Fresnel equations

When a light wave reaches a boundary between two media, part of its energy may be reflected and part transmitted. The Fresnel equations give the amplitude reflection and transmission coefficients at a flat interface, under specified optical conditions. Their inputs include the refractive indices of the media and the directions of the incident and transmitted waves; the latter are related by Snell's law. The equations follow from electromagnetic boundary conditions at the interface.

For a ray meeting a surface, the plane of incidence contains the incident ray and the surface normal. In s polarization, the electric field is perpendicular to this plane; in p polarization, it lies within it. The Fresnel equations give distinct coefficients for these two cases, so the amount of reflected light generally depends on polarization as well as angle and material. For unpolarized light, reflectance can be found by combining the two polarization contributions with the appropriate averaging.

The reflection coefficient describes the reflected electric-field amplitude relative to the incident amplitude. For a non-absorbing interface, the reflectance for a given polarization is the squared magnitude of its amplitude reflection coefficient. Converting a transmission amplitude into transmitted power also requires a factor accounting for the refractive indices and propagation angles. At normal incidence between transparent, non-absorbing media, the reflectance is R = ((n₁ − n₂)/(n₁ + n₂))², where n₁ and n₂ are the refractive indices.

For a transparent dielectric boundary, the p-polarized reflection coefficient becomes zero at Brewster's angle. At that angle, reflected unpolarized light is s-polarized. Fresnel calculations are used to estimate reflection losses and design optical systems, including lenses, windows and laser resonators. The basic transparent-medium formulas require care for absorbing materials: these can be treated using a complex refractive index, and transmitted power must be calculated with attention to absorption and energy flow.