Mean Field Theory with Only a Few Transverse Fourier Components of EM Fields for Low-Frequency Two-Dimensional Photonic Bands

We propose a mean-field band structure theory for low-lying two-dimensional photonic states based on the principle of plane wave expansion. Our theory reduces the complexities of a two-dimensional problem into that of an effective one-dimensional crystal, which provides two key advantages: a) simplification of dimensions assists in ease of calculation and b) delineation of the photonic state physics leads to a gain in its physical insights. Our method distinguishes itself from previous known mean field theories in its capability of including more than one Fourier component of EM fields decomposed along the direction perpendicular to propagation. Furthermore, the method applies for virtually any crystal structure and direction of propagation, and was discovered to function well for both E-polarization and H-polarization modes of states. We also attempt to demonstrate systematic improvement of the calculation with increasing number of Fourier components. Satisfactory numerical accuracy is obtained particularly for the states of the lowest two bands.