Abstract

The approximations of Penndorf and Shifrin-Punina to the Mie solution at forward scattering angles are extended to smaller size parameter values. The present approximation, Eq. (7), is found to represent accurately the Mie result down to x ~ 0.5–1.0 for refractive index m = 1.33, and to x ~ 2.0 for much larger index values. The implications of this result are discussed relative to the reconstruction of particle size distributions utilizing the Shifrin-Fymat analytical inversion formula of forward scattered intensities.

© 1981 Optical Society of America

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References

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  1. K. S. Shifrin, A. Y. Perelman, in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1967).
  2. A. L. Fymat, in Remote Monitoring of the Terrestrial Atmosphere from Space (Rassegna, Rome, 1973), p. 183; also in Proceedings of the International Colloquium on Drops and Bubbles, Vol. 2, D. J. Collins, M. S. Plesset, M. M. Saffren, Eds. (1974), p. 572. (U.S. Government Publication 1976-685-197/2).
  3. A. L. Fymat, Appl. Opt. 17, 1677 (1978).
    [CrossRef] [PubMed]
  4. A. L. Fymat, K. D. Mease, in Remote Sensing of the Atmosphere: Inversion Methods and Applications, A. L. Fymat, V. E. Zuev, Eds. (Elsevier, Amsterdam, 1978), p. 195.
  5. J. R. Hodkinson, Appl. Opt. 5, 839 (1966).
    [CrossRef] [PubMed]
  6. R. O. Gumprecht, C. M. Slipcevich, J. Phys. Chem. 57, 90 (1953).
    [CrossRef]
  7. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  8. R. Penndorf, J. Opt. Soc. Am. 52, 797 (1962).
    [CrossRef]
  9. K. S. Shifrin, V. A. Punina, Bull. Izv. Acad. Sci. USSR Atmos. Oceanic. Phys. 4, No. 7, 450 (1968).

1978

1968

K. S. Shifrin, V. A. Punina, Bull. Izv. Acad. Sci. USSR Atmos. Oceanic. Phys. 4, No. 7, 450 (1968).

1966

1962

1953

R. O. Gumprecht, C. M. Slipcevich, J. Phys. Chem. 57, 90 (1953).
[CrossRef]

Fymat, A. L.

A. L. Fymat, Appl. Opt. 17, 1677 (1978).
[CrossRef] [PubMed]

A. L. Fymat, K. D. Mease, in Remote Sensing of the Atmosphere: Inversion Methods and Applications, A. L. Fymat, V. E. Zuev, Eds. (Elsevier, Amsterdam, 1978), p. 195.

A. L. Fymat, in Remote Monitoring of the Terrestrial Atmosphere from Space (Rassegna, Rome, 1973), p. 183; also in Proceedings of the International Colloquium on Drops and Bubbles, Vol. 2, D. J. Collins, M. S. Plesset, M. M. Saffren, Eds. (1974), p. 572. (U.S. Government Publication 1976-685-197/2).

Gumprecht, R. O.

R. O. Gumprecht, C. M. Slipcevich, J. Phys. Chem. 57, 90 (1953).
[CrossRef]

Hodkinson, J. R.

Mease, K. D.

A. L. Fymat, K. D. Mease, in Remote Sensing of the Atmosphere: Inversion Methods and Applications, A. L. Fymat, V. E. Zuev, Eds. (Elsevier, Amsterdam, 1978), p. 195.

Penndorf, R.

Perelman, A. Y.

K. S. Shifrin, A. Y. Perelman, in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1967).

Punina, V. A.

K. S. Shifrin, V. A. Punina, Bull. Izv. Acad. Sci. USSR Atmos. Oceanic. Phys. 4, No. 7, 450 (1968).

Shifrin, K. S.

K. S. Shifrin, V. A. Punina, Bull. Izv. Acad. Sci. USSR Atmos. Oceanic. Phys. 4, No. 7, 450 (1968).

K. S. Shifrin, A. Y. Perelman, in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1967).

Slipcevich, C. M.

R. O. Gumprecht, C. M. Slipcevich, J. Phys. Chem. 57, 90 (1953).
[CrossRef]

van de Hulst, H. C.

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

Appl. Opt.

Bull. Izv. Acad. Sci. USSR Atmos. Oceanic. Phys.

K. S. Shifrin, V. A. Punina, Bull. Izv. Acad. Sci. USSR Atmos. Oceanic. Phys. 4, No. 7, 450 (1968).

J. Opt. Soc. Am.

J. Phys. Chem.

R. O. Gumprecht, C. M. Slipcevich, J. Phys. Chem. 57, 90 (1953).
[CrossRef]

Other

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

A. L. Fymat, K. D. Mease, in Remote Sensing of the Atmosphere: Inversion Methods and Applications, A. L. Fymat, V. E. Zuev, Eds. (Elsevier, Amsterdam, 1978), p. 195.

K. S. Shifrin, A. Y. Perelman, in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1967).

A. L. Fymat, in Remote Monitoring of the Terrestrial Atmosphere from Space (Rassegna, Rome, 1973), p. 183; also in Proceedings of the International Colloquium on Drops and Bubbles, Vol. 2, D. J. Collins, M. S. Plesset, M. M. Saffren, Eds. (1974), p. 572. (U.S. Government Publication 1976-685-197/2).

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Figures (2)

Fig. 1
Fig. 1

(a) Percent errors relative to the Mie exact forward scattering for refractive index m = 1.33. Results are illustrated as a function of the particle size parameter for the F-, the PSP-, and the FM-approximations. (b) Same as Fig. 1(a) for refractive index m = 1.50. (c) Same as Fig. 1(a) for refractive index m = 1.60.

Fig. 2
Fig. 2

(a) Normalized forward scattered intensities for size parameter x = 2. Results are illustrated for the exact Mie (M) scattering and for the F-, the PSP-, and the FM-approximations. (Numbers on the curves are percent errors relative to Mie scattering.) (b) Same as Fig. 2(a) for size parameter x = 5. (c) Same as Fig. 2(a) for size parameter x = 14.

Tables (2)

Tables Icon

Table I Forward Scattered Intensities for the Penndorf-Shifrin-Punina (PSP) and Fymat-Mease (FM) Approximations Compared With the Mie (M) Exact Solution for a Transparent Water Aerosol

Tables Icon

Table II Percentage of Relative Errors of the PSP- and FM-Approximations Relative to the Mie Result for Selected Size Parameter Ranges and Various Refractive Indices

Equations (14)

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I M ( 0 ) = k - 2 { [ Re S ( 0 ) ] 2 + [ Im S ( 0 ) ] 2 ] ,
Re S ( 0 ) = ( x 2 / 4 ) Q ext ,
I p = x 4 4 k 2 ( Q ext 2 4 ) ,
I F ( θ ) = x 2 J 1 2 ( x sin θ ) k 2 sin 2 θ ,
I F ( 0 ) = x 4 4 k 2 .
I P I P ( 0 ) = I F ( 0 ) ( Q ext 2 4 ) .
I P S P ( θ ) = I F ( θ ) ( Q ext 2 4 ) .
I F M ( θ ) = I P S P ( θ ) f 1 f 2 ,
f 1 - 1 f - 1 [ ( m - 1 ) x ] = 1 - J 0 2 [ ( m - 1 ) x ] for ( m - 1 ) x a or b ( m - 1 ) x c = 1 , otherwise f 2 - 1 f - 1 [ 2 ( m - 1 ) x ] = 1 - J 0 2 [ 2 ( m - 1 ) x ] for 2 ( m - 1 ) x d = 1 , otherwise .
a = 3.63 ,             b = 5.52 ,             c = 6.6 ,             d = 2.40.
I θ = k - 1 0 I M ( θ , x ) n ( x ) d x ,
I θ k - 1 0 I F M ( θ , x ) n ( c ) d x = 1 k 3 θ 2 0 x 2 J 1 2 ( y ) n * ( x ) d x ,
n * ( x ) = [ Q ext 2 ( m , x ) f 1 f 2 / 4 ] n ( x ) .
n * ( x ) = - 2 π k 3 x 2 0 J 1 ( y ) / Y 1 ( y ) y d d θ ( θ 3 I θ ) d θ .

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