Validity of the Classical Theory of Spontaneous Emission and the Fast Multipole Method for Electromagnetic Scattering

The interaction of the electromagnetic field with material boundaries has long been a subject of intense investigation. On the theoretical side are problems concerning the quantum-mechanical properties of the electromagnetic field near material boundaries. Such problems are of interest to physicists in the field of quantum optics near surfaces. On the practical side are problems concerning the numerical techniques used to solve the equations of classical electrodynamics in various practical situations involving boundaries. Such problems are of interest to engineers in the field of electromagnetic scattering. This thesis provides quantitative solutions to specific theoretical and practical problems in the subject of the interaction between the electromagnetic field and material boundaries. First, the lifetime of an excited atom near a lossy dielectric surface is calculated from an exact solution of a microscopic Hamiltonian model, which includes the effects of dispersion, local field correction and near -field Coulomb interaction. Results for the total decay rate are shown to be in excellent agreement with those based on classical electromagnetic theory and to yield the well-known result for the rate of nonradiative energy transfer in the limit of very small distance from the surface. Because our calculation is based on a fully canonical quantum theory, it provides the first fundamental demonstration of the validity of the classical electromagnetic theory of the rate of spontaneous emission near a lossy dielectric surface. Next, two new numerical techniques for three-dimensional electromagnetic scattering are proposed. The first technique is based on the physical-optics approximation and is suitable for piecewise-linear topography. The formalism of generalized Sommerfeld integrals is used to treat the effects of intra -surface multiple scattering in the physical-optics approximation. The technique of multipole acceleration is used to reduce the CPU cost of intra-surface multiple-scattering computation to O(N^{3/2}), where N is the number of surface unknowns. This approximate numerical technique is suitable for use in the simulation of photoresist exposure over large, piecewise-linear 3-D topography. The second technique is a rigorous numerical technique based on an alternative formulation of the Fast Multipole Method (FMM) and is suitable for arbitrarily shaped, perfectly conducting objects. Our FMM algorithm differs from the standard FMM algorithm in that we represent the field in the far zone due to a localized group of sources by a sum of multipole waves, rather than by a sum of plane waves. A procedure involving a combination of coordinate rotations and translation was developed to speed up the transformation of the multipole expansions. The CPU cost of our algorithm is O(N^{5/3}) compared to O(N^{3/2}) for the standard FMM algorithm. However, our algorithm is numerically stable in the long-wavelength limit whereas the standard FMM algorithm is not. This rigorous numerical technique can be extended for use in many important 3-D problems such as the modeling of optical proximity probes and on -chip interconnects.