**8. Conclusions**

The presented sensor combines multiple physical phenomena and, therefore, its optimization is not straightforward. In this paper, we discussed multiple parameters which affect the sensor performances and we analyzed their contributions. Our model takes into account some optimization for (1) the acoustic force; (2) the system damping; (3) the mechanical displacement under photoacoustic force considering damping mechanism; (4) the electrical signal under capacitive transduction mechanism, and (5) the signal-tonoise ratio. This model provides a method to retrieve the optimized cantilever geometry depending on the required size of the sensor and other restrictions which might be imposed by the fabrication process or the operating conditions of the sensor.

(1) Our model includes the gas relaxation time. We proposed a cantilever geometry optimized for photoacoustic gas sensor with a capacitive transduction mechanism in the context of *CH*<sup>4</sup> absorption with a concentration of 1%. For the same temperature and pressure conditions, only the amplitude of the photoacoustic force will change with the concentration. For this reason, the geometric parameters of the cantilever will remain the same for all the concentrations. For different gasses, only the optimal laser modulation frequency and the absolute acoustic pressure will change; the trends presented in other sections will remain the same. Based on our model, the cantilever optimal geometry can be recalculated for any other gas by taking into account the relaxation time and absorption coefficient. However, the optimal sensor should be created for one specific gas.

(2) The study of the different damping mechanisms show that viscous damping, and particularly squeeze film effect, is fundamental. The impact of squeeze film effect is

visible in Figure 11 for *d* values up to 200 μm. For *d* above this value, the acoustic damping becomes the limiting damping mechanism.

(3)–(5) To improve the SNR, one can increase the gap *d* and/or the surface collecting the photoacoustic energy. However, increasing the gap *d* between the two electrodes will decrease the output signal amplitude. Depending on the read-out circuit and the parasitic capacitance it is possible to obtain a high SNR while maintaining a sufficient output signal. For example, with a gap of *d* = 10 μm which can be realized on a silicon-on-oxide (SOI) wafer, the signal-to-noise ratio will reach 150 and the amplitude of the output signal will be around 0.9 μ*V*. For different values of the gap, one can use Figure 11.

Finally, despite a complex multiphysical problem, we have proposed a complete analytic model able to find the optimum geometric parameters of a cantilever for photoacoustic sensing with capacitive transduction. Beyond the simple optimization, this study is intended to provide all the tools allowing understanding of all the mechanisms of this complex problem. The variety of these physical mechanisms, often incompatible with each other during a finite element simulation, gives all its strength to our analytical approach to the problem. This paper demonstrates that a simple cantilever with capacitive transduction mechanism will not reach the same performance in terms of limit of detection as the best QEPAS technique or best standard photoacoustic technique using a microphone. However, besides being the optimization tool, this work is intended to be an educational tool allowing a mechanical resonator to be developed with more complex geometry and other transduction mechanisms. This study paves the way to develop new mechanical resonators for compact, integrated, and sensitive gas sensors.

**Author Contributions:** W.T. was in charge of the conceptualization, methodology and formal analysis under the supervision of M.B. She was assisted by N.M., R.R., and D.A. M.B. was in charge of the Python code writing. The article was written by W.T. and M.B., reviewed and edited by M.B. and A.V. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was financially supported by the French Ministry of Defense (DGA-MRIS), Region Languedoc Roussillon, European Community (FEDER) and Renatech, Agence Nationale de la Recherche (MULTIPAS Project No. [ANR-16-CE04-0012], and NOMADE Project No. [ANR-18- CE04-0002-01]).

**Data Availability Statement:** Data and Python code developed for this article is available by contacting the correspoding author.

**Conflicts of Interest:** The authors declare no conflict of interest.

**Sample Availability:** The Python code developed for this article is available by contacting the correspoding author.
