Long-Lived Scattering Resonances and Bragg Structures
- Autores: Braxton Osting, Michael I. Weinstein
- Localización: Siam journal on applied mathematics, ISSN 0036-1399, Vol. 73, Nº 2, 2013, págs. 827-852
- Idioma: inglés
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Resumen
We consider a system governed by the wave equation with index of refraction $n({\bf x})$, 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--self-adjoint 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 $n_\star({\bf x})$ 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 $n({\bf x})-1$ and pointwise upper and lower (material) bounds on $n({\bf x})$ for ${\bf x} \in \Omega$, i.e., $0 < n_- \le n({\bf x}) \le n_+ < \infty$. We formulate this problem as a constrained optimization problem and prove that an optimal structure, $n_\star({\bf x})$, exists. Furthermore, $n_\star({\bf x})$ is piecewise constant and achieves the material bounds, i.e., $n_\star({\bf x})\in\{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
