THE THEORY OF OPTICS PAUL DRUDE Students, teachers, and researchers in physical optics, astrophysics, optometry, film optics, and related fields will weli:ome this unabridged reprinting of one of the finest fundamental texts in physical optics. This classic study continues to be unequalled for wide, thorough coverage and complete mathematical treatment of basic ideas. One of the first applications of Maxwell's electromagnetic theory as developed by Hertz to the problem of light, it offers a large amount of valuable material unavailable elsewhere in one volume. This treatise includes perhaps the fullest treatment of the application of thermodynamics to optics. Though one of the most powerful and significant approaches, no other text points out so fully the important practical and theoretical consequences which may be deduced simply and directly from elementary principles by these means. Drude investigates this in such areas as temperature radiation and luminescence, Kirchoff's law of emission and absorption, the sine law in the formation of images of surface elements, the effect of change of temperature on the spectrum of a black body, and the distribution of energy in the spectrum of such a body. In addition, there are particularly valuable sections on absorbing media, crystal optics, and interference. Partial contents, Fundamental Laws, Geometrical Theory of Optical Images, Physical Conditions for Image Formation, Apertures and Effects Depending Upon Them, Optical Instruments, Velocity of Light, Interference, Huygens' Principle, Diffraction, Polarization, Theory of Light, Transparent Isotropic Media, Optical Properties of Transparent Crystals, Absorbing Media, Dispersion, Optically Active Substances, Magnetically Active Substances, Bodies in Motion, Radiation, Application of the Second Law of Thermodynamics to Pure Temperature Radiation, Incandescent Vapors and Gases. "Remarkably original and consecutive presentation of the subject of Optics," A. A. Michelson, in his introduction. Unabridged, unaltered republication of last English translation. Translated by C. Riborg Mann and Robert A. Millikan. Introduction by A. A. Michelson. Index. 110 illustrations. + xxi 546pp. 5% x 8. 60532-9 Paperbound THE THEORY OF OPTICS BY PAUL DRUDE TRANSLATED FROM THE GERMAN BY C. RIBORG MANN AND ROBERT A. MILLIKAN DOVER PUBLICATIONS, INC. NEW YORK This new Dover edition first published in 1959 is an unabridged and unaltered republication of the last English translation. Standard Boole Number: 486-60532-9 Library of Congress Catalog Card Number: 59-65313 Manufactured in the United States of America Dover Publications, Inc. 180 Varick Street New York, N. Y. 10014 PREFACE TO THE ENGLISH TRANSLATION THERE does not exist to-day in the English language a general advanced text upon Optics which embodies the important advances in both theory and experiment which have been made within the last decade. Preston's " Theory of Light " is at present the only general text upon Optics in English. Satisfactory as this work is for the purposes of the general student, it approaches the subject from the historical standpoint and contains no fundamental development of some of the important theories which are fast becoming the basis of modern optics. Thus it touches but slightly upon the theory of optical instruments-a branch of optics which has received at the hands of Abbe and his followers a most extensive and beautiful development ; it gives a most meagre presentation of the electromagnetic theorya theory which has recently been brought into particular prominence by the work of Lorentz, Zeeman, and others ; and it contains no discussion whatever of the application of the laws of thermodynamics to the study of radiation. The book by Heath, the last edition of which appeared in 1895, well supplies the lack in the field of Geometrical Optics, and Basset's " Treatise on Physical Optics" ( I 892) is a valuable and advanced presentation of many aspects of the wave theory. But no complete development of the electromagnetic theory in all its bearings, and no comprehensive discussion of iii iv PREFACE TO THE ENGLISH TRANSLATION the relation between the laws of radiation and the principles of thermQdynamics, have yet been attempted in any general text in English. , It is in precisely these two respects that the" Lehrbuch der Optik" by Professor Paul Drude (Leipzig, 1900) particularly excels. Therefore in making this book, written by one who has contributed so largely to the progress which has been made in Optics within the last ten years, accessible to the English-speaking public, the translators have rendered a very important service to English and American students of Physics. No one who desires to gain an insight into the most mod- ern aspects of optical research can afford to be unfamiliar with this remarkably original and consecutive presentation of the subject of Optics. A. A. MICHELSON. UNIVBB.SITY OF CHICAGO, February, 1902. AUTHOR'S PREFACE THE purpose of the present book is to introduce the reader who is already familiar with the fundamental concepts of the differential and integral calculus into the domain of optics in such a way that he may be able both to understand the aims and results of the most recent investigation and, in addition, to follow the original works in detail. The book was written at the request of the pubiisher-a request to which I gladly responded, not only because I shared his view that a modern text embracing the entire domain was wanting, but also because I hoped to obtain for myself some new ideas from the deeper insight into the subject which writing in book form necessitates. In the second and third sections of the Physical Optics I have advanced some new theories. In the rest of the book I have merely endeavored to present in the simplest possible way results already published. Since I had a text-book in mind rather than a compendium, I have avoided the citation of such references as bear only upon the historical development of optics. The few references which I have included are merely intended to serve the reader for more complete information upon those points which can find only brief presentation in the text, especially in the case of the more recent investigations which have not yet found place in the text-books. V vi AUTHOR'S PREFACE In order to keep in touch with experiment and attain the simplest possible presentation of the subject I have chosen a synthetic method. The simplest experiments lead into the domain of geometrical optics, in which but few assumptions need to be made as to the nature of light. Hence I have begun with geometrical optics, following closely the excellent treatment given by Czapski in " Winkelmann's Handbuch der Physik " and by Lommer in the ninth edition of the " MiillerPouillet " text. The first section of the Physical Optics, which follows the Geometrical, treats of those general properties of light from which the conclusion is drawn that light consists in a periodic change of condition which is propagated with finite velocity in the form of transverse waves. In this section I have included, as an important advance upon most previous texts, Sommer' feld's rigorous solution of the simplest case of diffraction, Cornu's geometric representation of Fresnel's integrals, and, on the experimental side, Michelson's echelon spectroscope. In the second section, for the sake of the treatment of the optical properties of different bodies, an extension of the hypotheses as to the nature of light became for the first time necessary. In accordance with the purpose of the book I have merely mentioned the mechanical theories of light ; but the electromagnetic theory, which permits the simplest and most consistent treatment of optical relations, I have presented in the following form : Let X, Y, Z, and a, /J, y represent respectively the com- ponents of the electric and magnetic forces (the first measured in electrostatic units); also letJ~ ,J~ ,J~, and s,., s,, s. represent the components of the electric and magnetic current densities, I i.e. - times the number of electric or magnetic lines of force 4,r which pass in unit time through a unit surface at rest with reference to the ether ; then, if c represent the ratio of the AUTHOR:S PREFACE vii electromagnetic to the electrostatic unit, the following fundamental eqtttiti'ons always hold : = 41rs.,. oY oZ -c- oz - oy' etc. The number of lines of force is defined in the usual way. The particular optical properties of bodies first make their appearance in the equations which connect the electric and magnetic current densities with the electric and magnetic forces. Let these equations be called the sttbstance equations in order to distinguish them from the above fundamental equations. Since these substance equations are developed for non-homogeneous bodies, i.e. for bodies whose properties vary from point to point, and since the fundamental equations hold in all cases, both the differential equations of the electric and magnetic forces and the equations of condition which must be fulfilled at the surface of a body are immediately obtained. In the process of setting up " substance and fundamental equations " I have again proceeded synthetically in that I have deduced them from the simplest electric and magnetic experiments. Since the book is to treat mainly of optics this process can here be but briefly sketched. For a more complete development the reader is referred to my book "Physik des Aethers auf elektromagnetische Grundlage" (Enke, 1894). In this way however, no explanation of the phenomena of dispersion is obtained because pure electromagnetic experiments lead to conclusions in what may be called the domain of macrophysical properties only. For the explanation of optical dispersion a hypothesis as to the mi'crophysical properties of bodies must be made. As such I have made use of the ion-hypothesis introduced by Helmholtz because it seemed to me the simplest, most intelligible, and most consistent way of presenting not only dispersion, absorption, and rotary 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 formulre for the natural and magnetic rotation of the plane of polarization, formulre which are experimentally verified. Furthermore, in the case of the metals, the ionhypothesis leads to dispersion formulre which make the continuity of the optical and electrical properties of the metals depend essentially upon the inertia of the ions, and which have also been experimentally verified within the narrow limits thus far accessible to observation. The third section of the book is concerned with the relation of optics to thermodynamics and (in the third chapter) to the kinetic theory of gases. The pioneer theoretical work in these subjects was done by Kirchhoff, Clausius, Boltzmann, and W. Wien, and the many fruitful experimental investigations in radiation which have been more recently undertaken show clearly that theory and experiment reach most perfect development through their mutual support. AUTHOR'S PREFACE ix Imbued with this conviction, I have written this book in the endeavor to make the theory accessible to that wider circle of readers who have not the time to undertake the study of the original works. I can make no claim to such completeness as is aimed at in Mascart's excellent treatise, or in Winkelmann's Handbuch. For the sake of brevity I have passed over many interesting and important fields of optical investigation. My purpose is attained if these pages strengthen the reader in the view that optics is not an old and worn-out branch of Physics, but that in it also there pulses a new life whose further nourishing must be inviting to every one. Mr. F. Kiebitz has given me efficient assistance in the reading of the proof. LEIPZIG, January, lg<>o. INTRODUCTION MANY optical phenomena, among them those which have found the most extensive practical application, take place in accordance with the following fundamental laws: 1. The law of the rectilinear propagation of light; 2. The law of the independence of the different portions of a beam of light; 3. The law of reflection; 4. The law of refraction. Since these four fundamental laws relate only to the geometrical determination of the propagation of light, conclusions concerning certain geometrical relations in optics may be reached by making them the starting-point of the analysis without taking account of other properties of light. Hence these fundamental laws constitute a sufficient foundation for so-called geometrical optics, and no especial hypothesis which enters more closely into the nature of light is needed to make the superstructure complete. In contrast with geometrical optics stands physical optics, which deals with other than the purely geometrical properties, and which enters more closely into the relation of the physical properties of different bodies to light phenomena. The best success in making a convenient classification of the great multitude of these phenomena has been attained by devising particular hypotheses as to the nature of light. From the standpoint of physical optics the four above-mentioned fundamental laws appear only as very close approxima- x1 xii INTRODUCTION tions. However, it is possible to state within what limits the Jaws of geometrical optics are accurate, i.e. under what circumstances their consequences deviate from the actual facts. This circumstance must be borne in mind if geometrical optics is to be treated as a field for real discipline in physics rather than one for the practice of pure mathematics. The truly complete theory of optical ins.truments can only be developed from the standpoint of physical optics; but since, as has been already remarked, the laws of geometrical optics furnish in most cases very close approximations to the actual facts, it seems justifiable to follow out the consequences of these laws even in such complicated cases as arise in the theory of optical instruments. TABLE OF CONTENTS PART !.-GEOMETRICAL OPTICS CHAPTER I THE FUNDAMENTAL LAWS ART. PAGB 1. Direct Experiment.................... , ..................... . 2. Law of the Extreme Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3. LawofMalus ................................................ 11 CHAPTER II GEOMETRICAL THEORY OF OPTICAL IMAGES 1. The Concept of Optical Images .. . . .. . . .. . . . . .. .. . . .. . . . .. . . . 14 2. General Formulre for Imag<'s . . .. .. .. . . .. . . .. . . . .. . .. . . .. .. .. 15 3. Images Formed by Coaxial Surfaces.......................... 17 4. Construction of Conjugate Points ................ ,........... 24 5. Classification of the Different Kinds of Optical Systems......... 25 6. Telescopic Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 7. Combinations of Systems.................................... 28 CHAPTER III PHYSICAL CONDITIONS FOR IMAGE FORMATION r. Refraction at a Spherical Surface............................. 32 2. Reflection at a Spherical Surface...................... , . . . . . . . 36 3. Lenses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4. Thin Lenses.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5. Experimental Determination of Focal Length.. . . . . . . . . .. . . . . . . 44 6. Astigmatic Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7. Means of Widening the Limits of Image Formation............ 52 8. Spherical Aberration..................................... - . . . 54 xiii xiv TABLE OF CONTENTS ART. PA.Gil 9. The Law of Sines...................... , .. , . , , .. , , , , . . . . . • •• . 58 10. Images of Large Surfaces by Narrow Beams................... 63 11. Chromatic Aberration of Dioptric Systems.................... 66 CHAPTER IV APERTURES AND THE EFFECTS DEPENDING UPON THEM 1. Entrance- and Exit-Pupils...... . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. 73 2. Telecentric Systems.......................................... 7S 3. Field of View. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4. The Fundamental Laws of Photometry. . . . . . ........... _. . . . . 77 5. The Intensity of Radiation and the Intensity of Illumination of Optical Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6. Subjective Brightness of Optical Images............ . . . . . . . . . . 86 7. The Brightness of Point Sources................. . . . .. . . . . . 90 8. The Effect of the Aperture upon the Resolving Power of Optical Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 CHAPTER V OPTICAL INSTRUMENTS 1. Photographic Systems....................................... 93 2. Simple Magnifying-glasses................................... 9S 3. The Microscope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4. The Astronomical Telescope... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5. The Opera Glass.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6. The Terrestrial Telescope.... . . . . . . . . . . . . . . . . . . . . . . . . . . ..... 112 7. The Zeiss Binocular ......................................... 112 8. The Reflecting Telescope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 PART IL-PHYSICAL OPTICS SECTION I GENERAL .PROPERTIES OF UGHT CHAPTER I THE VELOCITY OF LIGHT I. Romer's Method .....................................•...... 114 2. Bradley's Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . 115 TABLE OF CONTENTS xv ART. PAC.& 3. Fizeau's Method ..................... , ...................... 116 4. Foucault's Method ... _............. , .... , ................... 118 5. Dependence of the Velocity of Light upon the Medium and the Color .................................................... 120 6. The Velocity of a Group of Waves............................ r:z1 CHAPTER II INTERFERENCE OF LIGHT 1. General Considerations ..........................••.......... 124 2. Hypotheses as to the Nature of Light......................... 124 3. Fresnel's Mirrors....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4. Modifications of the Fresnel Mirrors........ , ................. 134 5. Newton's Rings and the Colors of Thin Plates................. 136 6. Achromatic Interference Bands..... . . . . . . . . . . . . . . . . . . . . . . . . . . 144 7. The Interferometer.......................................... 144 8. Interference with Large Difference of Path .................... 148 9. Stationary Waves............................................ 154 10. Photography in Natural Colors ............................... 156 CHAPTER III HUYGENS' PRINCIPLE 1. Huygens' Principle as first Conceived......................... 159 2. Fresnel's Improvement of Huygens' Principle.................. 162 3. The Differential Equation of the Light Disturbance..... . . . . . . . 169 4. A Mathematical Theorem.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 5. Two General Equations ...................................... 174 6. Rigorous Formulation of Huygens' Principle .................. 179 CHAPTER IV DIFFRACTION OF LIGHT 1. General Treatment of Diffraction Phenomena.................. 185 2. Fresnel's Diffraction Phenomena, ......... , . . . . . . . . . . . . . . . . . . 188 3. Fresnel's Integrals........................................... 188 4. Diffraction by a Straight Edge................................ 192 5. Diffraction through a Narrow Slit................ ,............. 198 6. Diffraction by a Narrow Screen............................... 201 ,. Rigorous Treatment of Diffraction by a Straight Edge.. . . . . . . . . 203 xvi TABLE OF CONTENTS ART. PA.GK 8. Fraunhofer's Diffraction Phenomena.......................... 213 9. Diffraction through a Rectangular Opening................... 214 10. Diffraction through a Rhomboid ............................. 217 11, Diffraction through a Slit .................................... 217 12. Diffraction Openings of any Form ............................ 219 13. Several Diffraction Openings of like Form and Orientation..... 219 14. Babinet's Theorem .......................................... 221 15. The Diffraction Grating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 16. The Concave Grating ........................................ 225 17. Focal Properties of a Plane Grating .......................... 227 18. Resolving Power of a Grating ................................ 227 19. Michelson's Echelon ......................................... 228 20. The Resolving Power of a Prism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 :z1. Limit of Resolution of a Telescope..... . . . . . . . . . . . . . . . . . . . . . . . 235 22. The Limit of Resolution of the Human Eye ................... 236 23. The Limit of Resolution of the Microscope.................... 236 CHAPTER V POLARIZATION 1. Polarization by Double Refraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 2. The Nicol Prism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 3. Other Means of Producing Polarized Light.. . . . . . . . . . . . . . . . . . . 246 4. Interference of Polarized Light. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 247 5. Mathematical Discussion of Polarized Light. . . . . . . . . . . . . . . . . . . 247 6. Stationary Waves Produced by Obliquely Incident Polarized Light .................................................... 251 7. Position of the Determinative Vector in Crystals .............. 252 8. Natural and Partially Polarized Light ........................ 253 9. Experimental Investigation of Elliptically Polarized Light...... 25s SECTION II OPTICAL PROPERTIES OF BODIES CHAPTER I THEORY OF LIGHT 1. Mechanical Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 ,. Electromagnetic Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26o 3. The Definition of the Electric and of the Magnetic Force. . . . . . . 262 TABLE OF CONTENTS xvii AllT. l'AGS 4- Definition of the Electric Current in the Electrostatic and the Electromagnetic Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 5. Definition of the Magnetic Current............................ 265 6. The Ether. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 7. Isotropic Dielectrics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 8. The Boundary Conditions .................................... 271 9. The Energy of the Electromagnetic Field. . . . . . . . . . . . . . . . . . . . . 272 10. The Rays of Light as the Lines of Energy Flow................ 273 CHAPTER II TRANSPARENT ISOTROPIC MEDIA 1. The Velocity of Light........................................ 274 2. The Transverse Nature of Plane Waves........................ 278 3. Reflection and Refraction at the Boundary between two Trans- parent Isotropic Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 4- Perpendicular Incidence; Stationary Waves................... 284 5. Polarization of Natural Light by Passage through a Pile of Plates..................................................... 285 6. Experimental Verification of the Theory... . . . . . . . . . . . . . . . . . . . 286 7. Elliptic Polarization of the Reflected Light and the Surface or Transition Layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 8. Total Reflection. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 295 9. Penetration of the Light into the Second Medium in the Case of Total Reflection .......................................... 299 10. Application of Total Reflection to the Determination of Index of Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 11. The Intensity of Light in Newton's Rings ..................... 302 12. Non-homogeneous Media; Curved Rays . . . . . . . . . . . . . . . . . . . . . . 306 CHAPTER III OPTICAL PROPERTIES OF TRANSPARENT CRYSTALS r. Differential Equations and Boundary Conditions............... 3o8 2. Light-vectors and Light-rays................................. 3u 3. Fresnel's Law for the Velocity of Light . . . . . . . . . . . . . . . . . . . . . . . 314 4- The Directions of the Vibrations. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 316 5. The Normal Surface ......................................... 317 6. Geometrical Construction of the Wave Surface and of the Direc- tion of Vibration .. , , . , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 xviii TABLE OF CONTENTS ARTa PAGE 7. Uniaxial Crystals ............................................ 323 8. Determination of the Direction of the Ray from the Direction of the Wave Normal ........................................ 324 9. The Ray Surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 IO. Conical Refraction .......................................... 331 11. Passage of Light through Plates and Prisms of Crystal ......... 335 12. Total Reflection at the Surface of Crystalline Plates . . . . . . . . . . . . 339 13. Partial Reflection at the Surface of a Crystalline Plate ......... 344 14. Interference Phenomena Produced by Crystalline Plates in Polarized Light when the Incidence is Normal. ............ 344 15. Interference Phenomena in Crystalline Plates in Convergent Polarized Light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 CHAPTER IV ABSORBING MEDIA 1. Electromagnetic Theory.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 2. Metallic Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 3. The Optical Constants of the Metals. . . . . . . . . . . . . . . . . . . . . . . . . . 366 4. Absorbing Crystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 5. Interference Phenomena in Absorbing Biaxial Crystals ......... 374 6. Interference Phenomena in Absorbing Uniaxial Crystals........ 380 CHAPTER V DISPERSION 1. Theoretical Considerations.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 2. Normal Dispersion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 3. Anomalous Dispersion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 4. Dispersion of the Metals.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3g6 CHAPTER VI OPTICALLY ACTIVE SUBSTANCES r. General Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 2. Isotropic Media.............................................. 401 3. Rotation of the Plane of Polarization.. . . . . . . . . . . . . . . . . . . . . . . . 404 4. Crystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 5. Rotary Dispersion .......................................... 412 6. Absorbing Active Substances................................. 415 TABLE OF CONTENTS xix CHAPTER VII MAGNETICALLY ACTIVE SUBSTANCES A. Hypothesis of Molecular Currents ART. •AGS 1. General Considerations ..................................•... 418 2. Deduction of the Differential Equations ....................... 420 3. The Magnetic Rotation of the Plane of Polarization.. . . . . . . . . . . 426 4. Dispersion in Magnetic Rotation of the Plane of Polarization .. 429 5. Direction of Magnetization Perpendicular to the Ray........... 433 B. Hypothesis of the Hall Effect , . General Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 2. Deduction of the Differential Equations ....................... 435 3. Rays Parallel to the Direction of Magnetization. . . . . . . . . . . . . . . 437 4. Dispersion in the Magnetic Rotation of the Plane of Polarization. 438 5. The Impressed Period Close to a Natural Period .............. 440 6. Rays Perpendicular to the Direction of Magnetization ......... 443 7. The Impressed Period in the Neighborhood of a Natural Period. 444 8. The Zeeman Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 9. The Magneto-optical Properties of Iron, Nickel, and Cobalt... 449 10. The Effects of the Magnetic Field of the Ray of Light ......... 452 CHAPTER VIII BODIES IN MOTION 1. General Considerations ...................................... 457 2. The Differential Equations of the Electromagnetic Field Re- ferred to a Fixed System of Coordinates .................... 457 3. The Velocity of Lig-ht in Moving Media ....................... 465 4. The Differential Equations and the Boundary Conditions Re- ferred to a Moving System of Coordinates which is Fixed with Reference to the Moving Medium ..................... 467 5. The Determination of the Direction of the Ray by Huygens' Principle ................................................. 470 6. The Absolute Time Replaced by a Time which is a Function of the Coordinates........................................... 471 7. The Configuration of the Rays Independent of the Motion. . . . . 473 8. The Earth as a Moving System.............................. 474 9. The Aberration of Light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 S IO. Fizeau's Experiment with Polarized Light...................... 477 11. Michelson's Interference Experiment ......................... 478 xx TABLE OF CONTENTS PART III.-RADIATION CHAPTER I ENERGY OF RADIATION AJIT. PAGS 1. Emissive Power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 2, Intensity of Radiation of a Surface............................ 484 3. The Mechanical Equivalent of the Unit of Light .............. 485 4. The Radiation from the Sun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 5. The Efficiency of a Source of Light. . . . . . . . . . . . . . . . . . . . . . . . . . . 487 6. The Pressure of Radiation ................................... 488 7. Prevost's Theory of Exchanges ............... , ............... 491 CHAPTER II APPLICATION OF THE SECOND LAW OF THERMODYNAMICS TO PURE TEMPERATURE RADIATION 1. The Two Laws of Thermodynamics........................... 493 2. Temperature Radiation and Luminescence..................... 494 3. The Emissive Power of a Perfect Reflector or of a Perfectly Transparent Body is Zero.................................. 495 4- Kirchhoff's Law of Emission and Absorption, ................. 4¢ 5. Consequences of Kirchhoff's Law............................. 499 6. The Dependence of the Intensity of Radiation upon the Index of Refraction of the Surrounding Medium .................. 502 7. The Sine Law in the Formation of Optical Images of Surface Elements ................................................ 505 8. Absolute Temperature....................................... 5o6 9- Entropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 10. General Equations of Thermodynamics ...................... 5n 11. The Dependence of the Total Radiation of a Black upon its Ab- solute Temperature ....................................... 512 12. The Temperature of the Sun Calculated from its Total Emission 515 13. The Effect of Change in Temperature upon the Spectrum of a Black Body............................................. 516 14. The Temperature of the Sun Determined from the Distribution of Energy in the Solar Spectrum ............... , ........ , .. 523 15. The Distribution of the Energy in the Spectrum of a Black Body .................................................... 524 TABLE OF CONTENTS CHAPTER III INCANDESCENT VAPORS AND GASES ART. PAGS 1. Distinction between Temperature Radiation and Luminescence. 528 2. The Ion-hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 3. The Damping of Ionic Vibrations because of Radiation. . . . . . . . 534 4- The Radiation of the Ions under the Influence of External Radiation .................................... •.. • . . . . . . . . 535 5. Fluorescence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5J6 6. The Broadening of the Spectral Lines Due to Motion in the Line of Sight. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 7. Other Causes of the Broadening of the Spectral Lines ......... 541 INDEX ••••••••••••••••••••••• , •••••••••• , ••••••• , • • • • • • • • • • • 543 PART I GEOMETRICAL OPTICS CHAPTER I THE FUNDAMENTAL LAWS 1. Direct Experiment.-The four fundamental laws stated above are obtained by direct experiment. The rectilinear propagation of light is shown by the shadow of an opaque body which a point source of light P casts upon a screen S. If the opaque body contains an aperture L, then the edge of the shadow cast upon the screen is found to be the intersection of S with a cone whose vertex lies in the source P and whose surface passes through the periphery of the aperture L. If the aperture is made smaller, the boundary of the shadow upon the screen S contracts. Moreover it becomes indefinite when L is made very small (e.g. less than I mm.), for points upon the screen which lie within the geometrical shadow now receive light from P. However, it is to be observed that a true point source can never be realized, and, on account of the finite extent of the source, the edge of the shadow could never. be perfectly sharp even if light were propagated in straight lines (umbra and penumbra). Nevertheless, in the case of a very small opening L (say of about one tenth mm. diameter) the light is spread out behind L upon the screen so far that in this case the propagation cannot possibly /Je rectilinear. 2 THEORY OF OPTICS The same result is obtained if the shadow which an opaque body S' casts upon the screen S is studied, instead of the spreading out of the light which has passed through a hole in an opaque object. If S' is sufficiently small, rectilinear propagation of light from P does not take place. It is therefore necessary to bear in mind that the law of the rectilinear propagation of light holds only when the free opening through which the light passes, or the screens which prevent its passage, are not too small. In order to conveniently describf the propagation of light from a source P to a screen S, it is customary to say that P sends rays to S. The path of a ray of light is then defined by the fact that its effect upon 5 can be cut off only by an obstacle that lies in the path of the ray itself. When the propagation of light is rectilinear the rays are straight lines, as when light from P passes through a sufficiently large opening in an opaque body. In this case it is customary to say that P sends a beam of light through L. Since by diminishing L the result upon the screen S is the same as though the influence of certain of the rays proceeding from P were simply removed while that of the other rays remained unchanged, it follows that the different parts of a beam of light are independent of one another. This law too breaks down if the diminution of the opening L is carried too far. But in that case the conception of light rays propagated in straight lines is altogether untenable. The concept of light rays is then merely introduced for convenience. It is altogether impossible to isolate a single ray and prove its physical existence. For the more one tries to attain this end by narrowing the beam, the less does light proceed in straight lines, and the more does the con~ept of light rays lose its physical significance. If the homogeneity of the space in which the light rays exist is disturbed by the introduction of some substance, the rays undergo a sudden change of direction at its surface: each ray splits up into two, a reflected and a refracted ray. If the sur- THE FUNDAMENTAL LAWS 3 face of the body upon which the light falls is plane, then the plane of incidence is that plane which is defined by the incident ray and the normal N to the surface, and the angle of z'nddence ¢ is the angle included between these two direc- tions. The following laws hold: The re.fleeted and refracted rays both lie in the plane of incidence. The angle of reflection (the angle included between N and the reflected ray) is equal to the angle of incidence. The angle of refraction¢' (angle included between N and the refracted ray) bears to the angle of incidence the relation -ssiinn-q-/ -n' (I) in which n is a constant for any given color, and is called the index of refraction of the body with reference to the surrounding medium.-Unless otherwise specified the index ofrefraction with respect to air will be understood.-For all transparent liquids and solids n is greater than I. If a body A is separated from air by a thin plane parallel plate of some other body B, the light is refracted at both surfaces of the plate in accordance with equation (I); i.e. = sin sin n6 , and if (6) then sin I; i.e. there is no real angle of refraction C from those for which that sum < C. The former regions lie on the side of the aplanatic surface toward the less refractive medium (left in the figure), the latter on the side toward the more refractive medium (right in the figure). If now SOS' represents a section of the surface between the THE FUNDAMENTAL LAWS II two media, and PO, P' 0 the actual path which the light takes in accordance with the law of refraction, then the length of the path through O is a maximum or a minimum according as SOS' is more or less convex toward the less refracting medium .lt Fm.4. than the aplanatic surface AOA'. The proof appears at once from the figure. If, for example, SOS' is a plane, the length of the path is a minimum. In the case shown in the figure the length of the path is a maximum. Since, as will be shown later, the index of refraction is inversely proportional to the velocity, the optical path nl is proportional to the time which the light requires to travel the distance l. The principle of least path is then identical with Fermat's principle of least time, but it is evident from the above that, under certain circumstances, the time may also be a maximum. = Since o~nl o holds for each single reflection or refraction, the equation o~nl = o may at once be applied to the case of any number of reflections and refractions. 3. The Law of Malus.-Geometrically considered there are two different kinds of ray systems: those which may be cut at right angles by a properly constru;:ted surface F (ortho- 12 THEORY OF OPTICS tomic system), and those for which no such surface F can be found (anorthotomic system). With the help of the preceding principle the law of Malus can now be proved. This law is stated thus: An ortlzotomic system of rays remains orthotomz'c after any number of reflections and refractions. From the standpoint of the wave theory, which makes the rays the normals to the wave front, the law is self-evident. But it can also be deduced from the fundamental geometrical laws already used. Let (Fig. S) ABCDE and A'B'C'D'E' be two rays infinitely close together and let their initial direction be normal to a surface F. If L represents the total f-.........,,...- f optical distance from A to E, then it may be proved that every ray rt. whose total path, measured from its origin A, A', etc., has the same optical lc::ngth L, is normal to a sur- face F' which is the locus of the ends E, E', etc., of those paths. For the purpose of the proof let A'B and E 'D be drawn. FIG. 5. According to the law of extreme path stated above, the length of the path A'B'C'D'E' must be equal to that of the infinitely near path A'BCDE', i.e. equal to L, which is also the length of the path ABCDE. If now from the two optical distances A'BCDE' and ABCDE the common portion BCD be sub- tracted, it follows that n-AB+n'.DE= n-A'B+ n'-DE', in which n represents the .index of the medium between the surfaces F and B, and n' that of the medium between D and F'. But since AB= A'B, because AB is by hypothesis normal to F, it follows that DE= DE', THE FUNDAMENTAl LAWS 13 i.e. DE is perpendicular to the surface F'. In like manner it may be proved that any other ray D'E' is normal to F'. Rays which are emitted by a luminous point are normal to a surface F, which is the surface of any sphere described about the luminous point as a centre. Since every source of light may be looked upon as a complex of luminous points, it follows that light rays always form an orthotomic systmi. CHAPTER II G~OMETRICAL THEORY OF OPTICAL IMAGES 1. The Concept of Optical Images.-If in the neighborhood of a luminous point P there are refracting and reflecting bodies having any arbitrary arrangement, then, in general, there passes through any point P' in space one and only one ray of light, i.e. the direction which light takes from P to P' is completely determined. Nevertheless certain points P' may be found at which two or more of the rays emitted by Pintersect. If a large number of the rays emitted by P intersect in a point P', then P' is called the optical image of P. The intensity of the light at P' will clearly be a maximum. If the actual intersection of the rays is at P', the image is called real,· if P' is merely the intersection of the backward prolongation of the rays, the image is called vi'rtual. The simplest example of a virtual image is found in the reflection of a luminous point P in a plane mirror. The image P' lies at that point which is placed symmetrically to P with respect to the mirror. Real images may be distinguished from virtual by the direct illumination which they produce upon a suitably placed rough surface such as a piece of white paper. In the case of plane mirrors, for instance, no light whatever reaches the point P'. Nevertheless virtual images may be transformed into real by certain optical means. Thus a virtual image can be seen because it is transformed by the eye into a real image which illumines a certain spot on the retina. The cross-section of the bundle of rays which is brought together in the image may have finite length and breadth or may be infinitely narrow so as in the limit to have but one GEOMETRICAL THEORY OF OPTICAL IMAGES 15 dimension. Consider, for example, the case of a single refraction. If the surface of the refracting body is the aplanatic surface for the two points P and P', then a beam of any size which has its origin in P will be brought together in P'; for all rays which start from P and strike the aplanatic surface must intersect in P', since for all of them the total optical distance from P to P' is the same. If the surface of the refracting body has not the form of the aplanatic surface, then the number of rays which intersect in P is smaller the greater the difference in the form of the two surfaces (which are necessarily tangent to each other, see page 10). In order that an infinitely narrow, i.e. a plane, beam may come to intersection in P', the curvature of the surfaces at the point of tangency must be the same at least in one plane. If the curvature of the two surfaces is the same at 0 for two and therefore for all planes, then a solid elementary beam will come to intersection in P'; and if, finally, a finite section of the surface of the refracting body coincides with the aplanatic surface, then a beam of finite cross-section will come to intersection in P'. Since the direction of light may be reversed, it is possible to interchange the source P and its image P', i.e. a source at P' has its image at P. On account of this reciprocal relationship P and P' are called co1yi1gate po£nts. 2. General Formulm for Images.-Assume that by means of reflection or refraction all the points P of a given space are imaged in points P' of a second space. The former space will be called the ob.feet space; the latter, the £1nage space. From the definition of an optical image it follows that for every ray which passes through P there is a conjugate ray passing through P'. Two rays in the object space which intersect at P must correspond to two conjugate rays which intersect in the image space, the intersection being at the point P' which is conjugate to P. For every point P there is then but one conjugate point P'. If four points P1P.j'/'4 of the object space lie in a plane, then the rays which connect any two pairs of THEORY OF OPTICS these points intersect, e.g. the ray P 1P 2 cuts the ray PaP, in the point A. Therefore the conjugate rays P'1P'2 and P'8P'4 also intersect in a point, namely in A' the image of A. Hence the four images P/P 21Pa'P/ also lie in a plane. In other words, to every point, ray, or plane in the one space there corresponds one, and but one, point, ray, or plane in the other. Such a relation of two spaces is called in geometry a collz"near relations/tip. The analytical expression of the collinear relationship can be easily obtained. Let x, y, z be the coordinates of a point P of the object space referred to one rectangular system, and x', y', z' the coordinates of the point P' referred to another rectangular system chosen for the image space; then to every x, y, z there corresponds one and only one x', y', z', and vt"ce versa, This is only possible if + + + x, - -a1x- -b-1y- ~c1z- d1 + + - ax+ by cz d ' y , = a~+ b2y+ ax+ by+ ccz~++dd2' + ++ z, -- -aaaX-x-+b-ba-yy++~cc8zz- ~ dd3' () • 1 in which a, b, c, d are constants. That is, for any given x', y', z', the values of x, y, z may be calculated from the three linear equations (1); and inversely, given values of x, y, z determine x', y', z'. If the right-hand side of equations (1) were not the quotient of two linear functions of x, y, z, then for every x', y', z' there would be several values of x, y, z. Furthermore the denominator of this quotient must be one and + the same linear function (ax+ by+ cz d), since otherwise a plane in the image space A'x' + B'y' + C'z' +I)'= o would not again correspond to a plane Ax+By+cz+D= o in the object space. GEOMETRICAL THEORY OF OPTICAL IMAGES 17 If the equations (1) be solved for x, y, and z, forms analogous to (I) are obtained; thus + + + + + a/x' + x = a'x' b/y' b'y' c/z' d1' c'z' d' ' etc. From (1) it follows that for ax+ by + cz + d = o: x' = y' = z' = oo • Similarly from (2) for + + a'x' b'y' c'z' + d' = o: x = y = z = oo • + + + The plane ax by cz d = o is called the focal plane ~ of the object space. The images P' of its points P lie at infinity. Two rays which originate in a point P of this focal plane correspond to two parallel rays in the image space. + + + The plane a'x' b'y' c'z' d' = o is called the focal plane g:' of the £mage space. Parallel rays in the object space correspond to conjugate rays in the image space which inter- sect in some point of this focal plane g:'. In case a= b = c = o, equations (1) show that to finite values of x, y, z correspond finite values of x', y', z'; and, inversely, since, when a, b, and c are zero, a', b', c' are also zero, to finite values of x', y', z' correspond finite values of x, y, z. In this case, which is realized in telescopes, there are no focal planes at finite distances. 3. Images Formed by Coaxial Surfaces.-In optical in- struments it is often the case that the formation of the image takes place symmetrically with respect to an axis; e.g. this is true if the surfaces of the refracting or reflecting bodies are surfaces of revolution having a common axis, in particular, surfaces of spheres whose centres lie in a straight line. From symmetry the image P' of a point P must lie in the plane which passes through the point P and the axis of the system, and it is entirely sufficient, for the study of the image formation, if the relations between the object and image in such a meridian plane are known. 18 THEORY OF OPTICS If the xy plane of the object space and the x'y' plane of the image space be made to coincide with this meridian plane, and if the axis of symmetry be taken as both the x and the x' axis, then the z and z' coordinates no longer appear in equations (r ). They then reduce to + x' _ a1x+b1y+d1 - ar+by d, (3) The coordinate axes of the xy and the x'y' systems are then parallel and the .x and .x' axes lie in the same line. The origin O' for the image space is in general distinct from the origin O for the object space. The positive direction of .x will be taken as the direction of the incident light (from left to Y' 0 .:iC, O' FIG. 6. right); the positive direction of x', the opposite, i.e. from right to left. The positive direction of y and y' will be taken upward (see Fig. 6). From symmetry it is evident that x' does not change its value when y changes sign. Therefore in equations (3) b1 = b = o. It also follows from symmetry that a change in sign of y produces merely a change in sign of y'. Hence = = a2 d2 o and equations (3) reduce to ..., - A, - a~a1-xx--++--,dd_-1' y '-- b~ ax+-tf (4) Five constants thus remain, but their ratios alone are sufficient to determine the formation of the image. Hence GEOMETRICAL THEORY OF OPTICAL IMAGES 19 there are in general four characteristic constants which determine the formation of z"mages by coaxial surfaces. The solution of equations (4) for x and y gives = _ dx' - d1 x - a1 - ax'' y a1d - ad1 y' -b2-- • a1 - ax'" (S) +The equation of the focal plane of the object space is ax d = o, that of the focal plane of the image space ax' - a1 = o. The intersections F and F' of these planes with the axis of the system are called the principal foe£. If the principal focus F of the object space be taken as the origin of x, and likewise the principal focus F' of the image + space as the origin of x', then, if x0 , x0' represent the coordi- nates measured from the focal planes, ax0 will replace ax d and - ax0', a1 - ax'. Then from equations (4) (6) Hence only two characteristic constants remain in the equations. The other two were taken up in fixing the positions of the focal planes. For these two complex constants simpler expressions will be introduced by writing (dropping subscripts) xx'=ff', (7) In this equation x and x' are the distances of the object and the z"mage from the principal focal planes g: and g:1 respectively. The ratio y' :y is called the magnification. It is I for x = f, i.e. x' = f'. This relation defines two planes .p and .p' which are at right angles to the axis of the system. These planes are called the unit planes. Their points of intersection Hand H' with the axis of the system are called unit poz"nts. The unz"t planes are characterized by the fact that the distance from the axz"s of any poz"nt P i'n one unz"t plane i's equal to that of the conjugate poz"nt P' i'n the other unz"t plane. The two remaining constants/ andf' of equation (7) denote, in accord- zo THEORY OF OPTICS ance with the above, the distance of the unit planes op, op' from the focal planes ~. ~'. The constant f is called the focal length of the object space; f', the focal length of the z"mage space. The direction off is positive when the ray falls first upon the focal plane ~. then upon the unit plane .\); for f' the case is the reverse. In Fig. 7 both focal lengths are positive. The significance of the focal lengths can be made clear in the following way: Parallel rays in the object space must have conjugate rays in the image space which intersect in some point in the focal plane ~' distant, say, y' from the axis. The value ofy' evidently depends on the angle of inclination u of the incident ray with respect to the axis. If u = o, it follows = from symmetry that y' o, i.e. rays parallel to the axis have conjugate rays which intersect in the principal focus F'. But f FIG. 7. if u is not equal to zero, consider a ray PFA which passes through the first principal focus F, and cuts the unit plane .p in A (Fig. 7). The ray which is conjugate to it, A'P', must evidently be parallel to the axis since the first ray passes through F. Furthermore, from the property of the unit planes, A and A' are equally distant from the axis. Consequently the distance from the axis y' of the image which is formed by a parallel beam incident at an angle u is, as appears at once from Fig. 7, y' =f-tan u. (8) Hence the following law: Tlte focal length of the obj'ecl space i's equal to the ratz"o of the linear magnz"tude of an z"mage GEOMETRICAL THEORY OF OPTICAL IMAGES u formed t"n the focal plane of the t'mage space to the apparent (angular) magnitude of its in.finitely distant object. A similar definition holds of course for the focal length f' of the image space, as is seen by conceiving the incident beam of parallel rays to pass first through the image space and then to come to a focus in the focal plane ~- If in Fig. 7 A'P' be conceived as the incident ray, so that the functions of the image and object spaces are interchanged, then the following may be given as the definition of the focal Jength f, which will then mean the focal length of the image space: The focal length of the image space is equal to the distance between the axis and any ray of the object space which t"s parallel to the axis divided by the tangent of the inclinatz"on of its conjugate ray. Equation (8) may be obtained directly from (7) by making = = tan u y: x and tan u' y' : x'. Since x and x' are taken positive in opposite directions and y and y' in the same direction, it follows that u and u' are positive in different directions. The angle of i'ncli'nation u of a ray t"n the object space i's posi'#ve if the ray goes upward from left to right; the angle of inclination u' of a ray in the image space is positive if tlze ray goes downwardfrom left to right. The magnification depends, as equation. (7) shows, upon x, the distance of the object from the principal focus F, and upon f, the focal length. It is, however, independent of y, i.e. the image of a plane object which is perpendicular to the axis of the system is similar to the object. On the other hand the image of a solid object is not similar to the object, as is evident at once from the dependence of the magnification upon x. Furthermore it is easily shown from (7) that the magnification t"n depth, i.e. the ratio of the increment dx' of .i:-' to an increment dx of x, is proportional to the square of the lateral magnification. Let a ray in the object space intersect the unit plane .p in 22 THEORY OF OPTICS A and the axis in P (Fig. 8). Its angle of inclination u with respect to the axis is given by = = AH AH tan u PH f _ .i-' if .i- taken with the proper sign represents the distance of P from F. 96' FIG, 8. The angle of inclination u' of the conjugate ray with respect to the axis is given by tan u' = A'H' P' H' = A'H' f' _ .i-" if .i-' represent the distance of P' from F', and P 1 and A' are the points conjugate to P and A. On account of the property of the unit planes AH= A'H'; then by combination of the last two equations with (7), tan u' f - .i- .i- f tan u = f' - .i-' = - f' = - ?· (9) The ratio of the tangents of inclination of conjugate rays is called the convergence ratio or the angular magnification. It is seen from equation (9) that it is independent of u and u'. = - = The angular magnification I for .i- f' or .i-' = f. The two conjugate points Kand K' thus determined are called the nodal points of the system. They are characterised by the GEOMETRICAL THEORY OF OPTICAL IMAGES 23 fact that a ray tltrough one nodal point K is cot?fugate and parallel to a ray through the other nodal point K'. The position of the nodal points for positive focal lengths f and f' is fie FIG. 9. shown in Fig. 9. KA and K'A' are two conjugate rays. It follows from the figure that the distance between the two nodal points z"s the same as that between the two unit points. If f = f', the nodal points coincide with the unit points. Multiplication of the second of equations (7) by (9) gives y' tan u' f y tan u = - f'' (10) If e be the distance of an object P from the unit plane op, and e' the distance of its image from the unit plane op', e and e' being positive if P lies in front of (to the left of) op and P' behind (to the right of) .p', then e = f - x, e' = f' - x'. Hence the first of equations (7) gives e+ f = f' e' I. • • • • The same equation holds if e and e' are the distances of P and P' from any two conjugate planes which are perpendicular to the axis, and f and f' the distances of the principal foci from these planes. This result may be easily deduced from (7). THEORY OF OPTICS 4. Construction of Conjugate Points.-A simple graphical interpretation may be given to equation (11). If ABCD (Fig. 10) is a rectangle with the sides / and /', then any straight line ECE' intersects the pro- longations of/ and/' at such distances = from A that the conditions AE e and A E AE' = e' satisfy equation (r I). Fm. ro. It is also possible to use the unit plane and the principal focus to determine the point P' conju- gate to P. Draw (Fig. I I) from Pa ray PA parallel to the axis and a ray PF passing through the principal focus F. FIG. II. A'F' is conjugate to PA, A' being at the same distance from the axis as A; also P'B', parallel to the axis, is conjugate to PFB, B' being at the same distance from the axis as B. The intersection of these two rays is the conjugate point sought. The nodal points may also be conveniently used for this construction. The construction shown in Fig. I I cannot be used when P and P' lie upon the axis. Let a ray from P intersect the focal plane ~ at a distance g and the unit plane ~ at a distance h from the axis (Fig. 12). Let the conjugate ray intersect~' and ~ at the distances h'(= h) and g'. Then from the figure ,,~= g PF k=J--+- - x g' P'F' - x' < - r' 7i= f'+P'P= J'-x'; GEOMETRICAL THEORY OF OPTICAL IMAGES 25 and by addition, since from equation (7) xx'= ff', g + g' 2xx1 -fx' -f'x -,,-=ff' +xx -fx' -f'x = 1' (iz) P' may then be found by laying off in the focal plane ~, the = distance g' h - g, and in the unit plane op' the distance X JC' P' ,..___,. -f FIG. 12. h' = h, and drawing a straight line through the two points thus determined. g and g' are to be taken negative if they lie below the axis. 5. Classi:fication of the Different Kinds of Optical Sys- tems.-The different kinds of optical systems differ from one another only in the signs of the focal lengths f and /'. If the two focal lengths have the same sign, the system i's concurrent, i.e. if the object moves from left to right (x increases), the image likewise moves from left to right (.r' decreases). This follows at once from equation (7) by taking into account the directions in which x and x' are considered positive (see above, p. I 8 ). It will be seen later that this kind of image formation occurs if the image is due to refraction alone or to an even number of reflections or to a combination of the two. Since this kind of image formation is most frequently produced by refraction alone, it is also called dioptric. THEORY OF OPTICS If the t.wo focal lengths have opposite signs the system is contracurrent, i.e. if the object moves from left to right, the image moves from right to left, as appears from the formula xx' = ff'. This case occurs if the image is produced by an odd number of reflections or by a combination of an odd number of such with refractions. This kind of image formation is called katoptric. When it occurs the direction of propagation of the light in the image space is opposite to that in the object space, so that both cases may be included under the law: In all cases of image formation if a point P be conceived to move along a ray in the direction in which the light travels, the image P' of that point moves along the co,ifugate ray in the direction in whiclt the ltglzt travels. Among dioptric systems a distinction is made between those having positive and those having negative focal lengths. The former systems are called convergent, the latter divergent, because a bundle of parallel rays, after passing the unit plane ,O' of the image space, is rendered convergent by the former, di,vergent by the latter. No distinction between systems on the ground that their foci are real or virtual can be made, for it will be seen later that many divergent systems (e.g. the microscope) have real foci. By similar definition katoptric systems which have a nega- tive focal length in the image space are called convergent, - for in reflection the direction of propagation of the light is reversed. There are therefore the four following kinds of optical systems: + Dioptric ... { a. b. Convergent: Divergent: - f, f, + Katoptric .. { a. b. Convergent: Divergent: - f, f, +f'. -f'. -f'. +f'. 6. Telescopic Systems.-Thus far it has been assumed that the focal planes lie at finite distances. If they lie at infinity the case is that of a telescopic system, and the coeffi- GEOMETRICAL THEORY OF OPTICAL IMAGES 27 cient a vanishes from equations (4), which then reduce by a suitable choice cf the origin of the x coordinates to x' = ax, y' = fly. . (13) Since x' = o when x = o, it is evident that any two conjugate points may serve as origins from which x and x' are measured. It follows from equation ( I 3) that the magnification in breadth and depth are constant. The angular magnification is also constant, for, given any two conjugate rays OP and O'P', their intersections with the axis of the system may serve as the ongms. If then a point P of the first ray has the coordinates x, y, and its conjugate point P' the coordinates x', y', the tangents of the angles of inclination are tan u = y : .r, tan u' = y' : x'. Hence by (13) tan u' : tan u = fl : a. a must be positive for katoptric (contracurrent) systems, negative for dioptric (concurrent) systems. For the latter it is evident from (14) and a consideration of the way in which z, and u' are taken positive (see above, p. 21) that for positive fJ erect images of infinitely distant objects are formed, for nega- tive /J, inverted images. There are therefore four different kinds of telescopic systems depending upon the signs of a and fl. Equations (14) and (13) give y' tan u' fP y tan u a A comparison of this equation with (10) (p. 23) shows that for telescopic systems the two focal lengths, though both infinite, have a finite ratio. Thus f fJ2 f'= - a· = If f f', as is the case in telescopes and in all instru- ments in which the index of refraction of the object space is- THEORY OF OPTICS equal to that of the image space (cf. equation (9), Chapter III), then a= - fP. Hence from (14) tan tt' : tan u = - I : fl. This convergence ratio (angular magnification) is called in the case of telescopes merely the magnification r. From (13) y :y' = - r, (14') i.e. for telescopes the reciprocal of the lateral magnification is numerz''cally equal to the allgular magnijicatz"on. 7. Combinations of Systems.-A series of several systems must be equivalent to a single system. Here again attention will be confined to coaxial systems. If f. and f.' are the focal lengths of the first system alone, and /4 and /4,' those of the second, and f and f' those of the combination, then both the focal lengths and the positions of the principal foci of the combination can be calculated or constructed if the distance = F/F 2 L1 (Fig. I 3) is known. This distance will be called for brevity the separation of the two systems I and 2, and will be considered positive if F/ lies to the left of F2 , otherwise negative. A ray S (Fig. I 3), which is parallel to the axis and at a '!le; s s F, rsl '!I, FIG. 13. distance y from it, will be transformed by system I into the ray S1 , which passes through the principal focus F 1' of that system. S 1 will be transformed by system 2 into the ray S'. GEOMETRICAL THEORY OF OPTICAL IMAGES 29 The point of intersection of this ray with the axis is the prin- cipal focus of the image space of the combination. Its position can be calculated from the fact that F/ and F' are conjugate points of the second system, i.e. (cf. eq. 7) ·f/, F/F' =f2 (17) in which F/F is positive if F' lies to the right of F 2'. F' may be determined graphically from the construction given above on page 2 5, since the intersection of S1 and S' with the focal planes F2 and F 2' are at such distances g and g' from the axis + = that g g' y 1• The intersection A' of S' with S must lie in the unit plane .p' of the image space of the combination. Thus .p' is deter- mined, and, in consequence, the focal length f' of the com- bination, which is the distance from .p' of the principal focus F' of the combination. From the construction and the figure it follows that f' is negative when L1 is positive. f' may be determined analytically from the angle of incli- nation u' of the ray S'. For S1 the relation holds: tan = u1 y :f/, in which u1 is to be taken with the opposite sign if 5 1 is COnsidered the object ray of the second system. Now by (9), tan u' L1 tan u1 = h' or since tan u1 = - y : f.', tan u' = - y L1 ·f.'f./ = Further, since (cf. the law, p. 21) y : f' tan u', it follows that f - L1 • l - - [,.'.// (18) A similar consideration of a ray parallel to the axis in the image space and its conjugate ray in the object space gives f= - f.f, · (19) 30 THEORY OF OPTICS and for the distance of the principal focus F of the combination from the principal focus F 1 , FF1 = ff/, . in which FF1 is positive if Flies to the left of F 1. Equations (17), (18), (19), and (20) contain the character- istic constants of the combination calculated from those of the systems which unite to form it. Precisely the same process may be employed when the combination contains more than two systems. If the separation L1 of the two systems is zero, the focal lengths f and f' are infinitely great, i.e. the system is tele- scopit. The ratio of the focal lengths, which remains finite, is given by (18) and (19). Thus j,=f·i· (21) Fro!Il the consideration of an incident ray parallel to the axis the lateral magnification y' : y is seen to be y' :y = fJ = -/4 :J;_'. . (22) By means of (21), (22), and (16) the constant a, which repre- sents the magnification in depth (cf. equation (13)) is found. Thus x' --;; = a = - ff.Jf.;''" . . . . . (23) Hence by (14) the angular magnification is tan u': tan u = fJ: a =f1 :/4'. (24) The above considerations as to the graphical or analytical determination of the constants of a combination must be somewhat modified if the combination contains one or more telescopic systems. The result can, however, be easily obtained by constructing or calculating the path through the successive systems of an incident ray whic.lt is parallel to the axis. CHAPTER III PHYSICAL CONDITIONS FOR IMAGE FORMATION ABBE'S geometrical theory of the formation of optical images, which overlooks entirely the question of their physical realization, has been presented in the previous chapter, because the general laws thus obtained must be used for every special case of image formation no matter by what particular physical means the images are produced. The concept of focal points and focal lengths, for instance, is inherent in the concept of an image no matter whether the latter is produced by lenses or by mirrors or by any other means. In this chapter it will appear that the formation of optical images as described ideally and without limitations in the previous chapter is physically impossible, e.g. the image of an object of finite size cannot be formed when the rays have too great a divergence. It has already been shown on page I 5 that, whatever the divergence of the beam, the image of one point may be produced by reflection or refraction at an aplanatic surface. Images of other points are not produced by widely divergent rays, since the form of the aplanatic surface depends upon the position of the point. For this reason the more detailed treatment of special aplanatic surfaces has no particular physical interest. In what follows only the formation of images by refracting and reflecting spherical surfaces will be treated, since, on account of the ease of manufacture, these alone are used in optical instruments; and since, in any case, for the reason mentioned above, no other forms of reflecting or refracting surfaces furnish ideal optical images. 31 32 THEORY UP OPTICS It will appear that the formation of optical images can be practically accomplished by means of refracting or reflecting spherical surfaces if certain limitations are imposed, namely, limitations either upon the size of the object, or upon the divergence of the rays producing the image. 1. Refraction at a Spherical Surface.-In a medium of index n, let a ray PA fall upon a sphere of a more strongly refractive substance of index n' (Fig. 14). Let the radius of the sphere be r, its centre C. In order to find the path of th~ refracted ray, construct about C two spheres I and 2 of radii r1 = n' n- r and r2 = ,nnr (method of Weierstrass). Let PA meet sphere I in B; draw BC intersecting sphere 2 in D. Then AD is the refracted ray. This is at once evident from the fact that the triangles ADC and BA C are similar. For AC: CD = BC: CA = n' : n. Hence the 1'.'. DAC = ~ABC= ', the angle of refraction, and since 1'.'. BA C = , the angle of incidence, it follows that sin : sin ' = BC : AC = n' : n, which is the law of refraction. If in this way the paths of different rays from the point P PHYSICAL CONDITIONS FOR IMAGE FORMATION 33 be constructed, it becomes evident from the figure that these rays will not all intersect in the same point P'. Hence no image is formed by widely divergent rays. Further it appears from the above construction that all rays which intersect the sphere at any point, and whose prolongations pass through B, are refracted to the point D. Inversely all rays which start from D have their virtual intersection in B. Hence upon every straight line passing through the centre C of a sphere of radius r, there are two points at distances from C of I r'!n_ and r nn, respectt"vely which, for all rays, stand z"n the relation of object and virtual (not real) image. These two points are called the aplanatic points of the sphere. If u and u' represent the angles of inclination with respect to the axis BD of two rays which start from the aplanatic points Band D, i.e. if 1'.'.ABC= u, 1'.'.ADC= u', then, as was shown above, 1'.'. ABC= 1'.'. DA C = u. From a consideration of the triangle ADC it follows that sinu':sinu=AC:CD=n':n. (1) In this case then the ratio of the sines of the angles of inclination of the conjugate rays is independent of u, not, as in equation (9) on page 22, the ratio of the tangents. The difference between the two cases lies in this, that, before, the image of a portion of space was assumed to be formed, while now only the image of a surface formed by widely divergent rays is under consideration. The two concentric spherical surfaces I and 2 of Fig. 14 are the loci of all pairs of aplanatic points B and D. To be sure, the relation of these two surfaces is not collinear in the sense in which this term was used above, because the surfaces are not planes. If s and s' represent the areas of two conjugate elements of these surfaces, then, since their ratio must be the same as that of the entire spherical !;11rfaces I and 2, 34 THEORY OF OPTICS Hence equation (I) may be written: = sin2 u-s-n2 sin2 u' ,s' ,n'2• • It will be seen later that this equation always holds for two surface elements s and s' which have the relation of object and image no matter by what particular arrangement the image is produced. In order to obtain the image of a portion of space by means of refraction at a spherical surface, the divergence of the rays which form the image must be taken very small. Let PA (Fig. 15) be an incident ray, AP' the refracted ray, and PCP' p Tl, n' FIG. IS, the line joining P with the centre of the sphere C. Then from the triangle PAC, = sin ¢ : sin a PH+ r : PA, and from the triangle P'AC, sin ¢' : sin a = P'H - r : P'A. Hence by division, = n = sin n, a convex curvature (positive r) means a convergent system. Real images (e' > o) are formed so long as e > f. Such images are also inverted. Equation (10) on page 23 becomes y' tan u' n v tan u = - n1· • • • • • (6) By the former convention the angles of inclination u and u' of conjugate rays are taken positive in different ways. If they are taken positive in the same way the notation 'u will be used = - instead of u', i.e. 'u u'. Hence the last equation may be written: = ny tan u n'y' tan 'u. . . (7) THEORY OF OPTICS In this equation a quantity which is not changed by refraction appears,-an optical invariant. This quantity remains constant when refraction takes place at any number of coaxial spherical surfaces. For such a case let n be the index of refraction of the first medium, n' that of the last; then equation (7) holds. But since in general for every system, from equation (10), page 23, y' tan u' f ytan u - f" • (S) there results from a combination with (7) f:f' = n: n', (9) i.e. In the formation of i·mages by a system of coaxial refracting spherical surfaces the ratio of the focal lengths of the system is equal to the ra#o of the indices of refraction of the .irst and last media. If, for example, these two media are air, as is the case with lenses, mirrors, and most optical instruments, the two focal lengths are equal. 2. Reflection at a Spherical Surface.-Let the radius r be considered positive for a convex, negative for a concave mirror. p FIG. 16. = By the law of reflection (Fig. 16) ~ PAC ~ P'AC. Hence from geometry PA :P'A = PC:P'C. (10) If the ray PA makes a large angle with the axis PC, then the position of the point of intersection P' of the conjugate ray PHYSICAL CONDITIONS FOR IMAGE FORMATION 37 with the axis varies with the angle. In that case no image of the point P exists. But if the angle APC is so small that the angle itself may be used in place of its sine, then for every point P there exists a definite conjugate point P', i.e. an image is now formed. It is then permissible to set PA = PH, P'A = P'H, so that (IO) becomes PH:P'H= PC:P'C, = - or if PH= e, P'H e', then, since r in the figure is nega- tive, --eI +,eI =-r2 .. A comparison of this with equation ( I I) on page 2 3 shows that the focal lengths of the system are f= I --r, f'= I +-r; (13) 2 2 that the two unit planes .p and .p' coincide with the plane tangent to the sphere at the vertex H; that the two principal foci coincide in the mid-point between C and H; and that the nodal points coincide at the centre C of the sphere. The signs of e and e' are determined by the definition on page 23. Since f and f' have opposite signs, it follows, from the criterion given on page 25, that the system is katoptric or contracurrent. By the conventions on page 26 a negative r, i.e. a concave mirror, corresponds to a convergent system; on the other hand a convex mirror corresponds to a divergent system. A comparison of equations (13) and (5) shows that the results here obtained for reflection at a spherical surface may be deduced from the former results for refraction at such a sur- = - face by writing n': n I. In fact when n': n = - I, the law of refraction passes into the law of reflection. Use may be made of this fact when a combination of several refracting or reflecting surfaces is under consideration. Equation (9) holds for all such cases and shows that a positive ratio f: f' THEORY OF OPTICS always results from a combination of an even number of reflections from spherical surfaces or from a combination of any number of refractions, i.e. such systems are dioptric or concurrent (cf. page 25). The relation between image and object may be clearly brought out from Fig. 17, which relates to a concave mirror. The numbers I, 2, 3, ... 8 represent points of the object at a constant height above the axis of the system. The numbers 7 and 8 which lie behind the mirror correspond to virtual objects, i.e. the incident rays start toward these points, but fall upon the mirror and are reflected before coming to an intersection at them. Real rays are represented in Fig. 17 by ., ,,. f continuous lines, virtual rays by dotted lines. The points J', . . . I', 2', 8' are the images of the points I, 2, 3, ... 8. Since the latter lie in a straight line parallel to the axis, the former must also lie in a straight line which passes through the principal focus F and through point 6, the intersection of the object ray with the mirror, i.e. with the unit plane. The con- tinuous line denotes real images; the dotted line, virtual im- ages. Any image point 2' may be constructed (cf. page 24) by drawing through the object 2 and the principal focus F a straight line which intersects the mirror, i.e. the unit plane, in some point A 3• If now through A 2 a line be drawn parallel PHYSICAL CONDITIONS FOR IMAGE FORMATION 39 to the axis, this line will intersect the previously constructed image line in the point sought, namely 2 1• From the figure it may be clearly seen that the images of distant objects are real and inverted, those of objects which lie in front of the mirror within the focal length are virtual and erect, and those of virtual objects behind the mirror are real, erect, and lie in front of the mirror. Fig. I 8 shows the relative positions of object and image t 2 + __.s __ e_____ 7 _______ a FIG. 18. for a convex mirror. It is evident that the images of all real objects are virtual, erect, and reduced; that for virtual objects which lie within the focal length behind the mirror the images are real, erect, and enlarged; and that for more distant virtual objects the images are also virtual. p Equation (11) asserts that PCP'H are four harmonic points. The image of an object P may, with the aid of a proposition of synthetic geometry, be constructed in the following way: THEORY OF OP TICS From any point L (Fig. 19) draw two rays LC and LH, and then draw any other ray PDB. Let O be the intersection of DH with BC: then LO intersects the straight line PH in a point P' which is conjugate to P. For a convex mirror the construction is precisely the same, but the physical meaning of the points C and His interchanged. 3. Lenses.-The optical characteristics of systems composed of two coaxial spherical surfaces (lenses) can be directly deduced from § 7 of Chapter II. The radii of curvature r 1 and r 2 are taken positive in accordance with the conventions given above (§ 1); i.e. the radius of a spherical surface is considered positive if the surface is convex toward the incident ray (convex toward the left). Consider the case of a lens of index n surrounded by air. Let the thickness of the lens, i.e. the distance between its vertices S1 and 5 2 (Fig. 20), be - 6 ,: L1 fe F, s. F/ & s.. 6' -&' n FIG. 20. denoted by d. If the focal lengths of the first refracting surface are denoted by f 1 andfi.', those of the second surface by /4 andfz', then the separation L1 of the two systems (cf. page 28) is given by ,::::J=d-f,_'-/4, and, by (5), = .Ii= = 1 , r1n---1• f,_ n r 1n---1• .Is= r n s1---n• /,4 I ( r21---n• 15) PHYSICAL CONDITIONS FOR IMAGE FORMATION 41 Hence by equations (19) and (18) of Chapter II (page 29) the focal lengths of the combination are ' n r1r2 f=f=n-1°d(n-I)-nr1 +nr/ (16) while the positions of the principal foci F and F' of the com~ bination are given by equations ( I 7) and (20) of Chapter II (page 29). By these equations the distance o of the principal = + focus F in front of the vertex S1 , and the distance a-' of the principal focus F' behind the vertex S2 are, since o, r2 < o), Plano-convex lenses (r1 > o, r2 oo ) > Concavo-convex lenses (r1 > o, r2 > o, ,,2 r 1), in short all lenses which are thicker in the middle than at the e.dges. PHYSICAL CONDITIONS FOR IMAGE FORMATION 43 Lenses of negative focal length (divergent lenses) include Double-concave lenses (r1 < o, r 2 > o), Plano-concave lenses (r1 = oo , r 2 > o), Convexo-concave lenses (r1 > o, > 1·2 o, < r2 r1), i.e. all lenses which are thinner in the middle than at the edges.* The relation between image and object is shown diagrammatically in Figs. 2 I and 22, which are to be interpreted in Z 3 F F'~ .J• FIG. 21. the same way as Figs. I 7 and I 8. From these it appears that whether convergent lenses produce real or virtual images of 1 z F' FIG. 22. real objects depends upon the distance of the object from the lens; but divergent lenses produce only virtual images of real * The terms collective (dioptric), for systems of positive focal length, dispersive, for those of negative focal length, have been chogen on account of this property ol lenses. A lens of positive focal length renders an incident beam more convergent, one of negative focal length renders it more divergent. When images are formed by a system of lenses, or, in general, when the unit planes do not coincide, say, with the first refracting surface, the conclusion as to whether the system is convergent or divergent cannot be so immediately drawn. Then recourse must be had to the definition on page 26, 44 THEORY OF OPTICS objects. However, divergent lenses produce real, upright, and enlarged image'> of virtual objects which lie behind the lens and inside of the principal focus. If two thin lenses of focal lengths ft and /4 are united to form a coaxial system, then the separation L1 (cf. page 40) is L1 = - ( f1 + f,). Hence, from equation (19) of Chapter II (page 29), the focal length of the combination is f = li.l+i.hh= f'' or I I I f = f1+ h. It is customary to call the reciprocal of the focal length of a lens its power. Hence the law: The power of a combination of thin lenses is equal to the sum of the powers of the separate lenses. 5. Experimental Determination of Focal Length.-For thin lenses, in which the two unit planes are to be considered as practically coincident, it is sufficient to determine the positions of an object and its image in order to deduce the focal length. For example, equation (11) of Chapter II, page 23, reduces here, sincef=f, to Since the positions of real images are most conveniently determined by the aid of a screen. concave lenses, which furnish only virtual images of real objects, are often combined with a convex lens of known power so that the combination furnishes a real image. The focal length of the concave lens is then easily obtained from (24) when the focal length of the combination has been experimentally determined. This pro• cedure is not permissible for thick lenses nor for optical sy!'!tems generally. The positions of the principal foci are readily deter~ PHYSICAL CONDITIONS FOR IMAGE FORMATION 45 mined by means of an incident beam of parallel ray'>. If then the positions of an object and its image with respect to the principal foci be determined, equations (7), on page 19, or (9), on page 22, give at once the focal length/ ( =/'). Upon the definition of the focal length given in Chapter II, page 20 (cf. equation (8)), viz., f=y': tan u, . it is easy to base a rigorous method for the determination of focal length. Thus it is only necessary to measure the angular magnitude u of an infinitely distant object, and the linear magnitude y' of its image. This method is particularly convenient to apply to the objectives of telescopes which are mounted upon a graduated circle so that it is at once possible to read off the visual angle u. If the object of linear magnitude y is not at infinity, but is at a distance e from the unit plane .p, while its image of linear magnitude y' is at a distance e' from the unit plane .p', then y':y=-e':e, = because, when f f', the nodes coincide with the unit points, i,e. object and image subtend equal angles at the unit points. By eliminating e and e' from (25) and (27) it follows that f= e y I - - y' --e-' ,. I-~ y Now if either e or e' are chosen large, then without appreciable error the one so chosen may be measured from the centre of the optical system (e.g. the lens), at least unless the unit planes are very far from it. Then either of equations (28) may be used for the determination of the focal length f when e or e' and the magnification y': y have been measured. The location of the positions of the object or image may be avoided by finding the magnification for two positions of THEORY OF OPTICS the object which are a measured distance l apart. For, from (7), page 19, hence in which (y :y')1 denotes the reciprocal of the magnification for + the position x of the object, (y : y')2 the reciprocal of the mag- nification for a position x l of the object. l is positive if, in passing to its second position, the object has moved the distance / in the direction of the incident light (i.e. from left to right). Abbe's focometer, by means of which the focal lengths of microscope objectives can be determined, is based upon this principle. For the measurement of the size of the image y' a second microscope is used. Such a microscope, or even a simple magnifying-glass, may of course be used for the measurement of a real as well as of a virtual image, so that this method is also applicable to divergent lenses, in short to all cases.* 6. Astigmatic Systems.-In the previous sections it has been shown that elementary beams whose rays have but a small inclination to the axis and which proceed from points either on the axis or in its immediate neighborhood may be brought to a focus by means of coaxial spherical surfaces. In this case all the rays of the beam intersect in a single point of the image space, or, in short, the beam is homocentrz'c in the image space. What occurs when one of the limitations imposed above is dropped will now be considered, i.e. an • A more detailed account of the focometer and of the determination of focal lengths is given by Czapski in Winkelmann, Handbuch der Physik, Optik, 1'P· z85-2g6. PHYSICAL CONDITIONS FOR IMAGE FORMATION 47 elementary beam having any tizclination to the axis will now be assumed to proceed from a point P. In this case the beam is, in general, no longer homocentric in the image space. An elementary beam which has started from a luminous point P and has suffered reflections and refractions upon surfaces of any arbitrary form is so constituted that, by the law of Malus (cf. page 12), it must be classed as an orthotomic beam, i.e. it may be conceived as made up of the normals N to a certain elementary surface ~- These normals, however, do not in general intersect in a point. Nevertheless geometry shows that upon every surface ~ there are two systems of curves which intersect at right angles (the so-called lines of curvature) whose normals, which are also at right angles to the surface ~. intersect. If a plane elementary beam whose rays in the image space are normal to an element /1 of a line of curvature be alone considered, it is evident that an image will be formed. The image is located at the centre of curvature of this element /1 , since its normals intersect at that point. Since every element / 1 of a line of curvature is intersected at right angles by some other element /2 of another line of curvature, a second elementary beam always exists which also produces an image, but the positions of these two images do not coincide, since in general the curvature of /1 is different from that of /2• What sort of an image of an object P will then in general be formed by any elementary beam of three dimensions ? Let I, 2, 3, 4 (Fig. 23) represent the four intersections of the four lines of curvature which bound the element d~ of the -surface ~- Let the curves I-2 and 3-4 be horizontal, 2-3 and r-4 vertical. Let the normals at the points r and 2 intersect at r2, those at 3 and 4 at 34. Since the curvature of the line I-2 differs by an infinitely small amount from that of the line 3-4, the points of intersection z2 and 34 lie at almost the same distance from the surface ~- Hence the line p1 which connects the points r2 and 34 is also nearly perpendicular to the ray S which passes through the middle of d~ and is normal to it. THEORY OF OPTICS This ray is called the principal ray of that elementary beam which is composed of the normals to d~. From the symmetry of the figure it is also evident that the line p 1 must be parallel to the lines 2-3 and r--4, i.e. it is vertical. The normals to any horizontal line of curvature intersect at some point of the line p 1. FIG. 23. Likewise the normals to any vertical line of curvature intersect at some point of the line p2 which connects I 4 and 23. Also, p2 must be horizontal and at right angles to S. These two lines p 1 and p 2 , which are perpendicular both to one another and to the principal ray, are called the two focal lz"nes of the elementary beam. The planes determined by the principal ray Sand the two focal lines p 1 andp2 are called the focal planes of the beam. It can then be said that in general the image of a luminous point P, formed by any elementary beam, consists of two focal lines which are at right angles to each other and to the principal ray, and lie a certain distance apart. This distance is called the astigmatic difference. Only in special cases, as when the curvatures of the two systems of lines of curvature are the same, does a homocentric crossing of the rays and a true image formation take place. This present more general kind of image formation will be called astigma#c in order to distinguish it from that considered above.* A sharp, recognizable image of a collection of object points P is not formed by an astigmatic system. Only when the • Stigma means focus, hence an astigmatic beam is one which has no focus. PHYSICAL CONDITIONS FOR IMAGE FORMATION 49 object is a straight line can a straight-line image be formed; and only then when the line object is so placed that all the focal lines which are the images of all the points P of the line object coincide. Since the image of every point consists of two focal lines p1 and p2 which are at right angles to each other, there are also two positions of the line object 90° apart which give rise to a line image. These two images lie at different distances from the surface ~. Similarly there are two orientations of a system of parallel straight lines which give rise to an image consisting of parallel straight lines. If the object is a right-angled cross or a network of lines at right angles, there is one definite orientation for which an image of one line of the cross or of one system of parallel lines of the network is formed in a certain plane ~ 1 of the image space; while in another plane ~ 2 of the image space an image of the other line of the cross or of the other system of lines of the network is formed. This phenomenon is a good test for astigmatism. Astigmatic images must in general be formed when the elementary refracting or reflecting surface has two different curvatures. Thus cylindrical lenses, for example, show marked astigmatism. Reflection or refraction at a spherical surface also renders a homocentric elementary beam astigmatic when the incidence is oblique. In order to enter more fully into the consideration of this case. let the point object P, the centre C of the sphere, and the point A in which the principal ray of the elementary beam emitted by P strikes the spherical surface, lie in the plane of the figure (Fig. 24). Let the line PA be represented by s, the line AP2 by s2• Now since it follows that = + ss2 sin {, s!. du'= AB. cos', r. da AB. (31) But since = ¢ ff -f- 11.- cl>' :.-= ff - tt', PHYSICAL CONDITIONS FOR IMAGE FORMATION 51 it follows that r dcp = da + du = AB( I + cos c/J) -s- , (I dI = da - duI = AB -r - - coss- 1. • (34) !_ s1 = ~ s2 -r=c-o(-s1- cp - cos cp), or = - s - s - 2- s1s- 2 1 2 r sin cp tan ¢, . (35) *Fora convex mirror r is positive; for a concave, negative. 52 THEORY OF OPTICS an equation which shows clearly how the astigmatism increases with the angle of incidence. This increase is so rapid that the astigmatism caused by the curvature of the earth may, by suitable means, be detected in a beam reflected from the surface of a free liquid such as a mercury horizon. Thus if the reflected image of a distant rectangular network be observed in a telescope of 7. 5 m. focal length and ½ m. aperture, the astigmatic difference amounts to -lo mm., i.e. the positions in which the one or the other system of lines of the network is in sharp focus are lo mm. apart. In the giant telescope of the Lick Observatory in California this astigmatic difference amounts to j\, mm. Thus the phenomena of astigmatism may be made use of in testing the accuracy of the surface of a plane mirror. Instead of using the difference in the positions of the images of the two systems of lines of the network, the angle of incidence being as large as possible, the difference in the sharpness of the images of the two systems may be taken as the criterion. For this purpose a network of dotted lines may be used to advantage. 7. Means of Widening the Limits of Image Formation. -It has been shown above that an image can be formed by refraction or reflection at coaxial spherical surfaces only when the object consists of points lying close to the axis and the indination to the axis of the rays forming the image is small. If the elementary beam has too large an inclination to the axis, then, as was shown in the last paragraph, no image can be formed unless all the rays of the beam lie in one plane. Now such arrangements as have been thus far considered for the formation of images would in practice be utterly useless. For not only would the images be extremely faint if they were produced by single elementary beams, but also, as will be shown in the physical theory (cf. Section I, Chapter IV), single elementary beams can never produce sharp images, but only diffraction patterns. Hence it is necessary to look about for means of widening the limits hitherto set upon image formation. In the first place PHYSICAL CONDITIONS FOR IMAGE FORMATION 53 the limited sensitiveness of the eye comes to our assistance: we are unable to distinguish two luminous points as separate unless they subtend at the eye an angle of at least one minute. Hence a mathematically exact point image is not necessary, and for this reason alone the beam which produces the image does not need to be elementary in the mathematical sense, i.e. one of infinitely small divergence. By a certain compromise between the requirements it is possible to attain a still further widening of the limits. Thus it is possible to form an image with a broadly divergent beam if the object is an element upon the axis, or to form an image of an extended object if only beams of small divergence are used. The realization of the first case precludes the possibility of the realization of the second at the same time, and vz'ce versa. That the image of a point upon the axis can be formed by a widely divergent beam has been shown on page 33 in connection with the consideration of aplanatic surfaces. But this result can also be approximately attained by the use of a suitable arrangement of coaxial spherical surfaces. This may be shown from a theoretical consideration of so-called spherical aberration. To be sure the images of adjacent points would not in general be formed by beams of wide divergence. In fact the image of a surface element perpendicular to the axis can be formed by beams of wide divergence only if the socalled sine law is fulfilled. The objectives of microscopes and telescopes must be so constructed as to satisfy this law. The problem of forming an image of a large object by a relatively narrow beam must be solved in the construction of the eyepieces of optical instruments and of photographic systems. In the latter the beam may be quite divergent, since, under some circumstances (portrait photography), only fairly sharp images are required. These different problems in image formation will be more carefully considered later. The formation of images in the ideal sense first considered, i.e. when the objects have any size and the beams any divergence, is, to be 54 THEORY OF OPTICS sure, impossible, if for no other reason, simply because, as will be seen later, the sine law cannot be simultaneously fulfilled for more than one position of the object. 8. Spherical Aberration.-If from a point Pon the axis two rays S1 and S2 are emitted of which S1 makes a very small angle with the axis, while S2 makes a finite angle u, then, after refraction at coaxial spherical surfaces, the image rays S/ and S2' in general intersect the axis in two different points P 1' and Pt The distance between these two points is known as the spherical aberration (longitudinal aberration). In case the angle tt which the ray S2 makes with the axis is not too great, this aberration may be calculated with the aid of a series of ascending powers of tt. If, however, u is large, a direct trigonometrical determination of the path of each ray is to be preferred. This calculation wiil not be given here in detail.* For relatively thin convergent lenses, when the object is distant, the image P 1 formed by rays lying close to the axis is farther from the lt!ns than the image Pi formed by the more oblique rays. Such a lens, i.e. one for which P 2 lies nearer to the object than P 1 , is said to be undercorrectt·d. Inversely, a lens for which P 2 is more remote from the object than P 1 is said to be overcorrected. Neglecting all terms of the power series in tt save the first, which contains u2 as a factor, there results for this so-called aberration of the first order, if the object P is very distant, in which h represents the radius of the aperture of the lens, fits focal length, n its index of refraction, and