## 4 Methodology: Optical Design of the Prime System
### Design and Verification of the Underwater Optical Camera
We could obtain the area of each lens set on the wheel according to their fitness by using Equation (12). \[q(i)=\sum_{m=1}^{i}p(n)\ \ \textit{for\ }i=1,\ 2,\ \ldots,\ \textit{pop\_size} \tag{12}\]  
and  
\[q(0)=0 \tag{13}\]  
To randomly generate a number \(\alpha\) between 0 and 1, if \(q(i-1)<\alpha\leq q(i)\), we selected the genes of the \(i\)th lens for crossover. Moreover, the genes of two lens sets were required during a crossover; therefore, two selection mechanisms were required. Here, we let the genes of the two selected parent lens sets be \(\mathbf{X}\) = (\(x_{1}\), \(x_{2}\), \(\ldots\), \(x_{10}\)) and \(\mathbf{Y}\) = (\(y_{1}\), \(y_{2}\), \(\ldots\), \(y_{10}\)) and that of their children lens sets be \(\mathbf{Z}\) = (\(z_{1}\), \(z_{2}\), \(\ldots\), \(z_{10}\)). The crossover procedure for generating the next generation was subsequently implemented after the genes of two parent lens sets were selected. Since both selected parents would propagate their features to the new genes, we further used random crossover to select the glass material as follows:  
\[z_{i}=\beta x_{i}+(1-\beta)y_{i}\ \ \textit{for\ }i=1,2,\ldots,10 \tag{14}\]  
where \(\beta\) is a random number between 0 and 1, which indicated that the feature of new gene partially came from one of the parents and partially from the other. According to the preceding crossover calculation, we could obtain the genetics of the next-generation lenses. However, although living organisms adapt to their growth environment and pass on the better adapted genes to their next generation, their genes also mutate to make them more competitive in the environment. In terms of the application of the GA to the optimization of optical lenses, we defined a mutation rate \(p_{m}\) between 0 and 1 and selected a random value between 0 and 1 for each gene. When \(\gamma<p_{m}\), we implemented a mutation calculation for corresponding gene. Since the common Abbe number is from 25 to 80, the genetic changes of their subsequent generation are expressed as follows:  
\[z_{i}=\textit{random\ number\ from\ }25\textit{to\ }80 \tag{15}\]  
In the preceding selection, crossover, and mutation were the basic calculations of the genetic algorithm. Figure 23 presents the implementation procedure of the GA in this study. This GA is based on pop_size 200, crossover rate 0.8, and mutation rate 0.3 for running 200 generations. Methodology for Extended Optimization of Zoom Optics with Genetic Algorithm Specific for Chromatic Aberration  
Optimization was executed by CODE V basically for most aberrations in this system so that it played the role at modulation transfer function (MTF). Genetic algorithm in this research was close to so-called \"extended optimization\" specific for chromatic aberrations only and nothing to do with other aberrations in this system. Therefore, there was no obvious contribution to MTF (2%) of the whole system via genetic algorithm extended aberrations. Why is chromatic aberration so sensitive to lens for under water purposes although it might not play the role at system MTF? Chromatic aberration might cause focus shift due to various wavelengths. From 1 m to 100 m for under water applications, there is plenty of variety regarding wavelengths so that is a better choice to minimize chromatic aberration first in order to guarantee the best performance from 1 m to 100 mm for under-water image application. From the simulation of CODEV and Zemax, there might be only 2% improvement of simulation MTF. Why is chromatic aberration so critical in practical systems? The key point is that chromatic aberration might be strongly-emphasized with so called integral aberrations effects due to ocean currents under water, which can be seen as a kind of time-domain variance of the glass index, which seriously degrades system performance although CODE V or Zemax is not able to simulate that cases.