*Results and Discussion*

The parameters used in the simulation are detailed in Table 2. We chose a laser emitting at 1.65 μm to target a strong methane *CH*<sup>4</sup> absorption line. Based on our numerical simulation presented in Appendix A.1, Figure A1 illustrates how *xL* and *yL* coordinates maximize the acoustic force, while *zL* = 250 μm conserves the assumption that laser light does not interfere with the cantilever.


**Table 2.** Parameters used to describe the laser source and the acoustic wave.

Due to the thermal relaxation time, the modulation frequency strongly affects the heat production rate (Equation (2)) and subsequently the acoustic force. Indeed, to allow the molecules to thermalize efficiently, the laser modulation needs to be lower than the molecules relaxation time.

Each molecule exhibits a different relaxation time. To maximize the photoacoustic force, the optimisation needs to be made with respect to one type of gas. We chose *CH*<sup>4</sup> diluted in nitrogen *N*<sup>2</sup> for which the relaxation time is equal to 11.5 μs [24]. However, the relaxation time between the molecules might differ by several orders of magnitude.

Figure 3 presents the acoustic pressure and force for *CH*<sup>4</sup> diluted in *N*2, 1% and 0.5%, respectively. Only the acoustic force depends on cantilever geometry. To maintain a fixed frequency, the cantilever length is adjusted with the following equation:

$$f\_{\rm li} = \frac{\omega\_{\rm li}}{2\pi} = \frac{a\_n^2}{2\pi\sqrt{12}} \frac{h}{L^2} \sqrt{\frac{E}{\rho\_b}}\tag{7}$$

where *fn* is the resonance frequency of a clamped-free cantilever, *ρ<sup>b</sup>* = 2330 kg/m3 is the silicon density and *E* = 130 GPa is Young's modulus for silicon in [100] direction [25].

The values of the acoustic force and pressure clearly depend on the modulation frequency as it is presented in Figure 3. For each concentration, they increase with the frequency until reaching a maximum around 20 kHz for the acoustic pressure and around 11 kHz for the acoustic force. This maximum is related to *CH*<sup>4</sup> relaxation time value. The maximum shift to lower frequency between the acoustic pressure and the acoustic force is due to the cantilever length which appears only in the acoustic force, Equation (5). According to Equation (7), the length of the cantilever is longer for lower frequencies. Therefore, the surface exposed to the acoustic pressure is larger, which subsequently increases the acoustic force at low frequencies.

The maximum value of the acoustic force is at 11 kHz. To maximize the force applied on the cantilever, this frequency is used in the following numerical simulations of the cantilever geometry (width *b*, thickness *h*, and length *L*). However, the model is adaptable to any frequency with respect to the assumptions.

**Figure 3.** Acoustic force and acoustic pressure dependency on the modulation frequency for diluted *CH*<sup>4</sup> at 1% and 0.5% in nitrogen. Cantilever width *b* = 25 μm and thickness *h* = 100 μm.

Figure 4 represents the total photoacoustic force applied on cantilever for different cantilever geometries. It shows two general trends. Firstly, the photoacoustic force increases with the width *b* and the thickness *h*. Indeed, the surface enlargement increases the energy collection from the acoustic wave. Secondly, the thickness increment increases the pressure difference between the top and bottom sides of the cantilever, which enhances the acoustic force. For a fixed cantilever frequency, the increase of the thickness causes the length increment and enlarges the total surface (Equation (7)). The results presented in Figure 4 would change while using different gases, different volume mixing ratios, or different frequencies (Figure 3). Nevertheless, the general trend would remain constant.

**Figure 4.** Acoustic force for 1% of *CH*<sup>4</sup> in *N*<sup>2</sup> as a function of width *b* and thickness *h* of the cantilever. For different thickness, the length is adjusted to maintain constant frequency: 11 kHz. This frequency was chosen to maximize acoustic force. The area with the weakest acoustic force corresponds to cantilevers with the smallest surface.

The results presented in the following sections are further taken into the calculation to get the optimized geometry of the cantilever with regard to electrical signals and signal-tonoise ratio.
