Optical frequency combs, a light source of a series of discrete and equally spaced frequencies, have found wide applications in modern technology, such as optical clock [1], astronomical spectrograph calibration [2], precision time/frequency transfer [3], ultraviolet and infrared (IR) spectroscopy [4–6], precision distance measurement [7], and optical communication [8]. In the past, the main technologies developed to generate optical frequency combs have been confined to mode-locked femtosecond lasers [9,10], electro-optic modulation of a continuous-wave laser [11], and parametric frequency conversion through cascaded four-wave mixing [12–16]. Optomechanical nonlinearity in a micro-cavity [17–19], as well as the related magnotic realization [20], is a recently interesting approach to realize such a light source known as an optomechanic-optical frequency comb (OMOFC). OMOFCs are repeated pulses at the corresponding mechanical frequencies, which are relatively low, e.g., less than 1 GHz, and such low-repeating OMOFCs are highly relevant to some state-of-the-art technologies such as high-resolution spectroscopy [21,22], spectral measurements [23], multiphoton entangled states [24], and high-power continua in holey fibers [25].

A unique character of optomechanical nonlinearity is the generation of high-order sidebands, the field frequency components distanced exactly by the mechanical oscillation frequency, due to the energy transfer between the cavity field mode and the mechanical mode of a cavity optomechanical system (OMS) [17,18,26]. For practical applications OMOFCs must have more sidebands. However, the comb lines of the previously achieved OMOFCs are rather limited, only around dozens or hundreds as reported by Refs. [17,19,20]. A much more improved setup combining thermal-optic effect can make 938 comb lines at a laser power of 448 mW [18]. For many applications such as precision optical metrology, such limited numbers of comb lines are not enough. A straightforward method for broadening comb span is to use higher pump power. However, the OMS can enter the regimes of chaos under very strong drive intensity, and the tremendously increased intracavity photons would damage the micro-cavity after turning on the strong pump laser for a period of time. An effective way to beat the bottleneck is crucial to the next stage of developing practical OMOFCs.

As we will see later, the frequency span of an OMOFC is determined by how large the associated mechanical oscillation amplitude can be realized. Therefore, the more the energy that is transferred to the mechanical resonator, the wider the output cavity field spectrum will be. Here, we propose an approach of efficiently adding up the mechanical energy based on a newly discovered resonance mechanism [26]: the mechanical oscillation of an unresolved-sideband OMS will be significantly enhanced under a two-tone driving field, with the difference of its tone frequencies being matched to the intrinsic mechanical frequency of the OMS. In this scenario the pump power for generating OMOFCs can be reduced to the level of only microwatts (μW) for obtaining dozens of comb lines. On the level of milliwatts (mW), more than 10,000 comb lines can be generated with the currently available technologies. A system optimized further can realize the frequency combs with even broader spans.

Another important issue for frequency combs is their stability, i.e., to preserve the comb lines perfectly equidistant for a period as long as possible, for hours and even for days. In the past experiments optical spring effect is unavoidable, depending on the drive power and resulting in unpredictable mechanical frequency shift that changes the comb line distance. Our adopted nonlinear resonance also has a property of mechanical frequency locking, rendering the comb line distance be locked to be close to the intrinsic mechanical frequency of the system, totally independent of the applied drive power. It can thus practically guarantee the stability of the generated OMOFCs.

The rest of the paper is organized as follows. In Section 2, the mechanism of realizing large mechanical oscillation by a special two-tone driving field is detailed in comparison with the corresponding mechanism due to a single-tone laser field, to illustrate the important features of a broadband OMOFC and the associated oscillation frequency locking. The optimization of the concerned OMOFC by a parallel displacement of the drive tones is discussed in Section 3. All factors affecting the performance of the system, including the drive intensity and intrinsic mechanical frequency are illustrated in Section 4. Then the technical requirements for the used drive tones are sketched in Section 5, before the final part of conclusive remarks.

The system we consider is a generic OMS, which is represented by the Fabry–Perot cavity in Fig. 1: the movable mirror acts as a mechanical resonator with frequency ${\omega }_m$ and the cavity is pumped by a two-tone laser field with their frequencies ${\omega }_1$ and ${\omega }_2$, respectively. The radiation pressure of the cavity field, which is proportional to the photon number ${|a|}^2=(X_c^2+P_c^2)/2$ ($X_c$ and $P_c$ are two quadratures of the field), drives the mechanical resonator into oscillation once the pump field satisfies a certain condition. As the first-order correction to the cavity field energy by the mechanical displacement under radiation pressure, there is the interaction potential, $$V_{\mathrm{int}}=-g_m{X}_m{|a|}^2,$$(1)for the cavity field and mechanical resonator, where $g_m$ is the optomechanical coupling strength at the single photon level. Here we have introduced the dimensionless mechanical displacement $X_m$ by the scaling $x_m=\sqrt{\frac{\hslash }{m{\omega }_m}}{X}_m$ ($m$ is the effective mass of the mechanical resonator), and the conjugate dimensionless momentum $P_m$ is defined by another scaling $p_m=\sqrt{m\hslash {\omega }_m}{P}_m$. Then the dynamical equations of the system are given in the rotation frame with respect to the cavity frequency ${\omega }_c$ as follows: $$\begin{gathered} \dot{a}=-\kappa a+i{g}_m{X}_{m}a+\sum _{n=\mathrm{1,2}}{E}_n{e}^{i{\mathrm{\Delta }}_{n}t}, \\ {\dot{X}}_m={\omega }_m{P}_m, \\ {\dot{P}}_m=-{\omega }_m{X}_m-{\gamma }_m{P}_m+g_m{|a|}^2+\sqrt{2{\gamma }_m}{\xi }_m(t), \end{gathered}$$(2)where the detuning is defined as ${\mathrm{\Delta }}_{1(2)}={\omega }_c-{\omega }_{1(2)}$. The cavity damping rate $\kappa$ includes the coupling rate to the external field ${\kappa }_e$ and the intrinsic loss ${\kappa }_i$, so that $\kappa ={\kappa }_e+{\kappa }_i$. The drive amplitudes are determined by the pump power $P$ as $E_{1(2)}=\sqrt{2{\kappa }_{e}P/\hslash {\omega }_{1(2)}}$. Moreover, at the room temperature, the thermal noise can be treated as a white noise satisfying the correlation $⟨{\xi }_m(t){\xi }_m(t^{\prime })⟩=(2n_{\mathrm{th}}+1)\delta (t-t^{\prime })$ [27], where $n_{\mathrm{th}}=(e^{\hslash {\omega }_m/k_{B}T}-{1)}^{-1}$ is the thermal phonon number at the temperature $T$. This thermal noise can be simulated with a stochastic function [28], but it does not have a significant effect on the concerned dynamical processes [26], so we will ignore it and adopt a mean-field approximation in the following discussions.

First, we take a look at the scenario when an OMS is driven by a single-tone pump field. When the power of the pump field is higher than a certain threshold of Hopf bifurcation, the system will enter the coupled oscillations of the cavity field and mechanical resonator. The mechanical oscillation generally stabilizes in the form $$X_m(t)=A_m\cos ({\mathrm{\Omega }}_{m}t)+d_m,$$(3)where $A_m$ denotes the oscillation amplitude and $d_m$ a pure displacement with $d_m\ll A_m$. It should be noted that, due to the existence of optical spring effect, the frequency ${\mathrm{\Omega }}_m$ can considerably differ from the intrinsic mechanical frequency ${\omega }_m$, which is determined by the fabrication of the system. If one plugs this $X_m(t)$ into the first equation of Eq. (2), which is reduced to the situation of a single-tone drive, there will be the evolved cavity field in the form $a(t)={\sum }_n{a}_n{e}^{in{\mathrm{\Omega }}_{m}t}$ (up to a global phase), where $$a_n=E\sum _p\frac{J_{n-p}(g_m{A}_m/{\mathrm{\Omega }}_m)J_p(-g_m{A}_m/{\mathrm{\Omega }}_m)}{ip{\mathrm{\Omega }}_m+\kappa +i(g_m{d}_m-\mathrm{\Delta })},$$(4)with $J_p(x)$ being the Bessel function of the first kind (see the derivation in Appendix A). All these components $a_n=|a_n|e^{i{\phi }_n}$ are called sidebands of the cavity field, from which it can be seen that their phases ${\phi }_n$ are fixed under a set of given system parameters, but the different sidebands have their different oscillation phases. The variable $g_m{A}_m/{\omega }_m$ of the Bessel functions is highly relevant to the sideband structure. There is one property of the Bessel function: given a fixed $x$, the magnitude of the $p$th order Bessel function $J_p(x)$ will decay quickly if $|p|>|x|$; this property also guarantees the convergence of the summation in Eq. (4). Then, the value of $g_m{A}_m/{\mathrm{\Omega }}_m$ determines to which sideband $p$ the cavity field can be. In other words, the condition $p<g_m{A}_m/{\mathrm{\Omega }}_m$ or $-p>-g_m{A}_m/{\mathrm{\Omega }}_m$ specifies a range, $$-g_m{A}_m<\mathrm{\Delta }\omega <g_m{A}_m,$$(5)covering all possible sidebands with their frequency $p{\mathrm{\Omega }}_m$ ($p\in \mathbf{Z}$) in a reference frame rotating at the pump laser frequency. The argument $g_m{A}_m/{\mathrm{\Omega }}_m$ of the Bessel functions in Eq. (4) happens to be the half of the comb line number of the generated cavity field as the frequency comb (the extension to the negative integers $p$ gives the other half). Broad comb span should be certainly obtained by having a large mechanical amplitude $A_m$. The straightforward way for having a large $A_m$ is to enhance the drive amplitude $E$, but the drive power cannot be increased without a limit because the accompanying heat would damage the systems. As reported by a recent experiment [18], a large pump detuning to the blue-detuned side can also increase the mechanical amplitude due to a peculiar system dynamics of OMS (see the discussions in Section 3). Another approach for having more sidebands is to make of the unresolved sideband regime (${\omega }_m<\kappa$), where pulsed cavity field can be generated [29–31]. It is easy to reach the unresolved sideband regime by adjusting the pump-field coupling rate ${\kappa }_e$ for a system with fixed intrinsic mechanical frequency ${\omega }_m$. In what follows, we will consider this unresolved sideband regime for the designs of OMOFCs.

The sideband amplitude in Eq. (4) is based on an already realized mechanical amplitude $A_m$, which is through a complicated dynamical process that must consider the coupled nonlinear equations [Eq. (2)]. All similar processes vary with the applied pump field. If the single-tone pump is replaced by a two-tone driving field, the stabilized mechanical amplitude will have totally different patterns. The two tones with their corresponding detunings ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ satisfy the relation $$|{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2|={\omega }_m+\delta ,$$(6)where the deviation $\delta$ can be any real number. Once the above difference between the two tones gets close to the intrinsic mechanical frequency ${\omega }_m$, i.e., $\delta \to 0$, the system being operated in the unresolved sideband regime (${\omega }_m<\kappa$) will exhibit a resonance phenomenon shown in Fig. 2(a). Generally, various nonlinear resonances can occur when a particular nonlinear oscillator has the maximum response to an external drive [32]. Our concerned resonance is out of a self-organization of the system when it is operated under the two conditions, $|{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2|={\omega }_m$ and ${\omega }_m/\kappa <1$ [26]. In the unresolved sideband regime satisfying ${\omega }_m/\kappa <1$, two cavity field pulses are generated in each mechanical oscillation period; one of them emerges when the mechanical resonator approaches its positive top speed in the oscillation period, but the other manifests when it is close to its negative top speed (the radiation force of the pulse is opposite to the moving direction of the mechanical oscillator). Under a further condition $\delta =0$, the field pulse acting in the opposite direction of the mechanical resonator’s movement will be significantly suppressed, so the other pulse acting along with the movement of the mechanical resonator will successively add more and more energy, $${\mathcal{E}}_m=\frac{1}{2}{X}_m^2+\frac{1}{2}{P}_m^2\approx \frac{1}{2}{A}_m^2,$$(7)to it, in the form of a quasi-linear energy increase in Fig. 2(a). Such tremendously increased mechanical amplitude $A_m$ surely gives rise to a broadband $2g_m{A}_m$ of the cavity field, since, according to the reduced dynamics of the cavity field part [by plugging the stabilized mechanical oscillation $X_m(t)$ into the first equation of Eq. (2)], the stabilized mechanical amplitude $A_m$ still determines the cavity field frequency span so that the range in Eq. (5) is valid (such sideband ranges can be accurately verified by the numerical calculations with our concerned two-tone drives). More features of the used nonlinear resonance are detailed in Ref. [26]. The main point of the current work is how to utilize this mechanism to realize good-quality OMOFCs.

In the resolved sideband regime ${\omega }_m\gg \kappa$, a two-tone field satisfying the condition $\delta =0$ realizes a complete locking of an OMS: the mechanical oscillation amplitude, the mechanical oscillation frequency, as well as the oscillation phase, can be fixed without varying even after a considerable change of the pump power and the associated noise effects, once the system has stabilized [33,34]. These properties can be applied for various precise sensing [35–37]. The system operating in the unresolved sideband regime of ${\omega }_m/\kappa <1$ also inherits the property of frequency locking. Due to the existence of optical spring effect, there is an unavoidable and mostly obvious shift of the actual mechanical oscillation frequency ${\mathrm{\Omega }}_m$ from the intrinsic mechanical frequency ${\omega }_m$ determined by system fabrication, when the system is driven by a single-tone field. An intuitive view of the effect is that the difference between the two frequencies, ${\mathrm{\Omega }}_m$ and ${\omega }_m$, will accumulate a phase difference between the actual mechanical oscillation and a reference mechanical oscillation at the frequency ${\omega }_m$, after the oscillations proceed for a period of time; see Fig. 2(b1). This effect causes a problem of instability of frequency combs’ repetition rates, as it was observed by the measurement of the Allan deviations with changed pump powers [18]. However, if the same single-tone pump field is split into two components with the difference of their frequencies matching the intrinsic mechanical frequency ${\omega }_m$, the stabilized mechanical oscillation will well keep its frequency to the intrinsic one ${\omega }_m$, as seen from the exactly coinciding actual and reference oscillations in Fig. 2(b2). It is even clearer to see the oscillation frequency locking by comparing the mechanical spectrum induced by the single-tone and double-tone pumps, which are in the lower panel of Figs. 2(b1) and 2(b2), respectively. More generally, the mechanical oscillation frequency is locked to the frequency difference $|{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2|$ of two drive tones, as long as the error $\delta$ in Eq. (6) is within a certain range. Such a frequency locking is independent of the applied pump power and totally robust against noise perturbations.

The advantages of the two-tone pump satisfying $|{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2|={\omega }_m$ are clearer by a comparison with the corresponding single-tone pump; see Fig. 3. In the right part of Fig. 3, the corresponding pumps of a single tone (in the left part) are divided into two with one of them being shifted to a different frequency of ${\mathrm{\Delta }}_2=0$. Dozens of comb lines can be obtained with a rather low pump power proportional to $E^2$, while the system driven by a single-tone drive below its Hopf bifurcation point stabilizes to a static equilibrium; compare Figs. 3(a1) and 3(b1). The comb span in Fig. 3(b1) is close to that in Fig. 3(a2), where the pump power is, however, increased by 156 times. In Fig. 3(b3) the number of comb lines has been beyond 1000 and, given the experimental setup in Ref. [18], this OMOFC can be achieved with a pump power below 100 mW. Because we split an original pump into two tones, the corresponding comb teeth brightness will be lower due to the energy conservation. The brightness will be improved if both pump tones have the same power as in the corresponding single-tone scenario. For example, given the pump field of $E_1=E_2{=10}^4\kappa$ for the process in Fig. 3(b2), the amplitude of the highest comb tooth will be increased to $5.21\times {10}^3$, well close to the corresponding one with $5.28\times {10}^3$ in Fig. 3(a2).

Different patterns of adding up mechanical energy explain the difference in the generated OMOFCs. As seen from Figs. 3(a2) and 3(a3), the mechanical energy ${\mathcal{E}}_m$ under a single-tone pump increases in a manner of asymptotic stabilization with time. A stronger pump drives the same system to a higher mechanical energy, but the energy increase rate is not so significant. On the other hand, the system driven by a two-tone field satisfying $|{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2|={\omega }_m$ exhibits a rapid growth of mechanical energy due to a self-organized resonance at the beginning stage of time evolution [26]. Only after this stage will the system gradually tend to a stabilization in a process dominated by its mechanical damping. The asymptotic behaviors of the systems operated in the unresolved sideband regime (${\omega }_m<\kappa$) are very different from those in the resolved sideband regime (${\omega }_m\gg \kappa$), where the optomechanical interaction enhances the mechanical damping so that the total effective damping rate is increased with an extra term ${\mathrm{\Gamma }}_{\mathrm{opt}}$ [38]. Optomechanical cooling processes in the regime of resolved sideband are therefore much faster than the mechanical relaxation characterized by the time ${\gamma }_m^{-1}$ [39–41]. In contrast, a cavity field realized in an unresolved sideband regime becomes pulsed, affecting the mechanical resonator only within a short period of time. Therefore, the intrinsic mechanical damping rate ${\gamma }_m$ determines the stabilization in the regime of ${\omega }_m<\kappa$. From Fig. 3 one sees that the processes under a single-tone pump stabilize after a time of the order of ${\gamma }_m^{-1}$ but, due to the existence of a nonlinear resonance, the dynamical processes under the concerned two-tone drives are modified and take even longer time than ${\gamma }_m^{-1}$ when they become finally stabilized. An even clearer comparison between the single-tone scenario and our concerned two-tone scenario is displayed in Fig. 4, where the achieved mechanical amplitudes $A_m$ under the lower drive amplitude $E$ have the totally different tendencies—there is a definite Hopf bifurcation point to start optomechanical oscillation for the single-tone scenario (a behavior called “phonon laser” [38]), but the optomechanical oscillation induced by the concerned two-tone pump has a continuous existence toward $E=0$ [26]. The realized bandwidths $2g_m{A}_m$ of OMOFCs by the concerned two-tone pump fields, therefore, significantly surpass those by the corresponding single-tone pump fields.

The adopted exemplary system in Figs. 3 and 4 has a mechanical quality factor $Q=2\times {10}^4$. Due to the extremely slow stabilization processes caused by the optomechanical resonance, we have not simulated the dynamical evolution with a still smaller mechanical damping ${\gamma }_m$. OMOFCs covering much more sidebands are expected with the still improved mechanical quality factors. The continued discussions will be about how to achieve the better OMOFCs by the adjustment of other system parameters under the proper conditions.

Driven by a single-tone pump, there are two tendencies for a stabilized cavity field intensity, $|a(t{)|}^2={\sum }_n{A}_n·\cos (n{\mathrm{\Omega }}_{m}t+{\varphi }_n)$ (${\varphi }_n$ is the phase for the $n$th sideband), with the increased blue detuning of the pump. One is certainly that the overall cavity field will dwindle as the pump is off the resonance. But the other is an energy redistribution among the field sidebands so that the first sideband of $|a(t{)|}^2$ with its magnitude, $A_1=2|{\sum }_p{\alpha }_{p+1}{\alpha }_p^{\ast }|$, will be enhanced. This first sideband with the magnitude $A_1$ predominantly determines the mechanical amplitude $A_m$ since it is the closest to the mechanical resonance frequency ${\omega }_m$ among all those with their magnitudes $A_n$, and an enhanced magnitude $A_1$ by blue detuning will thus increase the bandwidth $2g_m{A}_m$ of the generated OMOFC. These two tendencies compete with each other in the determination of the sideband magnitude $A_1$, and give rise to an optimum pump detuning for the bandwidth of the OMOFC, beyond which the comb span will drop instead. An experimental demonstration of such enhancement by blue detuning is reported in Ref. [18], which shows the generation of the OMOFCs with about 1000 comb lines by a large detuning further increased by a thermal-optic effect.

When it comes to our concerned two-tone pump scenario, there is also a similar enhancement of comb span by varying drive tones. The previously mentioned optomechanical resonance still exists after the application of the following displacements [26]: $${\mathrm{\Delta }}_1\to {\mathrm{\Delta }}_1+\delta \omega ,{\mathrm{\Delta }}_2\to {\mathrm{\Delta }}_2+\delta \omega ,$$(8)for the two tones of the pump from their original frequencies of ${\mathrm{\Delta }}_1=0$, ${\mathrm{\Delta }}_2=-{\omega }_m$ we have adopted before. All features of the applied optomechanical resonance are well preserved as long as the two tones satisfy the condition $|{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2|={\omega }_m$. Now we demonstrate that an appropriate displacement $\delta {\omega }_m$ to the blue-detuned side can considerably broaden the comb span, as shown in Fig. 5 where a displacement of $\delta \omega =-20\kappa$ extends the original comb span by about 2.64 times.

Such spectrum broadening of the OMOFC by a parallel displacement of the drive tones can be understood with a slight asymmetry of the comb spectrum shown in Fig. 5(b). The parallel displacement will move the peak of the optical spectrum from the original position at $\omega ={\omega }_c$ to a displaced position at $\omega ={\omega }_c+|\delta \omega |$, and the sidebands with their frequencies less than the original peak frequency will be enhanced after the displacement; see Fig. 5(b). This asymmetry amplifies the first sideband magnitude $A_1$ in the optical field intensity spectrum ${|a|}^2(\omega )$, as indicated by the values numerically obtained in Fig. 5. Like in a single-tone scenario [18], a larger mechanical amplitude $A_m$ due to the enhanced sideband magnitude $A_1$ will therefore lead to a further extended comb span $2g_m{A}_m$.

In Fig. 6, we show the relations between the dimensionless mechanical amplitude $A_m$ and the frequency displacement $\delta \omega$ for two different systems with their mechanical frequencies ${\omega }_m=0.2\kappa$ and $0.9\kappa$, which are also respectively driven by one two-tone pump of $\sqrt{2}E=5\times {10}^5\kappa$ and another two-tone pump of $\sqrt{2}E=5\times {10}^6\kappa$. The comb span can be broadened further by about 3 times for the lower drive power, while it can be broadened by about 2.2 times for the higher drive amplitude. For the system described in Fig. 3, the comb line number will be more than 10,000 after the dimensionless mechanical amplitude is higher than $A_m{=10}^8$. In the upper panels of Fig. 6, such improvement will be stopped where the curve of $A_m$ is broken at the blue detuning with $|\delta \omega |=32\kappa$ for ${\omega }_m=0.2\kappa$, as well as at $|\delta \omega |=29\kappa$ for ${\omega }_m=0.9\kappa$. These points indicate the optimum comb spans around them. The applicable displacement range can be increased further with a still larger drive amplitude $E$, as shown in the two lower panels of Fig. 6. Overall, the associated dynamical processes modified by drive-tone displacements are rather complicated due to the relevance of various factors.

So far, we have discussed the figures-of-merit of the OMOFC generated by a specified two-tone drive on an OMS, which is operated by adjusting the pump–cavity coupling ${\kappa }_e$ to an unresolved sideband regime $\kappa >{\omega }_m$. In addition to locking the mechanical oscillation frequency to near the intrinsic one ${\omega }_m$, the comb span can be enlarged by times if one simply displaces the tones together to a proper detuning. A more obvious factor is the pump power $P=\hslash {\omega }_{1(2)}{E}^2/(2{\kappa }_e)$. In Fig. 7(a) we illustrate how the realized mechanical amplitude $A_m$ changes with the associated two-tone drive amplitude $E$. The realized amplitude $A_m$ for the different ${\omega }_m$ increases linearly with the drive amplitude [note that a logarithmic scale is used for the vertical axis of Fig. 7(a)]. If the system fabrication can enhance the optomechanical coupling $g_m$ to the level of ${10}^{-4}\kappa$, all systems with the used ${\omega }_m$ in Fig. 7(a) can realize the frequency comb line numbers $2g_m{A}_m/{\omega }_m>10,000$ under the low pump powers, which achieve the dimensionless mechanical amplitude $A_m$ beyond ${10}^7$. Like the systems driven by a single-tone pump in unresolved sideband ${\omega }_m/\kappa <1$ [17], those under a two-tone drive with the frequency difference $|{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2|$ close to ${\omega }_m$ encounter less problem of chaos when the pump power is extremely high, so the room for applying strong pump is big as long as the systems can withstand the heating by the intensified pulses in their cavities.

The mechanical frequency ratio ${\omega }_m/\kappa$ is another important factor to determine the comb span. Its relation with the realized mechanical amplitude is shown in Fig. 7(b). Especially, within the range of ${\omega }_m/\kappa <0.1$, the mechanical amplitude increases rapidly with this decreased ratio. For example, at the ratio ${\omega }_m/\kappa =0.05$, about 10,500 comb lines can be obtained with a relatively low drive amplitude $E{=10}^5\kappa$. More than 30,500 comb lines can be realized under the same pump power if the ratio ${\omega }_m/\kappa$ is decreased further to 0.01.

One method for a small ratio ${\omega }_m/\kappa$ is to have a higher rate ${\kappa }_e$ for the coupling between the pump and the cavity. However, it will also affect the other system parameters, especially the optomechanical coupling ratio $g_m/\kappa$. Then, it is better to have a mechanical mode with lower intrinsic frequency ${\omega }_m$, which can be realized by the fabrication to a different geometry of OMS. A smaller ${\omega }_m$ certainly implies a less spring restoring force proportional to $-{\omega }_m^2{X}_m$ and thus a larger oscillation amplitude $A_m$. From the view of photon–phonon interaction, a cavity photon with its energy $\hslash {\omega }_c$ will absorb (release) more phonons with a less energy $\hslash {\omega }_m$, so that it can be converted to more of those with the frequencies ${\omega }_c+n{\omega }_m$ (${\omega }_c-n{\omega }_m$), where $n$ is an integer, to have a broader spectrum in Fig. 1(b). On the other hand, the cavity damping rate $\kappa$ can be regarded as the line width of the cavity field. In a good regime of generating OMOFCs, one such line width encompasses a large quantity of the sidebands distanced by the mechanical frequency ${\omega }_m$.

For example, given the experimentally achievable parameters ${\omega }_m=0.05\kappa =5\pi \mathrm{MHz}$ and $Q=1500$, together with an optomechanical coupling tuned to $g_m=2.3\times {10}^{-5}\kappa$ by the total cavity field damping rate $\kappa =2\pi \times 50\mathrm{MHz}$, 9500 comb lines can be obtained by the use of a two-tone pump with the detunings ${\mathrm{\Delta }}_1=0$ and ${\mathrm{\Delta }}_2=-0.05\kappa$. The required pump power corresponding to the drive amplitude $\sqrt{2}E{=10}^5\kappa$ is 32.1 mW for the pump laser of 1550 nm. Furthermore, the comb lines can be increased to 22,500 by simply displacing the detunings to ${\mathrm{\Delta }}_1=-10\kappa$ and ${\mathrm{\Delta }}_2=-10.05\kappa$. When the pump power is decreased to 8.03 mW, there will be 11,150 comb lines realized by the pump with the detunings ${\mathrm{\Delta }}_1=-6.0\kappa$ and ${\mathrm{\Delta }}_2=-6.05\kappa$. It should be noted that, for the convenience of reaching the final stability more quickly (see the discussion in Section 2.C), the mechanical quality factor $Q$ used for our simulations is low, and the better OMOFCs are expected to realize with a slight improvement on the quality of the mechanical resonator. An even better improvement is with the optomechanical coupling strength $g_m$. If it can be increased by 1 order of magnitude, a much-enlarged comb span $2g_m{A}_m$ can be realized with the pump powers reduced by 2 orders of magnitude.

In realistic situations, there exits an unavoidable error $\delta$ in Eq. (6) for the two tones of the driving field. This error $\delta$ could come from the fluctuation of the frequencies of a pump laser. To have a clear view of how this error would affect the generated OMOFC, we illustrate the effects from various errors in Fig. 8, which is also based on the exemplary process in Fig. 3. From Figs. 8(a) and 8(b), one sees that the generated OMOFC will only lose some sidebands to a smaller span $2g_m{A}_m$, if the error $\delta$ is within ${10}^{-3}\kappa$. This defect can be compensated by a parallel displacement of the two tones as in Fig. 5(b). For a setup with $\kappa \sim 2\pi \times 10\mathrm{MHz}$, the requirement on $\delta \sim {10}^{-4}\kappa$ specifies that the distance between two drive tones should not deviate from the frequency ${\omega }_m$ by more than $2\pi \times 1\mathrm{kHz}$. It is feasible to apply an acoustic-optic modulator (AOM) [42] or single-sideband modulator (SSB) [43] to keep the error of the two tones within the range. If a spectrum analysis of the mechanical oscillation is performed, one will see that the mechanical oscillation frequency under such a small error $|\delta |$ will be locked to the difference $|{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2|$ of the two drive tones. Up to the error $\delta {=10}^{-2}\kappa$ in Fig. 8(c), the mechanical oscillation frequency, as well as the distance between the individual sidebands, is still locked to the difference $|{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2|$, totally independent of the applied pump power. Therefore, the stability of the generated OMOFCs relies on how well we can control the frequency fluctuations of two drive tones.

However, given a still larger error up to $\delta =2\times {10}^{-2}\kappa$ for the exemplary system in Fig. 8, the dynamics of the system will become totally different. The effect of self-organized resonance will be lost, but the system never behaves as if it were under two independent drives. In this situation the two drive tones will realize other cooperative effects. First of all, more mechanical frequency components, such as the fractional ones, will emerge while the peak of the mechanical spectrum $X_m(\omega )$ returns to the original mechanical frequency ${\omega }_m$ from $|{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2|$. The corresponding cavity field sidebands will therefore split into a number of different sets; the two main sets will be shifted to the frequency positions ${\mathrm{\Delta }}_1+n{\omega }_m$ and ${\mathrm{\Delta }}_2+n{\omega }_m$ ($n$ are integers), respectively, as shown in Fig. 8(d). By a further variation of the error $\delta$, the system will enter chaos or other complicated quasi-periodic oscillations. The dynamical patterns in the regimes of unmatched pump tones with $\delta \ne 0$ are determined by the ratio ${\omega }_m/\delta$; see more details in Ref. [26].

The comb span of OMOFCs generated by an OMS is simply proportional to the mechanical oscillation amplitude it can possibly realize. We have investigated an effective approach to add up the mechanical energy of oscillation for the OMS, so that it can realize broadband OMOFCs under feasible conditions. It is to operate the OMS in the regime of unresolved sideband and pump it by a two-tone field with the frequency difference of the drive tones being close to its intrinsic mechanical frequency. Then the system will be in a nonlinear resonance that quickly enlarges the mechanical oscillation amplitude and, purely through the intrinsic mechanical damping, it will stabilize to a final mechanical oscillation of significantly enhanced amplitude. Such enhanced mechanical oscillation amplitude can be increased further simply by choosing the proper detunings of the drive tones. A better operation point in the parameter space of the system will continually improve the generated OMOFC. With the experimentally feasible setups based on the mechanism, frequency comb lines of more than ${10}^4$ can be realized under pump powers of the order of mW. A very useful feature of the mentioned nonlinear resonance is that the mechanical oscillation frequency can be well locked to the frequency difference of the two drive tones. The resulting OMOFC pulse repetition rate and comb teeth spacing can be thus locked without any change by the pump power. It is therefore possible to achieve good stability of the generated OMOFC by controlling the frequency fluctuations of two drive tones. The currently illustrated scenario, which is implementable by a specified two-tone pump, may reduce the distance to the real applications of OMOFCs.

Acknowledgment. The authors thank Dr. Ming Li for helpful discussions.