## 6 Numerical Experiments
### Triple-Frequency Tuning for Gain Factor Optimization  
Assume that a 300 THz (\(\lambda_{free\ space}=\)1 \(\upmu\)m) infra-red stimulus wave \(E_{st}\) and a high-power pump wave \(E_{tp}\) that is composed of three ultrashort pulses (frequencies are to be determined), are generated to propagate in a micro-resonator with two reflecting walls. The wall on the left side can be thought as an optical isolator, which fully transmits from its left side and almost fully reflects from its right side. The wall on the right side represents an optical band-pass filter with a frequency-dependent reflection coefficient \(\Gamma(f)\)(see Figure 7). Both waves are generated at x = 0 \(\upmu\)m and at the time instant t = 0 ps. The waves and the parameters of the gain medium are as given below:  
\[E_{tp}(x=0\mu m,t)=\sum\
olimits_{i=1}^{3}A_{i}\cos(2\pi f_{i}t+\psi_{i})(u(t) -\ u(t-\Delta T_{i}))(u(t):\text{{\it Unit step function}})\]  
where \(A_{1}=1.5\times 10^{8},\ A_{2}=2\times 10^{8},\ A_{3}=2.5\times 10^{8},\ \ \Delta T_{1}=4 ps,\ \ \Delta T_{2}=2 ps,\ \ \Delta T_{2}=1 ps\)  
\[E_{st}(x=0\mu m,t)=1\times\sin\!\left(2\pi\!\left(3\times 10^{14}\right)\!\right) \ V/m,\ \ \ for\ \ \ 0\leq t\leq 40 ps\]  
\[\text{{\it Dielectric constant of the gain medium}}=\ \varepsilon_{\infty}=1+\chi=10\ \ \ \ (\mu_{r}=1)\]  
\[\text{{\it Resonance frequency of the gain medium}}:\ \ f_{0}=600 THz\]  
\[\text{{\it Damping rate of the gain medium}}:\ \ \gamma=1\times 10^{10}Hz\]  
\[\text{{\it Duration of simulation}}:\ \ 0\leq t\leq 40 ps;\ \text{{\it Range of the gain medium}}:0\mu m<x<10\mu m\]  
\[\text{{\it Right cavity wall location}}:\ \ x=10\mu m;\ \text{{\it Left cavity wall location}}:\ \ x=0\mu m\]  
\[\text{{\it Electron density of the medium}}:\ \ N=3.5\times 10^{28}/m^{3};\ \ \text{{\it Atomic diameter}}:d=0.3\ nm\]  
Figure 6: Stimulus wave amplification (in polychromatic form) inside the cavity at x = 5.73 \(\upmu\)m.  
**Our problem:** Find the optimal pump wave pulse frequencies \(f_{p1},\ f_{p2},\ f_{p3}\) that maximize the magnitude of the monochromatic stimulus wave in the cavity (\(\big{|}E_{st}(\ f_{st}=300 THz)\big{|}\)), for 50 THz \(<\big{\{}f_{p1},f_{p2},\ f_{p3}\big{\}}<500\ THz,\ 0\mu m<x<10\mu m,0\leq t\leq 40\ ps\), such that  
\[\
abla^{2}E_{hp}\Big{(}f_{p1},\ f_{p2},\ f_{p3}\Big{)}-\mu_{0}\varepsilon_{ \infty}\frac{\partial^{2}E_{hp}\Big{(}f_{p1},\ f_{p2},\ f_{p3}\Big{)}}{ \partial t^{2}}=\mu_{0}\sigma\frac{\partial E_{hp}\Big{(}f_{p1},\ f_{p2},\ f_{p3} \Big{)}}{\partial t}+\mu_{0}\frac{\partial^{2}P_{hp}}{\partial t^{2}}\]  
\[E_{st}(x=0\mu m,t)=1\times\sin\!\left(2\pi\!\left(3\times 10^{14}\right)\!\right) \ V/m,\ \ \ for\ \ \ 0\leq t\leq 40 ps\]0  
\[E_{st}(x=0\mu m,t)=1\times\sin\!\left(2\pi\!\left(3\times 10^{14}\right)\!\right) \ V/m,\ \ \ for\ \ \ 0\leq t\leq 40 ps\]1  
\[E_{st}(x=0\mu m,t)=1\times\sin\!\left(2\pi\!\left(3\times 10^{14}\right)\!\right) \ V/m,\ \ \ for\ \ \ 0\leq t\leq 40 ps\]2  
\[E_{st}(x=0\mu m,t)=1\times\sin\!\left(2\pi\!\left(3\times 10^{14}\right)\!\right) \ V/m,\ \ \ for\ \ \ 0\leq t\leq 40 ps\]3  
It is important to emphasize that our aim is to maximize the magnitude of the stimulus wave at its original frequency \(f_{st}=300\ THz\) (monochromatic form). This is a precaution against any degree of spectral broadening that the stimulus wave may go through while being amplified. A plain attempt to maximize the magnitude of the stimulus wave (\(|E_{st}|\)) independent of the original excitation frequency (\(f_{st}\)), may result in an amplified stimulus wave with different frequency components. In fact, these different frequency components might be even much more dominant than the original excitation frequency of the stimulus wave.