Abstracts

Asymptotic techniques give accurate approximationsin the modelling of wave propagation when the wavelength is muchsmaller than any other length scale in the system. These methodsrely principally on ray information. The optical properties of aninhomogeneous medium can be characterized by a function called thewave propagator, and the derivation of an asymptotic expression forthis function is given in this thesis. The limitations to thevalidity of the resulting expression are also treated here and theyrelate to the presence of caustics. The central contribution,however, is the development of a collection of new tools fordealing with the difficulties associated with caustics. Thefractional Legendre transformation (FLT) is among these tools, andgives a continuous transition between a function and its Legendretransform. Also proposed is a generalization of Hamilton'sformalism for geometrical optics. This generalization allows forthe definition of countless new representations for thecharacteristic functions that characterize the system in the raydomain. A subset of these new representations, associated with theFLT, is shown to have the fractional Fourier transform of a fieldas its counterpart in the wave domain. The asymptotic relationbetween these representations is the key to the derivation of a newasymptotic form for the wave propagator that has no difficultieswith caustics and automatically accounts for the caustic phaseshifts.