Transition operators in electromagnetic-wave diffraction theory. II. Applications to optics.

Abstract

This paper is the second of a series. The first [G. E. Hahne, Phys. Rev. A 45, 7526 (1992)] developed a general theory of the transition operator approach to diffraction of time-harmonic electromagnetic waves from fixed obstacles, such that the response of the obstacle, denoted by \ensuremath{\Omega}, to an impinging electromagnetic signal with wave number ${\mathit{k}}_{0}$ was simulated by nonlocal, homogeneous Leontovich (i.e., impedance) boundary conditions on the obstacle's surface, which surface is called \ensuremath{\partial}\ensuremath{\Omega}. Moreover, the exterior region, called ${\mathrm{\ensuremath{\Omega}}}^{\mathrm{ex}}$, was presumed to be unbounded empty space, and has an electromagnetic response that can be expressed in terms of the so-called radiation impedance operator, denoted Z${\mathrm{\ifmmode \breve{}\else \u{}\fi{}}}_{\mathit{k}0}^{+}$; Z${\mathrm{\ifmmode \breve{}\else \u{}\fi{}}}_{\mathit{k}0}^{+}$ is a certain invertible, linear functional operator that maps the space of complex tangent-vector fields on \ensuremath{\partial}\ensuremath{\Omega} into itself.The matching of the limiting tangential electric and magnetic fields on \ensuremath{\partial}\ensuremath{\Omega} yielded a functional-operator expression for the transition operator and thereby a formal reduction to quadratures of the entire direct-scattering problem. This paper is intended to serve as an illustration and elaboration of the formalism presented in the first paper. After a brief recapitulation of the theory, the following topics are dealt with. First, an infinite sum in terms of vector spherical harmonics is obtained for Z${\mathrm{\ifmmode \breve{}\else \u{}\fi{}}}_{\mathit{k}0}^{+}$ for \ensuremath{\partial}\ensuremath{\Omega} a sphere. Second, a quasiplanar approximation, based directly on the exact result for planar \ensuremath{\partial}\ensuremath{\Omega}, is obtained for Z${\mathrm{\ifmmode \breve{}\else \u{}\fi{}}}_{\mathit{k}0}^{+}$ when \ensuremath{\Vert}${\mathit{k}}_{0}$\ensuremath{\Vert} is large compared to the local surface curvatures of \ensuremath{\partial}\ensuremath{\Omega} and attention can be restricted to small neighborhoods on \ensuremath{\partial}\ensuremath{\Omega}; when augmented by the method of stationary phase, this approximation is shown to lead to the familiar physical optics method for smooth, convex surfaces. Third, the latter method, supported by further interventions of the method of stationary phase, is applied in a well-established manner to secure the results of geometrical optics for the complete Green's function for the time-harmonic Maxwell field in the presence of a smooth-surfaced, convex, perfectly electrically conducting obstacle.One feature of the latter computations is that the original source currents from which the free-space Maxwell Green's function is constructed are presumed to generate ordinary outgoing (``causal'') electromagnetic waves for electric current sources, and purely ingoing (``anticausal'') waves for magnetic current sources (i.e., sinks) of electromagnetic radiation. Fourth, a formal construction is derived for mapping the Leontovich boundary conditions on an inner surface into a set of Leontovich boundary conditions on an outer surface, in the circumstance that a layer comprised of a material medium, which has possibly nonuniform and nonsymmetric tensors representing its constitutive properties, fills the domain between the surfaces; the construction presumes that a Green's function is available for the corresponding Maxwell equations in the medium when the medium is extended appropriately to fill all space. The main body of the paper concludes with a brief discussion of directions of possible future work and applications. The first appendix shows how to apply the method of stationary phase in the present context, in particular for the mixed case that the original radiation source is anticausal and the response currents generated in the obstacle are causal sources of radiation. A second appendix develops an acoustic (Helmholtz equation) analog to the fourth topic mentioned above, and exhibits an interplay of the theory with symplectic transformations in the case that the principle of reciprocity holds; it is noted that a symplectic connection also exists in the electromagnetic case when the propagation medium's constitutive properties are such that reciprocity holds.