First-Order Design and the y, y Diagram

Erwin Delano

A new method is described for graphically representing the first-order layout of any optical system ofaxially symmetric refracting or reflecting surfaces. Such a system can be uniquely defined by theactual path of an arbitrary paraxial skew ray traversing it, provided the value of the Lagrange invariantis also known. This paper describes some interesting properties of the diagram obtained by projectingthe ray onto the object plane. The diagram has proved to be a very useful aid in first-order design,particularly in systems where lens diameters and locations are severely constrained.

Introduction

The idea of choosing a different set of independentparameters from the radii, thicknesses, and indices inorder to simplify the first-order theory of axially sym-metric optical systems is not new. Luneberg' hasshown how the path of an arbitrary paraxial ray to-gether with the indices completely defines any suchsystem, assuming all surfaces to be spheres. However,the path of a second paraxial ray cannot be independ-ent of the first, since the Lagrange relation must besatisfied. It proves desirable in some respects to beable to define the optical system in terms of arbitrarydata from both the paraxial marginal and paraxialprincipal rays and any additional independent para-meters required to impose uniqueness.

In his two articles on the theory of periscopes,Smith2' 3 describes a graphical construction which allowsone to define a certain class of periscopic systems. Thisspecial class of systems may be represented by polygonscircumscribed about a circle, and this representationhas some interesting properties which will not be dis-cussed here. The present paper describes a moregeneral representation than the one given by Smithand includes his method as a special case. It is quitepossible that the method to be described was known tohim, but no reference to it can be found in his works.This method has been found to be very useful in thefirst-order design of axially symmetric systems of thicklenses and mirrors, particularly where certain mechani-cal constraints must be satisfied.

The author was with Bausch & Lomb, Inc., Rochester, NewYork. His present address is St. John Fisher College, Rochester,New York.

Received 10 April 1963.

Basic Relationships

Given an arbitrary system of axially symmetricrefracting or reflecting surfaces, consider the path ofany two paraxial rays such that the Lagrange invariantQ for them does not vanish, for example, the paraxialmarginal ray and the paraxial principal ray. Since wewill only be concerned with first-order properties, thesubsequent discussion will be understood to be limitedto paraxial rays. Among the numerous ways in whichthe system can be uniquely defined there is one repre-sentation which is of unusual interest in first-ordertheory. In the discussion below, the data for themarginal ray will be distinguished from the data for theprincipal ray by using plain letters for the marginalray quantities and placing a bar (-) over each letter forthe principal ray quantities.

As defining parameters, choose the heights y of themarginal ray, the heights g, of the principal ray, thecoordinates xi of the vertices relative to some arbitraryorigin, and the Lagrange invariant Q for the two rays.In addition to the optical surfaces of the system, theobject and image planes must be included-or anyother two reference planes not coincident with the ex-ternal vertices of the system-one in the object spaceand the other in the image space.

It is easy to show that these are independent para-meters and are just enough in number uniquely todetermine not only all the radii, indices, and separationsbut also to define completely the paths of the tworays through the system. Let r be the radius ofcurvature of the ith surface, and let Nf, d, u , and vibe the index, axial thickness, and slope angles of themarginal and principal rays, respectively, for the preced-ing space. Then, all of these may be calculated by

December 1963 / Vol. 2, No. 12 / APPLIED OPTICS 1251

means of the following equations, which are derived inthe Appendix. Given the value of Q and the para-meters yt, gi, xf at every surface, then:

di xl- xi-,, Di = (i- ii - yipi-,)/ Q,Ni dilDi,ai = (yi - yi-I)Di = Nus, KI = (aiaL+1 - i+1ai)1Q,ati = (i -9i - WAD = Nt~i,(1ri =-(Ni1 +- Ni)/Ki, Q = yiai-iai

= iai + - iai+i.

The quantity D, is the equivalent air path for the thick-ness d1, and K, is the surface power of the ith surface.

General PropertiesIn general, any axially symmetric optical system

may be represented by the actual path of a singleparaxial skew ray traversing the system. To demon-strate this, choose a set of right-handed Cartesian co-ordinates (x, y, z such that the x axis is coincidentwith the optic axis, placing the origin at the axial pointof either the object or pupil planes. Because of thelinear nature of first-order optics, it is possible to choosea skew ray such that its projection on the x, y plane isthe marginal ray and its projection on the x, z planeis the principal ray. Therefore, this single skew rayuniquely determines the optical system according toEqs. (1) if one substitutes z for . To simplify thenotation, the z axis will hereafter be referred to as theyaxis.

Since the marginal and principal rays could have beenchosen arbitrarily except for meeting the conditionQ 5- 0, it follows that the path of an arbitrary skew rayuniquely determines the system it traverses. Theprojection of this skew ray onto the y, plane has thefollowing interesting property: as the ray travels fromsurface to surface, its projection winds about the originin a clockwise direction if Q > 0, and in a counterclock-wise direction if Q < 0. The viewer is assumed to belooking in the direction of increasing x.

To prove this, consider the projections onto the y, yplane of any two successive points 1 and 2 along theray path. If a triangle is formed with the origin asone vertex and the two projections as the other twovertices, then from elementary analytic geometry thearea of this triangle is

A = '/2(yg2-y2l) = /1QD2,

where D2 is the equivalent air distance from point 1 topoint 2. Since D is positive for any physically realiz-able case (including reflection, when both d and Nchange sign), A must have the same sign as Q, whichproves that the radius vector to the projected pointalways sweeps in a constant direction about the origin.For convenience, it will hereafter be assumed that Q= 2 so as to make area A equal to D directly. Thisimplies no loss of generality, since it can always beaccomplished by simply changing the unit of length,and has the advantage of simplifying the exposition.

It is clear that the projection onto the y, planedefines the system of thin lenses in air which is equiva-lent to the actual system in a first-order sense. There-fore, for a system of thin lenses in air the parametersx are redundant and the system is completely definedby only the y, projection. In general, however, thevalue of xf for the ith surface is needed and may berecorded next to the corresponding point (, ) on they, projection. In this way, all the necessary informa-tion is contained on a two-dimensional plot which willhereafter be referred to as the y, g diagram.

By means of an example, some properties of they, diagram which are very useful in practice will nowbe described. Figure 1 shows the y, diagram for anobjective of the telephoto type, consisting of two thinseparated components and working at finite conjugates.It is easy to show (see Appendix) that the polygon isalways concave toward the origin at surfaces of positivepower and convex toward the origin at surfaces ofnegative power.

Since the points J and A represent the positions ofthe object plane and first component, respectively, thearea OJA is equal to the axial distance between these;similarly, area OAB is equal to the separation betweenthe two components, and area OBM is equal to thedistance from the second component to the image plane.

If line OC is constructed parallel to JA and line ABis extended till it intersects it at point D, say, the areaOAD is equal to the focal length of the A lens, as willbe evident later. Similarly, constructing line OEparallel to AB and extending line BM, the negativearea OBE is equal to the focal length of the B lens. Tofind the magnifications and positions of the stop Sand lens B as imaged into the object space by lens A,one connects the origin 0 with points S and B, respec-tively, and extends lines OS and OB till they intersectline JA extended in points S' and B' as shown. Thisconstruction uses the fact that planes conjugate toeach other have the same value of /y. The areasOAS' and OAB' are equal to the distances from lensA to the images of S and B. The magnifications of Sand B as imaged by lens A are OS'/OS and OB'/OB,respectively. In a similar manner one may determinethe image positions and magnifications for the lens Aand stop S as imaged by lens B into the image space.These images are represented by the points of inter-section S" and A", respectively.

The point P represents the two principal planes of theobjective. It is located at the intersection of lines JAand BM extended, since, by definition, the principalplanes are imaged onto each other with a magnificationof plus unity. The negative areas OAP and OBP are,respectively, equal to the distances from lens A to thefirst principal plane and from lens B to the secondprincipal plane. The area OPF equals the focal lengthof the combination.

1252 APPLIED OPTICS / Vol. 2, No. 12 / December 1963

Fig. 1. y, g diagram for telephoto objective. Both componentsthin.

The y, diagram has the additional important

property that any conjugate shift may be rigorously

represented by a pure shear parallel to the y axis. Simi-

larly, any stop shift is represented by a shear parallel to

the g axis. This is easily demonstrated for the case of

a conjugate shift by noting that, since the principal ray

and the value of Q are unaltered during conjugateshift,

Q = Ya = a _

where y* and a* are the modified values of y and af.

Hence,

(Y* - W = (-* - cyg

so that

(y* - 0)/ = (a* - .)/a.

It is clear that the left-hand side of the last equation is

not changed by refraction and the right-hand side is

not changed by transfer to the next surface; hence,

a = (y* - 0/

has the same value at every surface, so that the entirey, g diagram is transformed by the shear:

y* = Y. y* = y + ag.

Alternatively, the diagram may be left intact and the

y, g coordinate system may be transformed by an

identical shear in the opposite direction to convert it

into the appropriate system of oblique coordinates.

Figure 2 illustrates this property for the case of conju-

gate shift. The diagram JABCM represents an ob-

jective consisting of three thin separated elements.

The quantity "a" measures the amount of conjugate

shift; the diagram J*A*B*C*M* results from trans-

forming the diagram JABCM by the shear

y* = y; y* = y + ay.

To perform this shear, a y* axis is drawn at an angle of

0 = - tan-' a to the y axis, and any point P on thediagram is transformed by drawing a line through P

parallel to the y axis till it meets the g axis in R and

the 9* axis in S. Then, the transformed point P* mustalso lie on this line at a position such that P*R = PS.

Since straight lines are preserved, transforming the

points J, A, B, C, and M is sufficient to determine the

new diagram. Note that the new object and imageplanes correspond to the points K and L at the inter-section of the lines JA and CM extended to meet the

y axis.The basis for the previously given graphical construc-

tion for focal length is now clear. In Fig. 1, the line

OC is the g* axis corresponding to a conjugate shift of

the object plane to infinity. Since a shear preservesareas, the area OAD equals the rear conjugate distance

corresponding to object at infinity, which is just the

focal length of lens A.Figure 3 gives six examples of the y, 7 diagram for

six optical systems made up of thin components in air.

Case (a) represents a Petzval lens with stop at the frontcomponent and object at infinity. Careful inspection

of the diagram in view of the properties already de-

scribed shows that the distance from front component

y*= y+ay

y* =y a = -tan e

Fig. 2. Shear of y, g diagram due to conjugate shift.

y

I0

PETZ VALLENS

a.

Y REVERSE-TELEPHOTO

0 YC.

Y ERECTINGPERISCOPE

0~~~~

f.

Fig. 3. Examples of different lens types.

December 1963 / Vol. 2, No. 12 / APPLIED OPTICS 1253

to

to image plane exceeds the focal length and also thatthe nodal points are crossed, the first one being to theright of the first component, the second being to theleft of the second component. Case (b) illustratesa three-component 1:1 relay with stop at the centralnegative component and having both entrance andexit pupils at infinity. Case (c) illustrates a reversetelephoto lens working at infinity with stop about two-thirds of the way from the first component to thesecond. The diagram shows the characteristic longback focal length and demonstrates that the nodalpoints cannot be crossed. Cases (d) and (e) show aGalilean and a Keplerian telescope, respectively, bothhaving 2X magnification and positive eye relief. Theentrance pupils are located behind the objective forthe Galilean and at the objective for the Kepleriantelescope. The latter employs a field lens. Finally,case (f) illustrates a 2 X erecting periscope employing afield lens at each of the two intermediate images andhaving positive eye relief. Since over-all length of thesystem is determined by the total area swept over by theradius vector on the diagram, additional circuits aboutthe origin would be required to obtain a substantiallylonger periscope having similar diameter constraints.

The y, diagram reduces to the construction givenby Smith 2 3 for the special case of systems for whichthe sides of the polygon, when extended, are tangent toa circle with center at the origin. In this case it is seenthat all separations and focal lengths can be measuredas distances along the sides of the polygon itself. Nowa central ellipse can always be transformed into a circleby combining a shear with a change of scale, both alongeither axis. Therefore, any system for which the sidesof the polygon are tangent to a central ellipse can bemore easily treated using the method of Smith.

Applications

As a first application, consider Fig. 4, which illus-trates the general condition which must be satisfied byan optical system, in air, having the requirement that

C

A

Fig. 4. , fJ diagram for system with externally situated nodalpoints.

Fig. 5. y, i7 diagram for system subject to constraints ondiameter and over-all length.

the front nodal point be to the left of the first vertexand the rear nodal point be to the right of the lastvertex. In addition, suppose that a real object and areal inverted image are required. Since the y, poly-gon must always wind clockwise about the origin forQ > 0, the point B corresponding to the front vertexmust lie clockwise from the node N, and the point Ccorresponding to the last vertex must lie counterclock-wise from N. Therefore, the only way to connect Bto C is to wind clockwise about 0, meanwhile crossingthe axis an even number of times, with a minimum oftwo times. This demonstrates that the system mustform an even number of intermediate real images.Similarly, it may be shown that for the case where theimage is required to be real and erect, an odd number ofintermediate real images is necessary.

A second example of a typical practical problem thatarises in the design of optical instruments is illustratedin Fig. 5. For the given system, the over-all object-to-image distance and the magnification are both specified.In addition, the position of the entrance and exit pupilsare given as well as the size of the entrance pupil. Thelens elements are constrained to lie within a cylinderof specified diameter, length, and location along theoptic axis. The problem is to determine the minimumrequired number of thin lenses as well as their focallengths and positions.

The lengths OJ and OM are equal to the object andimage heights, respectively. The height on the en-trance pupil is OR; the length OS is determined by thefact that the area OMS equals the distance from imageplane to exit pupil. Points A and B correspond to thefront and rear faces of the constraining cylinder andmust lie along the straight lines JR and SM extended,their locations along their respective lines being deter-mined from the known position of the cylinder facesalong the optical axis. The required optical systemmay be represented by some undetermined curve travel-ing clockwise from A to B in such a fashion that it in-

1254 APPLIED OPTICS / Vol. 2, No. 12 / December 1963

eludes an area (shown shaded) equal to the length ofthe cylinder. It must, moreover, be consistent withthe diameter constraint of the cylinder. Assuming novignetting, the diameter constraint may be representedby the condition

2(jyl + !gl)

where is the maximum permissible free aperture.In the present example, the shaded area must thereforelie to the left of the line 2(y + ) = . If somevignetting were permissible, this condition would bemodified. For instance, if vignetting were permissibleonly to the degree that both marginal and principalrays would at least barely get through, then the condi-tion becomes

21yI <, i0 Jy > Ii; 21gl <, , IyI < Ii.

Let the area of the triangle OAB be denoted by T.Then, if the shaded area is not less than T, the problemcan always be solved with at most two thin lenses(possibly one), provided the diameter constraint is notviolated. If the shaded area is less than T, the problemcan always be solved with three thin lenses of whichthe outer two are positive and the inner one negative.Once the exact curve connecting A and B is determined,the focal lengths and locations of the required thinlenses can be calculated using Eq. (1).

Most of the discussion above has dealt with thinlenses to avoid complexity. However, application tothick lenses is essentially the same, except that thereplacement of actual distances by air equivalent dis-tances requires the introduction of refractive indices.In fact, it is sometimes more convenient to regard theNf for every space as independent parameters insteadof the x for every surface. Depending on whatvalues of xi or Ni are chosen, every y, g diagram maybe interpreted as either a system of thin lenses or ofthick ones. Thus, Fig. 1 may alternatively be regardedas the diagram for a thick meniscus singlet for which thethickness would be determined once the value of N isgiven, or vice versa.

Finally, it is of interest to note that many of theabove results are valid even for the case of an axiallysymmetric medium with continuously varying index ofrefraction. The curved path of an arbitrary paraxialskew ray uniquely defines the first-order properties ofsuch a system, in the same manner as before, exceptthat Eqs. (1) must be written in differential form.Here again, it is possible to work only with the projec-tion on the y, y plane and treat the monotone increasingvariable x as a parameter along the curve.

Conclusion

The method of first-order design described above hasbeen applied to numerous problems at Bausch & Lomband has frequently been found to yield insights which

cannot readily be obtained using better known meth-ods. It has proved to be a direct and time-saving toolduring the early stages of the optical design of anyinstrument in which the location and diameter of thelenses and images are of primary importance. Alsobecause of the ease with which individual surfacescan be imaged into the object and image spaces viathe intervening components, the method has proveduseful for the determination of cosmetic specificationsas well as for relative illumination calculations.

AppendixIt will first be proven that Q = y - ga is invariant.

The well-known ray tracing equations for a paraxialray are:

ai+1 = i - Kiyi, yi+i = yi + Di+ia i+i, (2)

where Ki = (Ni± 1 - Ni)/ri, Di = di/N, Oaf = Niu.

Using bars for a second ray,

yia- ia = yi(aii+1 + Kig7) - gi(ai+1 + Kiyi)= yii+l - fliai+1 = (yi+1 - D+las+ )ai+

-(gi+l - Dil+i+i)ai+i = Yi+1&'i+1- al+ gos+ 1'

which shows, by mathematical induction, that Q isinvariant. Now

Q = yi - giai = [yi(- i-) - -V(yi-yi-1)]/Df= (yi-lg; -yigsl)1Di

= [i(ai - ai+i) - ai(ai - i+1)/K= (aiai+ -ari)1Ki

therefore,

Di = (yi- ii i - yigl- )/Q, Ki = (aiai+1 - ca+i+i)/Q. (3)

The other relations of Eqs. (1) follow immediately fromthe definitions of xi, Di, and Ki and simple rearrange-ment of Eqs. (2).

It will now be shown that any vertex V of the y, ypolygon is concave or convex toward the origin accord-ing to whether the corresponding power is positive ornegative respectively. Let I be the side preceding Vand R the side following, and let K be the surfacepower associated with V. If the angles after refractionare indicated by primes, the equations of I and R are:

Q = Ya - a

and

Q = ya' - ga"

where we regard the angles as constant. Letting 0'be the slope angle of R, then tan 0' = o'/' and differ-entiating,

sec2 0 ' ~da- _ ad' a'

dK dK dK/

But a' - a = -Ky, hence

da'I/dK= - y, da'/dK -;

December 1963 / Vol. 2, No. 12 / APPLIED OPTICS 1255

moreover,

sec2/ = (a' 2 + a12)//2,

therefore, substituting into the equation for d'/dK,one obtains:

do' _ Q

dK ai + '2 (4)

Equation (4) states that d0'/dK and Q differ in sign.Therefore, for a small increment AK > 0, line R is

deviated clockwise (A0' < 0) for Q > 0 and counter-clockwise (AO' > 0) if Q < 0, with exactly the reversesituation if AK < 0. Since R is parallel to I for K0, the truth of the proposition immediately follows.

References1. R. K. Luneberg, Mathematical Theory of Optics (Brown

University, Providence, Rhode Island, 1944), pp. 301-303.2. T. Smith, Trans. Opt. Soc. (London) 23, 217 (1921-22).3. T. Smith, Dictionary of Applied Physics, IV (Peter Smith,

New York, 1950), pp. 357-359.

From the EditorAt last some readers have been stirred to write us short notes

concerning units. We are happy to point out that these lettersare running two to one in favor of the metric system. (To becompletely frank, we should say that two of the three have beenin favor, and the other opposed.) One of them referred to thereport of a formal inquiry relative to the possible introduction ofthe metric system into England-as reported in the PhilosophicalMagazine, Series IV, Vol. 25, p. 74 (1863)-a Professor Miller wasbeing questioned. The interrogation went as follows:

Question: "Do you find in your learned pursuits that ourpresent system of weights and measures interferes with scientificinvestigation in any way?"

Answer: "Not in the least; they are so complicated that it isquite impossible to use them.... Since 1830 no chemist ever madeuse of any weights which were not decimally divided." (Deci-mally divided balances were made by Robinson and by Oertling.)

Question: "So far as scientific investigations are concerned,our present system is useless?"

Answer: "Entirely.....That was one hundred years ago, but at last it appears that

Christopher Robin's world of pounds, shillings, and ounces isbeginning to crumble. The New York Times of 22 and 24 Septem-ber 1963 presented the long-awaited recommendations of theRoyal Commission under Lord Halsbury concerning the decimali-zation of the pound into 100 cents. Gone will be the guineas,half crowns, and florins, the sixpence and the threepenny bit.But there will still be a five-cent piece to be called a shilling, sothat 20 shillings remains a pound. You may say that thismonetary change has little to do with the metric system, but notso: this is the first British unit or conversion factor involving 10since Henry Briggs proposed decimal logarithms in 1617, andfew Britons use logarithms in their daily life. (It is also reportedthat those who do prefer Napierian logarithms.) There is somehope that they may find it almost as easy to think in terms of tensand tenths as in terms of twelfths, fourteenths, and sixteenthsnow used in divisions of feet, stones, and pounds. (It is interestingto observe that the ounce, one-sixteenth of a pound, derives fromthe Latin uncia, meaning twelfth part. Our word inch alsoderives from uncia.)

If the host at a present-day British supper party desires tofigure quickly a ten percent tip on a bill of £25 18s 9/2 d, he mustwork his way first through the £: 2.5 or £2 10s, then the shillings:1.8s or Is 9.6d, finally the pence: 0.95; and then total to obtain£2 11s 10.55d, which, if he is from the southern parts of GreatBritain, he will round off to £2 12s. (In decimal pounds the billis £25.94, and a generous tip £2.60.)

Our nonmetric reader does not mind metric units being used forpurely scientific purposes, but demurs when common objects (the

kitchen sink, a sheet of paper, a '/4-in. pipe or a 6-ft man) aredimensioned this way. "If I spoke of a 183-cm man you wouldhave no idea whether he was a giant or a runt." We havechecked this particular point with several European friends, andthey assure us that they can still mentally picture stature in cm,although one can no longer sing "157.5 cm, eyes of blue". (Per-haps "157, eyes like heaven".) In fact it gives one a new goal toshoot for: the Russians are very proud of the fact that Peter theGreat was over 2 m tall. Our reader asks "Would you go intoour machine shop and ask for a piece of 6-mm tubing?" Prob-ably one would, if it were glass tubing; and in these days ofsmall foreign cars practically every mechanic, if not every ma-chinist, now has his collection of metric wrenches and metricnuts and bolts. He might even have some metric calipers.We agree with the reader, however, that even in our own machineshop the machinists might be dismayed by some of the dimensionsexpressed in our pages; but this journal is not really addressedto them, and few of them read it regularly. To those readers whodesire to construct in their own laboratory an exact copy of some-thing described in this journal, we would recommend that youwrite directly to the author and ask him for a set of his workingdrawings: the author would probably tell you many things thathe would do differently if he were starting again.

Our goal in these pages is not to convert your machine shop, butto make the optics audience more metric minded. In 1940the Director of Purchasing at Eli Lilly and Company, a leadingpharmaceutical supplier, began a campaign to convert the com-pany to metric operations. Many of their raw materials camefrom metric parts of the world, and suppliers in the U.S. variouslyused apothecary and avoirdupois systems, so that problems ofmaterials control, inventory management, and accounts payablebegan to tax even electronic computers. During the war a seriesof so-called wonder drugs was developed, and it became com-monplace to express dosage in metric units, usually in milligramsper dose. In early 1954 the Company began experimentally toinclude on the labeling the exact metric equivalent to the avoirdu-pois-apothecaries weights and measures. Over a period of fouryears all of the company's scales, balances, and other measuringequipments were converted to metric, as were the bookkeepingand supply records. Suppliers of such bulk commodities assugar, starch, and salt were asked to quote and supply in packagesof even kilograms, such as 30, 50, 70, or 100 kilos; and Lilly waspleasantly surprised to find that they did so readily and withoutcomplaint. Money talks. The only organization with whichthey had real difficulty was the U.S. Government, who insisted thattaxes on grain alcohol must be computed in terms of gallons.This company is now completely metric in its internal operationsand almost so externally. There is no reason why the opticalindustry and the optics community could not be similarly inadvance of its times.

JOHN N. HOWARD

1256 APPLIED OPTICS / Vol. 2, No. 12 / December 1963