Finite turn-on time effects on the transient phenomena in dispersive pulse propagation

Abstract

To study the effects of a finite turn-on time on the transient phenomena associated with dispersive pulse propagation, we consider the propagated field in a Lorentz medium due to an input hyperbolic-tangent modulated signal f(t) = u (t) sin(ωct) of carrier frequency ω c with initial pulse envelope where the parameter β, which is indicative of the rapidity of turn-on of the signal, is real and positive. In the limit as β → ∞ this initial envelope function approaches a unit step function. The dynamic evolution of the propagated field is described via the dynamics of the saddle points in the complex ω plane that are associated with the complex phase function appearing in the integral representation of the propagated field and their interaction with the simple pole singularities of the spectrum ũ (ω − ω c ) of the intial pulse envelope function. This analysis shows that the precursor fields that are characteristic of the input unit step-function modulated signal will persist nearly unchanged for the input hyperbolic-tangent modulated signal for values of β of the order of δ or greater, where δ is the damping constant of the Lorentz model medium. As β decreases below δ, the precursor fields become less important and the field becomes quasimonochromatic.