## 8 Discussion  
It is important to note that although the presented algorithm is very accurate in obtaining the highest gain factor and determining the optimal frequencies for the ultrashort pump wave pulses, the parameters of the gain medium may limit the maximum achievable gain factor. For example, in our presented numerical experiments, the background permittivity values were set as \(\varepsilon_{\infty}=10\) and \(\varepsilon_{\infty}=12\) respectively, these values correspond to the background permittivity of many solid dielectric media in the near infra-red and the visible frequency range. A very large gain factor is also achievable for smaller values of the background permittivity at the expense of a slight increase in ultrashort pump wave intensity to compensate for the decreased stored energy in the micro-resonator. Similarly, for a gain material with a higher background permittivity, the required pump wave intensity can be lowered in order to achieve the same gain factor. Another important parameter that heavily influences the gain factor is the polarization damping rate or simply the damping rate \(\gamma\). A damping rate that is greater than \(\gamma=10^{11}\) slows down the rate of amplification via non-linear wave mixing and results in a lower gain factor [21].  
The frequencies of the ultrashort pump wave pulses can be practically tuned from the far IR (infrared) range to the near UV (ultraviolet) range. Commercial IR laser sources such as the Neodymium-YAG laser, or the Helium-Neon laser can be used in the near IR range and also in the visible range via frequency doubling. A visible laser light can be used in combination with a near IR laser light to generate a UV laser light via the process of sum frequency generation. A far IR laser light and even a THz laser light can be generated using two near IR laser lights of slightly different frequencies, through difference frequency generation process.  
Figure 11: Comparison of the parametric amplification rates with respect to the pump wave amplitude.  
Optical amplification via non-linear wave mixing offers the advantage of wide-band amplification of a monochromatic stimulus wave, practically in a frequency interval ranging from the far IR to the near UV part of the spectrum [1; 4; 19]. Hence, the numerical algorithm presented in Figure 4, can be used to amplify a monochromatic stimulus wave of any frequency that lies between the far IR to the near UV part of the spectrum.  
Our main aim in this article is to show that super-gain optical parametric amplification can be achieved in a simple fabry-perot type micro-resonator. For more complicated resonators including microring and microdisc resonators, the one-dimensional model presented in this article must be extended to a two-dimensional model. Our algorithm, which is presented in Figure 4, can be extended to a two-dimensional or a three-dimensional micro-resonator analysis using the technique that is presented in the article referenced in [22]. This article explains how, using FDTD, [PERSON]'s 2-D and 3-D curl equations in vector form can be effectively stepped in time simultaneously with a system of auxiliary differential equations, in order to create an integrated model that is capable of accurately simulating a medium involving instantaneous non-linearity, dispersive non-linearity, and multiple linear dispersions. Concerning implementation, the Greene-Taflove algorithm given in [22] can easily be incorporated into our algorithm during the discretization of the equations via finite difference time domain method. Hence the algorithm presented in this article can be employed in the analysis of arbitrary-shaped 2-D and 3-D micro-resonators.