In an electromagnetic cloak based on a transformation approach, Reduced sets of
material properties are generally favored due to their easier implementation in reality,
although a seemingly inevitable drawback of undesired reflection exists in such cloaks.
Here we suggest using high-order transformations to create smooth moduli at the outer
boundary of the cloak, therefore completely eliminating the detrimental scattering
within the limit of geometric optics. We apply this scheme to a non-magnetic
cylindrical cloak and demonstrate that the scattered field is Reduced substantially in a
cloak with optimal quadratic transformation as compared to its linear counterpart.
