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 ~ μ_{0}, and n = (ε/ε_{0})^{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** = ε_{0}*χ*_{e}**E** = Nα_{e}**E**_{eff} = N**p**.
(SI units)

Here *χ*_{e} is the electric susceptibility,
α_{e} 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. **E**_{eff} is the microscopic
field due to everything else except that particular atom or molecule at the
location of that atom or molecule.

**
D** = ε_{0}**E** + **P**
= ε_{0} (1 + *χ*_{e})**E**
= ε**E**.

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

**F**_{e} = -q_{e}**E**_{eff} = m(d^{2}**r**/dt^{2}
+ γd**r**/dt + ω_{0}^{2}**r**).

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

**
E**_{eff} = **E**_{0}exp(-iωt).

We find **r** = **r**_{0}exp(-iωt),
with **r**_{0} = -(q_{e}/m)**E**_{0}/(ω_{0}^{2}
- ω^{2} - iγω).

The induced dipole moment of the atom is

**p** = -q_{e}**r** = (q_{e}^{2}/m)**E**_{eff}/(ω_{0}^{2}
- ω^{2} - iγω),

if the atom contains a single electron.

If there are f_{j} electrons per atom with binding frequencies ω_{j}
and damping constants γ_{j}, then

**p** = (q_{e}^{2}/m)**E**_{eff}∑_{j} (f_{j}/(ω_{j}^{2}
- ω^{2} - iγ_{j}ω)), ∑_{j} f_{j} = Z.

Here Z is the total number of electrons per atom and the f_{j} are
called the oscillator strengths.

The polarization **P** is now given by

**P **= N**p** = (Nq_{e}^{2}**E**_{eff}/m)
∑_{j} (f_{j}/(ω_{j}^{2} - ω^{2} - iγ_{j}ω))
= ε_{0}*χ*_{e}**E**.

In substance with low density **E**_{eff} ~
**E**. Then

*χ*_{e} = (Nq_{e}^{2}/(ε_{0}m)) ∑_{j}
(f_{j}/(ω_{j}^{2} - ω^{2} - iγ_{j}ω))

and

ε = ε_{0} + (Nq_{e}^{2}/m) ∑_{j} (f_{j}/(ω_{j}^{2}
- ω^{2} - iγ_{j}ω)).

In general in dielectric materials ω_{j}
>> γ_{j} and ε(ω) ~ real for most frequencies.

For metals some ω_{j}
<< γ_{j} and ε(ω) is a complex number.

If the angular frequency of the effective electric field ω
is less than all the ω_{j}, then ε
> ε_{0}, and if ω
is greater than all the ω_{j}, then ε
< ε_{0}.

- Normal dispersion: Re(ε(ω)) rises with ω.
- Anomalous dispersion: Re(ε(ω)) decreases with increasing ω.

Anomalous dispersion occurs when ω ~ ω_{j}, 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.

k^{2}
= (ω^{2}/v^{2}) = μεω^{2}.

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** = E_{x}**i**
we have

E_{x} = E_{0}exp(i(kz - ωt)) = E_{0}exp(-(α/2)z)
exp(i(βz - ωt)).

The intensity is proportional to E^{2} and falls off as exp(-αz).

The index of refraction is n = (ε/ε_{0})^{½}.

n = [1 + (Nq_{e}^{2}/(ε_{0}m)) ∑_{j} (f_{j}/(ω_{j}^{2}
- ω^{2} - iγ_{j}ω))]^{½}.

If ε_{ }~ ε_{0}
and n ~ 1, then we can write

n ≈ 1 + (Nq_{e}^{2}/(2ε_{0}m)) ∑_{j} (f_{j}/(ω_{j}^{2}
- ω^{2} - iγ_{j}ω)).

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