Fresnel equations

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The Fresnel equations (or Fresnel conditions), deduced by Augustin-Jean Fresnel (pronounced /freɪˈnɛl/), describe the behaviour of light when moving between media of differing refractive indices. The reflection of light that the equations predict is known as Fresnel reflection.



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 on the right, an incident light ray PO strikes at point O the interface between two media of refractive indices 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 θi, θr and θt, respectively. The relationship between these angles is given by the law of reflection: θi = θr ; and Snell's law: sin(θi) / sin(θt) = n2 / n1 .

The fraction of the incident power that is reflected from the interface is given by the reflectance R and the fraction that is refracted is given by the transmittance T.[1] The media are assumed to be non-magnetic.

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 θt can be derived from θi by Snell's law and is simplified using trigonometric identities.

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

As a consequence of the conservation of energy, the transmission coefficient in each case is given by Ts = 1 − Rs and Tp = 1 − Rp.[2]

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

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