Mueller calculus

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 ; and any optical element can be represented by a Mueller matrix (M).
If a beam of light is initially in the state formula_1 and then passes through an optical element M and comes out in a state formula_2, then it is written
If a beam of light passes through optical element M1 followed by M2 then M3 it is written
given that matrix multiplication is associative it can be written
Matrix multiplication is not commutative, so in general
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:
where * indicates the complex conjugate [sic], ["A" is:]
and ⊗ is the tensor (Kronecker) product.
While the Jones matrix has eight independent parameters [two Cartesian or polar components for each of the four complex values in the 2-by-2 matrix], the absolute phase information is lost in the [equation above], leading to only seven independent matrix elements for a Mueller matrix derived from a Jones matrix.
Mueller matrices.
Below are listed the Mueller matrices for some ideal common optical elements:
General expression for reference frame rotation from the local frame to the laboratory frame:
where formula_10 is the angle of rotation. For rotation from the laboratory frame to the local frame, the sign of the sine terms inverts.
The Mueller matrices for other polarizer rotation angles can be generated by reference frame rotation.
Mueller tensors.
The Mueller/Stokes architecture can also be used to describe non-linear optical processes, such as multi-photon excited fluorescence and second harmonic generation. The Mueller tensor can be connected back to the laboratory-frame Jones tensor by direct analogy with Mueller and Jones matrices.
where formula_23 is the rank three Mueller tensor describing the Stokes vector produced by a pair of incident Stokes vectors, and formula_24 is the 2×2×2 laboratory-frame Jones tensor.