The scattering of time-harmonic linear waves by periodic media arises in a wide array of applications from materials science and nondestructive testing to remote sensing and oceanography. In this work we have in mind applications in optics, more specifically plasmonics, and the surface plasmon polaritons that are at the heart of remarkable phenomena such as extraordinary optical transmission, surface-enhanced Raman scattering, and surface plasmon resonance biosensing. In this paper we develop robust, highly accurate, and extremely rapid numerical solvers for approximating solutions to grating scattering problems in the frequency regime where these are commonly used. For piecewise-constant dielectric constants, which are commonplace in these applications, surface formulations are clearly advantaged as they posit unknowns supported solely at the material interfaces. The algorithms we develop here are high-order perturbation of surfaces methods and generalize previous approaches to take advantage of the fact that these algorithms can be significantly accelerated when some or all of the interfaces are trivial (flat). More specifically, for configurations with one nontrivial interface (and one trivial interface) we describe an algorithm that has the same computational complexity as a two-layer solver. With numerical simulations and comparisons with experimental data, we demonstrate the speed, accuracy, and applicability of our new algorithms.
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