Title: On the Uniqueness of 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.