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Quantum Electronics

On the distortion of a wave packet propagating in an amplifying medium

Nikolai S. Bukhman

Published Apr 30, 2004• 9 citations
Publisher

Abstract

The propagation of a narrow-band signal (wave packet) in a medium with population inversion is considered. It is shown that in an optically thick medium layer the signal decomposes into the initial and amplified signals propagating at different velocities and having different durations. The initial signal propagates without distortions at the velocity c / n 0 ( n 0 is the refractive index of the medium away from the resonance frequency) and plays the role of a precursor of the amplified signal. The amplified signal moves at a lower velocity. It lags behind the initial signal during propagation and acquires a universal Gaussian shape (irrespective of the shape of the initial signal and the spectral line profile). The appearance of the amplified signal, which substantially differs from the initial signal in all the parameters, as well as the interference between the amplified and initial signals looks like a 'extra-distortion' of the initial signal.

DOI
10.1070/qe2004v034n04abeh002670

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The propagation of a narrow-band signal (wave packet) in a medium with population inversion is considered. It is shown that in an optically thick medium layer the signal decomposes into the initial and amplified signals propagating at different velocities and having different durations. The initial signal propagates without distortions at the velocity c / n 0 ( n 0 is the refractive index of the medium away from the resonance frequency) and plays the role of a precursor of the amplified signal. The amplified signal moves at a lower velocity. It lags behind the initial signal during propagation and acquires a universal Gaussian shape (irrespective of the shape of the initial signal and the spectral line profile). The appearance of the amplified signal, which substantially differs from the initial signal in all the parameters, as well as the interference between the amplified and initial signals looks like a 'extra-distortion' of the initial signal.
Propagation of a short quasi-monochromatic electromagnetic (light or radio) signal (wave packet) is considered. The carrier frequency of the signal is close to the frequency of the absorption spectral line (or group of lines) of a medium, and the spectrum width is large relative to the width of this spectral line (or group of lines). Analytic formulas for the temporal dependence of the signal are derived and numerically confirmed for paths of various lengths. It is shown that the signal is substantially distorted long before it noticeably decays owing to absorption. The signal’s temporal dependence turns out to be rather universal with respect to the form factor of the spectral line, the optical thickness of the substance layer, and the initial temporal dependence of the signal.
General relations are obtained that govern the evolution of the leading edge of a signal propagating in a selectively amplifying medium. The propagation of a pulse with a rectangular envelope, the carrier frequency of which is close to the frequency of one of the spectral amplification lines of a selective (in frequency) amplifying medium, is considered. The time dependence of the envelope of the initially rectangular signal is studied for various values of the optical thickness of the substance layer for the Gaussian and Lorentzian contours of the spectral line. It is shown that in the case of a shift of the carrier frequency of the signal relative to the center of the spectral line, the interference of the primary and amplified signal can lead to very significant oscillations of the total signal in time.
It is shown that, when a narrow-band signal with a non-Gaussian envelope propagates in a medium with a strong dispersion, the signal envelope being distorted due to the dispersion of the gain (or) absorption in the medium. Nevertheless, the amplitude-modulated signal having a symmetric (even) intensity envelope remains the same. The propagation velocity of the `middle point' (i.e., the symmetry centre) of the signal coincides with the real group velocity of the wave with a given carrier frequency in the medium and can be subluminal, superluminal or negative. Therefore, the concept of a real group velocity (including the superluminal or negative velocity) is also applicable in media with a strong dispersion of the gain (or absorption).
The propagation of a light pulse through a layer of a substance with population inversion and, correspondingly, with an anomalous dispersion in the transparency range is considered. It is shown that, when the wave packet approaches the layer of the substance with a strong anomalous dispersion, two additional wave packets arise at the rear boundary of the layer. One of these packets has a negative energy and moves in the layer from its rear to the front boundary, while the other packet moves away from the rear boundary of the layer into a vacuum. As the initial wave packet comes into contact with the front boundary of the layer of the substance, it meets the packet of the negative energy. As a result, the two packets annihilate one another, after which only one packet remains, which moves away from the layer.
We analytically study the linear propagation of arbitrarily shaped light pulses through an absorbing medium with a narrow transparency window or through a resonant amplifying medium. We point out that, under certain general conditions, the pulse acquires a nearly Gaussian shape, irrespective of its initial shape and of the spectral profile of the line. We explicitly derive in this case the pulse parameters, including its skewness, responsible for a deviation of the delay of the pulse maximum from the group delay. We illustrate our general results by analyzing the slow-light experiments having demonstrated the largest fractional pulse delays.
Abstract The paper is concerned with the distortion of a sharp leading edge of a quasi-monochromatic electromagnetic (light or radio) signal (wave packet) propagating in a medium with an absorption line (or band). It is shown that the frontal jump of the complex envelope of a radio signal propagates without changing its magnitude. Behind the leading edge, the signal level decreases with a rate increasing with the path length. As a result, the leading and trailing (if it exists) edges of the signal acquire tooth kinks, which become more and more acute with increasing path length. Such tooth kinks are formed not only at the points of discontinuity of the real amplitude of a signal but also at the points of discontinuity of its phase; therefore, the propagation of a phase-shift keyed radio signal in a resonantly absorbing medium is accompanied by its “self-disclosure”.