Mueller calculus is a matrix method for manipulating Stokes vectors, which represent the polarization of light. It was developed in 1943 by Hans Mueller. In this technique, the effect of a particular optical element is represented by a Mueller matrix—a 4×4 matrix that is an overlapping generalization of the Jones matrix.
Introduction Disregarding coherent wave superposition, any fully polarized, partially polarized, or unpolarized state of light can be represented by a Stokes vector ( S → {\displaystyle {\vec {S}}} ); and any optical element can be represented by a Mueller matrix (M). If a beam of light is initially in the state S → i {\displaystyle {\vec {S}}_{i}} and then passes through an optical element M and comes out in a state S → o {\displaystyle {\vec {S}}_{o}} , then it is written
S → o = M S → i . {\displaystyle {\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
S → o = M 3 ( M 2 ( M 1 S → i ) ) {\displaystyle {\vec {S}}_{o}=\mathrm {M} _{3}\left(\mathrm {M} _{2}\left(\mathrm {M} _{1}{\vec {S}}_{i}\right)\right)}
given that matrix multiplication is associative it can be written
S → o = M 3 M 2 M 1 S → i . {\displaystyle {\vec {S}}_{o}=\mathrm {M} _{3}\mathrm {M} _{2}\mathrm {M} _{1}{\vec {S}}_{i}\ .}
Matrix multiplication is not commutative, so in general
M 3 M 2 M 1 S → i ≠ M 1 M 2 M 3 S → i . {\displaystyle \mathrm {M} _{3}\mathrm {M} _{2}\mathrm {M} _{1}{\vec {S}}_{i}\neq \mathrm {M} _{1}\mathrm {M} _{2}\mathrm {M} _{3}{\vec {S}}_{i}\ .}
Mueller vs. Jones calculi With disregard for coherence, light which is unpolarized or partially polarized must be treated using the Mueller calculus, while fully polarized light can be treated with either the Mueller calculus or the simpler Jones calculus. Many problems involving coherent light (such as from a laser) must be treated with Jones calculus, however, because it works directly with the electric field of the light rather than with its intensity or power, and thereby retains information about the phase of the waves. More specifically, the following can be said about Mueller matrices and Jones matrices:
Stokes vectors and Mueller matrices operate on intensities and their differences, i.e. incoherent superpositions of light; they are not adequate to describe either interference or diffraction effects. (...) Any Jones matrix [J] can be transformed into the corresponding Mueller–Jones matrix, M, using the following relation:
M = A ( J ⊗ J ∗ ) A − 1 {\displaystyle \mathrm {M=A(J\otimes J^{*})A^{-1}} } , where * indicates the complex conjugate [sic], [A is:]
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