Abstract

The reciprocity theorem in light scattering is a general theorem that is verified theoretically and experimentally. However, violation of the reciprocity theorem has been encountered in previous investigations for simulation of light scattering from agglomerates. We demonstrate that the violations of the reciprocity theorem are due to inappropriate orientation averaging or the incorrect formulation of light-scattering quantities. In situ optical diagnostics of aggregated aerosols requires the calculation of the orientation averages of scattering quantities. Thus it is imperative to establish a criterion that can be used to determine a sufficient number of orientations for the reliable calculation of averages for the scattering quantities. It is demonstrated that the reciprocity theorem may serve as such a criterion for typical sizes of agglomerates such as flame soot with fractal dimensions D f = 1.8, primary particle size parameter x ≤ 0.3, and number of primary particles less than 260. It is shown that the use of 21 × 21 × 21 orientations will satisfy the reciprocity theorem to within 0.5%.

© 2000 Optical Society of America

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References

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  1. R. S. Krishnan, “Reciprocity theorem in colloid optics,” Proc. Indian Acad. Sci. Sect. A 7, 91–97 (1938).
  2. D. S. Saxon, “Tensor scattering matrix for the electromagnetic field,” Phys. Rev. 100, 1771–1775 (1955).
    [CrossRef]
  3. R. S. Krishnan, “The reciprocity theorem in colloid optics and its generalization,” Proc. Indian Acad. Sci. Sect. A 11, 21–35 (1938).
  4. A. D’Alessio, “Laser light scattering and fluorescence diagnostics in rich flames,” in Particulate Carbon, Formation during Combustion, D. C. Siegla, G. W. Smith, eds. (Plenum, New York, 1981).
  5. Ü. Ö. Köylü, G. M. Faeth, “Optical properties of soot in buoyant laminar diffusion flames,” J. Heat Transfer 116, 971–979 (1994).
    [CrossRef]
  6. J.-S. Wu, S. S. Krishnan, G. M. Faeth, “Refractive indices at visible wavelengths of soot emitted from buoyant turbulent diffusion flames,” J. Heat Transfer 119, 230–237 (1997).
    [CrossRef]
  7. A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London Ser. A 366, 111–127 (1979).
    [CrossRef]
  8. L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and its Application (Applied Science, London, 1981).
    [CrossRef]
  9. J. C. Ku, K. H. Shim, “A comparison of solution for light scattering and absorption by agglomerated or arbitrarily-shaped particles,” J. Quant. Spectrosc. Radiat. Transfer 47, 201–220 (1992).
    [CrossRef]
  10. M. F. Iskander, H. Y. Chen, J. E. Penner, “Optical scattering and absorption by branched chains of aerosols,” Appl. Opt. 28, 3083–3091 (1989).
    [CrossRef] [PubMed]
  11. W. Lou, T. T. Charalampopoulos, “On the electromagnetic scattering and absorption of agglomerated small spherical particles,” J. Phys. D 27, 2258–2270 (1994).
    [CrossRef]
  12. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980).
  13. D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
    [CrossRef]
  14. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge University, Cambridge, 1989).
  15. 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]
  16. R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
    [CrossRef]
  17. T. T. Charalampopoulos, “Morphology and dynamics of agglomerated particulates in combustion systems using light scattering techniques,” Prog. Energy Combust. Sci. 18, 13–45 (1992).
    [CrossRef]
  18. B. L. Drolen, C. L. Tien, “Absorption and scattering of agglomerated soot particulate,” J. Quant. Spectrosc. Radiat. Transfer 37, 433–448 (1987).
    [CrossRef]
  19. S. Kumar, C. L. Tien, “Effective diameter of agglomerates for radiative extinction and scattering,” Combust. Sci. Technol. 66, 199–216 (1989).
    [CrossRef]
  20. S. di Stasio, P. Massoli, “Morphology, monomer size and concentration of agglomerates constituted by Rayleigh particles as retrieved from scattering/extinction measurements,” Combust. Sci. Technol. 124, 219–247 (1997).
    [CrossRef]
  21. T. T. Charalampopoulos, P. K. Panigrahi, “Depolarization characteristics of agglomerated particles–reciprocity relations,” J. Phys. D 26, 2075–2081 (1993).
    [CrossRef]
  22. B. M. Vaglieco, O. Monda, F. E. Corcione, M. P. Mengüç, “Optical and radiative properties of particulates at diesel engine exhaust,” Combust. Sci. Technol. 102, 283–299 (1994).
    [CrossRef]
  23. T. L. Farias, Ü. Ö. Köylü, M. G. Carvalho, “Range of validity of the Rayleigh–Debye–Gans theory for optics of fractal aggregates,” Appl. Opt. 35, 6560–6567 (1996).
    [CrossRef] [PubMed]
  24. T. L. Farias, M. G. Carvalho, Ü. Ö. Köylü, “Radiative heat transfer in soot-containing combustion systems with aggregation,” Int. J. Heat Mass Transfer 41, 2581–2587 (1998).
    [CrossRef]
  25. G. Shu, T. T. Charalampopoulos, “Unified inversion scheme for the morphological parameters and optical properties of aggregated aerosols using light scattering,” Appl. Opt. (to be published).
  26. S. Manickavasagam, M. P. Mengüç, “Scattering matrix elements of fractal-like soot agglomerates,” Appl. Opt. 36, 1337–1351 (1997).
    [CrossRef] [PubMed]

1998

T. L. Farias, M. G. Carvalho, Ü. Ö. Köylü, “Radiative heat transfer in soot-containing combustion systems with aggregation,” Int. J. Heat Mass Transfer 41, 2581–2587 (1998).
[CrossRef]

1997

S. Manickavasagam, M. P. Mengüç, “Scattering matrix elements of fractal-like soot agglomerates,” Appl. Opt. 36, 1337–1351 (1997).
[CrossRef] [PubMed]

J.-S. Wu, S. S. Krishnan, G. M. Faeth, “Refractive indices at visible wavelengths of soot emitted from buoyant turbulent diffusion flames,” J. Heat Transfer 119, 230–237 (1997).
[CrossRef]

S. di Stasio, P. Massoli, “Morphology, monomer size and concentration of agglomerates constituted by Rayleigh particles as retrieved from scattering/extinction measurements,” Combust. Sci. Technol. 124, 219–247 (1997).
[CrossRef]

1996

1995

1994

W. Lou, T. T. Charalampopoulos, “On the electromagnetic scattering and absorption of agglomerated small spherical particles,” J. Phys. D 27, 2258–2270 (1994).
[CrossRef]

Ü. Ö. Köylü, G. M. Faeth, “Optical properties of soot in buoyant laminar diffusion flames,” J. Heat Transfer 116, 971–979 (1994).
[CrossRef]

B. M. Vaglieco, O. Monda, F. E. Corcione, M. P. Mengüç, “Optical and radiative properties of particulates at diesel engine exhaust,” Combust. Sci. Technol. 102, 283–299 (1994).
[CrossRef]

1993

T. T. Charalampopoulos, P. K. Panigrahi, “Depolarization characteristics of agglomerated particles–reciprocity relations,” J. Phys. D 26, 2075–2081 (1993).
[CrossRef]

1992

T. T. Charalampopoulos, “Morphology and dynamics of agglomerated particulates in combustion systems using light scattering techniques,” Prog. Energy Combust. Sci. 18, 13–45 (1992).
[CrossRef]

J. C. Ku, K. H. Shim, “A comparison of solution for light scattering and absorption by agglomerated or arbitrarily-shaped particles,” J. Quant. Spectrosc. Radiat. Transfer 47, 201–220 (1992).
[CrossRef]

1989

M. F. Iskander, H. Y. Chen, J. E. Penner, “Optical scattering and absorption by branched chains of aerosols,” Appl. Opt. 28, 3083–3091 (1989).
[CrossRef] [PubMed]

S. Kumar, C. L. Tien, “Effective diameter of agglomerates for radiative extinction and scattering,” Combust. Sci. Technol. 66, 199–216 (1989).
[CrossRef]

1988

R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
[CrossRef]

1987

B. L. Drolen, C. L. Tien, “Absorption and scattering of agglomerated soot particulate,” J. Quant. Spectrosc. Radiat. Transfer 37, 433–448 (1987).
[CrossRef]

1979

A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London Ser. A 366, 111–127 (1979).
[CrossRef]

1955

D. S. Saxon, “Tensor scattering matrix for the electromagnetic field,” Phys. Rev. 100, 1771–1775 (1955).
[CrossRef]

1938

R. S. Krishnan, “The reciprocity theorem in colloid optics and its generalization,” Proc. Indian Acad. Sci. Sect. A 11, 21–35 (1938).

R. S. Krishnan, “Reciprocity theorem in colloid optics,” Proc. Indian Acad. Sci. Sect. A 7, 91–97 (1938).

Bayvel, L. P.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and its Application (Applied Science, London, 1981).
[CrossRef]

Carvalho, M. G.

T. L. Farias, M. G. Carvalho, Ü. Ö. Köylü, “Radiative heat transfer in soot-containing combustion systems with aggregation,” Int. J. Heat Mass Transfer 41, 2581–2587 (1998).
[CrossRef]

T. L. Farias, Ü. Ö. Köylü, M. G. Carvalho, “Range of validity of the Rayleigh–Debye–Gans theory for optics of fractal aggregates,” Appl. Opt. 35, 6560–6567 (1996).
[CrossRef] [PubMed]

Charalampopoulos, T. T.

W. Lou, T. T. Charalampopoulos, “On the electromagnetic scattering and absorption of agglomerated small spherical particles,” J. Phys. D 27, 2258–2270 (1994).
[CrossRef]

T. T. Charalampopoulos, P. K. Panigrahi, “Depolarization characteristics of agglomerated particles–reciprocity relations,” J. Phys. D 26, 2075–2081 (1993).
[CrossRef]

T. T. Charalampopoulos, “Morphology and dynamics of agglomerated particulates in combustion systems using light scattering techniques,” Prog. Energy Combust. Sci. 18, 13–45 (1992).
[CrossRef]

G. Shu, T. T. Charalampopoulos, “Unified inversion scheme for the morphological parameters and optical properties of aggregated aerosols using light scattering,” Appl. Opt. (to be published).

Chen, H. Y.

Corcione, F. E.

B. M. Vaglieco, O. Monda, F. E. Corcione, M. P. Mengüç, “Optical and radiative properties of particulates at diesel engine exhaust,” Combust. Sci. Technol. 102, 283–299 (1994).
[CrossRef]

D’Alessio, A.

A. D’Alessio, “Laser light scattering and fluorescence diagnostics in rich flames,” in Particulate Carbon, Formation during Combustion, D. C. Siegla, G. W. Smith, eds. (Plenum, New York, 1981).

di Stasio, S.

S. di Stasio, P. Massoli, “Morphology, monomer size and concentration of agglomerates constituted by Rayleigh particles as retrieved from scattering/extinction measurements,” Combust. Sci. Technol. 124, 219–247 (1997).
[CrossRef]

Drolen, B. L.

B. L. Drolen, C. L. Tien, “Absorption and scattering of agglomerated soot particulate,” J. Quant. Spectrosc. Radiat. Transfer 37, 433–448 (1987).
[CrossRef]

Faeth, G. M.

J.-S. Wu, S. S. Krishnan, G. M. Faeth, “Refractive indices at visible wavelengths of soot emitted from buoyant turbulent diffusion flames,” J. Heat Transfer 119, 230–237 (1997).
[CrossRef]

Ü. Ö. Köylü, G. M. Faeth, “Optical properties of soot in buoyant laminar diffusion flames,” J. Heat Transfer 116, 971–979 (1994).
[CrossRef]

Farias, T. L.

T. L. Farias, M. G. Carvalho, Ü. Ö. Köylü, “Radiative heat transfer in soot-containing combustion systems with aggregation,” Int. J. Heat Mass Transfer 41, 2581–2587 (1998).
[CrossRef]

T. L. Farias, Ü. Ö. Köylü, M. G. Carvalho, “Range of validity of the Rayleigh–Debye–Gans theory for optics of fractal aggregates,” Appl. Opt. 35, 6560–6567 (1996).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge University, Cambridge, 1989).

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980).

Iskander, M. F.

Jones, A. R.

A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London Ser. A 366, 111–127 (1979).
[CrossRef]

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and its Application (Applied Science, London, 1981).
[CrossRef]

Khersonskii, V. K.

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
[CrossRef]

Köylü, Ü. Ö.

T. L. Farias, M. G. Carvalho, Ü. Ö. Köylü, “Radiative heat transfer in soot-containing combustion systems with aggregation,” Int. J. Heat Mass Transfer 41, 2581–2587 (1998).
[CrossRef]

T. L. Farias, Ü. Ö. Köylü, M. G. Carvalho, “Range of validity of the Rayleigh–Debye–Gans theory for optics of fractal aggregates,” Appl. Opt. 35, 6560–6567 (1996).
[CrossRef] [PubMed]

Ü. Ö. Köylü, G. M. Faeth, “Optical properties of soot in buoyant laminar diffusion flames,” J. Heat Transfer 116, 971–979 (1994).
[CrossRef]

Krishnan, R. S.

R. S. Krishnan, “Reciprocity theorem in colloid optics,” Proc. Indian Acad. Sci. Sect. A 7, 91–97 (1938).

R. S. Krishnan, “The reciprocity theorem in colloid optics and its generalization,” Proc. Indian Acad. Sci. Sect. A 11, 21–35 (1938).

Krishnan, S. S.

J.-S. Wu, S. S. Krishnan, G. M. Faeth, “Refractive indices at visible wavelengths of soot emitted from buoyant turbulent diffusion flames,” J. Heat Transfer 119, 230–237 (1997).
[CrossRef]

Ku, J. C.

J. C. Ku, K. H. Shim, “A comparison of solution for light scattering and absorption by agglomerated or arbitrarily-shaped particles,” J. Quant. Spectrosc. Radiat. Transfer 47, 201–220 (1992).
[CrossRef]

Kumar, S.

S. Kumar, C. L. Tien, “Effective diameter of agglomerates for radiative extinction and scattering,” Combust. Sci. Technol. 66, 199–216 (1989).
[CrossRef]

Lou, W.

W. Lou, T. T. Charalampopoulos, “On the electromagnetic scattering and absorption of agglomerated small spherical particles,” J. Phys. D 27, 2258–2270 (1994).
[CrossRef]

Mackowski, D. W.

Manickavasagam, S.

Massoli, P.

S. di Stasio, P. Massoli, “Morphology, monomer size and concentration of agglomerates constituted by Rayleigh particles as retrieved from scattering/extinction measurements,” Combust. Sci. Technol. 124, 219–247 (1997).
[CrossRef]

Mengüç, M. P.

S. Manickavasagam, M. P. Mengüç, “Scattering matrix elements of fractal-like soot agglomerates,” Appl. Opt. 36, 1337–1351 (1997).
[CrossRef] [PubMed]

B. M. Vaglieco, O. Monda, F. E. Corcione, M. P. Mengüç, “Optical and radiative properties of particulates at diesel engine exhaust,” Combust. Sci. Technol. 102, 283–299 (1994).
[CrossRef]

Monda, O.

B. M. Vaglieco, O. Monda, F. E. Corcione, M. P. Mengüç, “Optical and radiative properties of particulates at diesel engine exhaust,” Combust. Sci. Technol. 102, 283–299 (1994).
[CrossRef]

Moskalev, A. N.

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
[CrossRef]

Mountain, R. D.

R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
[CrossRef]

Mulholland, G. W.

R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
[CrossRef]

Panigrahi, P. K.

T. T. Charalampopoulos, P. K. Panigrahi, “Depolarization characteristics of agglomerated particles–reciprocity relations,” J. Phys. D 26, 2075–2081 (1993).
[CrossRef]

Penner, J. E.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge University, Cambridge, 1989).

Saxon, D. S.

D. S. Saxon, “Tensor scattering matrix for the electromagnetic field,” Phys. Rev. 100, 1771–1775 (1955).
[CrossRef]

Shim, K. H.

J. C. Ku, K. H. Shim, “A comparison of solution for light scattering and absorption by agglomerated or arbitrarily-shaped particles,” J. Quant. Spectrosc. Radiat. Transfer 47, 201–220 (1992).
[CrossRef]

Shu, G.

G. Shu, T. T. Charalampopoulos, “Unified inversion scheme for the morphological parameters and optical properties of aggregated aerosols using light scattering,” Appl. Opt. (to be published).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge University, Cambridge, 1989).

Tien, C. L.

S. Kumar, C. L. Tien, “Effective diameter of agglomerates for radiative extinction and scattering,” Combust. Sci. Technol. 66, 199–216 (1989).
[CrossRef]

B. L. Drolen, C. L. Tien, “Absorption and scattering of agglomerated soot particulate,” J. Quant. Spectrosc. Radiat. Transfer 37, 433–448 (1987).
[CrossRef]

Vaglieco, B. M.

B. M. Vaglieco, O. Monda, F. E. Corcione, M. P. Mengüç, “Optical and radiative properties of particulates at diesel engine exhaust,” Combust. Sci. Technol. 102, 283–299 (1994).
[CrossRef]

Varshalovich, D. A.

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge University, Cambridge, 1989).

Wu, J.-S.

J.-S. Wu, S. S. Krishnan, G. M. Faeth, “Refractive indices at visible wavelengths of soot emitted from buoyant turbulent diffusion flames,” J. Heat Transfer 119, 230–237 (1997).
[CrossRef]

Appl. Opt.

Combust. Sci. Technol.

B. M. Vaglieco, O. Monda, F. E. Corcione, M. P. Mengüç, “Optical and radiative properties of particulates at diesel engine exhaust,” Combust. Sci. Technol. 102, 283–299 (1994).
[CrossRef]

S. Kumar, C. L. Tien, “Effective diameter of agglomerates for radiative extinction and scattering,” Combust. Sci. Technol. 66, 199–216 (1989).
[CrossRef]

S. di Stasio, P. Massoli, “Morphology, monomer size and concentration of agglomerates constituted by Rayleigh particles as retrieved from scattering/extinction measurements,” Combust. Sci. Technol. 124, 219–247 (1997).
[CrossRef]

Int. J. Heat Mass Transfer

T. L. Farias, M. G. Carvalho, Ü. Ö. Köylü, “Radiative heat transfer in soot-containing combustion systems with aggregation,” Int. J. Heat Mass Transfer 41, 2581–2587 (1998).
[CrossRef]

J. Heat Transfer

Ü. Ö. Köylü, G. M. Faeth, “Optical properties of soot in buoyant laminar diffusion flames,” J. Heat Transfer 116, 971–979 (1994).
[CrossRef]

J.-S. Wu, S. S. Krishnan, G. M. Faeth, “Refractive indices at visible wavelengths of soot emitted from buoyant turbulent diffusion flames,” J. Heat Transfer 119, 230–237 (1997).
[CrossRef]

J. Phys. D

T. T. Charalampopoulos, P. K. Panigrahi, “Depolarization characteristics of agglomerated particles–reciprocity relations,” J. Phys. D 26, 2075–2081 (1993).
[CrossRef]

W. Lou, T. T. Charalampopoulos, “On the electromagnetic scattering and absorption of agglomerated small spherical particles,” J. Phys. D 27, 2258–2270 (1994).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

J. C. Ku, K. H. Shim, “A comparison of solution for light scattering and absorption by agglomerated or arbitrarily-shaped particles,” J. Quant. Spectrosc. Radiat. Transfer 47, 201–220 (1992).
[CrossRef]

B. L. Drolen, C. L. Tien, “Absorption and scattering of agglomerated soot particulate,” J. Quant. Spectrosc. Radiat. Transfer 37, 433–448 (1987).
[CrossRef]

Langmuir

R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
[CrossRef]

Phys. Rev.

D. S. Saxon, “Tensor scattering matrix for the electromagnetic field,” Phys. Rev. 100, 1771–1775 (1955).
[CrossRef]

Proc. Indian Acad. Sci. Sect. A

R. S. Krishnan, “The reciprocity theorem in colloid optics and its generalization,” Proc. Indian Acad. Sci. Sect. A 11, 21–35 (1938).

R. S. Krishnan, “Reciprocity theorem in colloid optics,” Proc. Indian Acad. Sci. Sect. A 7, 91–97 (1938).

Proc. R. Soc. London Ser. A

A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London Ser. A 366, 111–127 (1979).
[CrossRef]

Prog. Energy Combust. Sci.

T. T. Charalampopoulos, “Morphology and dynamics of agglomerated particulates in combustion systems using light scattering techniques,” Prog. Energy Combust. Sci. 18, 13–45 (1992).
[CrossRef]

Other

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980).

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
[CrossRef]

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge University, Cambridge, 1989).

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and its Application (Applied Science, London, 1981).
[CrossRef]

A. D’Alessio, “Laser light scattering and fluorescence diagnostics in rich flames,” in Particulate Carbon, Formation during Combustion, D. C. Siegla, G. W. Smith, eds. (Plenum, New York, 1981).

G. Shu, T. T. Charalampopoulos, “Unified inversion scheme for the morphological parameters and optical properties of aggregated aerosols using light scattering,” Appl. Opt. (to be published).

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Figures (10)

Fig. 1
Fig. 1

Coordinate system illustrating the implications of the reciprocity theorem.

Fig. 2
Fig. 2

Coordinate system for determination of the orientation of a scatterer.

Fig. 3
Fig. 3

hv and vh of straight chain with x = 0.3, m = 1.7 + i0.7, N p = 10; number of orientations, 12 × 12; equally spaced abscissas are used.

Fig. 4
Fig. 4

hv and vh of compact cluster with x = 0.3, m = 1.7 + i0.7, N p = 10; number of orientations, 12 × 12 × 12; Gaussian quadrature is used.

Fig. 5
Fig. 5

hv and vh of randomly branched chain with x = 0.3, m = 1.7 + i0.7, N p = 10; number of orientations, 12 × 12 × 12; Gaussian quadrature is used.

Fig. 6
Fig. 6

hv / vh ratio of two-array chain with x = 0.1, m = 1.7 + i0.7, N p = 16; number of orientations, 12 × 12 × 12 by use of Eq. (11); number of orientations, 12 × 12 by use of Eq. (10).

Fig. 7
Fig. 7

hv / vh ratio for linear chains with x = 0.3, m = 1.7 + i0.7, N p = 10, 15, 20, and 30; number of orientations is fixed at 10 × 10.

Fig. 8
Fig. 8

Comparison of hv and vh calculated from 10 × 10 orientations and 20 × 20 orientations for a straight chain with x = 0.3, m = 1.7 + i0.7, N p = 30.

Fig. 9
Fig. 9

Comparison of vv and hh calculated from 10 × 10 orientations and 20 × 20 orientations for a straight chain with x = 0.3, m = 1.7 + i0.7, N p = 30.

Fig. 10
Fig. 10

Number of orientations for one of the orientation angles (χ, ψ, ω) required for making hv and vh satisfy the reciprocity theorem. Different sizes of randomly branched chain with m = 1.7 + i0.7, D f = 1.8 are considered.

Tables (1)

Tables Icon

Table 1 hv/vh Ratios and the Maximum Errors in vv, hh, hv, and vh,a

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

1+ωC0Ei-ωC2j=1j1Np TijEj=Einc,i;i=1, 2, 3,, N,
ω=-1=n+ik2-1,
C0=1/33-2ix2h11x,
C2=i/3x2j1x,
Cvpθ=R2x2j12x|-1|2|V|2,
Chpθ=R2x2j12x|-1|2|H|2,
V=i=1Npexp-ikri cos βiEx,i,
H=i=1Npexp-ikri cos βicos θEy,i-sin θEz,i,
cos βi=cos θi cos θ+sin θi sin θ cosϕi-π/2,
C¯pp=18π202πdω 02πdψ 0π Cpp sin χdχ.
C¯pp=14π02πdψ 0π Cpp sin χdχ.
C¯pp=18π2i=1Mχj=1Mψk=1Mω Cppχi, ψj, ωksin χi2πMω2πMψπMχ,
C¯pp=18π2i=1Nj=1Nk=1N Cppχi, ψj, ωksin χiwiwjwk,
Np=kfRg/dpDf,

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