## 3 Results and Discussion  
The values of the numerical parameters are summarized in Table 1, and are applied throughout this section unless otherwise stated. The values of the branching rations and energy level lifetimes are given in Table 2[21]. The spontaneous emission parameter \(\gamma_{sp}\) has been approximated using an equation derived in [22]:  
\[\gamma_{sp}=2h\upsilon\ \Delta\upsilon\ \sigma_{SE} \tag{8}\]  
where \(\upsilon\) is the signal frequency corresponding to the wavelength of 2800 nm, while \(\Delta\upsilon\) is assumed to be equal to 10 nm, following the experimental results presented in [9]. First, a study of the steady state characteristics of the fiber laser is performed. Figure 5. shows the dependence of the output signal power on the pump power for a 2 m long fiber with 0.04 reflectivity for signal wave at \(z=L\). Both linear and logarithmic scales are used to present the results, in order to show in more detail the behavior of the fiber laser near threshold. The lasing threshold occurs near 1.8 W of pumping power. When the facet reflectivity for signal wave at z = L is increased to 0.2, the threshold pump power decreases significantly to nearly 1.3 W (Figure 6). Further, the laser differential and wall-plug efficiency increase. Figure 4: Flowchart showing the sequence followed during numerical calculations. Figure 5: Numerically calculated dependence of the output signal power on the pump power for a 2 m fiber length and 0.04 reflectivity for signal at \(z=L\): (**a**) Presented on a linear scale. The inset shows zoomed region near the lasing threshold; (**b**) Presented on a logarithmic scale. \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Unit** & **Value** \\ \hline \(b_{1}/b_{2}\) & & 0.1/0.16 \\ \(W_{11}\) & m\({}^{3}\)/s & \(1\times 10^{-24}\) \\ \(W_{22}\) & m\({}^{3}\)/s & \(0.3\times 10^{-24}\) \\ \(\lambda_{p}\) & m & \(976\times 10^{-9}\) \\ \(\lambda_{s}\) & m & \(2.8\times 10^{-6}\) \\ \(\sigma_{GSA}\) & m\({}^{2}\) & \(2.1\times 10^{-25}\) \\ \(\sigma_{SE}\) & m\({}^{2}\) & \(4.5\times 10^{-25}\) \\ \(N_{Er}\) & 1/m\({}^{3}\) & \(9.6\times 10^{26}\) \\ \(L\) & m & 2 \\ \(\Gamma_{p}\) & & 0.011 \\ \(\Gamma_{s}\) & & 1.0 \\ \(\alpha_{p}\) & 1/m & \(23\times 10^{-3}\) \\ \(\alpha_{s}\) & 1/m & \(3.0\times 10^{-3}\) \\ \(R_{p}(z=0)\) & & 0 \\ \(R_{p}(z=L)\) & & 0.04 \\ \(R_{s}(z=0)\) & & 0.04 \\ \(R_{s}(z=L)\) & & 0.04 \\ \hline \end{tabular}
\end{table} 
Table 1: Modelling parameters. Figure 6: Numerically calculated dependence of the output signal power on the pump power for a 2 m fiber length and 0.2 reflectivity for signal at \(z=L\): (**a**) Presented on a linear scale. The inset shows zoomed region near the lasing threshold; (**b**) Presented on a logarithmic scale. After discussing the static characteristics of the fiber laser, the discussion moves to the time domain analysis of the Q-switched pulses. Figure 7 shows the time dependence of the facet reflectivity on time, as used in the time domain simulations of the Q-switched laser operation. In the following simulations, the \(R_{sL}\) is equal to 0.04, while \(R_{sH}\) is equal to 0.2. Therefore, the laser cavity is effectively switched between the two steady states considered in simulation results presented in Figures 5 and 6. The value of the initial leading time \(T_{i}\) is assumed to be equal to zero. The time of the signal duration \(T_{d}\) is equal to 1 \(\upmu\)s, while the time of the entire period \(T_{p}\) is equal to 30 ms. The transition time \(T_{tr}\) is equal to 200 ns, unless otherwise stated. During the transition, the pulse follows the sine squared (rising edge) or cosine squared (descending edge) shape considered within the first quadrant. Figure 8 shows a numerically calculated dependence of the output power 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, at four selected values of the pump power: 1.2 W (a), 1.5 W (b), 2 W (c) and 8 W (d).