Abstract

The new Mie-scattering calculation is a robust and efficient algorithm used to compute light scattering from spheres. It calculates the ratio between Riccati-Bessel functions instead of the complicated logarithmic derivative. The Kapteyn inequality is used to estimate the number of significant digits of the calculated Riccati-Bessel functions and their ratio. This new algorithm is stable and accurate for both large and small particles. The implemented C++ code yields the same accurate results for both small and large particles compared with Wiscombe’s MIEV0 code in double precision. Suggestions are provided for the porting of the MIEV0 code.

© 2004 Optical Society of America

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References

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  1. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
  2. L. Infeld, “The influence of the width of the gap upon the theory of antennas,” Q. Appl. Math. 5, 113–132 (1947).
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    [CrossRef]
  4. W. J. Lentz, “Generating Bessel functions in Mie scattering calculations using continued fractions,” Appl. Opt. 15, 668–671 (1976).
    [CrossRef] [PubMed]
  5. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  6. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  7. W. J. Wiscombe, “Mie scattering calculations: advances in technique and fast, vector-speed computer codes,” NCAR/TN-140+STR, NCAR Tech. Note (National Center for Atmospheric Research, Boulder, Colo., 1979, revised 1996); ftp://climate.gsfc.nasa.gov/pub/wiscombe/Single_Scatt/Homogen_Sphere/Exact_Mie/NCARMieReport.pdf .
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    [CrossRef] [PubMed]
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    [CrossRef]
  10. M. I. Mishchenko, NASA Goddard Institute for Space Studies, New York, N.Y. (personal communication, 2002).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  18. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965).
  19. W. J. Wiscombe, “NoPMOM version of MIEV0” (1992), ftp://climate.gsfc.nasa.gov/pub/wiscombe/Single_Scatt/Homogen_Sphere/Exact_Mie/MVTstNew.out .
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    [CrossRef]
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2003

2002

1997

1996

C. Kim, N. Lior, K. Okuyama, “Simple mathematical expressions for spectral extinction and scattering properties of small size-parameter particles, including examples for soot and TiO2,” J. Quant. Spectrosc. Radiat. Transfer 55, 391–411 (1996).
[CrossRef]

1994

1991

R. T. Wang, H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106–117 (1991).
[CrossRef] [PubMed]

V. E. Cachorro, L. L. Salcedo, “New improvements for Mie scattering calculations,” J. Electromagn. Waves. Appl. 5, 913–926 (1991).
[CrossRef]

1984

W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

1980

1976

1969

J. V. Dave, “Scattering of electromagnetic radiation by a large, absorbing sphere,” IBM J. Res. Dev. 13, 302–313 (1969).
[CrossRef]

1967

1947

L. Infeld, “The influence of the width of the gap upon the theory of antennas,” Q. Appl. Math. 5, 113–132 (1947).

1908

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965).

Cachorro, V. E.

V. E. Cachorro, L. L. Salcedo, “New improvements for Mie scattering calculations,” J. Electromagn. Waves. Appl. 5, 913–926 (1991).
[CrossRef]

Cheng, L.

Dave, J. V.

J. V. Dave, “Scattering of electromagnetic radiation by a large, absorbing sphere,” IBM J. Res. Dev. 13, 302–313 (1969).
[CrossRef]

de Rooij, W. A.

W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

Gouesbet, G.

Gréhan, G.

Guo, L. X.

Infeld, L.

L. Infeld, “The influence of the width of the gap upon the theory of antennas,” Q. Appl. Math. 5, 113–132 (1947).

Kai, L.

Kattawar, G. W.

Kim, C.

C. Kim, N. Lior, K. Okuyama, “Simple mathematical expressions for spectral extinction and scattering properties of small size-parameter particles, including examples for soot and TiO2,” J. Quant. Spectrosc. Radiat. Transfer 55, 391–411 (1996).
[CrossRef]

Lentz, W. J.

Lior, N.

C. Kim, N. Lior, K. Okuyama, “Simple mathematical expressions for spectral extinction and scattering properties of small size-parameter particles, including examples for soot and TiO2,” J. Quant. Spectrosc. Radiat. Transfer 55, 391–411 (1996).
[CrossRef]

Massoli, P.

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Mishchenko, M. I.

M. I. Mishchenko, NASA Goddard Institute for Space Studies, New York, N.Y. (personal communication, 2002).

Okuyama, K.

C. Kim, N. Lior, K. Okuyama, “Simple mathematical expressions for spectral extinction and scattering properties of small size-parameter particles, including examples for soot and TiO2,” J. Quant. Spectrosc. Radiat. Transfer 55, 391–411 (1996).
[CrossRef]

Plass, G. N.

Ren, K. F.

Salcedo, L. L.

V. E. Cachorro, L. L. Salcedo, “New improvements for Mie scattering calculations,” J. Electromagn. Waves. Appl. 5, 913–926 (1991).
[CrossRef]

Siu, G.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965).

van de Hulst, H. C.

van der Stap, C. C. A. H.

W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

Wang, R. T.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, London1966).

Wiscombe, W. J.

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

W. J. Wiscombe, “Mie scattering calculations: advances in technique and fast, vector-speed computer codes,” NCAR/TN-140+STR, NCAR Tech. Note (National Center for Atmospheric Research, Boulder, Colo., 1979, revised 1996); ftp://climate.gsfc.nasa.gov/pub/wiscombe/Single_Scatt/Homogen_Sphere/Exact_Mie/NCARMieReport.pdf .

Wu, Z. S.

Yang, W.

Ann. Phys. (Leipzig)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Appl. Opt.

Astron. Astrophys.

W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

IBM J. Res. Dev.

J. V. Dave, “Scattering of electromagnetic radiation by a large, absorbing sphere,” IBM J. Res. Dev. 13, 302–313 (1969).
[CrossRef]

J. Electromagn. Waves. Appl.

V. E. Cachorro, L. L. Salcedo, “New improvements for Mie scattering calculations,” J. Electromagn. Waves. Appl. 5, 913–926 (1991).
[CrossRef]

J. Opt. Soc. Am. B

J. Quant. Spectrosc. Radiat. Transfer

C. Kim, N. Lior, K. Okuyama, “Simple mathematical expressions for spectral extinction and scattering properties of small size-parameter particles, including examples for soot and TiO2,” J. Quant. Spectrosc. Radiat. Transfer 55, 391–411 (1996).
[CrossRef]

Q. Appl. Math.

L. Infeld, “The influence of the width of the gap upon the theory of antennas,” Q. Appl. Math. 5, 113–132 (1947).

Other

M. I. Mishchenko, NASA Goddard Institute for Space Studies, New York, N.Y. (personal communication, 2002).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, London1966).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965).

W. J. Wiscombe, “NoPMOM version of MIEV0” (1992), ftp://climate.gsfc.nasa.gov/pub/wiscombe/Single_Scatt/Homogen_Sphere/Exact_Mie/MVTstNew.out .

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

W. J. Wiscombe, “Mie scattering calculations: advances in technique and fast, vector-speed computer codes,” NCAR/TN-140+STR, NCAR Tech. Note (National Center for Atmospheric Research, Boulder, Colo., 1979, revised 1996); ftp://climate.gsfc.nasa.gov/pub/wiscombe/Single_Scatt/Homogen_Sphere/Exact_Mie/NCARMieReport.pdf .

H. Du, H. Zhang, “Ultra high precision Mie scattering calculation” (2002), http://optics.physics.miami.edu/exp/du/UltraHighMie.pdf .

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Tables (2)

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Table 1 Comparison of Qext and Qsca Calculated by MIEV0 and MIECPP in Double Precision

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Table 2 Results of Amplitude Functions S1(0), S2(0), S1(π), and S2(π) Calculated by Double-Precision MIEV0a

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

an=Anmx/m+n/xψnx-ψn-1xAnmx/m+n/xζnx-ζn-1x,
bn=Anmxm+n/xψnx-ψn-1xAnmxm+n/xζnx-ζn-1x.
ψn+1x=2n+1ψnx/x-ψn-1x,
sμπnμ,
ts-πn-1μ,
πn+1μ=s+t+t/n,
τnμ=nt-πn-1μ.
an=rnmx/m+n1-1/m2/xψnx-ψn-1xrnmx/m+n1-1/m2/xζnx-ζn-1x,
bn=rnmxmψnx-ψn-1xrnmxmζn-ζn-1x
|Jnz|z exp1-z/n21/2n1+1-z/n21/2nψnz,
lnz |Imz|-ln2-nRelnz/n+1-z/n21/2-ln1+1-z/n21/2/ln10.
rn+1mx=2n+1mx-rnmx-1.
cotmx=i+tanRemx-exp-2ImmxtanRemx+i exp-2Immx-1+i tanRemx+i exp-2ImmxtanRemx+exp-2Immx
a+bic+di=ac+bdc2+d2+bc-adc2+d2 i
cotmx=cot*m*x

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