SHAPED NONIMAGING FRESNEL LENSES y 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 y 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, z, 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 2 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- 0 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 y 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.