The Mathematics of Geometrical and Physical Optics

Transcription

1 Orestes N. Stavroudis The Mathematics of Geometrical and Physical Optics The fc-function and its Ramifications WILEY- VCH WILEY-VCH Verlag GmbH & Co. KGaA

2 I Preliminaries 1 1 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 12 2 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 26 3 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'sLaw The Hamilton-Jacobi Partial Differential Equations The Eikonal Equation 43 4 The Differential Geometry of Surfaces Parametric Curves Surface Normals The Theorem of Meusnier The Theorem of Gauss Geodesies on a Surface The Weingarten Equations Transformation of Parameters 55 The Mathematics of Geometrical and Physical Optics: The k-function and its Ramifications. O.N. Stavroudis Copyright 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN:

3 XII 4.8 When the Parametric Curves are Conjugates WhenF^O The Structure of the Prolate Spheroid Other Ways of Representing Surfaces 64 5 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 81 II The fc-function 83 6 The Geometry of Wave Fronts Preliminary Calculations The Caustic Surface Special Surfaces I: Plane and Spherical Wavefronts Parameter Transformations Asymptotic Curves and Isotropic Directions 94 7 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 The Prolate Spheroid Rays in the Spheroid Aberrations in Finite Terms Herzberger's Diapoints Herzberger's Fundamental Optical Invariant The Lens Equation Aberrations in Finite Terms Half-Symmetric, Symmetric and Sharp Images 127

4 XIII 9 Refracting the k -Function Refraction The Refracting Surface The Partial Derivatives The Finite Object Point The Quest for C Developing the Solution Conclusions Maxwell Equations and the fc-function TheWavefront The Maxwell Equations Generalized Coordinates and the Nabla Operator Application to the Maxwell Equations Conditions on V Conditions on the Vector V Spherical Wavefronts 158 III Ramifications The Modern Schiefspiegler Background The Single Prolate Spheroid Coupled Spheroids The Condition for the Pseudo Axis Magnification and Distortion Conclusion The Cartesian Oval and its Kin The Algebraic Method The Object at Infinity The Prolate Spheroid The Hyperboloid of Two Sheets Other Surfaces that Make Perfect Images The Pseudo Maxwell Equations Maxwell Equations for Inhomogeneous Media The Frenet-Serret Equations Initial Calculations Divergence and Curl Establishing the Relationship The Perfect Lenses of Gauss and Maxwell Gauss'Approach 197

5 XIV 14.2 Maxwell's Approach 198 A Appendix. Vector Identities 205 A.I Algebraic Identities 206 A.2 Identities Involving First Derivatives 207 A.3 Identities Involving Second Derivatives 207 A.4 Gradient 208 A.5 Divergence 209 A.6 Curl 210 A.7 Lagrangian 212 A.8 Directional Derivative 212 A.9 Operations on W and its Derivatives 213 A. 10 An Additional Lemma 214 B Bibliography 217 Index 223