Journal of the

OPTICALof

VOLUME 52, NUMBER 4

SOCIETYAMERICA

APRIL, 1962

Two-Component Optically Compensated Varifocal System

LEONARD BERGSTEINPolytechnic Institute of Brooklyn, Brooklyn, New York

AND

LLOYD MOTZRutherford Observatory, Columbia University, New York, New York

(Received July 5, 1961)

The general theory of optically compensated varifocal systems is applied to the case of a two-componentsystem consisting of a single movable component placed behind a fixed component. This is the simplestcase of a varifocal system and one that can be treated in all its details by elementary analytical methods.It is shown that for a given focal range four different systems are possible in all of which the focal lengthof the movable component has the same magnitude. Of these four systems only two are of practical interest.It is demonstrated that if the optimum conditions are realized the image deviation can be reduced to asufficiently small value so that the two-component varifocal system can be of practical value in some specificapplications, as for example in projection systems and viewfinders, as well as in telescopes of small magnifica-tion range.

I. INTRODUCTION

IN a previous paper a general theory of opticallycompensated varifocal systems comprising any

number of components was developed by one of theauthors. In this and subsequent papers it will be shownhow that theory can be applied to specific cases suchas the two-, the three-, and the four-component systems.The simplest case of a varifocal system is the two-component system. Various aspects of two-componentvarifocal systems have been treated in patent disclosuresby Allen2 and by Gramatzki,3 4 but these investigationsdid not give a general discussion of the two-componentlens. In the present paper the two-component systemis analyzed in all its details by elementary analyticalmethods.

It may seem that the two-component varifocalobjective is much too simple a system to offer much inthe way of practical application, since there are onlytwo positions of the movable component for which theimage plane passes through the same predetermined

'L. Bergstein, J. Opt. Soc. Am. 48, 154 (1958).2 C. C. Allen, U. S. Patent 696,788 (1902).3 H. J. Gramatzki, Brit. Patent 449,434 (1953).4 H. J. Gramatzki, Probleme der Konstruktiven Optik (Akademie

Verlag, Berlin, 1954).

position in space. But this is not the case. If theoptimum conditions are realized, the deviation of theimage plane for intermediate positions of the movablecomponent can be reduced to sufficiently small valuesand the two-component varifocal system can be ofgreat practical use in some specific applications, as forexample, in projection systems and viewfinders, aswell as in telescopes of small magnification range.

II. APPLICATION OF THE GENERAL THEORY TOTHE TWO-COMPONENT SYSTEM

1. Two-Component Systems and theirGaussian Parameters

The two-lens varifocal system consists of a singlemovable component placed behind a fixed component.A third, fixed component is usually inserted behind themovable component, but as was already pointed out,'it is not considered as part of the varifocal systemproper (see Fig. 1).

In the analysis that follows, the designation of thelens elements is the same as that in reference 1, that is,in ascending order from the object side to the imageside. Thus, subscript 2 is assigned to the fixed front com-ponent, subscript 1 to the movable component and

353

LEONARD BERGSTEIN AND LLOYD MOTZ

VARIFOCAL SYSTEM AUXILIARYPROPER SYSTEM

2 BURL t- IL-IIMAGE

PLANEAPERTURE

STOP

FIG. 1. The two-component optically compensatedvarifocal system.

subscript 0 to the rear fixed (auxiliary) system. Thevarifocal system proper can be described by threeGaussian parameters, the two focal lengths F2 and F,of the two components and either the spacing S21between the rear and front principal planes of the twocomponents or the separation D21 between their rearand front focal planes (see Fig. 2).

Let Z be the displacement of the movable componentfrom its original position in the direction from theobject side to the image side and let Zm denote themaximum displacement. Following the theory developedin reference 1, we normalize all system parameters withrespect to Zm, abbreviate all normalized parameters bylower-case letters, i.e., with k=2,1,

A Fk a Dk,k-1fk=-, dk,k-1=

Zm Zm

A Skk-1Sk,kl= X

Zm(1)

where the distances dkk-I and Skk.1 refer to the originalposition of the movable component, and introduce thenormalized variable

Z= (Z/Zm). (2)

The only nonzero normalized "bracket" of the system is

(3)

Using the relations developed in reference 1, we findthat the normalized over-all focal length of the systemas a function of the displacement z is given by

The two-component varifocal system has two pointsof full compensation. Let these points be located atz, and Z2 With 0(<z 1<Z2<1.0. ntroducing

'Y1=Z1+-'2, (7)

'Y2= ZZ 2,

Eq. (ID 2-a) can be rewritten into

(Z- Z1) (z-Z2) z2- 'YI+72YWz) = - = . (ID2-b)

z+bi z+bi

The focal range f(z) and the image-plane deviationy(z) of the two-component system are shown in Fig. 2as functions of the displacement z of the movablecomponent.

Of the three Gaussian parameters of the system two,f2 and d2l, are determined by the predetermined pointsof full compensation and by the required relative focalrange.

To obtain the predetermined points of full compensa-tion we require that

(VF 2-I)

AUXILIARY SYSTEMELEMENT 2 ELEMENT I'

r-s2.1 lzitO -.l, ReferencelifI.#2; EfX o~x;--*-I position of the

image plane-x-- o x-i in the original

F F. F position of the2 V 1 ra~~~~~~~mvablecomponents

j j A2j rczrI Image planeI i | /in the position

.- x -- ----- z of theFl F,' F'4i F, , movable

2V 2 1 ' V .I components

v Z)=Aperture-Z+yp

Lola, Wz = VI-ty(z,oJ4f(z)= -flf2/(z+b,).

For the normalized final image distance (measuredfrom the rear focal plane of the rear movable compo-nent) we obtain

X11(z) =f11(z+b1),

_I(4)

and the normalized image-plane deviation, measuredfrom a reference plane located at a distance -y(O)from the image plane in the position z= 0 of the movablecomponent, becomes

y(z)= (z2 +C1z+C2 )/(z+b1),

cl = b- (x'- yo),

(ID 2-a)where

f (Z) = ibo ,(Nsystems)

i--- - (Psystems)

0( - (Z-ZI)(Z-Z2)

a z b1,,Z I I 4 ZS.,

ID-. z

/ ry() for Psysfems, -y(z) for Nsystems

yomy(O), and

(5) FIG. 2. The two-component optically compensated varifocalsystem and its Gaussian parameters, its over-all normalizedfocal length f(z) and image-plane deviation y(z) as functions of

(6) the (normalized) displacement z.

(FL2 )

I .1

. . . _ ..

354 Vol 52

bi- (x'-yo) = -yi,

,Pl (2, 1)-_b, = d,,.

1.0 -9- Z

x'=_x1'(0)=f121b1.

TWO-COMPONENT VARIFOCAL SYSTEM

where x'= fP2/A_ andyo ( 2 /b,). (8)

The location of the points of full compensation entersinto Eq. (VF2 -I). However, it is evident that theparameters of a two-component system are independentof the location of the points of full compensation ifthe image-plane deviation is the same at both ends ofthe operating range (see Fig. 2). This is obviouslyalways desirable.

Settingy(O)=y(1),

we obtain from Eq. (ID 2-b),

Y2= (1-y)b.This gives

y(1)-yo= ("y/b,) = -,and Eq. (VF2 -I) becomes

x'-bi= 1.0.

(9)

(9a)

(9b)

(VF2-Ia)

The operating range of focal lengths of the system isdetermined by the "relative focal range" R defined asthe ratio of maximum to minimum over-all focal lengths,i.e.,

A Fmax fmaxaR= =

Fmin fmin(10)

It was already pointed out' that two (optically com-pensated varifocal) lens types are possible for anyrequired focal range 6R. Either a lens system can bechosen which has the maximum over-all focal lengthwhen the movable components are in the front position,or a system can be chosen which has the maximumover-all focal length when the movable components arein the rear position (see Fig. 3). We will refer to thefirst type as the P system and to the latter as the Nsystem. We define the "focal ratio"

(Pa-2)

-mlSall'-52,

-- #1: 1XNa

(Na)

' . .~z I -

--- + - I." ,-(- -Pb) -(Nb)(P b) (N b)

fmax

0(Z* = 1 zR-l)Z

0 max-..f(z)- fMLax

I +TR-J I-z)fmax - ,,

0 1.0-*z

SYSTEM f I fmaxtw(z

(Pa-I) +FR SZ Vit-I PIR+ 1 _S I + K )

Pa-2) 7W-1 X F RI R+t+-l t-z))(P)i dI+5- g - /SH i

(Nb) i - + l 5 a 7'iR 521 , i'R -X-z)

FIG. 3. Possible two-component varifocal systems, the focallengths f, and fi of their components, their maximum over-allfocal length f(max), their final image distance l'(z) and focalrangef(z). The systems shown in (Pa-I), (Pa-2) and (Pb) are Psystems, the systems shown in (Na) and (Nb) are N systems. Onlythe systems shown in (Pa-i), (Pa-2), and (Na) are of practicalinterest.

The two equations (VF2-Ia) and (VF2-II) can bereadily solved for the two system parameters f, andd21. We obtain

1 1-rd21: b= = ' (11)

r-1 2rA F(0) f(0)r=-=

F (1) f(l)'(lOa)

where f(0) and f(1) are the over-all focal lengths in thepositions z= O and z= 1.0, respectively. For the Psystem r= (R, whereas = 1/(R for the N system. It isalso convenient to introduce the normalized range

Af(O)-f(l) r-1 61-1=__ - =-= hi , (lOb)f(O)+f(l) r+1 6+ 1

where the positive sign applies to the P system, thenegative sign to the N system. We note that 1.0 < R < o,O<r< , and -1.O<r<1.O.

To obtain the required relative focal range 61 adthe desired type of system, P or N, we require that

and

r 1+rx' = b+ 1 = -=-

r-1 2r

(r) I (1 - ')If, = - =-r-1 2r

(12)

(13)

In Eq. (13) the positive sign as well as the negativesign can be used for the focal length of the movablecomponent. For any given focal range four basicsystems are thus possible, two P systems (r= 61) andtwo N systems (r= 1/6R), in all of which the focal lengthof the movable component has the same magnitude

I f, I = ((R),/ (6- 1). (13*)

All four systems are shown schematically in Fig. 3.The focal length of the front component is given by

f 2= S21 -d 2l-fl, (VF2-IV)

April 1962 355

(r- )bi= 1.0. (VF2-II)

LEONARD BERGSTEIN AND LLOYD MOTZ

where s21 can be arbitrarily chosen except that it mustbe greater than the minimum physical separationrequired between the rear principal plane of the frontcomponent and the front principal plane of the movablecomponent in position z=O. [If no components areused whose principal planes are at an appreciabledistance from the respective physical boundaries,s21 (minzO.] Using Eqs. (11) and (13) we obtain

1f2= ~+3S21- (14)

1F (r)i

In Eq. (14), as in the following Eqs. (15)-(19),where a double sign appears, the upper sign is associatedwith the positive sign in Eq. (13) and the lower sign isassociated with the negative sign in Eq. (13).

The over-all focal length of the system is given byEq. (FL2). Using Eqs. (11) and (13) we find that themaximum over-all focal length is given by

fmax= F ((R)'f2, (15)

and depends only on the focal length of the frontcomponent.

The image formed by the fixed component servesthe purpose of "object" with respect to the movablecomponent. The normalized distance between this"object" plane and the (front principal plane of the)movable component is given by

1l(z) =-(d 2 l+fl+z)=-[ (r) (1-z)1. (16)

The normalized distance between the (rear principalplane of the) movable component and the final imageplane is

Il/(Z) = X()+f1= [r i -+y (zAO

(r)r rz(r)kF 1 1+ (-1)z

where y(z,O)=y(z)-yo, is the image-plane deviationmeasured from the position of the image plane at z= 0.The normalized distance loi(z) between the "object"and the image planes is 11 '(z)-11(z). Thus,

(r) ii 1oi (Z)=t(Z) - (Z) ) ~~+yzOM (18)

(r) 1F1

From the results of the next section it can be readilyshown that "object" and image of the movable compo-nent are disposed symmetrically with respect to theposition z= 2 of maximum image-plane deviation;

(1±1) (r) (19)

r-1

That is, in the position z= i, "object" and image havethe same size and are situated in planes which areequidistant from the movable component. This is truefor all four basic two-component systems.

We shall first consider the two systems obtained byusing the positive sign in Eq. (13). From Eqs. (16),(17), and (19) we observe that "object" and imageare always on opposite sides of the movable component.In position z= , of maximum image-plane deviation,the "object" and image distances are

2lo(2)=-A (z = l' z)= 2(r)/ (-1)= 2l.(19a)

For the P system, r = (R, the movable component ispositive and the final image formed is real and issituated in a plane behind the movable component.The image serving as "object" with respect to themovable component is situated in a plane in front ofthe movable component and can be either virtual orreal. It can therefore be formed either by a negativelens or by a positive lens placed at a proper distance infront of the movable component. In the first case the"object" is virtual and the distance S21 between thefront and rear component must be less than 1/[(R)'- 1];in the latter case the "object" is real and S21 must begreater than 1/[(QR)- 1]. In most cases it is obviouslyadvisable to use a negative front component, especiallyif the varifocal system is part of a photographicobjective. In this case the system consists of a negative,fixed component followed by a positive, movablecomponent. The final image formed is real and inverted.This system is shown in Fig. 3(Pa-1). In some applica-tions, however, a positive front component may beused. For example, if the varifocal system is used aspart of a sighting system the fixed front component canbe used as the objective and the movable componentas part of the erector-eyepiece system. Both componentsare in this case positive and the final image formed isreal and upright. This system is shown in Fig. 3 (Pa-2).

For the N system, r= 1/(R. The front component isnow positive and the movable component is negative.The final image is virtual and inverted. This system isshown in Fig. 3 (Na).

We shall now consider the two systems obtained byusing the negative sign in Eq. (13).

The P system is shown in Fig. 3(Pb). The movablecomponent is negative and the front component ispositive.

The N system is shown in Fig. 3(Nb). Both compo-nents are now positive.

We note from Eqs. (16), (17), and (19) that in thetwo systems shown in Figs. 3(Pb) and 3(Nb) "object"and image are always on the same side of the movablecomponent and that in position z= of maximumimage deviation

ioitio =-I(z) = 11'( h mvb (19b)

That is, in position z= 2, the movable component

Vol. 52356

TWO-COMPONENT VARIFOCAL SYSTEM

crosses the final image plane and acts as a field lens.The final image formed is thus real for the positionsO<z<z of the movable component and virtual for thepositions <z<1.0. Moreover, for the same relativefocal range 61 and separation s21 these two systemsrequire a front component of a higher power than theother two systems. The two systems shown in Figs.3(Pb) and 3(Nb) thus have a smaller over-all focallength and consequently a larger value of the maximumimage-plane deviation than the systems of Figs. 3 (Pa-i),3 (Pa-2), and 3 (Na).

It is thus obvious that of the four possible basicsystems only the two systems shown in Figs. 3 (Pa)and 3 (Na), respectively, are of practical interest. Thiswas already pointed out previously.' We will refer tothese systems as the optimum systems, or simply, asthe PN or NP systems, respectively. In Fig. 4 theparameters of the two optimum systems are shown asfunctions of the focal range 61 (or r).

Once the parameters of the varifocal system properare determined the parameters of the rear auxiliarysystem can be readily found.'

2. Focal Range and Image-Plane Deviation

The relative over-all focal length of the varifocalsystem proper is given by Eq. (FL2 ). Using Eqs. (11)and (13) we obtain

f (z) = (r f2 (FL2*)1±(r-1)z

where f2 is the focal length of the front component. Themaximum over-all focal length is equal to (61)1f2.This gives

fp(max)= fp(0)= 1 {-[(6)'-1]S21}, (2OPa)(6)1-1I

for the optimum PN system, and

fN(max)=fN(1)=- () [+ 1 (2ONa)

for the optimum NP system. In Fig. 4 the maximumover-all focal length f(max) is shown as a function of61 for s21 = 0 and 2 1 = 1.0, respectively.

The relative image plane deviation is given byEq. (ID 2-b). Using Eqs. (9b) and (11) we obtain

z(1- Z)y(z,0)_=y(z)-y= -(r- 1)l O-1) .(ID2)

I+ (r- 1)z

Differentiating y(z) with respect to z and settingdy(z)/dz = 0 we find that the position of maximum imagedeviation is given by

1 1 r (r1-11 =-- a- -- 2 (1-g), (21)( i1 2L (r) i+ 1I

r -Z'I 2 'r

(f2) f2-s 1 2r .f7

Cm() I+ + L f 2c -i±ri -4 -5--'maJo I+L) 2 T . 4

fmax (fmox)c 1t1_TL71 3

...., i 1 1o'1 _LJ -2

C- C I --I 1---

t _ _ __+ IFI

; !1 II-I I1,11 1 I I t H 1 I I I I

-4 im(o)lZ

-64(fm ~I' ___

I [ _ _-II I -

,f Md H

FIG. 4. The parameters of the optimum two-component varifocalsystems as functions of the focal range T. fi is the normalized focallength of the movable component; f2 is the normalized focallength of the front component, and (f 2 )0=f 2 -S 2 1 , is the focallength of the front component when the distance S21 between therear principal plane of the front component and the front principalplane of the movable component in its extreme front position iszero; l1'(0) is the final image distance measured from the rearprincipal plane of the movable component in position z = 0;fmax is the normalized maximum over-all focal length of thesystem; ()o=fm-+((R) 1

S21 , is the maximum over-all focallength for the case when s21=O; and (fma) =(fmax)o- (1) isthe maximum over-all focal length when S21 = 1.0.

whereA (r)i-1

(r)i+1

(61)12-1(an 1gl=

and 1g] =(61) 1 ± 1.(22)

Substituting Eq. (21) into Eq. (ID2 ) we find the imageplane deviation at z= z. Thus,

y(z)-yo= -(ri- 1)/(rap 1) -g. (23)

If full compensation at z=O and z= 1.0 is notrequired, optimum conditions are obviously obtainedwhen the focal plane is adjusted so that y(O) = -y(z)=y(l). Then,

y(O)l = ly(Z)| = Iy(1)| =y(max)= 1lg1 =21((R1)/((R1+1), (24)

and the system is fully compensated in the two positionsof the movable component given by

ZI,2 =[2-gF (2-g2 )i]. (25)

N

April 96i2 357

\

LEONARD BERGSTEIN AND LLOYD MOTZ

-. 0_ -8--.4 -- 0 2 4 . . .0_

_ I O -.8 .6 -.4 .2 0 .2 .4 .6 .8 1.0--,

FIG. 5. The maximum value y2 (max) of the normalized image-plane deviation y(z) of two-component varifocal systems withoutfull compensation at both ends of the operating range (and anoptimized image deviation function) as a function of the focalrange T. The image-plane deviation y(z) as a function of thedisplacement z of the movable component is shown above thegraph of y2 (max).

However, if full compensation at both ends of theoperating region is required, y (0) =y(1) = 0, and

I y(i) =y*(max) = gl = (&i-1)/ ((Ri+ 1). (24*)

The maximum value of the relative image-planedeviation is shown in Fig. 5 as a function of the focalrange (R (or T).

The relative image-plane deviation of the varifocalsystem proper depends only on the relative focal rangeand is independent on the particular design chosen.This is not true, however, for the image-plane deviationof the complete varifocal system (including the auxiliarycomponent).

The image-plane deviation of the complete varifocalsystem is given by'

FT(max) Y (z) 1 FT (max)YT(Z)=- I I-

Zm L'-(max)i Zm,(26)

where FT(max) is the maximum over-all focal lengthof the varifocal system and (z) I y(z)/f2 (max). UsingEqs. (20a) and (24) we obtain

YT(max) I- 1 - 1-( (max)= -* (27)

FT2 (max)/Zm L2(R1(Ri+1) )f 22

This gives (611-1)3 1 2

()3p(max) = _ - (27Pa)261((Ri+1) 1-(1-l)S21

for the optimum PN system, and

((R 1)3 1 2

ON (max)= (R1R+1) ±(6111)S21/611 , (27Na)

for the optimum NYP system.In Fig. 6 E) (max) is shown as a function of 6R (or of r)

for 21=0 and 21=1.0, respectively. It is seen thatE (max) of an N system is smaller than that of a Psystem. The image-plane deviation of an N\ system is

thus smaller than that of a P system having the samefocal range (, separation 2l and maximum displace-ment Z,,, of the movable component. This is especiallypronounced for large values of (R. Dividing Eq. (27Na)by Eq. (27Pa) we obtain

YTN(max) )N(max)

YTp(max) Ep(max)

I r (R- 1)s21 12=-I 1- 1-

Cq. (611)+ (y- 1)slJ(28a)

We also note that while E(p (max) increases with increasingfocal range, EN (max) reaches its maximum value at a cer-tain finite value of (R and then decreases with increasing(R. [When S21=0, ON (max) reaches its maximum valueof 0.0125 at R=7.464 (or r=-0.764); when 21=0.5,

N)N(max) reaches its maximum value of 0.0073 atR=6.156 (or =-0.721); when 21=1.0, N)N(max)

reaches its maximum value of 0.0049 when 61=5.462(or r= -0.691).]

3. Application to Sighting Systems

Sighting systems (telescopes, viewfinders, etc.) maybe divided into systems in which a real image is formedwithin the system and systems in which no real imageis formed. The first type includes the "astronomical"systems, in which the final image is inverted withrespect to the object, and standard "terrestrial"systems, in which the final image is erect. The astronom-ical system is composed of a front objective, which

IIII I - I . 1 1 11 l IIIII-~~~~~.05

-f X ---ru 911111

-71.0 -. 8 -.6 -.4 -2 0 .2 .4 .6 .8 I.0-r

FIG. 6. The maximum value, YT(max)= [FT2(max)/Z,,]X E) (max), of the image-plane deviation of optimum two-compo-nent varifocal systems without full compensation at both ends ofthe operating range (and an optimized image deviation function)as a function of the focal range T. [(max)]o is the value of0(max) for the case when S21=0 and [e)(max)]i is the value ofI O(max) for the case when S21= 1.0; S21 is the distance between therear principal plane of the front component and the front principalplane of the movable component in the position z=O. FT(max)is the maximum over-all focal length of the system and Zm is themaximum displacement of the movable component.

358 Vol. 52

TWO-COMPONENT VARIFOCAL SYSTEM

forms a real inverted image of a distant object, and arear positive lens, the eyepiece, which forms the finalinverted virtual image. In a standard terrestrialsystem an erecting system is placed between theobjective and the eyepiece in order to form a realerect image of the object which is then magnified by theeyepiece. Sighting systems in which no real images areformed within the system are referred to as Galileansystems. In a Galilean system the inverted virtualimage formed by the objective is reinverted by the useof a diverging eyepiece lens.

Any sighting system can thus be considered asconsisting of two parts, a front objective and a rearsystem, the eyepiece or erector-eyepiece combination.If FOB is the focal length of the objective and FE thatof the eyepiece or the erector-eyepiece combination,the magnification M of the system is given by

M= -FOE/FE. (29)

Variable magnifying power can be obtained by in-corporating a varifocal system in either one of the twoparts of the sighting system, the objective or theerector-eyepiece combination. If the simplest varifocalsystem, i.e., the two-component system, is used, thesighting system will consist of three components: afront, stationary component, a movable componentand a rear, stationary component. Either one of thethree systems shown in Figs. 3(Pa-1), 3(Pa-2), and3(Na), respectively, can be used. The system ofFig. 3(Pa-1) can be used as either one of the threepossible sighting system types, with the movablecomponent as part of the objective; the system ofFig. 3(Pa-2) can be used either as an astronomicalor as a terrestrial system, with the movable componentas part of the eyepiece-erector system, whereas thesystem shown in Fig. 3 (Na) can be used only as aGalilean system in which the movable component ispart of the eyepiece-erector system. (For the purpose ofanalysis we will in all cases consider the movablecomponent as part of the objective.)

Let M(max) be the maximum magnification of thesighting system and a its magnification range, i.e.,61=M(max)/M(min). The focal lengths, F 2 =Zmf 2and F=Zmfi, of the front objective and the movablecomponent are given by Eqs. (13) and (14), respec-tively. For the focal length of the rear fixed component,i.e., of the erector-eyepiece lens, we obtain [usingEqs. (FL2 *) and (29)],

A!(f2& ,=- Z_. (30~)

M (m x)

The distance between the rear principal planie of themovable component in the position z=O and the frontprincipal plane of the rear fixed component is given

by

(Si)P= [I+ (axf]{( 1 )6a- 1 M (max) I RI- 1

[M(max)- 1 (RGS21 1X 1 1± Z m,

LM(max) I M(max)(3lPa)

for the systems of Figs. 3 (Pa-1) and 3 (Pa-2), and

(S1)N=[- I +± (a jZm= ( i 1

R 61 _ R X L-1 1 Ma Zm,

M (max) M (max) (3lNa)

for the system of Fig. 3 (Na).Because of the image-plane deviation of the varifocal

component, the sighting system will be perfectlycollimated only for two positions of the movablecomponent. The deviation from perfect collimation atall other positions of the movable component can bereadily found.

Let Y(z)=Zmy(z) be the image-plane deviation ofthe varifocal objective. Then the light beam thatleaves the sighting system will not be collimated butwill either converge to, or diverge from a point that isat a distance Xo'(z) = -Fo 2 /Y(z) from the rear focalplane of the erector-eyepiece lens. If we express thisdeviation in diopters we have &0(z)=-Y(z)/Fo2 .Using Eqs. (15) and (28) we obtain

Z, y(z) [M2 (max)1[ y(z)]

F02 Zm f2 (max)

M2 (max)=- @(z),

Zmand

M2 (max)8q (max) = - -(3(max),

Zm

(32)

(33)

where ®(max) is given by Eq. (27Pa) for the systems ofFigs. 3(Pa-1) and 3(Pa-2) and by Eq. (27Na) for thesystem of Fig. 3(Na).

III. EXAMPLES

Example 1: Varifocal Projection System

As a first exampl)le we sliall consider a varifocalprojection system with all over-all focll lengthd varyingcontinuously between 20.0 and 30.() mm. We shallassume that the maximum permissible image-planedeviation is 0.05 mm.

April 1962 359

LEONARD BERGSTEIN AND LLOYD MOTZ

The relative focal range of the system is only 1.5.It should therefore be possible to keep the image-planedeviation within the permissible value by incorporatingjust a single movable component in the varifocal system.Of the two optimum systems the NP system will bechosen since it has a smaller image-plane deviationthan the PN system. The system will thus consist of apositive front component, a negative movable compo-nent, and an auxiliary positive rear component.

We have

(r)=1/6l=1/1.5 and r=-0.2.

Using Eqs. (24) and (26Na) we find that if full compen-sation at both ends of the operating range is notrequired

y(max) = 0.05051,and

YT(max) 0.001134_______85S1= ® (max) =

FT2 (max)/Zm (1+0.18350s22)2

The minimum separation required between the twoequivalent thin lenses of the varifocal system proper(in position z=O) is s21(min)=O. To account for thefinite thicknesses of the components we choose s21 = 0.3.We then obtain

FT2 (max)YT(max)=0.00102

Z m

The maximum displacement of the movable compo-nent must therefore be Z (0.00102)(900)/(0.05)mm= 18.40 mm, in order that the image-plane deviationdoes not exceed the permissible value of 0.05 mm.Choosing

we obtainZ.= 20.0 mm,

YT(max) = (0.00102) (45.0) mm= 0.046 mm.

Using Eqs. (13Na), (14Na), (FL2 *) and (17Nb), wefind the followlng values for the thin-lens parameters ofthe two-component system:

Fl=- (2.4495) (20.0) mm= -48.990 mm,

S 21 = (0.3) (20.0) mm= 6.000 mm,and

F2 = + (5.4495+0.3) (20.0) mm= + 114.990 mm.

The final image distance of the front system is

1.5(1-z) Ll'(z)=-5.4495- +.5(-z) (20.0) mm,

and the maximum over-all focal length is

F(max) = - (7.0417) (20.0) mm= -140.833 mm.

The complete system is to have a maximum over-all

focal length,FT(max)= 30.0 mm.

The reduction factor of the auxiliary system is therefore

W=- 140.833/30.0= -4.6944.

The minimum separation (in position z=O) betweenthe movable component and the auxiliary system is,assuming zero thickness of the components, so(min)= 1.0. Since the auxiliary system will be a compositesystem comprising more than one lens element wechoose

Si 0 = (1.5) (20.0) mm= 30.0 mm.

This gives Lo=-118.990 mm. The focal length ofthe auxiliary system becomes

Fo= (118.990/5.6944) mm= +20.896 mm,

and the final image distance is

Lo/= (118.990/4.6944) mm= + 25.347 mm.

The physical length of the system is

S= (1.8) (20.0) mm= 36.0 mm,

and the over-all distance between the front componentand the image plane is

L= (36.0+25.35) mm= 61.35 mm.

Assuming that the system has a relative aperture of0.25 and covers a field varying from 150 (in position ofminimum over-all focal length) to 100 (in position ofmaximum over-all focal length) we find that thediameters required for the aperture are (Dia2) A=7.50mm, (Dial)A=5.81 mm, and (DiaO)A=6.34 mm, andthat the diameters required for the field are (Dia2 )F=7.17 mm, (Dial)F=5.23 mm, and (Diao)F= 1.31 mm,respectively.

If a PlY system is chosen,

(r)p=6{=1.5 and r=+0.2,and

0.001701

(max) = (1-0.22475s2 ,)2

If we choose as before, s2 1 = 0.3 and Z,,,= 20.0 mm,we then obtain

FT2 (max)Yr(max)=0.00196 = 0.088 mm.

Zm

The thin-lens parameters of the system are now

1 1-+ (2.4495) (20.0) nwI- +48.990 inni,

S2 1 = (0.3) (20.0) = 6.000 mm,an(l

F.= - (4.4495-0.3) (20.0) mm= -82.990 mm.

360 Vol. 52

TWO-COMPONENT VARIFOCAL SYSTEM

Furthermore,

1.5zLI'(z)= (5.4495- )20.0 mm,

1+0.5z

and

F(max) = +5.0821(20.0) mm=+ 101.641 mm.

The reduction factor of the auxiliary system is now

W= (101.641/30.0) = 3.3880.

Choosing as previously, S10 = 30.0 mm, we obtainLo= +78.990 mm, and consequently

Fo= (78.990/2.3880) mm=33.077 mm,and

Lot= (78.990/3.3880) mm= 23.314 mm.

The physical length of the system is S=36.0 mm,the same as that of the NP system, and the over-alldistance between the front component and the imageplane is

L= (36.0+23.314) mm=59.314 mm.

For a relative aperture of 0.25 and a field varyingbetween 15° and 100, the diameters required for theeaperture are (Dia2)A= 7.50 mm, (Dial)A= 8.04 mm,and (Diao)A= 5.83 mm, and the diameters required forthe field are (Dia2)F=5.82 mm, (Dial)F=5.01 mm,and (DiaO)F= 1.26 mm, respectively.

Both systems, the NP and PN, are shown schemat-ically in Fig. 7. Also shown in the same figure are thefocal range and the image-plane deviations of the twosystems.

Example 2: Varifocal Viewfinder

As the second example we shall consider a varifocalviewfinder with a magnification varying continuouslybetween 0.5 and 2.0.

__q- O + _

104-30. 0

130 5(1 ,.1

Y(z) = - [. 0458- 4538 z(Z) O lmi0.I 5 0 (1-.)

NP SYM1.0N P SYSTEM

2. 0M(z) =

1 +3(1-z)

2. 01.5sF1. 0_0. 5

0

It l

Il

6 +Z) = [- x + 191J) 3 diopters

+0.

0

0 1.0

FIG. 8. The varifocal viewfinder of example 2, its magnificationrange M(z) and image plane deviation a+(z) in diopters.

We assume that the sighting system covers a smallfield so that a two-component system can be used.Using the relations developed in the preceding chapterit is readily shown that if 8(max) is the maximumpermissible deviation (from perfect collimation) indiopters, M(max) the maximum magnification of thesighting system, then

F22 I,-1 M2 (max)

Zm 2G+1) L84(max) I

where F2 is the focal length of the front component andZm is the maximum displacement of the movablecomponent. For

R=4.0and

this givesFT(5) = 30.0

M (max) = 2.0

F22 1

Z,, 6 (max)

The normal human eye can accommodate within-+0.5 diopter. We therefore set

YT(z) = [.0880 - .8711.(1 -) Im

-f l I -Zo I-. ioH YSTE

106PN SYSTEM

FIG. 7. The two optimum two-component projection systems ofexample 1, their focal range f(z) and image-plane deviation y(z).

This gives

Choosing

84 (max)= 0.5 diopter.

F22 /Zm ) 333.3 mm.

Z,= 30.0 mm

we obtain F2 > 100.0 mm. We set

F2 = + (3.333) (30.0) mm= + 100.00 mm.

r | @ I-

361April 1962

-7it . W _+_.1

362 LEONARD BERGSTrEIN AND LLOYD MOTZ Vol. 52

If we choose the system of Fig. 3 (Na) (assuming that image-plane deviation in diopters are shown schemat-the viewfinder can be of the Galilean type) we obtain ically in Fig. 8.

If the system of Fig. 3(Pa-1) is used we obtainF1= - (0.667) (30.0) mm= - 20.00 mm,

S21 = (1.333) (30.0) mm= 40.00 mm, F1 = + (0.667) (30.0) mm=+ 20.00 mm,Fo= + (3.333) (30.0) mm= + 100.00 mm, S21= (4.333) (30.0) mm= 130.00 mm,

S10= (2.333) (30.0) mm= 70.00 mm, Fo= + (3.333) (30.0) mm= + 100.00 mm,S10= (5.333) (30.0) mm= 160.00 mm,

and a physical length of approximately 110.0 mm.The sighting system, its magnification range and and a physical length of approximately 290.0 mm.