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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.
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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 [15–19]. 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 [20–23]. 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 [24–26]. 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]:
Fig. 1. Schematic of the vortex beam reflection in graphene hyperbolic metasurfaces. The GH and IF shifts are shown by dotted lines.
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
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 haveTransforming 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
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]
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].
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.
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.
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. (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.
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.
Fig. 5. The relative strength for $s$-polarized Laguerre-Guass modes. Parameters are the same as in Fig. 2.
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.
Fig. 6. The relative strength for $s$-polarized Laguerre-Guass modes. Parameters are the same as in Fig. 2.
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.
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