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)Eeff∑j (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.