## Abstract

A method using a freeform surface lens for LED secondary optic design is proposed in this paper. By Snell’s Law, the differential equations are given to build the relationship between the normal direction of a freeform surface and its input/output ray vectors. Runge–Kutta formulas are used to calculate the differential equations to design the freeform surface. Moreover, the optical model for uniform illumination is simulated and optical performance is analyzed. A practical freeform surface lens for LED uniform illumination is fabricated using an injection molding method. By the process, our system demonstrates a uniform illumination with a divergence half-angle of $6\text{}\mathrm{\xb0}$ and an efficiency of 78.6%.

© 2009 Optical Society of America

## 1. Introduction

LEDs have been regarded as the best potential light sources for next generation lighting because of their attractive characteristics [1]. In many cases, light flux from an LED is often redistributed to meet the needs of lighting. Secondary optics is commonly used in LEDs to ensure that the total light output from the LED dies meets the overall specification and creates the desired appearance and beam pattern. But LED dies are always described as Lambertian [2], and many researchers realized that using conventional optical elements in secondary optics is hard work because of their complexity and low light usage efficiency. However, a freeform lens using a nonimaging concept is considered a realizable method. In past years, much research on this area was done and methods were developed, such as the “Uniform B-Spline Surfaces Method” [3] and the “Tailored Freeform Surface Method” [4, 5]. Some research was developed using freeform surface reflectors [6, 7, 8], but it is impossible to cover the whole emergence spatial angle of the LED without any occlusion with this method, and thus the efficiency of reflective surface cannot be very high. The devices based on refraction and/or total internal reflection (TIR) with the source embedded in the lens have potential advantages to supply 100% usage. Some such devices [9] are listed in Lumileds’ catalog; Huang [10] also showed a similar design in his thesis, but the uniformity on the target plane is not good enough. And a compound parabolic concentrator (CPC) [11] is a selectable solution only when there is no limitation on the axial length of the device. For uniformity illumination, Tai and Schwarte [12] designed a plano-aspherical singlet with a large numerical aperture for LED to generate a homogenous irradiance distribution on a target. Also, many geometry methods are reported to be applied in laser beam shaping techniques [13]. These geometrical methods of beam shaping have generally involved solving differential equations for the contours of the optics. Although the application systems of these methods, including a 2-plano-aspherical lens system, a 2-mirror system, and a 3-element GRIN system with spherical surface lenses, can provide uniformity distribution in the target surface with lossless energy, it is difficult to design and fabricate the system to collect a widely divergent angle fully in a LED source.

In this paper, a method for designing a refractive freeform surface that can supply uniform illumination and achieve high light usage efficiency has been developed. By Snell’s Law, the differential equations are given to build the relationship between the normal direction of a freeform surface and its input/ output ray vectors. Runge–Kutta formulas [14] are used to calculate the differential equations to design the freeform surface. The optical model for uniform illumination is discussed, and simulation results are evaluated by ASAP software using Monte-Carlo ray tracing. Moreover, a practical freeform surface lens for LED uniform illumination is manufactured and tested. By this process, a uniform, narrow- divergence half-angle, high-usage illumination to a given area is achieved.

## 2. Method

In this paper, the illumination system is supposed to have axis symmetry, with the optical axis crossing the position of the source and the center of the target plane. A practical approximation for the luminous intensity distribution of LEDs is given by [15]

where*φ*is the viewing angle and ${I}_{0}$ is luminous intensity at the direction of the normal of the source surface. The number

*m*is given by the angle ${\phi}_{1/2}$ (a value typically provided by the manufacturer, defined as the view angle when irradiance is half of the value at $0\xb0$): So the luminous flux of the LED source is

Take the position of the source as the origin and build the coordinate, shown in Fig. 1; the direction of the maximum luminous intensity (${I}_{0}$) is defined as the *Z* axis. Consider a 2D mode first. Take the plane that includes the $+X$ and $+Z$ axes, for example, see Fig. 1; Snell’s Law of vector mode can be written as

*n*is the refractive index of the media of lens, $\overrightarrow{N}$ is normal of the freeform surface at the intersection point A, $\overrightarrow{\mathrm{IN}}$ is the input ray, and $\overrightarrow{\text{Out}}$ is the output ray.

If *A* is located at $(x,0,z)$ and the coordinate of point *B* of the target plane is $({x}_{d},0,H)$, one can get

*x*and/or

*z*;

*H*is the distance between the source and the target plane.

Applying Eq. (6) to Eq. (5) shows that

whereTo solve Eq. (7), an extensive equation ${x}_{d}=g(x,z)$ is needed. As illuminance *E* on the target plane is considered to be constant, one gets

*S*is the area of the plane;

*E*is the illuminance of target plane. To get the maximum usage of light, the equation below should be established:

Uniform illumination means that the fractional flux illuminating on the target plane within a circle of radius ${x}_{d}$ is simply $({x}_{d}^{2}/{R}^{2})$. Here *R* is the radius of the target plane, as shown in Fig. 2. On the other hand, the light coming out from the source into a cone-shaped solid angle with a divergence half-angle *φ* has a fractional flux [16] of $(1-{\mathrm{cos}}^{m+1}\phi )/(1-{\mathrm{cos}}^{m+1}{\phi}_{\mathrm{max}})$. There are two models that describe the corresponding relationship between them.

Figure 2a illustrates the “divergent illumination model.” In this model, the ray coming out from source whose direction is along the *Z* axis (such a ray is called the “central ray”) will keep its original direction until reaching the center of the target plane; while the rays that have the divergence half-angle of ${\phi}_{\mathrm{max}}$ (such rays are called the “boundary rays”) will refract at the freeform surface and reach the boundary of the target plane. In this case, the equation ${x}_{d}=g(x,z)$ can be written as below:

On the contrary, the light irradiation of “convergent illumination model” [see Fig. 2b] is the opposite: the central ray will finally reach the boundary of the target plane, and the boundary rays will be con centrated by the freeform surface to the center of the target plane. So the equation ${x}_{d}=g(x,z)$ becomes

According to the selected illumination model, one can apply Eq. (10) or Eq. (11) to Eq. (7) and get $dz/dx=f(x,z)$. Take $x=0$, $z=1$ as the initial value condition, then using Runge–Kutta formulas can solve the equation and get the freeform-surface locus in 2D mode. Runge–Kutta formulas are based on the Taylor equation, and the core of this is using the values of neighbor points to express the derivative at an assigned point. Runge–Kutta formulas have expressions with different order; the higher the order, the more accurate the result can be. But meanwhile, the expressions would be more difficult and users should have to spend more time to get the final solutions. Previous experiences show that the fourth- order Runge–Kutta formulas are a good compromise between accuracy and calculation time. Here, the fourth-order Runge–Kutta formulas expressions are given:

where*h*is the step length. For a more detailed description, please read [11].

An alternative way is using Euler formulas to advance calculation speed but with less accuracy. Rotating the locus around the *Z* axis for a circle, the design of the desired lens is completed.

## 3. Simulation

With the method mentioned above, a lens with a freeform surface collimates light of a Lambertian source using the maximum divergence half-angle of $90\xb0$, and an additional TIR reflector is needed to consummate the system.

In Fig. 3a a principle layout of the freeform surface lens is shown. Figure 3b is the partial enlarged detail of the freeform surface S1. All light rays coming out from source are divided by divergence half-angle *φ* into two parts: $0\xb0\le \phi \le 45\xb0$ and $45\xb0\le \phi \le 90\xb0$. To the rays of part $0\xb0\le \phi \le 45\xb0$, just using the “convergent illumination model” once can calculate out the locus; the part $45\xb0\le \phi \le 90\xb0$ is a little complex, so it is discussed in detail here.

Before exiting from the freeform lens, the rays of this part will first reach the surface S2 and the TIR will happen there. S2 is a parabolic whose focus is just the origin, or the source’s position. All rays reflected by S2 will run parallel to the *Z* axis until reaching S1.3 or S1.5. The positions of S1.3 and S1.5 are arranged higher than S1.1 to force all rays of this part to reach S2 first. The locus of S1.3 and S1.5 is calculated out by using “divergent illumination model”. In early experiments, these two surfaces were designed as a united one. But in that case, some rays whose divergence half-angle is smaller than $45\xb0$ might radiate into the lens again after exiting from S1.1. If so, the luminance of a ring area on a target plane would be much lower than expected and it looked as if some light were “stolen”. So the original one surface was split into S1.3 and S1.5, and the position of S1.3 was lowered, its inner boundary was still a little higher than the outer boundary of S1.1. Avoiding the occurrence of such a phenomenon is also the reason why the previous part uses the “convergent illumination model” and not the “divergent” one. Import the design into the software ASAP to do Monte-Carlo ray tracing to check if everything has done well as planned.

With regard to the existence of the surfaces s1.2 and s.1.4 in Fig. 3, there are steps between surfaces. It is not a good design because of the difficulty of manufacturing; also, even using the “convergent illumination model” to design the s1.1 surface, there are still some rays that will reach surfaces s1.2 and s1.4 because of the extended light source. To resolve the problem, some technical considerations were added to smooth the step of the design in Fig. 3. Figure 4 shows the principle model of a freeform surface lens. All light rays coming out from the source are still divided by $0\xb0\le \phi \le 45\xb0$ and $45\xb0\le \phi \le 90\xb0$. The S3 surface of Fig. 3 is split into two surfaces: S3.1 and S3.2. S1.1, which combined with S3.2, is calculated for $0\xb0\le \phi \le 45\xb0$. The S3.2 is the plane by considering the cost, and the S1.1 surface is calculated by the “divergent illumination model” to avoid concavity on the top. On the other hand, the S3.1 surface, calculated for $45\xb0\le \phi \le 90\xb0$ associating with S1.2, S2.1, S2.2, is a freedom surface to parallel the ray coming from the source. S2.1 and S2.2 are designed as a circular cone surface to just consider the TIR and manufacture. In this wide angle part design, the “convergent illumination model” is still used to calculate S1.2. By these considering methods, S1.1 will be connected with S1.2 smoothly, and the steps can be removed and fabrication will be reasonable.

Figure 5 shows the simulation result of the design shown in Fig. 4; the light usage efficiency of this example is 95.2% in software modulation. The degree of uniformity is 92% (here uniformity is defined as the ratio of minimum and maximum luminance on target plane).

All results mentioned above are figured out from the differential equations. The strict correspondence in mathematics between the source and the target plane guarantees the accuracy and performance of the result. Meanwhile, any deviation from the original position might influence the performance. So the analysis of the “field depth” is necessary. Figure 6 shows the result.

The trend of two curves shows the deviation will cause the decreases of rates on both usage efficiency and uniformity. If $z>{z}_{0}$ (${z}_{0}$ is the specified distance as the design and *z* is the real one), the decrease of uniformity is less serious, but the system has to afford the risk of less efficiency. Generally, if $0.95{z}_{0}<z<1.07{z}_{0}$, this system can fulfill most applications with strict requirements, such as the illumination module of the projector.

Another fact is in real applications, the dies of most high brightness LEDs (HB-LEDs) are not small enough to be considered as the point source. Most commercial HB-LEDs have a die with the size of $1\text{\hspace{0.17em}}\mathrm{mm}\times 1\text{\hspace{0.17em}}\mathrm{mm}$ or larger. So the intensity distribution on the target plane in extended source is another property to evaluate the approach. Figure 8 shows the simulation result of efficiency and homogeneity as a function of the diameter of the source. If the diameter of the source is less than $1.2\text{\hspace{0.17em}}\mathrm{mm}$ of the lens size, the efficiency is over 80% and the homogeneity is over 75%, and the illumination pattern is also acceptable (shown in Fig. 7), but if the size continues increasing, the “hot spot” in the center of the target plane will appear.

## 4. Molding and Measurement

We used an injection molding method to fabricate the freeform lens. First, a mold design and production are necessary for fulfilling the freeform shape in the injection molding process. Two half parts of mold were used; the optical element is overturned in the mold because smooth molding injection is taken into consideration and some technical considerations such as rounding chamfers are added. Figure 9 shows our experiment mold.

Mold part B is locked into the stationary side of the press; mold part A is mounted on part B. Both of the mold parts are manufactured from stainless steel (SUS405) taking into account the relatively high relative permeability and mechanism properties. The freeform surfaces need surface finishing with the design date. The molds are mounted with a molding press. During molding, the two halves of the mold are closed and clamped under high pressure; the epoxy resin is injected into the mold from the injection holes located on the part A side. To obtain an accurate finish and cleaning of the lens, a layer of $20\text{\hspace{0.17em}}\mathrm{nm}$ Cr was deposited on the steel substrate first. Before using, the molds were cleaned by ultrasonic cleaning for $10\text{\hspace{0.17em}}\mathrm{min}$, while using HD-303 of R&L Chemical as organic solvents. In the molding process, 20# agent of Dow Corning was used as the mold releasing agent. After continuing at a service temperature $120\text{\hspace{0.17em}}\xb0\mathrm{C}$ for $120\text{\hspace{0.17em}}\mathrm{min}$ and standing until room temperature by natural cooling, a lens with a freeform surfaces was manufactured. Figure 10 shows the appearance of the lens. The refractive index of the lens is 1.497. Figure 11 shows the measurement mechanical difference of the freeform surface compared with the theory date. The maximum of difference is $15\text{\hspace{0.17em}}\mathrm{\mu m}$ as tested by a Taylor–Hobson prolifometer. Figure 12 shows the illumination modular with the freeform surface lens. A pure surface emitting LED is adopted as the source and would be immersed in the freeform lens, the LED has a Lambertian radiation distribution, and the luminous surface is $1\text{\hspace{0.17em}}\mathrm{mm}\times 1\text{\hspace{0.17em}}\mathrm{mm}$.

A $1\text{\hspace{0.17em}}\mathrm{m}$ diameter integrating sphere is used to measure the output flux of the LED, shown as Fig. 13. At the input of the integrating sphere, an aperture was used to restrict the incident angle from the LED, which had a $6\text{}\mathrm{\xb0}$ half-angle by design. Four molding lenses have been molding and tested to calculate the average output flux. The total flux of the LED was $130.1\text{\hspace{0.17em}}\mathrm{lm}$, which was measured by only putting the LED into the integrating sphere. The average output flux with a freeform surface lens was $102.2\text{\hspace{0.17em}}\mathrm{lm}$, shown in Table 1.

As there is no antireflective coating on the lens, the collection efficiency of the lens in a half-angle of $6\text{}\mathrm{\xb0}$ is

Also, the uniformity was tested by the system shown in Fig. 14.Figure 15 shows the test result of uniformity and beam shape of a LED with a freeform surface lens. 78.4% of optical uniformity was achieved using a freeform surface lens. The uniformity is far away from the simulation result of 92%. Many reasons cause the difference, including fabrication error, the size of light source, the intensity distribution of light source, the angle distribution of light source, and so on. The intensity distribution result will be influenced by the size of the light source, as shown in Fig. 7. The homogeneity becomes worse and the hot spot is seen as the size of the light source is increased. Moreover, the simulation result is done based on the ideal situation that the intensity distribution of the LED is uniform and the angle distribution is Lambertian. Actually, there is a little difference between the factual status with the ideal condition. Also, the experiment result is greatly affected by the fabrication error. So, amending the prototype by the actual measurement data of the intensity/light distribution of an actual LED light source and improving the mechanical precision are the ways to advance the performance of the design.

## 5. Conclusion and Summary

In this paper, a method for designing a refractive freeform surface lens to supply uniform illumination to a given area was shown. The design models were discussed, and examples of the lens using this method were given. By the analysis of software, a good result was shown: uniform luminance distribution, narrow angular distribution, and high light usage efficiency could be expected. The design processes were based on mathematical calculation first. Then, simulation result were given by illumination software. For real manufacturing, error tolerance analysis of the surface discussion was analyzed. A practical freeform surface lens for LED uniform illumination was fabricated using an injection molding method. By the process, the system demonstrated uniform illumination with a divergence half-angle of $6\text{}\mathrm{\xb0}$, an efficiency of 78.6%, and a uniformity of 78.4%. Future work will focus on the application of the novel freeform lens for applications of LED light source to video projectors and mobile projectors.

This research is partially supported by the National Basic Research Program of China (973 Program) (No. 2009CB320803).

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