On the Uniqueness of FROG Methods

Tamir Bendory*, Pavel Sidorenko, Yonina C. Eldar

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


The problem of recovering a signal from its power spectrum, called phase retrieval, arises in many scientific fields. One of many examples is ultrashort laser pulse characterization, in which the electromagnetic field is oscillating with ∼1015 Hz and phase information cannot be measured directly due to limitations of the electronic sensors. Phase retrieval is ill-posed in most of the 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 ultrashort laser pulses is the frequency-resolved optical gating (FROG). In FROG, the acquired data are 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.

Original languageEnglish
Article number7891012
Pages (from-to)722-726
Number of pages5
JournalIEEE Signal Processing Letters
Issue number5
StatePublished - May 2017
Externally publishedYes


FundersFunder number
Horizon 2020 Framework Programme646804


    • Frequency-resolved optical gating (FROG)
    • phase retrieval
    • quartic system of equations
    • ultrashort laser pulse measurements


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