Fresnel equations

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The Fresnel equations, deduced by Augustin-Jean Fresnel (IPA pronunciation: [fɹeɪˈnɛl]), 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.

$A light ray striking the interface between two media is split into two - a reflected part and a refracted part.$

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 θ_i, θ_r and θ_t, respectively. The relationship between these angles is given by the law of reflection and Snell's law.

The fraction of the intensity of incident light that is reflected from the interface is given by the reflection coefficient R, and the fraction refracted by the transmission coefficient T. The Fresnel equations, which are based on the assumption that the two materials are both non-magnetic, may be used to calculate R and T in a given situation.

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:

$R_s = \left[ \frac{\sin (\theta_t - \theta_i)}{\sin (\theta_t + \theta_i)} \right]^2=\left[\frac{n_1\cos(\theta_i)-n_2\cos(\theta_t)}{n_1\cos(\theta_i)+n_2\cos(\theta_t)}\right]^2$

where θ_t can be derived from θ_i by Snell's law.

If the incident light is polarised in the plane of the diagram (p-polarised), the R is given by:

$R_p = \left[ \frac{\tan (\theta_t - \theta_i)}{\tan (\theta_t + \theta_i)} \right]^2=\left[\frac{n_1\cos(\theta_t)-n_2\cos(\theta_i)}{n_1\cos(\theta_t)+n_2\cos(\theta_i)}\right]^2$

The transmission coefficient in each case is given by T_s = 1 − R_s and T_p = 1 − R_p.

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

The reflection and transmission coefficients correspond to the ratio of the intensity of the incident ray to that of the reflected and transmitted rays. 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 R_p 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.

When moving from a more dense medium into a less dense one (i.e. n₁ > n₂), above an incidence angle known as the critical angle, all light is reflected and R_s = R_p = 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 (θ_i ≈ θ_t ≈ 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.