Tutorial on Fourier and Hankel Transforms for Ultrafast Optics

This tutorial is designed for individuals who are new to the field of ultrafast optics. It was written in response to the apparent lack of comprehensive introductions to the basic Fourier transform, extending beyond the flat-phase description. Additionally, there is a need for complete derivations of several relations involving the Fourier transform, maintaining its most general formulation. This approach avoids the arbitrary selection of Fourier-transform constants and ensures a complete understanding. It shows the importance of having Fourier-transform constants as parameters. Most important of all, there have been misuse of Fourier transform from my observations, which cannot be easily detected by checking the smoothness of the result of a numerical implementation or by seeing if the simulation duplicates the “overall physics.” This problem should be easily solved by a simple tutorial (see Sec. 2.2). I hope that this tutorial can help people understand more about the Fourier transform, especially in the context of ultrafast optics. In addition, a tutorial of the Hankel transform, which arises from the two-dimensional spatial Fourier transform of a radially-symmetric function, is provided. Its numerical implementation based on the fast Hankel transform with high accuracy is also provided, which is the core element of fast radially-symmetric full-field ultrafast propagation. Feel me to send me an email if there is any confusion, or you think that there is more to add to this tutorial. For a deeper understanding into the ultrafast pulse propagation that involves these transforms, please check out our publicly-shared Github code [https://github.com/AaHaHaa/MMTools].