## Abstract

We present the photonic spin Hall effect on an ellipsoidal Rayleigh particle, which amounts to a polarization-dependent shift in scattering far-field. Based on the dipole model, we demonstrate that such shift is unavoidable when the light incidence is inclined with respect to the main axis of the ellipsoidal Rayleigh particle. The result has general validity and can be applied to metal and dielectric materials. In addition, the photonic spin Hall effect also manifests itself in the optical force and torque exerted on the particle, which is promising for precision metrology, spin-optics devices and optical driven micro-machines. Due to wide existence of the Rayleigh particles in nature, we believe that our findings might provide a useful toolset for investigating polarization-dependent scattering of particles.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

There is a growing interest in phenomena related to photonic spin Hall effect which originates from optical spin-orbit interaction [1]. Optical spin-orbit interaction plays a key role in understanding many fundamental optical processes that can be specifically divided into two types: spin with internal orbital angular momentum (OAM) interaction and spin with external OAM interaction. The second type is named as photonic spin Hall effect due to the transverse spin-dependent redistribution of the light intensity [2–13]. Such effect results from the coupling between the spin and the trajectory of the optical field [14]. To date, various methods have been presented to realize photonic spin Hall effect via reflection or refraction at optical interfaces [3,6,7,15–20], inhomogeneous anisotropic media [2,21–27], tilted observation planes [14,28–32] and so on. The transverse spin-dependent shift in photonic spin Hall effect is promising for the manipulation of light [33]. Photonic spin Hall effect provides a powerful engine for precision metrology and spin-optics devices. Various relevant works have been reported in recent years [34–39]. Based on detecting the spin-dependent displacements, a new method has been proposed to measure the thickness of nanometal film and identify the graphene layers [34–36]. The wavelength-scale image error in optical localization has been presented to estimate the position of emitters by spin-orbit coupling of light [37]. The optical edge detecting technology based on spin-to-orbit interactions in image processing have been demonstrated by using air-glass interface reflection and high-efficiency dielectric metasurface [38,39].

In the last decade, the scattering of light by nanoparticles has been the subject of intense research activity [40–43]. Spherical silicon nanoparticles with strongly anisotropic scattering in visible spectral range have been experimentally demonstrated [40]. It has been shown that the far-field directivity of single silicon spheres is greatly dependent on the nanoparticle size and the incident wavelength. It has been reported that the spin-based resonance effect and the polarization-sensitive focusing can be observed by using dielectric nanoparticle clusters [41]. The physical origin of the phenomena is attributed to the geometric phase arising from the interaction between light and dielectric nanoparticle clusters. The relationship of the Rayleigh scattering properties of a single Au nanoparticle with its size, shape, and local dielectric environment has been reported in [42] and they also investigated the refractive index sensitivity of nanospheres, oval-shaped nanoparticles and nanorods. Recently, photonic spin Hall effect by a dipole scatterer [43] has been demonstrated, which indicates that the spin-split scattering is highly dependent on the position of the dipole scatterer relative to the beam center.

Rayleigh particles are particles with radii smaller than the tenth of light wavelength. For visible light, the size of the Rayleigh particles is about tens of nanometers. A deep understanding of the optical properties of the nanometer-sized Rayleigh particles has both fundamental and practical significance. The deformation of the particle shape could cause anisotropic polarizability, which might give rise to polarization-dependent scattering field. The interaction between light and the particle brings about the transfers of linear momentum and angular momentum, which will induce the polarization-dependent optical force and torque exerted on the particle. In this paper, we focus on the photonic spin Hall effect on an ellipsoidal Rayleigh particle. To the best of our knowledge, such effect in scattering far-field of a Rayleigh particle has not yet been reported. Based on the dipole model, we will give a general description of the relationship between the scattering far-field and the spin state of the incident light in theory and simulation. It is shown that a polarization-dependent light splitting occurs when a beam of light obliquely incidents on an ellipsoidal Rayleigh particle. The influence of the ellipsoid radius and the incident angle of light, on the direction of the centroid (or barycenter) [28] of scattering far-field are presented. Considering the interaction between light and the ellipsoidal Rayleigh particle, we also show the optical force and torque phenomena related to the photonic spin hall effect which is promising for related technologies of optical driven micro-machines.

## 2. Theory

The scheme of the problem we considered is shown in Fig. 1(a), in which circularly polarized light obliquely incidents on an ellipsoidal Rayleigh particle. We assume that the axis of the incident beam in the $y$-$z$ plane is tilted by an angle $\beta$ with respect to the $z$ axis (see Fig. 1(a)). We investigate the centroid (or barycenter) of the scattering far-field for an ellipsoidal Rayleigh particle. First, the obliquely input circularly polarized light field can be written as

## 3. Results and discussions

To confirm the theoretical analysis, we use the commercial software COMSOL Multiphysics to do some simulations. Figure 1(b) shows the far-field scattering diagram of a metallic (Au) ellipsoidal Rayleigh particle. The radii of the particle in the $x$, $y$ and $z$ directions are 10 nm, 3 nm and 30 nm, respectively. The wavelength of the incident light is 650 nm and the incident angle is $45^\circ$. The relative permittivity of gold at optical frequencies is modeled using Drude formula $\epsilon = 1-\omega _p^2/(\omega ^2 + i \omega _{\tau } \omega )$ with $\omega _p = 1.37 \times 10^{16} rad/s$ and $\omega _{\tau } = 1.215 \times 10^{14} rad/s$ . The angular distributions of the far-field scattering in ${Oxy}$ plane are symmetric about the plane of incidence for the left and right circularly polarized lights. As shown in Fig. 1(b), a clear polarization-dependent split can be seen in the scattering field.

As implied in Eq. (12), the transverse shift of the scattering far-field originates from the anisotropic polarizability in $y$ and $z$ directions. Here, we use $a_x$, $a_y$ and $a_z$ to denote the radii of the particle in $x$, $y$ and $z$ directions and try to change the ratio $a_y/a_z$ and $a_x/a_z$ to check its influence on the transverse shift. First, we set $a_x$ = 10 nm, $a_z$ = 30 nm and change $a_y$ from 5 nm (=1/6$a_z$) to 30 nm (=$a_z$), the cosine values of azimuthal angles and elevation angles for the centroids of scattering far-field are calculated and shown in Figs. 2(a) and 2(b). We also set $a_y$ = 10 nm, $a_z$ = 30 nm and change $a_x$ from 5 nm (=1/6$a_z$) to 30 nm (=$a_z$), the cosine values of azimuthal angles and elevation angles for the centroids of scattering far-field are calculated and shown in Figs. 2(c) and 2(d). The incident angle $\beta$ is fixed at $45^\circ$. It is clearly seen that the azimuthal angles for the incident left and right circularly polarized lights are opposite to each other but they are zero for the linearly polarized light in Figs. 2(a) and 2(c). Figures 2(b) and 2(d) show the elevation angles for the incident left and right circularly polarized lights are same, which is due to the elevation angle independent of the helicity of the incident light according to the Eq. (13). Therefore, the centroids of the scattering far-field for the incident left and right circularly polarized lights are always symmetric about the plane of incidence. The above results reveal the fact that a polarization-dependent split occurs when a beam of light obliquely incidents on an ellipsoidal Rayleigh particle, which are consistent with our theory. It is worth remarking that when the radii of the particle in the $y$ and $z$ directions are equal (both are 30 nm), the cosine values of the azimuthal angles for the linearly, left and right circularly polarized lights are all equal to zero, which indicates the polarization-dependent split disappears. The relevant works show the spin Hall effect of light in Mie spherical particles [49–51], which all focus on the transverse shift in perceived location of the source. It implies the phenomenon valid for the ellipsoidal particles and the broken symmetry of the ellipsoidal particles might be more conducive to the spin Hall effect. In this paper, we investigate the centroid (or barycenter) of the scattering far-field for an ellipsoidal Rayleigh particle to manifest the spin Hall effect. In addition, the polarization-dependent shift disappears for an isotropic and homogeneous spherical particle.

The dependences of the cosine values of azimuthal angles for the centroids of scattering far-field on the incident angle are shown in Fig. 3, where $a_x$ = 10 nm, $a_y$ = 20 nm and $a_z$ = 30 nm. It is easy to observe that the cosine values of the azimuthal angles for the left and right circularly polarized lights are odd functions with the incident angle. When the incident angle is $0^\circ$ and $\pm 90^\circ$, the cosine values of the azimuthal angles for the linearly, left and right circularly polarized lights are all equal to zero, which indicates the polarization-dependent split disappears. The relationship between the cosine value of azimuthal angle and the incident angle can be described by Eq. (12). This equation can be written in a more simplified form as

## 4. Optical force

The interaction between light and the ellipsoidal Rayleigh particle brings about the transfer of linear momentum, which will induce optical force exerted on the illuminated object. The photonic spin Hall effect in the scattering field suggests that the directions of the forces on the particle may also shift away from the regular directions. The time-averaged optical force $\langle {\textbf {F}}\rangle$ on the ellipsoidal Rayleigh particle can be calculated by integrating the Maxwell stress tensor over a surface $S$ enclosing the ellipsoidal Rayleigh particle, which is expressed as

## 5. Optical torque

The time-averaged optical torque $\langle {\mathbf {\Gamma }}\rangle$ on the ellipsoidal Rayleigh particle can be expressed as

## 6. Conclusion

In conclusion, we have explored the photonic spin Hall effect on an ellipsoidal Rayleigh particle in scattering far-field. The results show that a spin split scattering far-field behavior of a beam of light obliquely incident on an ellipsoidal Rayleigh particle. It should be emphasized that the radii of the ellipsoidal Rayleigh particle are different in the plane of incidence. Moreover, we present an analysis about the optical force and torque on a metallic (Au) ellipsoidal Rayleigh particle to get insight into the more application potential of the photonic spin Hall effect. These results will assist in the investigation of optical manipulation which can be used to design precision metrology, spin-optics devices and optical driven micro-machines.

## Funding

National Natural Science Foundation of China (11874132, 61307072, 61308017, 61377016, 61405056, 61575055); National Basic Research Program of China (973 Program) (2013CBA01702).

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