Focal switch of cosine-Gaussian beams focused by a thin lens without aperture

The focal switch of cosine-Gaussian beams focused by a thin lens without aperture is studied in detail.The third-order equation governing the position of axial maximum intensity and the formula of the relative transition height is derived.A large number of numerical calculation results are presented.It is shown that the focal switch of focused cosine-Gaussian beams can take place as the beam parameter β is larger than 1.