Abstract

A general method is developed to formulate extinction and absorption efficiency for nonspherical particles at arbitrary and random orientations by use of anomalous diffraction theory (ADT). An ADT for finite circular cylinders is evaluated as an example. Existing ADT’s for infinite cylinders at arbitrary orientations and for finite cylinders at the normal incidence are shown to be special cases of the new formulation. ADT solutions for finite cylinders are shown to approach the rigorous T-matrix results when the refractive indices approach unity. The importance of some physical processes that are neglected in the ADT approximation are evaluated by comparisons between ADT and rigorous calculations for different particle geometries. For spheres, van de Hulst’s ADT and Mie theory are used, whereas the ADT that we present and T-matrix calculations are used for cylinders of different diameter-to-length ratios. The results show that the differences in extinction between ADT and exact solutions generally decrease with nonsphericity. A similar decrease occurs for absorption at wavelengths of relatively strong absorption. The influence of complex refractive index is evaluated. Our results suggest that ADT may provide a useful approximation in parameterization and remote sensing of cirrus clouds in the Christiansen bands where the real part of the refractive index approaches unity and/or where relative absorption is strong.

© 1998 Optical Society of America

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    [CrossRef]
  2. D. L. Mitchell, A. Macke, Y. Liu, “Modeling cirrus clouds. Part II: Treatment of radiative properties,” J. Atmos. Sci. 53, 2967–2988 (1996).
    [CrossRef]
  3. Q. Fu, K. N. Liou, “Parameterization of the radiative properties of cirrus clouds,” J. Atmos. Sci. 50, 2008–2025 (1993).
    [CrossRef]
  4. W. P. Arnott, C. Schmitt, Y. Liu, J. Hallett, “Droplet size spectra and water-vapor concentration of laboratory water clouds: inversion of Fourier transform infrared (500–5000 cm-1) optical-depth measurement,” Appl. Opt. 36, 5205–5216 (1997).
    [CrossRef] [PubMed]
  5. M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
    [CrossRef]
  6. P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).
  7. P. W. Barber, H. Massoudi, “Recent advances in light scattering calculations for nonspherical particles,” Aerosol Sci. Technol. 1, 303–315 (1982).
    [CrossRef]
  8. M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. D102, 16831–16847 (1997).
    [CrossRef]
  9. 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]
  10. P. Yang, K. N. Liou, W. P. Arnott, “Extinction efficiency and single-scattering albedo for laboratory and natural cirrus clouds,” J. Geophys. Res. D102, 21825–21835 (1997).
    [CrossRef]
  11. A. J. Baran, J. S. Foot, D. L. Mitchell, “Ice crystal absorption: a comparison between theory and implications for remote sensing,” Appl. Opt. 37, 2207–2215 (1998).
    [CrossRef]
  12. W. P. Arnott, Y. Liu, J. Hallett, “Unreasonable effectiveness of mimicking measured infrared extinction by hexagonal ice crystals with Mie ice spheres,” in Optical Remote Sensing of the Atmosphere, Vol. 5 of 1997 Technical Digest Series (Optical Society of America, Washington, D.C., 1997), 216–218.
  13. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  14. G. L. Stephens, “Scattering of plane waves by soft obstacles: anomalous diffraction theory for circular cylinders,” Appl. Opt. 23, 954–959 (1984).
    [CrossRef] [PubMed]
  15. S. A. Ackerman, G. L. Stephens, “The absorption of solar radiation by cloud droplets: an application of anomalous diffraction theory,” J. Atmos. Sci. 44, 1574–1588 (1987).
    [CrossRef]
  16. G. R. Fournier, B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30, 2042–2048 (1991).
    [CrossRef] [PubMed]
  17. B. T. N. Evans, G. R. Fournier, “Analytic approximation to randomly oriented spheroid extinction,” Appl. Opt. 33, 5796–5805 (1994).
    [CrossRef] [PubMed]
  18. J. M. Breenberg, A. S. Meltzer, “The effects of orientation of non-spherical particles on interstellar extinction,” Astrophys. J. 132, 667–671 (1960).
    [CrossRef]
  19. F. D. Bryant, P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci. 30, 291–304 (1969).
    [CrossRef]
  20. D. A. Cross, P. Latimer, “General solutions for the extinction and absorption efficiencies of arbitrarily oriented cylinders by anomalous-diffraction methods,” J. Opt. Soc. Am. 60, 904–907 (1970).
    [CrossRef]
  21. P. Chylek, J. D. Klett, “Extinction cross sections of nonspherical particles in the anomalous diffraction approximation,” J. Opt. Soc. Am. A 8, 274–281 (1991).
    [CrossRef]
  22. 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]
  23. A. Maslowska, P. J. Flatau, G. L. Stephens, “Scattering of light by cubes: anomalous diffraction and discrete dipole approximations,” IRS’ 92 Current Problems in Atmospheric Radiation, S. Keevallik, O. Kärner, eds. (Deepak, Hampton, Va., 1992), 533–535.
  24. P. Chylek, G. Videen, “Longwave radiative properties of polydispersed hexagonal ice crystals,” J. Atmos. Sci. 51, 175–190 (1994).
    [CrossRef]
  25. S. Asano, M. Sato, “Light scattering by randomly oriented spherical particles,” Appl. Opt. 19, 962–974 (1980).
    [CrossRef] [PubMed]
  26. A. Bowyer, J. Woodwark, A Programmer’s Geometry (Butterworth, London, 1983).
  27. R. L. Burden, J. D. Faires, Numerical Analysis (PWS-Kert, Boston, Mass., 1993).
  28. M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
    [CrossRef] [PubMed]
  29. R. D. Haracz, L. D. Cohen, A. Cohen, “Scattering of linearly polarized light from randomly oriented cylinders and spheroids,” J. Appl. Phys. 58, 3322–3327 (1985).
    [CrossRef]
  30. P. C. Chylek, G. W. Grams, R. G. Pinnick, “Light scattering by irregular randomly oriented particles,” Science 193, 480–482 (1976).
    [CrossRef]
  31. S. Warren, “Optical constants of ice from the ultraviolet to the microwave,” Appl. Opt. 23, 1206–1225 (1984).
    [CrossRef] [PubMed]
  32. S. C. Hill, B. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, P. K. Change, eds. (World Scientific, Singapore, 1988).
  33. L. G. Guimaraes, H. M. Nussenzveig, “Theory of Mie resonances and ripple fluctuations,” Opt. Commun. 89, 363–369 (1992).
    [CrossRef]
  34. S. A. Ackerman, “Remote sensing aerosols using satellite infrared observations,” J. Geophys. Res. D 102, 17069–17079 (1997).
    [CrossRef]
  35. H. M. Steele, R. P. Turco, “Retrieval of aerosol size distributions from satellite extinction spectra using constrained linear inversion,” J. Geophys. Res. D 102, 16737–16747 (1997).
    [CrossRef]
  36. W. P. Arnott, Y. Dong, J. Hallett, “Extinction efficiency in the infrared (2–18 μm) of laboratory ice clouds: observations of scattering minima in the Christiansen bands of ice,” Appl. Opt. 34, 541–551 (1995).
    [CrossRef] [PubMed]

1998 (1)

1997 (5)

W. P. Arnott, C. Schmitt, Y. Liu, J. Hallett, “Droplet size spectra and water-vapor concentration of laboratory water clouds: inversion of Fourier transform infrared (500–5000 cm-1) optical-depth measurement,” Appl. Opt. 36, 5205–5216 (1997).
[CrossRef] [PubMed]

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. D102, 16831–16847 (1997).
[CrossRef]

P. Yang, K. N. Liou, W. P. Arnott, “Extinction efficiency and single-scattering albedo for laboratory and natural cirrus clouds,” J. Geophys. Res. D102, 21825–21835 (1997).
[CrossRef]

S. A. Ackerman, “Remote sensing aerosols using satellite infrared observations,” J. Geophys. Res. D 102, 17069–17079 (1997).
[CrossRef]

H. M. Steele, R. P. Turco, “Retrieval of aerosol size distributions from satellite extinction spectra using constrained linear inversion,” J. Geophys. Res. D 102, 16737–16747 (1997).
[CrossRef]

1996 (3)

D. L. Mitchell, A. Macke, Y. Liu, “Modeling cirrus clouds. Part II: Treatment of radiative properties,” J. Atmos. Sci. 53, 2967–2988 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

1995 (1)

1994 (3)

B. T. N. Evans, G. R. Fournier, “Analytic approximation to randomly oriented spheroid extinction,” Appl. Opt. 33, 5796–5805 (1994).
[CrossRef] [PubMed]

D. L. Mitchell, W. P. Arnott, “A model predicting the evolution of ice particle size spectra and radiative properties of cirrus clouds. Part II: Dependence of absorption and extinction on ice crystal morphology,” J. Atmos. Sci. 51, 817–832 (1994).
[CrossRef]

P. Chylek, G. Videen, “Longwave radiative properties of polydispersed hexagonal ice crystals,” J. Atmos. Sci. 51, 175–190 (1994).
[CrossRef]

1993 (1)

Q. Fu, K. N. Liou, “Parameterization of the radiative properties of cirrus clouds,” J. Atmos. Sci. 50, 2008–2025 (1993).
[CrossRef]

1992 (1)

L. G. Guimaraes, H. M. Nussenzveig, “Theory of Mie resonances and ripple fluctuations,” Opt. Commun. 89, 363–369 (1992).
[CrossRef]

1991 (3)

1989 (1)

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]

1987 (1)

S. A. Ackerman, G. L. Stephens, “The absorption of solar radiation by cloud droplets: an application of anomalous diffraction theory,” J. Atmos. Sci. 44, 1574–1588 (1987).
[CrossRef]

1985 (1)

R. D. Haracz, L. D. Cohen, A. Cohen, “Scattering of linearly polarized light from randomly oriented cylinders and spheroids,” J. Appl. Phys. 58, 3322–3327 (1985).
[CrossRef]

1984 (2)

1982 (1)

P. W. Barber, H. Massoudi, “Recent advances in light scattering calculations for nonspherical particles,” Aerosol Sci. Technol. 1, 303–315 (1982).
[CrossRef]

1980 (1)

1976 (1)

P. C. Chylek, G. W. Grams, R. G. Pinnick, “Light scattering by irregular randomly oriented particles,” Science 193, 480–482 (1976).
[CrossRef]

1970 (1)

1969 (1)

F. D. Bryant, P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci. 30, 291–304 (1969).
[CrossRef]

1960 (1)

J. M. Breenberg, A. S. Meltzer, “The effects of orientation of non-spherical particles on interstellar extinction,” Astrophys. J. 132, 667–671 (1960).
[CrossRef]

Ackerman, S. A.

S. A. Ackerman, “Remote sensing aerosols using satellite infrared observations,” J. Geophys. Res. D 102, 17069–17079 (1997).
[CrossRef]

S. A. Ackerman, G. L. Stephens, “The absorption of solar radiation by cloud droplets: an application of anomalous diffraction theory,” J. Atmos. Sci. 44, 1574–1588 (1987).
[CrossRef]

Arnott, W. P.

P. Yang, K. N. Liou, W. P. Arnott, “Extinction efficiency and single-scattering albedo for laboratory and natural cirrus clouds,” J. Geophys. Res. D102, 21825–21835 (1997).
[CrossRef]

W. P. Arnott, C. Schmitt, Y. Liu, J. Hallett, “Droplet size spectra and water-vapor concentration of laboratory water clouds: inversion of Fourier transform infrared (500–5000 cm-1) optical-depth measurement,” Appl. Opt. 36, 5205–5216 (1997).
[CrossRef] [PubMed]

W. P. Arnott, Y. Dong, J. Hallett, “Extinction efficiency in the infrared (2–18 μm) of laboratory ice clouds: observations of scattering minima in the Christiansen bands of ice,” Appl. Opt. 34, 541–551 (1995).
[CrossRef] [PubMed]

D. L. Mitchell, W. P. Arnott, “A model predicting the evolution of ice particle size spectra and radiative properties of cirrus clouds. Part II: Dependence of absorption and extinction on ice crystal morphology,” J. Atmos. Sci. 51, 817–832 (1994).
[CrossRef]

W. P. Arnott, Y. Liu, J. Hallett, “Unreasonable effectiveness of mimicking measured infrared extinction by hexagonal ice crystals with Mie ice spheres,” in Optical Remote Sensing of the Atmosphere, Vol. 5 of 1997 Technical Digest Series (Optical Society of America, Washington, D.C., 1997), 216–218.

Asano, S.

Baran, A. J.

Barber, P. W.

P. W. Barber, H. Massoudi, “Recent advances in light scattering calculations for nonspherical particles,” Aerosol Sci. Technol. 1, 303–315 (1982).
[CrossRef]

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

Benner, B. E.

S. C. Hill, B. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, P. K. Change, eds. (World Scientific, Singapore, 1988).

Bowyer, A.

A. Bowyer, J. Woodwark, A Programmer’s Geometry (Butterworth, London, 1983).

Breenberg, J. M.

J. M. Breenberg, A. S. Meltzer, “The effects of orientation of non-spherical particles on interstellar extinction,” Astrophys. J. 132, 667–671 (1960).
[CrossRef]

Bryant, F. D.

F. D. Bryant, P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci. 30, 291–304 (1969).
[CrossRef]

Burden, R. L.

R. L. Burden, J. D. Faires, Numerical Analysis (PWS-Kert, Boston, Mass., 1993).

Chylek, P.

Chylek, P. C.

P. C. Chylek, G. W. Grams, R. G. Pinnick, “Light scattering by irregular randomly oriented particles,” Science 193, 480–482 (1976).
[CrossRef]

Cohen, A.

R. D. Haracz, L. D. Cohen, A. Cohen, “Scattering of linearly polarized light from randomly oriented cylinders and spheroids,” J. Appl. Phys. 58, 3322–3327 (1985).
[CrossRef]

Cohen, L. D.

R. D. Haracz, L. D. Cohen, A. Cohen, “Scattering of linearly polarized light from randomly oriented cylinders and spheroids,” J. Appl. Phys. 58, 3322–3327 (1985).
[CrossRef]

Cross, D. A.

Dong, Y.

Evans, B. T. N.

Faires, J. D.

R. L. Burden, J. D. Faires, Numerical Analysis (PWS-Kert, Boston, Mass., 1993).

Flatau, P. J.

A. Maslowska, P. J. Flatau, G. L. Stephens, “Scattering of light by cubes: anomalous diffraction and discrete dipole approximations,” IRS’ 92 Current Problems in Atmospheric Radiation, S. Keevallik, O. Kärner, eds. (Deepak, Hampton, Va., 1992), 533–535.

Foot, J. S.

Fournier, G. R.

Fu, Q.

Q. Fu, K. N. Liou, “Parameterization of the radiative properties of cirrus clouds,” J. Atmos. Sci. 50, 2008–2025 (1993).
[CrossRef]

Grams, G. W.

P. C. Chylek, G. W. Grams, R. G. Pinnick, “Light scattering by irregular randomly oriented particles,” Science 193, 480–482 (1976).
[CrossRef]

Guimaraes, L. G.

L. G. Guimaraes, H. M. Nussenzveig, “Theory of Mie resonances and ripple fluctuations,” Opt. Commun. 89, 363–369 (1992).
[CrossRef]

Hallett, J.

Haracz, R. D.

R. D. Haracz, L. D. Cohen, A. Cohen, “Scattering of linearly polarized light from randomly oriented cylinders and spheroids,” J. Appl. Phys. 58, 3322–3327 (1985).
[CrossRef]

Hill, S. C.

S. C. Hill, B. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, P. K. Change, eds. (World Scientific, Singapore, 1988).

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

Kahn, R. A.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. D102, 16831–16847 (1997).
[CrossRef]

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.

Latimer, P.

D. A. Cross, P. Latimer, “General solutions for the extinction and absorption efficiencies of arbitrarily oriented cylinders by anomalous-diffraction methods,” J. Opt. Soc. Am. 60, 904–907 (1970).
[CrossRef]

F. D. Bryant, P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci. 30, 291–304 (1969).
[CrossRef]

Liou, K. N.

P. Yang, K. N. Liou, W. P. Arnott, “Extinction efficiency and single-scattering albedo for laboratory and natural cirrus clouds,” J. Geophys. Res. D102, 21825–21835 (1997).
[CrossRef]

Q. Fu, K. N. Liou, “Parameterization of the radiative properties of cirrus clouds,” J. Atmos. Sci. 50, 2008–2025 (1993).
[CrossRef]

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]

Liu, Y.

W. P. Arnott, C. Schmitt, Y. Liu, J. Hallett, “Droplet size spectra and water-vapor concentration of laboratory water clouds: inversion of Fourier transform infrared (500–5000 cm-1) optical-depth measurement,” Appl. Opt. 36, 5205–5216 (1997).
[CrossRef] [PubMed]

D. L. Mitchell, A. Macke, Y. Liu, “Modeling cirrus clouds. Part II: Treatment of radiative properties,” J. Atmos. Sci. 53, 2967–2988 (1996).
[CrossRef]

W. P. Arnott, Y. Liu, J. Hallett, “Unreasonable effectiveness of mimicking measured infrared extinction by hexagonal ice crystals with Mie ice spheres,” in Optical Remote Sensing of the Atmosphere, Vol. 5 of 1997 Technical Digest Series (Optical Society of America, Washington, D.C., 1997), 216–218.

Macke, A.

D. L. Mitchell, A. Macke, Y. Liu, “Modeling cirrus clouds. Part II: Treatment of radiative properties,” J. Atmos. Sci. 53, 2967–2988 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

Mackowski, D. W.

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

Maslowska, A.

A. Maslowska, P. J. Flatau, G. L. Stephens, “Scattering of light by cubes: anomalous diffraction and discrete dipole approximations,” IRS’ 92 Current Problems in Atmospheric Radiation, S. Keevallik, O. Kärner, eds. (Deepak, Hampton, Va., 1992), 533–535.

Massoudi, H.

P. W. Barber, H. Massoudi, “Recent advances in light scattering calculations for nonspherical particles,” Aerosol Sci. Technol. 1, 303–315 (1982).
[CrossRef]

Meltzer, A. S.

J. M. Breenberg, A. S. Meltzer, “The effects of orientation of non-spherical particles on interstellar extinction,” Astrophys. J. 132, 667–671 (1960).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. D102, 16831–16847 (1997).
[CrossRef]

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

Mitchell, D. L.

A. J. Baran, J. S. Foot, D. L. Mitchell, “Ice crystal absorption: a comparison between theory and implications for remote sensing,” Appl. Opt. 37, 2207–2215 (1998).
[CrossRef]

D. L. Mitchell, A. Macke, Y. Liu, “Modeling cirrus clouds. Part II: Treatment of radiative properties,” J. Atmos. Sci. 53, 2967–2988 (1996).
[CrossRef]

D. L. Mitchell, W. P. Arnott, “A model predicting the evolution of ice particle size spectra and radiative properties of cirrus clouds. Part II: Dependence of absorption and extinction on ice crystal morphology,” J. Atmos. Sci. 51, 817–832 (1994).
[CrossRef]

Nussenzveig, H. M.

L. G. Guimaraes, H. M. Nussenzveig, “Theory of Mie resonances and ripple fluctuations,” Opt. Commun. 89, 363–369 (1992).
[CrossRef]

Pinnick, R. G.

P. C. Chylek, G. W. Grams, R. G. Pinnick, “Light scattering by irregular randomly oriented particles,” Science 193, 480–482 (1976).
[CrossRef]

Sato, M.

Schmitt, C.

Steele, H. M.

H. M. Steele, R. P. Turco, “Retrieval of aerosol size distributions from satellite extinction spectra using constrained linear inversion,” J. Geophys. Res. D 102, 16737–16747 (1997).
[CrossRef]

Stephens, G. L.

S. A. Ackerman, G. L. Stephens, “The absorption of solar radiation by cloud droplets: an application of anomalous diffraction theory,” J. Atmos. Sci. 44, 1574–1588 (1987).
[CrossRef]

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

A. Maslowska, P. J. Flatau, G. L. Stephens, “Scattering of light by cubes: anomalous diffraction and discrete dipole approximations,” IRS’ 92 Current Problems in Atmospheric Radiation, S. Keevallik, O. Kärner, eds. (Deepak, Hampton, Va., 1992), 533–535.

Travis, L. D.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. D102, 16831–16847 (1997).
[CrossRef]

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

Turco, R. P.

H. M. Steele, R. P. Turco, “Retrieval of aerosol size distributions from satellite extinction spectra using constrained linear inversion,” J. Geophys. Res. D 102, 16737–16747 (1997).
[CrossRef]

van de Hulst, H. C.

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

Videen, G.

P. Chylek, G. Videen, “Longwave radiative properties of polydispersed hexagonal ice crystals,” J. Atmos. Sci. 51, 175–190 (1994).
[CrossRef]

Warren, S.

West, R. A.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. D102, 16831–16847 (1997).
[CrossRef]

Woodwark, J.

A. Bowyer, J. Woodwark, A Programmer’s Geometry (Butterworth, London, 1983).

Yang, P.

P. Yang, K. N. Liou, W. P. Arnott, “Extinction efficiency and single-scattering albedo for laboratory and natural cirrus clouds,” J. Geophys. Res. D102, 21825–21835 (1997).
[CrossRef]

Aerosol Sci. Technol. (1)

P. W. Barber, H. Massoudi, “Recent advances in light scattering calculations for nonspherical particles,” Aerosol Sci. Technol. 1, 303–315 (1982).
[CrossRef]

Appl. Opt. (9)

S. Asano, M. Sato, “Light scattering by randomly oriented spherical particles,” Appl. Opt. 19, 962–974 (1980).
[CrossRef] [PubMed]

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

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

G. R. Fournier, B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30, 2042–2048 (1991).
[CrossRef] [PubMed]

B. T. N. Evans, G. R. Fournier, “Analytic approximation to randomly oriented spheroid extinction,” Appl. Opt. 33, 5796–5805 (1994).
[CrossRef] [PubMed]

W. P. Arnott, C. Schmitt, Y. Liu, J. Hallett, “Droplet size spectra and water-vapor concentration of laboratory water clouds: inversion of Fourier transform infrared (500–5000 cm-1) optical-depth measurement,” Appl. Opt. 36, 5205–5216 (1997).
[CrossRef] [PubMed]

A. J. Baran, J. S. Foot, D. L. Mitchell, “Ice crystal absorption: a comparison between theory and implications for remote sensing,” Appl. Opt. 37, 2207–2215 (1998).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Arbitrarily shaped particle and light rays characterized by the direction n = (cos α, cos β, cos γ).

Fig. 2
Fig. 2

Finite circular cylinder and light rays characterized by the direction n = (sin γ, 0, cos γ).

Fig. 3
Fig. 3

ADT extinction and absorption efficiencies for randomly oriented cylinders having a variety of aspect ratios. Note that efficiencies approach those of infinite cylinders as the D/L ratio decreases. The curves for the 0.01 and 0.001 D/L ratios are almost the same as the infinite cylinder results. All the curves were computed by use of the new ADT.

Fig. 4
Fig. 4

ADT efficiencies for a randomly oriented cylinder. Note that cylinder ADT approaches T-matrix calculations when the real part of the refractive index is close to 1.0. The D/L ratio is 1.0.

Fig. 5
Fig. 5

Same as Fig. 4, except that the D/L ratio is 2.5.

Fig. 6
Fig. 6

Efficiencies calculated from cylinder ADT (solid curve), volume-equivalent spheres (dotted curve), area-equivalent spheres (dashed–dotted curve), and the T-matrix method (heavy solid curve). The D/L ratio is 1.0, the wavelength is 2 μm, and the refractive index is 1.274 - 0.001588i. Mie theory was used for the spheres.

Fig. 7
Fig. 7

Real refractive index and absorption angle for ice as a function of wavelength.

Fig. 8
Fig. 8

Error of ADT approximations. Mie theory and van de Hulst’s ADT for spheres were applied to spheres. The T-matrix method and the new ADT were used for cylinders.

Fig. 9
Fig. 9

Same as Fig. 8, except that the wavelength is 12.5 μm.

Fig. 10
Fig. 10

Same as Fig. 8, except that the wavelength is 2.2201 μm. Note that results of a size parameter less than 2, where ADT tends to overestimate the efficiencies, are not shown to emphasize the ripple structure.

Fig. 11
Fig. 11

Relative change of ADT error reduction when particle shapes change from spheres to compact cylinders with an aspect ratio of 1: (a) extinction, (b) absorption. See the text for the meaning of Δ a and Δ e .

Equations (62)

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Φ = 2 π d λ m - 1 .
C e = 2     Re 1 - exp - i Φ dP ,
C a =   1 - exp - 4 π m i d λ dP
n = n x ,   n y ,   n z ,
f 1 x ,   y ,   z = 0 .
x - x 1 = n x t ,
y - y 1 = n y t ,
z - z 1 = n z t ,
f 2 x ,   y ,   z = 0 ,
d = x 2 - x 1 2 + y 2 - y 1 2 + z 2 - z 1 2 1 / 2 .
d = t n x 2 + n y 2 + n z 2 1 / 2 = t ,
n s = f 1 p 1 f 1 p 1 ,
dP = n · d s = n · n s ds .
x 2 + y 2 = r 2 ,
z = 0 .
x 2 + y 2 = r 2 ,
z = L ,
x 2 + y 2 = R 2 ,
0 z L ,
d 1 1 = - n x x + n y y ± n x 2 + n y 2 R 2 - r 2 + n y x + n x y 2 1 / 2 n x 2 + n y 2 ,
d 1 1 = n x 2 + n y 2 R 2 - r 2 + n y x + n x y 2 1 / 2 - n x x + n y y n x 2 + n y 2 .
d 1 1 = r 2 n x cos   ϕ + n y sin   ϕ 2 + n x 2 + n y 2 R 2 - r 2 1 / 2 - r n x cos   ϕ + n y sin   ϕ n x 2 + n y 2 .
n y = 0 ,
n x = sin   γ ,
n z = cos   γ ,
d 1 1 = R 2 - r 2 sin 2   ϕ 1 / 2 - r   cos   ϕ sin   γ .
dP = n · n s ds = r   cos   ϕ d r d ϕ .
Ce 1 1 = 2   cos   γ   A 1 1 - exp - ik m - 1 d 1 1 · r d r d ϕ ,
d 1 2 = L n z = L cos   γ .
C e 1 2 = 2   cos   γ   A 2 1 - exp - ik m - 1 d 1 2 r d r d ϕ ,
d 2 1 = - 2 n x x + n y y n x 2 + n y 2 ,
d 2 1 = - 2 R n x cos   ϕ + n y sin   ϕ n x 2 + n y 2 .
d 2 1 = 2 R   cos   ϕ sin   γ .
- π 2 ϕ π 2 .
n s = x 1 R ,   y 1 R ,   0 .
dP = - n · n s ds = R   sin   γ   cos   ϕ d ϕ d z .
C e 2 1 = 2 R   sin   γ   A 3 1 - exp - ik m - 1 d 2 1 × cos   ϕ d z d ϕ ,
d 2 2 = L - z cos   γ .
C e 2 2 = 2 R   sin   γ   A 4 1 - exp - ik m - 1 d 2 2 × cos   ϕ d z d ϕ ,
C e 1 = j = 1 2   C e 1 j = 2   cos   γ   0 2 π d ϕ   0 R d r 1 - exp - ik m - 1 d 1 r ,
d 1 = d 1 1 ,   if   d 1 1 cos   γ L d 1 2 ,   if   d 1 1 cos   γ > L .
C e 2 = j = 1 2   C e 2 j = 2 R   sin   γ   0 L d z   - π / 2 π / 2 d ϕ × 1 - exp - ik m - 1 d 2 cos   ϕ ,
d 2 = d 2 1 ,   if   z + d 2 1 cos   γ L d 2 2 ,   if   z + d 2 1 cos   γ > L .
C e = j = 1 2   C ej .
C ¯ = 0 π / 2   C γ sin   γ d γ ,
P = π 2   RL + π 2   R 2 = π R 2 2 + ε 2 ε ,
ε = 2 R / L .
Q = C ¯ / P .
n z = cos   γ = 0 .
d 2 = d 2 1
C e = 4 RL Re 0 π / 2 1 - exp - i 2 Rk m - 1 × cos   ϕ cos   ϕ d ϕ .
Q e = 2 Re 0 π / 2 1 - exp - i ρ *   cos   ϕ cos   ϕ d ϕ ,
ρ * = 2 kR m - 1 .
C e = 4 RL   sin   γ Re 0 π / 2 1 - exp - i ρ *   cos   ϕ sin   γ × cos   ϕ d ϕ .
Q e γ = 2   Re 0 π / 2 1 - exp - i ρ *   cos   ϕ sin   γ cos   ϕ d ϕ .
b = Q 1 - 1 a ,
c = Q 2 b ,
c = Ta ,
T = Q 2 Q 1 - 1 ,
ζ = tan - 1 m i m r - 1 .
δ = 100   Q exact - Q ADT Q exact ,
Δ = 100   δ Mie - δ ADT δ Mie .

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