Communications in Mathematical Sciences

Volume 16 (2018)

Number 7

Non reflection and perfect reflection via Fano resonance in waveguides

Pages: 1779 – 1800



Lucas Chesnel (INRIA / Centre de mathématiques appliquées, École Polytechnique, Université Paris-Saclay, Palaiseau, France)

Sergei A. Nazarov (St. Petersburg State University, St. Petersburg, Russia)


We investigate a time-harmonic wave problem in a waveguide. By means of asymptotic analysis techniques, we justify the so-called Fano resonance phenomenon. More precisely, we show that the scattering matrix considered as a function of a geometrical parameter $\varepsilon$ and of the frequency $\lambda$ is in general not continuous at a point $(\varepsilon, \lambda) = (0, \lambda^0)$ where trapped modes exist. In particular, we prove that for a given $\varepsilon \neq 0$ small, the scattering matrix exhibits a rapid change for frequencies varying in a neighbourhood of $\lambda^0$. We use this property to construct examples of waveguides such that the energy of an incident wave propagating through the structure is perfectly transmitted (non reflection) or perfectly reflected in monomode regime. We provide numerical results to illustrate our theorems.


waveguides, Fano resonance, non reflection, perfect reflection, scattering matrix

2010 Mathematics Subject Classification

35J05, 35Q60, 65N21, 78A40, 78A46

Received 26 January 2018

Accepted 28 July 2018

Published 7 March 2019