Multiple hydrodynamical shocks induced by the Raman effect in photonic crystal fibers

We theoretically predict the occurrence of multiple hydrodynamical-like shock phenomena in the propagation of ultrashort intense pulses in a suitably engineered photonic crystal fiber. The shocks are due to the Raman effect, which acts as a nonlocal term favoring their generation in the focusing regime. It is shown that the problem is mapped to shock formation in the presence of a slope and a gravity-like potential. The signature of multiple shocks in cross-correlation frequency-resolved optical gating (XFROG) signals is unveiled.

Figure 1

(a) Positions of a particle falling inside potential $U(\xi )$ as described by Eq. (10), for different values of the initial position $\xi =s$ and for ${\tau }_R=0.1$. Multiple collisions between oscillating bound particles produce multiple values for the velocity field $v$ (i.e., shocks). Nonbound particles can also collide with bound particles,although more rarely, contributing to isolated shocks. (b) Shape of the effective potential for different values of ${\tau }_R$ with $\rho =\text{sech}(\xi ){}^2$. (c) Solution of the hydrodynamical system (11) for different values of $z$, showing multiple-shock occurrences. The velocity is plotted for clarity with an arbitrary vertical shift for the various $z$.

Figure 2

Pulse propagation and shock formation according to Eq. (13), using the GVD of Fig. 3. (a) Velocity field $v=\partial {}_t\varphi$ as a function of time for three different propagation distances, namely $z=0.4$ (red solid line, before the occurrence of shock), $z=0.62$ (black dotted line, at the moment of shock), and $z=1.5$ (blue dashed line, long distance dynamics), for ${\tau }_R=0.1$ and $\epsilon =0.2$. For long propagation dynamics the formation of clear undular bores is observed. (b) Amplitude $\left|\psi \right|$ at the same values of $z$ as in (a). At the moment of the first shock a sharp cusp is formed.

Figure 3

GVD curve of the PCF used in our simulations. The black dots represent the data measured experimentally through white-light interferometry while the solid curve is a polynomial fit. The zero dispersion wavelength (ZDW) is close to ${\lambda }_Z=690$ nm. The inset shows an SEM of the fiber cross section.

Figure 4

XFROG spectrograms for the pulses at $z$ equal to (a) $1.761$, (b) $2.192$, (c) $2.723$, and (d) $3.393$ cm. The peak power of the $t_0=130$ fs input pulse was $P=5$ kW ($N=3.1$), and they were launched at $\lambda =1$ $\mu$m. Figures (a) and (b) are snapshots of the first shock. Figures (c) and (d) show the second shock, coinciding with the second spectral oscillation.

Figure 5

(a)–(d) Amplitude (black dotted line) and velocity field (red solid line) in the same conditions as in Figs. 4a, 4b, 4c, 4d. (a) is the moment of the first shock, while (b)–(d) show the generation of undular bores from the strong velocity field oscillations of the singularities due to the quantum pressure potential (9).

Figure 6

Spectral evolution of $P=5$ kW pulses for Raman fractions (a) $f_R=0.18$ (${\tau }_R=0.0334$) and (b) $f_R=0.3$ (${\tau }_R=0.0557$), with $t_0=130$ fs and a pulse launched at $\lambda =800$ nm in the PCF of Fig. 3. For increasing values of ${\tau }_R$, the spectral oscillation rate rises, which can be interpreted by the hydrodynamical formulation of Eqs. (6)–(7). Horizontal dashed lines indicate the distance at which one has five spectral oscillations to help visualization.