Approximate description for Bessel, Bessel–Gauss, and Gaussian beams with finite aperture

An approximate analysis is derived for the propagation of Bessel, Bessel–Gauss, and Gaussian beams with a finite aperture. This treatment is based on the fact that the circ function can be expanded into an approximate sum of complex Gaussian functions, so that these three beams are typically expressed as a combination of a set of infinite-aperture Bessel–Gauss beams. Correspondingly, the evaluation of the diffracted field distribution of the beams is reduced to the summation of Bessel–Gauss functions. From analytical results, the present approach provides a good description of the diffracted beams in the region far (greater than a factor of the Fresnel distance) from the aperture. A possible extension of this method to other apertured beams is also discussed.