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Physical Review A

Why do superluminal pulses become subluminal once they go far enough?

Lipsa Nanda, Harshawardhan Wanare, S. Anantha Ramakrishna

Published Apr 29, 2009• 13 citations

Abstract

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.

DOI: 10.1103/physreva.79.041806

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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.

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.

We use the time moments of the Poynting vector associated with an electromagnetic pulse to characterize the traversal times and temporal pulse widths as the pulse propagates in highly dispersive media. The behaviour of these quantities with the propagation distance is analysed in three canonical cases: Lorentz absorptive medium, a Raman gain doublet amplifying medium and a medium exhibiting electromagnetically induced transparency. We find that superluminal pulse propagation in the first two cases with anomalous dispersion is usually accompanied by pulse compression and eventually the pulse becomes subluminal with increasing distance of propagation. In a medium with electromagnetically induced transparency with large normal dispersion, we identify a range of frequencies for which the pulse undergoes minimal temporal expansion while propagating with ultra-slow speed.

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.

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.

Anomalous dispersion cannot occur in a transparent passive medium where electromagnetic radiation is being absorbed at all frequencies, as pointed out by Landau and Lifshitz. Here we show, both theoretically and experimentally, that transparent linear anomalous dispersion can occur when a gain doublet is present. Therefore, a superluminal light-pulse propagation can be observed even at a negative 7group velocity through a transparent medium with almost no pulse distortion. Consequently, a negative transit time is experimentally observed resulting in the peak of the incident light pulse to exit the medium even before entering it. This counterintuitive effect is a direct result of the rephasing process owing to the wave nature of light and is not at odds with either causality or Einstein's theory of special relativity.