## 3 Methods
### Primary Design Approximation of Optical Lens
The equation for the aspherical surface is shown in Equation (8), where \(c\) represents the curvature and y represents the vertical distance from the optical axis when the ray falls on the refracting surface. \(A_{4}\) and \(A_{6}\) are \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{BFL} & **Before** & **R1** & **53.9982** & **R4** & **28.8431** & **D11** & **16.7154** \\  & **After** & & 37.235853 & & 18.078948 & & 13.959187 \\ \hline \multirow{2}{*}{EFL} & **Before** & **R2** & 25.7944 & **R3** & 45.1546 & **R25** & \(-\)88.95502 \\  & **After** & & 20.841198 & & 30.198396 & & \(-\)134.9441 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Combined optical layout BFL and EFL change value before and after correction. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **RMS (mm)** & **100\% (mm)** & **SA\({}_{\mathbf{A}}\) (mm)** & **SA\({}_{\mathbf{T}}\) (mm)** \\ \hline (a) & 3.571648 & 11.712321 & \(-\)0.873108 & \(-\)2.69256 \\ (b) & 2.343004 & 9.375108 & 0 & \(-\)1.155744 \\ (c) & 1.485043 & 7.553029 & 0.64045 & \(-\)0.000062 \\ (d) & 1.029953 & 6.184465 & 1.1 & 0.843992 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Spot size and the spherical aberration change due to the radiation correction of the synthesized optical system. Figure 4: Optical layout after combining the optical layout of JP20160012034A2P and the optical layout of KR1020140124286 AP. the fourth- and sixth-order aspherical coefficients, respectively, and the change of the Seidel aberration coefficient by the fourth-order aspherical coefficient can be expressed as Equations (9)-(12) [17]. \[ \begin{array}{ll}\mathit{Sag}_{AS}&=\frac{cv^{2}}{1+\sqrt{1-c^{2}y^{2}}}+A_{4} y^{4}+A_{6}y^{6}\cdots\\ &=\frac{y^{2}}{2r}+\frac{\left(1+8y^{2}A_{4}\right)y^{4}}{8\kappa^{3}}+\frac{ \left(1+16y^{2}A_{6}\right)y^{6}}{16y^{5}}\cdots,\end{array}  \tag{8}\]  
\[ \begin{array}{ll}\mathit{S}_{I\,\,total}=S_{I}+\delta(n-n^{\prime})A_{4}y^{4},\end{array}  \tag{9}\]  
\[ \begin{array}{ll}\mathit{S}_{IItotal}=S_{II}+\delta(n-n^{\prime})A_{4}y^{3} \overline{y},\end{array}  \tag{10}\]  
\[ \begin{array}{ll}\mathit{S}_{III\,\,total}=\mathit{S}_{III}+\delta(n-n^{ \prime})A_{4}y^{2}\overline{y}^{2},\end{array}  \tag{11}\]  
\[ \begin{array}{ll}\mathit{S}_{V\,\,total}=S_{V}+\delta(n-n^{\prime})A_{4}y \overline{y}^{3},\end{array}  \tag{12}\]  
where \(S_{I}\) represents the coefficient of the spherical aberration of the Seidel aberration coefficient, the subscript total represents the Seidel aberration coefficient including the aspherical surface, \(n\) represents the refractive index immediately before the refracting surface, \(n^{\prime}\) represents the refractive index of the refracting surface, \(y\) is the height of the axial ray of the refracting surface, \(S_{II}\), \(S_{III}\), and \(S_{V}\) are the coma aberration, astigmatism, and distortion aberration, and \(\hat{y}\) represents the principal ray height at the corresponding refracting surface. Moreover, as the aspherical surface becomes larger, the material and processing cost increase [18]. Therefore, the 18 th surface was selected as the aspherical surface to correct spherical aberration considering both the correction effect and the processing cost. In this study, a method to reduce the total number of aberrations by adjusting the balance with higher order aberrations was used. Using Equation (9), the fourth-order aspheric coefficient \(A_{4}\) when the Seidel spherical aberration is zero was calculated as \(-1.982156\mathrm{e}^{-5}\), and Figure 6b shows the order difference when it is applied to the 18 th surface. In this case, by comparing with Figure 6a, which is the order difference before the correction, it can be confirmed that the longitudinal spherical aberration shifted in the positive direction. Additionally, as shown in Table 3 line (b), \(\mathrm{SA}_{\mathrm{T}}\) and spot size decreased from 3.571648 to 2.343004 mm. Here, \(\mathrm{SA}_{\mathrm{T}}\) is the sum of the third-order, fifth-order, and seventh-order spherical aberrations.