Pulse stabilization

In this section we summarize the ways to achieve the stable single pulse operation in the system prone to the multiple pulse lasing. As the shortest pulse width for the fixed GDD is achieved by the $\sigma$ growth, the main goal of the system's optimization is to enhance the stability against the bounded perturbations. The obvious way in this direction is the decrease of the intracavity energy. Such decrease can be undesirable for some systems (the high-power pulse sources, for example) and can degrade the self-start ability for the Kerr-lens mode-locked lasers. Moreover, this decrease can reduce the gain saturation and enhance the destabilization due to the background growth. Let us describe the main approaches to the single pulse stabilization. 1) The increase of the modulation depth, i. e. the $\gamma$ parameter, expands the region of the single pulse generation with subsequent pulse shortening and reduces the zone of unstable operation (transition from b to a in Fig. ). It should be noted, that such scenario decreases the self-starting ability, and may be a bad choice for many solid-state lasers. However, this way is acceptable for the Cr$^{2+}$:ZnSe laser due to its large values of $n_{2}$ and $P$ (for given pump intensity $I_{p}$). 2) The increase of the linear loss, i. e. $\rho$, expands the stability region for the single pulse operation. However, the pulse shortening that can be achieved in this way is comparatively small (compare Fig. , a and Fig. , c). This is so because the background amplification is not suppressed in this case owing to the reduced gain saturation (pulse energy is lower in the case of higher $\rho$), so that there is only small improvement to the minimum GDD defining the minimal pulse duration. 3) The most appropriate choice is the increase of the $\epsilon$ parameter (Fig. , d in comparison to Fig. , a). The increase of the gain saturation in comparison to the self-phase modulation contribution prevents both the background and bounded perturbation growth. Unfortunately, this parameter is constant for the given laser medium with fixed length x. Taking shorter active medium with the same $\alpha _{max}$ enhances the stability. In this context, the Cr$^{2+}$:ZnSe is very attractive in comparison to other media due to the large value of $\epsilon$. For example: =1.3 $\times 10^{-3}$, =6.9 $\times10^{-4}$, =6.3 $\times10^{-4}$, $\epsilon _{Ti:sap}$=3.6 $\times10^{-4}$ The high value of $n_{2}$ for ZnSe is partially compensated by the large $\sigma _{g}$, $\lambda$ and n (see definition of $\epsilon$ in section ). 4) Another approach is to decrease $P$ by decreasing $I_{p}$ or $T_{cav}$ (Fig. , e in comparison to Fig. , a) or to decrease $T_r$ (Fig. , f). Unfortunately, the advantage of Cr$^{2+}$:ZnSe consisting in its large absorption cross-section turns into disadvantage: other parameters being equal, the larger $\sigma _{a}$ and smaller $\nu _{p}$ result in the larger $P$. For example, for 1.5 W of the absorbed pump power at the center of the corresponding absorption lines and 100 $\mu$m diameter of the pumping beam $P_{Cr:ZnSe}$ = 5.9 $\times10^{-4}$, $P_{Yb:KYW}$ = 6.3 $\times10^{-5}$, $P_{Ti:sap}$ = 5.4 $\times10^{-5}$, $P_{Cr:LiSGaF}$ = 1.2 $\times10^{-5}$. The upper-laser level lifetime $T_r$ is the material constant, and can be reduced (e.g. by heating or concentration quenching) only at the expense of higher laser threshold. It should be noted, that according to Fig. , the pulse intensity decrease provided by the methods 2) - 4) increases value of $\sigma$, at which the minimum pulse duration is achieved. This can demand a very thorough optimization of the laser design. Note also that the finger-like shape of the stability region for the small negative GDD providing the minimal pulse durations requires fine system optimization by the appropriate choice of $\sigma$. This feature is more critical for higher $\gamma$ (compare graphs a and b in Fig. ).

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