# Long-lived Scattering Resonances and Bragg Structures

We consider a system governed by the wave equation with index of refraction $nx$, taken to be variable within a bounded region $\Omega\subset \mathbb R^d$, and constant in $\mathbb R^d \setminus \Omega$. 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-selfadjoint eigenvalue problem governing the time-harmonic solutions of the wave Helmholtz equation which are outgoing at $\infty$. 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 $n *x$ 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 $nx-1$ and pointwise upper and lower material bounds on $nx$ for $x \in \Omega$, i.e., $0 n - \leq nx \leq n + \infty$. We formulate this problem as a constrained optimization problem and prove that an optimal structure, $n *x$ exists. Furthermore, $n *x$ is piecewise constant and achieves the material bounds, i.e., $n *x \in {n -, n +}$. In one dimension, we establish a connection between $n *x$ and the well-known class of Bragg structures, where $nx$ is constant on intervals whose length is one-quarter of the effective wavelength.

Author: Braxton Osting; Michael I. Weinstein

Source: https://archive.org/