Focal switch effect of focused Bessel-Gaussian beams

The focal switch effect of Bessel-Gaussian beams focused by a thin lens without aperture is studied in detail. The third-order algebraic equation determining the position of the axial maximum intensity and the formula of the relative transition height are derived. Numerical calculation results are presented to illustrate the theoretical predictions. It is shown that, if the beam parameter is bigger than 2, the focal switch of Bessel-Gaussian beams can appear. The relative transition height increases with increasing the beam parameter. The normalized axial minimum intensity decreases with increasing the beam parameter, and the normalized axial minimum intensity is zero, as the beam parameter is bigger than 3.9.