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Abstract
In this paper, the magnetically tunable and enhanced photonic spin Hall effect (PSHE) of reflected light beam at terahertz frequencies is achieved by using a multilayer structure where anisotropic graphene is inserted. This enhanced PSHE phenomenon results from the excitation of surface plasmon polariton (SPP) at the interface between two dielectric materials. By considering the 4×4 transfer matrix method and the quantum response of graphene, the PSHE of the reflected light can be enhanced by harnessing the anisotropic conductivity of graphene. Besides, the PSHE can be tuned through the external magnetic field and structural parameters. This enhanced and tunable PSHE approach is promising for fabricating anisotropic graphene-based terahertz spin devices and other applications in nanophotonics.
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1. Introduction
When a linear polarization beam propagates through media and is reflected or transmitted at the interface, the left-handed and right-handed circularly polarized components of the light will split in opposite directions perpendicular to the incident plane, and this called photonic spin Hall effect (PSHE) [1–3]. Due to the unique physical properties of PSHE at the interface of reflection or transmission, it has been widely used in the field of micro/ nano photonics. Relevant researchers proposed many applications about PSHE. Such as identifying the graphene layer numbers [4], precise measurement of optical conductivity of atomically thin crystal [5], multichannel switch [6], and detection of chemical reaction rate [7]. Exactly, the enhancement and amplification of spin shifts are not very important. On the one hand, small beam spin shifts are also used in optical differential calculation [8] and image edge detection [9]. On the other hand, weak measurement technology also provides a way to amplify and enhance the spin transverse displacement. However, it is worthwhile finding enhanced and even directly observable beam spin transverse displacement, because the equipment required for weak measurement technology is relatively complex. In recent years, various methods for enhancing PSHE have been proposed. For example, Wan et al. proposed a Kretschmann model based on N-type coherent atomic medium, which enhanced by excitation of surface plasmon resonance [10]; Wang et al. investigated the PSHE occurring on the surface of one-dimensional photonic crystal and found that the maximum spin-dependent transverse displacement is 37.94 times of incident wavelength [11]; In addition, Jiang et al. based on the sandwich structure of Epsilon-near-zero material greatly enhanced the spin displacement of light by generating the bound states in the continuum [12]. And there are also in-depth investigation and experimental observations on the optical spin Hall effect based on the weak measurement method. Qin et al. observed the optical spin Hall effect of the beam at the air-glass interface with weak measurement method, and the spin splitting reached several hundred nanometers [13]; Luo et al. found that a large optical spin shift of 3200 nm was generated when the beam was reflected near Brewster angle [14]. Recently, they also measured ion concentration by PSHE experimental setup [15]. Although many ways to enhance the PSHE have been continuously proposed, but seeking an enhanced and easily controlled PSHE approach is still worthwhile trying.
For the past few years, two-dimensional materials, represented by graphene, have attracted extensive attention in the field of optoelectronics due to their excellent photoelectric characteristics [16]. It is known that graphene possesses dynamically controllable conductivity [17], nonlinear optical properties [18], and high electron mobility [19]. These features have attracted great attention in the field of controllable and enhanced PSHE [20]. For example, Lin et al. investigated the PSHE on a monolayer of black phosphorus. Due to the in-plane anisotropic property of black phosphorus, the PSHE is accompanied with Goos-Hänchen and Imbert-Fedorov effects, resulting in an asymmetric spin splitting [21]; Dong et al. enhanced and controlled the PSHE of graphene-dielectric structure at the terahertz range by modulating graphene conductivity via weak optical pumping [22]; Su et al. proposed the PSHE of type II hyperbolic metamaterials is achieved due to near filed interference, which provides a way to decide the propagation direction of subwavelength beam [23]. Furthermore, people found that the surface conductivity of anisotropic graphene can be controlled by the uniform magnetic field vertically applied to this graphene [24,25]. This gives us a flexible approach to design elastic and controllable spin optoelectronic devices based on graphene. Graphene can effectively excite surface plasmon polariton (SPP) [26–28], and SPP has strong local field enhancement properties. So, we are considering the possibility of enhancing SPP-based PSHE through external magnetic field manipulation in the terahertz range. In this paper we excite SPP by combining anisotropic graphene with a typical Otto structure. On the one hand, the enhancement of spin displacement is realized based on the local field enhancement effect of SPP, and on the other hand, the dynamic adjustment of optical spin displacement is realized through the change of external magnetic field. The magnetic field can be directly added outside the device and thus avoid direct contact with the device when adjusting the PSHE. This is a distinct advantage of the proposed method compared with electronic control. In addition, we also pay attention to the influence of graphene Fermi energy, air layer thickness and other parameters on spin displacement. By optimizing these parameters, we obtain a spin displacement of 16 times of the incident wavelength which provides a new idea for the enhancement and modulation of PSHE.
2. Theoretical model and method
We consider a multilayer composite structure consisting of a coupled prism, a layer of graphene and a substrate, as shown in Fig. 1. It is assumed that the coupling prism and substrate of the structure are composed of Si and SiO2 respectively, and a uniform magnetic field is applied along the positive z axis. It should be pointed out that although the layered nanostructure in this work has relatively strict requirements on processing and transfer processes, as well as preparation of terahertz source and strong magnetic field, the current technology can basically support the realization of this scheme. Here, we set the permittivity of Si, air and SiO2 as ${\varepsilon _1} = 12$, ${\varepsilon _0} = 1$, and ${\varepsilon _2} = 4$, respectively. The thickness of air layer is $d = 5\textrm{ }\mu m$.
When the applied uniform magnetic field B acts vertically on the graphene layer along the z-axis, the surface conductivity $\bar{\sigma }$ of graphene can be written as 2×2 in the form of a tensor [24,25]:
Here, $\Re = {n_F}({{M_n}} )- {n_F}({{M_{n + 1}}} )$, $\Psi = {n_F}({ - {M_{n + 1}}} )- {n_F}({ - {M_n}} )$, $\Theta = {e^2}v_f^2|{eB} |/i\pi$, $\chi = {M_{n + 1}} - {M_n}$, and ${n_F}(\omega )= 1/1 + \textrm{exp} [{({\hbar \omega - {E_F}} )/{\kappa_0}T} ]$ is the Fermi-Dirac distribution, ${M_n} = \sqrt {{\Delta ^2} + 2n|eB|\hbar v_f^2}$ is the Landau level energy with the Landau level index n. $\omega$, ${v_f}$, e, c and $\hbar$ represent the incident light frequency, Fermi velocity, electron charge, light velocity in vacuum, and reduced Planck constant, respectively. ${E_F}$, $\Delta $, $\Gamma $, $T$ and $B$ are Fermi energy, excitonic gap, scattering rate, temperature and external magnetic field intensity, respectively. Next, through a the $4 \times 4$ transmission matrix D, we can connect the field coefficients of the left-side and right-side media on the single-layer graphene in the proposed structure. Consequently, ${D_{1,2}}$ and ${D_{2,3}}$ are respectively as follows:
Here, ${k_{0z}} = \sqrt {{\varepsilon _0}} \omega \cos {\theta _i}/c$ represents the component of the wave vector of the incident light in the vacuum along z direction, ${k_{1z}} = \sqrt {{\varepsilon _1}} \omega \cos {\theta _i}/c$ and ${k_{2z}} = \sqrt {{\varepsilon _2}} \omega \cos {\theta _i}/c$ represent the z-direction components of the wave vector of incident light in the prism and $\textrm{Si}{\textrm{O}_\textrm{2}}$ respectively, $\omega$ is the angular frequency. When the incident light propagates in homogeneous medium (i.e., air), the propagation matrix in our proposed structure be expressed as:
Next, from the transfer matrix M of the entire structure, we can get the reflection coefficient as follows:
Therefore, the reflectivity for p-polarization (horizontal (H) polarization) and s-polarization (vertical (V) polarization) can be expressed as follows:
We use the angular spectrum theory to calculate the PSHE and assume that the incident light is a linearly polarized Gaussian beam, then, its angular spectrum expression is:
Accordingly, the centroid displacements of H polarization light can be defined as:
3. Results and discussions
In this part, we will discuss the spin characteristics of the above structure in detail. Before investigating the influence of incident light frequency on spin displacement, we first evaluate the influence of different incident light frequencies on the reflectance of the entire structure for p and s-polarization, as shown in Figs. 2(a) and 2(b). In the two polarization of the incident light, the graphene SPP wave vector and the incident light wave vector are coupled to excite the graphene SPP, which then lead to a dip in reflectivity at a specific angle. With the increase of incident light frequency, both p-polarization and s-polarization reflection peaks are generated at larger angles. For example, when $f = 1.9\textrm{ THz}$, the reflection peak is generated at $49.3^\circ$. When $f = 2.2\textrm{ THz}$, the angle of the reflection peak increases to $54.0^\circ$. And it can be seen that the reflection peak is reduced more drastically for p-polarization. Figures 2(c) and 2(d) shows the variation of spin displacement with incident angle at different incident light frequencies in the structure. The optical spin displacement has an obvious enhancement phenomenon, and the angle at which the peak spin displacement is generated is consistent with the angle at which SPPs are excited. According to Figs. 2(a) and 2(b), this is due to the rapid changes in reflection coefficients toward 0 for p-polarization, and these reflection coefficients also change at the angle of SPP excitation for s-polarization. It can be seen from Eq. (14) and Eq. (15) that the drastic changes of the reflection coefficients will inevitably have a great influence on the spin displacement. The above results show that when the incident light frequency is 2 THz, the maximum optical spin transverse displacement approximates $16\textrm{ }\lambda$. Therefore, we simply proceed from theoretical research and do not consider the practical light source acquisition, we select the incident light frequency for subsequent research is 2.0 THz.
Fig. 2. The reflectivity changes with incident angle when frequency is 1.9 THz, 2.0 THz, 2.1 THz, and 2.2 THz (a) p-polarization state and (b) s-polarization state. Spin transverse displacement at four different incident frequencies (c) left-handed component and (d) right-handed.
In order to further investigate the method of adjusting the spin transverse displacement when the incident light frequency is 2 THz, we first consider the influence of the intrinsic parameter Fermi energy of graphene on SPP. As shown in Figs. 3(a) and 3(b), the angle of SPP excitation can be effectively controlled by adjusting the Fermi energy. It can be seen that with the increase of Fermi energy, the reflectivity peaks for p-polarization and s-polarization are generated at a smaller angle. For example, when ${E_F} = 0.60\textrm{ eV}$, the reflectivity peak appears at $57.5^\circ$, while when ${E_F} = 0.75\textrm{ eV}$, the angle of the reflectivity peak appears drops to $48.2^\circ$. It can be seen from Figs. 3(c) and 3(d) that the influence of Fermi energy on the change of the spin displacement peaks for left and right-handed is the same as that on reflectivity. This phenomenon indicates that Fermi energy can modulate optical spin displacement, and when ${E_F} = 0.70\textrm{ eV}$, the value of left and right optical spin displacement reaches the maximum. The rapid downward peak resulting from the spin transverse displacement of light in H-polarization is attributed to the dramatic change in reflectivity caused by the excitation of graphene SPP. Therefore, we can control and manipulate the spin displacement through modulating the Fermi energy of graphene. And the modulation of Fermi energy can be achieved by applying an external voltage to graphene. We know that the relationship between the Fermi energy ${E_F}$ and applied external electric field ${V_g}$ can be expressed as [30]: ${E_F} = \hbar {v_f}\sqrt {\eta \pi |{{V_g} + {V_{dirac}}} |}$. Where $\eta$ is derived from the single capacitor model, ${V_g}$ is offset bias which reflects graphene’s doping and its impurities. So, the modulation of Fermi energy ${E_F}$ can be achieved by applying an external voltage ${V_g}$ to graphene, thereby leading to voltage-controlled PSHE.
Fig. 3. The dependence of reflectance on incident angle in (a) p-polarization state and (b) s-polarization state at different Fermi energies. The dependence of displacements on incident angle for (c) left-handed component and (d) right-handed component at different Fermi energies, where ${E_F} = 0.60\textrm{ eV}$, $0.65\textrm{ eV}$, ${E_F} = 0.70\textrm{ eV}$ and ${E_F} = 0.75\textrm{ eV}$.
Next, we discuss the external magnetic field effect on PSHE. According to Eq. (2), we can find that the external magnetic field play an important role in the conductivity of graphene. So it is necessary to discuss the influence of external magnetic field on spin splitting. First of all, we draw the change of reflectivity with incident angle for p and s-polarization, as shown in Figs. 3(a) and 3(b). It can be seen that when the magnetic field intensity is 0 and the incident angle is greater than the critical angle, the reflected wave does not produce downward peak and the reflectivity is close to 1. It is indicates that the reflected light has total reflection at the interface, On the contrary, when adding an external magnetic field, we can see that the reflected wave produce an obvious falling peak at different incident angles. Moreover, with the increase of the magnetic field intensity, the peak reflectivity for the two polarization occurs at a lower angle, and different magnetic field intensities also have a great influence on the peak reflectivity. This shows that the external magnetic field plays an important role in the excitation of plasmons on the surface of magnetostically biased single graphene surrounded by different media [28]. For magneto plasmonic nanostructures, the excitation coupling between SPP and local surface plasmon will cause the resonance drop of reflection spectrum [31], which leads to the change of PSHE change under the action of external magnetic field. Furthermore, we can see from Figs. 4(c) and 4(d) that, within the magnetic field range of 0–5 T, the intensity of external magnetic field plays an important role in modulating the PSHE and the maximum optical spin displacement is close to 16 times of incident wavelength.
Fig. 4. The dependence of reflectance on incident angle in(a) p-polarization state and (b) s-polarization state at different magnetic field intensities, where $B = 0\textrm{ T}$, $\textrm{1 T}$, $\textrm{3 T}$, $\textrm{5 T}$. And the dependence of displacements on incident angle for(c) left-handed component and (d) right-handed component at different magnetic field intensities.
Next, in order to better understand the influence of magnetic field intensity on PSHE, we discuss the effect of different magnetic field intensities on the normalized electric field of H-polarized reflected light, as shown in Fig. 5. In order to further explore the influence of external magnetic field on PSHE, we draw the normalized electric field distribution of the reflected light for different magnetic fields. According to the results of Figs. 4(a) and 4(b) above, the optical spin displacement is also weak at the magnetic field intensity of B = 0 T, because the SPP of graphene cannot be excited. In Fig. 5(a), we can see that when B = 0 T, the centroid of the reflected beam hardly moves. Figures 5(b) and 5(c) shows that when B = 1 T and B = 3 T, the centroid of the reflected beam obviously moves to a negative value, which is consistent with the spin displacement in Figs. 4(c) and 4(d). The magnetic field can help us get a large spin transverse displacement, because magnetic field excited the SPPs at the reflected interface. And the SPP can enhance the reflected light electric field intensity, make the centroid of the reflected beam occur lager spin transverse displacement. According to the result, we can enhance and tune the PSHE through adjusting magnetic field intensity besides external electric field. Compared with the electric field control the spin displacement, magnetic field is more advantageous, because the magnetic field avoid direct contact with the proposed structure. Therefore, we believe that the approach can provide potential applications for us to design controllable optical spin devices.
Fig. 5. The distribution of normalized electric field of thereflected light electric field intensity for three different magnetic field intensities: (a)$B = 0\textrm{ T}$, (b)$\textrm{B = 1 T}$, (c)$\textrm{B = 3 T}$.
We know that the deviation in the thickness of each layer is inevitable in the manufacture of the nanostructure. The deviation would lead the experimental results of spin-dependent displacement are completely different from theoretical predictions. Therefore, it is very necessary to grasp the modulation relation of thickness to PSHE. At last, we find that the structural parameter of the multilayer structure-air layer thickness d also has an important influence on PSHE. As shown in Fig. 6, we found that the thickness of the air layer in this structure can also control the optical spin displacement and we set ${E_F} = 0.70\textrm{ eV}$ and $B = 1\;T$. It can be seen that with the increase of the air layer thickness, the optical spin displacement peak occurs at a smaller angle. For example, when the air layer thickness $d$ increases from $5\textrm{ }\mu \textrm{m}$ to $8\textrm{ }\mu \textrm{m}$, the angle at which the optical spin displacement peak occurs decreases by $2.8^\circ$, and the optical spin displacement is the largest when the air layer thickness satisfies $d = 5\;\mu \textrm{m}$.This is because the thickness of the air layer directly affects the excitation of SPP, which makes the thickness of the air layer become very sensitive to the optical spin characteristics. Therefore, the relation of the air layer thickness to the optical spin displacement, which provides an important reference for us to design a reasonable optical spin device.
Fig. 6. The change of displacements on incident angle for (a) left-handed component and (b) right-handed component for air layer of different thicknesses.
4. Conclusion
In conclusion, we have theoretically studied the enhancement and easier control of PSHE by external magnetic field at terahertz range through a prism coupling structure composed of anisotropic graphene and SiO2. Due to the excitation of the graphene SPP at the interface of the two dielectric materials, the obtained spin displacement is nearly 16 times of incident wavelength. Through the adjustments of the graphene Fermi energy, the intensity of the external magnetic field and the structural parameters, we found that PSHE can be effectively controlled. In addition, the influence of external magnetic field on SPP and the intensity of reflected light electric field are theoretically analyzed. We believe that enhancing and modulating the PSHE by external magnetic field will provide us a new way to design more convenient spin devices.
Funding
National College Students Innovation and Entrepreneurship Training Program (202110542014); Scientific Research Fund of Hunan Provincial Education Department (21B0048); Natural Science Foundation of Hunan Province (2018JJ3325); National Natural Science Foundation of China (11704119).
Acknowledgments
This work was supported by the National Natural Science Foundation of China, the Hunan Provincial Natural Science Foundation of China, Scientific Research Fund of Hunan Provincial Education Department, and National College Students’ innovation and entrepreneurship training program.
Disclosures
The authors declare no conflicts of interest.
Data availability
The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.
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