## 2 Design of Analytic Collimating Lens for Light Collimation
### Analysis of Collimated Light  
As the collimating lens is axially symmetrical, the two lens profiles shown in Figure 2 can rotate \(360^{\circ}\) around the optical axis (\(+z\) axis) to form two lens surfaces indicated as \(S_{1}\) and \(S_{2}\) in Figure 3. In Figure 3, the illuminating surface area of the UV-LED under evaluation located at the original \(x-y\) plane and a detector plane that collects all the UV light after collimating are shown. A ray-tracing algorithm has been developed in this work to carry out the evaluation. Generally speaking, the ray data file for each UV-LED light source contains millions of optical rays emitting from the LED luminance area. Each ray has 7 variables which will be indicated as \((x_{0j},y_{0j},z_{0j},tx_{0j},ty_{0j},tz_{0j},P_{0j})\) to describe the \(j\)th ray originated from the LED chip. \((x_{0j},y_{0j},z_{0j})\) represents the coordinates of the starting point of the ray, \((tx_{0j},ty_{0j},tz_{0j})\) represents the directional cosines of the ray with respect to the \((x,y,z)\) coordinate system, and \(P_{0j}\) represents the optical power (\(mW\)) carried by the \(j\)th ray from the UV-LED chip. Referring to Appendix A and Figure A1, the surfaces \(S_{1}\) and \(S_{2}\) of the analytic collimating lens are mathematically characterized by functions \(r_{1}(\theta)\) and \(r_{2}(\alpha)\), respectively. Since the property of axially symmetrical, the lens's surfaces \(S_{1}\) and \(S_{2}\) can then be characterized by rotating \(r_{1}(\theta)\) and \(r_{2}(\alpha)\)\(360^{\circ}\) with respect to the \(z\) axis, as shown in Figure 3. The ray emitted from \((x_{0j},y_{0j},z_{0j})\) travels along the \(\vec{V_{0j}}\) direction with \(\vec{V_{0j}}=(tx_{0j},ty_{0j},tz_{0j})\) intersects the lens surface \(S_{1}\) at the point \((x_{1j},y_{1j},z_{1j})\). Let \(\vec{V_{ij}}\) represent the directional vector of the \(j\)th ray after being refracted by the \(i\)th surface and \((x_{ij},y_{ij},z_{ij})\) represent the coordinates of interception point of the \(j\)th ray with the \(i\)th surface, in which \(i\) equals 1 or 2 for lens surfaces \(S_{1}\) and \(S_{2}\), respectively. Figure 2: The analytic collimating lens is designed by: (**a**) selecting \(r_{0}\) and \(\xi\) to determine \(S_{1}\) starting from point \(A\) and ended with an aperture angle of \(\theta_{FWHM}\), (**b**) constructing \(S_{2}\) starting from the symbol \(+\) and ended at the distance of \(D\) relative to point \(A\) on the optical axis. As both lens surfaces had been analytically characterized, all the \((x_{ij},y_{ij},z_{ij})\) and \(\vec{V_{ij}}\) can be numerical solved using parameter root-finding algorithm such as Newton-Raphson method. Furthermore, considering the vector analysis of the spherical coordinate system and the axisymmetric curve, the unit normal vectors \(\vec{N_{1j}}\) and \(\vec{N_{2j}}\) of the \(j\)th incident ray intercepting surfaces \(S_{1}\) and \(S_{2}\), respectively, can be determined by,  
\[ \begin{split}\vec{N_{1j}}&=\frac{(-r_{1}\cdot\sin \theta+r_{1}\cdot\cos\theta)\cdot\cos\phi}{\sqrt{r_{1}^{2}+r_{1}^{2}}}\cdot \epsilon_{x}+\frac{(-r_{1}\cdot\sin\theta+r_{1}\cdot\cos\theta)\cdot\sin\phi}{ \sqrt{r_{1}^{2}+r_{1}^{2}}}\cdot\epsilon_{y}\\ &-\frac{(-r_{1}\cdot\cos\theta+r_{1}\cdot\sin\theta)}{\sqrt{r_{1 }^{2}+r_{1}^{2}}}\cdot\epsilon_{z}\\ \end{split}  \tag{1}\]  
\[\vec{N_{1j}}=\frac{\vec{N_{1j}}}{\left|\vec{N_{1j}}\right|} \tag{2}\]  
\[\vec{N_{2j}}=\frac{(-r_{2}\cdot\sin\alpha+r_{2}\cdot\cos\alpha)\cdot\cos\phi} {\sqrt{r_{2}^{2}+r_{2}^{2}}}\cdot\epsilon_{x}+\frac{(-r_{2}\cdot\sin\alpha+r_ {2}\cdot\cos\alpha)\cdot\sin\phi}{\sqrt{r_{2}^{2}+r_{2}^{2}}}\cdot\epsilon_{y} \tag{3}\]  
\[-\frac{(-r_{2}\cdot\cos\alpha+r_{2}\cdot\sin\alpha)}{\sqrt{r_{2}^{2}+r_{2}^{2 }}}\cdot\epsilon_{z}\]  
\[N_{2j}=\frac{\vec{N_{2j}}}{\left|\vec{N_{2j}}\right|} \tag{4}\]  
in which, \(r_{1}(\theta)\) and \(r_{2}(\alpha)\) are the functions for characterizing surfaces \(S_{1}\) and \(S_{2}\), respectively.