optical design of a solar flux homogenizer for concentrator photovoltaics

Optical design of a solar flux homogenizer forconcentrator photovoltaics

Kathi Kreske

An optical solution is described for the redistribution of the light reflected from a 400-m2 paraboloidalsolar concentrating dish as uniformly as possible over an approximately 1-m2 plane. Concentratorphotovoltaic cells will be mounted at this plane, and they require a uniform light distribution for highefficiency. It is proposed that the solar cells will be mounted at the output of a rectangular receiver boxwith reflective sidewalls �i.e., a kaleidoscope�, which will redistribute the light. I discuss the receiver boxproperties that influence the light distribution reaching the solar cells. © 2002 Optical Society ofAmerica

OCIS codes: 360.6050, 220.1770, 120.4500.

1. Introduction

The Sun is our largest energy resource, and its po-tential has barely been tapped. The main reason iscost competitiveness with traditional energy sources.Solar energy conversion systems in use today areusually photovoltaic or solar thermal in type. Pho-tovoltaic systems directly convert the energy of thephotons from sunlight into electricity, whereas solarthermal systems employ the sunlight to producehigher temperatures than ambient, often as a inter-mediate step toward generating electricity. Theprice of large solar thermal systems has been reducedenough to make them cost competitive in certain cir-cumstances. But because of the large turbines gen-erally needed to produce cost-efficient solar thermalelectricity, smaller power plants have not yet beendemonstrated as a reliable option.

Standard photovoltaic systems are economicallycompetitive only in niche markets, such as areas thatare far from any electrical grid. Recent research hasproposed using the optics of concentrating solar ther-mal systems to work with newly developed photovol-taic cells designed for highly concentrated sunlight.If the amount of expensive photovoltaic material is

K. Kreske �[email protected]� is with the National Cen-ter for Solar Energy, Jacob Blaustein Institute for Desert Re-search, Ben-Gurion University of the Negev, Sede Boqer Campus84990, Israel.

Received 17 July 2001; revised manuscript received 28 Novem-ber 2001.

0003-6935�02�102053-06$15.00�0© 2002 Optical Society of America

minimized and replaced with cheaper mirrors, it maybe possible to bring down the costs of smaller solarenergy plants to within what the electricity market iswilling to pay.1

The main difficulty in this use of concentrator op-tics is that photovoltaic cells require a uniform fluxdistribution to work at high efficiency. This is be-cause the amount of current produced by a photovol-taic cell is directly proportional to the amount of solarflux incident on the cell, and cells wired together willbe limited by the lowest current. For an extendedsource such as the Sun, even an imaging system suchas a paraboloidal mirror will produce uneven fluxdistributions.2 As the solar irradiance at the focalplane is much too high for current concentrator pho-tovoltaic cells, we can reduce the concentration ratiowhile smoothing out the flux distribution.

The solar concentrator that we are concerned within this study is the Paraboloidal Energy Transformerand Astrophysics Laboratory �PETAL�, the 412.5-m2

paraboloidal mirror in Sede Boqer, Israel. PETALhas a hexagonal periphery, a focal length of 13.1 m,and tracks the Sun so that its optical axis is alwaysintersecting the center of the solar disk. The centerto midpoint of the hexagon edge distance is 10.9 m,and the center to vertex distance is 12.6 m �Fig. 1�.At its focus, PETAL has an average flux concentra-tion ratio of �10,000 Suns over an area of approxi-mately 0.04 m2. We are interested in convertingthis to a uniform flux distribution over a square orrectangular plane of approximately 1 m2, with anirradiance level in the range of 500 Suns. The pho-tovoltaic cells to be wired on this plane are 1 cm � 1cm squares.

1 April 2002 � Vol. 41, No. 10 � APPLIED OPTICS 2053

We propose placing a rectangular receiver box withreflective sidewalls �i.e., a kaleidoscope� near the fo-cal plane of the paraboloid. The sunlight reflectedfrom the paraboloid will travel through the receiverbox to the plane where the solar cells are mounted.In this paper I discuss the receiver design and themanner in which its dimensions influence the distri-bution of the light reaching the solar cells.

A kaleidoscope flux homogenizer in combinationwith a paraboloidal solar concentrating dish was firstproposed by Chen et al.3 in 1963 to prevent hot spotsin solar thermal applications. The idea was nextdiscussed in 1997 when Ries et al.4 proposed a hex-agonal kaleidoscope–hexagonal dish combination forphotovoltaic applications. In this study a square orrectangular kaleidoscope receiver box was preferredto accommodate an array of square photovoltaic cells,and the factors affecting flux uniformity for this setupare discussed in Section 3.

Two- and three-dimensional modeling of the sys-tem was performed to aide in the kaleidoscope design.Monte Carlo ray tracing was then used to determineexact flux distributions at different planes along thekaleidoscope for various kaleidoscope sizes and posi-tions. Different theories were developed to arrive atmore uniform flux distributions, and these theorieswere checked by ray tracing until better than thestated goal was achieved. For this project, the goalwas to limit flux inhomogeneities to less than �5% ofthe average flux, as this is the error between solarcells that is due to variations in their internal resis-tance.

2. Ray Analysis

To analyze the position of the light rays at differentpoints near the focal plane of the paraboloid, two-dimensional modeling was employed by use of thegeometry of a parabola and the laws of reflection.Both on-axis and off-axis incident rays were traced.The Sun was treated as having a pillbox shape withangular radius �max � �4.7 mrad. The z axis lies

along the optical axis of the dish, and the focal pointof the paraboloid is at z � 13.1 m �Fig. 1�a�. Theorigin is the point where the optical axis intersectsthe paraboloid. The initial kaleidoscope studied wasa 1-m2 square of 2 m in length, positioned as in Fig.1�b�.

Assuming perfect optics, the equation for a two-dimensional parabola is

z1 � x12�4f, (1)

where f is the focal length of the parabola and �x1, z1�is any point on the parabola. If a ray from a one-dimensional extended source Sun is reflected off theparabola at a point �x1, z1�, the ray reflected off aparabola will travel according to the equation

x � m� z � z1� � x1 , (2)

where

m � � � x1

f � z1� tan ��1 �

x1

f � z1tan �� . (3)

Fig. 1. Schematic view of the paraboloidal concentrator and its associated receiver box: �a� cross-section including the optical axis, �b�projection of the receiver box on the hexagonal periphery of the paraboloid.

Fig. 2. Off-axis rays reflected off of the paraboloidal concentratingmirror.

2054 APPLIED OPTICS � Vol. 41, No. 10 � 1 April 2002

Here � is the incident angle of the ray and �x,z� is anypoint on the reflected ray �Fig. 2�. Equation �3� canbe used for rays reflected off any point on a three-dimensional paraboloid when we define differentcross-sectional cuts across the paraboloid and kalei-doscope receiver.

Rays reflected from different points on the parab-oloid will reach the mirrors of a 1-m2 square kaleido-scope receiver box at different planes along the z axis.It is important to analyze the edge rays—those raysreflected off the hexagonal periphery of theparaboloid—as these rays will determine the outerpoints of the flux distribution. The rays to reach thekaleidoscope mirrors at a plane closest to the parab-oloid focal plane will be those impinging on the dishat �max � 4.7 mrad in the x–z plane and reflected offthe paraboloid at the points ��12.6 m, 0, z1�. Theserays will reach the kaleidoscope mirrors at the planez � 13.4 m. The closest plane where there will be noareas without flux is the z � 13.65-m plane, wherethe � � 4.7-mrad rays reflected from the paraboloid atthe points ��8 m, �8 m, z1� reach the four corners ofthe kaleidoscope box.

An ideal kaleidoscope entrance plane would havefew areas without flux. As can be seen in Fig. 3, thisis not the situation with a square-shaped kaleido-scope entrance plane at z � 13.4 m, as the flux dis-tribution has a hexagon shape. Rotating thekaleidoscope with respect to the hexagon does nothave any advantages, but rectangular-shaped kalei-doscopes can improve the situation. For our parab-oloid, at z � 13.4 m, the x-direction extreme rays fromthe two x-axis vertices are at �0.5 m. They-direction extreme rays are those reflected off thefour hexagon vertices that are not along the x axis�x � �6.3 m, y � �10.9 m�; and at z � 13.4 m, theserays will be at y � �0.43 m. Therefore the smallestrectangle at the z � 13.4-m plane that would includeall rays is 1 m � 0.86 m.

It was found through ray tracing that the receiverbox can be adjusted so that some of the edge rays areleft out, with a minimum amount of flux lost. One

method is to move the entrance plane further fromthe dish focus. It is also possible to keep the en-trance plane at z � 13.4 m, and instead to reduce thedimensions of the kaleidoscope. For a 0.96 m �0.8 m kaleidoscope with the entrance plane at z �13.4 m, 0.05% of the flux is lost. For a 0.88 m �0.8 m box, 0.1% of the flux is lost, all from rays inci-dent on the paraboloid at large angles ��max� andreflected off the six vertices of the hexagon.

3. Analyzing the Ray-Tracing Results

OptiCAD ray-tracing software was used to determinethe irradiance distribution at different planes alongthe kaleidoscope length. At z � 13.4 m, the shape ofthe flux distribution is essentially an image of thehexagonal paraboloid, with a peak flux of 1940 Sunsaround the optical axis �Fig. 3�. Obviously, the fluxnonuniformity will be at a maximum at this plane.After entering the kaleidoscope receiver box, theserays will bounce back and forth along parallel mir-rored walls until reaching the exit plane, where thearray of photovoltaic cells is located. As each raykeeps to its own two mirrors, the irradiance distribu-tion will have fourfold mirror symmetry. The angleof each ray with respect to the optical axis will remainconstant. It depends on the radial distance from thecenter of the paraboloid of the point from which theray is reflected and on the incident angle of the in-coming ray. Rays at different angles bounce off thekaleidoscope mirrors at different frequencies, caus-ing the image of the hexagon to become more andmore scrambled as the length of the kaleidoscope isincreased.

As the rays bounce back and forth through thekaleidoscope receiver, their energy is binned to de-termine the irradiance distribution and various sta-tistical parameters at different planes along theoptical axis. The irradiance distribution was firststudied at 0.1-m intervals and was later decreased to0.01-m intervals. The ray traces included differentshaped kaleidoscopes, varied entrance planes, rota-tion of the receiver box with respect to the dish, anddifferent kaleidoscope mirror reflectivity. A largenumber of ray traces were carried out to determinean optimal exit plane and kaleidoscope geometry fora reasonable kaleidoscope length of less than 2.5 m.

One of the statistical parameters used to determinethe uniformity of a specific z plane in the kaleidoscopereceiver was the normalized standard deviation,

� std1000

�s�, (4)

where std is the standard deviation of the averagenumber of Suns in the area of each bin and �s� is theaverage number of Suns over the entire plane.

When the normalized standard deviation wasgraphed against the kaleidoscope length, it was foundthat, in general, the flux inhomogeneities decreasedalong an oscillating curve �Fig. 4�. There are planeswith local standard deviation minima, and the dis-tance between them was observed to be approxi-

Fig. 3. Contour map of the irradiance distribution at the kalei-doscope entrance plane �z � 13.4 m� in number of Suns.


mately 0.4 m. Near the kaleidoscope entranceplane, the decrease in standard deviation was rela-tively large, down to 10–20% of its value at the en-trance plane by the time it reached its first localminima. By the third or fourth minima, the down-ward trend flattened out, and the planes of local min-ima have a value of less than 5% of the at thekaleidoscope entrance plane. After this point, it isquestionable whether a small improvement in uni-formity is worth an extra 0.4 m of kaleidoscope andextra reflection losses from the kaleidoscope mirrors�mirror absorption will most likely be between 5%and 10%�.

Another important parameter is , the relative per-cent of greatest nonuniformity:

�maximum flux � minimum flux

average flux (1002 ) . (5)

This parameter, � , provides the maximum devia-tions from the average flux across the plane ofinterest. Other parameters such as minimum flux�maximum flux were also analyzed.

The original goal was to achieve a relative percent-age of greatest nonuniformity of less than 5%. AsI show, this was achieved with a kaleidoscope lengthof 2.44 m. At shorter kaleidoscope lengths, the max-imum nonuniformity was between 5% and 10%, andthis greater nonuniformity was often caused by alarge dip around the optical axis.

4. Results

The goal of this research was to use optics to trans-form the solar radiation concentrated by a hexagon-shaped paraboloidal dish into an almost uniformirradiance distribution, which will be used to illumi-nate an array of 1-cm2 square photovoltaic cells.The base configuration for this project was the412.5-m2 Sede Boqer hexagon-shaped paraboloid anda 1 m � 1 m kaleidoscope receiver box with reflectivesidewalls and an entrance plane at z � 13.4 m �Fig.1�. Improved flux uniformity was then sought out bya change in the location of the kaleidoscope entranceplane and use of rectangular-shaped kaleidoscopes.Receiver box mirror reflectivity and shading from thereceiver box were also studied.

The shading from the receiver box itself had amuch greater effect on the irradiance distributionthan expected and is felt at every plane along thekaleidoscope length. Although the area of the pa-raboloid that is shaded by the receiver box is only 1m2�412.5 m2, or 0.24%, a relatively large effect fromthe shading is observed throughout the kaleidoscope.In Fig. 5, an x-axis cross section of the irradiancedistribution at the z � 13.4-m plane is compared fora shaded and unshaded paraboloid. When the pa-raboloid is shaded by the receiver box, the peak fluxis reduced from 1940 Suns to 1850 Suns �4.7%� overan area with a diameter of approximately 0.125 m.This reduction in flux will not be smoothed out alongthe length of the kaleidoscope. Looking at the oppo-site situation, we can place a bundle of parallel raysand rays with small angles at the center of a kalei-doscope box. These rays will not be smoothed out ina 2.5-m-length kaleidoscope. It follows that, if theserays that stay along the optical axis are missing, thismissing flux will not be smoothed out. Rays withhigher angles will pass in and out of the center area,but the rays that stay in the center area will bemissing. Exit planes that had a central peak with-out shading were sought out for their uniformity withshading.

Rotating of the kaleidoscope receiver box with re-spect to the paraboloidal dish did not improve theuniformity of the irradiance distributions. In fact itsmoothed out some of the local minima as the sym-metries of the hexagon and the square were no longerworking with one another.

Fig. 4. Normalized standard deviation versus kaleidoscopelength for a 0.88 m � 0.8 m rectangular-shaped kaleidoscope with95% mirror reflectivity and an entrance plane at z � 13.4 m.

Fig. 5. x-axis cross section of the irradiance distribution at the z � 13.4-m plane in number of Suns: �a� no shading by the kaleidoscope�all rays from the source in the collection area of the dish will reach the dish�, �b� center of the dish is shaded by a 1-m2 kaleidoscope of2.5 m in length.


Moving the entrance plane of the kaleidoscope fur-ther from the paraboloid focal plane to the z � 13.5-mplane showed some advantages. The areas with noflux at the kaleidoscope entrance plane are reduced,and the amount of rays that did not enter the kalei-doscope was less than 1%. An optimum plane withmaximum nonuniformity of 7% was achieved, andthe nonuniformity was largely caused by a dip at thecenter of the kaleidoscope exit plane �Fig. 6�. Anoption with this configuration is to leave out the pho-tovoltaic cells in the area of the center dip and con-centrate the cell wiring there, or to wire the centercells separately.

Rectangular-shaped kaleidoscope receiver boxesgave the most promising results. Again the areaswith no flux at the kaleidoscope entrance plane arereduced, and the amount of rays lost was less than1%. A smaller receiver box also has the advantageof reducing the shading of the paraboloidal dish.The goal of � 5% was achieved with a kaleidoscopelength of 2.44 m for a 0.88 m � 0.8 m rectangularkaleidoscope. Its normalized standard deviation ver-sus kaleidoscope length can be seen in Fig. 4. Theirradiance distribution of the z � 15.84-m plane, for200 million rays traced and 95% mirror reflectivity,can be seen in Fig. 7. The minimum flux�maximumflux is 0.955, and , the percent nonuniformity, is�2.3%.

The kaleidoscope length is kept under 2.5 m forpractical reasons �it is suspended 13.5 m above thecenter of the paraboloid with minimal supports� andto minimize reflection losses from the kaleidoscopemirrors. If we take a 0.88 m � 0.8 m receiver box asan example, for a mirror reflectivity of 95%, approx-imately 2% of the total flux is lost every 0.4 m �i.e., atz � 14.7 m, 8% of the total flux is lost to mirrorabsorption, 10% at z � 15.1 m, 12% at z � 15.5 m�.For 90% mirror reflectivity, the amount of the total

flux that is lost to mirror absorption is 15% at z �14.7 m, 18% at z � 15.1 m, and 22% at z � 15.5 m.

5. Future Research

A. Optical Errors

The ray tracing and analysis assumed a paraboloidaldish with perfect optics, which is not realistic. Theoptical errors of the Sede Boqer dish have not yetbeen determined, but are probably similar to thosemeasured on the 400-m2 hexagonal paraboloid inCanberra, Australia.5 Another potential opticalproblem could result from the accumulation of dust ordirt on the paraboloid and kaleidoscope mirrors,causing scattering of the reflected sunlight.

B. Sun Shape

I defined a pillbox-shaped solar disk above, but thereare more accurate models6 that gradually reduce therelative amount of off-axis edge rays. Whether thiswill change the accuracy of the actual flux distribu-tion depends on the optical errors of the dish.5 An-other issue is the day-to-day variation in the angularsize of the Sun. Hazy skies will increase the angularsize of the Sun, whereas clear skies will result in aminimum angular-sized Sun.

C. Tracking and Kaleidoscope Positioning Errors

Tracking errors7 will also affect the flux distribution.If the Sun is not precisely on the optical axis of thedish, the focal spot will also be decentered, affectingthe flux distribution throughout the length of thekaleidoscope. Other changes in flux distributioncould result from inaccurate positioning of the kalei-doscope with respect to the optical axis, or from ka-leidoscope rotation.

D. Additional Shading

The kaleidoscope receiver box will be held in place bysupports, which will also be shading the paraboloidaldish. The exact size and position of these supportsare not yet known. Also, the dimensions of the ka-leidoscope receiver box do not take into account the

Fig. 6. Irradiance distribution �in number of Suns� at the z �15.06-m plane by use of a 1 m � 1 m kaleidoscope with the entranceplane at z � 13.5 m. The center dip area can be used for cellwiring.

Fig. 7. Irradiance distribution �in number of Suns� at the z �15.84-m plane with a 0.88 m � 0.8 m kaleidoscope with the en-trance plane at z � 13.4 m. Percent nonuniformity is �2.6%.


mirror thickness and the cooling apparatus that willbe mounted within the receiver box walls, which willprobably be in the range of 0.1 m. This additionalshading must be included in the final analysis.

E. Shape and Position of the Kaleidoscope

Most importantly, all possible configurations of thekaleidoscope shape and position have not been ex-hausted, and the optimum plane that was achieved inthis project is not necessarily the best possible result.As each ray trace takes over a day of computing time,it is impossible to cover all the possibilities. Thestarting configurations were chosen randomly, andthe directions that seemed promising were probedfurther. The optimum plane achieved is the bestresult that was traced, within our desired goals, andis a good direction in which to continue to improveflux uniformity.

6. Conclusion

It was found that, with paraboloidal concentratingdishes and secondary optics, it is theoretically possi-ble to achieve flux uniformity within the limits nec-essary for photovoltaic applications, with aconcentration ratio in the range of 500 Suns. Closeto uniform flux distribution is necessary as the solarcells are wired in an array, the array current is lim-ited by the lowest cell current, and this is directlyproportional to the irradiant flux on the cell.Greater flux inhomogeneities will therefore result inlower output power and excess heat. Achieving fluxuniformity at high concentration ratios is of great

importance to bring down the cost of solar energy andphotovoltaics.

The author especially thanks her advisor DavidFaiman and the National Solar Energy Center inSede Boqer, Israel. Financial support was receivedfrom the Israel Ministry of National Infrastructuresand the Leon David Asseo Foundation. Specialthanks also go to Daniel Feuerman, Jeffrey Gordon,and Stanley Rotman.

References1. D. Faiman, “Concentrator PV: cost and materials issues,” in

Ninth Sede Boqer Symposium on Solar Electricity Production,D. Faiman, ed. �Sede Boqer, Israel, 1999�.

2. S. M. Jeter, “The distribution of concentrated solar radiation inparaboloidal collectors,” J. Solar Energy Sci. Eng. 108, 219–225�1986�.

3. M. M. Chen, J. B. Berkowitz-Mattuck, and P. E. Glaser, “Theuse of a kaleidoscope to obtain uniform flux over a large area ina solar or arc imaging furnace,” Appl. Opt. 2, 265–271 �1963�.

4. H. Ries, J. M. Gordon, and M. Laskin, “High-flux photovoltaicsolar concentrators with kaleidoscope-based optical designs,”Sol. Energy 60, 11–16 �1997�.

5. G. Johnston, “On the analysis of surface error distributions onconcentrated solar collectors,” J. Solar Energy Sci. Eng. 117,294–296 �1995�.

6. M. Schubnell, “Sunshape and its influence on the flux distribu-tions in imaging solar concentrators,” J. Solar Energy Sci. Eng.114, 260–266 �1992�.

7. K. Bammert, A. Hegazy, and H. Lange, “Determination of thedistribution of incident solar radiation in cavity receivers withapproximately real parabolic dish collectors,” J. Solar EnergySci. Eng. 112, 237–243 �1990�.


optical design of a solar flux homogenizer for concentrator photovoltaics

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