## 2 Design of Analytic Collimating Lens for Light Collimation
### Analytic Collimating Lens Design  
The theory of designing an analytic collimating lens was proposed earlier [25] and is schematically depicted in Figure 2, in which the \(+z\) axis represents the principle optical axis of the LED light source with its center located at point \(S\). As shown in Figure 2, the analytic-collimating lens contains two analog and non-spherical lens surfaces, \(S_{1}\) and \(S_{2}\), which are both axially symmetrical with respect the optical axis of \(+z\) axis. The on axis distance between the centers of the first lens surface \(S_{1}\) and the LED light source is denoted as \(r_{0}\). The working principle of this analytic collimating lens is to refract all light rays emitting from \(S\) by the lens surface \(S_{1}\) at the air/lens interface such that the transmitted rays in the lens material can be viewed as all radiating from a virtual light source located at \(S^{\prime}\). The distance between \(S\) and \(S^{\prime}\) is \(\xi r_{0}\), in which the \(\xi\) is a non-dimensional parameter one can determine in designing the analytic collimating lens. Finally, the lens surface \(S_{2}\) is designed to refract all the light rays again at the lens/air interfaced in such a way after refraction and all light rays in the air are parallel to the optical axis. Both the surface profiles of \(S_{1}\) and \(S_{2}\) can be analytically or mathematically derived with given information on \(r_{0}\), \(\xi\) and optical refraction index of the lens material (\(n_{1}\)). Details are given in the Appendix A and [25]. The goal of this research is to optimize the design of the analytic collimating lenses for each of the six chosen UV-LEDs by adjusting the two parameters of \(r_{0}\) and \(\xi\) to achieve a much-collimated UV light beam with smaller divergent angle and larger energy efficiency.  
Another parameter that needs to be chosen for finalizing the lens design is the full aperture angle of the lens. As shown in Figure 1, we will assign the directivity angle (\(\theta_{FWHM}\)) of each UV-LED to be the full aperture lens angle in designing its corresponding collimating lens. By doing so we can make each lens collect approximately the same percentage of total LED's power, which forms the basis for fair comparison of the designed collimating lenses. As shown in Figure 2b, both lens surfaces (\(S_{1}\) and \(S_{2}\)) can be analytically decided based on chosen parameters of \(r_{0}\) and \(\xi\), and \(D\) is the central thickness of this collimating lens which unit is mm.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline \multicolumn{4}{c}{**Optical Characteristics (Ths = 25 \({}^{\circ}\)C)**} \\ \hline \multicolumn{4}{c}{**Luminous Area**} & **Luminous Flux** & **Directivity** \\ \multicolumn{4}{c}{**Item**} & **(mm\({}^{2}\))** & **(mW)** & \(\theta\) (\(deg\)) \\ \hline \multicolumn{4}{c}{Luminous} \\ \multicolumn{4}{c}{**(I)** SST-10-UV} & \(1\times 1\) & 2200–2500 & 120 \\ \multicolumn{4}{c}{**(II)** CBT-120-UV} & 4.63 \(\times\) 2.60 & 14,600–19,500 & 100 \\ \hline \multicolumn{4}{c}{Nichia} \\ \multicolumn{4}{c}{**(III)** NVSU33A} & \(2\times 2\) & 4640 & 138 \\ \multicolumn{4}{c}{**(IV)** NVSU119C} & \(2\times 2\) & 1420 & 110 \\ \hline \multicolumn{4}{c}{Seoul Viosys} \\ \multicolumn{4}{c}{**(V)** CUN0 GB1A} & \(2.8\times 2.8\) & 1630 & 60 \\ \multicolumn{4}{c}{**(VI)** CUN06B1B\_140516} & \(2\times 2\) & 1160 & 46 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Specification of commercial UV-LED.  
Figure 1: Radiation patterns or optical intensity directivities of the six chosen UV-LEDs (**I–V**).