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Solar Energy 86 (2012) 877–885

Design of dome-shaped non-imaging Fresnel lenses takingchromatic aberration into account

Atsushi Akisawa a,⇑, Masao Hiramatsu b, Kouki Ozaki b

a Institute of Engineering, Tokyo University of Agriculture and Technology, Koganei, Tokyo, Japanb R&D Department, Technology Division, Daido Metal Co., Ltd., Inuyama, Aichi, Japan

Received 13 June 2011; received in revised form 8 September 2011; accepted 19 December 2011Available online 23 January 2012

Communicated by: Associate Editor Avi Kribus

Abstract

Concentration PV system is a technology for providing solar-based electricity at very high conversion efficiency of 40%. It needs solarconcentration of 500 suns or more, for which the authors developed dome-shaped non-imaging Fresnel lenses with a certain acceptancehalf angle. As conventional design method uses only one wave length, the performance suffers from chromatic aberration. In this paper, anew design method is proposed. One of the points is that it uses two kinds of design wave length which covers a given range of solarspectrum for the concentration. The other is that new design points are located on the corners of prisms while the conventional pointis at the center of prisms. Numerical examples with the concentration ratio of 500 were designed and optical efficiency was examined byray tracing simulation. The results indicate that the lens based on the conventional way has dish-like shape and the lenses designed by theproposed method have relatively deep dome shape in contrast. The optical efficiency of the new design is better than that of the conven-tional one at the incident angle equal to the acceptance half angle. It was concluded that the proposed method could produce more effec-tive solar concentrator with a certain tolerance of solar incident angle.� 2011 Elsevier Ltd. All rights reserved.

Keywords: Non-imaging Fresnel lens; Concentration photovoltaic system; Chromatic aberration; Dome shape; Optical efficiency

1. Introduction

Renewable energies are expected to increase the installa-tion to reduce fossil fuel consumption. Especially solar PVsystems have been adopted worldwide. However, conven-tional Si-based PV cells have the efficiency of approximately20% at most. To utilize much more solar energy, it is essentialto improve the PV efficiency significantly. One technologicalcandidate to attain such a high efficiency is concentration PVsystems (CPV) which concentrates solar irradiation onto thePV cell by lenses or mirrors with the concentration ratio of500 sums or more. The cell with quite high energy conversionefficiency of 40% has been developed for CPV (Kurt, 2009).

0038-092X/$ - see front matter � 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.solener.2011.12.017

⇑ Corresponding author. Tel./fax: +81 42 388 7226.E-mail address: akisawa@cc.tuat.ac.jp (A. Akisawa).

New Energy and Industrial Technology DevelopmentOrganization (NEDO), a governmental agency to supporttechnological development in Japan, once committedlaunching a project of developing CPV about 10 yearsago. In the project, the authors were involved in the designand the production of dome shaped lenses of 500 suns(Akisawa and Kashiwagi, 2005). One of the features is thatthe dome shaped lenses have undercut prisms which cannotbe produced by ordinary mold injection technique. One ofthe authors successfully developed a production process fordome shaped lenses in the project (Hiramatsu et al., 2003).While most of CPVs use flat shaped lenses, Japanese CPVsimplement dome shaped lenses thanks to the NEDOproject.

The design of shaped Fresnel lenses was proposed byLeutz et al. (1999) and Leutz and Suzuki (2001) based on

Fig. 2. Photo of a manufactured dome shaped lens (500 suns).

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the theory of non-imaging optics. It allows Fresnel lenseswith curved surface, for example, arch shape or dome shape.Dome shaped Fresnel lens array was also developed byPiszczor et al. (1991) for solar concentrator prior to them.In contrast, most of CPV use flat type Frenel lenses withpoint focus. Xie et al. (2011) surveys various types of Fresnellenses for solar concentrator applications. The advantage ofnon-imaging Fresnel lenses is to have acceptance half angleto collect sun light effectively. In other words, the lenses areinsensitive to the incident direction to some extent. Ryu et al.(2006) proposed a new type of Fresnel lens concentratorwhich unites many modular Fresnel lenses into one piecein flat shape. It has allowance of the incident angle withnot so high concentration ratio of 9-121 suns.

It is a nature of lenses to have chromatic aberration,which may cause degradation of the concentration whenthe lenses are applied for solar concentrators. In the shapedlens design method proposed by Leutz, wave length of 550nm is adopted to design prisms. It is likely that the lensperformance suffers from chromatic aberration. The objec-tive of this paper is to improve the design method for domeshaped non-imaging Fresnel lenses taking chromatic aberra-tion into account explicitly. Further techniques for improv-ing concentration performance are also discussed andexamined with ray-tracing simulations.

2. Non-imaging Fresnel lenses

2.1. Edge ray principle

The purpose of the proposed dome shaped lenses is to col-lect sun light as much as possible on the absorber. The objec-tive of ordinary lenses is to enlarge images, for instance, withdefinite focus for clear image formation. In contrast, imageformation is not required for the dome shaped lenses becausecollecting solar incident rays onto the absorber is essential,whatever the image is. This feature results in an advanta-geous characteristic of having acceptance half angle. Theacceptance half angle, h, is defined as the angle where solar

θθ

absorber

acceptance half angle

lens

ray

Fig. 1. Principle of designing non-imaging Fresnel lens with acceptancehalf angle.

incident rays coming in between +h and �h is captured onthe absorber. In other words, the ray at +h goes to an edgeof the absorber while the ray at �h reaches the other edge.Rays between +h and�h arrive at somewhere on the absor-ber, which is enough for the purpose of collecting sun light.Fig. 1 shows the principle of the lens design, which is so-called “Edge ray principle”. Because Fresnel lenses consistof many prisms, each prism is required to have appropriateshape incorporating this principle.

2.2. Dome shaped lens

Ordinary Fresnel lenses have flat surface and grooves onone side either upper face or lower face. However, theoreti-cally their acceptance half angle is considered zero becausethe main purpose is image formation. Contrary to theirshape, prisms of non-imaging Fresnel lenses basically haveinclined surface on both upper and lower sides to hold agiven acceptance half angle. It causes that the lenses havecurved shape looking like a dome in three dimensions ifthe lenses have smooth surface of the upper side. The authorsmanufactured dome shaped Fresnel lenses actually andtested the performance. Fig. 2 shows a photo of the domeshaped lens made of acrylic plastic material (PMMA).

3. New design method of dome shaped lenses

3.1. Coping with chromatic aberration

Although chromatic aberration is inevitable for lenses,conventional way of designing non-imaging Fresnel lensesdoes not take the effect into account. Rays of single wavelength are used for the design, which is regarded as neglect-ing chromatic aberration in the design process. Chromaticaberration eventually degrades the lens performance whensolar irradiation is applied to the lenses. For imagingoptics, some lenses are compounded so that the effect ofchromatic aberration is canceled, which is so-calledachromatic lens. Leutz and Ries (2003) examined anachromatic dome-shaped Fresnel lens which consists of

A. Akisawa et al. / Solar Energy 86 (2012) 877–885 879

two layers having different refractive indices. In contrast,the attempt of this study is to propose Fresnel lenses withone layer coping with chromatic aberration. because non-imaging lenses need no clear focus, non-imaging Fresnellenses is intrinsically considered insensitive to chromaticaberration. The requirement is that lights of different wavelength should arrive at somewhere on the absorber, notarrive at the focal point as is the case of imaging optics.

The proposed way of designing dome-shaped Fresnellenses consists of the following three steps.

(1) Determining upper and lower wave length to captureon the absorber. Each wave length is correspondingto different refractive index.

(2) Assigning the longer wave length coming at +h (cen-ter side) to get the edge of right hand side.

(3) Assigning the shorter wave length coming at �h(outer side) to get the edge of left hand side.

Fig. 3 indicates the behavior of the rays showing that theshorter wave length coming at +h is expected to arrive atthe edge of left hand side on the absorber. Similarly thelonger wave length coming at �h is predicted to get theother edge of the absorber. Therefore, light in the rangeof wave length can be fully captured by the absorber eventhough there occurs chromatic aberration. It should benoted that the proposed non-imaging Fresnel lenses candeal with chromatic aberration not by canceling the effectbut by accepting the effect.

absorber

single wave length

msirP

+θ -θ

Fig. 3. Edge rays considering chromatic aberration into acco

3.2. Decoupling lens height with acceptance half angle

Shaped non-imaging Fresnel lenses for solar collectingdevices were designed by Leutz et al. as mentioned before.According to the design method, lens height (distancebetween top of the lens and the absorber) is determinedby the given acceptance half angle as expressed in Eq. (1).

d ¼ H tanh ð1Þ

where d: half width of the absorber (m), H: lens height (m),h: acceptance half angle (�).

Based on Eq. (1), the lens height is increasing as the accep-tance half angle becomes smaller when the absorber width isconstant. Shorter lens height is preferable in order to reducethe volume of the CPV system. It requires larger acceptancehalf angle, which results in deeper curvature and grooves onthe lens. Finally it induces difficulties in manufacturingactual lenses. It is a kind of conflict to attain both smallacceptance half angle and short lens height in the designmethod.

The authors proposed decoupling of these two parame-ters. In other words, lens height and acceptance half angleare treated as independent parameters in the new designmethod, which means that designing dome-shaped non-imaging Fresnel lens gets one more freedom to make pref-erable lenses.

Based on the coupled design, all the prisms have commongiven acceptance half angle and curved surface starts from

absorber

dual wave length

short wave length

long wave length

msirP

+θ -θ

unt. (a) Conventional method and (b) proposed method.

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the center of the lens. In contrast, the decoupled designallows the lens height lower than that in the coupled design.

3.3. Separating design points for each of edge rays

In the current design method, the design point where therays go through is set on the center of each prism. How-ever, it is not necessarily effective because sun light spreadsover the upper surface of every prism. It can be observedthat rays which incident angle is the acceptance half angleand are not on the design point go out of the absorber. Tocollect light sifting from the center of prisms, the authorspropose to adopt two design points which are correspond-ing to each edge ray. Fig. 4 expresses the idea of two designpoints each of which is located at one of the corners on theupper surface of each prism. The outer corner is used forthe edge ray getting to the right hand side edge of theabsorber while the inner corner is for one arriving at theleft hand side edge. Sun light through the upper surfaceat the acceptance half angle goes parallel with the edgeray and finally gets to a point shifting from the edge, buton the absorber. Because the width of prisms is generallysmaller than the width of the absorber, most of sun lightis captured on the absorber. It should be noted that theedge ray at the outer corner may into the next prism anddoes not get the edge of the absorber as designed. However,assigning the corner as one of the design points is acceptedfor the simplicity of the design method.

3.4. Flattened dome shape

The conventional design method by Leutz assumes thatevery prism has the same acceptance half angle, which

absorber

-θ +θ

short wave length

long wave length

Fig. 4. Two design points of each prism.

results in curved upper surface. That is the reason whythe Fresnel lenses have dome shape. It should be noted thatprisms with flat upper surface can have a certain accep-tance half angle. Every prism has edge rays such that eachof them gets an edge of the absorber. The incident angle ofthem is called acceptable half angle here. Fig. 5 shows theacceptable half angle of each prism with monochromaticdesign on a flat Fresnel lens as an example (Akisawaet al., 2007). The horizontal axis indicates the position ofprisms in terms of non-dimensional distance from the cen-ter of the lens normalized by the half width of the absorber.The acceptable half angle changes with the position of theprism. Prisms located around the center have relativelylarge acceptable half angle compared with prisms on theouter side. It can be understood that some prisms can haveenough large acceptable half angle even though the uppersurface is flat. It suggests that the decoupled design withlower lens height than the coupled design allows largeracceptable half angle than the specified acceptance halfangle in the center part. Because the upper surface is flat,the lens holds the given lens height in that region.

On the other hand, prisms on the outer side do not satisfythe requirement of the given acceptance half angle. Theyneed to have declined upper surface to hold the acceptablehalf angle is equal to the acceptance half angle. In this outerpart both the decoupled design and the conventional coupleddesign satisfy the same edge ray constraints. It means that thecurved shape in this region is common for both cases.

It is suggested from the discussion that dome-shapedFresnel lenses are not necessarily fully domed and can haveflat part around the center. It is advantageous becauseprisms on the flat part have no undercut theoretically, whichis suitable for mold injection manufacturing processes. It

non-dimensional distance from the center x/d (-)

acce

ptab

le h

alf a

ngle

(deg

)

given acceptance half angle = 0.7deg

Fig. 5. Acceptable half angle of the prisms on a flat Fresnel lens (anexample).

Fig. 6. New shape of non-imaging Fresnel dome lenses.

A. Akisawa et al. / Solar Energy 86 (2012) 877–885 881

should be noted that prisms around the center of conven-tional dome-shaped Fresnel lenses are arranged to removeundercut shape for such manufacturing processes. The newdesign method allows dome-shaped Fresnel lenses to haveno need of non-undercut prisms as the ideal shape. Conse-quently prisms on the outer part need to have curved uppersurface, which is illustrated in Fig. 6.

In the proposed design method, two kinds of extremewave length of blue and red rays are adopted to investigatethe behavior of edge rays. Larger acceptable half angle isexplored as far as it is possible in the center flat part whilethe given acceptance half angle is kept in the outer curvedpart for both extreme edge rays.

4. Results of lens shape

4.1. Design parameters

In order to examine the new design method, investigationhere focuses on the effect of the dual wave length to cope withchromatic aberration. The parameters for the dome-shaped

Table 1Design parameters.

Item Value Note

Acceptance half angle 0.7�Lens height 60 Represented in terms ofWidth of prism 0.25 non-dimensional lengthThickness of prism 1.0 normalized with the half width of the

absorberGeometrical

concentration ratio506

Number of prisms 90 Lens radius = 22.5

Refractive index

Single wave length 1.49Dual wave length 1.48/

1.50

Fresnel lenses are summarized in Table 1. Concentrationratio is set to be 500 suns. PMMA is assumed for the lensmaterial as before. The dome-shaped lens is rotationallysymmetrical. The feature of the flattened dome shape is alsotaken into account. As mentioned before, the lens height isdecoupled with the acceptance half angle. Although the lensheight should be 81.8 (=1.0/tan0.7) based on the conven-tional design method, it is reduced to 60 in this simulation.

The refractive index of PMMA is formulated as the fol-lowing (Leutz and Suzuki, 2001).

nðkÞ ¼ 1:468þ 9:342=ðk� 123:5Þ ð2Þwhere n: refractive index, k: wave length (nm).

4.2. Result of dome shape

Here, three cases of dome-shaped lenses are examinedand compared to evaluate the effect of the proposedmethod separately.

Case 1: single wave length (n = 1.49) at the center ofprisms.

Case 2: double wave length (n = 1.48 and 1.50) at thecenter of prisms.

Case 3: double wave length (n = 1.48 and 1.50) at thecorners of prisms.

The results of each lens shape are shown in Fig. 7. It isslightly surprising that the lens of Case 1 is almost flatlooking like a dish. In contrast, the lenses of Case 2 andCase 3 seem bowls which has flat bottom at the centerand curvature around it. It indicates that the new designmethod considering chromatic aberration requires signifi-cant curvature to collect solar irradiation. Furthermore,the lens of Case 3 is deeper than that of Case 2. Shiftingdesign point from the center to the corners leads to largercurvature because incident rays have to be bent more,which might cause disadvantage from the manufacturingpoint of view.

It should be noted that prisms on the curved part haveundercut while prisms on the flat part have no undercut.These dome-shaped lenses cannot be produced by ordinaryinjection mold techniques due to the undercut prisms.

To capture larger dispersion, it is required to select widerange of refractive index in the design procedure. It ismathematically possible, but it will result in relatively deepdome-shape, which induces smaller concentration ratio. Orsmaller acceptance half angle is achievable to have a certainconcentration ratio.

4.3. Effect of the new design

In order to observe the effect of the new design methods,the optical efficiency was measured for specific wave lengthof beam light. The refractive indices of 1.48 and 1.50 repre-sent the wave length of 900 nm and 400 nm, respectively.The refractive index of 1.49 is corresponding to the wave

Fig. 7. Lens shapes corresponding to design methods: (a) Case 1, (b) Case 2, (c) Case 3.

Fig. 8. Lens performance to beam irradiation of specific wave length: (a)Case 1, (b) Case 2, (c) Case 3.

882 A. Akisawa et al. / Solar Energy 86 (2012) 877–885

length of 550 nm. Therefore, 420, 550, 700, 880 nm of thewave length are selected so that they are included in therange. Ray tracing simulation was employed to estimatethe optical efficiency with beam irradiance. The refractiveindex of Eq. (2) was incorporated in the simulation. Lightsource is assumed to have square shape because actualproducts of dome-shaped lenses are square as can be seenin Fig. 2. To evaluate the performance of the lens products,the light source is adjusted in that shape. The length of theone side is 45=

ffiffiffi

2p¼ 32. The absorber is also assumed

square with the one side length offfiffiffi

2p

.Fig. 8 shows the results of each lens with inclined incident

angle of 0–1.0�. The performance of Case 1 indicates that thewave length of 550 nm can be captured when the incidentangle is less than 0.5�. When the angle is equal to the designacceptance half angle of 0.7�, most of the light of 550 nmcannot get the absorber. The efficiency for the light of700 nm and 880 nm start decreasing when the angles becomelarger than 0.3 and 0.2, respectively. In contrast, the perfor-mances of Case 2 show steeper edge around the angle of 0.7�.The efficiency of 550 nm keeps high when the angle is lessthan 0.6� although the efficiency is as low as 0.3 at the angle

Fig. 9. Optical efficiency corresponding to design method (wave length of300–800 nm).

Fig. 10. Flux distribution on the absorber with the incident angle of 0.7�. (Broken line indicates the boundary of the absorber of 1.4 mm � 1.4 mm.) (a)Case 1, (b) Case 2, (c) Case 3.

A. Akisawa et al. / Solar Energy 86 (2012) 877–885 883

Fig 10. (continued)

884 A. Akisawa et al. / Solar Energy 86 (2012) 877–885

of 0.7�. The lens of Case 2 captures the light in the wavelength range at the efficiency of 0.9 when the angle is less than0.3�. The lens of Case 3 has the best performance amongthem. For the design acceptance half angle of 0.7�, theefficiency of every wave length is approximately 0.4-0.5.Furthermore, the efficiency holds as high as 0.9 for the anglerange of 0–0.4�. It is found out that the deviation of the effi-ciency behaviors is smaller than the others. The lens of Case3 can capture wider range of wave length even when the inci-dent angle is larger than the design acceptance half angle.Consequently the proposed design method using two wavelength with two design points is advantageous comparedwith the conventional method.

5. Performance evaluation

5.1. Optical efficiency

One of the indices of lens performance is optical efficiencywhich indicates how much solar inlet is captured on theabsorber. The concentration ratio of a lens is calculated bymeans of the geometrical concentration ratio multiplied withthe optical efficiency. Here, ray tracing simulation with solarirradiance was employed to estimate the optical efficiency.The range of solar wave length was decomposed into severalsegments so that they reflect the flux profile of AM1.5D.As seen in Fig. 8, the lenses do not collect infrared rays

effectively. So, in order to investigate the performance ofthe lenses, the authors focused on the visible wave lengthof sun light, that is, 300–800 nm for the comparison. Again,the light source is assumed to have square shape as theprevious analysis.

Fig. 9 shows the optical efficiency of the dome-shapedlenses when solar incident angle changes from 0� (normaldirection) to 1.0�. As shown in the graph, optical efficienciesare as high as 0.85–0.9 in the incident range of 0–0.4�. On theother hand, the efficiencies become smaller when the incidentangle is larger than 0.4�. Eventually the efficiencies areapproximately 0.3 and 0.5 for Case 1 and Case 3 respectivelywhen the incident angle is equal to the acceptance half angleof 0.7�. The lens of Case 3 has the highest optical efficiencyamong them when the incident rays are inclined. The effi-ciency of Case 2 is better than that of Case 1. The resultclearly reveals that the proposed method is effective toimprove the concentration performance of dome-shapedlenses.

5.2. Flux on the absorber

The prediction that the optical efficiency of Case 3 isabout 0.5 indicates half of solar input gets onto the absorbereven though the incident angle is 0.6–0.7�. Fig. 10 illustratesthe flux distribution on the absorber for these three lenses

A. Akisawa et al. / Solar Energy 86 (2012) 877–885 885

when the incident angle is 0.7�, the acceptance half angle.The difference among the three cases does not seem signifi-cant, but it can be observed that the flux distribution ofCase 3 is stretched into the absorber and the flux peak stillremains on the edge of the absorber. It is the reason whyCase 3 has higher optical efficiency than the others.

6. Conclusion

The study proposed new design methods of dome-shaped non-imaging Fresnel lenses for CPV systems. Thepoints to improve the concentration performance are thatusing two wave length to consider chromatic aberrationexplicitly and that two design points located at the cornersof each prism where the edge ray goes through. For solarconcentration, achromatic lenses are not needed in non-imaging optics because the essential thing is to collect solarirradiance on the absorber whatever the wave length is.This study presented that single Fresnel lens would be pos-sible to incorporate such function of capturing a certainrange of wave length by adjusting prism angles. Further-more, lenses designed in this study has flat part aroundthe center of the lens while curved part outside. The shapeis in contrast to that of fully curved dome-shaped lenseswhich the authors developed before. The new dome shapeis advantageous from the manufacturing point of viewbecause the flat part does not have undercut prisms.

Compared with conventional design method, i.e. usingone wave length and one design point located at the centerof prisms, the proposed methods improve the optical effi-ciency when the incident angle is equal to the acceptancehalf angle. On the other hand, numerical examples repre-sent that new design lenses have much deeper dome shapein contrast to conventionally designed dome-shaped lens. Itimplies that better concentration performance of dome-shaped lenses requires larger curvature looking like a bowl.Because the lenses investigated in this study do not capture

infrared rays very well, it will be a future task to design andanalyze lenses to collect much wider range of the wavelength.

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