Tutorial of Fourier and Hankel transforms for ultrafast optics

TL;DR

This tutorial clarifies the proper analytic-signal decomposition and the distinct roles of offset frequency and envelope in ultrafast optics, emphasizing a consistent Fourier-transform convention across theory and numerics. It presents a detailed, parameterized treatment of spectral transforms, convolution, aliasing, phase effects, and time–frequency analysis, and then extends to the Hankel transform with a focus on fast, energy-conserving computation (FHATHA) and its improvements. The work not only distills common misconceptions but also introduces a new numerical scheme that enhances FHATHA and discusses integration with discrete Hankel transforms for radially symmetric beam propagation. Together, these insights offer robust, physically consistent tooling for accurate simulations in ultrafast nonlinear optics and spatially structured fields.

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 offset-frequency analytic signal and slowly-varying envelope. (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.

Figures (22)

• Figure 1: Spectral domain of the Fourier-transform components of (a) the real-valued signal and (b) the envelope of its analytic signal [Eq. (\ref{['eq:E(t)']})]. PSD: power spectral density $\sim\abs{\mathfrak{F}\left[\cdot\right]}^2$
• Figure 2: Importance of applying the correct (same) Fourier-transform convention for the signal as the propagation equation. If the wrong convention is employed such that a signal in one convention is propagated with an equation of a different convention, the equation essentially sees a different signal. For example, the pulse, designed to be up-chirped (spectral components temporally vary from low at the leading edge to high frequencies at the trailing edge of a pulse), the equation with a different convention will treat it as down-chirped. Then the pulse propagation will be a propagation of a down-chirped pulse, which behaves differently from an up-chirped pulse. As an example, if the fiber exhibits normal dispersion, a down-chirped pulse will get dechirped and becomes shorter in time; however, an up-chirped pulse will be temporally stretched and up-chirped more.
• Figure 3: Signals in different conventions differ by complex conjugate. However, a mixed use between two conventions is forbidden (Fig. \ref{['fig:FT_convention_importance']}).
• Figure 4:
• Figure 5: DFT conversion. (a) is the "formal" use of DFT when the temporal profile $\abs{A(t)}^2$ is centered at $t=0$. However, in numerical simulations, it is common to place the pulse at the center of the time window for visualization purpose (b), resulting in a spectral phase shift that doesn't affect the spectral shape. (c) is the result after $\mathop{\mathrm{fftshift}}\nolimits$ centers the spectrum with respect to the frequency window. PSD: power spectral density $\sim\abs{A(\omega)}^2$. Here, the subscript "$x$" represents the coordinate of sampling points, rather than the actual time and frequency coordinates.