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Abstract
In this study, an ultra-thin reflective metasurface is proposed for polarization conversion in the terahertz band. Each unit cell of metasurface is composed of graphene ribbons lying diagonally on silicon substrate. A reflective metal is also placed at the bottom of the structure. Our polarization converter works as a linear polarization converter (LPC) and linear to circular polarization converter (LTC-PC) by variation of the chemical potential of graphene, which can actively be changed by chemical doping or electrical bias of the graphene. The working bandwidth of LPC changes by adjusting the chemical potential of the graphene. The LPC structure has more than 99% polarization conversion ratio in the frequency range of 0.83-0.92 THz, even by changing the angle of incident wave up to 45°, the results are still acceptable. The LTC-PC has less than 3dB axial ratio (AR) in the frequency range of 0.6-0.67 THz for left-handed circularly polarized (LHCP) waves and 0.72-0.97 THz for right-handed circularly polarized (RHCP) waves. To verify the simulation results, an equivalent circuit model based on the structure performance is proposed. Equivalent circuit model results agree very well with the simulation results. Due to the fabrication feasibility, ultra-thin thickness, incident angle insensitive, and high efficiency, our structure has great potential in state-of-the-art technologies such as imaging, sensing, communication, and other optical applications.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Terahertz (THz) region, the frequency range of 0.1 THz to 10 THz, has been unexplored for many decades. Recently, THz regime has attracted a great interest in detection, imaging, sensing, spectroscopy, and security devices [1–4]. Lack of natural materials which can interact with THz waves has made metasurfaces a good choice for practical terahertz devices [5]. Metasurface, a two-dimensional artificial material which can be an appropriate replacement for bulky metamaterials, consists of periodic subwavelength metal or dielectric patterns. By changing shape, size, and material of the unit cells, different optical properties, such as anomalous optical reflection and negative refractive indices, are achievable [6,7]. Graphene, a single layer of carbon atoms arranged in a honeycomb lattice, shows unique carrier mobility, mechanical, electrical, thermal, electromagnetic, and optical properties [8–10]. In spite of having a single atom thickness, graphene exhibits strong interaction with incident wave and supports surface plasmon polaritons (SPPs) with long propagation length from THz to mid-infrared (MIR), therefore, it is a good choice to be integrated into metamaterials and metasurfaces [9,11–14]. Controlling the chemical doping and electrical bias of graphene lead to a tunable surface conductivity of graphene which affects its electrical and optical response [7,12]. In addition, graphene is transparent in THz region [9,11]. Recently, graphene has been used in many optical and electrical devices such as electro-optic modulators, absorbers, couplers, polarization converters, solar cells, and transistors [12,14–17]. Periodic structure enhances the energy coupled between incident wave and SPPs of graphene [9]. The graphene metasurfaces introduce actively tunable applications which their properties can change not only by size and shape, but also by graphene conductivity. This means that we would have a reconfigurable structure which does not need to be refabricated.
Many practical devices are sensitive to polarization, therefore, manipulation of the polarization state of electromagnetic waves has got great importance [18]. Polarization converters (PCs) are capable of using in imaging, sensing, detection and communication applications including modulators and switches [8,19–22]. Conventional polarization converters use birefringent crystals or optical gratings to change the polarization state of the waves [18,22,23]. These PCs are not appropriate for THz regime because of their bulky structures, low efficiency and narrow bandwidth [6,8]. Recently, metamaterials and metasurfaces have suppressed this limitation and many structures have been proposed based on metamaterials and metasurfaces [14,18,24–28]. Metamaterials are still bulky and are not appropriate to be used in the state-of-the-art applications. Although metasurfaces eliminate this problem, many of them have narrow bandwidth, low efficiency, complicated structure which cannot be easily fabricated and are not tunable [14,24,27,28].
In this study, we propose a high-efficiency linear polarization converter (LPC) with graphene metasurface for THz regime. It is shown that by changing the graphene chemical potential, $\mu _c$, of the proposed LPC structure, linear to circular (LTC) polarization conversion is also achievable. To verify the proposed structure performance and to find a fast method to get the analysis results, an equivalent circuit model is proposed. Results of equivalent circuit model (ECM) and simulations are in very good agreement. Our proposed structure is ultra-thin and can be easily fabricated. By using graphene in the structure, the optical response and application of the metasurface can be varied by changing the graphene chemical potential, ${\mu _c}$, without re-fabricating the structure.
The remaining of this paper is organized as follows: In Section 2, the structure design and analysis method are described. The results are presented and discussed in Section 3. In Section 4 an equivalent circuit model is proposed for the structure. Finally, the paper is concluded in Section 5.
2. Structure design and analysis method
The schematic view of the proposed three-dimensional reflective polarization converter (PC) structure and its unit cell are shown in Fig. 1. Our proposed structure is composed of three layers, which from bottom to top are gold layer, silicon substrate and graphene layer, respectively. The bottom layer is a reflective gold which reflects all the incident wave and plays an important role in obtaining efficient polarization conversion. The silicon substrate with refractive index of 3.4 is deposited over the gold layer. Graphene layer is composed of a diagonal graphene ribbon in the middle of the unit cell and two small graphene triangles at corners of the unit cell as shown in Fig. 1(b). These two small triangles make the graphene ribbons continuous in the metasurface, so the PC can be easily fabricated. We believe that the fabrication of our proposed structure is feasible and straightforward. Many of metasurfaces have spirals, double rings, ginkgo-leafs, and other complicated geometrical patterns which are not easy for lithography. On the other hand, the graphene ribbons have simple and common structure in comparison with those complicated structures. To pattern a large-area graphene, the etching method has been widely employed by using a mask consisting of a resist pattern defined by electron beam lithography, pre-deposited nanowires, or self-assembled block copolymer (BCP) nanopatterns [29]. Graphene ribbons can also be written by lithography without a mask [29]. Also, there are plenty of methods for graphene ribbons fabrication [29].
Fig. 1. (a). The schematic view of the proposed three dimensional linear polarization converter (LPC) and linear to circular polarization converter (LTC- PC) and (b) the structure unit cell.
Graphene is modeled as a two-dimensional surface and its surface conductivity is calculated by Kubo formula, which is consisted of inter-band and intra-band conductivities [30]:
Intra-band conductivity can be evaluated as:
In special conditions, when $ {k_B T \ll |\mu _c|, \hbar \omega }$ inter-band can be approximated as [30]:
Some parameters of equations and the optimum dimensions for maximum efficiency of our proposed structure of Fig. 1 are given in Table 1. As can be seen, the optimum thickness of substrate is large, which would require a large bias voltage, $V_g$, to change the $\mu _c$, for excitation of graphene ribbons. As shown in Fig. 1, to address this issue and get a practical structure with applicable bias voltage, a thin layer of a transparent conductive oxide (TCO) can be applied between the ${t}_{s}$ and rest of the substrate, which due to the extreme thinness of the TCO, its effect in simulation is negligible [31,32]. Indium tin oxide (ITO), fluorine doped tin oxide (FTO), doped zinc oxide or networks of polymers such as polysilicon can be used as TCO.
The relation between ${\mu }_{c}$ and $V_g$ can be approximated to the below expression [31,33–35] :
The PC structure is analyzed by the finite difference time domain (FDTD) method. The boundary conditions are considered periodic in $ x$ and $ y$ directions and perfectly matched layer (PML) in $ z$ axis. The meshes in $ x$ and $ y$ directions on the graphene layer are considered 0.5 $\mu$m. The PC is illuminated by an $ x$-polarized normal plane wave from the top. When the PC is illuminated by an $ x$-polarized wave, a plasmonic resonance is excited along the graphene ribbon which leads to $ x$ and $ y$-polarized reflected wave.
To describe the reflected wave and its polarization state, we introduce the Stokes Parameters as follows [36–38]:
where $ {R_{xx}}$ and $ {R_{yx}}$ are respectively the amplitudes of the co-polarized and cross-polarized reflection coefficients when the structure is illuminated by a normal $ x$-polarized wave. ${\Delta \phi }$ is the difference of corresponding phases $ {\phi _{xx}}$ and $ {\phi _{yx}}$. The Stokes Parameters $ {I_n}$, $ {Q_n}$, $ {U_n}$, and $ {V_n}$ stand respectively for total reflection, the horizontal and vertical linear polarization state of the reflection, the linear +45 or -45 polarization state of the reflection, and circular polarization state. The subscript $ n$ denotes that the Stokes Parameters are for reflections, i.e. are normalized to incident wave. The axial ratio (AR) is defined as [36]: where $ \beta$ is the ellipticity angle defined as [36]:The axial ratio (AR) is used for evaluating the circular polarization performance. The analyzed wave is circularly polarized if the axial ratio is less than 3dB.
3. Results and discussion
Figure 2(a) illustrates the spectra of the reflection coefficients when the LPC structure is illuminated by a normal $ x$-polarized wave with central frequency of 0.83 THz (360 $\mu$m) amplitudes in $ z$- direction. As shown, most of the reflection is cross-polarized, while co-polarized reflected wave is negligible in the frequency range of 0.78-0.97 THz. $ {Q_n}$ parameter of Eq. (6) is depicted in Fig. 2(b). In the frequency range of 0.78 THz to 0.97 THz, $ {Q_n}$ is approximately equal to -1, which stands for linearly $ y$-polarized reflected wave, and when the reflected wave is mostly $ x$-polarized, $ {Q_n}$ is around +1. To approve LPC’s performance, polarization conversion ratio (PCR) is calculated as [7,39]:
Fig. 2. The spectra of (a) Co-polarized and cross-polarized reflection coefficient amplitudes, (b) Qn parameter, and (c) the PCR of the proposed structure of Fig. 1, as a linear polarization converter (LPC).
As shown in Fig. 2(c) PCR is more than 0.9 in the frequency range of 0.78-0.97 THz (308-384 $\mu$m), which implies that linear polarization is properly converted to its cross polarized wave, which the device performs as a linear polarization converter (LPC).
Figure 3(a) demonstrates the spectra of the reflection coefficients amplitudes for different height of the structure. The related PCR spectra are depicted in Fig. 3(b). As shown, PCR is not very ideal for $ {h}$=20 $\mu$m, but for $ {h}$=23 $\mu$m or 26 $\mu$m PCR is ideal with a little difference in bandwidth. Hence, $ {h}$=23 $\mu$m is chosen in this study.
Fig. 3. The spectra of (a) Reflection coefficient amplitude for three different heights of the substrate and (b) the related PCR of the structure of Fig. 1.
In practical applications, the transmitter and receiver block each other in normal incidence. Hence, oblique incidence of plane wave must be considered. For this purpose, we have simulated LPC structure for different angles of incidence of $ {\theta }$ = $15^{\circ }$, $30^{\circ }$, and $45^{\circ }$. The related PCRs are depicted in Fig. 4. As shown, even for the wave with $45^{\circ }$ incident angle PCR is more than 90% in a wide frequency range of 0.77 THz to 0.97 THz.
The spectra of PCR for different $ {\mu _c}$ of graphene is demonstrated in Fig. 5. As shown, altering $ {\mu _c}$ from 0.4 eV to 0.7 eV, changes the LPC’s working frequency range.
The cross-polarized reflection coefficient amplitude versus $ {\mu _c}$ at frequency of 0.83 THz (360 $\mu m$) is depicted in Fig. 6. As illustrated, for $ {\mu _c}$ more than 0.25, the structure behaves as linear polarization converter. Therefore, our proposed LPC is an active metasurface and one can find the best performance for the desired frequency range just by adjusting $ {\mu _c}$. Figure 6 indicates that when $ {\mu _c}$ changes in the range of 0.14-0.17 eV, the reflection coefficient amplitude of the cross-polarized wave is around 0.7 (3 dB), which means that the amplitude of the reflected $ {y}$-polarized wave is equal to the $ {x}$-polarized one. Thus, a circularly-polarized wave is achievable.
Figure 7 illustrates the spectra of the reflection coefficients amplitude and phases of the co-polarized and cross-polarized electrical fields reflected in $ {z}$ direction when the graphene chemical potential is set as 0.152 meV and the structure is illuminated by a normal $ {x}$-polarized wave with central frequency of 0.83 THz. As shown in Fig. 7(a), normalized reflection coefficient amplitudes of cross-polarized and co-polarized reflected waves at the frequency range of 0.61 THz to 0.67 THz (wavelength range of 457-492 $\mu m$) and 0.72-0.97 THz (wavelength range of 308-413 $\mu m$) are between 0.5 and 0.8 which can be considered approximately equal. As depicted in Fig. 7(b), in these ranges the cross-polarized and co-polarized reflected waves have $\pm 90^\circ$ phase difference. The equal amplitudes and $\pm 90^\circ$ phase differences approve that the reflected wave is a circularly polarized wave.
Fig. 6. Variation of the cross-polarized reflection coefficient amplitude of structure of Fig. 1 vs. chemical potential, $ {\mu _c}$, at the central frequency of 0.83 THz.
Fig. 7. The spectra of (a) amplitude and (b) phases of the reflection coefficients of the structure of Fig. 1.
Figure 8(a) illustrates the spectra of AR in desired frequency range. As shown, AR is less than 3dB in the frequency range of 0.6-0.67 THz (wavelength range of 457-492 $\mu m$) and 0.72-0.97 THz (wavelength range of 308-413 $\mu m$), which indicates that LTC polarization conversion has properly happened. The ellipticity of reflected wave is calculated as ${{V}_{n}}/{{I}_{n}}$ which is depicted in Fig. 8(b). As shown in Fig. 8(b), in the frequency range of 0.72-0.97 THz ellipticity is around +1, which indicates that the reflected wave is right-handed circularly polarized (RHCP) wave. Ellipticity is around -1 in the range of 0.6 to 0.67 which stands for a left-handed circularly polarized (LHCP) wave.
Fig. 8. The spectra of (a) axial ratio and (b) the ellipticity for proposed LTC-PC structure of Fig. 1.
Based on the results of the simulations, the proposed structure of Fig. 1 can be used as linear polarization converter (LPC) and also linear to circular polarization converter (LTC-PC) by changing the graphene chemical potential, $\mu _c$. Essential dimensions of the LPC and LTC-PC are given in Table 1. The dimensions have been optimized for obtaining maximum efficiency. By setting $\mu _c$ as 0.152 eV, our proposed structure works as LTC-PC which converts linearly polarized incident wave to a circularly polarized wave. For $\mu _c$ more than 0.25 eV the structure works as LPC and converts the linearly polarized incident wave ($ {x}$- or $ {y}$-polarized) to its cross polarized wave ($ {y}$- or $ {x}$-polarized). The best performance of LPC is achievable at $\mu _c$ = 0.5 eV. We have investigated that our proposed PC structure is more efficient and feasible to fabricate, compared to other similar structures reported in Table 2.
Table 2. Comparison of our proposed structure and some other similar structures
4. Equivalent circuit model (ECM)
To verify simulation results, due to the perforxmance of the PC structure of Fig. 1, we have proposed an equivalent circuit model (ECM). This equivalent circuit is shown in Fig. 9. Impedance of the silicon substrate is considered as ${{Z}_{Si}}={{\eta }_{0}}/{\sqrt {{\epsilon }_{{r}_{Si}}}}$, where ${{\eta }_{0}}\cong 120\pi$ which is the free space wave impedance. Gold layer behaves as a perfect electric conductor (PEC) because of its short skin depth in THz range, and therefore, its impedance value is about zero. As it can be seen in Fig. 1(a), graphene ribbons are lying on silicon substrate with 45-degree angle with $ {x}$ and $ {y}$ axis. The structure is illuminated by an $ {x}$-polarized normal plane wave which can be decomposed into $ {x^\prime }$ and $ {y^\prime }$ axis as:
Fig. 9. The equivalent circuit model for the structure of Fig. 1, in (a) $x^{\prime }$ and (b) $y^{\prime }$ directions.
We have considered two equivalent circuit models for $ {x^\prime }$ and $ {y^\prime }$ axis, individually. In $ {y^\prime }$ direction, periodic arrays of graphene ribbons (PAGR) can be modeled with parallel R-L-C circuits. The relative values of the resistance, inductance, and capacitance of the R-L-C are obtained as [41]:
where subscript $ GR$ stands for graphene, ${{{S}_{1}^2}}={8a}/{9}$, in which $ a$ is the width of the ribbons; and $q_1$ is the eigenvalue of the equation which is given in Table 1 of [41], $q_1$ is the first eigenvalue of the equation governing the current on the graphene sheet. $P$ and $\tau$ are respectively the period of the unit cell, and the carrier relaxation time. The effective permittivity of graphene’s surrounding area is calculated by ${{\epsilon }_{eff}}={({\epsilon }_{0}+{\epsilon }_{Si})}/{2}$, in which $\epsilon _0$ and $\epsilon _{Si}$ are the permittivity of the air and silicon, respectively. The relative permittivity of silicon is 3.4. The impedance of each graphene ribbon can be calculated as:In $ x^\prime$ direction, graphene is modeled as two parallel inductances, each representing the magnetic energy stored in the upper and lower media, as [41]:
The reflection coefficients are calculated by using equivalent circuit model and FDTD method which are illustrated in Fig. 10(a) and 10(b), respectively, for LPC and LTC-PC structure. The results of these two methods are in good agreement. As depicted in Fig. 10(a), equivalent circuit model of cross polarized reflection coefficient is approximately equal compared with the negligible co-polarized reflected wave, in the frequency range of 0.81-1 THz. Thus, we can either realize that linear polarization conversion is done properly using equivalent circuit model. Also, reflection coefficient amplitudes obtained by equivalent circuit model in Fig. 10(b), show that amplitude of co-polarized and cross-polarized reflection coefficients are approximately the same, in frequency range of 0.75-0.97 THz. This means that linear to circular polarization conversion could be done in this range. Equivalent circuit model and FDTD results have slight difference in both LPC and LTC-PC structures, which have occurred due to the approximations in ECM and different analysis methods used in ECM and FDTD methods.
Fig. 10. Reflection coefficient amplitudes calculated by equivalent circuit model theory (ECM) and the FDTD method for (a). LPC structure and (b). LTC-PC structure of Fig. 1.
5. Conclusion
In this study, we have proposed a multi-functional ultra-thin high efficiency reflective polarization converter (PC), using active graphene metasurface in terahertz (THz) band. Our proposed structure can work as linear polarization converter (LPC) and linear to circular polarization converter (LTC-PC) by changing the chemical potential of graphene. Both PCs are simulated by finite difference time domain method, and also equivalent circuit models are proposed for both structures. Results of these two methods were in a good agreement. A perfect linear polarization conversion was achieved with polarization conversion ratio (PCR) higher than 99% in the frequency range of 0.83–0.92 THz. LTC-PC is achieved in two different wavelength ranges. In the frequency range of 0.6 to 0.67 left-handed circularly polarized (LHCP) wave is achieved and in the frequency range of 0.72-0.97 THz the right-handed circularly polarized (RHCP) wave is achieved. Our proposed active PC structure is composed of graphene nanoribbons over silicon substrate which its fabrication is simple. This property in addition to its ultra-thin thickness and high efficiency are highly desirable properties in THz band to give this PC structure great application potentials in imaging, sensing, and communications.
Disclosures
The authors declare that there is no conflict of interest.
Data availability
No data were generated or analyzed in the presented research.
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