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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 16 — Aug. 3, 2009
  • pp: 13918–13927
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Analysis of the far-field region of LEDs

Ching-Cherng Sun, Wei-Ting Chien, Ivan Moreno, Chih-Ching Hsieh, and Yi-Chien Lo  »View Author Affiliations


Optics Express, Vol. 17, Issue 16, pp. 13918-13927 (2009)
http://dx.doi.org/10.1364/OE.17.013918


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Abstract

Versatility in the design of optical systems is one of the key features of light-emitting diodes (LEDs) that has attracted considerable attention. In the analysis of systems using LEDs, it is useful to know if the distance is far enough from the LED to allow the radiation pattern to be simulated by the point source approach. We propose three far-zone conditions for LED light modeling: the far-field distance, and for practical purposes the quasi far-field and minimum far-field distances. Different types of LEDs have different far-field ranges. We analyze these differences by modulating key parameters like geometrical structure of encapsulating lens, chip size, chip shape, chip position, and package errors. We find that far-field region considerably depends more on the shape of both lens and chip than all other parameters.

© 2009 OSA

1. Introduction

LEDs are energy efficient, more durable, non-toxic, and smartly controllable light sources. These properties will lead to many applications. However, there are special challenges in the optical design of a system containing LEDs [1–14

1. C. C. Sun, C. Y. Chen, H. Y. He, C. C. Chen, W. T. Chien, T. X. Lee, and T. H. Yang, “Precise optical modeling for silicate-based white LEDs,” Opt. Express 16(24), 20060–20066 (2008). [CrossRef] [PubMed]

]. Care must be taken to use the simplest LED model that more accurately predicts the performance of the system [1–5

1. C. C. Sun, C. Y. Chen, H. Y. He, C. C. Chen, W. T. Chien, T. X. Lee, and T. H. Yang, “Precise optical modeling for silicate-based white LEDs,” Opt. Express 16(24), 20060–20066 (2008). [CrossRef] [PubMed]

]. In the analysis of an optical system containing LEDs, it is very useful to know if the LED can be simulated as a directional point source. In most of the applications, a simple optical model is desirable because it is common to trace millions of rays for analyzing and testing the optical system. Near the LED, accuracy can only be obtained by modeling the LED as an extended source. In this region a precise model must include in detailed key component parts as chip, phosphor layer, cup, and lens [1–3

1. C. C. Sun, C. Y. Chen, H. Y. He, C. C. Chen, W. T. Chien, T. X. Lee, and T. H. Yang, “Precise optical modeling for silicate-based white LEDs,” Opt. Express 16(24), 20060–20066 (2008). [CrossRef] [PubMed]

]. Sufficiently far away from the LED, the same accuracy can be obtained by modeling the LED as a point source whose output is apodized as a function of angle [4

4. I. Moreno and C. C. Sun, “Modeling the radiation pattern of LEDs,” Opt. Express 16(3), 1808–1819 (2008). [CrossRef] [PubMed]

,5

5. H. Yang, J. W. M. Bergmans, T. C. W. Schenk, J. P. M. G. Linnartz, and R. Rietman, “An analytical model for the illuminance distribution of a power LED,” Opt. Express 16(26), 21641–21646 (2008). [CrossRef] [PubMed]

].

In some cases, secondary optics is designed by means of advanced methods that model the LED as a directional point source [6–8

6. Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008). [CrossRef] [PubMed]

]. In these applications is useful to know where far-field begins. In other applications the spatial radiation pattern of LED must be simulated as an extended source by using a precise model [9–13

9. D. Feng, J. Yoo, K. Nagatani, W. Kim, and H. C. Kim, “High Illumination Efficiency Linear Light Source Using Light Emitting Diodes,” Jpn. J. Appl. Phys. 46(2), 563–565 (2007). [CrossRef]

]. In such situations, it is useful to know within which distance range the extended source model must be used for accurate performance. Other applications are totally based on the point source approximation; in these cases it is necessary to know the minimum distance for which the analysis is valid [14–16

14. H. Yang, J. W. M. Bergmans, T. C. W. Schenk, J. P. M. G. Linnartz, and R. Rietman, “Uniform Illumination Rendering Using an Array of LEDs: A Signal Processing Perspective,” IEEE Trans. Signal Process. 57(3), 1044–1057 (2009). [CrossRef]

].

Recently, we developed a far-field condition for LED arrays, where each LED was considered to be a directional point source [16

16. I. Moreno, C. C. Sun, and R. Ivanov, “Far-field condition for light-emitting diode arrays,” Appl. Opt. 48(6), 1190–1197 (2009). [CrossRef]

]. Here we address the issue of the far field of a single LED. Depending on the working distance, the optical modeling of an LED must be performed in a different way. Basically, in LEDs there are three working conditions: near-field, mid-field, and far-field (Fig. 1). The near-field is a very short region near the chip, where the electro-magnetic wave effects strongly change the wave function with distance. In the midfield region, the light pattern significantly varies from one distance to another due to the finite size of chip and LED package structure. Thus mid-field field is a region for verifying optical models of extended light sources. Farther than the mid-field, in far-field the angular intensity distribution remains practically constant with distance. A clear definition of far-field is helpful to know if the measurement distance is located in the region of mid-field so light source model can be verified in such a region. However, there is not available a standard convention to delimitate these regions because LEDs come in many varieties. To alleviate this problem, we propose three conditions: a far-field (FF) condition, a quasi far-field (QFF) condition, and a minimum far-field (MFF) condition. These distances are determined from comparing the shape of radiation pattern at a finite distance with that at a very far distance (intensity pattern of reference). We analyze their variation by modulating key parameters like geometrical structure of encapsulating lens, chip size, chip shape, chip position, and package errors.

Fig. 1. There are three basic working conditions for the optical modeling and experimental characterization of the radiation pattern of an LED: near-field, mid-field and far-field.

2. Where does far-field begin ?

An answer to this question involves comparing the normalized angular intensity distribution at some distance with that at a very far distance (infinity in theory). Therefore, this requires simulating the angular intensity distribution for several distances or LED-detector separations D (see Fig. 2(a)). As an example, consider a batwing LED from Lumileds Lighting (Fig. 2(b)). The radiation pattern [arbitrary units per stereo radian] of this LED is plotted in Fig. 3 for several distances. The shape of both experimental and simulated light patterns varies from one distance to another, but after some distance tends to remain practically unchanged.

To evaluate the similarity between the radiation pattern at some distance and that of reference (at a very long distance), we compute the normalized cross correlation (NCC) [2–4

2. C. C. Sun, T. X. Lee, S. H. Ma, Y. L. Lee, and S. M. Huang, “Precise optical modeling for LED lighting verified by cross correlation in the midfield region,” Opt. Lett. 31(14), 2193–2195 (2006). [CrossRef] [PubMed]

]. NCC is a merit function that is frequently used for image recognition to compare the similarity between two images. The NCC of the radiation pattern ID at distance D with respect to the radiation pattern of reference IR is

NCC=n[I(θn)DI¯D][I(θn)RI¯R]n[I(θn)DI¯D]2n[I(θn)RI¯R]2,
(1)

where θ n is the n-th view angle, e.g. the angular displacement of the detector shown in Fig. 2(a). ĪD and ĪR are the mean value of ID and IR across the angular range. As distance D increases the NCC approaches 100%. An NCC equal to 100% means that ID and IR are exactly the same.

Figure 4 shows NCC vs. measurement distance for the radiation patterns of Fig. 3, i.e. the LED with batwing pattern depicted in Fig. 2(b). In both simulation and experiment, the radiation pattern of reference is that at distance D = 500mm.

In our experience in designing LED lighting systems for industry, depending on the NCC of the LED model the design predicts with different degree of precision the experimental results [2–4

2. C. C. Sun, T. X. Lee, S. H. Ma, Y. L. Lee, and S. M. Huang, “Precise optical modeling for LED lighting verified by cross correlation in the midfield region,” Opt. Lett. 31(14), 2193–2195 (2006). [CrossRef] [PubMed]

]. We choose three NCC values to delimitate a range of working distances for modeling an LED as a directional point source with different levels of precision. We propose, as the lower bound, the minimum far-field (MFF) distance as that distance at which NCC = 99%; and we define the upper bound as the far-field (FF) distance for NCC = 99.90%. We do not choose 100% as the upper bound because the manufacturing variability among LEDs of the same type limits the precision of any optical model [2

2. C. C. Sun, T. X. Lee, S. H. Ma, Y. L. Lee, and S. M. Huang, “Precise optical modeling for LED lighting verified by cross correlation in the midfield region,” Opt. Lett. 31(14), 2193–2195 (2006). [CrossRef] [PubMed]

,3

3. W. T. Chien, C. C. Sun, and I. Moreno, “Precise optical model of multi-chip white LEDs,” Opt. Express 15(12), 7572–7577 (2007). [CrossRef] [PubMed]

]. Modeling an LED as point source at MFF distance gives enough accuracy for some applications, e.g. for first order optical designs. The third condition is the quasi far-field (QFF) distance for NCC = 99.50%. We recommend for precise designs to use an extended source model if the distance is shorter than the QFF distance. In general, if the extended source model is used at longer distances than QFF, a little more precise optical design is obtained, but after the FF distance the extended source model will give the same results than the point source approach. Figure 4 shows the regions delimitated by MFF, QFF and FF distances (80, 100, and 250 mm) for an LED with batwing pattern.

These working distances significantly depend upon key LED parameters like geometrical structure of encapsulating lens, chip size, chip shape, chip position, and manufacturing errors. In what follows, we analyze the effects on FF, QFF and MFF conditions due to changes on these key parameters.

Fig. 2. (a) Schematic diagram used in the simulation of the angular intensity distribution of an LED. Distance D is measured from the LED tip. (b) Geometric model of batwing LED from Lumileds Lighting.
Fig. 3. The simulated and the measured radiation patterns of the LED depicted in Fig. 2(b). The radiation patterns are measured at (a) D = 15mm, (b) D = 30mm, (c) D = 60mm and (d) D = 300mm.
Fig. 4. NCC in function of distance D from the LED. This NCC is for the LED shown in Fig. 2(b) Also are shown the minimum far-field (MFF), quasi far-field (QFF), and far-field (FF) distances of this LED.

3. Analysis

In the following analysis each chip surface is considered to be a Lambertian emitter for Monte Carlo simulation. We traced ten million rays for the simulation of each radiation pattern. In order of simplify the analysis, we darkened the bottom of encapsulant. The refractive index of encapsulating lens is 1.5 in all simulations. The angular range to evaluate NCC is different in each case, it is the angular width of the radiation pattern of every modeled LED. With the exception of section 3.2, the chip is square shaped (1 × 1mm2) with flat faces. D is the distance from the LED tip as shown in Fig. 2(a).

3.1 Dependence upon encapsulating lens

From a classic point of view, we should expect that the far-field distance remains the same when the chip size does not change. However, the far field distance not only changes in function of the shape and size of emitting area, but also in function of the size and shape of lens.

First we consider a hemispherical encapsulating lens. Figure 5(a) shows the NCC of the LED radiation pattern in function of distance D for several lens diameters. Figure 5(b) shows FF, QFF, and MFF. As can be noted, the FF distance is about 10~20 times larger than the MFF distance. Taking into account that distance D is measured from the LED tip, when increasing the lens diameter the QFF and MFF distance from the chip is a little longer than that shown in Fig. 5(b). The analysis of the lens size dependence leads to a basic condition: once increasing the lens diameter, the MFF distance also will be increased by 3 times of the lens diameter, and QFF will be about 5 times the lens diameter. This fact is very useful for precise optical modeling of LED systems.

Now we consider a free form lens with revolution ellipsoid shape, in which its thickness is varied. By fixing the horizontal semi axis of the ellipsoid, we changed the vertical axis for changing the thickness of the encapsulation. The bottom of the encapsulation is circular. Figure 6(a) shows the NCC in function of distance D for several lens thicknesses. Figure 6(b) shows the FF, QFF, and MFF dependence on lens thickness. The behavior is not linear, and shows a maximum around L = 1.5 mm.

Figure 7 shows the dependence with lens width. This is an ellipsoid lens with circular bottom, but now the semi-major axis is varied. The FF, QFF and MFF distances have a minimum for a 1.5mm width, where MFF distance is zero. It is interesting to note that the width for this minimum coincides with the thickness of the maximum in Fig. 6(b). From observing the LED radiation patterns associated to Fig. 6, one can note that the far-field pattern for L = 1.5mm has a wide angular range, and the sides of the angular distribution reduce to zero with a pronounced slope. In contrast, when observing the LED radiation patterns associated to Fig. 7, one can note that the far-field pattern for r = 1.5mm also has a wide angular range, but the sides of the angular distribution reduce to zero very smoothly. For the sake of clarity, we do not show here the simulated radiation patterns. However, this is the basic dependence of the far-field region with the angular intensity distribution in most of the following cases, with the only exception of case analyzed in Fig. 11.

Fig. 5. Lens size dependence. (a) NCC in function of distance D from the LED. Inset shows geometry of the encapsulated LED. (b) Far field region for several lens sizes.
Fig. 6. Lens thickness dependence. (a) NCC in function of distance from the LED tip. (b) Far field region for several lens thicknesses.
Fig. 7. Lens width dependence. (a) NCC in function of distance from the tip of LED. (b) Far field range for several lens widths.

3.2 Dependence upon chip characteristics

In the following, we analyze the dependence upon chip characteristics of an LED with a hemispheric lens. From a classical point of view, we should expect that the far-field distance increases in function of the source size. However, as we show below, the QFF distance not always increases when the chip size increases. Furthermore, the QFF considerably increases if the chip or primary source is elliptically shaped.

First we consider a flat chip. Figure 8 shows the dependence on chip size. The chip dimension A changes from 1 mm to 3.5mm. We can see how MFF and QFF distances remain almost constant for several chip sizes. When A is small, the far-field light pattern is just like the Lambertian. However, as A increases more than 3.5mm, the width of the far-field pattern width remains almost the same but decreases more rapidly at large view angles.

Now we consider a chip with hemispheric shape. Figure 9 shows the dependence on the primary source size. The QFF and MFF distance increases nearly linear with the chip size.

Figure 10 shows the FF, QFF, and MFF distance for an LED with elliptic shaped primary source. These distances strongly increase with the source size. This case has the same trend in function of the far-field radiation pattern; as Ls increases the radiation pattern widens and the profile significantly sharpens at the sides of the angular distribution.

Fig. 8. Chip size dependence. (a) NCC in function of distance D from the LED. (b) Far field region for several chip sizes A.
Fig. 9. Diameter dependence for an hemispherical source. (a) NCC in function of distance D from the LED. (b) Far field range for several source sizes Rs.
Fig. 10. Diameter dependence for an elliptic source. (a) NCC in function of distance D from the LED. (b) Far field range for several ellipse semi-axis sizes Ls.

3.3 Dependence upon chip position or package manufacturing errors

Any LED suffers from tolerances like all optical devices. A common manufacturing error is the chip displacement from a nominal position. This variation leads to the fluctuation of the far-field range of an LED. In the following simulations, we keep constant both the lens size and shape (hemisphere).

First we consider a chip moved in backward direction. Figure 11(a) shows the NCC of the LED radiation pattern in function of distance D for several chip shifts. The QFF and MFF distance show a nearly linear increment with chip displacement (Fig. 11(b)). From observation of the far-field radiation patterns one can note that beam directionality increases as the chip is displaced in back direction. This is the only case in which the QFF and MFF increases as the far-field radiation pattern is more directional. The angular intensity distribution is narrowed when the chip is displaced in this way, as happens for LEDs with the typical 5mm package.

Now we consider an LED with the chip displaced in forward direction. Figure 12 shows how the MFF distance remains almost the same when the chip is moved to the front. However, the QFF increases linearly in function of this chip shift.

Finally, Fig. 13 shows the FF, QFF, and MFF dependence on chip displacement in transverse direction. Although the associated radiation patterns are twisted with the lateral shear, the MFF and QFF distance remain almost constant.

Fig. 11. Backward displacement of chip. (a) NCC in function of distance D from the LED. (b) Far field region for several chip displacements h.
Fig. 12. Forward displacement of chip. (a) NCC in function of distance D from the LED. (b) Far field range for several chip displacements H.
Fig. 13. Chip transverse displacement dependence. (a) NCC in function of distance D from the LED. (b) Far field range for several chip displacements t.

4. Summary

As far as we know, there is not available a standard convention to delimitate the far-field region of LEDs. However, in the optical analysis of systems containing LEDs, it is useful to know if the spatial radiation pattern of LEDs can be modeled as a directional point source. To alleviate this situation, we proposed three far-zone distances for LED optical modeling: the far-field distance, and for practical purposes the quasi far-field and minimum far-field distances. These working distances are determined from comparing the shape of radiation pattern at a finite distance with that (radiation pattern of reference) at a very far distance. We analyzed the far-field region dependence on important parameters, in particular the shape and size of the encapsulating lens, chip size, chip shape, chip position, and package errors. The analysis showed some interesting aspects about the effect of these parameters. For example, in all cases the FF distance was between 1.6 and 23 times larger than MFF distance. And the FF distance was between 1.3 and 11 times larger than QFF distance. Also, the far-field region considerably depends more on the shape of both lens and chip than all other parameters.

Acknowledgement

References and links

1.

C. C. Sun, C. Y. Chen, H. Y. He, C. C. Chen, W. T. Chien, T. X. Lee, and T. H. Yang, “Precise optical modeling for silicate-based white LEDs,” Opt. Express 16(24), 20060–20066 (2008). [CrossRef] [PubMed]

2.

C. C. Sun, T. X. Lee, S. H. Ma, Y. L. Lee, and S. M. Huang, “Precise optical modeling for LED lighting verified by cross correlation in the midfield region,” Opt. Lett. 31(14), 2193–2195 (2006). [CrossRef] [PubMed]

3.

W. T. Chien, C. C. Sun, and I. Moreno, “Precise optical model of multi-chip white LEDs,” Opt. Express 15(12), 7572–7577 (2007). [CrossRef] [PubMed]

4.

I. Moreno and C. C. Sun, “Modeling the radiation pattern of LEDs,” Opt. Express 16(3), 1808–1819 (2008). [CrossRef] [PubMed]

5.

H. Yang, J. W. M. Bergmans, T. C. W. Schenk, J. P. M. G. Linnartz, and R. Rietman, “An analytical model for the illuminance distribution of a power LED,” Opt. Express 16(26), 21641–21646 (2008). [CrossRef] [PubMed]

6.

Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008). [CrossRef] [PubMed]

7.

A. Pawlak and K. Zaremba, “Reflector luminaire with high power light-emitting diodes for general lighting,” Appl. Opt. 47(3), 467–473 (2008). [CrossRef] [PubMed]

8.

H. Xiang, Z. Zhenrong, L. Xu, and G. Peifu, “Freeform surface lens design for uniform illumination,” J. Opt. A, Pure Appl. Opt. 10(7), 075005 (2008). [CrossRef]

9.

D. Feng, J. Yoo, K. Nagatani, W. Kim, and H. C. Kim, “High Illumination Efficiency Linear Light Source Using Light Emitting Diodes,” Jpn. J. Appl. Phys. 46(2), 563–565 (2007). [CrossRef]

10.

C. H. Tsuei, J. W. Pen, and W. S. Sun, “Simulating the illuminance and the efficiency of the LED and fluorescent lights used in indoor lighting design,” Opt. Express 16(23), 18692–18701 (2008). [CrossRef]

11.

F. Muñoz, P. Benítez, O. Dross, J. C. Miñano, and B. Parkyn, “Simultaneous multiple surface design of compact air-gap collimators for light-emitting diodes,” Opt. Eng. 43, 1522 (2004). [CrossRef]

12.

P. C. P. Chao, C. Y. Shen, C. W. Chiu, J. S. Huang, Y. Y. Kao, T. Y. Tu, C. L. Wang, S. Chi, H. W. Lin, and S. Y. Tsai, “A novel lens cap designed for the RGB LEDs installed in an ultra-thin and directly lit backlight unit of large-sized LCD TVs,” J. Soc. Inf. Disp. 16(2), 317–327 (2008). [CrossRef]

13.

K. Wang, X. B. Luo, Z. Y. Liu, B. Zhou, Z. Y. Gan, and S. Liu, “Optical analysis of an 80-W light-emitting-diode street lamp,” Opt. Eng. 47(1), 013002 (2008). [CrossRef]

14.

H. Yang, J. W. M. Bergmans, T. C. W. Schenk, J. P. M. G. Linnartz, and R. Rietman, “Uniform Illumination Rendering Using an Array of LEDs: A Signal Processing Perspective,” IEEE Trans. Signal Process. 57(3), 1044–1057 (2009). [CrossRef]

15.

I. Moreno, M. Avendaño-Alejo, and R. I. Tzonchev, “Designing light-emitting diode arrays for uniform near-field irradiance,” Appl. Opt. 45(10), 2265–2272 (2006). [CrossRef] [PubMed]

16.

I. Moreno, C. C. Sun, and R. Ivanov, “Far-field condition for light-emitting diode arrays,” Appl. Opt. 48(6), 1190–1197 (2009). [CrossRef]

OCIS Codes
(230.3670) Optical devices : Light-emitting diodes
(350.4600) Other areas of optics : Optical engineering

ToC Category:
Optical Devices

History
Original Manuscript: April 24, 2009
Revised Manuscript: June 24, 2009
Manuscript Accepted: July 21, 2009
Published: July 27, 2009

Citation
Ching-Cherng Sun, Wei-Ting Chien, Ivan Moreno, Chih-Ching Hsieh, and Yi-Chien Lo, "Analysis of the far-field region of LEDs," Opt. Express 17, 13918-13927 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13918


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References

  1. C. C. Sun, C. Y. Chen, H. Y. He, C. C. Chen, W. T. Chien, T. X. Lee, and T. H. Yang, “Precise optical modeling for silicate-based white LEDs,” Opt. Express 16(24), 20060–20066 (2008). [CrossRef] [PubMed]
  2. C. C. Sun, T. X. Lee, S. H. Ma, Y. L. Lee, and S. M. Huang, “Precise optical modeling for LED lighting verified by cross correlation in the midfield region,” Opt. Lett. 31(14), 2193–2195 (2006). [CrossRef] [PubMed]
  3. W. T. Chien, C. C. Sun, and I. Moreno, “Precise optical model of multi-chip white LEDs,” Opt. Express 15(12), 7572–7577 (2007). [CrossRef] [PubMed]
  4. I. Moreno and C. C. Sun, “Modeling the radiation pattern of LEDs,” Opt. Express 16(3), 1808–1819 (2008). [CrossRef] [PubMed]
  5. H. Yang, J. W. M. Bergmans, T. C. W. Schenk, J. P. M. G. Linnartz, and R. Rietman, “An analytical model for the illuminance distribution of a power LED,” Opt. Express 16(26), 21641–21646 (2008). [CrossRef] [PubMed]
  6. Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008). [CrossRef] [PubMed]
  7. A. Pawlak and K. Zaremba, “Reflector luminaire with high power light-emitting diodes for general lighting,” Appl. Opt. 47(3), 467–473 (2008). [CrossRef] [PubMed]
  8. H. Xiang, Z. Zhenrong, L. Xu, and G. Peifu, “Freeform surface lens design for uniform illumination,” J. Opt. A, Pure Appl. Opt. 10(7), 075005 (2008). [CrossRef]
  9. D. Feng, J. Yoo, K. Nagatani, W. Kim, and H. C. Kim, “High Illumination Efficiency Linear Light Source Using Light Emitting Diodes,” Jpn. J. Appl. Phys. 46(2), 563–565 (2007). [CrossRef]
  10. C. H. Tsuei, J. W. Pen, and W. S. Sun, “Simulating the illuminance and the efficiency of the LED and fluorescent lights used in indoor lighting design,” Opt. Express 16(23), 18692–18701 (2008). [CrossRef]
  11. F. Muñoz, P. Benı́tez, O. Dross, J. C. Miñano, and B. Parkyn, “Simultaneous multiple surface design of compact air-gap collimators for light-emitting diodes,” Opt. Eng. 43, 1522 (2004). [CrossRef]
  12. P. C. P. Chao, C. Y. Shen, C. W. Chiu, J. S. Huang, Y. Y. Kao, T. Y. Tu, C. L. Wang, S. Chi, H. W. Lin, and S. Y. Tsai, “A novel lens cap designed for the RGB LEDs installed in an ultra-thin and directly lit backlight unit of large-sized LCD TVs,” J. Soc. Inf. Disp. 16(2), 317–327 (2008). [CrossRef]
  13. K. Wang, X. B. Luo, Z. Y. Liu, B. Zhou, Z. Y. Gan, and S. Liu, “Optical analysis of an 80-W light-emitting-diode street lamp,” Opt. Eng. 47(1), 013002 (2008). [CrossRef]
  14. H. Yang, J. W. M. Bergmans, T. C. W. Schenk, J. P. M. G. Linnartz, and R. Rietman, “Uniform Illumination Rendering Using an Array of LEDs: A Signal Processing Perspective,” IEEE Trans. Signal Process. 57(3), 1044–1057 (2009). [CrossRef]
  15. I. Moreno, M. Avendaño-Alejo, and R. I. Tzonchev, “Designing light-emitting diode arrays for uniform near-field irradiance,” Appl. Opt. 45(10), 2265–2272 (2006). [CrossRef] [PubMed]
  16. I. Moreno, C. C. Sun, and R. Ivanov, “Far-field condition for light-emitting diode arrays,” Appl. Opt. 48(6), 1190–1197 (2009). [CrossRef]

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