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Reliability of point source approximations in

compact LED lens designs

Thøger Kari,* Jesper Gadegaard, Thomas Søndergaard, Thomas Garm Pedersen,

and Kjeld Pedersen

Dept. of Physics and Nanotechnology, Aalborg University, Skjernvej 4A, DK-9220 Aalborg, Denmark

*tkj@nano.aau.dk

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 Optical Society of America

OCIS codes: (000.4430) Numerical approximation and analysis; (220.2740) Geometric optical

design; (220.4298) Nonimaging optics; (230.3670) Light-emitting diodes.

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).

3. A. Domhardt, S. Weingaertner, U. Rohlfing, and U. Lemmer, “TIR Optics for non-rotationally symmetric

illumination design,” Proc. SPIE 7103, 710304 (2008).

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).

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).

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).

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

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).

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.

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.

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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-

Fig. 1. (a) Polar emission profile of the simulated point source (black line) and the factory-

measured ray-file used for the LED (gray line). (b) Cross section of a TIR lens, showing the

rotational profile that defines the different lens types. Italics signify free optimization variables

while bold fonts signify constants. υ and ω are calculated from (R), e and j. Variables with a

square marker are only free for some lens types. In addition, k = (R)/2 and θ 2 = 0? for lens

type (I).

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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 θ2 locked at k = R/2, and θ2 = 0?, which is non-optimal for LED optimizations. These

are unlocked for lens types 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

Fig. 2. Selected convergence graphs for type III lenses. ‘PS’ and ‘RF’ signify point source and

ray-file optimizations respectively. The ratios in the legend are the lens diameter to the edge

length of the LED. Two graphs are shown for each lens/source category, in black and in gray.

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Fig. 3. Each point on the graphs represents the efficiency of a lens with a given size. ‘PS’ and

‘RF’ indicate source model and ‘I’, ‘II’ or ‘III’ indicate lens type. ‘PS2RF’ graphs in (a) show

the efficiency obtained by inserting an RFS into a PS optimized lens at a given size ratio and

reversely for ‘RF2PS’ in (b). The first 5 graphs of (b) show the relative efficiency of the RFIII

lenses compared with the RF(I), RFII, and PS lenses.

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 RFIII lenses compared with their RFI

and RFII counterparts as well as with the three PS optimized lenses.

Figure 4 compares the efficiency of RFIII lenses with PSIII 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.

Fig. 4. Efficiencies of RF optimized lenses at three FOV angles. ‘WOA’ is the window of

application. The ‘Chen’ graphs concern the results listed by Chen and Lin [6], and ‘RF vs.

PS2RF’ and ‘RF vs. Chen’ show the relative efficiency of the RF lenses. Each ‘PS2RF’ value

represents an average over 5 PS lenses.

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Fig. 5. Diagram of optimal lens shape as a function of lens-to-LED size ratio. At large ratios,

the lens designs approach those found by point source optimizations. The figures are cross

sections of the actual optimized 3D lenses.

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 EFOV ≈π2

R2sin2(8?), while the LED carries ELED ≈π9mm2 = 28.3mm2. 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≈12.2mm2, 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 RFI, RFII and RFIII 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

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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.

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