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Let us investigate an electromagnetic wave propagating
through a transparent, isotropic, nonconducting, medium.
The electric displacement inside the medium is given by

(631) 
where is the electric
polarization. Since electrons are much lighter than
ions (or atomic nuclei),
we would expect the former to displace further than the latter under
the influence of an electric field. Thus, to a first
approximation the polarization is determined by the electron
response to the wave. Suppose that the electrons displace a distance
from their rest positions in the presence of the wave. It follows
that

(632) 
where is the number density of electrons.
Let us assume that the electrons are bound ``quasielastically'' to
their rest positions, so that they seek to return to these positions
when displaced from them by a field . It follows that
satisfies the differential equation of the form

(633) 
where is the electron mass, is the restoring force,
and denotes a partial derivative with respect to time.
The above equation can also be written

(634) 
where

(635) 
is the characteristic oscillation
frequency of the electrons. In almost all dielectric
media this frequency lies in the far ultraviolet region of the
electromagnetic spectrum. In Eq. (4.12) we have added a phenomenological
damping term
, in order to take into account the
fact that an electron excited by an impulsive electric field does not
oscillate for ever. In general, however, electrons in dielectric media
can be regarded as highQ oscillators, which is another way of
saying that the dimensionless damping constant is typically much less than
unity. Thus, an electron ``rings'' for a long time after being excited by an
impulse.
Let us assume that the electrons oscillate in sympathy with the wave at the
wave frequency . It follows from Eq. (4.12) that

(636) 
Note that we have neglected the response of the electrons to the
magnetic component of the the wave. It is easily demonstrated that this
is a good approximation provided that the electrons do not oscillate with
relativistic velocities (i.e., provided that the amplitude of
the wave is sufficiently small).
Thus, Eq. (4.10) yields

(637) 
Since, by definition,

(638) 
it follows
that

(639) 
Thus, the index of refraction is frequency dependent. Since
typically lies in the ultraviolet region of the spectrum (and since
), it is clear that the denominator
is positive in the entire visible spectrum, and is
larger at the red end than at the blue end. This implies that
blue light is refracted more than red light. This is normal
dispersion. Incidentally, an expression, like the above, which specifies the
dispersion of waves propagating through some dielectric medium is usually
called a dispersion relation.
Let us now suppose that there are molecules per unit volume
with electrons per molecule, and that instead of a single oscillation
frequency for all electrons, there are electrons per molecule with
oscillation frequency and damping constant . It is
easily demonstrated that

(640) 
where the oscillator strengths satisfy the sum rule,

(641) 
A more exact quantum mechanical treatment of the response of an atom,
or molecule, to a low amplitude electromagnetic wave also leads to a
dispersion relation
of the form (4.18), except that the quantities
, , and can, in principle,
be calculated from first principles. In practice, this is too difficult except
for the very simplest cases.
Since the damping constants are generally small compared to unity,
it follows from Eq. (4.18) that is a predominately real
quantity at most wave frequencies. The factor
is positive for
and negative for
.
Thus, at low frequencies, below the smallest , all of the terms
in the sum in (4.18) are positive, and is consequently
greater than unity. As is raised so that
it passes successive values, more and more negative terms occur
in
the sum, until eventually the whole sum is negative and
is less than unity. Thus, at very high frequencies electromagnetic
waves propagate through dielectric media with phase velocities which exceed
the velocity of light in a vacuum. For
,
Eq. (4.18) predicts a rather violent variation of the refractive index with
frequency. Let us examine this phenomenon more closely.
Next: Anomalous dispersion and resonant
Up: Electromagnetic wave propagation in
Previous: Introduction
Richard Fitzpatrick
20020518