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An electromagnetic wave of real (positive) frequency can be written
The wave-vector, , indicates the direction of propagation of the
wave, and also its phase-velocity, , via
Since the wave is transverse in nature, we must have
. Finally, the familiar
leads us to the following relation between the constant vectors and :
is a unit vector pointing in the
direction of wave propagation.
Suppose that the plane forms the boundary between two different dielectric
media. Let medium 1, of refractive index , occupy the region ,
whilst medium 2, of refractive index , occupies the region .
Let us investigate what happens when an electromagnetic wave is incident
on this boundary from medium 1.
Consider, first of all, the simple case of incidence normal
to the boundary (see Fig. 55). In this case,
incident and transmitted waves, and
for the reflected wave. Without loss of generality, we can assume that
the incident wave is polarized in the -direction.
Hence, using Eq. (1214), the incident
wave can be written
where is the phase-velocity in medium 1, and
. Likewise, the reflected wave takes the form
Finally, the transmitted wave can be written
where is the phase-velocity in medium 2, and
For the case of normal incidence, the electric and magnetic
components of all three waves are parallel to the boundary between
the two dielectric media. Hence, the appropriate boundary conditions
to apply at are
The latter condition derives from the general boundary condition
, and the fact that
in both media (which are assumed to be non-magnetic).
Application of the boundary condition
Likewise, application of the boundary condition (1222) gives
Equations (1223) and (1225) can be solved to give
Thus, we have determined the amplitudes of the reflected and transmitted
waves in terms of the amplitude of the incident wave.
It can be seen, first of all, that if then and .
In other words, if the two media have the same indices of refraction then
there is no reflection at the boundary between them, and the transmitted
wave is consequently equal in amplitude to the incident wave. On the other
hand, if then there is some reflection at the boundary. Indeed,
the amplitude of the reflected wave is roughly proportional to the difference between and . This has important practical consequences.
We can only see a clean pane of glass in a window because some of the light incident
at an air/glass boundary is reflected, due to the different refractive indicies
of air and glass. As is well-known, it is a lot more difficult to see glass when it is submerged in water. This is because the refractive indices of glass and water are quite similar, and so there is very little reflection of light
incident on a water/glass boundary.
According to Eq. (1226), when .
The negative sign indicates a phase-shift of the reflected wave, with
respect to the incident wave. We conclude that there is a phase-shift of the reflected wave, relative to the incident wave, on reflection from a boundary with a
medium of greater refractive index. Conversely, there is no
on reflection from a boundary with a medium of lesser refractive index.
The mean electromagnetic energy flux, or intensity, in the -direction is simply
The coefficient of reflection, , is defined as the ratio
of the intensities of the reflected and incident waves:
Likewise, the coefficient of transmission, , is the ratio of
the intensities of the transmitted and incident waves:
Equations (1226), (1227), (1229), and (1230)
Note that . In other words, any wave energy which is not reflected
at the boundary is transmitted, and vice versa.
Let us now consider the case of incidence oblique to the boundary (see Fig. 56).
Suppose that the incident wave subtends an angle with the
normal to the boundary, whereas the reflected and transmitted
waves subtend angles and , respectively.
The incident wave can be written
with analogous expressions for the reflected and transmitted waves.
Since, in the case of oblique incidence, the electric and magnetic
components of the wave fields are no longer necessarily parallel to the
boundary, the boundary conditions (1221) and (1222) at must
be supplemented by the additional boundary conditions
Equation (1235) derives from the general
It follows from Eqs. (1222) and (1236) that both components
of the magnetic field are continuous at the boundary. Hence, we can write
at . Given that , , and are
constant vectors, the only way in which the above equation can be satisfied for all
values of and is if
throughout the plane. This, in turn, implies that
It immediately follows that if then
In other words, if the incident wave lies in the - plane then the
reflected and transmitted waves also lie in the - plane. Another
way of putting this is that the incident, reflected, and transmitted
waves all lie in the same plane, know as the plane
of incidence. This, of course, is one of the laws of geometric optics.
From now on, we shall assume that the plane of incidence is the -
with similar expressions for and . Hence, according
to Eq. (1239),
which implies that
Of course, the above expressions correspond to the law of reflection
and Snell's law of refraction, respectively.
For the case of oblique incidence, we need to consider two independent
wave polarizations separately. The first polarization
has all the wave electric fields perpendicular to the plane of incidence, whilst
the second has all the wave magnetic fields perpendicular to the plane
Let us consider the first wave polarization. We can write unit vectors
in the directions of propagation of the incident, reflected, and transmitted
The constant vectors associated with the incident wave are written
where use has been made of Eq. (1214). Likewise, the constant
vectors associated with the reflected and transmitted waves are
Now, the boundary condition (1221) yields
Likewise, the boundary condition (1236) gives
However, using Snell's law, (1243), the above expression reduces to Eq. (1253). Finally, the boundary condition (1222) yields
It is convenient to define the parameters
Equations (1253) and (1255) can be solved in terms
of these parameters to give
These relations are known as Fresnel equations.
The wave intensity in the -direction is given by
Hence, the coefficient of reflection is written
whereas the coefficient of transmission takes the form
Note that it is again the case that .
Let us now consider the second wave polarization. In this case, the
constant vectors associated with the incident, reflected, and transmitted
waves are written
The boundary condition (1222) yields
Likewise, the boundary condition (1221) gives
Finally, the boundary condition (1235) yields
Making use of Snell's law, and the fact that
above expression reduces to Eq. (1269).
Solving Eqs. (1239) and (1270), we obtain
The associated coefficients of reflection and transmission take the
respectively. As usual, .
Note that at oblique incidence the Fresnel equations, (1258) and
(1259), for the wave polarization in which the electric
field is parallel to the boundary are different to the Fresnel equations,
(1272) and (1273), for the wave polarization
in which the magnetic field is parallel to the boundary. This implies that
the coefficients of reflection and transmission for these two wave polarizations
are, in general, different.
Figure 57 shows the coefficients of reflection (solid curves) and transmission
(dashed curves) for oblique incidence from air () to
glass (). The left-hand panel shows the wave polarization
for which the electric field is parallel to the boundary, whereas the
right-hand panel shows the wave polarization for which the
magnetic field is parallel to the boundary. In general, it can
be seen that the coefficient of reflection rises, and the coefficient of
transmission falls, as the angle of incidence increases. Note, however,
that for the second wave polarization there is a particular angle of incidence,
know as the Brewster angle,
at which the reflected intensity is zero. There is no similar behaviour for
the first wave polarization.
It follows from Eq. (1272) that the Brewster angle corresponds
to the condition
where use has been made of Snell's law. The above expression
. Hence, the Brewster angle satisfies
If unpolarized light is incident on an air/glass (say) boundary at the Brewster angle
then the reflected beam is plane polarized.
Up: Electromagnetic radiation
Previous: Dielectric constant of a