I. INTRODUCTION
Frequency selective surfaces (FSS) are structures used as filters [^{1}], which are employed usually by periodically distributing optimized unit cells over surfaces. FSSs have a great number of applications in telecommunications, such as in designing of radomes, absorbers and electromagnetic shielding structures [^{1}]. In this context, FSSs operate as band-reject filters, of which rejection band(s) is(are) dependent on the geometry (unit cell configuration) and materials of their constitutive elements. Additionally, thickness and material parameters of substrate are also important design parameters [^{2}]. However, traditional (metallic) FSSs do not have dynamic adjustment of central frequencies of their rejection bands. In order to obtain such functionality, graphene has been incorporated into recently proposed FSS designs [^{3}].
Graphene is a two-dimensional material consisting of a planar arrangement of carbon atoms, of which electrical properties are of particular interest [^{4}], such as the dynamic control of the complex surface conductivity by regulating chemical potential μ_{c}, as described by Kubo formalism [^{5}]. Tuning of μ_{c} is performed by means of a controlled external transverse DC electric field, which in turn is controlled by a gate voltage V_{DC} applied between the graphene sheet and an electrode. According to [^{6}],
Hybrid FSSs designed with graphene and metallic parts have been proposed [^{7}]-[^{9}], providing reconfigurable rejection band. The electrical conductivity of graphene can also be modified by applying external magnetic field, increasing the degrees of freedom to dynamically reconfigure FSSs [^{10}], [^{11}].
In this work, a smart FSS composed only of graphene elements and a glass substrate is proposed. The main contribution of this paper consists on a novel class of smart reconfigurability, which is obtained by optimizing the proposed unity cell designed with a graphene ring and a graphene sheet placed in its aperture. By adjusting the chemical potentials of two graphene elements of the unit cell, the novel FSS device can operate as: a) reconfigurable single-band filter or b) reconfigurable dualband filter. When the proposed FSS operates in its single-band mode, smartness provides dynamical frequency reconfiguration of the rejection band. On the other hand, when the proposed FSS operates in its dual-band mode, smartness provides the following capabilities: b.1) dynamical frequency reconfiguration of either of the rejection bands individually, i.e., maintaining the other one fixed, b.2) dynamical frequency offset of both rejection bands simultaneously. It is important to notice that the ability to operate as single or dual band device is not among the capabilities of graphene FSSs designed in previously published papers such as [^{2}], [^{3}] and [^{10}]. Recently, Tasolamprou et al. conducted experiments in [^{12}] regarding measurements of THz waves interacting with graphene sheet set up over an SU-8 substrate, showing real feasibility of fabricating the device proposed in this paper.
In addition, an improved FDTD formulation, mainly based on Matrix Exponential technique [^{13}], is developed for calculating electric field and electric current density vectors on graphene sheets. It is demonstrated that the formulation produces a subcellular thin sheet of which thickness is that of graphene. The proposed formulation and smart FSS are validated by performing comparisons of FDTD solutions with those of commercial software such as HFSS [^{14}], COMSOL and CST.
II. MATHEMATICAL FORMULATION
In FDTD lattices, graphene sheets can be modeled as subcellular planar objects, similarly to the formulation developed in [^{15}]. Yee cell faces containing electric field components defining TE modes are used for performing the subcellular representation of graphene, such as illustrated by Fig. 1 for the TEz mode [^{13}], [^{15}].
The numerical representation of graphene is based on specific updating equations used to calculate the components of electric field
According to the Drude model, disregarding interband contributions, the conductivity of the graphene can be represented by the tensor [^{13}] [^{16}]
where
and
In (2) and (3),
Fig. 2 shows the conductivity
and
for when B_{0} = 0, where k_{SPP} is the plasmon wavenumber and f is the frequency. The parameters k_{0} and η_{0} are the well-known free space wavenumber and wave impedance, respectively.
On the region containing the graphene sheet, the frequency domain Ampère's law can be written as
where
Integrating (7) in the interval [z_{k} – Δz / 2, z_{k} + Δz / 2], where Δz is the length of the Yee cell edge parallel to z (Fig. 1), and applying the sampling property of the impulse function, one obtains
in which
Thus, we may write
and
Applying (2) and (3) to (10), using (11) and transforming (10) to time domain produce
Similar mathematical procedures executed applying (2) and (3) to (11) and using (10) yield
Equations (12) and (13) can be written using a compact matrix notation as
in which
In (15), Γ = 1/ (2τ) is the scattering rate.
In order to solve the matrix differential equation given in (14), the matrix exponential method is applied [^{13}], [^{19}]. At first, (14) is transformed to Laplace domain. Subsequently, after performing the proper mathematical manipulations, one sees that
Then, by applying the inverse Laplace transform in (16), we have
In the FDTD method,
where
and
Equations (19) and (20) can be simplified using the matrix exponential method. Therefore, both equations are written in terms of the eigenvalues of M as
and
in which
and
Therefore, the discrete equations for updating J_{x} and J_{y} are given by
and
Differently from what is proposed in [^{13}], in this work components of surface current density tangential to graphene sheets are calculated at the same spatial positions defined by Yee for calculating the corresponding components of electric field, such as Fig. 1 illustrates for a graphene sheet placed parallelly to the x-y plane. This is necessary for performing physically-appropriate calculations of the components of
and
Similar procedure is carried out for the field components
Once the scalar components of
In an analogous way, it is possible to obtain a FDTD equation for
III. VALIDATION OF THE DEVELOPED FORMULATION
In order to validate the FDTD formulation presented in Section II, the frequency response of the graphene FSS described in [^{10}] is reproduced in this paper. The numerical solutions obtained using the FDTD routine developed in this work are compared with results calculated employing HFSS.
The FSS modeled for validation purposes is illustrated by Fig. 3. The square unit cells have sides (period) measuring D = 5 μm. The edge length of each square graphene element of the periodic structure is D – g, where g = 0.5 μm. The parameters of the FSS graphene sheets are μ_{c} = 0.5 eV, τ = 0.5 ps and T = 300 K. For proper comparison of results with those provided by [^{10}], the graphene structure is simulated in free space (i.e., the substrate relative permittivity is ε_{r} = 1 and h is consequently irrelevant for this case). Finally, the structure is under influence of external magnetostatic field
The computational domain used in the FDTD method to represent the unit cell of the FSS under analysis has 20×20×400 cells. In this mesh, the Yee cells are cubic, with edges measuring Δ = 0.25 μm. Beneath and above the periodic structure, convolutional perfectly matched layer (CPML) formulation [^{22}] is used, absorbing electromagnetic waves propagating outwardly the FDTD lattice. Periodicity is achieved by applying the PBC (Periodic Boundary Condition) technique defined in [^{23}] at the side ends of a single period of the FSS. The FSS is excited by a x-polarized plane wave generated using the TF/SF technique (Total-Field/Scattered-Field) [^{20}], of which temporal profile is governed by a Gaussian pulse with a minimum significant spectral amplitude at f = 20 THz.
In HFSS, graphene sheets are modeled as anisotropic surface impedances (designation of the boundary condition in that software). Real and imaginary parts of each element of σ^{−1}(ω) (denoted by Z_{s}(ω) in the HFSS) over the entire frequency range under analysis are imported using an auxiliary file. The matrix σ^{−1}(ω) is the inverse of (1). The excitation element called Floquet port and the Master and Slave boundary conditions are used to enforce the periodicity to the problem.
Fig. 4 shows the co-polarization transmission coefficient obtained in this work using the proposed FDTD modeling of graphene and HFSS. Over the full band of interest (0.5 – 10 THz), the frequency responses obtained using FDTD and HFSS show good agreement with results presented in [^{10}], in which the minimum co-polarization transmission coefficient is 0.22 for the structure of Fig. 3, occurring approximately at 4.70 THz.
IV. THE PROPOSED SMART FSS: DESIGN, ANALYSIS AND RESULTS
In this section, a smart graphene FSS is proposed. The unit cell of the structure of Fig. 3 is replaced by the unit cell illustrated by Fig. 5. It is composed of two graphene elements: a square ring with chemical potential μ_{ce} and a graphene sheet placed coplanarly to the ring, in its aperture, with chemical potential μ_{ci}. The proposed FSS has the following fixed parameters: D = 5 μm, g = 0.5 μm and a = 100 nm. In addition, the magnetostatic field is not applied to the structure (B_{0} = 0). Notice that those fixed parameters are based on the FSS proposed in [^{11}], of which unit cell has solely the square graphene ring with aperture edges measuring d = l = 2.25 μm (lacking the internal graphene sheet proposed in this work). Every graphene element of the unit cell is set up with τ = 0.25 ps and T = 300 K. The substrate is characterized by the parameters ε_{r} = 3.9 and h = 1 μm.
A. Design of a smart graphene FSS operating as dual-band or single band filter
As a preliminary procedure to obtain the smart multiband FSS, we analyze the spectral response of an FSS of which unit cell is formed by a graphene ring set up with μ_{ce} = 1 eV and a rectangular aperture with the geometric parameters d = 0.25 μm, l = 2.25 μm and d_{e} = 1.59μm, as indicated in Fig. 5. For this initial analysis, the graphene sheet in the ring aperture is not included in the unit cell. This configuration is investigated via FDTD and HFSS to demonstrate that a rectangular opening in the graphene ring used as the FSS unit cell produces more than one rejection band. This analysis is a first step towards the design of the proposed smart FSS device. In FDTD simulations, the uniform computational mesh has 160×160×400 cubic Yee cells, of which edges measure 31.25 nm.
As it can be seen in results of Fig. 6, this periodic structure has two rejection bands (i.e., it is a dual-band FSS). By considering the rejection level of −4 dB as reference, the lower-band rejection window has relative bandwidth of 40 %, ranging from 2.70 THz to 4.05 THz. In this frequency range, the minimum transmission of −10.05 dB is seen at the frequency of 3.39 THz. The bandwidth of the higher-frequency rejection band (6.43 – 7.38 THz) is 13.76%. The minimum transmission is −7.71 dB at 6.93 THz.
Fig. 7(a) shows the spatial distribution of surface current density
Figs. 8(a) and 8(b) show distributions of
Thus, the main idea grounding the design of the proposed smart FSS is using the graphene sheet at the rectangular aperture of the ring to turn on or off the high-frequency rejection band, producing two reconfigurable operation modes. This is possible because setting the chemical potential of a graphene sheet to zero makes it practically transparent for electromagnetic waves and the increment of this parameter increases electrical conductivity of the graphene element (see Fig. 2). For the cases in which μ_{ci} = 0 and μ_{ce} ≅ 1, the FSS operates mainly due the intense currents seen on the ring of Fig. 8(c) (dipole mode), which is similar to the current distribution seen in Fig. 7(a) for the case based on the ring only. Additionally, when μ_{ci} approaches 1eV, a second resonance is created, working supported not only by the current distribution on the ring seen in Fig. 8(d), which is similar to the quadrupole mode of Fig. 8(b) of the rectangular aperture ring second resonance. It is also supported by the currents induced on the graphene sheet in the ring aperture. In summary, when μ_{ce} and μ_{ci} are much larger than zero simultaneously, the space a between the ring and the internal graphene sheet works similarly to the rectangular aperture of Fig. 8(b). Further, when μ_{ci} approaches zero, the inner sheet becomes nearly transparent and the FSS behaves analogously as the structure of Fig. 7(a).
For the proposed smart FSS, the dimensions associated to the aperture are d = l = 2.25 μm, and the distance between the border of the aperture and the graphene sheet is a = 100 nm. In the FDTD method, a uniform computational mesh with 200×200×400 cubic Yee cells is used (Δ_{x} = Δ_{y} = Δ_{z} = 25 nm). In the mesh created using HFSS, the minimum and maximum edges of the triangular elements are 13 nm and 270 nm, respectively. This finite element mesh is shown in Fig. 9.
The first operation state is the dual-band mode (mode on). As previously explained, it can be obtained by, for example, setting μ_{ce} = μ_{ci} = 1 eV. For this state setup, the first rejection band has a relative bandwidth of 29.30 % (2.36 – 3.17 THz), where the minimum transmission is −7.19 dB at 2.78 THz. The second transmission rejection band has a bandwidth of 23.30 % (5.50 – 6.95 THz), of which transmission minimum is −10.27 dB at f = 6.21 THz. As Fig. 10 shows, the transmission minimum in the higher-band rejection window is 3 dB lower than the minimum of the first rejection band. This is expected since the lower-frequency rejection band is mainly supported by the currents induced on the ring and the high-frequency rejection band is supported not only by induced currents on the graphene ring, but also by induction on the graphene sheet in the ring aperture. This physical behavior can be appreciated by inspecting Figs. 8(c) and 8(d), which illustrate
The second operation state is single-band mode (mode off), which can be obtained by fixing μ_{ce} = 0.75 eV and μ_{ci} = 1 meV. In this state, the device works with a single frequency band of low transmission levels since the graphene sheet placed in the square-aperture ring is virtually transparent. The parameter μ_{ce} is set to 0.75 eV in order to maximize the coincidence of the single spectral rejection range to the lower resonance band of the dual-band mode. Thus, the minimum transmission is −7.66 dB at f = 2.65 THz. The relative bandwidth in this single-mode configuration is approximately 37.50% (2.21 – 3.23 THz), slightly smaller than the bandwidth of the lower resonance band of the mode on, as it can be inspected using Fig. 10(a). Additionally, as shown in Fig. 7(b),
Lower rejection band | Higher rejection band | |
---|---|---|
FSS based on Graphene ring with rectangular aperture | 40 % (2.70 – 4.05 THz)−10.05 dB (3.39 THz) | 13.76 % (6.43 – 7.38 THz)−7.71 dB (6.93 THz) |
Smart FSS dual-band mode | 29.30 % (2.36 – 3.17 THz)−7.19 dB (2.78 THz) | 23.30 % (5.50 – 6.95 THz)−10.37 dB (6.21 THz) |
Smart FSS single-band mode | 37.50 % (2.21 – 3.23 THz) −7.76 dB (2.65 THz) | – |
For sake of full validation of the developed FDTD formulation and implemented software, Figs. 10(b) and 10(c) show comparisons of the FDTD results to numerical data obtained in this work using COMSOL, CST and HFSS. As it can be observed, in both operation modes in which the proposed FSS works, our FDTD results agree well with those calculated using the commercial simulators.
B. Fine-tuning of rejection band(s)
In this section, operation modes of the proposed device are presented along with results and physical analysis, grounding the functioning mechanisms of the FSS.
B.1. Tuning exclusively the higher rejection band (mode on)
In order to controllably obtain shifting exclusively of the higher rejection band in mode on, it is sufficient to regulate μ_{ci}. Thus, μ_{ce} is set to 1 eV and μ_{ci} can assume values between 0.4 eV and 1.0 eV. Fig. 11 shows the transmission coefficients for four configurations of chemical potentials, illustrating the shifting solely of the higher rejection band. For the demonstrated settings, the spectral sweeping band is 1.89 THz (from 4.33 to 6.22 THz). The first rejection band is not shifted, in such way that its minimum transmission tends to be around 2.75 THz as μ_{ci} is tuned.
For better understanding the physics governing this device operation mode, we further analyze two setups: μ_{ce} = μ_{ci} = 1 eV and μ_{ce} = 1 eV / μ_{ci} = 0.6 eV. Figs. 12(a) and 13(a) illustrate spatial distributions of
Chemical Potential | Frequency | λ_{0} (μm) | λ_{SPP} (μm) |
---|---|---|---|
μ_{c} = 1 eV | 2.71 THz | 110.70 | 88.38 |
2.73 THz | 109.89 | 87.49 | |
2.78 THz | 107.91 | 85.32 | |
4.91THz | 61.10 | 35.74 | |
5.09 THz | 58.94 | 33.65 | |
6.21 THz | 48.31 | 23.90 | |
μ_{c} = 0.65 eV | 2.07 THz | 144.92 | 108.88 |
4.88 THz | 61.47 | 26.20 | |
μ_{c} = 0.6 eV | 2.73 THz | 109.89 | 67.80 |
5.09 THz | 58.94 | 22.67 | |
μ_{c} = 0.55 eV | 2.07 THz | 144.92 | 100.32 |
2.71 THz | 110.70 | 64.88 | |
4.88 THz | 61.47 | 22.75 | |
4.91 THz | 61.10 | 22.50 |
By comparing Figs. 12(a) and 13(a), we additionally see that the current density produced on the aperture graphene sheet when μ_{ci} = 0.6 eV, at the first resonance, displays higher levels than the current density on that sheet configured with μ_{ci} = 1 eV. This feature can be explained by analyzing data in Table II. When μ_{ci} = 0.6 eV, at the first resonance (f = 2.73 THz), λ_{SPP} = 67.8μm. However, for μ_{ci} = 1 eV, also in the first resonance (2.78 THz), λ_{SPP} = 85.32 μm. Thus, the dimensions of edges of the sheet are closer to λ_{SPP} = 67.8 μm, facilitating the pointed out higher currents to flow. In addition, by comparing the levels of
The distributions of
Regarding the second resonance currents on the graphene ring, it is important to observe that currents are produced mostly aligned to the diagonal lines of the ring. In addition, currents aligned to x-axis are observed due to the polarization of the plane wave excitation. Diagonal currents are formed because of the anti-phase currents induced by the sheet currents on the ring's graphene strips which are parallel to the x-axis and, of course, due to the polarization of the plane wave excitation. As it can be seen in Figs. 13(a) and 13(b), the μ_{ce} = μ_{ci} = 1 eV configuration has higher diagonal currents than the μ_{ce} = 1 eV / μ_{ci} = 0.6 eV setup at their respective second resonance frequencies. The different current levels are related to the graphene diagonal parameter d_{e}, which measures 1.59 μm (Fig. 5). The length d_{e} is closer to λ_{SPP} = 23.90 μm (case of Fig. 12(b)) than to λ_{SPP} = 33.65 μm (case of Fig. 13(b)). Therefore, it is possible to say that higher diagonal currents on the ring for the case of Fig. 12(b) establish larger effective area for reflection of the incident wave, i.e., the ring has relevant contribution for reflection levels in this case. Thus, the transmission level for the μ_{ce} = μ_{ci} = 1 eV configuration is therefore smaller than that for the μ_{ce} = 1 eV / μ_{ci} = 0.6 eV setup, as one can see in Fig. 11. In conclusion, the main mechanism for shifting exclusively the second rejection band is based on the following features: 1) at the first resonance, reflection is mainly governed by the ring, as far as electrical dimensions of the sheet do not favor appreciable influence; 2) at the second resonance, reflections occur mainly on the aperture sheet. Dynamical adjustments of electrical dimensions of the sheet by tuning μ_{ci} are the most important mechanism of smartness. The ring can also contribute to define the reflection profile of second resonance band when relevant levels of currents are induced.
B.2. Tuning simultaneously both rejection band (mode on)
As it is clearly seen from Fig. 2(c) and (5.1)–(5.2), tuning μ_{ce} individually would alter the electrical dimensions of the ring, as far as λ_{SPP} is strongly dependent of graphene chemical potential. Central frequency of the lower rejection band is thus altered as a direct consequence. However, based on the previous discussion on the influence of the ring on the higher resonance band, one may expect that the higher rejection band is also affected by tuning μ_{ce}, while keeping μ_{ci} fixed. This is demonstrated in Fig. 14, which shows several curves of transmission coefficients obtained with μ_{ci} = 0.6 eV and μ_{ce} ranging between 0.45 eV and 1.0 eV. The fact that the shifting of resonance bands is to the right as μ_{ce} is increased can be understood by noticing from (5.1) and (5.2) that plasmon wave velocity on the graphene increases with μ_{ce} [^{17}], i.e., electric lengths of the graphene ring are reduced.
By considering the rejection level of −4 dB as reference, the spectral sweeping for the lower rejection band is 0.85 THz (1.87 – 2.72 THz) and for the higher rejection band is 0.55 THz (4.54 – 5.09 THz). Sweeping bandwidth for higher rejection band is noticeably narrower than that of the lower rejection band. This feature is clearly explained by the fact that the former depends much more on the coupling between sheet and ring than the later, as earlier discussed. Finally, Table III contains the relative bandwidth, also calculated by taking the reference rejection level of −4 dB, for each configuration of chemical potentials in Fig. 14.
Chemical potentials | Lower rejection band | Higher rejection band | |
---|---|---|---|
Shifting of the lower rejection band in mode on | μ_{ce} = 0.43 eV, μ_{ci}= 0.73 eV | 1.83 THz only | — |
μ_{ce} = 0.55 eV, μ_{ci} = 0.65 eV | 16.42 % (1.90 – 2.24 THz) | — | |
μ_{ce} = 0.75 eV, μ_{ci} = 0.58 eV | 22.6 % (2.12 – 2.66 THz) | — | |
μ_{ce} = 1 eV, μ_{ci} = 0.55 eV | 25.50 % (2.36 – 3.05 THz) | — | |
Shifting of the higher rejection band in mode on | μ_{ce} = 1 eV, μ_{ci} = 0.4 eV | — | 25.55 % (3.79 – 4.90 THz) |
μ_{ce} = 1 eV, μ_{ci} = 0.6 eV | — | 23.09 % (4.52 – 5.70 THz) | |
μ_{ce} = 1 eV, μ_{ci} = 0.7 eV | — | 22.80 % (4.82 – 6.06 THz) | |
μ_{ce} = 1 eV, μ_{ci} = 0.8 eV | — | 23.04 % (5.07 – 6.39 THz) | |
μ_{ce} = 1 eV, μ_{ci} = 1eV | — | 23.30 % (5.5 – 6.95 THz) | |
Shifting of the both rejection bands | μ_{ce} = 0.45 eV, μ_{ci} = 0.6 eV | 1.87 THz only | 17.8% (4.20 – 5.02 THz) |
μ_{ce} = 0.6 eV, μ_{ci} = 0.6 eV | 18.1% (1.96 – 2.35 THz) | 19% (4.38 – 5.3 THz) | |
μ_{ce} = 0.8 eV, μ_{ci} = 0.6 eV | 23.93 % (2.17 – 2.76 THz) | 20.97 % (4.48 – 5.53 THz) | |
μ_{ce} = 1 eV, μ_{ci} = 0.6 eV | 26.48 % (2.36 – 3.08 THz) | 23.04 % (4.52 – 5.7 THz) |
B.3. Tuning exclusively the lower rejection band (mode on)
Shifting exclusively the lower rejection band can be achieved by properly regulating μ_{ci} and μ_{ce}. Tuning of both parameters is necessary because, as previously discussed, modifying uniquely μ_{ci} produces shifts exclusively on the second rejection band and, as it is clearly seen from the previous analysis, tuning μ_{ce} alter both rejection bands concurrently, as Fig. 15 further illustrates. Therefore, electrical lengths of the sheet and of the ring must be modified simultaneously for producing the desired effect of tuning exclusively the lower rejection band.
As far as we are able to shift solely the second rejection band by simply tuning μ_{ci}, as previously demonstrated, the problem at hand can be solved in two steps. Referring to Fig. 15, suppose that one has the lower rejection band centered at approximately 2.0 THz (continuous line), which should be shifted to approximately 2.8 THz (dashed line), preserving however the central frequency of the higher rejection band at approximately 4.9 THz (continuous line). To this aim, the first step is to centralize the first rejection band at 2.8 THz. This can be done preliminarily by increasing μ_{ce} from 0.55 eV to 1 eV. However, as shown in Fig. 14, both rejection bands are shifted to the right, as expected. Consequently, the second step is reducing μ_{ci} in order to set the minimum transmission of the second resonance band back to 4.9 THz. By following the described procedure, the lower rejection band can be shifted controllably, as Fig. 16 illustrates.
Fig. 16 also shows the obtained chemical potential arrangements for solving the problem at hand, i.e., shifting the lower rejection band while preserving the higher rejection band centered at approximately 4.9 THz. As μ_{ce} is increased, μ_{ci} is reduced for compensating the undesired shifts of the higher rejection band. Considering the rejection level of −4 dB as reference for maximum tolerable transmission, the obtained spectral sweeping band for the lower rejection band is 870 GHz (from 1.84 THz to 2.71 THz). Table III provides the relative bandwidth for each arrangement of chemical potentials given in Fig. 16.
The shifting exclusively of the first resonance band can also be understood by analyzing the surface current distributions
Current densities for the higher rejection band (around 4.9 THz) are shown by Figs. 17(b) and 18(b) for the configurations μ_{ce} = 1 eV / μ_{ci} = 0.55 eV and μ_{ce} = 0.55 eV / μ_{ci} = 0.65 eV, respectively. Currents on the aperture sheet are moderately more intense for the configuration of Fig. 17(b) because the sheet is slightly electrically larger than in the case of Fig. 18(b), as it can be seen by inspecting Table II (λ_{SPP} on the sheet is 22.50 μm and 26.20 μm, respectively). Thus, in order to preserve the central frequency of the second rejection band at approximately 4.9 THz when the device operation mode is switched from μ_{ce} = 1 eV / μ_{ci} = 0.55 eV to μ_{ce} = 0.55 eV / μ_{ci} = 0.65 eV, reduction of electric length of the aperture sheet is compensated by increasing the electric length of the ring, as far as λ_{SPP} on the ring goes from 35.74 μm to 22.75 μm. This rise of the electrical length of the ring produces the considerable currents seen on the ring in Fig. 18(b), as it favors ring resonance, thus demonstrating the importance of electromagnetic coupling between the ring and the aperture sheet for preserving the central frequency of the higher rejection band.
B.4. Tuning the rejection band of mode off
In mode off, aperture graphene sheet must not favor current flows, thus suppressing the higher rejection band. As it can be seen in Figs. 2(a) and 2(b), this can be achieved by setting μ_{ci} to very small value, in such way that the real and imaginary parts of conductivity are negligible. In this work, μ_{ci} is fixed to 1 meV. Thus, the shifting of the rejection band in mode off is produced by tuning μ_{ce} on the graphene ring, as demonstrates Fig. 19. Table IV provides the relative bandwidth, considering the rejection level of −4 dB as reference, for each configuration of chemical potential shown in Fig. 19. For the present operational mode, the spectral sweeping band is 1.33 THz (1.76 – 3.09 THz). Thus, the tuning of the ring electric length allows the structure to have various rejection band central frequencies.
Configurations of chemical potentials | Rejection band | |
---|---|---|
Shifting of the rejection band in mode off | μ_{ce} = 0.3 eV, μ_{ci} = 1 meV | 1.76 THz only |
μ_{ce} = 0.5 eV, μ_{ci} = 1 meV | 28.7 % (1.91 – 2.55 THz) | |
μ_{ce} = 0.85 eV, μ_{ci} = 1 meV | 39.16% (2.32 – 3.45 THz) | |
μ_{ce} = 1 eV, μ_{ci} = 1 meV | 41.41% (2.47 – 3.76 THz) |
As μ_{ce} is incremented, central frequencies of rejection bands increase once more due to the augment of plasmon wave velocity given in (5). Additionally, increments on μ_{ce} causes a more metallic behavior of the graphene ring, as it is noticeable in Figs. 2(a) and 2(b). As a result, the FSS will reflect more electromagnetic power as μ_{ce} is increased.
V. FINAL REMARKS
In this paper, an FDTD formulation based on the matrix exponential method is developed. The graphene sheets are modeled considering only the intraband contribution of graphene conductivity. The results show good agreement with those obtained in commercial HFSS software, validating the developed FDTD formulation. The developed formulation presents numerical corrections for properly characterize the physical interdependence between J_{x} and J_{y} represented by the tensorial nature of the electrical conductivity of the graphene. Additionally, a novel intelligent FSS is proposed. It is formed only of graphene elements and is analyzed via FDTD and HFSS. The unity cell of the device contains a graphene ring and a coplanar aperture graphene sheet. with this geometry, the device can operate in single-band or dual band modes. In the single-band configuration, the structure has a fractional bandwidth of 52.3%, with its central frequency rejection band reconfigurable. In the dual-band operation, the first rejection band has a fractional bandwidth of 37.9 % and the second, 33.3 %. In addition, by operating in dual-band, the device may have the first, second or both rejection bands shifted properly tuning the chemical potentials of the graphene elements. For the shifting of the lower rejection band in mode on, the spectral sweeping band is 0.87 THz (1.84 – 2.71 THz). The shifting of the higher rejection band shows a spectral sweeping band of 1.89 THz (6.22 – 4.33 THz). Considering both the bands of rejections shifted, the spectral sweeping for lower rejection band is 0.85 THz (1.87 – 2.72 THz) and for higher rejection band is 0.55 THz (4.54 – 5.09 THz). Finally, for the mode off, the spectral sweeping band is 1.33 THz (1.76 – 3.09 THz).