Chapter 17 - The Fermat Principle and Chemical Waves

Abstract

The most common formulation of the Fermat principle states: light propagates in such a way that the propagation time is minimal. This chapter outlines the historical background as well as the different formulations of the Fermat principle. The Fermat principle is valid not only for light for any traveling wave. In optics, the underlying equation is the wave equation, which describes all details of propagation. Geometrical optics is a limiting case when wavelength tends to zero. The opposite limiting case—when wavelength tends to infinity—is the field of geometrical wave theory. That theory is based on the Fermat principle as well as on the dual concepts of rays and fronts. Chemical waves can be described by reaction–diffusion equations. These equations may have traveling-wave solutions. The essential features of the evolution of chemical wave fronts can also be derived from the geometric theory of waves. Some recent formulations of the Fermat principle require only stationarity of the extremals instead of the stricter requirement of minimal propagation time. This problem is discussed and it is concluded that for chemical waves only the requirement of minimum is relevant: maximum and “inflexion-type” local stationarity does not play any role. The chemical waves and their prairie-fire picture underlines: only the quickest rays have effects. Nevertheless, there are singularities, when there are several paths of rays with the same propagation time. For these cases the related fields of singularities and caustics are relevant. Special attention is paid to aplanatic surfaces, where all the reflected or refracted paths require the same time, that is stationarity holds globally. Finally, the shape of a chemical lens is derived: this ‘chemical lens' is able to perfect image formation with chemical waves that are circular fronts will be also circular after refraction.