Abstract

This bipartite comparative study aims at inspecting the similarities and differences between the Jones and Stokes–Mueller formalisms when modeling polarized light propagation with numerical simulations of the Monte Carlo type. In this first part, we review the theoretical concepts that concern light propagation and detection with both pure and partially/totally unpolarized states. The latter case involving fluctuations, or “depolarizing effects,” is of special interest here: Jones and Stokes–Mueller are equally apt to model such effects and are expected to yield identical results. In a second, ensuing paper, empirical evidence is provided by means of numerical experiments, using both formalisms.

© 2014 Optical Society of America

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OSA Recommended Articles
Monte Carlo modeling of polarized light propagation: Stokes vs. Jones. Part II

H. Günhan Akarçay, Ansgar Hohmann, Alwin Kienle, Martin Frenz, and Jaro Rička
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Historical revision of the differential Stokes–Mueller formalism: discussion

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J. Opt. Soc. Am. A 34(3) 410-414 (2017)

References

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  1. W. H. McMaster, “Matrix representation of polarization,” Rev. Mod. Phys. 33, 8–28 (1961).
    [Crossref]
  2. N. G. Parke, “Matrix optics,” Ph.D. dissertation (MIT, 1948).
  3. N. G. Parke, “Matrix algebra of electromagnetic waves,” (MIT Research Laboratory of Electronics, 1948).
  4. N. G. Parke, “Statistical optics: I. Radiation,” (MIT Research Laboratory of Electronics, 1949).
  5. N. G. Parke, “Statistical optics: II. Mueller phenomenological algebra,” (MIT Research Laboratory of Electronics, 1949).
  6. W. A. Shurcliff, “Mueller calculus and Jones calculus,” in Polarized Light (Harvard University, 1962), Chap. 8.
  7. Additional remarks, and references with respect to both matrix methods can be found in the last paragraph of J. M. Bennett, “Polarization,” in Handbook of Optics Volume I, M. Bass, E. W. Van Stryland, D. R. Williams, and W. L. Wolfe, eds., 2nd ed. (McGraw-Hill, 1995), Chap. 5.
  8. G. W. Kattawar and G. N. Plass, “Radiance and polarization of multiple scattered light from haze and clouds,” Appl. Opt. 7, 1519–1527 (1968).
    [Crossref]
  9. A. S. Martinez and R. Maynard, “Faraday effect and multiple scattering of light,” Phys. Rev. B 50, 3714–3732 (1994).
    [Crossref]
  10. R. Lenke and G. Maret, “Magnetic field effects on coherent backscattering of light,” Eur. Phys. J. B 17, 171–185 (2000).
    [Crossref]
  11. M. J. Raković, G. W. Kattawar, M. Mehrübeoğlu, B. D. Cameron, L. V. Wang, S. Rastegar, and G. L. Coté, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. Opt. 38, 3399–3408 (1999).
    [Crossref]
  12. S. Bartel and A. H. Hielscher, “Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media,” Appl. Opt. 39, 1580–1588 (2000).
    [Crossref]
  13. X. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: time-resolved simulations,” Opt. Express 9, 254–259 (2001).
    [Crossref]
  14. J. Ramella-Roman, S. Prahl, and S. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part I,” Opt. Express 13, 4420–4438 (2005).
    [Crossref]
  15. J. Ramella-Roman, S. Prahl, and S. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part II,” Opt. Express 13, 10392–10405 (2005).
    [Crossref]
  16. S. V. Gangnus, S. J. Matcher, and I. V. Meglinski, “Monte Carlo modeling of polarized light propagation in biological tissues,” Laser Phys. 14, 886–891 (2004).
  17. M. Xu, “Electric field Monte Carlo simulation of polarized light propagation in turbid media,” Opt. Express 12, 6530–6539 (2004).
    [Crossref]
  18. N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biophotonics 16, 110801 (2011).
  19. H. G. Akarçay, A. Hohmann, A. Kienle, M. Frenz, and J. Rička, “Monte Carlo modeling of polarized light propagation. Part II. Stokes versus Jones,” Appl. Opt.53, 7586–7602 (2014).
  20. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1999).
  21. J. Rička and M. Frenz, “Polarized light: electrodynamic fundamentals” and “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. 4, pp. 65–108 and Chap. 7, pp. 203–266, respectively.
  22. Despite the “quantum” notation, the present treatment remains essentially within classical optics; i.e., we only consider pure and mixed coherent states. For the so-called “nonclassical” polarization states, see, e.g., [23–25].
  23. A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
    [Crossref]
  24. N. Korolkova, “30.4 Stokes operators questioned: degree of polarization in quantum optics,” in Quantum Information Processing, T. Beth and G. Leuchs, eds., 2nd ed. (Wiley, 2005), pp. 413–416.
  25. J. Lehner, U. Leonhardt, and H. Paul, “Unpolarized light: classical and quantum states,” Phys. Rev. A 53, 2727–2735 (1996).
    [Crossref]
  26. R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, E. W. van Stryland, D. R. Williams, and W. L. Wolfe, eds., 2nd ed. (McGraw-Hill, 1995), Vol. 2, Chap. 22, 22.1–22.37.
  27. J. M. Bennett, “Polarizers,” in Handbook of Optics, M. Bass, E. W. van Stryland, D. R. Williams, and W. L. Wolfe, eds., 2nd ed. (McGraw-Hill, 1995), Vol. 2, Chap. 3, 3.1–3.12.
  28. S.-M. F. Nee, “Polarization measurement,” in The Measurement, Instrumentation, and Sensors: Handbook, J. G. Webster, ed. (CRC and IEEE, 1999), Chap. 60, 1655–1679.
  29. It is difficult to ascertain an accurate chronology regarding the emergence of models using the Stokes vector to describe polarized light propagation. Several physicists were particularly prolific to this end, though arguably, long after Sir Stokes’ 1852 paper [30]. Chandrasekhar (apparently unaware of Soleillet’s findings [31]) was the first to reintroduce the then forgotten four real Stokes parameters in the English-speaking community, as part of his pioneering work on radiative transfer in astrophysics [32,33]. A historical overview can also be found in [20].
  30. G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399–416 (1852).
  31. P. Soleillet, “Sur les paramètres caractérisant la polarisation partielle de la lumière dans les phénomènes de fluorescence,” Ph.D. dissertation (École Normale Supérieure, 1929) and published in Annales de Physique 12 (1929), pp. 23–59.
  32. S. Chandrasekhar, “Radiative transfer: a personal account,” in Selected Papers, Vol. 2 of Radiative Transfer and Negative Ion of Hydrogen (University of Chicago, 1989), pp. 511–542.
  33. Interview of S. Chandrasekhar by S. Weart on 18 May 1977, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD, USA, http://www.aip.org/history/ohilist/4551_2.html .
  34. Regrettably, such a confusion affected the V element in three equations in [21]. To comply with the present conventions, these equations should be corrected as follows: 4.100: S3=V=〈SL〉−〈SR〉=+i[〈ExEy*〉−〈Ex*Ey〉], 4.104: S0≔〈〈E|σ0|E〉〉S1≔〈〈E|σ1|E〉〉S2≔〈〈E|σ2|E〉〉S3≔−〈〈E|σ3|E〉〉, 4.116: s⃗3=12(s⃗L−s⃗R)=(0,0,0,1).
  35. Modern treatments of the topic prefer to work with the correlation matrix 〈|Ex|2〉T〈ExEy*〉T〈EyEx*〉T〈|Ex|2〉T, instead of the Stokes parameters. Particularly thorough discussions on polarization statistics are found in [20,36].
  36. M. Born and E. Wolf, “Statistical optics,” in Fundamentals of Photonics, B. E. A. Saleh and M. C. Teich, eds. (Wiley, 1989), Chap. 10, pp. 342–383.
  37. R. C. Jones, “A new calculus for the treatment of optical systems,” series of eight papers published in the J. Opt. Soc. Am. from 1941 to 1956.
  38. We prefer here the “Perrin–Mueller matrix” to the more common “Mueller matrix” not only to give due credit to Francis Perrin, but also because Hans Mueller’s publications on the subject are unfortunately impossible to find (as was confirmed to us by a representative of the libraries of the Massachusetts Institute of Technology, where Mueller worked as a professor). The only written trace of Mueller’s work is a little revealing conference abstract [39]; therefore, we strongly suggest consulting Perrin’s comprehensive 1942 paper [40] for more information on the PM matrix of arbitrary isotropic, symmetric, and asymmetric scatterers. (Credit should be given also to Paul Soleillet [31,41], who got the idea almost two decades before Perrin and Mueller.) Besides, the notation PM recalls the “phase matrix” introduced by Chandrasekhar to treat radiative transfer of polarization.
  39. H. Mueller, “The foundation of optics,” abstract for a contributed paper presented at the 1948 Winter Meeting of the OSA.
  40. F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415–427 (1942).
    [Crossref]
  41. More historical and background information on Paul Soleillet and Francis Perrin can be found on Oriol Arteaga’s home page: http://www.mmpolarimetry.com
  42. R. M. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier Science, 1987).
  43. K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433–437 (1987).
    [Crossref]
  44. F. Jaillon and H. Saint-Jalmes, “Description and time reduction of a Monte Carlo code to simulate propagation of polarized light through scattering media,” Appl. Opt. 42, 3290–3296 (2003).
    [Crossref]
  45. This low aperture approximation is also used in the wave-optical treatment in Section 7.5.6 of [21], but the suggested validity range is incorrect. The correct wave-optical approximation, that is valid for moderate apertures up to 0.6, is 〈f|e^D〉=[e^f*·e^D]κ(n^D,k^out)O(n^D,k^out). Wave optical aspects are included in the Fresnel inclination factor κ(n^D,k^out)=(1+k^out·n^D)/2 (as it occurs in Fresnel–Kirchhoff diffraction theory [46,47]) and O(n^D,k^out) is the scalar overlap that describes the effective aperture of the observation. The bracketed quantity [e^f*·e^D] is the same as in geometrical approximation, with e^D from Eq. (18).
  46. M. Born and E. Wolf, “Elements of the theory of diffraction,” in Principles of Optics (Pergamon, 1970), Chap. 8, pp. 370–455.
  47. A. Takada, M. Shibuya, T. Saitoh, A. Nishikata, K. Maehara, and S. Nakadate, “Incoming inclination factor for scalar imaging theory,” Opt. Eng. 49, 023202 (2010).
    [Crossref]
  48. D. Marcuse, J. Wait, and R. M. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: a differential 4 × 4 matrix calculus,” J. Opt. Soc. Am. 68, 1756–1767 (1979).
  49. H. Hurwitz, “The statistical properties of unpolarized light,” J. Opt. Soc. Am. 35, 525–531 (1945).
    [Crossref]
  50. J. S. Hendricks and T. E. Booth, “MCNP variance reduction overview,” in Monte-Carlo Methods and Applications in Neutronics, Photonics and Statistical Physics (Springer, 1985), pp. 83–92.
  51. I. Meglinski and V. L. Kuzmin, “Coherent backscattering of circularly polarized light from a disperse random medium,” Prog. Electromagn. Res. M 16, 47–61 (2011).
    [Crossref]

2011 (2)

N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biophotonics 16, 110801 (2011).

I. Meglinski and V. L. Kuzmin, “Coherent backscattering of circularly polarized light from a disperse random medium,” Prog. Electromagn. Res. M 16, 47–61 (2011).
[Crossref]

2010 (1)

A. Takada, M. Shibuya, T. Saitoh, A. Nishikata, K. Maehara, and S. Nakadate, “Incoming inclination factor for scalar imaging theory,” Opt. Eng. 49, 023202 (2010).
[Crossref]

2005 (2)

2004 (2)

S. V. Gangnus, S. J. Matcher, and I. V. Meglinski, “Monte Carlo modeling of polarized light propagation in biological tissues,” Laser Phys. 14, 886–891 (2004).

M. Xu, “Electric field Monte Carlo simulation of polarized light propagation in turbid media,” Opt. Express 12, 6530–6539 (2004).
[Crossref]

2003 (1)

2002 (1)

A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
[Crossref]

2001 (1)

2000 (2)

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

R. Lenke and G. Maret, “Magnetic field effects on coherent backscattering of light,” Eur. Phys. J. B 17, 171–185 (2000).
[Crossref]

1999 (1)

1996 (1)

J. Lehner, U. Leonhardt, and H. Paul, “Unpolarized light: classical and quantum states,” Phys. Rev. A 53, 2727–2735 (1996).
[Crossref]

1994 (1)

A. S. Martinez and R. Maynard, “Faraday effect and multiple scattering of light,” Phys. Rev. B 50, 3714–3732 (1994).
[Crossref]

1987 (1)

1979 (1)

1968 (1)

1961 (1)

W. H. McMaster, “Matrix representation of polarization,” Rev. Mod. Phys. 33, 8–28 (1961).
[Crossref]

1945 (1)

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, A. Hohmann, A. Kienle, M. Frenz, and J. Rička, “Monte Carlo modeling of polarized light propagation. Part II. Stokes versus Jones,” Appl. Opt.53, 7586–7602 (2014).

Azzam, R. M.

Bartel, S.

Bashara, N. M.

R. M. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier Science, 1987).

Bennett, J. M.

Additional remarks, and references with respect to both matrix methods can be found in the last paragraph of J. M. Bennett, “Polarization,” in Handbook of Optics Volume I, M. Bass, E. W. Van Stryland, D. R. Williams, and W. L. Wolfe, eds., 2nd ed. (McGraw-Hill, 1995), Chap. 5.

J. M. Bennett, “Polarizers,” in Handbook of Optics, M. Bass, E. W. van Stryland, D. R. Williams, and W. L. Wolfe, eds., 2nd ed. (McGraw-Hill, 1995), Vol. 2, Chap. 3, 3.1–3.12.

Booth, T. E.

J. S. Hendricks and T. E. Booth, “MCNP variance reduction overview,” in Monte-Carlo Methods and Applications in Neutronics, Photonics and Statistical Physics (Springer, 1985), pp. 83–92.

Born, M.

M. Born and E. Wolf, “Elements of the theory of diffraction,” in Principles of Optics (Pergamon, 1970), Chap. 8, pp. 370–455.

M. Born and E. Wolf, “Statistical optics,” in Fundamentals of Photonics, B. E. A. Saleh and M. C. Teich, eds. (Wiley, 1989), Chap. 10, pp. 342–383.

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1999).

Cameron, B. D.

Chandrasekhar, S.

S. Chandrasekhar, “Radiative transfer: a personal account,” in Selected Papers, Vol. 2 of Radiative Transfer and Negative Ion of Hydrogen (University of Chicago, 1989), pp. 511–542.

Chipman, R. A.

R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, E. W. van Stryland, D. R. Williams, and W. L. Wolfe, eds., 2nd ed. (McGraw-Hill, 1995), Vol. 2, Chap. 22, 22.1–22.37.

Coté, G. L.

Frenz, M.

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

J. Rička and M. Frenz, “Polarized light: electrodynamic fundamentals” and “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. 4, pp. 65–108 and Chap. 7, pp. 203–266, respectively.

Gangnus, S. V.

S. V. Gangnus, S. J. Matcher, and I. V. Meglinski, “Monte Carlo modeling of polarized light propagation in biological tissues,” Laser Phys. 14, 886–891 (2004).

Ghosh, N.

N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biophotonics 16, 110801 (2011).

Hendricks, J. S.

J. S. Hendricks and T. E. Booth, “MCNP variance reduction overview,” in Monte-Carlo Methods and Applications in Neutronics, Photonics and Statistical Physics (Springer, 1985), pp. 83–92.

Hielscher, A. H.

Hohmann, A.

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

Hurwitz, H.

Jacques, S.

Jaillon, F.

Jones, R. C.

R. C. Jones, “A new calculus for the treatment of optical systems,” series of eight papers published in the J. Opt. Soc. Am. from 1941 to 1956.

Kattawar, G. W.

Kienle, A.

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

Kim, K.

Korolkova, N.

N. Korolkova, “30.4 Stokes operators questioned: degree of polarization in quantum optics,” in Quantum Information Processing, T. Beth and G. Leuchs, eds., 2nd ed. (Wiley, 2005), pp. 413–416.

Kuzmin, V. L.

I. Meglinski and V. L. Kuzmin, “Coherent backscattering of circularly polarized light from a disperse random medium,” Prog. Electromagn. Res. M 16, 47–61 (2011).
[Crossref]

Lehner, J.

J. Lehner, U. Leonhardt, and H. Paul, “Unpolarized light: classical and quantum states,” Phys. Rev. A 53, 2727–2735 (1996).
[Crossref]

Lenke, R.

R. Lenke and G. Maret, “Magnetic field effects on coherent backscattering of light,” Eur. Phys. J. B 17, 171–185 (2000).
[Crossref]

Leonhardt, U.

J. Lehner, U. Leonhardt, and H. Paul, “Unpolarized light: classical and quantum states,” Phys. Rev. A 53, 2727–2735 (1996).
[Crossref]

Luis, A.

A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
[Crossref]

Maehara, K.

A. Takada, M. Shibuya, T. Saitoh, A. Nishikata, K. Maehara, and S. Nakadate, “Incoming inclination factor for scalar imaging theory,” Opt. Eng. 49, 023202 (2010).
[Crossref]

Mandel, L.

Marcuse, D.

Maret, G.

R. Lenke and G. Maret, “Magnetic field effects on coherent backscattering of light,” Eur. Phys. J. B 17, 171–185 (2000).
[Crossref]

Martinez, A. S.

A. S. Martinez and R. Maynard, “Faraday effect and multiple scattering of light,” Phys. Rev. B 50, 3714–3732 (1994).
[Crossref]

Matcher, S. J.

S. V. Gangnus, S. J. Matcher, and I. V. Meglinski, “Monte Carlo modeling of polarized light propagation in biological tissues,” Laser Phys. 14, 886–891 (2004).

Maynard, R.

A. S. Martinez and R. Maynard, “Faraday effect and multiple scattering of light,” Phys. Rev. B 50, 3714–3732 (1994).
[Crossref]

McMaster, W. H.

W. H. McMaster, “Matrix representation of polarization,” Rev. Mod. Phys. 33, 8–28 (1961).
[Crossref]

Meglinski, I.

I. Meglinski and V. L. Kuzmin, “Coherent backscattering of circularly polarized light from a disperse random medium,” Prog. Electromagn. Res. M 16, 47–61 (2011).
[Crossref]

Meglinski, I. V.

S. V. Gangnus, S. J. Matcher, and I. V. Meglinski, “Monte Carlo modeling of polarized light propagation in biological tissues,” Laser Phys. 14, 886–891 (2004).

Mehrübeoglu, M.

Mueller, H.

H. Mueller, “The foundation of optics,” abstract for a contributed paper presented at the 1948 Winter Meeting of the OSA.

Nakadate, S.

A. Takada, M. Shibuya, T. Saitoh, A. Nishikata, K. Maehara, and S. Nakadate, “Incoming inclination factor for scalar imaging theory,” Opt. Eng. 49, 023202 (2010).
[Crossref]

Nee, S.-M. F.

S.-M. F. Nee, “Polarization measurement,” in The Measurement, Instrumentation, and Sensors: Handbook, J. G. Webster, ed. (CRC and IEEE, 1999), Chap. 60, 1655–1679.

Nishikata, A.

A. Takada, M. Shibuya, T. Saitoh, A. Nishikata, K. Maehara, and S. Nakadate, “Incoming inclination factor for scalar imaging theory,” Opt. Eng. 49, 023202 (2010).
[Crossref]

Parke, N. G.

N. G. Parke, “Matrix optics,” Ph.D. dissertation (MIT, 1948).

N. G. Parke, “Matrix algebra of electromagnetic waves,” (MIT Research Laboratory of Electronics, 1948).

N. G. Parke, “Statistical optics: I. Radiation,” (MIT Research Laboratory of Electronics, 1949).

N. G. Parke, “Statistical optics: II. Mueller phenomenological algebra,” (MIT Research Laboratory of Electronics, 1949).

Paul, H.

J. Lehner, U. Leonhardt, and H. Paul, “Unpolarized light: classical and quantum states,” Phys. Rev. A 53, 2727–2735 (1996).
[Crossref]

Perrin, F.

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

Plass, G. N.

Prahl, S.

Rakovic, M. J.

Ramella-Roman, J.

Rastegar, S.

Ricka, J.

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

J. Rička and M. Frenz, “Polarized light: electrodynamic fundamentals” and “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. 4, pp. 65–108 and Chap. 7, pp. 203–266, respectively.

Saint-Jalmes, H.

Saitoh, T.

A. Takada, M. Shibuya, T. Saitoh, A. Nishikata, K. Maehara, and S. Nakadate, “Incoming inclination factor for scalar imaging theory,” Opt. Eng. 49, 023202 (2010).
[Crossref]

Shibuya, M.

A. Takada, M. Shibuya, T. Saitoh, A. Nishikata, K. Maehara, and S. Nakadate, “Incoming inclination factor for scalar imaging theory,” Opt. Eng. 49, 023202 (2010).
[Crossref]

Shurcliff, W. A.

W. A. Shurcliff, “Mueller calculus and Jones calculus,” in Polarized Light (Harvard University, 1962), Chap. 8.

Soleillet, P.

P. Soleillet, “Sur les paramètres caractérisant la polarisation partielle de la lumière dans les phénomènes de fluorescence,” Ph.D. dissertation (École Normale Supérieure, 1929) and published in Annales de Physique 12 (1929), pp. 23–59.

Stokes, G. G.

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

Takada, A.

A. Takada, M. Shibuya, T. Saitoh, A. Nishikata, K. Maehara, and S. Nakadate, “Incoming inclination factor for scalar imaging theory,” Opt. Eng. 49, 023202 (2010).
[Crossref]

Vitkin, I. A.

N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biophotonics 16, 110801 (2011).

Wait, J.

Wang, L. V.

Wang, X.

Wolf, E.

K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433–437 (1987).
[Crossref]

M. Born and E. Wolf, “Elements of the theory of diffraction,” in Principles of Optics (Pergamon, 1970), Chap. 8, pp. 370–455.

M. Born and E. Wolf, “Statistical optics,” in Fundamentals of Photonics, B. E. A. Saleh and M. C. Teich, eds. (Wiley, 1989), Chap. 10, pp. 342–383.

Xu, M.

Appl. Opt. (4)

Eur. Phys. J. B (1)

R. Lenke and G. Maret, “Magnetic field effects on coherent backscattering of light,” Eur. Phys. J. B 17, 171–185 (2000).
[Crossref]

J. Biophotonics (1)

N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biophotonics 16, 110801 (2011).

J. Chem. Phys. (1)

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

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Laser Phys. (1)

S. V. Gangnus, S. J. Matcher, and I. V. Meglinski, “Monte Carlo modeling of polarized light propagation in biological tissues,” Laser Phys. 14, 886–891 (2004).

Opt. Eng. (1)

A. Takada, M. Shibuya, T. Saitoh, A. Nishikata, K. Maehara, and S. Nakadate, “Incoming inclination factor for scalar imaging theory,” Opt. Eng. 49, 023202 (2010).
[Crossref]

Opt. Express (4)

Phys. Rev. A (2)

A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
[Crossref]

J. Lehner, U. Leonhardt, and H. Paul, “Unpolarized light: classical and quantum states,” Phys. Rev. A 53, 2727–2735 (1996).
[Crossref]

Phys. Rev. B (1)

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[Crossref]

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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 (29)

P. Soleillet, “Sur les paramètres caractérisant la polarisation partielle de la lumière dans les phénomènes de fluorescence,” Ph.D. dissertation (École Normale Supérieure, 1929) and published in Annales de Physique 12 (1929), pp. 23–59.

S. Chandrasekhar, “Radiative transfer: a personal account,” in Selected Papers, Vol. 2 of Radiative Transfer and Negative Ion of Hydrogen (University of Chicago, 1989), pp. 511–542.

Interview of S. Chandrasekhar by S. Weart on 18 May 1977, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD, USA, http://www.aip.org/history/ohilist/4551_2.html .

Regrettably, such a confusion affected the V element in three equations in [21]. To comply with the present conventions, these equations should be corrected as follows: 4.100: S3=V=〈SL〉−〈SR〉=+i[〈ExEy*〉−〈Ex*Ey〉], 4.104: S0≔〈〈E|σ0|E〉〉S1≔〈〈E|σ1|E〉〉S2≔〈〈E|σ2|E〉〉S3≔−〈〈E|σ3|E〉〉, 4.116: s⃗3=12(s⃗L−s⃗R)=(0,0,0,1).

Modern treatments of the topic prefer to work with the correlation matrix 〈|Ex|2〉T〈ExEy*〉T〈EyEx*〉T〈|Ex|2〉T, instead of the Stokes parameters. Particularly thorough discussions on polarization statistics are found in [20,36].

M. Born and E. Wolf, “Statistical optics,” in Fundamentals of Photonics, B. E. A. Saleh and M. C. Teich, eds. (Wiley, 1989), Chap. 10, pp. 342–383.

R. C. Jones, “A new calculus for the treatment of optical systems,” series of eight papers published in the J. Opt. Soc. Am. from 1941 to 1956.

We prefer here the “Perrin–Mueller matrix” to the more common “Mueller matrix” not only to give due credit to Francis Perrin, but also because Hans Mueller’s publications on the subject are unfortunately impossible to find (as was confirmed to us by a representative of the libraries of the Massachusetts Institute of Technology, where Mueller worked as a professor). The only written trace of Mueller’s work is a little revealing conference abstract [39]; therefore, we strongly suggest consulting Perrin’s comprehensive 1942 paper [40] for more information on the PM matrix of arbitrary isotropic, symmetric, and asymmetric scatterers. (Credit should be given also to Paul Soleillet [31,41], who got the idea almost two decades before Perrin and Mueller.) Besides, the notation PM recalls the “phase matrix” introduced by Chandrasekhar to treat radiative transfer of polarization.

H. Mueller, “The foundation of optics,” abstract for a contributed paper presented at the 1948 Winter Meeting of the OSA.

R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, E. W. van Stryland, D. R. Williams, and W. L. Wolfe, eds., 2nd ed. (McGraw-Hill, 1995), Vol. 2, Chap. 22, 22.1–22.37.

J. M. Bennett, “Polarizers,” in Handbook of Optics, M. Bass, E. W. van Stryland, D. R. Williams, and W. L. Wolfe, eds., 2nd ed. (McGraw-Hill, 1995), Vol. 2, Chap. 3, 3.1–3.12.

S.-M. F. Nee, “Polarization measurement,” in The Measurement, Instrumentation, and Sensors: Handbook, J. G. Webster, ed. (CRC and IEEE, 1999), Chap. 60, 1655–1679.

It is difficult to ascertain an accurate chronology regarding the emergence of models using the Stokes vector to describe polarized light propagation. Several physicists were particularly prolific to this end, though arguably, long after Sir Stokes’ 1852 paper [30]. Chandrasekhar (apparently unaware of Soleillet’s findings [31]) was the first to reintroduce the then forgotten four real Stokes parameters in the English-speaking community, as part of his pioneering work on radiative transfer in astrophysics [32,33]. A historical overview can also be found in [20].

N. Korolkova, “30.4 Stokes operators questioned: degree of polarization in quantum optics,” in Quantum Information Processing, T. Beth and G. Leuchs, eds., 2nd ed. (Wiley, 2005), pp. 413–416.

More historical and background information on Paul Soleillet and Francis Perrin can be found on Oriol Arteaga’s home page: http://www.mmpolarimetry.com

R. M. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier Science, 1987).

N. G. Parke, “Matrix optics,” Ph.D. dissertation (MIT, 1948).

N. G. Parke, “Matrix algebra of electromagnetic waves,” (MIT Research Laboratory of Electronics, 1948).

N. G. Parke, “Statistical optics: I. Radiation,” (MIT Research Laboratory of Electronics, 1949).

N. G. Parke, “Statistical optics: II. Mueller phenomenological algebra,” (MIT Research Laboratory of Electronics, 1949).

W. A. Shurcliff, “Mueller calculus and Jones calculus,” in Polarized Light (Harvard University, 1962), Chap. 8.

Additional remarks, and references with respect to both matrix methods can be found in the last paragraph of J. M. Bennett, “Polarization,” in Handbook of Optics Volume I, M. Bass, E. W. Van Stryland, D. R. Williams, and W. L. Wolfe, eds., 2nd ed. (McGraw-Hill, 1995), Chap. 5.

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

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1999).

J. Rička and M. Frenz, “Polarized light: electrodynamic fundamentals” and “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. 4, pp. 65–108 and Chap. 7, pp. 203–266, respectively.

Despite the “quantum” notation, the present treatment remains essentially within classical optics; i.e., we only consider pure and mixed coherent states. For the so-called “nonclassical” polarization states, see, e.g., [23–25].

J. S. Hendricks and T. E. Booth, “MCNP variance reduction overview,” in Monte-Carlo Methods and Applications in Neutronics, Photonics and Statistical Physics (Springer, 1985), pp. 83–92.

This low aperture approximation is also used in the wave-optical treatment in Section 7.5.6 of [21], but the suggested validity range is incorrect. The correct wave-optical approximation, that is valid for moderate apertures up to 0.6, is 〈f|e^D〉=[e^f*·e^D]κ(n^D,k^out)O(n^D,k^out). Wave optical aspects are included in the Fresnel inclination factor κ(n^D,k^out)=(1+k^out·n^D)/2 (as it occurs in Fresnel–Kirchhoff diffraction theory [46,47]) and O(n^D,k^out) is the scalar overlap that describes the effective aperture of the observation. The bracketed quantity [e^f*·e^D] is the same as in geometrical approximation, with e^D from Eq. (18).

M. Born and E. Wolf, “Elements of the theory of diffraction,” in Principles of Optics (Pergamon, 1970), Chap. 8, pp. 370–455.

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Figures (3)

Fig. 1.
Fig. 1.

Graphical representation of the four Stokes parameters as determined from measurements with the three fundamental filter pairs defined in Table 1. The arrow e ( t ) drawn in the x y plane of the transversal coordinate system represents the real part of the complex polarization vector of the light transmitted through a filter: e ( t , z ) = R [ e ^ f exp ( i ω t + i k z ) ] at z = 0 . The pictograms defining I , Q , U , and V represent the motion of the tip of this arrow in the x y plane, as seen when looking from the point of view of the receiver, against the beam propagation. To avoid confusion that arises from the “point of view” definition, we define right-circular polarization | C through the chirality: the loci of the tips of e ( t , z ) , at any instant t , form a right-handed spiral that is pushed (not rotated) through the x y plane.

Fig. 2.
Fig. 2.

Left: optical setup for polarization imaging. Polarization analysis is done in the collimated space of the telecentric lens pair. Right: geometrical representation of polarization analysis of the light stream received by one pixel of the imager. The collimating objective lens is replaced by the refractive plane σ D , which deflects a ray (dotted line) exiting from the observed pixel area on the sample surface in the direction n ^ D . The polarization analyzer is represented by its filter state e ^ f , here, for example, e ^ f = | L + . Further explanations in text.

Fig. 3.
Fig. 3.

Geometry of a scattering event. Notice here our convention: the x axis of the i th coordinate system is perpendicular to the scattering plane σ i .

Tables (1)

Tables Icon

Table 1. Fundamental Pairs of Polarization States

Equations (24)

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E ( r , t ) P ( t ) X ( r ) e ^ ( t ) exp [ i ( ω t + k · r + ϕ ) ] ,
| E k = ( E x ( t ) E y ( t ) ) k = P e i Φ ( e x ( t ) e y ( t ) ) k .
I = | E x | 2 T A + | E y | 2 T A ,
Q = | E x | 2 T A | E y | 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 = | E x | 2 + | E y | 2 ,
Q = | E x | 2 | E y | 2 ,
U = 2 | E x | | E y | cos ( ϕ x ϕ y ) ,
V = 2 | E x | | E y | sin ( ϕ x ϕ y ) .
| E out = O | E in ,
( E x out ( t ) E y out ( t ) ) = ( O x x ( t ) O x y ( t ) O y x ( t ) O y y ( t ) ) ( E x in E y in ) .
M ( t ) = ( M⃗ 1 ( t ) M⃗ 2 ( t ) M⃗ 3 ( t ) M⃗ 4 ( t ) )
= ( S⃗ + ( t ) + S⃗ ( t ) S⃗ L X ( t ) S⃗ L Y ( t ) S⃗ L + ( t ) S⃗ L ( t ) S⃗ C + ( t ) S⃗ C ( t ) )
= ( I + + I I L X I L Y I L + I L I C + I C Q + + Q Q L X Q L Y Q L + Q L Q C + Q C U + + U U L X U L Y U L + U L U C + U C V + + V V L X V L Y V L + V L V C + V C ) .
M = M ( t ) T = ( S⃗ + ( t ) + S⃗ ( t ) S⃗ L X ( t ) S⃗ L Y ( t ) S⃗ L + ( t ) S⃗ L ( t ) S⃗ C + ( t ) S⃗ C ( t ) ) T .
S⃗ out = S⃗ out ( t ) T = M ( t ) T S⃗ in ( t ) T = M S⃗ in .
e ^ D = e ^ out ( e ^ out · n ^ D ) [ k ^ out + n ^ D ] / ( 1 + k ^ out · n ^ D ) .
| E f = FR ( ϕ D ) R ( ϕ m ) | e out ,
S⃗ D = R M ( ϕ D ) R M ( ϕ m ) S⃗ out .
| E out = S n R n S i R i S 1 R 1 | e in .
S⃗ out = M S n M R n M S i M R i M S 1 M R 1 s⃗ in .
S⃗ out X + S⃗ out Y 2 = M S n M R n M S i M R i M S 1 M R 1 s⃗ X + s⃗ Y 2 .
S⃗ out O = M S n M R n M S i M R i M S 1 M R 1 s⃗ O ,

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