# Three-dimensional quasi-periodic shifted Green function throughout the spectrum--including Wood anomalies

## TL;DR

This paper introduces a novel Green function method for efficiently solving wave scattering problems involving doubly periodic gratings at and near Wood anomaly frequencies, overcoming previous convergence issues.

## Contribution

It presents a modified quasi-periodic Green function that ensures convergence and solvability at Wood anomalies, along with an efficient high-order numerical solver for these challenging problems.

## Key findings

- Successfully solves scattering problems at Wood frequencies.
- Achieves algebraic convergence with shifted Green functions.
- Demonstrates applicability to acoustic scattering simulations.

## Abstract

This work presents an efficient method for evaluation of wave scattering by doubly periodic diffraction gratings at or near "Wood anomaly frequencies". At these frequencies, one or more grazing Rayleigh waves exist, and the lattice sum for the quasi-periodic Green function ceases to exist. We present a modification of this sum by adding two types of terms to it. The first type adds linear combinations of "shifted" Green functions, ensuring that the spatial singularities introduced by these terms are located below the grating and therefore outside of the physical domain. With suitable coefficient choices these terms annihilate the growing contributions in the original lattice sum and yield algebraic convergence. Convergence of arbitrarily high order can be obtained by including sufficiently many shifts. The second type of added terms are quasi-periodic plane wave solutions of the Helmholtz equation which reinstate certain necessary grazing modes without leading to blow-up at Wood anomalies. Using the new quasi-periodic Green function, we establish, for the first time, that the Dirichlet problem of scattering by a smooth doubly periodic scattering surface at a Wood frequency is uniquely solvable. We also present an efficient high-order numerical method based on the this new Green function for the problem of scattering by doubly periodic three-dimensional surfaces at and around Wood frequencies. We believe this is the first solver in existence that is applicable to Wood-frequency doubly periodic scattering problems. We demonstrate the proposed approach by means of applications to problems of acoustic scattering by doubly periodic gratings at various frequencies, including frequencies away from, at, and near Wood anomalies.