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5-16-1975 Computer Predesign of Cooke Triplet Anastigmat Frank Vanek
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Recommended Citation Vanek, Frank, "Computer Predesign of Cooke Triplet Anastigmat" (1975). Thesis. Rochester Institute of Technology. Accessed from
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Title: Computer Predesign of Cooke Triplet Anastigmat
Submitted by: Frank Vanek
On: 16 May, 1975
To: ROCHESTER INSTITUTE OF TECHNOLOGY
Department of Photographic Science and Instrumentation
Thesis Advisor: John Carson yz^n
Abstract:
There are many methods for designing the Cooke Triplet
Anastigmat. In this thesis, one of them is programmed into a
computer in BASIC language. Because of the length of the program,
it had to be split into two separate programs using a data file
to save data produced by the first and needed by the second.
The first program uses the equation for system power, and
the equations for Petzval curvature, longitudinal and lateral
chromatic aberrations, space ratio to solve for the powers
and spacings of the elements. Solution is accomplished by the
Crout method of simultaneous equation solution and by iteration.
The second program solves for the bendings of the elements.
Assuming a value for the first curvature, it solves for the
second and third using the equations for coma and astigmatism,
respectivly. The value of spherical aberration is used to check
the result, and if not correct a new value is calculated for the
first cuvature.
The program was checked by carrying through an example that
was also solved by hand, and comparing the results. The program
has successfully solved two examples. Introduction:
Everyone in the photographic field knows that computers are used to design lenses. Those in optics know computers have been in use for about thirty years. Even so, the problem of program ming computers to design lenses is not a trivial one. Optical programming is still an area of extensive research for both government and industry.
Most optical programming works to optimize a design that *
has already been completed in rough form; the computer makes small
adjustments to refine the performance of the test lens. Very
often the final performance of the new design depends on the
quality of the rough design. A well-known lens is the Cooke
Triplet Anastigmat, which consists of a negative element between
two widely spaced positive elements. This lens is unique; it has
just enough degrees of freedom to allow the designer to control
the power, lateral and longitudinal chromatic aberrations, and the
Seidel aberrations. Algorithms for the solution of the design
have been published prior to and since the turn of the century;
the lens was invented in 1893 hy H. Dennis Taylor-
While there are almost as many methods of using these equa
tions as there are designers, any design formulated by the methods
is excellent for subsequent treatment in an optimization program.
Thousands of these lenses are in use today in moderately priced
cameras. UlLcuJL^C^
These methods are usually done with pencil and desk calcula
tor, and sometimes with some aid from a. computer. No one has published a computer program for this purpose. This is the swbstance of the thesis: to program the computer in BASIC language for predesign of the Cooke Triplet Anastigmat. The method used Is basically the one proposed by Schwarzschild about
1904, taken from Dr. Kingslake's notes.
A predesign deals only with first order optics and thin
elements; the equations are completely true only for rays dif
ferentially close to the axis, and elements so thin as to be
considered of zero thickness. This approximation to rays at
finite height and elements of finite thickness is good enough
that the first order predesign can be converted into a rough design
by assignment of thicknesses to the elements. The design v/ould
then be computer optimized. Background Theory and Plan of Attack:
The eight aberrations include lateral and longitudinal chromatic aberration, Petzval curvature, distortion, astigmatism, spherical aberration and coma. System power (inverse focal length) is not actually an aberration, but the equation is in the same form as aberration equations and is used in the same way. For the sake of brevity, system power will be referred to as an aber ration.
The lens has three elements, and thus three powers, one for each element; two spacings between the elements and three bendings, one for each element; for a total of eight degrees of freedom.
To greatly simplify calculations, the aperture stop is as sumed to be located at the second element: in other words, at zero distance from it.
The basic method used is to choose an aim value for each of the aberrations, then solve a convenient combination of them for some of the unknown variables. These aim or residual values would not be equal to zero; by using non-zero values it is pos sible to compensate for higher order aberrations that will appear when thicknesses are added to the elements. The only way of es timating these residuals is through experience with this lens, or by trial and error, following the design to completion each time. They would also depend on the purpose of the lens; in a
fast lens, for example, spherical aberration would have to be very carefully controlled. / Procedure: Fart One
The first task is to determine the powers and spacings.
Four aberrations are functions of element power: system power,
longitudinal chromatic, Petzval curvature and lateral chromatic.
To solve for five items, a fifth relation is needed, but there
is no aberration to supply it. Therefore, one is defined; the
ratio of the spacings must be some constant K; which is set to
an arbitrary value at first. When the design is completed, K
can be changed to reduce distortion.
The first step is to set up a panel of simultaneous equations:
(&Vl&(&X)fcWV07=-v
( i/H) &+ ( iA)4+ ( 1/ig#=Ptz
Where: y-ray height for the marginal ray (first order)
0=Tpower (reciprocal focal length) of each element
V=V number of the glass of each element
N=index of refraction of each glass NB
L'ch=longitudinal axial chromatic aberration contri bution
Ptz=Petzval curvature of the system
u0' =entering angle of the marginal ray from axial object point
u0* If the object is taken at infinity, then equals zero, and % can be calculated as one half of the focal length divided aperture. Then and are by the 0*; 0^ r0c yb , y. unknowns. Since the first element is converging, yh< ya and 3>yb. We C3n then choose "likely" and the path clear for solu values for yh y. , leaving tion of . The Crout method was used to perform the 0^ , 0^ t 0C simultaneous equation solution. Once the powers are known, the spacings may be calculated, then K and T'ch, the space ratio and lateral chromatic. Unless and these we have been extremely lucky in our choice of lj$, yc , will not be at their aim values. A better approximation can be calculated using a method used in electronic engineering: the Newton-Raphson method. This method uses two simultaneous equa tions in this case: A T'ch= kS& A v + ^J2Jl k Ay, ay- n By varying yb only, it is possible to determine v.fc. and ffT'ch. The same technique is used to find -4&- and to achieve the aim K and T'ch. To prevent the computer from overshooting the change is limited to , 0.001 times the y value it is applied to. This does not affect accuracy, but speeds con vergence. Since the functions are not straight lines, convergence would not be immediate. The new yb and y would be substituted for the arbitrary values, and the powers solved again. The whole process would be repeated as many times necessary until K and T'ch converge on the aim values. Here is a test example, where the aim ("E-04" values for K and T'ch are 1 and 0 respectively means "times IO"4"). ' YC 2) K YC2) T'CH - . 12694E-C2 .338339 2.41453 1.00000 1 -2. 12682E-02 .366229 1 .4523 2 .954026 -7.60220E-04 .362J36 1 .2213 1 .954990 -2.52056E704 .361771 1 . 12733 .955945 -1.21126E-04 .36141 1 1.09 416 .956901 > 53' -3.27574E-05 - .361.242 1 .07541 .9573 -6- 5- .361222 1 .05922 .953316 2451 1E-0 ' A -4.42434E-C5 .361212 1.04254 .9 5977 5 -2. - '\ZrA'- .8 61206 1.02306 .960725 77 2.26E-0 5 E- . 0 5 .361201 1 . 0 1 27 6 .961695 -1 22223 .962502 -1.81969E-07 . .361299 1.00002 , -9.6447 5E-12 .361293 1 .00000 .962502 ...--'-.--. r . With this process completed the powers and spacings are known. The program had by this point almost filled the memory core available. Instead of proceeding to Part Two, pertinent data, is stored under the reference name of the lens in a data file. Some of the data used in Part Two is calculated at this point and stored so that more data from Part One can be dis carded, reducing net storage space needed. Chart 1 is a simplified flow chart of the first program. Start TNPur Fo'AU uilMtrTn (=/ NUM3JR FtEl-D SNOLE , Gt-AVi 1>AT/" RESIO ui,i_'J -i/- COMPUTE iA ASSUME l^M. JL- SoU/e for. -4L CAL-CUI_ATt=. Y5.5 STOR.E DATA. NO -ik. f EKiO J To find A. -^ 8 The first task of the second program is to retrieve the data stored from the first. This is done automatically once the operator inputs the reference name. The total curvature of each element is a function of ele ment power and index of refraction. This specifies the rela tionship of the front surface to the rear surface curvature only. There are still an infinite number of front or rear curvatures that satisfy that power and index, but each of these possible bendings has different aberration contributions. For example, the first element could be made into a meniscus element, strongly bent; or a double convex element, weakly bent; both with the same power. Their spherical, coma, and astigmatism contributions "bend" would be different. The purpose of the second program is to the three elements until the sum of their individual aberration contributions equals the aim residual values. To make this task easier, the aberration equations are placed in quadratic coefficient form. For example, the equation for the spherical contribution is: SC*= ^ (qc3- G^c, + q<*v + qpcy- Gxjqv + QpV*) uk Where: G, through (^ are the G-sums dependent on index of re fraction c=total curvature for element in question Uj,'=final angle for the marginal ray = u^ v=the front curvature for that element noted that there are It can be terms in ct , , terms in c, , and constant terms. These can be collected and the equation placed in the form: SC*=3a + a,cf+ a^ cf Where: ap = ^M-7i(Glc? + Gr&v + Gov*) a.:=-~pl(qc^+ G^cv) A summary of the equations is in Appendix b. These coefficient expressions are of a convenient size for programming. Once calculated, they can be used to find the aberration value if c, is known, or solve for c, if the aberration contribution is known. After the data is removed from storage, constants used through the program are calculated. These include the G-sums, total curvatures, v's, and also the astigmatism contribution for element B: AG^. Since the stop is located at element B,the astigmatism there is independent of bending, and the coma ex pression is linear rather than quadratic. To start the process, an arbitrary value for the first curve of the first element/ qq, is assumed. If set equal to total curvature, the element is plano-convex, a reasonable starting then possible point. It is to compute SC*, , CQ*, and AC,*: spheri cal, coma, and astigmatism contributions for element A. The star superscript indicates that the contribution is at an element not located at the aperture stop. AC- Since total aim astigmatism, Asts, is known, is constant AC* and AQ* has been calculated, is known, since AQf+ACj+AC=Asts. The astigmatism equation for element C is placed in the form AC? =d+d^cK:+ dgc, so the well known quadratic formula may be used 10 to solve for the two values of Cit that are roots of the equation. Usually the smaller in magnitude of these is used, since strongly bent elements are generally not desirable. It is then possible SC* to calculate CC and . coma as the The same method is used to solve for Cjb , using CC* unknown, since CG = Comas - - CC*. Since the stop is at B, coma equation the CC- = + so the can be placed in form b, b^.c,6 , only SC- one root of the equation is necessary. is calculated, then LA'=SC* + + SC* SCb . LA' Since the value of cIA was assumed, is not likely to be LA' the aim value. Since is close to being a quadratic function v of a good approximation of a new can be reached C/a , c,fl by choosing LA' two more values of c/4 , finding new values for by repeating the process outline above. These can be arranged into a panel of simultaneous equations: LA,'=Ac^, + Bc{ai + C LAA'=Ad^+ BcKi+ C LA3,=Acffl, + Bclfl3+ C The Crout method is then used to solve for the coefficients LA* A, B, and C. Since the aim is known, the quadratic formula may be used again to find the two values of c for the next trial. Usually the smaller of these is picked to avoid strong bending. The entire process is repeated, starting with the calculation of CC^- AC|- and . Since this new is an approximation SC , , cl!V only of the true value, the process would have to be repeated a number LA' of times until equals the aim value. In the case of both c^and c|C the operator must choose the 11 smaller or larger root; the computer prints: ^* THE VALUES OF CC3-1) ARE:-. 628841 2.64839E-02 - PICK LARGE OR SiMALL ?S CC 1 ) C(3) LA* .237943 2.64839E-02 -5.27260E-02 , THE FIRST CURVE FOR THE NEXT TRIAL CAN BE .231370 OR .2963 62 PICK LARGE OR SMALL ?S . . The first line indicates the two roots for cic.- The next asks the operator to choose. If the operator had typed any character string starting with "L", the larger value v/ould be used. Otherwise the smaller in magnitude is automatically used, as in this case. The computer then displays the values used in that run through, and the resulting LA'. It computes the two new roots for the first curve, and asks again for a choice. In this way the operator interacts continuously with the program. For the purposes of the program cw is actually started at 0.0001 less than c, , then incremented twice in a loop, adding LA* 0.0001 each time. This produces the three cIK and values needed to fix the parabola at 0.0001 intervals of cIA . This same technique is used for the new c,a to provide three values aga in . Chart 2 is a simplified flow chart of the second program. 12 STAR.T j-NPlIT REFEBeNcg NAME -iki. RErKieve DATA FROM NAMED FIUS ^_ comstant^ ^-10. ~ C'* LET C^= c1CL- O.OOOI LA, - Ad*ai II =4 tBc11( - ia--Bc1(U.* YES LAi At C ift-="C,ft + 0.0OW DOE6 Xt =3 A^-3 + "Be4A3 + C, Il = Il tl Soi-VE PoK. TUJO Cva.VAU\S5 COMPUTE yes PRINTOUT AC* PICK, c.la. :ND Compute Ac=A,+*-A-Aq, solve t=o(- LA'= sc+sc^+scj i\ sy vccxc to-Apwre scb c)(_ >/_ CtfeT COMAS tOMPUTB SOLVE FoR. c. . 13 Summary: Here is a summary of which degrees of freedom control v/hich aberrations: Degrees of Freedom Method Item Three Powers Simultaneous Solution System Power Two Spacings Longitudinal Chromatic Petzval Curvature Iteration "Lateral Chromatic Distortion Three Bendings Iteration [Spherical Aberration Coma. Astigmatism In this program distortion is not calculated or corrected. Thicknesses should be added before distortion is determined, and then K would be changed and the predesign repeated with the new K. lb Program Check: , The checking and debugging process is threefold. First, the program must be run to see if it functions to completion. When it does not, and all obyious errors have been corrected, the next step is to start computing the test example by hand, comparing results to those generated by the computer until a significant difference is found and the erronious equation pinpointed. Once this first example is completed, and the pro gram operational, a. second example is run and the results checked against the hand calculated values. The program was checked using the example in Dr. Kingslake's notes, and the complete example can be found in Appendix 3 The answers were not completely identical, but the differences small enough to be attributed to differences in rounding. They gen- erally agreed to five decimal places. The program was then checked by an example from Mr. Carson; the correct values for the output were known only to him. Again, the answers agreed to a reasonable number of places. Therefore , the program works . 15 Use of Program in Design Process: Since the predesign is an approximate design with zero thickness assumed for the elements, the next step in the de sign process is to add thicknesses. Since thick elements have a Gauss seperation, the spacings must be adjusted to compensate. The thicknesses added must be sufficient to allow for trimming, centering and mounting of the elements with the -required clear apertures. These calculations would produce a design which could ac tually be built, but tracing exact rays would. reveal that is not yet good enough. The aberration values found would be used to correct the residuals used in the predesign, and the predesign would be repeated. If the original guess for the residuals was reasonably good, not too many repetitions of the process v/ould be needed to minimize the aberrations in the rough design. The design could then be computer optimized, and built if the finished design meets specifications. 16 APPENDIX 1 Program 17 10 REM COMPUTER PREDESIGN 0F C00KE TRIPLET ANASTI 3MAT 20 REM INPUT- P0WER AND SPACING PR0GRAM 30 REM FRANK VANEK- 5/3/75 - - - 40 DIM AC 3* 4) -GC3) -NC3>,P<2> SC 2) TC 3) > UC 3) VC 3) -YC 5> 50 F0R I 1 = J T0 10 60 PRINT 70 NEXT I I 30 PRINT "WHAT I S LENS 90 INPUT MJ 100 PRINT "WHAT IS THE DESIRED F0CAL LENGTH' 1 10 INPUT FJ 120 PRINT "WHAT IS THE DESIRED MAXIMUM APERTURE"; 130 INPUT F2 140 PRINT "WHAT IS THE DESIRED MAXIMUM FIELD ANGLE CDEG> 150 INPUT F3 160 F3R 11=] T0 3 170 PRINT "GLASS F0R ELEMENT"! M 130 INPUT GCI 1) 190 PRINT "NC"I ] 200 INPUT NCI 1) 210 PRINT "VC"I 1 >=< 220 INPUT VCI 1) 230 NEXT I ] 240 YCl)=Fl/2/F2 250 U2=-TANCRADCF3>) 260 Nl = l 270 D3=JE-6 28 0 D4=lE-6 290 PRINT "WHAT IS THE AIM VALUE F0R: 300 PRINT "PETZVAL CURVATURE "J 310 INPUT Al 320 PRINT "L ' CH ; 330 INPUT A2 340 PRINT "T'CH ; 350 INPUT A3 360 PRINT "AST'S"; 370 INPUT A5 380 PRINT- "C0MA *; 390 INPUT A6 400 PRINT "LA' 410 INPUT A7 420 PRINT "K 430 INPUT .Kl 440 Ul=l/2/F2 450 U3=0 460 PRINT 470 PRINT -"T'CH' 480 PRINT "YC2>"-"K"-"YC3> 490 YC2)=YCJ)*.8 500 YC3)=YC J)*.9 18 510 IF YC l)t 2/VC !)<>YC3) T2/VC3) THEN 530 520 YC3)=YC3)+.l 530 AC 1 -4)=UJ-U3 540 REM CR0UT METH0D STARTS HERE 550 AC2.4>=-A2*U1?2 560 AC3-4)-Al 570 F0R I !=^J T0 3 530 AC 1 - I 1)=YCI J ) 2- 590 AC I 1 ) =YC I 1 > t 2/VC I 1 ) 600 AC 3- 1 !)=1/NCI 1) 610 NEXT I 1 620 AC ! - 2)=AC J-2)/AC ), 1) 630 AC 1-3)=AC l-3)/AC 1- 1) 640 AC 1 - 4)=AC 1-4) /AC 1- J) 2- 2)= 2- J- 650 AC AC 2) -AC 2.. J) *AC 2) 660 AC3-2)=AC3-2)-AC3- J)*AC 1,2) 670 AC2, 3) = CAC2> 3)-AC2. J)*AC J,3))/AC2,2) 680 AC2, 4) = CAC2.4)-AC2, J)*AC 1-4)) /AC 2- 2) AC3-3)=AC3- - 2- 690 3)-AC3> 1 ) *AC 1 3) -AC 3* 2)*AC 3) 2- 700 AC 3.. 4) = C AC 3/ 4) -AC 3* J)*AC 1, 4) -AC 3* 2)* AC 4) ) /AC 3, 3) 710 PC3)=AC3*4) 720 PC 2)=AC2>4)-AC 2j 3)*PC3) 7 30 PC 1)=AC J>4)-AC 1 -2)*PC2)-AC 1 -3)*PC3) 740 U4=U3+YC J)*PC J) 750 U5=U4+YC2)*PC2) 760 SC 1) = CYC J)-YC2))/U4 770 SC2)=CYC2)-YC3))/U5 780 K2=SC2)/SC 1) 790 YC4)=SCJ)*U2/C J-SC 1)*PC1) ) 800 YC5)=-K2*YC4) 810 A4=-J/UJ*CYC J)*YC4)*PC 1 ) /VC 1 ) +YC 3)*YC 5)* PC 3)/VC3)) 8 20 IF N)<> 1 THEN 3 50 830 PRINT YC2) -K2,YC3)#A4 840 REM NEWT0N-RAPHS0N METH0D STARTS HERE 850 0N NJ G0T0 370*940., J020 86(5' REM NEXT TW0 LINES ARE EXIT 870 IF ABSCKJ-K2)>JE-3 THEN 390 880 IF ABSCA3-A4X JE-8 THEN J 230 890 Y = 900 NJ 2 , 910 TJ=K2 9 20 T2=A4 930 G0T0 530 940 D1=TJ-K2 950 D2=T2-A4 960 S2=DJ/D4 970 S4=D2/D4 980 Nl=3 990 YC2)=YC2)+D3 1000 YC3)=YC3)-D4 1010 G0T0 530 1020 S1=CTJ-K2)/D3 1030 S3=CT2-A4)/D3 1040 YC2)=YC2)-D4 1050 D1=T1-KJ > 19 1060 D2=T2-A3 1070 D3=CS2*D2-S4*DJ ) /C S2* S3-S J * S4) 1030 IF S2=0 THEN 1 1 JO J090 D4=CD1-SJ*D3),/S2' 1 100 G0T0 1 J 20 1110 D4=CD2-S3*D3)/S4 1120 IF D3<>D5 THEN J 1 60 1130 IF D4=D6 THEN 1230 1140 REM NEXT J J LINES LIMIT CHANGE IN Y'S AND RESET F0R 1150 REM NEXT RUN-THR0UGH 1160 IF D3<.001*YC2) THEN 1130 1170 D3=.001*YC2)*SGNCD3) 1 180 YC2)=YC2)+D3 1 J90 D5=D3 1200 D3=JE-6 1210 IF D4<.00I*YC3) THEN 1230 D4= 1220 .001*YC3)*SGNCD4) 1230 YC3)=YC3)+D4 1240 D6=D4 1250 D4=lE-6 1260 Nl=l 1270 G0T0 530 1280 REM EXIT: 0UTPUT AND DATA ST0RAGE START HERE 1290 PRINT 1300 PRINT 1310 PRINT "P0WERS"-PC J) -PC 2) -PC 3) 1320 PRINT "SPACINGS"-" "SC J ) " "SC2) 1330 H1=YC5)-YC3)*CU2+YC4)*PC J ) +YC 5) *PC 3) ) /U J 1340 QJ=YC4)/YC 1) 1350 V1 = U3/YC J ) 1360 Q3=YC5)/YC3) 1370 V3=U5/YC3) 1380 V2=U4/YC2) 1390 OPEN Mi; "007"- 0 1400 PUT J 1410 F0R I !=] T0 3 1420 UCI J)=YCI 1) 1430 NEXT I J 1440 PUT Fl* F2,F3 1450 MAT PUT N 1460 MAT PUT V 1470 MAT PUT U 1480 PUT VJ/V2/V3 1490 PUT QJ -Q3 1500 PUT Ul-HJ 1510 PUT A5-A6*A7 1520 MAT PUT P 1530 MAT PUT S 1540 MAT PUT G 1550 CL0SE 0 1560 F0R I 1 = 1 T0 10 1570 PRINT 1580 NEXT I 1 1590 END > 20 10 REM C0MPUTER PREDESIGN 3F C00KE TRIPLET ANASTIGMAT 20 REM BENDING PR03RAM 30 REM FRANK VANEK* 5/4/75 - - - - - 40 DIM AC3-4) CC 3 - 2) GC 3* 3) *LC 3) * NC 2).. PC 3) * SC 2) VC 3) YC 3) ZC 3) 50 F0R 11=1 T0 JO 60 PRINT 70 NEXT I 1 80 PRINT "WHAT IS DESIGNATION 0F LENS"; 90 INPUT Ml 100 0PEN Mj; "007"* I 1 10 GET Nl 1 20 IF Nl-1 TH EN J 50 130 PRINT "INC0RRECT DATA FILE ACCESS" 140 ST0P 150 GET F1*F2*F3 160 MAT GET N 170 MAT GET V 180 MAT GET Y 190 GET VI -V2-V3 200 GET QJ-Q3 210 GET Ul -HI 220 GET A5*A6*A7 230 MAT GET P 240 MAT GET S 250 MAT GET Z 260 CL0SE I 270 F0R I 1 = 1 T0 3 230 T1=NCI J) 290 GC 1*1 l)=TJt 2*CTl-J)/2 = -300 GC2*I l) C2*TJ + J)*CTJ-l)/2 310 GC3.-I J) = C3*TJ+ J)*CTJ-l)/2 320 GC4-I 1) = CTI + 2)*CTJ-1 )/2/Tl 330 GC5*I J) = 2*CTJ+I)*CT1-1)/T1 340 GC6-I J )=C3*TJ+2)*CT)-J )/2/TJ 350 GC7-I J)=C2*Tl+J)*CTl-J)/2/TJ 360 GCS-I J )=Tl*CTl-l)/2 370 NEXT I J 380 L1=PC D/CNC J )-!> 390 L2=PC2)/CNC2)-1) 400 L3=PC3)/CNC2)-1) 410 G2=-HH2*PC2)/2 420 CC1 - J)=LJ 430 CC J* 1)=CC J* J >-.O0OJ 440 F0R I 1 = 1 T0 3 450 CC I*2) = CC J* D-LJ 460 REM C0EFFICIENTS. F0R ELEMENT A 470 TJ=-YC J)t4*CGC 1* 1)*LJ t3+GC3* J)*LJt 2*V1+GC6- J )*L 1*V1 1 2) /Ul t 2 480 T2=YC l)t4*CGC2- 1)*LJ T2+GC5* 1)*L1*V1 )/UJt2 490 T3=-YC J)t4*GC4* 1)*LJ/UJ t2 500 T4=HJ*YC l)t 2*CGC7- J )*L J* VI +GCS - 1 )*L 1 t 2) + Ql*Ul*Tl > 21 510 T5=-.25*HJ*YC J)t2*3C5> J ) *L J + T2*Q1*UJ 520 T6=Q1*U!*T3 +GC3- t 530 T7=3Jt 2*T1 + 2*3!*H1*YC1)t 2* CGC 7/ 1 )*L 1*V J J )*L J 2) /UJ 540 T7=T7-HJ t 2*PC J )/2 550 T3 = Qlt 2*T2-.5*QJ*HJ*YC J.)t2*GC5- 1)*LJ/UJ 560 T9 = Q]t2*T3 570 REM SC*"-CC*-AC* F0R ELEMENT A 530 S1=TJ + T2*CC J - D + T3+CC J- J )2 590 C1=T4+T5*CC J - I)+T6*CC 1 - 1 )f 2 600 G1=T7 + T3*CC1 - D + T9+CC 1 - 1M2 610 REM C0EFFICIENTS F0R ELEMENT C 3)*L?t 3)*L3*V3 6 20 TJ=-YC2)T4*CGC J * 3)*L3 2+GC 3* 2* V3+GC 6> 2)/Ult2 630 T2=YC3)t 4*CGC2- 3)*L3*2+GC 5- 3)*L3*V3) /Ul t 2 640 T3=-YC3) t4*GC4* 3)*L3/U1 t2 650 T4=HJ*YC 3) t 2*CGC7- 2)*L3*V3+GCS - 3)*L3 2) + Q3*Ul *TJ 660 T5=-.25*Hl*YC3)t2*GC5-3)*L3+Q3*UJ*T2 670 T5=Q3*U1*T3 630 T7 = Q3t 2*T1 + 2*Q3*HI*YC3)2*CGC7- 3)*L3*V3+GC8 - 3)*L3t 2)/Ul 690 T7=T7-Hlt2*PC3)/2+Gl+G2-A5 700 T8 = Q3t 2*T2-.5*Q3*HJ*YC3)2*GC5-3)*L3/U1 7J0 T9 = Q3t2*T3 720 REM S0LUTI0N 0F CURVATURE F0R ELEMENT C 7 30 IF 4*T9*T7 > 22 1060 REM C0MPUTATI0N 0F LA' 1070 LCI + JXSJ S2+S3 , 1080 PRINT "CC !)"- "CC3) - 1090 PRINT CC 1 1) -CC 1100 REM NEXT LINE IS EXIT- 1110 IF ABS^CLCI D-A7XJE-3 THEN 1530 - = 1 120 CC J 1) CC J* D + .000J J 130 NEXT I 1 1 140 CC J* J) = CC 1* D-.0004 1150 REM S0LUTI0N F0R PARAB0LIC C0EFFI CI ENTS. 1160 F0R I J=J T0 3 1 170 AC I 1* J) = CCC 1* J)+I J*.000J)t2 1J30 ACI J* 2)=CC 1* 1)+I 1*.000I J 190 ACI 1 - 2) = J 1200 ACI J*4)=LCI I) 1210 NEXT I 1 1220 AC 1-2) = AC 1*2)/AC1 - J) 1230 AC J*3) = AC J*3)/AC 1* 1) 1240 AC 1*4)= AC 1-4) /AC 1* J ) 2)= 1250 AC 2* AC 2* 2) -AC 2* J) *AC 1*2) 1260 AC3*2)=AC3*2)-AC3* I)*AC J* 2) 1270 AC 2* 3) = C AC 2* 3) -AC 2* J)*AC J* 3)) /AC 2* 2) 1280 AC 2* 4) = C AC 2* 4) -AC 2* J ) *AC 1 * 4) ) /AC 2* 2) f290 AC3*3)=AC3*3)-AC3* 1)*AC J* 3) -AC 3* 2)*AC 2* 3) 1300 AC3*4) = CAC3*4)-AC3* 1)*AC J * 4) -. AC 3* 2)* AC 2* 4) ) /AC 3* 3) 1310 T3=AC3*4) 1320 T2=AC2*4)-AC2* 3)*T3 1330 T1=AC J* 4) -AC 1*2)*T2-AC 1*3)*T3 1340 T3=T3-A7 1350 IF 4*Tl*T3 1400 X2=T3/TJ/X1 / 1410 PRINT 1420 PRINT "THE FIRST CURVE F0R THE NEXT TRIAL CAN BE "XI" 0R "X2 1430 PRINT "PICK LARGE 0R SMALL"; 1440 INPUT XI 1450 CC J* 1)=XJ 1460 IF ABSCXJ)>ABSCX2) THEN 1480 1470 CC J* 1)=X2 J 480 IF XSC:J*1)="L" THEN 430 1490 CC J* 1)=X1 1500 IF ABSCXJ XABSCX2) THEN 430 15J0 CC J* J')=X2 1520 G0T0 430 1530 REM. EXIT: DATA SUMMARY 0UTPUT PRINTED 1540 FOR I 1=1 T0 10 1550 PRINT 1560 NEXT I 1 "ELEMENT" 1570 PRINT "VARIABLE"** 1580 PRINT , 1*2* 3 1590 PRINT "P0WER"*PC J)*PC2)*PC3) 1600 PRINT "CURVATURE"*L1*L2*L3 > 23 J6J0 PRINT "CURVE J "* CC J * I ) * CC 2* J ) * CC 3* 1 ) 2- 1620 PRINT "CURVE 2"* CC 1 * 2) * CC 2) * CC 3* 2) 1630 PRINT "A3R"*YC ) ) *YC 2) *YC 3) J 640 PRINT "SPACING"*" "SCD" "SC2) 1650 PRINT "GLASS"*ZC1)*ZC2)*ZC3> J 660 PRINT "INDEX"*NC J)*NC2)*NC 3) 1670 PRINT ^'V#"*VC J)*VC2)*VC 3) 1630 PRINT "F0CAL LEN3TH="F1 1690 PRINT "MAXIMUM APERTURE="F2 J 700 PRINT "MAXIMUM FIELD ANGLE="F3 1710 F0R I J = J T0 JO 17 20 PRINT 1730 NEXT I 1 1740 END 2h APPENDIX 2 List of Variables 25 / Program Variable Used In Description Al 1 Aim Petzval Curvature A2 1 Aim Longitudinal Chromatic A3 1 Aim Lateral Chromatic kk 1 Computed Lateral Chromatic A5 Both Aim Astigmatism=Asts A6 Both Aim Coma=Comas A7 Both Aim Spherical=LA' A(x .y) Both Array members for Crout method Cx 2 Coma contribution for element x C(x .y> 2 Curvature of surface y, element x For Newton-Raphson method: Dl 1 AK for run with Ayfc = 0 D2 1 A T'ch for run with A yb =0 D3 1 A y6 for run with Ayc =0 Bk 1 A yc for run with A y- =0 D5 1 D5=D3 from previous run iteration D6 1 B6=Dk from previous run iteration Fl Both Focal length F2 Both Maximum F/humber F3 Both Maximum field angle Gx 2 Astigmatism contribution for element x G(xi) 1 Glass name for element x 2 G(x,-y) G-sum number x for element y HI Both Image plane intersect height for paraxial principal ray 11 Both Loop variable Kl Both Aim Space Ratio, K 26 , Program Variable Used In Description K2 1 Computed space ratio Lx 2 Total curvature, c ,for element x LA' L(x) 2 for run through x Ml Both Reference name, called "Lens Des ignation" in computer output Nl Counter to keep track of the part of the Newton-Raphson procedure being executed Nl Check to verify that correct data file was being accessed N(x) Both D index of refraction for element x P(x) Both Power of element x Qx Both Q constant for element x SI 1 dK/by S2 1 bn/By S3 1 ciT'ch/^y sk 1 OT'ch/By Sx 2 Spherical contribution for element x S(x) Both Spacing between element x and element x+1 Both Tx Variable for temporary storage , also for aberration coefficients u0' Ul Both also written as u'K U2 1 Angle for incoming paraxial principal ray U3 1 Angle for incoming marginal ray \]b 1 Angle for marginal ray entering element 2 V5 Angle for marginal ray entering element 3 27 Program Variable Used In Description U(x) 1. Variable used to store only the Y(x) values needed in the second program Vx Both The v constant for element x V(x) Both V number for glass of element x Xx 2 Variable for temporary storage x$ 2 String variable for choice of "LARGE OR SMALL?" Y(x) Both For x=l to 3- ray height for marginal ray at "element X. For x=k or 5 i ray height of paraxial prin cipal ray at element 1 or 3. re spectively Z(x) Glass name for element x 28 APPENDIX 3 Example of Input/Output 29 Using the example from Dr- Kingslake's notes: WHAT I S LENS DESIGNATION 7KTEST WHAT IS THE DESIRED FOCAL LENGTH 7 10 UHAT IS- THE DESIRED MAXIMUM APERTURE 7 4.5 tjHAT IS THE 'DESIRED MAXIMUM. FIELD ANGLE CDEG) 20 CLASS F0R ELEMENT 1 7SKJ6 1)= NC ,?1 .6203 1)= V< .760.2 (LASS F0R' ELEMENT 2 ?F4 i ' ' 2)= NC ''? J .6164 2)= VC .736.6 GLASS F0R ELEMENT 3 7 SKI 6 = NC 3) ? J .5202 VC 2) = 7 60.3 WHAT IS THE AIM VALUE F0R: PETZVAL CURVATURE ., ?. 035 L' CH 7-.02 " T' CH L, 7 0 'AST'S ': 7-.09 C0MA. L 7.0025 LA1 7-.08, K 71 YC2) S;w * T*.OC " -AA'--** .383839 2.41453 i.doooo -lVt;369 4E-02 .366339 1.4523 3 -954036 -2. K2632E-0 3 .863136 1 ?22181 *9 54990; -^T.dOSSOE-Q^ ..861771 1.12788 .9 559 45 "-22Q56E-Q4 .86141 I 1.09 416 .956901 -1-21T36E-04 .361342 1.07541 19 578 53 -8.37574E-05 .361323 1.. 059 23 9588.16? -6.245RTE-05 J. 04 3 54 .'9 59 77 5 -4.4343AE-05 .861306 J .02306 .960735 -2.77 336E-05 - E- 36J30r 1. 0T 276 .961695 1.233 33 05 .861 299 J .00003 .962503 -J .31969S-07 1 .00000 .'? 6250-3 -9.6447 5E-I2 P0VERS. .16-3101 -.30 471 2. 194056 ' SPACINGS J 33743 . . 1 33743 30 WAT IS DESIGNATI0N 0F LENS ?KTE,ST ' ' / THE VALUES 0F CC3*1>. ARE: -.674664 7.23061E-02 ' PICK LARGE 0R. SMALL ?S LA' CCI) . CC3) .270900 7.23061E-02 2.61735E-02 - r ' ' - '"'m THE VALUES 0F CC 3* 1 ) ARE: -.674739 7.2431 5E-02 PICK LARGE 0R SMALL "7 S ... LA' CC1) CC3) . .271000 7.24315E-02. 2.63005E-02 THE VALUES 0F CC3*I) ARE: -.6749 14 7.25568E-02^ PICK LARGE 0R SMALL ?S CCJ) CC3) LA' .271100, 7.25568E-02 2.64270E-02 THE FIRST CURVE F0R THE NEXT TRIAL CAN BE .227703 0R_ .360523 PICK LARGE 0R SMALL! ?S -M_- THE VALUES 0F CC3/1) ARE:'-. 61 2252 9.89460E-03 SMALL;- M PICK LARGE 0R ?S . CXI) CC3> LA* M 9.89460E-03 -9.33489E-02 ARE:-. .0061 . THE VALUES 0F CC 3*1) 612419 I 2E-02 . " " PICK LARGE 0R SMALL ?S M CCI) CC3) LA* .227703 -9.78416E^-02 1.00612E-02 ,,_/>; >a THEoVALUES 0F CC3*1) ARE: -.'61 258 5 . 1.02276E-:02 L.:a,M/.- a . PICK LARGE 0R SMALL ?S .\ "27:: 'CCI) ': CC3) . -r^L -9.73357E-02 .227303 1.02276E-02 , BE. .231417- .THEFIRST-CURVE F0RaTHE. NEXT TRIAL CAN 0R.-;;>296O1 6 S,,:"" PICK LARGE 0R SMALL ".M 7 ^yy,. rry^^y -^ 1.59972E-02 T-IE VALUES 0F CC3* I > ARE: 613 355 , -PICK LARGE 0R SMALLa;? ISAUyyy CC 1 ) j'-yi$y: CC 3)#V;L#|tA!. .23131T't- 1.59972E-02:!M3.04264E-02 ""- '- - - . -^yy 7-yim^y sy , A^jixMi ^' ' 1L592E- 'THE VALUES- 0F CC 3*1 > ARE: -^613 517 1,^6 0.2 ' A CK' LARGE:' " - j: a M PI 0R SMALL ? S : _ LA*" : ; :.'M--:^'.-V.: -Z - CCI) ; Z-'-Zj CC3> . .231417 rL 1.6I592E-02 -7;99639E-02 VALUES 0F CC 3* 1> ARE: -.613679 1.63211E-02 PICK LARGE 0R SMALL ?S LA' CCI) CCS) ... 1.-63211E-02? .231517 -7.95127E-02 31 THE FIRST .303055 CURVE FOR THE NEXT TRIAL CAN BE .231410, 0R PICK LARGE 0R SMALL 7 5 THE VALUES 0F CC3*1) ARE: -.613344 L59862E-02 PICK LARGE 0R SMALL ?S - CCI) CC3) LA- .231310 I.59862E-02 -8.04576E-02 THE VALUES 0F CC3*1) ARE: -.618506 1. 61,48 2E-02 PICK LARGE 0R SMALL ?S' LA* CCI) . CC3) .231410 1.61482E-02 -8.00000E-02 VARIABLE:^ ELEMENTS _- - y . . . -.-;< 1 J2k? P3WER ja? .163101 -.304712 .194056 CURVATURE .271000 -V494341 .312342L CURVE -I Mi .231410 -.263741 lV6T482E;-02 CURVE 2.M -3.95900E-02. J-y 230 59 9 .296694 A3R 1 . 1 1 1 1 ILL \! .861298 : -.962502L SPACING- A: a 1 33748 .33748 GLASS SKI 6 F4 M SKI 6 INDEX 1 . 62030 1.61640 y.J:'7, J. 62030 V# .60.30003.6000 60.3000 EZfCAL LENGTH= 10 n&XIMUM APERTURE= 4.50000 MAXIMUM FIELD ANGLE=> 20 32 APPENDIX k Aberration Equations 33 Equations for Elements 1 and 3: For spherical aberration: u. SC*= Z2L (G,c9- G&, + Gj*V + Qcc^- Gpcy+ G^cv3) ^ can be placed in the form: SC*=%f^Q+aJcr (Gc3 Where: a^, = =2L + qx?v + Gcv1) u' ! +yf / For coma, contribution: CC*= - -hj^ (0.250,00, Gov -Ggc*)+ Qu (ac + a,c, + a^cf) can be placed in the form: CC^^+a^^+a^ Where: 83= H-hy3* (+G4CV + Ggc ) + Qu^ = a^ -hy*(0.25Gc) + Qu^a, af= Qu^a* For astigmatism contribution: AC*= d$L + 2iL_. (-0.250^0,+ + + CJpv q,ca) <$(a0 + cp, +a^) can be placed in the form: AC*=s^+aj.q+a^ Where: -^- SSIkL a, + (Gq^ + ^ ) + Q^ Constants: c = JZf/ (N-l) v - u/y Q = y/y 3* Equations for Element 2: For spherical aberration: if ^"-(qc3 SC= - Gj2c, + G-cV - G^qv + (|cv*) can be placed in the form: SC^+tjCi+b^cf y* Where: bc = l^L (G,c^+ ty*v + qcv*) UK (+(- b = ^ + 0v) UK For coma contribution: G* CC= - - -hyM0.25Gpc, Gp ) can be placed in the form: CC^h+hc, Where: b3 = -t-hy^+Gpv + (\cx) b+= -0.25hj%: For astigmatism contribution: AC- ifiL-l". 2- can be placed in the form: AC=bf 3-f-a Equations for G-sums: G1= N2(N-l)/2 G2= (2N+l)(M-l)/2 G3= (3N+l)(N-l)/2 Gk= (N+2)(N-1)/2N G5= 2(N+1)(N-1)/N &&= (3N+2)(N-1)/2N G?= (2N+1)(N-1)/2N Gg= N(N-l)/2 35 APPENDIX 5 The Crout Method 36 Crout Method : Given three simultaneous equations: ax + by + cz = d ex + fy + gz = h ix + jy + kz = m Where x, y and z are unknowns, the rest are constants Set up "A matrix": b c f g j k Then member A(2,3)=g, for example (1,1)" Then: A = A(l,l) A(2,l)' = A(2,l) A(3,D' = A(3,l) A(l,2)' = A(lf2)/A(i;i) A(l,3)' = A(l,3)/A(l,l) k(l,b)' = A(l,4)/A(l,l) A(2,2)' A(2,l),A(L;2X,^.a'-' = A(2,2) - A(3,2)' A(3,D'A(1,2)' = A(3,2) - A(2,3)' A(2,1)'A(1,3)' A(2,2)' = A(2,3) - / A(2,*0' A(2,l)'A(l,4)' A(2,2)' = A(2,4) - / A(3,3)' l)'A(lf3)' = - A(3,2)'A(2,3)' A(3,3) A(3, - k(3,b)' A(3,l)'A(l,4)' = - A(3,2)'A(2,^)' k(3,b) - / A(3-3) Finally: k(3,b)' z A(2,^)' y A(2,3)'z k(l,b)' x A(l,2)'y - A(l,3)z. 37 APPENDIX 6 References 38 References: John F. Carson, Thesis Advisor, Personal Correspondence Class notes, Design of Lenses and Optical System, Course number 0PT-472, Institute of Optics, University of Rochester, Dr. Rudolph Kingslake, 1973. Modern Optical Engineering, Warren J. Smith, McGraw-Hill Book Company, U.S..', 1966, pp. 280-282, pp. 340-34?. Class notes, Electrical Engineering II, Course number EEEE b62, Electrical Engineering Department, Rochester Institute of Tech nology, George L. Thompson, 1975* Type," "Design of Triplet Anastigmat Lenses of the Taylor Stephens, JOSA, v. 38. pp.1032-1039 (194-8) "Comment on Design of Triplet Anastigmat Lenses of the Taylor Type," Smith, JOSA, v.40, pp. 40 6-4-0 7 (1950) Triplets," "Design Study of Air Spaced Wallin, A0, v. 3, pp. 4-21- 426 (1964) 39 APPENDIX 7 Resources 40 Computer: Xerox model Sigma Six, property of Rochester Institute of Technology. Computer was accessed on line through teletype, using CP-V BASIC Language. Access to the computer is open to any registered R.I.T. students. Grant: This thesis was aided by a grant from the Central Intelligence Agency of the United States Government.