Abstract

The finite sum of the squares of the Mie coefficients is very useful for addressing problems of classical light scattering. An approximate formula available in the literature, and still in use today, has been developed to determine a priori the number of the most significant terms needed to evaluate the scattering cross section. Here, we obtain an improved formula, which includes the number of terms needed for determining the scattering cross section within a prescribed relative error. This is accomplished using extended precision computation for a wide range of commonly used size parameters and indices of refraction. The revised formula for the finite number of terms can be a promising and valuable approach for efficient modeling light scattering phenomena.

© 2012 Optical Society of America

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References

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  1. A. Mazolli, P. A. Maia Neto, and H. M. Nussenzveig, Proc. Roy. Soc. Lon. A. 459, 3021 (2003).
  2. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyl, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Opt. A: Pure Appl. Opt. 9, S196 (2007).
    [CrossRef]
  3. H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, Phys. Rev. Lett. 99, 149901 (2007).
    [CrossRef]
  4. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer, 1995).
  5. D. Jackèl and B. Walter, Comput. Graph. Forum 16, 201 (1997).
    [CrossRef]
  6. G. Mie, Ann. Phys. 330, 377 (1908).
    [CrossRef]
  7. W. J. Wiscombe, Appl. Opt. 19, 1505 (1980).
    [CrossRef]
  8. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).
  9. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).
  10. A. L. Aden, J. Appl. Phys. 22, 601 (1951).
    [CrossRef]
  11. V. E. Cachorro and L. L. Salcedo, J. Electromagn. Waves. Appl. 5, 913 (1991).
    [CrossRef]
  12. J. V. Dave, IBM J. Res. Dev. 13, 302 (1969).
    [CrossRef]
  13. H. C. van de Hulst, Light Scattering by Small Particles(Wiley, 1957).
  14. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University Press, 1992).
  15. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1965).
  16. C. C. Lam, P. T. Leung, and K. Young, J. Opt. Soc. Am. B 9, 1585 (1992).
    [CrossRef]
  17. P. Chýlek, J. T. Kiehl, and M. K. W. Ko, Phys. Rev. A 18, 2229 (1978).
    [CrossRef]
  18. A. A. R. Neves, A. Camposeo, S. Pagliara, R. Saija, F. Borghese, P. Denti, M. A. Iatì, R. Cingolani, O. M. Maragò, and D. Pisignano, Opt. Express 18, 822 (2010).
    [CrossRef]
  19. F. Borghese, P. Denti, R. Saija, M. A. Iatì, and O. M. Maragò, Phys. Rev. Lett. 100, 163903 (2008).
    [CrossRef]

2010 (1)

2008 (1)

F. Borghese, P. Denti, R. Saija, M. A. Iatì, and O. M. Maragò, Phys. Rev. Lett. 100, 163903 (2008).
[CrossRef]

2007 (2)

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyl, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Opt. A: Pure Appl. Opt. 9, S196 (2007).
[CrossRef]

H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, Phys. Rev. Lett. 99, 149901 (2007).
[CrossRef]

2003 (1)

A. Mazolli, P. A. Maia Neto, and H. M. Nussenzveig, Proc. Roy. Soc. Lon. A. 459, 3021 (2003).

1997 (1)

D. Jackèl and B. Walter, Comput. Graph. Forum 16, 201 (1997).
[CrossRef]

1992 (1)

1991 (1)

V. E. Cachorro and L. L. Salcedo, J. Electromagn. Waves. Appl. 5, 913 (1991).
[CrossRef]

1980 (1)

1978 (1)

P. Chýlek, J. T. Kiehl, and M. K. W. Ko, Phys. Rev. A 18, 2229 (1978).
[CrossRef]

1969 (1)

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

1951 (1)

A. L. Aden, J. Appl. Phys. 22, 601 (1951).
[CrossRef]

1908 (1)

G. Mie, Ann. Phys. 330, 377 (1908).
[CrossRef]

Abramowitz, M.

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

Aden, A. L.

A. L. Aden, J. Appl. Phys. 22, 601 (1951).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

Borghese, F.

Branczyl, A. M.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyl, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Opt. A: Pure Appl. Opt. 9, S196 (2007).
[CrossRef]

Cachorro, V. E.

V. E. Cachorro and L. L. Salcedo, J. Electromagn. Waves. Appl. 5, 913 (1991).
[CrossRef]

Camposeo, A.

Chen, H.

H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, Phys. Rev. Lett. 99, 149901 (2007).
[CrossRef]

Chýlek, P.

P. Chýlek, J. T. Kiehl, and M. K. W. Ko, Phys. Rev. A 18, 2229 (1978).
[CrossRef]

Cingolani, R.

Dave, J. V.

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

Denti, P.

Heckenberg, N. R.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyl, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Opt. A: Pure Appl. Opt. 9, S196 (2007).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

Iatì, M. A.

Jackèl, D.

D. Jackèl and B. Walter, Comput. Graph. Forum 16, 201 (1997).
[CrossRef]

Kerker, M.

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

Kiehl, J. T.

P. Chýlek, J. T. Kiehl, and M. K. W. Ko, Phys. Rev. A 18, 2229 (1978).
[CrossRef]

Knöner, G.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyl, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Opt. A: Pure Appl. Opt. 9, S196 (2007).
[CrossRef]

Ko, M. K. W.

P. Chýlek, J. T. Kiehl, and M. K. W. Ko, Phys. Rev. A 18, 2229 (1978).
[CrossRef]

Kong, J. A.

H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, Phys. Rev. Lett. 99, 149901 (2007).
[CrossRef]

Kreibig, U.

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer, 1995).

Lam, C. C.

Leung, P. T.

Loke, V. L. Y.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyl, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Opt. A: Pure Appl. Opt. 9, S196 (2007).
[CrossRef]

Maia Neto, P. A.

A. Mazolli, P. A. Maia Neto, and H. M. Nussenzveig, Proc. Roy. Soc. Lon. A. 459, 3021 (2003).

Maragò, O. M.

Mazolli, A.

A. Mazolli, P. A. Maia Neto, and H. M. Nussenzveig, Proc. Roy. Soc. Lon. A. 459, 3021 (2003).

Mie, G.

G. Mie, Ann. Phys. 330, 377 (1908).
[CrossRef]

Neves, A. A. R.

Nieminen, T. A.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyl, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Opt. A: Pure Appl. Opt. 9, S196 (2007).
[CrossRef]

Nussenzveig, H. M.

A. Mazolli, P. A. Maia Neto, and H. M. Nussenzveig, Proc. Roy. Soc. Lon. A. 459, 3021 (2003).

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University Press, 1992).

Pagliara, S.

Pisignano, D.

Rubinsztein-Dunlop, H.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyl, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Opt. A: Pure Appl. Opt. 9, S196 (2007).
[CrossRef]

Saija, R.

Salcedo, L. L.

V. E. Cachorro and L. L. Salcedo, J. Electromagn. Waves. Appl. 5, 913 (1991).
[CrossRef]

Stegun, I. A.

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

Stilgoe, A. B.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyl, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Opt. A: Pure Appl. Opt. 9, S196 (2007).
[CrossRef]

van de Hulst, H. C.

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

Vollmer, M.

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer, 1995).

Walter, B.

D. Jackèl and B. Walter, Comput. Graph. Forum 16, 201 (1997).
[CrossRef]

Wiscombe, W. J.

Wu, B. I.

H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, Phys. Rev. Lett. 99, 149901 (2007).
[CrossRef]

Young, K.

Zhang, B.

H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, Phys. Rev. Lett. 99, 149901 (2007).
[CrossRef]

Ann. Phys. (1)

G. Mie, Ann. Phys. 330, 377 (1908).
[CrossRef]

Appl. Opt. (1)

Comput. Graph. Forum (1)

D. Jackèl and B. Walter, Comput. Graph. Forum 16, 201 (1997).
[CrossRef]

IBM J. Res. Dev. (1)

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

J. Appl. Phys. (1)

A. L. Aden, J. Appl. Phys. 22, 601 (1951).
[CrossRef]

J. Electromagn. Waves. Appl. (1)

V. E. Cachorro and L. L. Salcedo, J. Electromagn. Waves. Appl. 5, 913 (1991).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyl, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Opt. A: Pure Appl. Opt. 9, S196 (2007).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Express (1)

Phys. Rev. A (1)

P. Chýlek, J. T. Kiehl, and M. K. W. Ko, Phys. Rev. A 18, 2229 (1978).
[CrossRef]

Phys. Rev. Lett. (2)

H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, Phys. Rev. Lett. 99, 149901 (2007).
[CrossRef]

F. Borghese, P. Denti, R. Saija, M. A. Iatì, and O. M. Maragò, Phys. Rev. Lett. 100, 163903 (2008).
[CrossRef]

Proc. Roy. Soc. Lon. A. (1)

A. Mazolli, P. A. Maia Neto, and H. M. Nussenzveig, Proc. Roy. Soc. Lon. A. 459, 3021 (2003).

Other (6)

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer, 1995).

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

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

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

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University Press, 1992).

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

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Figures (3)

Fig. 1.
Fig. 1.

Scattering cross section for different values of imaginary refractive index, for the b34 resonance, centered near x=40.32638, as previously described [17].

Fig. 2.
Fig. 2.

Log-log plot evidencing the functional dependence of the maximum number of terms (N) as a function of size factor (x) for m=2. Each curve is plotted for different truncation errors from 105 to 10155, from bottom to top, respectively. Dashed and dot-dash blue lines represent trends, and the dashed black line represents Wiscombe’s criterion Eq. (5).

Fig. 3.
Fig. 3.

Contour plot of the number of terms (N) as a function of size factor (x) and accuracy (ε) for m=2. First contour (left) corresponds to (N=50), and each successive contour adds another 50 terms to the series. Horizontal lines represent maximum accuracy for finite precision computation.

Equations (9)

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

Qs=2x2n=1(2n+1)(|an|2+|bn|2),
an={jn(x)hn(1)(x)}ln[mxjn(mx)]mln[xjn(x)]ln[mxjn(mx)]mln[xhn(1)(x)]
bn={jn(x)hn(1)(x)}ln[mxjn(mx)]m1ln[xjn(x)]ln[mxjn(mx)]m1ln[xhn(1)(x)].
ΔQs/Qs=12x2[Qs]1n=1N(2n+1)(|an|2+|bn|2).
N=x+4.05x1/3+2.
jn(x)(2x)1/2ν1/3(8c)1/4Exp(22c3/2/3)
yn(x)(x)1/2ν1/3(8c)1/4Exp(22c3/2/3),
Nx+αε2/3x1/3+β.
N=x+0.76ε2/3x1/34.1.

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