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Journal of Communications Technology and Electronics

On the Principle of Causality and Superluminal Signal Propagation Velocities

Nikolai S. Bukhman

Published Mar 1, 2021• 8 citations
Publisher

Abstract

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.

DOI
10.1134/s1064226921030049

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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.
Summary form only given. It is well known that the group velocity v/sub g/ of a light pulse can exceed c in an anomalously dispersive medium. As discussed many years ago by Sommerfeld and Brillouin, a group velocity greater than c does not violate causality because it is not the velocity of information transmission. They noted that the "frontal velocity," the velocity at which an infinitely sharp step-function-like disturbance of the light intensity propagates, can serve as a velocity of information transfer. We suggest here an operational definition of the signal velocity and apply it to the recently observed superluminal propagation of a light pulse in a gain medium.
An explanation for superluminal phenomena based on wave-particle duality of photons is suggested. A single photon may be regarded as a wave packet, whose spatial extension is its coherence volume. As a photon propagates as a wave train in vacuum, its velocity is just the speed of light. When it tunnels through a barrier as a particle, its wave function collapses and it will travel faster than light. Superluminal motion can occur only within the coherence length and the time constrained by uncertainty principle. A massive particle cannot be superluminal during the tunneling process. So superluminality does not violate causality. As for the superluminal and negative group velocities in anomalously dispersive medium, they are merely reshaping effect of the pulse, and they will become subluminal at large distances. A couple of experiments are proposed to test the superluminal phenomena.
\n Using pulses based on functions with compact support and perfectly smooth infinitely differentiable zero-intensity nonanalytical points, a new approach to model information propagating through an anomalous dispersive medium is demonstrated. Information can be unambiguously associated to the nonanalytical point in the pulse's leading edge, and its velocity investigated numerically and experimentally in fast-light media. We have clearly observed that the velocity of this nonanalytical point is bounded by Einstein causality and it is exactly equal to the light speed in vacuum c when the pulse's group velocity is superluminal. Moreover, we observed that causality can also cause steepening and a pulse shock-like effect.\n
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.