JOURNAL OF THE OPTICAL SOCIETY OF AMERICA Four-Component Optically Compensated Varifocal System* LEONARD BERGSTEIN Polylechnic Institute of Brooklyn, New York AND LLOYD MOTZ Rutherford Observatory, Columbia University, New York, New York (Received May 10, 1961) The general theory of optically compensated varifocal systems is applied to the case of a four-component system consisting of four alternate stationary and movable components. The second and fourth components, counting from the object side, are interconnected and displaced in unison to change the over-all focal length of the system. An iteration method for the solution of the varifocal equations is developed which enables the determination of the Gaussian parameters of the system for any predetermined focal range in a matter of minutes (without the use of computers). Using this method, explicit expressions are found for the approximate values of the parameters of the optimum four-component systems as functions of the focal range. A numerical example is given to illustrate the iteration method. I. INTRODUCTION ALTHOUGEl the analysis of the two- and three- componentl' 2 varifocal systems is quite important for illustrating in the most simple fashion the applica- tion of the general theory, 3 it is in the application of the theory to the four-component systems that we find its greatest practical usefulness. The reason for this lies in the fact that the additional point of full compensation that an added component brings with it results in an enormous reduction (by at least a factor of 20) in the maximum image-plane deviation of the varifocal system at no increased complexity or over-all length. Moreover, the four-component system is the simplest one for which it is possible to achieve a satisfactory correction of all the image aberrations over its entire operating range. These two factors enable the design of well-corrected four-component varifocal systems with a large operating range of focal lengths. Indeed, by using the theory developed in this work, four-component varifocal systems with a focal range of 6: 1 have been designed by one of the authors and are presently used in photog- raphy, television, and sighting systems. (A four- component optically compensated varifocal system that is almost diffraction limited over its entire focal range of 4:1 was recently designed for a maximum relative aperture of 0.5 and a maximum field of 300). The introduction of an additional component into the varifocal system will, of course, complicate the mathematical analysis considerably because one is led to a system of four simultaneous nonlinear algebraic equations. If one were to try to solve these equations by conventional methods, the attempts would be fruitless and the four-component systems would be of theoretical interest only. However, an iteration method is developed in this paper that does not require the * This work was submitted by L. Bergstein in partial fulfillment of requirements for the Ph.D. degree at Polytechnic Institute of Brooklyn, New York. 1 L. Bergstein and L. Motz, J. Opt. Soc. Am. 52, 353 (1962). 2 L. Bergstein and L. Motz, J. Opt. Soc. Am. 52, 363 (1962). 3 L. Bergstein, J. Opt. Soc. Am. 48, 154 (1958). solution of nonlinear equations and leads to a solution in a matter of minutes. Using this method, explicit expressions are found for the approximate values of the parameters of the optimum four-component systems as functions of the focal range. II. APPLICATION OF THE GENERAL THEORY TO THE FOUR-COMPONENT SYSTEM 1. The Varifocal Equations The four-component varifocal system consists of four alternate stationary and movable components. The second and fourth components, counting from the object side, are interconnected and displaced in unison to change the over-all focal length of the system. A fifth fixed component, not considered part of the verifocal system proper, is usually placed behind the last movable component. This is illustrated in Fig. 1. Using the same designation for the lens components as that in reference 3, we assign the subscript 4 to the front-fixed component, subscript 3 to the first movable component, subscript 2 to the second stationary component, subscript 1 to the rear movable component, and subscript 0 to the auxiliary component. The seven Gaussian parameters of the varifocal system proper are thus the focal lengths F 4 , F 3 , F 2 , and F, of the four components and either the spacings S 43 , S32, and S 2 1 between the rear and front principal planes of two consecutive components or the separations D 43 , D 32 , AUXILIARY I SYSTEM VARiFOCAL SYSTEM PROPER Y PLANE APERTURE STOP FIG. 1. The four-component optically compensated varifocal system. 376 VOLUME 52, NUMBER 4 APRIL, 1962
LLOYD MOTZRutherford Observatory, Columbia University, New York, New York
(Received May 10, 1961)
The general theory of optically compensated varifocal systems is applied to the case of a four-componentsystem consisting of four alternate stationary and movable components. The second and fourth components,counting from the object side, are interconnected and displaced in unison to change the over-all focal lengthof the system. An iteration method for the solution of the varifocal equations is developed which enablesthe determination of the Gaussian parameters of the system for any predetermined focal range in a matterof minutes (without the use of computers). Using this method, explicit expressions are found for theapproximate values of the parameters of the optimum four-component systems as functions of the focalrange. A numerical example is given to illustrate the iteration method.
I. INTRODUCTION
ALTHOUGEl the analysis of the two- and three-componentl'2 varifocal systems is quite important
for illustrating in the most simple fashion the applica-tion of the general theory,3 it is in the application of thetheory to the four-component systems that we find itsgreatest practical usefulness. The reason for this liesin the fact that the additional point of full compensationthat an added component brings with it results in anenormous reduction (by at least a factor of 20) in themaximum image-plane deviation of the varifocal systemat no increased complexity or over-all length. Moreover,the four-component system is the simplest one for whichit is possible to achieve a satisfactory correction of allthe image aberrations over its entire operating range.These two factors enable the design of well-correctedfour-component varifocal systems with a large operatingrange of focal lengths. Indeed, by using the theorydeveloped in this work, four-component varifocalsystems with a focal range of 6: 1 have been designed byone of the authors and are presently used in photog-raphy, television, and sighting systems. (A four-component optically compensated varifocal systemthat is almost diffraction limited over its entire focalrange of 4:1 was recently designed for a maximumrelative aperture of 0.5 and a maximum field of 300).
The introduction of an additional component intothe varifocal system will, of course, complicate themathematical analysis considerably because one isled to a system of four simultaneous nonlinear algebraicequations. If one were to try to solve these equationsby conventional methods, the attempts would befruitless and the four-component systems would be oftheoretical interest only. However, an iteration methodis developed in this paper that does not require the
* This work was submitted by L. Bergstein in partial fulfillmentof requirements for the Ph.D. degree at Polytechnic Institute ofBrooklyn, New York.
1 L. Bergstein and L. Motz, J. Opt. Soc. Am. 52, 353 (1962).2 L. Bergstein and L. Motz, J. Opt. Soc. Am. 52, 363 (1962).3 L. Bergstein, J. Opt. Soc. Am. 48, 154 (1958).
solution of nonlinear equations and leads to a solutionin a matter of minutes. Using this method, explicitexpressions are found for the approximate values ofthe parameters of the optimum four-componentsystems as functions of the focal range.
II. APPLICATION OF THE GENERAL THEORYTO THE FOUR-COMPONENT SYSTEM
1. The Varifocal Equations
The four-component varifocal system consists of fouralternate stationary and movable components. Thesecond and fourth components, counting from theobject side, are interconnected and displaced in unisonto change the over-all focal length of the system. Afifth fixed component, not considered part of theverifocal system proper, is usually placed behind thelast movable component. This is illustrated in Fig. 1.
Using the same designation for the lens componentsas that in reference 3, we assign the subscript 4 to thefront-fixed component, subscript 3 to the first movablecomponent, subscript 2 to the second stationarycomponent, subscript 1 to the rear movable component,and subscript 0 to the auxiliary component. The sevenGaussian parameters of the varifocal system properare thus the focal lengths F4, F3, F2, and F, of the fourcomponents and either the spacings S43, S32, and S 21between the rear and front principal planes of twoconsecutive components or the separations D43, D32,
AUXILIARYI SYSTEM
VARiFOCAL SYSTEM PROPER Y
PLANEAPERTURE
STOP
FIG. 1. The four-component optically compensatedvarifocal system.
376
VOLUME 52, NUMBER 4 APRIL, 1962
377FOUR-COMPONENT VARIFOCAL SYSTEMApril 1962
AUXILIARY SYSTEM
ELEMENT 4 ELEMENT 3 ELEMENT2 ELEMENT I
-4,3 t - 3,2 -- S2, bi - t1 o)lt-Al Referencef -- position of the
d ~ ~ ~ _~ -d I - bo
ef 54 3 rfZtf? 5d3,2-1z A-2.1111I 2-- J
4f -X43+ image plane---x-A-x--- 1 X x x.Al_ -1 i i Limageplne-- XI __X X__ -_ - - in the original
F~~' F , F' F2' Fj' F.' ~~~~~~~~positionl of the
4; 3 F4" F3 \ F~w F2 W F2 Fit \ F." \ I to W movableI- z- V components
.- dprur tl y (ZS4, 3+z~ ~~~~~~~~ S3.Z(Z - =2 -zty yzzo oA (z =Image plane
I ~~~~~~~~~~~~~in the position-xt-l -- z of the
F~~ F 1. F;' F F; F"~ ] movable
Aperture cmoet
Z .x~~~~~~~~I,(z) z+y(zo).~
f(z) f f 3ft z3+ bz2+b2z+b 3 (N system)
(P system)
O ~~1.0-Z( Z-Z,)(Z-Z2)(Z-Z;(Z-Z4)
zT+_b,_zz+bz+b., y(z) for Psysfems; 4 { y(z) for N systems
Z2 ,3
FIG. 2. The four-component optically compensated varifocal system and its Gaussian parameters, its over-all focal
length f(z) and image-plane deviation y(z) as functions of the normalized displacement.
and D21 between the rear and front focal planes oftwo consecutive components (see Fig. 2).
Let Z be the displacement of the movable componentsfrom their original position in the direction from theobject side to the image side, and let Zm denote themaximum displacement. We normalize all systemparameters with respect to Zm and abbreviate allnormalized parameters by lower case letters, i.e., withk=4, 3, 2,1,
A F A Dkk- 1 A Skk-Ifk=-, dk_1= Z , Zkkm1=
Z,, Zm Z""n(1)
where the distances dkk-, and Skk-1 refer to the originalposition of the movable components. We also introducethe normalized variable
Z=Z/Zm.
We now form the normalized "brackets"
I1(4,1)=b 1= d 4 3-d 3 2 +d 2 l,
lI2(4,1)b2= -d 4 3d3 2+d4 3d2l-4d 2d2l+fi 2+f22,
we obtainz2+alz+a2
xi' (Z)=f bz 3 +biz2 +b2 z+bi
(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 movablecomponents, is given by
Y(Z) =Z4 +CIZ3 +C2Z2 +C3Z+C4
z 3+biz 2 +b 2z+b 3
(ID 4-a)
where
c1=b- (x'-yo),
c2 = b2 - (x'-yo)bi+fl 2 ,
c3 =b3 - (x'-yo)b2+fi 2 ai,
C4=yob3;(2)
yomy(O), and
(5)
(6)a2
X, a X1/ (0) = fi2-.b3
1/3 (4, 1) b3= -d4 3d32d2 +d4 31f22+fa2d2l;
Ik (4,2) = a, = d 4 3 - d32,
4/2 (4,2) a2= -d 43d32+ f32.
(3)
Using the relations developed in reference 3 we findthat the normalized over-all focal length of the systemas a function of the displacement z is given by
f l f 2f3 f4
z3 +b 6z 2 +b 2 z+b3
For the normalized final image distance (measuredfrom the rear focal plane of the rear movable component)
(FL4)
The four-component varifocal system has fourpoints of full compensation. Equation (ID4-a) cantherefore be rewritten into
y (Z)(Z-ZI) (Z-Z2) (Z-Z3) (Z-z 4 )
z3+b 1z2+b 2z+b 3
Z4- YZ 3l iZ2 - Y3Z+Y4
z +biz 2 +b 2 z+b 3
(ID 4-b)
where z, Z2, Z3, and Z4 are the points of full compensa-tion, all chosen to be within the operating range of the
- R
- En, r is � z a._
s
=-
-
y
LEONARD BERGSTEIN AND LLOYD MOTZ
system, i.e., 0•z1<z 2<z4<1.0, and
7Y= Zi+Z 2+Z 3 +Z4,
72= ZIZ2+Z1Z3+ZlZ4+Z 2Z3+Z 2 Z4 +Z3Z4 , (7)
73= ZIZ2Z3+ZiZ2Z4 +ZZ 3Z4+Z3Z3Z4,
74 = Z1Z2Z3Z4.
If ful, compensation at both ends of the operatingrange is desired, y(0)=y(1)=0, z1=O and Z4 = 1.00, and
7i= l+z2+z3,72=z 2+z3 +z2 z3, (7a)73= Z2Z3,
we require that
f 3+d 32 +f 2 = s 32> 32 (min),
f 2 +d 21+fl s21> S2 1 (min), (VF 4-IJJ)
where s32 (min) and s21(min) are the physically smallestpossible spacings between the principal planes of thethird and second components and the second and firstcomponents, respectively, in the position z=0 of themovable components.
After the six equations (VF4-I), (VF4 -II) and(VF4-III) are solved for the six unknown systemparameters f, f2, f, d43, d32, and d2l, the focal lengthof the front component is found from
f 4=s43-d 43-f 3, (VF 4-IV)
The focal range f(z) and image plane deviationy(z) of the four-component system are shown inFig. 2 as functions of the displacement z of the movablecomponents.
Of the seven Gaussian parameters of the system six(namely f3, f2, fi, d43, d32, and d2j) are determined bythe predetermined points of full compensation, by therequired relative focal range and by the conditionsfor physical realizability.
The points of full compensation will be located inthe predetermined positions if
bj- (x'-yo)= 7-,b2- (x'-yo)bl +fi2 = 72, (VF4-I)
b3- (x'-yo)b2+f2aj -73,
where x'= f 2a2/b3 and
y0=Y41/b3. (8)
To obtain the required relative focal range GR andthe desired type of system, P or N,4 we require that
(r-1)b3=1+bj+b2. (VF4-II)
Furthermore, in order that the system is realizable
4 The "relative focal range" (R is defined as the ratio of maximumto minimum over-all focal lengths of the system, i.e.,
r ,3 = I. .i.Fmin fmin
For any required focal range (R two lens types are possible. Eithera lens system can be chosen which will have the maximum focallength when the movable components are in the extreme frontposition or a system can be chosen which will have its maximumfocal length when the movable components are in the rear position.We refer to the first system as the P system and the latter as theN system. We define the "focal ratio"
A F(0) f(0)
For the P system r=(R, whereas for the N system r 1/(6. Wealso introduce the normalized range
Af(O)-f(1) r-1 (R-17-f(0)+f (1) =~ = 1(R+1'
where the positive sign applies to the P system, the negativesign to the N system. We note that 1<R1<o, <r<co, and
where S43 must be greater than the minimum separationrequired between the fourth and third components butcan otherwise be chosen arbitrarily.
If no components are used whose principal planes areat an appreciable distance from the respective physicalboundaries
S2 (in)zt 0,
s32 (min) 1.0,
s43(min) - 0.(9)
2. Location of the Points of Full Compensation
To find the distribution of the points of full compensa-tion that optimizes the image deviation function wefollow the approximation method described in reference2.
The image deviation function y(z) of the four-component system is given by Eq. (ID 4-b). This func-tion (see Fig. 2) is obviously an optimum, i.e., itsmaxima within the operating range of the lens areminimized, when the absolute values y(0) I and y(1) Iat both ends of the operating region and the absolutevalues y1,2, 2,3, /3,4 of its three maxima are equal, thatis, when
IyY(O) I=Y1,2=Q2,3=/3,4= Iy(1) . (10)
Setting z=0 and z= 1.0 in Eq. (ID4-b) we obtain[(r-1)b 3 ]y(0) and [(r-1)b 3 ]y(1), respectively, interms of the points of full compensation. To find thevalues of the three maxima, y1,2, 2,3, and 3,4, weapproximate the denominator of the image deviationfunction by a straight line which coincides with theexact polynomial at the points z=0 and z= 1.0, thatis, we set
[(r_ 1)b3]Y(Z) : (Z-Z) (z-z 2 ) (z-z 3 ) (z-Z 4)
(1-r)/2r+z- (11)
where = (r-1)/(r+1), as defined previously. Wemake a further approximation and assume that themaxima positions (%1,2, z2,3, 3. 4) of the image deviationare located midway between the points of full compen-sation, i.e., Z1,2 2 (Zl+Z2 ), Z2,3 2 (Z 2+Z ), and z3 4
378 Vol. 52
379FOUR-COMPONENT VARIFOCAL SYSTEMApril 1962
nZ4(Z3+Z4 ). We can then in Eq. (11) set alternatelyZ=z1 ,2 Z= 2 ,3, and Z=Z3,4 and find [(r-1)b3]1i,2,[(r-l)b3] 2 ,3, and [(r-1)b3]y 3 ,4, respectively, interms of the points, z1, Z 2, Z3 , and Z4 , of full compensation.Equating the values of ly(O)I, y(1)1, 1,2, 2,3, andY3,4, we thus obtain four equations which can be solvednumerically for the four points of full compensation.
In most cases it is desirable that the system isfully compensated at both ends of the operating range.5
In that case, y(O)=y(l)=O, and optimum conditionsare obtained when
Y1,2= P2,3= Y3,4.
Setting, zi=0 and Z 4 = 1.0, Eq. (11) becomes
(lOa)
0 _.293- z2 O.707-Z _
_ _ _ ~~~~I OL_ F . - -
ret 1f rt .0 3 6 I
-. 0I l II6-.10~~~~~~~~~1
FIG. 3. The optimum distribution of the points (ZI,Z2,Z3,Z4) of
full compensation of a four-component varifocal system withfull compensation at both ends of the operating range as a functionof the focal range T. zI=O, z2=0. 2 9 3 -E, z3=0. 7 0 7 -e, Z4= 1.0.
We therefore introduce
Z(z-Z2) (Z-Z3) (Z- 1.0)I (1la)(1- r)/27+z
We must now determine the values of Z 2 and Z3
that Eq. (1Oa) is satisfied.First we note that when OR=1.0 (=0), Eq. (
becomes b3[Y(Z)],=O= z(z-Z2) (Z-Z 3) (z-1.0), andpoints of full compensation that optimize y(z) are giby
f cos(2i- 1)22.5]]l(i=24 1 -
cos22.5 )
(z)o= 1 (- tan22.50) = 1--= 0.293,2
(z3 ),= 4(1 +tan22. 5 0) =- 0.707,2
such
1 1.
Z2= 1 0---e'=0.293-e',2
V2 =-- _ e" = 0. 707 - "2
(15)
the where the absolute values of e' and e" depend only oniven the relative focal range iR; they are positive for P
systems and negative for N systems. Assuming, asbefore, that the maxima of the image-deviation function
(12) are located midway between the zeros, i.e.,
Z:-- 1 z2= 2 (1 Nr2_ e'
(12a)
(12b)
and 4 = 1.0. For values of (R greater than 1.0, r ispositive for P systems and negative for N systems. Itis thus evident from Eq. (la) that if (z 2 )p and (Z3 )p
are the points of full compensation that optimize theimage deviation function of P systems, (Z2)v and
(Z3)N the points of full compensation that optimize theimage deviation of N systems, then
I0< (Z2)P<0.293 I 1.293 < (Z2)N< 1.0 (13)
0< (3)P<0. 7 07 0.707< (Z3)N<1.0
Moreover, it was shown in reference 2 that
(Z 2)N=l (Z3 )P,
(Z3)N= 1-(Z 2)P. (14)
5 In reference 3 it is shown that under optimum conditions themaximum value of the image-plane deviation of a four-componentsystem with full compensation at both ends of the operating rangeis approximately i times the maximum value of the image-planedeviation obtained when full compensation at both ends of theoperating range is not required. The difference in the level of thetwo maximum values is thus small.
Setting P1,2= P2,3= P3,4 we obtain two equations whichcan be, for any given value of r, readily solved numer-ically for e' and e". It is found that e' and e" areapproximately equal for all T. This is illustrated inFig. 3, where 6'z 6" is shown as a function of r. If thisfunction is approximated by an odd polynomial offifth degree in r with points coinciding with the foundsolution at T=0(61= 1.0), r= -t0.3((R= 1.857), r=+0.6(a=4.0), and T= +0.9((R=19.0), respectively, we
[(r- )b3]y(z)
or, z=O,
--
LEONARD BERGSTEIN AND LLOYD MOTZ
obtain y(z)
em2 (' e") ;~-'z ' e"~: O. 100Tr
X (1+0.154T2+O.274r4). (18)
To find the maximum value of the image-planedeviation the values of e can be substituted into eitherone of the equations (17), or we may set
The result is illustrated in Fig. 4 where [(r-l)63]Xy(max) is plotted as a function of r. This function canbe closely approximated by an odd polynomial of fifthdegree in with points coinciding at r=, 0.3,+t0.6, and -+0.9, respectively. We obtain
The six system parameters, f3, f2, fl, d43, d32, and d21,are determined by the six varifocal equations. In termsof the unknown system parameters, the varifocalequations (VF4 -I), (VF4 -II) and (VF4 -III) read
-I
+ y5(max)iy3(max x) -0 Z -
[(r- )b3 ] Y4(max)
._024- _
.0 6
-.008---z
A-P-- - l n 4 a P ir
FIG. 4. The value of (3 3)Y4(max) as a function of the focalrange ; y4(max) is the maximum value of the normalized image-plane deviation of a four-component varifocal system with fullcompensation at both ends of the operating range and an optimizedimage deviation function, 63(r-l)b3=[2r/(1-r)]b3 and b isa Gaussian bracket of the system. The image-plane deviation v(z)as a function of the displacement z of the movable component isshown above the graph of ,83y4(max).
where Y4=0 in case the system is fully compensated atboth ends of the operating range, and S 3 2 and
2 1 canbe arbitrarily chosen except that 3 2>s32 (min) andS21> s 2 1(min).
An attempt to solve the above six simultaneous non-linear equations by standard methods appears to befruitless. [Eliminating two of the unknown parametersfrom the last two equations of (21) would only com-plicate the problem rather than simplify it.]
One way of solving the varifocal equations is to usea method similar to the one described previously forthe solution of the varifocal equation of a three-component system. The method is based on theproperties of the "Gaussian brackets" and on the factthat in solving the varifocal equations only Eqs. (VF-I)and (VF-II) must be satisfied exactly but not therealizability conditions (VF-III), since only approxi-mate values or a region of values for the separations s32and 21 will ever be specified. Thus, we can proceed asfollows.
The four varifocal equations (VF4-I) and (VF4 -II)contain explicitly six unknowns, b, b2, b3, a, x', andfi2. If we assign arbitrary values to two of the unknowns,to which we shall refer as U and U2, the four equationscan be readily solved for the remaining four unknowns.This determines the five "brackets" b, b2, b3, a, anda2 and the focal length fA. Using Eqs. (3) we can thencalculate the system parameters f3, f2, d43, d32, and d21.The resulting optical system will have the desired focalrange and the predetermined points of full compensa-tion. It will however, not necessarily be realizable andwill certainly not have the specified spacings 32 and 21unless the conditions (VF4-III) are satisfied. To obtainrealizable solutions and the specified values for thespacings 32 and s21 , we assign different values to oneof the parameters, say U2, while keeping the other, U1,fixed until positive values for 22 and f32 are obtained.Each non-negative value of f22 leads to two values forf2, a positive one and a negative one. Similarly, eachnon-negative value of f leads to two values for f3.For each of the four "mathematical" systems we can,
380 Vol. 52
I.V sV . .7 . - V .E . .w .V
f12 (f32 - d43d32) _74
FOUR-COMPONENT VARIFOCAL SYSTEM
using Eqs. (VF4 -III), calculate s32 and 21. Varying thevalue of U2 and repeating the calculations the specifiedvalue of, say, 21 can be obtained. This will, however,not yield the desired value of S32. Repeating the processwith different values assigned to U1, a set of parametersresulting in an optical system with the specified valuesof S21 and S32 and having the desired characteristics canbe found.
Although the above method does not require thesolution of nonlinear equations and always leads to thedesired results it is relatively time-consuming. It doesalso not give the designer any insight into the nature ofthe system. A modified method will therefore bedeveloped which leads to a solution of the varifocalequations in a matter of minutes.
To illustrate the new method we assume, as is mostlythe case, that the system is fully compensated at bothends of the operating range.
We have, from the varifocal equations (VF4 -I) and(VF 4 -II),
bi-x'== -yi, (22a)
b2-x'bi+fl2=72, (22b)
b3-X'02--feal= -73,
(r- 1)b3-b2-b, = 1.
Furthermore, from Eq. (6),
x'b 3 = f12 a2 .
Moreover, using Eqs. (3) we find that
51= al+d2 l,
b2 = a2+d21 aj+f22 ,
b 3 = d 2ja 2 + f22d4 3 .
Adding the second equation of (VF4 -III), we
f2+d21+fi= S21.
This gives a system of nine simultaneous equationswith ten unknowns, b1, b2, b3, a,, a2, x', f22, f1, d43, andd21. Since there is one more unknown than equations,we must assign an arbitrary value to one of the un-knowns. The unknown to which an arbitrary value isassigned must be properly chosen in order that theremaining nine unknowns can be (if at all possible)readily determined.
It is found convenient to introduce as a parameterthe final image distance measured from the rearprincipal plane of the movable component, i.e.,
11(0)=_I,,= x'+f . (23)
Using Eqs. (22) we then obtain,
e1o-e,1l+e12 12- 1
flw-2 (IS2 fi(-+ 21 )3
eoo- eOll-Ie02l 2 - eo3l1-I14
where
en- ' (yl+2S21)( 2+ _ 73 )- ,
ei,= [(,yl+2S2) i~+r_ t+ -
e12= [2-yi+3S21±+j
eoo= [ I(7+2s21)73],
r/-YI~,y-1r+l1eel= (71+221) K2+±-r)±v3j
r-1 r-1
eO2= [(eyl+2s21)( 1+ 1 I) r- 1)1,
andeO3 = [2y 1+2s 21 +2/(r-1)]
(25)
1/Cx) are known constants. Assigning an arbitrary value to I(22d) we can thus solve the quadratic equation (24) for the
focal length fi. The "brackets" of the system and x'are then readily calculated from the Eqs. (22). Choosing
(22e) one of the two roots of fl, we obtain,
x'=I-fl,(22f)
(22g)
(22h)
obtain
(22i)
(26)b2= x'b1- f12+72,
b3= [1/(r-1)](1+b,+b 2 ),
a1= (/f?)(xb2-b3-n),a2= (1/f12 ) (x'b3).
From Eqs. (3) and (22i) the system parameters are thenfound at once. Thus,
d21= b-aj,; ~f2 = S21-d2l- fl,
d4 3 = (1/f22) (b3-a2 d2 l), (27)
d32 = d4 3 - a,,f32= a2+d43d32.
This set of parameters results in an optical system thathas the required focal range, the predetermined pointsof full compensation, and the specified value for thespacing 21. However, since the first equation of(VF4-III) was not used in finding the system parametersthe system will not necessarily be physically realizableand will not have the specified spacing S32 between thethird and second components. In particular, someassigned values for I may lead to negative values off32 and thus to imaginary values of f3. However, as willbe shown, the region of values of 1 that lead to positive
381April 1962
(I+s2 1)2
LEONARD BERGSTEIN AND LLOYD MOTZ
(Pa)
- A I-9,L A(N)
(Pb)
'1 (Z) I-f () ,
0 I.0- -Z 0. 1 Z
FIG. 5. Optimum four-component varifocal systems. (Pa(Pb) show the two optimum PNP systems, and (N) shovoptimum NPN systems. Also shown is the focal range ofsystems.
values of f and thus to realizable systems is efound. For each value of I within this range and chroot of f that leads to positive values of f wecalculate
s32= f3 +d 3 2+f 2 ,
for each of the two possible signs of f. By varyingvalue of I an optical system having the desired cacteristics and the specified value of s32 is re-obtained. Since no exact value of 32 is ever prescribut only a range of such values, only a few trialsrequired until the final solution is found. The Ilength of the front component is then found from(VF4-IV). The maximum relative focal length ofsystem is given by
1-r ) ff2f3f4 / 2T fif2f3f4
1- -I b3 _-I I (r- )b3
Vol. 52
practical cases a negative front component will beused. All movable components are then positive, allstationary components negative. The final imageformed is real and inverted. However, if the varifocalsystem is used as part of a sighting system a positivefront component can be used as the objective and thevariable system behind it as part of the erector-eyepiecesystem. Both systems are shown in Figs. 5 (Pa) and5 (Pb), respectively.
In the AT system, to which we will refer as the NPTPNsystem, all movable components are negative, allstationary components positive. The final image formedis virtual and upright. This system is shown in Fig.5 (N).
and In order to find the range of values of 1, in whichsthehe realizable solutions of the PP and NPNT systems
exist and to reduce the number of trials which may benecessary to find a system having the desired character-
asily istics, we proceed as follows.osen We neglect first all component thicknesses and assumecan that the spacing between the (principal planes of the)
two movable components has the smallest possible(28) value. Thus, we assume 21 =0 and find (as a function
of the focal range) the value of the final image distancethe 1= ()ol for which the distance 32 becomes 1.0.
:har-adilybed,s arefocalEq.the
(29)
The resulting system will be a P system if we setr= 6R (T>O) and an N system if r= 1/c5 (T< 0). In eachcase a number of different lens configurations arepossible.
It was already pointed out' that of all possible lensconfigurations only the systems in which no real imagesare formed in the space between the movable compo-nents are of practical importance. There are only twosuch systems, one P system and one XT system, and wewill limit further consideration to only these twosystems.
In the P system, to which we will refer as the PNPsystem, the movable components are positive and thestationary component between them is negative. Theimage formed by the front stationary component islocated in a plane in front of the first movable compo-nent and can therefore be either virtual or real. It canaccordingly be formed either by a negative or by apositive lens placed at a proper distance in front of themovable component. It is obvious that in almost all
8
( 3) 5 3OL1~~~~~~~~(10
- 1' 1 1 ti 13 _1\ , W S- V 0
1 1.1 1 1 A d f f i I I I I I I _1 -.6 -.4 -.2 0 2 . .6 . .I
FIG. 6. The normalized final image distance (Q)oi-['(O)]oland the values of the parameter (f:s)oI[(r-1)b3] 01 of theoptimum four-comiponiet varifocal systems as functions of thefocal range r. (l)o, anl ()oi are the values of l11'(0) and133 (r-1 )b3 for systems in which in the position = 0 the distancesbetween the principal planes of the second and first, and thethird and second components are S21 = and 32 = 1.0, respectively.
382
____ TA 114 A---V Ae
AT-',' , AT -11 m ,V� 11 LJ
FOUR-COMPONENT VARIFOCAL SYSTEM
Setting in Eq. (28)
S32= 32(1,T) = 1.0, (30)
and USillg the values of Z2 aid Z:, as given hy Eq(Is. (15)
alrd (18), E(l. (30) cani be readily solved numerically
for (1),1j. The result is shown graphically in Fig. 6 where(1)ol is shown as a function of the normalized focalrange . Shown in the same figure is also the value ofthe parameter
(,33)ol- (r- 1) (b3)0l= 1.0+ (bl)ol+ (b2 )01 (31)
corresponding to (1)ol. Both functions, (1)ol and (13)01,
can be closely approximated by polynomials in T and1/r with the dominant terms proportional to 1/T and1/T2, respectively. If we let the approximating functionscoincide with the calculated values of (1)o and (3)01 atr= =t0.3, ±t0.6, and ±t0.9, respectively, we obtain
2.608(1) [(1- 0.290r2-0.233r4)+0.277T], (32)
and1.614
(63)01 (1- 0.435 T2-0. 166r) +0.063r]. (33)
The final image distance 1l'(0) I is positive for PNPsystems and negative for NPN systems. For both, thePNP and NPNV systems, an increase in the value of Iresults in a decrease of s32 if S21 is held constant or in anincrease of S21 if S32 is held constant [that is, (1)1> (1)o
and ()o < (1) o]. Optimum systems' are obtained whenall spacings have the physically smallest possiblevalues, i.e., when S21 is close to zero and S32 close to 1.0.
For the optimum systems, to which we shall refer asthe (PNP)0 and (NPN)o systems, respectively, thevalue of I which for a given value of S21 results in apredetermined value of S32 will thus' differ only slightlyfrom ()ol. One can, therefore, as a first trial, set1= (l)ol. If the resulting value of s32 is larger than therequired value we must in the next trial set 1> (l)ol; ifon the other hand the resulting value is smaller weset the second trial 1< (1)ol. Only two or three trialsare thus necessary to determine the system parametersfrom Eqs. (24), (26), (27), (28), and (VF4 -IV). On a
conventional desk calculator this takes no more than20 to 30 min. Often it is also found convenient to plot(for an assumed value of s21) the spacing s32 as a functionof 1. From the plot the value of I corresponding to anyspecified value of s32 can be determined and the norm-alized system parameters readily calculated. Theactual parameter values are obtained by multiplyingthe normalized parameters by Zt. The parameters ofthe auxiliary system are then readily found.3
An example illustrating the method described aboveand the general design procedure is given in this paper.
6 The choice between the two optimum systems is governed by anumber of considerations [L. Bergstein, J. Opt. Soc. Am. 48,154 (1958)]. (The PNP system is generally preferable.)
F -- 8-.-6 -2t 02- 4 .6.m oX17.X~~I 1 g
(fmaX >A7 a X
FIG. 7. The normalized focal lengths (fi)oi, (f2)ox, (f,)oi, and(f4)oio of the components and the normalized maximum over-allfocal length (f,,la)Oio of the optimum four-component varifocalsystems as functions of the focal range T. The index 01 indicatesthat the particular parameter was found under the assumptionthat s 2 1 =O, and s,1= 1.0, the index 010, that the parameter inquestion was found under the assumption that 1S.=0, S32=.0
and s43=0; S21, S32, and s43 are the distances between the principalplanes of the components 2 and 1, 3 and 2, and 4 and 3, respec-tively, in the position z=a of the movable components.
4. Focal Lengths of the Components, Final ImageDistance, Maximum Over-All Focal Length,
and Maximum Image-Plane Deviationof the (Optimum) Four-Component
Varifocal Systems
In the preceding section the value of the final imagedistance [ll'(0)]oim (l)ol was found for PNP and NPNsystems having the smallest possible separationsbetween the components under the assumption ofneglible component thicknesses, i.e., when s21=O and
S32= .0. For optimum systems 521 and s32 differ onlylittle from the values of zero and 1.0, respectively,and the corresponding value of I differs only littlefrom (1). The same is also true for the focal lengthsfi, f2, fa, and (f4 -s4 3 ) of the components of thevarifocal system. Their values for the optimum systemsdiffer only little from the values of (f)oe, (f2)il, (f),and (4)0l0 they assume when 521=° and S32= 1.0,
is the focal length of the front component for the case
383April 1962
LEONARD BERGSTEIN AND LLOYD MOTZ
when, in addition to 21 =° and s32 = 1.0, the distanceS43 has the smallest possible value under the assumptionof negligible component thicknesses, i.e., when s43=0.(Pl)ol, (f2)0l, (f3)01, and (f4)0o0 thus give approximatevalues for the focal lengths of the individual compo-nents of the two four-component optimum systems.[Generally, f1> (fl)ol, f2> (f2)0l, f3> (f3)01.]
The values of (fi)o, (f2)0l, and (f3)ol have alreadybeen found in solving Eq. (30) for (1)ol. Using Eq.(VF4-IV) also (f4)00 is readily calculated. The resultsare shown in Fig. 7, where (fl)ol, (f2)0l, (3)0l and(f4)010 are shown as functions of the focal range. Eachof these functions can be approximated by a polynomialin T and 1/r with the dominant term proportional to1/T. If we let the approximating polynomials coincidewith the calculated values at T= ±0.3, ±0.6, and ±t0.9,respectively, we obtain
0.996(fi)oi + (1 0.295T2-0.268T4)+0.118, (35)
T
0.805(f2)o- - (1- .2877T2-0.268T4)+0.138, (36)
0.996(13)0o1 ± (1- 0.2561.2-0.229T4)+0.119, (37)
Tand
2.609(f4)010, - (1- 0.220r2 0. 198T4)+0.276.
r(38)
For P systems f4 is negative and I (f4)0l 1 < I (f4)001;for N systems f4 is positive and (f4)0l> (f4)010.
The maximum over-all focal length of the varifocalsystem is given by Eq. (29). Thus, when S21 = 0, s32= 1.0and S43= O.
2(T 1 (fl) 01 (f2) 01 (f3) 01 (f4) 010)
(fm )lO I I ( 83) 01 J
Using the values of (fi)ol, (f2)01, (f3)01, (13)01, and(f4)010 found previously, (fma)ojo can be readilycalculated as a function of the focal range. The result isshown in Fig. 7. To the first approximation (f)ol,(f2)0l, (f3)0l, and (f4)010 are each proportional to 1/rand (3)01 is proportional to 1/r2. Therefore to the sameorder of approximation (fmax)Oio~o-' 1/r. Approximating(lmax)ojo by a polynomial in 7 and 1/r with pointscoinciding with the calculated values at = ±t0.3,±0.6, and ±0.9, respectively, we obtain
3.167(fmax)oloo [(1+0.951,2+0.639 4)
-0.109r(1+0.72r2+2.30r 4)]. (40)
The product of , 3 = (r- 1)b3 and the maximum value,y(max), of the normalized image plane deviation of thevarifocal system proper was found in Sec. II-2. Assuming
y4 (max)0 1
.2020-- -
- 4 02~ - - --
08] - - - -
- .0 -.8-.6 -.4 -. 0 2 4 .6 .8 .0-.r
FIG. 8. The maximum value y4(max)ol of the normalized image-plane deviation of the optimum four-component varifocal systemswith full compensation at both ends of the operating range (andan optimized image deviation function) as a function of the focalrange r. The distances between the principal planes of the compo-nents in the position z=O are assumed to be S21=O and S32= 1.0,respectively.
s 21 =0, and s32= 1.0, we obtain
y(max)ol= *[3 Y(max) (41)
(13)01
The maximum value YT(max) of the image-planedeviation of the complete varifocal system (includingthe auxiliary component) is given by Eq. (70) ofreference 3. Thus,
YT(max)olo y(max)ol- G(max)o 10= , (42)
F 2 (max)/Zm (fmax)oi02
where FT(max) is the maximum over-all focal lengthof the complete system. Using the values of [3y (max)],(/33)01 and (fm.)olo found previously, y(max)ol and0(max)o1 o are readily calculated (as functions of thefocal range). The results are shown in Figs. 8 and 9,respectively. To the first approximation [3 3y(max)]-0.021 | r, (3)01- 1.614/72, and (fmax)oio-3.167/r.Therefore, to the same order of approximation,
y (max)l- (0.021/1.614) 1 T3 1 = 0.013 1 T3 ,
and
0(max)oo- (0.013)/ (3.167)21 r51 = 0.00131 r.
Approximating y(max)ol and e (max)01 0 by polynomialsin r, starting with T3 and 5, respectively, with pointscoinciding with the calculated values at T= ±0.3,i0.6, and ±0.9, respectively, we obtain
Comparison of Eq. (44) with the corresponding equationfor a three-component system verifies the statementmade in reference 3 that the additional point of full
384 Vol. 52
FOUR-COMPONENT VARIFOCAL SYSTEM
compensation that an added component brings with itresults in a reduction by at least a factor of 20 in themaximum value of the image-plane deviation. We notethat, as in the case of two-component varifocal systems,the image-plane deviation of the PNP systems islarger than that of the NPN systems.
III. EXAMPLE
Let it be required to design a varifocal objectivewith an over-all focal length varying continuouslybetween, say, 25.00 and 100.00 mm.
We proceed with the design along the same lines asin reference 2 for the three-component system.
As the first step in the design we estimate the smallestvalue of the maximum image-plane deviation YT(max)that can be obtained with a four-component varifocalsystem.
The focal range of the system is 6R=FT(max)/
FT(min)=4.0 and (rI1)=(R-1)/(6R+1)=0.6. FromFig. 9 or Eq. (44) we find that if a P system is used(r= +0.6),
YT(max)10= 0.000072FT2 (max)/Zm,
and if an N system is used (T= -0.6),
YT(max)olo= 0.000060FT 2(max)/Zm.
Choosing Zm= 40.0 mm for the maximum displacementof the movable components we obtain with FT(max)= 100.00 mm,
[YT(max)ol]P=0.01 8 mm,and
EYT(max)O1]N=0.015 mm.
Assuming that these values are within the maximumpermissible value we can proceed with the design. Weshall consider both, the PNP and NPN, optimumsystems.
The approximate values for the thin-lens param-eters of the two optimum systems are found at once.Using the results of Fig. 7 or of Eqs. (35), (36), (37),(38), (40), and (32) we find that the focal lengths ofthe three components of the (PNP)o system (r= +0.6)are
(Fl)ol= + (1.544) (40.00) mm= +61.76 mm,
(F2)01=- (1.019) (40.00) mm= -40.76 mm,
(F3) 01= + (1.577) (40.00) mm= +63.08 mm,
and
(F4)010= - (3.618) (40.00) mm= - 144.72 mm.
The maximum over-all focal length of the variablefront system is
(Fmax))oio= + (6.988) (40.00) mm= +279.52 mm,
and the final image distance is
[L/(0)]ol= + (4.485) (40.00) mm= + 179.40 mm.
-1.0-.8-.6 -4 -.2 0 .2 .4 .f .8 i.u-*-,rFIG. 9. The maximum value YT(max)olo of the image-plane
deviation of optimum four-component varifocal systems withfull compensation at both ends of the operating range (and anoptimized image deviation function) as a function of the focalrange r. The distances between the principal planes of the compo-nents in the position z=O are assumed to be S21=0, S32=l.O,and s43=O, respectively. FT(max) is the maximum over-all focallength of the system and Zm is the maximum displacement of themovable components.
The parameters (fi)oi, (f2)ol, (f3)o1, (f4)olo, and thevalues of (fmrax)olo and [11'(0)]ol are found for the casewhen the separations between the components havethe smallest possible values under the assumption ofnegligible component thicknesses, i.e., when S21= 0,
s32= 1.0, and S43= O. In the actual case the finite thick-nesses of the components (i.e., the finite distancesbetween the principal planes and the correspondingcomponent boundaries) must be taken into accountand the separations must be larger than the minimumvalues required. We will now proceed to find the actualvalues of the parameters of the two optimum systems.
We shall first consider the (PNP)o system. We have,
(r)p=GL=4.0 and T=+0. 6 .
Assuming that the system is fully compensated at bothends of the operating range we find from Fig. 3 orEqs. (15) and (18) that for r= +0.6,
6 el'=0.065,
and the points of full compensation are thus located at
the value of 1. Assigning to I the value of 4.550 we findthat s32 = 1.2072 (see second column of Table I).Assuming that the desired value of 32 is 1.15 we thenfind, using linear interpolation, that the correspondingvalue of I is 4.562. From this value of I the systemparameters are readily calculated. The results are shownin the third column of the table. It is seen that thecalculated value of S32 is 1.1537 which is, for all practicalpurposes, equal to the desired distance. The parametersof the system are thus:
We shall assume that a distance of approximately6 mm is sufficient to account for the finite thicknessesof the rear stationary and movable components of thevarifocal system proper. We therefore assume thatS21= 0.15. [(0.15)(40.0) mm= 6.0 mm.] Equation (24)relating the final image distance 11'(0) =l and the focallength of the rear movable component thus becomes
f 12 +2K 1 f1-Ko= 0,
where
Ki - (-1.341998+5.0665271-4.85466712+13),(1+0.15)
and1
Ko- (0.422660- 3.0737241(1+0.15)2
+ 6.3 713422- 4.7046673+14).
To solve the above equation for f we must assign avalue to 1. The other Gaussian parameters of thesystem are then calculated from Eqs. (26), (27), (28),and (VF4-IV).
For r= +0.6, the value of I that results in a systemwith 21= 0 and s3 2= 1.0 is ()ol= 4.485. We will thereforefirst calculate tlhc paranieters of tlhe system fo 1 (1)ol=4.485. The results are shown in the first column ofTable I. It is seen that the resulting value of 32 is1.5133. To obtain a smaller value of 32 we must increase
None of the focal lengths differ by more than 3.6%from the approximate values found previously underthe assumption that 21= 0 and 32= 1.0.
The final image distance of the front system isL,'(z) (4.562-z) (40.00) mm= (182.48-40.0z) mm,
and the over-all focal length is given by
r 6.8515 1F(z)= 40.00 mm.
1+1.3832z+0.8499z2+0.7669Z3 0
The objective is to have an over-all focal lengthFT(Z) varying from 100.0 to 25.00 mm. The auxiliarysystem must therefore reduce the focal length of thefront system by a factor
W= (6.8515) (40.0)/100.00= 2.7406.The auxiliary system usually comprises a number oflens elements. We therefore assume
Slo= (1.30) (40.0) mm= 52.00 mm.
This givesLo= 130.48 mm.
The focal length of the auxiliary system becomesFo= + (130.48)/1.7406 mm= 74.96 mm,
and the final image distance is
Lo'= (130.48)/2.7406= 47.61 mm.The physical length of the system is
S= (2.7537) (40.0) mm= 110.15 mm,
and the over-all distance between the front componentand the image plane is
L= S+Lo'= 157.76 mm= 1.5776FT(max).The image-plane deviation of the objective is given by
YT (Z) = (250.00)
- z(z-0.228) (z-0.641) (z-1.000)-XI 0.0163 18mm.
L 1 + l.38 3 2 z+0.8499z 2 +0.7669z 3 _
386 Vol. 52
FOUR-COMPONENT VARIFOCAL SYSTEM
The maxima of YT(z) are located at 1,2=0-09, 2,3= 0.40, and 3,4= 0.84, and the corresponding values ofthe maxima are YT(max)==0.022, 0.023, and 0.021 mm,respectively. The values of the three maxima differ onlylittle from each other and are in close agreement withthe value of 0.018 mm given by Eq. (44).
Assuming that the objective is to have a relativeaperture of 0.25 and is to cover a field varying from 8°to 320, we find that the diameters required for theaperture are (Dia4)A=25.00 mm, (Dia3 )A=26.06 mm,(Dia2)A=15.62 mm, (Dial)A=16.65 mm, and (DiaO)A= 11.90 mm, and that the diameters required for thefield are (Dia4)F=31.44 mm, (Dia3 )F=29.23 mm,(Dia2)F=12.35 mm, (Dial)F=10.74 mm, and (DiaO)F=3.22 mm; respectively.
We shall now consider the (ATPN)o system. Now,
(r)N= 1/R= 1/4 and r= -0.6.
Assuming, as before, full compensation at both endsof the operating range we now have
e'/= e/= -0.065,
and the points of full compensation are located at
Z1 =0, Z2 =0.358, Z3 =0.773, Z4=1.000.
Forming the elementary functions of the zeros we obtain
For T= -0.6 the value of I that results in a systemwith S21= O and S32= 1.0 is (1)ol= -3.036. We thereforecalculate first the parameters of the system for 1= () ol= - 3.036. The results are shown in the first columnof Table II. The resulting value of S21 is +1.3392.Assigning to I the value of -3.000 we find that s3 2
=1.2229 (see second column of the table). Assumingthat the desired value of S32 is 1.15 we find, using linearinterpolation, that the corresponding value of I is
2.978. Using this value of I the system parameters arereadily.calculated. The results are shown in the thirdcolumn of the table. It is seen that the calculatedvalue of S32 is 1.1496 which agrees closely with thedesired value.
L,'(z) -3.978- (1.0-z)]40.0 mm=-[159.12-40.0(1.0-z)] mm,
and 8.2392\F(z) =}40.00 mm.
4.0- 5.4486z +3.313Z2- 0.8643z3)
We note again that none of the focal lengths differ bymore than 3.5% from the approximate values foundpreviously under the assumption that 21= 0 andS32= 1.0.
The reduction factor of the auxiliary system is now
W=-(8.2392)(40.00)/(100.00)= -3.2957.
Choosing, as before, Slo= 52.0 mm, we obtain L,=-171.12 mm, and consequently
FIG. 10. The two optimum four-component systems of the example of Sec. III, their image-plane deviation and focal range.
the same as that of the (PNP)o system, and the over-alldistance between the front component and the imageplane is
L= 161.90 mm= 1.619FT(max).
The image plane deviation of the system is given by
YT (z) = (250.0)
X[001273 z(z-0.358)(z-0.773)(z-1.000)
4.0- 5.4486z+3.3130z 2 - 0.8643z3
The maxima of YT(z) are located at il,2=0.16, Z2,3=0.59, and 3,4=0.91, and the corresponding valuesof the maxima are YT(max)=0.016, 0.019, and 0.017mm, respectively. Also now the values of the maximadiffer only little from each other and are approximatelyequal to the value of 0.015 mm given by Eq. (44).
Assuming that the objective is to have a relativeaperture of 0.25 and is to cover a field varying from 8°to 32° we now find that the component diametersrequired for the aperture are (Dia4)A= 25.00 mm,(Dia 3)A= 18.25 mm, (Dia 2)A= 19.46 mm, (Dial)A= 12.07 mm, and (Diao)A= 12.98 mm, and the diametersrequired for the field are (Dia4)F=30.23 mm, (Dia3)F
=25.81 mm, (Dia2 ) = 14.66 mm, (Dial) = 11.57 mm,and (Diao) =3.47 mm, respectively.
Both systems, the (PVP)o and (NPN)o, their focalrange, and image-plane deviations are shown inFig. 10.
It is of interest to compare the two four-componentsystems with the three-component systems of theexample given in reference 2. We note that the four-component system has a larger focal range (4:1 ascompared with 3:1) and a much smaller image-planedeviation. Yet, the increase in the focal range and thereduction in the image-plane deviation was achievedby the addition of only one single stationary element tothe (three-component) lens system without increasingits complexity or even its over-all length.
CONCLUSION
The iteration procedure developed enables one todetermine the optimum Gaussian parameters of afour-component varifocal system almost at a glance.The problem of correction of the image aberrations ofthe system over its entire operating range will bediscussed in a forthcoming paper.
