Enhanced Spin Hall Shift by Multipoles of Different Orders in Spherical Particles

Abstract

The spin–orbit interaction of light is universal in the process of light scattering, and an important aspect is the spin Hall effect. The spin Hall effect of light also exists in a three-dimensional (3D) system. When circularly polarized light is incident on a spherical particle, the transverse displacement of the particle relative to the scattering plane can be observed due to the spiraling of the Poynting vector in the far field. In general, the spin Hall shift of light is negligible and difficult to detect in experiments. In this paper, we use a high-refractive-index (HRI) core-shell structure to excite high-order multipoles and explore the interaction between different order multipoles to enhance the spin Hall shift in the microwave band. We show that there exist some angles that increase the spin Hall shift when two particular multipoles are equal and dominated. Our work provides a new perspective for understanding the interaction between light and particles and enhances the spin Hall shift of the sphere in the microwave band.

1. Introduction

Light carries both spin angular momentum (SAM) and orbit angular momentum (OAM). SAM and OAM interact in the process of light propagation called the spin–orbit interaction (SOI) of light [1,2,3]. Recently, the SOI of light has attracted enormous attention because it shows the dynamic characteristics of light and spin-to-orbit AM conversion, resulting in various interesting phenomena and applications. In the above phenomenon, the spin Hall effect of light (SHEL) is one of the most active research branches in the SOI of light.

When circularly polarized light is incident on the optical interface between two media, the scattered light will have a tiny displacement compared to the incident light. This tiny displacement is called the SHEL or photonic spin Hall effect (SHE) [4,5,6,7]. In 2004, Bliokh et al. used semiclassical theory to explain this phenomenon and named it the Hall effect of light [4]. In 2008, Hosten et al. designed a notable experiment to measure the spin Hall shift and verify the theory [6]. Generally, the SHEL is quite small, but it can be enhanced by metamaterial [8,9], metasurface [10,11], topological edge states [12], or the near Brewster angle [13]. The spin Hall effect of light is applied in identifying graphene layers [14], measuring 2D material optical constants [15], differential microscopy [16], precision metrology [17,18], and so on [1,19]. Under certain conditions, the longitudinal and transverse shifts of the optical beams may be highly anomalous, i.e., larger than the incident wavelength, such as when using space-varying PB phases in homogeneous media [20], exploring exceptional points in PT-symmetric structures [21], or using the singularity in the Berry phase appearing in the beam Fourier spectrum [22].

In the case of a three-dimensional (3D) scatterer, we can also define the spin Hall effect of light [23,24]. In the 3D case, the spin Hall shift [24,25,26] is caused by the spiral of the Poynting vector of the scattered light [24,27,28,29,30]. Generally, one can use Mie theory and multipole expansion to calculate the shift. Normally, the spin Hall shift of the sphere is negligible on small particles (r ≪ λ). In previous work, researchers used dual symmetry [25,31], the core-shell structure [32], the topological magnetoelectric effect [33], the graphene-wrapped sphere structure [26], the chiral structure [34], etc., to enhance the SHEL. In addition, researchers have also undertaken work on enhancing the spin Hall shift of the ellipsoidal structure [35,36] and using single closed elliptical nanoslits to observe optical SHE. However, little has been done to enhance the SHEL by exploiting higher multipole interactions.

In this paper, we use the core-shell structure to excite higher-order multipoles to enhance the spin Hall shift in multiple frequency bands in the microwave band. Our theory suggests that there will be some specific angles to enhance the spin Hall shift when two multipoles are equal and dominated. This study provides new insights into understanding the interaction between light and particles and offers new ways to enhance the spin Hall shift of a sphere in the microwave waveband.

2. Structure and Theoretical Formula

We consider a lossless core-shell particle embedded in a vacuum ($\epsilon =1$). The core radius $r_a$ is adjustable with a relative permittivity of ${\epsilon }_a=1$, i.e., the core-shell particle is a hollow particle. The total radius $r=12 \mathrm{m}\mathrm{m}$ and the permittivity of the shell ${\epsilon }_b=10.5$. Both the core and shell consist of homogeneous non-magnetic media ($\mu =1$). This structure can stimulate a higher electric/magnetic mode and strongly enhance the spin Hall shift reported in preview work [32,37,38,39]. The incident light is a circular plane wave with wave number ($k=\frac{2\pi }{\lambda }$), carrying a spin of $\sigma =\pm 1,$ depending on the left/right polarization state (a schematic diagram is illustrated in Figure 1).

We use multiple Mie expansions to solve this scattered problem. We consider a monochromatic left-handed circular plane wave of the form ${\overrightarrow{E}}_i^L=E_0{e}^{ikz}({\overrightarrow{e}}_x-i{\overrightarrow{e}}_y)e^{-i\omega t}$. The $e^{-i\omega t}$ is the time-harmonic factor and can be reduced in the calculation. We may expand the incident light in a vector spherical harmonics function.

$${\overrightarrow{E}}_i^L=\sum _{n=1}^{\infty }{E}_n\left({\overrightarrow{M}}_{o1n}^{\left(1\right)}-i{\overrightarrow{N}}_{e1n}^{\left(1\right)}-i{\overrightarrow{M}}_{e1n}^{\left(1\right)}+{\overrightarrow{N}}_{e1n}^{\left(1\right)}\right)$$

(1)

$${\overrightarrow{H}}_i^L=-\frac{k}{\omega \mu }\sum _{n=1}^{\infty }{E}_n\left({\overrightarrow{M}}_{o1n}^{\left(1\right)}+i{\overrightarrow{N}}_{o1n}^{\left(1\right)}+i{\overrightarrow{M}}_{o1n}^{\left(1\right)}+{\overrightarrow{N}}_{e1n}^{\left(1\right)}\right)$$

(2)

where $E_n=E_0{i}^n(2n+1)/\left(n\left(n+1\right)\right),$ ${\overrightarrow{M}}_{\mathrm{o}1\mathrm{n}}^{\left(1\right)},$ ${\overrightarrow{N}}_{\mathrm{e}1\mathrm{n}}^{\left(1\right)}$, ${\overrightarrow{M}}_{\mathrm{o}1\mathrm{n}}^{\left(1\right)}$, ${\overrightarrow{N}}_{\mathrm{o}1\mathrm{n}}^{\left(1\right)}$ are the vector spherical harmonics functions; the superscript (1) means the generating function is the spherical Bessel function of the first kind [40].

Similarly, we also expand the scattered field in the vector spherical harmonics function,

$${\overrightarrow{E}}_s^L=\sum _{n=1}^{\infty }{E}_n\left(i{a}_n{\overrightarrow{N}}_{e1n}^{\left(3\right)}-b_n{\overrightarrow{M}}_{o1n}^{\left(3\right)}-i{a}_n{\overrightarrow{N}}_{o1n}^{\left(3\right)}+i{b}_n{\overrightarrow{M}}_{e1n}^{\left(3\right)}\right)$$

(3)

$${\overrightarrow{H}}_s^L=\frac{k}{\omega \mu }\sum _{n=1}^{\infty }{E}_n\left(i{b}_n{\overrightarrow{N}}_{o1n}^{\left(3\right)}+i{a_n\overrightarrow{M}}_{e1n}^{\left(3\right)}+b_n{\overrightarrow{N}}_{e1n}^{\left(3\right)}+i{a}_n{\overrightarrow{M}}_{o1n}^{\left(3\right)}\right)$$

(4)

The superscript (3) means the generating function is the spherical Bessel function of the third kind, and $a_n,b_n$ are the scattering coefficients of the core-shell structure. By applying boundary conditions in the two interfaces ($r=r_a$ and $r=r_b$), we obtain the expression of the scattering coefficients $a_n$, $b_n$. We define the spin Hall shift of light as the transverse deflection observed in the far field, via the azimuthal component of the Poyning vector as [24]:

$${∆}_{\mathrm{S}\mathrm{H}}=\underset{r\to \infty }{\lim }r\left(\frac{S_{\varphi }{\overrightarrow{e}}_{\varphi }}{\left|S_r\right|}\right)$$

(5)

where $S_r$ and $S_{\varphi }$ are the azimuthal and radial components of the scattered Poynting vector $S=\frac{1}{2}*Re\{E_s\times H_s^{*}\}$. We can see in the expression that the direction of the spin Hall shift is perpendicular to the scattering plane. The physical origin of the spin Hall shift is the spiral of the Poynting vector. The incident CPL has a spin ($\sigma =\pm 1$) because of the SOI in the scattering process, the scattering Poynting vector has a component perpendicular to the radial direction causing a displacement of the spherical particles.

By solving the component of the Poynting vector, the spin Hall shift can be written as [19]:

$${∆}_{\mathrm{S}\mathrm{H}}=\frac{\sigma }{k}(\frac{\left|Re\left(S_1^{*}\left[{\sum }_{n=1}^{\infty }\left(2n+1\right)a_n{\pi }_n\right]+S_2\left[{\sum }_{n=1}^{\infty }\left(2n+1\right)b_n{\pi }_n\right]\right)\right|}{{\left|S_1\right|}^2+{\left|S_2\right|}^2})$$

(6)

where $\mathsf{\sigma }=\pm 1$ for the LCP and RCP state. ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ are the amplitude scattering matrix elements defined as:

$${\mathrm{S}}_1=\sum _{n=1}^{\infty }\frac{2n+1}{n\left(n+1\right)}\left(a_n{\pi }_n+b_n{\tau }_n\right)$$

(7)

$${\mathrm{S}}_2=\sum _{n=1}^{\infty }\frac{2n+1}{n\left(n+1\right)}\left(a_n{\tau }_n+b_n{\pi }_n\right)$$

(8)

${\pi }_n\left(cos\theta \right)$ and ${\tau }_n\left(cos\theta \right)$ are the angle-dependent functions defined as ${\pi }_n\left(cos\theta \right)=P_n^1\left(cos\theta \right)/sin\theta$, ${\tau }_n\left(cos\theta \right)={dP}_n^1\left(cos\theta \right)/d\theta$. It can be calculated using the associated Legendre function recursion.

3. Simulation Result

In this section, we investigate how core radius change causes change in the spin Hall shift and theoretically analyze how different multipole interactions induce enhancement of the spin Hall shift for a core-shell sphere. We also investigate how core radius change causes change in the spin Hall shift.

In previous work [23,25], we found that, when the electric or magnetic dipole is dominant, the maximum SH shift is $\lambda /\pi$. High refractive index (HRI) spherical particles can provide higher-order multipoles in different frequency bands [38,39,41,42]. We plot the first three scattering coefficients and the SH shift of the core-shell structure; the core radium $r_a$ is 4 mm. The value of the spin Hall shift (${∆}_{SH}$) is divided by the wavelength ($\lambda$).

Figure 2a illustrates that the electric and magnetic multipoles are enhanced at 3 GHz to 9 GHz. Due to the interaction between different poles, the spin Hall shift will be enhanced significantly around multiple frequencies at different scattering angles, as shown in Figure 2b. Thanks to the core-shell structure there is abundant freedom to adjust. Here, we plot the spin Hall shift as it varies with different core radii.

From Equation (6), we know that the spin Hall shift is related to the scattering coefficients of different modes. By changing the core radius, one can make the peak of the scattering coefficients move. The line shapes of the scattering coefficients are shifted differently for different modes causing the peak position of the spin Hall shift to appear, disappear or move. We observe that, in Figure 3d when $r_a=6 \mathrm{m}\mathrm{m}$, the frequency around 8.26 GHz and $\theta =\frac{\pi }{2}$, ${∆}_{SH}$ is about 1.5$\lambda$; it is easy to design experiments to observe the spin Hall shift. It is also noted that the enhancement of ${∆}_{\mathrm{s}\mathrm{h}}$ is sometimes due to the interaction of multipole modes; for example, the enhancement around 5.3 GHz is related to the electric dipole (${\mathrm{a}}_1$), the magnetic dipole (${\mathrm{b}}_1$) and the magnetic quadrupole (${\mathrm{b}}_2$). Such a case is complicated to show in equations, and we will discuss it in our future work.

3.1. Overlap of Electric Dipole and Magnetic Dipole

In the case that an electric dipole ($a_1$) and magnetic dipole ($b_1$) are dominated, substitute the value of ${\tau }_1$ and ${\pi }_1$ into Equations (7) and (8) (the denominator of the spin Hall shift). These reduce to:

$$\frac{3}{2}{a}_1+\frac{3}{2}{b}_{1}cos\theta =0$$

(9)

$$\frac{3}{2}{a}_{1}cos\theta +\frac{3}{2}{b}_1=0$$

(10)

The maximum spin Hall shift can be obtained if the denominator of the expression is close to 0. It is apparent when $a_1=b_1$ (dual symmetry) and $\mathsf{\theta }\approx \mathsf{\pi }$, the condition is satisfied. We plot the real part and the imaginary parts of $a_1$ and $b_1$ as well as the value of the spin Hall shift from 3 GHz to 3.5 GHz for different core radii.

Figure 4 illustrates when the scattering coefficients ${\mathrm{a}}_1$ and ${\mathrm{b}}_1$ are equal and the scattering angle ($\mathsf{\theta }$) is around $\mathsf{\theta }\approx \mathsf{\pi }$. Previous research work has found that, in dual symmetry (${\mathrm{a}}_1=b_1$), the spin Hall shift can be enhanced. Our work produced the same result—the spin Hall shift is enhanced significantly by about 1.5$\mathsf{\lambda }$. We also change the core radius of the core-shell structure. Figure 4a–c show that increase in the core radius causes the position of $a_1=b_1$ to undergo a blue shift. As a result, the enhanced position of the spin Hall shift also has a blue shift, while the enhancement of the spin Hall shift remains around 1.5$\mathsf{\lambda }$.

3.2. Overlap of Electric Quadrupole and Magnetic Octupole

In the case that the electric quadrupole ($a_2$) and magnetic octupole ($b_3$) are dominant, substituting the corresponding value of ${\tau }_n$ and ${\pi }_n$ into Equations (6) and (7), they reduce to:

$${\mathrm{S}}_1=\frac{5}{2}{a}_{2}cos\theta +\frac{7}{12}{b}_3\left(6cos\theta -\frac{45}{2}{\sin }^2\theta cos\theta \right)$$

(11)

$${\mathrm{S}}_2=\frac{5}{2}{a}_{2}cos2\theta +\frac{7}{12}{b}_3(6-\frac{15}{2}{\sin }^2\theta )$$

(12)

The denominator of the spin Hall shift $D={\left|S_1\right|}^2+{\left|S_2\right|}^2$. Considering $a_2\approx b_3$, when D is close to 0, the spin Hall shift has a maximum value. So, the denominator can be further simplified to the following form:

$$\mathrm{D}={\left|\frac{5}{2}cos\theta +\frac{7}{12}(6cos\theta -\frac{45}{2}{\sin }^2\theta cos\theta )\right|}^2+{\left|\frac{5}{2}cos2\theta +\frac{7}{12}(6-\frac{15}{2}{\sin }^2\theta )\right|}^2\approx 0$$

(13)

$$0\le \mathsf{\theta }\le \mathsf{\pi }$$

(14)

It is notable that $a_2$ and $b_3$ do not appear in the equations. In other words, if $a_2$ and $b_3$ are equal at a certain frequency and the equation has a solution, the spin Hall shift should be enhanced at a certain angle.

The scattering angles satisfying the conditions are about ${49}^{°}$ and ${132}^{°}$; the denominator has a local minimum in these two angles. We plot the real part and imaginary parts of $a_2$ and $b_3$ as well as the value of the spin Hall shift from 6.5 GHz to 7 GHz. We also plot the spin Hall shift as a function of the angle at 6.66 GHz, considering only $a_2$ and $b_3$ and considering the first six multipoles. We also plot the variation in the numerator and denominator of the spin Hall shift expression with an angle when only $a_2$ and $b_3$ are considered.

Figure 5a shows that the spin Hall shift enhances and splits around the scattering angles satisfying the equation (${49}^{°}$ and ${132}^{°}$) at the position where $a_2$ and $b_3$ are equal. The incident frequency in Figure 5b is 6.66 GHz. The above figure shows that only the considered $a_2$ and $b_3$ provide a good fit to the spin Hall shift as a function of angle. The figure below shows that, at the position where the denominator is the minimum, the numerator tends to zero faster than the parent, so the position where the spin Hall shift is the maximum is not exactly in the position where the denominator is the minimum. Notably, there is a conversion from positive to negative in the numerator at the position where the denominator is the minimum, causing the spin Hall shift to have a split around the denominator minimum position.

3.3. Overlap of Electric Octupole and Magnetic Dipole

In the case that the electric octupole ($a_3$) and the magnetic dipole ($b_1$) are dominated, substitute the corresponding value of ${\mathsf{\tau }}_{\mathrm{n}}$ and ${\mathsf{\pi }}_{\mathrm{n}}$ into Equations (6) and (7). This reduces to:

$${\mathrm{S}}_1\left(cos\theta \right)=\frac{7}{12}{a}_3\left(6-\frac{15}{2}{\sin }^2\theta \right)+\frac{3}{2}{b}_{1}cos\theta$$

(15)

$${\mathrm{S}}_2\left(cos\theta \right)=\frac{7}{12}{a}_3\left(6cos\theta -\frac{45}{2}{\sin }^2\theta cos\theta \right)+\frac{3}{2}{b}_1$$

(16)

The denominator of the spin Hall shift $D={\left|S_1\right|}^2+{\left|S_2\right|}^2$. Considering $a_3\approx b_1$, when D is close to the minimal value, the spin Hall shift has a maximum value. So, the denominator can be further simplified to the following form:

$$\mathrm{D}={{\left|a_3\right|}^2(\left|\frac{7}{12}\left(6-\frac{15}{2}{\sin }^2\theta \right)+\frac{3}{2}cos\theta \right|}^2+\left|\frac{5}{2}cos2\theta +\frac{7}{12}(6-\frac{15}{2}{\sin }^2\theta )\right|)$$

(17)

$$0\le \mathsf{\theta }\le \mathsf{\pi }$$

(18)

Similarly, the denominator minimum position is independent of the value of $a_3$ and $b_1$. The scattering angles satisfying the conditions are ${80}^{°}$ and ${155}^{°}$.The spin Hall shift should be enhanced around these two angles. We plot the values of the real and imaginary parts of $a_3$ and $b_1$ and the corresponding spin Hall shifts when the radii of the cores are 2 mm, 4 mm, and 8 mm. The frequency range is from 7.5 GHz to 8 GHz.

As is shown in Figure 6, the spin Hall shift is significantly enhanced around ${155}^{°}$ at 7.92 GHz ($a_3\approx b_1)$ when the core radii are 2 mm and 4 mm. However, the enhancement of the spin Hall shift is not obvious around ${80}^{°}$ because the values of the other multipoles are also large and contribute to the spin Hall shift. When the core radius is 8 mm, the values of ${\mathrm{a}}_3$ and ${\mathrm{b}}_1$ are not equal from 7.5 GHz to 8 GHz. Although the spin Hall shift enhances slightly, this is not caused by the dominance of $a_3$ and $b_1$.

4. Conclusions

In this paper, we investigate enhancement of the spin Hall shift of high refractive index core-shell spherical particles (hollow particles) in the microwave band based on Mie theory. We use the fact that high refractive index core-shell particles can excite higher-order multipoles and enhance the spin Hall shift of the core-shell structure through overlapping multipoles of different orders, generalizing the work of previous researchers. In the condition that certain multipoles are equal and dominant, we reveal that the enhanced position of the spin Hall shift is only related to the angle. Our work provides a theoretical basis for designing experiments to observe the spin Hall shift of microwave-band spherical particles. The spin Hall shift of light can also be observed in experiments in the far field. The change in the transversal shifts can be easily detected by flipping the helicity of the incident light [36]. We expect that our work will contribute to the design and optimization of microwave optical devices and to the understanding of spin–orbit interactions of light.