Abstract

Normalized scattering phase functions and extinction efficiencies for homogeneous dielectric spheres with size parameters 200 ≤ x ≤ 4520 and refractive index m = 1.333 − Oi are calculated directly from Mie theory. The results are compared to predictions cited in the literature. Averages of the angular structure and magnitude of the glory are given as a function of the dimensionless size parameter. The periods and amplitudes of regular oscillations in the backscatter phase function are determined. Distinct backscattering oscillations with periods Δx ≃ 0.41, 0.83, 1.1, and 14 are found.

© 1978 Optical Society of America

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References

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  1. S. T. Shipley, E. W. Eloranta, and J. A. Weinman, “Measurement of Rainfall Rates by Lidar,” J. Appl. Meteor. 13, 800–807 (1974).
    [CrossRef]
  2. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  3. H. C. Bryant and A. J. Cox, “Mie Theory and the Glory,” J. Opt. Soc. Am. 56, 1529–1532 (1966).
    [CrossRef]
  4. 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]
  5. G. Mie, “Beitrange zur Optik Trüber Medien, Speziell Kolloidaler Metallosungen,” Ann. Phys. 25, 377–445 (1908).
    [CrossRef]
  6. T. S. Fahlen and H. C. Bryant, “Optical Back Scattering from Single Water Droplets,” J. Opt. Soc. Am. 58, 304–310 (1968).
    [CrossRef]
  7. J. V. Dave, “Scattering of Visible Light by Large Water Spheres,” Appl. Opt. 8, 155–164 (1969).
    [CrossRef] [PubMed]
  8. V. Khare and H. M. Nussenzveig, “Theory of the Glory,” Phys. Rev. Lett. 38, 1279 (1977).
    [CrossRef]
  9. J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere,” , Palo Alto, (1968).
  10. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969).
  11. Ref. 2, p. 263.
  12. S. T. Shipley, Ph.D. Thesis, University of Wisconsin, Madison (1978) (to be published).
  13. K. S. Shifrin and Ya. I. Rabinovich, “The Spectral Indicatrices of Large Water Drops and the Spectral Polarization of Rainbows,” Bull. Acad. Sci. USSR 12, 73–89 (1957).
  14. F. Volz, “Der Regenbogen,” in Handbuch der Geophysik, 8 (Bornträger, Berlin, 1961), Vol. 8.
  15. Ref. 4, Eq. 6.73, p. 162.
  16. Ref. 4, Eq. 5.27, p. 152.
  17. Ref. 2, pp. 249–258.

1977 (1)

V. Khare and H. M. Nussenzveig, “Theory of the Glory,” Phys. Rev. Lett. 38, 1279 (1977).
[CrossRef]

1974 (1)

S. T. Shipley, E. W. Eloranta, and J. A. Weinman, “Measurement of Rainfall Rates by Lidar,” J. Appl. Meteor. 13, 800–807 (1974).
[CrossRef]

1969 (2)

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]

J. V. Dave, “Scattering of Visible Light by Large Water Spheres,” Appl. Opt. 8, 155–164 (1969).
[CrossRef] [PubMed]

1968 (1)

1966 (1)

1957 (1)

K. S. Shifrin and Ya. I. Rabinovich, “The Spectral Indicatrices of Large Water Drops and the Spectral Polarization of Rainbows,” Bull. Acad. Sci. USSR 12, 73–89 (1957).

1908 (1)

G. Mie, “Beitrange zur Optik Trüber Medien, Speziell Kolloidaler Metallosungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Bryant, H. C.

Cox, A. J.

Dave, J. V.

J. V. Dave, “Scattering of Visible Light by Large Water Spheres,” Appl. Opt. 8, 155–164 (1969).
[CrossRef] [PubMed]

J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere,” , Palo Alto, (1968).

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969).

Eloranta, E. W.

S. T. Shipley, E. W. Eloranta, and J. A. Weinman, “Measurement of Rainfall Rates by Lidar,” J. Appl. Meteor. 13, 800–807 (1974).
[CrossRef]

Fahlen, T. S.

Khare, V.

V. Khare and H. M. Nussenzveig, “Theory of the Glory,” Phys. Rev. Lett. 38, 1279 (1977).
[CrossRef]

Mie, G.

G. Mie, “Beitrange zur Optik Trüber Medien, Speziell Kolloidaler Metallosungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Nussenzveig, H. M.

V. Khare and H. M. Nussenzveig, “Theory of the Glory,” Phys. Rev. Lett. 38, 1279 (1977).
[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]

Rabinovich, Ya. I.

K. S. Shifrin and Ya. I. Rabinovich, “The Spectral Indicatrices of Large Water Drops and the Spectral Polarization of Rainbows,” Bull. Acad. Sci. USSR 12, 73–89 (1957).

Shifrin, K. S.

K. S. Shifrin and Ya. I. Rabinovich, “The Spectral Indicatrices of Large Water Drops and the Spectral Polarization of Rainbows,” Bull. Acad. Sci. USSR 12, 73–89 (1957).

Shipley, S. T.

S. T. Shipley, E. W. Eloranta, and J. A. Weinman, “Measurement of Rainfall Rates by Lidar,” J. Appl. Meteor. 13, 800–807 (1974).
[CrossRef]

S. T. Shipley, Ph.D. Thesis, University of Wisconsin, Madison (1978) (to be published).

van de Hulst, H. C.

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

Volz, F.

F. Volz, “Der Regenbogen,” in Handbuch der Geophysik, 8 (Bornträger, Berlin, 1961), Vol. 8.

Weinman, J. A.

S. T. Shipley, E. W. Eloranta, and J. A. Weinman, “Measurement of Rainfall Rates by Lidar,” J. Appl. Meteor. 13, 800–807 (1974).
[CrossRef]

Ann. Phys. (1)

G. Mie, “Beitrange zur Optik Trüber Medien, Speziell Kolloidaler Metallosungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Appl. Opt. (1)

Bull. Acad. Sci. USSR (1)

K. S. Shifrin and Ya. I. Rabinovich, “The Spectral Indicatrices of Large Water Drops and the Spectral Polarization of Rainbows,” Bull. Acad. Sci. USSR 12, 73–89 (1957).

J. Appl. Meteor. (1)

S. T. Shipley, E. W. Eloranta, and J. A. Weinman, “Measurement of Rainfall Rates by Lidar,” J. Appl. Meteor. 13, 800–807 (1974).
[CrossRef]

J. Math. Phys. (1)

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]

J. Opt. Soc. Am. (2)

Phys. Rev. Lett. (1)

V. Khare and H. M. Nussenzveig, “Theory of the Glory,” Phys. Rev. Lett. 38, 1279 (1977).
[CrossRef]

Other (9)

J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere,” , Palo Alto, (1968).

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969).

Ref. 2, p. 263.

S. T. Shipley, Ph.D. Thesis, University of Wisconsin, Madison (1978) (to be published).

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

F. Volz, “Der Regenbogen,” in Handbuch der Geophysik, 8 (Bornträger, Berlin, 1961), Vol. 8.

Ref. 4, Eq. 6.73, p. 162.

Ref. 4, Eq. 5.27, p. 152.

Ref. 2, pp. 249–258.

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Figures (10)

FIG. 1
FIG. 1

The extinction efficiency Kext(m,x) for a homogeneous dielectric sphere with refractive index m = 1.333 − 0i and size parameters near 500. The extinction efficiency cited as Eq. (2) is given by the dashed line.

FIG. 2
FIG. 2

The extinction efficiency Kext(m,x) for a homogeneous dielectric sphere with refractive index m = 1.333 − 0i and size parameters near 4520. The extinction efficiency cited as Eq. (2) is given by the dashed line.

FIG. 3
FIG. 3

A comparison of the modified phase function for diffraction, Eq. (3b), with the exact Mie theory results for size parameters near 500 (△) and 4520 (+). The numerical results were averaged over the extinction efficiency period Δx0 = π/(m − 1) with m = 1.333. These averages have fractional standard deviations less than 2%.

Fig. 4
Fig. 4

The normalized phase function from geometrical optics according to Shifrin and Rabinovich (Ref. 13) (▲) and Volz (Ref. 14) (dash) compared to the exact Mie theory results for x ~ 4520 (solid). The geometrical optics results have been divided by 2 to clarify the comparison. The Mie results were averaged over the Kext(m,x) oscillation period Δx0 = π/(m − 1).

FIG. 5
FIG. 5

The normalized backscatter phase function for m = 1.333 − 0i and size parameters near 500, showing the quasiperiodic structure with period Δxb ≃ 0.83. This data is an extension of the results of Bryant and Cox (Ref. 3).

FIG. 6
FIG. 6

The normalized backscatter phase function for m = 1.333 − 0i and size parameters near 4520, showing the quasiperiodic structure with period Δxb ≃ 0.83.

FIG. 7
FIG. 7

Running mean of the normalized backscatter phase function for size parameters near 500. The running mean was taken over the averaging interval Δxb = 0.83. Two periodic oscillations with periods Δxc ≃ 1.1 and Δxd ≃ 14 are apparent.

FIG. 8
FIG. 8

Average amplitude spectrum Cn as a function of period Δxn of the exact Mie theory results for backscattering with 500 ≤ x ≤ 528. This spectrum shows that the theoretically predicted periodic oscillations of Δxb ≃ 0.83, Δxc ≃ 1.1, and Δxd ≃ 14 are significant.

FIG. 9
FIG. 9

Magnitude of the normalized backscatter phase function averaged over the interval Δxd ≃ 14 as a function of size parameter. These results show a maximum in average backscattering near x ~ 103.

FIG. 10
FIG. 10

Angular variation of the averaged normalized backscatter phase function for x ~ 500 (dots) and x ~ 4520 (solid). The phase function was averaged over the backscattering oscillation period Δxd ≈ 14.

Tables (1)

Tables Icon

TABLE I Average amplitudes for the normalized backscatter phase function for m = 1.333 − 0i for several size parameter ranges.a

Equations (11)

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( m , x , θ ) = 2 x 2 K scat ( m , x ) [ i 1 ( m , x , θ ) + i 2 ( m , x , θ ) ] ,
K ext ( m , x ) = Re ( 2 + 4. ( 0.46 - 0.8 i ) x - 2 / 3 - 8 i m 2 exp [ - 2 i ( m - 1 ) x ] ( m + 1 ) ( m 2 - 1 ) x - 1 + " ripple " ) .
diff ( x , θ ) = 4 x 2 K scat ( J 1 ( u ) u ) 2 ,
diff ( x , θ ) K scat x 2 = 4 ( J 1 ( u ) u ) 2
( m , x , π ) = 1 2 A 0 + n = 1 N [ A n cos ( 2 π x Δ x n ) + B n sin ( 2 π x Δ x n ) ] ,
Δ x n = n δ x .
Δ x b π / [ 4 ( m 2 - 1 ) 1 / 2 + ζ 2 ] = 0.83 ,
f ( x - x 0 , π ) Â { 1 + B ˆ exp [ - 0.45 i ( x - x 0 ) ] + Ĉ exp [ - 5.77 i ( x - x 0 ) ] } ,
Δ x c 2 π / 5.77 = 1.1
Δ x d 2 π / 0.45 = 14.
( m , x , θ ) = g ( m , x , π ) + C [ ( 1 + k ) 2 J 0 2 ( u ) + ( 1 - k ) 2 J 2 2 ( u ) ] ,