Low Frequencies and the Brillouin Precursor

When an electromagnetic step-modulated sine wave propagates through a causal dielectric material, the signal evolves into a Brillouin precursor whose peak amplitude point decays algebraically for large, yet finite, propagation distances. This algebraic decay is an apparent contradiction to the Bouger-Lambert-Beer law, which states that each nonzero frequency component of the pulse decays exponentially with propagation distance. Hence, the Brillouin precursor is commonly attributed to the dc or low frequency content of the initial pulse. However, there have been no studies that give the dependence of the peak amplitude on the low frequency content of the initial pulse. We accomplish this here by application of a cascade of single-pole high-pass filters, each with cut-off frequency ω
H
, to a step-modulated sine wave with carrier frequency ω
c
≥ ω
H
. Saddle point methods are used to provide a closed-form asymptotic approximation to the propagated field in a Debye-type dielectric material. Our results show that the Brillouin precursor exists even with the dc and low frequency content suppressed and that a substantial amount of low frequencies must be removed in order to observe a significant decrease in the decay rate of the peak amplitude point of the Brillouin precursor.