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`

`

`Modern
`Lens Design
`
`A Resource Manual
`
`Warren J. Smith
`Chief Scientist
`Kaiser Electro-Optics, Inc.
`Csrlsbad, Csl"omla
`
`.;
`
`Genesee Optics Software, Inc.
`Rochester, New York
`
`McGraw-Hiii, Inc.
`New York St. Loura San Francftco Auckland Bogotj
`careen Ulbon London Madrid Mexico Miian
`Montreal New Delhl Perl• San Juan Slo Paulo
`Singapore Sydney Tokyo Toronto
`
`APPL-1006 / Page 2 of 88
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`

`

`Library of Congress Catalo~g-in-Publication Data
`
`Smith, Warren J .
`Modem lens design : a resource manual I Warren J. Smith and
`Genesee Optics Software, Inc.
`cm.-{Optical and electro-optical engineering series)
`p.
`Includes index.
`ISBN 0-07-059178-4
`1. Lenses-Design and construction-Handbooks, manuals, etc.
`ill. Series.
`I. Genesee Optics Software, Inc.
`II. Title.
`QC385.2.D• 7S65 1992
`681' .423-dc20
`
`92-20038
`CIP
`
`Copyright © 1992 by McGraw-Hill, Inc. All rights reserved. Printed
`in the United States of America. Except as permitted under the
`·united States Copyright Act of 1976, no part of this publication may
`be reproduced or distributed in any form or by any means, or stored
`in a data base or retrieval system, without the prior written
`permission of the publisher.
`
`1 2 3 4 5 6 7 8 9 0 DOC/DOC 9 8 7 6 5 4 3 2
`
`ISBN 0-07-059178-4
`
`The sport.soring editor for this book was Dani.el A Gonneau, the
`editing supervisor was David E. Fogarty, and the production
`supervisor was Suzanne W. Babeu{. It was set in Century SchoolQ<>ok
`by McGraw-Hill's Professional Book Group composition unit.
`
`Printed and bound by R. R. Donnelley & Sons Company.
`
`OPTICS TOOLBOX is a registered trademark of Genesee Optics
`Software, Inc.
`
`Information contained in this work has been obtained by McGraw·
`Hill, Inc., from sources believed to be reliable. However, neither
`McGraw-Hill nor its authors guarantee the accuracy or complete(cid:173)
`ness of any information published herein, and neither McGraw-Hill
`nor its authors shall be responsible for any erri>rs, omissions, or
`damages arising out of use of this information. This work is pub(cid:173)
`lished with the understanding that McGraw-Hill and its authors are
`supplying information but are not attempting to render engineering
`or other professional services. If such services are required, the as..
`sistance of an appropriate professional should be sought.
`
`APPL-1006 / Page 3 of 88
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`

`

`Contents
`
`Preface
`
`Ix
`
`Chapter 1. Introduction
`
`1
`
`Chapter 2. Automatic Lens Design: Managing the Lens Design Program 3
`2.1
`The Merit Function
`2.2 Optimization
`2.3
`Local Minima
`2.4 Types of Merit Functions
`2.5 Stagnation
`2.6 Generallzed Simulated Anneallng
`2.7 Considerations about Variables for Optimization
`2.8 How to Increase the Speed or Field of a System and Avoid Ray Failure
`Problems
`2.9 Test Plate Fits, Melt Fits, and Thickness Fits
`2.10 Spectral Weighting
`2.11 How to Get Started
`
`3
`5
`8
`e
`9
`10
`10
`
`14
`16
`1e
`19
`
`Chapter 3. Improving a Design
`3.1 Standard Improvement.Techniques
`3.2 Glass Changes (Index and VValue)
`3.3 Spllttlng Elements
`3.4 Separating a Cemented Doublet
`3.5 Compounding an Element
`3.6 Vignetting and Its Uses
`3.7 Ellmlnatlng a Weak Element; the Concentric Problem
`3.8 Balanclng Aberrations
`3.9 The Symmetrical Prlnclple
`3.1 o Aspherlc Surfaces
`
`I
`
`25
`
`25
`25
`27
`30
`30
`33
`34
`35
`39
`40
`
`v
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`·vi
`
`Contents
`
`Chapter 4. Evaluation: How Good rs This Design?
`4.1 The Uses of a Prellmlnary Evaluatlon
`4.2 OPD versus Measures of Performance
`4.3 Blur Spot Size versus Certain Aberrations
`4A MTF-The Modulailon Transfer Function
`
`Chapter s. Lens Design Data
`5.1 About the Sample Lenses
`5.2 Lens Preacrlptlons, Drawings, and Aberration Plots
`5.3 Estimating the Potential of a Design
`5.4 Scallng a Design, Its Aberrations, and Its MTF
`s.s Notes on the Interpretation of Ray Intercept Plots
`
`Chapter 6. Telescope Objectives
`6.1 The Thin Doublet
`8.2 Secondary Spectrum (Apochromatlc Systems)
`6.3 Spherochromatlsm
`6.4 Zonal Spherical Aberration
`6.5
`Induced Aberrations
`8.6 Three-Element Objectives
`
`Chapter 7. Eyepieces and Magnifiers
`
`7.1 Eyepieces
`7.2 Two Magnifier Designs
`7.3 Simple Two- and Three-Element Eyepieces
`7.4 Four-Element Eyepieces
`7.5 Five-Element Eyepieces
`7.8 Six· and Seven-Element Eyepieces
`
`Chapter 8. Cooke Triplet Anastlgmats
`
`8.1 Alrapaced Trlplet Anastlgmats
`8.2 01111 Choice
`8.3 Vertex Length and Residual Aberrations
`8.4 Other Design Considerations
`
`Chapter 9. Reverse Telephoto (Retrofocus and Fish-Eye) Lenses
`9.1 The Reverse Telephoto Prlnclple
`9.2 The Basic Retrofocus Lens
`9.3 The Fish-Eye, or Extreme Wide-Angle Reverse Telephoto, Lenses
`
`43
`
`43
`43
`47
`48
`
`49
`
`49
`so
`54
`57
`&B
`
`83
`63
`72
`75
`75
`79
`79
`
`87
`
`87
`89
`92
`92
`101
`101
`
`123
`
`123
`12s
`125
`121
`
`147
`
`147
`148
`1so
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`

`I•
`
`Contents
`
`vii
`
`Chapter 10. Telephoto Lenses
`10.1 The Basic Telephoto
`10.2 Close-up or Macro Lenses
`10.3 Sample Telephoto Designs
`
`Chapter 11. Double-Meniscus Anastlgmats
`
`11.1 Meniscus Components
`11.2 Hypergon, Topogon, and Metrogon
`11.3 Protar, Dagor, and Convertible Lenses
`11.4 The Spilt Dagor
`11.s The Dogmar
`
`Chapter 12. The Tessar, Heliar, and Other Compounded Triplets
`
`12.1 The Classlc Tessar
`12.2 The Hellar/Pentac
`12.3 Other Compounded Triplets
`
`Chapter 13. The Petzval Lens; Head-up Display Lenses
`13.1 The Petzval Portrait Lens
`13.2 The Petzval Projection Lens
`13.3 The Petzval with a Field Flattener
`13.4 Very High Speed Petzval Lenses
`13.5 Head·l!~ Display (HUD) Lenses; Blocular Lenses
`
`Chapter 14. Split Triplets
`
`Chapter 15. Microscope Objectives
`
`15.1 General Considerations
`15.2 Classical Objective Design Forms; the Aplanatlc Front
`15.3 Flat-Fleld Objectives
`15.4 Reflecting Objectives
`15.5 The Sample Lenses
`
`Chapter 16. Mirror and Catadloptrlc Systems
`16.1 The Good and the Bad Points of Mirrors
`16.2 The Classlcal Two-Mirror Systems
`16.3 Catadloptrlc Systems
`,
`16.4 Confocal Paraboloids
`16.5 Unobscured Systems
`
`Chapter 17. The Blotar or Double--Gauss Lens
`17 .1 The Basic Six-Element Version
`17.2 The Seven-Element Blotar-Spllt·Rear Slnglet
`17.3 The Seven-Element Blotar-Broken Contact Front Doublet
`
`169
`
`169
`110
`110
`
`183
`
`183
`183
`185
`190
`190
`
`197
`
`197
`210
`210
`
`221
`
`221
`221
`224
`228
`238
`
`239
`
`257
`
`257
`258
`261
`262
`263
`
`271
`
`211
`211
`285
`295
`296
`
`303
`
`303
`319
`319
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`viii
`
`Contents
`
`17.4 The Seven-Element Blotar-One Compounded Outer Element
`17.5 The Eight-Element Blotar
`17 .6 Mlscellaneous Blotara
`
`~o
`340
`340
`
`Chapter 18. Wlde .. Angle Lenses with Negative Outer Elements
`
`Chapter 19. Projection TV Lenses and Macro Lenses
`19.1 Projection TV Lenses
`19.2 Macro Lenses
`
`Chapter 20. Zoom Lenses
`
`Chapter 21. Infrared Systems
`21.1
`Infrared Optics
`21.2
`IR Objective Lenses
`IR Telescopes
`21.3
`
`Chapter 22. Scanner/f·8 and Laser Disk/Collimator ·Lenses
`22.1 Monochromatic Systems
`22.2 Scanner Lenses
`22.3 Laser Disk, Focusing, and Colllmator Lenses
`
`Chapter 23. Tolerance Budgeting
`The Tolerance Budget
`23.1
`23.2 Additive Tolerances
`23.3 Establlshlng the Tolerance Budget
`
`Formulary
`Sign Conventions, Symbols, and Definitions
`F.1
`F.2 The Cardlnal Points
`F.3
`Image Equations
`F.4 Paraxlal Ray Tracing (Surface by Surface)
`Invariants
`F.S
`F.6 Paraxfal Ray Tracing (Component by Component)
`F.7 Two-Component Relationships
`F.8 Third-Order Aberrations-Surface Contributions
`F.9 Third-Order Aberrations-Thin Lena Contributions
`F.10 Stop Shift Equations
`F.11 Third-Order Aberrations-Contributions from Aspherlc Surfaces
`F.12 Conversion of Abenatlons to Wavefront Deformation (OPD,
`Optical Path Difference)
`
`Appendix. Lens Listings
`
`Index 465
`
`355
`
`365
`365
`367
`
`373
`
`393
`
`393
`
`394
`
`406
`
`411
`
`411
`411
`412
`
`431
`
`431
`434
`438
`
`441
`
`441
`443
`443
`444
`445
`445
`
`446
`
`447
`449
`450
`451
`
`451
`
`453
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`

`Preface
`
`This book had its inception in the early 1980s, when Bob Fischer and
`I, as coeditors of the then Macmillan, now McGraw-Hill, SerieB on Op(cid:173)
`tical and Electro-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 staff of
`Genesee. In alphabetical order, the Genesee personnel who have been
`involved are Charles Dubois, Henry Gintner, Robert Macintyre,
`David Pixley, Lynn VanOrden, and Scott Weller. They have been re(cid:173)
`sponsible for the computerized lens data tables, lens drawings, and ab(cid:173)
`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(cid:173)
`tary within an expert-system, artificial-intelligence, relational data
`base.
`This author's optical design experience has spanned almost five de(cid:173)
`cades. In that period lens design has undergone many radical 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(cid:173)
`ous computation, combined with great understanding of lens design
`principles and dogged perseverance to produce what are now the clas(cid:173)
`sic lens design forms. Most of these designiorms 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(cid:173)
`ferent in both technique and philosophy. This change is, of course, the
`
`Ix
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`x
`
`Preface
`
`result of the vastly increased computational speed now available to
`the lens designer.
`In essence, much modern lens design consists of the selection of a
`starting lens form and its subsequent optimization by an automatic
`lens design program, which may or may not be guided or adjusted
`along the way by the lens designer. Since the function of the lens de·
`sign program is to drive the design form to the nearest local optimum
`(as defi~ed by a merit function), it is obvious that the starting design
`form and the merit function together uniquely define which local op·
`timum design will be the result of this process.
`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(cid:173)
`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 design principles. The design techniques
`described are those which the author has found to be useful in design(cid:173)
`ing with an optimization program. Many of the techniques have been
`developed or refined in the course of teaching lens design and optical
`system de~ign; indeed, a few of them were initially suggested or in(cid:173)
`spired by my students.
`In order to maximize their usefulness, the lens designs in this book
`are presented in three parts: the lens prescription, a drawing of the
`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 lenses are shown at a focal length of approx(cid:173)
`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 scaled. The aberration
`plots can be scaled, and in addition they indicate what aberrations are
`present and show which aberrations limit the performance of the lens.
`We have expanded on the usual longitudinal presentation of spherical
`aberration and curvature of field by adding ray intercept plots in
`three colors for the axial, 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(cid:173)
`cal optics and the basic elements of lens design. It is thus, in a sense,
`a companion volume to the author's Modem Optical Engi,neering,
`which covers such material at some length. Presumably the user of
`this text will already have at least a re~onable familiarity with this
`material.
`
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`

`Preface
`
`xi
`
`There are really only a few well-understood and widely utilized
`principles of optical design. If one can master a thorough understand(cid:173)
`ing of these principles, their effects, and their mechanisms, it is easy
`to recognize them in existing designs and also easy to apply them to
`one's own design work. It is our intent to promote such understanding
`by presenting both expositions and annotated design examples of
`these principles.
`Readers are free to use the designs contained in this book as starting
`points for their own design efforts, or in any other way they see fit. Most
`of the designs presented have, as noted, been patented; such designs may
`or may not be currently subject to legal prot.ection, although there may,
`of course, be differences of opinion as to the effectiveness of such protec(cid:173)
`tion. The reader must accept full responsibility for meeting whatever
`limitations are imposed on the use of these designs by any patent or
`copyright coverage (whether indicated herein or not).
`
`Warren J. Smith
`
`APPL-1006 / Page 10 of 88
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`

`

`Modern
`Lens Design
`
`t
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`APPL-1006 / Page 12 of 88
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`

`Chapter
`
`1
`
`Introduction
`
`Modern Lens Design is intended as an aid to lens designers who work
`with the many commercially available lens design computer pro(cid:173)
`grams. We assume that the reader understands basic optical princi(cid:173)
`ples and may, in fact, have a command of the fundamentals of classi(cid:173)
`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-Hill, 1990, is a comprehensive coverage of optical system de(cid:173)
`sign; it includes two full chapters which deal sp~cifically with lens de(cid:173)
`sign in considerable detail! Rudolf Kingslake's Optical System Design
`(1983), Fundamentals of Lens Design (1978), and A History of the Pho(cid:173)
`tographic Lens (1989), all by Academic Press, are complete, authori(cid:173)
`tative, and very well written. ·
`Authoritative books on lens design are rare, especially in English;
`there are only a few others available. The Kingslake series Applied
`Optics and Optical Engi,neering, Academic Press, contains several
`chapters of special interest to lens designers. Volume 3 (1965) has
`chapters on lens design, photographic objectives, and eyepieces. Vol(cid:173)
`ume 8 (1980) has chapters on camera lenses, aspherics, automatic de(cid:173)
`sign, and image quality. Volume 10 (1987) cont~s an extensiv:e
`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 differs in that no aberration plots are in(cid:173)
`cludep; instead, modulation transfer function (MTF) data for a specific
`focal length andfnumber are given. Now out of print, Arthur Cox's A
`System of Optical Design, Focitl Press, 1964, contains a complete, if
`unique, approach to lens design, plus prescriptions and Uongitudirtal)
`aberration plots for many lens design patents.
`This book has several primary aims. It is intended as a source book
`
`1
`
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`

`2
`
`Chapter One
`
`for a variety of designed lens types which can serve as suitable start(cid:173)
`ing points for a lens designer's efforts. 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(cid:173)
`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(cid:173)
`grams and how to use them. The merit function, optimization, vari(cid:173)
`ables, and the various techniques which are useful in connection with
`a program .. are covered. Chapter 3 details many specific improvement
`strategies which may be applied to .an existing design to improve its
`performance. The evaluation of a design is discussed from the stand(cid:173)
`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(cid:173)
`ginal and chief rays, and an aberration analysis consisting of ray in(cid:173)
`tercept plots for .three field angles, longitudinal plots of spherical ab(cid:173)
`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 desjgn
`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(cid:173)
`ning with the classic forms and continuing with several possible mod(cid:173)
`ifications which can be used to improve the aberration correction.
`These are treated in considerable· detail because they represent tech(cid:173)
`niques which are generally applicable to all types of designs. For sim(cid:173)
`ilar reasons, Chap. 8 deals with the basic principles of airspaced
`anastigmats in a rather extended treatment. The complexities of the
`interrelationships involved in the Cooke triplet anastigmat are impor(cid:173)
`tant to understand, as are the {almost universal) relationships be(cid:173)
`tween the vertex length of an ordinary anastigmat lens and its capa(cid:173)
`bilities as regards speed and angular coverage.
`
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`

`Chapter
`
`2
`
`Autornatic Lens Design:
`Managing the
`Lens Design Program
`
`2.1 The Merit Function
`
`What is usually referred to as automatic lens d~sign is, of course, noth(cid:173)
`ing of the sort. The computer programs which are so described are ac(cid:173)
`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(cid:173)
`claimers, we will use these commonly accepted terms in the discus(cid:173)
`sions which follow.
`Broadly speaking, the metj.t function can be described as a combi(cid:173)
`nation or function of calculated 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(cid:173)
`ative value of one defect does not offset a positive value of some other
`defect.
`The defects may be of many different kinds; usually most are re(cid:173)
`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(cid:173)
`mizing the image quality, but they do work well in correcting ordi(cid:173)
`nary lenses. Another type of merit function traces a large number of
`
`3
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`4
`
`Chapter Two
`
`rays from an object point. The radial distance of the image plane in(cid:173)
`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-mean-square (rms) spot sizes for several field angles. This
`type of merit function, while inefficient in that it requires many ray~
`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(cid:173)
`tion used in the various David Grey programs is of this 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(cid:173)
`trolled by the lens design program·. Specific construction parameters,
`such as radii, thicknesses, spaces, and the like, as well as focal length,
`working 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(cid:173)
`proach. One is that if the first-order characteristics which are targeted
`are not initially close to the target values, the program may correct
`the image aberrations without controlling these first-order character(cid:173)
`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(cid:173)
`tion has the effect of slowing the process of improving the imEJ.ge qual(cid:173)
`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(cid:173)
`trolled by features which are included in almost all programs, namely
`angle-solues 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. Witliout getting into
`the details of the mathematics involved, we can realize that the merit
`function is an n-dimensional space, where n is the number of the vari-
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`Automatic Lens Design
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`5
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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 (i.e., a lens prescrip(cid:173)
`tion or a solution vector) which minimizes the size of the merit func(cid:173)
`tion. In general, for a lens of reasonable complexity there will be many
`such locations in a typical merit function space. The automatic design
`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(cid:173)
`ical accuracy) and a small value (to get the local differential). 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(cid:173)
`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(cid:173)
`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(cid:173)
`ing the size of the changes in the solution. The mathematics of this
`process are described in Spencer, "A Flexible Automatic Lens Correc(cid:173)
`tion Program," Applied Optics, vol. 2, 1963, pp. 1257-1264, and by
`Smith in W. Driscoll (ed.), Handbook of Optics, ¥cGraw-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(cid:173)
`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(cid:173)
`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
`not be the best there is; it is a local optimum and there can be no as(cid:173)
`surance (except in very simple systems) that we have found a global
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`6
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`Chapter Two
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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(cid:173)
`man mind to comprehend; when 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 Yon the
`ridge at the lower left will go to the minimum at C. However, a start
`
`Figure 2;1 Topography of a hypothetical two-variable merit function, with three signif'.(cid:173)
`icant minima .CA, 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 points X, Y, and Z each lead to a different design minimum; other
`starting points can lead to one of the trivial minima.
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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(cid:173)
`ing point in the lower right quadrant of the map which would guar(cid:173)
`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 '~olting" the design, as described below.
`The fact that the automatic design program is severely limited and
`can find only the nearest optimum emphasizes the need for a knowl(cid:173)
`edge of lens design, in order that one can select a starting design form
`which is close to a good optimum. This is the only way that an auto(cid:173)
`matic program can systematically find a good design. If the program 'is
`started out near a poor local optimum, the result is a poor design.
`The mathematics of the damped least-squares.solution 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(cid:173)
`able which does not change (or which produces only a vecy small
`change in) the merit function items. (2) Two variables which have the
`same, nearly the same, or scaled effects on the items of the merit func(cid:173)
`tion. Fortunately, these conditions are rarely met exactly, and they
`can be e;:isily avoided.
`If the program settles into an unsatisfactory optimum (such as those
`at D, E, and Fin 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 lens .designs is virtually a necessity.) Some(cid:173)
`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~ change may cause rays to miss surfaces or to encounter total in(cid:173)
`ternal reflection, and the optimization process may break down. Con(cid:173)
`versely, too small a change may not be sufficient to allow the design to
`escape from a poor local optimum. Also, 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(cid:173)
`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 needed 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 obviously undesirable form (when you know 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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`8
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`Chapter Two
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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 plano-convex form by fixing
`the sec