Amplified total internal reflection

Totally internal reflected beams can be amplified if the lower-index medium has gain. We analyze the reflection and refraction of light, and analytically derive the expression for the Goos-Hänchen shifts of a Gaussian beam incident on a lower-index medium, both active and absorptive. We examine the energy flow and the Goos-Hänchen shifts for various cases. The analytical results are consistent with the numerical results. For the TE mode, the Goos-Hänchen shift for the transmitted beam is exactly half of that of the reflected beam, resulting in a "1/2" rule.