A model equation for the optical tunnelling problem using parabolic cylinder functions

Brazel, Neal
(1989)
A model equation for the optical tunnelling problem using parabolic cylinder functions.
Master of Science thesis, Dublin City University.

The fundamental purpose of this thesis is to estimate the
exponentially small imaginary part of the eigenvalue of a
second order ordinary differential equation subject to
certain stated boundary conditions. This problem is
modelled on a partial differential equation which arises
when examining wave losses m bent fibre optic waveguides.
In Chapter 1 we provide an overview of the thesis and
introduce the area of mathematics known as exponential
asymptotics. In Chapter 2 we investigate the physical
background to the problem of energy losses due to optical
tunnelling in fibre optic waveguides. We then derive the
partial differential equation upon which we base our model.
In Chapter 3 we commence by manipulating the partial
differential equation into a more convenient form. We then
outline the model problem we shall consider and obtain a
preliminary estimate for the eigenvalue of this problem.
In Chapter 4 we introduce the special function known as the
parabolic cylinder function and derive its asymptotic
behaviour. We also examine its connection with Stokes
phenomenon and deduce its Stokes and anti-Stokes lines. In
Chapter 5 , we finally solve the *model problem by
transforming it into one form of Weber's parabolic cylinder
equation. We then use the boundary conditions of the
problem together with properties of parabolic cylinder
functions to obtain a valid estimate for the imaginary part
of the eigenvalue. In Chapter 6 we conclude the thesis by
commenting on this result and indicating future developments in this area.