Self-trapping of light in a two-dimensional photonic lattice

Abstract

We study wave propagation in a two-dimensional photonic lattice with focusing Kerr nonlinearity, and report on the existence of various nonlinear localized structures in the form of fundamental, dipole, and vortex solitons. First, the linear bandgap structure induced by the two-dimensional photonic crystal is determined, and solitons are found to exist in the photonic bandgap. Next, structures of these solitons and their stability properties are analyzed in detail. When the propagation constant is not close to the edge of the bandgap, the fundamental soliton is largely confined to one lattice site; the dipole soliton consists of two π-out-of-phase, Gaussian-like humps, whereas the vortex comprises four fundamental modes superimposed in a square configuration with a phase structure that is topologically equivalent to the conventional homogeneous-bulk vortex. At high lattice potential, all these soliton states are stable against small perturbations. However, among the three states, the fundamental solitons are the most robust, whereas vortices are the least. If the propagation constant is close to the edge of the bandgap, then all three soliton states spread over many lattice sites and become linearly unstable as a result of the Vakhitov–Kolokolov instability.