Development of superluminal pulse propagation in a serial array of high-Q ring resonators

We experimentally examined the development of superluminal pulse propagation through a serial array of high-Q ring resonators that provides a dynamic recurrent loop. As the propagation distance, i.e., the number of ring resonators that the pulses passed through increased, the pulse advancement increased linearly, largely maintaining its Gaussian shape. The sharp edge encoded at the front of the pulse was, however, neither advanced nor delayed, in good accordance with the idea that information propagates at the speed of light. We also carried out a numerical simulation on the superluminal to subluminal transition of the pulse velocity, which appeared after the pulse had propagated a long distance. The time delays, which we calculated using the saddle point method and based on the net delay, were in good agreement with our results, even when predictions based on the traditional group delay failed completely. This demonstrates the superluminal to subluminal transition of the propagation velocity.


1 Pulse propagation under the off-resonance condition
In Fig. 2 in the main text, we experimentally examined superluminal pulse propagation in a serial array of ring resonators under the on-resonance condition. In this supplement, we examine pulse propagation under the off-resonance condition.
Supplementary Fig. 1 shows experimental observations of the smooth Gaussian-shaped pulse transmitted through the serial array of ring resonators under different tuning conditions of the incident laser frequency. The ring resonator and the dynamic recurrent system were the same as those shown in Fig. 2 in the main text (under-coupling conditions, x y  ). When the incident laser frequency was tuned within the anomalous dispersion region at the center of resonance, the advancement accumulated as the number of N increased (lines 1, 2), as discussed in the main text. On the other hand, when the laser frequency was tuned within the normal dispersion region at the wings of the resonance, the delay increased with the number of N (lines 4, 5). The

Front velocity in slow light
Many researchers have accepted that the true information is stored at the non-analytical points. The front edges of the pulses analyzed in the main text are one example of non-analytical points. We also investigated the edge propagation in other cases.
Supplementary Fig. 2 shows our experimental results. The left column shows the transmitted pulse profiles through the serial array of under-coupled (fast light) ring resonators, Supplementary Fig. 2 (a) shows the transmitted pulse profile of a smooth Gaussian-shaped input pulse and (b) Gaussian-shaped pulses on which the front edge was encoded on the leading side of the pulses, which correspond to the results shown in Figs. 2 (e1) and (e2), respectively, in the main text. Supplementary Fig.   2(c) shows the transmitted pulse profile, in which the edge was encoded in the trailing part of the pulse. In this case, the pulse peak was also advanced, but the sharp edge was neither advanced nor delayed.
The right column of Supplementary Fig. 2 shows the transmitted pulse profiles of the over-coupled resonator (slow light).
We used a 80:20 coupler to achieve the over-coupling condition. In this case, the pulse peak was delayed, reflecting normal dispersion in the over coupled ring resonator. Supplementary Fig. 2 (e) shows the transmitted pulse profile of the Gaussian pulse, where the edge was encoded on the leading part of the pulse. A small spike appeared at NA t = −62 ns, which was the position of the edge point. Supplementary Fig. 2 (f) shows the transmitted pulse when the edge was encoded on the trailing part of the pulse profile, The advancement and delay in pulse peaks were accumulated depending on the coupling conditions of the serial array of ring resonators; however, the non-analytical points were neither advanced nor delayed. The non-analytical points can be interpreted as information; therefore, the experimental results agreed well with the idea that information velocity is equal to the velocity of light in a vacuum or the background medium, and is independent of the group velocity.