Graphene, an atomically thin layer of carbon atoms arranged in a honeycomb lattice, has attracted significant interest owing to its superior electronic and optical properties ^[1–3]. The conductivity of graphene is very sensitive to external fields, such that its optoelectronic properties can be precisely tuned ^[4]. Owing to its unique properties, graphene has been suggested as an alternative to conventional metal-based structures to confine light ^[5], guide surface plasmon polaritons ^[6], and manipulate wavefronts ^[7,8]. The subwavelength metamaterial structures made of graphene sheets show advantages over those made of thin metal layers at frequency and amplitude tunable properties ^[9]. Recently, the graphene metamaterial has been experimentally realized in the mid-infrared range ^[10]. The metamaterial experiences an optical topological transition from elliptic to hyperbolic dispersion at a wavelength of 4.5 μm ^[10].

Spin splitting refers to the spatial separation of two opposite spin components of bounded light beams reflected from or transmitted through an interface between two different media ^[11–14]. The spin splitting phenomenon was first observed experimentally by Hosten and Kwiat in 2008 ^[11]. In their experiment, a weak measurement was used since the spin splitting of the transmitted beam through an air–glass interface was very small—only a fraction of a wavelength ^[11,12]. Lately, much larger spin splitting was found by reflected Gaussian beams near Brewster incidence ^[13]. Götte and coworkers have achieved a spin splitting of ten wavelengths near Brewster incidence by properly choosing the incident polarization state ^[14]. In 2015, a spin splitting equal nearly to the incident beam waist w0 was demonstrated when a one-dimensional Gaussian beam with w0=10.2μm was reflected from an air–glass interface ^[15]. For a two-dimensional Gaussian beam, however, the spin splitting could only reach 0.4w0. It was demonstrated that the spin splitting can also be enhanced by metal thin films ^[16] and metamaterials ^[17,18]. Recently, the upper bounds of the spin splitting of Gaussian incident beams were found, which were equal to the incident beam waists w0 ^[19].

Although less investigated, the beam shifts of higher-order Laguerre–Gaussian (LG) beams are very interesting ^[20–22]. When reflected by an interface between two different media, the complex vortex structures of LG beams will interact with the angular Goos–Hänchen (GH) and Imbert–Fedorov shifts, which leads to the orbital-angular-momentum- (OAM) dependent shifts along directions both parallel and perpendicular to the plane of incidence ^[23]. The OAM-dependent shifts increase linearly with the incident OAM, so that large beam shifts can be achieved ^[24]. These shifts have been used to steer asymmetric spin splitting ^[25].

Alternatively, here we theoretically show a symmetric spin splitting dependent on the incident OAM when transmitting higher-order LG beams through a graphene metamaterial slab. The metamaterial is based on the multilayer structure of alternating graphene sheets and Al2O3 layers. As a result of the interaction between graphene metamaterial and a light beam embedded with OAM, the two opposite spin components of the transmitted beam will undergo shifts toward opposite directions, and the shifts depend on the incident OAM ℓ. The OAM-dependent spin splitting is bounded by w0|ℓ|cosθt/[(|ℓ|+1)1/2cosθi], where θi and θt are the incident and transmitted angles, respectively. By modulating the Fermi energy of the graphene sheets, the spin splitting can be tuned from positive to negative. Also, the splitting can reach its upper bound. The Fermi energy of graphene can be tuned over a wide range by an externally applied bias (electrostatic gating) since the Fermi energy in graphene is related to the carrier concentration ^[26].

The graphene-dielectric multilayer structure can be viewed as a metamaterial with effective medium approximation. In the long-wavelength limit, the effective in-plane and out-of-plane permittivities of the metamaterial respectively are ^[10] where ϵd and h are the permittivity and thickness of the dielectric layer, λ is the wavelength in free space, and Z0 is the vacuum impedance. σ is the optical conductivity of graphene, which is in the form of ^[1,27]

Here, kB is the Boltzmann constant, ω is the frequency of light, EF is the Fermi energy, T=300K is the temperature, and τ is the carrier relaxation lifetime. τ=μEF/(evF2) with vF≈1×106m/s being the Fermi velocity and μ=10000cm2·V−1·s−1 being the mobility. As shown by Eq. (1), the real part of the optical conductivity σ raises the absorption of the graphene metamaterial. The image part of σ tunes the real part of the in-plane permittivity of the graphene metamaterial, which can be changed from positive value, zero, to negative value.

In order to investigate the OAM-dependent spin splitting, we launch a linearly polarized higher-order LG beam obliquely onto a graphene metamaterial from a CaF2 prism, as shown in Fig. 1(a). The incident and transmitted angles are θi and θt, respectively. The local coordinate systems attached to the incident and transmitted beams are (xi,yi,zi) and (xt,yt,zt), respectively. The angular spectrum of a vertically polarized LG beam is E˜i=A[w0(−ikix+sℓkiy)/21/2]|ℓ|exp[−(kix2+kiy2)w02/4]|V⟩, where A=w0/(2π|ℓ|!)1/2, kix and kiy are the transverse wave vector of the incident beam, |V⟩ is the vertical polarization state, and sℓ=sign[ℓ] denotes the sign of the OAM. According to Ref. ^[12], the angular spectrum of the transmitted beam is connected with that of the incident beam via Fresnel transmission coefficients. In the first order approximation, it is in form of where k=2πn/λ with n being the refractive index of the prism, ktx=ηkix, kty=kiy, and η=cosθt/cosθi. Xs=its′/ts and δs=(η−tp/ts)cotθi, with tp,s being Fresnel transmission coefficients for p and s waves and ts′ being the first derivative of ts. By making an inverse Fourier transformation, we obtain the transmitted light field in real space, which has the following form in circular polarization basis |±⟩=2−1/2[|H⟩±i|V⟩]:

The right- and the left-handed circular polarization (RCP and LCP, respectively) components of the transmitted beam are no longer LG modes. They might lose circular symmetry, thus their centroids might shift, as shown in Fig. 1(b). The displacements of the centroids of the RCP and LCP components of the transmitted beam along the xt axis are defined as Δ±=∬xt|Et±|2dxtdyt/∬|Et±|2dxtdyt ^[19]. After some straightforward calculations, we arrive at

The first term of Eq. (5) is the conventional GH shift originated from the Gaussian envelope ^[12]. It moves the RCP and LCP components of the transmitted beam together. However, the second term will shift the RCP and LCP components toward opposite directions. The term is OAM dependent, resulting from the coupling between incident OAM and the angular spin splitting σIm(δs) ^[28]. This OAM-dependent spin splitting is different from the in-plane photonic spin splitting ^[29], which vanishes for a vertically incident polarization. The spin splitting of the transmitted beam is defined as the distance between the centroids of two opposite spin components, Δ=Δ+−Δ−, which therefore is equal to the OAM-dependent spin splitting, independent from the GH shift. When the second term of the denominator in Eq. (5) is negligible, the spin splitting Δ is linearly proportional to the incident OAM ℓ. Otherwise, the Δ changes nonlinearly with ℓ. To maximize the spin splitting Δ, δs should be a pure imaginary number and |δs|≫|Xs|. When |δs|=kw0/(|ℓ|+1)1/2, the maximum Δ is obtained:

Therefore, Δup is the upper bound of the OAM-dependent spin splitting of the transmitted beam, which is determined by the incident OAM, beam waist, and the incident and transmitted angles. It has been already demonstrated that the upper bound of the spin splitting of a reflected Gaussian beam is w0 ^[19]. In the transmission case, the upper bound of spin splitting along the xt axis should be corrected as ηw0. Therefore, the upper bound of the spin splitting for the Gaussian beam is smaller than those of OAM-dependent spin splitting for LG beams when |ℓ|>1. Estimated by the second radial moment of the intensity, the beam size (beam width) of the LG beam is w0(|ℓ|+1)1/2 ^[30]. For a fixed w0, the size of the LG beam increases with |ℓ|. Thus, the ratio between the upper bound of the spin splitting and the beam size of the LG beam is η|ℓ|/(|ℓ|+1), which shows no advantage over the foundational Gaussian beam. However, the OAM-dependent spin splitting is not only physically interesting ^[12,22], but it also provides an alternative method for the control of optical spin ^[25]. As will be shown below, when the OAM-dependent spin splitting reaches its upper bound, the RCP and LCP components of the transmitted LG beam are well separated along the xt axis: the two intensity profiles RCP and LCP components are distinguishable according to the Rayleigh criterion ^[15]. However, the profiles are indistinguishable from each other for the case of Gaussian incident beams ^[19]. In the following, we will try to tune the spin splitting and approximate its upper bound by using graphene metamaterials.

Consider the graphene metamaterial composed of alternating Al2O3 layers and graphene sheets, with the thickness of Al2O3 layers being h=10nm [see Fig. 1(c)]. The real part of the in-plane permittivity Re[ϵeff,//] varies with Fermi energy and can take positive, zero, and negative values. When EF=0.335eV, Re[ϵeff,//] almost vanishes for a wavelength of λ=4.509μm. The graphene metamaterial is displaced on the CaF2 prism, whose reflective index is n=1.39 at λ=4.509μm ^[10]. The structure can be fabricated by the method provided by Ref. ^[10].

The spin splitting of the transmitted beam Δ will change with incident angle θi and the thickness of metamaterial d, as shown by Fig. 2(a), where Δ is normalized to its upper bound Δup. In the calculations, the incident OAM is ℓ=1, the beam waist is w0=180μm, the Fermi energy of the graphene sheets is EF=0.335eV, and the wavelength is λ=4.509μm. In this situation, the effective in-plane permittivity of the graphene metamaterial is ϵeff,//=−0.001+0.086i. When the incident angle θi changes from 10° to 45°, the spin splitting Δ can always approach the upper bound Δup by modulating d. When θi=33° and d=7.5μm, Δ is up to 0.994Δup. The dependences of normalized spin splitting Δ/Δup on the incident angle θi for w0=90μm (red color), 225 μm (blue color), and 450 μm (green color) are shown respectively in Fig. 2(b), where the thickness of metamaterial is fixed on 7.5 μm. The maximum value of Δ/Δup changes slightly with the beam waist w0. For the beam waist ranging from 90 to 450 μm, Δ/Δup is larger than 0.95, which indicates that Δ is close to its upper bound Δup over a wide range of w0.

For the incident beams carried with different values of OAM, the spin splitting of the transmitted beams Δ are different. According to Eq. (6), the upper bounds of the OAM-dependent spin splitting Δup increases with the OAM |ℓ|. The change of Δup with ℓ is shown in Fig. 3(a), where the spin splitting Δ is also shown. Δ is opposite in sign for negative and positive OAM ℓ, and Δ vanishes when ℓ=0. Δ and Δup are identical for the cases of ℓ=±1 since the parameters θi=33° and d=7.5μm are optimized for these cases, as mentioned above. However, Δ is smaller than Δup when |ℓ|>1, and the difference increases with |ℓ|, as shown by Fig. 3(a).

The spin splitting Δ for |ℓ|>1 can be increased by slightly changing the incident angle θi. Figure 3(b) shows the normalized spin splitting Δ/Δup changing with the incident angle θi for OAM ℓ=−10 (red lines), −5 (blue lines), −1 (green lines), and 0 (black line). For a Gaussian incident beam (ℓ=0), the spin splitting vanishes for all the incident angles. However, when ℓ≠0, the spin splitting |Δ| increases with the incident angle θi and decreases gradually after reaching peak. The incident angle of the splitting peak decreases with the increase of |ℓ|. For all ℓ, the spin splitting can reach more than 0.99 of their upper bounds.

The spin splitting of the transmitted beam can be flexibly tuned by modulating the Fermi energy of the graphene sheets. The normalized spin splitting Δ/Δup changing with Fermi energy EF for different thicknesses of the graphene metamaterial d are shown in Fig. 4(a), where the incident OAM ℓ=3 (solid lines) and ℓ=−3 (dashed lines). For each situation, there are two peaks in the pattern of the spin splitting: a positive peak and a negative peak. The positive and negative peaks are separated by a zero point, which is located around EF=0.31eV. When d=7.2 and 9.0 μm, the spin splitting |Δ| reach their upper bounds Δup at EF=0.336 and 0.297 eV, respectively. When d=8.3μm, however, the spin splitting Δ is smaller than Δup. Interestingly, the absolute values of the spin splitting |Δ| for positive and negative peaks are identically equal to 0.82Δup. To show the change of the spin splitting with the Fermi energy EF more clearly, the intensity profiles of the RCP (solid lines) and LCP (dotted lines) components of the transmitted beam along xt are shown in Fig. 4(b) for EF=0.275eV (red color), 0.299 eV (blue color), 0.318 eV (green color), 0.356 eV (pink color), and 0.4 eV (black color) when the thickness of the graphene metamaterial is d=8.3μm. The intensity profiles of the RCP and LCP components are overlapped when EF=0.318eV, indicating the spin splitting Δ=0. The giant spin splitting is evident in cases of EF=0.299 and 0.356 eV, where the two opposite spin components are well separated.

The spin splitting will change with the wavelength λ owing to the dispersion of the graphene metamaterial. Figure 5 plots the spin splitting Δ as a function of wavelength λ for different values of Fermi energy EF. From Fig. 5, one can see that the spin splitting for each EF almost vanishes in the short wavelength range (λ<3μm). However, Δ will increase suddenly and reach peak value. After that, it decreases sharply until reaching a negative peak. Then it will increase gradually. It is worth noticing that the magnitudes of the positive and negative peaks are different. The spin splitting undergoes blueshift when the Fermi energy EF increases. This phenomenon can be explained by the move of the zero points of the real part of the in-plane permittivity, Re[ϵeff,//], as shown in the inset of Fig. 5, where the changing of Re[ϵeff,//] with wavelength λ for different values of Fermi energy EF are shown.

From Fig. 5, one can conclude that the giant spin splitting is associated with the near-zero Re[ϵeff,//]. In these cases, however, the image part of the in-plane permittivity, Im[ϵeff,//], is nonzero, which causes loss. Specially, at the upper-bounded spin splitting, the energy transmissivity is of the order of 10−5. It is worth pointing out that the energy transmissivity is of the same order of magnitude when a reflected Gaussian beam reaches its upper-bounded spin splitting at Brewster incidence ^[19], and this upper-bounded spin splitting has been experimentally measured ^[29]. Therefore, the low-energy transmissivity will not prevent the experimental measurement of giant spin splitting of the transmitted beam through graphene metamaterial.

We have theoretically demonstrated the tunable OAM-dependent spin splitting by transmitting higher-order LG beams through graphene metamaterials. The upper bound of the OAM-dependent spin splitting is ηw0|ℓ|/[(|ℓ|+1)1/2], which increases with the incident OAM |ℓ|. The Fermi energy of the graphene sheets can change the effective permittivity of the graphene metamaterial, which therefore tune the spin splitting. The spin splitting can be tuned from positive to negative values and can reach its upper bound. These findings provide an effective method for the flexible control of the spin splitting and therefore facilitate the development of spin-based applications and the manipulation of the mid-infrared waves.