Stability of out-of-phase solitons and laser pulse self-compression in active multicore fibers

An out-of-phase soliton distribution of the wave field is found for a multicore fiber (MCF) from an even number of cores located in a ring. Its stability is proved with respect both to small wave-field perturbations, including azimuthal ones, and to small deformations of the MCF structure. As an example of using this soliton distribution, the problem of laser pulse compression in an active MCF is studied. The optimal fiber parameters, the minimum duration of the output pulse, and the compression length are found, and they are in good agreement with the results of numerical simulation. In order to achieve high energies in the output laser pulse, the requirements for MCF deformations are determined.

Figure 1

Schematic of the considered MCF with cores arranged in a ring. Color reflects the out-of-phase distribution of the wave field $u_n\propto {(-1)}^n$.

Figure 2

Dependence, (9), of the Hamiltonian $\mathcal{H}$ on the energy $W$. Curves correspond to the extrema, (12), for various $m$ values. Arrows show qualitative explanations of instability at $m=0$.

Figure 3

Dynamics of the wave-field envelope $|u_n|$ in a six-core MCF ($N=3$) with the coupling coefficient $\chi =1$. The initial distribution is $u_n=\sqrt{2}b/cosh\left(b\tau \right)$ with $b=0.7$.

Figure 4

Envelope dynamics of the wave field $\mathrm{Re}\left[u_n(z,\tau )\right]$ in six-core MCFs ($N=3$) in the case of fiber bending $h_n=h+\delta hsin(\pi n/N)$ for different values of the coupling coefficient $\chi$: (a, d, g) $\chi =10$; (b, e, h) $\chi =3$; and (c, f, i) $\chi =1$. The top row corresponds to $\delta h/\chi =0.1$; the middle row, to $\delta h/\chi =0.3$. The spatiotemporal soliton $u_n={(-1)}^n\sqrt{2}b/cosh\left(b\tau \right)$ was injected at the MCF input with $b=1$. (g, h, i) Dependence of the fraction of energy ${\eta }_{\pm }$ in the $\pm$ mode on the value of the parameter $\delta h/\chi$. Here, the dashed black line is the estimate, (26), found for the beam problem. The red line shows the results of numerical modeling. The yellow region represents the fraction of energy in the $\pm$ mode, when $\delta h_n$ is random and lies in the range $0\le \delta h_n\le \delta h$. Blue dots represent the fraction of energy in the $\pm$ mode after averaging over the ensemble.

Figure 5

Phase plane (${\tau }_p,\varpi$) of Eqs. (33) drawn for different values of the coefficients: (a) $\gamma =0.01$, $\mu ={10}^{-3}$; (b) $\gamma =0.1$, $\mu ={10}^{-3}$. The dashed vertical line represents the estimate of the minimum laser pulse duration, (38). Calculations were performed at $\mathrm{\Omega }=10$.

Figure 6

Dynamics of the wave packet $E_n(z,\tau )$ (a) and its spectrum $E_n(z,\omega )$ (b) along the propagation path $z$ (the electric-field strength and its spectrum are normalized to their maximal values), the maximal amplitude (c), and the laser pulse duration (f) depending on $z$. (d, e) Distribution of the total wave field ${|\sum (-1)}^n{E}_n\left(\tau \right)|$ and its spectrum for different $z$ values. The dashed horizontal line in (f) shows the value, (39b). The dashed vertical line in (e) is the boundary of the gain band of the active medium, $\omega =-\mathrm{\Omega }/2=-17\times {10}^{12}{\text{s}}^{-1}$. Dashed vertical lines in (a–c) and (f) are the medium length, $L_{\text{comp}}\approx 23$ m, at which the wave-packet duration reaches the minimal value, (39a), ${\tau }_{\text{lim}}^{\text{FWHM}}\approx 97$ fs. Calculations were performed for ${\tau }_{\text{in}}^{\text{FWHM}}=1$ ps, $\beta =100{\text{ps}}^2/\text{km}$, $1/\gamma =20$ m, $1/\mathrm{\Omega }=30$ fs, ${\chi }^{-1}=3$ mm, and $\mathrm{\Gamma }=1\text{(W}{\text{km)}}^{-1}$.

Figure 7

Same as Fig. 6. Calculations were performed for ${\tau }_{\text{in}}^{\text{FWHM}}=1$ ps, $\beta =100{\text{ps}}^2/\text{km}$, $1/\gamma =2$ m, $1/\mathrm{\Omega }=30$ fs, ${\chi }^{-1}=3$ mm, and $\mathrm{\Gamma }=1\text{(W}{\text{km)}}^{-1}$. The dashed vertical line in (a–c) and (f) is the medium length, $L_{\text{comp}}\approx 2.9$ m, at which the wave-packet duration reaches the minimal value, (39a), ${\tau }_{\text{lim}}^{\text{FWHM}}\approx 55$ fs.