Noninstantaneous, finite rise-time effects on the precursor field formation in linear dispersive pulse propagation

Abstract

The effects of a noninstantaneous finite rise time on the transient field phenomena associated with dispersive pulse propagation are considered by use of the asymptotic description of the propagated plane-wave field in a single-resonance Lorentz model dielectric. The asymptotic description is presented for both an input hyperbolic tangent modulated signal with initial pulse envelope u(t) = [1 + tanh(βt)]/2 and an input raised cosine envelope signal with initial pulse envelope u(t) = 0 for t ≤ 0, u(t) = [1 − cos(βt)]/2 for 0 ≤ t ≤ π/β, and u(t) = 1 for t ≥ π/β. In both cases the parameter β is indicative of the rapidity of the initial rise time of the signal f(t) = u(t)sin(ωct) with input carrier frequency ωc. In the limit as β → ∞ both of these initial envelope functions approach the unit step function. The dynamical evolution of the propagated field is described by means of 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 − ωc) of the initial pulse envelope function. The analysis shows that the Sommerfeld and Brillouin precursor fields that are characteristic of the propagated field that is due to an input unit step function modulated signal will persist nearly unchanged for values of the rise-time parameter β of the order of δ or greater, where δ is the damping constant of the Lorentz model dielectric. As β is decreased below δ, the precursor fields become less important in the overall field structure and the field becomes quasi-monochromatic.