Down scaling of micro-structured Fresnel lenses for solar
concentration - a quantitative investigation
Fabian Duerr*,a, Youri Meureta and Hugo Thienponta
a Vrije
Universiteit Brussel, Faculty of Engineering Sciences, Brussels Photonics Team
B-PHOT, TONA-FirW, Pleinlaan 2, 1050 Brussels, Belgium
ABSTRACT
Scaling down the dimensions of concentrating photovoltaic systems based on plane Fresnel lenses has several
promising advantages. By conserving a designed concentration ratio and reducing the aperture size of the lens,
the working distance decreases as well. This provides thinner modules and the dimensions of the used solar
cells can be scaled down to the millimeter range. An important benefit of this miniaturization process is the
avoidance of technically demanding cooling.
In this work the design of a plane Fresnel lens is introduced and the basic limitations concerning the achievable
concentration ratio are investigated based on geometrical optics. However, accompanied by the down scaling of
the prism dimensions, pure ray tracing based on the geometrical optics approximation may no longer be valid for
the determination of the concentration ratio. In terms of micro-structured Fresnel lenses for solar concentration,
only a qualitative description of this limit - typically a rule of thumb - is provided in the literature. For this
reason a quantitative investigation of the influence of the prisms’ down scaling and thus the appearing wave
optical effects on the obtained concentration ratio is presented. In a final step the introduced monochromatic
investigations are extended to a polychromatic analysis.
This allows for the prediction of the influence of miniaturization on the effective concentration ratio for a given
spectrum and thus the adequate size of the receiver. A better quantitative understanding of the impact of
diffraction in micro-structured Fresnel lenses might help to optimize the design of several applications in nonimaging optics.
Keywords: Micro-structured Fresnel lenses, concentrating photovoltaics, partial coherence, Gaussian beam
propagation, optical modeling
1. INTRODUCTION
Integral to photovoltaics is the need to provide improved economic viability. To become more economically
viable, photovoltaics has to be capable to harness more light at less cost. Besides optimizing photovoltaic cell
technology towards higher efficiencies and for potential cost reduction, different solar concentration concepts
have provided cause for pursuit. Due to their straightforward design and low cost, Fresnel lenses are widely used
in solar concentration systems.1 In the past, a plurality of different Fresnel lenses has been investigated. For a
detailed description of nonimaging Fresnel lenses we refer to the book by Leutz and Suzuki.2
Miniaturization of plane Fresnel lens concentrators leads to several advantages. By scaling down the aperture of
the lens to scale, the working distance decreases whilst maintaining concentration performance. This provides
thinner modules enabling dimension reduction of the photovoltaic cells to the millimeter range.3 A further aspect
of the miniaturization process is the avoidance of technically demanding cooling.4, 5
The assumption of an unchanged concentration performance when scaling down the dimensions of the solar
concentrator system neglects certain physical phenomena. Material properties like absorption favor thinner
modules. But even for a Fresnel lens with ideal optical properties the scale invariance of the concentration
performance is bounded below. When scaling a Fresnel lens down, the prisms of the Fresnel become smaller,
*E-mail correspondence: [email protected]; Telephone: +32 (0)2 6293570
grating effects which diffract the light away from the focal position will gain influence, and pure ray tracing
based on the geometrical optics approximation may no longer be valid for the determination of the concentration
performance.
In terms of micro-structured Fresnel lenses, a qualitative description of this limit is provided in the literature.6, 7
In a broader framework, the transition between refractive and diffractive micro-optical components was subject
of further analysis where the optical components were treated as coherently illuminated blaze gratings.8, 9
A better quantitative understanding of the impact of diffraction in micro-structured Fresnel lenses will help
to assess the lower bound for miniaturization of such concentrating photovoltaics systems and possibly other
nonimaging systems. For this reason a quantitative wave optical analysis of the miniaturization’s influence on
the obtained concentration 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 performance of these Fresnel lenses is investigated in section 3. The
concentration performance is quantified by simulations∗ . However, with a decreasing size of the prisms, the
question arises whether the applied 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 performance for spatial partially coherent illumination. In section 5,
this model is used for a monochromatic analysis of micro-structured Fresnel lenses that allows a quantitative
determination of the lower bound of the scalability due to wave optical effects. Section 6 completes these
considerations by taking the spectrum of the source also into account. Finally, in section 7, conclusions are
drawn and an outlook is given.
2. FRESNEL LENS DESIGN AND FIGURE OF MERIT
The design process of the Fresnel lenses investigated in this paper is based on a linear prism structure in order
to provide an analytical treatment. The prisms are aligned in such a manner that the equispaced center points
of the prisms’ slopes are on a straight line perpendicular to the optical axis. All prisms have the same width d.
Fig. 1 shows an exemplary schematic representation of the profile of such a component. The three-dimensional
Fresnel lens is then composed of concentric bands by rotating the profile around the optical axis.
Instead of the working distance W and the lens diameter D, it is feasible to use the f-number F/# = W/D, well
known from imaging optics. For a given f-number and a number of prisms p (whereas 2pd = D), the collecting
angle γ(i) between the optical axis and the central ray of the ith prism is calculated by Eq. (1).
2pF/#
◦
(1)
γ(i) = 90 − arctan
i − 1/2
The wedge angle α(i) for the ith prism is then calculated by Eq. (2) for a given index of refraction n at a certain
design wavelength λ.
!
n − cos γ(i)
α(i) = arccos
(2)
1/2
(1 + n2 − 2n cos γ(i))
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.10 Even though these
conditional equations are derived for perfect on-axis illumination, the equations approximately hold considering
the relatively small half divergence angle of approximately 4.7 mrad for sunlight.2 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 desired 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 half divergence angle of the considered source.
To provide comparability and cope with various Fresnel lenses with varying diameters and different numbers
∗
All simulations in this paper are performed with Advanced Systems Analysis Program (ASAP) version 2009 V2R2
from Breault Research OrganizationTM .
Figure 1. Exemplary schematic representation of the used Fresnel lens design for p = 5 prisms. Marked parameters are
the working distance W , lens diameter D, prism width d, wedge angle αi and collecting angle γi .
of prisms, the geometrical concentration ratio is chosen as a figure of merit to evaluate the concentration performance. The geometrical concentration ratio derived from simulated spot distributions depends on an exact
definition of the captured energy by the receiver. For instance, one possible definition would be that the receiver
plane captures all refracted rays. In this publication, 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 fraction 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 = 0.9
Aσ
Arec
(3)
Furthermore, optical losses due to material absorption and Fresnel reflections are neglected - only total internal
reflection is taken into account.
3. CONCENTRATION PERFORMANCE BASED ON GEOMETRICAL OPTICS
Based on the Fresnel lens design introduced in section 2, the concentration performance is investigated in terms
of pure geometrical optics (ray tracing). Assuming a perfect prism shape, the f-number and the number of equispaced prisms are sufficient for a full description of the Fresnel lens. As a consequence of the linear prism profiles,
the finite width d has an impact on the achievable concentration ratio. In addition, for a specified number of
prisms the corresponding f-number can be optimized for optimal concentration performance.
All ray tracing simulations were realized using an emitting disk source of one million rays for a design wavelength
of 600 nm and a half divergence angle of 4.7 mrad. Both, positions and directions of the particular rays were randomly chosen within the aperture of the lens and the divergence angle of the source using uniform distributions.
Fig. 2 shows the concentration ratio against the used number of prisms for three different f-numbers. With
an increasing number of prisms and thus a decreasing prism width d, the concentration performance for each
f-number increases monotonically and goes into saturation as p goes to infinity. These limiting cases correspond
to hyperbolic shaped lenses of the same f-numbers.
So far, all deduced simulation results only depend on the number of prisms and the f-number of the Fresnel lenses
- no matter what the actual Fresnel lens dimensions may be. As mentioned in the introduction, a miniaturization of plane solar concentrators provides promising advantages in concentrating photovoltaics. Based on this
Figure 2. Concentration ratio C against number of prisms p for three different f-numbers based on geometrical optics.
unrestricted scalability for geometrical optics, it is possible to design for a desired concentration performance
and link the correspondent Fresnel lens parameters to concrete system dimensions. However, the question arises
if the so far used geometrical optics approximation is valid for any scale.
4. SCALABILITY IN LIGHT OF COHERENCE THEORY
The geometrical optics approximation is justified as long as the coherence length of the incident light is negligible
in comparison to the dimensions of the illuminated elements.11 In solar engineering the sunlight is normally
considered as a spatial and temporal incoherent source, for which ray tracing is most suitable. Even though this
assumption holds for most applications in solar engineering it is not a priori always true.
Early works from van Cittert12 and Zernike13 could prove, that even a quasi-monochromatic spatially incoherent
source gives rise to a spatial partially coherent light field at a certain propagation distance† . The implications
for filtered sunlight on the surface of the earth were later deduced by Hopkins.14 The radius of the circle which
is quasi-coherently illuminated by a monochromatic source of an angular radius θ is given by Eq. (4):
rc =
0.16λ
sin(θ)
(4)
where rc denotes 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 ≈ 0.06 mm.11
In order to quantitatively investigate the concentration performance for miniaturized Fresnel lenses at any scale
it is desirable to adopt a universally valid optical model that works for both - the geometrical optics regime,
where full scalability is provided, as well as for the transition to smaller scales, where variations from geometrical
optics may arise due to partially coherent illumination by sunlight. 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 spatially 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 intensity distribution is therefore
obtained by adding up the intensities from each point source incoherently, at each detected point. Taking the
†
For a more detailed introduction to coherence theory we refer to the book of Mandel and Wolf.11
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. These plane wave fronts, incident on the
Fresnel lens aperture, are created within ASAP as flat top profiles composed of a superposition of approximately
125000 constituent Gaussian beams.15 The direction of each plane wave is randomly chosen within the sun’s
half divergence angle θ = 4.7 mrad.
For each wave front the constituent Gaussian beams are propagated through the optical system and added up
coherently in the receiver plane. Finally, 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 process. In the ideal case
an infinite number of these multi-directional, statistically independent plane wave fronts have to be superimposed.
To achieve adequate statistical accuracy each wave optical simulation in this work consists of 5000 statistically
independent directions. This total number was verified by the comparison of multiple simulation results whether
a stable result in terms of concentration performance for a given ensemble was achieved or not. Based on the
fact that the lenses are rotational symmetric it is sufficient to use a one dimensional detector. Therefore it was
possible to reduce the run-time of the simulations to an acceptable effort.
5. CONCENTRATION PERFORMANCE BASED ON WAVE OPTICS
Assuming that the simulated concentration performance based on geometrical optics in section 3 is not valid
for all scales, the simulations are performed again at a design wavelength of 600 nm, this time using the partial
coherence model introduced in the preceding section. In order to investigate the quantitative impact of wave
optical effects on the concentration performance, it is necessary to choose a fixed lens diameter plus a desired
f-number.
At first, a Fresnel lens with a diameter D = 10 mm and an f-number F/# = 1 is investigated. The intensity
cross sections for both the ray tracing and the partial coherent illumination are simulated and plotted together
which allow for a descriptive comparison of both optical models. Fig.3 shows the cross-section profiles for two
such Fresnel lenses consisting of p = 10 (a) and p = 50 (b) prisms, respectively. For a moderate number of
prisms, such as p = 10, the wave optical simulation shows excellent agreement with the intensity distribution
obtained by ray tracing. This result supports the validity of the chosen wave optical approach. However, for the
higher number of 50 prisms, the wave optical simulation result differs from the ray tracing result. Although this
difference suggests a minor impact on the concentration performance, it should be made clear that the enclosed
energy for the full two dimensional spot and thus the concentration performance is already significantly reduced.
Rather than comparing the intensity distributions, it is feasible to calculate the concentration ratios to obtain a
quantitative analysis.
To determine the lower bound of the scalability with regard to the concentration performance an initial Fresnel
is chosen and the concentration ratio is calculated by geometrical optics and wave optical simulations. Subsequently, this initial Fresnel lens is successively scaled down and each time the concentration ratio is calculated
Figure 3. Intensity cross sections in arbitrary units for p = 10 (a) and p = 50 (b) prisms: Comparison of geometrical
optics (GO) and wave optical (WO) simulations.
Figure 4. Comparison of the concentration ratio against the lens diameter for geometrical optics (GO) and wave optical
(WO) simulations. The number of prisms changes from p = 10 (a), 20 (b), 30 (c) and 40 (d), respectively. The f-number
F/# = 1 remains unchanged in all simulations.
by geometrical optics and wave optics. Fig. 4 shows the results of these simulations for Fresnel lens diameters
from D = 20 mm down to 6 mm with a step size of 1 mm, an f-number of F/# = 1 for p = 10 (a), 20 (b), 30
(c) and 40 (d) prisms, respectively.
In case of 10 prisms, the calculated concentration ratios of both simulation models coincide for all diameters.
However, with a higher number of 20, 30 or 40 prisms the concentration ratios for the wave optical simulations
begin to fall short of the ray tracing results due to partial coherent illumination. This discrepancy clearly increases with an increasing number of prisms and a decreasing Fresnel lens diameter.
With this simulation approach it is thus possible to analyze whether there is an impact due to partial coherent
illumination as well as to determine the lower bound of the scalability and estimate the changed concentration
ratio. The reduced concentration ratio is directly linked to an increased receiver size which again collects 90%
of the transmitted energy.
6. POLYCHROMATIC CONSIDERATIONS
Even though the results from section 5 are already providing interesting quantitative results about the transition
of geometrical optics towards regions where 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
design wavelength. Thus, the effective f-number and concentration ratio always depend on the chosen design
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
Figure 5. (a) Geometrical optics based concentration ratio against wavelength for three different Fresnel lenses with
p = 10, 30 and 50 prisms, respectively and F/# = 1. (b) Comparison of the concentration ratio against wavelength for
geometrical optics (GO) and wave optical (WO) simulations for p = 35 and F/# = 1.
wavelength of 600 nm, the wavelength dependence of the concentration ratio is illustrated based on ray tracing
simulations for a spectrum from 450 nm up to 1000 nm with a step size of 10 nm. These simulations are repeated
for different numbers of prisms.
Fig. 5 (a) shows the concentration ratio against the wavelength for three different prism numbers and a fixed
detector position at the design wavelength. Additionally, Fig. 5 (b) shows the comparison of the concentration
ratio against the wavelength for geometrical optics (GO) and wave optical (WO) simulations for p = 35 prisms
and a band width of 200 nm, again with a step size of 10 nm.
Although the peak concentration ratio at the design wavelength is highly dependent on the used number of
prisms, 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 concentration ratios are achievable with such a Fresnel lens. Recalling the reduction of
the concentration ratio due to wave optical effects in Fig. 4 it follows that the wave optical impact due to partial
coherent illumination is small compared with the limitations due to dispersion.
However, as Fig. 5 (b) emphasizes, in cases where either only a portion of the spectrum is used, the spectrum is
split using solar cells that are band-gapped in a particular wavelength range16, 17 or color-corrected Fresnel lenses
are used,18 the miniaturization is more sensitive to partially coherent illumination. 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
specific source-concentrator system.
7. CONCLUSIONS
Within the scope of this work the concentration performance of micro-structured 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 coherence 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 quantitative way.
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 comparison 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 quantitative investigation of the concentration performance it is therefore possible to
design diffraction limited micro-structured refractive concentrators that meet particular demands.
ACKNOWLEDGMENTS
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.
REFERENCES
[1] Swanson, R., “The promise of concentrators,” Prog. Photovoltaics Res. Appl. 8(1), 93–111 (2000).
[2] Leutz, R. and Suzuki, A., [Nonimaging Fresnel lenses: design and performance of solar concentrators ],
Springer Verlag (2001).
[3] Algora, C., Ortiz, E., Rey-Stolle, I., Diaz, V., Peña, R., Andreev, V., Khvostikov, V., and Rumyantsev, V.,
“A GaAs solar cell with an efficiency of 26.2% at 1000 suns and 25.0% at 2000 suns,” IEEE Trans. Electron
Devices 48(5), 840–844 (2001).
[4] Royne, A., Dey, C., and Mills, D., “Cooling of photovoltaic cells under concentrated illumination: a critical
review,” Sol. Energy Mater. Sol. Cells 86(4), 451–483 (2005).
[5] Araki, K., Uozumi, H., and Yamaguchi, M., “A simple passive cooling structure and its heat analysis for 500
x concentrator PV module,” in [Conference Record IEEE photovoltaic specialists conference ], 29, 1568–1571,
IEEE (2002).
[6] Davis, A., Kühnlenz, F., and Business, R., “Optical design using Fresnel lenses,” Optik and Photonik 4,
52–55 (2007).
[7] Egger, J., “Use of Fresnel lenses in optical systems: some advantages and limitations,” in [Society of PhotoOptical Instrumentation Engineers (SPIE) Conference Series ], 193, 63 (1979).
[8] Sinzinger, S. and Testorf, M., “Transition between diffractive and refractive micro-optical components,”
Appl. Opt. 34(26), 5970–5976 (1995).
[9] Rossi, M., Kunz, R., and Herzig, H., “Refractive and diffractive properties of planar micro-optical elements,”
Appl. Opt. 34(26), 5996–6007 (1995).
[10] Kasarova, S., Sultanova, N., Ivanov, C., and Nikolov, I., “Analysis of the dispersion of optical plastic
materials,” Opt. Mater. 29(11), 1481–1490 (2007).
[11] Mandel, L. and Wolf, E., [Optical coherence and quantum optics ], Cambridge University Press (1995).
[12] Van Cittert, P., “Die wahrscheinliche Schwingungsverteilung in einer von einer Lichtquelle direkt oder
mittels einer Linse beleuchteten Ebene,” Physica 1, 201–210 (1934).
[13] Zernike, F., “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795
(1938).
[14] Hopkins, H., “The concept of partial coherence in optics,” Proc. R. Soc. London, A 1, 263–277 (1951).
[15] Einziger, P., Raz, S., and Shapira, M., “Gabor representation and aperture theory,” J. Opt. Soc. Am.
A 3(4), 508–522 (1986).
[16] Imenes, A. and Mills, D., “Spectral beam splitting technology for increased conversion efficiency in solar
concentrating systems: a review,” Sol. Energy Mater. Sol. Cells 84(1-4), 19–69 (2004).
[17] Biles, J. R., “High concentration, spectrum spitting, broad bandwidth, hologram photovoltaic solar collector,” (2009).
[18] Kritchman, E., Friesem, A., and Yekutieli, G., “Highly concentrating Fresnel lenses,” Appl. Opt. 18(15),
2688–2695 (1979).