Light: Fresnel equations

Principle

The Fresnel equations, describe the behavior of light when moving between media of differing refractive indices (see Light: refraction). The reflection of light that the equations predict is known as Fresnel reflection and the refraction is described by Snell’s law.

When light moves from a medium of a given refractive index n₁ into a second medium with refractive index n₂, both reflection and refraction of the light may occur.

The fraction of the intensity of incident light that is reflected from the interface is given by the reflection coefficient R, and the fraction refracted by the transmission coefficient T. The Fresnel equations assume that the two media are both non-magnetic.

Fig. 1. a. Visualization of the decomposition of the unpolarized beam is the p and s-polarized parts. b. The angle of incidence is Brewster's angle. The reflected beam is purely s-polarized.

The calculations of R and T depend on the polarization of the incident ray. If the light is polarized with the electric field of the light perpendicular to the plane of Fig. 1a (s-polarized), the reflection coefficient is given by R_s. If the incident light is polarized in the plane of the diagram (p-polarized), the R is given by R_p.

A graphical example of the calculation of R_s and R_p is given in Fig. 2.

Fig. 2 Left, R_s and R_p for a light beam going from a medium with a low refractive index to one with a high index. Right, R_s and R_p for a light beam going from high to low refractive index.

At one particular angle for a given n₁ and n₂, the value of R_p goes to zero and a s-polarized incident ray is purely refracted. This angle is known as Brewster's angle, and is around 56° for a glass medium in air or vacuum.

When moving from a more dense medium into a less dense one (i.e. n₁ > n₂), exceeding an incidence angle known as the critical angle, all light is reflected and R_s = R_p = 1. This phenomenon is known as total internal reflection (see Snell’s law). The critical angle is approximately 41° for glass in air.

If the incident light is unpolarized (containing an equal mix of s- and p-polarizations), the reflection coefficient is R = (R_s + R_p)/2.

When the light is about perpendicular to the interface (θ_i ≈ θ_t ≈ 0), R and T are given by:

R = {(n₁ – n₂)/(n₁ + n₂)}², and

T = 1 – R.

For common (clean!) glass, R is about 4%. Note that reflection by a window is from the front side as well as the back side, and that some of the light bounces back and forth a number of times between the two sides. The combined reflection coefficient for this case is 2R/(1 + R).

Application

Applying Brewster's angle is a way to produce pure polarized light. Repeated reflection and refraction on thin, parallel layers is responsible for the colors seen in oil films on water, used in optics to make reflection free lenses and perfect mirrors, etc.

More info

The general equations for R_s and R_p are:

and:

where θ_t can be derived from θ_i by Snell's law.

In each case T is given by T_s = 1 − R_s and T_p = 1 − R_p.

R and S correspond to the ratio of the intensity of the incident ray to that of the reflected and transmitted rays. Equations for coefficients corresponding to ratios of the electric field amplitudes of the waves can also be derived, and these are also called "Fresnel equations". Here, we use squared amplitude (just as in the above equations n²). Therefore, the intensity is expressed in Watts, more precisely in W/steradian (see Light: photometric and radiometric units of measure). The above equations are approximations. The completely general Fresnel equations are more complex (see e.g. .Wikipedia).