## 4 Spectrogram
The recorded signal is obtained by sum or difference frequency generation of the two pulses. When we use intense reference pulses, this method can be very sensitive. Blue/UV ultrashort pulses or ultrafast fluorescence which is excited by the near-IR pulses are usually very weak. In this case, the intense near-IR pulses are available to serve as a reference pulse using difference-frequency configuration. Commonly, FROG is expensive relatively and difficult to alignment for the pulse splitting and scanning, phase-matching alignment, and spectral measurement. To simplify FROG, a new version called Grating-eliminated no-nonsense observation of ultrafast incident laser light E-fields (GRENOUILLE) [9] is proposed. As shown in Figure 3, a Fresnel biprism is introduced to replace the beam splitter, delay line, and the beam combiner, a thick crystal is introduced acts as not only the nonlinear medium but also a spectrograph for its angle-independent phase-matching angle and narrow bandwidth. The size of the commercial version is controlled at \(26\times 4.5\times 11.5\) cm\({}^{3}\)[40]. GRENOUILLE is an all-transmission device thus is simple to align and very stable. However, the large thickness of the crystal which required much longer than the group-velocity mismatch length may limit its capability in measuring complex pulse. The trace of FROG can be seen as a joint time-frequency distribution of the pulse, which makes it has some advantages in characterizing complex-pulses compared with autocorrelation or SPIDER that is based on 1D-measurement, such as when applies for \"coherent artifact\" [41; 42]. When multi-shot measurements are made for an unstable, complex pulse train, the autocorrelation trace consists of a narrow spike atop a broad structureless background. The spike, or coherent artifact, reflects the coherent, nonrandom component of the pulse and is usually narrower than the correct pulse length. The broad background indicates the instability, complexity in the pulse train. More important, coherent artifacts may also exist in a pump-probe measurement. [PERSON] et al. [41] compared several different  
Figure 2: Schematic of PG-FROG. Time-to-space encoded scanning is caused by cross incidence. techniques in measuring unstable pulse trains. The results show that FROGs (SHG-FROG, PG-FROG, and XFROG) perform much better than both Autocorrelator and SPIDER, which only yield coherent artifacts. FROGs retrieve the correct pulse length, distinguish a stable train of short pulses from an unstable train of much longer pulses, benefit by their time-frequency over-determine. PG-FROG and XFROG can even yield the structure roughly. As mentioned above, 1D phase-retrieval problem is unsolvable, to obtain the spectral phase, FROG introduces a delay scanning dimension \(\tau\) in spectral measurement, to construct a solvable 2D spectrogram \(S(\omega,\tau)\). Here, \(\tau\) is not irreplaceable. If appropriately replace \(\tau\) with another parameter \(\xi\), we may construct another solvable trace \(S(\omega,\xi)\), we call this trace generalized spectrogram. Nowadays, some generalized spectrogram methods are proposed, such as Multiphoton intrapulse interference phase scan (MIIPS) [43; 44; 45; 46], dispersion scan (d-scan) [47; 48; 49; 50; 51], Time-domain ptychography (TDP) [52; 53], chirp-scan [54], and so on. MIIPS, based on multiphoton intrapulse interference [55], uses a pulse shaper to scans the phase \(\varphi\) of a periodic function to modulate the SHG spectrum, and the unknown spectral phase can be retrieved from the trace \(S_{\text{MIIPS}}(\omega,\,\varphi)\). Simultaneously, the chirp in the pulse can also be compensated by the shaper. However, MIIPS with a 4f-shaper is complex and lowly efficient, meanwhile difficult to apply for pulses with very broad bandwidths. D-scan is an efficient method for the few-cycle pulse measurement and chirp compensation. Moreover, it has a very simple construction consisting of a double chirped mirror, a pair of wedges, an SHG crystal, and a spectrograph without any light split.