Abstract

A method is described for measuring the phase scattering function of a turbid medium in the near forward direction. The method is based on the use of a transmissometer with a variable field of view and is suitable for measurements on natural media, such as water or turbid atmospheres. Some precautions to be taken to avoid the effects of multiple scattering are indicated. Results of laboratory tests on suspensions of latex spheres in water are presented. An example of the use of the instrument in natural fog is also shown.

© 1989 Optical Society of America

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References

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  1. 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]
  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]
  3. 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).
  4. V. E. Zuev, M. V. Kabanov, B. A. Savelev, “Propagation of Laser Beams in Scattering Media,” Appl. Opt. 8, 137–141 (1969).
    [CrossRef] [PubMed]
  5. 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, 903–911 (1985).
    [CrossRef]
  6. P. Bruscaglioni, G. Zaccanti, A. Ismaelli, P. Pill, “Comparison Between Measured and Calculated Contributions of Multiply Scattered Radiation to the Transmittance of a Light Beam Through a Turbid Medium,” Radio Sci. 22, 899–905 (1987).
    [CrossRef]
  7. J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, C. G. McCreath, “A Laser Diagnostic Technique for the Measurement of Droplet and Particle Size Distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).
  8. L. G. Dodge, “Calibration of the Malvern Particle Sizer,” Appl. Opt. 23, 2415–2419 (1984).
    [CrossRef] [PubMed]
  9. J. Cornillault, “Particle Size Analyzer,” Appl. Opt. 11, 265–268 (1972).
    [CrossRef] [PubMed]
  10. R. W. Spinrad, J. R. V. Zaneveld, H. Pak, “Volume Scattering Function of Suspended Particulate Matter at Near-Forward Angles: A Comparison of Experimental and Theoretical Values,” Appl. Opt. 17, 1125–1130 (1978).
    [CrossRef] [PubMed]
  11. 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]
  12. H. E. Gerber, “Microstructure of a Radiation Fog,” J. Atmos. Sci. 38, 454–458 (1981).
    [CrossRef]
  13. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  14. N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).

1988 (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]

1987 (1)

P. Bruscaglioni, G. Zaccanti, A. Ismaelli, P. Pill, “Comparison Between Measured and Calculated Contributions of Multiply Scattered Radiation to the Transmittance of a Light Beam Through a Turbid Medium,” Radio Sci. 22, 899–905 (1987).
[CrossRef]

1986 (1)

1985 (1)

1984 (2)

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]

L. G. Dodge, “Calibration of the Malvern Particle Sizer,” Appl. Opt. 23, 2415–2419 (1984).
[CrossRef] [PubMed]

1981 (1)

H. E. Gerber, “Microstructure of a Radiation Fog,” J. Atmos. Sci. 38, 454–458 (1981).
[CrossRef]

1978 (1)

1977 (1)

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, C. G. McCreath, “A Laser Diagnostic Technique for the Measurement of Droplet and Particle Size Distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

1972 (1)

1969 (1)

Abbot, D.

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, C. G. McCreath, “A Laser Diagnostic Technique for the Measurement of Droplet and Particle Size Distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Battistelli, E.

Beer, J. M.

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, C. G. McCreath, “A Laser Diagnostic Technique for the Measurement of Droplet and Particle Size Distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Bruscaglioni, P.

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, A. Ismaelli, P. Pill, “Comparison Between Measured and Calculated Contributions of Multiply Scattered Radiation to the Transmittance of a Light Beam Through a Turbid Medium,” Radio Sci. 22, 899–905 (1987).
[CrossRef]

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, 903–911 (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]

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).

Cornillault, J.

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]

Dodge, L. G.

Gerber, H. E.

H. E. Gerber, “Microstructure of a Radiation Fog,” J. Atmos. Sci. 38, 454–458 (1981).
[CrossRef]

Ismaelli, A.

P. Bruscaglioni, G. Zaccanti, A. Ismaelli, P. Pill, “Comparison Between Measured and Calculated Contributions of Multiply Scattered Radiation to the Transmittance of a Light Beam Through a Turbid Medium,” Radio Sci. 22, 899–905 (1987).
[CrossRef]

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, 903–911 (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]

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).

Jerlov, N. G.

N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).

Kabanov, M. V.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

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]

McCreath, C. G.

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, C. G. McCreath, “A Laser Diagnostic Technique for the Measurement of Droplet and Particle Size Distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Pak, H.

Pill, P.

P. Bruscaglioni, G. Zaccanti, A. Ismaelli, P. Pill, “Comparison Between Measured and Calculated Contributions of Multiply Scattered Radiation to the Transmittance of a Light Beam Through a Turbid Medium,” Radio Sci. 22, 899–905 (1987).
[CrossRef]

Savelev, B. A.

Spinrad, R. W.

Swithenbank, J.

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, C. G. McCreath, “A Laser Diagnostic Technique for the Measurement of Droplet and Particle Size Distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Taylor, D. S.

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, C. G. McCreath, “A Laser Diagnostic Technique for the Measurement of Droplet and Particle Size Distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Zaccanti, G.

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, A. Ismaelli, P. Pill, “Comparison Between Measured and Calculated Contributions of Multiply Scattered Radiation to the Transmittance of a Light Beam Through a Turbid Medium,” Radio Sci. 22, 899–905 (1987).
[CrossRef]

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, 903–911 (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]

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).

Zaneveld, J. R. V.

Zuev, V. E.

Appl. Opt. (5)

J. Atmos. Sci. (1)

H. E. Gerber, “Microstructure of a Radiation Fog,” J. Atmos. Sci. 38, 454–458 (1981).
[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. A (1)

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]

Prog. Astronaut. Aeronaut. (1)

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, C. G. McCreath, “A Laser Diagnostic Technique for the Measurement of Droplet and Particle Size Distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Radio Sci. (1)

P. Bruscaglioni, G. Zaccanti, A. Ismaelli, P. Pill, “Comparison Between Measured and Calculated Contributions of Multiply Scattered Radiation to the Transmittance of a Light Beam Through a Turbid Medium,” Radio Sci. 22, 899–905 (1987).
[CrossRef]

Other (3)

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).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).

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Figures (11)

Fig. 1
Fig. 1

Scheme of the measurement apparatus. In our experimental apparatus the light beam source was a He–Ne laser and the lens system was an optical condenser with a focal length of 10.5 cm. The diaphragm D1 has a radius R. The variable radius diaphragm D2 is placed in the focal plane of the lens system. The detector is a photodiode.

Fig. 2
Fig. 2

δ = τ1P(α1)/τP0 calculated as a function of the optical depth τ for four values of α. With reference to Fig. 1, L = 20 cm, R = 1 cm, homogeneous medium with 〈Φ〉 = 15.8 μm; curve a, α = 2.25°, curve b, α = 1.5°, curve c, α = 0.75°, curve d, α = 0.375°.

Fig. 3
Fig. 3

Same as Fig. 2 except L = 50 m and R = 5 cm; curve a, α = 3°, curve b, α = 2°, curve c, α = 1°, curve d, α = 0.5°

Fig. 4
Fig. 4

Phase function p(θ) vs θ (degrees). Spheres with 〈Φ〉 = 15.8 μm in water at He–Ne wavelength: continuous curve, Mie theory; □, p ¯ i j evaluated using in Eq. (4) (case a) the results of Mie theory; ×, p ¯ i j evaluated by means of Eq. (8) (case b). The values of p ¯ i j and p ¯ i j are reported in correspondence to θ = (αi + αj)/2.

Fig. 5
Fig. 5

Calculated ratio r = (PsPs1)/Ps1 plotted vs τ: L = 20 cm; R = 1 cm for the homogeneous medium (case a). From upper to lowest curve, α = 2.25°, 1.5°, 0.75°, 0.375°, respectively; (a) 〈Φ〉 = 15.8 μm and (b) 〈Φ〉 = 1.091 μm.

Fig. 6
Fig. 6

Calculated ratio r = (PsPs1)/Ps1 plotted vs τ: L = 50 m, R = 5 cm for the homogeneous medium (case b). From upper to lowest curve, α = 3°, 2°, 1°, 0.5°, respectively. The scattering function is as in Fig. 5(a).

Fig. 7
Fig. 7

Comparison between calculated phase function p(θ) (continuous line) and values of the phase function ( , p ¯ i j ) measured in the geometry of case a. Angle θ is measured in degrees. Polystyrene spheres with 〈Φ〉 = 15.8 μm in water at He–Ne wavelength.

Fig. 8
Fig. 8

Measured effects of multiple scattering. The values of p ¯ i j, obtained inserting the measured values of Ps(α) in Eq. (4), are plotted vs τ. Each line of crosses refers to a different interval αiαj (0.75–0.375°, 1.125–0.75°, 1.5–1.125°, 1.875–1.5°, 2.25–1.875° from the upper line to the lowest line, respectively). Polystyrene spheres with 〈Φ〉 = 15.8 μm in water at He–Ne wavelength. Geometry of case a.

Fig. 9
Fig. 9

Measured values of p ¯ i j plotted vs τ for polystyrene spheres with 〈Φ〉 = 1.09 μm in water p ¯ i j refers to the interval αiαj = 2.25–0.375°

Fig. 10
Fig. 10

Measured values of p ¯ i j plotted vs τ, for polystyrene spheres with 〈Φ〉 = 0.305 μm in water. For each value of τ the results of two measurements for the interval αiαj = 2.25–0.375° and two measurements for the interval αiαj = 2.25–1.125° are reported.

Fig. 11
Fig. 11

Measurements in fog. Measured values of p ¯ i j [Eq. (8)] are plotted vs time during an ≈8 h interval. Curves a and b show p ¯ i j for αiαj = 1–0.5°, αiαj = 1.5–1°, respectively. Curve c shows the measured extinction coefficient of fog during the same time interval. Geometry of case b: L = 50 m, R = 5 cm.

Equations (13)

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P 0 = P e exp ( τ ) ,
d P s 1 ( α ) = a σ d x P e exp ( 0 x σ d x ) 0 θ ( x ) 2 π p ( θ ) sin θ × exp ( x L σ 1 cos θ d x ) d θ ,
P s 1 ( α ) = P 0 0 L a σ d x 0 θ ¯ ( x ) 2 π p ( θ ) sin θ d θ ,
P s 1 ( α ) = a τ P 0 0 α 2 π p ( θ ) sin θ d θ .
p ¯ i j = 1 Δ Ω α i α j 2 π p ( θ ) sin θ d θ = 1 2 π ( cos α i cos α j ) P s 1 ( α i ) P s 1 ( α i ) a τ P 0 .
P ( α i ) P ( α j ) = P s ( α i ) P s ( α j ) P s 1 ( α i ) P s 1 ( α j ) .
P s 1 ( α ) = 2 π a P 0 { 0 x 0 ( α ) σ d x 0 tan 1 [ R / ( L x ) ] p ( θ ) sin θ d θ + x 0 ( α ) L σ d x 0 α p ( θ ) sin θ d θ } .
P s 1 ( α i ) P s 1 ( α j ) = 2 π a σ P 0 { x 0 ( α j x 0 ( α i ) d x α j tan 1 [ R / ( L x ) ] p ( θ ) sin θ d θ + x 0 ( α i ) L d x α j α i p ( θ ) sin θ d θ } .
P s 1 ( α i ) P s 1 ( α j ) = 2 π a σ P 0 R p ¯ i j ( sin α i sin α j ) ,
p ¯ i j = L R 1 2 π ( sin α i sin α j ) P s 1 ( α i ) P s 1 ( α j ) a τ P 0 .
[ P s 1 ( α i ) P s 1 ( α j ) ] case a [ P s 1 ( α i ) P s 1 ( α j ) ] case b = p ¯ i j ( cos α j cos α i ) p ¯ i j ( sin α i sin α j ) × L R L R 1 2 ( α i + α j ) ,
P ( α i , t ) P ( α j , t ) = P s ( α i , t ) P s ( α j , t ) + P 0 ( t ) P 0 ( t ) .
P s 1 ( α i ) P s 1 ( α j ) τ P 0 = 1 2 N + 1 n = N N × P [ α i , t + ( i 1 ) Δ t + n T ] P [ α j , t + ( i 1 ) Δ t + n T ] P ( α 1 , t + n T ) τ 1 ( t + n T ) ,

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