Limits of ultrashort pulse stability

As it was shown in Section , there exist two basic mechanisms of the ultrashort pulse destabilization, which cause stable multipulse operation, viz., background amplification due to insufficient gain saturation by the pulse with reduced energy or excitation of the perturbation bounded within the high-energy ultrashort pulse. Hence there may exist a certain range of parameters providing a stable ultrashort pulse operation. The Figs. , a - f demonstrate such regions on the parametric plane, viz., the GDD and $\sigma$ coefficients. Increase of $\sigma$ corresponds to a shift towards the edge of the cavity stability region. The confined regions A correspond to the stable single pulse operation (for negative as well as positive values of the GDD coefficient). The B regions depict the stable multipulse operation. And, finally, the C regions are the domains of the unstable pulsed operation or the CW generation (on the lower boundaries of B). In Fig. , graph a is used as the reference and other pictures b - f are obtained by variation of only one of the parameters. In particular, graph b illustrates the decrease of the modulation depth $\gamma$, graph c - the increase of the output loss $\rho$, graph d - the increase of the $\epsilon$ parameter, graph e - the decrease of the pump $P$, and finally graph f - the decrease of the gain relaxation time $T_r$.

Figure: The regions of a pulse stability. The parameters are varied relatively the figure (a): (b) corresponds to the $\gamma$ decrease, (c) - the $\rho$ increase, (d) - the $\epsilon$ increase, (e) - the pump decrease, (f) - the $T_r$ decrease. P= $1.6\times 10^{-4}$ (a - d, f), $8\times 10^{-5}$ (e); =6 $\mu$s (a - e), 3 (f); $\epsilon$=0.0013 (a - c, e, f), 0.0026 (d); $\rho$=0.01 (a, b, d - f), 0.02 (c); $\gamma$=0.02 (a, c - f), 0.01 (b). A are the regions of the stable single pulse generation, B are the regions of the stable multiple pulse operation and C are the regions of the chaotical or CW lasing. Curve 1 shows the parameters providing the chirp-free pulse generation in the soliton model, curve 2 is the limit of the pulse stability from the soliton model.

As we can see, there always exist certain upper and lower limits on $\sigma$ for stable single pulse operation. The lower boundary (small $\sigma$, i. e. the absorber is too "hard" to be saturated) is caused by the background amplification and can separate the single pulse operation from the multiple pulses as well as also from CW lasing (for the positive and large negative GDD). The upper boundary (large $\sigma$, i. e. "soft" absorber with low saturation intensity) results from the destabilization due to the bounded perturbation growth. It should be noted, that as a result of the strong hysteresis, the upper stability boundary has a "fuzzy" character (Fig. )

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