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.