How to approach the design of a bilateral symmetric ... · How to approach the design of a bilateral symmetric optical system ... The study of these systems is rel- ... / OAR Image

How to approach the design of a bilateralsymmetric optical system

Jose M. SasianAT&T Bell Laboratories263 Shuman BoulevardNaperville, Illinois 60566

1 Introduction

symmetry has been for a century and continues to be a subjectof interest in lens design. The study of these systems is rel-evant for understanding the effects of fabrication and assem-bly errors in lenses nominally designed as axially symmetric,'for designing reflective systems without a central obstruc-tion,2 or for designing exotic systems that provide a practicalsolution to a difficult problem.3 The literature on the theoryof nonaxially symmetrical optical systems is extensive. Forexample, anamorphotic systems have been discussed,48 theeffects of decenters in lenses have been addressed,"9_'3 thetransverse ray aberrations of plane symmetric systems havebeen analyzed,'4 and many other relevant works on nonro-tationally symmetric systems have also been published.'527

The class ofoptical systems that possess a plane of bilateralsymmetry is of particular interest because such systems aresimple enough to be easily described, practical, and becausethey exhibit most of the significant differences between ax-ially symmetric systems and those that lack this symmetry.The purpose of this paper is to present a theoretical devel-opment for understanding and designing plane symmetricsystems with special emphasis on systems made with surfacesthat are spherical or slightly aspheric. This development canbe considered an extension of the wave theory of rotationallysymmetric systems. Although there are other methods fordesigning plane symmetric systems, the theory presented hasvalue because of the insight it gives2833 and because theconcepts used are familiar to the lens designer. Other methodsthat rely on the use of off-axis or eccentric sections of axiallysymmetric systems are very powerful because of the inherent

symmetry, but they do not provide many new insights. Thetheory presented in this paper can give a different point ofview in the analysis of a plane symmetric system. In the nextsection, we define several primordial entities of a bilateralsymmetric system. The third section establishes an aberrationfunction and consequently the size and position of the image,its defects, and the concept of aberration fields. The fourthsection presents an informal derivation of aberration coef-ficients. In the fifth section, simplifications for reflective con-focal systems are addressed, and finally, the last section dis-cusses and summarizes the paper contents.

2 DefinitionsIn a bilateral or plane symmetric optical system there is aplane of symmetry: that is, one half of the system is a mirrorimage ofthe other. Axially symmetric and double-plane sym-metric systems belong to the class of plane symmetric sys-tems. In bilateral symmetric systems, the optical components,mirrors and lenses, can be tilted in the plane of symmetryherewith providing an additional and useful degree of designfreedom. Another type of plane symmetry in an optical sys-tem that is not the subject of this paper is symmetry aboutthe aperture stop. A plane symmetrical optical system is ii-lustrated in Fig. 1.

In describing the imagery of an optical system it is indis-pensable to establish a reference. For the case of a planesymmetric system, this reference is a selected ray, called theoptical axis ray (OAR), lying in the plane of symmetry. TheOAR plays the same role that the optical axis does in thedescription of axially symmetric systems: it establishes acoordinate axis for referencing variables describing the sys-tem. In particular, these variables are the normalized aperturevector , which specifies any point in the system aperture;the normalized field vector H, which specifies any point inthe system field of view; and the unit i vector, which identifiesthe direction of plane symmetry. The field vector has its foot

OPTICAL ENGINEERING/June 1994/Vol. 33 No. 6/2045

Abstract. A theoretical development is presented for understanding anddesigning bilateral symmetric optical systems. This development usesconcepts familiar to the optical engineer and is an extension to the wavetheory of axially symmetric systems. An aberration function is developedand the image position and size, image defects, and aberration fieldsare discussed. The main dependence of aberrations as a function ofsystem parameters is established by a set of approximate aberrationcoefficients and some guidelines to approach the design of bilateral sym-metric systems are presented.

Subject terms: optical systems; lens design; geometrical optics; plane symmetry;optical aberrations.

Optical Engineering 33(6), 2045—2061 (June 1994).

The design of optical systems that lack an axis of rotational

Paper 14083 received Aug. 19, 1993; revised manuscript received Nov. 17, 1993;accepted for publication Nov. 18, 1993.

1994 Society of Photo-Optical Instrumentation Engineers. 009 l-3286/94/$6.0O.

Downloaded From: http://opticalengineering.spiedigitallibrary.org/ on 02/10/2014 Terms of Use: http://spiedl.org/terms

Fig. 1 Unobstructed Newtonian telescope consisting of an axiallysymmetric parabolic primary mirror and a double-curvature second-ary mirror: (a) a side view, showing the plane of symmetry of thetelescope coincident with the plane of the drawing, and (b) a topview.

at the intersection of the OAR with the image plane, and theaperture vector has its foot at the intersection of the OARwith the exit pupil plane. These three vectors are perpendic-ular to the OAR and are illustrated in Fig. 2. The optical axisray and the normal to a given surface at their intersectionpoint make an angle I as illustrated in Fig. 3.

In analogy with an axially symmetric system, the aperturethat limits the extent of light through a plane symmetric sys-tern is called the aperture stop and in this paper we assumeit to be circular and perpendicular to the OAR. Rays fromthe edge ofthe field that pass through the center ofthe aperturestop are called chief rays, and rays from the center of thefield that pass by the edge of the aperture stop are calledmarginal rays. By definition, the entrance and exit pupils arethe images ofthe aperture stop in the object and image spaces,respectively. The OAR is the ray that defines the center ofthe field of view, the center of the aperture stop, and thecenters of the pupils. Furthermore, the plane of symmetrycan also be called the tangential plane, and the plane per-pendicular to the symmetry plane and that contains the OARcan be called the sagittal plane. The sagittal plane is notphysically a plane but a set of planes since at each surfaceits orientation changes as dictated by the OAR; however,optically it is a continuous plane as the OAR is a straightline. An observer looking down the OAR of a plane sym-metric system would perceive the OAR as a straight line.

3 Wave Aberration FunctionTo determine the imagery of a plane symmetric system it iscritical to establish an aberration function. This function un-

Fig. 3 Geometry illustrating the OAR and other relevant parameters.The plane of the drawing is the plane of symmetry.

veils the imaging possibilities of an optical system, and there-fore it is crucial to determine it. In this development, theaberration function represents the wavefront deformationacross the field of view and aperture of the system. Theaberration function is a scalar subject to bilateral symmetry,so it must depend solely on combinations of the dot productof the field H, aperture , and symmetry unit i vectors. Thuswe can write

W(H,p) = n n + q,n,p,q(H . H)kk,m,n,p,q

X (. p)m(H. p)'(i . H)(i.p)q , (1)

where W2k+n n + q,n,p,q represents the coefficient of aparticular aberration form defined by the integers k, m, n, p,and q. By setting the sum of these integers equal to 0, 1, 2,

groups of aberrations are defined. In Tables 1 and 2 the

2046 / OPTICAL ENGINEERING / June 1994 / Vol. 33 No. 6

SASIAN

/ OAR

Image plane

(b)

(a)

(b)

Fig. 2 (a) Field and aperture vectors projected on the symmetryplane and (b) looking down the OAR.

(a)

Imageplane

ray

I'

Surface


DESIGN OF A BILATERAL SYMMETRIC OPTICAL SYSTEM

Table 1 Aberration groups.

First group

Second group

W02002 (ip)2

Wi1011 (1H)(ip)W2o020 (iH)2

Scalar form

Woiooi pcos(3)

Wiooio Hcos(a)

W02000 p2

Wi1100 Hpcos(4)

W20000 H2

Wo20o2 p2cos2(3)

W11011 Hpcos(a)cos()W20020 Hcos(ct)

Constant astigmatism

Anamorphism

Quadratic piston

w03001 (i•p)(p•p)

Wi2ioi (ip)(Hp)W12010 (iH)(pp)

W2iooi (ip)(HH)

W2iiio (iH)(Hp)

W300i0 (iH)(HH)

W03001 p3cos(f3)

Wi2101 Hp2 cos(4) cos(13)

Wi2oio Hp2 cos(cc)

W21001 H2pcos(3)

W21110 H2 pcos(4) cos(a)

W30010 H3cos(ct)

Constant coma

Linear astigmatism

Field tilt

Quadratic distortion I

Quadratic distortion II

Cubic piston

W04000 (p.p)2

W13100 (Hp)(pp)

W22200 (Hp)2

W22000 (HH)(pp)

W31100 (HH)(H.p)

W40000 (HH)2

W04 p4W13100 Hp3 cos()

W22200 H2p2cos2(4)2222000 " P

W31100 H3pcos()4

40OOO "

first four groups of aberrations are shown, each representinga degree of approximation to the wavefront deformation. Ineach group, the aberrations are organized in subgroups con-taming the aberrations characteristic of axial symmetry,double-plane symmetry, and plane symmetry. The parame-ters a and 1 are the angles between the symmetry unit vectorand the field and aperture vectors, respectively. The angle4: = a — 13 is the angle between the field and aperture vectors.The algebraic powers of these vectors H and and of theircorrespondent angles , a, and .3 are indicated by the integers2k + n +p, 2m + n + q, n, p, and q ; these integers are usedas subscripts and label the aberration coefficients. Thus, thesubscripts in the term W12101 (i. p)(H. p) for linear astig-matism indicate a linear dependence with respect to H, quad-ratic with respect to p, linear with respect to the cosine ofand 13, and none with respect to the cosine of a. In Table 1,the terminology constant, linear, quadratic, etc. refer to thefield dependence of the aberration in question.

As a function ofthe sum ofthe power ofthe field, aperture,and symmetry vector the first aberration group contains onlya term of zero order, the second group terms of second order,the third group terms of fourth order, and the fourth groupterms of sixth order. The discussion in this paper is centeredon the first three groups in Table 1 . The group arrangementthat results from the sum k + m + n +p + q is appropriate be-cause it best represents the aberrations that plane symmetricalsystems have when they are made with spherical surfaces; italso leads to a simpler grouping because of the reduced num-ber of terms to be treated. A different grouping of the termscould be defined by the sum 2k +2m + 2n +p + q. This al-ternative grouping results in forming groups with aberrationsof the same order as a function of the sum of the field andaperture vectors and it may be chosen, but it is not our bestchoice. The reason is that terms like line coma W03003(i p)3that are of third order on H and p are not more importantwhen spherical surfaces are used than spherical aberration

OPTICAL ENGINEERING / June 1994 /Vol. 33 No. 6 / 2047

Vector form

Woo000

W01001 i.p

W10010 iH

W02000 pp

W11100 Hp

W20000 HH

Third group

Name

Constant piston

Field displacement

Linear piston

Defocus

Magnification

Quadratic piston

Spherical aberration

Linear coma

Quadratic astigmatism

Field curvature

Cubic distortion

Quartic piston


W04000(p )2, which is of fourth order on . Line comaW03003(i p)3 is less significant than constant coma W03001(i• p)(p• p) because the former is of third order as a functionof the OAR angle of incidence land the latter is of first orderon I; this happens for surfaces that are spherical or axiallysymmetric when the OAR is incident in their pole. Inciden-tally, the aberration coefficients are a function of the OAR

2048 / OPTICAL ENGINEERING / June 1994 / Vol. 33No. 6

angle of incidence and the power of this angle for a particularterm is indicated by the power of the symmetry vector. Thus,the fact that we are emphasizing systems with spherical sur-faces leads to a simpler description. For systems involvingsurfaces that are plane symmetric but that have no restrictionin their shape it would be more appropriate to use the sum2k + 2m + 2n +p + q to define the aberration groups. For thecase of cylindrical surfaces,4 a different grouping or rectan-gular coordinates to define the aberration function would bemore suitable.

In this paper, we prefer polar coordinates because theaberrations of axially symmetric systems appear as asubgroup and because these coordinates seem to be the mostappropriate for describing systems made with spherical sur-faces. The presented bilateral symmetric aberration functionis a generalization of the classical wave aberration functionfor axially symmetric systems. If the integers p and q are setto zero, one obtains the well-known aberration function forrotationally symmetric systems:

W(H,p) = W2k+n2m±nn(H • H)k(p . p)m(H . p)fl • (2)k,m,n

3.1 Size and Position of the ImageThe main characteristics of an image are its size and its po-sition. The concern of this section is to provide a method tocalculate these image attributes. The first group of aberrationscomprises only a piston term that represents a uniform phasechange across the field of view. Since there is no aperturedependence, neither the image position nor the image qualityare affected by such a term. Thus, all the piston terms in theaberration function are irrelevant for this discussion. Thesecond group contains the terms field displacement (boresighterror), defocus, and magnification (scale change). Field dis-placement causes an offset of the center of the field of viewbut by definition the OAR intersects the image field centerand therefore the coefficient for field displacement is set tozero. The terms defocus and magnification in Table 1 rep-resent departures from an ideal in the longitudinal positionof the image and in its size. By properly defining and cal-culating these ideal image attributes, the size and position ofthe image along the OAR, the coefficients for defocus andmagnification can be set to zero. By nullifying by definitionthe coefficients of field displacement, defocus, and magni-fication we are stating that the position and size of the imageare known.

In this development, the ideal location of the image isdetermined along the optical axis ray by the intersection ofa paraxial marginal ray with the OAR traced in the sagittalplane. The ideal size ofthe image is defined by the intersectionof a chief paraxial ray in the sagittal plane with a planeperpendicular to the optical OAR; this plane intersects thepoint defined by the intersection of the paraxial marginal rayand the OAR. The ideal size and position of the image couldbe based on the tangential plane but we have preferred thesagittal plane because it leads to a simpler development.

To simplify this study, we associate with a given bilateralsymmetric system an axially symmetric system that is con-structed with the same surface separations along the OARand same refractive indices. The power of each surface inthe associated system is equal to the oblique power cI in theactual system; that is,

Table 2 Fourth group of aberrations.

SASIAN

Vector form Scalar form

w03003 (ip)

Wi2012 (iH)(ip)2w21021 (iH)2(ip)

w30030 (iH)3

w04002 (ip)2(pp)

Wi3011 (iH)(ip)(pp)w22002 (ip)2(HH)w22020 (iH)2(pp)

w31011 (iH)(ip)(HH)

w40020 (iH)2(HH)

Wi3102 (ip)2(Hp)

w22111 (iH)(ip)(Hp)w31120 (iH)2(Hp)w05001 (ip)(pp)2

Wi4010 (iH)(pp)2Wi4101 (ip)(Hp)(pp)

w23001 (ip)(HH)(pp)W23iio (iH)(Hp)(pp)

W232o1 (ip)(Hp)

W320io (iH)(HH)(pp)

W32101 (ip)(HH)(Hp)

W32210 (iH)(Hp)

W41110 (iH)(HH)(Hp)

W4iooi (ip)(HH)

Wsooio (iH)(HH)

w06000 (pp)3

W15100 (Hp)(pp)(HH)(pp)

W24200 (Hp)2(pp)

W33300 (Hp)3

w33100 (HH)(Hp)(pp)

w42200 (HH)(Hp)2

w42000 (HH)2(pp)

wsiioo (HH)2(Hp)

W60000 (HH)3

w03003 p3cos3(3)

Wi2012 Hp2cos(a)cos2(3)

w21021 H2pcos2(a)cos(3)

w30030 H3cos3(a)

w04002 p4cos2(3)

Wi3011 Hp3cos(a)cos(3)

w22002 H2p2cos2(3)

w22020 H2p2cos2(cx)

w31011 H3pcos(a)cos(3)

w40020 H4cos2(cx)

Wi3102 Hp3cos()cos2(3)

w22111 Hp2cos(4) cos(ci) cos(3)

w31120 H3pcos(4)cos2(a)w05001 p5cos(3)

Wi4010 Hp4cos(3)

Wi4101 Hp4 cos(4) cos(a)

w23001 H2p3cos(3)

w23110 Hp3 cos(4)cos(a)

w23201 112 3 cos2(4) cos(a)

w32010 H3p2cos(a)

W32ioi Hpcos(4)cos(3)

W32210 H2 p2 cos2 (4) cos(ct)

W41i10 Hpcos(4)cos(a)

W41001 Hpcos(3)

W50010 H5cos(a)

w06000 p6

W15100 Hp5 cos(4)24

24OOO P

W24200 H2p'cos2(4)

W3330o H3p3cos3(4)

W331 H3p3cos(4)

W42200 H4p2cos2(4)4242200 " P

W1 H5pcos(4)

W6000 H6



=n' cos(I') —n cos(I)R

where n, I, and R are, respectively, the index of refractionof the media preceding the surface, the OAR angle of mci-dence, and the vertex radius of curvature; n' and I' refer tothe same quantities after refraction. The resulting system isa design aid to calculate the position and size of the image,its defects, and other properties in the actual plane symmetricsystem. In this associated axially symmetric system, paraxialrays have the same properties as paraxial rays have in thesagittal plane of the actual bilateral symmetric system. Thisis so because the refraction and transfer equations for sagittalparaxial rays become the same in both systems. Therefore,we can use in the associated system any of the design methodsfor axially symmetric systems to calculate the ideal imagesize and its position. Furthermore, in the sagittal plane of theplane symmetric system we define the cardinal points to bethose of the associated system, whereby principal, nodal, andfocal points and planes are established.32

The anamorphism or magnification difference in the twoprincipal sections of a bilateral symmetric system has twocontributions. The first A is due to the difference in paraxialmagnification in the sagittal and tangential planes and is sim-ply given by

A fflcos(I)11=iLcos(1')i,

where j is the number of surfaces in the system and I and I'are the OAR angle of incidence and refraction relative tosurface i. The contribution A depends only on the structureof the system and is independent of how it is used; that is,it is independent of the object and image conjugate locationsand of the stop position. The factor cos(I)/cos(I' ) can beobtained by taking the derivative of Snell's law with respectto the OAR angle of incidence; this gives the refractionaround the OAR for paraxial rays in the tangential plane. Thesecond contribution arises as a consequence of the astig-matism introduced at each surface and it will be treated asan aberration term. Because of the intrinsic anamorphism ofa bilateral symmetric optical system, the principal planes areplanes of unit magnification in the sagittal plane and ana-morphic magnification A in the tangential plane.

3.2 Image AberrationsOnce the size and position of an image are determined, thenext concern is its quality. The third group of terms in theaberration function represent aberrations that must be cor-rected or balanced to obtain sharp imagery. In addition to theSeidel aberrations, one must consider within the third groupseven more aberrations in the design of a plane symmetricsystem. The correction or balancing of these aberrationsmakes the design of such systems more elaborate, but a newdimension of design forms is opened for exploration.

Within the third group in Table 1 the aberrations that havedouble-plane symmetry are uniform or constant astigmatismand image anamorphism. The aberrations with plane sym-metry are uniform or constant coma, linear astigmatism, fieldtilt, and quadratic distortions. The aberrations that exhibit

axial symmetry are the Seidel aberrations. We emphasize the(3) symmetry of the aberrations because its consideration sim-

plifies the analysis and correction of plane symmetric sys-tems. The wavefront deformation represented by each of theaberrations of a plane symmetric system does not have a newshape; it continues being linear as a function of the aperturefor distortion, quadratic for field tilt and curvature, cylindricalfor astigmatism, cubic for coma, and quartic for sphericalaberration. What is different in comparison to an axially sym-metric system is the aberration distribution across the fieldof view. For example, the point of correction, where an ab-erration vanishes or about which the aberration is symmetric,may have different location for different aberrations. This isin contrast to axially symmetric systems where all the off-axis aberrations have symmetry about the optical axis.

Other differences are that the image can be anamorphicand that the image plane can be tilted with respect to theoptical axis ray. In this development, image plane tilt is thetilt with respect to the OAR of the image formed by sagittalparaxial rays. More precisely defined, image plane tilt is theangle of the tangent line to the locus of sagittal paraxial fociformed as the chief ray is swept across the field of view inthe plane of symmetry. The tangent line to the sagittal locusof foci is taken at the OAR and the image plane tilt angle 0'is measured with respect to a perpendicular line to the OAR.The aberration linear astigmatism is a measure of the dif-ference between the sagittal and tangential image plane tilts.Within the third group of aberrations, there are two quadratic

(4) distortion terms. Quadratic distortion I produces an imageshift in the plane of symmetry when a circle is imaged; theshift is proportional to the square of the circle radius. Thisaberration causes the familiar line bowing observed in spec-troscopic instruments. Quadratic distortion II is the maincomponent of keystone distortion32 as can be verified byexpanding keystone distortion in a power series; the differ-ence between these two distortions are terms that belong toother groups of aberrations. Because keystone distortion,shown in Fig. 4, is often dominant we interpret quadraticdistortion II as keystone distortion.

The symmetrical subgrouping of aberrations presented inTable 1 suggests a strategy for correcting or balancing theaberrations of a plane symmetric system. That is, the cor-rection of a given system can be accomplished by correctingeach subgroup of aberrations with the system variables thatare in accordance with each subgroup symmetry. For ex-ample, the Seidel aberrations can be influenced with surfaceasphericities that are axially symmetric, the subgroup of ab-errations with two planes of symmetry can be influenced withasphericities that possess the same symmetry, and the ab-errations with one plane of symmetry can be influenced withsurface tilts or with plane symmetric asphericities. For planesymmetric systems made with surfaces that are spherical orslightly aspheric it can be observed that the Seidel aberrationcoefficients are almost independent of surface tilts. In otherwords, for small surface tilts, the Seidel aberration coeffi-cients remain approximately the same regardless of the sur-face tilts. This independence with respect to surface tilts, andtherefore with respect to the plane and double-plane sym-metric aberrations, simplifies the overall aberration correc-tion. One can divide the design of a plane symmetric systeminto two tasks. In the first, the Seidel aberrations are correctedwith standard design techniques and, in the second, the ab-

OPTICAL ENGINEERING I June 1994 I Vol. 33 No. 6 / 2049


= L1 i.— —

—\....

1T=

i: c LI I t

errations with plane and double-plane symmetry are thencorrected or balanced.

3.3 Chromatic Aberrations

Chromatic aberrations arise as a consequence of optical dis-persion. To the second order of approximation we are inter-ested in the variation with respect to the wavelength of fielddisplacement, magnification, and defocus. These three var-iations give rise to constant lateral chromatic abenationxwo1oo1, lateral chromatic abenation and longi-tudinal chromatic abenation W02000. The latter two chro-matic abenations conespond to the chromatic abenations ofsystems with rotational symmetry. Constant lateral chromaticabenation is the only additional effect that appears to secondorder of approximation. Its effects on the image are similarto those of lateral chromatic abenation, but they are uniformover the field of view.

3.4 Aberration Fields

In an abenation expansion, abenation fields are associatedwith abenation terms that have a particular aperture depen-dence. Within the third group of abenations there are sixfields that conespond to the Seidel abenations: spherical ab-enation, coma, astigmatism, field curvature, distortion, andpiston. An interesting and new feature of an abenation fieldin a system that is nonrotationally symmetric is the possibilityof having more than one point of conection (point node) ora line of correction (line node) where a particular abenationvanishes. Within the third group of abenations, the field ofspherical abenation is constant over the field and no newfeatures are developed. This field, shown in Fig. 5, is rep-resented over the field of view by the projection on the imageplane of the image envelopes of an anay of object points. Ina rotationally symmetric system, third-order coma is eitherconected over the entire field or linearly dependent with field.With plane symmetry, coma can be constant and nonzeroover the field. When constant coma and linear coma are pres-ent there remains one node, but this point of conection is nolonger at the center of the field. In Fig. 6, the componentsconstant and linear coma are represented over the field ofview by the projection on the image plane of the envelope

2050 I OPTICAL ENGINEERING I June 1 994 I Vol. 33 No. 6

00000000000000000000000000000000000000000000000000000000Fig. 5 Geometrical projection of the envelope of spherical aberrationon the image plane.

Fig. 6 Components of the field of coma: (a) constant and (b) linearcoma.

I .1

SASIAN

Fig. 4 Keystone distortion compared to a square grid.

999999999999999999999999999999999999999999

(a)

999999

QQ VPqQp

Q IQt(b)



of coma. Similarly, the field of astigmatism is now composedof constant astigmatism, linear astigmatism, and quadraticastigmatism. In this field, two nodes can coexist giving riseto the so-called binodal astigmatism.34 In a rotationally sym-metric system, the two nodes coincide at the center of thefield. The field of astigmatism is associated by forming thesum of all the astigmatism terms; that is

w —w 2+W 'astigmatism 02002k1 ) 121O11

+ W22200(H . )2 . (5)

This sum is of second order on the field vector and dependingon the aberration coefficients there may be two, one, or nofield points where it vanishes. In Figs. 7 and 8 the fieldcomponents constant, linear, quadratic, and binodal astig-matism are represented by the projection on the image planeof the sagittal and tangential foci as generated by an arrayof object points.

The field of curvature is formed by field tilt and fieldcurvature; this field has a node at the field center and mayhave a line or circular line node. The components of the fieldof curvature are shown in Fig. 9. The field of distortion iscomposed by anamorphism, quadratic distortion I and II, andcubic distortion; this field has a node at the field center andmay have other point nodes as well as line nodes. The fieldof piston is composed of all the piston terms and has a nodalbehavior that is similar to that of previously discussed fields.With regard to the system bilateral symmetry, the point nodescan be on the plane of symmetry or in lines perpendicular tothis plane. The node distribution must have the same bilateralsymmetry as the system.

To illustrate how the nodes are found let us consider thesimple case of the field of coma. By combining constant andlinear coma we obtain

— ( w031.\Wcoma W131001 H+ .p(. p) , (6)\ VV13100 /and therefore the coma node is found at

H=wi3'00

For other fields, the number of point nodes or the degree ofthe nodal line where a given field vanishes is related to thedegree on H of the polynomial representing the field. Thesubject of aberration fields and the singular vector18'3436treatment for their composition in the most general nonsym-metrical system provide a different perspective for lookingat aberrations.

4 Aberration Coefficients

By indicating how to find the position and size of an imageand the nature of its defects, the previous section gives insightinto the imagery of a given plane symmetric system. Theimage defects are described by aberration terms and, in turn,are quantified by aberration coefficients. The next step is toprovide the specific form of these coefficients as a functionof system structural parameters (surface radii, spacings, in-dices of refraction, etc.) and in terms of the configurationalparameters (object position, aperture, field of view, etc.). It

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

(a)

(7) not trivial to develop exact formulas for the aberrationcoefficients because quantities in both principal sections ofthe system are involved and because the wavefront defor-mation changes as it traverses the system.3738 However, itis possible to develop an approximate set of formulas thatdescribe the main dependence of the aberrations as a functionof the system parameters. These formulas are simple enoughto provide insight and to answer the question of what param-eters one must vary to correct a specific aberration. In ad-dition, the formulas allow one to make parametric analysisof simple systems to establish design trade-offs.

Aberration coefficients represent the maximum departuresfrom sphericity of the basic wavefront forms represented byaberrations. When these aberration forms are superimposedand added, their sum results in the overall wavefront defor-mation as measured at the exit pupil and across the field ofview. For the calculation of the approximate set of aberrationcoefficients it is necessary to trace a chief and a marginalparaxial ray in the associated system such as it is done in thestandard calculation of the Seidel aberrations.39 The La-grange 'I' invariant is given by

OPTICAL ENGINEERING / June 1994 I Vol. 33 No. 6 / 2051

xxx

>( >K

x x>K ><

Fig. 7 Representation of (a) constant astigmatism and (b) linearastigmatism using the projection of the astigmatic foci on the imageplane.

(b)


/ +k

:::.:+ F f)( )(+

(a)

__+:c:::+ :: :+-+—f-,(±)c + )(4k

where u and x are, respectively, the marginal ray slope andthe ray height at a particular system surface and where barredsymbols refer to the same quantities relative to the chief ray.The paraxial ray slope and height are measured with respectto the OAR, as illustrated in Fig. 10. Ray heights above theOAR are positive and ray slopes are positive if a counter-clockwise rotation of the OAR is required to reach the ray.The index of refraction of the media preceding a surface timesthe angle of incidence of the marginal and chief rays are,respectively,

/ x cos(I)\ — /_ cos(I)A=ni=nl u+ I and A=ni=nl u+RJ R

where I is the OAR angle of incidence and R is the radiusof curvature of the surface, which is assumed spherical. Theangle I is measured from the surface normal, and it is positiveif a counterclockwise rotation of the normal is required toreach the OAR. The radius of curvature is positive if thecenter of curvature lies on the right of the surface, negativeif on the left; light travels from left to right.

In terms of paraxial parameters we define the followingset of quantities:


000000000000000000000000o o 0 0 0 0 0 0o o 0 0 0 0 0 0000000000000000000000000

(a)

+

(b)

Fig. 8 (a) Quadratic astigmatism and (b) binodal astigmatism; thelatter field can be composed by combining constant and quadraticastigmatism.

SASIAN

'8 Fig. 9 Components of the field of curvature, (a) field tilt and (b) field) curvature, as represented by the projection of their envelope on the

image plane.

S1= !A(X, (10)8 \nj

11= 12(cos(I) (11)4R \ n /j= _!,2

sin2(I)()x, (12)

1 U

2 ()J11= ——n sin(I)AL — x (13)(9)

J111= —n sin(I)I'( , (14)

1 n sin(I)(!), (15)JIv= —— ______2 R n

J= -n sin(I)P2()!, (16)

(b)



=J (/_\3 /_\2 —

= () 11+ () (J+ J1) + Ji=1 \X/ X

(17) I /A\W22200 = 4( —) sj\A/ j

(18) i=j=

i1

OPTICAL ENGINEERING / June 1994 / Vol. 33 No. 6 / 2053

marginaln n'

H

H'

OAR

Surface

Cubic piston , (27)

i=jw04000= {S1}, Spherical aberration , (28)

i= I

sagittal plane. Wi3100= {4()s1} Linear coma (29)Fig. 10 Geometry illustrating paraxial ray heights and slopes on the

T1= — [u tan(0)]

Quadratic astigmatism , (30)

1

T11 'I'z[uH tan(O)J — , 2x

{/A\

I S1j Field curvature , (31)2( — ) Swhere z( ) is the Abbe difference operator that gives the / Jchange ofthe argument on refraction z(Wn) =u/n —u/n and

the OAR. The angles 0 and 0' are the tilt angles of the object 100 {4()35, + 2()SII} Cubic distortionH and H' are the object and image heights measured from

and image planes; they are positive if they require a coun-(32)terclockwise rotation to reach a plane perpendicular to the

OAR. 4 2/A\

Quartic piston . (33)In terms ofthe quantities defined, we present the following

w40000='{() s,+ )set of formulas for the aberration coefficients of a bilateral

symmetric system made off optical surfaces (quantities insidethe curly brackets pertain to surface number i): The contributions z.W that arise as a consequence of object

and image plane tilts 0 and 0' are1=]w02002 = {J1} Constant astigmatism , (19)

i=l 1=]

w12o1o= {T1},

Wi1011 = 12Jj Anamohism , (20)I 1 1 x j = — [uj tan(O') —u1 tan(O)] Field tilt , (34)

2

w20020 = • f( j11 Quadratic piston , (21)1=1 \XJ J w211Io=i=l L.

x1 =j

w03001 = : {J11} Constant coma , (22) J[l tan(O') — tan(0)]i= 1

Quadratic distortion II , (35)1=f I :ii=1 }j

Wi2101 = : 2 —J11 +J111 Linear astigmatism (23) .. 2

w3oo1o={ () T1

+ - II Cubic piston . (36)- xJ2:i 1

- I

w21001 = 11+_111+i= 1 L \X/ X J 4.1 Discussion of Aberration Coefficients

Quadratic distortion I , (24) In this section, we informally discuss how the coefficients,Eqs. (19) to (36), arise. These coefficients were derived23 byI J I

. . making an expansion about the OAR in a fashion similar to

i= 1 },,

context,37 coefficients similar to J1 and J11 have been previ-WI2010 = —JII + fly Field tilt (25) that done for axially symmetrical systems.4° In a different

2 ously derived. We have avoided presenting a rigorous math-

1 10 = 12(f) j11 + (J111 + 2J1)}ematical derivation ofthe coefficients because it is elaborated.

i= 1 t \X/ X . We preferred, instead, to try to show why the formulas have1

the form they have, because from the optical designer's pointQuadratic distortion II , (26) of view, it is more useful to know how the aberrations arise.


The formulas for the aberration coefficients depend onparaxial ray quantities calculated in the associated system,on the tilt angle I of the surfaces with respect to the OAR,and on the tilts 0 and 0' of the object and image planes. Itis appropriate to emphasize that the formulas presented inthe previous section apply to surfaces that are spherical; adifferent treatment would be required for cylindrical, toroidal,or other type of plane symmetric surfaces. When the tilt angleI of all surfaces is zero the system becomes axially symmetricand all the coefficients of the aberrations with double andplane symmetry, Eqs. (19) to (27), vanish; only the Seidel4°wave aberration coefficients, Eqs. (28) to (33), remain finiteas expected; these coefficients do not need to be discussed.The aberration coefficients, Eqs. (19) to (36), show the maindependence of the aberrations with respect to system param-eters. These coefficients are not exact but good approxima-tions to the exact coefficients when the surface tilts are small,in the order of a few degrees. It is interesting to note thatcoefficients, Eqs. (19) to (27), show a similar dependence on:Jx as stop shift equations for axially symmetric systems haveon Jx. The individual surface coefficients of the third groupof aberrations in Table 1 simply add with the conespondingcoefficients of other surfaces to form the abenation coeffi-cients of the complete system; this follows from the fact thatoptical paths add. We note that the first two groups of ab-enations in Table 1 do not contribute to the coefficients ofthe third group, since the abenation coefficients of theselower groups are zero for each surface.

To show how the abenation coefficients arise for on-axisastigmaftsm and coma, image anamorphism, linear astig-matism, field tilt, and quadratic distortions, we use the Seidelformulas. The coefficients, Eqs. (19) to (21), for abenationscharacteristic of double-plane symmetry depend on the con-stant astigmatism generated at each surface, which is givenby J1. This coefficient can be derived by replacing A byn sin(I) in the well-known Seidel formula, Eq. (30), for quad-ratic astigmatism. The coefficients, Eqs. (20) and (21), forimage anamorphism and quadratic piston result as a conse-quence of the eccentric passage of off-axis beams through asurface that is contributing constant astigmatism; this takesplace when the stop is not located at the surface in question.In analytical terms, this argument is written as

W02002[i . (g + H)]2 =

W02002[(i. p)2 + 2(i . H)(i . ) + (i H)2]

In this expression, the vector H conveys the chief ray dis-placement from the OAR. The terms in the right-hand mem-ber represent constant astigmatism, image anamorphism, andquadratic piston. The factors Jx and (/x)2 are needed in Eqs.(20) and (21) because the field and aperture vectors are nor-malized. The generation of image anamorphism and quad-ratic piston when the stop is not located at a given surfaceis similar to the well-known case of abenation generation byan aspheric surface in axially symmetric systems.

The coefficients, Eqs. (22) to (27), of abenations withplane symmetry depend on the quantities J11, J111, J1 and J.The coefficient for constant coma, Eq. (22), involves thequantity J11 and can be derived by replacing A by n sin(I) inthe Seidel formula for linear coma. All the terms involvingJ11 in the other coefficients, Eqs. (23) to (27), result as a


consequence of the eccentric passage of off-axis beamsthrough a surface contributing constant coma. In analyticalterms, this is expressed as

W03001[i. (+H)I[(p+H) . (+H)I =

W03001 [(j. p) + 2(i. i)(H. i) + (i H)(

+ (i. XH . H)+ 2(i . H)(H. )+(i H)(H. H)I

(38)

In this expression, H conveys the chief ray displacement fromthe OAR caused by having the stop away from the surface.The terms in the right-hand member represent, respectively:constant coma, linear astigmatism, field tilt, quadratic dis-tortion I and II, and cubic piston.

The coefficient of linear astigmatism can be derived byreplacing A by n sin(I) + A in the Seidel coefficient for quad-ratic astigmatism and retaining only the linear term. The resultof this is that quadratic astigmatism is described around adifferent field point. For a single surface, this is expressed as

i f\W22200=

= _![2 sin2 (I)+2n sin(I)A+A2I(x. (39)2 \nJBy taking only the linear term in A, which represents thecoefficient of linear astigmatism, and using the Lagrangeinvariant 'I' =Ax —A, we obtain the result

W12101 = —nsin(I)Az_J_n sin(I'I'—n n

=2J11+J111 . (40)

When the stop is not located at the surface the quantity J111generates some conesponding terms in the coefficients ofEqs. (24), (26), and (27) for quadratic distortion I and II, andcubic piston.

The abenation coefficients for quadratic distortion I and

(37) field tilt, Eqs. (24) and (25), contain terms that depend onthe stop position and on the quantities J and J1 respec-tively; these two quantities are derived in Appendix A(Sec. 7). When the stop is not located at the surface so thatxix is different from zero, some conesponding terms are gen-erated by J1 and J in the coefficients, Eqs. (26) and (27),for quadratic distortion and cubic piston.

The contributions that arise as the tilt of the object andimage planes change, are field tilt, quadratic distortion II, andcubic piston. The coefficient for field tilt, Eq. (34), is a state-ment of the Scheimpfiug condition and has been discussedin Ref. 30. When the stop is not located at the surface, thiscontribution to field tilt generates quadratic distortion II andcubic piston. The contribution T11 to quadratic distortion IIis also derived in Appendix A (Sec. 7). Quadratic distortionII represents keystone distortion and strongly depends on thetilts of the object and image planes; the expression,

SASIAN



tan(O')— tan(O) =0 , (41) zw02002 =Z Constant astigmatism , (50)

represents a condition for the absence of keystone distortion.This is similarto the Scheimpflug condition thatrelates object zi=2(

—

)Zc Anamorphism , (51)and image plane tilts:

I \2U' tan(O')—u1 tan(O)=O . (42) fx\ .

J w2oo2o = ( — I Z Quadratic piston , (52)

If in an axially symmetric system the object plane is tiltedand the Scheimpflug condition satisfied, then the coefficient W03o01 =Z Constant coma , (53)for quadratic distortion II becomes

0 2101 =2 Z Linear astigmatism , (54)w21110=

_hItmHt)

(43)X

where m andfare, respectively, the magnification and front W21001 ()2z Quadratic distortion I . (55)focal length of the system. Some understanding of Eq. (41) X

is gained by recalling that the image cast in a pinhole camerais free of keystone distortion when the object and image zW12010 = — Z Field tilt , (56)planes are parallel. For a pinhole camera we have ü' =ü and X

therefore to satisfy the condition we must have 0' =0. ,, 2

L1o =2(f) ; Quadratic distortion II , (57)

4.2 Contributions from Aspheric Surfaces X

In the previous section we assumed the optical surfaces to /3be spherical in shape. It is of interest to account for the LW30010 = ( — ) Z Cubic piston , (58)contributions that arise from surfaces that are slightly aspher-ic. We represent the sag Z of a bilateral symmetric surface as w04000 =z Spherical aberration , (59)z =

Zsphere + Zasphere , (44) :zW13100 =4— Z Linear coma , (60)

where Zsphere 5 the sag of the base sphere of radius R and X

Zasphere 5 the sag of an aspheric cap represented by 2

Zasphere =a(i . p)2 + (i p)(p . p) +(p . p)2 . (45) W222oo =4() Quadratic astigmatism , (61)

The coefficients a, 3, and -y determine the amount of as- 2phericity of each basic surface shape that conforms the as- (\

. . . w22ooo=2i — i Z Field curvature (62)pheric cap. These three basic aspheric shapes are, respec- \xJtively, a cylindrical paraboloid, a comatic surface, and afourth-order axially symmetric surface. The superposition of /\a spherical surface and these aspheric caps can result in de- L1oo 4(_) Z Cubic distortion , (63)scribing approximately other aspheric surfaces such as cyl-

X

inders, toroids, and conic sections. In Appendix B (Sec. 8) 4we discuss a technique for fabricating the first two aspher-icities. When the stop is located at the surface the contri- ZW40000 = Z Quartic piston . (64)butions L\Wto the aberration coefficients are simply given by

4.3 Chromatic CoefficientsL.W=L[n cos(I)]Za here (46) • .sp The coefficients for the chromatic aberrations areExcept for the cosine factor, this equation is similar to the .oneused to account for the wavefront deformation introduced . I 3n\byan aspheric plate such as in the case ofthe Schmidtcamera. wO10Ol L ' sin(I))xTo account for the situation of a distant stop, we define the

—

following quantities: Constant lateral chromatic , (65)

Za = thI[n cos(I)jx2 , (47) i__I i'\= AM — )x Lateral chromatic , (66)Z=3[n cos(I)]x3 , (48) i=' \ '1 /Z = yL[n cos(I)1x4 . (49) ='

1 \= —A( — )x Longitudinal chromaticWith these definitions, the contributions LWto the aberra- 1 2 \ fl /tions are (67)



The formulas for the aberration coefficients presentedhave been used in the design of unobstructed tele-scopes28'29'31'33 using surfaces with radii of curvature of theorder of a few meters and surface tilts of the order of 10 deg.Some of the aberration formulas were found to give errorsin the coefficients of the order of 5 or 10%, depending onthe particular design. Other formulas, mainly the Seidel ones,produced no significant errors.

5 Simplifications for Reflective ConfocalSystems

It is of interest to analyze the case of reflective systems con-structed with confocal surfaces4' such that the imaging alongthe OAR is stigmatic surface after surface, because thesetypes of systems have a reduced number of aberrations andpotentially can provide better imaging. To satisfy the re-quirement, the surfaces must be off-axis sections of coni-coids. These surfaces are nonspherical and contribute aspher-ic terms Z, Z, and Z that cancel exactly constantastigmatism and coma, spherical aberration, and in turn, allthe terms involving J1 and J11. In addition, because the angleof incidence of the OAR equals the angle of reflection forall the surfaces, the intrinsic anamorphismA becomes unity.For the case of a reflective surface L\(1/n2) is zero and J,

(68) vanishes as well. Furthermore, because L(u/n) 2x/nR, wewrite

JIII= —2J1 . (72)

This equality states that the tilt of the sagittal image plane isequal and opposite in sign to the tilt of the tangential imageplane. This is true for a mirror that does not contribute con-stant coma or that coincides with the stop or one ofits images.For the case of a mirror coinciding with the stop and the

(69) object at infinity, the equality is easily verified using theCoddington equations. In this case, the locus of the sagittalfoci is a straight line perpendicular to the optical axis of theuntilted surface. The locus is formed as the surface is tilteddifferent angles I, which become the OAR angle of incidence.The tangential locus is a circle passing by the surface vertex

(70) and the focal point of the untilted surface. This geometryreveals that the sagittal plane is exactly tilted an angle —Iand the tangential an angle I.

With the simplifications just discussed, the coefficients forthe aberrations characteristic of double and plane symmetryin a confocal reflective system are

W02002 = 0 Constant astigmatism , (73)

W11011 =0 Anamorphism , (74)

W20020 = 0 Quadratic piston , (75)

W03001 =0 Constant coma , (76)

W12101 = {J} Linear astigmatism , (77)

W21001 = Quadratic distortion I , (78)

SASIAN

where n = (n — 1)/v and v is the Abbe number. The coeffi-cients for lateral and longitudinal chromatic aberrations arethe standard wave aberration coefficients for axially sym-metric systems. To derive the coefficient for constant lateralchromatic aberration we replaced A by n sin(I) in the coef-ficient for lateral chromatic aberration.

4.4 Approximations in the CoefficientsIt has been mentioned that the aberration coefficients pre-sented are not exact, but good approximations to the exactquantities when the surface tilts are small. The approxima-tions have been done to simplify the formulas for the coef-ficients and still convey the main dependence of the aber-rations with respect to the parameters of a system. From apractical point of view, it is not critical to know the exactform of the coefficients because of the presence of higherorder aberrations and because a ray-tracing program can cal-culate with much better precision the wavefront deformation.

To illustrate the nature of the approximations involved,consider the following derivation of the J1 coefficient. Bytaking the difference ofthe sagittal and tangential Coddingtonformulas an exact expression that relates longitudinal astig-matism can be written

, I' i 1 \ I' 1 1 \ sin2(I) n' sin2(I')nt———I—nI——--J= —______

\s' t'J \s t,/ twhere s, s', t, and t' are the object and image distances in thesagittal and tangentialplanes measured along the OAR. If thereis no astigmatism before refraction, then s =t, and by approx-imating the sagittal and tangential distances s' t', we canwrite

n'(s'—t')=s'2 n2 sin2(l)(_\ns

To convert to a quadratic wavefront deformation, we multiplyboth members by u'2/2 to obtain

n'u'2 1J1=—(s'—t')= ——n2 sin2(I)M — )x

2 2

To keep this derivation simple, we also have used the sagittalparaxial ray height x ; using the tangential paraxial ray heightwould have been more appropriate because, by definition,the aberration coefficient gives the maximum wavefront de-formation and this occurs in the tangential plane. The ap-proximation that is made by replacing paraxial quantities inthe tangential plane by their correspondent ones in the sagittalplane becomes more accurate as the surface tilt angles becomesmall.

In the aberration coefficients presented, we have not ac-counted for the effects that arise as the wavefront deformswhen it traverses in free space. The change in deformationis given in Cartesian coordinates by38

2 2L I I iW(x,y) I I W(x,y)

W'(x,y) W(x,y) —4 L j + L ] , (71)

where W(x,y) is the starting wavefront and W'(x,y) is thewavefront after it has traversed a distance L.




1=J

WI2010= —

Jiiif Field tilt

I 10 0 Quadratic distortion II , (80)

w30010 = f—( 111T Cubic pistoni=I 2\x/ )

It is appropriate to note that the medial surface of the astig-matic field is perpendicular to the OAR and that linear astig-matism and field tilt vanish simultaneously. If the tilt of onemirror is left to correct these aberrations, then, and exceptfor the remaining quadratic distortion I and the piston term,the system behaves within the third group of aberrations asan axially symmetric system.

As an example of the use of these results consider a Cas-segrain telescope with specifications given in Table 3; it isan f/24 system with a 2400-mm aperture. The design taskwas to modify the telescope to make it unobstructed andsuppress the image plane tilt about the ray incident on thephysical center of the primary mirror and that becomes theOAR. To this end, the entrance pupil was decentered1800 mm and then the secondary minor was pivoted at thecommon foci of the mirrors by 0. 18 deg. This pivoting ma-neuver preserves the condition of stigmatic imaging alongthe OAR. The amount of rotation of the secondary is suchthat linear astigmatism vanish and therefore field tilt. Thissystem does not have an axis of rotational symmetry and isshown in Fig. 11.

In Fig. 12, spot diagrams are shown for the axially sym-metric [Fig. 12(a)] and unobstructed [Fig. 12(b)] telescopes;the spots were generated at the best combined focus for fivepositions around a semifield of view of 0.25 deg. In Fig. 13,a field display of the tangential astigmatic foci for the mod-ified system is presented; this was generated with CODEV(Optical Research Associates, Pasadena, California). Thespot diagrams for the modified system show that image qual-ity has not been degraded and show a distribution that issimilar to the one exhibited by the axially symmetric system.The field display of the tangential astigmatic foci also verifies

Table 3 Cassegrain specifications.

Mirror Radius Spacing Conic constant

Primary -11040mm -4906.6211 -1.0

Secondary -1358 mm 6298.4412 -1.474

Fig. 12 Spot diagrams for the (a) axially symmetric and (b) unob-structed Cassegrain telescopes. Despite the large offset of the mir-rors in the unobstructed system, the spot diagrams are very similar.Because by construction the center of the field is well corrected, thespots indicate that both systems are dominated by linear coma.

the symmetric behavior, which is essentially free of imageplane tilt and linear astigmatism with respect to the OAR.This supports the theoretical prediction and illustrates thevalue of the theory in providing useful insight.

6 Discussion and SummaryTomake the imaging behavior of a plane symmetrical systemunderstandable we have established the OAR as a reference,the wave aberration function for describing the imagery, andthe associated system as a calculation aid. Furthermore, anapproximate set of aberration coefficients has been devel-oped. The main imaging differences of a bilateral symmetricsystem in comparison to an axially symmetric system are theanamorphic magnification, the field tilt, and the aberrationdistribution across the field of view. We have divided thethird group of aberrations into subgroups that are character-ized by plane, double-plane, and axial symmetry. This clas-sification suggests that the aberration correction can be car-ned by correcting each subgroup at a time and by usingvariables according to the subgroup symmetry. The form ofthe coefficients indicate the main dependence of the aber-rations with respect to the system parameters and indicatehow to correct a given aberration. The aberration terms andcoefficients presented are suitable for systems made withspherical surfaces that may be slightly aspheric. As a functionof the ratio Jx the behavior of aberrations with plane anddouble-plane symmetry is reminiscent of the stop shift equa-tions for axially symmetric systems. For the latter systems,a condition for the absence of keystone distortion has beenestablished; this condition is similar to the Scheimpflug con-dition. The contribution from slightly aspheric surfaces andfrom optical dispersion have been accounted as well.

OPTICAL ENGINEERING I June 1 994 I Vol. 33 No. 6 I 2057

(79)

(81)

Field Position

H H0.00, -0.25 deg.

0.18, -0.18 deg.

0.25, 0.00 deg.

0.18, 0.l8deg.

0.00, 0.25 deg.

(a) (b)

Fig. 1 1 Unobstructed Cassegrain telescope derived from an axiallysymmetric one by displacing the entrance pupil and pivoting the sec-ondary about the common mirror foci.


n'nn' cos(I')—n cos(I)


s H(82) sin(w)= sin(I)

(s2 + H2)V2+ cos(I)

(s2 +(87)

H'

normal

surface

S

OAR H

LOU -///-N\\//// _ - _ NN\\////v-N\\\///// - -//ii'' - -

I Ii 1 1 ' - • - ' \ \

I I I / • • S \ \ \

\I I S I

\ \ \ \ S S 5 / / II\\\\ \ ' S S S / ////\\\ N _ - / ///\\N —_

-1.00

—1.00 1.00

Fig. 13 Display of the tangential line foci over a half degree field forthe unobstructed Cassegrain. The symmetry of the tangential focireveals that the telescope suffers quadratic astigmatism and no lin-ear astigmatism or field tilt.

Forsmall surface tilts the aberrations characteristic of axialsymmetry are uncoupled as a function of surface tilts fromthe aberrations characteristic of plane and double-plane sym-metry. This suggests that a good starting point to design aplane symmetric system is a well-corrected axially symmetricsystem. Then, by tilting the surfaces on a plane one generatesa plane symmetric form. Such a system will mainly need thecorrection of the aberrations with plane and double-planesymmetry and a slight reoptimization for the Seidel aberra-tions. These aberrations can be corrected by tilting the sur-faces and by using plane or double-plane symmetric as-phericities. Another way to design a plane symmetric systemis by starting with a system of confocal surfaces such thatthe imagery along the OAR is stigmatic. Design examplesthat illustrate how the concepts discussed in this paper havebeen applied in the design of plane symmetrical systems arepresented in Refs. 28 to 33.

With the insights generated from the theoretical devel-opment presented, the main imaging characteristics of bilat-eral symmetric systems have been established in terms fa-miliar to the optical engineer. The tools presented permit thedesigner to define a plane symmetric system, find the imageposition and size, and determine the main imaging defects.Furthermore, the aberration coefficients indicate how the im-agery behaves as a function of system parameters and suggesthow to approach the design of a plane symmetric system.

7 Appendix A

7.1 Derivation of J,With reference to Fig. 14, where the stop is located at thesurface, the sagittal Coddington equation is applied. This is

SASIAN

Fig. 14 Geometrical construction for deriving J and

The object distance s is kept constant (the object plane tilt iszero) and the change of the image distance s' is evaluatedwith respect to changes in the chief ray angle of incidence.This is

!5i:,=n' sin(I')SI'—n sin(I)I(83)s'2 R

Using the relations tan(O') =(s'/s'I'), n' cos(I')I' =n cos(I)I, and u' = — (xis'), the previous expression takesthe form

n' cos(I')—n cos(I)U' tan(O')=n tan(I) x . (84)nfl R

The tangent of the angle of tilt is approximately

n sin(I) /1\x—tan(O)= LI—J---7 . (85)

R \n/u

By multiplying by the image height H' to obtain the longi-tudinal defocus and then by n'u'2/2 to convert to a quadraticwavefront deformation, and using the identity 'I' = — n'u'H',we obtain J1:

n'u'2 1 n sin(I) 7 i\

JIv= —----i—H' tan(O')= — R I1L_)x. (86)

7.2 Derivation of J

Because there is no contribution to quadratic distortion II,when the stop is located at the surface we can derive J byexploring how the image height H' varies as a function ofthe object height H when the field vector lies on the planeof symmetry. With reference to Fig. 14, the angle of incidenceof the chief ray is

s' S R



or to second order on H, this can be rewritten as

H2\ Hsin(w)

sin(I)(1 —)+cos(I) (88)

Applying Snell's law, we can write n sin(co) = n' sin(w'), or

H2 Hn sin(I)—n sin(I)—j+n cos(I)—=2s s

H'2 H'n' sin(I')—n' sin(I')—j+n' cos(I')—2s s (89)

This relationship can be rearranged to express the differencebetween the image height H' and the object height multipliedby the paraxial magnification ns' cos(I)/n's cos(I') in theplane of symmetry:

ns' cos(I) s' n sin(I)/'H2 H2\H'— H= (———I (90)n's cos(J') n' cos(I') 2 \s'2 s2 J

where we have eliminated the terms n sin(J) and n' sin(I').By replacing in the right-hand member the image height bythe object height times the magnification, we obtain

ns' cos(I) s' n2H2 cos(I)2H'—

n's cos(I') n' cos(I') s2

n sin(I) I 1x M2 L2 cos(I)2

This difference represents quadratic distortion I expressed asa transverse quantity. To express the result as a linear wave-front deformation we multiply both members by —n'x/s',use the identity 'I' = nxH/s, and to further simplify we neglectthe cosine factors:

J — FH' flS' cos(I)Hl _n sin(I)2(11V [ n's cos(I') j 2 'sfl2)X

7.3 Derivation of T,,

To derive T11, it is sufficient to consider the case of an untiltedsurface. With reference to Fig. 15, the central projection Hof the field vector on the object plane and back on the planeperpendicular to the OAR is given by

H =H-KH21+KH (93)

where K= —tan(O)/s. The term —KH2 is the contributionfrom the tilt of the object plane to T11 written in transverseform. To compute the complete contribution we subtract asimilar term —K'H'2 due to the tilt of the image plane andmultiply by —nx/s and — nx/s to obtain the result as awavefront deformation:

T = flXK'H'2 — KH2 = 'V[uH tan(O)J!

Fig. 15 Geometrical construction for deriving T.

8 Appendix BIn this section, we discuss how cylindrical and comatic as-phericities can be generated on a spherical substrate. Previ-ously,42 polynomial asphericities have been discussed in re-

91 lation to the optimum design of nonaxially symmetricsystems. Nonrotationally symmetric surfaces can be madeby cutting off-axis sections of axially symmetric parent sur-faces, or by the stressed polishing method.43 These two ap-proaches are not suitable in the small optics shop where eitherit is not cost effective to produce a large parent mirror or theequipment to implement stress polishing is not available. Forthe occasional need of a nonaxially symmetric mirror it isdesirable to count on a technique that the master optician canimplement with tools available in the optics shop.

The idea44 we use to generate a nonaxially symmetricsurface is to conceptually decompose the desired surface in

92" elementary surface shapes. Then these shapes are generatedk I on the substrate one at a time and in such a way that their

superposition results in the overall surface asphericity. Threeelementary aspheric shapes of interest are the cylindrical, linecoma, and comatic. The sag Z of these shapes is describedin analytical terms by

Zcylinder = (i 1)2

Zcomatic =(i )(p . )2'7 —1'. \3

line coma

(95)

(96)

(97)

The cylindrical asphericity possesses two perpendicularplanes of symmetry and this makes its generation simple.The generation of a cylindrical asphericity on a sphericalsubstrate produces a surface of double curvature and severalmethods are known for this purpose.45 The fundamental prin-ciple to generate a surface of double curvature is that therelative movements between the surface and its grinding or

(94) polishing tools must be only translational; except for a ro-tation of 1 80 deg, which preserves symmetry. no rotational

OPTICAL ENGINEERING I June 1 994 I Vol. 33 No. 6 I 2059

0

object plane



toolmovements are allowed. The symmetry of the polishingmovements must be in accord with the symmetry of the sur-face to be generated. For relatively weak double-curvaturesurfaces, a double four-bar linkage,46 as illustrated in Fig. 16,can be used to constrain rotational grinding or polishingmovements. Details on how such a fixture has been used toproduce some precision double-curvature surfaces are givenin Refs. 28 and 31.

The generation of the comatic asphericity is of interestbecause it gives control over the constant coma of an opticalsystem. Let us consider first the generation of line-coma as-phericity. The sag Z of this asphericity in Cartesian coordi-nates is described by Z= Y3. The cross section along the Yaxis is a cubic line and along the X axis remains constant.For generating this shape, a narrow rectangular tool can bedirected with polishing movements increasing cubicly ontime. The movements with increasing time are made per-pendicularly to the longest side of the tool, as illustrated inFig. 17. For generating the comatic asphericity two line-comaasphericities are generated separated angularly by 60 deg.This, in polar coordinates, is expressed as

3\/p3[cos3(O —a) + cos3(O + a)] =

—a--—cos(O) , (98)

for a =30 deg. This technique has been used for generatinga weak comatic asphericity on a double-curvature surfaceand some fabrication details can be found in Ref. 28.

9 Appendix CForcompleteness, but without offering a derivation, we giveapproximate contributions from a spherical surface to the firstfour aberrations in Table 2.

1 /u2\W03003 = — n3

sin3(I)M-)x(99)

xW12012=3—W03003 , (100)x

Fixed link

pivot

SASIAN

polishingstroke

Polishingtime

tool position

Fig. 17 Polishing action of a rectangular tool for generating a line-coma asphericity.

W21021=3()W03003 , (101)

3

w30030= () w03003 . (102)

AcknowledgmentI would like to thank Kevin P. Thompson and the reviewersfor their valuable and helpful comments and suggestions,which led to improving the paper. The major part ofthis workwas done when the author was with the University of Arizona,Optical Sciences Center.

DedicationThis paper is dedicated to my friend Paco Cobos who intro-duced me to the art of optical design.

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KN

>>frn/tool link

Fig. 16 Four-bar linkage to constrain rotational grinding or polishingmovements and maintain a double curvature. Each bar can rotateabout its pivots; one bar is fixed in position and other is used to holdthe polishing tool.



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Jose M. Sasian received the BS degreefrom the Universidad Nacional Autonomade Mexico in 1982 and the MS and PhDdegrees from the University of Arizona in1987 and 1988, respectively. He was in-volved in optical fabrication and testingfrom 1 975 to 1 984 at the Institute of As-tronomy at the University of Mexico andfrom 1 985 to 1 988 at the Optical SciencesCenter at the University of Arizona. Sasianhas worked on the design, fabrication, and

testing of instruments for astronomical research. In 1990 he joinedthe Photonic Switching Technologies group at AT&T Bell Laborato-ries in Naperville, Illinois, where he is developing optical and opto-mechanical systems for photonic switching. His research interestsare in the areas of photonic switching, optical instrumentation in-cluding optical and mechanical design, and light propagation.



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