## 7 Verification of Our Computational Model
Coupled with the wave equation, the total wave \(E=E_{1}+E_{2}\) is computed from:  
\[\frac{E(i+1,j)-2E(i,j)+E(i-1,j)}{\Delta x^{2}}-\mu_{0}\varepsilon_{ \infty}(i,j)\frac{E(i,j+1)-2E(i,j)+E(i,j-1)}{\Delta x^{2}} \tag{21a}\] \[=\mu_{0}\sigma(i,j)\frac{E(i,j)-E(i,j-1)}{\Delta t}+\mu_{0}\frac{P(i,j+1)-2P(i )+P(i,j-1)}{\Delta t^{2}}.\] (21b) \[\frac{P(i,j+1)-2P(i,j)+P(i,j-1)}{\Delta t^{2}}+\gamma\frac{P(i,j)-P(i,j-1)}{ \Delta t}+\omega_{0}{}^{2}(P(i,j))-\frac{\omega_{0}{}^{2}}{N\omega^{2}}(P(i,j) )^{2}\] \[-\frac{\omega_{0}{}^{2}}{N^{2}e^{2}d^{2}}(P(i,j))^{3}=\frac{N^{2}}{m}(E(i,j )).\]  
For a time interval of \(0\leq t\leq t_{max}\), the computational formula for frequency up-conversion efficiency is:  
\[\eta_{computational}=\frac{\text{Intensity of the $\omega_{3}$ frequency component of the total wave at t}=t_{max}}{\text{Intensity of the $\omega_{2}$ frequency component of the total wave at t}=0} \tag{22}\]  
In this example, we have used the following values for each efficiency formula:  
\[\omega_{2}=(2\pi\times 180)THz,\omega_{1}=(2\pi\times 120)THz\]  
L = length of the interaction media = 3.33 \(\mu m\) (from x = 3.33 \(\mu m\) to 6.66 \(\mu m\))  
\(\omega_{3}=frequency\) of the upconverted wave = \(2\pi\times 300 THz\), n= \(Refractive\ index=\sqrt{10}\)  
\(d\) = material non-linearity coefficient = \(6.3\times 10^{-22}\) (The theoretical and the computational results agree for this value of \(d\) for a sample pump wave amplitude of \(A_{2}=10^{9}V/m\) (see Figure 9). Our aim is to see if the results also agree for all the other pump wave amplitudes for this value of d)  
\(A_{2}\) = pump wave amplitude (varied from \(5\times 10^{7}V/mto\ 2\times 10^{9}V/m\) )  
\(A_{1}\) = input wave amplitude = \(A_{2}/100\) (\(A_{1}\ll A_{2}\))  
**Example 2**. : _Optical amplification via non-linear wave mixing (  
Figure 9: Comparison of the frequency up-conversion efficiencies for \(f_{3}=300\) THz and d = \(6.3\times 10^{-22}\), versus the pump wave amplitude. The high-intensity pump wave \(\mathbf{E_{2}}\) is generated at x = 2.5 \(\upmu\)m. It has an amplitude of \(A_{2}\) V/m and a frequency of 220 THz. \[\mathbf{E_{2}}(x=2.5\mu m,t)=A_{2}\times\sin\!\left(2\pi\!\left(2.2\times 10^{14} \right)\!t+\varphi_{2}\right)\ V/m\]  
The input wave \(\mathbf{E_{1}}\) is generated at x = 2.5 \(\upmu\)m. It has an amplitude of \(A_{1}\) V/m and a frequency of 140 THz (see Figure 10). \[\mathbf{E_{1}}(x=2.5\mu m,t)=A_{1}\times\sin\!\left(2\pi\!\left(1.4\times 10^{14} \right)\!t+\varphi_{1}\right)\ V/m\ (\text{assume that}\ \varphi_{1}=0,\ \varphi_{2}=0)\]  
_Range of independent simulation variables: \(0\leq x\leq 13\mu m,\ \ 0\leq t\leq 60 ps\) Resonance frequency of the interaction medium_ : \(f_{0}=3\times 10^{14}Hz\) Damping coefficient of the interaction medium : \(\gamma=1\times 10^{11}Hz\) Dielectric constant of the interaction medium \((\varepsilon_{\infty})=1+\chi=10\ \ \ \ (\mu_{r}=1)\)_  
_Spatial range of the interaction medium : \(4\mu m<x<9\mu m\) Left perfectly matched layer \((\)left absorption layer\()\) is from \(x=0\) to \(x=2.25\mu m\) Right perfectly matched layer \((\)right absorption layer\()\) is from \(x=10.75\mu m\) to \(x=13\mu m\)_  
Our goal is to amplify the input wave via energy coupling from the high-power pump wave.