Abstract

The general properties of antisymmetrical solutions of the coupled-dipole equation are studied. This equation is used to describe the interaction of a cluster of small particles acting as elementary dipoles with an external electromagnetic wave. It is shown that antisymmetrical (with zero total dipole moment) eigenstates can be excited even in clusters that are much smaller in size than the wavelength of the incident radiation. In this case the quality of the collective optical resonance may be enhanced by the large parameter (λ/Rc)2 (Rc is the characteristic size of the cluster). This phenomenon, in contrast to superradiance, leads to an increased [by the factor (λ/Rc)2] lifetime of the system in the excited state and can be called antisuperradiance.

© 1995 Optical Society of America

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  1. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
  2. P. Chiappetta, “Multiple scattering approach to light scattering by arbitrary shaped particles,” J. Phys. A 13, 2101–2108 (1980).
    [CrossRef]
  3. M. K. Singham, S. B. Singham, and G. C. Salzman, “The scattering matrix for randomly oriented particles,” J. Chem. Phys. 85, 3807–3815 (1986).
    [CrossRef]
  4. S. B. Singham, C. W. Patterson, and G. S. Salzman, “Polarizabilities for light scattering from chiral particles,” J. Chem. Phys. 85, 763–770 (1986).
    [CrossRef]
  5. S. B. Singham and G. C. Salzman, “Evolution of the scattering matrix of an arbitrary particle using the coupled dipole approximation,” J. Chem. Phys. 84, 2658–2667 (1986).
    [CrossRef]
  6. S. B. Singham and C. F. Bohren, “Light scattering by an arbitrary particle: the scattering order formulation of the coupled-dipole method,” J. Opt. Soc. Am. A 5, 1867–1872 (1988).
    [CrossRef] [PubMed]
  7. B. T. Draine, “The discrete-dipole approximation and its application to the interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [CrossRef]
  8. P. J. Flatau, G. L. Stephens, and B. T. Draine, “Light scattering by rectangular solids in the discrete-dipole approximation: a new algorithm exploiting the Bloch–Teoplitz structure,” J. Opt. Soc. Am. A 7, 593–600 (1990).
    [CrossRef]
  9. M. V. Berry and I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
    [CrossRef]
  10. V. M. Shalaev and M. I. Stockman, “Fractals: optical susceptibility and giant Raman scattering,” Z. Phys. D 10, 71–79 (1988).
    [CrossRef]
  11. A. V. Butenko, V. M. Shalaev, and M. I. Stockman, “Fractals: giant impurity nonlinearities in optics of fractal clusters,” Z. Phys. D 10, 81–92 (1988).
    [CrossRef]
  12. V. A. Markel, L. S. Muratov, M. I. Stockman, and T. F. George, “Theory and numerical simulation of the optical properties of fractal clusters,” Phys. Rev. B 43, 8183–8195 (1991).
    [CrossRef]
  13. M. I. Stockman, T. F. George, and V. A. Shalaev, “Field work and dispersion relations of excitations of fractals,” Phys. Rev. B 44, 115–121 (1991).
    [CrossRef]
  14. V. M. Shalaev, R. Botet, and R. Jullien, “Resonant light scattering by fractal clusters,” Phys. Rev. B 44, 12216–12225 (1991).
    [CrossRef]
  15. V. M. Shalaev, R. Botet, and A. V. Butenko, “Localization of collective dipole excitations on fractals,” Phys. Rev. B 48, 6662–6664 (1993).
    [CrossRef]
  16. V. M. Shalaev and R. Botet, “Optical free-induction decay in fractal clusters,” Phys. Rev. B 50, 12987–12990 (1994).
    [CrossRef]
  17. D. Keller and C. Bustamante, “Theory of the interaction of light with large inhomogeneous molecular aggregates. I. Absorption,” J. Chem. Phys. 84, 2961–2971 (1986).
    [CrossRef]
  18. G. H. Goedecke and S. G. O’Brien, “Scattering by irregular inhomogeneous particles via the digitized Green’s function algorithm,” Appl. Opt. 27, 2431–2438 (1988).
    [CrossRef] [PubMed]
  19. A. Lakhtakia, “General theory of the Purcell–Pennypacker scattering approach and its extension to bianisotropic scatterers,” Astrophys. J. 394, 494–499 (1992).
    [CrossRef]
  20. A. Lakhtakia, “Strong and weak forms of the method of moments and the coupled-dipole method for scattering of time-harmonic electromagnetic fields,” Int. J. Mod. Phys. C 3, 583–603 (1992).
    [CrossRef]
  21. V. A. Markel, “Scattering of light from two interacting spherical particles,” J. Mod. Opt. 39, 853–863 (1992).
    [CrossRef]
  22. Except in the N= 2 case, when degeneracy can be broken by introducing slightly different polarizabilities χ1and χ2.
  23. The author has diagonalized a lot of matrices for random clusters with N up to 1000 but never encountered random degeneracy.
  24. There is a special case of the so-called isotropic eigenvector, i.e., |n〉 ≠ 0 and 〈n¯ |n〉 = 0. This case requires special consideration; however, |n〉 cannot be isotropic for a nondegenerate eigenstate. Besides, as it is shown in Section 4, this case cannot be isotropic if the size of the system is less than the wavelength. Therefore this case is not considered.
  25. The derivations of cross sections in Section 2 stay valid if one substitutes external fields E0exp(ikri) by some arbitrary fields Ei, but the amplitude E0appears to be undefined. However, one can always find the electromagnetic energy flux for the arbitrary distribution of the incident field and redefine E0in an appropriate way.
  26. R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
    [CrossRef]
  27. N. E. Rehler and J. H. Eberly, “Superradiance,” Phys. Rev. A 3, 1735–1751 (1971).
    [CrossRef]
  28. R. Bonifacio, P. Schwendimann, and F. Haake, “Quantum statistical theory of superradiance,” Phys. Rev. A 4, 302–313, 854–864 (1971).
    [CrossRef]
  29. A. I. Zaitsev, V. A. Malyshev, and E. D. Trifonov, “Superradiance of a many-atom system with Coulomb interactions,” Sov. JETP 84, 475–486 (1983).
  30. Yu. A. Avetisyan, A. I. Zaitsev, and V. A. Malyshev, “To the theory of superradiance of many-atom systems: account of resonance dipole–dipole interaction of atoms,” Opt. Spektrosk. 59, 967–974 (1985).
  31. R. A. Horn and C. A. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, 1985).
    [CrossRef]

1994 (1)

V. M. Shalaev and R. Botet, “Optical free-induction decay in fractal clusters,” Phys. Rev. B 50, 12987–12990 (1994).
[CrossRef]

1993 (1)

V. M. Shalaev, R. Botet, and A. V. Butenko, “Localization of collective dipole excitations on fractals,” Phys. Rev. B 48, 6662–6664 (1993).
[CrossRef]

1992 (3)

A. Lakhtakia, “General theory of the Purcell–Pennypacker scattering approach and its extension to bianisotropic scatterers,” Astrophys. J. 394, 494–499 (1992).
[CrossRef]

A. Lakhtakia, “Strong and weak forms of the method of moments and the coupled-dipole method for scattering of time-harmonic electromagnetic fields,” Int. J. Mod. Phys. C 3, 583–603 (1992).
[CrossRef]

V. A. Markel, “Scattering of light from two interacting spherical particles,” J. Mod. Opt. 39, 853–863 (1992).
[CrossRef]

1991 (3)

V. A. Markel, L. S. Muratov, M. I. Stockman, and T. F. George, “Theory and numerical simulation of the optical properties of fractal clusters,” Phys. Rev. B 43, 8183–8195 (1991).
[CrossRef]

M. I. Stockman, T. F. George, and V. A. Shalaev, “Field work and dispersion relations of excitations of fractals,” Phys. Rev. B 44, 115–121 (1991).
[CrossRef]

V. M. Shalaev, R. Botet, and R. Jullien, “Resonant light scattering by fractal clusters,” Phys. Rev. B 44, 12216–12225 (1991).
[CrossRef]

1990 (1)

1988 (5)

G. H. Goedecke and S. G. O’Brien, “Scattering by irregular inhomogeneous particles via the digitized Green’s function algorithm,” Appl. Opt. 27, 2431–2438 (1988).
[CrossRef] [PubMed]

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

S. B. Singham and C. F. Bohren, “Light scattering by an arbitrary particle: the scattering order formulation of the coupled-dipole method,” J. Opt. Soc. Am. A 5, 1867–1872 (1988).
[CrossRef] [PubMed]

V. M. Shalaev and M. I. Stockman, “Fractals: optical susceptibility and giant Raman scattering,” Z. Phys. D 10, 71–79 (1988).
[CrossRef]

A. V. Butenko, V. M. Shalaev, and M. I. Stockman, “Fractals: giant impurity nonlinearities in optics of fractal clusters,” Z. Phys. D 10, 81–92 (1988).
[CrossRef]

1986 (5)

M. V. Berry and I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

D. Keller and C. Bustamante, “Theory of the interaction of light with large inhomogeneous molecular aggregates. I. Absorption,” J. Chem. Phys. 84, 2961–2971 (1986).
[CrossRef]

M. K. Singham, S. B. Singham, and G. C. Salzman, “The scattering matrix for randomly oriented particles,” J. Chem. Phys. 85, 3807–3815 (1986).
[CrossRef]

S. B. Singham, C. W. Patterson, and G. S. Salzman, “Polarizabilities for light scattering from chiral particles,” J. Chem. Phys. 85, 763–770 (1986).
[CrossRef]

S. B. Singham and G. C. Salzman, “Evolution of the scattering matrix of an arbitrary particle using the coupled dipole approximation,” J. Chem. Phys. 84, 2658–2667 (1986).
[CrossRef]

1985 (1)

Yu. A. Avetisyan, A. I. Zaitsev, and V. A. Malyshev, “To the theory of superradiance of many-atom systems: account of resonance dipole–dipole interaction of atoms,” Opt. Spektrosk. 59, 967–974 (1985).

1983 (1)

A. I. Zaitsev, V. A. Malyshev, and E. D. Trifonov, “Superradiance of a many-atom system with Coulomb interactions,” Sov. JETP 84, 475–486 (1983).

1980 (1)

P. Chiappetta, “Multiple scattering approach to light scattering by arbitrary shaped particles,” J. Phys. A 13, 2101–2108 (1980).
[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 (2)

N. E. Rehler and J. H. Eberly, “Superradiance,” Phys. Rev. A 3, 1735–1751 (1971).
[CrossRef]

R. Bonifacio, P. Schwendimann, and F. Haake, “Quantum statistical theory of superradiance,” Phys. Rev. A 4, 302–313, 854–864 (1971).
[CrossRef]

1954 (1)

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
[CrossRef]

Avetisyan, Yu. A.

Yu. A. Avetisyan, A. I. Zaitsev, and V. A. Malyshev, “To the theory of superradiance of many-atom systems: account of resonance dipole–dipole interaction of atoms,” Opt. Spektrosk. 59, 967–974 (1985).

Berry, M. V.

M. V. Berry and I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

Bohren, C. F.

Bonifacio, R.

R. Bonifacio, P. Schwendimann, and F. Haake, “Quantum statistical theory of superradiance,” Phys. Rev. A 4, 302–313, 854–864 (1971).
[CrossRef]

Botet, R.

V. M. Shalaev and R. Botet, “Optical free-induction decay in fractal clusters,” Phys. Rev. B 50, 12987–12990 (1994).
[CrossRef]

V. M. Shalaev, R. Botet, and A. V. Butenko, “Localization of collective dipole excitations on fractals,” Phys. Rev. B 48, 6662–6664 (1993).
[CrossRef]

V. M. Shalaev, R. Botet, and R. Jullien, “Resonant light scattering by fractal clusters,” Phys. Rev. B 44, 12216–12225 (1991).
[CrossRef]

Bustamante, C.

D. Keller and C. Bustamante, “Theory of the interaction of light with large inhomogeneous molecular aggregates. I. Absorption,” J. Chem. Phys. 84, 2961–2971 (1986).
[CrossRef]

Butenko, A. V.

V. M. Shalaev, R. Botet, and A. V. Butenko, “Localization of collective dipole excitations on fractals,” Phys. Rev. B 48, 6662–6664 (1993).
[CrossRef]

A. V. Butenko, V. M. Shalaev, and M. I. Stockman, “Fractals: giant impurity nonlinearities in optics of fractal clusters,” Z. Phys. D 10, 81–92 (1988).
[CrossRef]

Chiappetta, P.

P. Chiappetta, “Multiple scattering approach to light scattering by arbitrary shaped particles,” J. Phys. A 13, 2101–2108 (1980).
[CrossRef]

Dicke, R. H.

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
[CrossRef]

Draine, B. T.

Eberly, J. H.

N. E. Rehler and J. H. Eberly, “Superradiance,” Phys. Rev. A 3, 1735–1751 (1971).
[CrossRef]

Flatau, P. J.

George, T. F.

V. A. Markel, L. S. Muratov, M. I. Stockman, and T. F. George, “Theory and numerical simulation of the optical properties of fractal clusters,” Phys. Rev. B 43, 8183–8195 (1991).
[CrossRef]

M. I. Stockman, T. F. George, and V. A. Shalaev, “Field work and dispersion relations of excitations of fractals,” Phys. Rev. B 44, 115–121 (1991).
[CrossRef]

Goedecke, G. H.

Haake, F.

R. Bonifacio, P. Schwendimann, and F. Haake, “Quantum statistical theory of superradiance,” Phys. Rev. A 4, 302–313, 854–864 (1971).
[CrossRef]

Horn, R. A.

R. A. Horn and C. A. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, 1985).
[CrossRef]

Johnson, C. A.

R. A. Horn and C. A. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, 1985).
[CrossRef]

Jullien, R.

V. M. Shalaev, R. Botet, and R. Jullien, “Resonant light scattering by fractal clusters,” Phys. Rev. B 44, 12216–12225 (1991).
[CrossRef]

Keller, D.

D. Keller and C. Bustamante, “Theory of the interaction of light with large inhomogeneous molecular aggregates. I. Absorption,” J. Chem. Phys. 84, 2961–2971 (1986).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia, “General theory of the Purcell–Pennypacker scattering approach and its extension to bianisotropic scatterers,” Astrophys. J. 394, 494–499 (1992).
[CrossRef]

A. Lakhtakia, “Strong and weak forms of the method of moments and the coupled-dipole method for scattering of time-harmonic electromagnetic fields,” Int. J. Mod. Phys. C 3, 583–603 (1992).
[CrossRef]

Malyshev, V. A.

Yu. A. Avetisyan, A. I. Zaitsev, and V. A. Malyshev, “To the theory of superradiance of many-atom systems: account of resonance dipole–dipole interaction of atoms,” Opt. Spektrosk. 59, 967–974 (1985).

A. I. Zaitsev, V. A. Malyshev, and E. D. Trifonov, “Superradiance of a many-atom system with Coulomb interactions,” Sov. JETP 84, 475–486 (1983).

Markel, V. A.

V. A. Markel, “Scattering of light from two interacting spherical particles,” J. Mod. Opt. 39, 853–863 (1992).
[CrossRef]

V. A. Markel, L. S. Muratov, M. I. Stockman, and T. F. George, “Theory and numerical simulation of the optical properties of fractal clusters,” Phys. Rev. B 43, 8183–8195 (1991).
[CrossRef]

Muratov, L. S.

V. A. Markel, L. S. Muratov, M. I. Stockman, and T. F. George, “Theory and numerical simulation of the optical properties of fractal clusters,” Phys. Rev. B 43, 8183–8195 (1991).
[CrossRef]

O’Brien, S. G.

Patterson, C. W.

S. B. Singham, C. W. Patterson, and G. S. Salzman, “Polarizabilities for light scattering from chiral particles,” J. Chem. Phys. 85, 763–770 (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]

Percival, I. C.

M. V. Berry and I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[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]

Rehler, N. E.

N. E. Rehler and J. H. Eberly, “Superradiance,” Phys. Rev. A 3, 1735–1751 (1971).
[CrossRef]

Salzman, G. C.

M. K. Singham, S. B. Singham, and G. C. Salzman, “The scattering matrix for randomly oriented particles,” J. Chem. Phys. 85, 3807–3815 (1986).
[CrossRef]

S. B. Singham and G. C. Salzman, “Evolution of the scattering matrix of an arbitrary particle using the coupled dipole approximation,” J. Chem. Phys. 84, 2658–2667 (1986).
[CrossRef]

Salzman, G. S.

S. B. Singham, C. W. Patterson, and G. S. Salzman, “Polarizabilities for light scattering from chiral particles,” J. Chem. Phys. 85, 763–770 (1986).
[CrossRef]

Schwendimann, P.

R. Bonifacio, P. Schwendimann, and F. Haake, “Quantum statistical theory of superradiance,” Phys. Rev. A 4, 302–313, 854–864 (1971).
[CrossRef]

Shalaev, V. A.

M. I. Stockman, T. F. George, and V. A. Shalaev, “Field work and dispersion relations of excitations of fractals,” Phys. Rev. B 44, 115–121 (1991).
[CrossRef]

Shalaev, V. M.

V. M. Shalaev and R. Botet, “Optical free-induction decay in fractal clusters,” Phys. Rev. B 50, 12987–12990 (1994).
[CrossRef]

V. M. Shalaev, R. Botet, and A. V. Butenko, “Localization of collective dipole excitations on fractals,” Phys. Rev. B 48, 6662–6664 (1993).
[CrossRef]

V. M. Shalaev, R. Botet, and R. Jullien, “Resonant light scattering by fractal clusters,” Phys. Rev. B 44, 12216–12225 (1991).
[CrossRef]

A. V. Butenko, V. M. Shalaev, and M. I. Stockman, “Fractals: giant impurity nonlinearities in optics of fractal clusters,” Z. Phys. D 10, 81–92 (1988).
[CrossRef]

V. M. Shalaev and M. I. Stockman, “Fractals: optical susceptibility and giant Raman scattering,” Z. Phys. D 10, 71–79 (1988).
[CrossRef]

Singham, M. K.

M. K. Singham, S. B. Singham, and G. C. Salzman, “The scattering matrix for randomly oriented particles,” J. Chem. Phys. 85, 3807–3815 (1986).
[CrossRef]

Singham, S. B.

S. B. Singham and C. F. Bohren, “Light scattering by an arbitrary particle: the scattering order formulation of the coupled-dipole method,” J. Opt. Soc. Am. A 5, 1867–1872 (1988).
[CrossRef] [PubMed]

M. K. Singham, S. B. Singham, and G. C. Salzman, “The scattering matrix for randomly oriented particles,” J. Chem. Phys. 85, 3807–3815 (1986).
[CrossRef]

S. B. Singham and G. C. Salzman, “Evolution of the scattering matrix of an arbitrary particle using the coupled dipole approximation,” J. Chem. Phys. 84, 2658–2667 (1986).
[CrossRef]

S. B. Singham, C. W. Patterson, and G. S. Salzman, “Polarizabilities for light scattering from chiral particles,” J. Chem. Phys. 85, 763–770 (1986).
[CrossRef]

Stephens, G. L.

Stockman, M. I.

V. A. Markel, L. S. Muratov, M. I. Stockman, and T. F. George, “Theory and numerical simulation of the optical properties of fractal clusters,” Phys. Rev. B 43, 8183–8195 (1991).
[CrossRef]

M. I. Stockman, T. F. George, and V. A. Shalaev, “Field work and dispersion relations of excitations of fractals,” Phys. Rev. B 44, 115–121 (1991).
[CrossRef]

A. V. Butenko, V. M. Shalaev, and M. I. Stockman, “Fractals: giant impurity nonlinearities in optics of fractal clusters,” Z. Phys. D 10, 81–92 (1988).
[CrossRef]

V. M. Shalaev and M. I. Stockman, “Fractals: optical susceptibility and giant Raman scattering,” Z. Phys. D 10, 71–79 (1988).
[CrossRef]

Trifonov, E. D.

A. I. Zaitsev, V. A. Malyshev, and E. D. Trifonov, “Superradiance of a many-atom system with Coulomb interactions,” Sov. JETP 84, 475–486 (1983).

Zaitsev, A. I.

Yu. A. Avetisyan, A. I. Zaitsev, and V. A. Malyshev, “To the theory of superradiance of many-atom systems: account of resonance dipole–dipole interaction of atoms,” Opt. Spektrosk. 59, 967–974 (1985).

A. I. Zaitsev, V. A. Malyshev, and E. D. Trifonov, “Superradiance of a many-atom system with Coulomb interactions,” Sov. JETP 84, 475–486 (1983).

Appl. Opt. (1)

Astrophys. J. (3)

A. Lakhtakia, “General theory of the Purcell–Pennypacker scattering approach and its extension to bianisotropic scatterers,” Astrophys. J. 394, 494–499 (1992).
[CrossRef]

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

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

Int. J. Mod. Phys. C (1)

A. Lakhtakia, “Strong and weak forms of the method of moments and the coupled-dipole method for scattering of time-harmonic electromagnetic fields,” Int. J. Mod. Phys. C 3, 583–603 (1992).
[CrossRef]

J. Chem. Phys. (4)

M. K. Singham, S. B. Singham, and G. C. Salzman, “The scattering matrix for randomly oriented particles,” J. Chem. Phys. 85, 3807–3815 (1986).
[CrossRef]

S. B. Singham, C. W. Patterson, and G. S. Salzman, “Polarizabilities for light scattering from chiral particles,” J. Chem. Phys. 85, 763–770 (1986).
[CrossRef]

S. B. Singham and G. C. Salzman, “Evolution of the scattering matrix of an arbitrary particle using the coupled dipole approximation,” J. Chem. Phys. 84, 2658–2667 (1986).
[CrossRef]

D. Keller and C. Bustamante, “Theory of the interaction of light with large inhomogeneous molecular aggregates. I. Absorption,” J. Chem. Phys. 84, 2961–2971 (1986).
[CrossRef]

J. Mod. Opt. (1)

V. A. Markel, “Scattering of light from two interacting spherical particles,” J. Mod. Opt. 39, 853–863 (1992).
[CrossRef]

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

J. Phys. A (1)

P. Chiappetta, “Multiple scattering approach to light scattering by arbitrary shaped particles,” J. Phys. A 13, 2101–2108 (1980).
[CrossRef]

Opt. Acta (1)

M. V. Berry and I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

Opt. Spektrosk. (1)

Yu. A. Avetisyan, A. I. Zaitsev, and V. A. Malyshev, “To the theory of superradiance of many-atom systems: account of resonance dipole–dipole interaction of atoms,” Opt. Spektrosk. 59, 967–974 (1985).

Phys. Rev. (1)

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
[CrossRef]

Phys. Rev. A (2)

N. E. Rehler and J. H. Eberly, “Superradiance,” Phys. Rev. A 3, 1735–1751 (1971).
[CrossRef]

R. Bonifacio, P. Schwendimann, and F. Haake, “Quantum statistical theory of superradiance,” Phys. Rev. A 4, 302–313, 854–864 (1971).
[CrossRef]

Phys. Rev. B (5)

V. A. Markel, L. S. Muratov, M. I. Stockman, and T. F. George, “Theory and numerical simulation of the optical properties of fractal clusters,” Phys. Rev. B 43, 8183–8195 (1991).
[CrossRef]

M. I. Stockman, T. F. George, and V. A. Shalaev, “Field work and dispersion relations of excitations of fractals,” Phys. Rev. B 44, 115–121 (1991).
[CrossRef]

V. M. Shalaev, R. Botet, and R. Jullien, “Resonant light scattering by fractal clusters,” Phys. Rev. B 44, 12216–12225 (1991).
[CrossRef]

V. M. Shalaev, R. Botet, and A. V. Butenko, “Localization of collective dipole excitations on fractals,” Phys. Rev. B 48, 6662–6664 (1993).
[CrossRef]

V. M. Shalaev and R. Botet, “Optical free-induction decay in fractal clusters,” Phys. Rev. B 50, 12987–12990 (1994).
[CrossRef]

Sov. JETP (1)

A. I. Zaitsev, V. A. Malyshev, and E. D. Trifonov, “Superradiance of a many-atom system with Coulomb interactions,” Sov. JETP 84, 475–486 (1983).

Z. Phys. D (2)

V. M. Shalaev and M. I. Stockman, “Fractals: optical susceptibility and giant Raman scattering,” Z. Phys. D 10, 71–79 (1988).
[CrossRef]

A. V. Butenko, V. M. Shalaev, and M. I. Stockman, “Fractals: giant impurity nonlinearities in optics of fractal clusters,” Z. Phys. D 10, 81–92 (1988).
[CrossRef]

Other (5)

Except in the N= 2 case, when degeneracy can be broken by introducing slightly different polarizabilities χ1and χ2.

The author has diagonalized a lot of matrices for random clusters with N up to 1000 but never encountered random degeneracy.

There is a special case of the so-called isotropic eigenvector, i.e., |n〉 ≠ 0 and 〈n¯ |n〉 = 0. This case requires special consideration; however, |n〉 cannot be isotropic for a nondegenerate eigenstate. Besides, as it is shown in Section 4, this case cannot be isotropic if the size of the system is less than the wavelength. Therefore this case is not considered.

The derivations of cross sections in Section 2 stay valid if one substitutes external fields E0exp(ikri) by some arbitrary fields Ei, but the amplitude E0appears to be undefined. However, one can always find the electromagnetic energy flux for the arbitrary distribution of the incident field and redefine E0in an appropriate way.

R. A. Horn and C. A. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, 1985).
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Equations (72)

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E inc ( r , t ) = E 0 exp ( i kr - i ω t )
d i = χ [ E 0 exp ( i k r i ) + j = 1 N W ^ ( r i - r j ) d j ] ,
W α β ( r ) = k 3 [ A ( k r ) δ α β + B ( k r ) r α r β r 2 ] ,
A ( x ) = ( x - 1 + i x - 2 - x - 3 ) exp ( i x ) ,
B ( x ) = ( - x - 1 - 3 i x - 2 + 3 x - 3 ) exp ( i x ) .
E s = k 2 i = 1 N d i - ( d i s ) s r i - R exp ( i k r i - R ) .
f ( s ) = k 2 i = 1 N [ d i - ( d i s ) s ] exp ( - i k s r i ) .
σ e = 4 π k Im [ f ( k / k ) E 0 * ] E 0 2 ,
σ s = 1 E 0 2 f ( s ) 2 d Ω ,
σ α = σ e - σ s .
σ e = 4 π k E 0 2 Im i = 1 N d i E 0 * exp ( - i k r i ) ,
σ s = k 4 E 0 2 i , j = 1 N [ d i d j * - ( d i s ) ( d j * s ) ] × exp [ i k s ( r i - r j ) ] d Ω .
σ s = i = 1 N σ s , i j + i j N σ s , i j ,
σ s , i i = 8 π k 4 3 d i 2 E 0 2 ,
σ s , i j = 4 π k 4 E 0 2 { [ sin ( k r i j ) k r i j + cos ( k r i j ) ( k r i j ) 2 - sin ( k r i j ) ( k r i j ) 3 ] d i d j * + [ - sin ( k r i j ) k r i j - 3 cos ( k r i j ) ( k r i j ) 2 + 3 sin ( k r i j ) ( k r i j ) 3 ] × ( d i n i j ) ( d j * n i j ) } ,
σ s , i j = 4 π k 4 E 0 2 [ d i d j * Im A ( k r i j ) + ( d i n i j ) ( d j * n i j ) × Im B ( k r i j ) ] .
i j N σ s , i j = 4 π k 4 E 0 2 Im i j N [ d i d j * A ( k r i j ) + ( d i n i j ) ( d j * n i j ) B ( k r i j ) ] .
i j N σ s , i j = 4 π k E 0 2 Im i j N j = 1 j i N [ d i * · W ^ ( r i - r j ) d j ] .
j = 1 N W ^ ( r i - r j ) d j = 1 χ d i - E 0 exp ( i k r i ) .
σ s = 4 π k E 0 2 i = j N { Im [ d i E 0 * exp ( - i k r i ) ] - y α d i 2 } ,
y α = - Im ( χ - 1 ) - 2 k 3 / 3.
σ α = 4 π k E 0 2 y α i = 1 N d i 2 .
d = χ ( E + W d ) .
W n = w n n .
m ¯ n = 0             if             m n ,
1 = n ¯ n n ¯ n .
d = n = 1 3 N n n ¯ E n ¯ n ( 1 / χ - w n ) .
i = 1 N d i E 0 exp ( - i k r i ) = E d ,             i = 1 N d i 2 = d d ,
σ e = 4 π k E 0 2 Im n = 1 3 N E n n ¯ E n ¯ n ( 1 / χ - ω n ) ,
σ α = 4 π k y a E 0 2 × m n 3 N E m ¯ m n n ¯ E m m ¯ n ¯ n ( 1 / χ - w n ) ( 1 / χ - w m ) * .
σ e = 4 π k Im ( 1 / χ - w M ) * 1 / χ - w M 2 ,             σ α = 4 π k y α 1 / χ - w M 2 .
Im w n - 2 k 3 / 3 n .
n = 1 3 N w n = 0 ,
Im w n ( 3 N - 1 ) 2 k 3 / 3 n .
Im w n ( N - 1 ) 2 k 3 / 3 ,             n [ 1 , N ] ( orthogonal polarization ) , Im w n ( 2 N - 1 ) 2 k 3 / 3 ,             n [ N + 1 , 3 N ] ( parallel polarization ) .
W = W r + i W i ,
W i = V 1 + V 2 + V 3 + .
i α V 1 j β = 2 k 3 3 ( 1 - δ i j ) δ α β ,
i α V 2 j β = 2 k 3 3 ( k r i j ) 2 10 ( - 2 δ α β + r i j α r i j β r i j 2 ) .
n = n ( 0 ) + n ( 1 ) + n ( 2 ) + ,
w n = w n ( 0 ) + w n ( 1 ) + w n ( 2 ) + .
σ e ( 0 ) = 4 π k E 0 2 n = 1 3 N E n ( 0 ) 2 ( y α + 2 k 3 / 3 + Im w n ) 1 / χ - w n 2 ,
σ α ( 0 ) = 4 π k E 0 2 n = 1 3 N E n ( 0 ) 2 y α 1 / χ - w n 2 ,
σ s ( 0 ) = 4 π k E 0 2 n = 1 3 N E n ( 0 ) 2 2 k 3 / 3 + Im w n 1 / χ - w n 2 .
w n ( 1 ) = i n ( 0 ) V 1 n ( 0 ) ,
w n ( 2 ) = - m n m ( 0 ) V 1 n ( 0 ) 2 w n ( 0 ) - w m ( 1 ) + i n ( 0 ) V 2 n ( 0 ) .
V 1 = 2 k 3 3 ( α = 1 3 0 α 0 α - I ) ,
m ( 0 ) V 1 n ( 0 ) = 2 k 3 3 α = 1 3 m ( 0 ) 0 α 0 α n ( 0 ) ,             for             m n ,
n ( 0 ) V 1 n ( 0 ) = 2 k 3 3 ( α = 1 3 n ( 0 ) 0 α 2 - 1 ) .
w n ( 1 ) = i 2 k 3 3 ( D n 2 - 1 ) ,
w n ( 2 ) = - ( 2 k 3 3 ) 2 m n D m D n 2 w n ( 0 ) - w n ( 0 ) + n ( 0 ) V 2 n ( 0 ) .
w n ( 1 ) = - i ( 2 k 3 / 3 ) ,             w n ( 2 ) = i n ( 0 ) V 2 n ( 0 ) .
σ s ( 0 ) = 8 π k 4 3 E 0 2 n = 1 3 N E n ( 0 ) 2 D n 2 1 / χ - w n 2 .
D tot 2 = n , m = 1 3 N n ( 0 ) E E m ( 0 ) D n D m ( 1 / χ - w n ) ( 1 / χ - w m ) * .
n ( 1 ) = m = 1 3 N m ( 0 ) m ( 0 ) V 1 n ( 0 ) w n ( 0 ) - w m ( 0 ) = 2 k 3 3 m = 1 3 N m ( 0 ) D m D n w n ( 0 ) - w m ( 0 ) .
m n = δ m n + i 4 k 3 3 ( 1 - δ m n ) D m D n w n ( 0 ) - w m ( 0 ) ,             n ¯ n = n n ¯ = 1.
σ e ( 1 ) = 16 π k 4 3 E 0 2 m n ( D n D m ) Re [ E n ( 0 ) m ( 0 ) E ] w n ( 0 ) - w m ( 0 ) × Re ( 1 1 / χ - w n ) ,
σ α ( 1 ) = 16 π k 4 y α 3 E 0 2 Im m n D n D m E n ( 0 ) m ( 0 ) E [ w n ( 0 ) - w m ( 0 ) ] ( 1 / χ - w n ) × ( 1 1 / χ - w n - 1 1 / χ - w m ) * .
1 / χ = - x - i y ,             y = 2 k 3 / 3 + y α
σ s ( 0 ) = 4 π k E 0 2 E M ( 0 ) 2 M ( 0 ) V 2 M ( 0 ) ( x + w M ( 0 ) ) 2 + [ y α + M ( 0 ) V 2 M ( 0 ) ] 2 ,
σ s ( 0 ) ( x = - w M ( 0 ) ) = 4 π k E 0 2 E M ( 0 ) 2 M ( 0 ) V 2 M ( 0 ) .
σ s ( 0 ) ( x = - w M ( 0 ) ) ~ k - 2 ~ λ 2 .
σ α ( 0 ) ( x = - w M ( 0 ) ) = 4 π k E 0 2 E M ( 0 ) 2 y α [ y α + M ( 0 ) V 2 M ( 0 ) ] 2 .
w 1 = - k 3 A ( k a ) ,             w 2 = - k 3 [ A ( k a ) + B ( k a ) ] ,
Im w 1 ( 1 ) = - 2 k 3 / 3 ,             Im w 1 ( 2 ) = ( 2 k 3 / 15 ) ( k a ) 2 ,
Im w 2 ( 1 ) = - 2 k 3 / 3 ,             Im w 2 ( 2 ) = ( k 2 / 15 ) ( k a ) 2 .
w 1 = - k 3 A ( 2 k a ) ,             w 2 = k 3 [ A ( 2 k a ) - 2 A ( k a ) ] .
Im w 1 ( 1 ) = - 2 k 3 / 3 ,             Im w 1 ( 2 ) = ( 4 k 3 / 15 ) ( k a ) 2 ,
Im w 2 ( 1 ) = - 2 k 3 / 3 ,             Im w 2 ( 2 ) = 0 ,             Im w 2 ( 3 ) = ( k 3 / 70 ) ( k a ) 4 .
m ¯ n = 0             if             m n ,
n ¯ W m = i , j n ¯ e i e i W e j e j m .
n ¯ W m = i , j m ¯ e j e j W e i e j n = m ¯ W n .

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