Abstract

Starting from first principles, we present a detailed analysis of the concept of single scattering of light by a small volume element filled with sparsely and randomly positioned particles. We first derive the formulas of the far-field single-scattering approximation, which treats the volume element as a single scatterer, and discuss its range of applicability, using for illustration exact T-matrix results for randomly oriented two-sphere clusters. Our second approach is to treat the volume element as a small cloud of particles and apply the so-called first-order-scattering approximation. We demonstrate that although the two approaches are based on somewhat different sets of assumptions, they give essentially the same result for the electromagnetic response of a sufficiently distant polarization-sensitive detector.

© 2004 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  2. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  3. L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
  4. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  5. M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds., Light Scattering by Nonspherical Particles (Academic, San Diego, Calif., 2000).
  6. A. A. Kokhanovsky, Optics of Light Scattering Media (Praxis, Chichester, UK, 2001).
  7. M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).
  8. F. Borghese, P. Denti, R. Saija, Scattering from Model Nonspherical Particles (Springer, Berlin, 2003).
  9. J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
    [CrossRef]
  10. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).
  11. J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).
  12. J. Lenoble, ed., Radiative Transfer in Scattering and Absorbing Atmospheres (Deepak, Hampton, Va., 1985).
  13. A. K. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, Boston, Mass., 1994).
  14. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, New York, 1997).
  15. E. G. Yanovitskij, Light Scattering in Inhomogeneous Atmospheres (Springer, Berlin, 1997).
  16. G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, Cambridge, UK, 1999).
  17. L. Tsang, J. A. Kong, K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, New York, 2000).
  18. K. N. Liou, An Introduction to Atmospheric Radiation (Academic, San Diego, Calif., 2002).
  19. M. I. Mishchenko, “Vector radiative transfer equation for arbitrarily shaped and arbitrarily oriented particles: a microphysical derivation from statistical electromagnetics,” Appl. Opt. 41, 7114–7134 (2002).
    [CrossRef] [PubMed]
  20. J. W. Hovenier, “Measuring scattering matrices of small particles at optical wavelengths,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), pp. 355–365.
  21. H. Volten, O. Muñoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17402 (2001).
    [CrossRef]
  22. O. Muñoz, H. Volten, J. F. de Haan, W. Vassen, J. W. Hovenier, “Experimental determination of scattering matrices of randomly oriented fly ash and clay particles at 442 and 633 nm,” J. Geophys. Res. 106, 22833–22844 (2001).
    [CrossRef]
  23. J. W. Hovenier, H. Volten, O. Muñoz, W. J. van der Zande, L. B. F. M. Waters, “Laboratory studies of scattering matrices for randomly oriented particles: potentials, problems, and perspectives,” J. Quant. Spectrosc. Radiat. Transf. 79/80, 741–755 (2003).
    [CrossRef]
  24. K. S. Shifrin, Scattering of Light in a Turbid Medium (National Aeronautics and Space Administration, Washington, D.C., 1968).
  25. P. Ivanov, A. Ya. Khairullina, “On the coherent scattering of light,” Opt. Spectrosc. 23, 83–86 (1967).
  26. M. I. Mishchenko, D. W. Mackowski, “Light scattering by randomly oriented bispheres,” Opt. Lett. 19, 1604–1606 (1994).
    [CrossRef] [PubMed]
  27. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).
  28. M. I. Mishchenko, D. W. Mackowski, L. D. Travis, “Scattering of light by bispheres with touching and separated components,” Appl. Opt. 34, 4589–4599 (1995).
    [CrossRef] [PubMed]
  29. A. Quirantes, F. Arroyo, J. Quirantes-Ros, “Multiple light scattering by spherical particle systems and its dependence on concentration: a T-matrix study,” J. Colloid Interface Sci. 240, 78–82 (2001).
    [CrossRef] [PubMed]
  30. D. S. Saxon, “Lectures on the scattering of light,” (Department of Meteorology, University of California at Los Angeles, Los Angeles, Calif., 1955).
  31. M. I. Mishchenko, “Microphysical approach to polarized radiative transfer: extension to the case of an external observation point,” Appl. Opt. 42, 4963–4967 (2003).
    [CrossRef] [PubMed]
  32. A. P. Ivanov, A. Ya. Khairullina, T. N. Kharkova, “Experimental detection of cooperative effects in a scattering volume,” Opt. Spectrosc. 28, 204–207 (1970).
  33. M. Hess, P. Koepke, I. Schult, “Optical properties of aerosols and clouds: the software package OPAC,” Bull. Am. Meteorol. Soc. 79, 831–844 (1998).
    [CrossRef]
  34. M. Min, University of Amsterdam, Amsterdam, The Netherlands (personal communication, 2003).

2003 (2)

J. W. Hovenier, H. Volten, O. Muñoz, W. J. van der Zande, L. B. F. M. Waters, “Laboratory studies of scattering matrices for randomly oriented particles: potentials, problems, and perspectives,” J. Quant. Spectrosc. Radiat. Transf. 79/80, 741–755 (2003).
[CrossRef]

M. I. Mishchenko, “Microphysical approach to polarized radiative transfer: extension to the case of an external observation point,” Appl. Opt. 42, 4963–4967 (2003).
[CrossRef] [PubMed]

2002 (1)

2001 (3)

H. Volten, O. Muñoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17402 (2001).
[CrossRef]

O. Muñoz, H. Volten, J. F. de Haan, W. Vassen, J. W. Hovenier, “Experimental determination of scattering matrices of randomly oriented fly ash and clay particles at 442 and 633 nm,” J. Geophys. Res. 106, 22833–22844 (2001).
[CrossRef]

A. Quirantes, F. Arroyo, J. Quirantes-Ros, “Multiple light scattering by spherical particle systems and its dependence on concentration: a T-matrix study,” J. Colloid Interface Sci. 240, 78–82 (2001).
[CrossRef] [PubMed]

1998 (1)

M. Hess, P. Koepke, I. Schult, “Optical properties of aerosols and clouds: the software package OPAC,” Bull. Am. Meteorol. Soc. 79, 831–844 (1998).
[CrossRef]

1995 (1)

1994 (1)

1983 (1)

J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).

1974 (1)

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

1970 (1)

A. P. Ivanov, A. Ya. Khairullina, T. N. Kharkova, “Experimental detection of cooperative effects in a scattering volume,” Opt. Spectrosc. 28, 204–207 (1970).

1967 (1)

P. Ivanov, A. Ya. Khairullina, “On the coherent scattering of light,” Opt. Spectrosc. 23, 83–86 (1967).

Arroyo, F.

A. Quirantes, F. Arroyo, J. Quirantes-Ros, “Multiple light scattering by spherical particle systems and its dependence on concentration: a T-matrix study,” J. Colloid Interface Sci. 240, 78–82 (2001).
[CrossRef] [PubMed]

Bayvel, L. P.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).

Bohren, C. F.

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

Borghese, F.

F. Borghese, P. Denti, R. Saija, Scattering from Model Nonspherical Particles (Springer, Berlin, 2003).

de Haan, J. F.

H. Volten, O. Muñoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17402 (2001).
[CrossRef]

O. Muñoz, H. Volten, J. F. de Haan, W. Vassen, J. W. Hovenier, “Experimental determination of scattering matrices of randomly oriented fly ash and clay particles at 442 and 633 nm,” J. Geophys. Res. 106, 22833–22844 (2001).
[CrossRef]

Denti, P.

F. Borghese, P. Denti, R. Saija, Scattering from Model Nonspherical Particles (Springer, Berlin, 2003).

Ding, K.-H.

L. Tsang, J. A. Kong, K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, New York, 2000).

Fung, A. K.

A. K. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, Boston, Mass., 1994).

Hansen, J. E.

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

Hess, M.

M. Hess, P. Koepke, I. Schult, “Optical properties of aerosols and clouds: the software package OPAC,” Bull. Am. Meteorol. Soc. 79, 831–844 (1998).
[CrossRef]

Hovenier, J. W.

J. W. Hovenier, H. Volten, O. Muñoz, W. J. van der Zande, L. B. F. M. Waters, “Laboratory studies of scattering matrices for randomly oriented particles: potentials, problems, and perspectives,” J. Quant. Spectrosc. Radiat. Transf. 79/80, 741–755 (2003).
[CrossRef]

O. Muñoz, H. Volten, J. F. de Haan, W. Vassen, J. W. Hovenier, “Experimental determination of scattering matrices of randomly oriented fly ash and clay particles at 442 and 633 nm,” J. Geophys. Res. 106, 22833–22844 (2001).
[CrossRef]

H. Volten, O. Muñoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17402 (2001).
[CrossRef]

J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).

J. W. Hovenier, “Measuring scattering matrices of small particles at optical wavelengths,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), pp. 355–365.

Huffman, D. R.

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

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, New York, 1997).

Ivanov, A. P.

A. P. Ivanov, A. Ya. Khairullina, T. N. Kharkova, “Experimental detection of cooperative effects in a scattering volume,” Opt. Spectrosc. 28, 204–207 (1970).

Ivanov, P.

P. Ivanov, A. Ya. Khairullina, “On the coherent scattering of light,” Opt. Spectrosc. 23, 83–86 (1967).

Jones, A. R.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).

Kerker, M.

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

Khairullina, A. Ya.

A. P. Ivanov, A. Ya. Khairullina, T. N. Kharkova, “Experimental detection of cooperative effects in a scattering volume,” Opt. Spectrosc. 28, 204–207 (1970).

P. Ivanov, A. Ya. Khairullina, “On the coherent scattering of light,” Opt. Spectrosc. 23, 83–86 (1967).

Kharkova, T. N.

A. P. Ivanov, A. Ya. Khairullina, T. N. Kharkova, “Experimental detection of cooperative effects in a scattering volume,” Opt. Spectrosc. 28, 204–207 (1970).

Koepke, P.

M. Hess, P. Koepke, I. Schult, “Optical properties of aerosols and clouds: the software package OPAC,” Bull. Am. Meteorol. Soc. 79, 831–844 (1998).
[CrossRef]

Kokhanovsky, A. A.

A. A. Kokhanovsky, Optics of Light Scattering Media (Praxis, Chichester, UK, 2001).

Kong, J. A.

L. Tsang, J. A. Kong, K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, New York, 2000).

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).

Liou, K. N.

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

Mackowski, D. W.

Min, M.

M. Min, University of Amsterdam, Amsterdam, The Netherlands (personal communication, 2003).

Mishchenko, M. I.

Muinonen, K.

H. Volten, O. Muñoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17402 (2001).
[CrossRef]

Muñoz, O.

J. W. Hovenier, H. Volten, O. Muñoz, W. J. van der Zande, L. B. F. M. Waters, “Laboratory studies of scattering matrices for randomly oriented particles: potentials, problems, and perspectives,” J. Quant. Spectrosc. Radiat. Transf. 79/80, 741–755 (2003).
[CrossRef]

O. Muñoz, H. Volten, J. F. de Haan, W. Vassen, J. W. Hovenier, “Experimental determination of scattering matrices of randomly oriented fly ash and clay particles at 442 and 633 nm,” J. Geophys. Res. 106, 22833–22844 (2001).
[CrossRef]

H. Volten, O. Muñoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17402 (2001).
[CrossRef]

Nousiainen, T.

H. Volten, O. Muñoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17402 (2001).
[CrossRef]

Quirantes, A.

A. Quirantes, F. Arroyo, J. Quirantes-Ros, “Multiple light scattering by spherical particle systems and its dependence on concentration: a T-matrix study,” J. Colloid Interface Sci. 240, 78–82 (2001).
[CrossRef] [PubMed]

Quirantes-Ros, J.

A. Quirantes, F. Arroyo, J. Quirantes-Ros, “Multiple light scattering by spherical particle systems and its dependence on concentration: a T-matrix study,” J. Colloid Interface Sci. 240, 78–82 (2001).
[CrossRef] [PubMed]

Rol, E.

H. Volten, O. Muñoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17402 (2001).
[CrossRef]

Saija, R.

F. Borghese, P. Denti, R. Saija, Scattering from Model Nonspherical Particles (Springer, Berlin, 2003).

Saxon, D. S.

D. S. Saxon, “Lectures on the scattering of light,” (Department of Meteorology, University of California at Los Angeles, Los Angeles, Calif., 1955).

Schult, I.

M. Hess, P. Koepke, I. Schult, “Optical properties of aerosols and clouds: the software package OPAC,” Bull. Am. Meteorol. Soc. 79, 831–844 (1998).
[CrossRef]

Shifrin, K. S.

K. S. Shifrin, Scattering of Light in a Turbid Medium (National Aeronautics and Space Administration, Washington, D.C., 1968).

Stamnes, K.

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, Cambridge, UK, 1999).

Thomas, G. E.

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, Cambridge, UK, 1999).

Travis, L. D.

M. I. Mishchenko, D. W. Mackowski, L. D. Travis, “Scattering of light by bispheres with touching and separated components,” Appl. Opt. 34, 4589–4599 (1995).
[CrossRef] [PubMed]

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).

Tsang, L.

L. Tsang, J. A. Kong, K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, New York, 2000).

van de Hulst, H. C.

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

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).

van der Mee, C. V. M.

J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).

van der Zande, W. J.

J. W. Hovenier, H. Volten, O. Muñoz, W. J. van der Zande, L. B. F. M. Waters, “Laboratory studies of scattering matrices for randomly oriented particles: potentials, problems, and perspectives,” J. Quant. Spectrosc. Radiat. Transf. 79/80, 741–755 (2003).
[CrossRef]

Vassen, W.

O. Muñoz, H. Volten, J. F. de Haan, W. Vassen, J. W. Hovenier, “Experimental determination of scattering matrices of randomly oriented fly ash and clay particles at 442 and 633 nm,” J. Geophys. Res. 106, 22833–22844 (2001).
[CrossRef]

H. Volten, O. Muñoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17402 (2001).
[CrossRef]

Volten, H.

J. W. Hovenier, H. Volten, O. Muñoz, W. J. van der Zande, L. B. F. M. Waters, “Laboratory studies of scattering matrices for randomly oriented particles: potentials, problems, and perspectives,” J. Quant. Spectrosc. Radiat. Transf. 79/80, 741–755 (2003).
[CrossRef]

O. Muñoz, H. Volten, J. F. de Haan, W. Vassen, J. W. Hovenier, “Experimental determination of scattering matrices of randomly oriented fly ash and clay particles at 442 and 633 nm,” J. Geophys. Res. 106, 22833–22844 (2001).
[CrossRef]

H. Volten, O. Muñoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17402 (2001).
[CrossRef]

Waters, L. B. F. M.

J. W. Hovenier, H. Volten, O. Muñoz, W. J. van der Zande, L. B. F. M. Waters, “Laboratory studies of scattering matrices for randomly oriented particles: potentials, problems, and perspectives,” J. Quant. Spectrosc. Radiat. Transf. 79/80, 741–755 (2003).
[CrossRef]

Yanovitskij, E. G.

E. G. Yanovitskij, Light Scattering in Inhomogeneous Atmospheres (Springer, Berlin, 1997).

Appl. Opt. (3)

Astron. Astrophys. (1)

J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).

Bull. Am. Meteorol. Soc. (1)

M. Hess, P. Koepke, I. Schult, “Optical properties of aerosols and clouds: the software package OPAC,” Bull. Am. Meteorol. Soc. 79, 831–844 (1998).
[CrossRef]

J. Colloid Interface Sci. (1)

A. Quirantes, F. Arroyo, J. Quirantes-Ros, “Multiple light scattering by spherical particle systems and its dependence on concentration: a T-matrix study,” J. Colloid Interface Sci. 240, 78–82 (2001).
[CrossRef] [PubMed]

J. Geophys. Res. (2)

H. Volten, O. Muñoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17402 (2001).
[CrossRef]

O. Muñoz, H. Volten, J. F. de Haan, W. Vassen, J. W. Hovenier, “Experimental determination of scattering matrices of randomly oriented fly ash and clay particles at 442 and 633 nm,” J. Geophys. Res. 106, 22833–22844 (2001).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf. (1)

J. W. Hovenier, H. Volten, O. Muñoz, W. J. van der Zande, L. B. F. M. Waters, “Laboratory studies of scattering matrices for randomly oriented particles: potentials, problems, and perspectives,” J. Quant. Spectrosc. Radiat. Transf. 79/80, 741–755 (2003).
[CrossRef]

Opt. Lett. (1)

Opt. Spectrosc. (2)

A. P. Ivanov, A. Ya. Khairullina, T. N. Kharkova, “Experimental detection of cooperative effects in a scattering volume,” Opt. Spectrosc. 28, 204–207 (1970).

P. Ivanov, A. Ya. Khairullina, “On the coherent scattering of light,” Opt. Spectrosc. 23, 83–86 (1967).

Space Sci. Rev. (1)

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

Other (21)

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

K. S. Shifrin, Scattering of Light in a Turbid Medium (National Aeronautics and Space Administration, Washington, D.C., 1968).

D. S. Saxon, “Lectures on the scattering of light,” (Department of Meteorology, University of California at Los Angeles, Los Angeles, Calif., 1955).

M. Min, University of Amsterdam, Amsterdam, The Netherlands (personal communication, 2003).

J. Lenoble, ed., Radiative Transfer in Scattering and Absorbing Atmospheres (Deepak, Hampton, Va., 1985).

A. K. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, Boston, Mass., 1994).

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, New York, 1997).

E. G. Yanovitskij, Light Scattering in Inhomogeneous Atmospheres (Springer, Berlin, 1997).

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, Cambridge, UK, 1999).

L. Tsang, J. A. Kong, K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, New York, 2000).

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

J. W. Hovenier, “Measuring scattering matrices of small particles at optical wavelengths,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), pp. 355–365.

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

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

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).

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

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds., Light Scattering by Nonspherical Particles (Academic, San Diego, Calif., 2000).

A. A. Kokhanovsky, Optics of Light Scattering Media (Praxis, Chichester, UK, 2001).

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

F. Borghese, P. Denti, R. Saija, Scattering from Model Nonspherical Particles (Springer, Berlin, 2003).

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Figures (16)

Fig. 1
Fig. 1

Local coordinate system used to describe the direction of propagation and the polarization state of a transverse electromagnetic wave at the observation point O.

Fig. 2
Fig. 2

Schematic representation of the electromagnetic scattering problem. The unshaded exterior region is unbounded in all directions, whereas the shaded area represents the interior region VINT.

Fig. 3
Fig. 3

Far-field scattering by a single particle.

Fig. 4
Fig. 4

The response of a collimated detector depends on the line of sight.

Fig. 5
Fig. 5

Scattering by a fixed configuration of N finite particles.

Fig. 6
Fig. 6

Far-field scattering by a collection of particles.

Fig. 7
Fig. 7

Results of exact T-matrix computations of the F11 element of the scattering matrix versus scattering angle Θ for a two-sphere cluster in random orientation. The distance d between the centers of the component spheres increases such that the product k1d grows from 15 to 60. The radius of each sphere is a=0.5 μm, their relative refractive index is m=1.5, and the wavelength in the surrounding medium is λ1=2π/k1=0.6283 μm. For comparison, the thick curves show the F11 element for two noninteracting spheres of the same size and relative refractive index.

Fig. 8
Fig. 8

Illustration of the equality |sˆ-rˆ|=2 sin(Θ/2).

Fig. 9
Fig. 9

The solid curve shows the results of exact T-matrix computations of the F11 element of the scattering matrix versus the scattering angle Θ for a two-sphere cluster in random orientation with k1d=60, a=0.5 μm, m=1.5, and λ1=0.6283 μm. For comparison, the dotted curve shows the result of using Eq. (90), whereas the dashed curve depicts the F11 element for two noninteracting spheres of the same size and relative refractive index.

Fig. 10
Fig. 10

f(Θ) versus Θ for k1d=15 and 60.

Fig. 11
Fig. 11

Ratio of the total scattering cross section for a two-particle cluster with identical touching components and in random orientation to the sum of the scattering cross sections of two noninteracting spheres of the same radius as a function of the sphere size parameter. The relative refractive index of the spheres is 1.5.

Fig. 12
Fig. 12

Computation of 〈G〉 for a randomly oriented cluster consisting of two identical touching spheres.

Fig. 13
Fig. 13

Ratio of the scattering cross section of a two-sphere cluster with equal components and in random orientation to the sum of the scattering cross sections of two noninteracting spheres of the same radius as a function of k1d. The relative refractive index of the spheres is 1.5, and their size parameter x=k1a varies from 1 to 10.

Fig. 14
Fig. 14

Same as Fig. 13, but for the ratio of the asymmetry parameter of a two-sphere cluster with equal components and in random orientation to the asymmetry parameter of two noninteracting spheres of the same radius.

Fig. 15
Fig. 15

Same as Fig. 7, but for the ratios F22/F11 and -F21/F11.

Fig. 16
Fig. 16

First-order scattering by a small volume element.

Equations (120)

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E(r)=E0 exp(iknˆ·r),E0·nˆ=0,
I=IQUV=12 εμ1/2E0θ E0θ*+E0φ E0φ*E0θ E0θ*-E0φ E0φ*-E0θ E0φ*-E0φ E0θ*i(E0φ E0θ*-E0θ E0φ*),
ρ=EE*=E0E0*,
I=12 εμ1/2θˆ·ρ·θˆ+φˆ·ρ·φˆθˆ·ρ·θˆ-φˆ·ρ·φˆ-θˆ·ρ·φˆ-φˆ·ρ·θˆ i(φˆ·ρ·θˆ-θˆ·ρ·φˆ),
E(r)=Einc(r)+k12VINTdr G(r, r)·E(r)[m2(r)-1],rR3,
G(r, r)=I+1k12 exp(ik1|r-r|)4π|r-r|
Esca(r)=VINTdr G(r, r)·VINTdr T(r, r)·Einc(r),rR3.
T(r, r)=k12[m2(r)-1]δ(r-r)I+k12[m2(r)-1]VINTdr G(r, r)·T(r, r),
r, rVINT,
ra,
rk1a2/2,
k1r1,
Esca(r)=exp(ik1r)r k124π(I-rˆrˆ)·VINTdr×[m2(r)-1]E(r)exp(-ik1rˆ·r).
Esca(r)=exp(ik1r)r E1sca(rˆ),rˆ·E1sca(rˆ)=0,
Einc(r)=E0inc exp(ik1nˆinc·r),
E1sca(nˆsca)=A(nˆsca, nˆinc)·E0inc,
nˆsca·A(nˆsca, nˆinc)=0.
A(nˆsca, nˆinc)·nˆinc=0.
Esca(rnˆsca)=exp(ik1r)r S(nˆsca, nˆinc)E0inc,
E=EϑEφ.
S11=θˆsca·A·θˆinc,S12=θˆsca·A·φinc,
S21=φsca·A·θˆinc,S22=φsca·A·φinc.
Iinc=IincQincUincVinc=12 ε1μ01/2E0θincE0θinc*+E0φincE0φinc*E0θincE0θinc*-E0φincE0φinc*-E0θincE0φinc*-E0φincE0θinc*i(E0φincE0θinc*-E0θincE0φinc*),
Isca(rnˆsca)=Isca(rnˆsca)Qsca(rnˆsca)Usca(rnˆsca)Vsca(rnˆsca)=1r2 12 ε1μ01/2E1θsca(nˆsca)[E1θsca(nˆsca)]*+E1φsca(nˆsca)[E1φsca(nˆsca)]*E1θsca(nˆsca)[E1θsca(nˆsca)]*-E1φsca(nˆsca)[E1φsca(nˆsca)]*-E1θsca(nˆsca)[E1φsca(nˆsca)]*-E1φsca(nˆsca)[E1θsca(nˆsca)]*i{E1φsca(nˆsca)[E1θsca(nˆsca)]*-E1θsca(nˆsca)[E1φsca(nˆsca)]*},
Signal2=ΔSIsca(rnˆsca),
Isca(rnˆsca)=1r2 Z(nˆsca, nˆinc)Iinc.
Z11=12(|S11|2+|S12|2+|S21|2+|S22|2),
Z12=12(|S11|2-|S12|2+|S21|2-|S22|2),
Z13=-Re(S11S12*+S22S21*),
Z14=-Im(S11S12*-S22S21*),
Z21=12(|S11|2+|S12|2-|S21|2-|S22|2),
Z22=12(|S11|2-|S12|2-|S21|2+|S22|2),
Z23=-Re(S11S12*-S22S21*),
Z24=-Im(S11S12*+S22S21*),
Z31=-Re(S11S21*+S22S12*),
Z32=-Re(S11S21*-S22S12*),
Z33=Re(S11S22*+S12S21*),
Z34=Im(S11S22*+S21S12*),
Z41=-Im(S21S11*+S22S12*),
Z42=-Im(S21S11*-S22S12*),
Z43=Im(S22S11*-S12S21*),
Z44=Re(S22S11*-S12S21*).
I(rrˆ)=12 ε1μ01/2Eθ(rrˆ)[Eθ(rrˆ)]*+Eφ(rrˆ)[Eφ(rrˆ)]*Eθ(rrˆ)[Eθ(rrˆ)]*-Eφ(rrˆ)[Eφ(rrˆ)]*-Eθ(rrˆ)[Eφ(rrˆ)]*-Eφ(rrˆ)[Eθ(rrˆ)]*i{Eφ(rrˆ)[Eθ(rrˆ)]*-Eθ(rrˆ)[Eφ(rrˆ)]*},
Signal1=ΔSdS I(rrˆ)=ΔSIinc-K(nˆinc)Iinc+ΔS 1r2 Z(nˆinc, nˆinc)Iinc,
Kjj=2πk1 Im(S11+S22),j=1,, 4,
K12=K21=2πk1 Im(S11-S22),
K13=K31=-2πk1 Im(S12+S21),
K14=K41=2πk1 Re(S21-S12),
K23=-K32=2πk1 Im(S21-S12),
K24=-K42=-2πk1 Re(S12+S21),
K34=-K43=2πk1 Re(S22-S11).
Signal1=rΔSIinc-K(nˆinc)Iinc.
π2k1aD2r,
Csca=1Iinc SdS Isca(r)=1Iinc 4πdrˆ[Z11(rˆ, nˆinc)Iinc+Z12(rˆ, nˆinc)Qinc+Z13(rˆ, nˆinc)Uinc+Z14(rˆ, nˆinc)Vinc],
Cext=1Iinc[K11(nˆinc)Iinc+K12(nˆinc)Qinc+K13(nˆinc)Uinc+K14(nˆinc)Vinc],
Cabs=Cext-Csca0.
E(r)=Einc(r)+i=1NVidr G(r, r)·Vidr Ti(r, r)·Ei(r),rR3,
Ei(r)=Einc(r)+j(i)=1NEijexc(r),
Eijexc(r)=Vjdr G(r, r)·Vjdr Tj(r, r)·Ej(r),
rVi,
E(r)=Einc(r)+Esca(r),rR3,
Esca(r)=i=1NEisca(r),
Eisca(r)=Vidr G(r, r)·Vidr Ti(r, r)·Einc(r).
Einc(r)=E0inc exp(ik1sˆ·r),E0inc·sˆ=0,
Einc(r)=E0inc exp(ik1sˆ·ri)exp(ik1sˆ·Ri),
Eisca(r)=exp(ik1sˆ·Ri) exp(ik1ri)riAi(rˆi, sˆ)·E0inc.
ri=|r-Ri|=r1-2rˆ·Rir+Ri2r21/2r-rˆ·Ri+Ri22r.
Eisca(r)=exp(ik1r)r exp(iΔi)Ai(rˆ, sˆ)·E0inc,
Δi=k1(sˆ-rˆ)·Ri,
Esca(r)=exp(ik1r)rA(rˆ, sˆ)·E0inc,
A(rˆ, sˆ)=i=1N exp(iΔi)Ai(rˆ, sˆ).
Esca(r)=exp(ik1r)r S(rˆ, sˆ)·E0inc,
S(rˆ, sˆ)=i=1N exp(iΔi)Si(rˆ, sˆ).
rmaxL2, Lk1aiπ orrmaxL2, 2Laiλ1,
i=1,, N,
rk1L28,
k1r1,
K=i=1NKi,
Cext=i=1N(Cext)i.
Re i=1Ni(i)=1N[Si(rˆ, sˆ)]kl[Si(rˆ, sˆ)]pq* exp[i(Δi-Δi)]Re i=1N[Si(rˆ, sˆ)]kl[Si(rˆ, sˆ)]pq*,
Im i=1Ni(i)=1N[Si(rˆ, sˆ)]kl[Si(rˆ, sˆ)]pq* exp[i(Δi-Δi)]Im i=1N[Si(rˆ, sˆ)]kl[Si(rˆ, sˆ)]pq*, k, l, p, q=1, 2,
Z=i=1NZi.
Csca=i=1N(Csca)i,
Cabs=i=1N(Cabs)i.
K=NK,
Z=NZ,
Cext=NCext,
Csca=NCsca,
Cabs=NCabs,
F(Θ)=Z(θsca=Θ, φsca=0;θinc=0, φinc=0).
14π 4πddˆ exp[-ik1d(sˆ-rˆ)·dˆ]
=14π 4πddˆ exp[ik1d(sˆ-rˆ)·dˆ]
=sin(k1d|sˆ-rˆ|)k1d|sˆ-rˆ|
=f(Θ),
f(Θ)=sin[2k1d sin(Θ/2)]2k1d sin(Θ/2),
Z(rˆ, sˆ)=2Z1(rˆ, sˆ)[1+f(Θ)],
F(Θ)=2F1(Θ)[1+f(Θ)],
da,
k1L1,
G=14π 4πddˆG(dˆ)=14π 02πdφ0πdθ sin θ G(θ)=0π/2dθ sin θ G(θ),
i=1N(Csca)iL2.
ρ(r)=E(r)E*(r)=Einc(r)[Einc(r)]*+Einc(r)i=1N[Eisca(r)]*+i=1N[Eisca(r)][Einc(r)]*+i=1N[Eisca(r)]j(i)=1N[Ejsca(r)]*+i=1N[Eisca(r)][Eisca(r)]*
pR(Ri)=1/Vif RiV0if RiV.
dai,i=1,, N.
ρ(r)=Einc(r)[Einc(r)]*+Einc(r)i=1N[Eisca(r)]*+i=1N[Eisca(r)][Einc(r)]*+i=1N[Eisca(r)]j(i)=1N[Ejsca(r)]*+i=1N[Eisca(r)][Eisca(r)]*.
Eisca(r)=R3dRi pR(Ri)Eisca(r)=R3dRi pR(Ri)exp(ik1sˆ·Ri)×exp(ik1ri)riAi(rˆi, sˆ)·E0inc
exp(ik1sˆ·Ri)=k1Ri i2πk1Ri[δ(sˆ+Rˆi)exp(-ik1Ri)-δ(sˆ-Rˆi)exp(ik1Ri)],
Eisca(r)=i2πk1 4πdRˆi0dRi[δ(sˆ+Rˆi)-δ(sˆ-Rˆi)exp(2ik1Ri)]pR(Ri)×Ai(-Rˆi, sˆ)·Einc(r).
Eisca(r1)=i2πk1VΔs(r1)Ai(sˆ, sˆ)·Einc(r1).
i=1N(Cext)iL2,
I(r)=Iinc-Δs(r)V i=1NKi(sˆ)Iinc+1r2 i=1NZi(sˆ, sˆ)Iinc
I(r)=1r2 i=1NZi(rˆ, sˆ)Iinc
Signal1=ΔSIinc-i=1NKi(sˆ)Iinc+ΔSr2 i=1NZi(sˆ, sˆ)Iinc,
Signal2=ΔSr2 i=1NZi(qˆ, sˆ)Iinc.
Signal1=ΔSIinc-i=1NKi(sˆ)Iinc.
Signal1=ΔSIinc-NK(sˆ)Iinc,
Signal2=1r2ΔSNZ(qˆ, sˆ)Iinc,
dk1a22,
k1d1
La2πN.

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