Abstract

Microwave analog measurements of the forward scattering produced by two spheres yield the extinction as a function of sphere separation for the case of one sphere's being shadowed by the other. The effects of dependent scattering are obvious up to a separation distance of about ten sphere diameters. Side-scattering measurements show a resonance when the axis of the two-sphere system is in the scattering plane and bisects the scattering angle. The magnitude of at least one measured resonance is a factor of 44 larger than the scattering that is due to a single sphere.

© 1981 Optical Society of America

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References

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  1. W. Trinks, “Zur Vielfachstreung an kleinen Kugeln,” Ann. Physik 22, 561–590 (1935).
    [CrossRef]
  2. P. C. Waterman, R. Truell, “Multiple scattering of waves,” J. Math. Phys. 2, 512–537 (1961).
    [CrossRef]
  3. O. A. Germogenova (Hermogenova), “The scattering of a plane electromagnetic wave by two spheres,” Akad. Nauk SSSR 4, 403–405 (1963).
  4. C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).
  5. V. Twersky, “Multiple scattering of electromagnetic waves by arbitrary configurations,” J. Math. Phys. 8, 589–610 (1967).
    [CrossRef]
  6. S. Levine, G. O. Olaofe, “Scattering of electromagnetic waves by two equal spherical particles,” J. Colloid Interface Sci. 27, 442–457 (1968).
    [CrossRef]
  7. J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. I, Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971);“Multiple scattering of EM waves by spheres. II, Numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19391–400 (1971).
    [CrossRef]
  8. V. I. Rozenberg, “Diffraction of electromagnetic waves from an arbitrary set of spheres,” Radio Eng. Electron. Phys. (USSR) 16, 394–404 (1971).
  9. K. Hongo, “Multiple scattering by two conducting circular cylinders,” IEEE Trans. Antennas Propag. AP-26, 748–751 (1978).
    [CrossRef]
  10. F. Borghese, P. Denti, G. Toscano, O. I. Sindoni, “Electromagnetic scattering by a cluster of spheres,” Appl. Opt. 18, 116–120 (1979).
    [CrossRef] [PubMed]
  11. G. W. Kattawar, T. J. Humphreys, “Electromagnetic scattering from two identical pseudospheres,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, ed. (Plenum, New York, 1980), pp. 177–190.
    [CrossRef]
  12. J. H. Bruning, Y. T. Lo, “Multiple scattering by spheres,” Tech. Rep. 69-5 (Antenna Laboratory, University of Illinois, Urbana, Ill., 1969).
  13. A. Goyette, A. Navon, “Two dielectric spheres in an electric field,” Phys. Rev. B 13, 4320–4327 (1976).
    [CrossRef]
  14. D. H. Woodward, “Multiple light scattering by spherical dielectric particles,” J. Opt. Soc. Am. 54, 1325–1331 (1964).
    [CrossRef]
  15. D. J. Angelakos, N. Kumagai, “High frequency scattering by multiple spheres,” IEEE Trans. Antennas Propag. AP-12, 105–109 (1964).
    [CrossRef]
  16. C. I. Beard, T. H. Kays, V. Twersky, “Scattering by random distribution of spheres vs. concentration,” IEEE Trans. Antennas Propag. AP-15, 99–118 (1967), and references therein.
    [CrossRef]
  17. S. W. Hawley, T. H. Kays, V. Twersky, “Comparison of distribution functions from scattering data on different sets of spheres,” IEEE Trans. Antennas Propag. AP-15, 118–135 (1967).
    [CrossRef]
  18. R. T. Wang, “Electromagnetic scattering by spheres of anisotropic refractive indices,” Ph.D. thesis (Rensselaer Polytechnic Institute, Troy, N.Y., 1968).
  19. A. C. Lind, R. T. Wang, J. M. Greenberg, “Microwave scattering by nonspherical particles,” Appl. Opt. 4, 1555–1561 (1965).
    [CrossRef]
  20. 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]
  21. D. W. Schuerman, R. T. Wang, “Experimental results of multiple scattering,” Final Rep., ARCSL-CR-81-003 (U.S. Army Chemical Systems Laboratory, Aberdeen Proving Ground, Md., July, 1980).
  22. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

1979

1978

K. Hongo, “Multiple scattering by two conducting circular cylinders,” IEEE Trans. Antennas Propag. AP-26, 748–751 (1978).
[CrossRef]

1976

A. Goyette, A. Navon, “Two dielectric spheres in an electric field,” Phys. Rev. B 13, 4320–4327 (1976).
[CrossRef]

1971

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. I, Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971);“Multiple scattering of EM waves by spheres. II, Numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19391–400 (1971).
[CrossRef]

V. I. Rozenberg, “Diffraction of electromagnetic waves from an arbitrary set of spheres,” Radio Eng. Electron. Phys. (USSR) 16, 394–404 (1971).

1968

S. Levine, G. O. Olaofe, “Scattering of electromagnetic waves by two equal spherical particles,” J. Colloid Interface Sci. 27, 442–457 (1968).
[CrossRef]

1967

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

V. Twersky, “Multiple scattering of electromagnetic waves by arbitrary configurations,” J. Math. Phys. 8, 589–610 (1967).
[CrossRef]

C. I. Beard, T. H. Kays, V. Twersky, “Scattering by random distribution of spheres vs. concentration,” IEEE Trans. Antennas Propag. AP-15, 99–118 (1967), and references therein.
[CrossRef]

S. W. Hawley, T. H. Kays, V. Twersky, “Comparison of distribution functions from scattering data on different sets of spheres,” IEEE Trans. Antennas Propag. AP-15, 118–135 (1967).
[CrossRef]

1965

1964

D. H. Woodward, “Multiple light scattering by spherical dielectric particles,” J. Opt. Soc. Am. 54, 1325–1331 (1964).
[CrossRef]

D. J. Angelakos, N. Kumagai, “High frequency scattering by multiple spheres,” IEEE Trans. Antennas Propag. AP-12, 105–109 (1964).
[CrossRef]

1963

O. A. Germogenova (Hermogenova), “The scattering of a plane electromagnetic wave by two spheres,” Akad. Nauk SSSR 4, 403–405 (1963).

1961

P. C. Waterman, R. Truell, “Multiple scattering of waves,” J. Math. Phys. 2, 512–537 (1961).
[CrossRef]

1935

W. Trinks, “Zur Vielfachstreung an kleinen Kugeln,” Ann. Physik 22, 561–590 (1935).
[CrossRef]

Angelakos, D. J.

D. J. Angelakos, N. Kumagai, “High frequency scattering by multiple spheres,” IEEE Trans. Antennas Propag. AP-12, 105–109 (1964).
[CrossRef]

Beard, C. I.

C. I. Beard, T. H. Kays, V. Twersky, “Scattering by random distribution of spheres vs. concentration,” IEEE Trans. Antennas Propag. AP-15, 99–118 (1967), and references therein.
[CrossRef]

Borghese, F.

Bruning, J. H.

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. I, Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971);“Multiple scattering of EM waves by spheres. II, Numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19391–400 (1971).
[CrossRef]

J. H. Bruning, Y. T. Lo, “Multiple scattering by spheres,” Tech. Rep. 69-5 (Antenna Laboratory, University of Illinois, Urbana, Ill., 1969).

Denti, P.

Germogenova, O. A.

O. A. Germogenova (Hermogenova), “The scattering of a plane electromagnetic wave by two spheres,” Akad. Nauk SSSR 4, 403–405 (1963).

Goyette, A.

A. Goyette, A. Navon, “Two dielectric spheres in an electric field,” Phys. Rev. B 13, 4320–4327 (1976).
[CrossRef]

Greenberg, J. M.

Hawley, S. W.

S. W. Hawley, T. H. Kays, V. Twersky, “Comparison of distribution functions from scattering data on different sets of spheres,” IEEE Trans. Antennas Propag. AP-15, 118–135 (1967).
[CrossRef]

Hongo, K.

K. Hongo, “Multiple scattering by two conducting circular cylinders,” IEEE Trans. Antennas Propag. AP-26, 748–751 (1978).
[CrossRef]

Humphreys, T. J.

G. W. Kattawar, T. J. Humphreys, “Electromagnetic scattering from two identical pseudospheres,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, ed. (Plenum, New York, 1980), pp. 177–190.
[CrossRef]

Kattawar, G. W.

G. W. Kattawar, T. J. Humphreys, “Electromagnetic scattering from two identical pseudospheres,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, ed. (Plenum, New York, 1980), pp. 177–190.
[CrossRef]

Kays, T. H.

C. I. Beard, T. H. Kays, V. Twersky, “Scattering by random distribution of spheres vs. concentration,” IEEE Trans. Antennas Propag. AP-15, 99–118 (1967), and references therein.
[CrossRef]

S. W. Hawley, T. H. Kays, V. Twersky, “Comparison of distribution functions from scattering data on different sets of spheres,” IEEE Trans. Antennas Propag. AP-15, 118–135 (1967).
[CrossRef]

Kumagai, N.

D. J. Angelakos, N. Kumagai, “High frequency scattering by multiple spheres,” IEEE Trans. Antennas Propag. AP-12, 105–109 (1964).
[CrossRef]

Levine, S.

S. Levine, G. O. Olaofe, “Scattering of electromagnetic waves by two equal spherical particles,” J. Colloid Interface Sci. 27, 442–457 (1968).
[CrossRef]

Liang, C.

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

Lind, A. C.

Lo, Y. T.

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. I, Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971);“Multiple scattering of EM waves by spheres. II, Numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19391–400 (1971).
[CrossRef]

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

J. H. Bruning, Y. T. Lo, “Multiple scattering by spheres,” Tech. Rep. 69-5 (Antenna Laboratory, University of Illinois, Urbana, Ill., 1969).

Navon, A.

A. Goyette, A. Navon, “Two dielectric spheres in an electric field,” Phys. Rev. B 13, 4320–4327 (1976).
[CrossRef]

Olaofe, G. O.

S. Levine, G. O. Olaofe, “Scattering of electromagnetic waves by two equal spherical particles,” J. Colloid Interface Sci. 27, 442–457 (1968).
[CrossRef]

Rozenberg, V. I.

V. I. Rozenberg, “Diffraction of electromagnetic waves from an arbitrary set of spheres,” Radio Eng. Electron. Phys. (USSR) 16, 394–404 (1971).

Schuerman, D. W.

D. W. Schuerman, R. T. Wang, “Experimental results of multiple scattering,” Final Rep., ARCSL-CR-81-003 (U.S. Army Chemical Systems Laboratory, Aberdeen Proving Ground, Md., July, 1980).

Sindoni, O. I.

Toscano, G.

Trinks, W.

W. Trinks, “Zur Vielfachstreung an kleinen Kugeln,” Ann. Physik 22, 561–590 (1935).
[CrossRef]

Truell, R.

P. C. Waterman, R. Truell, “Multiple scattering of waves,” J. Math. Phys. 2, 512–537 (1961).
[CrossRef]

Twersky, V.

S. W. Hawley, T. H. Kays, V. Twersky, “Comparison of distribution functions from scattering data on different sets of spheres,” IEEE Trans. Antennas Propag. AP-15, 118–135 (1967).
[CrossRef]

C. I. Beard, T. H. Kays, V. Twersky, “Scattering by random distribution of spheres vs. concentration,” IEEE Trans. Antennas Propag. AP-15, 99–118 (1967), and references therein.
[CrossRef]

V. Twersky, “Multiple scattering of electromagnetic waves by arbitrary configurations,” J. Math. Phys. 8, 589–610 (1967).
[CrossRef]

van de Hulst, H. C.

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

Wang, R. T.

A. C. Lind, R. T. Wang, J. M. Greenberg, “Microwave scattering by nonspherical particles,” Appl. Opt. 4, 1555–1561 (1965).
[CrossRef]

D. W. Schuerman, R. T. Wang, “Experimental results of multiple scattering,” Final Rep., ARCSL-CR-81-003 (U.S. Army Chemical Systems Laboratory, Aberdeen Proving Ground, Md., July, 1980).

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, “Electromagnetic scattering by spheres of anisotropic refractive indices,” Ph.D. thesis (Rensselaer Polytechnic Institute, Troy, N.Y., 1968).

Waterman, P. C.

P. C. Waterman, R. Truell, “Multiple scattering of waves,” J. Math. Phys. 2, 512–537 (1961).
[CrossRef]

Woodward, D. H.

Akad. Nauk SSSR

O. A. Germogenova (Hermogenova), “The scattering of a plane electromagnetic wave by two spheres,” Akad. Nauk SSSR 4, 403–405 (1963).

Ann. Physik

W. Trinks, “Zur Vielfachstreung an kleinen Kugeln,” Ann. Physik 22, 561–590 (1935).
[CrossRef]

Appl. Opt.

IEEE Trans. Antennas Propag.

K. Hongo, “Multiple scattering by two conducting circular cylinders,” IEEE Trans. Antennas Propag. AP-26, 748–751 (1978).
[CrossRef]

D. J. Angelakos, N. Kumagai, “High frequency scattering by multiple spheres,” IEEE Trans. Antennas Propag. AP-12, 105–109 (1964).
[CrossRef]

C. I. Beard, T. H. Kays, V. Twersky, “Scattering by random distribution of spheres vs. concentration,” IEEE Trans. Antennas Propag. AP-15, 99–118 (1967), and references therein.
[CrossRef]

S. W. Hawley, T. H. Kays, V. Twersky, “Comparison of distribution functions from scattering data on different sets of spheres,” IEEE Trans. Antennas Propag. AP-15, 118–135 (1967).
[CrossRef]

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. I, Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971);“Multiple scattering of EM waves by spheres. II, Numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19391–400 (1971).
[CrossRef]

J. Colloid Interface Sci.

S. Levine, G. O. Olaofe, “Scattering of electromagnetic waves by two equal spherical particles,” J. Colloid Interface Sci. 27, 442–457 (1968).
[CrossRef]

J. Math. Phys.

V. Twersky, “Multiple scattering of electromagnetic waves by arbitrary configurations,” J. Math. Phys. 8, 589–610 (1967).
[CrossRef]

P. C. Waterman, R. Truell, “Multiple scattering of waves,” J. Math. Phys. 2, 512–537 (1961).
[CrossRef]

J. Opt. Soc. Am.

Phys. Rev. B

A. Goyette, A. Navon, “Two dielectric spheres in an electric field,” Phys. Rev. B 13, 4320–4327 (1976).
[CrossRef]

Radio Eng. Electron. Phys. (USSR)

V. I. Rozenberg, “Diffraction of electromagnetic waves from an arbitrary set of spheres,” Radio Eng. Electron. Phys. (USSR) 16, 394–404 (1971).

Radio Sci.

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

Other

G. W. Kattawar, T. J. Humphreys, “Electromagnetic scattering from two identical pseudospheres,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, ed. (Plenum, New York, 1980), pp. 177–190.
[CrossRef]

J. H. Bruning, Y. T. Lo, “Multiple scattering by spheres,” Tech. Rep. 69-5 (Antenna Laboratory, University of Illinois, Urbana, Ill., 1969).

R. T. Wang, “Electromagnetic scattering by spheres of anisotropic refractive indices,” Ph.D. thesis (Rensselaer Polytechnic Institute, Troy, N.Y., 1968).

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]

D. W. Schuerman, R. T. Wang, “Experimental results of multiple scattering,” Final Rep., ARCSL-CR-81-003 (U.S. Army Chemical Systems Laboratory, Aberdeen Proving Ground, Md., July, 1980).

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

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Figures (2)

Fig. 1
Fig. 1

A–D: each figure is a P,Q plot of forward scattering by two identical spheres in continuous separation along the incident direction. Each curve is parameterized by the (dimensionless) mutual separation ks. The index of refraction (m) and the size parameter (x) of one component of each two-sphere array are indicated in their respective plots. A solid straight line marked SNGL SPHERE in each P,Q plot represents the complex scattering amplitude A, S(0) for one spherical component. The P and Q obtained from the projection of each of these solid lines are one half of their true values; the P, Q scales are defined with G equal to the area of the two spheres, not one. The dotted extensions of these lines indicate the scattering that is due to two spheres without interaction (independent scattering). In plot A, the S(0) vector for a calibration sphere (x = 4.993, m = 1.267) is also shown.

Fig. 2
Fig. 2

The scattering intensity i1(θ = 90°) versus orientation (χ) for two spheres in contact. When χ = 0, the axis of the two-sphere system, which is always confined to the scattering plane, is parallel to the incident radiation; χ = 90° when the axis is perpendicular to the incident radiation. The spheres used here have a size parameter of 4.341 and an index of refraction of 1.610–i0.004.

Equations (1)

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P = 4 π k 2 G Im [ S ( 0 ) ] , Q = 4 π k 2 G Re [ S ( 0 ) ] ,

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