I. INTRODUCTION

Terahertz frequency range (THz) of the electromagnetic spectrum is attracting a lot of interest during the last years. Among the several applications that rely on the advantages of using terahertz radiation, the development of terahertz communications systems with high performance stands out [^{1}], [^{2}].

Several materials can be used on the project of new components for terahertz applications. In this context, graphene emerge as an interesting alternative. High rigidity, good thermal and electrical conductivity, presence of plasmon resonances, and dynamic control of its physical properties by electrostatic and magnetostatic fields are some of the outstanding features of its two-dimensional structure [^{3}]-[^{6}]. These features enable the use of graphene, for example, in the design of field effect transistors (FETs) [^{7}], photodetectors [^{8}], and antennas [^{9}].

Besides, graphene-based resonators can be projected for operation at terahertz frequencies. They can be, for example, disk, ring, tape, square or cross shaped, and rely on the excitation of plasmon resonances. In the case where two graphene resonators are mutually coupled, one can verify, in the frequency response of the coupled system, a plasmonically induced transparency (PIT) band [^{5}]. The PIT region is located between the resonance frequencies of the individual resonators and its main characteristic is the presence of transmission levels close to 100%.

Analytical methods can be used for the study of coupled resonators systems. Among them, Temporal Coupled-Mode Theory (TCMT) [^{10}] is one of the most valuable, since it describes a given system in terms of the couplings that take place between its constitutive parts, like waveguides and resonators, and provides an easy way for the calculation of the frequency response of such structures.

These theoretical methods are very useful mainly because they allow the designer to quickly obtain several information of the system from a set of formulas. The knowledge obtained from theoretical analysis can provide, for example, a way to reduce the time spent on the realization of computational simulations or assembling experimental setups, speeding up the project of such components.

In this paper, we consider graphene-based double resonators consisting of two disk-shaped resonators located on opposite sides of a dielectric substrate. These structures can be used, for example, as switches or wavelength-selective filters in terahertz communication systems. In order to improve their performance, we have developed an analytical approach based on the application of Temporal Coupled-Mode Theory. This analytical model provides several insights that can be useful for tuning the quality factors of the coupled resonators and the bandwidth of the PIT region.

II. STRUCTURE AND DESIGN

In Fig. 1, we schematically show the unit cell of the array composed of two graphene disks with diameters *d* = 32 μm (disk 1) and *D =* 60 μm (disk 2) - see Appendix A for more details regarding the calculation of the disks diameters.

The resonator disks are placed on opposite sides of a dielectric substrate with unit cell size *A* = 100 μm, thickness *h* = 10 pm, relative permittivity *ε _{r}=* 3.75 and dielectric loss tangent

*tanδ*= 0.0184. In order to verify the influence of the dielectric substrate on the frequency response of the device, we have also performed some computational simulations with the two disks suspended in air.

We have considered only the intra-band contributions for the calculation of graphene conductivity *(σ _{g})* in this paper. The influence of inter-band transitions can be neglected in the considered operating frequency range, since the condition

*ħω<<2μ*is satisfied [

_{c}^{11}]. Therefore, the following formula derived from Kubo formalism [

^{6}] can be used for the calculation of

*σ*

_{g}.where *ω* is the angular frequency, *e* is the charge of electron, *k _{B}* is the Boltzmann constant,

*T*is the temperature (300 K),

*ħ*is the reduced Planck constant, Γ is the scattering rate (0.1 meV), and

*μ*is the chemical potential.

_{c}In Fig. 2, it is shown the variation of real and imaginary parts of graphene conductivity with frequency for different values of the chemical potential. As one can see, its imaginary part is negative, which in turn is the necessary condition for graphene based devices to support transverse magnetic (TM) surface plasmon polariton (SPP) waves [^{12}].

III. ANALYTICAL FORMULAS FOR THE CALCULATION OF THE FREQUENCY RESPONSE

An analytical description of the presented component based on the TCMT has been developed. By using the formalism presented in [^{13}], the TCMT based equations that describe the evolution of the electromagnetic field in the system comprising the two disk resonators can be written as follows:

where *a _{n}* is the amplitude of the resonant mode in the nth disk (n = 1, 2),

*ω*is the resonant frequency of the mode in the nth disk

_{n}*τ*is the decay rate of the mode in the nth disk due to intrinsic losses,

_{i,n}*τ*is the decay rate of the mode in the nth disk to the input/output,

_{w,n}*μ*is the mutual coupling coefficient between the two modes,

*S*is the amplitude of the incident wave,

_{i}*S*is the amplitude of the reflected wave,and

_{r}*S*is the amplitude of the transmitted wave.

_{t}From this set of equations, one can define the formulas for the transmission and reflection coefficients of the array with double disk resonators as follows:

Regarding transmittance (*T*) and reflectance (*R*) of the structure, they can be defined as follows:

IV. RESULTS AND DISCUSSIONS

In order to test our analytical model, we have performed computational simulations of the proposed component with the software product ANSYS HFSS, which in turn is based on the finite element method. We have considered only normal incidence (see Fig. 1) and the performance characteristics of the device are not dependent on the polarization state of the feeding.

In Figs. 3 and 4, it can be observed that, regarding the transmission and reflection spectra, the computational results are in good agreement with those provided by the presented TCMT-based approach, both for the case with graphene disks suspended in air (Fig. 3) and for the case with the dielectric substrate between the two disks (Fig. 4).

Besides, from Figs. 3 and 4 it is possible to verify that the dielectric substrate cause a significant red- shift of the resonant frequency for both disks. Such as in [^{14}], [^{15}], the increase in the average refractive index of the whole structure caused by the substrate is related to the lower resonant frequencies observed in Fig. 4.

A. Influence of the TCMT parameter μ for the case without dielectric substrate

The frequency response of the structure, considering the parameter *h* equal to 0.1, 10 and 30 μm, is shown in Fig. 5. One can see that it is highly dependent on the coupling between the two resonator disks. We have noticed that, by increasing the parameter *h* from 10 μm to 30 μm, there is a decrease in the maximum transmission levels and width of the window of plasmonically induced transparency (related to the Fano resonance effect), mostly because of the decrease on the coupling between both disks. For these two values of *h*, the transmission spectrum has two minimum points corresponding to the resonant frequencies of the two disks that also vary with this coupling. However, when the parameter is very small (0.1 μm), one can see that the frequency response of the device shows only one minimum point. We believe that, in the latter case, the coupling of the disks is very strong because they are quite close, so that they can be regarded as only one disk, that is, there is only one resonant frequency in the coupled system.

We have also calculated, for the considered values of the parameter h, the *E _{z}* field component at the resonant frequency 1.6 THz for both disks (see Fig. 6) and for all the considered values of h. Again, one can observe that the parameter

*h*directly influences the coupling force between the disks, which in turn influences the maximum and minimum transmission levels at the resonant frequencies of the structure.

In order to relate the computational results concerning the variation of the parameter *h* (or, equivalently, the coupling between both disks) to the analytical TCMT approach, we have calculated the transmission coefficient as a function of the parameter *μ* (see Fig. 7) by using formula (12).

One can see that, the higher the parameter *μ*, the higher the transmission levels at the Fano resonance frequency and the higher the frequency shift of the two resonances induced by the mutual coupling. By comparing Figs. 5 and 7, it is possible to state that this is equivalent to decreasing parameter *h*. Therefore, the geometrical parameter *h* is closely related to the analytical parameter *μ* included in the presented TCMT based description.

B. Influence of the TCMT parameter μ for the case with dielectric substrate

The same analysis has been performed for the case including the dielectric substrate between both disks. In Fig. 8, the resonant frequencies for the two coupled disks as a function of the dielectric thickness *h* are displayed. We have found that, due to the dielectric substrate in the structure, the resonant frequencies are shifted to smaller values when compared to those presented in Fig. 5.

Similarly to the case without the dielectric substrate, one can see that the higher the parameter *h* (considering the cases with *h* = 10 μm and *h* = *20* μm), the lower the transmission levels and the lower the width of the PIT region. Besides, for the maximum coupling case (h = 0.1 μm), it can be verified that the two resonant frequencies turn into only one (see Fig. 8), as previously seen in section IV.A.

For a better understanding of the relationship between the parameter *h* and the mutual coupling, we show in Fig. 9 the distribution of the electric field component *E _{z}* in the graphene disks at the resonant frequency 1.54 THz.

In Fig. 10, it is shown the influence of the TCMT parameter *μ* in the frequency response of the component. One can see again that the geometrical parameter *h* can be associated to the TCMT parameter *μ*, that is, it is possible to change the characteristics of the PIT region by modifying the value of*μ*.

C. Variation of the graphene chemical potential

We have also studied the influence of the chemical potential *(μ _{c})* on the frequency response of the component. Fig. 11 shows the transmission curves for different values of the graphene chemical potential for the case with dielectric substrate. The typical value

*μ*= 0.5 eV [

_{c}^{16}], [

^{17}] has been considered in the calculation of the transmission spectra presented in Figs. 3, 4, 5, and 8. One can verify that as the chemical potential gets higher, the resonant frequencies of both graphene disks are blue-shifted and the transmission levels at the resonant frequencies are decreased. These results show that, by changing the chemical potential, it is possible to shift the transparency window of the structure and use it as a frequency-tunable electromagnetic filter. Besides, this feature enables one the use of the proposed structure as a switch for THz applications.

V. CONCLUSIONS

We have suggested a new graphene-based structure that can be used as a frequency-tunable filter or as an electromagnetic switch in the THz frequency band. Also, a TCMT-based approach for the analysis and design of the proposed device has been developed andthe analytical results provided by the TCMT equations are in a good agreement with those obtained from full-wave numerical computations. By relating the analytical parameters of the TCMT description to the geometrical/material parameters of the graphene-based double disk system, we have shown how to use the analytical model in the analysis and design of the structure.