Solar Energy Materials & Solar Cells 95 (2011) 1654–1664

Contents lists available at ScienceDirect

Solar Energy Materials & Solar Cells

0927-02

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/solmat

Design and fabrication of freeform glass concentrating mirrors using a highvolume thermal slumping process

Yang Chen, Allen Y. Yi n

Department of Integrated Systems Engineering, The Ohio State University, 210 Baker Systems, 1971 Neil Ave, Columbus, Ohio 43210, USA

a r t i c l e i n f o

Article history:

Received 8 December 2009

Received in revised form

22 November 2010

Accepted 13 January 2011Available online 20 February 2011

Keywords:

Glass thermal slumping

Freeform

Concentrating photovoltaic

Primary mirror

48/$ - see front matter & 2011 Elsevier B.V. A

016/j.solmat.2011.01.024

esponding author. Tel.: +614 292 9984; fax:

ail address: yi.71@osu.edu (A.Y. Yi).

a b s t r a c t

Concentrated photovoltaics (CPV) power is a form of clean and renewable energy. However, the cost of

harvesting solar energy is still economically prohibitive as compared to more traditional electricity

generation methods such as hydroelectric or fossil fuel power. In this study, an innovative, high volume

but low cost thermal slumping process was proposed as an alternative method for manufacturing of

glass mirrors for high concentration photovoltaic system. In this paper, first a freeform optical design

was performed to create a two-stage concentrator with 711 acceptance angle and uniform output

irradiance. Ray-tracing simulation was performed to evaluate the optical design. A machinable ceramic,

MACORs, was tested as mold material for its preferred mechanical and chemical stability at high

temperature conditions. To assist the development of the slumping process, finite element method

(FEM) simulation was performed to compensate for the mold design for manufacturing errors in this

process. Moreover, surface profile and surface roughness were measured to characterize the thermal

slumping process. Different manufacturing parameters were tested to identify the proper slumping

conditions. It is discovered that surface roughness of the inner surface of the slumped glass mirror

remained unchanged after slumping under a pre-determined soaking temperature. This study

established a methodology for low cost, high volume glass optics for possible solar concentrator

applications.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

Concentrated photovoltaics (CPV) is becoming an alternativeapproach to production of electricity over the conventional fossil fuelbased energy generation approach. Solar photovoltaics can directlygenerate electricity without creating harmful emission during opera-tion [1]. However, the cost structure of solar energy today is lessfavorable as compared with more traditional electricity generationmethods, such as gas or hydro powered generators. Solar energy topsthe energy generation list at 25 to 30 cents per kilowatt-hour,compared with just 3 to 5 cents for coal or hydroelectric [2].

To improve photovoltaics systems’ efficiency and reducemanufacturing cost and complexity, different designs wereproposed. These designs include different energy conversionmethods, optical design, and fabrication and assembly meth-ods [3]. A photovoltaic solar system normally uses two typicaldesigns, i.e., non-concentrated flat plate and concentrated photo-voltaics (CPV). Concentrated collectors reduce the total area ofphotovoltaic receivers by reflecting or refracting the incident lightoff a large aperture optic onto a small absorbing area.

ll rights reserved.

+614 292 7852.

High concentration of sun light has two main advantages. First,it reduces the numbers or area of photovoltaic cells, and thereforereduces the cost of a solar system because lower number ofphotovoltaic cells is used. This is especially significant for the newmulti-junction III–V photovoltaic cells [4]. Second, it significantlyincreases the level of light intensity. At a high flux level,electricity generation efficiency of solar cells is drasticallyimproved, again particularly when multi-junction photovoltaiccells are used [5]. Therefore, there is a general agreement thatsome degree of concentration would be desirable for most solarphotovoltaic applications.

However, the cost and complexity associated with opticalconcentrators may outweigh the potential gains. Therefore, find-ing a high volume and low cost concentrator manufacturingprocess is critical to photovoltaic concentrator fabrication, espe-cially the primary mirror fabrication. To fabricate the primarymirrors for high concentration solar systems, injection molding isone of the popular manufacturing methods [6–8]. Hot and rollerembossing have also been tested to fabricate Fresnel primarylenses [9]. These methods are cost effective but the polymeroptics lack the performance due to the constraints from thepolymer materials used.

On the other hand, compared to polymer materials, glasses aremore durable, and thus can be used in harsher environments,

Fig. 1. Freeform secondary surface imaging of the primary surface on the

photovoltaic surface (Cs–Ce).

Y. Chen, A.Y. Yi / Solar Energy Materials & Solar Cells 95 (2011) 1654–1664 1655

such as in desert condition, or can work at high temperatureapplications because of their high glass transition temperature(Tg), good chemical stability, and robust mechanical proper-ties [10]. Unfortunately, traditional fabrication methods for glassoptical components were expensive, time consuming, and diffi-cult to be adapted for high volume manufacturing. To this end,precision glass thermal slumping process is a possible techniquethat derives from glass compression molding process and can beadopted for high volume precision glass optical elements fabrica-tion [11]. Recently, precision glass thermal slumping techniquehas been used to fabricate concentrating mirrors as a highvolume, low cost solution [12]. However there are still quite afew unsolved issues with this technology, mainly due to lack ofproper understanding about the precision glass forming.

In glass thermal slumping process, a raw glass sheet workpieceand mold are heated up to the working temperature (or soakingtemperature) and then slumped by its own weight or (negative)vacuum pressure. Controlled cooling of the slumped glass mirroris carried out immediately after slumping is completed to keepthe thermal shrinkage and residual stresses below the requiredlevels after the molding process. As compared to conventionalabrasive based process, thermal slumping is a high volume, lowcost, and one-step fabrication process. More recently, glassthermal slumping process has even been used to fabricate thesegment of X-ray telescope mirrors and other extremely highprecision glass optics [13,14]. Based on these successful applica-tions, glass thermal slumping process is becoming a promisingnew method for fabricating solar optical components at anaffordable cost.

In this paper, a two-stage nonimaging freeform concentratorwas designed. A ray-tracing simulation was performed to evalu-ate the system design. Thermal slumping process was proposed tofabricate glass mirror as the designed primary mirror for highconcentration photovoltaic systems. An easy to machineMACORs glass–ceramic was chosen as the mold material toreduce the total cost of fabrication process. Experiments and FEMsimulation were performed to improve the slumping process andcompensate for the mold design. Finally, the slumped glassmirrors were evaluated for surface quality and curvature accuracyusing coordinate measurement machine (CMM) and atomic forcemicroscope (AFM).

2. Freeform optical design

2.1. Optical design

In this section, the concentrating optical design is based on thefreeform optics using a geometrical approach. To satisfy therequirement of uniform irradiance on the receiver’s surface, atwo-surface design method of freeform surface based on theKohler integrator arrays was proposed. In the Kohler integratorsystem, the rays emitted by one point at the source mustilluminate the entire target [15,16]. For a concentrated solarphotovoltaic system, this means that the light rays impingingon the primary surface have to be redirected to the entirereceiving photovoltaic cell surface at any given incidence anglewithin the acceptance angle. In addition, to achieve a uniformirradiance distribution, the rays arriving at any point on the targetmust come from every point of the light source.

The integrated concentration Kohler optical design consists oftwo imaging optical reflective surfaces (primary and secondary).The secondary is placed at the focal plane of the primary, so thatthe primary surface images the sun on the secondary surface andthen the secondary images the primary on the target cell surface.Since the incident light from the sun can be treated as paralleled

rays with different incident angles, the primary surface wasselected as the parabolic segment to focus the incident light tothe secondary surface. To ensure uniform light distribution,ellipsoid surface segments are used to image the primary surfaceon the target surface. Fig. 1 illustrates the two-stage mirrorsurface system. The design algorithm detail is described below.

a.

The starting points of the primary mirror and secondary mirrorare given as the initial calculation starting position as Ps (Psx,Psy, Psz) and Ss (Ssx, Ssy, Ssz). At the same time, the position andsize of the photovoltaic cell are also defined by a pair of pre-defined start and end points (Cs and Ce).

b.

According to the rules of the edge ray principle, an ellipsoidsurface (Ss–Se) is initialized using points Ps and Ce as the focalpoints and passing point Ss. The size of this ellipsoid iscontrolled by the required acceptance angle at the primarysurface, shown as ray 2, 3 in Fig. 1. After being reflected by theellipsoid surface, all incident rays within the acceptance angleat point Ps are focused at target surface point Ce, according tothe reflective property of the ellipsoid surface.

c.

By defining that point Ss is passed through by the reflected line1 from point Ps with maximum incident angle a, the normaldirection Np of primary surface at point Ps can be calculated bythe vector form Snell’s law as expressed in Eq. (1)

r¼ i�2ðiUnÞn ð1Þ

In this equation, incident direction i and reflection direction rboth are known from the geometrical assumption, and then thenormal direction n can be derived.

d.

After the normal direction Np at point Ps is known, thedirection of the reflected incident ray along the normaldirection can be calculated using Eq. (1), shown as ray 2 inthe figure. The point of intersection between the reflected rayand the pre-defined ellipsoid Ss–Se is defined as the focal pointSf of the primary parabolic segment passing point Ps. Theintersection point can be found using the bisectionmethod [17]. Using the point Ps and the focal point Sf, theprimary parabolic segment profile can be derived as Ps–Pe.

e.

According to the incident light and reflected light path pair(line 2), the normal direction Ns at the point of intersection Sf

can be calculated using Eq. (1). In addition, the reflection of thenormal incident ray reaching the end point of the parabolicsegment is reflected to point Cs by the ellipsoid surface at thepoint Sf. Therefore, by calculating the direction of the reflectionof line Cs–Sf, the position of the end point Pe on the parabolic

Fig. 2. (a) Primary freeform mirror based on previous stated algorithm and (b) secondary freeform mirror.

Y. Chen, A.Y. Yi / Solar Energy Materials & Solar Cells 95 (2011) 1654–16641656

surface can be determined by bisection method, which can beused to find the intersection point between parabolic surfaceand ray 4.

f.

By calculating the reflected ray with maximum incidenceangle a at point Ps, the intersection point with the ellipsoidsurface could be determined, which was defined as the endpoint of the secondary segment.

g.

Storing the parabolic and ellipsoid segment, and setting pointPe, Se as the new start points Ps, Ss, and then go to the step a tostart another cycle until the position of Pe is on the left of theinitial start point Ss.

h.

Continuously performing the iterative algorithm until thecalculated point Pe is on the right of initial secondary startingpoint Ss, and then the calculation is terminated. This conditionis applied to avoid blockage of the secondary on the primarymirror.

This design algorithm provides a process to design 2D two-stage concentrator to obtain uniform irradiance distribution.When the 2D results of the primary and secondary mirrors areexpanded to a given width Wp and Ws, respectively, the lightdistribution on the target surface has a related width Wc. There-fore, the 2D design turns into a pseudo-3D surface similar to a‘‘stripe’’ pattern. Based on this idea, an array of the stripedprimary and secondary mirrors is combined and modified toilluminate the desired target cell surface. Each stripe of theprimary and secondary combination is optimized by the abovelisted algorithm. Using this method, a 3D profile is established.However, in this 3D design the integration is carried out in theradial direction, but not in the azimuthal direction. In addition, amerit function is applied to ensure the smoothness at theintersection of the adjacent ‘‘stripes’’ by minimizing the ‘‘jumpi-ness’’ between any two neighboring ‘‘stripes’’. Fig. 2 shows sur-face profiles of the primary and the secondary mirrors.

In the current design, the total area concentrating ratio can beestimated using Eq. (2) [3]

Cg ¼APrimary

Areceiverð2Þ

here the receiver is modeled as a unit with 5�5 mm sensing area,and the primary mirror is designed with a dimension of100�80 mm. Therefore, the final area concentrating ratio isapproximately 320 in this design. The thermodynamic limit in

this Kohler optical design can be calculated using Eq. (3) [5]

Cmax ¼NA2

NA1

� �2

¼sinð2y2Þ

sinð2y1Þ

� �2

ð3Þ

where NA1 and NA2 are the numerical apertures of the primaryreflector and the flat absorber exit respectively and y1 and y2 arethe incidence angle and the exit angle for the concentratorsystem, respectively. In the two-stage freeform design in thisresearch, for each given primary input numerical aperture, theexit numerical aperture is different for each primary–secondarysegment pair and changes with each iterative loop. For eachsegment, y2 is determined by the location of the primary/secondary reflectors and their geometrical parameters, and couldbe solved using cosine law and equations for ellipsoid/paraboloid.The total thermodynamic limit for this two-stage freeform systemis the summation of the thermodynamic limit of each segmentpair. Therefore, an integrated calculation formula is added in thealgorithm to find the thermodynamic limit. For instance, for thecase presented above, the calculated thermodynamic limit is 409based on the primary and secondary positions. Although thethermodynamic limit for this design is relative small, this two-stage freeform system does achieve a very good uniformity on thereceiver surface, which is important for the CPV application.

2.2. Ray-tracing simulation

To evaluate the result of the freeform optical design used in thisresearch, non-sequential ray-tracing is applied to trace the propaga-tion of light rays after the freeform reflector. Light rays are randomlygenerated from the point sources and then transmitted in a randomdirection constrained by the required flux distribution map. Afterreflection and/or refraction inside the optical system, the random rayswill finally reach the detector’s surface. If the number of rays is largeenough, the radiation pattern on the surface of receiver can be used toevaluate the system performance. In a beam shaping or illuminationdesign, ray-tracing is a useful tool to verify the final irradiancedistribution on the target detector. In this research, the freeformoptical mirrors were designed using a self written MATLAB codeprogrammed based on the algorithm described earlier and the opticalanalysis of the nonimaging optical system was performed using acommercial software, ZEMAX. To build the freeform surface in theZEMAX non-sequential mode, polygon objects were used in thisresearch.

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Primary and secondary mirrors were meshed into small seg-ments, which were defined as quadrangle reflective surfaces inZEMAX subroutine. A detector with 100�100 pixels was placedon the position of the solar photovoltaic cell. To simulate the

Fig. 3. Ray-trace simulation of the freeform CPV design: transmission versus

incidence angle.

Fig. 4. Ray-trace simulation: (a) light irradiance distribution on the photovoltaic cell w

along x- and y-axis.

Fig. 5. Ray-tracing simulation: (a) local light irradiance distribution on the photovo

photovoltaic cell along x- and y-axis.

incident light of the sun, uniform collimated light with 0.261 halfangle was introduced as the light source. We defined the ratiobetween the rays received by the detector and rays reaching theprimary mirror surface as the transmission ratio. Transmissionratio is a critical measure to evaluate the acceptance angle of theconcentrating system. Fig. 3 graphically shows the transmissionratio at different incident angles. As shown in this figure, whenthe tilted angle along x-axis of the incident ray is less than 711,there is no significant change in the transmission ratio. For thetilted angle along y-axis, that angle is 71.21. This result demon-strates that the freeform design satisfies the required condition ofthe acceptance angle. An acceptance angle of 711 is achievedaccording to the ray-tracing simulation result. The intensitydistribution is also evaluated by the ray-tracing simulation.

Fig. 4 shows the local irradiance distribution on the receiversurface when the incident light is along the normal direction. Thesimulation shows that the freeform mirror provides a vertical andhorizontal sharp cut-off gradient at the edge of solar cell receiver.Furthermore, the irradiance distribution inside the receiver bor-der is uniform as shown in Fig. 4. When the angle of the incidentlight is increased to 11, this two-stage freeform concentrator can

hen the sun is on the axis and (b) irradiance distribution on the photovoltaic cell

ltaic cell when the sun is off-axis by 11 and (b) irradiance distribution on the

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still provide sharp cut-off gradient at the edge, and the irradiancedistribution is also uniform, as shown in Fig. 5. This simulationresult demonstrates that this concentrator works well when theincident light angle is less than the acceptance angle, and canprovide uniform irradiance distribution on the receiver surface.This spatial uniform is achieved using the integrated Kohleroptical design method and freeform optical surface design. Thekey feature in this design is to allow the light to be transmitted toreceiver surface and achieve spatial uniformity. In addition, aniterative algorithm was used to minimize the least square devia-tion between calculated intensity distribution and ideal uniformintensity by changing the design parameters.

3. Thermal slumping

Thermal slumping process is a temperature dependent defor-mation method that can be used to replicate the pre-fabricatedmold surface profile onto the slumped glass workpiece. Glassthermal slumping is a low cost, high volume, and environmen-tally conscience process for fabricating glass optical surfaces.Previously, the thermal slumping process is a proven alternativefor ophthalmic applications and the x-ray telescope reflec-tors [13,14]. In glass thermal slumping process, the mold andglass workpiece are pre-heated to the slumping temperature untilthe glass becomes soft. Afterwards, the glass workpiece is gradu-ally deformed to the mold surface by gravity or by vacuum. Aftercontrolled cooling, the slumped glass workpiece is released atroom temperature. Fig. 6 shows the major steps of the thermalslumping process.

Fig. 6. Thermal slumping process used to duplicate the surface curvature of ceramic m

(c) releasing of the mirror.

Fig. 7. MACORs mold surface profile measured by CMM: (a) surface

3.1. Mold fabrication

The nature of the glass thermal slumping process requires that themold materials be able to work at high temperature with properchemical and thermal stabilities. In addition, to reduce the total costfor primary concentrator mirror fabrication, the mold material needsto be fabricated easily using conventional machine tools such as acomputer numerical controlled (CNC) lathe or milling machine. Attypical glass slumping temperature, glass viscosities are in the orderof 109–1010 Pa s [11]. For most glass materials used to fabricateoptical components, maximum forming temperature will be over600 1C. At this temperature, most mold materials used in plasticmolding process, such as aluminum, stainless steel, and nickel alloys,start deteriorating and potentially can stick to glass workpieces.Sometimes a thin layer material coating such as TiAlN/ZrN is appliedto the mold surface to reduce sticking and mold wear [18]. However,coating process will increase the complexity and cost of the thermalslumping process.

On the other hand, traditional glass compression moldinginsert materials, such as cemented tungsten carbide (WC), siliconcarbide (SiC), have good hardness, stability at high temperature,and wear resistance. However the fabrication processes aredesigned primarily for high precision optical lense manufacturing,and therefore can be relatively expensive [19]. In this research,MACORs (www.corning.com) ceramic was used as the thermalslumping mold material for its excellent chemical stability at hightemperature up to 1000 1C. MACORs is a machinable glass–ceramic, which can be machined into any desired shape usingstandard metal working tools [20]. According to the experimentsconducted in this research, there is no significant stickingbetween the MACORs mold and the soda lime glass mirror after

old to the glass workpiece: (a) heating of the mold assembly, (b) slumping, and

profile and (b) fabrication error compared with design surface.

Fig. 8. Temperature history of a selected thermal slumping process.

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thermal slumping when temperature was raised up to 660 1C.Furthermore, there were no visible wear and defects on MACORsmold after over 30 times thermal slumping cycles under varioustempeatures (from 600 to 660 1C) and holding times. In all theseexperiments, MACORs mold made contact with glass workpiece.These properties make MACORs ceramic suitable as a moldmaterial for high volume glass thermal slumping.

To fabricate the MACORs ceramic mold, first of all, the moldprofile was designed based on the optical performance requirement.Afterward numerical simulation and finitie element method wereperformed to compensate for the potential error that would beintroduced in the thermal slumping process. Mold design wasmodified by the compensation result. Based on the design surfaceprofile, a 3D drawing was created using a modeling software, andthen exported to CAE software FeatureCAM for CNC operation. In thisstudy, MACORs mold was fabricated using a CNC machine (Haas VF-3 CNC machining center) with a standard metal working carbideballend mill. The fabricated molds were measured using a coordinatemeasuring machine (CMM). Fig. 7a shows the surface profile of theMACORs ceramic mold for the freeform primary mirror, and Fig. 7bshows the deviation between measured results and the design surfaeprofile. In Fig. 7b, most curvature deviations are less than 40 mmexcept the edge of segments that may come from the misalignmentduring CMM measurement.

3.2. Thermal slumping process

The thermal slumping tests in this research were conducted ona home-built hot glass-molding machine and a thermal furnace(GRIEVE, BF-12128-HT thermal furnace). The details of the glass-molding machine design and its performance have been includedelsewhere [21]. This machine is capable of controlling tempera-ture inside the furnace up to 800 1C. To control the thermalslumping formation and optimize process time and performance,the following process was used to make the glass mirror:

�

A precut circular soda lime glass workpiece with 100 mmdiameter and 1.1 mm thickness was placed on top of theMACORs ceramic mold as shown in Fig. 6. � The glass disk and MACORs mold were loaded into the glass-

molding machine.

� The furnace temperature was raised up to the strain point

Tstrain, and then the furnace temperature was held for a fewminutes to ensure a uniform temperature distribution insidemold and glass workpiece.

� After temperature in the furnace reached a presumably uni-

form level, it was heated up again to the soaking temperatureat a lower heating rate.

� The furnace temperature reaches the soaking point and was

held for a fixed amount of time to allow the glass workpiece todeform to the MACORs mold surface.

� After soaking, the furnace was cooled down to room tempera-

ture, and the slumped glass mirror was removed.

� Fig. 8 shows an example of temperature history collected by

the thermal sensor inside the furnace during a cycle of thermalslumping process. The soaking temperature was 620 1C withduration of 120 min. Different soaking temperatures anddifferent holding times were also performed.

As stated in the previous section, at soaking temperature theviscosity of glass is in the order of 109–1010 Pa s. At this temperature,the viscosity is high enough such that the thickness of the glassworkpiece does not change appreciably during forming while thesurface tension can stretch the glass surface to the desired surfaceroughness. On the other hand, the viscosity of glass is still low enough

such that it can slump to the mold shape in finite working time. Athigh temperature, more specifically, around glass transition tempera-ture, glass exhibits viscoelastic behavior, which is strongly time andparticularly temperature dependent.

When temperature goes much higher than Tg, viscosity of glassbecomes extremely low such that stress relaxation times are in theorder of milliseconds or even shorter. These stress relaxation timesare much shorter as compared to the soaking time; thus the stressesintroduced from loading are negligible. Therefore, the glass materialcan be modeled as a 3D Newtonian fluid and the material behaviorcan then be described using Eq. (4) based on 3D Newtonianconstitutive law, where the stress tensor tij is a function of the strainrate _e and the viscosityZ related in the following equation [19,22]:

tij ¼ 2ZðTÞ_eij ð4Þ

The viscosity Z value of glass at different temperatures iscurve-fitted using the Vogel, Fulcher, and Tammann (VFT) equa-tion [22,23] give by Eq. (5)

logðZÞ ¼ AþB

T�T0ð5Þ

where A, B, and T0 are constants that are usually obtained fromthe glass vendor. In this research, soda lime glass disks with nearpolished surface on both sides with 100 mm diameter and1.1 mm thickness (Tg¼557 1C, S. I. Howard Glass Co. Inc, Worce-ster MA) were used as the glass workpiece. The surface finish wascreated during the manufacturing process when the glass sheetswere created from glass melt. No further polishing was performedon the glass samples prior to slumping. Using the exponentialviscosity curve for the soda lime glass, glass viscosity at othertemperatures in the molding range can be calculated for theexperiment and the matching numerical simulation.

3.3. Curvature compensation

In thermal slumping and compression molding process, one of themost important design criteria is to satisfy the requirement ofcurvature of glass surface to sufficient optical precision. Howevergeometrical deviation (or curve change as often referred to in opticalindustry) incurred during heating, molding, and cooling process is acritically important manufacturing quality parameter. In glass com-pression molding and thermal slumping process, there are manyfactors that could lead to curve change in the final products, such asthermal expansion, stress and structural relaxation, and inhomoge-neous temperature distribution inside the molding machine. To thisend, FEM simulation has been widely used in glass compressionmolding and thermal forming process to predict the final lens shapeand optimize the fabrication processes [23–25].

Following the same idea, thermal slumping of freeform primarymirrors was simulated using the same conditions as the experiments

Fig. 9. (a) Cross sectional view showing the mechanic and thermal load in the glass slumping process and the final shape of the glass workpiece after slumping. (b) A

slumped freeform glass mirror.

Fig. 10. FEM simulation result compared with measured glass mirror.

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in 2D condition. Due to the complexity of the freeform primarymirror surface contour, only one single line along x-axis was drawn tosimulate the slumping process in a 2D plane strain situation using theMSC/MARC FEM software. Fig. 9a shows the meshed geometry of aglass workpiece and the MACORs mold. The boundary conditions,including gravity, convection heat transfer, contact between glassworkpiece, and the MACORs mold, were also shown in this figure.The contact between glass workpiece and MACOR mold was definedas the Coulomb stick-slip model according to a previous research [18],and the shear friction coefficient is 0.5 in this simulation. Thesimulation includes two major steps: (1) the glass workpiece wasslumped down under its own weight at various levels of viscosityfrom the start of heating until the beginning of cooling; (2) theslumped glass mirror was cooled to room temperature under pre-determined cooling rates. As mentioned before, the glass workpiecewas defined as the Newtonian fluidic model in this simulation. Sincethe viscosity of soda lime glass is a function of temperature, a userdefined subroutine was introduced to calculate the viscosity values atdifferent temperatures. MSC/MACR can load the calculated viscosityvalues and use them in different slumping steps. In the second step,the Narayanaswamy model was applied to simulate the structuralrelaxation in cooling stage. The details of cooling and residual stressesof glass material can be found elsewhere [25]. Fig. 9b is a slumpedfreeform glass mirror.

Fig. 10 shows the FEM simulation results of the slumped glassmirror using the mold design before and after compensation. Asshown in this figure, the glass mirror slumped using the

compensated molds showed a much better agreement withdesigned value than the mirror molded without compensation.Therefore, new mold design was used in glass thermal slumpingtest in this research. The approach using numerical modelingprovides an opportunity for optical manufacturers to achieve alower production cost and a shorter cycle time.

4. Measurement

4.1. Surface contour

The inner surface contour of the slumped glass mirror wasmeasured using a coordinate measurement machine (CMM, Shef-field Cordax RS-30 DCC) to evaluate the manufacture result.The result of the CMM measurement is shown in Fig. 11. Themeasurement is also compared with the design surface profile. Asshown in the error map in Fig. 11b, the maximum curvaturedeviation at the edge is less than 100 mm, and at the area aroundthe center the curvature deviation is less than 50 mm.

Several important parameters were also investigated, includ-ing the relationship between surface contour error and thermalslumping temperature and holding time, and the relationshipbetween surface roughness and thermal slumping temperatureand holding time. The surface contour error is drawn from theCMM measurements. To evaluate the curvature error, the rootmean square (RMS) of the curvature deviation of each measure-ment is applied using Eq. (6)

RMS¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n

Xn

i ¼ 1

e2i

vuut ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n

Xn

i ¼ 1

ðxi�xiuÞ2

vuut ð6Þ

where n is the number of the total measurement points, ei is thedeviation at each point, xi is the design position, and xiu is themeasurement value from the CMM.

Fig. 12 shows the surface curvature deviation under differentthermal slumping conditions. First, we keep the slumping holdingtime as constant at 120 min to change the thermal slumpingtemperature from 600 to 660 1C every 10 1C, as shown in Fig. 12a.From the measurements, no further improvement was seen whenthe slumping temperature was higher than 640 1C. We canconsider that the glass workpiece has made full contact withthe MACORs mold at and beyond this temperature. We alsoinvestigated the holding time changing from 30 to 120 min atsame thermal slumping temperature 640 1C, as shown in Fig. 12b.Similarly, when the slumping time is more than 60 min, the glassworkpiece makes full contact with MACORs mold. Therefore, toensure full contact between the mold and the glass workpiece,640 1C slumping temperature and 90 min holding time were usedin all experiments reported in this paper.

Fig. 12. Surface contour error of the slumped freeform primary mirror under different thermal slumping parameters: (a) different thermal slumping temperature with 120

holding time and (b) different holding time at 640 1C slumping temperature.

Fig. 11. CMM measured results of the slumped freeform primary mirror: (a) surface profile and (b) error from the design shape.

Fig. 13. Surface roughness measured by AFM: (a) original glass blank surface, (b) thermal slumped glass mirror, and (c) thermal slumped glass mirror after coating with

100 nm aluminum film.

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4.2. Surface roughness

The surface roughness of the slumped freeform mirrors wasalso measured using AFM, as shown in Fig. 13. Obviously, the

surface roughness of the slumped glass mirror (Ra¼3.8 nm) ismuch better than the surface roughness of the MACORs moldsurface (Ra¼1.4 mm), and can be used as optical surfaces. Thesemeasurements demonstrated that glass thermal slumping process

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is capable of producing glass mirrors with desired shape as a lowcost one-step process.

The surface roughness of the glass workpiece before and afterslumping process was measured by AFM to evaluate the slumpingresults. To test the performance of surface reflectivity, a thin layerreflective material is coated on the slumped glass surface. In this

Fig. 14. Surface roughness of the slumped freeform primary mirror under different

120 min holding time and (b) different holding time at 640 1C slumping temperature.

Fig. 15. (a) Primary freeform mirror after scale up to 2160� , (b) secondary freefo

research, an aluminum film with 100 nm thickness was depositedon the slumped glass upper surface using an e-beam evaporator(Denton Vacuum EVP501). After the thin film deposition, thesurface roughness was again measured by AFM.

Fig. 13 shows the 3D surface texture of the AFM measure-ments. The surface roughness is calculated by the single line scan

thermal slumping conditions: (a) different thermal slumping temperature with

rm mirror, and (c) intensity distribution on the receiver simulated by ZEMAX.

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(10 mm length) from the AFM measurement. Fig. 13a shows theAFM scanning result of a blank glass workpiece before thermalslumping. Fig. 13b shows the AFM result of the thermal slumpedglass upper surface with 640 1C slumping temperature and120 min holding time. The surface roughness values are 6.0 nm(Ra) and 19.4 nm (Ra). Obviously, the surface quality of glassworkpiece became worse after the thermal slumping process.However, after the mirror was coated with 100 nm-thick alumi-num film, the surface roughness value improved to 6.9 nm (Ra) asshown in Fig. 13c as e-beam evaporation can improve surfaceroughness by filling the micro size cavities on glass. This surfaceroughness is considered to be adequate for the surface scatteringfor optical applications. The surface reflection at different surfaceroughnesses will be measured next.

Fig. 14 shows the surface roughness measurements underdifferent thermal slumping conditions measured by the MitutoyoS-3000 profiler under similar conditions shown in Fig. 12. Asshown in this result, the surface roughness increases with thethermal slumping temperature and the holding time. However,when compared with MACORs mold (Ra¼1.4 mm), the surfacequality can still be considered adequate for optical mirror appli-cations when coated with reflective material, such as aluminumfilms, as shown in Fig. 13c.

5. Scale up

This optical design and fabrication process can be easily scaledup for different size solar concentrator applications, such ashigher concentrating ratio and larger primary mirror surface. Anexample of the freeform concentrator with 2160� is presentedhere using the same algorithm above. As shown in Fig. 15a and b,the dimension of the primary mirror is about 300�180 mm, andthe solar cell is still 5�5 mm. According to the ZEMAX simula-tion, the new design can still allow uniform intensity distributionover the entire receiver surface, as shown in Fig. 15c.

6. Conclusion

A new high volume low cost thermal slumping process wasproposed to fabricate glass freeform primary mirrors for highconcentration solar energy. The surface profiles of the slumpedglass mirrors were measured on a CMM machine. Furthermore,the surface roughness was measured by AFM to evaluate thesurface quality after thermal slumping. Some of the key contribu-tions of this work are summarized as follows:

1.

A two-stage freeform concentrator was designed using aKohler integrated system. The optical design was simulatedusing ray-tracing method to evaluate its optical performance.As shown in the simulation, this freeform concentrationsystem has 711 acceptance angle and 320� concentrationratio. Furthermore, this concentration system can achieveuniform irradiance distribution over the entire acceptanceangle based on the ray-tracing simulation.

2.

MACORs, a machinable glass–ceramic material, was used asmold material for its excellent mechanical and chemicalstabilities at high temperature. All machining processes usedin creating the ceramic molds are widely available in machineshops in industry. No special tools were used in this study.Furthermore, it has been discovered that no significant stick-ing between the glass workpiece and the MACORs moldoccurred at the selected slumping temperature.

3.

FEM simulation could be used to optimize the slumpingprocess and compensate for the mold design in glass thermal

slumping process, providing industry with a powerful tool forhigh volume and low cost nonimaging glass optical manufac-turing. To increase productivity, vacuum or compressed air canbe used to speed up forming process. This has already beenutilized in industry for polymer mirror or dome manufactur-ing. However, when extra force is applied to deform the glassmaterial at high temperature, since glass will experience bothstress and structural relaxations, the final shape will be quitedifferent from the optics that was slumped at a slower rate. Tothis end, numerical modeling offers a perfect solution topredict and compensate for geometry deviation, allows man-ufacturers to maintain a high speed fabrication process with-out losing the proper shape control.

4.

If slumping temperature profiles were properly selected, thesurface roughness of slumped glass mirror would remainunchanged after slumping because of surface tension. Higherthe slumping temperature, shorter the slumping time. How-ever, the surface roughness starts deteriorating when soakingtemperature reaches a critical level. A proper temperature ofroughly 640 1C for soda lime glass was identified in thisresearch.

Future work would include performing the scaled up glassmirror fabrication process research for industrial applicationand identifying other mold materials and process optimization.At the same time, mold design needs to be modified to compen-sate for geometrical shape change due to different slumpingtemperatures, holding time used, and potential use of vacuumor compressed air, which has been demonstrated to be anefficient compensation method in thermal molding manufactur-ing process.

Acknowledgements

This material is partially based on work supported by theNational Science Foundation under Grants no. CMMI 0547311.Any opinions, findings, and conclusions or recommendationsexpressed in this material are those of the authors and do notnecessarily reflect the views of the National Science Foundation.Yang Chen acknowledges the support of the Presidential Fellow-ship from the Graduate School at The Ohio State University. Theauthors also acknowledge the help from Mary Hartzler at TheOhio State University for assisting in MACORs mold fabrication.

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