Microwave generation on an optical carrier in microresonator chains

We consider self-pulsing regimes in chains of Kerr nonlinear optical microresonators. By means of a supermodal diagonalization procedure of the conventional coupled-mode theory in time, we theoretically and numerically study the bifurcation diagrams of a singly pumped three-cavity and a doubly pumped four-cavity system: the latter allows us to predict thresholdless frequency tripling of a GHz modulation. These self-pulsing regimes are proven robust and will find applications in the generation and conversion of microwaves on an optical carrier.

Figure 1

The two different chains of nonlinear resonators under study. (a) Three-cavity configuration. (b) Four-cavity configuration; notice that the central coupling coefficient allows us to tune the supermode frequency spacing.

Figure 2

Comparison of the evolution toward the steady state of the three-cavity system and its three-mode truncation. $P=40,\delta =-3,\gamma =40$. Solid lines represent the solution of Eq. (2), while dashed lines represent the solution of Eq. (5): ${\eta }_0$ is in blue (dark-gray upper line), ${\eta }_1$ in red (gray at the bottom), $E$ in yellow (light gray at the top), and the imbalance $\alpha$ is in black (at the bottom) [only for Eq. (2)]. Notice that the last one exhibits small oscillations around zero. The other quantities reach the steady state more slowly in Eq. (5) than in the original model, but represent quite an accurate approximation of the system behavior.

Figure 3

Bifurcation diagram with $P=40$. The main panel shows the bistable curve of $E$ as a function of $\delta$ (bifurcation parameter) of fixed points of Eq. (5), with $\eta =1$, in black, and limit cycles, $\eta \ne 0,1$, in red (gray). Solid lines represent stable limit cycles, dashed lines represent unstable ones (saddle points), and dotted lines are delimited by Neimark-Sacker bifurcations. The inset shows the bifurcation of the other three variables: $\eta$ [red (gray), left axis], ${\varphi }_1$ [green (light gray), right axis], and $\psi$ [blue (dark gray), right axis].

Figure 4

Phase-space representations of the time evolution of Eq. (5). (a), (c) The plane $(Ecos{\varphi }_1,Esin{\varphi }_1)$, with the circles denoting steady $E$'s. (b), (d) The plane $(\eta cos\psi ,\eta sin\psi )$, with the unit circles included for reference. (a), (b) Correspond to Fig. 2. Cold-cavity (random noise in each mode) initial conditions (red crosses) are used, where the blue line represents the evolution and the red asterisk represents the attained steady state. (c), (d) The solutions for $\delta =-4.5$, at fixed $P=40$, for two different initial conditions: cold and hot cavity. The former [dashed line, with green (light gray) markers] collapses rapidly to the lower branch fixed point, while the latter (solid line) is attracted to an oscillating solution.

Figure 5

Same as Fig. 2, but with $\delta =-5.5$ and hot-cavity initial conditions $u_0=2.5$ and $u_{\pm 1}=0$. The limit cycles here are unstable and the system oscillates on a torus; the secondary frequency is much smaller the main one. The average conversion to sidebands is about $25\%$.

Figure 6

Bifurcation diagram of $E$ ($\eta \ne 0,1$) as a function of $\delta$ for different values of $P=60,100,140,180,220,260$. Solid lines represent stable limit cycles, dashed lines unstable ones (saddle points), and dotted lines are delimited by Neimark-Sacker bifurcations. Each curve is labeled according to its value of $P$. The absence of an intensity-dependent threshold makes it impossible to find precise values for triggering the oscillations. Instead, as before in Fig. 3, we have only a single branch of limit cycles for small $P<120$, while a saddle-node bifurcation is observed at higher $P>120$. An even more complex behavior is observed for larger injection power, but the supermode approach is less accurate.

Figure 7

Bifurcation diagrams of $\eta$ [red (gray), left ordinates; see arrow], $\psi$ [blue (dark gray), right ordinates], and $\varphi$ [green (light gray), right ordinates] for the values of $P$ of Fig. 6. Notice that $\eta >0.5$, i.e., more energy in lateral than in central modes.

Figure 8

Comparison of the evolution toward a steady state of the four-cavity system and its four-supermode truncation. $P=180,\delta =-4.5,\gamma =40$. Solid lines represent the solution of Eq. (2) ($M=4$), while dashed lines represent the solution of Eq. (7): ${\eta }_1$ is in blue (dark gray), ${\eta }_3$ in red (gray), $E$ in yellow (light gray), and the imbalances $\alpha$ ($\beta$) are at the bottom in black [green (light gray)] [only for Eq. (2)]. Notice that these last ones exhibit small oscillations around zero.

Figure 9

Same as Fig. 8, for $\delta =-10$, and hot-cavity initial conditions.

Figure 10

Phase-space representations of the time evolution of Eq. (7). (a),(c) The plane $(Ecos{\varphi }_1,Esin{\varphi }_1)$. (b),(d) The plane $(\eta cos\psi ,\eta sin\psi )$. (a),(b) Correspond to Fig. 8; (c),(d) correspond to Fig. 9, with hot-cavity initial conditions. The notation is the same as in Fig. 4, but we omit to show the evolution from a cold cavity.

Figure 11

Same as Fig. 8, for an adiabatic variation in $10<t<40$ of $\delta$ from $-3$ to $-15$. The right ordinate axis shows the values of $\delta$, plotted as a red (gray) dotted line.

Figure 12

Probability distribution of (a) $E^{\left(3\right)}$ and (b) ${\eta }^{\left(3\right)}{E}^{\left(3\right)}$ obtained by repeatedly solving 2500 times the system of Eq. (2) with $M=3,\delta =-3,\gamma =40$, and $P=40$. Every cavity parameter is normally distributed around their nominal value with a standard deviation of $2.5\%$. The thick red (gray) lines represent the nominal value.

Figure 13

Same as Fig. 12, for $M=4$ and a $1\%$ variance for parameters around their nominal values ($\delta =-4,P=180,\gamma =40,\kappa =\sqrt{4/3}$). (a) $E^{\left(4\right)}$, (b) ${\eta }^{\left(4\right)}{E}^{\left(4\right)}$. (c) The distribution of the ratio of eigenfrequencies ${\mathrm{\Omega }}^{\prime }/\mathrm{\Omega }$.