Abstract

We describe a numerical model for the interaction of light with large raindrops using realistic nonspherical drop shapes. We apply geometrical optics and a Monte Carlo technique to perform ray traces through the drops. We solve the problem of diffraction independently by approximating the drops with area-equivalent ellipsoids. Scattering patterns are obtained for different polarizations of the incident light. They exhibit varying degrees of asymmetry and depolarization that can be linked to the distortion and thus the size of the drops. The model is extended to give a simplified long-path integration.

© 2002 Optical Society of America

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References

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  1. P. Lenard, “Über Regen,” Meteorol. Z. 21, 249–260 (1904); for English translation, see Q. J. R. Meteorol. Soc. 31, 62–73 (1905).
  2. C. Magono, “On the shape of water drops falling in stagnant air,” J. Meteorol. 11, 77–79 (1954).
    [CrossRef]
  3. H. R. Pruppacher, K. V. Beard, “A wind tunnel investigation of the internal circulation and shape of water drops falling at terminal velocity in air,” Q. J. R. Meteorol. Soc. 96, 247–256 (1970).
    [CrossRef]
  4. K. V. Beard, C. Chuang, “A new model for the equilibrium shape of raindrops,” J. Atmos. Sci. 44, 1509–1524 (1987).
    [CrossRef]
  5. C. Chuang, K. V. Beard, “A numerical model for the equilibrium shape of electrified raindrops,” J. Atmos. Sci. 47, 1374–1389 (1990).
    [CrossRef]
  6. W. J. Glantschnig, S.-H. Chen, “Light scattering from water droplets in the geometrical optics approximation,” Appl. Opt. 20, 2499–2509 (1981).
    [CrossRef] [PubMed]
  7. L. G. Kazovsky, “Estimation of particle size distributions from forward scattering data,” Appl. Opt. 23, 455–464 (1984).
    [CrossRef] [PubMed]
  8. J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. II. Transmission and cross-polarization effects,” Appl. Opt. 35, 515–531 (1996).
    [CrossRef] [PubMed]
  9. G. S. Stamaskos, D. Yova, N. K. Uzunoglu, “Integral equation model of light scattering by an oriented monodisperse system of triaxial dielectric ellipsoids: application in ectacytometry,” Appl. Opt. 36, 6503–6512 (1997).
    [CrossRef]
  10. A. Macke, M. Großklaus, “Light scattering by nonspherical raindrops: implications for lidar remote sensing of rainrates,” J. Quant. Spectrosc. Radiat. Transfer 60, 355–363 (1998).
    [CrossRef]
  11. O. N. Ross, “Optical remote sensing of rainfall micro-structures,” Diplomarbeit thesis (Fachbereich Physik der Freien Universität Berlin, Berlin, 2000), http://www.soton.ac.uk/∼onr/MSc/Diplom.html .
  12. A. Macke, M. I. Mishchenko, K. Muinonen, B. E. Carlson, “Scattering of light by large nonspherical particles: ray-tracing approximation versus T-matrix method,” Opt. Lett. 20, 1934–1936 (1995).
    [CrossRef] [PubMed]
  13. G. A. Shah, “Geometrical optics and diffraction vis-à-vis Mie theory of scattering of electromagnetic radiation by a sphere,” Astrophys. Space Sci. 193, 317–328 (1992).
    [CrossRef]
  14. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  15. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  16. E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987).
  17. M. Born, E. Wolf, Principles of Optics, 6th ed. (with corrections) (Pergamon, Oxford, 1989), pp. 395–399.
  18. D. H. Towne, Wave Phenomena (Addison-Wesley, Reading, Mass., 1967), pp. 458–464.
  19. M. Françon, Diffraction, Coherence in Optics (Pergamon, London, 1966), pp. 36–37.
  20. J. S. Marshall, W. K. Palmer, “The distribution of raindrops with size,” J. Meteorol. 5, 165–166 (1948).
    [CrossRef]
  21. E. D. Hinkley, ed., Laser Monitoring of the Atmosphere, Vol. 14 of Topics in Applied Physics (Springer-Verlag, Berlin, 1976), p. 101.
  22. J. Joss, A. Waldvogel, “Raindrop size distribution and sampling size errors,” J. Atmos. Sci. 26, 566–569 (1967).
    [CrossRef]
  23. R. Gunn, G. D. Kinzer, “The terminal velocity of fall for water drops in stagnant air,” J. Meteorol. 6, 243–248 (1949).
    [CrossRef]

1998 (1)

A. Macke, M. Großklaus, “Light scattering by nonspherical raindrops: implications for lidar remote sensing of rainrates,” J. Quant. Spectrosc. Radiat. Transfer 60, 355–363 (1998).
[CrossRef]

1997 (1)

1996 (1)

1995 (1)

1992 (1)

G. A. Shah, “Geometrical optics and diffraction vis-à-vis Mie theory of scattering of electromagnetic radiation by a sphere,” Astrophys. Space Sci. 193, 317–328 (1992).
[CrossRef]

1990 (1)

C. Chuang, K. V. Beard, “A numerical model for the equilibrium shape of electrified raindrops,” J. Atmos. Sci. 47, 1374–1389 (1990).
[CrossRef]

1987 (1)

K. V. Beard, C. Chuang, “A new model for the equilibrium shape of raindrops,” J. Atmos. Sci. 44, 1509–1524 (1987).
[CrossRef]

1984 (1)

1981 (1)

1970 (1)

H. R. Pruppacher, K. V. Beard, “A wind tunnel investigation of the internal circulation and shape of water drops falling at terminal velocity in air,” Q. J. R. Meteorol. Soc. 96, 247–256 (1970).
[CrossRef]

1967 (1)

J. Joss, A. Waldvogel, “Raindrop size distribution and sampling size errors,” J. Atmos. Sci. 26, 566–569 (1967).
[CrossRef]

1954 (1)

C. Magono, “On the shape of water drops falling in stagnant air,” J. Meteorol. 11, 77–79 (1954).
[CrossRef]

1949 (1)

R. Gunn, G. D. Kinzer, “The terminal velocity of fall for water drops in stagnant air,” J. Meteorol. 6, 243–248 (1949).
[CrossRef]

1948 (1)

J. S. Marshall, W. K. Palmer, “The distribution of raindrops with size,” J. Meteorol. 5, 165–166 (1948).
[CrossRef]

1904 (1)

P. Lenard, “Über Regen,” Meteorol. Z. 21, 249–260 (1904); for English translation, see Q. J. R. Meteorol. Soc. 31, 62–73 (1905).

Beard, K. V.

C. Chuang, K. V. Beard, “A numerical model for the equilibrium shape of electrified raindrops,” J. Atmos. Sci. 47, 1374–1389 (1990).
[CrossRef]

K. V. Beard, C. Chuang, “A new model for the equilibrium shape of raindrops,” J. Atmos. Sci. 44, 1509–1524 (1987).
[CrossRef]

H. R. Pruppacher, K. V. Beard, “A wind tunnel investigation of the internal circulation and shape of water drops falling at terminal velocity in air,” Q. J. R. Meteorol. Soc. 96, 247–256 (1970).
[CrossRef]

Bohren, C. F.

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

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (with corrections) (Pergamon, Oxford, 1989), pp. 395–399.

Carlson, B. E.

Chen, S.-H.

Chuang, C.

C. Chuang, K. V. Beard, “A numerical model for the equilibrium shape of electrified raindrops,” J. Atmos. Sci. 47, 1374–1389 (1990).
[CrossRef]

K. V. Beard, C. Chuang, “A new model for the equilibrium shape of raindrops,” J. Atmos. Sci. 44, 1509–1524 (1987).
[CrossRef]

Françon, M.

M. Françon, Diffraction, Coherence in Optics (Pergamon, London, 1966), pp. 36–37.

Glantschnig, W. J.

Großklaus, M.

A. Macke, M. Großklaus, “Light scattering by nonspherical raindrops: implications for lidar remote sensing of rainrates,” J. Quant. Spectrosc. Radiat. Transfer 60, 355–363 (1998).
[CrossRef]

Gunn, R.

R. Gunn, G. D. Kinzer, “The terminal velocity of fall for water drops in stagnant air,” J. Meteorol. 6, 243–248 (1949).
[CrossRef]

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987).

Huffman, D. R.

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

Joss, J.

J. Joss, A. Waldvogel, “Raindrop size distribution and sampling size errors,” J. Atmos. Sci. 26, 566–569 (1967).
[CrossRef]

Kazovsky, L. G.

Kinzer, G. D.

R. Gunn, G. D. Kinzer, “The terminal velocity of fall for water drops in stagnant air,” J. Meteorol. 6, 243–248 (1949).
[CrossRef]

Lenard, P.

P. Lenard, “Über Regen,” Meteorol. Z. 21, 249–260 (1904); for English translation, see Q. J. R. Meteorol. Soc. 31, 62–73 (1905).

Lock, J. A.

Macke, A.

A. Macke, M. Großklaus, “Light scattering by nonspherical raindrops: implications for lidar remote sensing of rainrates,” J. Quant. Spectrosc. Radiat. Transfer 60, 355–363 (1998).
[CrossRef]

A. Macke, M. I. Mishchenko, K. Muinonen, B. E. Carlson, “Scattering of light by large nonspherical particles: ray-tracing approximation versus T-matrix method,” Opt. Lett. 20, 1934–1936 (1995).
[CrossRef] [PubMed]

Magono, C.

C. Magono, “On the shape of water drops falling in stagnant air,” J. Meteorol. 11, 77–79 (1954).
[CrossRef]

Marshall, J. S.

J. S. Marshall, W. K. Palmer, “The distribution of raindrops with size,” J. Meteorol. 5, 165–166 (1948).
[CrossRef]

Mishchenko, M. I.

Muinonen, K.

Palmer, W. K.

J. S. Marshall, W. K. Palmer, “The distribution of raindrops with size,” J. Meteorol. 5, 165–166 (1948).
[CrossRef]

Pruppacher, H. R.

H. R. Pruppacher, K. V. Beard, “A wind tunnel investigation of the internal circulation and shape of water drops falling at terminal velocity in air,” Q. J. R. Meteorol. Soc. 96, 247–256 (1970).
[CrossRef]

Ross, O. N.

O. N. Ross, “Optical remote sensing of rainfall micro-structures,” Diplomarbeit thesis (Fachbereich Physik der Freien Universität Berlin, Berlin, 2000), http://www.soton.ac.uk/∼onr/MSc/Diplom.html .

Shah, G. A.

G. A. Shah, “Geometrical optics and diffraction vis-à-vis Mie theory of scattering of electromagnetic radiation by a sphere,” Astrophys. Space Sci. 193, 317–328 (1992).
[CrossRef]

Stamaskos, G. S.

Towne, D. H.

D. H. Towne, Wave Phenomena (Addison-Wesley, Reading, Mass., 1967), pp. 458–464.

Uzunoglu, N. K.

van de Hulst, H. C.

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

Waldvogel, A.

J. Joss, A. Waldvogel, “Raindrop size distribution and sampling size errors,” J. Atmos. Sci. 26, 566–569 (1967).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (with corrections) (Pergamon, Oxford, 1989), pp. 395–399.

Yova, D.

Appl. Opt. (4)

Astrophys. Space Sci. (1)

G. A. Shah, “Geometrical optics and diffraction vis-à-vis Mie theory of scattering of electromagnetic radiation by a sphere,” Astrophys. Space Sci. 193, 317–328 (1992).
[CrossRef]

J. Atmos. Sci. (3)

K. V. Beard, C. Chuang, “A new model for the equilibrium shape of raindrops,” J. Atmos. Sci. 44, 1509–1524 (1987).
[CrossRef]

C. Chuang, K. V. Beard, “A numerical model for the equilibrium shape of electrified raindrops,” J. Atmos. Sci. 47, 1374–1389 (1990).
[CrossRef]

J. Joss, A. Waldvogel, “Raindrop size distribution and sampling size errors,” J. Atmos. Sci. 26, 566–569 (1967).
[CrossRef]

J. Meteorol. (3)

R. Gunn, G. D. Kinzer, “The terminal velocity of fall for water drops in stagnant air,” J. Meteorol. 6, 243–248 (1949).
[CrossRef]

J. S. Marshall, W. K. Palmer, “The distribution of raindrops with size,” J. Meteorol. 5, 165–166 (1948).
[CrossRef]

C. Magono, “On the shape of water drops falling in stagnant air,” J. Meteorol. 11, 77–79 (1954).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

A. Macke, M. Großklaus, “Light scattering by nonspherical raindrops: implications for lidar remote sensing of rainrates,” J. Quant. Spectrosc. Radiat. Transfer 60, 355–363 (1998).
[CrossRef]

Meteorol. Z. (1)

P. Lenard, “Über Regen,” Meteorol. Z. 21, 249–260 (1904); for English translation, see Q. J. R. Meteorol. Soc. 31, 62–73 (1905).

Opt. Lett. (1)

Q. J. R. Meteorol. Soc. (1)

H. R. Pruppacher, K. V. Beard, “A wind tunnel investigation of the internal circulation and shape of water drops falling at terminal velocity in air,” Q. J. R. Meteorol. Soc. 96, 247–256 (1970).
[CrossRef]

Other (8)

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

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

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987).

M. Born, E. Wolf, Principles of Optics, 6th ed. (with corrections) (Pergamon, Oxford, 1989), pp. 395–399.

D. H. Towne, Wave Phenomena (Addison-Wesley, Reading, Mass., 1967), pp. 458–464.

M. Françon, Diffraction, Coherence in Optics (Pergamon, London, 1966), pp. 36–37.

E. D. Hinkley, ed., Laser Monitoring of the Atmosphere, Vol. 14 of Topics in Applied Physics (Springer-Verlag, Berlin, 1976), p. 101.

O. N. Ross, “Optical remote sensing of rainfall micro-structures,” Diplomarbeit thesis (Fachbereich Physik der Freien Universität Berlin, Berlin, 2000), http://www.soton.ac.uk/∼onr/MSc/Diplom.html .

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Figures (16)

Fig. 1
Fig. 1

Drop shapes for diameters D = 2, 3, 4, 5, and 6 mm with dashed circles shown for comparison and the angle θ in the coordinate system used.

Fig. 2
Fig. 2

Flow chart for the ray trace.

Fig. 3
Fig. 3

Coordinate system used for the diffraction integral in Eq. (4).

Fig. 4
Fig. 4

Energy ring (left) and its projection onto the spherical detector.

Fig. 5
Fig. 5

Comparison between the ellipses used (dashed–dotted curve) and the actual drop shapes for drop diameters D = 2, 4, and 6 mm.

Fig. 6
Fig. 6

Location of the first three minima in the diffraction pattern from a drop with radius a = 3 mm.

Fig. 7
Fig. 7

Logarithmic intensity distribution for the directions (a) Σ = 90° and (b) Σ = 0°.

Fig. 8
Fig. 8

Contour plots of the intensity distribution obtained from a 3-mm drop for (a) unpolarized light, (b) y-polarized light, and (c) z-polarized light. (d) A plot of the difference in the obtained intensities for y-polarized and z-polarized light (nonlogarithmic). Note that the gray scale below (c) applies for (a)–(c) whereas (d), being the only nonlogarithmic plot, has a scale of its own.

Fig. 9
Fig. 9

Polar plots for a 3-mm drop showing the intensity distribution along (a) Θ = 82° for unpolarized light, (b) Φ = 90° for unpolarized light, (c) Θ = 82° for y-polarized light, (d) Φ = 90° for z-polarized light. The graphs in (e) and (f) show the difference between the results for y- and z-polarized light along Θ = 81° and Φ = 90°, respectively. Negative values are plotted into the lower half of the circles.

Fig. 10
Fig. 10

(a) Fraction of the total incident intensity that is forward scattered and (b) the fraction of all incident rays that are entirely internally reflected (regardless of their intensity).

Fig. 11
Fig. 11

Superposition of the polar plots of the logarithmic intensity including all components (solid curve) and without diffraction (dashed curve) along the 90° latitude for (a) a = 0.1 mm and (b) a = 3 mm.

Fig. 12
Fig. 12

Influence of diffraction on the overall scattering pattern. The percentile change between the ray trace only and the ray trace plus diffraction is shown along the 90° latitude for (a) a = 0.1 mm and (b) a = 3 mm.

Fig. 13
Fig. 13

Area presented to the incident light with drop diameter for three different rainfall intensities.

Fig. 14
Fig. 14

Optical depth as a function of rainfall intensity for four different path lengths.

Fig. 15
Fig. 15

Dependence of average received light intensity on rainfall rate R for (Θ, Φ) = (90°, 75°) and (Θ, Φ) = (90°, 27°).

Fig. 16
Fig. 16

Relative deviations from vertical symmetry in the scattering data for different rainfall intensities and different angular bins. The deviation is given as a percentage of the total received intensity at (Θ, Φ) = (90° + ΔΘ, 90°).

Tables (3)

Tables Icon

Table 1 Shape Coefficients for Cosine Distortion Fit [Eq. (1)] for Drop Radii between 0.5 and 4.5 mm from Chuang and Beard5

Tables Icon

Table 2 Separation into Forward and Backward Scattering for Polarizations 1 (First Number in Each Column) and 2

Tables Icon

Table 3 Comparison of Results from the Computer Model and van de Hulst14

Equations (28)

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rθ=a1+n=010 cn cosnθ,
P=rθ-sin ϕ sin θcos ϕ sin θ-cos θ,
N=Pθ × Pϕ.
q, Σ=εA expiωt-krrϱ=0aθ=02πexpikϱq/r×cosθ-Σϱdϱdθ,
I  1kaq/rkaq/r2=1ka sin Ψka sin Ψ2,
LΨ=1-02ak sinΨ-12ak sinΨ,
LΨsin Θ dΘ dΦ2π sin Ψ,
ĨΘΦ=12πΔΘΔΦ212akξsin2 Θ sin Φξ2dΘ dΦ,
A=12C r2θdθ=a20π1+n=010cn cosnθ2dθ=πa21+2c0+c02+n=110cn22,
q, Σ=A expiωt-krrCa=0Rθ=02π×expikqa/rcosθ-Σ×1+n=010cn cosnθ×a1+n=010cn cosnθ2da dθ.
χ=kqrcosθ-Σ, ξ=1+n=010cn cosnθ,
q, Σ=C a=0Rθ=02πexpiχξaaξ2da dθ
=-C θ=02πexpiχξaχ2ξ2iχξa-1a=0Rξ2dθ
=-C θ=02πexpiχξRiχξR-1+1χ2dθ.
q, Σ=Cθ=02πχξR sinχξR+cosχξR-1χ2dθ+i θ=02πsinχξR-χξR cosχξRχ2dθ,
qrcosθ-Σ=cos θ cos Θ - sin θ cos Φ.
ND=N0 exp-ΛD, N0=0.08 cm-4,
PD= 1 - exp-ΛD.
0D D2 exp-ΛDdD=-exp-ΛD×D2Λ+2DΛ2+2Λ3+2Λ3.
0D D2 exp-ΛDdD0 D2 exp-ΛDdD=-exp-ΛD×Λ2D22+DΛ+1+1.
τz=NeCz,
τz=2πa2Nez,
Ne=0 N0 exp-ΛDdD.
τz=2πN0z 0 a2 exp-ΛDdD
=4πN0z 0 a2 exp-2Λada
=πN0Λ3 z.
NV=V 0.020.8 N0 exp-ΛD dDVN0Λexp-0.02Λ,
D=-1Λln x, with x0, 1,

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