Theory of Superpotentials in Graded Index Dielectrics and Uniform Asymptotics

This monograph deals with the mathematical theory of electromagnetic wave propagation in a nonhomogeneous dielectric. The theory is first developed in terms of electric and magnetic potentials and a gauge condition which is a generalization of the Lorentz condition used in classical electromagnetic theory for an isotropic, homogeneous medium. The existence of these potentials under the generalized Lorentz condition is then established. Even though this generalized gauge simplifies the governing equations, it does not lead to partial differential equations which are readily tractable. To alleviate this problem, the theory of differential forms is used to demonstrate that the generalized Lorentz condition in E_4 implies the existence of superpotentials of the Hertzian variety, and it is shown that the field equations governing the superpotentials take a simple and beautiful form amenable to t standard analysis. Techniques of solution are then developed. The equation governing the superpotential for time-harmonic fields is then examined in relation to standard types of elliptic partial differential equations and the reduction of these equations to ordinary differential equations in standard form. A new asymptotic technique involving the method of global transformations for the solution of the resulting ordinary differential equations is then introduced. Necessary and sufficient conditions for existence of the global transformations are derived, and a full development of the method of solution far from the turning point is given. Finally, the surprising power of the asymptotic technique is demonstrated through two illustrative benchmark examples. It is shown with the help of two examples that the results obtained by using our new technique are exact and therefore far superior to those obtained by using existing techniques.