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# Mueller calculus

Mueller calculus is a matrix method for manipulating Stokes vectors, which represent the polarization of incoherent light. It was developed in 1943 by Hans Mueller, then a professor of physics at the Massachusetts Institute of Technology. Light which is unpolarized or partially polarized must be treated using Mueller calculus, while fully polarized light can be treated with either Mueller calculus or the simpler Jones calculus. Coherent light generally must be treated with Jones calculus because the latter works with amplitude rather than intensity of light. The effect of a particular optical element is represented by a Mueller matrix; which is a 4×4 matrix and a generalization of the Jones matrix.

Any fully polarized, partially polarized, or unpolarized state of light can be represented by a Stokes vector ($\vec S$). Any optical element can be represented by a Mueller matrix (M).

If a beam of light is initially in the state $\vec S_i$ and then passes through an optical element M and comes out in a state $\vec S_o$, then it is written

$\vec S_o = \mathrm M \vec S_i \ .$

If a beam of light passes through optical element M1 followed by M2 then M3 it is written

$\vec S_o = \bigg( \mathrm M_3 \Big( \mathrm M_2 ( \mathrm M_1 \vec S_i \ \big) \Big) \bigg) \ .$

given that matrix multiplication is associative it can be written

$\vec S_o = \mathrm M_3 \mathrm M_2 \mathrm M_1 \vec S_i \ .$

Beware, matrix multiplication is not commutative, so in general

$\mathrm M_3 \mathrm M_2 \mathrm M_1 \vec S_i \ne \mathrm M_1 \mathrm M_2 \mathrm M_3 \vec S_i \ .$

Below are listed the Mueller matrices for some ideal common optical elements:

${1 \over 2} \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \quad$ Linear polarizer (Horizontal Transmission)

${1 \over 2} \begin{pmatrix} 1 & -1 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \quad$ Linear polarizer (Vertical Transmission)

${1 \over 2} \begin{pmatrix} 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \quad$ Linear polarizer (+45° Transmission)

${1 \over 2} \begin{pmatrix} 1 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 \\ -1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \quad$ Linear polarizer (-45° Transmission)

$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{pmatrix} \quad$ Quarter wave plate (fast-axis vertical)

$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \end{pmatrix} \quad$ Quarter wave plate (fast-axis horizontal)

$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix} \quad$ Half wave plate (fast-axis vertical)

${1 \over 4} \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \quad$ Attenuating filter (25% Transmission)