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Physical Review E

Time-frequency dynamics of superluminal pulse transition to the subluminal regime

Ahmed H. Dorrah, Abhinav Ramakrishnan, Mo Mojahedi

Published Mar 24, 2015• 7 citations
Publisher

Abstract

Spectral reshaping and nonuniform phase delay associated with an electromagnetic pulse propagating in a temporally dispersive medium may lead to interesting observations in which the group velocity becomes superluminal or even negative. In such cases, the finite bandwidth of the superluminal region implies the inevitable existence of a cutoff distance beyond which a superluminal pulse becomes subluminal. In this paper, we derive a closed-form analytic expression to estimate this cutoff distance in abnormal dispersive media with gain. Moreover, the method of steepest descent is used to track the time-frequency dynamics associated with the evolution of the center of mass of a superluminal pulse to the subluminal regime. This evolution takes place at longer propagation depths as a result of the subluminal components affecting the behavior of the pulse. Finally, the analysis presents the fundamental limitations of superluminal propagation in light of factors such as the medium depth, pulse width, and the medium dispersion strength.

DOI
10.1103/physreve.91.033206

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Spectral reshaping and nonuniform phase delay associated with an electromagnetic pulse propagating in a temporally dispersive medium may lead to interesting observations in which the group velocity becomes superluminal or even negative. In such cases, the finite bandwidth of the superluminal region implies the inevitable existence of a cutoff distance beyond which a superluminal pulse becomes subluminal. In this paper, we derive a closed-form analytic expression to estimate this cutoff distance in abnormal dispersive media with gain. Moreover, the method of steepest descent is used to track the time-frequency dynamics associated with the evolution of the center of mass of a superluminal pulse to the subluminal regime. This evolution takes place at longer propagation depths as a result of the subluminal components affecting the behavior of the pulse. Finally, the analysis presents the fundamental limitations of superluminal propagation in light of factors such as the medium depth, pulse width, and the medium dispersion strength.
Anomalous dispersion is at the heart of superluminal pulse propagation in dispersive media. Most earlier analyses of these effects have been confined to discussions of pulse shapes or group velocity and group-velocity dispersion, while experiments typically measure the absorbed energy. We investigate the transport of energy in causal dispersive media including superluminal effects, via the time moments of the Poynting vector. We find that superluminal and negative traversal times are invariably accompanied by an initial temporal compression of the pulse. We predict a crossover distance at which superluminal behavior becomes subluminal even when the spectral width of the initial pulse lies well within the anomalous dispersion region. This effect is generic and we offer a theoretical basis for the experimental results of Talukder et al. [Phys. Rev. Lett. 86, 3546 (2001)]. As the pulse propagates, a strong spectral modification due to dissipation (gain) results in the dominance of normally dispersive spectral components that suffer from lower dissipation (higher gain) and is responsible for this crossover.
Energy exchange between an electromagnetic pulse and dispersive media may result in complicated, yet interesting, phenomena in which the group velocity becomes abnormal (i.e., superluminal or negative). For such cases, signal velocity (velocity of detectable information) remains debatable. In this chapter, we present a systematic study that can be applied to pulse propagation in any dispersive medium in order to quantify the detectable information content and calculate its speed, while accounting for pulse reshaping effects and noise generated in the medium and the detector. Accordingly, we present an operational context within which the constraints of superluminal signaling and its potential applications are shown. We provide scenarios in which the signal velocity is evaluated in microwave circuits with negative group delays and extend the method to include optical pulses in inverted media as well. Such analysis explores the fundamental limitations and capabilities of a broad range of superluminal signaling applications.
The propagation of pulses through dispersive media was investigated by solving Maxwell's equations without any approximation. It was demonstrated that the superluminal or subluminal pulse propagation originates essentially from the interference of the different frequency components of the pulse. The coherence of the different frequency components plays an important role in superluminal propagation. When the pulse becomes partially coherent, the propagation changes from superluminal to subluminal. Since superluminality always exists as long as the spectrum of the coherent pulse is within the anomalously dispersive region whatever the shape of the pulse is, such a coherent pulse must be transmitted superluminally.
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.
The velocity of detectable information (signal velocity) in a medium capable of supporting abnormal (superluminal or negative) group velocities is calculated. This is carried out by tracking the time instant at which the signal-to-noise ratio (SNR) at the detector output reaches a predetermined threshold while considering the total classical and quantum noise of the channel in addition to the detector noise. Furthermore, the method of steepest descent is incorporated to systematically study various forms of pulse reshaping associated with superluminal propagation and its effect on SNR. By studying the behavior of SNR as a function of both space and time, the present analysis predicts the existence of a cutoff distance beyond which signal velocity of a superluminal pulse is delayed as compared to a companion pulse traveling the same distance in vacuum. Finally, the interplay between the relative strength of the medium-generated noise and the detector noise and its effect on signal velocity is discussed.
The propagation of pulses through dispersive media was investigated by solving Maxwell's equations without any approximation. We show that the superluminal propagation of pulses through anomalous dispersive media is a result of the interference of different frequency components composed of the pulse. The coherence of the pulse plays an important role for the superluminal propagation. With the decrease of the coherence of the pulse, the propagation changes from superluminal to subluminal. We have shown that the anomalous dispersion (the real part of the susceptibility) not the amplification (the imaginary part of the susceptibility) plays the essential role in the superluminal propagation. Although the superluminality always exists as long as the spectrum of the coherent pulse is within the anomalously dispersive region, both the energy propagation velocity and the frontal velocity never exceed the light speed in the vacuum. The output pulse through the medium is not the original pulse; instead it carries the information of the original pulse and the information of the prepared medium.