# Wave scattering by a periodic perturbation: embedded Rayleigh-Bloch modes and resonances

## TL;DR

This paper investigates wave scattering in a 2D Helmholtz system with a periodic perturbation, deriving resonance formulas, identifying conditions for total transmission or reflection, and explicitly calculating embedded Rayleigh-Bloch modes.

## Contribution

It provides explicit formulas for resonances and embedded modes in a perturbed Helmholtz equation with periodic structures, advancing understanding of wave behavior near resonances.

## Key findings

- Formulas of Breit-Wigner and Fano type near resonances
- Conditions for total transmission and reflection
- Explicit calculation of embedded Rayleigh-Bloch modes

## Abstract

The scattering of quasiperiodic waves for a two-dimensional Helmholtz equation with a constant refractive index perturbed by a function which is periodic in one direction and of finite support in the other is considered. The scattering problem is uniquely solvable for almost all frequencies and formulas of Breit-Wigner and Fano type for the reflection and transmission coefficients are obtained in a neighborhood of the resonance (a pole of the reflection coefficient). We indicate also the values of the parameters involved which provide total transmission and reflection. For some exceptional frequencies and perturbations (when the imaginary part of the resonance vanishes) the scattering problem is not uniquely solvable and in the latter case there exist embedded Rayleigh-Bloch modes whose frequencies are explicitly calculated in terms of infinite convergent series in powers of the small parameter characterizing the magnitude of the perturbation.