## Abstract

Graphene can be utilized in designing tunable terahertz devices due to its tunability of sheet conductivity. In this paper, we combine the metamaterial having unit cell of cross-shaped metallic resonator with the double layer graphene wires to realize polarization independent absorber with spectral tuning at terahertz frequency. The absorption performance with a peak frequency tuning range of 15% and almost perfect peak absorption has been demonstrated by controlling the Fermi energy of the graphene that can be conveniently achieved by adjusting the bias voltage on the graphene double layers. The mechanism of the proposed absorber has been explored by a transmission line model and the tuning is explained by the changing of the effective inductance of the graphene wires under gate voltage biasing. Further more, we also propose a polarization modulation scheme of terahertz wave by applying similar polarization dependent absorbers. Through the proposed polarization modulator, it is able to electrically control the reflected wave with a linear polarization of continuously tunable azimuth angle of the major axis from 0° to 90° at the working frequency. These design approaches enable us to electrically control the absorption spectrum and the polarization state of terahertz waves more flexibly.

© 2014 Optical Society of America

## 1. Introduction

The terahertz (THz) range (0.1~10 THz) of the electromagnetic (EM) spectrum, which lies between the microwave and the far infrared frequencies and shares some properties from the two regimes, has attracted increasing attention due to the growing number of practical applications in astronomy, communication, imaging, spectroscopy, etc [1, 2]. For example, there are increasing research activities in the lower-terahertz band (< 2 THz) for future wireless communication with much higher data rate [3]. Many attempts have been made in developing THz devices [4–7], among which the THz absorber is believed to facilitate the development of more powerful THz detectors, stealth materials, thermal emitters, and sensors [8–12]. Recently, resonance based metamaterial absorber (MA) [13, 14], which can offer near unity absorption, has been successfully demonstrated to be a good candidate at terahertz frequencies where it is difficult to find strong absorbers in natural materials [6, 15, 16].

However, techniques to efficiently manipulate THz waves are still lagging behind, which are demanded in many applications. Metamaterial with artificial designed resonant unit cell structure enables that more functionalities can be incorporated, such as the MAs with absorbing frequency tuning, polarization modulating, and reflection/absorption switching [17–19]. These have been achieved by integrating diodes or varactor elements into the unit cells of MA at microwave frequencies [17–19]. Integrated Schottky diodes have also been successfully embedded into the THz metamaterial on an n-type GaAs substrate to dynamically modulate the THz transmission [20]. However, it is still required to find more paths to combine suitable material into metamaterial to achieve more engineered tunability, such as the frequency tuning or polarization modulation.

Graphene, a two-dimension material consisting of one monolayer of carbon atoms, has been recently applied in electronic and photonic devices due to its exotic properties, such as optical transparency, flexibility, high electron mobility [21, 22]. In addition, its sheet conductivity can be continuously tuned in a broad frequency range by shifting the electronic Fermi level via chemical or electronic doping, which enables fast electrical modulation and on-chip integration [23, 24]. This makes the continuous or structured graphene sheet a promising candidate for designing tunable THz metamaterial, and several works have been endeavored to design tunable THz MAs [25–30]. Most works result in MAs with tunable absorption by electrical control of the graphene [25–28]. Others utilize graphene nanostrips (or micro-ribbons) to construct MAs with frequencies tuning, however, these polarization dependent devices work at higher-THz frequencies (> 2 THz) and the frequency tuning is also accompanied with absorption reduction [29, 30].

In this paper we focus on the lower-THz band, and combine metallic resonator with structured double graphene sheets to design polarization independent tunable MA with nearly perfect absorption at tunable frequencies. Based on the similar mechanism, we then propose a polarization modulation scheme of THz waves by applying different voltage bias to control a polarization dependent reflector/absorber. The performances of the proposed devices have been verified with full wave EM simulation. Microstrip resonator model are employed to explore the physical mechanism of resonant frequencies tuning, which agreed with the simulation results.

## 2. Graphene based tunable metamaterial absorber

The sheet conductivity of graphene which can be derived using the well-known Kubo formula is described with interband and intraband contributions as [31]

*e*,

*ћ*and

*k*are universal constants representing the electron charge, Planck’s and Boltzmann’s constant, respectively.

_{B}*T*is the temperature and is fixed to 300K.

*μ*and 2

_{c}*Γ*(2

*Γ*=

*ћ/τ*,

*τ*is the electron-phonon relaxation time) are physical parameters of the graphene sheet accounting for the chemical potential (i.e. Fermi energy

*E*) and the intrinsic losses, respectively. We assumed

_{F}*Γ*= 0.1 meV, which is based on the theoretical estimation of maximum mobility in graphene [32]. In the THz region and below, where the photon energy

*ћω ≤ E*, the interband part (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 will be controlled by chemical potential via electrostatic gating, which provides an effective solution to tune the resonance of the MA.

_{F}The unit cell of the MA structure is composed with a stack of several layers including a metallic resonator and double layer of graphene sheets. To clearly show its structure, every layer is segregated from adjacent layer and schematically shown in the left part of Fig. 1. From top to bottom, they are a gold layer of cross-shaped resonator, double layers of cross shaped graphene sheets sandwiched with a thin silicon dioxide layer, the polymer dielectric substrate and the gold ground plane, respectively, tightly stacked to form the unit cell of the MA. The geometrical parameters of the unit cell are set after performance optimization through EM simulation. The width *w* and length *l* of the gold cross-wire are 15 μm and 92 μm, respectively. The width *w _{g}* of each graphene cross-wire is 5 μm, and length

*l*is equal to the period

_{g}*p*of the square unit cell which is 120 μm, in other words, graphene cross-wires in adjacent unit cells are connected together forming a grid structure. The thickness

*t*of silicon dioxide layer and

_{s}*t*of the polymer layer are 0.3 μm and 12.8 μm, respectively. The metallic layers including gold cross-wire and the ground plane have a thickness

_{p}*t*of 0.5 μm. As the thickness of the gold layer used here is much larger than the typical skin depth in the THz regime, the reflection is the only factor limiting the absorption. Here, a lossy polymer with permittivity

_{m}*ε*= 3.5 + 0.2

*i*(typical value for a polymer at THz [6, 15]) is used in our design. In addition, silicon dioxide is chosen as interlayer dielectric between two graphene cross-wires according to [33, 34]. If applying a DC bias voltage via the gated structure composed of two graphene layers, the chemical potential can be changed expediently, thus allow the control of the graphene conductivity. An approximate closed-form expression to relate

*μ*and

_{c}*V*is given by [35]

_{g}*ε*(taken as 3.9) and

_{r}*ε*

_{0}are the permittivity of silicon dioxide and vacuum respectively,

*V*is the bias voltage,

_{g}*e*and

*ν*are the electron charge and the Fermi velocity (1.1 × 10

_{f}^{6}m/s in graphene), respectively. Generally, the chemical potential can be tuned over a wide range (typically from −1 eV to 1 eV), but in the DC biased gated structure of our MA it is safely restricted within range from −0.5 eV to 0.5 eV according to [35].

We assume a plane wave normally impinges on the MA. By employing CST Microwave Studio for the full wave numerical simulations, the absorption spectrum for different Fermi energy (i.e. chemical potential) *E _{F}* has been calculated and displayed in Fig. 2. The frequency of the absorption resonance is blue-shifting with the increase of

*E*. The resonance frequency of the absorber can be tuned with a maximum relative tuning range of about 15% by varying

_{F}*E*. By applying a gate voltage (a static electric field) on the double graphene sheets, the chemical potential and thus the conductivity of graphene can be controlled on purpose. Furthermore, the absorption is polarization independent due to C4 symmetric of the unit cell structure. It should be mentioned that almost perfect absorption (≥ 95%) can be achieved in the tuning range, except a tendency of slight fall at higher frequencies (see Fig. 2) due to the increased loss in graphene which degrades the impedance match [25].

_{F}We have also investigated the robustness of the proposed MA under oblique incidence. Both the TM-polarized incidence (with magnetic field perpendicular to the incidence plane) and the TE-polarized incidence (with electric field perpendicular to the incidence plane) have been studied and the simulation results are displayed in Fig. 3. It is found that without biasing the graphene perfect absorption will not be affected until 70° for TM oblique incidence, while the absorption began to decline beyond 50° for TE oblique incidence, indicating a good robustness for oblique incidence. When tuning the MA with bias voltage on the graphene gate, the peak absorption frequency shift to higher frequency and the performance for oblique incidence becomes worse as the near perfect absorption is limited to incident angle within 25°.

## 3. Mechanism of the resonance frequency tuning

Now we focus on the underlying mechanism of the resonance frequency tuning for the proposed MA. To simplify the structure in order to obtain a clear model, we consider an MA structure for single polarization as shown in the inset of Fig. 4(a), i.e., a metallic cut-wire and double layers of graphene strips on top of the metal-backed dielectric substrate. Such structure works as a MA only for a particular polarized wave and can be modeled as a transmission line resonator (Fig. 4(b)). The simulated absorption for a *y*-polarized incidence is depicted in Fig. 4(a), which demonstrates tunable spectral response similar to that in Fig. 2.

Since the whole absorber is a periodic structure, we only need to analyze one unit cell using the periodic boundary condition according to Floquet theorem. Due to the symmetry with respect to both *x* and *y* axes, the PEC and PMC boundaries can be used as the periodic boundary condition for normal incidence. When there are no graphene strips in the unit cell, it represents a typical open-ended microstrip line resonator. The existence of metallic ground plane and PEC boundaries introduce the fringe capacitance at the terminals of the gold microstrip. Such capacitance loading can be equivalent to an additional transmission line with length Δ*l*, therefore, the cut-wire resonator with length *l* can be equivalent to an ideal open-ended transmission line whose length is *l _{m} = l +* 2Δ

*l*. Therefore, it can behave like a half wavelength microstrip resonator. For the

*y*-polarized incident wave, the fundamental half wavelength resonance mode will be excited with the current distribution as shown in Fig. 4(c). So the resonant condition of cut-wire is denoted as

*f*is the resonance frequency,

_{m}*λ*represents the guide wavelength,

_{g}*c*refers to the speed of light in vacuum, and

*ε*the effective permittivity of the microstrip resonator. Both Δ

_{eff}*l*and

*ε*can be easily obtained through standard microstrip-line theory. According to Eq. (5), we can estimate the resonance frequency to be 0.875 THz, which is consistent with the simulated value of 0.884 THz (see Fig. 4(a)).

_{eff}Next, we consider the cut-wire microstrip resonator with the double layer graphene strips underneath. It represents a microstrip resonator shorted at both ends due to the existence of the PEC boundaries, as illustrated in Fig. 4(b). The graphene strips underneath the gold microstrip have been connected to the neighboring gold resonators which prevents the occurrence of the fundamental half-wavelength resonance mode. The resonator now works in its second-order mode: the one-wavelength resonance mode. Figure 4(d) shows the current distribution of the second-order mode established in such resonator, which indicates the one-wavelength resonance having an asymmetric in phase and out of phase current distribution due to the different properties of gold and graphene strips. The current at the center is stronger and out of phase compared to those at the two terminals. This mode can be excited by the incident plane waves, which has a higher resonant frequency than that of the half-wavelength resonance mode. Therefore, its resonant condition can be expressed as

*E*. To interpret this, we can consider the segments of the graphene strip connected to the gold strip as the lumped elements because they are much shorter compared with the wavelength. Thus, the microstrip resonator can be modeled as an ideal transmission line of length

_{F}*l*terminated by the lumped elements at the two ends. The surface impedance of graphene is

*Z*1

_{S}=*/σ*, where

_{S}*σ*is the surface conductance and can be calculated by Eq. (1). The total impedance of the graphene strips at the two ends of gold resonator is obtained by multiplying the surface impedance with the length-width ratio of the graphene strip. It is then represented by a resistance

_{S}*R*(caused by the real part of

_{g}*σ*) and an inductance

_{S}*L*(caused by the imaginary part of

_{g}*σ*) in series (as shown in Fig. 4(b)). Therefore, decreasing the inductance will result in a reduction of Δ

_{S}*l*′ and an increase of the resonance frequency. According to the relationship between the Fermi energy and the effective inductance

*L*as shown in Fig. 5, increasing Fermi energy will decrease

_{g}*L*, and vice versa.

_{g}The resonance frequency can be estimated from the equivalent circuit model in Fig. 4(b) by solving the following resonant equation

*Z*

_{0}and

*β*indicate the characteristic impedance and the propagation constant of gold microstrip line, respectively. The circuit analysis and full wave simulation of the resonance frequency are compared in Fig. 5, which demonstrates a similar variation trend and the discrepancy between them is less than 5%. It is remarked that this circuit analysis is reasonable and also applies to the polarization independent design in Fig. 1. Therefore, by tuning the bias voltage applied on the graphene sheets, the Fermi energy of graphene is altered, thus the resonance absorbing peak can be shifted.

## 4. Polarization modulation

Utilizing gold cut-wire and graphene double layer strip as the unit cell of the absorber (as shown in Fig. 4(a)), it is able to absorb EM wave of one particular polarization, while has no effect on the cross-polarized wave. Based on this characteristic, we propose a polarization modulation scheme through EM wave reflection by integrating two polarization dependent MA structures together. The unit cell of such polarization modulator is schematically shown in the left part of Fig. 6 which is composed of multilayer of different materials. From top to bottom, they are gold cut-wire, graphene strip, thin silicon dioxide layer, graphene cross strip, second thin silicon dioxide layer, graphene strip, gold cut-wire, and finally polymer substrate backed with gold ground plane layer. These layers are tightly stacked together to form the polarization modulator (right part of Fig. 6). Its geometrical parameters are the same as that of the absorber in Fig. 1 except the thickness *t _{s}* of the silicon dioxide layer is reduced to 100 nm so as to reduce the performance difference for the two orthogonal polarized waves.

The two gold cut-wires in orthogonal orientations react to incident EM waves with the electric field polarized along *x*-axis or *y-*axis separately. To explore the response of the proposed modulator to different polarized waves, we calculate the reflection magnitude and phase and displayed in Fig. 7. The reflection magnitudes are almost the same, but the reflection phases are slightly different for the two orthogonally polarized waves at different Fermi energy. Hence the reflection coefficient of the *x*- or *y-*polarized wave can be controlled independently through tuning the bias voltage applied on the gated structure along *x*-axis or *y-*axis direction (*V _{x}* or

*V*). This feature allows us to modulate the polarization state of the reflected wave flexibly.

_{y}As a proof-of-concept example, we can assume a normal incident wave with the electric field polarized along 45° azimuth angle with respect to *x*-axis. It can be decomposed into two in phase *x*- and *y*-polarized components with equal amplitude. Hence the electric field of the reflected wave can be denoted as ${{\rm E}}_{r}(\overrightarrow{r},t)=\left|{E}_{x}\right|\mathrm{cos}(kz-\omega t)\widehat{x}+\left|{E}_{y}\right|\mathrm{cos}(kz-\omega t+\phi )\widehat{y}$, where *φ*, defined as *φ _{y} – φ_{x}* , is the phase difference between the

*y*and

*x*components. The values of

**|**

*E*

_{x}**|**and

**|**

*E*

_{y}**|**are proportional to the reflection magnitudes of the metamaterial for the orthogonal components as denoted in Figs. 7(a) and 7(c), and

*φ*or

_{x}*φ*takes the value of the reflection phase of the metamaterial for the corresponding component as denoted in Fig. 7(b) or 7(d), respectively. The reflected wave becomes linearly polarized if sin

_{y}*φ*= 0, or left-handed (right-handed) elliptically polarized if sin

*φ*> 0 (sin

*φ*< 0). The axial ratio (AR) of the elliptical polarization state can be calculated by 20 log

_{10}(

*|E*/

_{x}|_{max}*|E*) [19]. We also employ the polarization azimuth angle

_{y}|_{min}*θ*to describe the angle between the major polarization axis and the

*x*-axis, which can be defined as

*θ*= (1/2) arctan(2

*|**E*

_{x}

*| |**E*

_{y}**cos**

*|**φ*/(

*|**E*

_{x}

*|*^{2}-

*|**E*

_{y}

*|*^{2})) [36].

Based on the reflection magnitude and phase shown in Fig. 7, we calculate the azimuth angle *θ* and AR of the reflected waves at different bias voltage, under a 45° linearly polarized normal incident wave illuminating. The results are illustrated in Fig. 8. It is demonstrated in Fig. 8(a) that at 0.87 THz the polarization azimuth angle of the reflected wave can be controlled by bias voltage to change continuously from 0° to 90°. Above this frequency, the modulation range of the azimuth angle becomes to reduce from the full range of 0° to 90°. As the reflection phases of *x* and *y* components are different (see Figs. 7(b), and 7(d)), the reflected wave becomes elliptically polarized by the superposition of the two orthogonal components. It is found that the reflected wave is left-handed elliptical polarization (sin*φ* > 0) above the diagonal of Fig. 8(b), and right-handed elliptical polarization (sin*φ* < 0) below the diagonal of Fig. 8(b). However, the axial ratio is always better than 10 dB as shown in Fig. 8(b).

Therefore, by changing the bias voltage, i.e., tuning Fermi energy of graphene wires in *x*-axis and *y-*axis (see Eq. (4)), the amplitude and phase of the two orthogonally polarized components of the reflected wave can be adjusted independently so that the polarization state of the reflected wave can be well controlled. It should be noted that the change of amplitude and phase is mainly due to resonance frequency shifting. We also give a practical example to illustrate that the polarization of the reflected wave can be modulated by applying certain modulation bias voltages. We assume that two square wave (with certain rising and falling edges) modulation signals with *T*/2 time delay are applied on the gated structures along *x*- and *y-*axis as demonstrated in Fig. 9 (where *T* is the period of square wave modulation voltage). It can be seen that the EM wave reflected from the metamaterial at 0.87 THz can be modulated between *x* and *y* linear polarizations with the same frequency as the modulation bias signals. The presented polarization modulation scheme is based on the partial absorbing of certain polarized component by the MA, therefore the amplitude of reflected wave is a little attenuated. However, the reflected power is in principle no less than half of the incident EM wave power.

## 5. Conclusions

In summary, we proposed a polarization independent metamaterial absorber with tunable resonance frequency by employing graphene wires. Simulation results have shown that by tuning the bias voltage on the gated graphene wires, the absorption peak frequency can be adjusted actively in the lower terahertz frequencies. The tuning range of resonances of metamaterial absorbers can be up to 15%, with no degraded near perfect peak absorption. Based on the MA that can changes its reflectance by spectral shifting of its resonances, we also proposed a polarization modulation scheme through EM wave reflection by combining two polarization dependent MA structures together. The polarization modulator is demonstrated which is capable of generating linear polarized waves in the reflection with polarization azimuth angle continuously tunable from 0° to 90° at fixed working frequency by varying the Fermi energy of graphene. We believe that our study has demonstrated more ways to manipulate the terahertz wave through graphene and could be used in many future device applications, such as tunable absorber or switch, phase-shift surface, ellipsometer, senor and thermal emitter.

## Acknowledgments

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 Jiangsu Key Laboratory of Advanced Techniques for Manipulating Electromagnetic Waves.

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