# Fully Superconducting Josephson Bolometers for Gigahertz Astronomy
## 5 Performance Deduced from the Experimental Data
### Nano-TES Bolometer Experimental Deduced Performance  
The thermal fluctuations between the electrons and phonons in the active region are the limiting factor for the intrinsic noise equivalent power of a nano-TES [44], as described in Section 3.1. Indeed, the contributions related to the Johnson noise and the shunting resistor are negligibly small [33]. Despite that the \(R(T)\) characteristics are modulated by the bias current, Joule heating restricts the operation of the nano-TES at \(I\to 0\). As a consequence, the values of \(NEP_{TFN,nano-TES}\) can be only extracted for the lowest experimental values of \(I\).  
Table 1 shows the \(NEP_{TFN,nano-TES}\) of two different 1D-JJs. In particular, the thermal fluctuations limited \(NEP\) of sample 1 (\(NEP_{TFN,1}\simeq 5.2\times 10^{-20}\) W/\(\sqrt{\text{Hz}}\)) is lower than that of sample 2, since it can be operated at lower temperature (\(T_{C,1}<T_{C,2}\)) and its thermal losses are reduced (\(G_{th,1}\simeq 6.7\times 10^{-15}\) W/K and \(G_{th,2}\simeq 9.3\times 10^{-15}\) W/K). This sensitivity is one order of magnitude better than state-of-the-art transition edge sensors [20], since \(G_{th,nano-TES}\) is drastically reduced thanks to the small dimensions of \(A\) and the presence of efficient Andreev mirrors.  
The saturation power of the nano-TES can be written as  
\[P_{sat}=\left(1-\frac{R(T_{C})}{R_{N}}\right)P_{e-phr}, \tag{25}\]  
where \(R(T_{C})\simeq 40\)\(\Omega\) is the resistance of the active region at \(T_{C}\). The saturation power is of the order of tens of hundreds of aW. It is possible to increase the saturation power by increasing the heat losses through the phonons, that is by increasing the active region volume, with the simultaneous increase of the \(NEP\).  
The intrinsic time constant of the 1D-JJ (\(\tau\)) can be calculated from Equation (13). Here, the heat capacitance is given by Equation (14) taking the value \(C_{e,1}=\simeq 4\times 10^{-20}\) J/K and \(C_{e,2}=\simeq 4.2\times 10^{-20}\) J/K for samples 1 and 2, respectively, where the Sommerfeld coefficient of the active region is calculated as the average of the two components \(\gamma_{A}=(\gamma_{Cu}V_{Cu}+\gamma_{Al}\gamma_{Al})/\gamma_{A}\) (with \(\gamma_{Cu}=70.5\) JK\({}^{-2}\) m\({}^{-3}\), \(\gamma Cu=91\) JK\({}^{-2}\) m\({}^{-3}\)).  
The presence of the bias circuit producing the NETP speeds up the response of a nano-TES bolometer. In fact, effective detector time constant becomes \(\tau_{1}\simeq 10\) ns for sample 1 and \(\tau_{2}\simeq 200\) ns for sample 2. The difference between the two devices arises from the values of the electrothermal parameter (\(\alpha_{1}\simeq 2742\) and \(\alpha_{2}\simeq 121\), respectively), as shown by Equation (15).  
It is interesting to note that, in principle, the nano-TES is fully tunable through the current injection. In fact, the critical temperature, i.e., the temperature giving \(R(T_{C})=R_{N}/2\), can be varied by \(I\), as shown in Figure 2d. In order to be \(I\)-tunable, the electronic temperature of the active region should be not affected by Joule overheating. This condition is fulfilled when \(I^{2}R_{N}\ll P_{e-ph,n}\), that is when electron-phonon relaxation is able to fully balance the Joule dissipation in \(A\).