Multicore-fiber solitons and laser-pulse self-compression at light-bullet excitation in the central core of multicore fibers

The propagation of laser pulses in multicore fibers (MCFs) made of a central core and an even number of cores located in a ring around it is studied. Approximate quasisoliton homogeneous solutions of the wave field in the MCF considered are found. The stability of the in-phase soliton distribution is shown analytically and numerically. At low energies, its wave field is distributed over all MCF cores and has a duration that exceeds the duration of the nonlinear Schrödinger equation (NSE) soliton with the same energy by many (i.e., five to six) times. In contrast, almost all of the radiation at high energies is concentrated in the central core with a duration similar to the NSE soliton. The transition between the two types of distributions is very sharp and occurs at a critical energy, which is weakly dependent on the number of cores and on the coupling coefficient with the central core. The self-compression mechanism of laser pulses was proposed. It consists in injecting such MCFs with a wave packet being similar to the found soliton and having an energy larger than the critical value. It is shown that the compression ratio depends weakly on the energy and the number of cores and is approximately equal to six times with an energy efficiency of almost 100%. The use of longer laser pulses allows one to increase the compression ratio up to 30–40 times with an energy efficiency of more than 50%. The obtained analytical estimates of the compression ratio and its efficiency are in good agreement with the results of a numerical simulation.

Figure 1

Scheme of the MCF under consideration showing the cores arranged in a circle around the central core.

Figure 2

Dynamics of the wave-field envelope $|u_n|$ in a MCF consisting of six cores ($N=3$), in the absence of a central core ($\chi =0$). A laser pulse with initial distribution (8) with $b=0.7$ was injected into the fiber.

Figure 3

(a) Dynamics of the wave-field envelope in a MCF consisting of seven cores ($N=3$) with $\chi =1$. The initial distribution is given by Eq. (12). (b) The intensity distribution of a laser pulse in the central (${\left|a\right|}^2$, red lines), and the first ($|u_1{|}^2$, blue dotted line) cores in different MCF cross sections.

Figure 4

(a)–(d) Phase plane of Eqs. (16a) and (16b) allowing for relation (18) for $N=3, \chi =1$, and different values of energy $W$. Bold lines show separatrices. The dotted line corresponds to homogeneous filling $\left|a\right|=\left|f\right|$. (e) Equilibrium states depending on the energy $W$. Dashed and dash-dotted lines indicate unstable-equilibrium states.

Figure 5

Amplitude distributions of the laser pulse in the central $\left|a\right(\tau \left)\right|$ (solid red line) and in first $|u_1\left(\tau \right)|$ (blue dotted line) cores in different MCF cross sections. A laser pulse (27) was injected into the MCF input, corresponding to found branch II at $\theta =\pi , W=20$. The dispersion length is $L_{\text{dis}}\approx 0.2$.

Figure 6

Dynamics of the wave-field envelope in a seven-core MCF fiber ($N=3, \chi =1$). The initial distributions are (a) Eq. (28)—branch I with energy $W=10$; (b) Eq. (29)—branch I with energy $W=20$. Dispersion lengths are 1.5 and 0.015, respectively.

Figure 7

Compression ratio calculated by expression (35) with ${\eta }_{\text{est}}=1$ and $A=A_{\text{in}}$.

Figure 8

(a), (b) Dynamics of the wave-field envelope in a MCF consisting of seven cores ($N=3, \chi =1$) for the energy $W=15$. Panel (b) shows the intensity distribution of a laser pulse in the central core ${\left|a\right|}^2$ at different $z$. Panel (c) shows the dependencies of the compression ratio $Q$ (thick black line) and the fraction of energy $\eta$ in the central core (thin magenta line) on the $z$ coordinate. The axis for $Q$ is normalized to the maximum value (35) with ${\eta }_{\text{est}}=1$.

Figure 9

(a), (b) Dynamics of the wave-field envelope in 7-core MCF ($N=3, \chi =1$) for the energy $W=30$. The initial distribution is determined by Eq. (37). Captions for the figure are the same as in Fig. 8. The dotted line in panel (c) shows the efficiency estimated by Eq. (33).

Figure 10

Dependencies of the (a) compression ratio and (b) energy efficiency on the initial energy fraction in the central core $A$ and on the total pulse energy $W$ for a MCF consisting of seven cores ($N=3, \chi =1$). Solid curves show branch I [see Fig. 4]. Dashed lines denote $A=A_{\text{in}}$.

Figure 11

The same as Fig. 8 for MCF with bending $\delta h_n=Dsin(\pi n/3)$. Parameters are $D=1$ for panels (a), (c), (e), and $D=3$ for panels (b), (d), (f).

Figure 12

The green surface is the dependence (18) of the soliton duration ${\tau }_p^{\text{sol}}$ on $W$ and $A$; the red curve corresponds to the stationary values of $\{{\tau }_p^{\text{sol}},A_I\}$ on the stable branch $I$ [Fig. 4]; the blue dash is the duration ${\tau }_p^{\text{sol}}$ at $A=A_{\text{in}}$. The remaining curves show the dynamics of the wave-packet parameters within Eqs. (16) with the initial $A=A_{\text{in}}$ for three cases: (i) black line for ${\tau }_p={\tau }_p^{\text{sol}}$ and $W\approx W_{\text{cr}}$; (ii) blue line for ${\tau }_p=5{\tau }_p^{\text{sol}}$ and $W\approx W_{\text{cr}}$; (iii) magenta line for ${\tau }_p=5{\tau }_p^{\text{sol}}$ and $W=5<W_{\text{cr}}$.

Figure 13

(a), (b) Dynamics of the wave-field envelope in a 7-core MCF ($N=3, \chi =1$) for the energy $W=30$ and the duration ${\tau }_p=7.58$. The initial distribution is determined by Eq. (39). Captions for the figure are the same as in Fig. 8. The dotted line in panel (c) shows the efficiency estimated by Eq. (33). The axis for $Q$ is normalized with respect to the maximum value (38) with ${\eta }_{\text{est}}=1$.

Figure 14

Dependencies of (a) the minimum laser pulse duration, (b) compression efficiency $\eta ={\int \left|a\right|}^{2}d\tau /W$, and (c) compression ratio of the wave packet $Q={\tau }_p/{\tau }_p^{\mathrm{out}}$ on the energy $W$ for different initial durations ${\tau }_p$ (or energies $W_0\propto 1/{\tau }_p$). The calculations were performed for a 7-core MCF ($N=3, \chi =1$). The dotted line in panel (a) shows the duration of the NSE soliton with the energies $W=W_{\text{cr}}$ and $W=W_{\text{lim}}\approx 26$. The curves in panel (b) correspond to approximations (33) (solid line) and (34) (dotted line). The vertical dash-dotted line corresponds to boundary (32). The curves in panel (c) shows approximation (38).