# From zero transmission to trapped modes in waveguides

## TL;DR

This paper develops methods to design 2D waveguide geometries that achieve zero transmission and support trapped modes, using the unitarity of the scattering matrix and augmented scattering matrix, with numerical validation.

## Contribution

It introduces a novel approach to construct waveguide geometries with zero transmission without symmetry and to support trapped modes embedded in the continuous spectrum.

## Key findings

- Constructed geometries with zero transmission without symmetry.
- Built geometries supporting trapped modes with embedded eigenvalues.
- Validated results through numerical simulations.

## Abstract

We consider the time-harmonic scattering wave problem in a 2D waveguide at wavenumber $k$ such that one mode is propagating in the far field. In a first step, for a given $k$, playing with one scattering branch of finite length, we demonstrate how to construct geometries with zero transmission. The main novelty in this result is that the symmetry of the geometry is not needed: the proof relies on the unitary structure of the scattering matrix. Then, in a second step, from a waveguide with zero transmission, we show how to build geometries supporting trapped modes associated with eigenvalues embedded in the continuous spectrum. For this second construction, using the augmented scattering matrix and its unitarity, we play both with the geometry and the wavenumber. Finally, the mathematical analysis is supplemented by numerical illustrations of the results.