Goos-Hanchen shifts in frustrated total internal reflection studied with wave-packet propagation

We have investigated the Goos-Hanchen (GH) shifts in frustrated total internal reflection (FTIR) studied with wave-packet propagation. In the first-order approximation of the transmission coefficient, the GH shift is exactly the expression given by a stationary phase method, thus saturating an asymptotic constant in two different ways depending on the angle of incidence. Taking the second-order approximation into account, the GH shift does not saturate with increasing gap width when the small beam size is used. The GH shift increases by decreasing the beamwidth at the small incidence angles, while for the large incidence angles it reveals a strong decrease by decreasing the beamwidth. These phenomena offer the better understanding of the GH shift and tunneling delay time in FTIR.