Abstract

A Mie backscattering model for spherical particles with off-center inclusion has been developed and tested. The program is capable of dealing with size parameter values up to ∼1000, thus allowing one to simulate the optical behavior of a large variety of atmospheric aerosols, as well as cloud and precipitation particles. On the basis of this model, we simulated the optical properties of polydisperse composite atmospheric particles as observed by ground-based and airborne lidar systems. We have characterized optical properties in terms of host and inclusion radii, considering water particles with different composition inclusions. The performed modeling provides some insight into the so-called lidar bright- and dark-band phenomenon.

© 2004 Optical Society of America

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References

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  1. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chaps. 3 and 4.
  2. K. Sassen, “Contrail-cirrus and their potential for regional climate change,” Bull. Am. Meteorol. Soc. 78, 1885–1903 (1997).
    [CrossRef]
  3. P. Di Girolamo, B. B. Demoz, D. N. Whiteman, “Model simulations of melting hydrometeors: a new lidar bright band from melting frozen drops,” Geophys. Res. Lett. 30, 1626, doi: 10.1029/2002GL016825 (2003).
    [CrossRef]
  4. K. Sassen, T. Chen, “The lidar dark band: an oddity of the radar bright band,” Geophys. Res. Lett. 22, 3505–3508 (1995).
    [CrossRef]
  5. Q. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [CrossRef]
  6. T. Yokoyama, H. Tanaka, “Microphysical processes of melting snow-flakes detected by two-wavelength radar. Part I. Principle of measurement based on model calculation,” J. Meteorol. Soc. Jpn. 62, 650–666 (1984).
  7. K. Aydin, Y. Zhao, “A computational study of polarimetric radar observables in hail,” IEEE Trans. Geosci. Remote Sens. 28, 412–422 (1990).
    [CrossRef]
  8. S. K. Mitra, O. Vohl, M. Ahr, H. R. Pruppacher, “A wind tunnel and theoretical study of the melting behavior of atmospheric ice particles. IV: Experiment and theory for snow flakes,” J. Atmos. Sci. 47, 584–591 (1990).
    [CrossRef]
  9. R. Meneghini, L. Liao, “Effective dielectric constants of mixed-phase hydrometeors,” J. Atmos. Ocean. Technol. 17, 628–640 (2000).
    [CrossRef]
  10. J. G. Fikioris, N. K. Uzunoglu, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. 69, 1359–1366 (1979).
    [CrossRef]
  11. D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
    [CrossRef]
  12. F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Optical properties of spheres containing a spherical eccentric inclusion,” J. Opt. Soc. Am. A 9, 1327–1335 (1992).
    [CrossRef]
  13. N. C. Skaropoulos, M. P. Ioannidou, D. P. Chrissoulidis, “Indirect mode-matching solution to scattering from a dielectric sphere with an eccentric inclusion,” J. Opt. Soc. Am. A 11, 1859–1866 (1994).
    [CrossRef]
  14. K. A. Fuller, “Scattering and absorption cross sections of compounded spheres. III. Spheres containing arbitrarily located spherical inhomogeneities,” J. Opt. Soc. Am. A 12, 893–904 (1995).
    [CrossRef]
  15. D. Ngo, G. Videen, P. Chýlek, “A fortran code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun. 1077, 94–112 (1996).
    [CrossRef]
  16. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  17. W. M. Irvine, J. B. Pollack, “Infrared optical properties of water and ice spheres,” Icarus 8, 324–360 (1968).
    [CrossRef]
  18. L. J. Battan, Radar Observations of the Atmosphere (Univ. Chicago Press, Chicago, Ill., 1973).
  19. D. N. Whiteman, K. D. Evans, B. Demoz, D. O’C. Starr, E. W. Eloranta, D. Tobin, W. Feltz, G. J. Jedlovec, S. I. Gutman, G. K. Schwemmer, M. Cadirola, S. H. Melfi, F. J. Schmidlin, “Raman lidar measurements of water vapor and cirrus clouds during the passage of hurricane Bonnie,” J. Geophys. Res. 106, 5211–5225 (2001).
    [CrossRef]

2003 (1)

P. Di Girolamo, B. B. Demoz, D. N. Whiteman, “Model simulations of melting hydrometeors: a new lidar bright band from melting frozen drops,” Geophys. Res. Lett. 30, 1626, doi: 10.1029/2002GL016825 (2003).
[CrossRef]

2001 (1)

D. N. Whiteman, K. D. Evans, B. Demoz, D. O’C. Starr, E. W. Eloranta, D. Tobin, W. Feltz, G. J. Jedlovec, S. I. Gutman, G. K. Schwemmer, M. Cadirola, S. H. Melfi, F. J. Schmidlin, “Raman lidar measurements of water vapor and cirrus clouds during the passage of hurricane Bonnie,” J. Geophys. Res. 106, 5211–5225 (2001).
[CrossRef]

2000 (1)

R. Meneghini, L. Liao, “Effective dielectric constants of mixed-phase hydrometeors,” J. Atmos. Ocean. Technol. 17, 628–640 (2000).
[CrossRef]

1997 (1)

K. Sassen, “Contrail-cirrus and their potential for regional climate change,” Bull. Am. Meteorol. Soc. 78, 1885–1903 (1997).
[CrossRef]

1996 (1)

D. Ngo, G. Videen, P. Chýlek, “A fortran code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun. 1077, 94–112 (1996).
[CrossRef]

1995 (2)

1994 (1)

1992 (1)

1991 (1)

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
[CrossRef]

1990 (2)

K. Aydin, Y. Zhao, “A computational study of polarimetric radar observables in hail,” IEEE Trans. Geosci. Remote Sens. 28, 412–422 (1990).
[CrossRef]

S. K. Mitra, O. Vohl, M. Ahr, H. R. Pruppacher, “A wind tunnel and theoretical study of the melting behavior of atmospheric ice particles. IV: Experiment and theory for snow flakes,” J. Atmos. Sci. 47, 584–591 (1990).
[CrossRef]

1984 (1)

T. Yokoyama, H. Tanaka, “Microphysical processes of melting snow-flakes detected by two-wavelength radar. Part I. Principle of measurement based on model calculation,” J. Meteorol. Soc. Jpn. 62, 650–666 (1984).

1979 (1)

1968 (1)

W. M. Irvine, J. B. Pollack, “Infrared optical properties of water and ice spheres,” Icarus 8, 324–360 (1968).
[CrossRef]

1951 (1)

Q. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Aden, Q. L.

Q. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Ahr, M.

S. K. Mitra, O. Vohl, M. Ahr, H. R. Pruppacher, “A wind tunnel and theoretical study of the melting behavior of atmospheric ice particles. IV: Experiment and theory for snow flakes,” J. Atmos. Sci. 47, 584–591 (1990).
[CrossRef]

Aydin, K.

K. Aydin, Y. Zhao, “A computational study of polarimetric radar observables in hail,” IEEE Trans. Geosci. Remote Sens. 28, 412–422 (1990).
[CrossRef]

Battan, L. J.

L. J. Battan, Radar Observations of the Atmosphere (Univ. Chicago Press, Chicago, Ill., 1973).

Borghese, F.

Cadirola, M.

D. N. Whiteman, K. D. Evans, B. Demoz, D. O’C. Starr, E. W. Eloranta, D. Tobin, W. Feltz, G. J. Jedlovec, S. I. Gutman, G. K. Schwemmer, M. Cadirola, S. H. Melfi, F. J. Schmidlin, “Raman lidar measurements of water vapor and cirrus clouds during the passage of hurricane Bonnie,” J. Geophys. Res. 106, 5211–5225 (2001).
[CrossRef]

Chen, T.

K. Sassen, T. Chen, “The lidar dark band: an oddity of the radar bright band,” Geophys. Res. Lett. 22, 3505–3508 (1995).
[CrossRef]

Chrissoulidis, D. P.

Chýlek, P.

D. Ngo, G. Videen, P. Chýlek, “A fortran code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun. 1077, 94–112 (1996).
[CrossRef]

Demoz, B.

D. N. Whiteman, K. D. Evans, B. Demoz, D. O’C. Starr, E. W. Eloranta, D. Tobin, W. Feltz, G. J. Jedlovec, S. I. Gutman, G. K. Schwemmer, M. Cadirola, S. H. Melfi, F. J. Schmidlin, “Raman lidar measurements of water vapor and cirrus clouds during the passage of hurricane Bonnie,” J. Geophys. Res. 106, 5211–5225 (2001).
[CrossRef]

Demoz, B. B.

P. Di Girolamo, B. B. Demoz, D. N. Whiteman, “Model simulations of melting hydrometeors: a new lidar bright band from melting frozen drops,” Geophys. Res. Lett. 30, 1626, doi: 10.1029/2002GL016825 (2003).
[CrossRef]

Denti, P.

Di Girolamo, P.

P. Di Girolamo, B. B. Demoz, D. N. Whiteman, “Model simulations of melting hydrometeors: a new lidar bright band from melting frozen drops,” Geophys. Res. Lett. 30, 1626, doi: 10.1029/2002GL016825 (2003).
[CrossRef]

Eloranta, E. W.

D. N. Whiteman, K. D. Evans, B. Demoz, D. O’C. Starr, E. W. Eloranta, D. Tobin, W. Feltz, G. J. Jedlovec, S. I. Gutman, G. K. Schwemmer, M. Cadirola, S. H. Melfi, F. J. Schmidlin, “Raman lidar measurements of water vapor and cirrus clouds during the passage of hurricane Bonnie,” J. Geophys. Res. 106, 5211–5225 (2001).
[CrossRef]

Evans, K. D.

D. N. Whiteman, K. D. Evans, B. Demoz, D. O’C. Starr, E. W. Eloranta, D. Tobin, W. Feltz, G. J. Jedlovec, S. I. Gutman, G. K. Schwemmer, M. Cadirola, S. H. Melfi, F. J. Schmidlin, “Raman lidar measurements of water vapor and cirrus clouds during the passage of hurricane Bonnie,” J. Geophys. Res. 106, 5211–5225 (2001).
[CrossRef]

Feltz, W.

D. N. Whiteman, K. D. Evans, B. Demoz, D. O’C. Starr, E. W. Eloranta, D. Tobin, W. Feltz, G. J. Jedlovec, S. I. Gutman, G. K. Schwemmer, M. Cadirola, S. H. Melfi, F. J. Schmidlin, “Raman lidar measurements of water vapor and cirrus clouds during the passage of hurricane Bonnie,” J. Geophys. Res. 106, 5211–5225 (2001).
[CrossRef]

Fikioris, J. G.

Fuller, K. A.

Gutman, S. I.

D. N. Whiteman, K. D. Evans, B. Demoz, D. O’C. Starr, E. W. Eloranta, D. Tobin, W. Feltz, G. J. Jedlovec, S. I. Gutman, G. K. Schwemmer, M. Cadirola, S. H. Melfi, F. J. Schmidlin, “Raman lidar measurements of water vapor and cirrus clouds during the passage of hurricane Bonnie,” J. Geophys. Res. 106, 5211–5225 (2001).
[CrossRef]

Ioannidou, M. P.

Irvine, W. M.

W. M. Irvine, J. B. Pollack, “Infrared optical properties of water and ice spheres,” Icarus 8, 324–360 (1968).
[CrossRef]

Jackson, J. D.

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

Jedlovec, G. J.

D. N. Whiteman, K. D. Evans, B. Demoz, D. O’C. Starr, E. W. Eloranta, D. Tobin, W. Feltz, G. J. Jedlovec, S. I. Gutman, G. K. Schwemmer, M. Cadirola, S. H. Melfi, F. J. Schmidlin, “Raman lidar measurements of water vapor and cirrus clouds during the passage of hurricane Bonnie,” J. Geophys. Res. 106, 5211–5225 (2001).
[CrossRef]

Kerker, M.

Q. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chaps. 3 and 4.

Liao, L.

R. Meneghini, L. Liao, “Effective dielectric constants of mixed-phase hydrometeors,” J. Atmos. Ocean. Technol. 17, 628–640 (2000).
[CrossRef]

Mackowski, D. W.

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
[CrossRef]

Melfi, S. H.

D. N. Whiteman, K. D. Evans, B. Demoz, D. O’C. Starr, E. W. Eloranta, D. Tobin, W. Feltz, G. J. Jedlovec, S. I. Gutman, G. K. Schwemmer, M. Cadirola, S. H. Melfi, F. J. Schmidlin, “Raman lidar measurements of water vapor and cirrus clouds during the passage of hurricane Bonnie,” J. Geophys. Res. 106, 5211–5225 (2001).
[CrossRef]

Meneghini, R.

R. Meneghini, L. Liao, “Effective dielectric constants of mixed-phase hydrometeors,” J. Atmos. Ocean. Technol. 17, 628–640 (2000).
[CrossRef]

Mitra, S. K.

S. K. Mitra, O. Vohl, M. Ahr, H. R. Pruppacher, “A wind tunnel and theoretical study of the melting behavior of atmospheric ice particles. IV: Experiment and theory for snow flakes,” J. Atmos. Sci. 47, 584–591 (1990).
[CrossRef]

Ngo, D.

D. Ngo, G. Videen, P. Chýlek, “A fortran code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun. 1077, 94–112 (1996).
[CrossRef]

Pollack, J. B.

W. M. Irvine, J. B. Pollack, “Infrared optical properties of water and ice spheres,” Icarus 8, 324–360 (1968).
[CrossRef]

Pruppacher, H. R.

S. K. Mitra, O. Vohl, M. Ahr, H. R. Pruppacher, “A wind tunnel and theoretical study of the melting behavior of atmospheric ice particles. IV: Experiment and theory for snow flakes,” J. Atmos. Sci. 47, 584–591 (1990).
[CrossRef]

Saija, R.

Sassen, K.

K. Sassen, “Contrail-cirrus and their potential for regional climate change,” Bull. Am. Meteorol. Soc. 78, 1885–1903 (1997).
[CrossRef]

K. Sassen, T. Chen, “The lidar dark band: an oddity of the radar bright band,” Geophys. Res. Lett. 22, 3505–3508 (1995).
[CrossRef]

Schmidlin, F. J.

D. N. Whiteman, K. D. Evans, B. Demoz, D. O’C. Starr, E. W. Eloranta, D. Tobin, W. Feltz, G. J. Jedlovec, S. I. Gutman, G. K. Schwemmer, M. Cadirola, S. H. Melfi, F. J. Schmidlin, “Raman lidar measurements of water vapor and cirrus clouds during the passage of hurricane Bonnie,” J. Geophys. Res. 106, 5211–5225 (2001).
[CrossRef]

Schwemmer, G. K.

D. N. Whiteman, K. D. Evans, B. Demoz, D. O’C. Starr, E. W. Eloranta, D. Tobin, W. Feltz, G. J. Jedlovec, S. I. Gutman, G. K. Schwemmer, M. Cadirola, S. H. Melfi, F. J. Schmidlin, “Raman lidar measurements of water vapor and cirrus clouds during the passage of hurricane Bonnie,” J. Geophys. Res. 106, 5211–5225 (2001).
[CrossRef]

Sindoni, O. I.

Skaropoulos, N. C.

Starr, D. O’C.

D. N. Whiteman, K. D. Evans, B. Demoz, D. O’C. Starr, E. W. Eloranta, D. Tobin, W. Feltz, G. J. Jedlovec, S. I. Gutman, G. K. Schwemmer, M. Cadirola, S. H. Melfi, F. J. Schmidlin, “Raman lidar measurements of water vapor and cirrus clouds during the passage of hurricane Bonnie,” J. Geophys. Res. 106, 5211–5225 (2001).
[CrossRef]

Tanaka, H.

T. Yokoyama, H. Tanaka, “Microphysical processes of melting snow-flakes detected by two-wavelength radar. Part I. Principle of measurement based on model calculation,” J. Meteorol. Soc. Jpn. 62, 650–666 (1984).

Tobin, D.

D. N. Whiteman, K. D. Evans, B. Demoz, D. O’C. Starr, E. W. Eloranta, D. Tobin, W. Feltz, G. J. Jedlovec, S. I. Gutman, G. K. Schwemmer, M. Cadirola, S. H. Melfi, F. J. Schmidlin, “Raman lidar measurements of water vapor and cirrus clouds during the passage of hurricane Bonnie,” J. Geophys. Res. 106, 5211–5225 (2001).
[CrossRef]

Uzunoglu, N. K.

Videen, G.

D. Ngo, G. Videen, P. Chýlek, “A fortran code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun. 1077, 94–112 (1996).
[CrossRef]

Vohl, O.

S. K. Mitra, O. Vohl, M. Ahr, H. R. Pruppacher, “A wind tunnel and theoretical study of the melting behavior of atmospheric ice particles. IV: Experiment and theory for snow flakes,” J. Atmos. Sci. 47, 584–591 (1990).
[CrossRef]

Whiteman, D. N.

P. Di Girolamo, B. B. Demoz, D. N. Whiteman, “Model simulations of melting hydrometeors: a new lidar bright band from melting frozen drops,” Geophys. Res. Lett. 30, 1626, doi: 10.1029/2002GL016825 (2003).
[CrossRef]

D. N. Whiteman, K. D. Evans, B. Demoz, D. O’C. Starr, E. W. Eloranta, D. Tobin, W. Feltz, G. J. Jedlovec, S. I. Gutman, G. K. Schwemmer, M. Cadirola, S. H. Melfi, F. J. Schmidlin, “Raman lidar measurements of water vapor and cirrus clouds during the passage of hurricane Bonnie,” J. Geophys. Res. 106, 5211–5225 (2001).
[CrossRef]

Yokoyama, T.

T. Yokoyama, H. Tanaka, “Microphysical processes of melting snow-flakes detected by two-wavelength radar. Part I. Principle of measurement based on model calculation,” J. Meteorol. Soc. Jpn. 62, 650–666 (1984).

Zhao, Y.

K. Aydin, Y. Zhao, “A computational study of polarimetric radar observables in hail,” IEEE Trans. Geosci. Remote Sens. 28, 412–422 (1990).
[CrossRef]

Bull. Am. Meteorol. Soc. (1)

K. Sassen, “Contrail-cirrus and their potential for regional climate change,” Bull. Am. Meteorol. Soc. 78, 1885–1903 (1997).
[CrossRef]

Comput. Phys. Commun. (1)

D. Ngo, G. Videen, P. Chýlek, “A fortran code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun. 1077, 94–112 (1996).
[CrossRef]

Geophys. Res. Lett. (2)

P. Di Girolamo, B. B. Demoz, D. N. Whiteman, “Model simulations of melting hydrometeors: a new lidar bright band from melting frozen drops,” Geophys. Res. Lett. 30, 1626, doi: 10.1029/2002GL016825 (2003).
[CrossRef]

K. Sassen, T. Chen, “The lidar dark band: an oddity of the radar bright band,” Geophys. Res. Lett. 22, 3505–3508 (1995).
[CrossRef]

Icarus (1)

W. M. Irvine, J. B. Pollack, “Infrared optical properties of water and ice spheres,” Icarus 8, 324–360 (1968).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

K. Aydin, Y. Zhao, “A computational study of polarimetric radar observables in hail,” IEEE Trans. Geosci. Remote Sens. 28, 412–422 (1990).
[CrossRef]

J. Appl. Phys. (1)

Q. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

J. Atmos. Ocean. Technol. (1)

R. Meneghini, L. Liao, “Effective dielectric constants of mixed-phase hydrometeors,” J. Atmos. Ocean. Technol. 17, 628–640 (2000).
[CrossRef]

J. Atmos. Sci. (1)

S. K. Mitra, O. Vohl, M. Ahr, H. R. Pruppacher, “A wind tunnel and theoretical study of the melting behavior of atmospheric ice particles. IV: Experiment and theory for snow flakes,” J. Atmos. Sci. 47, 584–591 (1990).
[CrossRef]

J. Geophys. Res. (1)

D. N. Whiteman, K. D. Evans, B. Demoz, D. O’C. Starr, E. W. Eloranta, D. Tobin, W. Feltz, G. J. Jedlovec, S. I. Gutman, G. K. Schwemmer, M. Cadirola, S. H. Melfi, F. J. Schmidlin, “Raman lidar measurements of water vapor and cirrus clouds during the passage of hurricane Bonnie,” J. Geophys. Res. 106, 5211–5225 (2001).
[CrossRef]

J. Meteorol. Soc. Jpn. (1)

T. Yokoyama, H. Tanaka, “Microphysical processes of melting snow-flakes detected by two-wavelength radar. Part I. Principle of measurement based on model calculation,” J. Meteorol. Soc. Jpn. 62, 650–666 (1984).

J. Opt. Soc. Am. (1)

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

Proc. R. Soc. London Ser. A (1)

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
[CrossRef]

Other (3)

L. J. Battan, Radar Observations of the Atmosphere (Univ. Chicago Press, Chicago, Ill., 1973).

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

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chaps. 3 and 4.

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Figures (14)

Fig. 1
Fig. 1

Geometry of scattering.

Fig. 2
Fig. 2

Extinction and backscattering coefficients as a function of the inclusion relative radius. The inclusion with m i = 1.45 is fixed near the top of the host sphere with m = 1.348 and size parameter x = 4. Dashed lines show the extinction and backscattering for the host sphere without an inclusion.

Fig. 3
Fig. 3

Relative (a) backscattering and (b) extinction coefficients as a function of relative inclusion radius. Calculations are performed for a host sphere with x = 200, m h = 1.348, and m i = 1.45. The inclusion is fixed to the top (solid curve) or bottom (dotted curve) of the host sphere. The dashed-dotted curve presents the results obtained for concentric spheres. The calculation step is Δρ = 0.01.

Fig. 4
Fig. 4

Dependence of relative (a) backscattering and (b) extinction coefficients on the inclusion relative radius. The inclusion is fixed to the top (solid curve) or to the bottom (dotted curve) of the host particle, and the dashed-dotted curve shows the result for concentric spheres. Calculations are performed for log-normal distribution of host spheres’ radii with x mean = 300 and ln σ = 0.1. Refractive indices of the host particle and inclusion are m h = 1.348 and m i = 1.45, respectively. The calculation step is Δρ = 0.02, and the obtained results are smoothed with a 0.06 averaging interval.

Fig. 5
Fig. 5

Dependence of relative backscattering on the inclusion relative radius. The inclusion is fixed to the top (solid curve) or to the bottom (dotted curve) of the host particle, and the dashed-dotted curve shows the result for concentric spheres. Calculations are performed for log-normal distribution of host spheres’ radii with x mean = 300 and ln σ = 0.1. Refractive indices of the host particle and inclusion are m h = 1.348 and m i = 1.25, respectively. The calculation step is Δρ = 0.02, and the obtained results are smoothed with a 0.06 averaging interval.

Fig. 6
Fig. 6

Dependence of relative backscattering on the inclusion relative radius for (a) m i > m h and (b) m i < m h . The inclusion is fixed to the (a) top or (b) bottom of the host particle. Host spheres’ radii are log-normal distributed with x mean = 200 and ln σ = 0.1. The calculation step is Δρ = 0.02, and the obtained results are smoothed with a 0.06 averaging interval.

Fig. 7
Fig. 7

Dependence of maximal relative backscattering on the inclusion refractive index. Host spheres’ radii are log-normal distributed with x mean = 200 and ln σ = 0.1.

Fig. 8
Fig. 8

Relative backscattering as a function of the relative inclusion radius calculated for concentric spheres with m i = 1.15 (dashed-dotted curve), 1.25 (solid curve), 1.45 (dotted curve), and 1.55 (dashed-dotted-dotted curve). The calculation step is Δρ = 0.02, and the obtained results are smoothed with a 0.06 averaging interval.

Fig. 9
Fig. 9

Dependence of relative (a) backscattering and (b) extinction on the relative inclusion shift. Calculations are performed for monodisperse spheres with x = 200, ρ = 0.1, m h = 1.348, and m i = 1.45. The calculation step is Δδ = 0.01.

Fig. 10
Fig. 10

Dependence of particle relative backscattering on the inclusion shift for inclusion relative radii of 0.1 (solid curve), 0.5 (dotted curve), and 0.8 (dashed-dotted curve). Computations are performed for log-normal distribution of host spheres’ radii with x mean = 200 and ln σ = 0.1. Refractive indices of the host particle and inclusion are m h = 1.348 and m i = 1.45. The calculation step is Δδ = 0.02, and the obtained results are smoothed over a 0.06 averaging interval.

Fig. 11
Fig. 11

Dependence of particle backscattering on the inclusion shift for inclusion relative radii of 0.1 (solid curve), 0.5 (dotted curve), and 0.8 (dashed-dotted curve). Computations are performed for log-normal distribution of host spheres’ radii with x mean = 200 and ln σ = 0.1. Refractive indices of the host particle and inclusion are m h = 1.348 and m i = 1.25, respectively. The calculation step is Δδ = 0.02, and the obtained results are smoothed with a 0.06 averaging interval.

Fig. 12
Fig. 12

Dependence of relative backscattering on the inclusion relative radius for an ice-water-combined particle. An ice sphere is fixed to the top (solid curve) or to the bottom (dotted curve) of the host water sphere, and a dashed-dotted curve shows the result obtained for concentric spheres. Calculations are performed for log-normal distribution of host spheres’ radii with x mean = 400 and ln σ = 0.1. The calculation step is Δρ = 0.02, and the results for lidar ratio are smoothed with a 0.06 averaging interval.

Fig. 13
Fig. 13

Dependence of particle relative backscattering on the ice sphere relative shift for ρ = 0.1 (solid curve), 0.5 (dotted curve), and 0.85 (dashed-dotted curve). Computations are performed for log-normal distribution of host spheres’ radii with x mean = 400 and ln σ = 0.1. Refractive indices of the host particle and inclusion are m h = 1.348 and m i = 1.324. The calculation step is Δδ = 0.02.

Fig. 14
Fig. 14

Vertical profiles of temperature and scattering ratio illustrating lidar bright- and dark-band phenomena.

Equations (42)

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Einc,1=n=1m=-nnanmMnm1+bnmNnm1,
Esca,1=n=0m=-nncnmMnm3+dnmNnm3
k1rljC0,n0,0=2n+1ψnk1rlj, rjrlj,
k1rljC0,n0,0=-1n2n+1ψnk1rlj, rjrlj.
k1rljC0,n0,1=nψn-1k1rlj-n+1ψn+1k1rlj,
k1rljC0,n0,1=-1n-1nψn-1k1rlj-n+1ψn+1k1rlj,
Cm+1,nm+1,ν=k1ZljCm,n+1m,ν2n+3+k1ZljCm,n-1m,ν2n-1+Cm,nm,ν.
C1,n1,1=C0,n0,1-k1rljC0,n+10,12n+3+C0,n-10,12n-1.
C1,n1,1=C0,n0,1+k1rljC0,n+10,12n+3+C0,n-10,12n-1.
C1,n1,1=2n+1ψnk1rljkrlj2.
C1,n1,1=-1n-12n+1ψnk1rljkrlj2.
C1,n1,2=3n-12n-1 C1,n-11,1-n+22n+3 C1,n+11,1.
C1,n1,20°=-1nC1,n1,2180°.
C1,n1,ν=2ν-1ν-1ν2ν-1 C1,n1,ν-2-n+22n+3 C1,n+11,ν-1+n-12n-1 C1,n-11,ν-1,
C1,n1,ν0°=-1n+νC1,n1,ν180°.
A1,n1,ν=C1,n1,ν-k1rljn+2n+12n+3 C1,n+11,ν+n-1n2n-1 C1,n-11,ν,
B1,n1,ν=-ik1rljC1,n1,νnn+1
A1,n1,ν=C1,n1,ν+k1rljn+2n+12n+3 C1,n+11,ν+n-1n2n-1 C1,n-11,ν,
B1,n1,ν=+ik1rljC1,n1,νnn+1.
A1,n1,ν0°=-1n+νA1,n1,ν180°,
B1,n1,ν0°=--1n+νB1,n1,ν180°.
ν=1νmax tνmT1,m,νA+uνmU1,m,νB=a1mn1iν=1νmax tνmT2,m,νA+uνmU2,m,νB=a2mn1i.......................................................ν=1νmax tνmTnnax,m,νA+uνmUnnaxm,νA=annaxmn1iν=1νmax tνmT1,m,νB+uνmU1,m,νA=b1mn1iν=1νmax tνmT2,m,νB+uνmU2,m,νA=b2mn1i.......................................................ν=1νmax tνmTnnax,m,νA+uνmUnnax,m,νA=bnnaxmn1i,
Tn,m,νA=Am,nm,νξnka1ζnk1a1+Qνrξnk1a1-n1ξnka1ζnk1a1+Qνrξnk1a1,
Un,m,νB=Bm,nm,νξnka1ζnk1a1+Qνsξnk1a1-n1ξnka1ζnk1a1+Qνsξnk1a1,
Tn,m,νB=Bm,nm,νn1ξnka1ζnk1a1+Qνrξnk1a1-ξnka1ζnk1a1+Qνrξnk1a1,
Un,m,νA=Am,nm,νn1ξnka1ζnk1a1+Qνsξnk1a1-ξnka1ζnk1a1+Qνsξnk1a1,
Qνr=n1ζνk1a2ψνk2a2-n2ζνk1a2ψνk2a2n2ξνk1a2ψνk2a2-n1ξνk1a2ψνk2a2,
Qνs=n2ζνk1a2ψνk2a2-n1ζνk1a2ψνk2a2n1ξνk1a2ψνk2a2-n2ξνk1a2ψνk2a2.
cnm= 1n1ξn2ka1 ×ν=1νmaxtνmTn,m,νAξnka1ξnka1 -n1ζnk1a1+Qνrξnk1a1ζnk1a1+Qνrξnk1a1 + uνmUn,m,νBξnka1ξnka1 -n1ζnk1a1+Qνsξnk1a1ζnk1a1+Qνsξnk1a1-anmψnka1ξnka1,
dnm=1ξn2ka1×ν=1νmaxtνmTn,m,νBn1ξnka1ξnka1-ζnk1a1+Qνrξnk1a1ζnk1a1+Qnrξnk1a1+uνmUn,m,νAn1ξnka1ξnka1-ζnk1a1+Qνsξnk1a1ζnk1a1+Qνsξnk1a1-bnmψnka1ξnka1,
cn,-1=nn+1cn,1,
dn,-1=-nn+1dn,1.
dσdΩ=λ24π2n=1 innn+1cn,1-dn,12,
σext=-2πk2Ren=1 nn+1in+1cn,1*+-1n+1cn,1+-1n+1dn,1+dn,1*.
nNa1=12π a1 ln σexp-ln a1-ln a1,mean22 ln2 σ,
σext=-12k2E02Ren=1m=-nnnn+1Knm2anmcnm*+anm*cnm+bnm*dnm+bnmdnm*,
Knm=2n+14πn-m!n+m!1/2.
σext,m=-12k2E02Ren=1nn+1Knm2anmcnm*+anm*cnm+bnm*dnm+bnmdnm*.
σext,m=-12k2E02ReBTNC*+BT*NC,
ψB+ξC=AT,ψB+ξC=AT,
B2=DB1,
σext,1=σext,2.

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