## 3 Numerical Calculation
### Optimized Parameters after Evaluation  
Based on the numerically calculated data as shown in Figure 4, the optimized parameters of \(r_{0}\) and \(\xi\) with the smallest weighted mean divergence angle are selected for each of the six UV-LEDs and are shown in the 3 rd and the 4 th columns of Table 2. Moreover, the corresponding optical efficiency (OE) is evaluated by considering the numerical aperture of the first aspheric lens that could collect light rays from certain LEDs and Fresnel's loss of twice refraction. It is listed in the last column of Table 2. Additionally, a dimensionless physical quantity, the dimensional factor (DF) \(\eta\), is defined as the ratio of the UV-LED light-emitting area size to the parameter \({r_{0}}^{2}\) and is listed in the 5 th column of Table 2. The dimensional factor \(\eta\) is used to characterize the relative dimension size of the LED's light-emitting area in comparison with the aperture size of the collimating lens. The commonly accepted idea is that light collimating of LEDs can be much more easily achieved with a small value of dimensional factor \(\eta\). Figure 4: Contour maps of the weighted mean divergence angle, \(\overline{\gamma}\), as a function of the two collimating lens parameters, \(r_{0}\) and \(\xi\), for the six UV-LEDs (**I–VI**). Based on the chosen values of \(r_{0}\) and \(\xi\) shown in Table 2, the collimating lenses for the six UV-LEDs are now decided and their lens surface profiles are plotted and displayed at the left-hand side in Figure 5, in which the shaded area being the lens material of PMMA. The radiation patterns of the corresponding UV-LEDs are also shown in dash lines along with the designed collimating lenses' profiles in Figure 5. As being mentioned before, the aperture angle of each collimating lens is designed to match the directivity angle (\(\theta_{FWHM}\)) with a level of 50% normalized intensity in the radiation pattern of each UV-LED. To evaluate the over performance of the designed collimating lens for each UV-LED in detail, the designed lens' surface profiles are built into a CAD file by SolidWorks (Dassault Systemes SE, Velizy-Villacoublay, France) and is input along with the corresponding LED's ray data file to the optical simulation software Zemax OpticStudio (Kirkland, WA, USA) for light collimation and optical efficiency verification. In the simulation by Zemax software, the detector plane is placed at \(z=70\) mm away from the LED light source. Distributions of incoherent irradiance on the detector plane for each UV-LED with its corresponding collimating lens are shown in the central column in Figure 5. It is shown that the optical intensity distribution after collimating can vary significantly. Finally, the statistics on accumulated total optical energy as a function of divergent angle are also extracted from the data simulated by Zemax software and are displayed on the right-hand side in Figure 5. \(\gamma_{50\%}\) and \(\gamma_{90\%}\) are the divergence angles (DA) which can cover 50% and 90% of the totally accumulative intensity of the light after collimation. The obtained values of \(\gamma_{50\%}\) and \(\gamma_{90\%}\) for the six UV-LEDs with their corresponding lenses are listed in the 6 th and the 7 th columns in Table 2.  
Taking UV-LED (I) as an example, Table 2 shows that 50% of the light energy has a divergence angle within 1.56\({}^{\circ}\), and 90% of the light energy has a divergence angle within 4.16\({}^{\circ}\). With the analytic collimating lens design, the divergence angle is reduced from 120\({}^{\circ}\) to 1.56\({}^{\circ}\). Compared with the other five UV-LEDs, the UV-LED (I) has better light collimation partially due to a relatively smaller dimension factor \(\eta\) of 0.05. On the contrary, the UV-LED (III) has a relatively large dimensional factor which results in the worst collimation ability.