From Sommerfeld and Brillouin forerunners to optical precursors

TL;DR

This paper provides an analytical study of Sommerfeld and Brillouin optical precursors in a weak, narrow-resonance medium, revealing how overlapping forerunners form a resonant transient and examining ultra-short pulse effects.

Contribution

It derives explicit analytical expressions for optical precursors and the resonant transient in a single-resonance medium, including overlapping regimes and ultra-short pulse effects.

Findings

Explicit analytical expression for transmitted field at any time.

Overlapping forerunners lead to the formation of a resonant transient.

Analysis of ultra-short pulse induced dynamical beats.

Abstract

The Sommerfeld and Brillouin forerunners generated in a single-resonance absorbing medium by an incident step-modulated pulse are theoretically considered in the double limit where the susceptibility of the medium is weak and the resonance is narrow. Combining direct Laplace-Fourier integration and calculations by the saddle-point method, we establish an explicit analytical expression of the transmitted field valid at any time, even when the two forerunners significantly overlap. We examine how their complete overlapping, occurring for shorter propagation distances, originates the formation of the unique transient currently named resonant precursor or dynamical beat. We obtain an expression of this transient identical to that usually derived within the slowly varying envelope in spite of the initial discontinuity of the incident field envelope. The dynamical beats and 0Pi pulses generated by ultra-short incident pulses are also briefly examined