A third polaroid is inserted between two crossed polaroids (90° apart). If the middle polaroid's pass-axis is at angle θ to the first polaroid, the transmitted intensity (after the third polaroid) is maximum when:

After polaroid 1 (with unpolarised input I₀): I = I₀/2. After polaroid 2 at angle θ: I = (I₀/2) cos²θ. After polaroid 3 perpendicular to P₁ (so at angle 90°−θ to P₂): I = (I₀/2) cos²θ × cos²(90°−θ) = (I₀/2) cos²θ sin²θ = **(I₀/8) sin²(2θ)**. Max when 2θ = 90° → **θ = 45°**.