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- W2016507978 abstract "We consider a system governed by the wave equation with index of refraction $n({bf x})$, taken to be variable within a bounded region $Omegasubset mathbb R^d$ 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." @default.
- W2016507978 created "2016-06-24" @default.
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- W2016507978 date "2013-01-01" @default.
- W2016507978 modified "2023-10-18" @default.
- W2016507978 title "Long-Lived Scattering Resonances and Bragg Structures" @default.
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- W2016507978 doi "https://doi.org/10.1137/110856228" @default.
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