Virtual source for the fractional-order Bessel–Gauss beams

First, the nonparaxial integral expression of fractional-order Bessel–Gauss (FBG) beams during propagation is rigorously derived by using the virtual source technique and the principle of independent transmission and superposition of light, together with Weber integral formula and Fourier–Bessel transform pair. Then, the analytic expression of the important axial optical field amplitude of FBG beams is obtained. Finally, the on-axis intensity and phase of FBG beams are calculated on the basis of the analytic expression. Calculated results quantitatively revealed some important intensity information regarding the near-field propagation characteristic of FBG beams. •The nonparaxial integral expression of fractional-order Bessel-Gauss (FBG) beams during propagation and the analytic expression of the axial optical field amplitude of FBG beams is rigorously derived by using the virtual source technique, etc.•The concrete solution of third-, second- and first-order nonparaxial corrected fields and the zero-order paraxial solution light field can be obtained based on the analytic expression of the axial optical field amplitude of FBG beams.•The ‘fractional’ property of the “fractional nondiffracting” beams including the symmetry breakage of the FBG beams during propagation can also be explored based on the intensity distribution figures of different-order FBG beams varying with increased integer or fractional topological charge α.