# Double-Sided Freeform Lens for Light Collimation of Light Emitting Diodes
## 2 Double Freeform Lens Design for Collimating Light from Point Source  
This paper proposes a method for designing a LED light collimating lens with two analog freeform surfaces. As shown in Figure 1a, the LED light source was first approximated as a point source, \(S\), located at the origin of a \((x,z)\) coordinate system in which the \(+z\)-axis is the optical axis of the LED. The lens under consideration was axially symmetrical with respect to the \(z\)-axis. Therefore, the \(x\)-axis could be any direction perpendicular to the \(z\)-axis. It is noted that a real LED is not a point source and its radiation pattern may thus deviate slightly from axial-symmetry. However, after completing the lens design, a real LED light source was modeled with the designed freeform lens and the overall optical performance was evaluated. In addition, the designed lens was fabricated and characterized experimentally using a commercial UV-LED. As shown in Figure 1a, the two freeform lens surfaces are denoted as \(S_{1}\) and \(S_{2}\), respectively. The lens was designed such that the light emitted from the point source \(S\) was refracted at points \(P\) and \(P^{\prime}\) by surfaces \(S_{1}\) and \(S_{2}\), respectively, and became parallel to the \(z\)-axis in air as a result. The optical refractive index of the lens material is \(n\). The key element in the proposed freeform lens design method was to individually assign surfaces \(S_{1}\) and \(S_{2}\) with different functions such that they perfectly collimate the light emitted from the point source, \(S\). The function of surface \(S_{1}\) was to refract all rays emitted by \(S\) in such a way that they appear to originate from a new imaginary point source \(S^{\prime}\) (see Figure 1a,b). Let \(r_{o}\) be the distance between the point source \(S\) and the apex point \(A\) of surface \(S_{1}\). Note that the location of \(S^{\prime}\) is chosen to be at \(z\ =\ -\xi r_{o}\) where \(\xi\) is a parameter to be determined. As shown in Figure 1c, the function of surface \(S_{2}\) is to refract all of the rays emitted from point \(S^{\prime}\) at the lens/air interface such that they become parallel to the \(+z\) axis in air. To mathematically derive the equations and solutions for surfaces \(S_{1}\) and \(S_{2}\), the present analysis adopted the polar coordinate systems (\(r_{1}\), \(\theta\)) and (\(r_{2}\), \(\alpha\)) shown in Figure 1b,c, respectively. The lens design problem then becomes one of finding the functions of \(r_{1}\ =\ r_{1}(\theta)\) and \(r_{2}\ =\ r_{2}(\alpha)\) which uniquely characterize and represent surfaces \(S_{1}\) and \(S_{2}\), respectively, and fulfill the refractive functions assigned to them. As shown in Figure 1b, let \(\overrightarrow{n}\) and \(\overrightarrow{t}\) be the normal and tangential unit vectors of surface \(S_{1}\) at an arbitrary point \(P\), respectively, and let \(\overrightarrow{e}_{r}\) and \(\overrightarrow{e}_{\theta}\) be the two unit base vectors of polar coordinate \((r_{1},\ \theta)\). Furthermore, let the angle between the incident ray and the normal direction \(\overrightarrow{n}\) be denoted as \(\phi_{1}\). From simple geometry and [PERSON]'s law, it follows that:  
\[r_{1}\sin\theta\ =\ (\xi r_{0}+r_{1}\cos\theta)\tan\alpha \tag{1}\]  
and  
\[1\cdot\sin\phi_{1}\ =\ n\cdot\sin[\phi_{1}-(\theta-\alpha)] \tag{2}\]  
Since \(\overrightarrow{r}_{1}\ =\ r_{1}\overrightarrow{e}_{r}\), \(\overrightarrow{d}\overrightarrow{t}_{1}\ =\ \frac{d\overrightarrow{n}}{d \theta}\overrightarrow{e}_{r}+r_{1}\overrightarrow{e}_{\theta}\), and \(\overrightarrow{t}\) is parallel to \(\overrightarrow{d}\overrightarrow{t}_{1}\over d\theta\), one has  
\[\tan\phi_{1}\ =\ \frac{1}{r_{1}}(\
icefrac{{dr_{1}}}{{d\theta}}) \tag{3}\]  
Figure 1: Double freeform lens for collimating light from point source: (**a**) Overview, (**b**), and (**c**) details for constructing front (\(S_{1}\)) and back (\(S_{2}\)) surfaces of lens.