Mueller calculus

Remarks

Example: an ideal linear polarizer with horizontal transmission axis has $$M = \frac{1}{2}\begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}.$$ For unpolarized light $S_{\mathrm{in}} = (1, 0, 0, 0)$, it gives $S_{\mathrm{out}} = (1/2, 1/2, 0, 0)$: half the intensity, fully polarized horizontally.

Non-example: the matrix $\mathrm{diag}(1, 2, 0, 0)$ is not the Mueller matrix of a physical element. It sends the Stokes vector $(1, 1, 0, 0)$ to $(1, 2, 0, 0)$, which fails $S_0^2 \ge S_1^2 + S_2^2 + S_3^2$.

The ideal depolarizer has Mueller matrix $\mathrm{diag}(1, 0, 0, 0)$. Every Jones matrix gives a Mueller matrix, but the depolarizer shows that the converse fails.