Localized structures in dispersive and doubly resonant optical parametric oscillators

We study temporally localized structures in doubly resonant degenerate optical parametric oscillators in the absence of temporal walk-off. We focus on states formed through the locking of domain walls between the zero and a nonzero continuous-wave solution. We show that these states undergo collapsed snaking and we characterize their dynamics in the parameter space.

Figure 1

Schematic example of a doubly resonant DOPO. A ring resonator with a ${\chi }^{\left(2\right)}$ nonlinearity is driven by a CW field $B_{\mathrm{in}}$ at frequency $2f_0$. The quadratic interaction gives rise to a field $A$ centered around frequency $f_0$ that resonates together with a field $B$ centered around frequency $2f_0$.

Figure 2

In (a) the phase diagram in the $({\mathrm{\Delta }}_1,\rho )$-parameter space showing the principal bifurcation lines of the CW solutions: the pitchfork bifurcation ${\rho }_a$ (black line) and the fold or turning line ${\rho }_t$ corresponding to ${\mathrm{SN}}_t$ (green line). (b) The CW solutions for ${\mathrm{\Delta }}_1=-0.5$ and (c) those for ${\mathrm{\Delta }}_1=-2$. The linear stability with respect to homogeneous perturbations is shown using solid (dashed) lines for stable (unstable) states. The different regions are labeled by I–III and their description is given in the main text.

Figure 3

Panels (a) and (b) show the marginal instability curve and the bifurcation diagram associated with the CW solution for $({\mathrm{\Delta }}_1,{\eta }_2)=(-2,-0.8)$. Gray area in (a) shows the range of $I_s$ where the CW is unstable and correspond to the dashed lines plotted in (b). The CW solution is stable outside this region as shown with solid lines in (a). Panels (c) and (d) show the same type of diagrams but for $({\mathrm{\Delta }}_1,{\eta }_2)=(-2,-0.05)$. The MI occurs at the maximum of this curve and is signaled with a blue dot in (d). The vertical gray dashed lines correspond to the Maxwell point ${\rho }_M$ of the system for such values of the parameters.

Figure 4

[(a) and (b)] The real component of a DW of type I $\left({\mathsf{A}}_0\to {\mathsf{A}}^{+}\right)$ (a) and a DW of type II $\left(-{\mathsf{A}}^{+}\to {\mathsf{A}}^{+}\right)$ (b). (c) Sketch of the oscillatory interaction defined by Eq. (29) at the Maxwell point $\left(\nu =0\right)$ and two locations away from the Maxwell point (i.e., $\nu ={\nu }_1$ and ${\nu }_2)$. The stable (unstable) separations $D_s$ are labeled using $•(\circ )$; [(d) and (e)] examples of Type I (d) and Type II (e) LSs.

Figure 5

Weakly nonlinear solution around the pitchfork bifurcation ${\rho }_a$. Panels (a) and (b) show in blue the real and imaginary profiles of the weakly nonlinear state given by (44) for ${\mathrm{\Delta }}_1=-2$ and $\rho -{\rho }_a=0.01$. Red dashed lines represent the numerical solutions of Eq. (12) at the same point. Both lines are indistinguishable.

Figure 6

Bifurcation diagrams for LSs of type I at ${\mathrm{\Delta }}_1=-2$ and different values of ${\eta }_2$. Collapsed snaking [(a) and (d)] for ${\eta }_2=-0.8$, [(b) and (e)] for ${\eta }_2=-0.2$, and [(d) and (f)] corresponding to ${\eta }_2=-0.05$. Panels (d), (e), and (f) are close-up views of the bottom parts of the bifurcation diagrams shown in (a), (b), and (c). Solid (dashed) lines correspond to stable (unstable) solutions. The vertical gray point-dashed line denotes ${\rho }_M$, and red and orange vertical lines in panel (f) refer to Fig. 4. The different SNs of the LSs are labeled through ${\mathrm{SN}}_i^{l,r}$, and the red dots correspond to the LSs shown in the subpanels (i)–(xviii), where blue and green solid lines represent $U$ and $V$, respectively.

Figure 7

Phase diagram in the $({\eta }_2,\rho )$-parameter space for ${\mathrm{\Delta }}_1=-2$. The gray area limited by ${\mathrm{SN}}_1^r$ and ${\mathrm{SN}}_1^l$ corresponds to the parameter region where LSs of type I exist. The red and purple lines correspond to the Maxwell point ${\rho }_M$ and the MI ${\rho }_c$, respectively. The horizontal bifurcation lines in black and green are the pitchfork bifurcation ${\rho }_a$ and the saddle-node bifurcation ${\rho }_t$ of the CW solution. The inset shows a close-up view about the cusp bifurcation (C) where ${\mathrm{SN}}_1^r$ and ${\mathrm{SN}}_1^l$ collide and disappear. The pointed, dashed-pointed, and dashed vertical lines correspond to the bifurcation diagrams shown in Fig. 6 for ${\eta }_2=-0.8,-0.2$, and $-0.05$.

Figure 8

Bifurcation diagram for type-II LSs at $({\mathrm{\Delta }}_1,{\eta }_2)=(-2,-0.05)$. In panel (a) the collapsed snaking in green correspond to the type-I LSs [see profiles (i)–(v)] that has been added for comparison. The diagram in red correspond to the mixed structures shown in panels (vi)–(xiv) that eventually become a type-II LS as the one shown in panel (xv). The inset shows a close-up view of the bottom part of the bifurcation diagram including the stability of the branches, which alternates from unstable to stable between consecutive folds. In panel (b) we have removed the type-I bifurcation diagram and added the purple diagram corresponding to transition shown in panels (xvi)–(xx).

Figure 9

Phase diagram in the $({\mathrm{\Delta }}_1,\rho )$-parameter space for ${\eta }_2=-0.4$. The gray area between ${\mathrm{SN}}_1^l$ and ${\mathrm{SN}}_1^r$ corresponds to the region where LSs of type I exist. The pitchfork ${\rho }_a$ and saddle-node ${\rho }_t$ bifurcations of the CW solutions are plotted in black and green solid lines, respectively. The MI is the purple line labeled ${\rho }_c$, and the Maxwell point of the system ${\rho }_M$ is the red solid line. The inset shows a close-up of the phase diagram around the cusp bifurcation C. The labels (i) to (ii) and (iii) to (iv) correspond to the LSs shown in the panels below for ${\mathrm{\Delta }}_1=-2$ and ${\mathrm{\Delta }}_1=-6$, respectively.

Figure 10

Real and imaginary parts of of $A$ and $B$ for two different types of LSs of type I. Panels (i)–(iii) show the signal field $A$, and panels (ii)–(iv) the correspondent pump field $B$. Here $({\mathrm{\Delta }}_1,{\eta }_2)=(-6,-0.4)$.