### Mueller Matrix

**Mueller calculus** is a matrix method for manipulating Stokes vectors, which represent the polarization of light. It was developed in 1943 by Hans Mueller, a professor of physics at the Massachusetts Institute of Technology. In this technique, the effect of a particular optical element is represented by a Mueller matrix—a 4×4 matrix that is a generalization of the Jones matrix.

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. Many problems involving coherent light (such as from a laser) must be treated with Jones calculus, because it works with amplitude rather than intensity of light, and retains information about the phase of the waves.

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

If a beam of light is initially in the state $\backslash vec\; S\_i$ and then passes through an optical element M and comes out in a state $\backslash vec\; S\_o$, then it is written

- $\backslash vec\; S\_o\; =\; \backslash mathrm\; M\; \backslash vec\; S\_i\; \backslash \; .$

If a beam of light passes through optical element M_{1} followed by M_{2} then M_{3} it is written

- $\backslash vec\; S\_o\; =\; \backslash mathrm\; M\_3\; \backslash big(\backslash mathrm\; M\_2\; (\backslash mathrm\; M\_1\; \backslash vec\; S\_i)\; \backslash big)\; \backslash $

given that matrix multiplication is associative it can be written

- $\backslash vec\; S\_o\; =\; \backslash mathrm\; M\_3\; \backslash mathrm\; M\_2\; \backslash mathrm\; M\_1\; \backslash vec\; S\_i\; \backslash \; .$

Matrix multiplication is not commutative, so in general

- $\backslash mathrm\; M\_3\; \backslash mathrm\; M\_2\; \backslash mathrm\; M\_1\; \backslash vec\; S\_i\; \backslash ne\; \backslash mathrm\; M\_1\; \backslash mathrm\; M\_2\; \backslash mathrm\; M\_3\; \backslash vec\; S\_i\; \backslash \; .$

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)

## See also

## References

- E. Collett,
*Field Guide to Polarization*, SPIE Field Guides vol.**FG05**, SPIE (2005). ISBN 0-8194-5868-6. - E. Hecht,
*Optics*, 2nd ed., Addison-Wesley (1987). ISBN 0-201-11609-X.