Abstract

In the coupled-dipole method an arbitrary particle is modeled as an array of N polarizable subunits, each of which gives rise to only electric dipole radiation. The Clausius-Mosotti relation is widely used to calculate the polarizability of the subunits that correspond to the dielectric function of the particle that the array represents. We replace the Clausius-Mosotti relation with an exact expression for the electric dipole polarizability and find improvement in extinction calculations for spheres as compared with Mie theory. Near a Fröhlich frequency the coupled-dipole method yields extinction cross sections for spheres and spheroids that compare favorably with the continuous distribution of ellipsoids method and measured values.

© 1991 Optical Society of America

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References

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  1. E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
  2. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [CrossRef]
  3. G. H. Goedecke, A. G. O’Brien, “Scattering by irregular inhomogeneous particles via the digitized Green’s function algorithm,” Appl. Opt. 27, 2431–2438 (1988).
    [CrossRef] [PubMed]
  4. P. T. Flatau, G. L. Stephens, B. T. Draine, “Light scattering by rectangular solids in the discrete-dipole approximation: a new algorithm exploiting the Block-Toeplitz structure,” J. Opt. Soc. Am. A 7, 593–600 (1990).
    [CrossRef]
  5. S. B. Singham, C. W. Patterson, G. C. Salzman, “Polarizabilities for light scattering from chiral particles,”J. Chem. Phys. 85, 763–770 (1986).
    [CrossRef]
  6. V. V. Varadan, A. Lakhtakia, V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,”IEEE Trans. Antennas Propag. 37, 800–802 (1989).
    [CrossRef]
  7. A. Lakhtakia, “Macroscopic theory of the coupled dipole approximation method,” Opt. Commun. (to be published).
  8. M. K. Singham, S. B. Singham, G. C. Salzman, “The scattering matrix for randomly oriented particles,”J. Chem. Phys. 85, 3807–3815 (1986).
    [CrossRef]
  9. P. Chiapetta, “Multiple scattering approach to light scattering by arbitrarily shaped particles,”J. Phys. A 13, 2101–2108 (1980).
    [CrossRef]
  10. S. B. Singham, 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]
  11. S. B. Singham, G. C. Salzman, “Evaluation of the scattering matrix of an arbitrary particle using the coupled-dipole approximation,”J. Chem. Phys. 84, 2658–2667 (1986).
    [CrossRef]
  12. W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
    [CrossRef]
  13. D. R. Huffman, C. F. Bohren, “Infrared absorption spectra of non-spherical particles treated in the Rayleigh-ellipsoid approximation,” in Light Scattering by Irregularly Shaped Particles, D. Schuerman, ed. (Plenum, New York, 1980), pp. 103–111.
    [CrossRef]
  14. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).
  15. A. S. Barker, “Infrared absorption of localized longitudinal-optical phonons,” Phys. Rev. B 7, 2507–2520 (1973).
    [CrossRef]
  16. Y. L. Yung, “Variational principle for scattering light by dielectric particles,” Appl. Opt. 17, 3707–3707 (1978).
    [CrossRef] [PubMed]
  17. W. G. Spitzer, D. A. Kleinman, “Infrared lattice bands of quartz,” Phys. Rev. 121, 1324–1335 (1961).
    [CrossRef]

1990 (1)

1989 (2)

V. V. Varadan, A. Lakhtakia, V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,”IEEE Trans. Antennas Propag. 37, 800–802 (1989).
[CrossRef]

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

1988 (3)

1986 (3)

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

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

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

1980 (1)

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

1978 (1)

1973 (2)

A. S. Barker, “Infrared absorption of localized longitudinal-optical phonons,” Phys. Rev. B 7, 2507–2520 (1973).
[CrossRef]

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

1961 (1)

W. G. Spitzer, D. A. Kleinman, “Infrared lattice bands of quartz,” Phys. Rev. 121, 1324–1335 (1961).
[CrossRef]

Barker, A. S.

A. S. Barker, “Infrared absorption of localized longitudinal-optical phonons,” Phys. Rev. B 7, 2507–2520 (1973).
[CrossRef]

Bohren, C. F.

S. B. Singham, 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]

D. R. Huffman, C. F. Bohren, “Infrared absorption spectra of non-spherical particles treated in the Rayleigh-ellipsoid approximation,” in Light Scattering by Irregularly Shaped Particles, D. Schuerman, ed. (Plenum, New York, 1980), pp. 103–111.
[CrossRef]

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

Chiapetta, P.

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

Doyle, W. T.

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

Draine, B. T.

Flatau, P. T.

Goedecke, G. H.

Huffman, D. R.

D. R. Huffman, C. F. Bohren, “Infrared absorption spectra of non-spherical particles treated in the Rayleigh-ellipsoid approximation,” in Light Scattering by Irregularly Shaped Particles, D. Schuerman, ed. (Plenum, New York, 1980), pp. 103–111.
[CrossRef]

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

Kleinman, D. A.

W. G. Spitzer, D. A. Kleinman, “Infrared lattice bands of quartz,” Phys. Rev. 121, 1324–1335 (1961).
[CrossRef]

Lakhtakia, A.

V. V. Varadan, A. Lakhtakia, V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,”IEEE Trans. Antennas Propag. 37, 800–802 (1989).
[CrossRef]

A. Lakhtakia, “Macroscopic theory of the coupled dipole approximation method,” Opt. Commun. (to be published).

O’Brien, A. G.

Patterson, C. W.

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

Salzman, G. C.

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

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

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

Singham, M. K.

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

Singham, S. B.

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

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

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

Spitzer, W. G.

W. G. Spitzer, D. A. Kleinman, “Infrared lattice bands of quartz,” Phys. Rev. 121, 1324–1335 (1961).
[CrossRef]

Stephens, G. L.

Varadan, V. K.

V. V. Varadan, A. Lakhtakia, V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,”IEEE Trans. Antennas Propag. 37, 800–802 (1989).
[CrossRef]

Varadan, V. V.

V. V. Varadan, A. Lakhtakia, V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,”IEEE Trans. Antennas Propag. 37, 800–802 (1989).
[CrossRef]

Yung, Y. L.

Appl. Opt. (2)

Astrophys. J. (2)

E. M. Purcell, 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. (1)

V. V. Varadan, A. Lakhtakia, V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,”IEEE Trans. Antennas Propag. 37, 800–802 (1989).
[CrossRef]

J. Chem. Phys. (3)

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

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

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

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

J. Phys. A (1)

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

Phys. Rev. (1)

W. G. Spitzer, D. A. Kleinman, “Infrared lattice bands of quartz,” Phys. Rev. 121, 1324–1335 (1961).
[CrossRef]

Phys. Rev. B (2)

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

A. S. Barker, “Infrared absorption of localized longitudinal-optical phonons,” Phys. Rev. B 7, 2507–2520 (1973).
[CrossRef]

Other (3)

D. R. Huffman, C. F. Bohren, “Infrared absorption spectra of non-spherical particles treated in the Rayleigh-ellipsoid approximation,” in Light Scattering by Irregularly Shaped Particles, D. Schuerman, ed. (Plenum, New York, 1980), pp. 103–111.
[CrossRef]

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

A. Lakhtakia, “Macroscopic theory of the coupled dipole approximation method,” Opt. Commun. (to be published).

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Figures (8)

Fig. 1
Fig. 1

Spherical particle that has been represented by an array of 136 dipolar subunits. The effective radius of the sphere is determined by ae = (3N/4π)1/2, where the dipolar subunits are assumed to have unit volume.

Fig. 2
Fig. 2

Comparison between Mie theory and the coupled-dipole method when using the Doyle expression (—), Draine’s radiative-reaction term (⋯), and the CM relation (– – –). N varies from 27 to 251; the size of the dipolar subunit remains constant. The results for two refractive indices are shown.

Fig. 3
Fig. 3

Comparison between Mie theory and the coupled-dipole method when using the Doyle expression (—), Draine’s radiative-reaction term, (⋯), and the CM relation (– – –). N remains constant (136 dipolar subunits); the size of the dipolar subunit varies.

Fig. 4
Fig. 4

Complex refractive index and dielectric function as a function of wave number (cm−1) for quartz as obtained from the Lorentz oscillator model17: the uppermost two curves are the refractive index; the bottommost, the dielectric function for the real (—) and imaginary (– – –) parts.

Fig. 5
Fig. 5

Extinction efficiency as a function of particle radius (μm) obtained from Mie theory and the coupled-dipole method. Mie theory is represented by the continuous curves; each corresponds to a separate complex refractive index: of m = (0.1297, i1.444) or the Fröhlich mode (—), m = (1.33, i0.05) (⋯), m = (1.7, i0.1) (– – –). The coupled-dipole method is represented for using the CM relation (open squares) and using Doyle’s expression (open circles); both are at the Fröhlich mode.

Fig. 6
Fig. 6

Extinction cross section per unit volume (μm−1) as a function of wave number (cm−1) for spheres as obtained from Mie theory (—) and coupled-dipole method using the Doyle expression (– – –) Peaks correspond to the Fröhlich frequency for quartz.

Fig. 7
Fig. 7

Pictorial representation of small-particle extinction for quartz. The extinction cross section per unit volume (μm−1) is calculated by the coupled-dipole (Doyle expression) as a function of wave number (cm−1). Five shapes are represented: sphere (open circle), 2 × 1 prolate spheroid (vertical open rectangle), prolate spheroid, (vertical bar), 2 × 1 oblate spheroid (horizontal open rectangle), 4 × 1 oblate spheroid (horizontal bar).

Fig. 8
Fig. 8

Extinction cross section per unit volume (μm−1) as a function of wave number (cm−1) obtained from solid curve, continuous distribution of ellipsoids method; open circles, coupled-dipole method (Doyle expression) using five ellipsoids: sphere, 2 × 1 and 4 × 1 oblate and prolate spheroids.

Tables (2)

Tables Icon

Table 1 Comparison of Qext for Nine Spheres of Different Refractive Indices as Calculated by Mie Theory and by the Coupled-Dipole Method with the Doyle Expression, Draine’s Radiative-Reaction Term, or the CM Relationa

Tables Icon

Table 2 Comparison of Seven Dipole Arraysa

Equations (7)

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dip = 2 ( 1 - f ) m 2 - ( 2 + f ) m ( 1 - f ) - ( 1 + 2 f ) m ,
α = 3 ( - m ) N ( + 2 m ) ,
α = 4 π a 3 ( dip - m ) ( dip + 2 m ) ,
α = i 6 π k 3 α 1 ,
α 1 = m Ψ 1 ( m x ) Ψ 1 ( x ) - Ψ 1 ( x ) Ψ 1 ( m x ) m Ψ 1 ( m x ) ξ 1 ( x ) - ξ 1 ( x ) Ψ 1 ( m x ) ,
m 2 = - n + 1 n ,             n = 1 , 2 , .
C abs v = k Im ( 2 - 1 log ) .

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