Raman solitons in waveguides with simultaneous quadratic and Kerr nonlinearities

We analyze Raman-induced self-frequency shift in two-component solitons supported by both quadratic and cubic nonlinearities. Treating Raman terms as a perturbation, we derive expressions for soliton velocity and frequency shifts of the fundamental frequency and second-harmonic soliton components. We find these predictions compare well with simulations of soliton propagation. We also show that Raman shift can cause two-component solitons to approach the boundary of their own existence and subsequently trigger soliton instabilities. In some cases these instabilities are accompanied by an almost complete transfer of power to the second harmonic and emergence of a single-component Kerr solitonic pulse.

Figure 1

Real (dashed blue line) and imaginary (solid red line) parts of the Raman response spectra $R\left(\omega \right)$ for LN (top) and silica (bottom).

Figure 2

Frequency shift of soliton components as a function of $\nu$ and $\mu$, calculated by the numerical Newton-Raphson method. ${\delta }_f$ and ${\delta }_s/2$ are plotted by solid red and dashed blue lines, respectively. Panels are representative of solitons in the (a) quadratic limit with anomalous dispersion ($\mu =5, \kappa =0, s_1=0, r_2=1, s_2=1/4$) and (b) Kerr limit with normal dispersion ($\mu =-2, \kappa =300, s_1=0, r_2=-1, s_2=-1/4$). (c) and (d) show data for solitons in the quadratic limit with anomalous dispersion ($\kappa =0, s_1=0, r_2=1, s_2=1/16$). Inverse soliton velocity $\nu =0$ and $\nu =1$ in (c) and (d), respectively. The black dotted line marks the boundary of soliton existence for the given $\nu$ found using Eq. (9) [$\mu =0$ and $\mu =2$ in (c) and (d), respectively]. $t_0=100\mathrm{fs}$ throughout.

Figure 3

Rate of frequency shift ${\dot{\delta }}_f$ predicted by Eq. (16) for two-component solitons and Kerr solitons in LN nanowaveguides, shown as solid red and dotted black curves, respectively (for $\nu =0$). For the two-component solitons we plot the peak power as $|A_f{|}^2+{\left|A_s\right|}^2$. Insets compare soliton power profiles at high and low powers. FF and SH of the two-component solitons are plotted as solid red and dashed blue curves, respectively. The Kerr soliton power profile is shown as a dotted black curve with the same total peak power. We found that ${\dot{\delta }}_s/2$ very closely followed the same trend as ${\dot{\delta }}_f$ for the two-component soliton in this case.

Figure 4

Simulated soliton propagation in time and frequency domains shown in the top and bottom plots in each panel, respectively. The FF and SH are shown on the right and left sides of each panel, respectively. Dashed lines show time and frequency shifts predicted by Eqs. (15) and (16). Insets show the relative input soliton power plotted in the time domain for FF (solid red) and SH (dashed blue). (a) and (b) show solitons in the quadratic limit with anomalous and normal dispersion, respectively. (c) and (d) show solitons in the cascaded Kerr limit with anomalous and normal dispersion, respectively. Parameters used in each panel are as follows: (a) $\mu =5, \nu =0, \kappa =0, s_1=0, r_2=1, s_2=1/4$. (b) $\mu =-5, \nu =0, \kappa =0, s_1=0, r_2=-1, s_2=-1/4$. (c) $\mu =2, \nu =0, \kappa =-1000, s_1=10, r_2=1, s_2=1/4$. (d) $\mu =-2, \nu =0, \kappa =300, s_1=0, r_2=-1, s_2=-1/4$. For all of these simulations we used an input FF wavelength of 1500 nm (${\omega }_f/2\pi =200\mathrm{THz}$), characteristic timescale $t_0=100\mathrm{fs}$, and ${\alpha }_f=0.0007, {\alpha }_s=0.008, {\alpha }_c=0.005$.

Figure 5

Prediction and simulation of soliton instabilities. (a)–(c) show an example under anomalous dispersion with the same parameters as in Fig. 4 except with $s_2=1/16$. (d)–(f) show an example under normal dispersion with parameters as in Fig. 4 but with $s_2=-1/20$. Soliton existence analysis is shown in (a) and (d); shaded regions mark where each soliton existence criterion is met. The two-component soliton trajectory in the $\nu -\mu$ plane predicted by Eq. (14) is included as the dotted red curve. The solid blue curve in (a) shows the predicted trajectory of the Kerr soliton in the SH after the two-component soliton becomes unstable. (b) and (e) plot the fractional energy in each component of the soliton as a function of propagation distance. Data from the simulated soliton propagation are plotted as solid red and dashed blue curves for the FF and SH, respectively. Fractional energies predicted by our analysis for the FF and SH are shown as dotted red and dot-dashed blue lines, respectively, calculated from numerical solitons for the predicted $\nu$ and $\mu$ values. (c) and (f) plot the simulated propagation of the solitons in each example. The top and bottom rows of each panel show the spectral and temporal evolution, respectively; the SH and FF are shown in the left and right columns, respectively. The trajectories of the soliton predicted by Eqs. (16) and (15) are plotted as dashed black curves in the spectral and temporal plots, respectively. Solid black curves in (c) plot predictions for a Kerr soliton in the SH after the two-component soliton becomes unstable. Insets plot relative input soliton power in the time domain for the FF (solid red curve) and SH (dashed blue curve). Circles mark the initial soliton position; crosses mark a particular point of interest in (a)–(c) and (d)–(f).