Sistemas fotônicos PT-simétricos

Abstract

The spatial evolution of a pair of resonant Bragg modes through a medium characterized by a complex one-dimensional PT -symmetric periodic relative electric permittivity is thoroughly investigated. By using the two wave model, analytic solutions of Maxwell’s equations are derived in the nonparaxial regime in order to investigate the periodic energy exchange between the Bragg modes for the Hermitian optical lattices as well as for complex lattices and also to investigate the spatial evolution of the real part of the electric field that propagates through this medium. 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 complex eigenvalues below the symmetry breaking point, which collide and coalesce into a pair of complex doubly degenerate eigenvalues at the breaking point. Above the critical value each member of the pair bifurcates into a pair of complex values and now they have a nonzero real part. In the Hermitian case, it is demonstrated that a complete reciprocal energy transfer between the pair of Bragg modes, in a manner similiar to the Pendellösung effect known from X-ray diffraction by a crystal, takes place. When the optical lattice is complex, the dynamics of the energy transfer is completely different from the Hermitian case: below the symmetry breaking point there exist a very nonreciprocal beating-like oscillatory behavior of the energy transfer between the Bragg modes; above the symmetry breaking point the spatial evolution of the energy transfer grows unlimited but an oscillatory evolution still takes place; and at the symmetry breaking point the phenomenon of mode trapping does not occur in this nonparaxial regime (previously seen in PT -symmetric optical lattices in the paraxial regime). For the complex lattice, all these regimes share the common features: existence of a preferable mode for which the energy is transferred and a spatial evolution of this transfer in a nonreciprocal fashion, some of the characteristics very well known of PT -symmetric optical systems.