Abstract

For single scattering in a turbid medium, the Mueller matrix is the 4 × 4 matrix that multiplies the incident Stokes vector to yield the scattered Stokes vector. This matrix contains all the information that can be obtained from an elastic-scattering system. We have extended this concept to the multiple-scattering domain where we can define an effective Mueller matrix that, when operating on any incident state of light, will yield the output state. We have calculated this matrix using two completely different computational methods and compared the results for several simple two-layer turbid systems separated by a dielectric interface. We have shown that both methods give reliable results and therefore can be used to accurately predict the scattering properties of turbid media.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  2. S. Chandrasekhar, Radiative Transfer (Dover, Toronto, Ontario, 1960).
  3. J. Lenoble, Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (Deepak, Hampton, Va., 1985).
  4. G. W. Kattawar, C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
    [CrossRef]
  5. E. P. Zege, I. L. Katsev, I. N. Polonsky, “Multicomponent approach to light propagation in clouds and mists,” Appl. Opt. 32, 2803–2812 (1993).
    [CrossRef] [PubMed]
  6. E. P. Zege, L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transfer 55, 19–31 (1996).
    [CrossRef]
  7. A. P. Ivanov, E. P. Zege, I. L. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, Heidelberg, 1991).
  8. J. von Neumann, “Various techniques used in connection with random digits,” J. Res. Natl. Bur. Stand. 5, 36–38 (1951).
  9. G. W. Kattawar, G. N. Plass, J. A. Guinn, “Monte Carlo calculations of the polarization of radiation in the Earth’s atmosphere-ocean system,” J. Phys. Oceanogr. 3, 353–372 (1973).
    [CrossRef]
  10. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  11. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994).
  12. I. L. Katsev, E. P. Zege, A. S. Prikhach, I. N. Polonsky, “Efficient technique to determine backscattered light power for various atmospheric and oceanic sounding and imaging systems,” J. Opt. Soc. Am. A 14, 1338–1346 (1997).
    [CrossRef]
  13. J. L. Deueze, M. Herman, R. Santer, “Fourier series expansion of the transfer equation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 41, 483–494 (1989).
    [CrossRef]
  14. K. Masuda, T. Takashima, “Computational accuracy of radiation emerging from the ocean surface in the model atmosphere–ocean system,” Pap. Meteorol. Geophys. 37, 1–13 (1986).
    [CrossRef]
  15. G. N. Plass, T. J. Humphries, G. W. Kattawar, “Ocean-atmosphere interface: its influence on radiation,” Appl. Opt. 20, 917–931 (1981).
    [CrossRef] [PubMed]
  16. Concise Dictionary of Scientific Biography (Scribner, New York, 1981), p. 643. Although in the English-speaking community a convention of spelling Snel’s name with two l’s has arisen, Willebrord Snel von Royen used only one l in his last name.
  17. K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California, Berkeley, Calif., 1960).
  18. S. Chandrasekhar, D. D. Elbert, “The illumination and polarization of the sunlit sky on Rayleigh scattering,” Trans. Am. Phil. Soc. B 44, 643–728 (1954).
    [CrossRef]
  19. R. S. Fraser, “Atmospheric neutral points over water,” J. Opt. Soc. Am. 58, 1029–1031 (1968).
    [CrossRef]
  20. J. T. Adams, G. W. Kattawar, “Neutral points in an atmosphere–ocean system. 1: Upwelling light field,” Appl. Opt. 36, 1976–1986 (1997).
    [CrossRef] [PubMed]
  21. G. W. Kattawar, M. J. Rakovic, “Virtues of Mueller matrix imaging for underwater target detection,” Appl. Opt. 38, 6431–6438 (1999).
    [CrossRef]

1999

1997

1996

E. P. Zege, L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transfer 55, 19–31 (1996).
[CrossRef]

1993

1989

G. W. Kattawar, C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
[CrossRef]

J. L. Deueze, M. Herman, R. Santer, “Fourier series expansion of the transfer equation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 41, 483–494 (1989).
[CrossRef]

1986

K. Masuda, T. Takashima, “Computational accuracy of radiation emerging from the ocean surface in the model atmosphere–ocean system,” Pap. Meteorol. Geophys. 37, 1–13 (1986).
[CrossRef]

1981

1973

G. W. Kattawar, G. N. Plass, J. A. Guinn, “Monte Carlo calculations of the polarization of radiation in the Earth’s atmosphere-ocean system,” J. Phys. Oceanogr. 3, 353–372 (1973).
[CrossRef]

1968

1954

S. Chandrasekhar, D. D. Elbert, “The illumination and polarization of the sunlit sky on Rayleigh scattering,” Trans. Am. Phil. Soc. B 44, 643–728 (1954).
[CrossRef]

1951

J. von Neumann, “Various techniques used in connection with random digits,” J. Res. Natl. Bur. Stand. 5, 36–38 (1951).

Adams, C. N.

G. W. Kattawar, C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
[CrossRef]

Adams, J. T.

Chaikovskaya, L. I.

E. P. Zege, L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transfer 55, 19–31 (1996).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, D. D. Elbert, “The illumination and polarization of the sunlit sky on Rayleigh scattering,” Trans. Am. Phil. Soc. B 44, 643–728 (1954).
[CrossRef]

S. Chandrasekhar, Radiative Transfer (Dover, Toronto, Ontario, 1960).

Coulson, K. L.

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California, Berkeley, Calif., 1960).

Dave, J. V.

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California, Berkeley, Calif., 1960).

Deueze, J. L.

J. L. Deueze, M. Herman, R. Santer, “Fourier series expansion of the transfer equation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 41, 483–494 (1989).
[CrossRef]

Elbert, D. D.

S. Chandrasekhar, D. D. Elbert, “The illumination and polarization of the sunlit sky on Rayleigh scattering,” Trans. Am. Phil. Soc. B 44, 643–728 (1954).
[CrossRef]

Fraser, R. S.

Guinn, J. A.

G. W. Kattawar, G. N. Plass, J. A. Guinn, “Monte Carlo calculations of the polarization of radiation in the Earth’s atmosphere-ocean system,” J. Phys. Oceanogr. 3, 353–372 (1973).
[CrossRef]

Herman, M.

J. L. Deueze, M. Herman, R. Santer, “Fourier series expansion of the transfer equation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 41, 483–494 (1989).
[CrossRef]

Humphries, T. J.

Ivanov, A. P.

A. P. Ivanov, E. P. Zege, I. L. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, Heidelberg, 1991).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

Katsev, I. L.

Kattawar, G. W.

G. W. Kattawar, M. J. Rakovic, “Virtues of Mueller matrix imaging for underwater target detection,” Appl. Opt. 38, 6431–6438 (1999).
[CrossRef]

J. T. Adams, G. W. Kattawar, “Neutral points in an atmosphere–ocean system. 1: Upwelling light field,” Appl. Opt. 36, 1976–1986 (1997).
[CrossRef] [PubMed]

G. W. Kattawar, C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
[CrossRef]

G. N. Plass, T. J. Humphries, G. W. Kattawar, “Ocean-atmosphere interface: its influence on radiation,” Appl. Opt. 20, 917–931 (1981).
[CrossRef] [PubMed]

G. W. Kattawar, G. N. Plass, J. A. Guinn, “Monte Carlo calculations of the polarization of radiation in the Earth’s atmosphere-ocean system,” J. Phys. Oceanogr. 3, 353–372 (1973).
[CrossRef]

Lenoble, J.

J. Lenoble, Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (Deepak, Hampton, Va., 1985).

Masuda, K.

K. Masuda, T. Takashima, “Computational accuracy of radiation emerging from the ocean surface in the model atmosphere–ocean system,” Pap. Meteorol. Geophys. 37, 1–13 (1986).
[CrossRef]

Mobley, C. D.

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994).

Plass, G. N.

G. N. Plass, T. J. Humphries, G. W. Kattawar, “Ocean-atmosphere interface: its influence on radiation,” Appl. Opt. 20, 917–931 (1981).
[CrossRef] [PubMed]

G. W. Kattawar, G. N. Plass, J. A. Guinn, “Monte Carlo calculations of the polarization of radiation in the Earth’s atmosphere-ocean system,” J. Phys. Oceanogr. 3, 353–372 (1973).
[CrossRef]

Polonsky, I. N.

Prikhach, A. S.

Rakovic, M. J.

Santer, R.

J. L. Deueze, M. Herman, R. Santer, “Fourier series expansion of the transfer equation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 41, 483–494 (1989).
[CrossRef]

Sekera, Z.

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California, Berkeley, Calif., 1960).

Takashima, T.

K. Masuda, T. Takashima, “Computational accuracy of radiation emerging from the ocean surface in the model atmosphere–ocean system,” Pap. Meteorol. Geophys. 37, 1–13 (1986).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

von Neumann, J.

J. von Neumann, “Various techniques used in connection with random digits,” J. Res. Natl. Bur. Stand. 5, 36–38 (1951).

Zege, E. P.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. Oceanogr.

G. W. Kattawar, G. N. Plass, J. A. Guinn, “Monte Carlo calculations of the polarization of radiation in the Earth’s atmosphere-ocean system,” J. Phys. Oceanogr. 3, 353–372 (1973).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

E. P. Zege, L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transfer 55, 19–31 (1996).
[CrossRef]

J. L. Deueze, M. Herman, R. Santer, “Fourier series expansion of the transfer equation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 41, 483–494 (1989).
[CrossRef]

J. Res. Natl. Bur. Stand.

J. von Neumann, “Various techniques used in connection with random digits,” J. Res. Natl. Bur. Stand. 5, 36–38 (1951).

Limnol. Oceanogr.

G. W. Kattawar, C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
[CrossRef]

Pap. Meteorol. Geophys.

K. Masuda, T. Takashima, “Computational accuracy of radiation emerging from the ocean surface in the model atmosphere–ocean system,” Pap. Meteorol. Geophys. 37, 1–13 (1986).
[CrossRef]

Trans. Am. Phil. Soc. B

S. Chandrasekhar, D. D. Elbert, “The illumination and polarization of the sunlit sky on Rayleigh scattering,” Trans. Am. Phil. Soc. B 44, 643–728 (1954).
[CrossRef]

Other

Concise Dictionary of Scientific Biography (Scribner, New York, 1981), p. 643. Although in the English-speaking community a convention of spelling Snel’s name with two l’s has arisen, Willebrord Snel von Royen used only one l in his last name.

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California, Berkeley, Calif., 1960).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994).

A. P. Ivanov, E. P. Zege, I. L. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, Heidelberg, 1991).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

S. Chandrasekhar, Radiative Transfer (Dover, Toronto, Ontario, 1960).

J. Lenoble, Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (Deepak, Hampton, Va., 1985).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Geometry for rotations of the Stokes vector during a scattering event.

Fig. 2
Fig. 2

Geometry of the two-layer models used.

Fig. 3
Fig. 3

Vector radiance versus detector polar angle (semilog plot). Data for the Monte Carlo estimation and the MCA are compared for an atmosphere–ocean with an optical thickness of 1.15 (atmosphere thickness of 0.15 above dielectric interface, ocean thickness of 1.0 below), a smooth surface, and conservative scattering (single scatter albedo of 1). Radiation is normally incident on the system. Rayleigh and Henyey–Greenstein cases are shown. Filled symbols, Monte Carlo data; open symbols, MCA data.

Fig. 4
Fig. 4

Degree of polarization versus detector polar angle for the same system as in Fig. 3.

Fig. 5
Fig. 5

Vector radiance versus detector polar angle (semilog plot) for the same system as in Fig. 3 (Rayleigh case only) except with radiation incident on the system at 60°. Data shown are for the detector azimuth angles of ϕ = 0°, 90°, and 180°.

Fig. 6
Fig. 6

Vector radiance versus detector polar angle for the same system as in Fig. 5 except with Henyey–Greenstein scattering.

Fig. 7
Fig. 7

Degree of polarization versus detector polar angle for the same system as in Fig. 5.

Fig. 8
Fig. 8

Degree of polarization versus detector polar angle for the same system as in Fig. 6.

Fig. 9
Fig. 9

Degree of circular polarization versus detector polar angle at detector azimuth of ϕ = 90° for the case of radiation incident on the system at 60° only.

Equations (31)

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

μ dIτ, μ, ϕdτ =-Iτ, μ, ϕ+ω04π-1102π×Zμ, ϕ; μ, ϕIτ, μ, ϕdμdϕ,
I=IQUV=|E|2+|E|2|E|2-|E|2EE*+E*EiEE*-E*E,
Zμ, ϕ; μ, ϕ=RΦMΘscaRΨ,
MΘsca=a1b1b3b5c1a2b4b6c3c4a3b2c5c6c2a4
Rϕ=10000cos 2ϕsin 2ϕ00-sin 2ϕcos 2ϕ00001
fξ= fξpξdξ.
fξ= fξpξp˜ξ p˜ξdξ
pτ=exp-τ,
pτ=exp-τ1-exp-τm; 0ττm,
If=RΦMΘscaRΨI0.
If=a1I0+b1Q0 cos 2Ψ+U0 sin 2Ψ+b3-Q0 sin 2Ψ+U0 cos 2Ψ+b5V0.
pΘsca=2πa1Θsca.
pX|Y=pX, Y/pY,
pY=ab pX, YdX; aXb.
pΘsca=02π pΘsca, ΨdΨ=2πa1Θsca+b5ΘscaV0/I0,
pΨ|Θsca=a1I0+b1Q0+b3U0cos 2Ψ+b1U0-b3Q0sin 2Ψ+b5V02πa1+b5V0/I0.
pmaxΨ|Θsca=a1I0+b12Q02+U02+b32Q02+U021/2+b5V02πa1+b5V0/I0.
pΨ|Θsca=I0+b1a1Q0 cos 2Ψ+U0 sin 2Ψ.
Zμ, ϕ; μ, ϕ=RΦM˜ΘscaRΨ.
r=12R2+R2=12sin2θi-θtsin2θi+θt+tan2θi-θttan2θi+θt,
Ik=ω0wpΘscakexp-τdk|cos θdk|RΦkM˜ΘscakRΨkItot
Mk=ω0wpΘscakexp-τdk|cos θdk|RΦkM˜ΘscakRΨkMeff,
Iτ, n= Gτ, n, n0Iinn0dn0,
μ ddτGτ, n, n0=-Gτ, n, n0+ω0τ4π  Zn, nGτ, n, n×dn+Eδτδn-n0,
Z11x=A1Z11fx+1-A1Z11dx,
A1Z11fx=Z11x-Z11X; xX0; x<X.
Zn, n0=A1Zfn, n0+1-A1Zdn, n0,
Gτ, n, n0=Gfτ, n, n0+Gdτ, n, n0,
Zfn, n0=Z11fn, nZ˜n, n0; avx+Z11fn, n0Z˜n, n0; bjx,v=1, 2, 3, 4; j=1, 2.
pcos Θsca=316π1+cos2 Θsca.
pcos Θsca=1-g24π1+g2-2g cos Θsca3/2,

Metrics