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THE THEORY OF OPTICS

THE
THEORY OF OPTICS
BY
PAUL DRUDE
Professor of Physics at the University of Giessen
TRANSLATED FROM THE GERMAN
BY
C. RIBORG MANN AND ROBERT A. MILLIKAN
Assistant Professors of Physics at the University of Chicago
10528
LONGMANS, GREEN, AND CO.
91 AND 93 FIFTH AVENUE, NEW YORK
LONDON AND BOMBAY
1902

Copyright, 1901, BY
LONGMANS, GREEN, AND CO.
ROBERT DRUMMOND, PRINTER, NEW YORK.

5S

'REFACE 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 4t 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 funda-

mental 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 fol-

lowers a most extensive and beautiful development ; it gives

a most meagre presentation of the electromagnetic theory

a 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

v
Optics

(1892)

is

a

valua-

ble 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 thermodynamics, 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.

UNIVERSITY 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 publisher 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 endeav-
sd to present in the simplest possible way results already
Wished.
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.

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 " Miiller-
"
Pouillet 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, /?, y represent respectively the com-

ponents of the electric and magnetic forces (the first measured

in electrostatic

units);

also

\etjx ,jy ,jz , and

sx ,

s y

,

sg

represent

the components of the electric and magnetic current densities,

i.e.

times the number of electric or magnetic lines of force

47T

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 funda-
mental equations always hold :

^Y

ft ft

47fsx 3Y VZ

TT~

~ , etc . ,

-

TT~ 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 substance 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 equa-

tions 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 imme-

diately 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 com-

plete

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 experi-
ments lead to conclusions in what may be called the domain

of macrophysiccd properties only. For the explanation of

optical dispersion a hypothesis as to the microphysical proper-
ties 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

viii

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 formulae for the natural and magnetic rotation
of the plane of polarization, formulae which are experimentally verified. Furthermore, in the case of the metals, the ion-
hypothesis leads to dispersion formulas which make the con-
tinuity 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 rela-
tion 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 KirchhofT, Clausius, Boltzmann,
and W. Wien, and the many fruitful experimental investiga-
tions 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

I. mbued with this conviction, I have written this book in the :ndcavor to make the theory accessible to that wider circle of -eaders 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
My interesting" and important fields of optical investigation.
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 fuither ishing must be inviting to every one. Mr. F. Kiebitz has given me efficient assistance in the
ing of the proof.

iirziG, January, 1900.

INTRODUCTION

MANY optical phenomena, among them those which have

bund the most extensive practical application, take place in

iccordance 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

i 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, conclu-

sions 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-men-
tioned fundamental laws appear only as very close approxima-
vi

xii

INTRODUCTION

tions. However, it is possible to state within what limits the laws 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 instruments 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 I. GEOMETRICAL OPTICS
CHAPTER I
THE FUNDAMENTAL LAWS
PAGE
Direct Experiment ........................................... i
2. Law of the Extreme Path .................................... 6 3. Law of Malus ............................................. ... 1 1
CHAPTER II
GEOMETRICAL THEORY OF OPTICAL IMAGES
1. The Concept of Optical Images .............................. 14 2. General Formulae for Images ................................ 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
Refraction at a Spherical Surface ............................. 32 Reflection at a Spherical Surface .............................. 36 Lenses ........................ ............................. 40 Thin Lenses ........................................... . ---- 42 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

TABLE OF CONTENTS

9. The Law of Sines

. .. ..

58

10. linages of Large Surfaces by Narrow Beams

63

1 1 . Chromatic Aberration of Dioptric Systems

66

CHAPTER IV

APERTURES AND THE EFFECTS DEPENDING UPON THEM

1. Entrance- and Exit-Pupils

73

2. Telecentric Systems

75

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

95

3. The Microscope

97

4. The Astronomical Telescope

107

5. The Opera Glass

109

6. The Terrestrial Telescope

112

7. The Zeiss Binocular

8. The Reflecting Telescope

i

PART II. PHYSICAL. OPTICS

SECTION I

GENERAL PROPERTIES OF LIGHT

CHAPTER I

THE VELOCITY OF LIGHT

1. Romer's Method

114

2. Bradley 's Method

115

TABLE OF CONTENTS

xv

RT.

PACK

3. Fizeau's Method ............................................ 6 1 1

4. Foucault's Method .......................................... 1 1 8

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

Color .................................................... 1 20

6. The Velocity of a Group of Waves ............................ 121

CHAPTER II
INTERFERENCE OF LIGHT
i . General Considerations ...................................... 1 24 2. Hypotheses as to the Nature of Light ......................... 124
M 3. Fresnel's irrors ............................................ 1 30
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 ............................................ 1 54 0. Photography in Natural Colors ............................... 1 56

CHAPTER III
HUYGENS' PRINCIPLE
Huygens' Principle as first Conceived ......................... 1 59 Fresnel's Improvement of Huygens' Principle .................. 162 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 ................................ 1 92 5. Diffraction through a Narrow Slit ...................... . ...... 198 6. Diffraction by a Narrow Screen .......... ..................... 201 7. Rigorous Treatment of Diffraction by a Straight Edge .......... 203

xvi

TABLE OF CONTENTS

ART.

8. Fraunhofer's Diffraction Phenomena

213

9. Diffraction through a Rectangular Opening

214

10. Diffraction through a Rhomboid

217

1 1. 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

1 5. The Diffraction Grating

222

1 6. The Concave Grating

225

17. Focal Properties of a Plane Grating

227

1 8. Resolving Power of a Grating

227

19. Michelson's Echelon

228

20. The Resolving Power of a Prism

233

21. 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. .

255

SECTION II

OPTICAL PROPERTIES OF BODIES
CHAPTER I
THEORY OF LIGHT

1. Mechanical Theory

259

2. Electromagnetic Theory

260

The Definition of the Electric and of the Magnetic Force

262

3.

TABLE OF CONTENTS

xvii

of the Electric Current in the Electrostatic and the

Electromagnetic Systems

263

Definition of the Magnetic Current

;

265

The Ether

267

Isotropic Dielectrics

268

{DefTihneitBiouonndary Conditions

27 1

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-

I parent Isotropic Media

Perpendicular

Incidence ;

Stationary Waves

278 284

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

Equations and Boundary Conditions

308

(DifLfiegrhetn-vteicatlors and Light-rays

311

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

32

TABLE OF CONTENTS

ART.

P AGE

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

10. Conical Refraction

33 1

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
2. Normal Dispersion 3. Anomalous Dispersion 4. Dispersion of the Metals
CHAPTER VI
OPTICALLY ACTIVE SUBSTANCES
1 . General Considerations 2. Isotropic Media 3. Rotation of the Plane of Polarization 4. Crystals 5. Rotary Dispersion 6. Absorbing Active Substances

382 388 392 396

400

401

404

,

408

412

415

TABLE OF CONTENTS

xix

CHAPTER VII MAGNETICALLY ACTIVE SUBSTANCES

A ,

Hypothesis of Molecular Currents

ART.

PAGE

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

1. 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 Light 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

475

10. Fizeau's Experiment with Polarized Light

477

n. Michelson's Interference Experiment

478

xx

TABLE OF CONTENTS

PART III. RADIATION

CHAPTER I

ART.
1. Emissive Power

ENERGY OF RADIATION

2. Intensity of Radiation of a Surface
3. The Mechanical Equivalent of the Unit of Light 4. The Radiation from the Sun 5. The Efficiency of a Source of Light 6. The Pressure of Radiation

7. Prevost's Theory of Exchanges

PAGE 483
484 485 487 487 488 491

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

496

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

506

9. Entropy

510

10. General Equations of Thermodynamics

511

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

XXI

CHAPTER III

INCANDESCENT VAPORS AND GASES

urr.

PAGE

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

536

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

^% ^ ^N MNonn

OF

PART I
GEOMETRICAL OPTICS

CHAPTER I

THE FUNDAMENTAL LAWS

I. Direct Experiment. The four fundamental laws stated

above are obtained by direct experiment.
The rectilinear propagation of light is shown by the shadow
P of an opaque body which a point source of light casts upon
a screen 5. If the opaque body contains an aperture Z, then
the edge of the shadow cast upon the screen is found to be the
P intersection of S with a cone whose vertex lies in the source

and whose surface passes through the periphery of the aper-

ture 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

L case of a very small opening

(say of about one tenth ;;/;;/.

diameter) the light is spread out behind L upon the screen so

far that in this case the propagation cannot possibly be recti-

linear.

2

THEORY OF OPTICS

The same result is obtained if the shadow which an opaque body Sf 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, rectilineal-
P propagation of light from does not take place. It is there-
fore 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 describe the propagation of light
P P from a source to a screen 5, it is customary to say that
sends rays to S. The path of a ray of light is then defined
by the fact that its effect upon S 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,
P as when light from passes through a sufficiently large open-

ing in an opaque body. In this case it is customary to say
P that sends a beam of light through L.
Since by diminishing L the result upon the screen 5 is the

same as though the influence of certain of the rays proceeding

P from

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 open-
L ing 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 concept 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

N plane of incidence is that plane which is denned by the incident

ray and the normal

to the surface, and the angle of

incidence

is the angle included between these two direc-

tions.

The following laws hold : The reflected and refracted rays
botJi lie in tJie plane of incidence. Tlie angle of reflection (the angle included between yVand the reflected ray) is equal to the angle of incidence. The angle of refraction <p' (angle included between A^and the refracted ray) bears to the angle of incidence
the relation

sin ~d) =u.......

sin

(i)

in which ;/ 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 of refraction with respect to air will be understood. For all transparent liquids and solids n is greater than /.
A If a body 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

sin 0'

in which

represents the angle of incidence in air, 0' the

angle of refraction in the body B, <p" the angle of refraction in

B the

body A,

n b

the

index

of refraction

of

with respect to air,

A nab the index of refraction of with respect to B\ therefore

sin

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

4

THEORY OF OPTICS

last equation in combination with (i) then gives, na denoting
A the index of refraction of with respect to air,

A B i.e. the index of refraction of with respect to is equal to A B the ratio of the indices of and with respect to air.
If the case considered had been that of an infinitely thin
A plate placed upon the body B, the same process of reason-
ing would have given

Hence

= U ab

I : JI ba

A B i.e. the index of with respect to is the reciprocal of the B A index of with respect to .
The law of refraction stated in (i) permits, then, the con-
clusion that 0' may also be regarded as the angle of incidence
in the body, and 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 reversi-

bility also holds.

Therefore equation (i), 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

= ;/a .sin a

n sin

b

A,

(3)

N B in which a and (f>b denote the angles included between the

normal

A and the directions of the ray in and

respec-

tively, and

na

and

n b

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 n with respect to a vacuum is very small.

From (2)

n= ,

' :

(4)

THE FUNDAMENTAL LAWS

5

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

Its value at atmospheric pressure and o C. is

....... = n'

i : 1.00029

(5)

According to equation (3) there exists a refracted ray

to correspond to every possible incident ray a only when

< > na

n b;

for if na

n and if b,

rc

sm

> -b

,

n.

> then sin cpb

I ;

i.e.

there

is

no

real

angle

of refraction

A.

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 re-

fracted rays according to a law which will be more fully

considered later (Section 2, Chapter II). Here the observa-

tion 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 (i) 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

but

curved,

may it

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 invisi-J ble. Only objects which lie outside of the mirror, and whoseJ
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!
P from a point to a point P' experiences any number of reflec-
tions and refractions, then the sum of the products of the index of refraction of each medium by the distance traversed^
in it, i.e. 2nl, 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 d denotes the variation of the first order,

..... d^nl =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

OE POP' be the actual path of the light (Fig. i),

the inter-

PON section of the plane of incidence

with the surface (tan-

gent plane) of the refracting body, 0' a point on the surface of the refracting body infinitely near to 0, so that OO'

makes any angle 6 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'.OP'

n-PO'

n'-0'P r ,

.

.

(8)

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

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

upon PO' and a

perpendicular OR' upon P'O', then, to terms of the second

order,

= + - PO PO'

RO', OP' == OP' O'R'. . . (9)

Also, to the same degree of approximation,

= RO' == 00'. cos POO\ O'R' OO'-cos P'OO'. (10)

FIG. i.

OD In order to calculate cos POO' imagine an axis

perpen-

ON dicular to

and OE, and introduce the direction cosines of

PO the lines

and 00' referred to a rectangular system of

coordinates whose axes are ON, OE, and OD. If represent

the angle of incidence PON, then, disregarding the sign, the

PO direction cosines of

are

those of 00' are

cos 0, sin 0, o,

o, cos

sn

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.

and similarly

POO cos

1

sin 0-cos S,

cos P'OO' sin 0'-cos S,

in which <p' represents the angle of refraction. Then, from (9) and (10),

+ + n-PO' ri-O'P' = n.PO n-OO'-sin 0-cos

-j- n'-OP'

n'.OO'-s'm 0'-cos fl.

Since now from the law of refraction the relation exists

;/-sin

= w'-sin 0',

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

d(PO+OPf) =0,

(ii)

PO in which (Fig. 2)

and OP' denote the actual path of the

P P ray.

If

be that
l

point which is symmetrical

to

with

OE respect to the tangent plane

of the refracting body, then

= for every point O' in the tangent plane, PO' P^O' . The

P length of the path of the light from to P' for a single reflec-

THE FUNDAMENTAL LAWS 9

OE on at the tangent plane
+ P int O', equal to Pfi

is, then, for every position of the

Now O' .

this length is a mini-

m P if l , 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 (n), there-

equation (7) is proved for a single reflection.

It is to be observed that the vanishing of the first derivative

Kshe condition of a maximum as well as of a minimum. In 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

n called the law of least path.

When, however, the surface of the refracting or reflecting y is curved, then the path of the light is a minimum or a imuui according to the nature of the curvature. The

ishing of the first derivative is the only property which is
I 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 sur-
face such that from every point upon it the sum of the optical
P paths to two points 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 PA -f

= constant C,

(12)

P is an ellipsoid of revolution having the points and P' as foci.

If SOS' represents a section of a mirror (Fig. 3) and O

PO a point upon it such that

and P'O are incident and

reflected

rays, then

the

aplanatic

surface

AOA' which ',

io

THEORY OF OPTICS

passes through 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

SOSf is more concave than the aplanatic surface, then the

optical

PO path

-\-

OP'

is a

maximum,

otherwise

a

minimum.

FIG. 3.

The proof of this appears at once from the figure, since for al

points Of within the ellipsoid AOA' whose equation is given

+ PO in (12), the sum

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 O, 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 ;/ '), and indeed more
convex than a sphere described about P' as a centre.
This aplanatic surface also separates the regions for whose
points O' the sum of the optical paths n-PO' -\- n'-P'O' > 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

n

two media, and PO, P'O 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

FIG. 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 ;// is proportional to the time which the light requires to travel the distance /. The principle of least path is then identical witJi
Fermai's principle of least time, but it is evident from the
above that, under certain circumstances, the time may also be a maximum.
= Since d^nl o holds for each single reflection or refrac= tion, the equation d^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
F cut at right angles by a properly constructed surface (ortho-

12

THEORY OF OPTICS

F tomic system), and those for which no such surface 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 orfhotomic system of rays remains orthotomic

after any mimber 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.

ABODE Let (Fig. 5)

and A'B'C'D'E' be two rays infinitely

close together and let their initial direction be normal to a

L surface F. If represents the total

A optical distance from

to E, then

it may be proved that every ray

whose total path, measured from its

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

optical length Z, is normal to a sur-
F face f which is the locus of the ends

E, E', etc., of those paths. For
B the purpose of the proof let A' 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

in which ;/ represents the index of the medium between the

F D surfaces

and B, and ri that of the medium between

AB = A AB and F'. But since

B, because

is by hypothesis

normal to F, it follows that

DE = DE',

THE FUNDAMENTAL LAWS

13

DE i.e.

is perpendicular to the surface F' . In like manner

F it may be proved that any other ray D'E' is normal to

r .

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 system.

CHAPTER II

GEOMETRICAL THEORY OF OPTICAL IMAGES

I. The Concept of Optical Images. If in the neighb
P hood of a luminous point 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 Pto is completely determined. Nevertheless certain points P' maiay
P be found at which two or more of the rays emitted by interP sect. If a large number of the rays emitted by 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 virtual. The simplest exam-

ple of a virtual image is found in the reflection of a luminous
P point in a plane mirror. The image P' lies at that point P which is placed symmetrically to 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 be-

cause 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

tension. Consider, for example, the case of a single refrac-

>n. If the surface of the refracting body is the aplanatic

P race for the two points and P', then a beam of any size

P ich has its origin in

will

be

brought

together

in P' ;

for

P rays which start from

and strike the aplanatic surface

ist intersect in P', since for all of them the total optical dis-
P ce from to P' is the same.

If the surface of the refracting body has not the form of the anatic surface, then the number of rays which intersect in
is smaller the greater the difference in the form of the two
"aces (which are necessarily tangent to each other, see 10). In order that an infinitely narrow, i.e. a plane,
m may come to intersection in P\ the curvature of the sur-
es at the point of tangency must be the same at least in one ne. If the curvature of the two surfaces is the same at

two and therefore for all planes, then a solid elementary

,m will

come

to

intersection

in P' ;

and

if,

finally,

a

finite

tion of the surface of the refracting body coincides with the

lanatic surface, then a beam of finite cross-section will come

intersection in P''.

Since the direction of light may be reversed, it is possible

P P interchange the source

and its image

f i.e. a source at ,

has its image at P. On account of this reciprocal relation-

P p and P' are called conjugate points.

2. General Formulae for Images. Assume that by means
P reflection or refraction all the points of a given space are
P .aged in points of a second space. The former space will

called the object space ; the latter, the image space. From

i definition of an optical image it follows that for every ray

P ich passes through

there is a conjugate ray passing

ough 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
P is- conjugate to P. For every point there is then but one

conjugate point P'. If four points PJPff^ of the object space

lie in a plane, then the rays which connect any two pairs of

16

THEORY OF OPTICS

P P P P these points intersect, e.g. the ray

1 2 cuts the ray

in
34

f\ the point A. Therefore the conjugate rays P\P'2 and P'

also intersect in a point, namely in A' the image of A. Hence

the four images /YA'^Y^V also lie in a Plane - In otner

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

collinear relationship.

The analytical expression of the collinear relationship can

be easily obtained. Let x, j/, 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 vice

versa. This is only possible if

ax -\-by-\-cz-\-d

d ax -\- by -\- cz -\-

'

d ax -\- by -f- cz -\-

'
J

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 (i); and inversely, given values of x, y,

z

determine

x' ,

y' ,

z' .

If the right-hand side of equations (i)

were not the quotient of two linear functions of x, y, z, then

y for

every

x' ,

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 -f- d\ since otherwise

a plane in the image space

+ + + = A'x' B'y' Cz' D' o

would not again correspond to a plane

in the object space.

GEOMETRICAL THEORY OF OPTICAL IMAGES 17
If the equations (i) be solved for x, y, and z, forms analoms to (i) are obtained; thus

(2)

From (i) it follows that for
= -- d o:

x' =

= z = oo

lilarly from (2) for

cz d' =.

x = y = z = oo

+ = The plane ax -j- by -f- cz d o is called the focal plane

P of the object space. The images P' of its points

lie at

P Two nity.

rays which originate in a point of this focal

ne correspond to two parallel rays in the image space.

+ + + = The plane a'x'

b'y'

c' z'

df

o is called the focal

me g' of the image space. Parallel rays in the object space

respond to conjugate rays in the image space which inter-

t in some point of this focal plane gf'.

= = = In case a

b

c

o, equations (i) show that to finite

ues of x, y,

z

correspond

finite

values

of

x' ,

y' ,

z' ;

and,

in-

rsely, since, when

a, b,

and c are

zero,

a',

b' c' are also ,

zero, to finite values of JIT', y' , z' correspond finite values of

z.
',

In this case, which is realized in telescopes, there

no focal planes at finite distances.

3. Images Formed by Coaxial Surfaces. In optical in-

ments it is often the case that the formation of the image

es place symmetrically with respect to an axis; e.g. this

true if the surfaces of the refracting or reflecting bodies are

"aces of revolution having a common axis, in particular, sur-

s of spheres whose centres lie in a straight line.
P From symmetry the image P' of a point must lie in the P ne which passes through the point and the axis of the

tern, and it is entirely sufficient, for the study of the image

ormation, if the relations between the object and image in

h a meridian plane are known.

i8

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 (i).
They then reduce to

x=

+; by

=

ax -f- by -j- 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 0' for the image space is in general distinct from the
O origin for the object space. The positive direction of x will
be taken as the direction of the incident light (from left to

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)

= = b v

b

o. It also follows from symmetry that a change in

sign of y produces merely a change in sign of y' .

= a 2

d 2

o and equations (3) reduce to

Hence

_ "

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 images by coaxial surfaces.
The solution of equations (4) for x and y gives

dx' d.

a.d ad.

y

-- ,_,

1

*

'a,- ax"

*
l>,

- a, ax"

fC I

The equation of the focal plane of the object space is

= ax -{- d o, that of the focal plane of the image space

ax '

7<

F F o. The intersections

and f of these planes

i

with the axis of the system are called the principal foci.
F If the principal focus 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

X Q,

X' Q

represent

the

coordi-

nates measured from the focal planes, ax^ will replace ax -f- d

and

ax^

a l

ax' . Then from equations (4)

0-

y ax

(}

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

ibscripts)

L L XX' =ff

= = *.

(7)

In this equation x and x' are the distances of the object and

image from

the principal focal planes

^

and

f
$

respectively.

y The ratio y' \ is called the magnification. It is I for

= f /, i.e. x'

. This relation defines two planes and

T which are at right angles to the axis of the system. These

planes are called the unit planes. Their points of intersection
/f and H' with the axis of the system are called unit points.

The unit planes are characterized by the fact that the dis-
P tance from the axis of any point in one unit plane is equal to

that of the conjugate point P' in the other unit plane. The two

remaining constants /and/' of equation (7) denote, in accord-

20

THEORY OF OPTICS

Q ance with the above, the distance of the unit planes ),

from

f the focal planes g, g'. The constant is called the focal

length of the object space; f, the focal length of the image

space. The direction of/" is positive when the ray falls first

upon the focal plane g, then upon the unit plane

;

7
for/

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 g' distant, say, y' from the axis. The

value of y evidently depends on the angle of inclination u of

= the incident ray with respect to the axis. If u o, it follows

y = from symmetry thai

o, i.e. rays parallel to the axis have

conjugate rays which intersect in the principal focus F' . But

.pi

f

FTG. 7.

PFA if u is not equal to zero, consider a ray

which passes

through the first principal focus F, and cuts the unit plane )

A in

(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_/ of the image which is formed by

a parallel beam incident at an angle u is, as appears at once

from Fig. 7,

y' =/.tan u.

(8)

Hence the following law: The focal length of the object

space is equal to the ratio of the linear magnitude of an image

GEOMETRICAL THEORY OF OPTICAL IMAGES 21

formed in tJie focal plane of tJie image space to t/ie apparent

A (angular} magnitude of its infinitely distant object.

similar

f definition holds of course for the focal length 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

length /, which will then mean the focal length of the image

space :
The focal lengtJi of the image space is equal to the distance between the axis and any ray of the object space which is parallel to the axis divided by the tangent of the inclination of

its conjugate ray.
Equation (8) may be obtained directly from (7) by making
= = tan u y.x and tan it! y'\x'. Since x and x' are taken
positive in opposite directions and y and y' in the same direc-
tion, it follows that u and u' are positive in different directions. The angle of inclination u of a ray in the object space is positive
if the ray goes upward from left to rigJit; the angle of inclination u of a ray in tJie image space is positive if the ray goes
mvard from left to right. The magnification depends, as equation (7) shows, upon
the distance of the object from the principal focus F, and
m /, 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 in depth, i.e. the ratio of the increment dx' of x' 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 in

22

THEORY OF OPTICS

A P and the axis in (Fig. 8). Its angle of inclination u witl
respect to the axis is given by
AH AH

P if x taken with the proper sign represents the distance of
from F.
9C

H

FIG. 8.

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

tan u

A'H' A'H'

f P'H'

~ *"

if x' represent the distance of P' from F', and P' and A' are

P the points conjugate to and A . On account of the property

AH of the unit planes

A'H ;

then

by combination

of

the

last two equations with (7),

f tan u'

x

x

f

- tan u /' *'

(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

= f x I for

or x' f.

K The two conjugate points and K' thus determined are called

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

GEOMETRICAL THEORY OF OPTICAL IMAGES 23

K :t tJiat a ray tJirough one nodal point

is conjugate and

illel to a ray through the other nodal point K' . The posi-

of the nodal points for positive focal lengths /and/' is

K

FIG. 9.

KA >wn in Fig. 9.

and K'A' are two conjugate rays. It

lows from the figure that the distance between the two nodal

its is the same as tJiat between the two unit points. If

'=/', the nodal points coincide with the unit points.

Multiplication of the second of equations (7) by (9) gives

/ta""/

-.

y tan u

f

(io)

P If e be the distance of an object from the unit plane ,
e' the distance of its image from the unit plane $$, e and
P >eing positive if lies in front of (to the left of) JQ and P'
lind (to the right of) )', then
e=f-x, c'=f-x'.

mce the first of equations (7) gives

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

THEORY OF OPTICS

A Construction of Conjugate Points.

simple graphical

interpretation may be given to equation

ABCD (11). If

(Fig. 10) is a rectangle

/ with the sides

and /', then any

B f-

straight line ECE' intersects the pro-

f longations of/ and at such distances

AE = A from that the conditions

e and

E AE' =. e! satisfy equation (i i).

FIG. 10.

It is also possible to use the unit

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

P PA gate to P. Draw (Fig. n) from a ray

parallel to the

PF axis and a ray

passing through the principal focus F.

FIG. ii.

A'F' is conjugate to PA, A' being at the same distance from
A the axis as ; 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 con-

struction.

P The construction shown in Fig. 1 1 cannot be used when P and P' lie upon the axis. Let a ray from intersect the focal

g plane g at a distance and the unit plane at a distance h

from the axis (Fig. 12). Let the conjugate ray intersect $$

and $' at the distances k'(=. /i) and g' . Then from the figure

f ?.- PF

~*

~k~ f -YPF~J~^lc'

P F'

'

f *L-
li~~~-

f
~+~P

rF'

-l

x'
- x' ;

GEOMETRICAL THEORY OF OPTICAL IMAGES 25

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

g + g'

f - 2xx' fx ~~ x

h ff+xx' -fx -f'x

(12)

P' may then be found by laying off in the focal plane g' the

= Q distance g' h g, and in the unit plane

the distance

FIG. 12.

= h' h, and drawing a straight line through the two points thus

g determined,

and g' are to be taken negative if they lie

below the axis.

5. Classification of the Different Kinds of Optical Systems. The different kinds of optical systems differ from one

f another only in the signs of the focal lengths and /'.

If the two focal lengths have the same sign, the system is

concurrent i.e. i

if

the

object

moves

from

left

to

right

(x in-

creases), the image likewise moves from left to right

(x' decreases). This follows at once from equation (7) by
taking into account the directions in which x and x' are con-

sidered positive (see above,

p.

18 ).

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.

26

THEORY OF OPTICS

If tJie two focal lengths Jiave 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

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

P in the direction in which the light travels, the image

of that

point moves along the conjugate ray in the direction in which

the light 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 $$ of the image space, is rendered convergent by the former,
divergent 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 : DiWopptt>ric

K Convergent: +/, +/'
\b. Divergent: -/, -/'.

T^ ,

.

.

Katoptrtc. .

(a.

C~o.nvergent: -f f,

(6. Divergent:

/,

fJ .

+/,

.,

.

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'

Py. .

.

.

.

.

(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 (13) that the magnification in breadth

and depth are constant. The angular magnification is also

OP constant, for, given any two conjugate rays

and O'P' their ,

intersections with the axis of the system may serve as the

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

P x, y, and its conjugate point

1 the

coordinates

x' ,

y\

the

tangents of the angles of inclination are

tan u

y = : x, tan u'

y' : x' .

Hence by (13)

ttan u' : tan u

P:a

(14)

must be positive for katoptric (contracurrent) systems, nega-

tive for dioptric (concurrent) systems. For the latter it is

evident from (14) and a consideration of the way in which u

and u' are taken positive (see above, p. 2l) that for positive P

erect images of infinitely distant objects are formed, for nega-

tive P, inverted images. There are therefore four different

kinds of telescopic systems depending upon the signs of OL

p.
Equations (14) and (13) give

f
y tan u' p*

n i/ -f-a

it

/v

V ^/

(d

(16)

y = y If

/ ,

as

is

the

case

in

telescopes

and

in

all

instru-

cts in which the index of refraction of the object space is

28

THEORY OF OPTICS

equal to that of the image space (cf. equation (9), Chapter III]

then OL

2
ft .

Hence from (14)

= tan u' : tan u

I : ft.

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

i.e. for telescopes the reciprocal of the lateral magnification

numerically equal to tJie angular magnification.

A 7. Combinations of Systems.

series of several systems

must be equivalent to a single system. Here again attention

will be confined to coaxial systems. If/ and// are the focal

/ lengths of the first system alone, and 2 and // those of the

second, and /and/' those of the combination, then both the

focal lengths and the positions of the principal foci of the com-

bination can be calculated or constructed if the distance

= F^F 2 A (Fig. 13) is known. This distance will be called

for brevity the separation of the two systems I and 2, and will

F be considered positive if .F/'lies to the left of

otherwise
2,

negative.
A ray 5 (Fig. 13), which is parallel to the axis and at a

pr<

S
ff
K

FIG. 13.

distance y from it, will be transformed by system I into the

S ray l , which passes through the principal focus /</ of that

system.

S will l

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 prinipal focus of the image space of the combination. Its position
:an be calculated from the fact that F^ and F' are conjugate
>oints of the second system, i.e. (cf. eq. 7)

(.7)

F F F n which 2

is positive if F' lies to the right of

.
2

F' may

DC determined graphically from the construction given above

3n

page

25,

since the

intersection of S and 5' with the 1

focal

F F g planes

and
2

are at such distances
2

and g' from the axis

:hat g-\- g' yv The intersection A' of S' with 5 must lie in the unit plane

)' of the image space of the combination.

Thus

'
is deter-

f mined, and, in consequence, the focal length

of the com-

bination, which is the distance from $$ of the principal focus F'

of the combination. From the construction and the figure it

follows that/' is negative when A is positive.

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

nation

u'

of the

ray

Sf .

For S the relation holds : l

u^y tan

://,

r hich

?/ is to t

be taken with

the

opposite

S sign if

is con-
l

ired the object ray of the second system. Now by (9),

tan u r

A

= since tan u^

~~~

tan j

//

y : //,

tan ' = - y - j^/1/2

= irther, since (cf. the law, p. 21) y :/'

tan ?/, it follows

that

/'= -/^. .

(18)

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

3o

THEORY OF OPTICS

F and for the distance of the principal focus of the combination F from the principal focus l ,

(20)

FF F F in which

l is positive if lies to the left of r

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 A of the two systems is zero, the focal

f f lengths and

are infinitely great, i.e. the system is tele-

scopic. The ratio of the focal lengths, which remains finite,

is given by (18) and (19). Thus

From the consideration of an incident ray parallel to the axis

the

lateral

magnification y'

y :

is

seen

to

be

..... y'-.y

=

ft=-/1

'
:fl

(22)

By means of (21), (22), and (16) the constant a, which repre-
sents the magnification in depth (cf. equation (13)) is found.
Thus

Hence by (14) the angular magnification is

= = / tan u' : tan u

a ft :

x ://.

.

.

.

(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 which is parallel to
the axis.

x OF CMJFOMW*. .
Aiai^?

CHAPTER III

PHYSICAL CONDITIONS FOR IMAGE FORMATION

ABBE'S geometrical theory of the formation of optical

ages which overlooks entirely dization, has been presented in

the the

question previous

of their physical chapter, because

> general laws thus obtained must be used for every special

se of image formation no matter by what particular physical

*ans the images are produced. The concept of focal points

1 focal lengths, for instance, is inherent in the concept
image no matter whether the latter is produced by lense

by mirrors or by any other means.
In this chapter it will appear that the formation of optical
l4'aogobeujsescatscohfadpeftisnecirrtie bisseidzpehiydcseaianclnalloyltyabniedmpfowosisrtimhbeolduet,whleei.mgni.ttatthhieeonrsiaymsiangheatvhoeef

great a divergence.
It has already been shown

on

page

I5

that,

whatever

the

ivergence of the beam, the image of one point may be pro-

duced by reflection or refraction at an aplanatic surface. Images

f other points are not produced by widely ie form of the aplanatic surface depends

divergent
upon the

rays, since position of

,e point For this reason the more detailed treatment of

i

ecial aplanatic surfaces has no particular what follows only the formation of images

physical interest.
by refracting an

effecting spherical surfaces will be treated, since on account

jf the ease of manufacture, these alone are used in opti

nstruments; and since, in any case, for the reason mentioned

ibove, no other forms of reflecting or refracting surfaces farm

leal optical images.

32

THEORY OF OPTICS

It will appear that the formation of optical images can

practically accomplished by means of refracting or reflecting

spherical surfaces if certain limitations are imposed, nameh

limitations either upon the size of the object, or upon th<

divergence of the rays producing the image.

i. Refraction at a Spherical Surface, In a medium

PA index n, let a ray

fall upon a sphere of a more stronglj

refractive substance of index n' (Fig. 14). Let the radius oJ

FIG. 14.

the sphere be r, its centre C. In order to find the path of the
C refracted ray, construct about two spheres I and 2 of radii

= r and

r (method of Weierstrass).

PA BC Let

meet sphere I in B\ draw

intersecting sphere

AD 2 in D. Then

is the refracted ray. This is at once

ADC evident from the fact that the triangles

and BA C

= = CD BC A C are similar. For

:

CA :

n' : n. Hence the

^DAC = ABC = <

0', the angle of refraction, and since

BAC = <

0, the angle of incidence, it follows that

= BC A C = sin : sin <'

:

;/' :

,

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

PHYSICAL CONDITIONS FOR. IMAGE FORMATION 33

be constructed, it becomes evident from the figure that these
j
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

/>, are refracted to the point D. Inversely all rays which
D start from have their virtual intersection in B. Hence upon

every straigJit line passing through the centre C of a sphere

of radius r, there are two points at distances from C of

)i

n

r and r respectively ivJiich, for all rays, stand in the relation

of object and virtual (iiot real) image. These two points are

called the aplanatic points of the sphere.

If u and u' represent the angles of inclination with respect

BD to the axis

of two rays which start from the aplanatic

B points and D, i.e. if

ABC = ,

ADC = u',

^DAC = ABC then, as was shown above, <

= u. From

ADC a consideration of the triangle

it follows that

= AC CD = sin u' : sin u

:

n' : n.

.

.

.

(i)

In this case then the ratio of the sines of the angles of inclina-
tion of the conjugate rays is independent of u, not, as in equa-
tion (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
B and 2 of Fig. 14 are the loci of all pairs of aplanatic points
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

surfaces I and 2,

= s' : s

n* : n'*.

34

THEORY OF OPTICS

Hence equation (i) may be written:

= sin2 u-s-n2

sin 2 11 -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

AP (Fig. 15) be an incident ray,

r the refracted ray, and PCP'

P'

FIG. 15.

P the line joining with the centre of the sphere C.

PA the triangle

C,

PH + sin

: sin a

r : PA ,

Then from

and from the triangle P'A C,
= a sin 0' : sin P'H

r : P'A.

Hence by division,

PH+r sin

__

P'A

_'

" shT^7

n~~~ P'H- r' ~PA'

*

'

(3)
_,

Now A assume that lies infinitely near to //, i.e. that the angle

APH PA is very small, so that

may be considered equal to

PH, and P'A to P'H. Also let

PH = = P'H f

e,

e.

PHYSICAL CONDITIONS FOR IMAGE FORMATION 35
Then from (3)
er

nn n n
7+7=

which r is to be taken positive if the sphere is convex

C toward the incident light, i.e. if lies to the right of H. e is
P positive if 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

PC of a portion of space which lies close to the axis

is formed

by rays which lie close to PC.
A comparison of equation (4) with equation (n) on page

23 shows that the focal lengths of the system are

/= r^~n> f = %7^' ' -

(5)

and that the two unit planes and ' coincide and are tangent to the sphere at the point H. Since /"and/"' have the same sign, it follows, from the criterion on page 25 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 >/. Such images
re also inverted.
Equation (10) on page 23 becomes

y' tan u'

n

v tan u

(6)

\y 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 uf ,

i.e.

'u

u''. Hence the last equation may

be written:

= ny tan u riy tan 'u

(7)

36

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,

/_tanj/ ~- /
jtan u /"

there results from a combination with (7)

/:/'=:;/',

(9)

i.e. /;/ the formation of images by a system of coaxial refract-

ing spJierical surfaces tJie ratio of the focal lengths of the system is equal to the ratio of the indices of refraction of the
first 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.

FIG. 16.

^ PAC By the law of reflection (Fig. 1 6)
Hence from geometry

^ P'AC.

PA \P'A =PC :P'C

(10)

PA If the ray

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

ith the axis varies with the angle. In that case no image of

P APC point exists. But if the angle

is so small that the

igle itself may be used in place of its sine, then for every

P >int

there

exists

a

definite

conjugate

point

P' t

i.e.

an

image

PA = now formed. It is then permissible to set

PH,

'A P'H, so that (10) becomes

PH\P'H=PC\P'C, .... (n)

if PH = e, P'H =

e' ,

then,

since

r in

the

figure

is

nega-

te,

I

I

2

A comparison of this with equation (n) on page 23 shows " at the focal lengths of the system are

.... f=- f'=+ l

l

-r,

-r;

(13)

that the two unit planes

$ and

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 ef are determined by the definition on page 23.

f f Since and

have opposite signs, it follows,, from the

criterion given on page 25, that the system is katoptric or con-

tracurrent. 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=.

= \. 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/":/'

38

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 concur-
rent (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 7, 2, j, . . . 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

FIG. 17.

continuous lines, virtual rays by dotted lines. The points

/',

2* ',

3',

.

.

. 8'

are

the images

of the

points /,

2, j,

.

.

.

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

straight line which intersects -the mirror, i.e. the unit plane, in
some point A^. If now through A^ a line be drawn parallel

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

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.

H
FIG. 19.

PCP'H Equation (i i) asserts that

are four harmonic points.

P The image of an object may, with the aid of a proposition

of synthetic geometry, be constructed in the following way:

40

THEORY OF OPTICS

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 PH LO with BC\ then

intersects the straight line

in a

point P' which is conjugate to P. For a convex mirror the

construction is precisely the same, but the physical meaning of
H the points C and is interchanged.

3. Lenses. The optical characteristics of systems com-

posed of two coaxial spherical surfaces (lenses) can be directly

deduced from

7 of Chapter II.

The radii of curvature r^

and r^ are taken positive in accordance with the conventions

given above ( i); i.e. the radius of a spherical surface is considered positive if the surface is convex toward the inci-

dent 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 S l

and

5 2

(Fig.

20),

be

FIG. 20.

denoted by d. If the focal lengths of the first refracting sur-
face are denoted by /j and/', those of the second surface by
/2 and//, then the separation A of the two systems (cf. page
28) is given by -

J=W -//-/

(H)

and, by (5),

PHYSICAL CONDITIONS FOR IMAGE FORMATION 4 i

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

f-f- - ~

+ n

I d(n

I)

nr l

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

bination are given by equations (17) and (20) of Chapter II

(page 29). By these equations the distance 6 of the principal

F focus

in

front

of the vertex S lt

and

the

distance cr' of the

F principal focus r behind the vertex S2 are, since cr and cr' = FjF' +//,

+ FF l

f^

f

=^=T"^(-i)-r {r

+ 1

i

'

r,
l

'

'

(I7)

If h represents the distance of the first unit plane fe in front

of the vertex

S l,

and h' the

distance of the second unit

plane

/+ ' behind the vertex 5 then

h

2,

+ = cr and /'

h'

7
cr ,

and, from (16), (17), and (18), it follows that

d(n i) nrv -\-nr

(19)

-j

_2

(20)

Also, since the distance / between the two unit planes

= + $' is /

d-\~ h

//', it follows that

and

p

- d(n

\\-r-. J

d(n

-^

i)

+ nr -^

.
nr

l

2

.

.

.

(21)

f = Since

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 and v

r 2

alone,

but that

the thickness d of the lens is also an essential element. For

example,

a

double

convex

lens

(i\ positive,

r 2

negative),

of

42

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 d is very great.

4, Thin Lenses. In practice it often occurs that the thick-

ness d of the lens is so small that d\Ji

i) is negligible in

comparison with n(rl

r 2 ).

Excluding the case in which

/'j

r 2,

which

occurs

in

concavo-convex

lenses

of equal

radii,

equation (16) gives for the focal lengths of the lens

"~

r

(

l](

\ I

^~ ^

/-(* l

r)'

.
|.

.

.

(22)

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

nearly coincide with the nearly coincident tangent planes at

the two vertices 5 and S .

t

2

More accurately these equations give, when d(n

i) is

neglected in comparison to n(rl

r 2

),

h=

-
;/ i\

r, z

= h'

n -\'

1\

, /=d
;-

n

-

2

(23)

Thus the distance / between the two unit planes is indepen-

= = dent of the radii of the lens. For ;/

p i . 5,

d. For both

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

negative, the unit planes lie inside of the lens. For equal

= curvature r l

= = = r and for n 2,

1.5, /*

//

\d, i.e.

the distance of the unit planes from the surface is one third

the thickness of the lens.

When

i\

and

r 2

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 (rl > o,

r 2

o),

= Plano-convex lenses (i\ > o,

r 2

oo
)

Concavo-convex lenses (i\ > o,

r 2

>

o,

> r 2

rj,

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

edges.

PHYSICAL CONDITIONS FOR IMAGE FORMATION 43

Lenses of negative focal length (divergent lenses) include

Double-concave lenses (rL < o,

> r 2

o),

I

= Plano-concave lenses (7^

00,

> r 2

o),

< Convexo-concave lenses (rl > o,

> r 2

o,

r 2

r^),

i.e. all lenses which are thinner in the middle than at the
* edges.
The relation between image and object is shown diagram-
matically in Figs. 21 and 22, which are to be interpreted in

FIG. 21.
the same way as Figs. 17 and 18. From these it appears that
whether convergent lenses produce real or virtual images of

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,

those of negative focal length, have been chosen on account of this property of

A lenses.

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 con-

vergent or divergent cannot be so immediately drawn. Then recourse must be

to the definition on page 26.

44

THEORY OP OPTICS

objects. However, divergent lenses produce > real, upright,

and enlarged images of virtual objects which lie behind the

lens and inside of the principal focus.

f / If two thin lenses of focal lengths

and are united to

v

2

form a coaxial system, then the separation 21 (cf. page 40) is

= A

(fl -f-/2 )- Hence, from equation (19) of Chapter II

(page 29), the focal length of the combination is

fr_ /1/2

or

7,

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 tJie 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 posi-

tions of an object and its image in order to deduce the focal

length. For example, equation (11) of Chapter II, page 23,

f = f reduces here, since

to
t

7+7=7....... (25)

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 procedure is not permissible for thick lenses nor for optical systems
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 rays. 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.,

(26)

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 mag-

nitude 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 , while its image of linear

magnitude y' is at a distance e' from the unit plane ', then

..... = y y' \

e' : e,

(27)

because, when /==/', 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

/= - -= - -7. ...... (28)

i-^ l-^

y

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 /"when
and the magnification y' \y have been measured.
The location of the positions of the object or image may
avoided by finding the magnification for two positions of
re'

46

THEORY OF OPTICS

the object which are a measured distance / apart.

(7), page 19,

AI

-

v/'r /'

hence

For, from

/=

" (z

(\y^))2 T{\7y)1 1

in which (y : y'\ denotes the reciprocal of the magnification for
the position x of the object, (JF : j/)2 the reciprocal of the magx nification for a position -f- / of the object. / is positive if, in passing to its second position, the object has moved the dis-
tance /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 meas-
urement 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 hornocentric 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,
pp. 285-296,

PHYSICAL CONDITIONS FOR IMAGE FORMATION 47

48

THEORY OF OPTICS

This ray is called the principal ray of that elementary beam which is composed of the normals to d*2. From the symmetry
'
of the figure it is also evident that the line pl must be parallel to the lines 2-3 and i ,/, i.e. it is vertical. The normals to
any horizontal line of curvature intersect at some point of the
line pr

FIG. 23.
Likewise the normals to any vertical line of curvature
intersect at some point of the line p2 which connects /./ and 23. Also, /2 must be horizontal and at right angles to 5. These two lines p^ and/2 , which are perpendicular both to one another
and to the principal ray, are called the two focal lines of the elementary beam. The planes determined by the principal
ray 5 and the two focal lines pl and/2 are called ft\t 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 fo'cal 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 astigmatic 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

>ject

is

a straight

line

can a

straight-line

image

be

formed ;

d only then when the line object is so placed that all the

P :al lines which are the images of all the points of the line

)ject coincide. Since the image of every point consists of

or focal lines / 1 and /2 which are at right angles to each
;her, there are also two positions of the line object 90 apart

rhich give rise to a line image. These two images lie at

ifferent distances from the surface 2.

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 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

refracting or reflecting surface has two different

vatures. Thus cylindrical lenses, for example, show marked
Ementary igmatism. 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
P emitted by strikes the spherical surface, lie in the plane of

PA the figure (Fig. 24). Let the line

be

represented

by

s y

AP the line

Now by s .

2

2

since

APAP 2

=

APAC+

ACAP 2

,

it follows that

ss^ sin (0

= 0')

$r sin

-|- s^r sin 0',

5

THEORY OF OPTICS

in which and 0' denote the angles of incidence and refrac-

tion respectively, and r the radius of the sphere. Since now

= by the law of refraction sin

;/ sin 0', it follows from the

last equation that

ss2(n cos 0'

= + cos 0) srn sp, or

I

n
~

n cos 0'

cos

5

^2

r

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

FIG. 24.

P the axis at the same point

.
2

The beam made up of such

P rays is called a sagittal beam.

It has a focal point at

.
2

On the other hand a meridional beam, i.e. one whose rays

PA P all lie in the plane

C, has a different focal point r Let

PB be a ray infinitely near to PA, and let its angle of inclina-

+ tion to the axis be u du and its direction after refraction

^BP BPr Then

A is to be considered as the increment du'
V

^BCA of u', and

as the increment da of a. It is at once

evident that

= AB s. du

ABcos 0,

s . du' l

. cos<J>', r.da AB. (31)

But since

= ~ a -f- #, 0'

a

#',

PHYSICAL CONDITIONS FOR IMAGE FORMATION
follows that

^-^ = >'

= ^(i-^j. . . .

But a differentiation of the equation of refraction sin

n sin

r
gives

cos . d<p

n cos 0' . dcf>' .

(32)
=

Substituting in this the values of d(f> and d<p' taken from (32),
there results

cos2 ^

n cos2 0'
^

n cos 0' cos

r

'

'

'

( 33 '

From

(33) and (30) different values

s l

and

s 2

corresponding

to

P the same s are obtained, i.e. is imaged astigmatically. The

astigmatic difference is greater the greater the obliquity of the

incident beam, i.e. the greater the value of 0. It appears

from (30) and (33) that this astigmatic difference vanishes, i.e.

= = = s l

s 2

s', only when s

its'. 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

=

i, i.e.

= /

(cf. page 37). Thus for this case*

I

I

011 COS

2

~

s

s

2

r

s

s^~

r cos 0'

^

'

by subtraction,

12/1 -Ir\cos

\
cos 0J,
/

= - *

*y

-l

sin

tan 0,

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

(35)

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 sur-

face 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 m. -J-

aperture,

the

astigmatic difference amounts to -fa mm., i.e. the positions in

which the one or the other system of lines of the network is

in sharp focus are -fa 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

inclination 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 wideninj

the limits hitherto set upon image formation. In the first pla<

PHYSICAL CONDITIONS FOR IMAGE FORMATION 53
ie limited sensitiveness of the eye comes to our assistance: are unable to distinguish two luminous points as separate
iless they subtend at the eye an angle of at least one minute, fence a mathematically exact point image is not necessary,
id for this reason alone the beam which produces the image
js not need to be elementary in the mathematical sense, i.e. ie 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 vice
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 suit-
able 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 ful-

filled for more than one position of the object.
P 8. Spherical Aberration. If from a point on the axis

two

rays

5 X

and

S 2

are

emitted

of which

S l

makes

a

very

small

angle

with

the

axis,

while

5 2

makes a

finite

angle

u,

then,

after refraction at coaxial spherical surfaces, the image rays 5/

and

'
.$ 2

in

general

intersect

the

axis

in two different points P^

P and

'. 2

The distance between these two points is known as

the spherical aberration (longitudinal aberration). In case the

angle

u

which

the

ray

5 2

makes

with

the

axis

is

not

too

great,

this aberration may be calculated with the aid of a series of

ascending powers of u. If, however, u is large, a direct

trigonometrical determination of the path of each ray is to be

preferred. This calculation will not be given here in detail.*

For relatively thin convergent lenses, when the object is

P distant, the image

formed
l

by rays

lying

close to

the

axis

P is farther from the lens than the image L formed by the more

P oblique rays.

Such a lens, i.e. one for which

lies nearer
2

P to the object than

is said to be undercorrected.
l,

Inversely,

P P a lens for which

is 2

more

remote from the

object than

is
i

said to be overcorrected. Neglecting all terms of the power

series in u save the first, which contains ifi as a factor, there

results for this so-called aberration of the first order, if the
P object is very distant,

'i^2 _"

/. 2n(n

- 2

i) (i

of

in which h represents the radius of the aperture of the lens, /its focal length, n its index of refraction, and a 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. ;Muller-Pouillet's Lehrbuch d. Physik, 9th Ed. p. 487 ; or Heath,
Geometrical Optics.

PHYSICAL CONDITIONS FOR IMAGE FORMATION 55

'he

signs

of

1\

and

r 2

are

determined

by the

conventions

idopted on page 40; for example, for a double-convex lens

is

positive,

r 2

negative.

/Y/V is negative for an undercor-

rected lens, positive for an overcorrccted one. Further, the

-atio h :f is called the relative aperture of the lens. It

ippears then from (36) that if cr remains constant, the ratio

f of the aberration /Y^Y to tne f ca l length is directly pro-

portional to the square of the relative aperture of the lens.

For given values of /and h the aberration reaches a mini-
mum for a particular value u' of the ratio of the radii. * By

(36) this value is

(38)

For n

= 1.5, <r

I : 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

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

tat above considered, i.e. that of a distant object, by simply

iterchanging the roles of object and image. t For n 2,

= ^38) gives a'

-f- \. This condition is realized in a con-

rexo-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

f for different values of the ratio cr of the radii,

has been

assumed equal to I m. and h :f= TV i.e. h 10 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

A * This minimum is never zero.

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.

fit 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 /y, is obtained, as appears at once from a construction of the paths of the rays, by multi-
plication of the longitudinal aberration by the relative aperture h :f, i.e. in this case by T"L-. Thus the lateral aberration determines the radius of the illuminated disc which the outside
P rays from a luminous point form upon a screen placed in the P plane in which is sharply imaged by the axial rays.

f

m. k -- 10 cm.

PHYSICAL CONDITIONS FOR IMAGE FORMATION 57
len:ses.* By selecting for the compound system lenses of
different form, it is possible to cause the aberration not only
of the first but also of still higher orders to vanish, f 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 axis.
When the angle of inclination it is large, as in microscope
objectives in which u sometimes reaches a value of 90, the power series in it 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,
rhich are free from aberration.
Let lens i (Fig. 25) be plano-convex, for example, a hemi-

FIG. 25.

spherical

lens

of radius

r l,

and

let

its

plane surface be turned

P toward the object P. If the medium between and this lens

has. the

same

index

n l

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 combination of a convergent and a divergent lens.

58

THEORY OF OPTICS

I

proceeding from the object first takes place at the rear surface
P of the lens; and if the distance of from the centre of curva-

C ture

of the
l

back

surface

is

1\

:

;/ x,

then

the

emergent

rays

P C produce

at

a

distance

nr ]l

from

l a virtual image

free from
l

aberration. If now behind lens / there be placed a second

concavo-convex lens 2 whose front surface has its centre of

curvature in P^ and whose rear surface has such a radius r^ that

P lies l

in

the

aplanatic

point

of this

sphere

r^ (the

index

of

lens 2 being ;/2), then the rays are refracted only at this rear

surface, and indeed in such a way that they form a virtual
P image 2 which lies at a distance r ;/.7 9 from the centre of curva-

C ture 2 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
P P images 3 , 4 , 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 intro-

duced are too large to be compensated (cf. below).
9. The Law of Sines. In general it does not follow that
P if a \videly divergent beam from a point upon the axis gives

rise to an image P' which is free from aberration, a surface

P element d<? perpendicular to the axis at will be imaged in

P a surface

element

f
d<j

at

'. 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
P conjugate rays passing through and P', sin u : sin u' const.
According to Abbe systems which are free from aberra-
P P tion for two points and ! on the axis and which fulfil the
sine law for these points are called aplanatic systems. The
P points 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 34, 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 n : tan u' const. Unless u and u' 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-forpoint imaging of objects of any size by widely divergent beams
is physically impossible.
Only when u and u' are very small can both conditions be simultaneously fulfilled. In this case, whenever an image P' is formed of P, an image d<r' will be formed at P' of the surface element dcr at P. But if u 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 lagnification is, according to Abbe, the sine law.* The same inclusion was reached in different ways by Clausius t and v.
[elmholtz :. Their proofs, which rest upon considerations of tergy and photometry, will be presented in the third division
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.
|R. Clausius, Mechanische Warmetheorie, 1887, 3d Ed. I, p. 315. jv. Helmholtz, Pogg. Ann. Jubelbd. 1874, p. 557.
Hockin, Jour. Roy. Microsc. Soc. 1884, (2), 4. p. 337.

6o

THEORY OF OPTICS

P page 9).* Let the image of (Fig. 26) formed by ani axial

PA PS ray

and a ray

of inclination ?/ lie at the axial point P'.

P Also let the image of the infinitely near point 1 formed by a

P A P ray

l l parallel to the axis, and a ray

S
ll

parallel

to

PS,

P lie at the point P^.

The ray F'P^ conjugate to

A must

l

l

evidently pass through the principal focus F' of the image
P space. If now the optical distance between the points and

P' along the path through A be represented by (PAP'), that

FIG. 26.

along the path through SS' by (PSS'P'), and if a similar

notation be used for the optical lengths of the rays proceeding
P from l , then the principle of extreme path gives

= = (PAP')

(PSS'P') ; (P^F'PI)

(P S S 'P

l

l

l

l '),

and hence
- = - (PAP') (P.A.F'P,') (PSS'P') (P&St'PJ). . (39)

Now T since F' is conjugate to an infinitely distant object on

= = the axis, (TPAF f )

(TP^F'). But evidently TP

TP l,

since PP^ is perpendicular to the axis. Hence by subtraction

...... = (P1A 1F')

(40)

* 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 CONDITIONS FOR. IMAGE FORMATION 61
'urther, since P'P{ is perpendicular to the axis, it follows
= lat when P'P{ is small F'P' F'P^. Hence by addition

i.e. the left side of equation (39) vanishes. Thus

= (/WY/Y)

(40

Now if /Y is the intersection of the rays P'S' and /Y^/ tnen

T /Y is conjugate to an infinitely distant object l , the rays from

which make an angle u with the axis. Hence if a perpendic-

PN P P ular

be dropped from

upon

S
l l,

an

equation

similar

to

(40) is obtained; thus

By subtraction of this equation from (41),

+ (F^P'}= -(Ar/Y> (^//Y)-

(43)

!f now n is the index of the object space, n' that of the image
>pace, then, if the unbracketed letters signify geometrical

lengths,

= = (NPJ

ii-NP l

n-PP^s'm u. . . . (44)

'urther, if P'N' be drawn perpendicular to F^P', then, since

''/Y is infinitely small,

~

'

i

i

\ j)

Equation (43) in connection with (44) and (45) then gives
= n-PP^sm u n'-P'P^- sin u' .

PP y \{ denote the linear magnitude

of the
l

object,

andy the

P Pf
linear magnitude

of the image, then

sin u

sin

,
u'

n'y'
ny

(46)

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

62

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
P the axis. For if P' and /y (Fig. 27) are the images of and Plt then, by the principle of equal optical lengths,

= AP = (PAP')

(PSS'P'), (Pl

l ')

S S (P 'P 1 1 l

1 ^, .

(47)

PS P in which

and

S
ll

are

any

two

parallel

rays

of inclina-

tion u.

FIG. 27.

Subtraction of the two equations (47) and a process of

reasoning exactly like the above gives

- = + (/>//>') (Pf)

(/yV) (N'P') t

or

cos u) ri-P^P' (i cos '),
i.e.

sn

(48)

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 optical 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. It
is necessary in the first place to eliminate astigmatism (cf. page 46). But no law can be deduced theoretically for accom-
plishing 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.

64

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 PPf^ (Fig. 28) be an object

plane, P'P^P2 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 l , 2 , etc., pass through its centre. Likewise the beams in

B'

the image space are limited by a similar aperture

the
,

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

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

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

the figure,

= PP tan //j

l : /,

= tan a/ P'P{ : /',

= PP tan u 2

2 : I,

= tan ?//

P'P : /'.

If the magnification is to be constant, then the following rela-

tion must exist:

= PP P'P{ : PP, P'/Y :

2,

hence

..... tan u'
tan u

= tan u' const tan

(40)

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, ortho-
scopic points. Hence it may be said that, if the image is to be free from distortion, the centres of perspective of object and
image must be orthoscopic 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
R case of a projecting lens. Let (Fig. 29) be a stop on either
side of which two similar lens systems / and 2 are symmetrically placed. The whole system is then called a symmetrical double
objective. Let 5 and S' represent two conjugate principal
R rays. The optical image of the stop with respect to the

system / 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 5 must pass through

R the centre of B, the optical image of

with respect to /.

R Likewise B' ,

the

optical

image

of

with respect to 2, is the

exit-pupil. It follows at once from the symmetry of arrange-

ment that it is always equal

to u' ,

i.e.

the condition of orthos-

copy is fulfilled.

FIG. 29.

Such symmetrical double objectives possess, by virtue of

I eir symmetry, two other advantages: On the one hand, the

eridional

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.
Optik, p. 774.

Cf. Muller-Pouillet,

66

THEORY OF OPTICS

and hence also of the color of the light. If now each of the two systems / 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 / 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 large index, have astigmatic flat images become possible.
V This will be more fully considered in Chapter under the head
of Optical Instruments.
ii. 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. n c is the index for the red light corresponding to the Fraun-
*As will be seen later, this achromatizing can be attained with sufficient accuracy; on the other hand it is not possible at the same time to make the sizes of the
R different images of independent of the color.

PHYSICAL CONDITIONS FOR IMAGE FORMATION 67
ter line C of the solar spectrum (identical with the red
rogen line), nD that for the yellow sodium light, and np that the blue hydrogen line.
Glass.

Calcium-silicate-crown. Ordinary silicate-flint. .

The last column contains the so-called dispersive power

(v ., the substance.

It is defined by the relation

-

n,

(50)

It is practically immaterial whether 11 D or the index for any

ter 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, be identical for both colors. In many cases a

partial correction of the chromatic aberration is sufficient.

Thus a system may be 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.

system may be achromatized one way or the other according

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 of the
and F.

slit; for instance, for a given position of the table
spectrometer, let them bring together the rays C
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

of the focus. But in practice the cases in which thin lenses

PHYSICAL CONDITIONS FOR IMAGE FORMATION 69
ire used are more important. When such lenses are com>ined, the chromatic differences of the unit planes may be
leglected without appreciable error, since, in this case, these )lanes always lie within the lens (cf. page 42). If then the bcal lengths be achromatized, the system is almost perfectly ichrornatic, i.e. both for the position and magnitude of the
mage.
Now the focal length /j of a thin lens whose index for a
jiven color is n^ is given by the equation (cf. eq. (22), page 42)

in which k is an abbreviation for the difference of the curvav
tures of the faces of the lens.
Also, by (24) on page 44, the focal length /of a combination of two thin lenses whose separate focal lengths are/x and
f2 is given by

=

7 + 7 7 J

J\

J2

W)

For

an

increment

dn l

of the

index

;/ 1

corresponding

to

a

change of color, the increment of the reciprocal of the focal

length is, from (51),

d\-7\ =

dn,

I

K

in

wwhich

v l

represents

the

dispersive

power of the material

of

lens i between the two colors which are used. If the focal

ength/of the combination is to be the same for both colors,

llows from (52) and (53) that

{% ~^ ~j+

This equation contains the condition for achromatism. It

also

shows,

since

r
l

and

v 2

always

have the

same sign

no

matter what materials are used for I and 2, that the separate

70

THEORY OF OPTICS

focal length,s of a tJiiji double achromatic lens always have

opposite signs.
From (54) and (52) it follows that the expressions for the
separate focal lengths are

!.~ '

v i

-i.

/i />,-".' /.

I_IL_ /V.-n'

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. If/ is given and the two kinds of glass have been chosen,

then there are four radii of curvature at our disposal to make

/j and/2 correspond to (55). Hence two of these still remain

arbitrary. If the two lenses are to fit together, r^ must be

equal

to

r. 2

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 plano-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)

Hence, from (19) on page 29, the focal length of the combination is given by

..... + ^ I

i

- I

a

(56)

~~ If the focal length is to be achromatic, then, from (56) and (53), - ^i 4_ ?JL __

or