Global optimization using the y-ybar diagramDaniel M.BrownTeledyne Brown Engineering,Optical Systems DepartmentCummings Research Park,Huntsville,AL 35807ABSTRACTSoftware is under developement at Teledyne Brown Engineering to represent a lens configuration as a y-ybar or Delanodiagram.The program determines third-order Seidel and chromatic aberrations for each configuration.It performs a globalsearch through all valid permutations of configuration space and determines,to within a step increment of the space,theconfiguration with smallest third-order aberrations.The program was developed to generate first-order optical layouts whichpromised to reach global minima during subsequent conventional optimization.Other operations allowed by the program are:add or delete surfaces,couple surfaces(for Mangin mirrors),shift the stop position,and display first-order properties and theoptical layout (surface radii and thicknesses)for subsequent entry into a conventional lens design program with automaticoptimization.Algorithms for performing some of the key functions,not covered by previous authors,are discussed in thispaper.1.INTRODUCTIONThe advantages of representing a lens as a y-ybar or Delanol diagram are well understood.Like the high level abstraction ofan infinite series Bessel function to the symbol J,the y-ybar diagram is a higher level abstraction of surface radii andthicknesses.This symbolization of the lens hides unnecessary details from the user,often simplifying first-order design.When one wishes to globally search all of configuration space,data abstraction undoubtedly makes the task easier.Forexample,varying the y and ybar coordinates of a surface between two other surfaces is equivalent to varying three surfacecurvatures and two thicknesses.It may not be immediately apparent to the designer what the valid ranges are for surfacecurvatures and thicknesses.However,if the lens is represented as a y-ybar diagram with two independent coordinates for eachsurface,valid parameter ranges are often easier to visualize.generate promising initial configurations followed by a conventional optimization of selected points in configuration space.The YYBAR program was written at Teledyne Brown Engineering under IR&D to implement this design approach.Sturlesiand O'Shea point out that sampling configuration space with too small an interval produces many redundant configurationsthat all optimize to the same local minimum in the second phase.We attempted to design the algorithm of the global searchroutine in YYBAR to eliminate this problem.In this,we were only partially successful.The basic manipulations of the y-ybar diagram and the first-order information it provides are discussed in detail by Shack4 andLopez-Lopez5.6,and will not be repeated here.Lopez-Lopez also gives algorithms for computing the third-order aberrationsfrom the y-ybar diagram.This paper discusses some of the algorithms germane to using the y-ybar diagram for globallysearching configuration space for aberration minima.The global search routine in YYBAR uses a simple deterministic algorithm that completely searches all valid configurationswithin user-specified limits and increments.A practical minimum set of contraints for a global search are:1)maximumallowed diameter for each surface,2)fixed entrance pupil diameter,and 3)fixed effective focal length or magnification.Theseconstraints are easily handled by the y-ybar diagram.2.BASIC OPERATIONSThe user interface of YYBAR was designed to be simple and friendly.The user inputs basic specification data and an initialconfiguration via either its y-ybar coordinates or surface radii and thicknesses.If the configuration is input via y-ybarcoordinates,the entrance pupil diameter and optical invariant are computed when the first surface coordinates are input.Thefocal ratio is computed when the image plane ybar value is input.At each surface the user also indicates the surface type,refractor or reflector,and inputs the index of refraction and Abbe v number accordingly.After inputing an initial configuration various operations can be performed.These include:add or delete a surface,change the0-8194-06554/91/s4.00SPIE Vol.1527 Current Developments in Optical Design and Optical Engineering (1991)/19Downloaded From:http://proceedings.spiedigitallibrary.org/on 09220/2012 Terms of Use:http://spiedlorg/termsy-ybar coordinates of a surface,change the index of refraction,couple surfaces,shift the stop,change the optical invariant,andglobally search configuration space for aberration minima.The gobal search routine requires specifying aberration weights toconstruct a merit function.Three basic information displays include;first-order parameters (y-ybar coordinates,EFL,F/#,EPdiameter,etc.),the optical layout (radii and thicknesses),and the third order aberrations at each surface.3.ALGORITHMS3.1 Data StructuresSince several data items need to be stored for each surface and surfaces should be easily added or deleted at any position,it isnatural to choose a linked list data structure to represent the y-ybar diagram.The linked list consists of a headnode and at leastthree surface nodes.The headnode stores basic specification data and a pointer to the object surface node.The image surface isthe last surface in the list.Unlike an array,storage space for a linked list is allocated dynamically at run time,allowing anynumber of surfaces to be defined limited only by the available memory of the computer.A linked list also allows easyaddition or deletion of surfaces.The Pascal data structure for a surface in YYBAR is given by:Surface =RECORDstypeSurfaceType;y:REAL;(marginal ray height *ybar:REAL;(principal ray height*)indexREAL:(index of refraction *couple:INTEGER;(coupled surface number *nextSur:SurfacePtr;(pointer to next surface *END:3.2 Determining valid data entryIn the following discussion we use the convention that minus subscripts refer to surface/space parameters immediately beforenonsubsripted parameters.Plus subscripts refer to parameters immediately following nonsubscripted parameters.Thicknessesand refractive indices have the same subscript as the surface before themThe y-ybar diagram must rotate clockwise around the origin.This implies;yor in the notation of Lopez-Lopez;3,3>0where the two consecutive y-ybar surface coordinates are represented as position vectors from the origin.For spherical surfaces,ray heights cannot exceed the radius of curvature.(Ray heights can exceed the base radius of curvaturefor aspherics;e.g.in the x-ray telescope AXAF ray heights are almost 70 times the base radius.)Thus a second constraint is:n-nwhere n.and n are the indices of refraction before and after the surface and is the surface power.One can show that thisconstraint leads to20 SPIE Vol.1527 Current Developments in Optical Design and Optical Engineering (1991)Downloaded From:http://proceedings.spiedigitallibrary.org/on 09/20/2012 Terms of Use:http://spiedlorg/terms(3)where is the optical invariant and b and b are the y-axis intercepts of the two lines before and after the surface.Surface powers,and,of two consecutive surfaces are further constrained in order to maintain a positive edge thickness.Thus the difference between the two sags (to first order)must be less than the thickness between the two surfaces.Representedmathematically:2h.