Evolution of the Precursor Fields

Abstract

Based upon the foundational analyses just completed, the asymptotic behavior of pulse propagation in a Lorentz medium may now be developed. The analysis presented in this chapter begins with an examination of the exact field behavior for times t such that θ = ct/z < 1, for a fixed observation distance z. By applying the method Sommerfeld [7.1] used to treat the step-function modulated signal, it is shown here [7.2] that for fields with the initial envelopes u(t) that are zero for times t < θ, the propagated field is identically zero for all values of θ < 1, in agreement with the relativistic principle of causality [7.3]. The remainder of the chapter is devoted to determining the evolutionary properties of the precursor fields associated with the propagating pulse. The analysis follows the approach used by Brillouin [7.4, 5] in his treatment of the step function modulated signal except that the new approximations developed on Chap.6 for the saddle-point locations and the behavior of the complex phase function φ(ω,θ)at the saddle points are used and the advanced asymptotic techniques reviewed in Chap.5 are applied. Applications of the approximations from Chap.6 yields approximate expressions describing the dynamic behavior of the precursor fields which are much more accurate than any results available previously.