Equation (11) for Re δ > 0 indicates
vortex stability.
Vortex lattice solitons were initially introduced in discrete systems
(Aubry, 1997; Johansson, Aubry, Gaididei, Christiansen and Rasmussen,
1998; Malomed and Kevrekidis, 2001; Kevrekidis, Malomed, Bishop
and Frantzeskakis, 2001; Kevrekidis, Malomed, Chen and Frantzeskakis,
2004). Detailed theoretical analysis of properties and stability of vortex
solitons in continuous lattices induced in focusing Kerr media was
performed by Yang and Musslimani (2003). It was shown that even the
simplest lattice vortices exhibit strongly modulated intensity distributions
with four pronounced intensity maxima whose positions almost coincide
with the positions of the lattice maxima (Baizakov, Malomed and Salerno,
2003; Yang and Musslimani, 2003). In a focusing medium such vortices
originate from the semi-infinite gap of the lattice spectrum. Phase changes
by π/2 between neighboring vortex spots. Two types of charge-1 vortices
were found: off-site and on-site. Contrary to naive expectations, the
intensity distribution for high-amplitude vortices exhibits four very well
localized bright spots concentrated in the vicinity of the lattice maxima.
Both on-site and off-site vortex solitons that are oscillatory unstable near
104
Soliton Shape and Mobility Control in Optical Lattices
the lower cutoff become completely stable above a certain critical value
bcr . Note that in the isotropic photorefractive model widely used in the
literature, vortex solitons feature a limited stability domain since strong
nonlinearity saturation destabilizes them (Yang, 2004).
Experimental observation of off-site vortices in photorefractive optical
lattices was performed by Neshev, Alexander, Ostrovskaya, Kivshar,
Martin, Makasyuk and Chen (2004), while Fleischer, Bartal, Cohen,
Manela, Segev, Hudock and Christodoulides (2004) have reported the
generation of both on-site and off-site vortex configurations. Typically,
vortex lattice solitons can be excited with input ring-like beams carrying a
screw phase dislocation of unit charge. The interaction of a vortex soliton
with the surrounding photonic lattice can modify the vortex structure.
In particular, nontrivial topological transformations such as flipping of
vortex charge and inversion of its orbital angular momentum are possible
(Bezryadina, Neshev, Desyatnikov, Young, Chen and Kivshar, 2006).
Periodic media impose important restrictions on the available charges
of vortex solitons. Using general group theory arguments, Ferrando,
Zacares and Garcia-March (2005); Ferrando, Zacares, Garcia-March,
Monsoriu and Fernandez de Cordoba (2005) demonstrated that, unlike
in homogeneous media, no symmetric vortices of arbitrary high order
(or charge) can be generated in two-dimensional nonlinear systems
possessing a discrete-point rotational symmetry. In particular, in the case
of square optical lattices produced by interference of four plane waves, the
realization of discrete symmetry forbids the existence of symmetric vortex
solitons with charges higher than two. Still, dynamical rotating quasivortex double-charged solitons, reversing the topological charges and
the direction of rotation through a quadrupole-like transition state, can
exist in two-dimensional photonic lattices (Bezryadina, Eugenieva and
Chen, 2006). Stable stationary vortex solitons with topological charge 2,
introduced by Oster and Johansson (2006), incorporate eight main excited
lattice sites and feature a rhomboid-like intensity distribution. Vortex
solitons in discrete-symmetry systems are shown to behave as angular
Bloch modes, characterized by an angular Bloch momentum (Ferrando,
2005). Ideally, symmetric periodic lattices can support asymmetric
vortex solitons, including rhomboid, rectangular, and triangular vortices
(Alexander, Sukhorukov and Kivshar, 2004). Such nonlinear localized
structures describing elementary circular flows can be analyzed using
energy-balance relations. Higher-order asymmetric vortex solitons and
multipoles containing more than four lobes on a square lattice were
constructed by Sakaguchi and Malomed (2005).
Finite gaps of the lattice spectrum may give rise to specific gap
vortex solitons carrying vortex-like phase dislocation and existing due
to Bragg scattering. Vortices originating from the first gap were studied
Two-dimensional Lattice Solitons
105
in defocusing media by Ostrovskaya and Kivshar (2004b). Such vortices
experience delocalizing transition akin to flat-phase gap solitons when
their energy flow becomes too high. Due to their gap nature, such
vortex solitons develop very unusual chessboard-like phase structure. The
process of dynamical generation of spatially localized gap vortices was
studied by Ostrovskaya, Alexander and Kivshar (2006). The anisotropic
nature of the photorefractive nonlinearity may substantially affect the
properties of gap vortex solitons propagating in such media (see Richter
and Kaiser (2007)). In a focusing media, vortex lattice solitons arising from
the X symmetry points and residing in the first finite gap of the lattice
spectrum were suggested by Manela, Cohen, Bartal, Fleischer and Segev
(2004). Such solitons have the phase structure of a counter-rotating vortex
array and are stable for moderate lattice depths and soliton intensities.
Figure 10 illustrates such a gap vortex that was experimentally observed
by Bartal, Manela, Cohen, Fleischer and Segev (2005). Lattices with
more complicated geometries support vortex solitons featuring unusual
symmetries, such as multivortex solitons in triangular lattices (Alexander,
Desyatnikov and Kivshar, 2007). Kartashov, Ferrando and Garcı́a-March
(2007) predicted the existence of a new type of discrete-symmetry lattice
vortex soliton that can be considered as a coherent state of dipole solitons
carrying a nonzero topological charge.
Vectorial interactions between lattice vortices substantially enriches
their propagation dynamics. For example composite states of vortex
solitons originating from different gaps can be stable in certain parameter
regions, while the decay of unstable representatives of such families may
be accompanied by exchange of angular momentum between constituents
and transformation into other types of stable composite solitons (Manela,
Cohen, Bartal, Fleischer and Segev, 2004). Some novel scenarios of chargeflipping instability of incoherently coupled on-site and off-site vortices
were encountered by Rodas-Verde, Michinel and Kivshar (2006). Counterpropagating vortices in optical lattices are also the subject of current
studies (Petrovic, 2006; Petrovic, Jovic, Belic and Prvanovic, 2007).
6.6 Topological Soliton Dragging
Two-dimensional settings give rise to a new class of lattices possessing
higher topological complexity, e.g., regular lattices containing dislocations. In contrast to lattices that feature regular shapes, the guiding properties of lattices with dislocations can drastically change in the vicinity
of a dislocation, a possibility that leads to new phenomena. Such lattices
may be produced, e.g., by interference of a plane wave and a wave carrying one or several nested vortices (Kartashov, Vysloukh and Torner,
2005b). In the simplest case of a single-vortex wave, the lattice profile is
given by R(η, ζ ) = |exp(iαη) + exp(imφ + iφ0 )|2 , where α is the plane wave
106
Soliton Shape and Mobility Control in Optical Lattices
FIGURE 10 Experimental observation of a gap vortex lattice soliton. (a) Intensity
distribution of the input vortex ring-beam photographed (for size comparison) on the
background of the optically induced lattice. (b) Output intensity distribution of a low
intensity ring-beam experiencing linear diffraction in the lattice. (c) Output intensity
distribution of a high intensity ring-beam, which has evolved into a gap vortex soliton
in the same lattice as in (b). (d)–(f) Phase structure of a gap vortex lattice soliton. The
phase information is obtained by interference with a weakly diverging Gaussian
beam. (d) Phase distribution of the input vortex ring-beam. (e) Output phase
distribution of a high intensity ring-beam that has evolved into a gap vortex soliton
(interference pattern between the soliton whose intensity is presented in (c) and a
weakly diverging Gaussian beam). (f) Numerical validation of the phase information
in (e) (Bartal, Manela, Cohen, Fleischer and Segev, 2005)
propagation angle, φ is the azimuthal angle, m is the vortex charge, and
φ0 defines the orientation of the vortex origin. Such lattices are not stationary and distort upon propagation, but they can be implemented in
two-dimensional condensates that are strongly confined in the direction
of light propagation.
Representative profiles of lattices produced by interference of a plane
wave and a wave with nested vortices are shown in Figure 11. They
feature clearly pronounced channels where solitons can travel. The screw
phase dislocation results in appearance of the characteristic fork, so that
some channels may fuse or disappear in the dislocation. Notice that lattice
periodicity is broken only locally, in close proximity to the dislocation. The
higher the vortex charge, the stronger the lattice topological complexity
and the more channels that fuse or disappear in the dislocation. Lattice
Two-dimensional Lattice Solitons
107
FIGURE 11 (a) and (b) show lattices with single dislocations produced by the
interference pattern of a plane wave and a wave with nested charge-1 vortex for
various vortex orientations. Arrows show the direction of motion for solitons placed
into lattice channels in the vicinity of a dislocation. Positive (c) and negative (d)
topological traps created with two vortices with charges m = ±3. Map of Fζ force for
solitons launched in the vicinity of positive (e) and negative (f) topological traps
produced by two vortices with charges m = ±1. Arrows show the direction of soliton
motion. In (c)–(f) trap length δζ = π (Kartashov, Vysloukh and Torner, 2005b)
intensity gradient in the vicinity of a dislocation results in appearance
of forces acting on the soliton, that lead to the soliton’s fast transverse
displacement when it is launched into one of the lattice channels in the
vicinity of the dislocation (Figure 11(a),(b)). The dependence of sign and
magnitude of forces acting on solitons on the lattice topology makes
it possible to use combinations of several dislocations (produced by
spatially separated oppositely charged vortices) for creation of double
lattice defects (Figure 11(c),(d)), or soliton traps, which feature equilibrium
regions that are capable of supporting stationary solitons. Positive traps
form when several lattice rows fuse to form the region where R(η, ζ ) is
locally increased, while negative traps feature breakup of several lattice
channels.
The effective particle approach (see Section 5.2) yields the equations
d2
p
hηi =
U
dξ 2
Z Z
∞
−∞
|q|2
dR
dη dζ,
dη
108
Soliton Shape and Mobility Control in Optical Lattices
d2
p
hζ i =
2
U
dξ
Z Z
∞
|q|2
−∞
dR
dη dζ,
dζ
(12)
for soliton centre coordinates hηi, hζ i, which allow one to construct the
map of forces Fη ∼ d2 hηi /dξ 2 and Fζ ∼ d2 hζ i /dξ 2 acting on the solitons.
Such maps built for the simplest Gaussian ansatz for the soliton profile are
shown in Figure 11(e),(f) for both positive and negative traps. Both traps
repel all outermost solitons (some of the possible soliton positions are
indicated by circles in this figure) but support stationary solitons located
inside traps, which are stable for positive traps and unstable for negative
traps. This result confirms the crucial importance of lattice topology for
properties and stability of solitons that it supports. Notice that solitons in
two-dimensional lattices possessing defects, dislocations, and quasicrystal
structures were also investigated in details by Ablowitz, Ilan, Schonbrun
and Piestun (2006).
7. SOLITONS IN BESSEL LATTICES
The simplest optical lattices exhibit a square or honeycomb shape.
Lattices with different types of symmetry also exist, and offer new
opportunities for soliton existence, management, and control. Such lattices
can be imprinted with different types of nondiffracting beams, including
Bessel beams, parabolic beams or Mathieu beams. All these beams are
directly associated with the fundamental nondiffracting solutions of the
wave equation in circular, parabolic, or elliptical cylindrical coordinates
and thus exhibit a different symmetry. The techniques to generate
various types of nondiffracting beams can be found in Durnin, Miceli
and Eberly (1987), Gutierrez-Vega, Iturbe-Castillo, Ramirez, Tepichin,
Rodriguez-Dagnino, Chavez-Cerda and New (2001), Lopez-Mariscal,
Bandres, Gutierrez-Vega and Chavez-Cerda (2005). In this section we
describe the properties of solitons supported by lattices imprinted by
radially symmetric and azimuthally modulated Bessel beams, and by
Mathieu beams.
7.1 Rotary and Vortex Solitons in Radially Symmetric Bessel
Lattices
Radially symmetric Bessel lattices can be imprinted optically with zeroorder or higher order Bessel beams, whose field distribution can be
written as q(η, ζ, ξ ) = Jn [(2blin )1/2r ] exp(inφ) exp(−iblin ξ ), where r = (η2 +
ζ 2 )1/2 , φ is the azimuthal angle, n is the charge of the beam, and the
parameter blin sets the transverse extent of the beam core. Such beams
preserve their transverse intensity distribution upon propagation in a
linear medium. Rigorously speaking, such beams are infinitely extended
Solitons in Bessel Lattices
109
and thus carry infinite energy. However, accurate approximations can
be generated experimentally in a number of ways. Bessel lattices can
be imprinted in photorefractive crystals with the technique used for
the generation of harmonic patterns by incoherent vectorial interactions
(Efremidis, Sears, Christodoulides, Fleischer and Segev, 2002) and in
Bose–Einstein condensates. The simplest Bessel lattices, that exhibit an
intensity given by the zero-order Bessel beam, contain a central guiding
core surrounded by multiple bright rings. Lattices produced by higherorder beams have zero intensity at r = 0.
The field of the radially symmetric soliton in such lattices can be written
as q(η, ζ, ξ ) = w(r ) exp(imφ) exp(ibξ ), where m is the topological charge.
This yields the following equation for the soliton profile in the case of Kerr
nonlinear response:
1
2
d2 w 1 dw m 2 w
− 2
+
r dr
dr 2
r
!
− bw − σ w 3 + p R(r )w = 0.
(13)
To analyze the linear dynamical stability of the stationary soliton
families, one has to search for perturbed solutions in the form q(η, ζ, ξ ) =
[w(r ) + u(r, ξ ) exp(inφ) + v ∗ (r, ξ ) exp(−inφ)] exp(ibξ + imφ), where the
perturbation components u, v could grow with complex rate δ on
propagation, n is the azimuthal perturbation index. Linearization of the
nonlinear Schrödinger equation around a stationary solution yields the
eigenvalue problem:
1
iδu = −
2
1
−iδv = −
2
d2
(m + n)2
1 d
+
−
r dr
dr 2
r2
!
d2
(m − n)2
1 d
−
+
r dr
dr 2
r2
u + bu + σ w2 (v + 2u) − p Ru,
(14)
!
v + bv + σ w (u + 2v) − p Rv.
2
In the case of focusing nonlinear media (σ = −1), the simplest solitons
are located in the central core of the Bessel lattice (Kartashov, Vysloukh
and Torner, 2004c). At low energy flows such solitons are wide and
extend over several lattice rings, while at high energies they are narrow
and concentrate mostly within the core of the lattice. Solitons in lattices
produced by higher-order Bessel beams exhibit an intensity deep at r = 0.
Importantly, a Bessel lattice imprinted in a Kerr nonlinear medium can
suppress collapse and thus stabilize radially symmetric solitons in most
of their existence domain.
A fascinating example of localized structure supported by Bessel
lattices is given by solitons trapped in different rings of the lattice. Such
110
Soliton Shape and Mobility Control in Optical Lattices
solitons suffer from exponential instabilities close to low-amplitude cutoff
where they are broad, but they are completely stable above a threshold
b value. Solitons trapped inside the rings can be set into rotary motion
by launching them tangentially to the lattice rings. The important point
is that such solitons do not radiate upon rotation. A rich variety of
interaction scenarios is encountered for collective motion of solitons inside
the lattice rings (Figure 12).
Interactions between in-phase solitons located in the same ring may
cause their fusion into a high-amplitude slowly rotating soliton, while
collision of out-of-phase solitons results in formation of steadily rotating
soliton pairs with a fixed separation between components. Collisions of
out-of-phase solitons in different rings may be accompanied by reversal
of their rotation direction after each act of collision. Notice that radially
symmetric Bessel lattices may also support multipole solitons composed
of out-of-phase spots residing in different lattice rings.
Experimental demonstration of soliton formation in optically induced
ring-shaped radially symmetric photonic lattices was performed by Wang,
Chen and Kevrekidis (2006). The transition from diffraction (symmetric
in the case of excitation of the central guiding core and asymmetric in
the case of excitation of one of the outer lattice rings) to guidance of
the extraordinarily polarized probe beam by the lattice was observed
by tuning the lattice and nonlinearity strengths. In addition to immobile
solitons trapped in the central lattice core and in different lattice rings,
Wang et al observed controlled rotation of solitons launched at different
angles into lattice rings (see Figure 13).
The polarity of the biasing electric field applied to the photorefractive
crystal can be selected in such a way that Bessel-like beams with a bright
central core induce photonic lattices featuring a decreased refractive
index in the centre. Such a ring lattice with a low-refractive-index core
surrounded by concentric rings (akin to a photonic band-gap fiber) may
guide linear beams whose localization is achieved due to reflection on the
outer rings of the lattice. Guidance of linear beams in a Bessel-like lowrefractive-index core lattice was demonstrated experimentally by Wang,
Chen and Yang (2006).
Solitons supported by ring-shaped radially-periodic lattices were
studied by Hoq, Kevrekidis, Frantzeskakis and Malomed (2005) and by
Baizakov, Malomed and Salerno (2006b). In contrast to Bessel lattices,
whose intensity slowly decreases at r → ∞, radially-periodic lattices do
not decay at infinity, a property that substantially affects soliton behavior
and stability domains. In particular, no localized states exist in such
lattices at the linear limit and all localized states are truly nonlinear.
In focusing media, radially-periodic lattices support fundamental and
higher-order (i.e., concentrated in outer rings) axially uniform solitons.
Solitons in Bessel Lattices
111
FIGURE 12 (a) Interaction of out-of-phase solitons in the first ring of a zero-order
Bessel lattice. One of solitons is set into motion at the entrance of the medium by
imposing the phase twist ν = 0.1. (b) The same as in column (a) but for in-phase
solitons. (c) Interaction of out-of-phase solitons located in first and second lattice
rings. The soliton located in the first ring is set into motion by imposing the phase
twist ν = 0.2. All solitons correspond to the propagation constant b = 5 at p = 5.
Soliton angular rotation directions are depicted by arrows (Kartashov, Vysloukh and
Torner, 2004c)
In such settings all higher-order solitons are azimuthally unstable. In
contrast, solitons of the same type may be stable in defocusing media.
Lattices with defocusing nonlinearity also give rise to radial gap solitons.
112
Soliton Shape and Mobility Control in Optical Lattices
FIGURE 13 Soliton rotation in ring lattice. From (a) to (c), the probe soliton beam
was aimed at the far right side of the first, second, and third lattice ring with the same
tilting angle (about 0.4◦ ) in the y direction. From (d) to (f), the probe soliton beam was
aimed at the same second lattice ring but with different tilting angles of 0◦ , 0.4◦ , and
0.6◦ , respectively (Wang, Chen and Kevrekidis, 2006)
Surface waves localized at the edge of radially-periodic guiding
structures, consisting of several concentric rings, are also possible
(Kartashov, Vysloukh and Torner, 2007b). Such surface waves rotate
steadily upon propagation and, in contrast to non-rotating waves, for
high rotation frequencies they do not exhibit power thresholds for their
existence. There exists an upper limit for the surface wave rotation
frequency, which depends on the radius of the outer guiding ring and on
its depth.
Bessel optical lattices imprinted in defocusing media with higher-order
radially symmetric nondiffracting Bessel beams can also support stable
ring-profile vortex solitons (Kartashov, Vysloukh and Torner, 2005a).
Such solitons exist because a Bessel lattice compensates defocusing and
diffraction, thereby affording confinement of light that is impossible in
the uniform defocusing medium. In the low-energy limit, vortex solitons
transform into linear guided modes of the Bessel lattice, while high-energy
vortices greatly expand across the lattice and acquire multi-ring structure.
Since modulational instability is suppressed in defocusing media, ringshaped vortices in Bessel lattices may be stable. The higher the soliton
topological charge the deeper the lattice modulation necessary for vortex
stabilization.
Rotating quasi-one-dimensional and two-dimensional guiding structures are also interesting. In the limit of fast rotation, light beams propagating in such structures feel the average refractive index distribution
Solitons in Bessel Lattices
113
that may be equivalent, for example, to the refractive index distribution optically-induced by the Bessel beams. Solitons supported by twodimensional rotating lattices with focusing nonlinearity were introduced
by Sakaguchi and Malomed (2007). Rotating lattices, even quasi-onedimensional, were found to support localized ground-state and vortex
solitons in defocusing media, provided that the rotation frequency exceeds the critical value (Kartashov, Malomed and Torner, 2007). He, Malomed and Wang (2007) showed that rotary motion of a two-dimensional
soliton trapped in a Bessel lattice can be controlled by applying a finitetime push to the lattice, due to the transfer of linear momentum of the
lattice to the orbital soliton momentum. Solitons in discrete rotating lattices were considered by Cuevas, Malomed and Kevrekidis (2007).
7.2 Multipole and Vortex Solitons in Azimuthally Modulated
Bessel Lattices
Holographic techniques might be also used to produce higherorder azimuthally modulated beams and lattices. Lattices induced by
azimuthally modulated Bessel beams of order n have a functional shape:
R(η, ζ ) = Jn2 [(2blin )1/2r ] cos2 (nφ). The local lattice maxima situated closer
to the lattice centre are more pronounced than others, and form a ring of 2n
guiding channels where solitons can be located. Such lattices support both
individual solitons in any of the guiding channels and multipole solitons
arranged into necklaces (Kartashov, Egorov, Vysloukh and Torner, 2004).
Complex soliton configurations in focusing media may be stable when
the field changes its sign between neighboring channels. Thus, necklaces
consisting of 2n out-of-phase spots may form in the guiding ring of nth
order lattice. They are completely stable for high enough energy flow
levels (see Figure 14 for representative examples of necklace profiles).
Another intriguing opportunity afforded by azimuthally modulated
Bessel lattices is that a single soliton, initially located in one of the guiding
sites of a lattice ring and launched tangentially to the guiding ring, starts
traveling along the consecutive guiding sites, so that it can even return to
the input site. Since solitons have to overcome a potential barrier when
passing between sites, they radiate a small fraction of energy and can be
trapped in different sites, thus realizing azimuthal switching that can be
controlled by the input soliton angle and energy flow (Figure 14).
Localization of light in modulated Bessel optical lattices was
observed experimentally by Fisher, Neshev, Lopez-Aguayo, Desyatnikov,
Sukhorukov, Krolikowski and Kivshar (2006). In the experiment, higherorder lattices were generated in photorefractive crystals by employing a
phase-imprinting technique. The result of linear diffraction of low power
beams was shown to depend crucially on the specific location of the input
excitation. Thus, diffraction was observed to be strongest for excitation in
114
Soliton Shape and Mobility Control in Optical Lattices
FIGURE 14 Stable soliton complex supported by a third-order Bessel lattice at b = 3
and p = 20 (a) and by a sixth-order Bessel lattice at b = 5 and p = 40 (b). Azimuthal
switching of single tilted soliton to second (c) and fourth (d) channels of a third-order
Bessel lattice for tilt angles αζ = 0.49 and αζ = 0.626 at p = 2 and input energy flow
Uin = 8.26. Panels (e) and (f) show switching to third and sixth channels of a
sixth-order lattice for αζ = 0.8 and αζ = 0.93 at p = 5 and input energy flow Uin = 8.61.
Input and output intensity distributions are superimposed. Arrows show direction of
soliton motion and Sin , Sout denote input and output soliton positions (Kartashov,
Egorov, Vysloukh and Torner, 2004)
the origin of the lattice and almost suppressed for excitation of a single
lattice site. Nonlinearity resulted in strong light confinement in one of the
sites of a guiding lattice ring.
As discussed above, vortex solitons may be stabilized by imprinting
optical lattices in the focusing medium. However, the very refractive
index modulation causing stabilization of vortex solitons, simultaneously
imposes certain restrictions on the possible topological charges of vortices
dictated by the finite order of allowed discrete rotations (Ferrando,
Zacares and Garcia-March, 2005). A corollary of such a result is that
the maximum charge of a stable symmetric vortex in a two-dimensional
square lattice is equal to 1. Azimuthally modulated Bessel lattices offer
a wealth of new opportunities because the order of rotational symmetry
in such lattices may be higher than 4, in contrast to square lattices, a
property that has direct implications in the possible topological charges of
symmetric vortex solitons in Bessel lattices. By using general group theory
arguments Kartashov, Ferrando, Egorov and Torner (2005) have shown
Solitons in Bessel Lattices
115
that Bessel optical lattices may support vortex solitons with charges higher
than one and that the allowed value of vortex charge m in a lattice of
order n is dictated by the charge rule 0 < m 6 n − 1. A detailed stability
analysis has shown that vortex solitons in Bessel lattices may be stable
only if the topological charge satisfies the condition n/2 < m 6 n − 1, with
the exception that when n = 2, the vortex with m = 1 may be stable.
7.3 Soliton Wires and Networks
Arrays of lowest order Bessel beams might be used to produce
reconfigurable two-dimensional networks in Kerr-type nonlinear media.
The possibility of blocking and routing solitons in two-dimensional
networks was suggested by Christodoulides and Eugenieva (2001a,b),
Eugenieva, Efremidis and Christodoulides (2001). It was predicted that
solitons can be navigated in two-dimensional networks and that this
can be accomplished via vectorial interactions between two classes
of solitons: broad and highly mobile signal solitons and powerful
blockers. Interactions with signal and blocker beams combined with
the possibility for solitons to travel only along network wires might be
used for implementation of a rich variety of blocking, routing, and logic
operations. Solitons in such networks may propagate around sharp bends
with angles exceeding 90 degrees, while engineering of the corner site of
the bend allows minimization of bending losses.
While fabrication of such networks remains technically challenging,
they could be induced optically with arrays of Bessel beams in suitable
nonlinear material. Solitons launched into such networks can travel along
the complex predetermined paths (such as lines, corners, circular chains)
almost without radiation losses (Xu, Kartashov and Torner, 2005b), while
the whole network can be easily reconfigured just by blocking selected
Bessel beams in a network-inducing array. Several mutually incoherent
Bessel beams can induce reconfigurable directional couplers and Xjunctions (Xu, Kartashov, Torner and Vysloukh, 2005).
7.4 Mathieu and Parabolic Optical Lattices
As mentioned above, Bessel beams are associated with the fundamental
nondiffracting solutions of the wave equation in circular cylindrical
coordinates. Another important class of nondiffracting beams, so-called
Mathieu beams, stem from the elliptical cylindrical coordinate system
(Gutierrez-Vega, Iturbe-Castillo, Ramirez, Tepichin, Rodriguez-Dagnino,
Chavez-Cerda and New, 2001). The lattices induced by Mathieu beams
are particularly interesting because they afford smooth topological
transformation of a radially symmetric Bessel lattice into a quasi-onedimensional periodic lattice with modification of the so-called interfocal
parameter, thereby connecting two classes of optical lattices widely
116
Soliton Shape and Mobility Control in Optical Lattices
studied in the literature. The transformation of lattice topology finds
its manifestation in an important change in the shape and properties
of ground-state and dipole-mode solitons. In particular, solitons are
strongly pinned by almost radially symmetric lattices with small interfocal
parameters and can hardly jump into neighbouring lattice rings, while
in quasi-one-dimensional lattices with large interfocal parameters even
small input tilts cause considerable transverse displacements of groundstate and dipole-mode solitons (Kartashov, Egorov, Vysloukh and Torner,
2006a). The stability of dipole-mode solitons is determined by the very
shape of the Mathieu lattice. Transformation of the lattice into a quasi-onedimensional periodic structure destabilizes dipole solitons in the entire
domain of their existence.
The unique properties exhibited by solitons in lattices produced by
parabolic nondiffracting beams were introduced by Kartashov, Vysloukh
and Torner (2008). Parabolic lattices exhibit a nonzero curvature of their
channels that results in asymmetric shapes of higher-order solitons. It
was predicted that, despite such symmetry breaking, complex higherorder states can be stable, and it was shown that the specific topology
of parabolic lattices affords oscillatory-type soliton motion.
8. THREE-DIMENSIONAL LATTICE SOLITONS
In this section we briefly refer to the recent advances in the search
for three-dimensional lattice solitons. Such solitons may form in 3D
Bose–Einstein condensates loaded in optical lattices (for a review see
Morsch and Oberthaler (2006)) as well as in potentially suitable nonlinear
optical materials with a spatial refractive index modulation, when the full
spatiotemporal dynamics is considered (for a review on spatiotemporal
solitons, or light bullets, see Malomed, Mihalache, Wise and Torner
(2005)). Notice that in the absence of guiding structures, 3D solitons
in pure Kerr media experience strong collapse (for a review, see Berge
(1998)). Under appropriate conditions optical lattices may completely
stabilize ground-state and higher order 3D solitons, even in the case when
dimensionality of the lattice is lower than that of the soliton.
The dynamics of 3D excitations in 3D or lower-dimensional optical
lattices is described by the nonlinear Schrödinger equation that, in the case
of a focusing Kerr nonlinearity, takes the form:
∂q
1
i
=−
∂ξ
2
∂ 2q
∂ 2q
∂ 2q
+ 2+ 2
2
∂η
∂ζ
∂τ
!
− q |q|2 − p R(η, ζ, τ )q,
(15)
where for matter waves τ stands for the third spatial coordinate and
ξ stands for the evolution time, while for the spatiotemporal wave-
Three-dimensional Lattice Solitons
117
packets τ is the time (we assume anomalous group-velocity dispersion)
and ξ is the propagation distance. The function R(η, ζ, τ ) describes the
profile of the lattice, that can be of any dimensionality (up to 3D) in
the case of matter waves and two-dimensional or one-dimensional for
spatiotemporal solitons.
Three-dimensional periodic lattices with focusing nonlinearity can
stabilize fundamental 3D solitons (Baizakov, Malomed and Salerno, 2003;
Mihalache, Mazilu, Lederer, Malomed, Crasovan, Kartashov and Torner,
2005). The energy of three-dimensional solitons in optical lattices always
diverges at the cutoff and monotonically decreases as b → ∞. However,
a limited stability interval, where dU/db > 0, appears when the lattice
depth exceeds a critical value. The width of the stability domain as a
function of b and U increases with growing lattice depth. Illustrative
profiles of three-dimensional solitons in the three-dimensional lattice
R = cos(4η) + cos(4ζ ) + cos(4τ ) are depicted in Figure 15(a)–(c), where
transformation of a broad unstable soliton covering many lattice sites
into a stable localized soliton with increase of peak amplitude is clearly
visible. Importantly, the Hamiltonian-versus-energy diagram for threedimensional lattice solitons features two cuspidal points, resulting in a
typical swallowtail loop. Although this loop is one of the generic patterns
known in catastrophe theory, it rarely occurs in physical models.
Repulsive Bose–Einstein condensate confined in a three-dimensional
periodic optical lattice supports gap solitons and gap vortex states which
are spatially localized in all three dimensions and possess nontrivial
particle flow. Both on-site and off-site planar vortices with unit charge
that are confined in the third dimension by Bragg reflection were
encountered in the three-dimensional lattice (Alexander, Ostrovskaya,
Sukhorukov and Kivshar, 2005). Such planar states can be stable over
a wide range of their existence region. Besides planar vortices with
axes corresponding to the X symmetry point in the Brillouin zone,
solitons with other symmetry axes become possible in three-dimensional
lattices. Three-dimensional vortices in discrete systems were studied
by Kevrekidis, Malomed, Frantzeskakis and Carretero-Gonzalez (2004),
Carretero-Gonzalez, Kevrekidis, Malomed and Frantzeskakis (2005). In
such systems stable vortices with m = 1 and 3 were encountered, while
vortices with m = 2 are unstable and may spontaneously rearrange
themselves into charge-3 vortices. Multicomponent discrete systems allow
for existence of complex stable composite states, such as states consisting
of two vortices whose orientations are perpendicular to each other.
Low-dimensional lattices may also support a variety of stable soliton
states. Thus, quasi-two-dimensional lattices stabilize three-dimensional
solitons, while quasi-one-dimensional lattices may support stable twodimensional solitons in focusing cubic media (Baizakov, Malomed and
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Soliton Shape and Mobility Control in Optical Lattices
FIGURE 15 Isosurface plots of (a) unstable and (b), (c) stable 3D solitons in a 3D
periodic lattice. In (a) soliton energy U = 2.4 and its peak amplitude A = 1.8, in (b)
U = 2.04, A = 2.2, in (c) U = 2.4, A = 3. In (a)–(c) depth of periodic lattice p = 3. (d)–(f)
Isosurface plots of stable 3D solitons supported by two-dimensional Bessel lattice at
(d) p = 5, b = 0.11, (e) p = 5.5, b = 0.85, and (f) p = 6, b = 2 (Mihalache, Mazilu, Lederer,
Malomed, Crasovan, Kartashov and Torner, 2005; Mihalache, Mazilu, Lederer,
Malomed, Kartashov, Crasovan and Torner, 2005)
Salerno, 2004; Mihalache, Mazilu, Lederer, Kartashov, Crasovan and
Torner, 2004). Since a lattice is uniform in one direction the solitons
can freely move in that direction and experience head-on or tangential
collisions, which may be almost elastic (as in the case of head-on
collision of out-of-phase solitons) or may result in soliton fusion (for slow
tangential collisions). Solitons in such lattices typically acquire strongly
asymmetric shapes since they are better localized in the dimension where
the lattice is imprinted than in a uniform dimension. Notice that two
cuspidal points in the Hamiltonian-versus-energy diagram, resulting in a
swallowtail loop, also appear for three-dimensional solitons in quasi-twodimensional lattices. Such solitons feature a limited stability interval in b
or U when the lattice depth exceeds the critical level, which is much higher
than the critical depth for soliton stabilization in a fully-three-dimensional
lattice.
Cylindrical Bessel lattices imprinted in Kerr self-focusing media also
support stable three-dimensional solitons (Mihalache, Mazilu, Lederer,
Malomed, Kartashov, Crasovan and Torner, 2005). In J0 Bessel lattices, the
fundamental solitons might be stable within one or even two intervals
of their energy, depending on the depth of the lattice. In the latter case
Nonlinear Lattices and Soliton Arrays
119
the Hamiltonian-versus-energy diagram has a swallowtail shape with
three cuspidal points. Stability properties for solitons in Bessel lattices are
dictated to a large extent by the actual lattice shape and its asymptotical
behavior as r → ∞. Thus, J02 lattices allow for a single soliton stability
domain only. While high-amplitude solitons are mostly confined in the
central core of Bessel lattices, their low-amplitude counterparts acquire
clearly pronounced multi-ring structure (see Figure 15(d)–(f)). The basic
properties of three-dimensional vortex solitons in quasi-two dimensional
lattices have been analyzed by Leblond, Malomed and Mihalache (2007).
Quasi-one-dimensional lattices cannot stabilize three-dimensional
solitons in media with focusing Kerr nonlinearity (Baizakov, Malomed
and Salerno, 2004). However, it was predicted that a combination of
quasi-one-dimensional lattices with Feshbach resonance management
of scattering length in three-dimensional Bose–Einstein condensates
results in the existence of stable three-dimensional breather solitons
(Matuszewski, Infeld, Malomed and Trippenbach, 2005). Such stable
solitons may exist only if the average value of the nonlinear coefficient and
the lattice strength exceed certain minimum values. Note that both onedimensional and two-dimensional solitons in fully-dimensional optical
lattices with nonlinearity management have been studied too (Gubeskys,
Malomed and Merhasin, 2005). It was found, for example, that the energy
thresholds required for existence of all types of two-dimensional solitons
in dynamical systems is essentially higher than those in static systems. A
detailed description of the soliton properties in systems with nonlinearity
management is beyond the scope of this review. We refer to Malomed
(2005) for a comprehensive description of the relevant background and
methods.
9. NONLINEAR LATTICES AND SOLITON ARRAYS
One of the key ingredients required for the successful generation of
lattice solitons is the robustness of the periodic optical pattern that
effectively modulates the refractive index of the nonlinear medium,
thereby creating a shallow grating whose depth is comparable with the
nonlinear contribution to the refractive index produced by the soliton
beam. In all previous sections the robustness of the optically induced
lattice is achieved because the lattice propagates almost in the linear
regime, so that self-action of the lattice and cross-action from the copropagating soliton are negligible. In photorefractive media this can be
achieved by selecting proper (ordinary) polarization for lattice waves
(Fleischer, Segev, Efremidis and Christodoulides, 2003). At the same time,
extraordinarily polarized lattices would experience strong self-action and
may interact with soliton beams. The elucidation of conditions of robust
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Soliton Shape and Mobility Control in Optical Lattices
propagation of such patterns is of interest, since it allows extension of the
concept of optically induced gratings beyond the limit of weak material
nonlinearities. In this section we briefly describe the properties of the
nonlinear optical lattices and the techniques that can be used for their
stabilization.
9.1 Soliton Arrays and Pixels
A simple example of a periodic nonlinear pattern is provided by an array
of well-localized solitons. Such arrays can be created, for instance, by
launching a spatially modulated incoherent beam in a focusing medium
(Chen and McCarthy, 2002). The array of solitons generated in such a
way can be robust, provided that the coherence of the beam and the
nonlinearity strength are not too high. If the coherence is too high,
the array tends to break up into a disordered pattern rather than into
ordered soliton structures. Thus, the incoherence plays a crucial role for
stabilization of periodic patterns since it reduces the interaction forces
between neighboring soliton pixels in the array.
Soliton arrays imprinted in photorefractive crystals with partially
incoherent light might be used to guide probe beams at other wavelengths
or for transmission of images. Such arrays can be created dynamically
by seeding spatial noise on to a uniform partially spatially incoherent
beam due to the development of the induced modulational instability.
In strongly anisotropic photorefractive media, the dimensionality of
the emerging pattern depends on the strength of the nonlinearity
(Klinger, Martin and Chen, 2001). Large soliton arrays were also created
with coherent light (Petter, Schroder, Träger and Denz, 2003). It was
demonstrated that the attractive forces acting between in-phase solitons
in such arrays might be exploited for controlled fusion of several solitons
in an array caused by additional beams launched between array channels.
However, such attractive forces simultaneously put restrictions on the
minimal separation between in-phase solitons in the array. Decreasing
the separation results in increase of the interaction forces between
neighboring spots and enhances the probability of fast amplification of
asymmetries, introduced by input noise or slight deviations of the shapes
of separate spots, that could lead to decay of the whole array.
The stability of a soliton array can be substantially improved by
utilizing phase-dependent interactions between soliton rows (Petrovic,
Träger, Strinic, Belic, Schroder and Denz, 2003). In particular, changing
the phase difference between neighboring rows of an array by π makes
the interactions between them repulsive, prevents solitons in different
rows from fusion, and greatly enhances the stability of the entire array.
The technique of phase engineering allows achievement of a much denser
soliton packing rate.
Nonlinear Lattices and Soliton Arrays
121
9.2 Nonlinear Periodic Lattices
The properties and stability of truly infinite two-dimensional stationary
periodic waves in nonlinear media were analyzed by Kartashov, Vysloukh
and Torner (2003). Such patterns might be termed cnoidal waves, by
analogy with their one-dimensional counterparts. In saturable nonlinear
media such waves transform into harmonic patterns (akin to linear
periodic lattices discussed in Sections 5 and 6) in low- and high-power
limits, while at intermediate power levels they describe arrays of localized
bright spots in focusing media and arrays of dark solitons in defocusing
media. Different cnoidal waves were found that can be divided into
classes (three in focusing media and one in defocusing media) depending
on their shape (see Figure 16).
A stability analysis of such periodic patterns has shown that all
of them are unstable from a rigorous mathematical point of view,
but practically, the instability growth rates can be very small in the
two limiting cases of relatively low and high energy flows, indicating
that nonlinearity saturation has a strong stabilizing action on periodic
patterns. Importantly, stability of patterns comprising out-of-phase spots
and having chessboard phase structure (like cn-cn or sn-sn waves
of Figure 16) is substantially enhanced in comparison with that for
patterns built of in-phase spots. Two-dimensional nonlinear lattices with
chessboard phase structure were demonstrated in highly anisotropic
photorefractive media (Desyatnikov, Neshev, Kivshar, Sagemerten,
Träger, Jagers, Denz and Kartashov, 2005). The propagation of radiation
in such media under steady-state conditions is described by the system
of equations for dimensionless light field amplitude q and electrostatic
potential φ of the optically induced space-charge field Esc = ∇Usc :
!
∂ 2q
∂ 2q
∂φ
+ 2 + σq ,
2
∂η
∂η
∂ζ
(16)
2
2
∂ φ
∂φ ∂
∂ φ
∂φ ∂
2
2
+ 2 + E+
ln(1 + S |q| ) +
ln(1 + S |q| ) = 0,
∂η ∂η
∂ζ ∂ζ
∂η2
∂ζ
1
∂q
=−
i
∂ξ
2
−1/2
where q = AI0
, φ = (1/2)k 2 n 2r0 |reff | Usc , I0 is the input intensity,
σ = sgn(reff ), reff is the effective electro-optic coefficient involved, E =
(1/2)k 2 n 2r02 E 0 |reff | is the dimensionless static electric field applied to the
crystal, k is the wavenumber, r0 is the beam width, S = I0 /(Idark + Ibg )
is the saturation parameter. It was found that, due to anisotropy of
photorefractive response, refractive index modulation induced by the
periodic lattice is also highly anisotropic and nonlocal, and it depends
on the lattice orientation relative to the crystal axis. Moreover, stability
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Soliton Shape and Mobility Control in Optical Lattices
FIGURE 16 Cn-cn (a) and sn-sn (b) periodic waves in saturable medium with S = 0.1.
Plots in the first row show energy flow concentrated within one wave period T = 2π
versus propagation constant. Surface plots in each column show evolution of wave
shape with increase of energy flow U . Column (a) corresponds to focusing
nonlinearity, column (b) corresponds to defocusing nonlinearity (Kartashov, Vysloukh
and Torner, 2003)
Nonlinear Lattices and Soliton Arrays
123
properties of stationary periodic waves in such media are also strongly
affected by the pattern orientation: square patterns parallel to the crystal’s
c-axis are less robust than diamond patterns oriented diagonally.
As well as lattices with the simplest chessboard phase structure,
a variety of self-trapped periodic patterns was demonstrated by
Desyatnikov, Sagemerten, Fisher, Terhalle, Träger, Neshev, Dreischuh,
Denz, Krolikowski and Kivshar (2006), including triangular lattices
produced by interference of six plane waves (Figure 17) and vortex lattices
produced by waves with nested arrays of vortex-type phase dislocations.
A detailed analysis of two-dimensional nonlinear self-trapped photonic
lattices in anisotropic photorefractive media was carried out by Terhalle,
Träger, Tang, Imbrock and Denz (2006).
Nonlinear periodic structures may interact with localized soliton
beams. Localized beams cause strong deformations of periodic patterns
and under appropriate conditions they can form composite states
(Desyatnikov, Ostrovskaya, Kivshar and Denz, 2003). Thus, a periodic
nonlinear lattice propagating in a defocusing medium localizes other
component in the form of a stable gap soliton. A variety of stationary
composite lattice-soliton states of the vector Kerr nonlinear Schrödinger
equation were found via the Darboux transformation technique by Shin
(2004). In focusing media all such composite states were found to be
unstable due to the instability of periodic components, while the only
stable state was found in the defocusing medium.
Experimentally, one-dimensional composite gap solitons supported by
nonlinear lattices were observed in LiNbO3 crystals possessing saturable
defocusing nonlinearity. Such states form when a Gaussian beam is
launched at a Bragg angle into a nonlinear lattice (Song, Liu, Guo,
Liu, Zhu and Gao, 2006). In contrast to the case of fixed lattices,
transverse soliton mobility can be greatly enhanced in nonlinear lattices
that experience deformations at soliton locations due to cross-modulation
coupling (Sukhorukov, 2006a).
Interaction of two-dimensional localized solitons with partially
coherent nonlinear lattices was studied experimentally by Martin,
Eugenieva, Chen and Christodoulides (2004). Such lattices might be
produced with amplitude masks that modulate otherwise uniform
incoherent beams, while changing the degree of incoherence allows
control of the stability of the periodic pattern. When the intensity of the
soliton is comparable with that of the lattice and it is launched at a small
angle relative to the lattice propagation direction, the lattice dislocation is
created due to soliton dragging. In this case the transverse soliton shift is
much smaller in the presence of the lattice due to the strong soliton-lattice
interaction.
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Soliton Shape and Mobility Control in Optical Lattices
FIGURE 17 Theoretical (color) and experimental (grayscale) results for self-trapped
triangular lattices created by interference of six plane wave (see far field images in the
top panel). Two distinct orientations of the triangular lattices with respect to the
crystallographic c axis are compared (top and bottom panels) as well as low (left) and
high (right) saturation regimes. LW, lattice field (color) and intensity (grayscale) of the
lattice wave. GW, calculated profiles of the refractive index (color) and measured
intensity of the probe linear wave guided by the lattice (grayscale) (Desyatnikov,
Sagemerten, Fisher, Terhalle, Träger, Neshev, Dreischuh, Denz, Krolikowski and
Kivshar, 2006). (For interpretation of the references to color in this figure legend, the
reader is referred to the web version of this article.)
Polaron-like structures may form in a partially coherent nonlinear
lattice with small enough spacing between its spots. In such structures the
soliton drags toward it some of the lattice sites while pushing away the
others. Interaction between vortex beams and nonlinear partially coherent
lattices may result in lattice twisting driven by the angular momentum
carried by the vortex beam, with the direction of twisting being opposite
for vortices with opposite charges (Chen, Martin, Bezryadina, Neshev,
Kivshar and Christodoulides, 2005). The formation of two-dimensional
Defect Modes and Random Lattices
125
composite solitons upon interaction of mutually incoherent lattice and
stripe beams was observed by Neshev, Kivshar, Martin and Chen (2004).
Such interaction results in decrease of the spacing between the lattice rows
in the interaction region while moving stripes cause lattice compression
and deformation.
10. DEFECT MODES AND RANDOM LATTICES
In this section we summarize extensive theoretical and experimental
activity of different research groups in the field of so called defect
modes. Some structural imperfections are inevitable during fabrication
of waveguide arrays as well as upon optical lattice induction, so
their influence on the properties of solitons should be carefully
studied. Moreover, specially designed defects could even be applied for
controllable filtering, switching, and steering of optical beams in lattices.
10.1 Defect Modes in Waveguide Arrays and Optically-induced
Lattices
The propagation of discrete solitons in a waveguide array in the presence
of localized coupling constant perturbations was studied by Krolikowski
and Kivshar (1996). Some potential schemes for controllable soliton
switching and steering, utilizing array defects were discussed. Later,
suppression of defect-induced guiding due to focusing nonlinearity in
an AlGaAs waveguide array was demonstrated experimentally (Peschel,
Morandotti, Aitchison, Eisenberg and Silberberg, 1999). The existence
and stability of nonlinear localized waves in a one-dimensional periodic
medium described by the Kronig-Penney model with a nonlinear defect
was reported by Sukhorukov and Kivshar (2001). A novel type of stable
nonlinear band-gap localized state has been found. The existence and
stability of bright, dark, and twisted spatially localized modes in arrays of
thin-film nonlinear waveguides that can be viewed as multiple point-like
nonlinear defects in an otherwise linear medium (Dirac-comb nonlinear
lattice) was presented by the same authors (Sukhorukov and Kivshar,
2002b). An overview of the properties of nonlinear guided waves in such
structures can be found in the paper by Sukhorukov and Kivshar (2002c),
where the authors also discuss the similarities and differences between
discrete and continuous models.
Kominis (2006) and Kominis and Hizanidis (2006) applied phasespace methods for construction of analytical bright and dark soliton
solutions of the nonlinear Kronig-Penney model, describing the periodic
photonic structure with alternating linear and nonlinear layers. The
existence of stable nonlinear localized defect modes near the band edge
of a two-dimensional reduced-symmetry photonic crystal with Kerr
126
Soliton Shape and Mobility Control in Optical Lattices
nonlinearity was predicted by Mingaleev and Kivshar (2001). The physical
mechanism resulting in the mode stabilization was revealed. A new
type of waveguide lattice that relies on the effect of bandgap guidance
in the regions between waveguide defects was proposed by Efremidis
and Hizanidis (2005). It was shown that spatial solitons that emerge
from different spectral gaps of a periodic binary waveguide array can
be selectively reflected or transmitted through an engineered defect,
which acts as a low- or high-pass filter (Sukhorukov and Kivshar, 2005).
Defect modes in a one-dimensional optical lattice were discussed by
Brazhnyi, Konotop and Perez-Garcia (2006a) in the context of matter
waves transport in optical lattices. Such modes supported by the localized
defects in otherwise periodic lattices can be accurately described by an
expansion over Wannier functions, whose envelope is governed by the
coupled nonlinear Schrödinger equations with a δ-impurity (Brazhnyi,
Konotop and Perez-Garcia, 2006b).
The interaction of moving discrete solitons with defects in onedimensional fabricated arrays of semiconductor waveguides was investigated experimentally by Morandotti, Eisenberg, Mandelik, Silberberg,
Modotto, Sorel, Stanley and Aitchison (2003). Important changes of the
output soliton positions (a sudden switch across a narrow repulsive defect) were observed for relatively small changes of input conditions. The
formation of defect modes in polymer waveguide arrays was investigated
both theoretically and experimentally by Trompeter, Peschel, Pertsch, Lederer, Streppel, Michaelis and Brauer (2003). It was shown that variation in
the defect strength (or effective index) is accompanied by the modification of the character and number of modes bound to the defect. Although
the symmetric defect waveguide becomes multimode with the increase of
effective index it does not support antisymmetric modes.
The dynamical properties of one-dimensional discrete NLSE with
arbitrary distribution of on-site defects were considered theoretically by
Trombettoni, Smerzi and Bishop (2003). Several possible regimes of beam
propagation, including its complete reflection, oscillations around fixed
points and formation of self-trapped states, have been predicted. The
trapping of moving discrete solitons by linear and nonlinear impurities
in discrete arrays was discussed by Morales-Molina and Vicencio (2006).
Linear defect modes can also exist in photoinduced lattices with
both positive (locally increased refractive index) and negative (locally
decreased refractive index) defects. Guided modes supported by
relatively extended positive and negative defects in hexagonal optical
lattices and far-field power spectra of probe beams propagating in such
lattices that clearly reveal the physical mechanisms leading to localization
of linear modes (total internal or Bragg reflection) were presented by
Bartal, Cohen, Buljan, Fleischer, Manela and Segev (2005). The band-gap
Defect Modes and Random Lattices
127
FIGURE 18 Top panel: Intensity pattern of a two-dimensional optically induced lattice
with a single-site negative defect at crystal input (a) and output (b), (c). The defect
disappears in the linear regime (b) but can survive with weak nonlinearity (c) after
propagating through a 20-mm-long crystal. Bottom panel: Input (a) and output (b), (c)
of a probe beam showing band-gap guidance by the defect (c) and normal diffraction
without the lattice (b) under the same bias condition (Makasyuk, Chen and Yang,
2006)
guidance of light in optically induced 2D lattice with single-site negative
defect was observed by Makasyuk, Chen and Yang (2006). Defect modes
supported by such lattices feature long tails in the directions of the lattice
principal axes as shown in Figure 18.
In linear case the strongest defect mode confinement appears when
the lattice intensity at the defect site is nonzero rather than zero (Fedele,
Yang and Chen, 2005; Wang, Yang and Chen, 2007). It was shown that in
nonlinear regime defect solitons bifurcate from every infinitesimal linear
defect mode (Yang and Chen, 2006). At high powers in the medium with
focusing nonlinearity defect soliton modes become unstable in the case of
positive defects, but may remain stable in the case of negative defect.
Defect solitons in optically induced one-dimensional photonic lattices
in LiNbO3 :Fe crystals were observed experimentally by Qi, Liu, Guo,
Lu, Liu, Zhou and Li (2007). Wave and defect dynamics in twodimensional nonlinear photonic quasicrystals was studied experimentally
by Freedman, Bartal, Segev, Lifshitz, Christodoulides and Fleischer (2006).
The main properties of solitons supported by defects embedded in
superlattices have been analyzed by Chen, He and Wang (2007).
128
Soliton Shape and Mobility Control in Optical Lattices
10.2 Anderson Localization
The concept of Anderson localization was originally introduced in the
field of condensed matter physics for the phenomenon of disorderinduced metal–insulator transition in linear electronic systems. Anderson
localization refers to the situation where electrons, when released inside
a random medium, may stay close to the initial point (Anderson, 1978).
The mechanism behind this phenomenon has been attributed to multiple
scattering of electrons by random potential, a feature of the wave nature
of electrons. This concept may also be applied to classical linear wave
systems (Maynard, 2001).
Different linear regimes of light localization in disordered photonic
crystals were considered theoretically and experimentally by Vlasov,
Kaliteevski and Nikolaev (1999). In particular, it was shown that Bloch
states are disrupted only when local fluctuations of the band-edge
frequency (caused by randomization of the refractive index profile)
become as large as the band-gap width. Band theory of light localization in
one-dimensional linear disordered systems was presented by Vinogradov
and Merzlikin (2004). It was detected that Bragg reflection is responsible
not only for the appearance of band-gaps but also for Anderson
localization of light in the one-dimensional case.
Schwartz, Bartal, Fishman and Segev (2007) reported on the
experimental observation of Anderson localization in perturbed periodic
potentials. In the setup used by authors the transverse localization
of light was caused by random fluctuations on a two-dimensional
photonic lattice imprinted in a photorefractive crystal. The authors
demonstrated how ballistic transport becomes diffusive in the presence
of disorder, and that crossover to Anderson localization occurs at high
levels of disorder. In this experiment the controllable level of disorder
was achieved by combining random and regular lattices, while optical
induction techniques allowed easy creation of the required ensemble of
lattice realizations. Anderson localization in one-dimensional disordered
waveguide arrays has also been observed by Lahini, Avidan, Pozzi, Sorel,
Morandotti, Christodoulides and Silberberg (2008).
From the viewpoint of fundamental nonlinear physics the challenging
problem is the exploration of nonlinear analogues of Anderson
localization. The interplay between disorder and nonlinearity was studied
in systems described by the one-dimensional nonlinear Schrödinger
equation with random-point impurities (see, e.g., Kivshar, Gredeskul,
Sanchez and Vazquez (1990), Hopkins, Keat, Meegan, Zhang and
Maynard (1996)) as well as in one-dimensional (Kevrekidis, Kivshar and
Kovalev, 2003) and two-dimensional (Pertsch, Peschel, Kobelke, Schuster,
Bartlet, Nolte, Tünnermann and Lederer, 2004) discrete waveguide arrays.
Anderson-type localization of moving spatial solitons in optical lattices
Defect Modes and Random Lattices
129
with random frequency modulation was studied by Kartashov and
Vysloukh (2005). In this case dramatic enhancement of soliton trapping
probability on lattice inhomogeneities was detected with increase of
frequency fluctuation level. The localization process is strongly sensitive
to lattice depth, since in shallow lattices moving solitons experience
random refraction and/or multiple scattering, in contrast to relatively
deep lattices, where solitons are typically immobilized in the vicinity of
modulation frequency local minima. It is worth noticing that analogous
phenomena were encountered in trapped BECs (Greiner, Mandel,
Esslinger, Hänsch and Bloch, 2002; Abdullaev and Garnier, 2005; Kuhn,
Miniatura, Delande, Sigwarth and Muller, 2005; Schulte, Drenkelforth,
Kruse, Ertmer, Arlt, Sacha, Zakrzewski and Lewenstein, 2005; Gavish and
Castin, 2005).
10.3 Soliton Percolation
The combined effect of periodic and random potentials on the
transmission of moving and formation of stable stationary excitations has
been addressed in a number of studies and reviews (Bishop, Jimenez,
and Vazquez, 1995; Abdullaev, Bishop, Pnevmatikos and Economou,
1992; Abdullaev, 1994). A universal feature of wave packet and particle
dynamics in disordered media in different areas of physics is percolation
(Grimmett (1999); Shklovskii and Efros (1984)). Percolation occurs in
different types of physical settings, including high-mobility electron
systems, Josephson-junction arrays, two-dimensional GaAs structures
near the metal–insulator transition, or charge transfer between a
superconductor and a hoping insulator, to mention a few.
The nonlinear optical analog of biased percolation was introduced by
Kartashov, Vysloukh and Torner (2007c). The phenomenon analyzed is
the disorder-induced soliton transport in randomly modulated optical
lattices with Kerr-type focusing nonlinearity in the presence of a linear
variation of the refractive index in the transverse plane, thus generating a
constant deflecting force for light beams entering the medium. When such
a force is small enough, solitons in perfectly periodic lattices are trapped
in the vicinity of the launching point due to the Peierls–Nabarro potential
barriers, provided that the launching angle is smaller than a critical
value. Under such conditions soliton transport is suppressed, and thus
the lattice acts as a soliton insulator. However, random modulations of
the lattice parameters makes soliton mobility possible again, with the key
parameter determining the soliton current being the standard deviation
of the phase/amplitude fluctuations. Kartashov, Vysloukh and Torner
(2007c) discovered that the soliton current in lattices with amplitude and
phase fluctuations reaches its maximal value at intermediate disorder
levels and that it drastically reduces in both, almost regular and strongly
130
Soliton Shape and Mobility Control in Optical Lattices
disordered lattices. This suggests the possibility of a disorder-induced
transition between soliton insulator and soliton conductor lattice states.
11. CONCLUDING REMARKS
To conclude, we would like to add some comments on two important
fields emerging from theoretical and experimental research in the area of
lattice solitons, which fall out of the main stream of this review.
The first topic is the generalization of the lattice soliton concept to
partially coherent or incoherent (such as white-light) optical fields. From
a physical point of view, the diffraction spreading of a partially coherent
light beam is determined by the transverse coherence length rather than
by the beam radius. In shallow quasicontinuous lattices, this means
that self-trapping and soliton formation would normally require higher
power levels for incoherent radiation. In this case, the ratio between the
transverse coherence length and the lattice period becomes crucial, since
the soliton formation process is sensitive to the phase relations between
several neighboring lattice channels. Thus, the complex interplay between
coherence, nonlinearity, and waveguiding becomes central.
A powerful theoretical approach to describe the propagation and
self-focusing of partially spatially incoherent light beams in bulk
photorefractive media was developed by Christodoulides, Coskun,
Mitchell and Segev (1997), Mitchell, Segev, Coskun and Christodoulides
(1997). It was followed by the experimental observation of self-trapping
of white light in bulk media (Mitchell and Segev, 1997). Further
theoretical progress was reported by Buljan, Segev, Soljacic, Efremidis
and Christodoulides (2003). The existence of random phase solitons in
nonlinear periodic lattices was predicted by Buljan, Cohen, Fleischer,
Schwartz, Segev, Musslimani, Efremidis and Christodoulides (2004).
Importantly, such solitons exist when the characteristic response time
of the medium is much longer than the coherence time. The prediction
mentioned above was confirmed experimentally (Cohen, Bartal, Buljan,
Carmon, Fleischer, Segev and Christodoulides, 2005). Lately the existence
of lattice solitons made of incoherent white light, originating from an
ordinary incandescent light bulb, was analyzed by Pezer, Buljan, Bartal,
Segev and Fleisher (2006). Prospects of incoherent gap optical soliton
formation in a self-defocusing nonlinear periodic medium was analyzed
theoretically (Motzek, Sukhorukov, Kaiser and Kivshar, 2005; Pezer,
Buljan, Fleischer, Bartal, Cohen and Segev, 2005), and further supported
by experimental observation (Bartal, Cohen, Manela, Segev, Fleischer,
Pezer and Buljan, 2006). The phenomenon of modulational instability of
extended incoherent nonlinear waves in photonic lattices was addressed
by Jablan, Buljan, Manela, Bartal, and Segev (2007).
Concluding Remarks
131
A second intriguing and practically important topic directly linked
with lattice solitons is the physics of nonlinear surface wave formation
at the interface of periodic structures. Surface waves were originally
introduced by Igor E. Tamm in 1932 in the context of condensedmatter physics and have been intensively studied in different areas of
science. However the progress in experimental investigation of nonlinear
optical surface waves was severely limited by the high power thresholds
required for their formation. It has been predicted theoretically (Makris,
Suntsov, Christodoulides, Stegeman and Hache, 2005) and confirmed
experimentally (Suntsov, Makris, Christodoulides, Stegeman, Hache,
Morandotti, Yang, Salamo and Sorel, 2006) that surface waves formation
at the very edge of a one-dimensional waveguiding array is accessible for
moderate power levels. Formation of gap surface solitons at the edge of a
defocusing lattice is also possible as it was shown theoretically (Kartashov,
Vysloukh and Torner, 2006a) and confirmed experimentally in defocusing
LiNbO3 waveguiding arrays (Rosberg, Neshev, Krolikowski, Mitchell,
Vicencio, Molina and Kivshar, 2006; Smirnov, Stepic, Rüter, Kip and
Shandarov, 2006). Siviloglou, Makris, Iwanow, Schiek, Christodoulides,
Stegeman, Min and Sohler (2006) reported on experimental observation
of discrete quadratic surface solitons in self-focusing and defocusing
periodically poled lithium niobate waveguide arrays. Lattice interfaces
imprinted in nonlocal nonlinear media support interesting families of
multipole surface solitons (Kartashov, Torner and Vysloukh, 2006).
Two-dimensional geometries offers even richer possibilities for surface
wave formation (Kartashov and Torner, 2006; Kartashov, Vysloukh,
Mihalache and Torner, 2006; Kartashov, Egorov, Vysloukh, and Torner,
2006b; Makris, Hudock, Christodoulides, Stegeman, Manela and Segev,
2006). Solitons residing at the lattice edges, in the corners, and on specially
designed defects are illustrative examples. Experimental observations
of two-dimensional surface waves were realized at the boundaries of
a finite optically induced lattice (Wang, Bezryadina, Chen, Makris,
Christodoulides and Stegeman, 2007), and in laser-written waveguide
arrays (Szameit, Kartashov, Dreisow, Pertsch, Nolte, Tünnermann and
Torner, 2007). Many problems remain open to explore the combination of
surface solitons and optical lattices.
We thus conclude by stressing that we have presented a concise review
of the advances in optical soliton formation, manipulation, shaping, and
control in optical lattices, with special emphasis on optically-induced
lattices. Many concepts investigated in the context of optical lattices
have implications and applications in several branches of physics, such
as nonlinear optics, condensed-matter theory, and quantum mechanics.
Optical soliton control in optical lattices offers a unique laboratory
132
Soliton Shape and Mobility Control in Optical Lattices
to investigate universal phenomena, and as such provides a unique
exploration tool.
ACKNOWLEDGEMENTS
We thank all our colleagues and collaborators for invaluable discussions
in conducting joint research reported here and in completing this review.
Most especially we thank the members of our group, the co-authors
of joint papers described here, and G. Assanto, R. Carretero-Gonzalez,
Z. Chen, D. Christodoulides, L. Crasovan, C. Denz, A. Desyatnikov, A.
Ferrando, D. Kip, Y. Kivshar, F. Lederer, M. Lewenstein, B. Malomed, D.
Mazilu, D. Mihalache, M. Molina, D. Neshev, T. Pertsch, M. Segev, G.
Stegeman, A. Szameit, Z. Xu, F. Ye for contributing their knowledge and
figures included in this review. This work was produced with financial
support by the Generalitat de Catalunya and by the Government of Spain
through the Ramon-y-Cajal program and grant TEC2005-07815.
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