1. Design of dome-shaped non-imaging Fresnel lenses taking
chromatic aberration into account
Atsushi Akisawa a,⇑
, Masao Hiramatsu b
, Kouki Ozaki b
a
Institute of Engineering, Tokyo University of Agriculture and Technology, Koganei, Tokyo, Japan
b
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 2011
Available 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 solar
concentration of 500 suns or more, for which the authors developed dome-shaped non-imaging Fresnel lenses with a certain acceptance
half angle. As conventional design method uses only one wave length, the performance suffers from chromatic aberration. In this paper, a
new design method is proposed. One of the points is that it uses two kinds of design wave length which covers a given range of solar
spectrum for the concentration. The other is that new design points are located on the corners of prisms while the conventional point
is at the center of prisms. Numerical examples with the concentration ratio of 500 were designed and optical efficiency was examined by
ray tracing simulation. The results indicate that the lens based on the conventional way has dish-like shape and the lenses designed by the
proposed 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.
Ó 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 PV
systems have been adopted worldwide. However, conven-
tional Si-based PV cells have the efficiency of approximately
20% at most. To utilize much more solar energy, it is essential
to improve the PV efficiency significantly. One technological
candidate to attain such a high efficiency is concentration PV
systems (CPV) which concentrates solar irradiation onto the
PV cell by lenses or mirrors with the concentration ratio of
500 sums or more. The cell with quite high energy conversion
efficiency of 40% has been developed for CPV (Kurt, 2009).
New Energy and Industrial Technology Development
Organization (NEDO), a governmental agency to support
technological development in Japan, once committed
launching a project of developing CPV about 10 years
ago. In the project, the authors were involved in the design
and the production of dome shaped lenses of 500 suns
(Akisawa and Kashiwagi, 2005). One of the features is that
the dome shaped lenses have undercut prisms which cannot
be produced by ordinary mold injection technique. One of
the authors successfully developed a production process for
dome shaped lenses in the project (Hiramatsu et al., 2003).
While most of CPVs use flat shaped lenses, Japanese CPVs
implement dome shaped lenses thanks to the NEDO
project.
The design of shaped Fresnel lenses was proposed by
Leutz et al. (1999) and Leutz and Suzuki (2001) based on
0038-092X/$ - see front matter Ó 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).
www.elsevier.com/locate/solener
Available online at www.sciencedirect.com
Solar Energy 86 (2012) 877–885

2. the theory of non-imaging optics. It allows Fresnel lenses
with curved surface, for example, arch shape or dome shape.
Dome shaped Fresnel lens array was also developed by
Piszczor et al. (1991) for solar concentrator prior to them.
In contrast, most of CPV use flat type Frenel lenses with
point focus. Xie et al. (2011) surveys various types of Fresnel
lenses for solar concentrator applications. The advantage of
non-imaging Fresnel lenses is to have acceptance half angle
to collect sun light effectively. In other words, the lenses are
insensitive to the incident direction to some extent. Ryu et al.
(2006) proposed a new type of Fresnel lens concentrator
which unites many modular Fresnel lenses into one piece
in flat shape. It has allowance of the incident angle with
not 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 when
the lenses are applied for solar concentrators. In the shaped
lens design method proposed by Leutz, wave length of 550
nm is adopted to design prisms. It is likely that the lens
performance suffers from chromatic aberration. The objec-
tive of this paper is to improve the design method for dome
shaped non-imaging Fresnel lenses taking chromatic aberra-
tion into account explicitly. Further techniques for improv-
ing concentration performance are also discussed and
examined 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, with
definite focus for clear image formation. In contrast, image
formation is not required for the dome shaped lenses because
collecting 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. The
acceptance half angle, h, is defined as the angle where solar
incident rays coming in between +h and Àh is captured on
the absorber. In other words, the ray at +h goes to an edge
of the absorber while the ray at Àh reaches the other edge.
Rays between +h and Àh 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 consist
of many prisms, each prism is required to have appropriate
shape incorporating this principle.
2.2. Dome shaped lens
Ordinary Fresnel lenses have flat surface and grooves on
one side either upper face or lower face. However, theoreti-
cally their acceptance half angle is considered zero because
the main purpose is image formation. Contrary to their
shape, prisms of non-imaging Fresnel lenses basically have
inclined surface on both upper and lower sides to hold a
given acceptance half angle. It causes that the lenses have
curved shape looking like a dome in three dimensions if
the lenses have smooth surface of the upper side. The authors
manufactured dome shaped Fresnel lenses actually and
tested the performance. Fig. 2 shows a photo of the dome
shaped 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 lenses
does not take the effect into account. Rays of single wave
length are used for the design, which is regarded as neglect-
ing chromatic aberration in the design process. Chromatic
aberration eventually degrades the lens performance when
solar irradiation is applied to the lenses. For imaging
optics, some lenses are compounded so that the effect of
chromatic aberration is canceled, which is so-called
achromatic lens. Leutz and Ries (2003) examined an
achromatic dome-shaped Fresnel lens which consists of
θθ
absorber
acceptance half angle
lens
ray
Fig. 1. Principle of designing non-imaging Fresnel lens with acceptance
half angle.
Fig. 2. Photo of a manufactured dome shaped lens (500 suns).
878 A. Akisawa et al. / Solar Energy 86 (2012) 877–885

3. two layers having different refractive indices. In contrast,
the attempt of this study is to propose Fresnel lenses with
one layer coping with chromatic aberration. because non-
imaging lenses need no clear focus, non-imaging Fresnel
lenses is intrinsically considered insensitive to chromatic
aberration. The requirement is that lights of different wave
length should arrive at somewhere on the absorber, not
arrive at the focal point as is the case of imaging optics.
The proposed way of designing dome-shaped Fresnel
lenses consists of the following three steps.
(1) Determining upper and lower wave length to capture
on the absorber. Each wave length is corresponding
to 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 Àh
(outer side) to get the edge of left hand side.
Fig. 3 indicates the behavior of the rays showing that the
shorter wave length coming at +h is expected to arrive at
the edge of left hand side on the absorber. Similarly the
longer wave length coming at Àh is predicted to get the
other edge of the absorber. Therefore, light in the range
of wave length can be fully captured by the absorber even
though there occurs chromatic aberration. It should be
noted that the proposed non-imaging Fresnel lenses can
deal with chromatic aberration not by canceling the effect
but by accepting the effect.
3.2. Decoupling lens height with acceptance half angle
Shaped non-imaging Fresnel lenses for solar collecting
devices were designed by Leutz et al. as mentioned before.
According to the design method, lens height (distance
between top of the lens and the absorber) is determined
by 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 is
constant. Shorter lens height is preferable in order to reduce
the volume of the CPV system. It requires larger acceptance
half angle, which results in deeper curvature and grooves on
the lens. Finally it induces difficulties in manufacturing
actual lenses. It is a kind of conflict to attain both small
acceptance half angle and short lens height in the design
method.
The authors proposed decoupling of these two parame-
ters. In other words, lens height and acceptance half angle
are treated as independent parameters in the new design
method, 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 common
given acceptance half angle and curved surface starts from
absorber absorber
single wave length dual wave length
short wave
length
long wave
length
msirPmsirP
+θ -θ +θ -θ
Fig. 3. Edge rays considering chromatic aberration into account. (a) Conventional method and (b) proposed method.
A. Akisawa et al. / Solar Energy 86 (2012) 877–885 879

4. the center of the lens. In contrast, the decoupled design
allows 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 the
rays go through is set on the center of each prism. How-
ever, it is not necessarily effective because sun light spreads
over the upper surface of every prism. It can be observed
that rays which incident angle is the acceptance half angle
and are not on the design point go out of the absorber. To
collect light sifting from the center of prisms, the authors
propose to adopt two design points which are correspond-
ing to each edge ray. Fig. 4 expresses the idea of two design
points each of which is located at one of the corners on the
upper surface of each prism. The outer corner is used for
the edge ray getting to the right hand side edge of the
absorber while the inner corner is for one arriving at the
left hand side edge. Sun light through the upper surface
at the acceptance half angle goes parallel with the edge
ray and finally gets to a point shifting from the edge, but
on the absorber. Because the width of prisms is generally
smaller than the width of the absorber, most of sun light
is captured on the absorber. It should be noted that the
edge ray at the outer corner may into the next prism and
does not get the edge of the absorber as designed. However,
assigning the corner as one of the design points is accepted
for the simplicity of the design method.
3.4. Flattened dome shape
The conventional design method by Leutz assumes that
every prism has the same acceptance half angle, which
results in curved upper surface. That is the reason why
the Fresnel lenses have dome shape. It should be noted that
prisms with flat upper surface can have a certain accep-
tance half angle. Every prism has edge rays such that each
of them gets an edge of the absorber. The incident angle of
them is called acceptable half angle here. Fig. 5 shows the
acceptable half angle of each prism with monochromatic
design on a flat Fresnel lens as an example (Akisawa
et al., 2007). The horizontal axis indicates the position of
prisms 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 the
prism. Prisms located around the center have relatively
large acceptable half angle compared with prisms on the
outer side. It can be understood that some prisms can have
enough large acceptable half angle even though the upper
surface is flat. It suggests that the decoupled design with
lower lens height than the coupled design allows larger
acceptable half angle than the specified acceptance half
angle 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 satisfy
the requirement of the given acceptance half angle. They
need to have declined upper surface to hold the acceptable
half angle is equal to the acceptance half angle. In this outer
part both the decoupled design and the conventional coupled
design satisfy the same edge ray constraints. It means that the
curved shape in this region is common for both cases.
It is suggested from the discussion that dome-shaped
Fresnel lenses are not necessarily fully domed and can have
flat part around the center. It is advantageous because
prisms on the flat part have no undercut theoretically, which
is suitable for mold injection manufacturing processes. It
absorber
-θ +θ
short wave
length
long wave
length
Fig. 4. Two design points of each prism.
non-dimensional distance from the center x/d (-)
acceptablehalfangle(deg)
given acceptance half angle = 0.7deg
Fig. 5. Acceptable half angle of the prisms on a flat Fresnel lens (an
example).
880 A. Akisawa et al. / Solar Energy 86 (2012) 877–885

5. should be noted that prisms around the center of conven-
tional dome-shaped Fresnel lenses are arranged to remove
undercut shape for such manufacturing processes. The new
design method allows dome-shaped Fresnel lenses to have
no need of non-undercut prisms as the ideal shape. Conse-
quently prisms on the outer part need to have curved upper
surface, which is illustrated in Fig. 6.
In the proposed design method, two kinds of extreme
wave length of blue and red rays are adopted to investigate
the behavior of edge rays. Larger acceptable half angle is
explored as far as it is possible in the center flat part while
the given acceptance half angle is kept in the outer curved
part for both extreme edge rays.
4. Results of lens shape
4.1. Design parameters
In order to examine the new design method, investigation
here focuses on the effect of the dual wave length to cope with
chromatic aberration. The parameters for the dome-shaped
Fresnel lenses are summarized in Table 1. Concentration
ratio is set to be 500 suns. PMMA is assumed for the lens
material as before. The dome-shaped lens is rotationally
symmetrical. The feature of the flattened dome shape is also
taken into account. As mentioned before, the lens height is
decoupled with the acceptance half angle. Although the lens
height 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 1:468 þ 9:342=ðk À 123:5Þ ð2Þ
where n: refractive index, k: wave length (nm).
4.2. Result of dome shape
Here, three cases of dome-shaped lenses are examined
and compared to evaluate the effect of the proposed
method separately.
Case 1: single wave length (n = 1.49) at the center of
prisms.
Case 2: double wave length (n = 1.48 and 1.50) at the
center of prisms.
Case 3: double wave length (n = 1.48 and 1.50) at the
corners of prisms.
The results of each lens shape are shown in Fig. 7. It is
slightly surprising that the lens of Case 1 is almost flat
looking like a dish. In contrast, the lenses of Case 2 and
Case 3 seem bowls which has flat bottom at the center
and curvature around it. It indicates that the new design
method considering chromatic aberration requires signifi-
cant curvature to collect solar irradiation. Furthermore,
the lens of Case 3 is deeper than that of Case 2. Shifting
design point from the center to the corners leads to larger
curvature because incident rays have to be bent more,
which might cause disadvantage from the manufacturing
point of view.
It should be noted that prisms on the curved part have
undercut while prisms on the flat part have no undercut.
These dome-shaped lenses cannot be produced by ordinary
injection mold techniques due to the undercut prisms.
To capture larger dispersion, it is required to select wide
range of refractive index in the design procedure. It is
mathematically possible, but it will result in relatively deep
dome-shape, which induces smaller concentration ratio. Or
smaller acceptance half angle is achievable to have a certain
concentration 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 length
of 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. 6. New shape of non-imaging Fresnel dome lenses.
Table 1
Design parameters.
Item Value Note
Acceptance half angle 0.7°
Lens height 60 Represented in terms of
Width of prism 0.25 non-dimensional length
Thickness of prism 1.0 normalized with the half width of the
absorber
Geometrical
concentration ratio
506
Number of prisms 90 Lens radius = 22.5
Refractive index
Single wave length 1.49
Dual wave length 1.48/
1.50
A. Akisawa et al. / Solar Energy 86 (2012) 877–885 881

6. length of 550 nm. Therefore, 420, 550, 700, 880 nm of the
wave length are selected so that they are included in the
range. Ray tracing simulation was employed to estimate
the optical efficiency with beam irradiance. The refractive
index of Eq. (2) was incorporated in the simulation. Light
source is assumed to have square shape because actual
products of dome-shaped lenses are square as can be seen
in Fig. 2. To evaluate the performance of the lens products,
the light source is adjusted in that shape. The length of the
one side is 45=
ffiffiffi
2
p
¼ 32. The absorber is also assumed
square with the one side length of
ffiffiffi
2
p
.
Fig. 8 shows the results of each lens with inclined incident
angle of 0–1.0°. The performance of Case 1 indicates that the
wave length of 550 nm can be captured when the incident
angle is less than 0.5°. When the angle is equal to the design
acceptance half angle of 0.7°, most of the light of 550 nm
cannot get the absorber. The efficiency for the light of
700 nm and 880 nm start decreasing when the angles become
larger 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 less
than 0.6° although the efficiency is as low as 0.3 at the angle
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.
Fig. 9. Optical efficiency corresponding to design method (wave length of
300–800 nm).
882 A. Akisawa et al. / Solar Energy 86 (2012) 877–885

7. 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

8. of 0.7°. The lens of Case 2 captures the light in the wave
length range at the efficiency of 0.9 when the angle is less than
0.3°. The lens of Case 3 has the best performance among
them. For the design acceptance half angle of 0.7°, the
efficiency of every wave length is approximately 0.4-0.5.
Furthermore, the efficiency holds as high as 0.9 for the angle
range of 0–0.4°. It is found out that the deviation of the effi-
ciency behaviors is smaller than the others. The lens of Case
3 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 wave
length with two design points is advantageous compared
with the conventional method.
5. Performance evaluation
5.1. Optical efficiency
One of the indices of lens performance is optical efficiency
which indicates how much solar inlet is captured on the
absorber. The concentration ratio of a lens is calculated by
means of the geometrical concentration ratio multiplied with
the optical efficiency. Here, ray tracing simulation with solar
irradiance was employed to estimate the optical efficiency.
The range of solar wave length was decomposed into several
segments 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 of
the lenses, the authors focused on the visible wave length
of sun light, that is, 300–800 nm for the comparison. Again,
the light source is assumed to have square shape as the
previous analysis.
Fig. 9 shows the optical efficiency of the dome-shaped
lenses when solar incident angle changes from 0° (normal
direction) to 1.0°. As shown in the graph, optical efficiencies
are as high as 0.85–0.9 in the incident range of 0–0.4°. On the
other hand, the efficiencies become smaller when the incident
angle is larger than 0.4°. Eventually the efficiencies are
approximately 0.3 and 0.5 for Case 1 and Case 3 respectively
when the incident angle is equal to the acceptance half angle
of 0.7°. The lens of Case 3 has the highest optical efficiency
among them when the incident rays are inclined. The effi-
ciency of Case 2 is better than that of Case 1. The result
clearly reveals that the proposed method is effective to
improve the concentration performance of dome-shaped
lenses.
5.2. Flux on the absorber
The prediction that the optical efficiency of Case 3 is
about 0.5 indicates half of solar input gets onto the absorber
even though the incident angle is 0.6–0.7°. Fig. 10 illustrates
the flux distribution on the absorber for these three lenses
Fig 10. (continued)
884 A. Akisawa et al. / Solar Energy 86 (2012) 877–885

9. 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 of
Case 3 is stretched into the absorber and the flux peak still
remains on the edge of the absorber. It is the reason why
Case 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. The
points to improve the concentration performance are that
using two wave length to consider chromatic aberration
explicitly and that two design points located at the corners
of each prism where the edge ray goes through. For solar
concentration, achromatic lenses are not needed in non-
imaging optics because the essential thing is to collect solar
irradiance 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 certain
range of wave length by adjusting prism angles. Further-
more, lenses designed in this study has flat part around
the center of the lens while curved part outside. The shape
is in contrast to that of fully curved dome-shaped lenses
which the authors developed before. The new dome shape
is advantageous from the manufacturing point of view
because the flat part does not have undercut prisms.
Compared with conventional design method, i.e. using
one wave length and one design point located at the center
of prisms, the proposed methods improve the optical effi-
ciency when the incident angle is equal to the acceptance
half angle. On the other hand, numerical examples repre-
sent that new design lenses have much deeper dome shape
in contrast to conventionally designed dome-shaped lens. It
implies 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 and
analyze lenses to collect much wider range of the wave
length.
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