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Journal of the Optical Society of America B

Coherent transients: optical precursors and 0π pulses

Heejeong Jeong, Ulf L. Österberg

Published Mar 28, 2008• 16 citations
Publisher

Abstract

We discuss the transient effects associated with short-pulse propagation in a linearly dispersive medium by comparing optical precursors with 0π pulses. The classical and quantum effects were, in the past, treated independently due to the different time scales. Using results from recent studies [Phys. Rev. Lett.96, 143901 (2006)] as well as the discussions by Varoquaux et al. [Phys. Rev. B.34, 7617 (1989)] and Bürck [Hyperfine Interact. 123, 483 (1999)], we show analytically that precursors and 0π pulses describe the same physical phenomenon. We also show that most published results of transient pulse propagation can be expressed using one overarching equation. Finally, we give a physical interpretation of the analogy between precursors and 0π pulses as a medium-impulse response using a Green's function.

DOI
10.1364/josab.25.0000b1

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We discuss the transient effects associated with short-pulse propagation in a linearly dispersive medium by comparing optical precursors with 0π pulses. The classical and quantum effects were, in the past, treated independently due to the different time scales. Using results from recent studies [Phys. Rev. Lett.96, 143901 (2006)] as well as the discussions by Varoquaux et al. [Phys. Rev. B.34, 7617 (1989)] and Bürck [Hyperfine Interact. 123, 483 (1999)], we show analytically that precursors and 0π pulses describe the same physical phenomenon. We also show that most published results of transient pulse propagation can be expressed using one overarching equation. Finally, we give a physical interpretation of the analogy between precursors and 0π pulses as a medium-impulse response using a Green's function.
A theoretical analysis of the propagation of near resonant and linearly chirped light pulses through a resonant medium is presented under the restriction that the pulses are far enough off resonance or weak enough so that they do not significantly alter the atomic populations. If the pulses are of duration comparable to or less than the phase memory time T(2) of the atoms, it is found that the transient response of the medium gives rise to amplitude-and phase-modulating effects. Analytic solutions are presented to describe the propagation of an off-resonant pulse which is very short compared with T(2) and an off-resonant rectangular pulse. The evolution of a Gaussian pulse having a positive linear chirp is studied numerically.
We report theoretical and experimental studies of the propagation of a square-modulated laser pulse through a laser-cooled atomic ensemble of two-level absorbers, as well as through a three-level system with electromagnetically induced transparency. We find that the transmission characteristics over a wide range of optical depth can be fully accounted for as optical coherent transients excited by the wide spectral content of the square-modulated optical pulse. We show that both time-domain atom-field coupled equations and frequency-domain linear dispersion theory give precisely the same description of the optical transients at a weak power limit. At low optical depth with moderate absorption, resonant excitation dominates and free-induction decay (FID) contributes mostly to the transient field. At high optical depth when absorption and propagation effects become significant, lossless optical precursors start to dominate the transient response. By varying the optical depth from 0 to 45, we observe that optical transients evolve gradually from FID to optical precursors. We thus show that FID and optical precursors, which have been considered as two different optical transients for many decades, can be unified within a single theoretical frame and they are the manifestations of the same physical process in two different regimes.
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.
For pt.I see ibid., vol.8, no.3, 339 (1975). The transient behaviour of two short optical pulses in two-photon coherent propagation is obtained by calculating numerically the coherent propagation equations which couple the two electromagnetic pulses and the atomic system in self-consistent fashion. The coherent interaction of the two different frequency pulses gives rise to lossless propagation in the absorptive medium. This description is characterized by a two-photon area theorem. Novel coherent propagation effects of peak amplification and multiple pulse breakup are found. These effects are ascribed to two-photon self-induced transparency analogous to that of a one-photon transition in a two-level system. A difference of input intensity between the two different frequency pulses yields curious modulation effects involving sharp peak amplification and marked pulse breakup for the low-intensity input. In contrast, the high-intensity input pulse displays a smooth variation with distance and time. Low-loss pulse transmission above threshold, and pulse advance and delay are found for the present case in contrast to that for one-photon self-induced transparency.
Optical Sommerfeld-Brillouin precursors significantly ahead of a main field of comparable amplitude have been recently observed in an opaque medium with an electromagnetically induced transparency window [Wei et al., Phys. Rev. Lett. 103, 093602 (2009)]. We theoretically analyze in this article the somewhat similar results obtained when the transparency is induced by the propagating field itself and we establish an approximate analytic expression of the time delay of the main-field arrival, which fits fairly well the result obtained by numerically solving the Maxwell-Bloch equations.