I /\ 1 */X I > <# V,VA/\\///^A\^I H ,/ V ^*w / ^ /v 4% *|p % \ & /> i 'f I 1 \/ / f ^^ & \/Vx ^^^ 1 X \^ "^ I^ %>* . \ i* THE THEORY OF OPTICS THE THEORY OF OPTICS BY PAUL DRUDE Professor of Physics at the University of Giessen TRANSLATED FROM THE GERMAN BY C. RIBORG MANN AND ROBERT A. MILLIKAN Assistant Professors of Physics at the University of Chicago 10528 LONGMANS, GREEN, AND CO. 91 AND 93 FIFTH AVENUE, NEW YORK LONDON AND BOMBAY 1902 Copyright, 1901, BY LONGMANS, GREEN, AND CO. ROBERT DRUMMOND, PRINTER, NEW YORK. 5S 'REFACE TO THE ENGLISH TRANSLATION THERE does not exist to-day in the English language a general advanced text upon Optics which embodies the important advances in both theory and experiment which have been made within the last decade. Preston's 4t Theory of Light" is^at present the only general text upon Optics in English. Satisfactory as this work is for the purposes of the general student, it approaches the subject from the historical standpoint and contains no funda- mental development of some of the important theories which are fast becoming the basis of modern optics. Thus it touches but slightly upon the theory of optical instruments a branch of optics which has received at the hands of Abbe and his fol- lowers a most extensive and beautiful development ; it gives a most meagre presentation of the electromagnetic theory a theory which has recently been brought into particular prominence by the work of Lorentz, Zeeman, and others ; and it contains no discussion whatever of the application of the laws of thermodynamics to the study of radiation. The book by Heath, the last edition of which appeared in 1895, well supplies the lack in the field of Geometrical Optics, and Basset's " Treatise on Physical v Optics (1892) is a valua- ble and advanced presentation of many aspects of the wave theory. But no complete development of the electromagnetic theory in all its bearings, and no comprehensive discussion of iii iv PREFACE TO THE ENGLISH TRANSLATION the relation between the laws of radiation and the principles of thermodynamics, have yet been attempted in any general text in English. It is in precisely these two respects that the " Lehrbuch der " Optik by Professor Paul Drude (Leipzig, 1900) particularly excels. Therefore in making this book, written by one who has contributed so largely to the progress which has been made in Optics within the last ten years, accessible to the .English-speaking public, the translators have rendered a very important service to English and American students of Physics. No one who desires to gain an insight into the most mod- ern aspects of optical research can afford to be unfamiliar with this remarkably original and consecutive presentation of the subject of Optics. A. A. MICHELSON. UNIVERSITY OF CHICAGO, February, 1902. AUTHOR'S PREFACE THE purpose of the present book is to introduce the reader who is already familiar with the fundamental concepts of the differential and integral calculus into the domain of optics in such a way that he may be able both to understand the aims and results of the most recent investigation and, in addition, to follow the original works in detail. The book was written at the request of the publisher a request to which I gladly responded, not only because I shared his view that a modern text embracing the entire domain was wanting, but also because I hoped to obtain for myself some new ideas from the deeper insight into the subject which writing in book form necessitates. In the second and third sections of the Physical Optics I have advanced some new theories. In the rest of the book I have merely endeav- sd to present in the simplest possible way results already Wished. Since I had a text-book in mind rather than a compendium, I have avoided the citation of such references as bear only upon the historical development of optics. The few references which I have included are merely intended to serve the reader for more complete information upon those points which can find only brief presentation in the text, especially in the case of the more recent investigations which have not yet found place in the text-books. vi AUTHOR'S PREFACE In order to keep in touch with experiment and attain the simplest possible presentation of the subject I have chosen a synthetic method. The simplest experiments lead into the domain of geometrical optics, in which but few assumptions need to be made as to the nature of light. Hence I have begun with geometrical optics, following closely the excellent treatment given by Czapski in " Winkelmann's Handbuch der Physik " and by Lommer in the ninth edition of the " Miiller- " Pouillet text. The first section of the Physical Optics, which follows the Geometrical, treats of those general properties of light from which the conclusion is drawn that light consists in a periodic change of condition which is propagated with finite velocity in the form of transverse waves. In this section I have included, as an important advance upon most previous texts, Sommer- feld's rigorous solution of the simplest case of diffraction, Cornu's geometric representation of Fresnel's integrals, and, on the experimental side, Michelson's echelon spectroscope. In the second section, for the sake of the treatment of the optical properties of different bodies, an extension of the hypotheses as to the nature of light became for the first time necessary. In accordance with the purpose of the book I have merely mentioned the mechanical theories of light ; but the electromagnetic theory, which permits the simplest and most consistent treatment of optical relations, I have presented in the following form : Let X, Y, Z, and a, /?, y represent respectively the com- ponents of the electric and magnetic forces (the first measured in electrostatic units); also \etjx ,jy ,jz , and sx , s y , sg represent the components of the electric and magnetic current densities, i.e. times the number of electric or magnetic lines of force 47T which pass in unit time through a unit surface at rest with reference to the ether ; then, if c represent the ratio of the AUTHOR'S PREFACE vii .electromagnetic to the electrostatic unit, the following funda- mental equations always hold : ^Y ft ft 47fsx 3Y VZ TT~ ~ , etc . , - TT~ etc . , The number of lines of force is defined in the usual way. The particular optical properties of bodies first make their appearance in the equations which" connect the electric and magnetic current densities with the electric and magnetic forces. Let these equations be called the substance equations in order to distinguish them from the above fundamental equations. Since these substance equations are developed for non-homogeneous bodies, i.e. for bodies whose properties vary from point to point, and since the fundamental equa- tions hold in all cases, both the differential equations of the electric and magnetic forces and the equations of condition which must be fulfilled at the surface of a body are imme- diately obtained. In the process of setting up " substance and fundamental " equations I have again proceeded synthetically in that I have deduced them from the simplest electric and magnetic experiments. Since the book is to treat mainly of optics this process can here be but briefly sketched. For a more com- plete development the reader is referred to my book " Physik des Aethers auf elektromagnetische " Grundlage (Enke, 1894). In this way however, no explanation of the phenomena of dispersion is obtained because pure electromagnetic experi- ments lead to conclusions in what may be called the domain of macrophysiccd properties only. For the explanation of optical dispersion a hypothesis as to the microphysical proper- ties of bodies must be made. As such I have made use of the ion-hypothesis introduced by Helmholtz because it seemed to me the simplest, most intelligible, and most consistent way of presenting not only dispersion, absorption, and rotary viii AUTHOR'S PREFACE polarization, but also magneto-optical phenomena and the optical properties of bodies in motion. These two last-named subjects I have thought it especially necessary to consider because the first has acquired new interest from Zeeman's discovery, and the second has received at the hands of H. A. Lorentz a development as comprehensive as it is elegant. This theory of Lorentz I have attempted to simplify by the elimination of all quantities which are not necessary to optics. With respect to magneto-optical phenomena I have pointed out that it is, in general, impossible to explain them by the mere supposition that ions set in motion in a magnetic field are subject to a deflecting force, but that in the case of the strongly magnetic metals the ions must be in such a continuous motion as to produce Ampere's molecular currents. This supposition also disposes at once of the hitherto unanswered question as to why the permeability of iron and, in fact, of all other substances must be assumed equal to that of the free ether for those vibrations which produce light. The application of the ion-hypothesis leads also to some new dispersion formulae for the natural and magnetic rotation of the plane of polarization, formulae which are experimentally verified. Furthermore, in the case of the metals, the ion- hypothesis leads to dispersion formulas which make the con- tinuity of the optical and electrical properties of the metals depend essentially upon the inertia of the ions, and which have also been experimentally verified within the narrow limits thus far accessible to observation. The third section of the book is concerned with the rela- tion of optics to thermodynamics and (in the third chapter) to the kinetic theory of gases. The pioneer theoretical work in these subjects was done by KirchhofT, Clausius, Boltzmann, and W. Wien, and the many fruitful experimental investiga- tions in radiation which have been more recently undertaken show clearly that theory and experiment reach most perfect development through their mutual support. AUTHOR'S PREFACE ix I. mbued with this conviction, I have written this book in the :ndcavor to make the theory accessible to that wider circle of -eaders who have not the time to undertake the study of the original works. I can make no claim to such completeness as is aimed at in Mascart's excellent treatise, or in Winkelmann's : Handbuch. For the sake of brevity I have passed over many My interesting" and important fields of optical investigation. purpose is attained if these pages strengthen the reader in the view that optics is not an old and worn-out branch of Physics, but that in it also there pulses a new life whose fuither ishing must be inviting to every one. Mr. F. Kiebitz has given me efficient assistance in the ing of the proof. iirziG, January, 1900. INTRODUCTION MANY optical phenomena, among them those which have bund the most extensive practical application, take place in iccordance with. the following fundamental laws: 1. The law of the rectilinear propagation of light; 2. The law of the independence of the different portions of i beam of light ; 3. The law of reflection ; 4. The law of refraction. Since these four fundamental laws relate only to the geometrical determination of the propagation of light, conclu- sions concerning certain geometrical relations in optics may be reached by making them the starting-point of the analysis without taking account of other properties of light. Hence these fundamental laws constitute a sufficient foundation for so-called geometrical optics, and no especial hypothesis which enters more closely into the nature of light is needed to make the superstructure complete. In contrast with geometrical optics stands physical optics, which deals with other than the purely geometrical properties, and which enters more closely into the relation of the physical properties of different bodies to light phenomena. The best success in making a convenient classification of the great multitude of these phenomena has been attained by devising particular hypotheses as to the nature of light. From the standpoint of physical optics the four above-men- tioned fundamental laws appear only as very close approxima- vi xii INTRODUCTION tions. However, it is possible to state within what limits the laws of geometrical optics are accurate, i.e. under what circumstances their consequences deviate from the actual facts. This circumstance must be borne in mind if geometrical optics is to be treated as a field for real discipline in physics rather than one for the practice of pure mathematics. The truly complete theory of optical instruments can only be developed from the standpoint of physical optics ; but since, as has been already remarked, the laws of geometrical optics furnish in most cases very close approximations to the actual facts, it seems justifiable to follow out the consequences of these laws even in such complicated cases as arise in the theory of optical instruments. TABLE OF CONTENTS PART I. GEOMETRICAL OPTICS CHAPTER I THE FUNDAMENTAL LAWS PAGE Direct Experiment ........................................... i 2. Law of the Extreme Path .................................... 6 3. Law of Malus ............................................. ... 1 1 CHAPTER II GEOMETRICAL THEORY OF OPTICAL IMAGES 1. The Concept of Optical Images .............................. 14 2. General Formulae for Images ................................ 15 3. Images Formed by Coaxial Surfaces .......................... 17 4. Construction of Conjugate Points ............................ 24 5. Classification of the Different Kinds of Optical Systems ......... 25 6. Telescopic Systems .......................................... 26 7. Combinations of Systems ............................... . ---- 28 CHAPTER III PHYSICAL CONDITIONS FOR IMAGE FORMATION Refraction at a Spherical Surface ............................. 32 Reflection at a Spherical Surface .............................. 36 Lenses ........................ ............................. 40 Thin Lenses ........................................... . ---- 42 Experimental Determination of Focal Length .................. 44 6. Astigmatic Systems ........................ .... .............. 46 7. Means of Widening the Limits of Image Formation ............ 52 8. Spherical Aberration ......................................... 54 xiii TABLE OF CONTENTS 9. The Law of Sines . .. .. 58 10. linages of Large Surfaces by Narrow Beams 63 1 1 . Chromatic Aberration of Dioptric Systems 66 CHAPTER IV APERTURES AND THE EFFECTS DEPENDING UPON THEM 1. Entrance- and Exit-Pupils 73 2. Telecentric Systems 75 3. Field of View 76 4. The Fundamental Laws of Photometry 77 5. The Intensity of Radiation and the Intensity of Illumination of Optical Surfaces 84 6. Subjective Brightness of Optical Images 86 7. The Brightness of Point Sources 90 8. The Effect of the Aperture upon the Resolving Power of Optical Instruments 91 CHAPTER V OPTICAL INSTRUMENTS 1. Photographic Systems 93 2. Simple Magnifying-glasses 95 3. The Microscope 97 4. The Astronomical Telescope 107 5. The Opera Glass 109 6. The Terrestrial Telescope 112 7. The Zeiss Binocular 8. The Reflecting Telescope i PART II. PHYSICAL. OPTICS SECTION I GENERAL PROPERTIES OF LIGHT CHAPTER I THE VELOCITY OF LIGHT 1. Romer's Method 114 2. Bradley 's Method 115 TABLE OF CONTENTS xv RT. PACK 3. Fizeau's Method ............................................ 6 1 1 4. Foucault's Method .......................................... 1 1 8 5. Dependence of the Velocity of Light upon the Medium and the Color .................................................... 1 20 6. The Velocity of a Group of Waves ............................ 121 CHAPTER II INTERFERENCE OF LIGHT i . General Considerations ...................................... 1 24 2. Hypotheses as to the Nature of Light ......................... 124 M 3. Fresnel's irrors ............................................ 1 30 4. Modifications of the Fresnel Mirrors .......................... 134 5. Newton's Rings and the Colors of Thin Plates................. 136 6. Achromatic Interference Bands ............................... 144 7. The Interferometer......................................... , 144 8. Interference with Large Difference of Path .................... 148 9. Stationary Waves ............................................ 1 54 0. Photography in Natural Colors ............................... 1 56 CHAPTER III HUYGENS' PRINCIPLE Huygens' Principle as first Conceived ......................... 1 59 Fresnel's Improvement of Huygens' Principle .................. 162 The Differential Equation of the Light Disturbance ............ 169 4. A Mathematical Theorem .................................... 172 5. Two General Equations .......... . ........................... 174 6. Rigorous Formulation of Huygens' Principle .................. 179 CHAPTER IV DIFFRACTION OF LIGHT 1. General Treatment of Diffraction Phenomena .................. 185 2. Fresnel's Diffraction Phenomena ............................. 188 3. Fresnel's Integrals ............................... ............ 188 4. Diffraction by a Straight Edge ................................ 1 92 5. Diffraction through a Narrow Slit ...................... . ...... 198 6. Diffraction by a Narrow Screen .......... ..................... 201 7. Rigorous Treatment of Diffraction by a Straight Edge .......... 203 xvi TABLE OF CONTENTS ART. 8. Fraunhofer's Diffraction Phenomena 213 9. Diffraction through a Rectangular Opening 214 10. Diffraction through a Rhomboid 217 1 1. Diffraction through a Slit 217 12. Diffraction Openings of any Form 219 13. Several Diffraction Openings of like Form and Orientation 219 14. Babinet's Theorem 221 1 5. The Diffraction Grating 222 1 6. The Concave Grating 225 17. Focal Properties of a Plane Grating 227 1 8. Resolving Power of a Grating 227 19. Michelson's Echelon 228 20. The Resolving Power of a Prism 233 21. Limit of Resolution of a Telescope 235 22. The Limit of Resolution of the Human Eye 236 23. The Limit of Resolution of the Microscope 236 CHAPTER V POLARIZATION 1. Polarization by Double Refraction 242 2. The Nicol Prism 244 3. Other Means of Producing Polarized Light 246 4. Interference of Polarized Light 247 5. Mathematical Discussion of Polarized Light 247 6. Stationary Waves Produced by Obliquely Incident Polarized Light 251 7. Position of the Determinative Vector in Crystals 252 8. Natural and Partially Polarized Light 253 9. Experimental Investigation of Elliptically Polarized Light. . 255 SECTION II OPTICAL PROPERTIES OF BODIES CHAPTER I THEORY OF LIGHT 1. Mechanical Theory 259 2. Electromagnetic Theory 260 The Definition of the Electric and of the Magnetic Force 262 3. TABLE OF CONTENTS xvii of the Electric Current in the Electrostatic and the Electromagnetic Systems 263 Definition of the Magnetic Current ; 265 The Ether 267 Isotropic Dielectrics 268 {DefTihneitBiouonndary Conditions 27 1 9. The Energy of the Electromagnetic Field 272 10. The Rays of Light as the Lines of Energy Flow 273 CHAPTER II TRANSPARENT ISOTROPIC MEDIA 1 . The Velocity of Light 274 2. The Transverse Nature of Plane Waves 278 3. Reflection and Refraction at the Boundary between two Trans- I parent Isotropic Media Perpendicular Incidence ; Stationary Waves 278 284 5. Polarization of Natural Light by Passage through a Pile of Plates 285 6. Experimental Verification of the Theory 286 7. Elliptic Polarization of the Reflected Light and the Surface or Transition Layer 287 8. Total Reflection 295 9. Penetration of the Light into the Second Medium in the Case of Total Reflection 299 10. Application of Total Reflection to the Determination of Index of Refraction 301 11. The Intensity of Light in Newton's Rings 302 12. Non-homogeneous Media ; Curved Rays 306 CHAPTER III OPTICAL PROPERTIES OF TRANSPARENT CRYSTALS Equations and Boundary Conditions 308 (DifLfiegrhetn-vteicatlors and Light-rays 311 3. Fresnel's Law for the Velocity of Light 314 4. The Directions of the Vibrations 316 5. The Normal Surface 317 6. Geometrical Construction of the Wave Surface and of the Direc- tion of Vibration 32 TABLE OF CONTENTS ART. P AGE 7. Uniaxial Crystals 323 8. Determination of the Direction of the Ray from the Direction of the Wave Normal 324 9. The Ray Surface ; 326 10. Conical Refraction 33 1 11. Passage of Light through Plates and Prisms of Crystal 335 12. Total Reflection at the Surface of Crystalline Plates 339 13. Partial Reflection at the Surface of a Crystalline Plate 344 14. Interference Phenomena Produced by Crystalline Plates in Polarized Light when the Incidence is Normal 344 15. Interference Phenomena in Crystalline Plates in Convergent Polarized Light 349 CHAPTER IV ABSORBING MEDIA 1. Electromagnetic Theory 358 2. Metallic Reflection 361 3. The Optical Constants of the Metals 366 4. Absorbing Crystals 368 5. Interference Phenomena in Absorbing Biaxial Crystals 374 6. Interference Phenomena in Absorbing Uniaxial Crystals 380 CHAPTER V DISPERSION 1. Theoretical Considerations 2. Normal Dispersion 3. Anomalous Dispersion 4. Dispersion of the Metals CHAPTER VI OPTICALLY ACTIVE SUBSTANCES 1 . General Considerations 2. Isotropic Media 3. Rotation of the Plane of Polarization 4. Crystals 5. Rotary Dispersion 6. Absorbing Active Substances 382 388 392 396 400 401 404 , 408 412 415 TABLE OF CONTENTS xix CHAPTER VII MAGNETICALLY ACTIVE SUBSTANCES A , Hypothesis of Molecular Currents ART. PAGE 1. General Considerations 418 2. Deduction of the Differential Equations 420 3. The Magnetic Rotation of the Plane of Polarization 426 4. Dispersion in Magnetic Rotation of the Plane of Polarization. . 429 5. Direction of Magnetization Perpendicular to the Ray 433 B. Hypothesis of the Hall Effect 1. General Considerations 433 2. Deduction of the Differential Equations 435 3. Rays Parallel to the Direction of Magnetization 437 4. Dispersion in the Magnetic Rotation of the Plane of Polarization. 438 5. The Impressed Period Close to a Natural Period 440 6. Rays Perpendicular to the Direction of Magnetization 443 7. The Impressed Period in the Neighborhood of a Natural Period. 444 8. The Zeeman Effect 446 9. The Magneto-optical Properties of Iron, Nickel, and Cobalt... 449 10. The Effects of the Magnetic Field of the Ray of Light 452 CHAPTER VIII BODIES IN MOTION 1. General Considerations 457 2. The Differential Equations of the Electromagnetic Field Re- ferred to a Fixed System of Coordinates 457 3. The Velocity of Light in Moving Media 465 4. The Differential Equations and the Boundary Conditions Re- ferred to a Moving System of Coordinates which is Fixed with Reference to the Moving Medium 467 5. The Determination of the Direction of the Ray by Huygens' Principle 470 6. The Absolute Time Replaced by a Time which is a Function of the Coordinates 471 7. The Configuration of the Rays Independent of the Motion 473 8. The Earth as a Moving System 474 9. The Aberration of Light 475 10. Fizeau's Experiment with Polarized Light 477 n. Michelson's Interference Experiment 478 xx TABLE OF CONTENTS PART III. RADIATION CHAPTER I ART. 1. Emissive Power ENERGY OF RADIATION 2. Intensity of Radiation of a Surface 3. The Mechanical Equivalent of the Unit of Light 4. The Radiation from the Sun 5. The Efficiency of a Source of Light 6. The Pressure of Radiation 7. Prevost's Theory of Exchanges PAGE 483 484 485 487 487 488 491 CHAPTER II APPLICATION OF THE SECOND LAW OF THERMODYNAMICS TO PURE TEMPERATURE RADIATION 1 . The Two Laws of Thermodynamics 493 2. Temperature Radiation and Luminescence 494 3. The Emissive Power of a Perfect Reflector or of a Perfectly Transparent- Body is Zero 495 4. Kirchhoff's Law of Emission and Absorption 496 5. Consequences of Kirchhoff's Law 499 6. The Dependence of the Intensity of Radiation upon the Index of Refraction of the Surrounding Medium 502 7. The Sine Law in the Formation of Optical Images of Surface Elements 505 8. Absolute Temperature 506 9. Entropy 510 10. General Equations of Thermodynamics 511 11. The Dependence of the Total Radiation of a Black upon its Ab- solute Temperature , 512 12. The Temperature of the Sun Calculated from its Total Emission 515 13. The Effect of Change in Temperature upon the Spectrum of a Black Body 516 14. The Temperature of the Sun Determined from the Distribution of Energy in the Solar Spectrum 523 15. The Distribution of the Energy in the Spectrum of a Black Body 524 TABLE OF CONTENTS XXI CHAPTER III INCANDESCENT VAPORS AND GASES urr. PAGE 1. Distinction between Temperature Radiation and Luminescence. 528 2. The Ion-hypothesis 529 3. The Damping of Ionic Vibrations because of Radiation 534 4. The Radiation of the Ions under the Influence of External Radiation 535 5. Fluorescence 536 6. The Broadening of the Spectral Lines Due to Motion in the Line of Sight 537 7. Other Causes of the Broadening of the Spectral Lines 541 INDEX 543 ^% ^ ^N MNonn OF PART I GEOMETRICAL OPTICS CHAPTER I THE FUNDAMENTAL LAWS I. Direct Experiment. The four fundamental laws stated above are obtained by direct experiment. The rectilinear propagation of light is shown by the shadow P of an opaque body which a point source of light casts upon a screen 5. If the opaque body contains an aperture Z, then the edge of the shadow cast upon the screen is found to be the P intersection of S with a cone whose vertex lies in the source and whose surface passes through the periphery of the aper- ture L. If the aperture is made smaller, the boundary of the shadow upon the screen S contracts. Moreover it becomes indefinite when L is made very small (e.g. less than i mm.}, for points upon the screen which lie within the geometrical shadow now receive light from P. However, it is to be observed that a true point source can never be realized, and, on account of the finite extent of the source, the edge of the shadow could never be perfectly sharp even if light were propagated in straight lines (umbra and penumbra). Nevertheless, in the L case of a very small opening (say of about one tenth ;;/;;/. diameter) the light is spread out behind L upon the screen so far that in this case the propagation cannot possibly be recti- linear. 2 THEORY OF OPTICS The same result is obtained if the shadow which an opaque body Sf casts upon the screen S is studied, instead of the spreading out of the light which has passed through a hole in an opaque object. If S' is sufficiently small, rectilineal- P propagation of light from does not take place. It is there- fore necessary to bear in mind that the law of the rectilinear propagation of light holds only when the free opening through which the light passes, or the screens which prevent its passage, are not too small. In order to conveniently describe the propagation of light P P from a source to a screen 5, it is customary to say that sends rays to S. The path of a ray of light is then defined by the fact that its effect upon S can be cut off only by an obstacle that lies in the path of the ray itself. When the propagation of light is rectilinear the rays are straight lines, P as when light from passes through a sufficiently large open- ing in an opaque body. In this case it is customary to say P that sends a beam of light through L. Since by diminishing L the result upon the screen 5 is the same as though the influence of certain of the rays proceeding P from were simply removed while that of the other rays remained unchanged, it follows that the different parts of a beam of light are independent of one another. This law too breaks down if the diminution of the open- L ing is carried too far. But in that case the conception of light rays propagated in straight lines is altogether untenable. The concept of light rays is then merely introduced for convenience. It is altogether impossible to isolate a single ray and prove its physical existence. For the more one tries to attain this end by narrowing the beam, the less does light proceed in straight lines, and the more does the concept of light rays lose its physical significance. If the homogeneity of the space in which the light rays exist is disturbed by the introduction of some substance, the rays undergo a sudden change of direction at its surface: each ray splits up into two, a reflected and a refracted ray. If the sur- THE FUNDAMENTAL LAWS 3 face of the body upon which the light falls is plane, then the N plane of incidence is that plane which is denned by the incident ray and the normal to the surface, and the angle of incidence is the angle included between these two direc- tions. The following laws hold : The reflected and refracted rays botJi lie in tJie plane of incidence. Tlie angle of reflection (the angle included between yVand the reflected ray) is equal to the angle of incidence. The angle of refraction b denote the angles included between the normal A and the directions of the ray in and respec- tively, and na and n b the respective indices with respect to some medium like air or the free ether. The difference between the index n of a body with respect to air and its index n with respect to a vacuum is very small. From (2) n= , ' : (4) THE FUNDAMENTAL LAWS 5 in which ri denotes the index of a vacuum with respect to air. Its value at atmospheric pressure and o C. is ....... = n' i : 1.00029 (5) According to equation (3) there exists a refracted ray to correspond to every possible incident ray a only when < > na n b; for if na n and if b, rc sm > -b , n. > then sin cpb I ; i.e. there is no real angle of refraction A. In that case no refraction occurs at the surface, but reflection only. The whole intensity of the incident ray must then be contained in the reflected ray; i.e. there is total reflection. In all other cases (partial reflection) the intensity of the incident light is divided between the reflected and the re- fracted rays according to a law which will be more fully considered later (Section 2, Chapter II). Here the observa- tion must suffice that, in general, for transparent bodies the refracted ray contains much more light than the reflected. Only in the case of the metals does the latter contain almost the entire intensity of the incident light. It is also to be observed that the law of reflection holds for very opaque bodies, like the metals, but the law of refraction is no longer correct in the form given in (i) or (3). This point will be more fully discussed later (Section 2, Chapter IV). The different qualities perceptible in light are called colors. The refractive index depends on the color, and, when referred to air, increases, for transparent bodies, as the color changes from red through yellow to blue. The spreading out of white light into a spectrum by passage through a prism is due to this change of index with the color, and is called dispersion. If the surface of the body upon which the light falls is not plane but curved, may it still be looked upon as made up of very small elementary planes (the tangent planes), and the paths of the light rays may be constructed according to the 6 THEORY OF OPTICS above laws. However, this process is reliable only when the curvature of the surface does not exceed a certain limit, i.e. when the surface may be considered smooth. Rough surfaces exhibit irregular (diffuse) reflection and refraction and act as though they themselves emitted light. The surface of a body is visible only because of diffuse reflection and refraction. The surface of a perfect mirror is invisi-J ble. Only objects which lie outside of the mirror, and whoseJ rays are reflected by it, are seen. 2. Law of the Extreme Path.* All of these experi- mental facts as to the direction of light rays are comprehended: in the law of the extreme path. If a ray of light in passing! P from a point to a point P' experiences any number of reflec- tions and refractions, then the sum of the products of the index of refraction of each medium by the distance traversed^ in it, i.e. 2nl, has a maximum or minimum value; i.e. it differs from a like sum for all other paths which are infinitely close to the actual path by terms of the second or higher order. Thus if d denotes the variation of the first order, ..... d^nl =o. . (7) The product, index of refraction times distance traversed, is known as the optical length of the ray. In order to prove the proposition for a single refraction let OE POP' be the actual path of the light (Fig. i), the inter- PON section of the plane of incidence with the surface (tan- gent plane) of the refracting body, 0' a point on the surface of the refracting body infinitely near to 0, so that OO' makes any angle 6 with the plane of incidence, i.e. with the! line OE. Then it is to be proved that, to terms of the second or higher order, = + n'.OP' n-PO' n'-0'P r , . . (8) * ' Extreme ' is here used to denote either greatest or least (maximum or minimum). TR. THE FUNDAMENTAL LAWS 7 in which n and n' represent the indices of refraction of the adjoining media. If a perpendicular OR be dropped from upon PO' and a perpendicular OR' upon P'O', then, to terms of the second order, = + - PO PO' RO', OP' == OP' O'R'. . . (9) Also, to the same degree of approximation, = RO' == 00'. cos POO\ O'R' OO'-cos P'OO'. (10) FIG. i. OD In order to calculate cos POO' imagine an axis perpen- ON dicular to and OE, and introduce the direction cosines of PO the lines and 00' referred to a rectangular system of coordinates whose axes are ON, OE, and OD. If represent the angle of incidence PON, then, disregarding the sign, the PO direction cosines of are those of 00' are cos 0, sin 0, o, o, cos sn According to a principle of analytical geometry the cosine of the angle between any two lines is equal to the sum of the 8 THEORY OF OPTICS products of the corresponding direction cosines of the lines with reference to a system of rectangular coordinates, i.e. and similarly POO cos 1 sin 0-cos S, cos P'OO' sin 0'-cos S, in which C from those for which that sum < C. The former regions lie on the side of the aplanatic surface toward the less refractive medium (left in the figure), the latter on the side toward the more refractive medium (right in the figure). If now SOS' represents a section of the surface between the THE FUNDAMENTAL LAWS n two media, and PO, P'O the actual path which the light takes in accordance with the law of refraction, then the length of the path through O is a maximum or a minimum according as SOS' is more or less convex toward the less refracting medium FIG. 4. than the aplanatic surface AOA''. The proof appears at once from the figure. If, for example, SOS' is a plane, the length of the path is a minimum. In the case shown in the figure the length of the path is a maximum. Since, as will be shown later, the index of refraction is inversely proportional to the velocity, the optical path ;// is proportional to the time which the light requires to travel the distance /. The principle of least path is then identical witJi Fermai's principle of least time, but it is evident from the above that, under certain circumstances, the time may also be a maximum. = Since d^nl o holds for each single reflection or refrac= tion, the equation d^nl o may at once be applied to the case of any number of reflections and refractions. 3. The Law of Malus. Geometrically considered there are two different kinds of ray systems : those which may be F cut at right angles by a properly constructed surface (ortho- 12 THEORY OF OPTICS F tomic system), and those for which no such surface can be found (anorthotomic system). With the help of the preceding principle the law of Malus can now be proved. This law is stated thus: An orfhotomic system of rays remains orthotomic after any mimber of reflections and refractions. From the standpoint of the wave theory, which makes the rays the normals to the wave front, the law is self-evident. But it can also be deduced from the fundamental geometrical laws already used. ABODE Let (Fig. 5) and A'B'C'D'E' be two rays infinitely close together and let their initial direction be normal to a L surface F. If represents the total A optical distance from to E, then it may be proved that every ray whose total path, measured from its origin A, A', etc., has the same optical length Z, is normal to a sur- F face f which is the locus of the ends E, E', etc., of those paths. For B the purpose of the proof let A' and E'D be drawn. FIG. 5. According to the law of extreme path stated above, the length of the path A'B'C'D'E' must be equal to that of the infinitely near path A'BCDE' ', i.e. equal to L, which is also the length of the path ABCDE. If now from the two optical distances A'BCDE' and ABCDE the common portion BCD be sub- tracted, it follows that in which ;/ represents the index of the medium between the F D surfaces and B, and ri that of the medium between AB = A AB and F'. But since B, because is by hypothesis normal to F, it follows that DE = DE', THE FUNDAMENTAL LAWS 13 DE i.e. is perpendicular to the surface F' . In like manner F it may be proved that any other ray D'E' is normal to r . Rays which are emitted by a luminous point are normal to a surface F, which is the surface of any sphere described about the luminous point as a centre. Since every source of light may be looked upon as a complex of luminous points, it follows that light rays always form an orthotomic system. CHAPTER II GEOMETRICAL THEORY OF OPTICAL IMAGES I. The Concept of Optical Images. If in the neighb P hood of a luminous point there are refracting- and reflecting bodies having any arbitrary arrangement, then, in general, ? there passes through any point P' in space one and only one ray of light, i.e. the direction which light takes from Pto is completely determined. Nevertheless certain points P' maiay P be found at which two or more of the rays emitted by interP sect. If a large number of the rays emitted by intersect in a point P', then P' is called the optical image of P. The intensity of the light at P' will clearly be a maximum. If the actual intersection of the rays is at P' , the image is called real; if P' is merely the intersection of the backward prolongation of the rays, the image is called virtual. The simplest exam- ple of a virtual image is found in the reflection of a luminous P point in a plane mirror. The image P' lies at that point P which is placed symmetrically to with respect to the mirror. Real images may be distinguished from virtual by the direct illumination which they produce upon a suitably placed rough surface such as a piece of white paper. In the case of plane mirrors, for instance, no light whatever reaches the point P'. Nevertheless virtual images may be transformed into real by certain optical means. Thus a virtual image can be seen be- cause it is transformed by the eye into a real image which illumines a certain spot on the retina. The cross-section of the bundle of rays which is brought together in the image may have finite length and breadth or may be infinitely narrow so as in the limit to have but one GEOMETRICAL THEORY OF OPTICAL IMAGES 15 tension. Consider, for example, the case of a single refrac- >n. If the surface of the refracting body is the aplanatic P race for the two points and P', then a beam of any size P ich has its origin in will be brought together in P' ; for P rays which start from and strike the aplanatic surface ist intersect in P', since for all of them the total optical dis- P ce from to P' is the same. If the surface of the refracting body has not the form of the anatic surface, then the number of rays which intersect in is smaller the greater the difference in the form of the two "aces (which are necessarily tangent to each other, see 10). In order that an infinitely narrow, i.e. a plane, m may come to intersection in P\ the curvature of the sur- es at the point of tangency must be the same at least in one ne. If the curvature of the two surfaces is the same at two and therefore for all planes, then a solid elementary ,m will come to intersection in P' ; and if, finally, a finite tion of the surface of the refracting body coincides with the lanatic surface, then a beam of finite cross-section will come intersection in P''. Since the direction of light may be reversed, it is possible P P interchange the source and its image f i.e. a source at , has its image at P. On account of this reciprocal relation- P p and P' are called conjugate points. 2. General Formulae for Images. Assume that by means P reflection or refraction all the points of a given space are P .aged in points of a second space. The former space will called the object space ; the latter, the image space. From i definition of an optical image it follows that for every ray P ich passes through there is a conjugate ray passing ough P. Two rays in the object space which intersect at P must correspond to two conjugate rays which intersect in the image space, the intersection being at the point P' which P is- conjugate to P. For every point there is then but one conjugate point P'. If four points PJPff^ of the object space lie in a plane, then the rays which connect any two pairs of 16 THEORY OF OPTICS P P P P these points intersect, e.g. the ray 1 2 cuts the ray in 34 f\ the point A. Therefore the conjugate rays P\P'2 and P' also intersect in a point, namely in A' the image of A. Hence the four images /YA'^Y^V also lie in a Plane - In otner words, to every point, ray, or plane in the one space there corresponds one, and but one, point, ray, or plane in the other. Such a relation of two spaces is called in geometry a collinear relationship. The analytical expression of the collinear relationship can be easily obtained. Let x, j/, z be the coordinates of a point P of the object space referred to one rectangular system, and x' , y', z the coordinates of the point P' referred to another rectangular system chosen for the image space ; then to every x, y, z there corresponds one and only one x' , y' , z', and vice versa. This is only possible if ax -\-by-\-cz-\-d d ax -\- by -\- cz -\- ' d ax -\- by -f- cz -\- ' J in which a, b, c, d are constants. That is, for any given x', y' , z' , the values of x, y, z may be calculated from the three linear equations (i); and inversely, given values of x, y, z determine x' , y' , z' . If the right-hand side of equations (i) were not the quotient of two linear functions of x, y, z, then y for every x' , z' , there would be several values of x, y, z. Furthermore the denominator of this quotient must be one and + + the same linear function (ax by cz -f- d\ since otherwise a plane in the image space + + + = A'x' B'y' Cz' D' o would not again correspond to a plane in the object space. GEOMETRICAL THEORY OF OPTICAL IMAGES 17 If the equations (i) be solved for x, y, and z, forms analoms to (i) are obtained; thus (2) From (i) it follows that for = -- d o: x' = = z = oo lilarly from (2) for cz d' =. x = y = z = oo + = The plane ax -j- by -f- cz d o is called the focal plane P of the object space. The images P' of its points lie at P Two nity. rays which originate in a point of this focal ne correspond to two parallel rays in the image space. + + + = The plane a'x' b'y' c' z' df o is called the focal me g' of the image space. Parallel rays in the object space respond to conjugate rays in the image space which inter- t in some point of this focal plane gf'. = = = In case a b c o, equations (i) show that to finite ues of x, y, z correspond finite values of x' , y' , z' ; and, in- rsely, since, when a, b, and c are zero, a', b' c' are also , zero, to finite values of JIT', y' , z' correspond finite values of z. ', In this case, which is realized in telescopes, there no focal planes at finite distances. 3. Images Formed by Coaxial Surfaces. In optical in- ments it is often the case that the formation of the image es place symmetrically with respect to an axis; e.g. this true if the surfaces of the refracting or reflecting bodies are "aces of revolution having a common axis, in particular, sur- s of spheres whose centres lie in a straight line. P From symmetry the image P' of a point must lie in the P ne which passes through the point and the axis of the tern, and it is entirely sufficient, for the study of the image ormation, if the relations between the object and image in h a meridian plane are known. i8 THEORY OF OPTICS If the xy plane of the object space and the x'y' plane of the image space be made to coincide with this meridian plane, and if the axis of symmetry be taken as both the x and the x' axis, then the z and z' coordinates no longer appear in equations (i). They then reduce to x= +; by = ax -f- by -j- d (3) The coordinate axes of the xy and the x'y' systems are then parallel and the x and x' axes lie in the same line. The origin 0' for the image space is in general distinct from the O origin for the object space. The positive direction of x will be taken as the direction of the incident light (from left to O' FIG. 6. right); the positive direction of x' , the opposite, i.e. from right to left. The positive direction of y and y' will be taken upward (see Fig. 6). From symmetry it is evident that x' does not change its value when y changes sign. Therefore in equations (3) = = b v b o. It also follows from symmetry that a change in sign of y produces merely a change in sign of y' . = a 2 d 2 o and equations (3) reduce to Hence _ " Five constants thus remain, but their ratios alone are sufficient to determine the formation of the image. Hence GEOMETRICAL THEORY OF OPTICAL IMAGES 19 there are in general four characteristic constants which determine the formation of images by coaxial surfaces. The solution of equations (4) for x and y gives dx' d. a.d ad. y -- ,_, 1 * 'a,- ax" * l>, - a, ax" fC I The equation of the focal plane of the object space is = ax -{- d o, that of the focal plane of the image space ax ' 7< F F o. The intersections and f of these planes i with the axis of the system are called the principal foci. F If the principal focus of the object space be taken as the origin of x, and likewise the principal focus F' of the image space as the origin of x' , then, if X Q, X' Q represent the coordi- nates measured from the focal planes, ax^ will replace ax -f- d and ax^ a l ax' . Then from equations (4) 0- y ax (} Hence only two characteristic constants remain in the equations. The other two were taken up in fixing the positions of the focal planes. For these two complex constants simpler expressions will be introduced by writing (dropping ibscripts) L L XX' =ff = = *. (7) In this equation x and x' are the distances of the object and image from the principal focal planes ^ and f $ respectively. y The ratio y' \ is called the magnification. It is I for = f /, i.e. x' . This relation defines two planes and T which are at right angles to the axis of the system. These planes are called the unit planes. Their points of intersection /f and H' with the axis of the system are called unit points. The unit planes are characterized by the fact that the dis- P tance from the axis of any point in one unit plane is equal to that of the conjugate point P' in the other unit plane. The two remaining constants /and/' of equation (7) denote, in accord- 20 THEORY OF OPTICS Q ance with the above, the distance of the unit planes ), from f the focal planes g, g'. The constant is called the focal length of the object space; f, the focal length of the image space. The direction of/" is positive when the ray falls first upon the focal plane g, then upon the unit plane ; 7 for/ the case is the reverse. In Fig. 7 both focal lengths are positive. The significance of the focal lengths can be made clear in the following way: Parallel rays in the object space must have conjugate rays in the image space which intersect in some point in the focal plane g' distant, say, y' from the axis. The value of y evidently depends on the angle of inclination u of = the incident ray with respect to the axis. If u o, it follows y = from symmetry thai o, i.e. rays parallel to the axis have conjugate rays which intersect in the principal focus F' . But .pi f FTG. 7. PFA if u is not equal to zero, consider a ray which passes through the first principal focus F, and cuts the unit plane ) A in (Fig. 7). The ray which is conjugate to it, A'P', must evidently be parallel to the axis since the first ray passes through F. Furthermore, from the property of the unit planes, A and A' are equally distant from the axis. Consequently the distance from the axis_/ of the image which is formed by a parallel beam incident at an angle u is, as appears at once from Fig. 7, y' =/.tan u. (8) Hence the following law: The focal length of the object space is equal to the ratio of the linear magnitude of an image GEOMETRICAL THEORY OF OPTICAL IMAGES 21 formed in tJie focal plane of tJie image space to t/ie apparent A (angular} magnitude of its infinitely distant object. similar f definition holds of course for the focal length of the image space, as is seen by conceiving the incident beam of parallel rays to pass first through the image space and then to come to a focus in the focal plane $. If in Fig. 7 A'P' be conceived as the incident ray, so that the functions of the image and object spaces are interchanged, then the following may be given as the definition of the focal length /, which will then mean the focal length of the image space : The focal lengtJi of the image space is equal to the distance between the axis and any ray of the object space which is parallel to the axis divided by the tangent of the inclination of its conjugate ray. Equation (8) may be obtained directly from (7) by making = = tan u y.x and tan it! y'\x'. Since x and x' are taken positive in opposite directions and y and y' in the same direc- tion, it follows that u and u' are positive in different directions. The angle of inclination u of a ray in the object space is positive if the ray goes upward from left to rigJit; the angle of inclination u of a ray in tJie image space is positive if the ray goes mvard from left to right. The magnification depends, as equation (7) shows, upon the distance of the object from the principal focus F, and m /, the focal length. It is, however, independent of y, i.e. the image of a plane object which is perpendicular to the axis of the system is similar to the object. On the other hand the image of a solid object is not similar to the object, as is evident at once from the dependence of the magnification upon x. Furthermore it is easily shown from (7) that the magnification in depth, i.e. the ratio of the increment dx' of x' to an increment dx of x, is proportional to the square of the lateral magnification. ^ Let a ray in the object space intersect the unit plane in 22 THEORY OF OPTICS A P and the axis in (Fig. 8). Its angle of inclination u witl respect to the axis is given by AH AH P if x taken with the proper sign represents the distance of from F. 9C H FIG. 8. The angle of inclination u' of the conjugate ray with respect to the axis is given by tan u A'H' A'H' f P'H' ~ *" if x' represent the distance of P' from F', and P' and A' are P the points conjugate to and A . On account of the property AH of the unit planes A'H ; then by combination of the last two equations with (7), f tan u' x x f - tan u /' *' (9) The ratio of the tangents of inclination of conjugate rays is called the convergence ratio or the angular magnification. It is seen from equation (9) that it is independent of u and u' . = The angular magnification = f x I for or x' f. K The two conjugate points and K' thus determined are called the nodal points of the system. They are characterized by the GEOMETRICAL THEORY OF OPTICAL IMAGES 23 K :t tJiat a ray tJirough one nodal point is conjugate and illel to a ray through the other nodal point K' . The posi- of the nodal points for positive focal lengths /and/' is K FIG. 9. KA >wn in Fig. 9. and K'A' are two conjugate rays. It lows from the figure that the distance between the two nodal its is the same as tJiat between the two unit points. If '=/', the nodal points coincide with the unit points. Multiplication of the second of equations (7) by (9) gives /ta""/ -. y tan u f (io) P If e be the distance of an object from the unit plane , e' the distance of its image from the unit plane $$, e and P >eing positive if lies in front of (to the left of) JQ and P' lind (to the right of) )', then e=f-x, c'=f-x'. mce the first of equations (7) gives P The same equation holds if e and e' are the distances of and P' from any two conjugate planes which are perpendicular to the axis, and /and/' the distances of the principal foci from these planes. This result may be easily deduced from (7). THEORY OF OPTICS A Construction of Conjugate Points. simple graphical interpretation may be given to equation ABCD (11). If (Fig. 10) is a rectangle / with the sides and /', then any B f- straight line ECE' intersects the pro- f longations of/ and at such distances AE = A from that the conditions e and E AE' =. e! satisfy equation (i i). FIG. 10. It is also possible to use the unit plane and the principal focus to determine the point P' conju- P PA gate to P. Draw (Fig. n) from a ray parallel to the PF axis and a ray passing through the principal focus F. FIG. ii. A'F' is conjugate to PA, A' being at the same distance from A the axis as ; also P'B', parallel to the axis, is conjugate to PFB, B' being at the same distance from the axis as B. The intersection of these two rays is the conjugate point sought. The nodal points may also be conveniently used for this con- struction. P The construction shown in Fig. 1 1 cannot be used when P and P' lie upon the axis. Let a ray from intersect the focal g plane g at a distance and the unit plane at a distance h from the axis (Fig. 12). Let the conjugate ray intersect $$ and $' at the distances k'(=. /i) and g' . Then from the figure f ?.- PF ~* ~k~ f -YPF~J~^lc' P F' ' f *L- li~~~- f ~+~P rF' -l x' - x' ; GEOMETRICAL THEORY OF OPTICAL IMAGES 25 and by addition, since from equation (7) xx' ff, g + g' f - 2xx' fx ~~ x h ff+xx' -fx -f'x (12) P' may then be found by laying off in the focal plane g' the = Q distance g' h g, and in the unit plane the distance FIG. 12. = h' h, and drawing a straight line through the two points thus g determined, and g' are to be taken negative if they lie below the axis. 5. Classification of the Different Kinds of Optical Systems. The different kinds of optical systems differ from one f another only in the signs of the focal lengths and /'. If the two focal lengths have the same sign, the system is concurrent i.e. i if the object moves from left to right (x in- creases), the image likewise moves from left to right (x' decreases). This follows at once from equation (7) by taking into account the directions in which x and x' are con- sidered positive (see above, p. 18 ). It will be seen later that this kind of image formation occurs if the image is due to refraction alone or to an even number of reflections or to a combination of the two. Since this kind of image formation is most frequently produced by refraction alone, it is also called dioptric. 26 THEORY OF OPTICS If tJie two focal lengths Jiave opposite signs the system is contracurrent, i.e. if the object moves from left to right, the image moves from right to left, as appears from the formula xx' =. ff. This case occurs if the image is produced by an odd number of reflections or by a combination of an odd number of such with refractions. This kind of image formation is called katoptric. When it occurs the direction of propagation of the light in the image space is opposite to that in the object space, so that both cases may be included under the law: In all cases P of image formation if a point be conceived to move along a ray P in the direction in which the light travels, the image of that point moves along the conjugate ray in the direction in which the light travels. Among dioptric systems a distinction is made between those having positive and those having negative focal lengths. The former systems are called convergent, the latter divergent, because a bundle of parallel rays, after passing the unit plane $$ of the image space, is rendered convergent by the former, divergent by the latter. No distinction between systems on the ground that their foci are real or virtual can be made, for it will be seen later that many divergent systems (e.g. the microscope) have real foci. By similar definition katoptric systems which have a nega- tive focal length in the image space are called convergent, for in reflection the direction of propagation of the light is reversed. There are therefore the four following kinds of optical systems : DiWopptt>ric K Convergent: +/, +/' \b. Divergent: -/, -/'. T^ , . . Katoptrtc. . (a. C~o.nvergent: -f f, (6. Divergent: /, fJ . +/, ., . 6. Telescopic Systems. Thus far it has been assumed that the focal planes lie at finite distances. If they lie at infinity the case is that of a telescopic system, and the coeffi- GEOMETRICAL THEORY OF OPTICAL IMAGES 27 cient a vanishes from equations (4), which then reduce by a suitable choice cf the origin of the x coordinates to = = x' ax, y' Py. . . . . . (13) = = Since x' o when x o, it is evident that any two conjugate points may serve as origins from which x and x' are measured. It follows from equation (13) that the magnification in breadth and depth are constant. The angular magnification is also OP constant, for, given any two conjugate rays and O'P' their , intersections with the axis of the system may serve as the P origins. If then a point of the first ray has the coordinates P x, y, and its conjugate point 1 the coordinates x' , y\ the tangents of the angles of inclination are tan u y = : x, tan u' y' : x' . Hence by (13) ttan u' : tan u P:a (14) must be positive for katoptric (contracurrent) systems, nega- tive for dioptric (concurrent) systems. For the latter it is evident from (14) and a consideration of the way in which u and u' are taken positive (see above, p. 2l) that for positive P erect images of infinitely distant objects are formed, for nega- tive P, inverted images. There are therefore four different kinds of telescopic systems depending upon the signs of OL p. Equations (14) and (13) give f y tan u' p* n i/ -f-a it /v V ^/ (d (16) y = y If / , as is the case in telescopes and in all instru- cts in which the index of refraction of the object space is 28 THEORY OF OPTICS equal to that of the image space (cf. equation (9), Chapter III] then OL 2 ft . Hence from (14) = tan u' : tan u I : ft. This convergence ratio (angular magnification) is called in case of telescopes merely the magnification F. . From (13' i.e. for telescopes the reciprocal of the lateral magnification numerically equal to tJie angular magnification. A 7. Combinations of Systems. series of several systems must be equivalent to a single system. Here again attention will be confined to coaxial systems. If/ and// are the focal / lengths of the first system alone, and 2 and // those of the second, and /and/' those of the combination, then both the focal lengths and the positions of the principal foci of the com- bination can be calculated or constructed if the distance = F^F 2 A (Fig. 13) is known. This distance will be called for brevity the separation of the two systems I and 2, and will F be considered positive if .F/'lies to the left of otherwise 2, negative. A ray 5 (Fig. 13), which is parallel to the axis and at a pr< S ff K FIG. 13. distance y from it, will be transformed by system I into the S ray l , which passes through the principal focus /oints of the second system, i.e. (cf. eq. 7) (.7) F F F n which 2 is positive if F' lies to the right of . 2 F' may DC determined graphically from the construction given above 3n page 25, since the intersection of S and 5' with the 1 focal F F g planes and 2 are at such distances 2 and g' from the axis :hat g-\- g' yv The intersection A' of S' with 5 must lie in the unit plane )' of the image space of the combination. Thus ' is deter- f mined, and, in consequence, the focal length of the com- bination, which is the distance from $$ of the principal focus F' of the combination. From the construction and the figure it follows that/' is negative when A is positive. /' may be determined analytically from the angle of incli- nation u' of the ray Sf . For S the relation holds : l u^y tan ://, r hich ?/ is to t be taken with the opposite S sign if is con- l ired the object ray of the second system. Now by (9), tan u r A = since tan u^ ~~~ tan j // y : //, tan ' = - y - j^/1/2 = irther, since (cf. the law, p. 21) y :/' tan ?/, it follows that /'= -/^. . (18) similar consideration of a ray parallel to the axis in the ige space and its conjugate ray in the object space gives 3o THEORY OF OPTICS F and for the distance of the principal focus of the combination F from the principal focus l , (20) FF F F in which l is positive if lies to the left of r Equations (17), (18), (19), and (20) contain the character- istic constants of the combination calculated from those of the systems which unite to form it. Precisely the same process may be employed when the combination contains more than two systems. If the separation A of the two systems is zero, the focal f f lengths and are infinitely great, i.e. the system is tele- scopic. The ratio of the focal lengths, which remains finite, is given by (18) and (19). Thus From the consideration of an incident ray parallel to the axis the lateral magnification y' y : is seen to be ..... y'-.y = ft=-/1 ' :fl (22) By means of (21), (22), and (16) the constant a, which repre- sents the magnification in depth (cf. equation (13)) is found. Thus Hence by (14) the angular magnification is = = / tan u' : tan u a ft : x ://. . . . (24) The above considerations as to the graphical or analytical determination of the constants of a combination must be somewhat modified if the combination contains one or more telescopic systems. The result can, however, be easily obtained by constructing or calculating the path through the successive systems of an incident ray which is parallel to the axis. x OF CMJFOMW*. . Aiai^? CHAPTER III PHYSICAL CONDITIONS FOR IMAGE FORMATION ABBE'S geometrical theory of the formation of optical ages which overlooks entirely dization, has been presented in the the question previous of their physical chapter, because > general laws thus obtained must be used for every special se of image formation no matter by what particular physical *ans the images are produced. The concept of focal points 1 focal lengths, for instance, is inherent in the concept image no matter whether the latter is produced by lense by mirrors or by any other means. In this chapter it will appear that the formation of optical l4'aogobeujsescatscohfadpeftisnecirrtie bisseidzpehiydcseaianclnalloyltyabniedmpfowosisrtimhbeolduet,whleei.mgni.ttatthhieeonrsiaymsiangheatvhoeef great a divergence. It has already been shown on page I5 that, whatever the ivergence of the beam, the image of one point may be pro- duced by reflection or refraction at an aplanatic surface. Images f other points are not produced by widely ie form of the aplanatic surface depends divergent upon the rays, since position of ,e point For this reason the more detailed treatment of i ecial aplanatic surfaces has no particular what follows only the formation of images physical interest. by refracting an effecting spherical surfaces will be treated, since on account jf the ease of manufacture, these alone are used in opti nstruments; and since, in any case, for the reason mentioned ibove, no other forms of reflecting or refracting surfaces farm leal optical images. 32 THEORY OF OPTICS It will appear that the formation of optical images can practically accomplished by means of refracting or reflecting spherical surfaces if certain limitations are imposed, nameh limitations either upon the size of the object, or upon th< divergence of the rays producing the image. i. Refraction at a Spherical Surface, In a medium PA index n, let a ray fall upon a sphere of a more stronglj refractive substance of index n' (Fig. 14). Let the radius oJ FIG. 14. the sphere be r, its centre C. In order to find the path of the C refracted ray, construct about two spheres I and 2 of radii = r and r (method of Weierstrass). PA BC Let meet sphere I in B\ draw intersecting sphere AD 2 in D. Then is the refracted ray. This is at once ADC evident from the fact that the triangles and BA C = = CD BC A C are similar. For : CA : n' : n. Hence the ^DAC = ABC = < 0', the angle of refraction, and since BAC = < 0, the angle of incidence, it follows that = BC A C = sin : sin <' : ;/' : , which is the law of refraction. P If in this way the paths of different rays from the point PHYSICAL CONDITIONS FOR. IMAGE FORMATION 33 be constructed, it becomes evident from the figure that these j rays will not all intersect in the same point P' . Hence no image is formed by widely divergent rays. Further it appears from the above construction that all rays which intersect the sphere at any point, and whose prolongations pass through />, are refracted to the point D. Inversely all rays which D start from have their virtual intersection in B. Hence upon every straigJit line passing through the centre C of a sphere of radius r, there are two points at distances from C of )i n r and r respectively ivJiich, for all rays, stand in the relation of object and virtual (iiot real) image. These two points are called the aplanatic points of the sphere. If u and u' represent the angles of inclination with respect BD to the axis of two rays which start from the aplanatic B points and D, i.e. if ABC = , ADC = u', ^DAC = ABC then, as was shown above, < = u. From ADC a consideration of the triangle it follows that = AC CD = sin u' : sin u : n' : n. . . . (i) In this case then the ratio of the sines of the angles of inclina- tion of the conjugate rays is independent of u, not, as in equa- tion (9) on page 22, the ratio of the tangents. The difference between the two cases lies in this, that, before, the image of a portion of space was assumed to be formed, while now only the image of a surface formed by widely divergent rays is under consideration. The two concentric spherical surfaces I B and 2 of Fig. 14 are the loci of all pairs of aplanatic points and D. To be sure, the relation of these two surfaces is not collinear in the sense in which this term was used above, because the surfaces are not planes. If s and s' represent the areas of two conjugate elements of these surfaces, then, since their ratio must be the same as that of the entire spherical surfaces I and 2, = s' : s n* : n'*. 34 THEORY OF OPTICS Hence equation (i) may be written: = sin2 u-s-n2 sin 2 11 -s' -n'^ (2) It will be seen later that this equation always holds for two surface elements s and s' which have the relation of object and image no matter by what particular arrangement the image is produced. In order to obtain the image of a portion of space by means of refraction at a spherical surface, the divergence of the rays which form the image must be taken very small. Let PA AP (Fig. 15) be an incident ray, r the refracted ray, and PCP' P' FIG. 15. P the line joining with the centre of the sphere C. PA the triangle C, PH + sin : sin a r : PA , Then from and from the triangle P'A C, = a sin 0' : sin P'H r : P'A. Hence by division, PH+r sin __ P'A _' " shT^7 n~~~ P'H- r' ~PA' * ' (3) _, Now A assume that lies infinitely near to //, i.e. that the angle APH PA is very small, so that may be considered equal to PH, and P'A to P'H. Also let PH = = P'H f e, e. PHYSICAL CONDITIONS FOR IMAGE FORMATION 35 Then from (3) er nn n n 7+7= which r is to be taken positive if the sphere is convex C toward the incident light, i.e. if lies to the right of H. e is P positive if lies to the left of H\ e' is positive if P' lies to the right of H. To every e there corresponds a definite e' which is independent of the position of the ray PA, i.e. an image PC of a portion of space which lies close to the axis is formed by rays which lie close to PC. A comparison of equation (4) with equation (n) on page 23 shows that the focal lengths of the system are /= r^~n> f = %7^' ' - (5) and that the two unit planes and ' coincide and are tangent to the sphere at the point H. Since /"and/"' have the same sign, it follows, from the criterion on page 25 above, that the system is dioptric or concurrent. If n' > n, a convex curvature (positive r) means a convergent system. Real images > (e' o) are formed so long as e >/. Such images re also inverted. Equation (10) on page 23 becomes y' tan u' n v tan u (6) \y the former convention the angles of inclination u and u' of conjugate rays are taken positive in different ways. If they are taken positive in the same way the notation 'u will be used = instead of uf , i.e. 'u u''. Hence the last equation may be written: = ny tan u riy tan 'u (7) 36 THEORY OF OPTICS In this equation a quantity which is not changed by refraction appears, an optical invariant. This quantity remains constant when refraction takes place at any number of coaxial spherical surfaces. For such a case let n be the index of refraction of the first medium, n' that of the last; then equation (7) holds. But since in general for every system, from equation (10), page 23, /_tanj/ ~- / jtan u /" there results from a combination with (7) /:/'=:;/', (9) i.e. /;/ the formation of images by a system of coaxial refract- ing spJierical surfaces tJie ratio of the focal lengths of the system is equal to the ratio of the indices of refraction of the first and last media. If, for example, these two media are air, as is the case with lenses, mirrors, and most optical instruments, the two focal lengths are equal. 2. Reflection at a Spherical Surface. Let the radius r be considered positive for a convex, negative for a concave mirror. FIG. 16. ^ PAC By the law of reflection (Fig. 1 6) Hence from geometry ^ P'AC. PA \P'A =PC :P'C (10) PA If the ray makes a large angle with the axis PC, then the position of the point of intersection P' of the conjugate ray PHYSICAL CONDITIONS FOR IMAGE FORMATION 37 ith the axis varies with the angle. In that case no image of P APC point exists. But if the angle is so small that the igle itself may be used in place of its sine, then for every P >int there exists a definite conjugate point P' t i.e. an image PA = now formed. It is then permissible to set PH, 'A P'H, so that (10) becomes PH\P'H=PC\P'C, .... (n) if PH = e, P'H = e' , then, since r in the figure is nega- te, I I 2 A comparison of this with equation (n) on page 23 shows " at the focal lengths of the system are .... f=- f'=+ l l -r, -r; (13) that the two unit planes $ and coincide with the plane tangent to the sphere at the vertex H\ that the two principal foci coincide in the mid-point between C and H\ and that the nodal points coincide at the centre C of the sphere. The signs of e and ef are determined by the definition on page 23. f f Since and have opposite signs, it follows,, from the criterion given on page 25, that the system is katoptric or con- tracurrent. By the conventions on page 26 a negative r, i.e. a concave mirror, corresponds to a convergent system ; on the other hand a convex mirror corresponds to a divergent system. A comparison of equations (13) and (5) shows that the results here obtained for reflection at a spherical surface may be deduced from the former results for refraction at such a sur- face by writing n':n=. = \. In fact when n' \ n i, the law of refraction passes into the law of reflection. Use may be made of this fact when a combination of several refracting or reflecting surfaces is under consideration. Equation (9) holds for all such cases and shows that a positive ratio/":/' 38 THEORY OF OPTICS always results from a combination of an even number of reflections from spherical surfaces or from a combination of any number of refractions, i.e. such systems are dioptric or concur- rent (cf. page 25). The relation between image and object may be clearly brought out from Fig. 17, which relates to a concave mirror. The numbers 7, 2, j, . . . 8 represent points of the object at a constant height above the axis of the system. The numbers 7 and 8 which lie behind the mirror correspond to virtual objects, i.e. the incident rays start toward these points, but fall upon the mirror and are reflected before coming to an intersection at them. Real rays are represented in Fig. 17 by FIG. 17. continuous lines, virtual rays by dotted lines. The points /', 2* ', 3', . . . 8' are the images of the points /, 2, j, . . . 8. Since the latter lie in a straight line parallel to the axis, the former must also lie in a straight line which passes through the F principal focus and through point 6, the intersection of the object ray with the mirror, i.e. with the unit plane. The con- tinuous line denotes real images; the dotted line, virtual im- ages. Any image point 2' may be constructed (cf. page 24) F by drawing through the object 2 and the principal focus a straight line which intersects -the mirror, i.e. the unit plane, in some point A^. If now through A^ a line be drawn parallel PHYSICAL CONDITIONS FOR. IMAGE FORMATION 39 the axis, this line will intersect the previously constructed lage line in the point sought, namely 2'. From the figure it lay be clearly seen that the images of distant objects are real id inverted, those of objects which lie in front of the mirror 'ithin the focal length are virtual and erect, and those of virtual ejects behind the mirror are real, erect, and lie in front of the mirror. Fig. 1 8 shows the relative positions of object and image for a convex mirror. It is evident that the images of all real objects are virtual, erect, and reduced; that for virtual objects which lie within the focal length behind the mirror the images are real, erect, and enlarged; and that for more distant virtual objects the images are also virtual. H FIG. 19. PCP'H Equation (i i) asserts that are four harmonic points. P The image of an object may, with the aid of a proposition of synthetic geometry, be constructed in the following way: 40 THEORY OF OPTICS From any point L (Fig. 19) draw two rays LC and LH, and then draw any other ray PDB. Let O be the intersection of DH PH LO with BC\ then intersects the straight line in a point P' which is conjugate to P. For a convex mirror the construction is precisely the same, but the physical meaning of H the points C and is interchanged. 3. Lenses. The optical characteristics of systems com- posed of two coaxial spherical surfaces (lenses) can be directly deduced from 7 of Chapter II. The radii of curvature r^ and r^ are taken positive in accordance with the conventions given above ( i); i.e. the radius of a spherical surface is considered positive if the surface is convex toward the inci- dent ray (convex toward the left). Consider the case of a lens of index n surrounded by air. Let the thickness of the lens, i.e. the distance between its vertices S l and 5 2 (Fig. 20), be FIG. 20. denoted by d. If the focal lengths of the first refracting sur- face are denoted by /j and/', those of the second surface by /2 and//, then the separation A of the two systems (cf. page 28) is given by - J=W -//-/ (H) and, by (5), PHYSICAL CONDITIONS FOR IMAGE FORMATION 4 i Hence by equations (19) and (18) of Chapter II (page 29) the focal lengths of the combination are f-f- - ~ + n I d(n I) nr l F while the positions of the principal foci and F' of the com- bination are given by equations (17) and (20) of Chapter II (page 29). By these equations the distance 6 of the principal F focus in front of the vertex S lt and the distance cr' of the F principal focus r behind the vertex S2 are, since cr and cr' = FjF' +//, + FF l f^ f =^=T"^(-i)-r {r + 1 i ' r, l ' ' (I7) If h represents the distance of the first unit plane fe in front of the vertex S l, and h' the distance of the second unit plane /+ ' behind the vertex 5 then h 2, + = cr and /' h' 7 cr , and, from (16), (17), and (18), it follows that d(n i) nrv -\-nr (19) -j _2 (20) Also, since the distance / between the two unit planes = + $' is / d-\~ h //', it follows that and p - d(n \\-r-. J d(n -^ i) + nr -^ . nr l 2 . . . (21) f = Since f, the nodal and unit points coincide (cf. page 23). From these equations it appears that the character of the system is not determined by the radii r and v r 2 alone, but that the thickness d of the lens is also an essential element. For example, a double convex lens (i\ positive, r 2 negative), of 42 THEORY OF OPTICS not too great thickness d, acts as a convergent system, i.e. possesses a positive focal length; on the other hand it acts as a divergent system when d is very great. 4, Thin Lenses. In practice it often occurs that the thick- ness d of the lens is so small that d\Ji i) is negligible in comparison with n(rl r 2 ). Excluding the case in which /'j r 2, which occurs in concavo-convex lenses of equal radii, equation (16) gives for the focal lengths of the lens "~ r ( l]( \ I ^~ ^ /-(* l r)' . |. . . (22) while equations (19), (20), and (21) show that the unit planes nearly coincide with the nearly coincident tangent planes at the two vertices 5 and S . t 2 More accurately these equations give, when d(n i) is neglected in comparison to n(rl r 2 ), h= - ;/ i\ r, z = h' n -\' 1\ , /=d ;- n - 2 (23) Thus the distance / between the two unit planes is indepen- = = dent of the radii of the lens. For ;/ p i . 5, d. For both double-convex and double-concave lenses, since h and // are negative, the unit planes lie inside of the lens. For equal = curvature r l = = = r and for n 2, 1.5, /* // \d, i.e. the distance of the unit planes from the surface is one third the thickness of the lens. When i\ and r 2 have the same sign the lens is concavo-convex and the unit planes may lie outside of it. Lenses of positive focal lengths (convergent lenses) include < Double-convex lenses (rl > o, r 2 o), = Plano-convex lenses (i\ > o, r 2 oo ) Concavo-convex lenses (i\ > o, r 2 > o, > r 2 rj, in short all lenses which are thicker in the middle than at edges. PHYSICAL CONDITIONS FOR IMAGE FORMATION 43 Lenses of negative focal length (divergent lenses) include Double-concave lenses (rL < o, > r 2 o), I = Plano-concave lenses (7^ 00, > r 2 o), < Convexo-concave lenses (rl > o, > r 2 o, r 2 r^), i.e. all lenses which are thinner in the middle than at the * edges. The relation between image and object is shown diagram- matically in Figs. 21 and 22, which are to be interpreted in FIG. 21. the same way as Figs. 17 and 18. From these it appears that whether convergent lenses produce real or virtual images of FIG. 22. real objects depends upon the distance of the object from the lens; but divergent lenses produce only virtual images of real * The terms collective (dioptric), for systems of positive focal length, dispersive, those of negative focal length, have been chosen on account of this property of A lenses. lens of positive focal length renders an incident beam more convergent, one of negative focal length renders it more divergent. When images are formed by a system of lenses, or, in general, when the unit planes do not coincide, say, with the first refracting surface, the conclusion as to whether the system is con- vergent or divergent cannot be so immediately drawn. Then recourse must be to the definition on page 26. 44 THEORY OP OPTICS objects. However, divergent lenses produce > real, upright, and enlarged images of virtual objects which lie behind the lens and inside of the principal focus. f / If two thin lenses of focal lengths and are united to v 2 form a coaxial system, then the separation 21 (cf. page 40) is = A (fl -f-/2 )- Hence, from equation (19) of Chapter II (page 29), the focal length of the combination is fr_ /1/2 or 7, It is customary to call the reciprocal of the focal length of a lens its power. Hence the law: The power of a combination of thin lenses is equal to the sum of the powers of tJie separate lenses. 5. Experimental Determination of Focal Length. For thin lenses, in which the two unit planes are to be considered as practically coincident, it is sufficient to determine the posi- tions of an object and its image in order to deduce the focal length. For example, equation (11) of Chapter II, page 23, f = f reduces here, since to t 7+7=7....... (25) Since the positions of real images are most conveniently determined by the aid of a screen, concave lenses, which furnish only virtual images of real objects, are often combined with a convex lens of known power so that the combination furnishes a real image. The focal length of the concave lens is then easily obtained from (24) when the focal length of the combination has been experimentally determined. This procedure is not permissible for thick lenses nor for optical systems generally. The positions of the principal foci are readily deter- PHYSICAL CONDITIONS FOR IMAGE FORMATION 45 mined by means of an incident beam of parallel rays. If then the positions of an object and its image with respect to the principal foci be determined, equations (7), on page 19, or (9), = on page 22, give at once the focal length/ ( ./'). Upon the definition of the focal length given in Chapter II, page 20 (cf. equation (8)), viz., (26) it is easy to base a rigorous method for the determination of focal length. Thus it is only necessary to measure the angular magnitude u of an infinitely distant object, and the linear mag- nitude y' of its image. This method is particularly convenient to apply to the objectives of telescopes which are mounted upon a graduated circle so that it is at once possible to read off the visual angle u. If the object of linear magnitude y is not at infinity, but is at a distance e from the unit plane , while its image of linear magnitude y' is at a distance e' from the unit plane ', then ..... = y y' \ e' : e, (27) because, when /==/', the nodes coincide with the unit points, i.e. object and image subtend equal angles at the unit points. By eliminating e and e' from (25) and (27) it follows that /= - -= - -7. ...... (28) i-^ l-^ y y Now if either e or e' are chosen large, then without appreciable error the one so chosen may be measured from the centre of the optical system (e.g. the lens), at least unless the unit planes are very far from it. Then either of equations (28) may be used for the determination of the focal length /"when and the magnification y' \y have been measured. The location of the positions of the object or image may avoided by finding the magnification for two positions of re' 46 THEORY OF OPTICS the object which are a measured distance / apart. (7), page 19, AI - v/'r /' hence For, from /= " (z (\y^))2 T{\7y)1 1 in which (y : y'\ denotes the reciprocal of the magnification for the position x of the object, (JF : j/)2 the reciprocal of the magx nification for a position -f- / of the object. / is positive if, in passing to its second position, the object has moved the dis- tance /in the direction of the incident light (i.e. from left to right). Abbe's focometer, by means of which the focal lengths of microscope objectives can be determined, is based upon this principle. For the measurement of the size of the image y' a second microscope is used. Such a microscope, or even a simple magnifying-glass, may of course be used for the meas- urement of a real as well as of a virtual image, so that this method is also applicable to divergent lenses, in short to all cases.* 6. Astigmatic Systems. In the previous sections it has been shown that elementary beams whose rays have but a small inclination to the axis and which proceed from points either on the axis or in its immediate neighborhood may be brought to a focus by means of coaxial spherical surfaces. In this case all the rays of the beam intersect in a single point of the image space, or, in short, the beam is hornocentric in the image space. What occurs when one of the limitations imposed above is dropped will now be considered, i.e. an A * more detailed account of the focometer and of the determination of focal lengths is given by Czapski in Winkelmann, Handbuch der Physik, Optik, pp. 285-296, PHYSICAL CONDITIONS FOR IMAGE FORMATION 47 48 THEORY OF OPTICS This ray is called the principal ray of that elementary beam which is composed of the normals to d*2. From the symmetry ' of the figure it is also evident that the line pl must be parallel to the lines 2-3 and i ,/, i.e. it is vertical. The normals to any horizontal line of curvature intersect at some point of the line pr FIG. 23. Likewise the normals to any vertical line of curvature intersect at some point of the line p2 which connects /./ and 23. Also, /2 must be horizontal and at right angles to 5. These two lines p^ and/2 , which are perpendicular both to one another and to the principal ray, are called the two focal lines of the elementary beam. The planes determined by the principal ray 5 and the two focal lines pl and/2 are called ft\t focal planes of the beam. It can then be said that in general the image of a luminous point P, formed by any elementary beam, consists of two fo'cal lines which are at right angles to each other and to the principal ray, and lie a certain distance apart. This distance is called the astigmatic difference. Only in special cases, as when the curvatures of the two systems of lines of curvature are the same, does a homocentric crossing of the rays and a true image formation take place. This present more general kind of image formation will be called astigmatic in order to distinguish it from that considered above. * A sharp, recognizable image of a collection of object points P is not formed by an astigmatic system. Only when the * Stigma means focus, hence an astigmatic beam is one which has no focus. PHYSICAL CONDITIONS FOR IMAGE FORMATION 49 >ject is a straight line can a straight-line image be formed ; d only then when the line object is so placed that all the P :al lines which are the images of all the points of the line )ject coincide. Since the image of every point consists of or focal lines / 1 and /2 which are at right angles to each ;her, there are also two positions of the line object 90 apart rhich give rise to a line image. These two images lie at ifferent distances from the surface 2. Similarly there are two orientations of a system of parallel straight lines which give rise to an image consisting of parallel straight lines. If the object is a right-angled cross or a network of lines at right angles, there is one definite orientation for which an image of one line of the cross or of one system of parallel lines ^ of the network is formed in a certain plane of the image ^ space ; while in another plane 2 of the image space an image of the other line of the cross or of the other system of lines of the network is formed. This phenomenon is a good test for astigmatism. Astigmatic images must in general be formed when the refracting or reflecting surface has two different vatures. Thus cylindrical lenses, for example, show marked Ementary igmatism. Reflection or refraction at a spherical surface also renders a homocentric elementary beam astigmatic when the incidence is oblique. In order to enter more fully into the consideration of this case, let the point object P, the centre C of the sphere, and the point A in which the principal ray of the elementary beam P emitted by strikes the spherical surface, lie in the plane of PA the figure (Fig. 24). Let the line be represented by s y AP the line Now by s . 2 2 since APAP 2 = APAC+ ACAP 2 , it follows that ss^ sin (0 = 0') $r sin -|- s^r sin 0', 5 THEORY OF OPTICS in which and 0' denote the angles of incidence and refrac- tion respectively, and r the radius of the sphere. Since now = by the law of refraction sin ;/ sin 0', it follows from the last equation that ss2(n cos 0' = + cos 0) srn sp, or I n ~ n cos 0' cos 5 ^2 r P It is evident that all rays emitted by which have the same angle of inclination u with the axis must, after refraction, cross FIG. 24. P the axis at the same point . 2 The beam made up of such P rays is called a sagittal beam. It has a focal point at . 2 On the other hand a meridional beam, i.e. one whose rays PA P all lie in the plane C, has a different focal point r Let PB be a ray infinitely near to PA, and let its angle of inclina- + tion to the axis be u du and its direction after refraction ^BP BPr Then A is to be considered as the increment du' V ^BCA of u', and as the increment da of a. It is at once evident that = AB s. du ABcos 0, s . du' l . cos', r.da AB. (31) But since = ~ a -f- #, 0' a #', PHYSICAL CONDITIONS FOR IMAGE FORMATION follows that ^-^ = >' = ^(i-^j. . . . But a differentiation of the equation of refraction sin n sin r gives cos . d

' . (32) = Substituting in this the values of d(f> and d (^//Y)- (43) !f now n is the index of the object space, n' that of the image >pace, then, if the unbracketed letters signify geometrical lengths, = = (NPJ ii-NP l n-PP^s'm u. . . . (44) 'urther, if P'N' be drawn perpendicular to F^P', then, since ''/Y is infinitely small, ~ ' i i \ j) Equation (43) in connection with (44) and (45) then gives = n-PP^sm u n'-P'P^- sin u' . PP y \{ denote the linear magnitude of the l object, andy the P Pf linear magnitude of the image, then sin u sin , u' n'y' ny (46) Thus it is proved that if the linear magnification is con- >tant the ratio of the sines is constant, and, in addition, the 'alue of this constant is determined. This value agrees with 62 THEORY OF OPTICS that obtained in equation (2), page 34, for the aplanatic points of a sphere. The sine law cannot be fulfilled for two different points on P the axis. For if P' and /y (Fig. 27) are the images of and Plt then, by the principle of equal optical lengths, = AP = (PAP') (PSS'P'), (Pl l ') S S (P 'P 1 1 l 1 ^, . (47) PS P in which and S ll are any two parallel rays of inclina- tion u. FIG. 27. Subtraction of the two equations (47) and a process of reasoning exactly like the above gives - = + (/>//>') (Pf) (/yV) (N'P') t or cos u) ri-P^P' (i cos '), i.e. sn (48) This equation is then the condition for the formation, by a beam of large divergence, of the image of two neighboring points upon the axis, i.e. an image of an element of the axis. However this condition and the sine law cannot be fulfilled at the same time. Thus an optical system can be made aplanatic for but one position of the object. PHYSICAL CONDITIONS FOR IMAGE FORMATION 63 The fulfilment of the sine law is especially important in the case of microscope objectives. Although this was not known from theory when the earlier microscopes were made, it can be experimentally proved, as Abbe has shown, that these old microscope objectives which furnish good images actually satisfy the sine law although they were constructed from purely empirical principles. 10. Images of Large Surfaces by Narrow Beams. It is necessary in the first place to eliminate astigmatism (cf. page 46). But no law can be deduced theoretically for accom- plishing this, at least when the angle of inclination of the rays with respect to the axis is large. Recourse must then be had to practical experience and to trigonometric calculation. It is to be remarked that the astigmatism is dependent not only upon the form of the lenses, but also upon the position of the stop. Two further requirements, which are indeed not absolutely essential but are nevertheless very desirable, are usually im- FIG. 28. posed upon the image. First it must be plane, i.e. free from bulging, and second its separate parts must have the same magnification, i.e. it must be free from distortion. The first requirement is especially important for photographic objectives. 64 THEORY OF OPTICS For a complete treatment of the analytical conditions for this requirement cf. Czapski, in Winkelmann's Handbuch der Physik, Optik, page 124. The analytical condition for freedom from distortion may be readily determined. Let PPf^ (Fig. 28) be an object plane, P'P^P2 the conjugate image plane. The beams from the object are always limited by a stop of definite size which may be either the rim of a lens or some specially intro- duced diaphragm. This stop determines the position of a virtual aperture B, the so-called entrance-pupil, which is so situated that the principal rays of the beams from the objects P P l , 2 , etc., pass through its centre. Likewise the beams in B' the image space are limited by a similar aperture the , so-called exit-pupil, which is the image of the entrance-pupil.* If /and /' are the distances of the entrance-pupil and the exit- pupil from the object and image planes respectively, then, from the figure, = PP tan //j l : /, = tan a/ P'P{ : /', = PP tan u 2 2 : I, = tan ?// P'P : /'. If the magnification is to be constant, then the following rela- tion must exist: = PP P'P{ : PP, P'/Y : 2, hence ..... tan u' tan u = tan u' const tan (40) Hence for constant magnification the ratio of the tangents of the angles of inclination of the principal rays must be constant. In this case it is customary to call the intersections of the principal rays with the axis, i.e. the centres of the pupils, ortho- scopic points. Hence it may be said that, if the image is to be free from distortion, the centres of perspective of object and image must be orthoscopic points. Hence the positions of the pupils are of great importance. * For further treatment see Chapter IV. PHYSICAL CONDITIONS FOR IMAGE FORMATION 65 An example taken from photographic optics shows how the condition of orthoscopy may be most simply fulfilled for the R case of a projecting lens. Let (Fig. 29) be a stop on either side of which two similar lens systems / and 2 are symmetrically placed. The whole system is then called a symmetrical double objective. Let 5 and S' represent two conjugate principal R rays. The optical image of the stop with respect to the system / is evidently the entrance-pupil, for, since all principal rays must actually pass through the centre of the stop R, the prolongations of the incident principal rays 5 must pass through R the centre of B, the optical image of with respect to /. R Likewise B' , the optical image of with respect to 2, is the exit-pupil. It follows at once from the symmetry of arrange- ment that it is always equal to u' , i.e. the condition of orthos- copy is fulfilled. FIG. 29. Such symmetrical double objectives possess, by virtue of I eir symmetry, two other advantages: On the one hand, the eridional beams are brought to a sharper * focus, and, on the other, chromatic errors, which will be more fully treated in the = next paragraph, are more easily avoided. The result u u\ which means that conjugate principal rays are parallel, is altogether independent of the index of refraction of the system, * The elimination of the error of coma is here meant. Optik, p. 774. Cf. Muller-Pouillet, 66 THEORY OF OPTICS and hence also of the color of the light. If now each of the two systems / and 2 is achromatic with respect to the position of the image which it forms of the stop R, i.e. if the positions of the entrance- and exit-pupils are independent of the "color,* then the principal rays of one color coincide with those of every other color. But this means that the images formed in the image plane are the same size for all colors. To be sure, the position of sharpest focus is, strictly speaking, somewhat different for the different colors, but if a screen be placed in sharp focus for yellow, for instance, then the images of other colors, which lie at the intersections of the principal rays, are only slightly out of focus. If then the principal rays coincide for all colors, the image will be nearly free from chromatic error. The astigmatism and the bulging of the image depend upon the distance of the lenses / and 2 from the stop R. In general, as the distance apart of the two lenses increases the image becomes flatter, i.e. the bulging decreases, while the astigmatism increases. Only by the use of the new kinds of glass made by Schott in Jena, one of which combines large dispersion with small index and another small dispersion with large index, have astigmatic flat images become possible. V This will be more fully considered in Chapter under the head of Optical Instruments. ii. Chromatic Aberration of Dioptric Systems. Thus far the index of refraction of a substance has been treated as though it were a constant, but it is to be remembered that for a given substance it is different for each of the different colors contained in white light. For all transparent bodies the index continuously increases as the color changes from the red to the blue end of the spectrum. The following table contains the indices for three colors and for two different kinds of glass. n c is the index for the red light corresponding to the Fraun- *As will be seen later, this achromatizing can be attained with sufficient accuracy; on the other hand it is not possible at the same time to make the sizes of the R different images of independent of the color. PHYSICAL CONDITIONS FOR IMAGE FORMATION 67 ter line C of the solar spectrum (identical with the red rogen line), nD that for the yellow sodium light, and np that the blue hydrogen line. Glass. Calcium-silicate-crown. Ordinary silicate-flint. . The last column contains the so-called dispersive power (v ., the substance. It is defined by the relation - n, (50) It is practically immaterial whether 11 D or the index for any ter color be taken for the denominator, for such a change can never affect the value of v by more than 2 per cent. Since now the constants of a lens system depend upon the index, an image of a white object must in general show colors, i.e. the differently colored images of a white object differ from one another in position and size. In order to make the red and blue images coincide, i.e. in order to make the system achromatic for red and blue, it is necessary not only that the focal lengths, but also that the unit planes, be identical for both colors. In many cases a partial correction of the chromatic aberration is sufficient. Thus a system may be achromatized either by making the focal length, and hence the magnification, the same for all colors; or by making the rays of all colors come to a focus in the same plane. In the former case, though the magnification is the same, the images of all colors do not lie in one plane; in the latter, though these images lie in one plane, they differ in size. system may be achromatized one way or the other according the purpose for which it is intended, the choice depending upon whether the magnification or the position of the image is most important. 68 THEORY OF OPTICS A system which has been achromatized for two colors, e.g. red and blue, is not in general achromatic for all other colors, because the ratio of the dispersions of different substances in different parts of the spectrum is not constant. The chromatic errors which remain because of this and which give rise to the so-called secondary spectra are for the most part unimportant for practical purposes. Their influence can be still farther reduced either by choosing refracting bodies for which the lack of proportionality between the dispersions is as small as possible, or by achromatizing for three colors. The chromatic errors which remain after this correction are called spectra of the third order. The choice of the colors which are to be used in practice in the correction of the chromatic aberration depends upon the use for which the optical instrument is designed. For a system which is to be used for photography, in which the blue rays are most effective, the two colors chosen will be nearer the blue end of the spectrum than in the case of an instrument which is to be used in connection with the human eye, for which the yellow-green light is most effective. In the latter case it is easy to decide experimentally what two colors can be brought together with the best result. Thus two prisms of different kinds of glass are so arranged upon the table of a spectrometer that they furnish an almost achromatic image of the of the and F. slit; for instance, for a given position of the table spectrometer, let them bring together the rays C If now the table be turned, the image of the slit will in general appear colored ; but there will be one position in which the image has least color. From this position of the prism it is easy to calculate what two colors emerge from the prism exactly parallel. These, then, are the two colors which can be used with the best effect for achromatizing instruments intended for eye observations. Even a single thick lens may be achromatized either with reference to the focal length or with reference to the position of the focus. But in practice the cases in which thin lenses PHYSICAL CONDITIONS FOR IMAGE FORMATION 69 ire used are more important. When such lenses are com>ined, the chromatic differences of the unit planes may be leglected without appreciable error, since, in this case, these )lanes always lie within the lens (cf. page 42). If then the bcal lengths be achromatized, the system is almost perfectly ichrornatic, i.e. both for the position and magnitude of the mage. Now the focal length /j of a thin lens whose index for a jiven color is n^ is given by the equation (cf. eq. (22), page 42) in which k is an abbreviation for the difference of the curvav tures of the faces of the lens. Also, by (24) on page 44, the focal length /of a combination of two thin lenses whose separate focal lengths are/x and f2 is given by = 7 + 7 7 J J\ J2 W) For an increment dn l of the index ;/ 1 corresponding to a change of color, the increment of the reciprocal of the focal length is, from (51), d\-7\ = dn, I K in wwhich v l represents the dispersive power of the material of lens i between the two colors which are used. If the focal ength/of the combination is to be the same for both colors, llows from (52) and (53) that {% ~^ ~j+ This equation contains the condition for achromatism. It also shows, since r l and v 2 always have the same sign no matter what materials are used for I and 2, that the separate 70 THEORY OF OPTICS focal length,s of a tJiiji double achromatic lens always have opposite signs. From (54) and (52) it follows that the expressions for the separate focal lengths are !.~ ' v i -i. /i />,-".' /. I_IL_ /V.-n' Hence in a combination of positive focal length the lens with the smaller dispersive power has the positive, that with the larger dispersive power the negative, focal length. If/ is given and the two kinds of glass have been chosen, then there are four radii of curvature at our disposal to make /j and/2 correspond to (55). Hence two of these still remain arbitrary. If the two lenses are to fit together, r^ must be equal to r. 2 Hence one radius of curvature remains at our disposal. This may be so chosen as to make the spherical aberration as small as possible. In microscopic objectives achromatic pairs of this kind are very generally used. Each pair consists of a plano-concave lens of flint glass which is cemented to a double-convex lens of crown glass. The plane surface is turned toward the incident light. Sometimes it is desirable to use two thin lenses at a greater distance apart; then their optical separation is (cf. page 28) Hence, from (19) on page 29, the focal length of the combination is given by ..... + ^ I i - I a (56) ~~ If the focal length is to be achromatic, then, from (56) and (53), - ^i 4_ ?JL __ or