Radiation families emitted by a discrete soliton in parity-time-symmetric waveguide arrays

We investigate the dynamics of a spatial discrete soliton and the radiation families emitted by it inside a parity-time (PT)-symmetric waveguide array with alternate gain-loss channels. A strong spatial soliton that evolves inside the waveguide array due to the balance between discrete diffraction and Kerr nonlinearity excites linear waves in the form of diffractive radiation when launched with an angle. The PT-symmetric nature of the waveguide leads to additional radiations in Fourier space that have not been explored. In our paper, we mainly focus on the origin of these radiations and try to understand how to control them. Under strong PT symmetry, a discrete soliton launched normally to the waveguide array produces strong side lobes, which can lead to a population of the field at ±π/2 in momentum space. In addition, a strong soliton with initial phase gradient radiates a unique PT-symmetry-assisted linear wave. We establish a phase-matching condition to locate such radiation in momentum space. The periodic arrangement of the gain-loss channel also leads to radiations due to reflection and backscattering, which is prominent for a weak soliton. A linear Hamiltonian analysis for such a waveguide array is provided to identify the PT-phase-transition regime and to optimize the parameters for stable discrete soliton dynamics. We thoroughly investigate the origin of all the radiations that emerge in the PT-symmetric waveguide array and put forward the background theory, which is in good agreement with the full numerical results.