Fresnel’s equations
To calculate the relative intensities of reflected and refracted waves in terms of the energy of the incident wave, we introduce Fresnel's equations. We consider the two polarizations of electromagnetic radiation at the interface between two media of different index of refraction (n1
¹ n2). The derivation here applies for non-magnetic and non-conducting media (the magnetic permeability m1 = m2).The equations to be derived are given below. For electromagnetic radiation polarized normal to the plane of incidence (s polarized) we have
For electromagnetic radiation polarized parallel to the plane of incidence (p polarized) we have
The derivation of the Fresnel equations is straightforward. We assume that electromagnetic radiation propagates in two media. For the derivation below we will assume that the index of refraction of medium 2 is less than that of medium 1, n2 < n1.
A diagram of the electric and magnetic vectors for s polarized light is shown below. Note that the direction of the magnetic vector is shown in color and the direction of the electric vector is represented by dashed lines.
For electromagnetic radiation polarized normal to the plane of incidence (s polarized) we have
and
The relationship between the magnitude of the electric and magnetic field vectors is
where k is the wavevector and
w is the angular frequency of radiation. Thus, in terms of the electric vector we have
The wavevector k = n/
For light polarized parallel to the plane of incidence (p polarized) we can draw the following diagram.
For this configuration of the electric and magnetic field vectors we have
or
and
To understand the angular dependence we examine the electric vectors looking down on the plane of incidence.
The projection of Ei onto the interface is given by Ei cos
qi shown in violet in the figure. Equation IV simply states that the project of the electric field on the interface must be the same on each side.We can write equation (III) as
and substitute this equation into (IV). The Fresnel equations for p polarized light follow.
The Fresnel equations become
for s polarized light and
for p polarized light.
Refracted light does not change its polarization
The phase is not changed for refracted light for either s- or p-polarized radiation. (Et/Ei)s is always real and positive.
The change in polarization upon reflection
We assumed that n2 < n1 in the above derivation. For this case, the phase of s polarized light is unchanged upon reflection. We can see the change in phase by looking for a change in the sign of Er. According to the Fresnel equation for s-polarized radiation
We can use Snell’s law
to ask
Provided n2 < n1 Snell’s law states that sin
qt > sin qi and therefore qt > qi. Under these conditions sin(qt + qi) > 0 always and thus there is no change in phase upon reflection.If n1 < n2 holds then the phase of s-polarized radiation is changed by
p upon reflection. We can see this by noting that that sinqt < sin qi and therefore qt < qI in this case and thus sin(qt + qi) < 0. This means that the sign of Er changes upon reflection.In summary for s-polarized light the ratio (Er/Ei)s can be either positive or negative, depending on the value of n1/n2.
If n1/n2 > 1, then
qt > qi and cos qi > cos qt (no phase change).On the other hand if n1/n2 < 1, then
qi > qt and cos qt > cos qi (phase change).For p polarized light, the reflected beam the ratio Er/Ei can be either negative or positive depending on both the ratio n1/n2 and the incident angle
qi. The Er component is in phase with Ei at the interface if
Again we need Snell's law
.
Using the sum and difference angle formulae we can express this as
This inequality will be satisfied either if
The phase of the reflected wave in this case depends not only on the ratio of n1/n2, but also on
qi.If n2 > n1
s polarized light always undergoes a phase change upon reflection.
p polarized light undergoes a phase change upon reflection if either
