## 3 Simulations  
To analyze the resonant conditions and field enhancement in the hybrid structure, the electric field distributions at the frequency of the transmission peak are simulated firstly. When the incident THz wave is x polarized, the field distribution in the graphene layer is shown in Figure 2a. It indicates that the electric field is mainly located around the regions of the metallic slits. Along the electric polarization direction, the distribution reveals that the resonant is mainly excited along the side of the square slit. At the same frequency, Figure 2b is the electric field distributions in the x-z plane at the place of the blue dotted line in Figure 2a. It shows that the enhanced field caused by the resonance is located in an area along the z direction. And graphene layer at the placement of \(z\) = 0.15 \(\upmu\)m is immerged in the localized field, which leads to a strong interaction between the graphene and the localized THz field. As a result the modulation of the transmitted THz wave is enhanced under such strong interactions. The influence of the structure parameters on the resonance and the modulation enhancement are further discussed. In the hybrid structure, the side length of the square slit is simulated firstly, and the result is shown in Figure 3a. The black square dots display the relationship of the resonant frequency with the side length of the slit. The black solid fitting line indicates that the resonant frequency is exponentially decaying with the value of L. It illustrates that the resonant frequency decrease is fast,  
Figure 2: The electrical field distributions: (**a**) in the graphene layer; (**b**) in x–z plane. which is due to fact that the resonant is mainly excited along the side of the square slit as shown in Figure 2. The blue circular dots are the maximum modulation depth and the blue solid fitting line indicates that the modulation depth is approximately linearly decreasing with the side length. The result reveals that the decrease of the modulation depth is slow, and the minimum _MD_ is larger than 60%. It also reveals that by changing the side length of the square slit, the working frequency of the hybrid structured THz modulator could be tuned in a wideband and the modulation depth could be kept at a high level. Another key parameter is the slit width in the metallic metamaterial. Its influence on the resonant frequency and modulation depth is calculated and displayed in Figure 3b. The black square dots and the solid fitting curve show that the resonant frequency increases with the slit width slowly. However, the blue dots and the fitting curve standing for the modulation depth is exponentially decaying with the value of g. Further analyzing the transmission spectral curves, reveal that although the resonant frequency is increased with the slit width, the quality factor is getting lower. It is due to the fact that the transmission at the lower frequencies is kept at a high level with no change of the side length of the slit, which leads to a lower modulation depth when the slit width is larger. The results also reveal that a narrower slit is better in the hybrid structured modulator due to stronger confinement of the field, and the stronger interaction with the active graphene layer. In the hybrid structure, the dielectric layer between the metamaterial and graphene acts as a spacer to avoid the active material directly touching with the metallic layer. When increasing its thickness, as seen in Figure 3c, the resonant frequency and the modulation depth are all increased obviously when the thickness is lower than 20 nm. It is due to the coupling between the graphene layer and the metallic metamaterial layer being much stronger and more sensitive to the separation distance when the latter is relatively short. By further increasing the thickness of the dielectric layer, the resonant frequency and the modulation depth tend to be flat as the coupling between them remain stable.