A research team led by Associate Professor Lei Shao at the University of Michigan – Shanghai Jiao Tong University Joint Institute (UM-SJTU JI, JI hereafter), has published their latest research findings in Physical Review Letters, a leading journal in the field of physics. The paper, titled “Self-Injection Locked and Phase Offset-Free Micromechanical Frequency Combs,” introduces a self-injection locking mechanism for micromechanical frequency combs, offering significant advancements in nonlinear dynamics and precision clock signal sources. The study, for the first time, reveals the self-injection locking mechanism of micromechanical frequency combs, addressing fundamental mechanical science issues such as spectral expansion, phase coherence, self-locking and phase offset elimination in the time domain.
JI Ph.D. student Jiahao Wu is the first author of the paper, with Lei Shao and Professor Wenming Zhang from SJTU’s School of Mechanical Engineering serving as co-corresponding authors.
Figure 1: Schematic Illustration of Micromechanical Frequency Combs
Nonlinear micromechanical resonators integrated on a chip can generate micromechanical frequency combs, enabling chip-scale precision timing with the potential to surpass conventional quartz oscillators and MEMS clock sources (Figure 1). However, the underlying dynamics remain insufficiently explored. Current micromechanical frequency combs exhibit narrow spectra, unstable and incoherent comb tooth phases, and a lack of synchronization and frequency-locking mechanisms, resulting in poor frequency stability and limiting their practical application in precision timing.
To address these challenges, the research team designed and established a “translational-torsional” strongly nonlinear coupled resonator model. Based on model predictions, they achieved the excitation of multiple clusters of high-order harmonic frequency combs. By fine-tuning and aligning the frequencies of adjacent clusters, they discovered a sudden merging of comb teeth, which triggers the self-injection locking mechanism. This resulted in an ultra-broadband, over-ten-octave supercontinuum micromechanical frequency comb. Within the locking range, the system’s phase space significantly converged, and vibrations remained highly stable. The frequency jitter and Allan deviation were both reduced by more than an order of magnitude, achieving frequency stability far superior to the excitation signal (Figure 2).
Figure 2: Ultra-Broadband and Self-Injection Locked Micromechanical Frequency Comb
In the time domain, the system exhibited significant fluctuations in its phase space trajectory under unlocked conditions, with a closed Poincaré section curve indicative of quasi-periodic motion in nonlinear dynamics. Envelope analysis showed a constant periodic difference between the signal and the yellow carrier envelope, leading to cumulative phase offsets and poor frequency stability. However, when tuned to the locked state, the orange discrete points formed eight distinct straight lines, indicating significant phase-space convergence and stable vibration amplitude. The Poincaré section evolved into eight discrete points, consistent with periodic motion. Time-domain envelope analysis confirmed a constant phase offset of zero, ensuring that all comb spectral lines were locked at integer multiples of the comb tooth spacing, resulting in exceptional frequency stability (Figure 3).
Figure 3: Phase Offset-Free and Highly Stable Time-Domain Signal of the Frequency Comb
This study represents the first realization of self-injection locking and high-stability micromechanical frequency combs. It reveals a wealth of dynamic properties, including ultra-broadband cascading, ultra-stable frequencies, ultra-low phase noise, and zero time-domain phase offset. The findings hold significant theoretical implications for the development of chip-scale ultra-precise clocks and ultra-precision sensing technologies.
The scientific research by the JI team was supported by the National Natural Science Foundation of China and the Shanghai Science and Technology Commission.
Author Profiles
Jiahao Wu is a Ph.D. student at JI specializing in the nonlinear dynamics of MEMS. His research has been published in Physical Review Letters, Nonlinear Dynamics, IEEE Journal of Microelectromechanical Systems, and flagship international conferences such as IEEE MEMS and Transducers.
