Abstract

A ray-by-ray integration (RBRI) method has been developed for the solution of light scattering by nonspherical dielectric particles. The principles of geometric optics are applied to solve the internal electric field within the scattering particles (near field) with the inclusion of complete phase and polarization configurations. The scattered field at the radiation zone (far field) and the extinction and absorption cross sections are obtained by integrating the near field along the propagation paths of geometric rays inside the scatterers by using a number of rigorous electromagnetic integral equations. In the computations of extinction cross section and single-scattering albedo, we demonstrate that the well-known anomalous diffraction approximation is a special case of the RBRI method when the scatterers are optically tenuous. The RBRI method is employed to compute the single-scattering properties of hexagonal ice crystals at visible and near-infrared wavelengths. Based on the reference results computed by the finite-difference time domain (FDTD) technique, we show that the RBRI method is more accurate than the conventional geometric ray-tracing technique and the anomalous diffraction approximation. The extinction efficiency and the single-scattering albedo computed by the RBRI method converge to the reference results when the size parameters along the ice crystal maximum dimension are larger than approximately 15. Substantial differences in terms of relative errors, in comparison with the FDTD solutions, are still noted in the phase function and polarization patterns computed by the RBRI method for size parameters of the order of 10.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. N. Liou, “Influence of cirrus clouds on weather and climate process: a global perspective,” Mon. Weather Rev. 114, 1167–1199 (1986).
    [CrossRef]
  2. G. L. Stephens, S. C. Tsay, P. W. Stackhouse, P. J. Flatau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climate feedback,” J. Atmos. Sci. 46, 1742–1753 (1990).
    [CrossRef]
  3. K. N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climate implications,” Atmos. Res. 31, 271–298 (1994).
    [CrossRef]
  4. A. Ono, “The shape and riming properties of ice crystals in natural clouds,” J. Atmos. Sci. 26, 138–147 (1969).
    [CrossRef]
  5. A. J. Heymsfield, “Laboratory and field observations of the growth of columnar and plate crystals from frozen droplets,” J. Atmos. Sci. 30, 1650–1656 (1973).
    [CrossRef]
  6. A. J. Heymsfield, “Cirrus uncinus generating cells and the evolution of cirrusform clouds. Part I: Aircraft observations of the growth of the ice phase,” J. Atmos. Sci. 32, 799–808 (1975).
    [CrossRef]
  7. A. J. Heymsfield, K. M. Miller, J. D. Sphinhirne, “The 27–28 October 1986 FIRE IFO cirrus case study: cloud microstructure,” Mon. Weather Rev. 118, 2313–2328 (1990).
    [CrossRef]
  8. K. Sassen, “Air-truth lidar polarization studies of orographic clouds,” J. Appl. Meteorol. 17, 73–91 (1978).
    [CrossRef]
  9. R. T. Bruintjes, A. J. Heymsfield, T. W. Krauss, “An examination of double-plate ice crystals and initiation of precipitation in continental cumulus clouds,” J. Atmos. Sci. 44, 1331–1349 (1987).
    [CrossRef]
  10. S. C. Ou, K. N. Liou, W. Gooch, Y. Takano, “Remote sensing of cirrus cloud parameters using advantaged very-high-resolution radiometer 3.7- and 10.9-μm channels,” Appl. Opt. 32, 2171–2180 (1993).
    [CrossRef] [PubMed]
  11. S. Kinne, K. N. Liou, “The effects of the nonsphericity and size distribution of ice crystals on the radiative properties of cirrus clouds,” Atmos. Res. 24, 273–284 (1989).
    [CrossRef]
  12. P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: comparison of finite-difference time domain and geometric optics models,” J. Opt. Soc. Am. A 12, 162–176 (1995).
    [CrossRef]
  13. Q. Cai, K. N. Liou, “Polarized light scattering by hexagonal ice crystals: theory,” Appl. Opt. 21, 3569–3580 (1982).
    [CrossRef] [PubMed]
  14. Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I: Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 1–19 (1989).
    [CrossRef]
  15. A. Macke, “Scattering of light by polyhedral ice crystals,” Appl. Opt. 32, 2780–2788 (1993).
    [CrossRef] [PubMed]
  16. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  17. D. S. Saxon, “Lectures on the scattering of light,” in Proceedings of the UCLA International Conference on Radiation and Remote Probing of the Atmosphere, J. G. Kuriyan, ed. (Western Periodicals, North Hollywood, Calif., 1973), pp. 227–308.
  18. J. D. Jackson, Classic Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  19. G. H. Goedecks, S. G. O’Brien, “Scattering by irregular particles via the digitized Green’s function algorithm,” Appl. Opt. 27, 2431–2438 (1981).
    [CrossRef]
  20. E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 196, 705–714 (1973).
    [CrossRef]
  21. P. Yang, K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072–2085 (1996).
    [CrossRef]
  22. C. Acquista, “Light scattering by tenuous particles: a generalization of the Rayleigh–Gans–Rocad approach,” Appl. Opt. 15, 2932–2936 (1976).
    [CrossRef] [PubMed]
  23. K. S. Shifrin, “Scattering of light in a turbid medium,” (National Technical Information Service, Springfield, Va., 1968).
  24. J. D. Klett, A. Sutherland, “Approximate methods for modeling the scattering properties of nonspherical particles: evaluation of the Wentzel–Kramers–Brillouin method,” Appl. Opt. 31, 373–386 (1992).
    [CrossRef] [PubMed]
  25. T. W. Chen, “High energy light scattering in the generalized eikonal approximation,” Appl. Opt. 28, 4096–4102 (1989).
    [CrossRef] [PubMed]
  26. G. L. Stephens, “Scattering of plane wave by soft obstacles: anomalous diffraction theory for circular cylinders,” Appl. Opt. 23, 954–959 (1984).
    [CrossRef]
  27. P. Chylek, J. D. Klett, “Absorption and scattering of electromagnetic radiation by prismatic columns: anomalous diffraction approximation,” J. Opt. Soc. Am. A 8, 1713–1720 (1991).
    [CrossRef]
  28. P. Yang, K. N. Liou, “Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 53, 6568–6584 (1996).
    [CrossRef]
  29. S. A. Schekunoff, Electromagnetic Waves (Van Nostrand, New York, 1943).

1996 (2)

P. Yang, K. N. Liou, “Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 53, 6568–6584 (1996).
[CrossRef]

P. Yang, K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072–2085 (1996).
[CrossRef]

1995 (1)

1994 (1)

K. N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climate implications,” Atmos. Res. 31, 271–298 (1994).
[CrossRef]

1993 (2)

1992 (1)

1991 (1)

1990 (2)

G. L. Stephens, S. C. Tsay, P. W. Stackhouse, P. J. Flatau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climate feedback,” J. Atmos. Sci. 46, 1742–1753 (1990).
[CrossRef]

A. J. Heymsfield, K. M. Miller, J. D. Sphinhirne, “The 27–28 October 1986 FIRE IFO cirrus case study: cloud microstructure,” Mon. Weather Rev. 118, 2313–2328 (1990).
[CrossRef]

1989 (3)

S. Kinne, K. N. Liou, “The effects of the nonsphericity and size distribution of ice crystals on the radiative properties of cirrus clouds,” Atmos. Res. 24, 273–284 (1989).
[CrossRef]

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I: Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 1–19 (1989).
[CrossRef]

T. W. Chen, “High energy light scattering in the generalized eikonal approximation,” Appl. Opt. 28, 4096–4102 (1989).
[CrossRef] [PubMed]

1987 (1)

R. T. Bruintjes, A. J. Heymsfield, T. W. Krauss, “An examination of double-plate ice crystals and initiation of precipitation in continental cumulus clouds,” J. Atmos. Sci. 44, 1331–1349 (1987).
[CrossRef]

1986 (1)

K. N. Liou, “Influence of cirrus clouds on weather and climate process: a global perspective,” Mon. Weather Rev. 114, 1167–1199 (1986).
[CrossRef]

1984 (1)

1982 (1)

1981 (1)

1978 (1)

K. Sassen, “Air-truth lidar polarization studies of orographic clouds,” J. Appl. Meteorol. 17, 73–91 (1978).
[CrossRef]

1976 (1)

1975 (1)

A. J. Heymsfield, “Cirrus uncinus generating cells and the evolution of cirrusform clouds. Part I: Aircraft observations of the growth of the ice phase,” J. Atmos. Sci. 32, 799–808 (1975).
[CrossRef]

1973 (2)

A. J. Heymsfield, “Laboratory and field observations of the growth of columnar and plate crystals from frozen droplets,” J. Atmos. Sci. 30, 1650–1656 (1973).
[CrossRef]

E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 196, 705–714 (1973).
[CrossRef]

1969 (1)

A. Ono, “The shape and riming properties of ice crystals in natural clouds,” J. Atmos. Sci. 26, 138–147 (1969).
[CrossRef]

Acquista, C.

Bruintjes, R. T.

R. T. Bruintjes, A. J. Heymsfield, T. W. Krauss, “An examination of double-plate ice crystals and initiation of precipitation in continental cumulus clouds,” J. Atmos. Sci. 44, 1331–1349 (1987).
[CrossRef]

Cai, Q.

Chen, T. W.

Chylek, P.

Flatau, P. J.

G. L. Stephens, S. C. Tsay, P. W. Stackhouse, P. J. Flatau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climate feedback,” J. Atmos. Sci. 46, 1742–1753 (1990).
[CrossRef]

Goedecks, G. H.

Gooch, W.

Heymsfield, A. J.

A. J. Heymsfield, K. M. Miller, J. D. Sphinhirne, “The 27–28 October 1986 FIRE IFO cirrus case study: cloud microstructure,” Mon. Weather Rev. 118, 2313–2328 (1990).
[CrossRef]

R. T. Bruintjes, A. J. Heymsfield, T. W. Krauss, “An examination of double-plate ice crystals and initiation of precipitation in continental cumulus clouds,” J. Atmos. Sci. 44, 1331–1349 (1987).
[CrossRef]

A. J. Heymsfield, “Cirrus uncinus generating cells and the evolution of cirrusform clouds. Part I: Aircraft observations of the growth of the ice phase,” J. Atmos. Sci. 32, 799–808 (1975).
[CrossRef]

A. J. Heymsfield, “Laboratory and field observations of the growth of columnar and plate crystals from frozen droplets,” J. Atmos. Sci. 30, 1650–1656 (1973).
[CrossRef]

Jackson, J. D.

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

Kinne, S.

S. Kinne, K. N. Liou, “The effects of the nonsphericity and size distribution of ice crystals on the radiative properties of cirrus clouds,” Atmos. Res. 24, 273–284 (1989).
[CrossRef]

Klett, J. D.

Krauss, T. W.

R. T. Bruintjes, A. J. Heymsfield, T. W. Krauss, “An examination of double-plate ice crystals and initiation of precipitation in continental cumulus clouds,” J. Atmos. Sci. 44, 1331–1349 (1987).
[CrossRef]

Liou, K. N.

P. Yang, K. N. Liou, “Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 53, 6568–6584 (1996).
[CrossRef]

P. Yang, K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072–2085 (1996).
[CrossRef]

P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: comparison of finite-difference time domain and geometric optics models,” J. Opt. Soc. Am. A 12, 162–176 (1995).
[CrossRef]

K. N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climate implications,” Atmos. Res. 31, 271–298 (1994).
[CrossRef]

S. C. Ou, K. N. Liou, W. Gooch, Y. Takano, “Remote sensing of cirrus cloud parameters using advantaged very-high-resolution radiometer 3.7- and 10.9-μm channels,” Appl. Opt. 32, 2171–2180 (1993).
[CrossRef] [PubMed]

S. Kinne, K. N. Liou, “The effects of the nonsphericity and size distribution of ice crystals on the radiative properties of cirrus clouds,” Atmos. Res. 24, 273–284 (1989).
[CrossRef]

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I: Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 1–19 (1989).
[CrossRef]

K. N. Liou, “Influence of cirrus clouds on weather and climate process: a global perspective,” Mon. Weather Rev. 114, 1167–1199 (1986).
[CrossRef]

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

Macke, A.

Miller, K. M.

A. J. Heymsfield, K. M. Miller, J. D. Sphinhirne, “The 27–28 October 1986 FIRE IFO cirrus case study: cloud microstructure,” Mon. Weather Rev. 118, 2313–2328 (1990).
[CrossRef]

O’Brien, S. G.

Ono, A.

A. Ono, “The shape and riming properties of ice crystals in natural clouds,” J. Atmos. Sci. 26, 138–147 (1969).
[CrossRef]

Ou, S. C.

Pennypacker, C. P.

E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 196, 705–714 (1973).
[CrossRef]

Purcell, E. M.

E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 196, 705–714 (1973).
[CrossRef]

Sassen, K.

K. Sassen, “Air-truth lidar polarization studies of orographic clouds,” J. Appl. Meteorol. 17, 73–91 (1978).
[CrossRef]

Saxon, D. S.

D. S. Saxon, “Lectures on the scattering of light,” in Proceedings of the UCLA International Conference on Radiation and Remote Probing of the Atmosphere, J. G. Kuriyan, ed. (Western Periodicals, North Hollywood, Calif., 1973), pp. 227–308.

Schekunoff, S. A.

S. A. Schekunoff, Electromagnetic Waves (Van Nostrand, New York, 1943).

Shifrin, K. S.

K. S. Shifrin, “Scattering of light in a turbid medium,” (National Technical Information Service, Springfield, Va., 1968).

Sphinhirne, J. D.

A. J. Heymsfield, K. M. Miller, J. D. Sphinhirne, “The 27–28 October 1986 FIRE IFO cirrus case study: cloud microstructure,” Mon. Weather Rev. 118, 2313–2328 (1990).
[CrossRef]

Stackhouse, P. W.

G. L. Stephens, S. C. Tsay, P. W. Stackhouse, P. J. Flatau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climate feedback,” J. Atmos. Sci. 46, 1742–1753 (1990).
[CrossRef]

Stephens, G. L.

G. L. Stephens, S. C. Tsay, P. W. Stackhouse, P. J. Flatau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climate feedback,” J. Atmos. Sci. 46, 1742–1753 (1990).
[CrossRef]

G. L. Stephens, “Scattering of plane wave by soft obstacles: anomalous diffraction theory for circular cylinders,” Appl. Opt. 23, 954–959 (1984).
[CrossRef]

Sutherland, A.

Takano, Y.

K. N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climate implications,” Atmos. Res. 31, 271–298 (1994).
[CrossRef]

S. C. Ou, K. N. Liou, W. Gooch, Y. Takano, “Remote sensing of cirrus cloud parameters using advantaged very-high-resolution radiometer 3.7- and 10.9-μm channels,” Appl. Opt. 32, 2171–2180 (1993).
[CrossRef] [PubMed]

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I: Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 1–19 (1989).
[CrossRef]

Tsay, S. C.

G. L. Stephens, S. C. Tsay, P. W. Stackhouse, P. J. Flatau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climate feedback,” J. Atmos. Sci. 46, 1742–1753 (1990).
[CrossRef]

van de Hulst, H. C.

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

Yang, P.

Appl. Opt. (9)

P. Yang, K. N. Liou, “Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 53, 6568–6584 (1996).
[CrossRef]

C. Acquista, “Light scattering by tenuous particles: a generalization of the Rayleigh–Gans–Rocad approach,” Appl. Opt. 15, 2932–2936 (1976).
[CrossRef] [PubMed]

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

G. L. Stephens, “Scattering of plane wave by soft obstacles: anomalous diffraction theory for circular cylinders,” Appl. Opt. 23, 954–959 (1984).
[CrossRef]

G. H. Goedecks, S. G. O’Brien, “Scattering by irregular particles via the digitized Green’s function algorithm,” Appl. Opt. 27, 2431–2438 (1981).
[CrossRef]

T. W. Chen, “High energy light scattering in the generalized eikonal approximation,” Appl. Opt. 28, 4096–4102 (1989).
[CrossRef] [PubMed]

J. D. Klett, A. Sutherland, “Approximate methods for modeling the scattering properties of nonspherical particles: evaluation of the Wentzel–Kramers–Brillouin method,” Appl. Opt. 31, 373–386 (1992).
[CrossRef] [PubMed]

S. C. Ou, K. N. Liou, W. Gooch, Y. Takano, “Remote sensing of cirrus cloud parameters using advantaged very-high-resolution radiometer 3.7- and 10.9-μm channels,” Appl. Opt. 32, 2171–2180 (1993).
[CrossRef] [PubMed]

A. Macke, “Scattering of light by polyhedral ice crystals,” Appl. Opt. 32, 2780–2788 (1993).
[CrossRef] [PubMed]

Astrophys. J. (1)

E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 196, 705–714 (1973).
[CrossRef]

Atmos. Res. (2)

S. Kinne, K. N. Liou, “The effects of the nonsphericity and size distribution of ice crystals on the radiative properties of cirrus clouds,” Atmos. Res. 24, 273–284 (1989).
[CrossRef]

K. N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climate implications,” Atmos. Res. 31, 271–298 (1994).
[CrossRef]

J. Appl. Meteorol. (1)

K. Sassen, “Air-truth lidar polarization studies of orographic clouds,” J. Appl. Meteorol. 17, 73–91 (1978).
[CrossRef]

J. Atmos. Sci. (6)

R. T. Bruintjes, A. J. Heymsfield, T. W. Krauss, “An examination of double-plate ice crystals and initiation of precipitation in continental cumulus clouds,” J. Atmos. Sci. 44, 1331–1349 (1987).
[CrossRef]

G. L. Stephens, S. C. Tsay, P. W. Stackhouse, P. J. Flatau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climate feedback,” J. Atmos. Sci. 46, 1742–1753 (1990).
[CrossRef]

A. Ono, “The shape and riming properties of ice crystals in natural clouds,” J. Atmos. Sci. 26, 138–147 (1969).
[CrossRef]

A. J. Heymsfield, “Laboratory and field observations of the growth of columnar and plate crystals from frozen droplets,” J. Atmos. Sci. 30, 1650–1656 (1973).
[CrossRef]

A. J. Heymsfield, “Cirrus uncinus generating cells and the evolution of cirrusform clouds. Part I: Aircraft observations of the growth of the ice phase,” J. Atmos. Sci. 32, 799–808 (1975).
[CrossRef]

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I: Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 1–19 (1989).
[CrossRef]

J. Opt. Soc. Am. A (3)

Mon. Weather Rev. (2)

A. J. Heymsfield, K. M. Miller, J. D. Sphinhirne, “The 27–28 October 1986 FIRE IFO cirrus case study: cloud microstructure,” Mon. Weather Rev. 118, 2313–2328 (1990).
[CrossRef]

K. N. Liou, “Influence of cirrus clouds on weather and climate process: a global perspective,” Mon. Weather Rev. 114, 1167–1199 (1986).
[CrossRef]

Other (5)

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

D. S. Saxon, “Lectures on the scattering of light,” in Proceedings of the UCLA International Conference on Radiation and Remote Probing of the Atmosphere, J. G. Kuriyan, ed. (Western Periodicals, North Hollywood, Calif., 1973), pp. 227–308.

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

K. S. Shifrin, “Scattering of light in a turbid medium,” (National Technical Information Service, Springfield, Va., 1968).

S. A. Schekunoff, Electromagnetic Waves (Van Nostrand, New York, 1943).

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 (7)

Fig. 1
Fig. 1

(a) Geometry for the ray-by-ray integration (RBRI) algorithm for light scattering by a hexagonal ice crystal; (b) segment of the ray path within the ice crystal, in which a circular cross section has been assumed for the ray path.

Fig. 2
Fig. 2

Comparisons of the extinction efficiency and the single-scattering albedo computed by FDTD, RBRI, ADA, GOM2, and GOM1 at the 0.55-μm and 3.7-μm wavelengths.

Fig. 3
Fig. 3

Phase functions computed by FDTD, RBRI, GOM2, and GOM1 for hexagonal columns and plates. Also shown are the relative errors of the RBRI and GOM1 results in comparison with the reference results computed by FDTD. 100% has been added to the error values to avoid negative values in the logarithmic display.

Fig. 4
Fig. 4

Comparisons of the phase matrix elements computed by FDTD and RBRI for hexagonal columns.

Fig. 5
Fig. 5

Comparisons of the phase function computed by RBRI (GOM2) and GOM1 for size parameters kL=50 (L/a=6) and ka =50 (L/a=0.5). Also shown are the differences between RBRI (GOM2) and GOM1. 200% has been added to the error values so that negative values can be avoided in the logarithmic display.

Fig. 6
Fig. 6

Comparisons of the phase function computed by GOM1 and RBRI for infinitely long hexagonal cylinders randomly oriented around their axes with normal incidence.  

Fig. 7
Fig. 7

Geometry of the incident rays with respect to a hexagonal ice crystal.

Equations (50)

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

Es(r)|kr=k2 exp(ikr)4πr (-1)v {E(r)-rˆ[rˆE(r)]}exp(-ikrˆr)d3r,
σe=Imk|E0|2 (-1)vE(r)E0*(r)d3r,
σa=k|E0|2 ivE(r)E*(r)d3r,
e^1=Nr-1{e^0-(e^0n^1)n^1-[Nr2-1+(e^0n^1)2]1/2n^1},
e^p=e^p-1-2(e^p-1n^p)n^p,p=2, 3, 4, ,
β^p=(n^p×e^p)[1-(e^pn^p)2]-1/2,p=1, 2, 3, ,
α^p=e^p×β^p,p=1, 2, 3,  .
Ep(r)=α^pEp,α(r)+β^pEp,β(r).
Ep,α(r)Ep,β(r)=UpA exp[iδp(r)],rraypath0,rraypath
δp(r)=ke^0rQ1+Nj=1p-1dj+Ne^p(r-rQp),
dj=|rQj+1-rQj|,
rˆ=ξˆ×ηˆ.
Eps(r)=ηˆEp,ηs(r)+ξˆEp,ξs(r),p=1, 2, 3,  .
Ep,ηs(r)Ep,ξs(r)=exp(ikr)-ikr Sp(rˆ)As,
Sp(rˆ)=ik3(1-)4π qp(rˆ)KpUpΓ,
qp(rˆ)=v exp{i[δp(r)-krˆr]}d3r,
Kp=ηˆα^pηˆβ^pξˆα^pξˆβ^p,
r=rQp+ee^p+uu^p+vv^p,rv,
qp(rˆ)=expike^0rQ1+N j=1p-1 dj-rˆrQp×0dpexp[ik(N-rˆe^p)e]de ×raycrosssection exp[-ikrˆ(uu^p+vv^p)]dudv.
qp(rˆ)=Δσ˜p{exp[iζp+1(rˆ)]-exp[iζp(rˆ)]}/[ik(N-rˆe^p)],
ζp(rˆ)=ke^0rQ1+N j=1p-1 dj-rˆrQp
Δσ˜p=raycrosssection exp[-ikrˆ(uu^p+vv^p)]dudv=Δσp2J1(χ)/χ,
χ=k(Δσp/π)1/2 sin[cos-1(e^prˆ)].
Sp(rˆ)=k24π Δσ˜pN-rˆe^p (1-)KpUpΓ×{exp[iζp+1(rˆ)]-exp[iζp(rˆ)]}.
S(rˆ)=γp=1 Sp(rˆ),
S(rˆ)=γp=1 Sp(rˆ)=p=1 S˜p(rˆ),
S˜1(rˆ)=-k24π (1-) γ Δσ˜1N-rˆe^1 K1U1Γ exp(iζ1),
S˜p(rˆ)=k24π (1-) γΔσ˜p-1Kp-1Up-1N-rˆe^p-1-Δσ˜pKpUpN-rˆe^pΓ exp(iζp),p=2, 3, 4,  .
σe=2πk2 Re[S11(e^0)+S22(e^0)],
σa=12 γp=2 ΔσpNr exp-2kNi j=1p-1 dj×[1-exp(-2kNidp,γ)]×(ΛUp*UpTΛT+ΠUp*UpTΠT),
Λ=(10),Π=(01),
σe=2 P [1-exp(-τ)cos ρ]d2p,
τ=kdmi,ρ=kd(mr-1),
σa=P [1-exp(-2τ)]d2p.
e^1e^0,
K1U1Γ1001.
exp[iζ2(e^0)]-exp[iζ1(e^0)]exp[ikd1(N-1)]-1,
σe=Re-γ Δσ˜1(1-)N-1 {exp[ikd1(N-1)]-1}.
1-N-1=(1-m)(1+m)N-1-(1+m)-2.
Δσ˜1=Δσ1 limχ0 [2J1(χ)/χ]=Δσ1.
σe=2 γ Δσ1{1-cos[kd1(Nr-1)]exp(-kd1Ni)}.
σa=γ Δσ1[1-exp(-2kNid1)].
[P11(RBRI/GOM1)-P11(FDTD)]/P11(FDTD).
[P11(GOM1)-P11(RBRI/GOM2)]/P11(GOM1).
S(rˆ)S˜1(rˆ)=-k24π (1-)γ Δσ˜1N-rˆe^1 ×K1U1Γ exp(iζ1).
S˜1(rˆ)=-k24π (1-)jK1U1ΓN-rˆe^1jfjDj,
fj=[1-(e^1n^1)2]j1/2,
Dj=#jface exp[ik(e^0-rˆ)r]d2r,
Dj=aL exp[ik(e^0-rˆ)di] sin[k(e^0-rˆ)r4/2]k(e^0-rˆ)r4/2 ×sin[k(e^0-rˆ)ci/2]k(e^0-rˆ)ci/2,j=1, 2, 3,
D4=32 a2exp[ik(e^0-rˆ)(r4+r1+r2)/2] ×sin[k(e^0-rˆ)r1/2]k(e^0-rˆ)r1/2 sin[k(e^0-rˆ)r2/2]k(e^0-rˆ)r2/2+exp[ik(e^0-rˆ)(r4+r2+r3)/2] ×sin[k(e^0-rˆ)r2/2]k(e^0-rˆ)r2/2 sin[k(e^0-rˆ)r3/2]k(e^0-rˆ)r3/2+exp[ik(e^0-rˆ)(r4+r3+r1)/2] ×sin[k(e^0-rˆ)r3/2]k(e^0-rˆ)r3/2 sin[k(e^0-rˆ)r1/2]k(e^0-rˆ)r1/2,

Metrics