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Titre : The mathematics of geometrical and physical optics : The K-function and ramifications Type de document : texte imprimé Auteurs : Stavroudis,Orestes N Editeur : Weinheim : Wiley-Vch Année de publication : 2006 Importance : 1 vol (226 p.) Présentation : ill. Format : 24cm ISBN/ISSN/EAN : 978-3-527-40448-3 Note générale : 978-3-527-40448-3 Langues : Anglais (eng) Langues originales : Anglais (eng) Catégories : Physique Mots-clés : Physique
Optique géométrique : Mathématiques
Optique physique : Mathématiques
Optique géométrique
Optique physiqueIndex. décimale : 530.1 Physique mathématique Résumé :
Dans cette suite à son livre «L'optique des rayons, des fronts d'onde et des caustiques», Stavroudis ne se contente pas de couvrir ses propres résultats de recherche, mais inclut également des développements plus récents. Le livre est divisé en trois parties, en commençant par les concepts mathématiques de base qui sont ensuite appliqués dans le livre. La géométrie de surface est traitée avec des mathématiques classiques, tandis que la seconde partie couvre la fonction k, discutant et résolvant l'équation eikonale ainsi que les équations de Maxwell dans ce contexte. Une dernière partie sur les applications consiste en conclusions tirées ou développées dans les deux premières parties du livre, discutant de sujets tels que l'ovale cartésien, le Schiefspiegler moderne, le principe de Huygen, et le modèle de Maxwell de l'objectif parfait de Gauss. Tous ensemble pour devenir la référence dans ce domaine essentiel de l'optiqueNote de contenu :
Sommaire
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
Côte titre : Fs/12414-12416 The mathematics of geometrical and physical optics : The K-function and ramifications [texte imprimé] / Stavroudis,Orestes N . - Weinheim : Wiley-Vch, 2006 . - 1 vol (226 p.) : ill. ; 24cm.
ISBN : 978-3-527-40448-3
978-3-527-40448-3
Langues : Anglais (eng) Langues originales : Anglais (eng)
Catégories : Physique Mots-clés : Physique
Optique géométrique : Mathématiques
Optique physique : Mathématiques
Optique géométrique
Optique physiqueIndex. décimale : 530.1 Physique mathématique Résumé :
Dans cette suite à son livre «L'optique des rayons, des fronts d'onde et des caustiques», Stavroudis ne se contente pas de couvrir ses propres résultats de recherche, mais inclut également des développements plus récents. Le livre est divisé en trois parties, en commençant par les concepts mathématiques de base qui sont ensuite appliqués dans le livre. La géométrie de surface est traitée avec des mathématiques classiques, tandis que la seconde partie couvre la fonction k, discutant et résolvant l'équation eikonale ainsi que les équations de Maxwell dans ce contexte. Une dernière partie sur les applications consiste en conclusions tirées ou développées dans les deux premières parties du livre, discutant de sujets tels que l'ovale cartésien, le Schiefspiegler moderne, le principe de Huygen, et le modèle de Maxwell de l'objectif parfait de Gauss. Tous ensemble pour devenir la référence dans ce domaine essentiel de l'optiqueNote de contenu :
Sommaire
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
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