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Physical Review A

Propagation of phase nonanalytical points in fast- and slow-light media

Yuma Morita, Makoto Tomita

Published Aug 7, 2017• 2 citations
Publisher

Abstract

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, $c$, or in the background medium.

DOI
10.1103/physreva.96.023813

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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 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.
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 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 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.
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.
In early 1999, a news article in the prestigious journal Nature led off with the announcement, "€œAn experiment with atoms at nanokelvin temperatures has produced the remarkable observation of light pulses traveling at velocities of only 17 m/s."€ The review continued with the understatement, "Observation of light pulses propagating at a speed no faster than a swiftly moving bicycle...comes as a surprise."€ These findings (and their review) marked the beginning of the current wave of interest in the field that has come to be called "€œslow light.â€" When we refer to "the speed of light," we typically mean c, the phase velocity of light in a vacuum, or the speed of propagation of the phase fronts of monochromatic light. The phase fronts travel more slowly through a material, propagating at the speed c/n, where n is the index of refraction of the medium. However, this ordinary slowing of the phase velocity is not slow light. "€œSlow light"€ and "€œfast light"€ refer to changes in the group velocity of light in a medium. A pulse of light can be decomposed mathematically into a group of monochromatic waves at slightly different frequencies, as in Fig. 19.1. In a dispersive material, these monochromatic waves travel at different speeds. When one views the propagation of the pulse as a whole, its apparent velocity depends on the extent of the spread of individual monochromatic-wave velocities. Each monochromatic wave travels at its own phase velocity, while the pulse travels at the group velocity. Of course, the group velocity of a pulse of light is not a new concept. The field of slow and fast light has drawn on theory and developments from the work of Sommerfeld and Brillouin from 1907 to 1914, experiments with early laser amplifiers in 1966, and other work done through the end of the 20th century. (For more of the history behind slow and fast light, see Refs. 8 and 9.)