Phase velocity

A monochromatic wave $\cos(kz - \omega t)$ has crests that advance at the phase velocity

where $n$ is the refractive index or, for a guided mode, the effective index, and $k$ is the propagation constant. It is the velocity that sets wavelength in the medium, phase matching in nonlinear optics, and the phase accumulated in an interferometer arm.

A pulse or any modulated signal is a superposition of frequencies, and its envelope moves at the group velocity, $v_g = d\omega/dk = c/n_g$, with the group index $n_g = n - \lambda\,dn/d\lambda$. In normal dispersion, where the index falls with wavelength, $n_g > n$ and the envelope is slower than the crests, which slide forward through it. In fused silica at 1550 nm, $n \approx 1.444$ gives a phase velocity of about 207,600 km/s, while the bulk group index of 1.463 gives about 205,000 km/s (in standard fiber the group index is 1.468 and the group velocity 204,200 km/s); in a silicon wire waveguide the difference is much larger, with effective index about 2.4 and group index above 4 in typical designs.

Faster than light

The phase velocity can exceed $c$, and does whenever the index is below 1: for X-rays in matter, near an absorption line on its high-frequency side, and for any waveguide mode close to cutoff in a metallic waveguide. This carries no information, because a single frequency extending forever cannot signal. The speed of a signal front never exceeds $c$, which Sommerfeld and Brillouin showed in 1914; group velocity can also exceed $c$ in anomalously dispersive media without signalling faster than light, because the pulse reshapes rather than travelling undistorted.

Measurement

Phase velocity follows from a measured index or effective index: from the angle of refraction, from a prism at minimum deviation, from ellipsometry for thin films, or for waveguides from the angle at which a grating coupler couples light in. Interferometers measure phase delay, $n L$, directly. The group index is measured separately, from a pulse delay or from the fringe spacing of a resonator.

References: M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 1; L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).