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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 3 — Feb. 4, 2008
  • pp: 1808–1819
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Modeling the radiation pattern of LEDs

Ivan Moreno and Ching-Cherng Sun  »View Author Affiliations


Optics Express, Vol. 16, Issue 3, pp. 1808-1819 (2008)
http://dx.doi.org/10.1364/OE.16.001808


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Abstract

Light-emitting diodes (LEDs) come in many varieties and with a wide range of radiation patterns. We propose a general, simple but accurate analytic representation for the radiation pattern of the light emitted from an LED. To accurately render both the angular intensity distribution and the irradiance spatial pattern, a simple phenomenological model takes into account the emitting surfaces (chip, chip array, or phosphor surface), and the light redirected by both the reflecting cup and the encapsulating lens. Mathematically, the pattern is described as the sum of a maximum of two or three Gaussian or cosine-power functions. The resulting equation is widely applicable for any kind of LED of practical interest. We accurately model a wide variety of radiation patterns from several world-class manufacturers.

© 2008 Optical Society of America

1. Introduction

Today, LEDs are everywhere, in many shapes and forms, and cover a wide range of applications from indicator lights to solid-state lighting. They are gradually taking over traditional radiation sources because of their attractive characteristics [1–3

1. E. F. Schubert and J. K. Kim, “Solid-state light sources getting smart,” Science 308, 1274–1278 (2005). [CrossRef] [PubMed]

]. The need for realistic lighting designs is growing, but LEDs are far more difficult to model than the conventional sources. This is worsened because there is a wide variety of LED radiation patterns.

A radiation pattern describes the relative light strength in any direction from the light source. Several approaches are used to model this in real light sources. The models currently employed can be classified as analytical approximations or Monte Carlo ray tracing [4–7

4. M. S. Kaminski, K. J. Garcia, M. A. Stevenson, M. Frate, and R. J. Koshel, “Advanced Topics in Source Modeling,” Proc. SPIE 4775, 46 (2002). [CrossRef]

]. A ray tracing model is very useful when analyzing and designing the package and the secondary optics for the source. The analytic models are very useful when studying and optimizing the radiation transfer from the source to a target without intermediate complex optics. This is very common with LEDs because they have integrated optics. Secondary optics can also be analytically designed by non-imaging optics methods [8

8. P. Benitez and J. C. Min͂ano, “The Future of illumination design,” Optics & Photonics News 18, 20–25 (2007). [CrossRef]

]. Additionally, an analytic equation of the radiation pattern gives researchers and lighting designers more flexibility in analyzing the light in their application.

Radiometric and photometric modeling of a light source can be separated into two classes: near field and far field. In near field a source is modeled as an extended area, and it is usually assumed that the distance to the illuminated target is shorter than 5 times the maximum source dimension [9–11

9. M. W. Siegel and R. D. Stock, “General near-zone light source model and its application to computerautomated reflector design,” Opt. Eng. 35, 2661–2679 (1996). [CrossRef]

]. A far-field model assumes that the target is farther from the source than this nominal separation, and models the source as an emitting point. However, this rule of thumb is not so easily applied to LEDs because of the radiance complexity and wave effects [12–14

12. I. Moreno, “Spatial distribution of LED radiation,” Proc. SPIE 6342, 634216 (2006). [CrossRef]

]. Then the concept of the mid field was proposed [14

14. 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, 2193–2195 (2006). [CrossRef] [PubMed]

], which is similar but not equal to the radiometric near field. The rapid variation of the light pattern in the mid field opens a window for verification of an optical model for a light source like an LED. Using the radiometric definition, a high-power LED with a 1mm2 chip should have a 5-mm near field, but in practice up to 20-mm mid fields are found.

In this paper we consider the region after the mid field, i.e. the far field region, where the normalized radiation pattern does not change with the LED-to-target distance. In most cases, the LED’s far field began at a short distance. For example, measurements on high-power LEDs (with a 1mm2 chip) in our laboratory indicate that this distance (from the tip of the LED) is about 20mm for some power LEDs, and a little longer in multi-chip LEDs [15

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

].

The radiation pattern in far field is characterized by the angular intensity distribution. Radiant (or luminous) intensity is the radiant (or luminous) flux per solid angle in a given direction from the source. In this paper, we realistically model the radiant (or luminous) intensity of LEDs, which is useful in many applications (see Sec. 5).

2. Model development

LEDs are available in many different beam patterns which makes it difficult to find a general model for the multiple LED types. Based on our experience in radiometric theory and ray tracing simulations [12

12. I. Moreno, “Spatial distribution of LED radiation,” Proc. SPIE 6342, 634216 (2006). [CrossRef]

,14

14. 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, 2193–2195 (2006). [CrossRef] [PubMed]

,15

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

], we developed a phenomenological model based on light propagation characteristics from its generation inside the chip to its propagation through the packaging lens. Phenomenological models are of utmost importance for real applications in almost all branches of scientific practice, because they relate the real phenomenon to the underlying fundamental theory.

In LEDs, light is produced by spontaneous emission from the light-emitting region of chip. Therefore, the generated light is incoherent and its superposition is a linear combination in intensity terms. From the radiometric point of view, the emitting region is a Lambertian light source. Before exiting the chip, the light generated in this region propagates through different media and types of surfaces. The radiation pattern emitted by every chip face will generally be Lambertian, although in some LEDs, this pattern is modified by special microstructures on the chip and by wave guide effects inside the chip.

The final light pattern generated by an LED is the result of the sum of three terms: the light directly refracted by the encapsulating lens, the light internally reflected inside the lens, and the light reflected by the reflecting cup. These contributions generally are continuous functions that depend on the geometry of both the reflective cup and the encapsulating lens. In other words, the radiation pattern emitted by the chip is modified by the internal reflection inside the encapsulating lens, the reflection in the back mirror, and by the refraction through lens.

In particular, there are three structural factors that strongly modify the radiation pattern of the emitting region [16

16. Application brief AB20-5, “Secondary Optics Design Considerations for SuperFlux LEDs” Philips Lumileds.

]. The first is the roughness of the chip faces which causes every chip face to emit a Lambertian pattern (simple cosine function) or a narrow pattern (narrowed by microstructures on the chip and wave guide effects). Physical models of rough surfaces will often result in a surface reflection and transmission consisting of Gaussian and cosine-power functions [17

17. E. F. Schubert, J. K. Kim, H. Luo, and J. Q. Xi, “Solid-state lighting—a benevolent technology,” Rep. Prog. Phys. 69, 3069–3099 (2006). [CrossRef]

,18

18. C. C. Sun, C. Y. Lin, T. X. Lee, and T. H. Yang, “Enhancement of light extraction of GaN-based light-emitting diodes with a microstructure array,” Opt. Eng. 43, 1700–1701 (2004). [CrossRef]

]. The second factor is the geometry and roughness of the reflective cup. And, the third, geometry of the encapsulant lens, which distorts the radiation pattern produced by the chip and the reflected by the mirror. Also, due to reflections in the internal surface of the lens, another pattern appears, but is weaker than the others. In most cases, the lens angularly narrows the radiation pattern (due to the chip and the mirror), though sometimes it enlarges the pattern.

Taking into account all the above mentioned phenomena, one could assume that the light pattern is a linear combination of certain Lambertian functions and of some other functions generated by diffuse reflections and diffuse refractions (mainly Gaussian and cosine-power angular functions). From the ray-tracing point of view, every ray of light that is diffusely reflected or refracted spreads in a Gaussian or a cosine-power distribution. Therefore, the final radiation pattern should be a linear superposition of this type of functions, which are angularly shifted in function of the angle of incidence of every traced ray. This function overlapping (in a continuous angular range) in turn generates a single Gaussian or cosine-power function, which is:

θ1θ2exp(b1(θα)2)dαAexp(b2θ2) ,

and

θ1θ2[cos(θα)]m1dαBcosm2θ ,

this approximation should surely simplify the resulting radiation pattern.

Thus, taking into account all these criteria, we propose the following general and compact analytical representation of the intensity pattern of LEDs as a sum of Gaussian functions:

I(θ)=ig1iexp(ln2(θg2ig3i)2),
(1)

or as one sum of cosine-power functions:

I(θ)=ic1icos(θc2i)c3i.
(2)

Here intensity I(θ) [Arbitrary Units/sr] is defined as the flux emitted per unit solid angle in a given direction. θ is the polar angle in a coordinate system centered in the LED. We choose a one-dimensional equation because most of the LED radiation patterns are rotationally symmetric in far field. Otherwise, we could not do this if some rotational asymmetries in the radiation pattern appear due to emission non-homogeneities on the chip surface and some package obstructions (e.g. wire bondings), which is more significant in the mid field.

Some LEDs show a quite different radiation pattern in two perpendicular azimuthal directions mainly due to a rotationally asymmetric package. In such a case, an appropriate angular variation should be introduced. A useful example with Gaussian functions is:

I(θ,ϕ)=ig1iexp[(ln2)(θg2i)2(cos2ϕ(g3i)2+sin2ϕ(g4i)2)],
(3)

and with cosine-power functions

I(θ,ϕ)=ic1icos[θ(c2icos2ϕ+c3isin2ϕ)]c4i.
(4)

where ϕ is the azimuthal angle. Equations (1)–(4) are quite accurate for most LED types, and can be used as a generic LED source model. The radiation pattern can also be represented as a combination of the functions of these equations depending on the type of distribution. In some special cases, one term of the equation can be substituted by two twin terms (see Eq. (8)).

These analytical representations are quite compact. For example, an intensity pattern reproduced with the sum of 2 Gaussian (or cosine power) functions requires about 15 sinusoidal functions to be reproduced with a similar accuracy. Equations (1)–(4) have a number of advantages (see Sec. 5), e.g., they can be used for easily determining derivative values, definite integrals, analytic optimization of LED arrays, etc.

If I(θ,ϕ) is a normalized intensity distribution, the absolute intensity can be computed by measuring the LED flux Φ0 [19

19. A. Estrada-Hernandez, L. P. Gonzalez-Galvan, H. Zarate-Hernandez, R. Cardoso, and E. Rosas, “Luminous flux and correlated color temperature determination for LED sources,” SPIE 6422, 64220O (2007). [CrossRef]

,20

20. R. Young, “Measuring light emission from LEDs,” SPIE 6355, 63550H (2006). [CrossRef]

], or with the value provided by the manufacturer. The absolute intensity is then:

I(θ,ϕ)A=[Φ04πI(θ,ϕ)dΩ]I(θ,ϕ).
(5)

I(θ,ϕ)A is the radiant intensity [W/sr] if Φ0 is the radiant flux [Watts], and it is the luminous intensity [lm/sr] when Φ0 is the luminous flux [lumens].

3. Numerical fitting

Fitting equations to data is a very common mathematical process in different disciplines of science and technology [21

21. C. Daniel and F. S. Wood, Fitting equations to data: computer analysis of multifactor data (2nd ed.), (Wiley, New York, 1999).

]. The numerical adjustment has been very useful to obtain realistic models for a great variety of light sources in both near and far field regions [9

9. M. W. Siegel and R. D. Stock, “General near-zone light source model and its application to computerautomated reflector design,” Opt. Eng. 35, 2661–2679 (1996). [CrossRef]

,10

10. R. D. Stock and M. W. Siegel, “Orientation invariant light source parameters,” Opt. Eng. 35, 2651 (1996). [CrossRef]

,22–24

22. L. Fu, R. Leutz, and H. Ries, “Physical modeling of filament light sources,” J. Appl. Phys. 100, 103528 (2006). [CrossRef]

]. Here, we fit a compact analytical representation that optically models LEDs, in particular Eqs. (1)–(4) which are nonlinear in angle.

3.1 Nonlinear regression

A nonlinear regression is an iterative procedure, and the program must start with estimated values for each coefficient. It then adjusts these initial values to improve the fit. Not all mathematical programs are suited to efficiently compute a nonlinear fitting. However, some software packages provide initial values automatically if a built-in equation is used. Fortunately, a linear sum of Gaussians is commonly a built-in equation. In the case of mixed equations with cosine powers and Gaussians, initial values must be chosen carefully. For example, the initial value of g2i or c2i may be that of an intensity peak. It can be useful to estimate initial values after looking at the data graph and understanding the model.

A nonlinear regression program has no common sense. If some problems arise when estimating initial values, it is convenient to set aside the fitting program and continue fitting by trial and error. Depending on the type of radiation pattern, this process can be quickly performed. However, one must think about the optics of the LED, and decide which parameters should be fixed. Then, change the key coefficients one at a time, and see how they influence the shape of the curve and the accuracy of fitting.

3.2 Reconstruction accuracy

To demonstrate the accuracy of reconstruction, the difference between experimental data and modeled equation must be compared by computing both the root mean square (RMS) error and the normalized cross correlation (NCC) [14

14. 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, 2193–2195 (2006). [CrossRef] [PubMed]

,15

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

].

The RMS error between experiment (Ie) and modeled equation (I m) can be calculated on a range of M points over the domain. That is

RMS=1Mij[I(θi,ϕj)m-I(θi,ϕj)e]2.
(6)

The reconstructed pattern must be sufficiently accurate, regardless of the type of LED. The RMS error must be less than the standard limit of 5% [21

21. C. Daniel and F. S. Wood, Fitting equations to data: computer analysis of multifactor data (2nd ed.), (Wiley, New York, 1999).

].

Using NCC, the similarity between the simulated pattern (I m) and the measured pattern (I e) is

NCC=ijI(θi,ϕj)mI¯mI(θi,ϕj)eI¯eij[I(θi,ϕj)mI¯m]2ij[I(θi,ϕj)eI¯e]2,
(7)

Īm and Ī e are the mean values of simulation and experimental data across the angular range, respectively. From our experience in designing LED lighting systems for industry, we know that an LED model with a NCC higher than 99% gives enough accuracy for many applications [14

14. 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, 2193–2195 (2006). [CrossRef] [PubMed]

,15

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

].

3.3 Other details

It is important to know which of Eqs. (1)–(4) should be fitted. This depends on the type of radiation pattern. In particular, cosine-power functions are more suitable than Gaussians when the intensity decreases rapidly with the view angle at the sides of the radiation pattern.

Also, the number of Gaussians or cosine-powers in the equation depends on the type of LED. However, all distributions that we reconstructed (next section) basically required a maximum of 3 terms. This can be linked to the contributions of the three basic optical elements of LEDs: chip, mirror, and lens.

4. LED radiation pattern

4.1 Intensity distribution tables

Tables of coefficients for equations are very common in optics (e.g. constants for glass dispersion formulas). These tables would be very useful for the scientific and technological communities that use LEDs. This is because LEDs are available in a wide variety of beam patterns, and frequently new beam distributions are reported in academics and industry [25

25. S. Preuss, D. Potthoff, T. Preuss, and K. Lischka, “LED encapsulation- a new approach of rear light design,” SPIE 6198, 61980I (2006). [CrossRef]

,26

26. S. C. Allen and A. J. Steckl, “ELiXIR—Solid-State Luminaire With Enhanced Light Extraction by Internal Reflection,” J. Display Technol. 3, 155–159 (2007). [CrossRef]

].

Figures 1–3 show some modeled radiation patterns of some of the major manufacturers like Nichia, Philips Lumileds, Cree, Edison, Osram, and Seoul Semiconductor. Insets (a), (c), and (e) in Figs. 1–3 are the modeled three-dimensional radiation patterns in polar coordinates (θ,ϕ). A comparison between the modeled angular distribution and the one reported by the manufacturer is shown in the insets (b), (d), and (f) of Figs. 1–3.

Figures 2(a) and 2(b) show the angular distribution of the LUXEON® K2 (green, cyan, blue and royal-blue) from Lumileds Philips. Again, the only difference between UB and LB is in the coefficients of the relative intensity g11, g12.y g13. Figures 2(c) and 2(d) show the radiation pattern of a EdiPower™ Emitter 3-RD-01-H0001 LED from Edison. This pattern fits well as a perfect Lambertian emitter, and a first order analysis can be performed by using a simple cosine function, but with a 5.1% RMS error. However, using Eq. (1) with coefficients in Fig. 2(d) improves the analysis accuracy. Figures 2(e) and 2(f) show the distribution of the FIREFLY® Hyper-Bright LB-V19G LED from OSRAM. Its pattern is rotationally asymmetric, thus we used Eq. (4). Only one cosine-power function is required; this fits the pattern better than a Gaussian function because the intensity pattern decreases faster at the sides. The vertical (V) direction is for Φ=0° or 180°, and the horizontal (H) direction is for Φ=90° or 270°.

Figures 3(a) and 3(b) show the angular distribution of the LUXEON® Side Emitter from Lumileds Philips (white, green, cyan, blue and royal-blue). We did not simulate the small central peaks because this requires additional Gaussian functions in Eq. (1), and their contribution to NCC and RMS error is insignificant. Figures 3(c) and 3(d) show the radiation pattern of a LUXEON® Batwing from Lumileds Philips (green, cyan, blue and royal-blue). Instead of Eq. (1) we used

I(θ)=g11[exp(ln2(θ+g21g31)2)+exp(ln2(θg21g31)2)]+g12exp(ln2(θg22g32)2)
=h11exp[ln2(θh31)2]cosh(h21θ)+g12exp(ln2(θg22g32)2),
(8)

to fit this pattern because it only requires two terms, avoiding the need of additional functions when using Eqs. (1) or (2). For this pattern, the difference between UB and LB is in almost all the coefficients of Eq. (8). This is in contrast with the recent K2 and Rebel models (Figs. 1(f) and 2(b)), which lead us to believe that the manufacturing quality of Lumileds Philips has been improving over time. Figures 3(e) and 3(f) show the distribution of the Z-Power Side Emitter LED from Seoul Semiconductor. To fit this pattern, we had to displace the entire pattern 2.55° to the right on the radiation angle axis, and then perform the fit. Finally, to plot Fig. 3(f), the resulting Eq. was shifted by 2.55° to the left. This angular deviation of the pattern is probably due to a misalignment of the optical axis of the LEDs tested by the manufacturer.

The advantage of rendering the radiation pattern of a technical data sheet is that the pattern comes from several source samples. Therefore, this angular distribution is representative of that particular LED type. As seen in Figs. 1–3, the rendering accuracy of our model (with respect to manufacturers) is very high; the accuracy threshold, in RMS error and NCC (Sec. 3.2), was surpassed in all cases.

4.2 Irradiance distribution

Irradiance [W/m2] in far field can be expressed in function of intensity [W/sr]; over a flat surface it is simply:

E(r,θ,ϕ)=I(θ,ϕ)cosθr2,
(9)

where r is the distance from the source to a target point. Typically, Cartesian coordinates (x,y,z) are used to analyze irradiance over a plane. In such a case, the irradiance of an LED, with an intensity distribution given by Eqs. (1)–(5) or (8), is

E(x,y,z)=z(x2+y2+z2)32I[θ(x,y,z),ϕ(x,y)],
(10)

where

θ(x,y,z)=arctan[x2+y2z],
(11)
ϕ(x,y)=arctan(yx).
(12)

If I is the absolute intensity, Eq. (5), then E is the absolute irradiance [W/m2]. If I is the relative intensity (i.e., Eqs. (1)–(4) and (8)), then E is the relative irradiance [(Arbitrary Units)/m2]. Equation (10) can be practically used to analyze the irradiance produced by LED arrays, because the relative position of the j-th LED can be taken into account by simply changing coordinates (x,y) by (x-xj,y-yj) [27

27. I. Moreno and U. Contreras, “Color distribution of multicolor LED arrays,” Opt. Express 15, 3607–3618 (2007). [CrossRef] [PubMed]

,28

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

].

For example, using the UB radiation pattern of a LUXEON® K2 (green, cyan, blue and royal-blue) from Lumileds Philips, ∫I(θ)dΩ=4.78 can be obtained to calculate the absolute irradiance. And with a typical radiant flux Φ0=575 mW at 1000mA (royal-blue), using Eqs. (5) and (10), the absolute irradiance is

E(x,y,z)A=120z(x2+y2+z2)32i=12g1iexp{ln2[arctan(z1x2+y2)g2ig3i]2},
(13)

where E A is in [mW/m2] if the Cartesian coordinates (x,y,z) are in meters. The coefficients g are those given in Fig. 2(b).

Fig. 1. Three radiation patterns of some major manufacturers. (a) and (b) NSPW345CS from Nichia. (c) and (d) XLamp® XR-E LED from Cree. (e) and (f) LUXEON® Rebel from Lumileds Philips. (a), (c), and (e) are the modeled three-dimensional radiation patterns. (b), (d), and (f) are a comparison between modeled and manufacturer reported radiation pattern. UB-Upper bound, LB-Lower bound, and XX and YY are two perpendicular azimuthal directions.
Fig. 2. Other radiation patterns of some major manufacturers. (a) and (b) LUXEON® K2 from Lumileds Philips. (c) and (d) EdiPower Emitter 3-RD-01-H0001 LED from Edison. (e) and (f) FIREFLY® Hyper-Bright LB-V19G LED from OSRAM. (a), (c), and (e) are the modeled three-dimensional radiation patterns. (b), (d), and (f) are a comparison between modeled and manufacturer reported radiation pattern. UB-Upper bound, LB-Lower bound, and V and H are the vertical and horizontal directions.
Fig. 3. Other radiation patterns of some major manufacturers. (a) and (b) LUXEON® Side Emitter from Lumileds Philips. (c) and (d) LUXEON® Batwing from Lumileds Philips. (e) and (f) Z-Power Side Emitter LED from Seoul Semiconductor. (a), (c), and (e) are the modeled three-dimensional radiation patterns. (b), (d), and (f) are a comparison between modeled and manufacturer reported radiation pattern. UB-Upper bound, LB-Lower bound.

5. Applications

Analytical far-field approximations (single Gaussian or single cosine power) of the LED radiation pattern have been used in a wide variety of applications [13

13. P. Manninen, J. Hovila, P. Kärhä, and E. Ikonen, “Method for analysing luminous intensity of light-emitting diodes,” Meas. Sci. Technol. 18, 223–229 (2007). [CrossRef]

,28–51

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

]. For example, to determine and optimize the radiation flux reaching detectors, collectors or irradiated targets from arrays of LEDs [29–33

29. S. Tryka, “Radiative flux from a planar multiple point source within a cylindrical enclosure reaching a coaxial circular plane,” Opt. Express 15, 3777–3790 (2007) [CrossRef] [PubMed]

]. They are also used in the design of wireless communication systems [33–36

33. T. Matsumoto, N. Inoue, and M. Suzuki, “Optimum arrangement of LEDs in base station of optical wireless LANs,” SPIE 6134, 613403 (2006). [CrossRef]

], and for using LED traffic lights as transmitters [37

37. M. Akanegawa, Y. Tanaka, and M. Nakagawa, “Basic study on traffic information system using LED traffic lights,” IEEE Transactions on Intelligent Transportation Systems 2, 197–203 (2001). [CrossRef]

]. Analytical functions have been employed in optimizing illumination and color uniformity from LED assemblies in the far and near (with respect to the array size) field [28

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

,38–40

38. I. Moreno, J. Muñoz, and R. Ivanov, “Uniform illumination of distant targets using a spherical light-emitting diode array,” Opt. Eng. 46, 033001 (2007). [CrossRef]

]. Also, analytic expressions have been used together with ray trace techniques to design a color mixing device for a multicolor LED assembly [41

41. C. Deller, G. Smith, and J. Franklin, “Colour mixing LEDs with short microsphere doped acrylic rods,” Opt. Express 12, 3327–3333 (2004). [CrossRef] [PubMed]

,42

42. C. A. Deller, J. B. Franklin, and G. B. Smith, “Lighting simulations using smoothed LED profiles compared with measured profiles,” SPIE 6337, 63370X (2006). [CrossRef]

]. Another application is analyzing the LED performance as a light source for a microscope [43–45

43. M. A. Volkova, S. V. Zlatina, S. N. Natarovskii, O. N. Nemkova, T. F. Selezneva, N. B. Skobeleva, D. N. Frolov, L. M. Kogan, and B. P. Papchenko, “Prospects of using LEDs in the illuminating systems of microscopes,” J. Opt. Technol. 72, 186–190 (2005). [CrossRef]

]. These simple equations are also used in some stages of the optical design of a LED-based backlight for LCD TVs [46–48

46. C. C. Sun, I. Moreno, S. H. Chung, W. T. Chien, C. T. Hsieh, and T. H. Yang, “Brightness management in a direct LED backlight for LCD TVs,” J. Soc. Inf. Disp. In press (2007).

]. And of course, such equations are used in the design of general illumination systems with white LEDs [49

49. Y. Uchida and T. Taguchi, “Lighting theory and luminous characteristics of white light-emitting diodes,” Opt. Eng. 44, 124003 (2005). [CrossRef]

,50

50. L. Svilainis and V. Dumbrava, “LED Far Field Pattern Approximation Performance Study,” Information Technology Interfaces, 2007. ITI 2007. 29th International Conference on, 645–649 (2007).

].

A single Gaussian or single cosine power is a particular case of our model, as Lambertian (single cosine with c31=1 and c21=0), as imperfect Lambertian (single cosine power with c31>1 and c21=0) [50

50. L. Svilainis and V. Dumbrava, “LED Far Field Pattern Approximation Performance Study,” Information Technology Interfaces, 2007. ITI 2007. 29th International Conference on, 645–649 (2007).

,51

51. M. Bennahmias, E. Arik, K. Yu, D. Voloshenko, K. Chua, R. Pradhan, T. Forrester, and T. Jannson, “Modeling of non-Lambertian sources in lighting applications,” SPIE 6669, 66691A (2007). [CrossRef]

], or as perfect Gaussian. The imperfect Lambertian and the perfect Gaussian approaches are very similar, and become practically the same pattern for θ1/2<30°.

Depending on the accuracy required in simulation, for some applications the uniqueness of individual LED patterns needs to be taken into account [7

7. W. J. Cassarly, “LED modelling: pros and cons of common methods,” Photon. Tech Briefs IIa-2a (April 2002), special supplement to NASA Tech Briefs.

,41

41. C. Deller, G. Smith, and J. Franklin, “Colour mixing LEDs with short microsphere doped acrylic rods,” Opt. Express 12, 3327–3333 (2004). [CrossRef] [PubMed]

,42

42. C. A. Deller, J. B. Franklin, and G. B. Smith, “Lighting simulations using smoothed LED profiles compared with measured profiles,” SPIE 6337, 63370X (2006). [CrossRef]

,46

46. C. C. Sun, I. Moreno, S. H. Chung, W. T. Chien, C. T. Hsieh, and T. H. Yang, “Brightness management in a direct LED backlight for LCD TVs,” J. Soc. Inf. Disp. In press (2007).

]. In particular, this is important when the optical system combines the radiation pattern from an array of LEDs. Recently, with our realistic approach, we developed an all analytical method for color pattern determination from multicolor LED arrays [27

27. I. Moreno and U. Contreras, “Color distribution of multicolor LED arrays,” Opt. Express 15, 3607–3618 (2007). [CrossRef] [PubMed]

,52

52. A. L. Fischer, “LEDs and displays: Analytical method for computing color patterns in LEDs,” Photonics Spectra 41, 87–88 (June 2007).

]. Additionally, the intensity pattern of several sample LEDs could be measured to get a set of model coefficients, and then to simulate the optical system by randomly using the model values in the LED array.

6. Conclusion

A simple analytic representation for the radiation pattern of the light emitted from an LED was proposed. The radiation distribution was determined by adding a Gaussian or a power cosine expression for contributions from the emitting surfaces, and the light redirected by both the reflecting cup and the encapsulating lens. To render the intensity distributions, the sum of a maximum of 2 or 3 terms was necessary, which is closely linked to the contributions of the three main parts of LEDs (chip, mirror, and lens). The resulting equation is widely applicable for any kind of LED of practical interest, including Lambertian-type, batwing, and side emitting LEDs. We tested this model against LED datasheets of several major manufacturers, and found that it gives a highly accurate description of the radiation pattern. Moreover, the very simple mathematical representation of the model makes it easy to understand, apply, and perhaps improve it. For example, this model may be useful to introduce random manufacturing variations in the radiation pattern of each LED to realistically design lighting systems consisting of multiple LEDs. Additionally, because the LED emission spectra can be a realistically represented by Gaussian functions, the spectral and spatial distributions may be easily incorporated in a compact analytic representation [53

53. K. Man and I. Ashdown, “Accurate colorimetric feedback for RGB LED clusters,” SPIE 6337, 633702 (2006). [CrossRef]

].

Acknowledgments

References and links

1.

E. F. Schubert and J. K. Kim, “Solid-state light sources getting smart,” Science 308, 1274–1278 (2005). [CrossRef] [PubMed]

2.

Y. Narukawa, “White-light LEDs,” Opt. Photon. News 15, 24–29 (2004).

3.

A. Zukauskas, M. S. Schur, and R. Gaska, Introduction to Solid State Lighting (Wiley-Interscience, NY, 2002).

4.

M. S. Kaminski, K. J. Garcia, M. A. Stevenson, M. Frate, and R. J. Koshel, “Advanced Topics in Source Modeling,” Proc. SPIE 4775, 46 (2002). [CrossRef]

5.

H. Zerfhau-Dreihöfer, U. Haack, T. Weber, and D. Wendt, “Light source modeling for automotive lighting devices,” Proc. SPIE 4775, 58 (2002). [CrossRef]

6.

M. Jongewaard, “Guide to selecting the appropriate type of light source model,” Proc. SPIE 4775, 86–98 (2002). [CrossRef]

7.

W. J. Cassarly, “LED modelling: pros and cons of common methods,” Photon. Tech Briefs IIa-2a (April 2002), special supplement to NASA Tech Briefs.

8.

P. Benitez and J. C. Min͂ano, “The Future of illumination design,” Optics & Photonics News 18, 20–25 (2007). [CrossRef]

9.

M. W. Siegel and R. D. Stock, “General near-zone light source model and its application to computerautomated reflector design,” Opt. Eng. 35, 2661–2679 (1996). [CrossRef]

10.

R. D. Stock and M. W. Siegel, “Orientation invariant light source parameters,” Opt. Eng. 35, 2651 (1996). [CrossRef]

11.

I. Ashdown, “Near-field photometry: a new approach,” J. Illum. Eng. Soc. 22, 163–180 (1993).

12.

I. Moreno, “Spatial distribution of LED radiation,” Proc. SPIE 6342, 634216 (2006). [CrossRef]

13.

P. Manninen, J. Hovila, P. Kärhä, and E. Ikonen, “Method for analysing luminous intensity of light-emitting diodes,” Meas. Sci. Technol. 18, 223–229 (2007). [CrossRef]

14.

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, 2193–2195 (2006). [CrossRef] [PubMed]

15.

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

16.

Application brief AB20-5, “Secondary Optics Design Considerations for SuperFlux LEDs” Philips Lumileds.

17.

E. F. Schubert, J. K. Kim, H. Luo, and J. Q. Xi, “Solid-state lighting—a benevolent technology,” Rep. Prog. Phys. 69, 3069–3099 (2006). [CrossRef]

18.

C. C. Sun, C. Y. Lin, T. X. Lee, and T. H. Yang, “Enhancement of light extraction of GaN-based light-emitting diodes with a microstructure array,” Opt. Eng. 43, 1700–1701 (2004). [CrossRef]

19.

A. Estrada-Hernandez, L. P. Gonzalez-Galvan, H. Zarate-Hernandez, R. Cardoso, and E. Rosas, “Luminous flux and correlated color temperature determination for LED sources,” SPIE 6422, 64220O (2007). [CrossRef]

20.

R. Young, “Measuring light emission from LEDs,” SPIE 6355, 63550H (2006). [CrossRef]

21.

C. Daniel and F. S. Wood, Fitting equations to data: computer analysis of multifactor data (2nd ed.), (Wiley, New York, 1999).

22.

L. Fu, R. Leutz, and H. Ries, “Physical modeling of filament light sources,” J. Appl. Phys. 100, 103528 (2006). [CrossRef]

23.

J. Arasa, S. Royo, C. Pizarro, and J. Martinez, “Flux spatial emission obtained from technical specifications for a general filament light source,” Appl. Opt. 38, 7009–7017 (1999). [CrossRef]

24.

D. R. Jenkins and H. Monch, “Source Imaging Goniometer Method of Light Source Characterization for Accurate Projection System Design,” SID Symposium Digest 31, 862–865 (2000). [CrossRef]

25.

S. Preuss, D. Potthoff, T. Preuss, and K. Lischka, “LED encapsulation- a new approach of rear light design,” SPIE 6198, 61980I (2006). [CrossRef]

26.

S. C. Allen and A. J. Steckl, “ELiXIR—Solid-State Luminaire With Enhanced Light Extraction by Internal Reflection,” J. Display Technol. 3, 155–159 (2007). [CrossRef]

27.

I. Moreno and U. Contreras, “Color distribution of multicolor LED arrays,” Opt. Express 15, 3607–3618 (2007). [CrossRef] [PubMed]

28.

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

29.

S. Tryka, “Radiative flux from a planar multiple point source within a cylindrical enclosure reaching a coaxial circular plane,” Opt. Express 15, 3777–3790 (2007) [CrossRef] [PubMed]

30.

S. Tryka, “Spherical object in radiation field from a point source,” Opt. Express 12, 512–517 (2004). [CrossRef] [PubMed]

31.

J. L. Balenzategui and A. Marti, “Design of hemispherical cavities for LED-based illumination devices,” Appl. Phys. B 82, 75–80 (2006). [CrossRef]

32.

A. L. Dubovikov, S. S. Repin, and S. N. Natarovskii, “Features of the use of LEDs in artificial-vision systems,” J. Opt. Technol. 72, 40–42 (2005). [CrossRef]

33.

T. Matsumoto, N. Inoue, and M. Suzuki, “Optimum arrangement of LEDs in base station of optical wireless LANs,” SPIE 6134, 613403 (2006). [CrossRef]

34.

C. G. Lee, C. S. Park, J. H. Kim, and D. H. Kim, “Experimental verification of optical wireless communication link using high-brightness illumination light-emitting diodes,” Opt. Eng. 46, 125005 (2007). [CrossRef]

35.

J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE 85, 265–298 (1997). [CrossRef]

36.

D. W. K. Wong and G. Chen, “Illumination design of a white-light-emitting diode wireless transmission system,” Opt. Eng. 46, 085002 (2007). [CrossRef]

37.

M. Akanegawa, Y. Tanaka, and M. Nakagawa, “Basic study on traffic information system using LED traffic lights,” IEEE Transactions on Intelligent Transportation Systems 2, 197–203 (2001). [CrossRef]

38.

I. Moreno, J. Muñoz, and R. Ivanov, “Uniform illumination of distant targets using a spherical light-emitting diode array,” Opt. Eng. 46, 033001 (2007). [CrossRef]

39.

I. Moreno, “Color tunable hybrid lamp: LED-incandescent and LED-fluorescent,” SPIE 6422, 64220N (2007). [CrossRef]

40.

Y. Tu, S. Jin, Y. Wang, and L. Dou, “Color uniformity and data simulation in High-Power RGB LED modules using different LED-chips arrays,” SPIE 6828, 682816 (2007). [CrossRef]

41.

C. Deller, G. Smith, and J. Franklin, “Colour mixing LEDs with short microsphere doped acrylic rods,” Opt. Express 12, 3327–3333 (2004). [CrossRef] [PubMed]

42.

C. A. Deller, J. B. Franklin, and G. B. Smith, “Lighting simulations using smoothed LED profiles compared with measured profiles,” SPIE 6337, 63370X (2006). [CrossRef]

43.

M. A. Volkova, S. V. Zlatina, S. N. Natarovskii, O. N. Nemkova, T. F. Selezneva, N. B. Skobeleva, D. N. Frolov, L. M. Kogan, and B. P. Papchenko, “Prospects of using LEDs in the illuminating systems of microscopes,” J. Opt. Technol. 72, 186–190 (2005). [CrossRef]

44.

C. Ye, Y. Liu, and F. Yu, “New Illumination Patterns in Microscopes” SPIE 6033, 60330L (2005). [CrossRef]

45.

J. Xu, Z. Xiao, and T. Lin, “The Design of Microscope Field Illumination System Based on LED” SPIE 6841, 68410U (2007). [CrossRef]

46.

C. C. Sun, I. Moreno, S. H. Chung, W. T. Chien, C. T. Hsieh, and T. H. Yang, “Brightness management in a direct LED backlight for LCD TVs,” J. Soc. Inf. Disp. In press (2007).

47.

G. Harbers, S. J. Bierhuizen, and M. R. Krames, “Performance of high power light emitting diodes in display illumination applications” J. Display Technol. 3, 98–109 (2007). [CrossRef]

48.

J. W. Whang and Y. T. Teng, “Uniform illumination system with desired emitting angle,” SID Conf. Rec. Int. Display Res. Conf., 100–103 (2006).

49.

Y. Uchida and T. Taguchi, “Lighting theory and luminous characteristics of white light-emitting diodes,” Opt. Eng. 44, 124003 (2005). [CrossRef]

50.

L. Svilainis and V. Dumbrava, “LED Far Field Pattern Approximation Performance Study,” Information Technology Interfaces, 2007. ITI 2007. 29th International Conference on, 645–649 (2007).

51.

M. Bennahmias, E. Arik, K. Yu, D. Voloshenko, K. Chua, R. Pradhan, T. Forrester, and T. Jannson, “Modeling of non-Lambertian sources in lighting applications,” SPIE 6669, 66691A (2007). [CrossRef]

52.

A. L. Fischer, “LEDs and displays: Analytical method for computing color patterns in LEDs,” Photonics Spectra 41, 87–88 (June 2007).

53.

K. Man and I. Ashdown, “Accurate colorimetric feedback for RGB LED clusters,” SPIE 6337, 633702 (2006). [CrossRef]

OCIS Codes
(120.5630) Instrumentation, measurement, and metrology : Radiometry
(150.2950) Machine vision : Illumination
(220.4830) Optical design and fabrication : Systems design
(230.3670) Optical devices : Light-emitting diodes
(350.4600) Other areas of optics : Optical engineering

ToC Category:
Optical Devices

History
Original Manuscript: December 3, 2007
Revised Manuscript: January 20, 2008
Manuscript Accepted: January 21, 2008
Published: January 25, 2008

Citation
Ivan Moreno and Ching-Cherng Sun, "Modeling the radiation pattern of LEDs," Opt. Express 16, 1808-1819 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-3-1808


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References

  1. E. F. Schubert, J. K. Kim, "Solid-state light sources getting smart," Science 308, 1274-1278 (2005). [CrossRef] [PubMed]
  2. Y. Narukawa, "White-light LEDs," Opt. Photon. News 15, 24-29 (2004).
  3. A. Zukauskas, M. S. Schur, R. Gaska, Introduction to Solid State Lighting (Wiley-Interscience, NY, 2002).
  4. M. S. Kaminski, K. J. Garcia, M. A. Stevenson, M. Frate, and R. J. Koshel, "Advanced Topics in Source Modeling," Proc. SPIE 4775, 46 (2002). [CrossRef]
  5. H. Zerfhau-Dreihöfer, U. Haack, T. Weber, and D. Wendt, "Light source modeling for automotive lighting devices," Proc. SPIE 4775, 58 (2002). [CrossRef]
  6. M. Jongewaard, "Guide to selecting the appropriate type of light source model," Proc. SPIE 4775, 86-98 (2002). [CrossRef]
  7. W. J. Cassarly, "LED modelling: pros and cons of common methods," Photon. Tech Briefs IIa-2a (April 2002), special supplement to NASA Tech Briefs.
  8. P. Benitez and J. C. Miñano, "The future of illumination design," Opt. Photonics News 18, 20-25 (2007). [CrossRef]
  9. M. W. Siegel and R. D. Stock, "General near-zone light source model and its application to computer-automated reflector design," Opt. Eng. 35, 2661-2679 (1996). [CrossRef]
  10. R. D. Stock and M. W. Siegel, "Orientation invariant light source parameters," Opt. Eng. 35, 2651 (1996). [CrossRef]
  11. I. Ashdown, "Near-field photometry: a new approach," J. Illum. Eng. Soc. 22, 163-180 (1993).
  12. I. Moreno, "Spatial distribution of LED radiation," Proc. SPIE 6342, 634216 (2006). [CrossRef]
  13. P. Manninen, J. Hovila, P. Kärhä, and E. Ikonen, "Method for analysing luminous intensity of light-emitting diodes," Meas. Sci. Technol. 18, 223-229 (2007). [CrossRef]
  14. 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, 2193-2195 (2006). [CrossRef] [PubMed]
  15. W. T. Chien, C. C. Sun, and I. Moreno, "Precise optical model of multi-chip white LEDs," Opt. Express 15, 7572-7577 (2007). [CrossRef] [PubMed]
  16. Application brief AB20-5, "Secondary Optics Design Considerations for SuperFlux LEDs" Philips Lumileds.
  17. E. F. Schubert, J. K. Kim, H. Luo, and J. Q. Xi, "Solid-state lighting—a benevolent technology," Rep. Prog. Phys. 69, 3069-3099 (2006). [CrossRef]
  18. C. C. Sun, C. Y. Lin, T. X. Lee, and T. H. Yang, "Enhancement of light extraction of GaN-based light-emitting diodes with a microstructure array," Opt. Eng. 43, 1700-1701 (2004). [CrossRef]
  19. A. Estrada-Hernandez, L. P. Gonzalez-Galvan, H. Zarate-Hernandez, R. Cardoso, and E. Rosas, "Luminous flux and correlated color temperature determination for LED sources," SPIE 6422, 64220O (2007). [CrossRef]
  20. R. Young, "Measuring light emission from LEDs," SPIE 6355, 63550H (2006). [CrossRef]
  21. C. Daniel and F. S. Wood, Fitting equations to data: computer analysis of multifactor data (2nd ed.), (Wiley, New York, 1999).
  22. L. Fu, R. Leutz, and H. Ries, "Physical modeling of filament light sources," J. Appl. Phys. 100, 103528 (2006). [CrossRef]
  23. J. Arasa, S. Royo, C. Pizarro, and J. Martinez, "Flux spatial emission obtained from technical specifications for a general filament light source," Appl. Opt. 38, 7009-7017 (1999). [CrossRef]
  24. D. R. Jenkins and H. Monch, "Source Imaging Goniometer Method of Light Source Characterization for Accurate Projection System Design," SID Symposium Digest 31, 862-865 (2000). [CrossRef]
  25. S. Preuss, D. Potthoff, T. Preuss, and K. Lischka, "LED encapsulation- a new approach of rear light design," SPIE 6198, 61980I (2006). [CrossRef]
  26. S. C. Allen and A. J. Steckl, "ELiXIR—Solid-State Luminaire With Enhanced Light Extraction by Internal Reflection," J. Display Technol. 3, 155-159 (2007). [CrossRef]
  27. I. Moreno and U. Contreras, "Color distribution of multicolor LED arrays," Opt. Express 15, 3607-3618 (2007). [CrossRef] [PubMed]
  28. I. Moreno, M. Avendaño-Alejo, and R. I. Tzonchev, "Designing light-emitting diode arrays for uniform near-field irradiance," Appl. Opt. 45, 2265-2272 (2006). [CrossRef] [PubMed]
  29. S. Tryka, "Radiative flux from a planar multiple point source within a cylindrical enclosure reaching a coaxial circular plane," Opt. Express 15, 3777-3790 (2007) [CrossRef] [PubMed]
  30. S. Tryka, "Spherical object in radiation field from a point source," Opt. Express 12, 512-517 (2004). [CrossRef] [PubMed]
  31. J. L. Balenzategui and A. Marti, "Design of hemispherical cavities for LED-based illumination devices," Appl. Phys. B 82, 75-80 (2006). [CrossRef]
  32. A. L. Dubovikov, S. S. Repin, and S. N. Natarovskii, "Features of the use of LEDs in artificial-vision systems," J. Opt. Technol. 72, 40-42 (2005). [CrossRef]
  33. T. Matsumoto, N. Inoue, and M. Suzuki, "Optimum arrangement of LEDs in base station of optical wireless LANs," SPIE 6134, 613403 (2006). [CrossRef]
  34. C. G. Lee, C. S. Park, J. H. Kim, and D. H. Kim, "Experimental verification of optical wireless communication link using high-brightness illumination light-emitting diodes," Opt. Eng. 46, 125005 (2007). [CrossRef]
  35. J. M. Kahn and J. R. Barry, "Wireless infrared communications," Proc. IEEE 85, 265-298 (1997). [CrossRef]
  36. D. W. K. Wong and G. Chen, "Illumination design of a white-light-emitting diode wireless transmission system," Opt. Eng. 46, 085002 (2007). [CrossRef]
  37. M. Akanegawa, Y. Tanaka, and M. Nakagawa, "Basic study on traffic information system using LED traffic lights," IEEE Transactions on Intelligent Transportation Systems 2, 197-203 (2001). [CrossRef]
  38. I. Moreno, J. Muñoz, and R. Ivanov, "Uniform illumination of distant targets using a spherical light-emitting diode array," Opt. Eng. 46, 033001 (2007). [CrossRef]
  39. I. Moreno, "Color tunable hybrid lamp: LED-incandescent and LED-fluorescent," SPIE 6422, 64220N (2007). [CrossRef]
  40. Y. Tu, S. Jin, Y. Wang, and L. Dou, "Color uniformity and data simulation in High-Power RGB LED modules using different LED-chips arrays," SPIE 6828, 682816 (2007). [CrossRef]
  41. C. Deller, G. Smith, and J. Franklin, "Colour mixing LEDs with short microsphere doped acrylic rods," Opt. Express 12, 3327-3333 (2004). [CrossRef] [PubMed]
  42. C. A. Deller, J. B. Franklin, and G. B. Smith, "Lighting simulations using smoothed LED profiles compared with measured profiles," SPIE 6337, 63370X (2006). [CrossRef]
  43. M. A. Volkova, S. V. Zlatina, S. N. Natarovskii, O. N. Nemkova, T. F. Selezneva, N. B. Skobeleva, D. N. Frolov, L. M. Kogan, and B. P. Papchenko, "Prospects of using LEDs in the illuminating systems of microscopes," J. Opt. Technol. 72, 186-190 (2005). [CrossRef]
  44. C. Ye, Y. Liu, and F. Yu, "New illumination patterns in microscopes" SPIE 6033, 60330L (2005). [CrossRef]
  45. J. Xu, Z. Xiao, and T. Lin, "The design of microscope field illumination system based on LED" SPIE 6841, 68410U (2007). [CrossRef]
  46. C. C. Sun, I. Moreno, S. H. Chung, W. T. Chien, C. T. Hsieh, and T. H. Yang, "Brightness management in a direct LED backlight for LCD TVs," J. Soc. Inf. Disp. In press (2007).
  47. G. Harbers, S. J. Bierhuizen, and M. R. Krames, "Performance of high power light emitting diodes in display illumination applications" J. Display Technol. 3, 98-109 (2007). [CrossRef]
  48. J. W. Whang and Y. T. Teng, "Uniform illumination system with desired emitting angle," SID Conf. Rec. Int. Display Res. Conf., 100-103 (2006).
  49. Y. Uchida and T. Taguchi, "Lighting theory and luminous characteristics of white light-emitting diodes," Opt. Eng. 44, 124003 (2005). [CrossRef]
  50. L. Svilainis and V. Dumbrava, "LED Far Field Pattern Approximation Performance Study," Information Technology Interfaces, 2007. ITI 2007. 29th International Conference on, 645-649 (2007).
  51. M. Bennahmias, E. Arik, K. Yu, D. Voloshenko, K. Chua, R. Pradhan, T. Forrester, and T. Jannson, "Modeling of non-Lambertian sources in lighting applications," SPIE 6669, 66691A (2007). [CrossRef]
  52. A. L. Fischer, "LEDs and displays: Analytical method for computing color patterns in LEDs," Photonics Spectra 41, 87-88 (June 2007).
  53. K. Man and I. Ashdown, "Accurate colorimetric feedback for RGB LED clusters," SPIE 6337, 633702 (2006). [CrossRef]

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