Waveguides

This is the third of three conversation recorded during the
Conference on mathematics of wave phenomena 23-27 July 2018 in
Karlsruhe.

Gudrun is in conversation with Anne-Sophie Bonnet-BenDhia from
ENSTA in Paris about transmission properties in perturbed
waveguides.


The spectral theory is essential to study wave phenomena. For
instance, everybody has experimented with resonating frequencies
in a bathtube filled with water. These resonant eigenfrequencies
are eigenvalues of some operator which models the flow behaviour
of the water. Eigenvalue problems are better known for matrices.
For wave problems, we have to study eigenvalue problems in
infinite dimension. Like the eigenvalues for a finite dimensional
matrix the Spectral theory gives access to intrinisic properties
of the operator and the corresponding wave phenomena.


Anne-Sophie is interested in waveguides. For example, optical
fibres can guide optical waves while wind instruments are guides
for acoustic waves. Electromagnetic waveguides also have
important applications.

A practical objective is to optimize the transmission in a
waveguide, even if there are some perturbations inside. It is
known that for certain frequencies, there is no reflection by the
perturbations but it is not apriori clear how to find these
frequencies.


Anne-Sophie uses complex analysis for that. The idea is to
complexify the (originally real) coordinates by analytic
extension. It is a classic idea for resonances that she adapts to
the problem of transmission.

This mathematical method of complex scaling is linked to the
method of perfectly matched layers in numerics. It is used to
solve problems set in unbounded domains on a computer by finite
elements. Thanks to the complex scaling, she can solve a problem
in a bounded domain, which reproduces the same behaviour as in
the infinite domain.


Finally, Anne-Sophie is able to get numerically a complex
spectrum of frequencies, related to the quality of the
transmission in a perturbed waveguide. The imaginary part of the
complex quantity gives an indication of the quality of the
transmission in the waveguide. The closer to the real axis the
better the transmission.



References

A-S. Bonnet-BenDhia, L. Chesnel and V. Pagneux:Trapped modes
and reflectionless modes as eigenfunctions of the same spectral
problem Proceedings of the Royal Society A, 2018, doi
10.1098/rspa.2018.0050

A-S. Bonnet-BenDhia: Mathematical and numerical treatment of
plasmonic waves at corners of metals and metamaterials Emerging
Topics in Optics, IMA, Minneapolis, 2017

A-S. Bonnet-BenDhia, L. Chesnel and S. Nazarov: Perfect
transmission invisibility for waveguides with sound hard walls
Journal de Mathématiques Pures et Appliquées, 2017, doi
10.1016/j.matpur.2017.07.020

A.-S. Bonnet-BenDhia e.a.: A method to build non-scattering
perturbations of two-dimensional acoustic waveguides Math. meth.
appl. sci., vol. 40, pp. 335–349, 2015 doi 10.1002/mma.3447




Podcasts

S. Fliss, G. Thäter: Transparent Boundaries. Conversation in
the Modellansatz Podcast episode 75, Department of Mathematics,
Karlsruhe Institute of Technology (KIT), 2015.

M. Kray, G. Thäter: Splitting Waves. Conversation in the
Modellansatz Podcast episode 62, Department of Mathematics,
Karlsruhe Institute of Technology (KIT), 2015.

F. Sayas, G. Thäter: Acoustic scattering. Conversation in the
Modellansatz Podcast episode 58, Department of Mathematics,
Karlsruhe Institute of Technology (KIT), 2015.