Active Exterior Cloaking for the 2D Laplace and Helmholtz Equations

A new cloaking method is presented for 2D quasistatics and the 2D Helmholtz equation that we speculate extends to other linear wave equations. For 2D quasistatics it is proven how a single active exterior cloaking device can be used to shield an object from surrounding fields, yet produce very small scattered fields. The problem is reduced to finding a polynomial which is close to 1 in a disk and close to 0 in another disk, and such a polynomial is constructed. For the 2D Helmholtz equation it is numerically shown that three exterior cloaking devices placed around the object suffice to hide it.

Figure 1

Magnitude of the Hermite polynomial $h(z)$ for ${\delta }_{*}=1$ and degree $n=10$. The color scale is logarithmic from 0.01 (dark blue) to 100 (dark red).

Figure 2

Real part of the total field with the quasistatic cloaking device (a) inactive and (b) active in the $s$ plane. The color scale is linear from $-10$ (dark blue) to 10 (dark red) and the scatterer appears in black.

Figure 3

Real part of the total field with the Helmholtz cloaking devices (a) inactive and (b) active. The color scale is linear from $-1$ (dark blue) to 1 (dark red) and the axis units are $\lambda$.