Long-lived Scattering Resonances and Bragg Structures

TL;DR

This paper investigates the design of refractive index profiles to minimize energy decay in wave systems, proving the existence of optimal structures and linking one-dimensional solutions to classical Bragg structures.

Contribution

It formulates and proves the existence of optimal refractive index profiles with minimal resonance decay, showing they are piecewise constant and relate to Bragg structures in 1D.

Findings

Optimal structures exist within the specified bounds.

These structures are piecewise constant and attain material bounds.

In 1D, optimal profiles relate to Bragg structures with quarter-wavelength intervals.

Abstract

We consider a system governed by the wave equation with index of refraction $n(x)$, taken to be variable within a bounded region $\Omega\subset \mathbb R^d$, 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 $n(x)-1$ and pointwise upper and lower (material) bounds on $n(x)$ for $x \in \Omega$, i.e., $0 < n_- \leq n(x) \leq n_+ < \infty$. 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 $n(x)$ is constant on intervals whose length is one-quarter of the effective wavelength.