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Abstract
Ultrathin tunable broadband terahertz transmission mode linear-to-circular polarizers are proposed and numerically validated. Each structure consists of a dielectric substrate with slotted graphene patches on one or both sides of the substrate. Both polarizers yield tunable broadband conversion with large transmission coefficients in terahertz frequency range. Compared to the previous polarizers, the proposed structure with a single-layer of graphene patches exhibits a 120% larger fractional bandwidth and 60% larger transmittance. The polarizer with two layers of graphene patches features a 138% larger fractional bandwidth and 25% larger transmittance, compared to the previous polarizers, with excellent axial ratio. The tuning frequency range for the proposed single- and double-layer structures is (2.4 to 4.9) and (1.8 to 5.8) THz, respectively. The tuning is achieved by varying the graphene Fermi energy (chemical potential) from 0.2 to 1 eV, by the application of a DC bias voltage. Moreover, the polarization conversion performance is well maintained under oblique incidence condition, over a wide range of incident angles up to 70°. The size of the square-shaped unit cell is λ0/10, much smaller than the free-space wavelength of the incident waves, λ0, while the thickness is λ0/22. With the above features, the present work offers a further step in developing controllable polarization converters with potential applications in imaging, sensing, and communications.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
To date, several types of electromagnetic polarization converters based on metamaterials and metasurfaces have been reported [1–11]. Practical implementation of a polarizer can be a challenging task as the frequency band of interest may alter. However, graphene has been used in many tunable devices because of its attractive properties such as flexibility, strong interact with light at THz and infrared frequencies, and its tunable surface conductivity [12–17].
According to the above features, recently, various tunable polarizers have been introduced [18–44]. Most of these polarizers operate within narrow bandwidths with low transmission efficiencies. For instance, H. Cheng et al. designed tunable infrared polarization converters, based on graphene metasurfaces with cross-shaped unit cells, but the bandwidth was limited [40]. T. Guo et al. designed a broadband polarizer graphene metasurface with a unit cell comprising a rectangular graphene patch [30]; however, the transmission was reduced. Therefore, a tunable broadband polarization converter of high transmission efficiency is an on-going challenging task.
In the present work, two graphene-based tunable broadband polarizers are proposed. Each structure consists of a dielectric substrate with slotted graphene patches, on one or both sides of the substrate. The structure with graphene patches on one side of the substrate is called the single-layer, and the one using two layers of graphene patches on both sides of the substrate is called the double-layer. It is numerically shown the proposed metasurface polarizers yield tunable broadband conversion with large transmission coefficients at terahertz frequencies.
Compared to the previous polarizers, the single-layer structure exhibits a 120% larger fractional bandwidth and 60% larger transmittance, and the double-layer structure features a 138% larger fractional bandwidth and 25% larger transmittance, with an excellent axial ratio (AR). AR is defined as the ratio of the magnitude of the major and minor axes of the polarization ellipse of the transmitted or reflected waves. AR is always a positive real number, often expressed in the logarithmic scale in decibels (dBs). For a linear to circular polarizer, it is desirable to achieve small AR over a broad frequency range of interest. The frequency range over which AR is less than 3 dBs is considered as the bandwidth of the polarizer.
The frequency and polarization states can be dynamically tuned by varying the graphene chemical potential (Fermi energy), using a variable DC bias voltage. The overall operation frequency over which the axial ratio is less than 3 dB, for the proposed single- and double-layer structures are (2.4 to 4.9) THz and (1.8 to 5.8) THz, respectively. In both cases, the wideband operation is well maintained up to an incident angle of 70°.
In Section 2, the proposed converters and their basic features are introduced. In Section 3, a rigorous discussion of the simulation results and comparisons with the previous polarizers are presented. Finally, the conclusions are presented in Section 4.
2. Structure and numerical simulations
Figure 1 shows the schematic of the proposed metasurface made of an array of slotted rectangular graphene patches on a dielectric substrate.
The patches may be placed on one or both sides of the substrate. The surface conductivity of graphene is the sum of the intraband and interband terms. In the low terahertz frequency region, the interband term is ignored and the intraband conductivity is modeled as the Drude expression [45],
3. Results and discussions
3.1 Single-layer structure
For the single-layer structure, the amplitude and phase of the transmission coefficients for the x- and y-polarizations are shown in Fig. 2, for different values of W. The phase difference $\mathrm{\Phi }\Delta = {\mathrm{\Phi }_{\textrm{xx}}} - {\mathrm{\Phi }_{\textrm{yy}}}$ is also shown in Fig. 2(b). The cross-polarization conversion has been negligible. As observed from the Fig. 2, the transmission resonance is shifted when the converter is illuminated by x- or y-polarized waves. The two distinct resonances arise from the asymmetry between the length and the width of the grapheme patches. For W = 0 ($y$=0), the structure simplifies to the one in [30]. In this case, the transmittance, for both polarizations are equal, ${T_{xx}} = {T_{yy}} = 0.55$, at 4.75 THz, while the phase difference is exactly 90°. Interestingly, the 90° phase difference is obtained in a broad frequency range, from 4.3 to 5.1 THz, as shown in Fig. 2(b) [30]. By increasing W, the transmittance can be increases by 60%, from ${T_{xx}} = {T_{yy}} = 0.55,$ for W = 0, to ${T_{xx}} = {T_{yy}} = 0.88,$ for W = 3000 nm. Also, the frequency range in which the phase difference is constant and equal to 90° increases. This behavior can be interpreted through the resonance condition. In general, the resonant frequencies of the periodic graphene patch structure have been demonstrated to be affected by the length of the slot L and the Fermi energy ${E_F}$ following the relationship [17,30],
Fig. 2. (a) Transmission coefficients and (b) phase difference for the x- and y-polarized normally incident waves for the proposed single-layer structure with different values of W, when p = 7.6 µm, y = 250 nm, L1 = 6.5 µm, L2 = 5.2 µm, t = 1.5 µm, ${\varepsilon _r} = 2.25$, and ${E_F} = 0.95\textrm{}eV$ and comparing them with [30]. The unit of W is nm.
According to this relation, by increasing the length of the slot W along the x direction, the resonant frequency associated with ${T_{xx}}$ graph decreases, and as a result, the magnitude and phase of ${T_{xx}}$ shift to the left. Therefore, the intersection of ${T_{xx}}$ with ${T_{yy}}$ curves occurs at a large value, also, the frequency range in which the phase difference is constant increases.
Next, the axial ratio of the transmitted wave is calculated for the proposed single-layer structure. The general formula for calculating AR is [46]:
By placing Eq. (6) in Eq. (7), and Eq. (7) in Eq. (5), AR is obtained as:
By inserting the amplitude and phase of the transmission coefficients of Fig. 2 in Eq. (8), the axial ratio of the transmitted wave is obtained, as shown in Fig. 3. It is observed that, we can increase the bandwidth, frequency range over which AR < 3 dB, from 0.8 to 1.5 THz by increasing W, from 0 to 3000 nm. Table 1 shows the center frequency, ${f_0}\; $, transmittance, ${T_{ij}},\; ({i.j = x.y} )$ at ${f_0}\; ,\; $ phase difference, ΔΦ, bandwidth, $\varDelta f$, the fractional bandwidth (FBW), axial ratio, AR, and the averaged axial ratio in the 3-dB bandwidth (AAR), in terms of W. According to the table, increasing W, the center frequency decreases while the transmittance and bandwidth increase. However, increasing W, the axial ratio within the 3-dB bandwidth, AAR, increases slightly, as shown in Fig. 3. This is because the phase difference in this range decreased significantly as W increases, as observed from Fig. 2(b). Increasing W, the frequency range in which the phase difference is near 90° increases, but the phase difference in this range decreases and this leads to an increase of the axial ratio in the 3-dB bandwidth. This indicates that there is a trade-off between the 3-dB bandwidths and the averaged axial ratio in the 3-dB bandwidth, AAR.
Fig. 3. Axial ratio for the proposed single-layer structure with different values of W when p = 7.6 µm, y = 250 nm, L1 = 6.5 µm, L2 = 5.2 µm, t = 1.5 µm, ${\varepsilon _r} = 2.25$, and ${E_F} = 0.95\textrm{}eV$ and comparing them with [30]. The unit of W is nm.
The operation of the polarizer is highly adjustable via the graphene Fermi energy ${E_F}$. The Fermi energy can be changed either by chemical doping or by the application of a bias voltage, ${V_B}$ [30,47]. The relation between ${E_F}\textrm{}$ and ${V_B}\textrm{}$ is given in [30,48–52].
The axial ratio for different values of ${E_F}$, 0.4, 0.6, 0.8 and 1 eV, is calculated and the results are shown in Fig. 4. It is observed that, by varying ${E_F}$, one may effectively vary ${f_0}\; $ and the frequency range over which the axial ratio is less than 3 dB. Table 2 shows the variation of the operational frequency range and the corresponding bandwidth, in terms of the graphene Fermi energy. According to the table, by decreasing the Fermi energy, ${E_F}$, the center frequency is decreased, but the axial ratio increased. We consider the frequency range in which the axial ratio is less than 3 dB from 2.4 THz, for ${E_F} = 0.4\textrm{eV}$, to 4.9 THz, for ${E_F} = 1\textrm{eV}$, as the overall operational bandwidth with tuning. The fractional bandwidth, corresponding to the above frequency range, 2.4 to 4.9 THz, is 68%.
Table 1. Central frequency, ${{\boldsymbol f}_0}$, transmittance at ${{\boldsymbol f}_0}$, phase difference at ${{\boldsymbol f}_0}$, bandwidth, ${\boldsymbol \Delta f}$, fractional bandwidth (FBW), and averaged axial ratio (AAR) over the bandwidths, for different values of ${\boldsymbol W\; }$ of the proposed single-layer structure. Comparison between Case 1 [30] and Case 4 is made in the fifth row. The parameters of the proposed structure are p = 7.6 µm, y = 250 nm, L1 = 6.5 µm, L2 = 5.2 µm, t = 1.5 µm, ${{\boldsymbol \varepsilon }_{\boldsymbol r}}$=2.25, and ${{\boldsymbol E}_{\boldsymbol F}}$= 0.95 eV.
Fig. 4. Axial ratio of the proposed single-layer structure calculated for different values of graphene Fermi energy, ${E_F} = $ 0.4, 0.6, 0.8 and 1 eV.
Table 2. Center frequency ${{\boldsymbol f}_0}$, 3-dB bandwidth $\Delta {f}$, fractional bandwidth FBW and axial ratio AR at different values of Fermi energy, ${{\boldsymbol E}_{\boldsymbol F}} = $1, 0.8, 0.6 and 0.4 eV for the proposed single-layer structure
In order to apply the electrostatic bias voltage to the graphene patches, narrow metallic ribbons (or transparent electrodes) can be placed under the substrate [30,48], as a DC ground plane. One end of the bias voltage ${V_B}\textrm{}$ is connected to the graphene plate and the other end is connected to the DC ground plate. The coplanar graphene patches are also interconnected with narrow metallic ribbons [49–51]. The thin wires are short circuited in the DC mode, and open circuited in the AC mode.
3.2 Double-layer structure
In order to increase the phase difference, and reduce the axial ratio, we used two identical graphene patch arrays on both sides of the dielectric substrate. The results are shown in Fig. 5, Fig. 6, and Table 3. The results of [30] are added for comparison. In Table 3, the center frequency ${f_0}$, transmittance at ${f_0}$, phase difference at ${f_0}$, bandwidth, and the average axial ratio over the bandwidth are shown. Comparison with [30] is added in the third row. Comparing the double-layer and single-layer structures, it is seen that by increasing the number of graphene layers, the transmission coefficient decreases by 22% (from 88% to 69%), but $\mathrm{\Phi }\Delta $ and bandwidth increase, and the averaged axial ratio (AAR) over the bandwidth decreases by 48% (from 2.3 to 1.2 dB). According to Fig. 5, Fig. 6, and Table 3, the bandwidth and transmission coefficient of the double-layer structure compared to [30], increase by 138% (from 16.8% to 40%) and 25% (from 55% to 69%), respectively, but the averaged axial ratio (AAR) increased slightly from 1.5 to 1.2 dB.
Fig. 5. (a) Transmission coefficients amplitude, for the x- and y-polarized normally incident waves, Insets: the z-component of the electric field distribution at the two resonances. (b) phase difference for the proposed double-layer structure, for $W = $2500 nm, p = 7.6 µm, y = 250 nm, L1 = 7.2 µm, L2 = 5.2 µm, t = 1.5 µm, ${\varepsilon _r} = 2.25$, and ${E_F} = 0.95\textrm{}eV$ and comparing them with [30].
Fig. 6. Axial ratio for the proposed double-layer structure, with $W$=2500 nm, p = 7.6 µm, y = 250 nm, L1 = 7.2 µm, L2 = 5.2 µm, t = 1.5 µm, ${\varepsilon _r}$=2.25, and ${E_F}$= 0.95 eV and comparing it with [30].
Table 3. Center frequency, ${{\boldsymbol f}_0}$, transmittance ${{\boldsymbol T}_{{xx}}} = {{\boldsymbol T}_{{yy}}}$ at ${{\boldsymbol f}_0}$, phase difference, ΔΦ, bandwidth, ${\boldsymbol \Delta f}$, fractional bandwidth (FBW), and averaged axial ratio in the passband (AAR), for the proposed double-layer structure with ${\boldsymbol W}$=2500 nm, ${\boldsymbol p}$=7.6 µm, ${\boldsymbol y}$=250 nm, ${{\boldsymbol L}_1}$=7.2 µm, L2 = 5.2 µm, ${\boldsymbol t}$=1.5 µm, ${{\boldsymbol \varepsilon }_{\boldsymbol r}}$=2.25, and ${{\boldsymbol E}_{\boldsymbol F}}$=0.95 eV and comparing them with [30]
The z-component of the electric field distribution at the two resonance modes are calculated and shown in the inset of Fig. 5(a). Dipolar resonant modes are excited along the edges of the graphene patches with different orientations, depending on the polarization of the incident wave. In this structure, W = 2500 nm and the other parameters are L1 = 7.2 µm, L2 = 5.2 µm, y = 250 nm, t = 1.5 µm, ${\varepsilon _r}$=2.25, and ${E_F}$=0.95 eV.
The transmission amplitude ratio ${{\boldsymbol T}_{xx}}$/${{\boldsymbol T}_{yy}}$ for the double-layer structure is shown in Fig. 7 for ${L_1}$= 6500 and 7200 nm, when $p$=7.6 µm, $y$=250 nm, ${L_2}$=5.2 µm, $t$=1.5 µm, ${\varepsilon _r} = 2.25$, $W$= 2.5 µm, and ${E_F}$= 0.95 eV. It is observed from the figure that increasing L1, the slope of the ${{\boldsymbol T}_{xx}}$/${{\boldsymbol T}_{yy}}$ curve decreases by 70% about the center frequency of f0= 4.75 THz, where, ΔΦ is close to 90°. Therefore, a wideband linear to circular polarization conversion is achieved with a great accuracy.
Fig. 7. Transmission amplitude ratio for the double-layer structure for L1 = 6500 and 7200 nm, with p = 7.6 µm, y = 250 nm, L2 = 5.2 µm, t = 1.5 µm, ${\varepsilon _r}$=2.25, W = 2.5 µm, and ${E_F}$=0.95 eV.
Another feature of the proposed polarizer is its robustness at oblique incidence. The transmittance and phase difference at oblique incident angles from 0° to 70° is shown in Fig. 8. As observed from the figure, the transmittance and phase difference do not vary much in a broad frequency range around 4.75 THz, when the incident angle is varied from 0° to 50°. Figure 9 shows that at the incident angle of 70°, the axial ratio is less than 3 dB at 4.75 THz. This indicates that even at large incidence angles, the conversion is performed well.
Fig. 8. (a) Transmission coefficients amplitude for the x- and y-polarized incident waves and (b) phase difference, under oblique incidence, for angles θ=0 to 70°, by steps of 10°, for the double-layer structure, with p = 7.6 µm, y = 250 nm, L1 = 6.5 µm, L2 = 5.2 µm, t = 1.5 µm, ${\varepsilon _r}$=2.25, W = 2.5 µm, and ${E_F}$=0.95 eV.
Fig. 9. Axial ratio under oblique incidence, for incident angles of θ=0, to 70°, for the double-layer structure, when p = 7.6 µm, y = 250 nm, L1 = 6.5 µm, L2 = 5.2 µm, t = 1.5 µm, ${\varepsilon _r}$=2.25, W = 2.5 µm and ${E_F}$=0.95 eV.
The phase difference of the proposed ultrathin polarization converter can be tuned by controlling the graphene Fermi energy, ${E_F}$, as shown in Fig. 10. It is observed that the proposed structure can be tuned to operate in a broad range of frequency, and its operational bandwidth blueshifts as the Fermi energy increases. In addition, the polarization state of the transmitted wave can be dynamically tuned by varying a DC bias voltage. These features can be interpreted through the relationship (4), according to which, the resonant frequency is directly related to ${E_F}$. Therefore, increasing ${E_F}$, $f_r^{{T_{xx}}}$ and $f_r^{{T_{yy}}}\; $ increase, and as a result, ${T_{xx}}$ and ${T_{yy}}$ transmission diagrams and their phase difference shift towards higher frequencies.
Fig. 10. Calculated phase difference $\Delta \Phi = {\Phi _{xx}} - {\Phi _{yy}}\textrm{}$ of the circular polarization converter as a function of the operation frequency and Fermi energy for the double-layer structure. The parameters are p = 7.6 µm, y = 250 nm, L1 = 6.5 µm, L2 = 5.2 µm, t = 1.5 µm, ${\varepsilon _r} = 2.25$, and W = 2.5 µm.
The frequency range in which the axial ratio is less than 3 dB may be adjusted by changing the Fermi energy, as shown in Fig. 11. For each value of ${E_F}$, the linear to circular polarization conversion occurs at five frequencies. According to Fig. 11, by decreasing the Fermi energy, the central frequency decreases, but the axial ratio increases. Table 4 shows the variation of center frequency, ${f_0}$, the 3-dB bandwidth, $\Delta f$, and the operational frequency range in which the axial ratio is less than 3 dB, for several values of $\textrm{}{E_F}$. According to the table, the operational frequency range blueshifts with ${E_F}$. The 3-dB bandwidth varies from 0.9 THz, for ${E_F}$=0.2 eV, to 1.9 THz, for ${E_F}$=1 eV; the center frequency varies from 2.25 THz, for ${E_F}$=0.2 eV, to 4.85 THz, for ${E_F}$=1 eV. The minimum fractional bandwidth of 39.2% is obtained for ${E_F}$=1 eV. The overall operation bandwidth with tuning is (1.8 to 5.8), for which an overall tuning fractional bandwidth may be calculated to be 105%.
Fig. 11. Axial ratio of the double-layer structure calculated for different values of graphene Fermi energy, ${E_F} = $ 0.2, 0.4, 0.6, 0.8 and 1 eV.
Table 4. Center frequency ${{\boldsymbol f}_0}$, 3-dB bandwidth $\Delta {\boldsymbol f}$, fractional bandwidth FBW and axial ratio AR at different values of Fermi energy, ${{\boldsymbol E}_{\boldsymbol F}} = $1, 0.8, 0.6, 0.4 and 0.2 eV for the double-layer structure
A comparison of the two polarizers of the present work with the previous ones is given in Table 5. The transmission and bandwidth of the present polarizers are well improved compared to the previous ones. Meanwhile, the proposed structures have simple geometries.
Table 5. Comparison of the proposed structures with previous transmission-mode polarization converters, considering the center frequency ${{\boldsymbol f}_0}$, transmittance at ${{\boldsymbol f}_0}$, fractional bandwidth (FBW), axial ratio (AR), structural complexity, and dependence on the incident angle (θ)
It is worth mentioning that the application of the bias voltage to the graphene in the double-layer structure is similar to that of the single-layer. In both cases, the co-planar graphene patches are interconnected with narrow metallic ribbons (transparent electrodes). The ribbons are short circuited in the DC mode, and open circuited in the AC mode. In the double-layer structure, the electrostatic bias voltage is applied between the two planes.
4. Conclusions
We numerically demonstrated two broadband tunable terahertz linear-to-circular polarization converters, in transmission mode, based on graphene metasurface. Each converter comprised a dielectric substrate with slotted graphene patches on one side (single-layer) or both sides (double-layer) of the substrate. Compared to the previous polarizers, the single-layer structure exhibited 120% larger fractional bandwidth and 60% larger transmittance, and the double-layer converter showed a 138% larger fractional bandwidth and 25% larger transmittance, with excellent axial ratio. Both structures are dynamically tuned via the graphene Fermi energy (chemical potential), by the application of a DC bias voltage. The tunable operational frequency range over which the axial ratio is less than 3 dB, for the single-layer polarizer is (2.4 to 4.9) THz, and for the double-layer structures is (1.8 to 5.8) THz, achieved by varying the graphene Fermi energy from 0.2 to 1 eV. The fractional bandwidth, corresponding to the above frequency ranges, is 68% and 105%, for the single-layer and double-layer structures, respectively. The 3-dB bandwidth of operation blueshifts as the Fermi level is increased, while the fractional bandwidth remains between 37 and 45 percent, for each value of Fermi level. The polarization conversion performance is well maintained under oblique incidence, for a wide range of incident angles from 0° to 70°, indicating robust operation in practice. The unit cell size is ${\lambda _0}/10,\; $ much smaller than the free-space wavelength of the incident waves, $\textrm{}$ while the thickness is ${\lambda _0}/22$.
Funding
Shiraz University.
Disclosures
The authors declared that there are no conflicts of interests.
Data Availability
Simulation data, information, and results are available upon request.
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