Nature of the stability boundaries

The $\sigma _{lower}$ decreases as a result of the $\vert D\vert$ increase. Such behavior can be obtained also on the basis of the soliton model. For example, in the weak-nonlinear limit one can obtain the following condition of the background amplification [2]:

(curve 2 in Fig. ). Here we keep the normalization of $\sigma$ and $D$ to $\beta$ and $t_f^2$, respectively. We have only qualitative agreement with the numerical results owing to the weak-nonlinear approximation and the simplified model of the gain behavior in the referenced model. However even this simplified model agrees with numerical calculation for the lower stability threshold by the order of $\sigma$ magnitude. As it can be seen from the Fig. , the behavior of the lower stability boundary in the region of positive GDD agrees with our qualitative treatment as well. We have to stress that the destabilization in the framework of the soliton model takes place only for the chirped pulses, while it is not a necessary condition in the numerical model in the region of negative GDD. The curve 1 in Fig. shows the location of the system parameters corresponding to the chirp-free sech-pulse for the weak-nonlinear approximation (this is [17]; we keep the usual normalizations). The vicinity of the stability boundary to this curve causes the destabilization of the nearly chirp-free pulse. The agreement with the soliton model results from the comparatively small loss saturation for small $\sigma$. The strong deviation from the weak-nonlinear model takes place for $\vert D\vert\rightarrow 0$, where the pulse intensities are sufficiently large for the strong loss saturation. The referenced expression for $\sigma _{lower}$ explains also the increase of the stability boundary as a result of the modulation depth decrease (see lower boundary of the A region for $D <$ 0 in Fig. , b) and the weak dependence of the lower stability boundary on other laser parameters (with the exception of $\epsilon$) for the large values of $\vert D\vert$ (see Figs. , c, e, f). Formally, the dependence of $\sigma _{lower}$ on $\gamma$ is obvious in the weak-nonlinear approximation: the expansion of Eq. () on $\vert a\vert^{2}$ gives $\gamma \sigma$ as the self-amplitude modulation coefficient in the first order. The $\gamma$ decrease reduces the contribution of the self-amplitude modulation, i. e. the difference between the pulse and background net-gain, thus favoring the pulse destabilization. The influence of the self-phase modulation on the pulse stability can be interpreted in the following way. Increasing $\beta$ favors the pulse destabilization because the pulse energy decreases with growing spectral loss as a result of the pulse spectrum expansion. Additionally, this spectral expansion reduces the gain saturation ( $\epsilon \propto 1/\beta$). This also leads to the pulse destabilization by the background. The $\epsilon$ growth intensifies the gain saturation and so stabilizes the pulse against the background amplification (Fig. , d). Here we do not consider the possible stabilization against automodulations produced by the self-phase modulation [17]. The contribution of the destabilization due to the bounded perturbation growth complicates the picture. Since the amplitude of such perturbation scales with the pulse energy [42], [44], the decrease of the pulse energy will result in the pulse stabilization against this instability. The pulse energy decreases for $\vert D\vert \rightarrow \infty$, but increases with $\sigma$. Hence the $\sigma _{upper}$ defining the pulse destabilization increases due to the $\vert D\vert$ increase (see Figs. ). For some $\vert D\vert _{min}$ switching between destabilization mechanisms is possible (see the transition from the positive to the negative net-gain and from the chirped to chirp-free pulses at the stability boundary presented in Fig. ). Since the pulse duration decreases with $\vert D\vert$ decrease, such switching (if it takes place) confines the minimal duration of the ultrashort pulses (the minimal pulse widths are shown by points in Figs. ). It should be noted that destabilization due to the spectral loss can occur also due to the chirp growth. This is illustrated by the Fig. , b, where the pulse spectral width at the stability boundary is shown to be as much as 50 times that of the bandwidth-limited pulse.

Figure: The net-gain (a) and time-bandwidth product related to the Schrödinger soliton one (b) at the single-pulse stability boundary shown in Fig. , a.

Thus, the existence of the minimal and maximal $\sigma$ defining the pulse stability results from the two different mechanisms of the ultrashort pulse destabilization, viz., the destabilization due to the continuum growth and pulse splitting due to the increase of the bounded perturbations. The twofold character of the destabilization complicates the laser optimization, as considered in the next section.

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