# Superconducting Electronic-Photonic Platform for HEB-Based Terahertz Spectrometers
## 3 Results and Discussion
### EM Simulations  
Figure 2 illustrates effective transition from a single-mode Si ribbon waveguide to a 50 \(\Omega\) microbridge developed via EM simulations. We use HFSS 15 as simulation software. Dielectric losses in Si are neglected, and its relative permittivity is set to 11.9. The waveguide side walls are surrounded by quarter-wave vacuum insets terminated by radiation boundaries. Front and rear ends of the waveguide are excited by two wave ports (ports 1 and 2). We can control coupling between the ports with the aid of a distributed Bragg reflector (DBR). The reflector is presented by a triplet of openings with diameters \(D_{1},D_{2},D_{3}\) in the waveguide at the THz signal input. The HEB-loaded tapered slot line is modeled as a golden sheet with a lumped port implemented in the center (port 3). Geometric parameters of the transition developed for 2.7 THz operation are detailed in Table 1.  
Figure 1: A 3D model of a nearly all-dielectric THz HEB mixer. We observe reflection and transmission coefficients with magnitudes \(S33=-16.1\) dB and \(S31=S32=-4.4\) dB for a DBR free transition. In the presence of DBR, \(S33=-32.3\) dB, \(S31=-2.4\) dB, and \(S32=-7.1\) dB. A further increase of \(S31\) up to approximately \(-1\) dB is possible at the expense of a drastic decrease of \(S32\). This option becomes attractive, for instance, if one designs an incoherent detector and LO is not required. For the transition from a ribbon waveguide to PCWG, the coupling efficiency can also be increased by a two-step-size tapered photonic crystal structure with a defect opening in the center [25]. To improve efficiency of an end-fire coupling, front and rear waveguide ends of the developed transition are terminated by Si forked antennas. Each antenna is presented by a four-element DRA array integrated with \(4\times 1\) ports MMIPC, as depicted in Figure 3.  
The MMIPC design relies on the use of an Si ribbon waveguide supporting propagation of multiple lateral modes. Such a condition is achieved for effective width of the waveguide defined as [26]:  
\[W_{eff}=\frac{\pi(m+1)}{k_{ym}}, \tag{1}\] \begin{table}
[\(\mu\)m] & [\(\mu\)m] & [\(\mu\)m] & [\(\mu\)m] & & [\(\mu\)m] & & [\(\mu\)m] & [\(\mu\)m] & [\(\mu\)m] \\ \hline
27 & 32 & 70.9 & 146 & 11.9 & 21 & 10 & 8 & 6 \\ \hline \end{tabular}
\end{table} 
Table 1: Geometry of a 2.7 THz Si ribbon waveguide integrated with a 50 \(\Omega\)-loaded TSL. Figure 3: EM model of a 2.7 THz Si DRA array integrated with MMIPC. Figure 2: EM model of a 2.7 THz Si ribbon waveguide integrated with a 50 \(\Omega\)-loaded TSL. where \(m\) is the mode number and \(k_{ym}\) is the lateral wavenumber of the \(m\)-th mode. Due to a blur of the interface between the dielectric medium and vacuum, the actual width of the waveguide, \(W_{pc}\), becomes smaller. Its deviation from \(W_{eff}\) is defined by Equation (2):  
\[W_{pc}=W_{eff}-\frac{\lambda_{0}}{\pi\epsilon_{r}^{r}\sqrt{\epsilon_{r}-1}}, \tag{2}\]  
where \(\lambda_{0}\) is the free space wavelength, \(\epsilon_{r}\) is the relative permittivity of a dielectric medium, and \(\sigma\) is the constant factor equal to 0 for transverse electric and 1 for transverse magnetic modes. The length of the waveguide, \(L_{pc}\), usually obeys Equation (3); however, it may experience a reduction of 17-23% in case of compact devices made of low-permittivity dielectrics [27]:  
\[L_{pc}=\frac{W_{eff}^{2}\sqrt{\epsilon_{r}}}{4\lambda_{0}}. \tag{3}\]  
Harvesting of each supported \(m\)-th mode at the output of the DRA array is maintained by MMIPC with a corresponding phase constant, defined as:  
\[\beta_{m}=\frac{2\pi\sqrt{\epsilon_{r}}}{\lambda_{0}}-\frac{\pi\lambda_{0}(m+ 1)^{2}}{4W_{eff}^{2}\sqrt{\epsilon_{r}}}. \tag{4}\]  
Superposition of the harvested modes results in four equally spaced power maxima at the MMIPC input. Almost 90\({}^{\circ}\) phase delays appear in the lateral ports of the combiner with respect to the central ones.