16 Wave Processes at the Boundary Between Two Media. Fresnel's Formulas, Reflection and Refraction

1. Laws of reflection and refraction of plane waves

The solution of the problem of reflection and refraction of plane waves at the interface between two media finds wide application in the theory of radio wave propagation and in optics.
Let a plane wave be incident on the interface between media. The disturbing action of the boundary, as one might suppose, will reduce to the appearance of reflected and refracted waves.
Any plane wave taken separately is a solution of Maxwell's equations. However, for these waves to represent a solution of the problem, it is necessary that, together with the incident
wave, they satisfy the boundary conditions at the interface of the media.
So, let the field intensity of the incident wave be

and the field intensities of the reflected and refracted waves are respectively equal to

where , (we consider the media to be dielectrics).
We will characterize the directions of propagation by angles
(fig. 1):
—angle of incidence; this angle is considered known;
and — the sought angles of reflection and refraction;

— known;
— unknown.
The boundary conditions, which all three waves must jointly satisfy,

(1)

or
(2)

By substituting the expressions for the fields of all three waves into one of the boundary conditions, we obtain a relation of the following form:

where r —is the radius vector of an arbitrary point of the boundary plane, drawn from the point of incidence (fig. 2). This equality must hold over the entire boundary plane for any value of the vector r. And this is possible if and only if all three phase factors are equal, that is, when the following conditions hold

Fig. 1 Fig. 2
Let us clarify the meaning of these two equalities.
For this, let us take into account the identity
(since ng = 0),
and obtain

Let us perform a cyclic permutation of the vectors here and arrive at the equalities

Since these equalities must hold for any value of the vector [ng], it follows that

These two vector equalities constitute the formulation of the laws of reflection and refraction of plane waves.
1. The incident ray, the reflected ray, and the refracted ray lie in one and the same plane — the plane of incidence.
The plane of incidence – is the plane containing the vector n, and the vector n — the normal to the interface, erected at the point of incidence.
2.