On the Nature of the Sommerfeld-Brillouin Forerunners (or Precursors)

TL;DR

This paper extends the Sommerfeld-Brillouin precursor theory to bandlimited signals, analyzing conditions for their existence and whether they exhibit super- or sub-oscillations, enhancing understanding of wave propagation in dispersive media.

Contribution

It introduces an extension of precursor theory to bandlimited signals, clarifying their conditions of existence and oscillation characteristics.

Findings

Precursor signals can persist under certain bandlimited conditions.

High-frequency precursors are not superoscillations.

Low-frequency precursors are not suboscillations.

Abstract

We present a brief overview of Sommerfeld's forerunner signal, which occurs when a monochromatic plane-wave (frequency {\omega}={\omega}_s) suddenly arrives, at time t=0 and at normal incidence, at the surface of a dispersive dielectric medium of refractive index n({\omega}). Deep inside the dielectric host at a distance z0 from the surface, no signal arrives until t=z0/c, where c is the speed of light in vacuum. Immediately after this point, a weak but extremely high frequency signal is observed at z=z0. This so-called Sommerfeld forerunner (or precursor) is highly chirped, meaning that its frequency, which is much greater than {\omega}_s immediately after t=z0/c, declines rapidly with the passage of time. The incident light with its characteristic frequency {\omega}_s eventually arrives at t~z0/vg , where vg is the group velocity of the incident light inside the host medium. Brillouin has identified a second forerunner that occupies the interval between the end of the Sommerfeld forerunner at t~n(0)z0/c and the beginning of the steady signal. This second forerunner is commonly referred to as the Brillouin forerunner (or precursor). Given that the incident wave has a sudden start at t=0, its frequency spectrum spans the entire range of frequencies from minus infinity to infinity. Consequently, the high-frequency first forerunner cannot be considered a superoscillation, nor can the low-frequency second forerunner be regarded as a suboscillation. The goal of the present paper is to extend the Sommerfeld-Brillouin theory of precursors to bandlimited incident signals, in an effort to determine the conditions under which these precursors would continue to exist, and to answer the question as to whether or not such precursors, upon arising from a bandlimited incident signal, constitute super- or sub-oscillations.