Designing LED array for uniform illumination distribution by simulated annealing algorithm

Zhouping Su; Donglin Xue; Zhicheng Ji

doi:10.1364/OE.20.00A843

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 [1–3]. 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 [1–4]. 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 [10–12], 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) [7–9,13–15].

I (θ) = I_{0} \cos^{m} θ

Where

θ

is the view angle and

I_{0}

is luminous intensity at the normal direction to the source surface. The number m depends on the angular half width

θ_{1 / 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) [5–9,13].

m = \frac{- l n 2}{\ln (\cos θ_{1/2})}

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

(x_{p}, y_{q}, 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 (x_{p}, y_{q}, z) = \frac{z^{m + 1} I_{0}}{{[{(x_{p} - X)}^{2} + {(y_{q} - Y)}^{2} + z^{2}]}^{\frac{m + 3}{2}}}

Therefore, the irradiance at the point A with an array of n LEDs can be expressed as:

E (x_{p}, y_{q}, z) = \sum_{i = 1}^{n} \frac{z^{m + 1} I_{0}}{{[{(x_{p} - X_{i})}^{2} + {(y_{q} - Y_{i})}^{2} + z^{2}]}^{\frac{m + 3}{2}}} \cdot

Here

(X_{i}, Y_{i}, 0)

are the coordinates of the i-th LED in the array.

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 $M \times N$ 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 (X_{1}, Y_{1}, ..., X_{i}, Y_{i}, ..., X_{n}, Y_{n}) = \frac{σ}{\bar{E}} \cdot

Here

\bar{E}

denotes the average irradiance of all grids on the target plane.

\bar{E} = \frac{1}{M \times N} \sum_{q = 1}^{N} \sum_{p = 1}^{M} E (x_{p}, y_{q}, z)

Where

σ

is the standard error of irradiance of all grids, which can be expressed by

σ = \sqrt{\frac{{\sum_{p = 1}^{N} \sum_{q = 1}^{M} (E (x_{p}, y_{q}, z) - \bar{E})}^{2}}{M \times N}} \cdot

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 [10–12,16,17]:

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 $s$ is 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 as $T_{h + 1} = {(0.95)}^{h} T_{h} \cdot$ 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 solution $s_{1}$ . If $f (s_{1}) < f (s)$ ,the new solution is always accepted, that is, $s = s_{1}$ . Otherwise, generate random number $w$ uniformly in the range [0,1].
  - (2.3.1) If $w < \exp {- \frac{[f (s_{1}) - f (s)]}{k T}}$ , then $s = s_{1}$ . 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 as $T_{h + 1} = {(0.95)}^{h} T_{h}$ , 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 .

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.

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.

Table 1. Initial Condition for Optimizing the First LED Array

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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.

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.

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

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

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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).

N C C = \frac{\sum_{v} [I {(θ_{v})}_{F} - {\bar{I}}_{F}] [I {(θ_{v})}_{O} - {\bar{I}}_{O}]}{\sqrt{\sum_{v} {[I {(θ_{v})}_{F} - {\bar{I}}_{F}]}^{2} \sum_{v} {[I {(θ_{v})}_{O} - {\bar{I}}_{O}]}^{2}}}

Where

I_{F}

and

I_{O}

are the value of the fitted and original intensity, respectively.

θ_{v}

is the v-th angular displacement.

{\bar{I}}_{F}

and

{\bar{I}}_{O}

are 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 (x_{p}, y_{q}, z) = \sum_{i = 1}^{n} \frac{z^{m + 1} I_{0}}{{[{(x_{p} - X_{1} \cos (\frac{(i - 1) .2 π}{n}))}^{2} + {(y_{q} - X_{1} \cos (\frac{(i - 1) .2 π}{n}))}^{2} + z^{2}]}^{\frac{m + 3}{2}}}

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.

Fig. 9 Schematic of circular array with n LEDs.

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Table 2. Initial Condition for Optimizing Circular LED Array

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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).

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

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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 of $\cos θ$ .

I (θ) = a_{8} \cos^{8} θ + a_{7} \cos^{7} θ + \cdot \cdot \cdot + a_{1} \cos θ + a_{0}

Where

a_{8}, a_{7} \cdot \cdot \cdot a_{0}

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 (x_{p}, y_{q}, z) = \sum_{i = 1}^{n} (\sum_{k = 0}^{9} \frac{z^{k + 1} a_{k}}{{[{(x_{p} - X_{i})}^{2} + {(y_{q} - Y_{i})}^{2} + z^{2}]}^{\frac{k + 3}{2}}}) \cdot

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.

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.

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

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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.

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Target plane size	Distance between LED and Target plane		The initial Temperature		Stop criteria
Target plane size	Distance between LED and Target plane		The initial Temperature		OFV COFV
40x40		50	5000	≤0.1 ≤10⁻¹⁰

Target plane size	Distance between LED and Target plane	The initial Temperature		Stop criteria
Target plane size	Distance between LED and Target plane	The initial Temperature		OFV COFV
40x40	100	5000	≤0.1 ≤10⁻¹⁰

Target plane size	Distance between LED and Target plane		The initial Temperature		Stop criteria
Target plane size	Distance between LED and Target plane		The initial Temperature		OFV COFV
40x40		50	5000	≤0.1 ≤10⁻¹⁰

Target plane size	Distance between LED and Target plane	The initial Temperature		Stop criteria
Target plane size	Distance between LED and Target plane	The initial Temperature		OFV COFV
40x40	100	5000	≤0.1 ≤10⁻¹⁰

Designing LED array for uniform illumination distribution by simulated annealing algorithm

Abstract

1. Introduction

2. The theory of high-uniform LED array design

3. Simulated annealing algorithm design

4. Results

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

4.2 Design array consisting of LEDs with imperfect Lambertian intensity distribution

4.3 Design array consisting of LEDs with special luminous intensity distribution

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (15)

Tables (2)

Equations (11)

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