Representation of Photonic Crystals and Their Localized Modes Through the Use of Fourier–Bessel Expansions

Photonic crystal (PhC) geometry is typically characterized by its translational symmetry. However, it can be treated based on the rotationally symmetry through the use of Fourier-Bessel expansions about the center of rotation. Fourier-Bessel expansions of the inverse dielectric of the structures and the transverse electric (TE) localized modes extract the rotational symmetry that is present. The results show that in PhCs and photonic quasi-crystals, the localized H_z field contains only the rotational orders that correspond to the rotational order of the dielectric plus and minus the rotational order of the corresponding perfect state. This relationship indicates the potential for simplification within the master equation in cylindrical coordinates that will require further examination.