Microwave-photonic Vernier effect enabled high-sensitivity fiber Bragg grating sensors for point-wise and quasi-distributed sensing

Chen Zhu; Osamah Alsalman; Jie Huang

doi:10.1364/OE.508158

1. Introduction

Since its first demonstration in the field of optical fiber sensors [1], the optical Vernier effect has gained a lot of interest as a tool for sensitivity enhancement of optical fiber point-wise interferometric sensors in the past ten years [2–4]. A Vernier effect-based sensor typically requires the integration of two interferometers in a single system, where one interferometer acts as the sensing device and the other acts as the reference device. A Vernier modulation envelope is then generated at the output spectrum of the system, which is used as the reference variable in response to external perturbations. The spectral shift of the Vernier envelope is significantly magnified, compared to that of the single interferometer. Thus, sensitivity-improved sensing can be realized.

The implementation of the Vernier effect has been realized using various interferometric configurations such as Fabry-Perot interferometers [5–10], Michelson interferometers [11–13], Mach-Zehnder interferometers [14–17], Sagnac interferometers [18–20], fiber rings [21,22], for measurements of a diverse array of physical, chemical, and biological parameters with significantly improved sensitivity. Recent progress of the Vernier effect-based optical fiber sensors can be found in [2–4].

Microwave photonics (MWP) has undergone tremendous growth and advancement in the past two decades [23,24]. MWP interferometry-based optical fiber sensors have also been demonstrated recently due to their high signal-to-noise ratio, the relieved requirement on sensor fabrication and alignment, and compatibility with multimode fibers [25–30]. Inspired by the optical Vernier effect, the so-called MWP Vernier effect was proposed very recently and has been employed to amplify the sensitivity of MWP interferometric sensors for point-wise and distributed sensing applications [31–34]. However, due to the long wavelength of the probing microwave signal, the gauge length of these sensors is on the order of meters, making it challenging for deployment in specific applications where localized measurements are required. Thus, fiber Bragg gratings have been incorporated into the MWP system for localized high-sensitivity sensing [35–38]. The optoelectronic oscillator (OEO)-based system represents a pathway to develop a high-resolution and fast FBG interrogator. The Vernier effect has also been applied to the FBG-OEO system for further sensitivity enhancement [39,40]. However, the systems are relatively complicated, requiring a dual-loop configuration. Also, the potential of using the Vernier effect to amplify the sensitivity of FBG array sensors for quasi-distributed sensing has not been explored yet.

In this article, we introduce and experimentally demonstrate a sensitivity-improved FBG sensor based on the MWP Vernier effect and interferometry. Specifically, an incoherent MWP system using a broadband light source is employed as the interrogator. Two FBGs, i.e., a reference FBG and a sensing FBG, are used as the two reflectors to construct a microwave-photonic MI. Changes in the Bragg wavelength of the sensing FBG are encoded into the variations of the phase delay of the formed MI based on a wavelength-to-delay mapping technique [41]. The interferogram of the MI is overlapped with the signal of an artificial interferometer with a slightly detuned optical path difference (OPD) to generate the virtual Vernier effect. Instead of directly tracking the change in the reflection spectrum of the MI (i.e., spectral shift), the envelope signal induced by the Vernier effect is considered as the variable for sensitivity-enhanced sensing when the sensing FBG experiences external perturbations. Compared to the previously reported Vernier effect-based MWP sensors [31–33], the gauge length of the present FBG sensor is only 10 mm, two orders of magnitude shorter. Additionally, thanks to the combination of wavelength-to-delay mapping and the Vernier effect, the strain sensitivity of the FBG-assisted MI is tens of times higher than that of the MWP interferometric sensors [31–33]. It is worth noting that although the two FBGs-assisted MWP sensing system has been reported previously, the measurement sensitivity is quite limited. In this work, we demonstrate the first implementation of the Vernier effect in such a system for sensitivity enhancement reaching a factor of 72, where no complexity has been added into the system hardware by taking advantage of the computational Vernier effect technique. More importantly, combining an FBG array and the proposed technique, we, for the first time, demonstrate sensitivity-enhanced quasi-distributed sensing based on cascaded FBG sensors. Additionally, the possibility of using the so-called harmonic Vernier effect for sensitivity improvement is investigated in detail, where a factor up to 685 is successfully demonstrated.

2. Methods

2.1 Principle

A schematic diagram of the proposed MWP-based sensitivity-improved FBG sensor system is presented in Fig. 1(a). The main part of the system consists of an incoherent MWP MI. Specifically, an amplified spontaneous emission (ASE) source is employed as the light source, generating a low-coherent probing light signal. The probing light is intensity modulated by a microwave signal at an electro-optic modulator (EOM), where the microwave signal is generated by a vector network analyzer (VNA). The intensity-modulated light is split into two paths at the fiber optic coupler, probing the sensing and the reference FBGs, thus, forming an MI. The reflected optical signals from the FBGs are combined and amplified by an erbium-doped fiber amplifier (EDFA), then pass through a dispersion element (e.g., a dispersion compensating fiber, DCF), and finally reach the photodetector (PD). The electrical signal acquired from the PD is then collected by the VNA. The complex transfer function of the system (i.e., the magnitude and phase spectra of the MI) can be obtained by sweeping the microwave modulation signal and recording the S₂₁ parameter from the VNA at each modulation frequency.

Fig. 1. Overview of the system. (a) Schematic of the sensitivity-improved FBG sensor system based on MWP. The sensing FBG and the reference FBG form an MI, whose interference pattern is highly dependent on the Bragg wavelength of the sensing FBG. Amp, amplifier; EOM, electro-optic modulator; EDFA, erbium-doped fiber amplifier; DCF, dispersion compensating fiber; PD, photodetector. (b) Conceptual illustration of the working principle for sensitivity-enhanced sensing. SI, sensing interferometer; RI, reference interferometer. The spectral shift Δf of the MI caused by a tensile strain of Δε is magnified by the Vernier effect to ΔF.

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The working principle of the FBG sensor is detailed as follows. The sensing FBG and the reference FBG are used as the point reflectors, which form an unbalanced MI. Since a low coherent ASE source is used, the normalized magnitude spectrum (i.e., the interference signal) of the MI can be expressed as [31]

(1)$$Mag = \sqrt {{A_1}^2 + {A_2}^2 + 2{A_1}{A_2}\cos \left( {2\pi f\frac{{OP{D_{sen}}}}{c}} \right)}$$

where A₁ and A₂ are the effective light intensity reflected by the two FBGs; OPD_sen is the OPD of the MI; f is the frequency of the microwave modulation signal; and, c is the speed of light in vacuum. The OPD of the MI is determined by the lead-in fiber length difference of the two FBGs and the Bragg wavelength difference of the two FBGs and is given by

(2)$$OP{D_{sen}} = 2n{L_0} + cD\Delta \lambda$$

where n is the refractive index of the optical fiber; D is the dispersion coefficient of the DCF; and, Δλ denotes the Bragg wavelength difference of the sensing FBG and the reference FBG. The resonance (dip) frequency can then be calculated by

(3)$${f_{res}} = \frac{{2k + 1}}{2}\frac{c}{{OP{D_{sen}}}}$$

where k is a non-negative integer, denoting the order of resonance. Thus, an external perturbation (e.g., a tensile strain, Δɛ) changes the Bragg wavelength of the sensing FBG as well as the Bragg wavelength difference between the sensing FBG and the reference FBG (Δλ, see Fig. 1(b)), leading to a variation in the OPD (phase delay) of the MI, and therefore changing the interference pattern of the MI (e.g., a spectral shift, Δf_res). The sensitivity of the MI can be expressed as

(4)$${K_0} = \frac{{\Delta {f_{res}}}}{{\Delta \varepsilon }} \approx{-} {f_{res}}\frac{{cD}}{{OP{D_{sen}}}}{\lambda _{sen}}({1 - {p_\varepsilon }} )={-} \frac{{2k + 1}}{2}\frac{{{c^2}}}{{OP{D_{sen}}^2}}D{\lambda _{sen}}({1 - {p_\varepsilon }} )$$

where λ_sen is the Bragg wavelength of the sensing FBG; p_ɛ is the effective photo-elastic coefficient of the FBG.

Here, instead of directly monitoring the spectral shift of the MI [42], we introduce the Vernier effect to amplify the measurement sensitivity of the FBG sensor. Different from the conventional Vernier effect-based sensor system that requires the physical integration of two interferometers in the system, a computational Vernier effect technique is adopted [43–45]. First, a virtual reference interferometer is built from the measured interference pattern of the MI at an initial state based on a computational microwave photonics technique [46]. Specifically, the magnitude spectrum of the virtual reference interferometer is given by

(5)$$Mag = \sqrt {{C_1}^2 + {C_2}^2 + 2{C_1}{C_2}\cos \left( {2\pi f\frac{{OP{D_{ref}}}}{c}} \right)}$$

where C₁ and C₂ are constants and are determined by using a nonlinear curve fit to the measured magnitude spectrum of the sensing interferometer given in Eq. (1); OPD_ref denotes the OPD of the virtual reference interferometer, which is slightly different from OPD_sen. Note that we chose to use a virtual reference interferometer instead of a physical one, because doing the latter would have added complexity to the system hardware. More importantly, based on the computational Vernier effect technique, the OPD of the reference interferometer can be flexibly chosen and is predicated on the desired sensitivity amplification factor according to [46]

(6)$$M \approx \frac{{OP{D_{sen}}}}{{OP{D_{sen}} - OP{D_{ref}}}}$$

By choosing different OPDs for the reference interferometer, the magnification factor can be easily adjusted, without changing any physical settings to the system hardware. A conceptual overview of the proposed approach for sensitivity-enhanced sensing is shown in Fig. 1(b). In addition to the conventional Vernier effect, by carefully choosing the OPD of the reference interferometer, the so-called harmonic Vernier effect can also be generated to obtain larger magnification factors [47], as will be shown in Section 4.

2.2 Quasi-distributed sensing

Considering that the wavelength-to-radio-frequency-delay mapping technique is able to interrogate cascaded FBG sensors [41], we further explore the possibility of using the proposed technique to enhance the measurement sensitivity of an FBG array sensor for quasi-distributed sensing. A schematic of the quasi-distributed sensor system is shown in Fig. 2(a). An optical fiber consisting of cascaded FBGs is used, and the rest of the system is the same as the one shown in Fig. 1(a).

Fig. 2. Sensitivity-enhanced quasi-distributed sensor system. (a) Schematic diagram of the system. (b) Flowchart of the signal demodulation. IFT, inverse Fourier transform; FT, Fourier transform; VE, Vernier effect.

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Since several FBGs are cascaded in the system, the frequency response of the system becomes more complicated compared to Eq. (1). The normalized frequency response can be expressed as

(7)$${S_{21}}(\Omega ) = {\Gamma _0}^2{e^{ - j\Omega {t_0}}} + \sum\limits_{i = 1}^N {{\Gamma _i}^2{e^{ - j\Omega {t_i}}}} \textrm{ }$$

where Γ₀ and t₀ are the effective reflectivity and delay of the reference FBG; Γ_i and t_i are the effective reflectivity and delay of the i-th sensing FBG, respectively; and, Ω denotes the angular frequency of the microwave modulation signal. By sweeping the modulation frequency and synchronized detection of the magnitude and phase at each frequency, the complex frequency response over a broad modulation frequency range can be obtained. The time-domain signal of the system can be subsequently obtained by applying an inverse Fourier transform to the complex frequency domain signal and can be simply expressed as

(8)$$T(t) = {\Gamma _0}^2\delta ({t - {t_0}} )+ \sum\limits_{i = 1}^N {{\Gamma _i}^2\delta ({t - {t_i}} )} \textrm{ }$$

A series of pulses would be observed in the time-domain signal, and each of the pulses corresponds to each of the FBGs in the system. Then, by applying a time gate to select the reference FBG pulse and the i-th sensing FBG pulse, followed by a Fourier transform, the magnitude spectrum (interferogram) of the i-th MI can be obtained, as given in Eq. (1). Next, the proposed computational Vernier effect can be employed to amplify the sensitivity of the i-th MI. By sliding the time gate to select each of the individual sensing FBG pulses and the reference FBG pulse, the interferograms of all the MIs can be reconstructed, and their sensitivity can all be enhanced using the computational Vernier effect. The flowchart of the signal demodulation is given in Fig. 2(b). As a result, sensitivity-enhanced quasi-distributed sensing based on an FBG array can be achieved.

3. Results

3.1 System validation

The key parameters of the components used in the experiment are listed as follows. EOM, iXblue MXAN-LN-10; ASE, 1528-1563 nm; VNA, R&SZNB8; PD, KG-PD-10 G. An optical fiber coupler with a coupling ratio of 50:50 is used. The length and the dispersion coefficient of the DCF are approximately 1870 m and -332 ps/nm@1545 nm, respectively. The Bragg wavelengths of the reference FBG and the sensing FBG are 1541.03 nm and 1541.07 nm, respectively. The reflectivity and 3-dB bandwidth of the two FBGs are larger than 80% and less than 0.2 nm. The measured reflection spectra for the two FBGs using an optical spectrum analyzer are shown in Fig. 3(a). The length of the FBGs is 10 mm, representing that the gauge length of the FBG sensor is two orders smaller than previously reported Vernier effect-based MWP sensors [31,32]. The lead-in fiber length difference between the two FBGs is ∼1.25 m.

Fig. 3. Construction of the sensing envelope based on the virtual Vernier effect. (a) Measured reflection spectra of the reference and sensing FBGs using an optical spectrum analyzer. (b) Measured interference signal of the sensing interferometer at an initial setting. (c) Calculated interference signal of the virtual reference interferometer. Overlapped signal with an expected magnification factor of (d) 80, (e) 20, and (f) 100. Note that variation of the magnification factor was achieved entirely in software, without modifying the system hardware.

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The measured interference signal of the MI system at a pre-stress state is given in Fig. 3(b). The free spectral range was found to be ∼79.93 MHz, corresponding to an OPD of ∼3.7533, which matched well with the predicted value based on Eq. (2), given the refractive index of the fiber to be 1.468. Consequently, an artificial reference interferometer could be constructed based on Eq. (5). Figure 3(c) shows the reconstructed signal of the virtual reference interferometer, where the OPD was set to 3.7064. The superimposed signal of the sensing interferometer and the reference interferometer is shown in Fig. 3(d), where the Vernier envelope modulation is generated as expected. The superimposed signal was obtained by directly adding the magnitude spectra of the sensing and reference interferometers. By tracking the spectral shift of the Vernier envelope, high-performance sensing with a sensitivity enhancement factor of 80 can be expected according to Eq. (6). To show the flexibility of the virtual Vernier effect-based approach, Fig. 3(e) gives the superimposed signal of the sensing interferometer with a virtual interferometer with an OPD of 3.5656, where a sensitivity enhancement factor of only 20 could be inferred. Figure 3(f) gives the superimposed spectrum where the OPD of the virtual interferometer was set to 3.7157. A sensitivity enhancement factor of 100 could be expected. A larger enhancement factor led to a larger free spectral range of the Vernier envelope. Limited by the frequency observation bandwidth, the largest enhancement factor that could be used is restricted, because the dip frequency of the Vernier envelope could be out of the frequency observation window, as revealed in Fig. 3(f). However, this issue could be partially overcome by the harmonic Vernier effect, as discussed later.

Tensile strains with a step-size of 4 µɛ were exerted on the sensing FBG employing the two translation stages-assisted approach, and the reflection signals of the system are recorded at each of the strain settings. The corresponding Vernier envelopes with a theoretical magnification factor of 80 were obtained using the procedures discussed above. Fourth-order polynomial curves were employed to fit these envelopes, as shown in Fig. 4(a). The measured original responses of the MI without applying the Vernier effect are given in Fig. 4(b) for comparison. Both the interference signal and the Vernier envelope shift to the lower-frequency regime, given that a positive magnification factor was assigned. As the applied tensile strain increased, the Bragg wavelength of the sensing FBG also increased, leading to an increment in the Bragg wavelength difference between the sensing FBG and the reference FBG, then leading to an increase in the OPD of the MI. Figure 4(c) shows the dip frequency of the Vernier envelope and the dip frequency of the original interference signal as a function of the tensile strain. Linear curve fits were used to determine the measurement sensitivity, as indicated in the figure. A sensitivity magnification factor (taking the sensitivity of the MI @6.27 GHz as a reference, see Fig. 4(b)) of approximately 72 was revealed. The slight difference between the measured magnification factor and the theoretical value is because the MI’s strain sensitivity (i.e., the reference sensitivity) is different at different frequency ranges due to the so-called frequency accumulation effect [32]. Note that in the strain experiments, the reference FBG was secured in a temperature-controlled box to avoid disturbances from external environments. The temperature response of the system to temperature variations was also characterized, as given in Supplement 1. It is shown that by using two identical FBGs as the reference FBG and sensing FBG, the temperature crosstalk could be eliminated through proper sensor installation.

Fig. 4. Sensitivity-improved sensing with an enhancement factor of 72. (a) Vernier envelopes and (b) original interference signals for different settings of tensile strains. (c) Dip frequencies of the Vernier envelopes and original interference signals as a function of tensile strains. Linear curve fits were applied to find out the measurement sensitivities.

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It is noted that the R-squared coefficient in both cases shown in Fig. 4(c) is found to be larger than 0.99, indicating that the linearity of the sensor’s response does not deteriorate after significant sensitivity enhancement. Meanwhile, the 95% confidence interval obtained in the linear curve fit for the original response from the MI without using the Vernier effect shown in Fig. 4(c) is found to be 20.3 kHz. The measurement errors are mainly attributed to the system noise. The 95% confidence interval for the Vernier effect-amplified case is 870 kHz, revealing that the measurement uncertainty is also magnified by a factor of 43. In other words, both the measurement sensitivity and measurement uncertainty are amplified by the Vernier effect, but with different factors, i.e., 72 and 43, respectively. Thus, the overall magnification factor for the measurement resolution is approximately 1.6. The increased error introduced in the Vernier case is mainly attributed to the complex signal processing involved in the demodulation, i.e., extraction of multiple fringe dips in the original interference signal and the fourth-order polynomial curve fit [48]. By employing more advanced algorithms, e.g., machine learning, we envision that the error could be significantly reduced [34]. Nevertheless, the possibility of employing the Vernier effect to boost the sensitivity of the FBG strain sensor is verified. Considering the enhanced sensitivity of -17.65 MHz/µɛ and the associated measurement uncertainty of 870 kHz, an estimated resolution of ∼50 nɛ is expected. The largest sensitivity magnification factor that can be realized is restricted by the limited frequency interrogation bandwidth, similar to the case of the optical Vernier effect. Also, it should be mentioned that the dynamic range of the system will be compromised accordingly if a large sensitivity magnification factor is introduced. What should also be reminded is that the sensitivity of the system to other parameters (e.g., temperature variations) is also magnified by the Vernier effect.

A stability test was also carried out to demonstrate the performance of the system. The magnitude spectrum (i.e., the original interference signal) of the system was continuously recorded every minute for a time period of 30 mins. Figure 5(a) shows the calculated dip frequency deviation @6 GHz (i.e., without applying the Vernier effect) during the stability test. The standard deviation is found to be 27.0 kHz during the 30-minute test. Figure 5(b) gives the calculated deviation of the dip frequency (@6 GHz) of the Vernier envelope after the Virtual Vernier effect was applied to the recorded interference signals. An increased standard deviation of 997 kHz is obtained, indicating that the stability of the system degrades with a degradation factor of 37 after the Vernier effect was applied in the demodulation. Considering that the sensitivity magnification factor is 70, the effective factor for the improvement of the detection limit of the system is calculated to be 1.9. Thus, the Virtual Vernier effect-based sensitivity magnification is accompanied by the degradation of the stability of the system, but with an overall slightly increased limit of detection.

Fig. 5. Stability test of the system (a) without and (b) with applying the Vernier effect.

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3.2 Quasi-distributed sensing

An optical fiber with 11 cascaded FBGs was used to demonstrate sensitivity-enhanced quasi-distributed sensing based on the proposed technique. Figure 6(a) gives the optical reflection spectrum of the 11-FBGs measured using an optical spectrum analyzer. The Bragg wavelengths are 1530.01, 1533.04, 1536.03, 1538.87, 1541.90, 1544.92, 1547.96, 1551.03, 1553.85, 1556.88, 1559.84 nm. The optical reflection spectrum of the reference FBG was also measured in the same test, as shown in Fig. 6(a). The peak wavelength of the reference FBG is 1530.98 nm. An additional delay fiber of ∼2 m was inserted before the lead-in fiber of the 11-FBG array to make sure that the reference FBG and the 11 FBGs are clearly separated in the time-domain signal. The 11 FBGs are equally spaced with an interval of 10 cm. The reflectivity of the reference FBG and the 11 sensing FBGs is larger than 80%. The measured frequency responses of the system, including the magnitude spectrum and the phase spectrum, are shown in Figs. 6(b) and 6(c), respectively. A multi-beam interference pattern is revealed, compared to the two-beam interference pattern shown in Fig. 3(b). The time-domain signal of the system was then calculated by performing an inverse Fourier transform to the complex frequency responses. Twelve distinct pulses are observed, corresponding to the 11 sensing FBGs and the reference FBG. Since an additional delay fiber was inserted into the sensing FBGs, the first pulse corresponds to the reference FBG. Note that the number of FBGs with different Bragg wavelengths that can be cascaded in the system is quite limited due to the restricted wavelength window and the fact that each FBG has to be allocated a small wavelength range for sensing, as encountered in conventional wavelength-division multiplexing. On the other hand, weak Bragg gratings with the same Bragg wavelengths can also be employed in the system, and the weak gratings are distinguished in the time-domain signal. Using weak Bragg grating arrays would significantly improve the multiplexing capacity of the system, enabling large-scale sensing. However, considering that the complex frequency response of the fiber under test is first measured, followed by an inverse Fourier transform to obtain the time-domain signal, it is crucial to fully resolve the time-domain signal without unambiguity of determining the locations of discrete FBGs (to avoid aliasing). A theoretical analysis can be found in Supplement 1, where a positive correlation between the number of sampling points in data acquisition and the number of cascaded FBGs is revealed.

Fig. 6. Characterization of the 11-FBG sensor device. (a) Optical reflection spectrum of the 11-FBG measured using an optical spectrum analyzer. The reflection spectrum of the reference FBG is also included in the plot. (b) Magnitude spectrum of the 11-FBG and the reference FBG. (c) Phase response. An enlarged view of the phase spectrum centered at 3 GHz is given as the inset. (d) Time-domain signal of the system.

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Following the flowchart shown in Fig. 2(b), the interferogram of each of the MIs constituted by the reference FBG and each of the sensing FBGs can be unambiguously reconstructed. Particularly, in the experiment, tensile strains were applied to FBG₅, FBG₆, and FBG₇ to demonstrate the sensitivity-enhanced quasi-distributed sensing capability of the system. The reconstructed interferograms of MI₅, MI₆, and MI₇ (formed by the reference FBG with the sensing FBGs of FBG₅, FBG₆, and FBG₇) are given in Fig. 7(a). The spectra of the corresponding artificial reference interferometers were calculated based on an expected sensitivity magnification factor of 30. The superimposed spectra are shown in Fig. 7(b). Vernier envelopes are revealed in all three spectra, as we expected. The Vernier dip frequencies exhibit slight deviations due to the fact that the OPD for the three interferometers is different. By tracking the spectral shift of the Vernier envelope in the corresponding superimposed spectrum, each FBG can achieve sensitivity-improved sensing. By integrating the information from all the sensing FBGs, quasi-distributed sensing can thus be realized with magnified sensitivity.

Fig. 7. (a) Reconstructed interference spectra of MI₅, MI₆, and MI₇. The OPDs of MI₅, MI₆, and MI₇ are determined to be 5.7638, 6.4085, and 7.0273 m. (b) Superimposed spectra of MI₅, MI₆, and MI₇ and their corresponding artificial reference interferometers. The OPDs of the reference interferometer for MI₅, MI₆, and MI₇ are 5.57167, 6.1949, and 6.7930 m to achieve a sensitivity magnification factor of ∼30 based on Eq. (6).

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As mentioned earlier, tensile strains were incrementally applied to FBG₅, FBG₆, and FBG₇ in a relatively large range of 500 µɛ with a step size of 100 µɛ. The rest FBGs were undisturbed in the experiment. The measured responses of the system are given in Fig. 8(a). Significant frequency shifts can be observed for the sensing interferometers of MI₅, MI₆, and MI₇, corresponding to sensing FBGs of FBG₅, FBG₆, and FBG₇, as expected. The responses of the system without applying the Vernier effect are given in Fig. 8(b) for the purpose of comparison. The frequency shifts of MI₅, MI₆, and MI₇ are amplified from 25 MHz, 22 MHz, and 21 MHz to 1063 MHz, 839 MHz, and 657 MHz due to the introduced Vernier effect when subject to a tensile strain of 500 µɛ, corresponding to magnification factors of 42, 38, and 31, respectively. The deviations in the frequency shift of FBG₅, FBG₆, and FBG₇ are due to the fact that their original OPDs are different. The obtained magnification factors deviate from the theoretical value of 30, which is due to the frequency accumulation effect, as discussed earlier. No significant frequency shifts for the other FBGs are observed in the experiment for both cases. The experimental results demonstrate the potential of the proposed technique for high-sensitivity quasi-distributed sensing applications.

Fig. 8. Sensitivity-improved quasi-distributed sensing. 3-D rendering of the responses of the FBG sensors (a) with the Vernier effect and (b) without employing the Vernier effect. The initial Vernier dip frequencies for the 11 sensing MIs that are tracked during the strain experiment are approximately 2.77, 2.32, 1.98, 3.52, 3.09, 2.79, 2.53, 2.32, 2.16, 3.02, and 2.81 GHz. The dip frequencies that are tracked for the case without using the Vernier effect are 3.00, 2.98, 3.01, 3.00, 2.99, 3.02, 3.00, 2.99, 2.98, 2.98, and 3.00 GHz.

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4. Further enhancement based on the harmonic Vernier effect

4.1 Principle

In the case of the optical Vernier effect, it is important to note that a key parameter that determines the performance of a Vernier sensor is the OPD deviation of the two interferometers that are included in the system; a smaller detuning leads to a larger sensitivity magnification factor. However, from the perspective of practical applications, the largest magnification factor that can be realized is always limited due to sensor fabrication errors and a limited wavelength interrogation range. To surpass the limitation, the optical harmonic Vernier effect was introduced, where the two interferometers in a system could have very different interference frequencies, and yet a large magnification factor could still be secured [47,49,50]. In this section, we further investigate the possibility of employing the harmonic MWP Vernier effect for sensitivity enhancement of FBG sensors. It is shown that considerable sensitivity enhancement factors can be sustained based on the harmonic Vernier effect, and the factor scales up with the harmonic orders, similar to the case of the optical harmonic Vernier effect.

Different from the fundamental Vernier effect as discussed in Section 2, in the case of the harmonic Vernier effect, the OPD for the reference interferometer is a few times larger than the OPD of the sensing interferometer,

(9)$$OP{D_{ref}} = OP{D_{ref0}} + i \cdot OP{D_{sen}},$$

where OPD_ref0 denotes the OPD of the reference interferometer in the case of the fundamental Vernier effect, and i is a non-negative integer, representing the order of harmonic. Instead of tracking the upper or lower envelopes as is the case for the fundamental Vernier effect shown in Section 2, internal envelopes are generated and are used for sensing in the case of the harmonic Vernier effect, due to the fact that the fringe visibility decreases significantly for upper and lower envelopes with increasing harmonic orders. The free spectral range (FSR) of the internal envelope can be expressed as

(10)$$FS{R_{\textrm{internal}}} = \left|{\frac{{({i + 1} )FS{R_{sen}}FSR_{ref}^i}}{{({i + 1} )FSR_{ref}^i - FS{R_{sen}}}}} \right|.$$

where FSR_internal, FSR_sen, and FSR_ref are the FSR of the internal envelope of the superimposed spectrum, the magnitude spectrum of the sensing interferometer, and the reference interferometer, respectively. The sensitivity magnification factor is then given by

(11)$${M^i} = \frac{{({i + 1} )FSR_{ref}^i}}{{({i + 1} )FSR_{ref}^i - FS{R_{sen}}}} = ({i + 1} )\frac{{OP{D_{sen}}}}{{OP{D_{sen}} - OP{D_{ref0}}}} = ({i + 1} )M.$$

Thus, for the harmonic Vernier effect, the sensitivity magnification further increases by a factor of i + 1, compared to the fundamental Vernier effect (see Eq. (6)). Meanwhile, the largest magnification factor that can be realized in the fundamental Vernier effect is limited by the FSR of the upper and lower envelopes due to the restricted frequency interrogation bandwidth. However, in the case of the harmonic Vernier effect, since the internal envelopes are tracked, the largest magnification factor is beyond the limitation imposed by the restricted frequency bandwidth.

4.2 System characterization

The same setup shown in Fig. 1 was used, including the same FBGs. The length of the lead-in fiber for the reference FBG was approximately 1.2 m longer than that of the sensing FBG. By stretching the grating area using two high-precision translation stages (OMTOOLS, HFA-XYZ), tensile strains were applied to the sensing FBG in the range of 0∼140 µε with an incremental step of 10 µε. The recorded frequency responses in magnitude are given in Fig. 9(a) for different settings of tensile strains. The spectrum shifts to the lower-frequency region with increasing tensile strains, as expected. Figure 9(b) shows the determined resonance frequencies at around 4.05 GHz as a function of applied tensile strains. The slope was determined by applying a linear curve fit to the measured discrete data points and was found to be -0.1372 MHz/µε, which agreed with the theoretical value of -0.1328 MHz/µε based on Eq. (4) given the resonance order to be 49 @∼4.05 GHz.

Fig. 9. Measured strain responses of the system. Tensile strains were applied to the sensing FBG. (a) Magnitude spectra for different settings of tensile strains. (b) Frequency shift as a function of tensile strain. A linear curve fit model was applied to find out the sensitivity of the measurement. The R-squared of the fitting was 0.9985.

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4.3 Harmonic Vernier effect

Next, the harmonic Vernier effect was employed to amplify the strain measurement sensitivity of the system. As discussed in Section 2, the computational Vernier effect was chosen. Based on the measured OPD of the sensing interferometer, the OPD of the virtual reference interferometer needs to be assigned for a desired sensitivity magnification factor as predicated by Eq. (11). It is worth noting that the sensitivity magnification factor given in Eq. (11) can be positive and negative, determined by the relationship between OPD_ref0 and OPD_sen, i.e.,

(12)$$\left\{ \begin{array}{l} if\textrm{ }OP{D_{ref0}} > OP{D_{sen}},\textrm{ }M < 0\\ if\textrm{ }OP{D_{ref0}} < OP{D_{sen}},\textrm{ }M > 0 \end{array} \right..$$

Considering that the strain sensitivity of the system without using the Vernier effect was a negative value as given in Fig. 9(b), OPD_ref0 was set to be larger than OPD_sen to obtain a negative magnification factor. This is because a negative magnification factor leads to a positive strain sensitivity after applying the Vernier effect, where the dip frequency will shift to the higher-frequency region with increasing strains, ultimately resulting in a higher sensitivity due to the frequency accumulation effect. Therefore, in the calculation, OPD_ref0 was set to be 3.695085, slightly larger than OPD_sen of 3.6585, corresponding to a magnification factor of -100 for the fundamental Vernier effect. Figures 10(a), (c), (e), and (g) give the numerically calculated magnitude spectra of the artificial interferometers that are expected to generate the fundamental, first harmonic, second harmonic, and third harmonic Vernier effect, respectively. Figures 10(b), (d), (f), and (h) show the corresponding superimposed magnitude spectra.

Fig. 10. Implementation of fundamental and harmonic Vernier effect based on the computational Vernier effect technique. Numerically calculated magnitude spectra for the artificial interferometers for (a) fundamental, (c) first harmonic, (e) second harmonic, and (g) third harmonic Vernier effect and the corresponding superimposed magnitude spectra for (b) fundamental, (d) first harmonic, (f) second harmonic, and (h) third harmonic Vernier effect. The parameters used to calculate the magnitude spectra included OPD_ref0 = 3.695085, and i = 0, 1, 2, and 3, corresponding to the first, second, and third harmonics. The insets show enlarged views of the spectra in the frequency range of 4∼4.2 GHz.

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As could be observed in the superimposed spectra, upper and lower envelopes were obtained in all four cases. However, due to the large sensitivity magnification factor, the FSR of the upper and lower envelopes was larger than the frequency observation window. At first glance, it would seem that such spectra cannot be used for sensing applications due to the loss of Vernier envelope dips, even in the case of the fundamental Vernier effect. We show in this work that this limitation can be easily resolved by tracking the internal envelopes for both the fundamental and harmonic Vernier effect. What can also be seen is that the visibility of the upper and lower envelopes decreased with increasing harmonic orders, similar to the case of the optical Vernier effect [47].

Let us first consider the fundamental Vernier effect. Different from the fundamental optical Vernier effect, internal envelopes were revealed in the fundamental MWP Vernier effect, as shown in Fig. 10(b). Similar internal envelopes are also observed in Figs. 3(d), 3(e), 3(f), and 7(b). The generation of internal envelopes in the fundamental Vernier effect might be due to the fact that the magnitude spectrum is in the form of the square root of a sinusoidal signal. Figure 11(a) gives several representative superimposed spectra for different settings of tensile strains. As can be seen, the intersection point generated by the internal envelopes significantly shifted to the higher-frequency region with increasing tensile strains. Second-order polynomial curve fit models were applied to the two internal envelopes, and then the intersection frequency was obtained. The intersection frequency as a function of the tensile strain is plotted in Fig. 11(b). The response of the single interferometer without applying the Vernier effect is also included for comparison. Magnified sensitivity can be observed from the plot. The nonlinear response is mainly attributed to the frequency accumulation effect; a larger resonance frequency leads to a higher sensitivity. An averaged sensitivity for the fundamental Vernier effect is estimated to be 22.12 MHz/µε, revealing a sensitivity magnification factor of -161. On the other hand, the high sensitivity detrimentally affects the dynamic range of the system, where the intersection point shifts out of the frequency window, limiting the strain range to 120 µε. It is worth mentioning that the nonlinear effect caused by the frequency accumulation effect can be corrected by converting the intersection frequency to wavelength and using the wavelength shift as a reference variable to strain variations. The inset in Fig. 11(b) gives the wavelength shift as a function of the tensile strain. A linear relationship was demonstrated by means of curve fitting, revealing a sensitivity of -0.0002397 m/µε (wavelength shift/strain).

Fig. 11. Fundamental Vernier effect. (a) Superimposed spectra for different settings of tensile strains. (b) The shift in the intersection frequency between internal envelopes as a function of tensile strain. The inset gives the shift in the intersection wavelength as a function of the tensile strain. The intersection wavelength was directly converted from the determined intersection frequency. A linear curve fit model was applied, and the slope and R-squared were -0.0002397 m/µε and 0.9983, respectively.

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Next, the harmonic Vernier effect was investigated. The OPD of the reference interferometer was chosen to be 7.353585 (i.e., 3.695085 + 1 × 3.6585), corresponding to the case of 1^st harmonic Vernier effect. Figure 12(a) plots the superimposed spectra for three different settings of tensile strains. The intersection frequency shifted to the high-frequency region, matching the results shown in Fig. 11(a). The relationship between the determined intersection frequency and the applied tensile strain is shown in Fig. 12(b), where the characteristics of the sensor without using the Vernier effect are also given for comparison. Nonlinear evolution of the frequency versus the tensile strain was again revealed with an averaged sensitivity of 41.57 MHz/µε, which is approximately 1.9 times larger than that of the fundamental Vernier effect, as expected. The inset gives the intersection wavelength shift as a function of tensile strain, and a linear curve fit was employed to verify the linear relation. Also, due to the significantly enhanced sensitivity and a limited frequency observation bandwidth, the strain measurement range was sacrificed accordingly.

Fig. 12. 1^st harmonic Vernier effect. (a) Superimposed spectra for different settings of tensile strains. (b) The shift in the intersection frequency between internal envelopes as a function of tensile strain. The inset shows the calculated shift in the intersection wavelength as a function of the tensile strain. A linear curve fit model was applied, from which the slope and R-squared were determined to be -0.0004534 m/µε and 0.9936, respectively.

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Following the same procedures, the responses of the sensor in the case of the 2^nd harmonic and 3^rd harmonic Vernier effect could be obtained and are given in Fig. 13. Significant shifts of the intersection frequency were observed in the two cases, as can be seen in Figs. 13(a) and (c). The relationship between the shift in the intersection frequency (wavelength) and the tensile strain was quantified for the 2^nd harmonic and 3^rd harmonic Vernier effect is given in Figs. 13(b) and 13(d) and the insets, respectively. The average sensitivity in terms of frequency shift was determined to be 63.89 MHz/µε and 94 MHz/µε for the 2^nd harmonic and 3^rd harmonic Vernier effect, where the wavelength sensitivity was calculated to be -0.000683 m/µε and -0.0009371 m/µε by means of linear curve fit. Thus, a higher order of harmonics leads to a larger measurement sensitivity, as predicted by Eq. (11). However, the measurement range is degraded accordingly. Thus, the OPD for the reference interferometer should be assigned properly in different applications to balance the measurement sensitivity and dynamic range.

Fig. 13. 2^nd and 3^rd harmonic Vernier effect. Superimposed spectra for different settings of tensile strains for (a) the 2^nd harmonic Vernier effect and (c) the 3^rd harmonic Vernier effect. The shift in the intersection frequency as a function of tensile strain for (b) the 2^nd harmonic Vernier effect and (d) the 3^rd harmonic Vernier effect. The insets give the intersection wavelength shift as a function of tensile strain and the corresponding linear curve fit model.

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To show the improved performance of the proposed sensor system, Table 1 is summarized to compare the state-of-the-art MWP Vernier effect-based sensors and the sensor demonstrated here. As can be seen, the measurement sensitivity of the proposed sensor is orders of magnitude higher than that of the previously reported sensors, and the gauge length of the proposed sensor is much smaller. The side effect of the significantly improved sensitivity is the restricted dynamic range limited by the system hardware.

Table 1. Comparison of different sensors based on the MWP Vernier effect

View Table

5. Conclusion

In conclusion, we have proposed and experimentally demonstrated highly sensitive FBG strain sensors based on MWP and the Vernier effect. Two FBGs (i.e., a sensing FBG and a reference FBG) were used to construct an incoherent MWP MI, whose interference signal was dependent on the Bragg wavelength of the sensing FBG. Taking advantage of the virtual Vernier effect, the Vernier envelope was generated as the sensing signal, where the strain sensitivity of the sensing FBG was found to be -17.65 MHz/µɛ, representing a sensitivity magnification factor of 72 based on the fundamental Vernier effect. The gauge length of the MWP-Vernier effect-based FBG sensor is only 10 mm, two orders of magnitude shorter than previous devices reported in the literature based on microwave-photonic interferometry. More importantly, we have further extended the system to distributed sensing by employing an FBG array as the sensing element. Additionally, the possibility of employing the so-called harmonic Vernier effect for further sensitivity enhancement has also been investigated, where a sensitivity magnification factor of 685 has been achieved. This work reports the first demonstration of using the optical Vernier effect as a potential tool to enhance the performance of FBG cascaded sensors and thus paves the way towards wider adoption of the Vernier effect in optical fiber sensing for various applications.

Funding

Research Initiation Project of Zhejiang Lab (2022ME0PI01).

Acknowledgment

The authors extend their appreciation to the Deputyship for Research and Innovation, ‘‘Ministry of Education’’ in Saudi Arabia for funding this research (IFKSUOR3-261-2).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Configuration	Sensitivity	Dynamic Range	Sensitivity magnification factor	Gauge length
Cascaded Fiber ring resonators [32]	-556.856 kHz/°C	36∼76 °C	30	200 m
Cascaded Fiber ring resonators [51]	−910.67,kHz/µε	0∼120 µε	20	200
Cascaded Fabry-Perot interferometers [31]	−266.1 kHz/°C	20∼80 °C	25	7.854 m
Single Fabry-Perot interferometer [34]	288 kHz/µε	0∼750µε	75	1.6 m
Cascaded Mach-Zehnder interferometers [52]	580.45 kHz/µε	360 °C	18	21 m
Dual-loop optoelectronic oscillators [39]	-11 kHz/µε	0∼945µε	35	N.A.
Parallel FBGs [This work]	22.12 MHz/µε	60µε	160	0.01 m
	63.89 MHz/µε	40µε	465
	94 MHz/µε	30µε	685

Microwave-photonic Vernier effect enabled high-sensitivity fiber Bragg grating sensors for point-wise and quasi-distributed sensing

Abstract

1. Introduction

2. Methods

2.1 Principle

2.2 Quasi-distributed sensing

3. Results

3.1 System validation

3.2 Quasi-distributed sensing

4. Further enhancement based on the harmonic Vernier effect

4.1 Principle

4.2 System characterization

4.3 Harmonic Vernier effect

5. Conclusion

Funding

Acknowledgment

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

Data availability

Cited By

Figures (13)

Tables (1)

Equations (12)

Optics Express