Abstract : |
We consider a waveguide, with one inlet and one outlet, and some arbitrary perturbation in between. In general, an ingoing wave in the inlet will produce a reflected wave, due to interaction with the perturbation. Our objective is to give an answer to the following important questions: what are the frequencies at which the transmission is the best one? And in particular, do they exist frequencies for which the transmission is perfect, in the sense that nothing is propagating back in the inlet?
Our approach relies on a simple idea, which consists in using a complex scaling in an original manner: while the same stretching parameter is classically used in the inlet and the outlet, here we take them as two complex conjugated parameters. As a result, we select ingoing waves in the inlet and outgoing waves in the outlet, which is exactly what arises when the transmission is perfect. This simple idea works very well, and provides useful information on the transmission qualities of the system, much faster than any traditional approach. More precisely, we define a new complex spectrum which contains as real eigenvalues both the frequencies where perfect transmission occurs and the frequencies corresponding to trapped modes (also known as bound states in the continuum). In addition, we also obtain complex eigenfrequencies which can be exploited to predict frequency ranges of good transmission. Let us finally mention that this new spectral problem is PT -symmetric for systems with mirror symmetry.
Several illustrations performed with finite elements in several simple 2D cases will be shown.
It is a common work with Lucas Chesnel (INRIA) and Vincent Pagneux (CNRS). |