## 2 Methods and Design
### Design of the Fundus Imaging System
#### 2.2.2 Aspheric Lens Design  
Considering the practicalities, the design adopted a single lens group, which had a simple structure, could achieve a compact design, and eliminated some of the aberrations of the human eye and the main aberration of the system. It could also avoid manufacturing defects as it is easier to accurately process lenses that meet requirements.  
In the design, the full field of view was 50\({}^{\circ}\). A large field of view would cause the fundus imaging to exceed the ideal range of the paraxial axis [18], which would inevitably increase the difficulty of aberration correction of the edge field of view of the system.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**No.** & **Surface Type** & **Radiums/mm** & **Thickness/mm** & **Material** & **Diameter** & **Cone Coefficient** \\ \hline OBJ(RETINA) & Standard & 11.000 & 16.580 & VITREOUS & 11.000 & 0.000 \\
1(LENS) & Standard & 6.000 & 3.700 & LENS & 5.000 & \(-3.000\) \\
2 & Standard & \(-10.000\) & 0.100 & AQUEOUS & 5.000 & 0.000 \\ STO(IRIS) & Standard & Infinity & 1.600 & AQUEOUS & 11.000 & 0.000 \\
4 & Standard & \(-11.000\) & 1.500 & AQUEOUS & 11.000 & 0.000 \\
5(CORNEA) & Standard & \(-6.700\) & 0.520 & CORNEA & 6.000 & \(-0.300\) \\
6 & Standard & \(-7.800\) & 2 & & 6.000 & \(-0.500\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of eye model.  
Figure 3: Eye model. (**a**) Standard eye model. (**b**) Optimized eye model.  
When the incident height of the ray on an aspheric surface is \(h\) and the casting height of the auxiliary rays is \(h_{z}\), the primary aberration coefficient [15] is as follows. In Equation (1), \(\Delta S_{I}\), \(\Delta S_{\Pi}\), \(\Delta S_{IV}\), and \(\Delta S_{V}\) are spherical aberration, coma, astigmatism, field curvature, and distortion. \(n\) is the object space refractive index, \(n^{\prime}\) is the image space refractive index, \(r\) is the radius of curvature, and \(b\) is the distance of the ray from the focal point of the optical axis to the aspheric surface. It can be seen that the aberration related to the field of view such as spherical aberration and comet changes due to the surface shape; on the contrary, we could improve the ability of the system to correct the aberration and simplify the system optical path by introducing the aspheric surface type.  
\[\begin{cases}\Delta S_{I}=\left(n^{\prime}-n\right)\frac{h}{h^{\prime}}h^{4} \\ \Delta S_{\Pi}=\Delta S_{I}\frac{h_{z}}{h}\\ \Delta S_{III}=\Delta S_{I}\left(\frac{h_{z}}{h}\right)^{2}\\ \Delta S_{IV}=0\\ \Delta S_{V}=\Delta S_{I}\left(\frac{h_{z}}{h}\right)^{3}\end{cases} \tag{1}\]  
The general rotationally symmetric aspheric expression is as follows in Equation (2), where \(k\) is the coefficient of the conic curve, \(c\) is the basic curvature of the vertex, \(r\) is the radial coordinate of the perpendicular optical axis [15], and \(a2,a4\), and \(a6\) are the higher-order coefficients of the aspheric equation.  
\[Z(r)=\frac{cr^{2}}{\left(1+\sqrt{1-(1+k)c^{2}r^{2}}\right)}+a_{2}r^{2}+a_{4}r^ {4}+a_{6}r^{6}+\ldots \tag{2}\]  
More optimization variables were introduced through aspheric surfaces, providing more freedom for the system design. So, the aspheric surface can reduce the aberration of the edge field of view. The ability of one aspheric surface to correct aberrations was equivalent to that of multiple spheres, which not only reduced the size and weight of the optical system but also achieved high resolution at large fields of view.