(1210) | |||

(1211) |

The wave-vector, , indicates the direction of propagation of the wave, and also its phase-velocity, , via

(1212) |

(1213) |

Here, 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,
for the
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

(1215) | |||

(1216) |

where is the phase-velocity in medium 1, and . Likewise, the reflected wave takes the form

(1217) | |||

(1218) |

Finally, the transmitted wave can be written

(1219) | |||

(1220) |

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
yields

(1224) |

since . 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
phase-shift
on reflection from a boundary with a medium of *lesser* refractive index.

The mean electromagnetic energy flux, or *intensity*, in the -direction is simply

(1228) |

Likewise, the

Equations (1226), (1227), (1229), and (1230) yield

(1231) | |||

(1232) |

Note that . In other words, any wave energy which is not reflected at the boundary is transmitted, and

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

(1233) | |||

(1234) |

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 boundary condition .

It follows from Eqs. (1222) and (1236) that both components
of the magnetic field are continuous at the boundary. Hence, we can write

(1237) |

(1238) |

and

(1240) |

Now,
and
. Moreover,

(1241) |

(1242) |

Of course, the above expressions correspond to the

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
of incidence.

Let us consider the first wave polarization. We can write unit vectors
in the directions of propagation of the incident, reflected, and transmitted
waves likso:

(1244) | |||

(1245) | |||

(1246) |

The constant vectors associated with the incident wave are written

(1247) | |||

(1248) |

where use has been made of Eq. (1214). Likewise, the constant vectors associated with the reflected and transmitted waves are

(1249) | |||

(1250) |

and

(1251) | |||

(1252) |

respectively.

Now, the boundary condition (1221) yields
,
or

(1254) |

It is convenient to define the parameters

(1256) |

(1257) |

These relations are known as

The wave intensity in the -direction is given by

(1260) |

(1261) |

(1262) |

Let us now consider the second wave polarization. In this case, the
constant vectors associated with the incident, reflected, and transmitted
waves are written

(1263) | |||

(1264) |

and

(1265) | |||

(1266) |

and

(1267) | |||

(1268) |

respectively. The boundary condition (1222) yields , or

Likewise, the boundary condition (1221) gives , or

Finally, the boundary condition (1235) yields , or

(1271) |

Solving Eqs. (1239) and (1270), we obtain

The associated coefficients of reflection and transmission take the form

(1274) | |||

(1275) |

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

(1276) |

(1277) |

(1278) |

(1279) |