# Fresnel Equations

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

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Author: Dr. RĂ¼diger Paschotta

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

For example, *t*_{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 θ_{1} and θ_{2} (see Figure 1).

For example, the amplitude transmission coefficient is *t*_{s} for s polarization, i.e., if the electric field vector is perpendicular to the plane of incidence.

The power reflection coefficients are obtained simply by taking the modulus squared of the corresponding amplitude coefficients.
For the transmission, one must add a factor (*n*_{2} cos θ_{2}) / (*n*_{1} cos θ_{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:

## 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 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.

## Questions and Comments from Users

2021-01-11

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

Answer from the author:

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?

Answer from the author:

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 transmission 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?

Answer from the author:

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

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

and other articles in the category general optics

2020-11-18

What are the Fresnel equations of unpolarized incident light?

Answer from the author:

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.