A polarizer dims light by the cosine-squared of the angle between the light and its axis. Two crossed polarizers go black — yet slip a third between them and light returns. Rendered, not quoted.
source É. L. Malus, «Sur une propriété de la lumière réfléchie», Mém. de physique et de chimie de la Société d’Arcueil, t. 2 (1809), pp. 145–158 — Wikisource amber pre-DOI historical memoir; link is a transcription, not a stable archival id.
Polarized light of intensity I0 meets an analyzer whose axis makes angle θ with the light. Only the projected component passes, and intensity goes as its square:
Consequences the engine holds exact: full at θ=0, half at 45°, zero at 90° (crossed). Unpolarized light has no preferred axis, so the first polarizer passes the angular average of cos², which is exactly ½.
Malus found this projection by reflection: at one incidence a glass surface reflects fully polarized light. That angle is the‑brewster‑angle — tan θB = n₂/n₁, reflected and refracted rays at a right angle. Brewster (1815) supplies the polarized beam; Malus (1809) supplies the law that dims it. Same cosine projection, two doors in.
Re-checks the engine live: transmission at 45° must equal 0.5·I0 and the three-polarizer stack must equal I0/8. If window 6 tampers the law, this flips red.
A beam of intensity I0 = 1, either polarized or unpolarized, and an analyzer angle θ you set below.
I / I0 = —
Curve: cos²θ over 0–180°. Dot = your angle. Stack at right: the three-polarizer paradox, live.
—
“Two crossed polarizers are already black. Inserting a third filter can only remove more light — adding an obstacle cannot brighten a dark field.”
Wall against intuition: a filter is not a sieve of fixed holes. It re-projects the polarization axis. The middle filter at 45° rebuilds a component the last filter can pass. Intensity out = I0/8 > 0. The intuition treats light as classical grains; the law treats it as a projected vector.
“Transmission falls linearly with angle, I = I0·cos θ.”
→ It falls as cos². At 45° that is 0.5, not 0.707. Malus measured the square.
“Unpolarized light loses a factor cos² at the first polarizer too.”
→ No axis to reference — it loses exactly ½, the average of cos².
“Crossed polarizers can never pass light.”
→ True only with nothing between. A 45° filter inserted gives I0/8.
Planted void: swap cos²θ for a linear cos θ. Crossed still darken, but 45° stops halving and the paradox reads I0/4, not I0/8. Witness (7) catches it.
law: cos²θ (intact)