Exact analytical approaches for elliptic cylindrical invisibility cloaks

Elliptic cylindrical cloaks are investigated analytically in elliptic cylindrical coordinates. It is shown that both the linear and higher-order transformations produce an imperfect cloaking due to the poor symmetry of the coordinate system. The imperfection being in concordance with that obtained by numerical simulations, it cannot be eliminated by improving the computer techniques. The cloaking becomes almost perfect in the limit case of nearly circular cloaks with the advantage that none of the parameters is singular in the cloak shell. In circular cylindrical coordinates instead, a perfect cloaking is achieved with elliptic cloaks, as has been proven in other studies by numerical simulations. Analytic solutions to Maxwell's equations are provided in these coordinates only in the limit case of nearly circular cloaks. Simple general expressions for material parameters are given also in circular cylindrical coordinates that can be then transformed in Cartesian coordinates. (C) 2009 Optical Society of America