## 5 Conclusions  
In this work, we studied the resonance properties of a T-shaped QTF and how they are affected by acoustic coupling with a pair of resonator tubes in a superrophone. This study was performed by analyzing the pressure-dependence of the resonance frequency and the \(Q\)-factor, both for the bare QTF and the superrophone. A theoretical model was proposed and validated by experimental results. The analysis showed that dependence of the resonance properties on air pressure is well described by the proposed model in the approximation of ideal gas law. This study allowed the authors to experimentally verify how, when the QTF is vibrating in air, the effect of the surrounding medium (modelled as an additive mass for the QTF itself) affects resonance frequency, whereas energy dissipation in air reduces \(Q\)-factor. Moreover, additive mass is a difficult parameter to be modeled for simple geometries. This work allowed the authors to evaluate it experimentally. A reduction in the additive mass was observed for a superrophone, when compared to the bare QTF. This means that tubes cause a shielding effect between QTF and air, leading to a reduction in the air effect on vibrating prongs. Comparing bare QTF results with those obtained with a superrophone, it was observed that at low pressures, the quality factor of a superrophone weakly deviates from that of the bare QTF (see Figure 6). At higher pressures, the acoustic coupling is stronger, leading to an increase in resonance frequency and a decrease in quality factor. No squeeze damping was observed when the resonator tubes were placed at a distance greater than 100 \(\upmu\)m from the QTF.  
The theoretical model developed in this work can be useful for QEPAS sensing to predict working pressure, which is usually determined experimentally and can vary from gas to gas. Indeed, the QEPAS signal strongly depends on gas pressure because both the superrophone \(Q\)-factor and the radiation-to-sound conversion efficiency (involving the acoustic wave generation within the gas) depends on gas pressure [6]. Although the dependence of the radiation-to-sound conversion efficiency on gas pressure has been well modelled [6], a theoretical description predicting the \(Q\)-factor behavior as a function of the pressure studied in this work is the fundamental step to converge for a combined theoretical model aimed at predicting the best operating pressure for QEPAS, avoiding a time-consuming experimental investigation.  
Moreover, a bare QTF has recently been used as a photodetector in tunable diode laser spectroscopy (TDLAS) setup [25], showing that the photo-induced electrical signal scales linearly with the QTF accumulation time, which is proportional to its quality factor. Thus, the \(Q\)-factor vs. pressure trend is the necessary step to predict the enhancement factor of the photo-induced signal as a function of pressure.  
Conceptualization, V.S. and P.P.; methodology, G.M., A.Z., V.M.N.P. and C.M.; validation, M.O. and M.G.; formal analysis, M.O. and M.D.G.; data curation, M.O. and G.M.; writing--original draft preparation, M.O. and P.P.; writing--review and editing, A.S., M.G. and V.S.; funding acquisition, P.P. All authors have read and agreed to the published version of the manuscript.  
P.P., V.S. and C.M. acknowledge funding from the European Union's Horizon 2020 research and innovation program under grant agreement No. 101016956 PASSEPARTOUT, in the context of the Photonics Public Private Partnership.  
Not applicable.  
This study did not report any data.  
The authors declare no conflict of interest.