LONG-LIVED SCATTERING RESONANCES AND BRAGG STRUCTURES

We consider a system governed by the wave equation with index of refraction n(x), taken to be variable within a bounded region Omega subset of R-d and constant in R-d \ Omega. The solution of the time-dependent wave equation with initial data, which is localized in Omega, spreads and decays with advancing time. This rate of decay can be measured (for d = 1, 3, and more generally, d odd) in terms of the eigenvalues of the scattering resonance problem, a non-self-adjoint eigenvalue problem governing the time-harmonic solutions of the wave (Helmholtz) equation which are outgoing at infinity. Specifically, the rate of energy escape from Omega is governed by the complex scattering eigenfrequency, which is closest to the real axis. We study the structural design problem: Find a refractive index profile n(star)(x) within an admissible class which has a scattering frequency with minimal imaginary part. The admissible class is defined in terms of the compact support of n(x) - 1 and pointwise upper and lower (material) bounds on n(x) for x is an element of Omega, i.e., 0 < n(-) <= n(x) <= n(+) < infinity. We formulate this problem as a constrained optimization problem and prove that an optimal structure, n(star)(x), exists. Furthermore, n(star)(x) is piecewise constant and achieves the material bounds, i.e., n(star)(x) is an element of {n(-), n(+)}. In one dimension, we establish a connection between n(star)(x) and the well-known class of Bragg structures, where n(x) is constant on intervals whose length is one quarter of the effective wavelength.