Abstract

The Mueller matrix (M) corresponding to the phase matrix in the backscattering region (scattering angles ranging from 175° to 180°) is investigated for light scattering at a 0.532-µm wavelength by hexagonal ice crystals, ice spheres, and water droplets. For hexagonal ice crystals we assume three aspect ratios (plates, compact columns, and columns). It is shown that the contour patterns of the backscattering Mueller matrix elements other than M11, M44, M14, and M41 depend on particle geometry; M22 and M33 are particularly sensitive to the aspect ratio of ice crystals. The Mueller matrix for spherical ice particles is different from those for nonspherical ice particles. In addition to discriminating between spherical and nonspherical particles, the Mueller matrix may offer some insight as to cloud thermodynamic phase. The contour patterns for large ice spheres with an effective size of 100 µm are substantially different from those associated with small water droplets with an effective size of 4 µm.

© 2003 Optical Society of America

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References

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  1. G. W. Kattawar, M. J. Rakovic, “Virtues of Mueller matrix imaging for underwater target detection,” Appl. Opt. 38, 6431–6438 (1999).
    [CrossRef]
  2. B. D. Cameron, M. J. Rakovic, M. Mehrubeoglu, G. W. Kattawar, S. Rastegar, L. V. Wang, G. L. Cote, “Measurement and calculation of the two-dimensional backscattering Mueller matrix of a turbid medium,” Opt. Lett. 23, 485–487 (1998).
    [CrossRef]
  3. T. Gehrels, ed., Planets, Stars and Nebulae (University of Arizona, Tucson, Ariz., 1974).
  4. K. Sassen, “The polarization lidar technique for cloud research: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1866 (1991).
    [CrossRef]
  5. S. R. Pal, A. I. Carswell, “Polarization anisotropy in lidar multiple scattering from atmospheric clouds,” Appl. Opt. 24, 3464–3471 (1985).
    [CrossRef] [PubMed]
  6. M. J. Rakovic, G. W. Kattawar, B. D. Cameron, M. Mehrubeoglu, S. Rastegar, L. V. Wang, G. L. Cot, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. Opt. 38, 3399–3408 (1999).
    [CrossRef]
  7. M. I. Mishchenko, K. Sassen, “Depolarization of lidar returns by small ice crystals: an application to contrails,” Geophys. Res. Lett. 25, 309–312 (1998).
    [CrossRef]
  8. Y. X. Hu, D. Winker, P. Yang, B. A. Baum, L. Poole, L. Vann, “Identification of cloud phase from PICASSO-CENA lidar depolarization: a multiple scattering sensitivity study,” J. Quant. Spectrosc. Radiat. Transfer 70, 569–579 (2001).
    [CrossRef]
  9. Y. X. Hu, P. Yang, B. Lin, C. Hostetler, “Discriminating between spherical and non-spherical scatters with lidar using circular component: a theoretical study,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 757–764 (2003).
    [CrossRef]
  10. D. M. Winker, B. A. Wielicki, “The PICASSO-CENA mission,” in Sensors, Systems, and Next-Generation Satellites III, H. Fujisada, J. B. Lurie, eds., Proc. SPIE3870, 26–36 (1999).
    [CrossRef]
  11. A. Ben-David, “Mueller matrices and information derived from linear polarization lidar measurements: theory,” Appl. Opt. 37, 2448–2463 (1998).
    [CrossRef]
  12. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  13. S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).
  14. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  15. M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).
  16. K. N. Liou, An Introduction to Atmospheric Radiation, 2nd ed. (Academic, San Diego, Calif., 2002).
  17. S. G. Warren, “Optical constants of ice from the ultraviolet to the microwave,” Appl. Opt. 23, 1206–1225 (1984).
    [CrossRef] [PubMed]
  18. P. Yang, K. N. Liou, “Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35, 6568–6584 (1996).
    [CrossRef] [PubMed]
  19. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  20. Q. M. Cai, K. N. Liou, “Theory of polarized light scattering by hexagonal ice crystals,” Appl. Opt. 21, 3569–3580 (1982).
    [CrossRef] [PubMed]
  21. Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
    [CrossRef]
  22. A. Macke, J. Muller, E. Rascke, “Single-scattering properties of atmopsheric crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
    [CrossRef]
  23. J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
    [CrossRef]
  24. M. D. King, “Determination of the scaled optical thickness of clouds from reflected solar radiation measurements,” J. Atmos. Sci. 44, 1734–1751 (1987).
    [CrossRef]
  25. J. W. Hovenier, “Structure of a general pure Mueller matrix,” Appl. Opt. 33, 8318–8324 (1994).
    [CrossRef] [PubMed]
  26. C. R. Hu, G. W. Kattawar, M. E. Parkin, P. Herb, “Symmetry theorems on the forward and backward scattering Mueller matrix for light scattering from a nonspherical dielectric scatter,” Appl. Opt. 26, 4159–4173 (1987).
    [CrossRef] [PubMed]

2003 (1)

Y. X. Hu, P. Yang, B. Lin, C. Hostetler, “Discriminating between spherical and non-spherical scatters with lidar using circular component: a theoretical study,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 757–764 (2003).
[CrossRef]

2001 (1)

Y. X. Hu, D. Winker, P. Yang, B. A. Baum, L. Poole, L. Vann, “Identification of cloud phase from PICASSO-CENA lidar depolarization: a multiple scattering sensitivity study,” J. Quant. Spectrosc. Radiat. Transfer 70, 569–579 (2001).
[CrossRef]

1999 (2)

1998 (3)

1996 (2)

A. Macke, J. Muller, E. Rascke, “Single-scattering properties of atmopsheric crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

P. Yang, K. N. Liou, “Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35, 6568–6584 (1996).
[CrossRef] [PubMed]

1994 (1)

1991 (1)

K. Sassen, “The polarization lidar technique for cloud research: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1866 (1991).
[CrossRef]

1989 (1)

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

1987 (2)

1985 (1)

1984 (1)

1982 (1)

1980 (1)

1974 (1)

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Baum, B. A.

Y. X. Hu, D. Winker, P. Yang, B. A. Baum, L. Poole, L. Vann, “Identification of cloud phase from PICASSO-CENA lidar depolarization: a multiple scattering sensitivity study,” J. Quant. Spectrosc. Radiat. Transfer 70, 569–579 (2001).
[CrossRef]

Ben-David, A.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Cai, Q. M.

Cameron, B. D.

Carswell, A. I.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).

Cot, G. L.

Cote, G. L.

Hansen, J. E.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Herb, P.

Hostetler, C.

Y. X. Hu, P. Yang, B. Lin, C. Hostetler, “Discriminating between spherical and non-spherical scatters with lidar using circular component: a theoretical study,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 757–764 (2003).
[CrossRef]

Hovenier, J. W.

Hu, C. R.

Hu, Y. X.

Y. X. Hu, P. Yang, B. Lin, C. Hostetler, “Discriminating between spherical and non-spherical scatters with lidar using circular component: a theoretical study,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 757–764 (2003).
[CrossRef]

Y. X. Hu, D. Winker, P. Yang, B. A. Baum, L. Poole, L. Vann, “Identification of cloud phase from PICASSO-CENA lidar depolarization: a multiple scattering sensitivity study,” J. Quant. Spectrosc. Radiat. Transfer 70, 569–579 (2001).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Kattawar, G. W.

King, M. D.

M. D. King, “Determination of the scaled optical thickness of clouds from reflected solar radiation measurements,” J. Atmos. Sci. 44, 1734–1751 (1987).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).

Lin, B.

Y. X. Hu, P. Yang, B. Lin, C. Hostetler, “Discriminating between spherical and non-spherical scatters with lidar using circular component: a theoretical study,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 757–764 (2003).
[CrossRef]

Liou, K. N.

P. Yang, K. N. Liou, “Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35, 6568–6584 (1996).
[CrossRef] [PubMed]

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

Q. M. Cai, K. N. Liou, “Theory of polarized light scattering by hexagonal ice crystals,” Appl. Opt. 21, 3569–3580 (1982).
[CrossRef] [PubMed]

K. N. Liou, An Introduction to Atmospheric Radiation, 2nd ed. (Academic, San Diego, Calif., 2002).

Macke, A.

A. Macke, J. Muller, E. Rascke, “Single-scattering properties of atmopsheric crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

Mehrubeoglu, M.

Mishchenko, M. I.

M. I. Mishchenko, K. Sassen, “Depolarization of lidar returns by small ice crystals: an application to contrails,” Geophys. Res. Lett. 25, 309–312 (1998).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).

Muller, J.

A. Macke, J. Muller, E. Rascke, “Single-scattering properties of atmopsheric crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

Pal, S. R.

Parkin, M. E.

Poole, L.

Y. X. Hu, D. Winker, P. Yang, B. A. Baum, L. Poole, L. Vann, “Identification of cloud phase from PICASSO-CENA lidar depolarization: a multiple scattering sensitivity study,” J. Quant. Spectrosc. Radiat. Transfer 70, 569–579 (2001).
[CrossRef]

Rakovic, M. J.

Rascke, E.

A. Macke, J. Muller, E. Rascke, “Single-scattering properties of atmopsheric crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

Rastegar, S.

Sassen, K.

M. I. Mishchenko, K. Sassen, “Depolarization of lidar returns by small ice crystals: an application to contrails,” Geophys. Res. Lett. 25, 309–312 (1998).
[CrossRef]

K. Sassen, “The polarization lidar technique for cloud research: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1866 (1991).
[CrossRef]

Takano, Y.

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

Travis, L. D.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).

van de Hulst, H. C.

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

Vann, L.

Y. X. Hu, D. Winker, P. Yang, B. A. Baum, L. Poole, L. Vann, “Identification of cloud phase from PICASSO-CENA lidar depolarization: a multiple scattering sensitivity study,” J. Quant. Spectrosc. Radiat. Transfer 70, 569–579 (2001).
[CrossRef]

Wang, L. V.

Warren, S. G.

Wielicki, B. A.

D. M. Winker, B. A. Wielicki, “The PICASSO-CENA mission,” in Sensors, Systems, and Next-Generation Satellites III, H. Fujisada, J. B. Lurie, eds., Proc. SPIE3870, 26–36 (1999).
[CrossRef]

Winker, D.

Y. X. Hu, D. Winker, P. Yang, B. A. Baum, L. Poole, L. Vann, “Identification of cloud phase from PICASSO-CENA lidar depolarization: a multiple scattering sensitivity study,” J. Quant. Spectrosc. Radiat. Transfer 70, 569–579 (2001).
[CrossRef]

Winker, D. M.

D. M. Winker, B. A. Wielicki, “The PICASSO-CENA mission,” in Sensors, Systems, and Next-Generation Satellites III, H. Fujisada, J. B. Lurie, eds., Proc. SPIE3870, 26–36 (1999).
[CrossRef]

Wiscombe, W. J.

Yang, P.

Y. X. Hu, P. Yang, B. Lin, C. Hostetler, “Discriminating between spherical and non-spherical scatters with lidar using circular component: a theoretical study,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 757–764 (2003).
[CrossRef]

Y. X. Hu, D. Winker, P. Yang, B. A. Baum, L. Poole, L. Vann, “Identification of cloud phase from PICASSO-CENA lidar depolarization: a multiple scattering sensitivity study,” J. Quant. Spectrosc. Radiat. Transfer 70, 569–579 (2001).
[CrossRef]

P. Yang, K. N. Liou, “Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35, 6568–6584 (1996).
[CrossRef] [PubMed]

Appl. Opt. (10)

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

Q. M. Cai, K. N. Liou, “Theory of polarized light scattering by hexagonal ice crystals,” Appl. Opt. 21, 3569–3580 (1982).
[CrossRef] [PubMed]

S. G. Warren, “Optical constants of ice from the ultraviolet to the microwave,” Appl. Opt. 23, 1206–1225 (1984).
[CrossRef] [PubMed]

S. R. Pal, A. I. Carswell, “Polarization anisotropy in lidar multiple scattering from atmospheric clouds,” Appl. Opt. 24, 3464–3471 (1985).
[CrossRef] [PubMed]

C. R. Hu, G. W. Kattawar, M. E. Parkin, P. Herb, “Symmetry theorems on the forward and backward scattering Mueller matrix for light scattering from a nonspherical dielectric scatter,” Appl. Opt. 26, 4159–4173 (1987).
[CrossRef] [PubMed]

J. W. Hovenier, “Structure of a general pure Mueller matrix,” Appl. Opt. 33, 8318–8324 (1994).
[CrossRef] [PubMed]

A. Ben-David, “Mueller matrices and information derived from linear polarization lidar measurements: theory,” Appl. Opt. 37, 2448–2463 (1998).
[CrossRef]

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

P. Yang, K. N. Liou, “Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35, 6568–6584 (1996).
[CrossRef] [PubMed]

M. J. Rakovic, G. W. Kattawar, B. D. Cameron, M. Mehrubeoglu, S. Rastegar, L. V. Wang, G. L. Cot, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. Opt. 38, 3399–3408 (1999).
[CrossRef]

Bull. Am. Meteorol. Soc. (1)

K. Sassen, “The polarization lidar technique for cloud research: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1866 (1991).
[CrossRef]

Geophys. Res. Lett. (1)

M. I. Mishchenko, K. Sassen, “Depolarization of lidar returns by small ice crystals: an application to contrails,” Geophys. Res. Lett. 25, 309–312 (1998).
[CrossRef]

J. Atmos. Sci. (3)

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

A. Macke, J. Muller, E. Rascke, “Single-scattering properties of atmopsheric crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

M. D. King, “Determination of the scaled optical thickness of clouds from reflected solar radiation measurements,” J. Atmos. Sci. 44, 1734–1751 (1987).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (2)

Y. X. Hu, D. Winker, P. Yang, B. A. Baum, L. Poole, L. Vann, “Identification of cloud phase from PICASSO-CENA lidar depolarization: a multiple scattering sensitivity study,” J. Quant. Spectrosc. Radiat. Transfer 70, 569–579 (2001).
[CrossRef]

Y. X. Hu, P. Yang, B. Lin, C. Hostetler, “Discriminating between spherical and non-spherical scatters with lidar using circular component: a theoretical study,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 757–764 (2003).
[CrossRef]

Opt. Lett. (1)

Space Sci. Rev. (1)

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Other (7)

T. Gehrels, ed., Planets, Stars and Nebulae (University of Arizona, Tucson, Ariz., 1974).

D. M. Winker, B. A. Wielicki, “The PICASSO-CENA mission,” in Sensors, Systems, and Next-Generation Satellites III, H. Fujisada, J. B. Lurie, eds., Proc. SPIE3870, 26–36 (1999).
[CrossRef]

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

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).

K. N. Liou, An Introduction to Atmospheric Radiation, 2nd ed. (Academic, San Diego, Calif., 2002).

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Figures (8)

Fig. 1
Fig. 1

Incident and scattering geometry for backscattering by a thin layer composed of scattering particles.

Fig. 2
Fig. 2

Nonzero elements of the phase matrix for hexagonal ice crystals with three aspect ratios. Here, L and a are, respectively, the length and the radius of the cylinder circumscribing the ice crystal.

Fig. 3
Fig. 3

Nonzero elements of the phase matrix for ice spheres and spherical water droplets.

Fig. 4
Fig. 4

Contours of the Mueller matrix for backscattering by randomly oriented hexagonal column ice crystals (2a/L = 100/600 µm, Reff = 113.5 µm). The X and Y coordinates for each of the Mueller images indicate the positions of the pixels on the image plane (same as in Figs. 58).

Fig. 5
Fig. 5

Contours of the Mueller matrix for backscattering by randomly oriented hexagonal compact ice crystals (2a/L = 200/200 µm, Reff = 139.1 µm).

Fig. 6
Fig. 6

Contours of the Mueller matrix for backscattering by randomly oriented hexagonal plate ice crystals (2a/L = 200/50 µm, Reff = 58.2 µm).

Fig. 7
Fig. 7

Contours of the Mueller matrix for backscattering by ice spheres (Reff = 100 µm).

Fig. 8
Fig. 8

Contours of the Mueller matrix for backscattering by spherical water droplets (Reff = 4 µm).

Equations (18)

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

I=<E E*>+<E E*>,
Q=<E E*>-<E E*>,
U=<E E*>+<E*E>,
V=i<E E*>-<E*E>,
IsQsUsVs=σs4πr2P11P1200P12P220000P33-P4300P43P44IiQiUiVi,
120π P11θ sin θ dθ=1.
Isx, yQsx, yUsx, yVsx, y=Mx, yIiQiUiVi,
M=M11M12M13M14M21M22M23M24M31M32M33M34M41M42M43M44.
IipQipUipVip=10000cos 2π/2-ϕsin 2π/2-ϕ00-sin 2π/2-ϕcos 2π/2-ϕ00001×IiyQiyUiyViy.
IspQspUspVsp=N<σs>4πr2P11P1200P12P220000P33-P4300P43P44×IipQipUipVip,
EspEsp=100cosπ-θEsEs.
IspQspUspVsp=cos2 θ+1/2cos2 θ-1/200cos2 θ-1/2cos2 θ+1/20000-cos θ0000-cos θIspQspUspVsp.
IsyQsyUsyVsy=10000cos 2π/2-ϕsin 2π/2-ϕ00-sin 2π/2-ϕcos 2π/2-ϕ00001×IspQspUspVsp.
IsyQsyUsyVsy=MIiyQiyUiyViy,
M=N<σs>4πr210000-cos2ϕsin2ϕ00-sin2ϕ-cos2ϕ00001cos2 θ+1/2cos2 θ-1/200cos2 θ-1/2cos2 θ+1/20000-cos θ0000-cos θ×P11θP12θ00P12θP22θ0000P33θP43θ00P43θP44θ10000-cos2ϕsin2ϕ00-sin2ϕ-cos2ϕ00001.
nr=N0reffVeffVeff-1/VeffΓ1-2Veff/Veff r1-3Veff/Veff×exp-r/reffVeff,
reff=r1r2 r3nrdrr1r2 r2nrdr,
Veff=r1r2r-reff2r2nrdrreff2r1r2 r2nrdr.

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