Automatic Lens Design for Nonexperts

Berlyn Brixner

Automatic lens design by nonexperts is practical with the 1962 LASL lens-designing code, a general-purpose iterative program for evaluating and designing optical lens and mirror systems with surfacesgenerated by conic sections. Using skew ray traces, it analyzes lens performance statistically, findsdifferences resulting from design-parameter alterations, and computes a linear combination which willimprove performance. Program characteristics, lens parameters, ray sampling, image evaluation, lens--performance weights, merit calculation, and parameter interactions are briefly described. Sample,calculations are given in detail for a Lister-type lens and in summary for a special-purpose zoom lens.

The purpose of this paper is to describe a lens-designing program for the IBM 7090 which is in use atthe Los Alamos Scientific Laboratory and to callattention to the fact that instrument designers withspecial optical problems can now learn to design theirown lenses. The paper is not aimed at the opticalspecialist.

For the optical specialist it may be well to mentionthat this computer program does not follow classicaldesigning methods. It does not make use of Gaussianand third-order optics or diffraction theory. It doestwo main things: it evaluates lens performance andimproves lens prescriptions. The method is so power-ful that the starting prescription need be no more thansheets of glass with plane surfaces. The possibilty ofmaking lens designs practically from scratch in thisway has apparently not been recognized by previousworkers1 - who have used the same least-squarestechnique to minimize spot sizes. These workersbased their solutions on the assumption that it is neces-sary to have a partially corrected prescription beforestarting designing. But actually the cruder the start-ing prescription the more rapid the rate at which theprogram converges on a solution.

To evaluate performance, the program analyzesimage defects precisely by calculating the sizes andpositions of image spots produced by bundles of skewrays traced through the lens. The size of the raybundle at the focal plane of interest is the main criterionof lens performance. The image spot is evaluated

The author is with the University of California, Los AlamosScientific Laboratory, Los Alamos, New Mexico.

Received 15 July 1963.This work was done under the auspices of the U.S. Atomic

Energy Commission.

from the rms radius of the rays from the centroid ofthe ray-bundle intersection on a plane normal to theoptical axis. To improve the lens, the programobtains a sharper focus by changing the parameters ofthe lens system to make the bundles of rays intersectthe image surface at the desired positions in patterns ofthe smallest size. The program performs this operationby first determining differences in lens performanceresulting from design parameter alterations. Then itapplies a least-squares technique to compute a linearcombination* which will give an improved lens pre-scription. At the conclusion of each design cycle amerit number is obtained. The smaller this number,the better the performance. A perfect lens would havea merit number of zero. To obtain this number theaverage deviation of the rays from each image point isfirst obtained. Next, each average deviation ismultiplied by the assigned weight. The merit numberis the sum of these products.

In addition to evaluating performance and improvingthe prescription, the program has an automatic merid-ional rim-ray trace, which gives the ray-intersectioncoordinates at each optical surface, and an automaticscaling routine, which gives the scaled ray-intersectioncoordinates at any focal plane of interest. Thesecoordinates are useful for preparing lens drawings andspot diagrams.

The program can investigate lens systems with asmany as 98 surfaces. Although the most used lens ormirror surfaces are either spheres or right-circularcylinders, surfaces generated from any conic sectionmay be used. Any of the surfaces of revolution maybe displaced or tilted as, desired. There is a wide

* This is a linearization procedure similar to that outlined byD. P. Feder, J. Opt. Soc. Am. 52, 180 (1962).

December 1963 / Vol.. 2, No. 12 / APPLIED OPTICS 1281

choice possible in the number and distribution of therays which may be traced through the lens. Thepositions and sizes of the entrance and exit pupils canbe readily controlled. The lens focus may be fittedto a specified focal surface, or the shape of the best focalsurface may be determined. Simultaneous design forseven image points, six wavelengths, and five con-ditions of use is possible. An example of differentconditions of use would be the positions of a zoom lens.

The rest of this paper will briefly give some charac-teristics of the program, will outline the method forusing it, and, in more detail, will give examples of twolenses designed by means of the program.4S5*

Selecting the Design

The instrument designer may find that one of themany standard lenses will be adequate for his needs.With the program he can readily evaluate a likely.prescription. Or, if the standard lens is not entirelyadequate, the machine can indicate minor alterationsof the surface radii and in the spacing of the elementswhich will give the best performance in his situation.If still better performance is required, he can let themachine determine the lens design most suitable forhis problem. One starts the design by making mono-chromatic runs on a series of air-spaced glass sheets.After making runs on a series of lenses of increasingcomplexity, he can then choose the lens which best suitshis needs. The monochromatic design is achromatizedby substituting crown and flint glasses in the prescrip-tion and readjusting by means of additional runs on themachine.

Preparing the Problem for the Machine

To evaluate an existing lens one needs, in additionto the prescription, the following information: thedistance from the object, the angle of view, the entrancepupil location, the relative aperture (f/number), theimage distance, and the shape of the focal surface(plane or curved). In the absence of specific con-jugates, the object distance is assumed to be infinity.To find the angle of view, one traces a series of raybundles from object points at increasing angles. Theentrance pupil distance may be specified or it may bedetermined by the designing program. If the imagedistance is specified, the ray-tracing program can setthe plane of interest at that distance. If the imagefield is curved, the additional values of the sagittafor the image points are also used. If the imagedistance is unknown, the ray-tracing program will setthe plane of interest at the back focus calculated from

* These reports are available at nominal cost from the Officeof Technical Services, Department of Commerce, Washington

25, D.C. They are also available for reference use by the general

public in 85 United States libraries and in 84 foreign libraries.

Table I. Summary of Lister Lens Designing Sequence andProgress

Increments Compu-used opu

Ray tationOperation pat- To- Per Cycles Figure timeperformed tern tal cycle run of merit (see)

Start 2 - - - 904.7 73Designing 2 4 1 29 24.9 18

Designing 2 6 1 25 18.0 17

Designing 2 6 2 25 0.26 22

Designing 2 8 2 25 0.23 22f/2.0 2 - - - 4.9 <1Designing 2 9 3 28 0.30 35

Designing 2 9 4 28 0.22 45Ray change 6 - - - 4.0 <1Designing 6 9 4 28 0.31 106

Designing 6 9 5 28 0.27 123

Designing 6 9 6 28 0.26 148Ray change 12 - - - 0.84 <1Designing 12 9 6 28 0.41 281

the automatic meridional ray trace with the firstrefractive index selected (plus the sagitta plus a fine-adjustment increment if these are assigned nonzerovalues).

For evaluating performance, the choice of inputrays is very important. Since the rays traced throughthe lens represent the behavior of light, they should beselected to function efficiently as agents for lens per-formance evaluation. Each ray may be consideredas a sample of the focusing ability of the lens region

MERIT 904 .7

24.9 ___ __ _

2.26 9- s0.26 -- - -----

Fig. 1. Five lenses in the Lister lens-design series.

1282 APPLIED OPTICS / Vol. 2, No. 12 / December 1963

Table II. Prescriptions for the Lister Lens Designs

Figure of merit

Glass 904.7 24.9 0.26 0.31 0.41

type t r t r t r t r t r

170.0 co 170.0 15.14 170.0 14.42 170.0 19.76 170.0 20.555518 596 2.7 2.7 2.7 2.7 2.7 _

621362 1.5 1 5 1 4.20 1.5 -13.63 -14.56818.0 o 18.0 1513 18.0 219.4 17.32 -94.52 19.299 -154.06

518596 1.8 1.8 43.62 1.8 16.85 1 8 13.94 1.8 12.183621362 1.0 1.0 -4 1.0 -6.310 1.0 -7.072 1.0 -8.0225

10.0 10.0 -144.2 10.0 -15.75 11.12 -22.50 11.525 -27.126Entrance

pupil -20.0 -20.0 -20.0 -36.0 -18.8

which surrounds the ray path. The larger the sur-rounding region, the less typical this sampling will be.The pattern of rays to be traced is placed in a unit-radius entrance pupil. The ray-tracing programscales the unit-pupil dimensions to the actual pupildimensions to obtain the coordinates of each ray to betraced. The rays usually sample the entrance pupilwith a grid-pattern array, but as many as 25 rays maybe positioned in any pattern desired. In this program,up to 100 rays can be traced from each object point foreach color.

When designing lenses, the designer can weight thedeviations of the rays from the desired image points toemphasize specific aspects of lens performance. Themore important the aspect, the higher the weightassigned. Optimum lens performance requires min-imum image-spot sizes on the focal surface. Theseimage spots are made up of a maximum of six colors,each from a maximum of seven object points. Theweights for the sizes of these spots are assigned in-dividually. Weights on the deviations caused by thevariation of magnification with color for each of theseseven images are also assigned individually. Theseven desired image heights, the individual weights foreach of these image heights, and the sagitta (for acurved field) are assigned by the programmer. Therequired principal focal length and the weight on thereciprocal focal length may be specified. The exit-pupil position is calculated from a ray traced from anobject point at a height selected by the programmer.The desired distance of the exit pupil from the lastoptical surface and the weight on this distance are alsospecified by the programmer. The above weightsare the total number which control the design program.Another factor of importance to lens performance isthe entrance pupil aperture radius or the operatingrelative aperture of the lens, one of which may befixed for designing.

Choosing the Design Program

The lens characteristics are specified by parameters,i.e., numbers which correspond to the physical charac-teristics of the lens. The designing program cancalculate a better number for a parameter by evaluatingthe effect of small changes of that parameter on theperformance of the lens. If two parameters arechanged and the effects computed simultaneously,there is an interaction which produces a much greaterimprovement of lens performance than that obtainedwith each taken alone. Up to a certain point, themore effects of parameter changes computed simul-taneously, the faster the improvement of lens perform-ance. However, if too many parameters are changedat once, the progress becomes erratic because of theprogram's inability to evaluate precisely the too com-plex problem. This program can work with a maxi-mum of ten simultaneous parameter changes.* Withten simultaneous changes, forty-five interacting pairs arepresent. To arrive at the minimum merit numberobtainable with a specific set of variable parameters,it is necessary to work with all possible pairs as often aspossible. An iterative sequencing procedure accom-plishes this result.

Boundary Conditions

Any variable parameter which shows a tendency toexceed its boundaries is manually fixed at the boundary.Since most problems have only a few boundary viola-tions and since the code is mainly used for short machineruns, this manual operation is not cumbersome.

* The selection of the ten-simultaneous-parameters maximumin this code was an arbitrary decision based on our experiencethat a larger number would not produce useful solutions at thepresent stage of the art. This conclusion is similar to that ofJ. Meiron.3

December 1963 / Vol. 2, No. 12 / APPLIED OPTICS 1283

Table ll. Weights for Images of Lister Lens Designs

Performance Image pointfeature

to be weighted First Second Third Other

Focal length - - - 0Exit pupil distance - - - 0rms spot size 100 100 100Lateral chromatism 0 0 0Image height 0 0 100(Image height desired 0.12 0.36 0.60)(Object height given 1.0 3.0 5.0)

Table IV. Analysis of Lister Lens Performance by Twelve-RayTraces

rms image-spot sizes

Image Light Height of Tan-point color centroid Sagittal gential Radius

First C 0.1202 0.0011 0.0011 0.0015First d 0.1201 0.0007 0.0007 0.0010First F 0.1199 0.0009 0.0009 0.0013Second C 0.3606 0.0009 0.0010 0.0014Second d 0.3603 0.0009 0.0007 0.0011Second F 0.3595 0.0010 0.0010 0.0014Third C 0.6007 0.0008 0.0009 0.0013Third d 0.6002 0.0013 0.0007 0.0015Third F 0.5989 0.0012 0.0013 0.0018

0o o0~~~~~

o T 00 0 D* , 0

0 0

E3 00 0 0

o o

0

0

0

*

0

o 0

o *

h = 5

0 0 *0

000 0 % e

El M0

aa

+ *

00 0

* 0

0

0

0 0 C LIGHT

, 126 0 d LIGHT

o 0 F LIGHTEl

0

Sample Calculation

The lens to be designed in the following samplecalculation consists of two cemented doublets in serieswhich produce an f/2 Lister-type lens6 set to work at amagnification of about one eighth. The objects are atan axial distance of 170 nun from the first lens surfaceand 1, 3, and 5 mm from the optical axis. The imagesare on a flat field which is started with the back focaldistance held at 10 mm. After the lens has beenstabilized, this distance is made a design parameter.The design was started with two cemented sheetsof 518 596 and 621 362 glass spaced 18 mm apart.The entrance pupil distance was fixed at -20 mm andthe aperture radius at 4 mm. Designing was startedwith three colors (C, d, F), a two-ray pattern, and aseries of twenty-nine single increments to the four glass-air surfaces. As a result, the prescription's figure ofmerit went from 905 to 25. A summary of the com-plete 15-min series of runs used to design this lens isshown in Table I. Figure 1 illustrates five of theprescriptions in the series to show the changes beingmade during the progress of the design. The prescrip-tions for five designs are given in Table II. Theweights used for all stages of the design are shown inTable III.

These runs produced an f/2.0 lens of 20.884-mm

h = 3

'00 0

0 0

on 0

*0 0 % 0 0

* 0%#.0 *

0 0 00 000 0 0

43 II

00 0

0 0

0

0

0

0

I 0.001 mm h = 1

Fig. 2. Spot diagrams of three images made by the Lister lens.

(C-light) focal length, which should give 700-lines/mmresolution over most of the 1.2-mm diam focal surfaceat 11.525-mm distance. There is a maximum of0.0018-mm variation of lateral magnification withcolor at the edge of the 1.70 half-field of view. TableIV shows the details of the twelve-ray-trace analysisof lens performance. The spot diagrams of the threeimages are shown in Fig. 2.

1284 APPLIED OPTICS / Vol. 2, No. 12 / December 1963

Table V. Prescription for the Zoom Lens

Thickness or space for zoom position Radius of

Glassa type One Two Three Four Five curvature

9000 9000 9000 9000 9000504.64

517 645 10 10 10 10 10295.86

617 366 15 15 15 15 15350.59

10 110 210 310 410350.75

517 645 20 20 20 20 20-462.72

617 366 10 10 10 10 105912.7

500 400 300 200 10045343.0

517645 10 10 10 10 10118.46

617366 15 15 15 15 15218.67

10 110 210 310 410581.80

517 645 20 20 20 20 20-224.34

617366 10 10 10 10 10-678.84

1000 900 800 700 600449.38

517 645 20 20 20 20 20-221.69

617366 10 10 10 10 10-649.20

376.532 376.532 376.532 376.532 376.532

Entrance pupil -800.0 -800.0 -800.0 -800.0 -800.0

a The indexes used are 1.000206, 1.000206, 1.000208, 1.51521, 1.51966, 1.52769, 1.61176, 1.62069, and 1.63839.

A Complex Design

An example of a complex design which can be madewith this program is a special-purpose zoom lens.This lens is an objective for a high-speed camera usedto photograph explosions through the 10.2-cm-diamviewing port of a concrete bunker. 20-mm images of150- to 450-mm explosive objects are required, withthe objects 9000 mm from the front lens element.To obtain this range of magnification, a zoom ratio ofapproximately 3.6:1 is needed. Between the viewingport and the objective's image, there is a distance ofmore than 2000 mm, which can be used for any desiredarrangement of lens elements.

In the search for a solution to this difficult imagingproblem, it was decided to let the lens-designing pro-gram find the optimum design for lens systems withthree or four or more elements so that we could selectthe simplest design which would give the requiredresolution. Using the program's substitution capa-bility, we let the machine design each lens systemsimultaneously for five equally spaced positions ofthe moving elements. Several monochromatic de-

signs were made with zoom arrangements whichprevious workers had found to be useful.7 The so-called four-lens system, which has five elements, wasfound to be the minimum which would give the desiredimage quality. The entire available length was usedbecause a long lens nearly always gives the best per-formance. The design was then restarted with fivecemented sheets of 517 645 and 617 366 glass (C, e, andg melt indexes adjusted for 2135-m elevation) in azoom system consisting of two coupled sliding lenseswhich move between three fixed lenses. These pairedcomponents quickly assumed shapes to give negative,

Table VI. Weights for Images of the Zoom Lens

Image pointPerformance feature

to be weighted First Second Third Other

Focal length - - - 0Exit pupil distance - - - 0rms spot size 100 100 100Lateral chromatism 0 50 100Image height 0 0 100(Image height desired 0 5 10)

December 1963 / Vol. 2, No. 12 / APPLIED OPTICS 1285

Table VII. Analysis of Zoom Lens Performance by Six-Ray Traces

Zoom position

Characteristic feature One Two Three Four Five

Element shift 0.0 100.0 200.0 300.0 400.0Object size, maximum 150.000 197.128 259.716 342.248 450.000Magnification 0.133 0.101 0.077 0.058 0.044Relative aperture 12.0 10.0 8.0 7.0 6.5Focal length, C 1592.04 1097.30 790.76 583.51 436.86Focal length, e 1595.01 1098.04 790.70 583.15 436.40Focal length, g 1592.16 1097.13 790.55 583.32 436.72Mean spot size, image 1 0.013 0.014 0.015 0.014 0.017Mean spot size, image 2 0.014 0.015 0.015 0.014 0.018Mean spot size, image 3 0.017 0.017 0.017 0.015 0.018Lateral chrom., image 2 0.009 0.006 0.005 0.005 0.006Lateral chrom., image 3 0.017 0.012 0.010 0.010 0.011

positive, negative, positive, and positive powers.Before the final design was achieved, we made severalreadjustments of the movements and spacings of theelements. Ten parameters were incremented si-multaneously during the final stages of the design.The parameters consisted of all the radii of curvature,the sizes of the three intermediate objects, and theback focal distance.

The prescription data for five zoom positions areshown in Table V. The weights used for designing areshown in Table VI. Table VII shows the details of thesix-ray trace analysis of lens performance for fivepositions. For each of these five positions the spotsizes and focal positions for C, d, e, F, g, and h lightwere determined. There was no discernible shift ofthe optimum focal plane for any of the positions tested,possibly because the variation of focus with color is theprincipal defect of this lens. Except for h light, themaximum spot sizes on the focal plane at the 376.532-mm back focus did not go above 0.027 mm, and the

average is aboutresolution of aboutof the zoom range.

0.013 mm. A visual white-light100 lines/mm is predicted for most

Thanks go to John C. Holladay for developingour automatic lens-designing program and instructingme in its use, to Charles A. Lehman for developing thisfaster enlarged program, and to William C. Davis forhelpful criticism of this paper.

References1. F. Wachendorf, J. Opt. Soc. Am. 43, 325 (1953).2. S. Rosen and C. Eldert, J. Opt. Soc. Am. 44, 250 (1954).3. J. Meiron, J. Opt. Soc. Am. 49, 293 (1959).4. C. A. Lehman, Los Alamos Scientific Laboratory Report

LA-n2837 (1963).5. B. Brixner, Los Alamos Scientific Laboratory Report

LA-2877 (1963).6. A. E. Conrady, Applied Optics and Optical Design, R. Kings-

lake, ed. (Dover, New York, 1960), Part 2, pp. 662-678.7. R. Kingslake, J. Soc. Motion Picture Televison Engrs. 69,

1 (1960).

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1286 APPLIED OPTICS / Vol. 2, No. 12 / December 1963