Parity-time and anti-parity-time states of Bragg gratings and their representation in spherical coordinates

Parity-time (PT) symmetry and anti-PT symmetry are two interesting types of non-Hermitian Hamiltonian systems. We propose a spherical graphical representation to concisely capture the (anti-)PT symmetry design space of Bragg gratings having phase-shifted real and imaginary refractive index perturbations. A general Bragg grating perturbation function is used such that designs can be arbitrarily changed to reach different (anti-)PT symmetric states. Special cases having simplified perturbations are explored, consisting of real index only, imaginary index only, and perturbations reaching (anti-)PT symmetry unbroken, broken, and exceptional point states. The evolution of the PT states of Bragg gratings and their interesting and unique properties lead to potential applications in sensing and lasing.