Jose Sasian
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`Introduction to Lens Design
`
`Optical lenses have many important applications, from telescopes and spectacles, to
`microscopes and lasers. This concise, introductory book provides an overview of the
`subtle art of lens design. It covers the fundamental optical theory, and the practical
`methods and tools employed in lens design, in a succinct and accessible manner. Topics
`covered include first-order optics, optical aberrations, achromatic doublets, optical
`relays, lens tolerances, designing with off-the-shelf lenses, miniature lenses, and zoom
`lenses. Covering all the key concepts of lens design, and providing suggestions for
`further reading at the end of each chapter, this book is an essential resource for graduate
`students working in optics and photonics, as well as for engineers and technicians
`working in the optics and imaging industries.
`
`is Professor of Optical Design at the James C. Wyant College of Optical
`JOSE SASIAN
`Sciences at the University of Arizona in Tucson, AZ. He has taught a course on lens
`design for more than 20 years and has published extensively in the field. He has worked
`as a ccmsultant in lens design for the optics industry, and has been responsible for the
`design of a variety of successful and novel lens systems.
`
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`Introduction to Lens Design
`
`JOSE SASIAN
`University of Arizona
`
`CAMBRIDGE
`UNIVERSITY PRESS
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`CAMBRIDGE
`UNIVERSITY PRESS
`
`University Printing House, Cambridge CB2 8BS, United Kingdom
`
`One Liberty Plaza, 20th Floor, New York, NY 10006, USA
`
`477 Williamstown Road, Port Melbourne, VIC 3207, Australia
`
`314-321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre,
`New Delhi - 110025, India
`
`79 Anson Road, #06-04/06, Singapore 079906
`
`Cambridge University Press is part of the University of Cambridge.
`
`It furthers the University's mission by disseminating knowledge in the pursuit of
`education, learning, and research at the highest international levels of excellence.
`
`www .cambridge.org
`Information on this title: www.cambridge.org/9781108494328
`DOI: 10.1017/9781108625388
`
`© Jose Sasian 2019
`
`This publication is in copyright. Subject to statutory exception
`and to the provisions of relevant collective licensing agreements,
`no reproduction of any part may take place without the written
`permission of Cambridge University Press.
`
`First published 2019
`
`Printed in the United Kingdom by TJ International Ltd, Padstow Cornwall
`
`A catalogue record for this publication is available from the British Library.
`
`Library of Congress Cataloging-in-Publication Data
`Names: Sasian, Jose M., author.
`Title: Introduction to lens design / Jose Sasian, University of Arizona.
`Description: Cambridge, United Kingdom; New York, NY, USA: University Printing
`House, 2019. I Includes bibliographical references and index.
`Identifiers: LCCN 20190194841 ISBN 9781108494328 (hardback)
`Subjects: LCSH: Lenses-Design and construction.
`Classification: LCC QC385.2.D47 S27 2019 I DDC 681/.423-dc23
`LC record available at https://lccn.loc.gov/2019019484
`
`ISBN 978-1-108-49432-8 Hardback
`
`Cambridge University Press has no responsibility for the persistence or accuracy
`of URLs for external or third-party internet websites referred to in this publication,
`and does not guarantee that any content on such websites is, or will remain,
`accurate or appropriate.
`
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`With appreciation to my lens design students.
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`This is evident even more when we realize that the combinations of
`lenses are very capricious entities, which in certain arrangements,
`probably because of laws deeply hidden in the building blocks of
`complicated functions, will give either not a good image at all, or one
`that is inevitably curved or distorted, and one understands easily that
`a lack of knowledge of these laws can lead to high costs and great
`useless efforts.
`
`Joseph Maxirnillian Petzval
`Bericht iiber die Ergebnisse einiger dioptrischen
`Untersuchungen (Pest, 1843)
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`Contents
`
`Preface
`
`page xiii
`
`1
`
`2
`
`3
`
`Introduction
`1.1 Aims of This Book
`1.2
`Topics Covered
`1.3
`The Art of Lens Design
`Further Reading
`
`Classical Imaging, First-Order Imaging, and
`Imaging Aberrations
`2.1 Classical Imaging
`2.2
`First-Order Optics
`2.3
`Imaging Aberrations
`2.4
`Computing Aberration Coefficients
`2.5
`Field of View and Relative Aperture
`2.6
`Lens Design Example
`2.7
`Stop Shifting
`2.8
`Parity of the Aberrations and the Principle of Symmetry
`Further Reading
`
`Aspheric Surfaces
`3.1
`Spherical Surfaces
`3.2 Conicoids
`3.3 Cartesian Ovals
`3.4
`Polynomial Surfaces
`3.5 Aberration Coefficients
`3.6
`Testing Aspheric Surfaces
`3.7 Control of Spherical Aberration
`3.8
`Freeform Surfaces
`
`Vll
`
`1
`1
`2
`3
`4
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`5
`5
`6
`10
`16
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`viii
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`Contents
`
`4
`
`5
`
`6
`
`3.9 User Defined Surfaces
`Further Reading
`
`Thin Lenses
`4.1 Thin Lens with the Aperture Stop at Lens
`4.2 Thin Lens with Remote Aperture Stop
`4.3
`Field Curves
`4.4 Optical Relay System
`4.5 Wollaston Periscopic Lens
`4.6
`Periskop Lens
`4.7 Criterion for Artificially Flattening the Field
`Further Reading
`
`Ray Tracing
`5.1
`Sequential Ray Tracing
`5.2 Non-Sequential Ray Tracing
`5.3 Ray Tracing Equations
`5.4 Ray Tracing Pitfalls
`5.5 Ray Definition
`5.6 Reverse Ray Tracing
`5.7 Zero Index of Refraction
`5.8 Zero Dispersion
`5.9
`Infinite Index of Refraction
`5.10 Negative Thickness
`5.11 Floating the Aperture Stop
`5.12 Dummy Surfaces
`5.13 Index Interpolation
`Further Reading
`
`Radiometry in a Lens System
`6.1 The Pinhole Camera
`6.2
`Pinhole Camera Relative Illumination
`6.3 Ratio of On-Axis lrradiance to Exitance in
`an Optical System
`6.4 Lens System Relative Illumination
`6.5 Light Vignetting
`6.6
`Irradiance at the Exit Pupil Plane
`6.7 Optical Etendue
`Further Reading
`
`7
`
`Achromatic and Athermal Lenses
`7.1 Chromatic Change of Focus and Magnification
`
`29
`29
`
`30
`30
`35
`37
`37
`38
`41
`41
`43
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`44
`44
`45
`46
`47
`47
`48
`49
`49
`50
`50
`51
`51
`52
`52
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`54
`54
`56
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`57
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`62
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`64
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`Contents
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`7.2 Optical Glass
`7.3
`Thin Achromatic Doublet
`7.4 Apochromatic Triplet
`7.5 Glass Selection
`7.6
`Thermal Change of Focus and Magnification
`7.7
`Techniques for Correcting Chromatic Aberration
`7.8 Diffractive Optical Elements
`Further Reading
`
`8
`
`Combinations of Achromatic Doublets
`8.1
`Structural Aberration Coefficients of a Thin Achromatic
`Doublet
`Field Curvature of a Thin Achromatic Doublet
`8.2
`Lister Microscope Objective
`8.3
`Petzval Portrait Objective
`8.4
`8.5 Rapid Rectilinear Lens
`8.6 Concentric Lens
`8.7 Anastigmatic Lens
`8.8
`Telephoto Lens
`8.9 Reverse Telephoto Lens
`Further Reading
`
`9
`
`Image Evaluation
`Image Evaluation of a Point Object
`9.1
`9.2
`Image Evaluation of an Extended Object
`Further Reading
`
`10 Lens Tolerancing
`10.1 Lens Dimensions and Tolerances
`10.2 Worst Case
`10.3 Sensitivity Analysis
`Inverse Sensitivity Analysis
`10.4
`10.5 Compensators
`10.6 Tolerancing Criterion Statistics
`10.7 RSS Rule
`10.8 Monte Carlo Simulation
`10.9 Monte Carlo Simulation Example
`10.10 Behavior of a Lens under Manufacturing Errors
`10.11 Desensitizing a Lens from Element Decenter, Tilt,
`or Wedge
`
`ix
`
`66
`67
`70
`71
`72
`74
`76
`80
`
`81
`
`81
`83
`84
`87
`89
`90
`91
`93
`96
`97
`
`98
`99
`104
`109
`
`110
`110
`113
`113
`114
`114
`115
`116
`117
`118
`120
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`122
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`Contents
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`10.12 Lens Drawings
`Further Reading
`
`11 Using Lens Design Software
`11.1 Utilities and Settings
`11.2 Merit Function
`11.3 Optimization Algorithm
`11.4 Analyzing a Lens
`11.5 Adjusting a Lens
`11.6 Modifying and Improving a Lens
`11.7 Designing a Lens
`11.8
`Inventing a New Lens
`11.9 Documenting a Lens
`Further Reading
`
`12 Petzval Portrait Objective, Cooke Triplet, and Double
`Gauss Lens
`12.1 Petzval Sum
`12.2 Lens Stress and Relaxation
`12.3 Petzval Portrait Objective
`12.4 Cooke Triplet Lens
`12.5 Double Gauss Lens
`Further Reading
`
`13 Lens System Combinations
`13.1
`Image Aberrations
`13.2 Pupil Aberrations
`13.3 Pupil Spherical Aberration
`13.4 Pupil Coma
`13.5 Pupil Distortion
`13.6 Chromatic Vignetting
`13.7 Optical Relays
`Further Reading
`
`14 Ghost Image Analysis
`14.1 Surface Reflectivity
`14.2 First-Order Analysis
`14.3 Real Ray Tracing Analysis
`14.4 Thin Lens Ghost Images
`14.5 Total Internal Reflection Ghost
`
`124
`125
`
`126
`127
`129
`131
`132
`132
`133
`134
`135
`135
`135
`
`137
`137
`139
`142
`144
`149
`151
`
`153
`153
`154
`156
`158
`159
`160
`161
`163
`
`164
`165
`166
`166
`168
`168
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`Contents
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`14.6 Narcissus Retro-Reflections
`14.7 Parallel and Concentric Surfaces
`Further Reading
`
`15 Designing with Off-the-Shelf Lenses
`15.1 Cooke Triplet Lens
`15.2 UV Lens
`15.3 Telecentric Lenses
`15.4 Relay Systems
`15.5 Off-the-Shelf Lens Suppliers
`
`16 Mirror Systems
`16.1 Single Mirrors
`16.2 Two-Mirror Systems
`16.3 Spherical Mirror Solutions
`16.4 Schwarzschild Flat-Field, Anastigmatic Solution
`16.5 Mersenne Telescopes
`16.6 Paul and Paul-Baker Systems
`16.7 Offner Unit Magnification Relay
`16.8 Meinel Two-Stage Telescope
`Further Reading
`
`17 Miniature Lenses
`17.1 Lens Specifications
`17.2 Lens Design Considerations
`17.3 Lens Manufacturing Considerations
`Further Reading
`
`18 Zoom Lenses
`18.1 Two-Group Zoom
`18.2 Example
`18.3 Three-Group Zoom
`18.4 Four-Group Zoom
`18.5 Zoom Lens Kernel
`18.6 Aberration Considerations
`Further Reading
`
`Appendix 1 Imaging Aberrations
`Appendix 2 Pupil Aberrations
`Appendix 3 Structural Aberration Coefficients
`
`xi
`
`169
`169
`170
`
`171
`171
`172
`173
`174
`175
`
`176
`176
`179
`181
`181
`182
`182
`183
`184
`186
`
`187
`187
`190
`194
`195
`
`196
`198
`199
`201
`203
`204
`205
`206
`
`207
`210
`211
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`xii
`
`Contents
`
`Appendix 4 Primary Aberrations of a Plane
`Symmetric System
`Further Reading
`Appendix 5 Sine Condition
`Further Reading
`Glossary
`Further Reading on Lens Design
`Index
`
`215
`217
`218
`220
`221
`229
`231
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`Preface
`
`I have been fortunate to have taught for many years the course Lens Design
`OPTI517 at the James C. Wyant College of Optical Sciences at the University of
`Arizona. The thrust of this course is to help graduate students to acquire the skill
`of lens design and obtain a solid foundation in the subject in the space of about
`16 weeks, which is the duration of the Fall academic term. Behind the scenes,
`the challenge has been how to completely and effectively achieve this thrust.
`This book is the result of teaching OPTl517 for about 20 years, and outlines the
`essential material interested students or optical engineers should know.
`I have had the support and help from many individuals and I would like to
`acknowledge and to thank them. Robert Shannon handed me OPTI517, which
`he initiated and taught for many years at the then Optical Sciences Center. My
`colleagues Russell Chipman, John Greivenkamp, Angus Macleod, Jim Burge,
`Yuzuru Takashima, Tom Milster, Ron Liang, Buddy Martin, Hong Hua, Jim
`Schwiegerling, Roland Shack, Masud Mansuripur, Roger Angel, Stanley Pau,
`Bill Wolfe, Roy Frieden, Brian Anderson, Arvind Marathay, Rolf Binder, Dae
`Woo Kim, Eustace Dereniak, Steve Jacobs, Harry Barrett, Charles Falco, Jack
`Gaskill, John Koshel, and Dan Vukobratovich have been helpful and inspir(cid:173)
`ational. I also would like to thank Richard Powell, Jim Wyant, and Tom Koch
`for the support they have provided me.
`I have been fortunate to receive a "yes" when I asked many experts to visit
`the University of Arizona and help me teach lectures in optical design. Among
`the many individuals that I can recall and would like to acknowledge and thank
`are Richard Juergens, Bill Cassarly, Rich Pfisterer, Michael Humphreys,
`Akash Arora, Vini Mahajan, Richie Youngworth, Richard Buchroeder, Donald
`Dilworth, Mary Turner, Michael Gauvin, Craig Pansing, Dave Shafer, Jay
`Wilson, Jake Jacobsen, and John Rogers.
`I would like to acknowledge and thank the lens design software companies,
`Lambda Research Corporation, Optenso™, Optical Systems Design, Inc.,
`
`xiii
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`XIV
`
`Preface
`
`Synopsis®, and Zemax for always providing excellent academic access to their
`lens design software and for their outstanding support.
`Richard Buchroeder, William Hicks, Craig Pansing, and Jim Schwiegerling
`provided many useful comments and suggestions to improve several chapters
`in this book. I am grateful for their help in this endeavor.
`I would like thank Nicholas Gibbons, Sarah Lambert, and Roisin Munnelly,
`at Cambridge University Press, Vinithan Sethumadhavan at SPi-Global, and
`Liz Steel for their excellent editorial work in publishing this book.
`
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`1
`Introduction
`
`Lens design is an exciting and important field of optics. This field provides
`designs for a great diversity of lens and mirror systems needed in many other
`fields, such as consumer optics, microscope optics, telescope optics, lenses for
`optical lithography, and photographic optics. Lens and mirror systems are
`ubiquitous. The work of a lens designer is to provide the constructional data
`and fabrication tolerances of all the optical elements that a given lens system
`requires to perform the intended function. Currently many students and engin(cid:173)
`eers are interested in lens design because the field by itself is of great interest,
`or because they have the need to analyze and design lens systems required in
`their engineering practice. An optical engineer should have at least some
`familiarity with how a lens system is designed so that he or she can effectively
`contribute to develop optical systems.
`
`1.1 Aims of This Book
`
`This book is an introduction to lens design, and has been written to provide an
`overview of topics that are indispensable to acquire the skill of lens design.
`Acquiring this skill, the skill of lens design, requires learning some theory,
`learning how to use lens design software, and gaining experience by designing
`actual lenses. This book will help the interested reader to understand the theory
`and methods used in lens design. The book does not have lengthy discussions
`but, rather, brief discussions to point out essential knowledge. A few refer(cid:173)
`ences are given for further reading, where the reader can deepen his or her
`knowledge about a topic.
`There are many excellent books about lens design, such as Lens Design
`Fundamentals by Rudolf Kingslake and Barry Johnson, and Modern Lens
`Design by Warren Smith. However, these and other comprehensive books
`
`1
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`2
`
`Introduction
`
`might not be appropriate for an introduction to lens design, as part of their
`main focus is the design and survey of a variety of specific lenses. Instead, this
`introductory book intends to be brief and also to give an overview of topics
`that a current optical engineer needs to know about lens system design.
`A graduate student or an optical engineer who understands the content of this
`book and models, in a lens design program, the different lens systems dis(cid:173)
`cussed in it, would then have a solid foundation to practice the skill of lens
`design. Another aim of this book is to provide an efficient introduction to lens
`design to an interested student or optical engineer, so that he or she is well
`positioned to analyze, combine, debug, adjust, or design lens systems.
`
`1.2 Topics Covered
`
`Essential to lens design, and to optical engineering, is an understanding about
`how optical aberrations are corrected, balanced, or minimized. The reader
`should have some familiarity with first-order optics and with the theory of
`optical aberrations, as many discussions revolve about the choices made in the
`layout of a lens system and how to correct the aberrations. In this book,
`structural aberration coefficients are used to determine primary aberrations
`and to understand how to correct, balance, or minimize them. Chapter 2
`provides a review of first-order optics and aberrations. Chapter 3 provides a
`brief discussion of aspheric surfaces. Chapter 4 provides a discussion of thin
`lenses and how aberrations are controlled in very simple lens systems. Chap(cid:173)
`ter 5 provides a discussion about how ray tracing takes place, and some useful
`techniques. Chapter 6 provides a discussion about radiometric aspects of a lens
`system, which are important for a more comprehensive understanding of how
`lenses work. Chapter 7 discusses achromatic and athermal lenses. Chapter 8
`provides a number of lens examples that use combinations of achromatic
`doublets. This chapter is insightful because it shows how lenses are combined.
`Chapter 9 discusses the tools used to determine image quality. Chapter 10
`discusses how to perform a tolerancing analysis for providing tolerances to the
`constructional parameters of a lens system that will be manufactured. Chap(cid:173)
`ter 11 comments on issues in using a lens design program. Chapter 12 dis(cid:173)
`cusses three classical lenses; the Petzval portrait objective, the Cooke triplet,
`and the double Gauss lens. Chapter 13 discusses issues that arise in combining
`lens systems; it also contributes to providing a more comprehensive view
`about lens systems. Chapter 14 discusses designing with off-the-shelf lenses.
`Chapter 15 discusses ghost images in a lens system. Chapter 16 discuses some
`basic mirror systems. Chapter 17 discusses miniature lenses. Chapter 18
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`1.3 The Art of Lens Design
`
`3
`
`provides basic concepts in zoom lens design. In addition to a glossary of terms,
`the book provides five Appendices, where several tables related to aberrations
`are provided, as well as a discussion of the sine condition. Thus, the book
`contents provide a shift in the way lens design is taught. In this introductory
`book there is more emphasis on providing a broader view of fundamentals and
`essential topics in lens design, rather than bringing attention to the detailed
`design of a survey of lenses. This shift responds to current needs in the optical
`industry, and modem approaches to learning. Yet, this book provides a solid
`introduction for those who would like to specialize in the art of lens design.
`
`1.3 The Art of Lens Design
`
`There are many types of lens systems, and their variety is increasing with
`advancements and the creation of new technological fields. Examples of lens
`types are projection lenses, telephoto lenses, convertible lenses, catadioptric
`lenses, zoom lenses, underwater lenses, lenses for aerial photography, ana(cid:173)
`morphic lenses, panoramic lenses, lenses for video and cinematography, lenses
`for scanning, relay lenses, periscope lenses, and lenses for endoscopes.
`The process of lens design starts with understanding the application the
`intended lens is to be designed for. From understanding the application,
`the lens specifications list follows. This list of specifications is not always
`complete or correct. A lens designer must make efforts to verify that the
`specifications list is as complete and correct as possible. The lens specifications
`may involve first-order, packaging, image quality, environmental, and lens
`fabrication constraints and requirements. Once the specifications are under(cid:173)
`stood, the lens designer may start a design from first principles, and by adding
`complexity to simple lenses. A first-order lens layout can help to visualize a
`given lens and determine, for example, lens size, element optical power, and
`type of lens configuration. From the first-order layout, considerations are made
`about how the aberrations could be corrected. Then a lens design program is
`used to model and optimize the lens system, and to find alternative lens
`solutions for comparison. A lens analysis is also made to determine tolerances
`that a lens manufacturer would need to make the lens elements. A lens design
`can also start from existing lenses in the patent literature. A lens designer
`should have effective communication with the opto-mechanical engineer and
`lens manufacturer to make sure that the designed lens can be mounted in a
`barrel, fabricated, and assembled. Lens drawings are then drafted. Some
`optical engineers may not actually design lenses, but would analyze, debug,
`adjust, and combine existing lens systems. A critical design review is often
`
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`4
`
`Introduction
`
`held to approve, or disapprove, a lens for fabrication. The overall process of
`lens design is also of exercising design creativity, and this in part is what
`makes lens design an exciting field.
`
`Further Reading
`
`Bentley, Julie L., Olson, Craig, Youngworth, Richard N. "In the era of global optimiza(cid:173)
`tion, the understanding of aberrations remains the key to designing superior optical
`systems," Proceedings of SPIE 7849, Optical Design and Testing IV, 78490C
`(2010); doi: 10.1117/12.871720.
`Kidger, M. J. "The importance of aberration theory in understanding
`Proceedings of SPIE, 3190 (1997), 26-33.
`Sasian, J. "Trends in teaching lens design," Proceedings of SPIE, 4588 (2001), 56-58.
`Sasian, Jose. "From the landscape lens to the planar lens: a reflection on teaching lens
`design," Proceedings of SPIE 5865, Tribute to Warren Smith: A Legacy in Lens
`Design and Optical Engineering, 586501 (2005); doi: 10.1117/12.624566.
`Shannon, Robert R. "Teaching of lens design," Proceedings of SPIE 1603, Education in
`Optics (1992); https://doi.org/10.1117 /12.57848.
`
`lens design,"
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`l
`
`1,
`
`2
`Classical Imaging, First-Order Imaging, and
`Imaging Aberrations
`
`This chapter provides a brief overview of essential imaging concepts used in
`lens design. Whether classical imaging, which is congruent with first-order
`optics, is required in a lens system, or any other type of imaging, depends on
`system application. Therefore, a clear understanding of what imaging is and of
`departures from such imaging, called aberrations, is essential for a lens design
`practice.
`
`2.1 Classical Imaging
`
`The main goal in lens design is the design of imaging lenses where images,
`particularly sharp, are formed. Then it is important to discuss the concept of an
`image. Depending on application, different imaging concepts can be devised.
`However, classical imaging, where the image is a scaled copy of the object, is
`often required for a lens system. The underlying mechanism for classical
`imaging is central projection. Object points are projected into image points
`on an image plane, by the line defined by an object point on the object plane
`and a central projection point pair as shown in Figure 2.1. The projection point
`pair is the center of perspective and in a lens system, which we assume to have
`axial symmetry, is represented by a nodal point in object space and its
`conjugate point, the nodal point in image space. The main attributes of a
`classical image are its location and its size. The Newtonian or Gaussian
`imaging equations shown in Table 2.1 permit calculating these attributes and
`represent central projection imaging.
`Ideal imaging as defined by central projection is often a designing goal. For
`an object at infinity that subtends a semi-field of view, 0, the image height, Y;,
`measured from the optical axis, is related to the focal length, f, by the
`mapping, Y; = f- tan ( 0). However, according to application, there are other
`
`5
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`6
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`Classical Imaging, 1st-Order Imaging & Imaging Aberrations
`
`Table 2.1 Imaging equations
`
`Newtonian equations
`j= -,!;
`i.-
`-m
`f' -u =ff'
`The object and image distances z and z'
`are measured, respectively, from the front
`and rear focal points. f and J' are the front
`and rear focal lengths.
`
`Gaussian equations
`
`m
`
`f_ +t = 1
`z'
`z
`I-1-l
`f-
`7=1-m
`The object and image distances z and z'
`are measured, respectively, from the front
`and rear principal points. The transverse
`magnification is m.
`
`Figure 2.1 Central projection imaging where the object on the left is imaged on
`the right. In this case the projection points coincide in space.
`possible mappings such as the equidistant mapping, Y; = f-0, or the ortho(cid:173)
`graphic mapping, Y; = f· sin (0). We are assuming that the object and image
`lay on planes perpendicular to the optical axis of the optical system. There are
`some applications that require the image to lay on a curved surface, and then
`the concept of classical imaging no longer applies.
`The point is that lenses are designed to produce images which require a lens
`designer to be clear about what imaging means. Imaging is and will continue to
`be an important subject which substantially impacts lens design. What imaging
`is depends on system application.
`
`2.2 First-Order Optics
`
`The concept of first-order imaging arises from a first-order approximation to
`the path of a real ray. A real ray in homogenous media travels in straight lines,
`
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`2.2 First-Order Optics
`
`7
`
`refracts according to Snell's law, n' sin (I') = n sin (I), and its tracing con(cid:173)
`siders the actual shape of the refracting surface. A first-order ray refracts
`according to a first-order approximation to Snell's law, n'i' = ni, and treats
`the optical surfaces as planar, but with refracting power, rjJ. To trace a first(cid:173)
`order ray, the refraction and transfer equations are used:
`n'u' = nu -y¢
`y' =y+u't,
`
`(2.1)
`
`(2.2)
`
`where u and u' are the slopes of the ray before and after refraction, y is the ray
`height at the surface which is assumed planar but with optical power¢, n is the
`index of refraction, and t is the distance to the next surface.
`In lens design we are concerned with first-order imaging, as obtained by tracing
`first-order rays, because it is equivalent to central projection imaging. In addition,
`first-order imaging establishes a model for a lens system where the cardinal
`points - these are the focal points, the nodal points, and the principal points -
`have specific ray properties and serve as useful references. Many calculations in
`lens design are done by tracing first-order rays and, therefore, an optical designer
`must be familiar with first-order optics. An example of a calculation in lens design
`software is what is known as a "solve" in which the program automatically sets the
`distance t from the last surface to the ideal image plane using t = -y / u'.
`The space where the object resides is called the object space and is infinite in
`extent. Similarly, the space where the image resides is called the image space
`and is infinite in extent. An important structure in a lens system is the aperture
`stop. The aperture stop is assumed to be circular, to lay on a plane perpendicu(cid:173)
`lar to the optical axis, and it solely limits the amount of light for the on-axis
`beam. The aperture stop helps to well define a lens system; this is, light beams
`for every field point become well defined after they pass through the aperture
`stop. The image of the aperture stop in object space is defined as the entrance
`pupil, and the image of the aperture stop in image space is defined as the exit
`pupil. The pupils and the stop are optically conjugated, meaning that their
`locations and sizes satisfy the Newtonian or Gaussian equations that are
`summarized in Table 2.1. Another aperture that contributes to well define a
`lens system is the field stop. The field stop limits the field of view of a lens
`system, and ideally it is located at an image plane.
`Rays that travel in a plane that contains the lens system axis of rotational
`symmetry are called meridional rays. Rays that do not travel in a meridional plane
`are called skew rays. Two important first-order rays are the marginal and chief
`rays. By definition, the marginal ray is a meridional ray that originates at the on(cid:173)
`axis object point and passes through the edge of the aperture stop. The chief ray is
`a meridional ray that originates at the edge of the field of view and passes through
`
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`Exhibit 2027
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`8
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`Classical Imaging, 1st-Order Imaging & Imaging Aberrations
`
`Table 2.2 First-order concepts
`
`Optical axis
`
`Object space
`
`Image space
`
`Aperture stop
`
`f
`J'
`Optical power or Refractive
`power(¢)
`
`Effective focal length (EFL)
`Fl#, F-number
`
`Lagrange invariant ()!{)
`
`Afocal
`Telecentricity in object space
`
`Telecentricity in image space
`
`Transverse magnification (m)
`
`The axis about which an optical system has rotational
`symmetry.
`The space where the object resides, which is assumed
`infinite in extent.
`The space where the image resides, which is assumed
`infinite in extent.
`The aperture that solely limits the amount of light for
`the axial light beam.
`Front focal length.
`Rear focal length.
`¢ = - J = f,; n is the index of refraction in object
`space, and n' is the index in image space. The unit of
`power is the diopter or I/meter.
`The inverse of the optical power.
`The effective focal length divided by the diameter of
`the entrance pupil. F / # = ~FL
`Y,
`It relates to the optical throughput or capacity of
`an optical system to transfer optical
`power. )K = nuy - nuy
`The focal lengths are not defined.
`The image of the aperture stop in object space is at
`infinity. Equivalently, the chief ray in object space is
`parallel to the optical axis.
`The image of the aperture stop in image space is at
`infinity. Equivalently, the chief ray in image space is
`parallel to the optical axis.
`The first-order ratio of the image size to the object
`size.
`
`Object plane
`
`Image plane
`
`Aperture stop ----+-
`
`Marginal ray
`
`i.,_,,,,,,-,-.;::---
`
`Field
`stop
`
`Figure 2.2 The marginal and chief rays (highlighted in bold) in relation to the
`aperture stop, the object and image planes, the field stop, and an ideal lens.
`
`the center of the aperture stop. The trace of these two rays permits obtaining useful
`information about the imaging of an optical system. Figure 2.2 shows an object
`plane, an aperture stop, a lens, an image plane, and two sets of rays defining two
`light beams for the on-axis object point and for an off-axis point. In particular,
`Figure 2.2 illustrates the marginal and chief rays using bold rays. Table 2.2
`provides a glossary of first-order concepts, and Table 2.3 provides a
`is defined by
`summary of first-order quantities. The Lagrange invariant, )K,
`
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`Exhibit 2027
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`2.2 First-Order Optics
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`9
`
`Table 2.3 Marginal and chief first-order rays' related quantities
`
`Item
`
`Marginal ray
`
`Object/pupil distance
`Image/pupil distance
`Ray slope of incidence
`Ray height at surface
`
`Ray slope
`Normal line slope
`Refraction invariant
`Surface radius
`Surface vertex curvature
`Thickness to next surface
`Surface optical power
`Lagrange invariant
`
`s
`s'
`i=u-a
`y
`Ye
`Ys
`u = -y/s
`a= -y/r = u - i
`A= ni = n(; - f)y
`r
`C
`t
`A.= n'-n
`'f
`,.
`JK = nuy - nuy = Ay - Ay
`
`Chief ray
`s
`s'
`i=u-a
`y
`Yo
`Y;
`u = -y/s
`a= -y/r = u-1
`A= n1 = n(; - ½)5i
`
`Quantities related to the chief ray carry a bar.
`Primed quantities refer to the image space and un-primed to the object space.
`
`Image
`Plane
`
`Figure 2.3 Model of an axially symmetric optical system showing the path of a
`first-order ray and the path of a real ray.
`
`)K = nuy - nuy, using the slope and height of the marginal and chief rays. Its
`value does not depend on the transverse plane where it is calculated. The amount
`of optical flux, or optical throughput, T = n2 )K 2
`, that can pass through an optical
`system is proportional to the square of the Lagrange invariant.
`Figure 2.3 provides a representation of an optical system where the object and
`image planes and the entrance and exit pupi