We theoretically study a tunable reflective focusing lens, based on graphene metasurface, which consists of rectangle aperture array. Dynamic control of either the focal intensity or focal length for terahertz circular polarized waves can be achieved by uniformly tuning the graphene Fermi energy. We demonstrate the graphene apertures with the same geometry; however, spatially varying orientations can only control the focal intensity. To change the focal length, the spatially varying aperture lengths are also required. A comparative study between the metalenses, which generate only geometric or both gradient and geometric phase changes, has shown that the apertures’ spatially varying length distribution is the key factor for determining the modulation level, rather than the focal length’s modulation range. This kind of metalens provides tunable, high-efficiency, broadband, and wide-angle off-axis focusing, thereby offering great application potential in lightweight and integrated terahertz devices.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Metasurfaces have fostered much interest in the optics and photonics communities in recent years. By spatially adjusting the geometrical parameters of metasurfaces, such as size, shape, and orientation of the building blocks, one can control the reflected or transmitted wavefront at will [1–3]. Owing to the design flexibility and versatility, metasurface-based optics provides opportunities to go beyond what is achievable by conventional diffractive optics. Among all the applications of metasurfaces, one intriguing development is metalens for concentrating light, because miniature and ultrathin flat lenses are essential optical components in nano-optics and on-chip photonic systems [4–6]. Since beam steering was first proposed and demonstrated as a unique function of metasurfaces by Capasso’s group in 2011 , the research on metalens has been booming. By engineering plasmonic resonances in metallic structures or Mie resonances in high-index dielectric structures, various metallic or dielectric resonators have been designed and fabricated as meta-atoms for focusing planar lenses [4–6]. However, most demonstrated metalenses so far are static in nature, whose functionalities cannot be dynamically tuned after being fabricated.
Metalens with a controlled focusing effect is important in active optical circuits and miniaturized integrated photonic components. To achieve the tunable flat lens, a number of tuning techniques have been developed. For instance, stretching the metalens on the flexible substrates [8–10], controlling the axially moving or angular orientation of the metalens via integrating with the microelectromechanical systems (MEMS) [11,12], and laterally actuating two separate cubic metasurfaces based on the Alvarez lens design . In addition to the mechanical tuning [8–13], there are also strategies of incorporating tunable materials into metasurfaces for changing the focal position or the hotspot, such as the use of anisotropic liquid crystals , phase-change materials [15–17], graphenes [18–21] or others . However, due to integrating microelectromechanical actuators or putting together several optical elements with different functions, these approaches typically require complex controlling systems, sophisticated surface morphologies and therefore complex fabrication processes.
Graphene has unique electrical and optical properties, whose optical conductivity can be altered by electrostatic or chemical doping . Compared with the noble metal plasmons in the visible and near-infrared regions, graphene plasmons have the stronger confinement and lower losses in the lower frequencies [24–26]. These features make graphene a promising candidate for tunable metasurfaces in the terahertz and mid-infrared regions [27–31], therefore having great application potentials in photonics and optoelectronics devices [24–31]. By integrating active graphenes with metal or dielectric metasurfaces, or directly utilizing the structured or patterned graphenes as the building blocks of metasurfaces, some graphene-based planar lenses have been proposed to control the wavefront [18–21,30–42] or the surface plasmon polaritons (SPPs) [43,44]. For example, assisted by graphenes, the active terahertz metalenses with tunable focal intensity or focal length have been demonstrated by metal metersurface-graphene hybrid systems, in which the graphene operates as a tunable Drude metal to control the radiation of the passive metasurface [18–21]. However, due to the Ohmic loss caused by metallic nanostructures, the metalens based on the hybrid systems are usually inefficient [18–21]. In addition, ultrathin graphene-based Fresnel zone plate lenses have been developed in the optical or infrared regime, whose focal intensity can be tuned by adjusting the graphene Fermi energy and the layer number [32–34]. Recently, graphene nanoribbon or patch arrays have been proposed to manipulate the amplitude and phase of the reflected or transmitted wave in the infrared and terahertz range, in which by changing the geometry or Fermi energy of graphene meta-atoms to engineer the plasmonic resonances, metalens with a tunable focal intensity or position can be achieved [31,35–42]. However, in these strategies, the tunable focal spots in the transversal or longitudinal directions can only be fulfilled by the individual control of each graphene resonator for different Fermi energies [31,35–39]. So it is extremely difficult to implement in the device fabrication and integration because of the deep subwavelength dimension of the graphene meta-atoms.
In this paper, we demonstrate the active focusing control of terahertz circular polarized waves by the graphene aperture-based planar lens with high efficiency. Compared with the metallic metasurface/graphene hybrid structure, the graphene metalens combines the focusing and modulation elements into one. By carefully designing the orientation or both the orientation and length of the graphene apertures, metalenses with tunable focal intensity or tunable focal length can be achieved respectively by changing graphene Fermi energy uniformly. The spatially varying length of the apertures is proved to be the key factor to realize variable focal length. The aperture orientations can be determined by Pancharatnam-Berry (PB) phase theory. A further orientation correction due to the nonuniform of aperture lengths has been demonstrated to have little influence on the focusing modulation level but great effect on the modulation range of the focal length. Due to the characteristic of electrically connected graphene layer, the graphene metasurface based on aperture array are more convenient for controlling the response of all units by applying gate voltage uniformly, consequently more suitable for tunable devices than that consisting of isolated graphene patches.
2. Structures and methods
The proposed tunable graphene metalens is schematically shown in Fig. 1, which consists of a graphene monolayer etched with rectangular apertures, a dielectric layer and a metal ground plate. The graphene apertures are arranged in the x and y direction with a period of 8 um. The width, length, and rotation angle of the rectangular aperture are denoted by a, b and φ, respectively, as the inset shown in Fig. 1. The pattern of the rectangular aperture array on the graphene monolayer can be obtained by photolithography or electron-beam lithography techniques [33,45]. A circularly polarized (CP) THz wave is normal incident on the graphene metasurface. The structure can be modeled as an asymmetric Fabry-Perot resonator, that is, a metasurface reflector as a partially reflecting mirror in the front and a metallic fully reflecting mirror in the back . To achieve the enhancing plasmon resonances in graphene apertures, the metal mirror below the dielectric layer should be separated at quarter wavelength distance from the graphene layer . After further optimization for high reflection at around the frequency of 5THz, we set the thickness of SiO2 spacing layer to be 8.5 um.
All calculations are carried out by the three-dimensional numerical simulation software of COMSOL Mutiphysics 5.3a based on the finite-element method. In simulations, the SiO2 layer has a relative permittivity of = 3.75 and a loss tangent of tanδ = 0.0184. Graphene is modeled by the surface conductivity approach [46,47], that is to assign a surface current () at the air/SiO2 interface instead of a graphene layer with finite thickness. The frequency-dependent surface conductivity is expressed in the local-RPA limit as :23]. The carrier density has a linear dependence on the external gate voltage, i.e. . is the applied gate voltage, is the Dirac point voltage and is the dielectric separation between the graphene and the gate electrode. So the Fermi energy Ef can be modulated by the gate voltage. For the proposed design of metalens, it does not suit to directly employ the metal reflector as a gate electrode , since the thickness of dielectric layer between the graphene layer and the metal ground plane is on the order of several micrometers, that requires high gating voltages of kV for high Ef. Therefore, ion-gel top gating technique should be employed, which has been demonstrated to effectively induce high carrier densities in graphenes [27,49,51]. The Fermi energy as high as Ef = 1–2 eV has been reported by the top gating schemes [51,52].
Firstly, we investigate the dependence of the reflection and phase on the geometry of the graphene aperture by simulating one unit-cell of the metalens under the normal incidence of a left-handed circularly polarized (LCP) THz wave. Floquet excitation port and periodic boundary condition are applied. S parameters are used to calculate the complex reflection coefficients of the co-polarized (RLL) and cross-polarized (RLR) waves. Then, we use 50 units of the graphene apertures to construct the metalens for simulating focusing effect. Perfectly matched layers (PML) are placed outside of the model and the scattering fields are investigated under the incidence of a LCP wave with f0 = 5 THz.
3. Results and discussion
Figure 2(a) shows the reflection spectrum of the graphene matesurface with uniform apertures under normal incidence of a LCP light. We select the geometric parameters of the aperture as a = 3 um, b = 6 um and φ = 0. The Fermi energy of the graphene is Ef = 1.0 eV. It can be seen that the reflected waves are decomposed into two circular polarization components, one component possessing the same helicity (RLL) as the incident wave and the other with opposite helicity (RLR) to the incident wave . A broad reflection band of RLL and simultaneously a reflection dip of RLR are shown in the frequency range from 4.8 to 6 THz, with the amplitude of RLL greater than 0.7 and RLR less than 0.01. These features show a polarization-keeping reflection with a high efficiency over a broadband can be achieved by the graphene metasurface . According to the PB phase theory, an abrupt phase change of 2φ will occur for the reflected light when the rectangular aperture has a rotation angle φ . Figure 2(b) shows the amplitudes and phase shifts of the LCP reflected wave at f0 = 5 THz as the aperture rotation angle φ changes from 0 to 180°. In Fig. 2(b), a phase variation across 2π can be achieved while the graphene apertures with different orientations have the similar ability to reflect light. The high reflectivity and the continuous 2π phase modulation enable the design of novel planar optical devices by graphene apertures for efficient control of THz wavefront.
3.1 Metalens with controlling focal intensity
To achieve the phase profile equivalent to a conventional cylindrical lens, the metasurface should carry a phase shift distribution that can be expressed as :Figure 3(a) gives the phase distributions ofobtained by Eq. (2) when the incident frequency is f0 = 5 THz and the focal length is F = 150 um and 190 um (black lines with squares), respectively. For convenience, we only provide the phase distributions of 50 aperture units along the x-axis. According to PB phase theory, we can obtain the corresponding aperture rotation angle distributions by:
Figure 3(b) shows the simulation result of the metalens with a designed focal length of F = 150 um at f0 = 5 THz, where all graphene apertures have the same geometry (a = 3 um and b = 6 um) and the rotation angle distributions satisfy the Eqs. (2) and (3), as the black line with solid squares shown in Fig. 3(a). The Fermi energy of graphene is Ef = 1.0 eV. In Fig. 3(b), the intensity distribution of the reflected field on the x-z plane indicates that the reflection wave is efficiently focused at the position of about z = 140 um as a LCP THz wave is incident vertically. The small discrepancy between the designed focal length (F = 150 um) and the simulated one (F = 140 um) is mainly due to the small Fresnel number of the lens [56,57].
Figure 3(c) shows the intensity of the reflected wave along the z-axis under different graphene Fermi energies. It can be seen that the focal length is almost no changed, but the focal intensity is significantly reduced by decreasing Ef. The focusing effect can be completely turned off when Ef is about 0.5 eV or smaller. This is because a low Fermi energy and consequently a small carrier density will weaken the plasmon resonance of graphene apertures. At the focal plane (z = 140 um), the calculated intensity distributions along the x axis direction are shown in Fig. 3(d), in which the full width at half maximum (FWHM) of the focal spot are about 34 μm () and the focusing efficiency is as high as 66.6% for Ef = 1.0 eV. The focusing efficiency is obtained by calculating the ratio of the reflected light intensity at the focal spot with a width of three times of the FWHM to the whole incident light intensity , which decreases from 66.6% to 15.8% as Ef changing from 1.0 eV to 0.6 eV. Due to the excellent plasmonic performance of graphene in the THz regime and the absence of Ohmic losses induced by metallic resonators, the graphene aperture metalens shows a significantly higher focusing efficiency than the metalens based on metallic aperture/graphene hybrid structure (only about 2.71%) .
3.2 Metalens with controlling focal length
As discussed above, the metalenses with tunable focal intensity can be achieved by carefully design of the orientation of each graphene aperture based on PB phase theory. To design the THz lens with a tunable focal length, besides the specific phase shift distribution determined by Eq. (2), the different phase change for apertures at different positions must be fulfilled as tuning Ef uniformly. As the example shown in Fig. 3(a), we give the phase shift distributions of and for the metalens with a focal length of F = 150 um and 190 um (black lines), respectively. In order to achieve the modulation of the focal length from 150 um to 190 um, the uneven phase change distribution of (red line) is required for the lens. Notably, the farther away from the center of the lens, the greater the reflected phase change. This can be achieved by using rectangular aperture antennas with different lengths, as described below.
Figures 4(a) and 4(b) show the dependence of the reflection and phase of the LCP reflected wave on the aperture length b for different Ef. The value range of b is from 4 um to 7 um. In Fig. 4(b), we further define to describe the phase change of the aperture with the length of b (um) as Ef changes from 0.6 to 1.0 eV. Figure 4(c) shows the normalized value of , which is calculated by for describing the difference of phase change between the aperture with the length of b and that with b = 7 um. From Fig. 4(c), we can see clearly that the value of normalized increases with the decrease of b, which means the apertures with different lengths can lead to different phase changes for the same change of Ef. The aperture with a shorter length contributes a greater phase change. So, in order to realize the uneven phase change distribution of under the uniform gate voltage tuning for fulfilling the designed focal shift from F1 to F2, as the red line shown in Fig. 3(a), the length of the aperture unit cells should decrease gradually from the lens center to the edge. In Fig. 5(a), we give the length distribution of the apertures along the x-axis of the metalens (black line). The aperture lengths are determined by having the normalized at all positions satisfy the phase change distribution of shown in Fig. 3(a).
Based on the distributions of aperture length b (black line) and rotation angle (blue line) shown in Fig. 5(a), an active THz metalens with a designed focal shift from F = 150 um to 190 um by changing Ef from 1.0 to 0.6 eV is firstly investigated. The rotation angle distributions are obtained by Eqs. (2) and (3) as f0 = 5 THz and F = 150 um. Except for the aperture length distribution, the lens investigated here has the same geometric parameters as that designed in Fig. 3(b)-3(d). Figures 5(b)-5(d) show the simulation results of the metalens with spatially varying aperture lengths. Clearly, the focusing effect is observed for different Ef. In Fig. 5(b), the focal length is increased from F = 116 um to 160 um as Ef is decreased from 1.0 to 0.6 eV, showing a focal shift of about 44 um. Although there is a similar modulation level between the simulated and the original designed metalens (44 um vs. 40 um), a significant deviation of modulation range between them exists ([116,160] um versus [150,190] um).
3.3 Improved metalens for controlling focal length
Then, an improved metalens for correcting the modulation range are proposed. Because of different aperture lengths in the varifocal metalens, the phase shift for a focal length F should include both the PB phase (also called geometric phase) and the gradient phase. The former results from the rotation of the apertures, and the latter is caused by the change of the aperture geometry. The deviation shown above on the focal length ([116,160] um vs. [150,190] um) during the modulation process is due to the neglect of the latter. According to Eq. (2), the phase shift distribution is fixed for the specific focal length and frequency. The nonuniform distribution of the aperture lengths leads to additional phase changes, as shown in Fig. 4(b), which may be offset or corrected by extra rotations of the apertures . In consequence, the aperture rotation angle at position x can be calculated by the modified equationFig. 4 (a). represents the phase difference between the aperture located at the position x and that at the lens center (x = 0). According to the Eq. (4), the rotation angle distribution of the apertures along the x-axis are recalculated and shown in Fig. 5(a) by a red line. Figure 6(a) shows the simulation results of the improved metalens with the corrected rotation angles. As illustrated in Fig. 6(a), when Ef is decreased from 1.0 to 0.6 eV, the focal length is increased from 136 um to 180 um, showing a similar modulation level (~44 um) but a different modulation ranges with respect to the unimproved lens ([136,180] um) vs. [116,160] um). If considering the deviation of about 10 um caused mainly by small Fresnel number of the lens (see Fig. 3 and the related discussion), the improved metalens agrees well with the original design that planning to tune the focal length in the range of [150,190] um. For clarity, we illustrate the tuning results of different lens in Fig. 6(b). A linear relationship between the focal position and the value of Ef can be observed for the two proposed variable-focus metalens. Figure 6(b) also provides the calculated focusing efficiency in the modulation process. When decreasing Ef from 1.0 to 0.6 eV for realizing a longer focal length, the focusing efficiency is decreased from 60% to 15%. Even so, the graphene aperture based metalens still has a much higher efficiency than the metalens based on metallic aperture/graphene hybrid structure (from 2.71% to 0.19% as Ef changes from 0.1 to 0.3 eV) [19,20].
3.4 Metalens with broadband focusing modulation
In the study above, we only investigated the focal length modulation at the designed working frequency of f0 = 5 THz. Actually the efficient and tunable focusing performance is still possible in a broadband range, overlapping with the broad reflection window of RLL shown in Fig. 2(a). Figure 7(a) shows the focal length modulation of the improved metalens at f0 = 4.8 THz. There is a focal shift of about 42 um as Ef is decreased from 1.0 to 0.6 eV. Figure 7(b) illustrates the broadband performance of focusing modulation in a frequency range from 4.2 to 5.4 THz. Clearly, the modulation range reaches its maximum at the working frequency of f0 = 5 THz. When the incident frequency deviates from the designed frequency, the modulation range will decrease. The decrease of focusing length with the decrease of the incident frequency can be understood by the axial chromatic aberration . In Fig. 7(b), due to the lower focusing efficiency (less than 15%) in some cases with Ef = 0.6 eV, we only provide the modulation range of the focus by decreasing Ef from 1.0 to 0.7 eV as f0 = 5.2 or 5.4 THz.
3.5 Metalens with wide-angle off-axis focusing
Finally, we also investigate the focusing effect of the improved metalens under different incident angle. As the electric field intensity distributions shown in Fig. 8, the focusing is still clear when the oblique incident angle becomes larger. Simultaneously, transverse and longitudinal movement of the focus spot can be realized by the oblique incidence.
Another point we would like to make is that the above designed varifocal metalens has the property that the focal length F is increased by decreasing the value of Ef. Based on the same principle, if the length distribution of the graphene aperture is opposite, that is, from the center of the lens to both sides, the aperture length b gradually increases, then a metalens with the tunability that the focal length increases with increasing Ef can be realized.
In summary, we have proposed a kind of graphene aperture-based metalens for active focusing control in terahertz regime. Our results demonstrate the graphene metalens can reach a high focusing efficiency of above 60%, superior to the metalens based on metallic metasurface/graphene hybrid structure. Due to the feature of electrically connected graphene layer, the metalens can achieve dynamic tuning by changing the Femi energy of the graphene metasurface uniformly, thus more suitable for tunable devices through electrostatic gating. In addition, we also demonstrate the graphene metalens with uniform aperture size and carefully designed aperture orientation can only achieve tunable focal intensity. For metalens with tunable focal position, both the orientation and the geometry of the apertures are required to be carefully designed. The nonuniform distribution of the aperture length is proved to be the key factor for the control of the focal length. The orientation correction resulting from the difference of the aperture length has great effect on the value of focal length but little influence on the focusing modulation range. The excellent agreements between the design and the simulation results are expected to provide a general design scheme for the tunable metasurface devices based on graphene apertures. In addition to adjusting focus intensity and focal length easily, this kind of graphene metalens has also the ability of high-efficiency, broadband, and wide-angle off-axis focusing, which make it have great application potential in lightweight and integrated terahertz devices.
Programs for Science & Technology Innovation Talents in Universities of Henan Province (No. 17HASTIT016); National Natural Science Foundations of China (Nos. 11404291, 11704344, and 11504333); Aeronautical Science Foundation of China (No. 2017ZC55006); Natural Science Foundations of Henan Province (No. 162300410314).
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