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arXiv:1302.2586 · Physical Review A

From Sommerfeld and Brillouin forerunners to optical precursors

B. Macke, Bernard Ségard

Published Apr 22, 2013• 21 citations
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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 approximation in spite of the initial discontinuity of the incident field envelope. The dynamical beats and $0\ensuremath{\pi}$ pulses generated by ultrashort incident pulses are also briefly examined.

DOI
10.1103/physreva.87.043830

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Annual Meeting Optical Society of America•Jan 1, 1987•0 citations
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 approximation in spite of the initial discontinuity of the incident field envelope. The dynamical beats and 0\ensuremathπ0\ensuremath{\pi}0\ensuremathπ pulses generated by ultrashort incident pulses are also briefly examined.
This article mainly deals with the propagation of step-modulated light pulses in a dense Lorentz medium at distances such that the medium is opaque in a broad spectral region including the carrier frequency. The transmitted field is then reduced to the celebrated precursors of Sommerfeld and Brillouin, far apart from each other. We obtain simple analytical expressions of the first (Sommerfeld) precursor, whose shape only depends on the order of the initial discontinuity of the incident field and whose amplitude rapidly decreases with this order (rise-time effects). We show that, in a strictly asymptotic limit, the second (Brillouin) precursor is entirely determined by the frequency dependence of the medium attenuation and has a Gaussian or Gaussian-derivative shape. We point out that this result applies to the precursor directly observed in a Debye medium at decimetric wavelengths. When attenuation and group-delay dispersion both contribute to its formation, we establish a more general expression of the Brillouin precursor, containing the previous one (dominant-attenuation limit) and that obtained by Brillouin (dominant-dispersion limit) as particular cases. We finally study the propagation of square or Gaussian pulses, and we determine the pulse parameters optimizing the Brillouin precursor. Obtained by standard Laplace-Fourier procedures, our results are explicit and contrast in their simplicity those derived by the uniform saddle-point methods, from which it is difficult to retrieve our asymptotic forms.
We study theoretically the propagation of a step-modulated optical field as it passes through a dispersive dielectric made up of a dilute collection of oscillators characterized by a single narrow-band resonance. The propagated field is given in terms of an integral of a Fourier type, which cannot be evaluated even for simple models of the dispersive dielectric. The fact that the oscillators have a low number density (dilute medium) and have a narrow-band resonance allows us to simplify the integrand. In this case, the integral can be evaluated exactly, although it is not possible using this method to separate out the transient part of the propagated field known as optical precursors. We also use an asymptotic method (saddle-point method) to evaluate the integral. The contributions to the integral related to the saddle points of the integrand give rise to the optical precursors. We obtain analytic expressions for the precursor fields and the domain over which the asymptotic method is valid. When combined to obtain the total transient field, we find that the agreement between the solutions obtained by the asymptotic and the exact methods is excellent. Our results demonstrate that precursors can persist for many nanoseconds and the chirp in the instantaneous frequency of the precursors can manifest itself in beats in the transmitted intensity. Our work strongly suggests that precursors have been observed in many previous experiments.
We observe interference between the optical precursors and the main signal for small optical depth α0 L ∼ 1, in which the main signal cannot be entirely absorbed. Since the main signal oscillates at the carrier frequency of the input pulse and precursors oscillate at medium resonance frequency, in our case carrier frequency dependence of the total transmitted field is observed as a form of modulation patterns oscillating at the detuning frequency. To distinguish between the Sommerfeld and Brillouin precursors for the case of weakly dispersive off-resonance medium, we utilize asymptotic precursor theory under the assumption of small detuning.
We theoretically study the linear propagation of a stepwise pulse through a dilute dispersive medium when the frequency of the optical carrier coincides with the center of a natural or electromagnetically induced transparency window of the medium (slow-light systems). We obtain fully analytical expressions of the entirety of the step response and show that, for parameters representative of real experiments, Sommerfeld-Brillouin precursors, main field and second precursors (``postcursors'') can be distinctly observed, all with amplitudes comparable to that of the incident step. This behavior strongly contrasts with that of the systems generally considered up to now.
To obtain a more complete understanding of the effects of temporal dispersion on the propagation dynamics of optical pulses in a physically realistic medium, we have examined the classical problem of dispersive pulse propagation in a Lorentz model medium that is characterized by two separate resonance frequencies. Both an input delta-function pulse and an input unit step-function modulated signal with carrier frequency ωc are considered. In each case the dynamic evolution of the propagated field is described in terms of the dynamics of the saddle points in the complex ω plane that are associated with the complex phase function appearing in the integral representation of the field. The appearance of a new precursor field, separate from the well-known Sommerfeld and Brillouin precursor fields, is shown to appear when the two resonance frequencies of the medium are sufficiently separated. Finally, the signal velocity that is characteristic of the field contribution oscillating at the applied carrier frequency ωc is obtained. These results are illustrated with numerical solutions that depict the entire evolution of the propagated field.
The uniform asymptotic description of the propagation of an input rectangle-modulated harmonic signal of angular frequency ω c and initial pulse width T into the half-space z ᐳ 0 that is occupied by a single resonance Lorentz medium is described. The asymptotic description is developed by representing the input rectangular pulse as the difference between two Heaviside unit step-function modulated signals that are separated in time by the initial pulse width T. This representation clearly shows that the resultant pulse distortion in the dispersive medium is solely due to the Sommerfeld and Brillouin precursor fields that are associated with the leading and trailing edges of the input pulse. The dynamic evolution of the pulse with increasing propagation distance z ᐳ 0 is described for both long and very short initial pulse widths T. In both cases it is shown that the pulse distortion becomes severe when the propagation distance z is such that the precursor fields associated with the trailing edge of the pulse interfere with the precursor fields associated with the leading edge.