Let us consider a plane wave that is incident at a planar interface of two lossless media, as illustrated in Figure 2.7. The wave comes from medium 1, which is characterized by €1 and /1, to medium 2 with €2 and /2. The planar interface is at z = 0. The angle of incidence is d1 and the propagation vector k 1 is in the xz-plane. Part E\ from the incident field is reflected at an angle d[ and part E2 is transmitted through the interface and leaves at an angle of 62.

According to the boundary conditions, the tangential components of the electric and magnetic field are equal on both sides of the interface in each point of plane z = 0. This is possible only if the phase of the incident, reflected, and transmitted waves change equally in the x direction, in other words, the phase velocities in the x direction are the same, or v 1 v 1 v 2 „

sin d1 sin Q[ sin d2

where v i and v2 (vj = colki) are the wave velocities in medium 1 and 2, respectively. From (2.79) it follows

which means the angle of incidence and angle of reflection are equal, and the angle of propagation of the transmitted wave is obtained from sin 02 / ui 61 „

Let us assume in the following that ¡¡1 = ¡l2 = ¡0, which is valid in most cases of interest in practice. Then (2.81), which is called Snell's law, can be rewritten as sin 02 n 1 „

where n 1 = -y er\ and n2 = ^J er2 are the refraction indices of the materials.

The reflection and transmission coefficients depend on the polarization of the incident wave. Important special cases are the so-called parallel and perpendicular polarizations; see Figure 2.7. The parallel polarization means that the electric field vector is in the same plane with kj and the normal n of the plane, that is, the field vector is in the xz-plane. The perpendicular polarization means that the electric field vector is perpendicular to the plane described previously, that is, it is parallel to the y-axis. The polarization of an arbitrary incident plane wave can be thought to be a superposition of the parallel and perpendicular polarizations.

In the case of the parallel polarization, the condition of continuity of the tangential electric field is

The magnetic field has only a component in the y direction. The continuity of the tangential magnetic field leads to

From (2.84) and (2.85) we can solve for the parallel polarization the reflection and transmission coefficients, pjj and r jj, respectively:

When the angle of incidence is 90°, that is, when the incident wave approaches perpendicularly to the surface, it holds for p and r that

In case of the perpendicular polarization, the boundary conditions lead to

from which we can solve for the perpendicular polarization

2 cos 01

Figure 2.8 shows the behavior of the reflection coefficient as a function of the angle of incidence for both polarizations, when n 1 < n2, that is, —1 < — 2. In case of the parallel polarization, the reflection coefficient is equal to zero at Brewster's angle

If —1 > — 2, a total reflection occurs at angles of incidence

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