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Register a new userApproximate cloaking using transformation optics for acoustic and electromagnetic waves
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This is a survey of approximate cloaking using transformation optics for acoustic and electromagnetic waves.
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We study the approximate cloaking via transformation optics for electromagnetic waves in the time harmonic regime in which the cloaking device {it only} consists of a layer constructed by the mapping technique. Due to the fact that no-lossy layer is required, resonance might appear and the analysis is delicate. We analyse both non-resonant and resonant cases. In particular, we show that the energy can blow up inside the cloaked region in the resonant case and/whereas cloaking is {it achieved} in {it both} cases. Moreover, the degree of visibility {it depends} on the compatibility of the source inside the cloaked region and the system. These facts are new and distinct from known mathematical results in the literature.
In this paper, we establish approximate cloaking for the heat equation via transformation optics. We show that the degree of visibility is of the order $epsilon$ in three dimensions and $|lnepsilon|^{-1}$ in two dimensions, where $epsilon$ is the regularization parameter.
We study approximate cloaking using transformation optics for electromagnetic waves in the time domain. Our approach is based on estimates of the degree of visibility in the frequency domain for all frequencies in which the frequency dependence is explicit. The difficulty and the novelty analysis parts are in the low and high frequency regimes. To this end, we implement a variational technique in the low frequency domain, and multiplier and duality techniques in the high frequency domain. Our approach is inspired by the work of Nguyen and Vogelius on the wave equation.
We consider the propagation of acoustic waves in a 2D waveguide unbounded in one direction and containing a compact obstacle. The wavenumber is fixed so that only one mode can propagate. The goal of this work is to propose a method to cloak the obstacle. More precisely, we add to the geometry thin outer resonators of width $varepsilon$ and we explain how to choose their positions as well as their lengths to get a transmission coefficient approximately equal to one as if there were no obstacle. In the process we also investigate several related problems. In particular, we explain how to get zero transmission and how to design phase shifters. The approach is based on asymptotic analysis in presence of thin resonators. An essential point is that we work around resonance lengths of the resonators. This allows us to obtain effects of order one with geometrical perturbations of width $varepsilon$. Various numerical experiments illustrate the theory.
We study the invisibility via anomalous localized resonance of a general source for electromagnetic waves in the setting of doubly complementary media. As a result, we show that cloaking is achieved if the power is blown up. We also reveal a critical length for the invisibility of a source that occurs when the plasmonic structure is complementary to an annulus of constant, isotropic medium.
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