Abstract

A numerical method was developed using geometrical optics to predict far-field optical scattering from particles that are symmetric about the optic axis. The diffractive component of scattering is calculated and combined with the reflective and refractive components to give the total scattering pattern. The phase terms of the scattered light are calculated as well. Verification of the method was achieved by assuming a spherical particle and comparing the results to Mie scattering theory. Agreement with the Mie theory was excellent in the forward-scattering direction. However, small-amplitude oscillations near the rainbow regions were not observed using the numerical method. Numerical data from spheroidal particles and hemispherical particles are also presented. The use of hemispherical particles as a calibration standard for intensity-type optical particle-sizing instruments is discussed.

© 1991 Optical Society of America

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References

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  1. E. A. Hovenac, “Droplet sizing instrumentation used for icing research: operation, calibration, and accuracy,” NASA Contract Rep. 182293 DOT/FAA/CD-89/13 (1989).
  2. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  3. A. Ungut, G. Grehan, G. Gouesbet, “Comparisons between geometrical optics and Lorenz–Mie theory,” Appl. Opt. 20, 2911–2918 (1981).
    [CrossRef] [PubMed]
  4. S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
    [PubMed]
  5. S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712–732 (1979).
    [CrossRef] [PubMed]
  6. S. Asano, M. Sato, “Light scattered by randomly oriented spheroidal particles,” Appl. Opt. 19, 962–974 (1980).
    [CrossRef] [PubMed]
  7. J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
    [CrossRef]
  8. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
    [CrossRef]
  9. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
    [CrossRef]

1981 (1)

1980 (1)

1979 (1)

1976 (1)

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

1975 (1)

1969 (2)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[CrossRef]

Asano, S.

Gouesbet, G.

Grehan, G.

Hovenac, E. A.

E. A. Hovenac, “Droplet sizing instrumentation used for icing research: operation, calibration, and accuracy,” NASA Contract Rep. 182293 DOT/FAA/CD-89/13 (1989).

Nussenzveig, H. M.

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

Sato, M.

Ungut, A.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Walker, J. D.

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

Yamamoto, G.

Am. J. Phys. (1)

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

Appl. Opt. (4)

J. Math. Phys. (2)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[CrossRef]

Other (2)

E. A. Hovenac, “Droplet sizing instrumentation used for icing research: operation, calibration, and accuracy,” NASA Contract Rep. 182293 DOT/FAA/CD-89/13 (1989).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

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Figures (10)

Fig. 1
Fig. 1

A 100-μm hemispherical particle attached to a quartz substrate.

Fig. 2
Fig. 2

Rays scattered by a spherical particle and the location of several foci.

Fig. 3
Fig. 3

Geometrical definition of the path length of a ray through a spherical particle.

Fig. 4
Fig. 4

Scattering boundary and reference circle used to determine the optical path length of a ray through a nonspherical particle.

Fig. 5
Fig. 5

Geometrical definition of the differential length dl.

Fig. 6
Fig. 6

Comparison of the numerical method to Mie theory for a spherical water droplet that is 20 μm in diameter.

Fig. 7
Fig. 7

Errors in the calculated scattering pattern can occur at large angles.

Fig. 8
Fig. 8

Numerical calculations of scattering from a spheroid compared to a spherical water droplet.

Fig. 9
Fig. 9

Numerical calculations of scattering from a hemispherical glass particle compared with a spherical water droplet.

Fig. 10
Fig. 10

Scattered intensity of the various component p fields from a 20-μm-diameter water droplet.

Equations (17)

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S 1 r = k a 1 D 1 / 2 e i σ ,
S 2 r = k a 2 D 1 / 2 e i σ ,
optical path length = k [ X 1 + ( n 2 L p ) + d ] .
δ = 2 k a k [ X 1 + ( n 2 L p ) + d ]
= 2 k a k [ ( a a sin τ ) + n 2 p ( 2 a sin τ ) + ( a a sin τ ) ]
= 2 k a ( sin τ n 2 p sin τ ) ,
σ = ( π / 2 ) + δ + ( π / 2 ) ( n a + n b ) .
S D = ( k a ) 2 J 1 ( k a sin θ ) k a sin θ ,
S 1 = p = 0 ( S 1 r ) + S D ,
S 2 = p = 0 ( S 2 r ) + S D .
d l = ( R 1 2 + R 2 2 2 R 1 R 2 cos θ d ) 1 / 2 ,
R 1 = ( x 1 + y 1 ) 1 / 2 , θ 1 = arctan ( y 1 / x 1 ) , R 2 = ( x 2 + y 2 ) 1 / 2 , θ 2 = arctan ( y 2 / x 2 ) , θ d = θ 2 θ 1 .
d p = R a sin θ a d ϕ ,
R a = ( R 1 + R 1 ) / 2 , θ a = ( θ 1 + θ 2 ) / 2 .
flux = I 0 d A
= I 0 sin ( β 1 + β 2 2 ) d l d p ,
d A = r 2 sin θ d θ d ϕ ,

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