A high-order perturbation of envelopes (HOPE) method for scattering by periodic inhomogeneous media

The interaction of linear waves with periodic structures arises in a broad range of scientific and engineering applications. For such problems it is often mandatory that numerical simulations be rapid, robust, and highly accurate. With such qualities in mind High-Order Spectral methods are often utilized, and in this paper we describe and test a perturbative method which fits into this class. Here we view the inhomogeneous (but laterally periodic) permittivity as a perturbation of a constant value and pursue (regular) perturbation theory. We demonstrate that not only does this lead to a fast and accurate numerical method, but also that the expansion of the field in this geometric parameter is valid for large deformations (up to topological obstruction). Finally, we show that, if the permittivity deformation is spatially analytic, then so is the field scattered by it.