High frequency limit of Helmholtz equations: refraction by sharp interfaces

Abstract

In this paper, we compute the high frequency limit of the Helmholtz equation with source term, in the case of a refraction index that is discontinuous along a sharp interface between two unbounded media. The asymptotic propagation of energy is studied using Wigner measures. Our result is twofold. First, in the general case, assuming some geometrical hypotheses on the index and assuming that the interface does not capture energy asymptotically, we prove that the limiting Wigner measure satisfies a stationary transport equation with source term. As a consequence, the Wigner measure is characterized as the integral, along the rays of geometrical optics and up to infinite time, of the energy source. This result encodes the refraction phenomenon. Second, we study the particular case when the index is constant in each media, for which the analysis goes further: we prove that the interface does not capture energy asymptotically in this case.