## 2 Methods and Design
### Design of the Fundus Illumination System
#### 2.3.1 Illumination Source Selection and Theoretical Calculation
Referring to the provisions of GB/T 7247.9-2016 \"Safety of Laser Products--Part 9: Maximum Allowable Exposure of Incoherent Optical Radiation\" [22] on the maximum allowable amount of irradiation that can be allowed by the human eye, the following formula was adopted [12]:  
\[L_{max}=\left(\frac{1\times 10^{6}}{t}\right)W/\left(m^{2}\cdot sr\right) \tag{4}\]  
\(t\) is the irradiation time, \(W/\left(m^{2}\cdot sr\right)\) is the unit of radiance, and \(sr\) is the unit of solid angle. Assuming illumination time \(t=1\) s, the maximum brightness of the LED can be calculated as \(L_{max}=10^{6}W/\left(m^{2}\cdot sr\right)\). Figure 7: Spectral distribution of light sources. (**a**) White LED spectrum. (**b**) Full-spectrum white LED spectrum. (**c**) The solar spectrum LED’s spectral distribution and color coordinates. Knowing the maximum brightness perceived by the eye, the maximum power of the light source \(Pmax\) can be calculated from Equation (5):  
\[P_{max}=L_{max}\cdot A_{fundus}\cdot\Omega_{source}/\eta \tag{5}\]  
\(Lmax\) is the maximum brightness of the light source, \(A_{fundus}\) is the illumination area of the light source in the fundus, \(\Omega_{source}\) is the solid angle of the illuminated beam, and \(\eta\) is the radiation efficiency of the light source [23]. The fundus illumination area was about 300 mm\({}^{2}\) = 0.0003 m\({}^{2}\), the beam solid angle was about 0.6\(sr\), and the radiation efficiency of the LED was about 0.3, so \(P_{max}=10^{6}\times 0.003\times 0.6/0.3=600\)\(w\). In order to ensure the safety of human eyes, when selecting a light source, its power should be within 1/100-1/10 of the maximum allowable power [23]. The solar spectrum LED met these requirements. Topology shows that the condition of mapping region A to region B and the boundary condition of mapping the boundary of region A to region B are sufficient and necessary conditions for each other. When the mapping is continuous, the edge rays of the light source must reach the edge of the target surface through the elements of the optical system. If the edge rays are already mapped within the required range, the other rays must be within the target spot. Based on the principle of edge light, the design reduced the consideration of other rays and simplified the calculation of optical systems with complex light distributions. The light source of the fundus illumination system is an LED light source, and its light-emitting surface is a certain size. The refractive index of the space where the light source is located is n, the area of the luminous surface is \(dA\), and the solid angle of the beam is \(d\Omega\)[23]. The angle between the normal of the luminescent surface element and the center line of the solid angle is \(\theta\), and the spatial properties of the beam of the light source can be described by the optical expansion, and the mathematical form of the optical expansion is given by Equation (6):  
\[E=n^{2}\iint\cos\theta dAd\Omega \tag{6}\]  
Assuming that the light source is an ideal point light source, in the direct coordinate system \(xyz\), the light source is located at the origin of the coordinate system, the optical axis is in the positive direction of the \(z\)-axis [23], the direction of light propagation is expressed by the angle \((\theta,\ \varphi)\), the angle between the ray and the positive direction of the \(z\)-axis is \(\theta\), the angle between the projection of the ray on the \(xyz\) plane and the positive direction of the \(x\)-axis is \(\varphi\), and the luminous intensity of the light source is \(I_{s}(\theta,\varphi)\). Then, the luminous flux of the light source is given by Equation (7):  
\[\phi_{s}=\int I_{s}(\theta,\varphi)d\Omega \tag{7}\]  
The illuminance of the light source reaching the fundus is \(E(x,y)\), and this value is related to the coordinates \((x.y)\).