Abstract

Light scattering and absorption by spherical particles is extended to aggregates of spheres with arbitrary shape and size. We applied the theory of Gérardy and Ausloos [Phys. Rev. B 25, 4204–4229 (1082)] to compute the total extinction loss spectra of several aggregates of nanometer-sized silver spheres from the near IR to the near UV. Silver was best suited to provide quantitative comparison with experiments concerning the scattering and absorption in the visible spectral region. Additional resonant extinction was obtained besides the resonant extinction of the single silver sphere. The spectra were discussed in detail to give general results that are independent of the particle material.

© 1993 Optical Society of America

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  1. G. Mie, “Beiträge zur” Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
    [CrossRef]
  2. P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
    [CrossRef]
  3. A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [CrossRef]
  4. A. Güttler, “Die Miesche Theorie der Beugung durch dielektrische Kugeln mit absorbierendem Kern und ihre Bedeutung für Probleme der interstellaren Materie und des atmosphärischen Aerosols,” Ann. Phys. 1, 65–98 (1952).
    [CrossRef]
  5. W. von Ignatowsky, “Reflexion elektromagnetischer Wellen an einem dünnen Draht,” Ann. Phys. 18, 495–522 (1905).
    [CrossRef]
  6. W. Seitz, “Die Beugung des Lichtes an einem dünnen, zylindrischen Drahte,” Ann. Phys. 21, 1013–1029 (1906).
    [CrossRef]
  7. S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
    [PubMed]
  8. W. Trinks, “Zur Vielfachstreuung an kleinen Kugeln,” Ann. Phys. 22, 561–590 (1935).
    [CrossRef]
  9. F. Borghese, P. Denti, G. Toscano, O. I. Sindoni, “Electromagnetic scattering by a cluster of spheres,” Appl. Opt. 18, 116–120 (1979).
    [CrossRef] [PubMed]
  10. J. M. Gérardy, 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. B25, 4204–4229 (1982).
  11. K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. I: Linear chains,” Opt. Lett. 13, 90–92 (1988).
    [CrossRef] [PubMed]
  12. K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. II: Clusters of arbitrary configuration,” Opt. Lett. 13, 1063–1065 (1988).
    [CrossRef] [PubMed]
  13. A.-K. Hamid, I. R. Ciric, M. Hamid, “Multiple scattering by a linear array of conducting spheres,” Can. J. Phys. 68, 1157–1165 (1990).
    [CrossRef]
  14. A.-K. Hamid, I. R. Ciric, M. Hamid, “Electromagnetic scattering by an arbitrary configuration of dielectric spheres,” Can. J. Phys. 68, 1419–1428 (1990).
    [CrossRef]
  15. A.-K. Hamid, I. R. Ciric, M. Hamid, “Iterative solution of the scattering by an arbitrary configuration of conducting or dielectric spheres,” Proc. Inst. Electr. Eng. Part H 138, 565–572 (1991).
  16. A.-K. Hamid, I. R. Ciric, M. Hamid, “Scattering by systems of small conducting spheres,” “.Backscattering cross section of many sphere systems,” presented at the Electromagnetic Research Symposium (PIERS), Massachusetts Institute of Technology, Cambridge, Mass., 1991.
  17. A.-K. Hamid, I. R. Ciric, M. Hamid, “Iterative technique for scattering by a linear array of spheres,” in Proceedings IEEE AP-S International Symposium (University of Western Ontario, London, Canada, 1991).
  18. J. Sotelo, G. A. Niklasson, “Optical properties of fractal clusters of small metallic particles,” Z. Phys. D 20, 321–323 (1991).
    [CrossRef]
  19. M. Quinten, U. Kreibig, “Optical properties of small metal particles,” Surf. Sci. 172, 557–577 (1986).
    [CrossRef]
  20. M. Quinten, U. Kreibig, “Optical extinction spectra of systems of small metal particles with aggregates,” in Proceedings of an Internationals Symposium on Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988), pp. 249–258.
  21. M. Quinten, D. Schönauer, U. Kreibig, “Electronic excitations in many-particle systems: a quantitative analysis,” Z. Phys. D 12, 521–525 (1989).
    [CrossRef]
  22. J. A. Stratton, Electrodynamic Theory (McGraw-Hill, New York, 1941), Chap. 7, p. 392.
  23. D. Langbein, Van der Waals Attraction, Vol. 72 of Springer Tracts in Modern Physics (Springer-Verlag, Berlin, 1972), Chap. 5, p. 72.
  24. B. Jeffreys, “Transformation of tesseral harmonics under rotation,” Geophys. J. R. Astron. Soc. 10, 141–145 (1965).
    [CrossRef]
  25. L. Genzel, T. P. Martin, U. Kreibig. “Dielectric function and plasma resonances of small metal particles,” Z. Phys. B 21, 339–346 (1975).
    [CrossRef]
  26. L. Genzel, U. Kreibig, “Dielectric function and infrared absorption of small metal particles,” Z. Phys. B 37, 93–101 (1980).
    [CrossRef]
  27. G. C. Papavassilion, T. Kokkinakis, “Optical absorption spectra of surface plasmons in small cooper particles,” J. Phys. F 4, L67–L68 (1974).
    [CrossRef]
  28. L. Garbowski, Ber. Dtsch. Chem. Ges. 36, 1215 (1903).
    [CrossRef]
  29. R. Zsigmondy, Das kolloide Gold (Akademie Verlagsgeschäft, Leipzig, 1925).
  30. P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]
  31. D. Schönauer, M. Quinten, U. Kreibig, “Precursor-states of percolation in quasi-fractal many-particle-systems,” Z. Phys. D 12, 527–532 (1989).
    [CrossRef]
  32. B. Dusemund, A. Hoffmann, T. Salzmann, U. Kreibig, G. Schmid, “Cluster matter: the transition of optical elastic scattering to regular reflection,” Z. Phys. D 20, 305–308 (1991).
    [CrossRef]
  33. E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]

1991 (3)

A.-K. Hamid, I. R. Ciric, M. Hamid, “Iterative solution of the scattering by an arbitrary configuration of conducting or dielectric spheres,” Proc. Inst. Electr. Eng. Part H 138, 565–572 (1991).

J. Sotelo, G. A. Niklasson, “Optical properties of fractal clusters of small metallic particles,” Z. Phys. D 20, 321–323 (1991).
[CrossRef]

B. Dusemund, A. Hoffmann, T. Salzmann, U. Kreibig, G. Schmid, “Cluster matter: the transition of optical elastic scattering to regular reflection,” Z. Phys. D 20, 305–308 (1991).
[CrossRef]

1990 (2)

A.-K. Hamid, I. R. Ciric, M. Hamid, “Multiple scattering by a linear array of conducting spheres,” Can. J. Phys. 68, 1157–1165 (1990).
[CrossRef]

A.-K. Hamid, I. R. Ciric, M. Hamid, “Electromagnetic scattering by an arbitrary configuration of dielectric spheres,” Can. J. Phys. 68, 1419–1428 (1990).
[CrossRef]

1989 (2)

M. Quinten, D. Schönauer, U. Kreibig, “Electronic excitations in many-particle systems: a quantitative analysis,” Z. Phys. D 12, 521–525 (1989).
[CrossRef]

D. Schönauer, M. Quinten, U. Kreibig, “Precursor-states of percolation in quasi-fractal many-particle-systems,” Z. Phys. D 12, 527–532 (1989).
[CrossRef]

1988 (2)

K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. I: Linear chains,” Opt. Lett. 13, 90–92 (1988).
[CrossRef] [PubMed]

K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. II: Clusters of arbitrary configuration,” Opt. Lett. 13, 1063–1065 (1988).
[CrossRef] [PubMed]

1986 (1)

M. Quinten, U. Kreibig, “Optical properties of small metal particles,” Surf. Sci. 172, 557–577 (1986).
[CrossRef]

1982 (1)

J. M. Gérardy, 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. B25, 4204–4229 (1982).

1980 (1)

L. Genzel, U. Kreibig, “Dielectric function and infrared absorption of small metal particles,” Z. Phys. B 37, 93–101 (1980).
[CrossRef]

1979 (1)

1975 (2)

S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
[PubMed]

L. Genzel, T. P. Martin, U. Kreibig. “Dielectric function and plasma resonances of small metal particles,” Z. Phys. B 21, 339–346 (1975).
[CrossRef]

1974 (1)

G. C. Papavassilion, T. Kokkinakis, “Optical absorption spectra of surface plasmons in small cooper particles,” J. Phys. F 4, L67–L68 (1974).
[CrossRef]

1973 (1)

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

1972 (1)

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

1965 (1)

B. Jeffreys, “Transformation of tesseral harmonics under rotation,” Geophys. J. R. Astron. Soc. 10, 141–145 (1965).
[CrossRef]

1952 (1)

A. Güttler, “Die Miesche Theorie der Beugung durch dielektrische Kugeln mit absorbierendem Kern und ihre Bedeutung für Probleme der interstellaren Materie und des atmosphärischen Aerosols,” Ann. Phys. 1, 65–98 (1952).
[CrossRef]

1951 (1)

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

1935 (1)

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

1909 (1)

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur” Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

1906 (1)

W. Seitz, “Die Beugung des Lichtes an einem dünnen, zylindrischen Drahte,” Ann. Phys. 21, 1013–1029 (1906).
[CrossRef]

1905 (1)

W. von Ignatowsky, “Reflexion elektromagnetischer Wellen an einem dünnen Draht,” Ann. Phys. 18, 495–522 (1905).
[CrossRef]

1903 (1)

L. Garbowski, Ber. Dtsch. Chem. Ges. 36, 1215 (1903).
[CrossRef]

Aden, A. L.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Asano, S.

Ausloos, M.

J. M. Gérardy, 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. B25, 4204–4229 (1982).

Borghese, F.

Christy, R. W.

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Ciric, I. R.

A.-K. Hamid, I. R. Ciric, M. Hamid, “Iterative solution of the scattering by an arbitrary configuration of conducting or dielectric spheres,” Proc. Inst. Electr. Eng. Part H 138, 565–572 (1991).

A.-K. Hamid, I. R. Ciric, M. Hamid, “Electromagnetic scattering by an arbitrary configuration of dielectric spheres,” Can. J. Phys. 68, 1419–1428 (1990).
[CrossRef]

A.-K. Hamid, I. R. Ciric, M. Hamid, “Multiple scattering by a linear array of conducting spheres,” Can. J. Phys. 68, 1157–1165 (1990).
[CrossRef]

A.-K. Hamid, I. R. Ciric, M. Hamid, “Scattering by systems of small conducting spheres,” “.Backscattering cross section of many sphere systems,” presented at the Electromagnetic Research Symposium (PIERS), Massachusetts Institute of Technology, Cambridge, Mass., 1991.

A.-K. Hamid, I. R. Ciric, M. Hamid, “Iterative technique for scattering by a linear array of spheres,” in Proceedings IEEE AP-S International Symposium (University of Western Ontario, London, Canada, 1991).

Debye, P.

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
[CrossRef]

Denti, P.

Dusemund, B.

B. Dusemund, A. Hoffmann, T. Salzmann, U. Kreibig, G. Schmid, “Cluster matter: the transition of optical elastic scattering to regular reflection,” Z. Phys. D 20, 305–308 (1991).
[CrossRef]

Fuller, K. A.

K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. I: Linear chains,” Opt. Lett. 13, 90–92 (1988).
[CrossRef] [PubMed]

K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. II: Clusters of arbitrary configuration,” Opt. Lett. 13, 1063–1065 (1988).
[CrossRef] [PubMed]

Garbowski, L.

L. Garbowski, Ber. Dtsch. Chem. Ges. 36, 1215 (1903).
[CrossRef]

Genzel, L.

L. Genzel, U. Kreibig, “Dielectric function and infrared absorption of small metal particles,” Z. Phys. B 37, 93–101 (1980).
[CrossRef]

L. Genzel, T. P. Martin, U. Kreibig. “Dielectric function and plasma resonances of small metal particles,” Z. Phys. B 21, 339–346 (1975).
[CrossRef]

Gérardy, J. M.

J. M. Gérardy, 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. B25, 4204–4229 (1982).

Güttler, A.

A. Güttler, “Die Miesche Theorie der Beugung durch dielektrische Kugeln mit absorbierendem Kern und ihre Bedeutung für Probleme der interstellaren Materie und des atmosphärischen Aerosols,” Ann. Phys. 1, 65–98 (1952).
[CrossRef]

Hamid, A.-K.

A.-K. Hamid, I. R. Ciric, M. Hamid, “Iterative solution of the scattering by an arbitrary configuration of conducting or dielectric spheres,” Proc. Inst. Electr. Eng. Part H 138, 565–572 (1991).

A.-K. Hamid, I. R. Ciric, M. Hamid, “Electromagnetic scattering by an arbitrary configuration of dielectric spheres,” Can. J. Phys. 68, 1419–1428 (1990).
[CrossRef]

A.-K. Hamid, I. R. Ciric, M. Hamid, “Multiple scattering by a linear array of conducting spheres,” Can. J. Phys. 68, 1157–1165 (1990).
[CrossRef]

A.-K. Hamid, I. R. Ciric, M. Hamid, “Iterative technique for scattering by a linear array of spheres,” in Proceedings IEEE AP-S International Symposium (University of Western Ontario, London, Canada, 1991).

A.-K. Hamid, I. R. Ciric, M. Hamid, “Scattering by systems of small conducting spheres,” “.Backscattering cross section of many sphere systems,” presented at the Electromagnetic Research Symposium (PIERS), Massachusetts Institute of Technology, Cambridge, Mass., 1991.

Hamid, M.

A.-K. Hamid, I. R. Ciric, M. Hamid, “Iterative solution of the scattering by an arbitrary configuration of conducting or dielectric spheres,” Proc. Inst. Electr. Eng. Part H 138, 565–572 (1991).

A.-K. Hamid, I. R. Ciric, M. Hamid, “Multiple scattering by a linear array of conducting spheres,” Can. J. Phys. 68, 1157–1165 (1990).
[CrossRef]

A.-K. Hamid, I. R. Ciric, M. Hamid, “Electromagnetic scattering by an arbitrary configuration of dielectric spheres,” Can. J. Phys. 68, 1419–1428 (1990).
[CrossRef]

A.-K. Hamid, I. R. Ciric, M. Hamid, “Iterative technique for scattering by a linear array of spheres,” in Proceedings IEEE AP-S International Symposium (University of Western Ontario, London, Canada, 1991).

A.-K. Hamid, I. R. Ciric, M. Hamid, “Scattering by systems of small conducting spheres,” “.Backscattering cross section of many sphere systems,” presented at the Electromagnetic Research Symposium (PIERS), Massachusetts Institute of Technology, Cambridge, Mass., 1991.

Hoffmann, A.

B. Dusemund, A. Hoffmann, T. Salzmann, U. Kreibig, G. Schmid, “Cluster matter: the transition of optical elastic scattering to regular reflection,” Z. Phys. D 20, 305–308 (1991).
[CrossRef]

Jeffreys, B.

B. Jeffreys, “Transformation of tesseral harmonics under rotation,” Geophys. J. R. Astron. Soc. 10, 141–145 (1965).
[CrossRef]

Johnson, P. B.

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Kattawar, G. W.

K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. II: Clusters of arbitrary configuration,” Opt. Lett. 13, 1063–1065 (1988).
[CrossRef] [PubMed]

K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. I: Linear chains,” Opt. Lett. 13, 90–92 (1988).
[CrossRef] [PubMed]

Kerker, M.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Kokkinakis, T.

G. C. Papavassilion, T. Kokkinakis, “Optical absorption spectra of surface plasmons in small cooper particles,” J. Phys. F 4, L67–L68 (1974).
[CrossRef]

Kreibig, U.

B. Dusemund, A. Hoffmann, T. Salzmann, U. Kreibig, G. Schmid, “Cluster matter: the transition of optical elastic scattering to regular reflection,” Z. Phys. D 20, 305–308 (1991).
[CrossRef]

D. Schönauer, M. Quinten, U. Kreibig, “Precursor-states of percolation in quasi-fractal many-particle-systems,” Z. Phys. D 12, 527–532 (1989).
[CrossRef]

M. Quinten, D. Schönauer, U. Kreibig, “Electronic excitations in many-particle systems: a quantitative analysis,” Z. Phys. D 12, 521–525 (1989).
[CrossRef]

M. Quinten, U. Kreibig, “Optical properties of small metal particles,” Surf. Sci. 172, 557–577 (1986).
[CrossRef]

L. Genzel, U. Kreibig, “Dielectric function and infrared absorption of small metal particles,” Z. Phys. B 37, 93–101 (1980).
[CrossRef]

L. Genzel, T. P. Martin, U. Kreibig. “Dielectric function and plasma resonances of small metal particles,” Z. Phys. B 21, 339–346 (1975).
[CrossRef]

M. Quinten, U. Kreibig, “Optical extinction spectra of systems of small metal particles with aggregates,” in Proceedings of an Internationals Symposium on Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988), pp. 249–258.

Langbein, D.

D. Langbein, Van der Waals Attraction, Vol. 72 of Springer Tracts in Modern Physics (Springer-Verlag, Berlin, 1972), Chap. 5, p. 72.

Martin, T. P.

L. Genzel, T. P. Martin, U. Kreibig. “Dielectric function and plasma resonances of small metal particles,” Z. Phys. B 21, 339–346 (1975).
[CrossRef]

Mie, G.

G. Mie, “Beiträge zur” Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Niklasson, G. A.

J. Sotelo, G. A. Niklasson, “Optical properties of fractal clusters of small metallic particles,” Z. Phys. D 20, 321–323 (1991).
[CrossRef]

Papavassilion, G. C.

G. C. Papavassilion, T. Kokkinakis, “Optical absorption spectra of surface plasmons in small cooper particles,” J. Phys. F 4, L67–L68 (1974).
[CrossRef]

Pennypacker, C. R.

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

Purcell, E. M.

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

Quinten, M.

D. Schönauer, M. Quinten, U. Kreibig, “Precursor-states of percolation in quasi-fractal many-particle-systems,” Z. Phys. D 12, 527–532 (1989).
[CrossRef]

M. Quinten, D. Schönauer, U. Kreibig, “Electronic excitations in many-particle systems: a quantitative analysis,” Z. Phys. D 12, 521–525 (1989).
[CrossRef]

M. Quinten, U. Kreibig, “Optical properties of small metal particles,” Surf. Sci. 172, 557–577 (1986).
[CrossRef]

M. Quinten, U. Kreibig, “Optical extinction spectra of systems of small metal particles with aggregates,” in Proceedings of an Internationals Symposium on Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988), pp. 249–258.

Salzmann, T.

B. Dusemund, A. Hoffmann, T. Salzmann, U. Kreibig, G. Schmid, “Cluster matter: the transition of optical elastic scattering to regular reflection,” Z. Phys. D 20, 305–308 (1991).
[CrossRef]

Schmid, G.

B. Dusemund, A. Hoffmann, T. Salzmann, U. Kreibig, G. Schmid, “Cluster matter: the transition of optical elastic scattering to regular reflection,” Z. Phys. D 20, 305–308 (1991).
[CrossRef]

Schönauer, D.

D. Schönauer, M. Quinten, U. Kreibig, “Precursor-states of percolation in quasi-fractal many-particle-systems,” Z. Phys. D 12, 527–532 (1989).
[CrossRef]

M. Quinten, D. Schönauer, U. Kreibig, “Electronic excitations in many-particle systems: a quantitative analysis,” Z. Phys. D 12, 521–525 (1989).
[CrossRef]

Seitz, W.

W. Seitz, “Die Beugung des Lichtes an einem dünnen, zylindrischen Drahte,” Ann. Phys. 21, 1013–1029 (1906).
[CrossRef]

Sindoni, O. I.

Sotelo, J.

J. Sotelo, G. A. Niklasson, “Optical properties of fractal clusters of small metallic particles,” Z. Phys. D 20, 321–323 (1991).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electrodynamic Theory (McGraw-Hill, New York, 1941), Chap. 7, p. 392.

Toscano, G.

Trinks, W.

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

von Ignatowsky, W.

W. von Ignatowsky, “Reflexion elektromagnetischer Wellen an einem dünnen Draht,” Ann. Phys. 18, 495–522 (1905).
[CrossRef]

Yamamoto, G.

Zsigmondy, R.

R. Zsigmondy, Das kolloide Gold (Akademie Verlagsgeschäft, Leipzig, 1925).

Ann. Phys. (2)

W. von Ignatowsky, “Reflexion elektromagnetischer Wellen an einem dünnen Draht,” Ann. Phys. 18, 495–522 (1905).
[CrossRef]

W. Seitz, “Die Beugung des Lichtes an einem dünnen, zylindrischen Drahte,” Ann. Phys. 21, 1013–1029 (1906).
[CrossRef]

Ann. Phys. (4)

G. Mie, “Beiträge zur” Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
[CrossRef]

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

A. Güttler, “Die Miesche Theorie der Beugung durch dielektrische Kugeln mit absorbierendem Kern und ihre Bedeutung für Probleme der interstellaren Materie und des atmosphärischen Aerosols,” Ann. Phys. 1, 65–98 (1952).
[CrossRef]

Appl. Opt. (2)

Astrophys. J. (1)

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

Ber. Dtsch. Chem. Ges. (1)

L. Garbowski, Ber. Dtsch. Chem. Ges. 36, 1215 (1903).
[CrossRef]

Can. J. Phys. (1)

A.-K. Hamid, I. R. Ciric, M. Hamid, “Multiple scattering by a linear array of conducting spheres,” Can. J. Phys. 68, 1157–1165 (1990).
[CrossRef]

Can. J. Phys. (1)

A.-K. Hamid, I. R. Ciric, M. Hamid, “Electromagnetic scattering by an arbitrary configuration of dielectric spheres,” Can. J. Phys. 68, 1419–1428 (1990).
[CrossRef]

Geophys. J. R. Astron. Soc. (1)

B. Jeffreys, “Transformation of tesseral harmonics under rotation,” Geophys. J. R. Astron. Soc. 10, 141–145 (1965).
[CrossRef]

J. Phys. F (1)

G. C. Papavassilion, T. Kokkinakis, “Optical absorption spectra of surface plasmons in small cooper particles,” J. Phys. F 4, L67–L68 (1974).
[CrossRef]

J. Appl. Phys. (1)

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Opt. Lett. (1)

K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. II: Clusters of arbitrary configuration,” Opt. Lett. 13, 1063–1065 (1988).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. B (2)

J. M. Gérardy, 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. B25, 4204–4229 (1982).

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Proc. Inst. Electr. Eng. Part H (1)

A.-K. Hamid, I. R. Ciric, M. Hamid, “Iterative solution of the scattering by an arbitrary configuration of conducting or dielectric spheres,” Proc. Inst. Electr. Eng. Part H 138, 565–572 (1991).

Surf. Sci. (1)

M. Quinten, U. Kreibig, “Optical properties of small metal particles,” Surf. Sci. 172, 557–577 (1986).
[CrossRef]

Z. Phys. B (2)

L. Genzel, T. P. Martin, U. Kreibig. “Dielectric function and plasma resonances of small metal particles,” Z. Phys. B 21, 339–346 (1975).
[CrossRef]

L. Genzel, U. Kreibig, “Dielectric function and infrared absorption of small metal particles,” Z. Phys. B 37, 93–101 (1980).
[CrossRef]

Z. Phys. D (1)

D. Schönauer, M. Quinten, U. Kreibig, “Precursor-states of percolation in quasi-fractal many-particle-systems,” Z. Phys. D 12, 527–532 (1989).
[CrossRef]

Z. Phys. D (2)

M. Quinten, D. Schönauer, U. Kreibig, “Electronic excitations in many-particle systems: a quantitative analysis,” Z. Phys. D 12, 521–525 (1989).
[CrossRef]

J. Sotelo, G. A. Niklasson, “Optical properties of fractal clusters of small metallic particles,” Z. Phys. D 20, 321–323 (1991).
[CrossRef]

Z. Phys. D (1)

B. Dusemund, A. Hoffmann, T. Salzmann, U. Kreibig, G. Schmid, “Cluster matter: the transition of optical elastic scattering to regular reflection,” Z. Phys. D 20, 305–308 (1991).
[CrossRef]

Other (6)

R. Zsigmondy, Das kolloide Gold (Akademie Verlagsgeschäft, Leipzig, 1925).

J. A. Stratton, Electrodynamic Theory (McGraw-Hill, New York, 1941), Chap. 7, p. 392.

D. Langbein, Van der Waals Attraction, Vol. 72 of Springer Tracts in Modern Physics (Springer-Verlag, Berlin, 1972), Chap. 5, p. 72.

M. Quinten, U. Kreibig, “Optical extinction spectra of systems of small metal particles with aggregates,” in Proceedings of an Internationals Symposium on Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988), pp. 249–258.

A.-K. Hamid, I. R. Ciric, M. Hamid, “Scattering by systems of small conducting spheres,” “.Backscattering cross section of many sphere systems,” presented at the Electromagnetic Research Symposium (PIERS), Massachusetts Institute of Technology, Cambridge, Mass., 1991.

A.-K. Hamid, I. R. Ciric, M. Hamid, “Iterative technique for scattering by a linear array of spheres,” in Proceedings IEEE AP-S International Symposium (University of Western Ontario, London, Canada, 1991).

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Figures (9)

Fig. 1
Fig. 1

Incident plane wave and scattered waves for a system of N neighboring spherical particles.

Fig. 2
Fig. 2

Definition of the relative coordinates Rij, Θij, and φij between two neighboring particles i and j.

Fig. 3
Fig. 3

Examples of highly symmetric, densely packed planar aggregates with different particle number N.

Fig. 4
Fig. 4

Calculated extinction spectra of different arrangements of N = 5 spherical silver particles with mean diameter 2R = 40 nm, embedded in vacuum. Computations were performed in (a) the dipolar approximation (L = 1) and (b) the quadrupolar approximation (L = 2) and for a next-neighbor distance RNN = 2.01R (nearly touching spheres). ▲ = single particle.

Fig. 5
Fig. 5

Dipolar eigenmodes of a linear chain of N = 3 identical spheres (the wave vector k is arbitrary to the given directions of the induced dipoles).

Fig. 6
Fig. 6

Extinction spectra of linear chains of N = 10 identical silver spheres embedded in vacuum with single-particle diameter varying from 2R = 10 nm (first spectrum) up to 2R = 100 nm (last spectrum) in steps of 10 nm.

Fig. 7
Fig. 7

Energy positions of the lowest energy maximum and the highest energy maximum in the spectra of linear chains with particle number N varying from N = 2 up to N = 30 versus the particle number N for different single-sphere diameters.

Fig. 8
Fig. 8

Extinction spectra of planar quadratic arrays of silver spheres (2R = 40 nm, RNN = 2.01R, ɛM = 1) with particle numbers N = 4 (open circles), N = 9 (solid circles), N = 16 (open squares) and N = 25 (solid squares), compared with the spectra of a thin silver film (open triangles) and a single sphere (solid triangles).

Fig. 9
Fig. 9

Extinction spectra of a cubelike shaped aggregate of N = 8 silver spheres (solid squares) and an icosahedron-like aggregate of N = 13 spheres (solid circles) with 2R = 40, RNN = 2.01R, and embedded in vacuum, compared with spectra of single spheres of diameters 2R = 100 nm (open squares) and 2R = 84 nm (open circles). For comparison, the spectrum of the single sphere is plotted (full rhombus).

Tables (1)

Tables Icon

Table 1 Peak Splitting (ΔE) of a Linear Chain of N = 10 Identical Silver Spheres Depending on the Next-Neighbor Distance (RNN)

Equations (31)

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E M ( i ) = E o ( i ) + E sca ( i ) + j i N E sca ( j ) = E o l = 1 m = - l l a l m ( i ) M l m 1 i ( k ) + b l m ( i ) N l m 1 i ( k ) + E o l = 1 m = - l l c l m ( i ) M l m 3 i ( k ) + d l m ( i ) N l m 3 i ( k ) + E o j i N q = 1 p = - q q c q p ( j ) M q p 3 j ( k ) + d q p ( j ) N q p 3 j ( k ) .
H M ( i ) = H o ( i ) + H sca ( i ) + j i N H sca ( j ) = H o l = 1 m = - l l b l m ( i ) M l m 1 i ( k ) + a l m ( i ) N l m 1 i ( k ) + H o l = 1 m = - l l d l m ( i ) M l m 3 i ( k ) + c l m ( i ) N l m 3 i ( k ) + H o j i N q = 1 p = - q q d q p ( j ) M q p 3 j ( k ) + c q p ( j ) N q p 3 j ( k ) .
E ( i ) = E o l = 1 m = - l l S l m ( i ) M l m 1 i ( K ) + T l m ( i ) N l m 1 i ( K ) .
H ( i ) = K / k H o l = 1 m = - l l T l m ( i ) M l m 1 ( K ) + S l m ( i ) N l m 1 i ( K ) .
rot rot Z - k 2 · Z = 0 ,
k ^ 2 = ω 2 ɛ o μ o μ [ ɛ + i σ / ( ω ɛ o ) ] = ( ω / c ) 2 μ ɛ ( ω ) .
M = × ( Φ c ) ,
N = k - 1 · × × ( Φ c ) ,
Φ l m ( r , Θ , φ ) = z l ( k r ) Y l m ( Θ , φ ) / [ l ( l + 1 ) ] 0.5 .
M q p 3 j = l = 1 m = - l l M q p 3 j M l m 1 i M l m 1 i + M q p 3 j N l m 1 i N l m 1 i , N q p 3 j = l = 1 m = - l l N q p 3 j M l m 1 i M l m 1 i + N q p 3 j N l m 1 i N l m 1 i ,
M q p 3 j M l m 1 i = M q p 3 j · M l m 1 i * sin Θ i d Θ i d φ i M l m 1 i 2 sin Θ i d Θ i d φ i ,
M q p 3 j M l m 1 i = N q p 3 j N l m 1 i A q p l m ( i , j ) , M q p 3 j N l m 1 i = N q p 3 j M l m 1 i B q p l m ( i , j ) .
{ E o ( i ) + E sca ( i ) + j i N E sca ( j ) - E ( i ) } × e r r = R i = 0 ,
{ H o ( i ) + H sca ( i ) + j i N H sca ( j ) - H ( i ) } × e r r = R i = 0 ,
c l m ( i ) - Γ l ( i ) j i N q = 1 p = - q q c q p ( j ) A q p l m ( i , j ) + d p q ( j ) B q p l m ( i , j ) = Γ l ( i ) a l m ( i ) ,
d l m ( i ) - Δ l ( i ) j i N q = 1 p = - q q d q p ( j ) A q p l m ( i , j ) + c q p ( j ) B q p l m ( i , j ) = Δ l ( i ) b l m ( i ) .
Δ l ( i ) = - K 2 j l ( K R i ) [ k R i j l ( k R i ) ] - k 2 j l ( k R i ) [ K R i j l ( K R i ) ] K 2 j l ( K R i ) [ k R i h l ( 1 ) ( k R i ) ] - k 2 h l ( 1 ) ( k R i ) [ K R i j l ( K R i ) ] ,
Γ l ( i ) = - j l ( K R i ) [ k R i j l ( k R i ) ] - j l ( k R i ) [ K R i j l ( K R i ) ] j l ( K R i ) [ k R i h l ( 1 ) ( k R i ) ] - h l ( 1 ) ( k R i ) [ K R i j l ( K R i ) ] ,
W abs = div S dV = S e r r 2 sin Θ d Θ d φ = W ext - W sca ,
S = 1 / 2 Re ( E M × H M * ) ,
W ext ( N ) = ( 2 π I o / k 2 ) Re i = 1 N l = 1 m = - l l a l m ( i ) × c l m * ( i ) + b l m ( i ) d l m * ( i ) ,
W sca ( N ) = ( 2 π I o / k 2 ) × Re i = 1 N l = 1 m = - l l { c l m * ( i ) × [ a l m ( i ) + c l m ( i ) ( 1 - 1 / Γ l ) ] + d l m * ( i ) [ b l m ( i ) + d l m ( i ) ( 1 - 1 / Δ l ) ] } ,
A q p l m ( i , j ) = exp [ i ( p - m ) φ i j ] s = - r r O ( l , m , s , Θ i j ) × O ( q , p , s , Θ i j ) V q l s ( k R i j ) ,
B q p l m ( i , j ) = exp [ i ( p - m ) φ i j ] s = - r r O ( l , m , s , Θ i j ) × O ( q , p , s , Θ i j ) W q l s ( k R i j ) ,
V q l s ( k r ) = { ( 2 q + 1 ) ( 2 l + 1 ) ( q - s ) ! ( 1 - s ) ! q ( q + 1 ) l ( l + 1 ) ( q + s ) ! ( 1 + s ) ! } 0.5 × { l ( l + 1 ) U q l s ( k r ) - l ( l + 1 - s ) / ( 2 l + l ) k r U q , l + 1 s ( k r ) - ( l + 1 ) ( l + s ) / ( 2 l + 1 ) k r U q , l - 1 s ( k r ) ,
W q l s ( k r ) = { ( 2 q + 1 ) ( 2 l + 1 ) ( q - s ) ! ( l - s ) ! q ( q + 1 ) l ( l + 1 ) ( q + s ) ! ( l + s ) ! } 0.5 × ( - i s k r ) U q l s ( k r ) ,
U q l s ( k r ) = ( - 1 ) q - s ( 2 / k r ) s n = 0 q - s ( - 1 ) n × Γ ( q - n + 0.5 ) Γ ( l - n + 0.5 ) Γ ( s + n + 0.5 ) Γ ( q + l - s - n + 1.5 ) Γ ( s + 0.5 ) Γ ( 0.5 ) × ( q + l - n ) ! ( q + l - s - 2 n + 0.5 ) ( q - s - n ) ! s ! ( l - s - n ) ! × h q + l - s - 2 n ( 1 ) ( k r )
O ( l , m , s , Θ ) = ( - 1 ) l + s { ( l + s ) ! ( l - s ) ! ( l + m ) ! ( l - m ) ! } 0.5 × r = max min ( - 1 ) r r ! · [ cos ( Θ / 2 ) ] 2 r + m + s [ sin ( Θ / 2 ) ] 2 ( l - r ) - m - s ( l - s - r ) ! ( l - m - r ) ! ( m + s + r ) !
U l q s ( k r ) = ( - 1 ) q + 1 U q l s ( k r ) ,
V q l - s ( k r ) = V q l s ( k r ) ,
W q l - s ( k r ) = - W q l s ( k r ) .

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