Bandwidth broadening of a graphene-based circular polarization converter by phase compensation

Xi Gao; Wanli Yang; Weiping Cao; Ming Chen; Yannan Jiang; Xinhua Yu; Haiou Li

doi:10.1364/OE.25.023945

1. Introduction

In recent years, graphene has attracted extensive attention from physicists, engineers and material scientists. Thanks to the special arrangement of atoms in graphene, that is, the hexagonal lattice, graphene exhibits many exotic physical phenomena, such as high charge carrier mobility at room temperature, anomalous quantum Hall effect and tunable interband and intraband conductivities [1–3]. Such fascinating properties make graphene a promising material in many potential applications ranging from quantum computers to new type of optoelectronic devices.

One of the valuable applications is the development of active devices in the terahertz regime, including the dynamically controlling of polarization converters. In recent years, many graphene-based polarization devices with diverse tunable functionalities, such as frequency tuning and polarizer switch, have been demonstrated. In these active devices, graphene is used to replace the conventional metal metasurfaces [4–9], or acts as an active medium under the metal metasurfaces [10–12]. However, these devices are usually backed with metallic ground plane and therefore operate in reflective mode, in which the metallic ground plane and metasurface form a Fabry-Perot-like cavity [13–15]. Due to the characteristic of Fabry-Perot resonance, the interference condition at the surface of metasurface may be destroyed when the operation frequency of polarization device shifts with a small frequency space, thus deteriorating the device performance. Therefore, these tunable polarization converters backed with metallic ground plane usually have a disadvantage of narrow tunable bandwidth.

In this paper, we present a reflective circular polarization converter consisting of a single graphene sheet patterned with butterfly-shaped holes, a dielectric spacer, and a 7-layer graphene ground plane. We discover that the 7-layer graphene not only can reflect the electromagnetic (EM) wave with high reflectivity but also can produce an additional phase for its reflected wave. By controlling the Femi energy of the 7-layer graphene, the additional phase can be modulated in a broad bandwidth to compensate the phase shift introduced by the frequency tuning of device, resulting in significant bandwidth broadening. We theoretically demonstrate that the proposed circular polarization converter can be tunable in the frequency range from 0.44 to 0.96 THz. Compared to other types of polarization device, the tunable bandwidth of the proposed polarizer is significantly extended.

2. Design and the electric characteristic of multi-layer graphene

A schematic configuration of the proposed circular polarization converter is described in Fig. 1(a). It is composed of a complementarily graphene-based metasurface, a dielectric spacer and a 7-layer graphene. The dielectric spacer is selected as silicon (Si) with thickness h = 30 μm, relative dielectric constant of 11.9, and loss tangent of 1e-6. The 7-layer graphene is used as a reflective plane like the metallic ground in the conventionally reflective polarization converter. On one hand, the 7-layer graphene has high conductivity so that it can reflect the electromagnetic (EM) wave propagating in dielectric substrate with high reflectivity; on the other hand, the 7-layer graphene can also modulate the phase of reflective EM wave by controlling its Femi energy. In the following content, we find that this characteristic of the 7-layer graphene is the main contributor that leads to enhancement of the tunable bandwidth of the proposed polarization converter. Figure 1(b) presents the front view of the unit cell of the complementarily graphene-based metasurface, which is formed by periodically perforating a single layer graphene sheet with butterfly-shaped holes. The parameters of the butterfly-shaped holes are shown in the caption of Fig. 1. The periodicity of the graphene-based metasurface is $p_{x} = p_{y} = 40 μm$ .

Fig. 1 (a) Schematic diagram of the proposed polarization converter consisting of a graphene metasurface patterned with butterfly-shaped holes, a dielectric spacer, and a 7-layer graphene ground. (b) Front View of the metasurface unit cell. The Femi energy of graphene is tuned by applying gate-voltages to the 7-layer graphene (V1) and to the graphene-based metasurface (V2).

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Before investigating the performance of the proposed polarization converter, we first analyze the conductivity of the single- and multi-layer graphene. As we know, the monolayer graphene can be electrically modeled as an infinitesimally thin conductivity layer characterized by a complex-valued surface conductivity $σ_{s} (ω, μ_{c}, Γ, T)$ which is derived from the Kubo formula consisting of both intraband and interband contributions [16]:

\begin{matrix} σ_{s} = σ_{i n t r a} (ω, μ_{c}, Γ, T) + σ_{i n t e r} (ω, μ_{c}, Γ, T) \\ σ_{i n t r a} (ω, μ_{c}, Γ, T) = - j \frac{e^{2} k_{B} T}{π ℏ^{2} (ω - j 2 Γ)} (\frac{μ_{c}}{k_{B} T} + 2 \ln (e^{- \frac{μ_{c}}{k_{B} T}} + 1)), \\ σ_{i n t e r} (ω, μ_{c}, Γ, T) ≅ - \frac{j e^{2}}{4 π ℏ} \ln (\frac{2 | μ_{c} | - (ω - j 2 Γ ℏ)}{2 | μ_{c} | + (ω - j 2 Γ ℏ)}) \end{matrix}

where e,

ℏ

,

k_{B}

, T,

μ_{c}

are the Planck’s constant, Boltzmann’s constant, temperature and chemical potential (i.e. Fermi energy

E_{F}

), respectively. The electron scattering time and the temperature are assumed to be

τ = 2 ps

and

T = 300 K

, respectively. For N-layer graphene, it is formed by randomly stacking mono-layer graphene. Furthermore, we assume all the graphene layers have the same relaxation time and the same Fermi energy for simplicity. According to ref [17], when the THz wave is incident to the N-layer graphene on substrate of silicon, the sheet conductivity of the N-layer (N>1) graphene can be denoted by the sheet conductivity of the monolayer graphene, which is described as follows [18]:

\begin{matrix} σ_{t o t a l}^{(N)} = [(M_{1} - M_{2}) \exp (i k_{s} t_{g r a}) + (M_{3} - M_{4}) \exp (- i k_{s} t_{g r a})] \frac{(n_{1} + z_{0} σ_{s}) (n_{3} + z_{0} σ_{s})}{2 z_{0} \sqrt{ε_{s}}} \\ + [(M_{1} + M_{2}) \exp (i k_{s} t_{g r a}) + (M_{3} + M_{4}) \exp (- i k_{s} t_{g r a})] \frac{n_{1} + z_{0} σ_{s}}{2 z_{0}} \\ + [(M_{1} - M_{2}) \exp (i k_{s} t_{g r a}) - (M_{3} - M_{4}) \exp (- i k_{s} t_{g r a})] \frac{n_{3} + z_{0} σ_{s}}{2 z_{0}}, \\ + [(M_{1} - M_{2}) \exp (i k_{s} t_{g r a}) - (M_{3} - M_{4}) \exp (- i k_{s} t_{g r a})] \frac{\sqrt{ε_{s}}}{2 z_{0}} - \frac{n_{1} + n_{3}}{z_{0}} \end{matrix}

where

t_{g r a}, σ_{s}, k_{s}, z_{0}, ε_{s}, n_{1}

and

n_{3}

are the thickness of single layer graphene, the conductivity of single layer graphene, the propagating vector in graphene, the wave impedance in the vacuum,the relative dielectric constant of single layer graphene, and the refractive indices of air and silicon, respectively. The coefficients

M_{1}, M_{2}, M_{3}, M_{4}

are expressed as follows:

(\begin{matrix} M_{1} & M_{2} \\ M_{3} & M_{4} \end{matrix}) = {(\begin{matrix} (1 + \frac{z_{0} σ_{s}}{2 \sqrt{ε_{s}}}) \exp (i k_{s} t_{g r a}) & \frac{z_{0} σ_{s}}{2 \sqrt{ε_{s}}} \exp (- i k_{s} t_{g r a}) \\ - \frac{z_{0} σ_{s}}{2 \sqrt{ε_{s}}} \exp (i k_{s} t_{g r a}) & (1 - \frac{z_{0} σ_{s}}{2 \sqrt{ε_{s}}}) \exp (- i k_{s} t_{g r a}) \end{matrix})}^{N - 2} .

Based on Eq. (2), we can calculate the conductivity of N-layer graphene for different N and different Fermi energy. We first investigate the variation of multi-layer graphene conductivity on the parameter N while fixing $τ = 2 ps$ and $E_{F} = 0 eV$ . Figure 2(a) and 2(b) shows the conductivity plotted as a function of frequency for different parameter N.

Fig. 2 (a) and (b) The calculated conductivities plotted as a function of frequency for different N when the relaxation time and the Femi energy are fixed to 2 ps and 0 eV, respectively. (c) and (d) The theoretically calculated conductivities of 7-layer graphene for different E_F2 when the relaxation time is fixed to 2 ps.

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It is observed that the conductivity is sensitive to N. Both the real and image parts of conductivity increase with increasing N. When N = 7, the real part of conductivity reaches to maximum value (about 3.8mS) at f = 0.4 THz as you can see in Fig. 2(a). The maximum conductivity can be further increased with the increase of N, but the loss may be increased accordingly. Considering this factor, we choose N = 7 and further study the variation of 7-layer graphene conductivity on frequency for different $E_{F}$ . Figure 2(c) and 2(d) plots the calculated conductivity when $E_{F}$ ranges from 0eV to 0.5 eV. We see that both the real and image parts of the 7-layer graphene conductivity increase with the increasing $E_{F}$ . For $E_{F} = 0.5 eV$ , the maximum real part of graphene conductivity is about 40 mS, as shown in Fig. 2(c). This high conductivity would make the 7-layer graphene reflect EM wave with high reflectivity. Therefore, it can be used as ground plane in our proposed reflective polarization converter.

3. Results and discussions

In order to investigate the performance of the proposed polarization converter, we conducted simulations using the CST Microwave Studio 2016. In the simulation, a single unit cell with periodic boundary condition along the x- and y-direction is employed, and a linearly polarized EM wave whose polarization direction is along the u direction (45° relative to the y axis) is used to excite the unit cell. The Femi energies of the single graphene patterned with butterfly-shaped holes and the 7-layer graphene are denoted as $E_{F 1}$ and $E_{F 2}$ , respectively, and $E_{F 1} = E_{F 2} = 0.4 eV$ . For convenience, we first define the $R_{u u}$ and $R_{v u}$ to denote the reflection coefficients of u- and v-polarized reflective waves when a u-polarized incident wave is used. The phase difference is defined as $Δ φ_{v u} = \arg (R_{u u}) - \arg (R_{v u})$ . If $| R_{u u} | = | R_{v u} |$ and $Δ φ_{v u} = 90 °$ , the reflective wave will be a circularly polarized wave. In other words, a circular polarization converter is realized. Based on this theory, we simulated the amplitudes of $R_{u u}$ and $R_{v u}$ , and their phase difference $Δ φ_{v u}$ for u-polarized incident wave. The results are presented in Fig. 3(a), which show that the amplitude ratio ( $| R_{u u} | / | R_{v u} |$ ) and the phase difference $Δ φ_{v u}$ are approximately equal to 1 and $90 °$ , respectively, in the frequency range from 0.76 THz to 0.9 THz, implying the ability of circular polarization conversion in this frequency range.

Fig. 3 (a) The simulated amplitude and phase difference of $R_{u u}$ and $R_{v u}$ for the proposed polarization converter when a u-polarized incident wave is used. (b) The simulated reflection amplitude and phase difference for $R_{x x}$ and $R_{y y}$ when the proposed polarization converter is excited with x- and y-polarized wave, respectively. (c) The simulated amplitude and phase difference of $R_{u u}$ and $R_{v u}$ for the polarization converter backed with metallic ground when a u-polarized incident wave is used. The Femi energy of graphene is $E_{F 1} = E_{F 2} = 0.4 eV$ .

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Actually, the physical origin of the proposed circular polarization conversion is related to the two eigenmodes excited at two orthogonal incident polarizations along the optical axis (x- and y-axis) of metasurface. To further get a physical insight, we carried out numerical simulations of the reflection amplitude and phase for the proposed device illuminated with polarization along the x- and y-axis. Figure 3(b) shows the reflection coefficients $R_{x x}$ and $R_{y y}$ , from which we see that the amplitude ratio ( $| R_{x x} | / | R_{y y} |$ ) and the phase difference $Δ φ_{y x}$ for $R_{x x}$ and $R_{y y}$ are approximately equal to 1 and $90 °$ near 0.8THz. Hence, we can clearly understand the physical mechanism of the proposed polarization conversion. Namely, when the incident u-polarization illuminates the device, it can be decomposed into two orthogonal components (x- and y-polarization). The two components can be simultaneously reflected to form $R_{x x}$ and $R_{y y}$ which have approximate amplitude and a near $90 °$ phase difference, resulting in a circularly polarized reflection wave.

For comparison, we also investigate the performance of the polarization converter with metallic ground plane, as shown the inset in Fig. 3(c). It is acted as a counterpart of the proposed polarization converter [see the inset in Fig. 3(a)]. The parameters of the counterpart are the same as that of the proposed polarization converter except that the ground plane is placed by gold. Figure 3(c) displays the simulated amplitudes of $R_{u u}$ and $R_{v u}$ , and their phase difference $Δ φ_{v u}$ . We deduce that the reflected wave is circularly polarized wave in the frequency range from 0.85 to 0.95 THz. Comparing Fig. 3(a) with 3(c), we find that the proposed polarization converter has the circular polarization conversion function and occupies high conversion efficiency, which is similar to the polarized device backed with metallic ground. Thus the 7-layer graphene in the proposed polarization converter behaves as the metallic ground plane in its counterpart. To better understand the physical property of the 7-layer graphene, we let a linearly polarized wave perpendicularly illuminate the 7-layer graphene on silicon substrate [see the inset in Fig. 4] and examine the reflective coefficients for different Femi energy ( $E_{F 2}$ ), as shown in Fig. 4. It is observed that a transmission window appears at 0.46 THz for $E_{F 2} = 0 eV$ and at 0.53 THz for $E_{F 2} = 0.05 eV$ , respectively. However, when the Femi energy is higher than 0.2 eV, the reflective coefficients are greater than 0.8, demonstrating the high reflectivity of the 7-layer graphene. The high reflection property leads to the high polarization conversion efficiency of the proposed polarization converter.

Fig. 4 Simulated reflection coefficients of a 7-layer graphene on silicon substrate.

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An interesting characteristic of the proposed polarization converter is that its operation frequency is tunable in a broad wideband. To investigate this characteristic in detail, we analyze the axial ration (AR) of the EM wave reflected by the proposed polarized device for different Femi energy and compare it with that of the polarization converter backed with metallic ground plane. The AR can be calculated by the amplitude and phase values of two orthogonal direction electric fields as follows:

\begin{matrix} A R = 10 \log_{10} \frac{{(E_{u} \cos α + E_{v} \cos φ \sin α)}^{2} + E_{v}^{2} \sin^{2} φ \sin^{2} α}{{(E_{u} \cos α + E_{v} \cos φ \sin α)}^{2} - E_{v}^{2} \sin^{2} φ \sin^{2} α}, \\ α = \frac{1}{2} \tan^{- 1} (\frac{2 E_{u} E_{v} \cos φ}{E_{u}^{2} - E_{v}^{2}}) \end{matrix}

where

E_{u}

and

E_{v}

are u- and v-polarized wave amplitudes, respectively, and

φ

denotes the phase difference between u- and v-polarized waves. The calculated results are shown in Fig. 5. For the polarization converter with metallic ground, when

E_{F 1}

changes from 0.2 eV to 0.6 eV, its operation frequency with AR being less than 3 dB is tunable in the frequency range from 0.78 THz to 1.01 THz, leading the relatively tunable bandwidth of circular polarization conversion to 26.8%. For the proposed polarization converter, there are two Femi energies (

E_{F 1}

and

E_{F 2}

) to be controlled to tune its operation frequency. From Fig. 5(b), it is observed that when

E_{F 1}

changes from 0.12 eV to 0.4 eV and

E_{F 2}

changes from 0 eV to 0.4 eV, the operation frequency of circular polarization conversion (AR less than 3 dB) is tuned in the frequency range from 0.44 THz to 0.96 THz, with the relatively tunable bandwidth of 74.3%.

Fig. 5 (a) The Calculated AR verus different Femi energy for the devices backed with metallic ground. (b) The Calculated AR verus different Femi energy for the proposed polarization converter. $E_{F 1}$ denotes the Femi energy of the single graphene patterned with butterfly-shaped holes and $E_{F 2}$ is the Femi energy of the 7-layer graphene ground.

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To exhibit the advantage of the proposed polarization converter, we further compare it with other graphene-based polarization converters that are reported in recent years. Table 1 illustrates the comparative results, in which the PCR is defined as $P C R = R_{v u}^{2} / (R_{u u}^{2} + R_{v u}^{2})$ . Here, we should note that the other graphene-based polarization converts represented in Table 1 are all backed with metallic ground and the spacer are dielectric plate with low dielectric constant. From Table 1, we see that the proposed polarization converter has the widest tunable bandwidth even if a dielectric spacer with high dielectric constant, implying a good performance.

Table 1. Comparison with other reflective graphene-based polarization converters

View Table

From the above comparison, we deduce that the 7-layer graphene ground play a crucial role in significantly expanding the tunable bandwidth of the proposed polarized device. To understand it, we study the physical process based on interference theory [19]. As we kown, when a u-polarized plane wave [see Fig. 1(b)] illuminates the device, both co-polarized and cross-polarized reflected and transmitted waves are generated due to the anisotropic characteristic of the metasurface. The transmitted waves undergo multiple reflections between the graphene-based metasurface and the 7-layer graphee ground plane, where they interfere with one another to create the final reflected wave, as shown in Fig. 6(a). The polarization state of the final reflected wave is determined by the amplitudes and phases of the co-polarized and cross-polarized waves undergoing multiple reflections. According to Fig. 6(a), the propagation phase of the EM wave running back and forth in the dielectric plate (Si) can be denoted as follows [20]:

Δ φ =2 π n_{s i} h / λ_{0} + θ,

where

λ_{0}

is the wavelength of the transmitted EM wave,

n_{s i}

and

h

are the relative index and the thickness of the dielectric plate, and

θ

is the additional phase produced by ground plane. The propagation phase

Δ φ

plays a key role on the performance of device. When

Δ φ

creates a constructive interference condition at the surface of graphene-based metasurface, the device will occupy a good performance, or else the device performance will be deteriorated.

Fig. 6 (a) The diagrammatic sketch of the EM wave path in the proposed polarized device. (b) The simulated additional phase $θ$ produced by a 7-layer graphene with different $E_{F}$ . (c) The phase $θ$ plotted as a function of Femi energy for a fixed frequency (f = 0.6 THz).

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For the graphene-based polarization device with metallic ground plane, the additional phase $θ$ is fixed to 180° and the propagation phase $Δ φ$ shown in Eq. (5) is dependent on the operation wavelength $λ_{0}$ . So, when the operation frequency of the device is tuned by the Femi energy of graphene, the propagation phase $Δ φ$ is accordingly varied. The varied $Δ φ$ may destroy the interference condition at the surface of metasurface, resulting in a narrow tunable bandwidth. However, this phenomenon is greatly relieved in the proposed polarization converter due to the 7-layer graphene providing a tunable additional phase $θ$ for its reflective wave.

To exhibit this characteristic for detail, we simulate the phase $θ$ of the reflective wave for different Femi energy when a plane EM wave perpendicularly illuminates the 7-layer graphene, as shown in Fig. 6(b). We see that the phase $θ$ is changed with the Femi energy in a broad band from 0.4 to 1.0 THz, demonstrating the tunable property of the $θ$ . To quantitatively analyze the ability of phase tuning, we plot the phase $θ$ as a function of E_F for a fixed frequency (f = 0.6 THz), as shown in Fig. 6(c). It is observed that the phase $θ$ is dynamically modulated from −79.38° to 119.26° when the Fermi energy $E_{F}$ changes from 0.0 eV to 0.6 eV. Since the additional phase $θ$ is tunable in a broad bandwidth, it can compensate the phase shift, that is the phase changing of $2 π n_{s i} h / λ_{0}$ in Eq. (6), produced by frequency tuning of device. Therefore, the constructive interference condition at the surface of graphene-based metasurface can be kept in a broad bandwidth, which dramatically broadens the tunable bandwidth of the proposed polarization converter.

4. Conclusion

In summary, we have theoretically investigated the electric properties of a 7-layer graphene formed by randomly stacking mono-layer graphene. The 7-layer graphene can reflect EM wave with high reflectivity and can provide an additional phase $θ$ for its reflected wave, over a broad bandwidth (from 0.4 to 1.0 THz). Based on these characteristics, we further proposed a reflective circular polarization converter, in which the 7-layer graphene was used as ground plane. By controlling the Femi energy of the 7-layer graphene, the additional phase $θ$ can be modulated to be an appropriate value to compensate the phase shift produced by the frequency tuning of device and then keeps the constructive interference condition at the surface of metasurface, resulting in good device performance in a broad bandwidth. Simulated results show that the tunable relative bandwidth of the proposed device is significantly increased to 74.3% (the absolute bandwidth is from 0.44 to 0.96 THz), which is much wider than the polarized device backed with metallic ground. The proposed graphene-based polarization converter is an essential step to develop active polarization device with high performance and broad tunable bandwidth.

Funding

National Natural Science Foundation of China (Grant Nos. 61761010, 61461016, 61640409, and 61661012); Natural Science Foundation of Guangxi Province (Grant Nos. 2014GXNSFAA118366 and 2015jjBB7002).

References and links

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References	Relative permittivity of dielectric spacer	Tunable bandwidth	Relative tunable bandwidth
[4]	3.9	0.91-1.1THz (PCR)	19%
[14]	2.1	27.5-33.5THz (PCR)		19.7%
[15]	2.1	30-50THz (PCR)		50%
Present study	11.9	0.44-0.96THz (AR)	74.3%

Bandwidth broadening of a graphene-based circular polarization converter by phase compensation

Abstract

1. Introduction

2. Design and the electric characteristic of multi-layer graphene

3. Results and discussions

4. Conclusion

Funding

References and links

Cited By

Figures (6)

Tables (1)

Equations (5)

Optics Express