Spherical representation of (anti-)parity-time states of Bragg gratings

Parity-time (PT) symmetry and anti-PT symmetry are two types of non-Hermitian Hamiltonian systems. Waveguide Bragg gratings with phase-shifted perturbations in real and imaginary refractive index can reach any (anti-)PT symmetry state. A spherical surface of the grating design space is proposed, illustrating (un)broken and exceptional point (anti-)PT symmetry states. Grating design examples are presented and discussed.