Abstract

A method for calculating the angular and optical depth dependencies of diffuse radiation in light scattering and absorbing particulate media is established. From the Lorenz–Mie theory, the coefficients involved in the expansion of the single particle phase function in terms of Legendre polynomials are obtained. Then the angular dependence of diffuse radiation is described by means of generalized phase functions, corresponding to the different scattering orders. The optical depth dependence is given in terms of weighting factors, which depend on average path-length parameters and forward-scattering ratios of the different scattering orders. Intensity patterns are obtained for different particle sizes, concentrations, and relative refractive indices, and the behavior of the forward and backward diffuse intensities in terms of optical depth is displayed. Comparisons with the forward-multiple-scattering theory of Hartel are carried out. A generalization of the Beer–Lambert law in the case of light-scattering materials is obtained from the present approach, which is derived from the radiative transfer equation.

© 1997 Optical Society of America

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References

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  1. A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
    [CrossRef]
  2. J. W. Ryde, “The scattering of light by turbid media. Part I,” Proc. R. Soc. London, Ser. A 131, 451–464 (1931).
    [CrossRef]
  3. P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).
  4. W. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten besonders Trübgläser,” Licht 10, 141–143, 165, 190, 191, 214, 215, 232–234 (1940).
  5. H. H. Theissing, “Macrodistribution of light scattered by dispersion of spherical dielectric particles,” J. Opt. Soc. Am. 40, 232–243 (1950).
    [CrossRef]
  6. H. G. Hecht, “The a priori calculation of the diffuse reflectance of a turbid medium,” Opt. Acta 30, 659–668 (1983).
    [CrossRef]
  7. H. C. Van de Hulst, Multiple Light Scattering (Academic, New York, 1980).
  8. J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
    [CrossRef]
  9. F. B. Yurevich, L. A. Konyukh, “Radiation attenuation by disperse media,” Int. J. Heat Mass Transfer 18, 819–829 (1975).
    [CrossRef]
  10. W. M. Irvine, “Multiple scattering by large particles,” Astrophys. J. 142, 1563–1575 (1965).
    [CrossRef]
  11. Y. Kuga, A. Ishimaru, H. W. Chang, L. Tsang, “Comparisons between the small-angle approximation and the numerical solution for radiative transfer theory,” Appl. Opt. 25, 3803–3806 (1986).
    [CrossRef] [PubMed]
  12. A. Zardecki, S. A. W. Gerstl, “Multi-Gaussian phase function model for off-axis laser beam scattering,” Appl. Opt. 26, 3000–3004 (1987).
    [CrossRef] [PubMed]
  13. S. A. W. Gerstl, A. Zardecki, W. P. Unruh, D. M. Stupin, G. H. Stokes, N. E. Elliot“On-axis multiple scattering theory of a laser beam in turbid media: comparison of theory and experiment,” Appl. Opt. 26, 779–785 (1987).
    [CrossRef]
  14. M. H. Eddowes, T. N. Mills, D. T. Delpy, “Monte Carlo simulations of coherent backscatter for identification of the optical coefficients of biological tissues in vivo,” Appl. Opt. 34, 2261–2267 (1995).
    [CrossRef] [PubMed]
  15. L. Bergougnoux, J. Misguich-Ripault, J. L. Firpo, J. André, “Monte Carlo calculation of backscattered light intensity by suspension: comparison with experimental data,” Appl. Opt. 35, 1735–1741 (1996).
    [CrossRef] [PubMed]
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    [CrossRef]
  18. W. E. Vargas, G. A. Niklasson, “Forward average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 36, 3735–3738 (1997).
    [CrossRef] [PubMed]
  19. D. H. Woodward, “Multiple light scattering by spherical dielectric particles,” J. Opt. Soc. Am. 54, 1325–1331 (1964).
    [CrossRef]
  20. C. Smart, R. Jacobsen, M. Kerker, J. P. Kratohvil, E. Matijevic, “Experimental study of multiple light scattering,” J. Opt. Soc. Am. 55, 947–955 (1965).
    [CrossRef]
  21. H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Dalzell, “Multiple scatter: comparison of theory with experiment,” J. Heat Transfer 92, 285–291 (1970).
    [CrossRef]
  22. P. Latimer, “Light scattering and absorption as methods of studying cell population parameters,” Ann. Rev. Biophys. Bioeng. 11, 129–150 (1982).
    [CrossRef]
  23. P. Latimer, “Blood platelet aggregometer: predicted effects of aggregation, photometer geometry, and multiple scattering,” Appl. Opt. 22, 1136–1143 (1983).
    [CrossRef] [PubMed]
  24. H. Schnablegger, O. Glatter, “Sizing of colloidal particles with light scattering: corrections for beginning multiple scattering,” Appl. Opt. 34, 3489–3501 (1995).
    [CrossRef] [PubMed]
  25. H. Schnablegger, D. Lehner, O. Glatter, “Static light scattering on dense colloidal systems: computational techniques, new instrumentation and experimental results,” in Electromagnetic and Light Scattering: Theory and Applications, T. Wriedt, M. Quinten, K. Bauckhage, eds. (Universität Bremen, Bremen, Germany, 1996), pp. 133–137.
  26. S. E. Orchard, “Multiple scattering by spherical dielectric particles,” J. Opt. Soc. Am. 55, 737 (1965).
    [CrossRef]
  27. P. Chylek, “Mie scattering into the backward hemisphere,” J. Opt. Soc. Am. 63, 1467–1471 (1973).
    [CrossRef]
  28. S. E. Orchard, “Reflection and transmission of light by diffusing suspensions,” J. Opt. Soc. Am. 59, 1584–1597 (1969).
    [CrossRef]
  29. J. Reichman, “Determination of absorption and scattering coefficients for nonhomogeneous media: 1: theory,” Appl. Opt. 12, 1811–1815 (1973).
    [CrossRef] [PubMed]
  30. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  31. C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw–Hill, New York, 1978).
  32. J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).
  33. A. Ben-David, “Multiple-scattering transmission and an effective average photon path length of a plane-parallel beam in a homogeneous medium,” Appl. Opt. 34, 2802–2810 (1995).
    [CrossRef] [PubMed]
  34. N. A. Voishvillo, “Dependence of the transmission coefficients of a scattering layer on its thickness,” Opt. Spektrosk. 66, 390–393 (1989).
  35. L. R. Bissonnette, R. B. Smith, A. Ulitsky, J. D. Houston, A. I. Carswell, “Transmitted beam profiles, integrated backscatter, and range-resolved backscatter in inhomogeneous laboratory water droplet clouds,” Appl. Opt. 27, 2485–2494 (1988).
    [CrossRef] [PubMed]

1997

1996

1995

1989

N. A. Voishvillo, “Dependence of the transmission coefficients of a scattering layer on its thickness,” Opt. Spektrosk. 66, 390–393 (1989).

1988

1987

1986

1983

1982

P. Latimer, “Light scattering and absorption as methods of studying cell population parameters,” Ann. Rev. Biophys. Bioeng. 11, 129–150 (1982).
[CrossRef]

1975

F. B. Yurevich, L. A. Konyukh, “Radiation attenuation by disperse media,” Int. J. Heat Mass Transfer 18, 819–829 (1975).
[CrossRef]

1974

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

1973

1970

H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Dalzell, “Multiple scatter: comparison of theory with experiment,” J. Heat Transfer 92, 285–291 (1970).
[CrossRef]

1969

1965

1964

1955

1950

1940

W. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten besonders Trübgläser,” Licht 10, 141–143, 165, 190, 191, 214, 215, 232–234 (1940).

1931

J. W. Ryde, “The scattering of light by turbid media. Part I,” Proc. R. Soc. London, Ser. A 131, 451–464 (1931).
[CrossRef]

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

1905

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

André, J.

Ben-David, A.

Bender, C. M.

C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw–Hill, New York, 1978).

Bergougnoux, L.

Bissonnette, L. R.

Carswell, A. I.

Chang, H. W.

Chu, C. M.

Churchill, S. W.

Chylek, P.

Dalzell, W. H.

H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Dalzell, “Multiple scatter: comparison of theory with experiment,” J. Heat Transfer 92, 285–291 (1970).
[CrossRef]

Delpy, D. T.

Duderstadt, J. J.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).

Eddowes, M. H.

Elliot, N. E.

Firpo, J. L.

Gerstl, S. A. W.

Glatter, O.

H. Schnablegger, O. Glatter, “Sizing of colloidal particles with light scattering: corrections for beginning multiple scattering,” Appl. Opt. 34, 3489–3501 (1995).
[CrossRef] [PubMed]

H. Schnablegger, D. Lehner, O. Glatter, “Static light scattering on dense colloidal systems: computational techniques, new instrumentation and experimental results,” in Electromagnetic and Light Scattering: Theory and Applications, T. Wriedt, M. Quinten, K. Bauckhage, eds. (Universität Bremen, Bremen, Germany, 1996), pp. 133–137.

Hansen, J. E.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Hartel, W.

W. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten besonders Trübgläser,” Licht 10, 141–143, 165, 190, 191, 214, 215, 232–234 (1940).

Hecht, H. G.

H. G. Hecht, “The a priori calculation of the diffuse reflectance of a turbid medium,” Opt. Acta 30, 659–668 (1983).
[CrossRef]

Hottel, H. C.

H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Dalzell, “Multiple scatter: comparison of theory with experiment,” J. Heat Transfer 92, 285–291 (1970).
[CrossRef]

Houston, J. D.

Irvine, W. M.

W. M. Irvine, “Multiple scattering by large particles,” Astrophys. J. 142, 1563–1575 (1965).
[CrossRef]

Ishimaru, A.

Jacobsen, R.

Kerker, M.

Konyukh, L. A.

F. B. Yurevich, L. A. Konyukh, “Radiation attenuation by disperse media,” Int. J. Heat Mass Transfer 18, 819–829 (1975).
[CrossRef]

Kratohvil, J. P.

Kubelka, P.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Kuga, Y.

Latimer, P.

P. Latimer, “Blood platelet aggregometer: predicted effects of aggregation, photometer geometry, and multiple scattering,” Appl. Opt. 22, 1136–1143 (1983).
[CrossRef] [PubMed]

P. Latimer, “Light scattering and absorption as methods of studying cell population parameters,” Ann. Rev. Biophys. Bioeng. 11, 129–150 (1982).
[CrossRef]

Lehner, D.

H. Schnablegger, D. Lehner, O. Glatter, “Static light scattering on dense colloidal systems: computational techniques, new instrumentation and experimental results,” in Electromagnetic and Light Scattering: Theory and Applications, T. Wriedt, M. Quinten, K. Bauckhage, eds. (Universität Bremen, Bremen, Germany, 1996), pp. 133–137.

Martin, W. R.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).

Matijevic, E.

Mills, T. N.

Misguich-Ripault, J.

Munk, F.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Niklasson, G. A.

Orchard, S. E.

Orszag, S. A.

C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw–Hill, New York, 1978).

Reichman, J.

Ryde, J. W.

J. W. Ryde, “The scattering of light by turbid media. Part I,” Proc. R. Soc. London, Ser. A 131, 451–464 (1931).
[CrossRef]

Sarofim, A. F.

H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Dalzell, “Multiple scatter: comparison of theory with experiment,” J. Heat Transfer 92, 285–291 (1970).
[CrossRef]

Schnablegger, H.

H. Schnablegger, O. Glatter, “Sizing of colloidal particles with light scattering: corrections for beginning multiple scattering,” Appl. Opt. 34, 3489–3501 (1995).
[CrossRef] [PubMed]

H. Schnablegger, D. Lehner, O. Glatter, “Static light scattering on dense colloidal systems: computational techniques, new instrumentation and experimental results,” in Electromagnetic and Light Scattering: Theory and Applications, T. Wriedt, M. Quinten, K. Bauckhage, eds. (Universität Bremen, Bremen, Germany, 1996), pp. 133–137.

Schuster, A.

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

Smart, C.

Smith, R. B.

Stokes, G. H.

Stupin, D. M.

Theissing, H. H.

Travis, L. D.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Tsang, L.

Ulitsky, A.

Unruh, W. P.

Van de Hulst, H. C.

H. C. Van de Hulst, Multiple Light Scattering (Academic, New York, 1980).

Vargas, W. E.

Vasalos, I. A.

H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Dalzell, “Multiple scatter: comparison of theory with experiment,” J. Heat Transfer 92, 285–291 (1970).
[CrossRef]

Voishvillo, N. A.

N. A. Voishvillo, “Dependence of the transmission coefficients of a scattering layer on its thickness,” Opt. Spektrosk. 66, 390–393 (1989).

Woodward, D. H.

Yurevich, F. B.

F. B. Yurevich, L. A. Konyukh, “Radiation attenuation by disperse media,” Int. J. Heat Mass Transfer 18, 819–829 (1975).
[CrossRef]

Zardecki, A.

Ann. Rev. Biophys. Bioeng.

P. Latimer, “Light scattering and absorption as methods of studying cell population parameters,” Ann. Rev. Biophys. Bioeng. 11, 129–150 (1982).
[CrossRef]

Appl. Opt.

P. Latimer, “Blood platelet aggregometer: predicted effects of aggregation, photometer geometry, and multiple scattering,” Appl. Opt. 22, 1136–1143 (1983).
[CrossRef] [PubMed]

H. Schnablegger, O. Glatter, “Sizing of colloidal particles with light scattering: corrections for beginning multiple scattering,” Appl. Opt. 34, 3489–3501 (1995).
[CrossRef] [PubMed]

J. Reichman, “Determination of absorption and scattering coefficients for nonhomogeneous media: 1: theory,” Appl. Opt. 12, 1811–1815 (1973).
[CrossRef] [PubMed]

A. Ben-David, “Multiple-scattering transmission and an effective average photon path length of a plane-parallel beam in a homogeneous medium,” Appl. Opt. 34, 2802–2810 (1995).
[CrossRef] [PubMed]

L. R. Bissonnette, R. B. Smith, A. Ulitsky, J. D. Houston, A. I. Carswell, “Transmitted beam profiles, integrated backscatter, and range-resolved backscatter in inhomogeneous laboratory water droplet clouds,” Appl. Opt. 27, 2485–2494 (1988).
[CrossRef] [PubMed]

Y. Kuga, A. Ishimaru, H. W. Chang, L. Tsang, “Comparisons between the small-angle approximation and the numerical solution for radiative transfer theory,” Appl. Opt. 25, 3803–3806 (1986).
[CrossRef] [PubMed]

A. Zardecki, S. A. W. Gerstl, “Multi-Gaussian phase function model for off-axis laser beam scattering,” Appl. Opt. 26, 3000–3004 (1987).
[CrossRef] [PubMed]

S. A. W. Gerstl, A. Zardecki, W. P. Unruh, D. M. Stupin, G. H. Stokes, N. E. Elliot“On-axis multiple scattering theory of a laser beam in turbid media: comparison of theory and experiment,” Appl. Opt. 26, 779–785 (1987).
[CrossRef]

M. H. Eddowes, T. N. Mills, D. T. Delpy, “Monte Carlo simulations of coherent backscatter for identification of the optical coefficients of biological tissues in vivo,” Appl. Opt. 34, 2261–2267 (1995).
[CrossRef] [PubMed]

L. Bergougnoux, J. Misguich-Ripault, J. L. Firpo, J. André, “Monte Carlo calculation of backscattered light intensity by suspension: comparison with experimental data,” Appl. Opt. 35, 1735–1741 (1996).
[CrossRef] [PubMed]

W. E. Vargas, G. A. Niklasson, “Forward average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 36, 3735–3738 (1997).
[CrossRef] [PubMed]

Astrophys. J.

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

W. M. Irvine, “Multiple scattering by large particles,” Astrophys. J. 142, 1563–1575 (1965).
[CrossRef]

Int. J. Heat Mass Transfer

F. B. Yurevich, L. A. Konyukh, “Radiation attenuation by disperse media,” Int. J. Heat Mass Transfer 18, 819–829 (1975).
[CrossRef]

J. Heat Transfer

H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Dalzell, “Multiple scatter: comparison of theory with experiment,” J. Heat Transfer 92, 285–291 (1970).
[CrossRef]

J. Opt. Soc. Am.

Licht

W. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten besonders Trübgläser,” Licht 10, 141–143, 165, 190, 191, 214, 215, 232–234 (1940).

Opt. Acta

H. G. Hecht, “The a priori calculation of the diffuse reflectance of a turbid medium,” Opt. Acta 30, 659–668 (1983).
[CrossRef]

Opt. Spektrosk.

N. A. Voishvillo, “Dependence of the transmission coefficients of a scattering layer on its thickness,” Opt. Spektrosk. 66, 390–393 (1989).

Proc. R. Soc. London, Ser. A

J. W. Ryde, “The scattering of light by turbid media. Part I,” Proc. R. Soc. London, Ser. A 131, 451–464 (1931).
[CrossRef]

Space Sci. Rev.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Z. Tech. Phys. (Leipzig)

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Other

H. C. Van de Hulst, Multiple Light Scattering (Academic, New York, 1980).

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

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw–Hill, New York, 1978).

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).

H. Schnablegger, D. Lehner, O. Glatter, “Static light scattering on dense colloidal systems: computational techniques, new instrumentation and experimental results,” in Electromagnetic and Light Scattering: Theory and Applications, T. Wriedt, M. Quinten, K. Bauckhage, eds. (Universität Bremen, Bremen, Germany, 1996), pp. 133–137.

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Figures (3)

Fig. 1
Fig. 1

Polar plot of diffuse radiation intensity patterns, in arbitrary units, for different values of the particle size parameter x (as indicated in the figures) and relative refractive index: (a) m=(2.75+i0.0)/1.50 and (b) m=(4.08+i0.04)/1.61. The particle volume fraction and free-space wavelength of the incident radiation were set to 0.05 and 0.55 μm, respectively. The optical depth was set to 6. The values of the forward diffuse intensity have been multiplied by Ω, with the corresponding values indicated in the figures.

Fig. 2
Fig. 2

Profiles of the forward and backward integrated intensities in terms of optical depth. Different size parameter values and particle relative refractive indices (as indicated in the figures) have been considered. The particle volume fraction and free-space wavelength of the incident radiation were put to 0.05 and 0.55 μm, respectively. Our approach is compared with results obtained by Hartel theory, with ξk=1 and with a general ξk.

Fig. 3
Fig. 3

Polar plot of diffuse reflectance for two different particle relative refractive indices: (a) m=(2.75+i0.0)/1.50 and (b) m=(4.08+i0.04)/1.61. The particle volume fraction and the free-space wavelength of the incident radiation were set to 0.05 and 0.55 μm, respectively. The particle size parameters x are indicated in the figures.

Equations (83)

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

p(μ, μ)=n=0ωnPn(μ)Pn(μ)
p1(μ, μ)=12ω0 n=0ωnPn(μ)Pn(μ)
f2(μ)=N2-11dμf1(μ)p1(μ, μ),
f2(μ)=14π n=0(2n+1)ωn/ω02n+12Pn(μ),
fk(μ)=Nk-11dμfk-1(μ)p1(μ, μ)=14π n=0(2n+1)ωn/ω02n+1kPn(μ).
I(z, μ)=k=1Qk(z)fk(μ),
dQ1dz=αQ0-(α+β)Q1,
dQkdz=αQk-1-(α+β)Qk,k=2, 3, ,
Qk(z)=(αz)kk! exp[-(α+β)z].
ξk(+)=01fk(μ)dμ01μfk(μ)dμ,ξk(-)=--10fk(μ)dμ-10μfk(μ)dμ.
ξk(±)=21±n=1(2n+1)gn[Ψn]k1±2ω13ω0k+2n=2(2n+1)hn[Ψn]k,
dQ1dz=αQ0-ξ1(α+β)Q1,
dQkdz=ξk-1αQk-1-ξk(α+β)Qk,
k=2, 3, ,
Q1(τ)={1-exp[-(ξ1-1)τ]} α exp(-τ)(α+β)(ξ1-1),
dpkdz+(ξk-1)(α+β)pk(z)
=αξk-1pk-1(z),k2,
pk(τ)=Fk{1-exp[-(ξk-1)τ]}+i=1k-1Gi,k{exp[-(ξk-1)τ]-exp[-(ξi-1)τ]},
Fk=ω0 ξk-1ξk-1 Fk-1,
Gk-1,k=ω0 ξk-1ξk-ξk-1 Fk-1-i=1k-2Gi,k-1,
Gi,k=ω0 ξk-1ξk-ξi Gi,k-1,
F1=ω0ξ1-1,G1,2=ω0ξ2-ξ1 F1,
pk(τ)=Fk{1-exp[-(ξk-1)τ]},
σ1(+)=σc,
σk(+)=01dμ01dμfk-1(μ)f1(μ,μ)01dμ-11dμfk-1(μ)f1(μ,μ)dμ
k=2, 3, ,
σ1(-)=σc,σk(-)=-10dμ-10dμfk-1(μ)f1(μ,μ)-10dμ-11dμfk-1(μ)f1(μ,μ)
k=2, 3, ,
σk(±)=2σd(i)±n=1(2n+1)gnηnk+1ω0 ±n=1gnηnkωn+n=2(2n+1)ηnkm=1gmχnmωm21±n=1(2n+1)gnηnk,
χnm=12n+m k=0[n/2]j=0[m/2] (-1)j+k[n+m+1-2(j+k)]× (2n-2k)!(2m-2j)!(n-k)!k!(n-2k)!(m-j)!j!(m-2j)!,
I(z, μ)=k=1Ik(z, u)k=1Qk(+)(z)fk(μ),
J(z, μ)=k=1Jk(z, μ)k=1Qk(-)(z)fk(μ),
μf1(μ) dQ1(±)dz=-(α+β)f1(μ)Q1(±)+αQ001p1(μ, μ)δ(μ-1)dμ,
μfk(μ) dQk(±)dz=-(α+β)fk(μ)Qk(±)+αQk-1(+)01p1(μ, μ)fk-1(μ)dμ+Qk-1(-)-10p1(μ, μ)fk-1(μ)dμ,
dQ1(+)dz=-ξ1(+)(α+β)Q1(+)+ξ1(+)αQ0,
dQk(+)dz=-ξk(+)(α+β)Qk(+)+akξk-1(+)σk-1(+)αQk-1(+)+bkξk-1(-)[1-σk-1(-)]αQk-1(-),
-dQ1(-)dz=-ξ1(-)(α+β)Q1(-)+ξ1(-)αQ0,
-dQk(-)dz=-ξk(-)(α+β)Qk(-)+ckξk-1(+)[1-σk-1(+)]αQk-1(+)+dkξk-1(-)σk-1(-)αQk-1(-),
ak=σk(+)wk-1(+)σk-1(+)wk(+),bk=[1-σk(-)]wk-1(-)[1-σk-1(-)]wk(+),
ck=[1-σk(+)]wk-1(+)[1-σk-1(+)]wk(-),dk=σk(-)wk-1(-)σk-1(-)wk(-),
wk+=01μfk(μ)dμ=14 1+2ω13ω0k+2n=2(2n+1)hn[Ψn]k,
wk(-)=--10μfk(μ)dμ=14 1-2ω13ω0k+2n=2(2n+1)hn[Ψn]k.
Q1(+)(τ)={1-exp[-(ξ1(+)-1)τ]} ξ1(+)α exp(-τ)(α+β)(ξ1(+)-1),
Q1(-)(τ)=ξ1(-)α exp(-τ)(α+β)(ξ1(-)+1).
pk(+)(τ)=Fk(+)[1-exp(-(ξk(+)-1)τ)]+i=1k-1Gi,k(+)×[exp(-(ξk(+)-1)τ)-exp(-(ξi(+)-1)τ)],
pk(-)(τ)=Fk(-)-i=1k-1Gi,k(-) exp(-(ξi(+)-1)τ),
Fk(+)=ω0akξk-1(+)σk-1(+)Fk-1(+)+bkξk-1(-)(1-σk-1(-))Fk-1(-)ξk(+)-1,
Fk(-)=ω0ckξk-1(+)(1-σk-1(+))Fk-1(+)+dkξk-1(-)σk-1(-)Fk-1(-)ξk(-)+1,
Gk-1,k(+)=ω0akξk-1(+)σk-1(+)Fk-1(+)-i=1k-2 G1,k-1(+)ξk(+)-ξk-1(+),
Gk-1,k(-)=ω0ckξk-1(+)[1-σk-1(+)]Fk-1(+)-i=1k-2Gi,k-1(+)ξk(-)+ξk-1(+),
Gi,k(+)=ω0akξk-1(+)σk-1(+)Gi,k-1(+)+bkξk-1(-)[1-σk-1(-)]Gi,k-1(-)ξk(+)-ξi(+),
 
Gi,k(-)=ω0ckξk-1(+)(1-σk-1(+))Gi,k-1(+)+dkξk-1(-)σk-1(-)Gi,k-1(-)ξk(-)+ξi(+),
 
F1(+)=ω0ξ1(+)ξ1(+)-1,F1(-)=ω0ξ1(-)ξ1(-)+1,
G1,2(+)=ω0a2ξ1(+)σ1(+)ξ2(+)-ξ1(+) F1(+),
G1,2(-)=ω0c2ξ1(+)(1-σ1(+))ξ2(-)+ξ1(+) F1(+).
pk(+)(τ)=Fk(+)[1-exp(-(ξk(+)-1)τ)],
pk(-)(τ)=Fk(-)-Gk(-) exp(-(ξk(+)-1)τ).
Gk(-)=ω0ckξk-1(+)(1-σk-1(+))Fk-1(+)+dkξk-1(-)σk-1(-)Gk-1(-)ξk(-)+ξk(+),
Gk-1(-)=-akξk-1(+)σk-1(+)bkξk-1(-)(1-σk-1(-)) Fk-1(+).
I(z, μ)=k=1Qk(+)(z)fk(μ)n=0cn(+)(z)Pn(μ),
J(z, μ)=k=1Qk(-)(z)fk(μ)n=0cn(-)(z)Pn(μ),
cn(±)(z)=2n+14π k=1 Qk(±)(z)k! ωn/ω02n+1k.
i(z)=2π01I(z, μ)dμ=2πc0(+)+n=1cn(+)gn,
j(z)=2π-10J(z, μ)dμ=2πc0(-)-n=1cn(-)gn,
q(+)(z)=2π01μI(z, μ)dμ=2πc0(+)2+c1(+)3+n=2cn(+)χn1,
q(-)(z)=-2π-10μJ(z, μ)dμ=2πc0(-)2-c1(-)3+n=2cn(-)χn1.
q(+)(z)=b0(+)2+b1(+)3+n=2bn(+)χn1exp(-βz),
q(-)(z)=b0(-)2-b1(-)3+n=2bn(-)χn1exp(-βz),
bn(±)(z)=(2n+1)exp(-αz)2(α+β)× k=1 ξk(±)pk(±)(z)ak(±)k!(ξk(±)-(±1)) ωn/ω02n+1k.
q(±)(z)=k=1qk(±)(z),
qk(+)(z)=2πQk(+)(z)01μfk(μ)dμ
qk(-)(z)=-2πQk(-)(z)-10μfk(μ)dμ.
dq1(+)dz=-ξ1(+)(α+β)q1(+)+ασ1(+)q0,
dqk(+)dz=-ξk(+)(α+β)qk(+)+ξk-1(+)σk(+)αqk-1(+)+ξk-1(-)(1-σk(-))αqk-1(-),
-dq1(-)dz=-ξ1(-)(α+β)q1(-)+ασ1(-)q0,
-dqk(-)dz=-ξk(-)(α+β)qk(-)+ξk-1(+)(1-σk(+))αqk-1(+)+ξk-1(-)σk(-)αqk-1(-),
dq1(+)dz=-(α+β)q1(+)+αv1q0,
dqk(+)dz=-(α+β)qk(+)+vkαqk-1(+)+hkαqk-1(-),
-dq1(-)dz=-(α+β)q1(-)+αh1q0,
-dqk(-)dz=-(α+β)qk(-)+hkαqk-1(+)+vkαqk-1(-),
vk=2π01fk(μ)dμ,hk=2π-10fk(μ)dμ

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