Optical properties of planar colloidal crystals: Dynamical diffraction and the scalar wave approximation

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Abstract
We present a quantitative comparison between two analytic theories for the propagation of electromagnetic waves in periodic dielectric structures. These theories have both been used extensively in the modeling of optical spectra of colloidal crystals exhibiting photonic band gap behavior. We demonstrate that dynamical diffraction theory is equivalent to the scalar wave approximation, in the limit of small dielectric contrast. This equivalence allows us to place quantitative limits on the validity of dynamical diffraction, relative to the predictions of the more accurate scalar wave theory. We also note that dynamical diffraction is often applied with boundary conditions which neglect the strong interference between the incident and diffracted waves within the periodic medium. These boundary conditions lead to expressions for the transmission spectrum which cannot be generalized to the case of normal-incidence propagation. We provide a corrected form for these expressions, and use them in comparisons with experimental spectra. Excellent agreement between theory and experiment is obtained for the widths of optical stop bands, for both positive and negative values of the dielectric contrast. These are among the first quantitative comparisons between theoretical and experimental optical spectra of colloidal photonic crystals.