1. Introduction

In recent years, the theoretical and experimental possibility of controlling the propagation dynamics of optical pulses have been widely explored by many researchers [1]. Many different methods have been proposed and several propagation characteristics have been successfully modified; in particular extremely low group velocities (“slow” light) [2–9] and superluminal or even negative group velocities (“fast” light) [10–15] have been reported in literature. More recently the ability to manipulate the dispersion of an optical fiber-based system has been exploited to demonstrate the possibility of a temporal cloak that is able to cancel the detectability of an event occurring in a window of 50 ps. [16].

A similar approach for a periodic signal at telecommunication data rate has been also reported [17]. Many of the experimental schemes proposed to manipulate the propagation properties of optical media involve coherent effects, such as electromagnetically-induced transparency (EIT) [5, 18–20], coherent population oscillations [13, 21], Raman gain in atomic gas [10, 11] or Brillouin scattering in optical fibers [3,12,22,23]. Slow or fast light propagation in the same medium has been obtained in a waveguide overcoupled to microring resonators [24], by coherent population oscillations in chromium ions in an alexandrite crystal at room temperature [13], in erbium-doped fibers [25] and by optically controlled stimulated Brillouin scattering [26].

A question arises from these results: to which extent can the velocity of light be manipulated, and in particular to which extent can an optical pulse be advanced during propagation in a superluminal medium with respect to vacuum propagation in vacuum? Causality and special relativity pose an upper limit to this advance, as was theoretically shown by A. Sommerfeld and L. Brillouin in their pioneering work in the first half of the 20-th century [27], and recently demonstrated in dedicated experiments [28, 29].

In this paper we show that this limit can be circumvented if the pulse, before entering the superluminal medium, is retarded by letting it propagate through a normal dispersive medium. In our experiment we mimic an usual transmission line in matter where a first stage characterized by slow light propagation (with delay respect to the vacuum velocity c) and absorption is followed by a necessary second stage with amplification. We then show how such amplification stage can be tailored to provide also the recover of the delay, thus resulting in the fastest transmission line. The recover is continuously tuned by an optical control field. The obtained results show an experimental proof of principle evidence that, we believe, can open exciting prospects for optical data processing and communication.

In a recent theoretical work we showed that a quite simple scheme of optical control of the group velocity can be adopted, making use of incoherent interactions only [30]. According to this picture, a control upon the propagation dynamics of a probe optical pulse can be achieved by sending it into a medium that has been previously excited by a pump optical pulse, without any temporal overlapping. Following this approach, we experimentally demonstrated incoherent optical control of propagation and compression over probe pulses of a few nanoseconds in duration propagating through a very low pressure atomic sodium vapor in the slow-light regime [31]. We also applied such scheme to the control of the propagation in the fast-light regime of a probe pulse whose time duration (3 ns) is much shorter than in the case previously reported in literature [32].

In the present experiment we utilize our incoherent scheme to manipulate the dispersion of an atomic medium and to demonstrate that the delay experimented by an optical signal propagating in a slow-light medium can be recovered during the propagation in a second stage where an inversion of population for the same scheme of levels has been created. Indeed, in such superluminal medium the optical pulse propagates faster if it has been previously retarded in the normal dispersion medium. The amount of pulse advance induced by the inverted medium can be optically controlled, leading to a partial or a complete recover of the delay; in the latter case a complete cancellation of the propagation history of the optical pulse into the slow-light medium can be achieved. In Fig. 1 the logical scheme of the experiment and diagram of interactions between optical beams and sodium atomic levels are shown. Levels are labeled as |0〉 = 2S_1/2, |1〉 = 2P_3/2 and |2〉 = 2D-doublet of closely spaced resonances.

 

Fig. 1 Logical diagram of the experimental procedure. The Slow (Fast) Light stage is labeled SL (FL), ON means that the stage is active for the probe field. The corresponding interaction schemes are also indicated. In case a) the probe pulse experiments only fast light propagation while in case b) the probe field experiments both slow and fast light interactions. In both cases the probe pulse is showed at the output of each single stage together with a reference pulse (dashed line) propagating at velocity c.

Download Full Size | PDF

2. Experimental set-up and results

The experimental scheme consists of two different stages, slow-light (SL) and fast-light (FL), that are driven by two different control pulses, provided by two independently tunable dye lasers pumped by a frequency-doubled Q-switched Nd:YAG laser at a repetition rate of 10 Hz. Each stage includes a heated cell (30 cm long for SL and 100 cm for FL) containing the sodium vapor, whose saturation pressure can be selected by varying the temperature up to a value of 280 °C. The control pulse for the SL stage, whose frequency is indicated as ${\omega }_C^{\mathit{SL}}$, connects the ground level |0〉 to the intermediate level |1〉, with an intensity high enough to saturate the |0〉 → |1〉 transition, whereas the FL stage is controlled by a pulse of frequency ${\omega }_C^{\mathit{FL}}$ tuned at the two-photons transition |0〉 → |2〉. The probe beam is obtained from a single-mode cw extended-cavity diode laser (ECDL), whose frequency ω_P is slightly detuned from the transition. The continuous emission of the diode laser is modulated by an electro-optic modulator in order to obtain a pair of identical pulses of 3.0 ns duration and selectable delays respect to the pump pulses. The first probe pulse passes through the cells prior to any pump pulse and is used as a time reference. The second probe pulse, sent after excitation of the sodium vapor, finds a passive medium and a normal dispersion zone of refractive index in the SL stage, with absorption and induced delay in propagation, and an active medium and anomalous dispersion zone in the FL stage, with amplification and induced advance. The strength of each stage interaction can be individually modulated by varying the temperature of the sodium cells, in order to tune the total number of available atoms, the pump intensity level, the control-probe pulse delays, and the detuning of the probe pulse with respect to the |1〉 → |2〉 transition frequency. An optical filter prevents pulses at ${\omega }_C^{\mathit{SL}}$ from entering the second cell while, at the second cell output, pulses at frequency ω_P are spatially separated from pulses at ${\omega }_C^{\mathit{FL}}$ by a prism and collected by a 12 GHz-bandwidth photodiode connected to a 8 GHz-bandwidth digital oscilloscope. Inhibition of a pump pulse allows to measure the effects of a single stage on the probe pulse propagation, whereas switching-off both stages leads to a propagation at near velocity c; the effect of a low pressure of sodium is indeed negligible on the probe pulse propagation when the state |1〉 is unpopulated.

In Fig. 2 the temporal shapes of the probe pulse after the propagation through the two stages are shown. The wavelength of the probe is 819.709 nm and the sodium temperature of SL stage is 165 °C, while in the FL stage this parameter reaches 270 °C. The SL stage induces a maximum delay of 519±5 ps with respect to the propagation in vacuum, while the FL stage induces advances up to 500 ps. When both stages are switched on, the latter stage is able to produce a complete recover of the previously induced delay with a pump pulse energy of about 80 mJ; a further increase of the pumping energy also allows an advance with respect to vacuum propagation eventually reaching the limit value of about 500 ps, that is the same limit that can be reached when the SL stage is switched off.

 

Fig. 2 Temporal shapes of the probe pulse after propagation through the SL and the FL stages. The red and the blue curves refer respectively to case (a) and (b) of Fig. 1. The black curve is the reference for propagation at velocity c and the green is the result for the SL stage on and FL stage off. In the SL stage, in case (a) the peak value is reduced to 21% with respect to the reference, whereas in case (b) the peak value reaches 390% with the only FL stage on and 79% when both stages are on.

Download Full Size | PDF

These results suggest that, with only the FL stage on, the trailing edge of the pulse travels very close to the precursor, thus limiting the maximum obtainable advance. When the SL stage is also activated, the acquired delay with respect to the precursor makes the subsequent advance in the second stage much more efficient. The interesting thing is that, after traversing both stages, it is impossible to distinguish if the pulse has been or not delayed in the first stage: a complete cancellation of the propagation history of the pulse through the SL stage is achieved.

In order to perform a more detailed study of the process and to show how the delay recover can be optically controlled we have realized several measurements of the temporal shape of the probe pulse by varying the energy of the control laser pulse for the FL stage.

In Fig. 3 we report the value of the advance (or delay) Δt, defined as the temporal difference between the rise time of the probe pulse and that of the reference pulse, as a function of the control pulse energy of the FL stage (all rise times are measured at half maximum; see also Fig. 2). In the figure are reported the results for a pulse that has been previously delayed (SL stage ON) and those without induced delay (SL stage OFF). Figure 3 clearly shows that the recovered delay can be controlled by the FL pump pulse energy and that the delay induced by the SL stage can be completely canceled out in the FL stage. It is interesting to note that the overall advance can be larger than 1 ns.

 

Fig. 3 Temporal delay (advance) Δt of the probe pulse respect to vacuum propagation as a function of the FL stage pumping energy. Positive values are delays, while negative are advances. Red points are the results for FL stage on and SL stage off, while blue crosses are temporal shifts with both stages active.

Download Full Size | PDF

3. Conclusion

We have experimentally shown the possibility, for a laser pulse traversing an optically prepared medium in the fast-light regime, to recover the delay (respect to propagation at c) previously acquired in the propagation through a normal dispersive medium. Besides a complete recovering of the induced delay (about 500 ps for a pulse 3 ns long), for a sufficient excitation level of the FL stage, the probe pulse can also experiment an advance in propagation time that approaches a limit value (up to 500 ps), which is the same obtained with only the FL stage on. This process can be continuously controlled in order to reach a partial or a complete recover of the delay. Besides an unavoidable temporal delay due to the dispersion properties of a material system, any transmission line is also necessarily affected by losses and then an amplification stage is needed to allow data transmission over long distances. In our experiment, combining amplification to fast-light propagation, we have demonstrated for the first time that the amplification stage can also be able to completely recover the delay with respect to propagation at velocity c. Hence our experiment is a proof of principle demonstration showing this, up to now, uncovered possibility that can lead to an actual propagation at the maximum reachable velocity for a signal.

In conclusion we believe that the recovering of the delay can be an exciting prospect for communication technology in optical transmission lines.