Hit glass at just the right slant and the reflection turns perfectly polarized — the physics behind polarized sunglasses and no-glare camera filters. At one special angle the reflected and refracted rays cross at a right angle and the p‑polarized reflectance drops to exactly zero. Rendered, not quoted.
source Brewster, D. “On the laws which regulate the polarisation of light by reflexion from transparent bodies,” Phil. Trans. R. Soc. Lond. 105 (1815) 125–159 · doi:10.1098/rstl.1815.0010
Light meets a flat boundary between two media of refractive index n1 (air) and n2 (glass). Part reflects, part refracts. Reflectance depends on polarization: s (perpendicular to the plane of incidence) and p (parallel).
tan(θB) = n₂ / n₁At the Brewster angle θB, the p‑channel Fresnel reflectance Rp falls to 0. Only s‑polarized light reflects — the glare is fully polarized, so a filter aligned to block it kills the reflection.
Fresnel amplitude: rp = (n₂cosθi − n₁cosθt) / (n₂cosθi + n₁cosθt), Rp = rp2.
Reflection turns polarized — Brewster 1815. He measured that the polarizing angle is set by the refractive index alone: tan(θB) = n₂/n₁, and there the reflected and refracted rays meet at a right angle.
This is one special angle of the-fresnel-equations (the full Rs/Rp vs. angle laws): the exact incidence where the p‑curve touches zero. The fully polarized beam it produces is the input the-malus-law then acts on with I = I0cos²θ. Fresnel R+T = 1 conserves the energy that does not reflect.
Re-runs the invariants live against the engine below. It confirms θB = atan(n₂/n₁), the reflected+refracted right angle, and Rp(θB) = 0. If the adversary tampers the ratio, the witness flips red on the next tick.
Two refractive indices and an incidence angle. Default: air–glass, n1 = 1.00, n2 = 1.50. Drag n2 to see the Brewster angle track it.
Curves: Rp dips to zero at θB; Rs climbs monotonically. Ray inset shows the reflected (up‑left) and refracted (down‑right) rays meeting at a right angle when θi = θB.
Proven at boot for air–glass (n2 = 1.5):
“Brewster’s angle is where reflection is strongest.” False — it is where the p‑channel reflection is weakest (zero). Total reflected power actually dips near θB for unpolarized light before rising to 1 at grazing.
“All the reflected light is gone at θB.” No — only the p‑component vanishes. The s‑component still reflects (Rs ≈ 0.148 for glass), which is why the glare is polarized, not absent.
Real failure modes: at grazing incidence R→1 for both polarizations; a metal (complex n) never nulls Rp to exactly zero; and beyond the critical angle from the dense side there is total internal reflection, no Brewster null at all.
tan(θB) = n₁/n₂ (ratio inverted).
→ tan(θB) = n₂/n₁. The inverted form gives 33.7° for glass and breaks the right‑angle law.
θB depends on wavelength as strongly as the color splits.
→ only weakly — through dispersion n(λ). For crown glass θB shifts <0.2° across the visible.
Brewster’s angle needs the Fresnel equations to derive.
→ the right‑angle condition θB+θt = 90° plus Snell gives tanθB = n₂/n₁ directly.
Planted void: swap the ratio to n1/n2. Brewster’s angle collapses to ~33.7°, the reflected+refracted rays no longer cross at 90°, and Rp at the reported angle is no longer zero. The Witness (7) catches it.