Abstract

We consider the multiple-scattering and forward-scattering corrections to the transmission measurements in the case of a detector with a variable field of view. The transmission functions for predicting the angular distribution of forward-scattering transmittance are proposed. We present results of measurements of transmission functions for polystyrene spheres. Numerical estimations of the forward-multiple-scattering corrections are in good agreement with the experimental results.

© 1998 Optical Society of America

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References

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  1. P. C. Ariesson, S. A. Self, R. H. Eustis, “Two-wavelength laser transmissometer for measurements of the mean size and concentration of coal ash droplets in combustion flows,” Appl. Opt. 19, 3775–3781 (1980).
    [CrossRef]
  2. K. L. Cashdollar, C. K. Lee, J. M. Singer, “Three-wavelength light transmission technique to measure smoke particle size and concentration,” Appl. Opt. 18, 1763–1769 (1979).
    [CrossRef] [PubMed]
  3. C. S. Wang, J. S. Lindner, “Investigations of particle size and number density in advanced energy systems,” J. Propul. Power 6, 552–558 (1990).
    [CrossRef]
  4. F. D. Bryant, P. Latimer, “Real-time particle sizing by a computer-controlled transmittance photometer,” Appl. Opt. 24, 4280–4282 (1985).
    [CrossRef] [PubMed]
  5. A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “Analysis of particles sizes, concentration, and refractive index in measurements of light transmittance in the forward-scattering-angle range,” Appl. Opt. 36, 1357–1366 (1997).
    [CrossRef] [PubMed]
  6. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  7. J. B. Guidt, G. Gouesbet, J. N. Le Toulouzan, “Accurate validation of visible infrared double extinction simultaneous measurements of particle sizes and number densities by using densely laden standard media,” Appl. Opt. 29, 1011–1022 (1990).
    [CrossRef] [PubMed]
  8. A. Zardecki, W. G. Tam, “Multiple scattering corrections to the Beer-Lambert law. 1: Open detector,” Appl. Opt. 21, 2405–2412 (1982).
    [CrossRef] [PubMed]
  9. W. G. Tam, A. Zardecki, “Multiple scattering corrections to the Beer-Lambert law. 2: Detector with a variable field of view,” Appl. Opt. 21, 2413–2420 (1982).
    [CrossRef] [PubMed]
  10. G. Zaccanti, P. Bruscaglioni, “Method of measuring the phase function of a turbid medium in the small scattering angle range,” Appl. Opt. 28, 2156–2164 (1989).
    [CrossRef] [PubMed]
  11. A. Deepak, M. A. Box, “Forwardscattering corrections for optical extinction measurements in aerosol media. 1: Monodispersions,” Appl. Opt. 17, 2900–2908 (1978).
    [CrossRef] [PubMed]
  12. A. Deepak, M. A. Box, “Forwardscattering corrections for optical extinction measurements in aerosol media. 2: Polydispersions,” Appl. Opt. 17, 3169–3176 (1978).
    [CrossRef] [PubMed]
  13. H. C. van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications (Academic, New York, 1980), Vol. 1, p. 2.
  14. A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “Analysis of radiant energy emission from high temperature medium with scattering and absorbing particles,” J. Quant. Spectrosc. Radiat. Transfer 54, 453–470 (1995).
    [CrossRef]
  15. O. S. Vaulina, A. P. Nefedov, O. F. Petrov, “Calculation of the scattering and absorbing characteristics of spherical particles in high-temperature flows,” High Temp. 30, 817–825 (1992).
  16. O. S. Vaulina, O. F. Petrov, A. A. Samarian, A. V. Chernyschev, “Extinction measurements in optical dense medium at different aperture angles of photodetector,” High Temp. (USSR) 33, 707–713 (1995).

1997

1995

A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “Analysis of radiant energy emission from high temperature medium with scattering and absorbing particles,” J. Quant. Spectrosc. Radiat. Transfer 54, 453–470 (1995).
[CrossRef]

O. S. Vaulina, O. F. Petrov, A. A. Samarian, A. V. Chernyschev, “Extinction measurements in optical dense medium at different aperture angles of photodetector,” High Temp. (USSR) 33, 707–713 (1995).

1992

O. S. Vaulina, A. P. Nefedov, O. F. Petrov, “Calculation of the scattering and absorbing characteristics of spherical particles in high-temperature flows,” High Temp. 30, 817–825 (1992).

1990

1989

1985

1982

1980

1979

1978

Ariesson, P. C.

Bohren, C. F.

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

Box, M. A.

Bruscaglioni, P.

Bryant, F. D.

Cashdollar, K. L.

Chernyschev, A. V.

O. S. Vaulina, O. F. Petrov, A. A. Samarian, A. V. Chernyschev, “Extinction measurements in optical dense medium at different aperture angles of photodetector,” High Temp. (USSR) 33, 707–713 (1995).

Deepak, A.

Eustis, R. H.

Gouesbet, G.

Guidt, J. B.

Huffman, D. R.

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

Latimer, P.

Le Toulouzan, J. N.

Lee, C. K.

Lindner, J. S.

C. S. Wang, J. S. Lindner, “Investigations of particle size and number density in advanced energy systems,” J. Propul. Power 6, 552–558 (1990).
[CrossRef]

Nefedov, A. P.

A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “Analysis of particles sizes, concentration, and refractive index in measurements of light transmittance in the forward-scattering-angle range,” Appl. Opt. 36, 1357–1366 (1997).
[CrossRef] [PubMed]

A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “Analysis of radiant energy emission from high temperature medium with scattering and absorbing particles,” J. Quant. Spectrosc. Radiat. Transfer 54, 453–470 (1995).
[CrossRef]

O. S. Vaulina, A. P. Nefedov, O. F. Petrov, “Calculation of the scattering and absorbing characteristics of spherical particles in high-temperature flows,” High Temp. 30, 817–825 (1992).

Petrov, O. F.

A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “Analysis of particles sizes, concentration, and refractive index in measurements of light transmittance in the forward-scattering-angle range,” Appl. Opt. 36, 1357–1366 (1997).
[CrossRef] [PubMed]

A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “Analysis of radiant energy emission from high temperature medium with scattering and absorbing particles,” J. Quant. Spectrosc. Radiat. Transfer 54, 453–470 (1995).
[CrossRef]

O. S. Vaulina, O. F. Petrov, A. A. Samarian, A. V. Chernyschev, “Extinction measurements in optical dense medium at different aperture angles of photodetector,” High Temp. (USSR) 33, 707–713 (1995).

O. S. Vaulina, A. P. Nefedov, O. F. Petrov, “Calculation of the scattering and absorbing characteristics of spherical particles in high-temperature flows,” High Temp. 30, 817–825 (1992).

Samarian, A. A.

O. S. Vaulina, O. F. Petrov, A. A. Samarian, A. V. Chernyschev, “Extinction measurements in optical dense medium at different aperture angles of photodetector,” High Temp. (USSR) 33, 707–713 (1995).

Self, S. A.

Singer, J. M.

Tam, W. G.

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications (Academic, New York, 1980), Vol. 1, p. 2.

Vaulina, O. S.

A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “Analysis of particles sizes, concentration, and refractive index in measurements of light transmittance in the forward-scattering-angle range,” Appl. Opt. 36, 1357–1366 (1997).
[CrossRef] [PubMed]

O. S. Vaulina, O. F. Petrov, A. A. Samarian, A. V. Chernyschev, “Extinction measurements in optical dense medium at different aperture angles of photodetector,” High Temp. (USSR) 33, 707–713 (1995).

A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “Analysis of radiant energy emission from high temperature medium with scattering and absorbing particles,” J. Quant. Spectrosc. Radiat. Transfer 54, 453–470 (1995).
[CrossRef]

O. S. Vaulina, A. P. Nefedov, O. F. Petrov, “Calculation of the scattering and absorbing characteristics of spherical particles in high-temperature flows,” High Temp. 30, 817–825 (1992).

Wang, C. S.

C. S. Wang, J. S. Lindner, “Investigations of particle size and number density in advanced energy systems,” J. Propul. Power 6, 552–558 (1990).
[CrossRef]

Zaccanti, G.

Zardecki, A.

Appl. Opt.

P. C. Ariesson, S. A. Self, R. H. Eustis, “Two-wavelength laser transmissometer for measurements of the mean size and concentration of coal ash droplets in combustion flows,” Appl. Opt. 19, 3775–3781 (1980).
[CrossRef]

K. L. Cashdollar, C. K. Lee, J. M. Singer, “Three-wavelength light transmission technique to measure smoke particle size and concentration,” Appl. Opt. 18, 1763–1769 (1979).
[CrossRef] [PubMed]

F. D. Bryant, P. Latimer, “Real-time particle sizing by a computer-controlled transmittance photometer,” Appl. Opt. 24, 4280–4282 (1985).
[CrossRef] [PubMed]

A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “Analysis of particles sizes, concentration, and refractive index in measurements of light transmittance in the forward-scattering-angle range,” Appl. Opt. 36, 1357–1366 (1997).
[CrossRef] [PubMed]

J. B. Guidt, G. Gouesbet, J. N. Le Toulouzan, “Accurate validation of visible infrared double extinction simultaneous measurements of particle sizes and number densities by using densely laden standard media,” Appl. Opt. 29, 1011–1022 (1990).
[CrossRef] [PubMed]

A. Zardecki, W. G. Tam, “Multiple scattering corrections to the Beer-Lambert law. 1: Open detector,” Appl. Opt. 21, 2405–2412 (1982).
[CrossRef] [PubMed]

W. G. Tam, A. Zardecki, “Multiple scattering corrections to the Beer-Lambert law. 2: Detector with a variable field of view,” Appl. Opt. 21, 2413–2420 (1982).
[CrossRef] [PubMed]

G. Zaccanti, P. Bruscaglioni, “Method of measuring the phase function of a turbid medium in the small scattering angle range,” Appl. Opt. 28, 2156–2164 (1989).
[CrossRef] [PubMed]

A. Deepak, M. A. Box, “Forwardscattering corrections for optical extinction measurements in aerosol media. 1: Monodispersions,” Appl. Opt. 17, 2900–2908 (1978).
[CrossRef] [PubMed]

A. Deepak, M. A. Box, “Forwardscattering corrections for optical extinction measurements in aerosol media. 2: Polydispersions,” Appl. Opt. 17, 3169–3176 (1978).
[CrossRef] [PubMed]

High Temp.

O. S. Vaulina, A. P. Nefedov, O. F. Petrov, “Calculation of the scattering and absorbing characteristics of spherical particles in high-temperature flows,” High Temp. 30, 817–825 (1992).

High Temp. (USSR)

O. S. Vaulina, O. F. Petrov, A. A. Samarian, A. V. Chernyschev, “Extinction measurements in optical dense medium at different aperture angles of photodetector,” High Temp. (USSR) 33, 707–713 (1995).

J. Propul. Power

C. S. Wang, J. S. Lindner, “Investigations of particle size and number density in advanced energy systems,” J. Propul. Power 6, 552–558 (1990).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “Analysis of radiant energy emission from high temperature medium with scattering and absorbing particles,” J. Quant. Spectrosc. Radiat. Transfer 54, 453–470 (1995).
[CrossRef]

Other

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

H. C. van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications (Academic, New York, 1980), Vol. 1, p. 2.

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Figures (7)

Fig. 1
Fig. 1

Optical scheme of the transmission measurements: 1, laser source; 2, particles; 3, aperture diaphragm; 4, aperture lens; 5, photodetector.

Fig. 2
Fig. 2

Angular distributions of the relative cross sections q d ) = σ(θ d )/σext for latex spheres in water: 1, r = 1; 2, r = 2; 3, r = 1.5 (λ = 0.633 μm).

Fig. 3
Fig. 3

Experimental setup: LG, laser; AD, aperture diaphragms; RD, rotating disk; LC, lens condenser; PRU, photoreceiver unit; IF, narrow bandwidth filter; BSP, beam-splitter glass plate; PD b , PD a , and PD r , photodiodes. GP, glass plate.

Fig. 4
Fig. 4

Measured optical depth τ = N p σ(θ d )l versus dilution extent N d and number density N p of particles with diameters of (a) d m = 0.442 μm and (b) d m = 2.032 μm for different angles of the detector’s field of view: ◆, θ d = 0.39°; ■, θ d = 1.36°; ▲, θ d = 2.04°; , θ d = 3.09°; , θ d = 4.64°; ●, θ d = 5.83°.

Fig. 5
Fig. 5

Measured dependencies of q ms d ) (dashed curves) on the detector angles θ d for particles with a diameter of d m = 1.306 μm at different optical depths: 1, τ = 0.32; 2, τ = 1.02; 3, τ = 2.95; 4, τ = 5.01. The solid curves mark q ms d ) calculated from Eq. (6) by use of C q = C q min.

Fig. 6
Fig. 6

Measured correction factors C q meas (dashed curves) and the calculated values of C q min (solid curves) versus optical depth τ for particles with diameters of (a) d m = 0.442 μm, (b) d m = 0.944 μm, and (c) d m = 2.032 μm and for different values of the detector angles θ d : 1, θ d = 1.36°; 2, θ d = 3.09°; 3, θ d = 5.83°.

Fig. 7
Fig. 7

Measured correction factors C q meas versus detector angles at optical depth τ for particles with a diameter of d m = 2.032 μm: ◆, long 200-mm cell, τ = 3.09; ■, short cell, τ = 2.95.

Tables (3)

Tables Icon

Table 1 Measured Correction Factors Cq meas and the Calculated Values of Cq = Cq min for Latex Particles with Different Diameters dm (Short Cell)

Tables Icon

Table 2 Measured Correction Factors Cq min and the Calculated Values of Cq = Cqmin for Latex Particles at the Different Detector Angles θ d with τ = 4.95 (Short Cell)

Tables Icon

Table 3 Errors of the Retrieved Diameter dr of Particles with a Manufactured Diameter of dm = 2.032 μm for Measurement Angles of θ d ≤ 3.09° and θ d ≤ 5.83°

Equations (21)

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T I / I 0 = exp - τ ,
σ ¯ ext = 0   σ ext f r d r ,
T 1 = exp - τ * ,
σ * θ d = 0   σ ext f r d r - 1 / 2 × 0   σ s f r d r   0 θ d   p θ sin   θ d θ ,
σ l * θ d = 0   σ ext f r d r - 1 / 2 K n     σ s f r d r × 0 l d x   0 θ x exp - τ x / l   cos   θ p θ sin   θ d θ ,
q θ d σ * θ d / σ ext = τ * / τ .
q ms θ d = q θ d 1 - C q ,
T 2 = T 1 1 + C t ,
C t = 1 - exp - τ ms * / 1 - C q
C t = 1 - exp - τ * C q .
C t C t max = 1 / 1 - b ,
C t C t min = τ   exp - τ s μ 0 1 - q θ d 2 1 - s exp - q θ d τ ,
C q C q min = ln 1 - C t min / q θ d τ .
C q = C q min + Δ .
q ms θ di = ln I b 0 / I r 0 / I b / I r ln I a 0 / I r 0 / I a / I r ,
q ms θ di = q θ di σ * θ di / σ ext ,
N p = N p min / N d ,
C q meas = q θ di ,   τ min - q θ di ,   τ / q θ di ,   τ min ,
ε d = d m - d r / d m .
θ d 0.956 / ρ .
C ql meas = 1 - q ms θ d / q cal θ d .

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