Page 1

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: fduerr@vub.ac.be

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

OCIS codes:

030.1640, 080.0080, 080.4298, 220.0220, 220.1770, 350.6050.

1.

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 [1]. 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 [2].

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,

enablingdimensionreductionofthephotovoltaiccells

to the millimeter range [3]. A further aspect of the

miniaturization process is the avoidance of techni-

Introduction

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 [6]. Furthermore, the question arises

whether there is a lower limit to this miniaturization

fromamanufacturingpointofview.Withadecreasing

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

appropriate.

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

0003-6935/10/122339-08$15.00/0

© 2010 Optical Society of America

20 April 2010 / Vol. 49, No. 12 / APPLIED OPTICS 2339

Page 2

may no longer be valid for the determination of the

concentration performance.

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 [9] and

diffractive optical elements for beam shaping [10].

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.

2.

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

calculated by

γðiÞ ¼ 90° − arctan

?2pF=#

i − 1=2

?

:

ð1Þ

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

?

:

ð2Þ

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 [13].

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 [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 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

Fig. 1.

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

Page 3

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

Arec

:

ð3Þ

Furthermore, optical losses due to material absorp-

tion and Fresnel reflections are neglected—only total

internal reflection is taken into account.

3.

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

A.

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

special cases.

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

calculated by

Fundamental (Geometric) System Parameters Limiting

DpðiÞ ¼D

2p

cosαðiÞcosβðiÞ

cosðβðiÞ − αðiÞÞ:

ð4Þ

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

?

F=# −1

4ptanαðiÞ

?

ðtanγþðiÞ − tanγ−ðiÞÞ:

ð5Þ

The two collecting angles γ?ðiÞ are derived from

the incident angles αðiÞ ? θ with regard to the

perpendicular:

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

inthedimensionless characteristic

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-

mately correct.

Forthesetwospecialcasesitisnowpossibletocom-

paretheimpactoftheprismsizeanddivergenceangle

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

numbersofprisms.Althoughtheseresultsdonotpro-

vide the concentration performance in three dimen-

sions, they are already valuable to estimate the

ð6Þ

spotratio

Fig. 2.

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

Page 4

predominantfactorontheconcentrationperformance

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.

B.

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.

4.

Thegeometricalopticsapproximationusedinraytra-

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 [14]. 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 [15] and Zernike [16]

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 [14]. The im-

plication for filtered sunlight on the surface of the

Earth was later deduced by Hopkins [17]. 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

rc¼0:16λ

sinðθÞ;

ð7Þ

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 [14].

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.

Fig. 3.

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

Fig. 4.

prisms for three different f-numbers.

(Color online) Concentration ratio against number of

2342 APPLIED OPTICS / Vol. 49, No. 12 / 20 April 2010

Page 5

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 [18]. 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

process.

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.

5.

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

Z

Monochromatic Wave Optical Simulations

EðrÞ ¼

r

0

r0Iðr0Þdr0;

ð8Þ

Fig. 5.

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

Page 6

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.

6.

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

of 10nm.

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

to dispersion.

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

Polychromatic Considerations

Fig. 6.

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

Page 7

color-corrected Fresnel lenses are used [24], 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.

7.

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-

tive 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 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.

Conclusions

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heat analysisfor

500 ×

(IEEE,2002),Vol.29,

Fig. 7.

respectively, and F=# ¼ 1. (b) Comparison of the concentration ratio against wavelength for geometrical optics (GO) and wave optical

(WO) simulations for p ¼ 35 and F=# ¼ 1.

(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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