Characteristic wave velocities in spherical electromagnetic cloaks

Abstract

We investigate the characteristic wave velocities in spherical electromagnetic cloaks, namely, phase, ray, group and energy-transport velocities. After deriving explicit expressions for the phase and ray velocities (the latter defined as the phase velocity along the direction of the Poynting vector), special attention is given to the determination of group and energy-transport velocities, because a cursory application of conventional formulae for local group and energy-transport velocities can lead to a discrepancy between these velocities if the permittivity and permeability dyadics are not equal over a frequency range about the center frequency. In contrast, a general theorem can be proven from Maxwell's equations that the local group and energy-transport velocities are equal in linear, lossless, frequency dispersive, source-free bianisotropic material. This apparent paradox is explained by showing that the local fields of the spherical cloak uncouple into an E wave and an H wave, each with its own group and energy-transport velocities, and that the group and energy-transport velocities of either the E wave or the H wave are equal and thus satisfy the general theorem.