## 3 Results and Discussion
### Test Pulse Characterization
#### 3.2.2 Characterization of an Optical Beat  
When two color pulses temporally overlap, an optical beat is generated to form a sinusoidal wave with a period determined by the frequency separation. The XFROG trace measured for the test pulse (Figure 6a) indicates that the two femtosecond pulses in the test pulse had temporally overlapped with each other. The temporal intensity profile of an optical beat is determined by the following five parameters: (a) frequency separation; (b) temporal separation; (c) relative intensities at the two frequencies; (d) spectral intensity profile and phase of the two pulses; and (e) relative spectral phase. The four parameters (a-d) are given by the present XFROG measurement. The frequency separation, ca. 4155 cm\({}^{-1}\), can be evaluated from the location of the two spectral components. The temporal separation between the two pulses forming the test pulse is evaluated from the relative time delay of the two components in the FROG trace (Figure 6a). The time delays are slightly different from each other, indicating that the two pulses at 800 nm and 1180 nm in the test pulse had not completely overlapped. The relative intensities at the two frequencies are evaluated from the relative intensities of the two signals in the XFROG trace. The spectral intensity profile and the phase of the two pulses are retrieved from the FROG trace as discussed in the previous section. Unfortunately, the relative spectral phase cannot be measured from the FROG trace because of the longer duration of the reference pulse than the spacing of the optical beat. Such a relative spectral phase would provide information of the temporal intensity profile of the optical beat; the envelope of the pulse is determined by the temporal intensity profiles of the two color pulses (Figure 7a,b). This is because the two waves are periodically in phase somewhere within the envelope as long as the width of the envelope is longer than the period of the optical beat. The spacing of the beat is determined by the frequency separation of pulses A and B, not by the relative phase. This would not be the case for an ultrashort optical pulse synthesized using more than two discrete spectral components [(26)], for instance a pulse train synthesized using multicolor laser emissions generated via FWM [(5; 6; 7; 19)]. The characterization of an optical beat formed by two-color laser beams is useful for the evaluation of the bandwidth available in the diagnostics. The accuracy in the relative spectral intensities of the two spectral components retrieved in the FROG analysis allows us to judge whether the diagnostic tool has a broader bandwidth than the frequency separation between the two color pulses or not. The present result suggests that the spectral range is covered from 800 nm to 1180 nm. The FROG error in the phase retrieval was 0.4% at a grid size of 1024 \(\times\) 1024. The retrieved XFROG trace (Figure 6b) was in good agreement with the measured XFROG trace (Figure 6a). In the retrieved spectral intensity profile (Figure 9a), there are two spectral components corresponding to pulses A and B. The spectral intensity distribution of the two components agreed well with the shape of the spectral intensity distribution obtained in the phase retrieval, in which the FROG signals of pulses A and B were separately analyzed (see the first paragraph of this section). The ratio of the integrated spectral intensities of these pulses observed in the retrieved spectrum was calculated to be 1.0, which was in good agreement with the energy ratio (0.96) measured using a power meter. This indicates that the relative spectral intensities (Figure 9a) reflect relative intensities of the two pulses. In other words, the bandwidth of the XFROG system was sufficiently broad to support the width corresponding to the frequency separation of the two pulses, i.e., 4155 cm\({}^{-1}\). Figure 10 shows the temporal intensity profiles retrieved from two XFROG traces measured independently.