Dispersion relation general
With Amperes law, Faraday’s law and the electric displacement asD = εEas well asB = μHthe
following equation which is a more general dispersion relation forωandkcan be derived (for plane
waves):
ε(ω,k)μ 0 ε 0 ω^2 = k^2 (124)On the left side (for positiveε) isc^2 ω^2 , withcthe speed of light in the material, therefore the waves
are travelling at the speed of light in the material. From the Diffusion-Ballistic regime we now that
the dielectric constant for a metal can be negative.
ε ≈ 1 −ω^2 P
ω^2Hencek(eqn. (124)) can get complex and a complexkmeans that the wave decays exponentially, this
shows fig. 72. This phenomenon is similar to electrons shooten into a semiconductor with an energy
in-between the band gap of the semiconductor, which are reflected because there are no allowed states.
For metals there is a forbidden frequency gap for light.
Figure 72: Dispersion relationωvs.k