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A detailed study of the focal shift and accompanying focal switch of flattened Gaussian beams (FGBs) passing through an aperture bifocal lens is presented. It is shown that by suitably choosing the parameters of the optical system and beam, there exists two axial irradiance maxima for focused FGBs, which vary with the Fresnel number of the aperture. The focal switch can take place when the Fresnel number decreases gradually. The focal switch of Gaussian beams is treated as a special case of FGBs. From the axial irradiance distribution of focused FGBs, a physical explanation of the focal switch is given. A comparison with previous work is also made.