Abstract

We propose a numerical optimization method designing LED array for achieving a good uniform illumination distribution on target plane. Simulated annealing algorithm is employed to optimize LED array arrangement. Using the method, we optimized three LED arrays with various luminous intensity profiles. In order to exhibit the design freedom of the method, we use some LEDs with different intensity value in the first and third array, respectively. By optimizing, the three arrays all produced highly uniform illumination distribution with the uniformity of 0.12, 0.23 and 0.13, respectively. It indicates our method can design various luminous intensity distribution LED arrays and design array consisting of LEDs with different intensity value. In addition, the method is simple and can optimize the LED array automatically by computer program. To the best of our knowledge, it is first time to use numerical optimization method to design the optimal LED array arrangement for uniform irradiance.

©2012 Optical Society of America

1. Introduction

Light-emitting diodes (LEDs) are potential light sources for many applications such as traffic and automotive lighting, road lighting, tunnel lighting and backlight, etc. because of environmental benefits, long lifetime, high reliability, and low power consumption [13]. However, owing to the Lambertian radiation distribution, LEDs cannot meet the requirements of illumination in many cases. Secondary optics are often used to redistribute light flux from LEDs. In most cases, secondary optical component is designed for single LED [14]. However, due to limited power output from single LED, it is necessary to use the LED array as light source in most cases. For the applications, it is very important to achieve a good uniform distribution for LED array sources. I. Moreno reported the maximization of illumination uniformity using LED arrays in direct lighting for the first time [5]. In I. Moreno’s report, analytical method was used to obtain the optimum LED-to-LED spacing in different array configurations [5]. In Ref [6], I. Moreno designed a spherical LED array which can distribute light over a large area with uniform illumination. Hongming Yang demonstrated different uniform illumination pattern of LED array across a range of distances [7]. Whang presented a method designing LED array with arbitrary view angle for uniform illumination distribution [8]. Zong Qin studied the uniform illumination condition for LED array with large view angle [9]. In the previous work, the analytic method was used to design LED array. Due to great complexity of the analytic method, the designed arrays are regular arrays such as circular array, rectangular array, etc. In the regular array, all LED are assumed same. In this paper, a numerical optimization method is developed to optimize the LED array arrangement for uniform illumination distribution on a target plane. Firstly, an object function is constructed to effectively reflect the uniformity of illumination distribution. In order to obtain the highly uniform illumination distribution on target plane, the object function was minimized. In this paper, the simulated annealing (SA) algorithm was used to optimize the coordinates of each LED in the array so that the object function can reach the minimum value.

SA is a global optimization algorithm [1012], which is often used to seek a global minimum of multi-variable objective function. The SA, just its name, originates from annealing techniques in metallurgy, which include heating and controlled cooling of a solid so as to reduce its defects. In a process of annealing, a melt is heated to a high temperature and become disordered. Then the temperature of metal is slowly decreased. Thus the process at any time can be approximately regarded as thermodynamic equilibrium process. As cooling proceeds, the structure of metal become more ordered and eventually is frozen, this happen at the lowest energy state. If the cooling is done sufficiently slowly, the system can form a configuration almost without defects. In the 1980s, Kirkpatrick [11] and Cerny [12] found that there are many similarities between some optimization problems and the physical process of annealing. (1) The current solution to the optimization problem is associated with the current energy state of the thermodynamic system. (2) The objective function of an optimization problem corresponds with the energy equation for the thermodynamic system. (3)The global minimum is analogous to the ground state.

By using the method, three LED arrays were optimized. In the first array, each LED has a perfect Lambertian luminous intensity distribution. The second array is made up of 7 imperfect Lambertian intensity distribution LEDs. In the third array, each LED exhibits a special luminous intensity distribution. In addition, we use some LEDs with different intensity value in the first and third array, respectively. The three optimized arrays all produced high uniform illumination distribution. Compared with traditional analytical method, the numerical optimization method has much more design freedom and can be implemented automatically by computer program.

2. The theory of high-uniform LED array design

The LED can be treated as a Lambertian source approximately with the luminous intensity profile described by Eq. (1) [79,1315].

I(θ)=I0cosmθ
Where θ is the view angle and I0 is luminous intensity at the normal direction to the source surface. The number m depends on the angular half width θ1/2 (a value typically provided by the manufacturer, defined as the view angle when the irradiance decreases to the half of the value at the normal direction), and this is given by Eq. (2) [59,13].
m=ln2ln(cosθ1/2)
In Fig. 1(a) , the LEDs are shown to mount on the S–plane (z = 0). The target plane (T-plane, see Fig. 1(b)) is at a distance z from the S-plane. For a point A on the target plane T with coordinates (xp,yq,z) and a LED at the coordinates (X,Y,0) on the S-plane, the irradiance at A generated by the LED can be expressed as Eq. (3) [5,8,9].
E(xp,yq,z)=zm+1I0[(xpX)2+(yqY)2+z2]m+32
Therefore, the irradiance at the point A with an array of n LEDs can be expressed as:
E(xp,yq,z)=i=1nzm+1I0[(xpXi)2+(yqYi)2+z2]m+32
Here (Xi,Yi,0) are the coordinates of the i-th LED in the array.

 figure: Fig. 1

Fig. 1 (a) Diagram of LED illumination. (b) Target plane with M × N grids.

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As shown Fig. 1(b), we divide the target plane into M×N grids. The irradiance of each grid from LED array can be calculated by Eq. (4). Various approaches are used to reflect the illumination uniformity of LED array [11,12].In this paper, to evaluate the illumination uniformity on the target plane, the CV(RMSE), as the abbreviation of coefficient of variation of root mean square error [9], is used as an objective function given by

f(X1,Y1,...,Xi,Yi,...,Xn,Yn)=σE¯
Here E¯ denotes the average irradiance of all grids on the target plane.
E¯=1M×Nq=1Np=1ME(xp,yq,z)
Where σis the standard error of irradiance of all grids, which can be expressed by
σ=p=1Nq=1M(E(xp,yq,z)E¯)2M×N
The independent variables of the objective function are the coordinates of all LEDs in the array. To achieve highly uniform illumination distribution, we minimize the object function.

3. Simulated annealing algorithm design

In this section, we will elaborate on how to optimize LED array by SA algorithm. The implementation of SA algorithm is expatiated as following [1012,16,17]:

  • 1. Initialization
    • (1.1) Construct an objective function as shown in Eq. (5), which reflects the illumination uniformity of LED array.
    • (1.2) A vector sis set as a initial solution. The vector consists of 2n elements, which are the coordinates of n LED.
    • (1.3) Select an initial temperature T = 2000.
    • (1.4) Define a temperature reduction function asTh+1=(0.95)hTh Where h is an iteration count that indicates the times of temperature decrement.
    • (1.5) Set iteration count IT and the maximum iteration count L at each fixed temperature.
  • 2. Repeat
    • (2.1) Set h = 0.
    • (2.2) Set IT = 0.
      • (2.2.1) IT = IT + 1.
    • (2.3) Randomly generate a new solutions1. Iff(s1)<f(s),the new solution is always accepted, that is, s=s1. Otherwise, generate random number w uniformly in the range [0,1].
      • (2.3.1) Ifw<exp{[f(s1)f(s)]kT}, thens=s1. Where k is Boltzmann constant and T is the current temperature.
    • (2.4) If IT<L, then the program will go back to step (2.2.1).
    • (2.5) If the termination condition is met, then go to step 3.
    • (2.6) h = h + 1
    • (2.7) Update the temperature asTh+1=(0.95)hTh, then the program will go back to step (2.2)
  • 3. Stop
  • 4. Output the best solution.

From the step (2.3) of the above program, we can see clearly that the algorithm allows the acceptance of the worse solution than the current solution .This can help the program escape the local minima. The current temperature decides what probability is used to accept the worse solution than current solution. With the lower temperature, the probability accepted the unfavourable solution becomes lower. So the sufficiently high initial temperature is conducive to approach the global minimum. The algorithm is terminated when one of the following stop conditions is met. (1) The objective function value (OFV) is less than the predetermined value. (2) The change in objective function value (COFV) is less than the predetermined value. Figure 2 shows the flowing chart of optimizing the LED array .

 figure: Fig. 2

Fig. 2 The flowing chart of optimization.

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4. Results

4.1 Design array consisting of LEDs with perfect Lambertian luminous intensity distribution

In this section, we optimize the first array which is made up of 7 LEDs by SA algorithm. In order to exhibit the design freedom of the method, we use two types of LEDs in this array. One type contains 4 identical LEDs with the luminous intensity distribution shown in Fig. 3 (blue solid curve) and the other type consists of the other 3 LEDs with the same luminous distribution as shown Fig. 3 (red dashed curve). The luminous intensity distribution of all LEDs is perfect Lambertian distribution. The only difference between the two types of LEDs is intensity value. The intensity value represented by red dashed curve is 1.5 times as much as that of blue solid curve at the same view angle.

 figure: Fig. 3

Fig. 3 The luminous intensity distribution of different types of LEDs.

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Before optimizing the array, some initial conditions are set as shown in Table 1 . The optimization is stopped when the OFV reaches 0.096 which is less than the predetermined value. The optimized arrangement of the first array is shown in Fig. 4 . The blue diamonds and the red circles represent the LEDs corresponding to the blue solid intensity distribution curve and the red dashed distribution curve in Fig. 3, respectively.

Tables Icon

Table 1. Initial Condition for Optimizing the First LED Array

 figure: Fig. 4

Fig. 4 The optimized arrangement of the first LED array.

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Figure 5 shows the irradiance map (left) and profile (right) of the first arrays after optimization, which is simulated by the optical software Tracepro [18]. Utilizing the irradiance data obtained by Tracepro simulation, the calculated illumination uniformity of the first array is 0.12. It is clearly that the uniformity value and the OFV are the same order of magnitude. This indicates the algorithm is quite effective.

 figure: Fig. 5

Fig. 5 The irradiance map (left) and profile (right) of the first LED array after optimization.

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4.2 Design array consisting of LEDs with imperfect Lambertian intensity distribution

The second array we designed is a circular array with 7 identical LEDs. Each LED in the array has a free-form lens which can change the view angle of LED. The free-form lens is designed by utilizing the method similar to Ref [8]. The luminous intensity distributions of LED without and with free-form lens are shown in Fig. 6 and Fig. 7 , respectively. As seen clearly from Fig. 6 and Fig. 7, the free-form lens can reduce the view angle to ± 50 degree from ± 90 degree. The intensity distribution of LED with free-form lens can be fitted by Eq. (1). Here m is calculated as m = 4.82 according to Eq. (2). The fitted intensity distribution curve is represented by the red dashed curve in Fig. 8 . The fitting range is between −60 degree and 60 degree. The original intensity distribution curve is shown in Fig. 7. Figure 8 shows the original (blue solid) and fitted (red dashed) curves together with a displaying region of between −60 degree and 60 degree.

 figure: Fig. 6

Fig. 6 The luminous intensity distribution of LED without free-form lens.

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 figure: Fig. 7

Fig. 7 The luminous intensity distribution of LED with free-form lens.

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 figure: Fig. 8

Fig. 8 The original and fitted intensity distribution curve of LED in the circular array.

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In order to evaluate how closely the fitted curve approaches the original curve, the normalized cross-correlation (NCC) [19] is adopted. The NCC is given by Eq. (8).

NCC=v[I(θv)FI¯F][I(θv)OI¯O]v[I(θv)FI¯F]2v[I(θv)OI¯O]2
Where IF and IO are the value of the fitted and original intensity, respectively. θv is the v-th angular displacement.I¯F and I¯Oare the mean value of fitted and original intensity of all sample angles. Here, the calculated NCC is 99.5%. It is clearly that the fitted curve is very close the original curve. For the original curve shown in Fig. 7, the ratio of intensity value within the range of −60 to 60 degree to the total intensity is about 96.5%. So the intensity distribution of LED with this free-form lens can be considered as an imperfect Lambertian intensity distribution.

Figure 9 shows a schematic of circular array in which all LEDs are arranged in a circle with the same angle separation. For the circular array, the irradiance of any point on target plane can be calculated by

E(xp,yq,z)=i=1nzm+1I0[(xpX1cos((i1).2πn))2+(yqX1cos((i1).2πn))2+z2]m+32
Where X1 is the abscissa of the first LED, which is equal to the radius of circular array. Thus, we optimize the 7-LED circular array by SA algorithm with the initial conditions listed in Table 2 . In the optimization, the COFV met the stop criteria firstly. By optimizing, the optimal radius of the array is obtained as r = 34.492mm. With the optimal radius, the circular LED array generates uniform illumination distribution. Figure 10 shows the irradiance map and profile of the optimized circular array, which is simulated by optical software Tracepro.

 figure: Fig. 9

Fig. 9 Schematic of circular array with n LEDs.

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Tables Icon

Table 2. Initial Condition for Optimizing Circular LED Array

 figure: Fig. 10

Fig. 10 The irradiance map (left) and profile (right) of the optimized circular array.

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The calculated illumination uniformity is 0.23. It seems the uniformity value is slightly large. This is because the optimized array is circular. However the target plane is a rectangle plane. As can be seen from the irradiance map in Fig. 10 (left), the illumination uniformity is poor in the four corners of the target plane. In fact, the illumination distribution is very uniform for the other region in the rectangle target plane.

4.3 Design array consisting of LEDs with special luminous intensity distribution

To illustrate the potential of the method, we optimized the third array consisting of 7 LEDs, in which each LED has a special luminous intensity distribution curve (dashed) shown in Fig. 11 . The luminous intensity distribution is generated by applying a special lens to LED. The special lens is formed by digging a conical cavity in a plate PMMA with a circular aperture as shown in Fig. 12 . Figure 12(a) and 12(b) show 2-D and 3-D layout of the special lens, respectively. In order to see the layout of the special lens clearly, we give a 3-D perspective of the special lens in Fig. 12(c).

 figure: Fig. 11

Fig. 11 The intensity distribution of LED with a special lens.

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 figure: Fig. 12

Fig. 12 2-D and 3-D layout of the special lens.

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It can be seen from Fig. 11 that the intensity distribution is not Lambertian distribution. As a result, the special intensity distribution cannot be fitted by Eq. (1). We fit the intensity distribution by the following polynomial ofcosθ.

I(θ)=a8cos8θ+a7cos7θ++a1cosθ+a0
Where a8,a7a0 are polynomial fitting coefficients. In Fig. 11, the red dashed curve represents the fitted intensity distribution curve by Eq. (10). We also calculate the NCC of the fitted and original intensity value. The calculated NCC is 99.99%. It is evident that there is fine coincidence between fitting and original intensity distribution curve. With the luminous intensity expression as shown in Eq. (10), the irradiance is given by
E(xp,yq,z)=i=1n(k=09zk+1ak[(xpXi)2+(yqYi)2+z2]k+32)
Thus, we still can use the same method to optimize this array with the objective function as shown in Eq. (5). In this array, there are 4 identical LEDs with the intensity distribution shown in Fig. 13 (blue solid curve) and the other 3 LEDs have same intensity distribution shown in Fig. 13 (red dashed curve). At the same view angle, the intensity value represented by red dashed curve is 1.5 times as much as that of blue solid curve. In the optimization, we use the same initial condition shown in Table 1.

 figure: Fig. 13

Fig. 13 The luminous intensity distribution curve of different types of LED in the third array.

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The optimization is terminated when the objective function value reaches 0.083. After optimizing, the third LED array is arranged as shown in Fig. 14 . The blue diamonds and the red circles represent the LEDs corresponding to the blue solid intensity distribution curve and the red dashed distribution curve in Fig. 14, respectively. Figure 15 shows the irradiance map (left) and profile (right) of the optimized LED array. The calculated uniformity is 0.13.

 figure: Fig. 14

Fig. 14 The optimized arrangement of the third LED array.

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 figure: Fig. 15

Fig. 15 The irradiance map (left) and profile (right) of the third optimized LED array .

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5. Conclusion

In conclusion, a numerical method is used to optimize the LED array arrangement for uniform illumination distribution on target plane. In the optimization, an objective function is constructed to evaluate the uniformity of illumination. The objective function value depends on the coordinates of all LED in the array. The SA algorithm is used to optimize the coordinates of all LED in the array so that the objective function can reach the minimum value. By the method, three various luminous intensity distribution LED arrays were optimized. In the first and third array, some LEDs with higher intensity value are included. The illumination distribution of the three optimized array is quite uniform which is verified with a simulation using the optical software Tracepro. Using the irradiance data obtained by the simulation, the calculated uniformity is 0.12, 0.23 and 0.13, respectively. With more designing freedom, it is expected that the method can also design array where every LED has a slightly different intensity profile due to manufacturing errors. With automated optimization, it is simple and time-saving to design the optimal LED array arrangement for highly uniform illumination distribution.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No.60908041), the Fundamental Research Funds for the Central Universities (JUSRP31005), Wuxi Construction Bureau science and technology project (WX2012006).We really appreciate Lambda Research Corporation for Tracepro software help. The authors also really appreciate Jung Y. Huang who is with Department of Photonics and Institute of Electro-Optical Engineering, National Chiao Tung University for helpful discussions.

References and links

1. 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]  

2. K. Wang, S. Liu, F. Chen, Z. Qin, Z. Y. Liu, and X. B. Luo, “Freeform LED lens for rectangularly prescribed illumination,” J. Opt. A, Pure Appl. Opt. 11(10), 105501 (2009). [CrossRef]  

3. B. Kim, H. Kim, and S. Kang, “Reverse functional design of discontinuous refractive optics using an extended light source for flat illuminance distributions and high color uniformity,” Opt. Express 19(3), 1794–1807 (2011). [CrossRef]   [PubMed]  

4. Z. X. Feng, Y. Luo, and Y. J. Han, “Design of LED freeform optical system for road lighting with high luminance/illuminance ratio,” Opt. Express 18(21), 22020–22031 (2010). [CrossRef]   [PubMed]  

5. 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]  

6. I. Moreno, J. Muñoz, and R. Ivanov, “Uniform illumination of distant targets using a spherical light-emitting diode array,” Opt. Eng. 46(3), 033001 (2007). [CrossRef]  

7. 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]  

8. A. J.-W. Whang, Y.-Y. Chen, and Y.-T. Teng, “Designing uniform illuminance systems by surface-tailored lens and configurations of LED arrays,” J. Disp. Technol. 5(3), 94–103 (2009). [CrossRef]  

9. Z. Qin, K. Wang, F. Chen, X. Luo, and S. Liu, “Analysis of condition for uniform lighting generated by array of light emitting diodes with large view angle,” Opt. Express 18(16), 17460–17476 (2010). [CrossRef]   [PubMed]  

10. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21(6), 1087–1092 (1953). [CrossRef]  

11. S. Kirkpatrick, C. D. Gelatt Jr, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983). [CrossRef]   [PubMed]  

12. V. Cerny, “Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm,” J. Optim. Theory Appl. 45(1), 41–51 (1985). [CrossRef]  

13. Z. Zhenrong, H. Xiang, and L. Xu, “Freeform surface lens for LED uniform illumination,” Appl. Opt. 48(35), 6627–6634 (2009). [CrossRef]   [PubMed]  

14. I. Moreno, “Illumination uniformity assessment based on human vision,” Opt. Lett. 35(23), 4030–4032 (2010). [CrossRef]   [PubMed]  

15. K. Wang, D. Wu, Z. Qin, F. Chen, X. B. Luo, and S. Liu, “New reversing design method for LED uniform illumination,” Opt. Express 19(S4Suppl 4), A830–A840 (2011). [CrossRef]   [PubMed]  

16. P. J. van Laarhoven and E. H. Aarts, “Simulated annealing,” in Simulated Annealing:Theory and Applications.(Kluwer Academic Publishers, Dordrecht,1987)

17. L. Wang, H. Y. Zhang, and X. P. Zheng, “Inter-domain routing based on simulated annealing algorithm in optical mesh networks,” Opt. Express 12(14), 3095–3107 (2004). [CrossRef]   [PubMed]  

18. http://lambdares.com/software_products/tracepro/.

19. 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]  

References

  • View by:

  1. 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]
  2. K. Wang, S. Liu, F. Chen, Z. Qin, Z. Y. Liu, and X. B. Luo, “Freeform LED lens for rectangularly prescribed illumination,” J. Opt. A, Pure Appl. Opt. 11(10), 105501 (2009).
    [Crossref]
  3. B. Kim, H. Kim, and S. Kang, “Reverse functional design of discontinuous refractive optics using an extended light source for flat illuminance distributions and high color uniformity,” Opt. Express 19(3), 1794–1807 (2011).
    [Crossref] [PubMed]
  4. Z. X. Feng, Y. Luo, and Y. J. Han, “Design of LED freeform optical system for road lighting with high luminance/illuminance ratio,” Opt. Express 18(21), 22020–22031 (2010).
    [Crossref] [PubMed]
  5. 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]
  6. I. Moreno, J. Muñoz, and R. Ivanov, “Uniform illumination of distant targets using a spherical light-emitting diode array,” Opt. Eng. 46(3), 033001 (2007).
    [Crossref]
  7. 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]
  8. A. J.-W. Whang, Y.-Y. Chen, and Y.-T. Teng, “Designing uniform illuminance systems by surface-tailored lens and configurations of LED arrays,” J. Disp. Technol. 5(3), 94–103 (2009).
    [Crossref]
  9. Z. Qin, K. Wang, F. Chen, X. Luo, and S. Liu, “Analysis of condition for uniform lighting generated by array of light emitting diodes with large view angle,” Opt. Express 18(16), 17460–17476 (2010).
    [Crossref] [PubMed]
  10. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21(6), 1087–1092 (1953).
    [Crossref]
  11. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
    [Crossref] [PubMed]
  12. V. Cerny, “Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm,” J. Optim. Theory Appl. 45(1), 41–51 (1985).
    [Crossref]
  13. Z. Zhenrong, H. Xiang, and L. Xu, “Freeform surface lens for LED uniform illumination,” Appl. Opt. 48(35), 6627–6634 (2009).
    [Crossref] [PubMed]
  14. I. Moreno, “Illumination uniformity assessment based on human vision,” Opt. Lett. 35(23), 4030–4032 (2010).
    [Crossref] [PubMed]
  15. K. Wang, D. Wu, Z. Qin, F. Chen, X. B. Luo, and S. Liu, “New reversing design method for LED uniform illumination,” Opt. Express 19(S4Suppl 4), A830–A840 (2011).
    [Crossref] [PubMed]
  16. P. J. van Laarhoven and E. H. Aarts, “Simulated annealing,” in Simulated Annealing:Theory and Applications.(Kluwer Academic Publishers, Dordrecht,1987)
  17. L. Wang, H. Y. Zhang, and X. P. Zheng, “Inter-domain routing based on simulated annealing algorithm in optical mesh networks,” Opt. Express 12(14), 3095–3107 (2004).
    [Crossref] [PubMed]
  18. http://lambdares.com/software_products/tracepro/ .
  19. 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]

2011 (2)

2010 (3)

2009 (4)

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]

A. J.-W. Whang, Y.-Y. Chen, and Y.-T. Teng, “Designing uniform illuminance systems by surface-tailored lens and configurations of LED arrays,” J. Disp. Technol. 5(3), 94–103 (2009).
[Crossref]

K. Wang, S. Liu, F. Chen, Z. Qin, Z. Y. Liu, and X. B. Luo, “Freeform LED lens for rectangularly prescribed illumination,” J. Opt. A, Pure Appl. Opt. 11(10), 105501 (2009).
[Crossref]

Z. Zhenrong, H. Xiang, and L. Xu, “Freeform surface lens for LED uniform illumination,” Appl. Opt. 48(35), 6627–6634 (2009).
[Crossref] [PubMed]

2008 (1)

2007 (2)

I. Moreno, J. Muñoz, and R. Ivanov, “Uniform illumination of distant targets using a spherical light-emitting diode array,” Opt. Eng. 46(3), 033001 (2007).
[Crossref]

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]

2006 (1)

2004 (1)

1985 (1)

V. Cerny, “Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm,” J. Optim. Theory Appl. 45(1), 41–51 (1985).
[Crossref]

1983 (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
[Crossref] [PubMed]

1953 (1)

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21(6), 1087–1092 (1953).
[Crossref]

Avendaño-Alejo, M.

Bergmans, J. W. M.

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]

Cerny, V.

V. Cerny, “Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm,” J. Optim. Theory Appl. 45(1), 41–51 (1985).
[Crossref]

Chen, F.

Chen, Y.-Y.

A. J.-W. Whang, Y.-Y. Chen, and Y.-T. Teng, “Designing uniform illuminance systems by surface-tailored lens and configurations of LED arrays,” J. Disp. Technol. 5(3), 94–103 (2009).
[Crossref]

Chien, W. T.

Ding, Y.

Feng, Z. X.

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
[Crossref] [PubMed]

Gu, P. F.

Han, Y. J.

Ivanov, R.

I. Moreno, J. Muñoz, and R. Ivanov, “Uniform illumination of distant targets using a spherical light-emitting diode array,” Opt. Eng. 46(3), 033001 (2007).
[Crossref]

Kang, S.

Kim, B.

Kim, H.

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
[Crossref] [PubMed]

Linnartz, J. P. M. G.

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]

Liu, S.

Liu, X.

Liu, Z. Y.

K. Wang, S. Liu, F. Chen, Z. Qin, Z. Y. Liu, and X. B. Luo, “Freeform LED lens for rectangularly prescribed illumination,” J. Opt. A, Pure Appl. Opt. 11(10), 105501 (2009).
[Crossref]

Luo, X.

Luo, X. B.

K. Wang, D. Wu, Z. Qin, F. Chen, X. B. Luo, and S. Liu, “New reversing design method for LED uniform illumination,” Opt. Express 19(S4Suppl 4), A830–A840 (2011).
[Crossref] [PubMed]

K. Wang, S. Liu, F. Chen, Z. Qin, Z. Y. Liu, and X. B. Luo, “Freeform LED lens for rectangularly prescribed illumination,” J. Opt. A, Pure Appl. Opt. 11(10), 105501 (2009).
[Crossref]

Luo, Y.

Metropolis, N.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21(6), 1087–1092 (1953).
[Crossref]

Moreno, I.

Muñoz, J.

I. Moreno, J. Muñoz, and R. Ivanov, “Uniform illumination of distant targets using a spherical light-emitting diode array,” Opt. Eng. 46(3), 033001 (2007).
[Crossref]

Qin, Z.

Rietman, R.

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]

Rosenbluth, A. W.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21(6), 1087–1092 (1953).
[Crossref]

Rosenbluth, M. N.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21(6), 1087–1092 (1953).
[Crossref]

Schenk, T. C. W.

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]

Sun, C. C.

Teller, A. H.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21(6), 1087–1092 (1953).
[Crossref]

Teller, E.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21(6), 1087–1092 (1953).
[Crossref]

Teng, Y.-T.

A. J.-W. Whang, Y.-Y. Chen, and Y.-T. Teng, “Designing uniform illuminance systems by surface-tailored lens and configurations of LED arrays,” J. Disp. Technol. 5(3), 94–103 (2009).
[Crossref]

Tzonchev, R. I.

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
[Crossref] [PubMed]

Wang, K.

Wang, L.

Whang, A. J.-W.

A. J.-W. Whang, Y.-Y. Chen, and Y.-T. Teng, “Designing uniform illuminance systems by surface-tailored lens and configurations of LED arrays,” J. Disp. Technol. 5(3), 94–103 (2009).
[Crossref]

Wu, D.

Xiang, H.

Xu, L.

Yang, H.

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]

Zhang, H. Y.

Zheng, X. P.

Zheng, Z. R.

Zhenrong, Z.

Appl. Opt. (2)

IEEE Trans. Signal Process. (1)

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]

J. Chem. Phys. (1)

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21(6), 1087–1092 (1953).
[Crossref]

J. Disp. Technol. (1)

A. J.-W. Whang, Y.-Y. Chen, and Y.-T. Teng, “Designing uniform illuminance systems by surface-tailored lens and configurations of LED arrays,” J. Disp. Technol. 5(3), 94–103 (2009).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

K. Wang, S. Liu, F. Chen, Z. Qin, Z. Y. Liu, and X. B. Luo, “Freeform LED lens for rectangularly prescribed illumination,” J. Opt. A, Pure Appl. Opt. 11(10), 105501 (2009).
[Crossref]

J. Optim. Theory Appl. (1)

V. Cerny, “Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm,” J. Optim. Theory Appl. 45(1), 41–51 (1985).
[Crossref]

Opt. Eng. (1)

I. Moreno, J. Muñoz, and R. Ivanov, “Uniform illumination of distant targets using a spherical light-emitting diode array,” Opt. Eng. 46(3), 033001 (2007).
[Crossref]

Opt. Express (7)

Opt. Lett. (1)

Science (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
[Crossref] [PubMed]

Other (2)

http://lambdares.com/software_products/tracepro/ .

P. J. van Laarhoven and E. H. Aarts, “Simulated annealing,” in Simulated Annealing:Theory and Applications.(Kluwer Academic Publishers, Dordrecht,1987)

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Figures (15)

Fig. 1
Fig. 1 (a) Diagram of LED illumination. (b) Target plane with M × N grids.
Fig. 2
Fig. 2 The flowing chart of optimization.
Fig. 3
Fig. 3 The luminous intensity distribution of different types of LEDs.
Fig. 4
Fig. 4 The optimized arrangement of the first LED array.
Fig. 5
Fig. 5 The irradiance map (left) and profile (right) of the first LED array after optimization.
Fig. 6
Fig. 6 The luminous intensity distribution of LED without free-form lens.
Fig. 7
Fig. 7 The luminous intensity distribution of LED with free-form lens.
Fig. 8
Fig. 8 The original and fitted intensity distribution curve of LED in the circular array.
Fig. 9
Fig. 9 Schematic of circular array with n LEDs.
Fig. 10
Fig. 10 The irradiance map (left) and profile (right) of the optimized circular array.
Fig. 11
Fig. 11 The intensity distribution of LED with a special lens.
Fig. 12
Fig. 12 2-D and 3-D layout of the special lens.
Fig. 13
Fig. 13 The luminous intensity distribution curve of different types of LED in the third array.
Fig. 14
Fig. 14 The optimized arrangement of the third LED array.
Fig. 15
Fig. 15 The irradiance map (left) and profile (right) of the third optimized LED array .

Tables (2)

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Table 1 Initial Condition for Optimizing the First LED Array

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Table 2 Initial Condition for Optimizing Circular LED Array

Equations (11)

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I(θ)= I 0 cos m θ
m= ln2 ln(cos θ 1/2 )
E( x p , y q ,z)= z m+1 I 0 [ ( x p X) 2 + ( y q Y) 2 + z 2 ] m+3 2
E( x p , y q ,z)= i=1 n z m+1 I 0 [ ( x p X i ) 2 + ( y q Y i ) 2 + z 2 ] m+3 2
f( X 1 , Y 1 ,..., X i , Y i ,..., X n , Y n )= σ E ¯
E ¯ = 1 M×N q=1 N p=1 M E( x p , y q ,z)
σ= p=1 N q=1 M (E( x p , y q ,z) E ¯ ) 2 M×N
NCC= v [I ( θ v ) F I ¯ F ][I ( θ v ) O I ¯ O ] v [I ( θ v ) F I ¯ F ] 2 v [I ( θ v ) O I ¯ O ] 2
E( x p , y q ,z)= i=1 n z m+1 I 0 [ ( x p X 1 cos( (i1).2π n )) 2 + ( y q X 1 cos( (i1).2π n )) 2 + z 2 ] m+3 2
I(θ)= a 8 cos 8 θ+ a 7 cos 7 θ++ a 1 cosθ+ a 0
E( x p , y q ,z)= i=1 n ( k=0 9 z k+1 a k [ ( x p X i ) 2 + ( y q Y i ) 2 + z 2 ] k+3 2 )

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