Automatic lens design with pseudo-second-derivative matrix:a contribution
Andrea Faggiano
An optical design computer program has been developed. Principally it makes the use of the method de-fined by D. C. Dilworth as "pseudo-second-derivative," which is an extension of the well-known dampedleast squares method. In the following exposition of the program a special procedure for the selection of thedamping factor, which significantly improves the convergence of the method, is emphasized. In additionsome sample problems are illustrated.
1. Introduction
An optical design computer program has been de-veloped. Principally it makes use of the method sug-gested by Buchele 1 and defined by Dilworth, 2 who ap-plied it to the optical systems optimization, as"pseudo-second-derivative" (PSD). Compared withthe well-known damped least squares method (DLS)it has the advantage of holding in due consideration thedifference of linearity of the independent variables and,therefore, presenting a more rapid convergence, prin-cipally when great nonlinearities are present as is thecase in optical systems. Moreover a special procedurefor the selection of the damping factor can further im-prove the convergence, as shown in Sec. III.
II. Mathematical MethodFor a more detailed description of the two methods,
DLS and PSD, Refs. 1-6 should be consulted; in thisarticle only a few specific references will be made.
Given an initial optical design described by n inde-pendent variables x (curvatures, air and glass axialthicknesses, glass refractive indices and dispersions,etc.), the goal is to discover a set of increments Axj suchthat the new set of parameters,
x = x + Axj,
will result in a minimum value of a merit function q,defined as the sum of the squares of m weighted aber-rations fi.
The author is with Aeritalia, Gruppo Equipaggiamenti, 20014Nerviano, C. P. 21, Italy.
Each f may be an optical aberration or any othercharacteristic that is to be controlled (focal length, backfocus, edge thickness, etc.).
In both methods the merit function is minimized bysolving the following matrix equation:
(AT-A+ C) Ax= -AT.F,
where A is the matrix of the first derivatives aij =bfiloxj;
Ax is the solution vector with elements Axj;and
F is the vector with elements f.This optical system then becomes the starting point forthe next iteration.
The difference between the PSD1,2 and the DLS3-6
methods consists of the matrix C, which takes into ac-count the fact that the relation between the aberrationsand the independent variables is nonlinear. For thePSD: C = p 2B, where B is the diagonal matrix withelements
m 2fi
bjj= i fo x?2
where the second homogeneous derivatives (2 fi/XJ are
approximated to the Dilworth method2 and p is thedamping factor. The cross derivatives, which are notincluded owing to the considerable computation effortinvolved, are likely to be unimportant in optical sys-tems. 2 5 7 For the DLS: C = p2 Q, where p is thedamping factor, and Q is the diagonal matrix, the ele-ments of which are suitable weights . A stated valueof c is usually given for the curvatures, another for thethicknesses and so on. According to Meiron4: c = ai2A comparison between the two methods of selecting c;is reported by Kidger and Wynne.8
In practice, the DLS method deals with the nonlin-earity of the system by either adding damping factorsto the principal diagonal elements of the resolutive
system matrix or by multiplying them by dampingfactors, thus limiting the steps of the independentvariables. In this way, the least linear variable limitsthe step of all the others. This problem is nQt presentin the PSD method in that the presence of the secondderivatives in the resolutive system ensures that eachindependent variable has a step depending also on itslinearity: if a variable is less linear than the others itsstep will be reduced without limiting the others. Theresult is a more rapid convergence of the PSD methodthan is the case with the DLS one.
The greeter the difference between the nonlinearityof the variables, the more rapid the convergence of thePSD method, compared with the DLS one, will be. Inoptical systems the matrix B elements differ consider-ably from each other.
We can also avoid the inconvenience caused by theexisting differences among different kinds of variables(e.g., curvatures and thicknesses) by means of the DLSmethod, provided that suitable weights c; are chosen.Generally, however, even the elements concerningvariables of the same kind differ greatly: in that casethe PSD method will converge much more rapidly thanthe DLS one. A practical comparison between the PSDand the DLS methods is reported by Dilworth2 ; anotherone is reported below in Sec. VI.
Ill. Selection of the Damping Factor
Before being added to the corresponding elements ofthe matrix (AT . A) principal diagonal, the diagonalmatrix B elements are multiplied by a factor p2. Thehigher the value of p2 the more the step made by thesystem in an iteration is reduced. The choice of thedamping factor p 2 is much less critical with the PSDmethod than with the DLS one.
In fact, (see also Dilworth2 ) there is generally littledisparity between the predicted and the actual meritfunction improvement, provided p 2 is constant andequal to one; however, as will be shown, a suitable choiceof p 2 may significantly reduce the number of iterationsand the running time.
There are various criteria for choosing the p 2 factor,e.g.:
(A) Kidger and Wynne,8 using the DLS, start with p2
initially equal to one and then double or halve it ac-cording to the agreement between the predicted and theactual improvement of the merit function. Dilworth2
also applies this method to the PSD. Experience showsthat this method often leads to p 2 being equal to one(see also Dilworth 2 ).
(B) Others 3 choose, by curve fitting, the value of p 2
that allows the best possible improvement of the meritfunction in a single step. Experimentation shows thatby using this method p 2 is almost invariably below onewith medium value ranging about 0.3.
A method that is very simple and saves running timebut nevertheless gives good results, has been developedas follows: starting from a very low p 2 value (actually0.005) we solve the system (1); if the merit function kdecreases we take the step as good, whereas if 0 in-creases we double p2 until 4 decreases. This method
SlI'3 11AM1R,.mm
..
~_7
11I SI ORI I ON,
SPOT IIAMETlR,mm PI STORT ICN,C
Fig. 1. Infrared triplet; initial and final configuration.
consists of allowing the system to develop as fast aspossible provided that q decreases, without taking intoaccount the agreement between the predicted and theactual improvement of 0. In practice, the value of p 2
remains very low, often 0.005.These methods (A, B, and the new method outlined
above and henceforth referred to as method C) havebeen compared in designing an IR triplet (3-5-,um band,Si-Ge-Si) with the following features: object at infinity,f/1.5, focal length = 150 mm, 60 field, and absence ofvignetting. All radii and air thicknesses were allowedto vary. Such a simple optical system has been chosento compare the methods without, as far as possible, in-terference caused by violations of the boundary condi-tions. In fact, only in the initial iterations did the firstair axial thickness tend to become negative.
Proceeding from a rough initial configuration com-.posed of plain-convex or plain-concave elements, thethree methods lead to the same final configuration (Fig.1), which has a lower merit function by a 106 factor, butwith a different number of iterations and runningtime.
The results are shown in Table I and Fig. 2. As canbe seen, method C reaches the solution more rapidly,followed by method B and then method A. At first, ascan be seen in Fig. 2, showing the dependence of themerit function q on the iteration number, method C hasa less rapid convergence than method B, but then itreaches the final configuration within a lower numberof iterations. This seems to depend on a more rapiddevelopment of the system towards points from whichthe merit function may decrease more rapidly, though
Table I. Comparison Between the Three Methods of Choosing theDamping Factor
Method A Method B Method C
No. of iterations, % 260 155 100Running time, % 190 170 100
NO'. F ITERATIONS
C' 1 IC 'O IO 4C C '' 70 "'IL '(C IC'
method A
- * _ thod C
- _..
Fig. 2. Comparison of the rates of convergence of the three methodsof choosing the damping factor for optimizing an IR triplet.
the single step does not optimize the merit functionreduction.
The lower convergence of method B compared withmethod C is not due to the approximations made insearch of p, which minimizes the merit function in thesingle step. In fact, by using a very careful and time-consuming method for the p search, an even slowerconvergence has been found. This probably resultsfrom the fact that the parabolic interpolation initiallyused, in the p search, leads to p values nearly alwayslower than those corresponding to the actual minimumin the single step (e.g., closer to the p search in methodC). Similar results have been obtained with differentoptical systems.
IV. Merit Function
The present program uses a merit function based onthe spot size. The periphery of the vignetted pupil isrepresented by an ellipse 9 that is divided by a rectan-gular mesh.'0 Each ray traced from the object pointthrough a mesh point at the center of a rectangle is re-garded as representative of the energy passing throughthat rectangle. The code minimizes the rms deviationof the rays from the centroid of the ray bundle on theimage surface. As shown by Brixner," this procedureminimizes the optical path difference also.
V. Boundary Conditions and AberrationsThe actual code is very flexible. In fact every de-
pendent variable fi-such as aberrations, optical fea-tures of the system (for example, focal length, magni-fication, entrance and exit pupil position and size, andso forth), mechanical conditions (axial and edge thick-nesses, overall length, and so forth)-may be placed inone of the following categories:
(1) variables with target value and weight;(2) variables with lower limit;(3) variables with upper limit; and(4) variables of which a precise value is requested.The sum of the squares of the differences between the
first category variables and their target values is mini-mized by the PSD method. The variables of the fourth
category are brought and held to their target values bythe Lagrange multipliers method.4 6 The variables ofcategories 2 and 3 may be treated in two ways: (a) bythe Lagrange multipliers method and (b) by consideringthem as aberrations and giving them suitable weights.Method (a) requires more computation time but it ismore precise and does not involve the weight choiceproblem.
VI. Sample Design
As an example of optical design the lens called byHopkins and Feder12 the Symposium lens has beenchosen. The system has been designed in unit focallength. The initial data (Table II and Fig. 3) remain thesame as those used by Hopkins and Feder, but theglasses are those used by Hopkins in his ORD-2 design(the automatic choice of glasses has not been used in thisdesign). Owing to the glass change the initial focallength is very different from the required one.
The following restraints have been imposed: mini-mum axial and edge glass thickness, 0.05 mm; minimumaxial and edge air thickness, 0.001 mm; and minimumback focus, 0.20 mm. The negative element axial
Table II. Prescriptions for the Starting Symposium Lens Designa
C t N V1.1500
0.1200 1.5004 65.760.0000
0.0010 11.8000
0.1200 1.5004 65.760.5000
0.2100 1-1.5000
0.0500 1.7213 29.241.5000
0.2500 1-0.5000
0.1200 1.6968 55.43-1.8000
a Entrance pupil is 0.45 to the right of the first surface; focal lengthis 1.93; back focal length is 1.3146; the f/number is 2.90; half-field angleis 6; corner illumination is 75%.
Table . Prescriptions for the Final Symposium Lens Design a
C t N V
1.608980.15926 1.5004 65.76
-0.235430.00099
1.964400.10853 1.5004 65.76
0.846310.03524
-0.129650.47425 1.7213 29.24
4.529600.09657
3.297070.18935 1.6968 55.43
0.10347
a Entrance pupil is 0.7070 to the right of the first surface; focallength is 1.00; back focal length is 0.1992; the f/No. is 1.5; half-field
angle is 6°; corner illumination is 75%.
UL~~f -
O(} C .0001 . C'2 - . -0.
-, SPOI RAPIIIS, l) I TORT I
Symposium lens; final design.
NO. OF II[RAIIONS
e 1 2 10 t 0 it) 9 0 hby , . . . I I
, I
P'S) mP-thod with m-th d C ,el-'tii q
th, ddipiiW a Idt tars -d00p ri u, iiii tlm
..-.. ILS m'thd with Miir', m-thod I'f
.- I lutiriq e; dd methd 8 oh. -cleutiny
the dmpiiy di tot - Ah0y rnii i iq f itm
Comparison of the rates of convergence of the PSD ai
methods for optimizing the Symposium lens.
thickness has been limited to the same value of theORD-2 design (0.47 mm) because of its tendency tobecome too thick. Observing the results obtained by
_X Hopkins and Feder, the distortion has been inserted inthe aberrations of the first category with target valueof +0.6%.
A better result is given by a second run in which thedistortion has been allowed to vary between -0.6 and+0.6%. In some iterations the distortion remained+0.6; afterward in the last iterations it reached the value
0.6%, and the system got a better minimum.The final system (obtained with twenty-six itera-
tions) and its optical characteristics are shown in TableIII and Fig. 4. The data concerning the encircled en-ergy distribution and rms image-spot radius have been
ON g obtained by tracing 384 rays for each of the wavelengthsC, d, and F; each one weighted one.
Finally, Fig. 5 shows a comparison between the PSDand the DLS method in the Symposium lens design. Ascan be seen, the first method has a more rapid conver-gence. In both cases the final configuration is thesame.
I wish to thank A. Canfora whose experience has beenuseful during the development of the present work andP. Consolati for our valuable exchange of ideas.
References1. D. R. Buchele, Appl. Opt. 7, 2433 (1968).
2. D. C. Dilworth, Appl. Opt. 17, 3372 (1978).3. D. P. Feder, Appl. Opt. 2, 1209 (1963).
4. J. Meiron, J. Opt. Soc. Am. 55, 1105 (1965).
5. C. G. Wynne and P. M. J. H. Wormell, Appl. Opt. 2, 1233
(1963).6. G. H. Spencer, Appl. Opt. 2, 1257 (1963).
7. D. S. Grey, J. Opt. Soc. Am. 53, 672 (1963).
8. M. J. Kidger and C. G. Wynne, Opt. Acta 14, 279 (1967).
9. W. B. King, Appl. Opt. 7, 197 (1968).
10. T. C. Doyle, "Automatic Lens Design by Nonlinear Least Squares
Optimization of a Continuous N-Parameter System," PreprintLA-UR-73-518 (LASL, Los Alamos, N.M., 1973).
id DLS 11. B. Brixner, Appl. Opt. 17, 715 (1978).12. R. E. Hopkins and D. P. Feder, Appl. Opt. 2, 1227 (1963).
