  We study theoretically the propagation of a step-modulated optical field as
it passes through a dispersive dielectric made up of a dilute collection of
oscillators characterized by a single narrow-band resonance. The propagated
field is given in terms of an integral of a Fourier type, which cannot be
evaluated even for simple models of the dispersive dielectric. The fact that
the oscillators have a low number density (dilute medium) and have a
narrow-band resonance allows us to simplify the integrand. In this case, the
integral can be evaluated exactly, although it is not possible using this
method to separate out the transient part of the propagated field known as
optical precursors. We also use an asymptotic method (saddle-point method) to
evaluate the integral. The contributions to the integral related to the
saddle-points of the integrand give rise to the optical precursors. We obtain
analytic expressions for the precursor fields and the domain over which the
asymptotic method is valid. When combined to obtain the total transient field,
we find that the agreement between the solutions obtained by the asymptotic and
the exact methods is excellent. Our results demonstrate that precursors can
persist for many nanoseconds and the chirp in the instantaneous frequency of
the precursors can manifest itself in beats in the transmitted intensity. Our
work strongly suggests that precursors have been observed in many previous
experiments.
