| This thesis presents novel concepts for electromagetic field simulations via partial differential equation (PDE) solvers. A vital aspect for any successful general implementation of a PDE solver is the use of an efficient absorbing boundary condition (ABC). The perfectly matched layer (PML) is a recently introduced ABC in Cartesian coordinates which provides reflection errors orders of magnitude smaller than previously employed ABCs. In this work, a new interpretation of the PML as an analytic continuation of the coordinate space is used to extend the PML to other coordinate systems. Modified equations replace the original Maxwell's equations, mapping propagating solutions into exponentially decaying solutions. Alternative (Maxwellian) formulations are also put forth, where the PML is represented as an artificial media with complex constitutive tensors, and the form of Maxwell's equations is retained. The causality and dynamic stability of the PML is characterized through a spectral analysis. In addition, a rationale is presented to extend the PML to complex media, e.g., dispersive and/or (bi-)anisotropic. For the Maxwellian formulation, the general expressions for the PML tensors matched to any interior dispersive and/or (bi-)anisotropic linear media are obtained.;A finite-difference time-domain (FDTD) algorithm in Cartesian coordinates which combines the PML ABC with piecewise-linear recursive convolution (PLRC) is proposed and implemented, allowing the simulation of electromagnetic fields in inhomogeneous and dispersive media with conductive loss. Two PML-PLRC-FDTD algorithms in cylindrical coordinates are also proposed and implemented. The first is developed through a split-field PML formulation, and the second through a Maxwellian (unsplit) PML formulation. A comparison is made between numerical properties of these two algorithms.;The PML concept is then studied within the language of differential forms to unify the various PML formulations. Finally, the language of differential forms is also utilized to provide a coordinate-free description and analyze consistency properties of the electromagnetic theory on lattice, for PDE solvers such as the finite-difference, finite-volume or finite-element methods. |
