Comparative study on the paraxial approximation errors of tightly focused fundamental Gaussian beams

摘要

When the beam waist size is on the order of or smaller than a wavelength, the traditional Gaussian solution exhibits noticeable errors in describing a tightly focused laser beam of the fundamental mode. The spatial distributions of the paraxial approximate errors of the Gaussian solution to the paraxial Helmholtz equation, as compared with Sapozhnikov's exact solution to the scalar Helmholtz equation, are illustrated, tabulated, and discussed systematically. The maximum and average errors of the Gaussian solution are presented for a list of different specified waist sizes of laser beams. We found that the maximum error distribution is of an annular structure and contains the largest error ring on the waist plane. The laser beam waist should be >1.2 wavelengths for a 1% error, while the beam waist should be >3.5 wavelengths for a 0.1 % error. This study provides a useful criterion to ascertain whether the traditional Gaussian solution or the exact solution is to be used to describe the fundamental laser beam in optical engineering. The results exhibit potential applications in micronano technology, near-field optical technology, and other fields where tightly focused laser beams are Utilized.