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 French engineer and physicist Augustin-Jean Fresnel () who was the first to understand that light is a transverse wave, when no one realized that the waves 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 give the ratio of the reflected wave's electric field to the incident wave's electric field, and the ratio of the transmitted wave's electric field to the incident wave's electric field, for each of two components of polarization. (The magnetic fields can also be related using similar coefficients.) These ratios are generally complex, describing not only the relative amplitudes but also the phase shifts at the interface. The equations assume the interface between the media is flat and that the media are homogeneous and isotropic. 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
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. The names "s" and "p" for the polarization components refer to German "senkrecht" (perpendicular or normal) and "parallel" (parallel to the plane of incidence). Although the reflection 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).
Configuration
In the diagram, an incident plane wave in the direction of the ray IO strikes the interface between two media of refractive indices n1 and n2 at point O. Part of the wave is reflected in the direction OR, and part refracted in the direction OT. 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: θ i = θ r , {\displaystyle \theta _{\mathrm {i} }=\theta _{\mathrm {r} },} and Snell's law: n 1 sin θ i = n 2 sin θ t . {\displaystyle n_{1}\sin \theta _{\mathrm {i} }=n_{2}\sin \theta _{\mathrm {t} }.}
The behavior of light striking the interface is explained by considering the electric and magnetic fields that constitute an electromagnetic wave, and the laws of electromagnetism, as shown below. The ratio of waves' electric field (or magnetic field) amplitudes are obtained, but in practice one is more often interested in formulae which determine power coefficients, since power (or irradiance) is what can be directly measured at optical frequencies. The power of a wave is generally proportional to the square of the electric (or magnetic) field amplitude.
Power (intensity) reflection and transmission coefficients
We call the fraction of the incident power that is reflected from the interface the reflectance (or reflectivity, or power reflection coefficient) R, and the fraction that is refracted into the second medium is called the transmittance (or transmissivity, or power transmission coefficient) T. Note that these are what would be measured right at each side of an interface and do not account for attenuation of a wave in an absorbing medium following transmission or reflection. The reflectance for s-polarized light is
R s = | Z 2 cos θ i − Z 1 cos θ t Z 2 cos θ i + Z 1 cos θ t | 2 , {\displaystyle R_{\mathrm {s} }=\left|{\frac {Z_{2}\cos \theta _{\mathrm {i} }-Z_{1}\cos \theta _{\mathrm {t} }}{Z_{2}\cos \theta _{\mathrm {i} }+Z_{1}\cos \theta _{\mathrm {t} }}}\right|^{2},}
while the reflectance for p-polarized light is
R p = | Z 2 cos θ t − Z 1 cos θ i Z 2 cos θ t + Z 1 cos θ i | 2 , {\displaystyle R_{\mathrm {p} }=\left|{\frac {Z_{2}\cos \theta _{\mathrm {t} }-Z_{1}\cos \theta _{\mathrm {i} }}{Z_{2}\cos \theta _{\mathrm {t} }+Z_{1}\cos \theta _{\mathrm {i} }}}\right|^{2},}
where Z1 and Z2 are the wave impedances of media 1 and 2, respectively. We assume that the media are non-magnetic (i.e., μ1 = μ2 = μ0), which is typically a good approximation at optical frequencies (and for transparent media at other frequencies). Then the wave impedances are determined solely by the refractive indices n1 and n2:
Z i = Z 0 n i , {\displaystyle Z_{i}={\frac {Z_{0}}{n_{i}}}\,,}
where Z0 is the impedance of free space and i = 1, 2. Making this substitution, we obtain equations using the refractive indices:
R s = | n 1 cos θ i − n 2 cos θ t n 1 cos θ i + n 2 cos θ t | 2 = | n 1 cos θ i − n 2 1 − ( n 1 n 2 sin θ i ) 2 n 1 cos θ i + n 2 1 − ( n 1 n 2 sin θ i ) 2 | 2 , {\displaystyle R_{\mathrm {s} }=\left|{\frac {n_{1}\cos \theta _{\mathrm {i} }-n_{2}\cos \theta _{\mathrm {t} }}{n_{1}\cos \theta _{\mathrm {i} }+n_{2}\cos \theta _{\mathrm {t} }}}\right|^{2}=\left|{\frac {n_{1}\cos \theta _{\mathrm {i} }-n_{2}{\sqrt {1-\left({\frac {n_{1}}{n_{2}}}\sin \theta _{\mathrm {i} }\right)^{2}}}}{n_{1}\cos \theta _{\mathrm {i} }+n_{2}{\sqrt {1-\left({\frac {n_{1}}{n_{2}}}\sin \theta _{\mathrm {i} }\right)^{2}}}}}\right|^{2}\!,}
R p = | n 1 cos θ t − n 2 cos θ i n 1 cos θ t + n 2 cos θ i | 2 = | n 1 1 − ( n 1 n 2 sin θ i ) 2 − n 2 cos θ i n 1 1 − ( n 1 n 2 sin θ i ) 2 + n 2 cos θ i | 2 . {\displaystyle R_{\mathrm {p} }=\left|{\frac {n_{1}\cos \theta _{\mathrm {t} }-n_{2}\cos \theta _{\mathrm {i} }}{n_{1}\cos \theta _{\mathrm {t} }+n_{2}\cos \theta _{\mathrm {i} }}}\right|^{2}=\left|{\frac {n_{1}{\sqrt {1-\left({\frac {n_{1}}{n_{2}}}\sin \theta _{\mathrm {i} }\right)^{2}}}-n_{2}\cos \theta _{\mathrm {i} }}{n_{1}{\sqrt {1-\left({\frac {n_{1}}{n_{2}}}\sin \theta _{\mathrm {i} }\right)^{2}}}+n_{2}\cos \theta _{\mathrm {i} }}}\right|^{2}\!.}
The second form of each equation is derived from the first by eliminating θt using Snell's law and trigonometric identities. As a consequence of conservation of energy, one can find the transmitted power (or more correctly, irradiance: power per unit area) simply as the portion of the incident power that isn't reflected:
T s = 1 − R s {\displaystyle T_{\mathrm {s} }=1-R_{\mathrm {s} }}
and
T p = 1 − R p {\displaystyle T_{\mathrm {p} }=1-R_{\mathrm {p} }}
Note that all such intensities are measured in terms of a wave's irradiance in the direction normal to the interface; this is also what is measured in typical experiments. That number could be obtained from irradiances in the direction of an incident or reflected wave (given by the magnitude of a wave's Poynting vector) multiplied by cos θ for a wave at an angle θ to the normal direction (or equivalently, taking the dot product of the Poynting vector with the unit vector normal to the interface). This complication can be ignored in the case of the reflection coefficient, since cos θi = cos θr, so that the ratio of reflected to incident irradiance in the wave's direction is the same as in the direction normal to the interface. Although these relationships describe the basic physics, in many practical applications one is concerned with "natural light" that can be described as unpolarized. That means that there is an equal amount of power in the s and p polarizations, so that the effective reflectivity of the material is just the average of the two reflectivities:
R e f f = 1 2 ( R s + R p ) . {\displaystyle R_{\mathrm {eff} }={\frac {1}{2}}\left(R_{\mathrm {s} }+R_{\mathrm {p} }\right).}
For low-precision applications involving unpolarized light, such as computer graphics, rather than rigorously computing the effective reflection coefficient for each angle, Schlick's approximation is often used.
Special cases
Normal incidence For the case of normal incidence, θi = θt = 0, and there is no distinction between s and p polarization. Thus, the reflectance simplifies to
R 0 = | n 1 − n 2 n 1 + n 2 | 2 . {\displaystyle R_{0}=\left|{\frac {n_{1}-n_{2}}{n_{1}+n_{2}}}\right|^{2}\,.}
For common glass (n2 ≈ 1.5) surrounded by air (n1 = 1), the power reflectance at normal incidence can be seen to be about 4%, or 8% accounting for both sides of a glass pane.
Brewster's angle
At a dielectric interface from n1 to n2, there is a particular angle of incidence at which Rp goes to zero and a p-polarised incident wave is purely refracted, thus all reflected light is s-polarised. This angle is known as Brewster's angle, and is around 56° for n1 = 1 and n2 = 1.5 (typical glass).
Total internal reflection
When light travelling in a denser medium strikes the surface of a less dense medium (i.e., n1 > n2), beyond a particular incidence angle known as the critical angle, all light is reflected and Rs = Rp = 1. This phenomenon, known as total internal reflection, occurs at incidence angles for which Snell's law predicts that the sine of the angle of refraction would exceed unity (whereas in fact sin θ ≤ 1 for all real θ). For glass with n = 1.5 surrounded by air, the critical angle is approximately 42°.
45° incidence Reflection at 45° incidence is very commonly used for making 90° turns. For the case of light traversing from a less dense medium into a denser one at 45° incidence (θ = 45°), it follows algebraically from the above equations that Rp equals the square of Rs:
R p = R s 2 {\displaystyle R_{\text{p}}=R_{\text{s}}^{2}}
This can be used to either verify the consistency of the measurements of Rs and Rp, or to derive one of them when the other is known. This relationship is only valid for the simple case of a single plane interface between two homogeneous materials, not for films on substrates, where a more complex analysis is required. Measurements of Rs and Rp at 45° can be used to estimate the reflectivity at normal incidence. The "average of averages" obtained by calculating first the arithmetic as well as the geometric average of Rs and Rp, and then averaging these two averages again arithmetically, gives a value for R0 with an error of less than about 3% for most common optical materials. This is useful because measurements at normal incidence can be difficult to achieve in an experimental setup since the incoming beam and the detector will obstruct each other. However, since the dependence of Rs and Rp on the angle of incidence for angles below 10° is very small, a measurement at about 5° will usually be a good approximation for normal incidence, while allowing for a separation of the incoming and reflected beam.
Complex amplitude reflection and transmission coefficients The above equations relating powers (which could be measured with a photometer for instance) are derived from the Fresnel equations which solve the physical problem in terms of electromagnetic field complex amplitudes, i.e., considering phase shifts in addition to their amplitudes. Those underlying equations supply generally complex-valued ratios of those EM fields and may take several different forms, depending on the formalism used. The complex amplitude coefficients for reflection and transmission are usually represented by lower case r and t (whereas the power coefficients are capitalized). As before, we are assuming the magnetic permeability, µ of both media to be equal to the permeability of free space µ0 as is essentially true of all dielectrics at optical frequencies.
In the following equations and graphs, we adopt the following conventions. For s polarization, the reflection coefficient r is defined as the ratio of the reflected wave's complex electric field amplitude to that of the incident wave, whereas for p polarization r is the ratio of the waves complex magnetic field amplitudes (or equivalently, the negative of the ratio of their electric field amplitudes). The transmission coefficient t is the ratio of the transmitted wave's complex electric field amplitude to that of the incident wave, for either polarization. The coefficients r and t are generally different between the s and p polarizations, and even at normal incidence (where the designations s and p do not even apply!) the sign of r is reversed depending on whether the wave is considered to be s or p polarized, an artifact of the adopted sign convention (see graph for an air-glass interface at 0° incidence). The equations consider a plane wave incident on a plane interface at angle of incidence θ i {\displaystyle \theta _{\mathrm {i} }} , a wave reflected at angle θ r = θ i {\displaystyle \theta _{\mathrm {r} }=\theta _{\mathrm {i} }} , and a wave transmitted at angle θ t {\displaystyle \theta _{\mathrm {t} }} . In the case of an interface into an absorbing material (where n is complex) or total internal reflection, the angle of transmission does not generally evaluate to a real number. In that case, however, meaningful results can be obtained using formulations of these relationships in which trigonometric functions and geometric angles are avoided; the inhomogeneous waves launched into the second medium cannot be described using a single propagation angle. Using this convention,
r s = n 1 cos θ i − n 2 cos θ t n 1 cos θ i + n 2 cos θ t , t s = 2 n 1 cos θ i n 1 cos θ i + n 2 cos θ t , r p = n 2 cos θ i − n 1 cos θ t n 2 cos θ i + n 1 cos θ t , t p = 2 n 1 cos θ i n 2 cos θ i + n 1 cos θ t . {\displaystyle {\begin{aligned}r_{\text{s}}&={\frac {n_{1}\cos \theta _{\text{i}}-n_{2}\cos \theta _{\text{t}}}{n_{1}\cos \theta _{\text{i}}+n_{2}\cos \theta _{\text{t}}}},\\[3pt]t_{\text{s}}&={\frac {2n_{1}\cos \theta _{\text{i}}}{n_{1}\cos \theta _{\text{i}}+n_{2}\cos \theta _{\text{t}}}},\\[3pt]r_{\text{p}}&={\frac {n_{2}\cos \theta _{\text{i}}-n_{1}\cos \theta _{\text{t}}}{n_{2}\cos \theta _{\text{i}}+n_{1}\cos \theta _{\text{t}}}},\\[3pt]t_{\text{p}}&={\frac {2n_{1}\cos \theta _{\text{i}}}{n_{2}\cos \theta _{\text{i}}+n_{1}\cos \theta _{\text{t}}}}.\end{aligned}}}
For the case where the magnetic permeabilities are non-negligible, the equations change such that every appearance of n i {\displaystyle n_{i}} is replaced by n i / μ i {\displaystyle n_{i}/\mu _{i}} (for both i = 1 , 2 {\displaystyle i=1,2} ). One can see that ts = rs + 1 and n2/n1tp = rp + 1. One can write very similar equations applying to the ratio of the waves' magnetic fi