Variation of the self-amplitude modulation

The variation of the $\sigma$ parameter, which corresponds to the change of the loss saturation intensity, is the common way to adjust the mode-locking efficiency in a typical Kerr-lens mode locking setup. For example, the growth of $\sigma$, corresponding to the saturation intensity decrease, can be achieved by setting the laser operating point closer to the edge of the cavity stability range [40]. As it was shown in [2,40], the typical normalized to the self-phase modulation coefficient values of $\sigma$ lie in the range of 1$\div$100 for the four-mirror Ti:sapphire laser. As Fig. , $a$ shows, the increase of $\sigma$ shortens the ultrashort pulse. Such shortening is accompanied by the growth of the phase retardation, the pulse energy and the intensity due to the saturation of the fast absorber. The energy rise decreases the net-gain outside the pulse due to the gain saturation (Fig. , $b$). It should be noted that for the given parameters the single pulse is almost chirp-free for the increasing $\sigma$ (Fig. , c).

Figure: The pulse width (a), the net-gain coefficient outside the pulse (b) and the time-bandwidth product related to the one for the Schrödinger soliton c for the single, double and triple pulses versus $\sigma$. Captions at points denote the inter-pulse distances. P= $8\times 10^{-5}$, $D$=-5000 $fs^{2}$.

The most interesting phenomenon caused by the $\sigma$ change is the existence of the mechanism of ultrashort pulse destabilization, which differs from the one considered above. We can see from Fig. that there exists the minimum and the maximum of $\sigma$ protected from the pulse destabilization. The small $\sigma$ does not allow to saturate the fast absorber. The absorber loss stay high, the pulse energy does not grow and as a consequence, the gain cannot saturate. Hence the net-gain becomes positive. This results in the pulse destabilization due to the background radiation growth. The existence of the maximum $\sigma$ providing the pulse stabilization is less trivial. As we can see from the Figure, the transition to the multiple pulse generation due to the increasing $\sigma$ is not accompanied by the sign change of the net-gain coefficient. In this case, there is no satellite growing outside the pulse: the pulse dissociates by itself. In the framework of the linear perturbation analysis (weak nonlinear limit $\sigma \gamma \ll$ 1) the satellite growth can be described by the excitation of the perturbation modes with continuous spectrum, while the pulse dissociation corresponds to the discrete spectrum of the perturbations. The loss saturation forms the ``potential well'', which can contain the nondecaying bounded ``states'' corresponding to the perturbations. This property is enhanced by the potential deepening and widening, i. e. with the pulse energy growth. For the examples of slow saturable absorber case, see [41,42]. The numerical analysis outside the weak nonlinear approximation also demonstrates the pulse destabilization due to the self-amplitude modulation growth in the absence of the background amplification [43]. The wings of such perturbations are visible in Fig. . Here the solid and dash-dotted curves show the logarithm of the field intensity in the vicinity of the stability threshold and far from it, respectively. The dashed line corresponds to the profile of the sech-pulse. The slowly decaying exponential wings corresponds to the ``bounded'' perturbations, which increase as a result of the approach to the stability boundary (transition from the dash-dotted to the solid curve). As the increase of the pulse intensity saturates the fast absorber, its discrimination strength decreases [13], thus favoring the perturbation growth and the pulse dissociation (dotted curve). The described picture agrees qualitatively with the analytical analysis presented in [45], where the complex amplitude of the perturbation is proportional to the pulse energy and the parameter of the saturation of the self-amplitude modulation. The profile of this perturbation is close to the dotted curve in Fig. [45].

Figure: The logarithm of the pulse intensity for $\sigma$=38 (solid), 30 (dash-dotted), 39 (dotted). Dashed curve correspond to the soliton profile.

We have to note also the existence of the strong multistability as a result of the $\sigma$ increase (see Figure ). The region of the single and double (and even triple) pulses coexistence is wide and becomes apparent as a result of the variation of the initial field in the simulations. If we start from the arbitrary initial (regular) field, the probability of the multiple pulse generation is increased by the $\sigma$ increase.

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