Asymptotic description of ultrashort electromagnetic pulse propagation in a linear, causally dispersive medium

Abstract

The asymptotic description of the propagation of an input pulse‐modulated harmonic signal of fixed angular frequency ωc and initial pulse width T into the half‐space z > 0 that is occupied by a single resonance Lorentz medium with resonance frequency ω0 and damping constant δ is described. Particular attention is given to both the input rectangle‐modulated and Gaussian‐modulated fields. The dynamical evolution of the pulse with increasing propagation distance z > 0 is described for both long and very short initial pulse widths T when the signal frequency is either below, near, or above the medium resonance frequency. For the input rectangular pulse envelope the asymptotic theory clearly shows that the resultant pulse distortion in the dispersive medium is due primarily to the Sommerfeld and Brillouin precursor fields that are excited by the leading and trailing edges of the input pulse. It is shown that the pulse distortion becomes severe when the propagation distance z is such that the precursor fields that are associated with the trailing edge of the pulse interfere with the precursor fields associated with the leading edge. Such is not the case, however, for the input Gaussian pulse envelope. For this pulse shape the asymptotic theory shows that the propagated field can be expressed solely in terms of a generalized Sommerfeld and Brillouin precursor field. If the applied signal frequency ωc is either below or near the medium resonance frequency, the propagated field is found to be dominated by a generalized Brillouin precursor field structure, while if ωc is well above the medium resonance frequency, the propagated field is dominated by a generalized Sommerfeld precursor field structure. The pulse distortion is shown to be due solely to the manner in which the initial Gaussian pulse‐envelope spectrum modifies the precursor field amplitude. Finally, the frequency dependence of the pulse velocity in the single resonance Lorentz medium is compared for these two different pulse types.