Perfect cloaking of ellipsoidal regions

Most of the existing explicit forms of cloaking devices concern spherical regions which exhibit the following behavior. If the cloaking region comes from blowing up a point we achieve perfect cloaking, but the conductivity tensor becomes singular and therefore hard to realize. A nonsingular conductivity tensor can be achieved by blowing up a small sphere, but then the cloaking is not perfect, since the interior field is controlled by the square of the radius of the initial small sphere. This behavior reflects the highly focusing effect of the spherical system where the 2-D manifold of a sphere degenerates to the 0-D manifold of its center as the radius diminishes to zero. In the present work, we demonstrate a cloaking region in the shape of an ellipsoid which achieves perfect interior cloaking and at the same time preserves the regularity of the conductivity tensor. This is possible since the confocal ellipsoidal system does not exhibit any loss of dimensionality as the ellipsoid collapses down to its focal ellipse. This is another example where high symmetry has a high price to pay. Furthermore, from the practical point of view, the most “economical” cloaking structure, for three dimensional objects, is provided by the ellipsoid which has three degrees of freedom and therefore can best fit any reasonable geometrical object. Cloaks in the shape of a prolate or an oblate spheroid, an almost disk, an almost needle, as well as a sphere are all cases of ellipsoidal degeneracies.