Graphene is a good candidate material in designing tunable terahertz devices due to its tunability of sheet conductivity. In this paper, we propose a scheme to design switchable quarter-wave plate for terahertz wave that is composed of graphene based grating and metallic grating structures. The proposed active device can dynamically switch the transmission wave among left-handed, right-handed circular polarization and linear polarization states by electrically controlling the Fermi energy of the graphene grating. The device is analyzed with grating circular polarizer theory and its performance is investigated through full wave simulations on practically realizable geometry. The proposed quarter-wave plate having a subwavelength thickness demonstrates a wide angle of incidence tolerance, and a broad bandwidth operation. This device concept offers a further step in developing tunable polarizers and polarization switchers, which may be applied in practical terahertz image and communication systems.
© 2015 Optical Society of America
The terahertz (THz) part of the electromagnetic spectrum, defined as that from 0.1 to 10 THz, has found numerous potential applications in astronomy, communication, imaging, and spectroscopy, therefore, the THz science and technology has become increasingly important over the past decade [1–3 ]. However, many natural materials have weak responses to THz wave, so that the techniques to efficiently manipulate THz waves are still lagging behind, which are demanded in many practical applications. On the other hand, artificial metamaterials can enable more flexible manipulation of the EM waves and are easily scaled to work at THz frequency [4–8 ]. As structure defined artificial materials, more functionalities can be incorporated into the metamaterials, such as the tunable metamaterials with amplitude modulation, frequency tuning, polarization modulation, and reflection/absorption switching [9–14 ], etc. As such, considerable attention at THz frequencies has been focused on metamaterials, especially the tunable THz metamaterials [15–17 ].
While among all tunable techniques, voltage control is one of the simplest ways in practical operations. Among the different tuning schemes for THz devices, graphene, the newly discovered two-dimensional material, has attracted remarkable attention due to its potential use in high-performance tunable THz and infrared devices, since it produces a largely-tunable surface conductivity with respect to the external electrostatic biasing [18–20 ]. The graphene has also been successfully applied into the metamaterials to realize various tunable functional devices in THz band [21–26 ].
The polarization state is an important characteristic of EM wave and has been widely used in microwave communication systems and optical instruments, etc. Consequently, much effort has been devoted to apply the metamaterial concept in the manipulation of the polarization states in a variety of device applications [27–34 ]. However, in most of these polarization manipulating devices the functionalities are neither dynamically tunable nor electrically switchable. Although an active metamaterial is reported which can dynamically control the polarization states of transmitted EM wave, but such concept is limited in microwave band due to the use of microwave PIN diode . In this paper, we incorporate the graphene into the metallic wire grating structure commonly used in EM wave plates [27–30 ] to realize broadband and dynamical control of THz wave polarization. In section 2, we will first propose a switchable metamaterial in the THz regime composed of double layers of graphene strips and a single patterned bottom layer with gold grating structure. We will demonstrate its functionality as a switchable quarter-wave plate (QWP) that can switch the transmitted wave between linear polarization (LP) and circular polarizations (CPs) (both the right-handed circularly polarization (LCP) and the left-handed circularly polarization (RCP)) in section 3. We will also investigate the wide angle of incidence tolerance with full-wave simulations and present an extensive analysis of the physical mechanism of circular polarization conversion and polarization state manipulation. Then in section 4, we will extend the previous concept to construct a dual-function switchable QWP. Finally, in Section 5 we will draw the conclusions of our work.
2. Graphene based switchable QWP
The graphene monolayer can be electrically modeled as an infinitesimally thin conductive layer characterized by a complex-valued surface conductivity σs (ω, μc, Γ, T), where ω is the working radian frequency, μc is the chemical potential (i.e. Fermi energy EF) related to the electrostatic biasing, and Γ (Γ = ћ/2τ, τ is the electron-phonon relaxation time) is the physical parameter of the graphene accounting for the intrinsic loss. Throughout this work, we assume τ = 0.2 ps which is in agreement with measured data from the chemical vapor deposited (CVD) graphene . T is the room temperature and is fixed to 300 K in this paper. The sheet conductivity of graphene which can be derived using the well-known Kubo formula is described with interband and intraband contributions as Eq. (3)] is negligible comparing to the intraband part. Therefore, in the THz region graphene is well described by the Drude-like surface conductivity with Eq. (2). Its sheet conductivity can be controlled by chemical potential via electrostatic gating, which provides an effective solution to tune its electromagnetic property upon bias voltage control.
The unit cell (or a small square part cut from the whole structure) of the proposed switchable QWP is schematically depicted in Fig. 1 . It consists of a top grating structure composed of graphene double strip and a bottom metallic wire grids which are spatially separated by a flexible dielectric spacer. The graphene grating is oriented at −45° with respect to the y-axis and the metallic grating is parallel to y-axis. The graphene strips is placed on both sides of a silicon dioxide insulating film to form a gated structure. This is due to that the tuning of the graphene properties by electrostatic gating usually requires at least two layers of graphene separated by a thin dielectric film , and the right candidate of the separating film is silicon dioxide according to . Each of the graphene sheets plays the role of a gate electrode, and applying bias voltage will control the Fermi level in both graphene layers . The metallic wire grids on the bottom are made of gold film (with a thickness of t m) having a conductivity σ = 4 × 107 S/m which behaves as perfectly reflecting layer in the THz regime. The flexible dielectric spacer can be chosen as the TOPAS polymer which is an ideal substrate material for broadband THz components due to its very low absorption and stable refraction index (about 1.53) across the THz band . The optimal geometrical parameters are shown in Table 1 .
If applying a DC bias voltage via the gated structure of the double layers of graphene grating, the chemical potential can be changed expediently, thus allows the control of the graphene conductivity. An approximate closed-form expression to relate μc and Vg is given by 39]. Thus the active metamaterial can be turned ON or OFF upon it is biased or unbiased, respectively. Therefore, we can use it to realize a switchable QWP. Next, we will give detailed numerical analysis to explain its working mechanism.
3. Analysis and discussions
As a proof-of-concept example, we consider that a y-polarized THz wave propagating along + z direction normally impinge into the switchable QWP as schematically shown in Fig. 1. Commercial software CST microwave studio is employed to simulate the transmission and polarization manipulation of this metamaterial structure. The co-polarized or cross-polarized transmission coefficient or as well as the phase difference between co-polarization and cross-polarization components are calculated and displayed in Fig. 2 . To determine the polarization state of the THz wave, the four Stokes parameters are introduced as 
The figure of merit for the switchable QWP is indicated through the relative electric field intensity S 0, polarization azimuth angle α and the normalized ellipticity χ. The polarization azimuth angle defined as α = 0.5arctan(S 2/S 1) presents the polarization rotation of the transmission wave respecting to the incidence wave, while the ellipticity defined as χ = S 3/S 0 equates to 1 or −1 indicating a perfect left-handed or right-handed circularly polarized wave, respectively.
The results in Fig. 2(a), which are calculated when applying a gate voltage on the graphene double layer strips, reveal similar amplitude transmission behavior from 1.1 to 1.65 THz, with the normalized transmission of around 0.45 for both orthogonal components. Furthermore, the phase difference exhibits a value around 90°, and the corresponding ellipticity χ is over 0.94 (or the axial ratio is below 3dB). The output relative electric field intensity S 0 is around 0.4, which is mainly due to the electromagnetic waves reflection caused by the large equivalent surface impedance of graphene leading to impedance mismatch to the free space. All these demonstrate the broadband functionality of a left-handed circular polarizer with a relative frequency bandwidth reaching 40%. Therefore, the whole structure becomes a LCP QWP if its graphene grating is switched ON. When the graphene strips are unbiased, the whole structure is switched OFF. As shown in Fig. 2(b), is around 0.8 but is close to zero in the band from 1.1 to 1.65 THz, leading to a co-polarization output. The ellipticity χ and polarization rotation angle in Fig. 2(b) also illustrate that the polarization property of the incident wave will be kept when passing through the metamaterial in the same working band. So in this way the whole structure behaves like a switchable QWP that outputs LCP or LP wave when biasing the graphene layers to switch it ON or OFF, respectively. The value of τ may change due to different fabrication processes of CVD graphene. However, it will not obviously change the performance of the device except tiny shrink of the working band in high frequency end.
It is also possible to realize a switchable QWP with a RCP output if we rotate the double layer graphene grating along the center of the structure by 90°, which results in a mirror-symmetry counterpart (see inset of Fig. 3(a) ) of the original one in Fig. 1. Figure 3(a) shows the simulated transmission results of such QWP for RCP. When it works in biased state, similar amplitude transmission behaviors as those of the previous QWP for LCP can be observed as indicated in Fig. 3(a), but with a phase difference around −90°. The ellipticity χ is less than −0.94 from 1.1 to 1.65 THz, indicating a broadband RCP output. Furthermore, without biasing the QWP can be switched OFF, and keeps the polarization state of the incident wave unchanged in the same frequency range as shown in Fig. 3(b).
The switching behavior of the above QWP comes from the biasing of the graphene strips. At zero bias, the Fermi level is at the Dirac point of the graphene layer resulting in near-unity terahertz transmission. With an applied bias, the Fermi level moves into the valence and conduction band, the graphene behaves like a conducting layer, leading to near-zero terahertz transmission . In the THz region graphene is well described by the Drude-like surface conductivity as described previously, so the graphene grating approximately works as a metal grating with an applied bias. Thus the graphene grating can be switched ON or OFF upon biasing or not biasing, respectively.
To elucidate the broadband circular polarizer effect of switchable QWP, we consider the whole structure as a three layer composite: the graphene sandwich as the first grating layer, the polymer separation layer as the second layer and the gold grating as the third layer, as shown in Fig. 1. The transmission of each layer is investigated separately and displayed in Fig. 4 . According to the grating circular polarizer theory described in , for the first layer, as the grating period D 1 = w 1 + g 1 is less than the wavelength (λ) and g 1/D 1 is close to 0.5, only the zero-order mode contributes, and leads to an equal transmission amplitude of the two components parallel to and at right angle to the strips, but the phase retardances are sensitive to D 1/λ resulting in different x and y components of field transmission (see Fig. 4(b)). As shown in Fig. 4(a), the linear phase difference exhibits an obvious frequency-dependent characteristic and increases with the increasing of the frequency due to the birefringence of the first layer. Therefore, in order to achieve an approximate phase retardance achromatism, it requires the third layer not only affecting the transmission amplitude of the x and y components of field transmission but also compensating the phase difference in broadband. As shown in Fig. 4(a), the third layer of gold grating can provide a similar birefringence property which introduces negative phase difference dispersion and compensates with that of the first layer. The waves travel back and forth between the first and third layers forming a Fabry-Perot like cavity, and result in polarization coupling and constructive interference facilitating equal transmission amplitude of Ex and Ey as indicated in Figs. 4(b) and 4(c). Therefore, through such a composite of three layers, an approximate phase retardance achromatism is realized and the amplitude transmission matches the spectral feature of a QWP. So an incident wave linearly polarized at 45° (−45°) with respect to the graphene strips will generate a LCP (RCP) output wave in broadband.
In addition, we have investigated the robustness of the proposed switchable QWP under oblique incidence. Incidence in xoz-plane with different angle φ or in yoz-plane with different angle θ has been studied and the simulation results are displayed in Fig. 5 . Without biasing the device shows near zero ellipticity χ up to 70° of either φ or θ in an extremely broad band. While applying bias voltage on the graphene gate, for incidence angle φ large ellipticity χ (> 0.94) is achieved up to 70°, but the performance of oblique incidence in yoz-plane becomes slightly degraded as the large ellipticity χ is limited to incident angle within 40° in the working band. Over all the results reveal insensitivity to the wide angle of oblique incidence, suggesting good tolerance to the misalignment issue in practical application when the switchable QWP works as either a circular polarization or a co-polarization transmission metamaterial.
4. Dual-function switchable QWP
The previously proposed switchable QWP concept can be extended with more degrees of freedom for polarization manipulation. For example, we can construct switchable device with three different output polarization states. Fig. 6 schematically shows the device which is composed by adding an orthogonal oriented graphene double strip grating on top of the original switchable QWP of Fig. 1. The two graphene sandwiches (denoted as “1” and “2”), which are separated by a silicon dioxide layer of thickness t s, are mirror-symmetric with each other. The two graphene double strips have same geometrical parameters and their values are shown in Table 1. For a y-polarized incidence wave, such device can realize three output states, which can be categorized as the follows. In the first state, the sandwich “1” is biased (ON) while the sandwich “2” is unbiased (OFF). So the device switches to a QWP with an output of LCP. In the second state, the sandwich “2” is biased (ON) while the sandwich “1” is unbiased (OFF). The device switches to a QWP with an output of RCP. When the two graphene sandwiches are all unbiased (OFF), the device is working in the third state, and the QWP function by either “1” or “2” is switched OFF, resulting in an output of co-polarized LP.
The performance of the proposed dual-function switchable QWP is verified through full wave simulation and the results is displayed in Fig. 7 . It is found that the two graphene gratings which are placed together do not interfere with each other very much. When the device is biased in state 1, the first graphene grating works, which convert the incident linearly polarized wave into the LCP wave with an ellipticity of around 1, as shown in Fig. 7(a). On the contrary, when it is biased in state 2, the second graphene grating works and RCP feature dominates the structure, thus the output EM wave is reversed to RCP with an ellipticity of around −1, as shown in Fig. 7(b). When it works in state 3, both graphene gratings are shut off, the incident wave preserves its polarization state, leading to co-polarization transmission, as shown in Fig. 7(c). Hence, this switchable QWP can conveniently realize multiple polarization property by switching its feature among LCP, RCP or co-polarized LP transmission in a broadband frequency from 1.0 to 1.5 THz.
We remark that there may be slight performance deviation between the simulation model and the practical device. The limitation of the numerical modeling lies in that the parameter variation generated in the device processing cannot be fully considered. Deviations may also come from the change of τ due to the impurity infiltration in the fabrication processes of CVD graphene. However, such deviations mainly cause some tiny shift or shrink of the working band, which does not affect the basic function of the proposed devices.
We have proposed a general scheme to design graphene based switchable quarter-wave plate for THz wave, which combines graphene grating and metallic grating structures together. It can switch its output EM wave property not only between linear polarization and circular polarization transmission, but also between left-handed and right-handed chirality by controlling the bias voltage on each graphene grating. Our designs may provide a large degree of dynamical manipulation of THz wave with polarization property between LCP, RCP or LP over a broad frequency bandwidth. It can be utilized as a tunable polarizer or wave plate and find applications in THz imaging or communication systems. The geometry can be implemented within currently available CVD techniques for graphene sheet and easily integrated with other THz devices for polarization manipulation, detection, and sensing at the nanoscale.
This work is partially supported by the National Nature Science Foundation of China (61371034, 61301017, 61101011), the Key Grant Project of Ministry of Education of China (313029), the Ph.D. Programs Foundation of Ministry of Education of China (20120091110032), and partially supported by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), Jiangsu Key Laboratory of Advanced Techniques for Manipulating Electromagnetic Waves.
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