||Controlling and understanding the propagation of optical pulses through dispersive
media forms the basis for optical communication, medical imaging, and other modern
technological advances. Integral to this control and understanding is the ability
describe the transients which occur immediately after the onset of a signal. This
thesis examines the transients of such a system when a unit step function is applied.
The electromagnetic field is described by an integral resulting from Maxwell’s
Equations. It was previously believed that optical precursors, a specific transient
existed only for only a few optical cycles and contributed only small magnitudes
to the field. The main results of this thesis show that the transients arising from
integral are entirely precursors and that they may exist on longer time scales and
contribute larger magnitudes to the field.
The experimental detection of precursors has previously been recognized only
through success comparison to the transient field resulting from an application of
method of steepest descent to that field integral. For any parameter regime where
steepest descents may be applied, this work gives iterative methods to determine
saddle points which are both more accurate than the accepted results and to extend
into regimes where the current theory has failed. Furthermore, asymptotic formulae
have been derived for regions where previous attempts at steepest descent have failed.
Theory is also presented which evaluates the applicability of steepest descents in
represention of precursor behavior for any set of parameters. Lastly, the existence
of other theoretical models for precursor behavior who may operate beyond the
reach of steepest descent is validated through successful comparisons of the transient
prediction of those methods to the steepest descent based results of this work.