International Journal of Antennas and Propagation

Volume 2015 (2015), Article ID 478580, 10 pages

http://dx.doi.org/10.1155/2015/478580

## A Simple Proof of Damien’s Theorem and Duality in Theory of the Zero-Distance Phase Front

Institut für Optik und Atomare Physik, Technische Universität Berlin, Sekretariat ER 1-1, Strasse des 17. Juni 135, 10623 Berlin, Germany

Received 9 April 2015; Accepted 31 May 2015

Academic Editor: Giuseppe Castaldi

Copyright © 2015 Andrey Valerievich Gitin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Duality plays the main role in all mathematical theories. In wave optics based on the concept of “the zero-distance phase front,” duality takes the form of Damien’s theorem and the mirror symmetry between the conic refracting surfaces with the plane zero-distance phase front and the plane refracting surfaces with the conic zero-distance phase front. A systematic study of these dualities has been performed.

#### 1. Introduction

In 1825 Joseph Gergonne [1] noticed the “symmetry” of roles played by points and lines in definitions and theorems of geometry: if we assume that all parallel lines intersect at infinity (the axiom of projective geometry), any theorem or definition one-to-one corresponds to another theorem or definition (the so-called dual statement), when we substitute “point” for “line,” “lie on” for “pass through,” “collinear” for “concurrent,” “intersection” for “join,” or vice versa. At present, various types of duality have been discovered in various branches of mathematics and theoretical physics (including optics [2, 3]).

It is known that the application of the law of refraction together with the axioms of geometry gives rise to “optical geometry,” implemented in design of aspheric lenses [4]. In optical geometry based on the concept of “the zero-distance phase front” [5] duality takes the form of Damien’s theorem and of the mirror symmetry between the conic refracting surfaces with the plane zero-distance phase front and the plane refracting surfaces with the conic zero-distance phase front [4, 6–11].

We propose a simple proof of Damien’s theorem and describe dual pairs: “the refracting (reflecting) surface-the zero-distance phase front.”

#### 2. Refraction and Reflection

A monochromatic wave can be described by a locus of points of the wave with the same phase of oscillation, the so-called phase front [12]. The wave propagation can be described by a family of the phase fronts. An alternative way of describing of wave propagation is given by orthogonal trajectories to the phase fronts, the so-called light rays. The shape of the phase fronts (and light rays) is usually determined by the geometry of the source and the properties of the medium. In a homogeneous medium a point source generates a family of concentric spherical (circular) phase fronts with a common center at (or a homocentric beam of light rays with a vertex at ) (Figure 1(a)), and a plane source generates plane phase fronts (or parallel rays) (Figure 1(b)). In an inhomogeneous medium phase fronts can be very complicated, however the principle of equal optical paths [12] states that, in an optical system consisting of refracting and reflecting elements, the optical path length (OPL) between any two phase fronts is identically the same for all rays (Figure 2). The OPL is defined by the line integral along any ray path :where is the refraction index at each point along the ray path and .