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

  • P. K. Jakobsen
  • M. MansuripurEmail author
Regular Paper
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Abstract

We present a brief overview of Sommerfeld’s forerunner signal, which occurs when a monochromatic plane-wave (frequency \(\omega =\omega _{\text {s}})\) suddenly arrives, at time \(t=0\) and at normal incidence, at the surface of a dispersive dielectric medium of refractive index \(n\left( \omega \right) \). Deep inside the dielectric host at a distance \(z_0 \) from the surface, no signal arrives until \(t=z_0/c\), 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 \(z=z_0 \). This so-called Sommerfeld forerunner (or precursor) is highly chirped, meaning that its frequency, which is much greater than \(\omega _{\text {s}} \) immediately after \(t=z_0/c\), declines rapidly with the passage of time. The incident light with its characteristic frequency \(\omega _{\text {s}} \) eventually arrives at \(t\cong z_0 /v_{\text {g}} \), where \(v_{\text {g}} \) is the group velocity of the incident light inside the host medium—it is being assumed here that \(\omega _{\text {s}} \) 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 \(t\cong n\left( 0 \right) z_0 /c\) and the beginning of the steady signal (i.e., that which has the incident frequency \(\omega _{\text {s}}\)) at \(t=z_0 /v_{\text {g}} \). This second forerunner, which is also weak and chirped, having a frequency that is well below \(\omega _{\text {s}} \) at first, then grows rapidly in time to reach \(\omega _{\text {s}} \), 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 \(-\infty \) to \(\infty \). 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.

Keywords

Sommerfeld forerunner Brillouin forerunner Electromagnetic precursors Superoscillations Suboscillations 

Notes

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Copyright information

© Chapman University 2019

Authors and Affiliations

  1. 1.College of Optical SciencesThe University of ArizonaTucsonUSA
  2. 2.Department of Mathematics and StatisticsUIT The Arctic University of NorwayTromsoNorway

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