Spatial optical solitons in nonlinear photonic crystals

Moving Bragg grating solitons in a semilinear dual-core system with dispersive reflectivity

The existence, stability and collision dynamics of moving Bragg grating solitons in a semilinear dual-core system where one core has the Kerr nonlinearity and is equipped with a Bragg grating with dispersive reflectivity, and the other core is linear are investigated. It is found that moving soliton solutions exist as a continuous family of solutions in the upper and lower gaps of the system's linear spectrum. The stability of the moving solitons are investigated by means of systematic numerical stability analysis, and the effect and interplay of various parameters on soliton stability are analyzed. We have also systematically investigated the characteristics of collisions of counter-propagating solitons. In-phase collisions can lead to a variety of outcomes such as passage of solitons through each other with increased, reduced or unchanged velocities, asymmetric separation of solitons, merger of solitons into a quiescent one, formation of three solitons (one quiescent and two moving ones) and destruction of both solitons. The outcome regions of in-phase collisions are identified in the plane of dispersive reflectivity versus frequency. The effects of coupling coefficient, relative group velocity in the linear core, soliton velocity and dispersive reflectivity and the initial phase difference on the outcomes of collisions are studied.

Solitons supported by singular modulation of the cubic nonlinearity

A model of the optical media with a spatially structured Kerr nonlinearity is introduced. The nonlinearity strength is modulated by a set of singular peaks on top of a self-focusing or defocusing uniform background. The peaks may include a repulsive or attractive linear potential too. We find that a pair of mutually symmetric peaks readily gives rise to the spontaneous symmetry breaking (SSB) of modes pinned to individual peaks, while antisymmetric pinned modes are always unstable, transforming into robust spatially odd breathers. Three- and five-peak structures support symmetric modes, with in-phase or twisted profiles, and do not give rise to asymmetric states. A stability area is found for the twisted states pinned to the triple peaks, while the corresponding in-phase modes are unstable, unless the three modulation peaks are set very close to each other, covered by a single-peak pinned mode. All patterns pinned to five peaks are unstable too. Collisions of moving solitons with the singular-modulation peak are studied too. Slowly moving solitons bounce back from the peak, while the collisions are quasi-elastic for fast solitons. In the intermediate case, the soliton is destroyed by the collision. In a special case, the condition of a resonance of the incident soliton with a trapped mode supported by the peak leads to capture of the soliton.

Modulated zero-area solitary pulses: properties and applications

This paper investigates a specific kind of solitary pulse, the modulated zero-area (MZA) solitary pulse, when propagating within MgO photonic bandgap medium doped with silver nanoparticles (NPs). It will be shown that two coupled MZA pulses do propagate unattenuated within this medium but for a certain combination of the dipole moments and the density of the NPs. More important, and in contrast to the other kinds of solitary pulses, one of the two MZA pulses exhibits a slowing in its group velocity in comparison to the other one, depending on the amplitudes of the components of the dipole moments of the NPs that are in resonance with the two MZA pulses. With this particular feature, the system has the potential of working as an all-optical switch.

Bright solitons in defocusing media with spatial modulation of the quintic nonlinearity

It has been recently demonstrated that self-defocusing (SDF) media with cubic nonlinearity, whose local coefficient grows from the center to the periphery fast enough, support stable bright solitons without the use of any linear potential. Our objective is to test the genericity of this mechanism for other nonlinearities, by applying it to one- and two-dimensional (1D and 2D) quintic SDF media. The models may be implemented in optics (in particular, in colloidal suspensions of nanoparticles), and the 1D model may be applied to the description of the Tonks-Girardeau gas of ultracold bosons. In 1D, the nonlinearity-modulation function is taken as g0+sinh2(βx). This model admits a subfamily of exact solutions for fundamental solitons. Generic soliton solutions are constructed in a numerical form and also by means of the Thomas-Fermi and variational approximations (TFA and VA). In particular, a new ansatz for the VA is proposed, in the form of "raised sech," which provides for an essentially better accuracy than the usual Gaussian ansatz. The stability of all the fundamental (nodeless) 1D solitons is established through the computation of the corresponding eigenvalues for small perturbations and also verified by direct simulations. Higher-order 1D solitons with two nodes have a limited stability region, all the modes with more than two nodes being unstable. It is concluded that the recently proposed inverted Vakhitov-Kolokolov stability criterion for fundamental bright solitons in systems with SDF nonlinearities holds here too. Particular exact solutions for 2D solitons are produced as well.

Algebraic bright and vortex solitons in defocusing media

We demonstrate that spatially inhomogeneous defocusing nonlinear landscapes with the nonlinearity coefficient growing toward the periphery as (1+|r|(α)) support one- and two-dimensional fundamental and higher-order bright solitons, as well as vortex solitons, with algebraically decaying tails. The energy flow of the solitons converges as long as nonlinearity growth rate exceeds the dimensionality, i.e., α>D. Fundamental solitons are always stable, while multipoles and vortices are stable if the nonlinearity growth rate is large enough.

Tunable reflectance spectra of multilayered cholesteric photonic structures with anisotropic defect layers

In this paper, we investigate the spectral characteristics of normal incident light reflected by a multilayered structure composed of an alternated sequence of single-pitch cholesteric liquid-crystal (ChLC) and anisotropic layers. Using the Berreman 4x4 matrix formalism, we numerically obtain the reflection spectrum and the chromaticity diagram as a function of the anisotropic layers thickness d. For d-->0 , the structure behaves like a single ChLC layer, showing a single reflection band. As the anisotropic layer thickness increases, the reflection band shifts toward high-wavelength spectral regions, while new reflection bands appear. As a consequence, the reflection chromaticity continuously changes with d . It is observed that a suitable choice of the anisotropic layer thickness can produce a threefold reflection band with a red-green-blue associated color for both polarized and unpolarized incident lights.

Soliton excitations in a one-dimensional nonlinear diatomic chain of split-ring resonators

We present a systematic analytical study of the dynamics of nonlinear magnetoinductive waves in a one-dimensional diatomic lattice of split ring resonators (SRRs) with Kerr nonlinear interaction between nearest neighbors. The linear spectrum of this model have two branches and exhibits a gap, which is proportional to the difference between two types of SRRs. We analyze the nonlinear excitations genuine of the discreteness and nonlinearity in such a diatomic chain based on an extended quasidiscreteness approach. Gap solitons (with vibrating frequency lying in the gap), resonant kinks (with the vibrating frequency lying in the frequency bands), and intrinsic localized modes (with the vibrating frequency being above all the frequency bands) are obtained explicitly. It is also shown that the existence of different localized structures depend strongly on the type of nonlinearity of the embedded medium (a self-focusing or defocusing one).

Interlaced linear-nonlinear optical waveguide arrays

The system of coupled discrete equations describing a two-component superlattice with interlaced linear and nonlinear constituents is studied as a basis for investigating binary waveguide arrays, such as ribbed AlGaAs structures, among others. Compared to the single nonlinear lattice, the interlaced system exhibits an extra band-gap controlled by the, suitably chosen by design, relative detuning. In more general physics settings, this system represents a discretization scheme for the single-equation-based continuous models in media with transversely modulated linear and nonlinear properties. Continuous wave solutions and the associated modulational instability are fully analytically investigated and numerically tested for focusing and defocusing nonlinearity. The propagation dynamics and the stability of periodic modes are also analytically investigated for the case of zero Bloch momentum. In the band-gaps a variety of stable discrete solitary modes, dipole or otherwise, in-phase or of staggered type are found and discussed.

Two-dimensional discrete solitons in rotating lattices

We introduce a two-dimensional discrete nonlinear Schrödinger (DNLS) equation with self-attractive cubic nonlinearity in a rotating reference frame. The model applies to a Bose-Einstein condensate stirred by a rotating strong optical lattice, or light propagation in a twisted bundle of nonlinear fibers. Two types of localized states are constructed: off-axis fundamental solitons (FSs), placed at distance R from the rotation pivot, and on-axis (R=0) vortex solitons (VSs), with vorticities S=1 and 2 . At a fixed value of rotation frequency Omega , a stability interval for the FSs is found in terms of the lattice coupling constant C , 0<C<C_{cr}(R) , with monotonically decreasing C_{cr}(R) . VSs with S=1 have a stability interval, C[over ]_{cr};{(S=1)}(Omega)<C<C_{cr};{(S=1)}(Omega) , which exists for Omega below a certain critical value, Omega_{cr};{(S=1)} . This implies that the VSs with S=1 are destabilized in the weak-coupling limit by the rotation. On the contrary, VSs with S=2 , that are known to be unstable in the standard DNLS equation, with Omega=0 , are stabilized by the rotation in region 0<C<C_{cr};{(S=2)} , with C_{cr};{(S=2)} growing as a function of Omega . Quadrupole and octupole on-axis solitons are considered too, their stability regions being weakly affected by Omega not equal 0 .

Nonparaxial effects on the propagation and scattering of a polarized optical pulse

Propagation characteristics of a polarized optical solitary pulse are analyzed by taking into account the effect of nonparaxiality and mutual interaction. To start with, a pair of generalized nonlinear Schrodinger equations is deduced through an operator approach. Stationary solutions of such a system are then analyzed numerically through a boundary value problem in two stages, with and without the nonparaxial effect. In the second stage, the propagating form of the corresponding spatial soliton is studied by an extended split step algorithm ETDRK. The initial profile is considered to be both a one- and two-soliton solution, to visualize the event of scattering and fusion. From this data, we have computed the intensity, root mean square spectral width, and chirp of a single soliton as it propagates. In the case of the two-soliton solution, we observe that for source parameter values, the fusion is more favored than scattering. It is observed that nonparaxiality and the interaction between A(x) and A(y) tends to destroy the periodic behaviors of these parameters. Lastly, we have investigated the modulational instability of the system as function of frequency detuning and nonparaxiality. The form of the gain is discussed as a function of nonparaxiality.

Analytical solitary wave solutions of the nonlinear Kronig-Penney model in photonic structures

A phase space method is employed for the construction of analytical solitary wave solutions of the nonlinear Kronig-Penney model in a photonic structure. This class of solutions is obtained under quite generic conditions, while the method is applicable to a large variety of systems. The location of the solutions on the spectral band gap structure as well as on the low dimensional space of system's conserved quantities is studied, and robust solitary wave propagation is shown.

Discrete dark solitons with multiple holes

We consider staggered dark solitons admitted by the discrete nonlinear Schrödinger equation with focusing cubic nonlinearity. In particular, we focus on the study of dark solitons with several holes characterized by the number of zeros in the uncoupled case. Such structures reveal interesting behaviors, such as stable intersite dark solitons. All of the structures have no counterpart in the strong coupling limit since they disappear in a saddle-node bifurcation. We also consider the evolution of structures with multiple holes representing an interaction between multiple dark solitons in a very discrete case.

Formation of discrete solitons in light-induced photonic lattices

We present both experimental and theoretical results on discrete solitons in two-dimensional optically-induced photonic lattices in a variety of settings, including fundamental discrete solitons, vector-like discrete solitons, discrete dipole solitons, and discrete soliton trains. In each case, a clear transition from two-dimensional discrete diffraction to discrete trapping is demonstrated with a waveguide lattice induced by partially coherent light in a bulk photorefractive crystal. Our experimental results are in good agreement with the theoretical analysis of these effects.

Bifurcations and stability of gap solitons in periodic potentials

We analyze the existence, stability, and internal modes of gap solitons in nonlinear periodic systems described by the nonlinear Schrödinger equation with a sinusoidal potential, such as photonic crystals, waveguide arrays, optically-induced photonic lattices, and Bose-Einstein condensates loaded onto an optical lattice. We study bifurcations of gap solitons from the band edges of the Floquet-Bloch spectrum, and show that gap solitons can appear near all lower or upper band edges of the spectrum, for focusing or defocusing nonlinearity, respectively. We show that, in general, two types of gap solitons can bifurcate from each band edge, and one of those two is always unstable. A gap soliton corresponding to a given band edge is shown to possess a number of internal modes that bifurcate from all band edges of the same polarity. We demonstrate that stability of gap solitons is determined by location of the internal modes with respect to the spectral bands of the inverted spectrum and, when they overlap, complex eigenvalues give rise to oscillatory instabilities of gap solitons.

Photonic band-gap inhibition of modulational instabilities

Spatial structures as a result of a modulational instability are studied in a nonlinear cavity with a photonic crystal. The interaction of the modulated refractive index with the nonlinearity inhibits the instability via the creation of a photonic band gap. A novel mechanism of light localization due to defects and pattern inhibition is also described.

Perturbation-induced radiation by the Ablowitz-Ladik soliton

An efficient formalism is elaborated to analytically describe dynamics of the Ablowitz-Ladik soliton in the presence of perturbations. This formalism is based on using the Riemann-Hilbert problem and provides the means of calculating evolution of the discrete soliton parameters, as well as shape distortion and perturbation-induced radiation effects. As an example, soliton characteristics are calculated for linear damping and quintic perturbations.

Composite band-gap solitons in nonlinear optically induced lattices

We introduce novel optical solitons that consist of a periodic and a spatially localized component coupled nonlinearly via cross-phase modulation. The spatially localized optical field can be treated as a gap soliton supported by the optically induced nonlinear grating. We find different types of these band-gap composite solitons and demonstrate their dynamical stability.

Spatial solitons in optically induced gratings

We study experimentally nonlinear localization effects in optically induced gratings created by interfering plane waves in a photorefractive crystal. We demonstrate the generation of spatial bright solitons similar to those observed in arrays of coupled optical waveguides. We also create pairs of out-of-phase solitons, which resemble twisted localized states in nonlinear lattices.

Instabilities and bifurcations of nonlinear impurity modes

We study the structure and stability of nonlinear impurity modes in the discrete nonlinear Schrödinger equation with a single on-site nonlinear impurity emphasizing the effects of interplay between discreteness, nonlinearity, and disorder. We show how the interaction of a nonlinear localized mode (a discrete soliton or discrete breather) with a repulsive impurity generates a family of stationary states near the impurity site, as well as examine both theoretical and numerical criteria for the transition between different localized states via a cascade of bifurcations.

Discrete vector spatial solitons in a nonlinear waveguide array

A vector discrete diffraction managed soliton system is introduced. The vector model describes propagation of two polarization modes interacting in a nonlinear waveguide array with varying diffraction via the cross-phase modulation coupling. In the limit of strong diffraction we derive averaged equations governing the slow dynamics of the beam's amplitudes, and their stationary (in the form of bright-bright vector bound state) and traveling wave solutions are found. Through an extensive series of direct numerical simulations, interactions between diffraction-managed solitons for different values of velocities, diffraction, and cross-phase modulation coefficient are studied. We compare each collision case with its classical counterpart (constant diffraction) and find that in both the scalar and vector diffraction management cases, the interaction picture involves beam shaping, fusion, fission, nearly elastic collisions, and, in some cases, multihump structures. The collision scenario is found, in both the scalar and vector diffraction managed cases, to be rather different from the classical case.