Aberration considerations in lens tolerancingEric Herman*and Jose SasianCollege of Optical Sciences,University of Arizona,Tucson,Arizona 85721,USA"Corresponding author:eherman @optics.arizona.eduReceived 20 September 2013;revised 11 December 2013;accepted 11 December 2013;posted 16 December 2013(Doc.ID 198021);published 14 January 2014Often the tolerancing of an optical system is performed by treating the optical system as a black box inwhich the designer sets tolerances for perturbations and then runs a Monte Carlo analysis to determinethe as-built performance.When the effects of the perturbations are not considered,the tolerances mightresult tighter than necessary,proper compensation might be missed,and manufacturing cost can be in-creased.By acquiring aberration sensitivity for each type of perturbation,an optical engineer can in-crease tolerances by ad hoc compensation.An aberration sensitivity evaluation can be performedquickly and can be incorporated into theinitial lens design phase.A lens designer can find what surfacesor elements within the opticalsystem will be problematic before any time-consuming Monte Carlo run isperformed.In this paper we use aberration theory of plane symmetric systems to remove,tosome usefulextent,the black-box tolerancing approach and to provide some insights into tolerancing.The tolerancesensitivities that are analyzed are with respect to surface tilt,center thickness,index value,and radius.To analyze these perturbations,exact wavefront calculations are performed for the following aberrations:uniform astigmatism,uniform coma,linear astigmatism,distortion I,distortion II,spherical aberration,linear coma,quadratic astigmatism,and cubic distortion.We provide a discussion about how theaberration tolerancing analysis is useful.2014 Optical Society of AmericaOCIS codes:(080.2208)Fabrication,tolerancing:(220.1010)Aberrations (global):(080.1005)Aberration expansions.http:/k.doi.org10.1364/A0.53.0003411.IntroductionCurrently a perception is that most of the engi-Traditionally,optical tolerancing is performed as aneer's design time should be spent tolerancingblind process.This has become prominent because[3-6].By understanding aberration effects,it is pos-of the ease of accessibility of high-power computing.sible to speed up the design process.Reducing theDesigners now have the opportunity to run thou-amount of time the designer spends tolerancing issands of Monte Carlo iterations of a system and finda step in the right direction.out what the possible lens manufacturing yield is.As perturbations are introduced in an axially sym-While being highly accurate,this blind method oftol-metric system,this symmetry is lost.The perturbederancing creates a curtain in the tolerance process.system can be analyzed as having an axially symmet-Taking advantage of aberration theory can substan-ric part and a plane symmetric part,when elementtially help to reduce tolerances and remove thedecentration and tilts are in one direction.Thus incurtain somewhat [1,2].By using aberration theoryour approach we define and determine the aberra-the designer can understand the main effects of thetion sensitivities of a lens to both axial [4]andperturbations and devise ways to compensate,miti-nonaxial perturbations [7-9].This approach couldgate,or remove them.be extended to perturbations in two planes;however,only one plane is necessary to substantially under-stand the lens under perturbation.1559-128XW/14/030341-06315.00/0As shown below we analyzed a set of objective2014 Optical Society of Americalenses [8]using the tolerancing approach discussed20 January 2014 Vol.53,No.3/APPLED OPTICS341in this paper.Our analysis presents results fromsymmetryand subsequently to one plane of sym-each individual lens perturbation in a stacked barmetry.Evaluating the aberration coefficients ofchart format.This presentation ofdata gives an easeTable 1provides understanding about how theof breakdown of tolerance sensitivities that providesnominal lens is perturbed.For example,becauseuseful insight.uniform astigmatism depends on the square of thesurface tilt angle,it is not an aberration that can2.Motivationbe expected to be significant in a perturbed lensTolerancing an optical system requires a vast knowl-system.Uniform coma and linear astigmatism areedge of not just the lens system itself,but also howthe main aberrations to be expected to degrade athose optics will work in conjunction with the re-sharp image.Clearly,by adding the tilt of the imagemainder of the system.Being able to classically ana-plane as a compensator,one can expect increasedlyze an optical system for the standard aberrations istolerances.If the sensitivity to uniform coma fromtypically not sufficient when considering the toler-a lens is high,then this lens can be used to compen-ance and how to build a system.sate uniform coma.Thus,knowledge of the aberra-An extension of primary aberrations considers thetion forms that result from fabrication errors isoptical system as a plane symmetric system.Whenuseful.this type of aberration theory is dissected,the actualOne way to determine the aberration coefficientsminutiae of an optical design can be understood fromis by analytical formulas [8,10].However,there area tolerance perspective.The designer is then able toinduced aberrations [11]that are not consideredproceed toward getting an end product built.Thein a simple summation of aberrations.In thiscombination of an understanding of tolerances withpaper,to increase accuracy,we define aberrationsclassic aberration theory gives the lens designer anby symmetry and calculate them with real rayunderstanding of what is actually occurring in thetracing.optical system as an as-built system.Without under-standing of the plane symmetric aberration theory,itmay be possible for a lens designer to make a lens,4.Aberration Coefficient Definedbut the tolerances associated with it may not beFor the purposes of accurate tolerancing,we rede-appropriate.fined the aberration coefficients in terms of the aber-ration symmetry and by using real ray tracing.We3.Theorytrace real rays through the lens system and calculateIn a tolerancing analysis,a nominally axially sym-the optical path difference (OPD)and the real raymetric system is perturbed and the axial symmetryheight for a given ray.By using the OPD of the opticalis lost.If surface tilts are in one plane,the lens sys-system,we are in contrast to classic aberrationtem becomes plane symmetric [8].Table 1 providestheory,which breaks the aberrations out by specificthe primary aberrations ofa plane symmetric systemorders.This ray information for simplicity is writtenorganized in groups and subgroups according to aber-in matrix form as follows:ration symmetry.Examination of Table 1 shows thatit consists of the primary aberrations of axially sym-metric systems and new aberrations that result from[](1)reducing the system symmetry to two planes ofTable 1.Tilted Component Optical System Aberrations to Fourth OrderRay HeighWavefrontFunctionalCoefficientFormAberration NameUniform astigmatismwhere H and H,are the normalized field coordi-Anamorphism(i.)H)Quadratic pistonnates,p andpy are the normalized pupil coordinates,W21110(Hp)(iH)Quadratic distortion II (smile)and x andy are the ray coordinates.These definitionsW12010Field tiltassume that there is no focus error.W30010Cubic pistonWe assume that a perturbation is a surface tilt in aW12101Linear astigmatismdirection and define aberration coefficients forUniform comauniform astigmatism,uniform coma,linear astigma-Quadratic distortion Itism,quadratic distortion I,and quadratic distortion(keystone)Ⅱ.Specifically,Quadratic astigmatismLinear coma0000Cubic distortionOPDSpherical aberrationField curvatureUniform ComaQuartic piston(3)342APPLIED OPTICS/Vol.53,No.3/20 January 2014Uniform Astigmatism+HAp(6)Linear Astigmatism(72The two quadratic distortion aberrations are calculated interms of percentage.Quadratic distortion I is alsoreferred to as keystone distortion,and quadratic distortion II is also referred to as smile distortion.The formulas used to calculate these aberrations are Eqs.(8)and (9)for smile and keystone distortion,respectively.Similar formulas,Egs.(10)through(13),are defined to calculate the aberrations when axial symmetry isretained,that is,for spherical aberration,linear coma,quadratic astigmatism,and cubic distortion.All ofthese aberrations when calculated are found for their complete terms.As previously stated,this is an entireaberration value,not just a specific ordered term.Ray Height10000Ray Height0SmileRay Height00Keystone-Smile.Spherical Aberration =(10)]-oP99]Linear Coma(11)Quadratic Astigmatism(12)20 January 2014/Vol.53,No.3 /APPLED OPTICS343