From optical clocks [1] and the 2005 Nobel Prize for measurement of optical frequencies with unprecedented precision [2] to ultraviolet spectroscopy [3], astronomical spectrograph calibration [4], optical waveform synthesis [5], and high-precision ranging [6], frequency comb generation has revolutionized many areas of modern optics. With the current focus on reduced device size, robust performance, and on-chip integration, one of the most promising research directions in this field is the optical comb formation in high-$Q$ optical microcavities [7–10].

Until recently, the comb generation in optical microcavities has been generally described in terms of the standard route [11], where new comb modes are formed through parametric frequency conversion: Kerr nonlinearity enables cascaded four-wave mixing (both degenerate and nondegenerate), which leads to the comb formation [12–16]. In this work, we are studying comb generation in optical microcavities in the near-zero dispersion regime with the following motivation: lower dispersion leads to a broader modulation instability gain spectrum [17], which in turn may lead to a wider frequency comb with an octave-spanning spectral range.

Treated as a dynamic system, the pumped optical microresonator undergoes a transition from integrability to chaos with the increase of the pump power. In the course of this transition, it follows the universal pattern [18]: the periodic states of the dynamical system lose their stability in the cascade of period-doubling bifurcations, leading to the formation of new nonlinear oscillation modes. With octave-spanning frequency bandwidth, this mechanism represents a sought-for direction in optical comb generation.

Neglecting the generally low material loss in a resonator (compared to the loss due to outcoupling), the wave propagation inside the microring (see Fig. 1) can be described by the nonlinear Schrödinger’s equation

 

Fig. 1. Schematics of the microring resonator.

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As material and waveguide dispersion tend to inhibit nonlinear mixing due to deviations from phase matching, we focus on the case of zero to low dispersion (${\beta }_2/T_c^2\ll \gamma P$, where $T_c\equiv L/v_g$ and $L$ are the round-trip time and the circumference of the microring resonator, respectively), when the system dynamics governed by Eqs. (1) and (2) can be described by the equation

Fixed points $b^{(1)}$ of period-1 of the field amplitudes inside the ring can be found from the nonlinear map (4) for the case $b^{n+1}=b^n=b^{(1)}$:

Fixed points $P^{(1)}={|b^{(1)}|}^2$ of period-1 for steady states of the power $P$ inside the microring with $\tau =0.96$ at different values of the pump power $P_0$ are shown in Fig. 2. Originally stable orbits at low values of $P_0$ (shown in red), lose their stability at higher $P_0$ (bifurcation points), transforming to unstable orbits (shown in blue). Note that there are quite a few stable fixed points coexisting at the same chosen value of $P_0$ (at $\gamma {\mathrm{LP}}_0\approx 1$, for example) that differ in the values of $P^{(1)}$ (different cascades). In other words, there could exist mutiple allowed steady states $P^{(1)}$ inside the ring at the given $P_0$. Therefore, it is theoretically possible to achieve higher power steady states (or switch between them) at the chosen pump power.

 

Fig. 2. Fixed points of period-1 for steady states of the power $P$ inside the microring with $\tau =0.96$ at different values of the pump power $P_0$. Red solid lines represent stable orbits, blue unstable. The green arrow indicates the point of the first period-doubling bifurcation in the first cascade.

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Figure 3 illustrates the case of an extremely low transmission coefficient in the ring ($\tau =0.30$). It has a much lower density of fixed points compared to the considered earlier case $\tau =0.96$. However, it is useful in the sense that it shows distinctive branches of stable/unstable fixed points and transitions between them and follows the same period-1 bifurcation pattern as do high-transmission cases. The pattern starts with a stable orbit at near-zero pump power (see Fig. 3). At $\gamma {\mathrm{LP}}_0\approx 2$, it loses stability and undergoes a period-doubling bifurcation (which will be considered later). Near $\gamma {\mathrm{LP}}_0\approx 5$, the system regains stability via a reverse bifurcation, and then at $\gamma {\mathrm{LP}}_0\approx 7$, it undergoes a tangent bifurcation [20] and loses stability again, leading to an unstable branch. This completes the first cascade. All the following period-1 cascades behave similarly.

 

Fig. 3. Fixed points of period-1 for steady states of the power $P$ inside the microring with $\tau =0.30$ at different values of the pump power $P_0$. Red solid lines represent stable orbits, blue unstable.

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In agreement with the standard scenario of the transition from integrability to chaos [18], the steady states of the system (4) follow the period-doubling bifurcation cascade (Figs. 4 and 5) where the originally stable period-1 mode corresponding to the time-independent power inside the ring (the red curve in Fig. 4 and the red waveform in Fig. 6) loses its stability (at the point indicated by the green arrow in Fig. 2 and Fig. 4) and “gives birth” to the new state with the period $2T_c$ corresponding to two ring round trips. At this point, the steady-state power in the microring is no longer time independent (the green curve in Fig. 4 and the green waveform in Fig. 6), leading to multiple subbands in its frequency spectrum (bottom panel in Fig. 6). This period-2 state will in turn bifurcate (blue arrow in Fig. 4), leading to the formation of the period-4 mode (with the period of $4T_c$). A zoom-in on the period-4 orbits is shown in Fig. 5. The period-4 state undergoes a reverse bifurcation (cyan arrow in Fig. 5), which returns the system to a stable period-2 state. Then, the stable period-2 state undergoes a tangent bifurcation, loses stability, and leads to the formation of an unstable period-2 state (black arrow in Fig. 4).

 

Fig. 4. Bifurcation diagram for steady states of the power $P$ inside the microring with $\tau =0.96$ at different values of the pump power $P_0$ (the first cascade). Solid lines represent stable orbits, dotted unstable. Arrows indicate the positions of bifurcations (green and blue, period-doubling; cyan, reverse; black, tangent bifurcation).

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Fig. 5. Period-4 orbits of the power $P$ inside the microring with $\tau =0.96$. Solid lines represent stable orbits, dotted unstable. Arrows indicate the positions of bifurcations (blue, period-doubling; cyan, reverse).

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Fig. 6. (a) Waveforms and (b) power spectra inside the microring without dispersion (${\beta }_2=0$) at different pump power levels: $\gamma {\mathrm{LP}}_0=20$ (shown in red) and $\gamma {\mathrm{LP}}_0=25$ (shown in green).

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While the transition between different power levels of the same mode is instantaneous in the limit of zero group velocity dispersion, a finite ${\beta }_2$ in the actual resonator replaces these discontinuities with a smooth variation, as shown in Fig. 7(a), and effectively limits the power spectrum cutting off the higher frequencies, while the overall spectrum scales as $1/\mathrm{\Delta }{\omega }^2$ away from the pump frequency [see Fig. 7(b)].

 

Fig. 7. (a) Waveforms and (b) power spectra inside the microring without dispersion (${\beta }_2=0$, shown in green) and with anomalous dispersion (${\beta }_2<0$, shown in dark green) at the pump power $\gamma {\mathrm{LP}}_0=25$.

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As a result, the system produces a well-defined frequency comb at pump powers above that of the first period-doubling bifurcation, $P^{(1)}$. Note that while subsequent bifurcation quantitatively changes the spectrum, it still follows the overall $1/\mathrm{\Delta }{\omega }^2$ profile as long as ${\beta }_2\mathrm{\Delta }{\omega }^2\lesssim \gamma P$.

It should be noted that the frequency comb generation mechanism described here is qualitatively different from that of the soliton-based frequency comb formation theory [16]. The resonator system described in this work does not support solitons, as it is operating in the regime of near-zero dispersion and high powers (${\beta }_2/T_c^2\ll \gamma P$), and the comb generation is caused by switching between stable power states. The existence of stable power states is governed exclusively by self-phase modulation. First-order dispersion (${\beta }_1$) only determines the switching period $T_c$, while (small) second-order correction ${\beta }_2$ affects the steepness of the switching. In contrast, the soliton-based approach [16] relies on generation of solitons and Turing patterns inside the microresonator for producing a frequency comb. Thus, a single soliton traveling inside the resonator induces a comb with spectral lines separated by a single free spectral range (FSR), while multiple Turing rolls lead to a comb with multiple-FSR spacing.

Note that, as opposed to our nonlinear dynamics approach valid for any $\kappa$, Lugiato–Lefever-equation-based continuous approximation [12] can only be used here in the extreme limit $|\kappa |\ll 1$.

Theoretically, the resonator system described in the current work could be used, for example, for producing square waveforms with finite rise and fall times due to dispersion. However, practical implementation of such a device would be a complicated task, at least, for two reasons. First of all, in order to achieve the generation regime, the system should pass over the point of the first period-doubling bifurcation $P_0^{(1)}$, which requires the pump power to be of the order of $P_0\gtrsim (\gamma L{)}^{-1}$ (for example, in Fig. 2, $\gamma {\mathrm{LP}}_0^{(1)}\approx 25$ for the first cascade and at least $\gamma {\mathrm{LP}}_0^{(1)}\approx 2$ for higher cascades). Finally, holding the overall (material and waveguide) dispersion at the near-zero level during the generation presents another problem.

In conclusion, for the regime of near-zero dispersion, we have studied the route to chaos and the frequency comb formation in the microresonators related to the transition from integrability to chaos in these systems at the increase of the optical pump power. The theoretical findings may lead to potential applications, given that the underlying implementation problems will be addressed.