Problem 1 [20 points]
Consider as a simplified model for the frequency dependent permittivity function
| ε(ω)/ε0 = 1 + | ωp2
ω02 − ω2 |
(This is the model discussed in lecture in the limit that the damping force vanishes.) Assume ω0 < ωp.
a) (6pts) Using the dispersion relation k2 = (ω2/c2)(ε(ω)/ε0), make a sketch of k vs ω for electromagnetic wave propagation.
b) (6pts) Re-draw this sketch as ω vs k, and show that for each value of k > 0 there are two allowed values (modes) of ω for the electromagnetic waves. Show that at both small k and at large k one of these two modes has a dispersion relation characteristic of an electromagnetic wave in the vacuum, i.e. ω = vpk where vp is only weakly dependent on k. Show that the other mode has a frequency ω that is, to lowest order, independent of k and so like some internal atomic mode of vibration. Show that these two modes exchange their characteristic behavior as one crosses from small k to large k. In this intermediate region the modes have a mixed character and are referred to as "polaritons".
c) (8 pts) Show that, for a given value of k, the higher frequency mode has a phase velocity that satisfies vp > c, while the lower frequency mode satisfies vp < c. Show that for both modes the group velocity always satisfies vg < c. Hint: You might find that it is easier to demonstrate this graphically rather than algebraically.