Impact of time-delayed feedback on spatiotemporal dynamics in the Lugiato-Lefever model

We analyze the impact of delayed optical feedback (OF) on the spatiotemporal dynamics of the Lugiato-Lefever model. First, we carry out linear stability analysis and reveal the role of the OF strength and phase on the shape of the bistable curve as well as on Turing, Andronov-Hopf, and traveling-wave instability regions. Further, we demonstrate how the OF impacts the spatial dynamics by shifting the regions with different spatial eigenvalue spectra. In addition, we reveal a clustering behavior of cavity solitons as a function of the OF strength at fixed OF phase. Depending on the feedback parameters, OF can also induce a drift bifurcation of a stationary cavity soliton, as well as an Andronov-Hopf bifurcation of a drifting soliton. We present an analytical expression for the threshold of the drift bifurcation and show that above a certain value of the OF strength the system enters a region of spatiotemporal chaos.

Figure 1

Steady-state curves of the homogeneous solution of the Lugiato-Lefever model [Eq. (1)] for $\theta =2$ and different phases of the optical feedback: $\varphi =0$ (dashed blue line) and $\varphi =\pi$ (dotted red line). Top (bottom) panel corresponds to the feedback strength $\eta =0.1$ ($\eta =0.5$). Black solid lines in two panels are obtained for the case without optical feedback. Saddle-node bifurcations [Eq. (4)] are denoted by $\square$ and coincide with the onset of the Turing instability $I_S^{T(k^2=0)}=I_S^{S{N}_{h1,2}}$ for the limit $k^2\to 0$ [cf. Eq. (8)].

Figure 2

Wave-vector square $k^2$ at the onset of modulation instabilities (real eigenvalue $\lambda =0$) as a function of the steady-state intensity $I_S$ for different phase of the optical feedback: $\varphi =0$ (blue, medium-dot curve) and $\varphi =\pi$ (red, large-dot curve). Top panel is for feedback strength of $\eta =0.1$ and the bottom panel is for $\eta =0.5$ (see Fig. 1). Black, small-dot curves in the two panels are for the case without optical feedback.

Figure 3

Maps of the saddle-node bifurcations in the $\theta -E_i$ plane. Black, small-dot (red, open-circle) curves are for $\eta =0.1$ ($\eta =0.5$) and solid (dashed) curves are for feedback phase $\varphi =0$ ($\varphi =\pi$).

Figure 4

Time-delayed feedback-induced Andronov-Hopf bifurcation in the Lugiato-Lefever model (1) for $\eta =0.5, \tau =10$, and $\varphi =\pi /3$. (a) Homogeneous steady-state solution $I_S\left(E_i\right)$; (b) real and (c) imaginary parts of the eigenvalues calculated from Eq. (6) for $k^2=0$. Saddle-node and Andronov-Hopf bifurcations are denoted by $\square$ and $\diamond$, respectively.

Figure 5

Map of the fold (dashed blue lines) and Andronov-Hopf (solid red lines) bifurcations in the Lugiato-Lefever model in the plane of ($E_i, \eta$) for fixed OF phase of $\varphi =\pi$ and $\tau =10$.

Figure 6

Time-delayed feedback-induced traveling-wave (TW) instability in the Lugiato-Lefever model for $\eta =0.5$ and $\varphi =\pi /3$. (a) Steady-state homogeneous solution $E_i\left(I_S\right), I_S={\left|E_S\right|}^2$; (b) imaginary parts of the eigenvalue $\mathrm{\Omega }$ calculated from Eq. (6) for $k^2>0$; and (c) corresponding wave-vector square, $k^2$.

Figure 7

Regions of different spatial eigenvalue spectra denoted by pictograms: dots denote single and crosses double (degenerate) eigenvalues with $x$-axis ($y$-axis) real (imaginary) value. Transition (bifurcation) lines between different regions are denoted by $S{N}_{h1,2}$ for saddle-node bifurcation and MI for modulation instability. QZ denotes quadruple-zero eigenvalue. Feedback strength and phase are (a) $\eta =0$, (b) $\eta =0.5, \varphi =0$, and (c) $\eta =0.5, \varphi =\pi$.

Figure 8

Same map as in Fig. 7 presented in the plane $\theta -E_i$.

Figure 9

Three cavity soliton branches of single CS (blue), and double (green) and triple (red) CS clusters as a function of the strength $\eta$ of the time-delayed feedback for $E_i=1.3, \tau =10$, and $\varphi =\pi /3$. Cavity soliton branches are denoted by the corresponding cavity soliton profiles. The bottom black (grey) solid line is the homogeneous stable (unstable) steady sate.

Figure 10

CS drift for $\theta =2,E_i=1.3$ and feedback with different strengths $\eta$ and fixed delay time of $\tau =10$ and phase of $\varphi =\pi$. The CS is launched at $t=0$ and starts moving after a certain delay, which decreases as the feedback strength is increased [22]. Notice also the CS moves faster as $\eta$ is increased.

Figure 11

Threshold values of the product of the feedback strength and delay time, ${\eta }_0\tau$, at the onset of CS drift instability as a function of feedback phase $\varphi$. The LL parameters are $\theta =2$ and $E_i=1.3$.

Figure 12

CS evolution from a stable CS through moving and moving-oscillating CS until developed spatiotemporal chaos. $\theta =2, E_i=1.3$, and the feedback strength $\eta$ increases each 200 time units from 0 to $0.5$ with step of $0.02$. Feedback delay is $\tau =10$ and phase is $\varphi =\pi$.

Figure 13

Space-time dynamics in the LL model for fixed optical feedback parameters in the four representative regions denoted in Fig. 12 by A, B, C, and D. The OF strength is (a) $\eta =0.08$, (b) $\eta =0.2$, (c) $\eta =0.34$, and (d) $\eta =0.4$.