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Abstract
We propose an all-fiber reflective sensing scheme to simultaneously measure temperature and strain. A length of polarization-maintaining fiber serves as the sensing element, and a piece of hollow-core fiber assists with introducing Vernier effect. Both theoretical deductions and simulative studies have demonstrated the feasibility of the proposed Vernier sensor. Experimental results have shown that the sensor can deliver sensitivities of −88.73 nm/°C and 1.61 nm/με for temperature and strain, respectively. Further, Both theoretical analyses and experimental results have suggested the capability of simultaneous measurement for such a sensor. Significantly, the proposed Vernier sensor not only presents high sensitivities, but also exhibits a simple structure, compact size and light weight, as well as demonstrates ease of fabrication and hence high repeatability, thus holding great promise for widespread applications in daily life and industry world.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical fiber sensors have many admirable merits, e. g., compact size, light weight, high resolution and excellent immunity to electromagnetic interference. Hence they have attracted considerable attention for decades, and inspired widespread applications in physics [1,2], chemistry [3,4], biomedicine [5,6], etc. To date, various types of fiber sensor have been proposed to measure refractive index [7,8], temperature [9,10], strain [11,12], transverse load [13], etc., among which the temperature and strain may be the most common physical quantities involved in daily life and industrial world. Along with the rapid development of modern society, people have made growing demands for higher sensitivity in these measurements.
In recent years, a great deal of fiber sensors have been proposed for temperature and/or strain measurements, and they can, in the main, fall into two categories, which are based on fiber gratings [14–16] and interferometers [17–21], respectively. Fiber gratings exhibit compact sizes and light weights, but present low sensitivities and would suffer from intractable crosstalk. Moreover, the equipment used to fabricate fiber gratings is expensive. With respect to fiber interferometers, there exist four typical configurations, including Michelson interferometer, Sagnac interferometer, Mach-Zehnder interferometer and Fabry-Perot interferometer (FPI). In comparison with fiber gratings, interferometer based fiber sensors can achieve higher sensitivities, and have potential for tackling the crosstalk problem. In addition, the well-known Vernier mechanism can be exploited in interferometer based fiber sensors to further improve the sensitivity significantly, which has thus been brought into sharp focus in recent years [22]. Such a mechanism consists in integrating a pair of interferometers, referred to as the reference interferometer (RI) and sensing interferometer (SI), respectively, which produce two spectra with different free spectral ranges (FSRs) to mimic the pair of Vernier scales. Then a superimposed spectrum can be derived from the integration of such an interferometer pair, yielding a Vernier envelope with its FSR much broader than that in a single interferometer. Accordingly, the envelope shift in response to environmental changes is also much more remarkable, which exactly demonstrates the effect of sensitivity amplification.
Based on the Vernier mechanism, a wide variety of sensing schemes have been proposed to measure the temperature or strain exclusively. To name a few, many schemes for temperature measurements were designed with an interferometer pair both being thermally sensitive [23–27], in which the RI should be kept thermostatic rigorously to mimic the fixed Vernier scale, in order for the whole scheme to achieve high sensitivity and accuracy. Alternatively, an air-cavity FPI was used to serve as the RI [28–30], since it has proven insensitive to temperature. Besides, another remedy is to engineer a RI with reverse thermal response against the SI [31,32], or rather, the spectra of the two interferometers shift in opposite directions with varying the temperature, thereby enhancing the Vernier effect. While for strain measurements, the scheme with a pair of interferometers which are both thermo-sensitive turns out to be fairly preferable [33–35], since such a counterbalanced structure can effectively compensate for the temperature crosstalk. Moreover, an additional fiber Bragg grating has been introduced in a Vernier scheme to further improve the sensing precision [36], however at the cost of increasing the fabrication difficult.
More to the point, some Vernier schemes can measure the temperature and strain both, but in a separate manner [37–40]. Yet still, these designs were equipped with a thermo-sensitive RI and thus should be thermostatically shielded for temperature measurements. Significantly, another design employed a hollow-core fiber (HCF) based FPI to act as a thermo-stable RI, and the temperature and strain can be simultaneously measured by demodulation of an obtained coefficient matrix [41], but the overall sensitivities seem not high despite the presence of Vernier effect, which could be ascribed to a necessarily short length in the sensing element. It is still worth mentioning that such a design has two advantages: not only has the sensor function been augmented, but also the crosstalk problem can be cured.
In this paper, we propose a Vernier sensor with an all-fiber reflective scheme to measure the temperature and strain simultaneously. A section of polarization-maintaining fiber (PMF) and a piece of HCF are directly spliced to serve the key function of such a sensor. Besides, a length of single-polarization fiber (SPF) is adopted to act as the polarizer, which can not only produce a linearly polarized beam to interrogate the sensing element, but also assist with a Lyot filter [42] to induce polarization interference. Based on such a scheme, a Vernier spectrum can be generated, and the spectral shift can be tracked to monitor temperature and strain variations. To elucidate how it works, theoretically rigorous deductions are performed in Sect. 2, which comes complete with mathematical simulations. Then experiments are demonstrated in Sect. 3 to test the sensing performance. Both theoretical and experimental results show that the proposed sensor can deliver ultra-high sensitivities in simultaneous measurements of temperature and strain. Finally, the conclusion is given in Sect. 4.
2. Principle and simulation
Figure 1(a) schematically shows the framework of the proposed reflective sensing system. It can count as an all-fiber cascaded scheme incorporating four types of commercially available fibers, i. e., the SPF, PMF, HCF and coreless termination fiber (CTF), which can be readily connected in order just by fusion-splicing operations. Specifically, the PMF operates as the sensing element, while the HCF is intended to assist with introducing the Vernier effect. Here the SPF has been adopted as a polarizer in view of its many advantages. In comparison with traditional fiber polarizers, the SPF is lighter and more flexible. More important, the SPF can radically rule out the cross coupling since only one polarized mode exists in such a fiber [43], thereby improving the stability of the sensing system against external disturbances. In terms of the optical path, a beam from a supercontinuum broadband optical source (SBOS) is first fed into the SPF based polarizer, at the end of which the beam turns out to be a linearly polarized one. Then it will split into two orthogonally polarized beams and propagate via the slow and fast axes in the PMF, respectively, when the two front fibers are aligned with an angle bias between their axes, about 45$^\circ$ as the best. For both the two polarized beams, a portion of them will be reflected back at the PMF-HCF interface, while their remainders will transmit through the HCF, until they touch the HCF-CTF interface, and then partly bounce back. The residual light will still transmit through the CTF, but its return loss can reach as high as 65 dB for a 0.25-m length [44], which can hence be neglected. On the whole, there are mainly four linearly polarized beams reflected back, and polarization interference should occur after they transmit through the SPF, which can be recorded by an optical spectrum analyzer (OSA). Whereupon, the reflected spectrum in terms of the electric field can be modeled as
From Eq. (1) or Eq. (2), the reflected spectrum in terms of light intensity can be derived as
At dips of the envelope, the following equation holds:
So the $n$-th dip lies at
It is known that, when the temperature $T$ and/or strain $\epsilon$ varies, so do the birefringence and length of the PMF, both of which contribute to OPD changes in this fiber, hence yielding a net result that the interferometric spectrum shifts over wavelengths.Whereupon, the variation of temperature and/or strain can be measured by monitoring the shifting amount of spectrum, which exactly demonstrates the working principle of the sensing system. From Eq. (6), the sensitivity, referring to the shifting rate of the $n$-th dip, can be derived as
In respect of strain, the term ${\rm {d}}\delta /{\rm {d}}p$ turns out to be
Based on the foregoing analyses, it can be seen that the sensitivities proves inversely proportional to the value of $B - \rho$. In other words, the sensitivities can be improved by controlling the quantity $\rho$ that involves the length comparison between PMF and HCF.
To provide more insight into the benefit from Vernier effect, one can envisage a single-interferometer based sensor that only contains the SPF and PMF, while leaving out the HCF and CTF. Apparently, this sensor works as a typical Lyot filter, while rules out the Vernier effect. For such a sensor, the corresponding wavelength at the $m$-th dip can be given as
Then the sensitivity of this sensor can be deduced as
By comparing Eqs. (7) and (13), one can find that the Vernier effect has magnified the sensitivity by a factor
Notice ${\lambda _n} = {\lambda _m}$ holds when the same wavelength is taken for the two compared sensors.
In the light of the above principle, simulative studies have been performed for the proposed Vernier sensor as well as the single-interferometer based sensor. In a normalized context, there is $\left ( {1 + \eta } \right )\left ( {X + Y} \right ) = 1$. We assumed $\eta = 0.6$, $X = 0.2$ and $Y = 0.425$. The initial birefringence and length of the PMF were set to be $B = 4.45 \times {10^{ - 4}}$ and $L = 0.88\;{\rm {m}}$, respectively, and the length of HCF was assigned with $l = 3.8 \times {10^{ - 4}}\;{\rm {m}}$. Based on these given parameters, there is $\rho =4.32\times {10^{ - 4}}$ and hence $B - \rho > 0$. In respect of the Vernier scheme, its initial spectrum can be obtained from Eq. (3), as shown in Fig. 2(a). Significantly, the blue curve derived from Eq. (4) has perfectly outlined the spectral envelope, hence testifying to the effectiveness of the deductions following this formula. Whereas for the single interferometer, its spectrum at the initial state can be easily worked out as shown in Fig. 2(b).
Fig. 2. Simulative reflected spectra for the proposed Vernier sensor and single-interferometer based sensor. (a) $\&$ (b): initial states; (c) $\&$ (d): varied spectra in response to a temperature rise of 1 $^{\circ }$C; (e) $\&$ (f): varied spectra in response to a strain increase of 100 $\mu \epsilon$.
When the temperature is raised by $\Delta T = 1{}^{\rm {o}}{\rm {C}}$, the OPD grows with a variation $\Delta {\delta _{\rm {T}}} = 2L\left ( {\gamma + \alpha B} \right )\Delta T$ based on Eq. (8). For the calculation of this variation, the TCB of PMF was assumed as $\gamma = - 5.6 \times {10^{ - 7}}/{}^{\rm {o}}{\rm {C}}$ [46]. Since the TCE of fused silica has a value of the same order of TCB while the birefringence of the PMF is on the order of ${10^{ - 4}}$, the contribution of the term $\alpha B$ can be neglected in the calculation. Then the varied OPD $\delta + \Delta {\delta _{\rm {T}}}$ was applied in Eqs. (3) and (4) for the Vernier scheme to generate the varied spectrum and envelope, respectively, as shown in Fig. 2(c). It is observed that the spectrum shifts towards shorter wavelengths with raising the temperature, which can be ascribed, from Eq. (9), to the combination of $B-\rho > 0$ and $\gamma < 0$. Substitute the assumed parameters into this equation, the theoretical sensitivity to temperature was calculated to be 64.67 nm/$^{\rm {o}}{\rm {C}}$ at $\lambda _n = 1550$ nm. It is worth noting that, in principle, the wavelength shifting direction can be adjusted by controlling the length comparison between the PMF and HCF. When it comes to the single interferometer, the varied spectrum corresponding to the varied OPD is shown in Fig. 2(d), which reveals that the spectrum also shifts towards shorter wavelengths with raising the temperature. In particular, the blue-shift behavior must be always so for the single interferometer.
Next, we examined the output response to strain variation. With a strain increment of $\Delta \epsilon = 100\;\mu \epsilon$, the variation of OPD can be derived as $\Delta {\delta _{\rm {S}}} = 2L\left ( {\kappa + B} \right )\Delta \epsilon$ from Eq. (10). Here the SCB was assumed to be $\kappa = 1.6 \times {10^{ - 2}}/\epsilon$ [47]. Then the varied OPD $\delta + \Delta {\delta _{\rm {S}}}$ was applied in Eqs. (3) and (4) for the Vernier scheme, yielding a spectrum with its envelope shifting towards longer wavelengths versus the initial state, as shown in Fig. 2(e). The red shift in response to strain increase can be explained by Eq. (11). Substitute the assumed parameters into this equation, the theoretical sensitivity to strain was calculated to be 1.93 nm/$\mu \epsilon$ at $\lambda _n = 1550$ nm. Note that the wavelength shifting direction for strain increase should always go into reverse against that for temperature rise due to the opposite algebraic signs of TCB and SCB, and so do the behaviors of the single interferometer. The spectral output of the single interferometer is shown in Fig. 2(f), which exhibits a red shift versus the initial state, however with a shifting direction opposite to that for temperature rise.
3. Experimental results
In the light of the foregoing principle, experiments were implemented thoroughly. A supercontinuum laser (OYSL SC-5-FC) was used to provide the broadband light source, and reflected spectra were recorded by an OSA (Yokogawa, AQ6370D) with a wavelength resolution of 0.02 nm. For sensor fabrications, only splicing operations were needed and performed by a splicer (Fujikura, FSM100P+), suggesting very simple fabrication processes.
We first tested the performance of the single-interferometer based sensor, which only comprises an SPF of 2 m in length and PMF of 0.88 m in length, with their cross-sectional images shown in Fig. 3(a) and (b), respectively, while Fig. 3(c) displays the alignment in the splicer, with about 45$^{\circ }$-angle bias between their polarization axes. In Fig. 4, the blue curve exemplifies a reflected spectrum of this sensor placed in the air, validating the polarization interference therein. Note that the length of SPF should be adequately long. To illustrate this note, another two lengths of SPF have also been tested, with obtained results represented by yellow and red curves in Fig. 4. It can be seen that the interferometric extinction ratio at shorter wavelengths decrease with reducing the length of SPF, which means the undesirable mode has not been sufficiently filtered out, and the SPF behaves more like a PMF.
Fig. 3. Cross-sectional images of the (a) SPF and (b) PMF. (c) Splicing alignment between SPF and PMF.
To work out the sensitivity of such a sensor to temperature, the whole PMF was placed in a highly thermostatic water bath, as schematically shown in Fig. 5(a). Reflected spectra were obtained for temperatures from 25 $^{\circ }$C to 31 $^{\circ }$C in increments of 1 $^{\circ }$C, which were measured by a digital thermometer. as shown in Fig. 5(b). Notice that the reflection occurred at the silica-water interface, which has thus decreased in comparison to that occurred at the silica-air interface as shown in Fig. 4. As expected, the spectrum shifts towards shorter wavelengths with raising the temperature, on account of the negativity of TCB ($\gamma < 0$). We have monitored the shifts of two different wavelength dips, with obtained results shown in Fig. 5(c). After linear fittings with ${R^2} > 0.99$, the sensitivities at the shorter and longer wavelengths were determined as −2.03 nm/$^{\circ }$C and −2.12 nm/$^{\circ }$C, respectively. It can be seen that the longer-wavelength dip exhibits a higher sensitivity.
Fig. 5. (a) Experimental setup of single-interferometer based sensor for temperature measurements. (b) Reflected spectra under different temperatures. (c) Sensitivity determinations of temperature for two wavelength dips.
As for the strain sensing, the whole PMF together with a portion of the SPF were taken to have a total length of ${L_{\rm {T}}} = 1\;{\rm {m}}$. Then the total section was straightened, with its two endpoints clipped to a fixed stage and a translation stage, respectively, as schematically shown in Fig. 6(a). One endpoint was kept immobile, while the other was translated away to apply axial strain. The translated distance $\Delta {L_{\rm {T}}}$ is related to the applied strain through $\epsilon = \Delta {L_{\rm {T}}}/{L_{\rm {T}}} =\Delta L/L$. Fig. 6(b) shows the reflected spectra for applied strains from 0 to 180 $\mu \epsilon$ in increments of 30 $\mu \epsilon$. Contrary to the thermal response, increasing the strain leads the spectrum to shift towards longer wavelengths, on account of the positivity of SCB ($\kappa > 0$). Again, two dips were tracked to quantify their sensitivities, and the results with linear fittings (${R^2} > 0.99$) are plotted in Fig. 6(c), which presents 36.59 pm/$\mu \epsilon$ and 37.62 pm/$\mu \epsilon$ for the two dips, respectively. Again, the longer-wavelength dip favors a higher sensitivity to strain.
Fig. 6. (a) Experimental setup of the single-interferometer based sensor for strain measurements. (b) Reflected spectra under different strains. (c) Sensitivity determinations of strain for two wavelength dips.
Afterwards, a 380-$\mu$m long HCF and a 2-cm long CTF were spliced sequentially following the PMF, to produce the Vernier sensor as schematically shown in Fig. 1(a). The HCF has inner and outer diameters of 25 $\mu$m and 125 $\mu$m, respectively. Figure 7(a) exemplifies a reflected spectrum of this sensor, which shows an interferometic fringe in line with the foregoing simulative result. To give more insight into the interference, Fourier transformation was applied to the $k$-dependent spectrum, yielding a space spectrum shown in Fig. 7(b). Evidently, five dominant components can be observed in accord with the terms expressed by Eq. (3). Specifically, peak 2 and peak 3 correspond to space frequencies of 757.64 $\mu$m and 777.48 $\mu$m, which are no other than the dominant OPDs for round trips in the HCF and PMF, i. e., $2l$ and $2BL$, respectively, while peak 1 and peak 4 are right the beat and sum frequencies of them. Based on the dominant OPDs, the theoretical magnification was predicted to be 39.18 on the basis of Eq. (14). Nevertheless, it should be noted that, since the birefringence is factually wavelength-dependent, so is the actual magnification, which will be verified in following experiments.
Fig. 7. (a) An exemplified spectrum of the proposed Vernier sensor. (b) The corresponding space spectrum.
To work out the temperature sensitivity of the Vernier sensor, the whole probe including the CTF was placed in the highly thermostatic water bath, as schematically shown in Fig. 8(a). Reflected spectra together with their envelopes were obtained for temperatures from 25.5 $^{\circ }$C to 26.1 $^{\circ }$C in increments of 0.1 $^{\circ }$C, as shown in Fig. 8(b). These envelopes were generated by applying the Matlab function "envelope" to the recorded spectra, and then their lower extrema were taken as Vernier dips to be tracked. Since the OPD in the PMF was arranged to be larger than in the HCF, i. e., $\delta > 2l$, there is $B-\rho >0$. And because of $\gamma <0$, the Vernier spectrum shifts, as expected, towards shorter wavelengths with raising the temperature. Two different wavelength dips were tracked against temperature variations, with obtained results shown in Fig. 8(c). After linear fittings with ${R^2} > 0.99$, the sensitivities for the two wavelength dips were determined as ${K_{\rm {t}}}({\lambda _n}) = - 88.73$ nm/$^{\circ }$C and ${K_{\rm {t}}}({\lambda _{n + 1}}) = - 71.85$ nm/$^{\circ }$C, respectively, which clearly suggests that higher sensitivity can be achieved at shorter wavelengths. To explain this, one can look back on the sensitivity expression given by Eq. (7), which contains two multiplying terms. In the case of $\delta - 2l > 0$, the first term can be rewritten as
Since the dip at a longer wavelength corresponds to a smaller value of $n$, the first term for this dip is bound to be larger than that for a shorter-wavelength dip. However, the results shown in Fig. 8(c) have reflected a fact that the longer-wavelength dip presents a lower sensitivity. Consequently, the second term for a longer-wavelength dip must be smaller than a shorter-wavelength one and outweigh the reverse effect of the first term. Therefore, it can be extrapolated that, based on Eq. (8), the absolute value of TCB $\gamma$ should be wavelength-dependent, and exhibit a decreasing trend with increasing the wavelength.
Fig. 8. (a) Experimental setup of the Vernier sensor for temperature measurements. (b) Reflected spectra under different temperatures. (c) Sensitivity determinations of temperature for two wavelength dips.
Further, a setup was built to determine the strain sensitivity, as illustrated in Fig. 9(a). Then a test process analogous to that for the single-interferometer based sensor has been implemented on the Vernier sensor, yielding reflected spectra for applied strains from 0 to 60 $\mu \epsilon$ in increments of 10 $\mu \epsilon$ while under a fixed temperature 26.1 $^{\circ }$C, as shown in Fig. 9(b). It can be seen that, contrary to the temperature response, increasing the strain leads the spectrum to shift towards longer wavelengths, which stands to reason on account of an opposite algebraic sign of SCB against the TCB. Again, the same dips as those for temperature sensing were tracked to determine the sensitivities of strain, with obtained data shown in Fig. 9(c), and the linear fitting results with ${R^2} > 0.99$ have quantified strain sensitivities of ${K_{\rm {s}}}({\lambda _n}) = 1.61$ nm/$\mu \epsilon$ and ${K_{\rm {s}}}({\lambda _{n + 1}}) = 1.24$ nm/$\mu \epsilon$, respectively. Likewise, shorter wavelengths prove to be in favour of higher sensitivity to strain. By an analysis analogous to that for temperature and based on Eq. (10), the SCB $\kappa$ should also be wavelength-dependent, and exhibit a decreasing trend with increasing the wavelength.
Fig. 9. (a) Experimental setup of the Vernier sensor for strain measurements. (b) Reflected spectra under different strains. (c) Sensitivity determinations of strain for two wavelength dips.
The above experimental results have tangibly demonstrated the highly sensitive performance of the proposed Vernier sensor. In principle, the Vernier effect can improve sensitivities to tend towards infinity. However, the actually achievable performance depends on the practical matching manipulation for the RI and SI. In terms of the proposed Vernier sensor, the required matching condition can be easily calculated and implemented just by controlling of fiber lengths. It is exactly the simple structure and easy of fabrication that allow such high sensitivities to be readily achieved. On the other hand, it should be noted that higher sensitivities will adversely narrower the measurable range, which is in fact a common issue in broadband-interference based sensing systems. The possible application of the the proposed Vernier sensor involves those scenarios that make demand for high sensitivities rather than broad measurable ranges. In the biomedical science, the temperature variation of a living body is very limited, and the temperature measurement with a high sensitivity is good for scientific research [48], therapy management [49] and wearable temperature sensor [50]. For monitoring of structural health [51] and battery state [52], people would pay more attention to a critical threshold of the applied strain, whose measurement with a high sensitivity is quite favorable in decreasing false-alarm rates. More to the point, it is worth mentioning that an effective method has recently been proposed to extent the measurable range for such sensing systems [53].
The errors between the practically tested sensitivities and the foregoing calculated ones using Eqs. (9) and (11) can be blamed on the wavelength-dependent characteristics of $B$, $\gamma$ and $\kappa$. Besides, there should have uncertainties in the measurements of fiber lengths and hence in the $\rho$. So such errors are reasonable in that those calculations were performed by using assumed parameters with constant values, while small changes in $B$ and $\rho$ can give rise to comparatively large variations of sensitivities. Moreover, an interesting phenomenon can be observed that, for the single-interferometer based sensor, the dip at a shorter wavelength exhibits lower sensitivities, but the situation goes into reverse for the Vernier sensor, which presents higher sensitivities at the shorter-wavelength dip. To find out the reason, one can look back on the magnification expression as shown by Eq. (14). Given $B - \rho > 0$, the $M$ turns out to be a decreasing function of birefringence $B$. On the other hand, it is known that the birefringence of a PMF generally grows with increasing the wavelength, i. e., the $B$ is an increasing function of wavelength. On the whole, the $M$ proves to be a decreasing function of wavelength. In other words, a shorter-wavelength dip should favours a larger magnification. Based on the experimental results, the magnifications for temperature were worked out as 43.71 and 33.89 at the shorter and longer wavelengths, respectively, as well as 42.8 and 33.89 for the strain, testifying to the above argument. Therefore, although the sensitivity at the shorter-wavelength dip is lower for the single-interferometer based sensor, it can be favourably magnified by Vernier effect to overtake the magnified one at the longer-wavelength dip.
Given the sensitivity at two different wavelength dips, simultaneous measurements of temperature and strain would be possible by matrix demodulation. So we first demonstrate the feasibility of such a method for the proposed Vernier sensor. Here the matrix is denoted $\mathcal {K}$, which can be formulated by
Whereupon, the shifting amounts of the two wavelength dips in response to variations of temperature and strain can be derived as
If the condition rank($\mathcal {K}$) = 2 holds, then one can have
To verify this condition, the $\mathcal {K}$ was rewritten on the basis of Eqs. (7), (8) and (10) as
In order to verify the above analyses, we have conducted the experiment on simultaneous measurements temperature and strain for the proposed Vernier sensor. The setup is the same as that for strain measurements. Over a period of time, the applied temperature was slowly and smoothly increased by the air-conditioning system while practically monitored by the digital thermometer, and the applied strain was regulated to follow a step function versus the serial number of time. The two corresponding wavelength dips were tracked as shown in Fig. 10(a), with their shifting amounts $\Delta \lambda$ plotted in Fig. 10(b). Then the variations of temperature and strain against their baselines can be demodulated by using Eq. (20), and the final calculated results for temperature and strain are shown in Fig. 10(c) and (d), respectively. It can be seen that these results are in good agreement, which confirms that the proposed Vernier sensor has the capability to simultaneously measure the temperature and strain. The small divergences between the calculated results and the applied quantities can be blamed on the reading uncertainties from the digital thermometer and translation stage due to their lower resolutions.
Fig. 10. Experimental results for simultaneous measurements of temperature and strain. (a) Two wavelength dips tracked with varying the temperature and strain at different times. (b) Shifting amounts of wavelength dips. (c) Applied and calculated temperatures. (d) Applied and calculated strains.
4. Conclusion
In conclusion, this paper has proposed a Vernier sensor with an all-fiber reflective scheme to measure temperature and strain simultaneously. Theoretical analyses demonstrate that the interferometric spectrum provides a Vernier slope which can be exploited for temperature and strain sensing, with its sensitivity appreciably magnified versus the single-interferometer based sensor. Besides, the magnification as well as the shifting direction of spectrum can be controlled by a length comparison between the PMF and HCF. Further, simulative studies have preliminary validated theoretical predictions. Subsequently, experiments were carried out to test the performance of the proposed sensor. Obtained results show that it can achieve high sensitivities of −88.73 nm/$^{\circ }$C and 1.61 nm/$\mu \epsilon$ for temperature and strain, respectively, which outperform most of the existing ones. Significantly, It has been theoretically illustrated that the wavelength-dependent characteristics of TCB and SCB allow the temperature and strain to be simultaneously measured by matrix demodulation. And the experimental results have confirmed the capability of simultaneous measurement for such a sensor. It is worth saying that the proposed Vernier sensor not only presents high sensitivities, but also exhibits a simple structure, compact size and light weight, as well as demonstrates ease of fabrication and hence high repeatability, thus holding great promise for widespread applications in daily life and industry world.
Funding
Natural Science Foundation of Guangdong Province (2022A1515011352, 2022A1515011481, 2023A1515011196, 2023A1515011756).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available from the corresponding authors upon reasonable request.
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