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 $\Omega\subset \mathbb R^d$,andconstantin$\mathbb R^d \setminus \Omega$. The solution of the time-dependent wave equation with initial data, which is localized in $\mathrm{\Omega }$, 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 $\mathrm{\infty }$. Specifically, the rate of energy escape from $\mathrm{\Omega }$ 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 $n_{\ast }(x)$ 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 $x \in \Omega$,i\mathrm{.}e\mathrm{.},$0 < n_- \leq n(x) \leq n_+ < \infty$. We formulate this problem as a constrained optimization problem and prove that an optimal structure, $n_{\ast }(x)$ exists. Furthermore, $n_{\ast }(x)$ is piecewise constant and achieves the material bounds, i.e., $n_{\ast }(x)\in n_{-},n_{+}$. In one dimension, we establish a connection between $n_{\ast }(x)$ and the well-known class of Bragg structures, where $n(x)$ is constant on intervals whose length is one-quarter of the effective wavelength.