Transmission through a sequence of linear polarizers ($I_{\mathrm{out}}=(I_{\mathrm{in}}/2){\prod }_{j=2}^N{\cos }^2({\theta }_j-{\theta }_{j-1})$)

For ideal polarizers with successive transverse transmission directions ${\theta }_j$, the Jones matrix is the rank-one orthogonal projection $P_j={\mathbf{u}}_j{\mathbf{u}}_j^{\mathsf{T}}$. The identity ${\mathbf{u}}_j^{\mathsf{T}}{\mathbf{u}}_{j-1}=\cos ({\theta }_j-{\theta }_{j-1})$ reduces the transmitted probability amplitude analogue, here the optical complex amplitude, to a product of projections. For unpolarized light, the first element transmits half the incident light intensity, giving the displayed product. For a specified incident linear polarization, replace the first half-factor with its actual projection.