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This monograph is about two methods of calculating the electromagnetic fields due to radiation scattering by a single scatterer. Both methods yield valid results for all wavelengths of the incident radiation as well as a wide variety of scatterer configurations. Only the theory is discussed; numerical consequences of the theory are not presented.
Ruling out any changes of state owing to high-energy processes, two essential features of radiation scattering are: (1) the shape of the scatterer, and (2) the wavelength of the incident radiation. Taking these accurately into account in scattering analyses is a goal that has challenged researchers since the formulation of optical science. Many approximate methods have been devised to this end.
One of the methods to be generalized and discussed in this book was originated by Arnold Sommerfeld in two fundamental papers published in 1896 and 1897. The promising simplicity and accuracy of the expressions derived, well-verified by experiment, led to expectations of further progress in the use of Sommerfeld’s method that were not realized in the years that followed. However, the need for exact solutions has been demonstrated by frequent references to the implications of Sommerfeld’s work, even when the scatterers did not have Sommerfeld’s original configuration. The lack of success in generalizing Sommerfeld’s analysis has led to a general belief that it cannot be done.
The author was fortunate to find an alternate formulation of Sommerfeld’s method—a logical continuation of research originated under the direction of Professor Griffith C. Evans at the University of California, Berkeley. This research was directed toward a generalization of Sommerfeld’s method of constructing multiple-valued potentials defined on a three-dimensional multileaved space, i.e., solutions to Laplace’s equation. Eventually, such a generalization was achieved and published in 1970. It was then evident that a similar generalization could be applied to construct multiple-valued Green’s functions for the equation of wave propagation. The results constitute part of the work in the following text, indicating a variety of ways in which Green’s functions can be constructed for a variety of configurations. For this introductory work, the method and the numerous consequences that follow from it render the theory more significant than the possible resulting calculations. However, it is hoped that such calculations will soon be carried out.
Although Sommerfeld’s method delivers solutions in a convenient, closed, and analytic form, these solutions have the drawback of being applicable only to surfaces spanning given space curves; it is not applicable to solids bounded by closed surfaces. This limitation is remedied in a method of analysis originated by the author and discussed following the exposition of Sommerfeld’s method. The new method provides a means of solving boundary-value problems for solids bounded by a large variety of surface configurations.
Although the analyses presented are primarily mathematical, they are not strictly rigorous. In this respect, the discussion follows the example set in the historical development of optics theory in which physical models and sufficient, rather than necessary, conditions have played an important role. The use of sufficient conditions, for example, is illustrated by superpositions of elementary solutions of the wave equation based on a physical model; it is then assumed that the series will converge to the correct solution of the boundary-value problem. Further, if the physical model is defined well enough to serve as a guide to any approximation, even a divergent series can be used as a solution, e.g., an asymptotic expansion. Indeed, Sommerfeld’s original solution for diffraction by a semi-infinite plane was given as the first term of an asymptotic expansion. A similar expansion is used in the second method of this book, although it is not an essential part of the analysis.
It will be seen that the following work can be considerably extended.
Frederick E. Alzofon
September 2003
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