Fresnel Polarization Calculator
Fresnel reflection and transmission coefficients, their phases, and the Brewster and critical angles at an interface, including absorbing media and total internal reflection. r_p = −r_s at normal incidence (Born & Wolf; the thin-film pages use Macleod's opposite sign).
rs = (n₁cos θ₁ − N₂cos θ₂)/(n₁cos θ₁ + N₂cos θ₂), rp = (N₂cos θ₁ − n₁cos θ₂)/(N₂cos θ₁ + n₁cos θ₂)
n₁ sin θ₁ = N₂ sin θ₂, N₂ = n₂ − iκ₂; in total internal reflection N₂cos θ₂ = −i√(n₁²sin²θ₁ − n₂²)
Ts = Re(N₂cos θ₂)/(n₁cos θ₁)·|ts|², R + T = 1
Convention: time factor ei(ωt − kz), so absorption is N = n − iκ and a positive phase is an advance; rp = −rsat normal incidence (Born & Wolf, Hecht; the ellipsometry convention, Δ = 180° there).
Refraction angle θ₂
19.20°
Brewster / critical angle
56.66° / —
Rₚ = 0 at Brewster; TIR above the critical angle
R (unpolarized light)
4.41 %
reflected light polarized 38.7 %
Δ = arg(rp/rs), ψ
180.0°
tan ψ = |rp|/|rs|: ψ = 33.63°
Amplitude coefficients (magnitude ∠ phase)
rₛ
0.2474 ∠ 180.0°
rₚ
0.1646 ∠ 0.0°
tₛ
0.7526 ∠ 0.0°
tₚ
0.7662 ∠ 0.0°
Power
Rs
6.121 %
Rp
2.708 %
Ts
93.879 %
Tp
97.292 %
Reflectance and transmittance against the angle of incidence
Reflection phases (°)
Model and references
Exact. The equations are exact within the stated assumptions, or are a standard's tabulated values. The results are as good as the inputs. About the tiers
Plane wave at a flat interface from a lossless medium onto N₂ = n₂ − iκ₂; complex coefficients in total reflection; time factor e^i(ωt − kz), r_p = −r_s at normal incidence. Not modelled: films, roughness.
- Born M., Wolf E. (1999). Principles of Optics, 7th ed., §1.5 Reflection and refraction of a plane wave (Fresnel formulae, total reflection). Cambridge University Press. doi:10.1017/CBO9781139644181.010
- Goldstein D. H. (2011). Polarized Light, 3rd ed., ch. Fresnel Equations, pp. 117–147. CRC Press. doi:10.1201/b10436