Malus's Law

Étienne-Louis Malus worked this out in 1809 while looking through a calcite crystal at sunlight reflecting off the windows of the Luxembourg Palace, and it remains the only law you need to predict how much light gets through a pair of polarisers. The second sheet — the analyser — passes only the component of the electric field lying along its own axis, and that component is E₀ cos θ. Intensity goes as the square of the field, so the transmitted intensity is I₀ cos²θ. At 0° everything passes; at 45° exactly half does; at 90° the axes are crossed and the field has no component to give, so nothing does.

Two consequences catch people out. The first is that unpolarised light hitting the first polariser loses half its intensity regardless of orientation — Malus's law does not apply there, because unpolarised light has no single θ, and averaging cos²θ over all angles gives ½. The second is the three-polariser trick: cross two sheets so no light passes, then slide a third between them at 45°, and light reappears. Each stage passes cos²45° = ½, so ¼ of what reached the middle sheet comes out the far side of a pair that was completely dark a moment earlier. Nothing was added; the middle sheet re-aimed the polarisation it transmitted. LCD screens run this arithmetic in every pixel, rotating the polarisation electrically to dial each subpixel between the two extremes.