## 3 Results and Discussion
### Discussions  
The time-bandwidth products in Figure 7 show that in the short wavelength region of the optical spectrum, there is a balance between optical nonlinearity and the negative intracavity GVD. In semiconductors, nonlinear optical effects are associated with photons with energies either below or above the band-gap energy. The former are instantaneous nonlinearities such as the Kerr effect. In semiconductor laser diodes, the cladding layers are transparent at the lasing wavelength so that the Kerr nonlinearity can occur. However, the nonlinear refractive index of GaN for photons with energies just below the band-gap energy has been reported to be large but negative [16]. This is due to the large two-photon absorption coefficient for photons close to the band-gap energy [17]. Therefore, a negative nonlinear refractive index is expected in the cladding layers, so that we consider that the instantaneous optical nonlinearity cannot balance the negative intracavity GVD in the GaInN MLLD. The other type of optical nonlinearity to be considered is saturation in the GaInN BS-LD. In the SA and gain sections, saturation leads to the opposite sign of change in the refractive index. In the gain section, the refractive index increases with decreasing carrier density during pulse propagation. In contrast, in the SA section, the refractive index decreases as the absorption saturates [18]. Because the frequency shift increases with the degree of absorption saturation [19], this can balance the negative intracavity GVD. In the model of the saturation dynamics [19], the nonzero phase-relaxation time of the SA, T\({}_{2}\), was not included assuming that pulse durations were longer than T\({}_{2}\). The carrier-carrier scattering time in GaN is reported to be less than 15 fs [20], so that the saturation dynamics in the SA can be applied to explain the frequency shift. From Figure 7, the width of the spectral region with the minimum time-bandwidth product is as large as about 2 THz. Since the absorption coefficient for the GaInN active layer has been reported to be larger than that for other semiconductors, the large frequency shift can be inherent to GaN-based materials [12]. After the steep leading edge is formed, the gain saturation can mainly contribute to the trailing edge of the optical pulses. Since gain saturation is accompanied by carrier depletion, the refractive index increases. The temporal change in the gain is characterized by the carrier lifetime and the carrier depletion due to the stimulated emission [21]. The optical pulses from the GaInN MLLD are considered to be much shorter than the sub-ns carrier lifetime, so that the trailing edge is formed by the spectral components that are red-shifted due to the gain saturation and are delayed due to the negative intracavity GVD. This can be a reason why the peak position in Figure 5 follows the time delay of the  
Figure 7: Time-bandwidth product of the optical pulses shown in Figure 6 as a function of the peak wavelength of the spectral components. Gray trace: the entire spectrum after the GaInN MLLD without the BPF. dispersion compensation optics. Temporal-broadening of the spectral components is more significant at longer wavelengths, as shown in Figure 6, due to the accumulation of these effects. This broadening behavior can also explain the asymmetric shape of the optical spectrum in Figure 2b. The spectrally-resolved autocorrelation traces in Figure 6 do not show coherence spikes, so the entire spectrum is coherent. This is consistent with the signal-to-noise ratio of greater than 60 dB for the RF spectrum shown in Figure 4b of Reference [10]. In our previous reports, we speculated that the femtosecond pulse generation mechanism involved soliton mode-locking, because the femtosecond optical pulses were produced by a GaInN MLLD with a negative intracavity GVD and an SA with slow absorption recovery [10,11].