Electromagnetic Theory and Geometrical Optics (Classic Reprint)

It may seem unnecessary at this late date to discuss the relationship of electromagnetic theory to geometrical optics. The content of both fields is well known and everyone knows also that geometrical optics is the limit for vanishing wave length of electromagnetic theory. Moreover, since Maxwell's theory supersedes the older geometrical optics, presumably, then, geometrical optics could be discarded. The optical industry continues to use it but perhaps that is because it is behind the times.

There are, however, at least three major reasons for pursuing and clarifying the relationship in question. The first is the purely theoretical or academic problem of building a mathematical bridge between the two domains, electromagnetic theory and geometrical optics. The older bases for asserting that geometrical optics is a limiting case of electromagnetic theory are vague and from a mathematical standpoint highly unsatisfactory.

The second major reason for the investigation is a practical one. To solve problems of electromagnetic theory, whether in the range of radio frequencies or visible light frequencies, one should solve Maxwell's equations with the appropriate initial and boundary conditions. However, as is well known, Maxwell's equations can be solved exactly in only a few problems. Hence physicists and engineers, especially those concerned with ultra-high frequency problems, have resorted to the simpler methods of geometrical optics. Although these methods have proved remarkably efficacious in the optical domain, they are intrinsically limited; they do not furnish information about some of the most important phenomena such as diffraction, polarization, and interference, to say nothing about the numerical accuracy of what geometrical optics does yield. Hence the practical question becomes whether the establishment of a better link between Maxwell's theory and geometrical optics will provide more accurate approximate methods of solving electromagnetic problems. Insofar as ultra-high frequency problems are concerned, the answer, based on work of the last ten years, can already be given affirmatively.

Tags: optics geometrical theory wave equations equation electromagnetic light field time asymptotic waves differential media solution rays solutions discontinuities space mathematical
Category: Science - Quantum Mechanics

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