Photonic spin Hall effect on an ellipsoidal Rayleigh particle in scattering far-field

Wenjia Li; Jianlong Liu; Jianlong Liu; Yang Gao; Keya Zhou; Shutian Liu; Shutian Liu

doi:10.1364/OE.27.028194

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

(1)$$\textbf{E}_{\textrm{inc}} = \textrm{E}_{0}(-\sigma i \hat{\textbf{x}} +\textrm{cos}\beta \hat{\textbf{y}} + \textrm{sin}\beta \hat{\textbf{z}})\textrm{exp}[i k_0(-\textrm{sin}\beta y + \textrm{cos}\beta z)],$$

(2)$$\textbf{H}_\textrm{inc} ={-}\textrm{H}_{0}(\hat{\textbf{x}} +\sigma i \textrm{cos}\beta \hat{\textbf{y}} + \sigma i \textrm{sin}\beta \hat{\textbf{z}})\textrm{exp}[i k_0(-\textrm{sin}\beta y + \textrm{cos}\beta z)],$$

where $\sigma$ denotes the helicity of the beam, $k_0$ is the wave number in vacuum, ${\textrm{E}_0}$ and ${\textrm{H}_0}$ are the amplitudes of incident electric and magnetic fields. In the following, the helicity is chosen as $\sigma$ $\equiv$ 0, +1, and -1 indicating the linear, right-circular, and left-circular polarizations, respectively. In the following analysis, we adopt the dipole model in which the ellipsoidal Rayleigh particle has electric and magnetic dipole responses. Following the dipole model, the scattering far-field can be described by the equations [44]

(3)$$\textbf{E}_\textrm{sc} = {\frac{k_0^2} {4 \pi \epsilon_0}} {\frac{e^{i k_0 r}} {r}}{[(\hat{\textbf{n}}\times{\textbf{p}})\times{\hat{\textbf{n}}}-{\frac{1} {c}}\hat{\textbf{n}}\times{\textbf{m}}]},$$

(4)$$\textbf{H}_\textrm{sc} = {\frac{1} {\eta_0}} {(\hat{\textbf{n}}\times{\textbf{E}_\textrm{sc}})},$$

where $\hat {\textbf {n}}$ is the unit direction vector, $r$ is the distance away from the scatterer, $\eta _0$ is the impedance of free space, $\textbf {p}$ and $\textbf {m}$ are the electric and magnetic dipole moments. For an ellipsoidal Rayleigh particle, we consider $\overline {\alpha }_\textrm{em}$ = $\overline {\alpha }_\textrm{me}$ = 0 for this achiral geometry so the electric and magnetic dipole moments can be expressed as

(5)$$\textbf{p} = \overline{\alpha}_\textrm{ee}\cdot{\textbf{E}_\textrm{inc}},$$

(6)$$\textbf{m} = \overline{\alpha}_\textrm{mm}\cdot{\textbf{H}_\textrm{inc}},$$

where $\overline {\alpha }_\textrm{ee}$ and $\overline {\alpha }_\textrm{mm}$ denote 3 $\times$ 3 polarizability tensors which can be written as

(7)$$\overline{\alpha}_\textrm{ee} = \left(\begin{array}{ccc} \alpha_\textrm{ee1}^{xx}+i \alpha_\textrm{ee2}^{xx} & 0 & 0 \\ 0 & \alpha_\textrm{ee1}^{yy}+i \alpha_\textrm{ee2}^{yy} & 0 \\ 0 & 0 & \alpha_\textrm{ee1}^{zz}+i \alpha_\textrm{ee2}^{zz}\end{array} \right),$$

(8)$$\overline{\alpha}_\textrm{mm} = \left(\begin{array}{ccc} \alpha_\textrm{mm1}^{xx}+i \alpha_\textrm{mm2}^{xx} & 0 & 0 \\ 0 & \alpha_\textrm{mm1}^{yy}+i \alpha_\textrm{mm2}^{yy} & 0 \\ 0 & 0 & \alpha_\textrm{mm1}^{zz}+i \alpha_\textrm{mm2}^{zz}\end{array} \right).$$

It is worth noticing that in the case of lossless or loss ellipsoidal Rayleigh particle, the quantities contained in $\overline {\alpha }_\textrm{ee}$ and $\overline {\alpha }_\textrm{mm}$ are complex due to the radiation loss correction to polarizabilities [45–47]. The time-averaged intensity of the scattering field for a monochromatic electromagnetic beam in vacuum is defined as

(9)$$\langle\textbf{S}\rangle = {\frac{1} {2}}{\textrm{Re}} ({\textbf{E}_\textrm{sc}}\times{\textbf{H}_\textrm{sc}^*}).$$

The centroid of scattering far-field can be described by azimuthal angle ($\varphi$) and elevation angle ($\theta$) in a spherical coordinate system. In the spherical coordinate system, the cosine functions of the azimuthal angle and elevation angle can be calculated as [48]

(10)$$\langle{\textrm{cos}\varphi}\rangle = \frac{\iint{\textrm{cos}\varphi} S_r r^2 {\textrm{sin}\theta} d\theta d\varphi} {\iint S_r r^2 {\textrm{sin}\theta} d\theta d\varphi},$$

(11)$$\langle{\textrm{cos}\theta}\rangle = \frac{\iint{\textrm{cos}\theta} S_r r^2 {\textrm{sin}\theta} d\theta d\varphi} {\iint S_r r^2 {\textrm{sin}\theta} d\theta d\varphi},$$

where $S_r$ is the radial component of the time-averaged intensity of the beam. By applying Eqs. (1)–(11), we can obtain

(12)$$\langle{\textrm{cos}\varphi}\rangle \propto \sigma \textrm{sin}{2\beta} \frac{\eta_0 \textrm{W}_3} {\eta_0^2 \textrm{W}_1 + (1/{c^2}) \textrm{W}_2},$$

(13)$$\langle{\textrm{cos}\theta}\rangle \propto \textrm{cos}{\beta} \frac{\eta_0 \textrm{W}_4} {\eta_0^2 \textrm{W}_1 + (1/{c^2}) \textrm{W}_2},$$

with

(14a)$$\textrm{W}_1 = ({\alpha_\textrm{ee1}^{xx}{^2}} + {\alpha_\textrm{ee2}^{xx}{^2}}) + ({\alpha_\textrm{ee1}^{yy}{^2}} + {\alpha_\textrm{ee2}^{yy}{^2}}) \textrm{cos}^2\beta + ({\alpha_\textrm{ee1}^{zz}{^2}} + {\alpha_\textrm{ee2}^{zz}{^2}}) \textrm{sin}^2\beta,$$

(14b)$$\textrm{W}_2 = ({\alpha_\textrm{mm1}^{xx}{^2}} + {\alpha_\textrm{mm2}^{xx}{^2}}) + ({\alpha_\textrm{mm1}^{yy}{^2}} + {\alpha_\textrm{mm2}^{yy}{^2}}) \textrm{cos}^2\beta + ({\alpha_\textrm{mm1}^{zz}{^2}} + {\alpha_\textrm{mm2}^{zz}{^2}}) \textrm{sin}^2\beta,$$

(14c)$$\textrm{W}_3 = ({\alpha_\textrm{mm2}^{zz}} {\alpha_\textrm{ee1}^{yy}} - {\alpha_\textrm{mm1}^{zz}} {\alpha_\textrm{ee2}^{yy}}) - ({\alpha_\textrm{mm2}^{yy}} {\alpha_\textrm{ee1}^{zz}} - {\alpha_\textrm{mm1}^{yy}} {\alpha_\textrm{ee2}^{zz}}),$$

(14d)$$\textrm{W}_4 = ({\alpha_\textrm{mm1}^{yy}} {\alpha_\textrm{ee1}^{xx}} + {\alpha_\textrm{mm2}^{yy}} {\alpha_\textrm{ee2}^{xx}}) + ({\alpha_\textrm{mm1}^{xx}} {\alpha_\textrm{ee1}^{yy}} + {\alpha_\textrm{mm2}^{xx}} {\alpha_\textrm{ee2}^{yy}}).$$

From Eqs. (12) and (13), we can see that the helicity of the incident light has no influence on the elevation angle $\theta$ but it strongly affect the azimuthal angle $\varphi$ . The azimuthal angles $\varphi$ for the incident left ( $\sigma$ = -1) and right ( $\sigma$ = +1) circularly polarized lights are symmetric about the plane of incidence. Such symmetric transverse shift for light with opposite handedness is called photonic spin Hall effects [14, 28-30]. As shown in Eq. (12), such photonic spin Hall effect is unavoidable if the incidence is inclined ($\beta \neq 0^\circ , 90^\circ , 180^\circ$) and the value of $\textrm {W}_3$ is not zero. The later condition can be satisfied if the shape of the particle is anisotropic in $y$ and $z$ directions. It is also worth noting that the elevation angle $\theta$ is not rarely equal to the incident angle $\beta$ unless $\beta$ is $90^\circ$, which is indicated in Eq. (13).

Fig. 1. The three-dimensional schematic of the system and polarization dependence of light scattering. (a) The blue and red rotatory arrows represent the incident left and right circularly polarized lights, respectively. The incident directions are all in the $y$-$z$ plane. $\beta$ is the angle between the incident light and the $z$-axis. The blue and red lines represent the scattering directions of left and right circularly polarized lights, respectively. (b) The far-field scattering diagram of a metallic (Au) ellipsoidal Rayleigh particle in ${Oxy}$ plane simulated in COMSOL Multiphysics software for different incident polarized lights. 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 $\beta$ is $45^\circ$. $\sigma \equiv 0$, +1, and -1 indicate the linear, right-circular and left-circular polarizations, respectively.

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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.

Fig. 2. The simulated cosine values of azimuthal angles and elevation angles for the centroids of scattering far-field. The material of the ellipsoidal Rayleigh particle is metallic (Au). The wavelength of the incident light is 650 nm. The incident angle is $45^\circ$. (a) and (b) The changes in the cosine values of azimuthal angles and elevation angles for the centroids of scattering far-field with the radius of the particle in $y$ direction ($a_y$) through simulation. The radii of the particle in the $x$ and $z$ directions are 10 nm and 30 nm, respectively. (c) and (d) The changes in the cosine values of azimuthal angles and elevation angles for the centroids of scattering far-field with the radius of the particle in $x$ direction ($a_x$) through simulation. The radii of the particle in the $y$ and $z$ directions are 10 nm and 30 nm, respectively.

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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

(15)$$\langle{\textrm{cos}\varphi}\rangle = \sigma \textrm{sin}{2\beta} \frac{C_1} {C_2 + C_3 \textrm{cos}^2 \beta},$$

where $C_1$, $C_2$ and $C_3$ are parameters which are mainly determined by the polarizability properties of the particle. Using Eq. (15), we obtain the fitting curves as shown in Fig. 3. The curves and the simulated data coincide well. The fitted values of parameters $C_1$, $C_2$ and $C_3$ are -0.000445, 4.4415, and -3.2810, respectively.

Fig. 3. The dependences of the cosine values of azimuthal angles for the centroids of scattering far-field on the incident angle. The radii of the particle in the $x$, $y$ and $z$ directions are 10 nm, 20 nm and 30 nm, respectively.

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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

(16)$$\langle{\textbf{F}}\rangle = \oint_S \hat{\textbf{n}}\cdot\langle{\overline{\textbf{T}}}\rangle dS,$$

where $\hat {\textbf {n}}$ denotes the unit outward normal vector at surface $S$ and ${\overline {\textbf {T}}}$ is the time-averaged Maxwell stress tensor can be written as

(17)$$\langle{\overline{\textbf{T}}}\rangle = {\frac{1} {2}} \textrm{Re} [\epsilon \textbf{E} \textbf{E}^* + \mu \textbf{H} \textbf{H}^* - {\frac{1} {2}}(\epsilon \textbf{E}\cdot\textbf{E}^* + \mu \textbf{H}\cdot\textbf{H}^*)\overline{\textbf{I}}],$$

with $\overline {\textbf {I}}$ is the unit tensor, $\epsilon$ and $\mu$ denoting the permittivity and permeability of the medium surrounding the particle. Figures 4(a)–4(f) show the simulated optical forces exerted on the metallic (Au) ellipsoidal Rayleigh particle. The wavelength of the incident light is 650 nm. Within the $a_y$ interval between 5 nm and 30 nm, the changes in the $x$, $y$ and $z$ components of optical forces with $a_y$ are shown in Figs. 4(a)–4(c) where $a_x$ = 10 nm, $a_z$ = 30 nm and the incident angle is $45^\circ$. Figures 4(d)–4(f) illustrate the changes in the $x$, $y$ and $z$ components of optical force with the incident angle where $a_x$ = 10 nm, $a_y$ = 20 nm and $a_z$ = 30 nm. It is easy to identify that the $x$ and $y$ components of optical force are odd functions with the incident angle but the $z$ component of optical force is an even function with the incident angle in Figs. 4(d)–4(f). It is seen that the $x$ components of optical forces for the incident left and right circularly polarized lights are always opposite to each other while the handedness of the incident light does not affect the $y$ and $z$ components of optical forces. This result implies that the direction of the optical force is shifted away from the propagation direction of the incident light. The $x$ components of the optical forces for the incident left and right circularly polarized lights are equal to zero when $a_y$ is equal to $a_z$. Because the polarization-dependent split disappears when the radii of the particle in the $y$ and $z$ directions are equal. The simulations are in good agreement with the analytical results. This phenomenon of the optical force is a direct result of the photonic spin Hall effect. It indicates the lateral optical force exerted on the ellipsoidal Rayleigh particle can be generated by the circularly polarized light and the direction of the force can be readily modulated by the handedness of the incident light. It may provide a potential pathway to optical manipulation of Rayleigh particles.

Fig. 4. The simulated optical forces exerted on the metallic (Au) ellipsoidal Rayleigh particle. The wavelength of the incident light is 650 nm. (a) - (c) The changes in the $x$, $y$ and $z$ components of optical forces with the radius of the particle in $y$ direction ($a_y$). The radii of the particle in the $x$ and $z$ directions are 10 nm and 30 nm, respectively. The incident angle is $45^\circ$. (d) - (f) The changes in the $x$, $y$ and $z$ components of optical forces with the incident angle. The radii of the particle in the $x$, $y$ and $z$ directions are 10 nm, 20 nm and 30 nm, respectively.

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5. Optical torque

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

(18)$$\langle{\mathbf{\Gamma}}\rangle = \oint_S \hat{\textbf{n}}\cdot(\langle{\overline{\textbf{T}}}\times{\textbf{r}}\rangle) dS,$$

where $\textbf {r}$ is the position vector originating from the center of mass. Utilizing the changing of the $a_y$ and incident angle, we obtain the changing of optical torques exerted on the metallic (Au) ellipsoidal Rayleigh particle, as shown in Figs. 5(a)–5(f). Figures 5(a)–5(c) depict the $x$, $y$ and $z$ components of optical torque varying with $a_y$ where $a_x$ = 10 nm and $a_z$ = 30 nm. The incident angle is fixed at $45^\circ$. In Figs. 5(d)–5(f), we see that the $x$ and $y$ components of optical torque are odd functions with the incident angle and the $z$ component of optical torque is even function with the incident angle. The $y$ ($z$) components of optical torques for the incident left and right circularly polarized lights are the opposite of each other, which originates from the transfers of the angular momenta from the incident photonic spins. The incident left and right circularly polarized lights have opposite angular momentum along the propagation direction which is in the $y$-$z$ plane in our case. The transfer of the angular momentum between the light and the ellipsoidal Rayleigh particle takes place in $y$-$z$ plane and can be projected to the $y$ and $z$ directions, respectively. As a result, the $y$ ($z$) components of the torques for the incident left and right circularly polarized lights are opposite to each other. The nonzero angular momentum of electromagnetic wave interacts with the ellipsoidal Rayleigh particle, which generates the optical torque exerted on the particle due to the conservation of angular momentum in the whole system. If there is a transfer of the angular momentum between the light and the ordinary particle, the change rule of the optical torque of the ellipsoidal particle is also applicable to the ordinary particle. The results can be clearly understood by the relationship between optical angular momentum and optical torque.

Fig. 5. The simulated optical torques exerted on the metallic (Au) ellipsoidal Rayleigh particle. The wavelength of the incident light is 650 nm. (a) - (c) The changes in the $x$, $y$ and $z$ components of optical torques with the radius of the particle in $y$ direction ($a_y$). The radii of the particle in the $x$ and $z$ directions are 10 nm and 30 nm, respectively. The incident angle is $45^\circ$. (d) - (f) The changes in the $x$, $y$ and $z$ components of optical torques with the incident angle. The radii of the particle in the $x$, $y$ and $z$ directions are 10 nm, 20 nm and 30 nm, respectively.

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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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Photonic spin Hall effect on an ellipsoidal Rayleigh particle in scattering far-field

Abstract

1. Introduction

2. Theory

3. Results and discussions

4. Optical force

5. Optical torque

6. Conclusion

Funding

References

Cited By

Figures (5)

Equations (21)

Optics Express