LIGHT TRANSMITTED BY VERY SMALL PINHOLES

illuminated by a lens with a rectangular aperture, asin a prism spectrograph. From these it can be deduced,as in Sec. 2, that, if a slit is illuminated by a condenserwith a rectangular aperture and if the slit is narrowenough to give an image corresponding to that of anincoherent line source, the flux in the objective will beabout 80% of the maximum possible. Thus for spectro-scopic purposes the usual rule for the condenser apertureis valid. In fact, the use of a condenser aperture corre-

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

sponding exactly to that of the dispersing element,together with a field lens at the slit, is a well-known wayof reducing stray light in spectrographs. This techniqueis useless with a point source and the use of diaphragmsafter the pinhole is necessary.

ACKNOWLEDGMENT

Acknowledgment is made to the National CoalBoard for a grant-in-aid of this research.

VOLUME 48, NUMBER 3 MARCH, 1958

Generalized Theory of Zoomar Systems

FRANK G. BACK AND HERBERT LwENZoomar, Inc., Glen Cove, New York

(Received May 2, 1956)

A Zoomar is a varifocal lens system consisting of alternately fixed and movable lenses or lens groups inwhich all movable components are contained in one common, linearly movable barrel. The basic theory ofsuch a Zoomar system has been illustrated in a previous paper for a three lens system.

The present paper will generalize this theory to cover any Zoomar system and will also illustrate thedevelopment of the Zoomar system by selected examples of applications.

INTRODUCTION

NEARLY ten years ago the first varifocal lens withsingle-barrel linear movement and optical com-

pensation was introduced under the name "Zoomar".Though this name was primarily meant as a trade

mark for optical instruments it must also be regardedas the name of a basic lens-type like the Tessar, theCooke Triplet, the Petzval lens, etc. Like these lenses itallows numerous variations and modifications withoutlosing its basic characteristics. The Zoomar was by nomeans the first varifocal lens; but it introduced for thefirst time a new principle of image-shift compensation,which justifies its claim to be regarded as a lens typein its own right.

There are two distinctly different types of varifocaloptical systems possible, mechanically compensatedones and self-compensating (optically compensated)systems (Fig. 1).

Mechanical image-shift compensation can be achievedin an infinite number of ways. The only commonmechanical design feature of such lenses is some kindof nonlinear linkage.

In theory a mechanically compensated lens is capableof a greater zoom range than a self-compensating one,because there should be no image shift at all. Inpractice, however, small deviations from the mathe-matical shape of the nonlinear movement and especiallyany slack caused by natural wear will cause themechanical system to deviate from its theoretical path.

Consequently, the mechanically compensated lenshas no real advantage over the self-compensating oneas far as image deviation is concerned. Furthermore, as

will be explained later, the zoom range is mainlydetermined by the limits of aberrational correction.The greater the zoom range the harder it is to getsatisfactory image quality throughout the zoom.

BASIC CHARACTERISTICS OF ZOOMAR SYSTEMS

A Zoomar-type varifocal lens consists of a sequenceof alternately fixed and movable lenses or lens groups.The movable lens elements are all contained in onecommon barrel, fixedly spaced from each other, andtherefore moving always in the same direction and bythe same distance when the focal length is changed.The first lens may be either fixed or movable, whilethe last lens of the Zoomar system proper always hasto be a movable element. The image formed by this

MECHANICALCOMPENSATION

WIDE ANGLE

- _iiT_ f MEDIUM

X__ IiX r TELE PHOTC

A

z~~~~~~~~~00

E.F.L

OPTICAL(ZOOMAR)COMPENSATION

{ Ie

_.T

i

) ,..

E.F.L.

FIG. 1. Image shift compensation in varifocal lenses.

March 1958 149

F. G. BACK AND H. LOWEN

TABLE I. Self-compensating varifocal systems.

Number Numberof lens of zero Zoom

Name Manufacturer groupsa Year points range

Forerunner of Zoomar systems

Transfocator Siemens 2 1928 2 2:1Varifocal viewfinder Research &

Development Lab 2 1944 2 10:1

Zoomar systems

Zoomar "A" Zoomar 4 1946 3 3:1Television Zoomar Zoomar 4 1947 3 4:1Newsreel Zoomar Zoomar 4 1947 3 4:1Superzoomar Zoomar 4 1949 4 5:1Pan-Cinor 16 S.O.M. 3 1951 3 3:1Zoomtelescope Zoomar 4 1952 4 10:1Zoomar 16 Zoomar 4 1952 3 3:1T. V. Studio Zoomar Zoomar 3 1953 3 3:1Zoomar 8 Zoomar 3 1953 3 3:1Pan-Cinor 8 S.O.M. 3 1953 3 3:1Zoomar 35 Zoomar 3 1954 3 3:1Pan-Cinor 70 S.O.M. 4 1955 4 4:1Universal TV Zoomar Zoomar 4 1956 4 6:1I. T. V. Zoomar Zoomar 4 1956 4 6:1

a Lens groups counted in the direction of the light path up to and in-cluding the last movable lens group, but not including any lenses betweenthe last movable lens group and the final image.

last movable lens, wherever it may be positioned,remains substantially stationary in space. Though fullcompensation is only achievable for a finite number ofdiscrete barrel positions, the deviation can be keptwithin the depth of field tolerances. The image cantherefore form the object for any conventional typeof optical system. It can be collimated, inverted,reduced, enlarged, or otherwise optically changed tosuit the purposes for which the system is used. Thesame Zoomar in conbination with a relay can be usedas a camera objective by itself. With a colimating rearsystem it forms an auxiliary lens attachment and withan appropriate eye piece it may be used as a varifocaltelescope.1

Self-compensating lenses by their very nature haveto show a certain uniformity in their mechanicaldesign. Every Zoomar system has to have a fixed outerbarrel, a movable intermediate barrel and one or morefixed inner barrels. The outer barrel carries the rearsystem, adapted to the specific use of the Zoomar, andsometimes a front lens. The intermediate barrel containsall the movable lenses. The fixed inner barrel or barrelsare for the stationary lenses positioned between themovable elements. The movable intermediate barrelhas to have openings or windows through which pinsor other members extend, which connect the innerbarrels to the outer barrel.

TABLE II.

I Lens mn=02 Lenses m2+an=0

3 Lenses 113+an12+b11=04 Lenses mn0+ain3+brn2+cn=On Lenses ant+am-1+bnf<+cn3+ * * =0.

1 Frank G. Back, J. Opt. Soc. Am. 43, 1195 (1953).

HISTORICAL DEVELOPMENT

The forerunners of the Zoomar were two lens systemsconsisting of a stationary front lens and a movable rearlens namely, Gramatzki's transfokator2 and the vari-focal view finders of the U. S. Signal Corps.3 Both werebased on the principle of interchangeability betweenobject and image. If two conjugate points of a lens areeither both real or both virtual, there are always twolens positions in which the distance between the objectand image is the same. The magnification in one lensposition is the reciprocal of the magnification in theother position. The focal range of such a system is equalto the square of the ratio of long conjugate to shortconjugate. Though selfcompensating, such systemscannot be regarded as Zoomars, because the deviationbetween the two compensation points is too great tomake them usable as camera lenses, except within aninsignificant magnification range.

In the nineteen-thirties the late Robert Richter ofCarl Zeiss, had a microscope illuminating devicepatented, which in one of its examples contains avarifocal device consisting of three lenses. Though to acertain extent it looks like a Zoomar-system, theinventor himself states in his disclosure that thecompensation of this system is not even good enoughfor a pancratic telescope, where the accommodationof the human eye permits an image deviation, whichcould not be tolerated in a camera lens.4

In order to combine satisfactory image shift com-pensation with sufficient focal range at least threeelements are required. In a previous paper' we havedemonstrated how such a three component Zoomarworks, and we have proven that such a lens can befully compensated for three-barrel positions. We havealso explained how the addition of a fourth component,namely a stationary front lens improves the deviationcurve of the three-lens system.

We developed already in 1949 a 4-componentZoomar with a zoom range of 1: 5 and 4 points of fullimage shift compensation. Unfortunately, in those days,the new rare earth glasses with their high refractiveindex and low dispersion were not available to us. Itwas therefore impossible to keep the lens aberrationsunder control and the system never got beyond theprototype stage. Table I shows all Zoomar type lensesand their forerunners in chronological order.

GENERALIZED THEORY OF ZOOMAR SYSTEMS

In a previous paper we described the derivation of thedeviation equation for a three lens Zoomar system asshown again in Fig. 2. A, the deviation, is a function of

2 H. J. Gramatzki, Probleme Der Konstruktiven Optik (Akademie-Verlag, Berlin, 1954).

3 Frank G. Back, J. Soc. Motion Picture Engrs. 45, 466 (1945).4 U. S. Patent No. 2,078,586, April 27, 1937, German Patent

920537, October 3, 1935.6 Frank G. Back and Herbert Lowen, J. Opt. Soc. Am. 44, 684

(1954).

Vol. 48150

GENERALIZED THEORY OF ZOOMAR SYSTEMS

Compensafor X X~

_- - '_1- I:RI I

__ 7 _ _ __.'

)( &£i__

Xc=Xi d - fdXxXg

-C - XIXX=- Xm d7 , X==X X

X& X+ M. d= d.+ m, X'= Xe* M+ A:

I;^ m+= fa ( X.+m)f + (. ra) ( X.+ m)

fc(X.+m)f,' (d.+ m) (X.+ m) fX. + 4dX M

m3 + (d.+X.- rrim + (d.X,+ f.(f X. ) M

i=~~~~~~~V d X)fd.

=n-i- (d;> X.) m + ( f,+ d.X.)

rnS+ant+.bma = M 3--a0i

Mrna c m + dFIG. 2.

the barrel movement "m", the independent variable.If /v becomes 0 the equation can be reduced to a verysimple form, namely,

m3+ am2+ bm = 0.

This is, as mentioned before, the equation for zerodeviation in a three lens Zoomar system. In a similarway the zero deviation equation can be deduced for anynumber of lenses in a Zoomar system. Table II showsthese equations, correlated to the number of lensgroups in a Zoomar system. From this table it can beclearly seen that for a one lens system the compensationequation is of first degree-for two lenses it is of second

degree, for three lenses of third degree, etc. Or, in otherwords, the degree of these equations equals the numberof lenses in a Zoomar system.

Since the number of roots of an equation is equal toits degree it can be concluded that the maximum numberof possible compensation points in a Zoomar systemequals the number of fixed and movable lenses up toand including the last movable lens. This, of course,does not mean that a four lens system always has tohave four zero deviation points. On the contrary,practical considerations in designing zoom lenses maygive a better over-all picture quality if one possiblecompensation point is sacrificed in favor of aberrationalcorrections.

151March 1958

F. G. BACK AND H. LOWEN

THE FIRST LENS

As can be seen from Fig. 2 the first lens of the Zoomarsystem regardless if it is movable or fixed does notenter into the compensation equation and thereforehas no influence on the image deviation.

Already the first Zoomar lens had two interchangeablefront lenses. The zoom characteristics of that lens wasnot disturbed by interchanging them because the focalpoints of these two front lenses were in the same positionwith relation to the following lens elements.

The difference in focal lengths of the front lensesthough, is in direct relation to the absolute equivalentfocal length of the entire zoom system, according to thewell-known classical formula

SIGN OF COMPONENT POWERS

From the previously shown derivation of the com-pensation equation

Lfc2 (Xo+m) fC 2Xo

I .

fi2+ (dolm) (Xo+m) fj2+doXo

it can be seen that the focal lengths of the lens elementswhich determine the zoom deviation enter only assquares. Positive and negative elements are thereforein this respect equivalent.

Summarizing, we can deduce three conclusions aboutthe basic characteristics of any Zoomar system. Thethree conclusions are graphically illustrated in Figs. 3-5.

I LENS

2 LENSES

3LENSES

4 LENSES

5 LENSES

ALm

A

l m

OPTIMATION VS STABILIZATION

Besides the problems connected with the computationof a single focus lens the designer of the Zoomar systemhas to cope with additional difficulties. In a single focuslens the aberration contribution of each component isfixed. In a zoom lens most ot the elements change notonly their aperture, but also their conjugate pointsthroughout the zoom. The error contributions of eachsurface therefore change continuously.

In a single focus lens the definition of the final imageis already determined by a great number of componentsthe most important being geometrical aberrations,diffraction, and macrocontrast. In a zoom lens thesesame difficulties are present to a higher degree (due tochanging contributions) and are compounded by theimage deviation between the zero points.

The main difference in designing a single focus lensand a zoom lens lies in the goal which the designer hasto achieve. In a single focus lens one has to try for thebest image obtainable with the given system wherebyit is still a moot question what the best image really is.In the design of a zoom lens it is much more importantthat the image aberrations remain as stable as possiblethroughout the zoom. This is also the reason why veryoften not all possible compensation points can be usedbecause in the final effect it does not make muchdifference if the image loses definition because of firstor higher order defects.

Once image stabilization is achieved a zoom systemhas one redeeming feature over a single focus lens. Thislatter is usually one integral unit and therefore thefinal image after the last element has to show the bestpossible correction. A zoom system is usually followedby some kind of a rear system. This rear system canbe used to improve the stabilized image of the zoomsystem proper.

APERTURE STOP

In an ordinary lens system the position of the aperturestop is determined mainly by the considerations of bestimage correction. In a zoom system the designer hasto keep the light output, i.e., the relative apertureconstant throughout the zoom. This can be achievedeither by placing the aperture stop behind the lastmovable lens element or by introducing a mechanicalvariation of the iris diaphragm. Furthermore the chang-

A m~~~t A

1Sr CONCLUSION

THE NUMBER OF LENSES USED INA ZOOMAR SYSTEM DETERMINESTHE MAXIMUM POSSIBLE NUMBEROF ZERO COMPENSATION POINTS

Fio. 3.

2" CONCLU SIONTHE FIRST LENS OF ANY ZOOMAR SYSTEM HAS NO INFLUENCE ON THEIMAGE DEVIATION WT OETERMINES ONLY THE ABSOLUTE FOCAL LENGTHS

FIG. 4.

I~~~~~~~~~~ At

am

Vol. 48152

l t

153GENERALIZED THEORY OF ZOOMAR SYSTEMS

TABLE III. Differences in design considerations.

Single focus lenses Zoom lenses

1. Aberration contribution of each 1. Aberration contribution of zoomcomponent fixed. components variable.

2. Definition determined by aber- 2. Definition determined by aber-rations, diffraction, and macro- rations, diffraction, macrocon-contrast. trast, and zoom deviation.

3. Optimation of image quality. 3. Stabilization of image quality.4. After last element best possible 4. After last element best stabili-

correction. zation. Correction can beachieved by independent rearsystem.

5. Stop position is determined by 5. Stop position is determined bycorrection requirements only. speed, stability and physical size

of the system.

ing size and location of the entrance pupil and itsinfluence on the physical dimensions of the systemhave to be considered to avoid unnecessary bulkinessof the system on one hand and vignetting or loss ofspeed during the zoom on the other (see Table III).

CLOSING SUGGESTIONS

We want to offer respectfully the following suggestion:In his excellent classification table of Photographic

Lens Types,6 R. Kingslake lists under "U" "Zoom"lenses, (a) complete zoom lens systems and (b) afocalzoom attachments. In our opinion this subdivision

6 Rudolf Kingslake, J. Opt. Soc. Am. 36, 251 (1946).

I

3.' CONCLUSION

------ , m -F.. + d.+X.- e- -)y + (f.+d.X.+ f,~-d.(.

m'u (d.+ X.)m + (f,'+d.X.)

f2 ONLY . .- f)' =(4)2IN A ZOOMAR SYSTEM NEGATIVE OR POSITIVE COMPONENTS

ARE EOUIVALENT WITH RESPECT TO ZOOM COMPENSATION

FIG. 5.

should be changed into: Mechanically CompensatedLenses and Self-Compensating Systems. Whether alens can be used by itself or as an auxiliary attachmentis not an essential quality of the varifocal system itselfbut rather an accidental one. The same varifocalarrangement with a relay as rear system is a completezoom lens, while, when used with a collimator as rearsystem, it becomes an "afocal zoom attachment."Furthermore, there is always the possibility to addanother subdivision to class "X" Lens Attachmentsand call it X(f) "Afocal Zoom Attachments."

March 1958