Trefftz DG Approximation of the T-Matrix for Scattering by Periodic Layered Structures

We study the scattering of time-harmonic electromagnetic waves by periodic layered gratings, modelled by the 2D Helmholtz equation. The periodic obstacle may include penetrable and impenetrable regions, and consists of a finite number of stacked layers. The boundary value problem is formulated on a single periodic cell using quasi-periodic boundary conditions. The radiation condition in the vertical directions is imposed through Dirichlet-to-Neumann (DtN) operators. To efficiently treat multilayer configurations, we adopt a formulation based on the T-matrix method. The global scattering problem is decomposed into boundary value problems posed on individual layers. On the layer boundaries, the field is expressed in terms of quasi-periodic modal expansions, and the layer T-matrix describes the map between incoming and outgoing wave modes. Each local T-matrix is approximated numerically using a plane-wave based Trefftz Discontinuous Galerkin (TDG) method, which provides an efficient discretization of the layer scattering response. The T-matrix technique leads to linear computational complexity in the number of layers in the grating.