**4. Displacement**

The deflection *Wn*(*x*, *ω*) at the position *x* of the n-th mode of the beam under a photoacoustic driving force *FPA*(*ω*) is given by:

$$W\_n(\mathbf{x}, \omega) = \chi\_n(\omega) F\_{\text{PA}}(\omega) \phi\_n(\mathbf{x}) = w\_n(\omega) \phi\_n(\mathbf{x}) \tag{20}$$

where *χn*(*ω*), *FPA*(*ω*), *φn*(*x*), *wn*(*ω*) are the mechanical susceptibility, photoacoustic driving force, mode shape function normalized with *max*(*φm*(*x*)) = 1, and the maximal displacement, respectively. For the fundamental mode *wn*(*ω*) denotes the displacement amplitude at the extremity of the beam. The susceptibility represents the frequencydependent response of the cantilever under an external force and can be expressed as:

$$\chi\_n(\omega) = \frac{1}{m\_n(\omega\_n^2 - \omega^2) + i(\frac{\omega\_n\omega\_{\text{W}\_n}}{Q\_{\text{total}}})} $$

where *Qtotal*, *mn* are the total mechanical quality factor and the effective mass, respectively. The effective mass represents the part of structure actually involved in the movement.

The structure is subjected to two opposite forces: the photoacoustic force *FPA* which is periodic and drives the beam into motion and the resistance caused by damping of the structure. The damping is given by the total quality factor *Qtotal*. Both forces are presented in previous sections. The effective mass is described by:

$$m\_{\rm ll} = \rho\_{\rm b} \text{lbf} \int\_0^L \phi\_n^2(\mathbf{x}) \, d\mathbf{x} \tag{21}$$

It is related to the resistance of the resonator for motion changes. Consequently, it decreases the susceptibility and amplitude displacement of the resonator.
