ANALYTICAL ASPECTS OF THE y-y DIAGRAMFernando Jose Lopez-LopezGeneral DynamicsConvair Aerospace DivisionSan Diego,CaliforniaAbstractAn analytical description of the y-y diagram is given in terms of plane vectorsrepresenting the points and lines in the diagram.This allows for a very powerful andelegant tool for deriving and using the design and symthesis properties of the y-y dia-gram.Exact analytical expressions are given to derive the first-order properties ofan optical system from its representation in the diagram.The Seidel(third order)aberation coefficients are expressed in terms of variables better suited to the y-ydiagram,and the description of a computer program that implements these ideas,isincludedAt the request of a number of participants in the Seminar,a complete bibliographyon the y-y diagram is included as an Appendix.Introductionin the y-y plane.This is a tutorial paper on the more analyti-The advantage of using vectors lies in the con-cal aspects of the y-y diagram and its applicationsciseness and elegance of the notation.Vectorto the analysis and design of optical systems.Asgeometry in the plane is ideally suited to the analy-such,it complements the geometric descriptiontical treatment of the diagram.The complexgiven by Dr.Roland V.Shack in these samenumber representation is better suited for numer-Proceedings.ical calculations,as most computers handle themas single entities.A single point or line in theAs a matter of fact,the geometric approachdiagram may be represented by a single vector oris essential as the first step in the analysis orcomplex number and handled accordingly.design of a system due to the inherently graphicalcharacter of the diagram.However,when moreIn the present paper we shall give a shortquantitative information is needed to achieve andescription of this vectorial formulation-as wellanalytical or numerical solution to the problem atas illustrating its use to derive the first-orderhand,a more thorough analysis of the diagram isproperties of an optical system from its repre-needed.sentation in the y-y diagram.Fortunately,analytic geometry provides theWe shall also quote expressions for the Seidelnecessary tools to convert the pencil and ruleraberration coefficients in terms of these first-geometrical constructions into algebraic quanti-order variables,as well as describing the opera-ties that can be handled with more accuracy andtions of generalized bending and thickening ofgenerality.Moreover,as an optical system islenses,so useful in the design of optical systems.represented in the diagram by a set of points andconnecting lines,we may in turn represent theseWe also describe the structure and operationpoints and lines by vectors or as complex numbersof the computer program YYRANCH (y-y Repre-151sentation And Necessary Computational Handling)that implements all these ideas into a serviceoriented program that allows the user to analyzeand/or design an optical system using the y-ywhere k is the unit vector perpendicular to thediagram.The variables in this program are they-y plane.Eq.(5)is valid at any surface or spacemarginal and chief ray heights (y,y);and the morebetween surfaces along the optical system.(Fromtraditional ones,thicknesses and curvatures,areradiometric arguments it is possible to show thatobtained as a result.The program has been de-the flux of energy through the system,is propor-signed to analyze the first and third order prop-erties of an optical system and to produce a work-able third-order design that will maintain thebeing a numerical constant,we may de-first-order constraints imposed on such a systemfine new quantities scaled by this constant.TheThe resulting design may then be improved byadvantage of doing this,is that the quantities thusmore sophisticated programs.defined are derivable directly from the points andlines in the diagram and do not depend on the par-The Vectorial Representationticular choice of the value of It is easy toshow that this quantity is determined by a choiceThe y-y diagram is a plot of the heights ofof the focal length f',f-number N,and fieldthe chief ray (as abscissa)and the marginal rayangle a,of the system,through the expression:(as ordinate)at each surface of an optical system.We define a plane vector at each surface,given(6bywhere n,is the index of refraction in object space.Using H as a scaling factor we define theIn the spaces between the surfaces,charac-quantitiesterized by an index of refraction n,each raymakes an angle with the optical axis.We definea vector for each of those spaces(7which we shall call by extension line vector,sep-where u,u are the angles between the rays andaration,and power.These quantities may bethe optical axis.expressed in terms of the point vectors ateach surface:In terms of these vectors,the paraxial ray-trace equations of refraction and transfer take thefollowing forms;for refraction at surface of powerand for the transfer from surface i to surface(10)Note that the notation used in Eqs.(8)-(10)isconsistent with the fact that these are "oriented"magnitudes,and should not be expressed as abso-where z =t/n,is the reduced separation be-lute values alone.tween the two surfaces.2We could also choose the lines w as ourBy eliminating either or from Eqs.(3),basic elements in which case,the a vectors are(4)by appropriate vector products we find thatgiven bythe Lagrange invariant takes the vectorial form152Downloaded From:http://proceedings.spiedigitallibrary.org/on 06/11/2013 Terms of Use:http://spiedlorg/termsNote the reciprocity between Eqs.(9)andtive we may write Eq.(12)as(11),which express a duality between the vectors言andW：In the y-y diagram the vectorsrepresent points,and clearly Eq.(9 is the(13)vectorial equation of the straight line defined bytwo of these points.Therefore,the w vectorsand we see that in this case the Wi must lierepresent indeed lines.Similarly,in a diagramcounterclockwise to the wi in order to preservein which the W vectors represent points,Eq.(11)the proper sign for the unit vector k.Finally,tells us that the vectors represent lines de-when the power is zero there is no deflection andfined by such w vectors;we call such a diagramthe two vectors are parallel to each other.Thisan n-n diagram.These two diagrams areis the case of an afocal system or of a plane inter-dual of each other in the projective geometryface.sense:The points of one diagram transforminto lines in the other,and vice versa.ApartFirst-Order Propertiesfrom the mathematics,this property proves use-ful in the solution of several problems.ForAs we know,each point a in the y-y diagramexample,a figure defined in the y-y diagram inrepresents a plane perpendicular to the optical axisterms of lines is difficult to analyze;by trans-forming it to the n-diagram we get the equiv-in a physical space of index of refraction n,whichis represented in the diagram by the line w.Wealent figure defined by points,which is the usualsay that two points,located each on a different wway of studying most curves.Once the analysisis finished in the n-n diagram we may returnline,are conjugate to each other when they are re-to the y-y diagram to get a "feeling"for it.lated byWe may also note that Eq.(8)explains a(14peculiarity of the y-y diagram that puzzles first-time users of it.We may see from this expres-where the quantity mr,defined by this equation,is the transverse magnification.sion that the reduced separation is equal to thenumerical value of the cross product of two con-From its definition we see that the magnifica-secutive vectors,i.e.,to the area enclosedby the parallelogram formed by the two vectors.tion is represented by a line through the originIn other words,the reduced separation betweenjoining the two conjugate points;this is the "con-jugate"line.two surfaces is proportional to the area of thetriangle formed by the two points representing theFor the purposes of this section,we assumesurfaces and the origin of the diagram.that the point is on the line w (which we call"object line')and that the point is located onOn the other hand,remembering that thevectors represent lines in the y-y diagram,wethe line w,("image line),and that these twomay see from Eq.(10),written in the formlines are the only ones that comprise the systemunder consideration.In this case we may write(12)(15)that the power of a given surface is representedby the change of direction of the two w vectorsdefining it,and in fact it is proportional to theangle between such vectors.When the power ispositive,we see that the vector W lies clock-From these equations we may write the rela-wise to the Wi vector;when the power is nega-tionships between two points conjugate to each153Downloaded From:http://proceedings.spiedigitallibrary.org/on 06/11/2013 Terms of Use:http://spiedlorg/terms