## Abstract

In many applications, compact concentrator lenses are used for collimating light from LEDs into high output beams. When optimizing lens designs, the LED is often approximated as a point source. At small lens-to-LED size ratios this is known to be inaccurate, but the performance compared to optimizations with more realistic models is rarely addressed. This paper examines the reliability of a point source model in compact lens design by comparing with optimisations that use a factory measured LED ray-file. The point source is shown to cause significant, unnecessary efficiency loss even at large lens sizes, while the use of a ray-file allows for a >55% reduction in the footprint area of the lens. The use of point source approximations in compact lens designs is therefore generally discouraged.

© 2011 OSA

## 1. Introduction

LEDs are an up-and-coming feature in a range of applications, and already the preferred light source in some fields. This includes low depth spotlights, residential light bulb replacements, architectural and decorative light displays, headlamps, and stage lighting [1–5]. In many applications, concentrator lenses are used for the purpose of collimating the LED light into a desired cone or spot. Some usage of Total-Internal-Reflection (TIR) is usually a desired feature for radially compact lenses and narrow beam angles [1–4, 6].

Fast analytical methods for lens optimizations have been developed, such as [1–3, 6, 7], where the LED die is approximated by a point source (PS). However, this approximation is questionable at small lens-to-LED size ratios [5–8], where a PS optimized lens might diverge significantly from what would be truly optimal for an LED. Using PS optimizations to speed up the design process could therefore impact both the luminous efficacy and optical precision. Other analytical methods such as Simultaneous Multiple Surfaces (SMS) [5, 7, 8] can offer some degree of control over the shape of the light source, but are currently limited in 3D due to the lack of stable optimization algorithms for more than 2 surfaces [7, 8]. Wang et al. [5], for instance, present a compact SMS optimized LED lens that is convincing for the intended application at beam angles of 80–129°, but this paper targets lenses with more acute beam angles, below ≈20°, requiring design features that would most likely make SMS infeasible. A much greater similarity is found in the paper by Chen and Lin [6]. They also acknowledge the size-effect issues and therefore include a chart of luminous efficiency vs. die size at 5 field of view (FOV) angles, in order to provide a window of application (WOA) for their method. As expected, the efficiency is significantly decreased when exchanging the PS with an LED, especially for large dies and narrow FOVs. However, the results are not compared with lenses optimized specifically for LEDs, which could significantly improve the efficiency and the WOA. This paper is therefore focused on a thorough investigation of this issue by using a factory measured ray-file (RF) that more accurately represents an LED emission profile.

## 2. Method

The following investigation scheme was performed: **(1)** Use three different lens types: **I**: A simple lens model with non-optimal variable restrictions. **II**: Same as type **I**, with the restrictions removed by unlocking the 2 associated variables. **III**: Same as type **II**, with 3 additional variables unlocked. **(2)** Optimize each lens type using a PS. **(3)** Optimize each lens type using an RF source (RFS) at a range of size ratios, defined as the lens diameter to the edge length of the die. **(4)** Investigate the reliability of the PS approximation for each size ratio and lens type by exchanging the PS with the RFS. **(5)** Investigate if the RFS optimized lenses are also optimal PS lenses by exchanging the RFS with the PS.

Lens types **I** and **II** were used to test the hypothesis that PS optimizations could achieve similar performances with both simple and complex lens models, while RFS optimizations would show model dependence. This would be a strong indication of the unreliability of PS in this type of design optimization, since an algorithm would have no reason to favor the most proper design. Step **(5)** is relevant since it is most likely impossible to find optimal LED lenses using a PS if RFS optimized lenses are shown not to be optimal PS lenses.

The exact same algorithms and models were used for all optimizations. The LED was represented by a ray-file of a green Luminus CBT-90G with a 3×3 mm die [9], measured by Luminus, at a wavelength of 530nm. The PS was therefore modelled as a ray set with a quasi-random, Lambertian emission profile, and results were calculated by recursive ray tracing; splitting a ray into a refracted and a reflected ray at each intersection, and thus taking Fresnell losses into account. Figure 1**(a)**
compares the emission profiles of both source models. Note that the most important difference is not visible, namely that the RFS rays originate from a range of different positions on the die. Figure 1**(b)** shows the rotational profile that determines the three lens types, which are similar to those described in [4]. Free variables are indicated by italics, constants (**R** and **θ**) by bold fonts, while square markers indicate variables that are only free for specific lens types. To facilitate injection molding **θ** is set to 88°. Lens type **I** has *k* and *θ _{2}* locked at

*k =*

**R/**2, and

*θ*0°, which is non-optimal for LED optimizations. These are unlocked for lens types

_{2}=**II**and

**III**. The peripheral TIR section of the lens (

*e*to

*j*) is calculated as a parabola via

**R**,

*e*, and

*j*, and is deformable by use of

*f*,

*g*, and

*h*for lens type

**III**. The number of free variables is therefore: 10 for type

**I**, 12 for type

**II,**and 15 for type

**III**.

The optimization objective was the maximization of luminous output within a 16° FOV since FOVs are used by Chen and Lin and is common in LED lens specification. Results can also be doubly interpreted since loss of efficiency within a FOV implies loss of beam control. Simulated Annealing Monte Carlo (SAMC) was used for optimizations due to previous experience with the algorithm [4]. SAMC takes random steps in variable-space, allowing “bad” steps according to some probability-function that depends on the severity. Over time, both the step size and probability-function shrinks. With this scheme, the algorithm will tend to initially jump over local minima, but fine-tune and converge later in the optimization process. Since narrower FOVs are intrinsically more problematic, two additional angles, 6° and 10°, were investigated with lens type **III.** These angles, along with the choice of acrylic glass as lens material, make it possible to compare results with those of Chen and Lin [6].

## 3. Results

Optimizations were performed in an in-house programmed software, and results were verified with Zemax, a commercial ray tracer. For the RFS, 7 optimization runs were performed per lens type at 8 lens-to-LED size ratios between 18:3 and 48:3. For the PS, 5 optimizations per type sufficed due to more stable convergence, and the lens size was kept fixed since scaling would have no effect on the angular distribution. The result was a total of 183 individual lens optimizations, each starting with a randomized initial geometry. The maximum number of steps allowed for each optimization was limited to 3000 for type **I**, 3200 for type **II**, and 3500 for type **III** lenses. Each source model was assigned 200.000 initial rays.

#### 3.1 Efficiency

Figure 2
visualizes the convergence of the SAMC algorithm for type **III** lenses in the 16° FOV. For visualization purposes only steps that improve the design are shown. Two optimizations are represented in each lens/source category (one in black and one in grey), and

RF optimizations cover size ratios between 18:3 and 40:3. All PS optimizations converged quickly toward 91.9±0.1% luminous efficiency while RF optimizations converged slightly less stably overall.

Figures 3**(a)**
and 3**(b)** compare the efficiency of the best lenses found according to the investigation scheme at the 16° FOV. Each point on the ‘RF’ graphs in 3**(a)** represents RFS optimized lenses of a given type and size ratio, while the single ‘PS’ graph covers all three lens types due to the results being very close. The ‘PS2RF’ graphs represent a PS optimized lens, with the PS replaced by an RFS and the lens scaled in order to calculate the size-dependence. ‘RF2PS’ in 3**(b)** is the reverse: the PS is inserted into RFS optimized lenses. The first 5 graphs in 3**(b)** show the relative efficiency of the RF**III** lenses compared with their RF**I** and RF**II** counterparts as well as with the three PS optimized lenses.

Figure 4
compares the efficiency of RF**III** lenses with PS**III** lenses and with the results listed by Chen and Lin [6], at FOV angles 6° and 10°. The results at 16° shown in Fig. 3 are included for comparison. The narrow FOVs required an extension of the range of size ratios to 150:3. Two windows of application are shown: 80% for the 10° FOV as suggested by Chen and Lin, and 72.5%, which was deemed appropriate for a 6° FOV. Also, the PS2RF graphs were averaged over 5 different lenses to show the trend instead of just the superior result. The maximum efficiency of the RF lenses was found to be 91.9% at the 10° FOV.

#### 3.2 Lens Shape

In addition to the convergence in efficiency, the lens shapes also converged as a function of size ratio. This is visualized in Fig. 5
, which is a diagram of the most optimal type **III** lenses found. At small ratios, the top center lens was highly prominent and extended, while the top of the peripheral TIR section was almost flat in direction toward the center. At larger ratios, the center lens receded, while the top peripheral part inclined and became nearly tangential to the topmost lens, which approaches the typical shape found for PS optimizations.

## 4. Discussion

It is clear from the similarity of the emission profiles in Fig. 1**(a)** why LEDs are assumed to generally behave like point sources. However, all other results clearly indicate the opposite. The poor performance of the PS approximation even at size ratios above 32:3, is perhaps the most surprising: Optimizing with an RFS raised the efficiency by 9-18%. This is most likely a consequence of the lens design. The 16° FOV and lens radius constraints can be considered an aperture with an 8° acceptance angle and radius R, which carries an etendue of *E _{FOV} ≈π*

^{2}

*R*

^{2}

*sin*

^{2}

*(*8

*°)*, while the LED carries

*E*9mm

_{LED}≈π^{2}

*=*28.3mm

^{2}. In an unconstrained, lossless system, it should therefore be theoretically possible to deliver all lumens from the LED to the 16

*°*FOV, down to a radius of

*R≈*12.2mm

^{2}, or a size ratio of 24.3:3 [10]. The compact lens design, however, is quite constrained and is essentially a 5-body problem concerning etendue matching of the small individual surfaces. Especially noteworthy is the entry surface directly above the LED. It has a small area that scales quadratically with radius, and the incoming angles also change as a function of radius. Its etendue is therefore subject to large variation, suggesting that it requires particularly reliable optimization, especially since it lies in the direction of the peak of the emission profile. In this case it seems to have the consequence that RFS optimized lenses have a larger curvature on this particular surface than PS optimized lenses, a fact that can be visually observed in Fig. 5.

At size ratios below 36:3 the performance of PS optimized lenses was very unpredictable. No notable trends were found except that the size-effects of type **I** lenses were consistently worse, while type **III** and **II** lenses were much more equal. In general it was very much up to chance, on account of the stochastic SAMC algorithm used. More well defined methods such as [1] and [6] might be more stable. From this, one could argue that the results might be due to unreliability of the algorithm, but this is clearly not the case. All PS optimizations converged uniformly to ≈92.0% efficiency; very close to the transmission through two acrylic glass interfaces at normal incidence (92.3%). The slightly lower efficiency can therefore be explained solely by Fresnel losses. Also, Fig. 2 shows that lenses converge reliably, independent of initial geometry; the RF**I**, RF**II** and RF**III** graphs in Fig. 3 do not cross at any point, with the efficiency scaling with degrees of freedom; and Figs. 3–5 show that the RF lenses converge in both performance and shape to the PS lenses as the size ratio increases, up to a maximum efficiency of 91.9%. The crossed graphs in Fig, 3**(b)** show another important trend: At small size ratios, lenses optimized properly for LEDs are not optimal for point sources. Although not conclusive, this strongly indicates that it may be wholly impossible to find optimal LED lenses using PS optimizations in this range. The restricted type **I** lenses also converged to almost the exact same efficiency as type **III** with a PS, but performed consistently worse with the more realistic RFS. These facts should be reason enough for anyone to thoroughly test their models with a more realistic source model such as an RFS, even though this often makes the optimization process slower and more cumbersome.

As Fig. 4 shows, the RF optimized lenses also perform better at FOV angles 6° and 10°, also compared with the results of Chen and Lin [6] which incidentally coincide well with the averaged PS2RF graphs. At the selected WOAs, the lens footprint area is reduced by >55% at 6° FOV and >72% at 10°. These figures increase greatly if the WOAs are raised, and can be considered lower bounds since the efficiency could probably be raised slightly by increasing the degrees of freedom and optimizing further. The RFS optimizations thereby allow for at least twice as many lumen per area or a correspondingly lower etendue, which could make it possible to significantly improve the efficacy in applications with an aperture. A justification for continued development of fast PS-based methods – aside from algorithmic and mathematic advances – can be found in Fig. 5. It shows that PS optimizations could possibly be useful as seeds for more accurate optimizations, and thereby speed up the overall process. However, this would only make sense for relatively large lenses.

In general all results point to the same fact: Point source approximations are highly unreliable in small-dimensional TIR lenses and can cause significant efficiency losses even at larger dimensions. And while this paper presents a specific design, and results may vary from case to case, all similar compact LED lens designs can be expected to have roughly the same proportions in terms of surface sizes and curvatures. It would therefore be prudent to assume up front that the size-effects could be quite significant before using PS optimized designs for an end product, and it should probably be completely avoided when possible.

## 5. Conclusion

A large number of optimizations were performed in order to investigate how well a point source (PS) approximation performed compared with a more accurate LED representation in a compact LED lens design process. An LED was modeled as a factory-measured ray-file (RF), and a PS as a ray set with a Lambertian emission profile. Three lens models, named **I**, **II**, and **III**, were investigated. Lens type **I** had a minimum of free variables (10), while type **II** and **III** had 12 and 15, respectively. This was used to test whether the lens complexity would affect the end result of both the PS and RF optimizations. The main objective was to maximize the luminous efficiency within a 16° field of view (FOV). The optimization algorithm (Simulated Annealing Monte Carlo) was thoroughly tested for reliability, and additional results at FOV angles 6° and 10° were calculated in order to compare with results from another paper [6].

PS optimized lenses had consistently lower efficiency than RF optimized lenses, with unpredictable performance at small lens-to-LED size ratios. Additionally, all three lens types would perform equally well with a PS, while RF optimizations would create superior results according to the degrees of freedom of the lenses. RF optimizations would perform >9% better than PS optimizations, even at large size ratios up to 48:3, and up to 55% better at small ratios. In the 16° FOV an efficiency of 91.3% was achieved with the RF at size ratios up to 48:3, while up to 91.9% was reached within the 6° and 10° FOVs at larger ratios. In addition, the window of application of the lenses was increased significantly, reducing the lens footprint area by >55% at the 6° FOV and by >72% at 10°, at a fixed efficiency. It was also shown that optimal LED lenses were not optimal PS lenses, especially not at the small scale and for the more appropriate type **II** and **III** lenses, indicating that it might not be at all possible to find optimal LED lenses using PS optimizations. In conclusion, using PS models for compact lens-designs is generally advised against when precision and energy efficiency are important factors, unless the soundness of the approximation has been sufficiently verified by a more realistic model.

## Acknowledgments

This research was funded by Aalborg University and the Danish National Advanced Technology Foundation project nr. 053-2007-3.

## References and links

**1. **J. Jiang, S. To, W. B. Lee, and B. Cheung, “Optical design of a freeform TIR lens for LED streetlight,” Optik (Jena) **121**, 1761–1765 (2010).

**2. **A. Domhardt, U. Rohlfing, S. Weingaertner, K. Klinger, D. Kooß, K. Manz, and U. Lemmer, “New design tools for LED headlamps,” Proc. SPIE **7003**, 70032C, 70032C-10 (2008). [CrossRef]

**3. **A. Domhardt, S. Weingaertner, U. Rohlfing, and U. Lemmer, “TIR Optics for non-rotationally symmetric illumination design,” Proc. SPIE **7103**, 710304 (2008). [CrossRef]

**4. **T. Kari, J. Gadegaard, D. T. Jørgensen, T. Søndergaard, T. G. Pedersen, and K. Pedersen, “Compact lens with circular spot profile for square die LEDs in multi-LED projectors,” Appl. Opt. **50**(24), 4860–4867 (2011). [CrossRef] [PubMed]

**5. **K. Wang, F. Chen, Z.-Y. Liu, X.-B. Luo, and S. Liu, “Design of compact freeform lens for application specific Light-Emitting Diode packaging,” Opt. Express **18**(2), 413–425 (2010). [CrossRef] [PubMed]

**6. **J.-J. Chen and C.-T. Lin, “Freeform surface design for a light-emitting diode-based collimating lens,” Opt. Eng. **49**(9), 093001 (2010). [CrossRef]

**7. **H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A **19**(3), 590–595 (2002). [CrossRef] [PubMed]

**8. **P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. **43**(7), 1489–1502 (2004). [CrossRef]

**9. **CBT-90 series LEDs, Luminus Devices, Inc., http://www.luminus.com.

**10. **N. Shatz and J. C. Bortz, “Consequence of symmetry” in *Nonimaging Optics*, R. Winston, J. C. Miňano, and P. Benítes, (Elsevier, 2005), pp. 235–264.