# On Signal Reconstruction from FROG Measurements

## TL;DR

This paper proves that reconstructing signals from FROG measurements, which involve phaseless quartic Fourier data, can be achieved with minimal translations, improving the understanding of phase retrieval in optical applications.

## Contribution

The paper demonstrates that only three translations are needed to recover almost all bandlimited signals from FROG data, and only two when Fourier magnitude is known, advancing phase retrieval methods.

## Key findings

- Three translations suffice for almost all signals
- Two translations are enough if Fourier magnitude is available
- Significant improvement over previous phase retrieval results

## Abstract

Phase retrieval refers to recovering a signal from its Fourier magnitude. This problem arises naturally in many scientific applications, such as ultra-short laser pulse characterization and diffraction imaging. Unfortunately, phase retrieval is ill-posed for almost all one-dimensional signals. In order to characterize a laser pulse and overcome the ill-posedness, it is common to use a technique called Frequency-Resolved Optical Gating (FROG). In FROG, the measured data, referred to as FROG trace, is the Fourier magnitude of the product of the underlying signal with several translated versions of itself. The FROG trace results in a system of phaseless quartic Fourier measurements. In this paper, we prove that it suffices to consider only three translations of the signal to determine almost all bandlimited signals, up to trivial ambiguities. In practice, one usually also has access to the signal's Fourier magnitude. We show that in this case only two translations suffice. Our results significantly improve upon earlier work.