Faster LASL Lens Design Program

Faster LASL Lens Design Program

Berlyn Brixner

The following improvements have made the LASL lens design system much faster, though less versatile,and have maintained the calculation accuracy: an increment-vector damping technique optimized by asearch procedure, analytic differentiation, simultaneous design on all variables, reliable convergence cri-teria, vignette control by biased violation errors, bounds for the variables, and an enlarged weighting pro-cedure. The designing system optimizes the sizes and positions of many-ray image spots without consid-ering the classical aberrations. As a sample problem the symposium lens has been redesigned.

History

This paper will describe some new improvementsthat have speeded the LASL lens design program,1 2

which during the course of its 14-year history hasbeen variously designated as the Holladay code, theLehman code, the JPL code, and the Brixner code,and which may in the future also come to be knownas the Doyle code. T. C. Doyle's version is the latestin a series that goes back ultimately to 1951 when acolleague made for me an easy-to-use ray-tracingcode3 that employed for the first time the now-essen-tial floating decimal calculation.4 It was while wewere designing lenses with this program on the IBMcard-programmed electronic calculator, makingmanual changes of the lens parameters and at thesame time getting data for drawing the rays inter-secting the focal plane,5 that we discovered that asimple graphic analysis helped reduce the spot sizes,which, in turn, made us envision the possibility ofautomating the process through a least-squares min-imizing technique. Even after large fast memoriesbecame available, however, there was the usual con-servative resistance to change; and it was some timebefore two mathematicians interested in innovativeschemes-J. C. Holladay and C. A. Lehman-devel-oped my proposals into an effective code.

This code, devised by Holladay6 in 1958-59 andfurther developed by Lehman until his death in1966, is the only basic program that designs onmany-ray image-spot size and position data, makingno use of classical aberration theory. Since 1960 thecode has been widely distributed and modified bymany other people to meet their special require-ments (B. J. Howell, R. L. Hughes, K. B. O'Brien,L. F. Schmidt, G. W. Wilkerson, etc.). Until the

The author is with Los Alamos Scientific Laboratory, Universi-ty of California, Los Alamos, New Mexico 87544.

Received 1 March 1973.

Holladay-Lehman FAP-language code was rewrittenin FORTRAN by P. J. Firnett and L. A. Wilson in1967 under JPL-NASA contract,7 it could only berun on IBM.

The LASL design system, which enables both theprofessional optical designer8 and the less-specializedengineer or scientist with a modest knowledge of op-tics9 to produce excellent lens designs in a shortertime than was previously possible,10 has been usedsuccessfully by other designers8 -15 mainly becausethe program's versatile control system makes it easyto emphasize particular lens-performance character-istics.

Designing Procedure

Since Doyle's improvements need to be seen in thecontext of the program as a whole, a brief summaryof the designing procedure follows. This system,which analyzes the many-ray spots statistically inorder to measure lens performance in terms of eighteasy-to-use characteristics that are also require-ments, fills a long-felt need' 6 for a relatively simplemethod of tracing large numbers of specified skewrays quickly, of determining where the rays aregoing, of specifying where they should go, and of get-ting them to go where they should go.

(1) Focus. Each image is formed by a bundle ofrays traced from an object point, through the lens,and on to make a spot on the focal surface of inter-est. The best focus is found by moving the focalsurface to the region of smallest size along the raybundle.

(2) The image position is found by measuringfrom the optical axis to the position of the spot'scentroid on the focal plane of interest.

(3) The image size is found by measuring the rmsradius of the rays in the spot from the centroid.

(4) The distortion is found by measuring thevariation of the centroid position from the positionrequired by rectilinear geometry.

November 1973 / Vol. 12, No. 11 / APPLIED OPTICS 2703

(5) The lateral chromatism is found by measur-ing the variation of the centroid positions over theselected spectral range.

(6) The longitudinal chromatism is found bymeasuring the variation of the best focus over the se-lected spectral range.

(7) The f/number is found by measuring the ray-to-axis angle of each ray in the axial image-formingcone and then scaling it to the entrance pupil diame-ter.

(8) The entrance and exit pupils are found bymeasuring the size and location of the aperturesthrough which the light appears to enter and leavethe lens.

Since the ideal image error of zero for all the aboveperformance characteristics is seldom approximated,it is usually necessary to set up a system of prioritiesto meet the specialized needs of the lens user and tocompromise the quality of some performance charac-teristics in order to enhance that of others. Beforeimage errors are put into the least-squares minimiz-ing system, they are weighted individually in any de-sired combination in such a way as to minimize theimage errors for the important performance charac-teristics. The above list of eight conflicting lens per-formance characteristics is a great help in determin-ing the weight adjustment.

Program ImprovementsThe new improvements that have speeded up the

design program are the result of several years of workby T. C. Doyle, a mathematician who has beeninvestigating nonlinear least-squares optimizing pro-cedures for continuous many-parameter systems andwho regarded his own version of the lens-design pro-gram'7 as a severe test problem for the techniquesthat he was studying. His error function, which in-cludes most of the image characteristics describedabove, adds some effective schemes for controllingboundary conditions and for controlling the loss ofrays within the lens. The present increment-vectordamping technique is the best of several procedureshe tested. However, one compromise of the newprogram is that all surfaces must be axially centered.

Doyle's increment-vector damping technique is themost effective method that we know of for reducingthe size of the sum of the squared image errors,hereafter called the error function. Through thistechnique, which is a refinement of the Levenbergdamping technique as analyzed by Doyle,18 an effi-cient search procedure for finding the optimumdamping number has been worked out. In order tofind which parameter changes are needed to make alens with a smaller error function, a positive (damp-ing) number is first added to each of the diagonalterms in the matrix of the linear system beingsolved. The size of the initial number used is basedon the size of the largest number found in the matrixdiagonal. Next, in order to find a number that givesan even smaller error function than was obtainedwhen the initial number was used, a series of larger

numbers adjacent to the starting number are tried.The parameter set that gives the smallest errorfunction is accepted for the next iteration and thewhole sequence restarted. From the results of theprevious iteration a new search interval is calculated.

The second improvement is Doyle's use of analyticdifferentiation to calculate the rate of change of theobserved image errors with respect to the variableparameters. Derivatives formed by analytic differ-entiation, which take less computing time to evalu-ate than the numerical differentiation used previous-ly, are available for the following parameters: thedistance from the object to the first surface, the dis-tance from the object to the entrance pupil, the radi-us of the entrance pupil, the surface curvature, theconic eccentricity of the surface, and the distance tothe following surface. (When a parameter is bound-ed, the code transforms the bounded parameter to arelated unbounded parameter that it can process.)Analytic differentiation is also available for fourconditions, called violations, that generally cannotbe corrected by using parameter boundaries. Theseviolations terminate the tracing of the ray and gener-ate a violation error that is used in place of theimage error that would be used if the ray reached thefocal surface. The code will detect the following fourtypes of violations: failure of the ray to intersect anoptical surface, total internal reflection occurring ata surface, featheredge occurring between adjacentsurfaces, or a ray intersecting the surface fartherthan a specified distance from the axis. Finally, an-alytic differentiation is available for the exit pupildistance from the focal surface and for the radius ofthe exit pupil.

The third improvement is that Doyle's refined cal-culation of the parameter increment vector makes itpossible to design on all the variable parameters si-multaneously. A simple numerical flag scheme se-lects the parameters to be varied, the dependent pa-rameters, and the upper and/or lower bounds of eachparameter. Much of the program's speed during allstages of design results from the simultaneous opera-tion of all the variables. As many as thirty-three in-dependent parameters have been varied at the sametime with no indication of difficulty.

The fourth improvement is that three reliable anduseful convergence criteria are used to terminate cal-culation. These criteria are comparison numbers forthe error function, the percentage change in the errorfunction per iteration, and the absolute value of thesquared magnitude of the parameter increment vec-tor, which last is very small at convergence or whenoverdamped. By experience the designer learnswhich comparison values to use to avoid wasting ma-chine time. The iteration count and the calculationtime limit also make good stopping places.

The fifth improvement comes from minimizing theloss of rays within the lens, called violations. Theseviolations are caused by total reflection, failure ofthe ray to intersect the surface, an aperture obstruc-tion, or lens-element featheredge, which last alsocauses an obstruction. Whenever a ray is lost, the

2704 APPLIED OPTICS / Vol. 12, No. 11 / November 1973

function that measures the defect generates a nega-tive number, which is added to a negative bias num-ber assigned by the designer before the sum isweighted to form the violation error. Minimizingthe violation error tends to overcorrect the ray defectto a positive value, thus causing the violation to dis-appear. The scheme is very effective.

The sixth improvement comes from enlarging theweighting procedure to include seven classes ofweights that are available for each image point andeach color. These weights are on the X componentsof the image spots, the Y components, the imageheight, the violations, the f/number, the exit pupildistance, and the exit pupil radius. A compromisehas been made by including the lateral-chromatic er-rors among the image-height errors. This compro-mise is justified because most lenses must make arectilinear image that has little lateral chromatism.A final new weighting capability allows each ray ineach of the ten different ray patterns available forperformance analysis with this code to be weightedindividually. The ray patterns range from 2 to 158rays.

The seventh improvement is that a lens-prescrip-tion scale factor is included to control the mobility ofthe axial-distance parameters in relation to the sur-face curvature parameters. Before performanceanalysis and designing are begun, the code multipliesthe input lens prescription by the scale factor. Achange in scale factor changes the parameter mobili-ty because the derivative of the error function rela-tive to the axial distances varies as the first power ofthe scale factor, whereas relative to the surface curv-atures it varies as the third power. It should benoted that the error function varies as the scale fac-tor squared.

The eighth improvement is Doyle's optional incre-ment-vector coefficient search technique for furtherreducing the size of the error function after the opti-mum damping number has been found when thecoefficient is one. This search is usually most help-ful during the first few iterations. The coefficientthat gives the smallest error function generates a pa-rameter set that is accepted for the next iteration,and the whole sequence is started again. For reduc-ing the size of image errors, the coefficient search isnot usually as fast as the damping number search.Although optimum coefficients ranging from 0.1 to10.0 have been observed, most were in the 0.7-1.1 in-terval.

The new program can trace up to 10,000 rays persurface per second on the CDC-7600, a rate that in-cludes all accessory computations. The increasedspeed, which is to be compared with the 1,000-rayrate of the 1962 FAP code, results partly from theprogram improvements described above, which havespeeded convergence during designing, but mainlyfrom machine and language differences. Althoughan exact comparison of design speeds cannot bemade, one way to measure the improvement is tocompare the number of rays traced to achieve a de-sign that has specified performance characteristics;

and the sample problem described below, which is arerun of an earlier problem, gives a measure of theconvergence improvement. Where the new lens de-sign, LASL-3, used about 110,000 rays, the LASL-2design, rerun with hindsight efficiency, used about720,000 rays, larger by a factor of 6 for this sixteen-variable-parameter problem. Work on a thirty-three-parameter problem suggests that for largeproblems the program improvements have yielded atenfold increase in convergence speed.

For very large problems the new program, writtenin FORTRAN IV, requires about 33,000 words in faststorage and up to 155,000 words of slow storage. Forsmall problems the storage reservations can be re-duced. The program is available at nominal costfrom the Argonne Code Center, Argonne NationalLaboratory, 9700 S. Cass Avenue, Argonne, Illinois60439.

Sample Problem

For the sample problem I chose the symposiumlens19 so that the results could be directly comparedwith those obtained from the earlier program.' Thenew program's design, which gives performancepractically indistinguishable from that of the 1963LASL-2 design, 2 0 was discovered in two runs total-ing 22 sec of computation.

During all iterations sixteen variables were usedsimultaneously-eight surface curvatures and eightaxial distances. To make the axial-distance param-eters change during designing, the lens size wasscaled up by a factor of 2.5. During all iterationsthe entrance-pupil size and the first air space wereheld constant. The first run used 6-ray patterns foreach object point and each color; the second used26-ray patterns. Figure 1 shows the starting lensand the new design, which has only a modest thick-ening of the lens elements and no unusual shape fea-tures.

Fig. 1. Starting and improved symposium lens designs.


wa:

I-

M

0

0

enM

I . I . . I . - J _ l l I

0 1 2 3 4 5 6 7 8 9 10 11 1 2 3

6-RAY ITERATION 26-RAY

Fig. 2. Surface curvature

Table l. Prescriptions for the Symposium Lens Designsa

Glass Param. SYM-0 6-ray LASL-3 LASL-2

EP -1.250 -1.286 -1.2868 -2.860Ri 2.174 2.6759 2.7278 28.100

620603 DI 0.300 0.3887 0.3892 0.625R2 o -6.9035 -6.4116 -3.7923D2 0.0025 0.0025 0.0025 0.036R3 1.390 1.7183 1.7167 1.6422

620603 D3 0.300 0.2937 0.2930 0.475R4 5.000 8.1637 8.1349 21.033D4 0.525 0.1415 0.1349 0.142R5 -1.667 -2.6231 -26914 -2.8286

649338 D5 0.125 0.2191 0.2374 0.625R6 1.667 1.3425 1.3176 1.0763D6 0.625 0.4933 0.4909 0.174R7 -5.000 7.2588 7.0110 2.4209

620603 D7 0.300 0.2358 0.2352 0.7353R8 -1.390 -1.4655 -1.4857 -1.3463BF 1.188 1.5534 1.5589 1.3513

a Note that all prescriptions are scaled up by a factor of 2.5.

changes during symposium lens de-signing.

Li0

en

-_J

4

o L1 2

6-RAY ITERATION 26-RAY

Fig. 3. Thickness changes during symposium lens designing.

The changes of the lens parameters during design-ing are shown in the next two illustrations, with eachdesign in the series giving better performance thanthe preceding design. Figure 2 shows the radius ofcurvature of each lens surface at each iteration forboth runs. As is expected, the greatest changesoccur during the first few iterations. Note thatthere is little change during the last two iterations,an indication that a stable design has been achieved.Figure 3 shows the axial distances between adjacentlens surfaces at each iteration for both runs. Againwe see that the greatest changes occur during thefirst few iterations and that there is little change

during the last two iterations. Also shown is themovement of the entrance pupil during designing.The graph shows all the surface positions measuredfrom the focal plane. Table I gives the prescriptionsfor four lens designs.

Figure 4 shows design progress in the first run,during which the error function drops three orders ofmagnitude in a series of twelve iterations. In each ofthe above least-squares iterations the automaticsearch procedure tries a series of damping numbersthat produce an improved lens prescription, which inturn is used as the starting prescription for the nextleast-squares iteration. Although the first 11-sec rungave a very good lens design, the second run gaveslightly better over-all image quality without makingmuch change in the lens shape.

Figure 5 analyzes in detail one damping-numbersearch curve that was selected from another seriesbecause it shows the random occurrence of localminima when small damping numbers are used on alens near convergence. The numbered data pointsare the ones found by the program during one itera-tion. To get the data for this illustration the codewas later directed to evaluate the error function for alarge number of additional damping numbers. Ascan be seen, at least four local minima exist in theregion explored. We usually find that if the intervalis explored in still finer detail, additional minimawill be discovered.

Table II gives the performance characteristics ofthe initial design (SYM-0) and of the 6-ray and 26-ray (LASL-3) designs described here, as well as thoseof LASL-2, all as evaluated by the new program. Itcan be seen that although the LASL-2 and LASL-3designs are quite different in appearance, they havesimilar performance characteristics. By additionalweight adjustment closer performance agreementcould probably be obtained.


so _5 START SYM-O

H ~~6-RAY0 I TE RAT

-6

3

-8 EXIT 12I l l I

-6 -5 -4 -3 -2LOG DAMPING NUMBER

Fig. 4. Design progress during first symposium lens run.

-2

-3 -

0

o 211-4_

0

-6 -3

LO-7 , TRIAL 56 789

-5 -4 -3 -2 -ILOG DAMPING NUMBER

Fig. 5. Local minima on damping-number search curve.

Discussion

We have found that the general appearance of anoptimized lens design will vary considerably with thefreedom of movement allowed for the axial-distanceparameters, a variation well illustrated by theLASL-2 and LASL-3 lenses, which have very similarperformance characteristics. We have also foundthat when the different parameters of a lens systemare varied simultaneously, their effectiveness in re-ducing the size of the error function differs greatly.Generally, the surface-curvature parameters aremost effective, while the axial distances are least ef-fective. However, designs with long axial distancesgenerally show the best performance characteristicsif there is no need to consider such other factors asconvenience or practicability or difficulty of con-struction.

By enlarging the lens being designed from a 1.0-unit to a 2.5-unit size the changes in the axial dis-tance parameters were increased from insignificantto significant amounts, which changes led to theLASL-3 design. For LASL-2 this size control wasnot used, and it was not possible to test all parame-ters operating simultaneously. It may be mentionedthat for the other symposium lens designs the sizecontrol apparently was not used.

In an attempt to find out if there is somethingunique about the LASL-2 lens design, I restarted de-signing with eight plane surfaces, using the LASL-2axial distances, and then designed on the surfacecurvatures only. In twenty-eight iterations the pro-gram found curvatures approximating the LASL-2curvatures, with an average difference of about 3%,while the error function of this design turned out tobe trivially smaller than the LASL-2 error function(as evaluated by the new code).

Tests of the new program with more complex lens-design problems indicate that its gain in speed overthe early program will rise with the increase in thenumber of parameters varied. Because no rays werelost within the lens and no boundaries were passed,these program controls were not tested during thedesigning of LASL-3.

Table II. Characteristics of the Symposium Lens Designsa

Performance feature SYM-0 6-ray LASL-3 LASL-2

Ray pattern 6 6 26 26f/number 1.96 1.41 1.41 1.42Error function X 10-9 8140 7.9 6.4 4.7Spot radius, avg rms (0.0625 0.0234 0.00076 0. 00076' 0.00071

for image hgt 0.1625 0.0259 0.00080 0.00063 0.000640.2625 0.0314 0.00135 0.00114 0.00086

Lateral chromatism, (0.0625 0.0010 0.00002 0.00003 0.00009forhgt 0.1625 0.0028 0.00005 0.00008 0.00022

0.2625 0.0048 0.00007 0.00011 0.00032Distortion error for 0.0625 +0.0200 -0.00005 -0.00012 -0.00018

image hgt 0.1625 +0.0536 -0.00006 -0.00025 -0.000300.2625 +0.0919 +0.00012 +0.00016 +0.00008

a Note that all characteristics are scaled up by a factor of 2.5.


Because of the program's great speed, more newdesign territory can be explored when searching forbetter lenses, and at LASL we find that this capabil-ity has often resulted in greatly improved lenses. Afew apparently new and certainly excellent designshave been discovered and their performance hasbeen confirmed by construction. For example, a142-mm, f/4.0 copy lens designed for 20:1 gives ahigh contrast white-light visual resolution of 400lines/mm throughout a 30-mm-diameter flat imagefield.

Conclusion

The increased speed of the LASL lens design pro-gram, which maintains the calculation accuracy ofthe earlier program while being easier to use, wasdemonstrated by redesigning the symposium lens.We have also demonstrated that lenses of differentappearances may still have similar performancecharacteristics. The net benefit of the new fast pro-gram is that one can now explore a larger design ter-ritory than was possible formerly while at the sametime producing excellent lenses, usually at less com-puter cost, despite generally rising costs.

In addition to Thomas C. Doyle, who has so great-ly speeded the LASL lens design program by refiningthe optimization procedure, thanks go to WilliamAnderson, whose code for another problem demon-strated the possibilities of the damping technique,and to Morris M. Klein, who devised the un-bounded-parameter boundary control scheme.

This work was performed under the auspices of theU.S. Atomic Energy Commission.

References1. B. Brixner, Appl. Opt. 2, 1281 (1963).2. B. Brixner, C. A. Lehman, and J. C. Holladay, in Proceedings

of Panel Discussion on Some Automatic Methods of Lens De-sign (Washington, D.C., 1966), J. S. Courtney-Pratt, Ed.(SMPTE, New York, 1967), pp. 186-191, 198-200.

3. W. A. Allen and R. H. Stark, J. Opt. Soc. Am. 41, 636 (1951).4. D. B. MacMillan and R. H. Stark, IBM Tech. Newsletter 2, 16

(1951).5. I was greatly assisted in this work by Albert J. Lipinski (died

1953), a Rochester Institute of Optics graduate.6. J. C. Holladay, in Computer Applications-1960, B. Mittman

and A. Ungar, Eds. (Macmillan, New York, 1961), pp. 112-127.

7. P. J. Firnett and L. A. Wilson, Fortran Optical Lens DesignProgram (Informatics, Inc., Los Angeles, 1967).

8. B. J. Howell, in Proceedings of the Conference on Lens Designwith Large Computers (Rochester, N.Y., 1966), W. L. Hyde,Ed. (Institute of Optics, University of Rochester, 1967), pp.21-1 to 21-17.

9. M. A. Winkler, Appl. Opt. 5, 1019 (1966).10. A. B. Meinel, in Ref. 2, pp. 201-203, Figs. 1 and 7.11. K. B. O'Brien, J. Opt. Soc. Am. 54, 1252 (1964).12. D. H. Schulte, Appl. Opt. 5, 313 (1966).13. G. W. Wilkerson, Design History and Specifications for the

Mariner 1969 High Resolution Telescope, Tech. Rpt. 51 (Op-tical Sciences Center, University of Arizona, Tucson, 1970).

14. U. Ludwig, J. Opt. Soc. Am. 62, 1393 (1972).15. R. B. Gallipeau, Opt. Spectra 6 (2), 29 (1972).16. 0. N. Stavroudis, The Optics of Rays, Wavefronts, and Caus-

tics (Academic Press, New York, 1972), pp. 81, 161.17. T. C. Doyle, "Automatic Lens Design by Nonlinear Least

Squares Optimization of a Continuous N-Parameter Sys-tem," Preprint LA-UR-73-518 (LASL, Los Alamos, NewMexico, 1973).

18. T. C. Doyle, "Nonlinear Least Squares Optimization of aContinuous N-Parameter System," Preprint LA-DC-72-1018(LASL, Los Alamos, New Mexico, 1972).

19. D. P. Feder, Appl. Opt. 2, 272 (1963).20. B. Brixner, Appl. Opt. 2, 1331 (1963).

Itek people photographed by F. Corbett: J. Wyant (left), Berge Tatian, and Kent Bowker.


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Faster LASL Lens Design Program