By Banerjee, Partha P.; Jarem, John M
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Although the trend is similar, this is a considerable improvement over the convergence test results in Fibich and Ilan . 13 shows the growth of the onaxis intensity using AFHTSS technique for the test function above . 1487 written above. 1581 . This implies that the critical distance zc is a little bit over estimated and the maximum intensity reached is also less than the AFHTSS method. 11 The AFHTSS algorithm, a symmetrized version of the split-step FFT using Hankel transform instead, and using adaptive longitudinal stepping and transverse grid management.
Consequently, AFHTSS permits us to track peak intensities higher, faster, and more accurate than what is achievable by AFFTSS. 1581 for ε = 10 −8, and for an adaptive ε varying as ε = (λ/4πr)2, using AFFTSS with 10242 samples. P. , Opt. , 249, 293, 2005. ) non-paraxiality parameter ε in the scalar nonparaxial equation, we obtain results similar to those of the more complex vector method  and superior to those when ε is constant . Finally, we compare computation speeds of the AFHTSS and AFFTSS.
24) where n0 denotes the intrinsic refractive index of the medium, and n(2) is a measure of the gradation in the refractive index. 3 Contour plots showing periodic focusing of an initial Gaussian profile. 3. The contour plots show the initial (Gaussian) beam profile, the beam profile where the initial Gaussian attains its minimum waist during propagation before returning back to its original shape again, due to periodic focusing by the graded index distribution. Note that there exists a specific eigenmode (a Gaussian of a specific width, related to the refractive index gradient) for which the beam propagates through the material without a change in shape as a result of a balance between the diffraction of the beam and the guiding due to the parabolic gradient index profile.