Problem 1 [15 points] Consider as a very simplified model for the d...

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Problem 1 [15 points] Consider as a very simplified model for the dielectric function E(W) wo2-w (This is the model discussed in lecture in the limit that the damping force vanishes.) Assume wo < Wp. a) Using the dispersion relation k2 = W2E(W)IC2, make a sketch of k vs w for electromagnetic wave propagation. b) Re-draw this sketch as w vs k, and show that for each value of k > 0 there are two allowed values (modes) of W 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. w = Vpk where Vp is only weakly dependent on k. Show that the other mode has a frequency W 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) Show that, at 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.

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