Question

1. Investigate the difference between "phase velocity" and "group velocity." Why does an index of refraction of less than 1 (so v>c) not violate relativity, which says no information can move faster than the speed of light?

Answer 1

Phase velocity

The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave travels

Group Velocity

The group velocity of a wave is the velocity which the overall shape of the waves' amplitudes known as the modulation or envelope  of the wave propagates through space.

phase velocity vp = ω/k and

the group velocity vg = dω/dk depend on the frequency

and on the medium.

An interesting example is :

one must have seen the large group of dancers in a republic day parade in the celebrations.

the group usually has about 50 performers and have their own phase speeds as per the dance movements..their individual speeds are phase speeds

but as the whole group advances with a group velocity ahead which is the modulated wave.

More difference-

1. In an anomalous medium the group velocity is greater than the phase velocity.

2. In a normal medium phase velocity is greater than the group velocity.

3. In a wave packet the envelope travels with the group velocity while career wave travels with phase velocity.

Why does an index of less than 1 not violate relativity-

Beacuse phase velocity is speed at which the Crests of the wave move and it can be faster than the speed of light in vacuum and therefore given a refractive index less than 1.