Fresnel Equations

Equivalent forms

R_p = \left|\frac{n_1\cos\theta_t - n_2\cos\theta_i}{n_1\cos\theta_t + n_2\cos\theta_i}\right|^2

R_0 = \left(\frac{n_1-n_2}{n_1+n_2}\right)^2

Boundary continuity of one field yields the entire angle-and-polarization behavior of every reflecting surface — mirrors, lenses, lakes, and lasers alike.

Symbol	Name	SI unit	Dimension	Range
$theta_i$	Incidence angle Angle from the surface normal	deg	1	0 – 90
$n1$	Index in Refractive index of the incident medium	dimensionless	1	1 – 2.5
$n2$	Index out Refractive index of the second medium	dimensionless	1	1 – 2.5
$R_s$	s-reflectanceoutput Reflected power fraction, s-polarization	dimensionless	1	0 – 1

Unit systems

SI:: angles in radians, R dimensionless
CGS:: same
Imperial:: degrees

Where it holds

Smooth, non-magnetic, non-absorbing interface. For metals/absorbers n becomes complex; surface roughness comparable to lambda adds scattering not captured here.

Discovery

Augustin-Jean Fresnel · 1823

Fresnel derived the reflection and transmission amplitudes from his elastic-ether wave theory, correctly predicting the polarization dependence and Brewster's angle decades before Maxwell showed light was electromagnetic and rederived the same formulas.

Try this

Why does a lake act like a mirror at a low angle but a window at noon?

The Fresnel equations give the exact fraction of light reflected at a surface as a function of angle and polarization — from a few percent at normal incidence to nearly 100% at grazing angles.

Research status: stable

Common misconceptions

Reflectance is not a single number for a surface — it depends strongly on angle and polarization. At normal incidence both polarizations coincide; they diverge as the angle grows.

Derivation

Matching tangential E and H across the interface for each polarization yields the amplitude coefficients

r_s = (n_{1} \cos ti - n_{2} \cos tt)/(n_{1} \cos ti + n_{2} \cos tt)

and

r_p = (n_{2} \cos ti - n_{1} \cos tt)/(n_{2} \cos ti + n_{1} \cos tt)

, with

\sin ti/\sin tt = n_{2}/n_{1}

.

Reflectance is

R = |r|^2

;

r_p = 0

defines Brewster's angle

\tan tB = n_{2}/n_{1}

.

Limiting cases

\theta _i = 0

(normal)⟶

R = ((n_{1}-n_{2})/(n_{1}+n_{2}))^2

—

\sim 4

% for air-glass

\theta _i =

Brewster⟶

R_p = 0

— reflected light is fully s-polarized

theta_i -> 90 (grazing)⟶ R -> 1 — the surface becomes a near-perfect mirror

Brewster's Angle→Snell's Law→Malus's Law→Critical Angle (Total Internal Reflection)→