On the Uniqueness of FROG Methods

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 $\sim{\text{10}}^{15}$ 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.