Inverse dispersion method for calculation of complex photonic band diagram and $\cal{PT}$-symmetry

Abstract: We suggest an inverse dispersion method for calculating photonic band diagram for materials with arbitrary frequency-dependent dielectric functions. The method is able to calculate the complex wave vector for a given frequency by solving the eigenvalue problem with a non-Hermitian operator. The analogy with $\cal{PT}$-symmetric Hamiltonians reveals that the operator corresponds to the momentum as a physical quantity and the singularities at the band edges are related to the branch points and responses for the features on the band edges. The method is realized using plane wave expansion technique for two-dimensional periodical structure in the case of TE- and TM-polarization. We illustrate the applicability of the method by calculation of the photonic band diagrams of an infinite two-dimension square lattice composed of dielectric cylinders using the measured frequency dependent dielectric functions of different materials (amorphous hydrogenated carbon, silicon, and chalcogenide glass). We show that the method allows to distinguish unambiguously between Bragg and Mie gaps in the spectra.