Book contents
- Frontmatter
- Preface to corrected reprint of the seventh edition
- Preface to the first edition
- Preface to the second edition
- Preface to the third edition
- Preface to the fourth edition
- Preface to the fifth edition
- Preface to the sixth edition
- Preface to the seventh edition
- Contents
- Historical introduction
- I Basic properties of the electromagnetic field
- II Electromagnetic potentials and polarization
- III Foundations of geometrical optics
- IV Geometrical theory of optical imaging
- V Geometrical theory of aberrations
- VI Image-forming instruments
- VII Elements of the theory of interference and interferometers
- VIII Elements of the theory of diffraction
- IX The diffraction theory of aberrations
- X Interference and diffraction with partially coherent light
- XI Rigorous diffraction theory
- XII Diffraction of light by ultrasonic waves
- XIII Scattering from inhomogeneous media
- XIV Optics of metals
- XV Optics of crystals
- Appendices
- Author index
- Subject index
IV - Geometrical theory of optical imaging
- Frontmatter
- Preface to corrected reprint of the seventh edition
- Preface to the first edition
- Preface to the second edition
- Preface to the third edition
- Preface to the fourth edition
- Preface to the fifth edition
- Preface to the sixth edition
- Preface to the seventh edition
- Contents
- Historical introduction
- I Basic properties of the electromagnetic field
- II Electromagnetic potentials and polarization
- III Foundations of geometrical optics
- IV Geometrical theory of optical imaging
- V Geometrical theory of aberrations
- VI Image-forming instruments
- VII Elements of the theory of interference and interferometers
- VIII Elements of the theory of diffraction
- IX The diffraction theory of aberrations
- X Interference and diffraction with partially coherent light
- XI Rigorous diffraction theory
- XII Diffraction of light by ultrasonic waves
- XIII Scattering from inhomogeneous media
- XIV Optics of metals
- XV Optics of crystals
- Appendices
- Author index
- Subject index
Summary
The characteristic functions of Hamilton
IN §3.1 it was shown that, within the approximations of geometrical optics, the field may be characterized by a single scalar function S(r). Since S(r) satisfies the eikonal equation §3.1 (15), this function is fully specified by the refractive index function (r) alone, together with the appropriate boundary conditions.
Instead of the function S(r), closely related functions known as characteristic functions of the medium are often used. They were introduced into optics by W. R. Hamilton, in a series of classical papers. Although on account of algebraic complexity it is impossible to determine the characteristic functions explicitly for all but the simplest media, Hamilton's methods nevertheless form a very powerful tool for systematic analytical investigations of the general properties of optical systems.
In discussing the properties of these functions and their applications, an isotropic but generally heterogeneous medium will be assumed.
The point characteristic
Let (x0, y0, z0) and (x1, y1, z1) be respectively the coordinates of two points PQ and P\ each referred to a different set of mutually parallel, rectangular axes (Fig. 4.1). If the two points are imagined to be joined by all possible curves, there will, in general, be some amongst them, the optical rays, which satisfy Fermat's principle. Assume for the present that not more than one ray joins any two arbitrary points.
- Type
- Chapter
- Information
- Principles of OpticsElectromagnetic Theory of Propagation, Interference and Diffraction of Light, pp. 142 - 227Publisher: Cambridge University PressPrint publication year: 1999
- 2
- Cited by