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Telescopemirror supports: plate deflectionson point supports Jerry E. Nelson Lawrence BerkeleyLaboratory,Space Sciences Laboratory,University of California,Berkeley,California 94720 Jacob Lubliner Departmentof Civil Engineering, University of California, Berkeley,Calitornia94720 Terry S, Mast Lawrence BerkeleyLaboratory,Space Sciences Laboratory,University of California,Berkeley,CaliÍornia94720 Abstract 'ùy'edescribethe deflectionsof uniform-thickness plates supportedby discretepoints and by continuousrings. The calculationsare basedon the theory of deflectionsof thin plates. In somecasesthe effectof shearon the deflectionsis also included. The optimum locationsof the support points for a wide variety of simple geometriesare given. The deflectionsand methodsfor estimatingthe deflectionsfor the limiting caseof a large number of supportpoints are also desc¡ibed.Sincethe slopeerrorsinducedin a mirror by its supportmay be relevantto opticalimagequality,we describe a method of relating the surfacedeflectionsto the surface slopes,and hencethe geometricopticsimageblur that results from a support system. Theseresultsare particularly useful for the supportof very thin mirrors,wherethe optimum support is needed. l. Introduction problemsin the designand One of the most challenging constructionof largetelescopes is that of properlysupporting the primarymirror so that the largeforcesdue to gravity do alter its shape.High qualitytelescopes not objectionably typically requirethat the mirror deformarions be limited to some small fractionof the wavelengthof light that the telescopeis designedto use. For optical telescopes in particular, the deflectiontolerancesare sufficientlysmall that great caremust be taken in the mirror supportif thesetolerancesare to be met. As a telescopepointsto differentzenithanglesthe direction of gravity relativeto the primarymirror changes.As a result of this, a mirror supportsystemmust be able to carry this force both in the axial direction(normal ro the mirror surface)and in a radialdirection(parallelto the mirror surface). Thesetwo components typicallyhavedifferentsupport systems,and it is often the casetha! the axial supportis the more delicateand difficultsupportproblem.For largemirrors this axial supportis usuallyprovidedby some distributionof forcesappliedalong the backside of the mirror, or at some points locatedalong the centerof gravityof the mirror. The radialsupportsystemis similarin form, typicallybeing a system of forces applied either around the periphery of the mirror, or at discretepointslocatedalongthe midplaneof the mirror. In this paperwe will not dealexplicitlywith the problems of radial support, but will only addressthe nature of axial supports. This generalproblem has been of great interestin the past,and a detailedsrudyof axial supportsystems in theory and practicewas made by Couder (1931). More recently,the entire field of the supportand testingof astronomical mirrors wasreviewedin a symposium(Crawford et al., 1968). Several specificmirror supports are also describedby Pearson(1980). The largesttelescopes have useda discreteset of point supportsfor their axialsupport.The Soviet6m telescopehas 60 backsupports,while the Hale200" (5m) has 36 backsupports, and smaller telescopes such as the Lick 3m have 18 back strpportpoints. Thesepointsof supportare points at 2l 2 / SPIE Vol. 332 Advanced Technology Optical Telescopes (1982) which specifredforcesare applied,often with counterweight systems.Other methodsof applyingtheseforcesincludeair pads and whifrletreesystems.In other cases,back support has been achievedby usingring supportswith, for example, fluid filled tubes to achievea uniform distributionof force alongthe circumferential lengthof the tubes. Althoughaxialsupportsfor telescope mirrors havebeen dealt with in some detail by previousauthorsand telescope designers,the currentdesireto build extremelythin and large mirrors makesthe subjectof great interest. Generaldesign guidelinesfor possiblesupportsof thesevery flexiblemirrors will be most useful. In addition,the recentinterestin telescopemirrors that are segmentedor composedof multiple telescopeprimaries, makesthe uniformityof focallengtha new and criticalissue. With the supportof only a singlemirror concernover the accuratecontrol of the radiusof curvatureof the mirror is minimal. In this case,changesin the focallengthof the mirror due to imperfectionsin the support only require a refocusingof the telescope, somethingwhich is often required as a resultof thermaleffectsin any case. For telescopes that have primarieswith multiple mirror components,the focal lengthsof the mirrors shouldeither not changeat all, or at least all changetogether. Particularlyfor segmentedpriboth obsermaries,whereone may attemptto accommodate vationsin the visible(seeingtimited)and infrared (diffraction limited) regimes,all segmentsmust have the same focal length to very high accuracy.Of coursefor the supportof flatsthe propercontrolover the curvatureis essentialas well. In section2 we will developsomescalinglawsand other designof axial semiquantitativerulesto aid in the conceptual support systems. In keepingwith our conceÍn over focal length,we will considerany platedeformationsthat alter the focal lengthas detrimental.A varietyof examplesof optimized supportsystemswill also be given as further aid in the designof supports.Because the goalsof this work are rather general, simple idealizedcasesare discussed,rather than specificrealisticcasesthat must be dealtwith in the detailed designof a specificsupportsystem.In this spirit we will typicallyignorethe effectsof centralholes,variablemirror thickness,shelleffects,and the effectsof shearin mirror deformageneralscalinglawsthat applyto the tion. Section2 describes the deflection deformationof thin plates.Section3 discusses of plates that are infinite in extent. These have no edge effectsand thus representthe limiting caseof a mirror that is supportedat a great many points. Section 4 gives for a varietyof casesthe optimizedsupportconfigurationsfor circular plates.Thesecoverboth point supportsand continuous ring supports.In the final sectionwe brieflyaddressthe issue of supporttolerances and sensitivity In keepingwith the opticalmotivationfor thesestudies, we evaluatethe effect on the image quality of imperfections in the mirror support. There are two common ways of evaluatingmirror distortions. l) Using the geometricoptics approximationthe imageblur is evaluatedfrom slopeerrors the in the mirror surface.2) The other extremecharacterizes mirror surfaceby the root mean squaredeviation (rms) of the the mirror from its desiredshape,and thus characterizes imageby an rms wavefronterror. This approach assumes that the wavefront error can be characterizedby a singlenumber and that this determinesthe optical effects. Although this is an excellentapproximation when the wavefront errors are small compared to the wavelengthof light, and the geometriclimit is a Soodapproximation when the wavefront errors âre largecomparedto the wavelengthof light, the intermediateøse requires the full and complexapplicationof the physicaltheory of light. Some tools for understandingthis intermediateregionare contained in our report to this conference"The Effectsof Mirror Segmentationon TelescopeImage Quality (Mast et al.)." Conceivablythe "optimumnsupportsupportwill dependon which approachone choosesfor analyzingimageblur, but as we relationcan be show in the next section,a fairly quantitative made betweenrms surfaceerror and maximum slope errors, identicalresults. should give essentially so both approaches rrVetypicallywill optimize on the rms surfacedeviation from the mean deflection rathe¡ than the slopeerrors. The peakto-valleyerror is also closelyrelatedto the rms surfaceerror. error is about It is commonlythe casethat the peak-to-valley five times the rms, and hencethey are essentiallyequivalent. However, since the peak-to-valleyonly gives information from two extrelne points, whereasthe rms is averagedover the entire surface.the rms is normallythe more useful surfacecharacterization. 2. ScaiingLaws In the theo¡y of thin plateswhereonly flexuralbending contributes to the plate deflection and shear effects are ignored,deflectionsof circularplatestakethe form: (1) ^-rms: k "q a 4 D Ìvnere q - the appliedforce per unit area a - the plateradius. O - nnjll2(1-vz), the flexuralrigidity, E : the materialelasticmodulus. v - the materialPoissonsratio, h - the platethickness,and k is a parameterthat dependson the natureof the supportconfiguration. For self-weightdeflectionsthis gives the well known scaling law that the deflectionsvary as: ô-4 h¿ e) form of this relationis For non-circularplatesa generalized more useful. ô,,":t4,* 4, ' D h2' (3) where A is the plate area and (, the supportefficiency,is a dimensionlessparameter that depends on the support configuration. This form is useful for estimatingfor example the deflectionsof hexagonalplates,a shapedesired in segmentedmirrors. When one considersthe deflectionwith N support points,it is evidentthat a pertinentparameteris the areaper supportpoint, ratherthan the areaitself. Thus one cancouch the rms deflectionof a platesupportedby N points in the followingform: ô.^:tNfitftlz (4) In this form, once 7¡ is known, the deflection for a mirror of any size can be found. The virtue of this form of the deflection law is that the parameter y¡¡ becomesa fixed constant for an optimized support configuration, at least in the limit of large N. For small values of N there will be noticeable edge effects that will make 7¡ a function of the number of support points. As the number of support points increases/¡y will decrease to an asymptotic limit y-. Improperly optimized support systems will of course have a larger y than otherwise. Recognizing that 7 is a measure of the efficiency of a given support coofiguration, we call y the support point efficiency. We will typically ignore the effects of shear. This eases the calculations and allows us to ignore the effects of plate thickness, except in D. However, a rough estimate of the additional deflectionsthat will result from the inclusion of the effects of shear can be found by use of the formula r + 6¡oat: Earn¿¡nell "(!)21 [ " l t (5) where u is an effective length between support points, and a is a constant. If one deflnesu from the relation Nru;¿: A (6) then for a variety of casesnumericallyevaluatedone obtains an approximatevaluefor a of o : 2. Note that this rough formula only appliesto platesthat have optimizedsupports. A non-optimizedsupportmay have a much larger bending deflection but essentiallythe same shear deflection, and the inclusionof sheareffectsthenresultsin a smallerchangethan indicated. We emphasizethat the effects of shear scale spanratio, rather accordingto the thickness-to-support-point ratio. than accordingto the thickness-to-plate-radius As mentionedearlier,the optic¿leffectsdue to mirror deflectionsare often estimatedusing surfaceslopesrather than the ¡ms deviationof the surfacefrom its desiredshape. The slope,the rms deflection,and the effectivelength must haveat leasta roughrelationship, (7) Sn õr^r/ u This gives the scalingrelation of l=14=yn ùc - 8- Nat 1\ I4.¡uz :/ t?rñ,' (8) One can test these relationsin a variety of casesto flnd a value for the parameterg. We have found for a surprising variety of cases that, for systems optimized to give the minimum rms deviationfrom the desiredsurface,the maximum slopecanbe found (within-20V0)by using (9) g9:9v ¡,t The maximum imagediameteris four times the maximum relaof thiswell definedslopevs displacement slope. Because tion, and becausewe are particularlyinterestedin deflections that are sufficientlysmall to allow diffraction limited perforthe surfaceby its rms deviation mance,we will characterize from the idealone for the rest of the paper. This meansthat the averagedeflectionon a given supportsystemhas been removed. We includeall other effectsincludingchangesin the overallradiusof curvature(focus). 3. InflnitePlates Real finite plates have a variety of edge effects that tend to increase the plate deflections and appreciably complicate the analysis of their deflection. If a plate is simply supported SPIE Vot. 332 Advanced Technotogy Optical Telescopes (1982) / 213 ,i; neara givensupportpoint will at many points,the deflections be dominatedby the locationsof the neighboringsupport points and will be only weaklyeffectedby the locationand shape of the free edge of the plate. For these reasons' infinite platesare worth studying. We also expectthat no frnite plate can be supportedas efficientlyas an infinite plate, thus, the efficiencycoeñcient for an infinite plate will set a lower bound on the emciencycoefficientof any point support systemfor a frnite plate. Finally,the studyof infinite plates cån serveas a useful guidefor the optimaltopologyof support point locationsfor frniteplates. If we restrict ourselvesto point supportgrids that are are posidenticalfor all supportpoints,,only threetopologies sible. These are shown in Figure I and are the ones with three neighboringpoints (hexagonal),with four neighboring points (triangupoints (square),and one with six neighboring iar). Becausethese support topologiesare different, we expectthat their efficiencycoefficientswill differ as well. The deflectionsof an infinite plate supportedby rectangularrows of supportpostsis a classicproblemin plate theory. A discussionof this problem and its solution are given by Timoshenko(1959). This solutioncan be readily appliedto the squaregrid case. The deflectionsof the plate on triangularand hexagonal supportpoint grids can also be readilyfound' Followingan of a ideaby Medwadowski(1981),we considerthe deflections plate supportedby a rectangulargrid of supportpoints and to flnd the triangular use the principleof linearsuperposition and hexagonalcasedeflections.Detailsof this derivation,and the resultsare given in AppendixA. The resultsfor these grids can be summarized by determining 7 as defined in Equation4. We find ^l : 2'36 x 10{ hexasonal - l'33 x 10-r I square ^f triansutar:1'19 x 10-r ratio v. of Poissons Note that theseresultsareindependent Not surprisingly,the triangulargrid is the most efficient, and the hexagonalgrid is decidedlythe worst' It followsfrom this that the triangulargrid is the basictopologicalpatternthat shouldbe usedas a startingpoint in the optimizationof support point forcesand locationsfor finite plates. 4. CircularPlates We now turn to the most common shapefor primary mirrors, the circle. As mentionedbefore,we will not explicitly discussannular platesor variablethicknessplates,or include the effects of shear or the curvature of the plate. These effectsshould be includedin any detailedanalysisof a specifrcsystem. We will first discuss the deflections that result from point supports. We then evaluatedeflectionsdue to ring supportswhich for some configurationsform a limiting caseof point supports. In analyzingand optimizingsupportsystemsin general, we strongly emphasizethat the supportsprovide specified forces,but do not by themselvesdefinethe positionof the mirror. In the caseof N supportpoints,threeof thesepoints can be specifiedto definethe rigid body motion of the mirror, but the other supportpointsshouldin practicebe allowedto float in position and only provide the specifiedforce. It fol' lows from this that the deflectionsat the support points are not required to be zero for the overall optimizedsupport. for these This is importantin developinga physical-intuition systemsand differs markedlyfrom the intuition one has of conventionalsupportsof plateson fixedpoints. 214 / SPIE Vot. 332 Advanced Technology Optical Telescopes (1982) To find the deflectionsof a circularplate supportedby a and seriesof pointswe will use the principleof superposition divide thè supportpointsup into groups,eachgroupcontaining k pointsúnifortirtvspacèdarounda circleof a fixed radius as-shownin Figure2. The derivationfor the deflectionsdue tõ a singlesucli groupis givenin AppendixB. The deflection has the form w(p,o): iw^$)cos(kmg) (lo) m-0 where the formulae for w^(p) aregivenin AppendixB' To find the deflectionsfrom ihe full supportsystem'we simply udd th. deflectionsfrom eachgroup,weightingthe deflections by the fractionof the loadbeingcarriedby that group' Thus w(p,Ð : f e¡w¡(n¡,ß¡,p,9-þ), (12) i-I where w(p.0) - Plate deflection n - number of groups or rings e, : the fractional load carried by i't ring )rrj : ¿, : B¡ : é¡ : Et': I platedeflectionswith a singlering of point support: numberof supportpointson the irn ring radiusof the l¡n ring azimuttraloffsetof pointson ifi ring A given supportsystemis optimizedby varying the forces, radii, and rotationanglesof eachgroupto minimizethe rms defleition. For the õasesdescribedbelow the optimization was made by numericallyevaluatingthe rms deflectionwith a computer,and using a standardoptimizingpackagefor non' lineaì functions. In addition,someof theseresultshavebeen checkedwith a finiteelementanalysisprogram' independently (Tayìor,Asfura,and Budiansky,1982) A singlegroupof supportpointsgivesan rms deflection that variesïitñ ttré raOiusof the ring and with the numberof supportpointson the ring. The rms deflection(or ratherthe tupport èffñciency)as a functionof thesetwo parametersis shoin in Figure 3. Note that as the radius of the ring one, and the numberof support,pointsbecomes approaches tárge, ttrat the deflectionsapproachthose of a plate simply suõportedalong its rim' It is interestingthat six points is to a continuousring' As atreä¿ya uery ðtoseapproximation zero,the deflectionsbecome the raãiusof the ring approaches thoseof a platesupforié¿Uy a singlepoint at its center' The deflectionsof the piatesupportedby a ring at the outer edge define a kind of "àatural"scalefor the deflections. The rms deflectionnearthe optimumis shownin Figure4 and is a factor of about 25 smallerthan the naturalscale. This dramatic improvementis a reflectionof the fact that deflectionsvary as a4'. This also leads to great sensitivityto support errors' Exceptvery near the minimum itself' the rms variesessen' tially linearlywith the radiusof the supportring. When one analyzessupportswith multiple rings of becomepossible'[n other pointsa varietyof point topologies words,specifyingthe numberof supportpointsdoesnot pro' vide enoughinformationto readilyoptimizethe supportsyscharaaterof the rms tem. If oñe considersthe mathematical numberof points,the funcdeflectionfunctionfor a specified tion will have multiple extremain general. These extrema and thus the topologies typicallyexhibit differentsymmetries, fôì eaitr extrema should be found and then each used as a startingpoint for the optimization. The problemof optimizingthe positionsand forcesof N : oointshas 3N variablesand 3 constraints( net force l, net -o-ents : 0). In additionrotationof the coordinategrid about the plate center has no effect. Thus we seek minimum value of a functionin a 3N-4 dimensional the Sincethis can be a very largespace,and thË-ïunction space. is non_ Iinear in the variables,the matLematical difficutties... errãi_ mous. We will reduce..the difñcultybV addingsymmeÍy con_ ditions^whenever possible.For most casesoi int"r.st, t9q of support with reflectionsymmetryãUout ¡ " lvï_ ä*"r'i. desired. This reducesthe problemto Zñtl-_ 1 variaUlei. We now divide.rhis spacein_ioregionswiitr'¿istinct propertiesrelaliveto the reflectioìaxes. Theie ,yãããö can be exhibited on one sixth of rhe circularpl"te. Àñìxumple 'ftreof this decomposirion for lS.pointsis shown¡n ¡igure S. noìã_ tion usesthe superscripr for the numbei oi"point, on the first symmetryaxis, the subscriptfor the second;ymmetry axis. _ Some topologies are obviously not candidatesfor efficientsuppoit syitems. otlers a¡å, .nã-trr.r" have been optimizedwith the computer.The number'of local mrnrrnu for eachtopologyis belþvedto Ueveiv smaä, -*" probablyone in many cases.When multipleminimueilsi, u." nor cer_ tain thar the lowestminimumhasbeenfounã, Uutexpect that this is the case. We have found. thar the topologiesthat produce the lowest extrema (smallesrrms deflèctiö)-.iä-ìr,òr. .ppr*i_ matlng a triangula¡grid, as we wouldexpectfrom the discussion on- infinite plates. As the number óf-support points grows, the number of possibletopologiesgroru, ., well, for largenumbersof supporrpointss-electäitt. U"rt and täpology may be a difficult task. We have foünà in fact ihat seve¡al topologiesmay- have essentiallyidentical ,uppãii efficiencies.Particularlyfor circularp-lãiËi'"nJlo-. values of N, the triangulargrid may not ahvaysproviãe an obvious naturalropology. For example,.fo¡ N : 9, thelriangulãr is unsuitableand the supportis quite inemc¡ent.iVe LriO tñãw llrat n-o support sy-stemcàn have'an .m¿i"r;t ractor uéiier than that of the infinite triangulargriO. ftrui,'in practice the effciency of a given optimiz;d topäloiy ä-i,; comparedro the triangular grid and .n uppé, bã"rd"; the possible improvementby choosinganoth;; topoloÀicarr'Ue assessed. A variety of supportshave been optimized. For these caseswe have assumed v - 0.25. Thè simplest cases shown in Figure 6, indicatingttre numuei anã iåcat¡on aÍe or trre supportpoints,and the alrernatetopologies considered.Cón_ tour plo_ts of the plaredeflections foi t¡õse ..s.j.re shownin actualoptimumparameters are given in Table .rhe frS,I9./. r rvntcngtvesthe numberof supportpoints,the number of oprimized, the.ring radii, the forces for e.ò¡ ,lng !1t:T:t.* oI supports,the azimuthalrotationfor eachring of suppoiisl th_eachievedefficiency from which ttre rms åËflectio;'for; gl:: 1*.? can readityþe fouqd using Equationq), unO-i¡ð rauo or tne peak-to-valley deflectionto the rms deflection. The efficiencycoefficientsî*;¿;;;;jr;pl'oti.ä'in Figures I 5 1n^d.16 along with the efficiencyfunciions for the three geometries. supporr We now give brief descrip:l^r1T,:"ll"r. uons ol the supportsystems. . The-singlesupportpointsystemhasno parameters to be optimized,and has rms defleðtion,,orn. zii-imes greatãi than rhe limit suggested by the innn¡æiiiangulargrid. The two.point sysremhas a radiusthat can Ue uãiËã, ãnOwhãn optimized,..this systemgivesan effic¡ency iunciiä tnat is 7.2 trmesthe limit. The threepointsupportis the most common of and its optimum'oc."*äiti,å well known .supportsystems, radius of ß : 0.645. Ir is iomewhats"rp¡ri"s (anà'ålõ': pointing) rhat the best rhreepoint ãmcienli,'li:'?.A ti..r-trî" limit Continuingon the singiering d;;;; see that a six ':,_op_ti-ir:9.for B : 0.ó83and has an efficiency Î".'i::lT:T z.r umes rne ltmrt. Addingmore pointsto a singlering now becomesless efficient than creatingn"* iingî of süppãrt points. The simplestadditionis a neú ring withîero radius.a point at the center. forln of sevenpoint supportsystem lhi¡ is notice3blymore efficienr than the ,if iäììrt system,as is seenin Figure 16. This naturallyraisestnl-lúestion whether six point systemwith one poini .t tt" surroundedby I nve points can be more efficientthan tire "óntJi Cónventionats¡nglä ring six point sysrem. .Optimilatiãni.r..l, ti"t the single6 point ring is betterby about 130/0. ring systemsare neededwhen we move ro -,_^ pornt supports.Here^sym_metry nrnelll_jhr.e ' allowsthe rwo iãpãlå_ gies shown in Ficure g.- Both 'po¡nL,r,år, nãu"'"uirost identical efficiencies.Note tñat t¡e nine have much lower efficiencythan rne slx or sevenpoint supports. be explainedas due to.rhefact that ,¿;m;i;;;'riungutar It can grids cannotbe madewith nine points. A completeriangutargrid can be made with and the resulrsfor the besi topologi aiä-gluen 12 points Table l. The topotosies and locarions ôf qf", ãntiüirço inpoints are shownin Figure9. The best t*o, 12¡l ^írä'tT¿, -it i,ãu. èrsrn_ tially the same efficiency.As expecteJ, grid _ givessuperiorperformance., "-t.¡.ngular with an em"iLnåi.uout worse rhan the infinite.rriangulargriO. OpiimizeO1.6times systemsare shownin Figure.10. TopotogieåtS,tån¿15 poini isolîiË about equal,but sinceno triangular iri¿îãnngurution is pos_ sible,the efûciencyis poor. ìVfovingup to the,next completetriangulargrid . -18.point we find the system (we omit it. .rnifai-pìlnt). topotogiesgive equivalentefrciencies(t-gg; 1E";;d Several lgrr'); indicatingthat strict triangulargrl¿ topofoiìe.-"r. no, .rr"ntial, and rhe circularnature oftiri ;úËË;;r. importanr. optimizedtopologies,r"-r¡J*i-in Éiäìr. lr. l-1_v_erat Contour mapsof the deflections..are shownin Fijui. lZ. À.tuãilv the.grid byjd¿ing the lgrh polni'rt the center !?-Tll:t¡ne gves an even higher efficiencyas is seenin Figures 15 and Iô. As we increasethe numberof supportpoints, probably it becomeslessimportanrto choose points consisrenr_with triangulargrids. The ";ñ'b;;;'J';upporr n.if iriunöunr grid sys_ 27.supportpoints,.with an 1"T.þ.r L¿l times that of the infinite triangular "ptir*rn-åfr.t"nõy-ãuãut Severaloptim_ -euite ized topoloeiesare shorvq in-Ed*"'iã:Àii¿. similar efficiencies are found for t t6 ana¿tf . . We completeour examplesof disc¡etesupportswith a 36 point supportsysrem. The'200" t.l.ììopãïr'iiì.ry has a 36 point support system, and the propoi.o r,r'pïãit system for the indivídualmirror segment,oïtli" uni"-;r-r'irü of California T^enÌvleter Telescopetr.s ttris numue, ãirrîiäí,r. There are 28 different.topologies that satisfy-ìhe -beenrt i.O'!v-*.try condi_ t¡ons. The bestcandidates have ôpti.¡iËä,.n0 rhe support point^locations areshownin Figure,la. Ãiìsual, several systems( 36d, 36i, 36j) areequa[] efficieni.-"fne -*årrî'ìr,* optimized ppporr efficiency'is on-lyauoui +õã¡o rhat of the infinitetriangulargrid. When so manysupportpointsare used, the is so complexthar a varietyof similarsupport;o;ii;irssystem givessimi_ lar results. For example,_ifone assumeìthat all 36 points must lie on three rings of 6,.12, and iS-pó-i";, one greaily reducesthe number of variablei,U"t tr,Jï"åi'optimum is onty about 150/o worsethan the fú,ty ;pi;;i;;d -itris-'iÃ-"ãr"ti.uf system. This does nor mean that one can actr¡eíe -* perfor_ mancewithout great care. .On the .ontrrry, Lne goes to more and more supports,the carean¿ oàtäilln tr¡e ãn"rysii requiredro achievethe optimumgo., up áiá_iiicaily. euite minor changesin the parameters èanlea'dto largè the deflecrions.Consideredi" tf,"ärìrr-.ilävïàver,changesin it ooes seem to be the caserhat.the multiple ãi..^ion.f spacein which one searchesfor the minimïm does ffi; rather flat "valleys".where more than one of p"rumete.swill give _set rather similar rms deflecrions.eìen-soi-itr'îãlr, of rhese SPIE Vol. 332 Advanced Technology Optical Telescopes lt9B2),r 215 I I t t\ l i I I I I I ¡ I I "valleys"are exceedinglysteep,and greatcareis neededto achievethe performanceindicatedhere. Based on our experiencewith the supports analyzed here, it appearslikely that if one goesto even more support points, concentricrings of point supportsare an adequate approximationto the ideal, but more complex optimum geometry. The most efficientuse of supportpoints in this situationis likely to be one with the numberof supportsper ring increasingoutwardsin the series6, 12, 18, 24, etc. As noted before,the degreeto which one hasoptimizedthe use of the support poínts can be checkedby comparingthe efficiencycoefÊcientagainstthat of an infinite triangulargrid. system,as the numberof supportpoints For a well-designed becomeslarge, the edge effectsbecomenegligible,and the efficiencycoefficientshould equalthat of the infinite plate. Another way to evaluatethe qualityof the point support systemsis to comparethem with specialcaseshaving an infinitenumberof supportpointson eachring. Thus.weturn to the analysisof continuousring supports.Theseare noticeably simplerto analyzeand optimizethan discretepoint supports, since the number of parametersfor optimizationis for the deflections imaller and the actualanalyticexpressions are substantiallysimpler. One again uses the principle of superposition,and uses the formula for a ring support as given in AppendixB. Sincethe numberof supportpointsis not defined, \r'e return to the notation of Equation 3 and study the parameterf as a function of the number and radii of the ring supports.As with points,we do not insistthat the deflectionsvanish at the rings but rather let the deflectionsat the rings be determinedfrom the elasticequationsgoverning the platedeflections. We have optimizedthe radiusand force for eachring, and the results are shown in Figure 15. We have also included the theoreticallimits for the three infinite plate cases,and some of the optimizedpoint supportsystemsin this figure. It is interestingto use this graph to comparea ring system's efficiency against the point support system efficiencies. For example, notice that the deflectionachievable from three continuousrings is only slightlybetter than that achievedwith three "¡ings"of discretesupportpointsas seen in the 36 point support system. This provides confirmationfor the idea that addingmore pointsto a given ring is not productivebeyonda certainpoint, and that one benefitsmuch more by addingnew ringsof supportpoints. The supportpoint efficiencyis shownin Figure 16. As expected,as the number of support points grows, the efficiencyapproachesthat of a triangulargrid. 5. Sensitivityof Deflectionsto SupportPerturbations In the precedingdiscussion,we havegiven the theoretical limits for plate deflectionsthat can be achieved. In pracand fabrication tice, various approximations,simplifrcations' of the supor alignmenterrors will degradethe performance port system. Particularlywhen the number of supportsis associated of the deflections superposition large,the necessary with eachsupportpoint that will allowalmostperfectcancellationsof thesedeflectionsbecomesa very delicateprocedure. When one is supportinga singlecurved mirror, maintainingthe exactradiusof curvatureis not alwayscritical,and is often possiin this case,some relaxationof the tolerances ble, sincethe flrst order effectintroducedby analysiserrors, etc.,is typicallya changein the curvatureof the plate. One is fortunatewhen this can be focusedout. For caseswherean opticalflat is beingsupported,,or when a seriesof segments 216 / SPIE Vol. 332 Advanced Technology Optical lelescopes (1982) are being supportedthat must haveidenticalradii of curvacritical. ture, any error in the mirror shapebecomes If we assumethat the optimumsupportis known' the effect of displacementsof the supportsfin the plane o[ the mirror) from their optimum locationcan be understoodby of a point is equivalentto the fotlowing. The displacement the supportforce beingin the correctpositionplus a moment beingappliedat the supportpointwith a sizeequalto the proof the duct-of ih" fot." at that point timesthe displacement of the resultingdeflections supportpoint. The superposition from eachof the supportpoint errorsgivesthe additionalsurfaceerror introducedby the supportdeflections. Perturbationsof support positionsand forces must satisfythe constraintsof overallequilibrium;the sum of the forcesequalsthe weightof the mir¡or, and the net moments about the mirror centermust vanish' The true optimization of a support system is an extremelypainstakingprocedure.However,we conjectureon the basis òf severalnumericalèxperiments,that the exact supportpositionscan be adjustedover a small rangewitlout gròisty increasingthe deflections,as long as one carefully õptimizesthe fories appliedat thosepoints. Fortunately,the mathematicaloptimizationof the forcesis a linear problem, hencea uniqueoptimumexists. It canbe easilyfound if the deflectionfoi eachgroup of supportpoints (evaluatedas sole supports) is known. As indicatedearlier, the deflectionsat vanish,but will follow thè'supportpointswill not necessarily from tiù análysis. An exampleshowingtwo configurationsof the sametopologyis shownin Figure17. Thesehaveessentially identicalefficiencies. 6. Conclusions We have describedthe generalconcepts,generalscaling laws, and given some specificexamples0nfinite plates and circular pÈtes) for supportingthin plates on a number canserv-eas the basisfor discretepoints. Thesedescriptions beginning -Inthe designof supportsystemsfor thin telescope additionthey providea basisfor evaluatingthe miirors. of a givensystem.For any real mirror suprelativeeffÊciency port system, an analysis that includesthe detailed shape inon-circulargeometry,centralholes,etc.)and the curvature is essential. A careful examinationof tolerances,although difficult,is alsoinvaluable. Acknowledgments rùy'ethank Michael Budiansky,Alex Asfura, Robert Taylor, Karl Pister,and Enzo Ciampifor their excellentcounsel the work and aid with the computing.We gratefullyappreciate of BarbaraSchaeferon the illust¡ations' References of Large Mirro-rs Couder, A. Researchon the Deformations Bulletin AstronomiqueVII Observations UsedFor Asrronomical (1931) MemoiresEt Varietesp. 201 ff (translatedby E' T' Pearson) Crawford,D. L., Meinel, A. B' , and Stockton,M. W' Supoort and Testiw of Large AstonomicalMirrors. Symposium Center (Tucsón,Arizona,1968)OpticalSciences Þroceedings TechnicalReport# 30 S. , (1981)privatecommunication Medwadowski, AÞPendixA for-.the1990'sKitt Pearson,8., Opticaland InfraredTelescopes (1980) Piocèedings Observatory"òõJ"ttnt" National Peak M'' (1982)private Taylor, R. ,. Asfura, 4., and Budiansky' communlcatlon S'' Theoryof Plaæs Timoshenko, S. and Woinow-sþ-Krieger' (1959) edition S"cond ;';; lì';ß. Deflectionsof Semi-InfinitePlates Points Supportedon ManYSuPPort plate can The questionof efficiencyof supportof a large deviation square mean root the u" uO¿iãtt-"0W àetermining since the plates ;i tdJ;fÈ¿tiãn ftorn the ñreundeflection' points'edge support with manv u".v large, ;";;;;lõ-b. dis' best^support the õoísequentlv, ;ffeJiJä; ilìãgtecte¿. will be amóng the class of geometries ;,Ñil-;¡;óin-is where all supportpointslook alike' If we conThere are three geometriesthat are possible' -Ñi itian have either 3, 4' o¡ 6 nearestneighsiOerïiiùn point, ottrei ãistribution of points will have the Ë.iil ñi"tt. points took atike' These three topologies all Jimriöli;t are shownin Figurel. plate supnow wish to calculatethe deflectionsof a we aÍe Since types' support three thtt. portediv- each-ãf 'ioi"."rt"O the.sup' arrange must we in the supportefficiency, int-.iea Canie¿bv eachsupportpoint is ö;îö;il;tð1ñ"f cases' ln this way, we cån directly comthree ltté r"'*" ii all parethe deflections. The deflectionsof a rectangulargrid of.support-points and tp..itgt ïi- ä .n¿ å are dJscriueãbv Timoshenko of -":n;i;ü:ir;åeãt "itft tlssõl' to the deflections.forthe case can be readilyfound' supportpoints four neigh-boring 'fVe : Drectorstr(a,b;x,y) #r_' t +)2 + Ao -' '1-¡mlz"orWt a oa3b S 2n3D ^-?1,t.. n3sinhc.tanha. .---. mTJ-(c'*tanhc m¡ t rin¡ I t rnno -) l \ u where t t " I ) cosn@l * m ' a a " ' a l a'*tanha . oåå s - |7 \ a/ ^ - - = - - ' As--:T " L tann'dm " Zt" D ^_T¡.t... r¡l and mrb a^- T Setting4 - ä givesthe squaregrid case' T h e d e f l e c t i o n s f o r t h e s i x p o i n t c a s . e ( a t r i a n g uAn largrid) can Uï'iounO by using the principleof superposition'tiis pròblem ìvas..obtainedbv-S' J' ìoiuìion'for ;Ëñ difrerent i,r"?¡"îoãirii ü sb1) utïng irt. superpositionof two we- prèsent,here a somewhat ;;ú;t.ti";t' ;ffi"ilü' the JJ"rion b-asedon the sãme idea: Consider íiäil* geometrysnown ln {eure A1'^ We can *:rr1"-th" deflections grid of supports ã. ir,. ,ú* of the deflótions of a rectangular grid of supports. recrangular a displaced of äirJtr,Jãånã.tiðns and If we assumethe spacingof the triangr¡largrid is ä' ö' we can assumethe rectangular-lridhãsa spacingof a and write the rectangulargrid deflectionsas Drr*ongt"(a,b;x,Y) with a - Jîu, of The triangular grid will then give deflections Telescopes(1 982) / 21 7 SPIEVol. 332 Advanced TechnologvOpticat D,rions!"( b, x, Ð : AppendixB or"rr*t"( a, bix,t) I Defìectionsof a Circularplateon a Ring of point Supports I * lD,"r,oner"(a,b;x i,, *r. The t!1ee noin! supporrsystemcan be handledin a simi_ . rar way. ttgure AZ shows the hexagonalgrid of supports. Assuming the point separationis ã, onã ãn- write the deflections as the sum. of fgur ãirpf"..¿ rectangular deflectionsas is indicatedin the figure. 'iV" ifr.n obtain for the hexagonal grid, deflections of ( b;x,Ð : Dhuoson y) IIo,,,,*¡"(a,B; x, + Dr"r,onglr(o,A;*-ll $) + Dr"rto nsr,(a,B;x- y,y) * Drrr**¡" (a,A,.!,, lll The deflectionsof thin, constantthicknessplatesare of great,interestin opticals.upportsystems.Many problems can oe solvedby the appropriate superpositions of soiutionsof the caseot_a crrcularplateaxiallysupportedby a singleconcentric point (as supports shoùà ¡n F¡dure-ã)l- we ¿erivì Sins of here the solutionto this problem __ Assume a uniformly loaded plate of radius ¿ and stiffnessD, with total load suipoitgã-ul-i point -...7. sup_ porrs locatedat r - b (- a) ana''0-2 jt¡l ß",,is tc,j : ¡; The deflectionw(r,0) is governedby - 4* 4f a(':¿r Dyaw: ,[r_+1. ta2 kíi, á "t" kl TheFourier series for É tfr- +l isoftheform /cJ i-t t Q where a:3b, Ê:ß8, and q) n æ ++E ^t: -b. a^coskm9, with +É,'['þ-Tl**^,* "^- Note that for thesecases,the areasper point are: At iorgle : f,, k 17 hencethe loadingis . ¿ : 62 " square , ../3 âhexagon: tT - \*$atr - al* 7ra" ,7 t-, P ô( r :á) û b i- c os * m e. .ar- Assumethedeflection hastheform _ æ w\r,0): f, 4(r) coskmï, We write the solution in the form m:0 where rir¡ is governedby . .l t t ìlì ¡- : -, !D .t lNt z' where _ø - the appliedpressureon the plate, D" - the modulusof rigidiry otitré pUie, - the plateareaper supportpoint. l: We_can define ô as either the peak deflection, the mean deflection,or the root meansquaredeviationoi nb ¿en..tion from the mean. Fo¡ each of ihese we ..n õo*pãr, the rela'[* fti:l.ies üãïu", õl , lXt, ^of the. three confi_gurationsLv') aÍe grven fbr eachof the definitionsand configurations in the followingtable. triangular square hexagonal ¿ltD peak mean 4.95 3.71 e) L dt rtr dt rþlll__ P=+å¡(r_¿), ; d , l ' d , l ; d , l ' d , J J-J- l ¿ r ç ¡ t i.e., w¡.is ju-stthe deflectionof a platesupportedon a ring of radius á. We can obtain w6 by superposingtne ãeflectioni ãi thãt are uniformli loadãdand ring:lTf]I :gp"fed.ptates loaoeo, wtth totalloadsequaland opposite,to obtain (3) rms p>p. For m2 1, we have l¿ I d È,,?12 p Iof" ,-ã-Tl "': hõG-Ð' In other words, [æ,r l..-+ ldl d rdr tt^212 ¿ | If we define w-(o) ang of wry^$.) ^, the resrrictions ø (b,al and [0, å),'iespectivelíl tn.o *ã i,"u. irrLcontinuiry equations 21 8 / SPIE Vot. 332 Advanced Technotogy Optical felescopes (t 982) \ '.1 w m{ a" ) ( 6 . l ) : w ^ ( Ð ( b - ) , d - - r o-G) 16+¡ ar +(1-v) #lr#-Ëll,:. (5) : fiw^$) Ø-)' so that & w^(ù{b+): (b-), #w^(b) ¿P ê w^(ò(b+)-ã tt-l : -1. *n^r, ar Thesolutiontakestheform (with n : km) n^G) (r) : A^rì + B^r'+2+ cman + D^r-n+2, w ^ ( Ð( ò : A ', r ' + B ' ^ r n + z , l+l- !"r"'l ¿ n ^ ( ò +r l 4 n ^ ' " ' - 4 n ^ < o t dr"^ ¿P (6) and the continuity equations become (A^- A' ,) bn+ (Bn- B'^) 6n+z 0) + C ^ b - ' + D ^ b - n + 2: 0 n ( A . - A ' ^ ) b n+ ( n + Ð ( B ^ - B ' , ) 6 n +-2 n C ^ b - , - (n-2) D^b-n+2- g n b - l ) ( A ^ - A '^ ) b n+ ( n + Ð G + l ) G ^ B '^ ) 6 n + z * n ( n * 7 ) C ^ b - n+ ( n - 2 ) ( n - l ) D ^ b - ' + z: g n ( n - I ) ( n - 2 ) ( A , - A ' ^ )b ' + G + Ð ( ¡ t+ 1 )n ( B ^ - B ' ^ ) 6 n + z - n ( n 1 1 )( z + 2 )C ^ b - ' - n(n-Ð (n-2)D^6-n+2: nOL Thesolutions are (13) *(1-,' #ll r,t [å#n^,",- Substituting(6) into (10) and (13), we see that 1., Bn, Cm, D. must satisfy (14) (1-z) n(n-1)A^a ( ( * n + D l,n + 2 v n 2)l B, a'+2 +(1-z) nGll)C^a-n +G- 1)[¿- 2-vG+2)]D^o-n+2: g, ( l - v ) n z ( n - l ) A ^ a n * n ( n * I ) ( n - 4 - v n )B ^ a n + z - ( 1- v ) , ? G + l ) C ^ a - n - n(n-I) (n+4-vòQ^a-'+z - 0' Since C- ar'd D^ are known alreadyfrom (8) we can solvefor A^ and B^: t ^ a ^P_a 2f3fßþ in t . - , f t ïolt *ì ,f f i8](,1 + y ) ì [ r r _t ," -r ' _ Pa2 B' /1 .f ¡ - + ) -;;õfr(r-,,[;B^an*'- I ß2 I ffi|. (15) (8) We rewrite the remainingsolutionsas (B^-B'^)6n+2: #{*, a - m "h - n : - | n(n*l) 1 "m' Pb2 8¡ D' Pb2 Paz ßn+2 ïnD n(n*I)' g n Dßnf f i , un r -e"-+i- : -Pa2 D^6-'--:ffiffi A ' . a ' - A ^ a n * â " _3 f f i , In addition, ur(o) must satisfy the conditions of zero moment and shear at r - a. Now, using standardexpressions. P - n, . B , - a n + z :B - a , + 2 - y 8¡D n(n*l)' [^r (. lðl I 06) o^2 -1 ì'l : o. La.: l4*,1!9s*+.È+ll D r ör I ð0¿ (e) p-n+2 Thus the solutionto (1) has been obtained. It is given by usins(15)and (16)in (6), and (6) in (2). ll,-o so æ I I , \ *,1: . l t . a : (ro) w^G) ftw^G) fr I "^{a)lo; furthermore *r,,:(r-,,+ #[*-+ì, 0l) -f,o,:'fi'[#.+#. ,#1, and -f,v,It-*n,*+f¡ll,-a (,2) :[*[#.+#. i#l SPIE Vot. 332 Advanced Technology Optical Telescopes (l 982) / 21 9 =. -È-!+i-- -+rL+ llvar Yr(xlO3¡ 2.60 3.8 8.56 3.9 3 ), t6 6\ 2.93 2.36 ni, ßí, eÍ, Oí P-p rms ß = .3s38 ß = .64s .t.J ß = .681 4.9 n = I I 7r o ß = 0 E = .1183 o 0 = 6 . 73 7 .8817 0 n = 3 ß = .282s e = .2309 ,Þ= o 3 .7936 .3637 0 n = 3 ß = .3lsl e = ,2783 3 .7662 .2843 r .0472 3 .8257 .2187 .3497 3 .BZs1 .2187 _.3497 3 .7765 .2046 .7833 3 .8412 .1538 .2615 3 .7165 .2046 _.7833 2 3.76 t' - 2l 5.0 t 1.94 0 = I l5o 2.32 s.4 t l8t r.89 5.5 n = ß = e = 3 ,3192 .2833 0 = o n = ß = e = A = 2 .ut1 L r. 65 9 I . OJ 3 .4741 .1625 3 .3195 s .817r 3 .8536 3 .84r2 , l53B -.zol) o .2071 1.0472 ,.t731 .7820 .1421 .2663 3 .Bl7l . I 73r -.7820 3 .8536 .1421 -.2663 n ß e A = = = = 3 .2044 .1152 0 3 .8740 . LO49 o 3 .4777 . ß47 1.0472 3 .s55s . rt62 .33gg 3 ,8787 .o95s .4280 3 .8574 . tl10 .8383 n ß e 9 = = = = 6 .2s6g .1671 0 6 .5771 . l8l2 .2649 6 .5771 .1812 .7823 6 .8830 .t54g .1703 6 .8834 .1607 .5236 6 .8830 .1549 .8769 I 361 o 3 .7700 .4054 1.0472 TABLE I 3 .5555 . Ir62 -.3399 J .ötó/ .0955 -.4280 'ì . 8 5 74 .tlt0 -.8383 : -..:tr:È*æi.:i.@ h T T a a a TRIANGULARGRID ,' Ztr- i -<-r , a \ t a I t SQUAREGRID \ \ b n= Eh3 l2(l-vz) a a HEXAGONAL GRID x8L 823-8449 x B L8 2 3 - 8 4 4 6 2. The geometry and notation for supportinga circular plate on a_single ring of support points. P is the appliedforce. In the figuresix supportpointsare used. l. The three kinds of point symmetric infinite grids. These grids are shown with equal areasper point. As describedin the text, the triangulargrid is the most efficient. SPIEVol.332 Advanced TechnologyOpticatTetescopes (t982) / 221 iif lii arr.=Ç-.9f €rrrxlO4 €rr.xtO4 4 N=4 N=6 N=æ 02 o.4 o.6 p o.8 t.o o 060 xBL en-8445 3. The support efüciencyas a function . of the ring radius for po_ints on a single ring. Caseswitn 2, 3, ãnó? lippo.t po¡nt, are shown. More than 6 points i, i"dirìi;;ùJhiule rrom e points. Note the dramatic improvement *îth-increasing N, and the great sensitivitvof theïfficiånl;';,il1# normalized radiusB. 0.70 xBL 823-8448 a, Jhe region of Figure 3 nearJhe minimum is expandedfor clarity. Note againthe small differen* üñ;eliñ : 6 and N : infinitv. 222 / SplE Vot. 332 Advanced lechnology Optical Telescopes (l 9g2) . - t83 xsl 823-8447 :1,,,,_** 5. The possiblesupporrtopologies-foran -ìuppãr, lg point supporr. Three fold ¡eflectiôn s¿mmetri of tire ,yrt"* i, 6. The optimal locationsof supportpoints ass¡med.inestablishing given fgr the are -iuai'i¡-ethe posiibletopolog¡åi.Tf,e nom"nsupport configurationsihown. note ctarure.gtvesln the superscriptthe numbe¡ of points ¿j' ;i; i:i"]"l on one force appti^e$ axts or symmetry and the subscript gives thê number to the central point. --' T¡e" plãts l-"1 .. l"g"j1ue of are orderedin the senseof increasing pointson the other axis. "mr¡"n'.v. SPIEVol.332 Advanced Technotogy Optical Tetescopes lt9B2),/ 223 @ ( l Jlli',\¡, t/)¡¡\",, ?,ê @* 3å ttl[tî''\t,u zgN Ð)l-i @ äwAñrrn'ìÈaátr â 4¿ 6; xBL 823_8590 7. ^Contour maps of conngurationsof Figure ^ tfr-o^"li.lions for the support isonè,harr iiñ The iï,;"i mean dellection. ;:.* heavy tine is the mean deflection. ll"; "åfi:J,""i"jF,'nj"li?i":î 224 / SplE Vot. 332 Advanced Technotogy Opticat Tetescopes llgg2) \ a a a a o a a a O a a O a a o a a a a a a o a a a a a a o a a t2R a a a xBL823-8436 9. The optimal locations of support points for the general supportconfigurations shown. o t a a a a a a a a a a a a o a O så a a x B L8 2 3 - 8 4 3 5 a 8. The optimal locationsof supportpoints for the general supportconfigurationsshown. a a o a a a r lo rI e x B L8 2 3 - 8 4 3 7 10. The optimal locationsof supportpointsfor the general supportconfigurationsshown. SPIE Vol. 332 Advanced Technotogy Optical Tetescopeslt 982) / 225 _- - . \ t a a a a o a a a a a a a a a a a o a a a a a o a t8? pon?o t \ / t a . . a a t a ' a a o a a Eiffi '/@.ffil@'', o O a a a o a a o ' o a t8i a a a a o a a a xBL823-8438 11. The optimal locationsof supportpoints for the general supportconfigurationsshown, O xBL 823-8589 12. _ Contour maps of the deflections for the support configurationsof Figure I l. The contourinterval for eachþlot is one half times îhe rms deflectionof the surfacefrom the mean deflections. The heavyline is the meandeflection. a a ' a a o o a O O a a a a a o o . l a ' a o ¡ oo"'o363 ' o a ' a a a o t a a a 363 a a a r a a a ' ' a a a a a a O a a ì Ét'.;',.', I (i:':.:': \t/ 27? ' a ' a \:..'..'.. J \i;,/ a a o a 561 ,l]\ /' .' ..'. ì a a t7 366 a a 36É Æ\36?/-.'-\ a a a ¡e3 27t f8L 8?3-8440 x 8 L8 2 3 - 8 4 3 9 13. The optimal locationso[ supportpoints for the general supportconfigurationsshown. 226 / SPIE Vol. 332 Advanced Technology Optical Tetescopes (19g2) 14. The op-timallocationsof supportpointsfor the general supportconfigurationsshown. %.=r"f,t$tz = 7¿ lJ9 xlO-3 Tì 7a a o - - - _ lElqGgVÀ!.qR!D_1,!¡qr_a a a O o SOUAREGRID LIMIT TRIANGULARGRIOLIMIT 3 4 20 3 4 5 678910 NUMBER OF SUPPORTPOINTS 5 678910 NUMBEROF SUPPORT POINTS 30 40 50 xBL BZ3_S¡¡41 xBL823_8442 15. The support efficiencyas a function of the number of supportpoints. For eachnumber the bestresult is given, and the-optimum for other topologiesis also occasionallyindicated. The limiting value for each of the three infrnite g4ds (triangular, square, and hexagonal) is also given. The efficienciesfor optimizedcontinuousrings are shown--bythe dashedhorizontal lines . As N becomeslarge the efficiency that of a triangulargrid. of the best topologyapproaches 16. The efficiency per support point as a function of the number of supporipoints' The vertical scaleis normalizedto the limiting álcienóy, that of an infrnite triangulargrid' As triangulargrid as the expected,Ihe efficiencyapproaches-the the best op-timumfor pointsgrows. Only nu'.u"iäf iupport each number-ii shown. The limits set by each of the three infinite grid systemsare alsoindicated. SPIE Vol.332 Advanced TechnotogY Optical Tetescopes (1982) / 227 þ--o-_! Tt b ' x J-. . a a X a ' vî i , rl I i I \ t t8i x B L8 2 3 - 8 4 4 3 A.l: T.ry infinite triangular.gridjs shown. points are labeled with different symbots-to.strã* irã* tnË;r:irilil be decom_ rectansutargrids,mî'äi'Ë.-inatyzed L:::_d_l11q.two. superposed to give the deflectionsof tfre tiiã¡u-uräigria. and o x o o x A O yôt A a t8i o a l-bl o x A o o ' e l Á o o o o o o ô o a o x8L 823-8444 x B L8 2 3 _ 8 5 8 8 17. Two different ootimizations for the same topologyare shown. one wasrestiictedt"^h*.i'h. io¡nts tte 'rwo s,ame radius- The rms deflections ilnää . of t¡eiè "t rather ornerentoptimizations are.essentially i¿"ìt¡..1.iti, example itrdicatesthe difficuttiesin finoini ìïã Ëöhil. A2' The infinite hexaronar.grid-is shown. pointsare labered with different symbotito,shó* h";-'rhË"sr¡¿ -äî'-uJînalyzea äi u. decom_ four recrangutargrios thaf L:::-d_lll". an¿ _ superposed to give the deflectionsroiiir"iå*"goial gri¿. 229 ,/ SplE Vot. 532 Advanced Technotogy Opticat Telescopes fl9g21 -___,._ *:- _