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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 19 — Sep. 23, 2013
  • pp: 21879–21888
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Analysis of rainbow scattering by a chiral sphere

Qing-Chao Shang, Zhen-Sen Wu, Tan Qu, Zheng-Jun Li, Lu Bai, and Lei Gong  »View Author Affiliations


Optics Express, Vol. 21, Issue 19, pp. 21879-21888 (2013)
http://dx.doi.org/10.1364/OE.21.021879


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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

There has been a long history of research on rainbow phenomenon. The earliest theoretical study can date back to several centuries ago when Descartes and other researchers explained the rainbow by using the classical geometrical optics. Later, Yong and Airy, respectively, proposed their theories after taking into account the effect of optical interference. With the help of electromagnetic theory, rainbow phenomenon can be described precisely by using the Lorenz-Mie theory [1

1. V. D. Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

3

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]

]. In recent decades, study on rainbows has been expanded to cases of Gaussian laser beams incidence [4

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]

], multilayered particles [5

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

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]

], and ellipsoids [7

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]

]. Based on the rainbow phenomenon, rainbow techniques [8

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

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]

] and global rainbow techniques [11

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

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]

] have been developed and applied in measurements of particle sizes and temperature.

Natural chiral media are called “optical active media”, due to that linearly polarized light changes its polarization plane after traveling through them. Representatives of natural chiral media are solutions of substances with handed microstructure, such as grape sugar and tartaric acid [14

14. D. Jaggard, A. Mickelson, and C. Papas, “On electromagnetic waves in chiral media,” Appl. Phys., A Mater. Sci. Process. 18, 211–216 (1979).

]. Researches on optical activity help a lot in exploring the structure of some biological molecules [15

15. L. D. Barron, Molecular light scattering and optical activity (Cambridge Univ Pr, 2004).

]. Later, scientists explained optical activity by using electromagnetic theory and determined chiral media by defining the constitutive relations [16

16. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics (Springer, 1989), Vol. 335.

]. Interactions of electromagnetic waves with chiral media, including reflection and transmission [17

17. D. L. Jaggard and X. Sun, “Theory of chiral multilayers,” J. Opt. Soc. Am. A 9(5), 804–813 (1992). [CrossRef]

20

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 3(6), 830–837 (1986). [CrossRef]

], radiation and scattering [21

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 5(2), 175–184 (1988). [CrossRef]

, 22

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. 43(3), 232–238 (1995). [CrossRef]

] are well studied by many researchers. Bohren first solved light scattering by chiral spheres in Lorenz-Mie framework [23

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

]. Based on his work, the authors researched on scattering from large chiral spheres and found very different rainbow phenomenon for chiral particles [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]

].

This paper is devoted to studying the rainbow phenomenon of a chiral sphere by analyzing the numerical results generated by Lorenz-Mie theory. We mainly focus on the new character of rainbows and the effects of chirality on them, as the other parameters, such as size and loss of the sphere, affect rainbows of chiral spheres and isotropic spheres in similar ways. The following is the arrangement of this paper. In section 2, we review the work of electromagnetic scattering from a large chiral sphere. Circularly polarized plane wave incidences are considered here as the rainbow phenomenon is different for these cases. In section 3, the scattering intensity distributions of rainbows for chiral spheres are presented. We examine the effects of chirality parameters on rainbow structures and make an attempt to analyze their spectrum. Then rainbows for circularly polarized wave incidence are analyzed. In section 4, we analyze rainbows of chiral spheres with slight chirality as common chiral media at optical frequencies in nature, i.e., the optical active solutions, seem to have very slight chirality parameters. Section 5 is a summary of our work. In the following analysis, a time dependence of exp(iωt) is assumed.

2. Scattering by a large chiral sphere

2.1. Scattering coefficients

In electromagnetics, chiral medium is characterized by their special constitutive relations. The constitutive relations of chiral medium in this paper are adopt as D=εrε0E+iκε0μ0H and B=iκε0μ0E+μrμ0H, where εr, μr, and κ are the relative permittivity, relative permeability, and chirality parameter of the medium, respectively. ε0 and μ0 represent the permittivity and permeability of free space, respectively. The problem of plane wave scattering by a chiral sphere was solved by Bohren [23

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

]. Based on his work, we extended the theory to calculate scattering by a large chiral sphere [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]

]. Consider a chiral sphere of radius 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

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]

], the incident field, scattered field and internal field of a chiral sphere can be expanded in terms of spherical vector wave functions (SVWFS) [26

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]

], respectively, as follows:
Eip=E0n=1m=nn[amnipMmn(1)(r,k)+bmnipNmn(1)(r,k)],
(1)
Hip=kE0iωμn=1m=nn[amnipNmn(1)(r,k)+bmnipMmn(1)(r,k)],
(2)
Es=E0n=1m=nn[AmnsMmn(3)(r,k)+BmnsNmn(3)(r,k)],
(3)
Hs=kE0iωμn=1m=nn[AmnsNmn(3)(r,k)+BmnsMmn(3)(r,k)],
(4)
Eint=n=1m=nn[AmnMmn(1)(r,k1)+AmnNmn(1)(r,k1)+BmnMmn(1)(r,k2)BmnNmn(1)(r,k2)],
(5)
Hint=iεrε0μrμ0n=1m=nn[AmnNmn(1)(r,k1)+AmnMmn(1)(r,k1)+BmnNmn(1)(r,k2)BmnMmn(1)(r,k2)],
(6)
where E0 represents the amplitude of electric field; ω is the angular frequency of the incident wave; ε, μ and k=ωεμ denote the permittivity, permeability and wave number in the surrounding medium, respectively. k1=ω(μrεr+κ)ε0μ0 and k2=ω(μrεrκ)ε0μ0 represent, respectively, wave number of the right-handed circularly polarized (RCP) wave and the left-handed circularly polarized (LCP) wave in chiral medium. The superscript ip in the equations above indicates the 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. amnip and bmnip represent expansion coefficients of the incident wave. Amn and Bmn are expansion coefficients of internal field of the chiral sphere.

According to the boundary conditions at the spherical surface, scattering coefficients of scattered field Amns and Bmns can be obtained as [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]

]:
Amns=Ansaamnip+Ansbbmnip,Bmns=Bnsaamnip+Bnsbbmnip,
(7)
where,
Ansa=ψn(x0)ξn(x0)Dn(1)(x1)ηrDn(1)(x0)ηrDn(1)(x1)Dn(3)(x0)+Dn(1)(x2)ηrDn(1)(x0)ηrDn(1)(x2)Dn(3)(x0)ηrDn(3)(x0)Dn(1)(x1)ηrDn(1)(x1)Dn(3)(x0)+ηrDn(3)(x0)Dn(1)(x2)ηrDn(1)(x2)Dn(3)(x0),
(8)
Ansb=ψn(x0)ξn(x0)ηrDn(1)(x1)Dn(1)(x0)ηrDn(1)(x1)Dn(3)(x0)ηrDn(1)(x2)Dn(1)(x0)ηrDn(1)(x2)Dn(3)(x0)ηrDn(3)(x0)Dn(1)(x1)ηrDn(1)(x1)Dn(3)(x0)+ηrDn(3)(x0)Dn(1)(x2)ηrDn(1)(x2)Dn(3)(x0),
(9)
Bnsa=Ansb,
(10)
Bnsb=ψn(x0)ξn(x0)ηrDn(1)(x1)Dn(1)(x0)Dn(1)(x1)ηrDn(3)(x0)+ηrDn(1)(x2)Dn(1)(x0)Dn(1)(x2)ηrDn(3)(x0)Dn(3)(x0)ηrDn(1)(x1)Dn(1)(x1)ηrDn(3)(x0)+Dn(3)(x0)ηrDn(1)(x2)Dn(1)(x2)ηrDn(3)(x0).
(11)
In the formulas above, x0=ka, x1=k1a, x2=k2a, and ηr=ε/μ/εrεr/(μrμ0); Dn(i)(z) and ψn(z)/ξn(z) are functions associated with Bessel functions and both can be calculated efficiently by using the recurrence relations [27

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

, 28

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

By substituting the scattering coefficients Amns and Bmns into Eq. (3), the scattered field in the far field can be readily obtained as:
Eθs=E0exp(ikr)krn=1m=nn(i)n[Amnsmπmn+Bmnsτmn]eimϕ,
(12)
Eϕs=iE0exp(ikr)krn=1m=nn(i)n[Amnsτmn+Bmnsmπmn]eimϕ,
(13)
where πmn and τmn are defined as:

πmn=Pnm(cosθ)sinθ,τmn=dPnm(cosθ)dθ.
(14)

For the case of plane wave incidence, Eqs. (12) and (13) can be simplified a lot as the expansion coefficients of plane waves are much easier than those of beams. The following presents the expansions of differently polarized plane waves and the simplification of scattered field. For an x-polarized plane wave propagating along the z-axis with the form Eix=E0eikzx^, the expansion coefficients amnix and bmnix are [29

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]

]:
amnix=in+12n+12n(n+1)δm,1+in+12n+12δm,1,bmnix=in+12n+12n(n+1)δm,1in+12n+12δm,1,
(15)
with δs,l being the Kronecker delta. Thus, for an x-polarized plane wave incidence, scattered field can be simplified after substituting Eq. (15) into Eqs. (12) and (13) (See appendix 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]

] for details), as follows:
Eθs=iE0exp(ikr)krn=12n+1n(n+1)[cosϕ(Ansaπn+Bnsbτn)+isinϕ(Ansbπn+Bnsaτn)],
(16)
Eϕs=E0exp(ikr)krn=12n+1n(n+1)[cosϕ(Bnsaπn+Ansbτn)+isinϕ(Bnsbπn+Ansaτn)],
(17)
where πn and τn are the angle-dependent functions with the following relations with Legendre functions:

πn=π1n=Pn1(cosθ)sinθ,τn=τ1n=dPn1(cosθ)dθ.
(18)

Now consider the case of circularly polarized plane wave incidence. A circularly polarized plane wave can be regarded as a superposition of two linearly polarized waves with vibration direction perpendicular to each other. A right-handed circularly polarized plane wave with the form EiR=E0eikz(x^+iy^) can be decomposed into two parts: Eix=E0eikzx^ and Eiy=E0eikzy^. By taking a curl of Eq. (1), for the case of incidence Eix=E0eikzx^, we have
×n=1m=nn[amnixMmn(1)(r,k)+bmnixNmn(1)(r,k)]=×(eikzx^).
(19)
As ×(eikzx^)=ikeikzy^, N=(×M)/k and M=(×N)/k, we get expansion coefficients of a y-polarized plane wave from Eq. (19): amniy=ibmnix and bmniy=iamnix. Hence, the expansion coefficients of RCP wave with the form EiR=E0eikz(x^+iy^) can be readily obtained as
amniR=amnix+iamniy=amnix+bmnix,bmniR=bmnix+ibmniy=bmnix+amnix.
(20)
By substituting Eq. (15) into Eq. (20), the expressions of expansion coefficients amniR and bmniR are obtained as:
amniR=bmniR=in+12n+1n(n+1)δm,1
(21)
Hence, the scattered fields for a RCP wave incidence can be simplified as

Eθs=E0exp(ikr)krn=1[ieiϕ2n+1n(n+1)(Ansaπn+Ansbπn+Bnsaτn+Bnsbτn)],
(22)
Eϕs=E0exp(ikr)krn=1[eiϕ2n+1n(n+1)(Ansaτn+Ansbτn+Bnsaπn+Bnsbπn)].
(23)

For the case of a LCP plane wave with the formEiL=E0eikz(x^iy^), the expansions coefficients can be obtained in a similar way to RCP case:
amniL=bmniL=in+1(2n+1)δm,1.
(24)
Hence, we can derive the simplified expressions for a LCP incident wave.
Eθs=E0exp(ikr)krn=1[i2n+1n(n+1)eiϕ(AnsaπnAnsbπnBnsaτn+Bnsbτn)],
(25)
Eϕs=E0exp(ikr)krn=1[2n+1n(n+1)eiϕ(AnsaτnAnsbτnBnsaπn+Bnsbπn)].
(26)
Finally, the scattering intensity can be calculated by using the following expressions:

Is=limrk2r2(|Eθs|2+|Eϕs|2)/|E0|2.
(27)

3. Rainbow phenomenon of chiral spheres

3.1. Rainbow structures of chiral spheres

We calculated the scattering intensity distributions of a large chiral sphere and gave a brief introduction to its rainbow phenomenon 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]

]. Figure 1(a)
Fig. 1 Rainbow phenomenon of a chiral sphere. (a) κ = 0.10; (b) κ = 0.01.
and Fig. 1(b) show rainbow scattering intensities by chiral spheres with chirality parameter 0.10 and 0.01 for an x-polarized incident plane wave with wavelength λ, respectively. As the radius is as large as 500λ, step of the scattering angle θ is set to 0.01 to avoid the distortion. As we know, rainbow phenomenon of isotropic spheres illuminated by an x-polarized plane wave can be observed in only H-plane (ϕ=90). However, three first-order rainbows and slight second-order rainbows can be found in both E-plane (ϕ=0) and H-plane in Fig. 1(a) for a chiral sphere with chirality 0.10. In the following depiction, we name the three rainbow structures, respectively, the Left, the Middle, and the Right rainbow, according to their relative positions. It can be seen that the Left rainbow and the Middle rainbow have a good Airy structure. However, the Airy structure of the Right rainbow is not so obvious, which might be due to that the Right rainbow is affected by backward scattering or by the two rainbows before it. For a chiral sphere with chirality 0.01, as shown in Fig. 1(b), rainbow phenomenon can be also observed in both E-plane and H-plane. However, it’s difficult to identify Airy structure from the curves. Neither the three Airy structures as shown in Fig. 1(a), nor a single Airy structure as the isotropic one’s can be observed. We infer that the three rainbows are too close to each other and superposed seriously by each other. The following numerical results will account for the assumption and make it more understandable.

3.2. Variation of rainbow structure with chirality parameters

A group of results for a chiral sphere with different chirality parameters is calculated to examine the effects of chirality on rainbow structures. Figure 2(a)
Fig. 2 Effects of chirality on (a) Peak angle; (b) Peak intensity.
and Fig. 2(b) show, respectively, the variation of peak angles (the scattering angle corresponding to the peak) and peak intensities (the scattering intensity of the peak) of the three rainbows with the chirality parameters. The peak angle can generally represent rainbow angle as it is just a little larger than rainbow angle. It can be seen from Fig. 2(a) that as the chirality approaches zero, the three peak angles come close to each other and approach the peak angle of the isotropic one. While as the chirality increases, peak angle of the Left rainbow decreases rapidly; peak angle of the Middle rainbow slightly decreases; and peak angle of the Right rainbow increases, moving towards backward direction. Thus, it’s quite understandable that for a small chirality, such as 0.01 in Fig. 1(b), the three rainbows are too close that their structures are affected seriously by each other. In Fig. 2(b), the intensities of the Left rainbow and the Middle rainbow decrease roughly to a very small value; and intensity of the Right rainbow does not seem to reduce too much. However, for a larger chirality, the Right rainbow is closer to backward direction and affected more by the strong backward scattering. Finally for a chiral sphere with chirality large enough, the Left rainbow and Middle rainbow disappear; the Right rainbow is buried in backward scattering; and no rainbow phenomenon can be observed. It seems quite reasonable to associate the three rainbows of a chiral sphere with its refractive indices. There are two refractive indices for a chiral medium [2

2. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).

] as only RCP and LCP waves propagate in it. The refractive index of a chiral medium is nR=n+κ for the RCP wave and nL=nκ for the LCP wave, where n=εrμr. Considering the relation between rainbow angle and refractive index of an isotropic sphere, it’s readily to associate the Left rainbow with nL, and the Right rainbow with nR. We presented a rough physical interpretation 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]

] that after once internal reflection in the chiral sphere there may be three emitted rays for a linearly polarized ray incidence. However, more work is necessary if we want to explain the curves in Fig. 2(a) from viewpoint of geometric optics.

3.3. Spectrums of the rainbow structures

Rainbow phenomenon can be applied to measuring the size and temperature of a particle [8

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]

, 9

9. 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).

]. One method to achieve it is to analyze angular spectrum of the rainbow structures. The spectrum of the three rainbow structures in E-plane of Fig. 1(a) is calculated and shown in Fig. 3
Fig. 3 Rainbow structures and spectrums.
, respectively. To suppress effects of the mean value and low-frequency component, we conduct fast Fourier transform (FFT) on derivation of the rainbow structure. According to the properties of Fourier transform, the processed result is proportional to |fF(f)|, where F(f) is the original spectrum. It can be seen that the spectrum structure of the Left rainbow and Middle rainbow are similar to that of an isotropic one, which can be generally divided into three sections [30

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).

]. As shown in Fig. 3, section A corresponds to spectrum of the Airy structure; section B corresponds to spectrum of the ripple structure; and section C results from the interference between surface waves. It is found that ripple frequency of the Left rainbow and Middle rainbow is 12.5977 and 11.3281, respectively. As the Right rainbow structure is different from the ordinary rainbows, its spectrum is strange compared with the other two. However, the ripple frequency of Right rainbow can still be identified at 9.9609 easily.

To make a simple analysis on the three ripple frequencies, we assume that the Left rainbow, Middle rainbow and Right rainbow correspond to refractive index 1.23, 1.33 and 1.43, respectively. Then we can use the empirical relation in [30

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).

] proposed by Han to estimate the size of the chiral sphere. The diameter d can be calculated by the following formulas after considering the effect of wavelength: d/λ=58.5505n1.4944fripple, where n represents the refractive index and fripple 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, 1005.0λ, 1015.7λ, and 995.3λ. It can be seen that results estimated according to the Left rainbow and Right rainbow are very close to1000λ, 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

Scattering characteristics of a chiral sphere illuminated by a RCP wave and LCP wave are different. Therefore, different circularly polarized waves generate different rainbow phenomenon for a chiral sphere. Figure 4(a)
Fig. 4 Rainbow structures for circularly polarized wave incidences. (a) κ = 0.05; (b) κ = −0.05.
and Fig. 4(b) show, respectively, rainbows of a chiral sphere with chirality 0.05 and −0.05 illuminated by circularly polarized plane waves. As scattering intensity distributions in E-plane and H-plane are the same for a circularly polarized plane wave incidence, rainbows in E-plane are presented in Fig. 4. Compared with an x-polarized wave incidence, only two rainbow structures can be found. As shown in Fig. 4(a), for a chiral sphere with chirality κ=0.05 illuminated by a RCP plane wave, the Left rainbow disappears and only the Middle and Right rainbows exist. Conversely, for a LCP incidence case, the Left and Middle rainbows remain, and the Right rainbow disappears. In Fig. 4(b), for a chiral sphere with negative chirality parameter −0.05, the results are converse to those with chirality 0.05 in Fig. 4(a). Symmetry can be found between rainbows with opposite chirality. It is quite reasonable if we note that the scattering intensity distributions of 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

Section 3 and 4 focus on the rainbow phenomenon of chiral spheres with sufficient chirality, which is more accordant with the chirality of the man-made chiral material in the microwave region. However, most of chiral media at optical frequencies, such as the optically active media in nature, do not possess so large a chirality parameter. According to the electromagnetic characteristics of chiral media, it can be readily to derive the relationship between the optical activity and chirality parameters. For a linearly polarized plane wave propagating through a chiral slab of thickness d with chirality κ, the polarization plane of the incident wave is rotated an angle of κk0d, where k0 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)
Fig. 5 Rainbows for sphere with slight chirality. (a) κ = 5 × 10−5; (b) κ = −1.5 × 10−4.
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

The authors gratefully acknowledge supports from the National Natural Science Foundation of China under Grant No. 61172031, No. 61308025, No. 61308071 and the Fundamental Research Funds for the Central Universities.

References and links

1.

V. D. Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

2.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).

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]

9.

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).

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]

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. 43(18), 3600–3607 (2004). [CrossRef] [PubMed]

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]

14.

D. Jaggard, A. Mickelson, and C. Papas, “On electromagnetic waves in chiral media,” Appl. Phys., A Mater. Sci. Process. 18, 211–216 (1979).

15.

L. D. Barron, Molecular light scattering and optical activity (Cambridge Univ Pr, 2004).

16.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics (Springer, 1989), Vol. 335.

17.

D. L. Jaggard and X. Sun, “Theory of chiral multilayers,” J. Opt. Soc. Am. A 9(5), 804–813 (1992). [CrossRef]

18.

L. John, “Optical properties of isotropic chiral media,” Pure and Applied Optics: Journal of the European Optical Society Part A 5(4), 417–443 (1996). [CrossRef]

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 5(9), 1450–1459 (1988). [CrossRef]

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 3(6), 830–837 (1986). [CrossRef]

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 5(2), 175–184 (1988). [CrossRef]

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. 43(3), 232–238 (1995). [CrossRef]

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]

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]

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]

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]

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

  1. V. D. Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  2. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
  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. A10(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]
  9. 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).
  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 Characterization18(4), 196–204 (2001). [CrossRef]
  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.43(18), 3600–3607 (2004). [CrossRef] [PubMed]
  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. Fluids51(1), 149–159 (2011). [CrossRef]
  14. D. Jaggard, A. Mickelson, and C. Papas, “On electromagnetic waves in chiral media,” Appl. Phys., A Mater. Sci. Process.18, 211–216 (1979).
  15. L. D. Barron, Molecular light scattering and optical activity (Cambridge Univ Pr, 2004).
  16. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics (Springer, 1989), Vol. 335.
  17. D. L. Jaggard and X. Sun, “Theory of chiral multilayers,” J. Opt. Soc. Am. A9(5), 804–813 (1992). [CrossRef]
  18. 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]
  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. A5(9), 1450–1459 (1988). [CrossRef]
  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. A3(6), 830–837 (1986). [CrossRef]
  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. A5(2), 175–184 (1988). [CrossRef]
  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.43(3), 232–238 (1995). [CrossRef]
  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]
  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. Express21(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. Topics56(1), 1102–1112 (1997). [CrossRef]
  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]
  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]
  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).

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