Time delays and energy transport velocities in three dimensional ideal cloaking devices

We obtain the energy transport velocity distribution for a three dimensional ideal cloak explicitly. Near the operation frequency, the energy transport velocity has a rather peculiar distribution. The transport velocity along the direction pointing at the center of the cloak is found to be a constant, while the transport velocity approaches zero at the inner boundary of the cloak. A ray pointing right into the center of the cloak will experience abrupt changes in velocities when it impinges on the inner surface of the cloak. This peculiar distribution causes long time delays for beams passing through the ideal cloak within a geometric optics description.