A differential forms approach to electromagnetics in anisotropic media
The behavior of electromagnetic fields in an inhomogeneous, anisotropic medium can be characterized by a tensor Green function for the electric field. In this dissertation, a new formalism for tensor Green functions using the calculus of differential forms is proposed. Using this formalism, the scalar Green function for isotropic media is generalized to an anisotropic, inhomogeneous medium. An integral equation is obtained relating this simpler Green function to the desired Green function for the electric field, generalizing the standard technique for construction of the Green function for the isotropic case from the scalar Green function. This treatment also leads to a new integral equation for the electric field which is a direct generalization of a standard free space result. For the special case of a biaxial medium, a paraxial approximation for the Green function is used to obtain the Gaussian beam solutions. A straightforward analysis breaks down for beams propagating along two singular directions, or optical axes, so these directions are investigated specially. The associated phenomenon of internal conical refraction is known to yield a circular intensity pattern with a dark ring in its center; this analysis predicts the appearance of additional dark rings in the pattern.
0544: Electrical engineering