Abstract

Approximation formulas are derived for small spherical aerosols (r < λ) by using the series expansion of the exact solution (Mie theory). The first three terms of the series are given for real, complex, and infinite indices of refraction. The new formulas allow for computing the scattering and absorption coefficients for small aerosols more simply and faster than the Mie formulas. The error of our approximation formulas is determined, and it is found that the useful size range is extended by about a factor 2.

© 1962 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (J. Wiley & Sons, Inc., New York, 1957), p. 143.
  2. If K(s)≡ K(e), the notation K only will be used because it is unequivocal.
  3. R. Penndorf, “Scattering coefficients for absorbing and non-absorbing spheres,” AFCRL-TN60-667 Technical Report RAD-TR-60-27, Avco Co., Wilmington, Massachusetts. October1960, 92 p.
  4. G. Mie, Ann. Physik 4, 25, 377 (1908).
  5. The subscript R indicates Rayleigh approximation. Higher approximations are indicated by the use of RR and RRR as subscripts.
  6. H. Blumer, Z. Physik 32, 119 (1925); Z. Physik 38, 304 (1926).
    [Crossref]
  7. J. C. Johnson and J. R. Terrell, J. Opt. Soc. Am. 45, 451 (1955).
    [Crossref]
  8. F. C. Chromey, J. Opt. Soc. Am. 50, 730 (1960).
    [Crossref]
  9. Chromey lists in his table the equivalent value of K(s)and not K(e)− K(s) as he has stated in the text and in the heading of his Table I.

1960 (1)

1955 (1)

1925 (1)

H. Blumer, Z. Physik 32, 119 (1925); Z. Physik 38, 304 (1926).
[Crossref]

1908 (1)

G. Mie, Ann. Physik 4, 25, 377 (1908).

Blumer, H.

H. Blumer, Z. Physik 32, 119 (1925); Z. Physik 38, 304 (1926).
[Crossref]

Chromey, F. C.

Johnson, J. C.

Mie, G.

G. Mie, Ann. Physik 4, 25, 377 (1908).

Penndorf, R.

R. Penndorf, “Scattering coefficients for absorbing and non-absorbing spheres,” AFCRL-TN60-667 Technical Report RAD-TR-60-27, Avco Co., Wilmington, Massachusetts. October1960, 92 p.

Terrell, J. R.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (J. Wiley & Sons, Inc., New York, 1957), p. 143.

Ann. Physik (1)

G. Mie, Ann. Physik 4, 25, 377 (1908).

J. Opt. Soc. Am. (2)

Z. Physik (1)

H. Blumer, Z. Physik 32, 119 (1925); Z. Physik 38, 304 (1926).
[Crossref]

Other (5)

Chromey lists in his table the equivalent value of K(s)and not K(e)− K(s) as he has stated in the text and in the heading of his Table I.

The subscript R indicates Rayleigh approximation. Higher approximations are indicated by the use of RR and RRR as subscripts.

H. C. van de Hulst, Light Scattering by Small Particles (J. Wiley & Sons, Inc., New York, 1957), p. 143.

If K(s)≡ K(e), the notation K only will be used because it is unequivocal.

R. Penndorf, “Scattering coefficients for absorbing and non-absorbing spheres,” AFCRL-TN60-667 Technical Report RAD-TR-60-27, Avco Co., Wilmington, Massachusetts. October1960, 92 p.

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Figures (6)

Fig. 1
Fig. 1

Error in % of the total Mie-scattering coefficient for three approximation formulas. KR represents the leading term only in Eq. (4): KRR represents two terms; and KRRR all three terms in Eq. (4). At the right-hand side, the equivalent radius r (in μ) of the spherical aerosol can be read for four selected wavelengths λ.

Fig. 2
Fig. 2

Total Mie extinction coefficient K(e) (crosses and dashed lines) and KRR(e) (full circles and solid lines) for n ¯ = 1.29 i κ. (a) κ = 0 to 1.29; (b) κ = 1.50 to 5.16.

Fig. 3
Fig. 3

Error in % of the total Mie-extinction coefficient of the two approximations KR(e) and KRR(e) for n ¯ = 1.29 i κ. (a) KR(e), leading term in Eq. (8); (b) KR(e), Eq. (8). Solid lines mean positive error; dashed negative errors.

Fig. 4
Fig. 4

Total Mie-extinction coefficient K(e) (full circles and solid line) and KRR(e) (crosses and broken lines). (a) n ˜ = 1.25 i κ: (b) n ˜ = 1.50 i κ; (c) n ˜ = 1.75 i κ.

Fig. 5
Fig. 5

Ratio K(s)/K(e) for three complex indices of refraction in % as function of the size parameter α and the complex part of the refractive index κ. At the right-hand side, the equivalent radius (in μ) of a spherical aerosol can be read for four-selected wavelengths.

Fig. 6
Fig. 6

Error in % of the total Mie-extinction coefficient for KRR(e) (upper three figures) and the ratio KRR(s)/KRR(e) (lower three figures) for three refractive indices, namely n ˜ = 1.25 i κ at the left, n ˜ = 1.50 i κ in the middle and n ˜ = 1.75 i κ at the right. Solid lines mean positive error (KRR(e) > K(e)), dashed lines a negative error (KRR(e) < K(e)).

Tables (5)

Tables Icon

Table I Real and imaginary parts of am and bm for real n.

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Table II Real and imaginary parts of the factors containing n ˜.

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Table III Real and imaginary parts of am and bm for complex n.

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Table IV Real and imaginary parts of am and bm for n = ∞.

Tables Icon

Table V Total extinction coefficient KRRR(e) for infinite index of refraction and error of the approximation.

Equations (17)

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K ( s ) = 2 α 2 1 ( 2 m + 1 ) [ | a m | 2 + | b m | 2 ] ,
K ( e ) = + 2 α 2 Re [ 1 ( 2 m + 1 ) ( a m + b m ) ] .
K ( a ) = K ( e ) K ( s ) .
K R ( s ) = 8 α 4 3 ( n 2 1 n 2 + 1 ) 2 [ 1 + 6 5 α 2 ( n 2 2 n 2 + 2 ) + α 4 { 3 175 ( n 6 + 41 n 4 284 n 2 + 284 ( n 2 + 2 ) 2 ) + 1 900 ( n 2 + 2 2 n 2 + 3 ) 2 ( 15 + ( 2 n 2 + 3 ) 2 ) } ] .
σ R = π r 2 K R
n ˜ = n i κ ,
k = 4 π κ / λ ,
K R ( e ) = 4 P 2 α + [ 2.4 ( P 1 Q 2 + P 2 Q 1 ) + ( 2 / 3 ) S 2 + ( 2 / 15 ) V 2 ] α 3 + ( 8 / 3 ) ( P 1 2 P 2 2 ) α 4 + [ ( 6 / 175 ) ( P 1 R 2 + P 2 R 1 ) ( 10 / 21 ) T 2 + ( 2 / 225 ) U 2 + ( 2 / 315 ) ( V 2 + W 2 ) ] α 5 .
K R ( e ) = 24 n κ Z 1 α + { 4 15 n κ + 20 n κ 3 Z 2 + 4.8 n κ [ 7 ( n 2 + κ 2 ) 2 + 4 ( n 2 κ 2 5 ) Z 1 2 ] } α 3 + 8 3 { [ ( n 2 + κ 2 ) 2 + ( n 2 κ 2 2 ) ] 2 36 n 2 κ 2 Z 1 2 } α 4 ,
Z 1 = ( n 2 + κ 2 ) 2 + 4 ( n 2 κ 2 ) + 4 , Z 2 = 4 ( n 2 + κ 2 ) 2 + 12 ( n 2 κ 2 ) + 9.
K R ( s ) = ( 8 / 3 ) ( P 1 2 + P 2 2 ) α 4 { 1 + ( 6 / 5 ) Q 1 α 2 ( 4 / 3 ) P 2 α 3 + [ ( 9 / 25 ) ( Q 1 2 + Q 2 2 ) + ( 3 / 175 ) R 1 ] α 4 } + [ ( 2 / 45 ) ( S 1 2 + S 2 2 ) + ( 2 / 675 ) ( V 1 2 + V 2 2 ) ] α 8 .
K R ( s ) = ( 8 / 3 Z 1 2 ) { [ ( n 2 + κ 2 ) 2 + n 2 κ 2 2 ] 2 + 36 n 2 κ 2 } α 4 × { 1 + ( 6 / 5 Z 1 ) [ ( n 2 + κ 2 ) 2 4 ] α 2 ( 24 / 3 Z 1 ) n κ α 3 } .
K R ( e ) = 10 3 α 4 + 4 5 α 6 + ( 1 90 16 175 + 2 405 ) α 8 = ( 10 / 3 ) α 4 ( 1 + 0.24 α 2 0.022614 α 4 ) .
σ R ( e ) = ( 10 / 3 ) π r 2 α 4 = ( 160 π 5 r 6 ) / 3 λ 4 .
Error in % = [ ( K R K ) / K ] × 100.
K R ( e ) = A α + B α 3 + C α 4 ,
E = [ ( K R ( e ) K ( e ) ) / K ( e ) ] × 100