Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantum Electronics

On the propagation velocity of a wave packet in an amplifying medium

Nikolai S. Bukhman

Published Sep 30, 2001• 17 citations
Publisher

Abstract

It is shown that the delay time of a weak signal propagating in an amplifying medium on the wings of the spectral amplification line may be shorter than the time of propagation of the signal with the velocity of light in vacuum. It is found that in this case, the time dependence of the signal is exactly 'reconstructed' at the point of detection, and the detection of the signal continues even if it is abruptly terminated at the point of transmission. It is also shown that using the complex time of group delay of the signal, it is possible to improve the accuracy of the results in the first order of dispersion theory within this approximation.

DOI
10.1070/qe2001v031n09abeh002043

Related papers

On the Principle of Causality and Superluminal Signal Propagation Velocities

Nikolai S. Bukhman

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.

Journal of Communications Technology and Electronics•Mar 1, 2021•8 citations

On the reality of supraluminal group velocity and negative delay time for a wave packet in a dispersion medium

Nikolai S. Bukhman

It is shown that, in the case of the supraluminal group velocity of a wave packet in a dispersion medium, a wave packet with a smooth (analytical) envelope does propagate with a supraluminal velocity. In the case of a negative group velocity, the signal maximum does arrive at the detector earlier than at the transmitter. These facts are consistent with both the finiteness of the velocity of light in free space for information transfer (in the case of supraluminal propagation velocity) and the principle of causality (in the case of negative delay time). Basically, the effect of negative delay time may be employed for predicting an observable effect.

Technical Physics•Jan 1, 2002•7 citations

On the relationship between the delay time of a narrowband signal |in a dispersing medium and its attenuation

Nikolai S. Bukhman

The 1D problem of propagation of a narrowband signal in a homogeneous dispersing medium is considered. It is shown that there is a direct and immediate relationship between the additional (to vacuum) delay time of the signal and its attenuation during propagation in an absorbing medium. It is shown that with an additional delay time of more than several tens of coherence times of uniformly broadened spectral lines of the medium, the signal weakens by hundreds of decibels and therefore practically disappears. On the other hand, when the delay of the signal is less than the coherence time of the medium, its absorption is small. A similar situation arises in an amplifying (thermodynamically nonequilibrium) environment – of course, with the replacement of signal attenuation with its amplification and the replacement of signal delay with its advance. As an example, the propagation of a narrowband signal, the carrier frequency of which is outside the gain band of the space maser, is considered. It is shown that in this case, due to the long lifetime of the excited states of interstellar gas molecules, the signal propagates at superluminal velocity with practically no change in amplitude, which is very rare in problems of this type.

Journal of Radio Electronics•May 1, 2024•0 citations

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

Nikolai S. Bukhman

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.

Journal of Radio Electronics•Jul 1, 2025•0 citations

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

Nikolai S. Bukhman

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.

Quantum Electronics•Apr 30, 2004•9 citations

On the velocity of propagation of a frequency-modulated wave packet in a dispersive absorbing medium

Nikolai S. Bukhman

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.

Optics and Spectroscopy•Jul 1, 2004•9 citations
It is shown that the delay time of a weak signal propagating in an amplifying medium on the wings of the spectral amplification line may be shorter than the time of propagation of the signal with the velocity of light in vacuum. It is found that in this case, the time dependence of the signal is exactly 'reconstructed' at the point of detection, and the detection of the signal continues even if it is abruptly terminated at the point of transmission. It is also shown that using the complex time of group delay of the signal, it is possible to improve the accuracy of the results in the first order of dispersion theory within this approximation.
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, in the case of the supraluminal group velocity of a wave packet in a dispersion medium, a wave packet with a smooth (analytical) envelope does propagate with a supraluminal velocity. In the case of a negative group velocity, the signal maximum does arrive at the detector earlier than at the transmitter. These facts are consistent with both the finiteness of the velocity of light in free space for information transfer (in the case of supraluminal propagation velocity) and the principle of causality (in the case of negative delay time). Basically, the effect of negative delay time may be employed for predicting an observable effect.
The 1D problem of propagation of a narrowband signal in a homogeneous dispersing medium is considered. It is shown that there is a direct and immediate relationship between the additional (to vacuum) delay time of the signal and its attenuation during propagation in an absorbing medium. It is shown that with an additional delay time of more than several tens of coherence times of uniformly broadened spectral lines of the medium, the signal weakens by hundreds of decibels and therefore practically disappears. On the other hand, when the delay of the signal is less than the coherence time of the medium, its absorption is small. A similar situation arises in an amplifying (thermodynamically nonequilibrium) environment – of course, with the replacement of signal attenuation with its amplification and the replacement of signal delay with its advance. As an example, the propagation of a narrowband signal, the carrier frequency of which is outside the gain band of the space maser, is considered. It is shown that in this case, due to the long lifetime of the excited states of interstellar gas molecules, the signal propagates at superluminal velocity with practically no change in amplitude, which is very rare in problems of this type.
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.
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 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.