Approximate Cloaking via Transformation Optics for Electromagnetic Waves

Cloaking via transformation optics was introduced by Pendry, Schurig, and Smith for the Maxwell system and Leonhardt in the geometric optics setting. They used a singular change of variables which blows up a point into a cloaked region. The same transformation had been used by Greenleaf, Lassas, and Uhlmann in an inverse context. This singular structure implies difficulties not only in practice but also in analysis. To avoid using the singular structure, regularized schemes have been proposed. One of them was suggested by Kohn, Shen, Vogelius, and Weinstein for which they used a transformation which blows up a small ball instead of a point into the cloaked region. In this thesis, we study the approximate cloaking via transformation optics for electromagnetic waves in both the time harmonic regime and time regime. In the time-harmonic regime, the cloaking device only consists of a layer constructed by the mapping technique, no (damping) lossy-layer is required. Due to the fact that nolossy layer is required, resonance might appear. The analysis is therefore delicate and the phenomena are complex. In particular, we show that the energy can blow up inside the cloaked region in the resonant case and/whereas cloaking is achieved in both non-resonant and resonant cases. Moreover, the degree of visibility 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 the time regime, the cloaking device also consists of a fixed lossy layer. Our approach is based on estimates on the degree of visibility in the frequency domain for all frequency in which the frequency dependence is explicit. The difficulty and the novelty in the analysis are in the low and high frequency regimes. To this end, we implement the variational technique in low frequency and the multiplier and duality techniques in high frequency domain. The first part of the thesis is inspired by the work of Nguyen and the second part by the work of Nguyen and Vogelius on the wave equation.