Superluminal $k$-Gap Solitons in Nonlinear Photonic Time Crystals

We propose superluminal solitons residing in the momentum gap ($k$ gap) of nonlinear photonic time crystals. These gap solitons are structured as plane waves in space while being periodically self-reconstructing wave packets in time. The solitons emerge from modes with infinite group velocity causing superluminal evolution, which is the opposite of the stationary nature of the analogous Bragg gap soliton residing at the edge of an energy gap (or a spatial gap) with zero group velocity. We explore the faster-than-light pulsed propagation of these $k$-gap solitons in view of Einstein’s causality by introducing a truncated input seed as a precursor of a signal velocity forerunner, and find that the superluminal propagation of $k$-gap solitons does not break causality.

Figure 1

(a) Band structure of a SPC created by spatial periodic modulation $\epsilon (x)=\epsilon (x+\mathrm{\Lambda })$. (b) Band structure of a PTC created by the temporal periodic modulation $\epsilon (t)=\epsilon (t+T)$. (c) Self-focusing Kerr nonlinearity gives rise to a Bragg ($\omega$-gap) soliton emerging at the band edge where the group velocity ($v_g$) is zero. (d) Self-focusing Kerr nonlinearity gives rise to a $k$-gap soliton emerging at the band edge where the group velocity is infinite.

Figure 2

(a)–(c) Generation of a $k$-gap soliton from an input of (a) narrow, (b) mid-, and (c) wide beams, respectively. (d) The temporal profiles of the $k$-gap solitons at different locations ($x=0$, 50, 100 $[2cT]$) found numerically from the seed beam in (b). This soliton exhibits superluminal behavior with effective average group velocity of $v_g=14.7c$. (e) Evolution of a truncated beam with a central wave vector ($k_0$) at the middle of the $k$ gap. The truncation mimics the forerunner of signal velocity, instead of the center-of-mass light cone. Once formed, the $k$-gap soliton travels faster than light in medium, but still slower than the forerunner of the information in the leading edge. The peak (which is also the “center of mass”) of the $k$-gap soliton always slows down before it hits the forerunner, as indicated in (d) by the slowing-down of the effective group velocity from its value in the section $x=0$ to $x=50$, to a smaller value in the section $x=50$ to $x=100$.