## 2 Materials and Methods  
### Josephson Junction as an RF Element in Cascade Reception Chain  
In order to develop the model for the simulations of a superconducting antenna system, we investigated the circuit representation of the detector. The current flowing through the junction \(I_{S}\) is the superposition between the DC current supplied to the junction, and the current generated by the RF radiation. Neglecting noise, \(I_{S}\) can be expressed in terms of Bessel functions of the first kind \(J_{m}\)[9]:  
\[I_{S}=I_{C}\sum\
olimits_{m=-\infty}^{\infty}(-1)^{m}J_{m}\Big{(}\frac{2 eV_{ RF}}{\hbar\omega_{RF}}\Big{)}\sin\Big{(}\Big{(}\frac{2 eV_{DC}}{\hbar}-ma\omega_{ RF}\Big{)}\Big{\}}+\varphi_{0}\Big{)} \tag{1}\]  
The RF radiation generates steps in the I-V characteristics, known as the Shapiro steps. The height of the steps is proportional to the voltage amplitude of the radiation \(V_{RF}\)[34; 35]. It reflects the detection efficiency, as affected by the integration between the antenna and the Josephson junction detection element. The voltage separation \(V_{m}\) between the steps gives an accurate measure of the radiation frequency \(\omega_{RF}\), obtained from \(V_{m}=m\frac{\hbar}{2e}\omega_{RF}\). The current flowing through the junction is a superposition of three components: the normal current, \(I_{R}\), the charging current, \(I_{ch}\), and the Josephson supercurrent, \(I_{I\!I}\). Since the current supplied to the junction is larger than the critical current \(I_{c}\), the superconductor is not in its classic superconducting state, i.e., the junction has a parallel normal component that behaves as a resistor \(R_{n}\). We analyze the interrelations between the input impedance, \(Z_{in}\), the antenna impedance, \(Z_{A}\), and the Josephson junction equivalent impedance, \(Z_{I\!I}\). This relation is derived through a cascade chain connection of two \"2 port network\" models, using an equivalent ABCD matrix [27]. Figure 1 shows the layout of the networks. Network\({}^{\alpha}\) represents the junction network, and network\({}^{\beta}\) represents the antenna network. One can express the integration between these networks through the voltage/current matrix relationship. \[[\alpha]=\left[ \begin{array}{cc}1&0\\ \frac{1}{Z_{I\!I}}&1\end{array} \right]=\left[ \begin{array}{cc}1&0\\ \frac{1}{R_{n}}+j\omega_{T\!I}&1\end{array} \right] \tag{2}\]  
Figure 1: Cascade, chain connection, of two “2 port network”. \[[\beta]=Z_{A} \tag{3}\]  
Equations (2) and (3) describe the network matrices, the junction ABCD matrix \([\alpha]\), and the antenna impedance matrix \([\beta]\). From these equations, we derive an expression for the input impedance,  
\[Z_{in}=\frac{V_{1}}{I_{1}}=\frac{AV_{2}+BI_{2}}{CV_{2}+DI_{2}}=\frac{Z_{A}}{ \frac{Z_{A}}{R_{n}}+1+j\omega C_{J}Z_{A}} \tag{4}\]  
\(Z_{JI}\), can be written in terms of electromagnetic wave propagation as [8]:  
\[Z_{JI}=\frac{c}{\widetilde{c}}\bigg{(}\frac{l}{we_{rs}}\bigg{)}Z_{0} \tag{5}\]  
where \(Z_{0}\) = 377 \(\Omega\) is the free space impedance. \(w\) and \(l\) are the width and length of the junction, \(\varepsilon_{rs}\) is the relative dielectric constant of the junction, \(c\) is the speed of light in free space. The speed of light in a medium \(\widetilde{c}\), is given by: \(\widetilde{c}=c\sqrt{\frac{l}{\varepsilon_{in}}}\), where the effective length of the junction, is given by \(l^{\prime}=l+\lambda_{L1}+\lambda_{L2}\). The London penetration depth for both sides of the junction is given by \(\lambda_{L}=\sqrt{\frac{m_{e}}{\mu_{0}n_{e}a^{2}}}\), where \(m_{e}\) is the electron mass, \(\mu_{0}\) is the permeability of free space, \(n_{s}\) is the density of superconducting charge carriers, and \(\mathrm{e}\) is the electron charge.