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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 16 — Aug. 6, 2007
  • pp: 9918–9935
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Optimal design of an irregular Fresnel lens for multiple light sources using a three-layered Hierarchical Genetic Algorithm

Wen-Gong Chen, Chii-Maw Uang, and Chen-Hai Jou  »View Author Affiliations


Optics Express, Vol. 15, Issue 16, pp. 9918-9935 (2007)
http://dx.doi.org/10.1364/OE.15.009918


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Abstract

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

1. Introduction

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-1Nsub_seg+1)th, ((i-1Nsub_seg+2)th,…, and ((i-1Nsub_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.

Fig. 1. The fixed-length segment division mechanism
Fig. 2. The corresponding genes of the ith gene in the first control layer for the FLSDM.

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 Bgi-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.

Fig. 3. The varied-length segment division mechanism.
Fig. 4. The corresponding genes of the ith gene in the first control layer for the VLSDM.

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≤Rcl0.66, 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.

Fig. 5. A cycled loading mechanism.

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≤fiNga/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.

Fig. 6. A general structure of a chromosome in HGA3L for the case of FLSDM and Nsub_seg=3.

Table 1. Distribution of 330 groove angles on 33 segments

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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-1FSseg+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-1-1FSseg+1)th angle to the (fi-1×FSseg)th angle, second from the ((fi-1FSseg+1)th angle to the (fi×FSseg)th angle, and third from the ((fi+1-1FSseg+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.

Fig. 7. Part of a chromosome of a special example with FLSDM, Nseg=33, and Nsub_seg=3.

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

Ci(1:Nseq)=Ci(1:k11)+Cj(k1:k21)+Ci(k2:Nseg),
(1)

and

Cj(1:Nseg)=Cj(1:k11)+Ci(k1:k21)+Cj(k2:Nseg).
(2)

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

Ci(x:y)=Ci(x:x+Nsub_seg×(k11)1)+Cj(x+Nsub_seg×(k11):x+Nsub_seg×(k21)1)+Ci(x+Nsub_seg×(k21):y),
(3)

and

Cj(x:y)=Cj(x:x+Nsub_seg×(k11)1)+Cj(x+Nsub_seg×(k11):x+Nsub_seg×(k21)1)+Cj(x+Nsub_seg×(k21):y)
(4)

if r3 is less than or equal to 0.5, otherwise by

Ci(1:Nseq)=Cj(1:k11)+Ci(k1:k21)+Cj(k2:Nseg),
(5)
Cj(1:Nseg)=Ci(1:k11)+Cj(k1:k21)+Ci(k2:Nseg),
(6)
Ci(x:y)=Cj(x:x+Nsub_seg×(k11)1)+Ci(x+Nsub_seg×(k11):x+Nsub_seg×(k21)1)+Cj(x+Nsub_seg×(k21):y),
(7)
Cj(x:y)=Ci(x:x+Nsub_seg×(k11)1)+Cj(x+Nsub_seg×(k11):x+Nsub_seg×(k21)1)+Ci(x+Nsub_seg×(k21):y).
(8)

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

3. A design example

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

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

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

Rd=r=1Nrs=1NsRrs,
(9)

and the average number of light rays incident to a sector is given by

Ra=RdNr×Ns.
(10)

Now, in order to take the illuminance and uniformity into account simultaneously, we define the performance index I to be maximized as follows:

I=GL,
(11)

where

G=(RdRc)×Gw,
(12)
L=r=1Nrs=1NsLrs,
(13)

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:

Lrs={(1RaRrs)×LwforRrsRa(1RrsRa)×Lwotherwise,
(14)

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.

Table 2. Parameters used in HGA3L

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Fig. 12. Performances and convergence statuses of PAHGA3L_33_FLSD and PAHGA2L_33.

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.

Fig. 13. An irradiance map of a reading light system through FLHGA3L_33_FLSD.
Fig. 14. An irradiance map of a reading light system through FLHGA2L_33.

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).

Table 3. Distribution of light rays on the reading surface through FLHGA3L_33_FLSD.

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

Table 4. Distribution of light rays on the reading surface through FLHGA2L_33.

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

Fig. 15. (a). The cross section of FLHGA3L_33_FLSD.
Fig. 15. (b). The cross section of PAHGA2L_33

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.

Fig. 16. Performances and convergence statuses of PAHGA3L_33_FLSD and PAHGA3L_33_VLSD compared to PAHGA2L_33.
Fig. 17. An irradiance map through FLHGA3L_33_VLSD up to 2400 generations.

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.

Acknowledgments

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.

References and links

1.

T. S. Yu and X. Y. Yang, Introduction to Optical Engineering, Chap. 1–3, 1997.

2.

Reflexite Display Optics, NY, USA. Available: http://www.display-optics.com/.

3.

A. Davis, R. C. Bush, J. C. Harvey, and M. F. Foley, “Fresnel lenses in rear projection displays,” SID 2001 Digest, Volume XXXII, pp. 934–937, June 2001

4.

C. C. Sun, T.-X. Lee, S.-H. Ma, Y.-L. Lee, and S. Huang, “Precise optical modeling for LED lighting verified by cross correlation in the midfield region,” Opt. Lett. 31, 2193–2195 (2006). [CrossRef] [PubMed]

5.

W.-T. Chien, C. C. Sun, and I. Moreno, “Precise optical modeling of multi-chip white LEDs,” Opt. Express. 15, 7572–7577 (2007). [CrossRef] [PubMed]

6.

C. C. Sun, T. X. Lee, S. H. Ma, Y. L. Lee, and S. M Huang, “Precise optical modeling for LED lighting verified by cross correlation in the midfield region,” Opt. Lett. 31, 2193–2195 (2006). [CrossRef] [PubMed]

7.

F. Munoz, P. Benitez, O. Dross, J. C. Minano, and B. Parkyn, “Simultaneous multiple surface design of compact air-gap collimators for light-emitting diodes,” Opt. Eng. 43, 1522–1530 (2004). [CrossRef]

8.

P. Benitez, J. C. Minano, J. Blen, R. Mohendano, J. Chaves, O. Dross, M. Hemandez, and W. Falicoff, “Simultaneous multiple surface optical design,” Opt. Eng. 43, 1489–1502 (2004). [CrossRef]

9.

D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, (Addison Wesley, 1989).

10.

M. Gen and R. W. Cheng, Genetic Algorithms and Engineering Design, (Addison Wesley, 1997).

11.

W. G. Chen and C. M. Uang, “The design of a reading light system with RGB white light LED by Fresnel lens and evolutionary algorithms,” Proc. SPIE 5560, 370–379 (2005).

12.

W. G. Chen and C. M. Uang, “Better Reading Light System with light-emitting diodes using optimized Fresnel lens,” Opt. Eng. 45, 063001-1-7 (2006). [CrossRef]

13.

Lambda Research Corporation, MA, USA, TRACEPRO Reference Manual, Available: http://www.lambdares.com/.

14.

W. G. Chen and C. M. Uang, “Hierarchical Genetic Algorithm based design of a large scale Fresnel lens for a reading light system with multiple LED sources,” Appl. Opt. 45, 7832–7840 (2006). [CrossRef] [PubMed]

OCIS Codes
(200.4560) Optics in computing : Optical data processing
(220.3620) Optical design and fabrication : Lens system design
(330.4460) Vision, color, and visual optics : Ophthalmic optics and devices

ToC Category:
Optical Design and Fabrication

History
Original Manuscript: May 9, 2007
Revised Manuscript: June 24, 2007
Manuscript Accepted: July 6, 2007
Published: July 23, 2007

Virtual Issues
Vol. 2, Iss. 9 Virtual Journal for Biomedical Optics

Citation
Wen-Gong Chen, Chii-Maw Uang, and Chen-Hai Jou, "Optimal design of an irregular Fresnel lens for multiple light sources using a three-layered Hierarchical Genetic Algorithm," Opt. Express 15, 9918-9935 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-16-9918


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References

  1. T. S. Yu and X. Y. Yang, Introduction to Optical Engineering, Chap. 1-3, 1997.
  2. Reflexite Display Optics, NY , USA. Available: http://www.display-optics.com/.
  3. A. Davis, R. C. Bush, J. C. Harvey and M. F. Foley, "Fresnel lenses in rear projection displays," SID 2001 Digest, Volume XXXII, pp. 934-937, June 2001
  4. C. C. Sun, T.-X. Lee, S.-H. Ma, Y.-L. Lee, and S. Huang, "Precise optical modeling for LED lighting verified by cross correlation in the midfield region," Opt. Lett. 31, 2193-2195 (2006). [CrossRef] [PubMed]
  5. W.-T. Chien, C. C. Sun, and I. Moreno, "Precise optical modeling of multi-chip white LEDs," Opt. Express. 15, 7572-7577 (2007). [CrossRef] [PubMed]
  6. C. C. Sun, T. X. Lee, S. H. Ma, Y. L. Lee, and S. M Huang, "Precise optical modeling for LED lighting verified by cross correlation in the midfield region," Opt. Lett. 31, 2193-2195 (2006). [CrossRef] [PubMed]
  7. F. Munoz, P. Benitez, O. Dross, J. C. Minano, and B. Parkyn, "Simultaneous multiple surface design of compact air-gap collimators for light-emitting diodes," Opt. Eng. 43,1522-1530 (2004). [CrossRef]
  8. P. Benitez, J. C. Minano, J. Blen, R. Mohendano, J. Chaves, O. Dross, M. Hemandez, and W. Falicoff, "Simultaneous multiple surface optical design," Opt. Eng. 43, 1489-1502 (2004). [CrossRef]
  9. D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, (Addison Wesley, 1989).
  10. M. Gen and R. W. Cheng, Genetic Algorithms and Engineering Design, (Addison Wesley, 1997).
  11. W. G. Chen and C. M. Uang, "The design of a reading light system with RGB white light LED by Fresnel lens and evolutionary algorithms," Proc. SPIE 5560,370-379 (2005).
  12. W. G. Chen and C. M. Uang, "Better Reading Light System with light-emitting diodes using optimized Fresnel lens," Opt. Eng. 45, 063001-1-7 (2006). [CrossRef]
  13. Lambda Research Corporation, MA, USA, TRACEPRO Reference Manual, Available: http://www.lambdares.com/.
  14. W. G. Chen and C. M. Uang, "Hierarchical Genetic Algorithm based design of a large scale Fresnel lens for a reading light system with multiple LED sources," Appl. Opt. 45, 7832-7840 (2006). [CrossRef] [PubMed]

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