Exactly how much light a surface throws back and how much it lets through — split by polarization, angle by angle. Two amplitude laws set the whole budget: R + T = 1, a faint 4% off plain glass at head-on, and one angle where p-polarized reflection vanishes to nothing. Rendered, not quoted.
source · amber Fresnel, Mémoire sur la loi des modifications que la réflexion imprime à la lumière polarisée (read 7 Jan 1823), Mém. Acad. Sci. 1823 — Gallica 1823 vol. (memoir has no per-article DOI; AMBER = volume link, article pp. 393–433)
Light of amplitude 1 hits the boundary between index n₁ and n₂ at incidence θᵢ. Snell fixes the refracted angle: n₁ sinθᵢ = n₂ sinθₜ. Fresnel gives the reflected amplitude for each polarization:
Reflectance is amplitude squared. Transmittance carries a geometric weight — power spreads across the refracted beam:
That weight is the load-bearing part. Without it, energy does not close.
This is the amplitude law under every neighbour. the-brewster-angle is one point on the p-curve — the zero. the-thin-lens loses 4% per glass face to exactly this reflection; anti-reflection coatings are engineered to cancel Rs, Rp at one wavelength. Same equations feed the-fabry-perot, whose finesse π√R/(1−R) is built from this very R.
One index step → the whole reflected/transmitted split. Fresnel, 1823.
Re-runs the sealed selfcheck() against the engine right now. Green = energy closes, Brewster zero holds, grazing → mirror. If Red Team (6) breaks the transmittance weight, this flips red and names the failure.
witness idle
Same pure functions verify headless in node — no canvas, no DOM.
Air → crown glass: n₁ = 1.00, n₂ = 1.50. Sweep the incidence angle and watch the split.
booting…
Head-on glass reflects 4.0% (R = 0.0400, both polarizations identical). p-pol reflection hits exactly zero at Brewster θ_B = 56.31°. Energy closes to 1e-9 at every angle. At grazing, both → 1: the mirror in the road.
Metals & absorbers. These pure real-index equations assume a lossless dielectric. A metal has complex n = n − ik; R never reaches 0 and Brewster becomes a shallow "pseudo-Brewster" minimum. Real-only math is wrong for gold, silver, silicon-at-UV.
Total internal reflection. Going glass→air past the critical angle, cosθt goes imaginary — Math.sqrt(1−s²) returns NaN. R = 1 but the phase (evanescent wave) is lost. This engine is scoped n₁<n₂, no TIR.
Absorption & roughness. R+T=1 only if nothing is absorbed or scattered. Real surfaces bury a few % in loss.
"Glass reflects the same fraction at every angle."
→ No. 4% only at head-on; Rₛ climbs to 100% at grazing.
"Reflected light is never polarized."
→ At Brewster the reflection is purely s-polarized — R_p = 0. That is polarized sunglasses' whole trick.
"T is just 1 − r²."
→ T needs the (n₂cosθₜ)/(n₁cosθᵢ) weight. Drop it and energy overshoots or undershoots 1.
Disclosed sabotage: strip the (n·cosθ) weighting from transmittance T. Reflectance is untouched and the curves barely move — but R + T no longer equals 1. The Witness (7) catches it live.
Weighting: intact — R+T = 1.