Graphene-enabled reconfigurable terahertz wavefront modulator based on complete Fermi level modulated phase

Although great achievements have been obtained in metasurfaces so far, the functionalities of these devices are almost static. The dynamically adjustable devices are far less explored. Here we theoretically and numerically demonstrate a veritable reconfigurable terahertz wavefront modulator (TWM). The designed TWM can dynamically shape the wavefront at will via imposing different Fermi levels on the constituent graphene ribbons. By adopting the Dirac brackets and Matrix analyze method, the correlation between the phase shift and Fermi level is theoretically established, which offers a general scheme for designing dynamically switchable devices. As a proof of concept, three different sets of pre-calculated Fermi levels are imposed on the graphene ribbons. The TWM can be dynamically switched among back reflector, varifocal metalens and Airy beam generator, which has never been demonstrated before as far as we know. The proposed reconfigurable TWM owns the capability of dynamically steering terahertz wavefront, indicating great significance for the development of THz reconfigurable devices.

Despite great achievements having been obtained in metasurfaces so far, most of the newly demonstrated metasurface-based devices are naturally static. The most important reason is the fixed geometrical dimensions and dispersion relations of the metasurface. The tunability and reconfigurability of the devices, that have emerged as urgent industrial requirements, show vital importance in practical applications like dynamic holography and active focusing. To achieve devices with active tunability, active materials with adjustable electromagnetic properties in real time under external stimuli are needed. Some materials, such as liquid crystal metasurfaces [22,23], phase change material [24,25] and electrically driven carrier accumulation [26][27][28], have recently been employed to construct the adjustable devices. Graphene, a monolayer material consisting of honeycomb arranged carbon atoms, has been verified to be a promising candidate for dynamic wavefront control at terahertz (THz) and mid-infrared frequencies [29][30][31][32][33]. As we know, the Fermi level and surface conductivity of graphene can be easily tuned by the external stimuli (gate voltage or chemical doping), which can induce a controllable light-matter interaction [34]. Owing to this impressive characteristic, significant efforts have been devoted to investigate dynamically adjustable devices based on graphene or graphene-assisted structures [35][36][37][38][39][40][41][42]. Ding et al proposed a dynamic metalens constructed of graphene apertures, which possesses the capacity to modulate the focal intensity, length, and bandwidth by changing the graphene's Fermi level [41]. Liu et al designed a graphene-based grating constructed of graphene ribbons with different widths and obtained a tunable metalens [42]. Although the previous proposed devices can achieve dynamically switchable wavefront manipulation, almost all of them cannot get rid of restricted tunability by considering its limited phase shift (they cannot achieve phase coverage of 0-2 π in response to the external stimulus).
In this paper, we theoretically and numerically demonstrate a veritable reconfigurable terahertz wavefront modulator (TWM) by taking advantage of Fermi level modulated complete phase manipulation. Firstly, the graphene unit cell is optimized to obtain a Fermi level modulated phase coverage of 0-2 π, which is the basis of the reconfigurable TWM. Then, by adopting the Dirac brackets and matrix analyze method, the underlying physics of the TWM is investigated. In this process, the relationship between the phase shifts and the Fermi levels is established, providing a general scheme for designing dynamically switchable devices. As a proof of concept, three different sets of pre-calculated Fermi levels are imposed on the graphene ribbons. The TWM can be dynamically switched among back reflector, varifocal metalens and Airy beam generator. With the capability of dynamically steering the THz wavefront, the proposed TWM will be extended to design other active adjustable devices. Figure 1 shows the schematic of proposed reconfigurable TWM, which consists of a list of graphene ribbons, a silica spacing layer and an optically thick silver film. As shown in figure 1, the designed TWM exhibits Fermi level modulated reconfigurable functionality: it functions as a metalens or an Airy beam generator. The inset of figure 1 depicts the unit cell of design, from which one can see the detail parameters (the period along x-axis P x , the width of graphene ribbon w, the thickness of spacing layer t 1 and the thickness of the silver film t 2 ). The frequency dependent effective permittivity of silver is considered with the Drude model: ω p = 2 π × 2.2 × 10 15 rad s −1 , ω c = 2 π × 4.35 × 10 12 rad s −1 . The silica space layer (with refractive index of 1.45 and thickness of t 1 = 10 μm) is utilized to function as an FP resonant cavity, aiming at enhancing the reflected phase shift. The silver film at the bottom, which serves as a perfect electromagnetic conductor (PEC), with the thickness be set as t 2 = 2 μm. The period of the unit cell along x-axis is P x = 5 μm, and the period along y-axis is infinite. The graphene is modeled as a flat two-dimensional (2D) sheet. Its surface conductivity is dominated by the inter-band contribution in the THz regime:

Metasurface design and theoretical model
In order to obtain a planar device with reconfigurable functionality, the electromagnetic properties of the unit cell should be actively adjustable. For this purpose, the Fermi level modulated characteristic of the unit cell is investigated. The simulated reflected phase shifts and reflectivity of the unit cell as functions of frequency and Fermi level are shown in figures 2(a) and (b). The three-dimensional finite difference time domain (FDTD) method is utilized to characterize the electromagnetic property of the unit cell. In these simulations, periodic boundary conditions and perfectly matched players are utilized along the x and z directions, respectively. The incident THz wave is x-polarized that normally illuminated on the metasurface along the negative direction of z-axis. The other parameters are the thickness of the silica layer t 1 = 10 μm, the width of graphene ribbon w = 2 μm, the period of the unit cell P x = 5 μm and the thickness of the silver film t 2 = 2 μm. The simulation results show that, at some certain frequencies (from 2 THz to 6 THz), the reflected phase shifts show active adjustable characteristic, which vary from almost -π to π as the Fermi level ranges from 0 eV to 1 eV. It should be pointed out that this E f modulation can be experimentally realized by applying a bias voltage. With this method [44,45], a pair of ultrathin (e.g. 30 nm) and transparent conductive layers (e.g. doped silicon) as negative back electrode can be proposed upon the graphene grating in our device. The graphene films connect to the positive electrode, and there is a thin  dielectric layer between the back electrode and the graphene film. As shown in figures 2(c) and (d), the working frequency is selected as f = 5.75 THz for its full phase coverage of 2 π and relatively high reflectivity (with average reflectivity over 60%).
The Dirac brackets and Matrix analyze method are adopted to investigate the underlying physics of the Fermi level modulated characteristic of graphene ribbon [46,47]. For a graphene-based metasurfaces device without external stimuli, the interaction between the incident THz wave and the designed graphene-enabled device (constructed of n graphene ribbons) can be expressed as where |L> is the Dirac bracket of the linearly polarized (LP) light and the operatorĜ i represents the graphene ribbons' modulation effect on the amplitude and phase of the incident THz wave. a i and ψ i represent the amplitude and phase introduced by the ith graphene ribbons. Equation (1) also indicates that the reflected light is co-polarized with the incident one (due to the structure symmetry of graphene ribbons). For the device without incident THz wave, the Fermi level's modulation effect on the graphene ribbons can be expressed as where E mn represents the imposed Fermi level on the graphene ribbon. b mn and ϕ mn represent the corresponding amplitude and phase changes. The sign '·' defined as the Fermi level's modulation effect on graphene ribbons. Utilizing equations (1) and (2), the total modulation effect can be derived as Therefore, there is a correspondence relationship between the Fermi level and the phase shifts, namely, a set of imposed Fermi levels would undoubtedly results in a set of modulated phase shifts (see figure S1 in the supporting information) (stacks.iop.org/NJP/22/063054/mmedia). Such a corresponding relationship will give dynamically adjustable phase shifts, providing unprecedented capacity in designing reconfigurable devices.

Reconfigurable TWM with Fermi level modulated varifocal focusing
As a proof of concept, a reconfigurable TWM constructed of 161 identical graphene ribbons is designed, with a total width of 805 μm. For the designed TWM, it functions as a back reflector (with the incident THz wave back reflected) without or with identical Fermi levels (see part 2 in the Supporting Information).
To function as a dynamically varifocal metalens, the endowed phase shifts should not only fulfill the equal paths principle for focusing but also be independently modulated. The former can be achieved by introducing a parabolic-shaped phase shift and the latter can be achieved by Fermi levels modulation. For a cylindrical metalens, the required phase shifts ϕ(x) along the interface should satisfy where λ is the designed THz wavelength, x is the coordinate of the graphene ribbon and x c is the coordinate of focusing point on x-axis. f represents the vertical distance from the focal point to the metasurface plane and f c is the focal length which is defined as the distance from metasurface plane to the focal spot and can be calculated as f c =(x 2 + f 2 ) 1/2 . Hence, a dynamically varifocal cylindrical metalens can be designed by linearly arraying the identical graphene ribbons. The varifocal functionality can be achieved by imposing different pre-calculated sets of Fermi levels on the graphene ribbons.
To show that, the TWM is assigned to function as an on-center metalens (with focal length f c = 500 μm and the coordinate of focal spot on x-axis x c = 0). As shown in figure 3(a), the required phase shifts can be endowed by imposing a certain Fermi level on the graphene ribbons. Moreover, the required Fermi levels are almost in the range of 0.0-1.0 eV, which are experimentally attainable. For the Fermi levels go beyond the range in a few small areas, they are approximately set to be 0.0 eV or 1.0 eV in the simulations. Figure 3(b) shows the simulated intensity (|E| 2 ) profile at x-z plane, from which one can see clearly that the reflected THz wave is efficiently focused, with a focal spot exhibits at location (0 μm, 550 μm). The slight discrepancy between the simulated and designed focal lengths can be attributed to the finite sizes of the constructed unit cells with respect to the incident THz wavelength. The cross section of the focal spots along x-axis are depicted in figure 3(c), where the full-width at half-maximum (FWHM) is 54 μm, indicating a nearly diffraction-limited focal spot of the designed graphene-based metalens. Figure 3(d) shows the simulated phase profile at x-z plane, in which the phase profile exhibits parabolic shape.
As in the above-mentioned discussion, the graphene ribbon exhibits Fermi level modulated phase shift, providing a scheme to rearrange the phase profile of the designed device. Owing to such a characteristic, the designed TWM can be utilized to function as a reconfigurable metalens. Dynamically tuning the on-center focal spot along y-axis by Fermi level modulation is demonstrated in figure 4. It can be observed that the focal spot can be actively adjusted to the expected locations. Moreover, the working principle of our designed reconfigurable TWM is also visualized: a certain set of Fermi levels would result in specific phase profiles, and it finally leads to a specific functionality. Using the same strategy, dynamically tuning the off-center focal spot along x-axis by Fermi level modulation is also demonstrated and the corresponding results are depicted in figure 5. It can be observed that the focal spot can be dynamically adjust to the expected locations ((−250 μm, 400 μm), (0 μm, 400 μm), (250 μm, 400 μm)) via Fermi level modulation. Therefore, with a proper set of Fermi levels imposed on the proposed TWM, the focal spot can be adjusted at will, indicating more practical applications.

Reconfigurable TWM with its functionality reconstructed from focusing to Airy beam generating
Furthermore, by virtue of Fermi level modulated complete phase manipulation, the proposed reconfigurable TWM capable of switching from one functionality to another. Here the Airy beam generating is chosen as an example due to its unique properties (such as non-diffraction, self-accelerating, and self-healing) and interesting applications in optical micro-manipulation. Such a dynamically switching could be accomplished by imposing a specific Fermi levels on the graphene ribbons, without reconstituting the TWM's configuration. The amplitude of the 1D Airy beam can be described as follows [48] ψ(x, θ) = Ai(bx) exp(ax + ikbx sin θ), where Ai(bx) = 1 π ∞ 0 cos( t 3 3 + bxt) dt is the Airy function with x representing the transverse coordinate and k is the wave number. a, b, and θ are performance parameters of Airy beam which represent a positive number to truncate the Airy beam, transverse scale, and bending direction, respectively. The desired phase shift can be calculated as ϕ(x, θ) = arg(ψ(x, θ)).
Figures 6(a) and (b) depict the required amplitude and phase shift for the Airy beam, in which the performance parameters are set as a = 0.004, b = 0.021 and θ = 0 • . Here the phase modulated Airy beam is investigated due to the unattainable amplitude variation for our elaborately designed graphene ribbon [49].  To fulfill the required phase as shown in figure 6(b), two Fermi levels E f = 0.5068 eV and 0.3428 eV are selected. The two Fermi levels could provide a π phase difference along with identical magnitude (see part 3 in the supporting information). By imposing a set of Fermi levels constructed of the two selected ones, the TWM could be reconstructed from focusing to Airy beam generating. Figure 6(c) depicts the simulated electric field distribution (x-z plane at y = 0) of the TWM, showing clearly non-diffracting and self-bending characteristics. In addition, the self-healing property is investigated as well by placing a PEC obstacle with a size of 85 μm × 85 μm (x-z plane) in front of the main lobe centered (at x = 5 μm and z = 200 μm). The simulated electric field distribution is depicted in figure 6(d). It is obvious that the field distribution can be locally modified by the obstacle. However, the disturbed beam profiles could be automatically recovered to an Airy beam after passing by the obstacle.

Conclusions
In conclusion, we theoretically and numerically demonstrated a graphene-based veritable reconfigurable TMW, which possesses the capacity of dynamically shaping the wavefront of incident THz wave at will. The proposed TWM consists of identical graphene ribbons, which can get rid of the complex parameters design process. Moreover, the graphene ribbon is able to obtain Fermi level modulated phase coverage of 0-2 π. By utilizing the Dirac brackets and matrix analyze method, the correlation between the phase and Fermi level is established, provising a general scheme for designing reconfigurable devices. To verify such a strategy, different sets of pre-calculated Fermi levels are imposed on the graphene ribbons. The TWM can exhibit excellent reconfigurability with the functionality can be reconstructed among back reflecting, varifocal focusing and Airy beam generating. With Fermi level modulated full phase manipulation, the proposed reconfigurable TWM will be extended to achieve other dynamically switchable functionalities, opening up new doors for designing reconfigurable devices.

Conflicts of interest
The authors declare no conflict of interest.