This work presents a refined theoretical and numerical framework for shaping the axial intensity of finite-energy Bessel–Gaussian beams through programmable nonlinear phase modulation. Starting from the scalar Fresnel diffraction integral, we reformulate the propagation of a Gaussian-apodized axicon beam using a dimensionally consistent stationary-phase method. This analysis directly relates the radial phase gradient to the saddle-point trajectory, phase curvature, and on-axis intensity distribution. A Gaussian phase modulation (GPM) serves as a reference design to achieve a flattop axial profile while preserving the characteristic transverse Bessel ring structure. This work is validated against beam propagation simulations and previously reported spatial light modulator (SLM) experiments, confirming its accuracy within the paraxial regime. A parametric study then clarifies the scaling of wavelength, beam waist, axicon angle, and refractive index for extended focusing. Beyond standard GPM, several alternative nonlinear phase functions are systematically compared. High-performing profiles must replicate not only the amplitude scale but, more importantly, the radial phase-gradient structure of the Gaussian reference, which governs energy redistribution from annular regions to the axis. The results identify smooth, localized nonlinear functions as promising candidates for stable flattop Bessel beam generation. The proposed framework offers a flexible optical design for applications such as through-glass via (TGV) micromachining and light-sheet illumination, while prospective high-intensity laser plasma uses remain beyond the present linear model.
Elsharkawi et al. (Sat,) studied this question.