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Abstract
Multiplexed fiber optic Fabry–Perot interferometer (FPI) sensors are well known for their precision, simple construction, simpler wiring, and high sensing qualities. However, the limitations on existing demodulation methods degrade the measurement accuracy of multiplexed FPI sensors and necessitate large cavity length differences. In this paper, we propose an adaptive high-precision demodulation method based on vector matching and cluster-competitive particle swarm optimization (CCPSO), which transforms cavity length demodulation into searching for the global extreme. The proposed CCPSO, which uses agglomeration within clusters and competition between clusters simultaneously, enables the improvement of the global extreme search capabilities. The theoretical analysis and experimental results show that the proposed demodulation method decreases the lower limit of the needed cavity length differences to 22 μm, which is reduced by 76.9% compared with the fast Fourier transform-based method. An accuracy of 1.05 nm is achieved with a cavity length difference of 27.5 μm and a signal-to-noise ratio of 36.0 dB for the noise.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, with the rapid growth of the Internet of Things and industrial intelligence, the importance and needs of sensor technology have grown prominently. Fiberoptic sensors have significant potential research and application value due to their advantages of small size, corrosion resistance, and immunity to electromagnetic fields. As one of the important branches of fiberoptic sensors, Fabry–Perot interferometer (FPI) sensors have demonstrated the advantages of a great dynamic measurement range [1] and high resolution [2]. In studying the high-precision demodulation of single FPI sensors, several methods based on the fast Fourier transform (FFT) have been proposed. Zhen Wang et al. [3] proposed an improved method based on the FFT white-light interferometry. They eliminated the chirp in the optical spectrum, and the standard deviation of the method was decreased to 4 nm. Zhihao Yu et al. [4] developed a white-light interferometry demodulation algorithm for low-finesse FPIs. By computational simulation, the algorithm standard deviation was calculated to be 143 pm. Based on low-finesse Fabry–Perot interferometry, Ke Chen et al. [5] proposed a fiberoptic Fabry–Perot cantilever microphone; the pressure responsivity was 211.2 nm/Pa, and its signal-to-noise ratio (SNR) was over ten times higher than that of a reference condenser. In addition to the FFT, other demodulation methods have been developed. Ying Wu et al. [6] proposed a demodulation method based on spectrum sampling and reconstruction with a resolution of 4.59 nm. Wenyi Liu et al. [7] proposed another method based on least-squares fitting, which interrogated in a wide range. Its demodulation error was less than 12 nm. Algorithms based on discrete gap transforms (DGTs) [8, 9] and wavelets [10] have also been researched.
In addition to the advantages of single FPI sensors, multiplexed FPIs sensors on a single optical fiber are able to share one light source and one demodulation system, which will significantly reduce the hardware cost compared with the system with the same number of individual FPI sensors. Besides, simpler wiring is needed in multiplexed FPIs sensing system, which is important for applications with strict size restrictions, such as artificial satellite and aircraft engine. Moreover, multiplexed FPIs sensors are helpful for distributed sensing. Therefore, the multiplexing capability of FPI sensors has attracted great interest among researchers and has demonstrated their remarkable potential application value. For example, Rongtao Cao et al. [11] proposed a multiplexable FPI chemical sensor to sense hydrogen concentrations. Mohan Wang et al. [12] proposed a method of inscribing low-loss FPIs with femtosecond laser to sense high temperature and explored their spatial multiplexability. Hongchun Gao et al. [13] proposed a dual-cavity FPI sensor to sense high temperature and pressure. However, although some demodulation methods have been proposed for multiplexed Fabry–Perot sensors, these methods require a long cavity length difference. For example, Yi Jiang et al. [14] proposed a demodulation method for two multiplexed FPIs and achieved a resolution of 1 nm. However, this method needed a 2000 μm-long cavity difference. Zhihao Yu et al. [15] proposed a demodulation method for multiplexed FPIs and achieved a standard deviation of 0.44 nm. However, this method required the cavity length difference in the FPIs to reach at least 95.4 μm. According to fiber optics [16, 17], the beam propagating in the FPI cavity is a divergent beam, which can be approximated as a Gaussian beam. When light passes through an FPI cavity, its intensity loss can be expressed as:
According to Eq. (1), the light intensity loss curve of an FPI, which is made of single-mode fibers, can be obtained, as shown in Fig. 1. The single-mode fibers have a mold field diameter of 10 μm. The ${\lambda _0}$ and ${n_g}$ are set as 1550 nm and 1, respectively. It can be seen that the light intensity loss increases with the cavity length. The light intensity loss of an FPI will not only reduce its own signal intensity but also reduce the signal intensity of FPIs connected in series after it. Therefore, the longer the FPI cavities, the less the number of FPIs can be connected in series. Therefore, short cavity length differences between multiplexed FPIs are needed to enhance their multiplexing capability. Besides, because the white noise caused by the sensing system is less affected by the cavity length, the SNR of the FPI signal will reduce and the measurement error will increase as the light intensity loss increasing. Overall, a demodulation method that can accurately demodulate multiplexed FPIs with short cavity length differences is urgently needed.
In this paper, a high-precision demodulation method based on vector matching and cluster-competitive particle swarm optimization (CCPSO) is proposed for multiplexed FPIs. Vector matching in multidimensional spaces, including representation spaces and hidden spaces, transforms cavity length demodulation into searching for global extremes. We have improved the method of particle swarm optimization (PSO) by adding cluster competition to improve its capability of searching for the global extreme. This improved method is called CCPSO.
To reduce the computational effort of demodulation, we divide the demodulation process into two steps. First, the DGT is used to coarsely demodulate cavity lengths and determine the number of multiplexed FPIs; it adaptively adjusts the dimensions of the multidimensional space in the next step. The dimension error of the DGT is demonstrated to be less than 4.25 μm when the differences in the cavity lengths are no less than 22 μm. Second, within the error range of the DGT, CCPSO is used to precisely match the cavity lengths of the FPIs in the created vector multidimensional space. After these two steps, an adaptive and accurate demodulation of the multiplexed FPI cavity lengths, which has a large dynamic demodulation range, is achieved. This paper next analyzes the effect factors of the demodulation accuracy, including the important parameters of CCPSO and noise. Finally, different numbers of FPIs are multiplexed and demodulated experimentally to verify the accuracy of the proposed method. The results agree well with those of the theoretical analysis, affirming the demodulation ability of this method. The proposed demodulation method holds great promise in the universal applications of multiplexed FPIs.
2. Demodulation method based on vector matching and CCPSO
2.1 Principle of vector matching
The output signal of a single FPI can be expressed as:
When multiple FPIs are connected in parallel, the reflected signals from the FPIs are transmitted without interference, so the output spectrum is the sum of the reflected signals from each FPI. However, this is not the case when multiple FPIs are connected in series, as white light is modulated to varying degrees before reaching the next FPI cavity. Figure 2 shows a series and parallel diagram of multiple FPIs. ${I_k}$ represents the light incident on the k-th parallel branch. ${T_{k,n}}$ represents the light incident on the n-th FPI after the previous $n - 1$ FPIs in the k-th parallel branch. $R{^{\prime}_{k,n}}$ represents the reflection spectrum of the n-th FPI in the k-th parallel branch. ${R_{k,n}}$ represents the spectrum incident on the splitter after $R{^{\prime}_{k,n}}$ is transmitted through the $n - 1$ FPIs. ${R_k}$ represents the superposition of the reflection spectra of all the FPIs in the k-th parallel tributary. There are ${N_k}$ FPIs in series in the k-th parallel branch.
After analyzing the series of FPIs, the expression for ${R_{k,n}}$ can be computed as:
For a low-finesse FPI, ${\gamma _{k,i}} < < 1,$ so Eq. (3) can be simplified to:
After filtering and normalizing the amplitude, the expression for the superimposed spectra of multiple FPIs reflected in series and parallel is shown in Eq. (7).According to Eqs. (7) and (8), each point in the multidimensional space can construct a spectral $\overline {{R^V}} .$ We define the matching degree for the points in the space by Eq. (9):
When the matching degree ${F_v}$ reaches its maximum in the multidimensional space $V,$ the $L{^{\prime}_{k,n}}$ value corresponding to the point v is the cavity length of the FPIs.
2.2 DGT method and error analysis
The demodulation process was divided into two steps. First, the DGT was used to identify the number of the multiplexed FPIs and coarsely demodulate their cavity length. The range of the cavity length of the multiplexed FPIs is located by error analysis of the DGT demodulation. Then, within that range, exact demodulation was performed based on the CCPSO and the matching degree ${F_v}$ obtained by the vector matching method.
The principle of the DGT can be expressed as Eq. (10). It is based on the correlation between the simulated FPI spectra of different cavity lengths (${L_m}$) and the spectra ($R$) to be demodulated.
In the demodulation of a single FPI, there is one main peak and several side peaks in $I({L_m}),$ and the main peak corresponds to the cavity length. However, in the demodulation of multiplexed FPIs, there are several main peaks in $I({L_m}),$ and the peaks will overlap, which will reduce the demodulation accuracy. The ability of the DGT to recognize the number of multiplexed FPIs is constrained by the similarity of the FPI cavity lengths. When two FPIs have very similar cavity lengths, their main peaks merge into one, resulting in misidentification, as shown in Fig. 3(a). To determine the minimum value of the difference between the cavity lengths, we first need to determine the width of the main peak. $I({L_m})$ is squared and then derived. By solving for the zero point, Eq. (11) is obtained:
Fig. 3. DGT demodulating multiplexed FPIs. (a) DGT of two FPIs. (b) Miscalculation the number of FPIs.
Equation (11) indicates that ${W_{\textrm{main}}}$ is independent of the cavity length and is related only to the demodulation wavelength range. In this study, the demodulator wavelength bandwidth used was 1510-1590 nm, and the main peak width was calculated to be 30.03 μm. MATLAB calculations verified this conclusion. Figure 3(b) shows the widths of the two main FPI peaks with different height ratios (0.6-1.4) and cavity length differences (0-50 μm). The minimum cavity length difference without peak mismatch is 22 μm, which is lightly smaller than ${W_{\textrm{main}}}.$ The PMD in Fig. 3(b) represents for the peak mismatch divider. The upper right region of the PMD is the area where the miscalculation occurs. In this area, the main peak is wider due to the interaction between the main peaks. We set the cavity length difference divider (CLDD) perpendicular to the coordinate axis with 22 μm as the demarcation point. On the left side of the CLDD, the number of FPIs can be accurately identified. On this side, the maximum demodulation error of the cavity length is 4.25 μm. After the number of FPIs accurately identified by the DGT, the cavity lengths of FPIs can be exactly demodulated by the CCPSO in the next section. 22 μm, the lower limit of the cavity length differences achieved by the DGT, is 76.9% less than the FFT-based demodulation method [15].
2.3 Process of CCPSO
After the coarse demodulation by the DGT, the range of the multidimensional space ${V_K}$ is reduced, which is marked as $\overline {{V_K}} .$ However, even in the reduced range, the computational effort is undoubtedly great if traversal calculations are used at 1 nm intervals to achieve demodulation with nanoscale accuracy. Furthermore, there are multiple extremes of the matching degree ${F_i}$ in $\overline {{V_K}} .$ Taking the demodulation of three FPIs as an example, the distribution of extreme points is shown in Fig. 4. On the manifestation dimensions, there are multiple extremes with similar values, with the smallest difference being only 2.9%. In contrast, there is only one extreme point on each hidden dimension. To reduce the complexity of the search for the global extreme, we distinguish between the manifestation dimension and the hidden dimension. The manifestation space is constructed from the manifestation dimensions (Fig. 4(a)), denoted by ${V_L}.$ Each point in the manifestation space has a hidden space constructed from the hidden dimensions (Fig. 4(b)), denoted by ${V_A}.$ With CCPSO, multiple particle points are set in the representation space to find all the extremes. Only one particle point is set in each hidden space, which greatly reduces the total number of particles needed and significantly reduces the computational effort.
Fig. 4. Parts of multiple extremes generated by three multiplexed FPIs. (a) Manifestation space. (b) Hidden space.
The existence of multiple extreme points with similar values in the demodulation space poses a challenge for an accurate search for the global maximum. Therefore, we need a demodulation method with a strong global search capability. In this paper, we improve the PSO method to obtain the CCPSO method, which can be adapted to the demodulation of multiplexed FPIs. The expression for the CCPSO is as follows:
${J^t}$ creates a barrier band in regions of low adaptation between the extremes. When the velocities are low, the particles cannot cross the barrier band to reach the other extremes. ${c_3}\mathop{{\delta _{\textrm{pks}}}}\limits^\rightharpoonup \odot \mathop{{G_{\textrm{pks}}}}\limits^\rightharpoonup$ in Eq. (12) produces a small oscillation in the hidden domain ${V_A},$ which we call the hidden oscillation. The hidden oscillation and the only extreme ensure that we can use one particle to determine exactly where the maximum value is within the hidden domain. $\mathop{Q_{i,n}^t}\limits^\rightharpoonup $ can cause the particles to form multiple dispersed clusters and to be bound to the clusters. $\mathop{Q_{\textrm{all}}^t}\limits^\rightharpoonup ,$ on the other hand, indicates that the particles are also attracted to the global maximum. Thus, during every iteration, some of the particles are pulled towards the inner maximum of the cluster, while others attempt to break away and approach the global maximum. While approaching the global maximum, some of them will be blocked by the barrier band and return to their cluster. These processes can be seen as global and local extremes competing for particles, and the competition runs through the entire CCPSO. This competition allows the continuous improvement of the demodulation accuracy of each extreme so that each one has the potential to exceed and replace the current global extreme. As a result, premature convergence of the particle swarm is avoided, and the global search capability of the method is significantly improved.
The demodulation of 3 multiplexed FPIs is taken as an example. As the iterations proceed, the CCPSO can be divided into three stages: cluster formation (Fig. 6(a) and (b)), searching for new clusters (Fig. 6(c)), and particle swarm convergence (Fig. 6(d) and (e)). CNOI in the figures represents the current number of iterations.
Figure 6(a) shows the initialized particles in the spatial domain $\overline V .$ Some of the particles are uniformly scattered in the spatial domain, while the others are randomly distributed. In the cluster formation stage, due to the small inertia factor ${w^t},$ the particles around the local extremes converge rapidly and form a cluster, as shown in Fig. 6(b). In the stage of searching for new clusters, ${w^t}$ grows as the iterations proceed, and more particles begin to cross the barrier band to reach another cluster or wander in the spatial domain $\overline V ,$ as shown in Fig. 6(c). These particles have the opportunity to discover and record new extremes hidden in the blocking band and form new clusters. This process compensates for the possibility that there are undiscovered extremes during the cluster formation stage, further enhancing the global search capability of CCPSO. Even if some of the clusters disappear in the process due to the constant escape of the particles, their extremes are noted. Thereafter, in the stage of particle swarm convergence, ${w^t}$ decreases, and the particles converge again (Fig. 6(d)). The accuracy of the search for the global extreme is improved by iteration (Fig. 6(e)).
Fig. 6. Process of CCPSO. (a) Initialized particles. (b) Cluster formation. (c) Searching for new clusters. (d) Particle swarm convergence. (e) End of search.
3. Improvement of the global extreme search ability by cluster competition
In the proposed demodulation method, ${J^t}$ in Eq. (12) represents the barrier band between the clusters of particles. The comparison of the attraction of the global and local extremes to the particles can be represented by the logarithm of the ratio of ${c_1}$ and ${c_2},$ the parameters in Eq. (12), which is ${C_{1/2}} = \lg ({c_1}/{c_2}).$ ${C_{1/2}}$ is greater than 0 when the local extremes are more attractive to the particles. Conversely, ${C_{1/2}}$ is less than 0 when the global extreme is more attractive to the particles. The cluster competition is mainly determined by ${J^t}$ and ${C_{1/2}}$ The cluster competition slows the convergence of the particles, allowing the particles to fully search for the global extreme.
If ${J^t}$ equals 1, instead of following the procedure shown in Fig. 5, there will be no barrier bands between the clusters. The DOP represents for the density of particles. The triangles in Fig. 7(a) represent the probability (${P_{\textrm{GOS}}}$) of finding the correct global extreme in this case when ${C_{1/2}}$ takes different values. In contrast, the circles in the figure represent ${P_{\textrm{GOS}}}$ with barrier bands between the clusters. The higher ${P_{\textrm{GOS}}}$ is, the better the ability to search for the global extreme. From the comparison in Fig. 7(a), it can be concluded that the barrier band significantly improves the ability to search for the global extreme. Furthermore, when ${C_{1/2}}$ is slightly less than 0, the proposed demodulation method maintains a high ${P_{\textrm{GOS}}}.$ The increase in the DOP helps to increase the ${P_{\textrm{GOS}}}.$ Fig. 7(b) shows that ${P_{\textrm{GOS}}}$ with barrier bands increases with the increasing DOP. When the DOP is 6 μm-3 and ${C_{1/2}}$ is near -0.4, ${P_{\textrm{GOS}}}$ is higher than 90%. Each data point in Fig. 7 is obtained statistically by 500 simulation experiments.
Fig. 7. The probability of finding the global extreme in 500 simulation experiments. (a) ${P_{\textrm{GOS}}}$ with a DOP of 6 μm-3. (b) ${P_{\textrm{GOS}}}$ with DOPs of 2 μm-3, 6 μm-3, and 10 μm-3.
4. Effect of parameters and noise on the demodulation accuracy
To analyze the influence of the number of multiplexed FPIs on the demodulation error, accuracy stability, ${S_v},$ is calculated, as shown in Eq. (15). ${S_v}$ describes the ability of a single particle to approach the global extreme. Because the relative largeness, instead of the absolute difference, among matching degrees of particle swarm is used in the demodulation to search for the global extreme, the ${S_v}$ is defined as the ratios between matching degrees.
where ${F_v}$ represents for the matching degree of point v in the multidimensional space $V;$ ${F_V}$ represents for the maximum matching degree in the multidimensional space $V.$According to Eq. (15), ${S_v}$ of multiplexed FPIs with different numbers at different errors are obtained and shown in Fig. 8. Figure 8(b) shows the red dash-line region in Fig. 8(a). “N-FPIs” in Fig. 8 means that FPIs with a number of N is multiplexed. It can be seen that the ${S_v}$ changes with the error, and ${S_v}$ remains positive. The greater the ${S_v},$ the faster the particle approaches the global extreme. Furthermore, the ${S_v}$ of the multiplexed FPIs with different numbers overlap, which means that they have the same accuracy stability. Therefore, the demodulation errors of multiplexed FPIs with different numbers tend to be the same.
Fig. 8. Analysis of accuracy stability. (a) Accuracy stability of multiplexed FPIs with different numbers. (b) Accuracy stability of the red dash-line region in (a).
The factors affecting the demodulation accuracy can be distinguished as intrinsic and extrinsic factors. The intrinsic factors include the DOP and the number of iterations (NOI) during CCPSO. The extrinsic factors are the noise contained in the spectrum. In this section, we build a model of three multiplexed FPIs to explore the influence of each factor on the accuracy of the proposed demodulation method. The cavity lengths of the three FPIs are set as 49.5 μm (FPI-1), 76.9 μm (FPI-2) and 163.5 μm (FPI-3).
We explore the effect of the DOP and NOI on the demodulation accuracy. Figure 9(a) shows that the demodulation error of the proposed method decreases with increasing NOI. Figure 9(b) shows the corresponding standard deviations of the demodulation error at different NOIs and demonstrates the stability of the demodulation accuracy. It can be seen from Fig. 9(a) and (b) that the demodulation error stabilizes after 250 iterations, and this point is called the inflection point of the error. The demodulation error is less than 1 nm when the NOI is greater than 250 iterations. The probability of mistaking the global extreme is close to 0% when NOI is greater than 120 iterations. By contrast, when the NOI is greater than 120 iterations, the probability reaches about 71.0% if the original PSO algorithm is used instead of CCPSO, which makes it unsuitable for the demodulation of multiplexed FPIs. Furthermore, even if the gross errors caused by mistaking the global extreme are removed, its demodulation error is still higher than 5 nm, as shown in Fig. 11(a), which is four times bigger than that of CCPSO.
Fig. 9. The effect of the DOP and NOI on the demodulation accuracy. (a) The demodulation error changes with the increase in the NOI. (b) The standard deviations of the demodulation error at different NOIs. (c) The demodulation error changes with the increase in the DOP. (d) The standard deviations of the demodulation error at different DOPs.
According to Fig. 7(b), when the DOP is less than 6 μm-3, the probability of mistaking the extreme is greater than 10%. The mistake of the extreme will result in a demodulation error greater than 700 nm. To reduce the probability, the DOPs analyzed in this section are greater than 6 μm-3. When the DOP is greater than 6.25 μm-3, the probability of mistaking the global extreme is close to 0%. By contrast, the probability reaches 64.6% if the original PSO algorithm is used instead of CCPSO. The high probability makes the original method unsuitable for the demodulation of multiplexed FPIs. Figure 9(c) shows the relationship between the demodulation errors and the DOP of the proposed demodulation method. Figure 9(d) shows the corresponding standard deviations. There are no significant inflection points in Fig. 9(c) or (d). The demodulation error is less than 1 nm when the DOP is greater than 12.5 μm-3. By contrast, after removed the gross errors, which are caused by mistaking the global extreme, the demodulation error by original PSO method is still above 5 nm, as shown in Fig. 11(b), which is four times bigger than that of CCPSO. Figure 9 illustrates that the NOI has a more significant effect than the DOP on the demodulation accuracy.
We explored the effect of noise at different SNRs on the demodulation accuracy. Figure 10(a) shows the original spectrum of the three multiplexed FPIs without noise. Figure 10(b) shows the blurred interference fringes caused by spectrum noise. Figure 10(c) shows the demodulation error caused by noise. The demodulation error decreases with increasing SNR and is almost evenly distributed around zero. When the SNR is greater than 25 dB, the demodulation error remains less than 1 nm. By contrast, the demodulation error of the original PSO algorithm is over 5 nm when the SNR is greater than 25 dB after the gross errors caused by mistaking the global extreme is removed, as shown in Fig. 11(c). In terms of the probability of mistaking the global extreme, the probability of CCPSO is close to 0% when SNR is greater than 10.5 dB, while the probability of the original PSO method reaches 68.8% at the same SNR. The high probability makes the original method unsuitable for the demodulation of multiplexed FPIs.
Fig. 10. The effect of noise on the spectrum. (a) The original spectrum. (b) The blurred interference fringes. (c) Demodulation error caused by noise.
Fig. 11. The demodulation errors of the unimproved PSO method. (a) The demodulation error changes with the increase in the NOI. (b) The demodulation error changes with the increase in the DOP. (c) Demodulation error caused by noise.
Overall, to achieve high-accuracy demodulation results, the DOP and NOI needs to be greater than 12.5 μm-3 and 250 iterations, respectively. However, as the search for the correct global extreme is a probabilistic problem, there is still a small probability that a local extreme will be mistaken for the global extreme. The way to avoid such mistakes is to repeat the demodulation several times, eliminating the coarse error, and then take the average value.
5. Experimental verification
To verify the accuracy of the proposed demodulation method, two FPIs, three FPIs, four FPIs, and five FPIs were multiplexed, respectively. An experimental setup with a nanoscale motorized stage was used in this experiment, as shown in Fig. 12(a). Figure 12(b) shows a simplified diagram of the experimental setup. FPI-N represents the N-th multiplexed FPI. FPI-1 and FPI-2 were connected in series to form a branch, which was then connected in parallel with other FPIs. FPI-1 was adjusted and fixed by a motorized stage of a fiber fusion splicer. Other FPIs were pasted on the nanoscale motorized stage, and their cavity lengths could be varied in 1 nm steps by moving the stage. To minimize the effect of the mounting angle error of the motorized stage, a small measurement range of 12 nm was used in this experiment. The multiplexed FPIs are shown in Fig. 13. The DOP and NOI were set as 25 μm-3 and 400, respectively. The SNR of the noise was measured as 36.0 dB. An optical demodulator, with an integrated light source inside, was used to receive and display the spectrum of multiplexed FPIs.
Fig. 12. Experimental verification of the demodulation accuracy. (a) Experimental setup for the cavity length measurement of the three multiplexed FPIs. (b) Simplified diagram of the experimental setup.
Fig. 13. The multiplexed FPIs in experiment. (a) Two multiplexed FPIs. (b) Three multiplexed FPIs. (c) Four multiplexed FPIs. (d) Five multiplexed FPIs.
The demodulation results are shown in Fig. 14. It can be seen that the demodulation results of multiplexed FPIs match the ideal values. The maximum errors of two, three, four, and five multiplexed FPIs are 0.73 nm, 1.05 nm, 0.97 nm, and 1.05 nm, respectively. Their standard deviations are 0.41 nm, 0.52 nm, 0.5 nm, and 0.63 nm, respectively. The above analyze of the influence of the number of multiplexed FPIs on the demodulation errors in Fig. 8 shows that multiplexed FPIs with different numbers tend to have the same demodulation error. Therefore, the demodulation error in the verification experiment can be used to represent the demodulation accuracy of multiplexed FPIs. The maximum error in the experiment is slightly higher than the errors shown in Fig. 10(c) when the SNR is 36.0 dB. This phenomenon is because the experimental setup inevitably introduces errors into the FPI cavity measurement. Among the causes of errors, the angular misalignment of pasted FPIs, the slope of fiber reflective surface, and the gap between the optical fiber and the hollow capillary glass tube are the three main factors. In general, the direction of a pasted FPI does not coincide with the displacement direction of the motorized stage. They differ by an angle of ${\theta _D},$ as shown in Fig. 15(a). When the motorized stage is moved by $\Delta {L_m},$ the increase of the FPI cavity length is $\Delta {L_{\textrm{FPI}}}\textrm{ = }\Delta {L_m}\cos {\theta _D},$ which is slightly smaller than $\Delta {L_m}.$ In terms of the slope of fiber reflective surface. It is very challenging to make an absolutely right-angled reflective surface on an optical fiber. Generally, the reflective surface has a slope of ${\theta _f},$ as shown in Fig. 15(b). The optical path length will change to $2{L_{\textrm{FPI}}}\cos ({\theta _f})$ because of ${\theta _f}.$ When the fiber reflection surface moves $\Delta {L_{\textrm{FPI}}}$ along the axis of the fiber, the increase of cavity length of the FPI will be $\Delta {L_{\textrm{FPI}}}\cos ({\theta _f})$ instead of $\Delta {L_{\textrm{FPI}}}.$ The FPIs used in this research were collimated by hollow capillary glass tubes. The gap (${d_{\textrm{FPI}}}$) between the optical fiber and the hollow capillary glass tube will introduce error into the FPI cavity measurement in the presence of the slope ${\theta _f}.$ When the fiber reflection surface deviates from the axis of the glass tube, the cavity length of FPI will change accordingly. The errors mentioned above can be integrated into one equation, as shown in Eq. (16).
Fig. 14. Demodulation results of multiplexed FPIs in experiment. (a) Two multiplexed FPIs. (b) Three multiplexed FPIs. (c) Four multiplexed FPIs. (d) Five multiplexed FPIs.
Fig. 15. Main error factors in experimental setup. (a) Angular misalignment of pasted FPI. (b) The slope (${\theta _f}$) of fiber reflective surface and the gap (${d_{\textrm{FPI}}}$) between the optical fiber and the hollow capillary glass tube.
6. Conclusion
This study developed a high-precision demodulation method for multiplexed FPIs. Representation spaces and hidden spaces are proposed to transform FPI cavity length demodulation into global extreme searching. By adding cluster competition, the global extreme search capability of the proposed method is improved, leading to a high demodulation accuracy. Moreover, the spectrum of multiplexed FPIs is preprocessed with the DGT before demodulated by CCPSO. This approach can automatically identify the number of FPIs and initially determine the range of each cavity length, which makes the proposed method adaptive. The factors, including the number of multiplexed FPIs DOP, NOI, and noise, that affect the demodulation accuracy of the proposed method are analyzed. The results show that multiplexed FPIs with different numbers tend to have the same demodulation error. The NOI has a greater effect on the demodulation accuracy than the DOP. When the NOI is greater than 250, increasing the NOI or the DOP has a very limited effect on reducing the demodulation error. When the SNR of the noise is greater than 25 dB, the demodulation error tends to be less than 1 nm. An experimental setup based on a nanoscale motorized stage is constructed, different numbers of FPIs are multiplexed to verify the demodulation accuracy of the proposed method, and an accuracy of 1.05 nm is obtained with an SNR of 36.0 dB in the cavity change range of 12 nm.
Funding
National Natural Science Foundation of China (51805421, 51720105016, 91748207); National Key Research and Development Program of China (2018YFF0212301).
Acknowledgments
We thank the funding from National Natural Science Foundation of China (Nos.51805421, 51720105016, 91748207); National Key Research and development program of china (No.2018YFF0212301). We also thank the support from the International Joint Laboratory for Micro/Nano Manufacturing and Measurement Technologies.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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