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

• Published in: Zeitschrift für angewandte Mathematik und Physik Vol. 71; no. 3
• Main Authors: Chesnel, Lucas, Nazarov, Sergei A.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-06-2020; Springer Nature B.V; Springer Verlag
• Subjects: Analysis of PDEs; Backscattering; Engineering; Fano resonance; Mathematical Methods in Physics; Mathematical Physics; Mathematics; Symmetry; Theoretical and Applied Mechanics; Waveguides; Zero transmission; Fano resonance; 78A46; Waveguides; 35J05; 65N21; Scattering matrix; 78A40; 35Q60; scattering matrix; zero transmission
• ISSN: 0044-2275; 1420-9039
• DOI: 10.1007/s00033-020-01305-9

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.