A two-layered Hierarchical Genetic Algorithm (HGA) was proposed in a previous paper to solve the design problem of a large scale Fresnel lens used in a multiple-source lighting system. The research objective of this paper is to extend the previous work by utilizing a three-layered HGA. The goal of the suggested approach is to decrease the reliance on deciding the number of groove segments for the designed Fresnel lenses, as well as to increase the variety of groove angles in a segment to improve the performance of the designed Fresnel lens. The proposed algorithm will be applied on a simulated reading light system, and the simulation results demonstrate that the proposed approach not only makes the design of a large scale Fresnel lens more feasible but also works better than the previous one in both illuminance and uniformity for a simulated reading light system.
© 2007 Optical Society of America
A conventional Fresnel lens is not suitable to a lighting system with multiple light sources since it is designed in such a way that each groove is at a slightly different angle from the next but with the same focal length . Contrarily, due to its features of directing and collecting light rays, as well as its properties of being made essentially flat, of plastic, light weight, and cost efficient [2, 3], it is essential to design an irregular Fresnel lens to match multiple light sources as simultaneously as possible to improve the performance in both illuminance and uniformity for a multiple-source lighting system. There are many researchers [4–8] dealing with the light behavior and interactions among multiple-chips LED. In [4–6], they based on the mid-field concept to model the optical behavior of LEDs, which is not only useful in LED lighting behavior study but also useful in package design. While in [7–8], they dealt with the multiple interactions of light rays on one complex surface. It thus built a protocol for defining where prescribed rays were sent. However, it is still difficult to design such an irregular Fresnel lens that can directly and efficiently achieve the performance of both illuminance and uniformity in an LED-based lighting system.
Unlike a conventional one, an irregular Fresnel lens generally owns a set of groove angles with irregular order to fit different light sources. In fact, designing such a Fresnel lens is a task to search for a set of irregular groove angles to fit multiple light sources. In our earlier works, we proposed a conventional Genetic Algorithm-based approach [9–12] to design an irregular Fresnel lens by combining an optical software package  to construct a simulated optical environment and by evolving the light rays they were from all light sources and through the designed Fresnel lens. The simulated results demonstrated that the designed Fresnel lens indeed offered better performance in illuminance and uniformity than a conventional Fresnel lens. However, with the grow of a Fresnel lens’s scale, it is getting harder to design such an irregular Fresnel lens to match multiple light sources by using a conventional GA. This is the reason that the solution space formed by grooves and a wide range of degrees for each groove will also be getting larger, especially as the number of grooves increases. In our design opinion, a Fresnel lens is classified as small scale if its radius is less than 5 centimeters and its groove width is 0.5 millimeters, i.e. the number of its grooves is less than 100. It is classified as medium scale if its radius is greater than 5 centimeters but less than 15 centimeters, and as a large scale if its radius is greater than 15 centimeters. For a medium or large scale lighting system, a conventional GA will encounter the problem of a so called “NP-complete problem” due to the enormous solution space.
To solve the design problem caused by the enormous solution space, we proposed a two-layered Hierarchical Genetic Algorithm (HGA) called HGA2L . In that work, the idea was based on the phenomenon that a conventional Fresnel lens with a large amount of grooves is designed in such a way that each angle is slightly different from the next. In such a situation, it is reasonable to bind several grooves as a unit called a segment. The segment size refers to the number of grooves a segment contains. By binding the grooves into a segment and letting segment numbers be evolutionary parameters, the solution space formed by segments will be smaller than the solution space directly formed by the original grooves and their angles. Therefore, the problem of the developed algorithm’s convergence speed caused by the enormous search space can be simplified. Although the simulated results demonstrated that the proposed HGA2L was indeed superior to a conventional GA, two improvements can be made to improve the performance of HGA2L: a. to efficiently decide a suitable segment size, and b. to increase the variety of groove angles in a segment. The segment size should be different in terms of different scale of Fresnel lenses, and should not be fixed as in the proposed HGA2L. A suitable segment size to a specified Fresnel lens won’t deteriorate the convergence speed. On the contrary, it will improve the performance of the designed Fresnel lens. As for the variety of groove angles in a segment, it also plays an important part on the performance of the designed Fresnel lens. In the proposed HGA2L, the groove angles in a segment were directly derived from a conventional Fresnel lens database in order to simplify future fabrication of the designed Fresnel lens. In that case, they have the same sequence as those in a conventional Fresnel lens and lack variety. Empirically, lacking variety will have a negative influence on the performance of the designed Fresnel lens. Moreover, the bigger the segment size, the more serious the lack in variety.
In this paper, a three-layered HGA, called HGA3L, is proposed to improve the previously proposed HGA2L. Actually, when a designed Fresnel lens’s scale is getting larger, a fixed-length segment will not make sense anymore since it will cause opposite effect if an unsuitable segment size is applied. However, for a newly designed large-scale Fresnel lens, deciding on a suitable segment size is time-consuming and virtually impossible. In addition to the segment size, the variety of groove angles in a segment, i.e., the sequences of their degrees, also plays an important role to influence the performance of a designed Fresnel lens and the convergence speed of the related algorithm. As mentioned above, if a segment size is bigger, the variety of groove angles in that segment becomes increasingly necessary. Hence, the question of deciding on a suitable segment size and of increasing the variety of groove angles to improve the performance of a designed large scale Fresnel lens - without deteriorating the convergence speed of a developed algorithm - is a key issue in this paper.
The remainder of this paper is organized as follows: Section 2 describes the proposed three-layered HGA. In section 3, a reading light system is introduced as a design example. The simulated results are presented in section 4. Finally, we draw our conclusions in section 5.
2. The proposed three-layered HGA
To decide a suitable segment size and further increase the variety of the groove angles to improve the performance of a designed Fresnel lens with a large scale, the authors put forward an idea of creating varied-length sub-segments and using a cycled loading mechanism for those sub-segments. The varied-length sub-segments construct a new control layer for a chromosome structure: the second control layer. The control layer representing segments in HGA2L becomes the first control layer in the proposed HGA3L, and the parametric layer representing groove angles plays the same role in both HGA3L and HGA2L. As a whole, these layers form a three-layered structure of a chromosome in the proposed HGA3L. As for the problem of deciding on a suitable segment size, we roughly start with a set of fixed-length segments and store them in the first control layer. During the evolution, every segment is then divided into varied-length sub-segments. These sub-segments can, therefore, be considered to be a set of suitably-sized segments as if they are directly derived from the original conventional Fresnel lens database. By segmenting grooves in this way, not only is it unnecessary to spend a considerable time in establishing the suitable segment size for a newly designed large scale Fresnel lens, but also a suitable set of varied-length sub-segments can be evolved to improve the designed Fresnel lens. As for the variety of the designed groove angles, it can be increased by using a cycled loading mechanism on each varied-length sub-segment to change a segment’s original ascending sequences of groove angles extracted from a conventional Fresnel lens database.
Varied-length sub-segments come from a fixed-length segment or a varied-length segment. Their lengths are stored in the genes of the second control layer. For the case from a fixed-length segment, they are obtained by randomly dividing a fixed-length segment into several varied-length sub-segments. This is known as the fixed-length segment division mechanism (FLSDM), shown in Fig. 1. Let Nsub_seg denote the number of the divided varied-length sub-segments, in the second control layer, the genes that are activated by the ith gene of the first control layer are the ((i-1)×Nsub_seg+1)th, ((i-1)×Nsub_seg+2)th,…, and ((i-1)×Nsub_seg+Nsub_seg)th genes as shown in Fig. 2, where Nseg denotes the number of total segments of the designed grooves. Let FSseg denote the size of a fixed-length segment, i.e., the number of the grooves a fixed-length segment contains, and Ssub_seg denote the size of a sub-segment, then, 1≦Ssub_seg≦FSseg-Nsub_seg+1. This division mechanism is simpler but the designed groove angles are less variable.
As for the other case that variable-length sub-segments come from a variable-length segment, we propose that several fixed-length segments are first composed into a bigger segment and that this bigger segment is then decomposed into some varied-length segments. The goal is to further increase the variety of the designed groove angles. Usually, the number of the composed fixed-length segments is the same as that of the decomposed varied-length segments. Every varied-length segment is then divided into several varied-length sub-segments as the FLSDM does. It is called the varied-length segment division mechanism (VLSDM) as shown in Fig. 3. In such a case, the size of a varied-length sub-segment has the relation 1≦Ssub-seg≦VSseg-Nsub_seg+1, where VSseg denotes the size of a varied-length segment and has the relation of Nsub_seg≦VSseg≦Sc×FSseg-(Sd-1)×Nsub_seg where Sc represents the number of fixed-length segments that are composed into a bigger segment, and Sd represents the number of varied-length segments that a bigger segment is decomposed into. In the second control layer, the corresponded genes by the ith gene in the first control layer are the (Bg+1)th, (Bg+2)th,…, and (Bg+Nsub_seg)th genes where Bg=Σi-1j=1VSjseg for i=2,3,…,Nseg but Bg=0 if i=1. This is shown in Fig. 4. For this segment division mechanism, the amount of fixed-length segments a bigger segment should be composed of as well as the number of varied-length segments a bigger segment should be decomposed into are problem-dependent and difficult to be decided on. For the sake of simplicity, both Sc and Sd are set to three in the proposed HGA3L. Although this division mechanism is more complicated, the designed groove angles are expectedly more variable.
Irrespective of which segment division is applied, the size of a segment or a sub-segment plays an important role in influencing the performance of a designed Fresnel lens. Unfortunately, it is hard to be decided on.
In order to further increase the variety of the designed groove angles, we adopt a cycled loading mechanism to put groove angles into the parametric genes. It is such a way that, a segment of groove angles represented by a gene in the first control layer are extracted from a Fresnel lens database and put randomly into parametric genes according to a cycled sequence of sub-segments of groove angles. These sub-segments of groove angles are represented by that gene’s corresponding genes in the second control layer. To describe this mechanism in more detail, let Rcl represent a random number, 0 ≤ Rcl ≤ 1, Gfst represent the groove angles in the first sub-segment, Gsnd in the second sub-segment, Gtrd in the third sub-segment, and Glst in the last sub-segment. The groove angles represented by the corresponded genes in the second control layer are put into the parametric genes according to the following sequences in terms of the value of Rcl: Gfst, Gsnd, Gtrd, …, and Glst ; or Gsnd, Gtrd, …, Glst, and Gfst ; or Gtrd, …, Glst, Gfst, and Gsnd, etc. For example, suppose a fixed-length segment is divided into three varied-length sub-segments, the groove angles in the parametric genes will be one of the following three sequences according a random number, Rcl. First, they come from the first sub-segment, the second sub-segment, and the third sub-segment if 0≤Rcl‹0.33, as shown in part (a) of Fig. 5. Second, they com from the second sub-segment, the third sub-segment, and the first sub-segment if 0.33≤Rcl‹0.66, as the part (b) of Fig. 5. Third, they come from the third sub-segment, the first sub-segment, and the second sub-segment if 0.66≤Rcl‹1, represented in (c) of Fig. 5. With this loading mechanism, the original ascending sequence of the groove angles in a segment can be changed and the variety of the groove angles in the designed Fresnel lens can be increased.
2.1 The structure of a chromosome in HGA3L
As mentioned above, the chromosome of HGA3L consists of three layers, two layers of control genes and one layer of parametric genes. Its general structure for the case of FLSDM and Nsub_seg=3 is shown in Fig. 6.
In the first layer, the control genes are coded as integers, each ranging from 1 to Nga/FSseg, i.e., 1≤fi≤Nga/FSseg, i=1,2,3,…,Nseg, where fi in Fig. 6 denotes the value of the ith control gene, Nga denotes the number of the total groove angles of a conventional Fresnel lens database. Furthermore, let Ntg denote the number of the total grooves of a designed Fresnel lens, Nseg is equal to Ntg/FSseg. Take Nseg=33 as an example, the distribution of the Nga groove angles on 33 segments in ascending sequence is shown in Table 1.
In the second layer, the control genes are used to represent the lengths of sub-segments and also coded as integers. The lengths of sub-segments are obtained from two kinds of segment division mechanisms described above. Hence, their values are between 1 and FSseg-Nsub_seg+1 if the FLSDM is applied. Otherwise, they are between 1 and VSseg-Nsub_seg+1. In Fig. 6, the variable S11 represents the size of the first sub-segment, and the variable S12 represents the size of the second sub-segment, etc.
The parametric genes in the third layer are used to represent the groove angles derived from a conventional Fresnel lens database according to the values of control genes in the first and second layer and the mentioned cycled loading mechanism. The groove angles of every fixed-length segment with size FSseg are extracted from that database, ranging from the ((fi-1)×FSseg+1)th angle to the (fi×FSseg)th angle if a FLSDM is applied. On the contrary, if a VLSDM is applied and Sc=3, the FSseg×Sc groove angles are extracted from that database, first ranging from the ((fi-1-1)×FSseg+1)th angle to the (fi-1×FSseg)th angle, second from the ((fi-1)×FSseg+1)th angle to the (fi×FSseg)th angle, and third from the ((fi+1-1)×FSseg+1)th angle to the (fi+1×FSseg)th angle, respectively for i=2,3,..,Nseg-1. For the situation that i=1 or i=Nseg, let i be i+1 if i=1 or i-1 if i=Nseg in advance. After being extracted, they are composed and decomposed, and then put into the parametric genes in terms of the gene values of the second control layer as well as the mentioned cycled loading mechanism. Given a special example with Nseg=33 and Nsub_seg=3 where the FLSDM is applied, if the value of the first control gene in the first layer is 3 and the values of the first three control genes in the second control layer are 3, 3, and 4, respectively, then those ten groove angles in the third segment of Table 1 are extracted from the mentioned database and put into parametric genes sub-segment by sub-segment according to the previously mentioned cycled loading mechanism. Part of a special chromosome for this example is shown in Fig. 7. The figures in Fig. 7 are taken to two decimal places.
2.2 Implementation of HGA3L
At the initial stage of the proposed HGA3L, an initial population with Np chromosomes is generated. First, the values of the control genes of the first layer in every chromosome are set from 1 to Nga/FSseg in sequence to let the parametric genes start with the groove angles as approximately identical as those in a conventional Fresnel lens, irrespective of which segment division mechanism is applied. Then, the segment division mechanisms and a cycled loading mechanism are applied to create the control genes of the second layer and the parametric genes of the third layer, respectively. As mentioned above, the design goal of those mechanisms is to increase the variety of the designed groove angles to improve the performance of the designed Fresnel lens. After generating the initial population, every chromosome is evaluated for its fitness according to the performance index defined in section 3. Then, all chromosomes are sorted according to their fitness values. The sorted chromosomes Ci and Ci+ 1 have the property F(Ci)▫F(Ci+1) for i=1,2,…,Np, F(Ci) represents the fitness value of chromosome Ci.
With the initial sorted chromosomes, the offspring chromosomes are generated through first a mutation operation and then a crossover operation generation by generation. Since the genes in the second control layer are created by the genes in the first control layer and the parametric genes are created by the genes in the second control layer, the mutation operation is only performed on the first layer’s control genes of each chromosome. In a chromosome Ci, let gij denote the jth control gene of the first layer for j=1,2,3,…, Nseg and gik, gi(k+1), and gi(k+2) denote the kth, (k+1)th, and (k+2)th control genes of the second layer, where k=(j-1)×Nsub_seg+1. For the control gene gij, we generate a random number r. If r is less than the defined mutation rate, Mr, then the control gene gi,j is mutated to a new one called g′ij which has a value between 1 and Nga/FSseg, otherwise the control gene is not mutated, i.e., g′=gij. If the control gene gij is mutated to g′ij, the corresponding three control genes in the second layer are updated with three new values to represent three new sub-segments, i.e., gik is replaced with g′, gi(k+1) is replaced with g′i (k+1), and gi(k+2) is replaced with g′i(k+2). g′ik, g′i(k+1), and g′i(k+2) represent the lengths of the newly divided sub-segments. Otherwise, gik, gi(k+1) and gi(k+2) are not changed, i.e.,g′ik=gik, g′i(k+1)=gi(k+1), and g′i(k+2)=gi(k+2). A mutated chromosome C′i is therefore formed from the first layer’s control genes g′ij, j=1, 2, 3,…, Nseg, the second layer’s control genes g′ik, g′f and g′i(k+2), k=(j-1)×Nsub_seg+1, and the original and newly created parametric genes. The chromosome Ci, is replaced by the chromosome C′i, if the mutated chromosome C′i has a higher fitness value than Ci. Having performed the mutation operations on all chromosomes of the current generation, we make crossover operation to further improve the fitness of the chromosomes. The crossover operation is also merely performed on the control genes of all chromosome pairs (Ci, Ci+Np/2), i=1,2,3,…,Np/2. For a given chromosome pair (Ci, Cj), we generate three different random numbers r1, r2, and r3 over the interval (0,1) and obtain two integers 1 seg k1=[Nseg×r1] and k2=[Nseg×r2], which represent the crossover points and lie in the range [2,Nseg-1]. Assume that k1 is less than k2. For the chromosome Ci, let Ci(a:b) denote a control gene section from the ath control gene to the bth control gene in the first or second layer. Then, the first layer’s control genes of offspring chromosomes generated by crossover are given by
Let x and y denote the beginning and end position of the second layer’s control genes in a chromosome, respectively, i.e., x=Nseg+1, y=Nseg+Nsub_seg×Nseg. The second layer’s control genes of offspring chromosomes generated by crossover are given by
if r3 is less than or equal to 0.5, otherwise by
If the VLSDM is adopted, the neighboring control genes of k1 and k2 in the first layer need to be composed and decomposed again to update the corresponding control genes in the second layer. Finally, the newly created parametric genes of C′i and C′j are obtained through the mated control genes in the first and second layers and the cycled loading mechanism. The control genes of the parent chromosome Ci are replaced by the control genes of the offspring chromosome C′i if C′i has a higher fitness value than Ci. Cj and C′j follow the same replacement policy as Ci and C′i. After finishing the crossover operations, we obtain a new generation of chromosomes with increased fitness values compared to the previous generation. This evolution continues until a specified number of generations have been performed.
We use TracePro macro language  to implement the proposed HGA3L. By means of this language, we can combine the related TracePro macros into the developed algorithm to construct a simulated reading light system, to trace and collect light rays, to process the input and output of light rays, and to transform the parameters evolved in HGA3L into the groove angles of the designed Fresnel lens, etc. Finally, the main procedures of the proposed HGA3L are illustrated by a flow diagram shown in Fig. 8. The simulation results will be described and discussed in the following section.
3. A design example
In order to significantly compare the performance of the proposed HGA3L with that of the previously proposed HGA2L, it is necessary to construct an identical simulated environment as well as the same performance index for these two approaches. Hence, a simulated multiple-LED reading light system and the performance index defined in the previous paper  are summarized and re-described for reference as follows.
3.1 A Simulated Multiple-LED Reading Light System
A simulated multiple-LED reading light system shown in Fig. 9 consists of three coaxially placed parts: an LED light source set, a designed Fresnel lens, and a circular reading surface. Each simulated part is constructed at the beginning of the developed program by applying TracePro macros. Its detailed configurations are described as follows.
1). The light source set owns five identical LEDs, each of which is embedded in a reflector. The reflector has the shape of a frustum of a cone with a view angle of 114.62 degrees. It has an altitude of 9 mm, a circular ceiling with a radius of 6.5 mm, and a circular base with a radius of 12 mm. Its inner wall is assumed to have a perfect mirror surface property. An LED point light source is placed at the center of the ceiling of the cone-frustum shaped reflector. The configuration of the combination of the cone-frustum shaped reflector and an LED point light source is enlarged and shown in Fig. 10. This configuration ensures that none of the light rays emitted from an LED escapes upwards. The arrangement of five LEDs in the light source set is symmetrical.
2). A Fresnel lens to be designed is put at the floor of the cone-frustum shaped reflector and used to guide the light rays from the light sources to the reading surface. It is a piece of lightweight plastic sheet with a radius of 165 mm and a height of 0.5 mm, located between the LED light sources and the reading surface. More precisely, its top is about 107.25 mm from LED light sources and 761.5 mm from the reading surface.
3). The reading surface is constructed with a thin cylinder with a radius of 495 mm and a height of 1 mm and placed below the Fresnel lens with a distance of 761.5 mm. Its topside is simply designed to have the property of perfect absorption.
In order to ensure that the light rays emitted from the LED light sources all pass through the designed Fresnel lens, a bigger cone-frustum shaped reflector is added to enclose five LED light sources. The light source set of five LEDs is placed on the ceiling of this reflector while a designed Fresnel lens is placed at its base.
3.2 Performance index
For a reading light system, the illuminance and uniformity are two equally important factors to be considered. Hence, to design a Fresnel lens for a multiple-source lighting system, it is taken for granted that they are chosen as two important performance indices. For the sake of simplicity, we assume that each LED is a point light source and emits N light rays uniformly distributed over a 2π steradians of solid angle. Due to the fact that some of the light rays do not arrive at the reading surface, the illuminance performance of a reading light system can be measured according to the number of light rays incident to the reading surface. As for the distribution uniformity of light rays over a reading surface, a good reading light system requires that the incident light rays have a specified distribution over the reading surface. For a circular reading surface, the illuminance is strongest in the center area and is gradually reduced in an outward direction.
In order to measure the actual illuminance and uniformity of a reading light system, the circular reading surface is divided into Nr equal-area rings and each ring is further divided into Ns equal-area sectors, as shown in Fig. 11. Let Rrs indicate the number of light rays incident to the sth sector of the rth ring. Then, the total number of light rays incident to the reading surface through a designed Fresnel lens is given by
and the average number of light rays incident to a sector is given by
Now, in order to take the illuminance and uniformity into account simultaneously, we define the performance index I to be maximized as follows:
Equation (12) stands for the gain of the performance index whereas Eq. (13) stands for the loss of the performance index. In Eq. (12), Rc denotes the total number of light rays incident to the reading surface through a conventional Fresnel lens in the same reading light system, while Gw denotes the gain weight. The reason for deducting Rc from Rd is to let a conventional Fresnel lens be an initial condition in this paper. In Eq. (13), Lrs denotes the loss of the performance index of the sth sector in the rth ring. It is used to quantify the penalty of non-uniformity distribution for a sector that has too few or too many incident light rays and expressed as follows:
where Lw denotes the loss weight.
4. Simulation Results
Based on the simulated reading light system and the performance index described in section 3, we present two experiments to show the improvement of the designed Fresnel lenses by the proposed HGA3L. Table 2 shows the parameters used in this work. In the sequel, PAHGA3L_33_FLSD represents the proposed approach HGA3L by FLSDM with Nseg=33, PAHGA3L_33_VLSD represents the proposed approach HGA3L by VLSDM with Nseg=33, and PAHGA2L_33 represents the proposed approach HGA2L with Nseg=33. Moreover, the designed Fresnel lens is designated as FLHGA3L_33_FLSD if it is designed by PAHGA3L_33_FLSD, as FLHGA3L_33_VLSD if it is designed by PAHGA3L_33_VLSD, or as FLHGA2L_33 if it is designed by PAHGA2L_33.
In the first experiment, we intend to present the superiority of the proposed HGA3L over the previous HGA2L on the design of a large scale Fresnel lens for the reading light system simulated in section 3. For the sake of simplicity, we take Nseg=33 and choose the FLSDM. The experimental results are shown in Figs. 12, 13, and 14. From Fig. 12, it is clear that PAHGA3L_33_FLSD converges faster and has better performance than PAHGA2L_33 within a long period of evolutions. Since every chromosome’s fitness evaluation is based on the consideration of both illuminance and uniformity simultaneously, better fitness means better performance. The purpose of double runs of PAHGA3L_33_FLSD (one is represented by a real line and the other by a dotted line) is to further demonstrate the reliability of the proposed HGA3L algorithm.
Compare Fig. 13 to Fig. 14 for the illumination of the simulated reading light system, it is obvious that PAHGA3L_33_FLSD indeed works better than PAHGA2L_33 since more light rays are directed into the reading surface by the former. The fact is reflected from the 4738 and 4631incident rays shown at the bottom footnotes of Fig. 13 and Fig. 14, respectively. They represent the numbers of light rays incident to the reading surface of a reading light system through the designed Fresnel lenses by PAHGA3L_33_FLSD and PAHGA2L_33, respectively. It is clear that the illumination of the reading light system by PAHGA3L_33_FLSD has increased by 2.31% as compared to that of the reading light system by PAHGA2L_33. Although the increasing percentage seems not prominent, it is worth mentioning because the rate of 4631 incident light rays out of 5000 total emitted light rays is high enough and more expected improvements are not easily reached.
As for the uniformity of the reading light system, it is also obvious that there is a conspicuous difference between the two irradiance maps shown in Fig. 13 and Fig 14. In Fig. 14, more light rays are concentrated toward the center of the reading surface. Except for recognizing the uniformity from the irradiance maps, we can further quantify the uniformity by calculating the number of the light-rays incident into each sector of the reading surface shown in Fig. 11 for PAHGA3L_33_FLSD and PAHGA2L_33 respectively. Those numbers in all sectors are tabulated into Table 3 and 4. In comparing Tables 3 and 4, we find that even though more light rays are directed to the reading surface by PAHGA3L_33_FLSD, they don’t concentrate toward the center of the reading surface in a large proportion. On the contrary, they are effectively directed more outwards. The phenomenon of uniformity increase will be more obvious by presenting the percentage of the light-rays incident to each ring of the reading surface over the total light-rays incident into the reading surface. For instance, in the reading surface, the outmost ring by PAHGA3L_33_FLSD has a higher percentage than the same ring by PAHGA2L_33. On the other hand, the innermost ring by PAHGA3L_33_FLSD has a lower percentage than the same ring by PAHGA2L_33, shown in the last column of Table 3 and 4. Finally, we show the cross-section of the designed Fresnel lens, FLHGA3L_33_FLSD, in Fig. 15(a) and the compared cross-section of the designed Fresnel lens by PAHGA2L_33 in Fig. 15(b).
r : ring, s : sector, sub : subtotal, per : percent
ring, s : sector, sub : subtotal, per : percent
In the second experiment, the main goal is to demonstrate the idea that PAHGA3L_33_VLSD works better but converges slower than PAHGA3L_33_FLSD because the variety of the groove angles generated by VLSDM is greater than that by FLSDM. The result is shown in Fig. 16. The blue line with the symbol PAHGA3L_33_FLSD above it denotes the performance and convergence status of PAHGA3L_33_FLSD. The black line with the symbol PAHGA3L_33_VLSD below it denotes the performance and convergence status of PAHGA3L_33_VLSD. It is obvious that PAHGA3L_33_VLSD surpasses PAHGA3L_33_FLSD in performance about up to 2400 generations. An irradiance map through FLHGA3L_33_VLSD up to 2400 generations is shown in Fig. 17. There are 4751 incident rays on the map. Moreover, PAHGA3L_33_VLSD is also far superior to PAHGA2L_33. The superiority is presented in Fig. 16 by an additional blue line with the symbol PAHGA2L_33 below it.
We have presented a three-layered HGA and provided a model on how to improve the performance. Through the segment division and cycled loading mechanisms, not only do the varieties of the designed groove angles increase, but also the problem of searching for a suitable size of segment for a large set of grooves can be solved. However, the size of a segment or a sub-segment plays an important role in the performance of a designed Fresnel lens. Unfortunately, it is very difficult to be accurately decided. Hence, it will be better if there are more efficient ways to do the decision in the future. Besides, in the VLSDM, the number of fixed-length segments within a bigger segment as well as the amount of varied-length segments within a bigger segment equally form two important problems for future research.
This work was supported in part by I-Shou University under contract ISU-93-01-03, and in part by the National Science Council under grant NSC 94-2215-E-214-005.
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