# Mueller calculus

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)

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Jones calculus — In optics one can describe polarization using the Jones calculus, invented by R. C. Jones in 1941. Polarized light is represented by a Jones vector , and linear optical elements are represented by Jones matrices . When light crosses an optical… …   Wikipedia

• Event calculus — The event calculus is a logical language for representing and reasoning about actions and their effects first presented by Robert Kowalski and Marek Sergot in 1986.It was extended by Murray Shanahan and Rob Miller in the 1990s.The basic… …   Wikipedia

• Stokes parameters — The Stokes parameters are a set of values that describe the polarization state of electromagnetic radiation (including visible light). They were defined by George Gabriel Stokes in 1852, as a mathematically convenient alternative to the more… …   Wikipedia

• List of mathematics articles (M) — NOTOC M M estimator M group M matrix M separation M set M. C. Escher s legacy M. Riesz extension theorem M/M/1 model Maass wave form Mac Lane s planarity criterion Macaulay brackets Macbeath surface MacCormack method Macdonald polynomial Machin… …   Wikipedia

• St. Xavier High School (Cincinnati) — Saint Xavier High School Academia Sancti Xaverii Cincinnatensis School seal Vidit Mirabilia Magna; Ad Majorem Dei Gloriam He has seen great wonders; For the Greater Glory of God …   Wikipedia

• Heim theory — is a collection of ideas about the fundamental laws of physics proposed by Burkhard Heim [http://www.igw resch verlag.at/resch verlag/burkhard heim/band1.html] , [http://www.igw resch verlag.at/resch verlag/burkhard heim/band2.html] and further… …   Wikipedia

• Precalculus — In American mathematics education, precalculus (or Algebra 3 in some areas), an advanced form of secondary school algebra, is a foundational mathematical discipline. It is also called Introduction to Analysis. In many schools, precalculus is… …   Wikipedia

• Escape velocity — In physics, escape velocity is the speed where the kinetic energy of an object is equal to the magnitude of its gravitational potential energy, as calculated by the equation,:U g = frac{ Gm 1m 2}{r}.It is commonly described as the speed needed to …   Wikipedia

• Queen's Medal — Die Royal Medal, auch The Queen s Medal genannt, ist eine von der Royal Society verliehene Auszeichnung für Wissenschaftler, die innerhalb des Commonwealth of Nations besonders wichtige Beiträge zur Weiterentwicklung der Wissenschaften geleistet… …   Deutsch Wikipedia

• Formalisme de Jones — Pour les articles homonymes, voir Jones. Le formalisme de Jones est un formalisme matriciel permettant de décrire l état de polarisation de la lumière, ou de manière générale d une onde électromagnétique, et son évolution à travers un système… …   Wikipédia en Français