# Selecting a Proper Microsphere to Combine Optical Trapping with microsphere-Assisted Microscopy
## 2 The Selection of Microsphere Types
### Optical Trapping Simulations  
In simulations, we consider the trapping properties of index-different microspheres with varied diameters in single-beam optical tweezers. In stable optical tweezers, the gradient force, pointing to the trapping beam focus, is necessary to overcome the scattering force, which pushes the microsphere in the direction of beam propagation [22]. The T-matrix method is often used to calculate the forces acting on a particle illuminated by a focused beam [32], and an optical tweezers toolbox [33] has been developed to investigate the forces on particles in optical trapping. The typical Gaussian beam is commonly used as the trapping beam and it has a symmetric scattering profile for the radial displacement away from the beam axis for the microsphere. A scheme of optical trapping is shown in Figure 1a and the typical axial and radial force acting on a PS microsphere as a function of sphere position are shown in Figure 1b,c. The maximum reverse axial force A is important because it characterizes the strength of the trap. We focus on the maximum reverse axial force, which determines whether or not the microsphere can be trapped. Next, we use the optical tweezers toolbox to calculate the forces as a function of two major parameters for optical trapping: the refractive index of the microsphere and the size of the microsphere. The results are useful when choosing a proper microsphere for the combination of optical trapping with microsphere-assisted microscopy. The dependence of the trap strength (the maximum reverse axial force) on refractive index and microsphere diameter is simulated to form the optical trapping landscapes, as shown in Figure 2. The range of refractive index \(n\) is 1.4-2.0. The diameter of microsphere in simulations is in a range of 1-15 um for an enough field of view (FOV) in microsphere-assisted microscopy, also for saving computational time. The numerical aperture (NA) of the objective used to focus the 1064 laser beam in water is set to 0.9 which is relatively low compared to that in the normal optical tweezers (NA 1.25 or 1.3), but such a NA is high enough in microsphere-assisted microscopy  
Figure 1: Force on a polystyrene (PS) sphere in a Gaussian beam trap. The PS sphere has a radius of \(1\lambda\), and has a relative refractive index of \(n_{rel}=1.59/1.33=1.20\), trapped at 1064 nm, using an objective with numerical aperture of 1.02. (**a**) A scheme of optical trapping, in which the sphere in a focused Gaussian beam is dragged toward a region of high intensity. (**b**) The axial trapping efficiency as a function of axial displacement and (**c**) the transverse trapping efficiency as a function of transverse displacement from the equilibrium point. The maximum reverse axial force A is shown in (**b**) and it characterizes the strength of the trap. (normal NA 0.5-0.9). From the trapping landscapes, we can conclude that there is an upper limit of refractive index 1.8 for the used NA of 0.9 approximately, and a higher NA will increase the upper limit of the refractive index [34]. Microspheres with \(n<1.6\) can be trapped for diameters from 1 um into 15 um. For the high refractive indices (\(n>1.8\)), we can see that the microspheres cannot be trapped. With refractive indices between 1.6 and 1.8, most microspheres can be trapped, and the trapping efficiency decreases as the \(n\) increases. The trapping landscapes guide us when choosing a proper microsphere for the combination of optical tweezers and microscopy-assisted microscopy. For the typical microspheres used in microsphere-assisted microscopy like SiO\({}_{2}\) (\(n\sim 1.46\)), PS (\(n\sim 1.59\)), MF (\(n\sim 1.68\)) and BTG (\(n\sim 1.9\)) spheres, these index-different spheres have different trapping performances. From the trapping landscapes with the used 0.9 NA, we know that the SiO\({}_{2}\) and PS microsphere can be trapped stably for the \(n<1.6\). For the MF sphere, except for some small spheres of particular diameters, we can also trap the MF sphere with single-beam optical tweezers.