Abstract

In electromagnetic multisphere-scattering calculations the reexpansion method for seeking a single-field representation of the total scattered field is found impracticable because of severe numerical problems. We present a simple single-field expansion of the total scattered far field based on an asymptotic form of vector translational addition theorems. With this asymptotic expansion of the far field, we derive analytical expressions for the scattering properties of an arbitrary aggregate of spheres. Resulting formulas are free from numerical problems in practical applications. Theoretical predictions from this far-field solution for various aggregates of spheres that we tested agree favorably with laboratory microwave scattering measurements. Some numerical results are presented and compared with experimental data.

© 1997 Optical Society of America

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  8. J. P. Bradley, “Physical and mineralogical properties of anhydrous interplanetary dust particles in the analytical electron microscope,” in Origin and Evolution of Interplanetary Dust, A. C. Levasseur-Regourd and H. Hasegawa, eds (Kluwer Academic, Dordrecht, The Netherlands, 1991), pp. 63–70.
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  9. K. Tomeoka, “Aqueous alteration in hydrated interplanetary dust particles,” in Origin and Evolution of Interplanetary Dust, A. C. Levasseur-Regourd and H. Hasegawa, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1991), pp. 71–78.
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    [CrossRef]
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    [CrossRef] [PubMed]
  23. F. Borghese, P. Denti, R. Saija, G. Toscano, and O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 4, 227–235 (1984).
    [CrossRef]
  24. F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, “Reliability of the theoretical description of electromagnetic scattering by nonspherical particles,” J. Aerosol Sci. 20, 1079–1081 (1989).
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  33. J. D. Cartigny, Y. Yamada, and C. L. Tien, “Radiative transfer with dependent scattering by particles: Part 1–Theoretical investigation,” Trans. ASME 108, 608–613 (1982).
    [CrossRef]
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    [CrossRef]
  35. Y.-l. Xu and B. Å. S. Gustafson, “A complete and efficient multisphere scattering theory for modeling the optical properties of interplanetary dust,” in Physics, Chemistry, and Dynamics of Interplanetary Dust, B. Å. S. Gustafson and M. Hanner, eds., Vol. 104 of ASP Conference Series (Astronomical Society of the Pacific, San Francisco, 1996), pp. 419–422.
  36. R. T. Wang, “Extinction signatures of non-spherical/non-isotropic particles,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, ed. (Plenum, New York, 1980), pp. 255–272.
    [CrossRef]
  37. R. H. Zerull, B. Å. S. Gustafson, K. Schulz, and E. Thiele-Corbach, “Scattering by aggregates with and without an absorbing mantle: microwave analog experiments,” Appl. Opt. 32, 4088–4100 (1993).
    [CrossRef] [PubMed]
  38. B. Å. S. Gustafson, “Microwave analog to light scattering measurements: a modern implementation of a proven method to achieve precise control,” J. Quant. Spectrosc. Radiat. Transfer 55, 663–672 (1996).
    [CrossRef]

1996 (1)

B. Å. S. Gustafson, “Microwave analog to light scattering measurements: a modern implementation of a proven method to achieve precise control,” J. Quant. Spectrosc. Radiat. Transfer 55, 663–672 (1996).
[CrossRef]

1995 (2)

P. Rannou, M. Cabane, E. Chassefière, R. Botet, C. P. McKay, and R. Courtin, “Titan’s geometric albedo: role of the fractal structure of the aerosols,” Icarus 118, 355–372 (1995).
[CrossRef]

Y.-l. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995).
[CrossRef] [PubMed]

1994 (2)

1993 (2)

1991 (2)

R. A. West and P. H. Smith, “Evidence for aggregate particles in the atmospheres of Titan and Jupiter,” Icarus 90, 330–333 (1991).
[CrossRef]

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

1989 (1)

F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, “Reliability of the theoretical description of electromagnetic scattering by nonspherical particles,” J. Aerosol Sci. 20, 1079–1081 (1989).
[CrossRef]

1988 (2)

1984 (1)

F. Borghese, P. Denti, R. Saija, G. Toscano, and O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 4, 227–235 (1984).
[CrossRef]

1982 (2)

J. M. Gérardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. II. Optical properties of aggregated metal spheres,” Phys. Rev. B 25, 4204–4229 (1982).
[CrossRef]

J. D. Cartigny, Y. Yamada, and C. L. Tien, “Radiative transfer with dependent scattering by particles: Part 1–Theoretical investigation,” Trans. ASME 108, 608–613 (1982).
[CrossRef]

1980 (1)

1979 (1)

1971 (1)

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres, Part I—Multiple expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

1962 (1)

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–40 (1962).

1961 (1)

S. Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1961).

1955 (1)

D. S. Saxon, “Tensor scattering matrix for the electromagnetic field,” Phys. Rev. 100, 1771–1775 (1955).
[CrossRef]

1954 (1)

B. Friedman and J. Russek, “Addition theorems for spherical waves,” Quart. Appl. Math. 12, 13–23 (1954).

1933 (1)

W. Trinks, “Zur Vielfachstreung an kleinen Kugeln,” Ann. Phys. Dtsch. 22, 561–590 (1933).

1908 (1)

G. Mie, “Beiträage zue Optik trüber Medien speziell kolloidaler Metalläsungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

Å. S. Gustafson, B.

B. Å. S. Gustafson, “Microwave analog to light scattering measurements: a modern implementation of a proven method to achieve precise control,” J. Quant. Spectrosc. Radiat. Transfer 55, 663–672 (1996).
[CrossRef]

R. H. Zerull, B. Å. S. Gustafson, K. Schulz, and E. Thiele-Corbach, “Scattering by aggregates with and without an absorbing mantle: microwave analog experiments,” Appl. Opt. 32, 4088–4100 (1993).
[CrossRef] [PubMed]

Ausloos, M.

J. M. Gérardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. II. Optical properties of aggregated metal spheres,” Phys. Rev. B 25, 4204–4229 (1982).
[CrossRef]

Bohren, C. F.

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

Borghese, F.

F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, “Reliability of the theoretical description of electromagnetic scattering by nonspherical particles,” J. Aerosol Sci. 20, 1079–1081 (1989).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, and O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 4, 227–235 (1984).
[CrossRef]

F. Borghese, P. Denti, G. Toscano, and O. I. Sindoni, “Electromagnetic scattering by a cluster of spheres,” Appl. Opt. 18, 116–120 (1979).
[CrossRef] [PubMed]

Botet, R.

P. Rannou, M. Cabane, E. Chassefière, R. Botet, C. P. McKay, and R. Courtin, “Titan’s geometric albedo: role of the fractal structure of the aerosols,” Icarus 118, 355–372 (1995).
[CrossRef]

Bradley, J. P.

J. P. Bradley, “Physical and mineralogical properties of anhydrous interplanetary dust particles in the analytical electron microscope,” in Origin and Evolution of Interplanetary Dust, A. C. Levasseur-Regourd and H. Hasegawa, eds (Kluwer Academic, Dordrecht, The Netherlands, 1991), pp. 63–70.
[CrossRef]

Brownlee, D. E.

D. E. Brownlee, “Microparticle studies by sampling techniques,” in Cosmic Dust, J. A. M. McDonnel, ed. (Wiley, Chichester, UK, 1978), pp. 275–336.

Bruning, J. H.

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres, Part I—Multiple expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

Cabane, M.

P. Rannou, M. Cabane, E. Chassefière, R. Botet, C. P. McKay, and R. Courtin, “Titan’s geometric albedo: role of the fractal structure of the aerosols,” Icarus 118, 355–372 (1995).
[CrossRef]

M. Cabane, P. Rannou, E. Chassefière, and G. Israel, “Fractal aggregates in Titan’s atmosphere,” Planet. Space Sci. 41, 257–267 (1993).
[CrossRef]

Cartigny, J. D.

J. D. Cartigny, Y. Yamada, and C. L. Tien, “Radiative transfer with dependent scattering by particles: Part 1–Theoretical investigation,” Trans. ASME 108, 608–613 (1982).
[CrossRef]

Chassefière, E.

P. Rannou, M. Cabane, E. Chassefière, R. Botet, C. P. McKay, and R. Courtin, “Titan’s geometric albedo: role of the fractal structure of the aerosols,” Icarus 118, 355–372 (1995).
[CrossRef]

M. Cabane, P. Rannou, E. Chassefière, and G. Israel, “Fractal aggregates in Titan’s atmosphere,” Planet. Space Sci. 41, 257–267 (1993).
[CrossRef]

Courtin, R.

P. Rannou, M. Cabane, E. Chassefière, R. Botet, C. P. McKay, and R. Courtin, “Titan’s geometric albedo: role of the fractal structure of the aerosols,” Icarus 118, 355–372 (1995).
[CrossRef]

Cruzan, O. R.

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–40 (1962).

Denti, P.

F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, “Reliability of the theoretical description of electromagnetic scattering by nonspherical particles,” J. Aerosol Sci. 20, 1079–1081 (1989).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, and O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 4, 227–235 (1984).
[CrossRef]

F. Borghese, P. Denti, G. Toscano, and O. I. Sindoni, “Electromagnetic scattering by a cluster of spheres,” Appl. Opt. 18, 116–120 (1979).
[CrossRef] [PubMed]

Friedman, B.

B. Friedman and J. Russek, “Addition theorems for spherical waves,” Quart. Appl. Math. 12, 13–23 (1954).

Fuller, K. A.

Gérardy, J. M.

J. M. Gérardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. II. Optical properties of aggregated metal spheres,” Phys. Rev. B 25, 4204–4229 (1982).
[CrossRef]

Greenberg, R.

D. S. McKay, T. D. Swindle, and R. Greenberg, “Asteroidal regoliths: what we do not know,” in Asteroids II, R. P. Binzel, T. Gehrels, and M. S. Matthews, eds. (University of Arizona Press, Tucson, Ariz., 1989), pp. 617–642.

Gustafson, B. A. S.

R. T. Wang and B. A. S. Gustafson, “Angular scattering and polarization by randomly oriented dumbbells and chains of spheres,” in Proceedings of the 1983 Scientific Conference on Obscuration and Aerosol Research, J. Farmer and R. Kohl, eds. (U.S. Army Chemical Research, Development, and Engineering Center, Aberdeen, Md., 1984), pp. 237–247.

Huffman, D. R.

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

Israel, G.

M. Cabane, P. Rannou, E. Chassefière, and G. Israel, “Fractal aggregates in Titan’s atmosphere,” Planet. Space Sci. 41, 257–267 (1993).
[CrossRef]

Kattawar, G. W.

Kerker, M.

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

Lo, Y. T.

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres, Part I—Multiple expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

Mackowski, D. W.

D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994).
[CrossRef]

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

McKay, C. P.

P. Rannou, M. Cabane, E. Chassefière, R. Botet, C. P. McKay, and R. Courtin, “Titan’s geometric albedo: role of the fractal structure of the aerosols,” Icarus 118, 355–372 (1995).
[CrossRef]

McKay, D. S.

D. S. McKay, T. D. Swindle, and R. Greenberg, “Asteroidal regoliths: what we do not know,” in Asteroids II, R. P. Binzel, T. Gehrels, and M. S. Matthews, eds. (University of Arizona Press, Tucson, Ariz., 1989), pp. 617–642.

Mie, G.

G. Mie, “Beiträage zue Optik trüber Medien speziell kolloidaler Metalläsungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

Rannou, P.

P. Rannou, M. Cabane, E. Chassefière, R. Botet, C. P. McKay, and R. Courtin, “Titan’s geometric albedo: role of the fractal structure of the aerosols,” Icarus 118, 355–372 (1995).
[CrossRef]

M. Cabane, P. Rannou, E. Chassefière, and G. Israel, “Fractal aggregates in Titan’s atmosphere,” Planet. Space Sci. 41, 257–267 (1993).
[CrossRef]

Russek, J.

B. Friedman and J. Russek, “Addition theorems for spherical waves,” Quart. Appl. Math. 12, 13–23 (1954).

Saija, R.

F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, “Reliability of the theoretical description of electromagnetic scattering by nonspherical particles,” J. Aerosol Sci. 20, 1079–1081 (1989).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, and O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 4, 227–235 (1984).
[CrossRef]

Saxon, D. S.

D. S. Saxon, “Tensor scattering matrix for the electromagnetic field,” Phys. Rev. 100, 1771–1775 (1955).
[CrossRef]

Schulz, K.

Sindoni, O. I.

F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, “Reliability of the theoretical description of electromagnetic scattering by nonspherical particles,” J. Aerosol Sci. 20, 1079–1081 (1989).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, and O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 4, 227–235 (1984).
[CrossRef]

F. Borghese, P. Denti, G. Toscano, and O. I. Sindoni, “Electromagnetic scattering by a cluster of spheres,” Appl. Opt. 18, 116–120 (1979).
[CrossRef] [PubMed]

Smith, P. H.

R. A. West and P. H. Smith, “Evidence for aggregate particles in the atmospheres of Titan and Jupiter,” Icarus 90, 330–333 (1991).
[CrossRef]

Stein, S.

S. Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1961).

Swindle, T. D.

D. S. McKay, T. D. Swindle, and R. Greenberg, “Asteroidal regoliths: what we do not know,” in Asteroids II, R. P. Binzel, T. Gehrels, and M. S. Matthews, eds. (University of Arizona Press, Tucson, Ariz., 1989), pp. 617–642.

Thiele-Corbach, E.

Tien, C. L.

J. D. Cartigny, Y. Yamada, and C. L. Tien, “Radiative transfer with dependent scattering by particles: Part 1–Theoretical investigation,” Trans. ASME 108, 608–613 (1982).
[CrossRef]

Tomeoka, K.

K. Tomeoka, “Aqueous alteration in hydrated interplanetary dust particles,” in Origin and Evolution of Interplanetary Dust, A. C. Levasseur-Regourd and H. Hasegawa, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1991), pp. 71–78.
[CrossRef]

Toscano, G.

F. Borghese, P. Denti, R. Saija, G. Toscano, and O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 4, 227–235 (1984).
[CrossRef]

F. Borghese, P. Denti, G. Toscano, and O. I. Sindoni, “Electromagnetic scattering by a cluster of spheres,” Appl. Opt. 18, 116–120 (1979).
[CrossRef] [PubMed]

Trinks, W.

W. Trinks, “Zur Vielfachstreung an kleinen Kugeln,” Ann. Phys. Dtsch. 22, 561–590 (1933).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Wang, R. T.

R. T. Wang, “Extinction signatures of non-spherical/non-isotropic particles,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, ed. (Plenum, New York, 1980), pp. 255–272.
[CrossRef]

R. T. Wang and B. A. S. Gustafson, “Angular scattering and polarization by randomly oriented dumbbells and chains of spheres,” in Proceedings of the 1983 Scientific Conference on Obscuration and Aerosol Research, J. Farmer and R. Kohl, eds. (U.S. Army Chemical Research, Development, and Engineering Center, Aberdeen, Md., 1984), pp. 237–247.

West, R. A.

R. A. West and P. H. Smith, “Evidence for aggregate particles in the atmospheres of Titan and Jupiter,” Icarus 90, 330–333 (1991).
[CrossRef]

Wiscombe, W. J.

Xu, Y.-l.

Yamada, Y.

J. D. Cartigny, Y. Yamada, and C. L. Tien, “Radiative transfer with dependent scattering by particles: Part 1–Theoretical investigation,” Trans. ASME 108, 608–613 (1982).
[CrossRef]

Zerull, R. H.

Aerosol Sci. Technol. (1)

F. Borghese, P. Denti, R. Saija, G. Toscano, and O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 4, 227–235 (1984).
[CrossRef]

Ann. Phys. (1)

G. Mie, “Beiträage zue Optik trüber Medien speziell kolloidaler Metalläsungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

Ann. Phys. Dtsch. (1)

W. Trinks, “Zur Vielfachstreung an kleinen Kugeln,” Ann. Phys. Dtsch. 22, 561–590 (1933).

Appl. Opt. (4)

Icarus (2)

R. A. West and P. H. Smith, “Evidence for aggregate particles in the atmospheres of Titan and Jupiter,” Icarus 90, 330–333 (1991).
[CrossRef]

P. Rannou, M. Cabane, E. Chassefière, R. Botet, C. P. McKay, and R. Courtin, “Titan’s geometric albedo: role of the fractal structure of the aerosols,” Icarus 118, 355–372 (1995).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres, Part I—Multiple expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

J. Aerosol Sci. (1)

F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, “Reliability of the theoretical description of electromagnetic scattering by nonspherical particles,” J. Aerosol Sci. 20, 1079–1081 (1989).
[CrossRef]

J. Opt. Soc. Am. A (2)

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

B. Å. S. Gustafson, “Microwave analog to light scattering measurements: a modern implementation of a proven method to achieve precise control,” J. Quant. Spectrosc. Radiat. Transfer 55, 663–672 (1996).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. (1)

D. S. Saxon, “Tensor scattering matrix for the electromagnetic field,” Phys. Rev. 100, 1771–1775 (1955).
[CrossRef]

Phys. Rev. B (1)

J. M. Gérardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. II. Optical properties of aggregated metal spheres,” Phys. Rev. B 25, 4204–4229 (1982).
[CrossRef]

Planet. Space Sci. (1)

M. Cabane, P. Rannou, E. Chassefière, and G. Israel, “Fractal aggregates in Titan’s atmosphere,” Planet. Space Sci. 41, 257–267 (1993).
[CrossRef]

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]

Quart. Appl. Math. (3)

B. Friedman and J. Russek, “Addition theorems for spherical waves,” Quart. Appl. Math. 12, 13–23 (1954).

S. Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1961).

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–40 (1962).

Trans. ASME (1)

J. D. Cartigny, Y. Yamada, and C. L. Tien, “Radiative transfer with dependent scattering by particles: Part 1–Theoretical investigation,” Trans. ASME 108, 608–613 (1982).
[CrossRef]

Other (14)

Y.-l. Xu and B. Å. S. Gustafson, “A complete and efficient multisphere scattering theory for modeling the optical properties of interplanetary dust,” in Physics, Chemistry, and Dynamics of Interplanetary Dust, B. Å. S. Gustafson and M. Hanner, eds., Vol. 104 of ASP Conference Series (Astronomical Society of the Pacific, San Francisco, 1996), pp. 419–422.

R. T. Wang, “Extinction signatures of non-spherical/non-isotropic particles,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, ed. (Plenum, New York, 1980), pp. 255–272.
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Y.-l. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996); erratum, 134, 200 (1997).

Y.-l. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. (to be published).

R. T. Wang and B. A. S. Gustafson, “Angular scattering and polarization by randomly oriented dumbbells and chains of spheres,” in Proceedings of the 1983 Scientific Conference on Obscuration and Aerosol Research, J. Farmer and R. Kohl, eds. (U.S. Army Chemical Research, Development, and Engineering Center, Aberdeen, Md., 1984), pp. 237–247.

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres, Part II—Numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19, 391–400 (1971).

L. V. Lorenz, “Sur la lumière réfléchie et réfractée par une sphère transparente,” in Qeuvres Scientifiques de L. Lorentz, revues et annotées par H. Valentiner, (Librairie Lehman et Stage, Copenhagen, Denmark, 1898), pp. 405–529.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

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

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

D. S. McKay, T. D. Swindle, and R. Greenberg, “Asteroidal regoliths: what we do not know,” in Asteroids II, R. P. Binzel, T. Gehrels, and M. S. Matthews, eds. (University of Arizona Press, Tucson, Ariz., 1989), pp. 617–642.

D. E. Brownlee, “Microparticle studies by sampling techniques,” in Cosmic Dust, J. A. M. McDonnel, ed. (Wiley, Chichester, UK, 1978), pp. 275–336.

J. P. Bradley, “Physical and mineralogical properties of anhydrous interplanetary dust particles in the analytical electron microscope,” in Origin and Evolution of Interplanetary Dust, A. C. Levasseur-Regourd and H. Hasegawa, eds (Kluwer Academic, Dordrecht, The Netherlands, 1991), pp. 63–70.
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K. Tomeoka, “Aqueous alteration in hydrated interplanetary dust particles,” in Origin and Evolution of Interplanetary Dust, A. C. Levasseur-Regourd and H. Hasegawa, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1991), pp. 71–78.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Comparison between theoretical predictions from multisphere-scattering calculations through the use of the reexpansion method and laboratory scattering measurements for two bisphere systems with identical constituents. The two spheres in the first system are in contact, and in the second system they have a center-to-center separation of s = 2.6a, where a is the radius of the constituent sphere that has a size parameter of 3.083 and a refractive index of 1.61- i 0.004. The scattering calculations use a different number of scattering terms in the reexpansion of individual scattered fields. Dotted curves are theoretical predictions for i 11 and solid curves, for i 22. Open circles are the laboratory scattering measurements for i 11 and filled circles, for i 22. Orientation, intersphere separation distance of the two-sphere chains, and the highest degree of N in the field-expansion truncation are shown in the upper right corner label. The axis of symmetry of the bisphere systems is aligned along the x axis. (The xz plane is the scattering plane, and the z axis is parallel to the plane incident wave vector.)

Fig. 2
Fig. 2

Same as in Fig. 1, except the bisphere system shown here has the intersphere separation of s = 4a.

Fig. 3
Fig. 3

Comparison of theoretical predictions for angular distributions of i 11 and i 22 between multisphere-scattering calculations and noninteracting solutions. The bisphere system shown here is similar to those in Figs. 1 and 2, except that the intersphere separation is s = 10a. Dotted curves are the noninteracting solutions, and solid curves are the multisphere-scattering calculations through the use of the reexpansion method.

Fig. 4
Fig. 4

Comparison between the theoretical predictions from the far-field solution described here and laboratory scattering measurements for angular distributions of i 11 and i 22. The three bisphere systems shown here are the same as those in Figs. 1 and 2, but, in addition to the orientation of x alignment (h), the orientation of y alignment (ν) is also shown. Dotted curves are theoretical predictions for i 11 and solid curves, for i 22. Open circles are the laboratory scattering measurements for i 11 and filled circles, for i 22.

Fig. 5
Fig. 5

Six theoretical and experimental (P,Q) plots. The upper three figures refer to three square arrays of four contacting identical spheres that have the refractive indexes of 1.365, 1,366, and 1.363 and size parameters of 3.12, 3.752, and 4.678 (from left to right), respectively. The bottom three figures refer to three cubic arrays of eight contacting identical spheres that have the same refractive index and the same size parameter as in the square four-sphere array shown in the same column. For all six sphere arrays, two side surfaces are parallel and the other four, perpendicular, to the scattering plane. The arrays are continuously rotated in the scattering plane by 90° from the initial orientation with two side surfaces perpendicular to the incident plane wave vector. Solid curves are theoretical predictions from the far-field solution described here, and dotted curves are laboratory measurements.

Fig. 6
Fig. 6

Comparison of angular distributions i 11 between theoretical predictions from the far-field solution and noninteracting solutions when the intersphere separation in the bisphere system changes from contacting to s = 100a. Dotted curves correspond to the noninteracting solutions and solid curves, to the far-field solution.

Equations (46)

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Es=n=1m=-nn iEmnamnNmn3kr, θ, ϕ+bmnMmn3kr, θ, ϕ,Hs=kωμn=1m=-nn EmnbmnNmn3kr, θ, ϕ+amnMmn3kr, θ, ϕ,
Emn=E0in2n+1n-m!n+m!,
Mμν3krl, θl, ϕl=n=1m=-nnEmnEμνAmnμνljMmn1krj, θj, ϕj+BmnμνljNmn1krj, θj, ϕj, Nμν3krl, θl, ϕl=n=1m=-nnEmnEμνAmnμνljNmn1krj, θj, ϕj+BmnμνljMmn1krj, θj, ϕj,
Mμν3krl, θl, ϕl=n=1m=-nnEmnEμνAmnμνlj0Mmn3kr, θ, ϕ+Bmnμνlj0Nmn3kr, θ, ϕ,Nμν3krl, θl, ϕl=n=1m=-nnEmnEμνAmnμνlj0Nmn3kr, θ, ϕ+Bmnμνlj0Mmn3kr, θ, ϕ.
amnj=anjpmnj-lj1, Lν=1μ=-ννAmnμνljaμνl+Bmnμνljbμνl,bmnj=bnjqmnj-lj1, Lν=1μ=-ννBmnμνljaμνl+Amnμνljbμνl,
Eslj0=n=1m=-nniEmnamnlj0Nmn3kr, θ, ϕ+bmnlj0Mmn3kr, θ, ϕ, Hslj0=kωμn=1m=-nn Emnbmnlj0Nmn3kr, θ, ϕ+amnlj0Mmn3kr, θ, ϕ,
amnlj0=ν=1μ=-ννAmnμνlj0aμνl+Bmnμνlj0bμνl,bmnlj0=ν=1μ=-ννBmnμνlj0aμνl+Amnμνlj0bμνl.
amn=l=1Lamnlj0=l=1Lν=1μ=-ννAmnμνlj0aμνl+Bmnμνlj0bμνl, bmn=l=1Lbmnlj0=l=1Lν=1μ=-ννBmnμνlj0aμνl+Amnμνlj0bμνl.
i11=S1yθ, 02,i22=S2xθ,02.
Mmn3=iθiπmncos θ-iϕτmncos θhn1krexpim ϕ, Nmn3=irnn+1Pnmcos θhn1krkrexpim ϕ,+iθτmncos θ+iϕiπmncos θ×1krddrrhn1krexpim ϕ,
πmncos θ=msinθPnmcos θ,τmncos θ=ddθPnmcos θ.
Mmn3=iθiπmncos θ-iϕ τmncos θ×ξnρρexpimϕ, r  , Nmn3=iθτmncos θ+iϕiπmncos θ×ξnρρexpimϕ, r  ,
ξnρ-in+1 expiρ, ξnρ -in expiρ,
Mmn3=-iniθπmncos θ+iϕiτmncos θ×expiρρexpim ϕ, r , Nmn3=-iniθτmncos θ+iϕiπmncos θ×expiρρexpim ϕ, r ,
rl=r-Δl,
Δl=ir·dl=Xl sin θ cos ϕ+Yl sin θ sin ϕ+Zl cos θ.
Mmn3krl, θl, ϕl=exp-ikΔlMmn3kr, θ, ϕ, r  , Nmn3krl, θl, ϕl=exp-ikΔlNmn3kr, θ, ϕ, r  ,
Amnμνlj0=δmμδnν exp-ikΔl,Bmnμνlj0 0, r  ,
Amnμνlj0=δmμδnν,Bmnμνlj00,
amnlj0=exp-ikΔlamnl, bmnlj0=exp-ikΔlbmnl,
Eslj0=exp-ikΔln=1m=-nniEmn×amnlNmn3kr, θ, ϕ+bmnlMmn3kr, θ, ϕ, r , Hslj0=kωμexp-ikΔln=1m=-nn Emn×bmnlNmn3kr, θ, ϕ+amnlMmn3kr,θ, ϕ, r  .
amn=l=1Lexp-ikΔlamnl, bmn=l=1Lexp-ikΔlbmnl.
Es=l=1Lexp-ikΔln=1m=-nn iEmnamnlNmn3kr, θ, ϕ+bmnlMmn3kr,θ,ϕ,Hs=kωμl=1L-ikΔln=1m=-nn EmnbmnlNmn3kr,θ,ϕ+amnlMmn3kr,θ,ϕ.
amn=amn1+amn2 exp-ikd12 cosα, bmn=bmn1+bmn2 exp-ikd12 cosα,
amn=amn1+amn2 exp-ikd12 cos θ, bmn=bmn1+bmn2 exp-ikd12 cos θ.
amn=amn1+amn2 exp-ikd12,bmn=bmn1+bmn2 exp-ikd12,
amn=amn1+amn2, bmn=bmn1+bmn2.
Esθ=l=1Lexp-ikΔln=1m=-nnEmn×iamnlξnτmn-bmnlξnπmnexpimϕkr, Esϕ=l=1Lexp-ikΔln=1m=-nnEmn×-ibmnlξnτmn-amnlξnπmnexpimϕkr, Hsθ=kωμ0l=1Lexp-ikΔln=1m=-nnEmn×iamnlξnπmn+bmnlξnτmnexpimϕkr, Hsϕ=kωμ0l=1Lexp-ikΔln=1m=-nnEmn×ibmnlξnπmn-amnlξnτmnexpimϕkr,
S2θ,ϕ=n=1m=0nΨmn cosm-1ϕ+β+iΦmn sinm-1ϕ+β,S3θ,ϕ=-n=1m=0nΨmn sinm-1ϕ+β-iΦmn cosm-1ϕ+β,S4θ,ϕ=-n=1m=0niΘmn cosm-1ϕ+β-Ξmn sinm-1ϕ+β,S1θ,ϕ=n=1m=0niΘmn sinm-1ϕ+β+Ξmn cosm-1ϕ+β,
Ψmn=2n+11+δ0mΓmn+Γ-mn, Φmn=2n+11+δ0mΓmn-Γ-mn, Θmn=2n+11+δ0mΛmn-Λ-mn, Ξmn=2n+11+δ0mΛmn+Λ-mn,
Γmn=n-m!n+m!l=1Lexp-ikΔlamnlτmn+bmnlπmn,Γ-mn=-1ml=1Lexp-ikΔla-mnlτmn-b-mnlπmn, Λmn=n-m!n+m!l=1Lexp-ikΔlamnlπmn+bmnlτmn,Λ-mn=-1ml=1Lexp-ikΔla-mnlπmn-b-mnlτmn.
Cext=4πk2Rel=1Lexp-ikZln=12n+1×p1n0*a1nl+q1n0*b1nl+n2n+12p-1n0*a-1,nl+q-1n0*b-1nl,
p1n0=q1n0=exp-iβ2, p-1n0=-q-1n0=-expiβ2nn+1.
Cextx=2πk2Rel=1Lexp-ikZln=12n+1a1nl+b1nl-nn+1a-1nl-b-1nl, Cexty=2πk2Rel=1L i exp-ikZln=12n+1a1nl+b1nl+nn+1a-1nl-b-1nl.
Cext=-12I0ReAE0×Hs*+Es×H0*·irdA=-12I0ReAE0×1LHsl*+1LEsl×H0*·irdA,
E0θl=n=1m=-nn Emn-ipmnlψnτmn+qmnlψnπmnexpim ϕlkrl, E0ϕl=n=1m=-nn Emniqmnlψnτmn+pmnlψnπmnexpimϕlkrl, H0θl=kωμ0n=1m=-nn Emn-ipmnlψnπmn-qmnlψnτmn×expimϕlkrlH0ϕl=kωμ0n=1m=-nn Emn-iqmnlψnπmn+pmnlψnτmn×expimϕlkrl,
Esθl=n=1m=-nn Emniamnlξnτmn-bmnlξnπmnexpimϕlkrl,Esϕl=n=1m=-nn Emn-ibmnlξnτmn-amnlξnπmnexpimϕlkrl,Hsθl=kωμ0n=1m=-nn Emniamnlξnπmn+bmnlξnτmn×expimϕlkrl,Hsϕl=kωμ0n=1m=-nn Emnibmnlξnπmn-amnlξnτmn×expIm ϕlkrl.
Cext=4πk2l=1Ln=1m=-nn nn+12n+1×n-m!n+m!Repmnl* amnl+qmnl* bmnl.
Cext=4πk2l=1Ln=12n+1×Rep1nl*a1nl+q1nl*b1nl+n2n+12×p-1nl*a-1nl+q-1nl*b-1nl.
pmnl=expikZlpmn0, qmnl=expikZlqmn0,
Cabsl=4πk2n=1m=-nnnn+12n+1n-m!n+m!×Reiμmlμlψnψn*cmnl2-ψnψn*dmnl2,
anldmnl=dnlamnl, bnlcmnl=cnlbmnl,
Re-iml*ψn*ψn=Reimlψnψn*,
Cabs=4πk21Ln=1m=-nn nn+12n+1×n-m!n+m!Dnlamnl2+Cnlbmnl2,
Dnl=Reimlμμlψnylψn*ylμmlψnylψnxl-μlψnxlψnyl2, Cnl=Reiml* μμlψnylψn*ylμlψnylψnxl-μmlψnxlψnyl2.
amnlj0=exp-ikΔlanlpmnl,bmnlj0=exp-ikΔlbnlqmnl.

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