Abstract
We investigate the behavior of the solutions of the ray equation in isotropic media. We discuss local and global transversals and wavefronts and give examples of rays without wavefronts.
Highlights
Given the shape of a wavefront, the direction of any ray crossing the wavefront can be immediately calculated via the eikonal equation
We revisit here that problem: taking the ray equation as our starting point, we address the simple and unexplored question of whether given a family of rays one can always find the associated wavefronts
The unforeseen answer we find is that while in a two-dimensional world, the wavefronts always exist, it is not generally the case for three-dimensional rays
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Its fundamental ingredients are rays, which do not exist (except as a mathematical idealization), and wavefronts, which do exist, but are not directly observable [1]. It works: even with such an unsophisticated background, it maintains a unique position in modern technology [2]. With a small amount of calculation, one can show that the rays are normal to the wavefronts. It is not surprising that the authoritative textbook by Born and Wolf [5] states that “only normal congruences are of interest for geometrical optics”. Given the shape of a wavefront, the direction of any ray crossing the wavefront can be immediately calculated via the eikonal equation. The unforeseen answer we find is that while in a two-dimensional world, the wavefronts always exist (so the ray-wavefront duality is correct), it is not generally the case for three-dimensional rays
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