1. Introduction

Optical fiber sensing (OFS) system is widely used in bridge health detection, aerospace and other fields due to its advantages of light weight, small size and anti-electromagnetic interference [1–3]. At present, the OFS systems mainly include Fabry-Perot (FP) cavity interference system [4–6], Mach-Zehnder (MZ) interference system [7,8] and Sagnac loop system [9,10], etc. In general, the sensing mechanism of these systems is that their absolute optical path differences (OPDs) are modulated by external parameters, such as strain [11], temperature [12], refractive index [13] and so on. Therefore, the absolute OPD demodulation is of great significance in OFS systems.

For a single-interference system, the absolute OPD demodulation methods mainly include fast Fourier transform (FFT) [14], fundamental frequency cross-correlation (FFCC) algorithm [15] and high-order harmonic-frequency cross-correlation (HHCC) algorithm [16]. But these methods are difficult to accurately demodulate the absolute OPDs of multi-interference systems. The resolution accuracy of FFT depends on both the spectral interval and OPD interval. In multi-interference systems with narrow spectral interval and small OPD interval, FFT can only roughly demodulate the absolute OPDs. This is because there is a crosstalk between the demodulation results of two adjacent OPDs, leading to the demodulation accuracy decline. Similar to FFT, the demodulation result of FFCC algorithm is also limited by the spectral interval and the OPD interval. And the crosstalk between two adjacent OPDs also exists in the demodulation of FFCC algorithm. The HHCC algorithm is only suitable for the single-interference systems, since the high frequency signal orthogonal to one cosine signal is not orthogonal to the other cosine signal.

Moreover, the absolute OPD demodulation methods of multi-interference systems also include time-division multiplexing (TDM) [17], wavelength-division multiplexing (WDM) [18] and cepstrum-division multiplexing (CDM) [19]. The TDM method results in a very redundant system, which requires ultra-narrow line-width laser, multiple delay lines, photodetectors, high-speed acquisition cards, etc. The WDM method usually requires WDM devices and a wide spectrum of many different wavelength ranges. But neither of these two demodulation methods directly processes the mixed spectrum, which is superimposed by the output spectra of several single- interference systems. The CDM is a good scheme for demodulating the OPDs of multi-interference systems, whose key is decoding sensors in the optical cepstrum domain using a Capon estimator. Although this scheme has a strong multiplexing capability, it is realized in the case of wide spectral interval and large OPD interval.

Demodulating the absolute OPDs of the dual-interference system is based on that of the multi-interference system. Therefore, this paper proposes an iterative normalized cross-correlation (INCC) algorithm for demodulating the absolute OPDs of dual-interference systems. As far as we know, this algorithm is firstly proposed and applied to the OFS system. On the basis of FFCC algorithm, we construct a combined template function whose form is consistent with the mixed spectral form of the dual-interference system, and propose a reconstruction matrix to reconstruct its components. Then, the calculated results approach the absolute OPDs of the interferometer step by step via the INCC algorithm. The simulation and experiment show that the proposed algorithm has high demodulation accuracies of up to 5 nm near OPDs of 560 µm and 660 µm in the spectral interval from 45 nm to 80 nm. The relationship between OPDs and temperature is verified and the measured accuracy is comparable to the control accuracy. Finally, the convergence, limitation, signal to noise ratio (SNR), sampling interval, short or multiple OPD detection of the proposed algorithm are discussed in detail.

2. Principles

2.1 FFCC algorithm and FFT method

Ideally, the mixed spectrum of dual parallel interferometers can be assumed to be the superposition of two double-beam interference spectra with two different OPDs, which are denoted as L₁ and L₂, respectively. For a light with wavelength λ, the mixed spectrum can be expressed as

The FFCC algorithm is used to interrogate the L₁ and L₂. The template function

The cross-correlation coefficient C(l, Δλ) between the mixed spectrum and the template function is expressed as follows.

2.2 Problem description

It can be seen from Eq. (5) that, the maximum position of the eigenfunction Φ(l, L_i) occurs at l = L_i. But from Eq. (4), the local maximum position of C(l, Δλ) near L_i may not appear at l = L_i. This is because Φ(l, L₁) has an influence on the maximum position of C(l, Δλ) near the L₂. Similarly, Φ(l, L₂) has the same effect on that near L₁. The above effect between L₁ and L₂ is collectively referred to as the cross-crosstalk effect. In addition, Φ(l, L_-1) and Φ(l, L_-2) have effects on the maximum position of C(l, Δλ) near L₁ and L₂, respectively. They are called as the self-crosstalk effect. It should be noted that, similar to FFCC algorithm, the results of FFT calculation also have the crosstalk effect.

In order to describe the above crosstalk effect, it is assumed that the true OPDs L₁ and L₂ are 560 µm and 660 µm, respectively. Figure 1(a) shows the mixed spectrum of two interferometers with different OPDs. The initial wavelength λ₀ is 1520 nm, and the spectral interval Δλ varies from 45 nm to 90 nm. Firstly, FFT method is used to calculate the two different OPDs of the mixed spectrum, as shown in Fig. 1(b). The color bar on the right in Fig. 1(b) represents the amplitude of each OPD, and green * represents the optimal OPD calculated by FFT under different spectral intervals. It can be seen that the maximum error between the results calculated by FFT and the true OPDs is over 10 µm. Then the FFCC algorithm is used to calculate the OPDs of the mixed spectrum, and the results are shown in Figs. 1(c) and 1(d). The template OPD consists of two segmented OPD ranges; i.e., l_t1 and l_t2, which are 550-570 µm and 650-670 µm, respectively. Figures 1(c) and 1(d) are the 3D plane graph obtained by Eq. (4). The colorbar on the right side represents the correlation coefficient relative to the maximum. The darker the color is, the smaller the C(l, Δλ) is. The maximum points of C(l, Δλ) under different spectral intervals are marked with *. In Figs. 1(c) and 1(d), only a few OPDs (L_c1 and L_c2) determined by the local maximum position of C(l, Δλ) are consistent with L₁ and L₂ (shown by the green dotted line). When the spectral interval is smaller, the self-crosstalk effect and cross-crosstalk effect become more and more serious, leading to a greater error of the calculated L_c1 and L_c2. As a result of crosstalk effect, the OPD corresponding to the local maximum becomes uncertain, which brings a great challenge to obtain L₁ and L₂. Therefore, we propose a new algorithm to solve the problem caused by the crosstalk effect.

 

Fig. 1. A 3D plan of the relationship between the cross-correlation coefficient, the template OPD l and spectral interval. (a) The mixed spectrum, (b) the results calculated by FFT method, whose colorbar represents the amplitude of OPD, (c) and (d) collectively represent the demodulation results of FFCC algorithm. The colorbar represents the cross-correlation coefficient relative to the maximum, whose maximum in a spectral interval is marked as *, and the dotted green line represents the true OPD of 560 µm and 660 µm.

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2.3 INCC algorithm

The cross-correlation function represents the degree of match between one signal and another, and the template function is assumed as the following combined form.

When R(l, λ) is proportional to the mixed spectrum I(λ), the cross-correlation coefficient between them is the largest. By observing Eqs. (4) and (5), it can be found that once L₁ and L₂ are known, their eigenfunctions Φ(l, L₁) and Φ(l, L₂) are determined, and the components b₁ and b₂ of the eigenfunctions can be then solved by simultaneous C(L₁, Δλ) and C(L₂, Δλ). It's a pity that L₁ and L₂ are unknown. In Eq. (5), when Δl=2Nπ/k₀ (N is integer and not big), Φ(l, L_i)≈Φ(l+Δl, L_i). In this case, C(L_i+Δl, Δλ) instead of C(L_i, Δλ) is used to approximate the b₁ and b₂, which can be approximately considered as the r₁ and r₂. Therefore, finding appropriate L_i+Δl and C(L_i+Δl, Δλ) is the key to solute the proposed algorithm.

Based on FFCC algorithm, the demodulated L_c1 and L_c2 have a certain deviation from L₁ and L₂, while C(L_ci, Δλ) is approximately equal to C(L_i, Δλ). Thus, multiple cross-correlation calculations are used to solve the optimal solution l_ri^m of L_i, and l_ri^m is the m-th optimal OPD calculated near L_i (i=1, 2, and m=0, 1, 2, …). It is assumed that the maximum deviation does not exceed δ and l varies in the range of (l_ri^m-δ, l_ri^m+δ). The template function constructed near L_i is

According to the previous analysis, L_j≈l_rj^m+Δl and Ω(l_ri^m, L_j)≈Ω(l_ri^m, l_rj^m) (i, j=1, 2). Let r_i^m replaces b_i, then the Eqs. (9) and (10) can be rewritten as

Let

Usually, the normalized cross-correlation (NCC) is used as a similarity evaluation function. From the Cauchy-Schwarz inequality [20],

When l varies in (l_ri^m-δ, l_ri^m+δ), the abscissa corresponding to the local maximum of ρ^2m+i(l, Δλ) is denoted as l_ri^m+1. The l_ri^m+1 can be substituted into Eqs. (8), (14), and (15) to iteratively calculate l_ri^m+2 until it is stable. To better illustrate the iterative process, it is explained in steps below. Figure 2 is a demodulation flowchart of the proposed INCC algorithm, whose process is described as follows:

 

Fig. 2. The flow chart of INCC algorithm demodulation. The dotted green box represents the internal recycle (3→4→5→3), and the red dotted box represents the extrinsic cycle (1 or 6 →2→internal recycle→6).

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3. Simulations and experiments

In order to verify the effectiveness of the proposed algorithm, the two situations need to be discussed in detail: (a) whether the crosstalk effect in different spectral intervals has been resolved, and (b) whether the change of one OPD affects the other OPD when the spectral interval is small. Based on the above theory, the INCC algorithm is used to perform the numerical simulations of the mixed spectrum with two different OPDs.

3.1 Simulation results of different spectral intervals

Considering the case that the two OPDs L₁ and L₂ of mixed spectrum are fixed, which are 560 µm and 660 µm, respectively. The initial wavelength is 1520 nm, and the terminal wavelength varies from 1565 nm to 1610 nm in 0.5 nm steps. The sampling interval of the spectrum is assumed to be 0.1 nm. That is to say, the spectral interval Δλ is from 45 nm to 90 nm. Before applying INCC algorithm, the FFCC algorithm is utilized to obtain L_c1, L_c2, C(L_c1, Δλ) and C(L_c2, Δλ). From Figs. 1(a) and 1(b), both the deviations between the initial OPDs (L_c1 and L_c2) and the true OPDs (L₁ and L₂) are no more than 5 µm, which can be considered as the maximum deviation δ. And the r₁⁰ and r₂⁰ of R¹(l, λ) are then obtained by substituting the initial OPDs L_c1 and L_c2 into A^-1.

In the first iteration (m=1, i=1), L₂ is assumed to be a constant of l_r2⁰, and the l of R¹(l, Δλ) are scanned in the interval (l_r1⁰-δ, l_r1⁰+δ). When R¹(l, Δλ) is cross-correlated with the mixed spectrum, a new l_r1¹ can be obtained. When the spectral interval changes, a new series of l_r1¹ can be obtained. Figure 3(a) shows the cross-correlation spectrum between the mixed spectrum and the R¹(l, Δλ) during the first iteration. It can be seen that when the spectral interval changes, the l_r1¹ in the first iteration are unstable and many of them are far from L₁ of 560 µm.

 

Fig. 3. The relationship between the NCC coefficient and the template OPD under the spectral intervals from 45 nm to 90 nm. (a), (c) and (e) are the results of the first three iterations near L₁. (b), (d) and (f) are the results of the first three iterations near L₂. (g) and (h) are the details of the 3rd iteration, respectively.

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Then, the obtained l_r1¹ is taken as the invariant in the second iteration (m=1, i=2), and the l of R²(l, Δλ) are scanned in the interval (l_r2⁰-δ, l_r2⁰+δ). When the mixed spectra of different spectral intervals are cross-related with R²(l, Δλ), a series of new l_r2¹ are obtained. As shown in Fig. 3(b), most of l_r2¹ are close to the L₂ of 660 µm while a few are deviated from the L₂, which indicates that the l_r2¹ is converging after one iteration.

Next, the obtained l_r2¹ is taken as the invariant of R³(l, Δλ) during the next iteration (m=2, i=1), and the scanning interval l is returned to (l_r1¹-δ, l_r1¹+δ). After a similar cross-correlation operation as the previous step, a new series of l_r1² are obtained, as shown in Fig. 3(c). At this time, all l_r1² are concentrated around L₁, which indicates that all the results of this iteration converge. Then, the similar cross-correlation operations are performed for the case of m=2 and i=2 to obtain new OPDs l_r2² in different spectral intervals. The results with a few deviation values in Fig. 3(b) have been corrected in Fig. 3(d), indicating that l_r2² obtained after further iteration (m=2, i=2) completely converge to the L₂. Obviously, more iterations can be carried out to verify whether the obtained l_r1^m and l_r2^m converge.

Finally, the converged l_r1^m and l_r2^m can be resubstituted into A^-1 to obtain new r₁^m and r₂^m of R^2m+i(l, λ). The INCC operations above are repeated until the new l_r1^m and l_r2^m converge again. In this way, l_r1^m and l_r2^m can also converge to the b₁ and b₂ of the mixed spectrum, so as to improve the accuracy of the calculated OPDs and reduce the errors. Figures 3(e) and 3(f) are the cross-correlation spectra of the third iteration (m=3) near L₁ and L₂, respectively. It can be seen that both the obtained l_r1³ and l_r2³ converge to L₁ and L₂, which are the same as the results in Figs. 3(c) and 3(d). Seen from the further enlarged details of the third iteration in Figs. 3(g) and 3(h), both the l_r1³ and l_r2³ are about 5 nm away from the true L₁ and L₂, which indicates that the crosstalk effect between different OPDs is resolved after several iterations.

It is worth noting that only the results of the first three iterations are given here, because they have been converged and the fourth results are consistent with the third results. Similarly, the following sections also show only the results of the first three iterations.

3.2 Simulation results of different OPDs

In addition to considering the stability and accuracy of demodulation results of different spectral intervals, the effectiveness of the proposed algorithm should also be considered when either L₁ or L₂ changes in a small spectral interval. In the simulation, it is assumed that the L₁ of the mixed spectrum is fixed at 560 µm and L₂ varies from 650 µm to 674 µm. The given spectral interval is 55 nm equivalent to the spectral range from 1520 nm to 1575 nm, and the sampling interval is assumed to be 0.1 nm. Before performing the INCC algorithm, the L_c1 and L_c2 together with their corresponding C(L_c1, Δλ) and C(L_c2, Δλ) are obtained based on the FFCC algorithm. Both the maximum deviations δ between the initial OPDs (L_c1 and L_c2) and the true values (L₁ and L₂) are still set to 5 µm. Based on the above and the reconstructed matrix A^-1, the r₁⁰ and r₂⁰ of the R¹(l, λ) are obtained.

Since the iterative computation process has been described above in detail, the final calculation results are given here. Figures 4(a), 4(c) and 4(e) are the results of the first three iterations for demodulating L₁ based on the INCC algorithm. When the L₂ changes, the calculation results of the first iteration of L₁ are not stable. On the contrary, the calculation results of the second and third iterations are all concentrated around L₁, which is consistent with the situation with fixed L₁. Figures 4(b), 4(d), and 4(f) are the results of the first three iterations of demodulating L₂. When L₂ varies from 650 µm to 674 µm, the results of the first iteration still have a large deviation from the L₂, while the results of the second and third iteration are consistent with the L₂. This indicates that the proposed algorithm is still effective for demodulating the changed OPD. Figure 4(g) as a larger version of Fig. 4(e) reveals that the precision of demodulation results for the invariant L₁ is 4 nm. The demodulation results of the varying L₂ are shown in Fig. 4(h). The slope of the true value is 1 with the R² of 0.9999, which means that the demodulation accuracy of the varying L₂ remains high. Therefore, when a certain OPD changes, the proposed algorithm can still solve the crosstalk between different OPDs.

 

Fig. 4. The relationship between the NCC coefficient and the template OPD under the different L₂ from 650 µm to 675 µm, and the spectral interval is 55 nm. (a), (c) and (e) are the results of the first three iterations near L₁. (b), (d) and (f) are the results of the first three iterations near L₂. (g) and (h) are the details of the 3rd iteration, respectively.

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3.3 Experimental setup

In order to verify the practical feasibility of the proposed demodulation method, the mixed spectrum is generated by two PMF-Sagnac systems. The Sagnac loop with a PMF length of 130 cm is denoted as sensor 1, and another loop with a PMF length of 150 cm is denoted as sensor 2. All optical couplers (OCs) adopt 50:50 splitting ratio. The sampling spectrum of OSA ranges from 1523 to 1603 nm. The sampling resolution is set as 0.1 nm. In the experiment, sensor 1 is placed at room temperature and sensor 2 is heated by high-precision thermoelectric cooler (TEC). The exerted temperature ranges from 22.0 $^\circ $C to 30.0 $^\circ $C in 0.5 $^\circ $C steps. The temperature accuracy of TEC is 0.1 $^\circ $C and the temperature is kept for 10 minutes before recording to reduce the measurement error of sensor 2 caused by temperature fluctuation. The experimental spectra measured at different temperatures are shown in Fig. 5(b), and the color represents the intensity of the spectrum.

 

Fig. 5. (a) The experimental device that produces a mixed spectrum, in which the length of one PMF is 130 cm (Sensor 1) and the other is 150 cm (Sensor 2). (b) The mixed spectra at different temperatures. The colorbar indicates the intensity of the spectra.

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3.4 Experimental demonstration of different spectral intervals

Based on the proposed theory, the OPDs of sensor 1 and sensor 2 under different spectral intervals are calculated. Since it is very complicated to describe the calculation results of different spectral intervals at different temperatures, the mixed spectrum at 26.0 $^\circ $C is chosen to illustrate. The initial wavelength of the spectrum is 1523 nm, and the terminal wavelength is from 1568 nm to 1603 nm in 0.5 nm steps.

With the different spectral intervals, the OPDs of sensor 1 and sensor 2 are iterated for four times in turn, and the results of the first three iterations are presented in Figs. 6(a), 6(c), and 6(e). Due to the influence of spectral noise, the convergence rate of the iteration is slower than that of the ideal case. The calculated OPDs of the first iteration are very dispersed. After the first iteration of sensor 2, most results of the second iteration of sensor 1 are convergent. But in small spectral intervals, some calculated OPDs are scattered outside the convergent value. This is because the smaller the spectral interval is, the more serious the crosstalk effect is, and the smaller the convergence rate is. In addition, there also exists the influence of spectral noise. Figure 6(g) shows the details of the third iteration results of sensor 1. It can be seen that after three iterations, the calculated OPDs of sensor 1 converge to about 567.640 µm with an error of less than 5 nm.

 

Fig. 6. The relationship between the NCC coefficient and the template OPD under the spectral intervals from 45 nm to 80 nm. (a), (c) and (e) are the results of the first three iterations of sensor 1. (b), (d) and (f) are the results of the first three iterations of sensor 2. (g) and (h) are the details of the 3rd iteration, respectively.

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Figures 6(b), 6(d), and 6(f) are the results of the first three iterations of sensor 2. It can be seen that after the first iteration of sensor 1, the iteration results of sensor 2 are convergent, but there are still many calculated OPD values scattered outside the convergent value. After the second iteration of sensor 1, the calculation results of sensor 2 are significantly improved. However, there are still some points that do not converge and fall outside the convergence OPD, which is related to the noise and small spectral interval. After the third iteration, the calculated results of sensor 2 completely converge. As can be seen from its details in Fig. 6(h), the convergence OPD of sensor 2 is 671.458 µm, with an error of less than 5 nm. Then, the proposed algorithm is verified with the sensor 2 at different experimental temperatures in a small spectral interval. In theory, a narrower spectral interval should be chosen to verify the proposed theory. But in practice, the spectral noise in the narrow spectral interval will affect the demodulation results more. Since the purpose of this paper is to demonstrate the effectiveness of the proposed algorithm, the small spectral interval of 55 nm is sufficient to illustrate it. In the experiment, a spectrum range from 1523 nm to 1578 nm is selected. After the spectral noise reduction, the selected spectral interval can be narrower.

3.5 Experimental demonstration of different OPDs

 Figures 7(a), 7(c), and 7(e) are the results of the first three iterations of sensor 1 in the temperature range of 22.0 $^\circ $C to 30.0 $^\circ $C. In the first iteration, because of a large deviation between the initial OPD and the true one, the calculated OPD are not convergent. After the first iteration of sensor 2, most of calculated OPDs of sensor 1 are convergent, but parts of them are not. After the third iteration of sensor 1, it can be seen that all the computed OPDs are very concentrated. The green line in Fig. 7(g) is the detail of Fig. 7(e). It can be seen that the maximum OPD fluctuation of sensor 1 does not exceed 0.2 µm. According to 95% confidence of T-distribution in Fig. 7(h), the temperature accuracy measured by sensor 1 is 0.26 $^\circ $C, which can be approximately regarded as the local ambient temperature fluctuation within 170 min

 

Fig. 7. The relationship between the NCC coefficient and the template OPD under the different temperatures from 22.0 °C to 30.0 °C, and the spectral interval is 55 nm. (a), (c) and (e) are the results of the first three iterations of sensor 1. (b), (d) and (f) are the results of the first three iterations of sensor 2. (g) is the details of the 3rd iteration, respectively. (h) is the temperature fluctuation measured by sensor 1 and sensor 2 under the different confidence.

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Figures 7(b), 7(d) and 7(f) are the results of the first three iterations of sensor 2. Similarly, the calculated OPDs of the first iteration have no obvious linear relationship with temperature, while the linear relationship between the results of the second or third iteration and temperature is relatively obvious. From the dark red line in Fig. 7(g), the sensitivity between OPD of sensor 2 and temperature in the third iteration is -707.2 nm/$^\circ $C with R² of 0.9995 and a standard error of estimate (SEE) of 0.042 µm. Meanwhile, the TEC temperature fluctuation measured by sensor 2 is 0.125 $^\circ $C with 95% confidence of T-distribution, which is very close to the control accuracy 0.1 $^\circ $C of TEC.

Thus, in a small spectral interval, a slowly varying OPD can still be accurately demodulated by INCC algorithm, and the demodulation results of other OPDs are hardly affected.

4. Discussions

4.1 Convergence of INCC algorithm

In the process of simulation and experiment, only the results of the first three iterations of L₁ of 560 µm and L₂ of 660 µm are given. To illustrate the convergence of INCC algorithm, the results of the first twenty iterations are investigated as shown in Figs. 8(a) and 8(b). The illustrations show the detailed results from the 3^rd-iteration to 20^th-iteration, which reveals that the smaller the spectral interval is, the farther the results of the first iteration are from the true OPDs. And the results of the second iteration rapidly converge to the true OPDs, which indicates that the INCC algorithm is convergent and the true OPDs can be obtained via only two iterations theoretically. Figures 8(c) and 8(d) show the time spent per iteration running on Matlab with an Intel i5-7200U processor. The average iteration times of the spectra with 40 nm, 45 nm and 50 nm intervals are 0.17 s, 0.17 s and 0.16 s, respectively. The entire code takes about 2 s to run. Therefore, this algorithm is suitable for steady state measurement or slow variable measurement. If a faster processor is available, it can be used for real-time monitoring.

 

Fig. 8. Results of the first twenty iterations at different spectral intervals (40 nm, 45 nm, and 50 nm). (a) L₁ of 560 µm, (b) L₂ of 660 µm. (c) and (d) The time taken for each iteration of L₁ and L₂.

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4.2 Limit of INCC algorithm

Compared with FFCC method, the proposed algorithm can still achieve high demodulation accuracy when the spectral interval or the interval between two adjacent OPDs is small. But it should be emphasized that the algorithm is still limited to the spectral interval and OPD interval, because the proposed algorithm is based on FFCC algorithm. As shown in Fig. 9(a), when the interval (L₂-L₁) between two adjacent OPDs exceeds π/Δk, the calculation result of FFCC algorithm has several local maxima, at this time, the algorithm can accurately demodulate the OPDs of the mixed spectrum. When the interval between adjacent OPDs is less than π/Δk [Fig. 9(c)], there may not exist two local maxima, which thereby invalidate the INCC algorithm. When the interval is close to π/Δk, there may be local maxima as shown in Fig. 9(b). If it exists, this algorithm is still applicable. Therefore, within a given spectral interval, the theoretical limit of INCC algorithm to distinguish adjacent OPDs is

Equation (18) shows that the spectral interval can be enlarged to improve the ability of the proposed algorithm to distinguish adjacent OPDs. Besides, the resolution ability with a small spectral interval can be improved by increasing the difference between adjacent OPDs. In addition, if the amplitude of the two original spectra is quite different, the complexity of the proposed algorithm will be increased. In their cross-correlation curves, the main peak amplitude of the weak spectrum may be equal to the sidelobe amplitude of the strong spectrum, or even smaller, which will make the process of extracting multiple local maxima very difficult. For such cases, we will take remedial measures and discuss them in detail in future studies.

 

Fig. 9. The resolution ability of the INCC algorithm in three different situations: (a) L₂-L₁>π/Δk, (b) L₂-L₁≈π/Δk, (c) L₂-L₁<π/Δk

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4.3 Signal to noise ratio

The following is to analyze the influence of SNR on the demodulation results. In the test process, the sampling interval of the spectrum is set as 0.2 nm, and the noise is set as additive white Gaussian noise (AWGN). The test data are divided into three groups with the spectral interval of 90 nm, 75 nm and 60 nm, respectively. And the SNR of each group varies from 20 dB to 50 dB with a step of 5 dB. 50 groups of AWGN functions are generated to simulate the repeating measurements during the validation for each SNR value of the testing data. Figures 10(a) and 10(b) are demodulation results and root mean square errors (RMSEs) of each SNR data calculated by the proposed algorithm. The results show that the demodulation precision of the test spectra with high SNR is almost independent of the spectral interval. And the demodulation precision of the test spectra with large spectral interval is almost not affected by spectral noise, while the demodulation precision of the test spectra with small spectral interval is easily affected by spectral noise. Therefore, the demodulation accuracy can be improved by increasing the SNR or spectral interval.

 

Fig. 10. Demodulation results and RMSEs of the test spectrum with different SNR. (a) L₁ of 560 µm, (b) L₂ of 660 µm.

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4.4 Sampling interval

Both of the spectral sampling interval and spectral interval will affect the demodulation speed. In this part, the demodulation accuracies of the proposed algorithm under different sampling intervals and spectral interval are specifically analyzed. During this test, the SNR of the mixed spectrum is 35 dB. The test data are divided into three groups with spectral intervals of 90 nm, 75 nm and 60 nm, respectively. The sampling interval for each group varies from 0.1 nm to 0.5 nm, and the step is 0.1nm. And the 50 groups of each test data with different sampling interval and spectral interval are used to be simulated. Figures 11(a) and 11(b) are demodulation results and RMSEs of L₁ and L₂ at different sampling intervals, respectively. The solid and dashed lines represent the average OPD and RMSE obtained by INCC algorithm, respectively. The blue, red and green lines represent the results of the spectral intervals with 90 nm, 75 nm and 60 nm, respectively. It can be seen that when the sampling interval varies, the demodulation results of L₁ and L₂ are kept around 560 µm and 660 µm, respectively. And the RMSEs of this test are no more than 20 nm. Therefore, the variation of sampling interval and spectral range has little effect on the demodulation accuracy, and the demodulation speed can be improved by increasing the sampling interval or decreasing the spectral interval.

 

Fig. 11. Demodulation results and RMSEs of the test spectrum with different spectral interval. (a) L₁ of 560 µm, (b) L₂ of 660 µm.

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4.5 Short OPD detection

The OPDs used in the above study are relatively large, but the INCC algorithm is also suitable for the mixed spectrum with short OPDs. In this test, the two OPDs of the mixed spectrum are 80 µm and 130 µm, respectively. The initial wavelength of the test spectrum is 1520 nm, and the terminated wavelength of the spectrum is 1610 nm and 1595 nm, respectively. The corresponding spectral intervals are 90 nm and 75 nm, respectively. And the spectral sampling interval is 0.2 nm. Then the SNR of the test data for each spectral interval ranges from 25 dB to 50 dB, whose step is 5 dB. And the test data for each SNR is generated by 50 groups of AWGN functions. Figures 12(a) and 12(b) show the average OPDs and RMSEs of 50 repeats demodulated by INCC algorithm under different SNR. The demodulation accuracy of the spectrum with large spectral interval and high SNR is higher than that with small spectral interval and low SNR, which is the same as the result of the long OPD detection.

 

Fig. 12. Demodulation results and RMSE of the test spectrum with different SNR. (a) L₁ of 80 µm, (b) L₂ of 130 µm.

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4.6 Multiple OPD detection

The INCC algorithm is also applicable to multiple parallel double-beam interferometers. Figure 13 is the simulation results of a two-beam interferometer with six different OPDs (L1 ∼ L6), which are 160 µm, 220 µm, 298 µm, 356 µm, 420 µm and 502 µm, respectively. The spectrum ranges from 1520 nm to 1600 nm equivalent to a spectral interval of 80 nm. And the sampling interval is 0.2 nm. The SNR of the applied white Gaussian noise ranges from 30 dB to 50 dB with a step of 5 dB. The spectra of each SNR are repeated 50 times. The spectral tests of each SNR are repeated 50 times, denoted as a group, and the mean value and RMSE of demodulation results of each group are obtained, shown in Fig. 13. The results show that the six demodulated OPDs are kept at 160 µm, 220 µm, 298 µm, 356 µm, 420 µm and 502 µm, respectively. And the RMSEs are no more than 30 nm. Therefore, it can be shown that the INCC algorithm is universal for multiple parallel double beam interferometers.

 

Fig. 13. Simulation results of a mixed spectrum with six different OPDs (160 µm, 220 µm, 298 µm, 356 µm, 420 µm and 502 µm).

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It is important to note that since the template function of this algorithm is cosine, this algorithm is only applicable to the interferometer which can be approximated as a two-beam interference.

5. Conclusion

In conclusion, an INCC algorithm is proposed for absolute OPD demodulation of dual-interference system. Based on the reconstructed matrix, the components of the combined template function are optimized, and the optimal solution of the absolute OPDs of mixed spectrum is obtained by INCC algorithm. The simulation and experiment demonstrate the effectiveness of the algorithm and the demodulation precision of up to 5 nm. Compared with FFCC algorithm and FFT method, the proposed algorithm effectively solves the crosstalk effect between OPDs, which can improve the multiplexing ability of the interference system, optimize the fiber sensing network and reduce the laying cost.