Path-integral analysis of passive, graded-index waveguides applicable to integrated optics
The Feynman path integral is used to describe paraxial, scalar wave propagation in weakly inhomogeneous media of the type encountered in passive integrated-optical communication devices. Most of the devices considered in this work are simple models for graded-index waveguide structures, such as tapered and coupled waveguides of a wide variety of geometries. Tapered and coupled graded-index waveguides are the building blocks of waveguide junctions and tapered couplers, and have been mainly studied in the past through numerical simulations. Closed form expressions for the propagator and the coupling efficiency of symmetrically tapered graded-index waveguide sections are presented in this thesis for the first time. The tapered waveguide geometries considered are the general power-law geometry, the linear, parabolic, inverse-square-law, and exponential tapers. Closed form expressions describing the propagation of a centred Gaussian beam in these tapers have also been derived. The approximate propagator of two parallel, coupled graded-index waveguides has also been derived in closed form. An expression for the beat length of this system of coupled waveguides has also been obtained for the cases of strong and intermediate strength coupling. The propagator of two coupled waveguides with a variable spacing was also obtained in terms of an unknown function specified by a second order differential equation with simple boundary conditions. The technique of path integration is finally used to study wave propagation in a number of dielectric media whose refractive index has a random component. A refractive index model of this type is relevant to dielectric waveguides formed using a process of diffusion, and is thus of interest in the study of integrated optical waveguides. We obtained closed form results for the average propagator and the density of propagation modes for Gaussian random media having either zero or infinite refractive-index-inhomogeneity correlation-length along the direction of wave propagation.