Abstract: d is presented for describing the propagation of axisymmetric beams by expansion into Gaussian beams which are displaced axially relative to the entrance plane. If the position of the Gaussian beams' waists and their widths are properly chosen, the coefficients of the expansion can be found by means of fast Fourier transform (FFT) procedures. One disadvantage of propagation algorithms based on FFTs is that their range is limited by aliasing effects. The propagation length is then enhanced usually by enlarging the entrance plane and padding with zeros the transverse field range thus enlarging the size of vectors without increasing field details. An alternative approach is that of subdividing the propagation length into smaller steps and eliminating the leaking aliased field by introducing absorbing layers close to the boundaries of the transverse range. The method described here has significantly reduced aliasing effects due to the bound character of the Gaussian functions, thus enlarging the range of propagation of a single step. Another advantage of the method is that the sampling points are evenly distributed in the r$+2$/ coordinate, following linearly the power distribution across the radius of axisymmetric beams. !7
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