Focal shift in Sinh-Gaussian beams by using the encircled-power method

Starting from the Collins formula, focusing properties of unapertured sinhGaussian beam are studied by using the encircled-power method. It is found that the axial intensity of focused sinh-Gaussian beams vanishes, so that Li-Wolf's approach to the focal shift fails. The encircled-power method is introduced to define the beam width and real focal plane of focused sinh-Gaussian beams, thus their focal shift can be determined. Numerical calculations are performed and show that the focal shift in focused sinh-Gaussian beams depends on the Fresnel number and decentered parameter, and increases with decreasing them. The optimal focusing of sinh-Gaussian beams is also analyzed.