Nonlinearity-Tuned Optical Spin-Orbit Interaction of Graphene-Wrapped Nanoparticles

Nonlinear control of light-matter interaction plays an important role in various optical phenomena, including spin-orbit interaction of light. In this paper, we demonstrate that the spin-orbit interaction of graphene-wrapped nanoparticles can be tuned by the input intensity of light. Typical optical bistable responses are observed in the scattering efficiency and the local electric fields. In the optical bistable regime, the strength of spin-orbit interaction in the near-field is significantly tuned in the switching-up and switching-down branches. The tunability of spin-orbit interaction enables promising nonlinear control schemes in nanophotonic devices.

, [16], and all-optical image edge detection [17], [18], [19]. Among them, Luo et al. have done a series of interesting research works. For example, they have proposed the first unified description of spin Hall effect of light by the spin redirection and Pancharatnam-Berry phases, and directly observed giant photonic spin Hall effect in a dielectric-based metamaterial [11]. And then, with the spin-orbit interaction of light in the photonic Dirac metacrystal, they found a giant photonic spin Hall effect near the Dirac points in 2020 [14]. Meanwhile, they have reported a full optical differentiator at a simple optical interface, and the optical differential operation is independent of the wavelength [15]. In addition, Luo et al. have proposed a mechanism of edge detection based on a highly efficient dielectric metasurface and experimentally demonstrated broadband edge detection in 2019 [18]. Recently, He et al. have successfully further reduced the complexity of edge detection systems by integrating focusing and differentiation capabilities onto a monolithic metasurface through asymmetric photonic spin-orbit interactions [19]. Various methods have been developed to manipulate the spin-orbit coupling in nanophotonic devices. For example, induced singularity by interference is proposed to enhance the SOI of light in a uniaxial slab [20]. Nanostructures with graphene are widely used to tune and control SOI as well as spin Hall effect of light [21], [22]. Indeed, doping can confer various new or improved optical, electromagnetic, and structural properties on graphene, thus greatly extending the arsenal of graphene materials and their potential for a spectrum of applications. Different dopants, doping configurations and their relative ratios, and compositions of co-dopants give graphene distinct properties [23].
Recently, nonlinear control has attracted much attention in tuning parity-time symmetry [24], photonic topological insulator [25], and optical bound states in the continuum [26]. Nonlinear spin-orbit interaction of light can offer additional degree of freedom to engineer structured light [27], [28]. In the linear optical processes, the conversion of SAM to OAM is independent of the incident light intensity. In contrast, in the nonlinear optical processes, the SOI can be controlled by tuning the input intensity of light. Nonlinearity shows great potentials in the application of optical angular momentum [29], [30], [31]. What's more, high-order nonlinear SOI is demonstrated by plasmonic metasurfaces in nonlinear waves [32].
In this paper, we study the spin-orbit interaction of graphenewrapped nanoparticles under the incidence of circularly polarized light. We develop the nonlinear quasistatic theory on graphene-wrapped nanosphere with right circularly polarized (RCP) light. Typical optical bistable responses are found in the far-field scattering and near-field intensity, which are verified by nonlinear full-wave simulation. Our results show that the SOI effect can be tuned by the input intensity in the optical bistable regime. The circular polarization degree (CPD) in the near field, which characters the conversion between SAM and OAM, demonstrates obvious different strengths for switching-up and switching-down input intensity of light. This tunability of SOI is realized by controlling the surface current density of graphene, whose nonlinear optical response occurs when the particle is in an intense incident light [33], [34], [35], [36]. Our work offers an alternative way to control the spin-orbit interaction of near-field [37]. Fig. 1 illustrates the considered graphene-wrapped nanoparticle with radius a and relative permittivity ε. For simplicity, we set the host medium as vacuum with relative permittivity ε h . In this paper, the size of nanoparticle (a = 50 nm) that we studied is much smaller than the incident wavelength (above 10 μm). Hence the quasistatic approximation (QSA) can be adopted to obtain the local fields of the nanoparticles. The electric potentials satisfy the Laplace equation: ∇ 2 φ n = 0, where the subscript n = c stands for inside the particle and n = h in the host medium. For the incidence of RCP light [38], one can derive the general solutions of the above Laplace equation as,

II. THEORETICAL MODELS
The graphene layer on the nanoparticle's surface is considered as a thin conducting shell, which contributes to the conductivity σ in the boundary conditions [39].
where J = σE ≈ σE c represents the surface current density.
Here, the operator ∇ s stands for surface divergence, and E c is the linear local electric field inside the nanoparticle.
Solving (1) and (2), one can obtain the unknown coefficients A and B, where Θ = iσ/(ωaε 0 ) and B is the dipole polarizability of the graphene-wrapped sphere.
In this paper, the nonlinear surface conductivity of graphene are simplified and described by the random-phase approximation [40], [41]σ where the simplified linear term σ g and the Kerr nonlinear surface conductivity σ 3 are shown as follows [40], [42] where e, E F , , υ F , and τ are the electron charge, Fermi energy, reduced Planck constant, Fermi velocity of electrons, and the electron-phonon relaxation time, respectively. According to quasistatic approximation, the local electric field E c inside the nanoparticle and the external field E 0 are related by Equation (6) is a nonlinear equation since it contains the nonlinear surface conductivityσ g . Hence the optical bistable responses can be achieved. In addition, we calculate the normalized scattering efficiency Q sca of the graphene-wrapped nanoparticle in the far field, where k = 2π √ ε h /λ 0 is the wave number, and α is the electric polarizability including the radiation correction with the approximate static polarizability of the graphene-wrapped nanoparticle To verify the above theoretical analysis, we also performed the numerical simulation of this geometry with the finite element method (FEM) by COMSOL Multiphysics.
III. RESULTS Fig. 2(a) shows the normalized scattering efficiency Q sca versus the incident wavelength λ. The continuous curves correspond to solutions obtained by the QSA. Specifically, the black line is the result when only linear term σ g in (4) is considered, while the red and green curves are plotted considering the nonlinear surface conductivityσ g for various values of the incident electric field: E 0 = 0.5Mev(red line), and E 0 = 1.5Mev(green line). The resonant peak in scattering efficiency curves is due to the "metallic" graphene layer in the terahertz frequencies [44].
It clearly indicates that graphene's analogous Kerr nonlinearity on conductivity will generally result in a redshift of the resonant peak and decreased strength of the resonance. This is due to the fact that when the incident light's intensity increases, the eigen wavelengths will redshift for the Kerr-type metallic nonlinearity [45]. Compared the results obtained by the FEM with those of the QSA, the results are quite similar but with a minor difference. It is due to the theoretical approximations of the quasistatic limit, which is not applied in FEM. In the FEM, when we calculate the surface current density and nonlinear surface conductivity of graphene, we use the local electric field on the graphene's surface, instead of the electric field inside the nanoparticle, i.e., J = σE local andσ g = σ g + σ 3 |E local | 2 . Interestingly, when the optical field is relatively high (E 0 = 1.5Mev), optical bistable responses are observed, with two stable branches and an abrupt switching effect at the design wavelength. Fig. 2(b) shows the optical bistable responses of E c and the incident field E 0 for both XP and RCP light within the QSA and FEM, respectively. In the appropriate range of driving electric field, bistable responses are indeed found. As can be seen that when the incident field increases from a relatively low value (i.e., E 0 = 0.8Mev), the electric field E c within the nanoparticle first increases, and then jumps to the upper stable branch at the switching-up threshold field E 0,up . After that, the electric field E c will monotonically increase with the incident field. Conversely, when the incident field decreases from a relatively high value (i.e., E 0 = 1.4Mev), E c will discontinuously jump to the lower stable branch at the switching-down threshold field E 0,down . It should be noted that the dashed line in the plot represents the calculated unstable branch which is a mathematical solution of the nonlinear problem, but it is not reachable in practice. The effects of the nanoparticle parameters, such as particle size and relative permittivity, are consistent with previous work [46], [47], with details as follows. A larger particle size or larger ε, will generally result in a lower switching threshold field accompanied by a narrower bistable region.
From Fig. 2(b), one observes that the relationships between the internal E c and incident electric fields E 0 are identical for the incidence of linear x-polarized light (XP with the form E = E 0êx e ikz e −iωt ) and circularly polarized light in the case of the QSA. This is because the polarizability of the graphene-wrapped nanoparticle is independent of the incident polarization in the quasistatic limit. In the case of the FEM, however, the results are significantly distinguished. More importantly, for both polarized waves, the threshold fields calculated based on the FEM are all smaller than the result within the QSA. We reasonably speculate that this is because the QSA only considers the inner field E c , but in fact the electric fields at the boundary are effectively enhanced which have been ignored in the calculation, as shown in Fig. 3. As expected, RCP waves require a smaller threshold field than XP waves, since they excite a stronger localized field and larger enhanced region in the surface of graphene (see the x-y plane of XP and RCP in Fig. 3(a) and (d)).
Bistability not only affects the far-field scattering efficiency described above, but also affects the near-field, such as spinorbit interaction, as is shown in Fig. 4. The inserts in the plot are the distributions of circular polarization degree (CPD) for the near-field of different positions (marked as I, II, III, and IV) in two stable branches. Among them, positions II and IV share the same strength of incident field (0.93Mev). Here, we define CPD as |ε 0 E * × E + μ 0 H * × H|/(ε 0 |E 0 | 2 + μ 0 |H| 2 ), fulfilling CPD = 1 for the circular polarization [48].  According to the characteristics of the bistable response, I→II→III and III→IV→I are through switching-up and switching-down branches, respectively. First, we focus on the process of increasing the incident field gradually. Remarkably, since both points I and II are in the lower stable branch, they yielded almost identical results. Meanwhile, the CPD drops to high values for the field in position III after the abrupt discontinuous jump at the threshold field E 0,up . The high CPD of the field in position III, as compared to the field in position II, implies that most of the OAM of the photons is converted to the SAM by means of the SOI in this region. Similarly, when the applied field is continuously decreased, the CPD first remains the same (III-IV) and then drops to low values for the field in position I after the abrupt discontinuous jump at the threshold field E 0,down . Conversely, the low CPD of the field in position I, as compared to the field in position IV, implies that most of the SAM of the photons is converted to the OAM by means of the SOI. Interestingly, the distributions of CPD in positions II and IV are quite different, although they have the same incident field intensity. This stems from the fact that the distributions and intensity of surface current density J on the surface of graphene, are completely different at these two points, as shown in Fig. 5. Since the boundary condition in (2) contains the surface current density J, when the surface current changes, the boundary condition will change accordingly. The Ohm's law J = σE connects the surface current density and the field distribution around the nanoparticles. Hence, the CPD difference can be related to the difference of graphene surface current.

IV. CONCLUSION
In conclusion, we have performed the theoretical investigations on graphene-wrapped nanosphere. We develop the nonlinear quasistatic theory with RCP incidence. As expected, typical optical bistable responses are found in the far-field scattering and near-field intensity. For comparison, we also performed a nonlinear full-wave simulation and good agreement is found. Furthermore, we study the SOI of graphene-wrapped nanoparticles under the incidence of RCP light. Our results show that the SOI effect can be tuned by the input intensity in the optical bistable regime. The CPD in the near field, which characters the conversion between SAM and OAM, demonstrates obvious different strengths for switching-up and switching-down input intensity of light. Intrinsically, this tunability of SOI is realized by controlling the surface current density of the monolayer graphene.