# Fresnel Equations

Definition: equations for the amplitude coefficients of transmission and reflection at the interface between two transparent homogeneous media

Author: Dr. Rüdiger Paschotta

Cite the article using its DOI: https://doi.org/10.61835/tql

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Fresnel equations specify the amplitude coefficients for transmission and reflection at a perfectly flat and clean interface between two transparent homogeneous media – for two different polarization directions:

- s polarization: electric field vector perpendicular to the plane of incidence (the plane containing all three beams, which is the drawing plane of Fig. 1)
- p polarization: electric field vector in the plane of incidence

The equations are:

$${t_{\rm{s}}} = \frac{{2{n_1}\cos {\theta _1}}}{{{n_1}\cos {\theta _1} + {n_2}\cos {\theta _2}}}$$ $${r_{\rm{s}}} = \frac{{{n_1}\cos {\theta _1} - {n_2}\cos {\theta _2}}}{{{n_1}\cos {\theta _1} + {n_2}\cos {\theta _2}}}$$ $${t_{\rm{p}}} = \frac{{2{n_1}\cos {\theta _1}}}{{{n_1}\cos {\theta _2} + {n_2}\cos {\theta _1}}}$$ $${r_{\rm{p}}} = \frac{{{n_1}\cos {\theta _2} - {n_2}\cos {\theta _1}}}{{{n_1}\cos {\theta _2} + {n_2}\cos {\theta _1}}}$$For example, <$t_\textrm{s}$> is the amplitude transmission coefficient for s polarization; the transmitted amplitude is that factor times the incident amplitude in that case (disregarding any phase changes for transmission in the media). <$n_1$> and <$n_2$> are the refractive indices of the two media. The corresponding propagation angles (measured against the normal direction) are <$\theta_1$> and <$\theta_2$> (see Figure 1).

For example, the amplitude transmission coefficient is <$t_\textrm{s}$> for s polarization, i.e., if the electric field vector is perpendicular to the plane of incidence.

The power reflection coefficients (reflectivity or reflectance values) are obtained simply by taking the modulus squared of the corresponding amplitude coefficients. For the transmissivity, one must add a factor <$(n_2 \cos \theta_2) / (n_1 \cos \theta_1)$> in order to take into account the different propagation angles.

Figure 2 shows in an example case how the reflectivity of the interface depends on the angle of incidence and the polarization. The reflection coefficient vanishes for p polarization if the angle of incidence is Brewster's angle (here: ≈55.4°).

For the simplest case with normal incidence on the interface, the power reflectivity (which is the modulus squared of the amplitude reflectivity) can be calculated with the following equation:

$$R = {\left( {\frac{{{n_1} - {n_2}}}{{{n_1} + {n_2}}}} \right)^2}$$## Application to Absorbing Media

Fresnel equations can also be applied to absorbing media, including metals. In that case, a complex refractive index needs to be used; the imaginary part is related to the absorbance. The reflection and transmission coefficients are then generally no more real numbers.

A difficulty occurs concerning the calculation of power transmission for absorbing media in cases with non-normal incidence. The above-mentioned correction factor for taking into account the different propagation angles will be complex, but the power transmission factor would of course have to be a real number. The problem is related to the fact that the optical intensity is no longer constant along the wavefronts.

## Surface Quality

In practice, Fresnel equations may not completely describe the optical properties of an interface since the surface quality is not high enough. For example, surface irregularities even only on a sub-micrometer scale may lead to substantial irregular wavefront distortions. However, very high optical quality can be achieved with careful surface preparation. For example, the optical quality is generally excellent when optical contact bonding is applied. There are also other methods for making optical contacts.

See also: Fresnel reflections, reflectivity, transmissivity, refraction, total internal reflection, Brewster's angle, refractive index

## Questions and Comments from Users

2021-01-11

Is the Fresnel transmission coefficient dependent on the incident ray amplitude?

The author's answer:

Fresnel theory is done in context of linear optics, where ray amplitudes or optical intensities cannot have such an influence. In principle, however, transmission coefficients can change due to nonlinear effects. In practice, such changes will usually be rather small, since nonlinear refractive index changes are usually small.

2021-02-21

Can you add phase and amplitude changes to your calculator?

The author's answer:

In principle yes, but I think that would be useful only for a small minority of readers while making the calculator more complex to use. The amplitude issue is quite trivial when you have the factor for the optical power, except for the mentioned correction factor concerning the propagation angles. The phase change in reflection can be only 0° or 180°.

2021-03-24

In absorbing media, the reflection and transmission coefficients can be complex numbers. Does that mean that there is a non-trivial phase change in the reflected/transmitted light?

The author's answer:

I am not sure what you mean with non-trivial, but indeed phase changes can occur in such situations.

2021-11-29

What are Fresnel losses?

The author's answer:

These are losses of light caused by Fresnel reflection at (typically uncoated) optical surfaces, e.g. on optical windows or fiber ends.

2022-01-10

Thank you very much for sharing your knowledge. I wrote a simple Matlab code to compute the reflectances and transmittances of an air – lossy dielectric (with complex refractive index) interface using the Fresnel equations. Given a 'p' polarization and a non-zero incidence angle, computing the transmittivity (<$t$>), I get a value of transmittance (<$T = |t|^2$>) which is different from the same transmittance computed as <$T = 1 - R$>. My question would be: why do Fresnel equations not work properly here?

The author's answer:

That's a very tricky issue. Fresnel equations do work, but you need to be more careful calculating the transmitted intensity or power from those. Note that you have constant optical intensity along the interface, but not over the cross-section of the transmitted beam in the absorbing medium, if it has some angle against normal direction. It is not even obvious how to calculate the beam direction in that case, where Fresnel equations predict complex angles. The details are too sophisticated to be discussed here.

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2020-11-18

What are the Fresnel equations of unpolarized incident light?

The author's answer:

Fresnel equations are always for polarized light. However, you may, for example, calculate the reflectivity for both polarizations and take the arithmetic mean value as the reflectivity for unpolarized light.