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Physical Review A

Nonanalytic pulse discontinuities as carriers of information

Ahmed H. Dorrah, Mo Mojahedi

Published Jan 14, 2016• 3 citations
Publisher

Abstract

In this paper, we consider optical pulses encoded with two nonanalytic points, and we evaluate the detectable information of these pulses in media supporting slow- and fast-light propagation. It is shown that, in some configurations of slow- (subluminal) light propagation, the signal is not readily detectable, albeit the arrival of the encoded nonanalytic points at the receiver. It is thus argued that, from a practical point of view, information should not be entirely associated with the pulse discontinuities. On the other hand, it is confirmed that for propagation in vacuum or a fast-light medium, detectable information is bounded by the nonanalytic points, which create a space-time window within which detectable information cannot escape. As such, the distinction between the nonanalytic points of a signal, its energy transport, and detectable information is demonstrated.

DOI
10.1103/physreva.93.013823

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In this paper, we consider optical pulses encoded with two nonanalytic points, and we evaluate the detectable information of these pulses in media supporting slow- and fast-light propagation. It is shown that, in some configurations of slow- (subluminal) light propagation, the signal is not readily detectable, albeit the arrival of the encoded nonanalytic points at the receiver. It is thus argued that, from a practical point of view, information should not be entirely associated with the pulse discontinuities. On the other hand, it is confirmed that for propagation in vacuum or a fast-light medium, detectable information is bounded by the nonanalytic points, which create a space-time window within which detectable information cannot escape. As such, the distinction between the nonanalytic points of a signal, its energy transport, and detectable information is demonstrated.
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.
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.
We examined the propagation of nonanalytical points encoded on temporally Gaussian-shaped optical pulses in fast and slow light in an optical ring resonator at $\ensuremath{\lambda}$ = 1.5 \ensuremath{\mu}m. The temporal peak of the Gaussian pulse was either advanced or delayed, reflecting anomalous or normal dispersions in the ring resonator, relevant to under- or overcoupling conditions, respectively. The nonanalytical points were neither advanced nor delayed but appeared as they entered the ring resonator. The nonanalytical points could be interpreted as information; therefore, the experimental results suggested that information velocity is equal to the light velocity in vacuum or the background medium, independent of the group velocity. The transient behaviors at the leading and trailing edges of the nonanalytical points are discussed in terms of optical precursors.
\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
We study the physical mechanism and the information velocity of fast light based on the time-domain propagator governing light propagation in a dispersive system. We show that the output pulse can be regarded as the superposition of multiple ``copies'' of the input pulse, each traveling at a well-defined signal velocity no greater than the speed of light in vacuum, and the interference between the front edges of these copies generates the peak of the output pulse. The information velocity is defined as the ratio of the propagation distance to the mini-mum time required to complete any given information transfer process, directly reflecting how fast the signal of interest, i.e., the part of the pulse that carries the information of interest, propagates. With the peak of the output pulse representing the information, as is typical in slow- and fast-light applications, we find that the information velocity is significantly smaller than the speed of a discontinuity in the pulse, which was traditionally proposed to be the signal velocity of fast light. The analysis can also be applied to slow-light propagation.