# The phase velocity of laser light

What are the limitations on the phase velocity of laser light? Is it true to say that the phase velocity of laser light is limited to <c?

## Answers and Replies

That's true in vacuum, where is exactly c. Phase velocity in a medium has no meaning, I think; I've seen some cases where phase velocity is greater than c.

Andy Resnick
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The phase velocity is defined as v = w/k(w), where w is the frequency and k(w) is the wavenumber, similar to 1/wavelength. I've allowed the medium to be dispersive, so k = k(w). The 'group velocity' is dw/dk.

One way to picture this is that a pulsed carrier wave will have two relevant velocities- the phase velocity is the velocity of the carrier wave, while the pulse envelope moves as per the group velocity.

If a pulse moves through resonant media, strange things happen to the group velocity but not to the phase velocity- the phase velocity is simply v = c/(1+X'(w)/2), where X'(w) is the frequency-dependent real part of the susceptibility. The group velocity V =v(w)/(1-(w/v)(dv/dw)) and can become negative or exceed c0 near strong transitions.