Manufacturability estimates for optical aspheresG.W.Forbes"Abstract:Some simple measures of the difficulty of a variety of steps inasphere fabrication are defined by reference to fundamental geometricconsiderations.It is shown that effective approximations can then beexploited when an asphere's shape is characterized by using a particularorthogonal basis.The efficiency of the results allows them to be used notonly as quick manufacturability estimates at the production end,but moreimportantly as part of an efficient design process that can boost the resultingoptical systems'cost-effectiveness.2011 Optical Society of AmericaOCIS codes:(220.1250)Aspherics;(220.4830)Optical systems design;(220.4840)Opticaltesting;(220.4610)Optical fabrication.References and links1.A.Epple and H.Wang,""Design to manufacture'from the perspective of optical design and fabrication,"inOprical Fabrication and Testing,OSA Technical Digest (Optical Socicty of America,2008),OFB1.2.J.P.McGuire,Jr."Manufacturable mobile phone optics:higher order aspheres are not always better,"ProcSPIE7652765210,765210-8(2010).3.R.N.Youngworth and B.D.Stone,"Simple estimates for the effects of mid-spatial-frequency surface errors onimage quality,”Appl.0pt.3%13),2198-2209(2000).4.J.W.Foreman,Jr.,"Simple numerical measure of the manufacturability of aspheric optical surfaces,"Appl.Opt.5J.W.Foreman,Jr.,"Mercier's aspheric manufacturability index,"Appl.Opt.26(22),4711-4712(1987).6.J.Kumler,"Designing and specifying aspheres for manufacturability,"Proc.SPIE 5874,121(2005),do:10.1117/12.615197.7.C.du Jeu,"Criterion to appreciate difficulties of aspherical polishing,"Proc.SPIE 5494,113-121 (2004).8.G.W.Forbes and C.P.Brophy,"Designing cost-effective systems that incorporate high-precision asphericoptics,"SPIE Optifab (2009),paper TD06-25 (1),hrp://www.qedmrf.com.1819),19700-19712(2010).10.G.W.Forbes,"Shape specification for axially symmetric optical surfaces,"Opt.Express 15(8),5218-5226(2007).11.P.Murphy,J.Fleig,G.Forbes,D.Miladinovic,G.DeVries,and S.O'Donohue,"Subaperture stitchinginterferometry for testing mild aspheres,"Proc.SPIE 6293,62930J (2006).1.IntroductionAspheric optical surfaces deliver higher performing,more compact,and lighter systems in arange of applications.Aspheres become of even greater value as their manufacturability isintegrated more tightly into the design phase.Depending on the production processes andvolume,cost-effectiveness can be boosted by accounting for the difficulty of steps likepolishing,measuring,molding,and/or assembling the aspheric components.Explicit recipesfor estimating cost/difficulty are currently not easy to come by,however.For such estimatesto be embedded usefully within the design process,computational efficiency is critical.Aparticular class of highly efficient estimates of manufacturability is considered in this work.Brute-force analysis of "as-built performance"is conceptually straightforward.SuchMonte-Carlo-style processes enable designers either to determine or account for tolerances onparameters such as alignment,surface figure,mid-spatial frequency errors (MSF),homogeneity,etc.A variety of related aspects that are specific to aspheres was discussedrecently by Epple and Wang [1],and by McGuire [2].In the former,the key idea is to exploitaspheres to loosen system assembly tolerances,and the latter treats a particular examplewhere the removal of high-order terms from the aspheres is also used to loosen tolerances.By#144118-$15.00 USD Received 16 Mar 2011;revised 27 Apr 2011;accepted 30 Apr 201 1;published 5 May 2011(C)2011OSA9 May 2011 Vol.19,No.10/OPTICS EXPRESS 9923using perturbation methods like those applied by Youngworth and Stone [3],these processesare accelerated significantly because no additional ray tracing is needed to evaluate theperturbed system.(These same methods are invoked below in Sections 3 and 7.)Many of the asphere-related efforts in the area of"design for manufacturability"are basedon the vague notion that the less deviation from a sphere,the better.Foreman [4,5]was moreexplicit when he hypothesized that it is the radial rate of change of an asphere's meridionalradius of curvature that offers a useful relative measure of manufacturability.Absolutereference values and justifications for such measures of difficulty are hard to find.Kumler [6]presented a variety of practical considerations related to polishing and testing,and treated anexample where adding higher order terms to aspheres can reduce their difficulty for themanufacturer.He takes a significant step towards coupling measures of manufacturabilitydirectly to the capabilities of current fabrication and metrology tools.This allows him to gobeyond relative statements,such as his guideline that"the greater the slope of the asphericdeparture from a best-fit sphere,the more difficult the asphere,"to state absolute values asreference points for this slope,namely 2 microns per mm.His reference value is driven byboth polishing and metrology considerations,and is in keeping with his other machine-specific absolute numbers for edge thicknesses,margins for diameters of lens blanks,etc.Such work offers important empirical quantification of the fact that an asphere's cost andachievable figure and MSF tolerances depend strongly on its shape.It is part of an essentiallyunending project,of course,because innovations mean that such reference values areconstantly evolving,and they vary between production technologies.The slope discussed by Kumler directly yields the difficulty of full-apertureinterferometric tests because that slope is proportional to fringe density.In that context,theallowed maximal fraction of the Nyquist sampling rate is the natural measure.In relation tosub-aperture pad polishing,du Jeu [7]uses a prescribed level of maximal tool misfit to selectpad size,and the relative pad size then serves as an effective measure of difficulty.He givescompact equations for the case of the standard rotated conic sections.The same ideas areapplicable for more complex aspheres,but the equations become burdensome.Further,thesurface must then be analyzed at all radial zones since the outer edge need no longer be themost challenging zone.The main objective of what follows is to derive computationallyefficient estimates of geometrically based measures of difficulty like the two just mentionedSuch estimates can serve as the foundation for determining whether an asphere can plausiblybe made within spec and,if so,identifying the types of processes that may be required andhence its approximate cost.The method of specifying an asphere's nominal shape turns out tohold the key to efficiently evaluating the associated measures of difficulty.Although initialsteps have been made in this direction [8],they did not exploit some powerful algorithms thatwere reported recently [9].The specification of aspheric shape is reviewed in Section 2 where a generalization is alsointroduced to extend applicability.Even for the cases of conic null tests and simple CGHnulls,the estimation of whether such full-aperture interferometric tests are workable is shownin Section 3 to follow incredibly simply with this particular characterization of shape.Asdiscussed in Section 4,more sophisticated measures of manufacturability are coupled tovariation in the surface's local principal curvatures.Stitched interferometric metrology is usedas one example,and it happens to be closely related to the analysis of difficulty that wasdiscussed by du Jeu.The computation of the elements that are required for efficientlyestimating such measures of manufacturability is treated in Sections 5 and 6.2.A tailored characterization of shapeThe standard characterization of a rotationally symmetric asphere's shape is to express its sagin cylindrical polar coordinates as z=f(p).For a conventional conic section of axialcurvature c and conic constant K,this takes the form#144118-$15.00 USD Received 16 Mar 2011;revised 27 Apr 2011;accepted 30 Apr 201 1;published 5 May 2011(C)2011OSA9 May 2011/Vol.19,No.10/OPTICS EXPRESS 9924