Reference Section

Mueller Matrices
Common Defects of Polarization Elements
Polarimetry Definitions
Bibliography

Mueller Matrices

The Mueller matrix M for a polarization-altering device is the matrix which transforms an incident Stokes vector S into the exiting (reflected, transmitted, or scattered) Stokes vector S'

.

The Mueller matrix is a four-by-four matrix with real valued elements.

The Mueller matrix is a great formalism for characterizing polarization elements because it contains within its elements all of the polarization properties: diattenuation, retardance, depolarization, and their form, either linear, circular, or elliptical.   When the Mueller matrix is known, then the exiting polarization state is known for all incident polarization states.


The Mueller matrix M(k, l) for a device is always a function of the direction of propagation k and wavelength l.

The Mueller matrix M associated with a beam path through a sequence (cascade) of *polarization elements q = 1 , 2 , ... , Q is the right-to-left product of the individual matrices M = M_QM_Q-1...M₃M₂M₁.

Common Defects of Polarization Elements

1. Polarizers have nonideal diattenuation since T_max<1 and T_min>0.

2. Retarders have the incorrect retardance.  Thus, there will be some deviation from a quarter-wave or a half-wave of retardance, for example, because of fabrication errors or a change in wavelength.

3. Retarders usually have some diattenuation because of differences in absorption coefficients (dichroism) and due to different transmission and reflection coefficients at the interfaces.  For example, birefringent retarders have diattenuation due to the difference of the Fresnel coefficients at normal incidence for the two eigenpolarizations since n₁ ¹ n₂.  This can be reduced by good anti-reflection coatings.

4. Polarizers usually have some retardance; there is a difference in optical path length between the transmitted (principal) eigenpolarization and the small amount of the extinguished (secondary) eigenpolarization.  For example, sheet polarizers and wire-grid polarizers show substantial retardance when the secondary state is not completely extinguished.

5. The polarization properties vary with angle of incidence; for example, Glan-Thompson polarizers polarize over only a 4° field of view.   Birefringent retarders commonly show a quadratic variation of retardance with angle of incidence which increases along one axis and decreases along the orthogonal axis.   For polarizing beam splitter cubes, the axis of linear polarization rotates for incident light out of its normal plane (the plane defined by the face normals and the beam-splitting interface normal).

7. The polarization properties vary with wavelength; for example, for simple retarders made from a single birefringent plate, the retardance varies approximately linearly with wavelength.

8. For polarizers, the accepted state and the transmitted state can be different.   Consider a polarizing device formed from a linear polarizer oriented at 0° followed by a linear polarizer oriented at 2°.  Incident light linearly polarized at 0° has the highest transmittance for all possible polarization states and is the accepted state.   The corresponding exiting beam is linearly polarized at 2°, which is the only state exiting the device.  In this example, the transmitted state is also an eigenpolarization.  This "rotation" between the accepted and transmitted states of a polarizer frequently occurs, for example, when the crystal axes are misaligned in a birefringent polarizing prism assembly such as a Glan-Thompson polarizer.

9. A nominally "linear" element may be slightly elliptical (have elliptical eigenpolarizations).  For example, a *quartz linear retarder with the crystal axis misaligned becomes an elliptical retarder.  Similarly a circular element may be slightly elliptical.  For example, a circular polarizer formed from a linear polarizer followed by a quarter wave linear retarder at 45° becomes an elliptical polarizer as the retarder's fast axis is rotated and misaligned.

10. The eigenpolarizations of the polarization element may not be orthogonal;  i. e. a polarizer may transmit linearly polarized light at 0° without change of polarization while extinguishing linearly polarized light oriented at 88°.  Such a polarization element is referred to as inhomogeneous.  Sequences of polarization elements, such as optical isolator assemblies, often are inhomogeneous.

11. A polarization element may depolarize, coupling polarized light into unpolarized light.  A polarizer or retarder with a small amount of depolarization, when illuminated by a completely polarized beam, will have a small amount of unpolarized light in the transmitted beam.  Such a transmitted beam can no longer be extinguished by an ideal polarizer.  Depolarization results from fabrication errors such as surface roughness, bulk scattering, random strains and dislocations, and thin-film microstructure.

12. Multiply reflected beams and other "secondary" beams may be present with undesired polarization properties.  For example, the multiply reflected beams from a birefringent plate have various values for their retardance.  Antireflection coatings will reduce this effect in one waveband, but may increase these problems with multiple reflections in other wavebands.

Polarimetry Definitions

Analyzer - an element whose intensity transmission is proportional to the content of a specific polarization state in the incident beam.  Analyzers are placed before the detector in polarimeters.  The transmitted polarization state emerging from an analyzer is not necessarily the same as the state which is being analyzing.

Birefringence - a material property, the retardance associated with propagation through an anisotropic medium.  For each propagation direction within a birefringent medium there are two modes of propagation with different refractive indices n₁ and n₂.  The birefringence D_n is D_n = | n₁ - n₂ | .

Depolarization - a process which couples polarized light into unpolarized light.  Depolarization is intrinsically associated with scattering and with diattenuation and retardance which vary in space, time, and/or wavelength.

Diattenuation - the property of an optical element or system whereby the intensity transmittance of the exiting beam depends on the polarization state of the incident beam.  The intensity transmittance is a maximum P_max for one incident state, and a minimum P_min for the orthogonal state.  The diattenuation is defined as (P_max - P_min) / (P_max + P_min).

Diattenuator - any homogeneous polarization element which displays significant diattenuation and minimal retardance.  Polarizers have a diattenuation close to one, but nearly all optical interfaces are weak diattenuators.  Examples of diattenuators include the following: polarizers and dichroic materials, as well as metal and dielectric interfaces with reflection and transmission differences described by Fresnel equations; thin films (homogeneous and isotropic); and diffraction gratings.

Dichroism - the material property of displaying diattenuation during propagation.  For each direction of propagation, dichroic media have two modes of propagation with different absorption coefficients.  Examples of dichroic materials include sheet polarizers and dichroic crystals such as tourmaline.  The numerical value of the dichroism is the difference between two absorptions coefficients in Beers law for two modes.

Eigenpolarization - a polarization state transmitted unaltered by a polarization element except for a change of amplitude and phase.  Every polarization element has two eigenpolarizations.  Any incident light not in an eigenpolarization state is transmitted in a polarization state different from the incident state.   Eigenpolarizations are the eigenvectors of the corresponding Mueller or Jones matrix.

Ellipsometry - a polarimetric technique which uses the change in the state of polarization of light upon reflection for the characterization of surfaces, interfaces, and thin films.

Homogeneous polarization element - an element whose eigenpolarizations are orthogonal.  Then, the eigenpolarizations are the states of maximum and minimum transmittance and also of maximum and minimum optical path length.  A homogeneous element is classified as linear, circular or elliptical depending on the form of the eigenpolarizations.

Inhomogeneous polarization element - an element whose eigenpolarizations are not orthogonal.  Such an element will display different polarization characteristics for forward and backward propagating beams.  The eigenpolarizations are generally not the states of maximum and minimum transmittance.  Often inhomogeneous elements cannot be simply classified as linear, circular, or elliptical.

Ideal polarizer - a polarizer with an intensity transmittance of one for its principal state and an intensity transmittance of zero for the orthogonal state.

Linear polarizer - a device which when placed in an incident unpolarized beam produces a beam of light whose electric field vector is oscillating primarily in one plane, with only a small component in the perpendicular plane.

Non-polarizing element - an element which does not change the polarization state for arbitrary incident states.  The polarization state of the output light is equal to the polarization state of the incident light for all possible input polarization states.

Partially polarized light - light containing an unpolarized component; cannot be extinguished by an ideal polarizer.

Polarimeter - an optical instrument for the determination of the polarization state of a light beam, or the polarization altering properties of a sample.

Polarimetry - the science of measuring the polarization state of a light beam and the diattenuating, retarding, and depolarizing properties of materials.

Polarizance - the property of an optical element or system whereby unpolarized light is transformed into polarized light.  The polarizance is described by its magnitude (equal to the degree of polarization of light exiting the system when unpolarized light is input) and the Stokes vector of the output light.

Polarization - any process which alters the polarization state of a beam of light, including diattenuation, retardance, depolarization, and scattering.

Polarization coupling - any conversion of light from one polarization state into another state.

Polarized light - light in a fixed, elliptically (including linearly or circularly) polarized state.  A fully polarized beam can be extinguished by an ideal polarizer.  For polychromatic light, the polarization ellipses associated with each spectral component have identical ellipticity, orientation, and helicity.

Polarizer - a strongly diattenuating optical element designed to transmit light in a specified polarization state independent of the incident polarization state.   The transmission of one of the eigenpolarizations is very nearly zero.

Polarization element - any optical element which alters the polarization state of light.  This includes polarizers, retarders, mirrors, thin films, and nearly all optical elements.

Pure diattenuator - a diattenuator with zero retardance and no depolarization.

Pure retarder - a retarder with zero diattenuation and no depolarization.

Retardance - a polarization-dependent phase change associated with a polarization element or system.  The phase (optical path length) of the output beam depends upon the polarization state of the input beam.  The transmitted phase is a maximum for one eigenpolarization, and a minimum for the other eigenpolarization.   Other states show polarization coupling and an intermediate phase.

Retardation plate - a retarder constructed from a plane parallel plate or plates of linearly birefringent material.

Retarder - a polarization element designed to produce a specified phase difference between the exiting beams for two orthogonal incident polarization states (the eigenpolarizations of the element).  For example, a quarter-wave linear retarder has as its eigenpolarizations two orthogonal linearly polarized states which are transmitted in their incident polarization states but with a 90° (quarter-wavelength) relative phase difference introduced.

Spectropolarimetry - the spectroscopic study of the polarization properties of materials.  Spectropolarimetry is a generalization of conventional optical spectroscopy.  Where conventional spectroscopy endeavors to measure the reflectance or transmission of a sample as a function of wavelength, spectropolarimetry also determines the diattenuating, retarding, and depolarizing properties of the sample. Complete characterization of these properties is accomplished by measuring the Mueller matrix of the sample as a function of wavelength.

Waveplate - a retarder.