Discrete surface solitons in two-dimensional anisotropic photonic lattices

Abstract

We study nonlinear surface modes in two-dimensional anisotropic periodic photonic lattices and demonstrate that, in a sharp contrast to one-dimensional discrete surface solitons, the mode threshold power is lower at the surface, and two-dimensional discrete solitons can be generated easier near the lattice corners and edges. We analyze the crossover between effectively one-and two-dimensional regimes of the surface-mediated beam localization in the lattice. PACS numbers:Surface modes have been studied in different branches of physics; in guided wave optics surface states were predicted to exist at interfaces separating periodic and homogeneous dielectric media [1]. The interest in studying surface waves has been renewed recently because the interplay of discreteness and nonlinearity can facilitate the formation of discrete surface solitons [2,3] at the edge of the waveguide array. That can be understood as the localization of a discrete optical soliton near the surface [4] for powers exceeding a certain threshold value, for which the repulsive effect of the surface is balanced. A similar effect of light localization near the edge of the waveguide array and the formation of surface gap solitons have been predicted and observed for defocusing nonlinear media [5,6].It is important to analyze how the properties of nonlinear surface waves are modified by the lattice dimensionality, and the first studies of different types of discrete surface solitons in two-dimensional lattices [7,8,9,10] revealed, in particular, that the presence of a surface increases the stability region for two-dimensional (2D) discrete solitons [10] and the threshold power for the edge surface state is slightly higher than that for the corner soliton [9].In this Letter we consider anisotropic semi-infinite twodimensional photonic lattices and study the crossover between one-and two-dimensional surface solitons emphasizing the crucial effect of the lattice dimensionality on the formation of surface solitons.We consider a semi-infinite 2D lattice [shown schematically in Fig.2(a) below], described by the system of coupled-mode equations for the normalized amplitudes u n,m [11,12],where ξ is the normalized propagation distance. We de- fine the lattice coupling as follows:where α characterizes the lattice anisotropy.Linear lattice waves of the form u n,m (ξ) = u 0 sin(kn) sin(qm) exp(iβξ) satisfy the dispersion relation β kq = 2(cos k + α cos q). In the nonlinear case, we look for localized stationary solutions of the form u n,m (ξ) = u n,m exp(iλξ), where the amplitudes u n,m are real, and λ is the nonlinear propagation constant. For a given λ, localized solutions are found in a 15 × 15 lattice by using the Newton-Raphson method.We calculate the power threshold P th that characterizes the discrete solitons in 2D lattices [13]. We study three different modes: corner [ Fig. 1(a)], edge surface [ Fig. 1(b,c)] and central [ Fig. 1(d)] localized modes. The corner and edge modes represent 2D surface localized

Nonlinear Tamm states and surface effects in periodic photonic structures

“…8. Theoretical studies of discrete surface solitons localized in the corners or at the edges of two-dimensional photonic lattices have been performed by several research groups [41][42][43]. In particular, in a sharp contrast with onedimensional discrete surface solitons, it was shown that the power threshold is lower at the surface than in a bulk making the mode excitation easier [42].…”

“…To elucidate a qualitative different between oneand two-dimensional surface states, Vicencio et al [42] analyzed the properties of discrete surface solitons in two-dimensional anisotropic photonic lattices and studied the crossover between the isotropic (effective twodimensional) and high-anisotropic (effectively quasi-onedimensional) regimes of the surface-mediated beam localization in the lattice. They found that, unlike onedimensional discrete surface solitons, the threshold power [45] of the two-dimensional discrete soliton is lowered by the surface, so that two-dimensional solitons can be generated easier near the lattice corners and edges.…”

Interface solitons in two-dimensional photonic lattices

“…These twodimensional nonlinear surface modes demonstrate novel features in comparison with their counterparts in truncated one-dimensional waveguide arrays [7,8,9]. In particular, in a sharp contrast to one-dimensional discrete surface solitons, the mode threshold is lower at the surface than in a bulk making the mode excitation easier [2].…”

“…These twodimensional nonlinear surface modes demonstrate novel features in comparison with their counterparts in truncated one-dimensional waveguide arrays [7,8,9]. In particular, in a sharp contrast to one-dimensional discrete surface solitons, the mode threshold is lower at the surface than in a bulk making the mode excitation easier [2].Recently, Szameit et al[10] reported on the first experimental observation of two-dimensional interface solitons, i.e. spatial optical solitons generated at the interface separating square and hexagonal optical lattices with different refractive index modulation depths.…”

Two-color surface solitons in two-dimensional quadratic photonic lattices

“…4420; 190.5530; 190.5940 Two-dimensional surface solitons have been recently predicted to exist as novel types of discrete solitons localized in the corners or at the edges of two-dimensional photonic lattices 1,2,3 . These theoretical predictions were followed by the experimental observation of twodimensional surface solitons in optically-induced photonic lattices 4 and waveguide arrays laser-written in fused silica 5 .…”