JTu1C.1.pdfOrigins and Fundamentals of Nodal Aberration TheoryJohn R.RogersAbtract:Nodal Aberration Theory,developed by Kevin Thompson and Roland Shack,predictsseveral important aberration phenomena but remains poorly understood.To de-mystify the theory,we describe the origins and fundamental concepts of the theory.0 CIS codes:080.0080,080.1005,080.1010,220.1010,220.1140,220.27401.Fundamental conceptThe theory now known as Nodal Aberration Theory began with the work of Kevin Thompson[1-5]and his dissertationadvisor,Roland Shack.Shack attributed the fundamental concept to Dick Buchroeder,who had written an earlierdissertation [6]on the topic of tilted-component optical systems that had introduced the concept,shown in Figure 1,of mutually decentered aberration fields.It was the vector development ofthis concept that led to what is now knownas Nodal Aberration Theory.Aberrat ion contributionFig.1.Misaligned aberration contributionsAssuming for a moment that an optical system can be decomposed into subsystems that are each rotationallysymmetric(but possibly not mutually aligned),then the aberrations of each subsystem are fully understood accordingto the aberration theory for rotationally symmetric systems,e.g.,the wavefront expansion of Hopkins [7].In such acase,the difficulty is not to understand the aberrations of the individual subsystems,but to understand how the(mutually decentered)aberration contributions from the various subsystems combine to yield an overall aberrationpattern.It is precisely this question of how the mutually decentered aberration contributions interact with each otherthat lies at the heart of Nodal Aberration Theory.It is worth noting that most systems can be decomposed into rotationally symmetric subsystems.A sphericalsurface,even though tilted or decentered,is rotationally symmetric about the line joining its center of curvature andits entrance pupil,i.e.,the pupil in the space immediately before the sphere.A rotationally symmetric aspheric surface-even a tilted and/or off-axis section thereof-may be considered as a spherical surface plus a thin aspheric"shell"whose aberration contribution is (to third-order accuracy)rotationally symmetric about the line joining the asphericvertex and the pupil in the space immediately before the asphere.Freeform surfaces cannot be decomposed as described above;Nodal Aberration Theory has more recently beenextended by Fuerschbach [8]to include Freeform surfaces.2.Development of the theoryIn the Hopkins notation,the wavefront aberrations may be written to third order as:Having recognized that the problem to be solved involved combining decentered aberrational components,Shackelected to rewrite the aberration expansion of Hopkins,treating the image field as(what appears to be)a 2-dimensonalvector space.In this way,a decentered aberration contribution can be described as residing at a position,where H represents the normalized vector expression for the image field and represents the decentration of theJTu1C.1.pdfaberration contribution within the field.For reasons of symmetry,the terms in the Hopkins wavefront expansion areall products and powers of H2,p2,and Hpcos.The equivalent expressions in the vector expression for the field areAt this point,it is evident that the wavefront expansion for a subsystem whose aberration contribution is decenteredin the field by a normalized vector may be obtained by substituting (H-)for every occurrence of in Eq.2.The wavefront expression for a system containing multiple subsystems having mutually decentered contributions,expression over the index i.Such a sum is shown in Eq.3,where for clarity we have indicated a separate summationfor each aberration type:(3)This straightforward set of summations gives the total wavefront aberration for the system (up to third order),on aper-aberration-type basis;however,it provides little insight as to the structure of the aberrations in the image.A more useful exercise(which served as Thompson's dissertation topic)is to solve the individual sums ofEq.3 tofind the locations in the field where the various aberrations would vanish,i.e..the nodes of the aberrations.(Note:there is no requirement,or even expectation,that the nodes for the various aberrations should coincide.)Finding theresultant node for coma from Equation 3 is straightforward;however,the solutions become more interesting for theaberrations with more complex field dependences.In solving the equations for astigmatism,Shack found it necessaryto define a"new"vector operator,which he called"vector multiplication."The operation is most easily described asfollows:interpret the two vectors as representing locations in the complex plane,multiply the two complex numbers,and reinterpret the resulting complex number as representing a vector location in the image field.By similar analogy,it is possible to take the square-root of a vector,and indeed the expressions for the nodes of the aberrations containroots of vectors.In fact,Shack's vector multiplication was actually a rediscovery of an operation defined in a CliffordAlgebra [9,also known as a Geometric Algebra [10.The result of solving for the nodes produces not only the node locations,but also a description of the aberrationbehavior around those nodes.Two useful results from this analysis are that the coma node shifts in the field but comaremains linear about this node;astigmatism generally develops two nodes,with the magnitude of astigmatism at anyfield point depending on the product of the distances to the two nodes.If the sum of the astigmatism coefficients iszero,one node moves to infinity,leaving astigmatism that depends linearly with the distance to the remaining node.4.References[1]K.P.Thompson,"Aberration fields in tilted and decentered optical systems,Ph.D.dissertation (University of Arizona,Tucson,Arizona,1980)[2]K.P.Thompson,"Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry",J.Opt.Soc.AmA22.p1389-14012005).[3]K.P.Thompson,"Multinodal fifth-order optical aberrations of optical systems without rotational symmetry;spherical aberration""J.Opt.Soc.AmA26,p1090-1100(2009).[4]K.P.Thompson,"Multinodal fifth-order optical aberrations of optical systems without rotational symmetry;the comatic aberrations",",J.0pt.Soc.AmA27(6,1490-1504(2010).[5]K.P.Thompson,"Multinodal fifth-order optical aberrations of optical systcms without rotational symmetry:the astigmatic aberrations","J.[6]R.A.Buchroeder,"Tilted component optical systems,Ph.D.dissertation,(University of Arizona,Tucson,Arizona,1976).[7]H.H.Hopkins,The Wave Theary of Aberrations (Oxford at the Clarendon Press,Oxford,UK,1950).[8]K.Fuerschbach,J.P.Rolland,and K.P.Thompson,"Theory of aberration fields for general optical systems with freeform surfaces"OptExpress20(18)20139-20155[9]William K.Clifford,Applications of Grassmann's extensive algebra,Amer.Jour.Math 1(1878),350-358.[10]D.Hestenes,"Oersted Medal Lecture 2002:Reforming the Mathematical Language of Physics"Am.J.Phys.71 (2),104-121 (2003)