Direct Observation of a Pulse Peak Using a Peak-Removed Gaussian Optical Pulse in a Superluminal Medium

A series of experiments is performed to examine the arrival of a pulse peak, using a Gaussian-shaped temporal wave packet as the input pulse and truncating it at various positions on or before the peak of the packet. When the truncating point is within the negative group delay limit of the fast light medium, a smooth Gaussian peak is observed at the exit port, despite the absence of an input pulse peak. The experimental results explicitly demonstrate that the superluminal propagation of a smooth Gaussian-shaped pulse peak is an analytic continuation over time of the earlier portion of the input pulse envelope. To investigate the physical meaning of the pulse peak further, we also examine the propagation of triangular-shaped pulses, for which the pulse peak can be recognized as a nonanalytical point.

Figure 1

Schematic diagram of the experimental setup. $C$ is the fiber coupler. In and Out show the schematic input and output pulses, respectively, where the dashed-black lines correspond to the original Gaussian pulse and the blue solid lines represent the sharply truncated pulses.

Figure 2

(a) Resonance spectrum of the ring resonator with a spectral width of 6.4 MHz. (b) Experimentally observed negative delay of the output (solid red curve) peak when a Gaussian input (solid black curve) is injected into the resonator. Intensity is normalized to the peak of the input Gaussian pulse, whereas the output intensity is magnified ($2\times$) in the graph. The green dashed vertical line corresponds to the peak of the input pulse. The blue-dashed vertical line corresponds to that of output pulse, showing a negative delay of ${\tau }_p\cong {\tau }_g=-160\mathrm{ns}$.

Figure 3

Observed input (left) and output (right) pulses through the resonance shown in Fig. 2. All intensities are normalized to the maximum of the input Gaussian pulse. In the left column, colored-solid curves show the input pulses truncated at different positions: (a) $t_R=0\mathrm{ns}$, (b) $t_R=-40\mathrm{ns}$, (c) $t_R=-60\mathrm{ns}$, (d) $t_R=-80\mathrm{ns}$, (e) $t_R=-150\mathrm{ns}$, and (f) $t_R=-210\mathrm{ns}$, before the peak of the original Gaussian pulse shown by dashed-black curves. The colored-solid curves in the right column show the output pulses that correspond to the truncated input in the left column. For comparison, the dashed-black curve in each graph represents the transmitted pulse when the original Gaussian pulse was injected. The hatched region indicates the ${\tau }_g$ range ($-160\mathrm{ns}$), i.e., the peak advancement of the on-resonance pulse peak, relative to the off-resonance Gaussian pulse peak.

Figure 4

Calculated input (left) and output (right) pulses through the resonator corresponding to the graphs in Figs. 3 and 3. All intensities are normalized with respect to the maximum of the input Gaussian pulse.

Figure 5

Peak and kick positions observed in the transmitted pulse as a function of peak removal time in the input pulses. The open and solid circles represent the experimentally observed peak ${\tau }_p$ and kick positions $t_K$, respectively; the solid and dashed lines represent the corresponding simulations. The colors of the circles indicate different truncating position of input pulses shown in Fig. 3.

Figure 6

Left column: experimental results for triangular-shaped pulse propagation through the same negative velocity medium shown in Fig. 2. Solid-red curve in (a) and solid-blue curve in (b) represent the transmitted output pulses after being injected with the original triangular [dashed-black curves in (a)] and the truncated triangular pulses [dashed-black curves in (b)] at $t_R\le 0\mathrm{ns}$, respectively. Right column: plots (${\mathrm{a}}^{\prime }$) and (${\mathrm{b}}^{\prime }$) are the calculated results that correspond to plots (a) and (b) in the left column. All intensities are normalized with respect to the maximum of the original triangular pulse.