Fresnel equations

The Fresnel equations, deduced by Augustin-Jean Fresnel, describe the behaviour of light when moving between media of differing refractive indices.

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.

In the diagram above, an incident light ray PO strikes at point O the interface between two media of refractive indexes n₁ and n₂. Part of the ray is reflected as ray OQ and part refracted as ray OS. The angles that the incident, reflected and refracted rays make to the normal of the interface are given as θ_i, θ_r and θ_t, respectively. The relationship between these angles is given by the law of reflection and Snell's law.

The fraction of the 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 may be used to calculateR and T in a given situation.

The calculations of R and T depend on polarisation of the incident ray. If the light is polarised with the electric field of the light perpendicular to the plane of the diagram above (s-polarised), the reflection coefficient is given by:

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

If the incident light is polarised in the plane of the diagram (p-polarised), the R is given by:

.

The transmission coefficient in each case is given by T_s = 1 - R_s and T_p = 1 - R_p.

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

At one particular angle for a given n₁ and n₂, the value of R_p goes to zero and an p-polarised incident ray is purely refracted. This is known as Brewster's angle.

When moving from a more dense medium into a less dense one (i.e. n₁ > n₂), above 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.

When the light is at near-normal incidence to the interface(θ_i ≈ θ_t ≈ 0) , the reflection coefficient is given by:

.

Note that reflection by a window is from the front side as well as the back side, and that the latter also includes light that goes back and forth a number of times between the two sides. The total is 2R/(1+R).