Malus' law and crossed polaroids

Place a second identical polaroid in the beam. Now rotating one of them has a striking effect: at one setting almost no light gets through the pair, and 90° from that setting nearly all the light leaving the first polaroid passes the second.

Why: the light leaving the first polaroid has its electric field along that polaroid's pass-axis. If the second pass-axis makes an angle θ with the first, only the component E cos θ along it gets through.

Intensity goes as the square of the field, so I = I₀ cos²θ. This is Malus' law, where I₀ is the intensity of the polarised light that left the first polaroid.

θ = 0 passes all of I₀; θ = 90° (crossed polaroids) passes nothing; θ = 45° passes I₀/2. In one full turn of the second polaroid, through 2π, the intensity goes through two maxima and two minima.

One polaroid halves unpolarised light, so a pair can set the transmitted light anywhere from 50% of the original intensity down to zero just by changing the angle between their axes.

Example 10.2: a polaroid P₂ is rotated between two crossed polaroids P₁ and P₃. After P₂ the intensity is I₀cos²θ. The angle between P₂ and P₃ is π/2 − θ, so the light leaving P₃ is I₀cos²θ sin²θ = (I₀/4) sin²2θ.

That is zero when P₂ lines up with either crossed polaroid (θ = 0 or π/2) and greatest, I₀/4, when θ = π/4. Inserting a third sheet lets light through a pair that on its own would block it completely.

Polaroids control intensity in sunglasses and window panes, and are used in photographic cameras and 3D movie cameras.