1142JACOB BECK V.

accommodation. These results suggest that accommo-dative astigmatism may be the basis for the pooreracuity that has been found for diagonal lines whencompared to vertical and horizontal lines.2 This followsfrom the fact that points on a line parallel to a givenfocal line converge to that line. Thus, if the astigmaticaxes in the dynamic eye tend to be vertical and hori-zontal, the eye can focus for both vertical and horizontallines; diagonal lines, however can never be broughtinto sharp focus. An eye with regular astigmatism, infact, may exhibit marked irregularities in its ability toresolve a grating target at various orientations owingto the overlapping blurred images.A

J. R. Hamblin and T. H. Winser, Ref. 2.

While the present findings indicate that it is probablyrisky to decide that the effect of orientation on patternacuity can not be accounted for by dioptric factorsunless accommodative astigmatism is controlled, theexperiments do not specify the physiological changeswhich produce it. The cornea, the lens, and alignmentof the optical system in the eye all contribute to thetotal astigmatism. Because the astigmatism due to thelens may either intensify or compensate for the astigma-tism due to the cornea and alignment of the opticalsystem, an increase in astigmatism with accommoda-tion is compatible with either an increase or a decreasein lenticular astigmatism. Moreover, a change in theamount of lenticular astigmatism during accommoda-tion may itself be the result of a number of factors.'

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 55, NUMBER 9 SEPTEMBER 1965

Transfer Function of an Optical System in the Presence of Off-Axis AberrationsRICHARD BARAKAT AND AGNES HouSTON

Itek Corporation, Lexington, Mlassachusetts 02173(Received 9 February 1965)

The transfer function of an optical system in the presence of off-axis aberrations is evaluated via Gaussquadrature. The transfer function in the presence of third-order coma is studied and the phase of the transferfunction is shown to be a smooth function. The variance of the wavefront is obtained for all third- and fifth-order aberrations and its use is illustrated for an actual system along with the transfer function.

1. INTRODUCTION

ONE of the main purposes of this paper is to presenta numerical method of evaluating the transfer

function in the presence of off-axis aberrations whichemploys Gauss quadrature and hence can be consideredas a continuation of previous work,1' 2 which was con-fined to on-axis (rotationally symmetric) aberrations.If we are using the transfer function solely for lens-design purposes, then approximate methods such as thatof Hopkins 3 are sufficient since only a few percentaccuracy is generally required. However, the transferfunction is also employed as an auxiliary function forthe determination of the images of incoherently illumi-nated extended objects by the appropriate integrationof the product of the object spectrum and the transferfunction. Generally, the integrand is rapidly oscillatingowing to the object spectrum and thus the transferfunction must be known to great accuracy to obtaineven moderate accuracy in the image function. Thesepoints have been discussed in our previous work4-7

as well as in Refs. 1 and 2.

lR. Barakat, J. Opt. Soc. Am. 52, 985 (1962).2 R. Barakat and 1M. V. Morello, J. Opt. Soc. Am. 52, 992 (1962).3 H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).4R. Barakat and A. Houston, J. Opt. Soc. Am. 53, 1244 (1963).

R. Barakat and A. Houston, J. Opt. Soc. Am. 54, 768 (1964).R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 538, (1965).

7R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 881 (1965).

We summarize in Sec. 2 the basic formulas, however,employing the Luneburg formulation of Kirchhoff'sdiffraction integral.8 This procedure has the advantageof directly introducing the aberration function as theHamilton mixed characteristic which is an exact ex-pression for the optical path difference 9 in contrast tothe approximate procedure initiated by Nijboer10 andfollowed by others.'1 The numerical integration schemeis outlined in Sec. 3. Section 4 is devoted to some prob-lems involving third-order coma. The variance of thewavefront is obtained in Sec. 5 and its use is illustratedalong with representative transfer functions for anactual optical system in Sec. 6.

2. FORMULATION OF THE PROBLEM

According to Luneburg's 5 modified Kirchhoff dif-fraction theory, the complex diffracted amplitude

8 R. K. Luneburg, Alathematical Theory of Optics (University ofCalifornia Press, Berkeley, 1964) Chap. 6. This is a correctedreprint of Luenburg's original notes published by Brown Univer-sity Press, 1944.

The senior author wishes to thank M. Herzberger for astimulating series of conversations in 1960 on the implications ofLuneburg's work with regard to the mixed characteristic and theusual treatment of wavefront aberrations.

1' B. R. A. Nijboer, "The diffraction theory of aberrations"doctoral thesis, Groningen, 1942.

11 H. H. Hopkins, Wavs Theory of Aberrations (Oxford UniversityPress, London, 1950).

1 142 Vol. 55'

September1965 TRANSFER FUNCTION WITH OFF-AXIS ABERRATIONS 1143

a(xy) at a point (xy) in the Fraunhofer image planedue to a point source located at (xo,y0 ,z0) in the objectspace is given by

a(xy) = f (p,q)eikV(xou yor q)eik(Px+qy)dpdq. (2.1)

aperture

Here p,q are the optical direction cosines of the normalsto the converging wavefront and

p

q (p,q) = A o(p,q)/ (n-p2 - q')l, (2.2)

where Ao(p,q) is the amplitude distribution over thewavefront. In the special, but important, case of lownumerical-aperture objectives, it can be shown that' 2

0(p,q)'-Ao(pq). (2.3)

In particular, Ao (p,q)const characterizes the so-calledAiry objective. Only Airy objectives are considered inthis paper, because they are excellent approximations tothe usual types of low numerical-aperture systems thatoccur in practice.

The Hamilton mixed characteristic W(xo,yo,zo,p,q)represents the optical length of the light ray from itsorigin (xo,yozo) to the foot Q of the perpendiculardropped from 0 upon the tangent of the light ray atP.1 13 The mixed characteristic W is a function of theobject plane coordinates x0, yo, zo and the exit pupilcoordinates p,q. Only p,q enter because the thirddirection cosine is determined from p,q by the direction-cosine law. The intersection of the wavefront with theparaxial image plane (z= 0) is given by

FIG. 1. Region of integration (shaded area) in the transfer-functionintegral for a circular aperture.

variables. The term WO) is a constant which may beset equal to zero. The term W(2) can be shown to havethe form

W'= PXo2+ W020(p2+q2)+M(xop+yoq), (2.7)

where W020 is the defocusing parameter, M is themagnification ratio, and P is a pupil aberration. Thepupil aberration does not affect the diffraction imageor the transfer function. It is constant (at fixed xo),and is therefore omitted. The terms in W(4) are calledthe third-order aberrations and those in W(6W arefifth-order aberrations. If we omit the pupil aberrations,then the necessary aberrations are listed below alongwith their names. Incidently, the names of the aber-rations probably differ somewhat from current usageamong lens designers. The subscript numbering of theaberration coefficients follows that of Hopkins."The third-order aberrations are:

x=-aW/op; y=-aW/aq. (2.4)

Following Hamilton, let us develop W into a series ofthe form:

- W= 4)(+ W6) + .. (2.5)where W(V) is homogeneous and of degree n in(xoyop,q). Because the system is rotationally sym-metric, the appropriate combination of the independentvariables isl4,15

U1= xo'+yo 2

U2= p2+±q2, (2.6)

"3U XoP+yoq.

However, no loss of generality results from settingyo= 0, so that XO2, p2+q2, xop are the independent

12 A. H. Bennett, J. Jupnik, H. Osterberg, and 0. W. Richards,Phase Microscopy, Principles and Applications (John Wiley &Sons, Inc., New York, 1951), p. 241.

13 See Ref. 8, p. 103.14 See Ref. 8, p. 270, as well as M. Herzberger, Modern Geo-

metrical Optics (Interscience Publishers, Inc., New York, 1958)Chaps. 26, 27.

15 The generalization to nonrotationally symmetric wavefrontshas been treated by us and preliminary results communicated atthe 1964 Fall Meeting of the Optical Society of America, J.Opt. Soc. Am. 54, 1407A (1964). We hope to publish the finalresults shortly.

C31nxo3p= W311P

C040 (P'+ q')'= W040 (p2+q2)2

C131Xop(p'+q2)= W1 31p (p2+q2)

C222XO2p'= W222 p2

C 22oX2 (p2+q2) = W22 o(p2+q2)

The fifth-order aberrations are:

C42o0Xo4(p+±q2) = W42o(p2+q2)

C2406 2 (p2+ q2)' = W240 (p2+q2)2

C060(p1+q2)3= W060 (p2+ q2) 3

C51xo05p= W 511p

C33,X03p (p2+q2) = W311p (p2+ q2)C1s5xop (p2+q2)2== W1,p (p2+q2)2

C4 2 2XOp2= C4 2 2p2

(primary distortion),

(primary sphericalaberration),

(primary coma),

(primaryastigmatism),

(primary fieldcurvature).

(secondary fieldcurvature),

(secondary sphericalaberration),

(primary sphericalaberration),

(secondarydistortion),

(secondary coma),

(primary coma),

(primaryastigmatism),

1 143

1 RICHARD BARAKAT AND AGNES HOUSTO N

C24 2xO2p2 (p 2 + q2) =W 2. 2p2 (p 2 + q2) (secondaryastigmatism),

C3 33 xo3 p3= W 333p3 (trefoil).

We also include the seventh-order spherical aberration

C080 (p 2 + q2 )4 = W08os(p2+q2 )4 (special aberration),

since it can be obtained easily from Buchdahl's work.'Some of the fifth-order aberrations differ by only a

constant coefficient (at fixed xo) from some of the

third-order aberrations; when we collect terms themixed characteristic collapses to

-W = (Wl3l+ WVs1)q+ (W 22 2+ W 422)q2+ W333q3

+ (W 220+ Wv420+ W020) (p2 + q2) + (W040+ 14W2l0)X (p2 +q2 )2 + W060(p

2 +q 2) 3+ (W 131+W331)

Xq(p 2+q 2 )+W,5,q(p 2 +q2)2+ W242q2(p2+q2)

+ Woso (p 2+ q 2)4 . (2.8)

It is now convenient to employ slightly differentvariables in the diffraction integral (2.1). The terms ofV(2) are combined with the linear terms in the other

exponential

k (x-Mxo)p+k (y-Myo)q= vpp+vaq, (2.9)

so that

a(vpvq)= f eikIV (roopq)ei(vpP+vQq)dpdq. (2.10)

If Pmax, qnax are the values of p, q at the edge of theaperture, we define dimensionless coordinates

p = p/p1ax q' = q/qnax,

V,= =VpPinax, Vq =vVqqmax-.

Then (2.10) becomes

alv,',v,')= f eikWV(xo,yo,p',q')

aperture

Of course a(v,',v,1 ) is not an observable, but its absolutesquare, the point spread function, is. We define thepoint spread function t(vp',vq') by

1(v,,vq') = I a(vp',vq')/a(0,0)Ai 12, (2.13)

where a(O,0)Ai is the disturbance due to an aberration-

16 There are a number of ways of obtaining WV for an actualoptical system. One common method is to trace rays and thencurve fit via least squares: E. Marchand and R. Phillips, Appl.Opt. 2, 359 (1963); W. Brouwer, E. L. O'Neill, and A. Walther,Appl. Opt, 2, 1239 (1963). Another procedure developed by us andas yet unpublished utilizes certain interpolating properties of theZernike polynomials. The calculations in this paper for an actualsystemii (see Sec. 6) employed the Buchdahl aberration coefficients.Our lens-design computing facilities directly calculate all thethird- and fifth-order aberrations as well as seventh-order sphericalaberration as a matter of routine using an expanded version ofBuchdahl's work developed by M. Rimmer of our design depart-ment. It was a simple matter to adapt these coefficients to ourrequirements.

(2.11)

free Airy objective. Note that 0 t(vp',vq') 1. Wenow omit the primes on the dimensionless variables,keeping in mind (2.11).

The transfer function T(wp,w,,) in incoherent light isdefined as the two-dimensional Fourier transform of thepoint spread function t(vp,v,), or equivalently as theconvolution of the pupil function over the exit pupil;thus

T(wp,wq)= [T(0,0)f-1f A (p+4wp,q+Y+ q)

XA*(p-cpq-1wq)dpdq, (2.14)

where the region of integration is confined to the area of

overlap of two unit circles centered at -4-wp,4-co.q,

respectively (see Fig. 1). Here,

A =0 p 2 +q2> 1,

A ikq IV ess(p, q) p2+ q2< 1.(2.15)

It is convenient, following Hopkins,3 to reduce the prob-lem to a single-frequency variable by the transformation

p= a cos9- 3 sino,

q=a sin0+0 cos4. (2.16)Thus,

T(w,o)= [T(0,0)]-Thf A (a+2ci,0)

XA*(ae-1w,,O)daedf, (2.17)

where A (a,/3) is the pupil function expressed withreference to the new set of spatial-frequency coordinatesco,q given by

W= (wP 2+ cq 2)1i

q = arctan(w,/1w). (2.18)

The spatial frequency co is limited to the interval (0,2),because when co> 2, the convolution integral vanishes.

(2.12) The dimensionless spatial frequency co is related to thephysical parameters of the system by the equation

Q =co/2XF, (2.19)

where Q is the dimensional spatial frequency expressedin lines/mm, X is the wavelength of light in mm, and Fis the f/number of the system. The physical cutoff ofthe system occurs at Q= 1/XF.

The transfer function is thus given by

T (wo) = Tr ((),) + iTi (u,,) , (2.20)where

ra rb

Tr(io¢) = [T(0,0)]-1f f cos[27rV(a+1o,,0)-a -b

-27rW (a- w,3)]dad0,

Ti (,)= E[T(0,0)]-if j sin[27rW(a+±w,0)-b -b

-27rW(ae-1w,,)]dadO,

Vol. 551144

September1965 TRANSFER FUNCTION WITH OFF-AXIS ABERRATIONS 1145

and the limits of integration a, b are

a= (1 - 42)

b= (t- 2911

M0

0.8

I-

0

C.)

(see Fig. 1). The problem thus reduces to evaluatingthese two integrals as functions of the aberrations,spatial frequency, and azimuth. When only the rota-tionally symmetric aberrations are present, then it canbe shown that the imaginary part Ti(co,o) vanishesand Tr (wqo) = Tr (co).

3. COMPUTATION OF TRANSFER FUNCTIONUSING GAUSS QUADRATURE

In this section we outline the numerical integrationscheme in enough detail so that anyone wishing toemploy the method can easily fill in the details andadapt the technique to available computers. One verydesirable feature of the method is that when convolvingthe integral the same number of quadrature points arealways available. This is not the case with the Hop-kins method with which, in order to maintain uniformaccuracy, successively more quadrature points arerequired as the convolution progresses. We haveemployed 20-point Gauss quadrature; this is sufficientto yield six-decimal accuracy for the vast majorityof situations. Again we point out that this accuracy isnot required in lens design, but is definitely necessarywhen evaluating the images of extended objects. 4-7

Consider the integration over the inner integrals

Ib sin b

I [ ]da= f F(c,0,c,0)da,Jb COS Jb

(3.1)

and change the interval of integration (-b,b) to thestandard Gauss-quadrature interval (-1,1) by theformula

b

f F (wsbaf)da = bj F(w,sb,s,3)ds, (3.2)-b -1

where a= bs. Employing N-point Gauss quadrature toevaluate (3.2) yields

Nb E HnF(wckanf3). (3.3)

n-1

Here

H.= N-point Gauss weight factorsa,,= bsn= N-point Gauss-quadrature points.

Both Hn, s,, are extensively tabulated. 7", 8 Omittingthe constant factor [T(0,0)}-1, we find the double

17 Z. Kopal, Numerical Analysis (John Wiley & Sons, Inc.,New York, 1955).

18 V. I. Krylov, Approximate Calculation of Integrals (TheMacmillan Co., New York, 1962).

'3a:as

en .2z'cr

"-C0 .4 .8 I 1.2 1.6

DIMENSIONLESS SPATIAL FREQUENCY2.0

FIG. 2. Modulus of transfer function of circular aperture in thepresence of third-order coma W1 31 =1.0OX in the paraxial planeW020=0; -*- - c=0 , - - -=45°, -- q =90°.

integrals

ra 20

I 2 E H.F(.,qOa.,O)d3.-a n=1

(3.4)

Repeat the same procedure with respect to the ,variable; call the integrand of (34), G(co,o,#); then

Ga

XG(w,O,O)dO=-a-a

fJ G(co,O,l)dt,

where t3= at; consequently, we have

N(1 -42)i E HmG(cw,9#m).

m=-

(3.5)

(3.6)

The entire procedure for N= 20 was programmed on amoderate-size computer. The output data consistedof Ti(wo,4), Tr(@ f), |T(w,,0)|, and 6(cw,)=phase ofT(w,4). Any combination of W coefficients can beinserted and the transfer function can be evaluated atarbitrary values of both w are q5 if so desired.

4. TRANSFER FUNCTION FOR THIRD-ORDER COMA

One of the most intriguing problems has been thebehavior of the phase of the transfer function in thepresence of third-order coma. The first calculationswere performed by Marathay' 9 using Hopkins2 0 geo-metrical-optics treatment of the transfer function andby De and Nath,2 ' who employed series expansions.In particular, the phase curves of De and Nath showabrupt changes contrary to what would be expectedfrom the one-dimensional problem.22 The validity oftheir results has been questioned by many workers,23

and for this reason we present a few results from ourcalculations. The modulus of the transfer function

19 A. S. Marathay, Proc. Phys. Soc. (London) 74, 721 (1959).20 H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).21 M. De and B. K. Nath, Optik 15, 739 (1958).22 E. L. O'Neill, Introduction to Statistical Optics (Addison-

Wesley Publishing Co., Inc., Reading, Massachusetts, 1963).23 The senior author wishes to thank E. L. O'Neill for a series of

discussions concerning De and Nath's work shortly after itspublication. It is largely due to O'Neill that this problem hascome to be re-examined by us and independently by DeVelis.DeVelis has kindly informed us (private communication) that hehas located the error in their calculations.

mz Io -

1 145

en .

-1 - - I I - I I I I I I I .

RICHARD BARAI(AT AND AGNES HOUSTON

lDKJ0

z0P .6

La

aL .2

za- :

a .4 .8 1.2 1.6

DIMENSIONLESS SPATIAL FREQUENCY2.0

FIG. 3. Modulus of transfer function of circular aperture in thepresence of third-order coma fV, 31=2.OX in the paraxial planeW020=0: - * - 0°=0, - - - 0=45°, f =90°.

for W131= 1A, 2X in the paraxial plane 1020=0 areshown in Figs. 2 and 3. Little need be said of thesecurves except to point out the fact that when W131= IXthe modulus shows a smooth change as 0 varies, sothat A= 45° lies between A= 0° and = 90°. This is nottrue for W131= 2X. The phase curves are illustrated inFigs. 4 and 5, both with the linear phase included(curves marked A) and subtracted out (curves markedB). When j=0°, the phase shift vanishes. One of thesurprising features of these curves is their sensitivityto the azimuth q for fixed W131.

5. VARIANCE OF THE WAVEFRONT

The idea of a single merit function to characterizethe performance of an optical system has now beenlargely abandoned as unrealistic. Nevertheless, infairly well corrected systems it is useful to employ thevariance of the wavefront Eo as a design parameter.Mar6chal originally developed an aberration-balancingtheory which seeks to minimize Eo and hence maximizethe Strehl criterion SC (maximum illuminance indiffraction image). When the condition SC) 0.8 isimposed, Marechal has shown2 4

SC>, [1-1k 2 EJ]2 . (5.1)

- 'K- i'i---

1 r27r 1Eo=- [WV(p,0)]2pdpd0

J 2[1 f W(PO)pdpd ],(5.2)

where p=p sinO, q=p cosO. In order to effect theintegrations, we have to change the aberration functionto polar coordinates p, 0 and then integrate; a typicalterm is

Wi5up (p 2+ q2 ) = W1 5ip1 COS6. (5.3)

12

a,za

<r6

4

M 2(1

0 4 .8 1.2 1.6DIMENSIONLESS SPATIAL FREQUENCY

2.0

FIG. 4. Phase of transfer function of circular aperture in thepresence of third-order coma TV,31 =1.OX in the paraxial planeW02o=0. 'The phase vanishes idenLically in the azimiuLh 4=0o.The curves labeled A are the computed curves: - - -=45-- ==90°. The curves labeled B are the curves A with alinear phase shift subtracted.

24 A. Marechal, Rev. Opt. 26, 257 (1947).

Every term in W becomes the product of terms in pand 0 so that the integrations are simple. In view of thelarge amount of analysis required, only the final resultis quoted in (5.4).

At this point we can, following Mar6chal, minimizethe variance Eo by appropriate choices of the selectedaberration coefficients. Since Eo is a quadratic form, itcan always be minimized by using Lagrange's procedure.Unfortunately, this procedure is not generally possiblein practice, for two reasons. The first is that even whenwe solve the mathematical problem and obtain thedesired functional relations between the aberrationcoefficients, it is rarely possible to achieve these relationsexactly in actual lens design. This is simply because allthe lower-order aberration coefficients are not com-pletely controllable. Therefore, the practical balancingproblem reduces to achieving the best partial balancing,in the sense that Eo is made as small as possible by use of

FIG. 5. Phase of6 ,, transfer function of

circular aperture in,' /the presence of third-

'n2 -/ order coma W131=/A 2.OX in the paraxial

plane W0 20=0. Thephase vanishes iden-

Z ,tically in the azimuthW 0/ ,= 00 . The curves

labeled A are thea. 4 -/ computed curves:

--- / - - -=45°,-- -/ p=90'. The curves

C /, labeled B are theA /curves A with a

-" - linear phase shift-4 - subtracted.

0 4 .8 1.2 1.6 2.0

DIMENSIONLESS SPATIAL FREQUENCY

Thus for small aberrations a minimization of Eo ob-viously maximizes the Strehl criterion. We, however,regard Eo as the primary quantity and work directlywith it.

The variance Eo of the wavefront is given for acircular aperture by

A a .. . .... ....

1 146 Vol. 55

September1965 TRANSFER FUNCTION WITH OFF-AXIS ABERRATIONS 1147

whatever control is available. Secondly, the aberrationcoefficients Wjmr, are functions of xo. This means thateven when the mathematical problem has been solved,it may not be possible to apply the balancing relationsat a given field-point xo.

One possibility is to abandon any attempt to mini-mize Eo and just consider it as a useful criterion ofimage quality. Since E0 is positive we can also take intoaccount the entire field by averaging Eo over the fieldand thereby define a new merit function M

Eo= 1/4(W3 11 + W 511) 2+ 1/16(W 222 + W42 2)2+5/64(W333)2+ 1/12 (W020 + W220+ W 42 0 )2+4/45 (WO40 + W240)2

+9/112(Wo 60)2 + 16/225 (W080)2+ 1/8(Wl3l+W331)2+ 1/12(Wl5l) 2 + 17/360(W 242 )2+ 1/4W33 3(W311 +W5ll)

+-1/3 (W3 11 + W511 ) (Wl3 l + W 331)+ 1/4WT 51 (W31 1 + W511)+ 1/12 (W 222 + W42 2) (W 0 20 +W 2 2 0 + W4 20)

+ 1/12 (W 22 2 + W4 2 2) (W0 4 0 + W 2 40 )+3/40Wo 6 0 (W 2 22 + W4 22 )+ 1/151WV800 (W 22 2 + W 4 22 )+5/48W 2 4 2 (W 22 2 + W42 2)

+3/16W3 3 3 (W5 31 + W 3 3 1 )+3/20W 3 3 3WI 51 + 1/6 (W0 20 + W 22 0 + W 4 20) (W 0 4 0 + W 24 0)

+3/20WO6 0(Wo 2 o+ W 220 + W 420 )+ 2/15WoVo (W 020+ W 220+ W 4 20)+ 1/12W 242(Wo 2 0+ W 220+ W420)

+ 1/6Wo60 (W 040+ W240)+ 16/105Wo 80 (W 040+ W 240)+4/45W 242(W 040+ W 240)+3/20W 0 60 W080

+ 1/12W 0 60W2 42+8/105Wo 8oW242+ 1/5W1 51(Wl3l+ W331). (5.4)

M= f Eo(xo)Q(xo)dxo, (5.5)

where Q(xo) is some assigned weighting factor.2 5

Of course Eo itself is still useful as a quantity thatindicates the "plane of best focus." Suppose that wehave designed an optical system so that all the Wcoefficients are known, but, however, the defocusingparameter W020 is still at our disposal and can be varied.Therefore we pose the following question: Given that allthe aberration coefficients are fixed, what value (orvalues) of W02 0 minimize Eo for a specified field pointxo? A discussion of this point is undertaken in the nextsection in conjunction with the behavior of the resultanttransfer functions. It turns out that Eo does seem topredict the plane of best focus from a transfer-functionanalysis of an actual design.

An additional use of Eo as a merit function in lensdesign is also worth detailed investigation. The varianceEo is a quadratic function of the mixed characteristic(aberration function) which itself is a complicatedimplicit function of the optical-system design data.Since Eo is a single number, it is straightforward todevelop differential correction formulas with which to

w

0-1.0 -.8 -. 6 -. 4 -.2 0 . 2 .4 .6 .8 1.0

. W 20 IN WAVELENGTH UNITS

FIG. 6. Eo for f/3.5 lens operating at 6° off axis as a functionof W020 for the wavelengths: (A) 7000 A, (B) 6600 &, (C) 6000 ,(D) 5400 K. The approximate minima of Eo are at: (A) W020 = 0.6x,(B) W02 0=0.4X, (C) W0 20 =0, (D) W020=-0.7X.

21 We are presently investigating the use of this merit factor inlens design.

study the behavior of Eo with respect to constructionalparameters such as curvature or refractive index. Thecorrelation of Eo with the corresponding transfer func-tions should yield valuable information.26

TABLE I. Values of the aberration coefficients in wavelengthunits for f/3.5 lens operating at 60 off axis in four colors. All datain paraxial plane, W020=0-

7000 A 6600.A 6000 A 5400 A

W040 -1.3359 -1.2757 -1.0860 -0.6806W131 -0.4357 -0.3855 -0.2149 0.1740W220 -0.5429 -0.5381 -0.5186 -0.4744W222 -0.2550 -0.2276 -0.1726 -0.0956W311 0.1809 0.1275 0.0086 -0.1829W060 0.5221 0.5606 0.6320 0.7282W151 0.0091 0.0168 0.0356 0.0696W 2 4 0 0.2837 0.3035 0.3394 0.3863W242 -0.2546 -0.2661 -0.2839 -0.3012W331 0.3068 0.3261 0.3617 0.4092W333 0.0961 0.1028 0.1161 0.1361W 4 2 0 0.3232 0.3429 0.3771 0.4192W 4 2 2 0.9011 0.9550 1.0482 1.1607W511 0.0378 0.0365 0.0315 0.0201W080 0.0222 0.0237 0.0265 0.0301

6. NUMERICAL RESULTS FOR TEST LENS

A 10-in. focal-length f/3.5 Petzval-type lens wasemployed as a test objective and analyzed for four

8.8

z

LUx

Z.

n S

A 04UU

SPATIAL FREQUENCY IN LINES/MM

FIG. 7. Modulus of transfer function of f/3.5 lens operating60 off axis at 7000 X in plane of best focus W020 =0.6 X(minimumEo): -*- =0°,--- 0=4 5 0 , , =90Q

.1 2

,10 0

08 -

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RICHARD BARAKIAT AND AGNES HOUSTON

SPATIAL FREQUENCY IN LINES/MM

FIG. 8. Modulus of transfer function of f/3.5 lens operating 60off axis at 6600 A in plane of best focus W020 = 0.4x (minimum Eo):- -f =00,--- f p=450, 1 =900

wavelenths: 7000, 6600, 6000, and 5400 A at fieldangles of 40 and 60. The final data for 60 are listed inTable I. It turned out that the lens behaved at 40almost as if it were on axis (00) and consequently thesedata are omitted. Note that W040 (primary sphericalaberration) changes by a factor of two at the extremeends of the spectrum, whereas W422 (primary astigma-tism) is relatively insensitive to wavelength.

These aberration coefficients were inserted into (5.4)and calculations were performed to ascertain the mini-mum of Eo as a function of W1020 (defocusing) for thefour wavelengths. The computations are summarizedin Fig. 6. Next, the transfer functions were evaluated inthe planes where Eo was minimum as well as in adjacentplanes. The transfer functions were appreciably better

238

SPATIAL FREQUENCY IN LINES/MM

FIG. 9. Modulus of transfer function of f/3.5 lens operating 60off axis at 6000 A in plane of best focus W020=0 (minimum Eo):- - 0=0°, --- + = 45°, 4 =900.

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Vol. 55

265

SPATIAL FREQUENCY IN LINES/MM

FIG. 10. Modulus of transfer function of f/3.5 lens operating 60off axis at 5400 A in plane of best focus W020= O.7X (minimum Eo):- *-0=0,---=450 4=90.in the planes where Eo was minimum than in the otherplanes. In order not to unduly lengthen the paper, weshow only the curves of I T in the plane of minimumEo (for q5=0°, 450, 900); these curves are shown inFigs. 7-10. Test calculations on other optical systemsindicated an analogous behavior and correlation of Eowith the plane of best focus.

It is common practice in lens design (and to someextent in lens analysis) to compute the transfer functiononly in the azimuths q=00, 90° and to average themto obtain the so-called average angular transfer functionat =45° that lies half-way between 00 and 900.Examination of Figs. 7-10 shows the fallacy of thisargument. In fact we have studied a number of otheroptical systems with the same results occurring. Werecommend the inclusion of the transfer function ato = 450 in any future calculations of the angular-averagetransfer function in order to obtain a more realisticaverage. For example, consider the curves in Fig. 8;if we deal only with '= 0°, 900, then the angular averageis very high; however, the curve at 0 = 450 is lower andits inclusion lowers the average materially. In the firstcase we would be led to an overly optimistic perform-ance for this particular situation.

ACKNOWLEDGMENT

We are indebted to B. Tatian for his continued en-couragement as well as for his kindness in designing thePetzval lens for the test calculations reported in Sec. 6.

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