Self-synchronization phenomena in the Lugiato-Lefever equation

The damped driven nonlinear Schrödinger equation (NLSE) has been used to understand a range of physical phenomena in diverse systems. Studying this equation in the context of optical hyperparametric oscillators in anomalous-dispersion dissipative cavities, where NLSE is usually referred to as the Lugiato-Lefever equation, we are led to a reduced nonlinear oscillator model that uncovers the essence of the spontaneous creation of sharply peaked pulses in optical resonators. We identify attracting solutions for this model, which correspond to stable cavity solitons and Turing patterns, and study their degree of stability. The reduced model embodies the fundamental connection between mode synchronization and spatiotemporal pattern formation and represents a class of self-synchronization processes in which coupling between nonlinear oscillators is governed by energy and momentum conservation.

Figure 1

Phase locking after randomizing the phase profile of a single-soliton LLE solution. A dissipative soliton is propagated in time under the LLE. The soliton phase profile is suddenly randomized at $\tau =1$. (a) Integration of the LLE is continued normally after phase randomization. As a result of the interplay between power and phase evolution, the single-soliton state is lost but multiple solitons may appear, pointing at the existence of a phase-locking mechanism. (b) To suppress the influence of comb power evolution on the phase recovery, the power spectrum of a single soliton is enforced in every integration step after phase randomization and the single-soliton peak is seen to be recovered. The top panels illustrate the temporal evolution of the intracavity waveform by color density, while the middle and bottom panels show the frequency comb power and phase profiles, respectively.

Figure 2

Frequency-domain interpretation of the LLE. The LLE [Eq. (1)] defines a grid (dotted lines) with spacing equal to the resonator FSR at the pump frequency $D_1$ and is written in a rotating reference frame [12]. In the frequency domain, moving to the rotating frame translates into removing the spacing between the grid sites by folding the schematic of (a) such that all the dotted lines coincide. Because of modal dispersion, the modal resonances (green) will not all fall at the same position. Before phase locking, the different comb teeth may be at any spectral position around their corresponding resonance. Phase locking is established when all of the comb teeth align with the pump and, additionally, oscillate in synchrony. Here ${\sigma }_{\eta }\propto {\alpha }_{\eta }$ is the dimensional detuning of comb tooth $\eta$ from its nearest resonance.

Figure 3

Numerical solutions of Eq. (6). (a) Two examples of the PDs at the onset of integration (initial conditions). These initial PDs have uniform probability density over $(-\pi ,\pi ]$. (b) Steady-state PDs for the initial conditions shown in (a). It can be seen that the final PDs lie on straight lines irrespective of the initial conditions, but different initial values lead to different slopes for these lines. The upper blue (dark gray) line corresponds to the blue (dark gray) initial values, while the lower red (light gray) line corresponds to the red (light gray) initial values in (a). We verify the linearity of the final PDs through fitting a straight line to them (equations on the plot) and calculating the coefficient of determination ($R^2$). Here $R^2=1$ shows that the PDs do indeed lie on straight lines.

Figure 4

Phase alignment in (a) solitons and (b) Turing rolls in the steady-state solutions of Eqs. (1) and (2). The inset curves in red (top corners) show the spatiotemporal waveforms and those in black (bottom corners) are the frequency spectra. For both solitons and rolls the phases lie on straight lines of arbitrary slope (see Fig. 3). The parameter values are (a) $\alpha =2, d_2=-0.0124$, and $F=1.41$ and (b) $\alpha =0, d_2=-0.0124$, and $F=1.63$. The phase profile has been unwrapped in (b).

Figure 5

Nonzero eigenvalues of the equilibrium (the Jacobian matrix $\mathrm{J}$) vs comb span for Eq. (6) for (a) uniform and (b) mode-number-dependent comb teeth magnitude profile of $a_{\eta }\propto exp(-k_0\left|\eta \right|)$ ($k_0=0.1$). The $N$ distinct eigenvalues for each comb span are plotted with points of the same color except the negative eigenvalue of smallest magnitude, which is shown in black. The closest eigenvalues to zero for different spans form the black curves. Because these eigenvalues are dominant in determining stability, the black curves show that as the comb widens its stability improves for constant power spectrum, as shown in (a). For the realistic magnitude profile, on the other hand, the stability is not expected to improve, as shown in (b). The stability of a steady-state solution of the LLE is therefore affected by its power profile. The effect of amplitude can be taken into account through the coupling coefficients in Eq. (6).