1204.1518 (Hoai-Minh Nguyen)

A study of negative index materials using transformation optics with
applications to super lenses, cloaking, and illusion optics: the scalar case    [PDF]

Hoai-Minh Nguyen

This paper is devoted to the study of the behavior of the solution $u_\delta \in H^1_0(\Omega)$, as $\delta$ goes to 0, to the equation \dive(\eps_\delta A \nabla u) + k^2 \eps_0 \Sigma u = \eps_0 f in \Omega, where $\Omega$ is a smooth open subset of $\mR^d$ with $d=2$ or 3, $f \in L^2(\Omega)$, $k$ is a non-negative constant, $A$ is a uniformly elliptic matrix function, $\Sigma$ is a real function bounded above and below by positive constants, and $\eps_\delta$ is a complex function whose the real part takes the value 1 and -1, and the imaginary part is positive and converges to 0 as $\delta$ goes to 0. Under some additional general assumptions on $A$ and $\Sigma$, we characterize conditions on $f$ under which $\|u_\delta\|_{H^1(\Omega)}$ remains bounded as $\delta$ goes to 0. Under these conditions, we also show that $u_\delta$ converges weakly in $H^1(\Omega)$ to a limit which is the solution to the limit equation; moreover, we obtain a formula for computing the limit. The applications of these results for perfect lens, cloaking, and illusion optics using negative index materials will be given.

View original: http://arxiv.org/abs/1204.1518