Thawatchai Mayteevarunyoo

Spontaneous symmetry breaking of photonic and matter waves in two-dimensional pseudopotentials

We introduce the two-dimensional Gross–Pitaevskii/nonlinear-Schrödinger (GP/NLS) equation with the self-focusing nonlinearity confined to two identical circles, separated or overlapped. The model can be realised in terms of Bose–Einstein condensates (BECs) and photonic-crystal fibers. Following the recent analysis of the spontaneous symmetry breaking (SSB) of localized modes trapped in 1D and 2D double-well nonlinear potentials (also known as pseudopotentials), we aim to find 2D solitons in the two-circle setting, using numerical methods and the variational approximation (VA). Well-separated circles support stable symmetric and antisymmetric solitons. The decrease of separation L between the circles leads to destabilisation of the solitons. The symmetric modes undergo two SSB transitions. First, they are transformed into weakly asymmetric breathers, which is followed by a transition to single-peak modes trapped in one circle. The antisymmetric solitons perform a direct transition to the single-peak mode. The symmetric solitons are described reasonably well by the VA. For touching (L = 0) and overlapping (L < 0) circles, single-peak solitons are found – asymmetric ones, trapped in either circle, and symmetric solitons centered at the midpoint of the bi-circle configuration. If the overlap is weak, the symmetric soliton is unstable. It may spontaneously leap into either circle and perform shuttle motion in it. A region of stability of the symmetric solitons appears with the increase of overlap degree. In the case of a moderately strong overlap, another SSB effect is found, in the form of a pair of symmetry-breaking and restoring bifurcations which link families of the symmetric and asymmetric solitons.

Gap solitons in grating superstructures.

We report results of the investigation of gap solitons (GSs) in the generic model of a periodically modulated Bragg grating (BG), which includes periodic modulation of the BG chirp or local refractive index, and periodic variation of the local reflectivity. We demonstrate that, while the previously studied reflectivity modulation strongly destabilizes all solitons, the periodic chirp modulation, which is a novel feature, stabilizes a new family of double-peak fundamental BGs in the side bandgap at negative frequencies (gap No. -1), and keeps solitons stable in the central bandgap (No. 0). The two soliton families demonstrate bistability, coexisting at equal values of energy. In addition, stable 4-peak bound states are formed by pairs of fundamental GSs in bandgap -1. Self-trapping and mobility of the solitons are studied too.

Solitons and vortices in nonlinear two-dimensional photonic crystals of the Kronig-Penney type

Solitons in the model of nonlinear photonic crystals with the transverse structure based on two-dimensional (2D) quadratic- or rhombic-shaped Kronig-Penney (KP) lattices are studied by means of numerical methods. The model can also applies to a Bose-Einstein condensate (BEC) trapped in a superposition of linear and nonlinear 2D periodic potentials. The analysis is chiefly presented for the self-repulsive nonlinearity, which gives rise to several species of stable fundamental gap solitons, dipoles, four-peak complexes, and vortices in two finite bandgaps of the underlying spectrum. Stable solitons with complex shapes are found, in particular, in the second bandgap of the KP lattice with the rhombic structure. The stability of the localized modes is analyzed in terms of eigenvalues of small perturbations, and tested in direct simulations. Depending on the value of the KPs duty cycle (DC, i.e., the ratio of the voids width to the lattice period), an internal stability boundary for the solitons and vortices may exist inside of the first bandgap. Otherwise, the families of the localized modes are entirely stable or unstable in the bandgaps. With the self-attractive nonlinearity, only unstable solitons and vortices are found in the semi-infinite gap.

The interaction of Airy waves and solitons in a three-wave system

We employ the generic three-wave system, with the χ interaction between two components of the fundamental-frequency (FF) wave and second-harmonic (SH) one, to consider collisions of truncated Airy waves (TAWs) and three-wave solitons in a setting which is not available in other nonlinear systems. The advantage is that the single-wave TAWs, carried by either one of the FF component, are not distorted by the nonlinearity and are stable, three-wave solitons being stable too in the same system. The collision between mutually symmetric TAWs, carried by the different FF components, transforms them into a set of solitons, the number of which decreases with the increase of the total power. The TAW absorbs an incident small-power soliton, and a high-power soliton absorbs the TAW. Between these limits, the collision with an incident soliton converts the TAW into two solitons, with a remnant of the TAW attached to one of them, or leads to formation of a complex TAW-soliton bound state. At large velocities, the collisions become quasi-elastic.

Two-dimensional χ 2 solitons generated by the downconversion of Airy waves

Conversion of truncated Airy waves (AWs) carried by the second-harmonic (SH) component into axisymmetric χ2 solitons is considered in a 2D system with quadratic nonlinearity. The spontaneous conversion is driven by the parametric instability of the SH wave. The input in the form of the AW vortex is also considered. As a result, one, two, or three stable solitons emerge in a well-defined form, unlike the recently studied 1D setting, where the picture is obscured by radiation jets. Shares of the total power captured by the emerging solitons and conversion efficiency are found as functions of parameters of the AW input.

Gap solitons attached to a gapless layer

We consider linear and nonlinear modes pinned to a grating-free (gapless) layer placed between two symmetric or asymmetric semi-infinite Bragg gratings (BGs), with a possible phase shift between them, in a medium with the uniform Kerr nonlinearity. The asymmetry is defined by a difference between bandgap widths in the two BGs. In the linear system, exact defect modes (DMs) are found. Composite gap solitons pinned to the central layer are also found in analytical and numerical forms in the nonlinear model. In the asymmetric system existence boundaries for the DMs and gap solitons, due to the competition between attraction to the gapless layer and repulsion from the reflectivity step, are obtained analytically. Stability boundaries for solitons in the asymmetric system are identified by means of direct simulations. Collisions of moving BG solitons with the gapless layer are also studied.

Interaction of spatial solitons with a gapless stripe embedded into a Bragg-grating area

We introduce a model in which the grating is absent in a finite-width stripe in the waveguide, thus creating a gapless channel in the gapped medium. Two semi-infinite grating separated by the plain stripe may have a relative phase shift. This system modifies the Bragg bandgap, creating intra-gap defect modes (DFs) which are pinned to the gapless channel. A DF solution in the linear system is found analytically. Further, numerical analysis of the full nonlinear system demonstrates that the shape and stability of Bragg solitons are also strongly affected by the presence of the gapless channel, and by the possible phase shift between the two semi-infinite gratings. In particular, asymmetric and flat-top solitons appear.

Two-Component Gap Solitons in Self-Defocusing Photonic Crystals

Studies of solitons in spatially periodic (lattice) potentials have grown into a vast area of research, with profoundly important applications to nonlinear optics, plasmonics, and matter waves in quantum gases, as outlined in recent reviews [1]-[4]. In ultracold bosonic and fermionic gases, periodic potentials can be created, in the form of optical lattices (OLs), by coherent laser beams illuminating the gas in opposite directions [5]-[7]. Effective lattice potentials for optical waves are induced by photonic crystals (PhCs), which are built as permanent structures by means of various techniques [2, 8, 9], or as laser-induced virtual structures in photorefractive crystals [10]. Reconfigurable PhCs can be also based on liquid crystals [11]. Parallel to the progress in the experiments, the study of the interplay between the nonlinearity and periodic potentials has been an incentive for the rapid developments of theoretical methods [12, 13]. Both the experimental and theoretical results reveal that solitons can be created in lattice potentials, if they do not exist in the uniform space [this is the case of gap solitons (GSs) supported by the self-defocusing nonlinearity, see original works [14]-[17] and reviews [7, 18]], and solitons may be stabilized, if they are unstable without the lattice (multidimensional solitons in the case of self-focusing, as shown in Refs. [17], [19]-[25], see also reviews [1, 3, 4]). The stability of GSs has been studied in detail too—chiefly, close to edges of the corresponding bandgaps—in one [27]-[29] and two [30] dimensions alike.