Research on Crosstalk Suppression Technology of Wavelength Division Multiplexing for Fiber-Optic Sensing Based on Fabry–Perot Interference

Abstract

Aiming at the severe inter-channel crosstalk and degraded detection accuracy existing in wavelength division multiplexing (WDM)-based fiber Fabry–Perot (F-P) sensor arrays, this paper proposes a multi-channel WDM crosstalk suppression technique based on phase-shifting diversity demodulation. A theoretical crosstalk model is constructed, and analysis reveals that WDM crosstalk depends on four key parameters: wavelength diversity group number k, channel extinction ratio ε, amplitude of the measured signal A_s, and wavelength interval m. Numerical simulations and experiments are performed on a dual-channel WDM sensing system to evaluate the influences of these parameters on the crosstalk suppression performance. Results verify that the proposed crosstalk suppression technique can remarkably suppress inter-channel crosstalk compared with the system without crosstalk suppression, and the fundamental-frequency crosstalk suppression level is improved by 33.8 dB. For engineering optimization, there exist an optimal wavelength diversity group number and an optimal wavelength interval to achieve maximum crosstalk attenuation. Moreover, the proposed technique relaxes the strict extinction ratio requirements for WDM devices and mitigates the hardware limitations in large-scale array systems. This method effectively addresses the crosstalk bottleneck of fiber F-P sensor arrays under WDM configuration, and provides theoretical guidance and an engineering optimization strategy for large-scale underwater fiber F-P WDM sensing arrays.

1. Introduction

Fiber-optic Fabry–Perot (F-P) sensor arrays [1,2,3] typically employ multiplexing schemes including frequency division multiplexing (FDM) [4], wavelength division multiplexing (WDM) [5], and time division multiplexing (TDM) [6]. However, as the multiplexing scale of arrays continues to expand, the crosstalk issue becomes increasingly prominent [7]. It not only reduces the system multiplexing capacity but also introduces significant signal detection errors, making it one of the key bottlenecks restricting the practical application of fiber-optic F-P sensor array technology. In the field of optical sensing, structural optimization and machine-learning strategies [8] have been widely adopted in various photothermal [9], photoacoustic [10,11] and other sensing systems to suppress signal crosstalk, laying a general theoretical and methodological foundation for the development of crosstalk-suppression techniques in optical sensing.

At present, multiplexing research on fiber-optic F-P sensor arrays mainly focuses on TDM and space division multiplexing (SDM) architectures, while studies on crosstalk suppression for WDM systems are relatively scarce. Chen et al. applied Fourier transform demodulation to series and parallel multiplexed F-P sensor systems, achieving inter-channel crosstalk levels below 5% and 2.4%, respectively [12,13]. Yao et al. proposed an adaptive FFT demodulation algorithm based on white light interference for parallel multiplexed fiber-optic F-P sensor systems, which suppressed the fundamental frequency crosstalk by approximately 20 dB [14]. Xie proposed a cross-correlation demodulation method based on the Kaiser window function for a two-element FDM fiber-optic F-P sensor system, realizing a crosstalk level of −48 dB at 200 Hz [15]. Compared with TDM and SDM architectures, crosstalk-suppression techniques for WDM-based F-P sensor arrays have not yet formed a systematic research framework. Existing relevant achievements are mostly focused on hybrid multiplexing systems of conventional fiber-optic hydrophones. Cao et al. presented a crosstalk suppression structure based on wavelength-staggered transmission for fiber-optic hydrophones with dense wavelength division multiplexing (DWDM) and TDM architectures, which effectively suppressed nonlinear crosstalk [16]. Ren et al. addressed the crosstalk problem in hybrid DWDM/TDM fiber-optic sensor arrays by proposing an ultra-low crosstalk architecture based on wavelength cross-combination, which suppressed crosstalk between adjacent sensors to below −60 dB [17]. Overall, various crosstalk mitigation methods have been developed for typical multiplexing arrays such as TDM and SDM [18,19] systems. However, crosstalk suppression methods for pure WDM F-P sensor arrays remain insufficient. Existing WDM crosstalk-suppression techniques can hardly adapt to the detection scenario of pure WDM F-P sensor arrays. To tackle the above-mentioned issue, the proposed method targets spectral-leakage-induced crosstalk in pure WDM F-P arrays. Without modifying the front-end WDM optical hardware, it achieves a 33.8 dB improvement in fundamental-frequency crosstalk suppression merely by upgrading the back-end FPGA demodulation algorithm. This system features low hardware overhead; multi-channel expansion only requires additional back-end computing resources, which renders it well-suited for pure WDM sensor-array applications. This work extends the application scope of F-P sensing in the WDM domain and provides a feasible crosstalk-suppression technical solution for such systems.

Against this background, this paper proposes a multi-channel WDM crosstalk suppression technique based on a phase-shifting diversity demodulation scheme. On the basis of the traditional phase-shifting demodulation algorithm, this scheme introduces a diversity averaging mechanism for multi-group phase-shifted signals and fully utilizes the reflection spectrum information of extrinsic fiber-optic F-P sensors to achieve effective suppression of WDM crosstalk. Based on the theory of the phase-shifting diversity demodulation scheme, a WDM crosstalk suppression model for extrinsic fiber-optic F-P sensors is established. The influences of key parameters, including the wavelength diversity group number k, on the crosstalk performance of the WDM system are analyzed in detail through numerical simulations and experimental verification. The proposed technique can effectively suppress inter-channel crosstalk in WDM arrays of extrinsic fiber-optic F-P sensors under the conditions of lightweight implementation, no additional devices, low-extinction-ratio devices, and real-time demodulation. It breaks the WDM crosstalk bottleneck in large-scale multiplexing scenarios and provides a theoretical basis and technical support for the large-scale multiplexing of fiber-optic F-P sensor arrays.

2. Fundamental Principles

An M-channel WDM fiber-optic F-P sensor array is taken as the research object, and its structure is shown in Figure 1. An amplified spontaneous emission (ASE) source is adopted as the light source. The broadband light output from the source passes through an isolator and a circulator (CIR) sequentially and then is fed into a wavelength division multiplexer (WDM). The WDM demultiplexes the broadband light into multiple narrowband signals, which are injected into the corresponding sensing elements S₁, S₂, …, S_M, respectively. The signal light reflected by each element carrying sensing information returns along the original path and is combined into one beam by the WDM again. After passing through the circulator, the light is converted into electrical signals by a photodetector (PD) and transmitted to a personal computer (PC) for real-time demodulation and processing of the sensing signals.

Figure 1. Schematic diagram of a multi-channel WDM system.

The sensing element is an extrinsic fiber-optic F-P sensor. Its two reflective surfaces are usually a fiber end face and a reflective film with low reflectivity, and its interference can be approximated as two-beam interference. The interference spectrum signal of a single element can be expressed as

$$I=A+B\cos \left({\displaystyle \frac{4\pi L}{\lambda }}\right)$$

(1)

where A and B denote the DC and AC amplitudes of the interference signal, respectively; L is the initial cavity length of the sensing element, and λ is the wavelength of the incident light.

In the simulation, acoustic signals are loaded by dynamically modulating the cavity length of extrinsic fiber-optic F-P sensors. The frequency and amplitude of the acoustic signal are mapped to the oscillation frequency and deformation amplitude of cavity-length perturbation, respectively. By setting $L\left(t\right)=L_0+\Delta L\cos \left(2\mathsf{\pi }ft\right)$, changing the parameters f and ΔL enables the application of acoustic-field excitations with different frequencies and amplitudes to each extrinsic fiber-optic F-P sensing element, generating interference spectra containing channel crosstalk.

Since the system uses an ASE source, the optical path between typical extrinsic fiber-optic F-P sensing elements is on the meter scale, whereas the coherence length of the light source is at the millimeter scale. There exists a three-order-of-magnitude difference between them. The optical path difference between different sensing elements is much larger than the coherence length of the light source, so no interference occurs between the reflected light from different sensing elements. The total optical intensity received by the detector is

$$\begin{gathered} I_{\mathrm{total}}={\displaystyle \sum _{i=1}^M\left[A_i+B_i\cos \left(\frac{4\pi L_i}{\lambda }\right)\right]} \\ +{\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j\ne i}{\epsilon }_{ij}\left(\lambda \right)\left[A_j+B_j\cos \left(\frac{4\pi L_j}{\lambda }\right)\right]}} \end{gathered}$$

(2)

In the above equation, the first term on the right-hand side is the self-signal of each sensing element, and the second term is the crosstalk term. Here, ${\epsilon }_{ij}$ represents the channel extinction ratio describing the signal leakage from the interfering element S_j to the target element S_i. For the target element S_i, the optical intensity received by the detector is

$$\begin{gathered} I_i=\left[A_i\left(\lambda \right)+B_i\left(\lambda \right)\cos \left({\displaystyle \frac{4\pi L_i\left(t\right)}{\lambda }}\right)\right] \\ +{\displaystyle \sum _{j\ne i}{\epsilon }_{ij}\left(\lambda \right)\left[A_j\left(\lambda \right)+B_j\left(\lambda \right)\cos \left(\frac{4\pi L_j\left(t\right)}{\lambda }\right)\right]} \end{gathered}$$

(3)

The spectrum is sampled at equal wavelength intervals, with a sampling interval of Δλ and a sampling length of N.

Phase-shifting interferometry can be categorized into three-step, four-step, five-step, and n-step phase-shifting algorithms according to the phase-shift number n. In general, the value of the phase-shift number n is determined by the noise sensitivity of the phase-shift demodulation algorithm. Theoretically, a larger phase-shift number n yields lower phase-measurement error, better algorithm repeatability, and less stringent environmental requirements. Nevertheless, increasing the number of phase-shift steps in practical applications prolongs signal-processing time and may introduce additional errors. Benefiting from redundant sampling points, the five-step phase-shifting algorithm can effectively suppress demodulation deviations induced by random noise and phase-shift error. It features stronger robustness with only a moderate increase in computational cost and balances demodulation stability and real-time processing capability. Considering the detection requirements for weak underwater acoustic signals, the five-step phase-shifting algorithm is adopted in this work. Using the five-step phase-shifting demodulation algorithm, the demodulated signal of S_i can be obtained as follows:

$${\varphi }_i=\arctan \left({\displaystyle \frac{S{R}_i}{C{R}_i}}\right)=\arctan \left({\displaystyle \frac{I_{i2}-I_{i4}}{2I_{i3}-I_{i1}-I_{i5}}}·2\sin {\theta }_i\right)$$

(4)

where SR_i and CR_i are the orthogonal sine and cosine components, respectively; ${\theta }_j=-4\mathsf{\pi }{L}_j·m\Delta \lambda /{\lambda }^2$ is the demodulation parameter of the five-step phase-shifting demodulation scheme; m is the wavelength interval, representing the number of sampling wavelength intervals between the phase-shifted signals of the five-step scheme. $I_{ik}\left(k=1,2,\dots ,5\right)$ is the five-step phase-shifted signal extracted from the interference spectrum in the channel of the target element S_i, and each group of five-step phase-shifted signals contains five interference intensity signals with different initial phases.

Through analysis, it can be obtained that the phase error imposed on element S_i by any interfering element S_j satisfies:

$$\left\{\begin{array}{c}\delta {\varphi }_j\approx \cos {\varphi }_i·\delta {\varphi }_{SR,j}-\sin {\varphi }_i·\delta {\varphi }_{CR,j}\\ \delta {\varphi }_{SR,j}={\displaystyle \frac{B_j\sin {\theta }_j}{B_i\sin {\theta }_i}}\sin {\varphi }_j\\ \delta {\varphi }_{CR,j}={\displaystyle \frac{B_j{\sin }^2{\theta }_j}{B_i{\sin }^2{\theta }_i}}\cos {\varphi }_j\end{array}\right.$$

(5)

To simplify the analysis, it is assumed that no signal is loaded on the target element S_i, and each interfering element S_j is loaded with an independent single-frequency signal to be measured. Let ${\varphi }_{0i}$ and ${\varphi }_{0j}$ be the initial phases of the target element S_i and the interfering element S_j, respectively, and let ${\omega }_{s,j}$ and $A_{s,j}$ be the frequency and amplitude of the measured phase signal, respectively. $\cos {\phi }_s$ and $\sin {\phi }_s$ can be expressed using Bessel functions. Assuming that $\delta \varphi$ is the demodulation phase error caused by crosstalk from other elements to element S_i, it can be expressed as

$$\{\begin{array}{c}\delta \varphi ={\displaystyle \sum _{j=1,j\ne i}^M}[A_{DC,j}+A_{\mathrm{FFC},j}\cos {\omega }_{s,j}t+{\displaystyle \sum _{z=1}^{\infty }}{(-1)}^z{A}_{\mathrm{AHC}o,j}\cos (2z+1){\omega }_{s,j}t\\ +{\displaystyle \sum _{z=1}^{\infty }}{(-1)}^z{A}_{\mathrm{AHC}e,j}\cos 2z{\omega }_{s,j}t]\\ A_{\mathrm{FFC},j}=-2{\displaystyle \frac{{\epsilon }_{ij}·B_j}{B_i}}({\displaystyle \frac{\sin {\theta }_j}{\sin {\theta }_i}}\cos {\varphi }_{0j}\cos {\varphi }_{0i}+{\displaystyle \frac{{\sin }^2{\theta }_j}{{\sin }^2{\theta }_i}}\sin {\varphi }_{0j}\sin {\varphi }_{0i})J_1(A_{s,j})\\ A_{\mathrm{THC}\mathrm{o},j}=-2{\displaystyle \frac{{\epsilon }_{ij}·B_j}{B_i}}({\displaystyle \frac{\sin {\theta }_j}{\sin {\theta }_i}}\cos {\varphi }_{0j}\cos {\varphi }_{0i}+{\displaystyle \frac{{\sin }^2{\theta }_j}{{\sin }^2{\theta }_i}}\sin {\varphi }_{0j}\sin {\varphi }_{0i})J_{2z+1}(A_{s,j})\\ A_{\mathrm{THC}\mathrm{e},j}=2{\displaystyle \frac{{\epsilon }_{ij}·B_j}{B_i}}({\displaystyle \frac{\sin {\theta }_j}{\sin {\theta }_i}}\sin {\varphi }_{0j}\cos {\varphi }_{0i}+{\displaystyle \frac{{\sin }^2{\theta }_j}{{\sin }^2{\theta }_i}}\sin {\varphi }_{0i}\cos {\varphi }_{0j})J_{2z}(A_{s,j})\end{array}$$

(6)

Combined with the above equations, the signal from the interfering element S_j introduces three types of crosstalk components to the target element S_i: direct current (DC) crosstalk, fundamental frequency crosstalk (FFC), and total harmonic crosstalk (THC). The demodulation phase error is the superposition of contributions from all interfering elements, so the crosstalk generated by a single interfering element is taken as an example for derivation and analysis below. The DC crosstalk $A_{DC}$ contains the zero-order Bessel component and can be removed by a pre-stage DC filtering algorithm, so it will not be considered in the following. The FFC $A_{FFC}$ contains the first-order Bessel component and has the same frequency as the loaded signal. The THC $A_{THC}$ contains all high-order Bessel components, which can be further divided into odd-order harmonic crosstalk $A_{THCo}$ and even-order harmonic crosstalk $A_{THCe}$.

One interfering element S_j is selected for analysis. Assuming that the cavity length difference between the interfering element S_j and the target element S_i is ΔL, the starting wavelength of the interference spectrum is λ₀, the wavelength diversity group number is k, and the a-th wavelength sampling point is ${\lambda }_a={\lambda }_0+a\Delta \lambda \left(a\in \left[1,k\right]\right)$, the following expression can be obtained combined with the phase approximation under a small wavelength offset:

$$\{\begin{array}{l}\cos {\varphi }_{0i}\cos {\varphi }_{0j}\approx {\displaystyle \frac{1}{2}}[\cos ({\displaystyle \frac{4\pi }{{\lambda }_0}}\Delta L-{\displaystyle \frac{4\pi \Delta \lambda }{{\lambda }_0^2}}a\Delta L)+\cos ({\displaystyle \frac{4\pi }{{\lambda }_0}}(L_i+L_j)-{\displaystyle \frac{4\pi \Delta \lambda }{{\lambda }_0^2}}a(L_i+L_j))]\\ \sin {\varphi }_{0i}\sin {\varphi }_{0j}\approx {\displaystyle \frac{1}{2}}[\cos ({\displaystyle \frac{4\pi }{{\lambda }_0}}\Delta L-{\displaystyle \frac{4\pi \Delta \lambda }{{\lambda }_0^2}}a\Delta L)-\cos ({\displaystyle \frac{4\pi }{{\lambda }_0}}(L_i+L_j)-{\displaystyle \frac{4\pi \Delta \lambda }{{\lambda }_0^2}}a(L_i+L_j))]\\ \cos {\varphi }_{0i}\sin {\varphi }_{0j}\approx {\displaystyle \frac{1}{2}}[-\sin ({\displaystyle \frac{4\pi }{{\lambda }_0}}\Delta L-{\displaystyle \frac{4\pi \Delta \lambda }{{\lambda }_0^2}}a\Delta L)+\sin ({\displaystyle \frac{4\pi }{{\lambda }_0}}(L_i+L_j)-{\displaystyle \frac{4\pi \Delta \lambda }{{\lambda }_0^2}}a(L_i+L_j))]\\ \sin {\varphi }_{0i}\cos {\varphi }_{0j}\approx {\displaystyle \frac{1}{2}}[\sin ({\displaystyle \frac{4\pi }{{\lambda }_0}}\Delta L-{\displaystyle \frac{4\pi \Delta \lambda }{{\lambda }_0^2}}a\Delta L)+\sin ({\displaystyle \frac{4\pi }{{\lambda }_0}}(L_i+L_j)-{\displaystyle \frac{4\pi \Delta \lambda }{{\lambda }_0^2}}a(L_i+L_j))]\end{array}$$

(7)

Figure 2 shows the variation curves of four Group of trigonometric cross-product terms with the wavelength diversity group number k. For any sine or cosine function f(t) with period T, the sum over the entire period is 0, i.e., ${\displaystyle {\int }_0^{T}f\left(t\right)dt=0}$. According to the quasi-periodic fluctuation characteristics of the curves, when the wavelength diversity group number k for demodulation of the target channel S_i satisfies the integer-period requirement of the sine and cosine functions:

$$\{\begin{array}{c}{\displaystyle \frac{4\pi \Delta \lambda }{{\lambda }_0^2}}{k}_1\Delta L=m·2\pi ⟹k_1={\left[{\displaystyle \frac{m{{\lambda }_0}^2}{2\Delta \lambda ·\Delta L}}\right]}_{\mathrm{int}}\\ {\displaystyle \frac{4\pi \Delta \lambda }{{\lambda }_0^2}}{k}_2(L_i+L_j)=m·2\pi ⟹k_2={\left[{\displaystyle \frac{m{{\lambda }_0}^2}{2\Delta \lambda ·(L_i+L_j)}}\right]}_{\mathrm{int}}\end{array}m=\mathrm{1,2},3\cdots$$

(8)

Figure 2. Variation in trigonometric cross-coupling terms versus wavelength diversity group number k.

Averaging the demodulated phases of k Group can effectively suppress inter-channel crosstalk. Analysis of the above equations shows that the crosstalk amplitude is jointly determined by the extinction ratio ${\epsilon }_{ij}$, $\sin {\theta }_j/\sin {\theta }_i$, measured signal amplitude $A_{s,j}$, initial phases of the two elements, AC amplitude ratio of the reflection spectra of the two elements $B_j/B_i$ and wavelength diversity group number k. Combined with the five-step phase-shifting demodulation parameter formula $\theta =-4\mathsf{\pi }L·\Delta {\lambda }_m/{\lambda }^2$, it can be further derived that $\sin {\theta }_j/\sin {\theta }_i$ is related to the wavelength interval m and the cavity lengths of the two elements. The above analysis clarifies the key parameters that regulate crosstalk, laying a theoretical foundation and providing optimization ideas for the subsequent numerical simulation and experimental verification.

3. Simulation and Experiment

To clarify the regulation law of key system parameters on the suppression performance and guide the hardware selection and parameter optimization of practical sensor arrays, simulations and experiments are carried out in this section using the control variable method. As can be seen from the WDM crosstalk model, crosstalk from different WDM channels is linearly superimposed. Therefore, to reduce the complexity of theoretical verification, a dual-channel WDM sensing architecture is adopted for both numerical simulation and experimental verification in this paper. The key parameters of the experiment and system are listed as follows. The wavelength range of the ASE source is 1525–1565 nm. The central wavelengths of the WDM are 1530 nm and 1550 nm, with an isolation of 28 dB. The cavity lengths of Element-1 and Element-2 are 610 μm and 210 μm, respectively. The front-end hardware of the system discretizes the operating wavelength band of 1525–1565 nm into 512 fixed spectral sampling points. Spectral-frame acquisition is implemented on FPGA hardware, and the maximum spectral sampling rate reaches 5 kHz. The diversity group number k is 64, and the spectral wavelength interval is 5 sampling steps. Figure 3 demonstrates the spectra and crosstalk-suppression performance of the dual-channel WDM sensing system. Subplot (a) shows the original spectrum of the dual-channel WDM architecture, where the envelope is extracted by the extremum method to facilitate the subsequent elimination of envelope effects. Subplot (b) presents the time-domain crosstalk suppression results of the multi-channel wavelength crosstalk suppression technique under this sensing architecture. It demonstrates that the proposed algorithm can continuously suppress crosstalk components and possesses favorable real-time operation capability. Subplot (c) illustrates its frequency-domain results. Compared with the traditional phase-shifting demodulation algorithm, the proposed technique improves the FFC suppression effect by 33.8 dB. In the system, optical-path disturbances in the beam-combining section mainly stem from temperature drift. Given that this work targets underwater-detection scenarios, the magnitude of temperature-induced optical-path fluctuations is relatively low, and such disturbances can be reasonably neglected. Furthermore, for the F-P cavity sensing architecture, the uplink and downlink share the same optical path. Optical-path disturbances exert consistent influences on both signals, so separate analysis is unnecessary. Based on the above analysis, various effects originating from the non-ideal characteristics of WDM devices constitute the core research object of this paper.

Figure 3. Spectrum and crosstalk suppression performance of dual-channel WDM sensing system. (a) System spectrum; (b) time-domain results; (c) frequency domain results.

As derived from the above theoretical analysis, the channel crosstalk amplitude is affected by multiple parameters: extinction ratio ${\epsilon }_{ij}$, wavelength interval m, measured signal amplitude $A_{s,j}$, initial phases of the two elements, cavity lengths of the two elements, AC amplitude ratio of the reflection spectra of the two elements $B_j/B_i$, and wavelength diversity group number k. For any extrinsic fiber-optic F-P sensing element S_i in the system, the AC amplitude of the interference signal can be expressed as $B_i=2\sqrt{I_{i1}{I}_{i2}}=2\sqrt{R_{i1}{R}_{i2}{I_{i0}}^2}$, R_i1 and R_i2 are the reflection coefficients of the two end faces of the F-P cavity, respectively, and I_i0 is the incident light intensity. Since the reflection coefficients are fixed by the coating process and the incident light intensity is stable, $B_j/B_i$ can be reasonably assumed to be constants and are not taken as variables for subsequent analysis. The initial phase of the sensing element can be compensated by optical path calibration, so it is not an inherent crosstalk source of the WDM system. Meanwhile, the cavity length of the experimental elements is a fixed parameter after fabrication and cannot be continuously adjusted. Therefore, the channel extinction ratio ${\epsilon }_{ij}$, wavelength interval m, measured signal amplitude $A_{s,j}$, and wavelength diversity group number k are taken as research variables to explore the influences of these four key parameters on FFC and THC.

The same working conditions are set for both experiments and simulations: the system spectral band corresponds to two sensing elements, where element S₁ is loaded with a 63 Hz sinusoidal excitation signal, and element S₂ is packaged in a closed sound-insulation box lined with sound-absorbing cotton for inter-channel crosstalk analysis.

3.1. Wavelength Diversity Group Number

Diversity averaging is a unique technical feature of phase-shifting diversity demodulation schemes based on swept lasers or broadband light sources. The wavelength diversity group number directly determines the cancelation degree of periodic crosstalk components. According to the theoretical model in Equation (7), when k satisfies the integer-period requirement of sine and cosine functions, crosstalk in the sensor array can be effectively suppressed, corresponding to two calculation formulas for characteristic wavelength diversity group numbers:

$$\{\begin{array}{c}{k}_1={[n{{\lambda }_0}^2/2\Delta L\Delta \lambda ]}_{\mathrm{int}}\\ k_2={[n{{\lambda }_0}^2/2(L_1+L_2\left)\Delta \lambda \right]}_{\mathrm{int}}\end{array}$$

(9)

Theoretical derivation shows that the variation law of FFC is mainly determined by k₁, while THC is jointly determined by the two characteristic wavelength diversity group numbers. Limited by the bandwidth of the WDM system, the variation range of the wavelength diversity group number is set to 0–128. The variation laws of FFC and THC with the wavelength diversity group number k are comparatively analyzed through numerical simulations and experiments, and the test results are shown in Figure 4.

Figure 4. Variation curves of crosstalk versus wavelength diversity group number. (a) Fundamental frequency crosstalk (FFC); (b) total harmonic crosstalk (THC). Note that the vertical axis scales of simulation and experimental subplots are different. The comparison focuses on the curve trend rather than absolute amplitude.

By comparing the simulation and experimental results, it can be concluded that the variation laws of simulation and experiment are in good agreement. The traditional demodulation scheme does not involve diversity averaging, so its FFC and THC do not change with the wavelength diversity group number (i.e., k = 1). The crosstalk suppression capability of the phase-shifting diversity demodulation scheme decreases in an oscillatory manner as the wavelength diversity group number k increases. Affected by the Bessel function modulation in the theoretical model, the simulation curves show small periodic oscillations. In contrast, restricted by non-ideal factors such as non-uniform device loss and random environmental disturbances in the experimental system, the measured curves show a slightly oscillatory decreasing trend, with an overall oscillation amplitude smaller than that of the simulation results. The overall experimental crosstalk amplitude is higher than the simulation result because the actual optical path has multi-stage transmission losses, resulting in a significantly lower demodulated signal amplitude of element S₁ than the simulated value.

According to the baseline simulation parameters, the two types of characteristic average wavelength numbers within the limited range are calculated as k₁ = 50, 99 and k₂ = 29, 58, 87, 116, respectively. By comparing with the simulation curves, the variation law of FFC is completely consistent with the theoretical derivation: crosstalk minima appear at the k₁ positions, and other minima are generated by Bessel function modulation. The THC is regulated by the coupling of the two characteristic average wavelength numbers, and the curve shows a significant drop near k₁, proving that k₁ is also the dominant characteristic parameter for suppressing harmonic crosstalk. Therefore, the theoretical expression for the optimal average wavelength number in engineering practice is determined as $k_{opt}={\left[n{{\lambda }_0}^2/2\Delta L\Delta \lambda \right]}_{\mathrm{int}}$.

In this system, the reflected spectra are acquired by a photodetector (PD) and transmitted to the FPGA for spectral sampling. The sampled spectra are then uploaded to a personal computer (PC) for back-end signal processing. The front-end hardware discretizes the 1525–1565 nm operating band into 512 fixed spectral sampling points. For high-density WDM sensor array scenarios, the increase in total array channel capacity will compress the number of effective spectral sampling points allocated to a single channel, which further limits the available wavelength diversity group number. In practical engineering deployment, the optimal average wavelength number k_opt should be calculated first by substituting measured parameters such as device cavity length and light source wavelength interval. If the spectral sampling point resources are limited by hardware conditions, the optimal average wavelength number can be selected in a compromise manner within the neighborhood of k_opt to achieve an optimal trade-off between array channel multiplexing capacity and crosstalk suppression performance.

3.2. Channel Extinction Ratio

The extinction ratio ε determines the spectral isolation between adjacent WDM channels and is a core indicator affecting spectral leakage and channel crosstalk of WDM devices. To simplify the analysis, it is uniformly assumed in simulations and experiments that the channel extinction ratio is a constant value independent of wavelength. According to Equation (6), the extinction ratio ε exists as a proportional coefficient in the crosstalk expression, and theoretically, the crosstalk amplitude is negatively correlated with the extinction ratio ε. Numerical simulations are used to analyze the crosstalk evolution law caused by changes in channel extinction ratio under the phase-shifting diversity demodulation scheme, and the results are shown in Figure 5. Combined with the simulation results, as the extinction ratio increases, the degree of channel spectral leakage weakens, and both FFC and THC of the phase-shifting diversity demodulation scheme show a gradual decreasing trend, which matches the negative correlation derived from theory.

Figure 5. Variation curves of crosstalk versus extinction ratio. (a) Fundamental frequency crosstalk (FFC); (b) total harmonic crosstalk (THC).

Since the channel extinction ratio of conventional WDM systems is a fixed value, a tunable optical attenuator is added to the branch to analyze the influence of different extinction ratio values on the crosstalk suppression performance of the WDM system. The channel extinction ratio can be equivalently adjusted continuously by regulating the optical power of the branch. This approach introduces wavelength-independent flat amplitude attenuation, which can only yield an average extinction ratio and fails to reproduce the real spectral-leakage profile where the extinction ratio varies with wavelength at the passband edges of practical WDM filters. In this experiment, leaky light with a fixed amplitude is introduced in a controllable manner to qualitatively verify the suppression principle of the proposed technique against energy-type crosstalk. Limited by manual adjustment errors, two cases of ε = 0 dB and ε = 4 dB are selected for comparative testing and characteristic analysis. The experimental results under the two working conditions are shown in Figure 6.

Figure 6. Experimental comparison of crosstalk spectra under different extinction ratios. (a) ε = 0 dB; (b) ε = 4 dB.

The above figure shows the comparison results of crosstalk spectra introduced by element S₁ to element S₂ under the two extinction ratio conditions. By comparing the two sets of experimental data, the amplitudes of both FFC and THC decrease with the increase in extinction ratio, and this variation law is consistent with the simulation results. Both experimental and simulation tests confirm that under the same extinction ratio, the crosstalk suppression effect of the phase-shifting diversity demodulation scheme is stably better than 30 dB, which is significantly improved compared with the traditional scheme. This result indicates that the proposed crosstalk suppression technique can relax the hardware index requirement for the extinction ratio of WDM devices, which has positive engineering significance for reducing the hardware construction cost of large-scale sensor arrays.

3.3. Measured Signal Amplitude

The amplitude variation in the measured excitation signal directly determines the modulation depth of the extrinsic F-P fiber-optic sensor and is a key parameter that determines the nonlinearity of the system and induces harmonic components and cross-channel crosstalk. Large-scale fluctuations in signal amplitude can easily aggravate demodulation distortion and increase channel crosstalk. Therefore, exploring the crosstalk variation law under amplitude changes has important guiding significance for the design of dynamic sensing systems. According to the theoretical model in Equation (6), the measured signal amplitude A_s exists in the crosstalk analytical expression in the form of Bessel functions, and the crosstalk amplitude is directly related to the values of Bessel functions of each order, where FFC is proportional to the first-order Bessel function. The variation laws of 1st to 5th order Bessel functions with signal amplitude are shown in Figure 7.

Figure 7. Variation in Bessel functions versus signal amplitude.

Figure 8 shows the simulation and experimental variation curves of FFC and THC under different measured signal amplitudes. Comparing the simulation and experimental data, the overall variation trends of the two are in good agreement. As the measured signal amplitude gradually increases, the FFC of the phase-shifting diversity demodulation scheme shows periodic oscillatory changes, and the oscillation peak shows a decreasing trend with the increase in signal amplitude. The curve shape is highly consistent with the variation law of the first-order Bessel function, verifying the correctness of the previous theoretical derivation.

Figure 8. Variation curves of crosstalk versus signal amplitude. (a) Fundamental frequency crosstalk (FFC); (b) total harmonic crosstalk (THC). Note that the vertical axis scales of simulation and experimental subplots are different. The comparison focuses on the curve trend rather than absolute amplitude.

The THC curves all show a trend of first rising and then falling. Among them, the experimental curve shows small oscillations due to the temperature drift effect caused by long-term operation under non-ideal working conditions, which changes the equivalent cavity length, but the overall variation trend is consistent with the simulation results. Under the condition of large signal excitation, the phase-shifting diversity demodulation scheme still maintains excellent crosstalk suppression performance. From the perspective of engineering design, the proposed crosstalk suppression technique can effectively suppress nonlinear distortion and spectral leakage crosstalk caused by large signals, improve the upper limit of system dynamic measurement, and meet the requirements of dynamic sensing applications.

3.4. Wavelength Interval

As indicated by the five-step phase-shifting demodulation parameter formula $\theta =-4\mathsf{\pi }L·m\Delta \lambda /{\lambda }^2$, the wavelength interval m changes the demodulation signal θ, thereby adjusting the sinθ₁/sinθ₂ term, and ultimately affecting the crosstalk analytical expression given by Equation (6). Let sinθ₁/sinθ₂ = x, then the sum of FFC, odd and even harmonics can be simplified as a quadratic equation with respect to x. The theoretical derivation is carried out taking FFC as an example:

$$\begin{gathered} A_{FFC}=-2{\displaystyle \frac{\epsilon ·B_1}{B_2}}({\displaystyle \frac{\sin {\theta }_1}{\sin {\theta }_2}}\cos {\varphi }_{01}\cos {\varphi }_{02}+{\displaystyle \frac{{\sin }^2{\theta }_1}{{\sin }^2{\theta }_2}}\sin {\varphi }_{01}\sin {\varphi }_{02}{)J}_1\left(A_s\right) \\ \Rightarrow A_{FFC}=ax+b{x}^2 \end{gathered}$$

(10)

where $a=-2\epsilon B_1{J}_1\left(A_s\right)\cos {\varphi }_{01}\cos {\varphi }_{02}/B_2$, $b=-2\epsilon B_1{J}_1\left(A_s\right)\sin {\varphi }_{01}\sin {\varphi }_{02}/B_2$.

Figure 9 shows the variation curve of sinθ₁/sinθ₂ with the wavelength interval m. In the range of (0, 16], sinθ₁/sinθ₂ shows an approximately exponential growth trend with m. Combined with the quadratic function relationship between FFC/THC amplitudes and sinθ₁/sinθ₂, it can be inferred that both FFC and THC show an overall exponential growth characteristic with the increase in m. Restricted by the characteristics of the quadratic function, there theoretically exists a minimum position on the curves.

Figure 9. Variation curves of sinθ₁/sinθ₂ versus wavelength interval m.

To clarify the regulation law of wavelength interval on crosstalk performance, multiple sets of numerical simulations and experimental verifications are carried out. The simulation and experimental test results of FFC and THC under different wavelength intervals m are shown in Figure 10. Constrained by the number of channel sampling points, the maximum value of the wavelength interval in the experimental test is 8. According to the curve variation laws, as the wavelength interval gradually increases, the FFC and THC curves show an overall upward trend, matching the exponential growth law derived from theory; both the simulated and measured FFC and THC curves show obvious minima near m = 3, and this extremum characteristic is highly consistent with the theoretical prediction that there exists an optimal extreme point on the crosstalk curves.

Figure 10. Variation curves of crosstalk versus wavelength interval m. (a) Fundamental frequency crosstalk (FFC); (b) total harmonic crosstalk (THC). Note that the vertical axis scales of simulation and experimental subplots are different. The comparison focuses on the curve trend rather than absolute amplitude.

The optimal selection of the wavelength interval is carried out from the perspective of engineering design. If the system conditions satisfy the extremum existence condition, the wavelength interval corresponding to $\sin {\theta }_1/\sin {\theta }_2=-{\displaystyle \frac{\cos {\varphi }_{01}\cos {\varphi }_{02}}{2\sin {\varphi }_{01}\sin {\varphi }_{02}}}$ is taken as the optimal value for FFC. Since THC is subject to two sets of extremum conditions $\sin {\theta }_1/\sin {\theta }_2=-{\displaystyle \frac{\cos {\varphi }_{01}\cos {\varphi }_{02}}{2\sin {\varphi }_{01}\sin {\varphi }_{02}}}$ and $\sin {\theta }_1/\sin {\theta }_2=-{\displaystyle \frac{\cos {\varphi }_{01}\cos {\varphi }_{02}}{2\cos {\varphi }_{01}\sin {\varphi }_{02}}}$ coupling constraints, the optimal wavelength interval needs to be selected by synthesizing the two sets of extrema. When there is no valid minimum in the calculation interval, theoretically, the smaller the wavelength interval, the better the crosstalk suppression effect.

Figure 11 presents the curves of crosstalk versus wavelength diversity group number under different wavelength intervals. Three typical values of m = 2, 5, 8 are selected for comparison. It can be observed that the variation trend of crosstalk against the wavelength-diversity group number remains basically consistent when the wavelength interval is altered. Theoretically, optimal crosstalk-suppression performance can be achieved by simultaneously adopting the optimal wavelength interval and characteristic wavelength-diversity group number. Nevertheless, constrained by the available number of spectral sampling points in practical systems, it is difficult for both parameters to reach their theoretical optima simultaneously, and a trade-off analysis is therefore indispensable.

Figure 11. Variation curves of crosstalk versus wavelength diversity group number at different wavelength intervals. (a) Fundamental frequency crosstalk (FFC); (b) total harmonic crosstalk (THC).

The wavelength interval m affects the crosstalk value by modulating the demodulated signal θ. According to $\theta =-4\mathsf{\pi }·m\Delta \lambda /{\lambda }^2$, the optimal wavelength interval m is related to the element cavity length. Meanwhile, there exists a characteristic wavelength-diversity group number that delivers optimal crosstalk-suppression performance, which is also strongly dependent on the cavity length. For practical applications with fixed cavity length, the optimal wavelength interval can be accordingly determined. The feasible value range of the wavelength-diversity group number k is jointly defined by the system channel capacity and the total number of spectral sampling points. A resource-coupling effect exists between wavelength interval m and wavelength diversity group number k: increasing the wavelength interval reduces the valid sampling points available for diversity averaging, which limits the upper bound of the configurable wavelength diversity group number. Consequently, these two parameters cannot be selected independently and require a joint trade-off under the sampling-point budget. Accordingly, this work proposes an optimization strategy: subject to the constraints of system capacity and valid sampling-point quantity, the optimal parameter pair is selected within the feasible domain of wavelength interval and wavelength diversity group number targeting maximum crosstalk-suppression performance, so as to achieve a comprehensive balance among crosstalk suppression, sampling resources and system channel capacity.

4. Comprehensive Performance Evaluation of the Crosstalk Suppression Technique

To fully verify the engineering applicability of the proposed crosstalk suppression technique for large-scale optical-fiber sensing arrays, comprehensive evaluations are carried out in this section from three dimensions: multi-channel expandability, tolerance against non-ideal operating conditions of the light source, and dynamic working performance. These correspond to three typical non-ideal working conditions in practical engineering, namely array channel capacity expansion, spectral distortion of the light source, and time-varying perturbation of the optical-path environment. Using both simulation and experimental methods, the influences of WDM channel scale, ASE light-source spectral distortion, and external optical-path perturbations on crosstalk suppression performance are systematically investigated to clarify the application scope and practical working effectiveness of the proposed technique.

Practical large-scale sensing arrays are rarely implemented relying on a single multiplexing scheme; hybrid multiplexing architectures combining WDM, TDM and SDM are commonly adopted. Taking a 256-channel array as an example, system capacity expansion can be realized by combining 8-channel WDM, 4-channel TDM and 4-channel SDM. In view of this, comparative simulation analyses under 2-, 4- and 8-channel WDM scenarios are carried out in this work, and the simulation results are shown in Figure 12. The crosstalk-suppression levels for the three operating conditions are −57.3 dB, −43.2 dB and −33.6 dB, corresponding to improvements of 38.8 dB, 38.1 dB and 29.2 dB compared with the conventional phase-shift demodulation algorithm. It can be observed that the crosstalk-suppression performance degrades to some extent as the number of WDM channels increases. Nevertheless, the overall suppression level can still be maintained above 29 dB, which verifies the favorable applicability of the proposed technique for large-scale array applications.

Figure 12. Crosstalk suppression level under different channel numbers.

In practical sensing systems, an ASE light source can hardly produce an ideally flat spectrum. Spectral distortion introduces additional interference, which may potentially degrade the crosstalk suppression performance. Accordingly, it is necessary to quantitatively evaluate the robustness of the algorithm against non-ideal source spectra. Three typical ASE spectral-envelope models are constructed: Gaussian-shaped, Lorentzian-shaped and multi-peak superposition models, which correspond to ideal smooth spectra, spectra with tail distortion, and uneven spectra disturbed by side-peak components, respectively. Simulations are performed under identical system parameters, as illustrated in Figure 13. The simulation results show that the fundamental-frequency crosstalk-suppression levels under the three spectral conditions are 38.8 dB, 36.4 dB and 35.0 dB in sequence. As the spectral non-flatness of the ASE source becomes more severe, the crosstalk-suppression performance degrades slightly with limited overall deterioration. This can be attributed to the extreme-value envelope elimination mechanism introduced in the proposed demodulation algorithm. It can effectively suppress inherent spectral fluctuations of the light source and slowly varying spectral distortions originating from the transmission link, hence granting certain robustness against ASE spectral non-flatness.

Figure 13. Simulation results under different ASE spectral-envelope conditions. (a) Gaussian spectral envelope; (b) Lorentz-type spectral envelope; (c) multi-peak superimposed envelope; (d) corrected spectrum corresponding to subplot (a); (e) corrected spectrum corresponding to subplot (b); (f) corrected spectrum corresponding to subplot (c). The orange line denotes the upper envelope, and the gray line denotes the lower envelope.

Apart from the non-ideal effects originating from system architecture, optical-path perturbations and variations in device characteristics during field-site application cause the crosstalk amplitude to fluctuate dynamically over time, which directly impairs the long-term operational reliability of the system. For this purpose, experiments concerning stability and real-time demodulation performance are further carried out. According to the established theoretical model in Equation (6), under external conditions such as optical-path perturbation and non-ideal device characteristics, each crosstalk component behaves as a cosine-type periodically time-varying function. Consequently, the crosstalk amplitude fluctuates over time, as shown in Figure 14.

Figure 14. Measured crosstalk as a function of time.

To accurately assess the maximum interference level during the dynamic operation of the system, the maximum crosstalk value within a certain time period is selected as the experimental evaluation result. This maximum-value metric can remain relatively stable under long-term observation. Therefore, this measurement method possesses favorable repeatability and environmental adaptability.

The test results of real-time demodulation performance are shown in Figure 15. Continuous multi-frame time-domain demodulation phases are acquired under dynamic experimental conditions with optical-path perturbations. The phase output of the traditional algorithm suffers severe fluctuations with a peak value of approximately 8 × 10⁻³ rad, and the time-varying crosstalk introduces considerable measurement errors. In contrast, the proposed technique can track and suppress dynamic crosstalk frame-by-frame. Its demodulated phase oscillates slightly around the zero phase with a peak value of only about 1.2 × 10⁻³ rad, and the phase fluctuations are significantly suppressed. These time-domain comparison results demonstrate that the proposed algorithm possesses favorable real-time dynamic processing capability. It can continuously suppress time-varying crosstalk under external perturbations, effectively improve the accuracy of phase demodulation, and guarantee the long-term operational reliability of the system.

Figure 15. Real-time demodulation results of the proposed algorithm.

Combining the above simulation and experimental results, the following conclusion can be drawn: the crosstalk suppression technique proposed in this work can maintain high crosstalk-suppression gain under various typical non-ideal engineering conditions, including multi-channel array capacity expansion, spectral distortion of the light source, and external optical-path perturbations. Meanwhile, it delivers reliable real-time demodulation capability and can satisfy the practical engineering application requirements of large-scale optical-fiber sensing arrays.

5. Conclusions

To address the spectral leakage crosstalk problem in WDM arrays of extrinsic fiber-optic F-P hydrophones, this paper proposes a crosstalk suppression technique based on a phase-shifting diversity demodulation scheme, establishes a theoretical model of WDM channel crosstalk, and clarifies its crosstalk cancelation mechanism. Parameter analysis shows that the proposed technique can relax the hardware index requirement for device extinction ratio, which helps reduce the array construction cost. Meanwhile, the theoretical expression for the optimal average wavelength number is obtained as $k_{\mathrm{opt}}={\left[n{{\lambda }_0}^2/2\Delta L\Delta \lambda \right]}_{\mathrm{int}}$. When hardware conditions are limited, a compromise value can be selected in the neighborhood of k_opt to balance the array channel multiplexing capacity and crosstalk suppression performance. The technique can also increase the upper limit of signal dynamic measurement to adapt to underwater acoustic dynamic detection scenarios. There exists an optimal value for the wavelength interval. In practical design, we should take crosstalk-suppression performance, sampling resources and system channel capacity into consideration and conduct joint trade-off selection together with the wavelength-diversity group number. The proposed technique achieves crosstalk suppression only through algorithm optimization without modifying the hardware optical path, which can effectively break through the crosstalk bottleneck of WDM arrays. Targeting spectral-leakage crosstalk in pure WDM FP sensor arrays, the proposed technique achieves a 33.8 dB improvement in FFC suppression solely by optimizing the back-end FPGA demodulation algorithm, without modifying the front-end WDM optical hardware. This scheme features low hardware overhead. Multi-channel expansion only consumes back-end computing resources, and it exhibits favorable adaptability for all WDM sensing arrays. It not only expands the application scope of FP sensors in WDM systems but also provides an effective implementation approach for crosstalk suppression of such systems. With a clear principle, simple engineering implementation, and support for real-time demodulation of sensing signals, this technique can provide important technical support for the engineering application of large-scale extrinsic fiber-optic F-P sensor arrays in the field of underwater acoustic detection.