## 3 Results and Discussion  
### Reference Pulse Characterization  
The SD and SHG FROG traces measured for the reference pulse are shown in Figure 3a,b, respectively. The SD FROG trace is slightly tilted with respect to a time delay, indicating that the reference pulse was slightly negatively chirped. In contrast, the shape of the SHG FROG trace indicates the presence of the third-order spectral phase distortion [24]. The SHG FROG trace is nearly symmetric; the slight asymmetry is considered to be caused by the slight spatial chirp of the pulses or by the nature of the optics used in this study. To evaluate the reliability in the characterization of the reference pulse, the two temporal intensity profiles of the reference pulse retrieved from the SHG and SD FROG were compared. The FROG errors [13] in the SD FROG and the SHG FROG were 0.4% and 0.5%, respectively, at a grid size of 256 \(\times\) 256. The small FROG errors suggest that the traces are completely retrieved. In general, two kinds of electric fields are randomly given from a single SHG FROG trace, one of which is the complex conjugated time-reversed replica of the other [13]. One of them was in good agreement with the field retrieved from the SD FROG trace. The temporal intensity profiles (Figure 4a)  
Figure 2: Phase matching curve when a type 15 \(\upmu\)m BBO crystal is used and an 800 nm pulse is used as a reference pulse. retrieved from the SHG and SD FROG traces coincide with each other. The duration of the reference pulse determined by SHG FROG was 58 fs, whereas its duration as characterized by SD FROG was 59 fs; these results are nearly identical. Although the FROG medium and the optical path to the point of measurement are different in these techniques, the temporal phases retrieved from the two FROG traces are in good agreement in the temporal range between \(-\)50 fs and 50 fs where 97% of the total energy of the reference pulse is contained. As shown in Figure 4b, the spectral intensity profiles retrieved from the SHG and SD FROG traces are nearly identical to each other. Also, these two FROG analyses converged after 50 to 70 iterations and are reproducible. Note that the SD FROG system allows the characterization of the reference pulse without the ambiguity in the direction of time. On the other hand, it is necessary to check the direction of time by inserting a thin glass plate in the beam path for the SHG FROG measurements [13]. If the time-reversed replica instead of the correct temporal profile was used in the phase retrieval from an XFROG trace, the analysis would lead to a large error in the evaluation of the test pulse [25]. For this reason, SD FROG is preferred for the characterization of a reference pulse. The characterized reference pulse was slightly negatively chirped. A fused-silica plate with a thickness of 19 mm was then placed in the beam path of the reference pulse to minimize the pulse duration. The SD FROG trace measured for the compressed reference pulse (Figure 5a) yields the temporal profile of the reference pulse after phase retrieval (Figure 5b). This compressed reference pulse (43 fs) was then used in the following XFROG measurements. Note that the temporal intensity of the pulse (Figure 5b) is slightly steeper in the trailing edge than the leading edge, which is in contrast to the reference pulse of Figure 4. The feature of the temporal profile would be related to the ratio between the second-order and third-order spectral phases in the pulse. A numerical simulation performed assuming a positive cubic spectral phase in the pulse qualitatively reproduced this temporal behavior. The temporal intensity profile of a laser pulse with a negligibly small second-order phase distortion and a large amount of positive third-order spectral phase distortion is noted to have a steep trailing edge, whereas that with a non-negligible second-order phase distortion and a large positive third-order spectral phase distortion has a steep leading edge.