Index of refraction

For a monochromatic plane wave in a material v = 1/(με)½ = c/n.  The index of refraction is n = c(με)½.  For most dielectric materials μ ~ μ0, and n = (ε/ε0)½
For many materials ε depends on ω.  Can we understand how ε(ω) depends on ω?

Let us look at a simple model of a lih material.
P = ε0χeE = NαeEeff = Np. (SI units)

Here  χe is the electric susceptibility, αe is the atomic or molecular polarizability, N is the number of atoms or molecules per unit volume, and p is the dipole moment per atom or molecule.  E is the average macroscopic field in the medium.  Eeff is the microscopic field due to everything else except that particular atom or molecule at the location of that atom or molecule.
D = ε0E + P = ε0 (1 + χe)E = εE.

If in a material we model the electrons bound by a harmonic force to the atomic cores, and we include a damping force in the atomic oscillators, then the equation of motion for an electron in the presence of an electric field is
Fe = -qeEeff = m(d2r/dt2 + γdr/dt + ω02r).

If the electric field varies sinusoidally in time and the wavelength is long enough so that spatial variations over the dimensions of an atom can be neglected, we write
Eeff = E0exp(-iωt).

We find r = r0exp(-iωt), with r0 = -(qe/m)E0/(ω02 - ω2 - iγω).
The induced dipole moment of the atom is 
p = -qer = (qe2/m)Eeff/(ω02 - ω2 - iγω),
if the atom contains a single electron.

If there are fj electrons per atom with binding frequencies ωj and damping constants γj, then
p = (qe2/m)Eeffj (fj/(ωj2 - ω2 - iγjω)),  ∑j fj = Z.

Here Z is the total number of electrons per atom and the fj are called the oscillator strengths.
The polarization P is now given by
P = Np =  (Nqe2Eeff/m) ∑j (fj/(ωj2 - ω2 - iγjω)) = ε0χeE.

In substance with low density Eeff ~ E.  Then
χe = (Nqe2/(ε0m)) ∑j (fj/(ωj2 - ω2 - iγjω))
and
ε = ε0 + (Nqe2/m) ∑j (fj/(ωj2 - ω2 - iγjω)).

In general in dielectric materials ωj >> γj and ε(ω) ~ real for most frequencies.
For metals some ωj << γj and ε(ω) is a complex number.
If the angular frequency of the effective electric field ω is less than all the ωj, then ε > ε0, and if ω is greater than all the ωj, then ε < ε0.

Anomalous dispersion occurs when ω ~ ωj, in the neighborhood of resonant frequencies.  Here Im(ε(ω)) becomes large.
A complex ε(ω) in the wave equation implies a complex wave number k for plane wave solutions.
k2 = (ω2/v2) = μεω2.
If we write k = β + iβ/2, then α is called the attenuation constant.
For a plane wave propagating along the z-direction with E = Exi we have
Ex = E0exp(i(kz - ωt)) = E0exp(-(α/2)z) exp(i(βz - ωt)).
The intensity is proportional to E2 and falls off as exp(-αz).

The index of refraction is n = (ε/ε0)½.
n = [1 + (Nqe2/(ε0m)) ∑j (fj/(ωj2 - ω2 - iγjω))]½.
If ε ~ ε0 and n ~ 1, then we can write
n ≈ 1 + (Nqe2/(2ε0m)) ∑j (fj/(ωj2 - ω2 - iγjω)).

The index of refreaction, in general, is a complex number.