The Rayleigh Hypothesis and Scattering at Photonic Crystal Surfaces

We examine the Rayleigh hypothesis in the context of scattering of light off photonic crystal interfaces. First, the hypothesis -- which was initially suggested for scattering of waves off rough surfaces between homogeneous media -- is rephrased to apply to photonic crystal interfaces. Next, some exact and explicit functional forms are presented that map plane photonic crystal surfaces to periodic rough surfaces in free space, so that known criteria for the validity of the Rayleigh hypothesis for scattering at rough surfaces can be applied directly to scattering at the photonic crystal surfaces. The same maps also allow the scattering problem to be solved exactly. We present numerical results for scattering amplitudes at a photonic crystal surface using both the exact method (based on a surface integral formulation) and an approximate method (based on the Rayleigh hypothesis). The results are found to be consistent with the analytical criteria for the range of validity of the approximate method.