## 3 Results and Discussion
### Controlling the Particle Distribution with Cylindrical Vector Beams  
The particle distribution in the transverse plane can be controlled by changing the properties of the focused beam, which can be obtained by changing the ratio of the waist of the laser beam to the objective aperture [23], or using structured beams as the optical trapping beam [24; 25]. In this paper, we use the cylindrical vector beam to vary the intensity of the focused beam and study the optical assembling of the particles under the diffraction patterns. Cylindrical vector beams are the class of laser beams that arise from the solution to the vector Helmholtz equation with cylindrical boundary conditions. Cylindrical vector beams possess azimuthal symmetry in their amplitude distribution and a spatially inhomogeneous polarization direction [24]. There has been a rapid increase in investigations on cylindrical vector beams because of their interesting properties and potential applications in optical  
Figure 4: Images of the 0.5 \(\upmu\)m particles assembly formed by the diffraction pattern. (**a**) particle patterns at low concentration (4 \(\times\) 10\({}^{9}\) cm\({}^{-3}\)); (**b**–**f**) particle patterns at high concentration (10\({}^{11}\) cm\({}^{-3}\)). The power is 820 mW; scale bar, 10 \(\upmu\)m.  
trapping and laser machining [26; 27; 28; 29]. In this paper, the linearly polarized beam becomes a cylindrical vector beam after passing through the s-waveplate. The polarization angle, \(\psi\), is varied by rotating the s-waveplate. Beams with \(\psi=0\) and \(\psi=\pi/2\) are radially and azimuthally polarized, respectively. The transmission of the s-waveplate for radially and azimuthally polarized beams at 1064 nm are 77.6% and 68.3%, respectively. The laser power before the objective lens are 450 mW for the two cylindrical vector beams in the experiments. There is a dark line in the center of the diffraction patterns of the azimuthally polarized beam, which is consistent with the previous investigation [30]. After passing through the polarization filter in Figure 1a, the shape of intensity of the azimuthally polarized beam becomes a two-fan shape. There are two bright spots in the center, which are separated by a black line. The diffraction pattern is divided into two parts as fan-shaped distribution. When the trapping laser is working, the particles are also attracted and confined in the bright fringes. As the number of trapped particles increases, the particles are arranged as two fan-shaped distribution, as shown in Figure 5a, due to the fan-shaped distribution of the diffraction patterns. Although the center of the diffraction patterns is a dark line, the particles are attracted to the center of the fans because of the existence of two bright spots in the central region. The central dark line is covered by the particles and can not be observed in the particle patterns. The fan-shaped formation can be rotated by rotating the polarization direction of the polarization filter. We repeated the assembling experiment with a radially polarized beam. In the experiments, the radially polarized beam is no longer passing through the polarization filter in Figure 1a. When the trapping laser starts to work, the particles can be trapped and confined by the bright diffraction fringes. However, as time goes on, most of the particles are gradually attracted into the central diffraction spot. The trapped particles in the center gradually form a compact structure, and the result is shown in Figure 5b. The intensity distributions of the diffraction fringes are measured in the experiment when the trapping depth is 1 \(\upmu\)m from the interface. The results are listed in Table 1. The main peak in Table 1 is the brightest fringe of the central diffraction spot. For the linearly polarized beam, the main peak is the intensity of the bright central spot. There is a dark annulus in the intensity distributions of the diffraction field of the azimuthally and radially polarized beam.