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Journal of Communications Technology and Electronics

Distortion of a short quasi-monochromatic signal in a resonantly absorbing medium

Nikolai S. Bukhman

Published May 1, 2007• 10 citations
Publisher

Abstract

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.

DOI
10.1134/s1064226907050099

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Physical Review A•Dec 1, 1986•50 citations
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.
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”.
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.
It is shown that the average group velocity over the signal spectrum (both subluminal and superluminal) describes the movement of the «center of gravity» of not all, but a wide class of narrowband signals for any length of the path, including beyond the range of applicability of the first order of the classical theory of dispersion, where the time dependence of the signal is usually significantly distorted. The absorption dispersion inevitably distorts the frequency distribution of the signal spectrum as the length of the path increases, so its group velocity changes accordingly. So, in particular, the initial superluminal group velocity may change to sublight as the length of the route increases. Moreover, for any signals with an unlimited spectrum (including for any signals of artificial origin), with a sufficient length of the route, this is inevitable. The reverse transition from sublight group velocity to superluminal velocity is impossible. It is shown that any signal with a carrier frequency located near the central frequency of the spectral absorption line travels at the initial section of the path with a superluminal group velocity, and its absorption is insignificant. Of course, as the length of the route increases, this superluminal group speed usually changes to sublight.
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 the velocity of propagation of a frequency-modulated wave packet through a strongly dispersive absorbing medium can be significantly different from (either higher or lower than) that of a non-frequency-modulated wave packet. This difference is attributed to the absorption dispersion of the medium. The easiest way to take the absorption dispersion into account is to use the formalism of the complex group velocity of a wave packet. This paper considers the propagation of a linear frequency-modulated wave packet, whose carrier frequency is close to the frequency of a spectral absorption line of the medium.
Propagation of a Gaussian wave packet in an absorbing medium is examined in order to shed light on the physical basis of group velocity which exhibits a singular behavior. It is found that the velocity of the wave packet, defined as the traveling distance of the peak amplitude divided by its flight time, decreases in the absorption range of frequency, although the group velocity becomes infinite in the same range. Fast pulse propagation, which was observed by Chu and Wong and is characterized by a packet velocity faster than the light velocity, turns out to be a characteristic in the early stage of the flight and is understood in terms of packet distortion due to damping of Fourier-component waves in an anomalous dispersion region. It also turns out that slow pulse propagation characterized by a packet velocity less than the light velocity appears for long traveling distance. The results provided a unified picture of wave-packet propagation in an absorbing medium.