MATHEMATICAL THEORY OF OPTICS by R. K. LUNEBUR6 Foreword by EMILE WOLF Supplementary Notes by M.HERZBER6ER UNIVERSITY OF CALIFORNIA PRESS BERKELEY AND LOS ANGELES 1964 R. K. Luneburg University of California Press Berkeley and Los Angeles, California Cambridge University Press London, England @ 1964 by The Regents of the University of California Library of Congress Catalog Card Number: 64-19010. Printed in the United States of America FOREWORD During the Summer of 1944, Dr. Rudolf K. Luneburg presented a course of lectures on the Mathematical Theory of Optics at Brown University. The lecture material was later collected in a volume which was issued by Brown University in the form of mimeographed notes. These notes were by no means a compilation of generally available knowledge. They contained a highly original, thorough, and systematic account of the foundations of several branches of optics and numerous new and important results. The supply of copies of Luneburg's notes was soon exhausted, but demand for them has continued. The University of California Press is providing a real service to the scientific community by issuing a printed version of these notes. Fate has prevented Dr. Luneburg from seeing this volume. He died in 1949, at a time when the importance of his work was just beginning to be generally recognized. The chief contribution which Luneburg has made through these notes lies in having shown how the two main mathematical disciplines of instrumental optics, namely geometrical optics and the scalar diffraction optics, may be developed in a systematic manner from the basic equations of Maxwell's electromagnetic theory. Prior to Luneburg's work these two disciplines were, by and large, treated as self-contained fields, with little or no contact with electromagnetic theory. The starting point of Luneburg's investigation was the observation of the formal equivalence of the basic equation of geometrical optics (the eikonal equation) and the equation that governs the propagation of discontinuous solutions of Maxwell's equations (the equation of characteristics). By boldly identifying the geometrical optics field with the electromagnetic field on a moving discontinuity surface, Luneburg was led to a complete formulation of geometrical optics as a particular class of exact solutions of Maxwell's equations; This formulation is by no means based on traditional ideas; for traditionally geometrical optics is regarded as the short wavelength limit (or, more precisely, as the asymptotic approximation for large wave numbers) of the monochromatic solution of the wave equation. Luneburg was, of course, aware of this more traditional viewpoint and he touches briefly on it in §16. In fact, in a course of lectures which he later presented at New York University (during the academic year 1947-1948) Luneburg devoted considerable time to. the interrelation between the two approaches. Some of the ideas outlined in the two courses have become the nucleus from which a systematic theory of asymptotic series solutions of Maxwell's equations is gradually being developed. An account of the material presented by Luneburg in his New York V vi FOREWORD lectures and of related more recent developments will soon be published by Drs. M. Kline and I.W. Kay in a book entitled Electromagnetic Theory and Geometrical Optics (J. Wiley and Sons, New York). Chapter I of the present work contains the derivation of the basic laws of geometrical optics from Maxwell's equations. Amongst the many new results which this chapter contains, the transport equations, eq. (11.38), relating to propagation of the electric and the magnetic field vectors along geometrical rays, are of particular significance. In Chapter II Hamilton's theory of geometrical optics is formulated and in the chapter which follows it is applied to special problems . Amongst results which seem to make their first appearance in the scientific literature are some of the formulae of §24 relating to final corrections of optical instruments by aspheric surfaces; some new theorems relating to perfect optical instruments (§28.4); and the introduction in §29· of a new "perfect lens," which images stigmatically onto each other two spherical surfaces which are situated in a homogeneous medium. This is the now well known "Luneburg lens" which has found valuable applications as a microwave antenna. First and third order theories of optical systems are discussed in Chapters IVand V and, like all the other chapters, they contain a wealth of information. Chapter VI deals with the diffraction theory of optical instruments. The first sections of this chapter are devoted to the derivation, in a mathematically consistent way, of expressions for the electromagnetic field in the image region of an optical system suffering from any prescribed aberrations. A solution of this difficult problem (naturally somewhat idealized) is embodied in formulae (47.33), now known as the Luneburg diffraction integrals. These formulae are an important and elegant generalization of certain classical results of P. Debye and J. Picht. Section 48 deals with another important problem, often ignored in other treatises, namely with a systematic derivation of the scalar theory for the description of certain diffraction phenomena with unpolarized light. The concluding sections deal with problems of resolution and contain a discussion of the possibility of improvements in resolution by a suitable choice of the pupil function. These investigations are amongst the first in a field that has attracted a good deal of attention in recent years. In two appendices formulae are summarized relating to vector analysis and to ray tracing in a system of plane surfaces. They are followed by supplementary notes on electron optics, prepared by Dr. A. Blank and based on lectures of Dr. N. Chako. The volume concludes with supplementary notes by Dr. M. Herzberger, based on his lectures dealing with optical qualities of glass, with mathematics and geometrical optics and with symmetry and asymmetry in optical images. FOREWORD vii It is evident that Luneburg's Mathematical Theory of Optics is a highly original contribution to the optical literature. I consider it to be one of the most important publications on optical theory that has appeared within the last few decades. Department of Physics and Astronomy University of Rochester, Rochester 27, New York May, 1964 Emil Wolf PUBLISHER'S NOTE The present edition has been reproduced from mimeographed notes issued by Brown University in 1944. It is reprinted by permission of the Brown University Press. The University of California Press extends gratitude for help in making this edition possible to Dr. A. A. Blank, Dr. Max Herzberger, Mrs. R. K. Luneburg, Dr. Gordon L. Walker, and Dr. Emil Wolf. The author's name was misspelled in the original edition. This has, of course, been corrected, and a number of typographical errors, almost all of which were listed originally in the Errata of the mimeographed version, have also been corrected. Dr. Blank has clarified the last section of Chapter Von the basis of the Errata. No other changes have been made in this edition, which presents Luneburg's work as he left it. BIOGRAPHICAL NOTE Dr. Rudolf Karl Luneburg Born in Volkersheim, Germany, June 30, 1903; resident in United States of America since 1935; naturalized U. S. citizen 1944. Ph.D. University of Gottingen, 1930. Research Associate in Mathematics, University of Gottingen, 1930-1933; Research Fellow in Physics, University of Leiden, 1934-1935. Research Associate in Mathematics, New York University, 1935-1938. Mathematician, Research Department of Spencer Lens Company (subsidiary of American Optical Company), Buffalo, New York, 1938-1945. Visiting Lecturer, Brown University, Summer, 1944. Mathematical Consultant, Dartmouth Eye Institute 1946. Research Mathematician, Institute of Mathematics and Mechanics (now Courant Institute of Mathematical Sciences), New York University, 1946-1948. Visiting Lecturer, University of Marburg and Darmstadt Institute of Technology, 1948-1949. Associate Professor of Mathematics, University of Southern California, 1949. Died at Great Falls, Montana, August 19, 1949. PUBLICATIONS t 1. t Das Problem der Irrfahrt ohne Richtungbeschrankung und die Randwertaufgabe der Potentialtheorie. Math. Annalen 104, 45 (1931). 2. Eine Bemerkung zum Beweise eines Satzes iiber fastperiodische Funktionen. Copenhagen, Hovedkommissionaer: Levin & Munksgaard, B. Lunos boktrykkeri a/s, 1932. 3. On multiple scattering of neutrons. I. Theory of albedo and of a plane boundary (with 0. Halpern and 0. Clark). Phys. Rev. 53, 173 (1938). 4. Mathematical theory of optics (mimeographed lecture notes). Brown University, Providence, Rhode Island, 1944. 5. Mathematical analysis of binocular vision. Princeton, New Jersey: Princeton University Press, for the Hanover Institute, 1947. Research reports are not included in this bibliography. ix x BIOGRAPHICAL NOTE 6. Metric studies in binocular vision perception (Studies and Essays presented to R. Courant on his 60th birthday, January 8, 1948). New York: Jnterscience Publishers, Inc., 1948. 7. Propagation of electromagnetic waves (mimeographed lecture notes). New York University, New York, New York, 1948. 8. Multiple scattering of neutrons. II. Diffusion in a plane of finite thickness (with 0. Halpern). Phys. Rev. 76, 1811 (1949). 9. The metric of binocular visual space. J. Opt. Soc. Amer. 40,627 (1950). ACKNOWLEDGMENT These notes cover a course in Optics given at Brown University in the summer of 1944. The preparation for mimeographing [the original copies] was possible only through the assistance of Dr. Nicholas Chako and of my students, Miss Helen Clarkson, Mr. Albert Blank and Mr. Herschel Weil. I wish to express my thanks for their excellent cooperation. The supplementary note on Electron Optics has been written by Mr. Blank with the aim of giving a short derivatio!! of the main results by methods similar to those applied in the other parts of the course. This note has its source in lectures given by Dr. Chako on the physical side of this topic; in the mathematical approach it differs, however, from his presentation. To Dr. Max Herzberger of the Eastman Kodak Company I am greatly indebted for contributing the supplementary notes II, III, IV from his recent research. These notes are a record of his three lectures. I wish also to express my gratitude to Dean R.G.D. Richardson for his interest in the course and to his staff for unfailing help and cooperation in the task of preparing these notes. R. K. L. xi CONTENTS CHAPTER I WAVE OPTICS AND GEOMETRICAL OPTICS §1. The electromagnetic equations 1.1 Definitions and notations 1.2 Maxwell's differential equations 1.3 Energy relations 1.4 Boundary conditions Periodic fields 2.1 Special form of E and H in periodic fields 2.2 Complex vectors 2.3 Differential equations for u and v 2.4 Avj:lrage energy 2.5 Average :r.~x 2.6 Polarization §3. Differential equations for E and H 3.1 Second order equations for E and H 3,2 stratified media C ,' ~ '' §4. Integral form of Maxwell's equations 4.1 Eliminations of derivatives by integration 4.2 Integral conditions instead of div e:E = div µH = 0 4.3 Integral conditions instead of Maxwell's equations §5. General conditions for discontinuities ► II," 5.1 statement of the problem 5.2 First condition for E and H 5.3 Second condition for E 5,4 Complete set of difference equations for discontinuities of the electromagnetic field xiii Pages 1-5 1-2 2-3 3-5 5 5-11 5-6 6-7 7 7-8 8-9 9-11 12-14 12-13 13-14 15-18 15-16 16-17 17-18 18-20 18 18-19 20 20 xiv §6. §8. §9. §10. §11. CONTENTS Discontinuities of the optical properties 6.1 Conditions for E and H if E and µ are discontinuous Pages 20-21 20-21 Propagation of Discontinuities; Wave fronts 7.1 Homogeneous Media 7.2 Characteristic equation and characteristic hypersurfaces 7.3 Equation of the wave fronts 21-25 21-22 22-23 23-25 Bicharacteristics; Light rays 8.1 Introduction of light rays as orthogonal trajectories 8.2 Differential equations for the light rays 8.3 Interpretation of rays as paths of corpuscles 8.4 Fermat's problem of variation 25-29 25 25-26 26 26-29 Construction of wave fronts with the.~!~-~f light rays 9.1 Boundary value problem for 1/J(x,y,z) 9.2 Expression for the solution of the problem 9.3 Proof that the problem is solved by the expression in 9.2 9.4 Problem of finding the wave fronts which belong to a given special wave front 9.5 Spherical wave fronts; wavelets 9. 6 Huyghens' construction 29-36 29-30 30-31 31-33 33 33-34 34-36 Jacobi's theorem 10.1 Construction of the light rays with the aid of a complete integral of the equation of the wave fronts 10.2 Proof of Jacobi's theorem 10.3 Example: Light rays in stratified medium 36-38 36-37 37-38 38 Transport equation for discontinuities in continuous optical media 11.1 Differentiation along a light ray 11.2 Conditions for E and H on a characteristic hyper surface 11.3 Simplification of the above conditions; differential equations for discontinuities on a light ray 39-44 39 39-42 42-44 CONTENTS §12. Transport of discontinuities (Continued) 12.1 Equations for directions of the discontinuities U,V and their absolute values 12.2 Geometric interpretation of the quantities A€1/J and Aµl/J 12,3 Energy and flux on a wave front 12.4 The directions P and Q of the discontinuities depend only on the light ray and not on the wave fronts 12.5 The discontinuities U and V determine two applicable surfaces through a light ray 12.6 Non-euclidean parallelism of the discontinuities U and V on a light ray 12.7 Integration of the transport equations §13. Spherical waves in a homogeneous medium 13.1 Simplest type of solutions of the wave equation with spherical wave fronts 13.2 Method of obtaining solutions of this type by differentiation; multipole waves 13.3 Multipole _solutions of Maxwell's equations 13.4 Dipole solution of Maxwell's equations 13.5 Another form of the expressions 13.4 13.6 The vectors E and H on the spherical wave fronts §14. Wave fronts in media of discontinuous optical properties 14.1 Snell's law of refraction 14.2 The law of reflection §15. Transport of signals in media of discontinuous optical properties. Fresnel's formulae 15.1 Conditions for discontinuities before and after refraction 15.2 Orthogonal unit vectors on the rays 15.3 First set of equations for the components of U and V 15.4 Complete set of equations for U and V; Solution of these equations. Fresnel's formulae xv Pages 44-56 44-46 46-48 48-49 49-50 50-51 51-55 55-57 58-64 58-59 59 59-60 60-62 62-63 63-64 64-67 64-66 66-67 67-72 67-69 69-70 70 71-72 xvi §16. CONTENTS Periodic waves of small wave length 16.1 Radiation of a periodic dipole in a homogeneous medium 16.2 Radiation of a periodic dipole in a nonhomogeneous medium 16,3 Equations for the amplitude vectors UO and V0 for small wave lengths 16.4 Discussion and solution of the differential equations for U0 and V0 16.5 Media of discontinuous optical properties 16.6 Electromagnetic fields associated with geometrical optics Pages 73-81 73-74 74-75 75-77 77-79 79-80 81 CHAPTER II HAMILTON'S THEORY OF GEOMETRICAL OPTICS §17. Principles of geometrical optics 17.1 Huyghens' Principle 17.2 Light rays as paths of corpuscles 17.3 Electrons in an electrostatic field 17.4 Light rays in a medium with n* = Vn 2 + C 17.5 Fermat's principle 17.6 Example in which light rays are not curves of shortest optical path 82-88 82-84 84-85 85 85-86 86-87 87-88 §18. The canonical equations 18.1 Light rays in the form x = x(z), y = y(z) 18.2 Derivation of the canonical equations 18,3 Canonical equations for problems of variation in general 88-94 88-89 89-91 91-94 §19. Hamilton's characteristic function V(x 0 ,y0 ,z 0 ,x,y,z) 94-100 19,1 Problem of numerical investigation of an optical instrument 94-95 19.2 Definition of the point characteristic V for canonical equations in general 95-97 19.3 Principal theorem of Hamilton's theory regarding the differential dV 97 19.4 Proof of the theorem 98-99 19.5 Hamilton's theorem for geometrical optics 99-100 CONTENTS §20. Hamilton's characteristic functions, W and T 20.1 The mixed characteristic W 20.2 Geometric interpretation of W 20.3 The mixed characteristic W* 20.4 The angular characteristic T 20.5 Special significance of T §21. Integral invariants 21.1 Fields of light rays: necessary and sufficient conditions. Theorem of Malus. The integral !Pc n cos 0 ds is zero for any closed curve C 21.2 Non-normal congruences of rays. Poincare's invariant. Geometrical interpretation §22. Examples 22 .1 Mixed characteristic W for stratified media. Systems of plane parallel plates. Spherical aberration of stratified media 22.2 The angular characteristic in the case of a spherical mirror. First order development of T. Mirror equation 22.3 Angular characteristic for a refracting spherical surface. First order development of T. Lens equation xvii Pages 100-107 100-102 102-103 103-104 104-106 106-107 107-116 107-112 112-116 116-128 116-120 120-124 124-128 CHAPTER Ill APPLICATION OF THE THEORY TO SPECIAL PROBLEMS §23. Perfect conjugate points. Cartesian ovals 129-138 23.1 Light rays through perfect conjugate points have equal optical length 129-130 23.2 Cartesian ovals 130-132 23.3 Object point at infinity: The Cartesian ovals are ellipsoids and hyperboloids 132-133 23.4 Virtual conjugate points. Cartesian ovals for this case 133-134 23.5 An object point at infinity having a perfect virtual conjugate point. The Cartesian ovals are again ellipsoids and hyperboloids 135 23.6 The aplanatic points of a sphere. Graphical construction of light rays refracted on a sphere. Aplanatic surfaces in the front part of micro- scopic objectives 136-138 xviii §24. §25. §26. §27. CONTENTS Pages Final correction of optical instruments by aspheric surfaces 139-151 24.1 Justification of this method of correction 139 24.2 Solution of the problem with the aid of the point characteristic 140 24.3 Solution with the aid of the mixed characteristic W 140-142 24.4 Solution with the aid of the angular characteristic T 143 24.5 Example: A lens with a plane and an aspheric surface, which transforms a spherical wave into a plane wave 143-145 24.6 Example: A lens with a spherical and an aspheric surface which transforms a plane wave into a converging spherical wave 145 24.7 The principle of the Schmidt Camera 146-147 24.8 The correction plate of the Schmidt Camera 147-151 The angular characteristic for a single refracting surface 25.1 Equations for the point characteristic 25.2 Method of finding T by Legendre's trans- formation of the refracting surface 25.3 Surfaces of revolution 25.4 Example of a spheFical surface 25.5 The angular characteristic for an elliptic or hyperbolic paraboloid 151-156 151-152 152-154 154 155 156 The angular characteristic for systems of refracting surfaces 26.1 Variation problem for T for finitely many refracting surfaces 26.2 The corresponding problem of variation for a continuous medium 26.3 Another problem of variation of T 26.4 Equivalence of both problems 26.5 Systems of spherical surfaces 156-164 156-158 158-159 159-160 160-163 163-164 Media of radial symmetry 27.1 Equation of the lig4t_.rays 27.2 The form of the light rays in general 27.3 Example: n 2 = C + -r1 164-172 164-166 166-169 169-172 CONTENTS xix Pages §28. Maxwell's Fish eye 172-182 28.1 The wave fronts can be obtained by a Legendre transformation of the wave fronts of the potential field n 2 = C + .r! 172-173 28.2 The light rays are the circles through two points on opposite ends of a diameter of the unit circle. The medium n = 1/(l+r) represents a perfect optical system 173-175 28.3 Maxwell's fish eye obtained by a stereographic projection of the unit sphere 175-178 28.4 Perfect optical systems in the x,y plane which are obtained by other conformal projections of the unit sphere 178-180 28.5 Conformal projection of surfaces. Relation of optical design problem to the problem of perfect conjugate points on surfaces 180-182 § 29. Other optical media which image a sphere into a sphere }.82-188 29.1 Integral equation for the function p = nr 182-184 29.2 Transformation of the equation into an equation of Abel's type 184-185 29.3 Proof of the inversion theorem 185-186 29.4 Explicit solution of the problem 186-187 29.5 The special case r 0 = 00 , r 1 = 1 187-188 §30. Optical instruments of revolution 188-195 30.1 Euler's equations for optical media in which n = n(p,z); p = ...J x2 + y2 30.2 Transformation of Fermat's problem into a problem for p(z) alone 30.3 Skew rays can be treated as meridional rays if n(p,z) is replaced by 188-189 189-191 m(p,z) = ~ p 191-192 30.4 The characteristic functions in instruments of revolution. The mixed characteristic W depends only on u,v ,w 192-194 30.5 Similar statements for the other characteristic functions 194-195 xx §31. §32. CONTENTS Spherical Aberration and Coma. Condition for coma free instruments 31.1 31.2 31.3 31.4 31.5 31.6 31. 7 31.8 31.9 31.95 Spherical aberration L(p) of an axial bundle of rays Zonal magnification M(p) Development of W for small values of x 0 ,Yo but large values of p,q The ray intersection with the image plane. Construction of Coma flares. Abbe's sine condition Oblique bundles which are rotationally symmetrical to one of their rays Invariance of the characteristic W with respect to orthogonal transformations The characteristic W' in a coordinate system with the principal ray as the z' axis Condition of staeble-Lihotski for symmetrical bundles of rays The aplanatic points of a sphere are coma free. Graphical construction of two aspheric mirrors which possess two perfectly aplanatic points Modification for the object at infinity Pages 195-211 195-197 197-198 198 199-201 201-203 203-204 205-206 206-207 207-208 208-211 The condenser problem 32.1 statement of the problem. Intensity and illumination. Equation for the illumination of a given plane 32.2 Illumination of the center. Equation for the relative illumination 32.3 Condition for uniform illumination 32,4 Interpretation of the condition c/>(01) c/>(0) 1- for 0 0 but 211-215 211-213 213 213-214 214-215 CHAPTER IV FIRST ORDER OPTICS § 33. The first order problem in general 216-226 33.1 Canonical transformation of the rays by an optical instrument 216-217 33.2 Invariance of the canonical conditions with respect to orthogonal transformations of object and image space 217-218 CONTENTS xxi 33.3 Conditions for the matrices of the differentials dX i, dP i. Lagrangian brackets 33.4 Linear canonical transformations in first order optics 33.5 Linear equations obtained with the aid of the angular characteristic 33.6 Simplification of the equations Pages 218-221 221-222 222-224 224-226 §34. Gaussian optics 226-233 34.1 The matrices A,F ,C in systems of revolution. The linear transformations for Gaussian optics 226 34.2 Images of points of the plane z O = 0 227-228 34.3 General lens equation: Location of conjugate planes relative to a given pair of planes 228-229 34.4 Special lens equation; if the unit planes are chosen as reference planes. Unit points and nodal points 229-231 34.5 Focal points; Newton's lens equation 231-232 34.6 The point transformation of object into image space is a collineation. Images of inclined planes 232-233 §35. Orthogonal ray systems in first order optics 234-239 35.1 Definition of orthogonal systems 35.2 Primary or tangential focus; secondary or sagittal focus; astigmatism 35.3 The bundle through the point x 0 = Yo = 0. Focal lines. Primary and secondary magnifications. Astigmatic difference 35.4 The images of the points x 0 ,Yo of the plane Zo = 0 35.5 Dependence of the constants M,m,71. on the position z 0 of the object plane. Lens equation in orthogonal systems 234 234-235 235-236 236-237 237-239 §36. Non-orthogonal systems 240-243 36.1 The relations for the ray coordinates in non-orthogonal systems derived from the mixed characteristic W 240-241 36.2 The bundle through x 0 = y O = 0 241-242 36.3 The images of the points- x 0 ,Yo of the plane Zo = 0 242-243 36.4 Relation of the torsion of the image to the angle included by principal plane bundles of the axial bundles 243 xxii § 37. § 38. § 39. CONTENTS Differential equations of first order optics for systems of rotational symmetry 37.1 Formulation of the problem 37 .2 First order canonical equations. Paraxial rays and their geometric interpretation 37 .3 The corresponding problem of variation. Transversal surfaces. Jacobi's partial differential equation for paraxial rays 37.4 Interpretation of the function n1(z). Surface power. Final form of the paraxial differential equations 37.5 The general solution is expressed by two principal paraxial rays: axial ray and field ray. Conjugate planes 37 .6 Another definition of the field ray 37.7 Lagrange's invariant. The paraxial rays can be found by quadratures if one paraxial ray is known. Integral expressions for the focal length in terms of paraxial rays 37.8 The equivalent Riccati equation Pages 243-254 243-244 244-245 245-247 247-248 248-250 251 251-253 254 The path of electrons in the neighborhood of the axis of an instrument of revolution 38.1 Expressions for n 2(z,p) if n 2 (z,O) is known. Solution of the equation ~n 2 = 0 with given boundary values n 2(z,O) = f(z) 38.2 Surface power of an equipotential surface. Differential equations for paraxial electrons 38.3 The corresponding Hamilton-Jacobi equation and corresponding Riccati equation 255-257 255-256 256 257 Difference equations for a centered system of refracting surfaces of revolution 39.l Notations and definitions 39.2 The canonical difference equations 39.3 The associated problem of variation 39.4 The solution is expressed by two principal solutions: axial ray and field ray 39.5 Lagrange's invariant 39.6 The paraxial rays can be found by summations if one paraxial ray is known. Expressions for the focal length with the aid of invariant sums 39.7 The Gaussian quantities for a single lens 39.8 First order design of optical instruments 257-268 257-258 258-261 261 261-263 263-264 264-265 265-266 266-268 CONTENTS xxiii CHAPTER V THE THIRD ORDER ABERRATIONS IN SYSTEMS OF ROTATIONAL SYMMETRY Formulation of the problem Pages §40. General types of third order aberrations 269-281 40 .1 General expressions for the aberrations Axi, Ay1; Ap 0 , Aq 0 derived from the mixed characteristic W 40.2 Expressions for the third order aberrations Axi, Ay1; Ap 0 , Aq 0 obtained by developing W to the fourth order. Aberrations of meridional rays 40.3 Spherical aberrations, Coma, Astigmatism, Curvature of field, Distortion 40.4 The combined effect of third order aberrations 40.5 Dependence of the aberrations on the position of the diaphragm 269-270 271-272 273-276 277 277-281 § 41. The third order coefficients as functions of the position of object and pupil plane 281-286 41.1 The rays are determined by the intersection with two pairs of conjugate planes. Object and Image plane, Entrance and Exit pupil plane 281-282 41.2 General form of aberrations Axi, A~ 1 of meridional rays 282-283 41.3 Expression for the image aberrations Ax1 derived from the angular characteristic 283-284 41.4 Expression for the aberrations A~ 1 of the pupil plane 284-285 41.5 Simplified expressions for Ax1 and Ab 285 41.6 The case of the exit pupil plane at infinity_. A,B,C,E as functions of the magnification M of the object plane. Impossibility of correcting an optical instrument for all pairs of conjugate planes 285-286 §42. Integral expressions for the third order coefficients 287-299 42.1 Formulation of the problem 287-288 42.2 Development of the Hamiltonian function H(U,V ,z). Differential equations for the third order polynomials X ,Y P Q Corresponding ; . 3 3 3 3 boundary value problem 288-290 xxiv §43. CONTENTS 42.3 Integral expressions for Ax1 = X3 and Ay1 = Y 3 42.4 Integral expressions for the coefficients A,B,C,D,E 42.5 The quantities H ik in general 42.6 Petzval's theorem 42.7 The quantities H lk in the case of an electrostatic field 42.8 The case of spherical surfaces of refraction 42.9 Third order coefficients in the case of a finite number of spherical surfaces of refraction Pages 290-291 292-293 293-294 294-295 295-296 296-297 297-299 Chromatic Aberrations 299-304 43.1 The index of refraction varies with the wave length of light 299 43.2 Chromatic aberrations in the realm of Gaussian optics 299-301 43.3 Integral expressions for Axial Color and Lateral Color 301 43.4 Simplification of these expressions. Dispersion. Abbe's v-value. Summations formulae in the case of a finite number of refracting surfaces 301-304 43.5 "Chromatic" Aberrations in electron optics 304 CHAPTER VI DIFFRACTION THEORY OF OPTICAL INSTRUMENTS §44. Formulation of the problem 305-311 44.1 Differential equations for periodic electromagnetic fields 305 44.2 Radiation of a periodic dipole in a homogenous medium. Boundary values at infinity 305-306 44.3 The case of a nonhomogeneous medium. Directly transmitted waves 306-307 44.4 Mathematical form of the transmitted wave. The approximation of geometrical optics 307-309 44.5 Virtual extension of the transmitted wave in the image space 309-310 44.6 The transmitted wave in the image space is characterized by its boundary values at infinity and by the condition of regularity in the whole image space 310-311 CONTENTS § 45. The boundary value problem of the equation .6.u + k 2u = 0 for a plane boundary 45.1 45.2 45.3 45.4 45.5 45.6 45. 7 Formulation of the problem Integral representation of the solution Proof that the above integral satisfies the differential equation Proof that the boundary values are attained Completion of the proof The conditions at infinity The corresponding boundary value problem for Maxwell's equations § 46. Diffraction of converging spherical waves 46.1 The corresponding boundary values at infinity 46.2 The problem is solved by applying the integral formula of §45.2. Debye's integral. The associated electromagnetic field § 47. Diffraction of imperfect spherical waves 47.1 Boundary values at infinity 47 .2 Solution of the problem with the aid of §45.2. The electromagnetic wave is represented by integrals over a wave front. Introduction of Hamilton's mixed characteristic 47.3 Invariance of these integrals with respect to the choice of the wave front 47.4 Parametric representation of the wave fronts 47 .5 Simplified form of the general diffraction integrals §48. Diffraction of unpolarized light 48 .1 Introduction of the vector m(p,q) which determines the polarization of the wave at infinity 48.2 Linear polarization at infinity 48.3 Representation of the wave with the aid of the operator 'v = -27AT. .l grad. 48.4 Average electric energy of an unpolarized wave 48.5 Average magnetic energy of an unpolarized wave 48.6 Average flux vector of an unpolarized wave 48.7 Summary of the results for the case of unpolarized waves xxv Pages 311-320 311-313 313-316 316-317 317 318 318-319 319-320 321-324 321-322 322-324 324-333 324-326 326-329 329-330 330-331 331-333 333-339 333-334 334 334-335 335-336 336-337 337-338 338-339 xxvi §49. §50. §51. CONTENTS Diffraction patterns for different types of aberrations 49.1 Spherical aberration. Limit of resolution. Tolerance for spherical aberration. Raleigh limit 49.2 Coma 49.3 Astigmatism Pages 339-344 339-342 342 343-344 Resolution of two luminous points of equal intensity 50.1 Interpretation of F(x,y ,z) as Fourier integral 50.2 Diffraction integrals in the case of rotational symmetry 50.3 Absolute limit of resolution of two point sources. Case of coherence and incoherence 50.4 Inequality for I F(r) I2 . The normal pattern has the greatest value of the central maximum I F(0) 12 for a given total energy 50.5 Solution of the maximum problem: To find the diffraction pattern F of given total energy such that for a given a we have F(a) = 0 and such that the central maximum F(0) is as great as possible 50.6 Solution of the maximum problem: To find the diffraction pattern F of given total energy and given distance of the inflection points such that the central maximum F(0) is as great as possible 50. 7 The maximum problem: To find the diffraction pattern F of given total energy such that the energy in a given circle is as great as possible 344-353 344-345 345-346 346-348 348-349 349-351 351-352 352-353 Resolution of objects of periodical structure 354-359 51.1 Resolution of more than two points at equal distances 354 51.2 Light distribution in the image of a self- luminous object which has a given distribution U0 354 51.3 The light distribution in the image of, a non-self- luminous object 354-355 51.4 The light distributions of object and image are developed into Fourier series. Relation of the two series in the case of self luminous objects of periodic structure 355-356 CONTENTS 51.5 Limit of resolution for self luminous objects of periodic structure. Impossibility of decreasing this limit by coating the aperture with thin films. Appearance of detail which ' approaches the limit of resolution 51.6 Limit of resolution for non-self-luminous objects of periodic structure xxvii Pages 356-358 358-359 Appendix I Vector analysis: Definitions and theorems I.I Notation and definitions I.2 Vector identities I.3 Vector fields I.4 Vector identities: div fA, curl fA, curl(f curl A), div (A x B), curl (A x B), A x curl B + B x curl A 360-367 360-361 361-362 362-364 364-367 Appendix II Tracing of light rays in a system of plane reflecting or refracting surfaces II.I Vector form of the laws of reflection and refraction II.2 Vector recursion formulae in a system of plane surfaces II.3 Representation of the direction ·of the final ray with the aid of certain scalar quantities P;. Recursion formulae for P; II.4 The special case that all surfaces are reflectors II.5 Example of three mirrors at right angles to each other II.6 Example of a 90° roof prism II.7 Influence of the roof angle on the doubling of the image 368-372 368 368-369 369-370 370 370 370-371 371-372 xxviii CONTENTS SUPPLEMENTARY NOTES NOTE I ELECTRON OPTICS N. Chako and A. A. Blank Introduction § 1. The equation of movement 1.1 Notations and Definitions: The electromagnetic field. Force on the electron. Kinetic energy T of the electron 1.2 The generalized potential of the field. Existence of a Lagrangian kinetic potential L(x;,X;) for the electron. Explicit form of L 1.3 The corresponding problem of variation 1.4 Statement of conservation of energy Pages 373-410 373 374-377 374-375 375 376 376-377 § 2. The associated Fermat problem 377-383 2.1 Problem of finding the path of a particle through two given points if the energy of the particle is given. The corresponding problem of variation 377-378 2.2 Transformation of this problem of variation 378-379 2.3 Proof that the new integral is homogeneous in x; of order 1. Formulation of the general rule of finding the associated Fermat problem in mechanical systems where L is independent of t. 379-380 2.4 Example: L = ~gik X;Xk - U(x1) 380-381 2.5 The case of electron optics. Derivation of the "index of refraction" for electromagnetic fields. 381-382 2.6 Analogy to crystal optics. 382-383 § 3. The canonical equations of electron optics 383-384 3.1 The variable z as a parameter 3.2 Derivation of the Hamiltonian H(x,y ,z,p,q) for electron optics. The canonical equations for the functions x(z), y(z), p(z), q(z) 383 383-384 § 4. Electromagnetic fields of rotational symmetry 4.1 The electric potential cf> and the magnetic potential A in this case 4.2 Special form of H and of the canonical equations. The invariant xq - yp 384-388 384-385 385-386 CONTENTS 4.3 The potential cp for fields free of space charges. Integral formula and power series for cp(P,z) 4.4 Integral formula and power series for a(p,z) § 5. First order optics in systems of rotational symmetry 5.1 The paraxial canonical equations 5.2 Simplified form of the paraxial equation for the complex functions X,P 5.3 The coefficients D, n, w 5.4 Solution of the equations with the aid of axial ray and field ray 5,5 Conjugate planes. Rotation of the image by the magnetic field § 6. The Gaussian constants of an electron optical instrument 6.1 Equivalent focal length. The two focal points of an instrument. Newton's Lens equation 6.2 The relations between the ray coordinates at two planes z = 0 and z = P. 6.3 Integral equations for X and P. Solution by iteration. Explicit formulae for F and the position of the unit points and focal points of a system 6.4 Approximate formulae for short systems § 7. Third order electron optics in systems of rotational symmetry 7.1 Formulation of the problem for x,y,p,q 7.2 Corresponding problem for X,P 7.3 Differential equations for the 3rd order polynomials 7.4 Solution of these equations. Integral expressions for the aberrations Ai;, A7J 7.5 The third order polynomials. Integral expressions for the coefficients A,B,C,D,E;~,y,o,E 7.6 Explicit expressions for the quantities Hik (z) 7.7 Analog of Petzval's theorem xxix Pages 386-387 387-388 388-393 388-389 389-390 390-391 391-392 392-393 394-398 394 394-395 395-397 398 398-407 398-399 399-400 400-402 402-403 403-405 405-406 406-407 XXX § 8. CONTENTS Physical Discussion of the third order aberrations of an electron optical instrument 8.1 Spherical aberration 8.2 Coma 8.3 Astigmatism 8.4 Distortion Pages 407-410 407-408 408-409 409 409-410 NOTE II OPTICAL QUALITIES OF GLASS M. Herzberger 411-431 NOTE ID MATHEMATICS AND GEOMETRICAL OPTICS M. Herzberger 432-439 NOTE IV SYMMETRY AND ASYMMETRY IN OPTICAL IMAGES 440-448 M. Herzberger CHAPTER 1 WAVE OPTICS AND GEOMETRICAL OPTICS In this course we shall be concerned with the propagation of light in a transparent medium. We shall not consider absorbing media or non-isotropic media, such as metals or crystals; but we will allow the medium to be nonhomogeneous. The optical properties of such a medium can be characterized by a scalar function n = n(x, y, z) , the refractive index of the medium. In ordinary optical instruments this function is sectionally constant and discontinuous on certain surfaces. The mathematical treatrnen11 of the propagation of light can be based on two theories: The wave theory of light (Physical Optics) and the theory of light rays (Geometrical Optics), Both theories seem to be fundamentally different and can be developed independent of each other. Actually, however, they are intimately connected. Both points of view are needed, even in problems of practical optical design. The design of an optical objective is carried out in general on the basis of Geometrical Optics, but for the interpretation or prediction of the performance of the objective it becomes necessary to investigate the propagation of waves through the lens system. In view of this fact, these theories will be developed simultaneously. The wave theory is considered as the general theory, and Geometrical Optics will be shown to be that special part of the wave theory which describes the propagation of light signals, i.e., of sudden discontinuities. On the other hand, in the important case of periodic waves, it represents an approximate solution of the differential equations of wave optics. This approximate solution can be used in a method of successive approximation to develop the diffraction theory of optical instruments, as will be shown in the later parts of this course. §1. THE ELECTROMAGNETIC EQUATIONS. 1.1 The wave optical part of this course is based upon Maxwell's electromagnetic theory of light. The phenomenon of light is identified with an electromagnetic field. 2 MATHEMATICAL THEORY OF OPTICS The location in space is determined by three coordinates x, y, z, the unit of length being 1 cm. The time is determined by the coordinate t; the unit of this variable being 1 sec. The electromagnetic field is represented by two vectors: the electric vector: the magnetic vector: E(x,y ,z,t) H(x,y,z,t) (Ei,E 2,E 3) , (H1,H2 ,H3) . The components, (E 1,E 2,E 3), of the electric vector are functions of x,y,z,t; the unit of these components is 1 electrostatic unit of E. The unit of the components (Hi, H 2, H3) of the magnetic vector is 1 electromagnetic unit of H. The properties of the medium can be characterized by two scalar functions of x,y ,z (the medium thus is assumed not to change with the time): the dielectric constant: E = E (x,y,z) , the magnetic permeability µ µ(x,y,z) . (1.11) 1.2 The electromagnetic vectors satisfy a system of partial differential equations which, with the above choice of units, assumes the form: curl H - i Et = 0 , C curl E + I:!:. Ht = 0 . C (1.20) The constant c is the velocity of light, in our units numerically equal t.o C = 3 • 1010 • If the components of E and H are introduced, the above vector equations yield a system of six linear differential equations of first order: aH3 aH 2 E aE1 ay az C at 0 aH1 aH 3 E a E 2 0 az ax C at a E3 ay -a E-2 + az µ C aH1 at = 0 a E1 aE3 µ aH 2 az ax· + C at 0 , (1.21) aH 2 aH1 ax ay E a E3 C at 0 aE 2 aE1 + !!:. aH 3 ax ay C at 0 WAVE OPTICS AND GEOMETRICAL OPTICS 3 In the group of optical problems to be considered in this course, we can assume µ = 1, since our medium (glass) is not magnetic. The dielectric constant, E = E(x,y,z), will be replaced by the index of refraction of the sub- stance, according to the equation (1.22) This relation between two different properties of a medium is actually far from being satisfied by the substances we are mainly interested in. However, experience shows that the predictions of the electromagnetic theory are in excellent agreement with observation if in theoretical results the quantity -fE is replaced by the index of refraction, measured by optical methods. Furthermore it is possible to give a satisfactory explanation of the above discrepancy by molecular considerations. We prefer in the following sections to leave Maxwell's equations in the above forms, (1.20) and (1.21). The symmetrical structure of these equations will often allow us to find from one relation another one simply by inter- changing the letters E and µ, and replacing E by -H and H by E. It is customary to add two more equations to the equations (1.20), namely: (1.23) div(µH) = 0, or ...£_ (µHi) ax + ...£_ ay (µH 2 ) + ...£_ az (µH 3 ) = 0. These state that the electromagnetic field does not contain a source of electricity or magnetism. However, these equations are not independent of (1.20). Indeed, since div curl A = 0 for an arbitrary vector field A(x,y ,z,t), it follows 88t (div EE) = 88t. (div µH) = o , (1.24) i.e., both div(E E) and div(µH) are identically zero if they are zero at any particular time. 1.3 Energy. If we· form the scalar product of E with the first of the equations (1.20) and of H with the second one and subtract both results we obtain 1 E •curl H - H •curl E - - (EE• Et + µH, Ht) 0 . C (1.30) On account of the identity H·curl E - E·curl H div(E x H) (1.31) 4 this gives MATHEMATICAL THEORY OF OPTICS c div(E x H) + _! _Q_ (EE 2 + µH 2) 2 at 0 or (1.32) div 4c,r(E x H) + _£_ at _!_(EE 2 + µH 2) B1r 0. The function (1.33) W(x,y,z,t) (1.34) measures the distribution of electromagnetic energy in the field. It determines the light density in Optics. The vector S(x,y,z,t) = 4C,r (E x H) (1.35) is called Poynting's radiation vector, and the relation between W and S is given by the equation aw + div s = o at • (1.36) Let us integrate this equation over a domain D of the x,y ,z space enclosed by a closed surface r. From Gauss' integral theorem: aat fff W dx dy dz + ff Sv do = O, D r (1.37) Sv being the normal component of S on r. The first integral represents the change of the total energy of the domain D per unit time. The surface integral thus gives the amount of energy which has left the domain D through the surface. Hence we interpret the vector field S = -4C(,rExH) as the vector field (or better, tensor field) of energy flux. Let do be the area of a surface element at a point x,y,z, and N a unit vector normal to it. Then the energy flux through this surface element is given by WAVE OPTICS AND GEOMETRICAL OPTICS 5 where SN = S, N is the normal component of the Poynting vector, S. In Optics, the energy flux per unit area is called the illumination of the surface element. We have (1.38) 1.4 Boundary Conditions. The vector functions E and H are of course not uniquely determined by the differential equations (1.20), unless certain boundary conditions are added. For optical problems, the following problem types are significant: 1. To find a solution of the equations (1.20), i.e., two vector fields, E(x,y,z,t) and H(x,y,z,t), if the electromagnetic field E(x,y,z,0) and H(x,y,z,0), at the time t = 0 is given and satisfies at this time the conditions div(E E) = div(µH) = 0. 2. Let us assume that on the plane z = 0, the electromagnetic field is a known function of x,y and t when t > 0, and that certain homogeneous boundary conditions are satisfied on another plane, z = L; i.e., E = E(x,y,0,t) given for t > 0 and, for example, E(x,y ,L,t) = 0 on z = L (Figure 1). z Let furthermore E = H = 0 for t = 0. To find a solution, E and H, in the half- space z > 0 which satisfies these L boundary conditions. 3. Of greater practical importance is the case for which the electromagnetic field is established under the influence of Figure 1 a periodic oscillator. Let us assume that an electric dipole is oscillating at a given point in space, for example, in front of an optical objective (Figure 2). Under the influence of this point source, an electric field is established which represents the light wave which travels through the objective. The problem is to determine these forced vibrations of the space as solutions of Maxwell's equations. §2. PERIODIC FIELDS. 2,1 We can expect that the electromagnetic field which in the end is established by a point source periodic in the time, will be periodic in time itself and, that its frequency equals the frequency of the oscillator. On the 6 MATHEMATICAL THEORY OF OPTICS basis of this expectation, one is led to consider special solutions of Maxwell's equations which have the form E u(x,y,z)e- iwt H v(x,y,z)e- lwt , (2.11) Figure 2 where u and v are vectors independent of t. The quantity w 271" (2.12) is the frequency of the oscillator and 11. the wave length. 2.2 The above complex notation is chosen on account of its mathematical advantages. The vectors u and v are in general complex vectors u = a + ia*, V = b + ib*' i.e., complex combinations of real vectors a, a* and b, b*. Calculations involving these complex vectors can be carried out in the same way as those involving real vectors only, when i is considered a scalar quantity, with i 2 = -1. For example: Scalar product: u-v = (a•b - a*-b*) + i(a•b* + a*•b). (2.21) Vector product: u xv = (a x b - a* x b*) + i(a x b* + a* x b). (2.22) The absolute value of a complex vector: u-u = a 2 + (a*) 2 • Two complex vectors u and v are called orthogonal if U•V = u·v (2.23) 0, i.e., if a-b + a *·b* = 0 , ab* - a*·b = 0 . (2.24) WAVE OPTICS AND GEOMETRICAL OPTICS 7 If two complex vectors E and H satisfy Maxwell's equations then both the real and imaginary parts of E and H are solutions. The real parts of the vectors (2.11), for example, are given by E acoswt+a*sinwt, H b cos wt + b* sin wt , (2.25) and will be considered in the following as representing the electromagnetic field. 2.3 We now introduce the expressions (2.11) into Maxwell's equations. This yields curl v + -iw E u O , C curl u - ~ iw µ v = 0 , (2.31) i.e., a system of partial differential equations without the time variable. By introducing the constant k w C (2.32) we obtain curl v + ik E u 0 , curl u - ik µ v 0 . (2.33) It follows that div(Eu) = div(µv) = 0 (2.34) so that it is unnecessary to add these conditions explicitly, as in (1.23). 2A Energy. The period, T = 271" = ~ , of the functions (2.11) is so W C extremely short in optical problems that we are unable to observe the actual fluctuation of the electromagnetic field. Indeed, in case of sodium light, for example, we have 11. = 0.6 x 10-4 cm., hence T 2 x 10-15 sec. 8 MATHEMATICAL THEORY OF OPTICS The same is true for the extremely rapid fluctuations of the light density W(x,y,z,t) = 8~ [E(acoswt + a*sinwt) 2 +µ(bcoswt+b*sinwt) 2]. (2.41) We are, however, able to observe the average value of this energy, which is given by the integral (2.42) We can express this result in terms of the original complex vectors u and v and obtain -J -W = -l611r rL.E - U • U + µ V • V (2.43) as an expression for the observable light density at the point x,y ,z. 2.5 Flux. Similar considerations may be applied to the Poynting vector, S. By introducing the expressions (2.25) into the definition of S, (1.36), it follows that S = :'Ir (a cos wt + a*sin wt) x (b cos wt + b*sin wt) (2.51) which is also a periodic function with the small period, T. Again, only the average value, 1 -S=T1- rt, TSdt ' can be considered as physically significant. We obtain -S(x,y,z) = SC1r (a x b + a* x b*) , (2.52) or in terms of the complex vectors, u and v, S(x,y,z) = C 1671" (u x -v + -u x v) . (2.53) We can show that the vector field, S, of average flux is a solenoidal field, i.e., div S = 0. WAVE OPTICS AND GEOMETRICAL OPTICS 9 For the complex vectors u and v satisfy the equations curl v + ikE u 0' curl u - ikµv 0. (2.54) The conjugate complex vectors u and v, consequently, satisfy curl v - ikEu 0 ' curl u + ikµv 0. (2.55) It follows that ii curl v - v curl u + ik (Eu·ii ~ µv•v) 0' v curl u - u curl v + ik (Eu •iI - µv •v) 0 ' (2.56) or div (u X v) + ik (€ u. u - µv •V) 0 ' div (ii x v) - ik (EU • ii - µv •v) 0. Hence div (u xv + u x v) = O; i.e., div S O. (2.57) 2.6 Polarization. The vectors E and H given by (2.25) describe certain closed curves in space. The type of these curves determines the state of polarization of the wave at the point x,y,z, and this again represents an observable characteristic of the field. In general, the electric vector is considered as the vector which gives the polarization of the light. We have E = a cos wt + a* sin wt, or in components E1 a1 cos wt + a1* sin wt , E 2 = a 2cos wt + a 2* sin wt, E 3 a 3cos wt + a 3* sin wt . (2.61) The curve described by E is plane since E is a linear combination of the vectors a and a*. We can show easily that this curve is an ellipse. Let us introduce ~ = cos wt and 7) = sin wt. By squaring the components of E we find 10 MATHEMATICAL THEORY OF OPTICS These equations, together with the relation ~2 + 112 = 1 represent four linear equations for the three quantities ~2, 11 2, and 2~7). Their determinant thus must be zero: 1 1 1 0 Ei2 ai2 a1*2 a1a1* 0 E 22 a22 a2*2 a2a2* (2.62) Eg2 ag2 a3*2 a3a3* This is an equation of the type AEi2 + BE 22 + CEa2 = D, which means that the curve of E lies on a surface of second order. The intersection curve of a plane and a surface of second order, however, is a conic. It must be an ellipse because it is closed. The equations (2.61) show that the ellipse is symmetrical with respect to the origin, i.e., to the point x,y ,z in question. Thus we can find the length and direction of the axes by determining the extreme lengths of the vector, E, i.e., the extreme values of the quadratic form under the condition ~ 2 + 7J 2 = 1. In other words, the axes are equal to the characteristic values of the above quadratic form and are given by the two solutions, ;>..1 and ;>.. 2, of the quadratic equation la·a - ;>.. a•a* I a•a* a*-a* - ;>.. 0. (2.63) Hence, (2.64) The characteristic values, ;>.. , are real, since the expression under the radical is not negative. The characteristic values cannot be negative; for, with the aid of the inequality (a.a*) 2 ::§ a 2 • (a*) 2 , one can see that WAVE OPTICS AND GEOMETRICAL OPTICS 11 We illustrate three types of polarization: a* a Figure 3. Elliptical polarization. Two different characteristic values, Ai =/- A2 , both different from zero. The electric vector describes an ellipse. The characteristic values are equal, At = A2. This implies a* a-a a*• a*, a.a_'." 0. (2.65) a Figure 4. Circular polarization. The two components a and a* of u are orthogonal and equal in length. The electric vector describes a circle. a a* Figure 5. Linear polarization. The smaller one of the characteristic values, A, is zero. The electric vector describes a straight line. The two vector components of u have the same direction, a x a* = 0. (2.66) We can express these results again by using the complex vector, u directly. The quadratic equation for ll. may be written as follows: ¼ A. 2 - (u •u) A - (u X u) 2 = 0 ; a+ ia*, and this has the solution, (2.67) The ellipticity E of the polarization, i.e., the ratio of the lengths of the axes, is thus given by the expression U • U - v1u) 2 (u) 2 u•u + -ltu) 2 (U:) 2 (2.68) Hence, for linear polarization: for circular polarization: uxu = 0, u2 =u•u=0. (2.69) 12 MATHEMATICAL THEORY OF OPTICS §3. DIFFERENTIAL EQUATIONS FOR E AND H. 3.1 If we eliminate one of the vectors, E or H, from Maxwell's equations, (1.20), we obtain second order equations for either E or H. By differentiation with respect to t: curl Ht - -E Ett O , C curl Et +I!:. Htt 0. C We introduce Ht = - ~ curl E in the first of these equations, and Et C E curl H in the second. The results. are ~ µcurl(~ curl E) + Ett O, ?" E curl (¾ curl H) + H tt = 0 . (3.11) We apply the following vector identity, which holds for an arbitrary scalar function, f(x,y,z), and an arbitrary vector field, A(x,y,z), with continuous derivatives of the second order: curl (f curl A) = -f A A + f grad (div A) + (grad f) x (curl A) , (3.12) where A A = Axx + Ayy + Azz. Equations (3.11) become ? G t) - Ett - A E = (curl E) x grad grad (div E), ? Htt - AH = (curl H) x (e grad¾) - grad (div H) . (3.13) From the second pair of Maxwell's equations (1.23), it follows that i.e., div E defined by div E E E div E + E •grad E 0 ' divµ H µ div H + H • grad µ o,; (3.14) -E •p and div H = -H •q, where the vectors p and q are p 1 grade E q -1 µ gradµ grad (log E ) , grad (log µ) . (3.15) WAVE OPTICS AND GEOMETRICAL OPTICS We introduce and obtain from (3.11) the equations -;;zn2 E tt - 6. E = grad (p • E) + q x curl E , -;;zn2 Htt - 6. H = grad (q • H) + p x curl H . 13 (3.16) (3.17) The vectors p and q and the function n are given by the properties of the medium; they are not independent of each other but are related by the equation 21 (p + q) = grad (log n) . (3.18) In the special case of a homogeneous medium, both p and q are zero, and n = '/Eµ is a constant. The equations (3.17) become -;;z = n2 Ett - 6. E 0 , -n;;2z Rtt - 6. H = 0 . (3.19) Each component of E and H satsifies the ordinary wave equation. The velocity of the waves is given by the quantity v = c/n, which allows us to regard the quantity n = '/Eµ as the index of refraction of the medium, defined by the ratio n = c/v of the velocity of light in a vacuum to the velocity in the medium. In the case of a non-homogeneous medium, a more complicated set of equations is obtained. Since n is now a function of x,y ,z, the six equations, (3.17), no longer yield one equation in each component, for the first order operators on the right sides involve all the components of the vectors in each equation. However, it is still true that the wave velocity, v, is given by the ratio c/n. Indeed, we shall see that for the propagation of a light signal, i.e., a sudden disturbance of the electric field, only the second order terms in (3.17) are significant. These terms lead to a generalized wave equation in which the coefficient, n, is not constant. 3.2 Stratified media. Let us consider as an example the case of a stratified medium, in which the functions E and µ depend only on one variable, 14 MATHEMATICAL THEORY OF OPTICS for instance on z. This case is of considerable practical interest, since the propagation of waves through thin, multilayer films, evaporated on glass, leads to a problem of this type. Let µ = 1 and le = n(z). It follows that p = 2 grad (log n) = (o,o, 2 :•). , Hence q 0. p·E 2 n' E n 3 (3.21) grad (p •E) ( 2 !!..'._ n 8E3. ax ' 2 !!..'._ 8E3. n ay ' 2 _£. az (n' n E \) 3/ - p x curl H = 2 n' n (8H1 az _ 8H 3 ax ' -8aHy-3+ -8aHz-2 ' 0 ) The differential equations (3.17) become n2 a2E1 c 2 at2 -AE1 2n-'-B-E3 n ax n2 a2E2 c 2 at2 -AE 2 = 2n'-8E-3 nay n2 a2E3 c 2 at2 -AE 3 = 2.a£z...(nn' E~3 n2 a2H1 n' 8H1 c 2 at2 -AH1 + 2 n -az- +2nn-' -a8-xH3 + 2n-'-a-H3 nay (3.22) (3.23) We thus obtain two partial differential equations, namely, those for E 3 and H3, in which none of the other components appear. After E 3 and H3 have been determined from these two equations, they are substituted in the remaining equations of (3.23); and these equations then become modified wave equations for E 1, E 2, H1, and H2 modified in the sense that the right side is not zero, but a known function. As a result of this simplification it is possible to find explicit solutions for many problems connected with stratified media, especially with films producing low reflection. WAVE OPTICS AND GEOMETRICAL OPTICS 15 §4. INTEGRAL FORM OF MAXWELL'S EQUATIONS. The functions E (x,y,z) and µ(x,y ,z) are not necessarily continuous functions. We assume, however, that they are sectionally smooth, i.e., every finite domain of the x,y ,z space can be divided into a finite number of parts in which E and µ are continuous and have continuous derivatives. The differential equations (1.20) represent conditions for the electromagnetic field in every part of the space where E, µ, and E, H are continuous and have continuous derivatives. They are, however, not sufficient to establish conditions for the boundary values of E and H on a surface of discontinuity. This is the reason why it is advantageous to replace the differential equations (1.20) by certain integral relations. These integral equations are equivalent to the differential equations if E, µ, and E, H are continuous and have continuous derivatives. They are more general, on the other hand, since they apply equally well to the case of discontinuous functions E, µ; E, H and establish definite conditions for the electromagnetic field in this case. 4.1 Let us consider, in the four-dimensional x,y,z,t space,. a domain D which is bounded by a closed three-dimensional hypersurface r. We assume that the hypersurface r consists of a finite number of sections in which the outside normal N of the hypersurface varies continuously. This normal N is a unit vector in the x,y,z,t space given by A. if the surface r is represented by the equation 0 the vectors E and H are different from zero in a larger sphere of radius o + ct. In other words we expect that the surface which separates the parts of the space which are still at rest from those penetrated by the original impulse Figure 8 travels over the space. A surface of this type is called a wavefront. In the above example the wave fronts are spherical and given by the equation cp(x,y,z,t) =✓x2 + y 2 + z 2 - cS - ct = 0. (7 .11) If the boundary values of E(x,y,z,0) or H(x,y,z,0) on the original sphere of radius o are different from zero then this sphere is a surface on which the electromagnetic field is discontinuous. We must expect that at the time t > 0 22 MATHEMATICAL THEORY OF OPTICS the corresponding boundary values on the wavefront (7 .11) are likewise different from zero so that the electromagnetic field is also discontinuous on the new wave front. This consideration leads us to define a wave front more generally as any surface in the x,y ,z space on which, at a given time t, the electromagnetic field is discontinuous. An observer at a point x,y ,z will interpret such a discontinuity as a sudden signal which reaches him when the wave front goes through the point x,y,z. Instead of illustrating the equation (7 .11) by a set of surfaces in the threedimensional x,y ,z space depending on the parameter t, we can interpret such a relation rp(x,y ,z,t) = 0 as a hypersurface in the four-dimensional space x,y,z,t. In our example this hypersurface is the cone and the electromagnetic vectors are discontinuous on this cone. Its "contour lines", i.e., the cross sections of the hypercone, rp(x,y,z,t) = 0, with the hyperplanes t = const., then represent the above set of wave fronts in the x,y,z, space. of rp 7.2 We may expect from the above example that the hypersurfaces rp = 0 which determine the propagation of discontinuities are not arbitrary but must fulfill certain con- ditions. We can derive these conditions easily with the aid of the general relations (5.41). Let us assume that rp(x,y,z,t) = 0 represents a hypersurface on which the vectors E and H are Figure 9 discontinuous. The functions E (x,y ,z) and µ(x,y ,z) shall be continuous in the neighborhood 0. We introduce, on rp = 0, the vectors u [El V = [HJ E2 E1' H2 - H1' (7 .21) which measure the discontinuity of E and H on rp 0. It follows from (5.41): gradrpxV- i rpt u C 0. ' U • grad rp 0. ' grad rp x U + I!:. .. = 1 and denote the parameter O" by T. The equations (8.21) become ~ dz dT dT (8.31) and the equations (8.23) become: (8.32) The orthogonal trajectories (8.31) thus form a two-parameter manifold of solutions of the equations (8.32). According to (8.31) we have (8.33) The analogy of (8.32) to the equations of mechanics is obvious. If we interpret - ½n2 as a potential field our light rays can be regarded as paths of particles ½ moving in this field with energy (x2 + y2 + z2) - ~ 2 = o. 8.4 We choose next ;>.. = 1/n and denote the parameter O" by s. It follows: nd-x ds ,,, . n ~ = ,,, . n dz 'l'x ' ds 'f'Y ' ds !Jlz (8.41) and hence ( dxds)2 + (~ds)2 + (ddzs)2 = 1, i.e., the parameter s measures the length along the light rays. Equations (8.23) become d an ds (n :) ax' f) d an ds (n ay, (8.42) d an ds (n ::) az 28 MATHEMATICAL THEORY OF OPTICS These equations can easily be recognized as the Euler equations of the variation problem J P1 V n ds is an extremum. Po (8.43) Let us consider two points P0 and P 1 in the x,y,z space. Let x(s), y(s), z(s) be a continuous curve between P0 and P 1 which also shall have a continuous tangent. We define the optical length of this curve by the integral JP1 V n ds. Po (8.44) In case n = 1 this optical length coincides with the geometrical length. The problem is to find the curve for which the optical length is a minimum. Let us assume that a solution exists and is given in the form x = x(u), y = y(u), z = z(u) where u is a parameter, such that u = 0 at P0 and u = 1 at P 1 ; hence V = (8.45) Figure 12 is a minimum. The necessary conditions which the solution must satisfy are Euler's differential equations, i.e., in case of (8.45): ( ✓x•2 :u n;t ny ✓x•2 + y•2 + z'2 0 ' + y'2 + z'2 )- (8.46) ( ✓x•2 :u nz' nz ✓x•2 + y'2 + z'2 0 . + y•2 + z'2 )- If we introduce in these equations the geometric length s of the solution as a parameter we obtain the differential equations (8.42). WAVE OPTICS AND GEOMETRICAL OPTICS 29 The light ray between two points P0 and. P1 is the curve for which the optical path attains an extreme value. This result is known as Fermat's principle of geometrical optics. If the index of refraction is interpreted as the ratio c/v of the velocity of light in a vacuum to the velocity v(x,y,z) in the medium, the optical path J J J V = n ds = c !s = c dt becomes proportional to the time needed to travel from P0 to P 1 • The principle of Fermat states that the light ray is a curve on which this time is a minimum, or at least an extremum. §9. CONSTRUCTION OF WAVE FRONTS WITH THE AID OF LIGHT RAYS. 9.1 Every solution lf.,(x,y,z) of the equation (9.11) determines a two-parameter manifold of light rays, i.e., of orthogonal trajectories. We have seen that these light rays satisfy a system of ordinary. differential equations x n nx y n ny (9.12) z n nz and the condition x2 + y2 + z2 = n2 . (9.13) The dot means differentiation with regard to the parameter t. Our aim in the following is to show that the two problems of integrating the partial differential equation (9.11) or the system of ordinary differential equations (9.12) are equivalent. Let us first assume that the solutions of (9.12) are known. We show that it is possible then to solve the following problem simply by quadratures and eliminations. 30 MATHEMATICAL THEORY OF OPTICS Let r be an arbitrary surface section given in parametric form X f(L 7)) y g(L 7J) (9.14) z h(L 7J) To find a solution !Jl(x,y,z) of (9.11) ;which, on r, has given values !JI = F(L 7J ). 9.2 We know, if !JI is the desired solution, that the orthogonal trajectories of the surfaces !JI const. are solutions of (9.12) and (9.13) and the optical length between two points P0 and P 1 of such a trajectory is given by the difference (9.21) Figure 13 This leads to the following attempt to solve the above problem. We determine through every point ; , 7J of r a light ray, i.e., a solution x = x(L7Ji T), y y(L 7j; T) ' (9.22) z z(L 7J; T) , of (9.12) and (9.13) which satisfies the boundary conditions x(L 7J, O) f(;,7J), x(L 7J, O) a(;' 7J) ' y(;' 7j' 0) z(L 7J, 0) g(;' 7j) ' h(L 7J) , y(L 7J, O) z(L 7J, O) b(L 7J) , c(L7J). (9.23) The functions a,b,c must obey the condition a2 + b2 + c2 = n2(;,7J) (9.24) in order to insure that the functions (9.22) satisfy the condition (9.13) but are otherwise arbitrary. WAVE OPTICS AND GEOMETRICAL OPTICS 31 We now consider the expression T f 1/J(L T'/, T) = F(;, ri) + n 2 dT 0 (9.25) and expect that the solution of our problem can be obtained in this form when ; , T'/, T are expressed as functions of x,y ,z with the aid of (9.22). We assume that the Jacobian ~~;,:,:~ is not zero in the neighborhood of r in order to be able to carry out this elimination. 9.3 It is clear that 1/J(x,y ,z) has the correct boundary values on r; for T = 0, we have 1/J = F(;, T'/ ). We show next that if; satisfies the equation (9.11) if the functions a,b,c are chosen suitably. We determine the deriva- tives of 1/J(;, T'/, T). First, §!I!. = n2 = :x:2 + y2 + z2 aT Then, we write (9.25) in the form (9.31) I/! and obtain (9.32) TJ (. az) + O X ~ ax + y • £i. a; + • Z ~ d T , We introduce n nx x, n ny = y, n nz = z and find 1/!g I/Jg and similarly l/!11 ~ Fg + T d dT (x• ax+ a; • EX.+ Y a; • z az) a; dT Fi - ( a af a; + Qg + ah) b a; ca; +x. aa-x;+y• EX. a; + . az z~, (9.33) F ij - ( af aar i Qg +bari ah +cari ) +.xaa-rxi+ y• EX. ari + • z az ari • (9.34) 32 MATHEMATICAL THEORY OF OPTICS We now assume that the functions a,b,c satisfy the conditions a a8ff + b ~ + 8~ C a8hf = Ft ' a8-f+ bB=a- + c8-h 81'/ 8rJ 8rJ a2 + b2 + c2 = n2 . (9.35) These equations have two solutions (a,b,c) provided that not all of the subdeterminants of the matrix (9.36) are zero. Let (a,b,c) be one of these two solutions. With this choice of a,b,c we obtain • X a8xf + . y £l. 8; + . z a8zf ' x. -8+xy::.:..8.vz...+. z8z- 8rJ 8rJ 8rJ ' (9.37) On the other hand 1/Jx a8x[ + 1/iy £l. 8; + 1/Jz 8z 8; , 8x £l. 8z 1/Jx 8rJ + 1/Jy 8rJ + 1/Jz 8rJ • 8x £l. 8z 1/J X 8T + 1/J y 8T + 1/J z 8T ' (9.38) Since the determinant B(x,y,z) is different from zero it follows by comparing (9.37) and (9.38): 8(~, 7) ,T) 1/Jx = X, 1/Jy = Y, 1/Jz z. (9.39) WAVE OPTICS AND GEOMETRICAL OPTICS 33 Therefore, on account of (9.13) i.e., 1/J is a solution of (9.11). Actually, our method allows us to find two solutions of the above problem as we expect from the quadratic nature of (9.11). 9.4 We apply our result to the case F(~, 11) = O. The surface r itself is thus a wave front, namely, the wave front at the time t = 0. The problem is to find the position of the wave front at the time t. We calculate the two- parameter set of solutions of (9.12) which intersect- the surface r with direc- tions a(L 11), b(L 11), c(L 11), The quantities a,b,c can be found from (9.35), i.e., from a af 8~ + b £B: 8~ + c ~ah 0 ' 0 ' (9.41) a2 + b2 + c2 = n2 . From the first two of the above equations it follows that the light rays must be normal to the surface r. When these rays have been found: X X(~ , 1), T) , y y(L 1)' T) ' (9.42) z z(L 1J, T) , r we obtain the solution 1/J by the integral J T 1/J(L 1J, T) = n 2 (x,y ,z) dT , 0 (9.43) Figure 14 in which ~ , 7J , T have to be expressed in terms of x,y,z with the aid of (9.42). 9.5 The original wave surface. r of the preceding section may degen- erate into a point, (x 0 , Yo, z 0). The functions f, g, h are constant in this case, so that the conditions (9.41) reduce to only one condition: (9.51) 34 MATHEMATICAL THEORY OF OPTICS Let us determine the two-parameter set of light rays through the point (xo, Yo, z 0), i.e., solutions x x(a,b,c,T) , y y(a,b,c,T) , (9.52) z z(a,b,c,T), of (9.12) which satisfy the boundary conditions x(a,b,c,O) x(a,b,c,O) a, y(a,b,c,0) Yo z(a,b,c,O) y(a,b,c,O) b , z(a,b,c,O) C • We obtain a solution 1/J(x,y,z) of (9.11) in the form of the integral (9.53) T 1/J(a,b,c,T) = J n 2 (x,y,z) dT 0 (9.54) after a,b,c,T have been expressed by x,y,z with the aid of the relations (9.51) and (9.52). These special solutions are called "Spherical" Waves or simply wavelets. If x 0,y 0,z 0 are considered as variable parameters 1/J becomes a function of two points T V(x 0, Yo, z 0 ; x, y, z) = J n 2 dT 0 (9.55) It determines the optical distance of the two points (x 0,y 0 ,z 0) and (x,y,z). The spherical wave fronts around a point (x 0,y 0,z 0) then are given by the surfaces V(x 0,y 0 ,z 0 ; x,y,z) - ct = O. (9.56) 9.6 Huyghens' Construction. With the aid of the wavelet V(x y z x, y, z) another method can be obtained to determine the wave , O, ; O O fronts belonging to a given single wave front r. This method is of prime importance and is known as Huyghens' construction. We consider the wave- let functions V(~, 1): x,y,z) which belong to the points (~,TJ) of the surface r, At the time t a two-parameter set, V(~,T);x,y,z) - ct 0 (9.61) WAVE OPTICS AND GEOMETRICAL OPTICS 35 of spherical wave fronts is obtained; we show that the envelope of these wave fronts is the wave front 7/J(x,y ,z) - ct = 0 which, at t = O, coincides with the given surface r. Figure 15 We find the envelope of the surfaces (9.61) by eliminating the parameters (~ , 7J) from the three equations,__ V ~ (~, 7J ;x,y ,z) 0 , V1J (~, 7J ;x,y ,z) 0 ' V (~ , 7J ;x,y ,z) - ct = 0 . (9.62) V(!,77 1x,y,z) =ct Let ~ 7J A(x,y,z) , B(x,y ,z) (9.63) Figure 16 be the result of calculating ~, 7J from the first two equations (9.62). We introduce ~ and 7J in V(~, 7J ;x,y ,z) and show that 7/J(x,y,z) = V (A(x,y ,z) ,B(x,y,z);x,y ,z) (9.64) is a solution of (9.11). Indeed, or, on account of (9.62): and similarly, 7/Jz (9.65) 36 MATHEMATICAL THEORY OF OPTICS These equations state that, at the point (x,y ,z) the surface 1/J - ct = 0 has the same tangential plane as the wave V(i;, 71;x,y,z) - ct from the point(~, 71), by (9.63). Since V satisfies the equation (9.11), the same is true for the function (9.64). The function V(A,B;x,y,z) determines the optical distance of a point (x,y,z) from the corresponding point (~, 71) defined by (9.63). If the point (x,y ,z) approaches the surface r the corresponding point (~, 71) on r approaches the same limiting point. Instead of proving this analytically, we will refer to the geometric evidence. Hence 1/J = V(~, 71 ,x,y,z)-0 if (x,y,z) approaches the surface r. The function (9.64) thus is the desired solution. §10, JACOBI'S THEOREM. 10.l In this section we shall be concerned with the inverse problem. Suppose we are in a position to integrate the partial differential equation (10.11) To find the general solution of the differential equations of the light rays. The answer is given in a general theorem of Jacobi. This theorem, applied to the differential equations (10.11) states: Let 1/J(x,y,z; a,b) be a complete integral of the equation (10.11). A complete integral is defined as a set of solutions which depend on two arbitrary parameters a and b such that not all of the subdeterminants of the matrix 1/Jxa.• ( 1/Jza.•) 1/Jxb• 1/Jyb, 1/Jzb, (10.12) are zero. Then the light rays of the medium of refractive index n(x,y ,z) are given by the equations a aa 1/J(x,y,z; a,b) Cl! , a ab 1/J(x,y ,z; a,b) = {3 , (10.13) where Cl! and {3 are arbitrary constants. If, for example, the determinant 1/Jya. 1/Jza. =f. 0 ' 1/Jyb 1/Jzb WAVE OPTICS AND GEOMETRICAL OPTICS 37 then we may calculate y and z as functions of x: y y(x; a,b,O! ,/3) , z = z(x; a,b,a ,/3) . (10.14) These functions represent a four-parameter set of curves which, according to Jacobi's theorem, are the light rays of the medium. 10.2 For the proof of Jacobi's theorem let us assume that the curves (10.13) are given in parametric representation: x = x(cr; a,b,a,{3) , y y(cr; a,b,a,{3) , (10.21) z = z(cr; a,b,a,{3) . By introducing these functions in (10.13) we obtain identities in cr,a,b,a,{3. HE_mce by differentiation with respect to er: . I/Jax x+ I/Jay y+ 1/Jaz z 0' z 1/Jbx x + 1/Jby y + 1/Jbz 0. (10.22) The six quantities I/Jax• 1/Jbx• 1/Ja.y• 1/JbY• 1/Ja.z, 1/Jbz• (10.23) can be interpreted as two vectors which, on account of (10.12), are not linearly dependent and thus determine a plane. The equations (10.22) state that the vector (x,y ,z) is perpendicular to this plane. From (10.11) we have by differentiation with respect to a and b: (10.24) i.e., the vector (1/Jx, 1/Jy, 1/J.) is also normal to the above plane. Hence, x z = ?..1/Jx, y = ?..1/Jy, = ?..1/Jz, • (10.25) 38 MATHEMATICAL THEORY OF OPTICS By differentiating these last equations with respect to O" we have as in section 8.2: - -(!) 1 d • ;>,. dO" ;>,. nny' (10.26) which shows that the curves (10.21) are light rays. 10.3 Example. Let us consider the case of a stratified medium where n = n(z). We verify easily that l/J = ax + by + / ✓n 2 - a 2 - b 2 di; 0 is a complete integral of the equation (10.31) The light rays in such a medium thus are given by E!l!. ila t X - a di; o✓n2 (1:) - a2 - b2 Cl! ' E!l!. 8b y -b { di; o✓n2 (l;) - a2 - b2 (3 or J X = Cl!+ a z di; o ✓n 2 (1:) _ a 2 _ b2 y (10.32) (10.33) WAVE OPTICS AND GEOMETRICAL OPTICS 39 §11. TRANSPORT EQUATIONS FOR DISCONTINUITIES IN CONTINUOUS OPTICAL MEDIA. We shall derive certain differential relations in this section which allow us to calculate the discontinuities of an electromagnetic field along a given light ray if the discontinuity is known at one point of the ray. We assume explicitly that the functions E = E(x,y,z) and µ, = µ,(x,y,z) are continuous functions. The case of discontinuous optical media will be studied later and a principal difference between both cases will be found. 11.1 Differentiation along a light ray. Let F(x,y,z) be a differentiable function. Along a given light ray x = x(T), y = y(T), z = z(T), a function F(T) = F(x(T), y(T), Z(T)) , is obtained whose differential quotient is dF dT x On account of the relations = 1/Jx, etc. this becomes (11.11) Hence the differential operator a a a a 8T = 1/Jx 8X + 1/Jy 8y + 1/Jz 8z (11.12) can be interpreted as differentiation along a light ray provided that 1/J(x,y ,z) is a solution of the equation Let F(x,y ,z) 1/J(x,y,z), for example. It follows E!l!. = , = 8T 1, 2 + ,1, 2 + ,1, 2 'l'x 'l'Y 'l'Z n2 • (11.14) The operator (11.12) can also be applied to a vector field; for example (11.15) 11.2 We consider an electromagnetic field which is discontinuous on the hypersurface cp(x,y,z,t) 1/J(x,y,z) - ct 0. (11.21) 40 MATHEMATICAL THEORY OF OPTICS Let D and D' be two domains of the x,y,z,t space which are separated by the hypersurface cp = 0. We assume that E and H are continuous functions with continuous derivatives in the individual domains D and D'. The boundary values on cp = 0 which are assumed if this surface is approached from D and D' are denoted by Figure 17 E',H',Ex',Hx', ... , respectively. The discontinuities U and V of E and H are then given by U E 1 - E, V H' - H. First we consider the boundary values of E,H in the domain D. Both E and H(x,y,z,t) become functions of x,y,z on cp = 0 = 1/J - ct. We denote these vectors by ¾ E*(x,y,z) E (x,y,z; 1/J(x,y,z)) ¾ H*(x,y,z) = H (x,y,z; 1/J(x,y,z)) (11.22) The derivatives, for example of E *, are given by E*y E*z Therefore curl E* and similarly curl H* curl E + -1 grad 1/J x Et C curl H + -1 grad 1/J x Ht . C (11.23) (11.23) WAVE OPTICS AND GEOMETRICAL OPTICS 41 From Maxwell's equations: curl E = _ I± Ht and curl H C .f E . C t• hence: c curl E* c curl H* - µHt + grad 1/J x Et , (11.24) These equations can be considered as a system of six linear equations for the six components of the vectors Et and Ht . The matrix of these equations is the same as the matrix of the homogeneous equations for the discontinuities U and V on cp = 0: . grad 1/J x V + EU O , grad 1/J x U - µV O , (11.25) and we know that the determinant of this matrix is zero on 1/J - ct = 0. We conclude that the equations (11.24) are possible only if the left sides satisfy certain conditions. These conditions will now be derived. We form the vector product of grad 1/J with the second equation (11.24): c grad 1/J x curl H* = E grad 1/J x Et + grad 1/J x (grad 1/J x Ht) or on account of the vector identity (Appendix I.23) c grad 1/J x curl H* = E(grad 1/J x Et - µHt)+ (Ht· grad 1/J) grad 1/J . Hence with the aid of the first equation (11.24): 1 grad 1/J x curl H* - Ecurl E * = - (Ht· grad 1/J) grad 1/J . C (11.26) This equation states: The vector grad 1/J x curl H* - Ecurl E * has the direction of grad 1/J, i.e., is normal to the wave front 1/J = ct. The same considerations can be applied to the boundary values E'* = E'(x,y,z,.!1/J) , C H'* = H'(x,y,z,.!1/J) . C (11.27) If 1/J - ct 0 is approached from the domain D', we find that the vector grad 1/J x curl H'* - Ecurl E' * 42 MATHEMATICAL THEORY OF OPTICS is normal to the wave front ljJ = ct. Finally, by considering the differences U = E'* - E* and V = H' * - H* we have: The vector grad 1/J x curl V - Ecurl U is normal to the wave front 1/J = ct. We formulate this statement in the equation Ecurl U - grad ljJ x curl V = R grad ljJ (11.28) where R is a certain scalar function of x,y ,z. We can determine R explicitly by forming the scalar product of grad ljJ with equation (11.28). It follows R = 1 -µ (grad l/J • curl U) . (11,281) Let us, finally, introduce U = _.! (grad ljJ x V) with the aid of (11.25). The result is E ¾ curl (-; grad ljJ x V) + grad ljJ x curl V = - ~ grad ljJ (11.29) which is a differential equation of first order in the discontinuity V. 11.3 We can transform the equation (11.29) into a much simpler form. ¾ We remark that the vector V x curl ( grad ljJ) has the direction of grad l/J. Indeed, on account of the vector formula (Appendix I.412), we have (¾ ¾ ¾) curl grad ljJ) = curl grad ljJ + ( grad x grad l/J ¾) (grad x grad l/J • Hence, V x curl(¾ grad¢) = (V·grad ¢)grad¾ - (v·grad¾) grad l/J which proves our statement, for V • grad l/J = 0. Consequently, we can write (11.29) in the form ¾ ¾ ¾ curl ( grad ljJ x V) + grad l/J x curl V + V x curl ( grad l/J) = R' grad ljJ (11.31) WAVE OPTICS AND GEOMETRICAL OPTICS 43 where R' is a certain scalar function. The left side can be transformed with the aid of the vector identity (Appendix I.48) by introducing 1 A=-gradl/J, B=V E and (11.32) 1 a E aT Using A· B = 0, we obtain - -2 -av - V di.v (1- grad E aT E - R* grad 1/J, (11.33) R * being a new factor. We find R * by forming the scalar product of (11.33) with grad 1/J. We obtain n 2 R* 2 av E a:;:--•gradl/J - -2 a V·- grad 1/J E aT or, by (11.15) - -1V•grad n 2 E i.e., R* 2 V•grad n E n (11.34) Equation (11.33) becomes o. (11.35) Finally we introduce the notation i.e.' (11.36) 44 MATHEMATICAL THEORY OF OPTICS and we get the equation 2 av 87 + 1 f:J.dJV + ~1 (V • grad n)grad l/J = 0. (11.37) A similar relation can be found for the discontinuity U by replacing E and V by µ and -U in (11.37). Thus our complete result is: The discontinuities U and V satisfy the differential equations: (11.38) where the differential operators .6.€1/J and .6.µl/J are defined by [(¾ (¾ (¾ .6.€1/J = € l/Jxt + l/!yt + l/Jz)z] ' µ [(t (t (t .6.µl/J = l/Jx)x + l/!y)y + l/Jz\] (11.39) §12. TRANSPORT OF DISCONTINUITIES. (CONTINUED). 12.1 On a given light ray the equations (11.38) represent a system of ordinary differential equations. Indeed, we have shown in (11.1) that the dif- ferential operator a/aT differentiates a function in the direction of a light ray. Let us introduce, in (11.38), instead of U and V, the vectors u p (12.11) Figure 18 which have the same directions respectively as U and V, but different lengths. The differential equations (11.38) then assume an even simpler form: WAVE OPTICS AND GEOMETRICAL OPTICS 45 dP -d T + -n1 (P • grad n) grad 1/J = 0 , dQ -d T + -n1 (Q • grad n) grad z/! = 0 . (12.12) Since U and V are orthogonal to grad z/!, the same is true for P and Q. From U·V = 0 it follows that P •Q = 0. By forming the scalar product of P and Q with the equations (12.12) it follows that 0 , 0. - (12.13) This shows: The lengths of the vectors P and Q are not changed on a given light ray. Thus without loss of generality we can assume and interpret P and Q as unit vectors which determine the directions of the vectors U and V. If P and Q have been found as solutions of (12.12), we obtain U and V from u J 1 1" IUol P e 2 Aµz/!dT 0 1 J-r Aez/!dT V = IVol Q e 2 0 (12.14) These equations make it evident that U and V are zero on the whole light ray if they are zero on one particular point, T = 0, of the ray. The light rays thus determine the region of the space where directed signals can be seen. Let us assume that from the point 0 a light signal is released at the time t = 0. Let us furthermore assume r that the discontinuities U and V which repre- sent the signal are different from zero only on a section r 0 of the wave front z/! = ct 0• From (12.14) it follows that, at a time t > t 0, only on the corresponding section r of the wave front z/! = ct will discontinuities U,V be observed. Figure 19 This section is determined by the light rays 46 MATHEMATICAL THEORY OF OPTICS through r 0 • In other words, the light signal will be observed only in the part of the space which is covered by the light rays through r 0 • This does not exclude the possibility that light penetrates into other regions of the space. However, this "diffracted" excitation does not have a sudden discontinuous beginning. 12.2 The exponential factors in (12.14) have a simple geometric meaning. We have or Similarly a t:..1/) - aT (log E) , a t:..µ1/J = t:..1/) - aT (log µ) • (12.21) Let us now consider a "tube" of light rays, i.e., a domain D of the x,y,z space which is enclosed by a surface r consisting of two sections r 1 and r 2 of the wave fronts 1/) = Pt and 1/J = P2 and the cylindrical wall rs formed by the light rays through the circumference of r 2 and r 1. We apply the theorem of Gauss to this domain D: Figure 20 fff t:..1/) dx dy dz D ff ~do r (12.22) where Ea!Ev. is the derivative of ¢ in direction of the outside normals. However, E!E. av o on rs ' E!E. av n2 on r2 , E!E. av - n1 on r 1 • WAVE OPTICS AND GEOMETRICAL OPTICS 47 Hence, fff fllj) dx dy dz D (12.23) We now express the surface elements do 2 and do1 by the corresponding surface elements do of an arbitrarily chosen wavefront lj) = ct 0• We write (12.24) The factor K measures the expansion of an infinitesimally narrow tube of light rays. It follows that fff fllj) dx dy dz D The volume element dx dy dz can be expressed as follows: dx dy dz = K do ds = nK do dT . Hence, fff fllj) dx dy dz = fff ~ a~nK) dx dy dz . D D n T Since D is of arbitrary size, we find fllj) = _.!.. _.£_ (nK) nK 8T a 87 (log nK) and hence fl El/) -8aTl o gnK- E flµlJ! -8aTlog -nµK . (12.25) (12.26) 48 MATHEMATICAL THEORY OF OPTICS The exponential factors in equation (12.14) become (12.27) and we conclude from (12.14): K Ko -nElul2 = -no Eo 1u o12 ' (12.28) That is, the quantities ~ n E lu l 2 and ~n µ lvl 2 are constant along a light ray. This result allows us to determine the lengths of U and V along the 1° light rays of a given set of wave fronts !/J = ct without integration, simply by calculating the ratio : = of corresponding surface elements of the wave fronts. 0 00 12.3 Energy and flux on a wave front. Let us assume that the electro- magnetic field is zero on one side of the wave fronts !/J - ct = 0. In this case E = U and H = V; hence w u v s cu 1 = 871" (E 2 + µ 2), C = 471" x V). (12.31) E It follows from (11.25) that g rdd 1/1 E U2 U • (V X grad !/J) , (grad !/J x U) •V , and thus EU2 = µV 2 (12.32) Figure 21 i.e. Electric and magnetic energies are equal on a wave front. From (11.25) it follows further that WAVE OPTICS AND GEOMETRICAL OPTICS 49 U x V 1 = ~(V x grad l/1) x V = 1 n2 µV 2g rad 1/J 41r 2n W grad l/J. Hence, S = C 2n Wgradl/J. This yields for the absolute value of the Poynting vector: (12.33) 1s1 = ~n w. (12.34) This result allows us to interpret the equations (12.28). If we add both equations we find ~W n Ko no W0 and, introducing KK o -1 W n do -1 W no do 0 • Finally, on account of (12.34): ddo , we obtain oo Isl do = !Sol do 0 • (12.35) The flux through corresponding surface elements of a set of wave fronts is constant. 12.4 We continue the investigation of the differential relations (11.38) for U and V. The preceding results show that it is sufficient to consider the equations (12.12) for the directions P and Q of the vectors U and V. Let us represent the light ray in the vectorial form We have X grad l/J , X=ngradn and we can write (12.12) in the form . P + n1 2 (P ... • X)X O , Q + -n-¾- (Q •x)x o . (12.41) (12.42) 50 MATHEMATICAL THEORY OF OPTICS Instead of the parameter T we introduce the geometrical length s of the light ray, by using the relation ~: ~ We have and hence, X ndd-Xs nX' ' X n(nX1) I p nP', 0. (12.43) However, P • (nX1) 1 = P • (nX" + n'X') on account of P •X' = 0. This yields n(P •X") and similarly P 1 + (P • X")X' 0 , Q• + (Q • X")X' 0 . (12.44) The equations (12.44) demonstrate that the two unit vectors P and Q are determined by the light ray alone. The same light ray, of course, can be an orthogonal trajectory to many different sets of wave fronts. For example, in case n = 1, a given straight line can be orthogonal to systems of spherical wave fronts or to a system of plane wave fronts. Equations (12.44) however, state that the vectors U and V are submitted to the same rotation around the light ray no matter to which type of wave fronts the light ray is orthogonal. The wave fronts influence only the size of the discontinuities U and V. Figure 22 T' * P' - cT 12.5 The tangential vector T = X' of the light ray and the vectors P and Q define an orthog- onal system of unit vectors which travel along the light ray X = X(s). In general the c~ange of such a system along a curve is determined by three kinematic formulae: cP +bQ * +aQ (12.51) Q' - bT -aP * WAVE OPTICS AND GEOMETRICAL OPTICS 51 The coefficients a,b,c are functions of s. In our special case these formulae reduce to T' * cP +bQ pr - cT * * Q' -bT * * as is shown by (12.44). We have, incidentally, (12.52) a = 0, b = (Q • X"), c = (P ·X") . P (s) X (s) The fact that a = 0 for our system T ,P ,Q has a simple geometrical meaning. Let us consider the ruled surface which is formed by the straight lines through the vectors P on X(T), From dP = - cT ds it follows that P + dP = P - c ds T lies in the plane formed by the vectors P and T. This means that two neighboring unit vectors P and P + dP are not skew but intersect each other in a point of the plane of P and T. The total manifold of straight lines through the vectors P(s) thus envelopes a certain curve C in space and can be interpreted as the manifold of tangents of this curve. A ruled surface which consists of the Figure 23 tangents of a curve in space is called an applicable surface, since it is possible to apply it to a plane by bending without strain. The same consideration applies to the ruled surface of the vectors Q(s). Hence we can formulate the statement: The vector discontinuities U and V along a light ray determine a ruled surface which is applicable. 12,6 We can interpret the light rays as the geodetic lines in a space whose line element has the form (12.61) 52 MATHEMATICAL THEORY OF OPTICS J J By introducing, on a light ray X(o-) the parameter o- = n ds = n 2 dT which measures the optical length on the ray we have the relations dX do- 1 2 n grad I{!, dP 1 dP do- n 2 dT With this choice of the parameter the equations (12.12) become -ddoP- + -n1 (P • grad n)ddoX-- = 0 , -ddoQ- + -n1 (Q • grad n) -ddoX- = 0 . (12.62) We shall see that these equations characterize the vectors P or Q as being "parallel" along the light ray; parallelism being defined for the line element (12.61) in accordance with a definition which was introduced by Levy-Civita. The equations of the geodetic lines, i.e., of the light rays with o- as parameter, follow from (8 .23) letting X = 1/n2 and hence (:r + (?r + (~:r 1 n2 We find n2 d do- (n2 dx) do- nnx ' ~) n2 d do- (n2 do- nny , n2 d do- (n2 ~;) = nnz • Let us denote temporarily the components of the vector X(o-) X1(o-), X 2(o-), X3(o-), and the partial derivatives of n by an an an n1 ax ' n2 = ay ' Il3 = az • (12.63) (x,y,z) by Then (12.63) assumes the form n2 -d (n2 -dX-a) do- do- (12.64) WAVE OPTICS AND GEOMETRICAL OPTICS 53 or n2 d2xa du2 + 2ndX-C-l du (~ n1 dX1) du net n (~ d2XCl dXCl du2 + 2 du ni dX;) = net 1 n du n ~ -i" By introducing on the right side n ~ ( ddXu1) 2 , we obtain the differential equations of the geodetic lines in the form, (dX;) ( ~nn i -dX-;) du - -nne~t - du 2 0. (12.65) d 2X Thus, the second derivative dut of each component is equal to a quadratic form of the first derivatives ddXu1 ; we write 0 ' (12.66) where Cl rik = ~1 (. ,oai an axk + Oak an ax1 - Olk an ) axCl ' (12.67) as one can easily verify. The symbols o1k are Kronecker symbols Oik = 0 , i /- k Oik = 1 , i k. (12.671) In general the equations of the geodetic lines X(u) of any line element du2 = ~ g1k dx1 dxk 1,k (12.68) can be written in the above form (12.66). The matrices r~k are given by the coefficient g lk and are called Christoffel's symbols. In the special case of optics, i.e., for these coefficients are given by (12.67). 54 MATHEMATICAL THEORY OF OPTICS The theory of curvature of a space with the line element (12.68) can be developed in a similar way as the curvature of curves or surfaces, if one can compare the directions of vectors which are not attached to the same point. Obviously this involves a definition of parallelism in a non-Euclidean space, i.e., a criterion for the parallelism of two vectors at different points in space. Levy- Civita' s definition of parallelism is as follows: A vector A(u) is moved parallel on a given curve X(u) if it satisfies the differential relations Figure 24 dAa dXk du + l~k rf,k A1 du = o. (12.681) dX According to this definition, for example, the tangential vectors d a of a geodetic line are parallel. u In case of the optical line element (12.61) we find by using (12.67): +1-d-X-a n du = 0. (12,682) or In vector notation: dd(nA) + _! (nA. grad n) ddX - (nA. ddX) grad n = 0 . u n u u n which is the condition of parallelism in our optical medium. (12.683) Let us now consider a vector A(u) along a light ray; we have d(nA) + _! (nA . grad n) dX _ ( 0 . This means that the dipole begins to oscillate at the time t = 0. It follows that E = 0 and H = 0 for cp = r - ct > 0, i.e., for r > ct. At the time t, the sphere r = ct thus represents the wave front of the electromagnetic field, i.e., the boundary of the region of penetration. We assume furthermore that M(0) = M'(0) = 0 , but M"(0) = m I- 0, so that M(cp) is a function of the type indicated in Figure 27. On the wave fronts cp = r - ct = O; we thus have the boundary values E = -r1 (m X p) H = p X E = ;1_: p X (m X p) (13.62) which represent discontinuities of the electromagnetic field. We immediately verify the relations E •p = H • p = 0, and E • H = 0, as our former results required. Hence, only on the fO wavefront are the vectors E and H of the spherical wave normal to the light ray and perpendicular to each other. Figure 27 64 MATHEMATICAL THEORY OF OPTICS The quantities IE I 2 and IHI 2 on the wave front decrease in proportion to 1/r2; i.e., in proportion to the ratio of corresponding surface elements on the spherical wave fronts. §14. WAVE FRONTS IN MEDIA OF DISCONTINUOUS OPTICAL PROPERTIES. We shall not assume in the following that the functions E(x,y ,z) and µ(x,y,z) are continuous and have continuous derivatives. However, these functions will be sectionally continuous in any finite domain D of the (x,y ,z,t) space, i.e., it will be possible to divid.e D into a finite number of subdomains in which € and µ are continuous, and assume finite boundary values on the bounding surfaces. We furthermore assume that the derivatives of E and µ are also sectionally continuous. Our first aim is to find the laws according to which wavefronts pass through a refracting surface, i.e., a surface on which E or µ is discontinuous. The result will be Snell's law of refraction and the law of reflection. After that, we can answer the question of how signals, i.e., discontinuities of E and H, are influenced by such surfaces. We will find a system of formulae known as Fresnel's formulae. 14.1 Snell's Law of refraction. Let :r be a surface in the (x,y ,z) space which separates two media D and D' of indices of refraction n(x,y,z) and n'(x,y,z), such that on :r: [n] = n' - n -/- 0. Both n and n' shall have con- tinuous derivatives in their respective domains. A light signal may travel through the (x,y,z) space over a set of wave fronts cp(x,y ,z,t) = 1/J(x,y ,z) - ct = 0 . (14.11) In other words, a characteristic hypersurface cp = 0 in the (x,y,z,t) space is assumed on which E and H are discontinuous. This hypersurface is continu- ous but does not necessarily have continuous normals, as is indicated in Figure 28. Letting cp = 1jJ - ct we conclude that 1/J(x,y,z) must be a continuous solution of the equation y 1/J; + 1/Ji + 1/Ji = n2 (14.12) even on the surface :r where n is discontinuous. It's derivatives are X sectionally continuous. Char. surface cp = 0 in case n discontinuous on x = O. We assume that :r is given in the form Figure 28 x = f(~,1)), y = g(~;r,), z = h(~,TJ) (14.13) WAVE OPTICS AND GEOMETRICAL OPTICS 65 M n' and denote the values of 1/J on the two sides of :E by 1/J and 1/J'. 1/J and 1/J' have continuous derivatives in their domains. From this it follows that 1/J (f(~,7J), g(~,7J), h(~,7J)) 1/J' (f(~,7J), g(~,7J), h(~,7J)) Figure 29 and that both sides have continuous derivatives. Consequently (1/Jx' - 1/Jx)fs + (1/Jy' - 1/Jy)gi + (1/Jz' - 1/Jz}?s 0' (1/J; - 1/Jx)fll + (1/Jy' - 1/Jy)gll + (1/Jz' - 1/Jz)¾ 0. (14.14) These equations state that the vector is normal to the surface :E. If M is a unit vector in direction of the surface normal, we can write grad 1/J' - grad 1/J = rM . (14.15) Grad 1/J' and grad 1/J give the direction of the orthogonal trajectories of the surfaces 1/J' = const. and 1/J = const., i.e., of the light rays. Let T and T' be unit vectors along the light ray; then grad 1/J = nT and grad 1/J' = n'T'. It follows that n'T' - nT where r is a scalar factor. rM, (14.16) We conclude that the refracted ray leaves the surface :E in the plane formed by the incident ray and the surface normal M. From (14.16) it follows that n'(T' x M) = n(T x M) . (14.17) The length of the vector on the left side is n 'sin iJ '; on the right side, n sin iJ , where iJ is the angle of incidence and iJ' is the angle of refraction. This yields Snell's law of refraction n' sin iJ' n sin iJ . (14.18) 66 MATHEMATICAL THEORY OF OPTICS T' We find the factor r in (14.16) by forming the scalar product of M with (14.16): n'(T'-M) - n(T•M) = r' or r n' cos rJ ' - n cos rJ . (14.19) The relations Figure 30 n'T' - nT n' sin rJ' (n' cos rJ ' - n cos rJ )M , n sin rJ (14.191) allow us to find T' if T and M are given. 14.2 The law of reflection. We know by experience that a light signal when reaching a surface of discontinuity of n is not only transmitted, but also reflected. Mathematically, this possibility is suggested by the quadratic character of the equation We have seen in §9 that, on account of this, there exist two solutions ip which attain given boundary values on a given surface :E. That a surface of discontinuity must actually produce a set of reflected wave fronts will be seen in the next section by deriving Fresnel's formulae. Let us here assume the existence of a reflected signal. This means that the characteristic hypersurface cp = 0 consists of two branches cp = ip - ct = 0 and cp * = 1/1 * - ct in the neighborhood D of the surface :E. These two branches are joined together on :E, as is indicated in Figure 31 for the case when :E is the plane x = 0. This implies that the two functions 1/1 and ip * must have the same boundary values on :E, i.e., ip ( f(~,17), g(~. 7J), h(~. TJ)) 1/1* (rc~,TJ>,· g(~,7J>, iic~,7J>) (14.21) As above we conclude that Figure 31 grad 1/1* - grad ip = rM (14.22) WAVE OPTICS AND GEOMETRICAL OPTICS 67 or, by introducing the unit vector T * on the reflected ray, n(T* - T) = rM. (14.23) Figure 32 It follows: The reflected ray leaves the surface ~ in the plane determined by T and M. We find as above that T* X M = TX M' and hence sin {} * = sin {} , where {} * is the angle of reflection. For r we obtain r = n(cos {} * - cos {} ) . (14.24) (14.25) The equations T* - T sin {} * (cos {} * - cos {} )M , sin {} (14.26) can be satisfied by two vectors T * if T is given. From sin {} * sin {} it follows {} ' or cos {} * cos {} ' cos {} * - cos {} . (14.27) The first solutions give T* = T, i.e., the incident ray; the second solutions yield T* - T = - 2 cos rJ M . (14.28) This is the reflected ray. T* and T are symmetrical with respect to the tangent plane of ~ at the point of incidence. §15. TRANSPORT OF SIGNALS IN MEDIA OF DISCONTINUOUS OPTICAL PROPERTIES. FRESNEL'S FORMULAE. 15.1 In 14.2 we have seen that, on a surface ~: w(x,y ,z) = 0 on which the functions E and µ are discontinuous, a set of wave fronts 1/J = ct must be expected to split up into two sets of wave fronts; the transmitted wave fronts 1/J - ct = 0 and the reflected wave fronts 1/J* - ct = 0. The corresponding characteristic hypersurface then consists of three branches; the incident 68 MATHEMATICAL THEORY OF OPTICS branch