Objective The research on the electric field of ultrashort laser pulse has a wide application prospect. The ultrashort femtosecond laser pulse is used in many scientific and engineering domains, such as ultrashort spectroscopy, quantum coherent modulation, and ultra-intense laser physics. A typical method of measuring ultrashort laser pulse is the frequency-resolved optical gating (FROG). Using FROG, we first split the pulse into two replicas and then apply a time-delay to one of them. Next, we allow them to interact in a nonlinear process to generate a signal, in which we use spectroscopy to measure the spectral intensity. Finally, we tune the time delays to obtain the spectral intensities of a set of nonlinear signals, which constitute a spectrogram named FROG trace. A phase retrieval algorithm is always required to reconstruct the original laser pulse because only the intensities are recorded in FROG trace. As an efficient phase retrieval algorithm, the prominent component general projecting algorithm (PCGPA) is widely used for ultrashort pulse measurement. However, PCGA brings several practical problems for the FROG measuring process because PCGPA requires that the trace be a square one and that its frequency axis and delay axis coordinates are coupled by fast Fourier transform. According to the different applications of the nonlinear process, FROG can also be realized in different geometrical schemes, for example, second harmonic generation (SHG), polarization gate (PG), and cross-phase modulation (XPM). In this study, we aim to (i) build a non-square FROG trace by taking the features of the measuring devices into fully account to realize successful pulse reconstruction and avoid the drawbacks accompanied by PCGPA and (ii) compare the retrieving results in different geometries to select a more efficient and practical one.

Methods First, we build the non-square trace by applying the following three transformations on the 256×256 square FROG trace (Fig. 1): (i) low-pass filtering in the frequency axis (F_LPF), (ii) up-sampling in the frequency axis (F_US), and (iii) down-sampling in the delay axis (D_DS). To identify the relevant non-square traces, we also use the three aforementioned transformations. Then, we numerically generate 200 pulses, whose envelopes are the superposition of 2--6 Gaussian functions and phases are composed of chirps, quadratic chirps, and self-phase modulations. We also obtain few-cyclic experimental pulses for comparison. Next, we use the ptychography algorithm, originally developed in coherent diffraction imaging, to retrieve the FROG trace phase and the corresponding pulse field. Ptychography algorithm is accustomed to non-square traces without further adjustment because it uses only one column of a trace in its single iteration. To evaluate the results, we use the intersection angle θ between the original and the reconstructed pulses in multi-dimensional space, which is considered eligible when less than 0.1. Finally, we realize the pulse reconstruction from non-square FROG traces in different geometric schemes (i.e., SHG, PG, and XMP schemes) and compared the results.

Results and Discussions For SHG FROG (Table 1), pulse reconstruction can be realized from trace after F-LPF, which should be regarded as super-resolution one because of the absence of high-frequency information, and F_US application can improve the reconstruction. Although the results will be slightly worse for applying D_DS, it is still less than 0.1 when only 14-delay data are left. For PG and XPM FROG traces (Table 2 and Table 3, respectively), the effects of these three transformations on the retrieval result are similar to the SHG FROG trace; however, the minimum amount of delays for successful reconstruction is only 12 and 8, respectively. Ptychography algorithm from non-square FROG traces (Fig. 2 and Fig. 3) can restructure both the numerally generated pulse and the few-cyclic experimental pulse. For different geometries, XPM FROG can be deemed the most practical because it requires the least delays. SHG FROG trace is symmetrical; thus, only half of the delays in a trace are effective, leading to the requirement of more delay steps to make the algorithm be converged. The nonlinear signal in PG FROG is proportional to the product of the original pulse field and the delayed pulse field, so it vanishes and is useless when the time-delay is longer than the pulse duration. The nonlinear signal in XPM geometry is the original pulse under the same conditions because the delayed pulse replica only modulates the original pulse phase. In short, XPM FROG trace contains more effective information than the other two types of trace; therefore, few time delays are needed for efficient pulse reconstruction.

Conclusions In this study, we build non-square traces by applying three transformations on the corresponding square one: (i) F_LPF to reduce the requirement of large phase-matching bandwidth, (ii) F_US to utilize the high resolution of spectrometer, and (iii) D_DS to reduce the measuring time. Ptychograpy algorithm can reconstruct simulated and few-cyclic experimental pulses from the non-square traces in SHG, PG, and XPM FROG. After applying F_LPF and F_US for the XPM FROG trace, only eight delays are sufficient to retrieve the pulses successfully. Changing the delay is the most time-consuming step in FROG; hence, reducing delays will benefit FROG to realize the real-time measurement of ultrashort pulses.