Abstract

Considerable confusion exists regarding the applicability limits of the Bouguer–Lambert–Beer law of optical transmission. We review the derivation of the law and discuss its application to the optical thickness of the light-scattering medium. We demonstrate the range of applicability by presenting a method for determining particle size by measuring optical transmission at two wavelengths.

© 1999 Optical Society of America

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References

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  1. J. Lenoble, Atmospheric Radiative Transfer (Deepak, Hampton, Va., 1993), pp. 234–237.
  2. B. L. Oser, Hawk’s Physiological Chemistry, 14th ed. (McGraw-Hill, New York, 1965), p. 989.
  3. Ref. 1, p. 18.
  4. C. D. Mobley, Light and Water; Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994), p. 254.
  5. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 287.
  6. A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “Analysis of particle sizes, concentration, and refractive index in measurement of light transmittance in the forward-scattering-angle range,” Appl. Opt. 36, 1357–1366 (1997).
    [CrossRef] [PubMed]
  7. A. P. Nefedov, O. F. Petrov, O. S. Vaulina, A. M. Lipaev, “Application of a forward-angle-scattering transmissometer for simultaneous measurements of particle size and number density in an optically dense medium,” Appl. Opt. 37, 1682–1689 (1998).
    [CrossRef]
  8. R. M. Tabibian, W. Heller, J. N. Epel, “Experimental investigations on the light scattering of colloidal spheres,” J. Colloid Sci. 11, 195–213 (1956).
    [CrossRef]
  9. W. G. Tam, A. Zardecki, “Multiple scattering corrections to the Beer–Lambert law. 1: Open detector,” Appl. Opt. 21, 2405–2412 (1982).
    [CrossRef] [PubMed]
  10. A. Zardecki, W. G. Tam, “Multiple scattering corrections to the Beer–Lambert law. 2: Detector with a variable field of view,” Appl. Opt. 21, 2413–2420 (1982).
    [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. S. De Iuliis, F. Cignoli, S. Benecchi, G. Zizak, “Determination of soot parameters by a two-angle scattering-extinction technique in an ethylene diffusion flame,” Appl. Opt. 37, 7865–7874 (1998).
    [CrossRef]
  14. F. D. Bryant, P. Latimer, “Real-time particle sizing by a computer-controlled transmittance photometer,” Appl. Opt. 24, 4280–4282 (1985).
    [CrossRef] [PubMed]
  15. P. C. Ariessohn, 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] [PubMed]
  16. R. A. Dobbins, G. S. Jizmagian, “Optical scattering cross sections for polydispersions of dielectric spheres,” J. Opt. Soc. Am. 56, 1351–1354 (1966).
    [CrossRef]
  17. R. A. Dobbins, G. S. Jizmagian, “Particle size measurements based on use of mean scattering cross sections,” J. Opt. Soc. Am. 56, 1345–1350 (1966).
    [CrossRef]
  18. 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]
  19. J. R. Hodkinson, Aerosol Science, C. N. Davies, ed. (Academic, New York, 1966), Chap. 10, pp. 287–315.
  20. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 10.
  21. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  22. See, for example, A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 34, 34–40 (1995).
  23. K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media,” Phys. Rev. Lett. 64, 2647–2650 (1990).
    [CrossRef] [PubMed]
  24. Use of trade names does not constitute endorsement by the Naval Surface Warfare Center or the U.S. Navy.
  25. Ref. 5, pp. 317–318.
  26. A. Bassini, M. Menchise, S. Musazzi, E. Paganini, U. Perini, “Interferometric system for precise submicrometer particle sizing,” Appl. Opt. 36, 8121–8127 (1997).
    [CrossRef]
  27. A. J. Adorjan, J. A. Lock, T. W. Taylor, P. Tin, W. V. Meyer, A. E. Smart, “Particle sizing in strongly turbid suspensions with the one-beam cross-correlation dynamic light-scattering technique,” Appl. Opt. 38, 3409–3416 (1999).
    [CrossRef]

1999 (1)

1998 (2)

1997 (2)

1995 (1)

See, for example, A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 34, 34–40 (1995).

1990 (1)

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media,” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

1985 (1)

1982 (2)

1980 (1)

1979 (1)

1978 (2)

1966 (2)

1956 (1)

R. M. Tabibian, W. Heller, J. N. Epel, “Experimental investigations on the light scattering of colloidal spheres,” J. Colloid Sci. 11, 195–213 (1956).
[CrossRef]

Adorjan, A. J.

Alfano, R. R.

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media,” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Ariessohn, P. C.

Bassini, A.

Benecchi, S.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 287.

Box, M. A.

Bryant, F. D.

Cashdollar, K. L.

Chance, B.

See, for example, A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 34, 34–40 (1995).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Cignoli, F.

De Iuliis, S.

Deepak, A.

Dobbins, R. A.

Epel, J. N.

R. M. Tabibian, W. Heller, J. N. Epel, “Experimental investigations on the light scattering of colloidal spheres,” J. Colloid Sci. 11, 195–213 (1956).
[CrossRef]

Eustis, R. H.

Heller, W.

R. M. Tabibian, W. Heller, J. N. Epel, “Experimental investigations on the light scattering of colloidal spheres,” J. Colloid Sci. 11, 195–213 (1956).
[CrossRef]

Hodkinson, J. R.

J. R. Hodkinson, Aerosol Science, C. N. Davies, ed. (Academic, New York, 1966), Chap. 10, pp. 287–315.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 287.

Jizmagian, G. S.

Latimer, P.

Lee, C. K.

Lenoble, J.

J. Lenoble, Atmospheric Radiative Transfer (Deepak, Hampton, Va., 1993), pp. 234–237.

Lipaev, A. M.

Liu, F.

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media,” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Lock, J. A.

Menchise, M.

Meyer, W. V.

Mobley, C. D.

C. D. Mobley, Light and Water; Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994), p. 254.

Musazzi, S.

Nefedov, A. P.

Oser, B. L.

B. L. Oser, Hawk’s Physiological Chemistry, 14th ed. (McGraw-Hill, New York, 1965), p. 989.

Paganini, E.

Perini, U.

Petrov, O. F.

Self, S. A.

Singer, J. M.

Smart, A. E.

Tabibian, R. M.

R. M. Tabibian, W. Heller, J. N. Epel, “Experimental investigations on the light scattering of colloidal spheres,” J. Colloid Sci. 11, 195–213 (1956).
[CrossRef]

Tam, W. G.

Taylor, T. W.

Tin, P.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 10.

Vaulina, O. S.

Yodh, A.

See, for example, A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 34, 34–40 (1995).

Yoo, K. M.

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media,” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Zardecki, A.

Zizak, G.

Appl. Opt. (12)

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]

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]

P. C. Ariessohn, 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] [PubMed]

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

A. Zardecki, W. G. Tam, “Multiple scattering corrections to the Beer–Lambert law. 2: Detector with a variable field of view,” Appl. Opt. 21, 2413–2420 (1982).
[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 particle sizes, concentration, and refractive index in measurement of light transmittance in the forward-scattering-angle range,” Appl. Opt. 36, 1357–1366 (1997).
[CrossRef] [PubMed]

A. Bassini, M. Menchise, S. Musazzi, E. Paganini, U. Perini, “Interferometric system for precise submicrometer particle sizing,” Appl. Opt. 36, 8121–8127 (1997).
[CrossRef]

A. P. Nefedov, O. F. Petrov, O. S. Vaulina, A. M. Lipaev, “Application of a forward-angle-scattering transmissometer for simultaneous measurements of particle size and number density in an optically dense medium,” Appl. Opt. 37, 1682–1689 (1998).
[CrossRef]

S. De Iuliis, F. Cignoli, S. Benecchi, G. Zizak, “Determination of soot parameters by a two-angle scattering-extinction technique in an ethylene diffusion flame,” Appl. Opt. 37, 7865–7874 (1998).
[CrossRef]

A. J. Adorjan, J. A. Lock, T. W. Taylor, P. Tin, W. V. Meyer, A. E. Smart, “Particle sizing in strongly turbid suspensions with the one-beam cross-correlation dynamic light-scattering technique,” Appl. Opt. 38, 3409–3416 (1999).
[CrossRef]

J. Colloid Sci. (1)

R. M. Tabibian, W. Heller, J. N. Epel, “Experimental investigations on the light scattering of colloidal spheres,” J. Colloid Sci. 11, 195–213 (1956).
[CrossRef]

J. Opt. Soc. Am. (2)

Phys. Rev. Lett. (1)

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media,” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Phys. Today (1)

See, for example, A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 34, 34–40 (1995).

Other (10)

Use of trade names does not constitute endorsement by the Naval Surface Warfare Center or the U.S. Navy.

Ref. 5, pp. 317–318.

J. R. Hodkinson, Aerosol Science, C. N. Davies, ed. (Academic, New York, 1966), Chap. 10, pp. 287–315.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 10.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

J. Lenoble, Atmospheric Radiative Transfer (Deepak, Hampton, Va., 1993), pp. 234–237.

B. L. Oser, Hawk’s Physiological Chemistry, 14th ed. (McGraw-Hill, New York, 1965), p. 989.

Ref. 1, p. 18.

C. D. Mobley, Light and Water; Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994), p. 254.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 287.

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Figures (6)

Fig. 1
Fig. 1

Experimental setup for measuring the optical thickness τ.

Fig. 2
Fig. 2

Measured τ versus particle concentration for 1.0-µm spheres. The slope, intercept, and r-squared values are the result of the best linear fit for 0 < τ ≤ 9.95. The slope is the extinction Q = 3.135 ± 0.094; Q Mie = 3.151.

Fig. 3
Fig. 3

Calculated τ minus measured τ versus particle concentration. The measured τ diverges from the calculated value for concentrations resulting in τ > 10.

Fig. 4
Fig. 4

Results of particle sizing technique for d = 0.202 ± 0.006 µm, actual size. Possible values for 633 nm are 0.20 and 5.08 µm; for 543.5 nm, 0.20 and 4.01 µm. Measured value is d = 0.20 ± 0.02 µm.

Fig. 5
Fig. 5

Results of particle size technique for d = 1.771 ± 0.028 µm. Possible values for 633 nm are 0.64 and 1.80 µm; for 543.5 nm, 0.40 and 1.80 µm. Measured value is d = 1.80 ± 0.09 µm.

Fig. 6
Fig. 6

Ratio of optical thickness at two wavelengths as a function of particle size. The measured value most closely matches the larger 3.36-µm particle.

Tables (3)

Tables Icon

Table 1 Least-Square Fit Parameters for τ versus ρσgeom Z

Tables Icon

Table 2 Results of Particle Size Measurementsa

Tables Icon

Table 3 Results of Particle Size Measurements for Large Particlesa

Equations (13)

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

1νt+nˆ·Lr=-ξLr+s4π  Lrβθ, φdΩ,
Lz=L0exp-ξz.
θ1/27° λd,
τ=ξz=ρσ z.
Q=σσgeom,
τ=Qρσgeomz.
x=πdλ.
R=VspheresVT=ρVsphere,
τ=3zR2Qd.
ρ=fδVsphere,
R=fδ,
R=2τ3zdQ.
τ543.5τ633=Q543.5Q633,

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