Abstract

Measurements of the angular spectrum of light transmitted through turbid slabs with monodispersions of polystyrene spheres have been performed. The results obtained are compared with theoretical calculations, based on the small-angle approximation of the radiative transfer theory. The experimental data and the theoretical results coincide with a high accuracy, which allows us to develop the laser diffraction spectroscopy of optically thick light-scattering layers.

© 2001 Optical Society of America

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References

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  1. W. Witt, S. Röthele, “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. P. H. Kaye, K. Alexander-Buckley, E. Hirst, S. Saunders, J. M. Clark, “A real time monitoring system for airborne particle shape and size analysis,” J. Geophys. Res. D 101, 19215–19221 (1996).
    [CrossRef]
  4. 2600 Particle Sizer User Manual (Malvern Instruments, Malvern, UK, 1985).
  5. S. M. Puckhaber, S. Röthele, “Laser diffraction: millenium-link for particle size analysis,” Powder Handling Process. 11, 91–95 (1999).
  6. 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]
  7. V. F. Belov, A. G. Borovoi, N. I. Vagin, S. N. Volkov, “On small-angle method under single and multiple light scattering,” Izv. Acad. Nauk SSSR Fiz. Atmos. Okeana 20, 323–327 (1984).
  8. N. I. Vagin, V. V. Veretennikov, “Optical diagnostics of disperse media under multiple scattering in small angle apptoximation,” Izv. Acad. Nauk SSSR Fiz. Atmos. Okeana 25, 723–731 (1989).
  9. E. D. Hirleman, “Modelling of multiple scattering effects in Fraunhofer diffraction particle size analysis,” Part. Part. Syst. Charact. 5, 57–65 (1988).
    [CrossRef]
  10. 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]
  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. L. S. Dolin, “About scattering of light beam in a layer of a turbid medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. (Sov. Radiophys.) 7, 380–382 (1964).
  13. E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).
    [CrossRef]
  14. K. S. Shifrin, Introduction to Ocean Optics (Gidrometeoizdat, Leningrad, 1983).
  15. E. P. Zege, A. A. Kokhanovsky, “Analytical solution to the optical transfer function of a scattering medium with large particles,” Appl. Opt. 33, 6547–6554 (1994).
    [CrossRef] [PubMed]
  16. A. A. Kokhanovsky, Light Scattering Media Optics: Problems and Solutions (Wiley-Praxis, Chichester, UK, 1999).
  17. V. E. Zuev, V. V. Belov, V. V. Veretennikov, Linear Systems Theory in Optics of Disperse Media (Siberian Brunch of the Russian Academy of Sciences, Tomsk, Russia, 1997).
  18. K. S. Shifrin, Scattering of Light in a Turbid Medium (Gostekhteorizdat, Moscow, 1951) (English translation of NASA Tech. Trans. TT F-447, NASA, Washington, D.C., 1968).
  19. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

1999 (1)

S. M. Puckhaber, S. Röthele, “Laser diffraction: millenium-link for particle size analysis,” Powder Handling Process. 11, 91–95 (1999).

1996 (2)

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

P. H. Kaye, K. Alexander-Buckley, E. Hirst, S. Saunders, J. M. Clark, “A real time monitoring system for airborne particle shape and size analysis,” J. Geophys. Res. D 101, 19215–19221 (1996).
[CrossRef]

1995 (1)

1994 (1)

1993 (1)

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

1991 (1)

1989 (1)

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

1988 (1)

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

1984 (1)

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

1964 (1)

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

Alexander-Buckley, K.

P. H. Kaye, K. Alexander-Buckley, E. Hirst, S. Saunders, J. M. Clark, “A real time monitoring system for airborne particle shape and size analysis,” J. Geophys. Res. D 101, 19215–19221 (1996).
[CrossRef]

Bayvel, L. P.

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. Acad. Nauk SSSR Fiz. Atmos. Okeana 20, 323–327 (1984).

Belov, V. V.

V. E. Zuev, V. V. Belov, V. V. Veretennikov, Linear Systems Theory in Optics of Disperse Media (Siberian Brunch of the Russian Academy of Sciences, Tomsk, Russia, 1997).

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. Acad. Nauk SSSR Fiz. Atmos. Okeana 20, 323–327 (1984).

Clark, J. M.

P. H. Kaye, K. Alexander-Buckley, E. Hirst, S. Saunders, J. M. Clark, “A real time monitoring system for airborne particle shape and size analysis,” J. Geophys. Res. D 101, 19215–19221 (1996).
[CrossRef]

Dolin, L. S.

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

Glatter, O.

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

Hirst, E.

P. H. Kaye, K. Alexander-Buckley, E. Hirst, S. Saunders, J. M. Clark, “A real time monitoring system for airborne particle shape and size analysis,” J. Geophys. Res. D 101, 19215–19221 (1996).
[CrossRef]

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]

Kaye, P. H.

P. H. Kaye, K. Alexander-Buckley, E. Hirst, S. Saunders, J. M. Clark, “A real time monitoring system for airborne particle shape and size analysis,” J. Geophys. Res. D 101, 19215–19221 (1996).
[CrossRef]

Knight, J.

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.

Puckhaber, S. M.

S. M. Puckhaber, S. Röthele, “Laser diffraction: millenium-link for particle size analysis,” Powder Handling Process. 11, 91–95 (1999).

Robertson, G.

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]

Röthele, S.

S. M. Puckhaber, S. Röthele, “Laser diffraction: millenium-link for particle size analysis,” Powder Handling Process. 11, 91–95 (1999).

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

Saunders, S.

P. H. Kaye, K. Alexander-Buckley, E. Hirst, S. Saunders, J. M. Clark, “A real time monitoring system for airborne particle shape and size analysis,” J. Geophys. Res. D 101, 19215–19221 (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]

K. S. Shifrin, Introduction to Ocean Optics (Gidrometeoizdat, Leningrad, 1983).

K. S. Shifrin, Scattering of Light in a Turbid Medium (Gostekhteorizdat, Moscow, 1951) (English translation of NASA Tech. Trans. TT F-447, NASA, Washington, D.C., 1968).

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. I.

N. I. Vagin, V. V. Veretennikov, “Optical diagnostics of disperse media under multiple scattering in small angle apptoximation,” Izv. Acad. 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. Acad. Nauk SSSR Fiz. Atmos. Okeana 20, 323–327 (1984).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Veretennikov, V. V.

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

V. E. Zuev, V. V. Belov, V. V. Veretennikov, Linear Systems Theory in Optics of Disperse Media (Siberian Brunch of the Russian Academy of Sciences, Tomsk, Russia, 1997).

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. Acad. Nauk SSSR Fiz. Atmos. Okeana 20, 323–327 (1984).

Witt, W.

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

Zege, E. P.

Zuev, V. E.

V. E. Zuev, V. V. Belov, V. V. Veretennikov, Linear Systems Theory in Optics of Disperse Media (Siberian Brunch of the Russian Academy of Sciences, Tomsk, Russia, 1997).

Adv. Geophys. (1)

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. (3)

Izv. Acad. Nauk SSSR Fiz. Atmos. Okeana (2)

V. F. Belov, A. G. Borovoi, N. I. Vagin, S. N. Volkov, “On small-angle method under single and multiple light scattering,” Izv. Acad. 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 apptoximation,” Izv. Acad. Nauk SSSR Fiz. Atmos. Okeana 25, 723–731 (1989).

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (Sov. Radiophys.) (1)

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

J. Geophys. Res. D (1)

P. H. Kaye, K. Alexander-Buckley, E. Hirst, S. Saunders, J. M. Clark, “A real time monitoring system for airborne particle shape and size analysis,” J. Geophys. Res. D 101, 19215–19221 (1996).
[CrossRef]

Part. Part. Syst. Charact. (2)

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

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

Powder Handling Process. (1)

S. M. Puckhaber, S. Röthele, “Laser diffraction: millenium-link for particle size analysis,” Powder Handling Process. 11, 91–95 (1999).

Other (8)

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]

2600 Particle Sizer User Manual (Malvern Instruments, Malvern, UK, 1985).

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

K. S. Shifrin, Introduction to Ocean Optics (Gidrometeoizdat, Leningrad, 1983).

A. A. Kokhanovsky, Light Scattering Media Optics: Problems and Solutions (Wiley-Praxis, Chichester, UK, 1999).

V. E. Zuev, V. V. Belov, V. V. Veretennikov, Linear Systems Theory in Optics of Disperse Media (Siberian Brunch of the Russian Academy of Sciences, Tomsk, Russia, 1997).

K. S. Shifrin, Scattering of Light in a Turbid Medium (Gostekhteorizdat, Moscow, 1951) (English translation of NASA Tech. Trans. TT F-447, NASA, Washington, D.C., 1968).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

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Figures (6)

Fig. 1
Fig. 1

Experimental setup.

Fig. 2
Fig. 2

Angular dependence of the light transmitted through a sample with polystyrene spherical particles according to the experiment and the theory at the optical thickness equal to 0.144, 0.463, and 3.108. The diameter of spheres is equal to 9.685 µm.

Fig. 3
Fig. 3

Same as Fig. 2 but with diameter of spheres of 100 µm and optical thickness equal to 0.06, 3.257, and 5.345.

Fig. 4
Fig. 4

Dependence F(z) at different values of optical thickness equal to 0.1, 1, 3, and 5.

Fig. 5
Fig. 5

Dependence of transmitted diffused light intensity [see Eqs. (15) and (16)] on optical thickness at z = 1.5.

Fig. 6
Fig. 6

Angular dependence of light intensity according to the experiment and the theory for samples with polystyrene spheres of 100-µm diameter with accounting for the geometrical optical scattering at optical thickness equal to 3.257 and 5.345.

Equations (25)

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Iθ=Cθ-20 a2faJ12kaθda,
c=defτ/3L.
θd=3.83×180πx
cos ϑ dIτ, ϑdτ=-Iτ, ϑ+ω020π Iτ, ϑpϑ, ϑ×sin ϑdϑ,
cos ϑ dIτ, ϑdτdIτ, ϑdτ.
dIτ, ϑdτ=-Iτ, ϑ+ω020π Iτ, ϑpϑ, ϑsin ϑdϑ.
dIτ, ϑdτ=-Iτ, ϑ+ω020 Iτ, ϑpϑ, ϑϑdϑ,
120 pθθdθ=1.
Iτ, ϑ=I02π0exp-τ1-ω0PσJ0σϑσdσ,
I0, ϑ=I0δϑ,
δϑ=12π0 J0σϑσdσ
Pσ=120 pθJ0σθθdθ
Idτ, ϑ=I02πexp-τ0expω0Pστ-1×J0σϑσdσ.
Idτ, ϑ=I0/4πω0τpϑ,
pϑ=2 0 PσJ0σϑσdσ.
ω0=12,  pθ=4J12θxθ2.
Pσ=2πarccosσ2x-σ2x1-σ2x21/2uσ2x,
Idτ, ϑ=DFτ, α,
Fτ, z=exp-τ01expτπarccosy-y1-y21/2-1J0yzydy
pθ=σscadpdθ+σscagpgθσscad+σscag,
pθ=pdθ+pgθ2.
pgθ=4β exp-βθ2,
Pσ=Pdσ+Pgσ2,
Pgσ=exp-σ2/4β.
Pσ=0 a2faPσda0 a2fada,

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