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Physical Review A

Physical mechanism and information velocity of fast light: A time-domain analysis

Zhenshan S. Yang

Published Feb 4, 2013• 12 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.87.023801

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It is widely believed that the velocity of information ${\ensuremath{\upsilon}}_{i}$ encoded on an optical pulse is equal to the group velocity ${\ensuremath{\upsilon}}_{g}$, at least when ${\ensuremath{\upsilon}}_{g}$ is less than the speed of light in vacuum $c$. On the other hand, several authors suggest that ${\ensuremath{\upsilon}}_{i}=c$, although the size of the signal traveling at this velocity may be small, thereby making it difficult to measure. Here, we measure ${\ensuremath{\upsilon}}_{i}$ for pulses propagating through a resonant ``slow-light'' medium where ${\ensuremath{\upsilon}}_{g}\ensuremath{\approx}0.006c$. We find ${\ensuremath{\upsilon}}_{i}=1.03{c}_{\ensuremath{-}0.25c}^{+0.49c}$, or that ${\ensuremath{\upsilon}}_{i}\ensuremath{\approx}168{\ensuremath{\upsilon}}_{g}$, clearly demonstrating that the speed of information cannot be generally described by ${\ensuremath{\upsilon}}_{g}$, but is characterized by its own velocity.

Physical Review Letters•Feb 9, 2005•64 citations
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.
The phenomenon of fast-light pulse propagation has seen renewed interest in recent years, but its relationship to thespeed of information transferis still being debated. In this paper, we define thespeed of information transferas the propagation speed of a point of constant signal-to-noise ratio (SNR) on the leading edge of a signal pulse. We use standard telecommunication analysis to include the effects of noise so that the bit-error rate (BER) can be calculated as a function of a variable decision time. We introduce the concept of a time-dependent Q-factor so that pulse arrival times can be compared at equivalent BERs. We show that when receiver noise is included in a gain-assisted fast-light system, the measuredspeed of information transfercan exceed the speed of light for an equivalent pulse travelling through a vacuum.
In a dispersive medium, different monochromatic modes of light have different phase velocities. Under special circumstances, a superposition of these modes results in an interesting effect wherein the group velocity (the velocity at which the peak of the wavepacket propagates) could be greater than c or even negative although the phase velocities of the modes are all less than c . Can this superluminal group velocity be used for information transfer? We show here that the `superluminal' effect is due to a coherent optical wave superposition effect. Whatever the velocity of the `peak', the whole pulse cannot travel with a speed greater than the fastest phase velocity of its component modes. Thus the maximum speed for information transfer, which involves the sending of a finite pulse, cannot be greater than the maximum phase velocity in the medium.
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.
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.)
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.
It is widely believed that the velocity of information ${\ensuremath{\upsilon}}_{i}$ encoded on an optical pulse is equal to the group velocity ${\ensuremath{\upsilon}}_{g}$, at least when ${\ensuremath{\upsilon}}_{g}$ is less than the speed of light in vacuum ccc. On the other hand, several authors suggest that ${\ensuremath{\upsilon}}_{i}=c$, although the size of the signal traveling at this velocity may be small, thereby making it difficult to measure. Here, we measure ${\ensuremath{\upsilon}}_{i}$ for pulses propagating through a resonant ``slow-light'' medium where ${\ensuremath{\upsilon}}_{g}\ensuremath{\approx}0.006c$. We find ${\ensuremath{\upsilon}}_{i}=1.03{c}_{\ensuremath{-}0.25c}^{+0.49c}$, or that ${\ensuremath{\upsilon}}_{i}\ensuremath{\approx}168{\ensuremath{\upsilon}}_{g}$, clearly demonstrating that the speed of information cannot be generally described by ${\ensuremath{\upsilon}}_{g}$, but is characterized by its own velocity.