We report new models for out-of-plane focusing and manipulation of terahertz beams based on a silicon/copper grating covered by monolayer graphene. Dependences of focusing and manipulation of terahertz beams on the chemical potential and scattering rate of graphene are investigated. Based on the graphene/silicon grating model, we demonstrate that the focal distance and intensity are sensitively influenced by the chemical potential. Based on the graphene/copper grating model, we show how 2 to 1 and 3 to 1 modulation of terahertz beams can be efficiently realized through tuning the chemical potential of graphene. These tunable beam focusing and manipulation effects are well explained by the diffraction theory of optical images and the surface plasmon polariton theory of graphene. Our proposed devices are of compact structures, high electro-optical tunability and good repeatability, and they are expected to have prospective applications in terahertz communications, imaging, sensing, and so on.
© 2017 Optical Society of America
In the terahertz range, graphene behaves like Drude-type material and shows superior electronic and optical properties which can be controlled by changing its chemical potential via electrostatic gating or chemical doping [1–6]. In the past decade, many graphene-based tunable terahertz devices such as multimode modulators , absorbers [8–10], polarizers , amplitude and phase modulators [12, 13], filters, antennas and cloaking devices [14–16] have been intensively investigated. These simple, efficient and compact devices exhibit a wide variety of applications in terahertz communication, spectroscopy, biological sensing, military defense, and so on [17, 18].
Recently, graphene-based beam focusing and manipulation in the mid-infrared and terahertz regions have attracted a lot of attention [19–23]. For instance, Ashkan Vakil et al. demonstrated that a graphene sheet with designed conductivity distributions can act as a convex lens for focusing and collimating 30 THz surface waves propagating along the graphene . Jiu-Sheng Li proposed a graphene-based voltage-gated terahertz planar lens with tunable focal distance from 7.3 to 15.2 μm . However, these graphene-based models were designed for in-plane beam focusing and manipulation which are expected to show applications in integrated optics. Until quite recently, several reflective graphene metasurface lenses were reported for terahertz beam focusing in the out-of-plane free space [24–26]. As compared with the integrated optics, the out-of-plane transmission optics which has shown many important applications in modern industries is more practical. As compared with other tunable terahertz devices based on conventional phase transition materials (e.g. VO2)  and thermistor materials (e.g. SrTiO3) [28, 29], graphene-based tunable terahertz devices can be designed with more compact structures and operated in a more safe and convenient manner. In addition, as compared with the quite similar, ultra-thin and compact silicon tunable device , higher tunability is expected in graphene-based tunable devices due to much broader tuning range of graphene’s conductivity in the terahertz region. It is definitely worth to develop all kinds of graphene-based tunable terahertz devices for applications in the out-of-plane transmission optics.
In this paper, we would like to investigate the out-of-plane focusing and manipulation of terahertz beams based on Si/Cu gratings integrated with monolayer graphene. Numerical calculation and analysis will be made to demonstrate that our tunable models can work efficiently and stably for beam focusing and manipulation in the terahertz region.
2. Models and methods
The schematics of beam focusing and manipulation models investigated in this paper are depicted in Fig. 1. In the beam focusing model, as shown in Fig. 1(a), a simple silicon grating is covered by a monolayer graphene sheet. The thickness and period of the silicon grating are denoted as hSi and ΛSi, respectively. The filling factor of the silicon grating is 0.5, i.e., the width of the grating slit is equal to half of the grating period. In the beam manipulation model, as shown in Fig. 1(b), a graphene-covered copper grating with three air slits symmetrically surrounded by 2N (N = 10) grooves is used. hCu represents the thickness of the copper substrate. The widths of the center and side slits are denoted as wc and ws, respectively. l is the slit interspacing (center-to-center). The grooves have the same width of wg, the same depth of tg and the same period of Λg.
In these two cases, a transverse magnetic (TM) polarized plane wave is normally incident from air onto the bottom surface of the grating. The conductivity of the graphene sheet is derived from the Kubo formula which contains the intraband electron-phonon scattering term σintra and the interband electron transition term σinter :32, 33], e is the electron charge, ħ is the reduced Plank constant, and kB is the Boltzmann constant. In the terahertz region, the conductivity of graphene is dominated by the intraband transition, so the interband contribution can be ignored in our investigation. Quite recently, a dominated plasmonic electron-hole ratchet effect is predicted in a dual-grating-gate graphene model, which implies that the patterning on the substrate would result in modulated carrier density in graphene and affect its optical conductivity . In the current study, such potential influence can be ignored because our models do not contain any dual-grating-gate structures.
At the same time, graphene can be modeled by using a uniaxial anisotropic permittivity. The permittivity tensor contains two in-plane (x-y plane) components and one out-of-plane (z direction) component , which can be described by:
Tunable beam focusing and manipulation can be realized by changing the chemical potential of graphene, which can be controlled on purpose by applying a gate voltage (a static electric field) or by means of chemical doping. In this work, we employ the first method. As shown in Fig. 1, to apply a gate voltage on monolayer graphene, a thin Al2O3 gate dielectric film is used between graphene and the silicon/copper substrate. An approximate closed-form expression to relate the chemical potential μc to gate voltage Vg is , where νf is the Fermi velocity (~106 m/s in graphene), εd and d are the relative permittivity and thickness of the gate dielectric, respectively. A typical tuning range of is reported from 0 to 1.8 eV . Assuming that εd = 9 and d = 3 nm, a gate voltage from 0.045 to 3.540 V is required to change the chemical potential from 0.1 to 0.9 eV. The corresponding transverse field is required to be as large as 12 MV/cm, which is below the breakdown electric field of the 3nm-thick Al2O3 layer (~14 MV/cm) . Large-scale monolayer graphene is required for terahertz applications and it can be first fabricated by the chemical vapour deposition (CVD) method and then transferred onto the grating in organic solution. It should be noted that the properties of CVD grown graphene are currently not as promising for applications as mechanically exfoliated flakes. Due to unavoidable defects, the scattering rate of CVD grown graphene is relatively high. Meanwhile, the scattering rate is reported to grow proportionally to the Fermi energy . The influence of the scattering rate on the performance of our models was taken into account in our simulation.
A commercial three-dimensional finite-difference time-domain (3D-FDTD) software package, “Lumerical FDTD Solutions”, was employed to study the beam focusing and manipulation behaviors. In FDTD calculations, the perfectly matched layer (PML) absorbing boundary condition was used along both the x and z directions, and the periodic boundary condition was used along the y direction. The gird sizes (Δx, Δy, Δz) set in the beam focusing and manipulation cases are 1 μm and 0.1 μm, respectively. It is worth mentioning that the monolayer graphene is modeled by a surface conductivity material with vanishing thickness. The time step Δt satisfies the Courant stability limit, i.e., .
3. Results and discussion
3.1 Tunable focusing of terahertz beams
An optimized pure silicon grating in Fig. 1(a) with structural parameters of hSi = 36 μm and ΛSi = 50 μm is discussed here for beam focusing study at 3 THz. Based on this grating, we have investigated the focusing performance of graphene covered silicon grating under different control conditions. First, we considered a fixed scattering rate of 0.17 meV and a varied chemical potential from 0.1 to 0.9 eV in simulations. As shown in Fig. 2, an incident plane wave from the left side of the silicon grating is diffracted and focused on the right side of the graphene sheet when different chemical potentials are applied. In all cases, the focal distance is more than 26 wavelengths in the far field of propagation space. Second, we considered a fixed chemical potential of 0.5 eV and a varied scattering rate from 0.1 to 0.5 meV in simulations. The dependences of focal distance and intensity on the chemical potential and scattering rate are presented in Fig. 3.
From Fig. 3(a), we can find that the focal distance will become longer as the chemical potential increases. The tunable length (defined as the difference of focal distance between μc = 0.1 eV and μc = 0.9 eV) is 405 μm (~4λ). Here, we compare the performance of our device with other tunable out-of-plane THz focusing models. Details are shown in Table 1. Our graphene-based transmission focusing model shows a better tuning sensitivity than the reflection focusing models in Refs. [24 and 26] As compared with the transmission focusing model in Ref.  based on thermistor material InSb, our model is operated in a more safe and convenient manner.
Meanwhile, the focal intensity will gradually decrease as the chemical potential grows. When μc is 0.9 eV, the focal intensity is only one half of that at μc = 0.1 eV. From Fig. 3(b), it is found that the focal distance does not change at all as the scattering rate varies, while the focal intensity is only reduced by 10% when Γ increases from 0.1 to 0.5 meV. These phenomena can be explained in the following paragraphs.
Radiating fields that occur at the grating surface can be regarded as a row of individual point sources. For a designed focal distance f at a certain wavelength λ, the propagation phase difference between the center (x = 0) and side air slits (x≠0) should be equal to (m is integer) to satisfy the constructive interference condition at the focal point. A general phase match condition at the focal point can be written as:40–42]. εr1 and εr2 are the relative permittivities of the dielectric materials above and below the graphene sheet, respectively. c is the speed of light in vacuum. In our case, εr1 = 1 (for air) and εr2 can be defined as the effective permittivity of the silicon grating for the TM polarization, which is given by the effective medium theory for subwavelength gratings [43, 44], i.e.,
Based on Eqs. (2), (6) and (7), we calculated the σintra and nSPP by taking them as a function of the chemical potential and scattering rate at 3 THz. As shown in Figs. 4(a) and 4(b), when the scattering rate is fixed, both real and image parts of σintra and nSPP are sensitively controlled by the chemical potential. According to Eqs. (5) and (6), for fixed values of x and m, as μc increases, Re(nSPP) will decrease, and the corresponding ΔΨx will decrease too. As a result, the focal distance will gradually become longer because ΔΨx is inversely proportional to f. As a comparison, as shown in Figs. 4(c) and 4(d), when the chemical potential is fixed, both Im(σintra) and Re(nSPP) are almost not affected by the scattering rate, so the focal distance is almost not influenced by the scattering rate either. It should be noted that when μc is 0.9 eV, the scattering rate would reach as large as 6 meV. Our calculation results show that focal distance is only slightly decreased but the focal intensity is reduced by 50% when Γ grows from 0.17 to 6 meV. For simplicity, in the following simulations and discussion, the change of scattering rate will be ignored because the effect it takes in phase modulation is negligible.
There are two possible reasons for the decrease in focal intensity in Fig. 3(a). One is the reflection increase at the silicon/graphene interface. Since Eq. (4) has shown that the out-of-plane component of permittivity of graphene is not affected by the chemical potential, it is easy to know this reason is impossible if we calculate the Fresnel reflection coefficient. Another is the enhanced coupling of SPP at the graphene/silicon interface. By incorporating the SPP dispersion relationship in a continuous monolayer graphene with the phase match equation in silicon grating, the SPP resonant frequency ν0 can be derived as :
Here, θ is the incident angle. In our case, θ = 0. As the chemical potential grows from 0.1 to 0.9 eV, the calculated SPP resonant frequency will gradually increase from 0.31 to 2.15 THz. As a result, for an incident wave at 3 THz, the gradually enhanced SPP coupling will lead to a decrease in focal intensity.
In addition, the broadband frequency response of the optimized focusing model in Fig. 2 has been investigated. As shown in Fig. 5(a), the effective working frequency of beam focusing is demonstrated to be from 2.9 to 3.5 THz, where both the focal distance and the corresponding tunable length Δf will gradually decrease as the working frequency grows. From Fig. 5(b), it can be found that 3 THz is actually near the optimal working frequency (~3.1 THz), where the maximum focal intensity and a relatively low variation ratio of focal intensity can be obtained.
In order to further verify the stability of our focusing model, we also investigated another two cases with optimal working frequency at about 2 THz and 4 THz, respectively. The grating parameters in these two cases are linearly scaled according to the incident wavelength, i.e., the grating constant and slit width are magnified by 1.5 times in the 2 THz case and shrunk by 0.75 times in the 4 THz case, respectively. Figure 6 shows the focusing performance of these two cases, where similar tuning effects of focal distance can be observed. In 2 THz and 4 THz cases, the tunable lengths are 242 μm (~1.6λ) and 334 μm (~4.5λ), respectively. In agreement with the 3 THz case, the focal intensity in both 2 THz and 4 THz cases is also reduced by about 50% when μc is increased from 0.1 to 0.9 eV.
3.2 2 to 1 and 3 to 1 manipulation of terahertz beams
In this section, we used the model in Fig. 1(b) to study the beam diffraction phenomenon of a triple slit copper grating. An optimized pure copper grating with structural parameters of hCu = 36 μm, wc = 8 μm, ws = 14 μm, l = 93 μm, wg = 11 μm and tg = 14 μm is first discussed here for beam manipulation study at optimal working frequency of about 3 THz. Based on this grating, we will show how 2 to 1 and 3 to 1 beam manipulation effects can be realized when the grating is covered by monolayer graphene applied with different chemical potentials.
Figure 7 shows |P| distributions of 3 THz beam in x-z plane when Λg = 65 μm, Γ = 0.17 meV, and μc is increased from 0.1 to 0.9 eV. A cross-section profile of |P| along the dashed line (z = 350 μm) is presented on the top of each sub figure to clearly illustrate the evolution of beam manipulation. As we can see from Fig. 7(a), when the copper grating is not covered by graphene, an incident beam will be first diffracted into three main beams near the exit. After propagating a certain distance (around 2.5λ), these beams then degenerate into two main beams through far-field coupling. As shown in Figs. 7(b)-7(f), when the copper grating is covered by monolayer graphene, the two main beams will gradually degenerate into one main beam in the far field as the chemical potential grows. Such 2 to 1 beam manipulation effect is obviously observed when μc is below 0.7 eV. As μc grows further, saturation effect appears.
As shown in Fig. 8, when Λg is changed from 65 μm to 85 μm, a similar 3 to 1 beam manipulation effect can be obviously observed when μc varies from 0.1 to 0.7 eV. In the practical application, 0.7 eV can be set as the upper limit of μc for beam manipulation. Here we define the beam conversion efficiency η as the intensity ratio of the central beam to side beams. As shown in Fig. 9, the beam conversion efficiency of either 2 to 1 or 3 to 1 manipulation is proportional to the chemical potential, which is in good agreement with the phenomenological behaviors in Figs. 7 and 8. More importantly, the 2 to 1 and 3 to 1 beam manipulation effects are demonstrated to be stable for a longer propagation distance. For example, at propagation distance of 1000 µm, the initial diffraction pattern at μc = 0.1 eV is found to be more complicated but the final diffraction pattern at μc = 0.9 eV remains the same, i.e., only one main beam in the free space.
The robustness of our manipulation model can also be verified by shifting the optimal working frequency targeting from 3 THz to 2 THz or 4 THz. As shown in Fig. 10, in both 2 THz and 4 THz cases, good repeatability of beam manipulation is demonstrated.
The 2 to 1 and 3 to 1 beam manipulation effects can be simply explained based on the above discussion in Section 3.1. The coupling from either two or three beams to one beam is due to the degeneration of high-order diffraction. When the chemical potential of graphene is increased, the in-plane phase compensation Ψc is gradually decreased. As a result, more and more obvious degeneration of high-order diffraction and beam manipulation effects can be observed at a certain propagation distance (e.g. z = 350 μm).
In conclusion, we have investigated the graphene-based transmission grating models for terahertz beam focusing and manipulation in the out-of-plane propagation space. It is found that the beam focusing performance (e.g. focal distance and intensity) of a simple silicon grating, the beam diffraction phenomenon of a triple slit copper grating can be dynamically and efficiently controlled by changing the chemical potential of monolayer graphene on the top of gratings. These graphene-based electro-optical effects were well explained in theory. Our proposed models show high tunability and stable performance. Based on our models, more tunable THz devices are expected to be developed for beam steering, beam shaping, and holography.
Natural Science Foundation of China (No. 61675096, No. 61205042, and No. 61605087); Natural Science Foundation of Jiangsu Province (No. BK2014021828 and No. BK20160881); Jiangsu Provincial Natural Science Research Project (No. 16KJB140010).
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