## 3 Wave Propagation in Non-Linear Dispersive Media
In this case, the pair of equations that model the propagation of \(E_{2}\) is given as:  
\[\
abla^{2}(E_{2})-\mu_{0}\varepsilon_{\infty}\frac{\partial^{2}(E_{2})}{ \partial t^{2}}=\mu_{0}\sigma\frac{\partial(E_{2})}{\partial t}+\mu_{0}\frac{ \partial^{2}P_{2}}{\partial t^{2}}. \tag{5a}\] \[\frac{\partial^{2}P_{2}}{\partial t^{2}}+\gamma\frac{\partial P_{2}}{\partial t }+{\omega_{0}}^{2}(P_{2})-\frac{{\omega_{0}}^{2}}{Ned}(P_{2})^{2}-\frac{{ \omega_{0}}^{2}}{N^{2}e^{2}d^{2}}(P_{2})^{3}=\frac{Ne^{2}}{m}(E_{2}) \tag{5b}\]  
\(P_{2}\) : _Polarization density due to the electric field \(E_{2}\)_, \(P_{1}\) : _Polarization density due to the electric field \(E_{1}\)  
Now assume that \(E_{1}\) and \(E_{2}\) are simultaneously propagating in the same medium, in that case the pair of equations that describe the total electric field propagation is written as:  
\[\
abla^{2}(E_{1}+E_{2})-\mu_{0}\varepsilon_{\infty}\frac{\partial^{2}(E_{1}+E _{2})}{\partial t^{2}}=\mu_{0}\sigma\frac{\partial(E_{1}  
From Equation (7a), we can see that the stimulus wave \(E_{1}\) has a source term \(P_{1}\), which is the polarization density of the stimulus wave. The source term \(P_{1}\) is also dependent on the source term (polarization density) of the high-intensity pump wave \(P_{2}\). This means that in order to solve for the stimulus wave \(E_{1}\), we need to solve for the pump wave \(E_{2}\) as the equations for the stimulus wave and the pump wave are coupled to each other via the coupling term \(P_{2}\). Therefore, the stimulus wave can be solved by simultaneously solving all these four equations (Equations (5a,b) and (7a,b)).  
Our goal is to maximize the stimulus wave magnitude at a given stimulus wave frequency; \(\max\{\big{|}E_{1}(f_{st}=f^{\prime})\big{|}\}\). In order to achieve this, we will make a dispersion analysis that is based on the high-intensity pump wave frequency. By sweeping the excitation frequency of the pump wave (\(f_{p}\)) in a large spectral interval that extends from the far-infrared region to the near-ultraviolet region, we can investigate the pump wave frequency dependent variation of the maximum stimulus wave magnitude for \(f_{st}=f^{\prime}\), and we can select the optimal pump wave frequency value that maximizes the magnitude of the stimulus wave for \(f_{st}=f^{\prime}\). Mathematically our optimization problem can be stated as:  
\[\text{\emph{Given that }}\ f_{min}<f_{p}<f_{max}\ ;\ \ \ \mathbf{max}\ \big{|}E_{1}(f_{st}=f^{\prime})\big{|}\ \ \text{based on the following equations}\] \[\
abla^{2}\Big{(}E_{2}\Big{(}f_{p}\Big{)}\Big{)}-\mu_{0}\varepsilon_{\infty} \frac{\partial^{2}\Big{(}E_{2}\Big{(}f_{p}\Big{)}\Big{)}}{\partial t^{2}}=\mu_ {0}\sigma\frac{\partial\Big{(}E_{2}\Big{(}f_{p}\Big{)}\Big{)}}{\partial t}+\mu_ {0}\frac{\partial^{2}P_{2}}{\partial t^{2}} \tag{8a}\] \[\frac{\partial^{2}P_{2}}{\partial t^{2}}+\gamma\frac{\partial P_{2}}{\partial t }+{\omega_{0}}^{2}(P_{2})-\frac{{\omega_{0}}^{2}}{\text{Ned}}(P_{2})^{2}-\frac {{\omega_{0}}^{2}}{N^{2}e^{2}d^{2}}(P_{2})^{3}=\frac{Ne^{2}}{m}\Big{(}E_{2} \Big{(}f_{p}\Big{)}\Big{)}\] (8b) \[\
abla^{2}\Big{(}E_{1}\Big{(}f_{p}\Big{)}\Big{)}-\mu_{0}\varepsilon_{\infty} \frac{\partial^{2}\Big{(}E_{1}\Big{(}f_{p}\Big{)}\Big{)}}{\partial t^{2}}=\mu_ {0}\sigma\frac{\partial\Big{(}E_{1}\Big{(}f_{p}\Big{)}\Big{)}}{\partial t}+\mu_ {0}\frac{\partial^{2}(P_{1})}{\partial t^{2}}\] (8c) \[\frac{\partial^{2}(P_{1})}{\partial t^{2}}+\gamma\frac{\partial(P_{1})}{ \partial t}+{\omega_{0}}^{2}(P_{1})-\frac{{\omega_{0}}^{2}}{N\alpha}\Big{\{} \big{|}P_{1}}^{2}+2P_{1}P_{2}\Big{\}}-\frac{{\omega_{0}}^{2}}{N^{2}e^{2}d^{2} }\Big{\{|}P_{1}}^{3}+3{P_{1}}^{2}{P_{2}}+3{P_{1}}{P_{2}}^{2}\Big{\}}\] (8d) \[\frac{\partial^{2}P}{\partial t^{2}}+\gamma\frac{\partial P}{\partial t}+{ \omega_{0}}^{2}P-\frac{{\omega_{0}}^{2}}{Ned}P^{2}-\frac{{\omega_{0}}^{2}}{N^{ 2}e^{2}d^{2}}P^{3}=\frac{Ne^{2}}{m}E. \tag{4}\]0  
where  
\[\frac{\partial^{2}P}{\partial t^{2}}+\gamma\frac{\partial P}{\partial t}+{ \omega_{0}}^{2}P-\frac{{\omega_{0}}^{2}}{Ned}P^{2}-\frac{{\omega_{0}}^{2}}{N^{ 2}e^{2}d^{2}}P^{3}=\frac{Ne^{2}}{m}E. \tag{4}\]1  
\[\frac{\partial^{2}P}{\partial t^{2}}+\gamma\frac{\partial P}{\partial t}+{ \omega_{0}}^{2}P-\frac{{\omega_{0}}^{2}}{Ned}P^{2}-\frac{{\omega_{0}}^{2}}{N^{ 2}e^{2}d^{2}}P^{3}=\frac{Ne^{2}}{m}E. \tag{4}\]2  
If we are using \(N\) ultrashort high-intensity pulse excitations to amplify the low-intensity stimulus wave,  
\[\frac{\partial^{2}P}{\partial t^{2}}+\gamma\frac{\partial P}{\partial t}+{ \omega_{0}}^{2}P-\frac{{\omega_{0}}^{2}}{Ned}P^{2}-\frac{{\omega_{0}}^{2}}{N^{ 2}e^{2}d^{2}}P^{3}=\frac{Ne^{2}}{m}E. \tag{4}\]3  
\[\frac{\partial^{2}P}{\partial t^{2}}+\gamma\frac{\partial P}{\partial t}+{ \omega_{0}}^{2}P-\frac{{\omega_{0}}^{2}}{Ned}P^{2}-\frac{{\omega_{0}}^{2}}{N^{ 2}e^{2}d^{2}}P^{3}=\frac{Ne^{2}}{m}E. \tag{4}\]4  
\[\frac{\partial^{2}P}{\partial t^{2}}+\gamma\frac{\partial P}{\partial t}+{ \omega_{0}}^{2}P-\frac{{\omega_{0}}^{2}}{Ned}P^{2}-\frac{{\omega_{0}}^{2}}{N^{ 2}e^{2}d^{2}}P^{3}=\frac{Ne^{2}}{m}E. \tag{4}\]5 \[\frac{\partial^{2}P}{\partial t^{2}}+\gamma\frac{\partial P}{\partial t}+{ \omega_{0}}^{2}P-\frac{{\omega_{0}}^{2}}{Ned}P^{2}-\frac{{\omega_{0}}^{2}}{N^{ 2}e^{2}d^{2}}P^{3}=\frac{Ne^{2}}{m}E. \tag{4}\]6 \[\frac{\partial^{2}P}{\partial t^{2}}+\gamma\frac{\partial P}{\partial t}+{ \omega_{0}}^{2}P-\frac{{\omega_{0}}^{2}}{Ned}P^{2}-\frac{{\omega_{0}}^{2}}{N^{ 2}e^{2}d^{2}}P^{3}=\frac{Ne^{2}}{m}E. \tag{4}\]7  
\[\frac{\partial^{2}P}{\partial t^{2}}+\gamma\frac{\partial P}{\partial t}+{ \omega_{0}}^{2}P-\frac{{\omega_{0}}^{2}}{Ned}P^{2}-\frac{{\omega_{0}}^{2}}{N^{ 2}e^{2}d^{2}}P^{3}=\frac{Ne^{2}}{m}E. \tag{4}\]8  
\[\frac{\partial^{2}P}{\partial t^{2}}+\gamma\frac{\partial P}{\partial t}+{ \omega_{0}}^{2}P-\frac{{\omega_{0}}^{2}}{Ned}P^{2}-\frac{{\omega_{0}}^{2}}{N^{ 2}e^{2}d^{2}}P^{3}=\frac{Ne^{2}}{m}E. \tag{4}\]9  
We want to tune the frequencies (\(v_{1},\ v_{2},\ \ldots,v_{N}\)) of these ultrashort pulses so that the gain factor is maximized.