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 *n*_{1} into a second medium with refractive index *n*_{2}, 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 *n*_{1} and *n*_{2}. 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 calculate*R* 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 *n*_{1} and *n*_{2}, 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. *n*_{1} > *n*_{2}), 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).