An accurate approximation is presented of the focal shift in the diffraction field of a Gaussian laser beam that is focused by a thin lens that fills a circular aperture of any prescribed radius. The limiting cases of beam truncation by the aperture are discussed and formulas for nontruncated Gaussian beams and for uniform beams are obtained. The proposed formulas provide a convenient means for circumventing the transcendental equations in the focal-shift problem.
Yajun Li, Yajun Li,
"Accurate approximation of the focal shift in the transformation of truncated Gaussian beams," Optical Engineering 32(4), (1 April 1993). https://doi.org/10.1117/12.61196