## 3 Results and Discussion
The pump powers were selected to study a Q-switched laser operation in four different situations. Only the first microsecond is presented to show the shape of the generated pulse. At a pump power of 1.2 W, the fiber laser operates in a steady state below the threshold for the value of the facet reflectivity of both 0.04 and 0.2. At 1.5 W, the fiber laser operates below the threshold, with a facet reflectivity equal to 0.04 only, while at 0.2 it operates above the threshold. When the pump power equals 2 W, the fiber laser operates just above the threshold, both before and after switching. When the pump power is further raised to 8 W, one investigates the fiber laser's operation during Q-switching well above the lasing threshold. The results presented in Figure 8a show that when the laser operates below threshold, a low intensity pulse is formed. The pulse shape does not follow the switching pulse shape shown in Figure 7 exactly. The signal power increases approximately seven fold, while the facet reflectivity increases only five fold. When the pump power is increased to a value larger than the lasing threshold pump power for facet reflectivity, 0.04 (Figure 8b), a pulse is generated with a shape typical for Q-switching. The pulse has a fairly large delay when compared to the switching pulse raising edge (Figure 7). The delay results from the time needed to build up the pulse power within the cavity during subsequent round-trips after the raising edge of the facet reflectivity switching pulse. When the pump power is sufficient to \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Unit** & **Value** \\ \hline \(\tau_{1}\) & ms & 9 \\ \(\tau_{2}\) & ms & 6.9 \\ \(\tau_{3}\) & ms & 0.12 \\ \(\tau_{4}\) & ms & 0.57 \\ \(\beta_{21}\), \(\beta_{20}\) & & 0.37, 0.63 \\ \(\beta_{32}\), \(\beta_{31}\),\(\beta_{30}\) & & 0.856, 0.004, 0.014 \\ \(\beta_{43}\), \(\beta_{42}\), \(\beta_{41}\), \(\beta_{40}\) & & 0.34, 0.04, 0.18, 0.44 \\ \hline \end{tabular}
\end{table} 
Table 2: Branching ratios and level lifetimes. Figure 7: The time dependence of facet reflectivity at \(z=L\) for signal wave. reach the lasing threshold both before and after switching (Figure 8c), the pulse becomes narrower, and the delay of the pulse significantly reduces when compared to that presented in Figure 8b. Also, one can observe that pulse width narrowing is accompanied by a significant increase in the peak pulse power, from approximately 2.5 kW to nearly 10 kW. It is also noted that such output pulse peak power levels are observed in experiments. Further increasing of the pump power has only a negligible effect on the generated pulse shape (Figure 8d). There is only a small temporal shift of the pulse. This suggests that larger pump power allows for faster power build up within the cavity, which agrees with an intuitive guess. Finally, it is noted that with \(\gamma_{sp}\) in Equation (2) equal to zero, it is not possible to numerically model the pulse behavior below the lasing threshold, since a model without a spontaneous emission term would yield zero output power value. Thus, inclusion of spontaneous emission is essential in these simulations. The importance of the spontaneous emission will be further explored when discussing the behavior of the Q-switched laser cavity generating pulse trains. Figure 9 shows a numerically calculated dependence of the energy level populations on time for a 2 m long fiber, with the facet reflectivity for signal wave at \(z=L\) switched between 0.04 and 0.2. Four selected values of the pump power are considered: 1.2 W (a), 1.5 W (b), 2 W (c) and 8 W (d). The results from Figure 9 show that below the lasing threshold (Figure 9a), the generated pulse has practically no impact on the level populations due to its low photon density.