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Development of a numerical model for the luminous intensity distribution of a planar prism LED luminaire for applying an optimization algorithm.

1 INTRODUCTION

Light emitting diodes (LED) have the promise of lower power consumption, energy saving, high efficiency, long life, environmental friendliness, and a small volume and as a result have been getting the spotlight as a next-generation light source. LED lighting is often presented as the lighting technology of the future for object lighting and even for general lighting. Nevertheless, several technological problems have to be resolved for LEDs to be more widely used as a general purpose light source. For instance, LED luminaires have a property of high luminance due to a narrow beam angle and a small volume compared to general light sources. LED arrays are also necessary because the flux of a single power of LED is too low for most lighting applications. With these differences of luminous intensity distribution from conventional luminaires, it is difficult for LED luminaires to obtain the target luminous intensity distribution using the same optical design, optical parts, and with the same approach as conventional luminaires. It is hard to derive the desired target luminous intensity distribution only by applying the optical design as a general reflector. Thus, optical design is very important for the LED luminaires, and a new approach is necessary to conduct the optical design of planar LED luminaires.

The most applied shape to planar LED luminaires at present is that of an LED source installed downward with a diffuse plate or translucent plate is equipped to prevent the direct exposure of LED sources. Most LED sources have Lambertian emissionuthat is, the luminous intensity distribution is round, with a full beam angle of around 120[degrees]. Such planar LED luminaires use the Lambertian luminous intensity distribution curve. This differs from the bat-wing luminous intensity distribution curve spreading sideward found in current generally-used fluorescent luminaires. Consequently, the control of the Lambertian luminous intensity distribution curve is necessary to apply a planar LED luminaire to various spaces or to a nonuniform lighting environment [Vandeghinste 2008]. The control of the Lambertian distribution curve of a planar LED luminaire is practicable by installing optical parts, such as a lens and prism.

Previous studies about the optical design of optical parts of luminaires are on the vector for calculating laws of reflection and refraction [Bhattacharjee 2009; Bhattacharjee 2010] that is a basis for optical design, and the study on the lighting calculation using bidirectional reflectance distribution functions (BRDFs) used in computer graphics [Wandachowicz and others 2006]. Most studies about the optical design of LED luminaires are on the secondary optical design of LED sources. In particular, there are the studies on the optical control of LED sources by designing a reflector of LED sources [Vandeghinste 2008; Horng and others 2009; Pan and others 2011] and the studies on the optical design of a secondary optics element of LED sources [Chen and others 2011; Hsieh and others 2009; Sun and others 2009; Zhang and others 2009; Tran and others 2007; Joo and others 2009; Wang and others 2010]. In addition to these studies on the optical design of LEDs, there are studies on the optical design lens of LED sources to apply to streetlights [Lo and others 2011; Liu and others 2009; Jiang and others 2010] and the study on the optical design of the reflector of LED luminaires for automotive headlights [Huang and others 2010]. There is also a study on the prism pattern design for designing an LCD backlight module [Li and others 2010] and studies on the optical design of diffuse film for designing the backlight [Lin and others 2011; Mingyan and others 2011] as in the studies on the optical design of a prism.

As indicated above, most recent studies on the luminous intensity distribution control of LED luminaires are for the secondary optical design of a lens or reflector of the LED sources, and most of the ones on the optical design of a prism are related to backlight applications. In the studies on the optical design of luminaires, many studies on reflectors and light source have been conducted but the ones on prisms and lenses are insufficient at present.

The optical design of luminaires has been performed using a combination of ray-tracing techniques and trial-and-error experimentation to obtain a target luminous intensity distribution. The advent of faster computer capabilities allows luminaire optical designers to quickly perform the optical design process.However, time and a great deal of experience are still required. For the optical design of luminaires, in order to obtain a target luminous intensity distribution, the numbers of trials and errors can be significantly reduced by utilizing optimization algorithms. This study aims to develop a numerical model of planar LED luminaires for controlling luminous intensity distribution, which can be used later as a basis for applying optimization algorithms.

This numerical model can predict the change of luminous intensity distribution by tracing the behavior of photons which pass through a prism, emitted from a light source. The luminous intensity distribution of planar LED luminaires is entirely dependent on the angles of prism. Thus, this numerical model can forecast the new luminous intensity distribution curve according to the change of the prism angle and the photometric data of an LED source. It is possible to control the luminous intensity distribution of prism luminaires by using this algorithm. In fact, the fixed prism angles of the planar prism LED luminaire bring about a lack of various luminous intensity distributions. So, to control various luminous intensity distributions, this study proposed two methods: 1) applications of the unit prism angle zone, and 2) change in the width of the prism.

Eventually, this numerical model would be used as the basis of an optimized optical design algorithm using the genetic algorithm (GA) technique, which can derive the optimized angle of a prism to get the target luminous intensity distribution of planar prism LED luminaires. This study could be the preliminary step to achieve such an optimized optical design algorithm of planar prism LED luminaires.

2 NUMERICAL MODEL PROCEDURES AND METHODS

2.1 OVERVIEW OF THE NUMERICAL MODEL

The numerical model for luminous intensity distribution suggested in this study is comprised of the four phases. First, this numerical model used the luminous intensity of each vertical angle in the photometric data of an LED source applied to planar LED luminaires as the energy of photons for each vertical angle. Second, the numerical model set up the initial position of LED sources and direction vectors of photons for the coordinates of a planar LED luminaire. Next, the numerical model calculated the position and direction vector according to the optical behavior of the photons passing through the prism using the ray-tracing technique and the law of refraction. Finally, the numerical model produced a new luminous intensity distribution curve of a planar prism LED luminaire by the sum of photon energies in each vertical angle, which indicates the final position of photons. Figure 1 shows the overview of the numerical model of luminous intensity distribution.

In this case, this numerical model set the final position to the point distant as far as five times the length of the luminous area of the luminaire. This is the 5-times rule used in luminous flux transfer, one of the methods of calculating lighting. The rule is that the distance of the point where illuminance is obtained from the position of a luminaire should be at least 5-times the maximum dimension of the luminous area of the luminaire in order to regard the light source (luminaire) as a point light source (PLS) [DiLaura and others 2010]. The 5-times rule is used to calculate the direct component of illuminance from a luminaire.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

2.2 FIRST CALCULATION OF REFRACTION MECHANISM

As shown in Fig. 2, the 1st refractive process means the process that the photons emitted from LED sources enter at the prism and become refracted. The initial position ([x.sub.0],[y.sub.0]) of photons is the position where an LED source is installed, and moves by the initial direction vector ([[[vector].i].sub.x],[[[vector].i].sub.y]) calculated by the vertical angle of photometric data of an LED source as represented in (1). At this time, the luminous intensity in each vertical angle is used as the energy of photons for each direction. As represented in (2), the 2nd position ([x.sub.1], [y.sub.l=1] = 0) where photons reach at the top of the prism and enter the prism is calculated, and at this point the initial direction vector of photons becomes an incidence vector.

[FIGURE 3 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where

[[[vector].i].sub.x],[[[vector].i].sub.y]: the components of the initial direction vector

[x.sub.1],[x.sub.1]: the components of the 2nd position of photons

As shown in Fig. 3, the angle of incidence ([[theta].sub.i]) is calculated by the inner product of the unit direction vector (i) of the incidence photons and the unit normal vector ([[vector].n]), and the angle of refraction ([[theta].sub.r]) is calculated by "Snell's Law". Thus, after the angle of incidence ([[theta].sub.i]) and the angle of refraction ([[theta].sub.r]) are calculated, the 1st refraction vector ([^.r]) is calculated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where

[[theta].sub.i],[[theta].sub.r]: angle of incidence and angle of refraction

[n.sub.air],[n.sub.prism]: index of refraction of the air and prism

[[vector].i],[[vector].f],[[vector].n]: the unit direction vector, the unit refraction vector and the unit normal vector

Further we have

[^.P].sub.n] = [^.i] x [^.r]

where [^.P].sub.n], is calculated by the cross product of two unit vectors [^.i] and [^.r].

Next, the 1st refraction vector with the top surface of the prism is calculated, as shown in the following algorithm:

[[[vector].n].sub.x]=0

[[[vector].n].sub.y]=1

[[theta].sub.i]=[cos.sup.-1](-[[[vector].i].sub.x]x [[[vector]n].sub.x]-[[[vector].i].sub.y] x [[[vector].n].sub.y])

[[theta].sub.r]=[sin.sup.-1]([n.sub.air]/[n.sub.prism] x sin[[theta].sub.i])

det=(-[[[vector].i].sub.x] x [[[vector].n].sub.y]+[[[vector].i].sub.y] x ][[[vector].n].sub.x])

[[[vector].r].sub.x]=-[[[vector].n].sub.y] x cos(abs([[theta].sub.i]-[[theta].sub.r]))-[[[vector].i].sub.y] x cos[[theta].sub.r]/det

[[[vector].r].sub.y]=[[[vector].n].sub.x] x cos(abs([[theta].sub.i]-[[theta].sub.r]))+[[[vector].i].sub.x] x cos[[theta].sub.r]/det

where,

[[[vector].n].sub.x],[[[vector].n].sub.y]: the components of the normal vector

[[[vector].i].sub.x],[[[vector].i].sub.y]: the components of the initial direction vector 0i, 6r: the angle of incidence and angle of refraction

[[[vector].r].sub.x],[[[vector].r].sub.y]: the components of the 1st refraction vector

2.3 SECOND CALCULATION OF REFRACTION MECHANISM

As photons enter at the inside of the prism, they move with the direction changed by the 1st refraction vector([[[vector].r].sub.1]) as shown in Fig. 4. Then, the photons' incidence position into the prism faces (3rd position: [x.sub.2],[y.sub.2]) is calculated by the photons' incidence position into the prism (2nd position: [x.sub.1],[y.sub.1]) and the 1st refraction vector ([[[vector].r].sub.1]). Also, the normal vector necessary to calculate the 2nd refraction process is determined by the photons incidence position into the prism faces (3rd position: [x.sub.2],[y.sub.2]). In this numerical model, the imagination plane ([y.sub.3]) bordering the vertex of the prism (pointed angle) is calculated using (4):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

[FIGURE 4 OMITTED]

where [x.sub.3],[y.sub.3]: the components of the position of photons on the imagination plane.

[FIGURE 5 OMITTED]

And the side surface of the prism into which incidence photons are determined, and the point of intersection with the side surface of the prism are calculated, as shown in the following algorithm:

if ([x'.sub.2] * It * b) then

b = (-tan[alpha] x [x.sub.1] - [y.sub.2] + b x tan[alpha]/([[[vector].r].sub.y] + tan[alpha] x [[[vector].r].sub.x]

[[[vector].n].sub.x] = sin[alpha]

[[[vector].n].sub.y] = cos[alpha]

elseif ([x'.sub.2] * gt *b) then

b = (tan [alpha] x [x.sub.1] - [y.sub.2] - b x tan[alpha]/([[[vector].r].sub.y] - tan[alpha] x [[[vector].r].sub.x]

[[[vector].n].sub.x] = -sin[alpha]

[[[vector].n].sub.y] = cos[alpha]

where

[alpha]: the angle of the prism

[[[vector].n].sub.x],[[[vector].n].sub.y]: the components of the normal vector

[[[vector].r].sub.x],[[[vector].r].sub.y]: the components of the 1st refraction vector

[x.sub.1],[x.sub.1]: the components of the 2nd position of photons

[x.sub.2],[x.sub.2]: the components of the 3rd position of photons

As shown in Fig. 5, after the incidence vector, the point of intersection with the prism face and the ensuing normal vector are determined, the direction vector of photons that passed through the prism can be calculated by the second refraction process with the same method as the first refraction process. Thus, photons move at the refraction position in the prism face (3rd position) with the reaction vector ([[vector].n]) taken as the direction vector, and then the point of intersection meeting with the circle five times the dimension of the luminous area of the luminaire is to be the final position photons ([x.sub.4],[x.sub.4]). Finally, the global angle is calculated by the center of the luminaire and the final position of photons, and based on this, the new luminous intensity distribution curve of planar LED prism luminaires can be derived using (5):

Global angle = [cos.sup.-1][[vector].(final position] * [[[vector].N] (5)

where [[[vector].N] is global normal vector (0, -1).

[FIGURE 6 OMITTED]

2.4 DEVELOPMENT OF OPTICAL DESIGN ALGORITHM IN CONSIDERATION OF CRITICAL ANGLE AND DIFFUSE LIGHT

The algorithm of a numerical model for luminous intensity distribution suggested in this study traced the optical behavior of the photons by internal reflection, using the ray-tracing technique in case of internal reflection occurring in the prism faces with the angle of incidence getting greater than the critical angle. The refracted energy of photons varies with the angle of incidence and the refractive index of a medium [Hecht 2002]. Thus, this numerical model developed using the transmission coefficients of the Fresnel equation to find the ratio of transmission as represented with (6)

T=([n.sub.t]cos[[theta].sub.t]/[n.sub.i]cos[[theta].sub.i] x (2[n.sub.i]cos[[theta].sub.i]/[n.sub.i]cos[[theta].sub.t]+[n.sub.t]cos[[theta].sub.i] (6)

where ni and nt are the ratio of medium ([n.sub.i] > [n.sub.t], incidence from nt into [n.sub.t]). This study also developed the algorithm in consideration of diffuse light by calculating the direction vector of the photons refracted at the final photons refractive position in the prism faces, as well as the direction vector of the diffuse light as shown in Fig. 6.

This study conducted the simulation with the changes of the number of internal reflections and the ratio of the diffuse light to find out the change of luminous intensity distribution curve according to the number of internal reflections and the ratio of diffuse light. In this case, the simulation set up the prism angle to 30[degrees] and changed the number of internal reflections to 0 times, 5 times, 10 times, 20 times, 30 times, and 50 times and the ratio of diffuse light to 5 percent, 10 percent, 15 percent, 20 percent, and 25 percent (see Fig. 7). As a result of this simulation, this study found that the luminous intensity distribution curve varied a little according to the number of internal reflections up to 20 times, and the luminous intensity distribution curves appeared equally when the number of internal reflection was greater. This is because the internal reflection occurs within 20 times in the case of the prism angle being 30[degrees]. The shapes of luminous intensity distribution curves according to the changes of the ratio of diffuse light appeared equally, and this study shows that luminous intensity decreases at a regular ratio when the ratio of diffuse light is getting larger. This simulation result confirms the necessity of consideration the number of internal reflection and the ratio of diffuse light.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

3 SIMULATION RESULTS

3.1 CONDITION OF SIMULATION

This study used a practical type of planar LED luminaires to conduct the optical design simulation of an optical design of planar prism LED luminaires, using the photometric data of an LED source. The shape of the planar prism used in this study is an isosceles triangle with the vertex of the triangle downwards. The planar prism which was used in the numerical model was two dimensional, and its shape is not actually mini pyramid. The planar prism surface is also assumed to have zero thickness at the two vertices of the triangle. Furthermore, this numerical model should be able to perform the optical design in various widths (from wide to narrow) of the prism. Therefore, to analyze and compare the results of simulation by changing the widths of the prism, the two simulations were conducted by two different width of the prism (1 mm and 10 mm). The reflectance of the luminaire was set to 90 percent and the index of refraction of the prism was set to 1.491, respectively. These are the general properties of ordinary acrylic materials. In order to verify the accuracy of the simulation of the algorithm for the numerical model for luminous intensity distribution developed in the study, the results of the simulation were compared to a simulation using Photopia 2.0 which is the commercial optical design simulation software widely used in the lighting field. The simulation was performed under the same conditions. Figure 8 and Table 1 show the characteristics of the optical parts used in the simulation of this study.
TABLE 1. Characteristics of Optical Parts

Optical Parts                Characteristics

Material of the reflector   White paint

Reflectance                 90%

Material of the refractor   Standard acrylic

Index of the refractor      1.491

LED source                  2W, 31.1 lm


3.2 RESULTS OF SIMULATION

This study first conducted a simulation with no prism (index of refraction: 1.00, angle of prism: 0[degrees]) condition, and compared the simulation result with photometric data of an LED source. Figure 9(a) and (b) shows the simulation results of the numerical model for luminous intensity distribution and the results of Photopia 2.0 simulation. The shapes of intensity distribution from both simulations appear to be the same as input data of an LED source. The reason for the luminous intensity of the vertical angle 0 (appearing a little higher than the input data is that the photons which reflected within the luminaire moved to the vertical angle 0[degrees].

Figure 9(c) and (d) show the results of the Photopia simulations which were conducted by the width of the prism 10 mm and the random angles of the prism (20[degrees] and 45[degrees]). As shown in Fig. 9(c) and (d), the shapes of the horizontal angle 0[degrees] are not much different compared to the shapes of other horizontal angles.Furthermore, the distributions of the Photopia and the numerical model show almost quadrilaterals symmetry. So, this study compared the results of the horizontal angle 0[degrees] of the candela distributions from two simulations (the Photopia and the numerical model).

This study simulated changes in the angle of the prism (1[degrees], 5[degrees], 10[degrees], 20[degrees], 30[degrees], 40[degrees], 45[degrees], and 60[degrees]) to compare the change of the luminous intensity curves of a planar LED luminaire according to the angles of the prism. Then the simulation was conducted with the number of internal reflections set to 50 times, and the ratio of diffuse light set to 20 percent. Figure 10 shows a comparison between the simulation results using the numerical model for luminous intensity distribution and the simulation results using Photopia 2.0. When the size and the angle of the prism were small, two simulation results show a Lambertian luminous intensity distribution curve. And two simulation results show almost same shapes. The errors of the results of the numerical model with the Photopia are 7.81 percent (1 mm) and 8.31 percent (10 mm) by the width of prism. The error is the sum of the differences of each vertical angle between the results of the numerical model and the Photopia. The error was calculated with (7):

[FIGURE 9 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [C.sub.m] and [C.sub.p] are the values of the vertical angle of the luminous intensity distribution of the numerical model and the Photopia. And [C.sub.p-sum] and [C.sub.m-sum] are the summations of the values of vertical angle of the luminous intensity distribution of the numerical model and the Photopia.

There is a little difference according to the angles of the prism, which is because Photopia 2.0 cannot control the number of internal reflection and the ratio of transmission. In Photopia 2.0, the diffuse light of a prism used the measured characteristics of the medium. Overall, the accuracy of the developed numerical model was verified. And, in this algorithm, the measured optical property (the diffuse ratio & the transmittance) of the prism can be used as input data. Therefore, this numerical model can predict the new luminous intensity distribution curve according to the various prism materials.

3.3 THE UNIT PRISM ANGLE ZONES

The luminous intensity distribution of planar LED luminaires totally depends on the angles of prism. However, the fixed prism angles of the planar prism LED luminaire result in a lack of various luminous intensity distributions. Therefore, for various luminous intensity distributions, this study was applied to the unit prism angle zones of the planar prism LED luminaire. Figure 11 shows divisions of the unit prism angle zones. The number of the unit prism angle zones is determined by the input data.

The simulation set up the random angle of prism and changed the number of the unit prism angle zones to 3ea, 5ea, lea, and 10ea. Figure 12 shows a comparison between the simulation results using the numerical model for luminous intensity distribution and the simulation results using Photopia 2.0.As a result, the numerical model for luminous intensity distribution of planar prism LED luminaire with various unit prism angle zones parameters can be much more varied than the fixed prism angle.

4 CONCLUSIONS

The optical design of luminaires has been performed using a combination of ray-tracing techniques and trial-and-error experimentation to obtain a target luminous intensity distribution. The advent of faster computer capabilities allows luminaire optical designers to perform quickly in the optical design process. However, a great deal of time and experience are still required. For the optical design of luminaires, to obtain a target luminous intensity distribution, the numbers of trials and errors can be significantly reduced by utilizing optimization algorithm. This study aimed to develop a numerical model of planar LED luminaires for controlling luminous intensity distribution for optimization algorithm.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

This numerical model for luminous intensity distribution used the photometric data of an LED source and produced the new luminous intensity distribution curve controlled in terms of the changes of the prism angles by calculating the movement of photons with the use of the ray-tracing technique. In addition, this numerical model also took into account the internal reflection and the diffuse light which occurred in passing through the prism according to the angle of prism and the angle of incidence. In comparison between the simulation results of the developed numerical model and the simulation results of Photopia 2.0, their shapes of luminous intensity distribution curves appeared to be almost the same. The accuracy of the developed numerical model for luminous intensity distribution was verified.

Thus, the luminous intensity distribution curve of a planar prism LED luminaire can be forecasted through the optical design algorithm developed in this study. In addition, more various luminous intensity distribution controls can be made by applying various angles by setting up the unit prism angle zones than by conducting optical design in the same prism angle. On the basis of this numerical model, the next step is to develop an optimized optical design algorithm for planar prism LED luminaires that can derive the optimized angle of the prism to get a target luminous intensity distribution, using an optimization technique such as the genetic algorithm. Furthermore, the optimization algorithm of a numerical model for luminous intensity distribution would employ the unit prism angles zones of the planar prism LED luminaires as the optical design parameters for optimization.

[FIGURE 12 OMITTED]

ACKNOWLEDGMENTS

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0016737), and by the MKE (The Ministry of Knowledge Economy), Korea, under the Convergence-ITRC (Convergence Information Technology Research Center) support program (NIPA-2012-C6150-1101-0002) supervised by the NIPA (National IT Industry Promotion Agency).

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(1) Department of Architectural Engineering, Sejong University, Kunja-Dong, Kwangjin-Gu, Seoul, Korea 143-747

*Corresponding author: An-Seop Choi, E-mail: aschoi@sejong.ac.kr

Yu-Sin Kim (1), An-Seop Choi PhD (1) (*), and Jae-Weon Jeong PhD (1)

doi: 10.1582/LEUKOS.2012.09.01.004
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