Abstract

Results of computations are presented to show the variations of coefficients of four different Legendre series, one for each of the four scattering functions needed in describing directional dependence of the radiation scattered by a sphere. Values of the size parameter (x) covered for this purpose vary from 0.01 to 100.0. An adequate representation of the entire scattering function vs scattering angle curve is obtained after making use of about 2x + 10 terms of the series. It is shown that a section of a scattering function vs scattering angle curve can be adequately represented by a fourier series with less than 2x + 10 terms. The exact number of terms required for this purpose depends upon values of the size parameter and refractive index, as well as upon the values of the scattering angles defining the section under study. Necessary expressions for coefficients of such fourier series are derived with the help of the addition theorem of spherical harmonics.

© 1970 Optical Society of America

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References

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  1. G. Mie, Ann. Phys. 25, 377 (1908).
    [CrossRef]
  2. H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  3. H. H. Denman, W. J. Pangonis, W. Heller, Angular Scattering Functions for Spheres (Wayne State University Press, Detroit, Michigan, 1966).
  4. J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere,” Rep. No. 320–3237, IBM Scientific Center, Palo Alto, Calif. (1968).
  5. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier Publishing Company, Inc., New York, 1969).
  6. R. Hickling, J. Opt. Soc. Amer. 58, 455 (1968).
    [CrossRef]
  7. J. V. Dave, Appl. Opt. 8, 155 (1969).
    [CrossRef] [PubMed]
  8. F. S. Harris, Appl. Opt. 8, 143 (1969).
    [CrossRef] [PubMed]
  9. J. V. Dave, Appl. Opt. 8, 1161 (1969).
    [CrossRef] [PubMed]
  10. W. Hartel, Das Licht 40, 141 (1940).
  11. E. Hobson, Spherical and Ellipsoidal Harmonics (Cambridge University Press, London, 1931).
  12. S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, 1950).
  13. Z. Sekera, “Legendre Series of the Scattering Functions for Spherical Particles,” Rep. No. 5, Contr. No. AF 19(122)-239, Dept. of Meteorology, University of California, Los Angeles, Calif., ASTIA No. AD-3870 (1952).
  14. C. Chu, S. W. Churchill, J. Opt. Soc. Amer. 45, 958 (1955).
    [CrossRef]
  15. G. C. Clark, C. Chu, S. W. Churchill, J. Opt. Soc. Amer. 47, 81 (1957).
    [CrossRef]
  16. R. S. Fraser, “Scattering Properties of Atmospheric Aerosols,” Sci. Rep. No. 2, Contr. No. AF 19(604)-2429, Dept. of Meteorology, University of California, Los Angeles, Calif. (1959).
  17. J. V. Dave, J. Gazdag, “A Modified Fourier Transform Method for Atmospheric Multiple Scattering Calculations,” Rep. No. 320-3266, IBM Scientific Center, Palo Alto, Calif. (1969).
  18. R. B. Penndorf, J. Opt. Soc. Amer. 47, 1010 (1957).
    [CrossRef]
  19. J. V. Dave, B. H. Armstrong, J. Quant. Spectrosc. Rad. Transfer. 10, No. 6 (1970).
    [CrossRef]

1970

J. V. Dave, B. H. Armstrong, J. Quant. Spectrosc. Rad. Transfer. 10, No. 6 (1970).
[CrossRef]

1969

1968

R. Hickling, J. Opt. Soc. Amer. 58, 455 (1968).
[CrossRef]

1957

G. C. Clark, C. Chu, S. W. Churchill, J. Opt. Soc. Amer. 47, 81 (1957).
[CrossRef]

R. B. Penndorf, J. Opt. Soc. Amer. 47, 1010 (1957).
[CrossRef]

1955

C. Chu, S. W. Churchill, J. Opt. Soc. Amer. 45, 958 (1955).
[CrossRef]

1940

W. Hartel, Das Licht 40, 141 (1940).

1908

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Armstrong, B. H.

J. V. Dave, B. H. Armstrong, J. Quant. Spectrosc. Rad. Transfer. 10, No. 6 (1970).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, 1950).

Chu, C.

G. C. Clark, C. Chu, S. W. Churchill, J. Opt. Soc. Amer. 47, 81 (1957).
[CrossRef]

C. Chu, S. W. Churchill, J. Opt. Soc. Amer. 45, 958 (1955).
[CrossRef]

Churchill, S. W.

G. C. Clark, C. Chu, S. W. Churchill, J. Opt. Soc. Amer. 47, 81 (1957).
[CrossRef]

C. Chu, S. W. Churchill, J. Opt. Soc. Amer. 45, 958 (1955).
[CrossRef]

Clark, G. C.

G. C. Clark, C. Chu, S. W. Churchill, J. Opt. Soc. Amer. 47, 81 (1957).
[CrossRef]

Dave, J. V.

J. V. Dave, B. H. Armstrong, J. Quant. Spectrosc. Rad. Transfer. 10, No. 6 (1970).
[CrossRef]

J. V. Dave, Appl. Opt. 8, 155 (1969).
[CrossRef] [PubMed]

J. V. Dave, Appl. Opt. 8, 1161 (1969).
[CrossRef] [PubMed]

J. V. Dave, J. Gazdag, “A Modified Fourier Transform Method for Atmospheric Multiple Scattering Calculations,” Rep. No. 320-3266, IBM Scientific Center, Palo Alto, Calif. (1969).

J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere,” Rep. No. 320–3237, IBM Scientific Center, Palo Alto, Calif. (1968).

Deirmendjian, D.

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

Denman, H. H.

H. H. Denman, W. J. Pangonis, W. Heller, Angular Scattering Functions for Spheres (Wayne State University Press, Detroit, Michigan, 1966).

Fraser, R. S.

R. S. Fraser, “Scattering Properties of Atmospheric Aerosols,” Sci. Rep. No. 2, Contr. No. AF 19(604)-2429, Dept. of Meteorology, University of California, Los Angeles, Calif. (1959).

Gazdag, J.

J. V. Dave, J. Gazdag, “A Modified Fourier Transform Method for Atmospheric Multiple Scattering Calculations,” Rep. No. 320-3266, IBM Scientific Center, Palo Alto, Calif. (1969).

Harris, F. S.

Hartel, W.

W. Hartel, Das Licht 40, 141 (1940).

Heller, W.

H. H. Denman, W. J. Pangonis, W. Heller, Angular Scattering Functions for Spheres (Wayne State University Press, Detroit, Michigan, 1966).

Hickling, R.

R. Hickling, J. Opt. Soc. Amer. 58, 455 (1968).
[CrossRef]

Hobson, E.

E. Hobson, Spherical and Ellipsoidal Harmonics (Cambridge University Press, London, 1931).

Mie, G.

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Pangonis, W. J.

H. H. Denman, W. J. Pangonis, W. Heller, Angular Scattering Functions for Spheres (Wayne State University Press, Detroit, Michigan, 1966).

Penndorf, R. B.

R. B. Penndorf, J. Opt. Soc. Amer. 47, 1010 (1957).
[CrossRef]

Sekera, Z.

Z. Sekera, “Legendre Series of the Scattering Functions for Spherical Particles,” Rep. No. 5, Contr. No. AF 19(122)-239, Dept. of Meteorology, University of California, Los Angeles, Calif., ASTIA No. AD-3870 (1952).

Van de Hulst, H. C.

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

Ann. Phys.

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Appl. Opt.

Das Licht

W. Hartel, Das Licht 40, 141 (1940).

J. Opt. Soc. Amer.

C. Chu, S. W. Churchill, J. Opt. Soc. Amer. 45, 958 (1955).
[CrossRef]

G. C. Clark, C. Chu, S. W. Churchill, J. Opt. Soc. Amer. 47, 81 (1957).
[CrossRef]

R. B. Penndorf, J. Opt. Soc. Amer. 47, 1010 (1957).
[CrossRef]

R. Hickling, J. Opt. Soc. Amer. 58, 455 (1968).
[CrossRef]

J. Quant. Spectrosc. Rad. Transfer.

J. V. Dave, B. H. Armstrong, J. Quant. Spectrosc. Rad. Transfer. 10, No. 6 (1970).
[CrossRef]

Other

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

H. H. Denman, W. J. Pangonis, W. Heller, Angular Scattering Functions for Spheres (Wayne State University Press, Detroit, Michigan, 1966).

J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere,” Rep. No. 320–3237, IBM Scientific Center, Palo Alto, Calif. (1968).

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

R. S. Fraser, “Scattering Properties of Atmospheric Aerosols,” Sci. Rep. No. 2, Contr. No. AF 19(604)-2429, Dept. of Meteorology, University of California, Los Angeles, Calif. (1959).

J. V. Dave, J. Gazdag, “A Modified Fourier Transform Method for Atmospheric Multiple Scattering Calculations,” Rep. No. 320-3266, IBM Scientific Center, Palo Alto, Calif. (1969).

E. Hobson, Spherical and Ellipsoidal Harmonics (Cambridge University Press, London, 1931).

S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, 1950).

Z. Sekera, “Legendre Series of the Scattering Functions for Spherical Particles,” Rep. No. 5, Contr. No. AF 19(122)-239, Dept. of Meteorology, University of California, Los Angeles, Calif., ASTIA No. AD-3870 (1952).

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Figures (8)

Fig. 1
Fig. 1

Variations of the absolute values of normalized coefficients of the Legendre series for the scattering function M2(x,m,Ө) as a function of subscript k of the coefficient.

Fig. 2
Fig. 2

Variations of the absolute values of normalized coefficients of the Legendre series for the scattering function D21(x, m, Ө) as a function of subscript k of the coefficient. For clarity, coefficients with odd and even values of k are plotted separately.

Fig. 4
Fig. 4

Variations of the coefficients of the Legendre series for four different scattering functions (j = 1 through 4) as a function of size parameter of the sphere, k = 1.

Fig. 5
Fig. 5

Same as Fig. 4, but for k = 16.

Fig. 6
Fig. 6

Same as Fig. 4, but for j = 1, 2, and k = 26.

Fig. 7
Fig. 7

Same as Fig. 4, but for j = 3, 4, and k = 26.

Fig. 8
Fig. 8

Variations of the absolute values of coefficients Fn(1) (x, m, μ, μ) of the fourier series for the scattering function M2(x, m, Ө) as a function of subscript n; x = 100, m = 1.342, θ = 90°, θ = 10°, 50°, and 90°.

Tables (4)

Tables Icon

Table I Values of the First Five Normalized Coefficients of Legendre Series for the Scattering Function M2 (x,m,Ө); m = 1.342

Tables Icon

Table II Values of the Normalized Coefficients of the Legendre Series for Scattering Functions of a Sphere; x = 10.0, m = 1.342

Tables Icon

Table III Values of Fn(1)(x,m,μ,μ) for a Few Selected Values of μ; x = 10.0, m = 1.342, μ = 0.0

Tables Icon

Table IV Values of N(μ = cos θ, μ = cos θ) for Three Different Values of θ; x = 100.0, m = 1.342

Equations (29)

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M 2 ( x , m , Ө ) = k = 1 L k ( 1 ) ( x , m ) P k 1 ( cos Ө ) .
L k ( 1 ) ( x , m ) = ( k 0.5 ) m = k a m ( k 1 ) i = 0 k b i ( k 1 ) Δ i , k × Re [ D p ( x , m ) D q * ( x , m ) ] ,
L k ( 2 ) ( x , m ) = ( k 0.5 ) m = k a m ( k 1 ) i = 0 k b i ( k 1 ) Δ i , k × Re [ C p ( x , m ) C q * ( x , m ) ] ,
L k ( 3 ) ( x , m ) = ( 0.5 k 0.25 ) m = k a m ( k 1 ) i = 0 k b i ( k 1 ) Δ i , k × Re [ C p ( x , m ) D q * ( x , m ) + C p ( x , m ) D q ( x , m ) ] ,
L k ( 4 ) ( x , m ) = ( 0.5 k 0.25 ) m = k a m ( k 1 ) i = 0 k b i ( k 1 ) Δ i , k × Im [ C p ( x , m ) D q * ( x , m ) C p * ( x , m ) D q ( x , m ) ] .
k = { ( k 1 ) / 2 for odd k , ( k 2 ) / 2 for even k .
Δ i , k = { 1 for i = 0 , odd k , 2 for i > 0 , odd k , 2 for i 0 , even k .
a ( k 1 ) / 2 ( k 1 ) = 4 ( k 1 ) ( k 2 ) ( 2 k 1 ) ( 2 k 3 ) a ( k 3 ) / 2 ( k 3 ) ,
b 0 ( k 1 ) = ( k 2 k 1 ) 2 b 0 ( k 3 ) .
a m ( k 1 ) = ( 2 m k ) ( 2 m + k 1 ) ( 2 m + k ) ( 2 m k + 1 ) a m 1 ( k 1 ) ,
b i ( k 1 ) = ( k 2 i + 1 ) ( k + 2 i 2 ) ( k 2 i ) ( k + 2 i 1 ) b i 1 ( k 1 ) .
a ( k 2 ) / 2 ( k 1 ) = 4 ( k 1 ) ( k 2 ) ( 2 k 1 ) ( 2 k 3 ) a ( k 4 ) / 2 ( k 3 ) ,
b 0 ( k 1 ) = ( k 1 ) ( k 3 ) k ( k 2 ) b 0 ( k 3 ) .
a m ( k 1 ) = ( 2 m k + 1 ) ( 2 m + k ) ( 2 m + k + 1 ) ( 2 m k + 2 ) a m 1 ( k 1 ) ,
b i ( k 1 ) = ( 2 i + k 1 ) ( 2 i k ) ( 2 i k + 1 ) ( 2 i + k ) b i 1 ( k 1 ) .
C k ( x , m ) = ( 1 / k ) ( 2 k 1 ) ( k 1 ) b k 1 ( x , m ) + ( 2 k 1 ) i = 1 { [ p 1 + ( p + 1 ) 1 ] a p ( x , m ) [ ( p + 1 ) 1 + ( p + 2 ) 1 ] b p + 1 ( x , m ) } ,
D k ( x , m ) = ( 1 / k ) ( 2 k 1 ) ( k 1 ) a k 1 ( x , m ) + ( 2 k 1 ) i = 1 { [ p 1 + ( p + 1 ) 1 ] b p ( x , m ) [ ( p + 1 ) 1 + ( p + 2 ) 1 ] a p + 1 ( x , m ) } ,
Λ k ( j ) ( x , m ) = [ 4 / Q s ( x , m ) x 2 ] L k ( j ) ( x , m ) ,
cos Ө = μ μ + ( 1 μ 2 ) 1 2 ( 1 μ ) 1 2 cos ( φ φ ) .
4 x 2 Q s ( x , m ) M 2 ( x , m , Ө ) = k = 1 N Λ k ( 1 ) ( x , m ) n = 1 k ( 2 δ 1 n ) × ( k n ) ! ( k + n 2 ) ! P k 1 n 1 ( μ ) P k 1 n 1 ( μ ) cos ( n 1 ) ( φ φ ) ,
1 x 2 Q s ( x , m ) M 2 ( x , m , Ө ) = n = 1 N F n ( x , m , μ , μ ) cos ( n 1 ) ( φ φ ) ,
F n ( 1 ) ( x , m , μ , μ ) = ( 2 δ 1 n ) k = n N Λ k ( 1 ) ( x , m ) Y k 1 n 1 ( μ ) Y k 1 n 1 ( μ ) .
Y k 1 n 1 ( μ ) = [ ( k n ) ! / ( k + n 2 ) ! ] 1 2 P k 1 n 1 ( μ ) .
F 1 ( 1 ) ( x , m , μ , μ ) = 3 4 ( 1 μ 2 μ 2 + 3 μ 2 μ 2 ) ,
F 2 ( 1 ) ( x , m , μ , μ ) = 3 μ μ ( 1 μ 2 ) 1 2 ( 1 μ 2 ) 1 2 ,
F 3 ( 1 ) ( x , m , μ , μ ) = 3 4 ( 1 μ 2 ) ( 1 μ 2 ) ,
F 1 ( 2 ) ( x , m , μ , μ ) = 3 / 2 ,
F 1 ( 3 ) ( x , m , μ , μ ) = 2 3 μ μ ,
F 2 ( 3 ) ( x , m , μ , μ ) = 3 2 ( 1 μ 2 ) 1 2 ( 1 μ 2 ) 1 2 .

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