Injection Locking on a GHz NOEMS Platform

Introduction

Optoelectronic circuits support self-sustained oscillations resulting in high-Q resonances that gives rise to stable frequency reference with very low phase noise. The self-oscillations are generated through long optical fibers which induces large delays required for such implementation. The typical length of these fibers can extend up to few kilometers thus sacrificing the compactness of the devices. The group of Rémy Braive at C2N (Center for Nanoscience and Nanotechnology) at the University of Paris-Saclay are currently working on developing self-sustained oscillators based NOEMS platforms (Nano-Opto-Electro-Mechanical Systems) that can sustain modes at GHz frequency with low phase noise. The motivation of the work is to map low phase noise microwave reference on to optical domain using a GHz frequency mechanical resonance as the mediator, while being extremely compact in size. The transduction is based on the so-called principle of optomechanics [1], which allows coherent energy exchange between an optical and a mechanical element, thereby enabling mechanical resonance imprints on the optical signal. Recently, we had the privilege to visit the Braive group at C2N located in Saclay, where we were able to perform such measurements with the SHFLI 8.5 GHz Lock-in Amplifier.

Injection Locking

To achieve such stable frequency references, the group uses the concept of injection locking which follows the so-called Adler equation [2] describing an universal class of oscillators named self-sustained oscillators. The basis of this principle is shown in Fig. 1; A free running oscillator at \(f_0\) is “pulled“ when an AC signal is injected in the circuit. The injection locking has two principal effects: 

• Decreases the phase noise significantly compared to the free running oscillator. 
• A free running oscillator suffers from drifts due to environmental effects. While the injection locking helps improve this long term stability which can be demonstrated by variance measurements [1].

Typically a free running oscillator is susceptible to strong phase noise indicated in the Fig. 1 by wide dark blue Gaussian with a typical resonance frequency of \(f_0\). Injection locking to an external signal with a phase noise significantly less than the free running one allows to create a reference with remarkably lesser phase noise. The performance enhancement of the resonance is indicated by the light blue Gaussians in Fig. 1. Apart from the NOEMS platform, recent implementations include NEMS system with single electron tunneling [3] and Josephson junctions [4]. A greater locking-range results in a larger frequency range where the reference can be operated. This locking range is a function of the injected ac drive (also know as Arnolds Tongue) and can be determined by looking at either the amplitude or phase response of the circuit.

As mentioned previously, a signature of injection locking is to track the phase response of the resonator as the frequency of the injected drive is swept. Inside the locking regime, the phase of the resonator changes linearly with the oscillator frequency indicating a constant phase difference between the resonance and AC drive. The second signature is a constant resonance amplitude across the locking region which can of course be subjected to amplitude noise. Using a lock-in amplifier to measure these features would be the automatic choice, as it provides direct access to these two parameters. The resonance frequency optomechanical system used in the Braive group lies around 3 GHz; This typically is a frequency range that was not covered by commercial lock-in amplifiers before Zurich Instruments GHz lock-in amplifier, SHFLI.

Experimental Set-up and First Measurements

The sample consists of a heterogeneously integrated GaP one dimensional optomechanical crystal on silicon-on-insulator wafer (Figure 2). A Scanning Electron Micrograph (SEM) image of the chip is shown with a very simplistic experimental set-up. The optomechanical resonator is excited using the RF output of the SHFLI directly connected to the electrodes shown in yellow in the SEM image. Alternatively, it would also be possible to excite the resonator optically with an optical modulator controlled directly with the SHFLI. The readout is done by coupling the laser through optical grating below the beam resonator using transmission spectroscopy. The transmitted power is modulated due to the optomechanical coupling between the waveguide and the beam resonator. This modulation is at the mechanical frequency of about 3 GHz. The transmitted laser is then sent to a photodetector with appropriate bandwidth which is connected directly to the input of the SHFLI.

We start by looking at the Brownian motion of the fundamental breathing mode. This can be done either by using the Sweeper module to sweep a very wide range of frequency or by using the Spectrum Analyzer tool while defining a central frequency with an analysis bandwidth. The optomechanical interaction results in this Brownian motion converted into sidebands around the optical frequency. These sidebands can be directly measured by plugging the transmitted optical signal into a Photodetector followed by a lock-in detector set to the mechanical resonator eigen-frequency. In this particular case, since the frequency of the mechanical mode is well known, the spectrum analyzer tool is used with central frequency at around 3.1 GHz with a analysis bandwidth high enough to capture the eigenmode. The measured frequency response is shown in Figure 3. This makes possible to measure directly the motional sideband from the mechanical resonator around 3.081 GHz. Interested readers can follow this blog post on a more detailed discussion about such sideband measurements.

Injection Locking of the Optomechanical Resonator

The knowledge of the resonator frequency response makes it now possible to inject an AC drive in the vicinity of this resonance which directly locks to the injected signal. The RF drive is injected directly through the SHFLI output to the on-chip electrodes. When the drive crosses a certain threshold, the resonator is "locked" to the injected drive. This is usually observed by a narrowing linewidth of the resonator. To appreciate the locking range, sweeper tool inside the LabOne software is utilized. For a fixed drive amplitude, the frequency is swept around the resonance. For a RF drive of 100 mV, the locked regime can be seen both in amplitude and phase by the blue and the purple plots as shown in Fig. 4 (a) and (b). Inside the locking range, the amplitude response stays constant while the phase changes linearly as the frequency is swept. These are direct indication of injection locking. Increasing the AC drive to 120 mV results in a broader locking range as shown by the blue line in both amplitude and phase response. The wiggles around 3.077 GHz could be related to the very high Q-factor of the self-sustained resonance. Further discussion on this topic is out of scope of this blog.

Conclusion

In this blog post we discuss the measurement of injection locking of an optomechanical resonator with an injected AC drive. The SHFLI Lock-in Amplifier was used to generate the driving signal while simultaneously tracking the amplitude and phase response at GHz frequency allowing a full understanding of the injection locking phenomenon. The Sweeper tool enables a direct measurement of the amplitude and phase response of the optomechanical resonator as the frequency is swept. All these are done with a single instrument without any frequency up/down conversion and without the need of separate instruments for analysis such as VNA or Spectrum analyzer.

Acknowledgements

We thank Robert Horavth, Gregorio Beltramo and Rémy Braive for the possibility to perform the measurements in their lab and for sharing the data. The measurements were done together with my colleague Romain Stomp.

References

1. Horvath et al. “Sub-Hz Closed-Loop Electro-Optomechanical Oscillator with Gallium Phosphide Photonic Crystal Integrated on SoI Circuitry“. ACS Photonics 10, 2540 - 2548 (2023)
2. R. Adler. Proceedings of the IRE 34, 351 (1946).
3. Wen et al. “A coherent nanomechanical oscillator driven by single-electron tunneling“. Nature Physics 16, 75-82 (2020).
4. Danner et al. “Injection locking and synchronization in Josephson photonics devices“. Physical Review B 104, 054517 (2021).