This paper surveys recent analytic and numerical results on asymptotic models of spectra of electromagnetic (EM) waves in two-dimensional (2-D) thin high-contrast photonic bandgap (PBG) materials. These models lead to discovery of interesting phenomena, including extremely narrow bands that can be used for spontaneous emission enhancement, gaps in the long wave regions, and asymptotic periodicity of the spectrum. The asymptotic results provide unexpectedly good qualitative (and sometimes quantitative) description of spectral behavior for materials of finite contrast. In some cases, simple ordinary differential models can be derived that yield a good approximation of the spectra. In such situations, one can obtain approximate analytic formulas for the dispersion relations.
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