Fig. 1. Conceptual illustration of the coherent microwave interference-assisted interrogation technique for optical Vernier sensors. (a) Schematic diagram of the system. (b) Illustration of the signal evolution processes.

Fig. 2. Conceptual illustration of the working principle of the proposed interrogation scheme for optical Vernier sensors. (a) Superposition of two RF responses from two individual interferometers employed to constitute the Vernier sensor and the resultant RF response of the Vernier sensor. (b) Illustration of the evolution of the magnitude of the Vernier sensor’s RF response using the vector sum operation for phasors with respect to frequency. Three different cases are depicted, corresponding to three different frequencies indicated in (a) with the red circle, square, and triangle.

Fig. 3. System characterization. (a) Schematic diagram of the experimental setup. The inset gives an enlarged view of the structure diagram of the air-gap FPI fabricated by fusion splicing a short section of hollow-core fiber (HCF) in between two single-mode fibers (SMFs). The inner diameter of the HCF is 50 μm and the outer diameter is 125 μm, matching with the size of the SMF. A schematic diagram of the strain-applying apparatus is also given. (b) Measured optical interferograms of the sensing and reference FPIs. (c) Measured optical reflection of the Vernier sensor, where the two FPIs are arranged in parallel to generate the Vernier effect. A typical amplitude-modulation signal is obtained. (d) Measured magnitude spectra in the RF response of the sensing and reference FPIs. (e) Measured magnitude spectrum of the Vernier sensor in the RF response. Two passbands are observed, corresponding to the two FPIs. Importantly, a sharp notch is revealed in the magnitude spectrum of the Vernier sensor, where the dip is denoted as the E-point and exhibits ultra-high sensitivity, as demonstrated later.

Fig. 4. Static strain sensing. (a) Measured magnitude spectra of the Vernier sensor for different settings of tensile strains applied to the sensing FPI. (b) Dip magnitude as a function of strain. The fitting model is y=−12.79e−0.0436x−49.19e−0.004422x, with a correlation coefficient of 0.9997. (c) Magnitude sensitivity with respect to strain. (d) Measured relative phase spectra of the Vernier sensor for different settings of tensile strains. (e) Relative phase at frequency of 0.350 GHz as a function of strain. The fitting model is y=0.7131e0.004606x−0.7147e−0.04622x, with a correlation coefficient of 0.9999. (f) Phase sensitivity with respect to strain.

Fig. 5. Dynamic sensing using the CW mode of the VNA with an operating frequency of 0.349 GHz. (a) Measured magnitudes with increasing strain, with ten measurements performed at each strain setting. The inset displays the calculated standard deviation of the ten measurements for each strain setting. (b) Averaged magnitude as a function of strain, fitted using the mode y=−10.70e−0.0583x−51.15e−0.0048x, with a correlation coefficient of 0.9993. (c) Standard deviation of the magnitude and sampling time as functions of the VNA’s IF bandwidth. (d) Dynamic sensing of external perturbations, demonstrating the system’s response with a point sampling time of 0.01 s.

Fig. 6. Numerical validation of the proposed sensing scheme. (a) Calculated RF responses of the Vernier sensor for different tensile strains applied to the sensing FPI. (b) Dip magnitude of the notch as a function of strain. (c) Calculated change in magnitude of the notch as a function of the change in cavity length of the sensing FPI within an extended range of 3000 nm. (d) Calculated sensitivity as a function of the change in cavity length of the sensing FPI. The inset provides an enlarged view of the sensitivity evolution. The purple circle marks the point where the sign of the sensitivity changes, indicating the limited dynamic range of the scheme for sensing. This point corresponds to the point marked by a purple square in (c).

Fig. 7. Measured magnitude over a 30-min stability test at a strain setting of 50  με.

Fig. 8. Strain sensing using a single sensing FPI with conventional MWP filter interrogation. (a) Measured frequency responses for various tensile strain settings. (b) Enlarged view of the spectra at around 0.395 GHz. (c) Peak frequency of the passband plotted as a function of strain. (d) Magnitude at around 0.395 GHz as a function of strain, with the magnitude averaged over a 1.4 MHz frequency window.