Abstract

In this second part of our comparative study inspecting the (dis)similarities between “Stokes” and “Jones,” we present simulation results yielded by two independent Monte Carlo programs: (i) one developed in Bern with the Jones formalism and (ii) the other implemented in Ulm with the Stokes notation. The simulated polarimetric experiments involve suspensions of polystyrene spheres with varying size. Reflection and refraction at the sample/air interfaces are also considered. Both programs yield identical results when propagating pure polarization states, yet, with unpolarized illumination, second order statistical differences appear, thereby highlighting the pre-averaged nature of the Stokes parameters. This study serves as a validation for both programs and clarifies the misleading belief according to which “Jones cannot treat depolarizing effects.”

© 2014 Optical Society of America

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References

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  49. Perhaps, it would not be preposterous to claim that the “phenomenological” character attributed to the Stokes vector is an appropriate one, as linear states seem to be preponderant in nature, due not only to the passing of the sunlight through the atmosphere, but also to the diverse reflections that occur on various surfaces.
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2013 (1)

P. Sun, X. Cao, H. Sun, M. Sun, and M. He, “Spatial pattern characterization of linear polarization-sensitive backscattering Mueller matrix elements of human serum albumin sphere suspension,” J. Biol. Phys. 39, 501–514 (2013).
[Crossref]

2012 (1)

F. Voit, A. Hohmann, J. Schäfer, and A. Kienle, “Multiple scattering of polarized light: comparison of Maxwell theory and radiative transfer theory,” J. Biomed. Opt 17, 045003 (2012).
[Crossref]

2011 (1)

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

2010 (1)

2008 (2)

N. Ghosh, I. A. Vitkin, and M. F. G. Wood, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt. 13, 044036 (2008).
[Crossref]

S.-M. F. Nee and T.-W. Nee, “Polarization of dipole scattering by isotropic medium,” Proc. SPIE 7065, 70650P (2008).
[Crossref]

2007 (1)

2005 (2)

2003 (1)

2002 (2)

I. L. Maksimova, S. V. Romanov, and V. F. Izotova, “The effect of multiple scattering in disperse media on polarization characteristics of scattered light,” Opt. Spectra 92, 915–923 (2002).
[Crossref]

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
[Crossref]

2001 (1)

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7, 1245–1248 (2001).
[Crossref]

2000 (1)

1999 (1)

1998 (1)

1997 (2)

1994 (2)

G. W. Kattawar, “A search for circular polarization in nature,” Opt. Photon. News 5(9), 42–43 (1994).
[Crossref]

D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: influence of the size parameter,” Phys. Rev. E 49, 1767–1770 (1994).
[Crossref]

1990 (1)

R. Simon, “Nondepolarizing systems and degree of polarization,” Opt. Commun. 77, 349–354 (1990).
[Crossref]

1989 (1)

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

1985 (2)

J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259–261 (1985).
[Crossref]

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

1976 (2)

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

V. Maxia, “Light polarization problems,” Appl. Opt. 15, 2576–2578 (1976).
[Crossref]

1971 (1)

E. Collett, “Mueller-Stokes matrix formulation of Fresnel’s equations,” Am. J. Phys. 39, 517–528 (1971).
[Crossref]

1969 (1)

R. H. Muller, “Definitions and conventions in ellipsometry,” Surf. Sci. 16, 14–33 (1969).
[Crossref]

1942 (1)

F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415–427 (1942).
[Crossref]

1852 (1)

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399–416 (1852).

Akarçay, H. G.

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

H. G. Akarçay, J. Rička, and M. Frenz are preparing a manuscript to be called “jaMCp3: towards the realistic modeling of light propagation in biological tissues.”

H. G. Akarçay, “Polarized light propagation in biological tissue: towards realistic modeling,” Ph.D. dissertation (University of Bern, 2011), http://www.iap.unibe.ch/publications/pub-detail.php?lang=en&id=3706 .

H. G. Akarçay, A. Hohmann, A. Kienle, M. Frenz, and J. Rička, “Monte Carlo modeling of polarized light propagation. Part I. Stokes versus Jones,” Appl. Opt.53, 7576–7585 (2014).

Al-Qasimi, A.

Antonelli, M. R.

Azzam, R. M. A.

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

Backman, V.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7, 1245–1248 (2001).
[Crossref]

Badizadegan, K.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7, 1245–1248 (2001).
[Crossref]

Bailey, W. M.

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

Bartel, S.

Benali, A.

Bennett, J. M.

J. M. Bennett, “Polarization,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, 1995), Vol. 1, Chap. 2.5.

Bernabeu, E.

J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259–261 (1985).
[Crossref]

Bickel, W. S.

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

Bicout, D.

D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: influence of the size parameter,” Phys. Rev. E 49, 1767–1770 (1994).
[Crossref]

Bigio, I. J.

Bohren, C.

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

Brosseau, C.

D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: influence of the size parameter,” Phys. Rev. E 49, 1767–1770 (1994).
[Crossref]

Cameron, B. D.

Cao, X.

P. Sun, X. Cao, H. Sun, M. Sun, and M. He, “Spatial pattern characterization of linear polarization-sensitive backscattering Mueller matrix elements of human serum albumin sphere suspension,” J. Biol. Phys. 39, 501–514 (2013).
[Crossref]

Collett, E.

E. Collett, “Mueller-Stokes matrix formulation of Fresnel’s equations,” Am. J. Phys. 39, 517–528 (1971).
[Crossref]

Coté, G. L.

Côté, D.

Dasari, R. R.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7, 1245–1248 (2001).
[Crossref]

De Martino, A.

Eick, A. A.

Feld, M. S.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7, 1245–1248 (2001).
[Crossref]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal: The Art of Scientific Computing, 1st ed. (Cambridge University, 1991), Chap. 7.

Frenz, M.

J. Rička and M. Frenz, “Polarized light: electrodynamic fundamentals,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds., 2nd ed. (Springer, 2011), Chap. 4.

H. G. Akarçay, J. Rička, and M. Frenz are preparing a manuscript to be called “jaMCp3: towards the realistic modeling of light propagation in biological tissues.”

H. G. Akarçay, A. Hohmann, A. Kienle, M. Frenz, and J. Rička, “Monte Carlo modeling of polarized light propagation. Part I. Stokes versus Jones,” Appl. Opt.53, 7576–7585 (2014).

J. Rička and M. Frenz, “From electrodynamics to Monte Carlo simulations,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds., 2nd ed. (Springer, 2011), Chap. 7.

Freyer, J. P.

Gayet, B.

Georgakoudi, I.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7, 1245–1248 (2001).
[Crossref]

Ghosh, N.

N. Ghosh, I. A. Vitkin, and M. F. G. Wood, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt. 13, 044036 (2008).
[Crossref]

Gil, J. J.

J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259–261 (1985).
[Crossref]

Goldstein, D. H.

D. H. Goldstein, “Polarized Light: Second Edition, Revised and Expanded” (CRC Press, 2003).

Goldstein, H.

H. Goldstein, C. P. Poole, and J. L. Safko, Classical Mechanics, 3rd ed. (Addison Wesley, 2001), Chap. 4, Appendix A.

Gurjar, R. S.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7, 1245–1248 (2001).
[Crossref]

He, M.

P. Sun, X. Cao, H. Sun, M. Sun, and M. He, “Spatial pattern characterization of linear polarization-sensitive backscattering Mueller matrix elements of human serum albumin sphere suspension,” J. Biol. Phys. 39, 501–514 (2013).
[Crossref]

Hielscher, A. H.

Hohmann, A.

F. Voit, A. Hohmann, J. Schäfer, and A. Kienle, “Multiple scattering of polarized light: comparison of Maxwell theory and radiative transfer theory,” J. Biomed. Opt 17, 045003 (2012).
[Crossref]

H. G. Akarçay, A. Hohmann, A. Kienle, M. Frenz, and J. Rička, “Monte Carlo modeling of polarized light propagation. Part I. Stokes versus Jones,” Appl. Opt.53, 7576–7585 (2014).

Holm, R. T.

R. T. Holm, “Convention confusions,” in Handbook of Optical Constants of Solids: Index, E. D. Palik, ed. (Academic, 1998), Vol. 3, Chap. 2.

Huffman, D. R.

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

Itzkan, I.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7, 1245–1248 (2001).
[Crossref]

Izotova, V. F.

I. L. Maksimova, S. V. Romanov, and V. F. Izotova, “The effect of multiple scattering in disperse media on polarization characteristics of scattered light,” Opt. Spectra 92, 915–923 (2002).
[Crossref]

Jacques, S.

Jacques, S. L.

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
[Crossref]

S. L. Jacques, “Monte Carlo modeling of light transport in tissue (steady state and time of flight),” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds., 2nd ed. (Springer, 2011), Chap. 5.

Jaillon, F.

James, D.

Kattawar, G. W.

Kienle, A.

F. Voit, A. Hohmann, J. Schäfer, and A. Kienle, “Multiple scattering of polarized light: comparison of Maxwell theory and radiative transfer theory,” J. Biomed. Opt 17, 045003 (2012).
[Crossref]

H. G. Akarçay, A. Hohmann, A. Kienle, M. Frenz, and J. Rička, “Monte Carlo modeling of polarized light propagation. Part I. Stokes versus Jones,” Appl. Opt.53, 7576–7585 (2014).

Knuth, D. E.

D. E. Knuth, The Art of Computer Programming Vol. 2: Seminumerical Algorithms, 3rd ed. (Addison-Wesley, 1997), Chap. 3.

Korotkova, O.

Landi Degl’Innocenti, E.

E. Landi Degl’Innocenti, “The physics of polarization,” in Astrophysical Spectropolarimetry, J. Trujillo-Bueno, F. Moreno-Insertis, and F. Sánchez Martinez, eds. (Cambridge University, 2000).

Lee, K.

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
[Crossref]

MacKintosh, F. C.

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

Maksimova, I. L.

I. L. Maksimova, S. V. Romanov, and V. F. Izotova, “The effect of multiple scattering in disperse media on polarization characteristics of scattered light,” Opt. Spectra 92, 915–923 (2002).
[Crossref]

Martinez, A. S.

D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: influence of the size parameter,” Phys. Rev. E 49, 1767–1770 (1994).
[Crossref]

Maxia, V.

Mehrübeoglu, M.

Mourant, J. R.

Muller, R. H.

R. H. Muller, “Definitions and conventions in ellipsometry,” Surf. Sci. 16, 14–33 (1969).
[Crossref]

Nee, S.-M. F.

S.-M. F. Nee and T.-W. Nee, “Polarization of dipole scattering by isotropic medium,” Proc. SPIE 7065, 70650P (2008).
[Crossref]

Nee, T.-W.

S.-M. F. Nee and T.-W. Nee, “Polarization of dipole scattering by isotropic medium,” Proc. SPIE 7065, 70650P (2008).
[Crossref]

Novikova, T.

Perelman, L. T.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7, 1245–1248 (2001).
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F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[Crossref]

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H. Goldstein, C. P. Poole, and J. L. Safko, Classical Mechanics, 3rd ed. (Addison Wesley, 2001), Chap. 4, Appendix A.

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W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal: The Art of Scientific Computing, 1st ed. (Cambridge University, 1991), Chap. 7.

Rakovic, M. J.

Ramella-Roman, J.

Ramella-Roman, J. C.

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
[Crossref]

Rastegar, S.

Ricka, J.

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

J. Rička and M. Frenz, “Polarized light: electrodynamic fundamentals,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds., 2nd ed. (Springer, 2011), Chap. 4.

H. G. Akarçay, J. Rička, and M. Frenz are preparing a manuscript to be called “jaMCp3: towards the realistic modeling of light propagation in biological tissues.”

H. G. Akarçay, A. Hohmann, A. Kienle, M. Frenz, and J. Rička, “Monte Carlo modeling of polarized light propagation. Part I. Stokes versus Jones,” Appl. Opt.53, 7576–7585 (2014).

J. Rička and M. Frenz, “From electrodynamics to Monte Carlo simulations,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds., 2nd ed. (Springer, 2011), Chap. 7.

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[Crossref]

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H. Goldstein, C. P. Poole, and J. L. Safko, Classical Mechanics, 3rd ed. (Addison Wesley, 2001), Chap. 4, Appendix A.

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Schäfer, J.

F. Voit, A. Hohmann, J. Schäfer, and A. Kienle, “Multiple scattering of polarized light: comparison of Maxwell theory and radiative transfer theory,” J. Biomed. Opt 17, 045003 (2012).
[Crossref]

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D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: influence of the size parameter,” Phys. Rev. E 49, 1767–1770 (1994).
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P. Sun, X. Cao, H. Sun, M. Sun, and M. He, “Spatial pattern characterization of linear polarization-sensitive backscattering Mueller matrix elements of human serum albumin sphere suspension,” J. Biol. Phys. 39, 501–514 (2013).
[Crossref]

Sun, M.

P. Sun, X. Cao, H. Sun, M. Sun, and M. He, “Spatial pattern characterization of linear polarization-sensitive backscattering Mueller matrix elements of human serum albumin sphere suspension,” J. Biol. Phys. 39, 501–514 (2013).
[Crossref]

Sun, P.

P. Sun, X. Cao, H. Sun, M. Sun, and M. He, “Spatial pattern characterization of linear polarization-sensitive backscattering Mueller matrix elements of human serum albumin sphere suspension,” J. Biol. Phys. 39, 501–514 (2013).
[Crossref]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal: The Art of Scientific Computing, 1st ed. (Cambridge University, 1991), Chap. 7.

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van de Hulst, H. C.

H. C. van de Hulst, “Light Scattering by Small Particles” (Wiley, 1957).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal: The Art of Scientific Computing, 1st ed. (Cambridge University, 1991), Chap. 7.

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Vitkin, I. A.

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[Crossref]

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F. Voit, A. Hohmann, J. Schäfer, and A. Kienle, “Multiple scattering of polarized light: comparison of Maxwell theory and radiative transfer theory,” J. Biomed. Opt 17, 045003 (2012).
[Crossref]

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F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
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N. Ghosh, I. A. Vitkin, and M. F. G. Wood, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt. 13, 044036 (2008).
[Crossref]

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F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
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Appl. Opt. (5)

J. Biol. Phys. (1)

P. Sun, X. Cao, H. Sun, M. Sun, and M. He, “Spatial pattern characterization of linear polarization-sensitive backscattering Mueller matrix elements of human serum albumin sphere suspension,” J. Biol. Phys. 39, 501–514 (2013).
[Crossref]

J. Biomed. Opt (1)

F. Voit, A. Hohmann, J. Schäfer, and A. Kienle, “Multiple scattering of polarized light: comparison of Maxwell theory and radiative transfer theory,” J. Biomed. Opt 17, 045003 (2012).
[Crossref]

J. Biomed. Opt. (2)

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
[Crossref]

N. Ghosh, I. A. Vitkin, and M. F. G. Wood, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt. 13, 044036 (2008).
[Crossref]

J. Chem. Phys. (1)

F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415–427 (1942).
[Crossref]

Nat. Med. (1)

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7, 1245–1248 (2001).
[Crossref]

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[Crossref]

Opt. Commun. (1)

R. Simon, “Nondepolarizing systems and degree of polarization,” Opt. Commun. 77, 349–354 (1990).
[Crossref]

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[Crossref]

Opt. Spectra (1)

I. L. Maksimova, S. V. Romanov, and V. F. Izotova, “The effect of multiple scattering in disperse media on polarization characteristics of scattered light,” Opt. Spectra 92, 915–923 (2002).
[Crossref]

Phys. Rev. B (1)

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

Phys. Rev. E (1)

D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: influence of the size parameter,” Phys. Rev. E 49, 1767–1770 (1994).
[Crossref]

Proc. SPIE (2)

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

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[Crossref]

Trans. Cambridge Philos. Soc. (1)

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399–416 (1852).

Other (23)

H. G. Akarçay, A. Hohmann, A. Kienle, M. Frenz, and J. Rička, “Monte Carlo modeling of polarized light propagation. Part I. Stokes versus Jones,” Appl. Opt.53, 7576–7585 (2014).

S. L. Jacques, “Monte Carlo modeling of light transport in tissue (steady state and time of flight),” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds., 2nd ed. (Springer, 2011), Chap. 5.

H. C. van de Hulst, “Light Scattering by Small Particles” (Wiley, 1957).

C. Schwartz, “Probing random media with singular waves,” Ph.D. dissertation (University of Central Florida, 2006).

M. H. Smith, “Interpreting Mueller matrix images of tissues,” in The International Symposium on Biomedical Optics (International Society for Optics and Photonics, 2001).

Our simulated dataset is available on the following website (within the “Polarized light propagation in biological tissue” project): www.iapbp.unibe.ch/content.php/home/projects/ .

G. W. Rolfe, The Polariscope in the Chemical Laboratory: An Introduction to Polarimetry and Related Methods (Macmillan, 1919).

J. Rička and M. Frenz, “Polarized light: electrodynamic fundamentals,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds., 2nd ed. (Springer, 2011), Chap. 4.

J. Rička and M. Frenz, “From electrodynamics to Monte Carlo simulations,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds., 2nd ed. (Springer, 2011), Chap. 7.

H. G. Akarçay, “Polarized light propagation in biological tissue: towards realistic modeling,” Ph.D. dissertation (University of Bern, 2011), http://www.iap.unibe.ch/publications/pub-detail.php?lang=en&id=3706 .

H. G. Akarçay, J. Rička, and M. Frenz are preparing a manuscript to be called “jaMCp3: towards the realistic modeling of light propagation in biological tissues.”

R. T. Holm, “Convention confusions,” in Handbook of Optical Constants of Solids: Index, E. D. Palik, ed. (Academic, 1998), Vol. 3, Chap. 2.

E. Landi Degl’Innocenti, “The physics of polarization,” in Astrophysical Spectropolarimetry, J. Trujillo-Bueno, F. Moreno-Insertis, and F. Sánchez Martinez, eds. (Cambridge University, 2000).

J. M. Bennett, “Polarization,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, 1995), Vol. 1, Chap. 2.5.

D. H. Goldstein, “Polarized Light: Second Edition, Revised and Expanded” (CRC Press, 2003).

H. Goldstein, C. P. Poole, and J. L. Safko, Classical Mechanics, 3rd ed. (Addison Wesley, 2001), Chap. 4, Appendix A.

http://mathworld.wolfram.com/EulerAngles.html .

As stated above, ns=4 would actually be sufficient to construct a sample’s PM matrix. Here, we prefer to solve an overdetermined system primarily for didactic purposes, thereby remaining consistent with respect to our PM matrix definition given in [1].

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

Perhaps, it would not be preposterous to claim that the “phenomenological” character attributed to the Stokes vector is an appropriate one, as linear states seem to be preponderant in nature, due not only to the passing of the sunlight through the atmosphere, but also to the diverse reflections that occur on various surfaces.

Random number generators in C++ intended for Monte Carlo applications and released under the Gnu general public license can be found at: http://www.agner.org/random/ .

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal: The Art of Scientific Computing, 1st ed. (Cambridge University, 1991), Chap. 7.

D. E. Knuth, The Art of Computer Programming Vol. 2: Seminumerical Algorithms, 3rd ed. (Addison-Wesley, 1997), Chap. 3.

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Figures (11)

Fig. 1.
Fig. 1.

(a) Schematic drawing of the simulated polarimetric imaging experiment designed to record the spatial distribution of the light backscattered from a sample. The sample—a suspension of polystyrene spheres in water—is illuminated by a polarized or unpolarized laser beam. (Note that in our simulations, the illumination is done normal to the sample surface, and not with an oblique incidence as suggested here.) The backscattered light goes through the detection setup which is composed of a polarization analyzer and imaging optics, e.g., a CCD array coupled with a lens. This is implemented as shown in (b): a photon path that reaches the sample’s backscattering surface within the detection area is repositioned on the detector plane at z = 0 and is recorded by the imaging array, whose normal is given by n ^ D = ( 0 , 0 , 1 ) . Such a path is drawn in red (lighter color); the photon exits the sample with an angle, θ out , with respect to n ^ D . The Cartesian lab frame is denoted ( x ^ L , y ^ L , z ^ L ) and the coordinate system attached to the imager array is ( x ^ D , y ^ D , z ^ D ) . (c) Displays an example of a polarization sensitive intensity distribution image, J o | i , recorded by Program B (see also Fig. 3). (d) Shows an imaging PM matrix obtained from 6 × 6 J o | i measurements.

Fig. 2.
Fig. 2.

Polar plots of probability densities for the scattering angle, θ (see Section 3). (a) Samples of type §1 (radius a = 50 nm , size parameter, 0.50); the dipole-like scattering profile is characteristic of Rayleigh-type scatterers, (b) samples of type §2 (radius a = 200 nm size parameter 1.98), and (c) samples of type §3 (radius a = 800 nm , size parameter, 7.94); the forward scattering behavior is typical of the Mie regime.

Fig. 3.
Fig. 3.

“Primary data” obtained with Program B for the sample §3c (where the Fresnel interfaces were neglected); these ( 6 + 1 ) × 6 images constitute the intensity distributions, J o | i , as they would be recorded in a real experiment. Each column, i , corresponds to one of the n s = 6 + 1 input states, | e i ; each row, o , corresponds to a polarization analyzer, f | . J o | i a = J o | i / total ( J o | i ) , “max” stands for the overall maximum value, and γ = 3.3 . Images such as J L X | L X and J L X | L Y reveal co- and cross-polarization properties, respectively.

Fig. 4.
Fig. 4.

Top: n s = 6 + 1 output Stokes vectors calculated with Program B for the highly scattering sample §1c, composed of smaller polystyrene spheres. Like in Fig. 3, each column, i , corresponds to one of the input states, | e i . The first elements, which represent intensities, are displayed in a similar fashion to the distributions of Fig. 3 (thus, J o | i b here is analogous to J o | i a ). Bottom: Same series of S i out obtained with Program U. The helicity flip that occurs near the illumination point is visible in the V out elements of the vectors. S C out and S C + out . Note also that the U out elements of the S L + out / S L out vectors are identical to the Q out elements of the S L X out / S L Y out vectors, when rotated by ± 45 ° . This is a basic symmetry feature, useful for an assessment of the consistency of the data evaluation.

Fig. 5.
Fig. 5.

Top: n s = 6 + 1 output Stokes vectors calculated with Program B from the intensity distributions of Fig. 3, i.e., for the highly scattering sample §3c, composed of larger polystyrene spheres. Bottom: same series of S i out obtained with Program U. Stokes’ equivalence theorem [40] can be verified, e.g., S O out = S L X out + S L Y out .

Fig. 6.
Fig. 6.

We show here magnified versions of the V out elements from the vectors S C out (left) and S C + out (center) in Fig. 5, as well as the m 44 element from the imaging PM matrix at the right column of Fig. 7. These images were obtained with the strongly scattering sample §3c, composed of larger polystyrene spheres. The center of the images coincide with the light detected after single backscattering, i.e., with a helicity flip. The regions far from the center correspond to the light detected after multiply scattering; the forward scattering behavior of the spheres preserves the helicity of the input state.

Fig. 7.
Fig. 7.

Two sets of imaging PM matrices obtained with Program B (top) and Program U (bottom). The matrices on the left were calculated from the Stokes vectors of Fig. 4, while the matrices on the right were calculated from the data in Fig. 5. The first element, m 11 , is distinguished from the others; it represents the same intensity distribution as the element I out of S O out (and J o | i c here is analogous to J o | i a and J o | i b ). The remaining elements, m i j , are normalized with respect to m 11 , so as to represent polarization distributions.

Fig. 8.
Fig. 8.

Cross sections of the V out elements obtained with Programs B (in black) and U (in blue), for the samples §3a (left column) and §3c (right column). The three different rows correspond from top to bottom to (a) linear, | L X , (b) circular right, | C , and (c) unpolarized, | O , inputs. While the fluctuations are alike for inputs that are pure polarization states, second order statistical differences are visible for unpolarized input (third row), especially in the single scattering regime (left column).

Fig. 9.
Fig. 9.

Cross sections of the S O out vector’s V out element obtained with the sample §3a. Each curve corresponds to a different sampling method used for the modeling of the unpolarized light source: mixed sampling in black, orthogonal sampling from the linear pair ( | L X , | L Y ) in orange, and orthogonal sampling from the circular pair ( | C , | C + ) in green. In (a) the “classical” detection method is employed (as detailed in Section 3.D), whereas in (b) the modified detection scheme (where photons paths do not simultaneously pass through the different analyzers) was adopted.

Fig. 10.
Fig. 10.

S O out vector’s V out element obtained with the sample §3a. Each image corresponds to a different sampling method used for the modeling of the unpolarized light source. Left: orthogonal sampling from the linear pair ( | L X , | L Y ) . Center: orthogonal sampling from the circular pair ( | C , | C + ) . Right: mixed sampling. The “classical” detection method (as detailed in Section 3.D) leads to correlations in the resulting output Stokes vectors’ V out elements, as can be seen with the difference in patterns.

Fig. 11.
Fig. 11.

Convergence of the mean value over a small area ( 10 × 10 pixels) near the center of the S O out vector’s V out element, obtained with the sample §3c. Each curve corresponds to a different sampling method used for the modeling of the unpolarized light source: mixed sampling in black, orthogonal sampling from the linear pair ( | L X , | L Y ) in orange, and orthogonal sampling from the circular pair ( | C , | C + ) in green. The inset displays a magnified view of the curves in the region where the number of simulated photon paths exceeds 40 M.

Tables (5)

Tables Icon

Table 1. List of the 9 Samples Modeled in the Simulations, Together with Their Respective Specifications

Tables Icon

Table 2. Simulated Reflectance Values for the Different Samples a

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Table 3. Ratio σ B / σ U Given for All Elements of S O out , Where the Input s⃗ O = ( 1 0 0 0 ) in Program U

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Table 4. Jones and Stokes Vectors versus Polarization State

Tables Icon

Table 5. Specifications Relative to Both Programs

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

I out i = [ ( J L X | i + J L Y | i ) + ( J L + | i + J L | i ) + ( J C + | i + J C | i ) ] / 3 ,
Q out i = J L X | i J L Y | i ,
U out i = J L + | i J L | i ,
V out i = J C + | i J C | i .
M⃗ 1 = ( S⃗ out L X + S⃗ out L Y + S⃗ out D R + S⃗ out D L + S⃗ out C R + S⃗ out C L ) / 6 ,
M⃗ 2 = S⃗ out L X S⃗ out L Y ,
M⃗ 3 = S⃗ out L + S⃗ out L ,
M⃗ 4 = S⃗ out C + S⃗ out C .
| O = e ^ 0 e ^ 0 , with e ^ 0 = ( α 1 + i β 1 α 2 + i β 2 ) .
| e i + 1 = S U ( θ ) | e i = ( S 2 ( θ ) 0 0 S 1 ( θ ) ) ( e e ) i + 1 ,
| e i + 1 = S B ( θ ) | e i = ( S 1 ( θ ) 0 0 S 2 ( θ ) ) ( e x e y ) i + 1 ,
M iso = ( m 11 m 12 0 0 m 12 m 11 0 0 0 0 m 33 m 34 0 0 m 34 m 33 ) .

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