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A rigorous analysis of the propagation of transverse-electric modes in two-dimensionally periodic media is presented, based on an exact solution of the wave equation. The modulation functions studied lead to Mathieu and Hill equations. It is shown that the complete inhibition of the propagation of transverse-electric modes in a modulated plane is possible. Several modulation functions of the permittivity are studied in order to find the one for which the modulation depth necessary for the observation of a frequency band gap is minimized. In particular, thin grids and closely packed arrays of square rods are considered. The allowed angles of propagation of waves whose frequency is not contained within a forbidden band are also presented.
Villeneuve et al. (1991) studied this question.