Abstract

Multiple light scattering is an important issue in modern laser diffraction spectrometry. Most laser particle sizers do not account for multiple light scattering in a disperse medium under investigation. This causes an underestimation of the particle sizes in the case of high concentrations of scatterers. The retrieval accuracy is improved if the measured data are processed with multiple-scattering algorithms that treat multiple light scattering in a disperse medium. We evaluate the influence of multiple light scattering on light transmitted by scattering layers. The relationships among different theories to account for multiple light scattering in laser particle sizing are considered.

© 2001 Optical Society of America

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References

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  1. W. Witt, S. Rothele, “Laser diffraction—unlimited,” Part. Part. Syst. Charact. 13, 280–286 (1996).
    [CrossRef]
  2. K. S. Shifrin, G. Tonna, “Inverse problems related to light scattering in the atmosphere and ocean,” Adv. Geophys. 34, 175–252 (1993).
    [CrossRef]
  3. E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).
    [CrossRef]
  4. J. A. Lock, C. L. Chin, “Correlated light scattering by a dense distribution of condensation droplets on a window pane,” Appl. Opt. 33, 4663–4671 (1994).
    [CrossRef] [PubMed]
  5. A. A. Kokhanovsky, “On light scattering in random media with large densely packed particles,” J. Geophys. Res. D 103, 6089–6096 (1998).
    [CrossRef]
  6. P. G. Felton, A. A. Hamidi, A. K. Aigal, “Multiple scattering effects on particle sizing by laser diffraction,” (Department of Chemical Engineering, University of Sheffield, England, 1984).
  7. E. D. Hirleman, “Modeling of multiple scattering effects in Fraunhofer diffraction particle size analysis,” Part. Part. Syst. Charact. 5, 57–65 (1988).
    [CrossRef]
  8. E. D. Hirleman, “General solution to the inverse near-forward-scattering particle-sizing problem in multiple-scattering environments: theory,” Appl. Opt. 30, 4832–4838 (1991).
    [CrossRef] [PubMed]
  9. V. F. Belov, A. G. Borovoi, N. I. Vagin, S. N. Volkov, “On small-angle method under single and multiple light scattering,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 20, 323–327 (1984).
  10. N. I. Vagin, V. V. Veretennikov, “Optical diagnostics of disperse media under multiple scattering in small angle approximation,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 25, 723–731 (1989).
  11. H. Schnablegger, O. Glatter, “Sizing of colloidal particles with light scattering: corrections to beginning multiple scattering,” Appl. Opt. 34, 3489–3501 (1995).
    [CrossRef] [PubMed]
  12. W. Hartel, “Zur Theorie der Lichtstreuung durch trube Schichten, besonders Trubglaser,” Licht 10, 141–143, 165, 190–191, 214–215, 232–234 (1940).
  13. L. S. Dolin, “About scattering of light beam in a layer of a turbid medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 380–382 (1964).
  14. A. Borovoi, P. Bruscaglioni, A. Ismaeli, N. Vagin, V. Veretennikov, “Multiple Fraunhofer diffraction in optical particle sizing,” in Proceedings of the Fourth International Congress on Optical Particle Sizing, F. Durst, J. Domnick eds. (Nurnberg Messe GmbH, 1995), pp. 69–78.
  15. 2600 Particle Sizer User Manual (Malvern Instuments, Malvern, England, 1985).
  16. L. P. Bayvel, J. Knight, G. Robertson, “Alternative model-independent inversion program for Malvern particle sizer,” Part. Part. Syst. Charact. 4, 49–53 (1987).
    [CrossRef]
  17. L. P. Bayvel, J. Knight, G. Robertson, “Application of the Shifrin inversion to the Malvern particle sizer,” in Optical Particle Sizing, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988), pp. 311–319.
    [CrossRef]

1998

A. A. Kokhanovsky, “On light scattering in random media with large densely packed particles,” J. Geophys. Res. D 103, 6089–6096 (1998).
[CrossRef]

1996

W. Witt, S. Rothele, “Laser diffraction—unlimited,” Part. Part. Syst. Charact. 13, 280–286 (1996).
[CrossRef]

1995

1994

1993

K. S. Shifrin, G. Tonna, “Inverse problems related to light scattering in the atmosphere and ocean,” Adv. Geophys. 34, 175–252 (1993).
[CrossRef]

1991

1989

N. I. Vagin, V. V. Veretennikov, “Optical diagnostics of disperse media under multiple scattering in small angle approximation,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 25, 723–731 (1989).

1988

E. D. Hirleman, “Modeling of multiple scattering effects in Fraunhofer diffraction particle size analysis,” Part. Part. Syst. Charact. 5, 57–65 (1988).
[CrossRef]

1987

L. P. Bayvel, J. Knight, G. Robertson, “Alternative model-independent inversion program for Malvern particle sizer,” Part. Part. Syst. Charact. 4, 49–53 (1987).
[CrossRef]

1984

V. F. Belov, A. G. Borovoi, N. I. Vagin, S. N. Volkov, “On small-angle method under single and multiple light scattering,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 20, 323–327 (1984).

1964

L. S. Dolin, “About scattering of light beam in a layer of a turbid medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 380–382 (1964).

1940

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

Aigal, A. K.

P. G. Felton, A. A. Hamidi, A. K. Aigal, “Multiple scattering effects on particle sizing by laser diffraction,” (Department of Chemical Engineering, University of Sheffield, England, 1984).

Bayvel, L. P.

L. P. Bayvel, J. Knight, G. Robertson, “Alternative model-independent inversion program for Malvern particle sizer,” Part. Part. Syst. Charact. 4, 49–53 (1987).
[CrossRef]

L. P. Bayvel, J. Knight, G. Robertson, “Application of the Shifrin inversion to the Malvern particle sizer,” in Optical Particle Sizing, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988), pp. 311–319.
[CrossRef]

Belov, V. F.

V. F. Belov, A. G. Borovoi, N. I. Vagin, S. N. Volkov, “On small-angle method under single and multiple light scattering,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 20, 323–327 (1984).

Borovoi, A.

A. Borovoi, P. Bruscaglioni, A. Ismaeli, N. Vagin, V. Veretennikov, “Multiple Fraunhofer diffraction in optical particle sizing,” in Proceedings of the Fourth International Congress on Optical Particle Sizing, F. Durst, J. Domnick eds. (Nurnberg Messe GmbH, 1995), pp. 69–78.

Borovoi, A. G.

V. F. Belov, A. G. Borovoi, N. I. Vagin, S. N. Volkov, “On small-angle method under single and multiple light scattering,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 20, 323–327 (1984).

Bruscaglioni, P.

A. Borovoi, P. Bruscaglioni, A. Ismaeli, N. Vagin, V. Veretennikov, “Multiple Fraunhofer diffraction in optical particle sizing,” in Proceedings of the Fourth International Congress on Optical Particle Sizing, F. Durst, J. Domnick eds. (Nurnberg Messe GmbH, 1995), pp. 69–78.

Chin, C. L.

Dolin, L. S.

L. S. Dolin, “About scattering of light beam in a layer of a turbid medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 380–382 (1964).

Felton, P. G.

P. G. Felton, A. A. Hamidi, A. K. Aigal, “Multiple scattering effects on particle sizing by laser diffraction,” (Department of Chemical Engineering, University of Sheffield, England, 1984).

Glatter, O.

Hamidi, A. A.

P. G. Felton, A. A. Hamidi, A. K. Aigal, “Multiple scattering effects on particle sizing by laser diffraction,” (Department of Chemical Engineering, University of Sheffield, England, 1984).

Hartel, W.

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

Hirleman, E. D.

E. D. Hirleman, “General solution to the inverse near-forward-scattering particle-sizing problem in multiple-scattering environments: theory,” Appl. Opt. 30, 4832–4838 (1991).
[CrossRef] [PubMed]

E. D. Hirleman, “Modeling of multiple scattering effects in Fraunhofer diffraction particle size analysis,” Part. Part. Syst. Charact. 5, 57–65 (1988).
[CrossRef]

Ismaeli, A.

A. Borovoi, P. Bruscaglioni, A. Ismaeli, N. Vagin, V. Veretennikov, “Multiple Fraunhofer diffraction in optical particle sizing,” in Proceedings of the Fourth International Congress on Optical Particle Sizing, F. Durst, J. Domnick eds. (Nurnberg Messe GmbH, 1995), pp. 69–78.

Ivanov, A. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).
[CrossRef]

Katsev, I. L.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).
[CrossRef]

Knight, J.

L. P. Bayvel, J. Knight, G. Robertson, “Alternative model-independent inversion program for Malvern particle sizer,” Part. Part. Syst. Charact. 4, 49–53 (1987).
[CrossRef]

L. P. Bayvel, J. Knight, G. Robertson, “Application of the Shifrin inversion to the Malvern particle sizer,” in Optical Particle Sizing, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988), pp. 311–319.
[CrossRef]

Kokhanovsky, A. A.

A. A. Kokhanovsky, “On light scattering in random media with large densely packed particles,” J. Geophys. Res. D 103, 6089–6096 (1998).
[CrossRef]

Lock, J. A.

Robertson, G.

L. P. Bayvel, J. Knight, G. Robertson, “Alternative model-independent inversion program for Malvern particle sizer,” Part. Part. Syst. Charact. 4, 49–53 (1987).
[CrossRef]

L. P. Bayvel, J. Knight, G. Robertson, “Application of the Shifrin inversion to the Malvern particle sizer,” in Optical Particle Sizing, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988), pp. 311–319.
[CrossRef]

Rothele, S.

W. Witt, S. Rothele, “Laser diffraction—unlimited,” Part. Part. Syst. Charact. 13, 280–286 (1996).
[CrossRef]

Schnablegger, H.

Shifrin, K. S.

K. S. Shifrin, G. Tonna, “Inverse problems related to light scattering in the atmosphere and ocean,” Adv. Geophys. 34, 175–252 (1993).
[CrossRef]

Tonna, G.

K. S. Shifrin, G. Tonna, “Inverse problems related to light scattering in the atmosphere and ocean,” Adv. Geophys. 34, 175–252 (1993).
[CrossRef]

Vagin, N.

A. Borovoi, P. Bruscaglioni, A. Ismaeli, N. Vagin, V. Veretennikov, “Multiple Fraunhofer diffraction in optical particle sizing,” in Proceedings of the Fourth International Congress on Optical Particle Sizing, F. Durst, J. Domnick eds. (Nurnberg Messe GmbH, 1995), pp. 69–78.

Vagin, N. I.

N. I. Vagin, V. V. Veretennikov, “Optical diagnostics of disperse media under multiple scattering in small angle approximation,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 25, 723–731 (1989).

V. F. Belov, A. G. Borovoi, N. I. Vagin, S. N. Volkov, “On small-angle method under single and multiple light scattering,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 20, 323–327 (1984).

Veretennikov, V.

A. Borovoi, P. Bruscaglioni, A. Ismaeli, N. Vagin, V. Veretennikov, “Multiple Fraunhofer diffraction in optical particle sizing,” in Proceedings of the Fourth International Congress on Optical Particle Sizing, F. Durst, J. Domnick eds. (Nurnberg Messe GmbH, 1995), pp. 69–78.

Veretennikov, V. V.

N. I. Vagin, V. V. Veretennikov, “Optical diagnostics of disperse media under multiple scattering in small angle approximation,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 25, 723–731 (1989).

Volkov, S. N.

V. F. Belov, A. G. Borovoi, N. I. Vagin, S. N. Volkov, “On small-angle method under single and multiple light scattering,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 20, 323–327 (1984).

Witt, W.

W. Witt, S. Rothele, “Laser diffraction—unlimited,” Part. Part. Syst. Charact. 13, 280–286 (1996).
[CrossRef]

Zege, E. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).
[CrossRef]

Adv. Geophys.

K. S. Shifrin, G. Tonna, “Inverse problems related to light scattering in the atmosphere and ocean,” Adv. Geophys. 34, 175–252 (1993).
[CrossRef]

Appl. Opt.

Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana

V. F. Belov, A. G. Borovoi, N. I. Vagin, S. N. Volkov, “On small-angle method under single and multiple light scattering,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 20, 323–327 (1984).

N. I. Vagin, V. V. Veretennikov, “Optical diagnostics of disperse media under multiple scattering in small angle approximation,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 25, 723–731 (1989).

Izv. Vyssh. Uchebn. Zaved. Radiofiz.

L. S. Dolin, “About scattering of light beam in a layer of a turbid medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 380–382 (1964).

J. Geophys. Res. D

A. A. Kokhanovsky, “On light scattering in random media with large densely packed particles,” J. Geophys. Res. D 103, 6089–6096 (1998).
[CrossRef]

Licht

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

Part. Part. Syst. Charact.

L. P. Bayvel, J. Knight, G. Robertson, “Alternative model-independent inversion program for Malvern particle sizer,” Part. Part. Syst. Charact. 4, 49–53 (1987).
[CrossRef]

W. Witt, S. Rothele, “Laser diffraction—unlimited,” Part. Part. Syst. Charact. 13, 280–286 (1996).
[CrossRef]

E. D. Hirleman, “Modeling of multiple scattering effects in Fraunhofer diffraction particle size analysis,” Part. Part. Syst. Charact. 5, 57–65 (1988).
[CrossRef]

Other

P. G. Felton, A. A. Hamidi, A. K. Aigal, “Multiple scattering effects on particle sizing by laser diffraction,” (Department of Chemical Engineering, University of Sheffield, England, 1984).

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).
[CrossRef]

L. P. Bayvel, J. Knight, G. Robertson, “Application of the Shifrin inversion to the Malvern particle sizer,” in Optical Particle Sizing, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988), pp. 311–319.
[CrossRef]

A. Borovoi, P. Bruscaglioni, A. Ismaeli, N. Vagin, V. Veretennikov, “Multiple Fraunhofer diffraction in optical particle sizing,” in Proceedings of the Fourth International Congress on Optical Particle Sizing, F. Durst, J. Domnick eds. (Nurnberg Messe GmbH, 1995), pp. 69–78.

2600 Particle Sizer User Manual (Malvern Instuments, Malvern, England, 1985).

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Figures (5)

Fig. 1
Fig. 1

Fourier–Bessel transform of the Fraunhofer phase function [Eq. (24)] obtained with the exact solution [Eq. (26)] and the approximate formula [Eq. (34)].

Fig. 2
Fig. 2

Dependence of the normalized intensity on parameter b = 2kaϑ obtained with Eq. (28) at different values of optical thickness τ = 0.01, 1, 2, 5, 7.

Fig. 3
Fig. 3

Dependence of half-width h of the angular spectrum of the transmitted light on optical thickness τ.

Fig. 4
Fig. 4

Dependence of the optical thickness on obscuration.

Fig. 5
Fig. 5

Dependence of the normalized intensity on the observation angle obtained with the Mie theory (Mie) and Eqs. (47) and (43) for the gamma PSD f(a) = Aa 6 exp(-1.5a) at wavelength λ = 0.628 µm, the geometric thickness of a scattering layer L = 4 cm, and different volumetric concentrations of particles c = 0.000001, c = 0.00001, c = 0.0001, c = 0.0002, c = 0.0005, c = 0.0007.

Tables (1)

Tables Icon

Table 1 Dependence of the Half-Width Parameter h of the Angular Distribution of Transmitted Light on Optical Thickness τ for Disperse Media with Monodispersed Spheres

Equations (48)

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Iθ, a=N J12αθa2θ2R2 I0V,
Īθ, a=0 Iθ, aq0ada.
α  1, 2α|m-1|  1,
lL  1,
Dd  1,
cos ϑ dIτ, ϑdτ=-Iτ, ϑ+ω020π Iτ, ϑpθsin ϑdϑ,
θ=arccoscos ϑ cos ϑ+sin ϑ sin ϑ cosϑ-ϑ
cos ϑ dIτ, ϑdτdIτ, ϑdτ,
Iτ, ϑ=I02πj=0 ΞjτPjcos ϑ,
Ξjτ=j+12exp-1-ω0pjτ,
pj=120π pθPjcos θsin θdθ.
I0 exp-τ δcos ϑ-1=I02πexp-τ×j=0j+12Pjcos ϑ,
Iτ, ϑ=I02πj=0 ΞjτPjcos ϑ,
Ξjτ=j+12 e-τexpω0pjτ-1.
Pjcos θ  J0θj+12.
gσ=120 pθJ0σθθdθ,
l=0 fl+120fpdp
Iτ, ϑ=I02π e-τ0expτω0gσ-1J0σϑσdσ.
expτω0gσ=n=0τnω0ngnσn!.
Iτ, ϑ=I02π e-τn=1τnω0nn! Hnϑ,
Hnϑ=0 gnσJ0σϑσdσ.
Iτ, ϑ=I02π τω00 gσJ0σϑσdσ,
Iτ, ϑ=I02π σscaLpϑ,
pϑ=0 gσJ0σϑσdσ
pϑ=4J12ϑρϑ2.
Iτ, ϑ=pϑa2NLI0,
gz=2πarccosz-z1-z21/2z10z>1,
Iτ, ϑ=I02π e-τ0expτgσ/η2-1J0σϑσdσ
Iτ, ϑ=Ce-τ01expτgz2-1J0bzzdz,
ib=h=0.5.
a=hτ4πϑ0 λ.
1-hτ*h0.01ε,
τ=0 τaq0ada,
gσ=B gσ/2kaa2q0ada0 a2q0ada,
gz=1+j=03 sjz2j+1,
gσ=1+j=03 sjφ2j+1, B,
φn, B=BnB a2-nq0ada0 a2q0ada.
gσ=1+j=03 sjφ2j+1, β, φn, β=βnβ y2+μ-n exp-ydy0 y2+μ exp-ydy,
φn, β=βnΓμ+3-n1-Pμ+3-n, βΓμ+3,
Γα=0 tα-1 exp-tdt
Pα, x=1Γα0x tα-1 exp-tdt
Γμ=μ-1!,  Pμ, β=1-exp-βl=0μ-1βll!
φn, β=βn exp-βΓμ+3-nΓμ+3l=0μ+2-nβll!.
gσ=1+j=03l=0μ-2j+1 sjβ2j+l+1×exp-βΓμ-2j+2Γμ+3Γl+1.
τ=1.5cLaeff,
aeff=0 a3q0ada0 a2q0ada
aeff=a01+3μ.
Iϑ=I02πexp-3cL2aeff0exp3cLgσ4-1×J0σϑσdσ.

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