An efficient and stable spectral method for electromagnetic scattering from a layered periodic structure

The scattering of acoustic and electromagnetic waves by periodic structures plays an important role in a wide range of problems of scientific and technological interest. This contribution focuses upon the stable and high-order numerical simulation of the interaction of time-harmonic electromagnetic waves incident upon a periodic doubly layered dielectric media with sharp, irregular interface. We describe a boundary perturbation method for this problem which avoids not only the need for specialized quadrature rules but also the dense linear systems characteristic of boundary integral/element methods. Additionally, it is a provably stable algorithm as opposed to other boundary perturbation approaches such as Bruno and Reitich's ''method of field expansions'' or Milder's ''method of operator expansions''. Our spectrally accurate approach is a natural extension of the ''method of transformed field expansions'' originally described by Nicholls and Reitich (and later refined to other geometries by the authors) in the single-layer case.