Behavior of light at photonic crystal interfaces

Band structures and Bloch modes give a generalized description of light in infinite photonic crystals. We show that the band structure and Bloch modes also contain the information necessary to find the amplitude and phase of light reflected and transmitted from interfaces in systems made using finite and semi-infinite photonic crystals. We obtain the equivalent of the Fresnel coefficients for photonic crystals. We use these coefficients to derive the reflection of light from a photonic crystal of finite size and the resonant modes of photonic crystal cavities and line defects. Results are given for ideal two-dimensional crystals, as well as crystals etched in semiconductor slab waveguides.