Dielectric Media
When a light ray impinges on an interface between different media, we saw that some of the light
is reflected from that interface. However, in most cases some proportion of the light enters the second
mediumthe light is said to be transmitted. In this section we will concern ourselves only with transparent
or dielectric, media. The main change upon entering different media is that for any dielectric medium,
the permittivity, ε, and the permeability, μ, are different to those in free space. Thus the speed
of light in various media is different: v = 1/. We define the index of refraction of
the medium as:
nâÉá = 

Generally substances substances that are transparent are nonmagnetic, so
μ = mu_{0}, and in this case
n = .
Of course, this reasoning implies that light can travel faster or slower than c in different media; this seems
paradoxical since we have already said that photons can only exist at c. The
resolution to this is that light does propagate through the void between the atoms in a medium with speed c,
however when it encounters an atom it causes vibrations of the electrons which causes them to act like
spherical wave sources. It is these spherical waves which
cause the propagation through the medium, and they too travel at c between atoms. However, as you may know
from studying driven oscillations, the resulting oscillation usually lags behind the
driving oscillation by a factor between 90^{o} and 180^{o}, depending on whether the frequency
is above or below resonance. Thus the electron vibrations, and hence the spherical waves, are
out of phase with the incident light. As this spherical wave encounters another atom, there will
again be a phaselag for the light emitted from that atom, as so forth. Thus the light propagating through
the medium is constantly being retarded (or advanced) in phase. Since we measure the speed of light effectively
by watching the motion of a particular crest or trough, the continual phase shift amounts to a macroscopic
'slowing down' of the light.
Refraction
We are now ready to examine what happens to the transmitted ray. To put it simply, light entering a
medium at some nonzero angle with respect to the normal to the surface is bent. That is, upon entering a
different medium, light changes direction. It is not difficult to show that the change of speed involved in
moving from one medium to another implies this change in direction.
Figure %: Refraction of a wavefront.
The wavecrest
AB is just hitting the interface at
A. As soon as the wavefront enters the new medium it
must begin to travel with the new speed
v_{t}. The part of the wavefront at
B, however, must continue
traveling in a straight line to
D, with the speed
v_{i}; say that the time taken for the part of the
wavefront at
B to move to
D is
Δt. Thus the distance
BD is given by
v_{i}Δt. In this
same time the part of the wavefront at
A must continue in some direction a distance
v_{t}Δt. The
crucial point is that all the wavefronts must remain continuous; no wavefront (corresponding to a peak or
trough in the light wave) can get out of phase with itself. Indeed, this would contradict the notion of a
wavefront, which is just a line joining points of constant phase. If the transmitting medium is
denser that the incident medium,
v_{i} < v_{t}, and hence the distance
v_{i}Δt < v_{t}Δt. Thus
different parts of the wavefront have traveled different distances. It is not hard to see that it would be
impossible to maintain the continuity of the wavefront without bending it at some point. Since we know all
parts of the wavefront travel at a constant speed in one medium, the only place the bend could be is at the
interface. Clearly, the bending of the wavefront implies the bending of the light ray.
The two right triangles ABD and AED in share a common hypotenuse, AD, so we can write:
= 

But
BD = v_{i}Δt and
AE = v_{t}Δt, so we can write the law of refraction, or Snell's Law as:
n_{i}sinθ_{i} = n_{t}sinθ_{t} 

where
n_{i} = c/v_{i} and
n_{t} = c/v_{t}. Moreover, the incident, reflected and refracted rays all lie in
the same plane. From Snell's Law it is not hard to deduce that a light ray entering a denser (higher
index of refraction) medium bends towards the normal, while a ray entering a rarer (lower index of
refraction) medium bends away from the normal. In vector notation, for the normal to the surface
, and incident and transmitted waves
and
respectively,
the law of refraction can be written:
Dispersion
We saw that dielectric media have an index of refraction defined by n = .
For some media, however, this index is dependent on frequency. This effect causes different wavelengths of
light to bend by different amounts upon entering a medium, causing a light beam made up of different colors
to disperse. This phenomenon can be observed when white light enters a prism: the spectral pattern produced
is due to the dependence of the index of refraction of the glass on wavelength (or color). To examine this
effect we must once again consider the oscillations of the electron cloud caused by the oscillating electric
and magnetic fields of the light wave (see the discussion of scattering). It is also possible
that light waves can set up oscillations in ions or polar molecules of a substance, but their much greater
mass means that rapidly oscillating fields have little effect. We can assume that electrons are bound in
atoms in a fashion analogous to a mass on a spring, connecting the electron to the positive nucleus. Such
a system has a natural or resonant frequency σ_{0} = , where k will be determined by the
strength of the attraction between the electron and the nucleus. The timevarying electric field of the light
ray E(t) will cause a forced oscillation. The force due to this field will be F_{E} = q_{e}E(t) = q_{e}E_{0}cos(σt) where σ is the angular frequency of the light. From
Newton's Second Law we can write:
q_{e}E_{0}cosσt  kx = m_{e} = q_{e}E_{0}cosσt  m_{e}σ^{2}x 

You can check that the solution to this differential equation is
x = x_{0}cosσt. We can substitute
this back into the equation to solve for
x_{0} and we find:
x(t) = E_{0}cosσt 

Now, when a dielectric medium is subjected to an applied electric field,
, the internal charge
distribution is altered, creating a dipole moment. In other words, the
external field separates positive and negative charges, creating an additional electric field called
the electric polarization,
. The two fields are related by
(ε  ε_{0}). But
 is also given by the product of the magnitude
of the charge, the displacement, and the total number of displaced charge per unit volume, N:
, for displaced electrons. Substituting the (timeaveraged) value we found for
x,
above, we have:
 

We can then write the relationship between
and
as
ε = ε_{0} + = ε_{0} + = . Finally,
n^{2} = ε/ε_{0} so we can
express the index of refraction as a function of the angular frequency of the light as:
n^{2}(σ) = 1 + 

This is known as the dispersion equation. For frequencies above resonance
(σ_{0}^{2}  σ^{2}) < 0,
the oscillators are about 180
^{o} with the driving electric field (see the
discussion of phaselag) , and
n < 1. When the frequency is below resonance,
(σ_{0}^{2}  σ^{2}) > 0, the oscillation is in phase with the applied electric field and
n > 1.
In general, index of refraction increases with frequency, and hence blue light refracts more through glass
than red light. This situation is called normal dispersion. There are some cases of anomalous dispersion,
usually close to resonance, where index of refraction decreases for higher frequencies.