# Integrated Hybrid Tweezer for Particle Trapping with Combined Optical and Acoustic Forces
## 3 Optical Force and Acoustic Force
### Acoustic Force  
To efficiently excite acoustic waves by an IDT, we first evaluate the electromechanical coupling coefficient, which is an indicator of the efficiency with which a piezoelectric material converts electrical energy to mechanical energy. The electromechanical coupling coefficient can be obtained by comparing the acoustic velocity in the presence and absence of a perfect conductor on the piezoelectric material surface [49]. The highest electromechanical coupling coefficient occurs when the propagation direction of an acoustic wave rotates 64\({}^{\circ}\) from the \(y\)-axis. The thickness of LN was 20 \(\upmu\)m, which was optimized and is explained later. Calculation of the acoustic field uses finite element method (FEM). The boundary condition of the solid and acoustic field was PML to absorb acoustic waves. For the acoustic radiation force, the continuum equation of motion and strain-mechanical displacement relations are used to calculate the vibration displacement of the solid [50]. Simultaneously, based on the structure-acoustic interactive simulation [51], the continuity equation, the Navier-Stokes equation, and the energy equation were used to calculate the first-order pressures and first-order velocity of the acoustic field. For the Stokes drag force, the boundary condition of the solid-liquid interface was the no-slip boundary condition. Due to the equality of inflow rate and outflow rate, the total flow rate was set to 0. The second-order continuity equation and the Navier-Stokes equation were used to calculate the time-averaged second-order velocity of water. The time-averaged second-order velocity was obtained based on the first-order velocity by adding the weak contributions to the computational domain and boundaries [45]. Due to the relatively large aperture of the IDT, the acoustic wave was invariant in the \(z\)-direction. Therefore, two-dimensional simulations were performed. The displacement field is shown in Figure 3a. The enlarged view of the acoustic field is shown in Figure 3b. The red dotted box in Figure 3a highlights the position of the IDT where the acoustic wave was excited. Acoustic wave energy was well confined in the LN layer before leaving Part A. Due to the lack of silicon (the upper cladding layer) in Part B, a part of the acoustic wave propagates into water, as indicated by the black dashed box. Figure 3: (**a**) The displacement field distribution of solid vibration and pressure field distribution of the acoustic wave propagating into water, and (**b**) the enlarged view of the acoustic field in the black dotted box. (**c**) The distribution of time-averaged second-order velocity amplitude in the black dotted box region. The size and direction of the black arrows are used to indicate the amplitude and direction of the time-averaged second-order velocity. When the acoustic wave radiates into the water, its wavefront has a slant angle with respect to the waveguide surface. The acoustic waves propagating from the top surface of silicon nitride and the front facet of silicon at the position of \(\mathrm{y}=-\mathrm{d}\) (see Figure 1b) interfere with each other, causing the pressure field disturbance on the top of the silicon optical waveguide. The optical waveguide appears at a distance of 20 \(\mathrm{\SIUnitSymbolMicro m}\) from the facet of the upper cladding silicon. Since the waveguide height (220 nm) is much smaller than the acoustic wavelength (\(\sim\)7.5 \(\mathrm{\SIUnitSymbolMicro m}\)), the optical waveguide has negligible effect on the pressure field. The black dotted box specifies the region where particles are trapped, where the amplitude and direction of the time-averaged second-order velocity were calculated, as shown in Figure 3c. The size and direction of the black arrows are used to indicate the amplitude and direction of the time-averaged second-order velocity. Optimizing the thickness of each material layer in the device helps achieve the best performance.