# Integrated Hybrid Tweezer for Particle Trapping with Combined Optical and Acoustic Forces
## 3 Optical Force and Acoustic Force  
### Optical Force  
Light propagates in the fundamental TM mode (TM\({}_{00}\)) at 1.55 \(\upmu\)m, and its evanescent field above the optical waveguide is stronger than that of the fundamental TE mode (TE\({}_{00}\)), leading to a stronger optical force on the particle. Figure 2a shows the optical field distribution of TM\({}_{00}\) with a core size of 1 \(\upmu\)m \(\times\) 220 nm (width \(\times\) height). Calculation of the light field uses the finite-difference time-domain (FDTD) method. The boundary condition was PML.  
Figure 2b,c shows the optical force on a particle for different waveguide widths and heights. The refractive index and radius of the particle are selected to be 1.4 and 2.5 \(\upmu\)m, respectively, which are typical for yeast cells [47]. As the size of the optical waveguide gradually decreases, the evanescent field becomes stronger outside the waveguide core, and the particle feels a stronger optical force. However, when the waveguide width was below 1 \(\upmu\)m, e.g., around 0.9 \(\upmu\)m, the optical force decreased due to the hybridization of the TM\({}_{00}\) and the TE\({}_{10}\), which resulted in a smaller proportion of the optical field energy above the top surface and a smaller optical force at the same power. Therefore, to guarantee a strong evanescent field and relatively low waveguide loss, the waveguide width and height are selected to be 1 \(\upmu\)m and 220 nm, respectively.  
Figure 2d,e shows the x- and y-components of optical force along the \(x\)-axis at z = 0 \(\upmu\)m on the particles with different refractive indices (\(n\) = 1.4, 1.5, and 1.6), according to different kinds of cells ranging from 1.4 [47] to 1.54 [48]. Due to the strong x-component of the optical force, the particle was stably trapped on the top surface of the waveguide and pushed along it to the trapping region where the acoustic wave acts against the optical force.  
Figure 2f shows the x- and y-components of optical force on a particle on the top surface of the optical waveguide at z = 0 \(\upmu\)m, as the refractive index of the particle increases from 1.4 to 1.6, with the index contrast between the particle and water increased and thus a stronger optical force. For the particle with a radius of 2.5 \(\upmu\)m and a refractive index from 1.4 to 1.6, the y-component of optical force ranges from 1.37 pN/W to 34.9 pN/W, and meanwhile, the x-component of optical force was large, ensuring the stable trapping of particles on the top surface of the optical waveguide.  
Figure 2: (**a**) The optical field distribution of the TM\({}_{00}\) mode. (**b**) The x- and (**c**) y-component of optical force under different waveguide sizes. The red stars represent the selected waveguide size. (**d**) The x- and (**e**) y-components of optical force along the \(x\)-axis at z = 0 \(\upmu\)m on the particles with different refractive indices (\(n\) = 1.4, 1.5, and 1.6). (**f**) The x- and y-components of optical force on the particles with different refractive indices on the top surface of the optical waveguide at z = 0 \(\upmu\)m. The blue and red lines correspond to the x- and y-components of optical force, respectively.