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The Fresnel equations (or Fresnel coefficients) describe the reflection and transmission of light (or electromagnetic radiation in general) when incident on an interface between different optical media. They were deduced by Augustin-Jean Fresnel (/frˈnɛl/) who was the first to understand that light is a transverse wave, even though no one realized that the "vibrations" of the wave were electric and magnetic fields. For the first time, polarization could be understood quantitatively, as Fresnel's equations correctly predicted the differing behaviour of waves of the s and p polarizations incident upon a material interface.

Overview

When light strikes the interface between a medium with refractive index n1 and a second medium with refractive index n2, both reflection and refraction of the light may occur. The Fresnel equations describe the ratios of the reflected and transmitted waves' electric fields to the incident wave's electric field (the waves' magnetic fields can also be related using similar coefficients). Since these are complex ratios, they describe not only the relative amplitude, but phase shifts between the waves.

The equations assume the interface between the media is flat and that the media are homogeneous and isotropic.[1] The incident light is assumed to be a plane wave, which is sufficient to solve any problem since any incident light field can be decomposed into plane waves and polarizations.

S and P polarizations

The plane of incidence is defined by the incoming radiation's propagation vector and the normal vector of the surface.

There are two sets of Fresnel coefficients for two different linear polarization components of the incident wave. Since any polarization state can be resolved into a combination of two orthogonal linear polarizations, this is sufficient for any problem. Likewise, unpolarized (or "randomly polarized") light has an equal amount of power in each of two linear polarizations.

The s polarization refers to polarization of a wave's electric field normal to the plane of incidence (the z direction in the derivation below); then the magnetic field is in the plane of incidence. The p polarization refers to polarization of the electric field in the plane of incidence (the xy plane in the derivation below); then the magnetic field is normal to the plane of incidence.

Although the reflectivity and transmission are dependent on polarization, at normal incidence (θ = 0) there is no distinction between them so all polarization states are governed by a single set of Fresnel coefficients (and another special case is mentioned below in which that is true).

Power (intensity) reflection and transmission coefficients

Variables used in the Fresnel equationsWhen light strikes the interface between a medium with refractive index n1 and a second medium with refractive index n2, both reflection and refraction of the light may occur. The Fresnel equations describe the ratios of the reflected and transmitted waves' electric fields to the incident wave's electric field (the waves' magnetic fields can also be related using similar coefficients). Since these are complex ratios, they describe not only the relative amplitude, but phase shifts between the waves.

The equations assume the interface between the media is flat and that the media are homogeneous and isotropic.[1] The incident light is assumed to be a plane wave, which is sufficient to solve any problem since any incident light field can be decomposed into plane waves and polarizations.

S and P polarizations

The plane of incidence is defined by the incoming radiation's propagation vector and the normal vector of the surface.

There are two sets of Fresnel coefficients for two different linear polarization components of the incident wave. Since any polarization state can be resolved into a combination of two orthogonal linear polarizations, this is sufficient for any problem. Likewise, unpolarized (or "randomly polarized") light has an equal amount of power in each of two linear polarizations.

The s polarization refers to polarization of a wave's electric field normal to the plane of incidence (the z direction in the derivation below); then the magnetic field is in the plane of incidence. The p polarization refers to polarization of the electric field in the plane of incidence (the xy plane in the derivation below); then the magnetic field is normal to the plane of incidence.

Although the reflectivity and transmission are dependent on polarization, at normal incidence (θ = 0) there is no distinction between them so all polarization states are governed by a single set of Fresnel coefficients (and another special case is mentioned below in which that is true).

Power (intensity) reflection and transmission coefficients

Variables used in the Fresnel equations
The equations assume the interface between the media is flat and that the media are homogeneous and isotropic.[1] The incident light is assumed to be a plane wave, which is sufficient to solve any problem since any incident light field can be decomposed into plane waves and polarizations.

There are two sets of Fresnel coefficients for two different linear polarization components of the incident wave. Since any polarization state can be resolved into a combination of two orthogonal linear polarizations, this is sufficient for any problem. Likewise, unpolarized (or "randomly polarized") light has an equal amount of power in each of two linear polarizations.

The s polarization refers to polarization of a wave's electric field normal to the plane of incidence (the z direction in the derivation below); then the magnetic field is in the plane of incidence. The p polarization refers to polarization of the electric field in the plane of incidence (the xy plane in the derivation below); then the magnetic field is normal to the plane of incidence.

Although the reflectivity and transmission are dependent on polarization, at normal incidence (θ = 0) there is no distinction between them so all polarization states are governed by a single set of Fresnel coefficients (and another special case is mentioned below in which that is true).

Power (intensity) reflection and transmission coefficients i = θ r , {\displaystyle \theta _{\mathrm {i} }=\theta _{\mathrm {r} },}

and Snell's law: