# Analysis of Confinement in Dual Spherical-Tapered Ended Fiber Optical Trap
## 3 Experiment  
The optical layout for the dual spherical-tapered ended fiber optical trap is shown in Figure 5. A laser with a wavelength of 1436 nm was divided into two trapping beams by a 50/50 coupler. In each optical path, a variable optical attenuator (VOA) was used to ensure that the two beams incident on the trap had the same power. The trapping beams were ultimately transmitted to the dual spherical-tapered ended fiber optical trap chip to capture microspheres. The power input into the optical trap was precisely monitored by two power detectors (LD) in real-time. The phenomena were imaged by using a CCD camera. The schematic of the dual spherical-tapered ended fiber optical trap chip is shown in Figure 6. The particle could be trapped by counter-propagating beams emanating from the two spherical-tapered ended fibers. The diameter and depth of the columniform capturing room were 1 mm and 500 \(\upmu\)m, respectively. On the chip, the dual spherical-tapered ended fiber optical trap required high coupling accuracy of the two fibers. Therefore, the two fibers were fixed to V-grooves to increase coupling accuracy and stability. In the experiment, a polystyrene microsphere with a radius of 3 \(\upmu\)m was trapped using the simulation's configuration, see images in Figure 7. A polystyrene microsphere was trapped near the center of the dual spherical-tapered ended fiber trap as expected. The microsphere could be captured for a long time in the dual spherical-tapered ended fiber optical trap. Figure 4: The force on the microsphere along the (**a**) z-axis (**b**) x-axis on the optical trap chip. Figure 5: The experimental setup of dual-beam optical trap system. The particle trapping videos were recorded, and a particle tracking algorithm was used to extract the time-dependent particle positions from trapping videos. The displacement of the microspheres in the xz plane was collected from each video frame. The corresponding experimental results on the dual spherical-tapered ended fiber optical trap are shown in Figure 8. The axial and lateral displacements of the captured microspheres changed with time, shown, respectively, in Figure 8a,b. The probability distribution on the x-axis and the z-axis are shown in Figure 8c,d. The thermal displacement of the captured microsphere in the optical trap conformed to the Boltzmann distribution. The probability distribution of the x-axis displacement of the microsphere can be expressed as follows [25]:  
\[p(x)=Ce^{-\frac{E(x)}{k_{B}T}} \tag{8}\]  
where \(E(x)\) is the potential energy, \(k_{B}T\) is thermal energy (\(T\) is the temperature, \(k_{B}\) is the Boltzmann constant), and \(C\) is the normalizing constant of the probability distribution. According to the energy equipartition method, the stiffness of the optical trap can be directly expressed as [16]:  
\[k_{x}=\frac{\kappa_{B}\cdot T}{<x^{2}>} \tag{9}\]  
Figure 6: The schematic of dual spherical-tapered ended fiber optical trap chip. The small picture is the physical picture taken by the actual dual spherical-tapered ended fiber optical trap chip. Figure 7: Photomicrographs of the dual spherical-tapered ended fiber optical trap with 3 \(\upmu\)m radius polystyrene microsphere trapped near the center of the trap. where \(<\)x\({}^{2}\)\(>\) is the mean square deviation of the displacement, \(\kappa_{B}\) is the Boltzmann constant. The stiffness was calculated according to Equation (9). The stiffness along the x-axis was 1.76 pN/\(\upmu\)m, and along the z-axis it was 0.99 pN/\(\upmu\)m in the dual spherical-tapered ended fiber optical trap. Figure 9 shows the experimental results for the dual flat ended fibers optical trap. The stiffness along the x-axis was 1.68 pN/\(\upmu\)m, and along the z-axis it was 0.44 pN/\(\upmu\)m in the dual spherical-tapered ended fiber optical trap. Compared with the dual flat ended fibers optical trap, the dual spherical-tapered ended fiber optical trap effectively enhanced the stiffness of the dual-beam optical trap, which was consistent with the results of the simulation.