## 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 [1–3]. In recent decades, study on rainbows has been expanded to cases of Gaussian laser beams incidence [4], multilayered particles [5, 6], and ellipsoids [7]. Based on the rainbow phenomenon, rainbow techniques [8–10] and global rainbow techniques [11–13] 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]. Researches on optical activity help a lot in exploring the structure of some biological molecules [15]. Later, scientists explained optical activity by using electromagnetic theory and determined chiral media by defining the constitutive relations [16]. Interactions of electromagnetic waves with chiral media, including reflection and transmission [17–20], radiation and scattering [21, 22] are well studied by many researchers. Bohren first solved light scattering by chiral spheres in Lorenz-Mie framework [23]. Based on his work, the authors researched on scattering from large chiral spheres and found very different rainbow phenomenon for chiral particles [24].

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 $\mathrm{exp}(-i\omega t)$ 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 $D={\epsilon}_{r}{\epsilon}_{0}E+i\kappa \sqrt{{\epsilon}_{0}{\mu}_{0}}H$ and $B=-i\kappa \sqrt{{\epsilon}_{0}{\mu}_{0}}E+{\mu}_{r}{\mu}_{0}H$, where ${\epsilon}_{r}$, ${\mu}_{r}$, and $\kappa $ are the relative permittivity, relative permeability, and chirality parameter of the medium, respectively. ${\epsilon}_{0}$ and ${\mu}_{0}$ represent the permittivity and permeability of free space, respectively. The problem of plane wave scattering by a chiral sphere was solved by Bohren [23]. Based on his work, we extended the theory to calculate scattering by a large chiral sphere [24]. Consider a chiral sphere of radius *a* with chirality parameter *κ* illuminated by a beam. As we discussed in [24, 25], the incident field, scattered field and internal field of a chiral sphere can be expanded in terms of spherical vector wave functions (SVWFS) [26], respectively, as follows:

*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. ${a}_{mn}^{ip}$ and ${b}_{mn}^{ip}$ represent expansion coefficients of the incident wave. ${A}_{mn}^{}$ and ${B}_{mn}^{}$ are expansion coefficients of internal field of the chiral sphere.

According to the boundary conditions at the spherical surface, scattering coefficients of scattered field ${A}_{mn}^{s}$ and ${B}_{mn}^{s}$ can be obtained as [24]:

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

By substituting the scattering coefficients ${A}_{mn}^{s}$ and ${B}_{mn}^{s}$ into Eq. (3), the scattered field in the far field can be readily obtained as:

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 ${E}_{}^{ix}={E}_{0}{e}^{ikz}\widehat{x}$, the expansion coefficients ${a}_{mn}^{ix}$ and ${b}_{mn}^{ix}$ are [29]:

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

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 ${E}_{}^{iR}={E}_{0}{e}^{ikz}(\widehat{x}+i\widehat{y})$ can be decomposed into two parts: ${E}_{}^{ix}={E}_{0}{e}^{ikz}\widehat{x}$ and ${E}_{}^{iy}={E}_{0}{e}^{ikz}\widehat{y}$. By taking a curl of Eq. (1), for the case of incidence ${E}_{}^{ix}={E}_{0}{e}^{ikz}\widehat{x}$, we have

*y*-polarized plane wave from Eq. (19): ${a}_{mn}^{iy}=-i{b}_{mn}^{ix}$ and ${b}_{mn}^{iy}=-i{a}_{mn}^{ix}$. Hence, the expansion coefficients of RCP wave with the form ${E}_{}^{iR}={E}_{0}{e}^{ikz}(\widehat{x}+i\widehat{y})$ can be readily obtained as

For the case of a LCP plane wave with the form${E}_{}^{iL}={E}_{0}{e}^{ikz}(\widehat{x}-i\widehat{y})$, the expansions coefficients can be obtained in a similar way to RCP case:

Hence, we can derive the simplified expressions for a LCP incident wave.## 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]. Figure 1(a) 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 500*λ*, step of the scattering angle *θ* is set to ${0.01}^{\circ}$ 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 ($\varphi ={90}^{\circ}$). However, three first-order rainbows and slight second-order rainbows can be found in both E-plane ($\varphi ={0}^{\circ}$) 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) 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] as only RCP and LCP waves propagate in it. The refractive index of a chiral medium is ${n}_{R}=n+\kappa $ for the RCP wave and ${n}_{L}=n-\kappa $ for the LCP wave, where $n=\sqrt{{\epsilon}_{r}{\mu}_{r}}$. Considering the relation between rainbow angle and refractive index of an isotropic sphere, it’s readily to associate the Left rainbow with *n _{L}*, and the Right rainbow with

*n*. We presented a rough physical interpretation in [24] 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.

_{R}#### 3.3. Spectrums of the rainbow structures

Rainbow phenomenon can be applied to measuring the size and temperature of a particle [8, 9]. 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, 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 $\left|fF(f)\right|$, 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]. 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] 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: $d/\lambda =58.5505{n}^{1.4944}{f}_{ripple}$, where *n* represents the refractive index and *f _{ripple}* 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

*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

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) 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 $\kappa =0.05$ 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 *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.

In order to investigate the variation of rainbow intensity with chirality, a group of rainbows for chiral spheres with chirality from 1.0 × 10^{−5} to 8.0 × 10^{−4} are calculated, at an incidence of *x*-polarized plane wave. As peak angle and shape of the rainbow structure almost do not change with the slight chirality, the rainbow can be represented basically by the scattering intensities at the peak angle. Figure 6 presents the scattering intensities at peak angle in both E-plane and H-plane as a function of chirality, respectively. Phenomenon shown in Fig. 5 can be interpreted exactly by the curves presented in Fig. 6. It can be seen that the intensity varies periodically, just like a sine function of chirality. And a maximum intensity in E-plane corresponds to a minimum intensity in H-plane. Rainbow occurs and disappears alternately in E-plane and H-plane as the chirality slightly increases. Besides, the maximum intensity in E-plane is larger than the maximum value in H-plane. Though it’s difficult to give an interpretation, it’s obvious that the phenomenon is related to the rotation of the polarization plane after the incident wave propagating through the sphere.

## 5. Conclusion

Rainbow scattering by a chiral sphere is investigated in this paper based on the previous work about scattering by a large chiral sphere. It is found that for a chiral sphere with proper chirality, three rainbow structures with their peak angles varying with chirality occur in both E-plane and H-plane. All rainbows disappear when the chirality increases to a certain value. As the chirality decreases to zero, the three rainbow structures move close to each other and approach to structure of the isotropic ones. A FFT analysis of the three rainbow structures shows that there are different ripple frequencies for each rainbow structure. However, all of them can be used to estimate the size of the chiral sphere. Only two rainbow structures remain for a circularly polarized plane wave incidence. And symmetry on rainbow structures is found between chiral spheres with opposite chirality parameters. Finally the rainbows generated by chiral spheres with slight chirality are analyzed as the common optical active solutions may be regarded as chiral medium with slight chirality. The rainbows occur and disappear alternately in E-plane and H-plane and their intensities vary with the chirality periodically like sine functions.

## 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.

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