Abstract

A semiempirical approximation to the extinction efficiency based on a modification to the anomalous diffraction formula is given and compared to the exact Mie computation. This approximation has been verified for complex refractive indices m = niκ, where 1.01 ≤ n ≤ 2.00 and 0 ≤ κ ≤ 10. The approximation is uniformly valid over all size parameters and has the correct Rayleigh and large particle asymptotic behavior. The accuracy of this formula is discussed as well as its computational advantages. The formula is also applied to some of the lowtran aerosol models.

© 1990 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles, (Wiley, New York, 1957).
  2. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).
  3. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  4. H. M. Nussenzveig, W. J. Wiscombe, “Efficiency Factors in Mie Scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
    [CrossRef]
  5. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 71.
  6. D. Longtin, E. Shettle, Alternatives to Mie Theory, AFGLTR-88-0154, Scientific Report No. 7 (1988).
  7. S. G. Gathman, Optical Properties of the Marine Aerosol as Predicted by a BASIC Version of the Navy Aerosol Model, NRL Memorandum Report 5157 (1983).

1980

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

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 71.

Deirmendjian, D.

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

Gathman, S. G.

S. G. Gathman, Optical Properties of the Marine Aerosol as Predicted by a BASIC Version of the Navy Aerosol Model, NRL Memorandum Report 5157 (1983).

Kerker, M.

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

Longtin, D.

D. Longtin, E. Shettle, Alternatives to Mie Theory, AFGLTR-88-0154, Scientific Report No. 7 (1988).

Nussenzveig, H. M.

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

Shettle, E.

D. Longtin, E. Shettle, Alternatives to Mie Theory, AFGLTR-88-0154, Scientific Report No. 7 (1988).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 71.

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]

Phys. Rev. Lett.

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

Other

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 71.

D. Longtin, E. Shettle, Alternatives to Mie Theory, AFGLTR-88-0154, Scientific Report No. 7 (1988).

S. G. Gathman, Optical Properties of the Marine Aerosol as Predicted by a BASIC Version of the Navy Aerosol Model, NRL Memorandum Report 5157 (1983).

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

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

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

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Figures (6)

Fig. 1
Fig. 1

Extinction efficiency is plotted against ρ = 2(n − 1)r for a value of m, the complex index of refraction, corresponding to water at a wavelength of 0.55 μm. The solid line is given by the full Mie theory and the dashed line is given by the approximate formula.

Fig. 2
Fig. 2

Extinction efficiency is given as a function of ρ for a value of m where glory and surface wave effects become significant. The solid line is Mie theory and the dashed line is the approximate formula.

Fig. 3
Fig. 3

Extinction efficiency is shown as a function of ρ for a value of m corresponding to water at a wavelength of 10.59 μm. The solid line is given by Mie theory and the dashed line by the approximation.

Fig. 4
Fig. 4

Contour plot over the complex index of refraction plane of the maximum relative error in percentage units between the Mie theory and the proposed approximation, (QappQmie)/Qmie. Where this error occurs as a function of size parameter is discussed in the text.

Fig. 5
Fig. 5

Extinction cross section integrated over the zeroth order lognormal distribution given in the lowtran code for the rural aerosol component. Both the distribution and the index of refraction change as a function of relative humidity. The cross section is given in microns squared.

Fig. 6
Fig. 6

This plot is similar to Fig. 5 except that the zero order lognormal distributions and indices of refraction appropriate to the large oceanic particles have been used to carry out the average. The units are microns squared.

Equations (13)

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Q v = Re [ 2 + 4 exp ( - ω ) ω + 4 ( exp ( - ω ) - 1 ) ω 2 ] ,
ω = 2 κ x + i ρ , x = 2 π r / λ , ρ = 2 x ( n - 1 ) , m = n - i κ .
Q ext = 2 + 1.9923861 x - 2 / 3 + 2 x Im [ ( m 2 + 1 ) ( m 2 - 1 ) 1 / 2 ] - 0.7153537 x - 4 / 3 .
T = 2 - exp ( - x - 2 / 3 ) .
exp ( - x - 2 / 3 ) = 1 - 0.9664 x - 2 / 3 + 0.3536 x - 4 / 3 .
Q R = - Im { 4 m 2 - 1 m 2 + 2 x + [ 4 15 ( m 2 - 1 m 2 + 2 ) 2 m 4 + 27 m 2 + 38 2 m 2 + 3 ] x 3 } + Re [ 8 3 ( m 2 - 1 m 2 + 2 ) 2 ] x 4 +
Q v = 4 Re ( ω 3 - ω 2 8 + ω 3 30 - ω 4 144 + ) .
Q app = Q R [ 1 + ( Q R 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 .
N ( r ) = exp ( - σ g 2 / 2 ) 2 π r m σ g exp ( - ( ln r - ln r m ) 2 2 σ g 2 ) ,
σ = π 0 r 2 Q ext N ( r ) d r .

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