Abstract
The addition of the laws of reflection and refraction to basic Euclidean geometry gives rise to an optical geometry which extends into non-Euclidean spaces with the inclusion of non-uniform, isotropic media, for which the ray paths are curved. Transformation theorems arise which are stranger than can be expected from the mere addition of the laws of optics to ordinary geometry. A common characteristic of these transformations is their dependence on the concept of geometrical inversion; such inversions have otherwise been largely ignored in theoretical physics. Given that the geometrical optic field is the infinite frequency approximation of the exact electromagnetic field representation, it is anticipated that any transformations which apply to a geometrical optic system should also apply to the equivalent electromagnetic field system. The work of this thesis introduces the effects of transformations arising from geometrical optics by considering the effects of conformal transformation of the underling electromagnetic field. The geometrical optic transformations to be considered include, among others, Budden's and Bateman's inversion theorems and an inversion theorem applicable within a spherically symmetrical medium. The approach adopted is to proceed in the same manner as in the analogous two dimensional electrostatic/hydrodynamic case, where the conformal properties of complex variable theory are exploited. This approach anticipates that a similar process applies to the four dimensional case of electromagnetism through the use of functions of a hypercomplex variable. These arise in the form originally discovered by Hamilton as quaternion variables, which have been extended in this work through the inclusion of complex i to biquaternion variables. It is also demonstrated that biquaternion variables possess holomorphic properties which are highly relevant to the representation of the electromagnetic field, so that the complete set of Maxwell's electromagnetic field equations can be formulated algebraically and without reference to any physical entities. Most importantly, it is shown how the required transformations of the electromagnetic field arise immediately from biquaternion similarity transformations. Functions of a biquaternion variable are studied in much the same manner as functions of a complex variable. The work of this thesis investigates and makes extensive use of such biquaternion functions. It is necessary then to be able to generate such biquaternion functions. The generation of such functions is based on a technique attributable to Fueter who, for the case of quaternions, defined a generating function which produces a series of polynomials, commonly referred to as Fueter Polynomials. This technique has been extended to biquaternion in this work. The biquaternions used are also descriptive of physical entities, describing for example distance, velocity, force and potentials. These biquaternions are termed physical biquaternions. This work has defined and made use of a number of physical biquaternions which are particularly relevant to describing geometrical optics and electromagnetism. The customary potential descriptions of the electromagnetic field in terms of field vectors or in any of the usual potential descriptions have proved inadequate for applications analogous with the transformations of geometrical optics. An alternative representation of the electromagnetic field has been developed that allows such an association. This has been achieved through the introduction and reformulation of the electromagnetic field in terms of two scalar potentials. This reformulation of the electromagnetic field results in certain conditions being placed on these scalar potentials, which must be met if realisable electromagnetic fields are to be obtained. The two examples of plane wave and dipole generated fields are considered in this thesis. This reformulation has also shown, that, for the spherically symmetrical medium, it provides the orthogonal transformation (the inversion) for rays and for the medium refractive index.