Fresnel equations

The Fresnel equations are the four amplitude coefficients — r_s, r_p, t_s, t_p — giving what fraction of a plane electromagnetic wave's amplitude is reflected from or transmitted through a planar interface between two dielectric media of refractive indices n₁ and n₂. They are derived by applying the Maxwell boundary conditions — tangential continuity of E and H, normal continuity of D and B — to a plane wave hitting the interface at angle θ₁, decomposing separately for the two polarisations. For s-polarisation (E perpendicular to the plane of incidence), r_s = (n₁ cos θ₁ − n₂ cos θ₂)/(n₁ cos θ₁ + n₂ cos θ₂). For p-polarisation (E in the plane of incidence), r_p = (n₂ cos θ₁ − n₁ cos θ₂)/(n₂ cos θ₁ + n₁ cos θ₂). The intensity (power) reflectances are R_s = |r_s|², R_p = |r_p|².

The striking feature of r_p is that it passes through zero at Brewster's angle θ_B = arctan(n₂/n₁); at this angle the reflected wave is purely s-polarised, a fact Brewster measured systematically across dozens of materials in 1815. r_s, by contrast, is negative throughout and grows in magnitude monotonically toward 100% at grazing incidence. For n₂ < n₁, both coefficients develop complex phases beyond the critical angle θ_c = arcsin(n₂/n₁), producing total internal reflection with a polarisation-dependent phase shift (exploited in Fresnel rhombs for producing circular polarisation from linear). Applications: lens anti-reflective coatings (quarter-wave layers interfering at the right wavelength), polarising beam splitters (stacks of plates tilted toward Brewster), camera polarising filters, Brewster windows in gas-laser cavities, ellipsometry for thin-film analysis. Augustin-Jean Fresnel derived the equations in 1821–23 from his elastic-solid wave theory of light, decades before Maxwell showed light to be an electromagnetic wave.