Wikipedia: Fresnel equatations

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A set of equations deduced by Augustin-Jean Fresnel which describe the behaviour of light when moving between media of differing refractive indexes.

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.


                P      interface
                 *     |
                  *    |
                 . *   |
     n1        .    *  |               n2
              .  Ai  * |
              .       *|
   - - - - - - - - - - O - - - - - - - - - - - normal
              .       *|*   At .
              .  Ar  * |  *    . 
               .    *  |    * .
                 . *   |      *
                  *    |        *
                 *     |          *
                Q      |            S

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 A_i, A_r and A_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:

R_s = { sin(A_i - A_t) / sin(A_i + A_t) }² .

where A_t can be derived from A_i by Snell's law.

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

R_p = { tan(A_i - A_t) / tan(A_i + A_t) }² .

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), reflection coefficient is R = ( R_s + R_p ) / 2 .

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