Conclusion

The numerical analysis taking into account the saturable gain, the spectral filtering due to the gain band profile, the net-group-delay dispersion, the self-phase modulation and the fast loss saturation induced by the Kerr-lensing in the active medium allows to identify the main sources of the multiple pulse generation in the Kerr-lens mode-locked solid-state lasers. As it was shown, the nature of the single pulse destabilization leading to multipulsing is defined by the interplay between the gain and loss saturation in the combination with spectral filtering. The stable single pulse operation in the negative as well as positive GDD regions is limited by the saturation parameter, so that for the fixed GDD there exist its lower and upper values confining the single pulse stability region. The lower stability boundary corresponding to the transition to multipulsing is caused by the continuum amplification. This results from the lower gain saturation caused by the pulse energy decrease. The latter is induced by the growth of the pulse spectral loss due to the pulse shortening or its chirping. Since the multiple pulse generation originates from the continuum, the inter-pulse distances and phase differences are random. The upper stability boundary corresponding to the transition to the stable or unstable multipulse generation comes from the growth of the perturbation bounded within the pulse profile. The continuum amplification plays the minor role in this case because the net-gain for it is negative. As a result of the pulse energy growth accompanying the $\sigma$ increase, the pulse splits and the pulse satellites appear. The inter-pulse distance in this case has a good repeatability and the phase difference change is very slow so that there exists a set of attracting points, where an inter-pulse phase difference "stays" a longer time. This provides regular autocorrelation traces and spectral profiles. Our analysis suggests the existence of the inter-pulse interaction producing their binding without the usual inter-soliton interaction through the oscillating tails. This interaction is caused by a balance of the saturable gain, the spectral and the saturable loss. As a result of this balance, there exists a preferred inter-pulse distance and a phase difference providing the minimal net-loss for the propagating pulses. The revealed sources of the multiple pulse generation allow to formulate the main methods to suppress multipulsing. These are: reduction of the pump rate; decrease of the gain relaxation time; increase of the loss modulation depth, of the output loss or of the gain saturation. All these methods can shorten the pulse duration due to the extension of the single pulse stability zone into the vicinity of zero GDD. However, this occurs at the cost of the worse Kerr-lens mode locking self-starting ability or increase of the $\sigma$ parameter. The latter demands very thorough laser optimization because the high $\sigma$ values in Kerr-lens mode-locked lasers are achieved only at the very edge of the resonator stability range. The obtained numerical results are verified for the Cr$^{2+}$:ZnSe laser operating in the positive net-GDD region and for the Ti:sapphire laser operating in the regime of the chirp compensation. There is both qualitative and quantitative agreement between the theoretical and the experimental data. The presented analysis can be applied to the optimization of the lasers based on the various well known active media and to the estimation of the Kerr-lens mode locking stability of the new prospective active crystals.

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