Three Polarizer Paradox

Filter 1 at 0°, Filter 2 at θ (drag), Filter 3 at 90°. Without Filter 2: 0% transmission. With Filter 2 at 45°: 25% transmission. Malus's law: I → I·cos²(Δθ) at each filter.

Drag the middle filter to change θ

0.0%

Transmission (observed)

0.0%

Theory: sin²(2θ)/4

45°

θ (Filter 2 angle)

Theoretical curve: sin²(2θ)/4

θ = 0° to 90°; peak = 25% at 45°

P1: θ=45° → transmission converges to 25%.

P2: θ=0° or θ=90° → 0% transmission.