1. Introduction

Gradient metasurfaces (GMSs), consisting of rationally designed two-dimensional arrays of optical nanoantennas, offer exquisite control over the wavefronts of incident light for device applications such as beam steering, focusing and image formation, polarization control, holography, and vortex beam generation [1–3]. Their basic building blocks (meta-units) are usually metallic or dielectric nanoparticles whose scattering phase can be tuned across the entire 2π range by varying the size, shape, and/or orientation. With this prescription, ordered nanoparticle arrays can be designed to introduce any desired spatial phase profile on the incident wavefronts upon transmission or reflection. For operation at near-infrared wavelengths, metallic nanoantennas are particularly well suited by virtue of their pronounced plasmonic resonances featuring highly subwavelength confinement and strong coupling to radiation. For visible-range applications, dielectric nanostructures are instead preferable because of the rapidly increasing absorption losses of plasmonic resonances with decreasing wavelength. Both types of materials platforms have been developed extensively over the past decade [1–3], with recent work focusing on meta-units with actively tunable scattering response as a way to enable reconfigurable and dynamically switchable devices [4].

GMSs operating at terahertz wavelengths have also been reported, based on metallic meta-units modeled after resonant circuit elements or dielectric resonators [5–9]. Compared to their near-infrared and visible counterparts, these THz metasurfaces have so far received more limited attention. Their development, however, could help address the current lack of suitable devices for a wide range of well-established sensing and imaging applications of THz light [10,11]. Furthermore, the current push towards higher and higher carrier frequencies for wireless communications beyond 5G [12] provides a compelling new motivation for the exploration of novel THz devices for wavefront engineering.

In this context, graphene plasmonics can be envisioned to play a particularly significant future role. It is well established that graphene supports strongly confined plasmon polaritons with favorable propagation characteristics compared to surface plasmons in noble metals [13–18]. For typical graphene nano- and microstructures, the resulting plasmonic resonances can span a large portion of the mid-infrared and THz spectral regions, depending on the lateral dimensions. Furthermore, their resonance frequencies can be tuned actively by varying the carrier density with a gate voltage in a field-effect-transistor (FET) configuration. Graphene plasmonic resonators are therefore ideal candidates for the development of long-wavelength metasurfaces with particularly convenient electronic tuning capabilities.

For operation at THz wavelengths, several promising tunable graphene GMSs have been proposed and investigated numerically in recent years, based on ribbons of variable width and/or carrier density [19–22] or on meta-units of variable orientation [23,24]. However, all these theoretical studies assume extremely high carrier mobilities (typically 10,000 cm²/V/s), often combined with exceedingly large Fermi energies (up to 1 eV), to produce sufficiently strong coupling between the graphene plasmon polaritons (GPPs) and free-space radiation. Such mobility values are indeed accessible with exfoliated graphene, at least at more moderate carrier densities. Unfortunately, however, the resulting samples are generally limited to lateral dimensions of a few 10 μm (smaller than the target THz operation wavelengths), whereas the envisioned device applications require active areas of at least a few mm². At the same time, significantly lower mobilities (typically 1,000 to 2,000 cm²/V/s) are obtained with large-area graphene devices grown by chemical vapor deposition (CVD) and transferred on standard oxidized silicon substrates [25]. As a result, the only experimental demonstrations of graphene-tunable THz metasurfaces to date have relied on the gate modulation of free-carrier (rather than plasmonic) absorption in graphene films to tune the response of adjacent metallic nanoparticles [26–29]. This approach can be quite effective, but only at frequencies below ∼1 THz, beyond which the graphene free-carrier absorption becomes too weak to have an appreciable impact [30].

The goal of the present work is to devise a universal tunable THz metasurface platform based on graphene plasmonics, compatible with the electrical characteristics of realistic CVD graphene samples. To this purpose, we combine several ideas that have been investigated recently in other contexts, including the use of an open vertical cavity to enhance the graphene absorption cross-section [31,32], the introduction of metallic antennas to increase the coupling of GPPs to free-space radiation [33–38], and the use of double-layer graphene stacks to increase the carrier-density tuning range [39,40]. Our simulation results show that this design strategy can produce a practical device technology for active THz wave manipulation, and allow quantifying the expected performance metrics. Specifically, we demonstrate two representative configurations: (1) a beam steering device covering a broad angular range of over ±60° and (2) a cylindrical metalens with widely tunable focal length (from about 0.6 to over 4 mm for 2-mm lens size). In both cases, an operating frequency ν₀ = 3 THz (100-μm wavelength) is considered, near the low end of the spectral region accessible with graphene plasmonics [30]. Even though the quality factor of GPPs decreases linearly with decreasing frequency [18], a reasonably large efficiency in the range of 20 to over 30% is obtained for both device configurations. These results therefore indicate that the metasurface platform described in the present work is suitable for a wide range of THz technologies, including future wireless communications at carrier frequencies of a few THz.

2. Results and discussion

A schematic illustration of our GMS design is shown in Fig. 1. Structurally, this device consists of the periodic repetition of identical meta-units (delimited by the dashed lines in Fig. 1), each comprising two graphene ribbons of different widths interspersed with three Au rectangular antennas. The entire array is supported by an oxidized Si substrate of subwavelength thickness, with a reflective Au back mirror. The specific values of all geometrical parameters optimized for operation at 3 THz are listed in the figure caption. The Si substrate is taken to be lightly doped (with a resistivity of ∼10 Ohm cm) so that it can serve as the back conductor of the SiO₂ gate capacitor, while at the same time producing rather small free-carrier absorption losses at the target operation frequency. Each graphene ribbon partially overlaps with the Au antenna on its left-hand side, which can then be used to individually control its carrier density (and therefore plasmonic response and reflection phase) through the application of a separate gate voltage. With this configuration, the reflection phase profile along the x direction (the direction perpendicular to the ribbons in Fig. 1) can be tailored with a large but realistic number (∼ 100) of electrical control signals applied on a printed circuit board. In passing, it should be mentioned that a similar biasing configuration has been employed in the recent demonstration of a tunable near-infrared metasurface based on transparent conducting oxides [41].

 

Fig. 1. Schematic illustration of the metasurface design developed in this work. The two double-layer graphene ribbons in each repeat unit have widths w₁= 2.7 μm and w₂= 3.1 μm. The width and thickness of all the Au antennas are L = 6 μm and t = 100 nm, respectively. Half of each ribbon is covered by the antenna on its left-hand side, which acts as a perfect reflector producing a mirror image of the plasmonic oscillations in the uncovered ribbon area [37]. The size of the gaps between the two ribbons in each repeat unit and the antennas on their right-hand side are g₁= 200 nm and g₂= 400 nm. The SiO₂ and Si layers have thicknesses h₁= 300 nm and h₂= 4.5 μm, respectively. The GMS period is P = 22.5 μm.

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The key material parameters of graphene that affect its plasmonic response are the electronic mobility μ and the Fermi energy ${E_F} = \hbar {v_f}\sqrt {\pi N} $, where N is the carrier density and v_F ≈ 1×10⁶ m/s the Fermi velocity. The electronic scattering lifetime τ, which determines the GPP nonradiative decay rate Γ_nr = 1/τ and resonance linewidth, depends on both parameters through the relationship $\mu = q\textrm{v}_\textrm{f}^2\tau /{E_F}$ derived from the Drude conductivity model [42]. For standard CVD-grown graphene sheets transferred on Si/SiO₂ substrates with the commonly used poly(methyl methacrylate) (PMMA)-assisted process, mobilities in the range of 1,000 to 2,000 cm²/V/s are typically obtained from electrical transport measurements [25]. Larger values by a factor of ∼4 have also been reported recently with CVD-graphene samples produced with less established transfer methods [43] and carefully optimized growth processes [44], again measured with FET devices at low carrier densities. In nanoribbons, scattering from line-edge roughness can also generally cause some degradation, although for the ribbon widths of interest in this work (a few microns) this effect can be expected to be quite small [45]. With all these considerations in mind, in the present work we assume a mobility of 2,000 cm²/V/s, consistent with our recent measurement of plasmonic THz light emission from CVD-graphene ribbons, where the emission linewidth could be well reproduced by modeling with this value of μ [46,47]. It should also be noted that the lifetime and mobility values relevant to the plasmonic response of graphene can be somewhat larger than the values extracted from conductivity measurements, since the same scattering processes can have different impact on electrical transport and GPP damping [48].

The other key material parameter (the Fermi energy E_F) affects the GPP dispersion as well as damping, and therefore contributes to determine the resonance frequencies f_pl of graphene plasmonic resonators. For the ribbon geometry under study, a simple analytical expression can be derived from the Drude model, showing that ${f_{pl}} \propto \sqrt {{E_F}/w} $, where w is the ribbon width [18]. In a FET configuration, the carrier density N (and therefore E_F) can be tuned by varying the gate voltage V_g according to the gate capacitor relation. The maximum accessible Fermi level is therefore limited by dielectric breakdown in the gate oxide, and values up to about 0.45 eV have been reported with THz plasmonic devices on Si/SiO₂ substrates [49]. Recent work has also established that double-layer graphene stacks (assembled through the consecutive transfer of two single-layer sheets) can significantly enhance the doping range [39,40]. Specifically, it has been shown that the plasmonic response of a double-layer sample of total carrier density N is equivalent to that of a single layer with an effective Fermi energy ${\tilde{E}_F}$, which can be substantially larger (by up to a factor of $\sqrt 2 $) than that of an actual single-layer sheet with the same carrier density N. At the same time, the mobility determining the quality factor of the plasmonic resonances has been found to be essentially the same as in single-layer samples [39]. Following these reports, our GMS design is based on double-layer graphene ribbons with variable effective Fermi energy ${\tilde{E}_F}$ (computed with the model of ref. [40]) up to a maximum value of 0.6 eV. The minimum value is set at 0.1 eV to account for inevitable charge puddles from substrate impurities, which can wash out plasmonic effects at exceedingly low gate voltages [50].

In order to elucidate the specific role played by each element in the GMS design of Fig. 1 (graphene ribbons, vertical cavity, and Au antennas), in Fig. 2 we show numerical simulation results for different structures where these elements are introduced sequentially. All simulations are based on the finite difference time domain (FDTD) method and were performed using the Lumerical FDTD Solutions software package. The (two-dimensional) simulation window contains one repeat unit of the device under study, with periodic boundary conditions applied to the lateral boundaries and perfectly matched layers (PMLs) on the top and bottom surfaces. The incident light is a plane wave propagating towards the device along its surface normal, with 3-THz frequency and linear polarization perpendicular to the ribbons (for maximal coupling to the ribbon fundamental plasmonic resonance). Graphene is modeled with a complex frequency-dependent sheet conductivity based on the semi-classical Drude model. All other materials (Si, SiO₂, and Au) are described with their complex dielectric functions. For Si and SiO₂, we use the built-in database of the FDTD software. For Au, we use data from ref. [51], which cover the THz spectral range of interest in this work. The geometrical parameters of all structures analyzed in this figure (listed in the figure caption) were selected to produce a plasmonic resonance at 3 THz for an effective Fermi energy ${\tilde{E}_F}$ ≈ 0.3 eV (near the middle of the accessible 0.1-0.6 eV tuning range). In these simulations, we use a built-in function of the FDTD solver to compute the individual contributions to the total absorption of each structure from its various elements. In each case, we find that the contributions from the Au antennas and from the Si and SiO₂ layers are one order of magnitude smaller than the graphene plasmonic absorption, and therefore have negligible impact on the device operation.

 

Fig. 2. Numerical simulation results illustrating the role played by each element in the metasurface design of Fig. 1. (a) Periodic array of double-layer graphene ribbons with 3.2-μm width and 12.8-μm period on a semi-infinite oxidized Si substrate. The SiO₂ layer thickness (here and in all other structures considered in this figure) is 300 nm. (b) Absorption efficiency of the structure of (a) versus graphene mobility μ (bottom axis) and GPP nonradiative scattering rate Γ_nr (top axis), for an effective graphene Fermi energy ${\tilde{E}_F}\,\,$ 0.3 eV. The vertical dashed line indicates the mobility required for critical coupling. (c), (d) Reflectivity |r|² and reflection phase ϕ_r, respectively, of the structure of (a) versus ${\tilde{E}_F}$. In both plots, the black line was computed for μ = 2,000 cm²/V/s, while the red trace corresponds to an over-coupled mobility μ = 64,000 cm²/V/s. (e) Periodic array of double-layer graphene ribbons with 3.2-μm width and 12.8-μm period on a vertical cavity with 5.7-μm Si layer thickness. (f), (g), (h) Same as (b), (c), (d), respectively, for the structure of (e). The effective Fermi energy in (f) is ${\tilde{E}_F}\,\,$ 0.3 eV, while the over-coupled mobility for the red traces of (g) and (h) is μ = 32,000 cm²/V/s. (i) Periodic array of antenna-coupled double-layer graphene ribbons on a vertical cavity with 4.5-μm Si layer thickness. The Au antennas have 6-μm width and 100-nm thickness. The ribbon width is 2.7 μm (half of which is covered by the antenna on its left-hand side), with a gap of 200 nm from the edge of the antenna on the other side. The array period is 14.55 μm. (j), (k), (l) Same as (b), (c), (d), respectively, for the structure of (i). The effective Fermi energy in (j) is ${\tilde{E}_F}\,\,$ 0.31 eV, while the over-coupled mobility for the red traces of (k) and (l) is μ = 4,000 cm²/V/s. The detailed behavior of the reflectivity and reflection-phase traces of panels (c), (d), (g), (h), (k), and (l), and their transition from the under-coupled to the over-coupled regime, can be explained using coupled-mode theory [53]. All the reflection-phase traces are shifted vertically to be centered in the –π to π interval.

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First, we consider a simple periodic array of double-layer ribbons on a semi-infinite Si/SiO₂ substrate [Fig. 2(a)]. In Fig. 2(b) we plot the plasmonic absorption efficiency of these ribbons (i.e., absorption cross-section divided by physical area) versus graphene mobility μ (bottom axis) and GPP nonradiative scattering rate Γ_nr = 1/τ (top axis). The shape of this trace can be explained with a general argument based on coupled-mode theory [18,52]: the plasmonic absorption of free-space radiation is maximum when the nonradiative damping rate of the GPPs Γ_nr is equal to their radiative decay rate Γ_rad. For the structure of Fig. 2(a), we find that such optimal critical coupling occurs at an exceedingly large mobility of over 30,000 cm²/V/s (dashed vertical line). At lower (higher) mobilities, Γ_nr is larger (smaller) than Γ_rad so that the GPPs are under-coupled (over-coupled) to free-space radiation, and as a result the absorption decreases. Importantly, operation in the over-coupled regime is a key necessary condition to be able to vary the GPP reflection phase across the entire 2π range by tuning the resonance, which again can be explained using coupled-mode theory [52,53]. Furthermore, we notice from Fig. 2(b) that even at critical coupling the absorption efficiency is well below unity, indicating limited light-matter interaction caused by the ultrasmall graphene thickness. In Figs. 2(c) and 2(d) we plot, respectively, the reflectivity |r|² and reflection phase ϕ_r of the same structure versus ${\tilde{E}_F}$ for a practical mobility of 2,000 cm²/V/s (black traces) and for a representative value in the over-coupled regime of 64,000 cm²/v/s (red traces). Even in the latter case, ϕ_r can only be varied over a small interval well below 2π due to the limited light-matter interaction, whereby a substantial fraction of the incident light is reflected by the device without interacting with the GPPs. The structure of Fig. 2(a) is therefore clearly inadequate for the development of THz GMSs.

To address this limitation, in Figs. 2(e)-(h) we introduce a vertical cavity configuration with a Au back mirror under a subwavelength Si/SiO₂ layer supporting the ribbons. Under external illumination, this mirror can not only block transmission of the incident light but also, depending on the cavity thickness, decrease the overall reflection through interference [31,32], leading to enhanced plasmonic absorption [see Figs. 2(f), where the absorption efficiency at critical coupling is about 3.4]. Under conditions of over-coupling, the tuning range of the reflection phase ϕ_r can then be extended to nearly 2π, as shown by the red trace of Fig. 2(h). This basic idea has in fact been employed in most prior theoretical studies of graphene THz GMSs [19–23]. However, the mobility values needed to access the over-coupling regime in this type of structures [larger than 20,000 cm²/V/s for ${\tilde{E}_F}$ = 0.3 eV in Fig. 2(f)] are not compatible with CVD graphene samples. At our envisioned mobility μ = 2,000 cm²/V/s, the GPPs in the device of Fig. 2(e) are under-coupled and a rather small tuning range is once again obtained [black trace in Fig. 2(h)].

Further improvements can be achieved by adding metallic THz antennas in the near field of each ribbon, such as the Au rectangular patches in the structure of Fig. 2(i). These antennas support resonant modes at free-space wavelengths λ proportional to their lateral dimension L, with a moderate degree of spatial confinement λ/L ∼ 10 [54], as opposed to λ/w ∼ 100 for the ribbon GPPs. Therefore, they can be used to enhance the plasmonic radiative decay rate Γ_rad of the ribbons (and correspondingly decrease the required mobility for critical coupling), by mediating the large wavelength mismatch between GPPs and free-space radiation [37]. This expectation is confirmed by the absorption-efficiency simulation results of Fig. 2(j), showing that the over-coupling regime is already achieved at mobilities close to the envisioned value of 2,000 cm²/V/s. The resulting reflectivity and reflection phase are plotted versus Fermi energy in Figs. 2(k) and 2(l). In this case, the tuning range of ϕ_r at 2,000 cm²/V/s is no longer limited by under-coupling. However, it still falls short of the full 2π phase range due to a sizeable contribution to the reflected light from the Au antenna resonances, which of course are not tunable with varying ${\tilde{E}_F}$. In particular, the maximum phase change observed in Fig. 2(l) is about 1.3π, which is not large enough to enable the development of the metasurface devices presented below.

The optimized structure of Fig. 1 is based on all these considerations and further employs two ribbons per unit cell, so that their respective GPP resonances can be combined to enhance the usable phase tuning range [38,55]. Additionally, an extra antenna is introduced between neighboring ribbon pairs to minimize the electromagnetic coupling between adjacent meta-units, which would otherwise distort the phase profile. In this geometry, the reflection magnitude |r|² and phase ϕ_r of each unit cell depend on the effective Fermi energies ${\tilde{E}_{F1}}$ and ${\tilde{E}_{F2}}$ of both ribbons, and the resulting variations are plotted in the color maps of Fig. 3 for μ = 2,000 cm²/V/s. Two points of maximum absorption (dark blue for |r|² << 1) can be identified in the log-scale reflectivity map of Fig. 3(a), around which ϕ_r spans the full 2π interval as shown in the phase map of Fig. 3(b). The specific geometrical parameters of our final design (listed in the caption of Fig. 1) were selected to maximize the separation on the ${\tilde{E}_{F1}}$-${\tilde{E}_{F2}}$ plane between these two points, so as to open a path in between along which ϕ_r could be varied across the full range while maintaining a reasonably large reflectivity. This path is indicated by the dashed lines in both panels of Fig. 3, where each point corresponds to the combination of Fermi energies ${\tilde{E}_{F1}}$ and ${\tilde{E}_{F2}}$ (within the accessible 0.1-0.6 eV range) that produces each value of ϕ_r from 0 to 2π with the largest possible |r|². It should be noted that the low operating frequency considered in the present work provides a significant challenge in this respect, because of the decreasing quality factor of GPP resonances with decreasing frequency. As a result, large variations in reflectivity are still obtained even along the optimal path of Fig. 3, with minimum values in the few-percent range. Nevertheless, these meta-units are suitable for the development of tunable GMS devices with practical performance characteristics, as detailed below.

 

Fig. 3. Reflection properties of the meta-units of the structure of Fig. 1. (a) Reflectivity |r|² and (b) reflection phase ϕ_r plotted as a function of the effective Fermi energies ${\tilde{E}_{F1}}$ and ${\tilde{E}_{F2}}$ of the two double-layer graphene ribbons for μ = 2,000 cm²/V/s. The dashed lines in both maps show all the combinations of values of ${\tilde{E}_{F1}}$ and ${\tilde{E}_{F2}}$ (within the accessible 0.1-0.6 eV range) for which ϕ_r covers the full range from 0 to 2π for the largest possible |r|².

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As a first example of such devices, in Fig. 4 we consider a beam steering meta-mirror. In this case, the gate voltages (and therefore Fermi energies) of the different ribbons are selected to produce a discretized version of the linear phase profile ϕ_r(x) = ξx, so that the GMS behaves like a blazed diffraction grating with period 2π/ξ along the x direction [1–3]. Correspondingly, light incident along the device surface normal (the z direction) is reflected along the non-specular direction on the x-z plane at an angle $\mathrm{\bar{\theta }} = \textrm{arcsin}({\mathrm{c\xi }/2\mathrm{\pi }{\mathrm{\nu }_0}} )$ [Figure 4(a)]. The simulated device consists of 44 adjacent meta-units with a total lateral dimension of about 1 mm. As mentioned above, the reflection phase ϕ_r of each meta-unit can be set equal to any desired value between 0 and 2π by using the combinations of Fermi energies ${\tilde{E}_{F1}}$ and ${\tilde{E}_{F2}}$ indicated by the dashed lines of Fig. 3. However, for some of these combinations the reflection magnitude |r|² is still too low to produce an appreciable contribution to the GMS response. Therefore, for the meta-unit of center position x_n (n = 1, 2, …, 44), the effective Fermi energies ${\tilde{E}_{F1,n}}$ and ${\tilde{E}_{F2,n}}$ are selected to produce a reflection phase within a finite range δϕ_err from the target value ϕ_r(x_n) = ξx_n, while at the same time maximizing the reflectivity |r(x_n)|². With this approach, we found that optimal meta-mirror performance in terms of efficiency versus steering angle $\mathrm{\bar{\theta }}$ could be achieved for a maximum phase error δϕ_err of about 0.6 rad. The resulting optimized reflection phase and reflectivity profiles are illustrated in Fig. 4(b) for $\mathrm{\bar{\theta }}$ near 30°.

 

Fig. 4. Tunable beam steering meta-mirror based on the structure of Fig. 1. (a) Magnetic-field distribution of the reflected light for a set of Fermi energies {${\tilde{E}_{F1,n}}$, ${\tilde{E}_{F2,n}}$} optimized to produce a steering angle $\mathrm{\bar{\theta }}$ near 30°. The GMS location is indicated by the green bar near the bottom of the plot. (b) Reflection-phase and reflectivity profiles for the gating configuration of (a). (c) Far-field reflection patterns for four different sets of {${\tilde{E}_{F1,n}}$, ${\tilde{E}_{F2,n}}$} selected to produce four different values of $\mathrm{\bar{\theta }}$ from about 15° to over 60°. (d) FWHM divergence angle (red circles) and reflection efficiency (blue squares) of the four configurations of (c) versus steering angle.

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The meta-mirror operation was finally modeled with two-dimensional FDTD simulations with PMLs on all boundaries, where the device is illuminated at normal incidence with a Gaussian beam of 1-mm spot size and 3.6° divergence angle focused on the GMS. Figure 4(c) shows the far-field reflection patterns (angle-resolved power reflection coefficient) for four different sets of the Fermi energies {${\tilde{E}_{F1,n}}$, ${\tilde{E}_{F2,n}}$} selected to produce four different values of the steering angle $\mathrm{\bar{\theta }}$ from about 15° to over 60°. In each case, a pronounced peak centered along the target direction is obtained, illustrating the effectiveness of this GMS design to provide actively tunable anomalous reflection. Strong suppression of all undesired diffraction orders is also observed, indicating that the periodic reflectivity variations of these devices [e.g., see Fig. 4(b), bottom panel] do not add any significant diffraction. This result is a consequence of the more prominent role played by the phase profile (relative to the amplitude profile) in shaping the wavefronts of incident light. Importantly, we also note that the device tunability is continuous, only limited in practice by the control accuracy of the reflection phase of all meta-units.

The performance of all four configurations of Fig. 4(c) is summarized in Fig. 4(d), where we plot their full-width-at-half-maximum (FWHM) divergence angle δθ_FWHM and reflection efficiency (computed as the fraction of incident power reflected within an angular range of 3×δθ_FWHM around $\mathrm{\bar{\theta }}$). A rather small divergence angle of 6° is obtained for $\mathrm{\bar{\theta }}$ ≈ 15°, while at larger steering angles the peak in the reflection pattern broadens and δθ_FWHM increases (up to 11° at $\mathrm{\bar{\theta }}$ ≈ 60°), as a result of the corresponding decrease in the transverse cross-sectional area of the reflected beam. At the same time, reasonably large efficiencies in the range of 21% to 35% are obtained, which are comparable to typical values for infrared plasmonic GMSs [1,2,4,41]. The observed increase in efficiency with steering angle is due to the larger slope ξ of the phase profile, and thus smaller number of sampled phase values, in the high-$\mathrm{\bar{\theta }}$ configurations, which facilitates the task of avoiding meta-units with exceedingly low reflectivity.

The same device can also be reconfigured to act as a reflective metalens of tunable focal length $\bar{f}$, by choosing the Fermi energies {${\tilde{E}_{F1,n}}$, ${\tilde{E}_{F2,n}}$} so as to create a discretized version of the lens phase profile ${\phi _\textrm{r}}(x )\, = \,2\mathrm{\pi }{\mathrm{\nu }_0}\left( {\bar{f} - \sqrt {{{\bar{f}}^2} + {x^2}} } \right)/c$. Figures 5(a)-(c) show the field-intensity distributions of the reflected light on the x-z plane produced by three such configurations with $\bar{f}$ ranging from 0.9 mm to 2.2 mm (in these simulations the GMS size is 2 mm, so that the numerical aperture NA correspondingly varies from 0.74 to 0.41). Figure 5(d) shows similar results for a GMS phase profile that combines steering and focusing, leading to off-axis focusing with 1.2-mm focal length and 22° steering angle. These plots were computed with the same normal-incidence Gaussian-beam illumination described above, and again a maximum allowed phase error δϕ_err (up to 0.8 rad) was introduced to maximize the overall efficiency. The resulting phase and reflectivity profiles for the configuration of Fig. 5(b) are plotted in Fig. 5(e). It should be noted that the flat profiles observed in this figure near the center of the lens are a direct consequence of the approximation introduced by the finite value of δϕ_err.

 

Fig. 5. Tunable cylindrical metalens based the structure of Fig. 1. (a)-(c) Electric-field-intensity distributions of the reflected light on the x-z plane (the plane perpendicular to the ribbons) for three different sets of Fermi energies {${\tilde{E}_{F1,n}}$, ${\tilde{E}_{F2,n}}$} optimized to produce a focal length $\bar{f}\,\,$ 0.9 mm (a), 1.5 mm (b), and 2.2 mm (c). (d) Same as (a)-(c), for a gating configuration designed to produce off-axis focusing with $\bar{f}\,\,$ 1.2 mm and 22° steering angle. (e) Reflection-phase and reflectivity profiles for the gating configuration of (b). (f) Line cuts of the field-intensity distributions on the focal plane (z = $\bar{f}$) for five different on-axis metalens configurations (with focal lengths listed in the caption). (g) FWHM focused spot size (red circles) and focusing efficiency (blue squares) of all the configurations considered in this work versus focal length. The dashed line shows the diffraction-limited FWHM spot size.

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The widely tunable focusing capabilities of this device are well illustrated by our simulation results. For all configurations investigated in these simulations, the line cut of the field-intensity distribution on the focal plane (z = $\bar{f}$) shows a well-defined peak with minimal side lobes, as illustrated in Fig. 5(f). The FWHM focused spot size δx_FWHM [circles in Fig. 5(g)] increases with increasing focal length $\bar{f}$ in reasonably good agreement with the diffraction limit (shown by the dashed line in the same figure). Similar to the beam steering results of Fig. 4, the observed deviations from ideal behavior in this plot are mostly related to the reflectivity variations across the GMS as well as its limited diameter. Finally, the squares in Fig. 5(g) show the efficiency of each metalens configuration (computed as the fraction of incident power focused within a spot of lateral size 3×δx_FWHM on the focal plane) as a function of focal length. Values ranging between 21% and 31% are obtained, similar to the beam steering efficiencies discussed above, and again limited by the graphene plasmonic absorption.

3. Conclusion

We have introduced an actively tunable metasurface platform that allows for a wide range of THz wavefront manipulation functionalities in reflection mode. The underlying tuning mechanism is provided by the plasmonic resonances of graphene ribbons in a gated FET device geometry. The key challenge in the use of these resonances for GMS applications (their limited coupling to radiation compared to nonradiative damping) has been addressed with a comprehensive design strategy involving double-layer graphene stacks on an open cavity with metallic THz antennas. With this approach, the reflection phase of each meta-unit can be tuned electrically across the full 2π range while maintaining reasonably low absorption losses. Importantly, this capability is achieved based on the electrical characteristics of CVD-grown graphene samples, which can provide the required mm-scale dimensions unlike higher-mobility exfoliated graphene. To illustrate the effectiveness and versatility of the resulting GMS platform, we have investigated several gating configurations where the same device structure is used for beam steering and focusing at 3 THz. Our simulation results show satisfactory performance for both device functionalities with continuous broad-range tunability. The same device structure could be further reconfigured to enable additional wavefront shaping operations, including off-axis focusing and holography. Altogether, these capabilities are of interest for a wide range of THz imaging and sensing applications, e.g., for the development of novel cameras and spectrometers with enhanced miniaturization and functionality. Furthermore, they may play an enabling role in future wireless communications at THz frequencies for dynamically reconfigurable routing.