Abstract

In this study, the orbital angular momentum (OAM) sidebands of Laguerre-Gauss beams reflecting on graphene metasurfaces are investigated. Upon reflection, vortex beams carrying orbital angular momentum will acquire sidebands, whose relative intensity varies depending on the Fermi energy, the external magnetic field, and/or the wave frequency. The relative intensity of the sideband OAM modes locally has a small trough for s-polarized beams at the topological transition point between the hyperbolic and elliptic topology. Energy can transfer from the central mode to the neighboring OAM modes increasing the topologic charge l. When the electric field of the incident s-polarized light occurs along the low energy dissipation direction of the graphene metasurfaces, it is helpful for the mode transformation of vortex beams. When the electric field of the incident s-polarized light occurs along the high energy dissipation direction, it is beneficial to suppressing crosstalk of different sidemodes in terahertz communication.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Vortex beams have main characteristics of a singularity in phase and a topological structure on its wavefront originating from its helicoidal spatial wavefront around the phase singularity. At the singularity, the phase is indefinite and amplitude become zero, leading to dark center in the wave packet. In 1992, Allen and his co-workers [1] proposed that the optical vortex beam carrying quantized orbital angular momentum (OAM) when their Poynting vector rotates around the propagation axis. Since then, optical vortices and their propagation properties attract intensive investigations. As for optical reflection of vortex beams, unremitting efforts have been devoted to study the Goos-H$\ddot {\mbox {a}}$nchen (GH) and Imbert-Fedorov (IF) shifts of Laguerre-Gaussian (LG) beams. The theory about the GH and IF shifts of polarized vortex beams was established, first by Fedoseyev [2] and then by Bliokh [3]. Effects of the orbital angular momentum on beam shifts were theoretically and experimentally investigated by Merano [4]. The orbital angular momentum sidebands due to external reflection were observed experimentally at an air-glass interface [5], despite the intensity of sideband modes are weak and the OAM-dependent shifts are tiny. Subsequently, many results have been published about the GH and IF shifts of vortex beams upon reflection on various complex media interfaces, such as air-left-handed-materials interfaces [6], air-prism interface [7], air-anisotropic monolayer graphene [8], and air-gold interface [9], etc; nevertheless they did not consider orbital angular momentum sidebands. So far, there are only few results related to orbital angular momentum sidebands of vortex beams upon reflection on various interfaces, such as air-glass or glass-air interface (external reflection or total internal reflection) [5,10], air-graphene/hexagonal boron nitride/Au interfaces [11], air-thin metamaterial slab [12]. However, the mechanism of orbital angular momentum sidebands of vortex beams upon reflection on graphene hyperbolic metasurfaces is still unclear, which affects its potential application, such as intelligent reflecting surface for wireless network [13].

Metasurfaces are a two-dimensional analog of metamaterials [14]. They comprise a class of planar optical metamaterials with many subwavelength structural units, which have new functionalities allowing to control the phase, amplitude and polarization of light and thus to manage the propagation, reflection and refraction of light [1519]. When preserving most of the properties of three-dimensional metamaterials, they reduce energy losses, are simpler to fabricate, and are compatible with other photonics devices [2023]. Hyperbolic metasurfaces are uniaxial and extreme 2D anisotropy, in which the principle components of the effective permittivity tensor have opposite signs. They exhibit metal and dielectric properties at the same time. Thus the isofrequency surface becomes unbounded (hyperbolic), in the momentum space, and the optical density of states will be theoretically infinite [2426]. Hyperbolic metasurfaces could be created artificially by subwavelength metallic scatters [27], crystalline silver nanostructures [28], nanostructured van der Waals materials [29], or MAPbI$_3$ perovskite/Au [30], etc. As a two-dimensional materials, black phosphorous could be used to form a natural hyperbolic metasurfaces [31]. Using a densely packed set of graphene ribbons, hyperbolic metasurfaces operating at THz frequencies can also be realized. They utilize the inherent reconfigurable capabilities of graphene plasmonics and its topology can be adjusted though external electric field, magnetic field or frequency [24,32].

Motivated by the interesting properties of graphene hyperbolic metasurfaces and its potential application in intelligent reflecting surface, we study the orbital angular momentum sidebands of vortex beams impinging on graphene hyperbolic metasurfaces. Specially, we investigate the evolution of the orbital angular momentum sidebands versus the frequency, the Fermi energy and the external magnetic field, and determine the topological transition of graphene metasurfaces demonstrated by the Fresnel reflection coefficients and the relative strength for Laguerre-Gauss modes. We explore further the relation between the orbital angular momentum sidebands and the effective conductivity tensor of graphene metasurfaces which depends on the optical axis angle. Finally, we show that an overall picture of energy transfers from the central mode to the neighboring OAM modes with increasing the topologic charge $l$.

2. Theory and model

We start our analysis by considering the problem of an OAM-carrying light beam (Laguerre-Gaussian beam) with the frequency $\omega$ (or the wavelength $\lambda$) and an incident angle $\theta$, impinging on hyperbolic metasurfaces with a glass substrate (permittivity $\varepsilon$), as shown in Fig. 1. The plane of incidence is at an optical axis angle $\varphi$ to the dashed line which is perpendicular to graphene microribbons. An array of densely packed 2D graphene microribbons with the periodicity $L$ and the strip width $W$ make up hyperbolic metasurfaces that behave as dielectrics for waves propagating along one direction and as metals for waves travelling along the orthogonal one. This extreme anisotropy of hyperbolic metasurfaces can be well-described by an in-plane effective conductivity tensor [24]:

$$\begin{pmatrix} \sigma\!_{xx} & \sigma\!_{xy} \\ \sigma\!_{yx} & \sigma\!_{yy} \end{pmatrix} \!=\! \begin{pmatrix} \sigma\!_1\!\sin^2\!\varphi\!+\!\sigma\!_2\!\cos^2\!\varphi & \sin2\varphi(\sigma\!_1\!-\!\sigma\!_2)/2\!+\!\sigma\!_h^{eff}\\ \sin2\varphi(\sigma\!_1\!-\!\sigma\!_2)/2\!-\!\sigma\!_h^{eff} & \sigma\!_1\!\cos^2\!\varphi\!+\!\sigma\!_2\!\sin^2\!\varphi \end{pmatrix},$$
where $\sigma _1$ and $\sigma _2$ are effective conductivities along and across the main axis which is perpendicular to the dotted dashed line, and $\sigma _h^{eff}$ is an effective Hall conductivity related to an external magnetic field. $\sigma _1$ is Drude-like effective conductivity while $\sigma _2$ is Lorentz-like effective conductivity. The imaginary part of $\sigma _1$ and $\sigma _2$ is responsible for the polarizability. For Im$[\sigma _1]>0$ and Im$[\sigma _2]<0$, the hyperbolic topology of the equal-frequency contours of the light dispersion in metasurface is obtained; for Im$[\sigma _1]>0$ and Im$[\sigma _2]>0$, the elliptic topology in inductive metasurface is realized [27]. According to the effective medium theory [33], $\sigma _1=\tfrac {W}{L}(\sigma _g+\sigma _h^2/\sigma _g)-(\sigma _h^{eff})^2/\sigma _2$, $\sigma _2=\sigma _g \sigma _c/\left [\tfrac {W}{L}\sigma _c+(1-\tfrac {W}{L})\sigma _g\right ]$, $\sigma _h^{eff}=\tfrac {W}{L}\sigma _2\sigma _h/\sigma _g$. In the THz region, neglecting the interband transition and considering only intraband transitions in the local response approximation, the graphene conductivity and Hall conductivity of graphene can be described as $\sigma _g=\tfrac {e^2E_F}{\hbar ^2\pi }i(\omega +i/\tau )/[(\omega +i/\tau )^2-\omega _c^2]$, $\sigma _h=\tfrac {e^2E_F}{\hbar ^2\pi }\omega _c/[(\omega +i/\tau )^2-\omega _c^2]$, where $E_F$ is the Fermi level, $\tau$ is a relaxation time of the carrier, $\omega _c=eBv_F^2/E_F$ is the cyclotron frequency with $B$ being a static magnetic field applied perpendicularly to the hyperbolic metasurfaces and $v_F$ being the Fermi velocity. Based on the grid impedance method combined with the approximate Babinet principle, the equivalent conductivity $\sigma _c\approx -\frac {i\omega L}{\pi ^2}(\frac {1+\varepsilon }{2})\ln [\csc (\frac {\pi }{2}(1-\tfrac {W}{L}))]$ that considers the near-field coupling between adjacent strips.

 figure: Fig. 1.

Fig. 1. Schematic of the vortex beam reflection in graphene hyperbolic metasurfaces. The GH and IF shifts are shown by dotted lines.

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We use three Cartesian reference frames: the incident frame $(x_i, y, z_i)$, the laboratory frame $(x, y, z)$ attached to the hyperbolic metasurfaces, and the reflected frame $(x_r, y, z_r)$. The incident beam propagates in the $x$-$z$ plane which is at an angle $\varphi$ to the dotted dashed line across the strips array. Assuming the waist of the beam located at hyperbolic metasurfaces ($z=0$) and according to the angular spectrum method [6], the electric field of the vortex beam at hyperbolic metasurfaces can be determined as

$$\tilde{\mathbf{E}}_{i} =(f_p\hat{\mathbf{x}}_i+f_s\hat{\mathbf{y}})\tilde{\phi}_m^l,$$
where $f_p$ and $f_s$ fix the polarization of the incident beam and satisfy the relation $\left |f_p\right |^2+\left |f_s\right |^2=1$. $f_p=1$ and $f_s=0$ correspond to the p-polarized waves (or TM polarization) with the electric field parallel to the plane of incidence; $f_p=0$ and $f_s=1$ correspond to the s-polarized waves (or TE polarization) with the electric field orthogonal to the plane of incidence. $\tilde {\phi }_m^l$ is the angular spectrum of the vortex beam in momentum space with two independent parameters $l$ and $m$ denoting the azimuthal and radial indices of the vortex beam. We consider a incident vortex beam [1] with a purely azimuthal mode $(m=0)$ and then have
$$\tilde{\phi}_0^l=\frac{C_l w_0}{2} \exp{[{-}w_0^2(k_{xi}^2+k_{y}^2)/4]}[w_0({-}ik_{xi}+\mathrm{sgn}[l]k_y)/\sqrt{2}]^{\left| l \right|}.$$
Here $C_l$ is the normalization constant, $l$ is the topologic charge, $w_0$ is the beam waist of the Laguerre-Gaussian beam, $k_{xi}$ and $k_y$ are wave-vector components along the ${x_i}$ and $y$ axes, and sgn[] is the sign function. After the coordinate rotation [34], we get the electric fields of the reflected vortex beam:
$$\begin{aligned}\tilde{\mathbf{E}}_{r}&= \{f_p [\tilde{r}_{pp}+\frac{k_y}{k_0}(\tilde{r}_{sp}-\tilde{r}_{ps})\cot{\theta}]+f_s [\tilde{r}_{sp}-\frac{k_y}{k_0}(\tilde{r}_{pp}-\tilde{r}_{ss})\cot{\theta}]\}\tilde{\phi}_0^l \hat{\mathbf{x}}_r \\ &+\{f_p [\tilde{r}_{ps}+\frac{k_y}{k_0}(\tilde{r}_{pp}-\tilde{r}_{ss})\cot{\theta}]+f_s [\tilde{r}_{ss}+\frac{k_y}{k_0}(\tilde{r}_{sp}-\tilde{r}_{ps})\cot{\theta}]\}\tilde{\phi}_0^l \hat{\mathbf{y}}. \end{aligned}$$
Here $k_0=\omega /c$ is the vacuum wave number and $\tilde {r}_{i, j}$ ($i$, $j$ are $s$ or $p$) are the reflection coefficients of an arbitrary plane wave in momentum space. Considering the tiny in-plane spread of wave vectors and using a Taylor series expansion based on the arbitrary angular spectrum component, the Fresnel reflection coefficients around the central wave vector can be expanded as $\tilde {r}_{i, j}=r_{i, j}(\theta )+\frac {\partial r_{i, j}(\theta )}{\partial \theta }\frac {k_{xi}}{k_0}$, where $r_{i, j}(\theta )$ denotes the reflection coefficients of the central wave vector at the incident angle $\theta$. The reflection coefficients $r_{i, j}(\theta )$ can be determined by Maxwell’s equation using the metasurfaces boundary conditions:
$$[\mathbf{n},\mathbf{H}_{2}]-[\mathbf{n},\mathbf{H}_{1}]= \frac{4\pi}{c}\hat{\sigma}\mathbf{E}_{1}$$
$$[\mathbf{n},\mathbf{E}_{2}]-[\mathbf{n},\mathbf{E}_{1}]= 0$$
where $\mathbf {E}$ and $\mathbf {H}$ denote electric and magnetic fields, and $\mathbf {n}$ is a unit vector normal to the interface. After some calculation, we get
$$r_{ss}(\theta)=\frac{(\cos\theta-n\cos\theta-\sigma_{yy})(\sec\theta+n\sec\theta+\sigma_{xx})+\sigma_{xy}\sigma_{yx}} {(\cos\theta+n\cos\theta+\sigma_{yy})(\sec\theta+n\sec\theta+\sigma_{xx})-\sigma_{xy}\sigma_{yx}},$$
$$r_{sp}(\theta)=\frac{2\sigma_{xy}} {(\cos\theta+n\cos\theta+\sigma_{yy})(\sec\theta+n\sec\theta+\sigma_{xx})-\sigma_{xy}\sigma_{yx}},$$
$$r_{ps}(\theta)=\frac{2\sigma_{yx}} {(\cos\theta+n\cos\theta+\sigma_{yy})(\sec\theta+n\sec\theta+\sigma_{xx})-\sigma_{xy}\sigma_{yx}},$$
$$r_{pp}(\theta)=\frac{(\cos\theta+n\cos\theta+\sigma_{yy})(\sec\theta-n\sec\theta-\sigma_{xx})+\sigma_{xy}\sigma_{yx}} {(\cos\theta+n\cos\theta+\sigma_{yy})(\sec\theta+n\sec\theta+\sigma_{xx})-\sigma_{xy}\sigma_{yx}},$$
where the components of the conductivity tensor $\sigma _{ij}$ are normalized to $c/4\pi$. The similar results of the reflection coefficients can be found in Refs. [24,33]. The air and glass layers separated by the graphene hyperbolic metasuface are regarded as semi-infinite space and the refractive indexes of the air and glass layers are 1 and $n = \sqrt {\varepsilon }$, respectively.

Transforming the angular spectrum $\tilde {\mathbf {E}}_{r}$ in Eq. (4) into the position space form $\mathbf {E}_{r}$ by employing the inverse Fourier transformation, we get

$$\begin{aligned}\mathbf{E}_{r}=& \{f_p [r_{pp}-\frac{i}{k_0}(r_{sp}-r_{ps})\cot{\theta}\frac{\partial}{\partial y}]+f_s [r_{sp}+\frac{i}{k_0}(r_{pp}-r_{ss})\cot{\theta}\frac{\partial}{\partial y}]\}\phi_0^l \hat{\mathbf{x}}_r \\ &+\{f_p [r_{ps}-\frac{i}{k_0}(r_{pp}-r_{ss})\cot{\theta}\frac{\partial}{\partial y}]+f_s [r_{ss}-\frac{i}{k_0}(r_{sp}-r_{ps})\cot{\theta}\frac{\partial}{\partial y}]\}\phi_0^l \hat{\mathbf{y}}. \end{aligned}$$
Here $r_{i, j}=r_{i, j}(\theta )+\frac {i}{k_0}\frac {\partial r_{i, j}(\theta )}{\partial \theta }\frac {\partial }{\partial x_r}$ (subscripts $i$, $j$ are $s$ or $p$) are the reflection coefficients of an arbitrary plane wave in position space and $\phi _0^l=\frac {C_l}{w_0} \exp {[-(x_{r}^2+y_{r}^2)/w_0^2]}[\frac {\sqrt {2}}{w_0}(x_{r}-i\mathrm {sgn}[l]y_r)]^{\left | l \right |}$ is the complex amplitude of the vortex beam in position space.

Considering nonspecular reflection [35], the reflected beam experiences Goos-H$\ddot {\mbox {a}}$nchen $X$ and Imbert-Fedorov $Y$ shifts at hyperbolic metasurfaces: $X={\iint {\mathbf {E}}_r x {\mathbf {E}}_{r}^* \,dx\,dy}/{\iint {\mathbf {E}}_{r}{\mathbf {E}}_{r}^* \,dx\,dy}$, $Y={\iint {\mathbf {E}}_r y {\mathbf {E}}_{r}^* \,dx\,dy}/{\iint {\mathbf {E}}_{r}{\mathbf {E}}_{r}^* \,dx\,dy}$. According to Ref. [5], the spatial state of the input beam with the orbital angular momentum may not be preserved upon reflection, because Goos-H$\ddot {\mbox {a}}$nchen and Imbert-Fedorov effects induce mode-dependent displacement and coupling to neighboring OAM modes. For instance, a pure LG input mode $\{\phi _0^l\}$ can acquire sidebands and is transformed into a superposition of LG modes $\{\phi _0^{-l-1}, \phi _0^{-l}, \phi _0^{-l+1}\}$ upon reflection, where the minus sign comes from OAM reversal upon reflection. The sidebands intensity $C_{l,l'}$ determined by the spatial Fresnel coefficients $c_{l,l'}$ through the relation: $C_{l,l'}=|c_{l,l'}|^2$, where $l$ and $l'$ denote the input and output topologic charge (the OAM index), respectively. The spatial Fresnel coefficients can be defined as [10]

$$c_{l,l'}=\begin{cases} \pm Z^{{\pm}} \sqrt{|l|+1}\\ \mp Z^{{\mp}} \sqrt{|l|}\\ ({-}1)^l. \end{cases}$$
Here the complex-valued parameters
$$Z^{{\pm}}=\frac{\lambda}{2^{3/2}\pi w_0}({-}1)^l (X \pm i Y)$$
include main shifts, eg. longitudinal Goos-H$\ddot {\mbox {a}}$nchen $X$ and transverse Imbert-Fedorov $Y$ shifts.

3. Numerical results and analysis

Modulus and phase of the Fresnel reflection coefficients at the air-HMS-SiO$_2$ system are shown in Fig. 2. The non-diagonal reflection coefficients $\left |r_{sp}\right |$ and $\left |r_{ps}\right |$ are related to the cross conductivities $\sigma _{xy}$ and $\sigma _{yx}$ which lead to a coupling between $s$ and $p$ waves. According to Eqs. (8) and (9), $\sigma _{xy}=\sigma _{yx}=0$ at $\varphi =0^\circ$ and $B=0$ causes $\left |r_{sp}\right |=\left |r_{ps}\right |=0$ [Fig. 2(a,b)]; at $\varphi =0^\circ$ and $B \neq 0$ [Fig. 2(c)], $\sigma _{xy}=\sigma _h^{eff}$ and $\sigma _{yx}=-\sigma _h^{eff}$ which lead to little values of $\left |r_{sp}\right |$ and $\left |r_{ps}\right |$. Under the current condition, $s$ and $p$ waves only have the tiny coupling. Considering $\left |r_{pp}\right |=0.322$ at $B=0$ (T) and $\left |r_{pp}\right |=0.323$ at $B \approx 3$ (T), the variation of $\left |r_{pp}\right |$ can be neglected. The variation of $\left |r_{pp}\right |$ with the Fermi energy $E_F$ or the frequency $\omega$ is also not obvious. As a consequence, the reflection change of the vortex beam upon the air-HMS-SiO$_2$ system mainly depends on the term $\left |r_{ss}\right |$. In Fig. 2(d-f), the phase ${\Phi }_{ss}$ of the reflection coefficients $r_{ss}$ changes the sign at a certain point which corresponds to the topological transition point between the hyperbolic and elliptic topology. In the case of the effective conductivities with different signs of the imaginary parts Im$[\sigma _1]>0$ and Im$[\sigma _2]<0$, the hyperbolic topology of the equal-frequency contours in metasurface will be realized [27]; in the case of the effective conductivities with the positive imaginary parts Im$[\sigma _1]>0$ and Im$[\sigma _2]>0$, the elliptic topology will be obtained [see Fig. 2(g)]. The real parts of the effective conductivities, Re$[\sigma _1]$ and Re$[\sigma _2]$, account for energy dissipation; the imaginary parts Im$[\sigma _1]$ and Im$[\sigma _2]$ are responsible for the polarizability of the metasurface. As shown in Fig. 2(h,i), the variation of the Fermi energy $E_F$ or the external magnetic $B$ will change the imaginary part sign of the effective conductivities, and thus the hyperbolic or elliptic topology of the equal-frequency contours in metasurface will be affected. This case is similar to the variation of the effective conductivities versus the width of graphene ribbons [24].

 figure: Fig. 2.

Fig. 2. (a-c) Modulus and (d-f) phase of the Fresnel reflection coefficients $r_{ij}(\theta )$. (g-i) The imaginary part of the effective conductivity $\sigma _1$ and $\sigma _2$. Parameters are set as the strip width $W=0.5$ $\mu$m, the strip periodicity $L=1$ $\mu$m, the optical axis angle $\varphi =0^\circ$, the Fermi velocity $v_F=9.5\times 10^5$ m/s, the relaxation time of the carrier $\tau =0.1$ ps, the incident angle $\theta =85^\circ$, the beam waist $w_0=2\lambda$. (a,d,g) The Fermi energy $E_F$ and the external magnetic $B$ are 0.05 eV and 0 T, respectively. (b,e,h) The external magnetic $B$ and the frequency $\omega$ are 0 T and 8 THz, respectively. (g,h,i) The frequency $\omega$ and the Fermi energy $E_F$ are 8 THz and 0.05 eV, respectively.

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The relative intensity $I_{l,l'}$ of each Laguerre-Guass mode is given by $I_{l,l'}=C_{l,l'}/{\sum _{l'}C_{l,l'}}$, which is proportional to the photocurrent of a photodiode in the experiment [5]. As shown in Fig. 3(a,c,e), the relative intensity of the central OAM mode locally gets a small crest (the relative intensity of the sideband OAM modes has a small trough) for $s$-polarized beams at the topological transition point where the Fresnel reflection coefficient $\left |r_{ss}\right |$ nearly reaches a maximum. Upon reflection of the Laguerre-Guass beam, the OAM spectrum acquires sidebands. The increasing of the central OAM mode leads to the decreasing of the sideband OAM modes due to energy conservation. The overall variations of the OAM spectrum still depend on the frequency $\omega$, the Fermi energy $E_F$ and the external magnetic $B$. At $\omega \approx 0.1$ THz, the central OAM mode dominates with $I_{4,-4}\approx 0.995$ [Fig. 3(a)]. Similarly, the relative intensity of the central OAM mode is more than $92.4\%$ at $\omega > 50$ THz [Fig. 3(a)] and more than $98.6\%$ at $\omega > 100$ THz [not shown in Fig. 3(a)]. The GH $X$ and IF $Y$ shifts are proportional to the wave length $\lambda = 2\pi c/\omega$. From Eqs. (12) and (13) we can conclude that increasing the frequency will lead to decrease of the wave length $\lambda$, $X$, $Y$, $Z^\pm$, and $c_{l,l'}$, which will result in decrease of the relative intensity of the sideband OAM mode and increase of the relative intensity of the central OAM mode, as shown in Fig. 3(a). The domination of the central OAM mode is expected in communication [36], since crosstalk of the different signals can be reduced. When $0.8<\omega <20$ THz, the relative intensity of the central OAM mode is less than $50\%$, because energy is transferred from the central OAM mode to the sideband OAM modes. This property can be used for mode transformation [37]. At $E_F\approx 0$ (zero-bias) or $B>10$ THz, energy transfer to the sideband OAM modes can be suppressed, while energy can be efficiently transferred from the central OAM mode to the sideband OAM modes at $0.02<E_F<0.12$ eV or $B<5$ THz [Fig. 3(c,e)]. Because the relative intensity $I^p_{4,-5}$ or $I^p_{4,-3}$ approaches zero at the incident angle $\theta =85^\circ$[see Fig. 4(d)], the relative intensity of the $p$-polarized central OAM mode has little variation, and $I_{4,-4}^p \approx 1$ except the vicinity of the topological transition point in Fig. 3(b,d,f). Considering the properties of suppressing the sideband OAM modes in wide terahertz frequency band and little disturbing by external electrical and magnetic field, $p$-polarized beams are suitable for terahertz communication concerning reflection upon graphene metasurfaces.

 figure: Fig. 3.

Fig. 3. The relative strength for $s$-polarized and $p$-polarized Laguerre-Guass modes with $l=4$. Parameters are the same as in Fig. 2.

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 figure: Fig. 4.

Fig. 4. (a,c) The Goos-H$\ddot {\mbox {a}}$nchen $X$ and Imbert-Fedorov $Y$ shifts of the Laguerre-Guass beam with $l=4$. (b,d) The relative strength of Laguerre-Guass modes. (a,b) The incident angle $\theta = 85^\circ$. (c,d) The optical axis angle $\phi = 45^\circ$. The other parameters are the same as in Fig. 2.

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Considering nonspecular reflection, the reflected beam is shifted relative to the geometrical optics reflected ray due to diffractive correction upon Snell’s reflection law. The shifts parallel and perpendicular to the plane of incidence are called the Goos-H$\ddot {\mbox {a}}$nchen and Imbert-Fedorov shifts, respectively. The GH effect comes from the dispersion of the reflection coefficients, while the IF shift originates from the spin-orbit coupling of photons. Upon reflection, the spatial state of the OAM-carrying light beam are modified, because Goos-H$\ddot {\mbox {a}}$nchen and Imbert-Fedorov effects induce mode-dependent displacement and coupling to neighboring OAM modes. According to the Eqs. (12) and (13), the relative intensity of the sideband OAM modes depends on the GH $X_{s,p}$ and IF $Y_{s,p}$ shifts. As a result, the relative intensity of the sideband OAM modes $I^s_{4,-3}$ and $I^s_{4,-5}$ decreases when the IF shift $Y_s$ (a main term of the shifts) decreases with increasing the optical axis angle $\varphi$ [in Fig. 4(a,b)]. The $p$ polarized GH and IF shifts $X_p$ and $Y_p$ are tiny, causing $I^p_{4,-3}$ and $I^p_{4,-5}$ to be closed to zero after square calculation. As shown in Fig. 4(c), the GH and IF shifts $X_s$, $Y_s$, and $X_p$ achieve at maximum near the grazing angle, and $Y_p$ achieves at maximum at the grazing angle. The similar results have been theoretically predicted on graphene [38,39]. At the incident angle $\theta =85^\circ$, the relative intensity $I^p_{4,-5}$ or $I^p_{4,-3}$ approaches zero in Fig. 4(d). As a result, the sideband OAM modes are tiny in wide terahertz frequency band and little disturbed by external electrical or magnetic field in Fig. 3(b,d,f).

An array of densely-packed graphene ribbons constitutes the strongly anisotropic elliptic and hyperbolic metasurfaces, and any topology can be implemented by changing the ribbon width $W$ [24]. The relative intensity of the sideband OAM modes are also sensitive to the ribbon width $W$. For instance, the relative intensity $I^s_{4,-3}$ and $I^s_{4,-5}$ are quite different at $W = 0.1\mu$m and $W = 0.5\mu$m [in Fig. 5(a)]. $I^s_{l,l'}$ are proportional to ${|l|+1}$ or ${|l|}$, thus the relative intensity of the sideband OAM modes with $l=2$ is large than that with $l=0$ [in Fig. 5(b)]. The relative intensity of the sideband OAM modes are modulated by the optical axis angle $\varphi$ [Fig. 5(a,b)], because the effective conductivity tensor of graphene hyperbolic metasurfaces depends on the optical axis angle $\varphi$ as indicated in Eq. (1). At $\varphi =0^\circ$ and $B=0$, $\sigma _{xy}=\sigma _{yx}=0$, $\sigma _{xx}=\sigma _2$, $\sigma _{yy}=\sigma _1$, and the electric field direction of the incident $s$-polarized light is along graphene ribbons; while $\varphi =90^\circ$ and $B=0$, $\sigma _{xy}=\sigma _{yx}=0$, $\sigma _{xx}=\sigma _1$, $\sigma _{yy}=\sigma _2$, and the electric field direction of the $s$-polarized light is orthogonal to graphene ribbons. Here, the normalized effective conductivity $\sigma _1\approx 0.05+0.23i$ and $\sigma _2\approx 0.32-1.20i$. The real part of $[\sigma _1]$ and $[\sigma _2]$ is related to energy dissipation. Re$[\sigma _1]<$ Re$[\sigma _2]$ means the low energy dissipation direction is along graphene ribbons. Im$[\sigma _1]$Im$[\sigma _2]<0$, thus the behavior of graphene metasurfaces corresponds to the hyperbolic regime [27]. It can be concluded that the relative intensity of the sideband OAM modes reaches the maximum when the electric field direction of the incident $s$-polarized light is along the low energy dissipation direction, i.e., graphene ribbons. When the electric field direction of the incident $s$-polarized light is along the high energy dissipation direction, the relative intensity of the sideband OAM modes achieves the minimum.

 figure: Fig. 5.

Fig. 5. The relative strength for $s$-polarized Laguerre-Guass modes. Parameters are the same as in Fig. 2.

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Due to nonspecular reflection upon graphene hyperbolic metasurfaces, Goos-H$\ddot {\mbox {a}}$nchen and Imbert-Fedorov effects cause mode-dependent displacement, resulting in energy coupling between the central mode and neighboring OAM modes [see Fig. 6(a,b)]. The pure LG input mode $\{\phi _0^l\}$ will be transformed into a superposition of LG modes $\{\phi _0^{-l-1}, \phi _0^{-l}, \phi _0^{-l+1}\}$ upon nonspecular reflection. The relative intensity of the sideband OAM mode $\phi _0^{-l-1}$ are larger than that of $\phi _0^{-l+1}$. Some values of the relative intensity are as follows: $I^s_{0,-1}\approx 0.16\%$, $I^s_{0,0}\approx 98.84\%$, $I^s_{0,1}\approx 0$, $I^s_{3,-4}\approx 36.38\%$, $I^s_{3,-3}\approx 36.34\%$, $I^s_{3,-2}\approx 27.28\%$, $I^s_{5,-6}\approx 46.16\%$, $I^s_{5,-5}\approx 15.37\%$, $I^s_{5,-4}\approx 38.47\%$, etc. The spatial Fresnel coefficients $c_{l,l'}$, indicating the coupling between the central mode and neighboring OAM modes, are proportional to $\sqrt {|l|+1}$ or $\sqrt {|l|}$ in Eq. (12), thus the side OAM modes grow with increasing of the topologic charge $l$. This is in accordance with the theoretical analysis on vortex beams reflected on graphene surface [11] and vortex beams transmitted through a thin metamaterial slab [12]. The input LG beam with the low topologic charge $l$ can be utilized to suppress the neighboring OAM modes, while the input LG beam with high topologic charge $l$ is beneficial to mode transformation.

 figure: Fig. 6.

Fig. 6. The relative strength for $s$-polarized Laguerre-Guass modes. Parameters are the same as in Fig. 2.

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4. Conclusion

We have theoretically and numerically explored the OAM sidebands of vortex beams reflecting upon graphene metasurfaces. The phase ${\Phi }_{ss}$ of the Fresnel reflection coefficients can reflect the topological transition between the hyperbolic and elliptic topology via the sign change of ${\Phi }_{ss}$. For $s$-polarized beams, the relative intensity of the sideband OAM modes locally has a small trough at the topological transition point and energy couples between the central and sideband OAM modes in the certain region. For $p$-polarized beams, the sideband OAM modes are tiny in wide terahertz frequency band and little disturbed by external electrical or magnetic field. By rotating the graphene metasurfaces, it is shown that the mode transformation of vortex beams dominates when the electric field is along the low energy dissipation direction of the graphene metasurfaces, while the sideband OAM modes are suppressed in the direction of the high energy dissipation. With increasing the topologic charge $l$, energy transfers from the central mode to the neighboring OAM modes.

Funding

Fundamental Research Funds for the Central Universities (CZY20010, YZF20001); National Natural Science Foundation of China (62171487).

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the present research.

References

1. L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]  

2. V. Fedoseyev, “Spin-independent transverse shift of the centre of gravity of a reflected and of a refracted light beam,” Opt. Commun. 193(1-6), 9–18 (2001). [CrossRef]  

3. K. Y. Bliokh, I. V. Shadrivov, and Y. S. Kivshar, “Goos–Hänchen and Imbert–Fedorov shifts of polarized vortex beams,” Opt. Lett. 34(3), 389–391 (2009). [CrossRef]  

4. M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010). [CrossRef]  

5. W. Löffler, A. Aiello, and J. P. Woerdman, “Observation of orbital angular momentum sidebands due to optical reflection,” Phys. Rev. Lett. 109(11), 113602 (2012). [CrossRef]  

6. Z. Xiao, H. Luo, and S. Wen, “Goos-Hänchen and Imbert-Fedorov shifts of vortex beams at air–left-handed-material interfaces,” Phys. Rev. A 85(5), 053822 (2012). [CrossRef]  

7. W. Long, J. Pan, X. Guo, X. Liu, H. Lin, H. Zheng, J. Yu, H. Guan, H. Lu, Y. Zhong, S. Fu, L. Zhang, W. Zhu, and Z. Chen, “Optimized weak measurement of orbital angular momentum-induced beam shifts in optical reflection,” Photonics Res. 7(11), 1273–1278 (2019). [CrossRef]  

8. Z. Chen, H. Zhang, X. Zhang, H. Li, W. Zhang, and L. Xi, “Cross-coupling effect induced beam shifts for polarized vortex beam at two-dimensional anisotropic monolayer graphene surface,” Opt. Express 28(6), 8308–8323 (2020). [CrossRef]  

9. Z. Chen, H. Zhang, X. Zhang, H. Li, J. Yang, W. Zhang, L. Xi, and X. Tang, “Symmetric spin splitting of elliptically polarized vortex beams reflected at air-gold interface via pseudo-brewster angle,” Opt. Express 28(20), 29529–29539 (2020). [CrossRef]  

10. W. Löffler, N. Hermosa, A. Aiello, and J. P. Woerdman, “Total internal reflection of orbital angular momentum beams,” J. Opt. 15(1), 014012 (2013). [CrossRef]  

11. L. Zhuo, W. Long, M. Jiang, W. Zhu, H. Guan, J. Tang, J. Yu, H. Lu, J. Zhang, and Z. Chen, “Graphene-based tunable Imbert–Fedorov shifts and orbital angular momentum sidebands for reflected vortex beams in the terahertz region,” Opt. Lett. 43(12), 2823–2826 (2018). [CrossRef]  

12. W. Zhu, H. Guan, H. Lu, J. Tang, Z. Li, J. Yu, and Z. Chen, “Orbital angular momentum sidebands of vortex beams transmitted through a thin metamaterial slab,” Opt. Express 26(13), 17378–17387 (2018). [CrossRef]  

13. A. R. Ndjiongue, T. M. N. Ngatched, O. A. Dobre, A. G. Armada, and H. Haas, “Analysis of RIS-based terrestrial-FSO link over G-G turbulence with distance and jitter ratios,” J. Lightwave Technol. 39(21), 6746–6758 (2021). [CrossRef]  

14. P. Huo, S. Zhang, Y. Liang, Y. Lu, and T. Xu, “Hyperbolic metamaterials and metasurfaces: fundamentals and applications,” Adv. Opt. Mater. 7(14), 1801616 (2019). [CrossRef]  

15. A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10(11), 937–943 (2015). [CrossRef]  

16. S. Yu, L. Li, and N. Kou, “Generation, reception and separation of mixed-state orbital angular momentum vortex beams using metasurfaces,” Opt. Mater. Express 7(9), 3312–3321 (2017). [CrossRef]  

17. H. Ahmed, A. A. Rahim, H. Maab, M. M. Ali, N. Mahmood, and S. Naureen, “Phase engineering with all-dielectric metasurfaces for focused-optical-vortex beams with high cross-polarization efficiency,” Opt. Mater. Express 10(2), 434–448 (2020). [CrossRef]  

18. J. Han, X. Cao, J. Gao, J. Wei, S. Li, and H. Yang, “Broadband polarization-dependent coding metasurface for reflection manipulation,” Opt. Mater. Express 10(6), 1421–1431 (2020). [CrossRef]  

19. Z. Li, R. Yang, J. Wang, Y. Zhao, J. Tian, and W. Zhang, “Multifunctional metasurface for broadband absorption, linear and circular polarization conversions,” Opt. Mater. Express 11(10), 3507–3519 (2021). [CrossRef]  

20. A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013). [CrossRef]  

21. X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin hall effect at metasurfaces,” Science 339(6126), 1405–1407 (2013). [CrossRef]  

22. M. Wang, H. Zhang, T. Kovalevich, R. Salut, M.-S. Kim, M. A. Suarez, M. P. Bernal, H. P. Herzig, H. Lu, and T. Grosjean, “Magnetic spin-orbit interaction of light,” Light: Sci. Appl. 7(1), 24 (2018). [CrossRef]  

23. M. Wang, R. Salut, H. Lu, M. A. Suarez, N. Martin, and T. Grosjean, “Subwavelength polarization optics via individual and coupled helical traveling-wave nanoantennas,” Light: Sci. Appl. 8(1), 76 (2019). [CrossRef]  

24. J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic metasurfaces: surface plasmons, light-matter interactions, and physical implementation using graphene strips [invited],” Opt. Mater. Express 5(10), 2313–2329 (2015). [CrossRef]  

25. A. Mekawy and A. Alù, “Hyperbolic surface wave propagation in mid-infrared metasurfaces with extreme anisotropy,” JPhys Photonics 3(3), 034006 (2021). [CrossRef]  

26. O. Yermakov, V. Lenets, A. Sayanskiy, J. Baena, E. Martini, S. Glybovski, and S. Maci, “Surface waves on self-complementary metasurfaces: all-frequency hyperbolicity, extreme canalization, and TE-TM polarization degeneracy,” Phys. Rev. X 11(3), 031038 (2021). [CrossRef]  

27. O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Y. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015). [CrossRef]  

28. 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]  

29. P. Li, I. Dolado, F. J. A. Mozaz, F. Casanova, L. E. Hueso, S. Liu, J. H. Edgar, A. Y. Nikitin, S. Vélez, and R. Hillenbrand, “Infrared hyperbolic metasurface based on nanostructured van der Waals materials,” Science 359(6378), 892–896 (2018). [CrossRef]  

30. Z. Li, J. S. T. Smalley, R. Haroldson, D. Lin, R. Hawkins, A. Gharajeh, J. Moon, J. Hou, C. Zhang, W. Hu, A. Zakhidov, and Q. Gu, “Active perovskite hyperbolic metasurface,” ACS Photonics 7(7), 1754–1761 (2020). [CrossRef]  

31. A. Nemilentsau, T. Low, and G. Hanson, “Anisotropic 2D materials for tunable hyperbolic plasmonics,” Phys. Rev. Lett. 116(6), 066804 (2016). [CrossRef]  

32. J. S. Gomez-Diaz and A. Alù, “Flatland optics with hyperbolic metasurfaces,” ACS Photonics 3(12), 2211–2224 (2016). [CrossRef]  

33. O. V. Kotov and Y. E. Lozovik, “Enhanced optical activity in hyperbolic metasurfaces,” Phys. Rev. B 96(23), 235403 (2017). [CrossRef]  

34. X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88(5), 053840 (2013). [CrossRef]  

35. K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013). [CrossRef]  

36. S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020). [CrossRef]  

37. Y. Wen, I. Chremmos, Y. Chen, Y. Zhang, and S. Yu, “Arbitrary multiplication and division of the orbital angular momentum of light,” Phys. Rev. Lett. 124(21), 213901 (2020). [CrossRef]  

38. M. Merano, “Optical beam shifts in graphene and single-layer boron-nitride,” Opt. Lett. 41(24), 5780–5783 (2016). [CrossRef]  

39. M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95(6), 063827 (2017). [CrossRef]  

References

  • View by:

  1. L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref]
  2. V. Fedoseyev, “Spin-independent transverse shift of the centre of gravity of a reflected and of a refracted light beam,” Opt. Commun. 193(1-6), 9–18 (2001).
    [Crossref]
  3. K. Y. Bliokh, I. V. Shadrivov, and Y. S. Kivshar, “Goos–Hänchen and Imbert–Fedorov shifts of polarized vortex beams,” Opt. Lett. 34(3), 389–391 (2009).
    [Crossref]
  4. M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
    [Crossref]
  5. W. Löffler, A. Aiello, and J. P. Woerdman, “Observation of orbital angular momentum sidebands due to optical reflection,” Phys. Rev. Lett. 109(11), 113602 (2012).
    [Crossref]
  6. Z. Xiao, H. Luo, and S. Wen, “Goos-Hänchen and Imbert-Fedorov shifts of vortex beams at air–left-handed-material interfaces,” Phys. Rev. A 85(5), 053822 (2012).
    [Crossref]
  7. W. Long, J. Pan, X. Guo, X. Liu, H. Lin, H. Zheng, J. Yu, H. Guan, H. Lu, Y. Zhong, S. Fu, L. Zhang, W. Zhu, and Z. Chen, “Optimized weak measurement of orbital angular momentum-induced beam shifts in optical reflection,” Photonics Res. 7(11), 1273–1278 (2019).
    [Crossref]
  8. Z. Chen, H. Zhang, X. Zhang, H. Li, W. Zhang, and L. Xi, “Cross-coupling effect induced beam shifts for polarized vortex beam at two-dimensional anisotropic monolayer graphene surface,” Opt. Express 28(6), 8308–8323 (2020).
    [Crossref]
  9. Z. Chen, H. Zhang, X. Zhang, H. Li, J. Yang, W. Zhang, L. Xi, and X. Tang, “Symmetric spin splitting of elliptically polarized vortex beams reflected at air-gold interface via pseudo-brewster angle,” Opt. Express 28(20), 29529–29539 (2020).
    [Crossref]
  10. W. Löffler, N. Hermosa, A. Aiello, and J. P. Woerdman, “Total internal reflection of orbital angular momentum beams,” J. Opt. 15(1), 014012 (2013).
    [Crossref]
  11. L. Zhuo, W. Long, M. Jiang, W. Zhu, H. Guan, J. Tang, J. Yu, H. Lu, J. Zhang, and Z. Chen, “Graphene-based tunable Imbert–Fedorov shifts and orbital angular momentum sidebands for reflected vortex beams in the terahertz region,” Opt. Lett. 43(12), 2823–2826 (2018).
    [Crossref]
  12. W. Zhu, H. Guan, H. Lu, J. Tang, Z. Li, J. Yu, and Z. Chen, “Orbital angular momentum sidebands of vortex beams transmitted through a thin metamaterial slab,” Opt. Express 26(13), 17378–17387 (2018).
    [Crossref]
  13. A. R. Ndjiongue, T. M. N. Ngatched, O. A. Dobre, A. G. Armada, and H. Haas, “Analysis of RIS-based terrestrial-FSO link over G-G turbulence with distance and jitter ratios,” J. Lightwave Technol. 39(21), 6746–6758 (2021).
    [Crossref]
  14. P. Huo, S. Zhang, Y. Liang, Y. Lu, and T. Xu, “Hyperbolic metamaterials and metasurfaces: fundamentals and applications,” Adv. Opt. Mater. 7(14), 1801616 (2019).
    [Crossref]
  15. A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10(11), 937–943 (2015).
    [Crossref]
  16. S. Yu, L. Li, and N. Kou, “Generation, reception and separation of mixed-state orbital angular momentum vortex beams using metasurfaces,” Opt. Mater. Express 7(9), 3312–3321 (2017).
    [Crossref]
  17. H. Ahmed, A. A. Rahim, H. Maab, M. M. Ali, N. Mahmood, and S. Naureen, “Phase engineering with all-dielectric metasurfaces for focused-optical-vortex beams with high cross-polarization efficiency,” Opt. Mater. Express 10(2), 434–448 (2020).
    [Crossref]
  18. J. Han, X. Cao, J. Gao, J. Wei, S. Li, and H. Yang, “Broadband polarization-dependent coding metasurface for reflection manipulation,” Opt. Mater. Express 10(6), 1421–1431 (2020).
    [Crossref]
  19. Z. Li, R. Yang, J. Wang, Y. Zhao, J. Tian, and W. Zhang, “Multifunctional metasurface for broadband absorption, linear and circular polarization conversions,” Opt. Mater. Express 11(10), 3507–3519 (2021).
    [Crossref]
  20. A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
    [Crossref]
  21. X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin hall effect at metasurfaces,” Science 339(6126), 1405–1407 (2013).
    [Crossref]
  22. M. Wang, H. Zhang, T. Kovalevich, R. Salut, M.-S. Kim, M. A. Suarez, M. P. Bernal, H. P. Herzig, H. Lu, and T. Grosjean, “Magnetic spin-orbit interaction of light,” Light: Sci. Appl. 7(1), 24 (2018).
    [Crossref]
  23. M. Wang, R. Salut, H. Lu, M. A. Suarez, N. Martin, and T. Grosjean, “Subwavelength polarization optics via individual and coupled helical traveling-wave nanoantennas,” Light: Sci. Appl. 8(1), 76 (2019).
    [Crossref]
  24. J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic metasurfaces: surface plasmons, light-matter interactions, and physical implementation using graphene strips [invited],” Opt. Mater. Express 5(10), 2313–2329 (2015).
    [Crossref]
  25. A. Mekawy and A. Alù, “Hyperbolic surface wave propagation in mid-infrared metasurfaces with extreme anisotropy,” JPhys Photonics 3(3), 034006 (2021).
    [Crossref]
  26. O. Yermakov, V. Lenets, A. Sayanskiy, J. Baena, E. Martini, S. Glybovski, and S. Maci, “Surface waves on self-complementary metasurfaces: all-frequency hyperbolicity, extreme canalization, and TE-TM polarization degeneracy,” Phys. Rev. X 11(3), 031038 (2021).
    [Crossref]
  27. O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Y. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
    [Crossref]
  28. 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]
  29. P. Li, I. Dolado, F. J. A. Mozaz, F. Casanova, L. E. Hueso, S. Liu, J. H. Edgar, A. Y. Nikitin, S. Vélez, and R. Hillenbrand, “Infrared hyperbolic metasurface based on nanostructured van der Waals materials,” Science 359(6378), 892–896 (2018).
    [Crossref]
  30. Z. Li, J. S. T. Smalley, R. Haroldson, D. Lin, R. Hawkins, A. Gharajeh, J. Moon, J. Hou, C. Zhang, W. Hu, A. Zakhidov, and Q. Gu, “Active perovskite hyperbolic metasurface,” ACS Photonics 7(7), 1754–1761 (2020).
    [Crossref]
  31. A. Nemilentsau, T. Low, and G. Hanson, “Anisotropic 2D materials for tunable hyperbolic plasmonics,” Phys. Rev. Lett. 116(6), 066804 (2016).
    [Crossref]
  32. J. S. Gomez-Diaz and A. Alù, “Flatland optics with hyperbolic metasurfaces,” ACS Photonics 3(12), 2211–2224 (2016).
    [Crossref]
  33. O. V. Kotov and Y. E. Lozovik, “Enhanced optical activity in hyperbolic metasurfaces,” Phys. Rev. B 96(23), 235403 (2017).
    [Crossref]
  34. X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88(5), 053840 (2013).
    [Crossref]
  35. K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
    [Crossref]
  36. S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020).
    [Crossref]
  37. Y. Wen, I. Chremmos, Y. Chen, Y. Zhang, and S. Yu, “Arbitrary multiplication and division of the orbital angular momentum of light,” Phys. Rev. Lett. 124(21), 213901 (2020).
    [Crossref]
  38. M. Merano, “Optical beam shifts in graphene and single-layer boron-nitride,” Opt. Lett. 41(24), 5780–5783 (2016).
    [Crossref]
  39. M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95(6), 063827 (2017).
    [Crossref]

2021 (4)

A. R. Ndjiongue, T. M. N. Ngatched, O. A. Dobre, A. G. Armada, and H. Haas, “Analysis of RIS-based terrestrial-FSO link over G-G turbulence with distance and jitter ratios,” J. Lightwave Technol. 39(21), 6746–6758 (2021).
[Crossref]

Z. Li, R. Yang, J. Wang, Y. Zhao, J. Tian, and W. Zhang, “Multifunctional metasurface for broadband absorption, linear and circular polarization conversions,” Opt. Mater. Express 11(10), 3507–3519 (2021).
[Crossref]

A. Mekawy and A. Alù, “Hyperbolic surface wave propagation in mid-infrared metasurfaces with extreme anisotropy,” JPhys Photonics 3(3), 034006 (2021).
[Crossref]

O. Yermakov, V. Lenets, A. Sayanskiy, J. Baena, E. Martini, S. Glybovski, and S. Maci, “Surface waves on self-complementary metasurfaces: all-frequency hyperbolicity, extreme canalization, and TE-TM polarization degeneracy,” Phys. Rev. X 11(3), 031038 (2021).
[Crossref]

2020 (7)

Z. Li, J. S. T. Smalley, R. Haroldson, D. Lin, R. Hawkins, A. Gharajeh, J. Moon, J. Hou, C. Zhang, W. Hu, A. Zakhidov, and Q. Gu, “Active perovskite hyperbolic metasurface,” ACS Photonics 7(7), 1754–1761 (2020).
[Crossref]

S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020).
[Crossref]

Y. Wen, I. Chremmos, Y. Chen, Y. Zhang, and S. Yu, “Arbitrary multiplication and division of the orbital angular momentum of light,” Phys. Rev. Lett. 124(21), 213901 (2020).
[Crossref]

H. Ahmed, A. A. Rahim, H. Maab, M. M. Ali, N. Mahmood, and S. Naureen, “Phase engineering with all-dielectric metasurfaces for focused-optical-vortex beams with high cross-polarization efficiency,” Opt. Mater. Express 10(2), 434–448 (2020).
[Crossref]

J. Han, X. Cao, J. Gao, J. Wei, S. Li, and H. Yang, “Broadband polarization-dependent coding metasurface for reflection manipulation,” Opt. Mater. Express 10(6), 1421–1431 (2020).
[Crossref]

Z. Chen, H. Zhang, X. Zhang, H. Li, W. Zhang, and L. Xi, “Cross-coupling effect induced beam shifts for polarized vortex beam at two-dimensional anisotropic monolayer graphene surface,” Opt. Express 28(6), 8308–8323 (2020).
[Crossref]

Z. Chen, H. Zhang, X. Zhang, H. Li, J. Yang, W. Zhang, L. Xi, and X. Tang, “Symmetric spin splitting of elliptically polarized vortex beams reflected at air-gold interface via pseudo-brewster angle,” Opt. Express 28(20), 29529–29539 (2020).
[Crossref]

2019 (3)

W. Long, J. Pan, X. Guo, X. Liu, H. Lin, H. Zheng, J. Yu, H. Guan, H. Lu, Y. Zhong, S. Fu, L. Zhang, W. Zhu, and Z. Chen, “Optimized weak measurement of orbital angular momentum-induced beam shifts in optical reflection,” Photonics Res. 7(11), 1273–1278 (2019).
[Crossref]

P. Huo, S. Zhang, Y. Liang, Y. Lu, and T. Xu, “Hyperbolic metamaterials and metasurfaces: fundamentals and applications,” Adv. Opt. Mater. 7(14), 1801616 (2019).
[Crossref]

M. Wang, R. Salut, H. Lu, M. A. Suarez, N. Martin, and T. Grosjean, “Subwavelength polarization optics via individual and coupled helical traveling-wave nanoantennas,” Light: Sci. Appl. 8(1), 76 (2019).
[Crossref]

2018 (4)

M. Wang, H. Zhang, T. Kovalevich, R. Salut, M.-S. Kim, M. A. Suarez, M. P. Bernal, H. P. Herzig, H. Lu, and T. Grosjean, “Magnetic spin-orbit interaction of light,” Light: Sci. Appl. 7(1), 24 (2018).
[Crossref]

P. Li, I. Dolado, F. J. A. Mozaz, F. Casanova, L. E. Hueso, S. Liu, J. H. Edgar, A. Y. Nikitin, S. Vélez, and R. Hillenbrand, “Infrared hyperbolic metasurface based on nanostructured van der Waals materials,” Science 359(6378), 892–896 (2018).
[Crossref]

L. Zhuo, W. Long, M. Jiang, W. Zhu, H. Guan, J. Tang, J. Yu, H. Lu, J. Zhang, and Z. Chen, “Graphene-based tunable Imbert–Fedorov shifts and orbital angular momentum sidebands for reflected vortex beams in the terahertz region,” Opt. Lett. 43(12), 2823–2826 (2018).
[Crossref]

W. Zhu, H. Guan, H. Lu, J. Tang, Z. Li, J. Yu, and Z. Chen, “Orbital angular momentum sidebands of vortex beams transmitted through a thin metamaterial slab,” Opt. Express 26(13), 17378–17387 (2018).
[Crossref]

2017 (3)

S. Yu, L. Li, and N. Kou, “Generation, reception and separation of mixed-state orbital angular momentum vortex beams using metasurfaces,” Opt. Mater. Express 7(9), 3312–3321 (2017).
[Crossref]

O. V. Kotov and Y. E. Lozovik, “Enhanced optical activity in hyperbolic metasurfaces,” Phys. Rev. B 96(23), 235403 (2017).
[Crossref]

M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95(6), 063827 (2017).
[Crossref]

2016 (3)

M. Merano, “Optical beam shifts in graphene and single-layer boron-nitride,” Opt. Lett. 41(24), 5780–5783 (2016).
[Crossref]

A. Nemilentsau, T. Low, and G. Hanson, “Anisotropic 2D materials for tunable hyperbolic plasmonics,” Phys. Rev. Lett. 116(6), 066804 (2016).
[Crossref]

J. S. Gomez-Diaz and A. Alù, “Flatland optics with hyperbolic metasurfaces,” ACS Photonics 3(12), 2211–2224 (2016).
[Crossref]

2015 (4)

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic metasurfaces: surface plasmons, light-matter interactions, and physical implementation using graphene strips [invited],” Opt. Mater. Express 5(10), 2313–2329 (2015).
[Crossref]

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Y. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

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]

A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10(11), 937–943 (2015).
[Crossref]

2013 (5)

W. Löffler, N. Hermosa, A. Aiello, and J. P. Woerdman, “Total internal reflection of orbital angular momentum beams,” J. Opt. 15(1), 014012 (2013).
[Crossref]

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref]

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin hall effect at metasurfaces,” Science 339(6126), 1405–1407 (2013).
[Crossref]

X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88(5), 053840 (2013).
[Crossref]

K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
[Crossref]

2012 (2)

W. Löffler, A. Aiello, and J. P. Woerdman, “Observation of orbital angular momentum sidebands due to optical reflection,” Phys. Rev. Lett. 109(11), 113602 (2012).
[Crossref]

Z. Xiao, H. Luo, and S. Wen, “Goos-Hänchen and Imbert-Fedorov shifts of vortex beams at air–left-handed-material interfaces,” Phys. Rev. A 85(5), 053822 (2012).
[Crossref]

2010 (1)

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[Crossref]

2009 (1)

2001 (1)

V. Fedoseyev, “Spin-independent transverse shift of the centre of gravity of a reflected and of a refracted light beam,” Opt. Commun. 193(1-6), 9–18 (2001).
[Crossref]

1992 (1)

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Ahmed, H.

Aiello, A.

W. Löffler, N. Hermosa, A. Aiello, and J. P. Woerdman, “Total internal reflection of orbital angular momentum beams,” J. Opt. 15(1), 014012 (2013).
[Crossref]

K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
[Crossref]

W. Löffler, A. Aiello, and J. P. Woerdman, “Observation of orbital angular momentum sidebands due to optical reflection,” Phys. Rev. Lett. 109(11), 113602 (2012).
[Crossref]

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[Crossref]

Ali, M. M.

Allen, L.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Alù, A.

A. Mekawy and A. Alù, “Hyperbolic surface wave propagation in mid-infrared metasurfaces with extreme anisotropy,” JPhys Photonics 3(3), 034006 (2021).
[Crossref]

J. S. Gomez-Diaz and A. Alù, “Flatland optics with hyperbolic metasurfaces,” ACS Photonics 3(12), 2211–2224 (2016).
[Crossref]

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic metasurfaces: surface plasmons, light-matter interactions, and physical implementation using graphene strips [invited],” Opt. Mater. Express 5(10), 2313–2329 (2015).
[Crossref]

Arbabi, A.

A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10(11), 937–943 (2015).
[Crossref]

Armada, A. G.

Baena, J.

O. Yermakov, V. Lenets, A. Sayanskiy, J. Baena, E. Martini, S. Glybovski, and S. Maci, “Surface waves on self-complementary metasurfaces: all-frequency hyperbolicity, extreme canalization, and TE-TM polarization degeneracy,” Phys. Rev. X 11(3), 031038 (2021).
[Crossref]

Bagheri, M.

A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10(11), 937–943 (2015).
[Crossref]

Beijersbergen, M.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Bernal, M. P.

M. Wang, H. Zhang, T. Kovalevich, R. Salut, M.-S. Kim, M. A. Suarez, M. P. Bernal, H. P. Herzig, H. Lu, and T. Grosjean, “Magnetic spin-orbit interaction of light,” Light: Sci. Appl. 7(1), 24 (2018).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
[Crossref]

K. Y. Bliokh, I. V. Shadrivov, and Y. S. Kivshar, “Goos–Hänchen and Imbert–Fedorov shifts of polarized vortex beams,” Opt. Lett. 34(3), 389–391 (2009).
[Crossref]

Bogdanov, A. A.

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Y. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

Boltasseva, A.

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref]

Cai, L.

M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95(6), 063827 (2017).
[Crossref]

Cao, X.

Casanova, F.

P. Li, I. Dolado, F. J. A. Mozaz, F. Casanova, L. E. Hueso, S. Liu, J. H. Edgar, A. Y. Nikitin, S. Vélez, and R. Hillenbrand, “Infrared hyperbolic metasurface based on nanostructured van der Waals materials,” Science 359(6378), 892–896 (2018).
[Crossref]

Chen, S.

M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95(6), 063827 (2017).
[Crossref]

X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88(5), 053840 (2013).
[Crossref]

Chen, Y.

Y. Wen, I. Chremmos, Y. Chen, Y. Zhang, and S. Yu, “Arbitrary multiplication and division of the orbital angular momentum of light,” Phys. Rev. Lett. 124(21), 213901 (2020).
[Crossref]

Chen, Z.

Chremmos, I.

Y. Wen, I. Chremmos, Y. Chen, Y. Zhang, and S. Yu, “Arbitrary multiplication and division of the orbital angular momentum of light,” Phys. Rev. Lett. 124(21), 213901 (2020).
[Crossref]

de Leon, N. P.

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]

Devlin, R. C.

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]

Dibos, A.

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]

Dobre, O. A.

Dolado, I.

P. Li, I. Dolado, F. J. A. Mozaz, F. Casanova, L. E. Hueso, S. Liu, J. H. Edgar, A. Y. Nikitin, S. Vélez, and R. Hillenbrand, “Infrared hyperbolic metasurface based on nanostructured van der Waals materials,” Science 359(6378), 892–896 (2018).
[Crossref]

Edgar, J. H.

P. Li, I. Dolado, F. J. A. Mozaz, F. Casanova, L. E. Hueso, S. Liu, J. H. Edgar, A. Y. Nikitin, S. Vélez, and R. Hillenbrand, “Infrared hyperbolic metasurface based on nanostructured van der Waals materials,” Science 359(6378), 892–896 (2018).
[Crossref]

Faraon, A.

A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10(11), 937–943 (2015).
[Crossref]

Fedoseyev, V.

V. Fedoseyev, “Spin-independent transverse shift of the centre of gravity of a reflected and of a refracted light beam,” Opt. Commun. 193(1-6), 9–18 (2001).
[Crossref]

Fu, S.

W. Long, J. Pan, X. Guo, X. Liu, H. Lin, H. Zheng, J. Yu, H. Guan, H. Lu, Y. Zhong, S. Fu, L. Zhang, W. Zhu, and Z. Chen, “Optimized weak measurement of orbital angular momentum-induced beam shifts in optical reflection,” Photonics Res. 7(11), 1273–1278 (2019).
[Crossref]

Gao, J.

Gharajeh, A.

Z. Li, J. S. T. Smalley, R. Haroldson, D. Lin, R. Hawkins, A. Gharajeh, J. Moon, J. Hou, C. Zhang, W. Hu, A. Zakhidov, and Q. Gu, “Active perovskite hyperbolic metasurface,” ACS Photonics 7(7), 1754–1761 (2020).
[Crossref]

Glybovski, S.

O. Yermakov, V. Lenets, A. Sayanskiy, J. Baena, E. Martini, S. Glybovski, and S. Maci, “Surface waves on self-complementary metasurfaces: all-frequency hyperbolicity, extreme canalization, and TE-TM polarization degeneracy,” Phys. Rev. X 11(3), 031038 (2021).
[Crossref]

Gomez-Diaz, J. S.

Grosjean, T.

M. Wang, R. Salut, H. Lu, M. A. Suarez, N. Martin, and T. Grosjean, “Subwavelength polarization optics via individual and coupled helical traveling-wave nanoantennas,” Light: Sci. Appl. 8(1), 76 (2019).
[Crossref]

M. Wang, H. Zhang, T. Kovalevich, R. Salut, M.-S. Kim, M. A. Suarez, M. P. Bernal, H. P. Herzig, H. Lu, and T. Grosjean, “Magnetic spin-orbit interaction of light,” Light: Sci. Appl. 7(1), 24 (2018).
[Crossref]

Gu, Q.

Z. Li, J. S. T. Smalley, R. Haroldson, D. Lin, R. Hawkins, A. Gharajeh, J. Moon, J. Hou, C. Zhang, W. Hu, A. Zakhidov, and Q. Gu, “Active perovskite hyperbolic metasurface,” ACS Photonics 7(7), 1754–1761 (2020).
[Crossref]

Guan, H.

Guo, X.

W. Long, J. Pan, X. Guo, X. Liu, H. Lin, H. Zheng, J. Yu, H. Guan, H. Lu, Y. Zhong, S. Fu, L. Zhang, W. Zhu, and Z. Chen, “Optimized weak measurement of orbital angular momentum-induced beam shifts in optical reflection,” Photonics Res. 7(11), 1273–1278 (2019).
[Crossref]

Haas, H.

Han, J.

Hanson, G.

A. Nemilentsau, T. Low, and G. Hanson, “Anisotropic 2D materials for tunable hyperbolic plasmonics,” Phys. Rev. Lett. 116(6), 066804 (2016).
[Crossref]

Haroldson, R.

Z. Li, J. S. T. Smalley, R. Haroldson, D. Lin, R. Hawkins, A. Gharajeh, J. Moon, J. Hou, C. Zhang, W. Hu, A. Zakhidov, and Q. Gu, “Active perovskite hyperbolic metasurface,” ACS Photonics 7(7), 1754–1761 (2020).
[Crossref]

Hawkins, R.

Z. Li, J. S. T. Smalley, R. Haroldson, D. Lin, R. Hawkins, A. Gharajeh, J. Moon, J. Hou, C. Zhang, W. Hu, A. Zakhidov, and Q. Gu, “Active perovskite hyperbolic metasurface,” ACS Photonics 7(7), 1754–1761 (2020).
[Crossref]

Hermosa, N.

W. Löffler, N. Hermosa, A. Aiello, and J. P. Woerdman, “Total internal reflection of orbital angular momentum beams,” J. Opt. 15(1), 014012 (2013).
[Crossref]

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[Crossref]

Herzig, H. P.

M. Wang, H. Zhang, T. Kovalevich, R. Salut, M.-S. Kim, M. A. Suarez, M. P. Bernal, H. P. Herzig, H. Lu, and T. Grosjean, “Magnetic spin-orbit interaction of light,” Light: Sci. Appl. 7(1), 24 (2018).
[Crossref]

High, A. A.

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]

Hillenbrand, R.

P. Li, I. Dolado, F. J. A. Mozaz, F. Casanova, L. E. Hueso, S. Liu, J. H. Edgar, A. Y. Nikitin, S. Vélez, and R. Hillenbrand, “Infrared hyperbolic metasurface based on nanostructured van der Waals materials,” Science 359(6378), 892–896 (2018).
[Crossref]

Horie, Y.

A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10(11), 937–943 (2015).
[Crossref]

Hou, J.

Z. Li, J. S. T. Smalley, R. Haroldson, D. Lin, R. Hawkins, A. Gharajeh, J. Moon, J. Hou, C. Zhang, W. Hu, A. Zakhidov, and Q. Gu, “Active perovskite hyperbolic metasurface,” ACS Photonics 7(7), 1754–1761 (2020).
[Crossref]

Hu, W.

Z. Li, J. S. T. Smalley, R. Haroldson, D. Lin, R. Hawkins, A. Gharajeh, J. Moon, J. Hou, C. Zhang, W. Hu, A. Zakhidov, and Q. Gu, “Active perovskite hyperbolic metasurface,” ACS Photonics 7(7), 1754–1761 (2020).
[Crossref]

Hueso, L. E.

P. Li, I. Dolado, F. J. A. Mozaz, F. Casanova, L. E. Hueso, S. Liu, J. H. Edgar, A. Y. Nikitin, S. Vélez, and R. Hillenbrand, “Infrared hyperbolic metasurface based on nanostructured van der Waals materials,” Science 359(6378), 892–896 (2018).
[Crossref]

Huo, P.

P. Huo, S. Zhang, Y. Liang, Y. Lu, and T. Xu, “Hyperbolic metamaterials and metasurfaces: fundamentals and applications,” Adv. Opt. Mater. 7(14), 1801616 (2019).
[Crossref]

Iorsh, I. V.

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Y. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

Jiang, M.

Kildishev, A. V.

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref]

Kim, M.-S.

M. Wang, H. Zhang, T. Kovalevich, R. Salut, M.-S. Kim, M. A. Suarez, M. P. Bernal, H. P. Herzig, H. Lu, and T. Grosjean, “Magnetic spin-orbit interaction of light,” Light: Sci. Appl. 7(1), 24 (2018).
[Crossref]

Kivshar, Y. S.

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Y. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

K. Y. Bliokh, I. V. Shadrivov, and Y. S. Kivshar, “Goos–Hänchen and Imbert–Fedorov shifts of polarized vortex beams,” Opt. Lett. 34(3), 389–391 (2009).
[Crossref]

Kotov, O. V.

O. V. Kotov and Y. E. Lozovik, “Enhanced optical activity in hyperbolic metasurfaces,” Phys. Rev. B 96(23), 235403 (2017).
[Crossref]

Kou, N.

Kovalevich, T.

M. Wang, H. Zhang, T. Kovalevich, R. Salut, M.-S. Kim, M. A. Suarez, M. P. Bernal, H. P. Herzig, H. Lu, and T. Grosjean, “Magnetic spin-orbit interaction of light,” Light: Sci. Appl. 7(1), 24 (2018).
[Crossref]

Lenets, V.

O. Yermakov, V. Lenets, A. Sayanskiy, J. Baena, E. Martini, S. Glybovski, and S. Maci, “Surface waves on self-complementary metasurfaces: all-frequency hyperbolicity, extreme canalization, and TE-TM polarization degeneracy,” Phys. Rev. X 11(3), 031038 (2021).
[Crossref]

Li, H.

Li, L.

Li, P.

P. Li, I. Dolado, F. J. A. Mozaz, F. Casanova, L. E. Hueso, S. Liu, J. H. Edgar, A. Y. Nikitin, S. Vélez, and R. Hillenbrand, “Infrared hyperbolic metasurface based on nanostructured van der Waals materials,” Science 359(6378), 892–896 (2018).
[Crossref]

Li, S.

S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020).
[Crossref]

J. Han, X. Cao, J. Gao, J. Wei, S. Li, and H. Yang, “Broadband polarization-dependent coding metasurface for reflection manipulation,” Opt. Mater. Express 10(6), 1421–1431 (2020).
[Crossref]

Li, X.

S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020).
[Crossref]

Li, Z.

Liang, Y.

P. Huo, S. Zhang, Y. Liang, Y. Lu, and T. Xu, “Hyperbolic metamaterials and metasurfaces: fundamentals and applications,” Adv. Opt. Mater. 7(14), 1801616 (2019).
[Crossref]

Lin, D.

Z. Li, J. S. T. Smalley, R. Haroldson, D. Lin, R. Hawkins, A. Gharajeh, J. Moon, J. Hou, C. Zhang, W. Hu, A. Zakhidov, and Q. Gu, “Active perovskite hyperbolic metasurface,” ACS Photonics 7(7), 1754–1761 (2020).
[Crossref]

Lin, H.

W. Long, J. Pan, X. Guo, X. Liu, H. Lin, H. Zheng, J. Yu, H. Guan, H. Lu, Y. Zhong, S. Fu, L. Zhang, W. Zhu, and Z. Chen, “Optimized weak measurement of orbital angular momentum-induced beam shifts in optical reflection,” Photonics Res. 7(11), 1273–1278 (2019).
[Crossref]

Ling, X.

X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88(5), 053840 (2013).
[Crossref]

Liu, M.

S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020).
[Crossref]

M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95(6), 063827 (2017).
[Crossref]

Liu, S.

P. Li, I. Dolado, F. J. A. Mozaz, F. Casanova, L. E. Hueso, S. Liu, J. H. Edgar, A. Y. Nikitin, S. Vélez, and R. Hillenbrand, “Infrared hyperbolic metasurface based on nanostructured van der Waals materials,” Science 359(6378), 892–896 (2018).
[Crossref]

Liu, X.

W. Long, J. Pan, X. Guo, X. Liu, H. Lin, H. Zheng, J. Yu, H. Guan, H. Lu, Y. Zhong, S. Fu, L. Zhang, W. Zhu, and Z. Chen, “Optimized weak measurement of orbital angular momentum-induced beam shifts in optical reflection,” Photonics Res. 7(11), 1273–1278 (2019).
[Crossref]

Liu, Y.

M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95(6), 063827 (2017).
[Crossref]

Löffler, W.

W. Löffler, N. Hermosa, A. Aiello, and J. P. Woerdman, “Total internal reflection of orbital angular momentum beams,” J. Opt. 15(1), 014012 (2013).
[Crossref]

W. Löffler, A. Aiello, and J. P. Woerdman, “Observation of orbital angular momentum sidebands due to optical reflection,” Phys. Rev. Lett. 109(11), 113602 (2012).
[Crossref]

Long, W.

W. Long, J. Pan, X. Guo, X. Liu, H. Lin, H. Zheng, J. Yu, H. Guan, H. Lu, Y. Zhong, S. Fu, L. Zhang, W. Zhu, and Z. Chen, “Optimized weak measurement of orbital angular momentum-induced beam shifts in optical reflection,” Photonics Res. 7(11), 1273–1278 (2019).
[Crossref]

L. Zhuo, W. Long, M. Jiang, W. Zhu, H. Guan, J. Tang, J. Yu, H. Lu, J. Zhang, and Z. Chen, “Graphene-based tunable Imbert–Fedorov shifts and orbital angular momentum sidebands for reflected vortex beams in the terahertz region,” Opt. Lett. 43(12), 2823–2826 (2018).
[Crossref]

Low, T.

A. Nemilentsau, T. Low, and G. Hanson, “Anisotropic 2D materials for tunable hyperbolic plasmonics,” Phys. Rev. Lett. 116(6), 066804 (2016).
[Crossref]

Lozovik, Y. E.

O. V. Kotov and Y. E. Lozovik, “Enhanced optical activity in hyperbolic metasurfaces,” Phys. Rev. B 96(23), 235403 (2017).
[Crossref]

Lu, H.

W. Long, J. Pan, X. Guo, X. Liu, H. Lin, H. Zheng, J. Yu, H. Guan, H. Lu, Y. Zhong, S. Fu, L. Zhang, W. Zhu, and Z. Chen, “Optimized weak measurement of orbital angular momentum-induced beam shifts in optical reflection,” Photonics Res. 7(11), 1273–1278 (2019).
[Crossref]

M. Wang, R. Salut, H. Lu, M. A. Suarez, N. Martin, and T. Grosjean, “Subwavelength polarization optics via individual and coupled helical traveling-wave nanoantennas,” Light: Sci. Appl. 8(1), 76 (2019).
[Crossref]

M. Wang, H. Zhang, T. Kovalevich, R. Salut, M.-S. Kim, M. A. Suarez, M. P. Bernal, H. P. Herzig, H. Lu, and T. Grosjean, “Magnetic spin-orbit interaction of light,” Light: Sci. Appl. 7(1), 24 (2018).
[Crossref]

W. Zhu, H. Guan, H. Lu, J. Tang, Z. Li, J. Yu, and Z. Chen, “Orbital angular momentum sidebands of vortex beams transmitted through a thin metamaterial slab,” Opt. Express 26(13), 17378–17387 (2018).
[Crossref]

L. Zhuo, W. Long, M. Jiang, W. Zhu, H. Guan, J. Tang, J. Yu, H. Lu, J. Zhang, and Z. Chen, “Graphene-based tunable Imbert–Fedorov shifts and orbital angular momentum sidebands for reflected vortex beams in the terahertz region,” Opt. Lett. 43(12), 2823–2826 (2018).
[Crossref]

Lu, Y.

P. Huo, S. Zhang, Y. Liang, Y. Lu, and T. Xu, “Hyperbolic metamaterials and metasurfaces: fundamentals and applications,” Adv. Opt. Mater. 7(14), 1801616 (2019).
[Crossref]

Lukin, M. D.

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]

Luo, H.

M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95(6), 063827 (2017).
[Crossref]

X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88(5), 053840 (2013).
[Crossref]

Z. Xiao, H. Luo, and S. Wen, “Goos-Hänchen and Imbert-Fedorov shifts of vortex beams at air–left-handed-material interfaces,” Phys. Rev. A 85(5), 053822 (2012).
[Crossref]

Maab, H.

Maci, S.

O. Yermakov, V. Lenets, A. Sayanskiy, J. Baena, E. Martini, S. Glybovski, and S. Maci, “Surface waves on self-complementary metasurfaces: all-frequency hyperbolicity, extreme canalization, and TE-TM polarization degeneracy,” Phys. Rev. X 11(3), 031038 (2021).
[Crossref]

Mahmood, N.

Martin, N.

M. Wang, R. Salut, H. Lu, M. A. Suarez, N. Martin, and T. Grosjean, “Subwavelength polarization optics via individual and coupled helical traveling-wave nanoantennas,” Light: Sci. Appl. 8(1), 76 (2019).
[Crossref]

Martini, E.

O. Yermakov, V. Lenets, A. Sayanskiy, J. Baena, E. Martini, S. Glybovski, and S. Maci, “Surface waves on self-complementary metasurfaces: all-frequency hyperbolicity, extreme canalization, and TE-TM polarization degeneracy,” Phys. Rev. X 11(3), 031038 (2021).
[Crossref]

Mekawy, A.

A. Mekawy and A. Alù, “Hyperbolic surface wave propagation in mid-infrared metasurfaces with extreme anisotropy,” JPhys Photonics 3(3), 034006 (2021).
[Crossref]

Merano, M.

M. Merano, “Optical beam shifts in graphene and single-layer boron-nitride,” Opt. Lett. 41(24), 5780–5783 (2016).
[Crossref]

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[Crossref]

Moon, J.

Z. Li, J. S. T. Smalley, R. Haroldson, D. Lin, R. Hawkins, A. Gharajeh, J. Moon, J. Hou, C. Zhang, W. Hu, A. Zakhidov, and Q. Gu, “Active perovskite hyperbolic metasurface,” ACS Photonics 7(7), 1754–1761 (2020).
[Crossref]

Mozaz, F. J. A.

P. Li, I. Dolado, F. J. A. Mozaz, F. Casanova, L. E. Hueso, S. Liu, J. H. Edgar, A. Y. Nikitin, S. Vélez, and R. Hillenbrand, “Infrared hyperbolic metasurface based on nanostructured van der Waals materials,” Science 359(6378), 892–896 (2018).
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Naureen, S.

Ndjiongue, A. R.

Nemilentsau, A.

A. Nemilentsau, T. Low, and G. Hanson, “Anisotropic 2D materials for tunable hyperbolic plasmonics,” Phys. Rev. Lett. 116(6), 066804 (2016).
[Crossref]

Ngatched, T. M. N.

Nikitin, A. Y.

P. Li, I. Dolado, F. J. A. Mozaz, F. Casanova, L. E. Hueso, S. Liu, J. H. Edgar, A. Y. Nikitin, S. Vélez, and R. Hillenbrand, “Infrared hyperbolic metasurface based on nanostructured van der Waals materials,” Science 359(6378), 892–896 (2018).
[Crossref]

Ovcharenko, A. I.

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Y. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
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Pan, J.

W. Long, J. Pan, X. Guo, X. Liu, H. Lin, H. Zheng, J. Yu, H. Guan, H. Lu, Y. Zhong, S. Fu, L. Zhang, W. Zhu, and Z. Chen, “Optimized weak measurement of orbital angular momentum-induced beam shifts in optical reflection,” Photonics Res. 7(11), 1273–1278 (2019).
[Crossref]

Park, H.

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]

Perczel, J.

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]

Polking, M.

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]

Rahim, A. A.

Rho, J.

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin hall effect at metasurfaces,” Science 339(6126), 1405–1407 (2013).
[Crossref]

Salut, R.

M. Wang, R. Salut, H. Lu, M. A. Suarez, N. Martin, and T. Grosjean, “Subwavelength polarization optics via individual and coupled helical traveling-wave nanoantennas,” Light: Sci. Appl. 8(1), 76 (2019).
[Crossref]

M. Wang, H. Zhang, T. Kovalevich, R. Salut, M.-S. Kim, M. A. Suarez, M. P. Bernal, H. P. Herzig, H. Lu, and T. Grosjean, “Magnetic spin-orbit interaction of light,” Light: Sci. Appl. 7(1), 24 (2018).
[Crossref]

Sayanskiy, A.

O. Yermakov, V. Lenets, A. Sayanskiy, J. Baena, E. Martini, S. Glybovski, and S. Maci, “Surface waves on self-complementary metasurfaces: all-frequency hyperbolicity, extreme canalization, and TE-TM polarization degeneracy,” Phys. Rev. X 11(3), 031038 (2021).
[Crossref]

Shadrivov, I. V.

Shalaev, V. M.

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref]

Smalley, J. S. T.

Z. Li, J. S. T. Smalley, R. Haroldson, D. Lin, R. Hawkins, A. Gharajeh, J. Moon, J. Hou, C. Zhang, W. Hu, A. Zakhidov, and Q. Gu, “Active perovskite hyperbolic metasurface,” ACS Photonics 7(7), 1754–1761 (2020).
[Crossref]

Song, M.

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Y. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

Spreeuw, R.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Suarez, M. A.

M. Wang, R. Salut, H. Lu, M. A. Suarez, N. Martin, and T. Grosjean, “Subwavelength polarization optics via individual and coupled helical traveling-wave nanoantennas,” Light: Sci. Appl. 8(1), 76 (2019).
[Crossref]

M. Wang, H. Zhang, T. Kovalevich, R. Salut, M.-S. Kim, M. A. Suarez, M. P. Bernal, H. P. Herzig, H. Lu, and T. Grosjean, “Magnetic spin-orbit interaction of light,” Light: Sci. Appl. 7(1), 24 (2018).
[Crossref]

Sun, Q.

S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020).
[Crossref]

Tang, J.

Tang, X.

Tian, J.

Tymchenko, M.

Vélez, S.

P. Li, I. Dolado, F. J. A. Mozaz, F. Casanova, L. E. Hueso, S. Liu, J. H. Edgar, A. Y. Nikitin, S. Vélez, and R. Hillenbrand, “Infrared hyperbolic metasurface based on nanostructured van der Waals materials,” Science 359(6378), 892–896 (2018).
[Crossref]

Wang, G.

S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020).
[Crossref]

Wang, J.

Wang, L.

S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020).
[Crossref]

Wang, M.

M. Wang, R. Salut, H. Lu, M. A. Suarez, N. Martin, and T. Grosjean, “Subwavelength polarization optics via individual and coupled helical traveling-wave nanoantennas,” Light: Sci. Appl. 8(1), 76 (2019).
[Crossref]

M. Wang, H. Zhang, T. Kovalevich, R. Salut, M.-S. Kim, M. A. Suarez, M. P. Bernal, H. P. Herzig, H. Lu, and T. Grosjean, “Magnetic spin-orbit interaction of light,” Light: Sci. Appl. 7(1), 24 (2018).
[Crossref]

Wang, Y.

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin hall effect at metasurfaces,” Science 339(6126), 1405–1407 (2013).
[Crossref]

Wei, J.

Wen, S.

M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95(6), 063827 (2017).
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X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88(5), 053840 (2013).
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Z. Xiao, H. Luo, and S. Wen, “Goos-Hänchen and Imbert-Fedorov shifts of vortex beams at air–left-handed-material interfaces,” Phys. Rev. A 85(5), 053822 (2012).
[Crossref]

Wen, Y.

Y. Wen, I. Chremmos, Y. Chen, Y. Zhang, and S. Yu, “Arbitrary multiplication and division of the orbital angular momentum of light,” Phys. Rev. Lett. 124(21), 213901 (2020).
[Crossref]

Wild, D. S.

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).
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L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

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W. Löffler, N. Hermosa, A. Aiello, and J. P. Woerdman, “Total internal reflection of orbital angular momentum beams,” J. Opt. 15(1), 014012 (2013).
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W. Löffler, A. Aiello, and J. P. Woerdman, “Observation of orbital angular momentum sidebands due to optical reflection,” Phys. Rev. Lett. 109(11), 113602 (2012).
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M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[Crossref]

Xi, L.

Xiao, Z.

Z. Xiao, H. Luo, and S. Wen, “Goos-Hänchen and Imbert-Fedorov shifts of vortex beams at air–left-handed-material interfaces,” Phys. Rev. A 85(5), 053822 (2012).
[Crossref]

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P. Huo, S. Zhang, Y. Liang, Y. Lu, and T. Xu, “Hyperbolic metamaterials and metasurfaces: fundamentals and applications,” Adv. Opt. Mater. 7(14), 1801616 (2019).
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Yang, J.

Yang, R.

Ye, Z.

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin hall effect at metasurfaces,” Science 339(6126), 1405–1407 (2013).
[Crossref]

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O. Yermakov, V. Lenets, A. Sayanskiy, J. Baena, E. Martini, S. Glybovski, and S. Maci, “Surface waves on self-complementary metasurfaces: all-frequency hyperbolicity, extreme canalization, and TE-TM polarization degeneracy,” Phys. Rev. X 11(3), 031038 (2021).
[Crossref]

Yermakov, O. Y.

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Y. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

Yin, X.

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin hall effect at metasurfaces,” Science 339(6126), 1405–1407 (2013).
[Crossref]

Yu, J.

Yu, S.

Y. Wen, I. Chremmos, Y. Chen, Y. Zhang, and S. Yu, “Arbitrary multiplication and division of the orbital angular momentum of light,” Phys. Rev. Lett. 124(21), 213901 (2020).
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S. Yu, L. Li, and N. Kou, “Generation, reception and separation of mixed-state orbital angular momentum vortex beams using metasurfaces,” Opt. Mater. Express 7(9), 3312–3321 (2017).
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Z. Li, J. S. T. Smalley, R. Haroldson, D. Lin, R. Hawkins, A. Gharajeh, J. Moon, J. Hou, C. Zhang, W. Hu, A. Zakhidov, and Q. Gu, “Active perovskite hyperbolic metasurface,” ACS Photonics 7(7), 1754–1761 (2020).
[Crossref]

Zeng, C.

S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020).
[Crossref]

Zhang, C.

Z. Li, J. S. T. Smalley, R. Haroldson, D. Lin, R. Hawkins, A. Gharajeh, J. Moon, J. Hou, C. Zhang, W. Hu, A. Zakhidov, and Q. Gu, “Active perovskite hyperbolic metasurface,” ACS Photonics 7(7), 1754–1761 (2020).
[Crossref]

Zhang, H.

Zhang, J.

Zhang, L.

S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020).
[Crossref]

S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020).
[Crossref]

W. Long, J. Pan, X. Guo, X. Liu, H. Lin, H. Zheng, J. Yu, H. Guan, H. Lu, Y. Zhong, S. Fu, L. Zhang, W. Zhu, and Z. Chen, “Optimized weak measurement of orbital angular momentum-induced beam shifts in optical reflection,” Photonics Res. 7(11), 1273–1278 (2019).
[Crossref]

Zhang, S.

P. Huo, S. Zhang, Y. Liang, Y. Lu, and T. Xu, “Hyperbolic metamaterials and metasurfaces: fundamentals and applications,” Adv. Opt. Mater. 7(14), 1801616 (2019).
[Crossref]

Zhang, W.

Zhang, X.

Zhang, Y.

Y. Wen, I. Chremmos, Y. Chen, Y. Zhang, and S. Yu, “Arbitrary multiplication and division of the orbital angular momentum of light,” Phys. Rev. Lett. 124(21), 213901 (2020).
[Crossref]

Zhao, W.

S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020).
[Crossref]

Zhao, Y.

Zheng, H.

W. Long, J. Pan, X. Guo, X. Liu, H. Lin, H. Zheng, J. Yu, H. Guan, H. Lu, Y. Zhong, S. Fu, L. Zhang, W. Zhu, and Z. Chen, “Optimized weak measurement of orbital angular momentum-induced beam shifts in optical reflection,” Photonics Res. 7(11), 1273–1278 (2019).
[Crossref]

Zhong, Y.

W. Long, J. Pan, X. Guo, X. Liu, H. Lin, H. Zheng, J. Yu, H. Guan, H. Lu, Y. Zhong, S. Fu, L. Zhang, W. Zhu, and Z. Chen, “Optimized weak measurement of orbital angular momentum-induced beam shifts in optical reflection,” Photonics Res. 7(11), 1273–1278 (2019).
[Crossref]

Zhou, X.

X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88(5), 053840 (2013).
[Crossref]

Zhu, W.

Zhuo, L.

ACS Photonics (2)

Z. Li, J. S. T. Smalley, R. Haroldson, D. Lin, R. Hawkins, A. Gharajeh, J. Moon, J. Hou, C. Zhang, W. Hu, A. Zakhidov, and Q. Gu, “Active perovskite hyperbolic metasurface,” ACS Photonics 7(7), 1754–1761 (2020).
[Crossref]

J. S. Gomez-Diaz and A. Alù, “Flatland optics with hyperbolic metasurfaces,” ACS Photonics 3(12), 2211–2224 (2016).
[Crossref]

Adv. Opt. Mater. (2)

P. Huo, S. Zhang, Y. Liang, Y. Lu, and T. Xu, “Hyperbolic metamaterials and metasurfaces: fundamentals and applications,” Adv. Opt. Mater. 7(14), 1801616 (2019).
[Crossref]

S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. (2)

W. Löffler, N. Hermosa, A. Aiello, and J. P. Woerdman, “Total internal reflection of orbital angular momentum beams,” J. Opt. 15(1), 014012 (2013).
[Crossref]

K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
[Crossref]

JPhys Photonics (1)

A. Mekawy and A. Alù, “Hyperbolic surface wave propagation in mid-infrared metasurfaces with extreme anisotropy,” JPhys Photonics 3(3), 034006 (2021).
[Crossref]

Light: Sci. Appl. (2)

M. Wang, H. Zhang, T. Kovalevich, R. Salut, M.-S. Kim, M. A. Suarez, M. P. Bernal, H. P. Herzig, H. Lu, and T. Grosjean, “Magnetic spin-orbit interaction of light,” Light: Sci. Appl. 7(1), 24 (2018).
[Crossref]

M. Wang, R. Salut, H. Lu, M. A. Suarez, N. Martin, and T. Grosjean, “Subwavelength polarization optics via individual and coupled helical traveling-wave nanoantennas,” Light: Sci. Appl. 8(1), 76 (2019).
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Nat. Nanotechnol. (1)

A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10(11), 937–943 (2015).
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Nature (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]

Opt. Commun. (1)

V. Fedoseyev, “Spin-independent transverse shift of the centre of gravity of a reflected and of a refracted light beam,” Opt. Commun. 193(1-6), 9–18 (2001).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Opt. Mater. Express (5)

Photonics Res. (1)

W. Long, J. Pan, X. Guo, X. Liu, H. Lin, H. Zheng, J. Yu, H. Guan, H. Lu, Y. Zhong, S. Fu, L. Zhang, W. Zhu, and Z. Chen, “Optimized weak measurement of orbital angular momentum-induced beam shifts in optical reflection,” Photonics Res. 7(11), 1273–1278 (2019).
[Crossref]

Phys. Rev. A (5)

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[Crossref]

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Z. Xiao, H. Luo, and S. Wen, “Goos-Hänchen and Imbert-Fedorov shifts of vortex beams at air–left-handed-material interfaces,” Phys. Rev. A 85(5), 053822 (2012).
[Crossref]

M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95(6), 063827 (2017).
[Crossref]

X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88(5), 053840 (2013).
[Crossref]

Phys. Rev. B (2)

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Y. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

O. V. Kotov and Y. E. Lozovik, “Enhanced optical activity in hyperbolic metasurfaces,” Phys. Rev. B 96(23), 235403 (2017).
[Crossref]

Phys. Rev. Lett. (3)

A. Nemilentsau, T. Low, and G. Hanson, “Anisotropic 2D materials for tunable hyperbolic plasmonics,” Phys. Rev. Lett. 116(6), 066804 (2016).
[Crossref]

W. Löffler, A. Aiello, and J. P. Woerdman, “Observation of orbital angular momentum sidebands due to optical reflection,” Phys. Rev. Lett. 109(11), 113602 (2012).
[Crossref]

Y. Wen, I. Chremmos, Y. Chen, Y. Zhang, and S. Yu, “Arbitrary multiplication and division of the orbital angular momentum of light,” Phys. Rev. Lett. 124(21), 213901 (2020).
[Crossref]

Phys. Rev. X (1)

O. Yermakov, V. Lenets, A. Sayanskiy, J. Baena, E. Martini, S. Glybovski, and S. Maci, “Surface waves on self-complementary metasurfaces: all-frequency hyperbolicity, extreme canalization, and TE-TM polarization degeneracy,” Phys. Rev. X 11(3), 031038 (2021).
[Crossref]

Science (3)

P. Li, I. Dolado, F. J. A. Mozaz, F. Casanova, L. E. Hueso, S. Liu, J. H. Edgar, A. Y. Nikitin, S. Vélez, and R. Hillenbrand, “Infrared hyperbolic metasurface based on nanostructured van der Waals materials,” Science 359(6378), 892–896 (2018).
[Crossref]

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref]

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin hall effect at metasurfaces,” Science 339(6126), 1405–1407 (2013).
[Crossref]

Data availability

No data were generated or analyzed in the present research.

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Figures (6)

Fig. 1.
Fig. 1. Schematic of the vortex beam reflection in graphene hyperbolic metasurfaces. The GH and IF shifts are shown by dotted lines.
Fig. 2.
Fig. 2. (a-c) Modulus and (d-f) phase of the Fresnel reflection coefficients $r_{ij}(\theta )$. (g-i) The imaginary part of the effective conductivity $\sigma _1$ and $\sigma _2$. Parameters are set as the strip width $W=0.5$ $\mu$m, the strip periodicity $L=1$ $\mu$m, the optical axis angle $\varphi =0^\circ$, the Fermi velocity $v_F=9.5\times 10^5$ m/s, the relaxation time of the carrier $\tau =0.1$ ps, the incident angle $\theta =85^\circ$, the beam waist $w_0=2\lambda$. (a,d,g) The Fermi energy $E_F$ and the external magnetic $B$ are 0.05 eV and 0 T, respectively. (b,e,h) The external magnetic $B$ and the frequency $\omega$ are 0 T and 8 THz, respectively. (g,h,i) The frequency $\omega$ and the Fermi energy $E_F$ are 8 THz and 0.05 eV, respectively.
Fig. 3.
Fig. 3. The relative strength for $s$-polarized and $p$-polarized Laguerre-Guass modes with $l=4$. Parameters are the same as in Fig. 2.
Fig. 4.
Fig. 4. (a,c) The Goos-H$\ddot {\mbox {a}}$nchen $X$ and Imbert-Fedorov $Y$ shifts of the Laguerre-Guass beam with $l=4$. (b,d) The relative strength of Laguerre-Guass modes. (a,b) The incident angle $\theta = 85^\circ$. (c,d) The optical axis angle $\phi = 45^\circ$. The other parameters are the same as in Fig. 2.
Fig. 5.
Fig. 5. The relative strength for $s$-polarized Laguerre-Guass modes. Parameters are the same as in Fig. 2.
Fig. 6.
Fig. 6. The relative strength for $s$-polarized Laguerre-Guass modes. Parameters are the same as in Fig. 2.

Equations (13)

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( σ x x σ x y σ y x σ y y ) = ( σ 1 sin 2 φ + σ 2 cos 2 φ sin 2 φ ( σ 1 σ 2 ) / 2 + σ h e f f sin 2 φ ( σ 1 σ 2 ) / 2 σ h e f f σ 1 cos 2 φ + σ 2 sin 2 φ ) ,
E ~ i = ( f p x ^ i + f s y ^ ) ϕ ~ m l ,
ϕ ~ 0 l = C l w 0 2 exp [ w 0 2 ( k x i 2 + k y 2 ) / 4 ] [ w 0 ( i k x i + s g n [ l ] k y ) / 2 ] | l | .
E ~ r = { f p [ r ~ p p + k y k 0 ( r ~ s p r ~ p s ) cot θ ] + f s [ r ~ s p k y k 0 ( r ~ p p r ~ s s ) cot θ ] } ϕ ~ 0 l x ^ r + { f p [ r ~ p s + k y k 0 ( r ~ p p r ~ s s ) cot θ ] + f s [ r ~ s s + k y k 0 ( r ~ s p r ~ p s ) cot θ ] } ϕ ~ 0 l y ^ .
[ n , H 2 ] [ n , H 1 ] = 4 π c σ ^ E 1
[ n , E 2 ] [ n , E 1 ] = 0
r s s ( θ ) = ( cos θ n cos θ σ y y ) ( sec θ + n sec θ + σ x x ) + σ x y σ y x ( cos θ + n cos θ + σ y y ) ( sec θ + n sec θ + σ x x ) σ x y σ y x ,
r s p ( θ ) = 2 σ x y ( cos θ + n cos θ + σ y y ) ( sec θ + n sec θ + σ x x ) σ x y σ y x ,
r p s ( θ ) = 2 σ y x ( cos θ + n cos θ + σ y y ) ( sec θ + n sec θ + σ x x ) σ x y σ y x ,
r p p ( θ ) = ( cos θ + n cos θ + σ y y ) ( sec θ n sec θ σ x x ) + σ x y σ y x ( cos θ + n cos θ + σ y y ) ( sec θ + n sec θ + σ x x ) σ x y σ y x ,
E r = { f p [ r p p i k 0 ( r s p r p s ) cot θ y ] + f s [ r s p + i k 0 ( r p p r s s ) cot θ y ] } ϕ 0 l x ^ r + { f p [ r p s i k 0 ( r p p r s s ) cot θ y ] + f s [ r s s i k 0 ( r s p r p s ) cot θ y ] } ϕ 0 l y ^ .
c l , l = { ± Z ± | l | + 1 Z | l | ( 1 ) l .
Z ± = λ 2 3 / 2 π w 0 ( 1 ) l ( X ± i Y )