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Fermat Principle, Ramsey Theory and Metamaterials
Version 1
: Received: 21 November 2023 / Approved: 22 November 2023 / Online: 22 November 2023 (13:48:33 CET)
A peer-reviewed article of this Preprint also exists.
Frenkel, M.; Shoval, S.; Bormashenko, E. Fermat Principle, Ramsey Theory and Metamaterials. Materials 2023, 16, 7571. Frenkel, M.; Shoval, S.; Bormashenko, E. Fermat Principle, Ramsey Theory and Metamaterials. Materials 2023, 16, 7571.
Abstract
Re-interpretation of the Fermat principle governing propagation of light in media within the Ramsey Theory is suggested. Complete bi-colored graphs corresponding to the light propagation in media are considered. The vertices of the graphs correspond to the points of the real physical space, in which light sources or sensors are placed. Red links in the graphs correspond to the actual optical paths, emerging from the Fermat principle. A variety of optical events: refraction, reflection, etc. may be involved in the light propagation. Green links, in turn, denote the trial/virtual optical paths, which actually do not occur. The Ramsey Theorem states that within the graph containing six points, inevitably, actual or virtual optical cycle will be necessarily present. Implementation of the Ramsey Theorem to the light propagation in metamaterials is discussed. The Fermat Principle states that in metamaterials a light ray in going from point S to point P must traverse an optical path length L that is stationary with respect to variations of this path. Thus, bi-colored graphs built of links corresponding to maxima or minima of optical paths become possible. Graphs, comprising six vertices, inevitably will demonstrate optical cycles, built of the mono-colored links corresponding to the maxima or minima of the optical path. The notion of the “inverse graph” is introduced and discussed. The total number of triangles in the “direct” (source) and “inverse” Ramsey optical graphs is the same, Optical interpretation of the infinite Ramsey Theorem is suggested.
Keywords
Fermat principle; optical path; metamaterials; left-handed materials; graphs; Ramsey Theory; Ramsey Theorem; optical cycle.
Subject
Physical Sciences, Theoretical Physics
Copyright: 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.
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