Date of Award

Winter 1988

Document Type


Degree Name

Doctor of Philosophy (PhD)


Mathematics and Statistics


Computational and Applied Mathematics

Committee Director

J. Swetits

Committee Member

A. M. Buoncristiani

Committee Member

Stan Weinstein

Committee Member

John Heinbockel


This is a study of a mathematical model of the dynamics of an optically pumped four-level solid state laser system. A general mathematical model that describes the spatial and temporal evolution of the electron populations in the laser rod as well as the development of the left and right traveling photon fluxes in the cavity is developed. The model consists of a coupled set of first order semilinear partial differential equations. While the model was developed for Titanium-doped sapphire lasers, it is applicable to three and four level lasers in general.

The analysis of the model is conducted in two parts. First, by formally taking an average over the spatial variable, the system of partial differential equations is reduced to a system of ordinary differential equations describing the temporal behavior of the spatially-averaged dynamic quantities. Several qualitative properties of the solutions of this system are proved and stability of the solutions under various operating conditions is investigated. The rate equations are solved numerically and the effects on the solutions of changes in the physical parameters are discussed.

The second part of this study is concerned with the qualitative and numerical analysis of the spatial and temporal model of a Ti: Sapphire ring laser. Several qualitative properties of the solution are established. The system of partial differential equations is solved numerically by integration along the characteristic lines using an implicit integration scheme developed for this problem. The computed solutions are compared to those obtained by using a stable finite difference approximation. The results of the comparison demonstrate that the implicit integration scheme is viable as well as efficient for numerically solving the system of partial differential equations and can be considered a useful analytical tool for studying the dynamics of this type of laser system.

All computer codes are written in FORTRAN and currently run on a DEC VAX 11/750.