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Abstract
In this paper, the Karhunen-Loeve transform (KLT) and wavelength domain interferometric spectral singular value decomposition (SVD) are used for the first time to demodulate the pressure of an optical fiber Fabry-Perot (F-P) micro-pressure sensor, and the feasibility of the proposed method is demonstrated experimentally. The eigenvalue decomposition of the dominant frequency part of the beam-domain interferometric spectrum after the fast Fourier transform (FFT) is performed using KLT, and the singular value decomposition of the wavelength domain interferometric spectrum is additionally performed using SVD. Both methods use high-order eigenvalues as a new metric and then derive the relation between the new metric and the reference pressure. The two demodulation methods are experimentally compared, and we used an optical fiber F-P pressure sensor with unknown structure and material for pressure measurements. Even though the interferometric spectral signal is acquired using a coarse spectrometer (2.5 nm wavelength resolution), one can still achieve high demodulation accuracy with both algorithms. However, the SVD demodulation accuracy decreases significantly after reducing the spectral data points in the wavelength domain from 1566 to 783. KLT still has high demodulation accuracy and linearity after spectral data points are reduced from 1024 to 256 in the wavenumber domain. The satisfactory linearity of the measured pressure versus reference pressure and low reading errors validate the feasibility of the proposed demodulation algorithm.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Pressure monitoring inside human body is significant for assessing organ and tissue damage in the field of bio-medicine. Electromechanical pressure sensors are bulky and costly. Compared with the traditional electromechanical pressure sensor, the optical fiber Fabry-Perot (F-P) micro pressure sensor has various advantages, such as high sensitivity [1], anti-electromagnetic interference, and compact size [2]. Consequently, optical fiber F-P minute pressure sensing can realize precision medicine for many medical applications, e.g., cardiovascular assessment [3]. The performance of optical fiber F-P minute pressure sensors depends on the specific task assigned to the sensors. According to the collection standards for medical pressure analysis, in the field of cardiovascular assessment, the sensor’s measurement error shall be within 3$\%$ of reading over the range from $6.7$ kPa to $40$ kPa [4]. Therefore, high demodulation accuracy is required for pressure sensors.
The sensing core of the optical fiber F-P pressure sensor is to convert pressure changes into cavity length changes, so that the interferometric spectrum contains information about the F-P cavity length, which is then converted into the measured pressure by demodulating the cavity length. Therefore, research on interference spectrum demodulation technology is of great significance.
White-light interferometry (WLI) demodulates the interference spectrum to obtain physical quantity directly among the existing demodulation methods. WLI is widely accepted in the optical fiber sensing measurement area because of its high precision and absolute measurement. In recent years, the demodulation algorithm for WLI has developed sufficiently [5,6]. They can be classified into two types based on the treatment of the F-P cavity initial phase $\varphi_0$: Type I and Type II demodulation [7]. Fast Fourier transform (FFT) is a commonly used demodulation method to determine the optical path difference (OPD) by calculating the number of stripes through the FFT [8,9]. This kind of demodulation method based on direct frequency estimation is called Type I demodulation [10,11]. In Type I demodulation, the initial phase is either unknown or unused. Although FFT demodulation method can provide high demodulation speed, its accuracy and resolution are limited to hundred nanometers [12]. In addition, it has many other problems. For example, spectrum leakage and fence effect caused by the non-synchronized sampling of the interference spectrum also can reduce the demodulation precision during the FFT process [13]. If reference pressure changes during the process of interference spectrum acquisition, it will lead to the failure of Type I demodulation of optical fiber F-P sensor. Although improved methods for Type I demodulation have been proposed for the above problems, such as cubic spline interpolation [14], instantaneous frequency method [15] and window four-spectrum-line interpolation FFT [16], the coarse spectrometer limits the demodulation accuracy of Type I demodulation. In contrast, Type II demodulation utilizes a known initial phase with high resolution, large dynamic range, and robustness [17]. Optical fiber F-P sensors have used Type II demodulation to apply in many different application areas [18–21]. In practical applications, the value of the initial phase $\varphi _0$ of the sensor is usually determined by the structure of the sensor, and different surface materials or transmission media may produce a non-zero initial phase. In optical fiber F-P sensor, the initial phase $\varphi _0$ depends on multimode propagation [22] or beam divergence in an extrinsic F-P cavity [23], etc. Because of the low quality of the optical fiber F-P cavity, beam divergence or non-parallelism occurs, causing the initial phase to shift with cavity length. So, we need an accurate physical model of the sensor before demodulation to estimate the initial phase. Unfortunately estimating the initial phase requires a challenging level of experimental control and determination of the initial phase, so the experimental conditions are relatively harsh, which is difficult for us to achieve the initial phase in terms of our experimental conditions. Even if the initial phase is known, demodulation results will jump caused by the 2$\pi$ periodical total phase [24–26]. Type II demodulation accuracy depends on the initial phase [27]. Therefore, when the initial phase is not uncertain, it is impossible to achieve high-precision demodulation results with Type II demodulation.
We collected the interferometric spectrums by the coarse spectrometer at different pressures. Then use Type I demodulation to demodulate them. The result shows that the Type I demodulation is not suitable for micro pressure measurement using a coarse spectrometer and the initial phase of the F-P cavity cannot be derived from the acquired interference spectrum. In addition, we simulated the demodulation results at different F-P cavity initial phases. The simulation results show that the initial phase accuracy determines the accuracy of demodulation results. Li et al. proposed an algorithm based on maximum likelihood estimation to achieve absolute optical path length demodulation with high sensitivity and noise resistance [24]. Yu et al. proposed a fast white light interferometer demodulation algorithm using Buneman frequency estimation and total phase, which provided a high cavity length resolution of 0.071 nm [17]. Yang et al. proposed a high-speed and high-resolution demodulation algorithm that provides a $0.027$ nm cavity length resolution using 384 spectral data points [12]. However, it is quite difficult to obtain accurate initial phase by coarse spectra only when the sensor structure and material are unknown. The low quality of the optical fiber F-P cavity complicates demodulation, thus there is still a need to investigate a more effective demodulation approach.
In this paper, new mathematical tools for analyzing signals, Karhunen-Loeve transform (KLT) and singular value decomposition (SVD), are used to demodulate white light interferometric signals. KLT is a mathematical tool superior to FFT. KLT is similar to FFT but not limited by the conditions of FFT. The KLT can be applied to random functions that are non-stationary in time. This is a sheer advantage of the KLT over the FFT, inasmuch as the FFT rigorously applies to stationary processes. The KLT can detect signals embedded in noise down to unbelievably small values of the Signal-to-Noise Ratio, like $10^{-3}$ or so [28,29]. KLT rigorously applies to any finite bandwidth, rather than applying to infinitely small bandwidths only as the FFT does. K. Kinjalk et al. designed a tilt sensor and used KLT to analyze the spectrum. The measured angle’s linearity versus true angle is satisfactory, and the repeatability is excellent. The range of measurement of tilt angle is found to be $0^{\circ }\sim 3^{\circ }$ with a resolution of $0.0008^{\circ }$ [30]. M. Shaimerdenova et al. used a tilted fiber Bragg grating (TFBG) as a refractive index sensor and used KLT to demodulate the TFBG to measure the refractive index of the sucrose solution. KLT can successfully demodulate small refractive index changes, and yields a resolution in the order of $10^{-5}$ RIU [31]. On the other hand, SVD has a wide range of applications in signal processing and statistics. We can discard the characteristic information of the noise signal and reconstruct the useful characteristic signal [32]. Therefore, KLT and SVD can become mathematical tools for white light interference demodulation.
In this paper, KLT and SVD methods are used for optical fiber F-P pressure sensors demodulation. KLT decomposes the eigenvalue of the domain frequency part to convert the domain frequency variation with pressure into the variation of the new metric and pressure; SVD decomposes the wavelength domain interference spectrum singular values to convert the spectral variation with pressure into the variation of the new metric. The key of KLT is to perform eigenvalue decomposition of the signal [28,29,33,34]. Generally speaking, high-order eigenvalues contain the most of the energy in the interference spectra. As described by Maccone [35], KLT encodes key information in its high-tank eigenvalues; SVD distinguishes the effective features of the singular values from noise features, and the feature vector corresponding to the high-order singular value accounts for the most proportion of signal energy [32,36]. So we chose high-order eigenvalues as the new metric to establish a functional relation with the reference pressure for pressure demodulation. We compared the demodulation results of the two algorithms at different data points, and the results show that both KLT and SVD have excellent linearity when the spectral data points are high. After reducing the data points, the demodulation accuracy of SVD drops significantly, but KLT has better demodulation accuracy and linearity. So we chose KLT to demodulate the coarse spectra with pressure intervals of $1$ kPa and $0.1$ kPa. The demodulation results show that the demodulation accuracy is expected to meet the medical pressure analysis demonstration standard in the field of cardiovascular assessment.
2. Demodulation process
The interference spectrum of a typical F-P cavity applying a white light source in the wavenumber domain is expressed as:
Where $I_0$ is the power of the white light source, $\gamma$ is the fringe visibility, $L$ is the optical path length (OPL) of F-P cavity, $\varphi _0$ is the initial phase, $k$ is the wavenumber defined by $k=2\pi /\lambda$. $\cos (2Lk+\varphi _0)$ is the AC component after mean value removing and normalization.Theoretically, a change in reference pressure will cause a change in OPL certainly. Equation (1) shows that a change in OPL cause a change in the dominant frequency of the AC component and a phase shift. However, the wavelength resolution of the spectrometer is not high, and the reference pressure changes are small, which leads to insignificant changes in the dominant frequency.
We analyzed the influence of Type I and Type II demodulation on the measurement results of the optical fiber F-P pressure sensor when using the coarse spectrometer.
2.1 Type I demodulation result
We used an AvaSpec-ULS2048XL-EVO spectrometer with a wavelength resolution of $2.5$ nm to collect the interferometric signals of the $OPP-M200$ sensor produced by Opsens at different reference pressures, and then process the data with a system noise ratio (SNR) > $200$. Refer to Fig. 8 for a schematic diagram of data collection. We obtained discrete signals for N sampling points in the wavelength domain of the interference spectrum. Let $\lambda$ be the N wavelength samples, $I(\lambda )$ be the amplitude of the reflection spectrum corresponding to $\lambda$. Moreover, the data points in the wavelength domain are not uniformly sampled as well as the transformation in the wavenumber domain. The frequency spectrum after the FFT process in the wavenumber domain is shown in Fig. 1(a), the Full Width at Half Maximum (FWHM) of the dominant frequency is relatively large, reaching 7.1. The large FWHM value will greatly reduce the pressure measurement resolution. Figure 1(b) is an enlarged view of the peak position of Fig. 1(a). The dominant frequency shift is little within a range of pressure, so we obtained a specific shift value from the spectrum hardly. Figure 2 shows the Type I demodulation results at different spectral data points. As the spectral data points decrease, the variation of the dominant frequency peak with pressure becomes even less pronounced. Figures 1 and 2 show that for spectral signals acquired from low-resolution spectrometers, the Type I demodulation results are no longer able to distinguish the variation of the dominant frequency and it is also hard to accurately derive the initial phase from the interferometric spectrum. In summary, the Type I demodulation is not suitable for demodulating interference spectral signals acquired from a coarse spectrometer.
Fig. 1. (a) Dominant frequency part under different reference pressures. (b) A larger version of the dominant frequency peak.
2.2 Type II demodulation result
We obtained the demodulation result by demodulating the simulated spectrum with Type II demodulation. We first collected the spectrum of the light source, then used it in the simulations. The light was set to vertically incident into the simulated F-P cavity to obtain an interference spectrum, which is shown in Fig. 3. We set the cavity step size to $8.5$ nm and decrease it from $16\;\mathrm {\mu }\textrm {m}$ to $15.405\;\mathrm {\mu }\textrm {m}$ to simulate the pressure increases from $0$ kPa to $35$ kPa. As shown in Fig. 3, the thickness of metal chromium is $20$ nm, and it forms a low-finesse F-P cavity with the Pyrex7740, air, and SiO$_2$ substrate. Figure 4 shows the results of Type II demodulation with the clear initial phase . The total phase randomly causes the demodulation result to jump [12]. After that we eliminated the jump and changed the initial phase value to get the demodulation result, the specific results are shown in Figs. 5(a)–5(b). The actual cavity length $L_0$ is obtained from the reality initial phase $\varphi _0$, while the demodulation results $L_1$ and $L_2$ are obtained by adding and subtracting $1.2$ rad from $\varphi _0$, and the demodulation result $L_3$ is obtained by adding a random number to $\varphi _0$. As shown in Fig. 5(b), since the initial phase of $L_3$ fluctuates up and down $\varphi _0$, $L_3$ also fluctuates up and down $L_0$, the demodulation cavity length changes with the initial phase $\varphi _0$. We found that the demodulation accuracy is directly related to $\varphi _0$. The inaccurate initial phase leads to demodulation errors. In addition, we obtained the initial phase corresponding to the peak of the dominant frequency based on the experimentally collected spectra and obtained the demodulation results as shown in Fig. 6. The jumps in the initial phase $P_n$ deriving from the spectrum in Fig. 6 cause jumps in the demodulation cavity length $L_n$. The large error between $P_n$ and the actual initial phase leads to a large error in $L_n$ measurement. In summary, due to the non-synchronous sampling and fence effect of the coarse spectrometer, we can only observe the spectral line near the real spectral line, which would introduce a large measurement error in the measurement calculation [37,38].
Fig. 5. (a) Different initial phases and cavity length relations. (b) Type II demodulation results with different initial phases.
Fig. 6. The initial phase corresponding to the peak of the dominant frequency and the result of Type II demodulation using this initial phase.
In addition, white light interference spectrum is not strictly a periodic signal in the wavelength domain and the signal period varies with the wavelength. The FFT process broadens the dominant frequency components and introduces phase errors that lead to demodulation failure [39,40]. The interference spectrum at different reference pressures is shown in Fig. 7(a). Figure 7(b) is a partial enlarged view of Fig. 7(a). It can be seen in Fig. 7(a)-(b), the change of the spectrum contains the information of the change of the F-P cavity length, so the change of the reference pressure leads to the change of the wavelength domain interferometric spectrum. However, the spectral change is not obvious for a pressure change of $0.5$ kPa, and the low wavelength resolution of the spectrometer collecting the interferometric signal makes it difficult to obtain the accurate wavelength shift from the spectrum.
Fig. 7. (a) Wavelength domain interference spectrum under different reference pressures. (b) Local wavelength domain interference spectra under different reference pressures.
To improve the demodulation accuracy and avoid the problems of demodulation failure due to few spectral data points, poor quality of the optical fiber F-P cavity, unknown structure and material, we adopted more effective demodulation methods: KLT and SVD. We respectively used KLT and SVD to analyze the interferometric signal in the frequency and wavelength domains of the spectrum. Then, we demodulated the pressure directly and compared the demodulation of KLT and SVD results. We found that KLT has better pressure measurements than SVD with fewer spectral data points.
3. Theory of KLT and SVD
3.1 KLT method
KLT adopts the method of detecting carrier signal hidden in noise proposed by Maccone, which is particularly suitable for detecting the change from small-signal to noise [28,29]. Similarly, one can use KLT to convert the change of the interference spectrum in the wavelength domain into the change of the new metric.
First, we established the relation between a new metric of frequency change and reference pressure. We applied the FFT to $I(k)$, obtaining $G$:
Where for each $k=1, 2,\ldots, N$. Then, to apply the KLT, the new array is transformed into its symmetric Toeplitz matrix:3.2 SVD method
SVD is a mathematical theory [42], which performs an orthogonalized calculation on a real matrix $H\in R^{m\times n}$. Regardless of whether the rows and columns of the matrix are related, there must be two matrices $U\in R^{m\times m}$ and $V\in R^{n\times n}$ that make Eq. (7) hold.
Where $S$ is the singular value matrix of matrix $H$, $S=diag(\sigma _1, \ldots, \sigma _k, \ldots, \sigma _i, \ldots, \sigma _r)$, and $\sigma _i$ is any singular value of matrix $H$. $r=rank(H)$ is the rank of the matrix. $\sigma _r$ is the last eigenvalue of the eigenvalue sequence. To use SVD for the discrete wavelength domain interference spectrum $I((\lambda ))$, we must construct a Hankel matrix $H$ according to Eq. (8).4. Experiment and discussion
To test the capability of the proposed demodulation algorithm, we conducted experimental verification. The experiment setup is shown in Fig. 8. The light source is a broadband white light. The model of the spectrometer is: AvaSpec-ULS2048XL-USB2 (wavelength range: $200\sim 1160$ nm). Its grating line number is 300$/$mm and slit width is $50$ $\mu m$, so its wavelength resolution is $2.5$ nm. The pressure controller model is: FLUKE6270 with control accuracy of 0.001$\%$. The pressure set by FLUKE6270 is used as the reference pressure (pressure measurement range: $-50\sim +300$ mmHg). The electron microscope image of the sensor with a 100x magnification is shown in Fig. 9(a), which has a diameter of 0.3mm. The sensor model is OPP-M200 produced by Opsens and the internal CT of the sensor is shown in Fig. 9(b).
Fig. 9. (a) The electron microscope image of the sensor with a 100x magnification. (b) CT diagram of the sensor.
We collected the interference spectrum with the pressure increasing from $0$ kPa to $35$ kPa as pressure calibration data with a step length of $1$ kPa, and used KLT and SVD to calculate two new metrics and established the relation with reference pressures respectively. We then collected the interference spectrum with reference pressures from $0.5$ kPa to $34.5$ kPa, a step length of $1$ kPa, and an interference spectrum with pressure from $12.3$ kPa to $13$ kPa, a step length of $0.1$ kPa. Because the change in the interference spectrum for a pressure change of $1$ kPa is not obvious, the pressure interval is set at $10$ kPa and the pressure was increased from $0$ kPa to $30$ kPa. The interference spectra at different pressures are shown in Fig. 10.
To ensure a stable measurement environment, we have designed and manufactured a pressure tooling that connects the white light signal to the optical path in the pressure tooling via a connector; an air inlet screw is provided on the pressure tooling to connect the pressure controller to pressurize the pressure tooling. A schematic diagram of the optical connection of the micro pressure measurement system is shown in Fig. 11. After setting the FLUKE pressure controller to the assigned value, nitrogen gas will be fed into the pressure tooling until the inside pressure reaches the assigned value. At this point, the spectrum we have collected is the interference spectrum at the assigned pressure value. Then, we demodulated pressures corresponding to these interference spectrums and compared them with reference pressures to verify the algorithm’s feasibility.
In this section, the proposed demodulation algorithms are tested, analyzed, and discussed. Firstly, we used KLT and SVD severally to establish the functional relation between $\xi _1$, $\sigma _1$, and the reference pressure, as shown in Fig. 12. It is worth noting that $\xi _1$ and $\sigma _1$ have large big jumps in Fig. 12(a)-(b), the jump amplitude of $\sigma _1$ is greater than $\xi _1$. This is due to the different integration times of the spectrometer acquisitions. Therefore, the same integration time of the spectrograph should be guaranteed when collecting the interference spectrum. In addition, the curve in Fig. 12(b) has more fluctuations than the curve in Fig. 12(a), which may be caused by fluctuations in the light source’s light output. Then, we ignored the jump point and obtained two functions for demodulation. We acquired the spectra for the reference pressure interval of $1$ kPa and $0.1$ kPa and reduced the spectral data points, obtained the spectra respectively at different data points. The demodulation results of KLT are shown in Fig. 13 and 14. In Fig. 13, Linear Fit $R^2$ is 0.99988 at 1024 data points and Linear Fit $R^2$ is 0.99948 at 256 data points. In Fig. 14, Linear Fit $R^2$ is 0.96852 at 1024 data points and Linear Fit $R^2$ is 0.96508 at 256 data points. The linearity only slightly decreases in Fig. 14 as the 1024 data points are reduced to 256, and it is also able to distinguish changes in reference pressure of $0.1$ kPa. The optical fiber F-P sensor cavity length is very sensitive to changes in external physical quantities when measuring pressure, and factors such as external vibrations associated with the sensor cavity length, causing some of the errors. Figure 14 has a small full-scale output compared to Fig. 13, so the linearity of measuring $1$ kPa pressure interval is greater than that of measuring $0.1$ kPa pressure interval at the same data point.
Fig. 12. (a) The dominant frequency eigenvalue changes with the reference pressure (KLT). (b) Wavelength domain interference spectrum eigenvalue changes with the reference pressure (SVD).
Fig. 13. The demodulation results of KLT at reference pressure intervals of $1$ kPa (a) 1024 spectral data points. (b) 512 spectral data points. (c) 256 spectral data points.
Fig. 14. The demodulation results of KLT at reference pressure intervals of $0.11$ kPa (a) 1024 spectral data points. (b) 512 spectral data points. (c) 256 spectral data points.
The demodulation results of the reference pressure interval $1$ kPa and $0.1$ kPa spectra obtained by SVD at different spectral data points respectively, as shown in Fig. 15–16. The linearity in Fig. 15 decreases as the data points decrease. SVD has difficulty in resolving the $0.1$ kPa pressure change as seen in Fig. 16. Comparing Fig. 14 and Fig. 16, KLT’s ability to distinguish $0.1$ kPa pressure changes is significantly better than SVD’s ability. According to Fig. 12, SVD is more susceptible to the intensity of the interferometric spectrum compared to KLT. SVD is greatly affected by the intensity because it decomposes eigenvalues from the wavelength domain of the spectrum. The intensity of the interference spectrum is easily affected by environment and other influences, which reduce the SVD demodulation accuracy. The sensor is also susceptible to environmental influences such as temperature in the laboratory and vibration caused by pressure controlling equipment. Environmental effects are more pronounced when measuring pressure changes of $0.1$ kPa. SVD is particularly affected by fluctuations in the optical output of the light source and small variations in the intensity of the interference spectrum which are caused by multiple fusion points in the optical path. In contrast, KLT reflects the partial change of the dominant frequency, and the influence of interference spectrum intensity changes is relatively small. Therefore, KLT is more suitable for coarse spectrum demodulation than SVD.
Fig. 15. The demodulation results of SVD at reference pressure intervals of $1$ kPa (a) 1566 spectral data points. (b) 783 spectral data points. (c) 522 spectral data points.
Fig. 16. The demodulation results of SVD at reference pressure intervals of $0.1$ kPa (a) 1566 spectral data points. (b) 783 spectral data points. (c) 522 spectral data points.
We finally chose KLT to demodulate the interferometric spectral signal acquired by a coarse spectrometer. KLT is suitable for a low-cost demodulation scheme because of its advantages such as lower wavelength resolution requirement for the spectrometer and less influence by the light output fluctuation of the light source.
In Fig. 17, the pressure reading error at 1024 data points demodulated by KLT is analyzed and plotted as a percentage. In the reference pressure range of $6.5$ kPa$\sim 35$ kpa, the reading error at $7.5$ kPa is 3.147$\%$, and the reading error at other locations is less than $\pm {3\%}$. As shown in Fig. 13(a), the measured pressure curve and tested data points can be observed, in which the fitting curve can be acquired as $y=-0.033+1.001x$ with a standard error (SE) of 0.1121885. Taking twice the SE as the resolution [43], the estimated pressure resolution is $0.224377$ kPa. The demodulation error is shown in Fig. 18(a). The overall measurement error is less than $0.36$ kPa in the measurement range from $0$ kPa to $35$ kPa. By neglecting large errors represented by the blue measurement points in the figure, the measurement error of the remaining red measurement points can be as small as $0.13$ kPa. In Fig. 18(b), in the measurement range from $12$ kPa to $13$ kPa, the overall measurement error is less than $0.2$ kPa. It is believed that the reading error can reach $\pm {3\%}$ and the demodulation precision can reach $\pm 0.13$ kPa to meet the standards for medical pressure analysis in the cardiovascular field under a more stringent experimental environment [4]. Moreover, we are working with other teams to design and fabricate optical fiber F-P pressure sensors. In the next stage, we will analyse the influence of factors, such as sensor structure, on the pressure demodulation in detail.
Fig. 18. (a) Measurement error in the measuring range from $0$ kPa to $35$ kPa. (b) measurement error in the measuring range from $12$ kPa to $13$ kPa.
5. Conclusion
In this paper, we investigated and compared several demodulation algorithms for optical fiber F-P micro pressure sensors. Type I demodulation is unable to distinguish the variation of the dominant frequency, and the ability to distinguish the dominant frequency is further declined as the number of data points decreases. Although Type II demodulation can obtain high-precision demodulation results of the F-P cavity length, factors such as optical fiber F-P cavity structure and beam divergence affect the F-P cavity initial phase, causing large demodulation errors. The large FWHM makes the initial phase obtained from the spectrum inaccurate, resulting in disappointing Type II demodulation results. Therefore, the interferometric spectra of pressure signals collected by the coarse spectrometer are not suitable for Type I and Type II demodulation. We used KLT and SVD to demodulate pressure of a optical fiber Fabry-Perot micro pressure sensor for the first time. The high-order eigenvalues, $\xi _1$ and $\sigma _1$, are constructed by KLT and SVD in the frequency and wavelength domains, respectively, as a function of the reference pressure for pressure demodulation. Compared with Type I demodulation, KLT and SVD significantly improve the demodulation accuracy. They also overcomes the issue in Type II demodulation, i.e., need to know the the initial phase to increase the demodulation accuracy. High-accuracy pressure demodulation results can be obtained without the need for a high-resolution spectrometer. Fianlly, We compared the demodulation results of KLT and SVD at different spectral data points. KLT has higher demodulation accuracy and better ability to resolve pressure variations than SVD. It is expected that the KLT demodulation technique has high potential for application in the field of optical fiber F-P micro-pressure measurement.
Funding
Shenzhen Research Foundation (JSGG20201103091401005).
Acknowledgments
We thank the editors and the anonymous reviewers for their careful reading and valuable comments that were very helpful for revision and improving this paper.
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.
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