On the nature of the Sommerfeld–Brillouin forerunners (or precursors)

P. K. Jakobsen, M. Mansuripur

Research output: Contribution to journalArticlepeer-review

Abstract

We present a brief overview of Sommerfeld’s forerunner signal, which occurs when a monochromatic plane-wave (frequency ω= ωs) suddenly arrives, at time t= 0 and at normal incidence, at the surface of a dispersive dielectric medium of refractive index n(ω). Deep inside the dielectric host at a distance z from the surface, no signal arrives until t= z/ c, where c is the speed of light in vacuum. Immediately after this point in time, however, a weak but extremely high frequency signal is observed at z= z. This so-called Sommerfeld forerunner (or precursor) is highly chirped, meaning that its frequency, which is much greater than ωs immediately after t= z/ c, declines rapidly with the passage of time. The incident light with its characteristic frequency ωs eventually arrives at t≅ z/ vg, where vg is the group velocity of the incident light inside the host medium—it is being assumed here that ωs is outside the anomalous dispersion region of the host. Brillouin has identified a second forerunner that occupies the interval between the end of the Sommerfeld forerunner at t≅ n(0 ) z/ c and the beginning of the steady signal (i.e., that which has the incident frequency ωs) at t= z/ vg. This second forerunner, which is also weak and chirped, having a frequency that is well below ωs at first, then grows rapidly in time to reach ωs, 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 - ∞ to ∞. 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 suboscillations.

Original languageEnglish (US)
Pages (from-to)315-339
Number of pages25
JournalQuantum Studies: Mathematics and Foundations
Volume7
Issue number3
DOIs
StatePublished - Sep 1 2020

Keywords

  • Brillouin forerunner
  • Electromagnetic precursors
  • Sommerfeld forerunner
  • Suboscillations
  • Superoscillations

ASJC Scopus subject areas

  • Atomic and Molecular Physics, and Optics
  • Mathematical Physics

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