-rays 10-10 cm or shorter,
they are the same kind of wave traveling at the same velocity.
The term electromagnetic radiation encompasses a wide
variety of radiative phenomena which seem at first to be quite different from
one another. For example, radio waves, visible light, and the "gamma rays"
which emanate from radioactive substances all fall into this category. In
fact, they are manifestations of the same process, i.e., they all are
propagating waves of electric and magnetic field energy. The only way in which
they differ from each other is in their wavelengths; all electromagnetic
radiation propagates at a velocity c = 3 x 108 m/sec in vacuum.
Thus, while radio waves may have wavelengths of several thousand meters, light
waves 5 x 10-5 cm and -rays 10-10 cm or shorter,
they are the same kind of wave traveling at the same velocity.
The phenomena of reflection and refraction of e.m. waves are particularly familiar in the instance of visible light, where our means of detection is so very good. Reflection, of course, is what makes most objects visible to us, and refraction is responsible for the functioning of eyeglass and camera lenses.
To discuss these processes in more detail, we must establish the notion of
refractive index. This quantity, usually labelled n, is simply the factor by
which an e.m. wave is slowed upon entering a material medium from a vacuum
region. In other words, the velocity of the waves in this refractive medium is
c/n. Now, it is evident that the number of waves per second incident from a
vacuum region onto the surface of a refracting medium is the same as the number
per second entering that medium, i.e., the frequency (
) of the waves must
be the same in both regions. Therefore, because we know that 
=
c/n, ( is wavelength) we can conclude that in the refractive region the
wavelength is decreased such that
= 0/n.
Here, 0 is the vacuum wavelength.
However, the way in which refraction is most noticeable is in the bending of light rays. You are probably familiar with the way in which prisms and lenses bend beams of light, and nearly everyone has seen the apparent bending of a stick thrust into water at some oblique angle with the surface. We may relate the bending of rays quite easily to the refractive index by a geometrical construction, as follows.
Figure 1 Waves incident on an interface between index of refraction n1 and n2.
We imagine that a plane wave is traveling through a medium of index n1. By plane wave, we mean a disturbance uniform in a plane, which propagates along a ray direction perpendicular to the plane. The electric field and magnetic field vectors associated with the wave lie in the plane of the wave, and at right angles to each other; the space and time dependence of the electric field is given in vector notation by
= 
,
and the magnetic field by
= 
,
when the ray, or propagation vector, is along the z-axis.
Consider now a plane wave (Figure 1) incident upon a plane boundary separating
regions 1 and 2, which have indices n1 and n2,
respectively. We will assume for now that n2>n1, and
assign an angle 
1 between the wave and the interface. At a
time t1, the wave enters a portion of the surface of width x, and at
time t2 it has just crossed completely through it. We can see from
Fig. 1 that
=
x sin 1,
i.e., that in the time interval t2 - t1, the incident
wave has moved a distance x sin 1. But now one also sees
that the part of the wave which was just entering region 2 at time
t1 has moved a distance x sin 2 at time
t2. Therefore,
= t2 - t1 =
or
n1 sin 1 = n2 sin 2
.
This is Snell's law, which relates ray bending to a step in refractive index at a surface. Ordinarily, one draws the rays associated with the wave, in which we would have Fig. 2:
Figure 2 Rays incident on an interface between index of
refraction n1 and n2. Angles are measured with respect
to the normal to the interface. The refracted ray is at the new angle
2.
Our description of events at the interface is still incomplete, however, because in addition to the wave transmitted through the surface, some portion of the incident energy is reflected. The ray diagram is, therefore, the following
Figure 3 Rays incident on an interface between index of refraction n1 and n2. Angles are measured with respect to the normal to the interface. A reflected ray is shown as REFL.
where the angle of reflection is the same as the angle of incidence. (You may prove this result to yourself in the same manner as we proved Snell's law.)
We have not specified the fraction of the wave intensity transmitted, and the
fraction reflected. We shall not develop this subject here, since it involves
somewhat advanced concepts in electromagnetic theory. We will, however, state
without proof a result which applies to normal incidence (1 =
2 = 0); it is:
Etrans =
and
Erefl = 
where E0 is the incident electric field strength. Now, the intensity of an electromagnetic wave, or the energy transported by the wave across unit area each second, is proportional to E2; therefore, the intensity reflected back along a light beam incident on a refractive interface is
Irefl = 
Consider a light ray in air, whose index is essentially 1.00 (that of vacuum), incident on a sheet of glass, for which, typically, n2 = 1.5. Then
Irefl = 
= .04 I0.
This is a familiar result. Nearly all the light incident on a window passes through it while only enough is reflected to allow you to see dim images.
An interesting special situation arises for waves passing from a refracting medium into another region of smaller refractive index.
Figure 4 Rays incident on an interface between index of refraction n1 > n2.
We recall that n1sin 1 = n2sin
2, and so 1 < 2.
Suppose we increase 1, the angle of incidence, until sin
2 = 1; then,
sin1 = n2/n1
and the transmitted ray is moving just parallel to the interface. If we
increase 1 still further, a rather remarkable thing happens,
and it is called, appropriately enough, total internal reflection. The
interface now acts like a perfect mirror, and simply reflects all of the
incident energy back into region 1, at an angle equal to 1.
This is the general principle behind fiber optic cables and light guides.
Figure 5 Rays incident on an interface between
indices of refraction n1 > n2. The critical angle,
c, is shown for the case of total internal reflection.
The angle for which sin1 =
n2/n1 is called the critical angle, c =
1.
Consider a wave incident normally on a flat slab of paraffin. In addition to the wave reflected from the front surface there will be a refracted (transmitted) wave propagating into the paraffin slab. This transmitted wave will similarly give rise to reflected (back into the paraffin) and transmitted (out of the paraffin) waves at the rear surface of the slab. This second reflected wave will again be partially transmitted back through the front surface. If these two waves are in phase with each other they will add (constructive interference). If they are out of phase they will partially cancel (destructive interference).
Let us examine the conditions necessary for constructive interference of the
two reflected waves to occur. We have two waves, one (1) reflected from the
front surface and the other (2) from the rear surface of the slab. We will
suppose for now that upon reflection no change of phase occurs for either wave;
the direction of travel is simply reversed instantaneously. Recall that the
frequency of the wave is unchanged as it enters or leaves a medium of
different refractive index. Therefore, if the wave (2) transmitted into the
slab and reflected off its rear surface has just undergone an integral number
of oscillations when it reaches the front surface again, it will be exactly in
phase with the wave (1) which has just reflected off the front surface. These
two waves add together to give a strong reflection and we have that
constructive interference must depend upon the total time spent by wave (2)
within the slab since this determines the number of oscillations which have
occurred while the wave is within the slab. The time 
t is given in
terms of the wave velocity v = c/n within the material by
t =
=
The number m of oscillations in the time t is just
m = ft =
=
=
where is the wavelength within the slab of refractive index n. Note
that an alternative way of viewing this relation is that m is the number of
wavelengths contained in the total path length 2d. Thus, in the case
considered above where there is no phase change for any of the reflections, the
condition for constructive interference of the reflected waves is that m be an
integer.
The above conclusions also apply if upon reflection the waves (1) and (2) do undergo a phase change, so long as the phase change is the same for both waves.
It is possible, however, that a difference in phase shift can result from the two reflections. This should not be surprising since one reflection (at the front face) occurs for a beam moving from a medium of low refractive index (air) to one of high refractive index (paraffin), whereas the other reflection is for a beam in a high index medium passing to a low index medium. This is similar to the reflection of oscillations at the end of a rope. The reflections are different depending on whether or not the end of the rope is tied to a fixed support or is flapping free.
Suppose that one of the waves has its phase shifted by 180 and the other is not
shifted at all. In this case, the relation derived above, that 2d = m
(m an integer), will give a minimum in the reflected intensity rather than a
maximum. Thus, it is possible to distinguish the case of only one wave being
reflected with a 180 phase shift from that of either both or neither of the
waves being shifted in phase by 180.
A paraffin slab of variable thickness can be constructed in the lab by placing the two prisms together as shown below:
