Abstract

We develop a theoretical approach to calculating optical properties of carbonaceous soot in the long-wavelength limit. Our method is based on geometrical renormalization of clusters; it avoids both the inaccuracy of the dipole approximation in its pure form and the numerical complexity of rigorous direct methods of solving the EM boundary problem. The results are verified by comparison with the experimental measurements for specific extinction of diesel soot in the spectral region from 0.488 µm to 0.857 cm that were performed by Bruce et al. [Appl. Opt. 30, 1537 (1991)]. The theory leads to analytical expressions that are applicable to different soots, with various geometrical properties and optical constants. We show that the functional form of the long-wavelength asymptote of the specific extinction can depend critically on a parameter characterizing the sample geometry, and we identify the critical value of this parameter.

© 2001 Optical Society of America

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  1. S. R. Forrest, T. A. Witten, “Long-range correlations in smoke-particle aggregates,” J. Phys. A 12, L109–L117 (1979).
    [CrossRef]
  2. H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, J. F. Merklin, “In situ optical structure factor measurements of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
    [CrossRef]
  3. U. O. Koylu, G. M. Faeth, “Structure of overfire soot in buoyant turbulent diffusion flames at long residence times,” Combust. Flame 89, 140–156 (1992).
    [CrossRef]
  4. J. Cai, N. Lu, C. M. Sorensen, “Comparison of size and morphology of soot aggregates as determined by light scattering and electron microscope analysis,” Langmuir 9, 2861–2867 (1993).
    [CrossRef]
  5. M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
    [CrossRef]
  6. W. H. Dalzell, A. F. Sarofim, “Optical constants of soot and their application to heat-flux calculations,” Trans. ASME, Ser. C: J. Heat Transfer 91, 100–104 (1969).
    [CrossRef]
  7. G. W. Mulholland, C. F. Bohren, K. A. Fuller, “Light scattering by agglomerates: coupled electric and magnetic dipole method,” Langmuir 10, 2533–2546 (1994).
    [CrossRef]
  8. D. W. Mackowski, “Electrostatics analysis of radiative absorption by sphere clusters in the Rayleigh limit: application to soot particles,” Appl. Opt. 34, 3535–3545 (1995).
    [CrossRef] [PubMed]
  9. C. W. Bruce, T. F. Stromberg, K. P. Gurton, J. B. Mozer, “Trans-spectral absorption and scattering of electromagnetic radiation by diesel soot,” Appl. Opt. 30, 1537–1546 (1991).
    [CrossRef] [PubMed]
  10. L. S. Markel, V. A. Muratov, M. I. Stockman, “Optical properties of fractals: theory and numerical simulation,” Sov. Phys. JETP 71, 455–464 (1990).
  11. V. A. Markel, L. S. Muratov, M. I. Stockman, T. F. George, “Theory and numerical simulation of optical properties of fractal clusters,” Phys. Rev. B 43, 8183–8195 (1991).
    [CrossRef]
  12. V. M. Shalaev, R. Botet, R. Jullien, “Resonant light scattering by fractal clusters,” Phys. Rev. B 44, 12,216–12,225 (1991).
    [CrossRef]
  13. V. M. Shalaev, Nonlinear Optics of Random Media: Fractal Composites and Metal Dielectric Films (Springer-Verlag, Berlin, 2000).
  14. J. M. Gerardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. The long-wave limit,” Phys. Rev. B 22, 4950–4959 (1980).
    [CrossRef]
  15. F. Claro, “Absorption spectrum of neighboring dielectric grains,” Phys. Rev. B 25, 7875–7876 (1982).
    [CrossRef]
  16. R. Rojas, F. Claro, “Electromagnetic response of an array of particles: normal-mode theory,” Phys. Rev. B 34, 3730–3736 (1986).
    [CrossRef]
  17. K. A. Fuller, “Scattering and absorption cross sections of compounded spheres. I. Theory for external aggregation,” J. Opt. Soc. Am. A 11, 3251–3260 (1994).
    [CrossRef]
  18. F. J. G. de Abajo, “Interaction of radiation and fast electrons with clusters and dielectrics: a multiple scattering approach,” Phys. Rev. Lett. 82, 2776–2779 (1999).
    [CrossRef]
  19. F. J. G. de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B 60, 6086–6102 (1999).
    [CrossRef]
  20. J. E. Sansonetti, J. K. Furdyna, “Depolarization effects in arrays of spheres,” Phys. Rev. B 22, 2866–2874 (1980).
    [CrossRef]
  21. F. Claro, “Theory of resonant modes in particulate matter,” Phys. Rev. B 30, 4989–4999 (1984).
    [CrossRef]
  22. F. Claro, “Multipolar effects in particulate matter,” Solid State Commun. 49, 229–232 (1984).
    [CrossRef]
  23. V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, R. L. Armstrong, “Small-particle composites. I. Linear optical properties,” Phys. Rev. B 53, 2425–2436 (1996).
    [CrossRef]
  24. V. A. Markel, V. M. Shalaev, “Computational approaches in optics of fractal clusters,” in Computational Studies of New Materials, D. A. Jelski, T. F. George, eds. (World Scientific, Singapore, 1999), pp. 210–243.
  25. E. A. Taft, E. A. Philipp, “Optical properties of graphite,” Phys. Rep. 138, A197–A202 (1965).
  26. Z. G. Habib, P. Vervisch, “On the refractive index of soot at flame temperatures,” Combust. Sci. Technol. 59, 261–274 (1988).
    [CrossRef]
  27. S. C. Lee, C. L. Tien, “Optical constants of soot in hydrocarbon flames,” in Eighteenth Symposium (International) on Combustion, (The Combustion Institute, Pittsburgh, Pa., 1981), pp. 1159–1166.
  28. X and δ used in Refs. 11 and 10 differ from the dimensionless parameters defined below by a multiplicative factor with the dimensionality of length cubed.
  29. D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994).
    [CrossRef]
  30. P. Meakin, “Formation of fractal clusters and networks by irreversible diffusion-limited aggregation,” Phys. Rev. Lett. 51, 1119–1122 (1983).
    [CrossRef]
  31. R. Jullien, M. Kolb, R. Botet, “Aggregation by kinetic clustering of clusters in dimensions d>2,” J. Phys. (France) 45, L211–L216 (1984).
  32. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [CrossRef]
  33. E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
  34. B. Draine, P. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
    [CrossRef]
  35. V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole chain,” J. Mod. Opt. Soc. Am. B 40, 2281–2291 (1993).
    [CrossRef]
  36. A finite linear chain of touching spheres with multipole interaction was considered by Mackowski.8 It was found numerically that, in general, such a chain is not equivalent to a spheroid with the same aspect ratio. The intersection parameter that gives the same depolarization coefficient for an infinite chain of spheres as in an infinite cylinder may be different with the multipole interactions included from that in the dipole approximation.
  37. V. A. Markel, “Scattering of light from two interacting spherical particles,” J. Mod. Opt. 39, 853–861 (1992).
    [CrossRef]
  38. V. A. Markel, “Antisymmetrical optical states,” J. Opt. Soc. Am. B 12, 1783–1791 (1995).
    [CrossRef]
  39. T. A. Witten, L. M. Sander, “Diffusion-limited aggregation, a kinetic critical phenomenon,” Phys. Rev. Lett. 47, 1400–1403 (1981).
    [CrossRef]

1999 (2)

F. J. G. de Abajo, “Interaction of radiation and fast electrons with clusters and dielectrics: a multiple scattering approach,” Phys. Rev. Lett. 82, 2776–2779 (1999).
[CrossRef]

F. J. G. de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B 60, 6086–6102 (1999).
[CrossRef]

1996 (1)

V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, R. L. Armstrong, “Small-particle composites. I. Linear optical properties,” Phys. Rev. B 53, 2425–2436 (1996).
[CrossRef]

1995 (2)

1994 (4)

1993 (2)

J. Cai, N. Lu, C. M. Sorensen, “Comparison of size and morphology of soot aggregates as determined by light scattering and electron microscope analysis,” Langmuir 9, 2861–2867 (1993).
[CrossRef]

V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole chain,” J. Mod. Opt. Soc. Am. B 40, 2281–2291 (1993).
[CrossRef]

1992 (2)

V. A. Markel, “Scattering of light from two interacting spherical particles,” J. Mod. Opt. 39, 853–861 (1992).
[CrossRef]

U. O. Koylu, G. M. Faeth, “Structure of overfire soot in buoyant turbulent diffusion flames at long residence times,” Combust. Flame 89, 140–156 (1992).
[CrossRef]

1991 (3)

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

V. M. Shalaev, R. Botet, R. Jullien, “Resonant light scattering by fractal clusters,” Phys. Rev. B 44, 12,216–12,225 (1991).
[CrossRef]

C. W. Bruce, T. F. Stromberg, K. P. Gurton, J. B. Mozer, “Trans-spectral absorption and scattering of electromagnetic radiation by diesel soot,” Appl. Opt. 30, 1537–1546 (1991).
[CrossRef] [PubMed]

1990 (1)

L. S. Markel, V. A. Muratov, M. I. Stockman, “Optical properties of fractals: theory and numerical simulation,” Sov. Phys. JETP 71, 455–464 (1990).

1988 (3)

Z. G. Habib, P. Vervisch, “On the refractive index of soot at flame temperatures,” Combust. Sci. Technol. 59, 261–274 (1988).
[CrossRef]

H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, J. F. Merklin, “In situ optical structure factor measurements of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
[CrossRef]

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

1986 (2)

R. Rojas, F. Claro, “Electromagnetic response of an array of particles: normal-mode theory,” Phys. Rev. B 34, 3730–3736 (1986).
[CrossRef]

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

1984 (3)

F. Claro, “Theory of resonant modes in particulate matter,” Phys. Rev. B 30, 4989–4999 (1984).
[CrossRef]

F. Claro, “Multipolar effects in particulate matter,” Solid State Commun. 49, 229–232 (1984).
[CrossRef]

R. Jullien, M. Kolb, R. Botet, “Aggregation by kinetic clustering of clusters in dimensions d>2,” J. Phys. (France) 45, L211–L216 (1984).

1983 (1)

P. Meakin, “Formation of fractal clusters and networks by irreversible diffusion-limited aggregation,” Phys. Rev. Lett. 51, 1119–1122 (1983).
[CrossRef]

1982 (1)

F. Claro, “Absorption spectrum of neighboring dielectric grains,” Phys. Rev. B 25, 7875–7876 (1982).
[CrossRef]

1981 (1)

T. A. Witten, L. M. Sander, “Diffusion-limited aggregation, a kinetic critical phenomenon,” Phys. Rev. Lett. 47, 1400–1403 (1981).
[CrossRef]

1980 (2)

J. M. Gerardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. The long-wave limit,” Phys. Rev. B 22, 4950–4959 (1980).
[CrossRef]

J. E. Sansonetti, J. K. Furdyna, “Depolarization effects in arrays of spheres,” Phys. Rev. B 22, 2866–2874 (1980).
[CrossRef]

1979 (1)

S. R. Forrest, T. A. Witten, “Long-range correlations in smoke-particle aggregates,” J. Phys. A 12, L109–L117 (1979).
[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]

1969 (1)

W. H. Dalzell, A. F. Sarofim, “Optical constants of soot and their application to heat-flux calculations,” Trans. ASME, Ser. C: J. Heat Transfer 91, 100–104 (1969).
[CrossRef]

1965 (1)

E. A. Taft, E. A. Philipp, “Optical properties of graphite,” Phys. Rep. 138, A197–A202 (1965).

Armstrong, R. L.

V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, R. L. Armstrong, “Small-particle composites. I. Linear optical properties,” Phys. Rev. B 53, 2425–2436 (1996).
[CrossRef]

Ausloos, M.

J. M. Gerardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. The long-wave limit,” Phys. Rev. B 22, 4950–4959 (1980).
[CrossRef]

Berry, M. V.

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

Bohren, C. F.

G. W. Mulholland, C. F. Bohren, K. A. Fuller, “Light scattering by agglomerates: coupled electric and magnetic dipole method,” Langmuir 10, 2533–2546 (1994).
[CrossRef]

Botet, R.

V. M. Shalaev, R. Botet, R. Jullien, “Resonant light scattering by fractal clusters,” Phys. Rev. B 44, 12,216–12,225 (1991).
[CrossRef]

R. Jullien, M. Kolb, R. Botet, “Aggregation by kinetic clustering of clusters in dimensions d>2,” J. Phys. (France) 45, L211–L216 (1984).

Bruce, C. W.

Cai, J.

J. Cai, N. Lu, C. M. Sorensen, “Comparison of size and morphology of soot aggregates as determined by light scattering and electron microscope analysis,” Langmuir 9, 2861–2867 (1993).
[CrossRef]

Claro, F.

R. Rojas, F. Claro, “Electromagnetic response of an array of particles: normal-mode theory,” Phys. Rev. B 34, 3730–3736 (1986).
[CrossRef]

F. Claro, “Theory of resonant modes in particulate matter,” Phys. Rev. B 30, 4989–4999 (1984).
[CrossRef]

F. Claro, “Multipolar effects in particulate matter,” Solid State Commun. 49, 229–232 (1984).
[CrossRef]

F. Claro, “Absorption spectrum of neighboring dielectric grains,” Phys. Rev. B 25, 7875–7876 (1982).
[CrossRef]

Dalzell, W. H.

W. H. Dalzell, A. F. Sarofim, “Optical constants of soot and their application to heat-flux calculations,” Trans. ASME, Ser. C: J. Heat Transfer 91, 100–104 (1969).
[CrossRef]

de Abajo, F. J. G.

F. J. G. de Abajo, “Interaction of radiation and fast electrons with clusters and dielectrics: a multiple scattering approach,” Phys. Rev. Lett. 82, 2776–2779 (1999).
[CrossRef]

F. J. G. de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B 60, 6086–6102 (1999).
[CrossRef]

Draine, B.

Draine, B. T.

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

Faeth, G. M.

U. O. Koylu, G. M. Faeth, “Structure of overfire soot in buoyant turbulent diffusion flames at long residence times,” Combust. Flame 89, 140–156 (1992).
[CrossRef]

Flatau, P.

Forrest, S. R.

S. R. Forrest, T. A. Witten, “Long-range correlations in smoke-particle aggregates,” J. Phys. A 12, L109–L117 (1979).
[CrossRef]

Fuller, K. A.

G. W. Mulholland, C. F. Bohren, K. A. Fuller, “Light scattering by agglomerates: coupled electric and magnetic dipole method,” Langmuir 10, 2533–2546 (1994).
[CrossRef]

K. A. Fuller, “Scattering and absorption cross sections of compounded spheres. I. Theory for external aggregation,” J. Opt. Soc. Am. A 11, 3251–3260 (1994).
[CrossRef]

Furdyna, J. K.

J. E. Sansonetti, J. K. Furdyna, “Depolarization effects in arrays of spheres,” Phys. Rev. B 22, 2866–2874 (1980).
[CrossRef]

George, T. F.

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

Gerardy, J. M.

J. M. Gerardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. The long-wave limit,” Phys. Rev. B 22, 4950–4959 (1980).
[CrossRef]

Gurton, K. P.

Habib, Z. G.

Z. G. Habib, P. Vervisch, “On the refractive index of soot at flame temperatures,” Combust. Sci. Technol. 59, 261–274 (1988).
[CrossRef]

Jullien, R.

V. M. Shalaev, R. Botet, R. Jullien, “Resonant light scattering by fractal clusters,” Phys. Rev. B 44, 12,216–12,225 (1991).
[CrossRef]

R. Jullien, M. Kolb, R. Botet, “Aggregation by kinetic clustering of clusters in dimensions d>2,” J. Phys. (France) 45, L211–L216 (1984).

Kim, W.

V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, R. L. Armstrong, “Small-particle composites. I. Linear optical properties,” Phys. Rev. B 53, 2425–2436 (1996).
[CrossRef]

Kolb, M.

R. Jullien, M. Kolb, R. Botet, “Aggregation by kinetic clustering of clusters in dimensions d>2,” J. Phys. (France) 45, L211–L216 (1984).

Koylu, U. O.

U. O. Koylu, G. M. Faeth, “Structure of overfire soot in buoyant turbulent diffusion flames at long residence times,” Combust. Flame 89, 140–156 (1992).
[CrossRef]

Lee, S. C.

S. C. Lee, C. L. Tien, “Optical constants of soot in hydrocarbon flames,” in Eighteenth Symposium (International) on Combustion, (The Combustion Institute, Pittsburgh, Pa., 1981), pp. 1159–1166.

Lu, N.

J. Cai, N. Lu, C. M. Sorensen, “Comparison of size and morphology of soot aggregates as determined by light scattering and electron microscope analysis,” Langmuir 9, 2861–2867 (1993).
[CrossRef]

Mackowski, D. W.

Markel, L. S.

L. S. Markel, V. A. Muratov, M. I. Stockman, “Optical properties of fractals: theory and numerical simulation,” Sov. Phys. JETP 71, 455–464 (1990).

Markel, V. A.

V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, R. L. Armstrong, “Small-particle composites. I. Linear optical properties,” Phys. Rev. B 53, 2425–2436 (1996).
[CrossRef]

V. A. Markel, “Antisymmetrical optical states,” J. Opt. Soc. Am. B 12, 1783–1791 (1995).
[CrossRef]

V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole chain,” J. Mod. Opt. Soc. Am. B 40, 2281–2291 (1993).
[CrossRef]

V. A. Markel, “Scattering of light from two interacting spherical particles,” J. Mod. Opt. 39, 853–861 (1992).
[CrossRef]

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

V. A. Markel, V. M. Shalaev, “Computational approaches in optics of fractal clusters,” in Computational Studies of New Materials, D. A. Jelski, T. F. George, eds. (World Scientific, Singapore, 1999), pp. 210–243.

Meakin, P.

P. Meakin, “Formation of fractal clusters and networks by irreversible diffusion-limited aggregation,” Phys. Rev. Lett. 51, 1119–1122 (1983).
[CrossRef]

Merklin, J. F.

H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, J. F. Merklin, “In situ optical structure factor measurements of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
[CrossRef]

Mozer, J. B.

Mulholland, G. W.

G. W. Mulholland, C. F. Bohren, K. A. Fuller, “Light scattering by agglomerates: coupled electric and magnetic dipole method,” Langmuir 10, 2533–2546 (1994).
[CrossRef]

Muratov, L. S.

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

Muratov, V. A.

L. S. Markel, V. A. Muratov, M. I. Stockman, “Optical properties of fractals: theory and numerical simulation,” Sov. Phys. JETP 71, 455–464 (1990).

Olivier, B. J.

H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, J. F. Merklin, “In situ optical structure factor measurements of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
[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]

Percival, I. C.

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

Philipp, E. A.

E. A. Taft, E. A. Philipp, “Optical properties of graphite,” Phys. Rep. 138, A197–A202 (1965).

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]

Ramer, E. R.

H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, J. F. Merklin, “In situ optical structure factor measurements of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
[CrossRef]

Rojas, R.

R. Rojas, F. Claro, “Electromagnetic response of an array of particles: normal-mode theory,” Phys. Rev. B 34, 3730–3736 (1986).
[CrossRef]

Sander, L. M.

T. A. Witten, L. M. Sander, “Diffusion-limited aggregation, a kinetic critical phenomenon,” Phys. Rev. Lett. 47, 1400–1403 (1981).
[CrossRef]

Sansonetti, J. E.

J. E. Sansonetti, J. K. Furdyna, “Depolarization effects in arrays of spheres,” Phys. Rev. B 22, 2866–2874 (1980).
[CrossRef]

Sarofim, A. F.

W. H. Dalzell, A. F. Sarofim, “Optical constants of soot and their application to heat-flux calculations,” Trans. ASME, Ser. C: J. Heat Transfer 91, 100–104 (1969).
[CrossRef]

Shalaev, V. M.

V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, R. L. Armstrong, “Small-particle composites. I. Linear optical properties,” Phys. Rev. B 53, 2425–2436 (1996).
[CrossRef]

V. M. Shalaev, R. Botet, R. Jullien, “Resonant light scattering by fractal clusters,” Phys. Rev. B 44, 12,216–12,225 (1991).
[CrossRef]

V. M. Shalaev, Nonlinear Optics of Random Media: Fractal Composites and Metal Dielectric Films (Springer-Verlag, Berlin, 2000).

V. A. Markel, V. M. Shalaev, “Computational approaches in optics of fractal clusters,” in Computational Studies of New Materials, D. A. Jelski, T. F. George, eds. (World Scientific, Singapore, 1999), pp. 210–243.

Sorensen, C. M.

J. Cai, N. Lu, C. M. Sorensen, “Comparison of size and morphology of soot aggregates as determined by light scattering and electron microscope analysis,” Langmuir 9, 2861–2867 (1993).
[CrossRef]

H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, J. F. Merklin, “In situ optical structure factor measurements of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
[CrossRef]

Stechel, E. B.

V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, R. L. Armstrong, “Small-particle composites. I. Linear optical properties,” Phys. Rev. B 53, 2425–2436 (1996).
[CrossRef]

Stockman, M. I.

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

L. S. Markel, V. A. Muratov, M. I. Stockman, “Optical properties of fractals: theory and numerical simulation,” Sov. Phys. JETP 71, 455–464 (1990).

Stromberg, T. F.

Taft, E. A.

E. A. Taft, E. A. Philipp, “Optical properties of graphite,” Phys. Rep. 138, A197–A202 (1965).

Tien, C. L.

S. C. Lee, C. L. Tien, “Optical constants of soot in hydrocarbon flames,” in Eighteenth Symposium (International) on Combustion, (The Combustion Institute, Pittsburgh, Pa., 1981), pp. 1159–1166.

Vervisch, P.

Z. G. Habib, P. Vervisch, “On the refractive index of soot at flame temperatures,” Combust. Sci. Technol. 59, 261–274 (1988).
[CrossRef]

Witten, T. A.

T. A. Witten, L. M. Sander, “Diffusion-limited aggregation, a kinetic critical phenomenon,” Phys. Rev. Lett. 47, 1400–1403 (1981).
[CrossRef]

S. R. Forrest, T. A. Witten, “Long-range correlations in smoke-particle aggregates,” J. Phys. A 12, L109–L117 (1979).
[CrossRef]

Zhang, H. X.

H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, J. F. Merklin, “In situ optical structure factor measurements of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
[CrossRef]

Appl. Opt. (2)

Astrophys. J. (2)

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

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

Combust. Flame (1)

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X and δ used in Refs. 11 and 10 differ from the dimensionless parameters defined below by a multiplicative factor with the dimensionality of length cubed.

V. A. Markel, V. M. Shalaev, “Computational approaches in optics of fractal clusters,” in Computational Studies of New Materials, D. A. Jelski, T. F. George, eds. (World Scientific, Singapore, 1999), pp. 210–243.

V. M. Shalaev, Nonlinear Optics of Random Media: Fractal Composites and Metal Dielectric Films (Springer-Verlag, Berlin, 2000).

A finite linear chain of touching spheres with multipole interaction was considered by Mackowski.8 It was found numerically that, in general, such a chain is not equivalent to a spheroid with the same aspect ratio. The intersection parameter that gives the same depolarization coefficient for an infinite chain of spheres as in an infinite cylinder may be different with the multipole interactions included from that in the dipole approximation.

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Figures (7)

Fig. 1
Fig. 1

Real and imaginary parts of the complex refractive index m=ϵ=n+ik as functions of wavelength calculated from dispersion formula (1).

Fig. 2
Fig. 2

Spectral dependence of complex variable 1/χ=-(X+iδ) parameterized by wavelength λ, where χ=(3/4π)(ϵ-1)/(ϵ+2).

Fig. 3
Fig. 3

Specific extinction εe, multiplied by λ2 and averaged over spatial rotations, as a function of L (Fortran codes courtesy of D. Mackowski). The calculations were performed in the quasi-static limit for a three-dimensional cluster–cluster aggregate of N=100 touching spheres as shown in the inset.

Fig. 4
Fig. 4

Weighted density of states Γ(w) and its approximation by a step function with the equivalent normalization, first and second moments. The numerical diagonalization was performed for an ensemble of 10 clusters with N=1000. The values of the constants are vw0=2.29 and Γ0=1/2w0.

Fig. 5
Fig. 5

Specific extinction εe calculated numerically (solid curve) and according to analytical approximation (30) (circles). The noninteracting limit εe(noninteracting)=4πk Im χ is shown by the dashed curve.

Fig. 6
Fig. 6

Specific scattering εs normalized by k3Vtot calculated numerically and according to analytical approximation [Eq. (31)]. The noninteracting limit εs(noninteracting)/k3Vtot=(8π/3)k|χ|2 is shown by the dashed curve.

Fig. 7
Fig. 7

Specific extinction εe calculated according to analytical approximation (30) compared with experimental values adapted from Bruce et al.9 We used the mass density of black carbon, ρ=1.9 g/cm3, to convert the experimental data of Ref. 9 into the units shown here. The noninteracting limit εe(noninteracting)=4πk Im χ is shown by the dashed curve.

Equations (41)

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ϵ(ω)=1-n fn2ω2-ωn2+iγnω.
χ=34π ϵ-1ϵ+2.
εe=σeVtot,
di=αEinc(ri)+jiN Gˆ(ri-rj)dj,
Gαβ(r)=k3[A(kr)δαβ+B(kr)rαrβ/r2],
A(x)=(x-1+ix-2-x-3)exp(ix),
B(x)=(-x-1-3ix-2+3x-3)exp(ix),
σe=4πk|E0|2 Im i=1N di·Einc*(ri),
σa=4πk|E0|2ya i=1N|di|2,
ya=-Im1α-2k330.
1α=1vχ-i 2k33,
Rm=Rm(ξ/2)D/(3-D),
N=N(2/ξ)3D/(3-D),
l=ξRm,
RglN1/D,
VtotNRm3,
|d=α(|Einc+W|d),
iα|W|iβ=-δαβ|ri-rj|3+3(ri-rj)α(ri-rj)β|ri-rj|5.
|d=n |nn|Einc1/α-wn,
σe=4πk|E0|2 ImEinc|d=4πkv|E0|2 Im n Einc|nn|Einc1/χ-vwn,
σs=8πk43|E0|2|D|2,
σs=8πk4v23|E0|2 α,m,n Einc|mm|OαOα|nn|Einc(1/χ*-vwm)(1/χ-vwn).
σ¯e=4πkv3 Im n,α Oα|nn|Oα1/χ-vwn,
Γαβ(w)=1N nOα|nn|Oβδ(w-wn),
Γ(w)=13 αΓαα(w).
σ¯e=4πkVtot Im - Γ(w)dw1/χ-vw.
σ¯s=8πk4Vtot29 αβ - Γβα(w1)Γαβ(w2)dw1dw2(1/χ*-vw1)(1/χ-vw2).
- Γαβ(w)dw=δαβ.
σ¯s=8πk4Vtot23- Γ(w)dw1/χ-vw2.
σ¯e=2πkVtotvw0 arctan X+vw0δ-arctan X-vw0δ,
σ¯s=2πk4Vtot23(vw0)2 14 ln2 (X+vw0)2+δ2(X-vw0)2+δ2+arctan X+vw0δ-arctan X-vw0δ2.
ϵ=1-ωp2ω(ω+iγ)
X=-X=-4π/3,
δ=4πγω/ωp2.
εe=2πkC arctan C+Xδ+sgn(C-X)×arctan |C-X|δ.
εe=2πkC π2[1+sgn(C-X)]-δ1C+X+sgn(C-X)|C-X|.
εe=2π2kC1λ,C>X.
εe=(4π)2γkωωp2(X2-C2)1λ2,C<X,
(vw0)c=4π3 ξ23.
εsk3Vtot=2πk3C2 ln2 C-XC+X+π22,C>X,
εsk3Vtot=2πk3C2 ln2 X-CX+C+δ2 4C2(X2-C2)2,C<X.

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