Abstract

A comprehensive physically realizable space, namely, the overall purity index-components of purity (PI4DCP) space is proposed for the characterization of the depolarization caused by random (or deterministic) media. The overall purity index (PI4D)is obtained via indices of polarimetric purity which are incurred by the eigenvalues of the covariance matrix, whereas the components of purity (CP) are the functions of the elements of a Mueller matrix. On the one hand, the proposed space is useful in studying the depolarization caused by material media and on the other hand, it provides information on the diattenuation-polarizance properties of a Mueller matrix. Thus, it gives a remarkable physical insight of the depolarization problem.

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

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References

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2019 (5)

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, M. Durfort, J. C. Escalera, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Depolarizing metrics for plant samples imaging,” PLoS One 14(3), e0213909 (2019).
[Crossref] [PubMed]

Y. Chang and W. Gao, “Method of interpreting Mueller matrix of anisotropic medium,” Opt. Express 27(3), 3305–3323 (2019).
[Crossref] [PubMed]

D. Li, F. Chen, N. Zeng, Z. Qiu, H. He, Y. He, and H. Ma, “Study on polarization scattering applied in aerosol recognition in the air,” Opt. Express 27(12), A581–A595 (2019).
[Crossref] [PubMed]

W. Sheng, W. Li, J. Qi, T. Liu, H. He, Y. Dong, S. Liu, J. Wu, D. Elson, and H. Ma, “Quantitative Analysis of 4× 4 Mueller Matrix Transformation Parameters for Biomedical Imaging,” Photonics 6(1), 34 (2019).
[Crossref]

R. Ossikovski and J. Vizet, “Eigenvalue-based depolarization metric spaces for Mueller matrices,” J. Opt. Soc. Am. A 36(7), 1173–1186 (2019).
[Crossref]

2017 (1)

A. Tariq, P. Li, D. Chen, D. Lv, and H. Ma, “Physically realizable space for the purity-depolarization plane for polarized light scattering media,” Phys. Rev. Lett. 119(3), 033202 (2017).
[Crossref] [PubMed]

2016 (2)

J. J. Gil, “Structure of polarimetric purity of a Mueller matrix and sources of depolarization,” Opt. Commun. 368, 165–173 (2016).
[Crossref]

J. J. Gil, “Components of purity of a three-dimensional polarization state,” J. Opt. Soc. Am. A 33(1), 40–43 (2016).
[Crossref] [PubMed]

2014 (1)

2013 (2)

H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, and H. Ma, “A possible quantitative Mueller matrix transformation technique for anisotropic scattering media,” Photonics Lasers Med. 2(2), 129–137 (2013).
[Crossref]

H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, Y. He, and H. Ma, “Two-dimensional and surface backscattering Mueller matrices of anisotropic sphere-cylinder scattering media: a quantitative study of influence from fibrous scatterers,” J. Biomed. Opt. 18(4), 046002 (2013).
[Crossref] [PubMed]

2011 (2)

I. San José and J. J. Gil, “Invariant indices of polarimetric purity: generalized indices of purity for n× n covariance matrices,” Opt. Commun. 284(1), 38–47 (2011).
[Crossref]

J. J. Gil, “Components of purity of a Mueller matrix,” J. Opt. Soc. Am. A 28(8), 1578–1585 (2011).
[Crossref] [PubMed]

2010 (1)

2009 (1)

2008 (1)

2007 (1)

2005 (2)

A. Aiello and J. P. Woerdman, “Physical bounds to the entropy-depolarization relation in random light scattering,” Phys. Rev. Lett. 94(9), 090406 (2005).
[Crossref] [PubMed]

G. Puentes, D. Voigt, A. Aiello, and J. P. Woerdman, “Experimental observation of universality in depolarized light scattering,” Opt. Lett. 30(23), 3216–3218 (2005).
[Crossref] [PubMed]

2004 (1)

1996 (1)

1994 (1)

C. Brosseau and D. Bicout, “Entropy production in multiple scattering of light by a spatially random medium,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(6), 4997–5005 (1994).
[Crossref] [PubMed]

1993 (1)

1986 (1)

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta (Lond.) 33(2), 185–189 (1986).
[Crossref]

Aiello, A.

A. Aiello and J. P. Woerdman, “Physical bounds to the entropy-depolarization relation in random light scattering,” Phys. Rev. Lett. 94(9), 090406 (2005).
[Crossref] [PubMed]

G. Puentes, D. Voigt, A. Aiello, and J. P. Woerdman, “Experimental observation of universality in depolarized light scattering,” Opt. Lett. 30(23), 3216–3218 (2005).
[Crossref] [PubMed]

Barakat, R.

Bernabeu, E.

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta (Lond.) 33(2), 185–189 (1986).
[Crossref]

Bicout, D.

C. Brosseau and D. Bicout, “Entropy production in multiple scattering of light by a spatially random medium,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(6), 4997–5005 (1994).
[Crossref] [PubMed]

Brosseau, C.

C. Brosseau and D. Bicout, “Entropy production in multiple scattering of light by a spatially random medium,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(6), 4997–5005 (1994).
[Crossref] [PubMed]

R. Barakat and C. Brosseau, “Von Neumann entropy of N interacting pencils of radiation,” J. Opt. Soc. Am. A 10(3), 529–532 (1993).
[Crossref]

Campos, J.

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, M. Durfort, J. C. Escalera, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Depolarizing metrics for plant samples imaging,” PLoS One 14(3), e0213909 (2019).
[Crossref] [PubMed]

Chang, Y.

Chen, D.

A. Tariq, P. Li, D. Chen, D. Lv, and H. Ma, “Physically realizable space for the purity-depolarization plane for polarized light scattering media,” Phys. Rev. Lett. 119(3), 033202 (2017).
[Crossref] [PubMed]

Chen, F.

Chipman, R. A.

De Martino, A.

Dong, Y.

W. Sheng, W. Li, J. Qi, T. Liu, H. He, Y. Dong, S. Liu, J. Wu, D. Elson, and H. Ma, “Quantitative Analysis of 4× 4 Mueller Matrix Transformation Parameters for Biomedical Imaging,” Photonics 6(1), 34 (2019).
[Crossref]

Du, E.

H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, and H. Ma, “A possible quantitative Mueller matrix transformation technique for anisotropic scattering media,” Photonics Lasers Med. 2(2), 129–137 (2013).
[Crossref]

H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, Y. He, and H. Ma, “Two-dimensional and surface backscattering Mueller matrices of anisotropic sphere-cylinder scattering media: a quantitative study of influence from fibrous scatterers,” J. Biomed. Opt. 18(4), 046002 (2013).
[Crossref] [PubMed]

Durfort, M.

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, M. Durfort, J. C. Escalera, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Depolarizing metrics for plant samples imaging,” PLoS One 14(3), e0213909 (2019).
[Crossref] [PubMed]

Elson, D.

W. Sheng, W. Li, J. Qi, T. Liu, H. He, Y. Dong, S. Liu, J. Wu, D. Elson, and H. Ma, “Quantitative Analysis of 4× 4 Mueller Matrix Transformation Parameters for Biomedical Imaging,” Photonics 6(1), 34 (2019).
[Crossref]

Escalera, J. C.

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, M. Durfort, J. C. Escalera, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Depolarizing metrics for plant samples imaging,” PLoS One 14(3), e0213909 (2019).
[Crossref] [PubMed]

Gao, W.

Garcia-Caurel, E.

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, M. Durfort, J. C. Escalera, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Depolarizing metrics for plant samples imaging,” PLoS One 14(3), e0213909 (2019).
[Crossref] [PubMed]

Garnatje, T.

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, M. Durfort, J. C. Escalera, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Depolarizing metrics for plant samples imaging,” PLoS One 14(3), e0213909 (2019).
[Crossref] [PubMed]

Gil, J. J.

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, M. Durfort, J. C. Escalera, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Depolarizing metrics for plant samples imaging,” PLoS One 14(3), e0213909 (2019).
[Crossref] [PubMed]

J. J. Gil, “Structure of polarimetric purity of a Mueller matrix and sources of depolarization,” Opt. Commun. 368, 165–173 (2016).
[Crossref]

J. J. Gil, “Components of purity of a three-dimensional polarization state,” J. Opt. Soc. Am. A 33(1), 40–43 (2016).
[Crossref] [PubMed]

J. J. Gil, “Components of purity of a Mueller matrix,” J. Opt. Soc. Am. A 28(8), 1578–1585 (2011).
[Crossref] [PubMed]

I. San José and J. J. Gil, “Invariant indices of polarimetric purity: generalized indices of purity for n× n covariance matrices,” Opt. Commun. 284(1), 38–47 (2011).
[Crossref]

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta (Lond.) 33(2), 185–189 (1986).
[Crossref]

Guo, Y.

H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, and H. Ma, “A possible quantitative Mueller matrix transformation technique for anisotropic scattering media,” Photonics Lasers Med. 2(2), 129–137 (2013).
[Crossref]

H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, Y. He, and H. Ma, “Two-dimensional and surface backscattering Mueller matrices of anisotropic sphere-cylinder scattering media: a quantitative study of influence from fibrous scatterers,” J. Biomed. Opt. 18(4), 046002 (2013).
[Crossref] [PubMed]

Guyot, S.

He, H.

W. Sheng, W. Li, J. Qi, T. Liu, H. He, Y. Dong, S. Liu, J. Wu, D. Elson, and H. Ma, “Quantitative Analysis of 4× 4 Mueller Matrix Transformation Parameters for Biomedical Imaging,” Photonics 6(1), 34 (2019).
[Crossref]

D. Li, F. Chen, N. Zeng, Z. Qiu, H. He, Y. He, and H. Ma, “Study on polarization scattering applied in aerosol recognition in the air,” Opt. Express 27(12), A581–A595 (2019).
[Crossref] [PubMed]

H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, and H. Ma, “A possible quantitative Mueller matrix transformation technique for anisotropic scattering media,” Photonics Lasers Med. 2(2), 129–137 (2013).
[Crossref]

H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, Y. He, and H. Ma, “Two-dimensional and surface backscattering Mueller matrices of anisotropic sphere-cylinder scattering media: a quantitative study of influence from fibrous scatterers,” J. Biomed. Opt. 18(4), 046002 (2013).
[Crossref] [PubMed]

He, Y.

D. Li, F. Chen, N. Zeng, Z. Qiu, H. He, Y. He, and H. Ma, “Study on polarization scattering applied in aerosol recognition in the air,” Opt. Express 27(12), A581–A595 (2019).
[Crossref] [PubMed]

H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, Y. He, and H. Ma, “Two-dimensional and surface backscattering Mueller matrices of anisotropic sphere-cylinder scattering media: a quantitative study of influence from fibrous scatterers,” J. Biomed. Opt. 18(4), 046002 (2013).
[Crossref] [PubMed]

Huang, Y. S.

Jiang, X.

Kleinschmit, M. W.

Li, D.

D. Li, F. Chen, N. Zeng, Z. Qiu, H. He, Y. He, and H. Ma, “Study on polarization scattering applied in aerosol recognition in the air,” Opt. Express 27(12), A581–A595 (2019).
[Crossref] [PubMed]

H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, Y. He, and H. Ma, “Two-dimensional and surface backscattering Mueller matrices of anisotropic sphere-cylinder scattering media: a quantitative study of influence from fibrous scatterers,” J. Biomed. Opt. 18(4), 046002 (2013).
[Crossref] [PubMed]

H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, and H. Ma, “A possible quantitative Mueller matrix transformation technique for anisotropic scattering media,” Photonics Lasers Med. 2(2), 129–137 (2013).
[Crossref]

T. Yun, N. Zeng, W. Li, D. Li, X. Jiang, and H. Ma, “Monte Carlo simulation of polarized photon scattering in anisotropic media,” Opt. Express 17(19), 16590–16602 (2009).
[Crossref] [PubMed]

Li, P.

A. Tariq, P. Li, D. Chen, D. Lv, and H. Ma, “Physically realizable space for the purity-depolarization plane for polarized light scattering media,” Phys. Rev. Lett. 119(3), 033202 (2017).
[Crossref] [PubMed]

Li, W.

W. Sheng, W. Li, J. Qi, T. Liu, H. He, Y. Dong, S. Liu, J. Wu, D. Elson, and H. Ma, “Quantitative Analysis of 4× 4 Mueller Matrix Transformation Parameters for Biomedical Imaging,” Photonics 6(1), 34 (2019).
[Crossref]

T. Yun, N. Zeng, W. Li, D. Li, X. Jiang, and H. Ma, “Monte Carlo simulation of polarized photon scattering in anisotropic media,” Opt. Express 17(19), 16590–16602 (2009).
[Crossref] [PubMed]

Liao, R.

H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, Y. He, and H. Ma, “Two-dimensional and surface backscattering Mueller matrices of anisotropic sphere-cylinder scattering media: a quantitative study of influence from fibrous scatterers,” J. Biomed. Opt. 18(4), 046002 (2013).
[Crossref] [PubMed]

H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, and H. Ma, “A possible quantitative Mueller matrix transformation technique for anisotropic scattering media,” Photonics Lasers Med. 2(2), 129–137 (2013).
[Crossref]

Liu, S.

W. Sheng, W. Li, J. Qi, T. Liu, H. He, Y. Dong, S. Liu, J. Wu, D. Elson, and H. Ma, “Quantitative Analysis of 4× 4 Mueller Matrix Transformation Parameters for Biomedical Imaging,” Photonics 6(1), 34 (2019).
[Crossref]

Liu, T.

W. Sheng, W. Li, J. Qi, T. Liu, H. He, Y. Dong, S. Liu, J. Wu, D. Elson, and H. Ma, “Quantitative Analysis of 4× 4 Mueller Matrix Transformation Parameters for Biomedical Imaging,” Photonics 6(1), 34 (2019).
[Crossref]

Lizana, A.

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, M. Durfort, J. C. Escalera, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Depolarizing metrics for plant samples imaging,” PLoS One 14(3), e0213909 (2019).
[Crossref] [PubMed]

Lu, S. Y.

Lv, D.

A. Tariq, P. Li, D. Chen, D. Lv, and H. Ma, “Physically realizable space for the purity-depolarization plane for polarized light scattering media,” Phys. Rev. Lett. 119(3), 033202 (2017).
[Crossref] [PubMed]

Ma, H.

W. Sheng, W. Li, J. Qi, T. Liu, H. He, Y. Dong, S. Liu, J. Wu, D. Elson, and H. Ma, “Quantitative Analysis of 4× 4 Mueller Matrix Transformation Parameters for Biomedical Imaging,” Photonics 6(1), 34 (2019).
[Crossref]

D. Li, F. Chen, N. Zeng, Z. Qiu, H. He, Y. He, and H. Ma, “Study on polarization scattering applied in aerosol recognition in the air,” Opt. Express 27(12), A581–A595 (2019).
[Crossref] [PubMed]

A. Tariq, P. Li, D. Chen, D. Lv, and H. Ma, “Physically realizable space for the purity-depolarization plane for polarized light scattering media,” Phys. Rev. Lett. 119(3), 033202 (2017).
[Crossref] [PubMed]

H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, and H. Ma, “A possible quantitative Mueller matrix transformation technique for anisotropic scattering media,” Photonics Lasers Med. 2(2), 129–137 (2013).
[Crossref]

H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, Y. He, and H. Ma, “Two-dimensional and surface backscattering Mueller matrices of anisotropic sphere-cylinder scattering media: a quantitative study of influence from fibrous scatterers,” J. Biomed. Opt. 18(4), 046002 (2013).
[Crossref] [PubMed]

T. Yun, N. Zeng, W. Li, D. Li, X. Jiang, and H. Ma, “Monte Carlo simulation of polarized photon scattering in anisotropic media,” Opt. Express 17(19), 16590–16602 (2009).
[Crossref] [PubMed]

Nee, S. M. F.

Nee, T. W.

Ossikovski, R.

Puentes, G.

Qi, J.

W. Sheng, W. Li, J. Qi, T. Liu, H. He, Y. Dong, S. Liu, J. Wu, D. Elson, and H. Ma, “Quantitative Analysis of 4× 4 Mueller Matrix Transformation Parameters for Biomedical Imaging,” Photonics 6(1), 34 (2019).
[Crossref]

Qiu, Z.

San José, I.

I. San José and J. J. Gil, “Invariant indices of polarimetric purity: generalized indices of purity for n× n covariance matrices,” Opt. Commun. 284(1), 38–47 (2011).
[Crossref]

Shahriar, M. S.

Sheng, W.

W. Sheng, W. Li, J. Qi, T. Liu, H. He, Y. Dong, S. Liu, J. Wu, D. Elson, and H. Ma, “Quantitative Analysis of 4× 4 Mueller Matrix Transformation Parameters for Biomedical Imaging,” Photonics 6(1), 34 (2019).
[Crossref]

Tariq, A.

A. Tariq, P. Li, D. Chen, D. Lv, and H. Ma, “Physically realizable space for the purity-depolarization plane for polarized light scattering media,” Phys. Rev. Lett. 119(3), 033202 (2017).
[Crossref] [PubMed]

Van Eeckhout, A.

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, M. Durfort, J. C. Escalera, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Depolarizing metrics for plant samples imaging,” PLoS One 14(3), e0213909 (2019).
[Crossref] [PubMed]

Vidal, J.

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, M. Durfort, J. C. Escalera, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Depolarizing metrics for plant samples imaging,” PLoS One 14(3), e0213909 (2019).
[Crossref] [PubMed]

Vizet, J.

Voigt, D.

Woerdman, J. P.

G. Puentes, D. Voigt, A. Aiello, and J. P. Woerdman, “Experimental observation of universality in depolarized light scattering,” Opt. Lett. 30(23), 3216–3218 (2005).
[Crossref] [PubMed]

A. Aiello and J. P. Woerdman, “Physical bounds to the entropy-depolarization relation in random light scattering,” Phys. Rev. Lett. 94(9), 090406 (2005).
[Crossref] [PubMed]

Wu, J.

W. Sheng, W. Li, J. Qi, T. Liu, H. He, Y. Dong, S. Liu, J. Wu, D. Elson, and H. Ma, “Quantitative Analysis of 4× 4 Mueller Matrix Transformation Parameters for Biomedical Imaging,” Photonics 6(1), 34 (2019).
[Crossref]

Yang, D. M.

Yun, T.

Zeng, N.

D. Li, F. Chen, N. Zeng, Z. Qiu, H. He, Y. He, and H. Ma, “Study on polarization scattering applied in aerosol recognition in the air,” Opt. Express 27(12), A581–A595 (2019).
[Crossref] [PubMed]

H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, Y. He, and H. Ma, “Two-dimensional and surface backscattering Mueller matrices of anisotropic sphere-cylinder scattering media: a quantitative study of influence from fibrous scatterers,” J. Biomed. Opt. 18(4), 046002 (2013).
[Crossref] [PubMed]

H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, and H. Ma, “A possible quantitative Mueller matrix transformation technique for anisotropic scattering media,” Photonics Lasers Med. 2(2), 129–137 (2013).
[Crossref]

T. Yun, N. Zeng, W. Li, D. Li, X. Jiang, and H. Ma, “Monte Carlo simulation of polarized photon scattering in anisotropic media,” Opt. Express 17(19), 16590–16602 (2009).
[Crossref] [PubMed]

J. Biomed. Opt. (1)

H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, Y. He, and H. Ma, “Two-dimensional and surface backscattering Mueller matrices of anisotropic sphere-cylinder scattering media: a quantitative study of influence from fibrous scatterers,” J. Biomed. Opt. 18(4), 046002 (2013).
[Crossref] [PubMed]

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[Crossref]

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I. San José and J. J. Gil, “Invariant indices of polarimetric purity: generalized indices of purity for n× n covariance matrices,” Opt. Commun. 284(1), 38–47 (2011).
[Crossref]

J. J. Gil, “Structure of polarimetric purity of a Mueller matrix and sources of depolarization,” Opt. Commun. 368, 165–173 (2016).
[Crossref]

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W. Sheng, W. Li, J. Qi, T. Liu, H. He, Y. Dong, S. Liu, J. Wu, D. Elson, and H. Ma, “Quantitative Analysis of 4× 4 Mueller Matrix Transformation Parameters for Biomedical Imaging,” Photonics 6(1), 34 (2019).
[Crossref]

Photonics Lasers Med. (1)

H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, and H. Ma, “A possible quantitative Mueller matrix transformation technique for anisotropic scattering media,” Photonics Lasers Med. 2(2), 129–137 (2013).
[Crossref]

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Figures (3)

Fig. 1
Fig. 1 Graphical representation of polarimetric purity via P I 4D CP space
Fig. 2
Fig. 2 The points corresponding to M ^ a and M ^ b are represented by red filled markers o and □, respectively. The average experimental Mueller matrices with the direction of fibrous scatterers at 0°, 45°, and 90° are represented by marker Δ with blue, red, and magenta colors, respectively.
Fig. 3
Fig. 3 The backscattering Mueller matrices of some spheres (magenta), cylinders (red), and sphere-cylinders (blue) of diameters 0.1 (Rayleigh) and 1.1 (Mie) µm are represented by markers circle and triangle, respectively. The grey shaded region is the non-physical region of the space

Tables (2)

Tables Icon

Table 1 The regions of the P I 4D CP space characterized by the characteristic decomposition

Tables Icon

Table 2 The values of the components of purity ( P P and P S ) along with the overall purity index (P I 4D ) at the said three angles of the fibrous scatterers.

Equations (40)

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M= M 00 M ^ = M 00 [ 1 m 01 m 02 m 03 m 10 m 11 m 12 m 13 m 20 m 21 m 22 m 23 m 30 m 31 m 32 m 33 ].
M =[ 1 D T P m ].
D=[ m 01 m 02 m 03 ],
P=[ m 10 m 20 m 30 ].
m=[ m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 ].
m 2 = i=i 3 j=1 3 | m ij | 2 = tr( m T m) ,
P S = m 2 3 ,
P=| P |( i=1 3 m i0 2 ),
D=| D |( j=1 3 m 0j 2 ).
P P = | P | 2 + | D | 2 2 .
P 4D = i,j=0 3 m ij 2 1 3 .
P 4D = 2 P P 2 3 + P S 2 ,
P P 2 ( 3/2 P 4D ) 2 + P S 2 ( P 4D ) 2 =1.
P P 2 1+3 P S 2 2 .
H( M ^ )= 1 4 i,j=0 3 m ij E ij = 1 4 i,j=0 3 m ij ( σ i σ j ) .
1 λ 0 λ 1 λ 2 λ 3 0
P 1 = λ 0 λ 1 tr(H( M ^ )) ,
P 2 = λ 0 + λ 1 2 λ 2 tr(H( M ^ )) ,
P 3 = λ 0 + λ 1 + λ 2 3 λ 3 tr(H( M ^ )) ,
0 P 1 P 2 P 3 1.
P I 4D = P 1 2 + P 2 2 + P 3 2 3 .
P 4D = 1 3 ( 2 P 1 2 + 2 3 P 2 2 + 1 3 P 3 2 ) .
M ^ =( P 1 ) M ^ J1 +( P 2 P 1 ) M ^ 2 +( P 3 P 2 ) M ^ 3 +( 1 P 3 ) M ^ 4 .
M ^ =[ 1 0 T 0 0 3×3 ].
M ^ =[ 1 D T P 0 3×3 ],
M ^ =[ 1 0 T 0 m ].
2 P P 2 +3 P S 2 =1/3 .
[ 1/2 1/2 0 0 1/2 1/2 0 0 0 0 0 0 0 0 0 0 ][ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]= 1 2 [ 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ].
2 P P 2 +3 P S 2 =1.
M ^ =[ 1 0 T 0 m R ].
M ^ =[ 1 D T P P D T ].
M ^ = P 3 M ^ 3 +( 1 P 3 ) M ^ 4 .
M ^ = P 2 M ^ 2 +( 1 P 2 ) M ^ 3 .
12 P P 2 +3 P S 2 3,
M ^ = P 1 M ^ J1 +( 1 P 1 ) M ^ 2 .
M ^ a =[ 1 0.8817 0.2411 0.1021 0.7924 0.7871 0.4027 0.0125 0.4518 0.5552 0.1992 0.1014 0.1183 0.0610 0.1014 0.3924 ].
M ^ b =[ 1 0.2578 0.0716 0.1171 0.2577 0.9950 0.0540 0.0235 0.0148 0.0345 0.6790 0.4908 0.0849 0.0883 0.4752 0.6620 ].
m 00 m 11 m 22 + m 33 0.
M ^ b =[ 1 0.2423 0.0576 0.1051 0.2563 0.9514 0.0445 0.0334 0.0002 0.0357 0.6388 0.4651 0.0753 0.0713 0.4677 0.6596 ].
M ^ b =0.8244[ 1 0.2933 0.0305 0.0971 0.2994 0.9935 0.0426 0.0779 0.0515 0.0497 0.7695 0.5523 0.0584 0.0277 0.5517 0.7763 ]+0.1592[ 1 0.0077 0.1921 0.463 0.0662 0.8102 0.0526 0.1804 0.2466 0.0306 0.0124 0.0456 0.1612 0.2858 0.0882 0.1438 ]+0.0165[ 1 0.1058 0.1128 0.1093 0.0680 0.2028 0.0628 0.1303 0.1807 0.0259 0.1505 0.1546 0.907 0.1811 0.0660 0.1915 ].

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