1. Introduction

The Spin Hall effect of light (SHEL) is caused by the transformation of spin angular momentum and orbital angular momentum, i.e., spin-orbit interaction (SOI). When the light encounters refractive index gradient or interface during its propagation, SOI will happen due to the conservation of total angular momentum [1]. Hence SHEL is ubiquitous in light scattering or focusing, similar to the analog of spin Hall effect in electronic systems [2,3]. And SHEL is a good candidate for exploring the underlying physics in light-matter interaction. Recently, a unified perturbative approach is proposed to describe the SOI of both electrons and photons simultaneously in a waveguide [4].

It is more illustrative to understand the SHEL at the interface of two homogeneous slabs. When circularly polarized light is curved at the interface, its polarization will be rotated as well, which is related to the spin angular momentum. To maintain the total angular momentum, SOI takes place and deviates the light transversely from its geometrical path [5], i.e., spin Hall shift of light. While SHEL is usually very week, giant photonic spin-Hall shifts are reported in structured metamaterial [6] and metasurfaces [7]. What’s more, the SHEL can be observed in higher-dimensional systems, such as spherical spheres scattered by circularly polarized light. The scattered lights in far-field undergo a transverse shift, which is perpendicular to the scattering plane. The spin Hall shifts Δ are firstly discussed for dipolar scatters [8,9]. For the dipolar case, the spin Hall shift is independent of the optical properties of the dipole and only relates to the polar scattering angles θ. At the angle $\theta = \pi /2$ (perpendicular to the direction of incident light), there is a maximum value Δ = $\lambda /\pi$, where spin angular momentum fully transformed to orbital angular momentum [1]. Nevertheless, the shifts are very small even at the peak.

Enhanced spin Hall shifts are proposed by exploiting the strong SOI effects in plasmonic nanostructures [10,11]. Recently, the spin Hall shifts have been found to be large as several wavelengths in systems with dual symmetry [12,13], where strong SOI is associated with the interference of electric and magnetic dipolar modes. The enhanced spin Hall shifts can be realized in a broadband spectrum by using core-shell nanostructures [14]. Although the electric and magnetic responses can be spectrally tuned by changing the geometrical parameters of nanostructures [15,16]. Once the geometric parameters of the nanostructures are given, the spin Hall shifts of light cannot be tuned flexibly for a fixed incident wavelength.

On the other hand, SOI can be greatly enhanced by using doped extended graphene, which can excite plasmons with high-quality factors [17]. Graphene, being an outstanding optoelectronic material, has excellent electrical tunability in broad frequency regions. The inherent characters of graphene render it in engineering various applications, such as controlling spin-orbit scatterers [18], switching optical bistability [19], identifying graphene layers [20], and designing reconfigurable sensor [21]. Recently, we have proposed switchable optical nanoantennas via the dynamical modalities of graphene-warped particles [22,23]. It is crucial to determine the precise position of the nanoparticles, and it is a prerequisite for measuring the dynamical nonlinear behavior in further experiments. What is more, graphene-wrapped nanoparticles are also proposed for fluorescence bioimaging in vitro [24] and enhancement of photocatalytic activity [25]. However, wavelength-scale shifts [26] are found in optical super-resolution microscopy due to spin-orbit interaction and can compromise the precise position determination. The study of SHEL on graphene-wrapped spheres may facilitate the design of eliminating systematic errors in optical imaging techniques.

In this paper, we propose to use graphene to manipulate the SHEL of nanoparticles. A graphene coating is wrapped on the surface of the particles, and the electromagnetic scattering fields are solved by modified Mie theory. Two kinds of particles are discussed in this paper: one is high-index dielectric particle with simultaneously excited electric and magnetic modes; the second one is the grephene-wrapped Drude nanoparticle with excited electric dipolar and quadrupole modes. We demonstrate the tunability of graphene on the interferences between different modes, which modulate both spin-orbit interaction of light in the near field and the scattered spin Hall effect in far-field. In addition, we discuss the observation of such tunable spin Hall effect in detection plane in far-field. With the advantage of dynamical tuning of graphene, our results may shed light on removing the systematic positon errors in optical imaging systems.

2. Theoretical formulations

In this section, we modify the Mie theory to investigate the scattering behaviors of a graphene-wrapped sphere under the illumination of circularly polarized light. Without loss of generality, we consider a left-handed circularly polarized light ${\textbf E}_i^{} = {E_0}{\textrm{e}^{ikz}}(\hat{x}\textrm{ - }i\hat{y})$ and suppress the time-dependence factor $\exp ( - i\omega t)$. The radius of the spherical particle is a, and monolayer graphene is wrapped on the surface of the particle. Based on Mie theory [27], the incident electric and magnetic fields of circularly polarized light upon the graphene-wrapped nanoparticle have the form,

For a three-dimensional subwavelength scatterer, the spin Hall (SH) shift is defined as the transverse shift to the scatterer’s location in the perceived far-field, as is shown in Fig. 1. The red arrowed line that perpendicular to the scattering plane represents the SH shift. The definition expression is in the following form [1,10],

 

Fig. 1. Illustration of spin Hall shift of scattered light by a graphene-wrapped nanosphere under the circularly polarized light. The red arrowed line, which is perpendicular to the scattering plane, is the spin Hall shift ${\Delta _{SH}}$. The top right-hand inset shows the far-field intensity distribution at the detection plane. The white dot represents the center of the scattered light. ${\mu _\textrm{p}}$ is the measured displacement to the origin of the observation plane.

Download Full Size | PDF

The above-mentioned SH shifts are obtained from a single scattered line of Poynting vector. To experimentally observe the SH effect, one can consider the intensity distribution of the scattered light over the far-field detection plane where a bundle of field lines is sensed by the detector. One can set the scatterer as the origin, and the position of the detector is at $({r_0},{\theta _0},{\varphi _0})$, with the unit vectors (${\hat{{\textbf e}}_{{\theta _0}}}$ and ${\hat{{\textbf e}}_{{\phi _0}}}$) in the detection plane [31,32]. The position of the scattered light $(\lambda {\hat{{\textbf e}}_{{\theta _0}}},\mu {\hat{{\textbf e}}_{{\phi _0}}})$ is at the center of the intensity distribution $I({r_0};\lambda ,\mu ) = {\textbf S}(r)\cdot {\hat{r}_0}$ where the intensity is usually maximum (marked with a white dot), as is shown in the top right-hand corner of Fig. 1. Therefore, the measured displacement ${\mu _p}$ is the deviation of scattered lights’ center in ${\hat{{\textbf e}}_{{\phi _0}}}$ direction to the origin of the detection plane. Note that ${\Delta _{\textrm{SH}}}$ and ${\mu _p}$ have a common physical nature, which is due to the spin-orbit interaction of light. ${\Delta _{\textrm{SH}}}$ is the displacement of a Poynting vector line from the center of the nanoparticle. The reference point is the origin of coordinates. And ${\mu _p}$ is the displacement of intensity distribution of the radiation from the observation position. The reference point of ${\mu _p}$ is the position of the detection.

3. Numerical results

3.1 Tunable SH shift by a graphene-wrapped dielectric sphere

We now consider a high-index subwavelength sphere scattered by a circularly polarized light in a vacuum. High-index dielectric subwavelength structures are widely used to realize numerous novel effects, such as unidirectional scattering [33], generalized Brewster effect in all-dielectric metasurface [34], and nonlinear Fano resonance [35]. When the wavelength inside the structure is comparable to its size, the magnetic response can be excited and may reach the same magnitude as the electric dipole [36]. As is shown in Fig. 2(a), strong magnetic dipole modes are induced inside the high-index dielectric particle, when the wavelength inside the particle $\lambda /n$ ($n$ is the refractive index of the particle) is comparable to the particle’s diameter $2\textrm{a}$. The magnetic dipole response (${b_1}$) is sensitive to the variety of graphene’s Fermi level ${{\epsilon }_{{\tiny F}}}$, which can be controlled by external electrical gating field or chemical doping [19]. While the electric dipole response (${a_1}$) remains unchanged in this wavelength range. The tunability of graphene layer on optical responses originates from the induced surface current at the surface of graphene. It can also be understood from the additional graphene terms in the scattering coefficients (See Eq. (6) and (7)).

 

Fig. 2. (a) The Mie scattering coefficients ${a_1}$and${\textrm{b}_1}$varies with the incident wavelength and the warped graphene’s Fermi levels. The permittivity of the particle is $\varepsilon = 3.9$, and the radius is a=1200 nm. The solid lines represent electric dipole mode ${a_1}$ which are basically unchanged under the tuning of the Fermi levels of graphene. (b) The spin Hall shift of graphene-wrapped dielectric particle varies with the scattering angle for different graphene’s Fermi level at incident wavelength 5350 nm. The inset illustrates the transfer function T as a function of the Fermi level.

Download Full Size | PDF

Since the graphene could tune the electric and magnetic modes, it can effectively control the far-field scattering characters. In Fig. 2(b), we demonstrate the spin Hall shift of light for a graphene-wrapped sphere when ${a_1}$ and ${b_1}$ are of comparable magnitudes $(\lambda = 5350{\kern 1pt} {\kern 1pt} {\kern 1pt} nm)$. Compared to the case of Rayleigh dielectric particle [12], the enhanced spin Hall shift at around $\theta = {145^ \circ }$ is due to the interaction of simultaneously excited electric and magnetic dipoles in dual systems [12,13]. With the increase of graphene’s Fermi level ${{\epsilon }_{{\tiny F}}}$, the spin Hall shift first increases to the maximal at ${{\epsilon }_{\tiny F}} = 0.3{\kern 1pt} {\kern 1pt} \textrm{eV}$, then decrease with the increase of ${{\epsilon }_{{\tiny F}}}$. Note that the enhancement of ${\Delta _{SH}}$ mainly comes from the dual symmetry of the scattering system, not the wrapped graphene. For a dual particle, the geometric gradient phase of scattering light is greatly increased, which is directly associated with the spin Hall shift.

The role of wrapped graphene is merely adjusting the degree of the duality of the particle. One can evaluate the duality with the transfer function $\textrm{T}(\lambda ,{{\epsilon }_{{\tiny F}}}) = \textrm{W}_{ - p}^{sca}/\textrm{W}_p^{sca}$, which is the ratio between the energy scattering opposite polarization and the energy scattered in the same polarization as the incident light [37]. If we only consider the electric dipole (${a_1}$) and magnetic dipole (${b_1}$), the transfer function can be reduced to $\textrm{T}(\lambda ,{{\epsilon }_{{\tiny F}}}) = {|{a_1^{} - b_1^{}} |^2}/{|{a_1^{} + b_1^{}} |^2}$. The particle is dual when T tends to zero. From the inset of Fig. 2(b), we can see that the maximal enhancement of spin Hall shift corresponds to the minimal value of transfer function at ${{\epsilon }_{{\tiny F}}} = 0.3{\kern 1pt} {\kern 1pt} \textrm{eV}$.

From the view of light-matter interaction, the spin Hall shift of light is the result of spin-orbit interaction. We plot the near field of circular polarization to demonstrate the role of spin-orbit interaction. The circular polarization degree (CPD) is defined as follows, CPD $ = {\kern 1pt} {\kern 1pt} {\kern 1pt} |{\varepsilon _0}{{\textbf E}^\ast } \times {\textbf E} + {\mu _0}{{\textbf H}^\ast } \times {\textbf H}|/({\varepsilon _0}|{\textbf E}{|^2} + {\mu _0}|{\textbf H}{|^2})$[5]. For a circular polarization light, the CPD is 1, while for a linear polarization light, the CPD is 0. Hence the low value of CPD in the near field means that the spin angular momentum is transformed to orbital angular momentum. As is shown in Fig. 3(a), the enhanced spin Hall effect corresponds to the strong spin-orbit interaction near the particle. The simultaneously excited electric and magnetic dipoles play an important role in transforming the spin angular to orbital angular term. Compared to the case where only electric dipole mode dominates, the spin-orbit interaction is relatively weak near the particle, see Fig. 3(b).

 

Fig. 3. Near field distribution of circular polarization degree (CPD) for cases (a) electric and magnetic dipoles are equally excited, and (b) only electric dipole mode is excited. The blue areas show that the spin angular momentum is partly transformed to the orbital angular momentum. The yellow arrows shows the direction of incident light. Other parameters are the same as those in Fig. 2.

Download Full Size | PDF

3.2 Tunable SH shift by a graphene-wrapped Drude sphere

Many studies find that surface plasmon can dramatically enhance the spin-orbit interaction. Here we try to use graphene coating to modify the plasmons. We consider a graphene-wrapped Drude nanoparticle with dielectric permittivity, $\varepsilon (\omega ) = {\varepsilon _\infty } - {\omega _p}^2/(\omega (\omega + i{\gamma _D}))$, where ${\omega _p}$ is the free-electron plasma frequency and ${\gamma _D}$ is optical damping rate [38]. This kind of Drude nanoparticle can be made of doped semiconductor materials. By using Drude nanoparticles with weak dissipation [39]and optimizing particle size, not only electric dipole mode but also higher electric modes can be induced. As is shown in Fig. 4(a), quadrupole mode ${a_2}$ arises at about $\lambda = 2550{\kern 1pt} {\kern 1pt} nm$ and could have comparable magnitude with the dipole mode ${a_1}$. Under the tuning of the graphene coating the resonances blue shift for both dipole and quadrupole modes.

 

Fig. 4. (a) The Mie scattering coefficients ${a_1}$ and ${\textrm{a}_2}$ vary with the incident wavelength for different graphene’s Fermi levels. (b) and (c) show the distribution of Poynting fields and Poynting vector lines around Fano resonance with ${{\epsilon }_{{\tiny F}}}$ =0 eV and ${{\epsilon }_{{\tiny F}}}$ =0.9 eV, respectively. The singular points (vortex and saddle) represent the amplitude of the Poynting vector is zero. The parameters : $\lambda \textrm{ = }\textrm{2548}{\kern 1pt} {\kern 1pt} nm$ and $a = 300{\kern 1pt} {\kern 1pt} nm$.

Download Full Size | PDF

In the near field, the re-emitted light interferes with the incident light, giving rise to a complex near-field pattern around the particle [40]. This radiative coupling is usually controlled by the size parameter $x$($x = 2\pi a/\lambda$), which affects both the electric magnetic amplitudes (${a_n}$ and ${b_n}$, see Eq. (6) and (7)) and near field interference. Instead of tuning the incident wavelength, one can tune the Fermi level ${\epsilon}_{{\tiny F}}$ of graphene coating to vary the radiative coupling. Around the wavelength $\lambda = 2550{\kern 1pt} {\kern 1pt} nm$, sharp quadrupole modes overlap with the board dipolar modes that lead to Fano resonance [39]. The inherent sensitivity of Fano resonance renders graphene-tuned interference switching between strong enhancement (Fig. 4(b) with ${{\epsilon }_{{\tiny F}}} = 0{\kern 1pt} {\kern 1pt} \textrm{eV}$) and strong suppression (Fig. 4(c) with ${{\epsilon }_{{\tiny F}}} = 0.9{\kern 1pt} {\kern 1pt} \textrm{eV}$) of near-field energy flow. The energy flow magnitude is reduced by about one order of magnitude with ${{\epsilon }_{{\tiny F}}}$ increased from 0 to 0.9 $\textrm{eV}$. The energy flow streamline patterns are greatly changed as well. The singular points of both vortex and saddle points emerge, annihilate and move in the vicinity of the nanoparticle. It is a typical Fano resonance feature in the near field, resulting from constructive and destructive interference.

What’s more, the far-field scattering also exhibits a typical asymmetric lineshape of Fano resonance. For example, backward scattering and forward scattering are far-field observables that are sensitive to interference. Around Fano resonance, the dominant far-field scattering can switch swiftly from forward scattering to backward scattering by tuning the graphene and vice versa (results not shown in the text). These similar phenomena can also be found in topological systems with topological magnetoelectric effect [41].

It is interesting that not only far-field scattering intensities but SH shift of scattering light vary dramatically around Fano resonance. In Fig. 5(a), we show that the SH shift can be readily tuned via the Fermi level of graphene coating. The two peaks of enhanced SH shift for ${{\epsilon }_{{\tiny F}}} = 0{\kern 1pt} {\kern 1pt} \textrm{eV}$ are due to the constructive interference of dipolar and quadrupole modes, which are in phase and of the same magnitudes. The shifts are associated with the interference of dipolar and quadrupole modes, not directly related to graphene’s Fermi levels. The role of the wrapped graphene is to tune the interference and can enhance the peak of spin Hall shifts when the dipolar and quadrupole modes are in phase. With the increase of ${{\epsilon }_{{\tiny F}}}$, the dipolar and quadrupole modes are out of phase, and dipolar modes become dominant in the resonance. Then the SH shifts for ${{\epsilon }_{{\tiny F}}} = 0.9{\kern 1pt} {\kern 1pt} \textrm{eV}$ are decreased and only exhibit a single peak, which resembles the lineshape of a dipolar scatterer.

 

Fig. 5. The spin Hall shift of light and corresponding near field distribution of circular polarization degree for different graphene’s Fermi level. The blue areas show the full transformation from spin angular momentum to orbit angular momentum, indicating strong spin-orbit interaction. Other parameters are the same as those in Fig. 4.

Download Full Size | PDF

The corresponding CPD that indicates strong spin-orbit interaction in the near field explains the variance and enhancement of SH shift in the far-field, as is shown in Fig. 5(b) and (c). For the blue regions where CPD is close to 0, the spin-orbit interaction is strong. And the spin angular momentum of incident light is fully transformed to orbital angular momentum. In Fig. 5(b), the spin-orbit interaction extends to a large area and shows a typical quadrupole pattern. The different line profiles of quadrupole modes (two peaks) and dipolar modes (one peak) have a high correlation with the circular polarization degree (CPD) in the near field of graphene-warped particles. The typical quadrupole pattern of CPD shows two strong spin-orbit interaction regions in the left side or right side, which are corresponding to the two peaks of SH shift in the far-field for the scattering angle from 0 to 180 degree. In contrast, Fig. 5(c) shows relatively weak spin-orbit interaction and corresponds to the weak spin Hall effect case.

4. Discussion and conclusion

Last but not least, we discuss the possible observation of the spin Hall effect in experiment. The spin Hall shift that we considered comes from the transverse displacement of scattered Poynting vector line viewed from the far-field. The detection of the particle is not merely dependent on a single field line but on a bundle of field lines [32]. Hence we demonstrate the intensity distribution $I({r_0};\lambda ,\mu )$ over the detection plane (defined in Fig. 1). As we can see in Fig. 6(a), the scattered intensity distribution undergoes an observable displacement to the center of observation plane. The numerical results obtained by the finite element method (COMSOL Multiphysics V.5.4) also verify our prediction from theoretical model, as is shown in Fig. 6(b). Our calculations based on modified Mie theory are identical to the simulations from finite element method but are more efficient and can be done on a laptop.

 

Fig. 6. (a) The intensity distribution of scattered light field in the far-field detection plane at the observation direction (${\theta _0}\textrm{ = }\pi \textrm{/2}$ and ${\phi _0}\textrm{ = }0$). The white dot represents the position of the center of the scattered light in the detection plane and ${\mu _p}$ represents the displacement to the origin of the detection plane. (b) the measured displacement ${\mu _p}$ for different graphene’s Fermi level. Other parameters are the same as those in Fig. 4.

Download Full Size | PDF

In summary, we use modified Mie theory to study the spin Hall effect of light by graphene-wrapped nanoparticles. The scattered spin Hall shifts can be dynamically tuned by the graphene coating. Specifically, the enhanced spin Hall shifts in the far-field are due to the strong spin-orbit interaction around the particles, which can be realized either by overlapping the electric and magnetic dipolar modes or by overlapping the electric dipolar and electric quadrupole modes. The interferences of these modes are modulated through the conductivity of the graphene, which could be tuned in real-time by gate voltage in experiment. This feature of tunability can be used to manipulate the spin-orbit interaction of light in optical imaging techniques.