1. Introduction

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 [1]. 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 [13] 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 [14]. 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 N_{sub_seg} denote the number of the divided varied-length sub-segments, in the second control layer, the genes that are activated by the i^th gene of the first control layer are the ((i-1)×N_{sub_se}g+1)^th, ((i-1)×N_{sub_seg}+2)^th,…, and ((i-1)×N_{sub_seg}+N_{sub_seg})^th genes as shown in Fig. 2, where N_seg denotes the number of total segments of the designed grooves. Let FS_seg denote the size of a fixed-length segment, i.e., the number of the grooves a fixed-length segment contains, and S_{sub_seg} denote the size of a sub-segment, then, 1≦S_{sub_seg}≦FS_seg-N_{sub_seg}+1. This division mechanism is simpler but the designed groove angles are less variable.

 

Fig. 1. The fixed-length segment division mechanism

Download Full Size | PDF

 

Fig. 2. The corresponding genes of the i^th gene in the first control layer for the FLSDM.

Download Full Size | PDF

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≦S_sub-seg≦VS_seg-N_{sub_seg}+1, where VS_seg denotes the size of a varied-length segment and has the relation of N_{sub_seg}≦VS_seg≦S_c×FS_seg-(Sd-1)×N_{sub_seg} where S_c represents the number of fixed-length segments that are composed into a bigger segment, and S_d represents the number of varied-length segments that a bigger segment is decomposed into. In the second control layer, the corresponded genes by the i^th gene in the first control layer are the (B_g+1)^th, (B_g+2)^th,…, and (B_g+N_{sub_seg})^th genes where B_g=Σ^i-1_j=1VS^j_seg for i=2,3,…,N_seg but B_g=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 S_c and S_d are set to three in the proposed HGA3L. Although this division mechanism is more complicated, the designed groove angles are expectedly more variable.

 

Fig. 3. The varied-length segment division mechanism.

Download Full Size | PDF

 

Fig. 4. The corresponding genes of the i^th gene in the first control layer for the VLSDM.

Download Full Size | PDF

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 R_cl represent a random number, 0 ≤ R_cl ≤ 1, G_fst represent the groove angles in the first sub-segment, G_snd in the second sub-segment, G_trd in the third sub-segment, and G_lst 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 R_cl: G_fst, G_snd, G_trd, …, and G_lst ; or G_snd, G_trd, …, Glst, and G_fst ; or G_trd, …, G_lst, G_fst, and G_snd, 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, R_cl. First, they come from the first sub-segment, the second sub-segment, and the third sub-segment if 0≤R_cl‹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≤R_cl‹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.

 

Fig. 5. A cycled loading mechanism.

Download Full Size | PDF

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 N_{sub_seg}=3 is shown in Fig. 6.

In the first layer, the control genes are coded as integers, each ranging from 1 to N_ga/FS_seg, i.e., 1≤f_i≤N_ga/FS_seg, i=1,2,3,…,N_seg, where fi in Fig. 6 denotes the value of the i^th control gene, N_ga denotes the number of the total groove angles of a conventional Fresnel lens database. Furthermore, let N_tg denote the number of the total grooves of a designed Fresnel lens, N_seg is equal to N_tg/FS_seg. Take N_seg=33 as an example, the distribution of the N_ga 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 FS_seg-N_{sub_seg}+1 if the FLSDM is applied. Otherwise, they are between 1 and VS_seg-N_{sub_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.

 

Fig. 6. A general structure of a chromosome in HGA3L for the case of FLSDM and N_{sub_seg}=3.

Download Full Size | PDF

Table 1. Distribution of 330 groove angles on 33 segments

View Table | View all tables in this article

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 FS_seg are extracted from that database, ranging from the ((fi-1)×FS_seg+1)^th angle to the (fi×FS_seg)^th angle if a FLSDM is applied. On the contrary, if a VLSDM is applied and S_c=3, the FS_seg×S_c groove angles are extracted from that database, first ranging from the ((f_i-1-1)×FS_seg+1)^th angle to the (f_i-1×FS_seg)th angle, second from the ((f_i-1)×FS_seg+1)^th angle to the (fi×FS_seg)^th angle, and third from the ((_fi+1-1)×FS_seg+1)^th angle to the (fi+1×FS_seg)^th angle, respectively for i=2,3,..,Nseg-1. For the situation that i=1 or i=N_seg, let i be i+1 if i=1 or i-1 if i=N_seg 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 N_seg=33 and N_{sub_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.

 

Fig. 7. Part of a chromosome of a special example with FLSDM, N_seg=33, and N_{sub_seg}=3.

Download Full Size | PDF

2.2 Implementation of HGA3L

At the initial stage of the proposed HGA3L, an initial population with N_p chromosomes is generated. First, the values of the control genes of the first layer in every chromosome are set from 1 to N_ga/FS_seg 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 C_i and C_{i+ 1} have the property F(C_i)▫F(C_i+1) for i=1,2,…,N_p, F(C_i) represents the fitness value of chromosome C_i.

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 C_i, let g_ij denote the j^th control gene of the first layer for j=1,2,3,…, N_seg and g_ik, g_i(k+1), and g_i(k+2) denote the k^th, (k+1)^th, and (k+2)^th control genes of the second layer, where k=(j-1)×N_{sub_seg}+1. For the control gene g_ij, we generate a random number r. If r is less than the defined mutation rate, M_r, then the control gene g_i,j is mutated to a new one called g′_ij which has a value between 1 and N_ga/FS_seg, otherwise the control gene is not mutated, i.e., g′=g_ij. If the control gene g_ij 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., g_ik is replaced with g_′, gi_(k+1) is replaced with g_{′i (k+1}), and g_i(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, g_{ik, gi(k+1)} and g_i(k+2) are not changed, i.e.,g′ik=gik, g′_i(k+1)=g_i(k+1), and g_′i(k+2)=g_i(k+2). A mutated chromosome C′_i is therefore formed from the first layer’s control genes g′_ij, j=1, 2, 3,…, N_seg, the second layer’s control genes g′ik, _g′f and g′_i(k+2), k=(j-1)×N_{sub_seg}+1, and the original and newly created parametric genes. The chromosome C_i, is replaced by the chromosome C′_i, if the mutated chromosome C′_i has a higher fitness value than C_i. 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 (C_i, C_i+Np/2), i=1,2,3,…,N_p/2. For a given chromosome pair (C_i, C_j), we generate three different random numbers r₁, r₂, and r₃ over the interval (0,1) and obtain two integers 1 seg k₁=[N_seg×r1] and k₂=[N_seg×r₂], which represent the crossover points and lie in the range [2,N_seg-1]. Assume that k₁ is less than k₂. For the chromosome C_i, let C_i(a:b) denote a control gene section from the a^th control gene to the b^th control gene in the first or second layer. Then, the first layer’s control genes of offspring chromosomes generated by crossover are given by

and

Let x and y denote the beginning and end position of the second layer’s control genes in a chromosome, respectively, i.e., x=N_seg+1, y=N_seg+N_{sub_seg}×Nseg. The second layer’s control genes of offspring chromosomes generated by crossover are given by

and

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 C_i are replaced by the control genes of the offspring chromosome C′_i if C′_i has a higher fitness value than C_i. C_j and C′_j follow the same replacement policy as C_i 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 [13] 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.

 

Fig. 8. The flow diagram of the proposed HGA3L.

Download Full Size | PDF

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 [14] 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.

 

Fig. 9. A simulated reading light system.

Download Full Size | PDF

 

Fig. 10. A cone-frustum shaped reflector and an LED light source.

Download Full Size | PDF

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.

 

Fig. 11. A reading surface with 5 equal-area rings, 120 equal-area sectors

Download Full Size | PDF

In order to measure the actual illuminance and uniformity of a reading light system, the circular reading surface is divided into N_r equal-area rings and each ring is further divided into N_s equal-area sectors, as shown in Fig. 11. Let R_rs indicate the number of light rays incident to the s^th sector of the r^th 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:

where

Equation (12) stands for the gain of the performance index whereas Eq. (13) stands for the loss of the performance index. In Eq. (12), R_c denotes the total number of light rays incident to the reading surface through a conventional Fresnel lens in the same reading light system, while G_w denotes the gain weight. The reason for deducting R_c from R_d is to let a conventional Fresnel lens be an initial condition in this paper. In Eq. (13), L_rs denotes the loss of the performance index of the s^th sector in the r^th 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 L_w 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, PA_{HGA3L_33_FLSD} represents the proposed approach HGA3L by FLSDM with N_seg=33, PA_{HGA3L_33_}VLSD represents the proposed approach HGA3L by VLSDM with N_seg=33, and PA_{HGA2L_33} represents the proposed approach HGA2L with N_seg=33. Moreover, the designed Fresnel lens is designated as FL_{HGA3L_33_FLSD} if it is designed by PA_{HGA3L_33_FLSD}, as FL_{HGA3L_33_VLSD} if it is designed by PA_{HGA3L_33_VLSD}, or as FL_{HGA2L_33} if it is designed by PA_{HGA2L_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 N_seg=33 and choose the FLSDM. The experimental results are shown in Figs. 12, 13, and 14. From Fig. 12, it is clear that PA_{HGA3L_33_FLSD} converges faster and has better performance than PA_{HGA2L_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 PA_{HGA3L_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.

Table 2. Parameters used in HGA3L

View Table | View all tables in this article

 

Fig. 12. Performances and convergence statuses of PA_{HGA3L_33_FLSD} and PA_{HGA2L_33}.

Download Full Size | PDF

Compare Fig. 13 to Fig. 14 for the illumination of the simulated reading light system, it is obvious that PA_{HGA3L_33_FLSD} indeed works better than PA_{HGA2L_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 PA_{HGA3L_33_FLSD} and PA_{HGA2L_33}, respectively. It is clear that the illumination of the reading light system by PA_{HGA3L_33_FLSD} has increased by 2.31% as compared to that of the reading light system by PA_{HGA2L_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.

 

Fig. 13. An irradiance map of a reading light system through FLHGA3L_33_FLSD.

Download Full Size | PDF

 

Fig. 14. An irradiance map of a reading light system through FL_{HGA2L_33}.

Download Full Size | PDF

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 PA_{HGA3L_33_FLSD} and PA_{HGA2L_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 PA_{HGA3L_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 PA_{HGA3L_33_FLSD} has a higher percentage than the same ring by PA_{HGA2L_33}. On the other hand, the innermost ring by PA_{HGA3L_33_FLSD} has a lower percentage than the same ring by PA_{HGA2L_33}, shown in the last column of Table 3 and 4. Finally, we show the cross-section of the designed Fresnel lens, FL_{HGA3L_33_FLSD}, in Fig. 15(a) and the compared cross-section of the designed Fresnel lens by PA_{HGA2L_33} in Fig. 15(b).

Table 3. Distribution of light rays on the reading surface through FL_{HGA3L_33_FLSD}.

View Table | View all tables in this article

r : ring, s : sector, sub : subtotal, per : percent

Table 4. Distribution of light rays on the reading surface through FL_{HGA2L_33}.

View Table | View all tables in this article

ring, s : sector, sub : subtotal, per : percent

 

Fig. 15. (a). The cross section of FL_{HGA3L_33_FLSD}.

Download Full Size | PDF

 

Fig. 15. (b). The cross section of PA_{HGA2L_33}

Download Full Size | PDF

In the second experiment, the main goal is to demonstrate the idea that PA_{HGA3L_33_VLSD} works better but converges slower than PA_{HGA3L_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 PA_{HGA3L_33_FLSD} above it denotes the performance and convergence status of PA_{HGA3L_33_FLSD}. The black line with the symbol PA_{HGA3L_33_VLSD} below it denotes the performance and convergence status of PA_{HGA3L_33_VLSD}. It is obvious that PA_{HGA3L_33_VLSD} surpasses PA_{HGA3L_33_FLSD} in performance about up to 2400 generations. An irradiance map through FL_{HGA3L_33_VLSD} up to 2400 generations is shown in Fig. 17. There are 4751 incident rays on the map. Moreover, PA_{HGA3L_33_VLSD} is also far superior to PA_{HGA2L_33.} The superiority is presented in Fig. 16 by an additional blue line with the symbol PA_{HGA2L_33} below it.

 

Fig. 16. Performances and convergence statuses of PA_{HGA3L_33_FLSD} and PA_{HGA3L_33_VLSD} compared to PA_{HGA2L_33}.

Download Full Size | PDF

 

Fig. 17. An irradiance map through FL_{HGA3L_33_VLSD} up to 2400 generations.

Download Full Size | PDF

5. Conclusions

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.