Application of the algebraic aberration equations to optical design

S550

APPLICATION OF THE ALGEBRAIC ABERRATION EQUATIONS TO OPTICAL DESIGN By

I.

C. Gardner

ABSTRACT

The phase of optical engineering which deals with lens design and the measurement of the aberrations of a lens system is worthy of a more comprehensive treatment than is available in English. Even in our technical schools and universities seldom that one finds a course dealing adequately with optical imagery which goes beyond the first order or Gaussian equations. In geometrical optics there In the first, one has what may are two applications of the aberration equations. be termed the direct problem. The specifications of the lens system are given and the aberrations are to be determined. In the inverse problem one is to determine the specifications of the lens system which will have the desired aberration characteristics. Although the second problem is much the more important the

it is

literature dealing with it is relatively meager.

factory, either in

German

There

is

no

treatise entirely satis-

or English, which gives the third order aberrations in

a convenient form for the inverse solution Math a simple and consistent notation

and sign convention.

And yet the

is the central probthe aberrations, one must have had experience in computing and in measuring them. The need of a treatment of the aberrations covering a different field than that of the existing treatises has often been felt and has been recently well expressed by Mr. Emley ^ at a meeting of the Optical Society of London, in which he says, ''It is when the student attempts to get hold of expressions from which he can calculate these quantities (aberrations), however, that his difficulties begin. There is no standard English work leading him from his elementary geometrical and physical optics to problems of this kind. It has always seemed to me that there is a distinct gap in the subject which, looked There are many advanced works on at from any point of view, should be filled. instrument design and other specialized problems, but the ordinary type of student I have in mind can not follow them, partly because of the gap alluded to and partly because of the variety of symbols and constants used." It is hoped that the following treatment will partially fill this gap and at the same time serve as a reference book for the solution of problems in lens design. The laboratory measurement of the aberrations is not dealt with. The aberration equations are presented in such a manner as to permit their direct application to problems of lens design. Only the assumptions of geometrical optics in the restricted sense are applied, and all references to diffraction effects, resolving power, and kindred subjects are omitted. The derivations of the equations are omitted except for a brief reference in Appendix 2, and the physical interpretation and manner of application of the equations to problems is stressed. It is the intention thus to produce a grammar or handbook for reference which will contain the information necessary for the algebraic third order design of optical sj^stems composed of thin lenses.

lem

1

control of the aberrations of a projected sj^stem

of optical design.

To understand

H. H. Emley, Trans. Opt. Soc, 37,

p. 233; 1925-26.

73

74

Scientific

Papers of

the

Bureau of Standards

[

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For the third order equations there are two systems differing in the choice of parameters. The one termed the continental system (see p. 79) will be found treated, among others, by Schwarzschild,^ von Rohr,^ and Southall.^ The second system, termed the Taylor system, was originated by Coddington ^ and much extended by Taylor.^ Modifications have been introduced into the Taylor-Coddington equations which simplify them in appearance and which enable the two systems of equations to be carried along in parallel throughout the treatment. The reader can, accordingly, make his own choice as to which group of equations is to be used. The notation employed is substantially that of Schwarzschild and von Rohr, and its choice is justified by the wealth of material already published in which Some slight changes have been made to avoid conthis notation is employed. fusion when the notation shall be extended to projected publications dealing with the stage of optical design in which trigonometric ray tracing is employed. To avoid the attachment of two significations to the same symbol, it has been necessary to depart from the notation of Taylor in many instances. The sign convention adopted is the one believed to be most nearly universal in treatises

on apphed

The

optics.

order equations of imagery are dealt with only in such detail as is necessary to provide the basis of notation and sign convention to be used in the third order equations.^ The equations and a general description of each third order aberration for a single lens are given, after which the equations are extended to a system of thin lenses. A general discussion of the method of controlling the aberrations of an optical system follows with two numerical examples. In the first example the aberrations of a Ramsden eyepiece are determined. In the second example a Kellner eyepiece is designed to have given aberration characteristics. This last illustration is worked out in considerable detail in order to illustrate fully the different applications of the third-order equations. Following this there are given the equations for the third-order aberrations of thick plates or reflecting prisms and the method of their application to the design of optical systems which contain thin lenses and reflecting prisms or plane parallel first

plates.

There are four appendixes, which are as follows: Appendix 1. The notation and sign conventions used, with equivalent symbols

— — as given by Schwarzschild, with the method the thin lens equations. plates giving dimensional drawings —A the principal

by Taylor. Appendix 2. The Seidel equations

as applied

of derivation of

Appendix

3.

series of

of

prisms used in optical systems. These plates were prepared by Otto Kaspereit, of Frankford Arsenal, and were originally included in Elementary Optics and Applications to Fire Control Instruments, revision of January, 1924, Ordnance Department Publication No. 1065. Thanks are due the Ordnance Department, United States Army, for its courtesy in permitting the inclusion of these drawings. * Schwarzschild, K., Untersuchungen zur geometrischen Optik. I. Einleitung in die Fehlertbeorieoptischer Instrumente auf Grand des Eikonalbegrifis. Abh. der Koniglichen Geeecflschaft der Wissenschaften zu Qottingen. Math-Phys. Kl. Neue Folge, 4, No. 1; 1905. 3 von Rohr, M., DieBilderzeugunginoptiscbenlnstrumenten. Julius Springer, Berlin; 1904. English translation by Kanthack. H. M. Stationery Shop, London; 1920. < Southall, J. P. C, Principles and Methods of Geometrical Optics. The Macmillan Co., New York;

1910.

A Treatise on the Reflexion and Refraction of Light. Simkin & Marshall, London; A System of AppUed Optics. Macmillan & Co. (Ltd.), London; 1906.

«

Coddington, H.,

8

Taylor, H. D.,

1829.

7 If an aberration is measured by the angle subtended by the aberration disk at the pupil point all the monochromatic aberrations considered are of the third order. In a symmetrical system the aberrations of even order vanish. Hence, after the first-order equations the third-order equations offer the next approxi-

mation.

.

Opticol DesigU

Gardner]

75

—

Appendix 4. A table giving values of the functions of n used in the thirdorder equations for values of n from 1.4 to 1,75. This table has been computed by H. U. Graham and will be found very useful in connection with the thirdorder equations. In conclusion, the writer wishes to express his gratitude to C. D. Hillman, of Keuffel & Esser Co., and to Mr. Kaspereit for the care which they have taken in reading the manuscript and for their many constructive suggestions which have been adopted. Mr. Kaspereit has not only read the manuscript but has checked practically all the computations and compiled a list of errata which were used in revising the numerical parts.

CONTENTS I.

Page 77

Introduction 1.

2 3. 4.

5.

employed

imagery imagery for a single lens Variation of the index of glass with color Shape of the lens Convergence of the incident bundle of rays Convergence of the chief rays (a) Entrance and exit pupils and iris

II Parameters

in the third-order equations of

First order equations of

(6)

Pupil points and chief rays

Entrance and exit windows (d) Parameters giving the convergence of the chief rays. 6. Auxiliary ray by which height of incidence of marginal ray is determined 7. AuxiUiary ray by which angular distance of object from center of field is measured 8. Summary of different parameters Chromatic aberrations of a single thin lens 1. Longitudinal chromatic aberration of a single lens 2. Lateral chromatic aberration of a single lens 3. Relative importance of longitudinal and lateral chromatic (c)

III.

aberration IV. Monochromatic aberrations of a single thin lens 1. Spherical aberration of a single thin lens (a) General characteristics of spherical aberration (6) Shape of a single thin lens for minimum spherical aberration (c) Values of tt for which a component free from aberration can be designed (d) Aplanatic points 2. Coma of a single thin lens (a) Formation of coma in the absence of spherical aberration (6) Flare produced by normal coma and spherical aberration

Thin lens of minimum coma Curvature of image and astigmatism of a single thin lens (a) Four image surfaces (6) Distinction between curvature and astigmatism (c) Equations of curvature and astigmatism (c)

3.

(e)

Petzval curvature Normal curvature

(/)

Curvature arising from eccentric refraction plus

(g)

spherical aberration or coma Control of the curvature of a single lens

(d)

78 79 81 82 84 84 84 85 86 86

88 89 89 89 90 91 93 93

94 95 96

97 99 100 100 103 104 105 108 108 109 110 110

110 111

76

Scientific

Papers qf

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—

IV. Monochromatic aberrations of a single thin lens Continued. 4. Distortion of a single thin lens (a) Conditions necessary for freedom from distortion for all positions of object plane (6) Equations of distortion (c)

Thin

lens of zero distortion

V. First order equations extended to a system of thin lenses 1. Step-by-step method of locating the image formed by a system of thin lenses 2.

A A A

system composed of two lenses system of more than two lenses (&) system of thin lenses in contact (c) (d) Telescopic system Characteristics of paraxial imagery (a)

(a)

Lateral magnification

(6)

Longitudinal magnification

Angular magnification (d) Magnifications of an optical system (e) Definition of focal length in terms of tan a. (/) Magnification of a telescopic system Application of first order equations to determine the position of stops and the field of view (c)

4.

;

(a)

Method and

of identifying the entrance

and

iris

Entrance and exit windows and field diaphragm. _ VI. Aberration equations extended to a system of thin lenses VII. Application of the third order thin lens equations to optical design. _ 1. Direct determination of the aberrations of a Ramsden eyepiece by the third order equations

order equations (6) Determination of pupils and windows (c) Evaluation of the parameters (cO Application of the third order equations Design of a lens system which has given aberration (a)

Application of

first

characteristics (a)

Given conditions to be

(&)

Determination of the focal lengths and spacing of

satisfied

the components

Determination of the shapes of the components (d) Control exercised by a lens for which g is small (e) Control exercised by a lens for which }i is small (/) Petzval condition and chromatic aberration (gr) Symmetrical lens system Hemisymmetrical lens system {Jfi) (i) Aberration equations not always equated to zero __ (j) Choice of glass as an independent variable (fc) Choice of roots Application of the third order equations to the design of a (c)

3.

Page

112 113 114 116 117

117

Kellner eyepiece (a) (6) (c)

118 118 119 120 120 121 122 122 122 123 123 124 125

exit pupil

(6)

2.

n

Equivalent focal length and principal points of a system of lenses

3.

loi.

Determination of the 8^8, x's, g's, and h's Condition for freedom from lateral color Determination of the constants of the third order equations

125 127

128 131

131 132 133 134 134

136 136

137 137 138 139 140 142 143 144 144 145 145 146 147 147

Optical Design

Gardner]

77

VII. Application of the third order thin lens equations to optical design Continued. 3. Application of the third order equations to the design of a Kellner eyepiece Continued. (d) Equations for the determination of iC AXIS BUT DISPIACES THE SAME. TO THE AMOUNT OF "B". HEHSOLTerS^MS USETHll SYSTEM IN THEIR BINOCUIAR FIELD -eLASSBS,

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