Fig. 1. Generic setup for generating optomechanical frequency combs. (a) A two-tone pump laser field drives the optical cavity with a movable mirror as the mechanical resonator. If the frequency difference |ω1−ω2| of the two tones gets close to the mechanical frequency ωm, a special nonlinear resonance of optomechanics can be induced, leading to a significantly enhanced mechanical oscillation and the corresponding broadband cavity field as the frequency comb. (b) Typical spectrum of the driving field and the intracavity field with high-order sidebands generated by the consecutive interaction with the mechanical mode, which is quantitatively described by Eq. (1).

Fig. 2. Self-organized optomechanical resonance and its associated mechanical frequency locking. (a) Emergence of the resonance under the condition δ=0 in Eq. (6). The increased mechanical energy is at a rate much higher than the corresponding one under a single-tone pump of the same power. (b1), (b2) The stabilized mechanical oscillation (the red solid curve) is compared with a reference oscillation at the frequency ωm (the blue dashed curve)—the single-tone scenario shows their phase difference accumulated from a slight difference in their frequencies, but the two-tone scenario shows their exact overlap due to a frequency locking (the system has completely stabilized during the illustrated period of time, as the evolution time has been much larger than γm−1). Moreover, the mechanical frequency shifts due to optical spring effect are obvious in the scenario of single-tone pump, in contrast to the completely locked mechanical frequency at the intrinsic one ωm by the two-tone pump (the peaks of the mechanical spectrum for all different drive amplitudes E are exactly at the point ω=ωm). In the unresolved sideband regime where the stabilized cavity field is pulsed, a higher pump power gives rise to less spring effect for the system driven by a single-tone pump, because the cavity field pulse becomes narrower. The used parameters are gm=10−5κ, ωm=0.2κ, γm=10−5κ. In (a) the drive amplitude is 2E=105κ.

Fig. 3. Comparisons of the stabilized OMOFCs generated by a single-tone field at Δ=−ωm and a two-tone field with Δ1=−ωm and Δ2=0. Given the same pump power, the comb span of the latter is obviously much wider. The inset in each panel shows the stabilized cavity field intensity corresponding to the stabilized mechanical energy Em. As the pump amplitude E is increased, the illustrated field pulses become narrower, having a broader span of the generated OMOFC. The system parameters are chosen as gm=10−5κ, ωm=0.2κ, and γm=10−5κ.

Fig. 4. Comparison of the realized mechanical oscillation amplitudes by a single-tone pump and a corresponding two-tone pump. In the two-tone scenario the optomechanical oscillation exists with tiny amplitude near E=0. In the single-tone scenario, on the other hand, a Hopf bifurcation at E≈880κ starts the oscillation from Am=0, and it is evidenced by a critical slowing-down near the point [in Fig. 3(a1) the stabilization process close to the bifurcation has been rather slow]. Here the system parameters are the same as those in Fig. 3.

Fig. 5. Illustration of the effect by a parallel displacement of two tones. From (a) to (b), both tones are displaced to more blue-detuned side by an amount of 20κ, so that the comb span becomes broadened. The peak of the spectrum in (b) is shown to be shifted to ω=ωc+20κ, and the amplitudes of all sidebands less than this peak frequency are enhanced. One result of the process is that the indicated first sideband magnitude A1 on the right side of (a) is increased to the one on the right side of (b). Here the drive amplitude is 2E=4×105κ, with all other parameters being the same as those in Fig. 3.

Fig. 6. Induced mechanical amplitude Am under frequency displacement δω of two drive tones for two different mechanical frequencies and two drive amplitudes. A suitable displacement can efficiently enhance the mechanical amplitude. In the two upper panels, the amplitude Am reaches the maximum at δω=−32κ for ωm=0.2κ and at δω=−29κ for ωm=0.9κ. Given a higher pump power as in the lower panels, the possible displacement range can be extended further to reach an optimum point, for example, δω=−137κ for ωm=0.2κ and δω=−116κ for ωm=0.9κ (the curves extending to the left sides will be broken and drop down at these points). The fixed parameters are the same as those in Fig. 3.

Fig. 7. Realizable mechanical oscillation amplitudes Am due to varied drive amplitude (a) or different mechanical frequencies (b). A large mechanical amplitude is possible with a high pump power or a small mechanical frequency. The fixed system parameters adopted in these simulations are gm=10−5κ, Δ1=0, Δ2=−ωm, and Q=ωm/γm=1500.

Fig. 8. Possible consequences of imperfect frequency tone matching. The comb span or bandwidth is seen to be reduced by a larger error δ in Eq. (6). From (a) to (c), the distance between the comb teeth is ωm+δ. After the error becomes as large as δ=0.02κ, however, different sets of sidebands will be generated. These results are based on the system in Fig. 3 by fixing the drive amplitude of the two tones at 2E=4×105κ.