Spatial optical solitons in fifth order and seventh order weakly nonlocal nonlinear media

We analytically study the (1 + 1)-dimensional spatial optical solitons in weakly nonlocal nonlinear media with cubic–quintic nonlinearity (fifth order nonlinear media) and cubic–quintic–septic nonlinearity (seventh order nonlinear media). Explicit solutions are derived, which include optical bright solitons, singular solutions and singular triangular periodic solution.     

Solitons in PT–symmetric optical lattices with spatially periodic modulation of nonlinearity

We study the existence and stability of solitons forming in PT–symmetric optical lattices with spatially periodic modulation of the local strength of the nonlinear media. We found that the spatial modulation of the nonlinearity significantly affects the stability of solitons in PT–symmetric optical lattices. With the decrease of the strength of nonlinear refractive index modulation, the soliton's stability domain increases, whereas with the increase of the period of nonlinear refractive index modulation, the corresponding soliton's stability range narrows. In addition, we also investigate the influence of variation of the amplitude of the linear PT–symmetric lattice potential on soliton dynamics, in the presence of spatially periodic modulation of nonlinearity.     

Optical bistability in defective photonic multilayers doped by graphene

We study the optical bistability (OB) in photonic multilayers doped by graphene sheets, stacking two Bragg reflectors with a defect layer between the reflectors. OB stems from the nonlinear effect of graphene, so the local field of defect mode (DM) could enhance the nonlinearity and reduce the thresholds of bistability. The structure achieves the tunability of bistability due to that the DM frequency and transmittance could be modulated by the chemical potential. Bistability thresholds and interval of the two stable states could be remarkably reduced by decreasing the chemical potential. A lager Bragg periodic number could increase the localizing of field, but the graphene loss may decrease the intensity of transmission light. We have concluded an appropriate periodic number to achieve OB. The study suggests that the tunable bistability of the structure could be used for all-optical switches in optical communication systems.     

Spatial solitons in an optical lattice with varying diffraction and spatially inhomogeneous nonlinearity

In this paper, we present the (1+1)-dimensional inhomogeneous nonlinear Schrödinger (NLS) equation that describes the propagation of optical waves in nonlinear optical systems exhibiting optical lattice, inhomogeneous nonlinearity and varying diffraction at the same time. A series of interesting properties of spatial solitons are found from the numerical calculations, such as the stable propagation in the a nonperiodic optical lattice induced by periodic diffraction variations and periodic nonlinearity variations. Finally, the interaction of neighboring spatial solitons in a nonperiodic optical lattice is discussed, and the results reveal that two spatial solitons can propagate periodically and separately in the optical lattice without interaction.     

Hidden features of the soliton adaptation law to external potentials: Optical and matter-wave 3D nonautonomous soliton bullets

We show that the law of the soliton adaptation to varying-in-time external potentials indicates conclusively the way for solitonlike bullets generation in three-dimensional nonautonomous nonlinear and dispersive systems. It turns out that the generation of matter-wave soliton bullets can be realized if periodic variations of non-linearity and confining sign-reversal varying-in-time harmonic oscillator potential are opposite in phases so that peaks of nonlinearity inside the atomic cloud coincide in time with repulsive character of trapping potential during reversal periodic transformations from cigar-shaped to ball-shaped trapping structures. In nonlinear optical applications, periodic graded-index nonlinear structures with alternating waveguiding and antiwaveguiding segments can be used to simulate complicated processes of matter-wave soliton bullets generation.     

Nonlinear light propagation in the defect band of one-dimensional defective structures

In this paper we numerically investigate the nonlinear propagation of defect state in one-dimensional structures with defects. We investigate the nonlinear transmission spectra and the bistable response for defective structures with different index gradients. The results show that positive Kerr nonlinearity can suppress the Wannier-Stark localization. And the nonlinear response of defect states band exhibits an optical switch behavior, which may be applied to all-optical devices. And the gap solitons from these defect states are presented.     

Spatially Periodic Inhomogeneous States in a Nonlinear Crystal with a Nonlinear Defect

Spatially periodic inhomogeneous stationary states are shown to exist near a thin defect layer with nonlinear properties separating nonlinear Kerr-type crystals. The contacts of nonlinear self-focusing and defocusing crystals have been analyzed. The spatial field distribution obeys a time-independent nonlinear Schrödinger equation with a nonlinear (relative to the field) potential modeling the thin defect layer with nonlinear properties. Both symmetric and asymmetric states relative to the defect plane are shown to exist. It has been established that new states emerge in a self-focusing crystal, whose existence is attributable to the defect nonlinearity and which do not emerge in the case of a linear defect. The dispersion relations defining the energy of spatially periodic inhomogeneous stationary states have been derived. The expressions for the energies of such states have been derived in an explicit analytical form in special cases. The conditions for the existence of periodic states and their localization, depending on the defect and medium characteristics, have been determined.     

Analytical Periodic Wave and Soliton Solutions of the Generalized Nonautonomous Nonlinear Schrödinger Equation in Harmonic and Optical Lattice Potentials

We construct analytical periodic wave and soliton solutions to the generalized nonautonomous nonlinear Schrödinger equation with time- and space-dependent distributed coefficients in harmonic and optical lattice potentials. We utilize the similarity transformation technique to obtain these solutions. Constraints for the dispersion coefficient, the nonlinearity, and the gain (loss) coefficient are presented at the same time. Various shapes of periodic wave and soliton solutions are studied analytically and physically. Stability analysis of the solutions is discussed numerically.     

Attraction of nonlocal dark optical solitons

We study the formation and interaction of spatial dark optical solitons in materials with a nonlocal nonlinear response. We show that unlike in local materials, where dark solitons typically repel, the nonlocal nonlinearity leads to a long-range attraction and formation of stable bound states of dark solitons.     

Self-trapped spatially localized states in combined linear-nonlinear periodic potentials

We analyze the existence and stability of two kinds of self-trapped spatially localized gap modes,gap solitons and truncated nonlinear Bloch waves,in one-and two-dimensional optical or matter-wave media with self-focusing nonlinearity,supported by a combination of linear and nonlinear periodic lattice potentials.The former is found to be stable once placed inside a single well of the nonlinear lattice,it is unstable otherwise.Contrary to the case with constant self-focusing nonlinearity,where the latter solution is always unstable,here,we demonstrate that it nevertheless can be stabilized by the nonlinear lattice since the model under consideration combines the unique properties of both the linear and nonlinear lattices.The practical possibilities for experimental realization of the predicted solutions are also discussed.     

Quasi-phase-matched generation of optical intensity waves

The evolution of modulated light in a nonlinear medium, when described in terms of intensity waves, depends critically on a phase-matching condition for the intensity waves. We formally develop the conditions for quasi-phase matching of the interacting intensity waves and show that a periodic nonlinearity can be utilized to eliminate the dephasing between them. This is verified using stimulated Brillouin scattering with a periodically nonlinear optical fiber that has a period length equal to one-half of the (modulation) wavelength of the intensity waves.     

Nonlinear spectral-spatial control and localization of supercontinuum radiation

We present the first observation of spatiospectral control and localization of supercontinuum light through the nonlinear interaction of spectral components in extended periodic structures. We use an array of optical waveguides in a LiNbO3 crystal and employ the interplay between diffraction and nonlinearity to dynamically control the output spectrum of the supercontinuum radiation. This effect presents an efficient scheme for optically tunable spectral filtering of supercontinua.     

Nonlinear time-independent and time-dependent propagation of light in direct-gap semiconductors with paired excitons bound into biexcitons

We study a new class of nonlinear cooperative phenomena that occur when light propagates in direct-gap semiconductors. The nonlinearity here is due to a process, first discussed by A. L. Ivanov, L. V. Keldysh, and V. V. Panashchenko, in which two excitons are bound into a biexciton by virtue of their Coulomb interaction. For the geometry of a ring cavity, we derive a system of nonlinear differential equations describing the dynamical evolution of coherent excitons, photons, and biexcitons. For the time-independent case we arrive at the equation of state of optical bistability theory, and this equation is found to differ considerably from the equations of state in the two-level atom model and in the exciton region of the spectrum. We examine the stability of the steady states and determine the switchover times between the optical bistability branches. We also show that in the unstable sections of the equation of state, nonlinear periodic and chaotic self-pulsations may arise, with limit cycles and strange attractors being created in the phase space of the system. The scenario for the transition to the dynamical chaos mode is found. A computer experiment is used to study the dynamic optical bistability. Finally, we discuss the possibility of detecting these phenomena in experiments. Zh. éksp. Teor. Fiz. 112, 1778–1790 (November 1997)     

Nonlinear polarization bistability in optical nanowires

Using the full vectorial nonlinear Schr?dinger equations that describe nonlinear processes in isotropic optical nanowires, we show that there exist structural anisotropic nonlinearities that lead to unstable polarization states that exhibit periodic bistable behavior. We analyze and solve the nonlinear equations for continuous waves by means of a Lagrangian formulation and show that the system has bistable states and also kink solitons that are limiting forms of the bistable states.     

THE NONLINEAR SCHRÖDINGER EQUATION FOR THE DELTA-COMB POTENTIAL: QUASI-CLASSICAL CHAOS AND BIFURCATIONS OF PERIODIC STATIONARY SOLUTIONS

The nonlinear Schrödinger equation is studied for a periodic sequence of delta-potentials (a delta-comb) or narrow Gaussian potentials. For the delta-comb the time-independent nonlinear Schrödinger equation can be solved analytically in terms of Jacobi elliptic functions and thus provides useful insight into the features of nonlinear stationary states of periodic potentials. Phenomena well-known from classical chaos are found, such as a bifurcation of periodic stationary states and a transition to spatial chaos. The relation to new features of nonlinear Bloch bands, such as looped and period doubled bands, are analyzed in detail. An analytic expression for the critical nonlinearity for the emergence of looped bands is derived. The results for the delta-comb are generalized to a more realistic potential consisting of a periodic sequence of narrow Gaussian peaks and the dynamical stability of periodic solutions in a Gaussian comb is discussed.     

Optical nonlinearity enhancement for an anisotropic semiconductor–insulator composite

We investigate the effect of geometric anisotropy on the optical nonlinearity enhancement for a periodic composite with a rectangular array of elliptic semiconducting cylinders in an insulating host. By using a series expression of the space-dependent electric field obtained by a simple Fourier method in a periodic composite, we calculate the frequency dependence of the effective third-order nonlinear susceptibility as a function of anisotropy. The results show that the height of the nonlinearity enhancement peak may be increased by several orders of magnitude as the aspect ratio of the ellises is decreased or the lattice edge length ratio is increased. At resonance frequency, there exists a strong anomalous dispersion. We also investigate the effect of the volume fraction of the semiconductor phase for composites with a square array of circular semiconducting cylinders. Received: 23 November 2000 / Accepted: 2 August 2001 / Published online: 17 October 2001     

Asymptotics of large bound states of localized structures

We analyze stationary fronts connecting uniform and periodic states emerging from a pattern-forming instability. The size of the resulting periodic domains cannot be predicted with weakly nonlinear methods. We show that what determine this size are exponentially small (but exponentially growing in space) terms. These can only be computed by going beyond all orders of the usual multiple-scale expansion. We apply the method to the Swift-Hohenberg equation and derive analytically a snaking bifurcation curve. At each fold of this bifurcation curve, a new pair of peaks is added to the periodic domain, which can thus be seen as a bound state of localized structures. Such scenarios have been reported with optical localized structures in nonlinear cavities and localized buckling.     

Nonlinear wave interaction in photonic band gap materials

We present detailed analytical and numerical studies of nonlinear wave interaction processes in one-dimensional (1D) photonic band gap (PBG) materials with a Kerr nonlinearity. We demonstrate that some of these processes provide efficient mechanisms for dynamically controlling so-called gap-solitons. We derive analytical expressions that accurately determine the phase shifts experienced by nonlinear waves for a large class of non-resonant interaction processes. We also present comprehensive numerical studies of inelastic interactions, and show that rather distinct regimes of interaction exist. The predicted effects should be experimentally observable, and can be utilized for probing the existence and parameters of gap solitons. Our results are directly applicable to other nonlinear periodic structures such as Bose–Einstein condensates in optical lattices.     

Higher-order solitons in amplitude-disordered waveguide arrays

We investigate the existence and stability of different families of spatial solitons in optical waveguide arrays whose amplitudes obey a disordered distribution. The competition between focusing nonlinearity and linearly disordered refractive index modulation results in the formation of spatial localized nonlinear states. Solitons originating from Anderson modes with few nodes are robust during propagation. While multi-peaked solitons with in-phase neighboring components are completely unstable, multipole-mode solitons whose neighboring components are out-of-phase can propagate stably in wide parameter regions provided that their power exceeds a critical value. Our findings, thus, provide the first example of stable higher-order nonlinear states in disordered systems.     

Modulational instability and space-time dynamics in nonlinear parabolic-index optical fibers

Beam propagation in multimode graded-index parabolic optical fibers in the presence of group-velocity dispersion and Kerr nonlinearity is theoretically investigated. It is shown that a modulational instability arising from the periodic spatial focusing of the beam takes place regardless of the sign of fiber dispersion, leading to a highly nonlinear space-time dynamics and the generation of ultrashort optical pulses.