Mueller calculus
Encyclopedia
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 (
). Any optical element can be represented by a Mueller matrix (M).
If a beam of light is initially in the state
and then passes through an optical element M and comes out in a state 
, 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
Beware, matrix multiplication is not commutative, so in general
Below are listed the Mueller matrices for some ideal common optical elements:
Linear polarizer (Horizontal Transmission)
Linear polarizer (Vertical Transmission)
Linear polarizer (+45° Transmission)
Linear polarizer (-45° Transmission)
Quarter wave plate
(fast-axis vertical)
Quarter wave plate
(fast-axis horizontal)
Half wave plate
(fast-axis vertical)
Attenuating filter (25% Transmission)
Coherence (physics)
In physics, coherence is a property of waves that enables stationary interference. More generally, coherence describes all properties of the correlation between physical quantities of a wave....
light. It was developed in 1943 by Hans Mueller, then a professor of physics at the Massachusetts Institute of Technology
Massachusetts Institute of Technology
The Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...
. 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
Jones calculus
In optics, polarized light can be described 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...
. Coherent light generally must be treated with Jones calculus because the latter works with amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
rather than intensity
Intensity (physics)
In physics, intensity is a measure of the energy flux, averaged over the period of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech...
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
Jones calculus
In optics, polarized light can be described 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...
.
Any fully polarized, partially polarized, or unpolarized state of light can be represented by a Stokes vector (
). Any optical element can be represented by a Mueller matrix (M).If a beam of light is initially in the state
and then passes through an optical element M and comes out in a state , then it is writtenIf a beam of light passes through optical element M1 followed by M2 then M3 it is written
given that matrix multiplication
Matrix multiplication
In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...
is associative it can be written
Beware, matrix multiplication is not commutative, so in general
Below are listed the Mueller matrices for some ideal common optical elements:
Linear polarizer (Horizontal Transmission) Linear polarizer (Vertical Transmission) Linear polarizer (+45° Transmission) Linear polarizer (-45° Transmission) Quarter wave plateWave plate
A wave plate or retarder is an optical device that alters the polarization state of a light wave travelling through it.- Operation :A wave plate works by shifting the phase between two perpendicular polarization components of the light wave. A typical wave plate is simply a birefringent crystal...
(fast-axis vertical)
Quarter wave plateWave plate
A wave plate or retarder is an optical device that alters the polarization state of a light wave travelling through it.- Operation :A wave plate works by shifting the phase between two perpendicular polarization components of the light wave. A typical wave plate is simply a birefringent crystal...
(fast-axis horizontal)
Half wave plateWave plate
A wave plate or retarder is an optical device that alters the polarization state of a light wave travelling through it.- Operation :A wave plate works by shifting the phase between two perpendicular polarization components of the light wave. A typical wave plate is simply a birefringent crystal...
(fast-axis vertical)
Attenuating filter (25% Transmission)