# Theoretical Analysis and Experimental Verification of the Influence of Polarization on Counter-Propagating Optical Tweezers

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. T-Matrix Method

#### 2.2. Experimental Setup

_{2}sphere S in a vacuum chamber. The pressure in the vacuum chamber was set to 10 mBar. The NA of L1 and L2 were $0.0673$. The power of both trapping beams arriving at the sphere was $100\mathrm{mW}$. After being scattered by the sphere, the forward trapping laser then transmitted through the polarized beam splitter PBS2 and was detected by a homemade QPD. The half-wave plate HWP2, mounted on a rotation stage, was used to control the polarization state of the trapping laser. There was also a polarimeter we used to measure the polarization of the two trapping beams which is not shown in the figure.

## 3. Results

#### 3.1. Numerical Analysis

_{2}sphere levitated by two $1064\mathrm{nm}$ trapping beams with an NA of $0.0673$. For optical force in the x direction and y direction to be similar in the regime of this research, we chose the x direction force to represent the radial force. Figure 2a is the radial force distribution on the radial displacement of the sphere. When the polarization of the two trapping beams is orthogonal, ${P}_{1}=0$, the radial force is proportional to the radial displacement in the range of $\sim 1\mathsf{\mu}\mathrm{m}$, which is called the linear region. The optical tweezers can be described by the harmonic oscillator in this region. As shown in the figure, the power split ratio only slightly affects the maximum radial force. Figure 2b is the axial force distribution on the axial displacement of the sphere. When the polarization of the two beams is orthogonal, the axial force distribution has a similar trend compared with the radial force distribution, though the linear region of the axial force is larger because of a smaller light field gradient, about $\sim 10\mathsf{\mu}\mathrm{m}$, as shown in the inserted Figure 2b. We can learn from the figure that the power split ratio has a significant influence on the axial force. A non-zero ${P}_{1}$ can cause interference and create multiple axial balance points, making the linear region reduce to $\sim 0.1\mathsf{\mu}\mathrm{m}$, a quantity related to the laser wavelength. With the ${P}_{1}$ increase, the force gradient, or trap stiffness, becomes larger.

#### 3.2. Experimental Verification

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Experimental setup. The 1064 nm trapping laser was separated by a polarized beam splitter PBS1. We named the transmitted beam the forward trapping laser, and the reflected beam the backward trapping laser, indicated by the dashed line and solid line, respectively. The power of the two beams was adjusted to the same value by rotating the half-wave plate HWP1. The two beams were then aligned and focused with the numerical aperture (NA) of 0.0673. The 10-micron-diameter SiO

_{2}sphere was trapped by the counterpropagating beams. The forward trapping laser, after interacting with the levitated sphere, was detected by the quadrant photoelectric detectors.

**Figure 2.**The optical force distribution of different power split ratios. (

**a**) The radial force distribution and (

**b**) the axial force distribution in each direction of different power split ratios. The inserted figure in (

**b**) is the zoom-out axial force distribution of ${P}_{1}=0$.

**Figure 3.**The optical force distribution of different polarization phases. (

**a**) The radial force distribution and (

**b**) the axial force distribution in each direction in respect to phase differences in the first kind. (

**c**) The radial force distribution and (

**d**) the axial force distribution in each direction in respect to phase differences in the second kind. The power split ratio is ${P}_{1}={P}_{2}=\frac{1}{2}$.

**Figure 4.**The resonant frequency or trap stiffness of different polarizations. The axial (blue) and radial (magenta) resonant frequency or trap stiffness of (

**a**) different power split ratios with ${P}_{2}=0$ and (

**b**) the first kind phase difference with ${P}_{1}={P}_{2}=\frac{1}{2}$.

**Figure 5.**The experiment result. (

**a**) The power spectrum density of the radial axis of the different polarization angles. The coupling between the radial and axial axis is enlarged to make their signal appear to be at the same spectrum. (

**b**) The axial (blue) and radial (magenta) resonant frequency or trap stiffness of different angles between the polarization of two beams. The plus and star signs are experimentally detected by the QPD; the lines are theoretical calculations based on the trapping condition.

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**MDPI and ACS Style**

Chen, M.; Li, W.; Yang, J.; Hu, M.; Xu, S.; Zhu, X.; Li, N.; Hu, H. Theoretical Analysis and Experimental Verification of the Influence of Polarization on Counter-Propagating Optical Tweezers. *Micromachines* **2023**, *14*, 760.
https://doi.org/10.3390/mi14040760

**AMA Style**

Chen M, Li W, Yang J, Hu M, Xu S, Zhu X, Li N, Hu H. Theoretical Analysis and Experimental Verification of the Influence of Polarization on Counter-Propagating Optical Tweezers. *Micromachines*. 2023; 14(4):760.
https://doi.org/10.3390/mi14040760

**Chicago/Turabian Style**

Chen, Ming, Wenqiang Li, Jianyu Yang, Mengzhu Hu, Shidong Xu, Xunmin Zhu, Nan Li, and Huizhu Hu. 2023. "Theoretical Analysis and Experimental Verification of the Influence of Polarization on Counter-Propagating Optical Tweezers" *Micromachines* 14, no. 4: 760.
https://doi.org/10.3390/mi14040760