# Analysis of Confinement in Dual Spherical-Tapered Ended Fiber Optical Trap
## 2 Simulation
Due to the symmetry of the optical trap, the force distribution of the microspheres along the y-axis is similar to the x-axis. Therefore, only the x-axis and z-axis directions were analyzed in this paper. The z-axis direction is defined as the axial direction, and the x-axis direction is defined as the lateral direction. Near the center of the optical trap, the optical trapping force \(F_{x}\) and \(F_{z}\) acting on particles is a recovery force, which is proportional to the displacement of the microsphere offset from the center of the optical tweezers, expressed as:  
\[F_{x}=-k_{x}\cdot x \tag{6}\]  
\[F_{z}=-k_{z}\cdot z \tag{7}\]  
where \(k_{x}\) and \(k_{z}\) are the trapping stiffness on x-axis and z-axis, and \(x\) and \(z\) are the displacement deviating from the capture location on x-axis and z-axis. Additionally, a spherical particle has no torque in a dual-beam optical trap. When the two optical fibers experience misalignment [22; 23], the captured microsphere realizes optical rotation or optical orbital rotation. In the simulation, we set a polystyrene microsphere (refractive index \(n_{2}=1.59\), radius \(r_{0}=3\)\(\upmu\)m) to be captured in water (refractive index \(n_{1}=1.33\)). The output power of each fiber in the optical trap was \(P_{\text{L}}=P_{\text{R}}=18\) mW. Figure 3 shows the difference of force which acted on the microsphere between dual spherical-tapered ended fiber optical trap and flat ended dual-fiber optical trap along \(x\) and \(z\) axes. The distance between the two fibers was set as 25 \(\upmu\)m. According to Equations (6) and (7), in the spherical-tapered ended fiber optical trap, the trapping stiffness was 0.43 pN/\(\upmu\)m on the z-axis and 11.46 pN/\(\upmu\)m on the x-axis. In the flat ended fibers optical trap, the trapping stiffness was 0.15 pN/\(\upmu\)m on the z-axis and 3.61 pN/\(\upmu\)m on the x-axis. This showed that the polystyrene microsphere, with a radius of 3 \(\upmu\)m, could be captured more stably in the dual spherical-tapered ended fiber optical trap. Both the axial and transverse optical trap stiffnesses of the dual spherical-tapered ended fiber optical trap were greater than those of the dual flat ended fibers optical trap. The distance between two fiber ends is a key factor contributing to the trapping force. Figure 4 shows the trapping force acting on the microsphere for different distances between the two fiber ends. Due to the focus of the output optical field of the spherical-tapered ended fiber being near 12.5 \(\upmu\)m from the fiber end, the distance between the two fibers was set as 25 \(\upmu\)m. In addition, when the distance between the two ends was too close, the effective capture area became too small, so it would make capturing difficult in the experiment. Therefore, we discuss the impact of the distance between two ends on the optical trap from 25 \(\upmu\)m or greater distance. With the increase of the distance between the two fiber ends, the stiffness of the lateral optical trap changed little, and the stiffness of the axial optical trap decreased. When two points with a force of 0 and a negative slope of the tangent were found in the diagram, the microsphere could not be captured at the center of the optical trap. Instead, a bistable result appeared in the optical trap. The microsphere had two capture points symmetrically about the central position in the optical trap. This discovery could be applied to the study of bistable structures in the optical trap [24]. Figure 1:0 The simulated optical force on the microsphere along the (**a**) z-axis (**b**) x-axis on the optical trap chip. The distance between the two fibers is 25 \(\upmu\)m.