# Integrated Hybrid Tweezer for Particle Trapping with Combined Optical and Acoustic Forces
## 3 Optical Force and Acoustic Force
### Acoustic Force
First, to create an effective acoustic waveguide, the upper silicon cladding was set to be thick enough to confine acoustic waves, e.g., 15 \(\mathrm{\SIUnitSymbolMicro m}\), after taking into account the challenges of depositing a thick film. Second, to determine the thickness of the silicon nitride layer, which acts as both a cladding of the LN acoustic waveguide and controlling of acoustic radiation, we calculated the first-order pressure and the time-averaged second-order velocity to represent the acoustic radiation force and Stokes drag force based on Equations (3) and (5). By fixing the thickness of LN at 20 \(\mathrm{\SIUnitSymbolMicro m}\) and varying the thickness of silicon nitride, we obtained the amplitude of first-order pressures and the y-component of time-averaged second-order velocities along the \(y\)-axis at \(\mathrm{x}=2.5\ \mathrm{\SIUnitSymbolMicro m}\), as shown in Figure 4a,b. Since the upper silicon cladding ends near Part B, the acoustic wave in the LN waveguide was perturbed and radiated into the water from the top surface of the silicon nitride. As the silicon nitride became thicker, the pressure exhibited a weaker fluctuation, and its maximum amplitude decreased. The local oscillations of the first-order pressure stemmed from the interference between the acoustic waves radiating from the top surface in Part B and the facet at the position of \(\mathrm{y}=-\mathrm{d}\), as shown in Figure 4a. For thicker silicon nitride, the variation in the y-component of time-averaged second-order velocity is smaller, leading to a more uniform Stokes drag force. The amplitudes of first-order pressure and the y-component of time-averaged second-order velocity become smaller, as shown in Figure 4a,b. Even though the total acoustic force was relatively weaker, it can still be balanced with the optical force by adjusting the optical power or voltage because the total acoustic force on the particle was generally stronger than the optical force. Therefore, a thicker silicon nitride layer is more desirable in terms of acoustic performance. Considering the difficulty of depositing thick silicon nitride, the designed thickness of silicon nitride was 15 \(\mathrm{\SIUnitSymbolMicro m}\). Third, with the 15-\(\mathrm{\SIUnitSymbolMicro m}\)-thick silicon nitride, the amplitudes of first-order pressure and the y-component of time-averaged second-order velocity become smaller for thicker LN, as shown in Figure 4c,d because more acoustic energy is confined in LN. A thickness of 20 \(\mathrm{\SIUnitSymbolMicro m}\) is chosen for the relatively uniform distribution of the y-component of time-averaged second-order velocity along the \(y\)-axis and the higher first-order pressure. The acoustic force on the particle is calculated by assuming a particle density of 1110 kg/m\({}^{3}\) and a speed of sound of 1600 m/s, with reference to yeast cells [52]. Due to the similar material properties of cells and water, releasing the cells in water has negligible influence on the acoustic field distribution. Figure 4e,f show the acoustic radiation force, the Stokes drag force, and the total acoustic force on the particle along the \(y\)-axis with a voltage of 1 V applied to the IDT. Because of the oscillation of the acoustic field above the optical waveguide, the particle feels a periodically changing acoustic radiation force. The Stokes drag force is associated with the time-averaged second-order velocity. The total acoustic force can be obtained by summing the acoustic radiation force and the Stokes drag force. The thickness of the aluminum electrode and the upper silicon cladding were fixed at 200 nm and 15 \(\upmu\)m. After optimization, the thicknesses of the silicon waveguide, silicon nitride, and lithium niobate layers were 220 nm, 15 \(\upmu\)m, and 20 \(\upmu\)m, respectively, to achieve optimum performance. The thickness of the silicon substrate exceeded one hundred micrometers and had negligible effects on the optimized results. The optimized structural parameters are presented in Table 3.