Table Of ContentFUNDAMENTALS
OF PRACTICAL
ABERRATION
THEORY
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FUNDAMENTALS
OF PRACTICAL
ABERRATION
THEORY
Fundamental Knowledge
and Technics for
Optical Designers
Yoshiya Matsui
Advisory Scientist, Canon Inc.
Kyoji Nariai
Professor National Astronomical Observatory
Vfe W o r ld S c i e n t i f ic
Singapore • New Jersey • London • Hong Kong
Published by
World Scientific Publishing Co. Pie. Lid.
POBox 128, Fairer Road, Singapore 9128
USA office: Suite IB, 1060 Main Street, River Edge, NI 07661
UK office: 73 Lynlon Mead, Totteridge, London N20 SDH
FUNDAMENTALS OF PRACTICAL ABERRATION THEORY
Copyright © 1993 by World Scientific Publishing Co. Pte. Lid.
All righls reserved. This book, or parts thereof, may not be reproduced in any form
orbyanymeans, electronic or mechanical, including photocopying, recording orany
information storage and retrieval system now known or to be invented, without
written permission from the Publisher.
ISBN 981-02-1349-2
Printed in Singapore.
Preface
The aberration theory is one that makes clear the relation between the structure of
an optical system and the aberration in imaging. It can be relied on by an engineer
who works to develop optical systems. However, its full meaning may not have
been well understood by many optical engineers because of its complex formulae.
In this book, we tried to write explanations that may not have been included in the
textbooks of engineering related to the aberration theory hitherto published. Thus,
optical engineers can use the aberration theory more effectively.
This book was first written as a textbook in Japanese to be used in a course of
engineering of the Japan Opto electro-Mechanics Association. With the permission
of the Japan Opto electro-Mechanics Association, we have re-written it in English
as a monograph for engineers supplementing the original contents extensively.
We have received help from many people for this publication. Our sincere
thanks are due to Professor Junpei Tsujiuchi of the Chiba University who intro
duced us to the publishing companies, Professor K. K. Phua of World Scientific
Publishing Company who accepted the book voluntarily for publication, and Pro
fessor R. H. Koch of the University of Pennsylvania who checked the English text
extensively
Yoshiya Matsui, Kyoji Nariai
January 199S
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Contents
1 Introduction X
1.1 The role of aberration theory 1
1.2 Paraxial theory as the basis for aberration theory . . .. 2
1.2.1 The two roles of paraxial theory 2
1.2.2 Recursion formulae for meridional rays and recursion formulae
for paraxial rays 2
1.2.3 Paraxial imaging by a single thin lens 7
1.2.4 Paraxial imaging by an actual optical system . . 9
1.3 Application of paraxial theory to lens design 17
2 Derivation of aberration theory 23
2.1 Characteristic function of Hamilton . . .. 23
2.2 Outline of the Herzberger theory . 26
2.2.1 Expansion equation for aberration . 26
2.2.2 Transformation formulae for aberration coefficients 33
3 Practical aberration theory and its formulae 49
3.1 Transformation of the Herzberger aberration theory into practical form 49
3.1.1 Transformation of coordinates . . . . .. 49
3.1.2 Relations between the expansion equation of aberrations of an
individual surface and that of the entire optical system . . 55
3.1.3 The scale units and the Helmholtz- Lagrange invariant . . .. 56
3.1.4 Formulae for aberration coefficients for each surface 58
3.1.5 Chromatic aberration . . . . 62
3.2 Normalization of aberration coefficients . . .. 66
3.2.1 Meaning of normalization 66
3.2.2 Normalization for an imaging optical system (1) . .. 66
3.2.3 Normalization for an imaging optical system (2) 74
3.2.4 Normalization for an optical system used visually 83
3.2.5 Aberration coefficients for an imaging optical system and actual
aberrations . . . .. 90
3.2.6 Transformation formulae of aberration coefficients . . .. 94
3.3 Aberration coefficients for a thin lens system 99
3.3.1 Meaning of a thin lens approximation 99
3.3.2 Formulae of aberration coefficients of a thin lens system . . .. 99
vii
VI11
3.3.3 Implication of the theory of thin lens system 106
3.4 Intrinsic coefficients and the characteristic matrix . . .. . . . 107
3.4.1 Intrinsic coefficients and characteristic matrix 107
3.4.2 Structure of the characteristic matrix 108
3.4.3 Intrinsic coefficients and the characteristic matrix of a thin lens
system . . .. 116
3.4.4 Numerical examples 118
4 Case studies 127
4.1 Determination of the initial configuration of an optical system and the
application of aberration theory . 127
4.2 Determination of a configuration for a telephoto-type lens . . . .. 127
4.2.1 Procedures for determination of the configuration . . .. . 128
4.2.2 Numerical examples . 131
4.3 Determination of configuration of a triplet . .. 135
4.3.1 Procedures for the determination of shapes 135
4.3.2 Numerical examples 139
4.3.3 Triplets and their variations . 141
4.4 Design of a catadioptric system . .. 144
4.4.1 Optical system with reflecting surfaces 144
4.4.2 Distribution of power and the paraxial ray-traced values 144
4.4.3 Study of correction for aberrations 146
A Helmholtz-Lagrange invariant (when two imagings exist through the
same optical system) 151
B Derivation of formulae for calculating aberration coefficients for in
dividual surfaces in an optical system 155
B.l Aberration coefficients for special conditions as the starting point . . 155
B.2 Transfer to flat object and image surfaces . . . .. 158
B.3 Transformation related to shift of object position . . .. 161
B.4 Transformation related to shift of position of pupil . .. 164
B.5 Transformation by introduction of reduced coordinates . . 168
B.6 Another form for calculating formula of distortion . . . . 169
C Derivation of formulae for calculating chromatic aberration coeffi
cients for individual surfaces in an optical system 173
D Initial values of paraxial rays for calculating aberration coefficientsl77
D.l Normalization for imaging optical system (2) when sj —> oo 177
D.2 Normalization for visual observation and with (i —* co 179
Index 181
Chapter 1
Introduction
1.1 The role of aberration theory
Generally speaking, a design engineer cannot reach his goal in a straightforward way
and a trial and error effort is appropriate. In order to make steady progress, the
designer must be able to lorm an appropriate judgment about each problem that he
encounters. If a reliable theory exists, it will help him to make that judgment. For a
lens designer, the aberration theory is just such a reliable theory.
It is well known that the technique of lens design has been systematized on the
basis of ray tracing. From the point of view of desired accuracy, it may seem that
aberration theory is not a sufficient one. In general, too many people rely on computer
design and do not appreciate the value of an approximating theory. As a consequence,
on many occasions they waste time and labor trying to get an unattainable result,
which limitation they could easily have recognized had they understood the physics
underlying the phenomenon. It is true that aberration theory is only an approxi
mating one. However, that very fact makes it possible for an engineer to grasp the
fundamental relations between the structure of an optical system and its imaging
characteristics.
I believe that many engineers work on tasks characterized by challenges to break
through various difficult limitations. In carrying out work of this type, can an engineer
be confident of his results if he has no theory to rely on? In general, the applicability of
a theory or a technique becomes narrower as the accuracy of the theory or technique
increases. An engineer is, in fact, likely to realize the full accuracy of computer
calculation only if he has a deep understanding of optics based on aberration theory
In order to use aberration theory effectively, it is indispensable that an engineer
know its characteristics. The theory gives the analytical relationship between the
structure of an optical system and the aberrated image; thus, an engineer is able to
judge what is possible and what is impossible. It is important, therefore, that an
engineer, using aberration theory as the guide-line of his work of designing lenses,
does so with sufficient knowledge of what the formulae really mean although he can
use them only as convenient tools.
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