## 3 Results and Discussion
### Cross-Correlation Measurements of the Optical Pulses from a GaInN MLLD  
The autocorrelation trace in Figure 2a shows that the output optical pulses from the GaInN MLLD without spectral filtering cannot be reproduced using either sech\({}^{2}\) or Gaussian function. The fact that the corresponding optical spectrum shown in Figure 2b is highly asymmetric suggests that the pulse shape was asymmetric in time. However, the intensity autocorrelation measurement provides only a symmetric time trace. In order to investigate the pulse characteristics, we used a cross-correlation technique. Since the femtosecond optical pulses were obtained by extracting the short wavelength edge of the optical spectrum, we can obtain cross-correlation signals between the direct optical output from the MLLD and the femtosecond optical pulses by inserting a BPF into one of the arms of the Michelson interferometer in the intensity autocorrelation setup, as shown in Figure 3a [10]. Figure 3b shows an example of cross correlation of the output optical pulses. The leading edge of the optical pulse is as steep as a few hundred femtoseconds while the trailing edge shows exponential decay on a picosecond scale. This cross-correlation trace is consistent with the intensity autocorrelation trace without spectral filtering shown in Figure 2a. We carried out further cross-correlation measurements in order to determine the time delay between spectral components at different wavelengths. We inserted another BPF into the other interferometer arm for this purpose, as shown in Figure 4a. The incidence angle for one BPF was fixed to pass a spectral component at a particular wavelength, whereas that for the other BPF was varied to extract spectral components at different wavelengths. Figure 4b shows the spectral components used for the cross-correlation analysis. These spectra were measured by introducing the optical pulses before the \(\beta\)-BBO crystal into the optical spectrum analyzer. Figure 4c shows the cross correlation between the spectral component at 402.3 nm and those at other wavelengths. The peaks in the cross-correlation traces are seen to be more delayed and the traces become broader as the wavelength increases. The peak intensity of the traces slightly increases from 402.3 nm to 403.0 nm with the increase in the corresponding spectral components because the cross-correlation traces are not much broadened. For the spectral components longer than 403.3 nm, however, the peak intensity of the cross-correlation traces decreases despite the increase  
Figure 3: **(a)** Schematic representation of cross-correlation measurement. BPF: Bandpass filter, BBO: \(\beta\)-BaB\({}_{2}\)O\({}_{4}\) crystal; **(b)** Cross-correlation measurement trace. Inset: Cross-correlation trace in logarithmic intensity scale. in the intensity of the corresponding spectral components. This is due to the broadening of the cross-correlation traces. Figure 5 shows the time delay of the cross-correlation peaks as a function of the wavelength. The dashed line in the figure represents the group delay due to the intracavity dispersion-compensation optics. The time delay of the cross-correlation peaks is mainly determined by the intracavity GVD. Thus, the exponential decay of the trailing edge of the optical pulses can be attributed to the time delay and to pulse broadening of the spectral components at longer wavelengths. Figure 4: **(a)** Schematic representation of cross-correlation measurement setup to measure the correlation between spectral components at different wavelengths; **(b)** Spectra for cross-correlation measurement. The thick blue trace is the spectrum after the fixed BPF. The additional traces are spectra after the other BPF at different transmission wavelength. The gray trace represents the entire spectrum before the BPF. The wavelength is different from Figure 2b since the MLLD configuration is not the same; **(c)** Cross-correlation signals between the fixed spectral component at 402.3 nm and other spectral components.