## Abstract

We present a simple freeform lens design method for an application to LED collimating illumination. The method is derived from a basic geometric-optics analysis and construction approach. By using this method, a highly collimating lens with LED chip size of $1.0\text{mm}\times \text{1}\text{.0}\text{mm}$ and optical simulation efficiency of 86.5% under a view angle of ± 5 deg is constructed. To verify the practical performance of the lens, a prototype of the collimator lens is also made, and an optical efficiency of 90.3% with a beam angle of 4.75 deg is measured.

© 2012 OSA

## 1. Introduction

In recent years, due to the speedy growth of LED light sources in general illumination applications, such as LED bulbs, spotlights, street lights, vehicle headlamps, and etc., many lamp fabricators and designers have proposed various LED luminaire techniques. LED luminaires are generally and purposely designed to redirect light rays to produce a specific distribution, and many methods aim this goal with employing standard conic or aspherical optical surfaces. However, when high optical performance or compact volume is desired, LED luminaires should be designed with irregular or freeform surfaces. Thus, many freeform surface design methods [1–7] have been proposed in the past decade. Nevertheless, most methods [1, 3–5, 7] convert the design problems into suitable group differential equations through a mapping between the light source and the target, and therefore need to solve solution numerically. Other methods [2, 6], though not using the mapping approach, have complex mathematic derivation and operation, and even need further optimization.

As concerns the collimator lens, which has many versatile applications to spotlight, flashlight, vehicle headlamps, and LCD projector light, conventional methods [8–9] adopt multiple conic or aspherical optical elements to realize the lens. However, its volume is large, and light rays cannot be effectively utilized. Also, the composed elements need rigorous alignment to obtain good performance. Since the freeform total internal reflection (TIR) structure can concrete all elements into a single body and achieve high performance with compact volume, therefore, it has attracted many interesting applications [10–12] recently.

In this paper, we propose a method, which is derived from a basic and simple geometric-optics analysis and construction approach, to construct freeform surfaces without using complex derivations. Though this method has been used to design a TIR collimator lens [11], the detailed and complete outline of the method is given in this paper, and an application to a novel compact LED collimator lens with ellipsoidal and freeform profile is also given. In addition to the simulation results, a prototype is also made to practically verify the performance of the lens. As compared with the TIR lens in [11], the collimator lens in this paper has better optical efficiency and smaller spot size under same lens dimensions. In addition, the lens in this paper is more easily fabricated than that in [11] due to an acute angle existing in the latter.

## 2. Proposed methodology

The proposed freeform surface design method consists of two processes, the geometric-optics analysis and the freeform-surface construction. The purpose of the geometric-optics analysis is to find the tangential vector, which is used to calculate the associated reflective or refractive point on a freeform optical surface. Once the tangential vector at each reflective or refractive point is calculated, a two-dimensional (2D) contour of the surface can be constructed following the procedure given in Section 2.2. Finally, a three-dimensional (3D) freeform surface is obtained by rotating the 2D contour around the axis. Detailed analysis and construction processes are depicted in the follows.

#### 2.1 Geometric-optics analysis

Optical surfaces are normally classified into reflective and refractive surfaces, and therefore the associated geometric-optics analysis for the reflective and refractive surfaces will be given, respectively.

### 2.1.1 Analysis for the reflective surface

A reflective surface can shift the direction of an incident light ray to a specific direction or a point based on the reflection law. The basic geometric-optics relation for a reflective surface shifting a light ray emitted from a light source to a specific direction ${{\theta}^{\prime}}_{P}$ is depicted in Fig. 1(a)
, where *P* is an arbitrary point on the surface and the angle ${\theta}_{P}$ stands for the direction of the incident light ray to the *x* axis. Assume the incident and reflective angles to be ${\theta}_{i}$ and ${\theta}_{r}$, respectively; then, from Fig. 1(a), the intersection angle ${\theta}_{T}$ of a tangential vector $\overline{T}$at point *P* can be expressed as

In addition, from Fig. 1(a), we obtain the following relation:

According to the reflection law, the incident angle ${\theta}_{i}$ must be equal to the reflective angle ${\theta}_{r}$, and therefore we have

Substituting Eq. (3) into Eq. (1), we obtain

From Eq. (4), if ${\theta}_{P}$ and ${{\theta}^{\prime}}_{P}$ are known, the intersection angle ${\theta}_{T}$ can be solved, and the tangential slope and vector at point *P* can be given by

In the other case, when the incident light ray is redirected to a specific point *F* by the reflective surface, as shown in Fig. 1(b), the tangential slope at point *P* is also given by Eq. (5), whereas the angle ${{\theta}^{\prime}}_{P}$ is given by

Since point *F* is initially given and point *P* is found from the 2D-contour construction in Section 2.2; therefore, $\overline{FQ}$, $\overline{PQ}$, and hence the tangential slope at point *P* can be calculated.

### 2.1.2 Analysis for the refractive surface

Similarly, a refractive sureface can shift the direction of an incident light ray to a specific direction or a point based on the Snell’s law. The basic geometric-optics relation for a refractive surface shifting a light ray emitted from a source to a specific direction ${{\theta}^{\prime}}_{P}$ is depicted in Fig. 2(a) . Assume the incident and refractive angles to be ${\theta}_{i}$ and ${\theta}_{t}$, respectively; then from Fig. 2(a), the refractive angle can be expressed as

In addition, from this figure we obtain the following relation:

Since point *P* is a refractive point, thus ${\theta}_{i}$ and ${\theta}_{t}$ must satisfy Snell’s law, that is,

In Eq. (10), ${n}_{1}$ and ${n}_{2}$ are the refractive indices of the two media in Fig. 2. Substituting Eqs. (8) and (9) into Eq. (10) and rearranging the resultant equation, we obtain the intersection angle of the tangential vector $\overline{T}$ at point *P* as

The tangential slope and vector at point *P* can also be expressed as

In the other case, when a light ray emitted from the light source is redirected to a specific point *F* by the refractive surface, as shown in Fig. 2(b), the tangential slope at point *P* is also given by Eq. (12), whereas the angle ${{\theta}^{\prime}}_{P}$ is given by ${{\theta}^{\prime}}_{P}={\mathrm{tan}}^{-1}\left[\overline{FQ}/\overline{PQ}\right]$.

#### 2.2 Freeform surface construction

According to previous geometric-optics analysis, we can use the following procedure, which is shown in Fig. 3
, to find each reflective or refractive point on a freeform surface so that its 2D contour can be constructed step by step. Figure 3 illustrates light rays emitted from a light source and redirected by a freeform reflective surface to a prescribed direction parallel to the optical axis. As shown in Fig. 3, red lines marked with ${i}_{0}$, ${i}_{1}$, ${i}_{2}$ and ${i}_{3}$ stand for arbitrarily given light rays, which are emitted from the light source *S* and then hit the reflective surface at points of ${P}_{0}$, ${P}_{1}$, ${P}_{2}$, and ${P}_{3}$, respectively. Point ${P}_{0}$ is the initial point of the 2D contour, and is given by the designer according to the wanted surface dimension. The blue dashed lines ${\overline{T}}_{0}$, ${\overline{T}}_{1}$, and ${\overline{T}}_{2}$, which are used to construct the 2D contour, stand for the tangential vector at point ${P}_{0}$, ${P}_{1}$, and ${P}_{2}$, respectively.

The procedure to construct the 2D contour is stated as the follows. First, give the initial point ${P}_{0}$ and use Eq. (5) to find the tangential vector ${\overline{T}}_{0}$ at point ${P}_{0}$, where ${{\theta}^{\prime}}_{P}$ is assigned to be $\pi /2$ for the incident light ray ${i}_{0}$ reflected by the surface to a direction parallel to the optical axis. Next, find point ${P}_{1}$, which is located at the intersection point of the incident light ray ${i}_{1}$ and the tangential vector ${\overline{T}}_{0}$. The next, use Eq. (5) again to find the tangential vector ${\overline{T}}_{1}$ at the point ${P}_{1}$. Then find point ${P}_{2}$, which is located at the intersection point of the incident light ray ${i}_{2}$ and the tangential vector ${\overline{T}}_{1}$. Repeat the above process till all the *P*-points are found, and the whole 2D contour of the freeform surface can be constructed with the help of CAD software program and the obtained *P*-points. However, to retain the slope at each point, the contour between any two consecutive *P*-points should be constructed using a interpolation approach based on a high-degree polynomial [13] or B-spline functions [14].

For illustration, a third-degree polynomial is given by

If the coordinates of points ${P}_{n}$ and ${P}_{n+1}$ are given by $\left({P}_{n1}\text{,}{P}_{n2}\right)$and $\left({P}_{\left(n+1\right)1}\text{,}{P}_{\left(n+1\right)2}\right)$, then we have the following relations:

By simultaneously solving Eqs. (15) and (16) to give the value of *a*, *b*, *c*, and *d*, the contour between points ${P}_{n}$ and ${P}_{n+1}$ is thus given by $\left(x,p\left(x\right)\right)$ with ${P}_{\left(n+1\right)1}<x<{P}_{n1}$. Once the 2D contour is obtained, the 3D axis-symmetrical freeform surface can therefore be established by rotating the 2D contour about the optical axis.

## 3. Application to construct a compact collimator lens

For illustrating the application of the method, we employ it to construct a compact LED collimator lens. Both the construction process and computer simulation of the lens are stated in detail in the following subsections.

#### 3.1 Construction of the lens

The 2D plot of the proposed compact collimator lens is sketched in Fig. 4
, where only half of the lens is sketched because of the axis-symmetrical structure. As shown in Fig. 4, this lens consists of five reflective/refractive surfaces, which are respectively marked with a circled number. Surface [REMOVED EQ FIELD] is a spherical refractive surface, surface [REMOVED EQ FIELD] is an ellipsoidal reflective surface, and the other three surfaces are refractive surfaces with freeform profiles. In addition, the light source *S* is located at the first focal point of the ellipsoidal surface [REMOVED EQ FIELD], and point *F* is its secondary focal point. For conveniently explaining the function for each surface, we categorize light rays emitted from the source *S* into three groups, and the optical property of each group light rays is stated in the following:

- 1) Light rays of group 1 with a spread angle of ${\theta}_{a}$ will enter the zone 1 of the collimator lens through the spherical surface [REMOVED EQ FIELD] firstly, and then are reflected to point
*F*by the ellipsoidal surface [REMOVED EQ FIELD]. While before arriving point*F*, they will hit the freeform surface [REMOVED EQ FIELD] firstly and be diverted to parallel the optical axis. - 2) Light rays of group 2 with a spread angle of ${\theta}_{b}$ will enter the zone 2 of the collimator lens through the surface [REMOVED EQ FIELD] firstly. Then they are refracted to parallel the optical axis by the freeform surface [REMOVED EQ FIELD].
- 3) Light rays of group 3 with a spread angle of ${\theta}_{c}$ will enter the zone 3 of the collimator lens through the freeform surface [REMOVED EQ FIELD] and then are focused to point
*F*. While before arriving point*F*, they will hit the freeform surface [REMOVED EQ FIELD] firstly and be diverted to parallel the optical axis.

Therefore, all the light rays emitted from the light source *S* are finally collimated to the optical axis through the lens. In addition, all the surfaces, including surfaces [REMOVED EQ FIELD] and [REMOVED EQ FIELD], can be constructed by using the method depicted in Section 2.2.

The detailed construction process of the five surfaces is given in the following:

- 1) First, decide the dimension of the entrance aperture on the left of the lens. In other words, to give the initial point of the surface [REMOVED EQ FIELD].
- 2) Then, use the initial point to construct the surface [REMOVED EQ FIELD] that can focus light rays in zone 1 to point
*F*. - 3) Next, construct the spherical surface [REMOVED EQ FIELD] and let its center be located at the light source
*S*. Thus, light rays entering this surface will not be deflected. - 4) Next, construct the surface [REMOVED EQ FIELD] with its construction initial-point (or vertex) located at the optical axis to focus light rays in zone 3 to the point
*F*. - 5) Next, construct the surface [REMOVED EQ FIELD] with its construction initial-point located at the optical axis to make light rays in zone 1 and zone 3 parallel to the optical axis when passing through the surface. The point is the dimension of surface [REMOVED EQ FIELD] should be made larger as possibly to promote the optical efficiency but should not cross over the boundary between zone 2 and zone 3; otherwise, some light rays in zone 2 will hit the surface [REMOVED EQ FIELD] and encounter total internal reflection.
- 6) Finally, construct the surface [REMOVED EQ FIELD] with the end point of the surface [REMOVED EQ FIELD] as its construction initial-point so that light rays in zone 2 can become parallel to the optical axis when passing through this surface.

The spread angle for each group light rays is given by ${\theta}_{a}:{\theta}_{b}:{\theta}_{c}={56}^{\circ}:{10}^{\circ}:{24}^{\circ}$, which is determined by the intersection of surface [REMOVED EQ FIELD] and surface [REMOVED EQ FIELD] and also surface [REMOVED EQ FIELD] and surface [REMOVED EQ FIELD]. The constructed 2D contour of the collimator lens is shown in Fig. 5 . By rotating the contour about the optical axis, a 3D collimator lens is implemented as shown in Fig. 6 .

#### 3.2 Computer simulation

Though the compact collimator lens is constructed based on an ideal point source, to verify it also effective to a practical LED light source, a TracePro program is employed to trace one million light rays emitted from an LED through the lens. The associated simulation parameters of the collimator lens are given in Table 1 , and the ray trace is shown in Fig. 7 . To clearly see the deviation of the ray trace for a practical LED chip from that of a point source, both ray traces for the LED chip and point source are performed, and only five thousand rays are shown up, where Fig. 7(a) is the ray trace for the 1 mm × 1 mm LED chip and Fig. 7(b) is the ray trace for the point source. From Fig. 7(a), it can be seen that most of light rays emitted from the LED chip are collimated well by the lens; however, a small number of light rays deviate from the direction parallel to the optical axis. This phenomenon can be attributed to the non-ideal point source of the LED. However, Fig. 7(b) shows light rays from the point source can be collimated very well by this lens. Meanwhile, the main optical loss of the lens comes from the Fresnel loss and the absorption of the acrylic material, while some optical loss comes from the stray light as shown in Fig. 7(a). The associated illuminance map and distribution on a target plane at 6-m away are also shown in Fig. 8 , which shows the spot size at 6-m away to be about 0.4 m. The rectangular candela distribution plot is shown in Fig. 9 , which shows the beam angle is about 5 deg with total optical efficiency of 90.2% (referred to Fig. 8). In the computer simulation, the parameters, such as the material, the dimensions, and the shape of the lens, or the light distribution and chip size of the LED, affect the optical performance of the lens significantly, while the influence of the LED chip size is the most pronounced. Thus its effect on the optical efficiency and half view angles is investigated further. The simulation result is shown in Table 2 , where the optical efficiency is represented in relative percentage. In addition, the optical efficiency versus the half view angle for different LED chip sizes is also plotted in Fig. 10 . From the simulation results, it can be seen that the optical efficiency decreases with the LED chip size. For a 1.0 mm × 1.0 mm LED chip, the optical efficiency achieves 86.5% within a view angle of ± 5 deg. To obtain a reasonable optical efficiency, for example, 80% within a half view angle of 5 deg, the dimension of the LED chip should not be greater than 1.2 mm × 1.2 mm for a lens with aperture of 30 mm.

## 4. Practical measurement

In order to verify the optical performance of the collimator lens, a prototype in Fig. 11 is made; the candela distribution and the beam width versus distance are also measured. Both measurements proceed using an LED light source with measured flux of 47.89 lm and chip size less than1 mm × 1 mm, and the measurement results are shown in Figs. 12 and 13 . Figure 12 gives the measured candela distribution, and Fig. 13 gives the beam width and the illuminance (on the optical axis) versus the distance. The measurement results demonstrate that a beam angle of 4.75 deg with overall optical efficiency of 90.3% (not shown in Figs. 12 and 13) and a beam width of 0.5 m at 6-m away are measured. Both data are very close to the simulation results in Section 3.2.

## 5. Conclusion

A freeform lens design method with the methodology based on basic geometric-optics analysis and construction approach is depicted in detail in this paper. As compared with existing methods in literatures, the derivative process of the method is simple and straightforward. This method can be used to design various indoor/outdoor LED luminaires for general or specific illumination applications. By employing this method, a novel and highly collimating LED lens with an aperture of 30 mm is constructed. The computer simulation results demonstrate that this lens can achieve a collimated beam with optical efficiency up to 86.5% under a view angle of ± 5 deg and a 1.0 mm × 1.0 mm LED chip. The chip-size dependent character of the collimator lens is also investigated. To obtain a reasonable optical efficiency, the LED chip size should not be greater than 1.2 mm × 1.2 mm for a lens with aperture of 30 mm. In addition to the computer simulation, a practical measurement on a prototype of the lens is also conducted. The measurement results by using a LED with chip size less than 1 mm × 1 mm demonstrate that a beam angle of 4.75 deg with overall optical efficiency of 90.3% is obtained.

## Acknowledgment

The authors would like to thank the National Science Council in Taiwan for financially supporting this research under Grant No. NSC 100-2221-E-018-004.

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