🔬 optics

Tutorial of Fourier and Hankel transforms for ultrafast optics

This tutorial clarifies common misconceptions regarding analytic signals and Fourier transform conventions in ultrafast optics, provides complete derivations of transform relations, and introduces a novel, high-accuracy numerical scheme for the fast Hankel transform.

Original authors: Yi-Hao Chen

Published 2026-02-18

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Yi-Hao Chen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to describe a complex, flashing light show to a friend who has never seen one. You could try to describe every single flash, every color, and every movement in real-time. But that's overwhelming. Instead, you might describe the pattern of the lights, the rhythm, and the shape of the beams.

This paper by Yi-Hao Chen is essentially a "User Manual" for the mathematical tools scientists use to describe ultrafast light pulses (lasers that fire in trillionths of a second). The author is saying, "Hey, we've been using these tools a bit loosely, and sometimes we get the wrong answer because we're ignoring the fine print. Let's fix that."

Here is a breakdown of the paper's main points using simple analogies:

1. The "Ghost" in the Machine: The Analytic Signal

The Problem: Real light waves are like a seesaw: they go up (positive) and down (negative). In math, we often try to simplify this by just looking at the "up" part and ignoring the "down" part, assuming the "down" part is just a mirror image.
The Analogy: Imagine you are listening to a song. The sound waves go up and down in pressure. If you only recorded the "up" pressure, you'd lose half the information. But in physics, we often pretend the "down" part doesn't exist to make the math easier.
The Fix: The author explains that you can't just ignore the "down" part; you have to replace it with a specific "ghost" version called the Analytic Signal. If you don't do this correctly, you might miss half the frequencies in your laser, like listening to a song but only hearing the bass and missing the melody. This is crucial for calculating how lasers interact with materials (like in Raman scattering).

2. The "Zoom Lens": Frequency Offsets

The Problem: Lasers are incredibly fast. If you try to simulate a laser pulse on a computer, you have to take millions of "snapshots" per second to catch the rapid oscillations. This is computationally expensive (like trying to record a hummingbird's wingbeats with a slow camera).
The Analogy: Imagine you are filming a race car going 200 mph. If you try to film the whole car moving, you need a super-fast camera. But, what if you put the camera inside the car? Now, the car looks stationary, and you only need to film the tiny vibrations of the engine.
The Fix: The paper explains how to shift our mathematical "camera" to ride along with the laser's main frequency. This is called the Offset Frequency. By doing this, we only need to simulate the tiny wiggles (the envelope) rather than the massive, fast oscillations. This makes the computer simulation run 100 to 600 times faster!

3. The "Language Barrier": Fourier Transforms

The Problem: The Fourier Transform is a tool that turns a signal from "time" (what happens second by second) into "frequency" (what colors are in the light). The problem is that mathematicians and physicists use different "dialects" (conventions) for this tool. One group uses a specific formula, and another group uses a slightly different one.
The Analogy: Imagine a group of chefs. One group measures ingredients in "cups," and another uses "grams." If you follow a recipe written in grams but measure in cups without converting, your cake will be a disaster.
The Fix: The author warns that if you mix these "dialects" (using a math formula with a physics equation), you might get a result that looks smooth and pretty but is physically wrong. It's like baking a cake that looks great but tastes like soap. The paper provides a strict guide on how to translate between these dialects so your simulation actually matches reality.

4. The "Blurry Photo": Time-Frequency Analysis

The Problem: Sometimes a laser pulse changes its color while it is happening. You can't just look at the whole pulse to see the colors, because the colors change over time.
The Analogy: Think of a piano chord. If you play a C-major chord and then slide your hand to an F-major chord, a standard frequency analyzer just hears "C and F mixed together." It doesn't tell you when the switch happened.
The Fix: The paper discusses Spectrograms (like a musical score that shows notes changing over time). The author introduces a "High-Resolution Spectrogram" technique. Imagine taking many photos of the piano with different focus settings (some sharp on time, some sharp on pitch) and then combining them into one perfect image that shows exactly when the notes changed. This helps scientists see complex laser behaviors that were previously blurry.

5. The "Round Table": The Hankel Transform

The Problem: Most lasers are round (cylindrical). Standard math tools (Fourier transforms) are great for square grids (like a chessboard), but they are clumsy for round shapes. Using a square grid to simulate a round laser is like trying to fit a round table into a square room; you waste a lot of space and get jagged edges.
The Analogy: If you want to cut a perfect circle out of a square piece of paper, you have to cut off all the corners. It's inefficient.
The Fix: The Hankel Transform is a special tool designed specifically for round things. It's like having a circular cookie cutter instead of a square one.
The Innovation: The author points out that the existing "fast" versions of this tool (FHATHA) were a bit sloppy—they sometimes lost energy or missed the very center of the laser beam. The author proposes a new, improved version that:

Centers the beam: It makes sure the very middle of the laser (where it's brightest) is calculated perfectly.
Conserves energy: It ensures that the total "amount" of light stays the same during the calculation, just like a closed bank account shouldn't lose money just because you moved it between checking and savings.

Summary

This paper is a "Quality Control" guide for ultrafast optics. The author is saying:

Don't ignore the "ghost" parts of the light wave (Analytic Signal).
Zoom in to make simulations faster (Offset Frequency).
Speak the same language as your equations (Fourier Conventions).
Look at the details over time (High-Res Spectrograms).
Use the right shape for round lasers and make sure you don't lose energy in the math (Improved Hankel Transform).

By following these rules, scientists can build better simulations that actually predict what happens in the real world, rather than just looking pretty on a computer screen.

1. Problem Statement

The paper addresses critical misconceptions and implementation errors in the field of ultrafast optics, specifically regarding the mathematical foundations and numerical application of Fourier and Hankel transforms. The author identifies several recurring issues:

Analytic Signal Ambiguity: Textbooks often neglect the rigorous definition of the analytic signal (positive-frequency component), leading to confusion between propagation-constant-offset and frequency-offset formulations. This results in missing negative frequencies, incorrect factor-of-2 errors in nonlinear polarization calculations (e.g., Raman scattering), and improper handling of complex conjugates.
Fourier Transform Convention Confusion: There is a lack of comprehensive guidance on Fourier transform constants ( $C_F, C_{IF}$ ) and conventions (physical vs. mathematical). Misapplying these conventions (e.g., mixing $e^{i\omega t}$ and $e^{-i\omega t}$ ) leads to incorrect pulse propagation dynamics, such as misinterpreting chirp direction or dispersion effects.
Numerical Artifacts: Standard Discrete Fourier Transform (DFT) implementations suffer from aliasing, phase unwrapping errors due to temporal offsets, and convolution distortions when dealing with non-periodic or extended signals (e.g., Raman response functions).
Inefficiency in Radial Symmetry: While 2D spatial Fourier transforms are computationally expensive for radially symmetric systems (like multipass cells), existing Fast Hankel Transform (FHT) algorithms (specifically FHATHA) suffer from energy non-conservation and inaccuracies near the spatial center ( $r=0$ ), which is critical for high-intensity laser pulses.

2. Methodology

The paper employs a rigorous theoretical derivation followed by numerical verification to establish correct frameworks for optical simulations:

Analytic Signal Decomposition: The author derives the relationship between real-valued fields and their analytic signals using the Hilbert transform, emphasizing that the analytic signal is the unique positive-frequency representation.
Parametrized Fourier Analysis: Fourier transforms are defined with arbitrary constants ( $C_F, C_{IF}, c_s$ ) to unify physical and mathematical conventions. The paper derives conversion factors between continuous and discrete domains (DFT) to ensure physical units (e.g., $J/Hz$, $W$ ) are preserved.
Time-Frequency Analysis: The paper analyzes the Wigner distribution and Spectrogram, introducing a High-Resolution Spectrogram method that combines multiple spectrograms with varying window sizes to overcome the uncertainty principle limitations.
Hankel Transform Optimization: The paper critiques the Magni et al. Fast Hankel Transform of High Accuracy (FHATHA) algorithm. It proposes a modified numerical scheme that:
1. Incorporates the signal value at the spatial center ( $r=0$ ) to correct errors in Gaussian-like beams.
2. Implements an energy restoration step (scaling) to satisfy discrete Parseval's theorem.
3. Utilizes zero-padding and cross-correlation via FFT to maintain computational efficiency ( $O(N \log N)$ ).

3. Key Contributions

Clarification of Analytic Signals: The paper explicitly distinguishes between the "slowly-varying envelope assumption" (SVEA) and "frequency-offset analytic signals." It clarifies that SVEA is a specific case of the general propagation-constant-offset formulation and warns against deriving nonlinear equations directly from complex fields without accounting for missing negative frequency components.
Standardized Fourier Conventions: It provides a complete set of derivations for converting physical quantities (energy, power, spectral density) between continuous and discrete domains, resolving the "factor of 2" and unit inconsistencies common in literature.
High-Resolution Spectrogram: A novel algorithm is proposed to generate time-frequency representations with superior resolution by combining normalized spectrograms from different window sizes and selecting the maximum/minimum energy bounds, effectively bypassing the trade-off between temporal and spectral resolution.
Improved Fast Hankel Transform (FHATHA):
- New Numerical Scheme: The author proposes a modified FHATHA that includes the $r=0$ sampling point (using linear interpolation between $r=0$ and the first sampling point) and applies energy restoration.
- Performance: This new scheme outperforms the original FHATHA and the Discrete Hankel Transform (DHT) by significantly reducing Root-Mean-Squared Error (RMSE) and ensuring energy conservation, while retaining the computational speed of FFT-based methods.
Practical Implementation Guidelines: The paper offers specific advice on handling fftshift/ifftshift, phase unwrapping (removing temporal offsets before analysis), and zero-padding strategies to prevent aliasing in convolutions.

4. Results

Analytic Signal Verification: Demonstrated that failing to use the correct analytic signal decomposition leads to a factor-of-2 error in Raman-induced index change calculations and missing frequency components in four-wave mixing.
Convention Sensitivity: Simulations showed that using the wrong Fourier convention (e.g., applying a mathematical convention to a physical propagation equation) reverses the chirp sign, causing a pulse to de-chirp instead of chirp (or vice versa) under normal dispersion, leading to physically incorrect propagation dynamics.
High-Resolution Spectrogram: Successfully visualized complex time-frequency dynamics in optical wave breaking and supercontinuum generation, revealing fine temporal structures and energy transfer mechanisms that standard spectrograms missed.
FHATHA Accuracy:
- The modified FHATHA (with center-point inclusion and energy restoration) showed near-perfect overlap with analytical Gaussian solutions and ABCD matrix predictions for beam propagation.
- Original FHATHA exhibited significant energy drift and spatial distortion, particularly with low sampling points ( $N_\perp$ ).
- The modified scheme reduced RMSE by orders of magnitude compared to the original FHATHA, especially for small $N_\perp$ .

5. Significance

This tutorial serves as a foundational reference for researchers in ultrafast optics and nonlinear photonics. Its significance lies in:

Error Prevention: It prevents common, hard-to-detect numerical errors (like factor-of-2 discrepancies or chirp inversion) that can invalidate simulation results.
Computational Efficiency: The proposed modified FHATHA enables highly accurate, full-field, radially-symmetric simulations (essential for modeling high-power lasers, multipass cells, and filamentation) without the prohibitive cost of full 2D Cartesian FFTs.
Reproducibility: By providing explicit code snippets, GitHub repositories (MMTools, 3D-UPPE), and standardized unit conversions, the paper promotes reproducibility and standardization in the field.
Educational Value: It bridges the gap between abstract mathematical definitions and practical numerical implementation, clarifying the "why" and "how" of transform conventions for the next generation of optical physicists.