Nonparaxial electromagnetic Bragg scattering in periodic media with PT symmetry

Abstract

The evolution of a pair of resonant Bragg modes through a medium characterized by a complex one-dimensional $\mathcal{PT}$-symmetric periodic permittivity is thoroughly investigated. Analytic solutions of Maxwell's equations are derived beyond the paraxial approximation to investigate the periodic energy exchange that occurs between the Bragg modes for the Hermitian lattices as well as for complex lattices. Three regimes defined by the symmetry breaking point are discussed: below it, above it and at it. These regimes are determined by the existence of four real eigenvalues in the symmetric phase, which collide and coalesce into a pair at the breaking point. Above the critical value each member of the pair bifurcates into a pair of complex values. Therefore, the complex lattice reveals a variety of wave dynamics depending on the gain/loss balance. In all regimes of the transition the signature of $\mathcal{PT}$-symmetric systems is present, as the evolution is always nonreciprocal and unidirectional.