# A Back-to-Back Gap Waveguide-Based Packaging Structure for E-Band Radio Frequency Front-End
## 2 Unit Design of the Packaging Structure
### Hybrid Coupler and Bandpass Filter Design  
The hybrid coupler plays a critical role in the design of this packaging structure, as it must exhibit a wide bandwidth and low insertion loss and maximize port isolation. Consequently, a multi-branch hybrid coupler is employed in this design. As shown in Figure 5, the coupling between two adjacent GWs is achieved through five slots present in the middle metal plate. By adjusting the metal height between these coupling slots, different impedances can be obtained, and additional tunable variables can be introduced to enhance matching performance. In this design, the spacing distance between adjacent coupling slots corresponds to approximately one-quarter of the wavelength. Based on the theoretical analysis of odd- and even-mode methods [22], the ABCD matrix can be expressed as the multiplication of matrices of multiple networks:  
\[\left[ \begin{array}{cc}A&B\\ C&D\end{array} \right]_{e} = \left[ \begin{array}{cc}1&jw_{1}\tan\theta_{h}/2\\ 0&1\end{array} \right]\times\left[ \begin{array}{cc}0&\tan\theta_{s_{2}}\\ \tan\theta_{s_{2}}&0\end{array} \right]\times\left[ \begin{array}{cc}1&jw_{2} \tan\theta_{h_{1}}/2\\ 0&1\end{array} \right] \tag{1}\] \[\times \left[ \begin{array}{cc}0&\tan\theta_{s_{1}}\\ \tan\theta_{s_{1}}&0\end{array} \right]\times\left[ \begin{array}{cc}1&jw_{3} \tan\theta_{h_{2}}/2\\ 0&1\end{array} \right]\times\left[ \begin{array}{cc}0&\tan\theta_{s_{1}}\\ \tan\theta_{s_{1}}&0\end{array} \right]\] \[\times \left[ \begin{array}{cc}1&jw_{2}\tan\theta_{h_{1}}/2\\ 0&1\end{array} \right]\times\left[ \begin{array}{cc}0&\tan\theta_{s_{2}}\\ \tan\theta_{s_{2}}&0\end{array} \right]\times\left[ \begin{array}{cc}1&jw_{1} \tan\theta_{h}/2\\ 0&1\end{array} \right]\] \[\left[ \begin{array}{cc}A&B\\ C&D\end{array} \right]_{o}=\left[ \begin{array}{cc}A_{e}(-1/P)&B_{e}(-1/P)\\ C_{e}(-1/P)&D_{e}(-1/P)\end{array} \right] \tag{2}\]  
where \(\theta\) is the electrical length in the waveguide, \(\theta_{l}=\pi l\ [(2/\lambda_{0})^{2}-(1/\text{a})^{2}]^{1/2}\), \(a\) is the GW width, \(\lambda_{0}\) is the free-space wavelength, and \(P=\tan\theta/2\). Additionally, the scattering matrix can be expressed as  
\[ \begin{array}{ll}S_{11}=\frac{1}{2}\Gamma_{e}+\frac{1}{2}\Gamma_{o}&S_{21}= \frac{1}{2}T_{e}+\frac{1}{2}T_{o}\\ S_{31}=\frac{1}{2}T_{e}-\frac{1}{2}T_{o}&S_{41}=\frac{1}{2}\Gamma_{e}-\frac{1}{ 2}\Gamma_{o}\end{array}  \tag{3}\]  
where \(\Gamma_{e}/\Gamma_{o}\) and \(T_{e}/T_{o}\) are the reflection coefficient and transmission coefficient under even/odd mode excitation, respectively. Figure 5: Structure and dimensions of the hybrid coupler, P1 to P4 represent four ports. (where \(h1=0.9\) mm, \(h2=0.84\) mm, \(hs=1\) mm, \(w1=0.22\) mm, \(w2=0.34\) mm, \(w3=0.38\) mm, \(s1=1.29\) mm, \(s2=1.2\) mm, \(wc=3.1\) mm, and the radius of the chamfer is 0.1 mm). Initially, using the aforementioned formulas, the preliminary dimensions can be obtained. After simulation and optimization, the final dimensions and simulated results can be obtained. As shown in Figure 6, this designed coupler operates within a frequency range of 71-86 GHz while exhibiting return loss and isolation levels exceeding 23 dB, power distribution unbalance below 0.35 dB, and an output-port phase difference of 89.6 \(\pm\) 0.4\({}^{\circ}\). The performance of the filter determines the final frequency-selection capability of the module. Therefore, it is particularly important to design filters with low insertion loss, low-ripple, and high out-of-band rejection. In comparison to Butterworth and quasi-elliptic topologies, the Chebyshev filter balances the above indicators and is better suited for this module's design. Consequently, this design employs a fifth-order Chebyshev topology, as depicted in Figure 7, in which an electromagnetic coupling structure connects five resonators. The resonant frequency of the TE\({}_{101}\) mode in the GW resonator is presented in Figure 8, where \(w\) and \(l\) are the width and length of the resonator, respectively.