Computer derivation of the optical transformation in cylindrical and spherical coordinates

We present a symbolic analysis on the derivation of optical transformation in cylindrical and spherical coordinates using computer algebra system (CAS). In the design of electromagnetic cloak or mask, for symmetric and simple shapes, we can easily and directly obtain their analytical solutions. However, when we deal with arbitrary shapes, we must resort to numerical analysis methods (FDTD or FEM, etc.), in which it is inevitable to discretize the region into mesh. Since the formulas in Cartesian coordinate are available already, it is necessary to generalize them to cylindrical and spherical systems. As expected, they are finally given using computer algebra system-Maple.