Abstract

An analytic semi-empirical approximation to the extinction efficiency Qext for randomly oriented spheroids that is based on an extension of the anomalous diffraction formula is given and compared with the extended boundary condition method or T-matrix method. With this formula Qext can be evaluated over 104 times faster than by previous methods. This approximation has been verified for complex refractive indices m = nik, where 1 ≤ n ≤ ∞ and 0 ≤ k ≤ ∞, and aspect ratios from 0.2–5. We believe that the approximation is uniformly valid over all size parameters and aspect ratios. It has the correct Rayleigh, refractive-index, and large-particle asymptotic behaviors. The accuracy and limitations of this formula are discussed.

© 1994 Optical Society of America

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References

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  1. G. R. Fournier, B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30, 2042–2048 (1991).
    [CrossRef] [PubMed]
  2. G. R. Fournier, B. T. N. Evans, “Bridging the gap between the Rayleigh and Thomson limits for various spheres and spheroids,” Appl. Opt. 32, 6159–6166 (1993).
    [CrossRef] [PubMed]
  3. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 11, pp. 172–183.
  4. D. S. Jones, “High-frequency scattering of electromagnetic waves,” Proc. R. Soc. London Ser. A 240, 206–213 (1957).
    [CrossRef]
  5. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).
  6. P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).
  7. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 10, pp. 575–582.
  8. J. M. Greenberg, A. S. Meltzer, “The effect of orientation of nonspherical particles on interstellar extinction,” Astrophys. J. 132, 667–671 (1960).
    [CrossRef]
  9. B. T. N. Evans, G. R. Fournier, “Algebraic approximations to some integrals in optics,” J. Phys. A 26, 1–17 (1993).
    [CrossRef]
  10. S. Wolfram, Mathematica: A System for Doing Mathematics by Computer (Addison-Wesley, New York, 1989).
  11. H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
    [CrossRef]
  12. V. P. Beckmann, W. Franz, “Berechnung der Streuquer-schnitte von Kugel und Zylinder unter Anwedung einer modifizierten Watson-Transformation,” Z. Naturforschg. A 12, 533–537 (1957).
  13. Y. L. Luke, The Special Functions and Their Approximations (Academic, New York, 1969), Vol. 1, Chap. 3.5, pp. 48–57.
  14. B. T. N. Evans, G. R. Fournier, “A simple approximation to extinction efficiency valid over all size parameters,” Appl. Opt. 29, 4666–4670 (1990).
    [CrossRef] [PubMed]
  15. A. Cohen, E. Tirosh, “Absorption by a large sphere with an arbitrary complex refractive index,” J. Opt. Soc. Am. A 7, 323–325 (1990).
    [CrossRef]
  16. B. T. N. Evans, G. R. Fournier, “Analytic approximation to randomly oriented spheroid extinction,” rep. DREV R–4712/93 (Defence Research Establishment Valcartier, Quebec, 1993).

1993 (2)

1991 (1)

1990 (2)

1980 (1)

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

1960 (1)

J. M. Greenberg, A. S. Meltzer, “The effect of orientation of nonspherical particles on interstellar extinction,” Astrophys. J. 132, 667–671 (1960).
[CrossRef]

1957 (2)

D. S. Jones, “High-frequency scattering of electromagnetic waves,” Proc. R. Soc. London Ser. A 240, 206–213 (1957).
[CrossRef]

V. P. Beckmann, W. Franz, “Berechnung der Streuquer-schnitte von Kugel und Zylinder unter Anwedung einer modifizierten Watson-Transformation,” Z. Naturforschg. A 12, 533–537 (1957).

Barber, P. W.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

Beckmann, V. P.

V. P. Beckmann, W. Franz, “Berechnung der Streuquer-schnitte von Kugel und Zylinder unter Anwedung einer modifizierten Watson-Transformation,” Z. Naturforschg. A 12, 533–537 (1957).

Cohen, A.

Evans, B. T. N.

Fournier, G. R.

Franz, W.

V. P. Beckmann, W. Franz, “Berechnung der Streuquer-schnitte von Kugel und Zylinder unter Anwedung einer modifizierten Watson-Transformation,” Z. Naturforschg. A 12, 533–537 (1957).

Greenberg, J. M.

J. M. Greenberg, A. S. Meltzer, “The effect of orientation of nonspherical particles on interstellar extinction,” Astrophys. J. 132, 667–671 (1960).
[CrossRef]

Hill, S. C.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

Jones, D. S.

D. S. Jones, “High-frequency scattering of electromagnetic waves,” Proc. R. Soc. London Ser. A 240, 206–213 (1957).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 10, pp. 575–582.

Luke, Y. L.

Y. L. Luke, The Special Functions and Their Approximations (Academic, New York, 1969), Vol. 1, Chap. 3.5, pp. 48–57.

Meltzer, A. S.

J. M. Greenberg, A. S. Meltzer, “The effect of orientation of nonspherical particles on interstellar extinction,” Astrophys. J. 132, 667–671 (1960).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

Tirosh, E.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 11, pp. 172–183.

Wiscombe, W. J.

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

Wolfram, S.

S. Wolfram, Mathematica: A System for Doing Mathematics by Computer (Addison-Wesley, New York, 1989).

Appl. Opt. (3)

Astrophys. J. (1)

J. M. Greenberg, A. S. Meltzer, “The effect of orientation of nonspherical particles on interstellar extinction,” Astrophys. J. 132, 667–671 (1960).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. A (1)

B. T. N. Evans, G. R. Fournier, “Algebraic approximations to some integrals in optics,” J. Phys. A 26, 1–17 (1993).
[CrossRef]

Phys. Rev. Lett. (1)

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

D. S. Jones, “High-frequency scattering of electromagnetic waves,” Proc. R. Soc. London Ser. A 240, 206–213 (1957).
[CrossRef]

Z. Naturforschg. A (1)

V. P. Beckmann, W. Franz, “Berechnung der Streuquer-schnitte von Kugel und Zylinder unter Anwedung einer modifizierten Watson-Transformation,” Z. Naturforschg. A 12, 533–537 (1957).

Other (7)

Y. L. Luke, The Special Functions and Their Approximations (Academic, New York, 1969), Vol. 1, Chap. 3.5, pp. 48–57.

B. T. N. Evans, G. R. Fournier, “Analytic approximation to randomly oriented spheroid extinction,” rep. DREV R–4712/93 (Defence Research Establishment Valcartier, Quebec, 1993).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 10, pp. 575–582.

S. Wolfram, Mathematica: A System for Doing Mathematics by Computer (Addison-Wesley, New York, 1989).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 11, pp. 172–183.

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Figures (11)

Fig. 1
Fig. 1

Comparison between approximation and T-matrix methods for an index m = 1.3–0i and an aspect ratio r = 2.

Fig. 2
Fig. 2

Comparison between the approximation and T-matrix method for an index m = 1.3–0i and an aspect ratio r = 1/2.

Fig. 3
Fig. 3

Maximum relative error (percent) between the analytic approximation and Mie theory for r = 1.

Fig. 4
Fig. 4

Maximum relative error (percent) between the analytic approximation and T matrix for r = 2.

Fig. 5
Fig. 5

Comparison between approximation and Mie theory for an index m = 1.8–0i and an aspect ratio r = 1. There are significant surface waves.

Fig. 6
Fig. 6

Comparison between approximation and T-matrix method for an index m = 1–3i and an aspect ratio r = 1/2. There is a significant internal wave coherence effect.

Fig. 7
Fig. 7

Comparison between the approximation and T-matrix methods for an index m = 3–0i and an aspect ratio r = 2. There is an incipient MDR at b ≈ 1.

Fig. 8
Fig. 8

As in Fig. 7 (m = 3–0i, r = 2) but for T-matrix results low-pass-filtered for b > 1.

Fig. 9
Fig. 9

Comparison between the approximation and T-matrix methods for an index m = 35–35i and an aspect ratio r = 1/3.

Fig. 10
Fig. 10

Comparison between the approximation and T-matrix methods for an index m = 8.075–1.824i and an aspect ratio r = 2 for water at 9.4 GHz.

Fig. 11
Fig. 11

Comparison between the approximation and T-matrix methods for an index m = 8.075–1.824i and an aspect ratio r = 1/2 for water at 9.4 GHz.

Equations (44)

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Q small = Q sca + Q abs ,
Q sca = 16 9 b 4 r 2 A ¯ [ η ˜ 1 2 + η ˜ 1 2 + 2 ( η ˜ 2 2 + η ˜ 2 2 ) ] , Q abs = 8 3 b r A ¯ Re { i [ η ˜ 1 + η ˜ 1 + 2 ( η ˜ 2 + η ˜ 2 ) ] } ,
η ˜ i = 1 3 ( L i + 1 ɛ 1 i - 1 ) ,             η ˜ i = 1 3 ( L i + 1 u 1 i - 1 ) .
ɛ 1 i = [ 2 z ¯ i Ψ 1 ( z ¯ i ) Ψ 1 ( z ¯ i ) ] ζ ,             u 1 i = μ [ 2 z ¯ i Ψ 1 ( z ¯ i ) Ψ 1 ( z ¯ i ) ] ζ ,
z ¯ 1 = ( μ ) 1 / 2 b [ 1 + χ ( 1 - 1 / r 2 ) ] ,             prolates , = ( μ ) 1 / 2 b ( r 2 ) χ ,             oblates ,
z ¯ 2 = ( μ ) 1 / 2 b [ 1 + χ 1 / 3 ( 1 - 1 / r )             prolates , = ( μ ) 1 / 2 b ( r ) χ 1 / 3 ,             oblates ,
A ¯ = 1 + r 2 ( r 2 - 1 ) 1 / 2 sin - 1 [ ( r 2 - 1 ) 1 / 2 r ] ,             prolates , = 1 + r 2 ( 1 - r 2 ) 1 / 2 ln [ 1 + ( 1 - r 2 ) 1 / 2 r ] ,             oblates .
L 1 = ( 1 - g 2 ) g 2 { - 1 + 1 2 g ln ( 1 + g 1 - g ) } , L 2 = 1 - L 1 2 ,             g 2 = 1 - 1 r 2 ,
L 1 = 1 + f 2 f 2 { 1 - tan - 1 f f } , L 2 = 1 - L 1 2 ,             f 2 = 1 r 2 - 1.
Q ad = Re { 2 + 4 exp ( - ω ) ω + 4 [ ( exp ( - ω ) - 1 ) ] ω 2 } ,
ω = i Δ Ψ ,             Δ Ψ = 2 ( m - 1 ) r b p ,
p = ( cos 2 θ + r 2 sin 2 θ ) 1 / 2 ,             a = 2 π α / λ , b = 2 π β / λ ,             m = n - i k .
w = i Δ Ψ = i b { 2 r p [ p 2 cos ( ϕ ) + s sin ( ϕ ) p 2 cos 2 ( ϕ ) + q 2 sin 2 ( ϕ ) + 2 s cos ( ϕ ) sin ( ϕ ) ] } [ m - cos ( ϕ ) ] , cos ( ϕ ) = s 2 + p 2 Δ m ( p 4 + s 2 ) , sin ( ϕ ) = s ( Δ - p 2 ) m ( p 4 + s 2 ) , Δ = [ m 2 ( p 4 + s 2 ) - s 2 ] 1 / 2 , s = ( p 2 q 2 - r 2 ) 1 / 2 , q = [ r 2 cos 2 ( θ ) + sin 2 ( θ ) ] 1 / 2 .
Δ Ψ = 2 b { ( m 2 - cos 2 θ ) 1 / 2 - sin θ } .
Q ad ¯ = 0 π / 2 Q ad p sin θ d θ 0 π / 2 p sin θ d θ .
Q ad ¯ = 2 + 4 ( I 1 - I 2 ) / j ( 0 ) ,
I 1 = A [ e - C { ( 1 + 1 C ) F 2 ( C ) C - ( 1 + 2 C ) F 1 ( C ) C 2 } + ( 1 + 2 C ) F 1 ( 0 ) C 2 + F 2 ( 0 ) C 2 ] ,
I 2 = j ( 0 ) C 2 + A C 2 [ 1 ω ( 0 ) - 1 ω ( π / 2 ) ] + 2 A C 3 ln { B ω ( π / 2 ) [ B + j ( 0 ) ] ω ( 0 ) } ,
A = γ B [ B + j ( 0 ) ] , B = ω ( 0 ) - ω ( π / 4 ) γ + [ ω ( π / 4 ) - ω ( π / 2 ) ] / j ( π / 4 ) , C = ω ( 0 ) - γ B ,             γ = ω ( π / 2 ) - ω ( 0 ) j ( 0 ) ,
j ( θ ) = 1 2 { cos ( θ ) ( 1 - g 2 cos 2 ( θ ) ) 1 / 2 + sin - 1 [ g cos ( θ ) ] g } ,             prolates , = r 2 ( cos ( θ ) [ 1 + f 2 cos 2 ( θ ) 2 ] 1 / 2 + ln { f cos ( θ ) + [ 1 + f 2 cos 2 ( θ ) ] 1 / 2 } f ) ,             oblates ,
F n ( C ) = E n [ ω ( 0 ) - C ] ( ω ( 0 ) - C ) n - 1 - E n [ ω ( π / 2 ) - C ] [ ω ( π / 2 ) - C ] n - 1 ,
ω ( 0 ) = 2 b i [ ( m 2 - 1 ) 1 / 2 ( m - 1 ) ( 1 - α ) ( m - 1 ) + α ( m 2 - 1 ) 1 / 2 / r ] , ω ( π / 4 ) = 2 b i { g 2 [ ( m 2 - 1 / 2 ) 1 / 2 - 1 / 2 ] + m - 1 r 2 } , ω ( π / 2 ) = 2 b i ( m - 1 ) , 1 α = 1 - r c ( 1 - r ) ( 1 - r c ) ,             r c = | m - 1 m + 1 | 1 / 2 ,
ω ( 0 ) = 2 b r i ( m - 1 ) , ω ( π / 4 ) = 2 b r i ( m - 1 ) ( 2 1 + r 2 ) 1 / 2 , ω ( π / 2 ) = 2 b i ( m - 1 ) .
Q edge = 2 D r 2 / 3 p 2 / 3 F 2 1 [ - 2 / 3 , 1 / 2 ; 1 ; ( 1 - 1 / p 2 ) ] , prolates , = 2 D r 2 / 3 p 12 / 3 F 2 1 [ - 2 / 3 , 1 / 2 ; 1 ; ( 1 - p 2 ) ] , oblates ,
F 2 1 ( - 2 / 3 , 1 / 2 ; 1 ; z ) 0.999947 - 2.19081 z + 1.51871 z 2 - 0.325449 z 3 1 - 1.85884 z + 0.947705 z 2 - 0.0847327 z 3 , z 1 , D = c e b 2 / 3 .
D = c 0 [ b 2 β / 3 + 1 / m - 1 β ] 1 / β β = 4 / 25 m - 1 3 + 8 / 125 + | c 0 - c ln m - 1 | , = 20 ,             1.95 < m < 2.05
Q ext 2 + Q edge .
T = 2 - exp ( - Q edge / 2 ) .
T ¯ = 0 π / 2 T p sin θ d θ 0 π / 2 p sin θ d θ .
T ¯ = 2 - exp ( - δ C ) A F 2 / j ( 0 ) ,
A = γ B [ B + j ( 0 ) ] , B = 1 - v ( π / 4 ) γ + [ v ( π / 4 ) - v ( π / 2 ) / j ( π / 4 ) , C = 1 - γ B ,             γ = v ( π / 2 ) - 1 j ( 0 ) ,             δ = D r 2 / 3 ,
F 2 E 2 ( δ ) δ - E 2 ( δ v ( π / 2 ) ) δ v ( π / 2 ) .
j ( θ ) = r j ( θ ) , v ( π / 4 ) = F 2 1 [ - 2 3 , 1 2 ; 1 ; ( r 2 - 1 ) / ( r 2 + 1 ) ] × [ 2 / ( r 2 + 1 ) ] 3 , v ( π / 2 ) = F 2 1 [ - 2 3 , 1 2 ; 1 ; 1 - 1 / r 2 ] / r 2 / 3 ,
j ( θ ) = - j ( θ ) , v ( π / 4 ) = F 2 1 [ - 2 3 , 1 2 ; 1 ; 1 - r 2 / 2 ] / r , v ( π / 2 ) = F 2 1 [ - 2 3 , 1 2 ; 1 ; 1 - r 2 ] / r 2 .
Q large = Q ad T .
Q ¯ large = 0 π / 2 Q ad T p sin θ d θ 0 π / 2 p sin θ d θ .
Q ¯ large = Q ¯ ad T ¯ .
lim z 0 Γ ( b - a ) Γ ( b ) c a z a ν 1 F 1 [ a ; b , - c z ν ] Γ ( b - a ) Γ ( b ) c a z a ν [ 1 - c a z ν b + b a ( a + 1 ) z 2 ν 2 ! b ( b + 1 ) - ] , lim z Γ ( b - a ) Γ ( b ) c a z a ν 1 F 1 [ a ; b , - c z ν ] a ( 1 + a - b ) c z ν + .
B Q ¯ small F 1 1 [ 1 / ν ; b , - ( c z ) ν ] ,
c = Γ ( b ) / Γ ( b - 1 / ν ) ,             z = Q ¯ small / Q ¯ large ,
lim z 0 B Q ¯ small , lim z B Q ¯ large .
lim b B Q ¯ small [ 1 + z ν ] 1 / ν .
1 Q ¯ ex γ = 1 Q ¯ small γ + 1 Q ¯ large γ .
γ = γ l + ( 54 - γ l ) 1 + ( α r 1 / 3 b ) 4 ,             α = 5 2 + 3 k c k c + k , k c = ( n 2 - 1 ) 2 66 n , γ l = 1 + n - 1 [ ( n - 1 ) 1 / 3 + 1 ] 3 + 4 [ 4 n ( 16 + n 2 ) 1 / 2 - 1 ] 2 + 16 u 2 ( u 4 + 1 ) 1 / 2 ,             u = k [ 2 ( n - 1 ) ] 2

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