Abstract

The generalized multiparticle Mie-solution (GMM) is an extension of the well-known Mie-theory for single homogeneous spheres to the general case of an arbitrary ensemble of variously sized and shaped particles. The present work explores its specific application to periodic structures, starting from one- and two-dimensional regular arrays of identical particles. Emphasis is placed on particle arrays with a truncated periodic structure, i.e., periodic arrays (PAs) with finite overall dimensions. To predict radiative scattering characteristics of a PA with a large number of identical particles within the framework of the GMM, it is sufficient to solve interactive scattering for only one single component particle, unlike the general case where partial scattered fields must be solved for every individual constituent. The total scattering from an array as a whole is simply the convolution of the scattering from a single representative scattering center with the periodic spatial distribution of all replica constituent units, in the terminology of Fourier analysis. Implemented in practical calculations, both computing time and computer memory required by the special version of GMM formulation applicable to PAs are trivial for ordinary desktops and laptops. For illustration, the radiative scattering properties of several regular arrays of identical particles at a fixed spatial orientation are computed and analyzed. Numerical results obtained from the newly developed approach for PAs are compared with those calculated from the general GMM computer codes (that have been available online for about a decade). The two sets of numerical outputs show no significant relative deviations. However, the CPU time required by the specific approach for PAs could drop more than 10,000 times, in comparison with the general approach. In addition, an example PA is also presented, which consists of as large as 108 particles and the general solution process is unable to handle.

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  1. P. C. Waterman and N. E. Pedersen, “Electromagnetic scattering by periodic arrays of particles,” J. Appl. Phys. 59, 2609–2618 (1986).
    [CrossRef]
  2. I. E. Psarobas and N. Stefanov, “Scattering of elastic waves by periodic arrays of spherical bodies,” Phys. Rev. B 62, 278–291 (2000).
    [CrossRef]
  3. A. D. Yaghjian, “Scattering-matrix analysis of linear periodic arrays,” IEEE Trans. Antennas Propag. 50, 1050–1064(2002).
    [CrossRef]
  4. D. A. Genov, A. K. Sarychev, V. M. Shalaev, and A. Wei, “Resonant field enhancements from metal nanoparticle arrays,” Nano Lett. 4, 153–158 (2003).
    [CrossRef]
  5. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for periodic targets: theory and tests,” J. Opt. Soc. Am. A 25, 2693–2703 (2008).
    [CrossRef]
  6. F. J. García de Abajo, “Colloquium: light scattering by particle and hole arrays,” 2009, http://arXiv.org/pdf/0903.1671.pdf .
  7. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,”Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
  8. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [CrossRef]
  9. B. T. Draine and P. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
    [CrossRef]
  10. B. Friedman and J. Russek, “Addition theorems for spherical waves,” Quart. Appl. Math. 12, 13–23 (1954).
  11. S. Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1967).
  12. O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–40(1962).
  13. C. Liang and Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).
  14. J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres, part I & II,” IEEE Trans. Antennas Propag. AP-19, 378–400 (1971).
    [CrossRef]
  15. Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995).
    [CrossRef]
  16. Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres: far field,” Appl. Opt. 36, 9496–9508 (1997).
    [CrossRef]
  17. Y.-L. Xu and R. T. Wang, “Electromagnetic scattering by an aggregate of spheres: theoretical and experimental study of the amplitude scattering matrix,” Phys. Rev. E 58, 3931–3948 (1998).
    [CrossRef]
  18. Y.-L. Xu, “Electromagnetic by an aggregate of spheres: asymmetry parameter,” Phys. Lett. A 249, 30–36 (1998).
    [CrossRef]
  19. Y.-L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365 (1999).
    [CrossRef]
  20. Y.-L. Xu and B. Å. S. Gustafson, “A generalized multiparticle Mie-solution: further experimental verification,” J. Quant. Spectrosc. Radiat. Transfer 70, 395–419 (2001).
    [CrossRef]
  21. Y.-L. Xu and N. G. Khlebtsov, “Orientation-averaged radiative properties of an arbitrary configuration of scatterers,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1121–1137 (2003).
    [CrossRef]
  22. Y.-L. Xu, “Radiative scattering properties of an ensemble of variously shaped small particles,” Phys. Rev. E 67, 046620 (2003).
    [CrossRef]
  23. Y.-L. Xu, “Scattering Mueller matrix of an ensemble of variously shaped small particles,” J. Opt. Soc. Am. A 20, 2093–2105 (2003).
    [CrossRef]
  24. Y.-L. Xu and B. Å. S. Gustafson, “Light scattering by an ensemble of small particles,” in Recent Research Developments in Optics (Research Signpost, 2003), pp. 599–648.
  25. Y.-L. Xu, “Radiative-scattering signatures of an ensemble of nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 89, 385–419 (2004).
    [CrossRef]
  26. GMM public-domain FORTRAN codes are currently available at http://code.google.com/p/scatterlib .
  27. Y.-L. Xu, “Fast evaluation of the Gaunt coefficients,” Math. Comput. 65, 1601–1612 (1996).
    [CrossRef]
  28. Y.-L. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996).
    [CrossRef]
  29. Y.-L. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory” (erratum), J. Comput. Phys. 134, 200 (1997).
  30. Y.-L. Xu, “Fast evaluation of Gaunt coefficients: recursive approach,” J. Comput. Appl. Math. 85, 53–65 (1997).
    [CrossRef]
  31. Y.-L. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. 139, 137–165 (1998).
    [CrossRef]
  32. J. A. Gaunt, “On the triplets of helium,” Philos. Trans. R. Soc. London Ser. A 228, 151–196 (1929).
    [CrossRef]
  33. E. P. Wigner, “On the matrices which reduce the Kronecker products of representations of simply reducible groups,” in Quantum Theory of Angular Momentum, L. C. Biedenham and H. van Dam, eds. (Academic, 1965), pp. 89–132.
  34. H. A. van der Vorst, “BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 13, 631–644 (1992).
    [CrossRef]
  35. M. H. Gutknecht, “Variants of BICGSTAB for matrices with complex spectrum,” SIAM J. Sci. Comput. 14, 1020–1033(1993).
    [CrossRef]
  36. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).
  37. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

2008 (1)

2004 (1)

Y.-L. Xu, “Radiative-scattering signatures of an ensemble of nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 89, 385–419 (2004).
[CrossRef]

2003 (4)

D. A. Genov, A. K. Sarychev, V. M. Shalaev, and A. Wei, “Resonant field enhancements from metal nanoparticle arrays,” Nano Lett. 4, 153–158 (2003).
[CrossRef]

Y.-L. Xu, “Scattering Mueller matrix of an ensemble of variously shaped small particles,” J. Opt. Soc. Am. A 20, 2093–2105 (2003).
[CrossRef]

Y.-L. Xu and N. G. Khlebtsov, “Orientation-averaged radiative properties of an arbitrary configuration of scatterers,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1121–1137 (2003).
[CrossRef]

Y.-L. Xu, “Radiative scattering properties of an ensemble of variously shaped small particles,” Phys. Rev. E 67, 046620 (2003).
[CrossRef]

2002 (1)

A. D. Yaghjian, “Scattering-matrix analysis of linear periodic arrays,” IEEE Trans. Antennas Propag. 50, 1050–1064(2002).
[CrossRef]

2001 (1)

Y.-L. Xu and B. Å. S. Gustafson, “A generalized multiparticle Mie-solution: further experimental verification,” J. Quant. Spectrosc. Radiat. Transfer 70, 395–419 (2001).
[CrossRef]

2000 (1)

I. E. Psarobas and N. Stefanov, “Scattering of elastic waves by periodic arrays of spherical bodies,” Phys. Rev. B 62, 278–291 (2000).
[CrossRef]

1999 (1)

Y.-L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365 (1999).
[CrossRef]

1998 (3)

Y.-L. Xu and R. T. Wang, “Electromagnetic scattering by an aggregate of spheres: theoretical and experimental study of the amplitude scattering matrix,” Phys. Rev. E 58, 3931–3948 (1998).
[CrossRef]

Y.-L. Xu, “Electromagnetic by an aggregate of spheres: asymmetry parameter,” Phys. Lett. A 249, 30–36 (1998).
[CrossRef]

Y.-L. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. 139, 137–165 (1998).
[CrossRef]

1997 (3)

Y.-L. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory” (erratum), J. Comput. Phys. 134, 200 (1997).

Y.-L. Xu, “Fast evaluation of Gaunt coefficients: recursive approach,” J. Comput. Appl. Math. 85, 53–65 (1997).
[CrossRef]

Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres: far field,” Appl. Opt. 36, 9496–9508 (1997).
[CrossRef]

1996 (2)

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).
[CrossRef]

1995 (1)

1994 (1)

1993 (1)

M. H. Gutknecht, “Variants of BICGSTAB for matrices with complex spectrum,” SIAM J. Sci. Comput. 14, 1020–1033(1993).
[CrossRef]

1992 (1)

H. A. van der Vorst, “BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 13, 631–644 (1992).
[CrossRef]

1988 (1)

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

1986 (1)

P. C. Waterman and N. E. Pedersen, “Electromagnetic scattering by periodic arrays of particles,” J. Appl. Phys. 59, 2609–2618 (1986).
[CrossRef]

1973 (1)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,”Astrophys. J. 186, 705–714 (1973).
[CrossRef]

1971 (1)

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres, part I & II,” IEEE Trans. Antennas Propag. AP-19, 378–400 (1971).
[CrossRef]

1967 (2)

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

S. Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1967).

1962 (1)

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

1954 (1)

B. Friedman and J. Russek, “Addition theorems for spherical waves,” Quart. 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]

Blum, J.

Y.-L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365 (1999).
[CrossRef]

Bohren, C. F.

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

Bruning, J. H.

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres, part I & II,” IEEE Trans. Antennas Propag. AP-19, 378–400 (1971).
[CrossRef]

Cruzan, O. R.

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

Draine, B. T.

Flatau, P.

Flatau, P. J.

Friedman, B.

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

García de Abajo, F. J.

F. J. García de Abajo, “Colloquium: light scattering by particle and hole arrays,” 2009, http://arXiv.org/pdf/0903.1671.pdf .

Gaunt, J. A.

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

Genov, D. A.

D. A. Genov, A. K. Sarychev, V. M. Shalaev, and A. Wei, “Resonant field enhancements from metal nanoparticle arrays,” Nano Lett. 4, 153–158 (2003).
[CrossRef]

Giovane, F.

Y.-L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365 (1999).
[CrossRef]

Gustafson, B. Å. S.

Y.-L. Xu and B. Å. S. Gustafson, “A generalized multiparticle Mie-solution: further experimental verification,” J. Quant. Spectrosc. Radiat. Transfer 70, 395–419 (2001).
[CrossRef]

Y.-L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365 (1999).
[CrossRef]

Y.-L. Xu and B. Å. S. Gustafson, “Light scattering by an ensemble of small particles,” in Recent Research Developments in Optics (Research Signpost, 2003), pp. 599–648.

Gutknecht, M. H.

M. H. Gutknecht, “Variants of BICGSTAB for matrices with complex spectrum,” SIAM J. Sci. Comput. 14, 1020–1033(1993).
[CrossRef]

Huffman, D. R.

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

Khlebtsov, N. G.

Y.-L. Xu and N. G. Khlebtsov, “Orientation-averaged radiative properties of an arbitrary configuration of scatterers,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1121–1137 (2003).
[CrossRef]

Liang, C.

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

Lo, Y. T.

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres, part I & II,” IEEE Trans. Antennas Propag. AP-19, 378–400 (1971).
[CrossRef]

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

Pedersen, N. E.

P. C. Waterman and N. E. Pedersen, “Electromagnetic scattering by periodic arrays of particles,” J. Appl. Phys. 59, 2609–2618 (1986).
[CrossRef]

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,”Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Psarobas, I. E.

I. E. Psarobas and N. Stefanov, “Scattering of elastic waves by periodic arrays of spherical bodies,” Phys. Rev. B 62, 278–291 (2000).
[CrossRef]

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,”Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Russek, J.

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

Sarychev, A. K.

D. A. Genov, A. K. Sarychev, V. M. Shalaev, and A. Wei, “Resonant field enhancements from metal nanoparticle arrays,” Nano Lett. 4, 153–158 (2003).
[CrossRef]

Shalaev, V. M.

D. A. Genov, A. K. Sarychev, V. M. Shalaev, and A. Wei, “Resonant field enhancements from metal nanoparticle arrays,” Nano Lett. 4, 153–158 (2003).
[CrossRef]

Stefanov, N.

I. E. Psarobas and N. Stefanov, “Scattering of elastic waves by periodic arrays of spherical bodies,” Phys. Rev. B 62, 278–291 (2000).
[CrossRef]

Stein, S.

S. Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1967).

Tehranian, S.

Y.-L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365 (1999).
[CrossRef]

van de Hulst, H. C.

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

van der Vorst, H. A.

H. A. van der Vorst, “BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 13, 631–644 (1992).
[CrossRef]

Wang, R. T.

Y.-L. Xu and R. T. Wang, “Electromagnetic scattering by an aggregate of spheres: theoretical and experimental study of the amplitude scattering matrix,” Phys. Rev. E 58, 3931–3948 (1998).
[CrossRef]

Waterman, P. C.

P. C. Waterman and N. E. Pedersen, “Electromagnetic scattering by periodic arrays of particles,” J. Appl. Phys. 59, 2609–2618 (1986).
[CrossRef]

Wei, A.

D. A. Genov, A. K. Sarychev, V. M. Shalaev, and A. Wei, “Resonant field enhancements from metal nanoparticle arrays,” Nano Lett. 4, 153–158 (2003).
[CrossRef]

Wigner, E. P.

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

Xu, Y.-L.

Y.-L. Xu, “Radiative-scattering signatures of an ensemble of nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 89, 385–419 (2004).
[CrossRef]

Y.-L. Xu and N. G. Khlebtsov, “Orientation-averaged radiative properties of an arbitrary configuration of scatterers,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1121–1137 (2003).
[CrossRef]

Y.-L. Xu, “Radiative scattering properties of an ensemble of variously shaped small particles,” Phys. Rev. E 67, 046620 (2003).
[CrossRef]

Y.-L. Xu, “Scattering Mueller matrix of an ensemble of variously shaped small particles,” J. Opt. Soc. Am. A 20, 2093–2105 (2003).
[CrossRef]

Y.-L. Xu and B. Å. S. Gustafson, “A generalized multiparticle Mie-solution: further experimental verification,” J. Quant. Spectrosc. Radiat. Transfer 70, 395–419 (2001).
[CrossRef]

Y.-L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365 (1999).
[CrossRef]

Y.-L. Xu and R. T. Wang, “Electromagnetic scattering by an aggregate of spheres: theoretical and experimental study of the amplitude scattering matrix,” Phys. Rev. E 58, 3931–3948 (1998).
[CrossRef]

Y.-L. Xu, “Electromagnetic by an aggregate of spheres: asymmetry parameter,” Phys. Lett. A 249, 30–36 (1998).
[CrossRef]

Y.-L. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. 139, 137–165 (1998).
[CrossRef]

Y.-L. Xu, “Fast evaluation of Gaunt coefficients: recursive approach,” J. Comput. Appl. Math. 85, 53–65 (1997).
[CrossRef]

Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres: far field,” Appl. Opt. 36, 9496–9508 (1997).
[CrossRef]

Y.-L. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory” (erratum), J. Comput. Phys. 134, 200 (1997).

Y.-L. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996).
[CrossRef]

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

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

Y.-L. Xu and B. Å. S. Gustafson, “Light scattering by an ensemble of small particles,” in Recent Research Developments in Optics (Research Signpost, 2003), pp. 599–648.

Yaghjian, A. D.

A. D. Yaghjian, “Scattering-matrix analysis of linear periodic arrays,” IEEE Trans. Antennas Propag. 50, 1050–1064(2002).
[CrossRef]

Appl. Opt. (2)

Astrophys. J. (2)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,”Astrophys. J. 186, 705–714 (1973).
[CrossRef]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

A. D. Yaghjian, “Scattering-matrix analysis of linear periodic arrays,” IEEE Trans. Antennas Propag. 50, 1050–1064(2002).
[CrossRef]

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres, part I & II,” IEEE Trans. Antennas Propag. AP-19, 378–400 (1971).
[CrossRef]

J. Appl. Phys. (1)

P. C. Waterman and N. E. Pedersen, “Electromagnetic scattering by periodic arrays of particles,” J. Appl. Phys. 59, 2609–2618 (1986).
[CrossRef]

J. Comput. Appl. Math. (1)

Y.-L. Xu, “Fast evaluation of Gaunt coefficients: recursive approach,” J. Comput. Appl. Math. 85, 53–65 (1997).
[CrossRef]

J. Comput. Phys. (3)

Y.-L. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. 139, 137–165 (1998).
[CrossRef]

Y.-L. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996).
[CrossRef]

Y.-L. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory” (erratum), J. Comput. Phys. 134, 200 (1997).

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

J. Quant. Spectrosc. Radiat. Transfer (3)

Y.-L. Xu, “Radiative-scattering signatures of an ensemble of nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 89, 385–419 (2004).
[CrossRef]

Y.-L. Xu and B. Å. S. Gustafson, “A generalized multiparticle Mie-solution: further experimental verification,” J. Quant. Spectrosc. Radiat. Transfer 70, 395–419 (2001).
[CrossRef]

Y.-L. Xu and N. G. Khlebtsov, “Orientation-averaged radiative properties of an arbitrary configuration of scatterers,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1121–1137 (2003).
[CrossRef]

Math. Comput. (1)

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

Nano Lett. (1)

D. A. Genov, A. K. Sarychev, V. M. Shalaev, and A. Wei, “Resonant field enhancements from metal nanoparticle arrays,” Nano Lett. 4, 153–158 (2003).
[CrossRef]

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]

Phys. Lett. A (1)

Y.-L. Xu, “Electromagnetic by an aggregate of spheres: asymmetry parameter,” Phys. Lett. A 249, 30–36 (1998).
[CrossRef]

Phys. Rev. B (1)

I. E. Psarobas and N. Stefanov, “Scattering of elastic waves by periodic arrays of spherical bodies,” Phys. Rev. B 62, 278–291 (2000).
[CrossRef]

Phys. Rev. E (3)

Y.-L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365 (1999).
[CrossRef]

Y.-L. Xu and R. T. Wang, “Electromagnetic scattering by an aggregate of spheres: theoretical and experimental study of the amplitude scattering matrix,” Phys. Rev. E 58, 3931–3948 (1998).
[CrossRef]

Y.-L. Xu, “Radiative scattering properties of an ensemble of variously shaped small particles,” Phys. Rev. E 67, 046620 (2003).
[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 (1967).

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

Radio Sci. (1)

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

SIAM J. Sci. Comput. (1)

M. H. Gutknecht, “Variants of BICGSTAB for matrices with complex spectrum,” SIAM J. Sci. Comput. 14, 1020–1033(1993).
[CrossRef]

SIAM J. Sci. Stat. Comput. (1)

H. A. van der Vorst, “BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 13, 631–644 (1992).
[CrossRef]

Other (6)

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

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

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

GMM public-domain FORTRAN codes are currently available at http://code.google.com/p/scatterlib .

F. J. García de Abajo, “Colloquium: light scattering by particle and hole arrays,” 2009, http://arXiv.org/pdf/0903.1671.pdf .

Y.-L. Xu and B. Å. S. Gustafson, “Light scattering by an ensemble of small particles,” in Recent Research Developments in Optics (Research Signpost, 2003), pp. 599–648.

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Figures (12)

Fig. 1.
Fig. 1.

Dependence of the Mueller matrix element S11 on scattering angle θ and azimuth angle φ (shown as variation with x=sinθcosφ and y=sinθsinφ) for the particle array L1 (see Table 1), which is a linear chain of 3001 identical prolate spheroids of refractive index (1.6, 0.1). The incident plane wave is of wavelength 31.416 mm, propagating in the positive z direction. An individual spheroid has an aspect ratio (i.e., the minor-to-major axis ratio) of 0.5. Its volume-equivalent sphere diameter is 1 mm. Geometrical centers of the spheroids are aligned along the x axis, equally spaced by 1 mm. The major (rotational) axes of the spheroids are parallel to the z axis. In the four panels, the left two are for the forward hemisphere of 0°θ90° (with θ=0° being the exact forward scattering direction) and the right two are for the backward hemisphere of 90°θ180° (with θ=180° being the exact backward scattering direction). The lower two panels are equivalent to the upper two, simply presenting the same data in an alternative way.

Fig. 2.
Fig. 2.

Dependence of the Mueller matrix element ratio S22/S11 on scattering angle θ and azimuth angle φ (shown as variation with x=sinθcosφ and y=sinθsinφ) for the same linear spheroid array of L1 as shown in Fig. 1.

Fig. 3.
Fig. 3.

Dependence of the Mueller matrix element ratio S33/S11 on scattering angle θ and azimuth angle φ (shown as variation with x=sinθcosφ and y=sinθsinφ) for the same linear spheroid array of L1 as shown in Fig. 1.

Fig. 4.
Fig. 4.

Dependence of the Mueller matrix element ratio S44/S11 on scattering angle θ and azimuth angle φ (shown as variation with x=sinθcosφ and y=sinθsinφ) for the same linear spheroid array of L1 as shown in Fig. 1.

Fig. 5.
Fig. 5.

Extinction and absorption cross sections of the 3001 individual prolate spheroids in the linear array L1, which are identical for the majority of the component particles except for a few at both ends due to the edge effect.

Fig. 6.
Fig. 6.

Dependence of the Stokes parameters I and Q on scattering angle θ and azimuth angle φ (shown as variation with x=sinθcosφ and y=sinθsinφ) for an individual spheroid in an infinite array of identical prolate spheroids of refractive index (1.6, 0.1). The incident plane wave of wavelength 31.416 mm is x polarized, propagating in the positive z direction. An individual spheroid has an aspect ratio of 0.5. Its volume-equivalent sphere diameter is 1 mm. The major axes of the spheroids are parallel to the z axis. All particle centers are in the xy plane, equally spaced by 10 mm in the x direction and 1 mm in the y direction.

Fig. 7.
Fig. 7.

Same as Fig. 6 but for the Stokes parameters U and V when the incident plane wave is right-circularly polarized.

Fig. 8.
Fig. 8.

Same as Fig. 1 except the numerical results shown are obtained from the newly developed “gmm03_PA.f” instead of “gmm03s.f”.

Fig. 9.
Fig. 9.

Same as Fig. 2 except the numerical results shown are obtained from the newly developed “gmm03_PA.f” instead of “gmm03s.f”.

Fig. 10.
Fig. 10.

Same as Fig. 3 except the numerical results shown are obtained from the newly developed “gmm03_PA.f” instead of “gmm03s.f”.

Fig. 11.
Fig. 11.

Same as Fig. 4 except the numerical results shown are obtained from the newly developed “gmm03_PA.f” instead of “gmm03s.f”.

Fig. 12.
Fig. 12.

Dependence of the Stokes parameter Q on scattering angle θ and azimuth angle φ for a 10001×10001 rectangular array of identical spheres. The incident plane wave of wavelength 31.416 mm is unpolarized, propagating in the positive z direction. An individual sphere is of refractive index (1.6, 0.1) and of diameter 1 mm. All sphere centers are in the xy plane with rows parallel to the x axis and columns parallel to the y axis, equally spaced by 10 mm in the x direction and 1 mm in the y direction.

Tables (8)

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Table 1. Five Examples for Regular Arrays of Identical Particles

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Table 2. Computer Memory and CPU Time Required in the Scattering Calculations on an 8-Core 2.4 GHz DELL PRECISION T7500 Desktop Computer for the Regular Arrays of Identical Particles Listed in Table 1 (When Using the General GMM Computer Codes that are Currently Available Online)

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Table 3. Total Cross Sections for Extinction, Absorption, Scattering, Backscattering, and Radiation Pressure (mm2) Calculated from the General GMM Computer Codes “gmm01s.f” or “gmm03s.f” for the Regular Arrays of Identical Particles Listed in Table 1

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Table 4. Total Cross Sections (mm2) Calculated from the Newly Developed GMM Computer Codes for Periodic Structures (“gmm01_PA.f” or “gmm03_PA.f”) for the Regular Arrays of Identical Particles Listed in Table 1

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Table 5. Relative Deviations of the Total Cross Sections Calculated from the Newly Developed GMM Codes “gmm01_PA.f” or “gmm03_PA.f” Using the Results Shown in Table 3 as a Benchmark

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Table 6. Computer Memory and CPU Time Required for the Scattering Calculations on an 8-Core 2.4 GHz DELL PRECISION T7500 Computer for the Regular Arrays of Identical Particles Listed in Table 1 When Using the New PA-Type of GMM Codes

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Table 7. Ratios of the CPU and Computer Memory Required in the PA-type GMM Calculations to Those Shown in Table 2

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Table 8. Total Cross Sections (mm2) of a 10001×10001 Rectangular Array of Identical Homogeneous Spheres Calculated from the Newly Developed GMM Code “gmm01_PA.f”

Equations (45)

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Einc=E0exp(ik^·riωt),
dl=e^xXl+e^yYl+e^zZl.
k^=e^xsinϑinccosφinc+e^ysinϑincsinφinc+e^zcosϑinc.
Einc=in=1Nmaxm=nnp=12EmnpmnpNmnp(1)(r,θ,ϕ),
Cmn=[(2n+1)(nm)!n(n+1)(n+m)!]1/2.
Nmn1(1)={e^rn(n+1)Pnm(cosθ)jn(r)r+[e^θτmn1(θ)+e^ϕiτmn2(θ)]ψn(r)r}exp(imϕ),Nmn2(1)=[e^θiτmn2(θ)e^ϕτmn1(θ)]jn(r)exp(imϕ),
τmn1(θ)=ddθPnm(cosθ),τmn2(θ)=msinθPnm(cosθ).
pmnp=i02π0πexp(ik^·r)E0·Nmnp(1)*sinθdθdϕEmn02π0π|Nmnp(1)|2sinθdθdϕ.
pmnp=(1)m+1[τ˜mnp(ϑinc)cos(φincβp)+iτ˜mn3p(ϑinc)sin(φincβp)],
τ˜mnp=Cmnτmnp.
pmnp=0(|m|1),p1np=2n+12exp(iβp),p1np=(1)pp1np*.
Einc,l=iexp(ik^·dl)n=1Nmaxlm=nnp=12EmnpmnpNmnp(1)(rl,θl,ϕl),
Esca=l=1LEsca,l.
Esca=in=1Nmaxm=nnp=12EmnamnpNmnp(3)(r,θ,ϕ).
Nmn1(3)={e^rn(n+1)Pnm(cosθ)hn(1)(r)r+[e^θτmn1(θ)+e^ϕiτmn2(θ)]ξn(r)r}exp(imϕ),Nmn2(3)=[e^θiτmn2(θ)e^ϕτmn1(θ)]hn(1)(r)exp(imϕ),
Esca,l=iexp(ik^·dl)n=1Nmaxlm=nnp=12EmnamnplNmnp(3)(rl,θl,ϕl).
amnpl=a¯mnpl+a⃗mnpl.
a¯mnpl=ν=1Nmaxlμ=ννq=12T¯mnpμνqlpμνq.
a⃗mnpl=ν=1Nmaxlμ=ννq=12T¯mnpμνqlp⃗μνql,p⃗μνql=l=1Ln=1Nmaxlm=nnp=12(1δll)×exp(ik^·dll)Aμνqmnpllamnpl.
amnpl+l=1Ln=1Nmaxlm=nnp=12ν=1Nmaxlμ=ννq=12(1δll)×exp(ik^·dll)T¯mnpμνqlAμνqmnpllamnpl=a¯mnpl.
T¯mnpμνql=a¯nplδmμδnνδpq,
amnpl+a¯npll=1Ln=1Nmaxlm=nnp=12(1δll)×exp(ik^·dll)Amnpmnpllamnpl=a¯nplpmnp.
amnpl=j=1Lν=1Nmaxjμ=ννq=12exp(ik^·dlj)Tmnpμνqljpμνq,
Tmnpμνqlj=j0(djl)T¯mnpμνql+j=1Lj0(djj)[(δjj1)Tmnpμνqlj+ν=1Nmaxlμ=ννq=12l=1Ln=1Nmaxlm=nnp=12(δll1)×T¯mnpμνqlAμνqmnpllTmnpμνqlj],
Tmnpμνqlj=j0(dlj)a¯nplδmμδnνδpq+j=1Lj0(djj)×[(δjj1)Tmnpμνqlj+l=1Ln=1Nmaxlm=nnp=12(δll1)×a¯nplAμνqmnpllTmnpμνqlj].
ξn(r)(i)n+1exp(ir),ξn(r)(i)nexp(ir),
Nmn1(3)=(i)nexp(ir)r[e^θτmn1(θ)+e^ϕiτmn2(θ)]exp(imϕ),Nmn2(3)=(i)nexp(ir)r[e^θτmn2(θ)+e^ϕiτmn1(θ)]exp(imϕ).
Eθsca(θ,ϕ)=iE0exp(ir)rn=1Nmaxm=nnp=12amnpτ˜mnp(θ)exp(imϕ),Eϕsca(θ,ϕ)=E0exp(ir)rn=1Nmaxm=nnp=12amnpτ˜mn3p(θ)exp(imϕ).
(EscaEsca)=(EθscaEϕsca)=exp(ir)irS˜(EincEinc)=exp(ir)irS˜[cosϕsinϕsinϕcosϕ](E0cosβpE0sinβp)=exp(ir)ir[S˜2S˜3S˜4S˜1](E0cos(ϕβp)E0sin(ϕβp)).
S˜2l(θl,ϕl)=n=1Nmaxlm=0np=12fm[amnplexp(iϕm)(1)m+pamnplexp(iϕm)]τ˜mnp(θl),S˜3l(θl,ϕl)=in=1Nmaxlm=0np=12fm[amnplexp(iϕm)+(1)m+pamnplexp(iϕm)]τ˜mnp(θl),S˜4l(θl,ϕl)=in=1Nmaxlm=0np=12fm[amnplexp(iϕm)(1)m+pamnplexp(iϕm)]τ˜mn3p(θl),S˜1l(θl,ϕl)=n=1Nmaxlm=0np=12fm[amnplexp(iϕm)+(1)m+pamnplexp(iϕm)]τ˜mn3p(θl),
ϕm=(m1)ϕl+βp,fm=(1+δ0m)1.
(EθscaEϕsca)=exp(ir)irS(E0cosβpE0sinβp).
S=S˜[cosϕsinϕsinϕcosϕ].
S2l=n=1Nmaxlm=0np=12fm[amnplτ˜mnpexp(imϕl)exp(iβp)+amnplτ˜mnpexp(imϕl)exp(iβp)],S3l=in=1Nmaxlm=0np=12fm[amnplτ˜mnpexp(imϕl)exp(iβp)amnplτ˜mnpexp(imϕl)exp(iβp)],S4l=in=1Nmaxlm=0np=12fm[amnplτ˜mn3pexp(imϕl)exp(iβp)+amnplτ˜mn3pexp(imϕl)exp(iβp)],S1l=n=1Nmaxlm=0np=12fm[amnplτ˜mn3pexp(imϕl)exp(iβp)amnplτ˜mn3pexp(imϕl)exp(iβp)].
S2l=n=1Nmaxlm=nnp=12amnpl(0°)τ˜mnp(θl)exp(imϕl),S3l=n=1Nmaxlm=nnp=12amnpl(90°)τ˜mnp(θl)exp(imϕl),S4l=in=1Nmaxlm=nnp=12amnpl(0°)τ˜mn3p(θl)exp(imϕl),S1l=in=1Nmaxlm=nnp=12amnpl(90°)τ˜mn3p(θl)exp(imϕl).
amnp=l=1Lexp[idl·(k^r^)]amnpl,S=l=1Lexp[idl·(k^r^)]Sl.
a⃗mnpl=ν=1Nmaxlμ=ννq=12T¯mnpμνqlp⃗μνql,p⃗μνql=n=1Nmaxlm=nnp=12Cμνqmnplamnpl,Cμνqmnpl=l(1δll)exp(ik^·dll)Aμνqmnpll.
amnpl+n=1Nmaxlm=nnp=12ν=1Nmaxlμ=ννq=12T¯mnpμνql×Cμνqmnplamnpl=a¯mnpl.
amnpl+a¯npln=1Nmaxlm=nnp=12Cmnpmnplamnpl=a¯nplpmnp.
amnp=amnplj=1Lexp[idj·(k^r^)],S=Slj=1Lexp[idj·(k^r^)].
dj·(k^r^)=dj[cosηj(ϑinc,φinc)cosηj(θ,ϕ)],cosηj(β,α)=sinβsinϑjcos(αφj)+cosβcosϑj.
j=1Lexp[idj·(k^r^)]=sin(L·Δd·Φlin/2)sin(Δd·Φlin/2),Φlin=cosηlin(ϑinc,φinc)cosηlin(θ,ϕ),
j=1Lexp[idj·(k^r^)]=sin(L·Δx·sinθcosϕ/2)sin(Δx·sinθcosϕ/2).
j=1Lexp[idj·(k^r^)]=sin(Nrow·Δdrow·Φrow/2)sin(Δdrow·Φrow/2)·sin(Ncol·Δdcol·Φcol/2)sin(Δdcol·Φcol/2),Φrow=cosηrow(ϑinc,φinc)cosηrow(θ,ϕ),Φcol=cosηcol(ϑinc,φinc)cosηcol(θ,ϕ).
j=1Lexp[idj·(k^r^)]=sin(Nx·Δx·sinθcosϕ/2)sin(Δx·sinθcosϕ/2)·sin(Ny·Δy·sinθsinϕ/2)sin(Δy·sinθsinϕ/2).

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