## 6 Numerical Experiments
### Triple-Frequency Tuning for Gain Factor Optimization
Based on the above formulations, the maximum stimulus wave amplitude that has been reached in the cavity (for \(0<\text{t}<40\) ps) is determined as \(Gain_{max}=\left|E_{st}(f_{st}=300\textit{THz})\right|_{max}=6.34\times 10^{8}V /m\), which corresponds to the frequencies \(f_{p1}=290.8\textit{THz}\), \(f_{p2}=410.6\textit{THz}\), \(f_{p3}=209.7\textit{THz}\) (see Table 2), or to the free-space wavelengths \(\lambda_{p1}=1.032\mu\)m, \(\lambda_{p2}=0.731\mu\)m, \(\lambda_{p3}=1.431\mu\)m as the ultrashort pulses can be practically generated outside the resonator. In this simulation, the gain factor of the amplification is defined as:  
\[ \begin{array}{ll}Gain_{max}&=\left|E_{st}(\ f_{st}=300\textit{THz})\right|_{ max}\\ &=\frac{\textit{Amplitude of the stimulus wave at $t=t_{max}$ for $x=x^{\prime}$}}{\textit{Amplitude of the stimulus wave at $t=0$ for $x=x^{\prime}$}}\end{array} \]  
where \(x^{\prime}\) is chosen as an intra-cavity point (\(x^{\prime}=5\mu m\)) and \(t_{max}=40\textit{picoseconds}\). \[\textit{W}_{e,p}=\textit{Stored electric energy density via pump wave}=\frac{1}{2}\varepsilon_{\infty}E_{pump}^{2}+\frac{1}{2}E_{pump}P_{pump}.\ \ \left(\frac{\textit{Joules}}{m^{3}}\right)\]  
\[\textit{P}_{pump}:\ \textit{Polarization density created by the pump wave}\left(\frac{\textit{Coulomb}}{m^{2}}\right)\]  
\[\textit{E}_{pump}:\ \textit{P}_{pump wave electric field intensity},\ \varepsilon_{\infty}:\textit{Background permittivity}\]  
As we can see from Table 2, the optimal ultrashort pulse frequencies correspond to a very high stored electric energy density and a high polarization density. If we have a look at Table 2, we can see that the stored electric energy density and the polarization density must be simultaneously high for a significant stimulus wave amplification. The stored electric energy density indicates the achievable order of stimulus wave amplification. The polarization density serves as an energy-coupling coefficient and is a measure of the stored electric energy that can be coupled to the stimulus wave. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(Gain_{max}\) & \(f_{p1}\) & \(f_{p2}\) & \(f_{p3}\) & \(\textit{W}_{e,p}\ (\frac{I}{m^{3}})\) & \(\textit{P}_{pump}(\frac{C}{m^{3}})\) & **k (Iteration \#)** \\ \hline
2.84 & 200 THz & 150 THz & 100 THz & 1.41 \(\times\) 10\({}^{7}\) & 0.07 & 1 \\
5.47 & 205 THz & 153 Hz & 102 THz & 1.32 \(\times\) 10\({}^{7}\) & 0.09 & 2 \\
4.43 & 216.2 THz & 165.4 THz & 115.1 THz & 2.13 \(\times\) 10\({}^{7}\) & 0.06 & 5 \\
9.28 & 239.5 THz & 189.8 THz & 133.5 THz & 3.76 \(\times\) 10\({}^{7}\) & 0.08 & 8 \\
57.78 & 297.2 THz & 228.3 THz & 168.8 THz & 6.61 \(\times\) 10\({}^{7}\) & 0.09 & 11 \\
129.51 & 296.0 THz & 344.1 THz & 159.2 THz & 6.18 \(\times\) 10\({}^{7}\) & 0.10 & 14 \\
396.63 & 284.1 THz & 392.0 THz & 202.5 THz & 7.98 \(\times\) 10\({}^{7}\) & 0.10 & 17 \\
3049.7 & 287.6 THz & 399.3 THz & 204.6 THz & 8.53 \(\times\) 10\({}^{7}\) & 0.12 & 20 \\
4.2 \(\times\) 10\({}^{4}\) & 289.6 THz & 403.4 THz & 206.1 THz & 1.07 \(\times\) 10\({}^{8}\) & 0.17 & 23 \\
7.6 \(\times\) 10\({}^{5}\) & 290.7 THz & 408.7 THz & 208.9 THz & 1.72 \(\times\) 10\({}^{8}\) & 0.19 & 26 \\
1.4 \(\times\) 10\({}^{7}\) & 291.9 THz & 410.2 THz & 209.4 THz & 1.49 \(\times\) 10\({}^{8}\) & 0.23 & 31 \\
2.9 \(\times\) 10\({}^{8}\) & 291.1 THz & 410.8 THz & 209.9 THz & 2.36 \(\times\) 10\({}^{8}\) & 0.22 & 34 \\
6.3 \(\times\) 10\({}^{8}\) & 290.8 THz & 410.