Crosstalk suppression of extrinsic Fabry-Perot interferometric sensor array based on five-step phase shift demodulation scheme using multiwavelength averaging

Zheng Liu; Zheng Liu; Fuyin Wang; Ji Xia; Gang Liu; Gang Liu; Shuidong Xiong; Qiong Yao

doi:10.1364/OE.505368

1. Introduction

Optical fiber extrinsic Fabry-Perot interferometer (EFPI) sensors [1–3] have attracted significant attention due to their exceptional performance characteristics, including high sensitivity [4], compact size [5], and immunity to electromagnetic interference [6]. These sensors are capable of detecting variations in cavity length and refractive index of the medium, enabling the measurement of various physical factors, such as pressure [7–9], temperature [10], vibration [11], displacement [12], and acceleration [13].

In complex environments, detailed information about underwater acoustic signals is often challenging to obtain using an individual sensor, necessitating the integration of multiple sensors into a sensor array to achieve more accurate measurement results [14]. However, in optical fiber sensor arrays, crosstalk [15] occurs during the transmission and demodulation of optical signals, resulting in mutual interference between different channel signals. When the crosstalk in a multiplexing system exceeds -40 dB, it can reduce the multiplexing capacity and introduce deviations in signal detection [16,17]. Thus, crosstalk has emerged as a challenging issue that hinders the development and application of multiplexing techniques in optical fiber sensor arrays. Several methods and demodulation schemes have been proposed to reduce crosstalk in multiplexing systems. Chen [18–20] utilized a Fourier transform-based demodulation scheme to sequentially demodulate the signals of serially and parallelly multiplexing EFPI sensors, achieving levels of crosstalk below 5% and 2.4%, respectively, indicating effective crosstalk control. However, these results still fall short of meeting the crosstalk requirements for sensor arrays. Moreover, Cao [21,22] conducted a theoretical analysis of crosstalk between channels in time-division multiplexing (TDM) systems, revealing that crosstalk decreases approximately linearly with an increase in extinction ratio (ER) and increases with the number of channels in the array. Furthermore, Xie [23] employed a cross-correlation demodulation scheme based on the Kaiser window function to demodulate a frequency-division multiplexing (FDM) system of a two-element EFPI sensor array, achieving a crosstalk level of -48 dB at 200 Hz and effectively reducing crosstalk between channels in the multiplexing array. Similarly, Liu [24] implemented a dual-wavelength active quadrature demodulation scheme using a fast-tunable laser and ensured that the working wavelength was near the optimal point. They constructed a two-point sound velocity sensing system and a four-point sound source localization system, demonstrating the effective operation of different sensing elements without crosstalk. Although these demodulation schemes effectively suppress crosstalk in the EFPI sensor arrays, enhancing system multiplexing capacity and promoting signal demodulation stability, they still face limitations as crosstalk increases with the scale of multiplexing. Hence, meeting the requirements of large-scale EFPI sensor array multiplexing remains a challenge.

This paper proposes a five-step phase shift demodulation scheme based on a multiwavelength averaging method to effectively suppress crosstalk in the EFPI sensor arrays. Building upon the traditional five-step phase shift demodulation scheme, the proposed demodulation scheme introduces the concept of averaging the intensity information from multiple wavelengths to mitigate the impact of crosstalk on signal demodulation. The crosstalk of the multiwavelength demodulation scheme is analyzed, and its crosstalk suppression capability is verified through numerical simulations and experiments under different parameter conditions. The results demonstrate that the multiwavelength demodulation scheme achieves lower inter-channel crosstalk and higher accuracy in signal demodulation compared to the single-wavelength demodulation scheme. Therefore, this paper presents a feasible solution for the large-scale multiplexing applications of the EFPI sensor arrays.

2. Principle

Based on consideration of principles and experimental convenience, a parallel multiplexing system consisting of two EFPI sensors, namely S1 and S2, is selected as the research subject. The analysis primarily focuses on the occurrence of crosstalk resulting from the interference signal superposition between the signal light of S1 and the leakage light of S2. To provide a theoretical explanation of the demodulation scheme in crosstalk suppression, a double-beam interference model is employed for analysis. The interference spectrum of the EFPI sensor is obtained through white-light interferometry (WLI) technology. The constants N and Δλ represent the number and interval of wavelength sampling points, while L₁ and L₂ correspond to the initial cavity lengths of S1 and S2, respectively. Moreover, ER can be represented as ε. Thus, the superimposed interference spectral signal in the two-element sensing system can be expressed as

(1)$$I = {A_1} + {B_1}\cos ({{\varphi_1}} )+ {A_{20}} + {B_{20}}\cos ({{\varphi_2}} )$$

(2)$${\varphi _i} = \frac{{4\pi {L_i}}}{{{\lambda _k}}},i = 1,2$$

In the expression, λ_k=λ₀+ kΔλ represents the wavelength at the kth sampling point, where λ₀ is the initial wavelength. A₂₀=ε·A₂, B₂₀=ε·B₂, while A₁, A₂, B₁, and B₂ denote the direct current (DC) and alternating current (AC) components of the interference signal of S1 and S2. Then, the five-step phase shift signals can be written as

(3)$${I_m} = {A_1} + {B_1}\cos [{{\varphi_1} + (m - 3){\theta_1}} ]+ {A_{20}} + {B_{20}}\cos [{{\varphi_2} + (m - 3){\theta_2}} ],m = 1\sim 5$$

where θ_i= -4πL_i·MΔλ/λ_k² (i = 1, 2) represents the demodulation parameter. Two orthogonal signals, SR = (I₂-I₄)/2sinθ and CR = (2I₃-I₁-I₅)/4sin²θ, are generated by utilizing the phase connection between each group of five-step phase shift interference signals. The demodulation phase is obtained by the arctangent algorithm, which can be mathematically expressed as

(4)$$\varphi = \arctan \left( {\frac{{SR}}{{CR}}} \right)$$

To simplify the theoretical analysis, the assumption is made that there is no load signal present on element S1, while a single-frequency signal is being loaded on element S2. As a result, the phases of S1 and S2 can be expressed as φ₁=φ₀₁ and φ₂=φ₀₂+φ_s, where φ₀₁ and φ₀₂ represent the initial phases of S1 and S2, respectively. Therefore, the demodulated phase error induced by the interference signal superposition can be represented as

(5)$${n_\varphi } \approx \cos {\varphi _1}\delta SR - \sin {\varphi _1}\delta CR$$

where δSR = B₂₀sinθ₂sinφ₂/B₁sinθ₁, δCR = B₂₀sin²θ₂cosφ₂/B₁sin²θ₁. Assuming that the frequency of φ_s is ω_s and the signal amplitude is A_s. Both cosφ_s and sinφ_s can be rewritten with Bessel functions. Thus, the demodulation phase error can be rewritten as

(6)$${n_\varphi } = {A_{D\textrm{C}}} + {A_{FFC}}\cos {\omega _s}t + \sum\limits_{j = 1}^\infty {{{({ - 1} )}^j}{A_{THC - o}}\cos (2j + 1){\omega _s}t} + \sum\limits_{j = 1}^\infty {{{({ - 1} )}^j}} {A_{THC - e}}\cos 2j{\omega _s}t$$

In the given expression, the crosstalk can be categorized into different components, including the DC crosstalk, fundamental frequency crosstalk (FFC), and total harmonics crosstalk (THC). The crosstalk of the DC component is a constant that can be eliminated by employing DC removal techniques such as high-pass filtering or AC coupling. The proposed scheme is mainly aimed at suppressing crosstalk for multiplexing arrays of low-frequency underwater acoustic signals, so AC coupling is adopted to remove the DC component in the following signal demodulation. Then, the amplitude of FFC (A_FFC), odd-order harmonic crosstalk (A_THC-o), and even-order harmonic crosstalk (A_THC-e) can be expressed as

(7)$$\left\{ {\begin{array}{{c}} {{A_{FFC}} ={-} 2\frac{{\varepsilon \cdot {B_2}}}{{{B_1}}}\left( {\frac{{\sin {\theta_2}}}{{\sin {\theta_1}}}\cos {\varphi_{01}}\cos {\varphi_{02}} + \frac{{{{\sin }^2}{\theta_2}}}{{{{\sin }^2}{\theta_1}}}\sin {\varphi_{01}}\sin {\varphi_{02}}} \right){J_1}({{A_s}} )}\\ {{A_{THD - o}} ={-} 2\frac{{\varepsilon \cdot {B_2}}}{{{B_1}}}\left( {\frac{{\sin {\theta_2}}}{{\sin {\theta_1}}}\cos {\varphi_{01}}\cos {\varphi_{02}} + \frac{{{{\sin }^2}{\theta_2}}}{{{{\sin }^2}{\theta_1}}}\sin {\varphi_{01}}\sin {\varphi_{02}}} \right){J_{2j + 1}}({{A_s}} )}\\ {{A_{THD - e}} = 2\frac{{\varepsilon \cdot {B_2}}}{{{B_1}}}\left( {\frac{{\sin {\theta_2}}}{{\sin {\theta_1}}}\cos {\varphi_{01}}\sin {\varphi_{02}} + \frac{{{{\sin }^2}{\theta_2}}}{{{{\sin }^2}{\theta_1}}}\sin {\varphi_{01}}\cos {\varphi_{02}}} \right){J_{2j}}({{A_s}} )} \end{array}} \right.$$

According to Eq (7), the FFC and THC of the five-step phase shift demodulation scheme are related to ER, A_s, φ₀₁, φ₀₂, and sinθ₂/sinθ₁. In accordance with φ_i= 4πL_i/λ_k, the equation cosφ₀₁cosφ₀₂, sinφ₀₁sinφ₀₂, cosφ₀₁sinφ₀₂, sinφ₀₁cosφ₀₂ can be rewritten as

(8)$$\left\{ {\begin{array}{{c}} {\cos {\varphi_{01}}\cos {\varphi_{02}} \approx \frac{1}{2}\left[ {\cos \left( {\frac{{4\pi }}{{{\lambda_0}}}\Delta L - \frac{{4\pi \Delta \lambda }}{{\lambda_0^2}}k\Delta L} \right) + \cos \left( {\frac{{4\pi }}{{{\lambda_0}}}({{L_1} + {L_2}} )- \frac{{4\pi \Delta \lambda }}{{\lambda_0^2}}k({{L_1} + {L_2}} )} \right)} \right]}\\ {\sin {\varphi_{01}}\sin {\varphi_{02}} \approx \frac{1}{2}\left[ {\cos \left( {\frac{{4\pi }}{{{\lambda_0}}}\Delta L - \frac{{4\pi \Delta \lambda }}{{\lambda_0^2}}k\Delta L} \right) - \cos \left( {\frac{{4\pi }}{{{\lambda_0}}}({{L_1} + {L_2}} )- \frac{{4\pi \Delta \lambda }}{{\lambda_0^2}}k({{L_1} + {L_2}} )} \right)} \right]}\\ {\cos {\varphi_{01}}\sin {\varphi_{02}} \approx \frac{1}{2}\left[ { - \sin \left( {\frac{{4\pi }}{{{\lambda_0}}}\Delta L - \frac{{4\pi \Delta \lambda }}{{\lambda_0^2}}k\Delta L} \right) + \sin \left( {\frac{{4\pi }}{{{\lambda_0}}}({{L_1} + {L_2}} )- \frac{{4\pi \Delta \lambda }}{{\lambda_0^2}}k({{L_1} + {L_2}} )} \right)} \right]}\\ {\sin {\varphi_{01}}\cos {\varphi_{02}} \approx \frac{1}{2}\left[ {\sin \left( {\frac{{4\pi }}{{{\lambda_0}}}\Delta L - \frac{{4\pi \Delta \lambda }}{{\lambda_0^2}}k\Delta L} \right) + \sin \left( {\frac{{4\pi }}{{{\lambda_0}}}({{L_1} + {L_2}} )- \frac{{4\pi \Delta \lambda }}{{\lambda_0^2}}k({{L_1} + {L_2}} )} \right)} \right]} \end{array}} \right.$$

where ΔL is the difference between the cavity lengths of S1 and S2. As indicated in Eq (8), the level of crosstalk is significantly induced by the positions of the working points λ_k. For any sinusoidal function f(t) with a period T, the sum of the function over the entire period is zero, also known as $\int_0^\textrm{T} {f\left( t \right)} dt = 0$. Therefore, by ensuring that the number of sampled wavelengths between λ₀ and λ_k fulfills the condition for a complete period of the cosine function, averaging the demodulated results of the five-step phase shift signals between them can exhibit superior crosstalk suppression. In contrast, the conventional demodulation scheme for crosstalk suppression relies on information from only one working point, making it more susceptible to variations in the working point or errors in phase shift. This susceptibility can have a negative impact on crosstalk suppression. Additionally, it is crucial to consider the relationship between crosstalk and the cavity length difference. Thus, when the cavity length difference ΔL is zero, Eq (7) can be rewritten as

(9)$$\left\{ {\begin{array}{{c}} {{A_{FFC}} ={-} 2\frac{{{B_{20}}}}{{{B_1}}}{J_1}({{A_s}} )}\\ {{A_{THD - o}} ={-} 2\frac{{{B_{20}}}}{{{B_1}}}{J_{2j + 1}}({{A_s}} )}\\ {{A_{THD - e}} = 2\frac{{{B_{20}}}}{{{B_1}}}{J_{2j}}({{A_s}} )\sin \frac{{4\pi }}{{{\lambda_k}}} \cdot 2{L_1}} \end{array}} \right.$$

As shown in Eq (9), it is apparent that the proposed demodulation scheme does not effectively suppress FFC and odd harmonics crosstalk when the cavity length difference ΔL is zero. Although the amplitude of even-order harmonic crosstalk A_THC-e demonstrates sinusoidal fluctuations with changes in working point λ_k, the working point λ_k is restricted by the spectrum range. As a result, when the cavity lengths of the two elements are equal, the difference in crosstalk suppression between the multiwavelength demodulation scheme and the single-wavelength demodulation scheme is small enough to be neglected. Therefore, the analysis aims to evaluate the crosstalk suppression capability of the proposed demodulation scheme on a two-element sensing system with the cavity length difference ΔL unequal to zero.

3. Numerical simulation and experiment

It is reasonable to believe that the multiwavelength demodulation scheme has a better crosstalk suppression capability in contrast to the single-wavelength demodulation scheme from the previous theoretical analysis. To confirm this, numerical simulations and experiments are put forward to investigate the crosstalk suppression performance of both the multiwavelength and single-wavelength demodulation schemes. Various factors, including the ER of the system, the signal amplitude A_s, the number of average wavelengths Ns, and the demodulation parameter θ are taken into account during the investigation.

A two-element sensing system is constructed, consisting of a deaf EFPI sensor and a Michelson interferometer (MI) with an adjustable arm-length difference, as shown in Fig. 1. In the setup, the deaf EFPI sensor employed as probe S1, with a cavity length of L₁= 305 µm, while the MI serves as probe S2, with a cavity length of L₂= 605 µm. The system operates by transmitting signal light generated from amplified spontaneous emission (ASE, the spectrum range is 1520 nm to 1570 nm) into the optical fiber circulator through an isolator. The circulator directs the signal light towards the coupler, where it is split into two beams that enter the deaf EFPI sensor and the MI, respectively. An attenuator placed before the MI facilitates convenient control of the ER. As independent interferometric interactions occur in the deaf EFPI sensor and the MI, the reflected lights recombine upon passing through the coupler once again. The resulting interference signal is collected by a fiber Bragg grating analyzer (FBGA) and subsequently sent to a computer for real-time demodulation. Moreover, the sampling rate is 5 kHz. In the experiment, the deaf EFPI sensor is designed to isolate the sensitive region of the EFPI sensor by using UV adhesive, preventing it from receiving external signals. Therefore, it is not sensitive to external ambient signals and can effectively eliminate interference from external signals, allowing for a more accurate analysis of the crosstalk suppression of the proposed demodulation method. To adjust the arm-length difference of the MI, an optical delay line (ODL) is employed. Additionally, a signal with a specific frequency and amplitude is then loaded onto the MI through a signal source and a fiber phase shifter (FPS). The interference signal is generated by the light reflected from the Faraday rotator mirror (FRM). The advantage of using the MI with an adjustable arm-length difference lies in its capability to simulate various signal frequencies and amplitudes that can be loaded onto the EFPI sensor.

Fig. 1. The schematic diagram of the two-element sensing system based on a deaf EFPI sensor and the MI with an adjustable arm-length difference.

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To fulfill the array multiplexing requirements and ensure that no single parameter dominates the crosstalk suppression of the demodulation scheme. The controlled variable method is utilized to avoid overshadowing the crosstalk suppression during simulation and experimentation. This method enables the simulation and experimental analysis of the impact of different parameters on crosstalk. The parameter settings as indicated in Table 1.

Table 1. The parameter settings

View Table

3.1 Extinction ration

As extensively discussed by other researchers, it has been observed that the crosstalk between signals generally shows a consistent linear decrease as the ER is increased. This relationship implies that achieving lower levels of crosstalk requires higher ER, thereby significantly raising the requirements for arrayed waveguide multiplexing systems. In the experiment, a two-element sensing array is constructed, and an attenuator is used to control the ER. Here, the modulation signal frequency f_s in the experiment is set as 63 Hz to avoid the influence of the 50 Hz power frequency signal and its harmonic components. Moreover, the other parameters are as shown in Parameter 1 of Table 1. The results, depicted in Fig. 2, indicate that both FFC and THC decrease as the ER increases. Furthermore, the FFC and THC of the multiwavelength demodulation scheme are lower than those of the single-wavelength demodulation scheme, demonstrating the superior crosstalk suppression performance of the multiwavelength demodulation scheme. By employing the multiwavelength demodulation scheme, the EFPI sensor array can achieve FFC and THC below -50 dB and -40 dB respectively, with an ER of 25 dB. In contrast, the single-wavelength demodulation scheme requires an ER of over 35 dB to achieve similar performance. This highlights the advantage of the multiwavelength demodulation scheme in mitigating crosstalk, as it enables the array's crosstalk requirements to be met with a lower ER. As the number of multiplexing channels in the array increases, the crosstalk in the system tends to increase as well. However, the utilization of the multiwavelength demodulation scheme enables the construction of larger-scale EFPI sensor arrays. Nevertheless, there are some discrepancies between the experimental and simulated results in terms of the variation rates of FFC and THC concerning the ER. The cause of this discrepancy lies in the temperature drift effect resulting from the operation of the FPS, which causes variations in the equivalent cavity length of the MI.

Fig. 2. The crosstalk with ER variation ranging from 25 to 55 dB. The MW is multiwavelength demodulation, and the SW is single-wavelength demodulation. (a) FFC. (b) THC.

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3.2 Signal amplitude

As indicated in Eq (7), it is established that the FFC and THC are linked to the Bessel function values determined by the amplitude of the measured signal. The mathematical association allows for a deeper understanding of the relationship between crosstalk and the measured signal amplitude, facilitating the optimization of demodulation schemes and enhancing the performance of the entire system. To control the experimental variables, the other parameters are shown in Parameter 2 of Table 1. The simulation and experiment results are presented in Fig. 3. As shown in Fig. 3(a) and (c), it can be observed that FFC displays continuous fluctuations with variations in the signal amplitude A_s. The phase amplitude difference between the minimum of FFC is observed to be π rad, which aligns with the simulation results. The impact of the first-order Bessel function values of the modulated signal amplitude A_s on crosstalk can explain this behavior, resulting in periodic fluctuations with a period of π rad. Furthermore, the FFC of the multiwavelength demodulation scheme is approximately 30 dB lower than that of the single-wavelength demodulation scheme. Figure 3(b) and (d) illustrate that THC initially increases gradually with an increase in the signal amplitude. Once it reaches a maximum, the THC gradually stabilizes. However, in the experimental results, significant fluctuations in THC are observed for the multiwavelength demodulation scheme at A_s =8 rad, deviating from the simulation results. Insufficient preheating time for the MI in the second part is responsible for this discrepancy, which occurs due to the experiment being conducted in two parts. Indeed, the temperature drift effect can lead to changes in the equivalent cavity length, resulting in fluctuations in THC. The sensitivity of the multiwavelength demodulation scheme to cavity length changes exceeds that of the single-wavelength demodulation scheme. The phenomenon is due to the fact that the multiwavelength demodulation scheme relies on the information of multiple wavelengths within the EFPI sensor. Furthermore, by adjusting the cavity length difference, the multiwavelength demodulation scheme can effectively suppress crosstalk and enhance overall performance. This adjustment alleviates the requirement for consistent cavity length among the various elements in the sensing array. Therefore, careful design of the cavity length difference between elements in the EFPI array substantially strengthens the capability of the multiwavelength demodulation scheme to mitigate crosstalk and elevate performance.

Fig. 3. The crosstalk with signal amplitude A_s variation. (a) FFC of simulation results. (b) THC of simulation results. (c) FFC of experiment results. (d) THC of experiment results.

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3.3 Number of average wavelengths

To satisfy the requirement that the sum of the cosine function over the entire period is zero, the number of average wavelengths Ns in the multiwavelength demodulation scheme should be equal to the number of wavelengths between λ₀ and λ_k. The optimized number of average wavelengths Ns can be obtained from Eq (8).

(10)$$\frac{{4\pi \Delta \lambda }}{{\lambda _0^2}}Ns\Delta L = 2\pi \Rightarrow Ns = {\left[ {\frac{{m\lambda_0^2}}{{2\Delta \lambda \cdot \Delta L}}} \right]_{{\mathop{\rm int}} }}\begin{array}{{cc}} {}&{} \end{array},m = 1,2,3 \cdots$$

(11)$$\frac{{4\pi \Delta \lambda }}{{\lambda _0^2}}Ns({{L_1} + {L_2}} )= 2\pi \Rightarrow Ns = {\left[ {\frac{{m\lambda_0^2}}{{2\Delta \lambda \cdot ({{L_1} + {L_2}} )}}} \right]_{{\mathop{\rm int}} }}\begin{array}{{cc}} {}&{} \end{array},m = 1,2,3 \cdots$$

To analyze the impact of different numbers of average wavelengths Ns obtained from these two different calculation methods on crosstalk suppression, the number of average wavelengths is varied within the range of 1 to 128. The initial wavelength λ₀ is set to 1550 nm, with a sampling wavelength interval Δλ of 0.1 nm. Moreover, the other parameters are indicated in Parameter 3 of Table 1. The simulation and experimental results are presented in Fig. 4. Based on the simulated cavity lengths of 305 µm and 605 µm for the two elements, the optimized number of average wavelengths obtained from the two different calculation methods can be determined within the selected range of wavelength numbers. Specifically, for the first calculation method Ns = [mλ₀²/2ΔλΔL]_int, the optimized number of average wavelength Ns values are 40, 80, and 120. For the other method Ns = [mλ₀²/2Δλ(L₁+ L₂)]_int, the optimized number of average wavelength Ns values are 14, 26, 40, 53, 66, 79, 92, 106, and 119. Figure 4 illustrates that the first calculation method Ns = [mλ₀²/2ΔλΔL]_int showcases superior crosstalk suppression with the obtained average wavelength numbers. However, it is noteworthy that as the average wavelength number increases, the experimental results deviate more from the simulation results, as indicated by the blue dashed box in Fig. 4(a). This discrepancy can be attributed to the impact of temperature drift on the MI. Over time, with the temperature gradually rising, the effective cavity length of the MI increases. On the other hand, the second calculation method Ns = [mλ₀²/2Δλ(L₁+ L₂)]_int expresses a notable difference between the experimental and simulation results in terms of FFC suppression with the obtained average wavelength numbers. The black dashed box in Fig. 4(a) illustrates the discrepancy, which can also be linked to the influence of temperature drift on the MI. Furthermore, for certain average wavelength numbers (such as 26, 66, and 106), the crosstalk suppression effect is not adequately demonstrated. The fact that the crosstalk caused by the cosine and sine functions is not effectively suppressed at these specific average wavelength numbers can be linked to this observation.

Fig. 4. The crosstalk with the number of average wavelengths Ns variation. (a) FFC. (b) THC.

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3.4 Demodulation parameter

When analyzing the crosstalk of the two-element sensing system, it is crucial to thoroughly consider the correlation between the crosstalk and the demodulation parameter θ. Understanding and considering this relationship are key aspects in optimizing the performance of the system and ensuring accurate measurements. The analysis enables a deeper understanding of the relationship between the demodulation parameter θ and crosstalk, facilitating the development of effective strategies to minimize crosstalk and enhance the overall reliability and sensitivity of the system. According to θ₁=-4πL₁·MΔλ/λ_k² and θ₂=-4πL₂·MΔλ/λ_k², the demodulated parameter θ is related to the cavity lengths of two elements and the wavelength interval M. By utilizing the Eq (7), the crosstalk can be expressed as a simple quadratic equation, sinθ₂/sinθ₁ is the independent variable. The change of sinθ₂/sinθ₁ with θ₁ and θ₂ is shown in Fig. 5. As depicted in Fig. 5, the minimum values of the carve corresponding to each θ₂ occur at θ₁=π/2 rad. This is where the amplitude of FFC also reaches its minimum value. Similarly, the same conclusion can be obtained when analyzing harmonic crosstalk.

Fig. 5. The change of sinθ₂/sinθ₁ with θ₁ and θ₂.

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The simulation and experimental results are obtained using the parameters in parameter 4 of Table 1, as shown in Fig. 6. It can be observed that as the demodulation parameter θ₁ approaches π/2 rad, the corresponding FFC and THC gradually decrease. The experimental results of FFC suppression are consistent with the simulation results. Although the experimental results for THC show consistency with the simulation trend, demonstrate a discrepancy of approximately 20 dB compared to the theoretical results. The difference in THC can be ascribed to the excessive sensitivity of MI to external environmental noise, which leads to discordant outcomes between the simulation and experimental results.

Fig. 6. The crosstalk with the demodulation parameter θ₁ variation. (a) FFC. (b) THC.

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4. Conclusion

A five-step phase shift demodulation scheme using multiwavelength averaging is proposed to suppress the crosstalk in the EFPI sensor array. By averaging the demodulation results of five-step phase shift signals with multiple consecutive wavelengths, utilizing spectral information and selecting optimal parameters, it resolves the crosstalk in the EFPI sensor array. Numerical simulations and experiments are conducted to investigate the crosstalk suppression of the proposed demodulation scheme in this paper. On the one hand, the results indicate that the crosstalk of the multiwavelength demodulation scheme is 10 dB lower than that of the single-wavelength demodulation scheme at the same ER. This ensures crosstalk suppression capability even at the lower ER, greatly enhancing the optical multiplexing capability. Meanwhile, the multiwavelength demodulation scheme demonstrates excellent crosstalk suppression capability at the average wavelength numbers Ns = [mλ₀²/2ΔλΔL]_int. On the other hand, both the FFC and THC reach their minimum values when the demodulation parameter θ approaches π/2 rad, providing a solid theoretical foundation for the multiwavelength demodulation scheme. Moreover, the scheme effectively suppresses crosstalk between elements with different cavity lengths, successfully mitigating the challenge of inconsistent cavity lengths in EFPI sensor arrays. The proposed demodulation scheme, through multiwavelength averaging and parameter optimization, achieves low crosstalk and low fabrication requirements for large-scale optical fiber EFPI hydrophone multiplexing arrays.

Funding

National Defense Science and Technology Foundation Enhancement Program (173 Key Program) (2019-JCJQ-ZD-026-00).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Crosstalk suppression of extrinsic Fabry-Perot interferometric sensor array based on five-step phase shift demodulation scheme using multiwavelength averaging

Abstract

1. Introduction

2. Principle

3. Numerical simulation and experiment

3.1 Extinction ration

3.2 Signal amplitude

3.3 Number of average wavelengths

3.4 Demodulation parameter

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Tables (1)

Equations (11)

Optics Express

Parameter	Extinction ratio ε (dB)	The number of averaged wavelengths Ns	Wavelength interval M	Cavity length of S1 (µm)	Cavity length of S2 (µm)
Parameter 1	—	128	5	305	605
Parameter 2	40	128	5	305	605
Parameter 3	40	—	5	305	605
Parameter 4	40	128	—	305	605