# On the Uniqueness of FROG Methods

## TL;DR

This paper investigates the uniqueness of FROG methods in phase retrieval, demonstrating that almost all band-limited signals can be uniquely recovered from FROG measurements when combined with power spectrum data.

## Contribution

The paper introduces and formulates FROG-type techniques and proves the generic uniqueness of signal recovery in blind FROG with additional power spectrum information.

## Key findings

- Almost all band-limited signals are uniquely determined by blind FROG measurements.
- Uniqueness holds up to trivial ambiguities when the signal's power spectrum is also available.
- The results extend the theoretical understanding of phase retrieval via FROG methods.

## Abstract

The problem of recovering a signal from its power spectrum, called phase retrieval, arises in many scientific fields. One of many examples is ultra-short laser pulse characterization in which the electromagnetic field is oscillating with ~10^15 Hz and phase information cannot be measured directly due to limitations of the electronic sensors. Phase retrieval is ill-posed in most cases as there are many different signals with the same Fourier transform magnitude. To overcome this fundamental ill-posedness, several measurement techniques are used in practice. One of the most popular methods for complete characterization of ultra-short laser pulses is the Frequency-Resolved Optical Gating (FROG). In FROG, the acquired data is the power spectrum of the product of the unknown pulse with its delayed replica. Therefore the measured signal is a quartic function of the unknown pulse. A generalized version of FROG, where the delayed replica is replaced by a second unknown pulse, is called blind FROG. In this case, the measured signal is quadratic with respect to both pulses. In this letter we introduce and formulate FROG-type techniques. We then show that almost all band-limited signals are determined uniquely, up to trivial ambiguities, by blind FROG measurements (and thus also by FROG), if in addition we have access to the signals power spectrum.