Bulk vortices and half-vortex surface modes in parity-time-symmetric media

Abstract

We demonstrate that in-bulk vortex localized modes, and their surface half-vortex ("horseshoe") counterparts self-trap in two-dimensional (2D) nonlinear optical systems with PT -symmetric photonic lattices (PLs). The respective stability regions are identified in the underlying parameter space. The in-bulk states are related to truncated nonlinear Bloch waves in gaps of the PL-induced spectrum. The basic vortex and horseshoe modes are built, severally, of four and three beams with appropriate phase shifts between them. Their stable complex counterparts, built of up to 12 beams, are reported too.PACS numbers: 03.65. Ge, 11.30.Er, 42.65.Sf, 42.65.Tg Nonlinear spatially periodic systems support diverse types of self-trapped in-gap states. In particular, spatial gap solitons [1,2] originate from the interplay between the periodicity and nonlinearity. Further, surface gap solitons [4][5][6][7][8] appear at the interface between a uniform medium and a photonic lattice (PL) built into a nonlinear material. Extended self-trapped waves with steep edges also exist in these settings, being related to truncated nonlinear Bloch waves [9,10]. Modes of the latter type provide a link between extended nonlinear Bloch waves [11,12] and tightly localized gap solitons [1][2][3].Recently, a great deal of interest has been drawn to the realization of parity-time (PT ) symmetry in optics. Originally, this concept was developed in quantum mechanics, where it was demonstrated that, beyond the conventional Hermitian Hamiltonians, their PTsymmetric non-Hermitian counterparts may also give rise to purely real (hence physically relevant) spectra [13][14][15][16]. Following the similarity between quantum mechanics and paraxial optics [17,18], PT -symmetric optical systems with complex refractive indices [19,20] have been extensively studied theoretically [21][22][23][24][25][26][27][28][29][30][31][32][33] and experimentally [34][35][36][37][38][39][40][41]. In this context, PT -symmetric PLs play an important role. Taking into account the non-orthogonality of the respective eigenmodes, their coupled-mode description had to be reformulated via the variational principle [21]. The light propagation in PTsymmetric PLs embedded into linear media were analyzed preliminarily [24,42]. Further, it has been found that 1D and 2D spatial gap solitons exist in PLs built into a nonlinear material [23,[43][44][45][46][47].Although PT -symmetric systems combining lattices and nonlinearity was not reported yet. In particular, such self-trapped nonlinear states may serve as a necessary link between spatially localized gap solitons and extended nonlinear Bloch waves under the PT symmetry.Self-trapped vortices and surface modes are of great interest in the context of the PT -symmetric settings. Indeed, the study of nonlinear surface modes pinned on the interface of a PT -symmetric system opens a way to explore the interplay between surface effects, the nonlinearity, and the PT -symmetry. On the other hand, the analysis of localized vortices supported by PT ...

Discrete solitons and vortices on two-dimensional lattices of 𝒫𝒯 -symmetric couplers

“…There are fewer publications which reported PT -symmetric solitons in 2D settings [23,30,35]. The objective of the present work is to introduce a 2D network of horizontally coupled nonlinear dimers, which is a generalization of the 1D system from [39], and can also be realized in optics, using a photonic-crystal matrix into which nonlinear dual-core waveguides with the balanced gain and loss are embedded (see Fig.…”

Defect solitons in parity-time symmetric superlattices with focusing saturable nonlinearity

“…Since then, gap solitons in PT-symmetric media have drawn much attention. Li et al [13,14] even found stable PT surface gap solitons at the interface between a linear medium and a Kerr nonlinear medium. Cao et al [15] also reported stable PT gap solitons in a saturable self-defocusing nonlinear medium.…”

Stability of optical solitons in parity-time-symmetric optical lattices with competing cubic and quintic nonlinearities

“…It was also mentioned in [10] that PT symmetry is a necessary condition for the existence of continuous modes in one-dimensional complex potentials. Ever since, PT-symmetric optical solitons [11][12][13][14][15][16] have been widely studied, including bright spatial solitons [17], gray solitons [18], dark solitons and vortices [19,20], multipeak gap solitons [21], nonlocal gap solitons * gelijuan26@gmail.com [22][23][24][25], vector solitons [26], and defect solitons [27,28]. Soliton families in PT-symmetric nonlinear lattices [29], in PT-symmetric mixed linear-nonlinear optical lattices [30], in PT-symmetric systems with quadratic nonlinearity [31], and in PT-symmetric couplers (the dual-core waveguide with balanced gain and loss) [32] have also been investigated.…”