The Mathematics of Geometrical and Physical Optics – The K–function and its Ramifications

In this sequel to his book, "The Optics of Rays, Wavefronts, and Caustics," Stavroudis not only covers his own research results, but also includes more recent developments. The book is divided into three parts, starting with basic mathematical concepts that are further applied in the book. Surface geometry is treated with classical mathematics, while the second part covers the k-function, discussing and solving the eikonal equation as well as Maxwell equations in this context. A final part on applications consists of conclusions drawn or developed in the first two parts of the book, discussing such topics as the Cartesian oval, the modern Schiefspiegler, Huygen's principle, and Maxwell's model of Gauss' perfect lens.

Table of Contents:
I Preliminaries.1 Fermat's Principle and the Variational Calculus.1.1 Rays in Inhomogeneous Media.1.2 The Calculus of Variations.1.3 The Parametric Representation.1.4 The Vector Notation .1.5 The Inhomogeneous Optical Medium.1.6 The Maxwell Fish Eye.1.7 The Homogeneous Medium.1.8 Anisotropic Media.2 Space Curves and Ray Paths.2.1 Space Curves.2.2 The Vector Trihedron.2.3 The Frenet-Serret Equations.2.4 When the Parameter is Arbitrary.2.5 The Directional Derivative.2.6 The Cylindrical Helix.2.7 The Conic Section.2.8 The Ray Equation.2.9 More on the Fish Eye.3 The Hilbert Integral and the Hamilton-Jacobi Theory.3.1 A Digression on the Gradient.3.2 The Hilbert Integral. Parametric Case.3.3 Application to Geometrical Optics.3.4 The Condition for Transversality.3.5 The Total Differential Equation.3.6 More on the Helix.3.7 Snell's Law.3.8 The Hamilton-Jacobi Partial Differential Equations.3.9 The Eikonal Equation.4 The Differential Geometry of Surfaces.4.1 Parametric Curves.4.2 Surface Normals.4.3 The Theorem of Meusnier.4.4 The Theorem of Gauss.4.5 Geodesics on a Surface.4.6 TheWeingarten Equations.4.7 Transformation of Parameters.4.8 When the Parametric Curves are Conjugates.4.9 When F != 0.4.10 The Structure of the Prolate Spheroid.4.11 OtherWays of Representing Surfaces.5 Partial Differential Equations of the First Order.5.1 The Linear Equation. The Method of Characteristics.5.2 The Homogeneous Function.5.3 The Bilinear Concomitant.5.4 Non-Linear Equation: The Method of Lagrange and Charpit.5.5 The General Solution.5.6 The Extension to Three Independent Variables.5.7 The Eikonal Equation. The Complete Integral.5.8 The Eikonal Equation. The General Solution.5.9 The Eikonal Equation. Proof of the Pudding.II The k-function.6 The Geometry ofWave Fronts.6.1 Preliminary Calculations.6.2 The Caustic Surface.6.3 Special Surfaces I: Plane and SphericalWavefronts.6.4 Parameter Transformations.6.5 Asymptotic Curves and Isotropic Directions.7 Ray Tracing: Generalized and Otherwise.7.1 The Transfer Equations.7.2 The Ancillary Quantities.7.3 The Refraction Equations.7.4 Rotational Symmetry.7.5 The Paraxial Approximation.7.6 Generalized Ray Tracing - Transfer.7.7 Generalized Ray Tracing - Preliminary Calculations.7.8 Generalized Ray Tracing - Refraction.7.9 The Caustic.7.10 The Prolate Spheroid.7.11 Rays in the Spheroid.8 Aberrations in Finite Terms.8.1 Herzberger's Diapoints.8.2 Herzberger's Fundamental Optical Invariant.8.3 The Lens Equation.8.4 Aberrations in Finite Terms.8.5 Half-Symmetric, Symmetric and Sharp Images.9 Refracting the k-Function.9.1 Refraction.9.2 The Refracting Surface.9.3 The Partial Derivatives.9.4 The Finite Object Point.9.5 The Quest for C.9.6 Developing the Solution.9.7 Conclusions.10 Maxwell Equations and the k-Function.10.1 TheWavefront.10.2 The Maxwell Equations.10.3 Generalized Coordinates and the Nabla Operator.10.4 Application to the Maxwell Equations.10.5 Conditions on V.10.6 Conditions on the Vector V.10.7 SphericalWavefronts.III Ramifications.11 The Modern Schiefspiegler.11.1 Background.11.2 The Single Prolate Spheroid.11.3 Coupled Spheroids.11.4 The Condition for the Pseudo Axis.11.5 Magnification and Distortion.11.6 Conclusion.12 The Cartesian Oval and its Kin.12.1 The Algebraic Method.12.2 The Object at Infinity.12.3 The Prolate Spheroid.12.4 The Hyperboloid of Two Sheets.12.5 Other Surfaces that Make Perfect Images.13 The Pseudo Maxwell Equations.13.1 Maxwell Equations for Inhomogeneous Media.13.2 The Frenet-Serret Equations.13.3 Initial Calculations.13.4 Divergence and Curl.13.5 Establishing the Relationship.14 The Perfect Lenses of Gauss and Maxwell.14.1 Gauss'Approach.14.2 Maxwell's Approach.A Appendix. Vector Identities.A.1 Algebraic Identities.A.2 Identities Involving First Derivatives.A.3 Identities Involving Second Derivatives.A.4 Gradient.A.5 Divergence.A.6 Curl.A.7 Lagrangian.A.8 Directional Derivative.A.9 Operations on W and its Derivatives.A.10 An Additional Lemma.B Bibliography.Index.

About the Author :
O. N. Stavroudis has researched and lectured in the area of the mathematics of geometrical and physical optics and their application for 50 years. He is Professor Emeritus at the Optical Sciences Center, University of Arizona, Tucson, USA, and currently works as an Investigador Titular at the Centro de Investigaciones en Optics, Leon, Guanajuato, Mexico. His research field covers differential geometry of wavefronts, k-function and Maxwell equations, wildly off-axis telescopes, generalized ray tracing and modular optical design. He published numerous reviews and contributions for books, and authored several monographs.