Abstract

We present laboratory microwave scattering measurements for complex amplitude scattering matrices of three aggregates of 2, 8, and 27 identical spheres and compare them with theoretical predictions. Electromagnetic multiparticle-scattering calculations involve the determination of a large number of vector translation coefficients introduced by the addition theorems for vector spherical harmonics. For one of the two classes of vector translation coefficients there is an overall-sign discrepancy between two groups of formulations that exist in the literature. We compare our experimental data with the theoretical results from scattering calculations using the two different sets of formulas for computation of the translation coefficients. This comparison of experiment with theory reveals that Cruzan’s original research on the vector addition theorems [Q. Appl. Math. 20, 33–40 (1962)] is correct, although many authors believe that Cruzan’s formulation contains an overall-sign error.

© 1997 Optical Society of America

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References

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  1. B. Friedman, J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23 (1954).
  2. S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).
  3. O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).
  4. J. H. Bruning, “Multiple scattering by spheres,” Ph.D. dissertation (Department of Electrical Engineering, University of Illinois, Urbana, Ill., 1969).
  5. J. H. Bruning, 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]
  6. D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
    [Crossref]
  7. K. T. Kim, “The translation formula for vector multipole fields and the recurrence relations of the translation coefficients of scalar and vector multipole fields,” IEEE Trans. Antennas Propag. 44, 1482–1487 (1996).
    [Crossref]
  8. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1957), Chap. 3, p. 46.
  9. E. P. Wigner, “On the matrices which reduce the Krönecker products of representations of simply reducible groups,” in Quantum Theory of Angular Momentum, L. C. Biedenharn, H. van Dam, eds. (Academic, New York, 1965), pp. 89–132.
  10. Y.-l. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995).
    [Crossref] [PubMed]
  11. Y.-l. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996); erratum, J. Comput. Phys. 134, 200 (1997).
  12. J. A. Gaunt, “On the triplets of helium,” Philos. Trans. R. Soc. London Ser. A 228, 151–196 (1929).
    [Crossref]
  13. G. W. Kattawar, C. E. Dean, “Electromagnetic scattering from two dielectric spheres: comparison between theory and experiment,” Opt. Lett. 8, 48–51 (1983).
    [Crossref] [PubMed]
  14. K. A. Fuller, G. W. Kattawar, R. T. Wang, “Electromagnetic scattering from two dielectric spheres: further comparisons between theory and experiment,” Appl. Opt. 25, 2521–2529 (1986).
    [Crossref] [PubMed]
  15. K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by ensembles of spheres. I: Linear Chains,” Opt. Lett. 13, 90–92 (1988).
    [Crossref]
  16. K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by ensembles of spheres. II: Clusters of arbitrary configurations,” Opt. Lett. 13, 1063–1065 (1988).
    [Crossref] [PubMed]
  17. R. T. Wang, J. M. Greenberg, D. W. Schuerman, “Experimental results of dependent light scattering by two spheres,” Opt. Lett. 6, 543–545 (1981).
    [Crossref] [PubMed]
  18. R. T. Wang, “Extinction by dumbells and chains of spheres,” in Proceedings of the 1983 CRDC Scientific Conference on Obscuration and Aerosol Research, J. Farmer, R. H. Kohl, eds. (U.S. Army Chemical Research, Development, and Engineering Center, Aberdeen Proving Grounds, Md., 1984), pp. 237–247.
  19. 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]
  20. Y.-l. Xu, “Electromagnetic scattering by an aggregate of spheres: far field,” Appl. Opt. (to be published).
  21. Y.-l. Xu, “Fast evaluation of the Gaunt coefficients,” Math. Comput. 65, 1601–1612 (1996).
    [Crossref]
  22. Y.-l. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. (to be published).
  23. Y.-I. Xu, “Fast evaluation of Gaunt coefficients: recursive approach,” J. Comput. Appl. Math. (to be published).

1996 (4)

K. T. Kim, “The translation formula for vector multipole fields and the recurrence relations of the translation coefficients of scalar and vector multipole fields,” IEEE Trans. Antennas Propag. 44, 1482–1487 (1996).
[Crossref]

Y.-l. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996); erratum, J. Comput. Phys. 134, 200 (1997).

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]

Y.-l. Xu, “Fast evaluation of the Gaunt coefficients,” Math. Comput. 65, 1601–1612 (1996).
[Crossref]

1995 (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]

1988 (2)

1986 (1)

1983 (1)

1981 (1)

1971 (1)

J. H. Bruning, 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,” Q. Appl. Math. 20, 33–40 (1962).

1961 (1)

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

1954 (1)

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

1929 (1)

J. A. Gaunt, “On the triplets of helium,” Philos. Trans. R. Soc. London Ser. A 228, 151–196 (1929).
[Crossref]

Bruning, J. H.

J. H. Bruning, 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. H. Bruning, “Multiple scattering by spheres,” Ph.D. dissertation (Department of Electrical Engineering, University of Illinois, Urbana, Ill., 1969).

Cruzan, O. R.

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

Dean, C. E.

Edmonds, A. R.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1957), Chap. 3, p. 46.

Friedman, B.

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

Fuller, K. A.

Gaunt, J. A.

J. A. Gaunt, “On the triplets of helium,” Philos. Trans. R. Soc. London Ser. A 228, 151–196 (1929).
[Crossref]

Greenberg, J. M.

Gustafson, B. Å. S.

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]

Kattawar, G. W.

Kim, K. T.

K. T. Kim, “The translation formula for vector multipole fields and the recurrence relations of the translation coefficients of scalar and vector multipole fields,” IEEE Trans. Antennas Propag. 44, 1482–1487 (1996).
[Crossref]

Lo, Y. T.

J. H. Bruning, 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, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
[Crossref]

Russek, J.

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

Schuerman, D. W.

Stein, S.

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

Wang, R. T.

K. A. Fuller, G. W. Kattawar, R. T. Wang, “Electromagnetic scattering from two dielectric spheres: further comparisons between theory and experiment,” Appl. Opt. 25, 2521–2529 (1986).
[Crossref] [PubMed]

R. T. Wang, J. M. Greenberg, D. W. Schuerman, “Experimental results of dependent light scattering by two spheres,” Opt. Lett. 6, 543–545 (1981).
[Crossref] [PubMed]

R. T. Wang, “Extinction by dumbells and chains of spheres,” in Proceedings of the 1983 CRDC Scientific Conference on Obscuration and Aerosol Research, J. Farmer, R. H. Kohl, eds. (U.S. Army Chemical Research, Development, and Engineering Center, Aberdeen Proving Grounds, Md., 1984), pp. 237–247.

Wigner, E. P.

E. P. Wigner, “On the matrices which reduce the Krönecker products of representations of simply reducible groups,” in Quantum Theory of Angular Momentum, L. C. Biedenharn, H. van Dam, eds. (Academic, New York, 1965), pp. 89–132.

Xu, Y.-I.

Y.-I. Xu, “Fast evaluation of Gaunt coefficients: recursive approach,” J. Comput. Appl. Math. (to be published).

Xu, Y.-l.

Y.-l. Xu, “Fast evaluation of the Gaunt coefficients,” Math. Comput. 65, 1601–1612 (1996).
[Crossref]

Y.-l. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996); erratum, J. Comput. Phys. 134, 200 (1997).

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

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

Y.-l. Xu, “Electromagnetic scattering by an aggregate of spheres: far field,” Appl. Opt. (to be published).

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (2)

J. H. Bruning, 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]

K. T. Kim, “The translation formula for vector multipole fields and the recurrence relations of the translation coefficients of scalar and vector multipole fields,” IEEE Trans. Antennas Propag. 44, 1482–1487 (1996).
[Crossref]

J. Comput. Phys. (1)

Y.-l. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996); erratum, J. Comput. Phys. 134, 200 (1997).

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]

Math. Comput. (1)

Y.-l. Xu, “Fast evaluation of the Gaunt coefficients,” Math. Comput. 65, 1601–1612 (1996).
[Crossref]

Opt. Lett. (4)

Philos. Trans. R. Soc. London Ser. A (1)

J. A. Gaunt, “On the triplets of helium,” Philos. Trans. R. Soc. London Ser. A 228, 151–196 (1929).
[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]

Q. Appl. Math. (3)

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

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

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

Other (7)

J. H. Bruning, “Multiple scattering by spheres,” Ph.D. dissertation (Department of Electrical Engineering, University of Illinois, Urbana, Ill., 1969).

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1957), Chap. 3, p. 46.

E. P. Wigner, “On the matrices which reduce the Krönecker products of representations of simply reducible groups,” in Quantum Theory of Angular Momentum, L. C. Biedenharn, H. van Dam, eds. (Academic, New York, 1965), pp. 89–132.

Y.-l. Xu, “Electromagnetic scattering by an aggregate of spheres: far field,” Appl. Opt. (to be published).

R. T. Wang, “Extinction by dumbells and chains of spheres,” in Proceedings of the 1983 CRDC Scientific Conference on Obscuration and Aerosol Research, J. Farmer, R. H. Kohl, eds. (U.S. Army Chemical Research, Development, and Engineering Center, Aberdeen Proving Grounds, Md., 1984), pp. 237–247.

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

Y.-I. Xu, “Fast evaluation of Gaunt coefficients: recursive approach,” J. Comput. Appl. Math. (to be published).

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Figures (6)

Fig. 1
Fig. 1

Comparison of theoretical calculations with laboratory scattering measurements for angular distributions of polarization components of scattered intensity by two contacting identical BK7 glass spheres. Each BK7 sphere has a size parameter of 7.86 and a refractive index of 2.518 – 0.023i.

Fig. 2
Fig. 2

Comparison of theoretical calculations with laboratory scattering measurements for angular distributions of polarization components of scattered intensity by a cubic array of eight identical acrylic spheres. Each acrylic sphere has a size parameter of 5.03 and a refractive index of 1.615 - 0.008i.

Fig. 3
Fig. 3

Comparison of theoretical calculations with laboratory scattering measurements for angular distributions of polarization components of scattered intensity by a cubic array of 27 identical acrylic spheres. Each sphere has a size parameter of 5.03 and a refractive index of 1.615 - 0.008i.

Fig. 4
Fig. 4

Comparison of the multisphere-scattering calculations with an opposite (to that of Cruzan) overall sign for B translation coefficients with laboratory scattering measurements for the two-sphere system shown in Fig. 1.

Fig. 5
Fig. 5

Comparison of the multisphere-scattering calculations with an opposite (to that of Cruzan) overall sign for B translation coefficients with laboratory scattering measurements for the cubic eight-sphere array shown in Fig. 2.

Fig. 6
Fig. 6

Comparison of the multisphere-scattering calculations with an opposite (to that of Cruzan) overall sign for B translation coefficients with laboratory scattering measurements for the cubic 27-sphere array shown in Fig. 3.

Equations (11)

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Amnμνlj=-1m2ν+1n+m!ν-μ!2nn+1n-m!ν+μ!×expiμ-mϕljq=0qmax ipnn+1+νν+1-pp+1a-m, n, μ, ν, php1kdljPpμ-mcosθlj,
Bmnμνlj=-1m+12ν+1n+m!ν-μ!2nn+1n-m!ν+μ!×expiμ-mϕljq=1Θmaxip+1p+12-n-ν2×n+ν+12-p+121/2b-m, n, μ, ν, p+1, php+11kdljPp+1μ-mcos θlj,
a-m, n, μ, ν, p=-1μ-m2p+1×n-m!ν+μ!p+m-μ!n+m!ν-μ!p-m+μ!1/2×nνp-mμm-μnνp000,
b-m, n, μ, ν, p+1, p=-1μ-m2p+3×n-m!ν+μ!p+m-μ+1!n+m!ν-μ!p-m+μ+1!1/2×nνp+1-mμm-μnνp000,
p=n+ν-2q, qmax=minn, ν,n+ν-m-μ2, Θmax=minn, ν,n+ν+1-m-μ2,
j1j2j3m1m2m3
EsEs=expikr-z-ikrS2S3S4S1EiEi,
i11=S1y θ2, i22=S2x θ2,
Amnμνlj=-1miν+nn+2n-1ν+2ν+1n+ν+m-μ!4nn+ν+1n+νn-m!ν+μ!×expiμ-mϕlj×q=0qmax-1qnn+1+νν+1-pp+1a˜1qhp1kdljPpμ-mcos θlj,
Bmnμνlj=-1miν+n+1n+2n+1ν+2ν+1n+ν+m-μ+1!4nn+1n+m+1n+ν+2n+ν+1n-m!ν+μ!×expiμ-mϕljq=0Qmax-1q2n+1ν-μa˜2q-pp+3-νν+1-nn+3-2μn+1a˜3qhp+11kdljPp+1μ-mcos θlj,
Qmax=minn+1, ν,n+ν+1-m-μ2,

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