## High-frequency extinction efficiencies of spheroids: rigorous T-matrix solutions and semi-empirical approximations |

Optics Express, Vol. 22, Issue 9, pp. 10270-10293 (2014)

http://dx.doi.org/10.1364/OE.22.010270

Acrobat PDF (2245 KB)

### Abstract

A semi-empirical high-frequency formula is developed to efficiently and accurately compute the extinction efficiencies of spheroids in the cases of moderate and large size parameters under either fixed or random orientation condition. The formula incorporates the semi-classical scattering concepts formulated by extending the complex angular momentum approximation of the Lorenz-Mie theory to the spheroid case on the basis of the physical rationales associated with changing the particle morphology from a sphere to a spheroid. The asymptotic edge-effect expansion is truncated with an optimal number of terms based on *a priori* knowledge obtained from comparing the semi-classical Mie extinction efficiencies with the Lorenz-Mie solutions. The present formula is fully tested in comparison with the T-matrix results for spheroids with the aspect ratios from 0.5 to 2.0, and for various refractive indices m_{r} + im_{i}, with m_{r} from 1.0 to 2.0 and m_{i} from 0 to 0.5.

© 2014 Optical Society of America

## 1. Introduction

14. A. A. Kokhanovsky and E. P. Zege, “Optical properties of aerosol particles: a review of approximate analytical solutions,” J. Aerosol Sci. **28**(1), 1–21 (1997). [CrossRef]

15. H. M. Nussenzveig and W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. **45**(18), 1490–1494 (1980). [CrossRef]

18. S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. **14**(1), 29–49 (1975). [CrossRef] [PubMed]

9. G. R. Fournier and B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. **30**(15), 2042–2048 (1991). [CrossRef] [PubMed]

19. P. Yang, Q. Feng, G. Hong, G. W. Kattawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, and I. N. Sokolik, “Modeling of the scattering and radiative properties of nonspherical dust-like aerosols,” J. Aerosol Sci. **38**(10), 995–1014 (2007). [CrossRef]

22. K. N. Liou, Y. Takano, and P. Yang, “Light absorption and scattering by aggregates: application to black carbon and snow grains,” J. Quant. Spectrosc. Radiat. Transf. **112**(10), 1581–1594 (2011). [CrossRef]

9. G. R. Fournier and B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. **30**(15), 2042–2048 (1991). [CrossRef] [PubMed]

24. D. S. Jones, “Approximate methods in high-frequency scattering,” Proc. R. Soc. A **239**, 338–348 (1957). [CrossRef]

25. D. S. Jones, “High-frequency scattering of electromagnetic waves,” Proc. R. Soc. Lond. A Math. Phys. Sci. **240**(1221), 206–213 (1957). [CrossRef]

26. P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D Part. Fields **3**(4), 825–839 (1971). [CrossRef]

27. S. C. Hill, A. C. Hill, and P. W. Barber, “Light scattering by size/shape distributions of soil particles and spheroids,” Appl. Opt. **23**(7), 1025–1031 (1984). [CrossRef] [PubMed]

28. T. W. Chen, “High energy light scattering in the generalized eikonal approximation,” Appl. Opt. **28**(19), 4096–4102 (1989). [CrossRef] [PubMed]

5. L. Wang, X. Sun, and J. Xing, “Retrieval of spheroidal particle size distribution using the approximate method in spectral extinction technique,” Opt. Commun. **285**(7), 1646–1653 (2012). [CrossRef]

29. N. V. Voshchinnikov and V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. **204**(1), 19–86 (1993). [CrossRef]

5. L. Wang, X. Sun, and J. Xing, “Retrieval of spheroidal particle size distribution using the approximate method in spectral extinction technique,” Opt. Commun. **285**(7), 1646–1653 (2012). [CrossRef]

9. G. R. Fournier and B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. **30**(15), 2042–2048 (1991). [CrossRef] [PubMed]

30. B. R. Johnson, “Invariant imbedding T matrix approach to electromagnetic scattering,” Appl. Opt. **27**(23), 4861–4873 (1988). [CrossRef] [PubMed]

32. L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “A numerical combination of extended boundary condition method and invariant imbedding method applied to light scattering by large spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transf. **123**, 17–22 (2013). [CrossRef]

5. L. Wang, X. Sun, and J. Xing, “Retrieval of spheroidal particle size distribution using the approximate method in spectral extinction technique,” Opt. Commun. **285**(7), 1646–1653 (2012). [CrossRef]

**30**(15), 2042–2048 (1991). [CrossRef] [PubMed]

10. B. T. N. Evans and G. R. Fournier, “Analytic approximation to randomly oriented spheroid extinction,” Appl. Opt. **33**(24), 5796–5804 (1994). [CrossRef] [PubMed]

15. H. M. Nussenzveig and W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. **45**(18), 1490–1494 (1980). [CrossRef]

## 2. T-matrix method

## 3. Semi-classical Mie extinction efficiency

15. H. M. Nussenzveig and W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. **45**(18), 1490–1494 (1980). [CrossRef]

**45**(18), 1490–1494 (1980). [CrossRef]

_{r}is smaller than

**45**(18), 1490–1494 (1980). [CrossRef]

^{−7}to 10

^{−3}. By comparing the trend of the extinction curve with respect to the size parameter, we found that four terms are required when

## 4. Semi-empirical approach for spheroids

### 4.1 End-on Incidence

38. J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. II. Transmission and cross-polarization effects,” Appl. Opt. **35**(3), 515–531 (1996). [CrossRef] [PubMed]

40. F. Xu, K. F. Ren, X. Cai, and J. Shen, “Extension of geometrical-optics approximation to on-axis Gaussian beam scattering. II. By a spheroidal particle with end-on incidence,” Appl. Opt. **45**(20), 5000–5009 (2006). [CrossRef] [PubMed]

**30**(15), 2042–2048 (1991). [CrossRef] [PubMed]

38. J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. II. Transmission and cross-polarization effects,” Appl. Opt. **35**(3), 515–531 (1996). [CrossRef] [PubMed]

24. D. S. Jones, “Approximate methods in high-frequency scattering,” Proc. R. Soc. A **239**, 338–348 (1957). [CrossRef]

25. D. S. Jones, “High-frequency scattering of electromagnetic waves,” Proc. R. Soc. Lond. A Math. Phys. Sci. **240**(1221), 206–213 (1957). [CrossRef]

22. K. N. Liou, Y. Takano, and P. Yang, “Light absorption and scattering by aggregates: application to black carbon and snow grains,” J. Quant. Spectrosc. Radiat. Transf. **112**(10), 1581–1594 (2011). [CrossRef]

### 4.2 Oblique incidence

10. B. T. N. Evans and G. R. Fournier, “Analytic approximation to randomly oriented spheroid extinction,” Appl. Opt. **33**(24), 5796–5804 (1994). [CrossRef] [PubMed]

38. J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. II. Transmission and cross-polarization effects,” Appl. Opt. **35**(3), 515–531 (1996). [CrossRef] [PubMed]

**35**(3), 515–531 (1996). [CrossRef] [PubMed]

### 4.3 Random orientation

**30**(15), 2042–2048 (1991). [CrossRef] [PubMed]

## 5. Computational results

^{−7}, 10

^{−3}, 10

^{−2}, 10

^{−1}, and 0.5. In the II-TM simulations, the extinction efficiencies for randomly oriented spheroids and for 91 fixed orientations specified with

^{−7}. A small imaginary part of the refractive index corresponds to negligible absorption, and the interference between the diffraction and the transmission is evident. For prolate spheroids, because the phase shift is the same for all particle aspect ratios, the peaks and troughs are located at the same size parameter. However, the divergence of the central ray and the edge effect are different and the values of the extinction efficiencies differ. From the comparison, the II-TM agrees excellently with the approximations when the size parameter is larger than 10. For oblate spheroids, the phase shift varies for different aspect ratios because the size parameter is defined in terms of the larger semi-axis (namely,

**30**(15), 2042–2048 (1991). [CrossRef] [PubMed]

^{−7}. As in Fig. 5, the incident light is aligned with the symmetric axis. For oblate spheroids and prolate spheroids with the aspect ratio close to unity, the ripple structure of the extinction curve is obvious. However, for prolate spheroids with

**30**(15), 2042–2048 (1991). [CrossRef] [PubMed]

^{−7}and 1.3 + i0.001, the FE(a) and the FE(b) have close results because the contributions from higher-order terms are small. The accuracy of the present formula is better than the FE(a) and the FE(b) in comparison with the II-TM because the geometric optics term Eq. (19) is more accurate than the ADT formula Eq. (21). For the refractive index, 1.3 + i0.5, Eq. (19) and Eq. (21) are equal to two because the particle is highly absorptive such that the interference between diffraction and transmission is suppressed. However, the higher-order terms (in particular,

## 6. Summary and conclusion

^{−4}seconds).

*a priori*knowledge needs be obtained from the Lorenz-Mie theory regarding whether the higher-order edge effect terms should be included. To remove this inconvenience, further theoretical investigations on the semi-classical Mie extinction efficiency are most likely required or, practically, a table of optimal number of terms can be created with respect to typical refractive indices, although the operational principle behind this truncation is not clear. However, if no treatments are made (i.e., all higher-order edge-effect terms are included for higher real part of refractive indices), the additional resultant errors may not exceed ~5% (see Fig. 2) for moderate size parameters.

**30**(15), 2042–2048 (1991). [CrossRef] [PubMed]

44. L. Bi, P. Yang, and G. W. Kattawar, “Edge-effect contribution to the extinction of light by dielectric disks and cylindrical particles,” Appl. Opt. **49**(24), 4641–4646 (2010). [CrossRef] [PubMed]

## Appendix

45. J. A. Lock and T. A. McCollum, “Further thoughts on Newton’s zero–order rainbow,” Am. J. Phys. **62**(12), 1082–1089 (1994). [CrossRef]

## Acknowledgments

## References and links

1. | K. N. Liou, |

2. | G. E. Elicable and L. H. Garcia-Rubio, “Latex particle size distribution from turbidimetry using inversion techniques,” J. Colloid Interface Sci. |

3. | G. Crawley, M. Cournil, and D. D. Benedetto, “Size analysis of fine particle suspensions by spectral turbidimetry: potential and limits,” Powder Technol. |

4. | M. Z. Li and D. Wilkinson, “Particle size distribution determination from spectral extinction using evolutionary programming,” Chem. Eng. Sci. |

5. | L. Wang, X. Sun, and J. Xing, “Retrieval of spheroidal particle size distribution using the approximate method in spectral extinction technique,” Opt. Commun. |

6. | M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds., |

7. | A. A. Kokhanovsky, ed., |

8. | P. Chylek and J. D. Klett, “Extinction cross sections of nonspherical particles in the anomalous diffraction approximation,” J. Opt. Soc. Am. A |

9. | G. R. Fournier and B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. |

10. | B. T. N. Evans and G. R. Fournier, “Analytic approximation to randomly oriented spheroid extinction,” Appl. Opt. |

11. | A. J. Baran and S. Havemann, “Rapid computation of the optical properties of hexagonal columns using complex angular momentum theory,” J. Quant. Spectrosc. Radiat. Transf. |

12. | P. Yang, K. N. Liou, and W. P. Arnott, “Extinction efficiency and single-scattering albedo of ice crystals in laboratory and natural cirrus clouds,” J. Geophys. Res. |

13. | J. Q. Zhao and Y. Q. Hu, “Bridging technique for calculating the extinction efficiency of arbitrary shaped particles,” Appl. Opt. |

14. | A. A. Kokhanovsky and E. P. Zege, “Optical properties of aerosol particles: a review of approximate analytical solutions,” J. Aerosol Sci. |

15. | H. M. Nussenzveig and W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. |

16. | H. M. Nussenzveig, |

17. | P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. |

18. | S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. |

19. | P. Yang, Q. Feng, G. Hong, G. W. Kattawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, and I. N. Sokolik, “Modeling of the scattering and radiative properties of nonspherical dust-like aerosols,” J. Aerosol Sci. |

20. | L. Bi, P. Yang, G. W. Kattawar, and R. Kahn, “Single-scattering properties of triaxial ellipsoidal particles for a size parameter range from the Rayleigh to geometric-optics regimes,” Appl. Opt. |

21. | K. N. Liou, Y. Takano, and P. Yang, “On geometric optics and surface waves for light scattering by spheres,” J. Quant. Spectrosc. Radiat. Transf. |

22. | K. N. Liou, Y. Takano, and P. Yang, “Light absorption and scattering by aggregates: application to black carbon and snow grains,” J. Quant. Spectrosc. Radiat. Transf. |

23. | H. C. van de Hulst, |

24. | D. S. Jones, “Approximate methods in high-frequency scattering,” Proc. R. Soc. A |

25. | D. S. Jones, “High-frequency scattering of electromagnetic waves,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

26. | P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D Part. Fields |

27. | S. C. Hill, A. C. Hill, and P. W. Barber, “Light scattering by size/shape distributions of soil particles and spheroids,” Appl. Opt. |

28. | T. W. Chen, “High energy light scattering in the generalized eikonal approximation,” Appl. Opt. |

29. | N. V. Voshchinnikov and V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. |

30. | B. R. Johnson, “Invariant imbedding T matrix approach to electromagnetic scattering,” Appl. Opt. |

31. | L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transf. |

32. | L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “A numerical combination of extended boundary condition method and invariant imbedding method applied to light scattering by large spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transf. |

33. | M. I. Mishchenko, L. D. Travis, and A. A. Lacis, |

34. | M. I. Mishchenko, “Calculation of the amplitude matrix for a nonspherical particle in a fixed orientation,” Appl. Opt. |

35. | P. Yang, M. Wendisch, L. Bi, G. W. Kattawar, M. I. Mishchenko, and Y. Hu, “Dependence of extinction cross-section on incident polarization state and particle orientation,” J. Quant. Spectrosc. Radiat. Transf. |

36. | W. H. Beyer, ed., |

37. | W. T. Grandy, |

38. | J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. II. Transmission and cross-polarization effects,” Appl. Opt. |

39. | K. F. Ren, F. Onofri, C. Rozé, and T. Girasole, “Vectorial complex ray model and application to two-dimensional scattering of plane wave by a spheroidal particle,” Opt. Lett. |

40. | F. Xu, K. F. Ren, X. Cai, and J. Shen, “Extension of geometrical-optics approximation to on-axis Gaussian beam scattering. II. By a spheroidal particle with end-on incidence,” Appl. Opt. |

41. | M. Born and E. Wolf, |

42. | A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, |

43. | C. F. Bohren and D. R. Huffman, |

44. | L. Bi, P. Yang, and G. W. Kattawar, “Edge-effect contribution to the extinction of light by dielectric disks and cylindrical particles,” Appl. Opt. |

45. | J. A. Lock and T. A. McCollum, “Further thoughts on Newton’s zero–order rainbow,” Am. J. Phys. |

**OCIS Codes**

(290.0290) Scattering : Scattering

(290.2200) Scattering : Extinction

**ToC Category:**

Scattering

**History**

Original Manuscript: March 10, 2014

Revised Manuscript: April 8, 2014

Manuscript Accepted: April 14, 2014

Published: April 21, 2014

**Citation**

Lei Bi and Ping Yang, "High-frequency extinction efficiencies of spheroids: rigorous T-matrix solutions and semi-empirical approximations," Opt. Express **22**, 10270-10293 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-9-10270

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### References

- K. N. Liou, An Introduction to Atmospheric Radiation (Academic, 2002).
- G. E. Elicable, L. H. Garcia-Rubio, “Latex particle size distribution from turbidimetry using inversion techniques,” J. Colloid Interface Sci. 129(1), 192–200 (1989). [CrossRef]
- G. Crawley, M. Cournil, D. D. Benedetto, “Size analysis of fine particle suspensions by spectral turbidimetry: potential and limits,” Powder Technol. 91(3), 197–208 (1997). [CrossRef]
- M. Z. Li, D. Wilkinson, “Particle size distribution determination from spectral extinction using evolutionary programming,” Chem. Eng. Sci. 56(10), 3045–3052 (2001). [CrossRef]
- L. Wang, X. Sun, J. Xing, “Retrieval of spheroidal particle size distribution using the approximate method in spectral extinction technique,” Opt. Commun. 285(7), 1646–1653 (2012). [CrossRef]
- M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds., Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 1999).
- A. A. Kokhanovsky, ed., Light Scattering Reviews 8 (Springer-Praxis Publishing, Chichester, UK, 2013).
- P. Chylek, J. D. Klett, “Extinction cross sections of nonspherical particles in the anomalous diffraction approximation,” J. Opt. Soc. Am. A 8(2), 274–281 (1991). [CrossRef]
- G. R. Fournier, B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30(15), 2042–2048 (1991). [CrossRef] [PubMed]
- B. T. N. Evans, G. R. Fournier, “Analytic approximation to randomly oriented spheroid extinction,” Appl. Opt. 33(24), 5796–5804 (1994). [CrossRef] [PubMed]
- A. J. Baran, S. Havemann, “Rapid computation of the optical properties of hexagonal columns using complex angular momentum theory,” J. Quant. Spectrosc. Radiat. Transf. 63(2-6), 499–519 (1999). [CrossRef]
- P. Yang, K. N. Liou, W. P. Arnott, “Extinction efficiency and single-scattering albedo of ice crystals in laboratory and natural cirrus clouds,” J. Geophys. Res. 102(D18), 21825–21835 (1997). [CrossRef]
- J. Q. Zhao, Y. Q. Hu, “Bridging technique for calculating the extinction efficiency of arbitrary shaped particles,” Appl. Opt. 42(24), 4937–4945 (2003). [CrossRef] [PubMed]
- A. A. Kokhanovsky, E. P. Zege, “Optical properties of aerosol particles: a review of approximate analytical solutions,” J. Aerosol Sci. 28(1), 1–21 (1997). [CrossRef]
- H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45(18), 1490–1494 (1980). [CrossRef]
- H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, 1992).
- P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908).
- S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14(1), 29–49 (1975). [CrossRef] [PubMed]
- P. Yang, Q. Feng, G. Hong, G. W. Kattawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, I. N. Sokolik, “Modeling of the scattering and radiative properties of nonspherical dust-like aerosols,” J. Aerosol Sci. 38(10), 995–1014 (2007). [CrossRef]
- L. Bi, P. Yang, G. W. Kattawar, R. Kahn, “Single-scattering properties of triaxial ellipsoidal particles for a size parameter range from the Rayleigh to geometric-optics regimes,” Appl. Opt. 48(1), 114–126 (2009). [CrossRef] [PubMed]
- K. N. Liou, Y. Takano, P. Yang, “On geometric optics and surface waves for light scattering by spheres,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1980–1989 (2010). [CrossRef]
- K. N. Liou, Y. Takano, P. Yang, “Light absorption and scattering by aggregates: application to black carbon and snow grains,” J. Quant. Spectrosc. Radiat. Transf. 112(10), 1581–1594 (2011). [CrossRef]
- H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).
- D. S. Jones, “Approximate methods in high-frequency scattering,” Proc. R. Soc. A 239, 338–348 (1957). [CrossRef]
- D. S. Jones, “High-frequency scattering of electromagnetic waves,” Proc. R. Soc. Lond. A Math. Phys. Sci. 240(1221), 206–213 (1957). [CrossRef]
- P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D Part. Fields 3(4), 825–839 (1971). [CrossRef]
- S. C. Hill, A. C. Hill, P. W. Barber, “Light scattering by size/shape distributions of soil particles and spheroids,” Appl. Opt. 23(7), 1025–1031 (1984). [CrossRef] [PubMed]
- T. W. Chen, “High energy light scattering in the generalized eikonal approximation,” Appl. Opt. 28(19), 4096–4102 (1989). [CrossRef] [PubMed]
- N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204(1), 19–86 (1993). [CrossRef]
- B. R. Johnson, “Invariant imbedding T matrix approach to electromagnetic scattering,” Appl. Opt. 27(23), 4861–4873 (1988). [CrossRef] [PubMed]
- L. Bi, P. Yang, G. W. Kattawar, M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transf. 116, 169–183 (2013). [CrossRef]
- L. Bi, P. Yang, G. W. Kattawar, M. I. Mishchenko, “A numerical combination of extended boundary condition method and invariant imbedding method applied to light scattering by large spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 123, 17–22 (2013). [CrossRef]
- M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, absorption and emission of light by small particles (Cambridge University, 2002).
- M. I. Mishchenko, “Calculation of the amplitude matrix for a nonspherical particle in a fixed orientation,” Appl. Opt. 39(6), 1026–1031 (2000). [CrossRef] [PubMed]
- P. Yang, M. Wendisch, L. Bi, G. W. Kattawar, M. I. Mishchenko, Y. Hu, “Dependence of extinction cross-section on incident polarization state and particle orientation,” J. Quant. Spectrosc. Radiat. Transf. 112(12), 2035–2039 (2011). [CrossRef]
- W. H. Beyer, ed., CRC standard mathematical tables (CRC Press, 1981).
- W. T. Grandy, Scattering of waves from large spheres (Cambridge University, 2000).
- J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. II. Transmission and cross-polarization effects,” Appl. Opt. 35(3), 515–531 (1996). [CrossRef] [PubMed]
- K. F. Ren, F. Onofri, C. Rozé, T. Girasole, “Vectorial complex ray model and application to two-dimensional scattering of plane wave by a spheroidal particle,” Opt. Lett. 36(3), 370–372 (2011). [CrossRef] [PubMed]
- F. Xu, K. F. Ren, X. Cai, J. Shen, “Extension of geometrical-optics approximation to on-axis Gaussian beam scattering. II. By a spheroidal particle with end-on incidence,” Appl. Opt. 45(20), 5000–5009 (2006). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics (Pergamon, 2001).
- A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, 1954).
- C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, 1983)
- L. Bi, P. Yang, G. W. Kattawar, “Edge-effect contribution to the extinction of light by dielectric disks and cylindrical particles,” Appl. Opt. 49(24), 4641–4646 (2010). [CrossRef] [PubMed]
- J. A. Lock, T. A. McCollum, “Further thoughts on Newton’s zero–order rainbow,” Am. J. Phys. 62(12), 1082–1089 (1994). [CrossRef]

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