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the big one
2024-08-27 21:48:20 -05:00
MATHEMATICAL THEORY OF OPTICS
by R. K. LUNEBUR6
Foreword by EMILE WOLF Supplementary Notes by M.HERZBER6ER
UNIVERSITY OF CALIFORNIA PRESS
BERKELEY AND LOS ANGELES 1964
R. K. Luneburg
University of California Press Berkeley and Los Angeles, California
Cambridge University Press London, England
@ 1964 by The Regents of the University of California Library of Congress Catalog Card Number: 64-19010.
Printed in the United States of America
FOREWORD
During the Summer of 1944, Dr. Rudolf K. Luneburg presented a course of lectures on the Mathematical Theory of Optics at Brown University. The lecture material was later collected in a volume which was issued by Brown University in the form of mimeographed notes. These notes were by no means a compilation of generally available knowledge. They contained a highly original, thorough, and systematic account of the foundations of several branches of optics and numerous new and important results.
The supply of copies of Luneburg's notes was soon exhausted, but demand for them has continued. The University of California Press is providing a real service to the scientific community by issuing a printed version of these notes. Fate has prevented Dr. Luneburg from seeing this volume. He died in 1949, at a time when the importance of his work was just beginning to be generally recognized.
The chief contribution which Luneburg has made through these notes lies in having shown how the two main mathematical disciplines of instrumental optics, namely geometrical optics and the scalar diffraction optics, may be developed in a systematic manner from the basic equations of Maxwell's electromagnetic theory. Prior to Luneburg's work these two disciplines were, by and large, treated as self-contained fields, with little or no contact with electromagnetic theory.
The starting point of Luneburg's investigation was the observation of the formal equivalence of the basic equation of geometrical optics (the eikonal equation) and the equation that governs the propagation of discontinuous solutions of Maxwell's equations (the equation of characteristics). By boldly identifying the geometrical optics field with the electromagnetic field on a moving discontinuity surface, Luneburg was led to a complete formulation of geometrical optics as a particular class of exact solutions of Maxwell's equations; This formulation is by no means based on traditional ideas; for traditionally geometrical optics is regarded as the short wavelength limit (or, more precisely, as the asymptotic approximation for large wave numbers) of the monochromatic solution of the wave equation. Luneburg was, of course, aware of this more traditional viewpoint and he touches briefly on it in §16. In fact, in a course of lectures which he later presented at New York University (during the academic year 1947-1948) Luneburg devoted considerable time to. the interrelation between the two approaches. Some of the ideas outlined in the two courses have become the nucleus from which a systematic theory of asymptotic series solutions of Maxwell's equations is gradually being developed. An account of the material presented by Luneburg in his New York
V
vi
FOREWORD
lectures and of related more recent developments will soon be published by Drs. M. Kline and I.W. Kay in a book entitled Electromagnetic Theory and Geometrical Optics (J. Wiley and Sons, New York).
Chapter I of the present work contains the derivation of the basic laws of geometrical optics from Maxwell's equations. Amongst the many new results which this chapter contains, the transport equations, eq. (11.38), relating to propagation of the electric and the magnetic field vectors along geometrical rays, are of particular significance. In Chapter II Hamilton's theory of geometrical optics is formulated and in the chapter which follows it is applied to special problems . Amongst results which seem to make their first appearance in the scientific literature are some of the formulae of §24 relating to final corrections of optical instruments by aspheric surfaces; some new theorems relating to perfect optical instruments (§28.4); and the introduction in §29· of a new "perfect lens," which images stigmatically onto each other two spherical surfaces which are situated in a homogeneous medium. This is the now well known "Luneburg lens" which has found valuable applications as a microwave antenna. First and third order theories of optical systems are discussed in Chapters IVand V and, like all the other chapters, they contain a wealth of information.
Chapter VI deals with the diffraction theory of optical instruments. The first sections of this chapter are devoted to the derivation, in a mathematically consistent way, of expressions for the electromagnetic field in the image region of an optical system suffering from any prescribed aberrations. A solution of this difficult problem (naturally somewhat idealized) is embodied in formulae (47.33), now known as the Luneburg diffraction integrals. These formulae are an important and elegant generalization of certain classical results of P. Debye and J. Picht. Section 48 deals with another important problem, often ignored in other treatises, namely with a systematic derivation of the scalar theory for the description of certain diffraction phenomena with unpolarized light. The concluding sections deal with problems of resolution and contain a discussion of the possibility of improvements in resolution by a suitable choice of the pupil function. These investigations are amongst the first in a field that has attracted a good deal of attention in recent years.
In two appendices formulae are summarized relating to vector analysis and to ray tracing in a system of plane surfaces. They are followed by supplementary notes on electron optics, prepared by Dr. A. Blank and based on lectures of Dr. N. Chako. The volume concludes with supplementary notes by Dr. M. Herzberger, based on his lectures dealing with optical qualities of glass, with mathematics and geometrical optics and with symmetry and asymmetry in optical images.
FOREWORD
vii
It is evident that Luneburg's Mathematical Theory of Optics is a highly original contribution to the optical literature. I consider it to be one of the most important publications on optical theory that has appeared within the last few decades.
Department of Physics and Astronomy University of Rochester, Rochester 27, New York May, 1964
Emil Wolf
PUBLISHER'S NOTE
The present edition has been reproduced from mimeographed notes issued by Brown University in 1944. It is reprinted by permission of the Brown University Press.
The University of California Press extends gratitude for help in making this edition possible to Dr. A. A. Blank, Dr. Max Herzberger, Mrs. R. K. Luneburg, Dr. Gordon L. Walker, and Dr. Emil Wolf.
The author's name was misspelled in the original edition. This has, of course, been corrected, and a number of typographical errors, almost all of which were listed originally in the Errata of the mimeographed version, have also been corrected. Dr. Blank has clarified the last section of Chapter Von the basis of the Errata. No other changes have been made in this edition, which presents Luneburg's work as he left it.
BIOGRAPHICAL NOTE
Dr. Rudolf Karl Luneburg
Born in Volkersheim, Germany, June 30, 1903; resident in United States of America since 1935; naturalized U. S. citizen 1944. Ph.D. University of Gottingen, 1930.
Research Associate in Mathematics, University of Gottingen, 1930-1933; Research Fellow in Physics, University of Leiden, 1934-1935. Research Associate in Mathematics, New York University, 1935-1938. Mathematician, Research Department of Spencer Lens Company (subsidiary of American Optical Company), Buffalo, New York, 1938-1945. Visiting Lecturer, Brown University, Summer, 1944. Mathematical Consultant, Dartmouth Eye Institute 1946. Research Mathematician, Institute of Mathematics and Mechanics (now Courant Institute of Mathematical Sciences), New York University, 1946-1948. Visiting Lecturer, University of Marburg and Darmstadt Institute of Technology, 1948-1949. Associate Professor of Mathematics, University of Southern California, 1949.
Died at Great Falls, Montana, August 19, 1949.
PUBLICATIONS t 1. t Das Problem der Irrfahrt ohne Richtungbeschrankung und die
Randwertaufgabe der Potentialtheorie. Math. Annalen 104, 45 (1931). 2. Eine Bemerkung zum Beweise eines Satzes iiber fastperiodische
Funktionen. Copenhagen, Hovedkommissionaer: Levin & Munksgaard, B. Lunos boktrykkeri a/s, 1932. 3. On multiple scattering of neutrons. I. Theory of albedo and of a plane boundary (with 0. Halpern and 0. Clark). Phys. Rev. 53, 173 (1938). 4. Mathematical theory of optics (mimeographed lecture notes). Brown University, Providence, Rhode Island, 1944. 5. Mathematical analysis of binocular vision. Princeton, New Jersey: Princeton University Press, for the Hanover Institute, 1947.
Research reports are not included in this bibliography.
ix
x
BIOGRAPHICAL NOTE
6. Metric studies in binocular vision perception (Studies and Essays presented to R. Courant on his 60th birthday, January 8, 1948). New York: Jnterscience Publishers, Inc., 1948.
7. Propagation of electromagnetic waves (mimeographed lecture notes). New York University, New York, New York, 1948.
8. Multiple scattering of neutrons. II. Diffusion in a plane of finite thickness (with 0. Halpern). Phys. Rev. 76, 1811 (1949).
9. The metric of binocular visual space. J. Opt. Soc. Amer. 40,627 (1950).
ACKNOWLEDGMENT
These notes cover a course in Optics given at Brown University in the summer of 1944. The preparation for mimeographing [the original copies] was possible only through the assistance of Dr. Nicholas Chako and of my students, Miss Helen Clarkson, Mr. Albert Blank and Mr. Herschel Weil. I wish to express my thanks for their excellent cooperation.
The supplementary note on Electron Optics has been written by Mr. Blank with the aim of giving a short derivatio!! of the main results by methods similar to those applied in the other parts of the course. This note has its source in lectures given by Dr. Chako on the physical side of this topic; in the mathematical approach it differs, however, from his presentation.
To Dr. Max Herzberger of the Eastman Kodak Company I am greatly indebted for contributing the supplementary notes II, III, IV from his recent research. These notes are a record of his three lectures.
I wish also to express my gratitude to Dean R.G.D. Richardson for his interest in the course and to his staff for unfailing help and cooperation in the task of preparing these notes.
R. K. L.
xi
CONTENTS
CHAPTER I WAVE OPTICS AND GEOMETRICAL OPTICS
§1.
The electromagnetic equations
1.1 Definitions and notations 1.2 Maxwell's differential equations 1.3 Energy relations 1.4 Boundary conditions
Periodic fields
2.1 Special form of E and H in periodic fields 2.2 Complex vectors 2.3 Differential equations for u and v 2.4 Avj:lrage energy 2.5 Average :r.~x 2.6 Polarization
§3.
Differential equations for E and H
3.1 Second order equations for E and H 3,2 stratified media
C ,' ~ ''
§4.
Integral form of Maxwell's equations
4.1 Eliminations of derivatives by integration
4.2 Integral conditions instead of
div e:E = div µH = 0
4.3 Integral conditions instead of Maxwell's equations
§5.
General conditions for discontinuities
► II,"
5.1 statement of the problem
5.2 First condition for E and H
5.3 Second condition for E 5,4 Complete set of difference equations for
discontinuities of the electromagnetic field
xiii
Pages 1-5 1-2 2-3 3-5 5
5-11 5-6 6-7 7 7-8 8-9 9-11
12-14 12-13 13-14
15-18 15-16
16-17
17-18
18-20 18 18-19 20
20
xiv §6.
§8. §9.
§10. §11.
CONTENTS
Discontinuities of the optical properties
6.1 Conditions for E and H if E and µ are discontinuous
Pages 20-21
20-21
Propagation of Discontinuities; Wave fronts
7.1 Homogeneous Media 7.2 Characteristic equation and characteristic
hypersurfaces 7.3 Equation of the wave fronts
21-25 21-22
22-23 23-25
Bicharacteristics; Light rays
8.1 Introduction of light rays as orthogonal trajectories
8.2 Differential equations for the light rays 8.3 Interpretation of rays as paths of corpuscles 8.4 Fermat's problem of variation
25-29
25 25-26 26 26-29
Construction of wave fronts with the.~!~-~f light rays
9.1 Boundary value problem for 1/J(x,y,z) 9.2 Expression for the solution of the problem 9.3 Proof that the problem is solved by the
expression in 9.2 9.4 Problem of finding the wave fronts which belong
to a given special wave front 9.5 Spherical wave fronts; wavelets 9. 6 Huyghens' construction
29-36 29-30 30-31
31-33
33 33-34 34-36
Jacobi's theorem
10.1 Construction of the light rays with the aid of a complete integral of the equation of the wave fronts
10.2 Proof of Jacobi's theorem 10.3 Example: Light rays in stratified medium
36-38
36-37 37-38 38
Transport equation for discontinuities in continuous optical media
11.1 Differentiation along a light ray 11.2 Conditions for E and H on a characteristic
hyper surface 11.3 Simplification of the above conditions; differential
equations for discontinuities on a light ray
39-44 39 39-42 42-44
CONTENTS
§12.
Transport of discontinuities (Continued)
12.1 Equations for directions of the discontinuities U,V and their absolute values
12.2 Geometric interpretation of the quantities
A€1/J and Aµl/J 12,3 Energy and flux on a wave front 12.4 The directions P and Q of the discontinuities
depend only on the light ray and not on the wave fronts 12.5 The discontinuities U and V determine two applicable surfaces through a light ray 12.6 Non-euclidean parallelism of the discontinuities U and V on a light ray 12.7 Integration of the transport equations
§13.
Spherical waves in a homogeneous medium
13.1 Simplest type of solutions of the wave equation with spherical wave fronts
13.2 Method of obtaining solutions of this type by differentiation; multipole waves
13.3 Multipole _solutions of Maxwell's equations 13.4 Dipole solution of Maxwell's equations 13.5 Another form of the expressions 13.4 13.6 The vectors E and H on the spherical wave
fronts
§14.
Wave fronts in media of discontinuous optical properties
14.1 Snell's law of refraction 14.2 The law of reflection
§15.
Transport of signals in media of discontinuous optical
properties. Fresnel's formulae
15.1 Conditions for discontinuities before and after refraction
15.2 Orthogonal unit vectors on the rays 15.3 First set of equations for the components of
U and V 15.4 Complete set of equations for U and V; Solution
of these equations. Fresnel's formulae
xv Pages 44-56
44-46 46-48 48-49
49-50 50-51 51-55 55-57 58-64
58-59 59 59-60 60-62 62-63 63-64 64-67 64-66 66-67
67-72
67-69 69-70 70 71-72
xvi §16.
CONTENTS
Periodic waves of small wave length
16.1 Radiation of a periodic dipole in a homogeneous medium
16.2 Radiation of a periodic dipole in a nonhomogeneous medium
16,3 Equations for the amplitude vectors UO and V0 for small wave lengths
16.4 Discussion and solution of the differential equations for U0 and V0
16.5 Media of discontinuous optical properties 16.6 Electromagnetic fields associated with
geometrical optics
Pages 73-81
73-74 74-75 75-77 77-79 79-80 81
CHAPTER II HAMILTON'S THEORY OF GEOMETRICAL OPTICS
§17.
Principles of geometrical optics
17.1 Huyghens' Principle 17.2 Light rays as paths of corpuscles 17.3 Electrons in an electrostatic field
17.4 Light rays in a medium with n* = Vn 2 + C 17.5 Fermat's principle 17.6 Example in which light rays are not curves
of shortest optical path
82-88
82-84 84-85 85 85-86 86-87
87-88
§18.
The canonical equations
18.1 Light rays in the form x = x(z), y = y(z)
18.2 Derivation of the canonical equations 18,3 Canonical equations for problems of variation
in general
88-94 88-89 89-91
91-94
§19.
Hamilton's characteristic function V(x 0 ,y0 ,z 0 ,x,y,z)
94-100
19,1 Problem of numerical investigation of an optical
instrument
94-95
19.2 Definition of the point characteristic V for
canonical equations in general
95-97
19.3 Principal theorem of Hamilton's theory regarding
the differential dV
97
19.4 Proof of the theorem
98-99
19.5 Hamilton's theorem for geometrical optics
99-100
CONTENTS
§20.
Hamilton's characteristic functions, W and T
20.1 The mixed characteristic W 20.2 Geometric interpretation of W 20.3 The mixed characteristic W* 20.4 The angular characteristic T 20.5 Special significance of T
§21.
Integral invariants
21.1 Fields of light rays: necessary and sufficient conditions. Theorem of Malus. The integral
!Pc n cos 0 ds is zero for any closed curve C
21.2 Non-normal congruences of rays. Poincare's invariant. Geometrical interpretation
§22.
Examples
22 .1 Mixed characteristic W for stratified media. Systems of plane parallel plates. Spherical aberration of stratified media
22.2 The angular characteristic in the case of a spherical mirror. First order development of T. Mirror equation
22.3 Angular characteristic for a refracting spherical surface. First order development of T. Lens equation
xvii
Pages 100-107 100-102 102-103 103-104 104-106 106-107 107-116
107-112 112-116 116-128
116-120
120-124
124-128
CHAPTER Ill APPLICATION OF THE THEORY TO SPECIAL PROBLEMS
§23.
Perfect conjugate points. Cartesian ovals
129-138
23.1 Light rays through perfect conjugate points
have equal optical length
129-130
23.2 Cartesian ovals
130-132
23.3 Object point at infinity: The Cartesian ovals
are ellipsoids and hyperboloids
132-133
23.4 Virtual conjugate points. Cartesian ovals for
this case
133-134
23.5 An object point at infinity having a perfect
virtual conjugate point. The Cartesian ovals
are again ellipsoids and hyperboloids
135
23.6 The aplanatic points of a sphere. Graphical
construction of light rays refracted on a sphere.
Aplanatic surfaces in the front part of micro-
scopic objectives
136-138
xviii §24.
§25. §26. §27.
CONTENTS
Pages
Final correction of optical instruments by aspheric surfaces
139-151
24.1 Justification of this method of correction
139
24.2 Solution of the problem with the aid of the
point characteristic
140
24.3 Solution with the aid of the mixed
characteristic W
140-142
24.4 Solution with the aid of the angular
characteristic T
143
24.5 Example: A lens with a plane and an aspheric
surface, which transforms a spherical wave
into a plane wave
143-145
24.6 Example: A lens with a spherical and an
aspheric surface which transforms a plane wave
into a converging spherical wave
145
24.7 The principle of the Schmidt Camera
146-147
24.8 The correction plate of the Schmidt Camera
147-151
The angular characteristic for a single refracting surface
25.1 Equations for the point characteristic 25.2 Method of finding T by Legendre's trans-
formation of the refracting surface 25.3 Surfaces of revolution 25.4 Example of a spheFical surface 25.5 The angular characteristic for an elliptic or
hyperbolic paraboloid
151-156 151-152
152-154 154 155
156
The angular characteristic for systems of refracting surfaces
26.1 Variation problem for T for finitely many refracting surfaces
26.2 The corresponding problem of variation for a continuous medium
26.3 Another problem of variation of T 26.4 Equivalence of both problems 26.5 Systems of spherical surfaces
156-164
156-158
158-159 159-160 160-163 163-164
Media of radial symmetry
27.1 Equation of the lig4t_.rays 27.2 The form of the light rays in general 27.3 Example: n 2 = C + -r1
164-172
164-166 166-169 169-172
CONTENTS
xix
Pages
§28.
Maxwell's Fish eye
172-182
28.1 The wave fronts can be obtained by a Legendre
transformation of the wave fronts of the potential
field n 2 = C + .r!
172-173
28.2 The light rays are the circles through two points
on opposite ends of a diameter of the unit circle.
The medium n = 1/(l+r) represents a perfect
optical system
173-175
28.3 Maxwell's fish eye obtained by a stereographic
projection of the unit sphere
175-178
28.4 Perfect optical systems in the x,y plane which
are obtained by other conformal projections of
the unit sphere
178-180
28.5 Conformal projection of surfaces. Relation of
optical design problem to the problem of perfect
conjugate points on surfaces
180-182
§ 29.
Other optical media which image a sphere into a sphere }.82-188
29.1 Integral equation for the function p = nr
182-184
29.2 Transformation of the equation into an equation
of Abel's type
184-185
29.3 Proof of the inversion theorem
185-186
29.4 Explicit solution of the problem
186-187
29.5 The special case r 0 = 00 , r 1 = 1
187-188
§30.
Optical instruments of revolution
188-195
30.1 Euler's equations for optical media in which
n = n(p,z); p = ...J x2 + y2
30.2 Transformation of Fermat's problem into a problem for p(z) alone
30.3 Skew rays can be treated as meridional rays if n(p,z) is replaced by
188-189 189-191
m(p,z) = ~ p
191-192
30.4 The characteristic functions in instruments of
revolution. The mixed characteristic W
depends only on u,v ,w
192-194
30.5 Similar statements for the other characteristic
functions
194-195
xx §31.
§32.
CONTENTS
Spherical Aberration and Coma. Condition for coma free instruments
31.1 31.2 31.3 31.4
31.5 31.6 31. 7 31.8 31.9
31.95
Spherical aberration L(p) of an axial bundle of rays Zonal magnification M(p) Development of W for small values of x 0 ,Yo but large values of p,q The ray intersection with the image plane. Construction of Coma flares. Abbe's sine condition Oblique bundles which are rotationally symmetrical to one of their rays Invariance of the characteristic W with respect to orthogonal transformations The characteristic W' in a coordinate system with the principal ray as the z' axis Condition of staeble-Lihotski for symmetrical bundles of rays The aplanatic points of a sphere are coma free. Graphical construction of two aspheric mirrors which possess two perfectly aplanatic points Modification for the object at infinity
Pages 195-211 195-197 197-198 198
199-201 201-203 203-204 205-206 206-207
207-208 208-211
The condenser problem
32.1 statement of the problem. Intensity and
illumination. Equation for the illumination
of a given plane
32.2 Illumination of the center. Equation for the
relative illumination
32.3 Condition for uniform illumination
32,4 Interpretation of the condition c/>(01)
c/>(0) 1- for 0
0 but
211-215
211-213 213 213-214 214-215
CHAPTER IV FIRST ORDER OPTICS
§ 33.
The first order problem in general
216-226
33.1 Canonical transformation of the rays by an
optical instrument
216-217
33.2 Invariance of the canonical conditions with
respect to orthogonal transformations of object
and image space
217-218
CONTENTS
xxi
33.3 Conditions for the matrices of the differentials dX i, dP i. Lagrangian brackets
33.4 Linear canonical transformations in first order optics
33.5 Linear equations obtained with the aid of the angular characteristic
33.6 Simplification of the equations
Pages
218-221 221-222 222-224 224-226
§34.
Gaussian optics
226-233
34.1 The matrices A,F ,C in systems of revolution.
The linear transformations for Gaussian optics 226
34.2 Images of points of the plane z O = 0
227-228
34.3 General lens equation: Location of conjugate
planes relative to a given pair of planes
228-229
34.4 Special lens equation; if the unit planes are
chosen as reference planes. Unit points and
nodal points
229-231
34.5 Focal points; Newton's lens equation
231-232
34.6 The point transformation of object into image
space is a collineation. Images of inclined
planes
232-233
§35.
Orthogonal ray systems in first order optics
234-239
35.1 Definition of orthogonal systems
35.2 Primary or tangential focus; secondary or sagittal focus; astigmatism
35.3 The bundle through the point x 0 = Yo = 0.
Focal lines. Primary and secondary
magnifications. Astigmatic difference 35.4 The images of the points x 0 ,Yo of the plane
Zo = 0
35.5 Dependence of the constants M,m,71. on the
position z 0 of the object plane. Lens equation in orthogonal systems
234 234-235
235-236 236-237
237-239
§36.
Non-orthogonal systems
240-243
36.1 The relations for the ray coordinates in
non-orthogonal systems derived from the
mixed characteristic W
240-241
36.2 The bundle through x 0 = y O = 0
241-242
36.3 The images of the points- x 0 ,Yo of the plane
Zo = 0
242-243
36.4 Relation of the torsion of the image to the
angle included by principal plane bundles of the
axial bundles
243
xxii § 37.
§ 38. § 39.
CONTENTS
Differential equations of first order optics for systems of rotational symmetry
37.1 Formulation of the problem 37 .2 First order canonical equations. Paraxial
rays and their geometric interpretation 37 .3 The corresponding problem of variation.
Transversal surfaces. Jacobi's partial differential equation for paraxial rays 37.4 Interpretation of the function n1(z). Surface power. Final form of the paraxial differential equations 37.5 The general solution is expressed by two principal paraxial rays: axial ray and field ray. Conjugate planes 37 .6 Another definition of the field ray 37.7 Lagrange's invariant. The paraxial rays can be found by quadratures if one paraxial ray is known. Integral expressions for the focal length in terms of paraxial rays 37.8 The equivalent Riccati equation
Pages 243-254 243-244 244-245
245-247
247-248
248-250 251
251-253 254
The path of electrons in the neighborhood of the axis of an instrument of revolution
38.1 Expressions for n 2(z,p) if n 2 (z,O) is known.
Solution of the equation ~n 2 = 0 with given
boundary values n 2(z,O) = f(z) 38.2 Surface power of an equipotential surface.
Differential equations for paraxial electrons 38.3 The corresponding Hamilton-Jacobi equation
and corresponding Riccati equation
255-257
255-256 256 257
Difference equations for a centered system of refracting surfaces of revolution
39.l Notations and definitions 39.2 The canonical difference equations 39.3 The associated problem of variation 39.4 The solution is expressed by two principal
solutions: axial ray and field ray 39.5 Lagrange's invariant 39.6 The paraxial rays can be found by summations
if one paraxial ray is known. Expressions for the focal length with the aid of invariant sums 39.7 The Gaussian quantities for a single lens 39.8 First order design of optical instruments
257-268 257-258 258-261 261
261-263 263-264
264-265 265-266 266-268
CONTENTS
xxiii
CHAPTER V
THE THIRD ORDER ABERRATIONS IN SYSTEMS OF ROTATIONAL SYMMETRY
Formulation of the problem
Pages
§40.
General types of third order aberrations
269-281
40 .1 General expressions for the aberrations Axi, Ay1; Ap 0 , Aq 0 derived from the mixed characteristic W
40.2 Expressions for the third order aberrations Axi, Ay1; Ap 0 , Aq 0 obtained by developing W to the fourth order. Aberrations of meridional rays
40.3 Spherical aberrations, Coma, Astigmatism, Curvature of field, Distortion
40.4 The combined effect of third order aberrations 40.5 Dependence of the aberrations on the position
of the diaphragm
269-270
271-272 273-276 277 277-281
§ 41.
The third order coefficients as functions of the position of object and pupil plane
281-286
41.1 The rays are determined by the intersection
with two pairs of conjugate planes. Object
and Image plane, Entrance and Exit pupil plane 281-282
41.2 General form of aberrations Axi, A~ 1 of
meridional rays
282-283
41.3 Expression for the image aberrations Ax1
derived from the angular characteristic
283-284
41.4 Expression for the aberrations A~ 1 of the pupil
plane
284-285
41.5 Simplified expressions for Ax1 and Ab
285
41.6 The case of the exit pupil plane at infinity_.
A,B,C,E as functions of the magnification M
of the object plane. Impossibility of correcting
an optical instrument for all pairs of conjugate
planes
285-286
§42.
Integral expressions for the third order coefficients
287-299
42.1 Formulation of the problem
287-288
42.2 Development of the Hamiltonian function
H(U,V ,z). Differential equations for the third
order polynomials X ,Y P Q Corresponding
;
.
3
3
3
3
boundary value problem
288-290
xxiv §43.
CONTENTS
42.3 Integral expressions for Ax1 = X3 and
Ay1 = Y 3 42.4 Integral expressions for the coefficients
A,B,C,D,E 42.5 The quantities H ik in general 42.6 Petzval's theorem 42.7 The quantities H lk in the case of an
electrostatic field 42.8 The case of spherical surfaces of refraction 42.9 Third order coefficients in the case of a finite
number of spherical surfaces of refraction
Pages
290-291
292-293 293-294 294-295
295-296 296-297
297-299
Chromatic Aberrations
299-304
43.1 The index of refraction varies with the wave
length of light
299
43.2 Chromatic aberrations in the realm of
Gaussian optics
299-301
43.3 Integral expressions for Axial Color and
Lateral Color
301
43.4 Simplification of these expressions. Dispersion.
Abbe's v-value. Summations formulae in the
case of a finite number of refracting surfaces 301-304
43.5 "Chromatic" Aberrations in electron optics
304
CHAPTER VI DIFFRACTION THEORY OF OPTICAL INSTRUMENTS
§44.
Formulation of the problem
305-311
44.1 Differential equations for periodic
electromagnetic fields
305
44.2 Radiation of a periodic dipole in a homogenous
medium. Boundary values at infinity
305-306
44.3 The case of a nonhomogeneous medium.
Directly transmitted waves
306-307
44.4 Mathematical form of the transmitted wave.
The approximation of geometrical optics
307-309
44.5 Virtual extension of the transmitted wave in
the image space
309-310
44.6 The transmitted wave in the image space is
characterized by its boundary values at infinity
and by the condition of regularity in the whole
image space
310-311
CONTENTS
§ 45.
The boundary value problem of the equation
.6.u + k 2u = 0 for a plane boundary
45.1 45.2 45.3
45.4 45.5 45.6 45. 7
Formulation of the problem Integral representation of the solution Proof that the above integral satisfies the differential equation Proof that the boundary values are attained Completion of the proof The conditions at infinity The corresponding boundary value problem for Maxwell's equations
§ 46.
Diffraction of converging spherical waves
46.1 The corresponding boundary values at infinity 46.2 The problem is solved by applying the integral
formula of §45.2. Debye's integral. The associated electromagnetic field
§ 47.
Diffraction of imperfect spherical waves
47.1 Boundary values at infinity 47 .2 Solution of the problem with the aid of §45.2.
The electromagnetic wave is represented by integrals over a wave front. Introduction of Hamilton's mixed characteristic 47.3 Invariance of these integrals with respect to the choice of the wave front 47.4 Parametric representation of the wave fronts 47 .5 Simplified form of the general diffraction integrals
§48.
Diffraction of unpolarized light
48 .1 Introduction of the vector m(p,q) which determines the polarization of the wave at infinity
48.2 Linear polarization at infinity 48.3 Representation of the wave with the aid of
the operator 'v = -27AT. .l grad. 48.4 Average electric energy of an unpolarized
wave 48.5 Average magnetic energy of an unpolarized
wave 48.6 Average flux vector of an unpolarized wave 48.7 Summary of the results for the case of
unpolarized waves
xxv Pages
311-320 311-313 313-316 316-317 317 318 318-319 319-320 321-324 321-322
322-324 324-333 324-326
326-329 329-330 330-331 331-333 333-339
333-334 334
334-335
335-336 336-337 337-338 338-339
xxvi §49. §50.
§51.
CONTENTS
Diffraction patterns for different types of aberrations
49.1 Spherical aberration. Limit of resolution. Tolerance for spherical aberration. Raleigh limit
49.2 Coma 49.3 Astigmatism
Pages 339-344
339-342 342 343-344
Resolution of two luminous points of equal intensity
50.1 Interpretation of F(x,y ,z) as Fourier integral 50.2 Diffraction integrals in the case of rotational
symmetry 50.3 Absolute limit of resolution of two point
sources. Case of coherence and incoherence 50.4 Inequality for I F(r) I2 . The normal pattern has
the greatest value of the central maximum I F(0) 12 for a given total energy 50.5 Solution of the maximum problem: To find the diffraction pattern F of given total energy such
that for a given a we have F(a) = 0 and such
that the central maximum F(0) is as great as possible 50.6 Solution of the maximum problem: To find the diffraction pattern F of given total energy and given distance of the inflection points such that the central maximum F(0) is as great as possible 50. 7 The maximum problem: To find the diffraction pattern F of given total energy such that the energy in a given circle is as great as possible
344-353 344-345 345-346 346-348 348-349
349-351
351-352 352-353
Resolution of objects of periodical structure
354-359
51.1 Resolution of more than two points at equal
distances
354
51.2 Light distribution in the image of a self-
luminous object which has a given distribution
U0
354
51.3 The light distribution in the image of, a non-self-
luminous object
354-355
51.4 The light distributions of object and image are
developed into Fourier series. Relation of the
two series in the case of self luminous objects
of periodic structure
355-356
CONTENTS
51.5 Limit of resolution for self luminous objects of periodic structure. Impossibility of decreasing this limit by coating the aperture with thin films. Appearance of detail which ' approaches the limit of resolution
51.6 Limit of resolution for non-self-luminous objects of periodic structure
xxvii Pages
356-358 358-359
Appendix I Vector analysis: Definitions and theorems
I.I Notation and definitions I.2 Vector identities I.3 Vector fields I.4 Vector identities: div fA, curl fA,
curl(f curl A), div (A x B), curl (A x B), A x curl B + B x curl A
360-367 360-361 361-362 362-364
364-367
Appendix II Tracing of light rays in a system of plane reflecting or refracting surfaces
II.I Vector form of the laws of reflection and refraction
II.2 Vector recursion formulae in a system of plane surfaces
II.3 Representation of the direction ·of the final ray with the aid of certain scalar quantities P;. Recursion formulae for P;
II.4 The special case that all surfaces are reflectors
II.5 Example of three mirrors at right angles to each other
II.6 Example of a 90° roof prism II.7 Influence of the roof angle on the doubling
of the image
368-372
368 368-369
369-370 370 370 370-371 371-372
xxviii
CONTENTS
SUPPLEMENTARY NOTES
NOTE I
ELECTRON OPTICS N. Chako and A. A. Blank
Introduction
§ 1.
The equation of movement
1.1 Notations and Definitions: The electromagnetic field. Force on the electron. Kinetic energy T of the electron
1.2 The generalized potential of the field. Existence of a Lagrangian kinetic potential L(x;,X;) for the electron. Explicit form of L
1.3 The corresponding problem of variation 1.4 Statement of conservation of energy
Pages 373-410
373 374-377
374-375
375 376 376-377
§ 2.
The associated Fermat problem
377-383
2.1 Problem of finding the path of a particle
through two given points if the energy of the
particle is given. The corresponding problem
of variation
377-378
2.2 Transformation of this problem of variation
378-379
2.3 Proof that the new integral is homogeneous
in x; of order 1. Formulation of the general
rule of finding the associated Fermat problem
in mechanical systems where L is independent
of t.
379-380
2.4 Example: L = ~gik X;Xk - U(x1)
380-381
2.5 The case of electron optics. Derivation of
the "index of refraction" for electromagnetic
fields.
381-382
2.6 Analogy to crystal optics.
382-383
§ 3.
The canonical equations of electron optics
383-384
3.1 The variable z as a parameter 3.2 Derivation of the Hamiltonian H(x,y ,z,p,q)
for electron optics. The canonical equations for the functions x(z), y(z), p(z), q(z)
383 383-384
§ 4.
Electromagnetic fields of rotational symmetry
4.1 The electric potential cf> and the magnetic potential A in this case
4.2 Special form of H and of the canonical equations. The invariant xq - yp
384-388 384-385 385-386
CONTENTS
4.3 The potential cp for fields free of space charges. Integral formula and power series for cp(P,z)
4.4 Integral formula and power series for a(p,z)
§ 5.
First order optics in systems of rotational symmetry
5.1 The paraxial canonical equations 5.2 Simplified form of the paraxial equation for
the complex functions X,P
5.3 The coefficients D, n, w
5.4 Solution of the equations with the aid of axial ray and field ray
5,5 Conjugate planes. Rotation of the image by the magnetic field
§ 6.
The Gaussian constants of an electron optical
instrument
6.1 Equivalent focal length. The two focal points of an instrument. Newton's Lens equation
6.2 The relations between the ray coordinates at
two planes z = 0 and z = P.
6.3 Integral equations for X and P. Solution by iteration. Explicit formulae for F and the position of the unit points and focal points of a system
6.4 Approximate formulae for short systems
§ 7.
Third order electron optics in systems of rotational
symmetry
7.1 Formulation of the problem for x,y,p,q 7.2 Corresponding problem for X,P 7.3 Differential equations for the 3rd order
polynomials 7.4 Solution of these equations. Integral
expressions for the aberrations Ai;, A7J 7.5 The third order polynomials. Integral
expressions for the coefficients A,B,C,D,E;~,y,o,E 7.6 Explicit expressions for the quantities Hik (z) 7.7 Analog of Petzval's theorem
xxix Pages
386-387 387-388 388-393 388-389 389-390 390-391 391-392 392-393
394-398
394 394-395
395-397 398
398-407 398-399 399-400 400-402 402-403
403-405 405-406 406-407
XXX
§ 8.
CONTENTS
Physical Discussion of the third order aberrations of an electron optical instrument
8.1 Spherical aberration 8.2 Coma 8.3 Astigmatism 8.4 Distortion
Pages
407-410 407-408 408-409 409 409-410
NOTE II OPTICAL QUALITIES OF GLASS M. Herzberger
411-431
NOTE ID MATHEMATICS AND GEOMETRICAL OPTICS M. Herzberger
432-439
NOTE IV SYMMETRY AND ASYMMETRY IN OPTICAL IMAGES 440-448 M. Herzberger
CHAPTER 1
WAVE OPTICS AND GEOMETRICAL OPTICS
In this course we shall be concerned with the propagation of light in a transparent medium. We shall not consider absorbing media or non-isotropic media, such as metals or crystals; but we will allow the medium to be nonhomogeneous. The optical properties of such a medium can be characterized by a scalar function
n = n(x, y, z) ,
the refractive index of the medium. In ordinary optical instruments this function is sectionally constant and discontinuous on certain surfaces.
The mathematical treatrnen11 of the propagation of light can be based on two theories: The wave theory of light (Physical Optics) and the theory of light rays (Geometrical Optics), Both theories seem to be fundamentally different and can be developed independent of each other. Actually, however, they are intimately connected. Both points of view are needed, even in problems of practical optical design. The design of an optical objective is carried out in general on the basis of Geometrical Optics, but for the interpretation or prediction of the performance of the objective it becomes necessary to investigate the propagation of waves through the lens system.
In view of this fact, these theories will be developed simultaneously. The wave theory is considered as the general theory, and Geometrical Optics will be shown to be that special part of the wave theory which describes the propagation of light signals, i.e., of sudden discontinuities. On the other hand, in the important case of periodic waves, it represents an approximate solution of the differential equations of wave optics. This approximate solution can be used in a method of successive approximation to develop the diffraction theory of optical instruments, as will be shown in the later parts of this course.
§1. THE ELECTROMAGNETIC EQUATIONS.
1.1 The wave optical part of this course is based upon Maxwell's electromagnetic theory of light. The phenomenon of light is identified with an electromagnetic field.
2
MATHEMATICAL THEORY OF OPTICS
The location in space is determined by three coordinates x, y, z, the unit of length being 1 cm. The time is determined by the coordinate t; the unit of this variable being 1 sec. The electromagnetic field is represented by two vectors:
the electric vector: the magnetic vector:
E(x,y ,z,t) H(x,y,z,t)
(Ei,E 2,E 3) , (H1,H2 ,H3) .
The components, (E 1,E 2,E 3), of the electric vector are functions of x,y,z,t; the unit of these components is 1 electrostatic unit of E. The unit of the components (Hi, H 2, H3) of the magnetic vector is 1 electromagnetic unit of H.
The properties of the medium can be characterized by two scalar functions of x,y ,z (the medium thus is assumed not to change with the time):
the dielectric constant:
E = E (x,y,z) ,
the magnetic permeability µ µ(x,y,z) .
(1.11)
1.2 The electromagnetic vectors satisfy a system of partial differential equations which, with the above choice of units, assumes the form:
curl H - i Et = 0 , C
curl E + I:!:. Ht = 0 . C
(1.20)
The constant c is the velocity of light, in our units numerically equal t.o
C = 3 • 1010 •
If the components of E and H are introduced, the above vector equations yield a system of six linear differential equations of first order:
aH3 aH 2 E aE1
ay
az C at
0
aH1 aH 3 E a E 2 0
az
ax C at
a E3 ay
-a E-2 + az
µ
C
aH1 at
= 0
a E1 aE3 µ aH 2
az
ax· + C at
0
, (1.21)
aH 2 aH1
ax
ay
E a E3 C at
0
aE 2 aE1 + !!:. aH 3 ax ay C at
0
WAVE OPTICS AND GEOMETRICAL OPTICS
3
In the group of optical problems to be considered in this course, we can
assume µ = 1, since our medium (glass) is not magnetic. The dielectric
constant, E = E(x,y,z), will be replaced by the index of refraction of the sub-
stance, according to the equation
(1.22)
This relation between two different properties of a medium is actually far
from being satisfied by the substances we are mainly interested in. However, experience shows that the predictions of the electromagnetic theory are in
excellent agreement with observation if in theoretical results the quantity -fE
is replaced by the index of refraction, measured by optical methods. Furthermore it is possible to give a satisfactory explanation of the above discrepancy by molecular considerations.
We prefer in the following sections to leave Maxwell's equations in the above forms, (1.20) and (1.21). The symmetrical structure of these equations will often allow us to find from one relation another one simply by inter-
changing the letters E and µ, and replacing E by -H and H by E.
It is customary to add two more equations to the equations (1.20), namely:
(1.23)
div(µH)
= 0,
or
...£_ (µHi) ax
+
...£_ ay
(µH 2 )
+
...£_ az
(µH 3
)
= 0.
These state that the electromagnetic field does not contain a source of
electricity or magnetism. However, these equations are not independent of
(1.20). Indeed, since div curl A = 0 for an arbitrary vector field A(x,y ,z,t),
it follows
88t (div EE) = 88t. (div µH) = o ,
(1.24)
i.e., both div(E E) and div(µH) are identically zero if they are zero at any particular time.
1.3 Energy. If we· form the scalar product of E with the first of the equations (1.20) and of H with the second one and subtract both results we obtain
1 E •curl H - H •curl E - - (EE• Et + µH, Ht) 0 .
C
(1.30)
On account of the identity
H·curl E - E·curl H div(E x H)
(1.31)
4 this gives
MATHEMATICAL THEORY OF OPTICS
c div(E x H) + _! _Q_ (EE 2 + µH 2) 2 at
0
or
(1.32)
div 4c,r(E
x H)
+ _£_ at
_!_(EE 2 + µH 2) B1r
0.
The function
(1.33)
W(x,y,z,t)
(1.34)
measures the distribution of electromagnetic energy in the field. It determines the light density in Optics. The vector
S(x,y,z,t) = 4C,r (E x H)
(1.35)
is called Poynting's radiation vector, and the relation between W and S is given by the equation
aw + div s = o
at
•
(1.36)
Let us integrate this equation over a domain D of the x,y ,z space
enclosed by a closed surface r. From Gauss' integral theorem:
aat fff W dx dy dz + ff Sv do = O,
D
r
(1.37)
Sv being the normal component of S on r. The first integral represents the
change of the total energy of the domain D per unit time. The surface integral thus gives the amount of energy which has left the domain D through the surface. Hence we interpret the vector field
S = -4C(,rExH)
as the vector field (or better, tensor field) of energy flux.
Let do be the area of a surface element at a point x,y,z, and N a unit vector normal to it. Then the energy flux through this surface element is given by
WAVE OPTICS AND GEOMETRICAL OPTICS
5
where SN = S, N is the normal component of the Poynting vector, S.
In Optics, the energy flux per unit area is called the illumination of the surface element. We have
(1.38)
1.4 Boundary Conditions. The vector functions E and H are of course not uniquely determined by the differential equations (1.20), unless certain boundary conditions are added. For optical problems, the following problem types are significant:
1. To find a solution of the equations (1.20), i.e., two vector fields,
E(x,y,z,t) and H(x,y,z,t), if the electromagnetic field E(x,y,z,0) and
H(x,y,z,0), at the time t = 0 is given and satisfies at this time the conditions div(E E) = div(µH) = 0.
2. Let us assume that on the plane z = 0, the electromagnetic field is
a known function of x,y and t when t > 0, and that certain homogeneous
boundary conditions are satisfied on another plane, z = L; i.e.,
E = E(x,y,0,t) given for t > 0 and, for example,
E(x,y ,L,t) = 0 on z = L (Figure 1).
z
Let furthermore E = H = 0 for t = 0.
To find a solution, E and H, in the half-
space z > 0 which satisfies these
L
boundary conditions.
3. Of greater practical importance
is the case for which the electromagnetic
field is established under the influence of
Figure 1
a periodic oscillator. Let us assume that
an electric dipole is oscillating at a given
point in space, for example, in front of an
optical objective (Figure 2). Under the influence of this point source, an
electric field is established which represents the light wave which travels
through the objective. The problem is to determine these forced vibrations of
the space as solutions of Maxwell's equations.
§2. PERIODIC FIELDS.
2,1 We can expect that the electromagnetic field which in the end is established by a point source periodic in the time, will be periodic in time itself and, that its frequency equals the frequency of the oscillator. On the
6
MATHEMATICAL THEORY OF OPTICS
basis of this expectation, one is led to consider special solutions of Maxwell's equations which have the form
E u(x,y,z)e- iwt
H v(x,y,z)e- lwt , (2.11)
Figure 2
where u and v are vectors independent of t. The quantity
w
271"
(2.12)
is the frequency of the oscillator and 11. the wave length.
2.2 The above complex notation is chosen on account of its mathematical advantages. The vectors u and v are in general complex vectors
u = a + ia*,
V = b + ib*'
i.e., complex combinations of real vectors a, a* and b, b*. Calculations involving these complex vectors can be carried out in the same way as those involving real vectors only, when i is considered a scalar quantity, with
i 2 = -1. For example:
Scalar product: u-v = (a•b - a*-b*) + i(a•b* + a*•b).
(2.21)
Vector product: u xv = (a x b - a* x b*) + i(a x b* + a* x b). (2.22)
The absolute value of a complex vector: u-u = a 2 + (a*) 2 •
Two complex vectors u and v are called orthogonal if U•V = u·v
(2.23) 0, i.e., if
a-b + a *·b* = 0 , ab* - a*·b = 0 .
(2.24)
WAVE OPTICS AND GEOMETRICAL OPTICS
7
If two complex vectors E and H satisfy Maxwell's equations then both the real and imaginary parts of E and H are solutions. The real parts of the vectors (2.11), for example, are given by
E acoswt+a*sinwt, H b cos wt + b* sin wt ,
(2.25)
and will be considered in the following as representing the electromagnetic field.
2.3 We now introduce the expressions (2.11) into Maxwell's equations. This yields
curl v + -iw E u O ,
C
curl u - ~ iw µ v = 0 ,
(2.31)
i.e., a system of partial differential equations without the time variable. By introducing the constant
k w
C
(2.32)
we obtain
curl v + ik E u 0 , curl u - ik µ v 0 .
(2.33)
It follows that
div(Eu) = div(µv) = 0
(2.34)
so that it is unnecessary to add these conditions explicitly, as in (1.23).
2A Energy. The period, T = 271" = ~ , of the functions (2.11) is so
W
C
extremely short in optical problems that we are unable to observe the actual
fluctuation of the electromagnetic field. Indeed, in case of sodium light, for
example, we have
11. = 0.6 x 10-4 cm., hence T 2 x 10-15 sec.
8
MATHEMATICAL THEORY OF OPTICS
The same is true for the extremely rapid fluctuations of the light density
W(x,y,z,t) = 8~ [E(acoswt + a*sinwt) 2 +µ(bcoswt+b*sinwt) 2]. (2.41)
We are, however, able to observe the average value of this energy, which is given by the integral
(2.42)
We can express this result in terms of the original complex vectors u and v and obtain
-J -W = -l611r rL.E - U • U + µ V • V
(2.43)
as an expression for the observable light density at the point x,y ,z.
2.5 Flux. Similar considerations may be applied to the Poynting vector, S. By introducing the expressions (2.25) into the definition of S, (1.36), it follows that
S = :'Ir (a cos wt + a*sin wt) x (b cos wt + b*sin wt)
(2.51)
which is also a periodic function with the small period, T. Again, only the average value,
1 -S=T1- rt, TSdt '
can be considered as physically significant. We obtain
-S(x,y,z) = SC1r (a x b + a* x b*) ,
(2.52)
or in terms of the complex vectors, u and v,
S(x,y,z)
=
C
1671"
(u
x
-v
+
-u
x
v)
.
(2.53)
We can show that the vector field, S, of average flux is a solenoidal field, i.e.,
div S = 0.
WAVE OPTICS AND GEOMETRICAL OPTICS
9
For the complex vectors u and v satisfy the equations
curl v + ikE u 0' curl u - ikµv 0.
(2.54)
The conjugate complex vectors u and v, consequently, satisfy
curl v - ikEu 0 ' curl u + ikµv 0.
(2.55)
It follows that
ii curl v - v curl u + ik (Eu·ii ~ µv•v) 0'
v curl u - u curl v + ik (Eu •iI - µv •v) 0 '
(2.56)
or
div (u X v) + ik (€ u. u - µv •V) 0 ' div (ii x v) - ik (EU • ii - µv •v) 0.
Hence div (u xv + u x v) = O; i.e., div S O.
(2.57)
2.6 Polarization. The vectors E and H given by (2.25) describe certain closed curves in space. The type of these curves determines the state of polarization of the wave at the point x,y,z, and this again represents an observable characteristic of the field. In general, the electric vector is considered as the vector which gives the polarization of the light.
We have E = a cos wt + a* sin wt, or in components
E1 a1 cos wt + a1* sin wt , E 2 = a 2cos wt + a 2* sin wt, E 3 a 3cos wt + a 3* sin wt .
(2.61)
The curve described by E is plane since E is a linear combination of the vectors a and a*. We can show easily that this curve is an ellipse. Let us introduce ~ = cos wt and 7) = sin wt. By squaring the components of E we find
10
MATHEMATICAL THEORY OF OPTICS
These equations, together with the relation ~2 + 112 = 1 represent four
linear equations for the three quantities ~2, 11 2, and 2~7). Their determinant
thus must be zero:
1 1 1
0
Ei2 ai2 a1*2 a1a1* 0
E 22 a22 a2*2 a2a2*
(2.62)
Eg2 ag2 a3*2 a3a3*
This is an equation of the type AEi2 + BE 22 + CEa2 = D, which means that
the curve of E lies on a surface of second order. The intersection curve of a plane and a surface of second order, however, is a conic. It must be an ellipse because it is closed.
The equations (2.61) show that the ellipse is symmetrical with respect to the origin, i.e., to the point x,y ,z in question. Thus we can find the length and direction of the axes by determining the extreme lengths of the vector, E, i.e., the extreme values of the quadratic form
under the condition ~ 2 + 7J 2 = 1. In other words, the axes are equal to the
characteristic values of the above quadratic form and are given by the two
solutions, ;>..1 and ;>.. 2, of the quadratic equation
la·a - ;>..
a•a*
I a•a*
a*-a* - ;>..
0.
(2.63)
Hence,
(2.64)
The characteristic values, ;>.. , are real, since the expression under the radical
is not negative. The characteristic values cannot be negative; for, with the aid of the inequality (a.a*) 2 ::§ a 2 • (a*) 2 , one can see that
WAVE OPTICS AND GEOMETRICAL OPTICS
11
We illustrate three types of polarization:
a* a
Figure 3. Elliptical polarization.
Two different characteristic values, Ai =/- A2 , both different from zero. The electric vector describes an ellipse.
The characteristic values are
equal, At = A2. This implies a*
a-a a*• a*, a.a_'." 0.
(2.65)
a Figure 4. Circular polarization.
The two components a and a* of u are orthogonal and equal in length. The electric vector describes a circle.
a
a*
Figure 5. Linear polarization.
The smaller one of the characteristic values, A, is zero. The electric vector describes a straight line. The two vector components of u have the same direction,
a x a* = 0.
(2.66)
We can express these results again by using the complex vector, u directly. The quadratic equation for ll. may be written as follows:
¼ A. 2 - (u •u) A - (u X u) 2 = 0 ;
a+ ia*,
and this has the solution,
(2.67)
The ellipticity E of the polarization, i.e., the ratio of the lengths of the axes, is thus given by the expression
U • U - v1u) 2 (u) 2 u•u + -ltu) 2 (U:) 2
(2.68)
Hence,
for linear polarization: for circular polarization:
uxu = 0, u2 =u•u=0.
(2.69)
12
MATHEMATICAL THEORY OF OPTICS
§3. DIFFERENTIAL EQUATIONS FOR E AND H.
3.1 If we eliminate one of the vectors, E or H, from Maxwell's equations, (1.20), we obtain second order equations for either E or H. By differentiation with respect to t:
curl Ht - -E Ett O ,
C
curl Et +I!:. Htt 0.
C
We introduce Ht = - ~ curl E in the first of these equations, and Et
C E
curl H in the second. The results. are
~ µcurl(~ curl E) + Ett
O,
?" E curl (¾ curl H) + H tt = 0 .
(3.11)
We apply the following vector identity, which holds for an arbitrary scalar function, f(x,y,z), and an arbitrary vector field, A(x,y,z), with continuous derivatives of the second order:
curl (f curl A) = -f A A + f grad (div A) + (grad f) x (curl A) ,
(3.12)
where A A = Axx + Ayy + Azz. Equations (3.11) become
? G t) - Ett - A E = (curl E) x grad
grad (div E),
? Htt - AH = (curl H) x (e grad¾) - grad (div H) .
(3.13)
From the second pair of Maxwell's equations (1.23), it follows that
i.e., div E defined by
div E E E div E + E •grad E 0 ' divµ H µ div H + H • grad µ o,;
(3.14)
-E •p and div H = -H •q, where the vectors p and q are
p 1 grade
E
q
-1
µ
gradµ
grad (log E ) , grad (log µ) .
(3.15)
WAVE OPTICS AND GEOMETRICAL OPTICS We introduce
and obtain from (3.11) the equations
-;;zn2 E tt - 6. E = grad (p • E) + q x curl E , -;;zn2 Htt - 6. H = grad (q • H) + p x curl H .
13 (3.16) (3.17)
The vectors p and q and the function n are given by the properties of the medium; they are not independent of each other but are related by the equation
21 (p + q) = grad (log n) .
(3.18)
In the special case of a homogeneous medium, both p and q are zero,
and n = '/Eµ is a constant. The equations (3.17) become
-;;z = n2
Ett
-
6. E
0
,
-n;;2z Rtt - 6. H = 0 .
(3.19)
Each component of E and H satsifies the ordinary wave equation. The velocity of the waves is given by the quantity
v = c/n,
which allows us to regard the quantity n = '/Eµ as the index of refraction of the medium, defined by the ratio n = c/v of the velocity of light in a vacuum
to the velocity in the medium.
In the case of a non-homogeneous medium, a more complicated set of equations is obtained. Since n is now a function of x,y ,z, the six equations, (3.17), no longer yield one equation in each component, for the first order operators on the right sides involve all the components of the vectors in each equation. However, it is still true that the wave velocity, v, is given by the ratio c/n. Indeed, we shall see that for the propagation of a light signal, i.e., a sudden disturbance of the electric field, only the second order terms in (3.17) are significant. These terms lead to a generalized wave equation in which the coefficient, n, is not constant.
3.2 Stratified media. Let us consider as an example the case of a stratified medium, in which the functions E and µ depend only on one variable,
14
MATHEMATICAL THEORY OF OPTICS
for instance on z. This case is of considerable practical interest, since the propagation of waves through thin, multilayer films, evaporated on glass,
leads to a problem of this type. Let µ = 1 and le = n(z). It follows that
p = 2 grad (log n) = (o,o, 2 :•). ,
Hence
q 0.
p·E
2 n' E n 3
(3.21)
grad (p •E)
( 2 !!..'._ n
8E3.
ax '
2 !!..'._ 8E3. n ay '
2 _£.
az
(n' n
E \) 3/
-
p
x
curl H
=
2
n' n
(8H1
az
_
8H 3
ax '
-8aHy-3+ -8aHz-2 ' 0 )
The differential equations (3.17) become
n2 a2E1 c 2 at2 -AE1
2n-'-B-E3
n ax
n2 a2E2 c 2 at2
-AE
2
=
2n'-8E-3 nay
n2 a2E3
c 2 at2 -AE 3 = 2.a£z...(nn' E~3
n2 a2H1
n' 8H1
c 2 at2
-AH1
+
2 n
-az-
+2nn-' -a8-xH3
+ 2n-'-a-H3 nay
(3.22) (3.23)
We thus obtain two partial differential equations, namely, those for E 3 and H3, in which none of the other components appear. After E 3 and H3 have been determined from these two equations, they are substituted in the remaining equations of (3.23); and these equations then become modified wave equations for E 1, E 2, H1, and H2 modified in the sense that the right side is not zero, but a known function. As a result of this simplification it is possible to find explicit solutions for many problems connected with stratified media, especially with films producing low reflection.
WAVE OPTICS AND GEOMETRICAL OPTICS
15
§4. INTEGRAL FORM OF MAXWELL'S EQUATIONS.
The functions E (x,y,z) and µ(x,y ,z) are not necessarily continuous functions. We assume, however, that they are sectionally smooth, i.e., every finite domain of the x,y ,z space can be divided into a finite number of parts in which E and µ are continuous and have continuous derivatives.
The differential equations (1.20) represent conditions for the electromagnetic field in every part of the space where E, µ, and E, H are continuous and have continuous derivatives. They are, however, not sufficient to establish conditions for the boundary values of E and H on a surface of discontinuity. This is the reason why it is advantageous to replace the differential equations (1.20) by certain integral relations. These integral equations are equivalent to the differential equations if E, µ, and E, H are continuous and have continuous derivatives. They are more general, on the other hand, since they apply equally well to the case of discontinuous functions E, µ; E, H and establish definite conditions for the electromagnetic field in this case.
4.1 Let us consider, in the four-dimensional x,y,z,t space,. a domain D
which is bounded by a closed three-dimensional hypersurface r. We assume that the hypersurface r consists of a finite number of sections in which the
outside normal N of the hypersurface varies continuously. This normal N is
a unit vector in the x,y,z,t space given by
A.
if the surface r is represented by the equation <p(x,y,z,t)
denote the components of the unit vector N by
0. In general we
(4.11)
and call these components the direction cosines of N with respect to the four coordinate axes.
Let F(x,y,z,t) be a function which has continuous derivatives in D. We consider the integral of Fx over D and carry out the integration with respect to x:
dx dy dz dt
Figure 6
fff (F(P') - F(P)) dy dz dt.
16
MATHEMATICAL THEORY OF OPTICS
The integral. on the right side is a surface integral. over the hypersurface r.
We introduce
at P': dy dz dt
at P: dy dz dt - XN do
where x N' and xN are the x-components of the unit vector N at P' and P,
respectively, and do' and do, differentials independent of the choice of the
coordinate system. We call do the surface element of the hypersurface r.
With this notation, we obtain
ffff Fx dx dy dz dt
D
In the same way, we find
JJJ F XN do.
r
(4.12)
ffff Fy dx dy dz dt
D
ffff Fz dx dy dz dt
D
Jfff Ft dx dy dz dt
D
fff F YN do,
r
fff F zN do ,
r
fff F tN do.
r
(4.13)
These formulae allow us to transform equations which involve derivatives of a function F(x,y ,z,t) into conditions for the function F itself.
4.2 Let us apply these transformations to the equation div EE conclude first
0. We
ffff (div E E) dx dy dz dt
D
JJJ E(E1XN + E 2yN + E3ZN) do . (4.21)
r
The expression E1x N + E 2yN + E 3z N can be interpreted as the scalar product of the two three-dimensional. vectors E and
M = (xN ' y N ' z N)
(4.22)
The vector M is the projection of the four-dimensional. unit vector N into the x,y ,z space, i.e.,
M
(x N , y N , z N , 0) .
WAVE OPTICS AND GEOMETRICAL OPTICS
17
By using the vector M we can write (4.21) in the form
JJJJ JJJ div EE dx dy dz dt =
E(E •M) do.
D
r
(4.23)
This integral relation, of course, is nothing but the integral theorem of Gauss for four dimensions and applied to a vector EE for which the fourth component is zero.
Since div EE = div µH formulate the statement:
0, according to Maxwell's equations, we can
The surface integrals
JJJ J E (E • M) do and jf µ(H •M) do
r
r
(4.24)
are zero for any closed hypersurface r in the four-dimensional x,y,z,t space.
We have derived this result from Maxwell's equations under the assumption that E, µ; E, H are continuous and have continuous derivatives. In this case the integral relations (4.24) are equivalent to the differential equations
div EE = div µH = O, as we can see easily. However the relations (4.24) canbe
applied directly to discontinuous functions as long as they are integrable. We will see presently that explicit conditions for discontinuities can be derived from (4.24). In view of this we consider the integral equations as the original source of the differential equations to which we have to go back in case of doubt ..
4.3 We next apply our transformation to the equation
curl H - ~ Et = 0 .
C
From curl H = i x Hx + j x Hy + k x Hz follows:
JJJJ JJJ curl H dx dy dz dt =
[(ixN + jyN + kzN)] x H do
D
r
Jjf (M x H) do .
r
(4.31)
Hence
~ f.lJf (curl H - Et) dx dy dz dt
18
MATHEMATICAL THEORY OF OPTICS
and similarly,
~ f.!Jf (curl E +~Ht) dx dy dz dt = f£f (M x E + tNH) do (4.33)
Hence: The surface integrals
(4.34)
are zero for any closed hypersurface r in the four-dimensional x,y,z,t space.
Again we notice that these conditions involve only the vectors E and H and the functions E and µ, and not their derivatives. They are equivalent to Maxwell's equations (1.20) if the derivatives exist. We require, however, that the integral relations (4.34) must be satisfied also by discontinuous electromagnetic fields.
§5. GENERAL CONDITIONS FOR DISCONTINUITIES.
5.1 We apply the integral equations (4.24) and (4.34) to the following
problem. Let rp(x,y,z,t) = 0 represent a surface section on which E, µ or
E, H are discontinuous. What is the
relation of the boundary values of E
and H on the two sides of <P = 0 to
each other? We consider a closed
hypersurface r which is divided into two parts r 1 and r by the hyper-
2
surface rp = O. Let r be the part 0
of <P = 0 which lies inside of r. The
normal of the surface rp = 0 is
proportional to the vector (<Px• <Py, <Pz,
rpt). Let us assume that on r 0 this vector points toward r We denote
• 2
the boundary values of (E, µ, E, H) by
(Ei, µ1, Ei, H1) if r 0 is approached
Figure 7
from the domain Di, and by (E 2 , µ 2 ,
E H if r is approached from D
,
)
•
2
2
0
2
5.2 We now, apply the first
equation (4.34) to the closed surface
0.
(5.21)
WAVE OPTICS AND GEOMETRICAL OPTICS
19
However, this condition must also be satisfied if the closed surface r 1 + r0
is chosen. On r 0 we have in this case
M
grad cp
<Pt
; tN =--/-=;: 2====2===2=====2 V<Px + <Py + <Pz + <Pt
(5.22)
and hence
If the surface r2+ r0 is considered, in which case on r0
grad cp
M
-;:::::::~::::::::::::::::::::::::::::::::::::::::::- ;
✓ <Pt + <Pi + <Pz2 + <Pt2
tN =
we obtain
O . (5.25) We finally subtract the equations (5.25) and (5.23) from (5.21). The result is
0.
(5.26)
This relation must be true for any part r 0 of the surface cp
possible if the integrand in (5.26) is zero; hence
0. This is only
where
grad cp x I H ] - <Pt [ E E ] 0 C
(5.261) (5.27)
denotes the size of the discontinuity of the quantity inside the bracket.
20
MATHEMATICAL THEORY OF OPTICS
5.3 We next consider the first integral equation (4.24) and apply it to the
three closed surfaces r + r r + r r + r We obtain
;
;
.
1
2
1
0
2
0
fff E(E •M)do + fff E(E •M)do = 0 ,
r1
r2
(5.31)
fff
r1
E(E • M)do
+
fff
ro
(E1E1 • grad cp)✓cp} +
do <Pi +
cpJ
+
<Pt2
0, (5.32)
fff E(E •M)do
r 2
0, (5.33)
and by subtraction
0 .
(5.34)
This yields in the same way as above: [EE],grad cp = 0.
(5.35)
5.4 From (5.26) and (5.35) two more equations can be found by interchanging the letters E and µ,, and E and -H. We summarize our results as follows:
An electromagnetic field which is discontinuous on a hypersurface
cp(x,y,z,t) = 0 must satisfy the conditions:
<Pt grad cp x [H] - -[EE]
= 0,
[ EE ] • grad cp
C
0.
'
<Pt grad cp X [E] + -[µHJ
C
0, [µ,HJ • grad cp = o .
(5.41)
We notice that the second column of equations follows from the first column if cp t =/- O. The equations (5.41) may be considered as the counterpart of Maxwell's differential equations. They represent a system of linear difference equations which take the place of the differential equations (1.20) and (1.23).
§6. DISCONTINUITIES OF THE OPTICAL PROPERTIES.
We apply the general conditions (5.41) to the special case where discontinuities of E and H are introduced by discontinuities of the functions E or
µ. Any system of glass lenses gives an example for this case. Let 1/J(x,y ,z) = 0
represent a refracting surface on which E(x,y,z) and µ(x,y,z) are discontinuous. We consider the hypersurface
cp(x,y,z,t) = lf! (x,y,z) 0
WAVE OPTICS AND GEOMETRICAL OPTICS
21
in the four-dimensional x,y,z,t space. This is a cylindrical hypersurface the
generating lines of which are parallel to the t-axis. In this case, since <Pt = 0,
our conditions (5.41) become
grad 1/J x [H] 0, [EE]• grad 1/J 0, grad 1/J x [E] 0, [µH] • ~rad 1/J 0 .
(6.1)
The vectors g~;=a~ ; IH and g~~a~ ; I E are linearly related to the tangential components of H and E. The quantities El !r:aa:I 1/1 and µ~~:~~I 1/1 are the
normal components of EE and µH. Therefore we_may formulate the conditions (6.1) in the following customary way:
The tangential components of E and H and the normal components of EE and µH are continuous on a surface of discontinuity of E and µ.
§7. PROPAGATION OF DISCONTINUITIES; WAVEFRONTS.
7.1 Discontinuities of the electromagnetic field can appear without being
caused by a discontinuous distribution of substances. Let us, for example,
consider the case E = µ = 1 and
assume that at t = 0 the vectors
E(x,y,z,0) and H(x,y,z,0) are different
L
from zero only tn a small sphere of
radius o around the origin. We expect,
in analogy to other forms of wave
\\
0
motion, that this electromagnetic field
expands with increasing time such that
8
X
at a given time t > 0 the vectors E
and H are different from zero in a
larger sphere of radius o + ct. In
other words we expect that the surface
which separates the parts of the space
which are still at rest from those
penetrated by the original impulse
Figure 8
travels over the space. A surface of
this type is called a wavefront. In the
above example the wave fronts are
spherical and given by the equation
cp(x,y,z,t) =✓x2 + y 2 + z 2 - cS - ct = 0.
(7 .11)
If the boundary values of E(x,y,z,0) or H(x,y,z,0) on the original sphere
of radius o are different from zero then this sphere is a surface on which the
electromagnetic field is discontinuous. We must expect that at the time t > 0
22
MATHEMATICAL THEORY OF OPTICS
the corresponding boundary values on the wavefront (7 .11) are likewise different from zero so that the electromagnetic field is also discontinuous on the new wave front. This consideration leads us to define a wave front more generally as any surface in the x,y ,z space on which, at a given time t, the electromagnetic field is discontinuous.
An observer at a point x,y ,z will interpret such a discontinuity as a sudden signal which reaches him when the wave front goes through the point x,y,z.
Instead of illustrating the equation (7 .11) by a set of surfaces in the threedimensional x,y ,z space depending on the parameter t, we can interpret such
a relation rp(x,y ,z,t) = 0 as a hypersurface in the four-dimensional space
x,y,z,t. In our example this hypersurface is the cone
and the electromagnetic vectors are discontinuous on this cone. Its "contour
lines", i.e., the cross sections of the hypercone, rp(x,y,z,t) = 0, with the
hyperplanes t = const., then represent the above set of wave fronts in the
x,y,z, space.
of rp
7.2 We may expect from
the above example that the
hypersurfaces rp = 0 which
determine the propagation of
discontinuities are not arbitrary
but must fulfill certain con-
ditions. We can derive these
conditions easily with the aid
of the general relations (5.41).
Let us assume that rp(x,y,z,t) =
0 represents a hypersurface on
which the vectors E and H are
Figure 9
discontinuous. The functions
E (x,y ,z) and µ(x,y ,z) shall be
continuous in the neighborhood
0. We introduce, on rp = 0, the vectors
u [El
V = [HJ
E2 E1' H2 - H1'
(7 .21)
which measure the discontinuity of E and H on rp 0. It follows from (5.41):
gradrpxV- i rpt u C
0.
'
U • grad rp
0. '
grad rp x U + I!:. <pt V C
O· V • grad rp '
0.
(7.22)
WAVE OPTICS AND GEOMETRICAL OPTICS
23
The first column of these equations represents a system of six linear homogeneous equations for the six components U1,U 2 ,U 3 ; V1,V 2 ,V 3 • This system can have non-trivial solutions, U -=f- 0 and V -=f- 0, only if the determinant is zero. This establishes the desired condition for the function rp(x,y,z,t) = 0. We can derive this condition as follows: We form the vector product of grad rp with one of the equations (7.22), for example, with the second equation:
grad rp x (grad rp x U) + g_ <Pt grad rp x V = 0 C
and introduce grad rp x V = ~ <Pt U from the first equation. It follows
E½ grad rp x (grad rp x U) + r,oi2 U = 0 . C
If we apply the vector identity (Appendix: I.23) we obtain
(U • grad rp) grad rp - (grad rp) 2 U + ~ r,oi2 U C
0.
'
or, since U • grad rp = 0,
(7.23)
0.
(7 .24)
In a similar way we find
?" ((grad r,o) 2 -
<Pt2 )v = o
(7.25)
and conclude: If U and V are different from zero, i.e., if E and H are dis-
continuous on rp = 0, then rp(x,y,z,t) must satisfy the equation
?" (grad <,0)2 = <Px2 + r,o/ + r,o/ = <,0( .
(7.26)
This equation is called the characteristic equation of Maxwell's differential
equation. Every function rp(x,y ,z ,t) which, for rp(x,y ,z ,t) = 0, satisfies this
equation (7 .26) represents a hypersurface which is called a characteristic surface of the differential equations.
7,3 The characteristic equation (7.26) is not a true differential equation for rp(x,y,z,t); indeed it does not have to be satisfied identically in x,y,z,t but
only for those combinations x,y,z,t for which rp(x,y,z,t) = 0,
24
MATHEMATICAL THEORY OF OPTICS
We can, however, assume, without loss of generality, that the characteristic surface is given in the form
cp = 1/J(x,y,z) - ct = 0,
where 1/J(x,y,z) is independent of t. We obtain
(7 .31)
and this equation must be satisfied identically in x,y ,z.
We may formulate our result as follows: If E and H are discontinuous
on a set of wave fronts p(x,y,z) = ct then p(x,y,z) must be a solution of the
partial differential equation (7.31).
The equation (7.31) is called the equation of the wave fronts; in some
literature it is known as the Eiconal Equation. It is the basic equation of
Geometrical Optics; the greater part of this course is conc~~ed ;Nith'pro-
blems related to this equation.
\ '. ' • .
\ , l
,
If we introduce cp = 1/J - ct in the original equations (7 .22) we obtain
grad 1/J x V + EU O , gradl/JxU-µV 0.
(7 .32)
It is not necessary to add the other two equations (7 .2,2)-explicitly since both equations are a consequence of (7 .32):
U•grad 1/J O, V·gradlfJ O.
(7.33)
We furthermore conclude
U•V = 0.
(7.34)
Hence: The vectors U and V are tangential to the wave fronts and perpendicular to each other.
If 1/J - ct = 0 represents a set of wave fronts in the, sense of our
original definition, namely, boundaries of regions which have been penetrated by a light impulse, then U and V are equal to the vectors E and H on the
wave front (because E = H = 0 on one side of the surface
1/J - ct = 0).
WAVE OPTICS AND GEOMETRICAL OPTICS
25
We find: The electromagnetic vectors on
E
such a wave front are tangential to the
wave front and perpendicular to each
other.
§8. BICHARACTERISTICS; LIGHTRAYS.
Figure 10
8.1 The problem of integrating a partial differential equation of first order can be reduced to the problem of integration of a system of ordinary differential equations, the so-called characteristic differential equations. The integral curves of the characteristic equations are known as characteristics. The equation of the wave fronts
(8.11)
is itself a characteristic equation of Maxwell's differential equations. Therefore the characteristics of this first order equation are called Bicharacter.istics of Maxwell's equations.
For our purpose it is not necessary to introduce these bicharacteristics by general considerations which would apply to any partial differential
equation of first order. We would find that the bicharacteristics in our special case are nothing but the orthogonal trajectories of the wavefronts
if! = ct. Hence we prefer to introduce these bicharacteristics directly as orthogonal trajectories of a set of wavefronts 1/J = ct. We call these trajec-
tories the light rays of the optical medium and we will see in the following that this name is justified.
x, y t z
8,2 Let us consider a set of wavefronts 1/J(x,y ,z) = canst. An orthogonal trajectory of these surfaces at any point x,y ,z is normal to the wavefront through this point. The complete manifold of orthogonal trajectories through the given
set of wavefronts 1/J = canst. thus must be
identical with the solutions of the differential equations
Figure 11
dx dcr
dz dcr
(8.21)
26
MATHEMATICAL THEORY OF OPTICS
where l.. = l..(x,y,z,CT) is an arbitrary factor. The choice of l.. does not in-
fluence the geometrical form of the trajectories but only their parametric representation.
The orthogonal trajectories (8.21) depend of course upon the chosen set
of wavefronts, i.e., on the particular solution z/1 of (8.11). It is now significant
that it is possible to determine light rays, i.e., orthogonal trajectories of
surfaces z/1 = const. without reference to a particular solution z/1 of (8.11). Indeed, if we differentiate .! dx with respect to CT we obtain
l.. dCT
d dCT
(1~
dx) dCT
dx
~
dz
= lfixx dCT + z/!yx dCT + lfizx dCT '
A(z/!xx lfix + z/!yx z/!y + z/!zx !/Jz) ,
and hence, on account of (8.11)
1 d ( 1 dx) l.. dCT l.. dCT
1 an2 2 ax
(8.22)
By dealing similarly with the other equations (8.21) we find: The orthogonal trajectories (8.21) form a two-parameter manifold of solutions of the 2nd order equations:
1 d ( 1 dx) l.. dCT l.. dCT
-1 2
-aa(xn2)
'
-1 -a(n2)
2 ay '
(8.23)
We remark again that the choice of l.. does not affect the geometric form of the integral curves of (8.23). This can be seen from the following fact: Any particular solution x(CT), y(CT), z(CT) of (8.23) can be transformed by a transformation of the parameter CT into a solution x(CT'), y(CT'), Z(CT') of the equations
(8.24)
where l.. = 1. Such a transformation, however, does not affect the geometric
shape of the curve.
WAVE OPTICS AND GEOMETRICAL OPTICS
27
8.3 We choose first ;>.. = 1 and denote the parameter O" by T. The
equations (8.21) become
~
dz
dT
dT
(8.31)
and the equations (8.23) become:
(8.32)
The orthogonal trajectories (8.31) thus form a two-parameter manifold of solutions of the equations (8.32). According to (8.31) we have
(8.33)
The analogy of (8.32) to the equations of mechanics is obvious. If we interpret
- ½n2 as a potential field our light rays can be regarded as paths of particles ½ moving in this field with energy (x2 + y2 + z2) - ~ 2 = o.
8.4 We choose next ;>.. = 1/n and denote the parameter O" by s. It
follows:
nd-x ds
,,, . n ~ = ,,, . n dz
'l'x '
ds
'f'Y '
ds
!Jlz
(8.41)
and hence ( dxds)2 + (~ds)2 + (ddzs)2 = 1, i.e., the parameter s measures the
length along the light rays. Equations (8.23) become
d
an
ds (n :) ax'
f) d
an
ds (n
ay,
(8.42)
d
an
ds (n ::) az
28
MATHEMATICAL THEORY OF OPTICS
These equations can easily be recognized as the Euler equations of the variation problem
J P1
V
n ds is an extremum.
Po
(8.43)
Let us consider two points P0 and P 1 in the x,y,z space. Let x(s), y(s), z(s) be a continuous curve between P0 and P 1 which also shall have a continuous tangent. We define the optical length of this curve by the integral
JP1
V
n ds.
Po
(8.44)
In case n = 1 this optical length coincides with the geometrical length.
The problem is to find the curve for which the optical length is a minimum. Let us assume that a solution exists and is given in the form
x = x(u), y = y(u), z = z(u) where u is a parameter, such that u = 0 at P0 and u = 1 at P 1 ; hence
V =
(8.45)
Figure 12
is a minimum.
The necessary conditions which the solution must satisfy are Euler's differential equations, i.e., in case of (8.45):
( ✓x•2 :u
n;t
ny ✓x•2 + y•2 + z'2 0 '
+ y'2 + z'2 )-
(8.46)
( ✓x•2 :u
nz'
nz ✓x•2 + y'2 + z'2 0 .
+ y•2 + z'2 )-
If we introduce in these equations the geometric length s of the solution as a parameter we obtain the differential equations (8.42).
WAVE OPTICS AND GEOMETRICAL OPTICS
29
The light ray between two points P0 and. P1 is the curve for which the optical path attains an extreme value.
This result is known as Fermat's principle of geometrical optics.
If the index of refraction is interpreted as the ratio c/v of the velocity of light in a vacuum to the velocity v(x,y,z) in the medium, the optical path
J J J V = n ds = c !s = c dt
becomes proportional to the time needed to travel from P0 to P 1 • The principle of Fermat states that the light ray is a curve on which this time is a minimum, or at least an extremum.
§9. CONSTRUCTION OF WAVE FRONTS WITH THE AID OF LIGHT RAYS.
9.1 Every solution lf.,(x,y,z) of the equation
(9.11)
determines a two-parameter manifold of light rays, i.e., of orthogonal trajectories. We have seen that these light rays satisfy a system of ordinary. differential equations
x n nx
y n ny
(9.12)
z n nz
and the condition
x2 + y2 + z2 = n2 .
(9.13)
The dot means differentiation with regard to the parameter t.
Our aim in the following is to show that the two problems of integrating the partial differential equation (9.11) or the system of ordinary differential equations (9.12) are equivalent.
Let us first assume that the solutions of (9.12) are known. We show that it is possible then to solve the following problem simply by quadratures and eliminations.
30
MATHEMATICAL THEORY OF OPTICS
Let r be an arbitrary surface section given in parametric form
X f(L 7))
y g(L 7J)
(9.14)
z h(L 7J)
To find a solution !Jl(x,y,z) of (9.11) ;which, on r, has given values
!JI = F(L 7J ).
9.2 We know, if !JI is the desired solution, that the orthogonal trajectories of the surfaces !JI const. are solutions of (9.12) and (9.13) and the optical length
between two points P0 and P 1 of such a trajectory is given by the difference
(9.21)
Figure 13
This leads to the following attempt to solve the above problem. We determine
through every point ; , 7J of r a light ray,
i.e., a solution
x = x(L7Ji T),
y y(L 7j; T) '
(9.22)
z z(L 7J; T) ,
of (9.12) and (9.13) which satisfies the boundary conditions
x(L 7J, O) f(;,7J),
x(L 7J, O) a(;' 7J) '
y(;' 7j' 0) z(L 7J, 0)
g(;' 7j) ' h(L 7J) ,
y(L 7J, O) z(L 7J, O)
b(L 7J) , c(L7J).
(9.23)
The functions a,b,c must obey the condition
a2 + b2 + c2 = n2(;,7J)
(9.24)
in order to insure that the functions (9.22) satisfy the condition (9.13) but are otherwise arbitrary.
WAVE OPTICS AND GEOMETRICAL OPTICS
31
We now consider the expression
T
f 1/J(L T'/, T) = F(;, ri) + n 2 dT 0
(9.25)
and expect that the solution of our problem can be obtained in this form when ; , T'/, T are expressed as functions of x,y ,z with the aid of (9.22). We assume
that the Jacobian ~~;,:,:~ is not zero in the neighborhood of r in order to be
able to carry out this elimination.
9.3 It is clear that 1/J(x,y ,z) has the correct boundary values on r; for
T = 0, we have 1/J = F(;, T'/ ). We show next that if; satisfies the equation
(9.11) if the functions a,b,c are chosen suitably. We determine the deriva-
tives of 1/J(;, T'/, T). First,
§!I!. = n2 = :x:2 + y2 + z2
aT
Then, we write (9.25) in the form
(9.31)
I/! and obtain
(9.32)
TJ (. az) +
O
X
~ ax
+
y •
£i.
a;
+
•
Z
~
d
T ,
We introduce n nx x, n ny = y, n nz = z and find
1/!g I/Jg and similarly l/!11
~ Fg +
T d dT
(x•
ax+ a;
• EX.+ Y a;
• z
az) a;
dT
Fi
-
( a
af a;
+ Qg + ah) b a; ca;
+x. aa-x;+y•
EX. a;
+
. az z~,
(9.33)
F ij
-
(
af aar
i
Qg +bari
ah +cari
)
+.xaa-rxi+ y•
EX.
ari
+
• z
az
ari
•
(9.34)
32
MATHEMATICAL THEORY OF OPTICS
We now assume that the functions a,b,c satisfy the conditions
a a8ff +
b
~ + 8~
C
a8hf
=
Ft
'
a8-f+ bB=a- + c8-h
81'/ 8rJ 8rJ a2 + b2 + c2 = n2 .
(9.35)
These equations have two solutions (a,b,c) provided that not all of the subdeterminants of the matrix
(9.36)
are zero. Let (a,b,c) be one of these two solutions. With this choice of a,b,c we obtain
•
X
a8xf
+
. y
£l.
8;
+
.
z
a8zf
'
x. -8+xy::.:..8.vz...+. z8z-
8rJ 8rJ 8rJ '
(9.37)
On the other hand
1/Jx
a8x[
+
1/iy
£l.
8;
+
1/Jz
8z 8;
,
8x
£l.
8z
1/Jx 8rJ + 1/Jy 8rJ + 1/Jz 8rJ •
8x
£l.
8z
1/J X 8T + 1/J y 8T + 1/J z 8T '
(9.38)
Since the determinant B(x,y,z) is different from zero it follows by comparing
(9.37) and (9.38):
8(~, 7) ,T)
1/Jx = X, 1/Jy = Y, 1/Jz z.
(9.39)
WAVE OPTICS AND GEOMETRICAL OPTICS
33
Therefore, on account of (9.13)
i.e., 1/J is a solution of (9.11).
Actually, our method allows us to find two solutions of the above problem as we expect from the quadratic nature of (9.11).
9.4 We apply our result to the case F(~, 11) = O. The surface r itself
is thus a wave front, namely, the wave front at the time t = 0. The problem
is to find the position of the wave front at the time t. We calculate the two-
parameter set of solutions of (9.12) which intersect- the surface r with direc-
tions a(L 11), b(L 11), c(L 11), The quantities a,b,c can be found from (9.35), i.e., from
a
af 8~
+
b
£B:
8~
+
c
~ah
0 '
0 '
(9.41)
a2 + b2 + c2 = n2 .
From the first two of the above equations it follows that the light rays must
be normal to the surface r. When these rays have been found:
X
X(~ , 1), T) ,
y y(L 1)' T) '
(9.42)
z z(L 1J, T) ,
r
we obtain the solution 1/J by the integral
J T
1/J(L 1J, T) = n 2 (x,y ,z) dT ,
0
(9.43)
Figure 14
in which ~ , 7J , T have to be expressed in terms of x,y,z with the aid of (9.42).
9.5 The original wave surface. r of the preceding section may degen-
erate into a point, (x 0 , Yo, z 0). The functions f, g, h are constant in this case, so that the conditions (9.41) reduce to only one condition:
(9.51)
34
MATHEMATICAL THEORY OF OPTICS
Let us determine the two-parameter set of light rays through the point (xo, Yo, z 0), i.e., solutions
x x(a,b,c,T) ,
y y(a,b,c,T) ,
(9.52)
z z(a,b,c,T),
of (9.12) which satisfy the boundary conditions
x(a,b,c,O)
x(a,b,c,O) a,
y(a,b,c,0) Yo z(a,b,c,O)
y(a,b,c,O) b , z(a,b,c,O) C •
We obtain a solution 1/J(x,y,z) of (9.11) in the form of the integral
(9.53)
T
1/J(a,b,c,T) = J n 2 (x,y,z) dT
0
(9.54)
after a,b,c,T have been expressed by x,y,z with the aid of the relations (9.51) and (9.52).
These special solutions are called "Spherical" Waves or simply wavelets. If x 0,y 0,z 0 are considered as variable parameters 1/J becomes a function of two points
T
V(x 0, Yo, z 0 ; x, y, z) = J n 2 dT
0
(9.55)
It determines the optical distance of the two points (x 0,y 0 ,z 0) and (x,y,z). The spherical wave fronts around a point (x 0,y 0,z 0) then are given by the surfaces
V(x 0,y 0 ,z 0 ; x,y,z) - ct = O.
(9.56)
9.6 Huyghens' Construction. With the aid of the wavelet
V(x y z x, y, z) another method can be obtained to determine the wave
,
O,
;
O
O
fronts belonging to a given single wave front r. This method is of prime
importance and is known as Huyghens' construction. We consider the wave-
let functions V(~, 1): x,y,z) which belong to the points (~,TJ) of the surface r,
At the time t a two-parameter set,
V(~,T);x,y,z) - ct 0
(9.61)
WAVE OPTICS AND GEOMETRICAL OPTICS
35
of spherical wave fronts is obtained; we show that the envelope of these wave fronts is the wave front
7/J(x,y ,z) - ct = 0
which, at t = O, coincides with the given surface r.
Figure 15
We find the envelope of the surfaces (9.61) by eliminating the parameters (~ , 7J) from the three equations,__
V ~ (~, 7J ;x,y ,z) 0 ,
V1J (~, 7J ;x,y ,z) 0 ' V (~ , 7J ;x,y ,z) - ct = 0 .
(9.62)
V(!,77 1x,y,z) =ct
Let
~
7J
A(x,y,z) , B(x,y ,z)
(9.63)
Figure 16
be the result of calculating ~, 7J from the first two equations (9.62). We introduce ~ and 7J in V(~, 7J ;x,y ,z) and show that
7/J(x,y,z) =
V (A(x,y ,z) ,B(x,y,z);x,y ,z) (9.64)
is a solution of (9.11). Indeed,
or, on account of (9.62):
and similarly, 7/Jz
(9.65)
36
MATHEMATICAL THEORY OF OPTICS
These equations state that, at the point (x,y ,z) the surface 1/J - ct = 0 has the same tangential plane as the wave V(i;, 71;x,y,z) - ct from the point(~, 71), by (9.63). Since V satisfies the equation (9.11), the same is true for the function (9.64).
The function V(A,B;x,y,z) determines the optical distance of a point
(x,y,z) from the corresponding point (~, 71) defined by (9.63). If the point
(x,y ,z) approaches the surface r the corresponding point (~, 71) on r
approaches the same limiting point. Instead of proving this analytically, we
will refer to the geometric evidence. Hence 1/J = V(~, 71 ,x,y,z)-0 if (x,y,z) approaches the surface r. The function (9.64) thus is the desired solution.
§10, JACOBI'S THEOREM.
10.l In this section we shall be concerned with the inverse problem. Suppose we are in a position to integrate the partial differential equation
(10.11)
To find the general solution of the differential equations of the light rays.
The answer is given in a general theorem of Jacobi. This theorem, applied to the differential equations (10.11) states: Let 1/J(x,y,z; a,b) be a complete integral of the equation (10.11). A complete integral is defined as a set of solutions which depend on two arbitrary parameters a and b such that not all of the subdeterminants of the matrix
1/Jxa.• (
1/Jza.•)
1/Jxb•
1/Jyb,
1/Jzb,
(10.12)
are zero. Then the light rays of the medium of refractive index n(x,y ,z) are given by the equations
a
aa 1/J(x,y,z; a,b) Cl! ,
a
ab 1/J(x,y ,z; a,b)
= {3
,
(10.13)
where Cl! and {3 are arbitrary constants.
If, for example, the determinant
1/Jya.
1/Jza.
=f. 0 '
1/Jyb
1/Jzb
WAVE OPTICS AND GEOMETRICAL OPTICS
37
then we may calculate y and z as functions of x:
y y(x; a,b,O! ,/3) ,
z = z(x; a,b,a ,/3) .
(10.14)
These functions represent a four-parameter set of curves which, according to Jacobi's theorem, are the light rays of the medium.
10.2 For the proof of Jacobi's theorem let us assume that the curves (10.13) are given in parametric representation:
x = x(cr; a,b,a,{3) ,
y y(cr; a,b,a,{3) ,
(10.21)
z = z(cr; a,b,a,{3) .
By introducing these functions in (10.13) we obtain identities in cr,a,b,a,{3.
HE_mce by differentiation with respect to er:
. I/Jax x+ I/Jay y+ 1/Jaz z 0'
z 1/Jbx x + 1/Jby y + 1/Jbz
0.
(10.22)
The six quantities
I/Jax• 1/Jbx•
1/Ja.y• 1/JbY•
1/Ja.z, 1/Jbz•
(10.23)
can be interpreted as two vectors which, on account of (10.12), are not linearly dependent and thus determine a plane. The equations (10.22) state that the vector (x,y ,z) is perpendicular to this plane. From (10.11) we have by differentiation with respect to a and b:
(10.24)
i.e., the vector (1/Jx, 1/Jy, 1/J.) is also normal to the above plane. Hence,
x z = ?..1/Jx, y = ?..1/Jy, = ?..1/Jz, •
(10.25)
38
MATHEMATICAL THEORY OF OPTICS
By differentiating these last equations with respect to O" we have as in section 8.2:
- -(!) 1 d •
;>,. dO" ;>,.
nny'
(10.26)
which shows that the curves (10.21) are light rays.
10.3 Example. Let us consider the case of a stratified medium where
n = n(z). We verify easily that
l/J = ax + by + / ✓n 2 - a 2 - b 2 di; 0
is a complete integral of the equation
(10.31)
The light rays in such a medium thus are given by
E!l!.
ila
t X - a
di;
o✓n2 (1:) - a2 - b2
Cl! '
E!l!.
8b
y -b {
di;
o✓n2 (l;) - a2 - b2
(3
or
J X = Cl!+ a z
di;
o ✓n 2 (1:) _ a 2 _ b2
y
(10.32) (10.33)
WAVE OPTICS AND GEOMETRICAL OPTICS
39
§11. TRANSPORT EQUATIONS FOR DISCONTINUITIES IN CONTINUOUS OPTICAL MEDIA.
We shall derive certain differential relations in this section which allow us to calculate the discontinuities of an electromagnetic field along a given light ray if the discontinuity is known at one point of the ray. We assume
explicitly that the functions E = E(x,y,z) and µ, = µ,(x,y,z) are continuous
functions. The case of discontinuous optical media will be studied later and
a principal difference between both cases will be found.
11.1 Differentiation along a light ray. Let F(x,y,z) be a differentiable
function. Along a given light ray x = x(T), y = y(T), z = z(T), a function
F(T) = F(x(T), y(T), Z(T)) ,
is obtained whose differential quotient is
dF dT
x On account of the relations = 1/Jx, etc. this becomes
(11.11)
Hence the differential operator
a
a
a
a
8T = 1/Jx 8X + 1/Jy 8y + 1/Jz 8z
(11.12)
can be interpreted as differentiation along a light ray provided that 1/J(x,y ,z) is a solution of the equation
Let F(x,y ,z) 1/J(x,y,z), for example. It follows
E!l!. = , = 8T
1, 2 + ,1, 2 + ,1, 2
'l'x
'l'Y
'l'Z
n2
•
(11.14)
The operator (11.12) can also be applied to a vector field; for example
(11.15)
11.2 We consider an electromagnetic field which is discontinuous on the hypersurface
cp(x,y,z,t) 1/J(x,y,z) - ct 0.
(11.21)
40
MATHEMATICAL THEORY OF OPTICS
Let D and D' be two domains of the x,y,z,t space which are separated by the
hypersurface cp = 0. We assume that E
and H are continuous functions with continuous derivatives in the individual domains D and D'. The boundary values
on cp = 0 which are assumed if this
surface is approached from D and D' are denoted by
Figure 17
E',H',Ex',Hx', ... , respectively.
The discontinuities U and V of E and H are then given by
U E 1 - E,
V H' - H.
First we consider the boundary values of E,H in the domain D. Both
E and H(x,y,z,t) become functions of x,y,z on cp = 0 = 1/J - ct. We denote
these vectors by
¾ E*(x,y,z) E (x,y,z; 1/J(x,y,z)) ¾ H*(x,y,z) = H (x,y,z; 1/J(x,y,z))
(11.22)
The derivatives, for example of E *, are given by
E*y
E*z
Therefore
curl E*
and similarly curl H*
curl E + -1 grad 1/J x Et C
curl H + -1 grad 1/J x Ht . C
(11.23) (11.23)
WAVE OPTICS AND GEOMETRICAL OPTICS
41
From Maxwell's equations: curl E = _ I± Ht and curl H C
.f E .
C t•
hence:
c curl E* c curl H*
- µHt + grad 1/J x Et ,
(11.24)
These equations can be considered as a system of six linear equations
for the six components of the vectors Et and Ht . The matrix of these equations is the same as the matrix of the homogeneous equations for the
discontinuities U and V on cp = 0: .
grad 1/J x V + EU O , grad 1/J x U - µV O ,
(11.25)
and we know that the determinant of this matrix is zero on 1/J - ct = 0. We
conclude that the equations (11.24) are possible only if the left sides satisfy certain conditions. These conditions will now be derived.
We form the vector product of grad 1/J with the second equation (11.24):
c grad 1/J x curl H* = E grad 1/J x Et + grad 1/J x (grad 1/J x Ht)
or on account of the vector identity (Appendix I.23)
c grad 1/J x curl H* = E(grad 1/J x Et - µHt)+ (Ht· grad 1/J) grad 1/J .
Hence with the aid of the first equation (11.24):
1 grad 1/J x curl H* - Ecurl E * = - (Ht· grad 1/J) grad 1/J .
C
(11.26)
This equation states: The vector grad 1/J x curl H* - Ecurl E * has the direction of grad 1/J, i.e., is normal to the wave front 1/J = ct.
The same considerations can be applied to the boundary values
E'* = E'(x,y,z,.!1/J) , C
H'* = H'(x,y,z,.!1/J) . C
(11.27)
If 1/J - ct 0 is approached from the domain D', we find that the vector
grad 1/J x curl H'* - Ecurl E' *
42
MATHEMATICAL THEORY OF OPTICS
is normal to the wave front ljJ = ct. Finally, by considering the differences U = E'* - E* and V = H' * - H* we have: The vector
grad 1/J x curl V - Ecurl U is normal to the wave front 1/J = ct.
We formulate this statement in the equation
Ecurl U - grad ljJ x curl V = R grad ljJ
(11.28)
where R is a certain scalar function of x,y ,z. We can determine R explicitly by forming the scalar product of grad ljJ with equation (11.28). It follows
R
=
1 -µ
(grad
l/J • curl
U) .
(11,281)
Let us, finally, introduce U = _.! (grad ljJ x V) with the aid of (11.25). The
result is
E
¾ curl (-; grad ljJ x V) + grad ljJ x curl V = - ~ grad ljJ
(11.29)
which is a differential equation of first order in the discontinuity V.
11.3 We can transform the equation (11.29) into a much simpler form.
¾ We remark that the vector V x curl ( grad ljJ) has the direction of grad l/J.
Indeed, on account of the vector formula (Appendix I.412), we have
(¾ ¾ ¾) curl grad ljJ) = curl grad ljJ + ( grad x grad l/J
¾) (grad x grad l/J •
Hence,
V x curl(¾ grad¢) = (V·grad ¢)grad¾ - (v·grad¾) grad l/J
which proves our statement, for V • grad l/J = 0. Consequently, we can write
(11.29) in the form
¾ ¾ ¾ curl ( grad ljJ x V) + grad l/J x curl V + V x curl ( grad l/J)
= R' grad ljJ
(11.31)
WAVE OPTICS AND GEOMETRICAL OPTICS
43
where R' is a certain scalar function. The left side can be transformed with the aid of the vector identity (Appendix I.48) by introducing
1 A=-gradl/J, B=V
E
and
(11.32)
1 a
E aT
Using A· B = 0, we obtain
- -2 -av - V di.v (1- grad
E aT
E
- R* grad 1/J,
(11.33)
R * being a new factor. We find R * by forming the scalar product of (11.33) with grad 1/J. We obtain
n 2 R*
2 av
E a:;:--•gradl/J
-
-2
a
V·-
grad
1/J
E
aT
or, by (11.15)
- -1V•grad n 2
E
i.e.,
R*
2 V•grad n
E
n
(11.34)
Equation (11.33) becomes
o.
(11.35)
Finally we introduce the notation
i.e.' (11.36)
44
MATHEMATICAL THEORY OF OPTICS
and we get the equation
2 av
87
+
1
f:J.dJV
+
~1 (V • grad n)grad l/J
= 0.
(11.37)
A similar relation can be found for the discontinuity U by replacing E and V by µ and -U in (11.37).
Thus our complete result is: The discontinuities U and V satisfy the differential equations:
(11.38)
where the differential operators .6.€1/J and .6.µl/J are defined by
[(¾ (¾ (¾ .6.€1/J = €
l/Jxt + l/!yt + l/Jz)z] '
µ [(t (t (t .6.µl/J =
l/Jx)x + l/!y)y + l/Jz\]
(11.39)
§12. TRANSPORT OF DISCONTINUITIES. (CONTINUED).
12.1 On a given light ray the equations (11.38) represent a system of ordinary differential equations. Indeed, we have shown in (11.1) that the dif-
ferential operator a/aT differentiates a function in the direction of a light ray.
Let us introduce, in (11.38), instead of U and V, the vectors
u
p
(12.11)
Figure 18
which have the same directions respectively as U and V, but different lengths. The differential equations (11.38) then assume an even simpler form:
WAVE OPTICS AND GEOMETRICAL OPTICS
45
dP -d
T
+
-n1
(P
• grad
n)
grad
1/J
=
0 ,
dQ -d
T
+
-n1
(Q
• grad
n) grad
z/!
=
0 .
(12.12)
Since U and V are orthogonal to grad z/!, the same is true for P and Q. From U·V = 0 it follows that P •Q = 0.
By forming the scalar product of P and Q with the equations (12.12) it follows that
0 ,
0. -
(12.13)
This shows:
The lengths of the vectors P and Q are not changed on a given light ray. Thus without loss of generality we can assume
and interpret P and Q as unit vectors which determine the directions of the vectors U and V.
If P and Q have been found as solutions of (12.12), we obtain U and V from
u
J 1 1"
IUol P e
2
Aµz/!dT
0
1 J-r Aez/!dT
V = IVol Q e 2 0
(12.14)
These equations make it evident that U and V are zero on the whole light ray
if they are zero on one particular point, T = 0, of the ray. The light rays
thus determine the region of the space where
directed signals can be seen. Let us assume
that from the point 0 a light signal is released
at the time t = 0. Let us furthermore assume
r
that the discontinuities U and V which repre-
sent the signal are different from zero only on
a section r 0 of the wave front z/! = ct 0• From
(12.14) it follows that, at a time t > t 0, only on
the corresponding section r of the wave front
z/! = ct will discontinuities U,V be observed.
Figure 19
This section is determined by the light rays
46
MATHEMATICAL THEORY OF OPTICS
through r 0 • In other words, the light signal will be observed only in the part of the space which is covered by the light rays through r 0 •
This does not exclude the possibility that light penetrates into other regions of the space. However, this "diffracted" excitation does not have a sudden discontinuous beginning.
12.2 The exponential factors in (12.14) have a simple geometric meaning. We have
or Similarly
a
t:..1/) - aT (log E) ,
a
t:..µ1/J = t:..1/) - aT (log µ) •
(12.21)
Let us now consider a "tube" of light rays, i.e., a domain D of the x,y,z space which is enclosed by a surface r consisting of two sections r 1 and r 2 of the wave fronts
1/) = Pt and 1/J = P2
and the cylindrical wall rs formed by the light rays through the circumference of r 2 and r 1. We apply the theorem of Gauss to this domain D:
Figure 20
fff t:..1/) dx dy dz
D
ff ~do
r
(12.22)
where Ea!Ev. is the derivative of ¢ in direction of the outside normals. However,
E!E.
av
o on rs '
E!E.
av
n2 on r2 ,
E!E.
av
- n1 on r 1 •
WAVE OPTICS AND GEOMETRICAL OPTICS
47
Hence,
fff fllj) dx dy dz D
(12.23)
We now express the surface elements do 2 and do1 by the corresponding
surface elements do of an arbitrarily chosen wavefront lj) = ct 0• We write
(12.24)
The factor K measures the expansion of an infinitesimally narrow tube of light rays. It follows that
fff fllj) dx dy dz D
The volume element dx dy dz can be expressed as follows:
dx dy dz = K do ds = nK do dT .
Hence,
fff fllj) dx dy dz = fff ~ a~nK) dx dy dz .
D
D n
T
Since D is of arbitrary size, we find
fllj) = _.!.. _.£_ (nK)
nK 8T
a
87
(log
nK)
and hence
fl El/)
-8aTl
o
gnK-
E
flµlJ!
-8aTlog -nµK .
(12.25) (12.26)
48
MATHEMATICAL THEORY OF OPTICS
The exponential factors in equation (12.14) become
(12.27)
and we conclude from (12.14):
K
Ko
-nElul2 = -no Eo 1u o12 '
(12.28)
That is,
the quantities
~
n
E
lu
l
2
and
~n µ lvl 2
are constant along a light ray.
This result allows us to determine the lengths of U and V along the
1° light rays of a given set of wave fronts !/J = ct without integration, simply by
calculating the ratio : =
of corresponding surface elements of the
wave fronts.
0
00
12.3 Energy and flux on a wave front. Let us assume that the electro-
magnetic field is zero on one side of the wave fronts !/J - ct = 0. In this
case E = U and H = V; hence
w u v s cu 1 = 871" (E
2 + µ
2),
C
= 471"
x V).
(12.31)
E
It follows from (11.25) that
g rdd 1/1
E U2 U • (V X grad !/J) , (grad !/J x U) •V ,
and thus
EU2 = µV 2
(12.32)
Figure 21
i.e. Electric and magnetic energies are equal on a wave front. From (11.25) it follows further that
WAVE OPTICS AND GEOMETRICAL OPTICS
49
U x V
1 = ~(V x grad l/1) x V
=
1 n2
µV
2g
rad
1/J
41r
2n
W
grad
l/J.
Hence,
S
=
C
2n
Wgradl/J.
This yields for the absolute value of the Poynting vector:
(12.33)
1s1 = ~n w.
(12.34)
This result allows us to interpret the equations (12.28). If we add both
equations we find ~W n
Ko no
W0
and, introducing KK o
-1 W n
do
-1 W no
do 0 •
Finally, on account of (12.34):
ddo , we obtain oo
Isl do = !Sol do 0 •
(12.35)
The flux through corresponding surface elements of a set of wave fronts is constant.
12.4 We continue the investigation of the differential relations (11.38) for U and V. The preceding results show that it is sufficient to consider the equations (12.12) for the directions P and Q of the vectors U and V. Let us represent the light ray in the vectorial form
We have
X grad l/J ,
X=ngradn
and we can write (12.12) in the form
. P
+
n1 2
(P
... • X)X
O ,
Q + -n-¾- (Q •x)x o .
(12.41) (12.42)
50
MATHEMATICAL THEORY OF OPTICS
Instead of the parameter T we introduce the geometrical length s of the
light ray, by using the relation ~: ~ We have
and hence,
X ndd-Xs nX' ' X n(nX1) I p nP',
0.
(12.43)
However,
P • (nX1) 1 = P • (nX" + n'X') on account of P •X' = 0. This yields
n(P •X")
and similarly
P 1 + (P • X")X' 0 , Q• + (Q • X")X' 0 .
(12.44)
The equations (12.44) demonstrate that the two unit vectors P and Q are determined by the light ray alone. The same light ray, of course, can be an orthogonal trajectory to many different sets of wave fronts. For example,
in case n = 1, a given straight line can be orthogonal to systems of spherical
wave fronts or to a system of plane wave fronts. Equations (12.44) however, state that the vectors U and V are submitted to the same rotation around the light ray no matter to which type of wave fronts the light ray is orthogonal. The wave fronts influence only the size of the discontinuities U and V.
Figure 22
T'
*
P' - cT
12.5 The tangential vector
T = X' of the light ray and the
vectors P and Q define an orthog-
onal system of unit vectors which
travel along the light ray X = X(s).
In general the c~ange of such a
system along a curve is determined
by three kinematic formulae:
cP +bQ
* +aQ
(12.51)
Q' - bT
-aP *
WAVE OPTICS AND GEOMETRICAL OPTICS
51
The coefficients a,b,c are functions of s. In our special case these formulae reduce to
T' *
cP +bQ
pr - cT
* *
Q' -bT
*
*
as is shown by (12.44). We have, incidentally,
(12.52)
a = 0,
b = (Q • X"),
c = (P ·X") .
P (s) X (s)
The fact that a = 0 for our system T ,P ,Q has a simple geometrical meaning. Let us consider the ruled surface which is formed by the straight lines through the vectors P on X(T), From
dP = - cT ds
it follows that
P + dP = P - c ds T
lies in the plane formed by the
vectors P and T. This means that
two neighboring unit vectors P and
P + dP are not skew but intersect
each other in a point of the plane of
P and T. The total manifold of
straight lines through the vectors
P(s) thus envelopes a certain curve
C in space and can be interpreted as
the manifold of tangents of this curve.
A ruled surface which consists of the
Figure 23
tangents of a curve in space is called
an applicable surface, since it is
possible to apply it to a plane by
bending without strain. The same consideration applies to the ruled surface
of the vectors Q(s). Hence we can formulate the statement:
The vector discontinuities U and V along a light ray determine a ruled
surface which is applicable.
12,6 We can interpret the light rays as the geodetic lines in a space whose line element has the form
(12.61)
52
MATHEMATICAL THEORY OF OPTICS
J J By introducing, on a light ray X(o-) the parameter o- = n ds = n 2 dT
which measures the optical length on the ray we have the relations
dX do-
1
2 n
grad
I{!,
dP 1 dP do- n 2 dT
With this choice of the parameter the equations (12.12) become
-ddoP- + -n1 (P • grad n)ddoX-- = 0 , -ddoQ- + -n1 (Q • grad n) -ddoX- = 0 .
(12.62)
We shall see that these equations characterize the vectors P or Q as being "parallel" along the light ray; parallelism being defined for the line element (12.61) in accordance with a definition which was introduced by Levy-Civita. The equations of the geodetic lines, i.e., of the light rays with o- as parameter,
follow from (8 .23) letting X = 1/n2 and hence
(:r + (?r + (~:r
1 n2
We find
n2
d do-
(n2
dx) do-
nnx '
~) n2
d do-
(n2
do-
nny ,
n2 d do-
(n2 ~;)
= nnz
•
Let us denote temporarily the components of the vector X(o-) X1(o-), X 2(o-), X3(o-), and the partial derivatives of n by
an
an
an
n1 ax ' n2 = ay ' Il3 = az •
(12.63) (x,y,z) by
Then (12.63) assumes the form
n2 -d (n2 -dX-a)
do-
do-
(12.64)
WAVE OPTICS AND GEOMETRICAL OPTICS
53
or
n2
d2xa du2
+ 2ndX-C-l du
(~ n1
dX1) du
net n
(~ d2XCl
dXCl
du2
+ 2 du
ni dX;) = net 1
n du
n ~
-i" By introducing on the right side n
~
( ddXu1)
2 ,
we
obtain
the
differential
equations of the geodetic lines in the form,
(dX;) (
~nn i
-dX-;) du
-
-nne~t
-
du
2
0.
(12.65)
d 2X Thus, the second derivative dut of each component is equal to a quadratic form of the first derivatives ddXu1 ; we write
0 '
(12.66)
where
Cl
rik
= ~1
(. ,oai
an axk
+
Oak
an ax1
-
Olk
an )
axCl
'
(12.67)
as one can easily verify. The symbols o1k are Kronecker symbols
Oik = 0 , i /- k
Oik = 1 , i k.
(12.671)
In general the equations of the geodetic lines X(u) of any line element
du2 = ~ g1k dx1 dxk
1,k
(12.68)
can be written in the above form (12.66). The matrices r~k are given by the coefficient g lk and are called Christoffel's symbols. In the special case of optics, i.e., for
these coefficients are given by (12.67).
54
MATHEMATICAL THEORY OF OPTICS
The theory of curvature of a space with the line element (12.68) can be
developed in a similar way as the curvature of curves or surfaces, if one can
compare the directions of vectors which
are not attached to the same point.
Obviously this involves a definition of
parallelism in a non-Euclidean space,
i.e., a criterion for the parallelism of two
vectors at different points in space. Levy-
Civita' s definition of parallelism is as
follows: A vector A(u) is moved parallel
on a given curve X(u) if it satisfies the
differential relations
Figure 24
dAa
dXk
du + l~k rf,k A1 du = o.
(12.681)
dX According to this definition, for example, the tangential vectors d a of a
geodetic line are parallel.
u
In case of the optical line element (12.61) we find by using (12.67):
+1-d-X-a n du
= 0.
(12,682) or
In vector notation:
dd(nA) + _! (nA. grad n) ddX - (nA. ddX) grad n = 0 .
u
n
u
u
n
which is the condition of parallelism in our optical medium.
(12.683)
Let us now consider a vector A(u) along a light ray; we have
d(nA) + _! (nA . grad n) dX _ (<nA) . dX) grad I). = O
du
n
du
du n
and
_i_ (n dX) + .!(n dX •grad n) dX - n(dX)2 ~ = 0.
du du
ndu
du
dun
WAVE OPTICS AND GEOMETRICAL OPTICS
55
We multiply the first equation by n ~X and the second equation by nA. It follows, by adding the results that r;
0.
(12,684)
Hence: If A • ~: = 0 at one point of the ray, then it is zero at all points of
the ray. If a vector is normal to the ray then its parallel vectors on the ray are also normal to the ray.
Such normal vectors thus obey the condition
~
dr;
+
~ 1
(nA • grad
n)
dX dr;
= O.
(12.69)
.By comparing this with (12.62), we find that the vectors ¾p and ¾Q satisfy the above condition and thus demonstrate the parallelism of the directions P and Q on the light ray.
12. 7 Integration of the transport equations. We introduce the following
orthogonal system of unit vectors on the light ray, X = X(s):
Tangential vector: Principal normal: Binormal:
T x•.
N - -1- X " .
~ s = TX N.
(12,71)
The derivatives of these vectors, and the vectors themselves, are related by a system of formulae of the type (12.51) which, in this case, are known as Frenet' s formulae:
T'
N'
+-1 8
(12. 72)
T
8'
.! is the principal curvature, and .! is the torsion of the ray.
p
T
56
MATHEMATICAL THEORY OF OPTICS
Let us consider the equation
pr + (X" ·P)Xr = 0
(12. 73)
for the vector P. By introducing the notation (12.71), we obtain
pr + (Tr· P)T = 0 ,
or on account of (12. 72):
pr + -1 (N ·P)T = 0. p
(12.74)
Since P is normal to T, we can express it as a linear combination of N and S. We introduce
p aN + {3S
(12. 75)
in (12.74). This yields
a rN + aNr + 13rs + {3S' + ~ T
O ,
or, on account of (12.72):
It follows that
whence
Cl! r (!..
0 '
T
/3'
g_ +
=
0.
T
These two differential _equations can be written in the form
__Q_ (O! + i{3) + _Ti (O! + i/3) = 0 • ds
(12. 76) (12. 77)
WAVE OPTICS AND GEOMETRICAL OPTICS
57
and have a solution in the form
We introduce
Cl( + i{3
{ ds
0 T•
(12.78)
f 0
s ds
0 T
and
(12. 79)
Cl( cos .,,
cos 1' o
(3 sin 1' Equations (12. 78) become
(3 0
sin 1' 0
ei 1' ei( 1' 0 - 0)
whence,
(12. 791)
The vector P thus is given by
P = N cos(1' 0 - 0) + S sin("o - 0).
(12. 792)
P changes its position relative to the principal normal and binormal of the ray; the angle of rotation with respect to N being given by
s
Q
Figure 25
-0
In case .! = O, i.e., if the light ray reT
mains in one plane, then the vector P remains unchanged relative to the vectors N and S.
Finally we determine the vector Q by
Q T x P = - N sin (.,, 0 0) + S cos (1' o - 0) .
58
MATHEMATICAL THEORY OF OPTICS
§13. SPHERICAL WAVES IN A HOMOGENEOUS MEDIUM.
We illustrate the former results with the example of the spherical wave
which represents the electromagnetic field of a dipole. We assume the medium
is homogeneous. Without loss of generality we let µ = E = 1. In optics we
may consider this electromagnetic field as the simplest mathematical representation of the light wave which is radiated from a point source.
13.1 Maxwell's equations in vacuum are
curl H - -1 Et = 0 , C
curl E + -1 Ht = 0 . C
(13.11)
The vectors E and H satisfy the second order equations
1 2
Ett
-
AE
= 0,
C
1 2
Htt
-
AH
=
0.
C
(13.12)
However, only such vectors E or H are permitted for which
div E = div H = O •
In the case of the wave equation --\ Utt - Au = O for a scalar function C
u(x,y,z,t) it is easy to find spherical waves. We simply ask for solutions u
which depend only on r = ✓ x 2 + y 2 + z 2 and on t. We find the equation
1 2
(ru)tt
-
(ru)rr
=
0 ,
C
(13.13)
which possess the solution
u = .r! (f(r - ct) + g(r + ct)) ,
(13.14)
where f and g are arbitrary functions. Outgoing spherical waves are obtained
if g = o, i.e.,
u = -r1 f(r - ct) .
(13.15)
WAVE OPTICS AND GEOMETRICAL OPTICS
59
From this it follows that the first one of the equations (13.12) can be satisfied by a vector
E = -r1 F(r - ct)
(13.16)
where F = (F1,F 2 ,F 3) is a vector whose components are functions of (r - ct). However, from
div E = (grad r ) • aar (F(r r- ct))
(13.17)
it follows that div E = 0 is possible only in the case F 1 = F 2 = F 3 = 0, Physically, this means that electric fields are not possible in which the electric vector is constant for points on the same sphere at a given time, t.
13,2 In the case of the scalar wave equation
~ 1 Utt - 6.u = 0
we can find other spherical waves from (13.15) by differentiation. Let L be the d1'fferent1'al operator L = a..a.x! + b .a..y! + c aaz , where a,b ,c are constant s. Then
(13.21)
is a solution of the wave equations. It can be interpreted as the wave which is radiated from a dipole with an axis (a,b,c).
More general solutions can be found by repeated differentiation. Let L IJ =aI.J .a.x!+b.IJ.a.y! + cl/ aaz ' then
(13.22)
represents a wave from a 11multipole11 with K axes (a11 ,b11 ,c 11 ). 13.3 We proceed in a similar way for vector waves. Let
L
(13.31)
60
MATHEMATICAL THEORY OF OPTICS
be a differential operator in which A,B,C are matrices with constant elements
A1k, Btk , Ctk. We then obtain a variety of new solutions of ..!... E tt - .6.E in
the form
c2
~ ~ ~ E L F(r - ct)
(13.32)
or explicitly
t, a~) 3
E; =
(Aik a~ + Bik a~ + cik
;Fk (r - ct).
(13.33)
More complicated 11multipole 11 waves can be found by repeated differentiation:
(13.34)
of a vector -r1 F(r - ct).
13.4 All these solutions satisfy the equation --\ E tt - .6.E = 0 and the
C
remaining problem is to find operators Lu such that div E = O. For the dipole wave (13.32) this problem is solved by the operator L = curl. In fact, this operator is of the type (13.31), namely
A = (: : _:) , B = ( : : : ) , C = (: -: : ) •
0 1 0
-1 0 0
0 0 0
(13.41)
Furthermore, div L (; F) = div curl (; F) = 0 .
In order to obtain our solution in a suitable form, let us write the vector F as the derivative of a vector M = (Mi, M2, M3), i.e.,
F(r - ct) = M'(r - ct) .
(13.42)
We then know that E
curl -r1 M'(r - ct)
(13.43)
satisfies the equations --\ E tt - .6.E O; div E o. C
WAVE OPTICS AND GEOMETRICAL OPTICS
61
•Now we have to construct a vector H such that both equations (13.11)
¾ are satisfied. From the equation curl E + Ht = 0, it follows that
Ht = c curl curl -r1 M'(r - ct) .
Hence, if
H = - curl curl -1r M(r - ct) ,
(13.44)
we know that the second equation (13.11) is satisfied. Clearly, div H = O.
In order to show that the first equation (13.11) is also satisfied, we write with the aid of the vector identity (I.43)
H = Li..r!. M(r - ct) - grad div .r.!. M(r - ct)
It follows that
¾ curl H = curl ( M"(r - ct)) .
(13.45)
On the other hand, from (13.43):
Et = c curl (; M"(r - ct)) ,
(13.46)
and hence
curl H - -1 Et O . C
Our result is:
Let M(r - ct) be an arbitrary vector function of cp = r - ct with
continuous derivatives to the third order, M', M", M' ". Then a solution of Maxwell's equations is given by the vectors
¾ E = - curl ( M'(r - ct))
H
curl curl ( ~ M(r - ct))
(13.47)
62
MATHEMATICAL THEORY OF OPTICS
This electromagnetic field can be regarded as the field of a dipole at the point r = 0 whose momentum M = M(-ct) is given as a function of t.
Of course, a second solution can be obtained by replacing E by +H and H by -E. This yields
E curl curl (; M(r - ct)) H = - curl (;M'(r - ct)) .
(13.48)
From a mathematical point of view either one of these solutions may represent the radiation of a point source.
13.5 Let us consider, in the following, the solution (13.47). We have
(.! E = - _r! (ix + jy + zk) x .a..£r. r M ')
where p is the unit vector
X p'
(13.51)
(13.52)
The expression for H can be transformed as follows:
Hence
curl -Mr = .r... x -r12 ( M' - -Mr ) .
r X
-J 3 :Mr:T' + 3 5M r ) xr +2 (Mr:'r-?M")
This however is equal to
(13.53)
WAVE OPTICS AND GEOMETRICAL OPTICS Hence we get the final result:
E =
X p'
63 (13.54)
13.6 We conclude from (13.54):
E·p = O;
p
H •p
=
~ 2
(r
M'
-
M) • p ,
(13.61)
Figure 26
M (,,o)
i.e., the electric vector but, in general, not the magnetic vector, is tangent to the sphere.
Let us now assume that M(cp) = 0 for
cp > 0 . This means that the dipole begins to
oscillate at the time t = 0. It follows that E = 0 and H = 0 for cp = r - ct > 0, i.e., for r > ct. At the time t, the sphere r = ct
thus represents the wave front of the electromagnetic field, i.e., the boundary of the region of penetration. We assume furthermore that
M(0) = M'(0) = 0 ,
but M"(0) = m I- 0, so that M(cp) is a function
of the type indicated in Figure 27. On the wave
fronts cp = r - ct = O; we thus have the
boundary values
E = -r1 (m X p) H = p X E = ;1_: p X (m X p)
(13.62)
which represent discontinuities of the electromagnetic field. We immediately verify the
relations E •p = H • p = 0, and E • H = 0, as
our former results required. Hence, only on the
fO
wavefront are the vectors E and H of the spherical wave normal to the light ray and perpendicular to each other. Figure 27
64
MATHEMATICAL THEORY OF OPTICS
The quantities IE I 2 and IHI 2 on the wave front decrease in proportion to 1/r2; i.e., in proportion to the ratio of corresponding surface elements on
the spherical wave fronts.
§14. WAVE FRONTS IN MEDIA OF DISCONTINUOUS OPTICAL PROPERTIES.
We shall not assume in the following that the functions E(x,y ,z) and µ(x,y,z) are continuous and have continuous derivatives. However, these functions will be sectionally continuous in any finite domain D of the (x,y ,z,t) space, i.e., it will be possible to divid.e D into a finite number of subdomains in which € and µ are continuous, and assume finite boundary values on the bounding surfaces. We furthermore assume that the derivatives of E and µ are also sectionally continuous.
Our first aim is to find the laws according to which wavefronts pass through a refracting surface, i.e., a surface on which E or µ is discontinuous. The result will be Snell's law of refraction and the law of reflection. After that, we can answer the question of how signals, i.e., discontinuities of E and H, are influenced by such surfaces. We will find a system of formulae known as Fresnel's formulae.
14.1 Snell's Law of refraction. Let :r be a surface in the (x,y ,z) space
which separates two media D and D' of indices of refraction n(x,y,z) and
n'(x,y,z), such that on :r: [n] = n' - n -/- 0. Both n and n' shall have con-
tinuous derivatives in their respective domains. A light signal may travel through the (x,y,z) space over a set of wave fronts
cp(x,y ,z,t) = 1/J(x,y ,z) - ct = 0 .
(14.11)
In other words, a characteristic hypersurface cp = 0 in the (x,y,z,t) space is
assumed on which E and H are discontinuous. This hypersurface is continu-
ous but does not necessarily have
continuous normals, as is indicated
in Figure 28. Letting cp = 1jJ - ct
we conclude that 1/J(x,y,z) must be a
continuous solution of the equation
y
1/J; + 1/Ji + 1/Ji = n2
(14.12)
even on the surface :r where n is
discontinuous. It's derivatives are
X
sectionally continuous.
Char. surface cp = 0 in case n
discontinuous on x = O.
We assume that :r is given in
the form
Figure 28
x = f(~,1)), y = g(~;r,), z = h(~,TJ) (14.13)
WAVE OPTICS AND GEOMETRICAL OPTICS
65
M
n'
and denote the values of 1/J on the two sides of :E by 1/J and 1/J'. 1/J and 1/J' have continuous derivatives in their domains. From this it follows that
1/J (f(~,7J), g(~,7J), h(~,7J))
1/J' (f(~,7J), g(~,7J), h(~,7J))
Figure 29
and that both sides have continuous derivatives. Consequently
(1/Jx' - 1/Jx)fs + (1/Jy' - 1/Jy)gi + (1/Jz' - 1/Jz}?s 0' (1/J; - 1/Jx)fll + (1/Jy' - 1/Jy)gll + (1/Jz' - 1/Jz)¾ 0.
(14.14)
These equations state that the vector
is normal to the surface :E. If M is a unit vector in direction of the surface normal, we can write
grad 1/J' - grad 1/J = rM .
(14.15)
Grad 1/J' and grad 1/J give the direction of the orthogonal trajectories of the
surfaces 1/J' = const. and 1/J = const., i.e., of the light rays. Let T and T'
be unit vectors along the light ray; then grad 1/J = nT and grad 1/J' = n'T'.
It follows that
n'T' - nT
where r is a scalar factor.
rM,
(14.16)
We conclude that the refracted ray leaves the surface :E in the plane formed by the incident ray and the surface normal M.
From (14.16) it follows that
n'(T' x M) = n(T x M) .
(14.17)
The length of the vector on the left side is n 'sin iJ '; on the right side, n sin iJ , where iJ is the angle of incidence and iJ' is the angle of refraction. This yields Snell's law of refraction
n' sin iJ' n sin iJ .
(14.18)
66
MATHEMATICAL THEORY OF OPTICS
T'
We find the factor r in (14.16) by forming the
scalar product of M with (14.16):
n'(T'-M) - n(T•M) = r'
or
r n' cos rJ ' - n cos rJ .
(14.19)
The relations
Figure 30
n'T' - nT n' sin rJ'
(n' cos rJ ' - n cos rJ )M , n sin rJ
(14.191)
allow us to find T' if T and M are given.
14.2 The law of reflection. We know by experience that a light signal when reaching a surface of discontinuity of n is not only transmitted, but also reflected. Mathematically, this possibility is suggested by the quadratic character of the equation
We have seen in §9 that, on account of this, there exist two solutions ip which attain given boundary values on a given surface :E.
That a surface of discontinuity must actually produce a set of reflected
wave fronts will be seen in the next section by deriving Fresnel's formulae.
Let us here assume the existence of a reflected signal. This means that the
characteristic hypersurface cp = 0 consists of two branches cp = ip - ct = 0
and cp * = 1/1 * - ct in the neighborhood D of the surface :E. These two
branches are joined together on :E, as is indicated in Figure 31 for the case
when :E is the plane x = 0. This
implies that the two functions 1/1 and ip * must have the same boundary
values on :E, i.e.,
ip ( f(~,17), g(~. 7J), h(~. TJ))
1/1* (rc~,TJ>,· g(~,7J>, iic~,7J>)
(14.21)
As above we conclude that
Figure 31
grad 1/1* - grad ip = rM (14.22)
WAVE OPTICS AND GEOMETRICAL OPTICS
67
or, by introducing the unit vector T * on the reflected ray,
n(T* - T) = rM.
(14.23)
Figure 32
It follows: The reflected ray leaves the surface ~ in the plane determined by T and M. We find as above that
T* X M = TX M' and hence
sin {} * = sin {} , where {} * is the angle of reflection. For r we obtain
r = n(cos {} * - cos {} ) .
(14.24) (14.25)
The equations
T* - T sin {} *
(cos {} * - cos {} )M , sin {}
(14.26)
can be satisfied by two vectors T * if T is given. From sin {} * sin {} it follows
{} ' or
cos {} * cos {} '
cos {} * - cos {} .
(14.27)
The first solutions give T* = T, i.e., the incident ray; the second solutions yield
T* - T = - 2 cos rJ M .
(14.28)
This is the reflected ray. T* and T are symmetrical with respect to the tangent plane of ~ at the point of incidence.
§15. TRANSPORT OF SIGNALS IN MEDIA OF DISCONTINUOUS OPTICAL PROPERTIES. FRESNEL'S FORMULAE.
15.1 In 14.2 we have seen that, on a surface ~: w(x,y ,z) = 0 on which the functions E and µ are discontinuous, a set of wave fronts 1/J = ct must be expected to split up into two sets of wave fronts; the transmitted wave fronts 1/J - ct = 0 and the reflected wave fronts 1/J* - ct = 0. The corresponding characteristic hypersurface then consists of three branches; the incident
68
MATHEMATICAL THEORY OF OPTICS
branch <p = 1/J - ct = 0, the reflected branch <p* = 1/J*. - ct = 0, and the transmitted branch <p' = 1/J' - ct = 0. All these branches intersect each
other in a common manifold J which lies on the cylindrical hypersurface
w(x,y,z) = 0.
w=O , ~=O
M
X
Figure 33
In this section we study an electromagnetic field which is discontinuous on
these three branches and, of course, also on the cylindrical hypersurface
w = 0. The four-dimensional neighborhood of the common manifold of inter-
sections, J, is divided into five parts separated by the four hypersurfaces
<p = 0, <p* = 0, w = 0, <p' = 0. Let us assume that the vectors E and H
are continuous in these five parts and that they attain finite limits.
E 0 ,Ei,E 2 ,E 3 ,E 4,
(15.11)
if a point P of the manifold J is approached.
We denote the surface normal of w = 0 by M, i.e., a unit vector
proportional to the vector (wx, wy, Wz, 0) at P. By applying the conditions derived in §6 for E and H on a surface of refraction, we find
E3 X M; Eo x M
where E and E' are the boundary values of E on w = 0.
(15.12)
WAVE OPTICS AND GEOMETRICAL OPTICS
69
Similarly,
H 3 x M; H 0 x M
(15.13)
Since the boundary values of the discontinuities on J are given by
u E1 Eo
V H1 Ho
U* E2 E1
u• E3 E4
V* H2 H1 V' H3 H4
(15.14)
We find from (15.12) and (15.13) readily that
E(U + U*) ·M E'U'•M'
(U + U*) x M µ(V + V*) •M
U' x M; µ'V'•M,
(15.15)
(V + V*) X M V' x M.
15.2 We consider now the light rays which belong to the point P 0 on :r,
i.e., to the projection of P in the (x,y,z) space. The direction of these rays is given by
1
T
-n grad 1/1
T* -n1 grad 1/1*
(15.21)
T'
=
1 ,ng r a d
1/1'
.
These vectors and M are related by the equations
n'T' nT + (n' cos iJ' - n cos iJ )M T* T - 2 cos iJ M .
(15.22)
The four vectors T, T*, T', and M lie in one and the same plane of the x,y,z space. This plane is normal to the unit vector
S =T xM
sin i'J
T* X M sin i'J *
T' X M sin i'J'
(15.23)
70
MATHEMATICAL THEORY OF OPTICS
This fact suggests the introduction of the following orthogonal system of unit vectors attached to each of the three rays:
T,N s X T' s
T*, N* s X T*, s
(15.24)
T', N' s X T'' s
15.3 We consider the incident ray first. We know that the discontinuities U and V on this ray are related by the equations
grad ip x V + EU O , grad ip x U - µV O ,
(15.31)
or, on account of (15.21):
.,/µ (T x V) + VE U 0 VE(T x U) - .,/µv o.
(15.32)
From these equations it follows that U and V can be represented in the form
VEU = aN + {3S .,/µv = -f3N + as ,
(15.33)
as linear combinations of N and S.
In order to apply the conditions (15.15) let us determine the products U·M and U x M. We obtain
EU·M = aVEM•N = aVEsin ~
and
U X M = ; ; (N x M) + ~ (S x M) .
However,
NxM (S x T) x M - (M • T)S
cos ~ s.
Hence
EU•M UxM
a!E sin ~
¾ - ;; cos ~ s +
(S X M) .
(15.34)
WAVE OPTICS AND GEOMETRICAL OPTICS
71
15.4 Equations identical with (15.34) can be found for the products
EU*•M, U* X M, and EU'·M, u• X M. Since sands X Mare orthogonal,
and thus independent, we obtain from (15.15) the equations
.fE (a sin t'J + a* sin t'J *) = -.IE'"a ' sin t'J '
R r1;: (a cos t'J + a* cos t'J *) = -1-a ' cos t'J '
VE
(15.41)
- i (/3 + /3*) L
-..1£
-IE'
On account of t'J * 1r - t'J and {Eµ sin t'J = -..1£'µ' sin t'J ' these equations
become
a +a* ,J!µII'. Cl/ I
(3+{3* ,JIiT• /3' ,
(15.42)
Instead of carrying out analogous calculations for the vector V, we may
simply replace a by -/3, {3 by a, and E by µ in (15.42). The first equations
(15.42) do not give any new conditions. The last equation, however, yields
/3
_ /3 *
= ~ ,J µ'
COS t'J I cos "
/3 I
•
(15.43)
The four equations (15.42) and (15.43) allow us to express a', f3', and a*, {3 * as functions of a and {3. The result is
ff. ff. Cl/ I
2
a
+
~
cos"' cos"
H, ff,- a*=
/!:_ -
COS t'J I E' cos"
a
ff. COS t'J I + p E' cos"
P ff. if. =
/3
2
I +
cos"' cos"
JF,- I[. COS t'J I
£_=
cos "
ff. ff. /3
COS t'J I
+ µ' cos"
(15.44)