# Capture Dynamics of Dielectric Microparticles in Hollow-Core-Fiber-Based Optical Traps

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## Abstract

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## 1. Introduction

## 2. Optical Forces in Dual-Beam Trap in Front of HCF Endface

_{0,+z}= 2P

_{+}/($\pi {w}_{0,+z}^{2}$), where P

_{+}is the launched power through lens 2, w

_{0,+z}is the corresponding beam waist radius. For the beam emitted from HCF (−z direction), its waist diameter is related to the fiber core diameter D as w

_{0,−z}= 0.345D [20], giving a corresponding light intensity I

_{0,−z}= 5.33P

_{−}/D

^{2}, where P

_{-}is the transmitted power from HCF. In this situation, the configuration forms an asymmetric dual-beam trap due to the difference in the counter-propagating beam divergence. Under Rayleigh approximation, the scattering forces along the z axis generated from the +z and −z beam in front of the HCF endface read as:

_{p}is the particle diameter, n

_{p}and n

_{m}is the refractive index of the particle and the surrounding medium, respectively, and m = n

_{p}/n

_{m}. The correction factor α

_{z}takes into account the deviation of the Rayleigh approximation from the explicit solution of Maxwell equations when the particle diameter is approaching to wavelength λ. To estimate the correction factor, T-matrix method was also applied to calculate the optical forces, which is developed on the basis of generalized Lorentz–Mie theory to accurately solve the electromagnetic scattering problem [21]. Figure 2a–d compares the calculated axial optical force using Rayleigh approximation (F

_{RL}, z, red) and T-matrix approach (F

_{TM}, z, blue) for silica particles with diameters d

_{p}= 100 nm, 200 nm, 300 nm, and 400 nm. It can be seen the Rayleigh model agrees well with the T-matrix method when d

_{p}~λ/10, and the deviation grows as the particle size increases. Figure 2e,f (red dots) plots, respectively, the correction factor α

_{z}= F

_{RL,z}/F

_{TM,z}and α

_{x}= F

_{RL,x}/F

_{TM,x}, thus the ratio between the axial and radial optical forces, calculated using Rayleigh approximation and T-matrix method. It is found that α

_{z}more strongly depends on d

_{p}, and α

_{x}is close to unity for the considered range of d

_{p}. Given the particle refractive index, empirical equations relating the optical forces obtained from the Rayleigh approximation to the T-matrix method can be established for different particle diameters. The blue lines in Figure 2e,f are the corresponding polynomial fit to obtain the empirical equations. In the case of axial force, the fitting equation reads as:

_{grad,x}is the transverse stiffness of the trap.

## 3. Equilibrium Trapping Positions in Front of the HCF Endface

_{eq}) in front of the HCF endface can be predicted. Figure 3a plots the calculated F

_{scat,+z}, F

_{scat,−z}, F

_{scat,z}, F

_{grad,z}and F

_{z}= F

_{scat,z}+ F

_{grad,z}over z, where z = 0 denotes the position of fiber endface. The simulation parameters are set as follows: n

_{p}= 1.45 (silica), n

_{m}= 1 (air), d

_{p}= 0.4 μm, the numerical aperture of the focusing lens (NA

_{lens}) and the fundamental mode of HCF (NA

_{fiber}= λ/πw

_{0,−z}) is 0.08 and 0.1, respectively, D = 9.8 μm, λ = 1064 nm, and P

_{+}= P

_{−}= 100 mW. It can be seen that for d

_{p}= 0.4 μm silica particle, under the considered parameters, F

_{grad,z}is about one-tenth of F

_{z}, the contribution of which is negligible. The blue dot marks the stable axial trapping point in which the F

_{z}= 0 and the slope $\mathit{\partial}{F}_{z}/\mathit{\partial}z<0$. Figure 3b plots the simulation curve of F

_{z}over z under different ratios of NA

_{lens}/NA

_{fiber}. It can be seen that when NA

_{lens}/NA

_{fiber}is less than 1, stable trapping points (blue dots) are present and the positions of which become closer to the fiber endface when NA

_{lens}/NA

_{fiber}is approaching unity. In contrast, when NA

_{lens}/NA

_{fiber}is greater than 1, the slope $\partial {F}_{z}/\partial z$ is positive when F

_{z}= 0 (red dot), indicating that it is an unstable trapping position. Thus, when the counterpropagating beams have equal power, it is necessary to satisfy NA

_{lens}< NA

_{fiber}in order to achieve stable particle trapping in front of the HCF endface.

_{grad,x}with respect to particle gravity m

_{p}g (m

_{p}is the particle mass, g is the acceleration of gravity) at the axial stable trapping point for different particle diameters. It is interesting to note that F

_{grad,x}/m

_{p}g barely changes for particles of different sizes, and the maximum F

_{grad,x}can be two orders or magnitude stronger than m

_{p}g. This indicates that for the considered particle diameters, optical gradient forces are able to form a deep enough trapping potential to overcome the particle random Brownian motion and thus achieve a stable capture of particles in front of HCF. Figure 3d plots the corresponding transverse trapping stiffness k

_{grad}

_{,x}under different trapping powers (P = P

_{+}+ P

_{−}= 2P

_{+}) for silica particle with d

_{p}= 0.4 μm, showing that k

_{grad}

_{,x}increases linearly with the trapping power, as expected from the theory of optical tweezers.

_{eq}versus particle diameter d

_{p}and NA

_{lens}/NA

_{fiber}when NA

_{fiber}= 0.05 (D = 19.6 μm). It can be observed that the NA

_{lens}has a more significant impact on the value of z

_{eq}than particle diameter d

_{p}. As d

_{p}increases, z

_{eq}decreases, thus the stable trapping point becomes farther from the HCF endface, the dependence of z

_{eq}on d

_{p}also becomes weaker. This is because for smaller particles, the amplitude of axial scattering (F

_{scat,z}) and gradient forces (F

_{grad,z}) acted on the particle are comparable, both of which can affect the value of z

_{eq}. Since the beams propagating along +z and −z directions have distinct divergence, their difference in F

_{grad,z}can introduce a strong size-dependent effect. As the particle size increases, F

_{scat,z}turns to be much greater F

_{grad,z}(see Figure 4c), the impact of F

_{grad,z}on z

_{eq}diminishes. In this situation, the value of z

_{eq}mainly depends on the balance between P+ and P−, which is independent on d

_{p}. Figure 4b displays the results when NA

_{fiber}= 0.1 (D = 9.8 μm). Compared with Figure 4a, it can be seen that the stable equilibrium trapping position becomes closer to the HCF endface when NA

_{fiber}is larger (thus for a smaller core diameter).

_{eq}when NA

_{lens}= 0.08 and NA

_{lens}= 0.1, corresponding to the situation marked by the black-dashed line in Figure 4b. It can be observed that the stable trapping point moves farther from the endface when d

_{p}increases, and z

_{eq}varies sharply when d

_{p}< 0.3 μm as explained in the previous paragraph. When NA

_{lens}is varied, as shown in Figure 4d, the smaller the fiber core diameter, the closer the axial stable equilibrium point to the endface of the fiber. This is because the beam exiting the HCF has a larger divergence for a smaller core diameter, The corresponding outgoing beam intensity and the optical forces drop faster along z axis, causing the equilibrium trapping point to move towards the fiber endface.

## 4. Particle Capture Dynamics in Front of the HCF Endface

_{p}dz/dt and 3πηd

_{p}dx/dt are the corresponding viscous drag force acting on the particle in the z and x directions, η is the viscosity of air. The Runge–Kutta method was applied for numerically solving Equations (9) and (10), from which the particle trajectories and speed during the capture process can be explicitly obtained. The absolute and relative error tolerances of the iteration operation were both set as 10

^{−6}, which have been validated with the simulation convergence. The initial condition of the particle motion is given by the starting position (x

_{0}, z

_{0}) and the initial velocity (v

_{0x}, v

_{0z}). For convenience, the initial velocity is expressed in terms of its amplitude v

_{0}and angle of incident θ

_{0}= arctan (−v

_{0x}/v

_{0z}); thus, θ denotes the angle between the direction of the particle’s initial velocity with respective to +z axis (see inset of Figure 5b). The simulation parameters were set as x

_{0}= 10 μm, z

_{0}= −100 μm, n

_{p}= 1.45, n

_{m}= 1, d

_{p}= 0.4 μm, m

_{p}= 0.87 fg, P

_{+}= P

_{−}= 10 mW, NA

_{fiber}= 0.1, and NA

_{lens}= 0.08.

_{p}= 0.44 μm and 0.4 μm. It can be seen that larger particles are more difficult to be captured in the given v

_{0}-θ

_{0}parameter space due to their greater inertia. It is interesting to note that if the particles could be captured at one specific angle of incident (e.g., 0.51π when d

_{p}= 0.44 μm), it would be captured at any other incident angles. Figure 5a could be conveniently used to determine the suitable parameter space so as to increase the particle capture success rate.

_{p}= 0.4 μm. It can be seen that for cases 1 and 3, starting from the initial position (red dot), the optical gradient force is capable of reducing v

_{0x}, guiding the particle towards the fiber endface and finally to the equilibrium position (red cross) after several seconds. For case 2, since the particle has a relatively high initial velocity, it quickly passes through the beam with a short duration of light–particle interaction time. The particle therefore cannot be captured by the beam since the particle momentum is too large to balance and the particle’s trajectory is barely changed by the optical forces. When the particle speed is slow enough to be captured by the beam (cases 1 and 3), its trajectory can be rapidly varied by the optical forces within a short period of time, as shown in the insets of Figure 5b.

## 5. Discussions

_{p}= 0.4 μm, P

_{+}= P

_{−}= 10 mW, θ

_{0}= π/4, v

_{0}= 11.1 m/s. The red, blue, and cyan dots represent the situation when the initial position is (10 μm, −100 μm), (12 μm, −120 μm), and (1 μm, −100 μm), respectively. In case A, the particle can be stably captured by the beam. If the particle can be stably captured from a certain initial position (case A), under the same initial speed and angle of incident, it could also be captured when the incident from a position further from the optical axis (Case B). This is because air viscous drag could only reduce the kinetic energy of the particle and thus its speed. When the particle is incident from a position closer to the optical axis, it may escape from the trap due to a higher moving speed.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic diagram of HCF-based optical trap. Inset: the red solid and dashed lines represent the beam profile focused by Lens 2 and emitted from HCF, with arrows indicating the beam propagation directions. HWP, half-wave plate; PBS, polarization beam splitter.

**Figure 2.**Comparison of the calculated axial optical forces using Rayleigh approximation (red) and the T-matrix approach (blue) for silica particles with diameter of (

**a**) 100 nm, (

**b**) 200 nm, (

**c**) 300 nm, and (

**d**) 400 nm. (

**e**) Ratio of optical forces calculated using the Rayleigh approximation (F

_{RL}) and the T-matrix (F

_{TM}) method in the axial and (

**f**) radial directions.

**Figure 3.**(

**a**) Calculated optical gradient and scattering forces acted on the particle along the optical axis when d

_{p}= 0.4 μm, NA

_{lens}= 0.08 and NA

_{fiber}= 0.1. (

**b**) Calculated F

_{z}under different values of NA

_{lens}/NA

_{fiber}. (

**c**) Ratio of the transverse gradient force and particle gravity at the equilibrium trapping point for different particle diameters when NA

_{lens}= 0.08 and NA

_{fiber}= 0.1. (

**d**) Transverse optical stiffness versus optical powers when d

_{p}= 0.4 μm.

**Figure 4.**(

**a**) Contour lines of the axial equilibrium trapping position z

_{eq}versus particle diameter d

_{p}and NA

_{lens}/NA

_{fiber}when NA

_{fiber}= 0.05 and (

**b**) NA

_{fiber}= 0.1. (

**c**) The axial equilibrium trapping position versus d

_{p}when NA

_{lens}= 0.08 and NA

_{fiber}= 0.1, corresponding to the situation marked by the black-dashed line in (

**b**). (

**d**) The relationship between the equilibrium trapping position and NA

_{lens}under different core diameters when d

_{p}= 0.4 μm.

**Figure 5.**(

**a**) v

_{0}-θ

_{0}parameter space in the x–z plane determining the particle capture capability. The solid lines are the boundary separating the space that can or cannot achieve particle capture for different particle diameters. The area enclosed at the right-hand side of the boundary represents the case that the particle cannot be captured. (

**b**) Plots of particle trajectories in the vicinity of the HCF endface for the three cases marked in (

**a**) when d

_{p}= 0.4 μm. The red dot indicates the particle initial position, the arrows indicate the direction of trajectories, and the cross indicates the equilibrium trapping position. The two insets plot the zoom in of the regime in which the particle trajectories are rapidly changed.

**Figure 6.**Trajectories of the particles in front of the HCF endface for different initial positions (dots). The arrows indicate the direction of trajectories, and the crosses indicate the equilibrium trapping position (if possible).

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**MDPI and ACS Style**

Li, K.; Wang, R.; Shao, S.; Xie, F.; Jiang, Y.; Xie, S.
Capture Dynamics of Dielectric Microparticles in Hollow-Core-Fiber-Based Optical Traps. *Photonics* **2023**, *10*, 1154.
https://doi.org/10.3390/photonics10101154

**AMA Style**

Li K, Wang R, Shao S, Xie F, Jiang Y, Xie S.
Capture Dynamics of Dielectric Microparticles in Hollow-Core-Fiber-Based Optical Traps. *Photonics*. 2023; 10(10):1154.
https://doi.org/10.3390/photonics10101154

**Chicago/Turabian Style**

Li, Kun, Rui Wang, Shuangyun Shao, Fang Xie, Yi Jiang, and Shangran Xie.
2023. "Capture Dynamics of Dielectric Microparticles in Hollow-Core-Fiber-Based Optical Traps" *Photonics* 10, no. 10: 1154.
https://doi.org/10.3390/photonics10101154