zotero/storage/G5UU8FIL/.zotero-ft-cache

THE THEORY OF OPTICS
PAUL DRUDE

Students, teachers, and researchers in physical optics, astrophysics, optometry, film optics, and related fields will weli:ome this unabridged reprinting of one of the finest fundamental texts in physical optics. This classic study continues to be unequalled for wide, thorough coverage and complete mathematical treatment of basic ideas.
One of the first applications of Maxwell's electromagnetic theory as developed by Hertz to the problem of light, it offers a large amount of valuable material unavailable elsewhere in one volume. This treatise includes perhaps the fullest treatment of the application of thermodynamics to optics. Though one of the most powerful and significant approaches, no other text points out so fully the important practical and theoretical consequences which may be deduced simply and directly from elementary principles by these means. Drude investigates this in such areas as temperature radiation and luminescence, Kirchoff's law of emission and absorption, the sine law in the formation of images of surface elements, the effect of change of temperature on the spectrum of a black body, and the distribution of energy in the spectrum of such a body. In addition, there are particularly valuable sections on absorbing media, crystal optics, and interference.
Partial contents, Fundamental Laws, Geometrical Theory of Optical Images, Physical Conditions for Image Formation, Apertures and Effects Depending Upon Them, Optical Instruments, Velocity of Light, Interference, Huygens' Principle, Diffraction, Polarization, Theory of Light, Transparent Isotropic Media, Optical Properties of Transparent Crystals, Absorbing Media, Dispersion, Optically Active Substances, Magnetically Active Substances, Bodies in Motion, Radiation, Application of the Second Law of Thermodynamics to Pure Temperature Radiation, Incandescent Vapors and Gases.
"Remarkably original and consecutive presentation of the subject of Optics," A. A. Michelson, in his introduction.

Unabridged, unaltered republication of last English translation. Translated by C. Riborg

Mann and Robert A. Millikan. Introduction by A. A. Michelson. Index. 110 illustrations.

+ xxi 546pp. 5% x 8.

60532-9 Paperbound

THE
THEORY OF OPTICS
BY
PAUL DRUDE
TRANSLATED FROM THE GERMAN BY
C. RIBORG MANN AND ROBERT A. MILLIKAN
DOVER PUBLICATIONS, INC. NEW YORK

This new Dover edition first published in 1959 is an unabridged and unaltered republication of the last English translation.
Standard Boole Number: 486-60532-9 Library of Congress Catalog Card Number: 59-65313
Manufactured in the United States of America Dover Publications, Inc. 180 Varick Street New York, N. Y. 10014

PREFACE TO THE ENGLISH TRANSLATION
THERE does not exist to-day in the English language a general advanced text upon Optics which embodies the important advances in both theory and experiment which have been made within the last decade.
Preston's " Theory of Light " is at present the only general text upon Optics in English. Satisfactory as this work is for the purposes of the general student, it approaches the subject from the historical standpoint and contains no fundamental development of some of the important theories which are fast becoming the basis of modern optics. Thus it touches but slightly upon the theory of optical instruments-a branch of optics which has received at the hands of Abbe and his followers a most extensive and beautiful development ; it gives a most meagre presentation of the electromagnetic theorya theory which has recently been brought into particular prominence by the work of Lorentz, Zeeman, and others ; and it contains no discussion whatever of the application of the laws of thermodynamics to the study of radiation.
The book by Heath, the last edition of which appeared in 1895, well supplies the lack in the field of Geometrical Optics, and Basset's " Treatise on Physical Optics" ( I 892) is a valuable and advanced presentation of many aspects of the wave theory. But no complete development of the electromagnetic theory in all its bearings, and no comprehensive discussion of
iii

iv PREFACE TO THE ENGLISH TRANSLATION

the relation between the laws of radiation and the principles of

thermQdynamics, have yet been attempted in any general text

in English.

,

It is in precisely these two respects that the" Lehrbuch der

Optik" by Professor Paul Drude (Leipzig, 1900) particularly

excels. Therefore in making this book, written by one who

has contributed so largely to the progress which has been

made in Optics within the last ten years, accessible to the

English-speaking public, the translators have rendered a very

important service to English and American students of

Physics.

No one who desires to gain an insight into the most mod-

ern aspects of optical research can afford to be unfamiliar with

this remarkably original and consecutive presentation of the

subject of Optics.

A. A. MICHELSON.

UNIVBB.SITY OF CHICAGO,
February, 1902.

AUTHOR'S PREFACE
THE purpose of the present book is to introduce the reader who is already familiar with the fundamental concepts of the differential and integral calculus into the domain of optics in such a way that he may be able both to understand the aims and results of the most recent investigation and, in addition, to follow the original works in detail.
The book was written at the request of the pubiisher-a request to which I gladly responded, not only because I shared his view that a modern text embracing the entire domain was wanting, but also because I hoped to obtain for myself some new ideas from the deeper insight into the subject which writing in book form necessitates. In the second and third sections of the Physical Optics I have advanced some new theories. In the rest of the book I have merely endeavored to present in the simplest possible way results already published.
Since I had a text-book in mind rather than a compendium, I have avoided the citation of such references as bear only upon the historical development of optics. The few references which I have included are merely intended to serve the reader for more complete information upon those points which can find only brief presentation in the text, especially in the case of the more recent investigations which have not yet found place in the text-books.
V

vi

AUTHOR'S PREFACE

In order to keep in touch with experiment and attain the simplest possible presentation of the subject I have chosen a synthetic method. The simplest experiments lead into the domain of geometrical optics, in which but few assumptions need to be made as to the nature of light. Hence I have begun with geometrical optics, following closely the excellent treatment given by Czapski in " Winkelmann's Handbuch der Physik " and by Lommer in the ninth edition of the " MiillerPouillet " text.
The first section of the Physical Optics, which follows the Geometrical, treats of those general properties of light from which the conclusion is drawn that light consists in a periodic change of condition which is propagated with finite velocity in the form of transverse waves. In this section I have included, as an important advance upon most previous texts, Sommer' feld's rigorous solution of the simplest case of diffraction, Cornu's geometric representation of Fresnel's integrals, and, on the experimental side, Michelson's echelon spectroscope.
In the second section, for the sake of the treatment of the optical properties of different bodies, an extension of the hypotheses as to the nature of light became for the first time necessary. In accordance with the purpose of the book I have merely mentioned the mechanical theories of light ; but the electromagnetic theory, which permits the simplest and most consistent treatment of optical relations, I have presented in the following form :
Let X, Y, Z, and a, /J, y represent respectively the com-
ponents of the electric and magnetic forces (the first measured in electrostatic units); also letJ~ ,J~ ,J~, and s,., s,, s. represent the components of the electric and magnetic current densities,
I
i.e. - times the number of electric or magnetic lines of force 4,r
which pass in unit time through a unit surface at rest with reference to the ether ; then, if c represent the ratio of the

AUTHOR:S PREFACE

vii

electromagnetic to the electrostatic unit, the following fundamental eqtttiti'ons always hold :

= 41rs.,. oY oZ
-c- oz - oy' etc.

The number of lines of force is defined in the usual way. The particular optical properties of bodies first make their appearance in the equations which connect the electric and magnetic current densities with the electric and magnetic forces. Let these equations be called the sttbstance equations in order to distinguish them from the above fundamental equations. Since these substance equations are developed for non-homogeneous bodies, i.e. for bodies whose properties vary from point to point, and since the fundamental equations hold in all cases, both the differential equations of the electric and magnetic forces and the equations of condition which must be fulfilled at the surface of a body are immediately obtained.
In the process of setting up " substance and fundamental equations " I have again proceeded synthetically in that I have deduced them from the simplest electric and magnetic experiments. Since the book is to treat mainly of optics this process can here be but briefly sketched. For a more complete development the reader is referred to my book "Physik des Aethers auf elektromagnetische Grundlage" (Enke, 1894).
In this way however, no explanation of the phenomena of dispersion is obtained because pure electromagnetic experiments lead to conclusions in what may be called the domain of macrophysical properties only. For the explanation of optical dispersion a hypothesis as to the mi'crophysical properties of bodies must be made. As such I have made use of the ion-hypothesis introduced by Helmholtz because it seemed to me the simplest, most intelligible, and most consistent way of presenting not only dispersion, absorption, and rotary

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AUTHOR'S PREFACE

polarization, but also magneto-optical phenomena and the optical properties of bodies in motion. These two last-named subjects I have thought it especially necessary to consider because the first has acquired new interest from Zeeman's discovery, and the second has received at the hands of H. A. Lorentz a development as comprehensive as it is elegant. This theory of Lorentz I have attempted to simplify by the elimination of all quantities which are not necessary to optics. With respect to magneto-optical phenomena I have pointed out that it is, in general, impossible to explain them by the mere supposition that ions set in motion in a magnetic field are subject to a deflecting force, but that in the case of the strongly magnetic metals the ions must be in such a continuous motion as to produce Ampere's molecular currents. This supposition also disposes at once of the hitherto unanswered question as to why the permeability of iron and, in fact, of all other substances must be assumed equal to that of the free ether for those vibrations which produce light.
The application of the ion-hypothesis leads also to some new dispersion formulre for the natural and magnetic rotation of the plane of polarization, formulre which are experimentally verified. Furthermore, in the case of the metals, the ionhypothesis leads to dispersion formulre which make the continuity of the optical and electrical properties of the metals depend essentially upon the inertia of the ions, and which have also been experimentally verified within the narrow limits thus far accessible to observation.
The third section of the book is concerned with the relation of optics to thermodynamics and (in the third chapter) to the kinetic theory of gases. The pioneer theoretical work in these subjects was done by Kirchhoff, Clausius, Boltzmann, and W. Wien, and the many fruitful experimental investigations in radiation which have been more recently undertaken show clearly that theory and experiment reach most perfect development through their mutual support.

AUTHOR'S PREFACE

ix

Imbued with this conviction, I have written this book in the endeavor to make the theory accessible to that wider circle of readers who have not the time to undertake the study of the original works. I can make no claim to such completeness as is aimed at in Mascart's excellent treatise, or in Winkelmann's Handbuch. For the sake of brevity I have passed over many interesting and important fields of optical investigation. My purpose is attained if these pages strengthen the reader in the view that optics is not an old and worn-out branch of Physics, but that in it also there pulses a new life whose further nourishing must be inviting to every one.
Mr. F. Kiebitz has given me efficient assistance in the reading of the proof.
LEIPZIG, January, lg<>o.

INTRODUCTION
MANY optical phenomena, among them those which have found the most extensive practical application, take place in accordance with the following fundamental laws:
1. The law of the rectilinear propagation of light; 2. The law of the independence of the different portions of a beam of light; 3. The law of reflection; 4. The law of refraction. Since these four fundamental laws relate only to the geometrical determination of the propagation of light, conclusions concerning certain geometrical relations in optics may be reached by making them the starting-point of the analysis without taking account of other properties of light. Hence these fundamental laws constitute a sufficient foundation for so-called geometrical optics, and no especial hypothesis which enters more closely into the nature of light is needed to make the superstructure complete. In contrast with geometrical optics stands physical optics, which deals with other than the purely geometrical properties, and which enters more closely into the relation of the physical properties of different bodies to light phenomena. The best success in making a convenient classification of the great multitude of these phenomena has been attained by devising particular hypotheses as to the nature of light. From the standpoint of physical optics the four above-mentioned fundamental laws appear only as very close approxima-
x1

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INTRODUCTION

tions. However, it is possible to state within what limits the Jaws of geometrical optics are accurate, i.e. under what circumstances their consequences deviate from the actual facts.
This circumstance must be borne in mind if geometrical optics is to be treated as a field for real discipline in physics rather than one for the practice of pure mathematics. The truly complete theory of optical ins.truments can only be developed from the standpoint of physical optics; but since, as has been already remarked, the laws of geometrical optics furnish in most cases very close approximations to the actual facts, it seems justifiable to follow out the consequences of these laws even in such complicated cases as arise in the theory of optical instruments.

TABLE OF CONTENTS

PART !.-GEOMETRICAL OPTICS

CHAPTER I

THE FUNDAMENTAL LAWS

ART.

PAGB

1. Direct Experiment.................... , ..................... .

2. Law of the Extreme Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3. LawofMalus ................................................ 11

CHAPTER II
GEOMETRICAL THEORY OF OPTICAL IMAGES
1. The Concept of Optical Images .. . . .. . . .. . . . . .. .. . . .. . . . .. . . . 14 2. General Formulre for Imag<'s . . .. .. .. . . .. . . .. . . . .. . .. . . .. .. .. 15 3. Images Formed by Coaxial Surfaces.......................... 17 4. Construction of Conjugate Points ................ ,........... 24 5. Classification of the Different Kinds of Optical Systems......... 25 6. Telescopic Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 7. Combinations of Systems.................................... 28

CHAPTER III
PHYSICAL CONDITIONS FOR IMAGE FORMATION
r. Refraction at a Spherical Surface............................. 32 2. Reflection at a Spherical Surface...................... , . . . . . . . 36 3. Lenses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4. Thin Lenses.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5. Experimental Determination of Focal Length.. . . . . . . . . .. . . . . . . 44 6. Astigmatic Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7. Means of Widening the Limits of Image Formation............ 52 8. Spherical Aberration..................................... - . . . 54
xiii

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TABLE OF CONTENTS

ART.

PA.Gil

9. The Law of Sines...................... , .. , . , , .. , , , , . . . . . • •• . 58

10. Images of Large Surfaces by Narrow Beams................... 63

11. Chromatic Aberration of Dioptric Systems.................... 66

CHAPTER IV
APERTURES AND THE EFFECTS DEPENDING UPON THEM
1. Entrance- and Exit-Pupils...... . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. 73 2. Telecentric Systems.......................................... 7S 3. Field of View. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4. The Fundamental Laws of Photometry. . . . . . ........... _. . . . . 77 5. The Intensity of Radiation and the Intensity of Illumination of
Optical Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6. Subjective Brightness of Optical Images............ . . . . . . . . . . 86 7. The Brightness of Point Sources................. . . . .. . . . . . 90 8. The Effect of the Aperture upon the Resolving Power of Optical
Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
CHAPTER V
OPTICAL INSTRUMENTS
1. Photographic Systems....................................... 93 2. Simple Magnifying-glasses................................... 9S 3. The Microscope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4. The Astronomical Telescope... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5. The Opera Glass.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6. The Terrestrial Telescope.... . . . . . . . . . . . . . . . . . . . . . . . . . . ..... 112 7. The Zeiss Binocular ......................................... 112 8. The Reflecting Telescope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

PART IL-PHYSICAL OPTICS
SECTION I
GENERAL .PROPERTIES OF UGHT
CHAPTER I
THE VELOCITY OF LIGHT
I. Romer's Method .....................................•...... 114 2. Bradley's Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . 115

TABLE OF CONTENTS

xv

ART.

PAC.&

3. Fizeau's Method ..................... , ...................... 116

4. Foucault's Method ... _............. , .... , ................... 118

5. Dependence of the Velocity of Light upon the Medium and the

Color .................................................... 120

6. The Velocity of a Group of Waves............................ r:z1

CHAPTER II
INTERFERENCE OF LIGHT
1. General Considerations ..........................••.......... 124 2. Hypotheses as to the Nature of Light......................... 124 3. Fresnel's Mirrors....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4. Modifications of the Fresnel Mirrors........ , ................. 134 5. Newton's Rings and the Colors of Thin Plates................. 136 6. Achromatic Interference Bands..... . . . . . . . . . . . . . . . . . . . . . . . . . . 144 7. The Interferometer.......................................... 144 8. Interference with Large Difference of Path .................... 148 9. Stationary Waves............................................ 154 10. Photography in Natural Colors ............................... 156

CHAPTER III
HUYGENS' PRINCIPLE
1. Huygens' Principle as first Conceived......................... 159 2. Fresnel's Improvement of Huygens' Principle.................. 162 3. The Differential Equation of the Light Disturbance..... . . . . . . . 169 4. A Mathematical Theorem.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 5. Two General Equations ...................................... 174 6. Rigorous Formulation of Huygens' Principle .................. 179

CHAPTER IV
DIFFRACTION OF LIGHT
1. General Treatment of Diffraction Phenomena.................. 185 2. Fresnel's Diffraction Phenomena, ......... , . . . . . . . . . . . . . . . . . . 188 3. Fresnel's Integrals........................................... 188 4. Diffraction by a Straight Edge................................ 192 5. Diffraction through a Narrow Slit................ ,............. 198 6. Diffraction by a Narrow Screen............................... 201 ,. Rigorous Treatment of Diffraction by a Straight Edge.. . . . . . . . . 203

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TABLE OF CONTENTS

ART.

PA.GK

8. Fraunhofer's Diffraction Phenomena.......................... 213

9. Diffraction through a Rectangular Opening................... 214

10. Diffraction through a Rhomboid ............................. 217

11, Diffraction through a Slit .................................... 217

12. Diffraction Openings of any Form ............................ 219

13. Several Diffraction Openings of like Form and Orientation..... 219

14. Babinet's Theorem .......................................... 221

15. The Diffraction Grating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

16. The Concave Grating ........................................ 225

17. Focal Properties of a Plane Grating .......................... 227

18. Resolving Power of a Grating ................................ 227

19. Michelson's Echelon ......................................... 228

20. The Resolving Power of a Prism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

:z1. Limit of Resolution of a Telescope..... . . . . . . . . . . . . . . . . . . . . . . . 235

22. The Limit of Resolution of the Human Eye ................... 236

23. The Limit of Resolution of the Microscope.................... 236

CHAPTER V
POLARIZATION
1. Polarization by Double Refraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 2. The Nicol Prism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 3. Other Means of Producing Polarized Light.. . . . . . . . . . . . . . . . . . . 246 4. Interference of Polarized Light. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 247 5. Mathematical Discussion of Polarized Light. . . . . . . . . . . . . . . . . . . 247 6. Stationary Waves Produced by Obliquely Incident Polarized
Light .................................................... 251 7. Position of the Determinative Vector in Crystals .............. 252 8. Natural and Partially Polarized Light ........................ 253 9. Experimental Investigation of Elliptically Polarized Light...... 25s

SECTION II
OPTICAL PROPERTIES OF BODIES
CHAPTER I
THEORY OF LIGHT
1. Mechanical Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 ,. Electromagnetic Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26o 3. The Definition of the Electric and of the Magnetic Force. . . . . . . 262

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xvii

AllT.

l'AGS

4- Definition of the Electric Current in the Electrostatic and the

Electromagnetic Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

5. Definition of the Magnetic Current............................ 265

6. The Ether. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

7. Isotropic Dielectrics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

8. The Boundary Conditions .................................... 271

9. The Energy of the Electromagnetic Field. . . . . . . . . . . . . . . . . . . . . 272

10. The Rays of Light as the Lines of Energy Flow................ 273

CHAPTER II
TRANSPARENT ISOTROPIC MEDIA
1. The Velocity of Light........................................ 274 2. The Transverse Nature of Plane Waves........................ 278 3. Reflection and Refraction at the Boundary between two Trans-
parent Isotropic Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 4- Perpendicular Incidence; Stationary Waves................... 284
5. Polarization of Natural Light by Passage through a Pile of
Plates..................................................... 285 6. Experimental Verification of the Theory... . . . . . . . . . . . . . . . . . . . 286 7. Elliptic Polarization of the Reflected Light and the Surface or
Transition Layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 8. Total Reflection. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 295 9. Penetration of the Light into the Second Medium in the Case of
Total Reflection .......................................... 299 10. Application of Total Reflection to the Determination of Index
of Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 11. The Intensity of Light in Newton's Rings ..................... 302 12. Non-homogeneous Media; Curved Rays . . . . . . . . . . . . . . . . . . . . . . 306

CHAPTER III
OPTICAL PROPERTIES OF TRANSPARENT CRYSTALS
r. Differential Equations and Boundary Conditions............... 3o8 2. Light-vectors and Light-rays................................. 3u 3. Fresnel's Law for the Velocity of Light . . . . . . . . . . . . . . . . . . . . . . . 314 4- The Directions of the Vibrations. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 316 5. The Normal Surface ......................................... 317 6. Geometrical Construction of the Wave Surface and of the Direc-
tion of Vibration .. , , . , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

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TABLE OF CONTENTS

ARTa

PAGE

7. Uniaxial Crystals ............................................ 323

8. Determination of the Direction of the Ray from the Direction of

the Wave Normal ........................................ 324

9. The Ray Surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

IO. Conical Refraction .......................................... 331

11. Passage of Light through Plates and Prisms of Crystal ......... 335

12. Total Reflection at the Surface of Crystalline Plates . . . . . . . . . . . . 339

13. Partial Reflection at the Surface of a Crystalline Plate ......... 344

14. Interference Phenomena Produced by Crystalline Plates in

Polarized Light when the Incidence is Normal. ............ 344

15. Interference Phenomena in Crystalline Plates in Convergent

Polarized Light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

CHAPTER IV

ABSORBING MEDIA

1. Electromagnetic Theory.. . . . . . . . . . . . .

. . . . . . . . . . . . . . . . 358

2. Metallic Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

3. The Optical Constants of the Metals. . . . . . . . . . . . . . . . . . . . . . . . . . 366

4. Absorbing Crystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

5. Interference Phenomena in Absorbing Biaxial Crystals ......... 374

6. Interference Phenomena in Absorbing Uniaxial Crystals........ 380

CHAPTER V
DISPERSION
1. Theoretical Considerations.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 2. Normal Dispersion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 3. Anomalous Dispersion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 4. Dispersion of the Metals.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3g6
CHAPTER VI
OPTICALLY ACTIVE SUBSTANCES
r. General Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 2. Isotropic Media.............................................. 401 3. Rotation of the Plane of Polarization.. . . . . . . . . . . . . . . . . . . . . . . . 404 4. Crystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
5. Rotary Dispersion .......................................... 412 6. Absorbing Active Substances................................. 415

TABLE OF CONTENTS

xix

CHAPTER VII

MAGNETICALLY ACTIVE SUBSTANCES

A. Hypothesis of Molecular Currents

ART.

•AGS

1. General Considerations ..................................•... 418

2. Deduction of the Differential Equations ....................... 420

3. The Magnetic Rotation of the Plane of Polarization.. . . . . . . . . . . 426

4. Dispersion in Magnetic Rotation of the Plane of Polarization .. 429

5. Direction of Magnetization Perpendicular to the Ray........... 433

B. Hypothesis of the Hall Effect
, . General Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 2. Deduction of the Differential Equations ....................... 435 3. Rays Parallel to the Direction of Magnetization. . . . . . . . . . . . . . . 437 4. Dispersion in the Magnetic Rotation of the Plane of Polarization. 438 5. The Impressed Period Close to a Natural Period .............. 440 6. Rays Perpendicular to the Direction of Magnetization ......... 443 7. The Impressed Period in the Neighborhood of a Natural Period. 444 8. The Zeeman Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 9. The Magneto-optical Properties of Iron, Nickel, and Cobalt... 449 10. The Effects of the Magnetic Field of the Ray of Light ......... 452

CHAPTER VIII
BODIES IN MOTION
1. General Considerations ...................................... 457 2. The Differential Equations of the Electromagnetic Field Re-
ferred to a Fixed System of Coordinates .................... 457 3. The Velocity of Lig-ht in Moving Media ....................... 465 4. The Differential Equations and the Boundary Conditions Re-
ferred to a Moving System of Coordinates which is Fixed with Reference to the Moving Medium ..................... 467 5. The Determination of the Direction of the Ray by Huygens' Principle ................................................. 470 6. The Absolute Time Replaced by a Time which is a Function of the Coordinates........................................... 471 7. The Configuration of the Rays Independent of the Motion. . . . . 473 8. The Earth as a Moving System.............................. 474 9. The Aberration of Light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 S IO. Fizeau's Experiment with Polarized Light...................... 477 11. Michelson's Interference Experiment ......................... 478

xx

TABLE OF CONTENTS

PART III.-RADIATION

CHAPTER I

ENERGY OF RADIATION

AJIT.

PAGS

1. Emissive Power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

2, Intensity of Radiation of a Surface............................ 484

3. The Mechanical Equivalent of the Unit of Light .............. 485

4. The Radiation from the Sun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

5. The Efficiency of a Source of Light. . . . . . . . . . . . . . . . . . . . . . . . . . . 487

6. The Pressure of Radiation ................................... 488

7. Prevost's Theory of Exchanges ............... , ............... 491

CHAPTER II
APPLICATION OF THE SECOND LAW OF THERMODYNAMICS TO
PURE TEMPERATURE RADIATION
1. The Two Laws of Thermodynamics........................... 493 2. Temperature Radiation and Luminescence..................... 494 3. The Emissive Power of a Perfect Reflector or of a Perfectly
Transparent Body is Zero.................................. 495 4- Kirchhoff's Law of Emission and Absorption, ................. 4¢ 5. Consequences of Kirchhoff's Law............................. 499 6. The Dependence of the Intensity of Radiation upon the Index
of Refraction of the Surrounding Medium .................. 502 7. The Sine Law in the Formation of Optical Images of Surface
Elements ................................................ 505 8. Absolute Temperature....................................... 5o6 9- Entropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 10. General Equations of Thermodynamics ...................... 5n 11. The Dependence of the Total Radiation of a Black upon its Ab-
solute Temperature ....................................... 512 12. The Temperature of the Sun Calculated from its Total Emission 515 13. The Effect of Change in Temperature upon the Spectrum of
a Black Body............................................. 516 14. The Temperature of the Sun Determined from the Distribution
of Energy in the Solar Spectrum ............... , ........ , .. 523 15. The Distribution of the Energy in the Spectrum of a Black
Body .................................................... 524

TABLE OF CONTENTS

CHAPTER III

INCANDESCENT VAPORS AND GASES

ART.

PAGS

1. Distinction between Temperature Radiation and Luminescence. 528

2. The Ion-hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

3. The Damping of Ionic Vibrations because of Radiation. . . . . . . . 534

4- The Radiation of the Ions under the Influence of External

Radiation .................................... •.. • . . . . . . . . 535

5. Fluorescence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5J6

6. The Broadening of the Spectral Lines Due to Motion in the Line

of Sight. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

7. Other Causes of the Broadening of the Spectral Lines ......... 541

INDEX ••••••••••••••••••••••• , •••••••••• , ••••••• , • • • • • • • • • • • 543

PART I
GEOMETRICAL OPTICS
CHAPTER I
THE FUNDAMENTAL LAWS
1. Direct Experiment.-The four fundamental laws stated above are obtained by direct experiment.
The rectilinear propagation of light is shown by the shadow of an opaque body which a point source of light P casts upon a screen S. If the opaque body contains an aperture L, then the edge of the shadow cast upon the screen is found to be the intersection of S with a cone whose vertex lies in the source P and whose surface passes through the periphery of the aperture L.
If the aperture is made smaller, the boundary of the shadow upon the screen S contracts. Moreover it becomes indefinite when L is made very small (e.g. less than I mm.), for points upon the screen which lie within the geometrical shadow now receive light from P. However, it is to be observed that a true point source can never be realized, and, on account of the finite extent of the source, the edge of the shadow could never. be perfectly sharp even if light were propagated in straight lines (umbra and penumbra). Nevertheless, in the case of a very small opening L (say of about one tenth mm. diameter) the light is spread out behind L upon the screen so far that in this case the propagation cannot possibly /Je rectilinear.

2

THEORY OF OPTICS

The same result is obtained if the shadow which an opaque body S' casts upon the screen S is studied, instead of the spreading out of the light which has passed through a hole in an opaque object. If S' is sufficiently small, rectilinear propagation of light from P does not take place. It is therefore necessary to bear in mind that the law of the rectilinear propagation of light holds only when the free opening through which the light passes, or the screens which prevent its passage, are not too small.
In order to conveniently describf the propagation of light from a source P to a screen S, it is customary to say that P sends rays to S. The path of a ray of light is then defined by the fact that its effect upon 5 can be cut off only by an obstacle that lies in the path of the ray itself. When the propagation of light is rectilinear the rays are straight lines, as when light from P passes through a sufficiently large opening in an opaque body. In this case it is customary to say that P sends a beam of light through L.
Since by diminishing L the result upon the screen S is the same as though the influence of certain of the rays proceeding from P were simply removed while that of the other rays remained unchanged, it follows that the different parts of a beam of light are independent of one another.
This law too breaks down if the diminution of the opening L is carried too far. But in that case the conception of light rays propagated in straight lines is altogether untenable.
The concept of light rays is then merely introduced for convenience. It is altogether impossible to isolate a single ray and prove its physical existence. For the more one tries to attain this end by narrowing the beam, the less does light proceed in straight lines, and the more does the con~ept of light rays lose its physical significance.
If the homogeneity of the space in which the light rays exist is disturbed by the introduction of some substance, the rays undergo a sudden change of direction at its surface: each ray splits up into two, a reflected and a refracted ray. If the sur-

THE FUNDAMENTAL LAWS

3

face of the body upon which the light falls is plane, then the

plane of incidence is that plane which is defined by the incident

ray and the normal N to the surface, and the angle of

z'nddence ¢ is the angle included between these two direc-

tions. The following laws hold: The re.fleeted and refracted rays

both lie in the plane of incidence. The angle of reflection (the

angle included between N and the reflected ray) is equal to the

angle of incidence. The angle of refraction¢' (angle included

between N and the refracted ray) bears to the angle of incidence

the relation

-ssiinn-q<j>-/ -n'

(I)

in which n is a constant for any given color, and is called the index of refraction of the body with reference to the surrounding medium.-Unless otherwise specified the index ofrefraction with respect to air will be understood.-For all transparent liquids and solids n is greater than I.
If a body A is separated from air by a thin plane parallel plate of some other body B, the light is refracted at both surfaces of the plate in accordance with equation (I); i.e.

= sin <j>
sin <jl

nh,

-ssiinn-<qp-/11 -

n ab

•

in which ¢ represents the angle of incidence in air, ¢' the angle of refraction in the body B, ¢'' the angle of refraction in the body A, n0 the index of refraction of B with respect to air, n® the index of refraction of A with respect to B; therefore

If the plate Bis infinitely thin, the formula still holds. The case does not then differ from that at first considered, viz. that of simple refraction between the body A and air. The

4

THEORY OF OPTICS

last equation in combination with (1) then gives, n,. denoting the index of refraction of A with respect to air,

or

i.e. the index of refraction of A with respect to B is equal to the ratio of the i'ndict·s of A and B with respect to air.
If the case considered had been that of an infinitely thin plate A placed upon the body B, the same process of reasoning would have given
Hence

i.e. the index of A with respect to B i's the reciprocal of the index of B with respect to A.
The law of refraction stated in (1) permits, then, the conclusion that q,' may also be regarded as the angle of incidence
in the body, and q, as the angle of refraction in the surround-
ing medium; i.e. that the direction of propagation may be reversed without changing the path of the rays. For the case of reflection it is at once evident that this principle of reversibility also holds.
Therefore equation (1), which corresponds to the passage of light from a body A to a body B or the reverse, may be put in the symmetrical form
(3)
in which q,,. and q,6 denote the angles included between the
normal N and the directions of the ray in A and B respectively, and n,. and n6 the respective indices with respect to some medium like air or the free ether.
The difference between the index n of a body with respect to air and its index n0 with respect to a vacuum is very small. From (2)
(4)

THE FUNDAMENTAL LAWS

5

in which n' denotes the index of a vacuum with respect to air.

Its value at atmospheric pressure and 0° C. is

= n' I : I . 00029, ,

(5)

According to equation (3) there exists a refracted ray (¢6) to correspond to every possible incident ray </J,. only when
n,. < n6 ; for if n,. > n6 , and if

(6)

then sin <Po> I; i.e. there is no real angle of refraction <fJ6• In that case no refraction occurs at the surface, but reflection only. The whole intensity of the incident ray must then be contained in the reflected ray; i.e. there is total reflection.
In all other cases (partial reflection) the intensity of the incident light is divided between the reflected and the refracted rays according to a law which will be more fully considered later (Section 2, Chapter II). Here the observation must suffice that, in general, for transparent bodies the refracted ray contains much more light than the reflected. Only in the case of the metals does the latter contain almost the entire intensity of the incident light. It is also to be observed that the law of reflection holds for very opaque bodies, like the metals, but the law of refraction is no longer correct in the form given in (1) or (3). This point will be more fully discussed later (Section 2, Chapter IV).
The different qualities perceptible in light are called colors. The refractive index depends on the color, and, when referred to air, increases, for transparent bodies, as the color changes from red through yellow to blue. The spreading out of white light into a spectrum by passage through a prism is due to this change of index with the color, and is called dispersion.
If the surface of the body upon which the light falls is not plane hut curved, it may still be looked upon as made up of very small elementary planes (the tangent planes), and the paths of the light rays may be constructed according to the

6

THEORY OF OPTICS

above laws. However, this process is reliable only when the curvature of the surface does not exceed a certain limit, i.e. when the surface may be considered smooth.
Rough surfaces exhibit irregular (diffuse) reflection and refraction and act as though they themselves emitted light. The surface of a body is visible only because of diffuse reflection and refraction. The surface of a perfect mirror is invisible. Only objects which lie outside of the mirror, and whose rays are reflected by it, are seen.
2. Law of the Extreme Path.*-All of these experi-
mental facts as to the direction of light rays are comprehended in the law of the extreme path. If a ray of light in passing from a point P to a point P' experiences any number of reflections and refractions, then the sum of the products of the index of refraction of each medium by the distance traversed in it, i.e. ~nl, has a maximum or minimum value; i.e. it differs from a like sum for all other paths which are infinitely close to the actual path by terms of the second or higher order. Thus if ,S denotes the variation of the first order,

o-:Znl = o.

(7)

The product, index of refraction times distance traversed, is known as the optical length of the ray.
In order to prove the proposition for a single refraction let POP' be the actual path of the light (Fig. 1), OE the intersection of the plane of incidence PON with the surface (tangent plane) of the refracting body, 0' a point on the surface of the refracting body infinitely near to 0, so that 00' makes any angle (} with the plane of incidence, i.e. with the line OE. Then it is to be proved that, to terms of the second or higher order,

+ n-PO+ n'-OP' = n-PO' n'.O'P',

(8)

*•Extreme' is here used to denote either greatest or least (maximum or minimum).-Ta..

THE FUNDAMENTAL LAWS

7

in which n and n' represent the indices of refraction of the adjoining media.
If a perpendicular OR be dropped from O upon PO' and a perpendicular OR' upon P'O', then, to terms of the second order,
PO'= PO+ RO', O'P' = OP' - O'R'. . . (9)
Also, to the same degree of approximation,
RO'= 00'.cos POO', O'R' = OO'•cos P'OO'. (ro)

FIG. I.

In order to calculate cos POO' imagine an axis OD perpendicular to ON and OE, and introduce the direction cosines of the lines PO and 00' referred to a rectangular system of coordinates whose axes are ON, OE, and OD. If </J represent the angle of incidence PON. then, disregarding the sign, the direction cosines of PO are

cos ¢, sin ¢, o,

those of 00 are

o, cos 8, sin fJ.

According to a principle of analytical geometry the cosine of the angle between any two lines is equal to the sum of the

8

THEORY OF OPTICS

products of the corresponding direction cosines of the lines with

reference to a system of rectangular coordinates, i.e.

= cos POO' sin </J•cos -8,

and similarly

= cos P'OO' sin </J' •cos -8,

in which </J' represents the angle of refraction. Then, from (9) and (10),
+ = + n-PO' n' -O'P' n-PO n-00' -sin </J •cos -8 + n'.OP' - n'-00'-sin <1/•co5 -8.

Since now from the law of refraction the relation exists
= n-sin <P n' •sin </J',

it follows that equation (8) holds for any position whatever

of the point O' which is infinitely close to 0.

For the case of a single reflection equation (7) may be

more simply proved. It then takes the form

o(PO + OP') = o, .

(11)

in which (Fig. 2) PO and OP' denote the actual path of the ray. If P 1 be that point which is symmetrical to P with
p'

FIG. 2.
respect to the tangent plane OE of the refracting body, then for every point O' in the tangent plane, PO'= P 10'. The length of the path of the light from P to P' for a single reflec-

THE FUNDAMcNTAL LAWS

9

tion at the tangent plane OE is, then, for every position of the
+ point 0', equal to P 10' O'P'. Now this length is a mini-
mum if P 1 , O', and P' lie in a straight line. But in that case the point O' actually coincides with the point O which is determined by the law of reflection. But since the property of a minimum (as well as of a maximum) is expressed by the vanishing of the first derivative, i.e. by equation (11), therefore equation (7) is proved for a single reflection.
It is to be observed that the vanishing of the first derivative i3 the condition of a maximum as well as of a minimum. In the case in which the refracting body is actually bounded by a plane, it follows at once from the construction given that the
path of the light in reflection is a minimum. It may also be
proved, as will be more fully shown later on, that in the case of refraction the actual path is a minimum if the refracting body is bounded by a plane. Hence this principle has often been called the law of least path.
When, however, the surface of the refracting or reflecting body is curved, then the path o.f the light is a minimum or a maximum according to the nature o.f the curvature. The vanishing of the first derivative is the only property which is common to all cases, and this also is entirely sufficient for the determination of the path of the ray.
A clear comprehension of the subject is facilitated by the introduction of the so-called aplanatic surface, which is a surface such that from every point upon it the sum of the optical paths to two points P and P' is constant. For such a surface the derivative, not only of the first order, but also of any other order, of the sum of the optical paths vanishes.
In the case of reflection the aplanatic surface, defined by

+ = PA P'A constant C, .

(12)

is an ellipsoid of revolution having the points P and P' as foci. If SOS' represents a section of a mirror (Fig. 3) and 0
a point upon it such that PO and P'O are incident and reflected rays, then the aplanatic surface A OA ', which

JO

THEORY OF OPTICS

passes tlirough the point O and corresponds to the points P and P', must evidently be tangent to the mirror SOS' at 0, since at this point the first derivative of the optical paths vanishes for both surfaces. If now, as in the figure, the mirror SOS' is more concave than the aplanatic surface, then the optical path PO+ OP' is a maximum, otherwise a minimum.

FIG. 3.
The proof of this appears at once from the figure, since for all points (J within the ellipsoid A OA' whose equation is given in (12), the sum PO+ OP' is smaller than the constant C, while for all points outside, this sum is larger than C, and for the actual point of reflection 0, it is equal to C.
In the case of refraction the aplanatic surface, defined by
n-PA +n'•P'A = constant C,
is a so-called Cartesian oval which must be convex towards
the less refractive medium (in Fig. 4 n < n'), and indeed more
convex than a sphere described about P' as a centre. This aplanatic surface also separates the regions for whose
+ points 0' the sum of the optical paths n-PO' 11' -PO'> C
from those for which that sum < C. The former regions lie
on the side of the aplanatic surface toward the less refractive medium (left in the figure), the latter on the side toward the more refractive medium (right in the figure).
If now SOS' represents a section of the surface between the

THE FUNDAMENTAL LAWS

II

two media, and PO, P' 0 the actual path which the light takes in accordance with the law of refraction, then the length of the path through O is a maximum or a minimum according as SOS' is more or less convex toward the less refracting medium

.lt
Fm.4.
than the aplanatic surface AOA'. The proof appears at once from the figure.
If, for example, SOS' is a plane, the length of the path is a minimum. In the case shown in the figure the length of the path is a maximum.
Since, as will be shown later, the index of refraction is inversely proportional to the velocity, the optical path nl is proportional to the time which the light requires to travel the distance l. The principle of least path is then identical with Fermat's principle of least time, but it is evident from the above that, under certain circumstances, the time may also be a maximum.
= Since o~nl o holds for each single reflection or refraction, the equation o~nl = o may at once be applied to the
case of any number of reflections and refractions.
3. The Law of Malus.-Geometrically considered there
are two different kinds of ray systems: those which may be cut at right angles by a properly constru;:ted surface F (ortho-

12

THEORY OF OPTICS

tomic system), and those for which no such surface F can be

found (anorthotomic system). With the help of the preceding

principle the law of Malus can now be proved. This law is

stated thus: An ortlzotomic system of rays remains orthotomz'c

after any number of reflections and refractions. From the

standpoint of the wave theory, which makes the rays the

normals to the wave front, the law is self-evident. But it can

also be deduced from the fundamental geometrical laws already

used.

Let (Fig. S) ABCDE and A'B'C'D'E' be two rays infinitely

close together and let their initial direction be normal to a

surface F. If L represents the total

f-.........,,...-

f optical distance from A to E, then

it may be proved that every ray

rt.

whose total path, measured from its

origin A, A', etc., has the same

optical lc::ngth L, is normal to a sur-

face F' which is the locus of the ends

E, E', etc., of those paths. For

the purpose of the proof let A'B and

E 'D be drawn.

FIG. 5.

According to the law of extreme path stated above, the length of

the path A'B'C'D'E' must be equal to that of the infinitely

near path A'BCDE', i.e. equal to L, which is also the length

of the path ABCDE. If now from the two optical distances

A'BCDE' and ABCDE the common portion BCD be sub-

tracted, it follows that

n-AB+n'.DE= n-A'B+ n'-DE',

in which n represents the .index of the medium between the surfaces F and B, and n' that of the medium between D and F'. But since AB= A'B, because AB is by hypothesis normal to F, it follows that
DE= DE',

THE FUNDAMENTAl LAWS

13

i.e. DE is perpendicular to the surface F'. In like manner it may be proved that any other ray D'E' is normal to F'.
Rays which are emitted by a luminous point are normal to a surface F, which is the surface of any sphere described about the luminous point as a centre. Since every source of light may be looked upon as a complex of luminous points, it follows that light rays always form an orthotomic systmi.

CHAPTER II
G~OMETRICAL THEORY OF OPTICAL IMAGES
1. The Concept of Optical Images.-If in the neighborhood of a luminous point P there are refracting and reflecting bodies having any arbitrary arrangement, then, in general, there passes through any point P' in space one and only one ray of light, i.e. the direction which light takes from P to P' is completely determined. Nevertheless certain points P' may be found at which two or more of the rays emitted by Pintersect. If a large number of the rays emitted by P intersect in a point P', then P' is called the optical image of P. The intensity of the light at P' will clearly be a maximum. If the actual intersection of the rays is at P', the image is called real,· if P' is merely the intersection of the backward prolongation of the rays, the image is called vi'rtual. The simplest example of a virtual image is found in the reflection of a luminous point P in a plane mirror. The image P' lies at that point which is placed symmetrically to P with respect to the mirror. Real images may be distinguished from virtual by the direct illumination which they produce upon a suitably placed rough surface such as a piece of white paper. In the case of plane mirrors, for instance, no light whatever reaches the point P'. Nevertheless virtual images may be transformed into real by certain optical means. Thus a virtual image can be seen because it is transformed by the eye into a real image which illumines a certain spot on the retina.
The cross-section of the bundle of rays which is brought together in the image may have finite length and breadth or may be infinitely narrow so as in the limit to have but one

GEOMETRICAL THEORY OF OPTICAL IMAGES 15
dimension. Consider, for example, the case of a single refraction. If the surface of the refracting body is the aplanatic surface for the two points P and P', then a beam of any size which has its origin in P will be brought together in P'; for all rays which start from P and strike the aplanatic surface must intersect in P', since for all of them the total optical distance from P to P' is the same.
If the surface of the refracting body has not the form of the aplanatic surface, then the number of rays which intersect in P is smaller the greater the difference in the form of the two surfaces (which are necessarily tangent to each other, see page 10). In order that an infinitely narrow, i.e. a plane, beam may come to intersection in P', the curvature of the surfaces at the point of tangency must be the same at least in one plane. If the curvature of the two surfaces is the same at 0 for two and therefore for all planes, then a solid elementary beam will come to intersection in P'; and if, finally, a finite section of the surface of the refracting body coincides with the aplanatic surface, then a beam of finite cross-section will come to intersection in P'.
Since the direction of light may be reversed, it is possible to interchange the source P and its image P', i.e. a source at P' has its image at P. On account of this reciprocal relationship P and P' are called co1yi1gate po£nts.
2. General Formulm for Images.-Assume that by means of reflection or refraction all the points P of a given space are imaged in points P' of a second space. The former space will be called the ob.feet space; the latter, the £1nage space. From the definition of an optical image it follows that for every ray which passes through P there is a conjugate ray passing through P'. Two rays in the object space which intersect at P must correspond to two conjugate rays which intersect in
the image space, the intersection being at the point P' which is conjugate to P. For every point P there is then but one
conjugate point P'. If four points P1P.j'/'4 of the object space lie in a plane, then the rays which connect any two pairs of

THEORY OF OPTICS

these points intersect, e.g. the ray P 1P 2 cuts the ray PaP, in the point A. Therefore the conjugate rays P'1P'2 and P'8P'4
also intersect in a point, namely in A' the image of A. Hence

the four images P/P 21Pa'P/ also lie in a plane. In other words, to every point, ray, or plane in the one space there

corresponds one, and but one, point, ray, or plane in the

other. Such a relation of two spaces is called in geometry a

collz"near relations/tip.

The analytical expression of the collinear relationship can

be easily obtained. Let x, y, z be the coordinates of a point

P of the object space referred to one rectangular system, and

x', y', z' the coordinates of the point P' referred to another

rectangular system chosen for the image space; then to every

x, y, z there corresponds one and only one x', y', z', and vt"ce

versa, This is only possible if

+ + + x, - -a1x- -b-1y- ~c1z- d1 + + - ax+ by cz d '

y

,

=

a~+ b2y+
ax+ by+

ccz~++dd2'

+ ++ z, -- -aaaX-x-+b-ba-yy++~cc8zz- ~ dd3'

()
• 1

in which a, b, c, d are constants. That is, for any given x', y', z', the values of x, y, z may be calculated from the three linear equations (1); and inversely, given values of x, y, z determine x', y', z'. If the right-hand side of equations (1) were not the quotient of two linear functions of x, y, z, then for every x', y', z' there would be several values of x, y, z. Furthermore the denominator of this quotient must be one and
+ the same linear function (ax+ by+ cz d), since otherwise
a plane in the image space
A'x' + B'y' + C'z' +I)'= o

would not again correspond to a plane

Ax+By+cz+D= o in the object space.

GEOMETRICAL THEORY OF OPTICAL IMAGES 17

If the equations (1) be solved for x, y, and z, forms analogous to (I) are obtained; thus

+ + + + + a/x' + x = a'x'

b/y' b'y'

c/z' d1' c'z' d' ' etc.

From (1) it follows that for
ax+ by + cz + d = o: x' = y' = z' = oo •
Similarly from (2) for
+ + a'x' b'y' c'z' + d' = o: x = y = z = oo • + + + The plane ax by cz d = o is called the focal plane
~ of the object space. The images P' of its points P lie at infinity. Two rays which originate in a point P of this focal plane correspond to two parallel rays in the image space.
+ + + The plane a'x' b'y' c'z' d' = o is called the focal
plane g:' of the £mage space. Parallel rays in the object space
correspond to conjugate rays in the image space which inter-
sect in some point of this focal plane g:'.
In case a= b = c = o, equations (1) show that to finite
values of x, y, z correspond finite values of x', y', z'; and, inversely, since, when a, b, and c are zero, a', b', c' are also zero, to finite values of x', y', z' correspond finite values of x, y, z. In this case, which is realized in telescopes, there are no focal planes at finite distances.
3. Images Formed by Coaxial Surfaces.-In optical in-
struments it is often the case that the formation of the image takes place symmetrically with respect to an axis; e.g. this is true if the surfaces of the refracting or reflecting bodies are surfaces of revolution having a common axis, in particular, surfaces of spheres whose centres lie in a straight line.
From symmetry the image P' of a point P must lie in the plane which passes through the point P and the axis of the
system, and it is entirely sufficient, for the study of the image formation, if the relations between the object and image in such a meridian plane are known.

18

THEORY OF OPTICS

If the xy plane of the object space and the x'y' plane of the image space be made to coincide with this meridian plane, and if the axis of symmetry be taken as both the x and the x' axis, then the z and z' coordinates no longer appear in equations (r ). They then reduce to

+ x' _ a1x+b1y+d1 - ar+by d,

(3)

The coordinate axes of the xy and the x'y' systems are then parallel and the .x and .x' axes lie in the same line. The origin O' for the image space is in general distinct from the origin O for the object space. The positive direction of .x will be taken as the direction of the incident light (from left to

Y'

0

.:iC, O'

FIG. 6.

right); the positive direction of x', the opposite, i.e. from right to left. The positive direction of y and y' will be taken

upward (see Fig. 6). From symmetry it is evident that x' does not change its

value when y changes sign. Therefore in equations (3)
b1 = b = o. It also follows from symmetry that a change in
sign of y produces merely a change in sign of y'. Hence
= = a2 d2 o and equations (3) reduce to

..., -

A,

-

a~a1-xx--++--,dd_-1'

y '--

b~ ax+-tf

(4)

Five constants thus remain, but their ratios alone are sufficient to determine the formation of the image. Hence

GEOMETRICAL THEORY OF OPTICAL IMAGES 19

there are in general four characteristic constants which determine the formation of z"mages by coaxial surfaces.
The solution of equations (4) for x and y gives

= _ dx' - d1
x - a1 - ax'' y

a1d - ad1

y'

-b2-- • a1 - ax'"

(S)

+The equation of the focal plane of the object space is
ax d = o, that of the focal plane of the image space ax' - a1 = o. The intersections F and F' of these planes
with the axis of the system are called the principal foe£. If the principal focus F of the object space be taken as the
origin of x, and likewise the principal focus F' of the image
+ space as the origin of x', then, if x0 , x0' represent the coordi-
nates measured from the focal planes, ax0 will replace ax d and - ax0', a1 - ax'. Then from equations (4)

(6)

Hence only two characteristic constants remain in the equations. The other two were taken up in fixing the positions of the focal planes. For these two complex constants simpler expressions will be introduced by writing (dropping subscripts)

xx'=ff',

(7)

In this equation x and x' are the distances of the object and
the z"mage from the principal focal planes g: and g:1 respectively.
The ratio y' :y is called the magnification. It is I for
x = f, i.e. x' = f'. This relation defines two planes .p and
.p' which are at right angles to the axis of the system. These
planes are called the unit planes. Their points of intersection Hand H' with the axis of the system are called unit poz"nts.
The unz"t planes are characterized by the fact that the distance from the axz"s of any poz"nt P i'n one unz"t plane i's equal to that of the conjugate poz"nt P' i'n the other unz"t plane. The two
remaining constants/ andf' of equation (7) denote, in accord-

zo

THEORY OF OPTICS

ance with the above, the distance of the unit planes op, op' from
the focal planes ~. ~'. The constant f is called the focal length of the object space; f', the focal length of the z"mage space. The direction off is positive when the ray falls first upon the focal plane ~. then upon the unit plane .\); for f' the case is the reverse. In Fig. 7 both focal lengths are positive.
The significance of the focal lengths can be made clear in the following way: Parallel rays in the object space must have conjugate rays in the image space which intersect in some point in the focal plane ~' distant, say, y' from the axis. The value ofy' evidently depends on the angle of inclination u of the incident ray with respect to the axis. If u = o, it follows
= from symmetry that y' o, i.e. rays parallel to the axis have
conjugate rays which intersect in the principal focus F'. But

f

FIG. 7.
if u is not equal to zero, consider a ray PFA which passes
through the first principal focus F, and cuts the unit plane .p
in A (Fig. 7). The ray which is conjugate to it, A'P', must evidently be parallel to the axis since the first ray passes
through F. Furthermore, from the property of the unit planes, A and A' are equally distant from the axis. Consequently the distance from the axis y' of the image which is formed by a parallel beam incident at an angle u is, as appears at once
from Fig. 7,

y' =f-tan u.

(8)

Hence the following law: Tlte focal length of the obj'ecl space i's equal to the ratz"o of the linear magnz"tude of an z"mage

GEOMETRICAL THEORY OF OPTICAL IMAGES u
formed t"n the focal plane of the t'mage space to the apparent (angular) magnitude of its in.finitely distant object. A similar definition holds of course for the focal length f' of the image space, as is seen by conceiving the incident beam of parallel rays to pass first through the image space and then to come to a focus in the focal plane ~-
If in Fig. 7 A'P' be conceived as the incident ray, so that the functions of the image and object spaces are interchanged, then the following may be given as the definition of the focal Jength f, which will then mean the focal length of the image space:
The focal length of the image space is equal to the distance between the axis and any ray of the object space which t"s parallel to the axis divided by the tangent of the inclinatz"on of its conjugate ray.
Equation (8) may be obtained directly from (7) by making
= = tan u y: x and tan u' y' : x'. Since x and x' are taken
positive in opposite directions and y and y' in the same direction, it follows that u and u' are positive in different directions. The angle of i'ncli'nation u of a ray t"n the object space i's posi'#ve if the ray goes upward from left to right; the angle of inclination u' of a ray in the image space is positive if tlze ray goes downwardfrom left to right.
The magnification depends, as equation. (7) shows, upon x, the distance of the object from the principal focus F, and upon f, the focal length. It is, however, independent of y, i.e. the image of a plane object which is perpendicular to the axis of the system is similar to the object. On the other hand the image of a solid object is not similar to the object, as is evident at once from the dependence of the magnification upon x. Furthermore it is easily shown from (7) that the magnification t"n depth, i.e. the ratio of the increment dx' of .i:-' to an increment dx of x, is proportional to the square of the lateral magnification.
Let a ray in the object space intersect the unit plane .p in

22

THEORY OF OPTICS

A and the axis in P (Fig. 8). Its angle of inclination u with respect to the axis is given by
= = AH AH
tan u PH f _ .i-'

if .i- taken with the proper sign represents the distance of P from F.
96'

FIG, 8.

The angle of inclination u' of the conjugate ray with respect to the axis is given by

tan

u'

=

A'H' P' H'

=

A'H' f' _ .i-"

if .i-' represent the distance of P' from F', and P 1 and A' are

the points conjugate to P and A. On account of the property of the unit planes AH= A'H'; then by combination of the

last two equations with (7),

tan u' f - .i-

.i-

f

tan u = f' - .i-' = - f' = - ?·

(9)

The ratio of the tangents of inclination of conjugate rays is

called the convergence ratio or the angular magnification. It

is seen from equation (9) that it is independent of u and u'.

= - = The angular magnification

I for .i- f' or .i-' = f.

The two conjugate points Kand K' thus determined are called

the nodal points of the system. They are characterised by the

GEOMETRICAL THEORY OF OPTICAL IMAGES 23
fact that a ray tltrough one nodal point K is cot?fugate and parallel to a ray through the other nodal point K'. The position of the nodal points for positive focal lengths f and f' is
fie

FIG. 9.

shown in Fig. 9. KA and K'A' are two conjugate rays. It follows from the figure that the distance between the two nodal points z"s the same as that between the two unit points. If
f = f', the nodal points coincide with the unit points.
Multiplication of the second of equations (7) by (9) gives

y' tan u'

f

y tan u = - f''

(10)

If e be the distance of an object P from the unit plane op, and e' the distance of its image from the unit plane op', e and e' being positive if P lies in front of (to the left of) op and P'
behind (to the right of) .p', then
e = f - x, e' = f' - x'.

Hence the first of equations (7) gives

e+ f

= f'
e' I. • • • •

The same equation holds if e and e' are the distances of P and P' from any two conjugate planes which are perpendicular to the axis, and f and f' the distances of the principal foci from these planes. This result may be easily deduced from (7).

THEORY OF OPTICS

4. Construction of Conjugate Points.-A simple graphical

interpretation may be given to equation

(11). If ABCD (Fig. 10) is a rectangle

with the sides / and /', then any

straight line ECE' intersects the pro-

longations of/ and/' at such distances
= from A that the conditions AE e and

A

E AE' = e' satisfy equation (r I).

Fm. ro.

It is also possible to use the unit

plane and the principal focus to determine the point P' conju-

gate to P. Draw (Fig. I I) from Pa ray PA parallel to the

axis and a ray PF passing through the principal focus F.

FIG. II.

A'F' is conjugate to PA, A' being at the same distance from the axis as A; also P'B', parallel to the axis, is conjugate to PFB, B' being at the same distance from the axis as B. The intersection of these two rays is the conjugate point sought. The nodal points may also be conveniently used for this construction.
The construction shown in Fig. I I cannot be used when P and P' lie upon the axis. Let a ray from P intersect the focal plane ~ at a distance g and the unit plane ~ at a distance h from the axis (Fig. 12). Let the conjugate ray intersect~' and ~ at the distances h'(= h) and g'. Then from the figure

,,~= g

PF

k=J--+-

- x g'

P'F'

- x'

< - r' 7i= f'+P'P= J'-x';

GEOMETRICAL THEORY OF OPTICAL IMAGES 25

and by addition, since from equation (7) xx'= ff',

g + g'

2xx1 -fx' -f'x

-,,-=ff' +xx -fx' -f'x = 1'

(iz)

P' may then be found by laying off in the focal plane ~, the
= distance g' h - g, and in the unit plane op' the distance

X

JC'

P'
,..___,.
-f
FIG. 12.
h' = h, and drawing a straight line through the two points thus
determined. g and g' are to be taken negative if they lie below the axis.
5. Classi:fication of the Different Kinds of Optical Sys-
tems.-The different kinds of optical systems differ from one another only in the signs of the focal lengths f and /'.
If the two focal lengths have the same sign, the system i's concurrent, i.e. if the object moves from left to right (x increases), the image likewise moves from left to right (.r' decreases). This follows at once from equation (7) by taking into account the directions in which x and x' are considered positive (see above, p. I 8 ). It will be seen later that this kind of image formation occurs if the image is due to refraction alone or to an even number of reflections or to a combination of the two. Since this kind of image formation is most frequently produced by refraction alone, it is also called dioptric.

THEORY OF OPTICS

If the t.wo focal lengths have opposite signs the system is

contracurrent, i.e. if the object moves from left to right, the

image moves from right to left, as appears from the formula
xx' = ff'. This case occurs if the image is produced by an odd

number of reflections or by a combination of an odd number of

such with refractions. This kind of image formation is called

katoptric. When it occurs the direction of propagation of the

light in the image space is opposite to that in the object space,

so that both cases may be included under the law: In all cases

of image formation if a point P be conceived to move along a ray

in the direction in which the light travels, the image P' of that

point moves along the co,ifugate ray in the direction in whiclt

the ltglzt travels.

Among dioptric systems a distinction is made between those

having positive and those having negative focal lengths. The

former systems are called convergent, the latter divergent,

because a bundle of parallel rays, after passing the unit plane

,O' of the image space, is rendered convergent by the former,

di,vergent by the latter. No distinction between systems on

the ground that their foci are real or virtual can be made, for

it will be seen later that many divergent systems (e.g. the

microscope) have real foci.

By similar definition katoptric systems which have a nega-

tive focal length in the image space are called convergent, -

for in reflection the direction of propagation of the light is

reversed.

There are therefore the four following kinds of optical

systems:

+ Dioptric ...

{ a. b.

Convergent: Divergent:

-

f, f,

+ Katoptric ..

{

a.
b.

Convergent: Divergent:

-

f, f,

+f'.
-f'. -f'. +f'.

6. Telescopic Systems.-Thus far it has been assumed
that the focal planes lie at finite distances. If they lie at infinity the case is that of a telescopic system, and the coeffi-

GEOMETRICAL THEORY OF OPTICAL IMAGES 27

cient a vanishes from equations (4), which then reduce by a

suitable choice cf the origin of the x coordinates to

x' = ax, y' = fly. .

(13)

Since x' = o when x = o, it is evident that any two conjugate

points may serve as origins from which x and x' are measured.

It follows from equation ( I 3) that the magnification in breadth

and depth are constant. The angular magnification is also constant, for, given any two conjugate rays OP and O'P', their

intersections with the axis of the system may serve as the

ongms. If then a point P of the first ray has the coordinates

x, y, and its conjugate point P' the coordinates x', y', the

tangents of the angles of inclination are

tan u = y : .r, tan u' = y' : x'.

Hence by (13)

tan u' : tan u = fl : a.

a must be positive for katoptric (contracurrent) systems, negative for dioptric (concurrent) systems. For the latter it is evident from (14) and a consideration of the way in which z, and u' are taken positive (see above, p. 21) that for positive fJ erect images of infinitely distant objects are formed, for nega-
tive /J, inverted images. There are therefore four different
kinds of telescopic systems depending upon the signs of a
and fl.
Equations (14) and (13) give
y' tan u' fP y tan u a

A comparison of this equation with (10) (p. 23) shows that for telescopic systems the two focal lengths, though both infinite, have a finite ratio. Thus

f

fJ2

f'= - a·

= If f f', as is the case in telescopes and in all instru-
ments in which the index of refraction of the object space is-

THEORY OF OPTICS

equal to that of the image space (cf. equation (9), Chapter III),
then a= - fP. Hence from (14)
tan tt' : tan u = - I : fl.

This convergence ratio (angular magnification) is called in the
case of telescopes merely the magnification r. From (13)

y :y' = - r,

(14')

i.e. for telescopes the reciprocal of the lateral magnification is numerz''cally equal to the allgular magnijicatz"on.
7. Combinations of Systems.-A series of several systems must be equivalent to a single system. Here again attention will be confined to coaxial systems. If f. and f.' are the focal
lengths of the first system alone, and /4 and /4,' those of the
second, and f and f' those of the combination, then both the focal lengths and the positions of the principal foci of the combination can be calculated or constructed if the distance
= F/F 2 L1 (Fig. I 3) is known. This distance will be called
for brevity the separation of the two systems I and 2, and will
be considered positive if F/ lies to the left of F2 , otherwise
negative. A ray S (Fig. I 3), which is parallel to the axis and at a
'!le;
s

s

F,

rsl

'!I,

FIG. 13.
distance y from it, will be transformed by system I into the ray S1 , which passes through the principal focus F 1' of that
system. S 1 will be transformed by system 2 into the ray S'.

GEOMETRICAL THEORY OF OPTICAL IMAGES 29

The point of intersection of this ray with the axis is the prin-

cipal focus of the image space of the combination. Its position

can be calculated from the fact that F/ and F' are conjugate

points of the second system, i.e. (cf. eq. 7)

·f/, F/F' =f2

(17)

in which F/F is positive if F' lies to the right of F 2'. F' may be determined graphically from the construction given above

on page 2 5, since the intersection of S1 and S' with the focal
planes F2 and F 2' are at such distances g and g' from the axis
+ = that g g' y 1•
The intersection A' of S' with S must lie in the unit plane
.p' of the image space of the combination. Thus .p' is deter-
mined, and, in consequence, the focal length f' of the com-
bination, which is the distance from .p' of the principal focus F'

of the combination. From the construction and the figure it

follows that f' is negative when L1 is positive.

f' may be determined analytically from the angle of incli-

nation u' of the ray S'. For S1 the relation holds:
tan = u1 y :f/,

in which u1 is to be taken with the opposite sign if 5 1 is COnsidered the object ray of the second system. Now by (9),

tan u' L1

tan u1 = h'

or since tan u1 = - y : f.',

tan

u' =

-

y

L1 ·f.'f./

= Further, since (cf. the law, p. 21) y : f' tan u', it follows
that

f - L1 • l - - [,.'.//

(18)

A similar consideration of a ray parallel to the axis in the image space and its conjugate ray in the object space gives

f= - f.f, ·

(19)

30

THEORY OF OPTICS

and for the distance of the principal focus F of the combination from the principal focus F 1 ,
FF1 = ff/, .

in which FF1 is positive if Flies to the left of F 1. Equations (17), (18), (19), and (20) contain the character-
istic constants of the combination calculated from those of the systems which unite to form it.
Precisely the same process may be employed when the combination contains more than two systems.
If the separation L1 of the two systems is zero, the focal
lengths f and f' are infinitely great, i.e. the system is tele-
scopit. The ratio of the focal lengths, which remains finite, is given by (18) and (19). Thus

j,=f·i·

(21)

Fro!Il the consideration of an incident ray parallel to the axis

the lateral magnification y' : y is seen to be

y' :y = fJ = -/4 :J;_'.

. (22)

By means of (21), (22), and (16) the constant a, which repre-

sents the magnification in depth (cf. equation (13)) is found.

Thus

x'
--;; =

a =

-

ff.Jf.;''"

. . . . . (23)

Hence by (14) the angular magnification is

tan u': tan u = fJ: a =f1 :/4'.

(24)

The above considerations as to the graphical or analytical determination of the constants of a combination must be somewhat modified if the combination contains one or more telescopic systems. The result can, however, be easily obtained by constructing or calculating the path through the successive systems of an incident ray whic.lt is parallel to the axis.

CHAPTER III
PHYSICAL CONDITIONS FOR IMAGE FORMATION
ABBE'S geometrical theory of the formation of optical images, which overlooks entirely the question of their physical realization, has been presented in the previous chapter, because the general laws thus obtained must be used for every special case of image formation no matter by what particular physical means the images are produced. The concept of focal points and focal lengths, for instance, is inherent in the concept of an image no matter whether the latter is produced by lenses or by mirrors or by any other means.
In this chapter it will appear that the formation of optical images as described ideally and without limitations in the previous chapter is physically impossible, e.g. the image of an object of finite size cannot be formed when the rays have too great a divergence.
It has already been shown on page I 5 that, whatever the divergence of the beam, the image of one point may be produced by reflection or refraction at an aplanatic surface. Images of other points are not produced by widely divergent rays, since the form of the aplanatic surface depends upon the position of the point. For this reason the more detailed treatment of special aplanatic surfaces has no particular physical interest. In what follows only the formation of images by refracting and reflecting spherical surfaces will be treated, since, on account of the ease of manufacture, these alone are used in optical instruments; and since, in any case, for the reason mentioned above, no other forms of reflecting or refracting surfaces furnish ideal optical images.
31

32

THEORY UP OPTICS

It will appear that the formation of optical images can be practically accomplished by means of refracting or reflecting spherical surfaces if certain limitations are imposed, namely, limitations either upon the size of the object, or upon the divergence of the rays producing the image.
1. Refraction at a Spherical Surface.-In a medium of index n, let a ray PA fall upon a sphere of a more strongly refractive substance of index n' (Fig. 14). Let the radius of

the sphere be r, its centre C. In order to find the path of th~

refracted ray, construct about C two spheres I and 2 of radii

r1 =

n' n- r

and

r2 =

,nnr (method of Weierstrass).

Let PA meet sphere I in B; draw BC intersecting sphere

2 in D. Then AD is the refracted ray. This is at once

evident from the fact that the triangles ADC and BA C

are similar. For AC: CD = BC: CA = n' : n. Hence the

1'.'. DAC = ~ABC= </>', the angle of refraction, and since

1'.'. BA C = </>, the angle of incidence, it follows that

sin </> : sin </>' = BC : AC = n' : n,

which is the law of refraction. If in this way the paths of different rays from the point P

PHYSICAL CONDITIONS FOR IMAGE FORMATION 33

be constructed, it becomes evident from the figure that these rays will not all intersect in the same point P'. Hence no image is formed by widely divergent rays. Further it appears from the above construction that all rays which intersect the sphere at any point, and whose prolongations pass through B, are refracted to the point D. Inversely all rays which start from D have their virtual intersection in B. Hence upon every straight line passing through the centre C of a sphere of radius r, there are two points at distances from C of
I
r'!n_ and r nn, respectt"vely which, for all rays, stand z"n the relation of object and virtual (not real) image. These two points are called the aplanatic points of the sphere.
If u and u' represent the angles of inclination with respect to the axis BD of two rays which start from the aplanatic points Band D, i.e. if

1'.'.ABC= u, 1'.'.ADC= u',
then, as was shown above, 1'.'. ABC= 1'.'. DA C = u. From
a consideration of the triangle ADC it follows that

sinu':sinu=AC:CD=n':n.

(1)

In this case then the ratio of the sines of the angles of inclination of the conjugate rays is independent of u, not, as in equation (9) on page 22, the ratio of the tangents. The difference between the two cases lies in this, that, before, the image of a portion of space was assumed to be formed, while now only the image of a surface formed by widely divergent rays is under consideration. The two concentric spherical surfaces I
and 2 of Fig. 14 are the loci of all pairs of aplanatic points B and D. To be sure, the relation of these two surfaces is not
collinear in the sense in which this term was used above, because the surfaces are not planes. If s and s' represent the areas of two conjugate elements of these surfaces, then, since their ratio must be the same as that of the entire spherical !;11rfaces I and 2,

34

THEORY OF OPTICS

Hence equation (I) may be written:
= sin2 u-s-n2 sin2 u' ,s' ,n'2• •
It will be seen later that this equation always holds for two surface elements s and s' which have the relation of object and image no matter by what particular arrangement the image is produced.
In order to obtain the image of a portion of space by means of refraction at a spherical surface, the divergence of the rays which form the image must be taken very small. Let PA (Fig. 15) be an incident ray, AP' the refracted ray, and PCP'

p

Tl,

n'

FIG. IS,

the line joining P with the centre of the sphere C. Then from the triangle PAC,
= sin ¢ : sin a PH+ r : PA,

and from the triangle P'AC,
sin ¢' : sin a = P'H - r : P'A.

Hence by division,

= n = sin </J n' PH+ r P'A

sin ¢'

P'H - r. PA •

(3)

Now assume that A lies infinitely near to H, i.e. that the angle APH is very small, so that PA may be considered equal to PH, and P'A to P'H. Also let
PH= e, P'H= e'.

PHYSICAL CONDITIONS FOR IMAGE FORMATION 35

Then from (3) or

e+r e' n' e'-r"e-n'

en +

n' e'

=

n'-n -r-.

. (4)

In which r is to be taken positive if the sphere is convex toward the incident light, i.e. if C lies to the right of H. e is positive if P lies to the left of H; e' is positive if P' lies to the right of H. To every e there corresponds a definite e' which is independent of the position of the ray PA, i.e. an image of a portion of space which lies close to the axis PC is formed by rays which lie close to PC.
A comparison of equation (4) with equation (11) on page 23 shows that the focal lengths of the system are

= = f

n rn- ,--n- ,

f'

n' rn-,--n-, .

• (5)

and that the two unit planes ~ and ~' coincide and are tan-
gent to the sphere at the point H. Since f and f' have the
same sign, it follows, from the criterion on page 2 5 above,
that the system is dioptric or concurrent. If n' > n, a convex
curvature (positive r) means a convergent system. Real
images (e' > o) are formed so long as e > f. Such images
are also inverted.
Equation (10) on page 23 becomes

y' tan u'

n

v tan u = - n1· • • • • •

(6)

By the former convention the angles of inclination u and u' of

conjugate rays are taken positive in different ways. If they

are taken positive in the same way the notation 'u will be used

= - instead of u', i.e. 'u

u'. Hence the last equation may

be written:

= ny tan u n'y' tan 'u. .

. (7)

THEORY OF OPTICS

In this equation a quantity which is not changed by refraction appears,-an optical invariant. This quantity remains constant when refraction takes place at any number of coaxial spherical surfaces. For such a case let n be the index of refraction of the first medium, n' that of the last; then equation (7) holds. But since in general for every system, from equation (10), page 23,

y' tan u' f

ytan u - f" •

(S)

there results from a combination with (7)

f:f' = n: n',

(9)

i.e. In the formation of i·mages by a system of coaxial refracting spherical surfaces the ratio of the focal lengths of the system is equal to the ra#o of the indices of refraction of the .irst and last media. If, for example, these two media are air, as is the case with lenses, mirrors, and most optical instruments, the two focal lengths are equal.
2. Reflection at a Spherical Surface.-Let the radius r be considered positive for a convex, negative for a concave mirror.

p

FIG. 16.
= By the law of reflection (Fig. 16) ~ PAC ~ P'AC.

Hence from geometry

PA :P'A = PC:P'C.

(10)

If the ray PA makes a large angle with the axis PC, then the position of the point of intersection P' of the conjugate ray

PHYSICAL CONDITIONS FOR IMAGE FORMATION 37

with the axis varies with the angle. In that case no image of the point P exists. But if the angle APC is so small that the angle itself may be used in place of its sine, then for every point P there exists a definite conjugate point P', i.e. an image
is now formed. It is then permissible to set PA = PH, P'A = P'H, so that (IO) becomes

PH:P'H= PC:P'C,

= - or if PH= e, P'H

e', then, since r in the figure is nega-

tive,

--eI +,eI =-r2 ..

A comparison of this with equation ( I I) on page 2 3 shows that the focal lengths of the system are

f=

I
--r,

f'=

I
+-r;

(13)

2

2

that the two unit planes .p and .p' coincide with the plane
tangent to the sphere at the vertex H; that the two principal foci coincide in the mid-point between C and H; and that the nodal points coincide at the centre C of the sphere. The signs of e and e' are determined by the definition on page 23.
Since f and f' have opposite signs, it follows, from the criterion given on page 25, that the system is katoptric or contracurrent. By the conventions on page 26 a negative r, i.e. a concave mirror, corresponds to a convergent system; on the other hand a convex mirror corresponds to a divergent system.

A comparison of equations (13) and (5) shows that the

results here obtained for reflection at a spherical surface may

be deduced from the former results for refraction at such a sur-

= - face by writing n': n

I. In fact when n': n = - I, the

law of refraction passes into the law of reflection. Use may

be made of this fact when a combination of several refracting

or reflecting surfaces is under consideration. Equation (9)

holds for all such cases and shows that a positive ratio f: f'

THEORY OF OPTICS
always results from a combination of an even number of reflections from spherical surfaces or from a combination of any number of refractions, i.e. such systems are dioptric or concurrent (cf. page 25).
The relation between image and object may be clearly brought out from Fig. 17, which relates to a concave mirror. The numbers I, 2, 3, ... 8 represent points of the object at a constant height above the axis of the system. The numbers 7 and 8 which lie behind the mirror correspond to virtual objects, i.e. the incident rays start toward these points, but fall upon the mirror and are reflected before coming to an intersection at them. Real rays are represented in Fig. 17 by

.,
,,. f

continuous lines, virtual rays by dotted lines. The points

J', . . . I', 2',

8' are the images of the points I, 2, 3, ... 8.

Since the latter lie in a straight line parallel to the axis, the

former must also lie in a straight line which passes through the

principal focus F and through point 6, the intersection of the

object ray with the mirror, i.e. with the unit plane. The con-

tinuous line denotes real images; the dotted line, virtual im-

ages. Any image point 2' may be constructed (cf. page 24)

by drawing through the object 2 and the principal focus F a

straight line which intersects the mirror, i.e. the unit plane, in

some point A 3• If now through A 2 a line be drawn parallel

PHYSICAL CONDITIONS FOR IMAGE FORMATION 39
to the axis, this line will intersect the previously constructed image line in the point sought, namely 2 1• From the figure it may be clearly seen that the images of distant objects are real and inverted, those of objects which lie in front of the mirror within the focal length are virtual and erect, and those of virtual objects behind the mirror are real, erect, and lie in front of the mirror.
Fig. I 8 shows the relative positions of object and image

t

2

+ __.s __ e_____ 7 _______ a

FIG. 18.
for a convex mirror. It is evident that the images of all real objects are virtual, erect, and reduced; that for virtual objects which lie within the focal length behind the mirror the images are real, erect, and enlarged; and that for more distant virtual objects the images are also virtual.

p
Equation (11) asserts that PCP'H are four harmonic points. The image of an object P may, with the aid of a proposition of synthetic geometry, be constructed in the following way:

THEORY OF OP TICS
From any point L (Fig. 19) draw two rays LC and LH, and then draw any other ray PDB. Let O be the intersection of DH with BC: then LO intersects the straight line PH in a point P' which is conjugate to P. For a convex mirror the construction is precisely the same, but the physical meaning of the points C and His interchanged.
3. Lenses.-The optical characteristics of systems composed of two coaxial spherical surfaces (lenses) can be directly deduced from § 7 of Chapter II. The radii of curvature r 1 and r 2 are taken positive in accordance with the conventions given above (§ 1); i.e. the radius of a spherical surface is considered positive if the surface is convex toward the incident ray (convex toward the left). Consider the case of a lens of index n surrounded by air. Let the thickness of the lens, i.e. the distance between its vertices S1 and 5 2 (Fig. 20), be

- 6

,: L1 fe

F, s. F/ & s..

6'
-&'

n

FIG. 20.

denoted by d. If the focal lengths of the first refracting surface are denoted by f 1 andfi.', those of the second surface by
/4 andfz', then the separation L1 of the two systems (cf. page
28) is given by
,::::J=d-f,_'-/4,

and, by (5),

= .Ii=

= 1 ,
r1n---1• f,_

n
r 1n---1•

.Is=

r

n s1---n•

/,4

I ( r21---n• 15)

PHYSICAL CONDITIONS FOR IMAGE FORMATION 41

Hence by equations (19) and (18) of Chapter II (page 29) the focal lengths of the combination are

' n

r1r2

f=f=n-1°d(n-I)-nr1 +nr/

(16)

while the positions of the principal foci F and F' of the com~

bination are given by equations ( I 7) and (20) of Chapter II

(page 29). By these equations the distance o of the principal

= + focus F in front of the vertex S1 , and the distance a-' of the
principal focus F' behind the vertex S2 are, since <Y FF1 f, and a-'= F.'F' +f.',

+ <Y

d(n - nr r 1

•

1)

2

~~------c-~-~-

= + n - 1 d(n - 1) - nr1 nr/ •

(17) '

= + ,

r2

-d(n - 1) +nr1

<Y n - l • d(n - 1) - nr1 n~·

(lS)

If h represents the distance of the first unit plane .p in front

+ + of the vertex S1 , and !t' the distance of the second unit plane
.p' behind the vertex S2 , then f = h <Y and f' h' = a-',
and, from (16), (17), and (r8), it follows that

= + /2

r 1d

1 d(n - 1) - nr1 nr/

= + '

- r2d

h diln - 1) - nr1 nr2•

(20)

Also, since the distance p between the two unit planes .p and
.p' is p = d + h + h', it follows that

(21)

Sincef = f ' , the nodal and unit points coincide (cf. page 23). From these equations it appears that the character of the
system is not determined by the radii r 1 and r2 alone, but that the thickness d of the lens is also an essential element. For example, a double convex lens (r1 positive, r 2 negative), of

THEORY OF OPTICS
not too great thickness d, acts as a convergent system, i.e. possesses a positive focal length; on the other hand it acts as a divergent system when dis very great.
4, Thin Lenses. -In practice it often occurs that the thickness d of the lens is so small that d(n - I) is negligible in comparison with n(r1 - r 2). Excluding the case in which
= r 1 r 2 , which occurs in concavo-convex lenses of equal radii,
equation ( I 6) gives for the focal lengths of the lens

(22)

while equations (19), (20), and (21) show that the unit planes

nearly coincide with the nearly coincident tangent planes at

the two vertices sl and s2. More accurately these equations give, when d(n - 1) is

neglected in comparison to n(r1 - r 2),

l

z

=

d -n

-r·1

--r 1-r,1

,
lz

=

d +n

-r

·
1

--r 2-r

,
2

n- 1 P=d-n -·

(23)

Thus the distance p between the two unit planes is indepen-
= = dent of the radii of the lens. For n I. 5, p }d. For both

double-convex and double-concave lenses, since h and h' are

negative, the unit planes lie inside of the lens. For equal
curvature r 1 = - r 2 , and for n = 1.5, lz = h' = - }d, i.e.
the distance of the unit planes from the surface is one third

the thickness of the lens. When r 1 and r2 have the same sign the lens is concavo-convex and the unit planes may lie outside

of it.

Lenses of positive focal lengths (convergent lenses) include

= Double-convex lenses (r1 > o, r2 < o),
Plano-convex lenses (r1 > o, r2 oo )
> Concavo-convex lenses (r1 > o, r2 > o, ,,2 r 1),

in short all lenses which are thicker in the middle than at the

e.dges.

PHYSICAL CONDITIONS FOR IMAGE FORMATION 43
Lenses of negative focal length (divergent lenses) include
Double-concave lenses (r1 < o, r 2 > o),
Plano-concave lenses (r1 = oo , r 2 > o),
Convexo-concave lenses (r1 > o, > 1·2 o, < r2 r1),
i.e. all lenses which are thinner in the middle than at the edges.*
The relation between image and object is shown diagrammatically in Figs. 2 I and 22, which are to be interpreted in

Z

3

F

F'~

.J• FIG. 21.
the same way as Figs. I 7 and I 8. From these it appears that whether convergent lenses produce real or virtual images of

1

z

F'

FIG. 22.
real objects depends upon the distance of the object from the
lens; but divergent lenses produce only virtual images of real
* The terms collective (dioptric), for systems of positive focal length, dispersive,
for those of negative focal length, have been chogen on account of this property ol lenses. A lens of positive focal length renders an incident beam more convergent, one of negative focal length renders it more divergent. When images are formed by a system of lenses, or, in general, when the unit planes do not coincide, say, with the first refracting surface, the conclusion as to whether the system is convergent or divergent cannot be so immediately drawn. Then recourse must be had to the definition on page 26,

44

THEORY OF OPTICS

objects. However, divergent lenses produce real, upright,
and enlarged image'> of virtual objects which lie behind the
lens and inside of the principal focus.
If two thin lenses of focal lengths ft and /4 are united to
form a coaxial system, then the separation L1 (cf. page 40) is
L1 = - ( f1 + f,). Hence, from equation (19) of Chapter II
(page 29), the focal length of the combination is

f = li.l+i.hh= f''

or

I

I

I

f = f1+ h.

It is customary to call the reciprocal of the focal length of a lens its power. Hence the law: The power of a combination of thin lenses is equal to the sum of the powers of the separate lenses.
5. Experimental Determination of Focal Length.-For
thin lenses, in which the two unit planes are to be considered as practically coincident, it is sufficient to determine the positions of an object and its image in order to deduce the focal length. For example, equation (11) of Chapter II, page 23, reduces here, sincef=f, to

Since the positions of real images are most conveniently determined by the aid of a screen. concave lenses, which furnish only virtual images of real objects, are often combined with a convex lens of known power so that the combination furnishes a real image. The focal length of the concave lens is then easily obtained from (24) when the focal length of the combination has been experimentally determined. This pro• cedure is not permissible for thick lenses nor for optical sy!'!tems generally. The positions of the principal foci are readily deter~

PHYSICAL CONDITIONS FOR IMAGE FORMATION 45

mined by means of an incident beam of parallel ray'>. If then the positions of an object and its image with respect to the principal foci be determined, equations (7), on page 19, or (9), on page 22, give at once the focal length/ ( =/').
Upon the definition of the focal length given in Chapter II, page 20 (cf. equation (8)), viz.,

f=y': tan u, .

it is easy to base a rigorous method for the determination of focal length. Thus it is only necessary to measure the angular magnitude u of an infinitely distant object, and the linear magnitude y' of its image. This method is particularly convenient to apply to the objectives of telescopes which are mounted upon a graduated circle so that it is at once possible to read off the visual angle u.
If the object of linear magnitude y is not at infinity, but is
at a distance e from the unit plane .p, while its image of linear magnitude y' is at a distance e' from the unit plane .p', then

y':y=-e':e,
= because, when f f', the nodes coincide with the unit points,
i,e. object and image subtend equal angles at the unit points. By eliminating e and e' from (25) and (27) it follows that

f=

e
y

I - - y'

--e-' ,.
I-~
y

Now if either e or e' are chosen large, then without appreciable error the one so chosen may be measured from the centre of the optical system (e.g. the lens), at least unless the unit planes are very far from it. Then either of equations (28) may be used for the determination of the focal length f when e or e' and the magnification y': y have been measured.
The location of the positions of the object or image may be avoided by finding the magnification for two positions of

THEORY OF OPTICS
the object which are a measured distance l apart. For, from (7), page 19,
hence
in which (y :y')1 denotes the reciprocal of the magnification for
+ the position x of the object, (y : y')2 the reciprocal of the mag-
nification for a position x l of the object. l is positive if, in passing to its second position, the object has moved the distance / in the direction of the incident light (i.e. from left to right).
Abbe's focometer, by means of which the focal lengths of microscope objectives can be determined, is based upon this principle. For the measurement of the size of the image y' a second microscope is used. Such a microscope, or even a simple magnifying-glass, may of course be used for the measurement of a real as well as of a virtual image, so that this method is also applicable to divergent lenses, in short to all cases.*
6. Astigmatic Systems.-In the previous sections it has been shown that elementary beams whose rays have but a small inclination to the axis and which proceed from points either on the axis or in its immediate neighborhood may be brought to a focus by means of coaxial spherical surfaces.
In this case all the rays of the beam intersect in a single point
of the image space, or, in short, the beam is homocentrz'c in the image space. What occurs when one of the limitations imposed above is dropped will now be considered, i.e. an
• A more detailed account of the focometer and of the determination of focal lengths is given by Czapski in Winkelmann, Handbuch der Physik, Optik, 1'P· z85-2g6.

PHYSICAL CONDITIONS FOR IMAGE FORMATION 47
elementary beam having any tizclination to the axis will now be assumed to proceed from a point P.
In this case the beam is, in general, no longer homocentric in the image space. An elementary beam which has started from a luminous point P and has suffered reflections and refractions upon surfaces of any arbitrary form is so constituted that, by the law of Malus (cf. page 12), it must be classed as an orthotomic beam, i.e. it may be conceived as made up of the normals N to a certain elementary surface ~- These normals, however, do not in general intersect in a point. Nevertheless geometry shows that upon every surface ~ there are two systems of curves which intersect at right angles (the so-called lines of curvature) whose normals, which are also at right angles to the surface ~. intersect.
If a plane elementary beam whose rays in the image space are normal to an element /1 of a line of curvature be alone considered, it is evident that an image will be formed. The image is located at the centre of curvature of this element /1 , since its normals intersect at that point. Since every element / 1 of a line of curvature is intersected at right angles by some other element /2 of another line of curvature, a second elementary beam always exists which also produces an image, but the positions of these two images do not coincide, since in general the curvature of /1 is different from that of /2•
What sort of an image of an object P will then in general be formed by any elementary beam of three dimensions ? Let I, 2, 3, 4 (Fig. 23) represent the four intersections of the four lines of curvature which bound the element d~ of the -surface ~- Let the curves I-2 and 3-4 be horizontal, 2-3 and r-4 vertical. Let the normals at the points r and 2 intersect at r2, those at 3 and 4 at 34. Since the curvature of the line I-2 differs by an infinitely small amount from that of the line 3-4, the points of intersection z2 and 34 lie at almost the same distance from the surface ~- Hence the line p1 which connects the points r2 and 34 is also nearly perpendicular to the ray S which passes through the middle of d~ and is normal to it.

THEORY OF OPTICS
This ray is called the principal ray of that elementary beam which is composed of the normals to d~. From the symmetry of the figure it is also evident that the line p 1 must be parallel to the lines 2-3 and r--4, i.e. it is vertical. The normals to any horizontal line of curvature intersect at some point of the line p 1.
FIG. 23.
Likewise the normals to any vertical line of curvature
intersect at some point of the line p2 which connects I 4 and 23. Also, p2 must be horizontal and at right angles to S. These
two lines p 1 and p 2 , which are perpendicular both to one another and to the principal ray, are called the two focal lz"nes of the elementary beam. The planes determined by the principal ray Sand the two focal lines p 1 andp2 are called the focal planes of the beam. It can then be said that in general the image of a luminous point P, formed by any elementary beam, consists of two focal lines which are at right angles to each other and to the principal ray, and lie a certain distance apart. This distance is called the astigmatic difference. Only in special cases, as when the curvatures of the two systems of lines of curvature are the same, does a homocentric crossing of the rays and a true image formation take place. This present more general kind of image formation will be called astigma#c in order to distinguish it from that considered above.*
A sharp, recognizable image of a collection of object points
P is not formed by an astigmatic system. Only when the
• Stigma means focus, hence an astigmatic beam is one which has no focus.

PHYSICAL CONDITIONS FOR IMAGE FORMATION 49
object is a straight line can a straight-line image be formed; and only then when the line object is so placed that all the focal lines which are the images of all the points P of the line object coincide. Since the image of every point consists of
two focal lines p1 and p2 which are at right angles to each
other, there are also two positions of the line object 90° apart which give rise to a line image. These two images lie at different distances from the surface ~.
Similarly there are two orientations of a system of parallel straight lines which give rise to an image consisting of parallel straight lines.
If the object is a right-angled cross or a network of lines at right angles, there is one definite orientation for which an image of one line of the cross or of one system of parallel lines of the network is formed in a certain plane ~ 1 of the image space; while in another plane ~ 2 of the image space an image of the other line of the cross or of the other system of lines of the network is formed. This phenomenon is a good test for astigmatism.
Astigmatic images must in general be formed when the elementary refracting or reflecting surface has two different curvatures. Thus cylindrical lenses, for example, show marked astigmatism. Reflection or refraction at a spherical surface also renders a homocentric elementary beam astigmatic when the incidence is oblique.
In order to enter more fully into the consideration of this case. let the point object P, the centre C of the sphere, and the point A in which the principal ray of the elementary beam emitted by P strikes the spherical surface, lie in the plane of the figure (Fig. 24). Let the line PA be represented by s, the line AP2 by s2• Now since
it follows that
= + ss2 sin {</J - </J') sr sin </J s2r sin </J',

50

THEORY OF OP TICS

in which cp and cp' denote the angles of incidence and refraction respectively, and r the radius of the sphere. Since now
by the law of refraction sin ¢ = n sin cp', it follows from the
last equation that
= + ssin cos cp' - cos <P) srn s,r, or

-s I

+n ~2

-

= n co-s ¢-',r--c-os-cp

(~

It is evident that all rays emitted by P which have the same angle of inclination u with the axis must, after refraction, cross

Pz
p
Tl,

FIG. 24.
the axis at the same point P 2• The beam made up of such rays is called a sagittal beam. It has a focal point at P 2•
On the other hand a meridional beam, i.e. one whose rays all lie in the plane PAC, has a different focal point Pr Let E'B be a ray infinitely near to PA, and let its angle of inclina-
+ tion to the axis be u dtt and its direction after refraction
BP1. Then :?f.BP1A is to be considered as the increment du' of u', and :?f_BCA as the increment dtr. of a. It is at once evident that
= s. du= AB cos</>, s!. du'= AB. cos</>', r. da AB. (31)

But since

= ¢ ff -f- 11.- cl>' :.-= ff - tt',

PHYSICAL CONDITIONS FOR IMAGE FORMATION 51

it follows that

r dcp

=

da

+

du

=

AB( I

+

cos c/J)
-s- ,

(I d</>I = da - duI = AB -r - - coss- 1<P') .

(32)

= But a differentiation of the equation of refraction sin <P

n sin cp' gives

cos cp . dcp = n cos cp' . dcp'.

Substituting in this the values of dcp and dcp' taken from (32), there results

cos2 <P n cos2 cp' n cos <P' - cos cp

-s- + -s1 - = - -r - · ·

(33)

From (33) and (30) different values s1 and s2 corresponding to the sames are obtained, i.e. Pis imaged astigmatically. The
astigmatic difference is greater the greater the obliquity of the incident beam, i.e. the greater the value of cp. It appears from (30) and (33) that this astigmatic difference vanishes, i.e. s1 = s2 = s', only when s = - ns'. This condition determines the two aplanatic points of the sphere mentioned on page 33.
The equations for a reflecting spherical surface may be
deduced from equations (30) and (33) by substituting in them
n = - I, i.e. <P' = - <P (cf. page 37). Thus for this case*

I

I

COS c/J I

-s

-

-=
s2

-

2

-

r -

,

s

Or by subtraction,

2
r cos </>. • (34)

!_ s1

= ~
s2

-r=c-o(-s1- cp -

cos cp),

or

= - s - s
- 2- s1s- 2 1

2
r

sin cp tan ¢,

.

(35)

*Fora convex mirror r is positive; for a concave, negative.

52

THEORY OF OPTICS

an equation which shows clearly how the astigmatism increases with the angle of incidence. This increase is so rapid that the astigmatism caused by the curvature of the earth may, by suitable means, be detected in a beam reflected from the surface of a free liquid such as a mercury horizon. Thus if the reflected image of a distant rectangular network be observed in
a telescope of 7. 5 m. focal length and ½ m. aperture, the
astigmatic difference amounts to -lo mm., i.e. the positions in which the one or the other system of lines of the network is in sharp focus are lo mm. apart. In the giant telescope of the Lick Observatory in California this astigmatic difference
amounts to j\, mm. Thus the phenomena of astigmatism may
be made use of in testing the accuracy of the surface of a plane mirror. Instead of using the difference in the positions of the images of the two systems of lines of the network, the angle of incidence being as large as possible, the difference in the sharpness of the images of the two systems may be taken as the criterion. For this purpose a network of dotted lines may be used to advantage.
7. Means of Widening the Limits of Image Formation.
-It has been shown above that an image can be formed by refraction or reflection at coaxial spherical surfaces only when the object consists of points lying close to the axis and the indination to the axis of the rays forming the image is small. If the elementary beam has too large an inclination to the axis, then, as was shown in the last paragraph, no image can be formed unless all the rays of the beam lie in one plane.
Now such arrangements as have been thus far considered for the formation of images would in practice be utterly useless. For not only would the images be extremely faint if they were produced by single elementary beams, but also, as will be shown in the physical theory (cf. Section I, Chapter IV), single elementary beams can never produce sharp images, but only diffraction patterns.
Hence it is necessary to look about for means of widening the limits hitherto set upon image formation. In the first place

PHYSICAL CONDITIONS FOR IMAGE FORMATION 53
the limited sensitiveness of the eye comes to our assistance: we are unable to distinguish two luminous points as separate unless they subtend at the eye an angle of at least one minute. Hence a mathematically exact point image is not necessary, and for this reason alone the beam which produces the image does not need to be elementary in the mathematical sense, i.e. one of infinitely small divergence.
By a certain compromise between the requirements it is possible to attain a still further widening of the limits. Thus it is possible to form an image with a broadly divergent beam if the object is an element upon the axis, or to form an image of an extended object if only beams of small divergence are used. The realization of the first case precludes the possibility of the realization of the second at the same time, and vz'ce versa.
That the image of a point upon the axis can be formed by a widely divergent beam has been shown on page 33 in connection with the consideration of aplanatic surfaces. But this result can also be approximately attained by the use of a suitable arrangement of coaxial spherical surfaces. This may be shown from a theoretical consideration of so-called spherical aberration. To be sure the images of adjacent points would not in general be formed by beams of wide divergence. In fact the image of a surface element perpendicular to the axis can be formed by beams of wide divergence only if the socalled sine law is fulfilled. The objectives of microscopes and telescopes must be so constructed as to satisfy this law.
The problem of forming an image of a large object by a relatively narrow beam must be solved in the construction of the eyepieces of optical instruments and of photographic systems. In the latter the beam may be quite divergent, since, under some circumstances (portrait photography), only fairly sharp images are required. These different problems in image formation will be more carefully considered later. The formation of images in the ideal sense first considered, i.e. when the objects have any size and the beams any divergence, is, to be

54

THEORY OF OPTICS

sure, impossible, if for no other reason, simply because, as will be seen later, the sine law cannot be simultaneously fulfilled for more than one position of the object.
8. Spherical Aberration.-If from a point Pon the axis two rays S1 and S2 are emitted of which S1 makes a very small angle with the axis, while S2 makes a finite angle u, then, after refraction at coaxial spherical surfaces, the image rays S/ and S2' in general intersect the axis in two different points P 1' and Pt The distance between these two points is known as
the spherical aberration (longitudinal aberration). In case the angle tt which the ray S2 makes with the axis is not too great, this aberration may be calculated with the aid of a series of ascending powers of tt. If, however, u is large, a direct trigonometrical determination of the path of each ray is to be preferred. This calculation wiil not be given here in detail.* For relatively thin convergent lenses, when the object is distant, the image P 1 formed by rays lying close to the axis is farther from the lt!ns than the image Pi formed by the more oblique rays. Such a lens, i.e. one for which P 2 lies nearer to the object than P 1 , is said to be undercorrectt·d. Inversely, a lens for which P 2 is more remote from the object than P 1 is said to be overcorrected. Neglecting all terms of the power series in tt save the first, which contains u2 as a factor, there results for this so-called aberration of the first order, if the object P is very distant,

in which h represents the radius of the aperture of the lens, fits focal length, n its index of refraction, and <T the ratio of its radii of curvature, i.e.
(37)
* For a more complete discussion cf. Winkelmann's Handbuch der Physik,
Optik, p. 99 sq. ;Moller-Pouillet's Lehrbucb d. Physik, 9th Ed. p. 4,87 ; or Heath. Geometrical Optics.

PHYSICAL CONDITIONS FOR IMAGE FORMATION 55

The signs of r 1 and r 2 are determined by the conventions adopted on page 40; for example, for a double-convex lens r 1 is positive, r 2 negative. P/P/ is negative for an undercorrected lens, positive for an overcorrected one. Further, the
ratio h :f is called the relative aperture of the lens. It
appears then from (36) that if <T remains constant, the ratio of the aberration P/P 2' to the focal length f is directly proportional to the square of the relative aperture of the lens.
For given values off and h the aberration reaches a minimum for a particular value u' of the ratio of the radii.* By (36) this value is

= <T'

4+ n - 2n2
+ n(I 2nJ •

= = - For n 1.5, <T

l : 6. This condition may be realized

either with a double-convex or a double-concave lens. The

surface of greater curvature must be turned toward the incident

beam. But if the object lies near the principal focus of the

lens, the best image is formed if the surface of lesser curvature

is turned toward the object; for this case can be deduced from

that above considered, i.e. that of a distant object, by simply
= interchanging the roles of object and image.t For n 2,
+ (38) gives u' = !- This condition is realized in a con-

vexo-concave lens whose convex side is turned toward a dis-

tant object P.

The following table shows the magnitude of the longi-

tudinal aberration E for two different indices of refraction and

for different values of the ratio <T of the radii. f has been
assumed equal to I m. and h :f = io, i.e. = h IO cm. The

so-called lateral aberration C, i.e. the radius of the circle

which the rays passing through the edge of a lens form upon

* This minimum is never zero. A complete disappearance of the aberration
of the first order can only be attained by properly choosing the thickness of the
lens as well as the ratio of the radii.
t It follows at once that the form of the lens which gives minimum aberration
depends upon the position of the object.

THEORY OF OPTICS
a screen placed at the focal point P/, is obtained, as appears at once from a construction of the paths of the rays, by multiplication of the longitudinal aberration by the relative aperture h :f, i.e. in this case by T10 . Thus the lateral aberration determines the radius of the illuminated disc which the outside rays from a luminous point P form upon a screen placed in the plane in which Pis sharply imaged by the axial rays.
f = I m. h = 10 cm.

II= 1.5

•=2

Form of lens ............. .... CT - l '

C

CT

-l'

C

Front face plane .............. Both sides alike............... Rear face plane ............... Most advantageous form .....

. . 00 4.5 cm 4.5 mm
-I 1.67 1.67
. . 0 1.17 1.17 . -¼ 1.07 .. 1.07

. . 00 2 cm2 mm

. . -I I

I

0 0.5 0.5

.. +¼ 0.44 0.44"

That a plano-convex lens produces less aberration when its convex side is turned toward _a distant object than when the sides are reversed seems probable from the fact that in the first case the rays are refracted at both surfaces of the lens, in the second only at one; and it is at least plausible that the distribution of the refraction between two surfaces is unfavorable to aberration. The table further shows that the most favorable form of lens has but little advantage over a suitably placed plano-convex lens. Hence, on account of the greater ease of construction, the latter is generally used.
Finally the table shows that the aberration is very much less if, for a given focal length, the index of refraction is made large. This conclusion also holds when the aberration of a higher order than the first is considered, i.e. when the remaining terms of the power series in u are no longer neglected. Likewise the aberration is appreciably diminished when a single lens is replaced by an equivalent system of several

PHYSICAL CONDITIONS FOR IMAGE FORMATION 57
lenses.* By selecting for the compound system lenses of different form, it is possible to cause the aberration not only
t of the first but also of still higher orders to vanish. One
system can be made to accomplish this for more than one position of the object on the axis, but never for a finite length of the ax'is.
When the angle of inclination u is large, as in microscope objectives in which tt sometimes reaches a value of 90°, the power series in tt cannot be used for the determination of the
aberration. It is then more practicable to determine the paths
of several rays by trigonometrical calculation, and to find by trial the best form and arrangement of lenses. There is, how~ ever, a way, depending upon the use of the aplanatic points of a sphere mentioned on page 33, of diminishing the divergence of rays proceeding from near objects without introducing aberration, i.e. it is possible to produce virtual images of any size, which are free from aberration.
Let lens I (Fig. 25) be piano-convex, for example, a hemi-
FIG. 25.
spherical lens of radius r 1 , and let its plane surface be turned toward the object P. If the medium between P and this lens has the same index n1 as the lens, then refraction of the rays
* In this case, to be sure, the brightness of the image suffers somewhat on
account of the increased loss of light by reflection.
f Thus the aberration of the first order can be corrected by a suitable com-
bination of a amvergent and a divergent lens.

58

THEORY OF OPTICS

proceeding from the object first takes place at the rear surface
of the lens; and if the distance of P from the centre of curvature C1 of the back surface is r1 : n1 , then the emergent rays
produce at a distance n1r 1 from C1 a virtual image P 1 free from aberration. If now behind lens I there be placed a second concavo-convex lens 2 whose front surface has its centre of curvature in P 1 and whose rear surface has such a radius r 2 that P 1 lies in the aplanatic point of this sphere r2 (the index of lens 2 being n2), then the rays are refracted only at this rear surface, and indeed in such a way that they form a virtual image P 2 which lies at a distance n2r 2 from the centre of curvature C2 of the rear surface of lens 2, and which again is entirely free from aberration. By addition of a third, fourth, etc., concavo-convex lens it is possible to produce successive virtual images P8 , P4 , etc., lying farther and farther to the left, i.e. it is possible to diminish successively the divergence of the rays without introducing aberration.
This principle, due to Amici, is often actually employed in the construction of microscope objectives. Nevertheless no more than the first two lenses are constructed according to this principle, since otherwise the chromatic errors which are introduced are too large to be compensated (cf. below).
9. The Law of Sines.-In general it does not follow that if a widely divergent beam from a point P upon the axis gives rise to an image P' which is free from aberration, a surface element du perpendicular to the axis at P will be imaged in a surface element d<T' at P'. In order that this may be the case the so-called sine law must also be fulfilled. This law requires that if u and u' are the angles of inclination of any two
= conjugate rays passing through P and P', sin u: sin u' const.
According to Abbe systems which are free from aberration for two points P and P' on the axis and which fulfil the sine law for these points are called aplanatic systems. The points P and P' are called the aplanatic points of the system. The aplanatic points of a sphere mentioned on page 33 fulfil these conditions, since by equation (2), page u. the ratio of the

PHYSICAL CONDITIONS FOR IMAGE FORMATION 59
sines is constant. The two foci of a concave mirror whose surface is an ellipsoid of revolution are not aplanatic points although they are free from aberration.
It was shown above (page 22, equation (9), Chapter II) that when the image of an object of any size is formed by a
= collinear system, tan tt : tan u' const. Unless u and 111 are
very small, this condition is incompatible with the sine law, and, since the latter must always be fulfilled in the formation of the image of a surface element, it follows that a point-forpoi1tt imaging of objects of any size by widely di'l'ergent beams is pltysz'cally z'mpossible.
Only when 11 and u' are very small can both conditions be simultaneously fulfilled. In this case, whenever an image P' is formed of P, an image du' will be formed at P' of the surface element dcr at P. But if Y is large, even though the spherical aberration be entirely eliminated for points on the axis, unless the sine condition is fulfilled the images of points which lie to one side of the axis become discs of the same order of magnitude as the distances of the points from the axis. According to Abbe this blurring of the images of points lying off the axis is due to the fact that the different zones of a spherically corrected system produce images of a surface element of different linear magnifications.
The mathematical condition for the constancy of this linear magnification is, according to Abbe, the sine law.* The same
conclusion was reached in different ways by Clausius t and v.
Helmholtz t. Their proofs, which rest upon considerations of
energy and photometry, will be presented in the third division of the book. Here a simple proof due to Hockin § will be given which depends only on the law that the optical lengths of all rays between two conjugate points must be equal (cf.
* Carl's Repert. f. Physik, 1881, 16, p. 303.
t R. Clausius, Mechanische W!irrnetheorie, 1887, 3d Ed. 1, p. 315. t v. Helmholtz, Pogg. Ann. Jubelbd. 1874, p. 557.
§ Hockin, Jour. Roy. Microsc. Soc. 1884, (2), 4. p. 337•

60

THEORY OF OPTICS

page 9).* Let the image of P (Fig. 26) formed by an axial ray PA and a ray PS of inclination tt lie at the axial point P'. Also let the image of the infinitely near point P 1 formed by a ray P 1A 1 parallel to the axis, and a ray P 1S 1 parallel to PS, lie at the point P/. The ray F'P/ conjugate to P 1A1 must evidently pass through the principal focus F' of the image space. If now the optical distance between the points P and P' along the path through A be represented by (PAP'), that

Fm. 26.

along the path through SS' by (PSS'P'), and if a similar notation be used for the optical lengths of the rays proceeding from P 1 , then the principle of extreme path gives
= (PAP')= (PSS'P'); (P1A 1F'P/) (P1S1S/P/),

and hence

= (PAP') - (P1A 1F'P/) (PSS'P') - (P1S1S/P/).

(39)

Now since F' is conjugate to an infinitely distant object Ton the axis, ( TPAF') = ( TP1A 1F'). But evidently TP = TP1 , since PP1 is perpendicular to the axis. Hence by subtraction

* According to Bruns (Abh. d. sachs. Ges. d. Wiss. Bd. 21, p. 325) the sine law can be based upon still more general considerations, namely, upon the law of Malus (cf. p, 12) and the existence of conjugate rays.

PHYSICAL CONDJTIONS FOR IMAGE FORMATION 61

Further, since P'P/ is perpendicular to the axis, it follows
= that when P'P/ is small F'P' F'Pi', Hence by addition

(PAPP')= (P1A 1PP/), i.e. the left side of equation (39) vanishes. Thus

(PSS'P') = (P1S1S/P/). .

Now if Fi' is the intersection of the rays P'S' and P/S/, then F/ is conjugate to an infinitely distant object T1 , the rays from which make an angle u with the axis. Hence if a perpendicular PN be dropped from P upon P 1S1 , an equation similar to (40) is obtained; thus

= (PSS' F/) (NS1S/F/).

(42)

By subtraction of this equation from (41),

(43)

If now n is the index of the object space, n' that of the image space, then, if the unbracketed letters signify geometrical lengths,

(44)

Further, if P'N' be drawn perpendicular to F/P', then, since

P'P/ is infinitely small,

= = (F/P/) - (1'/P') n' .N'P/ n' -P'P/ • sin u'.

(45)

Equation (43) in connection with (44) and (45) then gives
n,PP1 -sin u = n' ,P'Pi' • sin u'.
Ify denote the linear magnitude PP 1 of the object, and y' the linear magnitude P'P/ of the image, then

sin u n'y'
sin u' = ny • •

Thus it is proved that if the linear magnification is constant the ratio of the sines is constant, and, in addition, the value of this constant is determined. This value agrees with

THEORY OF OPTICS that obtained in equation (2), page 34, for the aplanatic points of a sphere.
The sine law cannot be fulfilled for two different points on the axis. For if P' and P/ (Fig. 27) are the images of P and P1 , then, by the principle of equal optical lengths,
= (PAP')= (PSS'P'), (P1AP/) (P1S1S/P/), . (47)
in which PS and P/).1 are any two parallel rays of inclination u.
P'
n:
Subtraction of the two equations (47) and a process d reasoning exactly like the above gives
or
= n-P1P(1 - cos u) n' -P/P' (1 - cos u'),
i.e. sin2 tu n' -P'P/
sin2 ½it'= n .PP1- •
This equation is then the condition for the formation, by a beam of large divergence, of the image of two neighboring points upon the axis, i.e. an image of an element of the axis.
However this condition and the sine law cannot be fulfilled at the same time. Thus an op#cal system can be made aplanatic for but one position of the object

PHYSICAL CONDITIONS FOR IMAGE FORMATION 63 The fulfilment of the sine law is especially important in the case of microscope objectives. Although this was not known from theory when the earlier microscopes were made, it can be experimentally proved, as Abbe has shown, that these old microscope objectives which furnish good images actually satisfy the sine law although they were constructed from purely empirical principles. 10. Images of Large Surfaces by Narrow Beams.-lt is necessary in the first place to eliminate astigmatism (cf. page 46). But no law can be deduced theoretically for accomplishing this, at least when the angle of inclination of the rays with respect to the axis is large. Recourse must then be had to practical experience and to trigonometric calculation. It is to be remarked that the astigmatism is dependent not only upon the form of the lenses, but also upon the position of the stop. Two further requirements, which are indeed not absolutely essential but are nevertheless very desirable, are usually im-
FIG. 28.
posed upon the image. First it must be plane, i.e. free from bulging, and second its separate parts must have the same magnification, i.e. it must be free from distortion. The first requirement is especially important for photographic objectives.

THEORY OF OPTICS

For a complete treatment of the analytical conditions for this

requirement cf. Czapski, in Winkelmann's Handbuch der

Physik, Optik, page 124.

The analytical condition for freedom from distortion may

be readily determined. Let PP1P 2 (Fig. 28) be an object

plane, P'P/P/ the conjugate image plane. The beams from

the object are always limited by a stop of definite size

which may be either the rim of a lens or some specially intro-

duced diaphragm. This stop determines the position of a

virtual aperture B, the so-called entrance-pupil, which is so

situated that the principal rays of the beams from the objects

P P etc., pass through its centre. Likewise the beams in

,

,

1

2

the image space are limited by a similar aperture B', the

so-called exit-pupil, which is the image of the entrance-pupil.*

If l and l' are the distances of the entrance-pupil and the exit-

pupil from the object and image planes respectively, then, from

the figure,

tan u = PP l, tan u = PP I,

:

:

1

1

2

2

= tan u/ P'P/ : l', tan u2' = P'P/ : /'.

If the magnification is to be constant, then the following relation must exist:

hence

= tan u/
-tan-u- =

-ttaan-n uu-2'

const.

(49)

1

2

Hence for constant magnification the ratio of the tangents of the angles of inclination of the principal rays must be constant. In this case it is customary to call the intersections of the principal rays with the axis, i.e. the centres of the pupils, ortltoscopic points. Hence it may be said that, if the image i's to be free from distortion, the centres of perspective of object and image must be orthoscopi'c points. Hence the positions of the pupils are of great importance.

* For further treatment see Chapter IV.

PHYSICAL CONDITIONS FOR IMAGE FORMATION 65
An example taken from photographic optics shows how the condition of orthoscopy may be most simply fulfilled for the case of a projecting lens. Let R (Fig. 29) be a stop on either side of which two similar lens systems I and 2 are symmetrically placed. The whole system is then called a symmetrical double objective. Let S and S' represent two conjugate principal rays. The optical image of the stop R with respect to the system I is evidently the entrance-pupil, for, since all principal rays must actually pass through the centre of the stop R, the prolongations of the incident principal rays S must pass through the centre of B, the optical image of R with respect to I. Likewise B', the optical image of R with respect to 2, is the exit-pupil. It follows at once from the symmetry of arrangement that tt is always equal to tt', i.e. the condition of orthoscopy is fulfilled.
FIG. 29.
Such symmetrical double objectives possess, by virtue of their symmetry, two other advantages: On the one hand, the meridional beams are brought to a sharper focus,* and, on the other, chromatic errors, which will be more fully treated in the
= next paragraph, are more easily avoided, The result u u',
which means that conjugate principal rays are parallel, is altogether independent of the index of refraction of the system,
* The elimination of the error of coma is here meant. {;f. Mtlller-Pouillet,
Optik, p. 774-

66

THEORY OF OPTICS

and hence also of the color of the light. If now each of the two systems r and 2 is achromatic with respect to the position of the image which it forms of the stop R, i.e. if the positions of the entrance- and exit-pupils are independent of the color,* then the principal rays of one color coincide with those of every other color. But this means that the images formed in the image plane are the same size for all colors. To be sure, the position of sharpest focus is, strictly speaking, somewhat different for the different colors, but if a screen be placed in sharp focus for yellow, for instance, then the images of other colors, which lie at the intersections of the principal rays, are only slightly out of focus. If then the principal rays coincide for all colors, the image will be nearly free from chromatic error.
The astigmatism and the bulging of the image depend upon the distance of the lenses r and 2 from the stop R. In general, as the distance apart of the two lenses increases the image becomes flatter, i.e. the bulging decreases, while the astigmatism increases. Only by the use of the new kinds of glass made by Schott in Jena, one of which combines large dispersion with small index and another small dispersion with '1arge index, have astigmatic flat images become possible. This will be more fully considered in Chapter V under the head of Optical Instruments.
u. Chromatic Aberration of Dioptric Systems.-Thus
far the index of refraction of a substance has been treated as though it were a constant, but it is to be remembered that for a given substance it is different for each of the different colors contained in white light. For all transparent bodies the index continuously increases as the color changes from the red to the blue end of the spectrum. The following table contains the indices for three colors and for two different kinds of glass. nc is the index for the red light corresponding to the Fraun-
*As will be seen later, this achromatizing can be attained with sufficient accu-
racy; on the other hand it is not possible at the same time to make the sizes oftbe uift"erent images of R independent of the color.

PHYSICAL CONDITIONS FOR IMAGE FORMATION 67

hofer line C of the solar spectrum (identical with the red hydrogen line), nD that for the yellow sodium light, and n.F that for the blue hydrogen line.

Glass.

nF- •c

•c

nD

nF

v

=• »-

--

-
J.

Calcium-silicate-crown...... 1.5153 Ordinary silicate-flint....... 1.6143

1.5179 1.6202

1.5239 1.6314

0.0166 0.0276

The last column contains the so-called dispersive power v, of the substance. It is defined by the relation

=· y nF - nc

(50)

nD- l

It is practically immaterial whether nD or the index for any other color be taken for the denominator, for such a change

can never affect the value of v by more than 2 per cent.

Since now the constants of a lens system depend upon the

index, an image of a white object must in general show colors,

i.e. the differently colored images of a white object differ from one another in position and size.

In order to make the red and blue images coincide, i.e. in order to make the system achromatic for red and blue, it is necessary not only that the focal lengths, but also that the

unit planes, he identical for both colors. In many cases a partial correction of the chromatic aberration is sufficient.

Thus a system may he achromatized either by making the focal

length, and hence the magnification, the same for all colors;

or by making the rays of all colors come to a focus in the same

plane. In the former case, though the magnification is the

same, the images of all colors do not lie in one plane; in the

latter, though these images lie in one plane, they differ in size.

A system may be achromatized one way or the other according

to the purpose for which it is intended, the choice depending

upon whether the magnification or the position of the image is

most important.

68

THEORY OF OPTICS

A system which has been achromatized for two colors, e.g. red and blue, is not in general achromatic for all other colors, because the ratio of the dispersions of different substances in different parts of the spectrum is not constant. The chromatic errors which remain because of this and which give rise to the so-called secondary spectra are for the most part unimportant for practical purposes. Their influence can be still farther reduced either by choosing refracting bodies for which the lack of proportionality between the dispersions is as small as possible, or by achromatizing for three colors. The chromatic errors which remain after this correction are called spectra of the third order.
The choice of the colors which are to be used in practice in the correction of the chromatic aberration depends upon the use for which the optical instrument is designed. For a system which is to be used for photography, in which the blue rays are most effective, the two colors chosen will be nearer the blue end of the spectrum than in the case of an instrument which is to be used in connection with the human eye, for which the yellow-green light is most effective. In the latter case it is easy to decide experimentally what two colors can be brought together with the best result. Thus two prisms of different kinds of glass are so arranged upon the table of a spectrometer that they furnish an almost achromatic image of the slit; for instance, for a given position of the table of the spectrometer, let them bring together the rays C and F. If now the table be turned, the image of the slit will in general appear colored; but there will be one position in which the image has least color. From this position of the prism it is easy to calculate what two colors emerge from the prism exactly parallel. These, then, are the two colors which can be used with the best effect for achromatizing instruments intended for eye observations.
Even a single thick lens may be achromatized either with reference to the focal length or with reference to the position "lr the focus. But in practice the cases in which thin lenses

PHYSICAL CONDITIONS FOR IMAGE FORMATION 69
are used are more important. When such lenses are combined, the chromatic differences of the unit planes may be neglected without appreciable error, since, in this case, these planes always lie within the lens (cf. page 42). If then the focal lengths be achromatized, the system is almost perfectly achromatic, i.e. both for the position and magnitude of the image.
Now the focal length fi of a thin lens whose index for a
given color is n1 is given by the equation (cf. eq. (22), page 42)
(5 I)
in which k1 is an abbreviation for the difference of the curvatures of the faces of the lens.
Also, by (24) on page 44, the focal length f of a combina-
tion of two thin lenses whose separate focal lengths are J;. and
/2 is given by
For an increment dn1 of the index n1 corresponding to a change of color, the increment of the reciprocal of the focal length is, from (5 I),
(53)
in which Y 1 represents the dispersive power of the material of lens I between the two colors which are used. If the focal ength f of the combination is to be the same for both colors, it follows from (5 2) and (5 3) that
(54)
This equation contains the condition for achromatism. It also shows, since Y1 and Y2 always have the same sign no matter what materials are used for I and 2, that the separate

THEORY OF OPTICS

focal lengths of a thin double achromatic lens always lzaric opposi'te signs.
From (54) and (52) it follows that the expressions for the separate focal lengths are

___ ~

I

I Y1

f1-fv2-v/ .1z=-fv2-v1·

( 55 )

Hence in a combination of positive focal length the lens with the smaller dispersive power has the positive, that with the larger dispersive power the negative, focal length.
Iff is given and the two kinds of glass have been chosen, then there are four radii of curvature at our disposal to make
.Ii and.I; correspond to (55). Hence two of these still remain arbitrary. If the two lenses are to fit together, r/ must be
equal to r2• Hence one radius of curvature remains at our disposal. This may be so chosen as to make the spherical aberration as small as possible.
In microscopic objectives achromatic pairs of this kind are very generally used. Each pair consists of a piano-concave lens of flint glass which is cemented to a double-convex lens of crown glass. The plane surface is turned toward the incident light.
Sometimes it is desirable to use two thin lenses at a greater distance apart; then their optical separation is (cf. page 28)
L1 = a - (.t;_ +J;).
Hence, from (19) on page 29, the focal length of the combination is given by

If the focal length is to be achromatic, then, from (56) and (5 3),

+ + o - Y1

r2 - a( Y1

v2)

- .t.. /2 .li/2 '

or

• (57)

PHYSICAL CONDITIONS FOR IMAGE FORMATION i 1
= ,, If the two lenses are of the same material (v1 2), then, when
they are at the distance
.Ii+ fz
a= 2 ' • they form a system which i's achromatic with respect to the focal
= lcngtlt. Since v1 v2 , this achromatism holds for all colors.
If it is desired to achromatize the system not only with reference to the focal length. but completely, i.e. in respect to both position and magnification of the image, then it follows from Fig. 30 that
i.e. the ratio of the magnifications is (59)
11'

Frn. 30.

If, therefore, the image is to be achromatic both with

respect to magnitude and position, then, since e1 is constant for all colors,

= = d

\' e-1'-

e 2-

'

)

o, dez'

o.

(60)

e2
+ = But since e/ e2 a (distance between the lenses) is also

constant for all colors, it follows that de/= - de2 , while, from
= = = (6o), d(e//e2) o. Hence de/ o and de2 o, i.e. each of

the two separate lenses must be for itself achromatized, i.e.

must consist of an achromatic pair.

Hence the following general conclusion may be drawn:

A combiizatz"on which consists of several separated systems is

THEORY OF OPTICS
only perfectly aclwomatic (i.e. with respect to both position anti magnification of the image) when each system for itself is achromatic.
When the divergence of the pencils which form the image becomes greater, complete achromatism is not the only condition for a good image even with monochromatic light. The spherical aberration for two colors must also be corrected as far as possible; and, when the image of a surface element is to be formed, the aplanatic condition (the sine law) must be fuifilled for the two colors. Abbe calls systems which are free from secondary spectra and are also aplanatic for several colors "apoclwomatic" systems. Even such systems have a chromatic error with respect to magnification which may, however, be rendered harmless by other means (cf. below under the head Microscopes).

CHAPTER IV
APERTURES AND THE EFFECTS DEPENDING UPON THEM.
I. Entrance- and Exit-pupils.-The beam which passes through an optical system is of course limited either by the dimensions of the lenses or mirrors or by specially introduced diaphragms. Let P be a particular point of the object (Fig. 31); then, of the stops or lens rims which are present, that one which most limits the divergence of the beam is found in the following way: Construct for every stop B the optical image B1 formed by that part S1 of the optical system which lies between B and the object P. That one of these images B1 which subtends the smallest angle at the object point P is evidently the one which limits the divergence of the beam. This image is called the entrance-pupil of the whole system. The stop Bis itself called the aperture or iris.* The angle 2U which the entrance-pupil subtends at the object, i.e. the angle included between the two limiting rays in a meridian plane, is called the angular aperture of the system.
The optical image B/ which is formed of the entrance-
pupil by the entire system is called the exit-pupil. This evidently limits the size of the emergent beam which comes to a focus in P', the point conjugate to P. The angle 2U' which the exit-pupil subtends at P' is called the angle ofprojection of the system. Since object and image are interchangeable,
it follows at once that the exit-pupil Bi' is the image of the
* If th,. iris lies in front of the front lens of the system, it is identical with the
entrance-pu pi1. 73

74

THt.URY OF OPTICS

stop B formed by that part 5 2 of the optical system which lies between B and the image space. In telescopes the rim of the objective is often the stop, hence the image formed of this rim by the eyepiece is the exit-pupil. The exit-pupil may be seen, whether it be a real or a virtual image, by holding the

FIG. 31.
instrument at a distance from the eye and looking through it at a bright background.
Under certain circumstances the iris of the eye of the observer can be the stop. The so-called pupil of the eye is merely the image of the iris formed by the lens system of the eye. It is for this reason that the general terms entrancepupil and iris have been chosen.
As was seen on page 52, the position of the pupils is of importance in the formation of images of extended objects by beams of small divergence. If the image is to be similar to the object, the entrance- and exit-pupils must be orthoscopic points. Furthermore the position of the pupils is essential to the determination of the przitcipal rays, i.e. the central rays of the pencils which form the image. If, as will be assumed, the pupils are circles whose centres lie upon the axis of the system, then the rays which proceed from any object point P toward the centre of the entrance-pupil, or from the centre of the exit-pupil toward the image point P', are the principal rays of the object and image pencils respectively. When the

APERTURES AND THEIR EFFECTS

75

paths of the rays in any system are mentioned it will be understood that the paths of the principal rays are meant.
2. Telecentric Systems.-Certain positions of the iris can be chosen for which the entrance- or the exit-pupils lie at infinity (in telescopic systems both lie at infinity). To attain this it is only necessary to place the iris behind S1 at its principal focus or in front of S2 at its principal focus (Fig. 3I). The system is then called tclecentri'c,-in the first case, tclecentri'c on the side of the object; in the second, telecentri'c on the side of the image. In the former all the principal rays in the object space are parallel to the axis, in the latter all those of the image space. Fig. 32 represents a system which is telecentric on the side of the image. The iris B lies in front of and at the principal focus of the lens S which forms the real image P/P/ of the object P 1 and P 2• The principal rays

P,'

FIG. 32.
from the points P 1 and P 2 are drawn heavier than the limiting rays. This position of the stop is especially advantageous when the image P/Pz' is to be measured by any sort of a micrometer.
Thus the image P/Pz' always has the same size whether it
coincides with the plane of the cross-hairs or not. For even with imperfect focussing it is the intersection of the principal rays with the plane of the cross-hairs which determines for the observer the position of the (blurred) image. If then the principal rays of the image space are parallel to the axis, even with improper focussing the image must have the same size as if it lay exactly in the plane of the cross-hairs. But when the principal rays are not parallel in the image space, the apparent

THEORY OF OPTICS
size of the image changes rapidly with a change in the position of the image with respect to the plane of the cross-hairs.
If the system be made telecentric on the side of the object, then, for a similar reason, the size of the image is not dependent upon an exact focussing upon the object. This arrangement is therefore advantageous for micrometer microscopes, while the former is to be used for telescopes, in which the distance of the object is always given (infinitely great) and the adjustment must be made with the eyepiece.
3. Field of View.-In addition to the stop B (the iris), the images of which form the entrance- and exit-pupils, there are always present other stops or lens rims which limit the size of the object whose image can be formed, i.e. which limit the field ofview. That stop which determines the size of the field ofview may be found by constructing, as before, for all the stops the optical images which are formed of them by that part S1 of the entire lens system which lies between the object and each stop. Of these images, that one G1 which subtends the smallest angle 2w at the centre of the entrance-pupil is the one which determines the size of the field of view. 2w is called the angular field of view. The correctness of this assertion is evident at once from a drawing like Fig. 31. In this figure the iris B, the rims of the lenses S 1 and S2 , and the diaphragm G are all pictured as actual stops. The image of G formed by 5 1 is G1 ; and since it will be assumed that GI subtends at the centre of the entrance-pupil a smaller angle than the rim of 5 1 or the image which S 1 forms ofthe rim of the lens S 2 , it is evident that G acts as the field-of-view stop. The optical image G/ which
+ the entire system S 1 5 2 forms of G1 bounds the field of view
in the image space. The angle 2w' which G/ subtends at the centre of the exit-pupil is called the angle of the image.
In Fig. 3 I it is assumed that the image G1 of the field-ofview stop lies in the plane of the object. This case is characterized by the fact that the limits of the field of view are
perfectly sharp, for the reason that every object point P can
either completely fill the entrance-pupil with rays or el~e can.

APERTURES AND THEIR EFFECTS

77

send none to it because of the presence of the stop G1. If the plane of the object does not coincide with the image G1 , the boundary of the field of view is not sharp, but is a zone of continuously diminishing brightness. For in this case it is evident that there are object points about the edge of the field whose rays only partially fill the entrance-pupil.
In instruments which are intended for eye observation it is of advantage to have the pupil of the eye coincide with the exit-pupil of the instrument, because then the field of view is wholly utilized. For if the pupil of the eye is at some distance from the exit-pupil, it itself acts as the field-of-view stop, and the size of the field is thus sometimes greatly diminished. For this reason the exit-pupil is often called the eye-ring, and its centre is called the position of tlze eye.
Thus far the stops have been discussed only with reference to their influence upon the geometrical configuration of the rays, but in addition they have a very large effect upon the brightness of the image. The consideration of this subject is beyond the domain of geometrical optics; nevertheless it will be introduced here, since without it the description of the action of the different· optical instruments would be too imperfect.
4. The Fundamental Laws of Photometry.-By the total quantity of light M which is emitted by a source Q is meant the quantity which falls from Q upon any closed surface S completely surrounding Q. Smay have any form whatever, since the assumption, or better the definition, is made that the total quantity of light is neither diminished nor increased by propagation through a perfectly transparent medium.*
It is likewise assumed that the quantity of light remains constant for every cross-section of a tube whose sides are
made up of light rays (tube of light). t If Q be assumed

* In what follows perfect transparency of the medium is always assumed.
t The definitions here prc$ented appear as necessary as soon as light quantity
is conceived as the energy which passes through a cross-section of a tube in unit time. Such essentially physical concepts will here be avoided in order not to forsake entirely the doD"ain of geometrical optics.

THEORY OF OPTICS
to be a point source, then the light-rays are straight lines
radiating from the point Q. A tube of light is then a cone whose vertex lies at Q. By angle of aperture (or solid angle)
.a of the cone is meant the area of the surface which the cone cuts out upon a sphere of radius I ( I cm.) described about its apex as centre.
If an elementary cone of small solid at1.gle dD, be considered, the quantity of light contained in it is
dL = K d.a. .
The quantity K is called the candle-power of the source Q in the direction of the axis of the cone. It signifies physically that quantity of light which falls from Q upon unit surface at unit distance when this surface is normal to the rays, for in
= this case d[l 1.
The candle-power will in general depend upon the direction of the rays. Hence the expression for the total quantity of light is, by (61),
M=JK-d.a,
in which the integral is to be taken over the entire solid angle
about Q. If K were independent of the direction of the rays,
it would follow that M= 41tK,
since the integral of d .a taken over the entire solid angle about Q is equal to the surface of the unit sphere described about Q as a centre, i.e. is equal to 41r. The mean candle-power K,,. is defined by the equation
,[Kd.a M
j K,,,= • = -. d.a 41r
If now the elementary cone dll cuts from an arbitrary surface S an element dS, whose normal makes an angle S with the axis of the cone, and whose distance from the apex Q of

APERTURES AND THEIR EFFECTS

79

the cone, i.e. from the source of light, is r, then a simple geometrical consideration gives the relation
= dll•r2 dS.cos e. .

Then, by (61), the quantity of light which falls upon dS is

dL

=

K

dS-cos ~-

e
-

(65)

The quantity which falls upon unit surface 1s called the

intensity of illumination B. From (65) this intensity is

cos e

B=K~,.

(66)

i.e. the intensity of illumination is inversely proportional to the square of the distance from the point source and directly proportional to the cosine of the angle which the normal to the illuminated surface makes with the direction of the incident rays.
If the definitions here set up are to be of any practical value, it is necessary that all parts of a screen appear to the eye equally bright when they are illuminated with equal intensities. Experiment shows that this is actually the case. Thus it is found that one candle placed at a distance of I m. from a screen produces the same intensity of illumination as four similar candles placed close together at a distance of 2 m.
Hence a simple method is at hand for comparing light intensities. Let two sources Q1 and Q2 illuminate a screen from such distances r 1 and r 2 (@ being the same for both) that the intensity of the two illuminations is the same. Then the candle-powers K1 and K 2 of the two sources are to each other as the squares of the distances r 1 and r 2. A photometer is used for making such comparisons accurately. The most perfect form of this instrument is that constructed by Lammer and Brodhun.*
* A complete treatment of this instrument, as well as of all the laws of photometry, is given by Brod.hon in Winkelmann's Handbuch der Physik, Optik, p. 450 sq.