Phononic Frequency Comb via Intrinsic Three-Wave Mixing

Optical frequency combs have resulted in significant advances in optical frequency metrology and found wide applications in precise physical measurements and molecular fingerprinting. A direct analogue of frequency combs in the phononic or acoustic domain has not been reported to date. In this Letter, we report the first clear experimental evidence for a phononic frequency comb. We show that the phononic frequency comb is generated through the intrinsic coupling of a driven phonon mode with an autoparametrically excited subharmonic mode. The experiments depict the comb generation process evidenced by a spectral response consisting of equally spaced discrete and phase coherent comb lines. Through systematic experiments at different drive frequencies and amplitudes, we portray the well-connected process of phononic frequency comb formation and define the attributes to control the features associated with comb formation in such a system. In addition to the demonstration of frequency comb, the interplay between the nonlinear resonances and the well-known Duffing phenomenon is also observed.

Figure 1

Observation of the phononic frequency comb. (a) Left: Signal $S({\omega }_d)$ is applied on a free-free beam microstructure. Right: An intrinsic parametric excitation of the subharmonic mode (out of plane) at the drive power level 3 dBm of in-plane extensional mode. (b1) The pulse train corresponding to the phase coherent frequency comb at the drive power level 5 dBm, (b2) surface average displacement spectrum demonstrating comb formation, (b3) the subharmonic mode shape, (b4) the displacement at the node of the subharmonic mode indicating the absence of the frequency comb response, and (b5) the displacement at the antinode of the subharmonic mode indicating the presence of the frequency comb response.

Figure 2

High-order phononic frequency combs. (A) The drive power level-frequency contour indicating the thresholds $T_1$, $T_2$, $T_3$, $T_4$ for high-order frequency comb generation. (a)–(g) The out-of-plane displacement profiles at different frequencies on both sides of the dispersion curve. Here, Figs. 2 and 2 correspond to the nominal spatial vibration pattern of the comb mode, and Figs. 2 and 2 and 2 and 2 correspond to the spatial vibration pattern of the parametrically excited mode. rms: The rms displacement profiles in the frequency range 3.6–4.2 MHz at the drive frequency 3.862 MHz and drive power $\text{level}=4.5\mathrm{dBm}$. (B) The threshold $T_1$ as the drive frequency is detuned from resonance.

Figure 3

Interplay between the DNR and Duffing mechanisms. (a) The drive power level-frequency contour (drive $\text{frequency}=3.862\mathrm{MHz}$) indicating the DNR and DNR-Duffing regimes. (b), (c) The drive frequency-sense frequency contour (five comb lines) in the DNR and DNR-Duffing regimes, respectively. (d) The comb spacing with frequency renormalization (${\omega }_d-\widetilde{{\omega }_0}$) vs the comb spacing without frequency renormalization (${\omega }_d-{\omega }_0$) indicating the DNR and Duffing regimes.

Figure 4

Regime of the phononic comb formation. (a) Phononic comb regime falling just outside the dispersion band of the mechanical mode. (b), (c) The rms displacement profiles in the frequency range 3.6–4.2 MHz at different drive frequencies (drive power $\text{level}=6\mathrm{dBm}$) and drive amplitudes (${\omega }_d=3.862\mathrm{MHz}$), respectively.