# EM Group velocity & phase velocity in dispersive medium

by tabsquare
Tags: dispersive, phase, velocity
 Sci Advisor Thanks P: 2,351 One should emphasize that even in regions of anomalous dispersion, which occurs necessarily for frequencies of the em. wave close to a resonance of the medium, where both the group and the phase velocities are larger than the speed of light, no problems with causality occur. This has been solved already in 1907 by Sommerfeld, answering a question by W. Wien. In more detail this problem has been worked out by Sommerfeld and Brillouin in 1912. The reason is that the physical picture of the group velocity as the velocity of the propagation of the center of a breaks down close to a resonance. This picture is based on the validity of the stationary-phase approximation of the Fourier integral from the freqency to the time domain, which doesn't hold close to a resonance. There other more general approximation techniques have to be applied to get an understanding of signal propagation in the medium. In any case the front velocity of a wave packet of finite extension is always $\leq c_{\text{vac}}$, in the usual dispersion models it's $=c_{\text{vac}}$ since in the very first impact of the wave the medium has not yet responded to the incoming em. wave and thus, in this first moment, behaves like vacuum. Then very interesting transient effects set in, the socalled Sommerfeld and Brillouin precursors. The best read read about this is Sommerfelds famous textbooks (Lecture notes about Theoretical Physics, vol. IV, optics). Of course, also Jackson discusses these phenomena in his textbook (including a nice reference to the experimental verification of the classical dispersion theory with microwave experiments).