Focusing properties and focal shift in hyperbolic-cosine-Gaussian beams

Based on the generalized Huygens-Fresnel diffraction integral the expression of the focused hyperbolic-cosine-Gaussian (ChG) beams is derived for any transverse plane in the focal region. The properties of the focused ChG beams and the focal shift are studied, and their dependence on the Fresnel number and the decentered parameter β are illustrated with detailed numerical calculations.