# Exact zero transmission during the Fano resonance phenomenon in non symmetric waveguides

## TL;DR

This paper proves that in non-symmetric waveguides, there is always a real frequency at which the transmission coefficient is zero, meaning complete backscattering occurs, regardless of symmetry assumptions.

## Contribution

It demonstrates that zero transmission can occur in non-symmetric waveguides, extending previous results that required symmetry for such phenomena.

## Key findings

- Existence of a real frequency with zero transmission in non-symmetric waveguides
- Numerical validation of the theoretical results
- Energy is fully backscattered at the zero transmission frequency

## Abstract

We investigate a time-harmonic wave problem in a waveguide. We work at low frequency so that only one mode can propagate. It is known that the scattering matrix exhibits a rapid variation for real frequencies in a vicinity of a complex resonance located close to the real axis. This is the so-called Fano resonance phenomenon. And when the geometry presents certain properties of symmetry, there are two different real frequencies such that we have either $R=0$ or $T=0$, where $R$ and $T$ denote the reflection and transmission coefficients. In this work, we prove that without the assumption of symmetry of the geometry, quite surprisingly, there is always one real frequency for which we have $T=0$. In this situation, all the energy sent in the waveguide is backscattered. However in general, we do not have $R=0$ in the process. We provide numerical results to illustrate our theorems.