Optimal design of Fresnel lens for a reading light system with. multiple light sources using three-layered Hierarchical Genetic.

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1 Optimal desin of Fresnel lens for a readin liht system with multiple liht sources usin three-layered Hierarchical Genetic Alorithm Wen-Gon Chen, and Chii-Maw Uan Department of Electrical Enineerin, I Shou University, Kaohsiun 840, Taiwan, R.O.C. ABSTRACT A typical Fresnel lens is not suitable to a readin liht system with multiple liht sources since it is desined in such a way that each roove is at a slihtly different anle from the next but with the same focal lenth. Therefore, a Fresnel lens with suitably desined roove anles is needed for this kind of liht system. In this paper, a more efficient three-layered Hierarchical Genetic Alorithm (3LHGA) is proposed to find an optimal set of roove anles for a desined Fresnel lens to optimize both the illuminace and uniformity for a readin liht system with multiple liht sources. The roove anles of a desined Fresnel lens are directly derived from a Fresnel lens database by two layers of control enes in the proposed 3LHGA. The proposed 3LHGA not only makes it possible to evolve a lot of roove anles as parametric enes but also further improve the performance of a Fresnel lens and increase the speed of evolution. From the simulation results we can demonstrate that the desined Fresnel lens indeed offers improvement of liht-uidin performance for a multiple-led readin liht system. Keywords: Readin liht system, Liht Emittin Diode (LED), Fresnel lens, Genetic Alorithms (GAs). Hierarchical Genetic Alorithms (HGAs) 1. INTRODUCTION A typical Fresnel lens consists of a series of small, narrow concentric rooves on the surface of a lihtweiht plastic sheet 1 shown in Fi. 1. Its cross section with 330 rooves is shown in Fi. 2. Since it can be made thin, flat, and at low cost, it is ettin more and more popular in many applications such as lihthouses, overhead projectors, file Fresnel lenses, TV projections, condenser systems, automobile headlihts, solar enery, rear projections, passive motion detectors, traffic sins, solar concentrators, collimators, and LED applications 2-5. Take LED applications for example, LED-based readin liht systems may be seen everywhere nowadays. However, these systems are usually equipped with multiple LEDs as multiple liht sources due to the fact that a sinle LED does not provide enouh flux. To readin liht systems with multiple liht sources, a typical Fresnel lens becomes not so effective on both * Wen-Gon Chen. wchen93@yahoo.com.tw. Tel.: Ext. 6675, Fax.:

2 illuminance and uniformity because its every roove is desined as a slihtly different anle from the next but with the same focal lenth. In other words, it neither efficiently uides liht rays from liht sources to the readin surface, nor uniformly distributes them on the readin surface. To deal with the problem iven above, one way is to desin a particular Fresnel lens to fit a liht system with multiple liht sources, i.e. searchin for a special set of roove anles for a Fresnel lens 6,11,12. However, it seems to be not an easy task because the searchin space may become very lare if the number of Fresnel lens s rooves is bi and the rane of a roove s anles is wide. Take a Fresnel lens with 330 rooves as an example, the searchin space will contain possible solutions if each desined roove contains 330 anles ranin from 0.12 derees to derees. Therefore, how to find out a solution as an optimal set of roove anles from such an enormous searchin space is accurately a challenin task. Fi. 1 A typical Fresnel lens Fi. 2. The cross section of a typical Fresnel lens with 330 rooves. In the previous work 12, we proposed a two-layered Hierarchical Genetic Alorithm to solve the problem of an enormous searchin space. The first layer of a chromosome consisted of a series of control enes. It played an important role to reduce the searchin space since the number of its control enes is a lot less than that of desined rooves. The deree of reduction depended on how rooves were translated into sements. As for the second layer, it consisted of a series of parametric enes to keep a set of chosen roove anles for evaluatin a chromosome s fitness in population. Althouh that work indeed made the desin of a lare scale of Fresnel lens used in the multiple-led readin liht system feasible, two improvements are expected to improve the performance of that lens. One is varied-lenth sements instead of oriinal fixed-lenth sements when rooves are translated into sements. To any new Fresnel lens to be desined, it is not necessary to spend much time tryin to find better sementation any more. In fact, findin a better sementation is difficult if there are no efficient ways are applied. The other one is increasin the variation of roove anles sequence in a sement. In previous work, due to the reason that a sement s roove anles are directly picked from a Fresnel lens database accordin to the value of a control ene, they are in ascendin or descendin order sequence. In other words, the roove anles of a sement will be arraned in the desined Fresnel lens with the same sequence as they are oriinally stored in the Fresnel lens database. It is clear that such an arranement of roove anles is in shortae of variation. Lackin variation may result in bad influences not only on the performances of a desined Fresnel lens in illuminance and uniformity but also on the

3 speed of the developed alorithm s converence. To make matters worse, the bier the sement is, the reater the influence will be. To reach two expectations mentioned above, we propose a three-layered Hierarchical Genetic Alorithm (3LHGA) in this paper. By addin a control layer to the previous approach, 2LHGA, the oriinal sements can be divided into varied-lenth sub-sements and the arranements of roove anles in an oriinal sement can also be modified to varied arranements in terms of a cyclin mechanism. In order to compare with the previous work, a simulated readin liht system in this approach is identical to that of the previous work, as well as the performance index. Therefore, it is necessary to re-describe the simulated readin liht system and the performance index in this paper. The remainder of this paper is oranized as follows: Section 2 describes a simulated multiple-led readin liht system in advance. Section 3 defines the performance index used in the proposed 3LHGA. In section 4, introduction to 3LHGA and how to implement 3LHGA are presented. The simulated results are presented in section 5. Finally, we draw our conclusions in section A SIMULATED MULTIPLE-LED READING LIGHT SYSTEM A simulated LED-based readin liht system shown in Fi. 3 consists of three coaxially placed parts: an LED liht source set, a desined Fresnel lens, and a circular readin surface. Each simulated part is constructed at the beinnin of the developed prorams by applyin TRACEPRO 7 macros. Its detailed confiurations are described as follows. Fi. 3. A simulated readin liht system. Fi. 4(a). A cone-frustum shaped Fi. 4(b). The arranement of five LEDs reflector and LED liht source. in liht source set 1) The liht source set owns five identical LEDs and each of which is embedded in a reflector. The reflector has the shape of a frustum of a cone with a view anle of derees. It has an altitude of 9 mm and a circular ceilin with a radius of 3.25 mm. Also, its inner wall is assumed to have the perfect mirror surface property. An LED point liht source is placed at the center of the ceilin of the cone-frustum shaped reflector. The confiuration of the combination of the cone-frustum shaped reflector and LED point liht source is enlared and

4 shown in Fi. 4(a). This confiuration ensures that none of the liht rays emitted from an LED escapes upwards. The arranement of five LEDs in liht source set is symmetrical as shown in Fi. 4(b). 2) A Fresnel lens to be desined is put at the floor of the cone-frustum shaped reflector and used to uide liht rays from the liht sources to the readin surface. It is a piece of lihtweiht plastic sheet with a radius of 165 mm and a heiht of 0.5 mm and located between the LED liht sources and the readin surface. More precisely, its top is about mm from LED liht source and mm from the readin surface. The topside of the plastic sheet will be rooved as rided shapes with a series of N (in table 1) small narrow concentric rins with same width of 0.5 mm. The rided shapes form roove anles they are acted as the desin parameters in this paper. 3) The readin surface is constructed with a thin cylinder with a radius of 495 mm and a heiht of 1 mm and placed below the Fresnel lens with a distance of mm. Its topside is simply desined to have property of perfect absorption. In order to ensure that the liht rays emitted from LED liht sources all pass throuh the desined Fresnel lens, a bier cone-frustum shaped reflector is added to enclose five LED liht sources. The liht source set of five LEDs is placed at the ceilin of this reflector while a desined Fresnel lens is placed at its base. As a whole, the constructed simulated readin liht system leaves a Fresnel lens with a lare set of roove anles to be desined. The most important thin to think about is how to efficiently desin a set of roove anles to let more liht rays emitted from LEDs liht sources arrive at the readin surface and uniformly distribute over the readin surface. Before the desin of such an optimal Fresnel lens, we first define an appropriate performance index as an evolution basis for the developed alorithm in the followin section. 3. PERFORMANCE INDEX Since the illuminance and uniformity are equally important in a readin liht system, we have to take them into account simultaneously durin the desin of a Fresnel len. Hence, althouh the desin oal considered in this paper is to search for a set of variable roove anles to improve the performance of a desined Fresnel lens, the performance index of this approach is basically same as the previous approach 12 because two approaches are developed in the same simulated system. In such a way, it will be more sinificant to compare the simulated results of the two developed approaches. The performance indexes are re-described below. For simplicity, we assume each LED is a point liht source and emits N liht rays uniformly distributed over a 2 steradians of solid anle. Due to the fact that some of the liht rays do not arrive at the readin surface, the illuminance performance of a readin liht system can be measured accordin to the number of liht rays incident to the readin surface. As for the distribution uniformity of liht rays over a readin surface, a ood readin liht system requires that the incident liht rays have a specified distribution over the readin surface. For a circular readin surface, the illuminance is stronest in the center area and is radually reduced in the outward direction. In order to measure the actual illuminance and uniformity of distribution, the circular readin surface is divided into N r equal-area rins and each rin is further divided into N s equal-area sectors, as shown in Fi. 5. Let

5 R rs indicate the number of liht rays incident to the s th sector of the r th rin. Then, the total number of liht rays incident to the readin surface throuh a desined Fresnel lens is iven by R d Nr Ns R r1s 1 rs, and the averaed number of liht rays incident to a sector throuh a desined Fresnel lens is iven by (1) R a Rd. Nr Ns (2) Now, in order to take the illuminance and uniformity into account simultaneously, we define the performance index I to be maximized as follows: where IG L, G( R R ) G, (4) d t w Nr Ns L L r1 s1 rs, Eq. (4) stands for the ain of performance index whereas Eq. (5) stands for the loss of performance index. In Eq. (4), R t denotes the number of incident liht rays to the readin surface throuh a typical Fresnel lens in the readin liht system and G w denotes the ain weiht. The reason of deductin R t from R d is that a typical Fresnel lens is used as an initial condition in this approach. In Eq. (5), L rs denotes the loss of performance index of the s th sector of the r th rin. It is used to quantify the penalty of non-uniformity distribution for a sector that has too few or too many incident liht rays and expressed as follows: (3) (5) R ( a ) L for R rsr w a R L rs rs R ( 1 rs ) L otherwise R w a where L w denotes the loss weiht., (6) Fi. 5. A readin surface with 5 equal-area rins, 120 equal-area sectors Fi. 6 Fixed-lenth division mechanism

6 In this approach, the desin oal is to search for a better set of roove anles for a Fresnel lens to make the performance index better. Therefore, how to improve the desin of a Fresnel lens to reach the desin oal is the point of this approach. The developed 3LHGA is described in the followin section. 4. THE PROPOSED APPROACH Accordin to the desin concept of the previous work 12, the sementation of rooves played an important role in reducin the searchin space. However, the rooves in a Fresnel lens database were semented into fixed-lenth sements there. In addition, the sequence of roove anles in a sement is in the same sequence as they oriinally appeared in a Fresnel lens database. In such a desin, the sementation of rooves and the sequence of roove anles in a sement lack variety and probably restrict the performance of a desined Fresnel lens. To solve the problem, we propose a 3LHGA by addin a control layer to handle the problem of variation. In 3LHGA, the added control layer is acted as the second control layer and used to store the lenths of sub-sements that are divided from a sement. The sement to be divided may be a fixed-lenth sement or a varied-lenth sement. For a fixed-lenth sement, the division is simpler but less variable, as shown in Fi. 6. It is called the fixed-lenth division. For this way of division mechanism, sements divided into sub-sements are all fixed-lenths. On the other hand, for a varied-lenth sement, the division is a bit complicated but more variable, as shown in Fi. 7. It is an idea of composin some fixed-lenth sements into a bier sement first and then decomposin it into some varied-lenth sements. How many fixed-lenth sements are composed and how many varied-lenth sements are decomposed is problem-dependent. For simplicity, three sements are chosen as a composin and decomposin unit. In this way, all sements divided into sub-sements are possibly varied-lenths. We call the latter way the varied-lenth division. For this way of division mechanism, sements divided into sub-sements are possibly all varied-lenth. No matter which way is used, we can reach the oal of makin roove anles more variable.... Fi. 7 Varied-lenth division mechanism (a) (b) (c) Fi. 8 A cyclin loadin mechanism In order to further increase the variation of desined roove anles, we adopt a cyclin loadin mechanism to put roove anles into parametric enes accordin to the values of the second layer s control enes. It is such a way that rooves anles represented by sub-sements are put into parametric enes with cyclin sequence of sub-sements themselves. So, their oriinal sequence of those roove anles in Fresnel lens database can, therefore, be broken. For example, suppose that a sement is divided into three varied-lenth sub-sements, the roove anles represented by that sement will be put into parametric enes by (a) the order of the first sub-sement, the second sub-sement, and the third sub-sement, (b) the order of the second sub-sement, the third sub-sement, and the first

7 sub-sement, (c) the order of the third sub-sement, the first sub-sement, and the second sub-sement dependin on the value of a random number. It is shown in Fi. 8. As for how many sub-sements a sement will be divided, dependin on how bi a sement is. 4.1 The structure of chromosome in 3LHGA The chromosome of 3LHGA consists of three layers, two layers of control enes and on layer of parametric enes. Its structure is shown in Fi. 9. In Fi. 9, the variables S 11 represents the lenth of the first sub-sement, S 12 represents the lenth of the second sub-sement, etc. In the first layer, the control enes are used to represent the sement number and coded as inteers, each ranin from 1 to N se. N se denotes the number of sements and has the number of 33 in this paper. Here, a sement is a unit and includes a small roup of rooves in Fresnel lens database. For instance, a sement contains 10 rooves if N se is set to be 33. The distribution of 330 rooves on 33 sements and their correspondin anles are shown in Table 2. In the second layer, the control enes are used to represent the lenths of sub-sements and also coded as inteers, each ranin from 0 to L sub-se. L sub-se denotes the lenth of a sub-sement and has the number of 10. The lenths of sub-sements are obtained from two kinds of division mechanism described above. If N sub-se =3, the number of control enes in the second control layer is three times of N se. As for the parametric enes in the third layer, they are used to represent the roove anles derived from Fresnel lens database accordin to the values of control enes in the second layer. In order to increase the variety of roove anles sequences in the parametric enes, a described cyclin loadin mechanism is applied. Take N se =33 as an example and assume that fixed-lenth-sements-division is applied, if the value of the first control ene in the first layer is 3 and the values of the first three control enes in the second control layer are 3, 3, and 4, respectively, the ten roove anles in the third sement of Table 2 are extracted and put into parametric enes accordin to previous cyclin loadin mechanism to decide which sub-sement comes first, which one next, and so forth. A special structure of chromosome for this example is shown in Fi. 10. The fiures in Fi. 10 are taken to two decimal places. Table 1. Parameters used in the proposed approach Parameters Population size, N p Number of Groove anles, N Number of liht rays emittedfrom anled, N Number of sementsof 330roove anles, N se Mutation Rates, M r Number of equal-area rins of the readin surface, Nr Number of equal-area sectors of a rin, N s Gain weiht of performance index, G w Lossweiht of performance index, L w Lenth of a sub-sement, L su b-se Number of sub-sement ina sement, Nsub-se Value Table 2. Distributionof 330roove anles on 33 sements for N s e =33 Sement Groo ve an les ,0.33,0.53,0.73,0.94,1.14,1.34,1.55,1.75, ,3.37,3.57,3.78,3.98, 2.16,2.36,2.56,2.76, ,4.38,4.58,4.78,4.99,5.19,5.39,5.59,5.79, ,44.93,45.02,45.10,45.19,45.28,45.37,45.54,45.63, ,45.88,45.97,46.05,46.14, 46.22,46.30,46.39,46.47, ,46.72,46.80,46.88,46.96,47.04,47.13,47.21,47.29,47.37

8 N se Control enes of the first layer *Nse s 11 s 12 s 13 s 21 s 22 s 23 s 31 s 32 s 33 Control enes of the second layer Parametric enes 1... s 11 1 s s s 21 1 s 22 1 s s s 32 1 s Control enes of the first layer Control enes of thesecond layer Parametric enes Fi. 9 The eneral structure of a chromosome in 3LHGA Fi. 10 Part of a special chromosome in 3LHGA for N se = Implementation of 3LHGA In actual implementation of 3LHGA, an initial population with N p chromosomes is enerated first 8-10,12. The values of control enes of the first layer in every chromosome are set from 1 to N se in sequence to let parametric enes start with roove anles as the typical Fresnel lens database. In order to increase the variety of roove anles sequences in a desined Fresnel lens, the roove anles in database are extracted throuh two sement division mechanisms and a cyclin loadin mechanism. In fact, sement division mechanisms are mainly used to create the control enes of the second layer. It is easily known that different division ways produce different varieties of roove anles sequence in parametric enes. No matter which one is chosen, it can help to create roove anles with more variety. The cyclin mechanism for creatin parametric enes is also havin similar effect as sement division mechanisms. Those effects can be verified by observin every chromosome s fitness in practical implementation. After eneration of an initial population, fitness values of chromosomes are computed accordin to performance index defined in Eq. (3). Then, all chromosomes are sorted accordin to their fitness values. The sorted chromosomes C i and C i+ 1 have the property F(C i ) F(C i+1 ) for i=1,2,,n p, where F(C i ) represents the fitness value of chromosome C i. With the initial sorted chromosomes, the offsprin chromosomes are enerated throuh first mutation operation and then crossover operation eneration by eneration. The mutation operation with a mutation rate M r is only performed on the first layer s control enes of each chromosome. In a chromosome C i, let ij denote the j th control ene of the first layer and ik, i(k+1), and i(k+2) denote the k th, (k+1) th, and (k+2) th control enes of the second layer, where k=(j-1)*3+1. For the control ene ij, we enerate a random numbers r. If r is less than M r, then the control ene i,j is mutated to a new one called ij which has a value between 1 and N se, otherwise control ene is not mutated, i.e., ij = ij. If the control ene ij is mutated to ij, the correspondin three control enes of the second layer in a chromosome are updated with three new numbers to represent three new sub-sements, i.e., ik is

9 replaced with ik, i(k+1) is replaced with i ( k+1), and i(k+2) is replaced with i ( k+2). i k, i ( k+1), and i ( k+2) represent the lenths of newly divided sub-sements. Otherwise, ik, i(k+1) and i(k+2) are not chaned, i.e., ik = ik, i ( k+1) = i(k+1), and i ( k+2) = i(k+2). A mutated chromosome Ci is formed from the first layer s control enes ij, j=1, 2, 3,, N se, the second layer s control enes ik, i( k 1) and i( k 2), k=(j-1)*3+1, and the parametric enes throuh the cyclin loadin mechanism. The chromosome C i is replaced by the chromosome Ci if the mutated chromosome Ci has a hiher fitness value than C i. Havin performed the mutation operations on all chromosomes of the current eneration, we make crossover operation to further improve the fitness of chromosomes of the population. The crossover operation is also merely performed on control enes for all chromosome pairs (C i, C i+np/2 ), i=1,2,3,,n p /2. For a iven chromosome pair (C i, C j ), we enerate three different random numbers r 1, r 2, and r 3 over the interval (0,1) and obtain two inteers k Nser 1 1 and k N r 2 se 2, which represent the crossover points and lie in the rane [2,N se -1]. Assume that k 1 is less than k 2. In the chromosome C i, let C i(a:b) denote a control ene section from the a th control ene to the b th control ene in the first layer and C i(c:d) denote a control ene section from the c th control ene to the d th control ene in the second layer. Then, the first layer s control enes of offsprin chromosomes enerated by crossover are iven by i( 1: N, seq) i( : k1 ) j( k1: k2 ) i( k2: Nse ) C C C C (7) and j( 1: N ) ( : ) ( : ) ( : ). se jk 1 i k1 k2 j k2 Nse C C C C (8) Let x and y denote the beinnin and end position of the second layer s control enes in a chromosome, respectively, i.e., x=n se +1, y=n se +3*N se. The second layer s control enes of offsprin chromosomes enerated by crossover are iven by and i( x: y) i( x: x+ 3*( k - ) ) ( *( ): * ( * : ), 1 11 j x3 k 11 x3 k 21 i x3 k 2 y C C C C (9) j( x: y) j( x: x+ 3*( k1-1) 1 ) i( x3 *( k 11 ): x3 * k 21 j( x3 * k 2: y) C C C C (10) if r 3 is less than or equal to 0.5, otherwise by

10 i( 1: N ) ( : ) ( : ) ( : ), seq jk 1 i k1 k2 j k Nse 2 C C C C (11) j( 1: N ) ( : ) ( : ) ( : ), se ik 1 j k1 k2 i k2 Nse C C C C (12) i( x: y) j( x: x+ 3*( k - ) ) ( 3*( ): 3* ( 3* 2: ), 1 11 i x k 11 x k 21 j x k y C C C C (13) and j( x: y) i( x: x+ 3*( k - ) ) ( *( ): * ( * : ) j x3 k 11 x3 k 21 i x3 k 2 y C C C C (14) If varied-lenth division mechanism is adopted, the neihborin control enes of k 1 and k 2 are needed to divided aain. Finally, the parametric enes of Ci and C j are obtained throuh the cyclin loadin mechanism. The control enes of the parent chromosome C i, are replaced by the control enes of the offsprin chromosome C, if i C i has a hiher fitness value than C i. C j and C j follow the same replacement policy as C i and C i. After finishin the crossover operations, we obtain a new eneration of chromosomes, which have hiher fitness values than the previous eneration. By iteratin the above iterations, an evolution keeps oin on until a specified number of enerations have been performed. Finally, the detail steps of our proposed approach are drawn as a flow diaram shown in Fi. 11 and the simulation results will be described and discussed in the follow section. Start Fresnel lens Create a simulated readin liht system database Initialize a population with N p chromosomes : 1. initiallize control enes of the first layer Cyclin loadin 2. intitialize control enes of the second layer Sementsdivision mechanism usin sement-division mechanism for 3. initialize perametric enes mechanism parametric enes usin cyclin enerator for parametric enes Ray - tracin & Liht rays collectin Fitness evaluation and sortin Specified enerations? Mutate control enes of the first layer Do redivision and enerate new parametric enes Ray - tracin & Liht rays collectin Fitness evaluation and sortin Crossover control enes of the first layer Do redivision and enerate ne parametric enes for crossover point ene Ray - tracin & Liht rays collectin Fitness evaluation and sortin Stop Fi. 11 A block diaram of 3LHGA Fi. 12 Converence statuses for FL 3LHGA_33_FLSD and FL 2LHGA_33_FLSD.

11 5. SIMULATION RESULTS We have proposed a three-layered Hierarchical Genetic Alorithm to increase the variation of rooves anles in a desined Fresnel lens. There are two kinds of sement divisions to create different variations. They have been described before and shown in Fi. 6 and 7, respectively. The division way in Fi. 6 is called fixed-lenth division (FLD) and the other one is called varied-lenth division (VLD). No matter which division, the size of a sement plays an important role to influence the performance of a desined Fresnel lens finally. However, it belons to a NP-hard decision problem. In the sequel, the desined Fresnel lens is desinated as FL 3LHGA_33_FLD if it is desined by the approach of 3LHGA with a sement size of 33 and the division way of FLD, FL 3LHGA_33_VLD if same approach and sement size but the division way is chaned by VLD, FL 2LHGA_33_FLD if 3LHGA is chaned by 2LHGA. There are two experiments presented below and the parameters used in this approach are listed in Table 1. For the first experiment, we basically intend to present the performance of 3LHGA compared to 2LHGA for the readin liht system simulated in section 2. For simplicity, we just take N se = 33 as an example and use fixed-lenth sement division. The experimental results are shown in Fi. 12, 13, 14. From fi. 12, it is clear that 3LHGA converes faster and has better fitness within a lon period of evolutions. Since fitness is evaluated by considerin both illuminance and uniformity, better fitness means better performance. Compare Fi. 13 to Fi. 14, it is more obvious that 3LHGA indeed works better than 2LHGA since more liht rays are directed into the readin surface. They are 4738 and 4631 shown at the footnotes in Fi. 13 and Fi 14, respectively. It is obvious that the illumination of readin liht system with 3LHGA has increased by 2.31% as compared with that of the readin liht system with 2LHGA. The cross section of FL 3LHGA_33_FLSD is shown in Fi. 15. Fi. 13 An irradiance map of simulated readin liht system Fi. 14 An irradiance map of simulated readin liht system with FL 3LHGA_33_FLD. with FL 2LHGA_33_FLD.

12 Fi. 15 The cross section of FL 3LHGA_33_FLD. Fi. 16 Converences for FL 3LHGA_33_FLD and FL 3LHGA_33_VLD. For the second experiment, the comparison of results between FL 3LHGA_33_FLD and FL 3LHGA_33_VLD is shown in Fi. 16. The blue line denotes the converence status of FL 3LHGA_33_FLD and the black line denotes the status of FL 3LHGA_33_VLD. From their converence trends, it is clear that FL 3LHGA_33_VLD will create a better solution than FL 3LHGA_33_FLD if sufficient evolutions are performed. It stands to reason that the converence speed of FL 3LHGA_33_VLD is slower than FL 3LHGA_33_FLD since the varieties of roove anles by FL 3LHGA_33_VLD is reater than FL 3LHGA_33_FLD. Sufferin from the time-consumin of evolvin the developed alorithms, they are just evolved 1600 enerations. Therefore, how to improve the converence speed caused by the complicated structure and alorithm of 3LHGA with varied-lenth division mechanism is worth bein researched in the future. 6. CONCLUSIONS We have presented a three-layered HGA-based approach to improve a Fresnel lens for a readin liht system with multiple liht sources. By usin the scheme of three-layered HGA, we not only solve the complexity problem of the hue searchin space formed by a lot of rooves, but also increase the varieties of roove anles in a desined Fresnel lens to improve the performance compared to a two-layered HGA-based approach. However, some desined parameters are defined as default values such as the number of sub-sements and the number of composin factor when varied-lenth division mechanism is applied. The reasons that they are all set to be three is just for simplicity in proram codin. Since these numbers play an important role in this approach, it will be better if a more efficient or expectable way to decide those numbers in the future work. ACKNOWLEDGMENT This work was supported in part by I-Shou University under contract ISU , and in part by the National Science Council under rant NSC E

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