Uniform asymptotic description of ultrashort rectangular optical pulse propagation in a linear, causally dispersive medium

The uniform asymptotic description of the propagation of an input rectangle-modulated harmonic signal of fixed angular frequency ${\mathrm{\omega }}_{\mathit{c}}$ and initial pulse width T into the half-space z>0 that is occupied by a single-resonance Lorentz medium is presented. The asymptotic description is developed by representing the input rectangular pulse as the difference between two Heaviside-unit-step-function-modulated signals that are separated in time by the initial pulse width T. This representation clearly shows that the resultant pulse distortion in the dispersive medium is primarily due to the Sommerfeld and Brillouin precursor fields that are associated with the leading and trailing edges of the input pulse. The dynamical pulse evolution with increasing propagation distance z>0 is completely described for both long and very short initial pulse widths T. In both cases it is shown that the pulse distortion becomes severe when the propagation distance z is such that the precursor fields associated with the trailing edge of the pulse interfere with the precursor fields associated with the leading edge. Finally, the asymptotic theory clearly shows that the main body of the pulse propagates with the signal velocity in the dispersive medium.