THE MATHEMATICS OF GEOMETRICAL AND PHYSICAL OPTICS

By ROBERT SMITH

CONTENTS I PRELIMINARIES FERMAT’S PRINCIPLE AND THE VARIATIONAL CALCULUS RAYS IN INHOMOGENEOUS MEDIA THE CALCULUS OF VARIATIONS THE PARAMETRIC REPRESENTATION THE VECTOR NOTATION THE INHOMOGENEOUS OPTICAL MEDIUM THE MAXWELL FISH EYE THE HOMOGENEOUS MEDIUM ANISOTROPIC MEDIA SPACE CURVES AND RAY PATHS SPACE CURVES THE VECTOR TRIHEDRON THE FRENET-SERRET EQUATIONS WHEN THE PARAMETER IS ARBITRARY THE DIRECTIONAL DERIVATIVE THE CYLINDRICAL HELIX THE CONIC SECTION THE RAY EQUATION MORE ON THE FISH EYE THE HILBERT INTEGRAL AND THE HAMILTON-JACOBI THEORY A DIGRESSION ON THE GRADIENT THE HILBERT INTEGRAL PARAMETRIC CASE APPLICATION TO GEOMETRICAL OPTICS THE CONDITION FOR TRANSVERSALITY THE TOTAL DIFFERENTIAL EQUATION MORE ON THE HELIX SNELL’S LAW THE HAMILTON-JACOBI PARTIAL DIFFERENTIAL EQUATIONS THE EIKONAL EQUATION THE DIFFERENTIAL GEOMETRY OF SURFACES PARAMETRIC CURVES SURFACE NORMALS THE THEOREM OF MEUSNIER THE THEOREM OF GAUSS GEODESICS ON A SURFACE THE WEINGARTEN EQUATIONS TRANSFORMATION OF PARAMETERS WHEN THE PARAMETRIC CURVES ARE CONJUGATES WHEN F ?= 410 THE STRUCTURE OF THE PROLATE SPHEROID 411 OTHER WAYS OF REPRESENTING SURFACES PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER THE LINEAR EQUATION THE METHOD OF CHARACTERISTICS THE HOMOGENEOUS FUNCTION THE BILINEAR CONCOMITANT NON-LINEAR EQUATION: THE METHOD OF LAGRANGE AND CHARPIT THE GENERAL SOLUTION THE EXTENSION TO THREE INDEPENDENT VARIABLES THE EIKONAL EQUATION THE COMPLETE INTEGRAL THE EIKONAL EQUATION THE GENERAL SOLUTION THE EIKONAL EQUATION PROOF OF THE PUDDING II THE K-FUNCTION THE GEOMETRY OF WAVE FRONTS PRELIMINARY CALCULATIONS THE CAUSTIC SURFACE SPECIAL SURFACES I: PLANE AND SPHERICAL WAVEFRONTS PARAMETER TRANSFORMATIONS ASYMPTOTIC CURVES AND ISOTROPIC DIRECTIONS RAY TRACING: GENERALIZED AND OTHERWISE THE TRANSFER EQUATIONS THE ANCILLARY QUANTITIES THE REFRACTION EQUATIONS ROTATIONAL SYMMETRY THE PARAXIAL APPROXIMATION GENERALIZED RAY TRACING – TRANSFER GENERALIZED RAY TRACING – PRELIMINARY CALCULATIONS GENERALIZED RAY TRACING – REFRACTION THE CAUSTIC 710 THE PROLATE SPHEROID 711 RAYS IN THE SPHEROID ABERRATIONS IN FINITE TERMS HERZBERGER’S DIAPOINTS HERZBERGER’S FUNDAMENTAL OPTICAL INVARIANT THE LENS EQUATION HALF-SYMMETRIC, SYMMETRIC AND SHARP IMAGES REFRACTING THE K-FUNCTION REFRACTION THE REFRACTING SURFACE THE PARTIAL DERIVATIVES THE FINITE OBJECT POINT THE QUEST FOR C DEVELOPING THE SOLUTION CONCLUSIONS 10 MAXWELL EQUATIONS AND THE K-FUNCTION 101 THE WAVEFRONT 102 THE MAXWELL EQUATIONS 103 GENERALIZED COORDINATES AND THE NABLA OPERATOR 104 APPLICATION TO THE MAXWELL EQUATIONS 105 CONDITIONS ON V 106 CONDITIONS ON THE VECTOR V 107 SPHERICAL WAVEFRONTS III RAMI?CATIONS 11 THE MODERN SCHIEFSPIEGLER 111 BACKGROUND 112 THE SINGLE PROLATE SPHEROID 113 COUPLED SPHEROIDS 114 THE CONDITION FOR THE PSEUDO AXIS 115 MAGNI?CATION AND DISTORTION 116 CONCLUSION 12 THE CARTESIAN OVAL AND ITS KIN 121 THE ALGEBRAIC METHOD 122 THE OBJECT AT IN?NITY 123 THE PROLATE SPHEROID 124 THE HYPERBOLOID OF TWO SHEETS 125 OTHER SURFACES THAT MAKE PERFECT IMAGES 13 THE PSEUDO MAXWELL EQUATIONS 131 MAXWELL EQUATIONS FOR INHOMOGENEOUS MEDIA 132 THE FRENET-SERRET EQUATIONS 133 INITIAL CALCULATIONS 134 DIVERGENCE AND CURL 135 ESTABLISHING THE RELATIONSHIP 14 THE PERFECT LENSES OF GAUSS AND MAXWELL 141 GAUSS’APPROACH 142 MAXWELL’S APPROACH