Using phase retrieval to measure the intensity and phase of ultrashort pulses: frequency-resolved optical gating

TLDR

The inversion of a FROG trace to obtain pulse intensity and phase is a complex two‑dimensional phase‑retrieval problem. The study introduces an iterative Fourier‑transform algorithm to invert the FROG trace. FROG records the spectrum of two delayed pulse replicas as a function of delay, producing a trace that, together with an iterative Fourier‑transform algorithm employing a novel signal‑field constraint, directly retrieves the pulse’s intensity and phase. The FROG trace uniquely determines pulse intensity and phase, and the algorithm reliably retrieves them in practice—even with noisy data or distorted pulses—though it may stall for pulses with large intensity fluctuations.

Abstract

We recently introduced a new technique, frequency-resolved optical gating (FROG), for directly determining the full intensity I(t) and phase φ(t) of a single femtosecond pulse. By using almost any instantaneous nonlinear-optical interaction of two replicas of the ultrashort pulse to be measured, FROG involves measuring the spectrum of the signal pulse as a function of the delay between the replicas. The resulting trace of intensity versus frequency and delay yields an intuitive display of the pulse that is similar to the pulse spectrogram, except that the gate is a function of the pulse to be measured. The problem of inverting the FROG trace to obtain the pulse intensity and phase can also be considered a complex two-dimensional phase-retrieval problem. As a result, the FROG trace yields, in principle, an essentially unique pulse intensity and phase. We show that this is also the case in practice. We present an iterative-Fourier-transform algorithm for inverting the FROG trace. The algorithm is unusual in its use of a novel constraint: the mathematical form of the signal field. Without the use of a support constraint, the algorithm performs quite well in practice, even for pulses with serious phase distortions and for experimental data with noise, although it occasionally stagnates when pulses with large intensity fluctuations are used.