Evolution of the Precursor Fields

Abstract

On the basis of the foundational analysis just completed, the asymptotic description of dispersive pulse propagation in both Lorentz-type and Debye-type dielectrics as well as in conducting and semiconducting media may now be fully developed. The analysis presented in this chapter begins with an examination of the exact propagated wavefield behavior for superluminal times t such that θ = ct ∕ z < 1 for a fixed propagation distance z > 0. By applying the method Sommerfeld [1, 2] used to examine the wavefront evolution of a step-function modulated signal in a causally dispersive medium (the Lorentz medium in particular), it is shown here [3, 4] that for wavefields with an initial pulse function f(t) that identically vanishes for all times t < 0, the propagated wavefield is identically zero for all superluminal space–time points θ < 1, in complete agreement with the relativistic principle of causality [5]. The remainder of the chapter is devoted to the determination of the evolutionary properties of the precursor fields that, because of their intimate connection to the evolutionary properties of the saddle points, are a charcteristic of the dispersive medium. The analysis follows the now classic approach pioneered by Brillouin [60 7] in his treatment of the Heaviside step-function modulated signal with fixed angular carrier frequency ω c > 0 propagating in a single resonance Lorentz medium. That analysis was based upon the then recently developed method of steepest descent (see Sect. F.7 of Appendix F) due to Debye [8]. The analysis presented here is based upon the advanced saddle point methods described in Chap.10. When combined with the more accurate approximations of the saddle point locations and the complex phase behavior at them developed in Chap. 12 for both Lorentz- and Debye-type dielectrics as well as for Drude model conductors and semiconducting materials, accurate asymptotic approximations of the associated precursor fields result that are uniformly valid over the entire space–time domain of interest. If necessary, greatly improved accuracy can always be obtained by using numerically determined saddle point locations in the asymptotic expressions. As a result of this detailed analysis, each feature appearing in the propagated wavefield sequence illustrated in Fig. 13.1 may be traced back to the dynamical behavior of a particular saddle point (or points) together with their interaction with any pole singularity in the initial pulse envelope spectrum. The numerically determined propagated wavefield sequence presented in this figure is due to an initial Heaviside unit step function modulated signal with below resonance carrier frequency \({\omega }_{c} = {\omega }_{0}/2\) at 0, 1, 2, and 3 absorption depths [\({z}_{d} \equiv {\alpha }^{-1}({\omega }_{c})\)] in a single resonance Lorentz model dielectric. Notice that the steady-state wave structure oscillating at the input angular carrier frequency ω c at each propagation distance z has amplitude given by the attenuation factor \({e}^{-z/{z}_{d}}\). The complicated field structure preceding this steady-state behavior is then due to the saddle points and is referred to as the first and second precursor fields. Of particular interest here is the observation that the peak amplitude of the second precursor field attenuates with increasing propagation distance z at a significantly smaller rate than does the remainder of the propagated wavefield. This unique feature may then be exploited in both imaging and communications systems. In addition, its impact on health and safety issues concerning exposure to ultrawideband electromagnetic radiation may have far-reaching implications, particularly in regard to digital cellular telephony.KeywordsSaddle PointAsymptotic ApproximationCritical SpaceUniform Asymptotic ExpansionSteep Descent PathThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.