1. Introduction

Skyrmions, originally proposed by British physicist, Tony Skyrme in the 1960s [1], are particle-like spin textures which can form at the micro-nano scale, and which usually exhibit topological protection [2,3]. Recently it has been introduced to the realm of optics [4–14]. Surface plasmon polaritons (SPPs) have been widely employed to generate skyrmionic textures since the two-dimensional confinement of light in evanescent fields provide a smooth domain for topological textures, which can be constructed by either electric/magnetic fields or spin angular momentum. The versatile topological textures provide new degrees of freedom for shaping vectorial fields and encoding information, which has broad application prospects in ultrafast vector imaging [10], nanoscale metrology [15,16], topological hall devices [17], etc. Therefore, it is of great significance to realize flexible manipulation of optical skyrmions.

There are currently no easy ways to modulate the topologies of optical skyrmions, most of which are realized by designing different spatial structures on the substrate [18,19]. However, this method not only results in a limited-tuning range, but also a complex operating system, which hinders their utility in practical contexts. Besides, once the structure is fixed, only a specific topological state can be realized [18,20,21]: a direct transformation from one topological state to another lacks concrete transition process, which is not conducive to a clear understanding of the topological properties of optical skyrmions and their application in practice. Graphene plasmon (GP) [22,23], as a kind of surface wave, is essentially the collective oscillations of free Dirac electrons in graphene coupling to electromagnetic fields and has attracted a great deal of attention owing to their strong field confinement, short wavelengths and continuous electrical tunability [24–26]. GP can not only be regulated by the nanostructure deposited on the graphene, but also impressed voltage based on the material properities, which is not available in other metal surface plasmon [27–29].

In this paper, we constructed GP skyrmions based on the interaction between circular polarized light and graphene hexgonal structure. Dynamic regulation of optical skyrmions by both the material property and excitation beams was proposed. Numerical simulations are performed to verify the proposed technique. In section 2, the principle of GP skyrmions and topological transformation by continuity regulation by changing the excitation beams and the Fermi energy of graphene is discussed. In section 3, the theoretical analysis and full-wave electromagnetic simulations were carried out to demonstrate the feasibility of dynamic regulation of GP skyrmions based on vector field and the electrical tunable properties of graphene. Finally, the main conclusions are summarized in section 4. Our work opens an avenue for a continuous change of the topology of optical skyrmionic states and is promising for application in integrated photoelectric devices, including optical sensors and light modulators.

2. Working principle

The schematic diagram of continuous manipulation of optical skyrmions based on the electrical tunable properities of graphene is shown in Fig. 1(a) and Fig. 1(b). Six gratings creating a hexagon are deposited on the graphene layer which is placed on the Si substrate with a 300 nm SiO$_{2}$. Each pair of the parallel slits create a graphene plasmon standing wave that travels along the surface of the graphene as shown in Fig. 1(a). Therefore, the field at the center of the hexagon is the superposition of three pairs of the graphene plasmon standing waves passing through the slits. Changing the Fermi energy of graphene by applying gate-voltages, the graphene plasmon standing waves can be regulated as shown in Fig. 1(b). The z-component of the GP along the $x-y$ plane can be expressed as:

 

Fig. 1. Schematic diagram of dynamic manipulation of optical skyrmions by changing the Fermi energy of graphene by gate-voltages. (a) Circular polarized light with a wavelength of $\lambda _{0}$=10.653 $\mu$m is vertically incident on a graphene structure composed of three pairs of parallel slits, which numbered by 1, 4; 2, 5; and 3, 6, respectively. (b) Back-gate tunable the Fermi energy of a pair of parallel graphene slit. (c),(d) Néel-type optical skyrmions formed by GP standing waves at the center of the graphene hexagonal slits before and after the Fermi energy change.

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Therefore, the field in the center of the hexagon graphene slits can be regarded as the superposition of standing waves with a azimuth angle of -$\frac {\pi }{3}$, $0$, $\frac {\pi }{3}$, respectively, which are formed by three pairs of parallel graphene slits numbered by 1, 4; 2, 5; 3, 6 in Fig. 1(a) . The $z$ component of electric field at the center can be expressed as [30]:

The transverse electric field components satisfy the following form, which can be derived from Maxwell’s equations:

It is well known that, the topological properties of a skyrmionic configuration can be characterized by the skyrmion number (topological invariant) $S$, which can be defined as [30,31]:

Here, $\vec {e}$ is a real, normalized, three-component field; the area A covers one unit cell of the lattice as shown by the dotted boxes in Fig. 2(b) and Fig. 2(k); $x$ and $y$ are directions in the 2D plane. The skyrmion number $S$ is robust to deformations of the field $\vec {e}$ as long as $\vec {e}$ remains nonsingular and maintains the periodicity of the lattice [30].

 

Fig. 2. Calculated electric field distribution of GP skyrmions as changes the Fermi energy of one pair of the GP standing wave. (a) The dispersion relation of the GP as $E_{F14}$=0.20$\sim$0.38 eV. And $\theta _{14}$=$\theta _{25}$ =$\theta _{36}$=0, $E_{F25}$=$E_{F36}$=0.2 eV. The blue circles represent the data of the simulation result. (b)-(k) Axial electric field distribution at different Fermi energy, according to Eq. (2). The color scale indicates the value of $E_{z}{/}\left | E \right |$ and the dotted area in Fig. 2(b) and Fig. 2(k) represent one unit cell of the optical skyrmions lattice.

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From Eq. (2) and Eq. (3), it can be shown that the intensity of the GP standing wave at the center of the hexagonal slits can be controlled by changing the wave vector $k_{GP}$ of the GP. The dispersion relation of GP can be obtained by calculating the intrinsic vibration solution of passive electromagnetic waves at the interface of graphene. The expression satisfies the following form:

Here, $\varepsilon _{1}$, $\varepsilon _{2}$ are the dielectric constants of the top and bottom interface mediums of graphene, $k_{0}$ is the wave vector in vacuum and $\sigma$ is the conductivity of graphene, which satisfies [32]:

Therefore, when the excitation beams are fixed, the wave vector of the GP is related to the conductivity of graphene, which is the function of the Fermi energy. The Fermi energy $E_{F}$ in graphene with respect to the Dirac point can be easily tuned by changing the carrier concentration $n_{s}$ with $E_{F}=\hbar {\nu _{F}}\sqrt {\pi {n_{s}}}$ [25,26], proving an efficient method for active control of GP. Where $\nu _{F}=1\times {10}^{6}$m/s is the Fermi speed, $n_{s}$ is the carrier concentration. Figure 1(b) illustrates the back-gate configuration, where the voltage is applied between the Si substrate and the Au electrode connected to the graphene layer. The Au electrode is placed far away from the graphene slit structures to avoid the influence on the GP. In this way, the wave vector of each GP’s standing wave can be changed to control the superposition field distribution. Figure 1(c) and Fig. 1(d) is the vector representation of the local unit vector of the electric field before and after the Fermi energy changes in one of the three pairs of graphene slits, showing that a transition of skyrmion topology has taken place.

On the other hand, when the Fermi energy of the three pairs of graphene slits is fixed, the phase difference of each pair of the standing wave can be controlled by changing the phase and polarization of the excitation beams of the GP wave [26,33]. Considering the phase difference of the standing wave, the expression of the electric field distribution at the center of the graphene slits can be described as follows:

Here, $\phi _{\theta }$ is the phase difference of the GP standing wave formed by a pair of parallel slits with the azimuth of $\theta$. In order to obtain the maximum coupling effect, the polarization of the excitation beams need to be perpendicular to the slits. When a pair of the parallel graphene slits are excited by two excitation beams with zero phase and orthogonally polarized, the slits act as the reflection boundaries and the plasmonic wave propagates along opposite directions with a phase difference of $\pi$. Therefore, the GP standing wave can be formed by two separate parallel graphene slits, which interfere with each other in the middle region. So, the intensity at the center of the GP standing wave depends on the phase difference between two GP waves, which can be controlled by changing the phase and polarization of the excitation beams. Experimentally, the phase difference of the graphene standing wave can be obtained by shifting the distance from the excitation wave to the graphene slits.

3. Simulation and discussion

For convenience, the Fermi energy and the phase difference of the three standing waves, which formed by the parallel graphene slits in Fig. 1(a) are represented by $E_{F14}$, $E_{F25}$, $E_{F36}$, $\theta _{14}$, $\theta _{25}$ and $\theta _{36}$, respectively. Considering that the six excitation beams have the same phase and polarization, that is, $\theta _{14}$=$\theta _{25}$=$\theta _{36}$=0, we calculated the change of the optical skyrmions electric field at the center of the graphene hexagon when the Fermi energy $E_{F14}$ changes from 0.2 eV to 0.38 eV. Based on Eq. (6) and Eq. (7), the dispersion diagram of the GP wave as the changes of the Fermi energy is shown in Fig. 2(a) and the wavelength of excitation light is $\lambda _{0}$=10.653 $\mu$m. It can be shown that, with the increase of the Fermi energy, the real part of the wave vectors of the GP decreases gradually and the wave vector of $E_{F}$=0.38 eV is approximately half of that of $E_{F}$=0.20 eV shown as the blue circle in Fig. 2(a). Substituting the wave vectors of different Fermi energy into Eq. (2) and Eq. (3), the normalized results of the axial (out-of-plane) electric field were obtained by numerical analysis as shown in Fig. 2(b)-(k). Here, $E_{F25}$=$E_{F36}$=0 and $E_{F14}$ is continuously changed from 0.20 eV to 0.38 eV with a step of 0.02 eV. The color scale indicates the value of $E_{z}^{(\omega )}/\sqrt {(E_{x}^{(\omega )})^{2}+(E_{y}^{(\omega )})^{2}+(E_{z}^{(\omega )})^{2}}$.

The axial electric field has the form of a hexagonally symmetric lattice as the standing wave of the three pairs of parallel slits have the same Fermi energy $E_{F14}$=$E_{F25}$=$E_{F36}$=0.20 eV as shown in Fig. 2(b). Keeping $E_{F25}$=$E_{F36}$=0.20 eV unchanged and gradually increases $E_{F14}$ with a step of 0.02 eV, the shape of the optical skyrmions at the center along the vertical direction changes gradually from circle to square as shown in Fig. 2(c)-(k). At the same time, the shape of the optical skyrmions in the left and right areas along the vertical direction changes from circle to equilateral triangle and then the base of the equilateral triangles shrinks gradually to an isosceles triangle until it becomes a rectangle. To further illustrate the change of the topological states of GP skyrmions, the skyrmion number $S$ of each site was calculated based on Eq. (4) and Eq. (5). For the axial electric field has the form of a hexagonally symmetric lattice (the white hexagon dotted box in Fig. 2(b)) with skyrmion number $S$ of 0.9934 ($\approx$1) corresponding to $E_{F14}$=$E_{F25}$=$E_{F36}$=0.20 eV, while for that of the axial electric field which has the form of a rectangle lattice (the black rectangular dotted box in Fig. 2(k)), we attain a skyrmion number $S$ of 0.5097 ($\approx$0.5) corresponding to $E_{F14}$=0.38 eV, $E_{F25}$=$E_{F36}$=0.20 eV. This shows that the continuous manipulation of optical skyrmions topological states can be achieved by changing the Fermi energy of GP.

On the other hand, the Fermi energy of the three GP standing wave can be kept the same, here, $E_{F14}$=$E_{F25}$=$E_{F36}$=0.20 eV, the optical skyrmion’s electric field at the center of the graphene hexagon was simulated when the phase difference of one of the three standing waves were changed. Supposing that the excitation waves of the two pairs of the graphene slits with azimuths of -$\pi$/3 and $\pi$/3 have the same phase and polarization, that means $\theta _{25}=\theta _{36}$=0, while the other pair of graphene slit with azimuth of 0 has the same polarization but the phase difference of $\pi$. Therefore, the phase difference of the graphene hexagon is $\theta _{14}=\pi$. The electric field distribution at the center of the graphene hexagon structure satisfies Eq. (8) and Eq. (9). Therefore, the dynamic manipulation of the optical skyrmion’s electric field can also be achieved by changing the phase of the incident vector light.

The vector representation of the local unit vector of the electric field at the center of the graphene hexagon slits when changes the Fermi energy of graphene slits or the phase difference of the excitation wave as shown in Fig. 3. It is clear that symmetrical hexagon optical skyrmion appears in the electric field corresponding to $E_{F14}=E_{F25}=E_{F36}$=0.20 eV, $\theta _{14}=\theta _{25}=\theta _{36}$=0 of the three GP standing waves as shown in Fig. 3(a) with the three-dimensional electric field vector distribution at the bottom forming a N$\acute {\text {e}}$el-type skyrmion [34,35]. The direction of the vector field is from the outside to the inside and from bottom to top which is indicated by the direction of the arrows in Fig. 3. The length and the color of arrows indicate the field intensity. Then, only changed the Fermi energy of the GP standing wave with azimuthal angle of 0 to 0.38 eV, while others remains unchanged compared to Fig. 3(a). Hence, $E_{F14}$=0.38 eV, $E_{F25}=E_{F36}$=0.20 eV, $\theta _{14}=\theta _{25}=\theta _{36}$=0 and the shape of the optical skyrmions changes from circle to square shown in Fig. 3(b) with the three-dimensional electric field vector distribution at the bottom forming a N$\acute {\text {e}}$el-type skyrmion.

 

Fig. 3. Calculated electric field distribution of the optical skyrmion lattice when changing the Fermi energy $E_{F14}$ and the phase difference $\phi _{14}$, while $E_{F25}=E_{F36}=$0.20 eV, $\phi _{25}=\phi _{36}=0$ keeping unchanged. (a)-(d) Normalized $z$ component of electric field, according to Eq. (2) and Eq. (3) when (a) $E_{F14}=$0.20 eV, $\phi _{14}=0$; (b) $E_{F14}=$0.38 eV, $\phi _{14}=0$; (c) $E_{F14}=$0.20 eV, $\phi _{14}=\pi$; (d) $E_{F14}=$0.38 eV, $\phi _{14}=\pi$, respectively. The vector distributions of the electric field correspond to the intensity distribution in the bottom of (a)-(d) respectively.

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Using Eq. (4) and Eq. (5), the calculated skyrmion number of each unit cell’s of Fig. 3(a) and Fig. 3(b) is $S\approx$1 and $S\approx$0.5, respectively. And it further illustrated that the continuous change of the optical skyrmions topological states can be achieved based on the electrotunable properities of GP waves. Similarly, the vector representation of the local unit vector of the electric field and the intensity distribution at the center of the graphene hexagon slits was also calculated when only changed the phase of the excitation beams while the Fermi energy of the three GP standing waves was fixed at 0.2 eV. Changing the phase difference of $\theta _{14}$ from 0 to $\pi$, the axial electric field distribution of the optical skyrmions at the center of the graphene hexagon was shown in Fig. 3(c) and the three-dimensional electric field vector distribution at the bottom form a N$\acute {\text {e}}$el-type skyrmion, where $\theta _{25}=\theta _{36}$=0 and $E_{F14}=E_{F25}=E_{F36}$=0.20 eV. Compared with Fig. 3(a), the added phase difference of the GP standing wave not only caused the move of the optical skyrmions along the vertical direction, but also the direction of the electric field vector distribution become opposite as shown at the bottom of Fig. 3(c). Using the same method as above, the calculated skyrmion number of each unit cell’s of Fig. 3(c) is $S=-1.01$ ($\approx$-1). It proved that it is feasible to regulate the topological states of GP skyrmions by changing the phase difference of the excitation beams.

In addition, we further calculated the axial electric field distribution of the optical skyrmions at the center of the graphene hexagon slits while simultaneously changing the Fermi energy and the phase difference of one of the GP standing waves. The optical skyrmions electric field of the graphene hexagon slits when $E_{F14}$=0.38 eV, $E_{F25}=E_{F36}$=0.20 eV, $\phi _{14}=\pi$, $\phi _{25}=\phi _{36}=0$ is shown in Fig. 3(d). Compared to Fig. 3(c), Fig. 3(d) added the phase difference of $\pi$ of the GP standing wave at azimuth of 0. The shape of the optical skyrmions is similar to that in Fig. 3(b), but the position of a skyrmion in each unit cell of a lattice had shifted, besides, the direction of the vector distribution of the electric field was flipped. And the calculated skyrmion number of each site of Fig. 3(d) is $S=-0.5076$ ($\approx$-0.5). The change of the skyrmion number shows that the topological states of GP skyrmions can be changed by changing the Fermi energy of the GP and the excitation beams.

To further understand the dynamic manipulation of optical skyrmions enabled based on the electrotunable properities of GP and the tunable properties of the phase difference of the excitation beams, full-wave electromagnetic simulations is adopted to solve the Maxwell equations by using the commercial software COMSOL Multiphysics based on finite element method (FEM). The configuration of the calculation is shown in Fig. 1(a), the single layer graphene is placed above SiO$_{2}$(300 nm)/Si substrate. Six slits of 1.5 $\mu$m length and 0.05 $\mu$m width were set in the graphene layer with the permittivity of 1. The distance between a pair of parallel graphene slit is 3.2 $\mu$m. Here, graphene is modeled as a thin film with the thickness of 0.34 nm and the in-plane conductivity of the graphene is computed within the local-random phase approximation (RPA) [36,37]. The carrier mobility of the graphene is $\mu =8000\,cm^{2}/Vs$ [38] and ambient temperature is set as T=300 K. The permittivity of SiO$_{2}$ was taken from [39]. In the simulation, circular polarized light with the wavelength of $\lambda _{0}$=10.653 $\mu$m is normal-incidence in the graphene hexagon structure. The near-field signals were recorded as $E_{z}$ at the $x-y$ plane, 20 nm above the graphene. The optical skyrmions lattice is shown in Fig. 4 by plotting the normalized axial electric field distribution at the center of the hexagon graphene slits structure with size of 3$\lambda _{GP} \times$ 3$\lambda _{GP}$ at 20 nm above the graphene surface.

 

Fig. 4. The intensity distribution of optical skyrmions electric field based on the full-wave electromagnetic simulations. (a) $E_{F14}=E_{F25}=E_{F36}=0.40\,eV, \phi _{14}=\phi _{25}=\phi _{36}=0$. (b) $E_{F14}=0.20\,eV, E_{F25}=E_{F36}=0.40\,eV, \phi _{14}=\phi _{25}=\phi _{36}=0$. (c) $E_{F14}=E_{F25}=E_{F36}=0.40\,eV, \phi _{25}=\pi, \phi _{14}=\phi _{36}=0$. (d) $E_{F14}=0.20\,eV, E_{F25}=E_{F36}=0.40\,eV, \phi _{25}=\pi, \phi _{14}=\phi _{36}=0$.

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Using the same circular polarized light to excite the graphene hexagonal slits with the Fermi energy of 0.40 eV, that is, $E_{F14}=E_{F25}=E_{F36}=$0.40 eV, $\phi _{14}=\phi _{25}=\phi _{36}$=0 and the optical skyrmions electric field distribution at the center of the structure as shown in Fig. 4(a). In the actual simulation, to compensate for the phase of circular polarization at different locations, the graphene hexagonal structure is moved outward by a distance of $n\cdot \pi /6$, $n=1, 2, 3, 4, 5$, respectively, which is counterclockwise from the second slit. It can be shown that, the result is consistent with the analytical results obtained by Eq. (2) and Eq. (3) shown in Fig. 3(a). Moreover, the optical skyrmions topological states were also calculated when only changed the Fermi energy or the phase difference of the GP standing wave. Figure 4(b) and Fig. 4(c) corresponding to the optical skyrmions axial electric field distribution as only changed $E_{F14}$ from 0.40 eV to 0.20 eV, $E_{F25}=E_{F36}$=0.40 eV, $\phi _{14}=\phi _{25}=\theta _{36}$=0 and only changed $\theta _{14}$ from 0 to $\pi$, while $E_{F14}=E_{F25}=E_{F36}$=0.40 eV, $\theta _{25}=\theta _{36}$=0. In addition, changing the Fermi energy and the phase difference simultaneously, the distribution of the optical skyrmions axial electric field was shown in Fig. 4(d) with the three-dimensional electric field vector distribution at the bottom forming a N$\acute {\text {e}}$el-type skyrmion, where $\phi _{14}=\theta _{36}$=0, $\theta _{25}=\pi$, $E_{F25}=E_{F36}$=0.40 eV and $E_{F14}$=0.20 eV. By comparing the distribution of optical skyrmion electric fleld before and after the change of Fermi energy and the phase difference, we can get the following conclusions: the shape of the optical skyrmions remains unchanged, but the direction of the vector electric field becomes opposite when only change the phase from 0 to $\pi$ of the GP standing wave; the shape of the optical skyrmions changes from a circle to a square when only change the Fermi energy from 0.40 eV to 0.20 eV of the GP standing wave; in addition, when the phase difference and the Fermi energy changes simultaneously, the shape and the direction of the vector electric field of the optical skyrmions will change, which is equivalent to the superposition of the corresponding optical skyrmions distribution when changing the Fermi energy and the phase difference independently. These results are consistent with the theoretical analysis above.

4. Conclusion

In summary, we proposed the dynamic regulation of optical skyrmions topological states continously based on the electrical tunable properties of graphene. The GP skyrmions topological states is constructed by the interaction between incident circular polarization light and the graphene hexagonal structure. Therefore, the GP skyrmions topological states can be regulated from two aspects of the excitation beams and the graphene material, respectively. By adjusting the step of the Fermi energy changes of graphene, the continuous change of the topological states of the optical skyrmions can be achieved. Moreover, combining the optical regulation method based on excitation beams with the electrical regulation method based on graphene materials can realize the mutual conversion and the continuous regulation between different topological states of the optical skyrmions, which greatly broadens the scope of regulation. Our work demonstrates that it is feasible to achieve the continous regulation of optical skyrmions based on the electrotunable properities of graphene and increases the degree of freedom of regulation, which provides a new idea for future integrated photonics devices.