Generation of dissipative solitons in a doped optical fiber modeled by the higher-order dispersive cubic-quintic-septic complex Ginzburg-Landau equation

The generation of dissipative optical solitons is explored in doped fibers with correction effects under the activation of modulational instability (MI). The model, described by the cubic-quintic-septic complex Ginzburg-Landau equation, includes higher-order dispersion and nonlinear gradient terms. The Lange-Newell's criterion for MI of Stokes wave, boundary domains of MI, and integrated gain of MI are obtained via the linear stability analysis. Particular attention is given to the physical effect on the critical frequency detuning, especially in the normal regime, when varying the values of odd dispersion coefficients. Numerical simulations are undertaken and confronted with analytical predictions. Beyond the agreement between the linear stability analysis and trains of soliton generation, the soliton map induced by MI, along with the subsequent physical effects, is debated via bifurcation diagrams. This allows accurate prediction of transitions between various types of localized modes and well-calibrated generation of a wide variety of solitons with different energies. It is argued that knowing the center of mass and the energy of the generated structures can better characterize the long-time evolution of MI and, eventually, its nonlinear manifestations.

Figure 1

Instability gain spectra $[\text{Gain}\left(m^{-1}\right),\mathrm{\Omega }\left(\text{THz}\right)]$ of CQSCGLE for $P_0=1,{\gamma }_r=-9\times {10}^{-3}$: (a) $q_{3s}=1,$ (b) $q_{3r}=5\times {10}^{-1}.$

Figure 2

Influence of some physical effects on the frequency of the disturb wave using the sensitive analysis.

Figure 3

Second-order correction effect of Kerr nonlinearity on continuous wave propagation $\left[\xi \right(m),\tau (ps\left)\right]$. Focusing case: (a) Propagation of CW. (b) Distribution of intensity for $q_{3r}=3\times {10}^{-1}.$ Defocusing case: (c) Propagation of CW. (d) Comparison of energy and center of mass for $q_{3r}=-3\times {10}^{-1}.$

Figure 4

(a) 3D surface plot showing the integrated gain in the plane ($\mathrm{\Omega },d_{3r}$). (b) Integrated gain in the plane (0, $\mathrm{\Omega }$) for the various values of $d_{3r},$ for $P_0=5\times {10}^{-3},d_{3s}=4\times {10}^{-2},d_{4r}=0,d_{4s}=-8\times {10}^{-3}.$ The other parameters are the same as in Fig. 1.

Figure 5

(a) Propagation of the perturbed wave. (b) Effect of dopant through the energy $d_{3r}=-25\times {10}^{-3}$ (solid line), $d_{3r}=-15\times {10}^{-3}$ (dashed line); the other physical parameters are $\left[\xi \right(m),\tau (ps\left)\right]$: $P_0=1,q_{3r}=3\times {10}^{-1},q_{3s}=1,d_{3s}=0.04,d_{4r}=0,d_{4s}=-8\times {10}^{-3},{\gamma }_r=-99\times {10}^{-4}.$

Figure 6

Bifurcation diagram as a function of the dopant effect. The physical parameters are the same as in Fig. 5.

Figure 7

(a) 3D surface plot showing the integrated gain in the plane ($\mathrm{\Omega },d_{4r}$). (b) Integrated gain as a function of the frequency $\mathrm{\Omega }$ for various values of fourth-order dispersion coefficients for $P_0=5\times {10}^{-2},q_{3r}=5\times {10}^{-1},q_{3s}=1,d_{3s}=-1\times {10}^{-2},d_{3r}=-1\times {10}^{-2},d_{4s}=2\times {10}^{-3},{\gamma }_r=-9\times {10}^{-3}.$

Figure 8

(a) Evolution $\left[\xi \right(m),\tau (ps\left)\right]$ of the modulational instability when (a) $d_{4s}=-6\times {10}^{-2},$ and $P_0=1,q_{3r}=3\times {10}^{-1},q_{3s}=1,d_{3r}=d_{3s}=-1\times {10}^{-2},{\gamma }_r=-99\times {10}^{-4}.$ (b) Map soliton induced as a function of higher-order spectral filtering.

Figure 9

Effect of higher-order dopant on the instability spectrum. (a) Integrated gain as a function of $\mathrm{\Omega }.$ (b) Maximum integrated gain as a function of $d_{5r},$ with ${\mathrm{\Omega }}_{\text{max}}\left(\text{THz}\right)=1.6.$ The parameters used are $P_0=5\times {10}^{-1},q_{3r}=5\times {10}^{-1},d_{3r}=d_{3s}=-1\times {10}^{-2},d_{4r}=-5\times {10}^{-2},d_{4s}=-2\times {10}^{-2},d_{5s}=-1\times {10}^{-3},{\gamma }_r=-9\times {10}^{-3}.$

Figure 10

(a) Evolution $\left[\xi \right(m),\tau (ps\left)\right]$ of modulation instability for $d_{5r}=7\times {10}^{-3},$ and $P_0=1,q_{3r}=8\times {10}^{-1},q_{3s}=1.5,d_{3r}=d_{3s}=-1\times {10}^{-1},d_{4r}=-2\times {10}^{-2},d_{4s}=-2\times {10}^{-2},d_{5s}=-1\times {10}^{-3},{\gamma }_r=-99\times {10}^{-4}.$ (b) Map soliton induced as a function of higher-order dopant effect $d_{5r}.$

Figure 11

(a) 3D surface plot showing the integrated gain, in the plane ($\mathrm{\Omega },d_{6r}$); (b) integrated gain as a function of the frequency $\mathrm{\Omega },$ for the various of sixth-order of dispersion coefficient for $P_0=2\times {10}^{-1},q_{3r}=5\times {10}^{-1},q_{3s}=5\times {10}^{-1},d_{2r}=-5\times {10}^{-1},d_{2s}=5\times {10}^{-1},d_{3r}=d_{3s}=-1\times {10}^{-2},d_{4r}=-1\times {10}^{-2},d_{4s}=-18\times {10}^{-3},d_{5r}=-9\times {10}^{-3},d_{5s}=-1\times {10}^{-3},d_{6s}=55\times {10}^{-4},{\gamma }_r=-9\times {10}^{-4}.$

Figure 12

(a) The chart of stability. (b) Transversal distribution of the modulation instability at $\xi =117\times {10}^{1}m$ for modulation stable (solid line) and modulation unstable (dashed line). The physical parameters are $P_0=2\times {10}^{-1},q_{3s}=4\times {10}^{-1},d_{3s}=d_{3r}=-1\times {10}^{-2},d_{4r}=-2\times {10}^{-2},d_{4s}=-18\times {10}^{-3},d_{5r}=9\times {10}^{-3},d_{5s}=-4\times {10}^{-3},d_{6s}=-9\times {10}^{-4},{\gamma }_r=-99\times {10}^{-4},$ and (b) solid line $(d_{6r}=0;q_{3r}=0),$ dashed line $(d_{6r}=4\times {10}^{-1},q_{3r}=1).$

Figure 13

Transversal distribution of the modulation instability at $\xi =12\times {10}^{2}m$ for even dispersion (solid line) and full dispersion (dashed line). The physical parameters are $P_0=1,q_{3s}=1,q_{3r}=4\times {10}^{-3},d_{3s}=d_{3r}=-1\times {10}^{-2},d_{4r}=-2\times {10}^{-2},d_{4s}=-2\times {10}^{-2},d_{5r}=89\times {10}^{-4},d_{5s}=-1\times {10}^{-3},d_{6s}=-9\times {10}^{-4},d_{6r}=-1\times {10}^{-4},{\gamma }_r=-99\times {10}^{-4}.$

Figure 14

Comparison of map solitons as a function of higher-order $d_{6s}$ term through (a) energy and (b) center of mass.

Figure 15

(a) 3D surface plot showing the integrated gain, in the plane ($\mathrm{\Omega },P_0$). (b) Maximum gain as a function of $P_0,$ with ${\mathrm{\Omega }}_{\text{max}}\left(\text{THz}\right)=3.1,$ for different physical situations. The parameters are $q_{3r}=4\times {10}^{-1},q_{3s}=1,d_{3r}=d_{3s}=-1\times {10}^{-2},d_{4r}=-1\times {10}^{-2},d_{4s}=-2\times {10}^{-2},d_{5r}=6\times {10}^{-3},d_{5s}=-1\times {10}^{-3},d_{6s}=-6\times {10}^{-2},d_{6r}=6\times {10}^{-3},{\gamma }_r=-1\times {10}^{-3},m_{1r}=6\times {10}^{-1},m_{2r}=8\times {10}^{-2},m_{3r}=5\times {10}^{-3}.$

Figure 16

Evolution $\left[\xi \right(m),\tau (ps\left)\right]$ of modulational instability in the presence of self-frequency shift effect and their two correction terms. (a) Full dispersion. (b) Even dispersions. The other parameters are: $P_0=1,m_{1r}=6\times {10}^{-1},m_{2r}=8\times {10}^{-2},m_{3r}=5\times {10}^{-3}.$

Figure 17

The maximum gain as a function of power $P_0$ with ${\mathrm{\Omega }}_{\text{max}}\left(\text{THz}\right)=3.1,$ for different physical situations. The physical parameters are $n_{1s}=4\times {10}^{-1},n_{2s}=8\times {10}^{-2},n_{3s}=8\times {10}^{-3}.$

Figure 18

(a) Evolution $\left[\xi \right(m),\tau (ps\left)\right]$ of modulational instabilty in the presence of self-steepening effect and their two correction terms. (a) Full dispersion. (b) Even dispersions. The other parameters are $P_0=1,n_{1s}=2\times {10}^{-1},n_{2s}=1\times {10}^{-2},n_{3s}=1\times {10}^{-3}.$

Figure 19

Map soliton through the dips of the center-of-mass as a function of the self-steepening $n_{1s}.$

Figure 20

Influence of the physical parameters on the critical frequency using sensitive analyses.

Figure 21

(a) 2D surface plot showing the integrated gain, in the plane ($\mathrm{\Omega },P_0.$) (b) 2D plot for various values of $P_0.$ when $d_{4r}=-2\times {10}^{-2}.$ (c) 3D surface plot showing the integrated gain, in the plane ($\mathrm{\Omega },d_{4r}$), when $P_0=1.$ (d) 2D a plot as a function of $\mathrm{\Omega }$ for various values of $d_{4r}.$ Parameters are $q_{3s}=1,q_{3r}=4\times {10}^{-1},d_{3r}=-34\times {10}^{-3},d_{3s}=-1\times {10}^{-2},d_{4s}=-48\times {10}^{-3},d_{5r}=8\times {10}^{-3},d_{5s}=-1\times {10}^{-3},d_{6s}=-3\times {10}^{-3},d_{6r}=6\times {10}^{-3},{\gamma }_r=-99\times {10}^{-4},n_{1s}=2\times {10}^{-1},n_{2s}=5\times {10}^{-3},n_{3s}=2\times {10}^{-3},m_{1r}=6\times {10}^{-1},m_{2r}=8\times {10}^{-2},m_{3r}=1\times {10}^{-3}.$

Figure 22

(a) 2D surface plot showing the integrated gain, in the plane ($\mathrm{\Omega },d_{5r}$) when $d_{6r}=6\times {10}^{-3}.$ (b) 2D plot for the various value of $d_{5r}.$ (c) 2D surface plot showing the integrated gain, in the plane ($\mathrm{\Omega },d_{6r}$), when $d_{5r}=8\times {10}^{-3}.$ (d) 2D plot as a function of $\mathrm{\Omega }$ for various values $d_{6r}$ Parameters are as follows: $P_0=5\times {10}^{-1},d_{4r}=-2\times {10}^{-2},$ and the other parameters are the same as in Fig. 21.

Figure 23

Numerical evolution $\left[\xi \right(m),\tau (ps\left)\right]$ of the modulational instability in the full model: (a) Propagation of the CW when $P_0=1.$ (b) Transverse distributions of the intensity at position $\xi =1222m,$ considering full dispersion (solid line) and even dispersion (dashed line). (c) Propagation when $P_0=1.5.$ (d) Energy with full dispersion (solid line) and even dispersion (dashed line). Parameters are the same as in Fig. 21.