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MATHEMATICAL THEORY OF OPTICS
MATHEMATICAL THEORY OF OPTICS by R. K. L U N E B U R 6 Foreword by EMIL WOLF Supplementary Notes by M. HERZBERGER
UNIVERSITY OF CALIFORNIA PRESS BERKELEY A N D LOS ANGELES 1966
University of California P r e s s Berkeley and Los Angeles, California Second P r i n t i n g , 1966 Cambridge University P r e s s London, England
©
1964
by The Regents of the University of California
Library of Congress Catalog Card N u m b e r
6419010.
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 selfcontained 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 19471948) 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
l e c t u r e s and of related m o r e recent developments will soon be published by D r s . 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 p r e s e n t work contains the derivation of the b a s i c laws of geometrical optics f r o m Maxwell's equations. Amongst the many new r e s u l t s which this chapter contains, the t r a n s p o r t equations, eq. (11.38), relating to propagation of the e l e c t r i c and the magnetic field v e c t o r s along g e o m e t r i c a l r a y s , a r e of p a r t i c u l a r significance. In Chapter II Hamilton's theory of geometrical optics is formulated and in the chapter which follows it is applied to special p r o b l e m s . Amongst r e s u l t s which s e e m to make their f i r s t a p p e a r ance in the scientific l i t e r a t u r e a r e s o m e of the f o r m u l a e of §24 r e l a t i n g to final c o r r e c t i o n s of optical i n s t r u m e n t s by a s p h e r i c s u r f a c e s ; some new t h e o r e m s relating to p e r f e c t optical i n s t r u m e n t s (§28.4); and the introduction in §29 of a new "perfect l e n s , " which images stigmatically onto each other two spherical s u r f a c e s which a r e situated in a homogeneous medium. This is the now well known "Luneburg l e n s " which has found valuable applications as a microwave antenna. F i r s t and third o r d e r t h e o r i e s of optical s y s t e m s a r e discussed in Chapters IVand V and, like all the other c h a p t e r s , they contain a wealth of information. Chapter VI deals with the d i f f r a c t i o n theory of optical i n s t r u m e n t s . The f i r s t sections of this chapter a r e devoted to the derivation, in a mathematically consistent way, of e x p r e s s i o n s for the electromagnetic field in the image region of an optical s y s t e m suffering f r o m any p r e s c r i b e d a b e r r a t i o n s . A solution of this difficult problem (naturally somewhat idealized) is embodied in formulae (47.33), now known as the Luneburg diffraction i n t e g r a l s . T h e s e formulae a r e an important and elegant generalization of c e r t a i n c l a s s i c a l r e s u l t s of P . Debye and J . Picht. Section 48 deals with another important problem, often ignored in other t r e a t i s e s , namely with a systematic derivation of the s c a l a r theory for the description of c e r t a i n diffraction phenomena with unpolarized light. The concluding sections deal with p r o b l e m s of resolution and contain a discussion of the possibility of improvements in resolution by a suitable choice of the pupil function. T h e s e investigations a r e amongst the f i r s t in a field that has a t t r a c t e d a good deal of attention in r e c e n t y e a r s . In two appendices f o r m u l a e a r e s u m m a r i z e d relating to vector analysis and to ray t r a c i n g in a s y s t e m of plane s u r f a c e s . They a r e followed by supplementary notes on e l e c t r o n optics, prepared by Dr. A. Blank and based on lectures of Dr. N. Chako. The volume concludes with supplementary notes by Dr. M. H e r z b e r g e r , based on his l e c t u r e s dealing with optical qualities of g l a s s , with mathematics and geometrical optics and with s y m m e t r y and a s y m m e t r y in optical images.
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FOREWORD
It is evident that Luneburg's Mathematical Theory of Optics is a highly original contribution to the optical l i t e r a t u r e . I consider it to be one of the most important publications on optical theory that has appeared within the last few decades.
Emil Wolf
Department of Physics and Astronomy University of Rochester, Rochester 27, New York May, 1964
PUBLISHER'S NOTE
The p r e s e n t edition has been reproduced f r o m mimeographed notes issued by Brown University in 1944. It is reprinted by p e r m i s s i o n of the Brown University P r e s s . The University of California P r e s s extends gratitude for help in making this edition possible to Dr. A. A. Blank, Dr. Max H e r z b e r g e r , M r s . R. K. Luneburg, Dr. Gordon L. Walker, and Dr. Emil Wolf. The a u t h o r ' s name was misspelled in the original edition. This has, of c o u r s e , been c o r r e c t e d , and a number of typographical e r r o r s , almost all of which were listed originally in the E r r a t a of the mimeographed version, have also been c o r r e c t e d . Dr. Blank has clarified the last section of Chapter V on the basis of the E r r a t a . No other changes have been made in this edition, which p r e s e n t s Luneburg's work as hq left it.
vii
FOREWORD
It is evident that Luneburg's Mathematical Theory of Optics is a highly original contribution to the optical l i t e r a t u r e . I consider it to be one of the most important publications on optical theory that has appeared within the last few decades.
Emil Wolf
Department of Physics and Astronomy University of Rochester, Rochester 27, New York May, 1964
PUBLISHER'S NOTE
The p r e s e n t edition has been reproduced f r o m mimeographed notes issued by Brown University in 1944. It is reprinted by p e r m i s s i o n of the Brown University P r e s s . The University of California P r e s s extends gratitude for help in making this edition possible to Dr. A. A. Blank, Dr. Max H e r z b e r g e r , M r s . R. K. Luneburg, Dr. Gordon L. Walker, and Dr. Emil Wolf. The a u t h o r ' s name was misspelled in the original edition. This has, of c o u r s e , been c o r r e c t e d , and a number of typographical e r r o r s , almost all of which were listed originally in the E r r a t a of the mimeographed version, have also been c o r r e c t e d . Dr. Blank has clarified the last section of Chapter V on the basis of the E r r a t a . No other changes have been made in this edition, which p r e s e n t s Luneburg's work as hq left it.
BIOGRAPHICAL NOTE Di . Rudolf Karl Lüneburg
Born in Volkersheim, Germany, June 30, 1903; resident in United States of America since 1935; naturalized U. S. citizen 1944. Ph.D. University of Göttingen, 1930. R e s e a r c h Associate in Mathematics, University of Göttingen, 19301933; R e s e a r c h Fellow in Physics, University of Leiden, 19341935. Research Associate in Mathematics, New York University, 19351938. Mathematician, R e s e a r c h Department of Spencer Lens Company (subsidiary of American Optical Company), Buffalo, New York, 19381945. Visiting L e c t u r e r , Brown University, Summer, 1944. Mathematical Consultant, Dartmouth Eye Institute 1946. R e s e a r c h Mathematician, Institute of Mathematics and Mechanics (now Courant Institute of Mathematical Sciences), New York University, 19461948. Visiting Lecturer, University of Marburg and Darmstadt Institute of Technology, 19481949. Associate P r o f e s s o r of Mathematics, University of Southern California, 1949. Died at Great F a l l s , Montana, August 19, 1949. PUBLICATIONS t 1.
Das Problem der I r r f a h r t ohne Richtungbeschränkung und die Randwertaufgabe der Potentialtheorie. Math. Annalen 104, 45 (1931).
2.
Eine Bemerkung zum Beweise eines Satzes über 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 O. Halpern and O. Clark). Phys. Rev. 53, 173 (1938).
4.
Mathematical theory of optics (mimeographed lecture notes). University, Providence, Rhode Island, 1944.
5.
Mathematical analysis of binocular vision. Princeton, New J e r s e y : Princeton University P r e s s , for the Hanover Institute, 1947.
t l l e s e a r c h reports are not included In this bibliography.
ix
Brown
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: Inter science 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 O. 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 e x p r e s s 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 derivation 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 d i f f e r s , 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 f r o m his recent r e s e a r c h . 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 Pages il.
§2.
§3.
§4.
The electromagnetic equations
15
1.1 1.2 1.3 1.4
12 23 35 5
P e r i o d i c fields
511
2.1 2.2 2.3 2.4 2.5 2.6
56 67 7 78 89 911
Special f o r m of E and H in periodic fields Complex v e c t o r s Differential equations for u and v Average energy Average flux Polarization
Differential equations f o r E and H
1214
3.1 3.2
1213 1314
Second o r d e r equations for E and H Stratified media
Integral f o r m of Maxwell's equations
1518
4.1 4.2
1516
4.3
§5.
Definitions and notations Maxwell's differential equations Energy relations Boundary conditions
Eliminations of derivatives by integration Integral conditions instead of div eE = div/iH = 0 Integral conditions instead of Maxwell's equations
1617 1718
General conditions f o r discontinuities
1820
5.1 5.2 5.3 5.4
18 1819 20
Statement of the problem F i r s t condition f o r E and H Second condition f o r E Complete set of difference equations f o r discontinuities of the electromagnetic field
xiii
20
CONTENTS
V
Pages §6.
Discontinuities of the optical properties 6.1
§7.
2125
7.1 7.2
2122
8.2 8.3 8.4
2529 25 2526 26 2629 2936
9.1 9.2 9.3
2930 3031
9.5 9.6
Boundary value problem for ¡i(x,y,z) Expression f o r the solution of the problem Proof that the problem is solved by the expression in 9.2 Problem of finding the wave fronts which belong to a given special wave front Spherical wave fronts; wavelets Huyghens' construction
Jacobi's theorem 10.1
10.2 10.3 §11.
Introduction of light rays as orthogonal trajectories Differential equations for the light rays Interpretation of rays as paths of corpuscles Fermat's problem of variation
2223 2325
Construction of wave fronts with the aid of light rays
9.4
§10.
Homogeneous Media Characteristic equation and characteristic hypersurfaces Equation of the wave fronts
Bicharacteristics; Light rays 8.1
§9.
2021
Propagation of Discontinuities; Wave fronts
7.3 §8.
Conditions f o r E and H if e and fi are discontinuous
2021
Construction of the light rays with the aid of a complete integral of the equation of the wave fronts Proof of Jacobi's theorem Example: Light rays in stratified medium
Transport equation f o r discontinuities in continuous optical media 11.1 11.2 11.3
Differentiation along a light ray Conditions f o r E and H on a characteristic hypersurface Simplification of the above conditions; differential equations f o r discontinuities on a light ray
3133 33 3334 3436 3638
3637 3738 38
3944 39 3942 4244
CONTENTS
XV
Pages §12.
Transport of discontinuities (Continued) 12.1 12.2 12.3 12.4
12.5 12.6 12.7 §13.
Spherical waves in a homogeneous medium 13.1 13.2 13.3 13.4 13.5 13.6
§14.
§15.
Equations for directions of the discontinuities U,V and their absolute values Geometric interpretation of the quantities Aeil> and A^ip Energy and flux on a wave front The directions P and Q of the discontinuities depend only on the light ray and not on the wave fronts The discontinuities U and V determine two applicable surfaces through a light ray Noneuclidean parallelism of the discontinuities U and V on a light ray Integration of the transport equations
Simplest type of solutions of the wave equation with spherical wave fronts Method of obtaining solutions of this type by differentiation; multipole waves Multipole solutions of Maxwell's equations Dipole solution of Maxwell's equations Another form of the expressions 13.4 The vectors E and H on the spherical wave fronts
4456 4446 4648 4849
4950 5051 5155 5557 5864 5859 59 5960 6062 6263 6364
Wave fronts in media of discontinuous optical properties
6467
14.1 14.2
6466 6667
Snell's law of refraction The law of reflection
Transport of signals in media of discontinuous optical properties. F r e s n e l ' s formulae 15.1 15.2 15.3 15.4
Conditions for discontinuities before and after refraction Orthogonal unit vectors on the rays First set of equations for the components of U and V Complete set of equations for U and V; Solution of these equations. F r e s n e l ' s formulae
6772 6769 6970 70 7172
CONTENTS
xvi
Pages §16.
P e r i o d i c waves of small wave length 16.1 16.2 16.3 16.4 16.5 16.6
Radiation of a periodic dipole in a homogeneous medium Radiation of a periodic dipole in a nonhomogeneous medium Equations f o r the amplitude v e c t o r s U 0 and V0 f o r small wave lengths Discussion and solution of the differential equations f o r U 0 and V0 Media of discontinuous optical p r o p e r t i e s Electromagnetic fields associated with geometrical optics
7381 7374 7475 7577 7779 7980 81
CHAPTER II HAMILTON'S THEORY OF GEOMETRICAL OPTICS §17.
§18.
619.
P r i n c i p l e s of geometrical optics
8288
17.1 17.2 17.3 17.4 17.5 17.6
8284 8485 85 8586 8687
Huyghens' P r i n c i p l e Light r a y s a s paths of c o r p u s c l e s E l e c t r o n s in an e l e c t r o s t a t i c field Light r a y s in a medium with n* = Vn 2 + C F e r m a t ' s principle Example in which light r a y s a r e not c u r v e s of s h o r t e s t optical path
8788
The canonical equations
8894
18.1 18.2 18.3
8889 8991
Light r a y s in the f o r m x = x(z), y = y(z) Derivation of the canonical equations Canonical equations f o r p r o b l e m s of variation in general
Hamilton's c h a r a c t e r i s t i c function V ( x 0 , y 0 , z 0 , x , y , z ) 19.1 19.2 19.3 19.4 19.5
P r o b l e m of numerical investigation of an optical instrument Definition of the point c h a r a c t e r i s t i c V f o r canonical equations in general P r i n c i p a l t h e o r e m of Hamilton's theory regarding the differential dV Proof of the t h e o r e m Hamilton's t h e o r e m f o r geometrical optics
9194 94100 9495 9597 97 9899 99100
xvii
CONTENTS
Pages §20.
§21.
Hamilton's characteristic functions, W and T
100107
20.1 20.2 20.3 20.4 20.5
100102 102103 103104 104106 106107
Integral invariants 21.1
21.2 §22.
The mixed characteristic W Geometric interpretation of W The mixed characteristic W* The angular characteristic T Special significance of T
107116
Fields of light rays: necessary and sufficient conditions. Theorem of Malus. The integral ^ n cos 6 ds is zero for any closed curve C Nonnormal congruences of rays. Poincare's invariant. Geometrical interpretation
Examples 22.1 22.2 22.3
107112 112116 116128
Mixed characteristic W for stratified media. Systems of plane parallel plates. Spherical aberration of stratified media The angular characteristic in the case of a spherical mirror. First order development of T. Mirror equation Angular characteristic for a refracting spherical surface. First order development of T. Lens equation
116120
120124 124128
CHAPTER III APPLICATION OF THE THEORY TO SPECIAL PROBLEMS §23.
Perfect conjugate points. Cartesian ovals 23.1 23.2 23.3 23.4 23.5
23.6
Light rays through perfect conjugate points have equal optical length Cartesian ovals Object point at infinity: The Cartesian ovals are ellipsoids and hyperboloids Virtual conjugate points. Cartesian ovals for this case An object point at infinity having a perfect virtual conjugate point. The Cartesian ovals are again ellipsoids and hyperboloids The aplanatic points of a sphere. Graphical construction of light rays refracted on a sphere. Aplanatic surfaces in the front part of microscopic objectives
129138 129130 130132 132133 133134
135
136138
CONTENTS
XV i i Î
Pages §24.
Final correction of optical instruments by aspheric surfaces 24.1 24.2 24.3 24.4 24.5 24.6
24.7 24.8 §25.
The angular characteristic f o r a single refracting surface 25.1 25.2 25.3 25.4 25.5
§26.
Equations for the point characteristic Method of finding T by Legendre's t r a n s formation of the refracting surface Surfaces of revolution Example of a spherical surface The angular characteristic for an elliptic or hyperbolic paraboloid
The angular characteristic for systems of refracting surfaces 26.1 26.2 26.3 26.4 26.5
§27.
Justification of this method of correction Solution of the problem with the aid of the point characteristic Solution with the aid of the mixed characteristic W Solution with the aid of the angular characteristic T Example: A lens with a plane and an aspheric surface, which t r a n s f o r m s a spherical wave into a plane wave Example: A lens with a spherical and an aspheric surface which t r a n s f o r m s a plane wave into a converging spherical wave The principle of the Schmidt Camera The correction plate of the Schmidt Camera
Variation problem for T f o r finitely many refracting surfaces The corresponding problem of variation for a continuous medium Another problem of variation of T Equivalence of both problems Systems of spherical surfaces
139151 139 140 140142 143
143145
145 146147 147151
151156 151152 152•154 154 155 156 156164 156158 158159 159160 160163 163164
Media of radial symmetry
164172
27.1 27.2 27.3
164166 166169 169172
Equation of the light.xays The form of the light rays in general Example: n 2 = C + ^
xix
CONTENTS
Pages §28.
28.1
28.2
28.3 28.4 28.5
§29.
The wave fronts can be obtained by a Legendre transformation of the wave fronts of the potential field n2 = C + r The light rays are the circles through two points on opposite ends of a diameter of the unit circle. The medium n = 1/(1+ r) represents a perfect optical system Maxwell's fish eye obtained by a stereographic projection of the unit sphere Perfect optical systems in the x,y plane which are obtained by other conformal projections of the unit sphere Conformal projection of surfaces. Relation of optical design problem to the problem of perfect conjugate points on surfaces
172173
173175 175178 178180 180182
Other optical media which image a sphere into a sphere
?82188
29.1 29.2
182184
29.3 29.4 29.5 §30.
172182
Maxwell's Fish eye
Integral equation for the function p = nr Transformation of the equation into an equation of Abel's type Proof of the inversion theorem Explicit solution of the problem rt = 1 The special case r 0 =
Optical instruments of revolution 30.1 30.2 30.3
Euler's equations for optical media in which n = n(p,z); p = V x 2 + y 2 Transformation of Fermat's problem into a problem for p(z) alone Skew rays can be treated as meridional rays if n(p,z) is replaced by m(p,z) = ^ / n 2
30.4 30.5

The characteristic functions in instruments of revolution. The mixed characteristic W depends only on u,v,w Similar statements for the other characteristic functions
184185 185186 186187 187188 188195 188189 189191
191192
192194 194195
t
CONTENTS Pages §31.
Spherical A b e r r a t i o n and C o m a . Condition f o r coma free instruments 31.1 31.2 31.3 31.4
31.5 31.6 31.7 31.8 31.9
31.95 §32.
Spherical a b e r r a t i o n L ( p ) of an axial bundle of r a y s Zonal magnification M(p) Development of W f o r s m a l l v a l u e s of x 0 , y 0 but l a r g e v a l u e s of p,q The r a y i n t e r s e c t i o n with the i m a g e plane. Construction of Coma f l a r e s . A b b e ' s sine condition Oblique bundles which a r e rotationally s y m m e t r i c a l to one of t h e i r r a y s Invariance of the c h a r a c t e r i s t i c W with r e s p e c t to orthogonal t r a n s f o r m a t i o n s The c h a r a c t e r i s t i c W' in a c o o r d i n a t e system with the principal r a y a s the z ' axis Condition of StaebleLihotski f o r s y m m e t r i c a l bundles of r a y s The aplanatic points of a s p h e r e a r e coma f r e e . Graphical c o n s t r u c t i o n of two a s p h e r i c m i r r o r s which p o s s e s s two p e r f e c t l y aplanatic points Modification f o r the object at infinity
The c o n d e n s e r p r o b l e m 32.1
32.2 32.3 32.4
Statement of the p r o b l e m . Intensity and illumination. Equation f o r the illumination of a given plane Illumination of the c e n t e r . Equation f o r the r e l a t i v e illumination Condition f o r u n i f o r m illumination I n t e r p r e t a t i o n of the condition 4>(a) = 0 but (d) % f o r 9
195211 195197 197198 198
199201 201203 203204 205206 206207
207208 208211 211215
211213 213 213214 214215
CHAPTER IV FIRST ORDER OPTICS § 33.
The f i r s t o r d e r p r o b l e m in g e n e r a l 33.1 33.2
Canonical t r a n s f o r m a t i o n of the r a y s by an optical i n s t r u m e n t Invariance of the canonical conditions with r e s p e c t to orthogonal t r a n s f o r m a t i o n s of object and image space
216226 216217
217218
CONTENTS
xx i Pages 33.3 33.4 33.5 33.6
§34.
Gaussian optics 34.1 34.2 34.3 34.4 34.5 34.6
§35.
The matrices A,F,C in systems of revolution. The linear transformations for Gaussian optics Images of points of the plane z 0 = 0 General lens equation: Location of conjugate planes relative to a given pair of planes Special lens equation; if the unit planes are chosen as reference planes. Unit points and nodal points Focal points; Newton's lens equation The point transformation of object into image space is a collineation. Images of inclined planes
218221 221222
222224 224226 226233 226 227228 228229 229231 231232 232233
Orthogonal ray systems in first order optics
234239
35.1 35.2
234
35.3 35.4 35.5
§36.
Conditions for the matrices of the differentials dXi, dP,. Lagrangian brackets Linear canonical transformations in first order optics Linear equations obtained with the aid of the angular characteristic Simplification of the equations
Definition of orthogonal systems Primary or tangential focus; secondary or sagittal focus; astigmatism The bundle through the point x 0 = y 0 = 0. Focal lines. Primary and secondary magnifications. Astigmatic difference The images of the points x 0 ,y 0 of the plane z0 = 0 Dependence of the constants M,m,A on the position z 0 of the object plane. Lens equation in orthogonal systems
Nonorthogonal systems 36.1 36.2 36.3 36.4
The relations for the ray coordinates in nonorthogonal systems derived from the mixed characteristic W The bundle through x 0 = y 0 = 0 The images of the points x 0 ,y 0 of the plane z0 = 0 Relation of the torsion of the image to the angle included by principal plane bundles of the axial bundles
234235 235236 236237 237239 240243
240241 241242 242243 243
CONTENTS Pages )37.
Differential equations of f i r s t o r d e r optics f o r systems of rotational symmetry 37.1 37.2 37.3
37.4
37.5
37.6 37.7
37.8 i 38.
The path of electrons in the neighborhood of the axis of an instrument of revolution 38.1 38.2 38.3
i 39.
Formulation of the problem F i r s t order canonical equations. Paraxial rays and t h e i r geometric interpretation The corresponding problem of variation. T r a n s v e r s a l surfaces. Jacobi's partial differential equation for paraxial rays Interpretation of the function n t (z). Surface power. Final form of the paraxial differential equations The general solution is expressed by two principal paraxial rays: axial ray and field ray. Conjugate planes Another definition of the field ray Lagrange's invariant. The paraxial rays can be found by quadratures if one paraxial ray is known. Integral expressions f o r the focal length in t e r m s of paraxial rays The equivalent Riccati equation
2
39.5 39.6
39.7 39.8
243244 244245
245247 247248
248250 251
251253 254 255257
2
Expressions f o r n (z,p) if n (z,0) is known. Solution of the equation An 2 = 0 with given boundary values n 2 (z,0) = f(z) Surface power of an equipotential surface. Differential equations for paraxial electrons The corresponding HamiltonJacobi equation and corresponding Riccati equation
Difference equations for a centered system of refracting surfaces of revolution 39.1 39.2 39.3 39.4
243254
Notations and definitions The canonical difference equations The associated problem of variation The solution is expressed by two principal solutions: axial ray and field ray Lagrange's invariant The paraxial r a y s can be found by summations if one paraxial ray is known. Expressions for the focal length with the aid of invariant sums The Gaussian quantities f o r a single lens F i r s t order design of optical instruments
255256 256 257
257268 257258 258261 261 261263 263264
264265 265266 266268
xxiii
CONTENTS CHAPTER V THE THIRD ORDER ABERRATIONS I N SYSTEMS O F R O T A T I O N A L SYMMETRY
Pages Formulation of the problem ¡40.
General types of third o r d e r a b e r r a t i o n s 40.1
40.2
40.3 40.4 40.5
¡41.
The third o r d e r coefficients as functions of the position of object and pupil plane 41.1
41.2 41.3 41.4 41.5 41.6
¡42.
General e x p r e s s i o n s f o r the a b e r r a t i o n s Axj, Ay ( ; A p 0 , Aq 0 derived f r o m the mixed characteristic W Expressions f o r the third o r d e r a b e r r a t i o n s Ax ( , Ay); A p 0 , Aq 0 obtained by developing W to the fourth o r d e r . Aberrations of meridional rays Spherical a b e r r a t i o n s , Coma, A s t i g m a t i s m , Curvature of field, Distortion The combined effect of third o r d e r a b e r r a t i o n s Dependence of the a b e r r a t i o n s on the position of the diaphragm
The r a y s a r e determined by the intersection with two p a i r s of conjugate planes. Object and Image plane, Entrance and Exit pupil plane General f o r m of a b e r r a t i o n s Axi, A£j of meridional r a y s Expression f o r the image a b e r r a t i o n s Ax, derived f r o m the angular c h a r a c t e r i s t i c Expression for the a b e r r a t i o n s A£ j of the pupil plane Simplified e x p r e s s i o n s for Ax ( and A£ t The c a s e of the exit pupil plane at infinity. A,B,C,E a s functions of the magnification M of the object plane. Impossibility of c o r r e c t i n g an optical instrument f o r all p a i r s of conjugate planes
269281
269270
271272 273276 277 277281
281286
281282 282283 283284 284285 285
285286
Integral e x p r e s s i o n s f o r the third o r d e r coefficients
287299
42.1 42.2
287288
Formulation of the problem Development of the Hamiltonian function H(U,V,z). Differential equations for the third o r d e r polynomials X 3 , Y 3 ; P 3 Q 3 . Corresponding boundary value problem
288290
CONTENTS
XX iv
Pages 42.3 42.4 42.5 42.6 42.7 42.8 42.9 ¡43.
Integral expressions for Ax t = X 3 and Ay t = Y 3 Integral expressions for the coefficients A,B,C,D,E The quantities H l k in general Petzval's theorem The quantities H l k in the case of an electrostatic field The case of spherical surfaces of refraction Third order coefficients in the case of a finite number of spherical surfaces of refraction
Chromatic Aberrations 43.1 43.2 43.3 43.4 43.5
The index of refraction varies with the wave length of light Chromatic aberrations in the realm of Gaussian optics Integral expressions for Axial Color and Lateral Color Simplification of these expressions. Dispersion. Abbe's pvalue. Summations formulae in the case of a finite number of refracting surfaces "Chromatic" Aberrations in electron optics
290291 292293 293294 294295 295296 296297 297299 299304 299 299301 301 301304 304
CHAPTER VI DIFFRACTION THEORY OF OPTICAL INSTRUMENTS i 44.
Formulation of the problem 44.1 44.2 44.3 44.4 44.5 44.6
Differential equations for periodic electromagnetic fields Radiation of a periodic dipole in a homogenous medium. Boundary values at infinity The case of a nonhomogeneous medium. Directly transmitted waves Mathematical form of the transmitted wave. The approximation of geometrical optics Virtual extension of the transmitted wave in the image space 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
305311 305 305306 306307 307309 309310
310311
xxv
CONTENTS
Pages § 45.
The boundary value problem of the equation Au + k 2 u = 0 for a plane boundary 45.1 45.2 45.3 45.4 45.5 45.6 45.7
§ 46.
§ 47.
311313 313316 316317 317 318 318319 319320
Diffraction of converging spherical waves
321324
46.1 46.2
321322
The corresponding boundary values at infinity The problem is solved by applying the integral formula of §45.2. Debye's integral. The associated electromagnetic field
322324
Diffraction of imperfect spherical waves
324333
47.1 47.2
324326
47.3 47.4 47.5 § 48.
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
311320
Boundary values at infinity 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 Invariance of these integrals with respect to the choice of the wave front Parametric representation of the wave fronts Simplified form of the general diffraction integrals
Diffraction of unpolarized light 48.1 48.2 48.3
48.4 48.5 48.6 48.7
Introduction of the vector m(p,q) which determines the polarization of the wave at infinity Linear polarization at infinity Representation of the wave with the aid of the operator V =  ^ r grad. ¿7T1 Average electric energy of an unpolarized wave Average magnetic energy of an unpolarized wave Average flux vector of an unpolarized wave Summary of the results for the case of unpolarized waves
326329 329330 330331 331333 333339 333334 334 334335 335336 336337 337338 338339
CONTENTS
XXV ¡
Pages §49.
Diffraction patterns for different types of aberrations 49.1 49.2 49.3
§50.
339342 342 343344
Resolution of two luminous points of equal intensity
344353
50.1 50.2
344345
50.3 50.4 50.5
50.6
50.7
§51.
Spherical aberration. Limit of resolution. Tolerance for spherical aberration. Rayleigh limit Coma Astigmatism
339344
Interpretation of F(x,y,z) as Fourier integral Diffraction integrals in the case of rotational symmetry Absolute limit of resolution of two point sources. Case of coherence and incoherence Inequality for F(r) 2 . The normal pattern has the greatest value of the central maximum  F(0) 2 for a given total energy 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 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 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
Resolution of objects of periodical structure 51.1 51.2 51.3 51.4
345346 346348 348349
349351
351352 352353 354359
Resolution of more than two points at equal distances 354 Light distribution in the image of a selfluminous object which has a given distribution U0 354 The light distribution in the image of a nonselfluminous object 354355 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 355356
CONTENTS
xxv !i
Pages 51.5
51.6
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 Limit of resolution for nonselfluminous objects of periodic structure
Appendix I Vector analysis: Definitions and theorems 1.1 1.2 1.3 1.4
Notation and definitions Vector identities Vector fields 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
Appendix II Tracing of light rays in a system of plane reflecting or refracting surfaces 11.1 11.2 11.3 11.4 11.5 11.6 11.7
Vector form of the laws of reflection and refraction Vector recursion formulae in a system of plane surfaces Representation of the direction of the final ray with the aid of certain scalar quantities Pi. Recursion formulae for P4 The special case that all surfaces are reflectors Example of three mirrors at right angles to each other Example of a 90° roof prism Influence of the roof angle on the doubling of the image
356358 358359
360367 360361 361362 362364 364367
368372 368 368369 369370 370 370 370371 371372
CONTENTS
xxv i i i
SUPPLEMENTARY NOTES Pages NOTE I
ELECTRON OPTICS N. Chako and A. A. Blank
Introduction § 1.
1.2
1.3 1.4 §2.
2.2 2.3
2.4 2.5
2.6
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 Transformation of this problem of variation Proof that the new integral is homogeneous in Xj of order 1. Formulation of the general rule of finding the associated Fermat problem in mechanical systems where L is independent of t. Example: L = 2 g l k  U(Xi) The case of electron optics. Derivation of the "index of refraction" for electromagnetic fields. Analogy to crystal optics.
The canonical equations of electron optics 3.1 3.2
§4.
Notations and Definitions: The electromagnetic field. Force on the electron. Kinetic energy T of the electron The generalized potential of the field. Existence of a Lagrangian kinetic potential L(x l ( xi) for the electron. Explicit form of L The corresponding problem of variation Statement of conservation of energy
The associated Fermat problem 2.1
8 3.
373
The equation of movement 1.1
The variable z as a parameter 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)
Electromagnetic fields of rotational symmetry 4.1 4.2
373410
The electric potential
e Explicit expressions for the quantities Hik (z) Analog of Petzval's theorem
389390 390391 391392 392393 394398 394 394395
395397 398 398407 398399 399400 400402 402403 403405 405406 406407
CONTENTS
XXX
Pages Physical Discussion of the third o r d e r a b e r r a t i o n s of an electron optical instrument
407410
8.1 8.2 8.3 8.4
407408 408409 409 409410
Spherical aberration Coma Astigmatism Distortion
NOTE n
OPTICAL QUALITIES OF GLASS M. H e r z b e r g e r
411431
NOTE III
MATHEMATICS AND GEOMETRICAL OPTICS M. H e r z b e r g e r
432439
NOTE IV
SYMMETRY AND ASYMMETRY IN OPTICAL IMAGES M. H e r z b e r g e r
440448
CHAPTER 1 WAVE OPTICS A N D GEOMETRICAL OPTICS In this c o u r s e we shall be concerned with the propagation of light in a t r a n s p a r e n t medium. We shall not consider absorbing media o r nonisotropic m e d i a , such a s m e t a l s o r c r y s t a l s ; but we will allow the medium to be nonhomogeneous. The optical p r o p e r t i e s of such a medium can be c h a r a c t e r i z e d by a s c a l a r function n = n(x, y, z) , the r e f r a c t i v e index of the medium. In ordinary optical i n s t r u m e n t s t h i s function is sectionally constant and discontinuous on certain s u r f a c e s . The mathematical treatment» of the propagation of light can be based on two theories: The wave theory of light (Physical Optics) and the theory of light r a y s (Geometrical Optics). Both theories seem to be fundamentally diff e r e n t and can be developed independent of each other. Actually, however, they a r e intimately connected. Both points of view a r e needed, even in p r o b l e m s of p r a c t i c a l optical design. The design of an optical objective is c a r r i e d out in general on the b a s i s of Geometrical Optics, but f o r the i n t e r pretation or prediction of the p e r f o r m a n c e of the objective it becomes n e c e s s a r y to investigate the propagation of waves through the lens s y s t e m . In view of this fact, these t h e o r i e s will be developed simultaneously. The wave theory i s considered as the general theory, and Geometrical Optics will be shown to be that special p a r t of the wave theory which d e s c r i b e s the propagation of light signals, i . e . , of sudden discontinuities. On the other hand, in the important c a s e of periodic waves, it r e p r e s e n t s 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 i n s t r u m e n t s , as will be shown in the l a t e r p a r t s of this c o u r s e . 51. THE ELECTROMAGNETIC EQUATIONS. 1.1 The wave optical p a r t of this course is based upon Maxwell's electromagnetic theory of light. The phenomenon of light is identified with an electromagnetic field. 1
2
MATHEMATICAL THEORY O F 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:
E(x,y,z,t) = (Ej.E2.E3),
the magnetic vector:
H(x,y,z,t) = (H I ,H 2 ,H 3 ).
(1.10)
The components, (E 1 ,E2,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 (Hj, 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
n = n(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   E, = 0 , c curl E + ^ H , c
= 0 .
(1.20)
The constant c is the velocity of light, in our units numerically equal to c = 310 1 0 . If the components of E and H are introduced, the above vector equations yield a system of six linear differential equations of first order:
3y
3E2 9z
c
= 0
9 El 9z
9E3 9x
V 9H2 = 0 c 9t
€ 9E3 = 0 c 9t
9E2 3x
9 Ej 9H3 + M c 3t 3y
e 3 Ei = 0 c 31
9E3
3y
9H 2 9z
3 Hi 9z
9H3 3x
e 9E2 c at
9H2 ax
3 Hi
3H 3
ay
9 Hi = 0 at
= 0
(1.21)
WAVE OPTICS A N D GEOMETRICAL OPTICS
3
In the group of optical problems to be considered in this course, we can assume n = 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 substance, according to the equation n = /T .
(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 VT 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 interchanging the letters e and n, and replacing E by H and H by E. It is customary to add two more equations to the equations (1.20), namely: div(e E) = 0, or ^
(e E,) +
(e E 2 ) + ^
(e E 3 ) = 0 , (1.23)
divOxH) = 0, or ^
OxHj) + ^
(MH2) + ^
(MH3) = 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 y t (div eE) = ^
(div nH) = 0 ,
(1.24)
i.e., both div(e E) and div(pH) 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 E • curl H  H• curl E  ^ ( e E  E t + ^HH t ) = 0 .
(1.30)
On account of the identity H • curl E  E • curl H = div(E x H)
(1.31)
MATHEMATICAL THEORY O F OPTICS
4 this gives
c div(E x H) +  — ( e E 2 + ¿iH2) = 0 £i 9 t
(1.32)
div ~ ( E
(1.33)
x H) +
¿(eE2
+ nH 2 ) = 0 .
The function W(x,y,z,t) = ¿ ( e E 2
+ fiH 2 )
(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) = ¿ ( E
x H)
(1.35)
is called Poynting's radiation vector, and the relation between W and S is given by the equation 3W
—
„
+ div S = 0.
(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:
j ;
III D
w dx d y dz +
II
r
sv
do
=
0.
d37)
S„ being the normal component of S on r . The f i r s t 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 = t(EXH) 47T
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
dF = S N do where S N = 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 I = SN
(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(eE) = div(/iH) = 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). Let furthermore E = H = 0 for t = 0. To find a solution, E and H, in the halfspace z > 0 which satisfies these boundary conditions. 3. Of greater practical importance is the case for which the electromagnetic field is established under the influence of a periodic oscillator. Let us assume that Figure 1 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
MATHEMATICAL THEORY O F OPTICS
6
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~ lwi , H = vtx.y.zje" 1 " 1 , (2.11) where u and v are vectors independent of t. The quantity u> 2tt
c ~
\
(2.12)
is the frequency of the oscillator and A. 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:
uv
Vector product:
uxv
= ( a b  a*b*) + i ( a b * + a* b).
(2.21)
= (a x b  a* x b*) + i(a x b* + a* x b). (2.22)
The absolute value of a complex vector: uu = a 2 + (a*) 2 .
(2.23)
Two complex vectors u and v are called orthogonal if uv = uv = 0, i.e., if ab + 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 = a cos wt + a* sin wt , (2.25) H = b cos wt + b* sin wt , 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 iu> curl v + — e u = 0 , c (2.31) 1W curl u   ( j v
= 0,
i.e., a system of partial differential equations without the time variable. By introducing the constant k =  = — c X
(2.32)
we obtain curl v + ik e u = 0 , curl u  ik n v = 0 .
(2.33)
It follows that div(eu) = div(jiv) = 0
(2.34)
so that it is unnecessary to add these conditions explicitly, as in (1.23). 2.4 Energy. The period, T
=
~~
=
^ , of the functions (2.11) is so
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 \ = 0.6 x 10"4 cm., hence T = 2 x 10" 1 5 sec.
8
M A T H E M A T I C A L THEORY O F OPTICS
The same is true for the extremely rapid fluctuations of the light density W(x,y,z,t) = ~~~ L[e (a cos wt + a * s i n w t ) 2 + n(b cos wt + b* sin wt) 2Jl . 07T
(2.41)
We are, however, able to observe the average value of this energy, which is given by the integral
W = i
Wdt
= ^
[ e ( a 2 + a * 2 ) + n(b 2 + b * 2 ) ] .
(2.42)
We can express this result in terms of the original complex vectors u and v and obtain W =  7  [e uu + / i v v ]J lo7T ""
(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 c S = — (a cos wt + a*sin wt) x (b cos wt + b*sin wt) 47T
(2.51)
which is also a periodic function with the small period, T . Again, only the average value,
can be considered as physically significant. We obtain S(x,y,z) =
o7T
(a x b + a* x b*) ,
(2.52)
or in terms of the complex vectors, u and v, S(x,y,z) ~
J.07T
(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 A N D GEOMETRICAL OPTICS
9
For the complex vectors u and v satisfy the equations curl v + ike u = 0 , curl u  ikftv = 0 .
(2.54)
The conjugate complex vectors u and v, consequently, satisfy curl v  ikeu = 0 , curl u + ikjt v = 0 .
(2.55)
It follows that u curl v  v curl u + ik (eu*u  (ivv) = 0 , v curl u  u curl v + ik ( e u  u  fxvv) = 0 ,
(2.56)
or div (u x v) + ik (e u • u • v) = 0 , div (u x v)  ik ( e u  u  ( i v v ) = 0 .
(2.57)
Hence div (u x v + u x v) = 0; i.e., div S = 0. 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 E t = a t c o s wt + an* sin wt , E 2 = a 2 c o s wt + &2* sin wt , E 3 = a 3 c o s wt + a3* 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 rj = sin wt. By squaring the components of E we find Ei 2 = a t V
+ a,*V
+ Zajaj*
,
E 2 2 = a 2 2 1 2 + a 2 * V + 2a 2 a 2 * fr , E 3 2 = a 3 2 i 2 + a 3 * V + 2a 3 a 3 * rj .
10
MATHEMATICAL THEORY OF OPTICS
These equations, together with the relation t? + TJ2 = 1 represent four linear equations for the three quantities ? 2 , t?2> and 2£TJ. Their determinant thus must be zero: 1
1
1
0
Ei2
a,2
ai* 2
aja,*
E22
a22
A
2*
2
a2a2*
E32
A
32
A
3*2
a3a3*
= 0
(2.62)
This is an equation of the type AE t 2 + BE 2 2 + CE 3 2 = 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 l E12
a  a £ 2 + 2a • a* £ 7] + a *  a + r j 2 ,
under the condition £ 2 + TJ2 = 1. In other words, the axes are equal to the characteristic values of the above quadratic form and are given by the two solutions, and X 2 , the quadratic equation a•a  \ a • a*
a *  a *  A.
0 .
(2.63)
Hence, = \ [a 2 + a* 2 +
7 ( a 2  a*2)2 + 4 ( a  a * ) 2 ]
(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 >/(a 2  a * 2 ) 2 + 4(aa*) 2 £
" / ( a  a + a*  a*) 2 = a 2 + a* 2
11
WAVE OPTICS AND GEOMETRICAL OPTICS We illustrate three types of polarization:
Figure 3. Elliptical polarization.
Two different characteristic values, \i ^ X 2 , both different from zero. The electric vector describes an ellipse. The characteristic values are equal, = X 2  This implies a. a = a*•a* , a  a * = 0.
(2.65)
Figure 4. Circular polarization.
The two components a and a* of u are orthogonal and equal in length. The electric vector describes a circle.
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, axa*
= 0.
(2.66)
We can express these results again by using the complex vector, u = a + ia*, directly. The quadratic equation for X may be written as follows: \ 2  (uu) \  ^(u x u) 2 = 0 ; and this has the solution, X = I
[uïï + V(u) 2 (ü)2 ]
(2.67)
The ellipticity e of the polarization, i.e., the ratio of the lengths of the axes, is thus given by the expression 2
uïï  s/(u) 2 (u) 2
= x
2
uïï + V ( u ) 2 ( ï ï ) 2
(2.68)
Hence, for linear polarization: for circular polarization:
u x u = 0 , u 2 = uu = 0
(2.69)
12
MATHEMATICAL THEORY O F 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 H t 
Ett
= 0,
curl E t + ^ H t t c
= 0.
c We introduce H t =  — curl E in the first of these equations, and E t curl H in the second. The results are H curl ^
curl E^ + ^
Ett
= 0,
e curl ^
curl H^ +
Htt
= 0 .
c =—
(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) , where A A = A x x + A y y + Azz.
(3.12)
Equations (3.11) become
E t t  A E = (curl E) x ^u grad ^
 grad (div E), (3.13)
^ f Htt  A H
= (curl H) x ^e grad
 grad (div H) .
From the second pair of Maxwell's equations (1.23), it follows that div e E = e div E + E • grad e = 0 , div n H = fi div H + H • grad n = 0 ;
^
i.e., div E = E p and div H = H q, where the vectors p and q are defined by p = ^ grad e
= grad (log e ) , (3.15)
q = 
grad n = grad (log n) .
WAVE OPTICS A N D GEOMETRICAL OPTICS
13
We introduce n = VT^I ,
(3.16)
and obtain from (3.11) the equations n2
E t t  A E = grad (pE) + q x curl E (3.17) H t t  A H = grad (qH) + p x curl H
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  ( p + q) = grad (log n) .
(3.18)
In the special case of a homogeneous medium, both p and q are zero, and n = Ve^ is a constant. The equations (3.17) become n2 n2
Ett  A E = 0 , (3.19) Hn  A H = 0 .
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 = VTjI 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 nonhomogeneous 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 p. 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 y. = 1 and v T = n(z). It follows that p = 2 grad (log n)

(«.»• » ? )
•
q = 0 .
(3.21)
Hence pE = 2 — E i, n ( n' 9E, grad (pE) = ( 2 
 P
n' 9E 3 2 J
n \ 9z
9x
2
9y
3 /n' JL (jL
9z
Ej)
\\ j
(3 .22)
/
The differential equations (3.17) become n 2 9 2 Ei "^it2"_AEi n2 9 2 E 2 c 2 9t2 n2 92E c 2 at 2
n' 9E 3 =
n' 9E 3 _AE2
=
(H
 A E , = 2 JL/nl ""i'ezl
n2 9 2 H t n' 9Hi n' 9H,  AH,1 + 2 „ = + 2——i 2, 2 c 9t n 9z n 9x n 2 9 2 H,
n' 9H,
(3.23)
n1 9H,
2 n 232 99t H2 3  AH, = 0 .
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 Ej, E 2 , H), 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 (j(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 /j 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, n, 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, n, 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, n; E, H and establish definite conditions for the electromagnetic field in this case. 4.1 Let us consider, in the fourdimensional x,y,z,t space, a domain D which is bounded by a closed threedimensional 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 N = \( = ax + by + f «Jn 2  a 2  b 2 Jo
df
(10.31)
is a complete integral of the equation 4>x2
+
0
y 2
+
4>z2
=
n
2
(z)
.
The light rays in such a medium thus are given by
5  .
v V x, etc. this becomes f j
F(xz 9t 9x dy riz
(11.11)
(11.12) '
y
can be interpreted as differentiation along a light ray provided that ¡/j(x,y,z) is a solution of the equation 4>x + 4>y2 + 4>Z2 = n 2 .
Let F(x,y,z) = ¡/)(x,y,z), for example. It follows U
=
+ 4'y2 +
= n2 .
(11.14)
The operator (11.12) can also be applied to a vector field; for example ¿(grad^)
=
= f grad n 2 .
(11.15)
11.2 We consider an electromagnetic field which is discontinuous on the hypersurface cp(x,y,z,t) = 4>(x,y,z)  ct = 0 .
(11.21)
40
M A T H E M A T I C A L THEORY O F OPTICS L e t D and D' be two domains of the a r e separated by the *'y'Z,t sPace hypersurface
e or
2 a , , =  — V • — grad ip e 9T
or, by (11.15) n2 R* =  ^ Vgrad n 2 , i.e., R+
=
_ 2 Vgrad n e n
Equation (11.33) becomes +  e d i v ^ grad
V + (v •
Finally we introduce the notation A€ip i.e.,
= e d i v ^ grad ipj ,
n
) grad i/> = 0.
(11.35)
MATHEMATICAL THEORY OF OPTICS
44 and we get the equation
grad n)grad l = 0 .
(11.37)
A s i m i l a r relation can be found for the discontinuity U by replacing e and V by n and U in (11.37). Thus our complete result is: The discontinuities U and V satisfy the differential equations: +  A^U
+ i ( U  g r a d n ) g r a d ifr = 0 , (11.38)
+
l>\ + ^ (V • grad n ) g r a d ¡/j = 0
where the differential o p e r a t o r s A t $ and A^ip a r e defined by A t4>
(7 *"). + (7 4 + (7 4 +
a *>\ a 4 it 4 §12. TRANSPORT OF DISCONTINUITIES.
+
(11.39)
(CONTINUED).
12.1 On a given light ray the equations (11.38) r e p r e s e n t a system of ordinary differential equations. Indeed, we have shown in (11.1) that the diff e r e n t i a l o p e r a t o r 9 / 9 t differentiates a function in the direction of a light ray. Let us introduce, in (11.38), instead of U and V, the v e c t o r s
1
rT
f T A ^ dr 2 0 P = U e Uo
V
(12.11)
0 Q = V e
Figure 18
ì
Ì\1'àr
which have the s a m e directions r e s p e c tively a s U and V, but different lengths. The differential equations (11.38) then a s s u m e an even simpler f o r m :
45
WAVE OPTICS A N D GEOMETRICAL OPTICS dP dr~
+
1 n ^ ' Sra
dQ 1 — +  (Q • grad n) grad tp = 0 . UT n
(12.12)
Since U and V are orthogonal to grad ip, 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 dT
=
^
dT
(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 P = Q = l and interpret P and Q as unit vectors which determine the directions of the vectors U and V. from
If P and Q have been found as solutions of (12.12), we obtain U and V
U = ü„ P e
(12.14) 1
V = V0 Q e
 J
r
T
Aei/)dr
These equations make it evident that U and V are zero on the whole light ray if they are zero on one particular point, r = 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 that the discontinuities U and V which represent the signal are different from zero only on a section r 0 of the wave front tp = c t 0 . From (12.14) it follows that, at a time t > t 0 , only on the corresponding section r of the wave front !/» = ct will discontinuities U,V be observed. Figure 19 This section is determined by the light rays
MATHEMATICAL THEORY OF OPTICS
46
through Tq . In other words, the light signal will be observed only in the part of the space which is covered by the light r a y s through To. 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 = Ai/)  ~{exipx
+ €yipy +
ez!Pz)
Aei/> = Aip  — Gog e) (12.21) AMi/i = A>p  — (log n)
Similarly
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 Ti and r 2 of the wave fronts ip = pj and ip
P2
and the cylindrical wall r 3 formed by the light r a y s through the circumference of r 2 and Tj. We apply the theorem of Gauss to this domain D: = />
///A* dxdy dz = / /  J d o D
Figure 20 where ^
dl>
(12.22)
is the derivative of ip in direction of the outside normals. However, f v  0 on r 3 , M ^ = n 2 on r 2 , M •f =  n. on Ti . dv
47
WAVE OPTICS A N D GEOMETRICAL OPTICS Hence, f f fAip dx dy dz = f f n 2 do 2  f f nj do t .
D
(12.23)
Tx
r2
We now express the surface elements do2 and do] by the corresponding surface elements do of an arbitrarily chosen wavefront ip = c t 0 . We write do2 = K 2 do , dot
(12.24)
= Kjdo .
The factor K measures the expansion of an infinitesimally narrow tube of light rays. It follows that
JJJW
dz
d
= ff(n2K2
 njKOdo =
r0
JJf * ^ r 0 T,
dr do
.
The volume element dx dy dz can be expressed as follows: dx dy dz = K do ds = nK do dr . Hence,
IIIm
dx dy dz = f f f  ± M
dx dy dz .
Since D is of arbitrary size, we find A!p
=
i
£
(nK)
=
ihlo&nK>
(12 25)

and hence e
=
a . nK ^ ^ T ' 9 ,
nK
(12.26)
48
MATHEMATICAL THEORY OF OPTICS
The exponential factors in equation (12.14) become
/n„K 0*^0
nK
f0 (12.27) 1
r
"o J
T
Ae^dT
fnp K o 6
TSc
o ,
and we conclude from (12.14): elul*
J£o foU02 no
(12.28)
K K That is, the quantities — € u2 and —/jv2 are constant along a light ray. This result allows us to determine the lengths of U and V along the light rays of a given set of wave fronts ip = c t without integration, simply by calculating the ratio = of corresponding surface elements of the K„ dor. wave fronts. 12.3 Energy and flux on a wave front. Let us assume that the electromagnetic field is zero on one side of the wave fronts tp  ct = 0. In this case E = U and H = V; hence W = ¿ ( e U
!
+ ( i V ! ) , S = ¿ ( U x V) .
(12.31)
It follows from (11.25) that fU! grdd i/r
= U • (V X grad ip) ,
/JV2 = (grad ¡p x U) V , and thus eU 2
Figure 21
= (i V 2
(12.32)
i.e. Electric and magnetic energies are equal on a wave front. From (11.25) it follows further that
WAVE OPTICS A N D GEOMETRICAL
1 U x V =  ( V x grad ip)
4n
49
OPTICS
1 •> x V = ^ f A T g r a d 4>
W grad ip
Hence, W grad ip .
S =
(12.33)
This yields f o r the absolute value of the Poynting vector: S =  W . n
(12.34)
This result allows us to interpret the equations (12.28). If we add both equations we find —W = — W,, and, introducing 6 K n n0 0  W do =  W n n0
do«.
= t ^ " . we obtain do 0
Finally, on account of (12.34): S do = S0 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) f o r the directions P and Q of the vectors U and V. Let us represent the light ray in the vectorial form X(t) =
(x(T).y(T), z ( t ) ) .
We have X = grad ¡p , (12.41) X = n grad n and we can write (12.12) in the form P + \(P'X)X
= 0, (12.42)
MATHEMATICAL THEORY OF OPTICS
50 Instead of the p a r a m e t e r t we dr light ray, by using the relation — = ds
introduce the geometrical length s of the 1  . We have n
• dX „, X = n — = nX' , ds X = n(nX') , P = nP', and hence, P ' + ^ ( P . ( n X ' ) 1 ) X' = 0
(12.43)
However, P(nX')'
= P • (nX" + n'X') = n ( P  X " )
on account of P • X' = 0. This yields P ' + (P X")X' = 0 , (12.44) Q' + (Q .X")X' = 0 .
and similarly
The equations (12.44) d e m o n s t r a t e that the two unit v e c t o r s P and Q a r e determined by the light ray alone. The s a m e light ray, of c o u r s e , can be an orthogonal t r a j e c t o r y to many different s e t s of wave f r o n t s . F o r example, in c a s e n = 1, a given straight line can be orthogonal to s y s t e m s of spherical wave f r o n t s o r to a system of plane wave f r o n t s . Equations (12.44) however, state that the v e c t o r s U and V a r e submitted to the s a m e rotation around the light ray no m a t t e r to which type of wave f r o n t s the light r a y is orthogonal. The wave f r o n t s influence only the size of the discontinuities U and V. 12.5 The tangential v e c t o r T = X' of the light ray and the v e c t o r s P and Q define an orthogonal s y s t e m of unit vectors which t r a v e l along the light ray X = X(s). In general the change of such a system along a curve i s determined by t h r e e kinematic formulae:
X = X (s)
Figure 22 T' =
cP
+aQ
P ' =  cT Q' =  bT
+bQ
aP
(12.51)
51
WAVE OPTICS A N D GEOMETRICAL OPTICS
The coefficients a,b,c are functions of s. In our special case these formulae reduce to T» =
*
cP
+bQ
P 1 =  cT
*
*
Q' =  bT
*
*
(12.52)
as is shown by (12.44). We have, incidentally, a = 0,
b = (QX"),
P(s)
X(sj
c = (PX") . 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 dff2 = n z (dx* + dyz + dz z )
(12.61)
52
MATHEMATICAL THEORY O F OPTICS
By introducing, on a light ray X(rr) the parameter cr = / n ds = J n 2 dr which measures the optical length on the ray we have the relations
dP _ _1_ dP der n 2 dT With this choice of the parameter the equations (12.12) become
(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 LevyCivita. The equations of the geodetic lines, i.e., of the light rays with cr as parameter, follow from (8.23) letting A = 1/n 2 and hence
We find
(12.63)
Let us denote temporarily the components of the vector X(cr) = (x,y,z) by Xj(o), X2(Px)U + Wy'  ¡Mgri
+
(^z'  4>z)hn = 0 .
These equations state that the vector grad tp'  grad ^ =

¡py'  ¡py; tpz'
 ¡pz)
is normal to the surface 2. If M is a unit vector in direction of the surface normal, we can write grad ;y  grad ¡!> = TM .
(14.15)
Grad and grad ii give the direction of the orthogonal trajectories of the surfaces ip' = const, and ip = const., i.e., of the light rays. Let T and T ' be unit vectors along the light ray; then grad ip = nT and grad tp' = n'T'. It follows that n ' T '  nT = T M ,
(14.16)
where r is a scalar factor. We conclude that the refracted ray leaves the surface X 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 0'; on the right side, n sin 0, where d is the angle of incidence and t?' is the angle of refraction. This yields Snell's law of refraction n' sin 0 ' = n sin d
(14.18)
MATHEMATICAL THEORY OF OPTICS
66
We find the factor r in (14.16) by forming the scalar product of M with (14.16): n ' ( T ' . M )  n(TM) = r , or T = n ' cos tf '  n cos d .
(14.19)
The relations n ' T '  nT = (n' cos i?'  n cos d )M , (14.191) n' sin tV = n sin d allow us to find T ' if T and M a r e 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 + >py +
= n2 •
We have seen in §9 that, on account of this, there exist two solutions tp which attain given boundary values on a given surface 2 • That a surface of discontinuity must actually produce a set of reflected wave fronts will be seen in the next section by deriving F r e s n e l ' s formulae. Let us here assume the existence of a reflected signal. This means that the characteristic hypersurface
it follows t>* COS
=
d
d* = v  a ,
,
0 * = cos 0 ,
or
cos 0 * =  cos d .
(14.27)
The first solutions give T* = T , i.e., the incident ray; the second solutions yield T*  T =  2 cos 0 M .
(14.28)
This is the reflected ray. T * and T are symmetrical with respect to the tangent plane of 2 at the point of incidence. 815. TRANSPORT OF SIGNALS IN MEDIA OF DISCONTINUOUS OPTICAL PROPERTIES. FRESNEL'S FORMULAE. 15.1 In 14.2 we have seen that, on a surface 2: w(x,y,z) = 0 on which the functions e and ¡i are discontinuous, a set of wave fronts ip = ct must be expected to split up into two sets of wave fronts; the transmitted wave fronts ip  ct = 0 and the reflected wave fronts ip*  ct = 0. The corresponding characteristic hypersurface then consists of three branches; the incident
68
MATHEMATICAL THEORY OF OPTICS
branch
)U0 . The two right sides of these equations are conjugate complex numbers. They must be equal because they are real as are the left sides. Hence eU 0 U 0 = mV0V0 .
(16.33)
The average energy density, according to §2, is given by W
=
lfc" ( e U °
+
• V0 ) •
(16.34)
We conclude: The average electric energy is equal to the average magnetic energy. Hence W = ¿eU0U0
= ¿nVoVo .
The average flux vector S is given by the formula (2.53): S =
(U0 x V0 + U0 x V0) .
From (16.28) it follows that e(U0 x V0) = (V0 x grad 0) x V0 , = V0 • V0 grad tp .
(16.35)
WAVE OPTICS A N D GEOMETRICAL OPTICS
77
Hence U0 x V0 + U0 x V0 =
grad 4> =
0
0
grad i>
and s
= W
MV0V0 grad *
or S = c
W
grad 0 .
(16.36)
We conclude that the vector S of average flux is normal to the wave fronts. Its absolute value is related to the average energy W by the equation S = J W .
(16.37)
16.4 We consider next the equations (16.29) in the case v = 1; and assume continuity of e(x,y,z) and n(x,y,z). We have grad J x V  +
eUj =  curl V0 , (16.38)
grad ip x Uj  ^tVj =•  curl U0 . These equations are formally identical with the equations (11.24) for [ E t ] and [H t ] in §11. By the same argument we conclude that they are solvable only if the right sides satisfy certain conditions. The former method of deriving these conditions can be repeated literally. We obtain the result: The amplitude vectors Uc and V0 satisfy the following differential equations along the light rays: ^
+
fA^U
0
+
( u
0
. ^ ) g r a d *
= 0, (16.39)
^
+
+
grad*
= 0.
Again the only difference is that U0 and V0 are complex vectors. 16.4 We solve the equations (16.39) as follows. We introduce 1
U0(T) = U0(O)Pe
rT A J,&r 0
1 rT   ¡ A ^ d r
,
V,(R) = V(0)Qe *
0
,
(16.41)
MATHEMATICAL THEORY OF OPTICS
78 where P and Q satisfy the equations
dP 1 — +  (P • grad n) grad 0 = 0 , dr n (16.42) dQ 1 — + — (Q • grad n) grad i p = 0 . UT n The three vectors P , Q and T = — grad t p form a system of unitary v e c t o r s along the ray, i . e . , complex vectors which satisfy the relations T •P = T •P = 0 ,
T•T = 1 ,
T = P x Q ,
P . Q = P  Q = 0 ,
P P = 1 ,
P = Q x T ,
QT
Q•Q = 1 ,
Q =
= Q•T = 0 ,
(16.43)
T x P .
By introducing on the ray the orthogonal system of unit vectors T,N (Principal normal), and S (Binormal), and the geometrical length of the ray as p a r a m e t e r , the equations (16.41) become P ' +  (N • P)T
0 , (16.44)
Q' +  ( N  P ) T
= 0 .
A complex vector P , normal to T , for which P  P = 1 can be expressed in the following form as a linear combination of N and S; P = cos A (N cos s
2^ =
— + cos Ô' n1 cos ô
•
Of course, if we consider a given light ray which passes the surface ß = 0 we cannot expect that the flux through surface elements of the transmitted wave fronts is the same as the flux through the corresponding surface elements of the incident wave front. Part of the incident flux follows the reflected light ray. One also verifies readily that the polarization of the light on the transmitted or reflected ray is not the same as on the incident ray.
81
WAVE OPTICS A N D GEOMETRICAL OPTICS
16.6 An optical instrument consists in general of several refracting surfaces which separate media of different refractive indices. Consequently, multiple reflections must be expected, such that in every part of the medium infinitely many different sets of wave fronts are to be considered. The electromagnetic fields in each of the media are therefore considerably more complicated than the field given in the formula (16.51); namely an infinite sum E = s U(y) e i k ( * V c , ) Figure 36
(16.61)
with terms related to the different wave fronts.
However, on each surface of refraction, the boundary values of these different terms can be divided into groups of three corresponding to an incident, transmitted and reflected set of wave fronts, for which ip = i * = 4>' on the refracting surface. The boundary values of E and H given by infinite series of the type (16.61) then can satisfy the conditions (6.1) of §6 only if the conditions (16.53) and (16.54) are satisfied by each of the above groups individually. This means that Fresnel's formulae can be applied safely at every single step of the multiple reflection in the instrument. In practice the internal reflections are seldom of interest but are even carefully eliminated by the absorbing walls of the objective. Only the set of wave fronts which consists of transmitted wave fronts alone is of prime interest. With the aid of Fresnel's formulae — or in case of continuous variation of n — with the aid of the transport equations (16.39) we are now in the position to construct the electromagnetic field U0, V0 for small wave lengths, if the paths of light rays in the instrument are known. The field, which by this procedure is obtained in the image space, represents the actual field to a close approximation. Serious deviation from the exact
Figure 37
lution, however, must be expected in the neighborhood of conjugate points, i.e., at points where the bundle of light rays from a point source P contracts to a narrow region. In a later chapter we shall see how one can use the present first approximation to construct a second approximation which is also valid with great accuracy in the neighborhood of conjugate points.
CHAPTER II. H A M I L T O N ' S THEORY O F GEOMETRICAL OPTICS
§17.
PRINCIPLES OF GEOMETRICAL OPTICS.
The deduction of the principles of Geometrical Optics from the electromagnetic differential equations might tempt us to consider these principles as inherently connected with the special form of wave optics which Maxwell's theory represents. Actually only a few of the general premises of the wave theory are essential for the deduction of the laws of Geometrical Optics. Mathematically, this follows from the fact that the same characteristic equation can be obtained from various forms of wave equations, that, in other words, the same laws for the propagation of discontinuities can be found for quite different forms of wave motion. This applies especially if the field of geometrical optics is limited to the construction of wave fronts and light rays. In the next chapters we shall be concerned with this special part of optics. The concept of the electromagnetic field or any form of wave motion does not play any part in our investigation; it would, in fact, have been entirely possible to develop this theory from its own premises. However, in order to develop the diffraction theory of optical instruments, we then would be forced either to develop an entirely different theory or to carry out similar considerations to those in Chapter I. 17.1 Huyghens' Principle. Huyghens' construction of wave fronts as envelopes of wavelets, i.e., of spherical waves, can be considered as the principle of Geometrical Optics which is most closely related to the concept of wave motion. In Chapter I this principle has been formulated mathematically as follows: If ¡/)(x,y,z)  ct = 0 represents a set of wave fronts then ¡/>(x,y,z) must be a solution of the differential equation 4>x2
+ ipy2
+ ¡pz2
= n2
.
(17.11)
The direct connection with Huyghens1 construction was given by the theorem: If ¡/)(x,y,z; a,b) is an integral of (17.11) which depends on two arbitrary parameters a,b, then the envelope of the wave fronts i p ( \ , y , z ; a,b)  ct is also a wave front i p ( x , y , z )  ct = 0 and ¡/>(x,y,z) is a solution of (17.11). Indeed, Huyghens' wavelets V(x,y,z; x 0 , y 0 , z 0 ) where (x 0 ,y 0 ,z 0 ) is a point on a surface r represent integrals of the above type. It can be seen quite easily that Huyghens' construction leads to the above mathematical formulation. Let us assume that light is a wave motion of finite 82
HAMILTON'S THEORY OF GEOMETRICAL OPTICS
83
velocity. A light signal is given at a point (x 0 ,y 0 ,z 0 ) at the time t 0 and has penetrated at a time t > t 0 into a domain of the space which is enclosed by a surface given in the form V(x 0 ,y 0 ,z 0 ; x,y,z) = c(t  t 0 ) .
(17.12)
This set of wave fronts has a twoparameter manifold of orthogonal trajectories. We measure the velocity of the disturbance by the velocity of the wave fronts along the orthogonal trajectories. For two points P, and P 2 on such a trajectory we have V(x 0 ,y 0 ,z 0 ; x 2 , y 2 , z 2 )  V(x 0 ,y 0 ,z 0 ; x ^ . z , ) = c(t 2  t,) or in differential form
V Figure 38
Vx dx + Vy dy + Vz dz =• c dt ;
i.e., V V * dt
+
vVy ^ + V — = Cc Vz dt dt •
(17.13)
The velocity of the point P, along the trajectory is given by the vector /dx djf dz \ { dt ' dt ' dt / : dx dt
V. VV x 2 + Vy2 + Vz' dz dt
V„
dv dt
7V X 2 + Vy2 + Vz2 V.
(17.14)
V V X 2 + Vy2 + VJ
where v is the absolute value of the velocity. By introducing these expressions in (17.13) we find yvj
+ Vy2 + Vz2 = z * v
(17.15)
We finally assume that the medium is isotropic: the velocity v at the point (x,y,z) shall be independent of the direction of the particular trajectory, and also independent of (x 0 ,y 0 ,z 0 ), the point where the light signal was released We thus introduce ^ = n(x,y,z)
(17.16)
84
MATHEMATICAL THEORY OF OPTICS
and obtain Vx2 + Vy2 + Vz2
= n 2 (x,y,z) .
(17.17)
This means that V(x 0 > y 0 ,z 0 ; x,y,z) is a solution of (17.17) which depends on the arbitrary parameters x 0 , y 0 , z 0 . Huyghens' Principle, that all other wave fronts ¡/)(x,y,z)  ct = 0 can be found as envelopes of wavelets V(x,y,z; x 0 , y 0 , z 0 ) , then characterizes ip(x,y,z) also as a solution of this partial differential equation (17.17). We remark explicitly that, with this interpretation of v(x,y,z) as the velocity at points on the wave fronts, the velocity is smaller in an optical medium of greater n. 17.2 Light rays as paths of corpuscles. We have introduced the light rays of a medium as orthogonal trajectories of the wave fronts and we have found that it is possible to characterize the light rays, independently of the concept of wave fronts, as solutions of a set of ordinary differential equations, namely as those solutions of the system:
¿(H y = sr.
on

< 17  21 >
¿(H which satisfy the condition x 2 + y 2 + z 2 = n2 .
(17.22)
These equations allow us to employ a radically different interpretation of the phenomenon of light. By considering the parameter r as a time parameter we find that the light rays are nothing but the paths of corpuscles which move in a potential field = n 2 with a velocity w = n. This interpretation, however, forces us to admit that the velocity of the corpuscles is greater in the medium of greater n contrary to Huyghens' interpretation. The first derivation of Snell's Law was given by Descartes on the corpuscular basis by using the principle that the tangential component of the velocity is unchanged when a medium of different refractive index is entered. This means that the difference w '  w of the two velocity vectors w ' and w on the boundary of the medium has the direction of the normal unit vector
85
HAMILTON'S THEORY OF GEOMETRICAL OPTICS
M. Hence w1 = w + r M ; or, since w' = n'T' and w = nT on account of (17.22), where T and T ' are unit vectors: n ' T ' = nT + TM
Figure 39
.
(17.23)
This we recognize as our former formulation (14.16) of Snell's Law.
17.3 This second interpretation of Geometrical Optics is the one which we have to adopt in Electron Optics. If a charged particle moves in an electrostatic field of potential y 0 ,xi,yi; z) , (19.26) p = p ( z 0 , z j ; x 0 ,y 0 .xi.yi; z ) . q = q ( z 0 , z i ; x0yB,xt,yl;
z) ,
which represent the light ray which goes through the point x 0 , y 0 , z 0 object plane, and the point Xj.y^Zt of the image plane.
of the
HAMILTON'S THEORY OF GEOMETRICAL OPTICS
97
These functions now are introduced in the integral for the optical path zi
V
= J
V
= V(z 0 ,z,; x 0 jr 0 ,x 1 ,y 1 )
(xp + yq  H) dz ,
(19.27)
so that a function (19.28)
is found which determines the optical distance between the points (x 0 > y 0 > z 0 ) and (xt.y^z,). 19.3 Our aim is to show that the total differential dV of this function has the form dV = p,dxt + q,dy,  p 0 dx 0  q 0 dy 0  H,dz! + H 0 dz 0 ,
(19.31)
where the coefficients pj.qj; Po.qo a r e functions of (z 0 ,z,; Xg.yo.Xi.y,). From these functions the desired "image functions" (19.24) can be found by elimination. By Hj and H 0 we denote the expressions Hi
=
H(Xjjtj.Zi;
p,,qj) , (19.32)
H 0  H(x 0 ,y 0 ,z 0 ; p 0 ,q 0 ) . From (19.31) it follows that
av
Po =
9x0 '
av 9y0 '
qo
9V
Pi
9x, '
Qi
9V 9yi '
av
H, =
H0 =
9z (
av +
(19.33)
dz0
The first and second column of equations are equivalent to the equations (19.24) and demonstrate the fact that the four functions (19.24) can be found from one function V(zq,Zj; xq,yg, x],yj) by differentiation and elimination. The last column of (19.33) represents two partial differential equations for V which we obtain by introducing the partial derivatives of V for p 0 , q 0 , pj, q t in H0 and H t . We find 9V
dz0
_ u/v 9V_ 9V_\ V ° , y o , z ° ; " 9x0 ' " 8y 0 / '
=
9V _
—
/
9V
9V \
  H ^ j t l z , ; g ^  . a^J •
(19.34)
MATHEMATICAL THEORY OF OPTICS
98
19.4 The proof of the above theorem is not difficult. First we consider the integrand c/= xp + yq  H
(19.41)
of (19.27). This becomes a function of (ZQ.ZJ; x 0 , y 0 , xj.yj) and z if the functions (19.26) are introduced. Let
By analagous methods in case a  z 0 , we obtain from
av
the relation  — = + H 0 , which, together with (19.48) and (19.47), represents 8z.o the hypothesis (19.33). 19.5 In the case of the optical Hamiltonian H = ,/n 2  p2  q 2 , the function VizQ.Zj; x 0 ,y 0 ; Xj.y^ is identical with Huyghens' wavelet function. The surfaces V = const, are the spherical wave fronts around the object point x 0 , y 0 , z0. In our present interpretation it measures the optical distance
MATHEMATICAL THEORY OF OPTICS
100
of two points, one of which is lying on the object plane, the other on the image plane. V is completely determined by the instrument and the position z 0 , z ( of the two planes. Hence it is called the Characteristic of the instrument; especially the Point characteristic, since it is a function of two points. If the point characteristic V for a given pair of planes z 0 and z ( is known, we can find the image functions (19.24) by elimination from the formulae Po = Pi
=
q
=
°
qi
v
v
x„(zo.zi;
Xl
=
"
= q t , and xt = x 0 ,
aw_ y°
'
which yields (22.12). A solution W ( z 0 , z j ; x 0 ,yQ.p^qj) with these boundary values can be found easily. We introduce a function of the type W = a(z 0 )  x 0 p t  y 0 q,
(22.13)
in (22.11) and find the condition a'(zo) =  V n 2 ( z 0 ) 
Pl2
 q,2 ,
(22.14)
i.e., a (z0)
z°
=  J Zl
/
y n 2 ( z )  p,2  q,2dz + a(z,) .
(22.15)
The boundary condition (22.12) requires a(z t ) = 0; hence zi W = / v n 2 (z) zo
Pl2
 q,2 dz  x o P l  y 0 q,
(22.16)
is the desired solution. The image functions of our instrument are determined by the general equations (20.18). The result is: . rz' dz = x 0 + pi J — — ^ ^ z z z z z z z r . Po = Pi z ° V^n^z)  p,2  qi2 dz r az yi = yo + Qi J . 2 Zo > ( z )  P l 2  q,2
. ao = Qi
(22.17)
MATHEMATICAL THEORY OF OPTICS
118
The equation (22.16) is also valid in case n is discontinuous. For example, in the case of a system of parallel plates of thickness i i and index n( we find W = £ f t V n i 2 " Pi2 " Qi2 1= o  (xoPl + YoQi) • (22.18) This last result makes it obvious that the optical coordination of rays is completely independent of the order in which the plates are arranged.
Figure 60
We apply our result to the following problem. Let us consider a stratified medium such that n(z) is different from 1 only in the region Lo = z = L t . Outside the region we have n(z) = 1.
Figure 61 its intersection with the plane z
xi = Po
yi = qo
We consider all the light rays which originate at the point x 0 = y 0 = 0 of the plane z 0 and determine the c o r r e sponding rays after passing the region L 0 £ z £ L ^ The equations (22.17) demonstrate that the refracted ray is parallel to the incident ray: Pi = Po5 n 0 and a hyperbola, if nj < nq•
(23.36)
APPLICATION OF THE THEORY
133
T h e axes A and B of the ellipse, in case n, > n 0 , are given by .2 \NI + N0J
'
B2
= a2
and the center M and the eccentricity e = •S/A2  B 2 M = an, + nn
!
(23.37)
n, + n, by
Ì \ n( t — + nOJ
(23.371)
It follows that M + e = a t . Hence the point a coincides with the second focal point of the ellipse.
Ml
Figure 70
Similar formulae can be found f o r the hyperbola in case n( < n 0 . We have A 2 = a2 P M 2 . \nl + n o / '  "l n, + n„
n0
(23.38) M
= a
n, + nn
A 2 + B2
\«1 + n 0/ Figure 71
Hence M + e = a, i.e., the point a coincides with the focal point of the hyperbola, which is enclosed by the other branch. 23.4 Virtual conjugate points. If a bundle of rays, originating at a point P 0 , is transformed by refraction into a bundle such that the backwards extensions of the rays intersect at one and the same point P j , then we call P j a virtual conjugate point to P 0 . In order to determine a refracting surface w = 0 to which two given points P 0 and P , belong as virtual conjugate points we consider the diverging spherical waves
134
MATHEMATICAL THEORY OF OPTICS tp
= n0 V ^ 2 + y2 + z2
ip'
= n, V x 2 + y 2 + (z  a) 2 + C
(23.41)
in the two media separated by the surface u> = 0. The condition of continuity on w = 0 yields immediately the equation n o y x 2 + y 2 + z 2  n , , / x 2 + y 2 + (z  a) 2 = C .
(23.42)
By choosing the vertex of the surface at an arbitrary point z = A > a of the axis, we obtain n 0  / x 2 + y 2 + z 2  n l v / x 2 + y 2 + (z  a) 2 = n 0 A  n,(A  a) , (23.43) i.e., a surface of revolution generated by the curve n 0 > / x 2 + z 2  n l v / x 2 + (z  a) 2 = n 0 A  n,(A  a) .
(23.44)
Elimination of the radicals leads again to an algebraic curve of the fourth order. We notice that the same algebraic equation is obtained from
and
n0Vx2 n
+
z 2 " n,v^x 2 + (z  a) 2 = C
( h / x 2 + z 2 + n,^/x 2 + (z  a) 2 = C .
(23.45)
This demonstrates that our present problem is solved by surface sections which belong to the same algebraic surfaces as in the case of the problem (23.2). It is however another branch of these surfaces which we have to use now. An example is illustrated in Figure 72 for the case: no = 1, nj = 3/2; a = 1/2; A = 1. The construction is carried out similarly as in 23.2 with the 2 1 aid of two circles of radius r 0 and r, = — r 0  — about the points z = 0 2 o 6 and z = —, respectively.
APPLICATION OF THE THEORY
135
23.5 If the object is at infinity, i . e . , if p a r a l l e l wave f r o n t s have to be t r a n s f o r m e d into diverging spherical wave f r o n t s , we obtain by our principle of continuity the equation n0z  n ^ x
2
+ y 2 + (z + a) 2
=  n^
.
(23.51)
We take the point P ( at z =  a with a > 0. The curve n 0 z  njv/x 2 + (z + a) 2
=  n(a
(23.52)
which generates the s u r f a c e of revolution (23.51) is an ellipse in the c a s e n) > n j and a hyperbola if n ( < n 0 . The equation of the conic is given by
(\ z +, a a2
\
+ no/ T ( \ y n i + "oy
a2
r — "1  "0 n i + n0
= 1 .
(23.53)
which d i f f e r s f r o m (23.36) only in the quantity a which is replaced by  a. Consequently, the axes A, B and the eccentricity e a r e determined by the f o r m u l a e (23.37) and (23.38). The c e n t e r M i s found by replacing a by  a in these formulae, i.e., M =  a
"l n, + n 0
.
(23.54)
The point z =  a is one of the focal points as is illustrated in Figures 73 and 74.
Figure 73
Figure 74
136
M A T H E M A T I C A L THEORY O F OPTICS
23.6 The aplanatic points of a sphere. If the vertex A in (23.44) is chosen so that the right side is zero, i.e., if A =
"l
(23.61)
"l
then the curve (23.44) is a circle and therefore the surface (23.43) is a sphere. We find from (23.43) and (23.61): x2 + y2 + / _ V " n, + n 0
A\
2
J
=
( n0 y \n, + n0/
(23.62)
The sphere has its center at the point •l A < A n, + n 0
(23.63)
its radius is Figure 75
R =
n, + n„
A < A
(23.64)
This shows that the spherical surface is concave. Let us now consider a given concave spherical surface of radius R, which separates two optical media of index ng and ni, respectively. The center of the sphere is taken at z = 0. From the above it follows that there exist two points z = z 0 and z = zt on the zaxis such that a spherical wave from z q is transformed into a spherical wave diverging apparently from z t . In other words z 0 and z ( a r e virtual conjugate points. One calls these points the aplanatic points of the sphere. In our above notation we have z 0 =  m and zl follows from (23.61), (23.63), and (23.64) that
m. Hence it
n i z0 =  — R ,
(23.65)
APPLICATION OF THE THEORY
137
The point z 0 is called the aplanatic object point and z1 the aplanatic image point. The location of these points for the two cases n! > n 0 and < n0 is shown in Figures 77 and 78.
z0
Figure 77
Figure 78
3y reversing the direction of the light in the above figures, we see that convex spherical surfaces can be used to transform converging spherical waves into perfect spherical waves of the same type but with a different point of convergence. On account of the symmetry of the sphere we have not only two aplanatic points but infinitely many, located on two concentric spheres of radius n n i o — R aid — R, respectively. We can use this result for a simple graphical n0 n, method of constructing the refracted rays on a sphere if the incident rays a r e given. Let us consider a convex spherical surface S and let n, > n 0 . We nt n0 constrict the two aplanatic spheres of radius — R and — R. We extend the n0 nt incideit ray until it intersects the aplanatic object sphere at A 0 . We know that al. incident rays aiming towards A0 must be refracted towards the conjugate aplanatic point A1# We find this point by connecting A 0 with M. This n0 line intersects the aplanatic image sphere of radius — R at the point A ( . n i Hence PAj is the refracted ray. "he aplanatic points of a sphere are used in the construction of m i c r o scope >bjectives of high aperture. The object is submerged in oil which has nearly the same refractive index as the front lens of the objective. Hence we may cmsider the object as being located in a medium of an index of refraction greater than one. The front surface of the objective can be chosen arbitrarily but is n general made plane for practical reasons. The back surface of the front l;ns is chosen in such a manner that the object point P 0 coincides with
MATHEMATICAL THEORY OF OPTICS
138
the aplanatic object point of this sphere. The result is that a wide bundle of rays with a numerical aperture sin 0Q almost equal to 1 leaves the front lens without any spherical aberration but with a smaller aperture sin 0 t = l / n ( . The next surface is chosen concentric to the point P ( f r o m which this bundle seems to diverge such that the bundle enters the glass again without spherical aberration. We then insert again a spherical surface with Pi as the aplanatic object point. Hence the bundle
Figure 79
leaves this surface with decreased aperture sin 8
—  — . By this conn, n 2 struction one is in the position to decrease the steepness of the bundle without introducing spherical aberration until finally one or two cemented lenses transform the resulting bundle into a converging bundle. This succession of aplanatic lenses explains the characteristic construction of a microscope objective as shown in Figure 80. In practice this construction is not carried out rigorously. However the departures from the above principle a r e never considerable.
Figure 80
APPLICATION OF THE THEORY
139
§24. FINAL CORRECTION OF OPTICAL INSTRUMENTS BY ASPHERIC SURFACES. The usefulness of single a s p h e r i c s u r f a c e s of the type discussed in §23 is v e r y limited in p r a c t i c e . It i s t r u e that we a r e in the position to eliminate one a b e r r a t i o n with these s u r f a c e s , but the effect on other a b e r r a t i o n s is not favorable. In other words, if we study the r e f r a c t i o n of spherical wave f r o n t s which originate at a point (x 0 , y 0 ) of the object plane different f r o m x o = yo = we would find g r e a t d e p a r t u r e s f r o m perfectly converging spherical waves a f t e r r e f r a c t i o n , even if the s u r f a c e produces a perfectly s h a r p image of the point x 0 = y 0 = 0. However, an a s p h e r i c s u r f a c e can be used successfully in combination with spherical s u r f a c e s in o r d e r to eliminate the l a s t t r a c e of a remaining spherical aberration, f o r example. Let us a s s u m e that an optical system of spherical s u r f a c e s has been found which is nearly c o r r e c t in the sense that a considerable part of the object plane is sharply imaged if the system is used with small a p e r t u r e s . The use of l a r g e a p e r t u r e s is prohibited by a rapidly i n c r e a s i n g spherical a b e r r a t i o n . If it is then possible to eliminate this a b e r ration by replacing the last spherical s u r f a c e of the system by an a s p h e r i c s u r f a c e which departs only slightly f r o m the original spherical s u r f a c e , we can expect that no detrimental effect will be introduced with r e g a r d to the a b e r r a t i o n s of oblique bundles of r a y s . This p r o c e d u r e can be applied s u c cessfully, f o r example, in c e r t a i n simple types of photographic objectives known as Cooke Triplets. These objectives consist of two positive lenses with a negative lens between them. It is easily possible to find combinations of this type which would produce excellent images if the objective is used at a "speed" not g r e a t e r than F: The design of such objectives f o r g r e a t e r speeds however is difficult because of the spherical a b e r ration which rapidly a s s u m e s l a r g e values. On the other hand, it is of c o u r s e d e s i r a b l e d f o r the photographer to have an objective of as g r e a t a d i a m e t e r a s possible, in o r d e r to d e c r e a s e the time of e x p o s u r e . Hence we a r e led to the attempt to prevent the Figure 81 rapid i n c r e a s e of spherical a b e r r a t i o n by employing an aspheric s u r f a c e a s the boundary of the last lens. t T h e "speed" or Fnumber of a photoobjective is defined by the ratio f:d of focal length to diameter of the front lens.
MATHEMATICAL THEORY OF OPTICS
140
24.2 In principle the solution of the above problem is simple. Let us consider the spherical wave fronts originating at the point P ( on the axis. We remove the last surface of the instrument and determine the wave fronts of the bundle through P 0 in the still unlimited glass medium of the last lens. Let n be the index of refraction of this medium. These wave fronts are given by the function V = V(P 0 ; x,y,z) = ct ,
(24.21)
where V is nothing but Huyghens' wavelet function or Hamilton's point characteristic for the points P 0 and a point (x,y,z) in the glass. Let us assume that this funcp^.y.z tion is known. The problem is % A P] to determine a surface u>(x,y,z) = 0 such that, by refraction on this surface, the wave fronts V = V(x,y,z) are Figure 82 transformed into wave fronts / >
4>
1
.
C V x 2 + y 2 + (z 
Zl)2
(24.22)
which converge towards the point P t = (0,0,z t ). The condition of continuity of V and ip on w = 0 yields the equation of the aspheric surface directly: V(x,y,z) + x/x2~ + y 2 + (z  z,) 2 = C .
(24.23)
The constant C is determined when the vertex A of the aspheric surface has been chosen. We have C = V(0,0,A) + z,  A ,
(24.24)
and hence the equation V(x,y,z) + 7 x 2 + y 2 + (z  z t ) 2 = V(0,0,A) + z,  A .
(24.25)
It is quite clear that V(x,y,z) depends only on p = ^/x 2 + y 2 and z if the optical instrument is symmetrical with respect to the axis. In this case (24.25) represents a surface of revolution generated by the curve V(x,0,z) + 7 x 2 + (z  Zj)2 = V(0,0,A) + z,  A.
(24.26)
24.3 In general it is not easy to find explicit expressions for the function V(x,y,z). Therefore, another procedure of finding the surface w = 0, in which Hamilton's mixed characteristic W(z 0 , z; x 0 , y 0 , p,q) is used, is preferable. The surface, w = 0, is then obtained in parametric form. We have the relation
141
APPLICATION OF THE THEORY V = W + xp + yq
(24.31)
where W is a function of (z,p,q) since ( x 0 , y 0 , z 0 ) are considered as given. From 9 W
—
= yn. / n 2  p22  q 2r ,
(24.32)
it follows that W = W0 + z , / n 2  p 2  q 2
(24.33)
in which W 0 = W 0 (p,q) is independent of z. Furthermore, we have aw 9p
y

Ë^L 0q
"" 3p
aW
0
9q
+ zyn2 +
_ „2 p' _ „2 q'
(24.34)
z
y ^ T T r .p 2 _ n q2
Hence we obtain V = W0  p^f
 a
z
,
(24.35)
= C .
(24.36)
p2  q 2
as a function of p, q and z. We finally introduce (24.35) in (24.23) and get aw 0 » w0  p  r —  q  r — + Z Bp ^ 8q •v/n2  p 2  q 2 + \ / x 2 + y 2 + (z  z ( ) 2
The equations (24.34) and (24.36) represent two linear and one quadratic equation for x,y,z as functions of p and q. The solution gives the surface w = 0 in parametric representation. Let us now assume that our instrument is symmetrical with respect to the zaxis. The surface w = 0 is a surface of revolution, and it is sufficient to determine its cross section with the xzplane. We thus assume y = q = 0 in the above formulae and obtain
MATHEMATICAL THEORY OF OPTICS
142
W„(p)  p W0'(p) + z
+ y x 2 + (z  Z))2
= c , (24.37)
x =  W0'(p) + z i.e., one quadratic and one linear equation for x and z as a function of p. The functions x(p) and z(p) give the curve in parametric form. It is not difficult to determine the function W 0 (p) in any actual case. Let us choose the vertex of the aspheric surface as the point z = 0. By tracing rays from P 0 through the system we are able to calculate the intersection Xj with the plane z = 0 and the direction p of a ray at z = 0. By correlating the results Xj and p for a number of rays, we obtain a function x ( = x t (p) which can be fitted, for example, to a polynomial Figure 83 Xl
(p) = p(A0 + A,p2 + A 2 p 4 + ...) .
(24.38)
If we let W0(O) = 0, it follows from x, =  W0'(p) that W 0 (p) =  /
rp o
= P
x,(p) dp
+
At T p
+
A2 1TP
+
(24.381)
\ •••)•
The equations (24.37) become rp  J Xi(P) dp + pxi(p) + z
•v/x2 + (z  z,) 2
= C (24.39)
x = x,(p) + z Let us develop x = x(p) and y = y(p) in power series x = p(B 0 + Bip 2 + ...) , y =
C 0 + C l P 2 + ....
(24.391)
The coefficients of the power series can be readily determined from (24.39).
APPLICATION OF THE THEORY
143
24.4 Instead of the characteristic W we can also use the angular characteristic T. This is especially advantageous if the point P 0 is at infinity as in a photographic objective. We have V = T + xp + y q  x„p 0  y 0 q 0 , and hence, since p 0
= q0
= 0, = T + xp + yq
V
.
(24.41)
We can repeat the same considerations for T as for the case of W. The result is formally the same as before. The cross section of the aspheric surface with the xzplane is given by the equations
T 0 (p)  pT 0 '(p) + z
+ (z  Zi)2
= C ,
•v/n'  p^
(24.42)  T 0 '(p) + z y ^ V
2
In order to find T 0 (p) a number of parallel incident rays have to be traced through the system. Let X! and p be the intersection and direction of a ray at the plane z = 0. We obtain Figure 84
X
y
T 0 =  / P x , ( p ) dp ,
(24.43)
and hence the same formulae as in (24.39).
24.5 We shall apply the preceding methods to a number p 0 of simple examples. A lens of index of refraction n has to be found which consists of a plane surface and an aspheric s u r face. It is required that a Figure 85 spherical wave from a point P q be transformed into a plane wave after refraction. Let a be the distance of P 0 from the plane surface and t the thickness of the lens on the axis. a
t f
/
MATHEMATICAL THEORY OF OPTICS
144
The characteristic W(p,z) in the medium n is given by W = ay/l  p 2 + z^/n2  p 2
(24.51)
as we have seen in §22.1. The point characteristic V follows from V =
+ z•
Wpf^ 3P
n,2'
(24.52)
as a function of p and z. Since V = a constant represents the wave fronts in the medium n, and tp = z + C represents the required wave fronts in the final medium, we can apply the condition of continuity and find V  z = C
(24.53)
as the equation for the aspheric surface. With the aid of (24.52) this yields a

y r
Z
1

n,2'
\
v V
 p2
=c .
(24.54)
3W From x =  r— it follows that 3P ap
(24.55)
From (24.54) and (24.55) we find x = x(p) and z = z(p), i.e., the cross section of the aspheric surface in parametric form. The result is C n
2. y
n
2 .
p2 (24.56)
c  1 ,/n 2  p 2 The constant C is given by the condition that z = a + t for p = 0. We obtain C = n a + (n  1) t
(24.57)
APPLICATION OF THE THEORY
145
24.6 As the next example we consider a lens which consists of one spherical surface and one aspheric surface. It is required to find a lens which transforms a plane wave into a converging spherical wave. Let us assume that the center zl of the sphere is at z = 0. The • angular characteristic T(z,p) in the case where p 0 = 0 is ^R given by T = r V i + n 2  2,/n 2  p2
Figure 86
+ z^/n2  p2
(24.61)
which follows from (22.3). If z{ is the desired focal point, we obtain directly by (24.42) the equation of the aspheric surface: T„(P)  pTo'(p) + z . " y n 2  p2
+ 7 x 2 + (z  z,) 2
= C (24.62)
x =  T 0 '(p) + z p where T 0 (p) = Rv/l + n2  2,/n 2  p2 .
(24.621)
These equations can be simplified by introducing p = n sin a
(24.63)
and letting R = 1. With this we can write (24.62) in the form n(x sin a + z cos a) +
x2 + (z  z t ) 2 = C  \f\ + n 2  2n cos a , (24.64)
n(x cos a  z sin a) =
n sin a V l  n 2  2n
COS ft
The constant C is given by the position z = t  1 of the vertex of the aspheric surface. Letting a = x = 0 ; z = t  l i n the first equation of (24.64), we find C = (n  l)t + zi .
(24.65)
The formulae (24.64) allow us to find, without difficulty, the coefficients of the development of the functions x = x(a) and z = z(a) in powers of a .
MATHEMATICAL THEORY OF OPTICS
146
24.7 Schmidt Camera. A combination of an aspheric surface and a spherical m i r r o r characterizes the construction of the astronomical camera which is known as the Schmidt c a m e r a . This c a m e r a , f i r s t constructed by B. Schmidt in 1930, has become justly famous in recent y e a r s . This fact can be readily understood if one considers e a r l i e r astronomical c a m e r a s of high speed which could be used f o r angular fields of only a few degrees. With Schmidt's construction however, we can photograph an angular field of more than 15°. The fundamental principle of this c a m e r a can be derived as follows. Let us consider a spherical m i r r o r of radius R = 1 and of large angular opening 0O. A bundle of rays parallel to the zaxis is r e flected at the m i r r o r . Consider a ray of this bundle incident at the angle d with the zaxis. After reflection it intersects the zaxis at the point
z
Figure 87 AZ = Z
" P
cos 0S  a cos 2 0 ]
+
"
0, we obtain a concave paraboloid of revolution. The reflection on such a surface is determined by the characteristic 1 (Pi  Po) 2 + (Qi  q 4  z i s j ) ^ o + x ( s 0 ) p 0 + y(s0) q0 + z(s0)r0
(26.34)
We now consider the problem of variation of determining a stationary value of (26.34) with r e g a r d to the functions x(s), y(s), z(s) which s a t i s f y the condition z(s) = f ( x ( s ) , y(s), s ) .
(26.35)
26.4 Our aim is to show that the problem can be t r a n s f o r m e d into the problem (26.21) by the t r a n s f o r m a t i o n of F r i e d r i c h s . F o r this p u r p o s e we introduce four Lagrangian multipliers p(s), q(s), r(s), and A(s) ,
f CourantHilbert, Methoden der Math. Physik, Vol. 1, 2nd Ed. pp. 199209.
(26.41)
APPLICATION OF THE THEORY
161
and write (26.34) in the form
T
=
f
s, r
nvV
+ \(f  z)j
+ v 2 + w2 + p(x  u) + q(y  v) + r(z  w) dz + x(sq)pq + y ( s 0 ) q 0 + z ( s 0 ) r 0  x(s,)p!
 y(s,)qi  z ( S l ) r ,
(26.42)
in which x(s), y(s), z(s), u(s), v(s), w(s), and X(s), p(s), q(s), r(s) are now treated as variables in the variation problem. By carrying out the variation with regard to these functions, we obtain the conditions
^ / u 2 + V2 + w 2
 p = 0 ,
" q + v ' + w1
Xfx  p = 0 ,
= 0,
Af y  q = 0
 \  r = 0 (26.43)
and
P(s0)
= Po .
P(si)
= Pi .
q(s0)
= q0 .
q(st)
= Qi .
r(s0) = r 0 ,
r(s,) = r , ,
x  u = 0 , y  v = 0 ,
z  f(x,y,z) = 0
.
(26.44)
z  w = 0 , We can impose any of these conditions upon the problem (26.42) without influencing the solution since the solution of (26.42) must necessarily satisfy all conditions (26.43) and (26.44). If we impose the conditions (26.44), we obtain our original problem (26.34). The equations (26.43) then become the Euler equations of the problem (26.34). If, however, the conditions (26.43) are imposed upon (26.42), we obtain a new problem of variation which has the same
MATHEMATICAL THEORY OF OPTICS
162
extremum V as the problem (26.34). The Euler equations of this problem are precisely the equations (26.44), i.e., the conditions which were formerly imposed. This transformation of the original problem is the transformation of Friedrichs. We consider first the integral + \ i + w'  pu  qv  rw
ds .
(26.45)
By introducing p,q,r from (26.43), we obtain the value zero for this integral. The integral rs' J s (px + qy + rz) ds o
(26.46)
can be transformed into  fs
0
(xp + yq + zr)ds + [xp + yq + rzj
s0
(26.461)
Hence it follows, with the aid of the last three rows of (26.43), that the expression (26.42) assumes the form: T =  Jg
[xp + yq + zr  X(f  z)] ds .
(26.47)
We finally introduce X
=
r,
p =  r fx,
q =  r fy,
and obtain »1 T = J r (x f x + y f y  f) ds , 'So or with the aid of the function ÏÏ
(26.471)
,s) defined by (26.22):
,si T = J/ S l r a i  ? , Sn \ r
à)ds. r /
(26.48)
The functions p,q,r which are admissible in (26.48) must satisfy the relation p 2 + q 2 + r 2 = n2
(26.481)
APPLICATION OF THE THEORY
163
and the boundary conditions P(s 0 ) = Po .
P(si) = Pi ,
q(s 0 ) = q 0 .
q(si) = qi .
r(s 0 ) = r 0 ,
r(s,) = r , ,
(26.482)
as follows from (26.43). This however is our former variation problem (26.21) for the function T. It follows from the general theory of the above transformation that the stationary value of (26.48) is a maximum if the stationary value of (26.34) is a minimum and vice versa. 26.5 Systems of spherical surfaces. In the case of spherical surfaces z = a.  R i  x / 1 
x2 + y2 p 2
(26.51)
we have obtained, for fij (£,77), the expression «1(4,1) =  a , + R i v / l + i 2 + r , 2
(26.52)
and thus T,
=  a , ( r ,  Ti_i) + R , sign (r,  r ^ O ^ / i p ,  p
) 2 + (q t  q ii ) 2 + (r ,  r ,  , ) 2 .
By introducing the quantity K! = R1 sign ( r (  r , . , ) = R i S i g n ( n ,  n ,  ! )
(26.53)
we are led to the problem of variation of finding a stationary value of the sum k T = £
[K, T i p ,  P,1> 2 + (q,  q ,  , ) 2 + ( r ,  r ,  , ) 2
1=1
 a,(r,  r .  o j
(26.54)
under the conditions that Pi 2 + qi 2 + r , 2
=
ni2
,
and that p 0 , q 0 , r 0 ; p k , q k , r k have given values.
(26.541)
MATHEMATICAL THEORY OF OPTICS
164
For a continuous set of spherical surfaces
z = a(s)  R(s)
1 
x2 + y2 R (s)
(26.55)
the problem is to find a stationary value of the integral
T = /So'
[K(s)^Tqr
r2
 ai
ds
(26.56)
where (26.561)
p 2 + q 2 + r 2 = n 2 (s) , and where p(s), q(s), r(s) have given boundary values P(s 0 ) = Po .
P(Si) = Pi
q(s 0 ) = qo .
q ( s i ) = 9K
(27,12)
165
APPLICATION OF THE THEORY
where a is an arbitrary constant. We can easily find such a set of solutions by introducing in polar coordinates in (27.11) x = r cos 9 y = r sin 6
(27.13)
.
It follows that: 4>r + 72 ¡¿V = H »
(27.14)
which possesses the solution 4> = K0 ± J r y / n 2 "
Figure 108 Z =
Y =
2y 1 + r'
r2  1 r2 + 1
(28.33)
We can consider these formulae as a parametric representation of the unit sphere with the coordinates x,y as parameters. The line element of the sphere in these parameters is given by d s ' = dX2
+
dY2
+
dZ 2 = 4 ( d  ^ )
2
+
4 ( d ^ )
2
+
( d ^ )
2
which gives ds 2 =
2
(dx2 + dy 2 ) .
(28.34)
(1 + r ) We recognize that ds
has the form of an optical line element ds 2 = n 2 (dx 2 + dy 2 )
where n =
w is identical with the function n(r) which characterizes 1 + r Maxwell's fisheye with the exception of an insignificant factor. The light rays in this medium therefore must be those curves which are stereographic images of the geodetic lines of the unit sphere, that is, of great circles. We know that the great circles passing through a point (X 0 ,Y 0 ,Z 0 ) of the sphere intersect each other again at the point X, =  Xq,
Yj =  YQ,
Z, =  ZQ .
177
APPLICATION OF THE THEORY
The stereographic images of this set of circles are the light rays through the point x°
= r ^ V o•
y°
 r
1
^
= ( T  h ^ l f , l
= TTW
1 1
'
1
•
(28
44)
The light rays of this medium are the curves into which the great circles of the sphere are transformed by applying stereographic projection and the conformal transformation (28.41) in succession. Obviously it is still true that all the rays through a point (£ 0 ,tj 0 ) intersect each other at a conjugate point Ui.tj,). Hence it follows that every medium in the xyplane which has an index of refraction
n

i T u ' ^ i
:
z
=
x
+
iy
(28
45)
where f(z) is analytic, represents a perfect optical instrument in the following sense: Every point X0,y 0 of the xyplane has a perfect conjugate point x,,y, in the plane.
APPLICATION OF THE THEORY
179
The light rays in all these media can be obtained by conformal mapping of the sphere onto the plane, namely a s curves which correspond to the great c i r c l e s of the sphere. With the aid of (28.26) one can easily prove that these r a y s are given by the equation f 2 + (af + af)  1 = 0
(28.46)
where a = a + i/3 i s an arbitrary complex number. Let us consider the function f(z) = z r ,
y = 1
(28.47)
a s an example. We obtain a medium of radial symmetry, namely n(r)
=
iVr'i'r '
r
=
V x 2 + y2 •
(28.471)
The light rays in this medium are the curves
r 2y
+ r y A cos (0  ¡3)  1 = 0
(28.472)
where A and ¡3 are arbitrary real constants, a s follows from (28.46). Since n(r) i s a function of ^ / x 2 + y 2 in the xyplane we conclude that a medium of refractive index r
Y  t
1 + r2? in the xyzspace with r = v / x 2 + y 2 + z 2 i s perfect in the s e n s e of our above definition not only for points of the xyplane but for any points of the x,y,z space. Obviously, our result i s a direct generalization of Maxwell's fisheye which i s obtained by taking y  1. Another example i s given by the function f(z) = e l z .
(28.48)
This gives the refractive index n = —^r— . cosh y
(28.481)
The rays satisfy the equation sinh y = A sin(x + a ) with arbitrary constants A and a .
(28.482)
180
M A T H E M A T I C A L THEORY O F OPTICS
A straight line x = x 0 has a perfect image on the line x = x 0 + 7r; a point of the "object line" x = x 0 has a perfect image at the point y, =  y 0 of the "image line", x = x 0 + w. This example applies only to the xyplane. Contrary to the f o r m e r example it is not possible to find a medium in the x,y,zspace which c o r =y »1 o responds to this example and f o r m s a perfect optical instrument in the x,y,zspace. X *0
X0+7T
y0
Figure 109
28.5 Light rays which remain in one and the same plane play an important part in the investigation of optical instruments of revolution. Let us assume that a medium is symmetrical with respect to the xaxis so that n = n ( x , ^ / y 2 + z 2 ). Every ray which intersects the xaxis is plane, i.e., lies in a meridional plane through the axis of the instrument. We may assume that this plane is the xyplane. The rays in this plane, which is often called the primary plane of the instrument, are given by the geodetic lines of the line element ds 2
= n 2 (x,y) (dx2 + dy 2 ) . (28.51)
Figure 110 the x,y,z space. Its line element ds 2
Let x = x(u,v), y = y(u,v), z = z(u,v) be the parametric representation of a surface in has the form
ds 2 = E du 2 + 2F du dv + Gdv2 where E,F,G are functions of u and v.
(28.52)
APPLICATION OF THE THEORY
181
One can show that it is possible by a transformation of the p a r a m e t e r s u = u(x,y) v = v(x,y) to t r a n s f o r m the line element (28.51) into the form of an optical line element ds 2 = n 2 (x,y)(dx 2 + dy 2 ) .
(28.53)
The corresponding p a r a m e t r i c representation of the surface X = X(x,y);
Y = Y(x,y);
Z = Z(x,y)
(28.54)
can be considered as a conformal mapping of the surface onto the xyplane. Thus the light rays of a medium of refractive index n(x,y) in the xyplane are the images of the geodetic lines of a surface projected conformally onto the plane. As in the case of the sphere there exist many conformal projections (28.54) for a given surface. From one projection (28.54) we can find others by a conformal transformation of the plane onto itself. If x = u(£,r)), y = V(,T]) are the real and imaginary p a r t s of an analytic function f(5) = u($,i) +
1V(4,TJ);
f = £ + irj
we obtain a new conformal projection by introducing u and v in (28.54). The line element (28.53) becomes ds 2  n 2 (u,v) If'l 2 (d£2 + d7)2)
(28.55)
which corresponds to an optical medium of refractive index n*(i,n)  n ^ u U . n ) , v(i,ij))  f ' 
.
(28.56)
If the light rays of (28.53) are given in the form g(x,y,a,b) = 0
(28.57)
with arbitrary p a r a m e t e r s a,b then we obtain the light rays of the medium in the form g (u(i,rj), v(i,r) , a, b )
 0 .
(28.571)
The results derived in (28.4) are a special application of this theorem.
182
MATHEMATICAL THEORY OF OPTICS
Let us now consider the case of a homogeneous medium where n(x,y) = 1. The rays of this medium have the form x cos 0 + y sin d = C where C and d are arbitrary constants. From (28.56) and (28.571) it follows that the light Tays of a medium of refractive index n = f'(i + ir7 )
(28.58)
u cos d + v sin if = C
(28.581)
are given by the curves
in which u and v(,r)) are real and imaginary part of the arbitrary analytic function f = u + iv. The problem of optical design is to find a medium such that the light rays through an arbitrary point of a finite section of the object plane intersect each other at a conjugate point of the image plane. This necessarily implies the same condition for the plane rays of the xyplane and hence for the geodetic lines of the surface (28.54). We therefore recognize the close relationship of the problem of optical design to a problem in differential geometry which could be formulated as follows: To determine surfaces such that the geodetic lines through at least a oneparameter set of points on the surface intersect each other in a set of perfect conjugate points. The sphere is the simplest example of such a surface. Any surface of this type determines an instrument of revolution such that the rays of the primary plane produce a sharp image of a certain curve in this plane. Among these instruments we have to determine those in which the skew rays, i.e., the rays which do not intersect the axis focus at the same points as the primary rays. Only if this additional condition is satisfied can the instrument be considered as perfect. §29. OTHER OPTICAL MEDIA WHICH IMAGE A SPHERE ONTO A SPHERE. 29.1 We have found in Maxwell's fisheye an optical instrument which forms a perfect image of an arbitrary sphere about the origin. Both the object and the image sphere are located in a region where the index of refraction varies. This leads us to the question of whether it is possible to find a medium which is homogeneous inside this sphere such that two given spheres in the homogeneous part are perfect conjugate spheres. We shall see that the answer is in the affirmative and that the problem can be solved in many ways. We remark that our condition refers only to two given spheres and requires nothing of other spheres.
APPLICATION OF THE THEORY
183
Let us assume that the boundary of the nonhomogeneous medium is a sphere of radius 1. Let r 0 be the radius of the object sphere and r t the radius of the image sphere. The index of refraction shall be a function of •y.n = n(r), r < 1; n = 1, r = 1. We assume n(l) = 1 so that n(r) is a continuous function. On account of the radial symmetry of the medium it is sufficient to require that ail rays passing through a point x 0 =  r 0 ; y 0 = 0 which enter the glass sphere pass through the point xj = r j , y t = 0.
r
Figure 111 Since r = 1 at the points P 0 and P ( where a ray enters and leaves the unit sphere, we conclude that r(0) must reach a minimum r* at a certain angle 6*. Let us simplify our problem by excluding functions n(r) which introduce more than one extreme value r* along the rays. In other words, let us assume that the function p = rn(r) increases monotonically so that the equation P
2
n2r2
(29.11)
= K2 = r 0 2 s i n 2 a 0
has only one positive solution r* < 1. The equation of the light ray for 6 < 6* is given by equation (27.25) namely:
P
e K
r*
r
yP2  K2 (29.12)
in which r
r
Figure 112 (29.13)
184
MATHEMATICAL THEORY OF OPTICS
It follows that the ray i n t e r s e c t s the positive side of the x  a x i s at a point 8 = 0, r = r t where r, satisfies the equation x
=
K
dr
f
o
K2
r ^ P
K
dr
f * i\/p

. (29.14) K2
If p = nr is given then this equation determines the intersection r, with the axis as a function of K = r 0 sin ol q, i . e . , of the direction of the incident ray. If, however, r ( is a given constant then (29.14) represents an integral equation for the function p = p(r) f o r r < 1. F o r r = 1 we have p = r . Since . K arcsin — r
dr rVr2
K2
from (29.14) we obtain the condition 1 K
I r
*
dr r y p 2 ( r )  K2
= f(K)
(29.15)
where f(K) is the function 1 / K K f(K) = — ( ir + a r c s i n — + a r c s i n — r r * \ i o
 2 arcsin K
(29.16)
29.2 In order to transform (29.15) into an integral equation of a known type, let us f i r s t introduce the variable T
(29.21)
= log r
Then it follows that
K
f
dr
f(K)
(29.22)
V p 2 ( t )  K2
We now define the measure function S2(p) as the measure of all Tregions in the interval < t = 0 where p(r) > p. Since we have assumed that p = p(r) and hence p = p(r) is a monotonic function, we see that in our c a s e $2 (p) is nothing but « (p) =  r(p) =  log r(p) , (29.23)
t Figure 113
i . e . , the a b s c i s s a  r(p) where the curve p(t) reaches the distance p from the Taxis. Evidently, we have
A P P L I C A T I O N OF THE THEORY
185
£2(p) = 0 for p I 1. The difference «(pi)  £Mp2) m e a s u r e s the r interval where p t < p(r) ^ p 2 . With the aid of this function fi(p) we can write (29.22) in the form " K /'
= f(K) .
dn(p}
(29.24)
This is an integral equation of Abel's type. 29.3 theorem.
For the solution of this equation we shall prove the following
If the function f(K) is defined by the integral f(K) =  K
f'
d "
(29.31)
in the interval 0 ^ K ^ \ then ii(p) is determined by the integral S2(p) 
•
A
= 
IK " J
f z
n(l + ¿ 2 )
o V l
dz
(30
251)
+ p2 + p 2 u 2
in which u has to be expressed as a function of p and h with the aid of (30.23). We find h V 1 + p2 — t
r T5 r~7 V 1 + p2 + p2u2 =
;
\ / 2 n z ~7J V*
P
n v 1 + p2
V*2
P
,
(30.26)
~ IT
and hence /
z0
[nv/l + p 2 + p 2 u 2  hul dz = f L
J
z
1/n2
 ^ P
0 v
1
+ P2
dz
•
(30.27)
From the elimination of 6 in (30.22) by partial integration, we obtain the problem of variation r
V = J
z0
/ Vn
1
2
h2 )  — V 1 + P2 P
dz
+
( i " #o)
h 0
(30.28)
A P P L I C A T I O N O F THE THEORY
191
with = po ;
P( Z O)
P( Z 1) = PI
as the only boundary conditions. The variation with respect to h yields
9i  0O = h J
r Z
z, 1,1
+ Kp 2
y/1
°
2
p2Vn
A
dz ,

(30.281)
7
and the variation with respect to p(z) yields mp Vv
1
+
^ _ ^
y T V ^
= 0
(30.282)
p2
where m(p,z)
/ 2 h7 n (p,z) •
(30.29)
30.3 We can interpret this result as follows: Skew rays in a system of rotational symmetry can be treated as primary rays (h = 0) if, in the pzplane, the index of refraction n(p,z) is replaced by m(p,z) given by (30.29). Let us consider the following problem: To find a ray which intersects the plane z = z 0 at the point x0
= p 0 cos 0Q ,
y 0 = po sin 0O
(30.31)
with a direction p 0 , q 0 . F i r s t we determine the quantity by h = x 0 q 0  y 0 p„ .
(30.32)
The problem remains to find p(z) as a solution of 30.282 with the initial values p(z 0 ) = Po » •/„ \ p(z„) =
XpPo P o W
+
ypqp 2
" Po
• 2
" Qo
„n,,, (30.33)
MATHEMATICAL THEORY OF OPTICS
192 This last condition follows from
pp = xx + yy
(30.34)
with the aid of the definition (30.16) of p and q. Finally the functions x(z) and y(z) of the light ray are obtained in the form x = p(z) cos
(30.35)
6(7.)
y = p(z) sin 0(z) where 6(z) is given by the integral 9(z) = 6 0 + h /
z
^ \ + ^ p m
dz .
(30.36)
Our results allow us to formulate the condition for perfect optical instruments more precisely than at the end of §29. Let us assume that the line element ds 2 = n 2 (p,z) (dz 2 + dp 2 ) is that of a perfect optical instrument in the p,zplane. In order to have a perfect instrument of revolution we must also require that all line elements ds 2 = ^n 2 
(dz 2 + dp 2 ) with
arbitrary h be perfect with the same location of conjugate points on z = z 0 and z = z,. Furthermore, the integral (30.281) must have a constant value for all values of the parameter h. 30.4 The characteristic functions. We have found in examples that Hamilton's characteristic functions for systems of rotational symmetry depend only upon three combinations which are invariants of rotation about the zaxis. Let FiaobosajbjsZoZ]) be one of the four characteristics V, W, W*, and T. The variables a 0 , b 0 ; a^bj will represent the two pairs of the variables x 0 , y 0 ; po,q 0 ; Xj.y^ Pi.qi on which the particular function depends. We shall prove that F depends only on the combinations u = a 0 2 + b02 , v = aj 2 + b,2 ,
(30.41)
w = 2(a 0 a! + b 0 b,) . Let us first consider the function W(x 0 , y 0 , pi, q t ; z 0 , z t ) in which case u = x02 + y02 ,
v = P!2 + qj 2 ,
w = 2(x„p 1 + y 0 qi) .
(30.42)
193
APPLICATION OF THE THEORY We can characterize W as a solution of the partial differential equation of f i r s t order:   V n 2 ( x 0 , y „ z 0 )  WX()2  W yo 2 ,
^
(30.43)
which satisfies the boundary condition at z 0 = W(x 0 , y 0 ; p l t q,; z l t z,) =  (x„p, + y 0 qO =   w
.
(30.44)
We make the assumption that this boundary value problem has a unique solution. This assumption is certainly true for the function n(u,z) in the neighborhood of the boundary plane z ( under very general conditions. This follows from the general theory of partial differential equations of the f i r s t o r d e r . Let us now introduce in (30.43) a function W = «(u,w;z 0 ) .
(30.45)
We obtain the equation 2 2 2 dZ0 =  y / n ( u , z )  4(ufl„ + vn w + w f i u f i w )
(30.46)
by using the fact that n = n(x,y,z) is a function of u = x 0 2 + y 0 2 and z. The quantity v = p ( 2 + q ( 2 in this equation is a constant p a r a m e t e r . Let us again assume that the solutions of this equation a r e uniquely determined by the boundary values on z 0 = zx. Let fi(u,w,z0) be the solution with the boundary values S2 =  — w. This is a function of u,w,z 0 and the p a r a m e t e r s v and Z). On the other hand the function W = ii (u.v.w.Zq.Z!) which is obtained by introducing the expressions (30.42) is a solution of the problem (30.43) and (30.44) and therefore must be the desired characteristic W. This, however, proves our statement that W depends only on u,v,w and Zq.Zj. This result gives us another proof of the invariance of the expression xq  yp along a light ray. Indeed, we have x
i = " §oPl T
yi
=
" fqj"
=
"
2(x W
° w
+ W
vPi> •
=
"
2(y
°Ww
+ W
*qi) •
Po = " §OX T = " 2 W.Xo 0 q
°
=
" 9y^
=
"
2(Wuy
°
+ W
wPi) •
+ WwQl)
•
(30.46)
194
MATHEMATICAL THEORY OF OPTICS
It follows that x
lQl " yiPl
=  2 ( x 0 q ,  y 0 Pi)W w
x 0 q 0  y 0 p 0 =  2 ( x 0 q ,  y 0 Pi)W w
and hence x f l i  YiPl = x 0 q 0  y 0 p 0 .
(30.47)
30.5 The c h a r a c t e r i s t i c V(x 0 ,y 0 ; Xj.yt; z 0 , z i ) can be obtained f r o m W by the Legendre t r a n s f o r m a t i o n 9W Xl
yi
=
=
"
'
V  W = x,Pi + y,q! .
(30.51)
9W 857*
We use these relations to d e m o n s t r a t e that V depends only on u' = u = x 0 2 + y 0 2 , V
=
W =
x,2 + y, 2 ,
(30.52)
2(x 0 x f + y 0 y,) .
We have x, =  2(x 0 W w + p,W v ) , y, =  2(y 0 W w + q,W v ) , V  W = 
(30.53)
(wWw + 2vWv ) .
F r o m the f i r s t equations (30.53), we obtain the following relations: v' = 4(uW w 2 + vWv2 + wWvWw ) , w
(30.54)
=  2(uWw +  w W v ) .
These equations allow us to e x p r e s s v and w as functions of v ' , w ' , and u = u'. We introduce these expressions in V = W  wW,  2vWv
(30.55)
and obtain V as a function of u \ v ' , w'; z 0 , z t . We have to a s s u m e of course that the eliminations in (30.54) can be made.
APPLICATION OF THE THEORY
195
Similar considerations can be applied to the c a s e of the remaining c h a r a c t e r i s t i c s W* and T . We s u m m a r i z e the r e s u l t as follows: Point c h a r a c t e r i s t i c :
V = VÍU.V.WÍZQ.ZJ) xi2 + yj 2 ;
u = x02 + y02 Mixed C h a r a c t e r i s t i c :
W =
U = X02 + y 0 2 Mixed C h a r a c t e r i s t i c :
w = 2(x 0 x, + y 0 y,) .
W(u,v,w,z0,zt)
V = Pi2 + qi 2
w = 2(x 0 pi + y 0 q,) .
W* = W*(u,v,w;z 0 ,zi) ;
u = Po2 + qo2 Angular C h a r a c t e r i s t i c : u = Po2 + q 0 2
V = Xj2 + yj 2
w = 2(p 0 x, + q 0 y,)
TÎU.V.wîZo.ZJ) ; V = Pi2 + qi 2
w = 2(p 0 p, + q 0 q,) .
Our r e s u l t s a r e quite c l e a r f r o m a g e o m e t r i c a l point of view. In the c a s e of the point c h a r a c t e r i s t i c V, f o r example, it i s easy to see that the optical path between two points (x 0 ,y 0 ) and (xj.y,) depends only on the distances p 0 and p j = of the two points f r o m the axis and on the a b s o lute value 16]  0 O  of the angular difference. This last statement is equivalent to the relationship cos (0)  0 O ) = Figure 115
2VÜ7 (30.56)
so that V is determined by the three quantities u,v,w. In other words: The value of the function V(x 0 ,y 0 ,xi,yi) is unchanged if the coordinates (x 0 ,y 0 ) and (xj.yi) a r e submitted to simultaneous rotations o r reflections on planes through the z  a x i s . T h e r e f o r e it can only depend on the three invariants of these t r a n s f o r m a t i o n s u,v,w. §31. SPHERICAL ABERRATION AND COMA. CONDITION FOR COMA FREE INSTRUMENTS. 31.1 In this section we shall investigate the image of a small p a r t of the object plane located about the axial point x 0 = y 0 = 0. The instrument i s
196
MATHEMATICAL THEORY OF OPTICS
assumed to be symmetric with respect to the zaxis. It is a remarkable and important fact that it is possible to determine the image of such a surface element from the one bundle of rays which originates at the point xo = Yo = Let us first consider this bundle, Figure 116 the so called axial bundle. The intersections x t , y ( of a ray of this bundle with the image plane are functions of its optical direction cosines p l t q ( after refraction. These two functions are given by the derivatives X) = yi
9W0 3Pl
(31.11)
9W0 aq,
of the mixed characteristic W0 = W(0,0; Pi,q t ; Z0,Z[) belonging to the point x o = Yo = Since the instrument is symmetric with respect to the zaxis, we know that W ^ p ^ ) ) depends only on the expression q«
=
W
(31.12)
We introduce Pi = p cos
yi , z =  — X, =  r—
Pi
qi
or, from (31.13), by Z(p)  z, =
V " 2 " P2 •
(31.17)
We assume that Z(p) reaches a finite limit Z(0) if p — 0 and that the image plane is chosen at this point so that Z(0) = z ( . In general, we call the difference L(p) = Z(p)  Z(0)
(31.18)
the longitudinal spherical aberration. It is related to the lateral spherical aberration i(p) by the equation (31.17), i.e., by
P
31.2 Let (p 0 , q 0 ) and (pj, q t ) be the optical direction cosines of a ray of the axial bundle before and after refraction. Then we have the relation ^ = Pi qi
(31.21)
which follows from the symmetry of the ray bundle with respect to the zaxis. This ratio determines another function of p namely Po qo M(p) = — = — ^ Pi Qi
(31.22)
MATHEMATICAL THEORY O F OPTICS
198
which we shall call the zonal magnification of the bundle. In general this function converges to a finite limit M(0) = M 0 which, as we shall see in the next chapter, is equal to the Gaussian magnification of the object plane. M( p) and the difference AM = M ( p )  Mq
(31.23)
are explicitly known when the rays of the axial bundle have been traced. 31.3 We can now demonstrate that the two functions l(p) and M(p) which are given by the axial bundle of rays through x 0 = y 0 = 0 allow us to determine not only the image of the point x 0 = y 0 = 0 but also the image of points x 0 , y 0 / 0 near the axis. Let us consider the mixed characteristic W(x 0 ,y 0 ; Pi.qi) for the two planes z = z 0 and z = Z[. We develop this function in a Taylor series with respect to x 0 and y 0 with coefficients depending on P) and q t and disregard all terms which are nonlinear in x 0 and y 0 . Since W depends only on u = x02 + y 0 2 , v = pj2 + q ( 2 = p 2 , and w = 2(x0p1 + y 0 qi), we develop with respect to u and w and disregard all terms involving u, and all powers of w higher than the first power. It follows that W = W 0 ( p ) + 2(x 0 Pi + y 0 qi)W,(p) + . . .
.
(31.31)
The function W 0 obviously is identical with (31.16) given by the lateral spherical aberration of the axial bundle. We use the general relation
aw •=

k
aw
'
=

( 3 1

3 2 )
in order to find W t (p). It follows that p 0 =  2 p,Wt(p) + . .. ;
q0 =  2 q ^ p )
+ .. .
where the dots indicate terms which are homogeneous functions of x 0 and y 0 of at least first order. Letting x 0 = y 0 = 0, we obtain
 2 Wj(p) = ^
= ^
= M(p)
(31.33)
where M(p) is the zonal magnification defined for the axial bundle. We thus have the result W = W 0 ( p )  ( x o P l + y 0 q,) M ( p ) + . . .
(31.34)
which allows us to investigate the images of points x 0 , y 0 in the neighborhood of the axis, in other words the image of a surface element about the point
x 0 = y 0 = o.
APPLICATION OF THE THEORY
199
31.4 The ray intersection with the image plane is given by xi =  i(p) cos
(p) is defined by
lp{p)
=
¿7 (M(p) M(0))
=
m~ •
(31
44)
We readily find from (31.43) that Ax, =  JC (p) cos tp + M 0 x 0 {ip + pip' cos 2
y i
=
a q T
'
lead directly to the relations x
o = A n p 0 + A12q0 
ya = A12p0 + A22q„ 
z
F11P1
F
°

F
i2qi
+
— n 0

F
22qi
+
r ~ n 0
z
2lPl
Po
.
qo
.
°
224
MATHEMATICAL THEORY O F OPTICS zi  x t =  F n p 0  F21q0 + CuP! + C12q!  — ni
pt ,
yi =  f12PO " F22q0 + C 12 Pi + C 22 qj  — q, nl or in matrix notation /
z„
(»'
=7
\ E
Pn  FP, > (33.581)
 Xt =  F'P0 + where A and C are the symmetrical matrices 'Au
A 12 \
C 12 \
/Cn ;
C = I
A22/
,
\C12
C22
(33.582)
/
E the unit matrix
E = I \0
(33.583) 1/
and F a matrix /Fu
F 12 \
\F 21
F22 /
F =
(33.584)
which, in general, is not symmetric. The equations (33.581), however, express our above statement. We note in passing that one can use these equations in order to prove the matrix conditions (33.35) independently of the former derivations for systems in which C + 0. 33.6 We know that, by a rotation of the coordinate systems about the z 0 and Zj axes, it is possible to transform the two quadratic forms A[Po.qo]
= A llPo 2 + 2A12p0q0 + A 22 q 0 2 , (33.61)
C [Pi.qJ
=
C n p! 2
+ 2C 12 p iqi
+
C22qi2
FIRST ORDER OPTICS
225
into the normal form
A
[Po.qo]
= A iPo 2
+
A2q0z , (33.62)
C [Pi.Qi]
= C l P l 2 + C2qj2 .
Without loss of generality we can therefore assume that A and C have this form in our chosen coordinate system. Hence our problem is to investigate linear transformations of the general type: Po
yo = (A 2 +
"
F nPi
) y 0 of the plane z 0 = 0 intersect each other at the conjugate point c 1 " f
X
x
o » (34.23)
yi = f
yo
of the plane Zj = 0. Let us, therefore, introduce the notation M0 = y = !
(34.24)
as the magnification of the plane z 0 = 0. It follows from (34.13) that the coefficient of pj in the first equation (34.21) and of p 0 in the second has the value f independent of z 0 and Zj. This coefficient must therefore represent a physical property of the instrument which is invariant with respect to the choice of the reference planes. It is called the equivalent focal length of the system. By introducing these notations in (34.21) and (34.22) we obtain x
°
=
f
(5, pi)
•
(34.25)
 Xj = f ( Po + M o P l ) , and x x
i
Pi
o T
+
1 M^
Po
(34.26)
228
MATHEMATICAL THEORY OF OPTICS
and, of course, similar equations for y 0 , y t ; q 0 , q t . 34.3 We consider next another pair of reference planes z 0 and Zj. The equations (34.13) become f
+
z
o\
f
srr'Pi'
(34.31)
 xl =  fp0 + ( M 0 f  — ) Pi
The two planes z 0 and z1 are conjugate if the determinant of these equations is zero. This leads to the Lens equation nt I
0
M
iii o0 z ,
(\M„ tT

+ nf
i 77 M0
n0 z0
=f2 •
i = 7f
(34.32)
•
(34.33)
Equation (34.33) determines the position of any pair of conjugate planes relative to a given pair of conjugate planes. The magnification, M, of the plane z 0 is obtained by writing (34.31) in a form analogous to (34.25);
X
°
=
f M~ P o ~
Pl
'
(34.34)
 xj =  fp0 + Mfpt , and comparing coefficients with (34.31) it follows that M = M 0 ~ T" ~ ' »1
=
,—1~rr1 1 z0 M0 f n0
'
(34
35)
and hence M0  M M0M
f , (34.36)
n
i
= (M 0  M)f .
FIRST ORDER O P T I C S
229
The last equations give the positions of the object and image planes which belong to a given magnification M. From (34.36) we conclude the useful relation zi / zo M • M0  — — n t / n0
(34.37)
between the magnifications and positions of two pairs of conjugate planes. The coordination of the rays with z 0 and z1 as reference planes is given by equations similar to (34.26), namely Xi = MXQ ,
y j = My 0 , (34.38)
Pl =
xc
•T
+
i
Mp° •
Ql =
yo
T
+
i M
q
° •
These equations indicate a simple method of determining the focal length f of the system and the magnification M of a plane z 0 : For any ray which is parallel to the z 0 axis in the object space the ratio xo
 — gives the focal length f of the system. Po For any ray which originates at the point z 0 ; x 0 = y 0 = 0, the ratio — determines the magnification M of the plane z 0 . 34.4 It is customary to use as the original pair of conjugate planes the socalled unit planes of the system. This is the pair of conjugate planes for which M 0 = 1. The points z 0 = 0 and z4 = 0 are then the unit points of the system. We obtain the special lens equation
MATHEMATICAL THEORY O F OPTICS
230 The equations (34.36) become z
o
1  M M
z l — n" l = (1  M)f
(34.42) and the relation (34.37) becomes
«I M
z, = 0 
=
51 / (34.43) nl / n 0 "
Unit points /
However, it is often more convenient to use the general equation (34.33) and determine the posiFigure 132 tion of conjugate planes with respect to a known pair of magnification M 0 . Let us, for example, consider the nodal planes of the system as planes z 0 = z t = 0. The axial points of these planes are called the nodal points of the instrument and are defined as Po follows: A ray of direction — through the nodal point of the object space n o leaves the instrument with the same direction, i.e., for x 0 = 0 we have Po Pi — = — n0 nt
or
n Po „ o — =M0= — pt nt
(34.44)
The Lens equation with respect to the nodal points becomes (34.45)
zo Equation (34.37) yields M = z t / z0
(34.46)
Unit points and Nodal points are called the cardinal points of the lens system. In order to find the position of the cardinal points relative to each other, let us introduce M = 1 in (34.46). It follows that z, = z 0 , and hence from (34.45) Zl
(n0  nj)f
(34.47)
231
FIRST ORDER OPTICS
is the distance of the unit points from the corresponding nodal points. This equation shows that unit points and nodal points coincide if object and image space have the same index of refraction. 34.5 Focal points; Newton's Lens equation. Every lens system of revolution has two focal points F 0 and F^ An object ray through F 0 leaves the instrument parallel to the axis; an object ray parallel to the axis intersects it at Fj after refraction. Let us determine the locations of these points relative to the unit points. Letting z 0 = F 0 , Zj = and z 0 = z t = Fj, we obtain from (34.41): Figure 133
F 0 =  n 0 f;
Fj = n4f .
(34.51) The equation (34.32) for M0 = 1 assumes the form
(HK) = f2;
(34.52)
or with the aid of (34.51): Figure 134
( z 0  F 0 ) ( z t  F,) =  n 0 nif 2 . (34.53)
Hence, if we determine the position of conjugate planes by their distances Zo
=
z0  F0 ,
Z t = z t  Ft , from the respective focal points we find Newton's lens equation Z 0 Z, =  nonjf 2
(34.54)
MATHEMATICAL THEORY OF OPTICS
232
The correlation between conjugate points thus can be c h a r a c t e r i z e d a s an inversion on the points ± ^ / n ^ f. We mention finally that the r e l a t i o n s (34.42) also a s s u m e an e x t r e m e l y simple f o r m if the distances Z 0 and Z j a r e introduced; we readily find: Z0 f — = TT ; n0 M
Z. — =  Mf . nt
(34.55)
34.6 An optical i n s t r u m e n t of revolution can be considered a s a p e r f e c t optical i n s t r u m e n t in a f i r s t o r d e r approximation. Every r e a l or virtual o b ject point p o s s e s s e s a conjugate point in the image space, either in i t s real o r i t s virtual p a r t . The original t r a n s f o r m a t i o n of object r a y s into image r a y s thus introduces a t r a n s f o r m a t i o n of the points ( x 0 , y 0 , z 0 ) into the points (Xj, yt.Zi) of the image space. This t r a n s f o r m a t i o n i s given by the formulae x j = Mx 0 , y t = My 0 , if we introduce for M i t s e x p r e s s i o n in z 0 . With the focal points a s r e f e r e n c e points on the z  a x e s we obtain f r o m (34.54) and (34.55) n0f M = ~~ ¿0
,
(34.61)
and hence the t r a n s f o r m a t i o n formulae
¿0 yi = n 0 f ^ , o 1
(34.62)
_ n„ntf2 ~ " 7
The l a s t equation follows f r o m the lens equation (34.54). The t r a n s f o r m a t i o n (34.62) i s a collineation since planes a r e t r a n s f o r m e d into planes, and consequently straight lines a r e t r a n s f o r m e d into straight lines since any straight line i s formed by the intersection of two planes. We can use this r e s u l t to determine the image of a plane x 0 = (z 0 " £*o) t a n
0
(35.12)
f
qo +
c
2  r~
We still obtain two independent sets of equations for the coordinates (XQ.PQ.XJ, Pi) and (y 0 ,q 0 .yi>qi) However, these equations are no longer identical as in the Gaussian case. First order ray systems of the type (35.11) are called orthogonal systems. 35.2 Let us first consider the two plane bundles of rays q 0 = qj = 0 and Po = Pi = 0 which originate at the point x 0 = y 0 = z 0 = 0. We obtain 0 = ajPo  Fp! , (35.21) x, =  Fp 0 + (cj
Pi
235
FIRST ORDER OPTICS in the first case, and 0 = a 2 q 0  fqi ,
(35.22)
 yi =  fqo +
ni
Ql
in the other case. Every bundle clearly has a pair of conjugate points Zj and z 2 , which can be determined by letting the determinant of the equations (35.21) and (35.22) be zero. However, these two points zi and z2 are not identical in general. They are known as the primary or tangential and secondary or sagittal foci of the point z 0 = 0. The difference z2  z t is called the
Figure 137
astigmatic difference of the bundle. 35.3 We consider next all the rays through the point x 0 = y 0 = z 0 = 0. F f Since Po = + — Pi and q 0 = + — q t we obtain from (35.12): al a2 "
F2 z *  Pi
=
 yj =
c
2 — a
2
 —
j
(35.31) qi
If we introduce Pi = p cos 0 means that the surface i s convex towards the incident light. The expression "0 D = —
(37.45)
i s called the refracting power of the surface or simply the surface power. By using this notation in (37.26) we obtain the canonical equations x = ~ p , 1 y = ~ q .
p =  Dx , (37.46) q =  Dy ,
where n = n(z) is the refractive index on the zaxis and D(z) = of the refracting surface.
the power
The equivalent Hamiltonian equation (37.39) assumes the form 2ni/)z + ipx2 + i/'y2 + nD(x2 + y 2 ) = 0 .
(37.47)
The problem of solving this partial differential equation can be reduced to the corresponding problem for the equation 2nipz + 11>X2 + nDx2 = 0 .
(37.48)
In fact, if ip = ip (x,z,a) is a solution of (37.48) which depends upon an arbitrary parameter a, then ip(x,y,z; a,b) = tp(x,z,a) + ¡My,z,b)
(37.49)
i s a solution of (37.47) which depends on two arbitrary parameters a and b. 37.5 The functions x(z), p(z) and the functions y(z), q(z) satisfy the same differential equations. Therefore it i s sufficient to study one of the two sets, for example, the equations x = ~ p ,
p =  Dx .
(37.51)
FIRST ORDER OPTICS
249
The general solutions of these equations can be expressed as a linear combination of two linearly independent particular solutions. We choose these particular solutions as follows: (1) The Axial Ray: Let z 0 be the position of an arbitrary reference plane normal to the optic axis. We define the axial ray as the solution x = h(z) ,
(37.52)
p = 0 (z)
of (37.51) which, at z = z 0 assumes the boundary values h(z 0 ) = 0 ,
Figure 145
(37.521)
(z) ,
(37.54)
with arbitrary constants x 0 , p 0 , y 0 , q 0 . Indeed it follows from (37.521) and (37.531) that (37.54) is a ray which leaves the object plane z 0 at the point x 0 , y 0 with the direction p 0 , q 0 .
250
MATHEMATICAL THEORY OF OPTICS
Let us now assume that the axial ray x = h(z) intersects the axis again at a point z = zlt i.e., h(zt) = 0. From (37.54) it follows that the coordinates x t = xizj), y t = y(zj); pt = p(z t ), qt = qfz^ of any other paraxial ray on the plane z = z4 are given by x, = H(Zl)x„ ,
yi = H(Zj)y0 ,
y = 0(zi)xo + (Zi)Po .
q = 0(zi)yo
+
(37.55) t>(zi)qo • These equations demonstrate that the two planes z = z 0 and z = Zj are conjugate planes: Every paraxial ray x 0 ,y 0 of the object plane intersects the image plane z = zj at one and the same point
*o.yo'
X( = Mx0 yi = My0
Figure 146
where M is determined by the value H(zj) of the above field ray. By comparing the equations (37.55) with the equations which hold in general for Gaussian conjugate planes, namely Xj = Mx(o • Pi = "
(37.56) +
MP°
we obtain the Gaussian constants M and f of the medium between z 0 and zx from the boundary values of the axial and field ray: H(zj) = M;
6(Z!)=;
t>(z,)=^
(37.561)
The above considerations allow us to formulate the following statement: If an arbitrary paraxial ray x = h(z), p = 0(z) intersects the zaxis at two different points z 0 and zu then the planes z = z 0 and z = Zj are conjugate planes with magnification M=
J(zo)
(37.57)
FIRST ORDER OPTICS
251
37.6 In many cases it i s advantageous to choose the axial and field rays in a different manner. Let us assume that a diaphragm is placed at a point £ of the optical axis between z 0 and Zj. We define the axial ray h, d by the conditions Figure 147
h(z c ) = 0 h(0 = 1
(37.61)
and the field ray by H(z„) = 1 , H({) = 0 .
(37.62)
Hence the field ray goes through the center of the diaphragm. We now obtain the general solution of equations (37.46) in the form
Figure 148
x(z) = x 0 H(z) + £h(z)
y(z) = y 0 H(z) + rjh(z) ,
p(z) = x o 0(z) + £ i)( z )
q(z) = y o 0(z) + tj^(z) .
(37.63)
These equations represent a paraxial ray which originates at a point x 0 ,y0 of the object plane and intersects the plane of the diaphragm at the point f ,t). 37.7 Lagrange's invariant. Two solutions x,p and X , P of the equations (36.61) are related to each other by a simple expression which is known as Lagrange's invariant. The solutions satisfy the equations p =  Dx , (37.71) X =  P n
 DX
We readily conclude from these equations that Px  pX = 0 ,
xP  Xp = 0
(37.72)
252
MATHEMATICAL THEORY OF OPTICS
and hence (Xp
 xP) = 0 .
(37.73)
It follows that the determinant X
P
x
p
(37.74)
of two arbitrary solutions has a constant value r. We call this determinant the Lagrangian invariant. In the case of the axial and field rays, for example, we find with the aid of the boundary conditions (37.521) and (37.531) that H Ô  he = 1 .
(37.75)
We can use this result to show that it is possible to find the field ray and thus any other paraxial ray by a quadrature if the axial ray is known. The equations (37.72) applied to h, d and H,0 give 6>h  tf H = 0;
Htf  h0 = 0
(37.76)
From H(j  hé = 0 , H  he = 1 , it follows that
û
eo  tte
h
and hence ( 5 ) ' = hi> 2
(37.77)
Since 9(z 0 ) = 0 this yields j
6 = 0 f
htf
2
dz =
i i'O)
(37.78)
FIRST ORDER OPTICS
253
By introducing d =  Dh in the f i r s t integral, we find
9 =  d (z)
J z0
j2
(37.781)
dz .
The integrals (37.78) a r e i m p r o p e r since h ( z 0 ) = 0. The convergence at z = z 0 , however, i s ensured by the f o r m of (37.781) which i s a proper integral. If 9(z) i s found by these integrals we obtain H ( z ) with the aid of the invariant (37.75). We can easily g e n e r a l i z e the result expressed by the formulae (37.78) and (37.781). Let us consider an arbitrary solution x = x ( z ) , p = p(z). We have the relationships Hp  x0 = 0 , Hp  x0 = r , where T i s a constant. that
By letting z = z 0 we find r = p 0 . A s above it follows
( p )
=
P°
£
•
(37.79)
and hence
0(z) =  PoP
/
I
z0
d
=  p0p /
W
z0
dz . (37.791) P
This demonstrates that the integrals on the right side define the same function 6(z) f o r all solutions x ( z ) , p ( z ) of our differential equations, i . e . , they are invariant integrals. Let us assume that p ( z ) f 0 in the interval z 0 £ z < zi.  1/f we find the interesting relationship 1 J = PoP!
r1 /
z0
D(z)  1 2 ^ dz =
PoPl
J
z0
l /i\  dip .
Since 0(Zi)
=
(37.792)
\ /
The focal length of the medium between z 0 and zl can be found by the integrals (37.792) f r o m an arbitrary paraxial ray x = x ( z ) , p = p ( z ) , provided that P(z) i
0.
MATHEMATICAL THEORY OF OPTICS
254
37.8 The problem of integrating the two linear equations x = ^ p; p =  Dx
(37.81)
can be reduced to the problem of integrating one quadratic differential equation of Riccati' s type. Let us consider the ratio
Figure 149
x(z)
S(z)
(37.82)
P(z)
which determines by z  nS(z) the point where the tangent to the ray intersects the z  a x i s . We find
znS Figure 150
P
P
P
x
(37.83)
This gives because of (37.81) the Riccati equation S = ^ + DS 2 .
(37.84)
Let us assume that we know the solution S(z) which, for z = z 0 a s s u m e s the boundary value S 0 = jjjj . From (37.81) it follows that P P
DS ,
(37.85
and hence 1  J r DS dz
p = p0e
/ ,
x = p0Se
DS dz ,
(37.86)
i.e., the solutions x(z) and p(z) can be found by quadratures if we know the solution S(z) of (37.84) which s a t i s f i e s the boundary condition S(z 0 ) = jjJ
255
FIRST ORDER O P T I C S
§38. THE PATH O F ELECTRONS IN THE NEIGHBORHOOD OF THE AXIS OF AN INSTRUMENT OF REVOLUTION. 38.1 In the original definitions of the functions n(z) and D(z), the functions n 0 (z) and nj(z) are in general not related, but can be assigned arbitrarily. This, however, is not the c a s e for an electronoptical instrument of revolution. We assume that only e l e c t r o s t a t i c fields are used in the instrument. The path of the electrons in this c a s e can be found mathematically in the same manner as the light r a y s in a medium of rotational symmetry, provided that the index of refraction n(x,y,z) satisfies the differential equation
Let us now assume that the e l e c t r o s t a t i c potential ! ,
which represents a ray which intersects the object plane at the point x 0 ,y0 and the plane of the diaphragm at £ ,rj. In this form one can make good use of paraxial rays for approximately solving vignetting problems, i.e., the problem of determining the manifold of rays which passes through the instrument. Indeed, the quantities x, and y, approximate the points where the actual light rays intersect the lens surfaces. In many cases this approximation is sufficient. 39.5 Lagrange's invariant. Two solutions x, ,p, and X t , P, of the difference equations x
i + i " xil
= 6
Pl + l " Pil
=
iPi •
"
(39.51)
D x
i i
are related to each other by an invariant similar to that of solutions of the corresponding differential equations. We conclude from «1 =
i = 1,3,5,
Pi
Pl
Pl+l  Pi 1 x l
P. + 1  F11 x,
x
X,_i Pi
(39.52) i = 2,4,6,
that l + l Pl
x
x
l+l Pl
il
i = 1,3,
(39.53)
= lIì , I = 2,4,
(39.54)
Pi
and
X,
X,
P 1+ 1
Pl!
=
X
1 Pl + l
x
i
p
ll
264
MATHEMATICAL THEORY OF OPTICS
By introducing even subscripts i in (39.53) we obtain P,1
1
Pll
Xj 2 Pll
. i = 2,4, . . .
11
X, X
x
l2
Pll
(39.55)
and hence by comparing with (39.54) we obtain the identity X
i
Pi+1 Pl + 1
X,2
Pt1
l2
Pll
x
i = 2,4, . . .
(39.56)
or r4 = r,_ 2 . We conclude from this that the r, have the same value r for all i. Therefore our result is: The determinants X
i
P t +1
Xi
P,1
1
Pl+1
X
1
Pll
X
(39.57)
have the same value r for all surfaces, i. In the case of the rays (39.41) and (39.411), for example, we have H, h
i
i>i + i
H,
i + l
h,
0il
= 1 .
(39.58)
39.6 We can use the above invariant to express the data H1 ,0j of the field ray by the data h , , of the axial ray. We write (39.58) in the form ?i + i i+i
H.
»i l «V
Hi
ht
h
i ">i + i (39.61)
h
i «>ii
and obtain by subtraction fli+ 1 _ £tl_ i + l
1
= h
i
^ «Vl
\
«>i + i /
(39.62)
265
FIRST ORDER O P T I C S
This gives
H
= 2
¡
M
i
*
)
.
(39.63)
i.e., 01 + i is given by the data h,, of the axial ray. By introducing (39.63) and (39.61) we obtain Hj expressed by the data h 4 , . Letting i = k in (39.63) we find an expression for the total power D = of the k refracting surfaces. In fact, since D = j =  0 k + 1 we obtain K D
=
*k+1
I
iT„ ( ä h
 ¿ 7
) •
(39.64)
v=2
We can carry out the above considerations for an arbitrary paraxial ray, x,, p,, instead of the axial ray. The result is: K d
pipici
y v=2
By introducing p„_i formula:

— ( xv
\
p„+ 1
 
) •
p v _j /
(39.65)
p^t! = DyXy we can replace (39.65) by the equivalent
D PiPk+1
Y
p
f
(39.66)
v=2
39.7 We can use the difference equations x
i + i " xii
Pi + i  Pi1
»3 2
j
i'
1
1 Figure 156
»5
F
=
5
iPi • DlXl
(39.71)
to derive expressions for the Gaussian constants of a lens system, i.e., for the focal length f = 1/D and the location of the unit points. Let us illustrate this with the example of a single lens with two spherical surfaces. Through the lens we "trace" a paraxial ray from infinity with the initial data p4 = 0, x2 = 1.
266
MATHEMATICAL THEORY OF OPTICS
F r o m (39.71) it follows that P3 = " D2 , x 4 = 1 + 6 3 p 3 = 1  63D2 ,
(39.72)
P5 = P3 " D 4 x 4 =  D2  D 4 (1  63D2) . We know by the initial conditions that p 5 =  j =  D. Hence j = D = D2 + D4  6 3 D 2 D 4
(39.73)
where n  1 1  n to D2 = —  ; D4 = —  ; 63 = f k2 k4 n
.
(39.731)
The distance of the focal point F f r o m the s u r f a c e 4 i s given by the r a t i o h = "
=
•
(39.74)
Let j ' be the a b s c i s s a of the unit point of the image space relative to the s u r face 4. We know that the focal point has the distance F = n5f = f f r o m this unit point. Hence t 5  j ' = f or j ' = (x4  l)f. F r o m (39.72) it follows that D2 j ' =  53D2f =  63 —
.
(39.75)
By tracing a paraxial ray backwards through the system so that p 5 = 0; x 4 = 1 we can obtain the unit point of the object space by s i m i l a r considerations. The r e s u l t is j = 63 —
(39.76)
where j i s the a b s c i s s a of the unit point relative to the s u r f a c e 2. The above method can be applied, of c o u r s e , to optical s y s t e m s of any number of s u r f a c e s . However, the resulting f o r m u l a e rapidly become complicated and unsuited for numerical computation. A d i r e c t numerical computation of the quantities x t , p, with the aid of the linear equations (39.71) is actually incomparably s i m p l e r . 39.8 F i r s t O r d e r Design of Optical I n s t r u m e n t s . The equations (39.71) provide us with a simple method of constructing optical i n s t r u m e n t s which satisfy given f i r s t o r d e r conditions. These f i r s t o r d e r conditions, a s , f o r example, magnification, focal length, location of object and image plane, can
267
FIRST ORDER OPTICS
be expressed by quite simple conditions for the data of paraxial rays. Very complicated formulae are obtained, however, if the geometrical data of the lenses are introduced. This fact suggests the use of the paraxial rays as intermediate parameters of the system in optical design. Let us consider, for example, the k + 1 parameters h , , tJ, of the axial ray. We can assign these parameters and the refractive indices n, arbitrarily and then determine an optical instrument which possesses the assigned axial ray. Indeed, from h
i+i "
h
ii =
i .
i + i " " V l = "
D
(39.81)
ihi
we obtain a, =
hl+1h,_j — ; l
D =
J ii " .n i
+i
.
(39.82)
i.e., the optical separations and the surface powers of the system. The geometric separations and the radii of the surfaces are given by t, = n,5 4 , n
(39.83)
i+l " i  i n
Thus we find that an arbitrarily given paraxial ray represents an optical system which is uniquely determined when the indices of the media are chosen. We illustrate this method with a simple example. We want a lens which forms a real magnified image of a given plane on another given plane. Let M =  5 and L = 500 mm. be the distance from the object to the image plane. We also give the "working distance", i.e., the distance tj of the surface 2 from the object plane and the thickness t3 of the lens: t t  80 mm.; t 3 = 10 mm., t 5 = 410 mm. . Let n = 1.5. The problem has a unique solution. We arbitrarily choose d t = 5 and thus have d 5 =  1 in order to obtain the magnification M =  5. It follows that h2 = 400, I14 = 410 and hence =
Figure 157
Th
4 1 0
"
400
)
=
1 5
" *
The data of the axial ray are therefore known and we are in the position to determine
268
MATHEMATICAL THEORY OF OPTICS
the geometrical data of the lens. The results of the different steps are as follows *1 = 5
«1 = 80
h2 = 400
D2 = 0.00815
•>3 = 1.5
5 =  1
«S = 410
ti = 80 R 2 = 57.14 ts = 10
n3 = 1.5 .
R 4 =  82.0 t 5 = 410 .
If the lens consists of more than two surfaces our method can still be used but it leads to many solutions. Indeed, our conditions are satisfied as long as the axial ray has the values = 5 and =  1 and intersects the axis at two points which are 500 mm. apart. Any broken line curve which satisfies these conditions represents an optical 500 system which satisfies our conditions. We can impose other first order conditions, Figure 158 for example, that the optical system has a given focal length f. Then we can make use of the formula (39.64) which expresses D = 1/f by the quantities h , , 0 1 and take care to select only such sets h t , tf, which satisfy this condition (39.44). Any additional degree of freedom can be used by the designer to decrease the aberrations of the lens combination. The advantage of characterizing a lens system first by the intermediary parameters h 4 , t>, becomes even more evident in connection with the aberration problem. We shall see in the next chapter that it is possible to express the 3rd order aberrations as functions of the data of paraxial rays and that these functions are relatively simple and are well suited for numerical calculations.
CHAPTER V THE THIRD ORDER ABERRATIONS IN SYSTEMS O F ROTATIONAL SYMMETRY
The problem of this chapter is to develop the image functions xi = Xi(Xo.yo.Po.Qo) yi = yi(x 0 .yo.Po.qo) Pi = Pi(x 0 ,yo.Po.qo) qi = qi(x 0 .y 0 .Po.qo) one step beyond the first order development in the neighborhood of the principal ray. In general this leads to certain algebraic functions which are of second order in the initial ray coordinates x 0 ) y 0 , p 0 , q c . If we call the departure from the first order functions the aberrations of the rays, then the second order aberrations A x j , A y j ; Apj, A q j are given by homogeneous quadratic polynominals of x 0 ,y „ ,p 0 ,q 0 . In the case that the principal ray is the axis of an optical system of revolution we will obtain no second order aberrations. This follows from the fact that the above image functions are odd functions of x 0 , y 0 , p 0 , q 0 . The development of the image functions one step beyond the first order gives algebraic functions of third order. The departures A x ( , Ay t , Ap 1( Aq t from the Gaussian approximations are determined by certain homogeneous third order polynomials of x 0 , y 0 , p 0 , q 0 . These polynomials represent the third order aberrations of the instrument. In this chapter we shall be concerned with the theory of these aberrations. §40. GENERAL T Y P E S OF THIRD ORDER ABERRATIONS., 40.1 We first derive the possible types of aberrations which we must expect in the third order approximation. For this purpose we make use of Hamilton's mixed characteristic W(x 0 ,y0 ,Pi,qi) from which we obtain the image functions in the form 3W Xl
=
" 8Pt
p°=
aw
• sr. 9W
q° =
269
'Hi
(40.11)
270
MATHEMATICAL THEORY OF OPTICS
We know that W is a function, W(u,v,w) of the invariants of rotation, u = x02 + y02 (40.12)
v = Pi2 + qi2 w = 2(x 0 p, + y 0 qj) .
Let us assume that the reference planes z = z 0 and z = z, are conjugate in the Gaussian optics of the instrument. 7
This means, according to our results in §34, that the image functions of first order have the form
*
Figure 159
x t = Mx0 ,
M p 0 = — x 0 + Mpj ,
Yl = My0 ,
(40.13) M q 0 = y yo + Mq,
and hence, that W(u,v,w) can be developed in the form W
=
" 2
U + Mw
)
+
W
2(u,v'
w)
(40.14)
The expression W2(u,v,w) represents a homogeneous polynomial of second order in u,v,w. The third order aberrations of our system, i.e., the departures Ax, = x,  Mx0 ,
M Ap0 = p0  — x 0  Mpj , (40.15)
Ay, = yj  My„ ,
A
M
qo = qo  7" yo 
M
qi .
from the functions (40.13) are determined by the polynomial W2(u,v,w). We obtain /aw,
Ax,
A y
\aw
i
=
/aw2  2{^yo+
Xn +
\ 9W; dv p*J • A p ° = aw2
2
\ •Aq°= 
2
/aw2 U r x ° /aw2 U r y »
+
aw2 \ ^ p i J • aw2
+
\ •
(40.16)
THIRD ORDER ABERRATIONS
271
40.2 We a s s u m e that W2 (u,v,w) has the f o r m W2 =  [4FU 2 + iAv 2 +
C
~ D w2 + ^Duv + iEuw + ^ B u w j ,
(40.21)
where A,B.C,D,E,F a r e certain constants depending on the i n s t r u m e n t and on the choice of object and image planes. By applying (40.16) we obtain the f o l lowing e x p r e s s i o n s for the third o r d e r a b e r r a t i o n s : Axi = eu + p v +  ( C  D)w] x 0 + [du + Av + ~ B w j P l , Ay t = [eu + bv + i ( C  D)w] y 0 + üu + Av + ^Bw]
qi
, (40.22)
Ap 0 =  f u + Dv + E w j x 0 + eu + ^ B v + ^ ( C  D)wj p j , Aq 0  [fu + Dv + E w j y „ + [eu + ^ B v + ^ ( C  D)wj q t . The quality of the image is determined by the two functions Ax t and Ayj. We see that only the five coefficients A,B,C,D,E enter in these functions. This shows that, in the r e a l m of third order optics, five different types of a b e r r a tions have to be considered. The sixth coefficient F, however, i s n e c e s s a r y in order to c h a r a c t e r i z e the complete manifold of r a y s into which the r a y s of the object space a r e t r a n s f o r m e d by the optical instrument. By knowing F in addition to A,B,C,D,E we a r e , for example, in a position to determine the quality of images f o r any object plane, not m e r e l y for the object plane which was chosen originally. For the physical interpretation of the five coefficients A,B,C,D,E we may assume y 0 = 0 without l o s s of generality. Thus we introduce in (40.22): Yo = 0 • u
= x„ 2 ,
v
= Pi2 + qi 2 ,
w
= 2x 0 p t ,
272
MATHEMATICAL THEORY OF OPTICS
and obtain, by arranging with respect to powers of x 0 : Axt = A(P!2 + q i 2 ) P l +  b x 0 (3 P l 2 +
qi
2)
+ Cxo2P! + Ex 0 3 ,
2 Ayi = A(pj2 + qi 2 )qi +  B x 0 p , q i + D x 0 2 q i , (40.23) Ap0 = gB(pj 2 + q, 2 ) P l + (CP!2 + Dqi 2 )x 0 + 3Ex 0 2 p, + F x 0 3 , Aq0 = g B (Pl 2
+
qi2)qi
+
(C  D)XoPl1l
+
Ex 0 2 q t .
Let us first consider the special set of light rays which lie in the "primary plane", i.e., in the meridional plane through the point (x 0 ,O). Since yj = Qi = qo = 0> w e obtain from (40.23): Axj = A P l 3 + B x o P l 2 + Cx 0 2 P! + Ex 0 3 , (40.24) Ap0 =
bp! 3
+
CxqPj 2
+ 3Ex0
2
Pl
•+ F x 0
3
.
This equation demonstrates that five of the 3rd order aberrations, namely those connected with the coefficients A , B , C , E , F , can be observed by investigating meridional rays alone. Only one aberration is reserved for skew rays alone, namely that determined by the coefficient D. For an optical system which is perfectly corrected in 3rd order for meridional rays we have A = B = C = E = F = 0. In this case the image functions (40.23) become AXj = 0 ,
Ayt = Dx 0 2 q, ,
Ap0 = Dx 0 q, 2 ,
Aq0 =  DxoPiQi •
(40.25)
Figure 160
The points of the object plane are imaged as sections of lines which are normal to the optical axis. In many problems connected with third order aberrations we can limit ourselves to considering only the functions (40.24) for meridional rays. This is justified since we shall find that C and D are related to each other by a simple equation, the so called Petzval equation (§42.) which allows us to determine D as soon as C is known.
THIRD ORDER ABERRATIONS
273
40.3 For the interpretation of the quantities A,B.C,D,E let us consider the following optical instrument. A small diaphragm is placed at the f i r s t focal point F 0 of the objective. T h i s diaphragm selects a narrow manifold of rays which are nearly parallel to the axis after r e f r a c *i tion. If, in (40.23) we X let Pi = qi = 0, we obtain t a bundle of r a y s which » are parallel to the axis s after refraction. Obviously this can be i n t e r / y. preted as physically closing our diaphragm down to the l i m i t so that Figure 161 only a one parameter manifold of r a y s passes
*
the objective. We have called this bundle the bundle of f i e l d rays selected by the diaphragm. The aberrations Ax, and Ay ( in (40.23) can be considered as a superposition of different types of aberrations. Let us study these types separately. Spherical Aberration: by the expression
The image of the axial point x 0 = y 0 = 0 is given
(40.31) A y j = Ap* sin w , if we introduce polar coordinates p,w by P! = p cos (p (40.32) qi = p sin (p W e have represented the spherical aberration of the axial bundle 131.14 by the expressions Axi
 I (p)
COS
0
Hp)
sin o ,
,
(40.33)
and introduced i (p) as the lateral spherical aberration. and (40.31) we obtain I (p) =  Ap 3
By comparing (40.33)
(40.34)
274
MATHEMATICAL THEORY OF OPTICS
which represents the exact function I (p) for small values of p. The longitudinal spherical aberration L(p) is related to ¿(p) by (31.19). We find, therefore, L(p) =  n4Ap2
(40.341)
in the neighborhood of p = 0. Coma: We consider next the functions x t =  b x 0 (3 Pl 2 + qj 2 ) (40.35) yi = ^ B x ^ p ^ which represent the aberrations of our instrument for small quantities x 0 if the system is free of spherical aberration. In polar coordinates we have B 5 Axt = JX(iP ( 2
+
cos
2
(40.351)
B , Ayt = —XQp^sin 2w .
Let us consider the manifold of rays from (x0,O) for which p has a constant value. Physically we may obtain these rays by placing a ring diaphragm at the focal plane F 0 , i.e., a diaphragm which stops all rays except those through a narrow circular ring. The intersection figure with the image plane of these rays is then given by a circle which, according to (40.351), has its center at 2 , m = —Bx0p^ and the radius „
R =
(40.352)
B 2 m 3~X°P = 7
The superposition of the circles which belong to different values Figure 162 of p yields the characteristic coma figure which we have already studied in §31. By comparing the above expressions for m and R with those obtained in §31 for wide apertures, p, we find *X>2' the curvature of field. Figure 163 Distortion. By closing the diaphragm down to its limit so that only the field rays pass the objective, we obtain a certain optical p r o jection of the object plane. This projection, however, does not give a true image in general. From (40.23) we find for pj = q4 = 0 that (40.37) Ayt = 0
Xj = Mx 0 + Ex 0 3
(40.371)
Yl = 0 . The departure of the projected point x t from the point Mx0 of an undistorted projection increases as Ex 0 3 . From
xi
M
(40.372)
it follows that the image point is moved away from the axis if — > 0 and towards the axis if
< 0. The two cases are known as pin cushion and barrel d i s t o r M tion respectively, as is illustrated by the distorted images of a square in Fig. 164.
Figure 164
THIRD ORDER ABERRATIONS
277
40.4 The Combined Effect of Third Order Aberrations. Let us finally consider the case where all third order aberrations are present. We introduce polar coordinates in (40.23). This yields 2 i B Ax,= — Bp 2 x 0 + Ex 0 3 + (Ap3 + Cpx 0 2 ) cos  "rp 2 x 0 cos 2 co «3 3 Ay, =
(Ap + Dpx ) sin 0+ 3
02
sin
2
(40.41)
4 dz , L
g Zl r 1 B = J H n h 3 H + Hj2h ù (H ô + h0) + H 22 0tf 3 dz , Zo z4 r  Zi C=p J Hnh2H2 + 2H12Hh0t5 + H2202ô2\ dz + 2 r / H12dz , 6
(42.44
THIRD ORDER ABERRATIONS
293
?r h
hH3 + H12H6(H Ô + he) + H 22 0 3 0] dz ,
D=p /
zo
[HuH2h2 + H 12 (H 2 0 2 + h202) + H2202 iJ2l dz . L
The third order coefficients are thus given by certain integral forms of 4th order of the functions H(z), 0(z); h(z), 0 (z), i.e., of the paraxial rays which we have defined above. 42.5 The quantities H^ are given by the second derivatives of the Hamiltonian H =  Vn2(u,z)  v with respect to u and v for u = v = 0. Letting n(0,z) = n(z) , n u (0,z) = n t (z) ,
(42.51)
n^a(0,z) = n2(z) , we obtain Hu =  n2(z) , l(z) 2n'2 •
n
H,.
(42.52)
1 4n3
H.22
We can express n4 and n2 by the geometric characteristics of the surfaces n(u,z) = const., the refracting surfaces of the medium. Let Z = g(u) be the refracting surface which passes through the point z of the z axis. Then we have the identity
'Z=g(u)
n(u,g(u))=n(0,z) .
Figure 173
(42.53)
By differentiating and then letting u = 0 it follows that nj + ng' (0) = 0 ,
n2 + 2njg'(0) + H(g'(0)) 2 + ¿g"(0) = 0 .
(42.54)
MATHEMATICAL THEORY OF OPTICS
294
We represent Z = g(u) in the neighborhood of u = 0 by g(u)= z + ^
+u2 +...
(42.55)
where R(z) is the radius of curvature of the surface at its vertex, and a(z) = g"(0) is the coefficient of u2 in the development (42.55). We then obtain
(42.56)
n2 = K R 2 ) " A a ' and hence Hu = a n  i ( ^ 2 )
H'2 =
H
22 =
" H ( n )
4 n
3
,
'
A j j + xhAD^dz . z
(43.29)
o
By introducing the invariant r = Ht>  h0 = H(z t ) «> (Zj) = M «J(zj) we can write this in the form 1
^Ax1= p /
(pt>A^ + xIiAD) dz .
(43.291)
zo 43.3 The original paraxial ray x(z),p(z) is given by x = x 0 H(z) + £h(z); p = x o 0(z) + £, t> (z) as a linear combination of the two paraxial r a y s H(z),0(z) and h(z), tf (z) defined by the conditions (42.12). By introducing these expressions in (43.291) we obtain a linear polynomial in x 0 and £ , namely ¿Axt =
+ Lx0
.
(43.31)
where K and L a r e given by the integrals
K
J
2
'
(«>2Ai
+
h 2 AD)dz ,
Zo (43.32) Z j
L =p f
i
X
( f l t f A ^ + HhADj dz . z0
The quantity K i s called the Axial Color of the medium since it determines the chromatic aberrations of the axial bundle f r o m point x 0 = 0. The coefficient L m e a s u r e s the Lateral Color of the medium, namely the chromatic a b e r r a tions of the bundle of field r a y s £ = 0. 43.4 It is possible to simplify the expressions for K and L. We have
302
MATHEMATICAL THEORY OF OPTICS
Figure 175 and hence
K =p /
* (Dh 2 ^ 1 zo
An) dz .
(43.42)
We introduce D =  r" and — = h. It follows that h n K =  ~ /
+ l i ~ A n ) dz
1
Zo
n
(43.43)
n
By partial integration of the second term, using h(z 0 ) = h(z t ) = 0, we obtain
K
=F /
z
o
^ ( ( n ^ )
•
(43 44)

This however can be written as follows
K
=
1 r''.fiv(n) F / H n i T T V Ini
, ^ •
(43.45)
In a similar manner we obtain
T
l =
1 A F /
fe \ (' n") In/ "^ry
,
•
(43.46)
THIRD ORDER ABERRATIONS
303
The function An n
=
nMz)nMz) n \ 0 (z)
m e a s u r e s the dispersion of the optical medium. It i s closely related to Abbe's yvalue np  1 "
=
which is used in practice to characterize the dispersion of optical media. Indeed, letting X 0 = X D and X = X F and then X = X c we obtain nF  nc Ml " H e = —n n
1 nu  1 v nD
•
(43.481)
By introducing ¡x (z) in (43.46) we find
u (43.49)
•"jAffif ft* • If the optical instrument consists of a finite number of refracting surfaces with homogeneous media between them, we obtain, instead of (43.49), the summation formulae
(43.491)
These formulae can be used with advantage to determine the chromatic a b e r rations of an optical system numerically. We mention explicitly that h(z), i) (z), 0(z) in (43.49) and h, , t) , , 0, in (43.491) are the paraxial rays for the normal index of refraction for Dlight. By considering h 4 , d t as p a r a m e t e r s which determine an optical system (refer to §39.8) we can interpret the relations (43.491) for given values of K and L as conditions which must be
304
M A T H E M A T I C A L THEORY O F OPTICS
satisfied by these parameters. On the basis of these conditions and other conditions derived from the equations (42.93) for Seidel coefficients it is possible to solve many problems in optical design by direct algebraic methods. For details we refer to the extended literature of technical optics. We only mention that it seems to us that by using the parameters h , , t? j in any application one obtains these conditions in the simplest mathematical form. To introduce the geometric parameters R j , tj of the system in the above formulae invariably complicates the algebraic equations. 43.5 "Chromatic" Aberrations in electron optics. The same general method of correcting chromatic aberration can be applied to electron optics. If only electrostatic fields are present the index of refraction is defined by n2 = 2(C  k0)
(43.51)
where C is proportional to the original kinetic energy of the electrons. When the electrons have different original speeds we will have an analogue to chromatic aberration. We assume that the departures from a constant C are small. Unlike the optical case, the curvature l / R of the "refracting surfaces" depends on C and the derivation of (43.49) does not apply. Instead, from (43.51) we obtain nAn = AC, and from (43.51) and (38.22) together, nAD + DAn = 0. Proceeding as before, we obtain from the general formula (43.32)
(43.52)
The electronoptical instrument thus is "chromatically" corrected if the integrals in (43.52) are zero. In this case it is corrected for all values of C in the neighborhood of C 0 . The electron optical instrument differs in this respect from an achromatic optical instrument. If an optical instrument is corrected for two wave lengths it is in general not corrected for other wave lengths. It requires special glass combinations such as Flint, Crown, and Fluorite to obtain correction for even three colors simultaneously. The socalled Apochromatic microscope objectives are optical systems with this latter type of chromatic correction.
CHAPTER VI DIFFRACTION THEORY O F OPTICAL INSTRUMENTS
§44 FORMULATION OF THE PROBLEM. 44.1 The propagation of light waves in a nonhomogeneous optical medi um is called the diffraction theory of optical instruments. Let us assume that there is a periodic oscillator at a certain point in the medium. We expect that the electromagnetic field which is finally established by the oscillator will be periodic in time and will have the form E = ue ~ , u t , H = ve
.
(44.11)
u and v are complex vectors which satisfy the equations curl v ^ ikeu = 0 , curl u  ikxv = 0 .
(44.12)
The frequency of the oscillator is given by w c 2jt ~ X '
(44.13)
k is the quantity k =a = ^ c \
.
(44.14)
44.2 We have derived in §16 the simplest type of progressing wave which can be interpreted as the electromagnetic radiation from a dipole in a homogeneous medium e = n = 1 . The result was
U =
(r
v =p
+
k?)
xu
ei
" ( i SO (mp)p]eki r
(44.21)
where m = a + ia* is a given complex vector and p is the unit vector p (x,y,z).
305
306
MATHEMATICAL THEORY O F OPTICS
For small values of wave is given by
i.e., large values of k, the principal part of the
u0 = ^(m x p)e'kr v 0 = p x u0 .
(44.22)
This electromagnetic field (44.22) thus represents the approximate solution which belongs to geometrical optics. We observe, however, another fact which is of importance in the following: The vectors (44.22) give the principal term of the wave (44.21) not only for large values of k but also for large r. We can express this as follows: The wave (44.21) and the solution (44.22) of geometrical optics have the same boundary values at infinity in the sense that limno r(u  Ua) r— "' = 0 ,
(44.23)
jJlPJo r The index of refraction in these half spaces shall be n0 and n( respectively. In the domain z < i j we n(x,y,z) 0 assume n(x,y,z) to be 1 sectionally continuous. ' i The problem is to find a solution of the equations *o i Yo i 2 o (44.12) which represents the radiation from a periodic oscillator placed Figure 176 at the point (x 0 ,y 0 ,z 0 ) of the object space z < i 0 . Of special interest is the electromagnetic field in the image space z > £ t . A complete solution of this problem is very difficult. Even in comparatively simple geometrical configurations, for example a single lens, we obtain the vectors u(x,y,z) and v(x,y,z) as the superposition of infinitely many particular solutions which correspond to the internal reflections inside the lens. A similar complication is found if the medium is continuous. It is true, as we have seen in Chapter I, that the field of geometrical optics does not show any external or internal reflections. An equivalent result was obtained for the propagation of sudden discontinuities of the field, light signals. However this does not preclude the existence of scattered reflected waves as part of the complete solution of our problem. In
o
DIFFRACTION THEORY OF OPTICAL INSTRUMENTS
307 the case of a stratified medium the existence of these scattered reflected waves can be demonstrated directly by an explicit solution of the problem. In fact, these waves are used today to eliminate the reflections from a single a i r  g l a s s surface by interference with the aid of film layers evaporated on the s u r face.
In general, we are not interested in the complete solution u(x,y, z) and v(x,y,z) of the above problem. The wave t r a i n s which reach the image space after one or more internal reflections or by scattering do not contribute to the image f o r mation of the instrument. For this reason one carefully t r i e s to eliminate these waves either by absorption or today by coating the surfaces with thin films. Of prime interest, however, is the wave which is transmitted directly without any internal reflections. Our problem is to determine only those parts of the wave from an oscillator which represent transmitted waves of this type. Figure 177
44.4 The solution (44.21) for a homogeneous medium is a wave of the above type. We can write this solution in the form u = TI Ue
ikn10 r
= Ue
iki/> (44.41)
v = Ve
i k n u0 r
= Ve
where >p = n 0 r = n 0 ^/(xx 0 ) 2 + (yy 0 ) 2 • A
2
+^
• ^
+
ikip
(zz 0 ) 2
= n„
is
the solution of the equation (44.42)
which determines the spherical wave fronts about the point x 0 , y 0 , z 0 . We expect therefore that we can represent the transmitted waves in a nonhomogeneous medium by the expressions u = Ue ik v = Ve
ik !I>
(44.43)
308
MATHEMATICAL THEORY O F OPTICS
where ip (Xo>yo>zo> x >y> z ) satisfies the equation y/4>x2 + 4>y 2 +
= n(x,y,z)
(44.44)
and determines the "spherical" wave fronts in the medium about the point x 0 , y 0 , z 0 . By assuming (44.43) in §16 we have derived the electromagnetic field uc = U 0 e i k * (44.45) v0 = V 0 e l k * which is associated with geometrical optics. We have found that the complex vectors U n (x,y,z) and V 0 (x,y,z) satisfy a system of linear differential equations along a light ray, namely
(44.46) dV0 1 „, / grad n\ + ~ A i i p \ n + yVn • ~ J grad ip = 0 — where the parameter r is defined by ndr = ds where s is the geometrical length of the rays. If the light ray passes a surface of discontinuity in n(x,y,z) then these differential equations have to be replaced by difference equations namely the two of Fresnel's formulae (16.55) which give the transmitted part of the vectors U„ and V 0 . With the aid of these formulae we are in a position to determine the field U 0 (x,y,z) and V 0 (x,y,z) in the image space z > if the
field U 0 ,V„ in the object space z ¿j of the image space but also its virtual part z £ 1 W e also extend the electromagnetic wave E = U(x,y,z)e ik( +  ct) H = V(x,y,z)e
ik((.  ct)
(44.54)
backwards into the virtual part of the image space. In other words we regard (44.54) as part of a progressing wave which is defined in the image space as a whole.
1
1
1
1
1
1
\
j
'
1
»
if,
1

rt " p\ < *
Figure 179
310
MATHEMATICAL THEORY O F OPTICS
In a similar way we can supplement the electromagnetic field (44.47) by a virtual extension. Let us consider for example a wave front tp = ¡/i0 in the neighborhood of the plane z = / j. The light rays are given by the normals of this surface. We first construct the virtual extension of these rays and then the surfaces parallel to ¡p = ip0 which are normal to these ray extensions. With the aid of the differential relations dU0
1
(44.55) i =o dr 2' we finally obtain the vectors U„ and V n on the fictitious wave fronts in the virtual image space. dV_o
44.6 We are now in a position to formulate the hypothesis which is the basis of all the following considerations. The light wave which is directly transmitted into the image space can be expressed in the form E = ue" ikcl (44.61) IT
H = ve  i k c t where u and v are solutions of the equations (44.51) which satisfy the following conditions a) u(x,y,z) and v(x,y,z) are regular at any finite point of the x,y,z space. b) The boundary values of u and v at infinity are given by the functions u0 = U 0 e l k * (44.62) v„ = V „ e l k * which represent the electromagnetic field of geometrical optics. The boundary values are attained in the sense that lim i/)(u  u 0 ) = 0 ¡¡l— ± °o (44.63) lim ll>—±"O
¡/>(v  v 0 ) = 0 .
Both hypotheses can be made physically plausible. The assumption b) is supported by the evidence in the case of the spherical wave (44.21) in a homogeneous medium.
DIFFRACTION THEORY OF OPTICAL INSTRUMENTS
311
The condition a) however may be considered with an element of doubt. Let us, for example, assume that our optical instrument has two perfect conjugate points at P 0 and Pj. We consider P 0 as a point source which emits a light wave given by the expressions (44.21) for u and v. Both u and v tend to °° if the point P 0 is approached. After r e Figure 180 fraction however we exclude singularities by the condition a). In view of the fact that Pj is a perfect conjugate point to P 0 it does not seem to be a condition which is self evident. On the other hand, this condition has considerable optical consequences. Indeed, it excludes a complete reconcentration of radiated energy even at points which are perfectly conjugate in the sense of geometrical optics. The impossibility of perfect definition i.e., a limit for the resolution of any instrument therefore is introduced in our theory from the very beginning. We shall assume the two hypotheses in the following to be true and determine the solution of (44.51) which satisfies the conditions a) and b). We r e m a r k , however, that it must be possible either to prove or to disprove both conditions by mathematical considerations namely by a construction of the transmitted wave inside the optical medium. §45 THE BOUNDARY VALUE PROBLEM OF THE EQUATION Au + k2u = 0 FOR A PLANE BOUNDARY. 45.1 In order to solve the problem formulated in §44 we derive the following boundary value problem: To find a function u(x,y,z) which satisfies the equation Au + k2u = 0
(45.11)
in the half space z > 0 and which attains given boundary values u(x,y,0) = f(x,y) .
(45.12)
We assume that f(x,y) is sectionally continuous in the xy plane. Outside a c e r tain circle of radius R 0 f(x,y) shall be continuous and shall have continuous derivatives such that f(x,y) 0 and e > 0 and remains true in case R—oo ande—0 provided that the three different integrals converge to definite limits. First we consider the integral over r £ . Let v 0 = jre^ 1 " and Vj = ^ r e * ^ 1 and hence v = v 0  v t . Since v t is regular at (x 0 ,y 0 ,z 0 ) we have lim f f ( u  vl . — )do e—0 \ di> dvJ
= 0 .
(45.24)
Furthermore
where dco is the surface element of the unit sphere. It follows that this integral has the limit  4;ru(x0 ,y0 ,z 0 ). We thus obtain the equation 4,u=//fgdxdy r0
+
//(ugvg)do
(45.25)
r,
which is valid for all hemispheres which include the point x 0 ,y 0 ,z 0 . One verifies readily that the derivative 9v 2zo » = " T
9v
9 /1 ¡kr\ i?ÌFe )
on r 0 has the value „..
where r = J(xx 0 ) 2 + (yy 0 ) 2 + z 0 2 For r > 1 this yields the inequality 9v ai
0 and thus 1  cos tf ^ 1  cos 2 9 ^ r y • We divide the R domain of integration into two parts, u>i and o>2> separated by the conical surface of angular opening f  R 0 so that on a>i the conditions (45.14) are satisfied and on u)2 both conditions (45.14) and (45.141) are satisfied. Since — —1 if R  « w e conclude from (45.271) and the inequalities (45.14), (45.141) that I 0 can be estimated as follows: 11oI < A0 0 can be arbitrarily small. This means that I 0 —0 if R — .
MATHEMATICAL THEORY OF OPTICS
316
The same estimates can be c a r r i e d out in the case of the integral T I l =
r r f 9*1  y l U a T _
V l
9u\ ^ j j d u » where
Vl
1 ikrt =  e
with the result that 1 ,  0 if R *•*>. The formula (45.25) thus yields the following representation of u ( x 0 > y 0 , zo):
4™
= //f H tedy
o r with the aid of (45.26): 277u(x 0 ,y 0 ,z 0 ) =  z 0 / / f ( x , y ) ( ^    ^ 3 ) e i k r dxdy
(45.29)
where r = ^ / ( x  x 0 ) 2 + ( y  y 0 ) 2 + z 0 2 • This result demonstrates the uniqueness of the solution of our boundary value problem. Let us assume that u ( x , y , z ) is a solution of (45.11) which satisfies the conditions (45.14) and (45.141) and has the boundary values f = 0 . Since u must satisfy the relation (25.29) we find that u = 0 f o r z > 0. 45.3 It is not difficult to prove that the integral (45.29) represents the solution of our problem f o r any function f ( x , y ) which s a t i s f i e s the conditions (45.13). Let us write the integral in the f o r m
2*u(x,y,z) =  z J J
f ({,„)
 ^ y ) eikr
d^dr,
(45.31)
00
where r =\/(i x) 2 + (17  y ) 2 + z 2 . W e v e r i f y directly that K ( £ , v ;x,y,z) = 
1
/ik 1 \ ikr ^ e
1 =  ^
z r
8 /e' k r \ — )
¿¿Cr) is a solution of A K + k 2 K  0 f o r any values of £ ,tj. By differentiating with respect to x,y,z under the integral in (45.31) we obtain integrals which conv e r g e uniformly because of the f i r s t condition (45.13). T h i s insures that the
DIFFRACTION THEORY OF OPTICAL INSTRUMENTS
317
derivatives of u(x,y,z) in (45.31) can be obtained by differentiating under the integral signs. Hence Au + k2u = f f f ( t , v ) (AK + k2K)dd7] = 0
(45.33)
i.e., u is a solution of (45.11). 45.4 We show next that u(x,y,z) assumes the boundary values f(x,y) if z—0 at any point (x,y) where f(x,y) is continuous. We write (45.31) in the form 27ru(x,y(z) =  z / / f ( x + £ , y + r , ) F ^ ( ^ 7  ) d$dr,
(45.41)
00
where r =\/£ 2 + 772 + z2
and introduce the mean values
Q(x,y;p) =
[2nf(x
+ p costp, y + p sin«/;) dtp
(45.42)
of the function f on a circle of radius p about the point (x,y). It follows that u 1 we have
0 of Maxwell's equations curl v + iku = 0 curl u  ikv = 0
(45.73)
320
MATHEMATICAL THEORY OF OPTICS
which assumes given boundary values on z = 0. These boundary values are not arbitrary. However, we can give the components Ui(x,y,0) = f(x,y) (45.731) U2(x.y.0) = g(x,y) provided that f and g have continuous derivatives and satisfy the conditions for R > R 0 : Ifl I SI
B
R
i ff l I *l
B R
B I M < r2
=  (fx + g y ). With the aid of (45.31) and (45.72) we obtain the solution cf our problem by
= i U3 = +
i
ffe(( .V ) ~ ff(U
+ ër>)
di dr,
T~
(45.75)
did"
where r = )2 + (yrj )2 + z2 . It is easily verified that div u = 0 for all x,y,z. Thus if the vector u has been found we obtain v from ikv = curl u. If f(x,y) and g(x,y) are only assumed to be sectionally continuous with sectionally continuous derivatives then we obtain the vector u by the integrals
(45.76)
These formulae are identical with (45.75) if f(£ ,rj ) and g(£ ,17 ) have continuous derivatives. We remark that in (45.76) we can loosen the restrictions (45.74) and assume only that lfl
= (0 X , y,
T ^
where the prime denotes differentiation with respect to s. The solution of this problem is the solution of problem of variation: To find functions x ^ s ) satisfying the boundary conditions xt(0) = a, ;
x,(l)
=
ft
(2.12)
for which the integral V = j f ' [ l (X,, ^ f ) + c] t'ds
(2.13)
is an extremum. Note, in particular, that there are no boundary conditions upon t(s). Variation with respect to x s yields
378
MATHEMATICAL THEORY O F OPTICS
g where L ^ x , , ! ^ ) = tjt L ^ x ^ x , ) . If we divide by t' we recognize the 1 equations (2.11). Let us take the variation function of t, t + e f . This gives
J*' w
v
[L ( x " t )
+ c
]ds
=

0
(2 15)
for any function £(s). Integration by parts yields the equation t f e ^ ^ } ] «
 / ^ ^ ) ^ ]
1
d s  0 .
(2.16)
But £ is arbitrary, hence
=£H »
Q
(2.17)
and — (t ! L) + C = 0 for s = 0 and s = 1, therefore 3 ^ i' — (t'L) + C = L  2 — L , + C = 0 for all values of s. Xi ! = xi which gives the energy condition However, — =
c
(2.18)
•
The special condition (2.11) is therefore obtainable from (2.13) by variation with respect to the time t(s). 2.2 It will be shown now that the variation problem (2.13) can be transformed into an integral of Fermat's type V = J
o
F(xi,x,')ds
(2.21)
where F(xi,xi') is homogeneous of the first order in Xj'. In order to do this the function t(s) must be eliminated in (2.13). Consider the equivalent problem of variation V = J
l
— ) + p(s) (p  t') + Ct'j ds = Extremum
(2.22)
ELECTRON OPTICS
379
where p(s) is a Lagrangian multiplier.t Variation with respect to p gives L^x,, y )
 Z^L, Jxi.y)
+ p(s) = 0 .
(2.23)
Variation with respect to t gives p'(s) = 0
and
p(l) = p(0) = C .
(2.24)
Variation with respect to p(s) gives p  t' = 0 .
(2.25)
The conditions (2.23),.. .,(2.25) may be placed upon (2.22) without affecting the solution of the problem. The condition (2.25) gives the original problem (2.13), however we obtain a new problem by imposing the other conditions. From (2.24) p(s) = C
(2.26)
and hence by (2.23) L
(Xl'7L)
+ C
"
Z
7"
L
'(
X l ,
7")
=
c
+
L i
(Xl'
= 0 . (2.27)
This gives
v
= ^K^iM ds
(2.28)
where p may be eliminated by means of the condition (2.27). 2.3 The problem of variation has been expressed in the equivalent form
V = f
F(xi,xi')ds = Extremum
where F(Xi,Xi') = G( x , , x , \ pix^Xi'))
= p C + L^Xi,^^
and where p = p(Xi,Xi') is determined by the equation
t C o m p a r e the s i m i l a r p r o c e d u r e f o r the e l i m i n a t i o n of 0(z) in S30.2.
(2.31) (2.32)
MATHEMATICAL THEORY OF OPTICS
380 ~G(xilXl',p)
= 0 .
(2.33)
dp
It will now be shown that F(Xi,Xi') is homogeneous in x¡' to the first order. G(X,Xi', p) satisfies the relation 2 x j ' G x . = G  pGp as we may easily verify. Now from (2.32) and (2.33) it follows that FX
'
= V
+
=
Hence I x / F x > = Ex, 1 G X( > = G  pGp. we obtain
G
X
(2.34)
; •
Since G p = 0 and G = F ,
Zxj' F x t = F
(2.35)
i
which is Euler's condition for homogeneity of the first order in x t ' . In summary: The paths of particles of a given energy C in a mechanical system which possesses a Lagrangian function of the form L ( x i > X i ) are extremals of the Fermat problem V = f F(x1,xi')ds
(2.36)
where the function F ( x i , x i ' ) is homogeneous of the first order in x / and can be obtained by eliminating p from the equations F = p JC + M x i . x , ' ) ]
; J ^ P C + ( L x ,
= 0 .
(2.37)
We note that this definition of F is equivalent to the definition by the Legendre transformation ^^pL^Xi, — ^ j
=  C ; pL + F =  Cp.
2.4 Consider the example of the case where the kinetic energy is a quadratic form in the velocities T = 2 g l k x 1 x k , and the potential is a function only of the position, U(x,). Then L = T  U or L = S g l k x j x k  U(x,) which gives the function F through the equations (2.37)
(2.41)
381
ELECTRON OPTICS F = I SgiRx/x,,' + p(C  U)
(2.42)   \ g i k i k + (C  U) = 0 . P 2
x
x
We obtain the Jacobi principle of mechanics F = 2,/C  U , / Z g i k x ; x k '
(2.43)
2.5 The case of electron optics. The general result immediately yields the electron optical Fermat problem. From (1.26) we have the Lagrangian function L = mc 2 ( l  J l
~ P2) 
t>
ec
+
f (A X)
X i' 1 where 0 Z = ^ y S x j . The substitution of — for x, gives
G(xi,x,p)=mcV(
1 V
r  ~ Ex, 2 j + ^ (A • X') + p(C  e0)
(2.52)
which we write in the form ^
(aX')
= op (l
+
I peep .
(2.53)
The vector a and the scalar cp(x,y,z) are defined by a = ^ A
v
= (a^a^aj)
(2.54)
_ 2(C  e) mc 2 •
In these expressions the right sides are dimensionless. The extremals of the variation problem (2.26) are not changed by multiplying the function F i x j . x , ' ) by a constant. So that the index of r e fraction may be a dimensionless number, the function F is defined by the equations F =  J  G ( x i , x i ' , p) mc d
1
(2.55)
382
MATHEMATICAL THEORY O F OPTICS
through the elimination of p we obtain the equation F(xi>xi')
+ ± c p 2 y z x , ' 2 + (a x