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Fresnel equations

The Fresnel equations, deduced by Augustin-Jean Fresnel 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 indexes 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 and Snell's law.

The fraction of the incident power that is reflected from the interface is given by the reflection coefficient R, and the fraction that is refracted is given by the transmission coefficient T. 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:

R_s = left[frac{sin (theta_t - theta_i)}{sin (theta_t + theta_i)} right]^2
=left[frac{n_1cos(theta_i)-n_2cos(theta_t)}{n_1cos(theta_i)+n_2cos(theta_t)}right]^2 =left[frac{n_1cos(theta_i)-n_2sqrt{1-left(frac{n_1}{n_2} sintheta_iright)^2}}{n_1cos(theta_i)+n_2sqrt{1-left(frac{n_1}{n_2} sintheta_iright)^2}}right]^2

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:

R_p = left[frac{tan (theta_t - theta_i)}{tan (theta_t + theta_i)} right]^2
=left[frac{n_1cos(theta_t)-n_2cos(theta_i)}{n_1cos(theta_t)+n_2cos(theta_i)}right]^2 =left[frac{n_1sqrt{1-left(frac{n_1}{n_2} sintheta_iright)^2}-n_2cos(theta_i)}{n_1sqrt{1-left(frac{n_1}{n_2} sintheta_iright)^2}+n_2cos(theta_i)}right]^2

The transmission coefficient in each case is given by Ts = 1 − Rs and Tp = 1 − Rp.

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

Equations for coefficients corresponding to ratios of the electric field amplitudes of the waves can also be derived, and these are also called "Fresnel equations".

At one particular angle for a given n1 and n2, the value of Rp goes to zero and a p-polarised incident ray is purely refracted. This angle is known as Brewster's angle, and is around 56° for a glass medium in air or vacuum. Note that this statement is only true when the refractive indexes of both materials are real numbers, as is the case for materials like air and glass. For materials that absorb light, like metals and semiconductors, n is complex, and Rp does not generally go to zero.

When moving from a denser medium into a less dense one (i.e., n1 > n2), above an incidence angle known as the critical angle, all light is reflected and Rs = Rp = 1. This phenomenon is known as total internal reflection. The critical angle is approximately 41° for glass in air.

When the light is at near-normal incidence to the interface (θi ≈ θt ≈ 0), the reflection and transmission coefficient are given by:

R = R_s = R_p = left(frac{n_1 - n_2}{n_1 + n_2} right)^2
T = T_s = T_p = 1-R = frac{4 n_1 n_2}{left(n_1 + n_2 right)^2}

For common glass, the reflection coefficient is about 4%. Note that reflection by a window is from the front side as well as the back side, and that some of the light bounces back and forth a number of times between the two sides. The combined reflection coefficient for this case is 2R/(1 + R), when interference can be neglected.

In reality, when light makes multiple reflections between two parallel surfaces, the multiple beams of light generally interfere with one another, and the surfaces act as a Fabry-Perot interferometer. This effect is responsible for the colours seen in oil films on water, and it is used in optics to make optical coatings that can greatly lower the reflectivity or can be used as an optical filter.

It should be noted that the discussion given here assumes that the permeability μ is equal to the vacuum permeability μ0 in both media. This is approximately true for most dielectric materials, but not for some other types of material. The completely general Fresnel equations are more complicated.

See also


  • Hecht, Eugene (1987). Optics. 2nd ed., Addison Wesley. ISBN 0-201-11609-X.

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