Bichromatically pumped microresonator frequency combs

A study is made of the nonlinear dynamics of bichromatically pumped microresonator Kerr frequency combs described by a driven and damped nonlinear Schrödinger equation, with an additional degree of freedom in the form of the modulation frequency. A truncated four-wave model is derived for the pump modes and the dominant sideband pair, which is found to be able to describe much of the essential dynamical behavior of the full equation. The stability of stationary states within the four-wave model is investigated, and numerical simulations are made to demonstrate that a large range of solutions, including cavity solitons, are possible beyond previously considered low-intensity patterns.

Figure 1

Simplified schematic of a bichromatically pumped microring resonator. A laser source is modulated to create two frequencies that are used to pump the resonator. An optical frequency comb is generated through FWM, which can be measured using an optical spectrum analyzer.

Figure 2

Normal dispersion frequency comb for a dual-pump configuration. Numerical simulation of Eq. (1) showing a stationary background free comb corresponding to a pattern of four pulses for parameters $\kappa =1$, ${\delta }_0=3$, $|f_0|=3$, and $\Omega =2$. Normalized units.

Figure 3

Normal dispersion frequency comb for a dual-pump configuration with a finite-background component owing to the growth of a secondary comb which fills in the comb spectrum. Parameters are $\kappa =1$, ${\delta }_0=5$, $|f_0|=5$, and $\Omega =1$. Normalized units.

Figure 4

Fixed point curve showing normalized pump-mode intensity as a function of external pump amplitude for fixed dispersion $\kappa =-1$ and detuning ${\delta }_0=3$. Green (light-gray) curves are stable fixed points, while red (dark-gray) curves are unstable fixed points.

Figure 5

Fixed point curve showing normalized pump-mode intensity as a function of detuning for fixed dispersion $\kappa =-1$ and external pump amplitude $|f_0|=9$. Green (light-gray) curves are stable fixed points, while red (dark-gray) curves are unstable fixed points.

Figure 6

Stable anomalous dispersion frequency comb without a background component but with even modes filled in due to secondary comb generation. The dispersion parameter $\beta =-1$, the modulation frequency $\Omega =1$, and the external pumping is $|f_0|=9$, while the detuning has a negative value ${\delta }_0=-2$. Normalized units.

Figure 7

Numerical simulation of Eq. (1) showing cascaded comb generation in the anomalous dispersion regime. The comb is stable and the inset shows that the dc component is missing. Parameters are $\beta =-0.01$, $\Omega =1$, ${\delta }_0=-5$, and $|f_0|=9$. Normalized units.

Figure 8

Contour plot showing evolution of a stable hard excitation breather state with two oscillating bright pulses. Lighter color indicates higher normalized intensity. Parameters are $\kappa =-1$, $\Omega =1$, ${\delta }_0=14$, and $|f_0|=9$. Normalized units.

Figure 9

Numerical simulation of Eq. (1) showing coexistence of a single injected cavity soliton on top of a stationary 16-bit pattern of low-intensity clock pulses. Anomalous dispersion with $\beta =-0.001$, modulation frequency $\Omega =8$, detuning ${\delta }_0=1.7$, and external pump amplitude $|f_0|=1.3$. Normalized units.

Figure 10

Numerical simulation of Eq. (1) showing coexistence of multiple injected cavity solitons on top of a stationary 16-bit pattern of low-intensity clock pulses. Same parameters as in Fig. 9. Normalized units.