# Analysis of Confinement in Dual Spherical-Tapered Ended Fiber Optical Trap
## 2 Simulation  
We set a cone angle \(\theta=120^{\circ}\), a fiber cladding refractive index of 1.446, a core refractive index of 1.452, core radius \(d_{\text{core}}=8.2\) um, clad diameter \(d_{\text{clad}}=125\) um, and the radius of cone tip \(r_{\text{tip}}=2\) um. The schematic of a spherical-tapered ended fiber is shown in Figure 1.  
Compared with flat ended fiber, the light intensity distribution of spherical-tapered ended fiber can form a more focused spot. Figure 2 shows the light intensity distribution of the output light field by using a spherical-tapered ended fiber with a tip angle of \(120^{\circ}\) and flat ended fibers. The mode field emitted from the fiber has the characteristics of a small beam divergence angle, which lays the foundation for the high stiffness of the dual spherical-tapered ended fiber optical trap. Figure 1: Schematic of (**a**) the core (**b**) the clad of a spherical-tapered ended fiber. Figure 2: Simulation on the light intensity distribution with (**a**) spherical-tapered ended fiber (**b**) flat ended fibers in z-axis of the output light field in simulation. In this paper, Finite-difference time-domain (FDTD) was used to calculate the optical trapping force of microspheres in a dual spherical-tapered ended fiber optical trap and a dual flat ended fibers optical trap. FDTD is a numerical calculation method of electromagnetic field, first put forward by [PERSON] in 1966 [20]. It is based on the differential form of the Maxwell curl equation, and the computational domain is divided into grids containing electric and magnetic fields. This method has a wide range of applications, especially in the field of physics. When the electromagnetic field acts on the particles and changes the momentum of the particles, the momentum of the electromagnetic field also changes to some extent. Among the changes, the Lorentz force on the charge is expressed as:  
\[\mathbf{f}=\rho\mathbf{E}+\mathbf{J}\times\mathbf{B} \tag{1}\]  
where \(\mathbf{f}\) is the density of Lorentz force, \(\rho\) is the charge density, \(\mathbf{J}\) is current density, \(\mathbf{E}\) is electric field intensity, and \(\mathbf{B}\) is magnetic induction intensity. For the steady-state distribution of the electromagnetic field, the time change rate of the total amount of motion of the electromagnetic field is zero. The total momentum of the charging system in the region changes. The total force of the electromagnetic field in the volume is equal to the area of [PERSON]'s stress tension \(\mathbf{\tilde{T}}\) on the interface. By substituting Maxwell equations into Equation (1), [PERSON]'s stress tension \(\mathbf{\tilde{T}}\) and the total force \(\mathbf{F}\) can be obtained, expressed as:  
\[\mathbf{\tilde{T}}=-\varepsilon_{0}\mathbf{E}\mathbf{E}-\mu_{0}\mathbf{H} \mathbf{H}+\frac{1}{2}(\varepsilon_{0}E^{2}+\frac{1}{\mu_{0}}H^{2}) \tag{2}\]  
\[F=\int\limits_{V}\mathbf{f}dv=-\int\limits_{V}\
abla\cdot\mathbf{\tilde{T}}dv= -\oint\limits_{S}dS\cdot\mathbf{\tilde{T}} \tag{3}\]  
where \(\varepsilon_{0}\) is vacuum capacitance, \(\mu_{0}\) is vacuum magnetic permeability, and \(\mathbf{H}\) is magnetic field intensity. Considering the continuous wave of the two-dimensional TM model frequency, there are three field (\(H_{y}\), \(E_{x}\), \(E_{z}\)) components, which can be used to derive the force [21]:  
\[F_{z}=\frac{1}{2}\oint\limits_{S}[-\frac{1}{2}(\mu\mu_{0}|H_{y}|^{2}+ \varepsilon_{0}\varepsilon_{1}|E_{x}|^{2}-\varepsilon_{0}\varepsilon_{1}|E_{ z}|^{2})dx+\varepsilon_{0}\varepsilon_{1}\mathbf{Re}(E_{x}E_{z}^{*})dz] \tag{4}\]  
\[F_{x}=\frac{1}{2}\oint\limits_{S}[-\frac{1}{2}(\mu\mu_{0}|H_{y}|^{2}+ \varepsilon_{0}\varepsilon_{1}|E_{z}|^{2}-\varepsilon_{0}\varepsilon_{1}|E_{ x}|^{2})dz+\varepsilon_{0}\varepsilon_{1}\mathbf{Re}(E_{x}E_{z}^{*})dz] \tag{5}\]  
where \(S\) is the outer boundary of the target space in the two-dimensional XOZ plane, \(\varepsilon_{1}\) is the relative capacitance of the medium, and \(\mu\) is the relative magnetic permeability of the medium.