In recent years, significant endeavors have been dedicated to enhancing the measurement performance of optical fiber sensors [1–3]. These efforts encompass the utilization of new functional materials [4,5], specialty and micro-structured optical fibers [6,7], as well as the exploration of innovative sensing configurations [8,9]. Among these approaches, the optical Vernier effect has garnered substantial research attention and found widespread applications as a means to augment the measurement sensitivity of optical fiber interferometric sensors [10–12]. To generate the Vernier effect, it is necessary to integrate two interferometers with slightly different optical path lengths (OPLs) into a system, either in parallel or in series. Most of the research efforts have focused on the physical implementation of the optical Vernier effect, achieved by employing either two different types of interferometers or two instances of the same type of interferometer [12]. The literature showcases successful demonstrations of sensitivity-enhanced measurements for a wide range of physical, chemical, and biological parameters, achieving magnification factors of up to several hundred based on the Vernier effect [13–20].

Unlike an optical fiber interferometer-based sensor, where the interferogram shift is directly tracked for sensing purposes, the Vernier effect-based optical fiber sensor utilizes the shift of the Vernier envelope as the reference variable to enhance sensitivity. Notably, the free spectral range (FSR) of the Vernier envelope is considerably larger than the FSR of the single interferometer that comprises the Vernier sensor system. Consequently, a light source with a broad wavelength range (typically spanning tens of nanometers at a minimum) becomes a system requirement, with a high-resolution benchtop optical spectrum analyzer (OSA) serving as the detector. Furthermore, extracting the Vernier envelope necessitates multiple nonlinear approximations and curve fitting. Consequently, while the Vernier effect enhances sensitivity, it comes with increased system hardware costs and heightened complexity in signal demodulation. It is important to note that the resolution of a Vernier sensor may even degrade in certain scenarios where the Vernier envelope becomes distorted [21,22]. Thus, some recent efforts have focused on enhancing Vernier effect-based optical fiber sensors through two key approaches: simplifying the system hardware by introducing a virtual reference interferometer [23–25] and employing machine-learning techniques for signal analysis [26–28]. However, these approaches still rely on accurate optical measurement, and the fundamental limitations of the Vernier sensor system are not adequately addressed.

Microwave photonics (MWP), an interdisciplinary field that integrates the advantages of both optics and microwave technologies, has been explored for the interrogation of optical fiber sensors [29,30]. By converting variations in optical signals into corresponding changes in microwave signals using MWP and leveraging high-fidelity electronic instruments, it becomes feasible to achieve higher sensitivity, enhanced resolution, and faster demodulation speeds [31]. Recently, a variant of the optical Vernier effect, known as the MWP Vernier effect, has been reported to enhance the sensitivity of MWP interferometric sensors [32–37]. However, it is worth noting that these sensors typically have a sensor gauge length that is orders of magnitude larger than that of optical-domain fiber interferometric sensors. The relatively large size of these sensors limits their applicability primarily to scenarios where localized sensing is not required [15]. Also, the signal demodulation process is quite intricate and laborious [38], similar to the case of optical Vernier-effect-based sensors, as mentioned previously.

In this work, we propose and demonstrate an innovative interrogation approach for Vernier-effect-based optical fiber sensors using MWP. Instead of directly acquiring the optical spectrum response of a Vernier-effect-based optical fiber sensor, we measure and track the microwave frequency response of the Vernier sensor for sensing applications. Importantly, a coherent microwave interference-assisted detection scheme is utilized, generating a characteristic notch in the frequency response that is highly sensitive to external perturbations. Monitoring this notch in the frequency response is straightforward and avoids the intricate signal processing required for accurately extracting the Vernier envelope in conventional optical domain-based interrogation. This approach promises to streamline the measurement process while enhancing the accuracy and reliability of optical Vernier sensors. The underlying physics of the characteristic notch generation and the associated sensing principle are described in detail. As a proof of concept, a Vernier sensor composed of two parallel air-gap Fabry–Perot interferometers (FPIs) is employed for strain sensing, demonstrating high sensitivity and ease of signal readout.

Figure 1(a) illustrates the fundamental concept of the coherent microwave interference-assisted interrogation technique for sensitivity-enhanced fiber sensing. The core idea is to read the optical Vernier effect in the microwave domain. Probe light from a broadband light source is launched into a Vernier-effect-based optical fiber sensor, which typically comprises two individual interferometers with slightly detuned optical frequencies (i.e., different FSRs). One interferometer functions as the sensing device while the other serves as the reference to generate the Vernier effect, thereby improving sensitivity. The transmission or reflection (depending on the sensor configuration) from the sensor is directed into an electro-optic modulator (EOM), where a microwave signal is modulated onto the optical carrier. The amplitude-modulated light then passes through a delay line and reaches a high-speed photodetector (PD). The obtained electrical signal is synchronously detected at the microwave modulation frequency, allowing both amplitude and phase to be acquired. By tuning the microwave frequency over a desired range, the complex frequency response of the Vernier sensor can be determined and subsequently employed for sensing.

The fundamental basis of the proposed interrogation technique relies on photonic-assisted microwave processing, specifically the single-passband microwave filter [39]. The broadband optical signal is transmitted through the Vernier sensor to obtain optical samples with a continuous sample weight distribution across the wavelength observation span. Importantly, the obtained continuous optical spectrum contains two dominant frequencies determined by the OPLs of the two interferometers constituting the Vernier sensor. These two frequencies generate two passbands in the system’s frequency response. Notably, these passbands superimpose coherently, generating a notch in the overlapped frequency passband. By carefully tuning the difference in the OPL between the two interferometers to satisfy the phase-matching condition, destructive interference between the associated microwave signals occurs, and the notch’s dip magnitude reaches its minimum. When an external perturbation is applied to the sensing interferometer in the Vernier sensor, the optical transmission or reflection of the sensor (i.e., the Vernier spectrum) changes, breaking the phase-matching condition and causing a significant variation in the notch. Thus, by monitoring variations in the coherent-superposition-resultant frequency response of the Vernier sensor, high-sensitivity sensing can be achieved. Figure 1(b) conceptually shows the signal evolution of the sensing system.

The functionality of the proposed system can be divided into two components: the sensing end and the signal detection end. First, the system utilizes an interferometer as the sensor head, ensuring high sensitivity and a small footprint for an in-fiber device. Second, the sensitivity of the sensing interferometer is amplified by employing the optical Vernier effect. In this effect, the interferogram of the sensing interferometer incoherently overlaps that of a reference interferometer with a slightly detuned OPL. The superimposed spectrum contains a virtual envelope, showing a magnified shift in response to an external perturbation compared to the shift of the sensing interferometer, forming the basis for conventional optical domain-based interrogation. In the proposed MWP-enabled approach, instead of incoherently superimposing the optical reflection from the two interferometers to generate the virtual Vernier envelope, the frequency responses of the two interferometers in the microwave domain are coherently overlapped. This coherent microwave interference-assisted detection enhances the system’s sensitivity by converting a minute change in the optical spectrum into a significant change in the generated microwave signal. Thus, the system transforms the optical Vernier effect into coherent interference of the associated microwave signals, paving the way for ultra-highly sensitive sensing.

Consider a typical Vernier sensor, generated by the parallel arrangement of two FPIs [14]. The reflection from the Vernier sensor, including the reference and sensing interferometer, can be expressed as R(ω)=S(ω)·{C−V1cos(ωc·2L1)−V2cos(ωc·2L2)+V3cos[ωc·2(L1−L2)]},(1)where ω is the angular frequency of the probe light; S(ω) is the optical spectrum of the light source; C is a constant; c is the speed of light in vacuum; L1 and L2 are the cavity length of the reference and sensing interferometer, respectively; V1, V2, V3 are constants, representing the visibility of each interference component. The overall frequency response of the Vernier sensor can be written as [39] H(Ω)=∫R(ω)[mH*(ω)·H(ω+Ω)+mH(ω)·H*(ω−Ω)]dω,(2)where H(ω) denotes the transfer function of the delay line, Ω is the angular frequency of the microwave modulation signal, and m is the modulation depth. The delay time introduced by the dispersive element (e.g., a long length of dispersion-compensating fiber, DCF) is sufficiently larger than the coherence time of the source, ensuring an incoherent operation. By tuning the microwave frequency, the frequency response over a certain range can be quantified. Considering that the reflection from the Vernier sensor essentially carries three different frequencies induced by the two interferometers, three passbands should be observed in the magnitude spectrum of the frequency response, and the three central frequencies can be predicted by f1=1DDCFFSR1,f2=1DDCFFSR2,f3=|FSR1−FSR2|DDCFFSR1FSR2,(3)where FSR1 and FSR2 are the FSR of the reference and sensing interferometers, respectively; DDCF is the dispersion coefficient of the DCF. Additionally, there will be an additional passband, corresponding to the DC component. To generate the Vernier effect in the optical domain, the two interferometers are slightly detuned, meaning FSR1 and FSR2 are close, leading to a large FSR of the Vernier envelope. Thus, f3 is typically small and falls into the DC passband region in the frequency response. It is worth noting that conventionally, the two FPIs can be used for two-point sensing by separately monitoring the corresponding changes in the central frequencies of the two corresponding passbands in the frequency response [40,41]. Here, we use the two FPIs to generate the Vernier effect in the optical domain and demonstrate a novel MWP scheme for the interrogation of optical-domain Vernier sensors. A detailed analysis of the novel method is given below.

For theoretical analysis of the radiofrequency (RF) signal generation and associated superposition, the system model is simplified by assuming that the third-order dispersion in the system is negligible [42], considering that the frequency of interest is below 1 GHz. Consequently, the RF response in Eq. (2) can be simplified to the expression H(Ω,ω0)=(m·ejβL2Ω2+m·e−jβL2Ω2)·e−jΩ[τ(ω0)+βLω0]·HRF(Ω),(4)where HRF(Ω)=∫R(ω)·ejβLωΩdω,(5)where ω0 is the central angular frequency of the optical spectrum; τ(ω0) is the group delay time at the central frequency; β is the second-order dispersion coefficient of the DCF; and L is the length of the DCF. HRF(Ω) represents the Fourier transform characteristics of the optical spectrum reflected by the Vernier sensor, with βLΩ serving as the Fourier transform variable corresponding to frequency ω. To further conduct theoretical analysis, the spectrum envelope of the light source S is adjusted based on a known envelope S1, which is symmetric about the midpoint of the spectrum. Then an even function is defined as S1′(ω)=S1(ω+ω0).(6)

Equation (1) can then be rewritten as R(ω)=S1′(ω−ω0)·{C−V1cos(ωc·2L1)−V2cos(ωc·2L2)+V3cos[ωc·2(L1−L2)]}.(7)

Then the Fourier transform of R(ω), denoted as R(ξ), can be derived, as given in Appendix A. Considering the equivalence between positive and negative frequency components in the Fourier transform results and neglecting the DC passband region, we focus on the positive frequency components in R(ξ); then the corresponding RF response can be expressed as H+(Ω)=−2πmcos(βL2Ω2)·eiΩ[τ(ω0)+βLω0]·[V1S1f′(βLΩ−2L1c)e−iω0(βLΩ−2L1c)+V2S1f′(βLΩ−2L2c)e−iω0(βLΩ−2L2c)],(8)where the real function S1f′ is the Fourier transform function of S1′. Variations in the magnitude of the frequency response H+(Ω) (denoted as M) can be analyzed by considering the sum of two vectors M1 and M2 (see Appendix A for details), as given by $$\begin{gathered} M=|M|eiφ=M1+M2,|M|=A12+A22+2A1A2cos(φ2−φ1)=A12+A22+2A1A2cos[2ω0c(L2−L1)], \\ φ=arctanA1sinφ1+A2sinφ2A1cosφ1+A2cosφ2, \end{gathered}$$(9)where A1 and A2 are the amplitudes of the two vectors, and φ1 and φ2 are the phase terms.

Notably, the two vectors M1 and M2 correspond to the two passbands generated by the two FPIs used to construct the Vernier sensor system. As mentioned before, the two interferometers should be slightly detuned with different FSRs to generate the Vernier effect. Hence, the corresponding two passbands with central frequencies of f1 and f2 in the RF response partially overlap in the microwave domain, as conceptually illustrated in Fig. 2(a). For the superimposed RF response (i.e., the RF response of the Vernier sensor), the magnitude is a result of the coherent superposition of the two FPIs’ RF responses across different frequencies. Importantly, the two RF response vectors exhibit varying amplitudes characterized by the magnitude spectrum of the RF response. The amplitude peaks at the center of the passband and gradually decreases as the frequency deviates from the peak frequency in each case. However, the two RF response vectors exhibit a constant phase difference described by the cavity length difference of the two FPIs involved in the system, as revealed in Eq. (9). Specifically, in the region where the two passbands overlap, as the frequency keeps increasing, the amplitude of M1 keeps decreasing while the amplitude of M2 keeps increasing, as conceptually illustrated in Fig. 2(b) in the form of a vector-sum operation using phasors. This leads to the initial decrease in the magnitude of M, reaching its minimal value, and then it increases with increasing frequencies, as shown in Fig. 2(a). Hence, in the magnitude spectrum of the Vernier sensor’s RF response, a notch is sustained with a dip frequency denoted as f0. Additionally, the phase of M also accumulates with frequency.

Hence, the optical domain Vernier spectrum is converted into a characteristic spectrum with two passbands and a sandwiched notch in the microwave domain. In terms of sensing, one of the two interferometers that constitute the Vernier sensor serves as the reference device, and the other serves as the sensing device. Subtle changes in the optical Vernier spectrum induced by an external perturbation on the sensing interferometer are then translated into significant changes in the notch of the RF frequency response. Specifically, consider the interferometer with a cavity length of L2 as the sensing interferometer. When the sensing cavity is subjected to an external perturbation, such as a tensile strain, the optical interferogram of the sensing interferometer shifts, leading to a change in the optical Vernier spectrum. This change in the optical spectrum is transferred to a variation in the RF response, where the notch changes due to the modification of M2 [see Eq. (A3) in Appendix A]. Therefore, the magnitude of the notch varies with the external perturbation exerted on the sensing device. Essentially, the system translates a phase change in the sensing interferometer into a change in the associated microwave signal via a coherent microwave interference-assisted scheme with the help of the reference interferometer, thus facilitating high-sensitivity measurements.

Let us consider the measurement sensitivity of the proposed scheme. Since M2 (including magnitude and phase) is a function of the measurand of interest via a modification to L2, Eq. (9) has no analytical solution. Therefore, a simplified analysis is employed, where the magnitudes of M1 and M2 (i.e., A1 and A2) are assumed to be the same constants (i.e., A0) within a small varying range of L2. Thus, Eq. (9) can be rewritten as |M|=2A0|cosφ2−φ12|=2A0|cos(ω0c(L2−L1))|.(10)

Taking the derivative of Eq. (10), the sensitivity can be obtained as ∂|M|∂L2=±2A0ω0csin[ω0c(L2−L1)].(11)

Thus, the sensitivity reaches its maximum when the following condition is satisfied: L2−L1=2k+14λ0,(12)where k is a non-negative integer and λ0 is the center wavelength of the optical interferogram. In other words, the notch offers the highest sensitivity when the phase-matching condition for destructive interference of the two RF signals is satisfied. Conversely, the sensitivity diminishes as it decays away from the phase-matching condition, thus restricting the dynamic range for sensing.

An amplified spontaneous emission (ASE) source with a wavelength bandwidth of 37 nm (from 1528 to 1565 nm) was employed as the light source. A vector network analyzer was used as the microwave source and receiver. A long length of dispersion-compensating fiber packaged in a module with a total dispersion parameter of −1400ps/nm @1545 nm is used as the dispersive element in the system to serve as a broadband delay line. The long-length DCF introduces sufficient delay time to ensure coherent microwave interference. Additionally, as indicated in Eq. (3), the passbands of the frequency response are determined by the dispersion parameter of the DCF, with larger dispersion values leading to lower passband frequencies. The DCF with a dispersion parameter of −1400ps/nm sets the system’s operating frequency to approximately 0.4 GHz, enabling more cost-effective, low-frequency measurements without compromising performance. Meanwhile, an OSA was included in the system only for the purpose of characterizing and visualizing the optical reflection from the sensor. A schematic diagram of the experimental setup is depicted in Fig. 3(a). As a proof of concept, two air-gap FPIs were utilized to construct a reflection-type Vernier effect sensor. The interferometers were fabricated by fusion splicing a short section of hollow-core fiber with a desired length between two single-mode fibers, as illustrated in the inset of Fig. 3(a). The cavity lengths of the sensing and reference interferometers were measured using a microscope to be 635 and 553 μm, respectively.

The characterization results of the system are expanded upon below. Figure 3(b) shows the measured optical reflection spectra obtained from the OSA for the sensing FPI and reference FPI. Periodic patterns can be observed in both spectra, demonstrating the expected optical interference. The FSR for the sensing and reference interferometers is found to be 1.88 and 2.16 nm, respectively. Meanwhile, through a home-developed algorithm, the absolute cavity lengths of the sensing and reference interferometers are calculated to be 636.038 and 553.744 μm, respectively. The measured optical reflection of the Vernier sensor is shown in Fig. 3(c), where a typical amplitude-modulation spectrum is observed in the optical reflection from the sensor, with a Vernier envelope showing an enlarged FSR of approximately 14.50 nm. Thus, a sensitivity magnification factor of 7 can be expected if the conventional optical-domain demodulation technique is adopted. However, by reading the optical Vernier effect in the microwave domain, the optical-domain Vernier amplification factor is no longer applicable.

The measured RF frequency responses of the two individual FPIs are given in Fig. 3(d). Passbands are observed in both cases, with slight distortions due to the third-order dispersion introduced by the DCF module in the system. The central frequencies in the RF response of the sensing and reference interferometers are found to be 0.381 and 0.331 GHz, respectively. These measured frequencies slightly deviate from the predicted values of 0.380 and 0.330 GHz from Eq. (3). These differences are mainly attributed to the estimation errors of the FSR of the two interferometers and the fact that the dispersion coefficient varies with wavelength. The measured RF response in magnitude of the Vernier sensor is shown in Fig. 3(e). As expected, a sharp notch is revealed in the RF response due to the coherent superposition of the two passbands generated by the two interferometers. Notably, during the measurement, a minute strain was applied to the sensing interferometer to tune its cavity length (i.e., L2) so that the phase-matching condition for destructive interference could be satisfied, as given in Eq. (12). At this condition, the magnitude of the notch reached a minimum, denoted as the E-point in Fig. 3(e), offering ultra-high sensitivity for sensing applications. Additionally, the optical reflection from the sensing interferometer was measured using the OSA, revealing a cavity length of 636.0973 μm. Substituting the cavity lengths of the two interferometers into Eq. (12) verifies the satisfaction of the phase-matching condition with k to be 106.

Strain sensing was then performed using the sensing FPI via a two-stage assisted approach, as depicted in the inset of Fig. 3(a). The FPI sensor was placed between two stages with a separation distance of 1 m. Tensile strains were incrementally applied to the sensing interferometer in steps of 10με by translating the right stage outward with a step size of 10 μm, the highest readable resolution of the stage. The reference FPI was secured in a foam box to isolate it from environmental noise. The frequency response of the Vernier sensor was acquired around the E-point at each setting of tensile strains, as plotted in Fig. 4(a). The VNA was configured with 5000 sampling points and an intermediate frequency (IF) bandwidth of 500 Hz, resulting in an acquisition time of approximately 10 s per spectrum. The magnitude of the E-point drastically increased with increasing tensile strains. This is because as tensile strains were incrementally applied to the sensing FPI, the value of L2 increased, breaking the phase-matching condition given in Eq. (12) so that the destructive interference no longer held. Consequently, the magnitude of the superimposed spectrum kept increasing as L2 increased, deviating further from the destructive interference point. Meanwhile, the rate of magnitude increase diminished as L2 continued to increase, as expected. Interestingly, the passband corresponding to the sensing FPI showed little variation, demonstrating its low sensitivity to subtle changes in L2, as detailed in Section 4. The E-point magnitude as a function of applied tensile strain is plotted in Fig. 4(b). The discrete data points were fitted using an exponential model, as indicated in the figure, with a correlation coefficient of 0.9997. Although a nonlinear relationship is revealed, the Vernier sensor can be employed for sensing after proper calibration. Numerical investigations were also conducted, as detailed in Appendix B. The overall trends observed in both the experimental results and numerical simulations further validate the effectiveness of the proposed sensing scheme. Taking the derivative of the E-point magnitude with respect to the strain, the measurement sensitivity across the strain range can be obtained, as shown in Fig. 4(c).

As can be seen, the initial sensitivity at the destructive point reached 0.699 dB/με and gradually decreased. Considering that a 0.001 dB change in magnitude can be accurately quantified by a narrow-band detection-based VNA, the strain resolution of the system is estimated to be 1.4nε at the setting of destructive interference, 3 orders of magnitude higher than that of a typical fiber optic interferometer or fiber Bragg grating-based strain sensor device interrogated by an optical spectrometer with a wavelength resolution of 10 pm. Transforming the strain resolution to the cavity length resolution reveals a theoretical resolution of 0.89 pm for the FPI, with a cavity length of approximately 636 μm. However, in real measurements, system noise and environmental disturbances can cause fluctuations in the measured magnitude of the E-point, thereby increasing measurement uncertainty. To evaluate the system’s stability, we conducted a stability test in which the frequency response of the system was continuously recorded every minute over a 30-min period with a strain setting of 50 με. The standard deviation of the determined E-point magnitude was found to be 0.006 dB (see Appendix C). Considering the sensitivity of 0.274dB/με at a strain setting of 50με [see Fig. 4(c)], the measurement uncertainty is estimated to be 0.022με, which corresponds to a cavity length uncertainty of 0.014 nm. In previous work, two FPI-based Vernier sensors are employed for strain sensing using the conventional optical-domain interrogation approach. However, this method involves complex signal processing and imposes stringent requirements on sensor fabrication, particularly in controlling the length detuning between the sensing and reference FPIs to achieve high sensitivity magnification. The measurement accuracy for these devices has been rarely reported, with most studies focusing on sensitivity and magnification factors. In fact, applying the Vernier effect can even degrade measurement accuracy under certain conditions [21]. The proposed approach fundamentally differs from the conventional optical-domain method by eliminating the need for cumbersome signal processing, thereby enabling enhanced sensing performance. Table 1 compares the performance metrics of the proposed system with those of previous work. As shown, the proposed approach excels in both theoretical and experimentally measured resolution, making it a promising alternative to conventional optical-domain interrogation systems. By integrating more advanced sensing FPIs into the system, the strain sensitivity could be further enhanced. On the other hand, the unique advantage of the proposed method also lies in its ease of interrogation, which facilitates direct dynamic sensing, as demonstrated later. This capability could effectively broaden the application scope of Vernier sensors, enabling their use in highly sensitive dynamic sensing applications such as acoustic detection.Table 1.

Comparison of the Proposed Sensor System with Previous Optically Interrogated Vernier Systems^a

Additionally, it is worth noting that power fluctuations in the optical source can result in variations in the characteristic notch’s magnitude within the frequency response, potentially affecting the system’s measurement accuracy. To mitigate this issue, an additional reference variable can be introduced to compensate for these fluctuations. For example, the magnitude of the DC component in the frequency response could serve as a reliable reference to normalize the measured notch, ensuring consistent and accurate performance.

In addition to magnitude, the phase spectra of the Vernier sensor’s RF responses were also recorded by the VNA during the strain test. Figure 4(d) presents the measured relative phase spectra for different tensile strain settings. Each relative phase spectrum was obtained by subtracting the phase spectrum of the system at an initial setting (i.e., 0με). An abrupt change was observed in the relative phase spectrum, corresponding to the dip in the magnitude spectrum, attributed to the coherent superposition of the two RF passbands. In both low- and high-frequency regions relative to the E-point, the absolute value of the phase increased with increasing tensile strains, demonstrating the accumulation of phase in the notch. The phase in the high-frequency region shows higher sensitivity because a change in L2 directly modifies the phase information of the high-frequency passband. Taking the frequency 0.350 GHz as an example, Fig. 4(e) plots the relative phase as a function of tensile strain. A similar exponential dependence of the phase on the tensile strain is revealed through curve fitting, as indicated in the figure. Phase measurement sensitivities with respect to strain are shown in Fig. 4(f). Again, a decrease in sensitivity with increasing strains is demonstrated, similar to the magnitude response. Given the phase resolution of the VNA to be 0.01 deg (1.75×10−4rad), the theoretical strain resolution using phase measurements is estimated to be 6nε at the destructive interference point. The utilization of phase for sensing applications introduces new possibilities for further sensitivity amplification towards higher measurement resolution, for example, based on the phase-shift amplification technique [47,48]. Meanwhile, the phase information provides an additional reference variable that can potentially be leveraged to enhance the system’s functionality, such as enabling dual-parameter sensing by decoupling the magnitude and phase responses. However, as indicated in Eqs. (9), (A2), and (A3), the phase is also influenced by the group delay time of the DCF. Consequently, environmental factors that affect the group delay of the DCF, such as temperature or mechanical perturbations, may introduce variations in the phase information. These variations could increase the measurement uncertainty for phase-based sensing approaches.

In the calibration experiment just discussed, the frequency response over a broad range, including the notch, was measured in response to external perturbations applied to the sensing FPI. Acquiring such a wide-band frequency response is relatively time-consuming, comparable to obtaining the optical spectrum of a Vernier sensor using an OSA, which limits the system’s dynamic sensing capability. In this section, we demonstrate the effectiveness of single-frequency operation in enabling high-sensitivity dynamic sensing. The VNA was configured in a continuous wave (CW) mode, with the operating frequency set to the notch frequency at 0.349 GHz. The IF bandwidth of the VNA was reduced to 10 Hz, corresponding to a point sampling time of approximately 0.1 s. The strain test was repeated under this CW configuration, and at each strain setting, ten measurements of the magnitude at 0.349 GHz were recorded.

Figure 5(a) illustrates the measured magnitudes as the applied strain on the sensing FPI increased to 100με in increments of 10με. Clear step-like responses are observed, demonstrating the reliable performance of the CW-based interrogation method, which aligns well with the results shown in Fig. 4(a). The inset in Fig. 5(a) displays the calculated standard deviations for the ten measurements at each strain setting. The largest deviation, 0.011 dB, occurred at the pre-strain stage due to the high sensitivity and relatively low magnitude at the initial strain setting. Figure 5(b) shows the average magnitude of each group of ten measurements as a function of strain, along with an exponential curve fit. The fitted model differs slightly from that in Fig. 4(b), which may be attributed to experimental errors caused by the limited resolution and accuracy of the translation stage used in the experiment. It is worth noting that in CW mode, the accuracy of magnitude measurements depends heavily on the IF bandwidth setting of the VNA. A smaller IF bandwidth generally improves measurement precision but increases the acquisition time for each sampling point. Figure 5(c) presents the measured standard deviations in magnitude and the corresponding sampling times as a function of the IF bandwidth. At each IF bandwidth setting, 100 sampling points were recorded at a strain setting of 50με. The results highlight a trade-off between measurement precision and acquisition time. To evaluate the system’s dynamic sensing capability, an additional test was performed in which another piece of fiber was used to perturb the stressed sensing fiber to simulate dynamic events. The IF bandwidth of the VNA was increased to 100 Hz, resulting in a point sampling time of ∼0.01s, to capture these rapid perturbations. As shown in Fig. 5(d), two dynamic events were successfully recorded, with a peak strain of approximately 2.5με, demonstrating the system’s ability to perform high-sensitivity dynamic sensing. It is important to note that in real-world dynamic measurements, a VNA is not necessary. Once the operating frequency (i.e., the notch frequency) is determined, a single-frequency oscillator combined with a power meter can replace the VNA, simplifying the system and reducing costs. This dynamic sensing capability enabled by the proposed approach broadens the application scope of Vernier sensors, making them suitable for high-sensitivity, high-speed sensing tasks such as vibration and acoustic detection.

We have theoretically analyzed and experimentally validated the effectiveness of the microwave interference-assisted interrogation for a fiber optic Vernier sensor, demonstrating both high sensitivity and ease of signal tracking. On initial examination, this system may seem analogous to MWP single passband filter systems, which have recently been employed for the interrogation of single fiber optic interferometers in various sensing applications [40,49–52]. However, the underlying working principle is fundamentally different, leveraging the coherent superposition of two interferometers’ passbands in the frequency domain rather than the single interferometer’s passband resonance employed by the conventional MWP filter systems. More importantly, the sensitivity achieved by the introduced interrogation technique is orders of magnitude greater than that of MWP filter-based measurements. Table 2 summarizes the comparison between the proposed system and representative MWP filter-based FPI sensing systems.Table 2.

Comparison of the Proposed Sensor System with Previous MWP Filter-Based FPI Sensor Systems^a

These sensors operate by leveraging the dependence of the passband center frequency in the system’s frequency response on the FSR (or OPL) of the interferometer, tailored for different sensing applications. Let us re-examine Eq. (3), which describes the relationship between the center frequency and the FSR of the interferometer. Equation (3) can be rewritten as f=1DDCFFSR=1DDCF·OPLλ2,(13)where λ is the center wavelength of the optical interferogram. Taking the derivative of Eq. (13) with respect to OPL gives the frequency sensitivity, given by ∂f∂OPL=1DDCF·1λ2.(14)

Considering a dispersive module with a dispersion coefficient of −1400ps/nm at 1545 nm and a center wavelength of 1546.5 nm, the frequency sensitivity is calculated to be 300 Hz/nm. The associated third-order dispersion introduced by this dispersive element broadens and distorts the passband in the RF response, complicating the localization of the passband’s center frequency and limiting frequency accuracy to tens of kHz. Consequently, the resolution of frequency-tracking-based MWP filtering interrogation is quite limited.

To further illustrate the inefficiency of this method, we performed an additional strain test. We removed the reference interferometer and retained only the sensing FPI, applying tensile strains incrementally in steps of 10με over a range of 40με, which corresponds to a step-size cavity length change of 6.36 nm. We measured the RF response of the modified system and determined the peak frequencies of the passband for various tensile strain settings, as shown in Appendix D. The results revealed a non-monotonic relationship between peak frequency and applied strain, confirming the method’s limited resolution, which was worse than the investigated limit of 40με (i.e., 25.44 nm change in cavity length and 50.88 nm in terms of a change in OPL). On the other hand, the magnitude at the frequency of approximately 0.395 GHz showed a linear dependence on the applied strain, albeit with a limited sensitivity of −0.0011dB/με. Therefore, the proposed scheme offers a more efficient and effective approach to enhance the performance of conventional MWP single passband filter-based sensing systems. Instead of using a single interferometer to generate the RF response and also act as the sensing element, two interferometers should be employed in the improved system, where one serves as the sensing device and the other as the reference device. The coherent superposition of the RF responses from the two interferometers significantly improves sensitivity at the destructive interference point, as demonstrated earlier. Similar to the optical Vernier sensitivity amplification factor extensively reported in previous studies, the microwave interference-assisted system also achieves a significant sensitivity amplification compared to the single FPI-based MWP system. Based on the sensitivities shown in Fig. 4(c), the sensitivity amplification factor of the proposed system ranges from 122 to 635—orders of magnitude larger than the optical Vernier amplification factor of 7 mentioned earlier. Thus, the present novel approach is anticipated to advance MWP filter-based sensing techniques, opening new possibilities across a wide range of applications.

On the other hand, the optical Vernier effect has recently garnered extensive research interest and has been demonstrated as an effective modality to enhance the measurement sensitivity of fiber optic interferometric sensors. The interrogation of a Vernier sensor requires a broadband optical light source and a high-precision spectrometer to acquire the Vernier spectrum across a wide wavelength range. However, the associated cumbersome signal processing and potential degradation of sensing signal quality can lead to a real measurement resolution that is even lower than that of the individual interferometers constituting the Vernier sensor [22,25]. In this context, the proposed MWP-assisted interrogation approach provides an excellent solution to the complexities of existing optical Vernier sensor systems. First, the implementation of the interrogation system is cost-effective, as it eliminates the need for bulky OSAs. Moreover, the sensing signal is easy to observe and straightforward to demodulate, eliminating the complex and intricate signal-extraction procedures required in optical-domain interrogation and thereby reducing the potential for errors. Most importantly, the measurement sensitivity offered by the proposed MWP method is orders of magnitude higher than that of conventional methods, further pushing the limits of fiber optic interference-based sensing methodologies. Notably, the proposed scheme combines the advantages of both optical interferometry and microwave interferometry: the former provides higher sensitivity to the measurand of interest, while the latter offers greater stability and ease of control. Thus, transforming optical interference-based measurement into microwave interferometry-enabled coherent detection opens a new paradigm in ultra-sensitive sensing.

It is also worth noting that the highest sensitivity is achieved under the condition of destructive interference between two microwave signals. This property inherently limits the dynamic range of the proposed sensing system, as sensitivity gradually degrades when operating conditions deviate from the destructive interference point. Thus, the proposed approach is suitable for ultra-sensitive measurements over a small dynamic range. For example, in the prototype sensor system, the magnitude sensitivity reaches zero for the condition L2−L1=k2λ0.(15)

Given k to be 106, the measurable range for the change of L2 is calculated to be 386.625 nm, corresponding to a tensile strain of approximately 608με. Beyond this range, the magnitude sensitivity changes its sign and the magnitude starts to decrease, exhibiting a non-monotonic relationship in the extended range, as further validated by numerical simulations detailed in Appendix B. However, this property also provides an opportunity to optimize the system’s measurement sensitivity at the initial setting by adjusting the cavity length of the reference FPI (L1). By gradually tuning L1 to achieve destructive interference with the initial L2, the system can operate at its highest sensitivity point, making it particularly well-suited for applications requiring highly sensitive, small-range sensing.

An innovative approach to interrogating Vernier-effect-based optical fiber interferometric sensors is introduced and experimentally demonstrated in this work. Unlike conventional optical domain-based interrogation, the proposed scheme leverages the MWP technique and measures the frequency response of the optical Vernier sensor. The underlying principle involves converting a subtle optical phase change of the sensing interferometer, induced by the measurand of interest, into a significant change in the microwave signal magnitude via coherent microwave interference-enabled detection. In the proof-of-concept demonstration, a prototype Vernier sensor, constructed by arranging two FPIs in parallel, was employed. The frequency response of the Vernier sensor, resulting from the coherent superposition of the RF responses of the two interferometers, is acquired using a VNA. By tracking the magnitude of the notch that emerged in the superimposed frequency response, high-sensitivity strain sensing was experimentally verified.

This introduced microwave domain-based interrogation approach offers several distinct advantages over conventional optical domain methods, including easy implementation, straightforward sensing signal extraction, and significantly improved measurement sensitivity. Additionally, it is readily applicable to advanced fiber interferometric sensors for further performance enhancement [54–57]. The present scheme opens new avenues for the advancement of MWP sensing, leading to new possibilities in a variety of sensing applications. By transforming optical interference-based measurements into microwave interferometry-enabled coherent detection, the approach combines the high sensitivity of optical interferometry with the stability and control ease of microwave interferometry. This synergy results in a novel paradigm for ultra-sensitive sensing with broad implications for future sensor development.