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Journal of Radio Electronics

About the superluminal group velocity of a video signal in an amplifying medium

N.S. Bukhman, S.N. Bukhman

Published Dec 1, 2025• 0 citations
Publisher

Abstract

It is shown that holomorphic (infinitely differentiable) video signals of sufficient duration are, in principle, capable of propagating in an environment with population inversion (and gain dominance over absorption at least at low frequencies) without significant distortion and significant amplification at a superluminal group velocity. The practical implementation of this effect is limited not so much by the distortion or amplification of the video signal itself, as by the maximum size (and amplification of extraneous signals at the maximum gain frequency) of actually existing active media. At best, with reasonable amplification coefficients of real-world active media, one can hope for the amount of time ahead of the vacuum speed of light by the video signal on the order of its own duration.

DOI
10.30898/1684-1719.2025.12.13

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It is shown that holomorphic (infinitely differentiable) video signals of sufficient duration are, in principle, capable of propagating in an environment with population inversion (and gain dominance over absorption at least at low frequencies) without significant distortion and significant amplification at a superluminal group velocity. The practical implementation of this effect is limited not so much by the distortion or amplification of the video signal itself, as by the maximum size (and amplification of extraneous signals at the maximum gain frequency) of actually existing active media. At best, with reasonable amplification coefficients of real-world active media, one can hope for the amount of time ahead of the vacuum speed of light by the video signal on the order of its own duration.
Abstract This paper considers the question of superluminal (or negative) group velocities of signals in dispersive media, as well as negative group delay times of signals passing through linear filters. It is shown that information can be transmitted only with the help of jumps in the time dependence of a function or its derivatives and, therefore, the information propagation velocity in any (including dispersive, absorbing, or amplifying) medium exactly coincides with the speed of light in vacuum. As for the group velocity of a wave packet, it is only the propagation velocity of its infinitely differentiable envelope, by means of which the transmission of information, strictly speaking, is impossible. Therefore, in regions with anomalous dispersion, the group velocity can be superluminal or negative without violating the “light limitation” of the theory of relativity or the principle of causality. Examples of situations in which superluminal or negative group velocity occurs are given.
It is shown that there are no dispersing media with «subluminal only» or «superluminal only» real group velocity – if in some medium at some frequencies the real group velocity is subluminal, then in the same medium there are necessarily frequencies at which the real group velocity is superluminal. Conversely, if a medium has frequencies with a superluminal group velocity, then it necessarily has frequencies with a subluminal group velocity. In other words, there is a necessary connection between subluminal and superluminal group velocity – if any medium is capable of generating subluminal group velocity, then the same medium is capable of generating superluminal group velocity. Similarly, there are no media with «only positive» or «only negative» imaginary part of the complex group velocity – if at some frequencies the imaginary part of the group velocity is positive, then at other frequencies in the same medium it is negative and vice versa. This means that if there is a carrier frequency in the medium at which some linearly frequency-modulated signal propagates at a speed lower than the group speed, then in the same medium there is necessarily a carrier frequency at which the same signal will propagate at a speed higher than the group speed and vice versa.
In the region between two lines of a doublet of an inverted medium, there exists a point of zero group-velocity dispersion, where highly superluminal effects may be observable without significant gain, loss, distortion, or broadening. The results of this group-velocity analysis hold for sufficiently narrow-band, analytic pulses, and do not constitute a violation of causality, although the group, ``signal,'' and energy velocities as defined by Sommerfeld and Brillouin may all exceed c or even become negative. No sharp disturbance in the pulse (a real signal) could propagate faster than light, but the scheme offers an unusual noiseless amplification scheme for the leading edge of a pulse, both at the classical and at the single-photon level.
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.