## 2 Optical Design of the LED Light Source System
### Optical Design of the CTLAS System  
The light emitted from the quasi-Lambertian LEDs always diverges outward and requires convergence on the specimen to enhance the irradiation values. The CTLAS system we designed is composed of a TIR-lens array and a confocal system to efficiently collect the light energy of the LEDs array, as shown in Figure 4. In the CTLAS system, lens 1 is built up by attaching the TIR-lens array to the front lens of the confocal system as a one-piece lens. The left side of lens 1 is a \(3\times 3\) TIR-lens array used to assemble the 9 LEDs.  
Figure 4: Optical setup of the confocal total internal reflection lens array system (CTLAS) system.  
Figure 3: (**a**) The optical model of the LED, (**b**) the normalized intensity distribution comparison of the experimental result and the simulation result.  
The TIR lens is designed to collimate the divergent light of a single LED with a total internal reflection effect, as shown in Figure 5a. The height and diameter of the designed TIR lens are, respectively, 17.5 mm and 35.5 mm, and the material is PMMA with a 1.49 refractive index.  
By ASAP, the ray tracing the result of the TIR lens is shown in Figure 5b. We can see that, after passing through the TIR lens, the divergent light is well collimated. In light of the testing performance, we further acquired the result that the TIR-lens array can collimate the divergent light from the \(3\times 3\) LED array and turn it into quasi-collimating light with a larger beam size, as shown in Figure 6a. The side length of the TIR-lens array is 106.5 mm, and the normalized intensity distribution of the TIR-lens array is shown in Figure 6b. According to the simulation results, the half-width-half-magnitude (HWHM) value of the normalized intensity distribution of the TIR-lens array is \(\pm 2.8^{\circ}\). The optical efficiency, defined as the ratio of the luminous flux of the TIR-lens array to the total luminous flux of the LED, is 91.7%.  
In order to enhance the illuminance, the second surfaces of lens 1 and lens 2 form a confocal system. The quasi-collimating light of a larger beam size passing through the second surface of lens 1 is focused on its focal point, which is also the front focal point of lens 2. After that, the focal light will re-emerge from the focal point and form a small-size quasi-collimating light beam. As shown in Figure 7, lens 2 is a plano-convex lens.  
Figure 5: (**a**) The structure of the designed total internal reflection (TIR) lens (the values are in mm), (**b**) the ray tracing the result of the TIR lens.  
Figure 6: The designed \(3\times 3\) TIR-lens array, (**a**) the ray tracing simulation result, (**b**) the corresponding normalized intensity distribution.  
The back surfaces of lenses 1 and 2 are both aspherical in order to eliminate the spherical aberration. Those surfaces are also calculated by the sag function [48] which can represent the shape of the aspherical surface and which is obtained by  
\[z(r)=\frac{cr^{2}}{1+\sqrt{1-(1+k)c^{2}r^{2}}}+Ar^{A}, \tag{2}\]  
where \(z(r),k\), and A are the sag, conic constant, and the four-order aspherical coefficient of the aspherical surface, respectively. \(c\) is the curvature of the aspherical surface at its vertex, and \(r\) is the axial height on the aspherical surface. Thus, the second surfaces of lenses 1 and 2 are both designed and optimized by the Zemax software. The optimization values, \(c\), \(k\), and \(A\), of the second surfaces of lenses 1 and 2 are shown in Table 1. In the confocal system, the distance between the second surface of lens 1 and the bottom plane of lens 2 is 114 mm; moreover, a mutual focal point is located 24 mm away from the bottom plane of lens 2. The diameters of lens 1 and lens 2 are 150 mm and 28 mm, respectively. The size of lens 2 is set to satisfy the size of the observation area on the specimen of the fluorescence microscope because the diameter of the observation area size must be 27 mm.