In this paper, the self-reconstruction property of astigmatic Bessel beam is studied experimentally and theoretically. Based on the Fresnel diffraction integral theory and Babinet principle, the general expression of the intensity distribution of astigmatic Bessel beams passing through a circular obstacle is derived. The cross-section light intensity at transmission distance of, 10, 30, and 80 mm after astigmatism of the astigmatic Bessel beam are occluded by circular obstacles. The self-reconstruction process of the light field is observed and verified by using an specially designed experimental setup. In the experiment, we choose He-Ne laser as a light source, collimate and expand the beam through a telescope system, and a zero-order astigmatic Bessel beam is generated by a beam vertically incident on the tilted axicon after the diaphragm. A circular obstacle with a radius of 0.2 mm is placed at a distance of 200 mm behind the axicon. Finally, the cross-section intensities at different distances are observed and recorded by a microscope. The experimental phenomena are in good agreement with the theoretical prediction. The results show that the reconstruction of the zero-order astigmatic Bessel beams will occur after passing through the on-axis and off-axis obstacles. And as the transmission distance increases, the outer contour size of the astigmatic Bessel beam becomes larger, and the number of central spot arrays increases, and the complete beam is gradually reconstructed. Particularly, this feature is different from the behavior of the non-diffracting Bessel beam, which maintains the light field unchanged during transmission and has a single central spot. It is expected to be applied to multi-layer multi-particle control. And a new optical property is discovered in the experiments: the reconstruction speed of the beam in the horizontal and vertical direction are not consistent in the reconstruction process, and there is a certain speed difference. Further, we add a spiral phase plate between the diaphragm and the axicon to produce a high-order astigmatic Bessel beam. And it is verified that the high-order astigmatism Bessel beam has the same self-reconstruction characteristics after being shielded by obstacles. Compared with the zero-order aperture system, the high-order beam can not only expand the operating range, but also use the orbital angular momentum carried by the beam to achieve light rotation, which makes the particle manipulation more flexible. The research proves the self-reconstruction characteristics of astigmatic Bessel beams theoretically and experimentally, and broadens the research range of astigmatic Bessel beams. The research results have practical significance and application value in the field of optical micro-manipulation.

A novel technique to generate precisely size-controlled hollow beams by controlling the diameter of circular slit is proposed. Firstly, a laser beam is transformed into a quasi-monochromatic incoherent annular source by a rotating ground-glass disk and circular slit. Then, after passing through a thin converging lens, a J0-correlated Schell-model beam is synthesized by placing the annular incoherent source in the first focal plane of the thin lens. Finally, a partially coherent hollow beam is generated by focusing the J0-correlated Schell-model beam with an axicon. Based on the diffraction theory and the propagation law of partially coherent beams, the cross-spectral density function is derived to calculate the intensity distribution of the cross section and the radial intensity distribution along the propagation axis behind the axicon. By carrying out the theoretical calculation, the proposed optical system generates a partially hollow beam, and the size of the hollow beam expands continuously as the propagation distance increases. Before further investigating the effect of the diameter of incoherent annular source on the hollow beam behind the axicon, we also calculate the intensity distribution of the cross section and the size of hollow beams along the propagation axis at z=70 mm with the source diameters being 1, 2, 3, 4 and 5 mm, respectively. Results show that the size of the hollow beam also increases with the diameter of incoherent annular source increasing. In this case, the size of the hollow beam can be precisely controlled by tuning the diameter of incoherent annular source through circular slit. We also design and conduct an experimental generation of the hollow beam and investigate the propagation properties. In the experiment, we control the diameter of the annular source by tuning the diameter of the circular slit located before the rotating ground-glass disk. And the diameter of the annular source is equal to that of the circular slits. When the sizes of circular slits are 1, 2, 3, 4 and 5 mm, respectively, the corresponding hollow beams are measured by CCD. Experimental results show that the size of hollow beam can be controlled by the propagation distance and the diameter of the circular slit. The intensity profiles are in good agreement with theoretical predictions. Therefore, the size of hollow beams can be precisely generated and controlled by the proposed system so that the optical system can be flexibly employed in optical trapping and manipulation of particles with different sizes. The results may provide a powerful tool for manipulating the micro- and nano-particles.

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