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Journal of Radio Electronics

On the propagation velocity of a linearly frequency-modulated signal in a dispersing medium (using the method of moments)

Nikolai S. Bukhman

Published Jul 1, 2025• 0 citations
Publisher

Abstract

It is shown that in a selectively absorbing medium, the propagation velocity of a linearly frequency-modulated signal differs from its group velocity and depends on its modulation index, and the "modulation" addition to the group delay time reverses its sign when the sign of the modulation index changes. As a result, the propagation velocity of a linearly frequency-modulated signal can be superluminal at its subluminal group velocity and superluminal at its subluminal group velocity. When the center of the spectral line falls within the spectrum of a linearly frequency–modulated signal, a single linearly frequency-modulated signal is divided into two signals, one of which propagates at a speed higher than the group velocity (possibly higher than the vacuum light velocity), and the second is lower than the group velocity. The difference in the propagation speed of linearly frequency-modulated signals differing in the sign of the modulation index can lead to the fact that the initially amplitude-modulated signal, when propagating in a selectively absorbing medium, can be divided into several signals propagating at different speeds, and these signals, under certain conditions, can also be purely amplitude-modulated. Obviously, these effects can occur not only when a signal propagates in a selectively absorbing medium, but also when it is reflected from a selectively absorbing surface and propagated through linear filters in circuit theory.

DOI
10.30898/1684-1719.2025.7.12

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It is shown that in a selectively absorbing medium, the propagation velocity of a linearly frequency-modulated signal differs from its group velocity and depends on its modulation index, and the "modulation" addition to the group delay time reverses its sign when the sign of the modulation index changes. As a result, the propagation velocity of a linearly frequency-modulated signal can be superluminal at its subluminal group velocity and superluminal at its subluminal group velocity. When the center of the spectral line falls within the spectrum of a linearly frequency–modulated signal, a single linearly frequency-modulated signal is divided into two signals, one of which propagates at a speed higher than the group velocity (possibly higher than the vacuum light velocity), and the second is lower than the group velocity. The difference in the propagation speed of linearly frequency-modulated signals differing in the sign of the modulation index can lead to the fact that the initially amplitude-modulated signal, when propagating in a selectively absorbing medium, can be divided into several signals propagating at different speeds, and these signals, under certain conditions, can also be purely amplitude-modulated. Obviously, these effects can occur not only when a signal propagates in a selectively absorbing medium, but also when it is reflected from a selectively absorbing surface and propagated through linear filters in circuit theory.
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.
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.
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 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.
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).