Uniform illumination of distant targets using a spherical light-emitting diode array

Transcription

1 46 3, March 2007 Uniform illumination of distant targets using a spherical light-emitting diode array Ivan Moreno, MEMBER SPIE Universidad Autonoma de Zacatecas Unidad Academica de Fisica Apdo. Postal C Zacatecas, Zacatecas Mexico Jesús Muñoz, MEMBER SPIE Universidad de Guadalajara Centro Universitario de Los Lagos Lagos de Moreno, Jalisco, Mexico Rumen Ivanov Universidad Autonoma de Zacatecas Unidad Academica de Fisica Apdo. Postal C Zacatecas, Zacatecas Mexico Abstract. An array of light-emitting diodes LEDs assembled upon a spherical surface can produce a wider angle distribution of light than a typical array i.e., an array assembled by mounting LEDs into a flat surface. Arranging each single LED into an optimal placement, the uniformity of the illumination of a target can be improved. We derive approximate formulas and equations for the optimum LED-to-LED angular spacing of several spherical arrangements for uniform far-field irradiance. These design conditions are compact and simple tools that incorporate an explicit dependence on the half-intensity viewing angle half width half maximum angle of LEDs Society of Photo-Optical Instrumentation Engineers. DOI: / Subject terms: light-emitting diodes; LED lamps; lighting; uniform illumination. Paper R received May 5, 2006; revised manuscript received Sep. 13, 2006; accepted for publication Sep. 20, 2006; published online Mar. 20, This paper is a revision of a paper presented at the SPIE conference on Fifth Symposium Optics in Industry, Feb. 2006, Santiago de Queretaro, Mexico. This paper presented there appears unrefereed in SPIE Proceedings Vol Introduction Solid-state lighting technology is displacing traditional illumination sources due to unique advantages, mainly in luminous efficacy, compactness, and durability. 1 3 However, several individual light-emitting diodes LEDs must be mounted on a lamp to produce a practical power. 4 The precise power requirements depend on the particular application. LEDs emit light into one hemisphere with some degree of directionality. As a consequence, LED arrays can be designed to easily direct emission into specific lighting patterns without additional optics. Recently, we derived design formulas for typical arrays assemblies that lie upon a flat surface to achieve uniform near-field irradiance; 5 we then successfully applied these tools to design and assemble multicolor LED clusters to produce uniform color distributions. 6 One important practical problem is to produce a uniform illumination on distant targets. 7 9 Flat arrays are practical to achieve uniform near-field irradiance, but for the far-field case, the array size is as large as the uniformly illuminated area, 5 which is impractical for the size of a typical lamp. An array of LEDs assembled upon a spherical surface can produce a wider angle distribution of light than a typical plane array. Recently, a LED technology company introduced a new three-dimensional LED array assembled upon a spherical surface to produce a wide-angle distribution. 10 Several patents have registered circular and spherical arrays, 11,12 and a spherical array was recently designed to maximize the minimum intensity in the service area of optical wireless local area network systems. 13 However, in our /2007/$ SPIE knowledge, design tools concerning such arrays for uniform far-field irradiance have not yet been reported. In this paper, we propose a first order design of light sources consisting of multiple LEDs assembled upon a spherical surface to uniformly illuminate distant targets. We model each single source as an imperfect Lambertian emitter to derive simple approximate expressions for determining the optimal placement of each single LED. 2 General Considerations To create practical design tools, we assume for our analysis that 1. The illuminated object is a planar surface in front of the LED array. 2. Our approach considers LED arrays in the absence of any optical diffuser. 3. Because LEDs are sources that can emit infrared, visible, or ultraviolet radiation, we use the radiometric Fig. 1 Geometry of a LED displaced to position R,, and the illuminated target

2 terminology to create design tools for a wider variety of illumination systems. 4. We approximate the irradiance variation with distance with the inverse square law for a point source because the emitting region of LEDs is typically less than 1 mm on a side. This approximation is true even for the midfield region and for some part of the near zone. 5. The distance to the target is large enough compared to the spherical array radius to warrant the far-field approximation. 6. All LEDs of each array have equal values of radiant flux and equal distributions in space and wavelength. However, because distributions of irradiance and color of assembled LEDs are not exactly equal, even among LEDs of the same type, 14 this should be considered for an exact design. Traditionally, a single LED is optically modeled on realistic numerical models or on measurement-based models Although these time-consuming models can be useful for trial and error designs of light sources consisting of multiple LEDs, a practical model is required to analytically derive quick estimation formulas. We model each single LED as an imperfect Lambertian emitter, which means the radiant intensity distribution is a power law of the viewing-angle cosine function. 5,13,20 Because we assume that the illuminated object is a flat surface, we write the irradiance pattern for a LED displaced to position x 0 =R sin cos, y 0 =R sin sin, z 0 =R cos over the outer surface of a sphere with radius R in terms of Cartesian coordinates x,y,z see Fig. 1. The irradiance E Wm 2 over every point x,y on a flat screen at distance z from the LED array may be expressed as E x,y,z;r,, = A LED L LED x R sin cos sin cos + y R sin sin sin sin + z R cos cos m x R sin cos 2 + y R sin sin 2 + z R cos 2 m+2 /2, 1 where L LED is the radiance Wm 2 sr 1 of the LED chip and A LED is the LED emitting area m 2. The number m is given by the angle 1/2 a value typically provided by the manufacturer, defined as the view angle when radiant intensity is half of the value at 0 deg m = ln2 ln cos 1/2, where m depends on the relative position of the LED emitting region from the curvature center of the spherical encapsulant. 21 If the chip position coincides with the curvature center, the number m is nearly 1, and the source is nearly a perfect Lambertian e.g., some Lumileds and Lamina LEDs. Usual LEDs often have a larger value of m, and thus become more directional emitters e.g., some ichia LEDs. 2 3 Designs for Uniform Far-Field Irradiance We apply the theory of the optimal lamp placement to produce uniform illumination. 22 This theory consists of setting several terms in the two-dimensional Taylor series expansion of the resulting irradiance function to zero. Arranging the LEDs symmetrically, only one series term is sufficient, which is the equivalent to applying the Sparrow s criterion used in image resolution. 5 Using this criteria, the angle between each pair of LEDs can be adjusted so that the combined irradiance distribution is uniform, that is, the individual irradiance patterns are separated optimally to eliminate the minimum between the maxima from each pair of distribution. In other words, differentiating total irradiance twice and setting it to zero at the target center x=0, y=0 eventually yields the maximally flat condition for the angular spacing. 3.1 Array of Two LEDs For this array Fig. 2 a, the irradiance E is given by the sum of the irradiances for two LEDs the angle between LEDs is 2. E x,y,z = E x,y,z;r,,0 + E x,y,z;r,,0. 3 Differentiating E twice and setting 2 E/ x 2 =0 at x=0 and y=0 yields the maximally flat condition for m m 1 m +2 1 R 2 sin 2 + z R cos 2 2 z cos R 2 +2mR R 2 sin 2 + z R cos 2 z cos R 1 sin 2 R 2 sin 2 + z R cos 2 + m +4 R 2 =0, 4 If numerical values for z and R are provided, Eq. 4 may be easily solved for the optimum value of for both nearand far-field cases. For the far field z R, Eq. 4 can be significantly reduced giving the condition for uniform irradiance o = arctan m +2 5 m m 1. This design condition is independent of z the distance from the center of the array to the target center. Equation 5 and the following equations are approximately valid for panel-target distances 20R

3 Fig. 2 Two-LED array. a Diagram of the array with a screen at a distance z. b The uniform irradiance distribution normalized to its maximum value along the x direction at y=0, for m=0, z=250 cm, R=5 cm, and 2 =2 o =15.08 deg. Also, the irradiance distribution of each LED dotted lines is illustrated. Figure 2 b shows the irradiance distribution along the x direction for a selected value of m = 60, z = 250 cm, R =5 cm, and = o. As shown in this figure, the irradiance distribution is constant over the screen region in front of the LED array. 3.2 Ring Arrays A ring array Fig. 3 a has a rotationally symmetric configuration that can be particularly appropriate for LEDs with quasi-lambertian distributions. For this array, the total irradiance E is given by the sum of the contributions of 3 LEDs Fig. 3 Ring array of LEDs. a Schematic diagram of a spherical array with =6. b Two-dimensional irradiance distribution of this array for m=3 a representative value for high-power LEDs, z =300 cm, R=7 cm, and 2 =2 o = deg. c Resulting onedimensional distribution along the x direction at y=0. E x,y,z = n=1 E x,y,z;r,, 2 n, 6 2 m +2 o = arctan m m 1. 7 where 2 is the angle between each pair of LEDs. The symmetry makes the problem one dimensional so that without loss of generality we can calculate the maximally flat condition along any radial axis on screen, such as the x axis at y=0. Differentiating E twice and setting 2 E/ x 2 =0 at x=0 and y=0, yields the maximally flat condition. For the far field, the optimal angle o is This condition is again independent of z and curiously independent of the number of LEDs that assemble the ring. Therefore, the number of LEDs that assemble a ring does not appreciably affect the uniformity of the illumination. A similar conclusion for both point 22 and LED 5 sources, uniformly spaced along a coplanar circle, was previously proven. Figure 3 shows the design of a circular ring array

4 Fig. 4 Uniform irradiance pattern along the x direction at y=0 for a ring array with one LED in the center. The design parameters are =5, m=30, z=300 cm, R=7 cm, 2 =2 o =43.4 deg, and o =0.64. with 6 LEDs and the corresponding uniform irradiance pattern. For a ring array of LEDs with one LED in the center, the maximally flat condition, Eq. 7, does not give a flat irradiance pattern. To obtain a flat irradiance pattern, the relative flux of the middle LED must be adjusted. The optimal angle o then becomes 2 2m +1 o = arctan 8 m 2 m 1. This condition is again independent of the number of LEDs that assemble the ring; however, the optimum relative flux o = center / ring where center and ring are the radiant fluxes of the middle LED and of one LED over the ring, respectively linearly depends on o = m2 + m +4 m 2 cos m o. 9 4 Figure 4 illustrates the uniform irradiance pattern for a ring array with one LED in the center. We included the curve for m=30 of Fig. 3 c in Fig. 4 for comparison purposes. We can appreciate that for the ring with a centered LED, the irradiance distribution is uniform over a larger region than the pattern produced by a simple ring. 3.3 Linear Arrays For this array Fig. 5 a, the total irradiance E is given by the sum of the contributions of LEDs E x,y,z = E x,y,z;r, n,0, n=1 10 where n is the angular coordinate of the nth LED. In the far field, the maximally flat condition obtained by setting 2 E/ x 2 =0 at x=0, y=0 is n=1 m m 1 m 2 +2 cos 2 n cos m 2 n =0, 11 which yields n in functions of m and. Equation 11 gives the design condition for a two-led array when Fig. 5 Linear array of LEDs. a Schematic illustration of an array with =10. b The uniform irradiance distribution of this array, along the x direction at y=0, for m=6, z=250 cm, R=5 cm, and 0 =9.5 deg. =2. We proved several intuitive functions for n = f n, 0, where 0 is the angle between the optical axis and the nearest LED. However, due to the asymmetry between the geometries of array and target, the functions that yielded a flat pattern over the largest target region were n = arctan +1 2n tan 0, even, n = arctan n tan 0, odd. 12 Computation of the maximally flat condition is reduced to obtain 0 from Eqs. 11 and 12, but an analytical solution for this equation proved difficult. However, when numerical values for m and are provided, the solution can be easily obtained with any mathematical program. Figure 5 shows the design of a linear array with 10 LEDs and the corresponding uniform irradiance pattern. 3.4 Multiple-Ring Arrays The irradiance of a LED array with multiple-ring geometry Fig. 6 a is given by the sum of the irradiances for M rings, each ring with i LEDs where i=1...m and i 3 LEDs

5 M m m 1 tan 2 i 2 m +2 i cos m i =0, 14 i=1 which yields i in functions of m, i, and M. We proved several intuitive functions for i = f i, 0 for different combinations of i values 0 is the optimum angle between the optical axis and the first ring, that is, 0 is the optimum value of 1. A function that optimizes the LED distribution of LEDs over the array surface is i = arctan i k tan 0, 15 where k determines the fraction of the emitted power enclosed within the uniformly illuminated area. Therefore, computation of the maximally flat condition is reduced to obtain 0 from Eq. 14 by using Eq. 15. Figure 6 shows the design of a multiple-ring array with 18 LEDs and the corresponding uniform irradiance pattern. We included the curve for m=30 of Fig. 3 c in Fig. 6 c for the purpose of comparison. Figure 6 c indicates that both arrays, single and multiple rings, produce almost the same flat pattern. However, due to the finite size of each single LED, the multiple-ring array can incorporate in the same available space a larger number of LEDs than the single-ring array. This is particularly useful for high-power applications. One way to illuminate a larger area is to adjust the relative flux i = ring i / ring 1 of each ring. The irradiance for such an array is M i E x,y,z = i=1 n=1 i E x,y,z;r, i, 2 n. i 16 The optimum values of the relative radiant fluxes and the angle 0 can be obtained from Eq. 15 and M m m 1 tan 2 i 2 m +2 i i cos m i =0, i=1 17 Fig. 6 Multiple-ring array of LEDs. a Schematic illustration of an array with M=3 three rings, 1,2,3 =6. b The two-dimensional irradiance pattern for m=30, z=300 cm, R=7 cm, 1 = 0 =12.1 deg, 2 =16.9 deg, 3 =20.4 deg, and k=0.5. c Resulting onedimensional irradiance distribution along the x direction at y 0. M i E x,y,z = i=1 n=1 E x,y,z;r, i, 2 n. i 13 Setting 2 E/ x 2 =0 at x=0, y=0 gives the design condition for the far field M 2 2m +1 m 1 m 2 tan 2 i d i i=1 d 1 i i tan i cos m i =0. 18 Figure 7 shows the design of a two-ring array with 12 LEDs, and the corresponding uniform irradiance pattern. We included the curve of a two-ring array with equal fluxes for the purpose of comparison. Figure 7 indicates that by adjusting the relative fluxes, the uniformly illuminated area can be significantly increased, particularly when the values 1 and 2 given by Eqs. 17 and 18 are slightly increased through trial and error. However, for some applications, all LEDs should be operated at equal radiant flux. 3.5 Demonstrative Experiment We performed a simple demonstrative experiment with a two-led spherical array assembled with R=3.3 cm radius is measured from the origin of coordinates to the chipimage position, which in our LEDs coincides with the encapsulant base. These LEDs Steren 5/Ultra White emit white light with m=64.66 measured with respect to the

6 Fig. 7 Uniform irradiance pattern along the x direction at y=0 for a multiple-ring array of LEDs with different relative radiant flux for each ring. The design parameters are M=2 two rings, 1 =4, 2 =8, m=5, z=300 cm, R=7 cm. Using Eqs. 17 and 18, curve a shows the resulting distribution for 1 = 0 =15 deg, 2 =56.3 deg, 2 =2.2, and k=2.5. By using the values 1 and 2 of a as starting values, curve b shows the resulting pattern after a quick process of trial and error for 1 = 0 =18.9 deg, 2 =62.7 deg, 2 =2.71, and k =2.5. Curve c shows the resulting distribution for an array with equal values of flux for all LEDs, for 1 = 0 =34.3 deg, 2 =44 deg, and k= Conclusions LEDs emit light into only one hemisphere with some degree of directionality; therefore, a compact LED array must be assembled upon a convex surface to distribute light over a large area. We have analyzed the optimum LED-to-LED angular spacing of light sources consisting of multiple LEDs, assembled upon a spherical surface, to uniformly illuminate a distant target. Practical equations and formulas were derived for four representative array configurations. These design tools offer an easy way to estimate the performance of LED spherical lamps due to the explicit dependence on 1/2 typically provided by the manufacturer, the configuration geometry, and the number of LEDs that assemble the array. The derived analytical expressions can be a practical tool for both quick estimations first-order designs and as starting points to reduce the computation time for exact designs that must use a realistic LED model Depending on the application, our analysis can be extended to other array configurations to generate uniform irradiance over distant targets. chip-image position. The light transmitted through the screen a translucent diffuse sheet, placed at z=68.4 cm, was imaged with a charge-coupled device camera. Figure 8 shows the recorded and simulated irradiance patterns. For the purpose of comparison, Fig. 8 includes the simulation and measurement for the angles 2 =12, 14, and 16 deg. The experiment agrees with the theoretical predictions, particulary when 2 14 deg, confirming the above analysis. Fig. 8 a Experimental and b theoretical irradiance patterns of a two-led array with optimal angle 2 0 =14.5 deg with Eq. 5 and 13.8 deg with Eq. 4. Acknowledgments This research was supported by COACYT Consejo acional de Ciencia y Tecnologia Grant o. J48199-F. We thank Maureen Sophia Harkins for proofreading this paper. References 1. A. Zukauskas, M. Shur, and R. Caska, Introduction to Solid-State Lighting, Wiley, ew York M. G. Craford, LEDs for solid state lighting and other emerging applications: Status, trends, and challenges, in Fifth International Conference on Solid State Lighting, I. T. Ferguson, J. C. Carrano, T. Taguchi, and I. E. Ashdown, Eds., Proc. SPIE 5941, E. F. Schubert, Light-Emitting Diodes, Cambridge University Press, Cambridge O. Kuckmann, High power LED arrays: Special requirements on packaging technology, in Light-Emitting Diodes: Research, Manufacturing, and Applications X, K. P. Streubel, H. Walter Yao, and E. F. Schubert, Eds., Proc. SPIE 6134, I. Moreno, M. Avendaño-Alejo, and R. I. Tzonchev, Designing light-emitting diode arrays for uniform near-field irradiance, Appl. Opt. 45, I. Moreno and Luis M. Molinar, Color uniformity of the light distribution from several cluster configurations of multicolor LEDs, in Fifth International Conference on Solid State Lighting, I. T. Ferguson, J. C. Carrano, T. Taguchi, and I. E. Ashdown, Eds., Proc. SPIE 5941, P. T. Ong, J. M. Gordon, A. Rabl, and W. Cai, Tailored edge-ray designs for uniform illumination of distant targets, Opt. Eng. 34, J. M. Gordon and A. Rabl, Reflectors for uniform far-field irradiance: Fundamental limits and example of an axisymmetric solution, Appl. Opt. 37, W. J. Cassarly, S. R. David, D. G. Jenkins, A. P. Riser, and T. L. Davenport, Automated design of a uniform distribution using faceted reflectors, Opt. Eng. 39, D packages from Lednium provide wide-angle sources, LEDs Magazine, pp December Z. K. Zhang and Z. Q. Xiang, LED light bulb, U.S. Patent o B Z. Ishibashi, LED bulb, U.S. Patent o B T. Matsumoto,. Inoue, and M. Suzuki, Optimum arrangement of LEDs in base station of optical wireless LAs, in Light-Emitting Diodes: Research, Manufacturing, and Applications X, K. P. Streubel, H. W. Yao, and E. F. Schubert, Eds., Proc. SPIE 6134, J. M. Benavides and R. H. Webb, Optical characterization of ultrabright LEDs, Appl. Opt. 44,

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