On the limits of validity of the two-wave approximation in the dynamical theory of electromagnetic scattering by periodic dielectric media

We investigate the accuracy and limits of validity of the two-wave approximation in the dynamical theory of electromagnetic scattering by periodic dielectric media. The errors ensuing from the approximation are estimated by applying the dynamical theory to a scattering problem for which an alternative exact electromagnetic solution is available and comparing results. The conditions for applying the approximate theory and its accuracy are discussed in terms of concepts peculiar to the classical dynamical theory of the scattering of X-rays in crystals, such as the Ewald sphere in the reciprocal space and the resonance error. After introducing the basic equations of the dynamical theory of electromagnetic scattering by three dimensional periodic dielectric media, the theory is applied to the scattering by one-dimensional periodic layered structures where a rigorous analytical solution is available. The analysis of the errors involved in the two-wave approximation indicates that, in the general case, the quality of the approximation cannot be quantified in terms of just the resonance error but it is also strongly affected by the dielectric contrast. Simple formulae are reported yielding a reliable error estimate in many practical cases. An extension of the results to the two and three dimensional case is also provided. Finally, it is suggested that a modification of the boundary conditions which are usually enforced in the dynamical theory when solving the propagation equation could improve its accuracy and extend the limits of validity of the two-wave approximation.