Metamaterial cloaking

Remarks

Example: the spherical cloak of Pendry, Schurig and Smith. Let $0 < R_1 < R_2$, let $D$ be the ball $\abs{y} < R_1$ and $\Omega$ the ball $\abs{y} < R_2$. The radial map $r' = R_1 + r (R_2 - R_1)/R_2$ sends the punctured ball $0 < \abs{x} < R_2$ onto the shell $R_1 < \abs{y} < R_2$, and it is the identity on the sphere $\abs{x} = R_2$. By transformation optics, vacuum becomes the medium with, in spherical coordinates and relative to vacuum, $$\varepsilon_r = \mu_r = \frac{R_2}{R_2 - R_1} \frac{(r' - R_1)^2}{r'^2}, \qquad \varepsilon_\theta = \varepsilon_\phi = \mu_\theta = \mu_\phi = \frac{R_2}{R_2 - R_1}.$$ Outside $\Omega$ the map is the identity, so fields there are those of empty space, and waves never reach $D$. The radial parameters vanish at $r' = R_1$, so the medium is singular; the claim that it cloaks is made precise with weak solutions, or by a limit of regular approximate cloaks.

Non-example: a perfectly conducting shell hides what is inside it, but its scattered field outside $\Omega$ is not zero, since it scatters the incident wave like any metal ball.

The same construction for electrical impedance tomography was given by Greenleaf, Lassas and Uhlmann.