## Abstract

Photonic spin Hall effect (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. In this paper, we propose graphene-based hyperbolic metamaterials (GHMMs), which is composed of the alternating graphene/SiO_{2} multilayer. The numerical results show that when a dipole emitter is placed at the boundary of the GHMMs, the subwavelength beam with *λ*/40 full-with half maximum can be excited and propagates along the left or right channel, which is dependent on polarization handedness. In addition, we further demonstrate that the unidirectional propagation angle can be dynamically tuned by changing the external electric field bias applied to graphene.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Hyperbolic metamaterials (HMMs), strongly anisotropic uniaxial artificial engineered electromagnetic materials that enable custom-tailored electromagnetic responses of the medium [1–8], have drawn great attention in recent years. Such anisotropic metamaterials, whose equi-frequency surface (EFS) is an open hyperboloid due to different signs of the longitudinal (*ɛ*_{||}) and transverse (*ɛ*_{⊥}) components of the effective permittivity tensor, can be realized as nanowire arrays [9] and multilayer structures of alternating metal and dielectric [10–12]. Many interesting potential applications of HMMs have been demonstrated for negative refraction [11,13,14], sub-diffraction imaging [15], dark hollow light cone [16], biosensing [17–19] and photonic spin-hall effect (PSHE) [20]. Recently, graphene has attracted intensive attention in HMMs owing to its tunability of optical properties. Graphene-based HMMs (GHMMs), which is composed of multilayer structure of alternating graphene and dielectric, have been proposed and investigated [21–29]. GHMMs structure can be applied to the spontaneous emission enhancement [22], the negative refraction at THz frequencies [24], tunable broadband hyperlens [25], the perfect absorption [26], tunable terahertz amplification [27], epsilon-near-zero material [28], near-field radiative heat transfer [29], and so on.

PSHE is an interesting transport phenomenon that a transverse spin-dependent subwavelength shift happens for the reflection or refraction beam at an optical interface under a circularly polarized light incidence [30,31]. PSHE was first proposed by Onoda et al. in 2004 and experimentally verified by Hosten et al by using the preselection and postselection technique in 2008 [32,33]. This phenomenon originates from spin-orbit coupling of photons and the fundamental law of angular momentum conservation [34]. There has been intensive research about the PSHE, such as magneto-optical modulation in graphene [35], the symmetric and asymmetric spin splitting in monolayer black phosphorus [36], PSHE in metamaterials [37–39] and transmission [40–44]. However, the PSHE in GHMMs for tunable polarization-controlled routing of subwavelength beam has not yet been studied.

In this work, the GHMMs structure, which consists of alternating graphene/SiO_{2} multilayer, is proposed to realize dynamically tunable directional subwavelength beam propagation. When a dipole emitter is placed at the boundary of the GHMMs with Type II hyperbolic dispersion, whether excited beam propagates along the left or right channel is dependent on the polarization handedness. This kind of photonic spin Hall effect originates from the near field interference effect. Besides, the unidirectional propagation angle can be dynamically tuned by the external electric field bias applied to graphene. In the previous work of Zhu *et al.* [45], a symmetric spin splitting dependent on the incident orbital-angular-momentum is achieved by transmitting higher-order Laguerre–Gaussian beams through the GHMMs with epsilon-near-zero region. Differently, directional propagation of the subwavelength beam can be controlled by the dipole handedness in GHMMs with Type II hyperbolic dispersion in our work. Here, all the simulation results based on finite element method (FEM) are conducted to verify our method.

## 2. Design and theories

The GHMMs structure, which is proposed for tunable polarization-controlled directional propagation of subwavelength beam, is shown in Fig. 1. This structure consists of an alternating graphene/SiO_{2} multilayer. Bulk plasmon polaritons (BPPs) can be supported and strongly confined inside the GHMMs structure, which originates from the hybridization of short-ranged surface plasmon polaritons (SPPs) in each interface between graphene and SiO_{2}. For BPPs, the absolute value of modal indices is larger than that of SPPs. BPPs can be tailored by controlling the geometry of the metamaterials. This property can be utilized to achieve directional propagation of high-*k* waves [7]. Here, the period of the multilayer structure is *d* (= *t*_{g} + *t*_{d}), the thickness *t _{g}* of graphene is chosen as 0.35 nm, the thickness

*t*

_{d}of SiO

_{2}is chosen as 20 nm, and the working wavelength is 7 µm.

*ɛ*and

_{g}*ɛ*( = 1.24) represent the permittivities of graphene and SiO

_{d}_{2}, respectively. A dipole emitter (for example, an atom, molecule or classical antenna), which is placed in near-field proximity to the interface of the proposed GHMMs structure, emits into extraordinary modes of the metamaterial exhibiting high density of electromagnetic states (DOS) and strong spatial localization, with the directionality of energy propagation controlled by the dipole handedness [20].

In our work, graphene is characterized by the following surface conductivity *σ _{g}*, which is calculated according to the well-known Kubo formula [46]:

*E*is Fermi level of graphene,

_{f}*τ*is electron-phonon relaxation time,

*e*is the charge of an electron,

*k*is the Boltzmann’s constant,

_{B}*ω*is the radian frequency, and

*ħ*is the reduced Planck’s constant. Then the permittivity of monolayer graphene has in-plane component ${\varepsilon _{g,\textrm{||}}} = {\varepsilon _d} + \frac{{i{\sigma _g}{\eta _0}}}{{{k_0}{t_g}}}$ and out-plane component ${\varepsilon _{g, \bot }} = {\varepsilon _d}$, where

*η*

_{0}( = 377 Ω) is the impedance of air. In our study,

*T*is chosen as 300 K, relaxation time

*τ*is

*μE*/(

_{f}*ev*), DC mobility

_{f}^{2}*μ*is 10000 cm

^{2}/Vs, Fermi velocity

*v*is 1×10

_{f}^{6}m/s. Here, we consider monolayer graphene as a 2D surface current with

*σ*in FEM simulation [47]. Since the conductivity of monolayer graphene depends on chemical potential, the optical properties of our proposed GHMMs structure can be changed by controlling Fermi level of graphene. There are many ways to change the Fermi level of graphene. Such as chemical, molecular doping, and electrical or thermal stimulation [48]. Among the above-mentioned methods, the most efficient and popular way to change the Fermi level of graphene is to apply external electric field bias. For our alternating graphene/SiO

_{g}_{2}multilayer structure, we place a gate electrode on all the graphene layers with a bias voltage to tune the Fermi level of each graphene [49]. The relationship between the Fermi level

*E*

_{f}and applied external electric field bias

*V*

_{g}can be expressed as [50] Where

*η*≈ 9×10

^{16}m

^{-1}V

^{-1}is derived from the single capacitor model,

*V*

_{dirac}is offset bias which reflects graphene’s doping and its impurities. Thus, we can modulate the Fermi level of graphene by changing

*V*

_{g}for our multilayer structure.

If the period *d* of one unit is sufficiently small compared to the operating wavelength, the multilayer structure can be considered as an anisotropic metamaterial. The effective permittivity tensor of the multilayer structure is described as [51]

*ɛ*

_{||}and

*ɛ*

_{⊥}is given as follow:

Based on Eq. (4), we calculate the in-plane and out of plane components of effective permittivity under different *V*_{g} when the working wavelength *λ* is 7 µm. Here, we only consider the real part of effective permittivity, and the results are shown in Fig. 2. The black line and red line represent the in-plane and out of plane components of the effective permittivity, respectively. When *V*_{g} changes from 0 to 0.277 V, the signs of Re(ɛ_{||}) and Re(ɛ_{⊥}) are both positive and the multilayer structure shows the elliptical dispersion. When *V*_{g} changes from 0.277 to 2 V, multilayer structure shows Type II HMM dispersion (Re(ɛ_{||}) < 0 and Re(ɛ_{⊥}) > 0), Re(ɛ_{||}) decreases with increasing *V*_{g}. Thus, the GHMMs with Type II dispersion are realized. Besides, Re(ɛ_{⊥}) ( = 1.24) is almost unchanged with *V _{g}*. Especially, for

*V*= 0.745 V, Re(ɛ

_{g}_{||}) = -Re(ɛ

_{⊥}) = -1.24. In

*k*space, for anisotropic medium, the EFS of TM polarization can be expressed as:

*k*

_{0}is the wavevector of light in the vacuum.

*k*and

_{x}*k*

_{z}are the wavevector components along

*X*- and

*Z*-directions, respectively. Then, the calculated EFS are plotted in Fig. 2(b) when

*V*

_{g}is 0.2, 0.4, 0.745, and 1.6 V, respectively. The elliptical dispersion under

*V*

_{g}= 0.2 V indicates that high-frequency components (${k_x} > \sqrt {{\varepsilon _d}} {k_0}$) are prohibited to propagate in the multilayer structure; The Type II dispersion under

*V*= 0.4, 0.745, and 1.6 V indicates that low-frequency components (${k_x} < \sqrt {{\varepsilon _d}} {k_0}$) are prohibited to propagate in the GHMMs. And their asymptotes for Type II dispersion can be expressed as ${k_z} ={\pm} \sqrt { - \frac{{{\varepsilon _{\textrm{||}}}}}{{{\varepsilon _ \bot }}}} {k_x}$. The direction of light propagation depends on the group velocity. The group velocity demonstrates that the direction of the energy flow is perpendicular to the EFS and can be calculated by ${V_g}(\omega ) = {{\nabla }_k}\omega (k)$. Therefore, for Type II HMM in the

_{g}*k*–

_{x}*k*plane, electromagnetic radiation and energy travel preferentially along a cone. The cone angle

_{z}*θ*between the splitting beam and optical axis satisfies [52]

*θ*decreases with increasing

*V*

_{g}; and for

*V*

_{g}= 0.745 V, cone angle

*θ*is 45°.

To achieve unidirectional subwavelength beam propagation based on photonic spin Hall effect, we utilize GHMMs with Type II dispersion based on the near-field interference effect [53]. As shown in Fig. 1, if an elliptically polarized emitter is placed in near-field proximity to the interface of the proposed GHMMs structure, its emission will be coupled to the high local DOS modes. Here, we discuss a 2D case in which a 2D emitter with a dipolar moment is defined ${ \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {\textbf{p}}} }$ = [*p _{x}*,

*p*] for simplify considerations. Near-field interference effect will happen due to the linear superposition of the two orthogonal dipole orientations

_{z}*p*

_{x}and

*p*

_{z}. All possible plane wave modes (high-frequency components) in GHMMs can be excited by the dipole with an efficiency proportional to the strength of the interaction ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {\textbf {p}}} }{\; }\cdot{\; }\mathop{{{\textbf E}_{\textbf k}}}\limits^\rightharpoonup $, where $\mathop{{{\textbf E}_{\textbf k}}}\limits^\rightharpoonup$ (= [

*E*,

_{x}*E*]) is the electric field of

_{z}*k*-th mode at emitter location. The coupling efficiency

*C*to high LDOS mode is given by [20]

*y*axis. Under the paraxial propagation condition, the electric field of

*k*-th mode in the GHMMs close to the boundary satisfies ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {\textbf {E}}} } \approx {E_x}{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {x}} } - ({{{{k_x}} \mathord{\left/ {\vphantom {{{k_x}} {{k_z}}}} \right.} {{k_z}}}} ){E_x}{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {\boldsymbol z}} }$, where

*k*and

_{x}*k*(= $i\sqrt {{k_x}^2 - {k_0}^2}$) are the mode wave vectors in the

_{z}*x*and

*z*directions, respectively. The superposition of different components with $|{{k_x}} |\gg {k_0}$ make up the subwavelength high DOS modes which decide the propagation direction of beam in GHMMs with Type II HMM dispersion. Therefore, for GHMMs with Type II HMM dispersion, the electric field of the subwavelength mode near dipole can be approximated with ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {\textbf {E}}} } \approx {E_x}\approx {E_x}{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {x}} } \mp i{E_x}{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {\boldsymbol z}} }$, where the upper sign applies to the left channel (

*x*<

*x*

_{dipole}) and the lower sign applies to the right channel (

*x*>

*x*

_{dipole}) [53]. Thus, the coupling efficiency

*C*also can be expressed as

*C*is equal to zero when dipolar moment ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {\textbf {p}}} } = p[1,i]$ (left-handed circular (LHC) polarization) for right channel or dipolar moment ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {\textbf {p}}} } = p[1, - i]$ (right-handed circular (RHC) polarization) for left channel, due to destructive near-field interference. So, the emission of the dipole can be directionally guided along the opposite directions in GHMMs through controlling the spin of emitted photons and subwavelength beam directional propagation can be controlled by polarization handedness. Besides, the unidirectional propagation angle can also be tuned by applied external electric field bias based on Eq. (6).

## 3. Results and discussions

To verify the proposed way to dynamically tune directional subwavelength beam propagation, we put an elliptically polarized emitter in near-field proximity to the interface of the proposed GHMMs structure and conduct the FEM simulation based on commercial software COMSOL Multiphysics. When *V _{g}* = 0.745 V, as shown in Fig. 3, (

*y)*component of the magnetic field distributions

*H*and intensity of |

_{y}*H*|

_{y}^{2}are displayed under the dipole excitations with different polarizations. The distributions of

*H*under horizontal linear, vertical linear, LHC and RHC polarizations are depicted in Figs. 3(a), (c), (e) and (g), respectively. Figures 3(b), (d), (f), and (h) shows the intensity distributions of |

_{y}*H*

_{y}|

^{2}at

*z*= 0.143

*λ*from the GHMMs structure boundary in the

*z*direction corresponding to Figs. 3(a), (c), (e), and (g), respectively. When the dipole excites horizontally linear polarization as shown in Fig. 3(a), the field distribution of

*H*exhibits even symmetry (

_{y}*H*(x) =

_{y}*H*(-x)). When the dipole excites vertically linear polarization as shown in Fig. 3(c), the field distribution of

_{y}*H*exhibits odd symmetry (

_{y}*H*(x) = -

_{y}*H*(-x)). Based on Eq. (8), no near-field interference happens for the both linear polarizations. For the case of Figs. 3(a) and (c), dipolar moment ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {\textbf {p}}} }$ can be considered as [1,0] and [0,1], and the corresponding coupling efficiency

_{y}*C*are approximately equal to

*p*and ${\mp} $

_{x}E_{x}*ip*, respectively. Therefore, the field distribution of

_{z}E_{x}*H*for the left and right channels is symmetrical in Fig. 3(a), while the field distribution of

_{y}*H*is antisymmetric in Fig. 3(c). When the dipole excites LHC polarized light as shown in Fig. 3(e), the field distribution of

_{y}*H*shows that the left channel is open and the right channel is closed. Conversely, when the dipole excites RHC polarized light as shown in Fig. 3(g), the field distribution of

_{y}*H*shows the left channel is closed and the right channel is open. This is because the destructive interference condition is satisfied and the coupling efficiency

_{y}*C*of right or left channel is absent based on Eq. (8), respectively. So, the intensity of |

*H*|

_{y}^{2}in Figs. 3(f) and (h) clearly shows the strong unidirectionality of the confined subwavelength beam due to near field interference based on photonic spin Hall effect. Besides, the lateral confinement of the beam is about

*λ*/40 full-with half maximum.

To verify the unidirectional propagation angle can be tuned by applied external electric field bias, we calculate the field distribution of *H _{y}* under different

*V*: (a)

_{g}*V*= 0.2 V, (b)

_{g}*V*= 0.4 V, (c)

_{g}*V*= 0.8 V and (d)

_{g}*V*= 1.6 V, shown as Fig. 4. Here, we take the RHC polarized light (${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {\textrm {p}}} } = [{1, - i} ]$) as a case to illustrate. And other parameters are the same with Fig. 3(g). According to Fig. 2(a), when

_{g}*V*

_{g}= 0.2 V, the multilayer structure shows the elliptical dispersion; when

*V*

_{g}= 0.4, 0.8, and 1.6 V, the multilayer structure is Type II GHMMs. The electromagnetic wave is quickly dissipated in the elliptical dispersion region, and the electromagnetic wave can directionally propagate along the right channel in the Type II HMM region under RHC polarized light. And the unidirectional propagation angle is decided by Eq. (6). The propagation angles of subwavelength beam are

*θ*= 31.36°, 46.45°, and 56.01° when

*V*varies from 0.4 to 1.6 V, respectively. And the theoretical values of the angle are 30.32°, 46.25° and 55.35° based on the Eq. (6), which are almost identical to the simulation results. It is believed our proposed design pave a possible way to tune the propagation direction by changing

_{g}*V*.

_{g}Then, we calculate the offset angle in the range of 0.3 to 2 V, and other parameters are the same with Fig. 4. As shown in Fig. 5, the black curve represents the theoretical calculation result and the red dots represent the simulation results. It is obviously that the theoretical and the simulated values are almost identical. Therefore, the offset angle of the left or right channel beam can be well controlled by regulating the applied voltage of graphene. Besides, it is possible to achieve broadband directional subwavelength beam propagation based on PSHE, because graphene/SiO_{2} multilayer can be tuned to GHMMs with Type II hyperbolic dispersion by changing Fermi level under different wavelengths [25].

The proposed GHMMs multilayer structure consisting of alternating graphene/SiO_{2} is feasible at the current level of fabrication technology. The preparation procedure of the proposed multilayer structure is as follows: SiO_{2} layer is first coated on a dielectric substrate through thermal evaporation, and then the high-quality graphene sheets prepared using chemical vapor deposition (CVD) method are coated on the top of the SiO_{2} layer. For CVD, the number of layers can be controlled precisely by regulating the flow radio of CH_{4} and H_{2}, the reaction pressure, the temperature and reaction time [54]. And the large-area high-quality graphene films have also been demonstrated experimentally in other work by applying optimized liquid precursor chemical vapor deposition method [55,56]. Finally, the multilayer structure is realized by stacking of many graphene/SiO_{2} layers on the dielectric substrate.

## 4. Conclusion

In general, we propose the GHMMs structure, which consists of periodic stacking of graphene layers and SiO_{2} layers, to achieve directional subwavelength beam propagation based on PSHE. The PSHE effect is due to near field interference between high-frequency components in GHMMs with Type II hyperbolic dispersion. The propagation direction of subwavelength beam (about *λ*/40 full-with half maximum) is decided by polarization handedness, and the unidirectional propagation angles can be tuned by changing external electric field bias applied to graphene. All the design is verified by FEM and it is believed that our findings provide a new way to control the subwavelength beam unidirectional propagation dynamically.

## Funding

National Natural Science Foundation of China (1148081606193050); Fundamental Research Funds for the Central Universities (JUSRP115A15, JUSRP21935, JUSRP51628B); Undergraduate Innovation Training Program of Jiangnan University of China (1148088201191487).

## Disclosures

The authors declare no conflicts of interest.

## References

**1. **A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature **522**(7555), 192–196 (2015). [CrossRef]

**2. **Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express **14**(18), 8247–8256 (2006). [CrossRef]

**3. **D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. **84**(13), 2244–2246 (2004). [CrossRef]

**4. **A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics **7**(12), 948–957 (2013). [CrossRef]

**5. **P. Shekhar, J. Atkinson, and Z. Jacob, “Hyperbolic metamaterials: fundamentals and applications,” Nano Convergence **1**(1), 14 (2014). [CrossRef]

**6. **L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. **40**, 1–40 (2015). [CrossRef]

**7. **O. Takayama and A. V. Lavrinenko, “Optics with hyperbolic materials,” J. Opt. Soc. Am. B **36**(8), F38 (2019). [CrossRef]

**8. **Z. Guo, H. Jiang, and H. Chen, “Hyperbolic metamaterials: From dispersion manipulation to applications,” J. Appl. Phys. **127**(7), 071101 (2020). [CrossRef]

**9. **D. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. **90**(7), 077405 (2003). [CrossRef]

**10. **H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science **336**(6078), 205–209 (2012). [CrossRef]

**11. **A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. **6**(12), 946–950 (2007). [CrossRef]

**12. **O. Takayama, J. Sukham, R. Malureanu, A. V. Lavrinenko, and G. Puentes, “Photonic spin Hall effect in hyperbolic metamaterials at visible wavelengths,” Opt. Lett. **43**(19), 4602–4605 (2018). [CrossRef]

**13. **J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science **321**(5891), 930 (2008). [CrossRef]

**14. **J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**(18), 3966–3969 (2000). [CrossRef]

**15. **Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science **315**(5819), 1686 (2007). [CrossRef]

**16. **C. Yan, D. H. Zhang, D. Li, H. Bian, Z. Xu, and Y. Wang, “Metal nanorod-based metamaterials for beam splitting and a subdiffraction-limited dark hollow light cone,” J. Opt. **13**(8), 085102 (2011). [CrossRef]

**17. **A. Kabashin, P. Evans, S. Pastkovsky, W. Hendren, G. Wurtz, R. Atkinson, R. Pollard, V. Podolskiy, and A. Zayats, “Plasmonic nanorod metamaterials for biosensing,” Nat. Mater. **8**(11), 867–871 (2009). [CrossRef]

**18. **K. V. Sreekanth, Y. Alapan, M. ElKabbash, E. Ilker, M. Hinczewski, U. A. Gurkan, A. De Luca, and G. Strangi, “Extreme sensitivity biosensing platform based on hyperbolic metamaterials,” Nat. Mater. **15**(6), 621–627 (2016). [CrossRef]

**19. **E. Shkondin, T. Repän, M. E. Aryaee Panah, A. V. Lavrinenko, and O. Takayama, “High aspect ratio plasmonic nanotrench structures with large active surface area for label-free mid-infrared molecular absorption sensing,” ACS Appl. Nano Mater. **1**(3), 1212–1218 (2018). [CrossRef]

**20. **P. V. Kapitanova, P. Ginzburg, F. J. Rodríguez-Fortuño, D. S. Filonov, P. M. Voroshilov, P. A. Belov, A. N. Poddubny, Y. S. Kivshar, G. A. Wurtz, and A. V. Zayats, “Photonic spin Hall effect in hyperbolic metamaterials for polarization-controlled routing of subwavelength modes,” Nat. Commun. **5**(1), 3226 (2014). [CrossRef]

**21. **I. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B **87**(7), 075416 (2013). [CrossRef]

**22. **L.-M. Ye, X.-J. Yi, T.-B. Wang, Y. Zhao, T.-B. Yu, Q.-H. Liao, and N.-H. Liu, “Enhancement and modulation of spontaneous emission near graphene-based hyperbolic metamaterials,” Mater. Res. Express **6**(12), 125803 (2019). [CrossRef]

**23. **M. A. Othman, C. Guclu, and F. Capolino, “Graphene-based tunable hyperbolic metamaterials and enhanced near-field absorption,” Opt. Express **21**(6), 7614–7632 (2013). [CrossRef]

**24. **K. Sreekanth, A. De Luca, and G. Strangi, “Negative refraction in graphene-based hyperbolic metamaterials,” Appl. Phys. Lett. **103**(2), 023107 (2013). [CrossRef]

**25. **T. Zhang, L. Chen, and X. Li, “Graphene-based tunable broadband hyperlens for far-field subdiffraction imaging at mid-infrared frequencies,” Opt. Express **21**(18), 20888–20899 (2013). [CrossRef]

**26. **I. S. Nefedov, C. A. Valagiannopoulos, and L. A. Melnikov, “Perfect absorption in graphene multilayers,” J. Opt. **15**(11), 114003 (2013). [CrossRef]

**27. **T. Guo, L. Zhu, P.-Y. Chen, and C. Argyropoulos, “Tunable terahertz amplification based on photoexcited active graphene hyperbolic metamaterials,” Opt. Mater. Express **8**(12), 3941–3952 (2018). [CrossRef]

**28. **K. Halterman and J. M. Elson, “Near-perfect absorption in epsilon-near-zero structures with hyperbolic dispersion,” Opt. Express **22**(6), 7337–7348 (2014). [CrossRef]

**29. **Q.-M. Zhao, T.-B. Wang, D.-J. Zhang, W.-X. Liu, T.-B. Yu, Q.-H. Liao, and N.-H. Liu, “Contribution of terahertz waves to near-field radiative heat transfer between graphene-based hyperbolic metamaterials,” Chin. Phys. B **27**(9), 094401 (2018). [CrossRef]

**30. **K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics **9**(12), 796–808 (2015). [CrossRef]

**31. **X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. **80**(6), 066401 (2017). [CrossRef]

**32. **M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. **93**(8), 083901 (2004). [CrossRef]

**33. **O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science **319**(5864), 787–790 (2008). [CrossRef]

**34. **K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. **96**(7), 073903 (2006). [CrossRef]

**35. **T. Tang, J. Li, L. Luo, P. Sun, and J. Yao, “Magneto-Optical Modulation of Photonic Spin Hall Effect of Graphene in Terahertz Region,” Adv. Opt. Mater. **6**(7), 1701212 (2018). [CrossRef]

**36. **H. Lin, B. Chen, S. Yang, W. Zhu, J. Yu, H. Guan, H. Lu, Y. Luo, and Z. Chen, “Photonic spin Hall effect of monolayer black phosphorus in the Terahertz region,” Nanophotonics **7**(12), 1929–1937 (2018). [CrossRef]

**37. **H. Chen, S. Zhou, G. Rui, and Q. Zhan, “Magnified photonic spin-Hall effect with curved hyperbolic metamaterials,” J. Appl. Phys. **124**(23), 233104 (2018). [CrossRef]

**38. **T. Tang, C. Li, and L. Luo, “Enhanced spin Hall effect of tunneling light in hyperbolic metamaterial waveguide,” Sci. Rep. **6**(1), 30762 (2016). [CrossRef]

**39. **Y. Liu, Y. Ke, H. Luo, and S. Wen, “Photonic spin Hall effect in metasurfaces: a brief review,” Nanophotonics **6**(1), 51–70 (2017). [CrossRef]

**40. **L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. **96**(16), 163905 (2006). [CrossRef]

**41. **D. Haefner, S. Sukhov, and A. Dogariu, “Spin hall effect of light in spherical geometry,” Phys. Rev. Lett. **102**(12), 123903 (2009). [CrossRef]

**42. **X. Zhou, X. Ling, Z. Zhang, H. Luo, and S. Wen, “Observation of spin Hall effect in photon tunneling via weak measurements,” Sci. Rep. **4**(1), 7388 (2015). [CrossRef]

**43. **K. Y. Bliokh, C. Samlan, C. Prajapati, G. Puentes, N. K. Viswanathan, and F. Nori, “Spin-Hall effect and circular birefringence of a uniaxial crystal plate,” Optica **3**(10), 1039 (2016). [CrossRef]

**44. **O. Takayama and G. Puentes, “Enhanced spin Hall effect of light by transmission in a polymer,” Opt. Lett. **43**(6), 1343 (2018). [CrossRef]

**45. **W. Zhu, M. Jiang, H. Guan, J. Yu, H. Lu, J. Zhang, and Z. Chen, “Tunable spin splitting of Laguerre–Gaussian beams in graphene metamaterials,” Photonics Res. **5**(6), 684–688 (2017). [CrossRef]

**46. **Z. P. Su, Y. K. Wang, X. Luo, H. Luo, C. Zhang, M. X. Li, T. Sang, and G. F. Yang, “A tunable THz absorber consisting of an elliptical graphene disk array,” Phys. Chem. Chem. Phys. **20**, 14358 (2018). [CrossRef]

**47. **A. Y. Nikitin, F. Guinea, F. J. Garcia-Vidal, and L. Martin-Moreno, “Surface plasmon enhanced absorption and suppressed transmission in periodic arrays of graphene ribbons,” Phys. Rev. B **85**(8), 081405 (2012). [CrossRef]

**48. **B. Guo, L. Fang, B. Zhang, and J. R. Gong, “Graphene doping: a review,” Insci. J. **1**(2), 80–89 (2011). [CrossRef]

**49. **A. A. Sayem, M. R. C. Mahdy, I. Jahangir, and M. S. Rahman, “Ultrathin ultra-broadband electro-absorption modulator based on few-layer graphene based anisotropic metamaterial,” Opt. Commun. **384**, 50–58 (2017). [CrossRef]

**50. **B. Janaszek, A. Tyszka-Zawadzka, and P. Szczepański, “Tunable graphene-based hyperbolic metamaterial operating in SCLU telecom bands,” Opt. Express **24**(21), 24129–24136 (2016). [CrossRef]

**51. **A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B **74**(7), 075103 (2006). [CrossRef]

**52. **B. Wood, J. Pendry, and D. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B **74**(11), 115116 (2006). [CrossRef]

**53. **F. J. Rodríguez-Fortuño, G. Marino, P. Ginzburg, D. O’Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Near-field interference for the unidirectional excitation of electromagnetic guided modes,” Science **340**(6130), 328–330 (2013). [CrossRef]

**54. **Z. Q. Tu, Z. C. Liu, Y. F. Li, F. Yang, L. Q. Zhang, Z. Zhao, C. M. Xu, S. F. Wu, H. W. Liu, H. T. Yang, and P. Richard, “Controllable growth of 1-7 layers of graphene by chemical vapour deposition,” Carbon **73**, 252–258 (2014). [CrossRef]

**55. **Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. G. de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano **7**(3), 2388–2395 (2013). [CrossRef]

**56. **X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, A. Velamakanni, I. Jung, E. Tutuc, S. K. Banerjee, L. Colombo, and R. S. Ruoff, “Large-Area Synthesis of High-Quality and Uniform Graphene Films on Copper Foils,” Science **324**(5932), 1312–1314 (2009). [CrossRef]