Abstract

A simple and inexpensive method of measuring statistical parameters related to the time lengthening arising in the propagation of a light pulse in a turbid medium is presented. The method is based on the repetition of attenuation measurements of a light beam passing through the turbid medium when the absorption coefficient of the medium surrounding the diffusing particles is varied. The measurements are carried out using a cw source and a simple optical receiver with a common photodiode as a detector. The results of two measurements are reported together with the results of numerical simulations carried out using the scattering properties and geometric parameters corresponding to the experimental situation. Numerical results were obtained using a Monte Carlo based method. Good agreement between experimental and numerical results was found.

© 1990 Optical Society of America

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References

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  1. E. A. Bucher, R. M. Lerner, “Experiments on Light Pulse Communication and Propagation Through Atmospheric Clouds,” Appl. Opt. 12, 2401–2414 (1973).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  3. G. C. Mooradian, M. Geller, “Temporal and Angular Spreading of Blue-Green Pulses in Clouds,” Appl. Opt. 21, 1572–1577 (1982).
    [Crossref] [PubMed]
  4. Y. Kuga, A. Ishimau, A. P. Bruckner “Experiments on Picosecond Pulse Propagation in a Diffuse Medium,” J. Opt. Soc. Am. 73, 1812–1815 (1983).
    [Crossref]
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    [Crossref] [PubMed]
  6. A. K. Majumdar, “Transformation of Statistical Characteristics of Picosecond Laser Pulses by Multiple-Scattering Media,” Appl. Opt. 25, 4649–4655 (1986).
    [Crossref] [PubMed]
  7. P. Bruscaglioni, G. Zaccanti, M. Olivieri, “Laboratory Simulation of the Scattering of a Laser Beam in a Turbid Atmosphere,” J. Phys. D 21, S45–S48 (1988).
    [Crossref]
  8. E. A. Bucher, “Computer Simulation of Light Pulse Propagation for Communication Through Thick Clouds,” Appl. Opt. 12, 2391–2400 (1973).
    [Crossref] [PubMed]
  9. C. F. Bohren, D. P. Gilra, “Extinction by a Spherical Particle in an Absorbing Medium,” J. Colloid. Inteface Sci. 72, 215–221 (1979).
    [Crossref]
  10. G. Zaccanti, P. Bruscaglioni, “Deviation from the Lambert-Beer Law in the Transmittance of a Light Beam Through Diffusing Media: Experimental Results,” J. Mod. Opt. 35, 229–242 (1988).
    [Crossref]
  11. E. Battistelli, P. Bruscaglioni, A. Ismaelli, L. Lo Porto, G. Zaccanti, “Separation and Analysis of Forward Scattered Power in Laboratory Measurements of Light Beam Transmittance Through a Turbid Medium,” Appl. Opt. 25, 420–430 (1986).
    [Crossref] [PubMed]
  12. P. Bruscaglioni, G. Del Fante, A. Ismaelli, L. Lo Porto, G. Zaccanti, “A Variable Angular Field-of-View Transmissometer and Its Use to Monitor Fog Conditions,” Opt. Acta 31, 589–601 (1984).
    [Crossref]
  13. 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]
  14. P. Bruscaglioni, E. Battistelli, A. Ismaelli, G. Zaccanti, “Semoc: A Code for Monte Carlo Calculations of the Effects of Multiple Scattering on the Transmittance of a Light Beam Through a Turbid Medium,” Report Dipartimento di Fisica (Mar.1987).
  15. E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Use of Two Scaling Relations in the Study of Multiple-Scattering Effects on the Transmittance of Light Beams Through a Turbid Atmosphere,” J. Opt. Soc. Am. A 2, 903–912 (1985).
    [Crossref]

1989 (1)

1988 (2)

G. Zaccanti, P. Bruscaglioni, “Deviation from the Lambert-Beer Law in the Transmittance of a Light Beam Through Diffusing Media: Experimental Results,” J. Mod. Opt. 35, 229–242 (1988).
[Crossref]

P. Bruscaglioni, G. Zaccanti, M. Olivieri, “Laboratory Simulation of the Scattering of a Laser Beam in a Turbid Atmosphere,” J. Phys. D 21, S45–S48 (1988).
[Crossref]

1987 (1)

P. Bruscaglioni, E. Battistelli, A. Ismaelli, G. Zaccanti, “Semoc: A Code for Monte Carlo Calculations of the Effects of Multiple Scattering on the Transmittance of a Light Beam Through a Turbid Medium,” Report Dipartimento di Fisica (Mar.1987).

1986 (2)

1985 (1)

1984 (1)

P. Bruscaglioni, G. Del Fante, A. Ismaelli, L. Lo Porto, G. Zaccanti, “A Variable Angular Field-of-View Transmissometer and Its Use to Monitor Fog Conditions,” Opt. Acta 31, 589–601 (1984).
[Crossref]

1983 (2)

1982 (1)

1979 (2)

G. C. Mooradian, M. Geller, L. B. Stotts, D. H. Stephens, R. A. Krautwald, “Blue-Green Pulsed Propagation Through Fog,” Appl. Opt. 18, 429–441 (1979).
[Crossref] [PubMed]

C. F. Bohren, D. P. Gilra, “Extinction by a Spherical Particle in an Absorbing Medium,” J. Colloid. Inteface Sci. 72, 215–221 (1979).
[Crossref]

1973 (2)

Battistelli, E.

Bohren, C. F.

C. F. Bohren, D. P. Gilra, “Extinction by a Spherical Particle in an Absorbing Medium,” J. Colloid. Inteface Sci. 72, 215–221 (1979).
[Crossref]

Bruckner, A. P.

Bruscaglioni, P.

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]

G. Zaccanti, P. Bruscaglioni, “Deviation from the Lambert-Beer Law in the Transmittance of a Light Beam Through Diffusing Media: Experimental Results,” J. Mod. Opt. 35, 229–242 (1988).
[Crossref]

P. Bruscaglioni, G. Zaccanti, M. Olivieri, “Laboratory Simulation of the Scattering of a Laser Beam in a Turbid Atmosphere,” J. Phys. D 21, S45–S48 (1988).
[Crossref]

P. Bruscaglioni, E. Battistelli, A. Ismaelli, G. Zaccanti, “Semoc: A Code for Monte Carlo Calculations of the Effects of Multiple Scattering on the Transmittance of a Light Beam Through a Turbid Medium,” Report Dipartimento di Fisica (Mar.1987).

E. Battistelli, P. Bruscaglioni, A. Ismaelli, L. Lo Porto, G. Zaccanti, “Separation and Analysis of Forward Scattered Power in Laboratory Measurements of Light Beam Transmittance Through a Turbid Medium,” Appl. Opt. 25, 420–430 (1986).
[Crossref] [PubMed]

E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Use of Two Scaling Relations in the Study of Multiple-Scattering Effects on the Transmittance of Light Beams Through a Turbid Atmosphere,” J. Opt. Soc. Am. A 2, 903–912 (1985).
[Crossref]

P. Bruscaglioni, G. Del Fante, A. Ismaelli, L. Lo Porto, G. Zaccanti, “A Variable Angular Field-of-View Transmissometer and Its Use to Monitor Fog Conditions,” Opt. Acta 31, 589–601 (1984).
[Crossref]

Bucher, E. A.

Del Fante, G.

P. Bruscaglioni, G. Del Fante, A. Ismaelli, L. Lo Porto, G. Zaccanti, “A Variable Angular Field-of-View Transmissometer and Its Use to Monitor Fog Conditions,” Opt. Acta 31, 589–601 (1984).
[Crossref]

Elliott, R. A.

Geller, M.

Gilra, D. P.

C. F. Bohren, D. P. Gilra, “Extinction by a Spherical Particle in an Absorbing Medium,” J. Colloid. Inteface Sci. 72, 215–221 (1979).
[Crossref]

Ishimau, A.

Ismaelli, A.

P. Bruscaglioni, E. Battistelli, A. Ismaelli, G. Zaccanti, “Semoc: A Code for Monte Carlo Calculations of the Effects of Multiple Scattering on the Transmittance of a Light Beam Through a Turbid Medium,” Report Dipartimento di Fisica (Mar.1987).

E. Battistelli, P. Bruscaglioni, A. Ismaelli, L. Lo Porto, G. Zaccanti, “Separation and Analysis of Forward Scattered Power in Laboratory Measurements of Light Beam Transmittance Through a Turbid Medium,” Appl. Opt. 25, 420–430 (1986).
[Crossref] [PubMed]

E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Use of Two Scaling Relations in the Study of Multiple-Scattering Effects on the Transmittance of Light Beams Through a Turbid Atmosphere,” J. Opt. Soc. Am. A 2, 903–912 (1985).
[Crossref]

P. Bruscaglioni, G. Del Fante, A. Ismaelli, L. Lo Porto, G. Zaccanti, “A Variable Angular Field-of-View Transmissometer and Its Use to Monitor Fog Conditions,” Opt. Acta 31, 589–601 (1984).
[Crossref]

Krautwald, R. A.

Kuga, Y.

Lerner, R. M.

Lo Porto, L.

E. Battistelli, P. Bruscaglioni, A. Ismaelli, L. Lo Porto, G. Zaccanti, “Separation and Analysis of Forward Scattered Power in Laboratory Measurements of Light Beam Transmittance Through a Turbid Medium,” Appl. Opt. 25, 420–430 (1986).
[Crossref] [PubMed]

P. Bruscaglioni, G. Del Fante, A. Ismaelli, L. Lo Porto, G. Zaccanti, “A Variable Angular Field-of-View Transmissometer and Its Use to Monitor Fog Conditions,” Opt. Acta 31, 589–601 (1984).
[Crossref]

Majumdar, A. K.

Mooradian, G. C.

Olivieri, M.

P. Bruscaglioni, G. Zaccanti, M. Olivieri, “Laboratory Simulation of the Scattering of a Laser Beam in a Turbid Atmosphere,” J. Phys. D 21, S45–S48 (1988).
[Crossref]

Stephens, D. H.

Stotts, L. B.

Zaccanti, G.

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]

G. Zaccanti, P. Bruscaglioni, “Deviation from the Lambert-Beer Law in the Transmittance of a Light Beam Through Diffusing Media: Experimental Results,” J. Mod. Opt. 35, 229–242 (1988).
[Crossref]

P. Bruscaglioni, G. Zaccanti, M. Olivieri, “Laboratory Simulation of the Scattering of a Laser Beam in a Turbid Atmosphere,” J. Phys. D 21, S45–S48 (1988).
[Crossref]

P. Bruscaglioni, E. Battistelli, A. Ismaelli, G. Zaccanti, “Semoc: A Code for Monte Carlo Calculations of the Effects of Multiple Scattering on the Transmittance of a Light Beam Through a Turbid Medium,” Report Dipartimento di Fisica (Mar.1987).

E. Battistelli, P. Bruscaglioni, A. Ismaelli, L. Lo Porto, G. Zaccanti, “Separation and Analysis of Forward Scattered Power in Laboratory Measurements of Light Beam Transmittance Through a Turbid Medium,” Appl. Opt. 25, 420–430 (1986).
[Crossref] [PubMed]

E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Use of Two Scaling Relations in the Study of Multiple-Scattering Effects on the Transmittance of Light Beams Through a Turbid Atmosphere,” J. Opt. Soc. Am. A 2, 903–912 (1985).
[Crossref]

P. Bruscaglioni, G. Del Fante, A. Ismaelli, L. Lo Porto, G. Zaccanti, “A Variable Angular Field-of-View Transmissometer and Its Use to Monitor Fog Conditions,” Opt. Acta 31, 589–601 (1984).
[Crossref]

Appl. Opt. (8)

J. Colloid. Inteface Sci. (1)

C. F. Bohren, D. P. Gilra, “Extinction by a Spherical Particle in an Absorbing Medium,” J. Colloid. Inteface Sci. 72, 215–221 (1979).
[Crossref]

J. Mod. Opt. (1)

G. Zaccanti, P. Bruscaglioni, “Deviation from the Lambert-Beer Law in the Transmittance of a Light Beam Through Diffusing Media: Experimental Results,” J. Mod. Opt. 35, 229–242 (1988).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Phys. D (1)

P. Bruscaglioni, G. Zaccanti, M. Olivieri, “Laboratory Simulation of the Scattering of a Laser Beam in a Turbid Atmosphere,” J. Phys. D 21, S45–S48 (1988).
[Crossref]

Opt. Acta (1)

P. Bruscaglioni, G. Del Fante, A. Ismaelli, L. Lo Porto, G. Zaccanti, “A Variable Angular Field-of-View Transmissometer and Its Use to Monitor Fog Conditions,” Opt. Acta 31, 589–601 (1984).
[Crossref]

Report Dipartimento di Fisica (1)

P. Bruscaglioni, E. Battistelli, A. Ismaelli, G. Zaccanti, “Semoc: A Code for Monte Carlo Calculations of the Effects of Multiple Scattering on the Transmittance of a Light Beam Through a Turbid Medium,” Report Dipartimento di Fisica (Mar.1987).

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Figures (7)

Fig. 1
Fig. 1

Sketch of the geometric situation. Multiple scattering trajectories are indicated. Due to multiple scattering the received pulse P is broadened with respect to transmitted pulse Pe. The figure refers to a conical beam, but the proposed method of measuring is valid for any shape of emitted light beam.

Fig. 2
Fig. 2

Block diagram of the experimental setup. In our measurements the source was a cw 10-mW He–Ne laser. F1 is a calibrated neutral density filter, F2 is an interference bandpass filter. The optical receiver (radius R and angular FOV semiaperture α) uses a photodiode as a detector. In the scattering cell (filled with bidistilled water) the diffusing particles (polystyrene spheres in our case) are introduced and measurements are repeated with the addition of an absorbing medium (a blue dye).

Fig. 3
Fig. 3

Example of measurement at a high value of optical depth (τs = 31.0). The ratio P(τs + τd)/P(τs) between the total received power, when the optical depth due to the dye is τd and when no dye is present, is reported vs τd. The marks refer to the experimental results (+, □, × correspond to α = 3°, 2°, 1°, respectively), whereas the continuous line represents exp(−τd). The differences between the experimental results and the continuous line are ascribed to extra path attenuation.

Fig. 4
Fig. 4

The marks represent the same results reported in Fig. 3, pertaining to α = 3°. The continuous and dashed lines represent the quantity 〈exp(−τdδ)〉 [Eq. (2)] best fit with the results, which was calculated by using a γ function and an exponential function, respectively, for the probability function f(δ). The γ function fits the results very well.

Fig. 5
Fig. 5

Plot of the γ function that best fits with the results in Fig. 4. The f(δ) function has the same shape as the received pulse when a delta pulse is transmitted.

Fig. 6
Fig. 6

Measurement example at a small value of optical depth. The ratio between the scattered (Pd) and unscattered (P0) received power is reported vs τ for α = 3° 2°, 1° (+, □, ×, respectively). Data pertaining to τ > 7.06 refer to different concentrations of spheres only; those pertaining to τ > 7.06 refer to a fixed concentration of spheres and different concentrations of dye. The continuous curves represent the results of a numerical calculation based on the Monte Carlo method.

Fig. 7
Fig. 7

The same data as in Fig. 6 pertaining to α = 3° are reported as the ratio Pd(τs + τd)/Pd(τs) exp(τd) vs τd with τs = 7.06. The ratio is equal to Pd(τd + τs)/P0(τd + τs) divided by Pd(τs)/P0(τs) at τs = 7.06. The continuous line represents the second degree polynomial that best fits with the results.

Tables (1)

Tables Icon

Table 1 Comparison Between Calculated and Experimental Results for τs = 31.0; Particle Size 0.33 μm

Equations (3)

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P ( τ s + τ d ) = P ( τ s ) 0 exp [ - τ d ( 1 + δ ) ] f ( δ ) d δ = P ( τ s ) exp ( - τ d ) 0 exp ( - τ d δ ) f ( δ ) d δ .
exp ( - τ d δ ) = P ( τ s + τ d ) / [ P ( τ s ) exp ( - τ d ) ] .
exp ( - τ d δ ) = 1 - τ d δ + ½ τ d 2 δ 2 - τ d 3 δ 3 + = 1 - τ d δ + ½ τ d 2 δ 2 - τ d 3 δ 3 + .

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