Metamaterials are a class of man-made composites characterized by unique physical properties, which are widely utilized for the control of electromagnetic waves. Metasurfaces, the two-dimensional counterparts of metamaterials, are distinguished by their small size, low loss, and straightforward manufacturing processes. They can effectively modulate the amplitude, phase, and polarization of electromagnetic wavefronts, enabling the realization of abnormal electromagnetic phenomena^[1-2]. By designing the structural parameters and material properties of metasurfaces with arrays of meta-atoms, a variety of innovative devices in the field of electromagnetic waves have been developed, including anomalous reflections^[3-4], polarization conversion^[5], radar cross-section (RCS) reduction^[6], and vortex beams^[7-9]. In 2014, Cui et al. introduced the concept of coded metasurfaces, which integrate coding with digital signals. For instance, a 1-bit coded metasurface consists of two discrete meta-atom units, each with phase responses of 0° and 180°, labeled as “0” and “1”. These coding sequences are employed to achieve different functions. Furthermore, 2-bit and multi-bit coded metasurfaces can be realized, offering greater flexibility in the design of multifunctional metasurfaces^[10-11]. Generally, meta-atoms regulate the phase response under varying incident wave irradiation through two pathways: the propagation phase and the geometric phase. The propagation phase can be controlled by adjusting the size of the meta-atom under linearly polarized incidence, while the anisotropic meta-atom operates in two orthogonally polarized states, x-polarized and y-polarized, to achieve the desired functionality^[12]. In the context of geometrical phase, the meta-atom can achieve various orientation angles (α) to generate a specific phase shift (φ) under circularly polarized incidence through rotation. This relationship is expressed as φ = ± 2α, where the “+” and “−” symbols denote LCP and RCP incident waves, respectively^[13-14]. However, the intrinsic characteristics of the geometrical phase cause the metasurface to produce completely antisymmetric responses under LCP and RCP incidences, which cannot be modulated independently. This limitation results in a lack of functional diversity in the metasurface concerning geometrical phase, highlighting the urgent need to achieve independent manipulation of circularly polarized waves.

Two distinct angular momentum modes are observed in photons: spin angular momentum (SAM), which is associated with the spin state of the electromagnetic wave, and orbital angular momentum (OAM), which pertains to the spatial phase of the electromagnetic wave^[15-16]. The combination of geometrical and propagation phases can be employed to facilitate the conversion from SAM to OAM. The OAM carried by the vortex spin beam can significantly enhance the data capacity of communication systems, thereby providing new degrees of freedom that are crucial for multichannel optical and wireless communications. In 2018, Xu et al. proposed a strategy that integrates geometrical and propagation phases to overcome the intrinsic limitations of geometrical phases, thereby achieving the spin decoupling function. They designed two multifunctional circularly polarized bifunctional devices operating in the microwave region to validate their approach^[15]. In 2019, Zhang et al. introduced an efficient transport metasurface with switchable functions for the independent manipulation of LCP and RCP waves. They realized switchable focused beams and the ability to switch between vortex beams and focused beams within the microwave region^[17]. In 2022, Li et al. proposed a reflective spin-decoupled metasurface in the terahertz band that combines spin decoupling with the superposition theorem. Based on this concept, they successfully realized three types of metasurfaces: superpositions of multiple vortex beams and focusing beams, four-channel vortex beams, and four-channel focusing beams^[18]. Given that the functionality of purely structured metasurfaces is constrained by fixed dimensions, dynamically tunable electromagnetic waves have garnered considerable attention from researchers. They aim to incorporate tunable materials such as graphene^[19], indium antimonide^[20], vanadium dioxide^[21], and DSM^[22] into metasurfaces, leading to the design of actively tunable metasurfaces with enhanced performance by varying external conditions such as temperature and voltage^[23]. In 2021, Xu et al. proposed a graphene-based tunable metasurface capable of achieving two distinct functions: dynamic multibeam switching and dynamic diffusion switching, by modulating the E_F of graphene^[24]. In 2023, Ma et al. introduced a vanadium dioxide-based coded metasurface that facilitates the switching of vanadium dioxide between insulating and metallic states in response to temperature variations. This switching mechanism enables RCS scaling at different frequencies^[25]. These tunable materials offer versatile dynamic tuning within a single metamaterial device. Notably, DSM, as a novel topological quantum material akin to 3D graphene, exhibit high carrier mobility and provide additional structural degrees of freedom for the construction of functional devices. Furthermore, DSM boasts stable performance, ease of preparation, and rapid response times. Consequently, integrating DSM into coded metasurfaces enhances the dynamic modulation of electromagnetic waves.

In this study, a multilayer structure based on a DSM is designed. By adjusting the size and rotation angle of the cross-shaped meta-atom, we intergrate propagation and geometric phases to create a 2-bit meta-atom library, which serves as the foundation for designing a tunable, spin-decoupled coded metasurface operating in the terahertz band. When the Fermi energy level of the DSM is set to 6 meV, the system operates at 1.3 THz. By varying the incidence modes of LCP and RCP waves, we achieve the switching of double vortex beams with topological charges l = −2 and l = 1. Similarly, when the E_F of the DSM is increased to 80 meV, the metasurface operates at 1.4 THz, again facilitating the switching of double vortex beams with the same topological charges. In addition, the topological charge and deflection direction of the vortex beam can also be realized by orderly arrangement of meta-atoms. Consequently, the designed metasurface exhibits switchable characteristics, functioning effectively as a multichannel vortex beam generator to meet the needs of multifunctional applications, which is of great importance to the development of multiple fields.

When the incident wave illuminate on the metasurface normally, the electric field component of reflected wave can be described by formula^[26]

$ \begin{split} \left(\begin{array}{c}{E}_{{\mathrm{L}}}^{r}\\ {E}_{{\mathrm{R}}}^{r}\end{array}\right)=&\left(\begin{array}{c}{R}_{{\mathrm{LL}}}{R}_{{\mathrm{LR}}}\\ {R}_{{\mathrm{RL}}}{R}_{{\mathrm{RR}}}\end{array}\right)\cdot \left(\begin{array}{c}{E}_{{\mathrm{L}}}^{i}\\ {E}_{{\mathrm{R}}}^{i}\end{array}\right)=\\ &\left(\begin{array}{*{20}{c}}\eta {e}^{-i2\alpha }&\delta \\ \delta &\eta {e}^{-i2\alpha }\end{array}\right)\cdot \left(\begin{array}{c}{E}_{{\mathrm{L}}}^{i}\\ {E}_{{\mathrm{R}}}^{i}\end{array}\right)\quad, \end{split}$ (1)

where ${E}_{{\mathrm{L}}}^{r} $ and ${E}_{{\mathrm{R}}}^{r} $ are the orthogonal circular components of the reflected electric field, $ {E}_{{\mathrm{L}}}^{i} $ and ${E}_{{\mathrm{R}}}^{i} $ are the orthogonal circular components of the incident electric field. Where R_LL、R_LR、R_RL、R_RR are reflection coefficients, which also denote four different channels, subscripts are cross-polarization and co-polarization. The first subscript is the incident wave polarization mode while the second subscript is the reflected wave polarization mode. The equations for $ \eta $ and $\delta $ are given by $ \eta =({R}_{x}{e}^{i{\phi }_{x}}- $${R}_{y}{e}^{i{\phi }_{y}})/2 $, $ \delta =\left({R}_{x}{e}^{i{\phi }_{x}}+{R}_{y}{e}^{i{\phi }_{y}}\right)/2 $. ${R}_{x} $, $ {R}_{y} $, $ {\phi }_{x} $ and $ {\phi }_{y} $ denote the reflected amplitude and phase of the x- and y-polarized wave, respectively. α is the rotation angle of the metasurface resonant structure. When R_x = R_y = 1, $ {\phi }_{x}-{\phi }_{y} $ = 180°, $\eta $ and $\delta $ can be represented by $\eta =\left({R}_{x}{e}^{i{\phi }_{x}}-{R}_{y}{e}^{i{\phi }_{y}}\right)/2 $=${e}^{j{\phi }_{x}} $ and $ \delta =\left({R}_{x}{e}^{i{\phi }_{x}}+ {R}_{y}{e}^{i{\phi }_{y}}\right)/ 2=0 $, Therefore, the electric field of the reflected wave can be simplified as

$ \begin{split} \left(\begin{array}{c} {E}_{{\mathrm{L}}}^{r}\\ {E}_{{\mathrm{R}}}^{r}\end{array}\right)=& \left(\begin{array}{*{20}{c}} {e}^{i\left(\phi_x-2\alpha \right)}&0\\ 0&{e}^{i\left(\phi_x+2\alpha \right)}\end{array}\right)\cdot \left(\begin{array}{c}{E}_{{\mathrm{L}}}^{i}\\ {E}_{{\mathrm{R}}}^{i}\end{array}\right)=\\ &\left(\begin{array}{c}{e}^{i\left(\phi_x-2\alpha \right)}{E}_{{\mathrm{L}}}^{i}\\ {e}^{i\left(\phi_x+2\alpha \right)}{E}_{{\mathrm{R}}}^{i}\end{array}\right)\quad. \end{split} $ (2)

The reflected LCP and RCP waves are assigned additional phases denoted as $\varphi_{\mathrm{L}} $ and $\varphi_{\mathrm{R}} $:

$ \begin{split} &{\varphi }_{\rm L}={\varphi }_{x}-2\alpha \\ &{\varphi }_{\rm R}={\varphi }_{y}-2\alpha \quad, \end{split} $ (3)

where α represents the cell rotation angle associated with the geometric phase, and φ_x and φ_y denote the reflected amplitude and phase of the x- and y-polarized incidence. The equation above can be reorganized to yield the following

$ \begin{split} &{\varphi }_{x}=({\varphi }_{\rm R}+{\varphi }_{\rm L})/2\\ &{\varphi }_{y}=\left({\varphi }_{\rm R}+{\varphi }_{\rm L}\right)/2\\ &\alpha =\left({\varphi }_{\rm R}-{\varphi }_{\rm L}\right)/4\quad. \end{split} $ (4)

As there exists a reflection phase response characterized by a phase difference of π between the x- and y-polarizations (i.e., φ_x− φ_y= π), this indicates a distinct reflection phase for the x- and y-polarization incidences^[18]

$ \begin{split} &{\varphi }_{x}=({\varphi }_{\rm R}+{\varphi }_{\rm L})/2\\ &{\varphi }_{y}=\left({\varphi }_{\rm R}+{\varphi }_{\rm L}\right)/2-{\text{π}} \quad. \end{split} $ (5)

By the above equation, the two reflection phases, LCP and RCP, are independently controlled by designing the parameters φ_x, φ_y and α.

The structure of the designed meta-atom is illustrated in Fig. 1(a) (color online). The elements, arranged from top to bottom, consist of a cross-shaped DSM patch, a silica isolator with a dielectric constant of 3.85, a DSM film, a cross-shaped gold patch embedded within the DSM film, another silica isolator, and a gold substrate. The DSM patch, DSM film, and gold substrate each have a thickness of 0.5 μm, while the gold patch embedded in the DSM film has a thickness of 0.4 μm, and the silica isolator has a thickness of 20 μm. The meta-atom can also be viewed as having two metal-insulator-metal (MIM) structures. Each MIM structure has three layers, a cross-shaped metal patch, an intermediate SiO₂ spacer layer, and a metal substrate, in order from top to bottom. When the DSM is in the metallic state with E_F = 80 meV, the top MIM structure interacts with the linearly polarized incident wave, and the phase response is modulated by the DSM patch. When the DSM has an E_F =6 meV, it changes from a metallic state to an insulating state, and the upper MIM is regarded as a dielectric covering the lower MIM. The incident wave will pass through the upper MIM structure and interact with the lower MIM structure, at which point the phase response is modulated by the gold patch. The cross structure aids in reducing crosstalk by minimizing variations between the polarized reflection phases φ_x and φ_y. Consequently, by adjusting the lengths of l₁ and l₂, as well as l₃ and l₄, it is possible to select an appropriate coding unit at two distinct Fermi energy levels. In this study, the simulation software CST Microwave Studio is employed for modeling, utilizing periodic boundary conditions in the x and y directions，which can simulate the infinite array properties of the metasurface and facilitate the accurate calculation of electromagnetic interactions between meta-atoms. Open boundary conditions are set in the z-direction to simulate the actual situation of electromagnetic wave propagation in free space, preventing the boundary reflection triggered by improperly setting finite boundary conditions from affecting the accuracy of the simulation results. The complex conductivity of the DSM can be derived using Kubo's formula within the framework of random phase approximation (RPA) theory. In the low temperature limit (T<<E_F), the dynamic conductivity of the DSM can be expressed as^[27-28]

${{\mathrm{Re}}}\left(\sigma \left(\mathrm{\Omega }\right)\right)=\frac{{{\mathrm{e}}}^{2}}{\hslash }\frac{g{k}_{{\mathrm{F}}}}{24{\text{π}} }\mathrm{\Omega }\theta (\mathrm{\Omega }-2)\quad, $ (6)

$ {{\mathrm{Im}}}\left(\sigma \left(\mathrm{\Omega }\right)\right)=\frac{{{\mathrm{e}}}^{2}}{\hslash }\frac{g{k}_{{\mathrm{F}}}}{24{\text{π}} }\left[\frac{4}{\mathrm{\Omega }}-{{\mathrm{ln}}}\left(\frac{4{\varepsilon }_{{\mathrm{c}}}}{\left|{\mathrm{\Omega }}^{2}-4\right|}\right)\right] \quad,$ (7)

where e is the electron charge, $ \mathrm{\hslash } $ is the approximate Planck constant, and $ {k}_{{\mathrm{F}}}={E}_{{\mathrm{F}}}/{\hslash }{v}_{{\mathrm{F}}} $ is the Fermi momentum. $ {v}_{F}=1{0}^{6}\;{\mathrm{m/s}} $ is the Fermi velocity, E_F is the Fermi energy level, $ \mathrm{\Omega }=\mathrm{\hslash }\omega /{E}_{{\mathrm{F}}}+{\mathrm{i}}\mathrm{\hslash }{\tau }^{-1}/{E}_{{\mathrm{F}}} $, where $ \tau =\mu {E}_{{\mathrm{F}}}/{\mathrm{e}}{v}_{{\mathrm{F}}}^{2} $ is the relaxation time, $ \mu =3\times 1{0}^{4}\;{{\mathrm{cm}}}^{2}{{\mathrm{V}}}^{-1}{{\mathrm{s}}}^{-1} $, $ {\varepsilon }_{{\mathrm{c}}}={E}_{{\mathrm{c}}}/{E}_{{\mathrm{F}}}, $ and E_c is the cutoff energy. In this study, the AlCuFe quasicrystals that we have chosen as DSM,$ {g}=40 $ is the simplex factor. The complex relative permittivity $\varepsilon $ of the DSM can be expressed as:

$ \varepsilon ={\varepsilon }_{{\mathrm{b}}}+{\mathrm{i}}\sigma /\omega {\varepsilon }_{0}\quad, $ (8)

where $ {\varepsilon }_{{\mathrm{b}}}=1 $，$ {\varepsilon }_{0} $ is the vacuum dielectric constant.

Fig. 1(b)−1(c) present top views of the DSM patch and the gold patch, respectively. The period of the meta-atom is denoted as p = 100 μm, while the cross arm lengths of the DSM patch and the gold patch are represented as l₁,l₂,l₃, and l₄. The widths are defined as ω₁ = 15 μm and ω₂ = 11 μm. By adjusting the cross arm lengths l₁, l₂, l₃, and l₄, we can control the propagation phase, while the rotation angles α₁ and α₂ allow for the control of the geometric phase. Consequently, the design of our desired meta-atom, which meets the conditions for both the propagation phase and geometric phase, is achieved through the variation of the aforementioned parameters.

To obtain ideal meta-atoms, the amplitude and phase responses of meta-atoms at various DSM Fermi energy levels are investigated. The simulation software CST Microwave Studio is employed to model the designed meta-atoms, applying periodic boundary conditions in the x and y directions and open boundary conditions in the z direction. Applying a suitable voltage to the ends of a device containing a DSM can change the distribution of electrons within it, thereby modulating the Fermi energy levels. At a DSM E_F = 6 meV, the phase response is modulated by the gold patch. Consequently, the design of the gold patch parameters is streamlined, as there is no need to consider additional specific parameters or variations in the rotation angle of the DSM patch. Low crosstalk between x- and y-linear polarization is observed due to the cross structure; when the rotation angle α₂ in Fig. 1(c) is set to 0°, the parameters l₃ and l₄ determine the reflection phase response for x- and y-polarizations, respectively. With parameter l₄ fixed between 15 μm and 95 μm and α₂ held at 0°, varying parameter l₃ adjusts the reflected phase of the x-polarized waves, while the reflection amplitude and phase for y-polarization remain largely unaffected by changes in parameter l₃. Fig. 2(a) (color online) demonstrates that simulations of meta-atoms at a DSM E_F of 6 meV yield amplitude and phase results obtained by varying parameter l₃ of the gold patch. The eight values of parameter l₃ are 67.5 μm, 61.5 μm, 56.9 μm, 53.2 μm, 52.8 μm, 49.2 μm, 42.5 μm, and 15 μm, denoted by the numbers 1 to 8. As illustrated in the Fig 2(a), the reflection amplitudes of the meta-atoms exceed 0.6 at 1.3 THz, with reflection phases varying from 0° to 360°. Fig. 2(b) (color online) shows that eight coding units with a phase difference of approximately 180° between φ_xx and φ_yy are identified by adjusting the cross arm length at 1.3 THz, satisfying the requirement for 360° phase coverage in both x- and y-polarizations.

The objective of the meta-atom design is to facilitate the generation of two independent coding states for LCP and RCP light. It is evident from the previous derivation in Eq.(5) that RCP/LCP wave incidence must achieve 360° phase coverage to meet the phase requirement. The phases φ_LCP and φ_RCP are discretized into 0°, 90°, 180°, and 270° within a 2-bit code, corresponding to propagation phases of 0°, 45°, 90°, and 135°, respectively. By rotating eight meta-atoms at 22.5° intervals and combining the four states of LCP and RCP, we can create a library of 16 meta-atoms, as shown in Fig. 3 (color online).

Similar to Fig. 2, Fig. 4 (color online) presents the simulation results of the meta-atom at 1.4 THz, specifically at the DSM E_F=80 meV, during which the DSM exerts a dominant influence over the electromagnetic wave. When the rotation angle α₁, as shown in Fig. 1(b), is fixed at 0°, the parameters l₁ and l₂ dictate the reflection phase response for x- and y-polarizations, respectively. With the parameter l₂ fixed at one of the values ranging from 11 μm to 95 μm and α₁ set at 0°, varying the parameter l₁ allows for the control of the reflection phase of x-polarized waves. Fig. 4(a) illustrates that simulations of meta-atoms at a DSM Fermi energy level of 80 meV yield amplitudes and phases that are obtained by varying the DSM patch parameter l₁. The eight values of the parameter l₁ are 80 μm, 56.4 μm, 52 μm, 46.3 μm, 46 μm, 43 μm, 38.8 μm, and 17 μm, corresponding to the integers from 1 to 8. As illustrated in the Fig. 4(a), the amplitude phase of the coding unit exhibits significant changes under x-polarization when compared to 6 meV. The reflection amplitudes of the meta-atoms consistently exceed 0.6 at 1.4 THz, with the reflection phases varying correspondingly within the range of 0° to 360°. Fig. 4(b) demonstrates that at 1.4 THz, by adjusting the cross arm lengths, eight coding units are identified with a phase difference of approximately 180° between φ_xx and φ_yy. These eight coding units achieve 360° phase coverage in both x- and y-polarizations.

Utilizing the previously mentioned method, the four LCP and RCP states are combined to create an additional library of 16 meta-atom, as shown in Fig. 5 (color online).

The vortex beam enhances communication system performance by improving traffic volume and rate, while also facilitating multi-channel communications. To assess the feasibility of the model and elaborate on the design flow, several coded metasurfaces for generating helical OAM are demonstrated alongside their corresponding simulation results. The phase distribution of the vortex beam generator must conform to a spiral phase distribution, represented as φ(x,y) = lφ = larctan (y/x), where l denotes the topological charge of the vortex beam, φ is the azimuthal angle along the propagation direction^[29-30], and x and y represent the horizontal and vertical coordinates of the center of any meta-atom, respectively. For a 2-bit coded metasurface, the phase difference between neighboring coded areas is 90°. When the E_F is set to 6 meV for the DSM, Figs. 6(a)−6(b) illustrate the coding sequence with topological charge l = −2 is along the y-direction “0000222200002222”. Fig. 6(c) illustrates the LCP wave coding sequence plot after convolution of Fig. 6(a) with Fig. 6(b). Similarly, Figs. 6(e)−6(f) represent the coding sequences with topological charge l = 1 and along the x-direction is “0000222200002222”, respectively. Fig. 6(g) displays the coded sequence obtained after convolving Fig. 6(e) with Fig. 6(f) for the RCP wave design. Meta-atoms with suitable phase responses are selected from the library established in Fig. 3 to model the corresponding coding patterns. The designed coded metasurface comprises 16×16 cells, and simulations of the metasurface were conducted using CST Microwave Studio, applying open spatial boundary conditions in the x, y, and z directions. 3D normalized far-field scattering maps for the incidence of LCP and RCP waves are presented in Fig. 6(d) and 6(h). The vortex beam splitting with topological charge l = −2 occurs under LCP wave incidence, while the vortex beam splitting with topological charge l = 1 is observed under RCP wave incidence. This result indicates that it is possible to independently manipulate LCP or RCP waves by utilizing a pre-designed reflection-encoded metasurface, enabling the generation of distinct independent OAM modes for different circular polarization wave incidences. Fig. 6(i), 6(j) illustrate the corresponding 2D far-field scattering maps for the incidence of LCP and RCP waves, while Fig. 6(k), 6(l) present the corresponding vortex phases. From these figures, it is evident that the reflected beams achieve phase coverage of 720° and 360°, respectively, which aligns with the topological charges l = −2 and l = 1 associated with the OAM.

When the E_F is set to 80 meV in the DSM, Fig. 7(a) (color online) illustrates the convolution of the topological charge l = −1 with the coding sequence of the periodic pattern “0000222200002222” along the y-direction for the LCP wave design. Conversely, Fig. 7(e) (color online) depicts the RCP wave design for l = 0, where the coding sequence along the x-direction is also “0000222200002222”. Appropriate meta-atoms with the necessary phase response are selected from the library presented in Fig. 5 to model the corresponding coding patterns. The designed coded metasurface comprises 16×16 cells. The simulation of the metasurface was performed using the commercial software CST Microwave Studio, applying open and spatial boundary conditions in the x, y, and z directions. The 3D normalized far-field scattering maps for the incidence of LCP and RCP waves are presented in Fig. 7(b), 7(f) (color online). At LCP wave incidence, a splitting of the vortex beam with topological charge l = −1 occurs, while a similar splitting is observed for the RCP wave incidence. Figs. 7(c), 7(g) (color online) depict the corresponding 2D far-field scattering maps for LCP and RCP waves, respectively, while Figs. 7(d), 7(h) (color online) illustrate the associated vortex phases. Notably, Fig. 7(d) indicates that the reflected beam exhibits a 360° phase coverage, which aligns with the topological charge l = −1 of the OAM.

In summary, this paper presents a multi-functional metasurface. The switching of terahertz waveband functions is achieved by incorporating geometrical and propagation phases within the metasurface structure. At E_F of 6 meV for DSM, the gold patch effectively controls the electromagnetic wave, generating vortex beam splitting with a topological charge of l = −2 for the LCP wave, and vortex beam splitting with a topological charge of l = 1 for the RCP wave. At E_F of 80 meV for DSM, the DSM exerts complete control over electromagnetic waves, resulting in vortex beam splitting with a topological charge of l = 1 for the LCP wave, and beam splitting for the RCP wave. Thus by changing the polarization state of the incident wave and the E_F of the DSM, the function of the metasurface can be dynamically tuned. The findings of this study breaks the limitations of the spin-decoupled metasurface untunability in previous studies, which holds promising applications in the field of terahertz communication.

Data will be made available on request.