## 6 Numerical Experiments
### Double-Frequency Tuning for Gain Factor Optimization
In this simulation, the gain factor of the amplification is defined as:  
\[Gain_{max} =\left|E_{st}\left(\ f_{st}=250 THz\right)\right|_{max}\] \[=\frac{\text{Amplitude of the stimulus wave at }t=t_{max}\ for\ x=x^{\prime}}{\text{Amplitude of the stimulus wave at }t=0\ for\ x=x^{\prime}}.\]  
where \(x^{\prime}\) is chosen as an intra-cavity point (\(x^{\prime}=5.73\mu m\)) and \(t_{max}=10\)_picoseconds_. \[W_{e,p}=\text{Stored electric energy density via pump wave}=\frac{1}{2}\varepsilon_{\infty}E_{ pump}{}^{2}+\frac{1}{2}E_{pump}P_{pump}.\left(\frac{\text{Jules}}{m^{3}}\right)\]  
\[P_{pump}:\text{ Polarization density created by the pump wave}\left(\frac{\text{Coulomb}}{m^{2}} \right)\text{, \ }E_{pump}:\text{Pump wave electric field intensity}\] \[\varepsilon_{\infty}:\text{Background }(\text{infinite spectral band})\text{permittivity}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \(f_{\mathbf{p}1}\) & \(f_{\mathbf{p}2}\) & \(Gain_{max}\) & \(W_{e,p}\left(\frac{I}{m^{3}}\right)\) & \(P_{pump}(\frac{C}{m^{2}})\) & **k (Iteration \#)** \\ \hline
100 THz & 100 THz & 0.84 & 1.29 \(\times\) 10\({}^{7}\) & 0.08 & 1 \\
107.2 THz & 115.7 THz & 6.23 & 1.80 \(\times\) 10\({}^{7}\) & 0.09 & 4 \\
154.9 THz & 156.6 THz & 4.44 & 3.36 \(\times\) 10\({}^{7}\) & 0.08 & 7 \\
218.6 THz & 199.5 THz & 313.52 & 4.48 \(\times\) 10\({}^{7}\) & 0.10 & 10 \\
198.3 THz & 214.5 THz & 37.16 & 1.67 \(\times\) 10\({}^{7}\) & 0.10 & 13 \\
229.5 THz & 243.0 THz & 240.58 & 5.97 \(\times\) 10\({}^{7}\) & 0.09 & 16 \\
263.1 THz & 227.2 THz & 646.72 & 5.54 \(\times\) 10\({}^{7}\) & 0.10 & 19 \\
322.7 THz & 278.9 THz & 1.57 \(\times\) 10\({}^{3}\) & 6.12 \(\times\) 10\({}^{7}\) & 0.12 & 22 \\
396.0 THz & 299.8 THz & 4.28 \(\times\) 10\({}^{4}\) & 9.39 \(\times\) 10\({}^{8}\) & 0.15 & 25 \\
391.6 THz & 293.4 THz & 9.16 \(\times\) 10\({}^{4}\) & 1.76 \(\times\) 10\({}^{8}\) & 0.16 & 28 \\
380.7 THz & 311.7 THz & 3.85 \(\times\) 10\({}^{5}\) & 1.26 \(\times\) 10\({}^{8}\) & 0.20 & 30 \\
383.4 THz & 317.2 THz & 8.11 \(\times\) 10\({}^{4}\) & 6.40 \(\times\) 10\({}^{7}\) & 0.20 & 32 \\
386.0 THz & 318.4 THz & 6.32 \(\times\) 10\({}^{6}\) & 2.89 \(\times\) 10\({}^{8}\) & 0.23 & 34 \\
386.8 THz & 318.8 THz & 9.79 \(\times\) 10\({}^{7}\) & 2.63 \(\times\) 10\({}^{8}\) & 0.27 & 36 \\
387.2 THz & 319.3 THz & 3.96 \(\times\) 10\({}^{8}\) & 2.51 \(\times\) 10\({}^{8}\) & 0.29 & 38 \\
387.2 THz & 319.4 THz & 4.67 \(\times\) 10\({}^{8}\) & 2.95 \(\times\) 10\({}^{8}\) & 0.29 & 39 \\ \hline \end{tabular}
\end{table} 
Table 1: BFGS algorithm-based optimization process. As we can see from Table 1, the optimal ultrashort pulse frequencies correspond to very high stored electric energy density and high polarization density. 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 [14; 15], and the polarization density acts as a coupling coefficient, which is a measure of how much stored electric energy can be coupled to the stimulus wave. The time variation of the spectrally broadened (polychromatic) stimulus wave between t = 6.6 picoseconds and t = 10 picoseconds is shown in Figure 6. From the figure, we can see that the polychromatic stimulus wave reaches an amplitude of approximately \(8\times 10^{8}\ V/m\).