1968 J. Opt. Soc. Am. A/Vol. 18, No. 8 /August 2001 Adriaan Walther

Angle eikonals for a perfect zoom system

Adriaan Walther

Department of Physics, Worcester Polytechnic Institute, Worcester, Massachusetts 01609

Received November 6, 2000; accepted January 19, 2001

I show that it is possible to choose the eikonals for the lens groups of a zoom lens such that extended objects atinfinity are imaged perfectly, without any aberrations, at all zoom settings. © 2001 Optical Society of America

OCIS codes: 080.2720, 080.2740, 220.3620.

1. INTRODUCTIONIn a previous paper1 I discussed the eikonal theory of asymmetric three-group afocal zoom system known as aDonders system.2 The main result was that eikonalfunctions for the three groups can be constructed that,used together, yield a zoom telescope corrected at all zoomsettings for all third-, fifth-, and seventh-order aberra-tions. These eikonals can be combined with the eikonalof a perfect fixed-focal-length camera lens to create azoom lens free from all aberrations out to the seventh or-der, at least as long as the object is at infinity. The ques-tion now arises whether we can find eikonals for the threegroups that correct the aberrations of the Donders systemexactly, i.e., to all orders. In this paper I construct an ex-ample to show that this is indeed possible.

2. SPHERICAL ABERRATIONS AND COMAFigure 1 shows the afocal system and a ray coming fromthe on-axis object point at infinity. In practice the innergroup and the outer groups have powers of opposite sign,but in the interest of clarity the figure has been drawnwith all powers positive. The front group is always usedwith the object at infinity, and the rear group, which isthe mirror image of the front group, is always used withthe image at infinity. Clearly the aberrations of thesegroups are independent of the zoom setting, except forchanges in aperture and field. It is the center group thatis used at a variety of conjugates.

A simple way of correcting the spherical aberration ofthe Donders system at all zoom settings is to combineouter groups already corrected for spherical aberrationwith a center group corrected for spherical aberration atany magnification. This requires that the center groupsatisfy Herschel’s condition,

sin~u2/2!

sin~u3/2!5 G, (1)

for all points on axis. For the notation see Fig. 1. G isthe transverse magnification of the center group for theaxial points considered, which is equal to minus the an-gular magnification of the Donders telescope.

Equation (1) determines for the center group the pathsof all the meridional rays. It shows that there is a greatdeal of coma, because Abbe’s sine rule is violated on pur-

0740-3232/2001/081968-04$15.00 ©

pose. This is, however, a problem that can be easily over-come. By an extension of Abbe’s sine rule to afocal sys-tems, the entire telescope will be corrected for coma if theinput height x1 and the output height x4 (see Fig. 1) havea fixed ratio: x1 /x4 5 2G. This can be accomplished bydesigning the first group such that sin(u2/2) is propor-tional to x1 , so that, on account of the symmetry, sin(u3/2)is proportional to x4 . Together with Herschel’s rule thisguarantees that x4 is proportional to x1 , so that thethree-group system is corrected at all zoom settings notonly for spherical aberration but also for coma.

3. EIKONAL FUNCTIONSTo make further progress we need to construct the eikonalfunctions W1 , W2 , and W3 of the three groups.3 For thedirection cosines we write (L, M, N), with an index 1 forthe object space, 2 for the space between the first groupand the second group, 3 for the space between the secondgroup and the rear group, and 4 for the image space. Forthe time being we look at the meridional rays only, so Li5 sin ui , where u indicates the angle between the ray andthe axis. We shall also need the sines of u i/2, for whichwe write ti :

Li 5 sin u i , ti 5 sin~u i/2!, i 5 1 ... 4. (2)

We want to be able to deal with different types of dis-tortion. It may, for instance, be required that the en-trance and exit angles be proportional, or maybe thattheir tangents or their sines be proportional. To dealwith this complication we introduce modified directionvariables in the object and image space:

Hi 5 Li 1 a3Li3 1 a5Li

5 1 ... (3)

for i 5 1 and 4, and we require that H4 5 2GH1 . Notethat H is defined such that for rays close to the axis u,sin u, tan u, and H all have the same value.

Each of the three eikonals is taken between the frontfocal plane and the back focal plane of the lens group itrepresents. W1 depends on the directions of the ray inspace 1 before lens group 1, and in space 2 between group1 and group 2. We start by writing

W1 5 (n50

`

H1nFn~u2!. (4)

2001 Optical Society of America

Adriaan Walther Vol. 18, No. 8 /August 2001 /J. Opt. Soc. Am. A 1969

Group 1 must be corrected for spherical aberration, whichmeans that x2 5 2]W1 /]L2 must be zero when H15 L1 5 0. This means that ]W1 /]u2 must be zero, sothat in the series development the function F0(u2) mustbe constant. This constant is multiplied by H1 to thepower zero, which is also a constant. So the F0 term inthe series development is constant. As adding a constantto an eikonal has no effect on the image formation, we canjust as well set this constant equal to zero.

Moreover we saw in Section 2 that for L1 5 H1 5 0 theheight x1 of a ray in the object space must be proportionalto sin(u2/2) 5 t2 . To find the constant of proportionalitywe look at the paraxial relation, which is x1 5 2f1u2 .For finite heights and angles this clearly must be changedinto x1 5 22f1 sin(u2/2) 5 22f1t2 . Now

x1 5]W1

]L15 (

n51

`

nH1n21

dH1

dL1Fn~u2!. (5)

For H1 5 0 only the n 5 1 term survives. The deriva-tive of H1 by L1 for L1 5 H1 5 0 is unity, so

x1 5 22f1t2 5 F1~u2!, (6)

which gives us the function F1(u2). So we can write

W1 5 22f1H1t2 1 (n52

`

H1nFn~t2!. (7)

The rear group is the same as the front group, turnedbackward. We find W3 by a symmetry argument: L1changes into 2L4 and t2 changes into 2t3 . The sign ofthe angle eikonal does not change when the signs of boththe entrance angle and the exit angle are changed, so wecan write

W3 5 22f1H4t3 1 (n52

`

H4nFn~t3!. (8)

Note that Fn(t) is even when n is even and odd when n isodd.

Finally, the middle group images the entire axis per-fectly. The angle eikonal for such a lens is, at least as faras the meridional rays are concerned (see, e.g., Ref. 7),

W2 5 24f2 sin~u2/2!sin~u3/2! 5 24f2t2t3 . (9)

We now have expressions for the meridional parts of thethree eikonals and must join them together to get ahandle on the meridional aberrations of the system.

Fig. 1. Schematic of a Donders system with a ray coming fromthe axial point at infinity.

4. MERIDIONAL ABERRATIONS FORFINITE FIELD ANGLESConsider a parallel fan of rays entering the system. Thedirection of the rays is specified by the variable H1 . Wewould like these rays to emerge from the system parallelto each other, with the direction H4 5 2GH1 . To inves-tigate the aberrations we draw a plane perpendicular tothe incident rays passing through the front focal plane ofthe first group. This plane is a wave front for the inci-dent rays. We also draw a plane perpendicular to theparallel rays that we hope will emerge from the system,this plane passing through the back focal point of thethird group. This plane would be an ideal wave front inthe image space. We now calculate the optical path be-tween these two planes along the set of rays considered.If all is well, these path lengths will all be the same.

To find the optical path between these planes, we needthe distance z between F18 and F2 and the distance z8 be-tween F28 and F3 . These distances are given by

z 5 2f2 /G, z8 5 2Gf2 , (10)

where G is the transverse magnification of the secondgroup, which is equal to minus the angular magnificationof the telescope. The optical path E between the chosenreference planes can now be calculated. Using the defi-nition of the angle eikonals we see that, apart from an ad-ditive constant, E is given by

E 5 W1 1 z cos u2 2 z 1 W2 1 z8 cos u3 2 z8 1 W3 .(11)

The constants 2z and 2z8 have been added so that wecan simplify the shift terms. In the space between thefirst and the second group we can write

z cos u2 2 z 5 2z~1 2 cos u2!

5 22z sin2~u2/2! 5 22zt22.

Similarly, the shift term in the space between the secondand the third group can be written 22z8t3

2. Substitutingthe expressions for the eikonals that we derived previ-ously, we arrive at

E 5 22f1H1t2 1 (2

`

H1nFn~t2! 1 2

f2

Gt2

2 2 4f2t2t3

1 2Gf2t32 2 2f1H4t3 1 (

2

`

H4nFn~t3!. (12)

Actually, this is not quite the path length between thetwo selected planes. Fermat’s principle must be takeninto account. E is the path length only if, for the selectedvalues of the input and output directions H1 and H4 , E ismade stationary in the intermediate direction variables t2and t3 . And here an interesting and useful complicationarises. In most cases of this nature, one specifies the raydata in the initial and final space, and the stationarity de-termines the values of the intermediate variablesuniquely. Here we want to specify the value of H1 ,choose H4 equal to 2G times H1 , and then have the sta-tionarity conditions determine not a single ray throughthe system but an entire fan of rays. This is possibleonly when the two equations ]E/]t2 5 0 and ]E/]t3 5 0

1970 J. Opt. Soc. Am. A/Vol. 18, No. 8 /August 2001 Adriaan Walther

are dependent on each other, so that there really is onlyone stationarity condition. The two stationarity condi-tions are

22f1H1 1 (2

`

H1n

]Fn~t2!

]t21

4f2

Gt2 2 4f2t3 5 0, (13)

22f1H4 1 (2

`

H4n

]Fn~t3!

]t31 4Gf2t3 2 4f2t2 5 0. (14)

After multiplying the first condition by 2G and some re-arranging of the terms, it can be written in the form

22f1~2GH1! 2 G(2

`

H1n

]Fn~t2!

]t21 4Gf2t3 2 4f2t2 5 0.

(15)

Equations (14) and (15) can be made identical by first ofall leaving out the sums, i.e., by setting the functionsFn(t) equal to zero for n > 2. In addition, we have to setH4 equal to 2GH1 , or the two equations will not haveany roots at all. With these assumptions there is onlyone stationarity condition left, which means that there isnot just one ray with the assumed directions in the objectspace and image space but an entire fan of rays. So allthe incident rays with direction H1 emerge from the tele-scope with the direction H4 5 2GH1 . Note that this re-sult is not based on a small-angle approximation; it holdsfor all input directions for which the value of H45 2GH1 corresponds to a real direction in the imagespace.

So if we choose

W1 5 22f1H1t2 , W2 5 24f2t2t3 ,

W3 5 22f1H4t3 , (16)

we obtain a Donders system with no meridional aberra-tions at all. This can be further verified by rewriting Eq.(12). Straightforward algebra shows that the remainingterms can be cast into the form

E 52f2

G S t2 2 Gt3 2Gf1H1

2f2D 2

2Gf 1

2H12

2f2. (17)

Clearly both the derivative by t2 and the derivative by t3are zero when

t2 2 Gt3 2Gf1H1

2f25 0, (18)

and E has the same value no matter which pair of valuesfor t2 and t3 is chosen to satisfy this equation.

5. THIRD DIMENSIONThe next step is to discuss the problem in three dimen-sions. This requires a careful discussion of the variablesto be used. Starting with the middle group, its angle ei-konal would normally be specified in terms of the two di-rection cosines L2 and M2 in its object space and the twodirection cosines L3 and M3 in its image space. On ac-count of the axial symmetry, these variables are usuallycombined in the form

a 5 ~L22 1 M2

2!/2, b 5 L2L3 1 M2M3 ,

c 5 ~L32 1 M3

2!/2. (19)

To connect this notation with the work in the previoussections we first of all write the ray directions in terms ofthe angle with the axis and an azimuth angle:

L2 5 sin u2 cos f2 , M2 5 sin u2 sin f2 , (20)

L3 5 sin u3 cos f3 , M3 5 sin u3 sin f3 .(21)

It follows that

a 5 ~sin2 u2!/2, b 5 sin u2 sin u3 cos~ f2 2 f3!,(22)

c 5 ~sin2 u3!/2.

We can now express this in terms of the half-angles withthe axis:

a 5 2 sin2~u2/2!cos2~u2/2! 5 2t22~1 2 t2

2!, (23)

b 5 4 sin~u2/2!cos~u2/2!sin~u3/2!cos~u3/2!

3 cos~ f2 2 f3!

5 4t2t3A1 2 t22A1 2 t3

2 cos~ f2 2 f3!, (24)

c 5 2 sin2~u3/2!cos2~u3/2! 5 2t32~1 2 t3

2!. (25)

Now consider the power-series development of W2 interms of a, b, and c, and substitute for a, b, and c the ex-pressions just obtained. It is clear that the resulting se-ries can be written as a power series in terms of t2

2,t2t3 cos( f2 2 f3), and t2

2.By similar arguments it can be shown that the proper

variables to be used with the angle eikonal of the firstgroup are H1

2, H1t2 cos( f1 2 f2), and t22, and for the last

group H42, H4t3 cos( f4 2 f3), and t3

2.So the natural extension into three dimensions of Eqs.

(16) is

W1 5 22f1H1t2 cos~ f1 2 f2!, (26)

W2 5 24f2t2t3 cos~ f2 2 f3!, (27)

W3 5 22f1H4t3 cos~ f4 2 f3!, (28)

These terms are, in the language of Smith,8 all of classzero.

All sorts of terms belonging to other Smith classes canbe added to these eikonals. These other classes repre-sent skew-ray aberrations that are all zero for meridionalrays. Fortunately these additional terms are not needed.Let us use the eikonals just listed to calculate the pathfunction E introduced in Section 4. We assume that theincident rays are specified by the direction variable H1and are parallel to the (x, z) plane, so that f1 5 0. ThenH4 should be 2GH1 , and the azimuth angle f4 should bezero as well. This gives, for E,

Adriaan Walther Vol. 18, No. 8 /August 2001 /J. Opt. Soc. Am. A 1971

E 5 22f1H1t2 cos f2 1 2f2

Gt2

2 2 4f2t2t3 cos~ f2 2 f3!

1 2Gf2t32 2 2f1~2GH1!t3 cos f3 . (29)

Straightforward algebra shows that this expression canbe cast into the form

E 5 2f2

G S t2 cos f2 2 Gt3 cos f3 2Gf1

2f2H1D 2

1 2f2

G~t2 sin f2 2 Gt3 sin f3!2 2 G

f 12

2f2H1

2.

(30)

To fully specify a ray, one would normally choose its direc-tion in the object and image space and then determine thefour intermediate variables t2 , t3 , f2 , and f3 by usingFermat’s principle, i.e., by requiring that E be stationaryin these four variables. The fact that E can be written asthe sum of two squares has as a consequence that thesefour stationarity conditions lead to only two conditions:

t2 cos f2 2 Gt3 cos f3 2Gf1

2f2H1 5 0, (31)

t2 sin f2 2 Gt3 sin f3 5 0. (32)

So the input direction and the output direction determinenot just a single ray but a two-parameter fan of rays. Foreach of these rays we have

E 5 2Gf 1

2

2f2H1

2, (33)

independent of which of the rays we choose. So there areno aberrations at all; any ray with the specified input di-rection emerges with the required output direction.

6. CONCLUSIONSIt is straightforward to write the eikonals shown in Eqs.(26)–(28) in more conventional notation. The requisiterelations between the variables are given at the begin-ning of Section 5. If we use a 5 (L2 1 M2)/2, b 5 LL81 MM8, and c 5 (L82 1 M82)/2 as appropriate for eachof the three eikonals, straightforward algebra yields

W1 5 22f1

b

A1 2 2aA2~1 1 A1 2 2c !

, (34)

W2 5 22f2

b

A1 1 A1 2 2aA1 1 A1 2 2c

,

(35)

W3 5 22f1

b

A1 2 2cA2~1 1 A1 2 2a !

. (36)

Here it has been assumed that tan u4 5 2G tan u1 ; i.e.,the tangents of the directions in the object space and theimage space must be proportional to each other.

These eikonals need to be compared with the series de-velopments derived in Ref. 1. In that paper the eikonalswere taken between the principal planes, so shift termsf(A1 2 2a 1 A1 2 2c 2 2) have to be subtracted fromthe eikonals given above, with f 5 f1 for W1 and W3 andf 5 f2 for W2 . Furthermore f1 was called af in that pa-per, and f2 was called 2f. After these changes are made,the series developments can be carried out. The resultsare exactly as shown in Table 2 of Ref. 1, with P1 5 P75 af/2, Q1 5 Q7 5 2f/2 and Q4 5 Q10 5 0.

As a further test the eikonals shown above can be usedto ray trace the telescope.9 This calculation has been car-ried out, but the results are not worth showing because,except for miniscule rounding errors, the aberrationsturned out to be zero.

Several arbitrary decisions went into the constructionof this solution. It therefore cannot be expected that thissolution will be unique. Some modifications are trivial.One can, for instance, add spherical-aberration functionsA(t2) and A(t3) to the first and third groups and subtractA(t2) 1 A(t3) from the center group. All the groups nowhave spherical aberration, but the optical path function Eis unchanged. Other modifications, such as introducingterms of other Smith classes in the eikonals, are rathermore involved and have not been worked out. The maingoal of this paper, however, has been accomplished: Ithas been proved that eikonal functions can be constructedthat make the Donders system perfect. If we now allowthe usual assumption that any eikonal function can berealized,10 it follows that it is possible in principle to de-sign zoom lenses that are perfectly corrected for infinityat any zoom setting.

REFERENCES AND NOTES1. A. Walther, ‘‘Zoom lenses and computer algebra,’’ J. Opt.

Soc. Am. A 16, 198–204 (1999).2. R. Kingslake, A History of the Photographic Lens (Aca-

demic, New York, 1989), p. 161.3. For a general introduction to eikonal functions see Refs.

4–7.4. M. Herzberger, Strahlenoptik (Springer, Berlin, 1931).5. R. Luneburg, Mathematical Theory of Optics (University of

California Press, Los Angeles, Calif., 1964).6. H. A. Buchdahl, An Introduction to Hamiltonian Optics

(Cambridge University Press, Cambridge, UK, 1970).7. A. Walther, The Ray and Wave Theory of Lenses (Cambridge

University Press, Cambridge, UK, 1995).8. T. Smith, ‘‘The changes in aberrations when the object and

stop are moved,’’ Trans. Opt. Soc. 23, 139–153 (1921/1922).9. These calculations were carried out with the Oslo Six lens

design program. Oslo Six is a registered trade mark of Sin-clair Optics Inc., Fairport, New York.

10. See, e.g., Ref. 7, pp. 228–229.