## 3 Results and Discussion  
Figure 2 shows the simulated near-field distributions in the \(y=0\) or/and \(z=1.2\,\upmu\)m cross sections. It is clear that the terahertz electric field intensity in the antenna gap has been greatly enhanced, no matter the InSb-SiO\({}_{2}\) multilayers with/without the hole array is sandwiched between the gold dipole antenna and the substrate or not. For the conventional terahertz antenna which directly sits on the PI substrate, the electric field intensity in the gap is enhanced to be \(1.12\times 10^{4}\) of the incidence intensity, as shown by Figure 2a,d,g. By sandwiched with the HMM composed of the InSb-SiO\({}_{2}\) multilayers, Figure 2b,e,h shows that the enhancement is slightly increased to \(1.85\times 10^{4}\), corresponding to 50% improvement compared with the conventional terahertz dipole antenna. By further drilling periodic hole arrays in the HMM, as illustrated in Figure 1, Figure 2c,f,i shows that the enhancement can be further increased to \(3.65\times 10^{4}\), more than three times of that for the conventional antenna. Therefore, simulations reveal that by introducing the HMM with periodic hole arrays between the terahertz dipole antenna and the substrate, the near-field intensity localized in the antenna gap can be further enhanced by more than three times. Please note that quantitatively similar enhancement can also be achieved by using square-shaped periodic hole arrays. To understand the origins of this near-field enhancement, we further plot the dominant in-plane and out-of-plane components of the electric field in Figures 3 and 4. Results show that \(E_{y}\) is negligible, \(|E_{x}|^{2}\) in the gap of the conventional antenna is greatly enhanced to \(1\times 10^{4}\), whereas \(|E_{z}|^{2}\) is very weak (only 12). In other words, \(E_{x}\) dominates the near field, consistent with the literature [11]. However, for the proposed antenna structure, both the electric field components are greatly enhanced: \(|E_{x}|^{2}=2.4\times 10^{4}\), \(|E_{z}|^{2}=1.5\times 10^{4}\). Compared with the conventional antenna, \(|E_{x}|^{2}\) is doubled, whereas \(|E_{z}|^{2}\) is enhanced by three orders of magnitude such that it is now comparable to \(|E_{z}|^{2}\). Figure 3a,b shows that the \(|E_{x}|^{2}\) energy transmitted through the gap of the conventional dipole antenna is efficiently reflected back by the HMM with periodic hole arrays, resulting in double enhancement  
Figure 2: Comparison of three configurations on electric field intensity enhancement. (**a**,**d**,**g**) The gold dipolar antenna sits on the PI substrate directly. The antenna and the substrate is buffered by (**b**,**e**,**h**) the HMM composed of the multilayer structure, or (**c**,**f**,**i**) by the HMM with hole array structure. (**a**–**c**) side view at \(y=0\), (**d–**f**) top view at \(z=1.2\,\upmu\)m, and (**g–i**) line view at \(y=0\) and \(z=1.2\,\upmu\)m. The red horizontal lines indicate the value of field enhancement in the gap. of the near-field intensity in the gap. A similar reflection effect also occurs for the \(E_{z}\) component, as shown by Figure 3c,d. Figure 5a shows that the gap width \(g\) plays a key role on the near-field enhancement. In above discussions, we take \(g=1\,\upmu\)m and obtain three times further enhancement compared with the conventional antenna. As \(g\) increases, the near-field enhancement factors for both the proposed and the conventional antennas decrease, consistent with the literature. Their ratios also decrease from 3.3 to 2.2, indicating smaller enhancement due to the introduction of the HMM with periodic hole arrays. Figure 5b shows that the large near-field enhancement factor can be achieved only for \(t_{\text{m}}/t_{\text{d}}=1\):1, and that the enhancement factors are small for both \(t_{\text{m}}/t_{\text{d}}=1\):2 and \(t_{\text{m}}/t_{\text{d}}=2\):1.