The optics of rays, wavefronts, and caustics
The book provides a study of the mathematical foundation of geometrical optics. Included are: (a) Techniques for solving ray and wavefront problems in general inhomogeneous media-relevant particularly to the recently evolved concept of the 'inhomogeneous lens'. (b) Generalized ray tracing, a technique for calculating, in a ray neighborhood, the principal curvatures of a wavefront as it propagates through a lens. This process is immediately applicable to computer lens design programs. (c) A general solution for the eikonal equation for a homogeneous medium providing a general description of wavefronts in analytic terms. The book also treats several topics of considerable theoretical interest, including: (a) A system of equations, similar to the Maxwell equations but derived from the postulates of geometrical optics alone. (b) An algebraic theory of lens design in which lenses are represented as group elements. This treatise will be of particular importance to optical designers and optical physicists. For its formalistic treatment, mathematicians and theoretical physicists will find it considerably valuable. (Author).
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The Nineteenth Century
The Problem in Parametric Form
The Osculating Sphere
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angle arc length arc length parameter axis becomes calculate Calculus of Variations caustic surface Chapter coefficients constants of integration coordinate cos2 defined denote Descartes diapoints distance eikonal equation element equation of Eqs Euler equations expression Fermat's principle fish eye fundamental optical invariant geometrical optics given Hilbert integral image plane image space inhomogeneous medium Jacobian lens equation lenses linear matrix object point obtain optical path length optical system orthogonal orthotomic system pair partial derivatives partial differential equation perpendicular principal directions problem pupil plane quadratic ray equation ray path ray tracing reference plane refracting surface refractive index represent rotationally symmetric sin2 Snell's law solution space curve sphere Substituting surface normal symplectic group system of rays tangent vector theory total differential equation transfer transformation unit normal vector unit vector vanishes variables variational integral vector function velocity wavefront zero