MODERN
`LENS DESIGN
`
`A Resource Manual
`
`

`

`
`
`Modern
`
`Lens Deslgn
`
`A Resource Manual
`
`Warren J. Smith
`mm Salaam!
`
`Kaiser enema“, Inc.
`Carisbld. COME
`
`Genesee Optics Software, Inc.
`Ramadan New York
`
`McGraw-Hll. Inc.
`I'm-Yon OLLnqu unfit-cm Melina Bow
`canon when Location W m “In
`lam-I Hun-m Pun 8mm SloPlulo
`8W M 1” TM
`
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`

`Library of Congrats Catalog'ne-in-Pubfleadon om
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`
`
`Contents
`
`Preieoe IX
`
`Chapter 1. Introduction
`
`Chapter 2. Automatic Lene Deeign: Menegng the Lens Design Program
`
`2.1
`
`The Merit Function
`
`2.2 Optimization
`2.3
`Looel Minim-
`
`2.‘
`
`2.5
`
`types oi Merit Fmoiione
`
`smooc-
`
`2.6 Generalized Simulated Annealing
`
`2.7 Considerations thou Veflbiee tor minimum
`
`2.8 Howto Increase the Speeder Field oie System and Avoid Rey Failure
`Probim
`
`2.9 TMMOHMMflHm-ndmmnfih
`
`2.10 Spectral Weighting
`2.11 Howroeetsumod
`
`Chapter 3. improving a Design
`
`3.1
`
`Standout irnprovementjechnlqree
`
`3.2 Gieee Chance (Index end VVelue)
`3.3
`Splitting Elements
`
`3.4
`
`Separating e Contented Double!
`
`3.5 Conpoundng en Boment
`I
`3.6 Vignetthg and it lieu
`
`3.7
`
`Eliminating a Week Element; the Com-tie Problom
`
`u Balancing Aberration:
`
`3.9 no Symmetric» Principle
`
`3.10 Aepherio Surieoee
`
`1
`
`to
`
`to
`
`u
`
`II
`
`1.
`to
`
`h}OI
`
`8888888333
`
`<
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`VI
`
`Conunta
`
`Chaptar 4. Evaluation: How Good la Thla Daalgn?
`
`flnUaaaolaPtaflmlnaryEvakIann
`4.1
`4.2 0?!)vade
`
`4.3 Blur Spot Blaa varaua Canal" Ahanatlona
`4.4 MTF—Tha Modulation Tranam Function
`
`Chaptar 5. Lana Doalgn Data
`
`5.1 Ammo-mom
`
`0.2 mmmmmmmm
`
`0.3 WMMMaW
`
`0.4 SoulngaoaalmltaAbanauonaJndltam
`
`0.5 Notaaontha Wlonoiflaylmmp' Plota
`
`Guam 0. Tabaoopa Oblocuvaa
`
`6.1 Tho Thin Doublat
`
`0.2 Bacondary Sputum (Apochmatlo SW)
`
`0.3 BpnarochromaUam
`
`0.4 Zonal WodAbamfion
`0.0 lnduead Ahamuona
`
`0.0 Time-[Imam Oblactlvaa
`
`Chantal”). Evaptaoaa and locum
`
`7.1 Byaplaoaa
`
`7.2 Mflagnlflaroaalgna
`
`7.3 Slmphmo-mTwao-Emfiyaplma
`
`7.4 Four-Elamant lyaplaoaa
`
`7.0 Flva-Elamant 6mm
`
`1.0 81x- and Swan-Elana!!! Eyaplacaa
`
`CW 8. Cooka Tl'lplOt Anaaflgmau
`
`0.1 Alrapaoad Tflplat Anaatlgnlala
`0.2 om. Chem
`
`0.: Vamx Lang!!! and Raddual Abandon
`
`0.4 011m Daalgn Communion
`
`Chapur O. Havana Talaphoto (Hardocua and Flah-Eya) Lanaaa
`
`9.1 Tha Havana Talapm Pundpla
`0.2 Tho Baalc l-‘lalrofocua Lana
`
`93 ThaflaWwEmmWa-Anglaflmmhlaphommua
`
`£3888
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`§‘
`
`123
`
`m
`120
`
`119
`
`127
`
`147
`
`141
`14a
`
`100
`
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`Chapter 10. Telephoto Laneee
`
`10.1 The Beale Telephoto
`
`10.2 Cloee-up or Macro Laneee
`
`10.3 Sample Telephoto Deeigne
`
`Chapter 11. Double-Manleoue Anaetlgmate
`
`11.1 Meniecue Components
`
`11.2 Hyman. Topogon. end Metrogon
`
`11.3 Protar.Degor.andConvertlhleLeneee
`
`11.4 The Split Decor
`
`11.8 The Dagmar
`
`Chepter 12. The Teaear, Heller. and Other Compound“! Triplets
`
`12.1 The Claeelc Teeear
`
`12.2 The Heller/Pm
`
`12.3 Other Compouwed Triplet:
`
`Chapter 13. The Peuval Lone; Heed-up Dlepiay Leneee
`
`13.1 The Peuval Portrait Lens
`
`13.2 The Petzvel Prolection Lane
`13.3 The Peizval with a Field Fiettener
`
`13.4 Very High Speed Petzval Leneee
`
`13.8 Heed-up Dlepley (HUD) Leneee; Bloomer Leneee
`
`Chapter 14. Split Triplets
`
`Chapter 18. Mlcroecope Objectives
`
`18.1 General Coneideretlone
`
`18.2 Ciaeeloel Obleotive Design Forrne: the Apienetlo Mont
`
`15.3 Flat-Field Objective.
`
`18.8 Reflecting Objectives
`
`18.5 The Sample Leneee
`
`Chapter 18. ulnor and Catadioptrlo Syeteme
`
`18.1 maoodlndthoflldPohbdllmn
`
`16.2 The CIOWCCIMMW Syetente
`
`18.3 CatedioptrloSyeteme
`18.8 Conlooai Panboiolde
`18.8 UnobecmdSyeteme
`
`Chapter 17. The Blotar or Double-Game Lene
`
`11.1 The Baelo Six-Element Verelon
`
`17.2 The Moment NMpIh-Rm 8|nglet
`17.3 The Seven-Element Blow-Broken Contact Front Doublet
`
`189
`
`tee
`
`m
`
`170
`
`183
`
`183
`
`183
`
`185
`
`100
`
`1eo
`
`197
`
`101
`
`210
`
`210
`
`335§§§§§§8335933§5§8§13.
`
`DA .
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`523331H3355$
`
`411
`
`111
`«1
`
`412
`
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`
`1
`
`451
`
`45:
`
`VIII mm
`
`17.4
`
`17.5
`17.0
`
`Tho Swat-Elana: Blow-On. Compound Out-rm
`
`The Iain-llama! Blot-r
`uncommon. 3101'.
`
`Chapter 13. “Mo-Anglo Lama with mom Outer 51mm
`
`Chapter“. memmdmmm
`
`12.1 WNW
`12.2 mm
`
`Chum 2o. Zoom Loam
`
`Chapter 21. Infrared Systems
`
`21.1
`
`21.2
`
`21.3
`
`lnfnrnd Optics
`
`IR Objective Lama
`
`IR Toineopn
`
`Chapter 22. Summit-o and Lucr Disk/callnntor lama
`
`22.1 Hammond” SM
`22.2 Summons
`
`22.3 mmmmwm
`
`mamas. Tolerance autism
`
`20.1 mum-mow
`28.2 momma-mu
`
`23.2 Whiningmo‘l’mw
`
`Formulary
`
`F.1
`F1
`
`F3
`
`Sign conventions. mm and Mn!!!”
`The Cardinal Palm:
`
`Imago Squaw
`
`FA PM Ray Tnclno (Sutton by W)
`F:
`Invariant:
`
`EC Pam Rly Tracing (CW by Coma-nut)
`
`E1 MOWWin
`an
`Third-OranADMWon.
`
`P3
`
`Third-Order Aberration—111111 Lam cm
`
`no 8109 Still! lat-Horn
`
`F.11 Third~0rdor Abandon—Com from Alpha-Iom
`
`F.12 cmwmuwwmnmonm
`Opacalmmflm)
`
`Appondlx. Lona Lllflnga
`
`Index 465
`
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`
`
`Preface
`
`This book had its inception in the early 1980s, when Bob Fischer ami
`L as coeditors of the then Macmillan, now McGraw—Hill. Series on 0p-
`tical and Electra-Optical Engineering, were planning the sort of books
`we wanted in the series. The concept was outlined initially in 1982,
`and an extensive proposal was submitted to, and accepted by,
`Macmillan in 1986. At this point my proposed collaborators elected to
`pursue other interests, and the project was put on the shelf until it
`was revived by the present set of authors.
`My coauthor is Genesee Optics Software, Inc. Obviously the book is
`the product of the work of real people, i.e.. myself and the stafi' of
`Genesee. In alphabetical order, the Genesee personnel who have been
`involved are Charles Dubois, Henry Gintner, Robert Maclntyre,
`David Pixley, Lynn VanOrden, and Scott Weller. They have been re-
`sponsible for the computerized lens data tables, lens drawings, and ab
`erration plots which illustrate each lens design.
`Many of the lens designs included in this book are from OPTICS
`TOOLBOX° (a software product of Genesee Optics Software), which
`was originally authored by Robert E. Hopkins and Scott W. Weller.
`OPTICS TOOLBOX'is a collection of lens designs and design commen-
`taxy within an expert-system, artificial-intelligence, relafional data
`
`This author’s optical design experience has spanned almost five de-
`
`cades. In that period lens design has undergone many radiml changes.
`It has progressed from what was a semi-intuitive art practiced by a
`very small number of extremely patient and dedicated lovers of detail
`and precision. These designers used a very limited amount of labori-
`ous cbmputation, combined with great understanding of lens design
`principles and dogged perseverance to produce what are now the clas-
`sic lens design forms. Most of these design forms are still the best, and
`as such are the basis of many modern optical systems. However, the
`manner in which lenses are designed today is almost completely dif-
`ferent in both technique and philosophy. This change is, of course, the
`
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`1:
`
`Preface
`
`result of the vastly increased computational speed now available to
`the lens dedgner.
`In essence, much modern lens design consists of the selection of a
`starting lens form and its subsequent optimisation by an automatic
`lensdesignprogram,whichmavormaynotbeguidedoradiueted
`alongthewaybythelensdesigner. Sincethefunctionofthelensde-
`signprogrsmistodrivethedesign formtothenearestlocaloptimum
`(as deflnedbyamritfimction), itisobvious that thestartingdesign
`form and the merit function together uniquely define which local op-
`timumdesignwillbetheresultofthisprocess.
`Thus it is apparent that, in addition to a knowledge of the principles
`of optical design, a knowledge of appropriate starting-point designs
`and of techniques for guiding the design program have become essen-
`tial elements of modern lens design. The lens designs in this book
`have been chosen to provide a good selection of starting-point designs
`and to illustrate important dedgn principles. The design techniques
`described are those which the author has found to be useful in design-
`ing with an optimization program. Many of the techniques have been
`developed or refined in the course of teaching lens design and optical
`system design; indeed. a few of them were initially suggested or in-
`spired by my students.
`In ordertomaximizetheirusefiilness,thelensdesignsinthisbook
`arepresentedinthreeparts: thelensprescription,adrawingofthe
`lens which includes a marginal ray and a full-field principal ray, and
`a plot of the aberrations. The inclusion of these two rays allows the
`user to determine the approximate path of any other ray of interest.
`For easy comparison, all leans are shown at a focal length ofapprox-
`imately 100, regardless of their application. The performance data is
`shown as aberration plots; we chose this in preference to MTF plots
`because the MTF is valid only for the focal length for which it was
`calculated, and because the MTF cannot be sealed. The aberration
`plots can be scaled, and in addition they indicate what aberrations are
`pruent and show which aberrations limit the performance of the lens.
`We have expanded on the usual longitudinal presentation of spherical
`aberration and curvature offieldbyaddingrayintercept plotsin
`three colors for the aide], 0.7 zonal, and full-field positions. We feel
`that this presentation gives a much more complete, informative, and
`useful picture of the characteristics of a lens design.
`This book is intended to build on some knowledge of both geometri-
`cal optics and the basic elements of lens design. It is thus, in a sense,
`a companion volume to the author’s Modern Optical Engineering,
`which covers such material at some length. Presumably the user of
`thistextwillalreadyhaveatleastareasonablefamiliaritywiththis
`material.
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`Preteen
`
`all
`
`There are really only a few well~understood and widely utilized
`principles of optical design. Ifone can master athorough understand-
`ing of these principles, their effects, and their mechanisms, it is easy
`torecognizetheminecistingdeeignsandalsoeasytoapplythemto
`one’a own design work. It is our intent to promote such understanding
`by presenting both expositions and annotated dedgn examples of
`theeeprinciplee.
`Readmarefi-eetousethedeaigmcontainedinthisbookasetarting
`pointsfmthehowndedgnefl'odeminanyothermtheyseefltMost
`ofthedeeignspreeentedhaveasnoted,beenp%ted;suchdeeigmmay
`ormaynotbecurrentbrsubjeettolegalpmtectionmlthoughtheremay,
`ofwumebedifi'mnoeeofopinionaatotheefi‘ectiveneesofsuchpmteo-
`don. The reader must accept full responsibility for meeting whatever
`limitationsareimpoeedontheuseoftheeedeeignebyanypatentor
`wpyfightooveragflwhetherindicetedhereinornot).
`
`Warren J. Smith
`
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`
`
`Modern
`
`Lens Design
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`
`Chapter
`
`1
`
`Introductlon
`
`Modern Lens Design is intended as an aid to lens designers who work
`with the many commercially available lens design computer pro-
`grams. We assume that the reader understands basic optical princi-
`ples and may, in fact, have a command of the fundamentals of classi-
`cal optical design methods. For those who want or need information in
`these areas, the following books should prove helpful. This author's
`Modern Optical Engineering: The Design of Optical Systems, 2d ed.,
`McGraw-I-lill, 1990, is a comprehensive coverage of optical system de-
`sign; it includes two full chapters which deal specifically with lens de-
`sign in considerable detail. Rudolf Kingalake's Optical System Design
`(1983), Fundamentals ofLens Design (1978), and A History ofthe P130.
`tographic Lens (1989), all by Academic Press, are complete, authori~
`tative, and very well written.
`Authoritative books on lens design are rare, especialh' in English;
`there are only a few others available. The Kingslake series Applied
`Optics and Optical Engineering, Academic Press, contains several
`chapters of special interest to lens designers. Volume 3 (1965) has
`chapters on lens design, photographic objectives, and eyepieces. Vol-
`ume 8 (1980) has chapters on camera lenses, aspherics, automatic de-
`sign, and image quality. Volume 10 (1987) contains an extensive
`chapter on afocal systems. Milton Laikin’s Lens Design. Marcel
`Dekker, 1991. is a volume similar to this one. with prescriptions and
`lens drawings. Its format difl'ars in that no aberration plots are in-
`cluded; instead, modulation transfer function (MTF) data for a specific
`focal length and f number are given. Now out of print, Arthur Cox's A
`System of Optical Design, Focal Press, 1964, contains a complete, if
`unique, approach to lens design, plus prescriptions and (longitudinal)
`aberration plots for many lens design patents.
`This book has several primary aims. It is intended as a source book
`
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`2
`
`Chapter One
`
`for a variety of designed lens types which can serve as suitable start-
`ing points for a lens designer's efl'orts. A study of the comparative
`characteristics of the annotated designs contained herein should also
`illustrate the application of many of the classic lens design principles.
`It is also intended as a handy, if abridged, reference to many of the
`equations and relationships which find frequent use in lens design.
`Most of these are contained in the Formulary at the end of the book.
`And last, but not least, the text contains extensive discussions of de-
`sign techniques which are appropriate to modern optical design with
`an automatic lens design computer program.
`The book begins with a discussion of automatic lens design pro-
`grams and how to use them. The merit function, optimization, vari-
`ables, and the various wchniques which are useful in connection with
`a program'are covered. Chapter 8 details many specific improvement
`strategies which may be applied to an misting design to improve its
`performance. The evaluation of a design is discuswd from the stand-
`point of ray and wave aberrations, and integrated with such standard
`measures as MTF and Strehl ratio. The sample lens designs follow.
`Each presents the prescription data, a drawing of the lens with mar-
`ginal and chief rays, and an aberration analysis consisting of ray in-
`tercept plots for three field angles, longitudinal plots of spherical ab-
`erration and field curvature, and a plot of distortion. A discussion of
`the salient features of each design accompanies the sample designs,
`and comments (in some cases quite extensive) regarding the design
`approach are given for each class of lens. The Formulary, intended as
`a convenient reference, concludes the book.
`The design of the telescope objective is covered in Chap. 6, begin-
`ning with the classic forms and continuing with several possible mod-
`ifications which can be used to improve the aberration correction.
`These are treated in considerable detail because they represent tech-
`niques which are generally applicable to all types of designs. For sim-
`ilar reasons, Chap. 8 deals with the basic principles of airspaced
`anastigmats in a rather attended treatment The complemties of the
`interrelationships involved in the Cooke triplet anastigmat are impor'
`tant to understand, as are the (almost universal) relationships be-
`tween the vertex length of an ordinary enastigmat lens and its capa-
`bilities as regards speed and angular coverage.
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`Chapter
`
`2
`
`Automatic Lens Deslgn:
`Managing the
`Lens Design Program
`
`2.1 The Merit Function
`
`What is usually referred to as automatic lens design is, of course, noth-
`ing of the sort. The computer programs which are so described are ac—
`tually optimization programs which drive an optical design to a local
`optimum. as defined by a merit function (which is not a true merit
`function, but actually a defect function). In spite of the preceding dis-
`claimers, we will use these commonly accepted terms in the discus-
`sions which follow.
`
`Broadly speaking, the merit function can be described as a combi-
`nation or function of calculawd characteristics, which is intended to
`completely describe, with a single number, the value or quality of a
`given lens design. This is obviously an exceedingly difficult thing to
`do. The typical merit function is the sum of the squares of many image
`defects; usually these image defects are evaluated for three locations
`in the field of View (unless the system covers a very large or a very
`small angular field). The squares of the defects are used so that a neg
`ative value of one defect does not offset a positive value of some other
`defect.
`
`The defects may be of many difl'erent kinds; usually most are re
`lated to the quality of the image. However, any characteristic which
`can ‘be calculated may be assigned a target value and its departure
`from that target regarded as a defect. Some less elaborate programs
`utilize the third-order (Seidel) aberrations: these provide a rapid and
`efficient way of adjusting a design. These cannot be regarded as opti-
`mizing the image quality, but they do work well in correcting ordi-
`nary lenses. Another type of merit function traces a large number of
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`4
`
`Chapter Two
`
`raysfromanobjectpoint. Theradialdistanceoftheimageplanein—
`tersection of the ray from the centroid of all the ray intersections is
`then the image defect. Thus the merit function is effectively the sum
`of the root-mesn-square (rms) spot sizes for several field angles. This
`type of merit function, while inefficient in that it requires many rays
`to be traced. has the advantage that it is both versatile and in some
`ways relatively foolproof. Some merit functions calculate the values of
`the classical aberrations, and convert (or weight) them into their
`equivalent wavefront deformations. (See Formulary Sec. F-12 for the
`conversion factors for several common aberrations.) This approach is
`very efficient as regards computing time, but requires careful design
`of the merit function. Still another type of merit function uses the
`variance of the wavefront to define the defect items. The merit func-
`tion used in the various David Grey programs is ofthis type, and is
`certainly one of the best of the commercially available merit functions
`in producing a good balance of the aberrations.
`Characteristics which do not relate to image quality can also be con-
`trolled by the lens design program. Speciflc construction parameters,
`such as radii, thicknesses, spaces, and the like, as well as focal length,
`worldng distance, magnification, numerical aperture, required clear
`apertures, etc., can be controlled. Some programs include such items
`in the merit function along with the image defects. There are two
`drawbacks which somewhat offset the neat simplicity of this ap-
`proach. One is that if the first-order characteristics which are targeted
`are not initially close to the target values, the program may correct
`theimage aberrations without controlling these first-order character-
`istics; the result may be, for example, a well-corrected lens with the
`wrong focal length or numerical aperture. The program often finds
`this to be a local optimum and is unable to move away from it. The
`other drawback is that the inclusion of these items in the merit func-
`tion has the effect of slowing the process of improving the image qual-
`ity. An alternative approach is to use a system of constraints outside
`the merit function. Note also that many of these items can be con-
`trolled by features which are included in almost all programs, namely
`angle-solves and height-solves. These algebraically solve for a radius
`or space to produce a desired ray slope or height.
`In any case, the merit function is a summation of suitably weighted
`defect items which, it is hoped, describes in a single number the worth
`of the system. The smaller the value of the merit function, the better
`the lens. The numerical value of the merit function depends on the
`construction of the optical system; it is a function of the construction
`parameters which are designated as variables. Without getting into
`the details of the mathematics involved, we can realize that the merit
`function is an n-dimensionnl space, where n is the number of the vari-
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`able constructional parameters in the optical system. The task of the
`design program is to find a location in this space 6.9., a lens prescrip—
`tion or a solution vector) which minimizes the size of the merit func-
`tion. In general, for a lens of reasonable complexity there will be many
`such locations in a typical merit function space. The automatic desim
`program will simply drive the lens design to the nearest minimum in
`the merit function.
`
`2.2 Optimization
`
`The lens design program typically operates this way: Each variable
`parameter is changed (one at a time) by a small increment whose size
`is chosen as a compromise between a large value (to get good numer-
`ical accuracy) and a small value (to get the local difi'erential). The
`change produced in every item in the merit function is calculated. The
`result is a matrix of the partial derivatives of the defect items with
`respect to the parameters. Since there are usually many more defect
`items than variable parameters, the solution is a classical least.
`squares solution. It is based on the assumption that the relationships
`between the defect items and the variable parameters are linear.
`Since this is usually a false assumption, an ordinary least-squares so-
`lution will often produce an unrealizable lens or one which may in fact
`be worse than the starting design. The damped least-squares solution,
`in effect, adds the weighted squares of the parameter changes to the
`merit function, heavily penalizing any large changes and thus limit—
`ing the size of the changes in the solution. The mathematics of this
`process are described in Spencer, “A Flexible Automatic Lens Correc-
`tion Program," Applied Optics, vol. 2, 1963, pp. 1257-1264, and by
`Smith in W. Driscoll (ed.). Handbook of Optics, McGraw-Hill, New
`York, 1978.
`If the changes are small, the nonlinearity will not ruin the process,
`and the solution, although an approximate one, will be an improve-
`ment on the starting design. Continued repetition of the process will
`eventually drive the design to the nearest local optimum.
`One can visualize the situation by assuming that there are only two
`variable parameters. Then the merit function space can be compared
`to a landscape where latitude and longitude correspond to the vari-
`ables and the elevation represents the value of the merit function.
`Thus the starting lens design is represented by a particular location in
`the landscape and the optimization routine will move the lens design
`downhill until a minimum elevation is found. Since there may be
`many depressions in the terrain of the landscape, this optimum may
`notbethebestthereis; itisalocaloptimumandtherecanbenow
`surance (except in very simple systems) that we have found a global
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`optimum in the merit function. This simple topological analogy helps
`to understand the dominant limitations of the optimization process:
`the program finds the nearest minimum in the merit function, and
`that minimum is uniquely determined by the design coordinates at
`which the process is begun. The landscape analogy is easy for the hu—
`man mind to comprehend; wh- it is extended to a 10- or 20-
`dimension space, one can realize only that it is apt to be an extremely
`complex neighborhood.
`
`2.3 Local Minima
`
`Figure 2.1 shows a contour map of a hypothetical two-variable merit
`function, with three significant local minima at points A, B, and C;
`there are also three other minima at D, E, and F. It is immediately
`apparent that if we begin an optimization at point Z, the minimum at
`point B is the only one which the routine can find. A start at You the
`ridge at the lower left will go to the minimum at 0. However, a start
`
`
`
`Flgun 2.1 Topography of a hypothetical two-variable merit function, with three sigm-
`icant minima (A, B, C) and three trivial minima (D, E, F). The minimum to which a
`design program will go depends on the point at which the optimization process is
`started. Starting point: X. Y, and Z each lead to a different design minimum; other
`starting points can load to one of the trivial minims.
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`at X, which is only a short distance away from Y, will find the best
`minimum of the three, at point A. If we had even a vague knowledge
`of the topography of the merit function, we could easily choose a start-
`ing point in the lower right quadrant of the map which would guar-
`antee finding point A. Note also that a modest change in any of the
`three starting points could cause the program to stagnate in one of the
`trivial minima at D, E, or F. It is this sort of minimum from which one
`can escape by "jolting" the design, as described below.
`The fact that the automatic design progam is severely limited and
`can find only the nearest optimum emphasizes the need for a knowl-
`edgeoflens design, in orderthatonecan selectastartingdesign form
`which is close to a good optimum. This is the only way that an auto-
`matic program can systematically find a good design. Ifthe program is
`started out near a poor local optimum, the result is a poor design.
`The mathematics of the damped least-squaressolution involves the
`inversion of a matrix. In spite of the damping action. the process can
`be slowed or aborted by either of the following conditions: (1) A vari-
`able which does not change (or which produces only a very small
`change in) the merit function items. (2) Two variaqu which have the
`same, nearly the same, or scaled effects on the items of the merit func-
`tion. Fortunately, these conditions are rarely met exactly, and they
`can be easily avoided.
`If the program settles into an unsatisfactory optimum (such as those
`at D, E, and F in Fig. 2.1) it can often be jolted out of it by manually
`introducing a significant change in one or more parameters. The trick
`is to make a change which is in the direction of a better design form.
`(Again. a knowledge of lensdeeigns is virtually a necessity.) Some-
`times simply freezing a variable to a desirable form can be sufficient
`to force a move into a better neighborhood. The difficulty is that too
`big a change may cause rays to miss surfaces or to encounter total in-
`ternal reflection, and the optimization process may break down. Con-
`versely, too small a change may not be sufficient to allow the design to
`escape from a poor local optimum. Am. one should remember that if
`the program is one which adjusts (optimizes) the damping factor, the
`factor is usually made quite small near an optimum, because the pro-
`gram is taking small steps and the situation looks quite linear; after
`the system is jolted, it is probably in a highly nonlinear region and a
`big damping factor may be nwded to prevent a breakdown. A manual
`increase of the damping factor can often avoid this problem.
`Another often-encountered problem is a design which persists in
`moving to an obviome undesirable form (when you lmow that there is
`a much better, very different one—the one that you want). Freezing
`the form of one part of the lens for a few cycles of optimization will
`often allow the rest of the lens to settle into the neighborhood of the
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`desired optimum. For example, if one were to try to convert a Cooke
`triplet into a split front crown form, the process might produce either
`a form which is like the original triplet with a narrow airspaced crack
`in the front crown, or a form with rather wild meniscus elements. A
`technique which will usually avoid these unfortunate local optima in
`this case is to freeze the front element to a plane-convex form by fixing
`the second surface to a plane for a few cycles ofoptimization. Again,
`one must know which lens forms are the good ones.
`
`2.4 Types of Merit Functions
`
`Many programs allow the user to define the merit function. This can
`be a valuable feature because it is almost impossible to design a truly
`universal merit function. As an example, consider the design of a sim-
`ple Fraunhofer telescope objective: a merit function which controls the
`spherical and chromatic aberrations of the axial marginal ray and the
`coma of the oblique ray bundle (plus the focal length) is all that is nec-
`essary. If the design complexity is increased by allowing the airspace
`to vary and/or adding another element, the merit function may then
`profitably include entries which will control zonal
`spherical,
`spherochromatism, and/or fifth-order coma. But as long as the lens is
`thin and in contact with the aperture stop, it would be foolish to in-
`clude in the merit function entries to control field curvature and astig-
`matism. There is simply no way that a thin stop-in-contact lens can
`have any control over the inherent large negative astigmatism; the
`presence of a target for this aberration in the merit function will sim-
`ply slow down the solution process. It would be ridiculous to use a
`merit function of the type required for a photographic oljective to de-
`sign an ordinary telescope objective. (Indeed, an attempt to correct the
`field curvature may lead to a compromise design with a severely
`undercorrected axial spherical aberration which, in combination with
`coma, may fool the computer program into thinking that it has found
`a useful optimum.)
`There are many design tasks in this category, where the require-
`ments are effectively limited in number and a simple, equally limiwd
`merit function is clearly the best choice. In such cases, it is usually
`obvious that some specific state of correction will yield the best re-
`sults; there is no need to balance the correction of one aberration
`against another.
`More often, however, the situation is not so simple, compromises
`and balances are required and a more complex, suitably weighted
`merit function is necessary. This can be a delicate and somewhat
`tricky matter. For example, in the design of a lens with a significant
`aperture and field, there is almost always a (poor) local optimum in
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`which (1) the spherical aberration is left quite undercut-rm (2) a
`compromise focus is chomn well inside the par-axial foam, (3) the
`Petzval field is made inward-curving, and (4) overcorrected oblique
`spherical aberration is introduced to “balance" the design. A program
`which relies on the rms spot radius for its merit function is very likely
`to fall into this trap. A better design usually results if the spherical
`(both axial and oblique) aberrations are cormted, the Petzval curva~
`ture is reduced, and a small amount of overcorrected astigmatism is
`introduced. When one recognizes this sort of situation, it is a simple
`matter to adjust the weighting of the appropriate targets in the merit
`function to force the design into a form with the type of aberration bal-
`ance which is desired. Another way to avoid this problem is to force
`the system to be evaluated/designed at the paraxial focus rather than
`at a compromise focus, i.e., to not allow defocusing. As can be seen. the
`design of a general-purpose merit function which will optimally bal-
`ance a wide variety of applications is not a simple matter.
`Although it is not always newssary, there are occasions when it is
`helpful to begin the design process by controlling only the first-order
`properties (image size, image location, spatial limitations, etc). Then
`one proceeds to control the chromatic and perhaps the Petzval aberra-
`tions. (Things may even go better if the first-order and the chromatic
`are fai