Tutorial of Fourier and Hankel transforms for ultrafast optics

TL;DR

This tutorial clarifies misconceptions and provides comprehensive derivations of Fourier and Hankel transforms in ultrafast optics, including a novel fast Hankel transform scheme with improved accuracy and efficiency.

Contribution

It offers a detailed clarification of Fourier-transform conventions and introduces a new, more accurate numerical scheme for the Hankel transform in ultrafast optics.

Findings

Clarified Fourier-transform conventions for ultrafast optics

Derived comprehensive Fourier-transform relations and their parameters

Proposed a new high-accuracy fast Hankel transform scheme

Abstract

This tutorial is designed to clarify a few misconceptions in the field of ultrafast optics. (1) Analytic signal that underlies the complex-conjugate decomposition of the field is discussed, as well as the misunderstanding between propagation-constant-offset and offset-frequency analytic signal and slowly-varying envelope assumption. (2) It contains complete derivations of the general formulations of several Fourier-transform relations. It shows the importance of having Fourier-transform constants as parameters, and helps clarify the arbitrary selection of Fourier-transform constants and conventions. (3) It also clarifies the correct Fourier-transform convention to be employed in ultrafast optics. (4) Moreover, multiple Fourier-transform aspects are discussed, involving convolution, aliasing, phase effect, and short-time Fourier transform. (5) In addition to the Fourier transform, a tutorial on the Hankel transform is provided. Its numerical implementation based on the fast Hankel transform with high accuracy (FHATHA) is also provided. Despite being a tutorial, I, for the first time, propose a new numerical scheme for the fast Hankel transform that outperforms both the original FHATHA and the discrete Hankel transform.