Uniform Asymptotic Description of Gaussian Pulse Propagation of Arbitrary Initial Pulse Width in a Linear, Causally Dispersive Medium

Abstract

Modern asymptotic techniques^1–4 have recently been utilized^5–9 in order to obtain an approximate analytic evaluation of the classical, exact integral representation of the propagated field due to an input ultrashort, Gaussian-modulated harmonic wave in the mature dispersion region of a single resonance Lorentz medium. A straightforward consideration of the behavior of the classical complex phase function appearing in this integral representation showed^7,9 that these asymptotic techniques cannot be applied when the space-time parameter θ′ is less than unity. In order to circumvent this difficulty, the input ultrashort Gaussian envelope must be chosen to be centered around a time that is sufficiently larger than the initial pulse width. It was also shown^7–9 that the classical, analytic, nonuniform asymptotic approach to this problem, which was presented in References 5 and 6, is only qualitatively accurate in its description of the dynamical evolution of the propagated field as a result of the use of approximate analytic expressions for the saddle point locations and the derivatives of the classical complex phase function at them¹⁰. Although these approximate analytic expressions are adequate for instantaneous rise/fall-time input pulses, they are not of sufficient accuracy for input pulses with an exponentially-varying spectrum and consequently need to be improved using numerical techniques. However, even this improved classical asymptotic approach with numerically determined saddle point locations breaks down for two narrow θ′-ranges when two of the conditions of Olver’s theorem^1,2, upon which this approach is based, are violated. In order to overcome this difficulty, the appropriate uniform asymptotic techniques^3–4 have subsequently been employed. The accuracy of the classical asymptotic description of ultrashort Gaussian pulse propagation was completely verified^7–9 through a comparison with the corresponding results of two different numerical experiments: the first is based upon Hosono’s algorithm^11–12 for the numerical inversion of Fourier-Laplace transform-type integrals, while the second is a numerical implementation of the asymptotic method of steepest descents^2,13,14.

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