Introduction

Ultrashort-pulse laser oscillators demonstrate a tendency to destabilization and pulse splitting when the pulse duration approaches minimal values (see, for example, [1]). As the net group-delay dispersion (GDD) in the resonator approaches zero, a nonregular pulsing or a stable multiple pulse generation may occur. Both the unstable as well as stable multipulse oscillation can be generally considered as the obstacle for the pulse shortening. At the same time, it is possible to use the pulse splitting for an additional duration decrease as well as for an increase of the laser repetition rate. The analysis of this phenomenon is not straightforward because the ultrashort-pulse laser is a complex system with different interacting nonlinearities. At the moment, the following mechanisms of the multipulse operation have been considered: the appearance of a negative feedback due to the pulse chirp growth [2]; the radiation scattering growth accompanying the decrease of the soliton period [1]; the higher-order dispersions contribution to laser dynamics [3]; the laser continuum growth owing to the positive net-gain of the background radiation outside the pulse [4]; the dynamical gain saturation and recovery [5]. The multipulse operation can also be considered as a result of the higher-order soliton dissociation [6]. In the framework of the soliton approach [7,8] it was shown, that the multipulse oscillation can be caused by the formation of the bounded multisoliton complexes in resonators with both negative and positive net-GDD. The self-interaction of such bounded soliton-like complexes is governed by the relative phase of the constituent pulses [9]. The presence of the continuous local (i.e. through the pulse wings) interaction between the pulses distinguishes such bounded soliton-like complexes from the regimes created as a result of the repetition-rate multiplication due to the change of the gain and loss saturation balance in the system [10,11,12]. One should also distinguish the stable multipulse operation from the generation of the double pulses colliding in an active medium [13,14,15,16]. It should be noted, that the transition to the multipulse operation is not the sole scenario of the stability loss in the contiuous-wave (CW) mode-locked solid-state lasers: the automodulational instability can produce regular as well as nonregular oscillations of the single pulse or the so-called picosecond collapse with the abrupt transition from the femtosecond to the picosecond generation [17,18]. The stable multipulse operation in the negative GDD region was experimentally observed in Ti:sapphire [1,19,20,21], Cr:LiSGaF [22] and Yb:KYW [23] Kerr-lens mode-locked lasers as well as in Nd:glass [24], Ti:sapphire [4] and Cr$^{4+}$:YAG [25] lasers mode-locked by semiconductor saturable absorber mirror. In Ref. [26] the tendency to the multipulse generation was reported for the positive-GDD regime in the Cr$^{2+}$:ZnSe laser with passive mode locking initiated by acousto-optical modulation. The latter medium allowing diode pumping and possessing excellent lasing characteristics (see [27,28]) is of interest as tunable ultrashort pulsed mid-IR source. At this moment, a variety of generation regimes has been demonstrated: the efficient pulsed [29] and continuous-wave [30,31] diode-pumped operation, active mode locking [32], and the active modulator-assisted passive mode locking [26]. Above all that, Cr$^{2+}$:ZnSe is of interest as the model object for the study of the multiple pulse operation. This interest is explained by the combination of the unique characteristics: very large nonlinear index of refraction [33] as well as emission and absorption cross-sections. As it will be shown later on, these factors play a crucial role in the pulse destabilization and the transition to the multiple pulse operation. In order to explain the variety of the observed multipulse regimes one needs to have a basic understanding of the nature of the multiple pulse generation in Kerr-lens mode-locked lasers. Our model takes into account the strong saturation of the Kerr-lens induced fast absorber, the gain saturation and recovery dynamics, the GDD and the self-phase modulation, making it valid for the different femto- and picosecond Kerr-lens mode-locked lasers and sufficiently simple to obtain the physically significant conclusions. Our analysis reveal the existence of the two basic mechanisms causing the multiple pulse operation: the background amplification due to the gain saturation decrease and the increase of the "bounded" perturbation in the presence of the strong saturation of the fast absorber. The basic laser factors defining the pulse stability were found to be the gain and loss saturation in the combination with the spectral loss. As it will be demonstrated, the inter-pulse interaction is strong enough to produce the correlations of the inter-pulse distances and phases, which are governed by the GDD and the fast absorber parameters. The statistical and multistable properties of the bounded multiple pulse complexes will be considered as well. Finally, the limits of the single pulse operation and the methods of the stability enhancement will be analyzed.

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