# Centrosymmetric Optical Vortex
## 4 Results and Discussion  
First, we will compare the properties of the CSOV's intensity, OAM density, and gradient force between the PRF equaling to an odd and an even. The experimental intensity patterns and numerical simulation of the CSOVs (PRF = 2, 3) are displayed in the first and second columns of Figure 3, respectively. Their cone angle \(\alpha\) of the axicon is 0.06. In the following studies, we mainly studied the properties of the left-half part which is similar to the right-half part due to the CSOV's centrosymmetric light field. For PRF = 2, the local intensity at the joint Q1 is decreased. To characterize the magnitude of change, the intensity rate between the local extremu at Q1 and the uniform intensity position Q2 was calculated. The rate is greater than 65%, which is larger than 1/e of the uniform intensity at Q2. Therefore, the intensity at the joint Q1 can be considered as a smooth intensity distribution on the light ring. Furthermore, the intensity pattern of the CSOV (PRF = 3) has two gaps (Q3 and Q4) in the intensity ring due to the fractional phase jump at the joints, which is different to the CSOV (PRF = 2). However, two light petals formed in the left and right sides on the light ring of the CSOVs for PRF = 2 and PRF = 3, respectively. The reason is on account of the opposite azimuthal energy flow. As expected, the numerical simulation intensity patterns fit very well with the experimental results. Furthermore, the mode purity \(\varepsilon\) is a significant parameter which can reveal the quality of the generated CSOVs. In our experiments, the mode purity was estimated via the correlation coefficients between the experimental intensity pattern and the numerical simulation intensity pattern [16; 30]. As shown in Figure 3a1,a2, the values of the mode purity \(\varepsilon\) are both greater than 0.9, which indicates that the CSOVs still maintain a high beam quality. Figure 3c1,c2 depict the centrosymmetric OAM distributions of the CSOV and the computing method refers to References [11; 31]. For the local OAM density: Reds for positive values  
Figure 3: Comparison of CSOVs with phase reconstruction factor (PRF) equaling to an odd and an even, respectively. (**a1,a2**) are experimental intensity, (**b1,b2**) are numerical simulation intensity. (**c1,c2**) and (**S1–S4**) are the numerical simulation OAM distribution. In the fourth and fifth columns, the arrows visualize the gradient force. The \(\varepsilon\) values represent the mode purity of the generated CSOVs. The joints in the intensity rings mean that the border lines between quadrants in Figure 1a1,a2. (counter-clockwise), blues for negative values (clockwise), and greens for zero. The OAM distribution in the left-half part of the OAM ring provides a pair of opposite twist forces. Meanwhile, the OAM distribution in the right-half part is centrosymmetric with the left half part. In order to display more details of the OAM density distributions, the 3 \(\times\) magnification of the areas in the black boxes (S1-S4) in Figure 3c1,c2 are shown in Figure 3S1-S4, respectively. As Figure 3 shows, the OAM distributions in the area S1 are different from the area S3. The reason is the formation of the fractional phase jump when PRF is odd. Moreover, the twist forces provided by the OAM point to the center of the areas in S2 and S4, respectively. Furthermore, the arrows in Figure 3S1-S4 visualize the gradient force [32; 33]. In the particle manipulation field, gradient force offers the trap force. As shown in Figure 3S2,S4, larger gradient force in the black dashed frames provides a larger trap force which can realize particle stability trapping. Hence, the OAM and the gradient force provide a motion tendency for particle towards the black dashed frames in micro-manipulation field. Let us study the modulation properties of the CSOV's intensity, OAM distribution, and gradient force with the PRF increase from 2 to 10 in steps of 2.