Abstract

A semiempirical approximation to the extinction efficiency Qext for randomly oriented spheroids, based on an extension of the anomalous diffraction formula, is given and compared to the extended boundary condition method or T-matrix method. Using 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 1.01 ≤ n ≤ 2.00 and 0 ≤ k ≤ 1 and aspect ratios from 0.5 to 4. We believe the approximation is uniformly valid over all size parameters and aspect ratios. It has the correct Rayleigh and large particle asymptotic behavior. The accuracy and limitations to this formula are extensively discussed.

© 1991 Optical Society of America

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References

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  1. R. H. Kohl, Ed., in Proceedings, 1985 CRDEC Scientific Conference on Obscuration and Aerosol Research, CRDEC-SP-86019 (July1986).
  2. J. A. Morrison, M. J. Cross, “Scattering of a Plane Electromagnetic Wave by Axisymmetric Raindrops,” AT&T Tech. J. 53, 955–1019 (1974).
  3. J. M. Greenberg, A. S. Meltzer, “The Effect of Orientation of Non-Spherical Particles on Interstellar Extinction,” Astrophys. J. 132, 667–671 (1960).
    [CrossRef]
  4. T. P. Ackerman, O. B. Toon, “Absorption of Visible Radiation in Atmosphere Containing Mixtures of Absorbing and Non-absorbing Particles, Appl. Opt. 20, 3661–3668 (1981).
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  5. S. C. Hill, A. C. Hill, P. W. Barber, “Light Scattering by Size/Shape Distributions of Soil Particles and Spheroids,” Appl. Opt. 23, 1025–1031 (1984).
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    [CrossRef] [PubMed]
  7. S. Asano, G. Yamamoto, “Light Scattering by a Spheroidal Particle,” Appl. Opt. 14, 29–49 (1975).
    [PubMed]
  8. T. Wu, L. L. Tsai, “Scattering from Arbitrarily-Shaped Lossy Dielectric Bodies of Revolution,” Radio Sci. 2, 709–718 (1977).
    [CrossRef]
  9. R. Mittra, W. L. Ko, Y. Rahmat-Samii, “Transform Approach to Electromagnetic Scattering,” Proc. IEEE 67, 1486–1503 (1979).
    [CrossRef]
  10. B. T. N. Evans, G. R. Fournier, “Simple Approximation to Extinction Efficiency Valid Over All Size Parameters,” Appl. Opt. 29, 4666–4670 (1990).
    [CrossRef] [PubMed]
  11. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  12. G. L. Stephens, “Scattering of Plane Waves by Soft Obstacles: Anomalous Diffraction Theory for Circular Cylinders,” Appl. Opt. 23, 954–959 (1984).
    [CrossRef] [PubMed]
  13. D. S. Jones, “High-Frequency Scattering of Electromagnetic Waves,” Proc. R. Soc. London Ser. A 240, 206–213 (1957).
    [CrossRef]
  14. H. M. Nussenzveig, W. J. Wiscombe, “Efficiency Factors in Mie Scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
    [CrossRef]
  15. A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).
  16. S. Wolfram, Mathematica: A System for doing Mathematics by Computer (Addison-Wesley, New York, 1989).
  17. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  18. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1972).
  19. B. T. N. Evans, “An Interactive Program for Estimating Extinction and Scattering Properties of Most Particulate Clouds,” Materials Research Laboratory, Melbourne, Victoria, Australia, MRL-R-1123 (June1988).
  20. V. P. Beckmann, W. Franz, “Berechnung der Streuquerschnitte von Kugel und Zylinder unter Anwendung einer modifizierten Watson-Transformation,” Z. Naturforsh. Teil A 12, 533–537 (1957).

1990 (1)

1984 (2)

1981 (1)

1980 (1)

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

1979 (2)

1977 (1)

T. Wu, L. L. Tsai, “Scattering from Arbitrarily-Shaped Lossy Dielectric Bodies of Revolution,” Radio Sci. 2, 709–718 (1977).
[CrossRef]

1975 (1)

1974 (1)

J. A. Morrison, M. J. Cross, “Scattering of a Plane Electromagnetic Wave by Axisymmetric Raindrops,” AT&T Tech. J. 53, 955–1019 (1974).

1960 (1)

J. M. Greenberg, A. S. Meltzer, “The Effect of Orientation of Non-Spherical 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 Streuquerschnitte von Kugel und Zylinder unter Anwendung einer modifizierten Watson-Transformation,” Z. Naturforsh. Teil A 12, 533–537 (1957).

Ackerman, T. P.

Asano, S.

Barber, P. W.

Beckmann, V. P.

V. P. Beckmann, W. Franz, “Berechnung der Streuquerschnitte von Kugel und Zylinder unter Anwendung einer modifizierten Watson-Transformation,” Z. Naturforsh. Teil A 12, 533–537 (1957).

Chen, S.-H.

Cross, M. J.

J. A. Morrison, M. J. Cross, “Scattering of a Plane Electromagnetic Wave by Axisymmetric Raindrops,” AT&T Tech. J. 53, 955–1019 (1974).

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Evans, B. T. N.

B. T. N. Evans, G. R. Fournier, “Simple Approximation to Extinction Efficiency Valid Over All Size Parameters,” Appl. Opt. 29, 4666–4670 (1990).
[CrossRef] [PubMed]

B. T. N. Evans, “An Interactive Program for Estimating Extinction and Scattering Properties of Most Particulate Clouds,” Materials Research Laboratory, Melbourne, Victoria, Australia, MRL-R-1123 (June1988).

Fournier, G. R.

Franz, W.

V. P. Beckmann, W. Franz, “Berechnung der Streuquerschnitte von Kugel und Zylinder unter Anwendung einer modifizierten Watson-Transformation,” Z. Naturforsh. Teil A 12, 533–537 (1957).

Greenberg, J. M.

J. M. Greenberg, A. S. Meltzer, “The Effect of Orientation of Non-Spherical Particles on Interstellar Extinction,” Astrophys. J. 132, 667–671 (1960).
[CrossRef]

Hill, A. C.

Hill, S. C.

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).

Ko, W. L.

R. Mittra, W. L. Ko, Y. Rahmat-Samii, “Transform Approach to Electromagnetic Scattering,” Proc. IEEE 67, 1486–1503 (1979).
[CrossRef]

Kotlarchyk, M.

Magnus, W.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Meltzer, A. S.

J. M. Greenberg, A. S. Meltzer, “The Effect of Orientation of Non-Spherical Particles on Interstellar Extinction,” Astrophys. J. 132, 667–671 (1960).
[CrossRef]

Mittra, R.

R. Mittra, W. L. Ko, Y. Rahmat-Samii, “Transform Approach to Electromagnetic Scattering,” Proc. IEEE 67, 1486–1503 (1979).
[CrossRef]

Morrison, J. A.

J. A. Morrison, M. J. Cross, “Scattering of a Plane Electromagnetic Wave by Axisymmetric Raindrops,” AT&T Tech. J. 53, 955–1019 (1974).

Nussenzveig, H. M.

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

Oberhettinger, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Rahmat-Samii, Y.

R. Mittra, W. L. Ko, Y. Rahmat-Samii, “Transform Approach to Electromagnetic Scattering,” Proc. IEEE 67, 1486–1503 (1979).
[CrossRef]

Stephens, G. L.

Toon, O. B.

Tricomi, F. G.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Tsai, L. L.

T. Wu, L. L. Tsai, “Scattering from Arbitrarily-Shaped Lossy Dielectric Bodies of Revolution,” Radio Sci. 2, 709–718 (1977).
[CrossRef]

van de Hulst, H. C.

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

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).

Wu, T.

T. Wu, L. L. Tsai, “Scattering from Arbitrarily-Shaped Lossy Dielectric Bodies of Revolution,” Radio Sci. 2, 709–718 (1977).
[CrossRef]

Yamamoto, G.

Appl. Opt. (6)

Astrophys. J. (1)

J. M. Greenberg, A. S. Meltzer, “The Effect of Orientation of Non-Spherical Particles on Interstellar Extinction,” Astrophys. J. 132, 667–671 (1960).
[CrossRef]

AT&T Tech. J. (1)

J. A. Morrison, M. J. Cross, “Scattering of a Plane Electromagnetic Wave by Axisymmetric Raindrops,” AT&T Tech. J. 53, 955–1019 (1974).

Phys. Rev. Lett. (1)

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

Proc. IEEE (1)

R. Mittra, W. L. Ko, Y. Rahmat-Samii, “Transform Approach to Electromagnetic Scattering,” Proc. IEEE 67, 1486–1503 (1979).
[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]

Radio Sci. (1)

T. Wu, L. L. Tsai, “Scattering from Arbitrarily-Shaped Lossy Dielectric Bodies of Revolution,” Radio Sci. 2, 709–718 (1977).
[CrossRef]

Z. Naturforsh. Teil A (1)

V. P. Beckmann, W. Franz, “Berechnung der Streuquerschnitte von Kugel und Zylinder unter Anwendung einer modifizierten Watson-Transformation,” Z. Naturforsh. Teil A 12, 533–537 (1957).

Other (7)

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

R. H. Kohl, Ed., in Proceedings, 1985 CRDEC Scientific Conference on Obscuration and Aerosol Research, CRDEC-SP-86019 (July1986).

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

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

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

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

B. T. N. Evans, “An Interactive Program for Estimating Extinction and Scattering Properties of Most Particulate Clouds,” Materials Research Laboratory, Melbourne, Victoria, Australia, MRL-R-1123 (June1988).

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of the extended eikonal approach: Δψ, the phase difference, is given by L(m − cosϕ), where ϕ is the deflection angle of the central ray and θ is the spheroid orientation angle.

Fig. 2
Fig. 2

Comparison between approximation and T-matrix method for an index of 1.5 and an aspect ratio of 2.

Fig. 3
Fig. 3

Comparison between approximation and T-matrix method for an index of 1.8 and an aspect ratio of 2. The effect of surface waves is pronounced.

Fig. 4
Fig. 4

Comparison between approximation and T-matrix method index of 1.5–0.1i and an aspect ratio of 2. Surface waves are for an damped by absorption. Maximum relative error occurs in the transition region.

Fig. 5
Fig. 5

Comparison between approximation and T-matrix method for an index of 1.8 and an aspect ratio of 0.8. Oblate spheroid example.

Fig. 6
Fig. 6

Comparison between approximation and T-matrix method for an index of 1.5–1i and an aspect ratio of 2. Numerical instabilities appear in T-matrix code for b > 7.

Fig. 7
Fig. 7

Comparison between approximation and T-matrix method for an index of 1.5 and an aspect ratio of 4. Numerical instabilities appear in T-matrix code for about b > 3.

Fig. 8
Fig. 8

Comparison between randomly oriented infinite cylinders and randomly oriented spheroids with an aspect ratio of 100 and index of 1.3.

Fig. 9
Fig. 9

Contour plot of maximum relative error between approximation and T-matrix code for an aspect ratio of 2.

Equations (28)

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Q v = 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 , a = 2 π α / λ , b = 2 π β / λ , m = n - i k ,
w = i Δ ψ = i L ( m - cos ϕ ) = 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 2 ( p 2 - Δ ) 2 m 2 ( p 4 + s 2 ) 2 , Δ = [ m 2 ( p 4 + s 2 ) - s 2 ] 1 / 2 , s = p 2 q 2 - r 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 ] for prolates ,
Δ ψ = 2 a [ ( m 2 - sin 2 θ ) 1 / 2 - cos θ ] for oblates .
Q v ¯ = 0 π / 2 Q v p sin θ d θ 0 π / 2 p sin θ d θ .
Q edge = c 0 S P R 1 / 3 d s ,
S = π b 2 p , d s = b q d χ , R = b r 2 q p 3 ,
q = sin 2 χ + p 2 cos 2 χ ,
Q edge = 4 π c 0 r 2 / 3 b 2 / 3 p 2 0 π / 2 q 4 / 3 d χ ,
= 2 c 0 r 2 / 3 b 2 / 3 p 2 / 3 F 2 1 [ - 2 / 3 , 1 / 2 ; 1 ; ( 1 - 1 / p 2 ) ] ( prolates ) ,
= 2 c 0 r 2 / 3 b 2 / 3 p 2 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 + 1.51891 z 2 - 0.325449 z 3 1 - 1.85884 z + 0.947705 z 2 - 0.0847327 z 3 ,             z 1.
Q ext 2 + Q edge .
T = 2 - exp ( - Q edge / 2 ) .
Q ray = Q sca + Q abs ,
Q sca = 8 3 b 4 r 2 p [ sin 2 θ 2 η 1 2 + ( 1 + cos 2 θ ) 2 η 2 2 ] ,
Q abs = 4 b r p Re { i [ sin 2 θ 2 η 1 + ( 1 + cos 2 θ ) 2 η 2 ] } ,
η 1 = 1 3 ( L 1 + 1 m 2 - 1 ) , η 2 = 1 3 ( L 2 + 1 m 2 - 1 ) ,
L 1 = ( 1 - 2 ) 2 [ - 1 + 1 2 ln ( 1 + 1 - ) ] , L 2 = 1 - L 1 2 , 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 v = 4 Re { ω 3 - ω 2 8 + ω 3 30 - ω 4 144 + } .
Q app = Q ray [ 1 + ( Q ray Q v T ) μ ] 1 / μ ,
μ = α + γ x ,
α = 1 2 + [ ( n - 1 ) - 2 3 k - k 2 ] + [ ( n - 1 ) + 2 3 ( k - 5 k ) ] 2 ,
γ = [ 3 5 - 3 4 ( n - 1 ) 1 / 2 + 3 ( n - 1 ) 4 ] + 5 6 5 + ( n - 1 ) k .
Q app ¯ = 0 π / 2 Q app p sin θ d θ 0 π / 2 p sin θ d θ .

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