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

Jul 16, 2026

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This work investigates scattering of time-harmonic electromagnetic waves by periodic layered gratings modeled with the 2D Helmholtz equation. The obstacle may include penetrable or impenetrable regions and consists of finitely many stacked layers. A single periodic cell is treated via a quasi-periodic boundary value problem with quasi-periodic boundary conditions. Radiation in vertical directions is enforced using Dirichlet-to-Neumann operators. A T-matrix framework decomposes the global problem into layer problems using quasi-periodic modal expansions. Each layer’s T-matrix is numerically approximated with a plane-wave Trefftz discontinuous Galerkin method, yielding linear complexity in the number of layers and enabling scalable coupling of local responses.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages25

Reading Time1 h 3 min

Introduction

Helmholtz quasi-periodic boundary value problems and DtN operators

Layer formulation and local boundary value problems

T-matrix coupling for the quasi-periodic layered setting

DtN-TDG method to approximate layer T-matrices

Numerical results and comparisons

arXiv :2607 .06475v1 [math .NA] 7 Jul 2026 Trefftz DG Approximation of the T-Matrix for Scattering by Periodic Layered Structures Armando Maria Monforte∗, Andrea Moiola†, Simone Zanotto‡ July 8, 2026 Abstract 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 posedon 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. Keywords: Diffraction grating, Quasi-periodic, Helmholtz equation, T-matrix, Discontinuous Galerkin, Trefftz method, Plane wave basis Mathematics Subject Classification (2020): 65N30, 35J05, 35Q60, 78A45, 78M10 1 Introduction The scattering of time-harmonic electromagnetic waves by periodic structures, known as diffraction gratings, arises in a wide range of applications in photonics, optics, and metamaterial design [2, 12] . In the two-dimensional transverse-electric setting, Maxwell’s equations reduce to the Helmholtz equation with piecewise-constant, periodic relative permittivity ε . Classical numerical approximations of such diffraction gratings reduce the computational domain toa single periodic cell using quasi-periodic boundary conditions and Dirichlet-to-Neumann operators [3, 5 , 14] . To curb the computational cost of discretizing large and complex cells, we borrow the Tmatrix technique from multiple-scattering theory [17, 18 , 23] . To approximate the scattering by many obstacles, each of these is associated to a so-called T-matrix, which maps the ∗ Department of Mathematics, University of Pavia, Via Ferrata 5, Pavia, Italy (armandomaria.monforte01 @universitadipavia.it), ORCID: 0009-0000-7687-2217 †Department of Mathematics, University of Pavia, Via Ferrata 5, Pavia, Italy ([andrea.moiola@unipv.it](andrea.moiola@unipv.it)), ORCID: 0000-0002-6251-4440 ‡Istituto Nanoscienze – CNR, NEST-SNS, Piazza San Silvestro 12, Pisa, Italy ([simone.zanotto@nano.cnr.it](simone.zanotto@nano.cnr.it)), ORCID: 0000-0001-7180-3335 Fourier coefficients of any wave impinging on the obstacle into the Fourier coefficients of the corresponding scattered field. The size of a T-matrix only depends on the wavenumber and the obstacle size, but is independent of the obstacle shape and material properties. Multiple scattering problems coupling all obstacles are solved from a linear system built from the T-matrices of each obstacle, leading to considerable savings in the computational effort. We adapt this idea to the quasi-periodic setting. First, the domain Ω is split in N layers Ωj , which may contain inhomogeneities and impenetrable regions. Second, a T-matrix Tj , mapping incoming to outgoing fields, is computed for each layer, exploiting the natural Fourier expansion dictated by quasi-periodicity. Third, these layer T-matrices are coupled through propagation matrices into a global banded block linear system. A key feature of this formulation is the decoupling of scales: the global system is N × N block tridiagonal, with (4M + 2) × (4M + 2)-sized blocks, where M is the truncation

FAQs

What physical and mathematical model is used for the scattering problem?

Time-harmonic electromagnetic wave scattering is modeled through the 2D Helmholtz equation for a periodic layered grating. The periodic cell is governed by a quasi-periodic boundary value problem with Helmholtz-compatible radiation handling via DtN operators.

How is the radiation condition implemented in the formulation?

Radiation in the vertical directions is imposed using Dirichlet-to-Neumann (DtN) operators. These operators incorporate the influence of the exterior into the boundary value problem defined on the periodic cell.

How does the proposed T-matrix approach improve computational efficiency across layers?

The global scattering problem is decomposed into layer-wise boundary value problems and each layer is represented by a T-matrix mapping incoming to outgoing quasi-periodic modal expansions. Coupling yields a block tridiagonal system whose size depends on the truncation order of the DtN operators and the number of layers, enabling linear complexity in the number of layers.

Jul 16, 2026

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