Abstract

Scattering of electromagnetic radiation from a sphere, so-called Mie scattering, requires calculations that can become lengthy and even impossible for those with limited resources. At the same time, such calculations are required for the widest variety of optical applications, extending from the shortest UV to the longest microwave and radar wavelengths. This paper briefly describes new and thoroughly documented Mie scattering algorithms that result in considerable improvements in speed by employing more efficient formulations and vector structure. The algorithms are particularly fast on the Cray-1 and similar vector-processing computers.

© 1980 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. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  3. L. Infeld, Q. Appl. Math. 5, 113 (1947).
  4. J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere,” Report 320-3237 (IBM Scientific Center, Palo Alto, Calif., 1968).
  5. J. V. Dave, IBM J. Res. Dev. 13, 302 (1969).
    [CrossRef]
  6. W. J. Wiscombe, “Mie Scattering Calculations: Advances in Technique and Fast, Vector-Speed Computer Codes,” NCAR Technical Note NCAR/TN-140+STR (National Center for Atmospheric Research, Boulder, Colo. 80307,1979).
  7. G. W. Kattawar, G. N. Plass, Appl. Opt. 6, 1377 (1967).
    [CrossRef] [PubMed]
  8. W. J. Lentz, Appl. Opt. 15, 668 (1976).
    [CrossRef] [PubMed]
  9. V. Khare, “Short-Wavelength Scattering of Electromagnetic Waves by a Homogeneous Dielectric Sphere,” Ph.D. Thesis, U. Rochester, N.Y., 1976 (available from University Microfilms, Ann Arbor, Mich.).
  10. P. M. Johnson, Comput. Des. 17, 89 (1978).

1978

P. M. Johnson, Comput. Des. 17, 89 (1978).

1976

1969

J. V. Dave, IBM J. Res. Dev. 13, 302 (1969).
[CrossRef]

1967

1947

L. Infeld, Q. Appl. Math. 5, 113 (1947).

Dave, J. V.

J. V. Dave, IBM J. Res. Dev. 13, 302 (1969).
[CrossRef]

J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere,” Report 320-3237 (IBM Scientific Center, Palo Alto, Calif., 1968).

Infeld, L.

L. Infeld, Q. Appl. Math. 5, 113 (1947).

Johnson, P. M.

P. M. Johnson, Comput. Des. 17, 89 (1978).

Kattawar, G. W.

Kerker, M.

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

Khare, V.

V. Khare, “Short-Wavelength Scattering of Electromagnetic Waves by a Homogeneous Dielectric Sphere,” Ph.D. Thesis, U. Rochester, N.Y., 1976 (available from University Microfilms, Ann Arbor, Mich.).

Lentz, W. J.

Plass, G. N.

Van de Hulst, H. C.

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

Wiscombe, W. J.

W. J. Wiscombe, “Mie Scattering Calculations: Advances in Technique and Fast, Vector-Speed Computer Codes,” NCAR Technical Note NCAR/TN-140+STR (National Center for Atmospheric Research, Boulder, Colo. 80307,1979).

Appl. Opt.

Comput. Des.

P. M. Johnson, Comput. Des. 17, 89 (1978).

IBM J. Res. Dev.

J. V. Dave, IBM J. Res. Dev. 13, 302 (1969).
[CrossRef]

Q. Appl. Math.

L. Infeld, Q. Appl. Math. 5, 113 (1947).

Other

J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere,” Report 320-3237 (IBM Scientific Center, Palo Alto, Calif., 1968).

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

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

W. J. Wiscombe, “Mie Scattering Calculations: Advances in Technique and Fast, Vector-Speed Computer Codes,” NCAR Technical Note NCAR/TN-140+STR (National Center for Atmospheric Research, Boulder, Colo. 80307,1979).

V. Khare, “Short-Wavelength Scattering of Electromagnetic Waves by a Homogeneous Dielectric Sphere,” Ph.D. Thesis, U. Rochester, N.Y., 1976 (available from University Microfilms, Ann Arbor, Mich.).

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Figures (1)

Fig. 1
Fig. 1

Empirically determined values for the function f(mRe) described in the text, and polynomical fits to those data: (a) considers Qext. Qsca, and g only; (b) includes the angular functions S1 and S2 as well.

Tables (3)

Tables Icon

Table I Comparison of Number of Iterations Required by the Dave and Lentz Methods to Calculate AN(mx) in Situations Where Eq. (7) Requires Down Recurrence a

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Table II Execution Times for Dave5 and Wiscombe6 Algorithms for a Single Mie Calculation [Eqs. (1a)(1e)] with 182 Scattering Angles a

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Table III Cray-1 Times (in Milliseconds) to Execute the Vectorized miev1 Code6 for Various Combinations of Mie Size Parameters and Number of Angles a

Equations (18)

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Q ext = 2 x 2 n = 1 N ( 2 n + 1 ) Re ( a n + b n ) ,
Q sca = 2 x 2 n = 1 N ( 2 n + 1 ) ( | a n | 2 + | b n | 2 ) ,
g = 4 x 2 Q sca n = 1 N [ n ( n + 2 ) n + 1 Re ( a n a n + 1 * + b n b n + 1 * ) + 2 n + 1 n ( n + 1 ) Re ( a n b n * ) ] ,
S 1 ( μ ) = n = 1 N 2 n + 1 n ( n + 1 ) [ a n π n ( μ ) + b n τ n ( μ ) ] ,
S 2 ( μ ) = n = 1 N 2 n + 1 n ( n + 1 ) [ a n τ n ( μ ) + b n π n ( μ ) ] ,
A n ( mx ) = ψ n ( mx ) / ψ n ( mx ) ,
π n ( μ ) = P n ( μ ) ,
τ n ( μ ) = μ π n ( μ ) ( 1 μ 2 ) π n ( μ ) ,
m Im x f ( m Re ) .
x m Im * min all x ( x m Im * ) f ( m Re ) ,
f 1 ( m Re ) 8 + 26.22 m Re 2 0.4474 m Re 3 + 0.00204 m Re 6 0.000175 m Re 7 ,
f 2 ( m Re ) = 13.78 m Re 2 10.8 m Re + 3.9 .
f 3 ( m Re ) = 16.35 m Re 2 + 8.42 m Re 15.04 ,
τ n ( μ ) = nt π n 1 ( μ ) , τ n + 1 ( μ ) = s + ( n + 1 n ) t ,
S + S 1 + S 2 = n = 1 N 2 n + 1 n ( n + 1 ) ( a n + b n ) ( π n + τ n ) , S S 1 S 2 = n = 1 N 2 n + 1 n ( n + 1 ) ( a n b n ) ( π n τ n ) .
Q ext = 6 x Re ( â 1 + b ̂ 1 + 5 3 â 2 ) , Q scan = 6 x 4 T , g = 1 T Re [ â 1 ( â 2 + b ̂ 1 ) * ] , S 1 ( μ ) = 3 2 x 3 [ â 1 + ( b ̂ 1 + 5 3 â 2 ) μ ] , S 2 ( μ ) = 3 2 x 3 [ b ̂ 1 + â 1 μ + 5 3 â 2 ( 2 μ 2 1 ) ] ,
T | â 1 | 2 + | b ̂ 1 | 2 + 5 3 | â 2 | 2 , â 1 = 2 i m 2 1 3 1 1 10 x 2 + 4 m 2 + 5 1400 x 4 D , D = m 2 + 2 + ( 1 7 10 m 2 ) x 2 8 m 4 385 m 2 + 350 1400 x 4 + 2 i m 2 1 3 x 3 ( 1 7 10 x 2 ) , b ̂ 1 = i x 2 m 2 1 45 1 + 2 m 2 5 70 x 2 1 2 m 2 5 30 x 2 , â 2 = i x 2 m 2 1 15 1 1 14 x 2 2 m 2 + 3 2 m 2 7 14 x 2 .
N = { x + 4 x 1 / 3 + 1 0.02 x 8 x + 4.05 x 1 / 3 + 2 8 < x < 4200 x + 4 x 1 / 3 + 2 4200 x 20,000 .

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