Mueller matrix algebra for the analysis of polarimetric measurements:

• Significant physical quantities derived from a measured Mueller matrix

• Serial decompositions

• Parallel decompositions and polarimetric subtraction

• Characteristic ellipsoids of a Mueller matrix

J. J. Gil, "Review on Mueller matrix algebra for the analysis of polarimetric measurements," Journal of Applied Remote Sensing 8(1), 081599 (2014)   doi:10.1117/1.JRS.8.081599    Download

Physical quantities in a Mueller matrix. On obtaining birefringence, diattenuation and depolarization parameters from a Mueller matrix.

J. J. Gil, "Polarimetric characterization of light and media. Physical quantities involved in polarimetric phenomena" Eur. Phys. J. Appl. Phys. 40, 1–47 (2007)   http://dx.doi.org/10.1051/epjap:2007153

J. J. Gil, "Physical quantities involved in a Mueller matrix" Proc. SPIE 9853, 985302-16 (2016) 

General model for nondepolarizing systems

Based on the polar decomposition of its Mueller-Jones matrix.

A general method for extracting the seven physical parameters involved in the pure system: Diattenuation and birefringence quantities.

J. J. Gil, E. Bernabéu, "Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix" Optik, 76, 67-71 (1987)  Request_PDF file

3D model for polarized light.

J. J. Gil, "Interpretation of the coherency matrix for three-dimensional polarization states," Physical Review A 90, 043858 (2014).

J. J. Gil, "Intrinsic Stokes parameters for 3D and 2D polarization states," Journal of the European Optical Society-Rapid publications 10, 15054-5 (2014).

J. J. Gil, "Components of purity of a three-dimensional polarization state," Journal of the Optical Society of America A 33 (1), 40-43 (2016).

J. J. Gil, "Degrees of mutual coherence of a 3D polarization state," Journal of Modern Optics 63 (11), 1055-1058 (2016).

J. J. Gil, I. San José "3D Polarimetric purity". Opt. Commun. 22, 4430-4434 (2010)

The full characterization of the polarimetric purity by means of two indices was presented for the first time in

J.J. Gil, J.M. Correas, C. Ferreira, I. San José, P.A.
Melero, J. Delso 7 Reunión Nacional de Óptica, Santander, 2003.

• The group of Polarimetry. Zaragoza University (presentation)

• Coherency matrices. A unified model for the mathematical  representation of polarimetric phenomena(poster)

J. J. Gil, J. M. Correas, P. A. Melero, C. Ferreira. "Generalized polarization algebra" Monog. Semin. Mat. G. Galdeano 31, 161–167 (2004). (Download)

 

Purity space

Iso-purity curves

Depolarization criterion and Degree of Polarimetric Purity (DPP)

An universal criterion for polarimetric purity in Mueller matrices. See:

J. J. Gil, E. Bernabéu "A depolarization criterion in Mueller matrices" (Request PDF file)

Physical quantities characterizing the polarizing and depolarizing power of any optical system.

See also Depolarization and polarization indices of an optical system (Request PDF file)

See also Polarimetry (R. A. Chipman. Chap. 22. Handbook of Optics OSA)

The Degree of purity is re-defined in our work Characteristic properties of Mueller matrices (Request PDF file)

Polarimetric subtraction

Decomposition of Mueller matrices as a convex combination of pure Mueller matrices

On extracting the Mueller matrix of a pure component contained in the whole media: Calculus of the weighted components.

The polarimetric subtraction was presented for the first time in

• J. M. Correas, P. Melero, J. J. Gil "Decomposition of Mueller matrices into pure optical media" VII Jornadas Zaragoza-Pau de Matemática Aplicada y estadística : Jaca (Huesca), 17-18 september, 2001 (paper, presentation)

• J. J. Gil. J Correas "Polarimetric subtraction for obtaining the Mueller matrices of components which appear combined in a whole material sample under measurement". ICO Topical Meeting on Polarization Optics (ICOPO) Polvijärvi, July 2003. (poster) 

An updated an complete version of the decompostion procedure as well as the arbitrary decomposition can be found in Sec. 3.5.3 and 5.6.3 of the review paper "polarimetric characterization of light and media"

This topic was also dealt with in API'09 - First NanoCharM Workshop on Advanced Polarimetric Instrumentation:

J. J. Gil "Parallel decompositions of Mueller matrices and polarimetric subtraction". EPJ Web of Conferences 5, 04002 (2010) (view presentation)

A general and complete algebraic procedure for subtracting a component (pure or non-pure) from a system characterized by a given Mueller matrix is presented in

J. J. Gil, I. San José "Polarimetric subtraction of Mueller matrices". J. Opt. Soc. Am. A 30 (6), 1078-1088 (2013)"

Experimental application of the polarimetric subtraction:

Experimental applications are described in:

M. Foldyna, E. Garcia-Caurel, R. Ossikovski, A. De Martino, and J. J. Gil "Retrieval of a non-depolarizing component of experimentally determined depolarizing Mueller matricesRetrieval of a non-depolarizing component of experimentally determined depolarizing Mueller matrices" Opt. Express 17, 12794-12806 (2009) (Download)

R. Ossikovski, E. Garcia-Caurel, M. Foldyna, and J. J. Gil "Application of the arbitrary decomposition to finite spot size Mueller matrix measurements" Appl. Opt. 53 (26), 6030-6036 (2014)

We expect that the new results concerning the polarimetric subtraction will be very fruitful in various fields and especially in biological tissues and radar polarimetry

General characterization of Mueller matrices

The general conditions for a 4x4 real matrix to be a physical Mueller matrix are stated: Eigenvalue conditions and transmittance conditions.

J. J. Gil "Characteristic properties of Mueller matrices" J. Opt. Soc. Am. A  17, 328-334 (2000) (Request PDF file)

J. J. Gil "Mueller matrices" in Light Scattering from microstructures, Lecture Notes in Physics, Springer, 2000 (Request PDF file)

J. J. Gil "Polarization and depolarization of light in optical scattering: polarimetric indices of purity" Workshop on light scattering from microstructures, Laredo, Spain, 1998 (view presentation)

Indices of purity and general polarization algebra

The non-dimensional invariant quantities that characterize the quality of the polarimetric purity are defined from the coherency matrix. These indices are defined for light and material media. The model is generalized for nxn coherency matrices

I. San José, J. J. Gil "Invariant indices of polarimetric purity. Generalized indices of purity for nxn covariance matrices" . Opt. Commun. 284, 38–47 (2011). (Request PDF file) (you can find a preprint in arXiv:0807.2171, 2008).

The polarimetric properties of 2D light, 3D light and material media are presented under a unified model based on their corresponding coherency matrices.

Purity space

Indices of purity at SPIE NewsRoom (Purity indices yeld more information than entropy)

The indices of purity were presented for the first time in:

J. J. Gil "Polarization and depolarization of light in optical scattering: polarimetric indices of purity" Workshop on light scattering from microstructures. Laredo, Spain, 1998 (view presentation)

Generalized polarization algebra

A complete and updated version of this topic can be found in the review paper "polarimetric characterization of light and media"

Components of purity of a Mueller matrix

The degree of polarimetric purity of a Mueller matrix, also called “depolarization index” [J.J. Gil, E. Bernabéu, Opt. Acta 33, 185-189 (1986)] is expressed as a quadratic average of two contributions of different nature.

The contribution due to the polarizance and diattenuation properties is given by a unique parameter called “degree of polarizance” and the complementary contribution due to non-polarizing properties is given by a parameter called “degree of spherical purity”.

These two intrinsic quantities are useful in order to analyze the sources of the polarimetric purity of a material sample whose Mueller matrix has been measured and provide criteria for the classification of Mueller matrices.

J. J. Gil “Components of purity of a Mueller matrix” Journal Optical Society of America A 28, 1578-1585 (2011) Request PDF file

Feasible region of the components of purity of a Mueller matrix

Serial-parallel decompositions of Mueller matrices

Given a Mueller matrix, te possible parallel decompositions into a convex combinationof pure components are formulated through a previous serial symmetric decomposition [J.J. Gil, I. San José, R. Ossikovski J. Opt. Soc. Am. A 30, 32-50 (2013)]

Transmittance constraints in serial decompositions of Mueller matrices. The arrow form of a Mueller matrix  [J.J. Gil, J. Opt. Soc. Am. A 30, 701-707 (2013)]

Poincaré sphere mapping by Mueller matrices

[R. Ossikovski, J. J. Gil and I. San José J. Opt. Soc. Am. A 30 2291-2305 (2013)]

Any Mueller matrix M has associated three ellipsoids, which constitute a geometric representations of all the physical properties of M

Generalized Target Decomposition Theory (Remote sensing)

[S. R. Cloude, J. J. Gil, I. San José and R. Ossikovski. Proc. ‘PolInSAR 2013’, Frascati, Italy28 January – 1 February 2013 (ESA SP-713, August 2013)]

In this paper we present a generalized approach to decomposition theory in radar polarimetry (which also serves as an up-date of the review in [1]). This methodology not only unites all current approaches but extends them in two main directions. Firstly into bistatic scattering, where we show which decomposition methods are suitable for scaling into the more general scattering case (and which are not). Secondly we highlight several new ideas originally developed in optical polarimetry [2], which have direct and as yet unexplored application in radar sciences.

[1] Cloude S.R., E. Pottier, "A Review of Target Decomposition Theorems in Radar Polarimetry", IEEE Transactions on Geoscience and Remote Sensing, Vol. 34 No. 2, pp 498-518, March 1996.

[2] Gil J.J. “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40,1-47 (2007)

Design of tunable retarders from commercial retardation sheets (Spanish)

Serial combinations of retarders can be arranged in such a way that the whole system results in tunable retardation compensators or modulators based on the variation of the angle of the optical axis of one of the components.

Absolute authomatic polarimetry

A PCSCA Mueller dynamic Fourier polarimeter was implemented in 1979. The ratio between the angular velocities of the retarders was placed at 2/3 (see the Degree Thesis of J. J. Gil -Spanish-)

An absolute dynamic automatic PCSCA Mueller polarimeter was implemented in 1983. The ratio between the angular velocities of the retarders was placed at 5/2 (see the PhD Thesis of J. J. Gil -English-)

Several models of automatic polarimeters have been developed and applied to the analysis of different kinds of material samples as surfaces, aerosols, human eye, etc. Applications to imaging polarimetry; Mueller-optical-coherence-tomography; Cornea and biological tissues can be performed under agreement with other groups and industries.

See also:

E. Bernabéu, J. J. Gil "An experimental device for dynamic determination of Mueller matrices" Journal Optics, Paris, 16, 139-141 (1985)  (Request PDF file)

Geometric modeling of polarimetric transformations

Optical sensors by polarimetry (Spanish)

PhD Thesis of J. J. Gil (1983)

-English version 1-: "Determination of polarization parameters in matricial representation. Theoretical contribution and development of an automatic measurement device"

-Spanish version-: "Determinación de parámetros de polarización en representación matricial. Contribución teórica y realización de un dispositivo automático"

 Degree Thesis of J. J. Gil (1979): "Método dinámico de determinación de parámetros de Stokes y matrices de Mueller por análisis de Fourier" (Spanish version, to be translated to English)