Time-frequency dynamics of superluminal pulse transition to the subluminal regime

Spectral reshaping and nonuniform phase delay associated with an electromagnetic pulse propagating in a temporally dispersive medium may lead to interesting observations in which the group velocity becomes superluminal or even negative. In such cases, the finite bandwidth of the superluminal region implies the inevitable existence of a cutoff distance beyond which a superluminal pulse becomes subluminal. In this paper, we derive a closed-form analytic expression to estimate this cutoff distance in abnormal dispersive media with gain. Moreover, the method of steepest descent is used to track the time-frequency dynamics associated with the evolution of the center of mass of a superluminal pulse to the subluminal regime. This evolution takes place at longer propagation depths as a result of the subluminal components affecting the behavior of the pulse. Finally, the analysis presents the fundamental limitations of superluminal propagation in light of factors such as the medium depth, pulse width, and the medium dispersion strength.

Figure 1

Index of refraction as a function of detuning, the left axis represents the real part and the right axis represents the imaginary part of $n\left(\omega \right)$.

Figure 2

Spectral width and medium gain. (a) The medium gain $g\left(\omega \right)$ for $L=25$ (cm), 44.5 (cm), 54 (cm), and 63 (cm) and a Gaussian pulse width $2T=0.9$ (ns). (b) Medium gain $g\left(\omega \right)$ for $L=25$ (cm) and pulse widths $2T$ = 0.9 (ns), 0.7 (ns), 0.5 (ns), and 0.3 (ns), respectively. (c) The medium gain $g\left(\omega \right)$ at a fixed length $L=25$ (cm) and pulse width $2T=0.9$ (ns) for oscillator strengths ${\omega }_p$ equal to $4\delta ,4.8\delta ,5.6\delta$, and $6.4\delta$.

Figure 3

Comparison between the exact calculations of the cutoff distance versus the approximate closed form expression when: (a) The input spectral width is increased (or as the temporal pulse width is reduced). (b) The medium resonances $\left({\omega }_{0,2}-{\omega }_{0,1}\right)$ are tuned. (c) The plasma frequency (dispersion strength) is increased.

Figure 4

Subplots (a)–(d) correspond to propagation lengths of 25, 44.5, 54, and 63 (cm) inside the ammonia vapor cells, respectively. The arrows show the path of the saddle points in the complex frequency domain. The colored (green) contours show the real part of the phase function [$\mathrm{Re}\left\{\varphi (\omega ,{\theta }^{{}^{\prime }})\right\}]$ for $t=L/c$.

Figure 5

Saddle-point contributions. The dotted lines represent the path of the saddle points $({\omega }_{\mathrm{SP},M}$ and ${\omega }_{\mathrm{SP},L})$. The left column corresponds to the contributions of ${\omega }_{\mathrm{SP},M}$ and the right column corresponds to the contributions of ${\omega }_{\mathrm{SP},L}$. Cases [(a)-(e), (b)-(f), (c)-(g), and (d)-(h)] refer to propagation lengths of 25, 44.5, 54, and 63 (cm) inside the ammonia vapor cells, respectively.

Figure 6

The total output field $A(L,t)$. The black dotted curves correspond to the case of vacuum and $\langle t\rangle$ denotes the pulse arrival time. Cases (a)–(d) refer to propagation lengths of 25, 44.5, 54, and 63 (cm), respectively.