Lens design and local minima

Berlyn Brixner

The widespread belief that local minima exist in the least squares lens-design error function is not confirmedby the Los Alamos Scientific Laboratory (LASL) optimization program. LASL finds the optimum-mini-mum region, which is characterized by small parameter gradients of similar size, small performance improve-ment per iteration, and many designs that give similar performance. Local minima and unique prescrip-tions have not been found in many-parameter problems. The reason for these absences is that image errorscaused by a change in one parameter can be compensated by changes in the remaining parameters. Falselocal minima have been found, and four cases are discussed.

Nel mezzo del cammin di nostra vitami ritrovai per una selva oscura,

che la diritta via era smarrita.

In the midst of life's journeyI found myself in a dark woodwhere the right path was lost.

First stanza of Dante's Inferno

The author is with University of California, Los Alamos ScientificLaboratory, P.O. Box 1663, Los Alamos, New Mexico 87545.

Received 17 May 1980.0003-6935/81/030384-04$00.50/0.3 1981 Optical Society of America.

1. Introduction

There is a very reasonable concern that the lens-design error function may go through local minima inthe course of its reduction by least squares methods andthat designing may stop well away from the optimumminimum. One's intuition about local minima is basedon large experience with paths in 2-D and 3-D spaces.Lens design usually proceeds in spaces with many di-mensions, and it is not known whether local minima arelikely. In the absence of mathematical proof, a re-counting of experience with many minimizations isuseful.

Although there is widespread belief that local minimaexist in the least squares lens-design error function,1-5no examples are cited, and possible causes are not dis-cussed. Those beliefs range from the opinion that itwould be surprising if there were no local minima 3 to thedeclaration that the lens optimization problem has notbeen solved and that damped least squares only findslocal minima.5 No local minimum has been identifiedin the thousands of runs made during the 22-year his-tory of optimizing complex lenses with the Los AlamosScientific Laboratory (LASL) program. A lens designis in a local minimum when the error function can beimproved by large parameter changes but not by smallchanges.3 Because small parameter changes are the keyto identifying a local minimum, it is necessary to avoidextraneous changes that could cause the optimizationpath to branch and thus generate abrupt changes in thedesign or lead it into a dead end. One criterion foravoiding extraneous changes is to hold constant thelens's specific set of glass elements, variable parameters,image-quality targets, and rays traced to the imagesurface. Any violation of these four conditions willdelay or stop design progress and often simulates a localminimum. Violations are also caused by any of fourextraordinary situations that can occur during optimi-zation. When violations are avoided, local minima are

384 APPLIED OPTICS / Vol. 20, No. 3 / 1 February 1981

z00zE_ A \

0 o ~ ~~~~~~~ \CORRECTION PATH

Fig 1. Local minima along a hypothetical lens correction path.

not encountered, and the lens design that is found isbelieved to be in the optimum-minimum region. Aunique prescription has not been found for many-pa-rameter problems. The reason for this absence is thatimage errors caused by a change in one parameter canbe compensated by changes in the remaining parame-ters. Parameter compensation also accounts for theabsence of local minima.

II. Discussion

As long as the four constant conditions are main-tained, the least squares optimization procedure leadsto what we believe to be the optimum-minimum region,where the parameter gradients are very small (10-5-10-10), all of similar size, and all are slowly and irregu-larly approaching the limit of machine-computationaccuracy. At the same time, the error function gainsare also very small and become even smaller as opti-mization progresses toward convergence. For many-parameter problems, the minimum region appears tobe a vast shallow basin where absolute convergence hasnot been achieved and nothing like a unique prescrip-tion has been found, even when very long runs weremade. There are many similar, but slightly different,prescriptions that give about the same optical perfor-mance. Because the ability to correct image errors isdistributed among the available parameters, a slightchange in one parameter can be compensated by otherslight changes in those remaining. 6 7 The effects of alarge parameter change can be compensated in the earlystages of optimization.8 The belief that the region ofthe optimum minimum has been reached is reinforcedby the fact that no better region has ever been found inthe hundreds of complex lenses that have been opti-mized. In many cases a variety of starting prescriptionswere optimized. Additional evidence comes from somespecial few-parameter problems that do have uniquesolutions where the error function goes to machine zero.The unique solution is reached from many differentstarting prescriptions. Examples are the conic-sectionmirrors and the aplanatic lens. Unfortunately, nomethod exists to verify that the best possible many-parameter lens has been designed. 1 This problem isunresolved, because there is no analytic expression thatdescribes the overall lens performance in terms of itsparameters.

Figure 1 shows local minima along a hypotheticalcorrection path chosen by an optimization program.Such local minima are well known in mathematicalfunctions. 9 The concern is that a local minimum likethat shown at A might exist and that the program wouldstop there, thus failing to achieve a prescription thatmeets the system requirements indicated by the dashedline. Another concern is that, although the system re-quirements would be met if the program found the localminimum at B, the optimum minimum at C would notbe discovered. No such local minima have been ob-served during optimization with the LASL program.However, false local minima are generated under twoextraordinary situations, which are readily detectedbecause they distort the performance evaluation in anabrupt or even discontinuous manner. They should notbe considered local minima in the present context, be-cause they are on branches from the optimization path.The first extraordinary situation is a violation of theoptimization procedure that is caused by a loss of rays.The second extraordinary situation is the encounteringof a local minimum along the damping-number searchseries, not to be confused with a local minimum alongthe optimization path. Less obvious violations arecaused by two closely related extraordinary situations,a singularity in the program's matrix calculation and avariable parameter that does not move during optimi-zation. All four situations will be discussed.

The first and most frequent extraordinary situationthat can occur during optimization is the loss of raysthat are being traced to the image surface, a violationof the optimization procedure because it results in afalse evaluation of the image errors. The optimizationprocedure will generally favor the loss of rays becausea few rays can be directed toward each image target withless average error than can a large number of rays.Therefore, a ray-loss penalty must be imposed if thisfalse image-error evaluation is to be avoided duringoptimization. An effective penalty is a large ray-losserror to replace the ray's usual image error. The LASLprogram identifies the cause of the ray loss and gener-ates a ray-loss penalty that is proportional to the sizesof three quantities: the ray error; an assigned biasnumber; and the violation weight.'0 The ray loss iscaused by one of five possible conditions: total internalreflection; failure of the ray to intersect the optical

1 February 1981 / Vol. 20, No. 3 / APPLIED OPTICS 385

Table 1. Characteristics of Conrady and LASL Optimized Cemented-Doublet Lensesa

Lens Conrady LASL LASL LASL LASL ConradyCharacteristic left hand axial full field full field axial right hand

R1 +11.170 +12.609 +5.586 +5.596 +4.493 +4.464D1 0.250 0.250 0.250 0.250 0.250 0.250R2 -2.628 -2.646 -3.747 -3.744 -4.150 -4.065D2 0.200 0.200 0.200 0.200 0.200 0.200R3 -7.077 -6.965 -24.348 -24.166 +561.73 +431.71BF 9.492 9.864 9.764 9.764 9.716 9.713On-axis imageb 0.00018 0.00015 0.00044 0.00042 0.00014 0.00015Off-axis imageb (0.00096) (0.00099) 0.00055 0.00055 (0.00099) (0.00102)

a All calculations were made with C, e, and F indexes of 540597 and 622360 glasses.b The image size is the mean rms radius of the C, e, and F images at axial and 1.5° off-axis points.

-I

-2

s -320z

CD

0 -4

-J

-6

-7

LOG DAMPING NUMBER

Fig. 2. Local minima along a damping-number search curve.

surface; ray intersection at an aperture obstruction; rayintersection beyond a featheredge between adjacentoptical surfaces (another type of aperture obstruction);and ray intersection at a bounded parameter that haspassed its assigned maximum or minimum value.

The second extraordinary situation that can occurduring optimization is the encountering of a local min-imum along the damping-number search series gener-ated by the computer program. These local minima areartifacts introduced by the use of the damping number.Figure 2 shows the occurrence of local minima at severalvalues of the least squares damping numbers."1 Thecurve is based on one set of parameter derivatives. Thenumbered data points are the ones generated by theprogram during one iteration. By making a series oflarge steps, the search procedure avoids the local min-imum adjacent to trial 1 and proceeds to the optimumminimum near trial 9. Small steps would hold thesearch at the local minimum, where the optimum valueof the error function would not be found.

Figure 2 also illustrates the largely unexplained er-ratic fluctuation of the error function that occurs fre-quently in damped least squares, perhaps caused by thelimited number of significant figures in machine cal-culation. Undamped least squares are even more er-ratic. The use of damped or undamped least squares,to solve the lens design problem is the use of a linearprocedure to solve a very nonlinear problem. Becausethe linear least squares solution rapidly diverges fromthe nonlinear problem, each calculation iteration canonly generate an improved lens prescription that is notgreatly different from the starting prescription. Thedamping numbers cause the direction vector to sweepacross parameter space, thus increasing the discoveryof helpful solutions. By finding the best size for thedamping number,' 2 much larger lens performance gainscan be achieved than those given by an undamped sys-tem. Experience at LASL indicates that the sameminimum region is reached by both the damped and theundamped least squares procedures. In both cases the

386 APPLIED OPTICS / Vol. 20, No. 3 / 1 February 1981

gain at each optimization calculation is only a small partof the total gain that is finally achieved when the min-imum region is found after a sequence of repeated op-timizations in succession (iterations).

The third extraordinary situation that reduces orstops optimization gains is matrix singularity'3 or ill-conditioning,4"4 a computation difficulty caused by thecomputer's limited computation accuracy. A singu-larity occurs when the matrix coefficients for a param-eter go to computer zero, and the system solver thenproduces erratic results or comes to a halt. The addi-tion of a damping number to the matrix of the leastsquares system reduces the occurrence of singularities.Strict mathematical singularity probably does not occurin lens design.14

The fourth extraordinary situation that may simulatea local minimum is optimization with a variable pa-rameter that does not move a significant amount duringthe calculations. This occurs when the parametergradient is very small during the early stages of opti-mization. The gradient sizes of different types of pa-rameters (such as axial distances and surface curva-tures) can be approximately equalized by adjusting thesize of the lens being optimized.1 5 The adjustment ismade by assigning a suitable scale factor, which the codeuses to multiply the input lens prescription. Thisprocedure is effective because the derivative of the errorfunction relative to the axial distances varies as the firstpower of the scale factor, whereas relative to the surfacecurvatures it varies as the third power.15

Before concluding this discussion about local minima,it is necessary to mention the two spherical-aberrationminima that occur in classical lens design. Conradydescribes this condition in great detail and evaluates itfor the cemented-doublet achromatic objective, theairspaced doublet, the landscape lens, and the Cooketriplet.16 But the two spherical-aberration minima arenot observed when these lenses are optimized by theLASL program, which minimizes all seven aberrationssimultaneously by a least squares system that drives allthe lateral ray deviations toward zero.17 Althoughspherical aberration remains an important image error,the presence of the two minima is submerged orsmoothed over in the LASL procedure by the errors thatthe other six aberrations cause18 and by the errors fromthe many different rays that are used to sample theperformance of all parts of the lens.' 7 Table I illustratesthis procedure by presenting the results of six LASLoptimization runs that start with the two cemented-doublet solutions given in Conrady. The first two runswere made to find the last radius, the back focus, andthe image-spot sizes of the Conrady solutions, whichwere not given. The next two runs showed that the

LASL program could make small improvements in theperformance of the Conrady-minima solutions by smallchanges in the parameters. The last two runs optimizedperformance over a 3° image field and reached what isbelieved to be the optimum-minimum region. The twofull-field prescriptions are not close to either of thestarting prescriptions, but they are very similar to eachother.

Ill. Conclusions

The characteristics of many-parameter lenses opti-mized with the LASL program (small parameter gra-dients all of similar size, small performance improve-ment per iteration, and a multiplicity of designs withsimilar performance) suggest that the optimum-mini-mum region has been found in each case. No localminimum and no unique prescription have been foundin these optimization searches. The reason a uniquelens prescription is not found is that a change in oneparameter can be compensated by changes in the re-maining parameters. Parameter compensation alsoaccounts for the absence of local minima as the errorfunction is minimized.

References1. D. P. Feder, Appl. Opt. 2, 1209 (1963), p. 1210.2. A. B. Meinel, J. Soc. Motion Pict. Telev. Eng. 76, 209 (1967).3. J. B. Kruskal, J. Soc. Motion Pict. Telev. Eng. 76, 210 (1967).4. W. J. Smith, Opt. Spectra 8, 22 (Dec. 1974), p. 24.5. D. C. Sinclair, "Optical Design Using New Computer Technolo-

gy," in Proceedings, Los Alamos Conference on Optics '79, D. H.Liebenberg, Ed. (Society of Photo-Optical InstrumentationEngineers, Bellingham, Wash., 1979), Vol. 190, p. 497.

6. B. Brixner, Appl. Opt. 12, 2703 (1973), pp. 2706-2707, Tables I,II.

7. B. Brixner, Appl. Opt. 13, 2067 (1974), p. 2070, Tables I, II.8. Ref. 6, Figs. 2-4.9. H. Hancock, Theory of Maxima and Minima (Ginn and Co., New

York, 1917; Reprint, Dover, New York, 1960).10. Ref. 6, pp. 2704-2705.11. Ref. 6, Fig. 5.12. Ref. 6, Fig. 4.13. C. A. Lehman, "Treatment of Singularities Which Occur in the

Lens Design Problem," in Proceedings, Conference on LensDesign with Large Computers, W. L. Hyde, Ed. (Institute ofOptics, Rochester, New York, 1967), p. 20-1.

14. T. H. Jamieson, Optimization Techniques in Lens Design(American Elsevier, New York, 1971), pp. 14-17.

15. Ref. 6, p. 2705.16. A. E. Conrady, Applied Optics and Optical Design (Dover, New

York, 1957), Part 1 and (1960), part 2, pp. 213, 226, 782, 823.17. B. Brixner, in Ref. 5, p. 2.18. Ref. 16, p. 217.

It is a pleasure to thank William C. Davis and ThomasC. Doyle for their help in formulating the ideas pre-sented here and Max A. Winkler for making the LASLlens-design runs.

This work was supported by the U.S. Department ofEnergy.

1 February 1981 / Vol. 20, No. 3 / APPLIED OPTICS 387