UNIFORM ASYMPTOTIC DESCRIPTION OF ULTRASHORT GAUSSIAN-PULSE PROPAGATION IN A CAUSAL, DISPERSIVE DIELECTRIC

The complete asymptotic description of ultrashort Gaussian-pulse propagation in a single-resonance Lorentz medium in the mature-dispersion regime is presented and compared with the results of two independent numerical experiments of the propagated-field evolution. The nonuniform asymptotic method of Olver [Stud. Appl. Math. Rev. 12, 228 (1970)] (an extension of the method of steepest descents) is first applied to obtain the standard asymptotic description of the propagated field that is due to the given input Gaussian-modulated field. The description afforded by this asymptotic method, although nonuniform in certain space-time regions, is found to be in excellent agreement with purely numerical results, especially when exact, numerically determined saddle-point locations, and exact expressions for the derivatives of the phase function are used in the implementation of the theoretical approach. Modern uniform asymptotic techniques, which generalize Olver's nonuniform description, are then employed to obtain a rigorous description of ultrashort Gaussian-pulse propagation that is uniformly valid for all space-time points in the mature-dispersion regime. This asymptotic description clearly shows that the propagated field can be expressed solely in terms of a generalized Sommerfeld and a generalized Brillouin precursor field, the first of which dominates the total propagated field when the carrier frequency omega(c) is well above resonance while the second generalized precursor field dominates when omega(c) is below or near resonance. It is further shown that the pulse distortion is due solely to the manner in which the precursor field amplitude is modified by the initial Gaussian-pulse envelope spectrum. Finally, the frequency dependence of the signal velocity of the input ultrashort, Gaussian-modulated harmonic field in a single-resonance Lorentz medium is discussed.