Synchronization of limit-cycle oscillations in anomalously dispersive nonlinear fiber ring resonators

The synchronization of limit-cycle oscillations in anomalously dispersive nonlinear fiber ring resonators via unidirectional and mutual coupling is shown numerically. The limit-cycle oscillations can be synchronized in their phase as well as in their frequency within a respective synchronization interval. The synchronization of limit cycles establishes a further approach to tailor and modify frequency combs by optical means.

Figure 1

Schematic diagram of two passive nonlinear fiber ring resonators (FRR) coupled via mutual seeding (dashed lines). For details, see text.

Figure 2

Pulse-shape evolution for 14 iterations after a transition time of 100 iterations. The dashed line indicates the mean peak position at $T_m=18$ fs.

Figure 3

Phase space for the temporal electric field at $T_m=18$ fs. The light gray dots show the system states for 200 iterations, including a transition time of 100 iterations, where the system states spirally approach the orbit, indicated by the five branches. The blue (dark gray) dots describe the position of the system state $S_x$ on the cycle for 1.5 round trips on the cycle ($n=201$).

Figure 4

Perturbation of a limit-cycle oscillation after 100 iterations: The evolution of the peak power without (solid line) and with (dashed line) perturbation is shown. The perturbation was introduced by applying only $90\%$ of the pulse peak power for the 100th iteration.

Figure 5

Perturbation of a limit-cycle oscillation after 100 iterations: The phase space for 8 iterations after 200 iterations without [blue (dark gray) dots] and with (red crosses) perturbation (only $90\%$ of the pulse peak power for one single iteration) is shown. The dotted arrow depicts for a fixed iteration number of $n=201$ the change of the system state on the limit cycle from the case without perturbation to the case with perturbation.

Figure 6

(a) The system states in phase space after 200 iterations for five successive iterations are shown for different perturbation strengths (blue dots without and red crosses with perturbation). The gray shaded area illustrates the attractor of the limit cycle. (b) The limit-cycle oscillation phase as a function of the perturbation strength.

Figure 7

Phase synchronization of two limit-cycle oscillations via mutual coupling. (a) Peak-power evolution for FRR A (solid line) and FRR B (dashed line) for an interval from 100 to 300 iterations. The coupling is introduced after 200 iterations by seeding the output of each resonator synchronously with the pump pulse into the respective other resonator. (b) Pulse-shape evolution for the magnified interval from iteration 120 to iteration 145 (before coupling was introduced). (c) Pulse-shape evolution for the magnified interval from iteration 255 to iteration 280 (after coupling was introduced). (b) and (c) are color coded according to the color bar shown in Fig. 2.

Figure 8

Effect of coupling strength on transition time. The diagram shows the phase difference between both FRRs as a function of the coupling strength and the iteration number. The color bar is proportional to the phase difference between both FRRs. Mark A shows that for a coupling strength of $q_A=q_B=0.0175$ a transition time of 26 iterations is needed to obtain synchronization.

Figure 9

Limit-cycle phases of FRR A and FRR B before (solid and dashed lines, respectively) and after coupling (dots and crosses, respectively) for initially different phase differences (implemented by different perturbation strengths) in the case of (a) mutual coupling, (b) FRR A seeds FRR B, and (c) FRR B seeds FRR A.

Figure 10

Limit-cycle frequency as a function of the delay variation. The limit-cycle frequency can be changed by $0.1484\times f_{\mathrm{rep}}$; $f_A$ and $f_B$ indicate the two different frequencies used for the simulations.

Figure 11

Frequency synchronization of two limit-cycle oscillations via mutual coupling. (a) Peak-power evolution for FRR A (solid line) and FRR B (dashed line) for an interval from 50 to 350 iterations, where the coupling was introduced after 200 iterations. (b) Pulse-shape evolution for the magnified iteration interval from iteration 135 to iteration 160 (i.e., before coupling was introduced). (c) Pulse-shape evolution for the magnified iteration interval from iteration 240 to iteration 265 (i.e., after coupling was introduced). (b) and (c) are color-coded according to the color bar shown in Fig. 2. The radio frequency spectrum for (d) the non-coupled and (e) coupled cases are also shown.

Figure 12

Limit-cycle frequency for FRR A (solid line) and FRR B (dashed line) as a function of the delay variation for (a) the mutually coupled case, (b) FRR B seeds FRR A, and (c) FRR A seeds FRR B. The gray shading indicates the synchronization intervals.