Abstract

Extracting a system’s physical features from polarimetric experiments constitutes a challenging task, especially in the presence of multiple scattering. This can be attributed to the difficulty in interpreting the polarimetric measurements. In this study, we demonstrate that polarimetric images recorded in the backscattering geometry can be interpreted by analyzing the spatial variations of the backscattered light’s Stokes vectors and using symmetry/geometry arguments. To illustrate the applicability of our method, we examine experimental and simulation data collected by probing colloidal suspensions. We present an analytical model based on the coherency matrix and the geometric phase to describe the polarimetric behavior of the probed samples.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (1)

A. B. Tibbs, I. M. Daly, D. R. Bull, and N. W. Roberts, “Noise creates polarization artefacts,” Bioinspir. Biomim.  13, 015005 (2018).
[Crossref]

2017 (3)

L. S. Hirst and G. Charras, “Biological physics: Liquid crystals in living tissue,” Nature 544, 164–165 (2017).
[Crossref]

L. T. McDonald, E. D. Finlayson, B. D. Wilts, and P. Vukusic, “Circularly polarized reflection from the scarab beetle Chalcothea smaragdina: Light scattering by a dual photonic structure,” Interface Focus. 7, 20160129 (2017).
[Crossref]

J. Qi and D. S. Elson, “Mueller polarimetric imaging for surgical and diagnostic applications: a review,” J. Biophotonics 10, 950–982 (2017).
[Crossref] [PubMed]

2016 (2)

R. M. A. Azzam, “Stokes-vector and Mueller-matrix polarimetry [Invited],” J. Opt. Soc. Am. A 33, 1396–1408 (2016).
[Crossref]

S. Otsuki, “Symmetry relationships for multiple scattering of polarized light in turbid spherical samples: theory and a Monte Carlo simulation,” J. Opt. Soc. Am. A, Opt. image science, vision  33, 258–269 (2016).
[Crossref]

2015 (6)

C. M. Macdonald, S. L. Jacques, and I. V. Meglinski, “Circular polarization memory in polydisperse scattering media,” Phys. Rev. E -Stat. Nonlinear, Soft Matter Phys. 91, 1–5 (2015).
[Crossref]

M. V. Berry, “Nature’s optics and our understanding of light,” Contemp. Phys. 56, 2–16 (2015).

Y. Wang, H. He, J. Chang, N. Zeng, S. Liu, M. Li, and H. Ma, “Differentiating characteristic microstructural features of cancerous tissues using Mueller matrix microscope,” Micron 79, 8–15 (2015).
[Crossref] [PubMed]

S. Alali and A. Vitkin, “Polarized light imaging in biomedicine: emerging Mueller matrix methodologies for bulk tissue assessment,” J. Biomed. Opt.  20, 061104 (2015).
[Crossref]

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, G. Videen, and C. J. Grund, “The case for a modern multiwavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transf. 153, 131–143 (2015).
[Crossref]

S. Alali and A. Vitkin, “Polarized light imaging in biomedicine: emerging Mueller matrix methodologies for bulk tissue assessment,” J. biomedical optics 20, 61104 (2015).
[Crossref]

2014 (8)

S.-M. F. Nee, “Decomposition of Jones and Mueller matrices in terms of four basic polarization responses,” J. Opt. Soc. Am. A 31, 2518 (2014).
[Crossref]

H. G. Akarçay, A. Hohmann, A. Kienle, M. Frenz, and J. Rička, “Monte Carlo modeling of polarized light propagation: Stokes vs. Jones. Part I,” Appl. Opt. 53, 7586–7602 (2014).
[Crossref]

H. G. Akarçay, A. Hohmann, A. Kienle, M. Frenz, and J. Rička, “Monte Carlo modeling of polarized light propagation: Stokes vs. Jones. Part II,” Appl. Opt. 53, 7586–7602 (2014).
[Crossref]

A. J. Brown, “Equivalence relations and symmetries for laboratory, LIDAR, and planetary Müeller matrix scattering geometries,” J. Opt. Soc. Am. A 31, 2789–2794 (2014).
[Crossref]

F. Snik, J. Craven-Jones, M. Escuti, S. Fineschi, D. Harrington, A. De Martino, D. Mawet, J. Riedi, and J. S. Tyo, “An overview of polarimetric sensing techniques and technology with applications to different research fields,” Proc. SPIE 9099, 90990B (2014).

J. J. Gil, “Review on Mueller matrix algebra for the analysis of polarimetric measurements,” J. Appl. Remote. Sens.  8, 81599 (2014).
[Crossref]

M. K. Swami, H. Patel, M. R. Somyaji, P. K. Kushwaha, and P. K. Gupta, “Size-dependent patterns in depolarization maps from turbid medium and tissue,” Appl. Opt.  53, 6133 (2014).
[Crossref] [PubMed]

A. Doronin, C. Macdonald, and I. Meglinski, “Propagation of coherent polarized light in turbid highly scattering medium,” J. Biomed. Opt.  19, 025005 (2014).
[Crossref] [PubMed]

2013 (1)

H. Hu, E. Garcia-Caurel, G. Anna, and F. Goudail, “Maximum likelihood method for calibration of Mueller polarimeters in reflection configuration,” Appl. Opt.  52, 6350 (2013).
[Crossref] [PubMed]

2012 (4)

C. Macías-Romero and P. Török, “Eigenvalue calibration methods for polarimetry,” J. Eur. Opt. Soc.  7, 12004 (2012).
[Crossref]

A. J. Brown and Y. Xie, “Symmetry relations revealed in Mueller matrix hemispherical maps,” J. Quant. Spectrosc. Radiat. Transf. 113, 644–651 (2012).
[Crossref]

R. Ossikovski and I. A. Vitkin, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. biomedical optics 17, 105006 (2012).
[Crossref]

E. Avci, C. M. Macdonald, and I. Meglinski, “Helicity of circular polarized light backscattered from biological tissues influenced by optical clearing,” Proc. SPIE 8337, 833703 (2012).
[Crossref]

2011 (5)

C. Macdonald and I. Meglinski, “Backscattering of circular polarized light from a disperse random medium influenced by optical clearing,” Laser Phys. Lett. 8, 324–328 (2011).
[Crossref]

R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. letters 36, 2330–2332 (2011).
[Crossref]

R. M. A. Azzam, “The intertwined history of polarimetry and ellipsometry,” Thin Solid Films 519, 2584–2588 (2011).
[Crossref]

N. Ghosh and I. A. Vitkin, “Tissue polarimetry : concepts, challenges, applications, and outlook,” J. biomedical optics 16, 110801 (2011).
[Crossref]

H. G. Akarçay and J. Rička, “Simulating light propagation: towards realistic tissue models,” Proc. SPIE 8088, 80880K (2011).
[Crossref]

2010 (4)

G. Nirmalya, J. Soni, M. F. G. Wood, M. A. Wallenberg, and I. A. Vitkin, “Mueller matrix polarimetry for the characterization of complex random medium like biological tissues,” Pramana -J. Phys. 75, 1071–1086 (2010).
[Crossref]

O. Arteaga and A. Canillas, “Analytic inversion of the Mueller -Jones polarization matrices for homogeneous media,” Opt. letters 35, 559–561 (2010).
[Crossref]

M. Berry, “Geometric phase memories,” Nat. Phys. 6, 148–150 (2010).
[Crossref]

A. Beléndez, E. Fernndez, J. Francés, and C. Neipp, “Birefringence of cellotape: Jones representation and experimental analysis,” Eur. J. Phys. 31, 551–561 (2010).
[Crossref]

2008 (2)

C. Schwartz and A. Dogariu, “Backscattered polarization patterns determined by conservation of angular momentum,” J. Opt. Soc. Am. A, Opt. image science, vision  25, 431 (2008).
[Crossref]

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them?” Phys. Status Solidi (a) 205, 720–727 (2008).
[Crossref]

2007 (2)

O. Korotkova and D. James, “Definitions of the Degree of Polarization of a Light,” Opt. letters 32, 1015–1016 (2007).
[Crossref]

A. Parker, C. Baravian, F. Caton, J. Dillet, and J. Mougel, “Fast optical sizing without dilution,” Food Hydrocoll. 21, 831–837 (2007).
[Crossref]

2006 (3)

C. Schwartz and A. Dogariu, “Backscattered polarization patterns, optical vortices, and the angular momentum of light,” Opt. Lett.  31, 1121 (2006).
[Crossref] [PubMed]

W. Cai, X. Ni, S. K. Gayen, and R. R. Alfano, “Analytical cumulant solution of the vector radiative transfer equation investigates backscattering of circularly polarized light from turbid media,” Phys. Rev. E -Stat. Nonlinear, Soft Matter Phys. 74, 1–11 (2006).
[Crossref]

J. Dillet, C. Baravian, F. Caton, and A. Parker, “Size determination by use of two-dimensional Mueller matrices backscattered by optically thick random media,” Appl. Opt.  45, 4669 (2006).
[Crossref] [PubMed]

2005 (2)

P. Hariharan, “The geometric phase,” Prog. Opt. 48, 149–201 (2005).
[Crossref]

M. Xu and R. R. Alfano, “Circular polarization memory of light,” Phys. Rev. E -Stat. Nonlinear, Soft Matter Phys. 72, 1–4 (2005).
[Crossref]

2004 (1)

2001 (3)

A. Quirantes, F. Arroyo, and J. Quirantes-Ros, “Multiple Light Scattering by Spherical Particle Systems and Its Dependence on Concentration: A T-Matrix Study,” J. Colloid Interface Sci. 240, 78–82 (2001).
[Crossref] [PubMed]

L. Hespel, S. Mainguy, and J.-J. Greffet, “Theoretical and experimental investigation of the extinction in a dense distribution of particles: nonlocal effects Laurent,” Josa a 18, 3072–3076 (2001).
[Crossref]

A. C. Maggs and V. Rossetto, “Writhing Photons and Berry Phases in Polarized Multiple Scattering,” Phys. review letters 87, 253901 (2001).
[Crossref]

2000 (1)

S. Bartel and A. H. Hielscher, “Monte Carlo simulations of the diffuse backscattering mueller matrix for highly scattering media,” Appl. optics 39, 1580–1588 (2000).
[Crossref]

1999 (2)

M. J. Raković, G. W. Kattawar, M. B. Mehrubeoğlu, B. D. Cameron, L. V. Wang, S. Rastegar, and G. L. Coté, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. optics 38, 3399 (1999).
[Crossref]

E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers,” Appl. Opt.  38, 3490 (1999).
[Crossref]

1998 (1)

V. P. Dick, “Applicability Limits of Beer’s Law for Dispersion Media with a High Concentration of Particles,” Appl. optics 37, 4998–5004 (1998).
[Crossref]

1996 (1)

1994 (1)

C. Brosseau, D. Bicout, A. S. Martinez, and J. M. Schmitt, “Depolarization behavior of multiple scattered light from an optically dense random medium,” Phys. Rev. E 49, 1767–1770 (1994).

1990 (2)

S. Fraden and G. Maret, “Multiple light scattering from concentrated, interacting suspensions,” Phys. Rev. Lett. 65, 512–515 (1990).
[Crossref] [PubMed]

M. Berry, “Anticipations of the geometric phase,” Phys. Today 43, 34–40 (1990).
[Crossref]

1989 (1)

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattred light,” Phys. Rev. B -Rapid Commun. 40, 9342–9345 (1989).
[Crossref]

1988 (1)

M. Berry, “The Geometric Phase,” Sci. Am. 259, 46–55 (1988).
[Crossref]

1987 (1)

J. Gil and E. Bernabeu, “Obtainment of the polarizing and. retardation parameters of nondepolarizing optical system from. polar decomposition of its Mueller matrix,” Optik 76, 67 (1987).

1985 (1)

W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
[Crossref]

1976 (1)

R. M. A. Azzam, “Perspective on ellipsometry,” Surf. Sci. 56, 6–18 (1976).
[Crossref]

1948 (1)

1942 (1)

C. R. Jones, “A New Calculus for the Treatment of Optical Systems. IV,.” J. Opt. Soc. Am 32, 486–493 (1942).
[Crossref]

Akarçay, H. G.

H. G. Akarçay, A. Hohmann, A. Kienle, M. Frenz, and J. Rička, “Monte Carlo modeling of polarized light propagation: Stokes vs. Jones. Part I,” Appl. Opt. 53, 7586–7602 (2014).
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H. G. Akarçay, A. Hohmann, A. Kienle, M. Frenz, and J. Rička, “Monte Carlo modeling of polarized light propagation: Stokes vs. Jones. Part II,” Appl. Opt. 53, 7586–7602 (2014).
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H. G. Akarçay and J. Rička, “Simulating light propagation: towards realistic tissue models,” Proc. SPIE 8088, 80880K (2011).
[Crossref]

H. G. Akarçay, “Polarized light propagation in biological tissue: towards realistic modeling,” Ph.D. thesis, University of Bern (2011).

Alali, S.

S. Alali and A. Vitkin, “Polarized light imaging in biomedicine: emerging Mueller matrix methodologies for bulk tissue assessment,” J. biomedical optics 20, 61104 (2015).
[Crossref]

S. Alali and A. Vitkin, “Polarized light imaging in biomedicine: emerging Mueller matrix methodologies for bulk tissue assessment,” J. Biomed. Opt.  20, 061104 (2015).
[Crossref]

Alfano, R. R.

W. Cai, X. Ni, S. K. Gayen, and R. R. Alfano, “Analytical cumulant solution of the vector radiative transfer equation investigates backscattering of circularly polarized light from turbid media,” Phys. Rev. E -Stat. Nonlinear, Soft Matter Phys. 74, 1–11 (2006).
[Crossref]

M. Xu and R. R. Alfano, “Circular polarization memory of light,” Phys. Rev. E -Stat. Nonlinear, Soft Matter Phys. 72, 1–4 (2005).
[Crossref]

Anastasiadou, M.

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them?” Phys. Status Solidi (a) 205, 720–727 (2008).
[Crossref]

Anna, G.

H. Hu, E. Garcia-Caurel, G. Anna, and F. Goudail, “Maximum likelihood method for calibration of Mueller polarimeters in reflection configuration,” Appl. Opt.  52, 6350 (2013).
[Crossref] [PubMed]

Arroyo, F.

A. Quirantes, F. Arroyo, and J. Quirantes-Ros, “Multiple Light Scattering by Spherical Particle Systems and Its Dependence on Concentration: A T-Matrix Study,” J. Colloid Interface Sci. 240, 78–82 (2001).
[Crossref] [PubMed]

Arteaga, O.

O. Arteaga and A. Canillas, “Analytic inversion of the Mueller -Jones polarization matrices for homogeneous media,” Opt. letters 35, 559–561 (2010).
[Crossref]

Avci, E.

E. Avci, C. M. Macdonald, and I. Meglinski, “Helicity of circular polarized light backscattered from biological tissues influenced by optical clearing,” Proc. SPIE 8337, 833703 (2012).
[Crossref]

Azzam, R. M. A.

R. M. A. Azzam, “Stokes-vector and Mueller-matrix polarimetry [Invited],” J. Opt. Soc. Am. A 33, 1396–1408 (2016).
[Crossref]

R. M. A. Azzam, “The intertwined history of polarimetry and ellipsometry,” Thin Solid Films 519, 2584–2588 (2011).
[Crossref]

R. M. A. Azzam, “Perspective on ellipsometry,” Surf. Sci. 56, 6–18 (1976).
[Crossref]

Bailey, W. M.

W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
[Crossref]

Baravian, C.

A. Parker, C. Baravian, F. Caton, J. Dillet, and J. Mougel, “Fast optical sizing without dilution,” Food Hydrocoll. 21, 831–837 (2007).
[Crossref]

J. Dillet, C. Baravian, F. Caton, and A. Parker, “Size determination by use of two-dimensional Mueller matrices backscattered by optically thick random media,” Appl. Opt.  45, 4669 (2006).
[Crossref] [PubMed]

Bartel, S.

S. Bartel and A. H. Hielscher, “Monte Carlo simulations of the diffuse backscattering mueller matrix for highly scattering media,” Appl. optics 39, 1580–1588 (2000).
[Crossref]

Beléndez, A.

A. Beléndez, E. Fernndez, J. Francés, and C. Neipp, “Birefringence of cellotape: Jones representation and experimental analysis,” Eur. J. Phys. 31, 551–561 (2010).
[Crossref]

Ben Hatit, S.

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them?” Phys. Status Solidi (a) 205, 720–727 (2008).
[Crossref]

S. Ben Hatit, “Polarimétrie de Mueller résolue en angle,” Ph.D. thesis, Ecole Polytechnique X (2009).

Bernabeu, E.

J. Gil and E. Bernabeu, “Obtainment of the polarizing and. retardation parameters of nondepolarizing optical system from. polar decomposition of its Mueller matrix,” Optik 76, 67 (1987).

Berry, M.

M. Berry, “Geometric phase memories,” Nat. Phys. 6, 148–150 (2010).
[Crossref]

M. Berry, “Anticipations of the geometric phase,” Phys. Today 43, 34–40 (1990).
[Crossref]

M. Berry, “The Geometric Phase,” Sci. Am. 259, 46–55 (1988).
[Crossref]

Berry, M. V.

M. V. Berry, “Nature’s optics and our understanding of light,” Contemp. Phys. 56, 2–16 (2015).

Bickel, W. S.

W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
[Crossref]

Bicout, D.

C. Brosseau, D. Bicout, A. S. Martinez, and J. M. Schmitt, “Depolarization behavior of multiple scattered light from an optically dense random medium,” Phys. Rev. E 49, 1767–1770 (1994).

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).

Brosseau, C.

C. Brosseau, D. Bicout, A. S. Martinez, and J. M. Schmitt, “Depolarization behavior of multiple scattered light from an optically dense random medium,” Phys. Rev. E 49, 1767–1770 (1994).

Brown, A. J.

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, G. Videen, and C. J. Grund, “The case for a modern multiwavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transf. 153, 131–143 (2015).
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A. J. Brown, “Equivalence relations and symmetries for laboratory, LIDAR, and planetary Müeller matrix scattering geometries,” J. Opt. Soc. Am. A 31, 2789–2794 (2014).
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A. J. Brown and Y. Xie, “Symmetry relations revealed in Mueller matrix hemispherical maps,” J. Quant. Spectrosc. Radiat. Transf. 113, 644–651 (2012).
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Bull, D. R.

A. B. Tibbs, I. M. Daly, D. R. Bull, and N. W. Roberts, “Noise creates polarization artefacts,” Bioinspir. Biomim.  13, 015005 (2018).
[Crossref]

Byrne, S.

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, G. Videen, and C. J. Grund, “The case for a modern multiwavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transf. 153, 131–143 (2015).
[Crossref]

Cai, W.

W. Cai, X. Ni, S. K. Gayen, and R. R. Alfano, “Analytical cumulant solution of the vector radiative transfer equation investigates backscattering of circularly polarized light from turbid media,” Phys. Rev. E -Stat. Nonlinear, Soft Matter Phys. 74, 1–11 (2006).
[Crossref]

Cameron, B. D.

M. J. Raković, G. W. Kattawar, M. B. Mehrubeoğlu, B. D. Cameron, L. V. Wang, S. Rastegar, and G. L. Coté, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. optics 38, 3399 (1999).
[Crossref]

Canillas, A.

O. Arteaga and A. Canillas, “Analytic inversion of the Mueller -Jones polarization matrices for homogeneous media,” Opt. letters 35, 559–561 (2010).
[Crossref]

Caton, F.

A. Parker, C. Baravian, F. Caton, J. Dillet, and J. Mougel, “Fast optical sizing without dilution,” Food Hydrocoll. 21, 831–837 (2007).
[Crossref]

J. Dillet, C. Baravian, F. Caton, and A. Parker, “Size determination by use of two-dimensional Mueller matrices backscattered by optically thick random media,” Appl. Opt.  45, 4669 (2006).
[Crossref] [PubMed]

Chang, J.

Y. Wang, H. He, J. Chang, N. Zeng, S. Liu, M. Li, and H. Ma, “Differentiating characteristic microstructural features of cancerous tissues using Mueller matrix microscope,” Micron 79, 8–15 (2015).
[Crossref] [PubMed]

Charras, G.

L. S. Hirst and G. Charras, “Biological physics: Liquid crystals in living tissue,” Nature 544, 164–165 (2017).
[Crossref]

Chipman, R. A.

S.-Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106 (1996).
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R. A. Chipman, “Chapter 22: POLARIMETRY,” in Handbook of Optics Volume II: Devices, Measurements, and Properties, M. Bass, ed. (McGraw Hill, 2009), 2nd ed.

Colaprete, A.

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, G. Videen, and C. J. Grund, “The case for a modern multiwavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transf. 153, 131–143 (2015).
[Crossref]

Compain, E.

E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers,” Appl. Opt.  38, 3490 (1999).
[Crossref]

Coté, G. L.

M. J. Raković, G. W. Kattawar, M. B. Mehrubeoğlu, B. D. Cameron, L. V. Wang, S. Rastegar, and G. L. Coté, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. optics 38, 3399 (1999).
[Crossref]

Craven-Jones, J.

F. Snik, J. Craven-Jones, M. Escuti, S. Fineschi, D. Harrington, A. De Martino, D. Mawet, J. Riedi, and J. S. Tyo, “An overview of polarimetric sensing techniques and technology with applications to different research fields,” Proc. SPIE 9099, 90990B (2014).

Daly, I. M.

A. B. Tibbs, I. M. Daly, D. R. Bull, and N. W. Roberts, “Noise creates polarization artefacts,” Bioinspir. Biomim.  13, 015005 (2018).
[Crossref]

De Martino, A.

F. Snik, J. Craven-Jones, M. Escuti, S. Fineschi, D. Harrington, A. De Martino, D. Mawet, J. Riedi, and J. S. Tyo, “An overview of polarimetric sensing techniques and technology with applications to different research fields,” Proc. SPIE 9099, 90990B (2014).

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them?” Phys. Status Solidi (a) 205, 720–727 (2008).
[Crossref]

Dick, V. P.

V. P. Dick, “Applicability Limits of Beer’s Law for Dispersion Media with a High Concentration of Particles,” Appl. optics 37, 4998–5004 (1998).
[Crossref]

Dillet, J.

A. Parker, C. Baravian, F. Caton, J. Dillet, and J. Mougel, “Fast optical sizing without dilution,” Food Hydrocoll. 21, 831–837 (2007).
[Crossref]

J. Dillet, C. Baravian, F. Caton, and A. Parker, “Size determination by use of two-dimensional Mueller matrices backscattered by optically thick random media,” Appl. Opt.  45, 4669 (2006).
[Crossref] [PubMed]

Dogariu, A.

C. Schwartz and A. Dogariu, “Backscattered polarization patterns determined by conservation of angular momentum,” J. Opt. Soc. Am. A, Opt. image science, vision  25, 431 (2008).
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C. Schwartz and A. Dogariu, “Backscattered polarization patterns, optical vortices, and the angular momentum of light,” Opt. Lett.  31, 1121 (2006).
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Doronin, A.

A. Doronin, C. Macdonald, and I. Meglinski, “Propagation of coherent polarized light in turbid highly scattering medium,” J. Biomed. Opt.  19, 025005 (2014).
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Drevillon, B.

E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers,” Appl. Opt.  38, 3490 (1999).
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Elson, D. S.

J. Qi and D. S. Elson, “Mueller polarimetric imaging for surgical and diagnostic applications: a review,” J. Biophotonics 10, 950–982 (2017).
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Escuti, M.

F. Snik, J. Craven-Jones, M. Escuti, S. Fineschi, D. Harrington, A. De Martino, D. Mawet, J. Riedi, and J. S. Tyo, “An overview of polarimetric sensing techniques and technology with applications to different research fields,” Proc. SPIE 9099, 90990B (2014).

Fernndez, E.

A. Beléndez, E. Fernndez, J. Francés, and C. Neipp, “Birefringence of cellotape: Jones representation and experimental analysis,” Eur. J. Phys. 31, 551–561 (2010).
[Crossref]

Fineschi, S.

F. Snik, J. Craven-Jones, M. Escuti, S. Fineschi, D. Harrington, A. De Martino, D. Mawet, J. Riedi, and J. S. Tyo, “An overview of polarimetric sensing techniques and technology with applications to different research fields,” Proc. SPIE 9099, 90990B (2014).

Finlayson, E. D.

L. T. McDonald, E. D. Finlayson, B. D. Wilts, and P. Vukusic, “Circularly polarized reflection from the scarab beetle Chalcothea smaragdina: Light scattering by a dual photonic structure,” Interface Focus. 7, 20160129 (2017).
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Fraden, S.

S. Fraden and G. Maret, “Multiple light scattering from concentrated, interacting suspensions,” Phys. Rev. Lett. 65, 512–515 (1990).
[Crossref] [PubMed]

Francés, J.

A. Beléndez, E. Fernndez, J. Francés, and C. Neipp, “Birefringence of cellotape: Jones representation and experimental analysis,” Eur. J. Phys. 31, 551–561 (2010).
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Frenz, M.

Fujiwara, H.

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Garcia-Caurel, E.

H. Hu, E. Garcia-Caurel, G. Anna, and F. Goudail, “Maximum likelihood method for calibration of Mueller polarimeters in reflection configuration,” Appl. Opt.  52, 6350 (2013).
[Crossref] [PubMed]

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them?” Phys. Status Solidi (a) 205, 720–727 (2008).
[Crossref]

Gayen, S. K.

W. Cai, X. Ni, S. K. Gayen, and R. R. Alfano, “Analytical cumulant solution of the vector radiative transfer equation investigates backscattering of circularly polarized light from turbid media,” Phys. Rev. E -Stat. Nonlinear, Soft Matter Phys. 74, 1–11 (2006).
[Crossref]

Ghosh, N.

N. Ghosh and I. A. Vitkin, “Tissue polarimetry : concepts, challenges, applications, and outlook,” J. biomedical optics 16, 110801 (2011).
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D. Layden, N. Ghosh, and A. Vitkin, “Quantitative Polarimetry for Tissue Characterization and Diagnosis,” in Advanced Biophotonics: Tissue Optical Sectioning, R. K. Wang and V. V. Tuchin, eds. (CRC Press, 2016).

N. Ghosh, M. Wood, and A. Vitkin, “Polarized Light Assessment of Complex Turbid Media Such as Biological Tissues Using Mueller Matrix Decomposition,” in Handbook of Photonics for Biomedical Science, V. V. Tuchin, ed. (CRC Press, 2010), chap. 9, pp. 253–282.
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Gil, J.

J. Gil and E. Bernabeu, “Obtainment of the polarizing and. retardation parameters of nondepolarizing optical system from. polar decomposition of its Mueller matrix,” Optik 76, 67 (1987).

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J. J. Gil, “Review on Mueller matrix algebra for the analysis of polarimetric measurements,” J. Appl. Remote. Sens.  8, 81599 (2014).
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J. J. Gil and R. Ossikovski, Polarized Light and the Mueller Matrix Approach (CRC Press, 2016).

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D. H. Goldstein, Polarized Light, revised and expanded (CRC Press, 2003).
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Goudail, F.

H. Hu, E. Garcia-Caurel, G. Anna, and F. Goudail, “Maximum likelihood method for calibration of Mueller polarimeters in reflection configuration,” Appl. Opt.  52, 6350 (2013).
[Crossref] [PubMed]

Greffet, J.-J.

L. Hespel, S. Mainguy, and J.-J. Greffet, “Theoretical and experimental investigation of the extinction in a dense distribution of particles: nonlocal effects Laurent,” Josa a 18, 3072–3076 (2001).
[Crossref]

Grund, C. J.

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, G. Videen, and C. J. Grund, “The case for a modern multiwavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transf. 153, 131–143 (2015).
[Crossref]

Gupta, P. K.

M. K. Swami, H. Patel, M. R. Somyaji, P. K. Kushwaha, and P. K. Gupta, “Size-dependent patterns in depolarization maps from turbid medium and tissue,” Appl. Opt.  53, 6133 (2014).
[Crossref] [PubMed]

Hariharan, P.

P. Hariharan, “The geometric phase,” Prog. Opt. 48, 149–201 (2005).
[Crossref]

Harrington, D.

F. Snik, J. Craven-Jones, M. Escuti, S. Fineschi, D. Harrington, A. De Martino, D. Mawet, J. Riedi, and J. S. Tyo, “An overview of polarimetric sensing techniques and technology with applications to different research fields,” Proc. SPIE 9099, 90990B (2014).

He, H.

Y. Wang, H. He, J. Chang, N. Zeng, S. Liu, M. Li, and H. Ma, “Differentiating characteristic microstructural features of cancerous tissues using Mueller matrix microscope,” Micron 79, 8–15 (2015).
[Crossref] [PubMed]

Hecht, E.

E. Hecht, Optics (Pearson Education, 2002), 4th ed.

Hespel, L.

L. Hespel, S. Mainguy, and J.-J. Greffet, “Theoretical and experimental investigation of the extinction in a dense distribution of particles: nonlocal effects Laurent,” Josa a 18, 3072–3076 (2001).
[Crossref]

Hielscher, A. H.

S. Bartel and A. H. Hielscher, “Monte Carlo simulations of the diffuse backscattering mueller matrix for highly scattering media,” Appl. optics 39, 1580–1588 (2000).
[Crossref]

Hirst, L. S.

L. S. Hirst and G. Charras, “Biological physics: Liquid crystals in living tissue,” Nature 544, 164–165 (2017).
[Crossref]

Hohmann, A.

Hovenier, J. W.

J. W. Hovenier, “Symmetry Relationships for Scattering of Polarized Light in a Slab of Randomly Oriented Particles,” (1969).

Hu, H.

H. Hu, E. Garcia-Caurel, G. Anna, and F. Goudail, “Maximum likelihood method for calibration of Mueller polarimeters in reflection configuration,” Appl. Opt.  52, 6350 (2013).
[Crossref] [PubMed]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
[Crossref]

Jacques, S. L.

C. M. Macdonald, S. L. Jacques, and I. V. Meglinski, “Circular polarization memory in polydisperse scattering media,” Phys. Rev. E -Stat. Nonlinear, Soft Matter Phys. 91, 1–5 (2015).
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James, D.

O. Korotkova and D. James, “Definitions of the Degree of Polarization of a Light,” Opt. letters 32, 1015–1016 (2007).
[Crossref]

Jones, C. R.

C. R. Jones, “A New Calculus for the Treatment of Optical Systems. IV,.” J. Opt. Soc. Am 32, 486–493 (1942).
[Crossref]

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Kattawar, G. W.

M. J. Raković, G. W. Kattawar, M. B. Mehrubeoğlu, B. D. Cameron, L. V. Wang, S. Rastegar, and G. L. Coté, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. optics 38, 3399 (1999).
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Kienle, A.

Korotkova, O.

O. Korotkova and D. James, “Definitions of the Degree of Polarization of a Light,” Opt. letters 32, 1015–1016 (2007).
[Crossref]

Kushwaha, P. K.

M. K. Swami, H. Patel, M. R. Somyaji, P. K. Kushwaha, and P. K. Gupta, “Size-dependent patterns in depolarization maps from turbid medium and tissue,” Appl. Opt.  53, 6133 (2014).
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Kuzmin, V.

V. Kuzmin and I. Meglinski, “Helicity flip of the backscattered circular polarized light,” in Biomedical Applications of Light Scattering IV, Proc. of SPIE Vol. 7573, A. P. Wax and V. Backman, eds. (2010), February, p. 75730Z.
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Lacoste, D.

Layden, D.

D. Layden, N. Ghosh, and A. Vitkin, “Quantitative Polarimetry for Tissue Characterization and Diagnosis,” in Advanced Biophotonics: Tissue Optical Sectioning, R. K. Wang and V. V. Tuchin, eds. (CRC Press, 2016).

Lenke, R.

Li, M.

Y. Wang, H. He, J. Chang, N. Zeng, S. Liu, M. Li, and H. Ma, “Differentiating characteristic microstructural features of cancerous tissues using Mueller matrix microscope,” Micron 79, 8–15 (2015).
[Crossref] [PubMed]

Liu, S.

Y. Wang, H. He, J. Chang, N. Zeng, S. Liu, M. Li, and H. Ma, “Differentiating characteristic microstructural features of cancerous tissues using Mueller matrix microscope,” Micron 79, 8–15 (2015).
[Crossref] [PubMed]

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This was perhaps inspired by the seminal works of Jones (who, while introducing his matrix calculus, proposed a method to characterize non-depolarizing systems in terms of dichroism and birefringence [83,84]) that lies, by now, on solid mathematical grounds [85].

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In principle, the parameters δ1−4 could be resolved into statistical properties of the distributions of Δ, θc and ϕ. It would also be interesting to explore the effect of these newly defined semi-microscopic parameters on the predicted PEPs and polarization ellipses, analogous to what has been done in section 5.1 and pursue the analysis outlined in Appendix C. However, we shall leave such investigations to future studies, for they would exceed the capacities of this paper.

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Noise and speckle [86] reduction was achieved by taking the average of 100 images for each intensity distribution J〈o|i〉(ρ, ϕ). Moreover, the background noise/dark-current of the CCD is calculated by performing a full measurement set (again by taking an average over 100 images for each input/output pair) with the light source switched off and then subtracted from the J〈o|i〉(ρ, ϕ) images.

The matrix M(ρ, ϕ) can be constructed from 4×4 measurements, but as explained in [30], we prefer to use an over-determined system.

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We have ensured that we are working within the independent scattering regime [87–90].

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As a heuristic measure for the reliability of the presented measurements, we take the standard deviation σ of the de-rotated PM matrix. It is a measure for the variability of the averaged quantity, that is assumed constant. It includes statistical contributions from camera noise, but also systematic errors in the matrix elements not being rotationally symmetric (as it is clearly the case for sample §1). A matrix element is chosen to be significantly different from zero, if > 2σ. A rigorous error analysis would be necessary when probing more complex systems (see, e.g., [91]). Here, the focus lies on outlining our method and Monte Carlo simulations are sufficient to corroborate our results.

The origin of these differences is still subject to investigation (preliminary studies have shown that effects such as coherent backscattering, which are not modeled in the simulations, can be discarded). Moreover, in order to verify the reproducibility of our results, we have repeated the measurements five times on sample §2, by producing the sample anew for each measurement. We have established that the reproducibility error lies within the standard deviation given in the radial plots in Fig. 4.

M. Mischchenko, J. Hovenier, and L. Travis, eds., Light Scattering by Nonspherical Particles (Academic Press, 2000).

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[Crossref]

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[Crossref]

J. J. Gil and R. Ossikovski, Polarized Light and the Mueller Matrix Approach (CRC Press, 2016).

This input state dependence vanishes for d1(ρ) = d2(ρ), which is the case for a PM-matrix equivalent to Jones matrix that is diagonal, i.e., describing a non-depolarizing system.

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Figures (13)

Fig. 1
Fig. 1 Schematic drawing (not to scale) of the imaging polarimetric setup used in this study. This instrument is composed of two arms: one illumination arm containing the polarization state generator (PSG) and one detection arm containing the polarization state analyzer (PSA). LC, L1, and L2 designate lenses. LCR and LP are liquid crystal retarders and linear polarizers, respectively. The scattering system is probed at λ = 785 nm by a focused beam. The spatial distribution of the backscattered light intensity is recorded by a CCD chip placed at the end of the detection arm. Σlab denotes the laboratory frame.
Fig. 2
Fig. 2 Representation of all the right-handed coordinate systems involved in the description of the experiment. The laboratory frame Σlab corresponds to the one drawn in Fig. 1. Σin is the input frame in which the illumination beam’s polarization state (see Table 1) is defined. The detection frame Σout attached to the CCD camera coincides with Σlab. For the radial analysis carried out in this paper, we define for each point of interest at (ρ, ϕ) on the probed system’s surface a principal scattering plane. This plane is spanned by the k-vector (here collinear with Σin’s z-axis) of the probing light beam at (ρ = 0, ϕ = 0) and the k-vector (collinear with Σout ’s z-axis) of the backscattered light at the point at (ρ, ϕ). The polarization state of the incident and backscattered light can be defined with respect to this scattering plane, in the frames Σscat_in and Σscat_out, respectively: el and e denote the components that are parallel and perpendicular to this plane, respectively. The thick black line represents a photon path within the scattering plane.
Fig. 3
Fig. 3 Measured (top) and simulated (bottom) spatial PM-matrices for samples §1 (left), §2 (center), and §3 (right). All elements except for m11 are pixel-wise normalized by m11 (similarly to the plots shown in [30]). (Recall that the probing beam is focused at the center (ρ = 0,ϕ = 0) of the samples’ surface.)
Fig. 4
Fig. 4 The block-diagonal, de-rotated PM-matrices M ˜ ( ρ ) (top) obtained from measurements are shown for the samples §1 (left), §2 (center), and §3 (right). In the case of the weakly scattering sample §1, the tilt of the illumination/probing beam is visible at the center of the matrix elements. A more quantitative visualization is provided in the bottom row, where, for all samples, the matrix elements averaged over ϕ are plotted as a function of ρ: this allows for a better comparison between measurements (left) and simulations (right). The shaded areas represent the standard deviation calculated for each ϕ.
Fig. 5
Fig. 5 Measured polarization ellipses overplotted on inverse grayscale intensity maps for all three samples for the input states |LX〉 and |C+〉. The samples’ responses for the other input states can be derived from symmetry considerations. The ellipses are color coded with the helicity (red: right circular; blue: left circular; black: linear states) and scaled with the degree of polarization. Differences between the samples can be better observed as in Figs. 3 and 4.
Fig. 6
Fig. 6 Measured polarization ellipse parameters for all three samples for the input states |LX〉 and |C+〉 (first row: intensity distribution, second row: degree of polarization; third row: ellipticity/helicity; fourth row: orientation). These plots contain the same polarimetric information shown in Fig. 5, but organized differently. Again, this allows for a more detailed visualization of the differences between the samples, especially in the degree of polarization and the ellipticity plots. The samples’ responses for the other input states can be derived from symmetry considerations. In each image, a profile along ϕ gives the sample’s response obtained when applying the de-rotated matrix M ˜ ( ρ ) on a linear input state S i aligned with the profile at −ϕ. For instance, profiles along ϕ = 0 (1), ϕ = π/2 (2), and ϕ = π/4 (3) give radial responses corresponding to |LX〉, |LY〉, and |L−〉, respectively.
Fig. 7
Fig. 7 Diattenuation D(ρ) measured for different size parameters. The diattenuating properties of the colloidal suspensions can be read from the anisotropic intensity distributions in Figs. 5 and 6.
Fig. 8
Fig. 8 Degree of polarization Π(ρ) for various input states for the three probed samples. These radial plots coincide with the radial profiles along (1), (2), and (3) in the Π(ρ, ϕ) images in Fig. 6.
Fig. 9
Fig. 9 Phenomenological retardance R(ρ) calculated for circular |C±〉 and diagonal |L±〉 input states. As for specular reflection, the colloidal suspensions flip the helicity/orientation of the light polarization at the vicinity of the illumination point (ρ → 0). This gives R(ρ) ~ π. While R(ρ) remains constant in sample §1, it gradually decreases to 0 in samples §2 and §3. The trends observed here can be found in the radial profiles along (1), (2), and (3), both in Fig. 5 and in the ellipticity/helicity/orientation plots in Fig. 6.
Fig. 10
Fig. 10 PEPs obtained with the coherency matrix model of Eq. (43) for |LX〉 and |C+〉 input polarizations (first row: intensity distribution, second row: degree of polarization; third row: ellipticity/helicity; fourth row: orientation). The plots show the influence of the model parameters (helicity flip fraction α taken constant over ρ, semi-microscopic diattenuation parameter Δ ¯, and semi-microscopic retardance parameter θ ¯ c) on the polarization ellipses of the backscattered light. The field of view is set to 2 × 2 (arbitrary units). Profiles along ϕ = 0 (1), ϕ = π/2 (2), and ϕ = π/4 (3) are analogous to the ones given in Fig. 6 (the responses for other linear input states can be derived from symmetry considerations).
Fig. 11
Fig. 11 Polarization ellipses obtained with the coherency matrix model of Eq. (43) for |LX〉 input polarization. The plots here show the influence of the model parameters (helicity flip fraction α taken constant over ρ, semi-microscopic diattenuation parameter Δ ¯) on the polarization ellipses of the backscattered light in the absence of retardance. The field of view is set to 2 × 2 (arbitrary units). Profiles along ϕ = 0 (1), ϕ = π/2 (2), and ϕ = π/4 (3) are analogous to the ones given in Fig. 6 (the responses for other linear input states can be derived from symmetry considerations).
Fig. 12
Fig. 12 PEPs obtained with the coherency matrix model of Eq. (43) for |LX〉 and |C+〉 input polarizations (first row: intensity distribution, second row: degree of polarization; third row: ellipticity/helicity; fourth row: orientation). Like in Figs. 10 and 11, the field of view in these figures is set to 2×2 (arbitrary units). The models parameters {I(ρ), α(ρ), ∆(ρ), θc(ρ), BG(ρ)} have been manually fitted to describe the measurement results presented in Fig. 6 (the fitted parameters can be found in Table 3 in Appendix C).
Fig. 13
Fig. 13 Measurements performed on sample §2, where a cellophane tape has been added on the probed surface. As for Fig. 5, the polarization ellipses are overplotted on inverse grayscale intensity maps for the input states |LX〉, |L+〉 and |C+〉. The ellipses are color coded with the helicity (red: right circular; blue: left circular; black: linear states) and scaled with the degree of polarization. The fast and slow axes of the cellophane tape are along the profiles drawn at ϕ = ±π/4 (dashed lines).

Tables (3)

Tables Icon

Table 1 The three pairs of orthogonal polarization states that we used to probe the samples.

Tables Icon

Table 2 Features of the three samples probed in the experiments at λ =785 nm.

Tables Icon

Table 3 Coherency matrix model parameters determined empirically to describe the polarimetric experiments. These parameters are here compared to characteristics of the probed colloidal suspensions’ single scattering phase function.

Equations (89)

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| E k = ( E x ( t ) E y ( t ) ) k .
I = | E x | 2 T A + | E x | 2 T A
Q = | E x | 2 T A | E x | 2 T A
U = E x E y * T A + E x * E y T A
V = i ( E x E y * T A E x * E y T A ) ,
I o ( ρ , ϕ ) = ( J L X | i ( ρ , ϕ ) + J L Y | i ( ρ , ϕ ) ) / 3 + ( J L + | i ( ρ , ϕ ) + J L | i ( ρ , ϕ ) ) / 3 + ( J C + | i ( ρ , ϕ ) + J C | i ( ρ , ϕ ) ) / 3
Q o ( ρ , ϕ ) = J L X | i ( ρ , ϕ ) J L Y | i ( ρ , ϕ )
U o ( ρ , ϕ ) = J L + | i ( ρ , ϕ ) J L | i ( ρ , ϕ )
V o ( ρ , ϕ ) = J C + | i ( ρ , ϕ ) J C | i ( ρ , ϕ )
R ( ϕ ) = ( cos ( ϕ ) sin ( ϕ ) sin ( ϕ ) cos ( ϕ ) )
R ( ϕ ) = ( 1 0 0 0 0 cos ( ϕ ) sin ( ϕ ) 0 0 sin ( ϕ ) cos ( ϕ ) 0 0 0 0 1 )
J o | i ( ρ , ϕ ) = S o T M ( ρ , ϕ ) S i
S o ( ρ , ϕ ) = M ( ρ , ϕ ) S i ,
[ J o | i ( ρ , ϕ ) ] i , o = AM ( ρ , ϕ ) W .
M ( ρ , ϕ ) = A 1 [ J o | i ( ρ , ϕ ) ] i , o W 1 .
M ( ρ , ϕ ) = R ( ϕ ) M ˜ ( ρ ) R ( ϕ ) .
M ˜ ( ρ ) = ( a 1 ( ρ ) b ( ρ ) 0 0 b ( ρ ) a 2 ( ρ ) 0 0 0 0 d 1 ( ρ ) e ( ρ ) 0 0 e ( ρ ) d 2 ( ρ ) ) ,
M ˜ ( ρ ) = R ( ϕ ) M ( ρ , ϕ ) R ( ϕ ) .
tan [ 2 ψ ( ρ , ϕ ) ] = Q o ( ρ , ϕ ) U o ( ρ , ϕ ) with π / 2 ψ ( ρ , ϕ ) π / 2 .
ϵ ( ρ , ϕ ) = ( 1 M min ( ρ , ϕ ) M max ( ρ , ϕ ) ) h ( ρ , ϕ )
M max 2 ( ρ , ϕ ) = 1 2 Q o 2 ( ρ , ϕ ) + U o 2 ( ρ , ϕ ) + V o 2 ( ρ , ϕ ) + 1 2 Q o 2 ( ρ , ϕ ) + U o 2 ( ρ , ϕ )
M min 2 ( ρ , ϕ ) = 1 2 Q o 2 ( ρ , ϕ ) + U o 2 ( ρ , ϕ ) + V o 2 ( ρ , ϕ ) 1 2 Q o 2 ( ρ , ϕ ) + U o 2 ( ρ , ϕ ) .
h ( ρ , ϕ ) = sgn ( V o ( ρ , ϕ ) ) .
( ρ , ϕ ) = ( Q o 2 ( ρ , ϕ ) + U o 2 ( ρ , ϕ ) + V o 2 ( ρ , ϕ ) ) I o ( ρ , ϕ ) .
S o ( ρ ) = ( I o Q o U o V o ) T
= M ˜ ( ρ ) S i = M ˜ ( ρ ) ( I i Q i U i V i ) T
= ( a 1 ( ρ ) I i + b ( ρ ) Q i b ( ρ ) I i + a 2 ( ρ ) Q i d 1 ( ρ ) U i + e ( ρ ) V i e ( ρ ) U i + d 2 V i ) T .
I o ( ρ ) = a 1 ( ρ ) I i + b ( ρ ) Q i .
D ( ρ ) = | b ( ρ ) | a 1 ( ρ ) .
( ρ ) = ( a 2 ( ρ ) Q i + b ( ρ ) I i ) 2 + ( d 1 ( ρ ) U i e ( ρ ) V i ) 2 + ( d 2 ( ρ ) V i + e ( ρ ) U i ) 2 a 1 ( ρ ) I i + b ( ρ ) Q i .
( ρ ) | L X = a 2 ( ρ ) + b ( ρ ) a 1 ( ρ ) + b ( ρ ) ( ρ ) | L Y = a 2 ( ρ ) b ( ρ ) a 1 ( ρ ) b ( ρ )
( ρ ) | L ± = d 1 2 ( ρ ) + e 2 ( ρ ) a 1 ( ρ ) ( ρ ) | C ± = d 2 2 ( ρ ) + e 2 ( ρ ) a 1 ( ρ ) .
R ( ρ ) = arctan ( ( d 1 ( ρ ) d 2 ( ρ ) ) U i V i + e ( ρ ) ( U i 2 + V i 2 ) d 1 ( ρ ) ( U i 2 + V i 2 ) ( d 1 ( ρ ) d 2 ( ρ ) ) V i 2 ) .
R | L ± ( ρ ) = arctan ( e ( ρ ) d 1 ( ρ ) ) R | C ± ( ρ ) = arctan ( e ( ρ ) d 2 ( ρ ) ) .
C o ( ρ , ϕ ) = T ( ρ , ϕ ) C i T * ( ρ , ϕ ) .
C o ( ρ , ϕ ) = T ( ρ , ϕ ) T * ( ρ , ϕ ) C i ,
C o ( ρ , ϕ ) = ( ( 1 α ( ρ ) ) T c ( ρ , ϕ ) T c * ( ρ , ϕ ) + α ( ρ ) T f ( ρ , ϕ ) T f * ( ρ , ϕ ) ) C i ,
T c ( ρ , ϕ ) = R ( ϕ ) S c ( ρ ) R ( ϕ )
T f ( ρ , ϕ ) = R ( ϕ ) S f ( ρ ) R ( ϕ ) .
S c ( ρ ) = I ( ρ ) ( Δ c ( ρ ) e i θ c ( ρ ) 0 0 1 )
S f ( ρ ) = I ( ρ ) ( Δ f ( ρ ) e i θ f ( ρ ) 0 0 1 ) .
C o ( ρ , ϕ ) = I ( ρ ) ( 1 B G ( ρ ) ) ( ( 1 α ( ρ ) ) T c ( ρ , ϕ ) T c * ( ρ , ϕ ) + α ( ρ ) T f ( ρ , ϕ ) T f * ( ρ , ϕ ) ) C i + I ( ρ ) B G ( ρ ) T ˜ d e p o l
C o ( ρ , ϕ ) = T ˜ ( ρ , ϕ ) C i .
T ˜ d e p o l = ( 1 2 0 0 1 2 0 0 0 0 0 0 0 0 1 2 0 0 1 2 ) .
Δ ( ρ ) = { 10 Δ ¯ ρ ρ 0.1 Δ ¯ ρ > 0.1 .
M ( ρ , ϕ ) = T ˜ ( ρ , ϕ ) 1 ,
= ( 1 0 0 1 1 0 0 1 0 1 1 0 0 i i 0 ) .
M ˜ ( ρ ) = T ˜ ( ρ , ϕ = 0 ) 1 .
m 11 = a 1 = 1 2 ( 1 + 2 δ Δ 2 )
m 22 = a 2 = 1 2 ( 1 + Δ 2 )
m 12 = m 21 = b = 1 2 ( 1 Δ 2 )
m 33 = d 1 = Δ ( ( α 1 ) cos ( θ c ) + α )
m 44 = d 2 = Δ ( ( α 1 ) cos ( θ c ) + α )
m 34 = m 43 = e = Δ ( α 1 ) sin ( θ c ) .
δ = 1 B G 1 .
D = | b | a 1 = | 1 Δ 2 | 1 + 2 δ Δ 2 .
R | L ± = R | C ± = arctan e d 1 / 2 = arctan ( ( α 1 ) sin ( θ c ) ( α 1 ) cos ( θ c ) + α ) .
Π | L X = Δ 2 1 + δ Δ 2
Π | L Y = 1 δ
Π | L ± = 2 Δ ( α 1 ) 2 + α 2 + 2 α ( α 1 ) cos ( θ c ) 1 + 2 δ Δ 2
Π | C ± = Π | L ± .
Π | L ± = Π | C ± = 2 Δ | ( 2 α 1 ) | 1 + 2 δ Δ 2 .
C o = ( ( 1 α ) T c T c * + α T f T f * ) C i ,
T c / f T c / f * = T c / f T c / f * + Cov ( T c / f , T c / f ) .
Λ = ( δ 1 0 0 δ 4 0 δ 2 δ 4 0 0 δ 4 δ 2 * 0 δ 4 0 0 δ 3 )
M ˜ = ( ( 1 α ) T c T c * + α T f T f * + Λ ) 1 .
m 11 = a 1 = 1 2 ( 1 + δ 1 + δ 3 2 δ 4 + Δ 2 )
m 22 = a 2 = 1 2 ( 1 + δ 1 + δ 3 + 2 δ 4 + Δ 2 )
m 12 = m 21 = b = 1 2 ( 1 + δ 1 δ 3 + Δ 2 )
m 33 = d 1 = Re ( δ 2 ) + δ 4 Δ ( α + ( α 1 ) cos ( θ c ) )
m 44 = d 2 = Re ( δ 2 ) δ 4 Δ ( α + ( α 1 ) cos ( θ c ) )
m 34 = m 43 = e = Im ( δ 2 ) Δ ( α 1 ) sin ( θ c ) .
D = | b | a 1 = | 1 + δ 1 δ 3 + Δ 2 | 1 + δ 1 + δ 3 2 δ 4 + Δ 2
R | L ± = arctan ( Im ( δ 2 ) Δ ( α 1 ) sin ( θ c ) Re ( δ 2 ) + δ 4 Δ ( α + ( α 1 ) cos ( θ c ) ) )
R | C ± = arctan ( Im ( δ 2 ) Δ ( α 1 ) sin ( θ c ) Re ( δ 2 ) δ 4 Δ ( α + ( α 1 ) cos ( θ c ) ) ) .
Π | L X = δ 1 + δ 4 + Δ 2 δ 1 δ 4 + Δ 2
Π | L Y = 1 + δ 3 + δ 4 1 + δ 3 δ 4
Π | L ± = 2 ( Re ( δ 2 ) + δ 4 Δ ( α + ( α 1 ) cos ( θ c ) ) ) 2 + ( Im ( δ 2 ) Δ ( α 1 ) sin ( θ c ) ) 2 1 + δ 1 + δ 3 2 δ 4 + Δ 2
Π | C ± = 2 ( Re ( δ 2 ) δ 4 Δ ( α + ( α 1 ) cos ( θ c ) ) ) 2 + ( Im ( δ 2 ) Δ ( α 1 ) sin ( θ c ) ) 2 1 + δ 1 + δ 3 2 δ 4 + Δ 2
T | C ( ρ , ϕ ) = N A 2 ( ρ ) ( α ( ρ ) exp ( i 2 ϕ ) 1 α ( ρ ) 1 α ( ρ ) α ( ρ ) exp ( i 2 ϕ ) ) .
T | L ( ρ , ϕ ) = 1 2 ( i 1 i 1 ) J | C ( ρ , ϕ ) ( i 1 i 1 )
= N A 2 ( ρ ) ( 2 α sin ( ϕ ) 2 + 1 2 α cos ( ϕ ) sin ( ϕ ) 2 α cos ( ϕ ) sin ( ϕ ) 2 α cos ( ϕ ) 2 1 )
T | L ( ρ , ϕ ) = N A 2 ( ρ ) R ( ϕ ) ( 1 0 0 2 α ( ρ ) 1 ) R ( ϕ ) .
T c / f ( ρ , ϕ , η , ξ ) = R ( η ) T B R ( ξ ) R ( η ) T c / f ( ρ , ϕ ) R ( η ) T B R ( ξ ) R ( η ) ,
T B R ( ξ ) = ( 1 0 0 e i ξ ) .
S ( Φ ) = ( S ( Φ ) 0 0 S ( Φ ) )
Δ s = 1 π 0 π | S ( Φ ) | 2 | S ( Φ ) | 2 | S ( Φ ) | 2 + | S ( Φ ) | 2 d Φ .
θ s = 1 π 0 π min { | arg S ( Φ ) arg S ( Φ ) | | arg S ( Φ ) arg S ( Φ ) π | d Φ ,
g = 1 π 0 π cos ( Φ ) ( | S ( Φ ) | 2 + | S ( Φ ) | 2 ) d Φ .

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