## Analysis of rainbow scattering by a chiral sphere |

Optics Express, Vol. 21, Issue 19, pp. 21879-21888 (2013)

http://dx.doi.org/10.1364/OE.21.021879

Acrobat PDF (1257 KB)

### Abstract

Based on the scattering theory of a chiral sphere, rainbow phenomenon of a chiral sphere is numerically analyzed in this paper. For chiral spheres illuminated by a linearly polarized wave, there are three first-order rainbows, with whose rainbow angles varying with the chirality parameter. The spectrum of each rainbow structure is presented and the ripple frequencies are found associated with the size and refractive indices of the chiral sphere. Only two rainbow structures remain when the chiral sphere is illuminated by a circularly polarized plane wave. Finally, the rainbows of chiral spheres with slight chirality parameters are found appearing alternately in E-plane and H-plane with the variation of the chirality.

© 2013 OSA

## 1. Introduction

3. R. T. Wang and H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. **30**(1), 106–117 (1991). [CrossRef] [PubMed]

4. 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**(4), 693–706 (1993). [CrossRef]

5. J. A. Lock, J. M. Jamison, and C. Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. **33**(21), 4677–4690 (1994). [CrossRef] [PubMed]

6. Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. **36**(21), 5188–5198 (1997). [CrossRef] [PubMed]

7. G. Kaduchak, P. L. Marston, and H. J. Simpson, “E(6) diffraction catastrophe of the primary rainbow of oblate water drops: observations with white-light and laser illumination,” Appl. Opt. **33**(21), 4691–4696 (1994). [CrossRef] [PubMed]

8. J. P. A. J. van Beeck and M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. **35**(13), 2259–2266 (1996). [CrossRef] [PubMed]

10. X. Han, K. F. Ren, Z. Wu, F. Corbin, G. Gouesbet, and G. Gréhan, “Characterization of Initial Disturbances in a Liquid Jet by Rainbow Sizing,” Appl. Opt. **37**(36), 8498–8503 (1998). [CrossRef] [PubMed]

11. J. P. A. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global Rainbow Thermometry for Mean Temperature and Size Measurement of Spray Droplets,” Particle & Particle Systems Characterization **18**(4), 196–204 (2001). [CrossRef]

13. J. Wang, G. Gréhan, Y. Han, S. Saengkaew, and G. Gouesbet, “Numerical study of global rainbow technique: sensitivity to non-sphericity of droplets,” Exp. Fluids **51**(1), 149–159 (2011). [CrossRef]

## 2. Scattering by a large chiral sphere

### 2.1. Scattering coefficients

23. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. **29**(3), 458–462 (1974). [CrossRef]

24. Z.-S. Wu, Q.-C. Shang, and Z.-J. Li, “Calculation of electromagnetic scattering by a large chiral sphere,” Appl. Opt. **51**(27), 6661–6668 (2012). [CrossRef] [PubMed]

*a*with chirality parameter

*κ*illuminated by a beam. As we discussed in [24

24. Z.-S. Wu, Q.-C. Shang, and Z.-J. Li, “Calculation of electromagnetic scattering by a large chiral sphere,” Appl. Opt. **51**(27), 6661–6668 (2012). [CrossRef] [PubMed]

25. Q.-C. Shang, Z.-S. Wu, T. Qu, Z.-J. Li, L. Bai, and L. Gong, “Analysis of the radiation force and torque exerted on a chiral sphere by a Gaussian beam,” Opt. Express **21**(7), 8677–8688 (2013). [CrossRef] [PubMed]

26. D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell's equations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **56**(1), 1102–1112 (1997). [CrossRef]

*x*-polarized (linearly polarized in the

*x*-direction),

*y-*polarized (linearly polarized in the

*y*-direction), RCP and LCP wave incidences when

*ip*is

*ix*,

*iy*,

*iR*, and

*iL*, respectively.

24. Z.-S. Wu, Q.-C. Shang, and Z.-J. Li, “Calculation of electromagnetic scattering by a large chiral sphere,” Appl. Opt. **51**(27), 6661–6668 (2012). [CrossRef] [PubMed]

27. Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: Recursive algorithms,” Radio Sci. **26**(6), 1393–1401 (1991). [CrossRef]

28. A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys. **22**(10), 1242–1246 (1951). [CrossRef]

### 2.2. Scattered field for differently polarized incident plane waves

*x*-polarized plane wave propagating along the

*z*-axis with the form

29. Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antenn. Propag. **57**(2), 572–576 (2009). [CrossRef]

*x*-polarized plane wave incidence, scattered field can be simplified after substituting Eq. (15) into Eqs. (12) and (13) (See appendix in [24

**51**(27), 6661–6668 (2012). [CrossRef] [PubMed]

*y*-polarized plane wave from Eq. (19):

## 3. Rainbow phenomenon of chiral spheres

### 3.1. Rainbow structures of chiral spheres

**51**(27), 6661–6668 (2012). [CrossRef] [PubMed]

*x*-polarized incident plane wave with wavelength

*λ*, respectively. As the radius is as large as 500

*λ*, step of the scattering angle

*θ*is set to

*x*-polarized plane wave can be observed in only H-plane (

### 3.2. Variation of rainbow structure with chirality parameters

*n*, and the Right rainbow with

_{L}*n*. We presented a rough physical interpretation in [24

_{R}**51**(27), 6661–6668 (2012). [CrossRef] [PubMed]

### 3.3. Spectrums of the rainbow structures

8. J. P. A. J. van Beeck and M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. **35**(13), 2259–2266 (1996). [CrossRef] [PubMed]

*d*can be calculated by the following formulas after considering the effect of wavelength:

*n*represents the refractive index and

*f*is the corresponding ripple frequency. Our calculations show that the empirical relation is valid when the refractive index is in the range 1.23-1.43. By using the relation above, the diameter calculated by the ripple frequency of the Left rainbow, Middle rainbow, and Right rainbow is, respectively, 1005.0

_{ripple}*λ*, 1015.7

*λ*, and 995.3

*λ*. It can be seen that results estimated according to the Left rainbow and Right rainbow are very close to1000

*λ*, the actual diameter of the chiral sphere. Although the Right rainbow has a strange rainbow structure and spectrum structure, its ripple frequency can still be used to estimate the size of the chiral sphere. And the result is as good as that of the Left rainbow, which has an ordinary rainbow structure and spectrum structure as isotropic ones. For the Middle rainbow, the errors may be caused by the refractive index we adopted. In fact, according to the depiction in section 3.2, it is inappropriate to associate the Middle rainbow with a refractive index

*n*= 1.33. All the upper limits of the spectrums of the three rainbows are almost the same, which can be readily understood. The corresponding components result from the interferences of the surface waves, which depend on only the particle size and the wavelength, and have nothing to do with the medium of the particle.

### 3.4. Rainbows for circularly polarized plane wave incidences

*x*-polarized wave incidence, only two rainbow structures can be found. As shown in Fig. 4(a), for a chiral sphere with chirality

*κ*illuminated by a RCP wave are symmetric physically to those of a chiral sphere with chirality -

*κ*illuminated by a LCP wave. Additionally, no second order rainbow occurs for chirality 0.05 in the case of a LCP incidence and chirality −0.05 in the case of a RCP incidence.

## 4. Rainbows of chiral spheres with slight chirality

*d*with chirality

*κ*, the polarization plane of the incident wave is rotated an angle of

*κk*

_{0}

*d*, where

*k*

_{0}is wave number of the plane wave. Thus, it can be estimated that the common optically active media such as sugar solutions possess slight chirality parameters at about 10

^{−5}, which in fact are very close to isotropic media. Based on the curves presented in Fig. 2(a), there should be only one rainbow structure for these media and the rainbow angles are the same as those for isotropic spheres. Rainbows of a chiral sphere with slight chirality 5 × 10

^{−5}and 1.5 × 10

^{−4}are shown in Fig. 5(a) and Fig. 5(b), respectively. In Fig. 5(a), rainbow structures similar to isotropic ones can be observed in both E-plane and H-plane. However, the intensity of the rainbow in E-plane is weaker than that in H-plane. In Fig. 5(b) rainbow occurs in E-plane but almost disappears in H-plane. Obviously the intensity at peak angle in E-plane and H-plane depend on the chirality. Besides, all peak angles in Fig. 5 are 37.75°, identical to that of an isotropic one with the same parameters except the vanished chirality.

## 5. Conclusion

## Acknowledgment

## References and links

1. | V. D. Hulst, |

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

3. | R. T. Wang and H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. |

4. | 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 |

5. | J. A. Lock, J. M. Jamison, and C. Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. |

6. | Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. |

7. | G. Kaduchak, P. L. Marston, and H. J. Simpson, “E(6) diffraction catastrophe of the primary rainbow of oblate water drops: observations with white-light and laser illumination,” Appl. Opt. |

8. | J. P. A. J. van Beeck and M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. |

9. | J. P. A. J. van Beeck, |

10. | X. Han, K. F. Ren, Z. Wu, F. Corbin, G. Gouesbet, and G. Gréhan, “Characterization of Initial Disturbances in a Liquid Jet by Rainbow Sizing,” Appl. Opt. |

11. | J. P. A. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global Rainbow Thermometry for Mean Temperature and Size Measurement of Spray Droplets,” Particle & Particle Systems Characterization |

12. | M. R. Vetrano, J. P. van Beeck, and M. L. Riethmuller, “Global Rainbow Thermometry: Improvements in the Data Inversion Algorithm and Validation Technique in Liquid-Liquid Suspension,” Appl. Opt. |

13. | J. Wang, G. Gréhan, Y. Han, S. Saengkaew, and G. Gouesbet, “Numerical study of global rainbow technique: sensitivity to non-sphericity of droplets,” Exp. Fluids |

14. | D. Jaggard, A. Mickelson, and C. Papas, “On electromagnetic waves in chiral media,” Appl. Phys., A Mater. Sci. Process. |

15. | L. D. Barron, |

16. | A. Lakhtakia, V. K. Varadan, and V. V. Varadan, |

17. | D. L. Jaggard and X. Sun, “Theory of chiral multilayers,” J. Opt. Soc. Am. A |

18. | L. John, “Optical properties of isotropic chiral media,” Pure and Applied Optics: Journal of the European Optical Society Part A |

19. | S. Bassiri, C. Papas, and N. Engheta, “Electromagnetic wave propagation through a dielectric-chiral interface and through a chiral slab,” J. Opt. Soc. Am. A |

20. | M. Silverman, “Reflection and refraction at the surface of a chiral medium: comparison of gyrotropic constitutive relations invariant or noninvariant under a duality transformation,” J. Opt. Soc. Am. A |

21. | A. Lakhtakia, V. V. Varadan, and V. K. Varadan, “Field equations, Huygens's principle, integral equations, and theorems for radiation and scattering of electromagnetic waves in isotropic chiral media,” J. Opt. Soc. Am. A |

22. | L. Le-Wei, K. Pang-Shyan, L. Mook-Seng, and Y. Tat-Soon, “A general expression of dyadic Green's functions in radially multilayered chiral media,” IEEE Trans. Antenn. Propag. |

23. | F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. |

24. | Z.-S. Wu, Q.-C. Shang, and Z.-J. Li, “Calculation of electromagnetic scattering by a large chiral sphere,” Appl. Opt. |

25. | Q.-C. Shang, Z.-S. Wu, T. Qu, Z.-J. Li, L. Bai, and L. Gong, “Analysis of the radiation force and torque exerted on a chiral sphere by a Gaussian beam,” Opt. Express |

26. | D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell's equations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

27. | Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: Recursive algorithms,” Radio Sci. |

28. | A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys. |

29. | Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antenn. Propag. |

30. | X. e. Han, “Study of refractometry of rainbow and applications to the measurement of instability and temperature gradient of a liquid jet,” thesis (Rouen University, 2000). |

**OCIS Codes**

(290.4020) Scattering : Mie theory

(160.1585) Materials : Chiral media

(290.5825) Scattering : Scattering theory

**ToC Category:**

Scattering

**History**

Original Manuscript: July 9, 2013

Revised Manuscript: August 26, 2013

Manuscript Accepted: September 3, 2013

Published: September 10, 2013

**Citation**

Qing-Chao Shang, Zhen-Sen Wu, Tan Qu, Zheng-Jun Li, Lu Bai, and Lei Gong, "Analysis of rainbow scattering by a chiral sphere," Opt. Express **21**, 21879-21888 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-19-21879

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### References

- V. D. Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
- R. T. Wang and H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt.30(1), 106–117 (1991). [CrossRef] [PubMed]
- J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A10(4), 693–706 (1993). [CrossRef]
- J. A. Lock, J. M. Jamison, and C. Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt.33(21), 4677–4690 (1994). [CrossRef] [PubMed]
- Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt.36(21), 5188–5198 (1997). [CrossRef] [PubMed]
- G. Kaduchak, P. L. Marston, and H. J. Simpson, “E(6) diffraction catastrophe of the primary rainbow of oblate water drops: observations with white-light and laser illumination,” Appl. Opt.33(21), 4691–4696 (1994). [CrossRef] [PubMed]
- J. P. A. J. van Beeck and M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt.35(13), 2259–2266 (1996). [CrossRef] [PubMed]
- J. P. A. J. van Beeck, Rainbow Phenomena: Development of a Laser-Based, Non-Intrusive Technique for Measuring Droplet Size, Temperature and Velocity (Technische Universiteit Eindhoven, 1997).
- X. Han, K. F. Ren, Z. Wu, F. Corbin, G. Gouesbet, and G. Gréhan, “Characterization of Initial Disturbances in a Liquid Jet by Rainbow Sizing,” Appl. Opt.37(36), 8498–8503 (1998). [CrossRef] [PubMed]
- J. P. A. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global Rainbow Thermometry for Mean Temperature and Size Measurement of Spray Droplets,” Particle & Particle Systems Characterization18(4), 196–204 (2001). [CrossRef]
- M. R. Vetrano, J. P. van Beeck, and M. L. Riethmuller, “Global Rainbow Thermometry: Improvements in the Data Inversion Algorithm and Validation Technique in Liquid-Liquid Suspension,” Appl. Opt.43(18), 3600–3607 (2004). [CrossRef] [PubMed]
- J. Wang, G. Gréhan, Y. Han, S. Saengkaew, and G. Gouesbet, “Numerical study of global rainbow technique: sensitivity to non-sphericity of droplets,” Exp. Fluids51(1), 149–159 (2011). [CrossRef]
- D. Jaggard, A. Mickelson, and C. Papas, “On electromagnetic waves in chiral media,” Appl. Phys., A Mater. Sci. Process.18, 211–216 (1979).
- L. D. Barron, Molecular light scattering and optical activity (Cambridge Univ Pr, 2004).
- A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics (Springer, 1989), Vol. 335.
- D. L. Jaggard and X. Sun, “Theory of chiral multilayers,” J. Opt. Soc. Am. A9(5), 804–813 (1992). [CrossRef]
- L. John, “Optical properties of isotropic chiral media,” Pure and Applied Optics: Journal of the European Optical Society Part A5(4), 417–443 (1996). [CrossRef]
- S. Bassiri, C. Papas, and N. Engheta, “Electromagnetic wave propagation through a dielectric-chiral interface and through a chiral slab,” J. Opt. Soc. Am. A5(9), 1450–1459 (1988). [CrossRef]
- M. Silverman, “Reflection and refraction at the surface of a chiral medium: comparison of gyrotropic constitutive relations invariant or noninvariant under a duality transformation,” J. Opt. Soc. Am. A3(6), 830–837 (1986). [CrossRef]
- A. Lakhtakia, V. V. Varadan, and V. K. Varadan, “Field equations, Huygens's principle, integral equations, and theorems for radiation and scattering of electromagnetic waves in isotropic chiral media,” J. Opt. Soc. Am. A5(2), 175–184 (1988). [CrossRef]
- L. Le-Wei, K. Pang-Shyan, L. Mook-Seng, and Y. Tat-Soon, “A general expression of dyadic Green's functions in radially multilayered chiral media,” IEEE Trans. Antenn. Propag.43(3), 232–238 (1995). [CrossRef]
- F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett.29(3), 458–462 (1974). [CrossRef]
- Z.-S. Wu, Q.-C. Shang, and Z.-J. Li, “Calculation of electromagnetic scattering by a large chiral sphere,” Appl. Opt.51(27), 6661–6668 (2012). [CrossRef] [PubMed]
- Q.-C. Shang, Z.-S. Wu, T. Qu, Z.-J. Li, L. Bai, and L. Gong, “Analysis of the radiation force and torque exerted on a chiral sphere by a Gaussian beam,” Opt. Express21(7), 8677–8688 (2013). [CrossRef] [PubMed]
- D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell's equations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics56(1), 1102–1112 (1997). [CrossRef]
- Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: Recursive algorithms,” Radio Sci.26(6), 1393–1401 (1991). [CrossRef]
- A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys.22(10), 1242–1246 (1951). [CrossRef]
- Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antenn. Propag.57(2), 572–576 (2009). [CrossRef]
- X. e. Han, “Study of refractometry of rainbow and applications to the measurement of instability and temperature gradient of a liquid jet,” thesis (Rouen University, 2000).

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