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Physical Review A

Luminal pulse velocity in a superluminal medium

Heisuke Amano, Makoto Tomita

Published Dec 23, 2015• 2 citations
Publisher

Abstract

To investigate the physical meaning of pulse peak in fast and slow light media, we investigated propagation of differently shaped pulses experimentally, controlling the sharpness of the pulse peak. Symmetric behavior with respect to fast and slow light was observed in traditional Gaussian pulses; that is, propagated pulses were advanced or delayed, respectively, whereas the pulse shape remained unchanged. This symmetry broke down when the pulse peak was sharpened; in the fast light medium, the sharp pulse peak propagated with luminal velocity, and the transmitted pulse deformed into a characteristic asymmetric profile. In contrast, in the slow light medium, a time-delayed smooth peak appeared with a bending point at $t=0$. This symmetry breaking with respect to fast and slow light is a universal characteristic of pulse propagation in causal dispersive systems. The sharp pulse peak can be recognized as a bending nonanalytical point and may be capable of transferring information.

DOI
10.1103/physreva.92.063837

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To investigate the physical meaning of pulse peak in fast and slow light media, we investigated propagation of differently shaped pulses experimentally, controlling the sharpness of the pulse peak. Symmetric behavior with respect to fast and slow light was observed in traditional Gaussian pulses; that is, propagated pulses were advanced or delayed, respectively, whereas the pulse shape remained unchanged. This symmetry broke down when the pulse peak was sharpened; in the fast light medium, the sharp pulse peak propagated with luminal velocity, and the transmitted pulse deformed into a characteristic asymmetric profile. In contrast, in the slow light medium, a time-delayed smooth peak appeared with a bending point at t=0t=0t=0. This symmetry breaking with respect to fast and slow light is a universal characteristic of pulse propagation in causal dispersive systems. The sharp pulse peak can be recognized as a bending nonanalytical point and may be capable of transferring information.
We examined the propagation of information encoded as nonanalytical points on temporally Gaussian-shaped optical pulses in fast- and slow-light systems. The bandwidth of the input pulses determined the sharpness of the nonanalytical points. A sharp bending nonanalytical point propagated with luminal velocity in both fast- and slow-light systems, in good agreement with relativistic causality. As the bandwidth was reduced, the bending point became broad and propagated with the relevant group velocities. This transition was, however, qualitatively different in the fast- and slow-light systems. We also examined the predictability of the future pulse shape beyond the practical nonanalytical point on the basis of the expansion. When the bandwidth was reduced below a critical value, the expansion well predicted the future pulse shape.
We examined the propagation of bending nonanalytical points encoded on temporally smooth Gaussian-shaped optical pulses in fast- and slow-light media, using an optical ring resonator. For bending nonanalytical points, the pulse function is continuous, but the first-order derivative is discontinuous, in contrast to traditional nonanalytical points in which the pulse function itself is discontinuous. The nonanalytical points were neither advanced nor delayed, but appeared at the same instance as they entered the ring resonator, in good accordance with the fact that the information velocity was equal to the light velocity in vacuum or the background medium. The differences between the traditional discontinuous and the present bending nonanalytical points were investigated.
A series of experiments is performed to examine the arrival of a pulse peak, using a Gaussian-shaped temporal wave packet as the input pulse and truncating it at various positions on or before the peak of the packet. When the truncating point is within the negative group delay limit of the fast light medium, a smooth Gaussian peak is observed at the exit port, despite the absence of an input pulse peak. The experimental results explicitly demonstrate that the superluminal propagation of a smooth Gaussian-shaped pulse peak is an analytic continuation over time of the earlier portion of the input pulse envelope. To investigate the physical meaning of the pulse peak further, we also examine the propagation of triangular-shaped pulses, for which the pulse peak can be recognized as a nonanalytical point.
We performed a series of experiments to examine the arrival of phase nonanalytical points in fast- and slow-light media, using a Gaussian-shaped temporal pulse and encoding phase nonanalytical points at various positions within the pulse envelope. For the phase nonanalytical points, the amplitude of the slowly varying pulse envelope, as well as any order of the derivatives, is continuous, but the phase of the carrier wave is discontinuous. The phase nonanalytical points were neither advanced nor delayed, but appeared at the same instance as they entered the fast- and slow-light media, in good accordance with the idea that the information velocity was equal to the velocity of light in a vacuum, ccc, or in the background medium.
\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
Pulse propagation through 1-dimensional stratified media is studied. The general features of pulse propagation in relation to the delay or advancement of the pulse are discussed. It is shown that in a lossless system with spatial symmetry the reflected and the transmitted pulses have the same delay or advancement. The phase times and the explicit output pulse profiles for an input Gaussian pulse are calculated. While recalling recent results for a variety of systems (resonant absorbers or heterogeneous medium in a Fabry-Perot cavity, a slab of negative index medium, a left-handed/right-handed periodic structure, etc.), new results involving coupled waveguides and a high finesse cavity containing an electromagnetically induced transparency (EIT) medium are presented. Calculations clearly show that for pulses with a carrier tuned at the resonance of the stratified media, one generally has subluminal propagation. Alternatively, in stop gaps or away from the resonances one ends up with superluminal transit. In the case of the EIT cavity it is shown that the large dispersion offered by the intracavity medium can lead to substantial superluminality (ng≃−106) with peak transmission as high as 20%. This is in sharp contrast to large signal attenuation for usual superluminal transit. Note also that EIT, to the best of the authors knowledge, has so far been used to achieve only slow light.