## 2 Design and Theory  
Figure 1 depicts the proposed terahertz dipole antenna, which is composed of two gold (Au) strips on top of a polyimide (PI) substrate, sandwiched by multiple alternating thin films of Indium Antimonide (InSb) and silica (SiO\({}_{2}\)) that are dressed with periodic hole arrays. The two gold strips are of thickness \(t_{\rm Au}\), width \(w\) and length \(L\), and are separated by an air gap of width \(g\). The individual thicknesses of the InSb and SiO\({}_{2}\) layers are \(t_{\rm m}\) and \(t_{\rm d}\), respectively. The hole arrays in the InSb-SiO\({}_{2}\) multilayers, as illustrated in Figure 1a inset, have a radius of \(r\) for the holes and a period of \(p\) for the arrays. Compared with a conventional terahertz dipole antenna, which directly sits on the substrate, the sandwiched multilayers with hole arrays are novel. Figure 1: (**a**) Schematic of the proposed approach to further enhance the near fields of a terahertz dipole antenna composed of two metal stripes. A hyperbolic metamaterial (HMM) dressed with periodic hole arrays (inset) is sandwiched between the substrate and the dipole antenna. The HMM is composed of alternating layers of InSb (in blue) and SiO\({}_{2}\) (in green) with thicknesses of \(t_{\rm m}\) and \(t_{\rm d}\), respectively. A terahertz plane wave polarized in \(x\) direction with unitary amplitude normally impinges onto the structure. (**b**) Frequency-dependent complex relative permittivity of InSb at THz frequencies, which is calculated using the Drude model. (**c**) Real parts of the effective relative permittivity tensor components of the InSb-SiO\({}_{2}\) multilayer metamaterial according to the effective media theory. The purple region indicates that the metamaterial has hyperbolic dispersion characteristics. Because the individual InSb and SiO\({}_{2}\) layer dimensions satisfy the criteria of effective medium theory (EMT) [26], the effective dielectric tensor components of the anisotropic metamaterial composed of the InSb-SiO\({}_{2}\) multilayers can be calculated using the EMT,  
\[\bar{\varepsilon}=\left( \begin{array}{ccc}\varepsilon_{\perp}&0&0\\ 0&\varepsilon_{\perp}&0\\ 0&0&\varepsilon_{\parallel}\end{array} \right), \tag{1}\]  
where the subscripts \(\parallel\) and \(\perp\) indicate the components that are parallel and perpendicular to the \(x\)-\(y\) plane, respectively, and these components are given by [27],  
\[\varepsilon_{\parallel}=\frac{t_{\rm m}\varepsilon_{\rm m}+t_{\rm d} \varepsilon_{\rm d}}{t_{\rm m}+t_{\rm d}}, \tag{2}\]  
\[\varepsilon_{\perp}=\frac{\varepsilon_{\rm m}\varepsilon_{\rm d}(t_{\rm m}+t_ {\rm d})}{t_{\rm m}\varepsilon_{\rm d}+t_{\rm d}\varepsilon_{\rm m}}. \tag{3}\]  
The unique properties of such metamaterials stem from the isofrequency surface of extraordinary (transverse magnetic polarized) waves, which is given by [28]  
\[\frac{k_{x}^{2}+k_{y}^{2}}{\varepsilon_{\parallel}}+\frac{k_{z}^{2}}{ \varepsilon_{\perp}}=\left(\frac{\omega}{c}\right)^{2} \tag{4}\]  
where \(k_{x}\), \(k_{y}\) and \(k_{z}\) are wave vectors, \(\omega\) is the wave frequency and \(c\) is the speed of light. By tuning the parameters of \(\varepsilon_{\rm m}\), \(\varepsilon_{\rm d}\), \(t_{\rm m}\), and \(t_{\rm d}\) such that \(\varepsilon_{\parallel}\varepsilon_{\perp}<0\), one can attain the hyperbolic regime [28]. The frequency-dependent complex permittivities of semiconductor InSb at terahertz frequencies can be described by the Drude model [29]  
\[\varepsilon_{\rm m}=\varepsilon(\infty)-\frac{\omega_{p}^{2}}{\omega^{2}+i \omega\gamma}, \tag{5}\]  
where \(\varepsilon(\infty)\) is the high-frequency permittivity, \(\omega\) is the excitation frequency, \(\omega_{p}^{2}=ne^{2}/(m^{*}\varepsilon_{0})\) is the plasma frequency with \(n\) the free electron density, and \(e\) and \(m^{*}\) the electron's charge and effective mass, respectively. \(\gamma=1/\tau\) is the carrier momentum relaxation rate and \(\tau\) is the average collision time of the charge carriers.