Simple asymptotic forms for Sommerfeld and Brillouin precursors

Details Simple asymptotic forms for Sommerfeld and Brillouin precursors Simple asymptotic forms for Sommerfeld and Brillouin precursors

2012-03-20

arxiv.org/abs/1203.4461v1

Physics

Optics

Scientific paper

We examine from a physical viewpoint the classical problem of the propagation of a causal optical field in a dense Lorentz-medium when the propagation distance is such that the medium is opaque in a broad spectral region including the frequency of the optical carrier. The transmitted signal is then reduced to the celebrated precursors of Sommerfeld and Brillouin, well separated in time. In these conditions, we obtain explicit analytical expressions of the first (Sommerfeld) precursor, which only depend on the nature and the importance of the initial discontinuity of the incident field, and we show that the second (Brillouin) precursor has a Gaussian or Gaussian-derivative shape, depending whether the time-integral (algebraic area) of the incident field differs or not from zero. We demonstrate that the Brillouin precursor that has been actually observed in a Debye medium at decimetric wavelengths is also Gaussian. We complete these results by establishing a more general expression of the Brillouin precursor in the Lorentz medium, containing the previous Gaussian one and that obtained by Brillouin himself as particular cases. The propagation of pulses with a square or Gaussian envelope is also studied and we determine the pulse parameters optimizing the Brillouin precursor. Obtained by standard Laplace-Fourier procedures, our analytical results are explicit and contrast by their simplicity from those available in the abundant literature on precursors, the complexity of which often hides the asymptotic behaviors evidenced in the present article

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