In this paper, the focusing characteristics of Bessel-Gaussian beams are studied by means of vector diffraction theory. The vector field distribution of the axisymmetric Bessel-Gaussian beam of a cylindrical vector is derived by calculating and adding Airy mixing modulation to the Bessel-Gaussian beam. It is found that with a series of regular focusing change characteristics, the focusing presents strong stability of the optical chain structure, and the number of optical chain links can be adjusted. At the same time, it is pointed out that in the case of a tightly focused helically polarized beam, the polarization in the focal region is not uniform, but there was a similar horizontal shift in focus. Finally, the relevant practical application scenarios are briefly introduced. The correlation focus shift conversion can be widely used in electronic acceleration, optical sampling and operation, and biological imaging.
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