*Results and Discussion*

Figure 7 has been calculated with the mathematical expression of the previous section (Equation (4)). In this section, some approximations will be proposed to explain the shape of the graph.

The mechanical displacement *wn*(*ωn*) presented in Figure 7 is the product between the photoacoustic force and the mechanical susceptibility *χn*. At the resonance frequency 2*π fn* = *ωn*, the susceptibility can be approximated as *χ<sup>n</sup>* = *Q*/(*mnω*<sup>2</sup> *<sup>n</sup>*) and the displacement as *wn*(*ωn*) = *QtotalFPA*/(*ω*<sup>2</sup> *nmn*).

For the fundamental mode of the cantilever, we can write the acoustic force as *FPA* 0.39*bL*Δ*p*(*b*, *h*), where Δ*p*(*b*, *h*) is the pressure difference between the top and the back of the cantilever. The function Δ*p*(*b*, *h*) increases with the thickness *h* and in this approximation remains quite constant for various widths *b*. The effective mass of the cantilever fundamental mode is *mn* 0.25*ρbhbL* (i.e., 25% of the total mass). The displacement can then be approximated by:

$$w\_n(\omega\_n) = \frac{Q\_{total} F\_{PA}}{\omega\_n^2 m\_n} \simeq 1.56 \frac{Q\_{total}}{\omega\_n^2} \frac{\Delta p(b, h)}{\rho\_b h} \tag{22}$$

The simulations show that the fraction <sup>Δ</sup>*p*(*b*,*h*) *<sup>ρ</sup>bh* remains quite constant for different widths and is inversely proportional to the cantilever thickness: <sup>Δ</sup>*p*(*b*,*h*) *<sup>ρ</sup>bh* <sup>∝</sup> <sup>1</sup> *<sup>h</sup>* . Due to its homogeneity, this term is called "acceleration" in Figure 7. It is reducing the maximal displacement when the thickness increases. This region corresponds to weak acoustic force or/and heavy effective masses. Counterintuitively, the simplified Equation (22) shows that increasing the cantilever surface to collect more photoacoustic energy increases the effective mass, resulting in constant mechanical displacement. Indeed, a simplification by the surface *bL* appears between the term of the acoustic force and the effective mass.

**Figure 7.** Total displacement versus width and thickness for a cantilever with fundamental resonance frequency equal to 11 kHz (**a**) and 60 kHz (**b**) for a gap between support and cantilever equal to *d* = 10 μm.

The other limitations come from the viscous damping introduced by the *Qtotal* term, and are similar to those shown in the Figure 6.

Figure 7 shows a large difference in displacement amplitude between a modulation frequency of 11 kHz and 60 kHz. Despite the improvement of quality factor with increasing frequency, the acoustic force significantly drops at high frequency (Figure 3) and the susceptibility, as it is inversely proportional to *ω*<sup>2</sup> *<sup>n</sup>*. The results then show that photoacoustic force and susceptibility gain more importance with the change of frequency.
