Abstract

The Debye series expansion (DSE) is of importance for understanding light scattering features and for testing the validity of geometric optics approach to light scattering. We recast the partial-wave reflection and transmission coefficients so that all the related complex functions can be calculated with stability in all necessary orders. Numerical tests are performed for both the full scattering intensities and the components in a wide range of particle sizes and refractive indices. The results are compared with those from Mie calculation and from MiePlot v4.1, showing that the algorithm is stable, reliable, and robust in a wide range of particle sizes and refractive indices. The developed algorithm may also apply to the DSE calculation of light scattering by multilayered spheres or cylinders.

© 2010 Optical Society of America

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References

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  1. P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z.  9, 775–778 (1908).
  2. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig)  25, 377–445 (1908).
    [CrossRef]
  3. X. Han, H. Jiang, K. Ren, and G. Gréhan, “On rainbows of inhomogeneous spherical droplets,” Opt. Commun.  269, 291–298 (2007).
    [CrossRef]
  4. P. Laven, “The optics of a water drop: Mie scattering and the Debye series,” http://www.philiplaven.com/index1.html (2004).
  5. P. Laven, “Simulation of rainbows, coronas and glories using Mie theory and the Debye series,” J. Quant. Spectosc. Radiat. Transfer  89, 257–269 (2004).
    [CrossRef]
  6. P. Laven, “How are glories formed?,” Appl. Opt.  44, 5675–5683 (2005).
    [CrossRef] [PubMed]
  7. J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A  10, 693–706 (1993).
    [CrossRef]
  8. J. A. Lock, J. M. Jamison, and C. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt.  33, 4677–4690 (1994).
    [CrossRef] [PubMed]
  9. J. A. Lock and C. L. Adler, “Debye series analysis of the first-order rainbow produced in scattering of a diagonally incident plane wave by a circular cylinder,” J. Opt. Soc. Am. A  14, 1316–1328 (1997).
    [CrossRef]
  10. J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A  5, 2032–2044 (1988).
    [CrossRef]
  11. X. Li, X. Han, R. Li, and H. Jiang, “Geometrical-optics approximation of forward scattering by gradient-index spheres,” Appl. Opt.  46, 5241–5247 (2007).
    [CrossRef] [PubMed]
  12. L. Wu, H. Yang, X. Li, B. Yang, and G. Li, “Scattering by large bubbles: Comparisons between geometrical-optics theory and Debye series,” J. Quant. Spectrosc. Radiat. Transfer  108, 54–64 (2007).
    [CrossRef]
  13. F. Xu, K. Ren, and X. Cai, “Extension of geometrical optics approximation to on-axis Gaussian beam scattering. I. By a spherical particle,” Appl. Opt.  45, 4990–4999 (2006).
    [CrossRef] [PubMed]
  14. H. Yu, J. Shen, and Y. Wei, “Geometrical optics approximation for light scattering by absorbing spherical particles,” J. Quant. Spectosc. Radiat. Transfer  110, 1178–1189 (2009).
    [CrossRef]
  15. J. V. Dave, “Scattering of electromagnetic radiation by a large, absorbing sphere,” IBM J. Res. Dev.  13, 302–313 (1969).
    [CrossRef]
  16. Z. Wu and Y. Wang, “Electromagnetic scattering for multi-layered sphere: recursive algorithms,” Radio Sci.  26, 1393–1401 (1991).
    [CrossRef]
  17. Z. Wu, L. Guo, K. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt.  36, 5188–5198 (1997).
    [CrossRef] [PubMed]
  18. J. Shen, H. Wang, B. Wang, H. Yu, and B. Yu, “Stability in Debye series calculation for light scattering by absorbing particles and bubbles,” J. Quant. Spectrosc. Radiat. Transfer  111, 772–781 (2010).
    [CrossRef]
  19. E. A. Hovenac and J. A. Lock, “Assessing the contributions of surface waves and complex rays to far-field Mie scattering by use of the Debye series,” J. Opt. Soc. Am. A  9, 781–795 (1992).
    [CrossRef]
  20. R. Li, X. Han, H. Jiang, and K. Ren, “Debye series for light scattering by a multilayered sphere,” Appl. Opt.  45, 1260–1270 (2006).
    [CrossRef] [PubMed]
  21. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  22. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).
  23. H. C. van de Hulst, Light Scattering by Small Particles(Chapman & Hall, 1957).
  24. W. Yang, “Improved recursive algorithm for light scattering by a multilayered sphere,” Appl. Opt.  42, 1710–1720 (2003).
    [CrossRef] [PubMed]
  25. J. V. Dave, “Coefficients of the Legendre and Fourier series for the scattering functions of spherical particles,” Appl. Opt.  9, 1888–1896 (1970).
    [PubMed]
  26. W. J. Lentz, “Generating Bessel functions in Mie scattering calculations using continued fractions,” Appl. Opt.  15, 668–671 (1976).
    [CrossRef] [PubMed]
  27. J. Shen and X. Cai, “Algorithm of numerical calculation on Lorentz Mie theory,” PIERS Online  1, 691–694 (2005).
    [CrossRef]
  28. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys.  10 (1), 82–124 (1969).
    [CrossRef]
  29. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt.  19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  30. G. Roll, T. Kaiser, and G. Schweiger, “Controlled modification of the expansion order as a tool in Mie computations,” Appl. Opt.  37, 2483–2492 (1998).
    [CrossRef]
  31. P. Laven, “MiePlot: a computer program for scattering of light from a sphere using Mie theory & the Debye series,” http://www.philiplaven.com/mieplot.htm (2009).
  32. G. Gouesbet, “Debye series formulation for generalized Lorenz-Mie theory with the Bromwich method,” Part. Part. Syst. Charact.  20, 382–386 (2003).
    [CrossRef]
  33. R. Li, X. Han, H. Jiang, and K. Ren, “Debye series of normally incident plane-wave scattering by an infinite multilayered cylinder,” Appl. Opt.  45, 6255–6262 (2006).
    [CrossRef] [PubMed]
  34. R. Li, X. Han, L. Shi, K. Ren, and H. Jiang, “Debye series for Gaussian beam scattering by a multilayered sphere,” Appl. Opt.  46, 4804–4812 (2007).
    [CrossRef] [PubMed]
  35. R. Li and X. Han, “Generalized Debye series expansion of electromagnetic plane wave scattering by an infinite multilayered cylinder at oblique incidence,” Phys. Rev. E  79, 036602(2009).
    [CrossRef]
  36. R. Li, X. Han, and K. Ren, “Debye series expansion of shaped beam scattering by GI-POF,” Opt. Commun.  282, 4315–4321(2009).
    [CrossRef]

2010 (1)

J. Shen, H. Wang, B. Wang, H. Yu, and B. Yu, “Stability in Debye series calculation for light scattering by absorbing particles and bubbles,” J. Quant. Spectrosc. Radiat. Transfer  111, 772–781 (2010).
[CrossRef]

2009 (3)

H. Yu, J. Shen, and Y. Wei, “Geometrical optics approximation for light scattering by absorbing spherical particles,” J. Quant. Spectosc. Radiat. Transfer  110, 1178–1189 (2009).
[CrossRef]

R. Li and X. Han, “Generalized Debye series expansion of electromagnetic plane wave scattering by an infinite multilayered cylinder at oblique incidence,” Phys. Rev. E  79, 036602(2009).
[CrossRef]

R. Li, X. Han, and K. Ren, “Debye series expansion of shaped beam scattering by GI-POF,” Opt. Commun.  282, 4315–4321(2009).
[CrossRef]

2007 (4)

R. Li, X. Han, L. Shi, K. Ren, and H. Jiang, “Debye series for Gaussian beam scattering by a multilayered sphere,” Appl. Opt.  46, 4804–4812 (2007).
[CrossRef] [PubMed]

X. Li, X. Han, R. Li, and H. Jiang, “Geometrical-optics approximation of forward scattering by gradient-index spheres,” Appl. Opt.  46, 5241–5247 (2007).
[CrossRef] [PubMed]

L. Wu, H. Yang, X. Li, B. Yang, and G. Li, “Scattering by large bubbles: Comparisons between geometrical-optics theory and Debye series,” J. Quant. Spectrosc. Radiat. Transfer  108, 54–64 (2007).
[CrossRef]

X. Han, H. Jiang, K. Ren, and G. Gréhan, “On rainbows of inhomogeneous spherical droplets,” Opt. Commun.  269, 291–298 (2007).
[CrossRef]

2006 (3)

2005 (2)

J. Shen and X. Cai, “Algorithm of numerical calculation on Lorentz Mie theory,” PIERS Online  1, 691–694 (2005).
[CrossRef]

P. Laven, “How are glories formed?,” Appl. Opt.  44, 5675–5683 (2005).
[CrossRef] [PubMed]

2004 (2)

P. Laven, “The optics of a water drop: Mie scattering and the Debye series,” http://www.philiplaven.com/index1.html (2004).

P. Laven, “Simulation of rainbows, coronas and glories using Mie theory and the Debye series,” J. Quant. Spectosc. Radiat. Transfer  89, 257–269 (2004).
[CrossRef]

2003 (2)

W. Yang, “Improved recursive algorithm for light scattering by a multilayered sphere,” Appl. Opt.  42, 1710–1720 (2003).
[CrossRef] [PubMed]

G. Gouesbet, “Debye series formulation for generalized Lorenz-Mie theory with the Bromwich method,” Part. Part. Syst. Charact.  20, 382–386 (2003).
[CrossRef]

1998 (1)

1997 (2)

1994 (1)

1993 (1)

1992 (1)

1991 (1)

Z. Wu and Y. Wang, “Electromagnetic scattering for multi-layered sphere: recursive algorithms,” Radio Sci.  26, 1393–1401 (1991).
[CrossRef]

1988 (1)

1980 (1)

1976 (1)

1970 (1)

1969 (2)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys.  10 (1), 82–124 (1969).
[CrossRef]

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

1908 (2)

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z.  9, 775–778 (1908).

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

Adler, C. L.

Bohren, C. F.

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

Cai, X.

Dave, J. V.

J. V. Dave, “Coefficients of the Legendre and Fourier series for the scattering functions of spherical particles,” Appl. Opt.  9, 1888–1896 (1970).
[PubMed]

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

Debye, P.

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z.  9, 775–778 (1908).

Gouesbet, G.

G. Gouesbet, “Debye series formulation for generalized Lorenz-Mie theory with the Bromwich method,” Part. Part. Syst. Charact.  20, 382–386 (2003).
[CrossRef]

Z. Wu, L. Guo, K. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt.  36, 5188–5198 (1997).
[CrossRef] [PubMed]

Gréhan, G.

Guo, L.

Han, X.

Hovenac, E. A.

Huffman, D. R.

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

Jamison, J. M.

Jiang, H.

Kaiser, T.

Kerker, M.

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

Laven, P.

P. Laven, “How are glories formed?,” Appl. Opt.  44, 5675–5683 (2005).
[CrossRef] [PubMed]

P. Laven, “The optics of a water drop: Mie scattering and the Debye series,” http://www.philiplaven.com/index1.html (2004).

P. Laven, “Simulation of rainbows, coronas and glories using Mie theory and the Debye series,” J. Quant. Spectosc. Radiat. Transfer  89, 257–269 (2004).
[CrossRef]

P. Laven, “MiePlot: a computer program for scattering of light from a sphere using Mie theory & the Debye series,” http://www.philiplaven.com/mieplot.htm (2009).

Lentz, W. J.

Li, G.

L. Wu, H. Yang, X. Li, B. Yang, and G. Li, “Scattering by large bubbles: Comparisons between geometrical-optics theory and Debye series,” J. Quant. Spectrosc. Radiat. Transfer  108, 54–64 (2007).
[CrossRef]

Li, R.

Li, X.

X. Li, X. Han, R. Li, and H. Jiang, “Geometrical-optics approximation of forward scattering by gradient-index spheres,” Appl. Opt.  46, 5241–5247 (2007).
[CrossRef] [PubMed]

L. Wu, H. Yang, X. Li, B. Yang, and G. Li, “Scattering by large bubbles: Comparisons between geometrical-optics theory and Debye series,” J. Quant. Spectrosc. Radiat. Transfer  108, 54–64 (2007).
[CrossRef]

Lin, C.

Lock, J. A.

Mie, G.

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

Nussenzveig, H. M.

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys.  10 (1), 82–124 (1969).
[CrossRef]

Ren, K.

Roll, G.

Schweiger, G.

Shen, J.

J. Shen, H. Wang, B. Wang, H. Yu, and B. Yu, “Stability in Debye series calculation for light scattering by absorbing particles and bubbles,” J. Quant. Spectrosc. Radiat. Transfer  111, 772–781 (2010).
[CrossRef]

H. Yu, J. Shen, and Y. Wei, “Geometrical optics approximation for light scattering by absorbing spherical particles,” J. Quant. Spectosc. Radiat. Transfer  110, 1178–1189 (2009).
[CrossRef]

J. Shen and X. Cai, “Algorithm of numerical calculation on Lorentz Mie theory,” PIERS Online  1, 691–694 (2005).
[CrossRef]

Shi, L.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles(Chapman & Hall, 1957).

Wang, B.

J. Shen, H. Wang, B. Wang, H. Yu, and B. Yu, “Stability in Debye series calculation for light scattering by absorbing particles and bubbles,” J. Quant. Spectrosc. Radiat. Transfer  111, 772–781 (2010).
[CrossRef]

Wang, H.

J. Shen, H. Wang, B. Wang, H. Yu, and B. Yu, “Stability in Debye series calculation for light scattering by absorbing particles and bubbles,” J. Quant. Spectrosc. Radiat. Transfer  111, 772–781 (2010).
[CrossRef]

Wang, Y.

Z. Wu and Y. Wang, “Electromagnetic scattering for multi-layered sphere: recursive algorithms,” Radio Sci.  26, 1393–1401 (1991).
[CrossRef]

Wei, Y.

H. Yu, J. Shen, and Y. Wei, “Geometrical optics approximation for light scattering by absorbing spherical particles,” J. Quant. Spectosc. Radiat. Transfer  110, 1178–1189 (2009).
[CrossRef]

Wiscombe, W. J.

Wu, L.

L. Wu, H. Yang, X. Li, B. Yang, and G. Li, “Scattering by large bubbles: Comparisons between geometrical-optics theory and Debye series,” J. Quant. Spectrosc. Radiat. Transfer  108, 54–64 (2007).
[CrossRef]

Wu, Z.

Xu, F.

Yang, B.

L. Wu, H. Yang, X. Li, B. Yang, and G. Li, “Scattering by large bubbles: Comparisons between geometrical-optics theory and Debye series,” J. Quant. Spectrosc. Radiat. Transfer  108, 54–64 (2007).
[CrossRef]

Yang, H.

L. Wu, H. Yang, X. Li, B. Yang, and G. Li, “Scattering by large bubbles: Comparisons between geometrical-optics theory and Debye series,” J. Quant. Spectrosc. Radiat. Transfer  108, 54–64 (2007).
[CrossRef]

Yang, W.

Yu, B.

J. Shen, H. Wang, B. Wang, H. Yu, and B. Yu, “Stability in Debye series calculation for light scattering by absorbing particles and bubbles,” J. Quant. Spectrosc. Radiat. Transfer  111, 772–781 (2010).
[CrossRef]

Yu, H.

J. Shen, H. Wang, B. Wang, H. Yu, and B. Yu, “Stability in Debye series calculation for light scattering by absorbing particles and bubbles,” J. Quant. Spectrosc. Radiat. Transfer  111, 772–781 (2010).
[CrossRef]

H. Yu, J. Shen, and Y. Wei, “Geometrical optics approximation for light scattering by absorbing spherical particles,” J. Quant. Spectosc. Radiat. Transfer  110, 1178–1189 (2009).
[CrossRef]

Ann. Phys. (Leipzig) (1)

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

Appl. Opt. (13)

P. Laven, “How are glories formed?,” Appl. Opt.  44, 5675–5683 (2005).
[CrossRef] [PubMed]

J. A. Lock, J. M. Jamison, and C. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt.  33, 4677–4690 (1994).
[CrossRef] [PubMed]

X. Li, X. Han, R. Li, and H. Jiang, “Geometrical-optics approximation of forward scattering by gradient-index spheres,” Appl. Opt.  46, 5241–5247 (2007).
[CrossRef] [PubMed]

F. Xu, K. Ren, and X. Cai, “Extension of geometrical optics approximation to on-axis Gaussian beam scattering. I. By a spherical particle,” Appl. Opt.  45, 4990–4999 (2006).
[CrossRef] [PubMed]

Z. Wu, L. Guo, K. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt.  36, 5188–5198 (1997).
[CrossRef] [PubMed]

R. Li, X. Han, H. Jiang, and K. Ren, “Debye series for light scattering by a multilayered sphere,” Appl. Opt.  45, 1260–1270 (2006).
[CrossRef] [PubMed]

W. Yang, “Improved recursive algorithm for light scattering by a multilayered sphere,” Appl. Opt.  42, 1710–1720 (2003).
[CrossRef] [PubMed]

J. V. Dave, “Coefficients of the Legendre and Fourier series for the scattering functions of spherical particles,” Appl. Opt.  9, 1888–1896 (1970).
[PubMed]

W. J. Lentz, “Generating Bessel functions in Mie scattering calculations using continued fractions,” Appl. Opt.  15, 668–671 (1976).
[CrossRef] [PubMed]

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

G. Roll, T. Kaiser, and G. Schweiger, “Controlled modification of the expansion order as a tool in Mie computations,” Appl. Opt.  37, 2483–2492 (1998).
[CrossRef]

R. Li, X. Han, H. Jiang, and K. Ren, “Debye series of normally incident plane-wave scattering by an infinite multilayered cylinder,” Appl. Opt.  45, 6255–6262 (2006).
[CrossRef] [PubMed]

R. Li, X. Han, L. Shi, K. Ren, and H. Jiang, “Debye series for Gaussian beam scattering by a multilayered sphere,” Appl. Opt.  46, 4804–4812 (2007).
[CrossRef] [PubMed]

IBM J. Res. Dev. (1)

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

J. Math. Phys. (1)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys.  10 (1), 82–124 (1969).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Quant. Spectosc. Radiat. Transfer (2)

P. Laven, “Simulation of rainbows, coronas and glories using Mie theory and the Debye series,” J. Quant. Spectosc. Radiat. Transfer  89, 257–269 (2004).
[CrossRef]

H. Yu, J. Shen, and Y. Wei, “Geometrical optics approximation for light scattering by absorbing spherical particles,” J. Quant. Spectosc. Radiat. Transfer  110, 1178–1189 (2009).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (2)

L. Wu, H. Yang, X. Li, B. Yang, and G. Li, “Scattering by large bubbles: Comparisons between geometrical-optics theory and Debye series,” J. Quant. Spectrosc. Radiat. Transfer  108, 54–64 (2007).
[CrossRef]

J. Shen, H. Wang, B. Wang, H. Yu, and B. Yu, “Stability in Debye series calculation for light scattering by absorbing particles and bubbles,” J. Quant. Spectrosc. Radiat. Transfer  111, 772–781 (2010).
[CrossRef]

Opt. Commun. (2)

X. Han, H. Jiang, K. Ren, and G. Gréhan, “On rainbows of inhomogeneous spherical droplets,” Opt. Commun.  269, 291–298 (2007).
[CrossRef]

R. Li, X. Han, and K. Ren, “Debye series expansion of shaped beam scattering by GI-POF,” Opt. Commun.  282, 4315–4321(2009).
[CrossRef]

Part. Part. Syst. Charact. (1)

G. Gouesbet, “Debye series formulation for generalized Lorenz-Mie theory with the Bromwich method,” Part. Part. Syst. Charact.  20, 382–386 (2003).
[CrossRef]

Phys. Rev. E (1)

R. Li and X. Han, “Generalized Debye series expansion of electromagnetic plane wave scattering by an infinite multilayered cylinder at oblique incidence,” Phys. Rev. E  79, 036602(2009).
[CrossRef]

Phys. Z. (1)

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z.  9, 775–778 (1908).

PIERS Online (1)

J. Shen and X. Cai, “Algorithm of numerical calculation on Lorentz Mie theory,” PIERS Online  1, 691–694 (2005).
[CrossRef]

Radio Sci. (1)

Z. Wu and Y. Wang, “Electromagnetic scattering for multi-layered sphere: recursive algorithms,” Radio Sci.  26, 1393–1401 (1991).
[CrossRef]

Other (5)

P. Laven, “The optics of a water drop: Mie scattering and the Debye series,” http://www.philiplaven.com/index1.html (2004).

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

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

H. C. van de Hulst, Light Scattering by Small Particles(Chapman & Hall, 1957).

P. Laven, “MiePlot: a computer program for scattering of light from a sphere using Mie theory & the Debye series,” http://www.philiplaven.com/mieplot.htm (2009).

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Figures (6)

Fig. 1
Fig. 1

Schematic of the DSE.

Fig. 2
Fig. 2

Results of the DSE calculation for the full/components scattering intensities by homogeneous spheres with different refractive indices. The dimensionless particle size parameter is x = 100 , and the refractive indices are (a)  m = 1.33 , (b)  m = 1.33 + i 0.001 , (c)  m = 1.33 + i 0.01 , and (d)  m = 1.33 + i 0.1 . The curves for p = 1, 2 and 3 in (d) are shifted upward by 10 10 , 10 21 , and 10 25 , respectively, and the full scattering intensities of the DSE calculation are obtained for all p terms.

Fig. 3
Fig. 3

DSE results of the full scattering intensity ( p = 0 ) and the component of both diffraction and surface reflection ( p = 0 ) by a strongly absorbing sphere with x = 100 and m = 0.272 + i 3.24 .

Fig. 4
Fig. 4

Dependence of the absorption efficiency k abs and the surface reflection | R n = 60 212 | on the imaginary part of the refractive index. The particle size x is 100, and the real part of the refractive index Re ( m ) is 1.33.

Fig. 5
Fig. 5

Comparison of the component intensities obtained with different DSE calculation methods for x = 100 and m = 1.33 + i 0.1 : (a)  p = 0 , (b)  p = 1 , (c)  p = 2 , and (d)  p = 3 .

Fig. 6
Fig. 6

Comparison between the different DSE calculation methods and the Mie calculation for full scattering intensities ( p = 0 , x = 100 , and m = 0.667 + i 0.0777 ).

Equations (20)

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

a n b n } = 1 2 [ 1 R n 212 T n 21 T n 12 1 R n 121 ] = 1 2 [ 1 R n 212 p = 1 T n 21 ( R n 121 ) p 1 T n 12 ] ;
a n ( p = 0 ) b n ( p = 0 ) } = 1 2 ( 1 R n 212 ) ,
a n ( p ) b n ( p ) } = 1 2 T n 21 ( R n 121 ) p 1 T n 12 = 1 2 T n ( R n 121 ) p 1 ( p = 1 , 2 , ) .
T n 21 = m 1 m 2 2 i D n = m 2 i D n , T n 12 = 2 i D n , R n 212 = α ξ n ( 2 ) ( x ) ξ n ( 2 ) ( y ) β ξ n ( 2 ) ( x ) ξ n ( 2 ) ( y ) D n , R n 121 = α ξ n ( 1 ) ( x ) ξ n ( 1 ) ( y ) β ξ n ( 1 ) ( x ) ξ n ( 1 ) ( y ) D n ,
D n = α ξ n ( 1 ) ( x ) ξ n ( 2 ) ( y ) + β ξ n ( 1 ) ( x ) ξ n ( 2 ) ( y ) ,
α = { 1 TE   wave m TM   wave , β = { m TE   wave 1 TM   wave .
T n = m A n ( 34 ) ( y ) A n ( 34 ) ( x ) · [ D n ( 3 ) ( x ) D n ( 4 ) ( x ) ] · [ D n ( 3 ) ( y ) D n ( 4 ) ( y ) ] [ α D n ( 3 ) ( x ) β D n ( 4 ) ( y ) ] 2 , R n 212 = 1 A n ( 34 ) ( x ) · α D n ( 4 ) ( x ) β D n ( 4 ) ( y ) α D n ( 3 ) ( x ) β D n ( 4 ) ( y ) , R n 121 = A n ( 34 ) ( y ) · α D n ( 3 ) ( x ) β D n ( 3 ) ( y ) α D n ( 3 ) ( x ) β D n ( 4 ) ( y ) ,
i 1 ( θ ) = | Σ n = 1 2 n + 1 n ( n + 1 ) [ a n π n ( θ ) + b n τ n ( θ ) ] | 2 , i 2 ( θ ) = | Σ n = 1 2 n + 1 n ( n + 1 ) [ a n τ n ( θ ) + b n π n ( θ ) ] | 2 , i ( θ ) = i 1 ( θ ) + i 2 ( θ ) 2 .
π n ( θ ) = 2 n 1 n 1 cos θ π n 1 ( θ ) n n 1 π n 2 ( θ ) , τ n ( θ ) = n cos θ π n ( θ ) ( n + 1 ) π n 1 ( θ ) ,
D n ( 3 ) ( z ) = n z + 1 n z D n 1 ( 3 ) ( z ) .
T n = m A n ( 13 ) ( x ) A n ( 13 ) ( y ) u 0 { u 33 A n ( 13 ) ( y ) u 31 } 2 , 1 R n 212 = A n ( 13 ) ( x ) · u 13 A n ( 13 ) ( y ) u 11 u 33 A n ( 13 ) ( y ) u 31 , 1 R n 121 = A n ( 13 ) ( y ) u 31 u 33 A n ( 13 ) ( y ) u 31 ,
T n = m A n ( 13 ) ( x ) A n ( 31 ) ( y ) u 0 { A n ( 31 ) ( y ) u 33 u 31 } 2 , 1 R n 212 = A n ( 13 ) ( x ) · A n ( 31 ) ( y ) u 13 u 11 A n ( 31 ) ( y ) u 33 u 31 , 1 R n 121 = u 31 A n ( 31 ) ( y ) u 33 u 31 ,
u 0 = [ D n ( 1 ) ( x ) D n ( 3 ) ( x ) ] · [ D n ( 1 ) ( y ) D n ( 3 ) ( y ) ] , u i j = α D n ( i ) ( x ) β D n ( j ) ( y ) for     i , j = 1 , 3 ;
D n 1 ( 1 ) ( z ) = n z 1 n z + D n ( 1 ) ( z ) .
A n ( 13 ) ( z ) = [ A n ( 31 ) ( z ) ] 1 = 2 ψ n ( z ) ξ n ( 1 ) ( z ) .
A n ( 13 ) ( z ) = A n 1 ( 13 ) ( z ) · D n 1 ( 1 ) ( z ) n z D n 1 ( 3 ) ( z ) n z .
ln A n ( 13 ) ( z ) = ln A n 1 ( 13 ) ( z ) + ln D n 1 ( 1 ) ( z ) n z D n 1 ( 3 ) ( z ) n z ,
ln A 0 ( 13 ) ( z ) = 2 Im z + ln { exp ( 2 Im z ) cos ( 2 Re z ) + i sin ( 2 Re z ) } ,
n stop = x + 4.05 x 1 / 3 + 2.
R n 212 = ξ n ( 2 ) ( x ) ξ n ( 1 ) ( x ) r n 212 .

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