Shaped Nonimaging Fresnel Lenses

SHAPED NONIMAGING FRESNEL LENSES

Ralf Leutz , Akio Suzuki ,Atsushi Akisawa and Takao Kashiwagi

Tokyo University of Agriculture and Technology,

DepartmentofMechanical Systems Engineering

2-24-16 Naka{cho, Koganei{shi, Tokyo 184-8588, Japan

email [email protected], phone/fax +81-42-388-7076

UNESCO,

SC/EST, 1, rue Miollis, 75732 Paris Cedex 15, France

Abstract { Shap ed nonimaging Fresnel lenses have b een designed according to the edge ray princi-

ple, incorp orating any combination of two acceptance half angle pairs. A numerical solution yields

nonimaging lenses consisting of minimum deviation prisms. If the outer surface of the lens must

b e smo oth, the lens shap e will b e convex. A linear lens prototyp e intended as concentrator for the

collection of solar enery has b een designed, manufactured and tested. Results in terms of ux distri-

bution and optical concentration ratio are presented. Shap ed nonimaging Fresnel lenses are suited for

application as solar concentrators, or as collimators, or in lighting applications, where they can ful l

technological requirements as well as b eing adaptable to the necessities of fashionable design.

Nonimaging Fresnel Lenses

Lens height y Fresnel lens

Nonimaging optics have b ecome an integral part of

optical technologies. Various nonimaging solutions in

diverse elds such as optical transmission of commu-

nication, pumping of lasers, or the collection of so-

lar energy are widely employed. Virtually all present

Absorber

noimaging concentrators work on the principle of re-

volve mirrors to concentrate or

ection, i.e. they in x,

guide radiation.

This pap er presents shap ed nonimaging lenses,

i.e. refractive devices following a given form. Refrac-

Figure 1: Nonimaging linear Fresnel lens solar con-

tion, or the use of lenses do es suit itself for the use of

centrator. Schematic of acceptance half angles:

nonimaging design principles. In particular, an opti-

cross{sectional and p erp endicular .

mum nonimaging lens with smo oth outer surface has

b een created, and was manufactured and tested as so-

lar concentrator of medium concentration ratio.

Nonimaging design dep ends primarily on the edge Of great practical imp ortance is the design of non-

ray principle Welford and Winston, 1989, which de- imaging concentrator systems, where the exit ap er-

scrib es an optical system by means of rays entering ture is smaller that the entry ap erture by a concen-

its rst entry ap erture. Should the nonimaging sys- tration factor 1=C . It has b een shown that a lin-

tem b e ideal, the extreme rays of the entry ap erture ear Comp ound Parab olic Concentrator CPC can

are passing the optical system's second exit ap er- achieve ideal concentration at C =1=sin , where is

ture likewise as extreme rays. Ideal means that all the cross{sectional acceptance half angle, or the angle

radiation entering the system at angles smaller than between the rst edge ray and the collector's optical

those of the edge rays, will exit the system within the axis. Welford and Winston 1989 also pro ove that

b oundaries of the exiting edge rays. the CPC of rotational symmetry C =1=sin is not

ideal, on account of some skew rays b eing re ected tio is to b e achieved, accounting for tracking require-

back trough the entry ap erture. If the rst ap erture ments, and errors due to the size of the solar disc.

of the ideal concentrator is completely lled out by Minimum deviation prisms of nite size constitute for

uniform light a Lamb ertian source, the second ap er- the Fresnel lens. The absorb er has a nite size; two

ture will receive uniform ux. solid angles of light are translated into each other, a

fraction of the sky `seen' by the lens is refracted onto

Nonimaging systems do not create an image and

the solar receiver, or the size of the light source is re-

do not have a fo cus. Optical ab errations are similar,

fracted into an area to b e lit. The lens can b e used in

but not equal to those observed in imaging systems.

a reversible way.

Nonimaging optics are treated like geometrical optics,

and ray tracing is the to ol for their evaluation.

Refraction, or the design of lenses may take ad-

vantage of nonimaging principles. Lenses are sup erior

Nonimaging Lens Design

to mirrors in resp ect of the prisms' partial self correc-

tion of slop e errors, the refracted ray is a ected little,

Aschematic of the working principle of the nonimag-

while the re ected ray's direction angle is changed by

ing linear Fresnel lens depicting the cross{sectional ac-

double the slop e misorientation. New plastic materi-

ceptance half angle , and its p erp endicular counter-

als for lenses o er high transmittance and durability

part is shown in Fig. 1. The edge rays in the entry

for common wavelengths, while keeping the system in-

ap erture rays entering the lens surface at maximum

exp ensive and lightweight.

combinations of the design angles, are refracted by

Nonimaging lenses have rarely b een designed |

each prism so that they exit the second ap erture at

to our knowledge, only three times in the past: by

the corners of the illuminated eld on the absorb er.

Collares{Pereira 1979, at the same time by Kritch-

Graphically, an upside{down pyramid of lightenters

man et al. 1979, and slightly later by Lorenzo and

the lens, and is refracted towards the absorb er. The

Luque 1981. Most probably, the late W. T. Welford

lens can b e designed as b eing of rotational symmetry,

deserves the honour to b e credited with the initial idea

the p erp endicular design angle then is set to zero.

for the design of curved nonimaging lenses. All these

The design pro cess determining the prism inclina-

lenses were intended for the collection of solar energy,

tion , the prism angle , and the prism's lo cation

where photographic accuracy is of no imp ortance, as

relative to the absorb er has b een describ ed in detail

long as radiation can b e captured. None of the lenses

in Leutz et al. 1999a. The main design steps are as

had actually b een built, and their design lacks two

follows, refering to Fig. 2.

characteristics, namely a secondary acceptance half

As opp osed to imaging Fresnel lenses where the

angle p erp endicular to although the lenses were

distance b etween lens and fo cused image is determined

prop ositioned as linear concentrators; and the use of

by f/number and the lensmaker's formula, absorb er

minimum deviation prisms. The former is necessary

half width d and height of the nonimaging lens ab ove

to account for refraction in the p erp endicular plane,

the absorb er are related by y = d= tan . This as-

the latter minimizes optical losses, in particular the

sumes the prism on the optical axis to b e thin. In

chances for total internal re ection.

practice the innermost prisms are approximately at

O'Neill 1978 designed an imaging lens, but real-

plates.

ized that his lens could comp ensate for tracking errors

From this p oint, a prism is determined with the

and errors due to the size of the solar disk within nar-

help of the acceptance half angle pair describing

row limits. In contrast to nonimaging designs, this

asymmetrical incidence from the left and the right

lens is characterized by a delib erately `sloppy' ap-

on the prism and the symmetrical acceptance half an-

proach. O'Neill's lens is the only Fresnel lens for so-

gle . Only the right side of the lens is determined,

lar energy applications that, rst, had b een designed

the left side is later constructed as mirrored version.

solely for this purp ose, and second, has b een commer-

A set of vectors representing the incidence from left

cially somewhat successful.

and right on their paths of double refraction through

The nonimaging Fresnel lenses presented in the

the prism ~q and ~q is derived. These vectors

present pap er are designed Leutz et al. 1999a

represent the extreme rays of the edge ray principle.

as optimum lenses for linear concentration of solar

The two edge rayvectors dep end on the prism in-

energy, and for any systems where the lens should

clination , and the prism angle .Two conditions

roughly equal a collimator, such as lighting applica-

are to b e ful lled: the outer surface of the lens should

tions. The design incorp orates any acceptance half

b e smo oth or the lens shap e must follow an other-

angles and , dep ending on what concentration ra-

wise clearly de ned function; and the prisms designed 2

the level of the absorb er, or any other break criterion,

the simulation is stopp ed. In order to achieve nite

thickness of the lens, prisms are designed partly over- ying previous ones.

A la

The symmetry in the incidence angles and

B ~

ectors d reaching the edges of the C D determining the v

symmetrical absorb er, as well as the quasi{symmetry

in splitting the design of the prism into the calculation

of two dep endent angles and results in the creation

of prisms that are close to minimum deviation prisms.

q q

Of course, one prism can have only one angle of um deviation, but the design describ ed here

d d minim

yields paths for b oth edge rays that are reversible.

ys enter the prism at an angle , with its

absorber x The edge ra

cross{sectional comp onent dictated by the design half

angles , and the p erp endicular comp onent set by

the secondary acceptance half angle .

Figure 2: Evaluating the prism p osition relativeto

For the maximum angle of incidence on the rst

the absorb er. Refracted edge rays for incidence from

surface from the left side , an angle of refrac-

b oth sides, and dep ending on vectors ~q and

~q are compared with the prism p osition vectors

tion on the second surface is recorded, where

~ ~

the latter approximately coincides with the the angle

relative to either end of the absorb er d and d .

of maximum incidence from the right side , and

The vectors ~q are plotted to scale, the vectors d are

the former roughly equals the angle of refraction for

not. Pro jection into cross{sectional plane.

incidence from the right side on the second surface

should b e minimum deviation prisms for minimized

Although minimum deviation happ ens only for one

optical losses, and minimum disp ersion. The maxi-

angle of incidence on each prism, symmetrical paths

mum height y of the lens over the absorb er is found

and the principle of the reversibility of light are the

for the p ointofintersection of lens and optical axis of

basic concepts of minimum deviation, the `reversible'

the system: y =1=tan .

prisms describ ed here are to b e called minimum devi-

No analytical solution can b e found for prism angle

ation prisms.

and prism inclination, should the two conditions of

de ned surface and minimum deviation b e ful lled,

and no further assumptions b e made. In the numerical

Lens Shap es

solution, two nested in nity lo ops are programmed,

and solved with the help of Newton's Metho d. In the

The starting p oint A for each subsequent prism can

inner lo op, the prims angle is decided up on once

be chosen so that the prisms follow a prede ned lens

b oth directions of design incidence + ; and ; ;

shap e. Point A may b e found as y = f x, where

the edge rays yield prism angles identical within an

the mathematical function must ful l two conditions.

error margin.

First, the slop e of the function must allow the re-

The outer lo op compares the twovectors ~q and

fracted light to reach the absorb er, thus the tangens

~q pictured in Fig. 2 with the lo cation of the prism

at any p ointmust b e smaller than one, with the ab-

~ ~

stated by the vectors d and d . Once the vectors

sorb er assumed to b e small. The outermost prisms

d and ~q are found to b e parallel within a con dence

following function c in Fig. 3, and in lens c in Fig. 4

interval, the prism inclination is xed.

are problematic in this resp ect. Second, the function

It should b e noted that Fig. 2 is a two{dimensional, must pass through the previously found y , according

cross{sectional pro jection of the pro cedure outlined. to the edge ray principle.

The in uence of the p erp endicular angle is incorp o-

Though not imp ossible, three{dimensional func-

rated in the design, but can hardly b e traced in said

tions are to b e avoided for practical reasons. Prisms

gure.

would change their shap e according to y = f x; z ,

Subsequently, the next prism is found feeding its and manufacturing would b e a dicult task. One

numerical optimization pro cedure with the values of may understand the linear lenses in Fig. 4 as repre-

the previous prism. Once the next prism would reach senting various cross{sections at di erent depths z of 3

prototyp e has b een plotted as lens e in Fig. 4, with

yp e is truncated at y =

350 the di erence that the protot

1=2y .

300 d 0

or greater acceptance half-angles the lens moves 250 F

closer to the absorb er: i.e. if the distance b etween the

y 200 c

center of the lens and the absorb er is to b e kept con-

t, the absorb er width will increase. Likewise, the 150 stan

a: y = y0

curvature of an optimum lens with greater acceptance 100 b: y = b cos(x) + (y0 - b)

c: y = c sinh(x) + y 0 half angles will b ecome softer. 2

50 d: y = d x + y0

Shap es other than that of the optimum lens curved

ver the absorb er, like at, or concave forms are sub-

-4 -3 -2 -1 0 1 2 3 4 o

trating white light incident from

x, radians optimal in concen

b oth the left and right. The shap e of a lens maybe

desirable for on account of its smo oth integration into

existing shap es. Solar collectors may t architectural

Figure 3: Functions used to design shap ed nonimag-

designs, suchasdaylighting, or lenses for lamps should

ing Fresnel lenses.

suit lighting designs, for example. Erismann 1997

designed a Fresnel lens, which is inherently aspheri-

cal, following a spherical shap e for a infrared sensor,

350 for reasons of fashion. 300 c 250 a

200

Performance of Shap ed Lenses y, mm y, 150

b res- 100 The optical p erformances of the nonimaging F

d e

nel lenses are evaluated byray tracing. Geometrical

50 et absorber 2d = 15.8 mm losses at the lens are describ ed in detail in Leutz

. 1999a. Losses due to the refractive in uence of

-250 -200 -150 -100 -50 0 50 100 150 200 250 al

the p erp endicular acceptance half angle are shown

x, mm

in Fig. 5. Some rays incidentatcombinations of

and < are missing the absorb er, due to the re-

fractive in uence of the p erp endicular incidence. In

Figure 4: Shap ed nonimaging Fresnel lenses follow-

order to minimize absorb er misses, it is essential to

ing functions. Acceptance half angle pairs =2

keep the relation b etween the desig angles and

in the cross{sectional plane the plane of the pa-

reasonable, well within one order of magnitude.

p er, and =12 in the plane p erp endicular to

Incident lightmaybecovering a range of wave-

it. Oversized prisms. Lens e is the optimum lens

lengths, e.g. in the case of sunlight. Nonimaging Fres-

shap e, where a smo oth outer surface was required.

nel lenses are partially mixing the disp ersed refracted

The previously manufactured prototyp e of e is trun-

rays on the absorb er plane, due to incidence from b oth

cated at y =1=2y .

sides of the lens. This leads to a relatively homoge-

neous color distribution Leutz et al., 1999b.

one lens wavering over a at absorb er like a blanket

For a single combination of incidence within the ac-

in the wind. Three{dimensional lenses of rotational

ceptance half angles < ; <, there exists

in in

symmetry can easily b e obtained by setting the p er-

nonhomogeneous illumination on the absorb er, i.e. a

p endicular acceptance half angle =0.

`hot sp ot' is formed. This is exp ected as long as the

The lenses in Fig. 4 are all designed taking into entry ap erture is not lled out completely with ho-

account the same acceptance half angle pairs =2 mogeneous radiation. A typical distribution of ux

in the cross{sectional plane the plane of the pap er, on the absorb er at a prototyp e lens of =2 and

and =12 in the plane p erp endicular to it. The =12 has recently b een measured and simulated.

design of each prism follows the edge ray principle The prototyp e follows the discription of lens e ab ove,

represented by the vectors ~q and ~q . Acceptance including truncation. The exp erimental setup includ-

half angles and absorb er size have b een chosen to equal ing the lens and a device to measure radiation ux is

those of a prototyp e previously manufactured. This shown in Fig. 6. 4 Absorber Prism 6

-2

-4

Perpendicular Section z, Perpendicular Illumination from -6 right left -2 0 2 4 6 8 10 12 14

Cross-Section x,

Figure 5: Top view of examplatory prism and ab-

sorb er of the nonimaging Fresnel lens with accep-

tance half angle pairs =2 and =12.

Incident lightat ; lls out an upside{down

in in

pyramid on any p oint of the lens, is refracted twice

at the prism's faces, and shown intersecting the ab-

sorb er level, where rays form a curved band of light

due to p erp endicular refraction. Oversized prism, for

yellow light, refractive index n =1:49 Polymethyl-

methacrylate, PMMA.

The exp eriment is rather crude, the main prob-

lems b eing the exact orientation in relation to the

apparently moving sun, and the relatively large size

Figure 6: Exp erimental setup to measure the radia-

of the ux meter's window. In spite of that, a go o d

tion ux distribution of a linear nonimaging Fresnel

agreement of simulated and measured ux on the

lens with acceptance half angle pairs of =2 , and

absorb er of the prototyp e lens for normal incidence

=12 . The lens is intended as solar concentra-

= =0 can b e observed in Fig. 7. The mea-

in in

tor for photovoltaic applications. Lens length is 400

sured curve app ears shifted for reasons of misorienta-

mm. 30 July 1999, Tokyo, Japan.

tion. The `hot sp ot' of the simulated ux distribution

cannot b e followed by the ux meter due to its window

size.

tion of a smo oth outer surface do es not restrict this

Direct solar radiation during the ux exp eriment

parameter, disp ersion will act as a limit.

was approximately 750 W/m , while total radiation

The p erformance of any collector system incorp o-

did not exceed 915 W/m . Distortions due to di use

rating a concentrator may b e describ ed in terms of ra-

radiation which is not considered in the mo del are

diation reaching the absorb er. In contrast to the ux

negligible.

density which describ es intensity and distribution of

Interestingly, the lenses a d are characterized by

the radiation on the absorb er, the optical p erformance

a slightly more narrow `hot sp ot' than the optimum

of the lens itself may b e describ ed by the optical con-

shap ed lens e, due to the latter's design inherent non-

centration ratio. The optical concentration ratio is

ideal b ehaviour of the outermost prisms, which in the

understo o d as the ratio of angular radiation intensity

extreme case may b e shaded for part of the incidence.

after having passed the lens, and thereafter passing

The closer de nition of the fo cal area observed with

through the second ap erture, to the radiation inten-

the lenses a d is true only for mono chromatic light. If

sity of identical radiation that has not b een interfered

disp ersion o ccurs, the p erformance increasingly su ers

with. The optical concentration ratio is calculated as

for prisms that are relatively distant to the absorb er.

pro duct of geometrical concentration ratio and optical

Nonimaging Fresnel lenses can b e built as `fast' eciency, incorp orating geometrical losses, absorb er

lenses, i.e. their f/number < 1:0. Where the condi- misses, and optical losses rst order re ection. 5 Conclusions

The principles of nonimaging optics have b een success-

vel class of nonimag-

60 fully applied to the design of a no

resnel lenses. The edge ray principle allows for

simulated ing F

the creation of nonimaging lenses with any combina-

40 measured

tion of acceptance half angles in the cross{sectional

plane, and in the plane p erp endicular to it. Min-

Flux Density, - Flux Density, 20

imum deviation prisms constitute for the lens, solu-

tions are found in a numerical simulation. The con-

trator lens is nonideal for any design angle >0. 0 cen

Absorber

Optium nonimaging Fresnel lenses mayhaveany

-15 -10 -5 0 5 10 15

shap e. If the lens should have a smo oth outer surface,

Absorber Cross Section x, mm

its shap e is convex. The realization of lenses without

one smo oth side is likely to b e restricted by present

manufacturing technologies.

Figure 7: Simulated and measured relative radiation

A prototyp e linear concentrator intended for the

ux on the absorb er of a linear nonimaging Fresnel

collection of solar energy has b een manufactured. Test

lens with acceptance half angle pairs of =2 , and

results, and simulated p erformance show go o d agree-

=12 for normal incidence = =0 .30

in in

ment. The `hot sp ot' is of mo dest nature, and dep ends

July 1999, Tokyo, Japan.

strongly on the wavelength comp onents of the incident

radiation. If white light e.g. sunlight is available,

lens c disp ersed colors are mixed on the absorb er, and ux

ys, how- 15 distribution is more homogeneous. Some ra

10 er, will miss the absorb er; due to disp ersion and lens e ev 25 15

. 10 due to the in uence of the design angle 15

10 resnel lenses can b e designed as `fast' 5 Nonimaging F

0 lenses. Besides the initial success of their use in so-

20 15 lar energy applications as collector of medium concen- 10

voltaics, lenses of various shap es can -20 0 tration for photo -15 -10 -5 Optical Concentration Ratio, - Ratio, Optical Concentration -5 0 -10 in, °

5 -15

tegrated with architectural and technological re- 10 be in 15 -20

in, ° 20

quirements. A large eld of anticipated applications

is the use of the novel nonimaging lens as collimator,

or reversed concentrator for lighting.

Figure 8: Optical concentration ratios of nonimag-

ing Fresnel lenses c and e with acceptance half angle

pairs of =2 , and =12 .

Acknowledgement

The nancial supp ort of the Solar Fresnel Lens Pro ject

The optical concentration ratio of nonimaging de-

by the Japanese Science and Technology Corp oration

vices is characterized by a sharp drop once the inci-

is greatfully acknowledged.

dence angles exceed the design acceptance half angles,

and the refracted rays miss the absorb er, as shown in

Fig. 8. In this gure the optical concentration ratios

References

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