n₁ of the incident medium
n₂ of the transmission medium
θ₁ relative to the interface normal
Phase indices at λ = 589.3 nm (sodium D line). Liquids use representative values at 20 °C. Unpolarized light.
19.25
°56.59
°4.365
%Rs: 6.057% · Rp: 2.674%
95.63
%When a ray of light crosses the interface between two media with different refractive indices, it changes direction. The relationship between the angle of incidence (θ₁) and the angle of refraction (θ₂) is given by Snell's Law, independently formulated by Ibn Sahl (984) and Willebrord Snell (1621):
where n₁ and n₂ are the refractive indices of the media, defined as the ratio of the speed of light in vacuum (c) to the speed in the medium: n = c/v. On entering a higher-index medium, the ray bends toward the normal.
When light travels from a denser to a less dense medium (n₁ > n₂), there exists a critical angle θc. At that boundary the refracted ray travels along the interface (θ₂ = 90°) with no normal energy flux; above θc, total internal reflection occurs and no propagating transmitted ray exists:
TIR is the fundamental operating principle of optical fibers. Light remains "trapped" inside the fiber core (high-n core surrounded by a low-n cladding), enabling long-distance data transmission with minimal losses.
The Fresnel equations(Augustin-Jean Fresnel, 1823) describe what fraction of the light intensity is reflected and transmitted when light strikes a planar interface between two dielectric media. They are derived from the electromagnetic field boundary conditions (Maxwell's equations) at the interface.
s-Polarization (TE)
p-Polarization (TM)
For unpolarized light, the average reflectance is the arithmetic mean of both components:
At normal incidence (θ₁ = 0°), both polarizations are identical and reflectance simplifies to: R = ((n₁ − n₂)/(n₁ + n₂))². For an air-glass interface (n ≈ 1.52): R ≈ 4.3%.
There exists a particular angle, Brewster's angle (David Brewster, 1815), where the p-polarization reflectance (Rp) vanishes completely. The reflected light is then fully polarized in the s-plane (perpendicular to the plane of incidence):
Brewster's angle is the basis of Brewster windowsin laser cavities, where optical surfaces placed at Brewster's angle minimize reflection losses for a preferred polarization.