Miniaturization of Fresnel lenses for solar
concentration: a quantitative investigation
Fabian Duerr,* Youri Meuret, and Hugo Thienpont
Applied Physics and Photonics Department (TONA), Vrije Universiteit Brussel,
Pleinlaan 2, 1050 Brussels, Belgium
*Corresponding author: firstname.lastname@example.org
Received 21 January 2010; revised 14 March 2010; accepted 22 March 2010;
posted 24 March 2010 (Doc. ID 123080); published 14 April 2010
Sizing down the dimensions of solar concentrators for photovoltaic applications offers a number of pro-
mising advantages. It provides thinner modules and smaller solar cells, which reduces thermal issues. In
this work a plane Fresnel lens design is introduced that is first analyzed with geometrical optics. Because
of miniaturization, pure ray tracing may no longer be valid to determine the concentration performance.
Therefore, a quantitative wave optical analysis of the miniaturization’s influence on the obtained con-
centration performance is presented. This better quantitative understanding of the impact of diffraction
in microstructured Fresnel lenses might help to optimize the design of several applications in nonima-
ging optics.© 2010 Optical Society of America
030.1640, 080.0080, 080.4298, 220.0220, 220.1770, 350.6050.
Integral to photovoltaics is the need to provide im-
proved economic viability. To become more economic-
ally viable, photovoltaic technology has to be able to
harness more light at less cost. As well as optimizing
solar cell technology toward higher efficiencies and
for potential cost reduction, alternative solar concen-
tration concepts have provided cause for pursuit.
Due to their straightforward design and low cost,
Fresnel lenses are widely used in solar concentration
systems . In the past, a wide range of different
Fresnel lenses has been investigated. For a detailed
description of nonimaging Fresnel lenses we refer
the reader to the book by Leutz and Suzuki .
Miniaturization of plane Fresnel lens concentra-
tors leads to several advantages. By scaling down
the aperture of the lens, the working distance de-
creases while maintaining concentration perfor-
mance. Scaling down provides thinner modules,
to the millimeter range . A further aspect of the
miniaturization process is the avoidance of techni-
cally demanding cooling [4,5]. Potential drawbacks
of the miniaturization may occur from alignment er-
rors and current matching of a chain of small photo-
voltaic cells . Furthermore, the question arises
whether there is a lower limit to this miniaturization
structure size of the Fresnel lenses, effects of round-
ing of sharp edges and other imperfections will start
to impair the expected performance. With continued
work on nano-optics, however, it is difficult to predict
how serious these problems will be as research and
development proceeds. An analysis of the effects of
miniaturization on the concentration performance
from an optical modeling point of view is thus
The assumption of an unchanged concentration
performance when scaling the dimensions of the so-
lar concentrator system down neglects certain physi-
cal phenomena. Material properties like absorption
favor thinner modules, but the miniaturization of a
Fresnel lens also has a lower limit. When downsizing
a Fresnel lens the prisms of the Fresnel lens be-
come smaller, grating effects that diffract the light
away from the focal position will gain influence,
and pure ray tracing based on geometrical optics
© 2010 Optical Society of America
20 April 2010 / Vol. 49, No. 12 / APPLIED OPTICS 2339
may no longer be valid for the determination of the
In terms of microstructured Fresnel lenses, a qua-
litative description of this limit is provided in the
literature [7,8]. In a broader framework, microstruc-
tured Fresnel lenses were studied by others in the
domain of digital holographic microscopy  and
diffractive optical elements for beam shaping .
The transition between refractive and diffractive
micro-optical components was the subject of further
analysis where the optical components were treated
as coherently illuminated blaze gratings [11,12].
However, the quantitative impact of diffraction in
concentrating photovoltaics has not been investi-
gated yet to our knowledge. A better quantitative
understanding of the impact of diffraction in micro-
structured Fresnel lenses will help to optimize the
design of such concentrated photovoltaic applica-
tions and possibly other nonimaging systems. For
this reason a quantitative wave optical analysis of
the miniaturization’s influence on the obtained con-
centration performance based on the spatial partial
coherence of the sunlight is presented in this work.
In Section 2 the design of a plane Fresnel lens and
an adequate figure of merit are introduced. Based
on pure geometrical optics, the concentration per-
formance of these Fresnel lenses is investigated in
Section 3. The performance limits are analyzed anal-
ytically and quantified by simulations. (All simula-
tions in this paper are performed with Advanced
Systems Analysis Program (ASAP) version 2009
V1R1 from Breault Research Organization.) How-
ever, with a decreasing size of the prisms, the ques-
tion arises if the geometrical optics approach is still
valid. Therefore, a simulation model based on a
Gaussian beam propagation approach is presented
in Section 4 with the objective to make quantitative
conclusions in terms of the concentration perfor-
mance for spatial partially coherent illumination.
In Section 5, this model is used for a monochromatic
analysis of microstructured Fresnel lenses that al-
lows for a quantitative determination of the reduced
concentration performance due to wave optical ef-
fects. Section 6 completes these considerations by
also taking the spectrum of the source into account.
Finally, in Section 7, conclusions are drawn and an
outlook is given.
The (overall) design process of the Fresnel lenses in-
vestigated in this paper is based on a linear prism
structure in order to provide an analytical treatment.
The prisms are aligned in such a way that the equis-
paced center points of the prisms’ slopes are on a
straight line orthogonal to the optical axis. All prisms
have the same width d. Figure 1 shows a schematic
representation of the profile of such a component.
The three-dimensional Fresnel lens is then com-
posed of concentric bands by rotating the profile
around the optical axis.
Fresnel Lens Design Algorithm and Figure of Merit
Instead of the working distance W and the lens
diameter D, it is possible to use the f-number
F=# ¼ W=D, well known from imaging optics. For a
given f-number and a number of prisms p (whereas
2pd ¼ D), the collecting angle γðiÞ between the
optical axis and the central ray of the ith prism is
γðiÞ ¼ 90° − arctan
i − 1=2
The wedge angle αðiÞ for the ith prism is then calcu-
lated by Eq. (2) for a given index of refraction n at a
certain design wavelength λ:
αðiÞ ¼ arccos
n − cosγðiÞ
ð1 þ n2− 2ncosγðiÞÞ1=2
As base material for all Fresnel lenses in this work,
poly(methyl methacrylate) (PMMA) is used and the
refraction index as a function of the wavelength is
calculated using a modified Cauchy’s equation .
Even though these expressions are derived for per-
fect on-axis illumination, the equations approxi-
mately hold considering the relatively small half
divergence angle of approximately 4:7mrad for sun-
light . Subsequently it is assumed that the edges of
the prisms are perfectly parallel to the optical axis.
Of course, the validity of this assumption will depend
on the manufacturing process and hence on the de-
sired size of the structures. The lower bound of the
f-number is determined by the critical angle for total
internal reflection and is thus constrained by the
lens’ material as well as the divergence angle of
the considered source. To provide comparability
and to cope with diverse Fresnel lenses with varying
diameters and different numbers of prisms, the
Marked parameters are the working distance W, lens diameter D,
prism width d, wedge angle α, and collecting angle γ.
Schematic representation of the used Fresnel lens design.
2340APPLIED OPTICS / Vol. 49, No. 12 / 20 April 2010
geometrical concentration ratio is chosen as a figure
of merit to evaluate the concentration performance.
The geometrical concentration ratio derived from si-
mulated spot distributions depends on an exact defi-
nition of the captured energy by the receiver. For
instance, one possible definition would be that the re-
ceiver plane captures all refracted rays. In this pub-
lication, the circular receiver is defined in such a way
that 90% of the transmitted energy is collected.
Hence, the concentration ratio is defined by the frac-
tion of the entrance pupil Aσ(which is equivalent to
the area of the Fresnel lens) and the receiver area
Arec, defined before. This geometrical concentration
ratio is multiplied by 0.9 in order to take the 90%
energy transfer efficiency into account:
C ¼ 0:9Aσ
Furthermore, optical losses due to material absorp-
tion and Fresnel reflections are neglected—only total
internal reflection is taken into account.
Based on the Fresnel lens design algorithm intro-
duced in Section 2, the concentration performance
is investigated in terms of pure geometrical optics.
Assuming an idealized Fresnel lens shape, the f-
number and the number of equispaced prisms are
sufficient for a full description. First, the influence
of two fundamental system parameters on the con-
centration performance is investigated analytically,
before concluding simulation results complete this
geometrical optics investigation.
Geometrical Concentration Performance
the Concentration Performance
As a consequence of the linear prisms, the finite
width d has an impact on the achievable concentra-
tion ratio. In addition, the half divergence angle θmax
of the considered source limits the concentration per-
formance as well. In order to estimate the impact of
both factors separately, it is useful to consider two
For the impact of the finite prisms’ width d, the di-
vergence of the source is neglected and perfect on-
axis illumination is thus assumed. In this case, the
corresponding one-dimensional spot size for each
prism in the receiver plane is determined by trigono-
metric considerations. Figure 2(a) shows one exemp-
lary prism where the refracted edge rays and the
receiver build a trapezoid. The characteristic spot
size DpðiÞ for the ith prism in the receiver plane is
Fundamental (Geometric) System Parameters Limiting
cosðβðiÞ − αðiÞÞ:
Equation (4) can be normalized by the lens diameter
D, which results in the dimensionless characteristic
spot ratio DpðiÞ=D that is reciprocally proportional to
the number of prisms p. The wedge angle αðiÞ results
from Eqs. (1) and (2), and the refraction angle βðiÞ is
calculated by Snell’s law. For homogeneous illumina-
tion the characteristic spot possesses a homogeneous
intensity distribution as well.
Contrariwise, for the impact of the divergence an-
gle of the source it is useful to consider the prisms
having a hyperbolic shape. In this case the on-axis
illumination is perfectly imaged to the spot center
and the only reason for an extended spot size is
the divergence of the considered source. Figure 2(b)
shows an exemplary hyperbolic prism for two rays
with maximum divergence angles of ?θ. The corre-
spondent characteristic spot size is calculated by
DσðiÞ ¼ D
ðtanγþðiÞ − tanγ−ðiÞÞ:
The two collecting angles γ?ðiÞ are derived from
the incident angles αðiÞ ? θ with regard to the
γ?ðiÞ ¼ −αðiÞ þ arcsinðnsinðαðiÞ ? θÞÞ:
Again, the characteristic spot size of Eq. (5) can be
normalized by the lens diameter D, which results
DσðiÞ=D. The sinusoidal dependence of Snell’s law
causes an inhomogeneous intensity distribution for
the characteristic spot, but for small divergence an-
gles a homogeneous intensity distribution is approxi-
for different Fresnel lenses in order to find the predo-
minant term on the achievable concentration ratio.
Figure 3 shows the characteristic spot ratios Dp=D
and Dσ=D against the collecting angle γ for F=# ¼ 1,
the Sun’s divergence angle 4:7mrad, and different
vide the concentration performance in three dimen-
sions, they are already valuable to estimate the
divergence angle of the source (b) on characteristic spot sizes Dp
and Dσ, respectively.
Exemplary prisms for impact of finite sized prism (a) and
20 April 2010 / Vol. 49, No. 12 / APPLIED OPTICS2341
and provide an answer to the question of whether it
is useful to increase the number of prisms or not.
The three chosen Fresnel lenses clearly show the de-
creasing influence of the finite sized prisms with an
increasing number of prisms. With a mainly prism-
dominated concentration performance for p ¼ 10
prismsandatransitionzoneforp ¼ 20,thehalfdiver-
gence angle of the Sun is clearly the dominant factor
for p ¼ 50 prisms.
For a more quantitative estimation of the overall
concentration performance of a certain Fresnel lens
the two characteristic spot ratios have to be succes-
sively convolved for each prism and the resulting
function should be integrated. In two dimensionsthis
allows for an exact calculation of the resulting spot
ratio. A further step could be a three-dimensional ex-
tension of this model that also takes skewed rays into
account. Such an extended model might allow opti-
mization of the prisms’ widths for a constant number
of prisms in terms of the concentration performance.
However, that task is not a goal of this study.
Additional to the considerations above, the limita-
tions of the concentration performance are investi-
gated based on geometrical optics (ray tracing). All
simulations were realized with an emitting disk
source of one million rays for a design wavelength
of 600nm and a half divergence angle of 4:7mrad.
Both, positions and directions of the particular rays
are randomly chosen within the aperture of the lens
and the divergence angle of the source using uniform
distributions. Figure 4 shows the concentration ratio
for three different f-numbers against the used num-
ber of prisms.
As expected from previous results, the concentra-
tion performance increases with the number of
prisms and goes into saturation for all three graphs
as p goes to infinity. These limiting cases correspond
to hyperbolic shaped lenses of the same f-numbers.
So far, all deduced results only depend on the f-
number of the Fresnel lenses and the number of
prisms—no matter what the actual lens dimensions
Ray Tracing Results
may be. As mentioned in the introduction, a minia-
turization of plane solar concentrators provides pro-
mising advantages in concentrated photovoltaics. It
is possible to link the simulated concentration ratios
to a concrete system’s dimensions. However, the
question arises if the so far used geometrical optics
approximation is valid for all scales.
cing is justified as long as the coherence length of the
incident light is negligible in comparison to the di-
mensions of the illuminated elements . In solar
engineering the sunlight is normally considered as
a spatial and temporal incoherent source. Even
though this assumption holds for most applications
in solar engineering, it is not always appropriate.
Early works from van Cittert  and Zernike 
could prove that even a spatial incoherent quasi-
monochromaticsourcegives rise toa spatialpartially
coherent light field at a certain propagation distance.
For a more detailed introduction to coherence theory
we refer the reader to Mandel and Wolf . The im-
plication for filtered sunlight on the surface of the
Earth was later deduced by Hopkins . The radius
of the circle that is quasi-coherently illuminated by a
monochromatic source of angular radius θ is given by
Miniaturization in Light of Coherence Theory
where rcdenotes the transverse coherence radius for
which the degree of coherence drops to 0.880. From
this it follows that the transverse coherence length of
filtered sunlight on the surface of the Earth is of the
order of approximately rc;Sun≈ 0:06mm .
In order to quantitatively investigate the concen-
tration performance for miniaturized Fresnel lenses
it is desirable to adopt a universally valid optical si-
mulation model that works for both the geometrical
optics regime as well as for the transition to smaller
dimensions where variations from geometrical optics
may arise due to partially coherent illumination.
Sun’s half divergence angle 4:7mrad and Dp=D for three different
numbers of prisms against collecting angle γ.
(Color online) Characteristic spot ratios Dσ=D for the
prisms for three different f-numbers.
(Color online) Concentration ratio against number of
2342 APPLIED OPTICS / Vol. 49, No. 12 / 20 April 2010
For the wave optical simulations in this work a
Gaussian beam propagation approach is chosen. A
principal motivation may be obtained as follows:
The Sun as an extended source is composed of
point sources. Each point source gives rise to a spa-
tially coherent spherical wave. Since fluctuations in
the light from different source points of a thermal
source can be assumed to be mutually independent,
there is no fixed phase relationship. The overall in-
tensity distribution is therefore obtained by adding
up the intensities from each point source incoher-
ently, at each detected point. Taking the distance
of the Sun and the Earth into account, it is possible
to use a plane wave approximation for the wave
fronts coming from different point sources at the
position of the Earth.
In ASAP, these plane wave fronts coming from dif-
ferent directions are created as flattop profiles com-
posed of a superposition of approximately 125,000
constituent Gaussian beams . For each plane
wave front, the constituent Gaussian beams carrying
both phase and amplitude information are propa-
gated independently through the optical system by
geometrical optics methods [19–21]. After the ray
trace of the constituent Gaussian beams is complete,
ASAP sums these individual Gaussian beams coher-
ently in the receiver plane. The direction of each
plane wavefront is randomly chosen within the Sun’s
half divergence angle θ ¼ 4:7mrad. The intensity
patterns of all plane wave fronts are added up
incoherently and divided by the number of overlaid
intensity patterns, which concludes the averaging
In the ideal case an infinite number of these multi-
directional, statistically independent plane wave
fronts has to be superimposed. To achieve adequate
statistical accuracy, repeated simulations were per-
formed and compared to verify whether a stable re-
sult in terms of concentration ratio was achieved or
not. As an outcome of these repeated simulations,
each wave optical simulation in this work consists
of 5000 statistically independent superimposed wave
fronts. Based on the fact that the lenses are rota-
tional symmetric, it is sufficient to use a one dimen-
sional detector. This helped to reduce the runtime of
the simulations to an acceptable effort.
In order to investigate the quantitative impact of
wave optical effects on the concentration perfor-
mance of a miniaturized Fresnel lens, an explicit lens
with a fixed diameter as well as a favored f-number
is chosen. Using these fixed parameters, the number
of prisms of the Fresnel lens can now be varied, and
the corresponding concentration ratio is determined.
As already shown in Section 3 an increasing number
of prisms results in a monotonically increasing con-
centration ratio for geometrical optics. Next, these
simulations are performed again at a design wave-
length of 600nm, this time using the partial coher-
ence model introduced in Section 4.
At first, a Fresnel lens with a diameter D ¼ 10mm
and an f-number F=# ¼ 1 is assumed. The intensity
cross sections for both the ray tracing and the partial
coherent illumination are simulated and plotted to-
gether, which allow for a descriptive comparison of
both optical models. Figure 5 shows the cross-section
profiles for two such Fresnel lenses consisting of p ¼
10 (a) and p ¼ 50 (b) prisms, respectively.
For a moderate number of prisms, such as p ¼ 10,
the wave optical simulation shows excellent agree-
ment with the intensity distribution obtained by
ray tracing. This result supports the validity of the
chosen wave optical approach. However, for the high-
er number of 50 prisms, the wave optical simulation
result is different from the ray tracing result.
Although the wave optical cross section suggests a
minor impact on the concentration performance, it
should be made clear that instead of the cross sec-
tion, the enclosed energy of the full two dimensional
spot forms the basis of the calculated concentration
ratio. The total energy EðrÞ enclosed by the radius r
is given by
Monochromatic Wave Optical Simulations
(Color online) Intensity cross sections for p ¼ 10 (a) and p ¼ 50 (b): comparison of geometrical optics (GO) and wave optical (WO)
20 April 2010 / Vol. 49, No. 12 / APPLIED OPTICS 2343
with the integrand cross section Iðr0Þ weighted by the
radius r0. Rather than comparing the intensity cross
sections, it is feasible to calculate the concentration
ratios to obtain a quantitative analysis. Figure 6
shows the comparison of the geometrical (GO) and
wave optical (WO) simulated concentration ratios
against the number of prisms from 10 up to 50 with
a step size of 5 for D ¼ 10mm and three different f-
numbers, namely, F=# ¼ 0:7 (a), F=# ¼ 1 (b), and
F=# ¼ 1:3 (c).
For all three lenses both simulation approaches
show excellent agreement up to p ¼ 20 prisms. How-
ever, with a higher number of prisms, in all three
cases the concentration ratios for the wave optical si-
mulations fall short of the ray tracing results due to
the partial coherent illumination. This difference be-
tween geometrical and wave optical simulations
clearly increases with a decreasing f-number. With
this approach it is thus possible to analyze whether
there is an impact due to partial coherent illumina-
tion and additionally also to estimate the changed
concentration ratio itself. The reduced concentration
ratio is directly linked to an increased receiver size,
which again collects 90% of the transmitted energy.
Even though the results from Section 5 are already
providing interesting quantitative results about the
transition of geometrical optics toward regionswhere
wave optical effects are increasingly important, a
polychromatic investigation is essential for most
practical applications. For refractive concentrators,
dispersion plays a key role as a limiting factor for
the concentration performance. Due to dispersion,
the optimal receiver position moves closer to the
Fresnel lens for increasing wavelengths and in the
opposite direction for wavelengths lower than the de-
sign wavelength. Thus, the effective f-number and
concentration ratio always depend on the chosen de-
sign wavelength and the considered spectrum. For
this reason it is not sufficient to discuss the wave
optical impact on the concentration performance
without taking the spectral distribution into account.
For a given design wavelength of 600nm, the
wavelength dependence of the concentration ratio
is illustrated based on ray tracing simulations for
a spectrum from 450nm up to 1000nm with a step
size of 10nm. These simulations are repeated for dif-
ferent numbers of prisms. Figure 7(a) shows the con-
centration ratio against the wavelength for three
different prism numbers and a fixed detector position
at the design wavelength. Additionally, Fig. 7(b)
shows the comparison of the concentration ratio
against the wavelength for geometrical (GO) and
wave optical (WO) simulations for p ¼ 35 prisms
and a band width of 200nm, again with a step size
Although the peak concentration ratio at the de-
sign wavelength is highly dependent on the number
of prisms used, the effective concentration ratio is
particularly dependent on the considered spectral
band width for which the light should be collected
at the receiver. For a solar concentrator that collects
most of the Sun’s spectrum, only moderate concen-
tration ratios are achievable with such a Fresnel
lens. Recalling the reduction of the concentration ra-
tio due to wave optical effects in Fig. 6, it follows that
the wave optical impact due to partial coherent illu-
mination is small compared with the limitations due
However, as Fig. 7(b) emphasizes, in cases where
either only a portion of the spectrum is used, where
the spectrum is split using solar cells that are band-
gapped in a particular wavelength range [22,23], or
againstthe number of prisms for geometrical optics (GO) and wave
optical (WO) simulations. The f-number varies from F=# ¼ 0:7 (a),
F=# ¼ 1 (b), to F=# ¼ 1:3 (c).
(Color online) Comparison of the concentration ratio
2344APPLIED OPTICS / Vol. 49, No. 12 / 20 April 2010
color-corrected Fresnel lenses are used , the min-
iaturization is more sensitive to partially coherent il-
lumination. In any case, it is necessary to take that
part of the solar spectrum into account that will be
used to find the miniaturization limits for that spe-
cific source-concentrator system.
Within the scope of this work the concentration per-
formance of microstructured Fresnel lenses has been
investigated. Contrary to previous investigations,
the main objective was a quantitative analysis of
the concentration ratio in the transition region
where ray tracing is no longer valid. The wave optical
approach used to simulate the partial spatial coher-
ence of sunlight showed excellent agreement within
the regime of geometrical optics. With a decreasing
structure size of the Fresnel prisms it was possible
to determine the increasing influence of wave optics
on the concentration performance in a quantita-
Furthermore, it became evident that, besides the
structure size, the considered spectrum of the source
plays a key role. The reduction of the concentration
ratio due to wave optical effects is small in compar-
ison to dispersion effects, but it has to be taken into
account when Fresnel lenses are miniaturized and
designed for a reduced spectrum. Based on the quan-
titative investigation of the concentration perfor-
mance, it is therefore possible to design diffraction
limited microstructured refractive concentrators
that meet particular demands.
Our work reported in this paper was supported in
part by the Research Foundation—Flanders (FWO-
Vlaanderen) that provides a PhD grant for Fabian
Duerr (grant number FWOTM510) and in part by
the IAP BELSPO VI-10, the Industrial Research
Funding (IOF), Methusalem, VUB-GOA, and the
OZR of the Vrije Universiteit Brussel.
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(Color online) (a) Concentration ratio against wavelength for three different Fresnel lenses with p ¼ 10, 30, and 50 prisms,
20 April 2010 / Vol. 49, No. 12 / APPLIED OPTICS2345
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