Evaluation of beam convergence driven by spherical gradient refractive index lenses based on nonparaxial beam propagation

We utilize a theoretical method based on nonlinear beam propagation and finite difference eigenmode solver methods to precisely simulate Gaussian beam propagation in electrical fields through spherical gradient refractive index lenses. The theoretical computation uses second-order partial differentiation of propagation coordinates to generate microwave field propagation. Consequently, it offers accurate simulation results for any complex refractive index profile. The reliability of the proposed method is verified by comparing it with existing experimental and theoretical results. We employ the theoretical method to assess Gaussian beam convergence in terms of four key parameters: beam waist, maximum intensity, focal position, and Rayleigh range. The results indicate that gradient index spherical lenses have better convergence than convex thin lenses, as evidenced by a significant reduction in beam waist size. However, these lenses prompt an extremely short back focal length. Consequently, we propose a slight shift in the boundary and index distribution of spherical lenses to expand their back focal lengths.