Abstract

A robust scheme for characterizing chainlike aggregated aerosols by use of nonintrusive light-scattering measurements is presented. This scheme entails the selection of suitable scattering quantities and their optimal measurement angles; the development of an inversion algorithm to yield the complex refractive index of agglomerates m = n + ik, the primary particle diameter d p, the number of primary particles per agglomerate N p, the number density of agglomerates n A, and the volume fraction of agglomerates f v; and evaluation of the uncertainties of the inferred parameters that correspond to measuring uncertainties. The data-inversion algorithm is based on the exact formulation of light scattering for agglomerates that consist of primary particles in the Rayleigh limit and therefore has solid theoretical foundations. In addition, this approach yields all the desired parameters of the aggregated aerosols by using in situ light-scattering measurements with a minimum of possible uncertainties. Furthermore, the methodology developed here can be extended to aerosols with other types of morphology and optical property.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. T. Charalampopoulos, “Morphology and dynamics of agglomerated particulate in combustion systems using light scattering techniques,” Prog. Energy Combust. Sci. 18, 13–45 (1992).
    [CrossRef]
  2. T. T. Charalampopoulos, H. Chang, “In situ optical properties of soot particles in wavelength range from 340 nm to 600 nm,” Combust. Sci. Technol. 59, 401–421 (1988).
    [CrossRef]
  3. B. J. Stagg, T. T. Charalampopoulos, “Refractive Indices of pyrolytic graphite, amorphous carbon and flame soot in the temperature range 25 degrees to 600 degrees C,” Combust. Flame 94, 381–396 (1993).
    [CrossRef]
  4. G. Mie, “Beitrage zur Optik trüber Medien speziell kolliodaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
    [CrossRef]
  5. W. D. Erickson, G. C. Williams, H. C. Hottel, “Light scattering measurements on soot in a benzene–air flame,” Combust. Flame 8, 127–132 (1964).
    [CrossRef]
  6. M. Kunugi, H. Jinno, “Determination of size and concentration of soot particles in diffusion flames by a light scattering technique,” in Eleventh Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1966), p. 257.
  7. W. H. Dalzell, G. C. Williams, H. C. Hottel, “A light scattering method for soot concentration measurements,” Combust. Flame 14, 161–170 (1970).
    [CrossRef]
  8. A. D’Alessio, A. Di Lorenzo, A. F. Sarofim, F. Beretta, S. Masi, C. Venitozzi, “Soot formation in methane–oxygen flames,” in Fifteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1975), p. 1427.
    [CrossRef]
  9. A. D’Alessio, A. Di Lorenzo, A. Borghese, F. Beretta, S. Masi, “Study of soot nucleation zone of rich methane–oxygen flames,” in Sixteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1977), p. 695.
    [CrossRef]
  10. R. J. Santoro, H. G. Semerjian, R. A. Dobbins, “Soot particle measurements in diffusion flames,” Combust. Flame 51, 203–218 (1983).
    [CrossRef]
  11. Z. Ulanowski, Z. Wang, P. H. Kaye, I. K. Ludlow, “Application of neural networks to the inverse light scattering problem for spheres,” Appl. Opt. 37, 4027–4033 (1998).
    [CrossRef]
  12. M. Kerker, The Scattering of Light, and Other Electromagnetic Radiation (Academic, New York, 1969).
  13. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  14. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  15. R. Jullien, R. Botet, Aggregation and Fractal Aggregates (World Scientific, Singapore, 1987).
  16. J. E. Martin, A. J. Hurd, “Scattering from fractals,” J. Appl. Cryst. 20, 61–78 (1987).
    [CrossRef]
  17. R. A. Dobbins, C. M. Megaridis, “Absorption and scattering of light by polydisperse aggregates,” Appl. Opt. 30, 4747–4754 (1991).
    [CrossRef] [PubMed]
  18. C. M. Sorensen, J. Cai, N. Liu, “Light-scattering measurements of monomer size, monomers per aggregate, and fractal dimension for soot aggregates in flames,” Appl. Opt. 31, 6547–6557 (1992).
    [CrossRef] [PubMed]
  19. R. Puri, T. F. Richardson, R. J. Santoro, R. A. Dobbins, “Aerosol dynamic process of soot aggregates in a laminar ethane diffusion flame,” Combust. Flame 92, 320–333 (1993).
    [CrossRef]
  20. Ü. Ö. Köylü, G. M. Faeth, “Optical properties of overfire soot in buoyant turbulent diffusion flames at long residence times,” J. Heat Transfer 116, 152–159 (1994).
    [CrossRef]
  21. Ü. Ö. Köylü, “Quantitative analysis of in-situ optical diagnostics for inferring particle/aggregates parameters in flames: implications for soot surface growth and total emissivity,” Combust. Flame 109, 488–500 (1996).
    [CrossRef]
  22. G. M. Faeth, Ü. Ö. Köylü, “Soot morphology and optical properties in nonpremixed turbulent flame environments,” Combust. Sci. Technol. 108, 207–229 (1995).
    [CrossRef]
  23. M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
    [CrossRef]
  24. J. Nelson, “Test of a mean field theory for the optics of fractal clusters,” J. Mod. Opt. 36, 1031–1057 (1989).
    [CrossRef]
  25. H. Y. Chen, M. F. Iskander, J. E. Penner, “Light scattering and absorption by fractal agglomerates and coagulations of smoke aerosols,” J. Mod. Opt. 37, 171–181 (1990).
    [CrossRef]
  26. B. M. Vaglieco, F. Berretta, A. D’Alessio, “In situ evaluation of the soot refractive index in the UV–visible from the measurement of the scattering and extinction coefficients in rich flames,” Combust. Flame 79, 259–271 (1990).
    [CrossRef]
  27. H. Chang, T. T. Charalampopoulos, “Determination of the wavelength dependence of refractive indices of flame soot,” Proc. R. Soc. London Ser. A 430, 577–591 (1991).
    [CrossRef]
  28. R. Munoz, T. T. Charalampopoulos, “Evolution of compositional and structural properties of soot in premixed alkane flames,” in Twenty-Seventh Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1998), pp. 1471–1479.
    [CrossRef]
  29. E. M. Purcell, C. R. Pennypacker, “Scattering and absorption by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
  30. B. T. Draine, P. J. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
    [CrossRef]
  31. 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]
  32. A. R. Jones, “Electromagnetic wave scattering by assembles of particles in the Rayleigh approximation,” Proc. R. Soc. London Ser. A 366, 111–127 (1979).
    [CrossRef]
  33. A. R. Jones, “Scattering efficiency factors for agglomerates of small particles,” J. Phys. D 12, 1661–1671 (1979).
    [CrossRef]
  34. D. S. Saxon, Lecture on the Scattering of Light (University of California at Los Angeles, Los Angeles, Calif., 1974).
  35. T. T. Charalampopoulos, H. Chang, “Agglomerate parameters and fractal dimension of soot using light scattering—Effects on surface growth,” Combust. Flame 87, 88–99 (1991).
    [CrossRef]
  36. K. Kumar, C. L. Tien, “Effective diameter of agglomerates for radiative extinction and scattering,” Combust. Sci. Technol. 66, 199–216 (1989).
    [CrossRef]
  37. J. C. Ku, “Correction for the extinction efficiency factors given in the Jones solution for electromagnetic scattering by agglomerates of small spheres,” J. Phys. D 24, 71–75 (1991).
    [CrossRef]
  38. J. C. Ku, K. H. Shim, “A comparison of solutions for light scattering and absorption by agglomerated or arbitrarily-shaped particles,” J. Quant. Spectrosc. Radiat. Transfer 47, 201–220 (1992).
    [CrossRef]
  39. 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]
  40. W. Lou, T. T. Charalampopoulos, “On the electromagnetic scattering and absorption of agglomerated small spherical particles,” J. Phys. D 27, 2258–2270 (1995).
    [CrossRef]
  41. G. H. Goedecke, S. G. O’Brien, “Scattering by irregular inhomogeneous particles via the digitized Green’s function algorithm,” Appl. Opt. 27, 2431–2438 (1988).
    [CrossRef] [PubMed]
  42. W. Lou, T. T. Charalampopoulos, “On the inverse scattering problem for characterization of agglomerated particulates: partial derivative formulation,” J. Phys. D 28, 2585–2594 (1995).
    [CrossRef]
  43. Z. Zhang, T. T. Charalampopoulos, “Controlled combustion synthesis of nanosized iron oxide aggregates,” in Twenty-Sixth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1996), pp. 1851–1857.
    [CrossRef]
  44. 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), pp. 207–256.
  45. D. T. Venizelos, “A study of the radiative properties of agglomerated flame particulates using light scattering, Ph.D. dissertation (Louisiana State University, Baton Rouge, La., 1994).

1998 (1)

1996 (1)

Ü. Ö. Köylü, “Quantitative analysis of in-situ optical diagnostics for inferring particle/aggregates parameters in flames: implications for soot surface growth and total emissivity,” Combust. Flame 109, 488–500 (1996).
[CrossRef]

1995 (3)

G. M. Faeth, Ü. Ö. Köylü, “Soot morphology and optical properties in nonpremixed turbulent flame environments,” Combust. Sci. Technol. 108, 207–229 (1995).
[CrossRef]

W. Lou, T. T. Charalampopoulos, “On the electromagnetic scattering and absorption of agglomerated small spherical particles,” J. Phys. D 27, 2258–2270 (1995).
[CrossRef]

W. Lou, T. T. Charalampopoulos, “On the inverse scattering problem for characterization of agglomerated particulates: partial derivative formulation,” J. Phys. D 28, 2585–2594 (1995).
[CrossRef]

1994 (3)

B. T. Draine, P. J. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (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]

Ü. Ö. Köylü, G. M. Faeth, “Optical properties of overfire soot in buoyant turbulent diffusion flames at long residence times,” J. Heat Transfer 116, 152–159 (1994).
[CrossRef]

1993 (2)

B. J. Stagg, T. T. Charalampopoulos, “Refractive Indices of pyrolytic graphite, amorphous carbon and flame soot in the temperature range 25 degrees to 600 degrees C,” Combust. Flame 94, 381–396 (1993).
[CrossRef]

R. Puri, T. F. Richardson, R. J. Santoro, R. A. Dobbins, “Aerosol dynamic process of soot aggregates in a laminar ethane diffusion flame,” Combust. Flame 92, 320–333 (1993).
[CrossRef]

1992 (3)

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

C. M. Sorensen, J. Cai, N. Liu, “Light-scattering measurements of monomer size, monomers per aggregate, and fractal dimension for soot aggregates in flames,” Appl. Opt. 31, 6547–6557 (1992).
[CrossRef] [PubMed]

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

1991 (4)

J. C. Ku, “Correction for the extinction efficiency factors given in the Jones solution for electromagnetic scattering by agglomerates of small spheres,” J. Phys. D 24, 71–75 (1991).
[CrossRef]

R. A. Dobbins, C. M. Megaridis, “Absorption and scattering of light by polydisperse aggregates,” Appl. Opt. 30, 4747–4754 (1991).
[CrossRef] [PubMed]

T. T. Charalampopoulos, H. Chang, “Agglomerate parameters and fractal dimension of soot using light scattering—Effects on surface growth,” Combust. Flame 87, 88–99 (1991).
[CrossRef]

H. Chang, T. T. Charalampopoulos, “Determination of the wavelength dependence of refractive indices of flame soot,” Proc. R. Soc. London Ser. A 430, 577–591 (1991).
[CrossRef]

1990 (2)

H. Y. Chen, M. F. Iskander, J. E. Penner, “Light scattering and absorption by fractal agglomerates and coagulations of smoke aerosols,” J. Mod. Opt. 37, 171–181 (1990).
[CrossRef]

B. M. Vaglieco, F. Berretta, A. D’Alessio, “In situ evaluation of the soot refractive index in the UV–visible from the measurement of the scattering and extinction coefficients in rich flames,” Combust. Flame 79, 259–271 (1990).
[CrossRef]

1989 (3)

J. Nelson, “Test of a mean field theory for the optics of fractal clusters,” J. Mod. Opt. 36, 1031–1057 (1989).
[CrossRef]

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

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]

1988 (2)

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

T. T. Charalampopoulos, H. Chang, “In situ optical properties of soot particles in wavelength range from 340 nm to 600 nm,” Combust. Sci. Technol. 59, 401–421 (1988).
[CrossRef]

1987 (1)

J. E. Martin, A. J. Hurd, “Scattering from fractals,” J. Appl. Cryst. 20, 61–78 (1987).
[CrossRef]

1986 (1)

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

1983 (1)

R. J. Santoro, H. G. Semerjian, R. A. Dobbins, “Soot particle measurements in diffusion flames,” Combust. Flame 51, 203–218 (1983).
[CrossRef]

1979 (2)

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

A. R. Jones, “Scattering efficiency factors for agglomerates of small particles,” J. Phys. D 12, 1661–1671 (1979).
[CrossRef]

1973 (1)

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

1970 (1)

W. H. Dalzell, G. C. Williams, H. C. Hottel, “A light scattering method for soot concentration measurements,” Combust. Flame 14, 161–170 (1970).
[CrossRef]

1964 (1)

W. D. Erickson, G. C. Williams, H. C. Hottel, “Light scattering measurements on soot in a benzene–air flame,” Combust. Flame 8, 127–132 (1964).
[CrossRef]

1908 (1)

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

Beretta, F.

A. D’Alessio, A. Di Lorenzo, A. Borghese, F. Beretta, S. Masi, “Study of soot nucleation zone of rich methane–oxygen flames,” in Sixteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1977), p. 695.
[CrossRef]

A. D’Alessio, A. Di Lorenzo, A. F. Sarofim, F. Beretta, S. Masi, C. Venitozzi, “Soot formation in methane–oxygen flames,” in Fifteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1975), p. 1427.
[CrossRef]

Berretta, F.

B. M. Vaglieco, F. Berretta, A. D’Alessio, “In situ evaluation of the soot refractive index in the UV–visible from the measurement of the scattering and extinction coefficients in rich flames,” Combust. Flame 79, 259–271 (1990).
[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.

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

Borghese, A.

A. D’Alessio, A. Di Lorenzo, A. Borghese, F. Beretta, S. Masi, “Study of soot nucleation zone of rich methane–oxygen flames,” in Sixteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1977), p. 695.
[CrossRef]

Botet, R.

R. Jullien, R. Botet, Aggregation and Fractal Aggregates (World Scientific, Singapore, 1987).

Cai, J.

Chang, H.

H. Chang, T. T. Charalampopoulos, “Determination of the wavelength dependence of refractive indices of flame soot,” Proc. R. Soc. London Ser. A 430, 577–591 (1991).
[CrossRef]

T. T. Charalampopoulos, H. Chang, “Agglomerate parameters and fractal dimension of soot using light scattering—Effects on surface growth,” Combust. Flame 87, 88–99 (1991).
[CrossRef]

T. T. Charalampopoulos, H. Chang, “In situ optical properties of soot particles in wavelength range from 340 nm to 600 nm,” Combust. Sci. Technol. 59, 401–421 (1988).
[CrossRef]

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 (1995).
[CrossRef]

W. Lou, T. T. Charalampopoulos, “On the inverse scattering problem for characterization of agglomerated particulates: partial derivative formulation,” J. Phys. D 28, 2585–2594 (1995).
[CrossRef]

B. J. Stagg, T. T. Charalampopoulos, “Refractive Indices of pyrolytic graphite, amorphous carbon and flame soot in the temperature range 25 degrees to 600 degrees C,” Combust. Flame 94, 381–396 (1993).
[CrossRef]

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

T. T. Charalampopoulos, H. Chang, “Agglomerate parameters and fractal dimension of soot using light scattering—Effects on surface growth,” Combust. Flame 87, 88–99 (1991).
[CrossRef]

H. Chang, T. T. Charalampopoulos, “Determination of the wavelength dependence of refractive indices of flame soot,” Proc. R. Soc. London Ser. A 430, 577–591 (1991).
[CrossRef]

T. T. Charalampopoulos, H. Chang, “In situ optical properties of soot particles in wavelength range from 340 nm to 600 nm,” Combust. Sci. Technol. 59, 401–421 (1988).
[CrossRef]

Z. Zhang, T. T. Charalampopoulos, “Controlled combustion synthesis of nanosized iron oxide aggregates,” in Twenty-Sixth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1996), pp. 1851–1857.
[CrossRef]

R. Munoz, T. T. Charalampopoulos, “Evolution of compositional and structural properties of soot in premixed alkane flames,” in Twenty-Seventh Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1998), pp. 1471–1479.
[CrossRef]

Chen, H. Y.

H. Y. Chen, M. F. Iskander, J. E. Penner, “Light scattering and absorption by fractal agglomerates and coagulations of smoke aerosols,” J. Mod. Opt. 37, 171–181 (1990).
[CrossRef]

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]

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.

B. M. Vaglieco, F. Berretta, A. D’Alessio, “In situ evaluation of the soot refractive index in the UV–visible from the measurement of the scattering and extinction coefficients in rich flames,” Combust. Flame 79, 259–271 (1990).
[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), pp. 207–256.

A. D’Alessio, A. Di Lorenzo, A. F. Sarofim, F. Beretta, S. Masi, C. Venitozzi, “Soot formation in methane–oxygen flames,” in Fifteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1975), p. 1427.
[CrossRef]

A. D’Alessio, A. Di Lorenzo, A. Borghese, F. Beretta, S. Masi, “Study of soot nucleation zone of rich methane–oxygen flames,” in Sixteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1977), p. 695.
[CrossRef]

Dalzell, W. H.

W. H. Dalzell, G. C. Williams, H. C. Hottel, “A light scattering method for soot concentration measurements,” Combust. Flame 14, 161–170 (1970).
[CrossRef]

Di Lorenzo, A.

A. D’Alessio, A. Di Lorenzo, A. Borghese, F. Beretta, S. Masi, “Study of soot nucleation zone of rich methane–oxygen flames,” in Sixteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1977), p. 695.
[CrossRef]

A. D’Alessio, A. Di Lorenzo, A. F. Sarofim, F. Beretta, S. Masi, C. Venitozzi, “Soot formation in methane–oxygen flames,” in Fifteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1975), p. 1427.
[CrossRef]

Dobbins, R. A.

R. Puri, T. F. Richardson, R. J. Santoro, R. A. Dobbins, “Aerosol dynamic process of soot aggregates in a laminar ethane diffusion flame,” Combust. Flame 92, 320–333 (1993).
[CrossRef]

R. A. Dobbins, C. M. Megaridis, “Absorption and scattering of light by polydisperse aggregates,” Appl. Opt. 30, 4747–4754 (1991).
[CrossRef] [PubMed]

R. J. Santoro, H. G. Semerjian, R. A. Dobbins, “Soot particle measurements in diffusion flames,” Combust. Flame 51, 203–218 (1983).
[CrossRef]

Draine, B. T.

Erickson, W. D.

W. D. Erickson, G. C. Williams, H. C. Hottel, “Light scattering measurements on soot in a benzene–air flame,” Combust. Flame 8, 127–132 (1964).
[CrossRef]

Faeth, G. M.

G. M. Faeth, Ü. Ö. Köylü, “Soot morphology and optical properties in nonpremixed turbulent flame environments,” Combust. Sci. Technol. 108, 207–229 (1995).
[CrossRef]

Ü. Ö. Köylü, G. M. Faeth, “Optical properties of overfire soot in buoyant turbulent diffusion flames at long residence times,” J. Heat Transfer 116, 152–159 (1994).
[CrossRef]

Flatau, P. J.

Goedecke, G. H.

Hottel, H. C.

W. H. Dalzell, G. C. Williams, H. C. Hottel, “A light scattering method for soot concentration measurements,” Combust. Flame 14, 161–170 (1970).
[CrossRef]

W. D. Erickson, G. C. Williams, H. C. Hottel, “Light scattering measurements on soot in a benzene–air flame,” Combust. Flame 8, 127–132 (1964).
[CrossRef]

Huffman, D. R.

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

Hurd, A. J.

J. E. Martin, A. J. Hurd, “Scattering from fractals,” J. Appl. Cryst. 20, 61–78 (1987).
[CrossRef]

Iskander, M. F.

H. Y. Chen, M. F. Iskander, J. E. Penner, “Light scattering and absorption by fractal agglomerates and coagulations of smoke aerosols,” J. Mod. Opt. 37, 171–181 (1990).
[CrossRef]

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]

Jinno, H.

M. Kunugi, H. Jinno, “Determination of size and concentration of soot particles in diffusion flames by a light scattering technique,” in Eleventh Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1966), p. 257.

Jones, A. R.

A. R. Jones, “Scattering efficiency factors for agglomerates of small particles,” J. Phys. D 12, 1661–1671 (1979).
[CrossRef]

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

Jullien, R.

R. Jullien, R. Botet, Aggregation and Fractal Aggregates (World Scientific, Singapore, 1987).

Kaye, P. H.

Kerker, M.

M. Kerker, The Scattering of Light, and Other Electromagnetic Radiation (Academic, New York, 1969).

Köylü, Ü. Ö.

Ü. Ö. Köylü, “Quantitative analysis of in-situ optical diagnostics for inferring particle/aggregates parameters in flames: implications for soot surface growth and total emissivity,” Combust. Flame 109, 488–500 (1996).
[CrossRef]

G. M. Faeth, Ü. Ö. Köylü, “Soot morphology and optical properties in nonpremixed turbulent flame environments,” Combust. Sci. Technol. 108, 207–229 (1995).
[CrossRef]

Ü. Ö. Köylü, G. M. Faeth, “Optical properties of overfire soot in buoyant turbulent diffusion flames at long residence times,” J. Heat Transfer 116, 152–159 (1994).
[CrossRef]

Ku, J. C.

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

J. C. Ku, “Correction for the extinction efficiency factors given in the Jones solution for electromagnetic scattering by agglomerates of small spheres,” J. Phys. D 24, 71–75 (1991).
[CrossRef]

Kumar, K.

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

Kunugi, M.

M. Kunugi, H. Jinno, “Determination of size and concentration of soot particles in diffusion flames by a light scattering technique,” in Eleventh Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1966), p. 257.

Liu, N.

Lou, W.

W. Lou, T. T. Charalampopoulos, “On the inverse scattering problem for characterization of agglomerated particulates: partial derivative formulation,” J. Phys. D 28, 2585–2594 (1995).
[CrossRef]

W. Lou, T. T. Charalampopoulos, “On the electromagnetic scattering and absorption of agglomerated small spherical particles,” J. Phys. D 27, 2258–2270 (1995).
[CrossRef]

Ludlow, I. K.

Martin, J. E.

J. E. Martin, A. J. Hurd, “Scattering from fractals,” J. Appl. Cryst. 20, 61–78 (1987).
[CrossRef]

Masi, S.

A. D’Alessio, A. Di Lorenzo, A. Borghese, F. Beretta, S. Masi, “Study of soot nucleation zone of rich methane–oxygen flames,” in Sixteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1977), p. 695.
[CrossRef]

A. D’Alessio, A. Di Lorenzo, A. F. Sarofim, F. Beretta, S. Masi, C. Venitozzi, “Soot formation in methane–oxygen flames,” in Fifteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1975), p. 1427.
[CrossRef]

Megaridis, C. M.

Mengüç, M. P.

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]

Mie, G.

G. Mie, “Beitrage zur Optik trüber Medien speziell kolliodaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
[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]

Munoz, R.

R. Munoz, T. T. Charalampopoulos, “Evolution of compositional and structural properties of soot in premixed alkane flames,” in Twenty-Seventh Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1998), pp. 1471–1479.
[CrossRef]

Nelson, J.

J. Nelson, “Test of a mean field theory for the optics of fractal clusters,” J. Mod. Opt. 36, 1031–1057 (1989).
[CrossRef]

O’Brien, S. G.

Penner, J. E.

H. Y. Chen, M. F. Iskander, J. E. Penner, “Light scattering and absorption by fractal agglomerates and coagulations of smoke aerosols,” J. Mod. Opt. 37, 171–181 (1990).
[CrossRef]

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]

Pennypacker, C. R.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption 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]

Purcell, E. M.

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

Puri, R.

R. Puri, T. F. Richardson, R. J. Santoro, R. A. Dobbins, “Aerosol dynamic process of soot aggregates in a laminar ethane diffusion flame,” Combust. Flame 92, 320–333 (1993).
[CrossRef]

Richardson, T. F.

R. Puri, T. F. Richardson, R. J. Santoro, R. A. Dobbins, “Aerosol dynamic process of soot aggregates in a laminar ethane diffusion flame,” Combust. Flame 92, 320–333 (1993).
[CrossRef]

Santoro, R. J.

R. Puri, T. F. Richardson, R. J. Santoro, R. A. Dobbins, “Aerosol dynamic process of soot aggregates in a laminar ethane diffusion flame,” Combust. Flame 92, 320–333 (1993).
[CrossRef]

R. J. Santoro, H. G. Semerjian, R. A. Dobbins, “Soot particle measurements in diffusion flames,” Combust. Flame 51, 203–218 (1983).
[CrossRef]

Sarofim, A. F.

A. D’Alessio, A. Di Lorenzo, A. F. Sarofim, F. Beretta, S. Masi, C. Venitozzi, “Soot formation in methane–oxygen flames,” in Fifteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1975), p. 1427.
[CrossRef]

Saxon, D. S.

D. S. Saxon, Lecture on the Scattering of Light (University of California at Los Angeles, Los Angeles, Calif., 1974).

Semerjian, H. G.

R. J. Santoro, H. G. Semerjian, R. A. Dobbins, “Soot particle measurements in diffusion flames,” Combust. Flame 51, 203–218 (1983).
[CrossRef]

Shim, K. H.

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

Sorensen, C. M.

Stagg, B. J.

B. J. Stagg, T. T. Charalampopoulos, “Refractive Indices of pyrolytic graphite, amorphous carbon and flame soot in the temperature range 25 degrees to 600 degrees C,” Combust. Flame 94, 381–396 (1993).
[CrossRef]

Tien, C. L.

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

Ulanowski, Z.

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]

B. M. Vaglieco, F. Berretta, A. D’Alessio, “In situ evaluation of the soot refractive index in the UV–visible from the measurement of the scattering and extinction coefficients in rich flames,” Combust. Flame 79, 259–271 (1990).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Venitozzi, C.

A. D’Alessio, A. Di Lorenzo, A. F. Sarofim, F. Beretta, S. Masi, C. Venitozzi, “Soot formation in methane–oxygen flames,” in Fifteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1975), p. 1427.
[CrossRef]

Venizelos, D. T.

D. T. Venizelos, “A study of the radiative properties of agglomerated flame particulates using light scattering, Ph.D. dissertation (Louisiana State University, Baton Rouge, La., 1994).

Wang, Z.

Williams, G. C.

W. H. Dalzell, G. C. Williams, H. C. Hottel, “A light scattering method for soot concentration measurements,” Combust. Flame 14, 161–170 (1970).
[CrossRef]

W. D. Erickson, G. C. Williams, H. C. Hottel, “Light scattering measurements on soot in a benzene–air flame,” Combust. Flame 8, 127–132 (1964).
[CrossRef]

Zhang, Z.

Z. Zhang, T. T. Charalampopoulos, “Controlled combustion synthesis of nanosized iron oxide aggregates,” in Twenty-Sixth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1996), pp. 1851–1857.
[CrossRef]

Ann. Phys. (Leipzig) (1)

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

Appl. Opt. (5)

Astrophys. J. (1)

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

Combust. Flame (8)

B. M. Vaglieco, F. Berretta, A. D’Alessio, “In situ evaluation of the soot refractive index in the UV–visible from the measurement of the scattering and extinction coefficients in rich flames,” Combust. Flame 79, 259–271 (1990).
[CrossRef]

Ü. Ö. Köylü, “Quantitative analysis of in-situ optical diagnostics for inferring particle/aggregates parameters in flames: implications for soot surface growth and total emissivity,” Combust. Flame 109, 488–500 (1996).
[CrossRef]

W. D. Erickson, G. C. Williams, H. C. Hottel, “Light scattering measurements on soot in a benzene–air flame,” Combust. Flame 8, 127–132 (1964).
[CrossRef]

W. H. Dalzell, G. C. Williams, H. C. Hottel, “A light scattering method for soot concentration measurements,” Combust. Flame 14, 161–170 (1970).
[CrossRef]

R. J. Santoro, H. G. Semerjian, R. A. Dobbins, “Soot particle measurements in diffusion flames,” Combust. Flame 51, 203–218 (1983).
[CrossRef]

R. Puri, T. F. Richardson, R. J. Santoro, R. A. Dobbins, “Aerosol dynamic process of soot aggregates in a laminar ethane diffusion flame,” Combust. Flame 92, 320–333 (1993).
[CrossRef]

T. T. Charalampopoulos, H. Chang, “Agglomerate parameters and fractal dimension of soot using light scattering—Effects on surface growth,” Combust. Flame 87, 88–99 (1991).
[CrossRef]

B. J. Stagg, T. T. Charalampopoulos, “Refractive Indices of pyrolytic graphite, amorphous carbon and flame soot in the temperature range 25 degrees to 600 degrees C,” Combust. Flame 94, 381–396 (1993).
[CrossRef]

Combust. Sci. Technol. (4)

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

T. T. Charalampopoulos, H. Chang, “In situ optical properties of soot particles in wavelength range from 340 nm to 600 nm,” Combust. Sci. Technol. 59, 401–421 (1988).
[CrossRef]

G. M. Faeth, Ü. Ö. Köylü, “Soot morphology and optical properties in nonpremixed turbulent flame environments,” Combust. Sci. Technol. 108, 207–229 (1995).
[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]

J. Appl. Cryst. (1)

J. E. Martin, A. J. Hurd, “Scattering from fractals,” J. Appl. Cryst. 20, 61–78 (1987).
[CrossRef]

J. Heat Transfer (1)

Ü. Ö. Köylü, G. M. Faeth, “Optical properties of overfire soot in buoyant turbulent diffusion flames at long residence times,” J. Heat Transfer 116, 152–159 (1994).
[CrossRef]

J. Mod. Opt. (2)

J. Nelson, “Test of a mean field theory for the optics of fractal clusters,” J. Mod. Opt. 36, 1031–1057 (1989).
[CrossRef]

H. Y. Chen, M. F. Iskander, J. E. Penner, “Light scattering and absorption by fractal agglomerates and coagulations of smoke aerosols,” J. Mod. Opt. 37, 171–181 (1990).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. D (4)

W. Lou, T. T. Charalampopoulos, “On the inverse scattering problem for characterization of agglomerated particulates: partial derivative formulation,” J. Phys. D 28, 2585–2594 (1995).
[CrossRef]

W. Lou, T. T. Charalampopoulos, “On the electromagnetic scattering and absorption of agglomerated small spherical particles,” J. Phys. D 27, 2258–2270 (1995).
[CrossRef]

A. R. Jones, “Scattering efficiency factors for agglomerates of small particles,” J. Phys. D 12, 1661–1671 (1979).
[CrossRef]

J. C. Ku, “Correction for the extinction efficiency factors given in the Jones solution for electromagnetic scattering by agglomerates of small spheres,” J. Phys. D 24, 71–75 (1991).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

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

Opt. Acta (1)

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

Proc. R. Soc. London Ser. A (2)

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

H. Chang, T. T. Charalampopoulos, “Determination of the wavelength dependence of refractive indices of flame soot,” Proc. R. Soc. London Ser. A 430, 577–591 (1991).
[CrossRef]

Prog. Energy Combust. Sci. (1)

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

Other (12)

M. Kunugi, H. Jinno, “Determination of size and concentration of soot particles in diffusion flames by a light scattering technique,” in Eleventh Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1966), p. 257.

A. D’Alessio, A. Di Lorenzo, A. F. Sarofim, F. Beretta, S. Masi, C. Venitozzi, “Soot formation in methane–oxygen flames,” in Fifteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1975), p. 1427.
[CrossRef]

A. D’Alessio, A. Di Lorenzo, A. Borghese, F. Beretta, S. Masi, “Study of soot nucleation zone of rich methane–oxygen flames,” in Sixteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1977), p. 695.
[CrossRef]

R. Munoz, T. T. Charalampopoulos, “Evolution of compositional and structural properties of soot in premixed alkane flames,” in Twenty-Seventh Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1998), pp. 1471–1479.
[CrossRef]

M. Kerker, The Scattering of Light, and Other Electromagnetic Radiation (Academic, New York, 1969).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

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

R. Jullien, R. Botet, Aggregation and Fractal Aggregates (World Scientific, Singapore, 1987).

D. S. Saxon, Lecture on the Scattering of Light (University of California at Los Angeles, Los Angeles, Calif., 1974).

Z. Zhang, T. T. Charalampopoulos, “Controlled combustion synthesis of nanosized iron oxide aggregates,” in Twenty-Sixth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1996), pp. 1851–1857.
[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), pp. 207–256.

D. T. Venizelos, “A study of the radiative properties of agglomerated flame particulates using light scattering, Ph.D. dissertation (Louisiana State University, Baton Rouge, La., 1994).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Schematic of light-scattering system and the coordinate system for a linear chainlike aggregate: V-pol, vertical polarization orientation; H-pol, horizontal polarization orientation.

Fig. 2
Fig. 2

Data-inversion uncertainty at several measurement angles of R hv .

Fig. 3
Fig. 3

Data-inversion uncertainty at several measurement angles of K vve .

Fig. 4
Fig. 4

Data-inversion uncertainty at several measurement angles of R vv .

Fig. 5
Fig. 5

Data-inversion uncertainty at several measurement angles of R hh .

Fig. 6
Fig. 6

Data-inversion uncertainty of n for pairing sets of R vv (15°), R hv (75°), K vve (75°), and R hh (75°); chainlike aggregate; m = 1.7 + 0.7i.

Fig. 7
Fig. 7

Data-inversion uncertainty of k for pairing sets of R vv (15°), R hv (75°), K vve (75°), and R hh (75°); chainlike aggregate; m = 1.7 + 0.7i.

Fig. 8
Fig. 8

Data-inversion uncertainty of x for pairing sets of R vv (15°), R hv (75°), K vve (75°), and R hh (75°); chainlike aggregate; m = 1.7 + 0.7i.

Fig. 9
Fig. 9

Data-inversion uncertainty of N p for pairing sets of R vv (15°), R hv (75°), K vve (75°), and R hh (75°); chainlike aggregate; m = 1.7 + 0.7i.

Tables (3)

Tables Icon

Table 1 Pairing Sets and Their Inversion Uncertainties at Optimal Anglesa

Tables Icon

Table 2 Optimum Angles and Corresponding Uncertainty of n, k, x, and N p a

Tables Icon

Table 3 Examples of Data Inversion from Simulated Experimental Results

Equations (66)

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

1+ωC0Ei-ωC2j=1jiNp TijEj=Einc,i;  i=1, 2, 3,  Np,
ω=ε-1=n+ik2-1,
C0=3-2ix2h11x,
C2=i3 x2j1x.
Ei=3ε+2Einc,i+i3ε-1ε+2j=1jiNp 3x2j1xTijEj+2ε-1ε+2ix2h11x-1Ei;  i=1, 2, 3,  Np.
T=abcbadcde.
a=2h01kD-h21kDP2cos χ-1/2 cos2ψP22cos χ,
a=2h01kD-h21kDP2cos χ+1/2 cos2ψP22cos χ,
b=½h21kDP22cos χsin2ψ,
c=-h21kDP21cos χcos ψ,
d=-h21kDP21cos χsin ψ,
e=2h01kD+2h21kDP2cos χ,
cos χ=zj-ziD,
tan ψ=yj-yixj-xi,
D2=xj-xi2+yj-yi2+zj-zi2,
Escar=x2j1xε-1expikdkd×i=1Np exp-ikri cos βiΘieˆθ-Φieˆϕ,
Θi=Exricos θ cos ϕ+Eyricos θ sin ϕ-Ezrisin θ,
Φi=Exrisin ϕ+Eyricos ϕ,
cos βi=cos χi cos θ + sin χi sin θ cosψi-ϕ.
Csca=4π3 R2x2j12x|ε-1|Rei=1Npj=1Np Ei·TijEj*,
Cabs=4πR2j1xImε-1i=1Np |Ei|2,
Cext=4πR2j1xImε-1i=1Np Ei·Einc,i*,
Cvpθ=R2x2j12x|ε-1|2|V|2,
Chpθ=R2x2j12x|ε-1|2|H|2,
V=i=1Np exp-ikri cos βiEx,i,
H=i=1Np exp-ikri cos βicos θEy,i-sin θEz,i,
1+ωC0Ein-ωC2j=1jiNp TijEjn=1ωωnEi-Einc,i;  i=1, 2, 3,  Np,
1+ωC0Eix-ωC2j=1jiNp TijEjx=ikzix-1C2C2xEinc,i+1+ωC0C2C2x-ω C0xEi-ωC2j=1jiNp2xkDijh11-3h01+kDijh11h01h21Ej+j=1jiNpkDijh21-3×ωC2x TijEj;  i=1, 2, 3,  Np,
Eik=i Ein;  i=1, 2, 3,  Np.
Cextx=j1xxCextj1x+4πR2j1xIm×ω i=1Np Einc,i*Eix-ikzix Ei,
Cextk, Cextn=4πR2j1x×i=1Np Einc,i*ω Ein+ωn Ei.
Cvpn=1|ω|2|ω|2n Cvp+R2x2j12x|ω|2 ReV* Vn,
Cvpx=1xj12xj12x Cvp+R2|ω|2xj12 ReV* Vx.
C¯pp=18π202πdω 02πdψ 0π Cpp sin χdχ,
C¯pp=14π0πdψ 02π Cpp sin χdχ.
Ippθ=I0ΔΩVnAC¯ppθηoptτ,
τ=exp-nAC¯extL,
Kvveθ=C¯vvθC¯ext,
Khveθ=C¯hvθC¯ext,
Khheθ=C¯hhθC¯ext.
Rvvθ=C¯vvθC¯vv180°-θ,
Rhhθ=C¯hhθC¯hh180°-θ,
Rhvθ=C¯hvθC¯vvθ
Rvhθ=C¯vhθC¯hhθ.
Kvveθx=1C¯extC¯vvθx-KvveθC¯extx.
Kvveθ, NpNp=½Kvveθ, Np+1-Kvveθ, Np-1.
fv=nAπdp36.
Qiθ; n, k, x, Np=Miθ;  i=1, 2, 3, 4,
S=i=14Qiθ-MiθMiθ2
Miθ=Qiθ, n0, k0, x0, Np0+Qin Δn+Qik Δk+Qix Δx+QiNp ΔNp;  i=1, 2, 3, 4.
Qin Δn+Qik Δk+Qix Δx+QiNp ΔNp=Miθ-Qiθ, n0, k0, x0, Np0;  i=1, 2, 3, 4.
Miθ=Qiθ, n, k, x, Np;  i=1, 2, 3, 4.
Miθ+ΔMiθ=Qiθ, n+Δn, k+Δk, x+Δx, Np+ΔNp;  i=1, 2, 3, 4.
Qin Δn+Qik Δk+Qix Δx+QiNp ΔNp=ΔMi;  i=1, 2, 3, 4.
nQiQinΔnn+kQiQikΔkk+xQiQixΔxx+NpQiQiNpΔNpNp=ΔMiMi;  i=1, 2, 3, 4.
ΔM1M1, ΔM2M2, ΔM3M3, ΔM4M4T=AΔnn, Δkk, Δxx, ΔNpNpT,
A=nQ1Q1nkQ1Q1kxQ1Q1xNpQ1Q1NpnQ2Q2nkQ2Q2kxQ2Q2xNpQ2Q2NpnQ3Q3nkQ3Q3kxQ3Q3xNpQ3Q3NpnQ4Q4nkQ4Q4kxQ4Q4xNpQ4Q4Np.
Δnn=i=14 a1iΔMiMi,
Δkk=i=14 a2iΔMiMi,
Δxx=i=14 a3iΔMiMi,
ΔNpNp=i=14 a4iΔMiMi,
Δnn=i=14a1iΔMiMi2,
Δkk=i=14a2iΔMiMi2,
Δxx=i=14a3iΔMiMi2,
ΔNpNp=i=14a4iΔMiMi2.
Y=j=14i=14ajiΔMiMi2.

Metrics