1. Introduction

The interferometric fiber Bragg grating (FBG) array based on the Fabry-Pérot cavity, with its high sensitivity, minimal optical structure, and potential durability and reliability [1–3], has been considered as an attractive optical fiber sensing solution for various harsh environment applications, such as seabed permanent seismic monitoring [2–4] and compact small-diameter towed arrays [5,6].

Generally, the path-matching method is used to form effective interference [6–9], where the interrogating pulses are two laser beams with a time delay equaling to that of the roundtrip in the Fabry-Pérot cavity. As the two laser beams do not propagate in the leading fiber at the same time, there is a slight propagating phase discrimination between them, which can be demodulated from the interference. In practical applications, the sensing end (the wet end) is often far away from demodulation equipment. The sensor system always contains the leading fiber, which is hundreds of meters or even tens of kilometers in length. Limited by the sensing conditions, it is difficult to shorten the length of the leading fiber to reduce noise. When the leading fiber is disturbed by environmental vibration, torsion, bending, or temperature [10,11], the demodulated background phase noise can be greatly enhanced, i.e., the leading-fiber-induced phase noise, or the cable-induced noise. The leading-fiber-induced phase noise results from random fiber birefringence disturbance and random fiber length contraction and expansion. The former is usually considered as a part of polarization induced signal fading [12]. An excellent polarization-induced phase noise suppression method should suffice to eliminate the polarization disturbances both in and out the Fabry-Pérot cavity simultaneously. For instance, Peng Jiang et al. proposed a polarization switching (PS) and phase generated carrier (PGC) hybrid processing method to eliminate polarization-induced signal fading and then proved that the polarization synthesized signals from the PS method were independent of the leading fiber birefringence disturbance [13–15]. The latter noise owing to random fiber length contraction and expansion is also called the Doppler noise, which cannot be canceled without a specially designed method. Waagaard et al. proposed a reference interferometer based on a polarization resolved interrogation technique to achieve 22 dB suppression for the single-frequency Doppler noise [10]. It is noteworthy that the above two noise sources are coupled with each other in the demodulated background phase, so the research on Doppler noise and its suppression method when ignoring the birefringence disturbance is meaningless.

This paper focuses on the leading-fiber-induced Doppler noise using the PS and PGC hybrid processing method [15]. A model of the Doppler phase noise is presented, where the conversion from the arbitrary random stretching of the leading fiber to the demodulated sensor phase noise is demonstrated. Simulation results provide a full demonstration of the leading-fiber-induced Doppler noises in all the polarization channels and time-division multiplexing (TDM) channels. A Doppler phase noise suppression method based on the PS and PGC hybrid processing method is proposed in which the polarization synthesis should be completed before suppressing the Doppler noise. Experimental results verify the characteristics of the leading-fiber-induced Doppler noises and the noise suppression method, which achieved a maximum suppression of about 30 dB. To the best of our knowledge, this is the highest suppression for the Doppler noise in the interferometric FBG arrays.

This paper is arranged as follows. Section 2 is about the model of the Doppler phase noise. Section 3 is on the suppression method. Section 4 presents the experimental results with the sinusoidal and wideband Doppler noise excitations in the interferometric FBG sensor system and the evaluation of noise suppression effects. In Section 5, a brief summary is presented.

2. Theory of the Doppler phase noise

2.1 Modeling of the leading-fiber-induced Doppler noise

The principle of the interferometric FBG sensor array has been demonstrated in [13–15]. Here we present a similar sensing structure in which the laser pulses in the leading fiber are specially addressed, as shown in Fig. 1. The interrogation light laser pulses are injected into the leading fiber. If the pulse interval ${\tau _s}$ equals the light roundtrip between the adjacent FBGs, multiple interferences occur between the 2^nd pulse reflected by the i^th FBG and the 1^st one reflected by the (i+1)^th FBG (i=0,1). External physical quantity modulates interference lights’ phase by altering the fiber coils’ parameters, a valid sensing signal could be extracted according to the intensities. Since the interference pulses that characterize different fiber coils arrive in a time-sharing manner. Therefore, the multiple sensing signals could be obtained simultaneously via the time-division multiplexing (TDM) technique, as shown in Fig. 1, defined as TDM1 and TDM2.

 

Fig. 1. Schematic diagram of a two-TDM interferometric FBG sensor array

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The interrogation light pulse pairs with an interval ${\tau _s}$ are periodically injected into the leading fiber through an optical circulator (OC). After the 1^st pulse propagates through an optical path which contains in sequence the leading fiber, the FBG0, and the fiber coil 1, a portion reflected by FBG1 reversely propagates through the fiber coil 1 and FBG0 again, as shown by the red line in Fig. 1. At the same time, the 2^nd pulse injected ${\tau _s}$ behind the 1^st one passes through the leading fiber and is partially reflected by FBG0 as shown by the blue line. if ${\tau _s}$ equals to the light roundtrip between the adjacent FBGs, the interference of TDM1 occurs at the FBG0. Similarly, the interference of TDM2 occurs at FBG1. The two contributions of the interference beams’ phase-shift are the sensing fiber and the leading fiber. If random vibration is applied to the leading fiber, from point A to B in Fig. 1, a slight path-mismatch will occur in all interferometers via altering the interrogation pulse interval ${\tau _s}$. The resulting phase perturbance is called the Doppler phase-shift.

If we only focus on the leading fiber under arbitrary longitudinal stress, a model of the Doppler phase-shift is established as follows. Here, $p(t )$ is defined as the time required for light to propagate once in the leading fiber under the arbitrary disturbances. So, $p(t )$ can be expressed as

Here ${l_0}$ is the intrinsic length of the leading fiber, ${n_e}$ is the effective refractive index, and $c$ is the speed of light in vacuum. $n(t )$ is the increment in propagation time caused by the fluctuation of the fiber length and the effective refractive index. It is assumed that the two pulses are injected into the leading fiber at the time $t$ and $t + {\tau _s}$, respectively. Then, the propagation time for the two pulses to pass the leading fiber can be written as $p(t )$ and $p({t + {\tau_s}} )$, respectively. Here ${\tau _s}$ is the pulse-interval of the interrogation pulse pair at point A in Fig. 1. The moments when the two pulses arrive at point B, ${t_{B0}}$ and ${t_{B1}}$, could be written respectively as

The new pulse-interval $\tau _s^B$ at point B which is the actual input of the FBG array can be expressed as

Putting Eq. (1) into Eq. (3) and removing the static items, the pulse interval increment $\mathrm{\Delta }{\tau _s}$ can be obtained as

If the maximal angle frequency of $n(t )$ is ${\omega _h}$, as the pulse interval of the interrogation pulse pair is usually much smaller than the period of the mechanical stretching, i.e., ${\tau _s} \ll 2\pi /{\omega _h}$, Eq. (4) can be roughly rewritten as

Here $n^{\prime}(t )$ is the first derivative of $n(t )$. Then, the phase shift caused by the increment of the interrogation pulse interval, or the Doppler phase-shift ${\phi _D}(t )$, could be expressed as

Here $\upsilon $ is the light frequency of the interrogating light. Substituting Eq. (5) into Eq. (6), then the analytical expression of the Doppler phase-shift can be obtained as

Equation (7) gives the distribution model of the Doppler phase-shift induced by the leading fiber with arbitrary length stretching.

The Jones matrix of a single-mode (SM) fiber can be denoted by $\textbf{B}$, which is a 2×2 complex matrix. The Jones vectors ${\textbf{E}_{in1}} = {[{{E_{0x}}\; \; {E_{0y}}} ]^T}{e^{j{\phi _c}}}$ and ${\textbf{E}_{in2}} = {[{{E_{1x}}\; \; {E_{1y}}} ]^T}$ are used to describe the states of polarization (SOPs) for the two interrogation light beams. Here ${\phi _c} = C\textrm{cos}({{\omega_c}t} )$ is the phase modulation carrier, which is introduced to solve the random phase fading by using the PGC method [16]. $C$ is the PGC modulation depth, and ${\omega _c}$ is the angular frequency of the carrier.

Considering the fluctuation of propagation time in the leading fiber, the two laser beams of the TDM1 interferometer in Fig. 1 could be expressed in Jones vector representation as

Likewise, for the TDM2 channel, the two interference laser beams could be expressed as

Here ${\textbf{B}_0}$, ${\textbf{B}_1}$, ${\textbf{B}_2}$ are the Jones matrixes of the leading fiber, the TDM1 sensing fiber, and the TDM2 sensing fiber, respectively. ${\textbf{B}_0}(t )$ means that the first interrogation pulse is injected into the leading fiber at time t, ${\textbf{B}_0}({t + {\tau_s}} )$ means that the second one is injected at $t + {\tau _s}$. ${\textbf{B}_1}$, ${\textbf{B}_2}$ are considered to be constants and independent of time. The right arrows over them represent the forward transmission from the laser source to the array and the left ones represent reverse transmission. ${\rho _{0r}}$, ${\rho _{1r}}$, ${\rho _{2r}}$ are the reflection Jones matrixes of FBG0, FBG1, and FBG2. And ${\rho _{0t}}$, ${\rho _{1t}}$ are the transmission Jones matrixes of FBG0, FBG1. With the coordinate rotation, the forward and reverse transmission of a section fiber has the following relationship in the Jones matrix formalism

Here $k$ is the polarization-independent loss coefficient, ${\ast} $ denotes the conjugate, $a \cdot {a^\ast } + b \cdot {b^\ast } = 1$, and a, b is complex elements that depend on the birefringence properties [17].

According to the two laser beams of the interferometer in Eq. (8), the interference intensity of the TDM1 channel is given by

Here $\dag $ is the conjugate transpose operation. Similarly, for the TDM2 channel, the interference intensity is

To simplify Eq. (11) and Eq. (12), we make the following assumptions. Firstly, the FBGs employed in the array are far shorter than the sensing fiber, as in a typical long-baseline Fabry-Pérot cavity structure. So, polarization independence for both transmission and reflection can stand. Secondly, we assume that all the FBGs are identical and there is no energy loss. The reflection and transmission Jones matrixes can be written as ${\rho _{0r}} = {\rho _{1r}} = {\rho _{2r}} = r\textbf{I}$ and ${\rho _{0t}} = {\rho _{1t}} = t\textbf{I}$[13,14]. Here $\textbf{I}$ is the identity matrix, and the scalar $r$ and $t$ are the reflectivity and transmissivity of the FBG, respectively. Then, Eq. (11) and Eq. (12) can be simplified as

We assume that when the first laser pulse passes through the leading fiber at moment t, the Jones matrix of the leading fiber could be expressed as

Here ${\varphi _x}$ and ${\varphi _y}$ are the phase shifts of the two polarization eigenstates in the leading fiber, $\alpha $ is the rotatory angle of an optical rotator, and $\gamma $ is the angle between the phase plate and the eigenaxis of birefringence. Generally, a common phase offset ${e^{ - j({\varphi _x} + {\varphi _y})/2}}$ could be extracted, and only the differential phase shift term will be retained in the matrix.

Here $\delta (t )= ({{\varphi_y} - {\varphi_x}})/2$, $\overrightarrow {{\textbf{U}_0}(t )} $ is a complex matrix that represents the birefringence state of the leading fiber at time t, and ${\varphi _0} = ({\varphi _x} + {\varphi _y})/2$ is the intrinsic phase delay of leading fiber. Interestingly, $\textbf{U}$ was proved to be an arbitrary unitary matrix by Gregory et al. in [18]. Secondly, when the second laser pulse passes through the leading fiber at moment $t + {\tau _s}$, the Jones matrix of the leading fiber $\overrightarrow {{\textbf{B}_0}({t + {\tau_s}} )} $ could be expressed as

Here ${\varphi _{Dx}}$ and ${\varphi _{Dy}}$ are the Doppler phase shifts of the two polarization eigenstates caused by the longitudinal fiber strain. Similarly, a common phase offset ${e^{ - j({{\varphi }_x} + {{\varphi }_y})/2}} \cdot {e^{ - j({{\varphi }_{Dx}} + {{\varphi }_{Dy}})/2}}$ could be extracted as follows.

Here $\delta ({t + {\tau_s}} )= ({{\varphi_y} + {\varphi_{Dy}} - {\varphi_x} - {\varphi_{Dx}}})/2$, and ${e^{ - j({{\varphi }_{Dx}} + {{\varphi }_{Dy}})/2}}$ is the Doppler noise term ${e^{j[{ - {\phi_D}(t )} ]}}$. Substituting the Jones matrix of each fiber section into Eq. (13). Thus, the interference intensity in the two TDM sensor channels could be rewritten as

Here ${k_0},{k_1},{k_2}$ are the polarization-independent loss coefficients of the leading fiber and the two TDM channels’ fiber coils, and ${\textbf{U}_0}$, ${\textbf{U}_1}$, ${\textbf{U}_2}$ are the unitary matrixes which represent the birefringence characteristics of corresponding fibers. ${\phi _1}$ and ${\phi _2}$ are the intrinsic phase delays of the two TDM channels’ sensing fibers that will undergo random phase fading due to the external environment. Although ${e^{j[{ - {\phi_D}(t )} ]}}$ is a scalar phase factor, it should be noted that since the matrix ${\textbf{U}_0}$ also changes due to the time difference in the two interrogation pulses, $\overrightarrow {{\textbf{U}_0}(t )} $ to $\overrightarrow {{\textbf{U}_0}({t + {\tau_s}} )} $, the Doppler noise coupled with the fiber birefringence will appear in the demodulated signal.

2.2 Numerical simulation

A numerical simulation was implemented to further investigate the Doppler noise. In the simulation, the phase shift induced by leading fiber stretching was applied to ${\varphi _{Dx}}$ and ${\varphi _{Dy}}$ in Eq. (16) whereas the other fiber parameters, including phase shifts and rotatory angles of the sensing fibers, were set to be constants with a small white noise disturbance. The simulation parameters of the SM fibers in the array are given in Table 1.

Table 1. Parameters of the SM fibers in the array

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We provide the simulation results in four cases, where the Jones vectors of interrogation pulses, ${\textbf{E}_{in1}}$ and ${\textbf{E}_{in2}}$, were respectively set as shown in Table 2. Here ${e^{j{\phi _c}}}$ was applied to the 1^st pulse to introduce the PGC demodulation. The four cases were defined as $xx,xy,yy,yx$ polarization channels, consistent with the PS and PGC hybrid processing method in [14,15]. A 0.5% intensity noise was added to the interrogation pulses. The PGC modulation frequency was 10kHz and the modulation amplitude was 2.37. The interrogating frequency was 80kHz, and the pulse-interval of the interrogating pulse pair was fixed at 390ns.

Table 2. Jones vectors of the four interrogation pulse pairs

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In the beginning, a sinusoidal fiber stretching excitation was applied to the leading fiber with a frequency of 200Hz and an amplitude of 0.2$\mu \textrm{m}$. Time-domain demodulation signals and phase spectral densities (PSDs) of the two TDM sensors in the four polarization channels are shown in Fig. 2.

 

Fig. 2. The demodulation results of sinusoidal leading fiber stretching excitation at 200 Hz (a) time domain and (b) frequency domain

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Although the noise signal at 200Hz could be observed in all polarization channels, for a given polarization channel, nevertheless, there were significant amplitude and phase deviations in the two TDM channels. Besides, for a given TDM channel, the obvious amplitude differences also appeared in the four polarization channels, which was related to the rotatory angle setting of the sensing fibers. In Fig. 2(a), there is a 180 degrees out of phase scenario in different polarization channels, such as the $xy,yy$ channels of the TDM1. It depends on the parameter settings of SOP and which eigen-axis has more influence on a certain polarization channel.

Next, a multitone fiber stretching excitation which contained four frequencies of 200Hz, 400Hz, 600Hz, and 800Hz was applied to the leading fiber. The stretching amplitude at each frequency was constant at 0.2$\mu \textrm{m}$. The time-domain and PSD results are shown in Fig. 3.

 

Fig. 3. The demodulation results of multitone leading fiber stretching excitation (a) time domain and (b) frequency domain

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Two phenomena could be noticed from the results in Fig. 3. First, the noise was distributed in all polarization and TDM channels, which was consistent with the sinusoidal simulation. Second, despite the equal amplitude stretching at each frequency, the results showed that the noise amplitude increased with the stretching frequency. The higher the excitation frequency, the greater the noise amplitude. Substituting the sinusoidal excitation $n(t )= \textrm{sin}({{\omega_D}t} )$ into Eq. (7), the Doppler phase-shift could be expressed as

Here ${\omega _D}$ is the angular frequency of fiber stretching. It could be found that the noise amplitude is indeed proportional to the excitation frequency. Figure 4 shows the linear relationship between the Doppler noise amplitude and the stretching frequency in the $xx$ polarization channel, where the fiber stretching frequency was adjusted from 100Hz to 1400Hz in the simulation.

 

Fig. 4. The relationship between the Doppler noise amplitude and the stretching frequency

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Finally, a uniform distributed fiber stretching excitation with a bandwidth of 200Hz to 1000Hz was applied to the leading fiber, and the root mean square (RMS) of the stretching amplitude was approximately 0.1$\mu \textrm{m}$. The time-domain and PSD demodulation results are shown in Fig. 5. If the leading fiber was stretched randomly, the wideband Doppler noise was generated in all TDM and polarization channels. Moreover, the noise amplitude was also positively related to the frequency.

 

Fig. 5. The demodulation results of uniform distributed leading fiber stretching excitation (a) time domain and (b) frequency domain

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In summary, no matter what form of disturbance was applied to the leading fiber, two rules about the Doppler noise hold true. Firstly, the Doppler noise always appears in all polarization and TDM channels. Secondly, for the same polarization channel, the Doppler noises of all TDM channels are incomparable for a sharp distinction in amplitude and phase. The birefringence parameter of the two TDM channels is the major reason for this phenomenon. In other words, the Doppler noise is always coupled with the birefringence states of the fibers. Therefore, the polarization induced signal fading should be solved before suppressing the Doppler noise.

3. Doppler noise suppression method

3.1 Principle

Although the time variation of the matrix ${\textbf{U}_0}$ will cause the coupling of Doppler noise and birefringence, the $\overrightarrow {{\textbf{U}_0}({t + {\tau_s}} )} $ matrix, fortunately, is still a unitary matrix. Therefore, we adopt the PS and PGC hybrid processing method to eliminate the coupled Doppler noise and obtain the scalar phase ${\phi _D}(t )$. We define ${{\Re }_1}$, ${{\Re }_2}$ as the system response matrixes of the two TDM sensor channels [13], which could be expressed as

According to the properties of the unitary matrix, the product of multiple unitary matrices is still a unitary matrix, and the determinant of an arbitrary unitary matrix is 1 [19]. Performing the following operations on both sides of Eq. (20) obtains

Here $\textrm{det}({\cdot} )$ is the determinant of a matrix.

If the TDM1 is a reference sensor non-sensitive to external physical quantities, and the phase fading could be eliminated by PGC demodulation, i.e., ${\phi _1} = 0$, then the sensing phase of the TDM2 can be obtained by subtracting the two equations in Eq. (21) as

Here $\angle $ is the angle operation of a complex scalar. The demodulation results are no longer affected by the leading-fiber-induced Doppler noise and independent of the polarization state. This means that the two kinds of leading fiber noises, caused by birefringence or disturbance, could both be effectively suppressed.

Without loss of generality, the system response matrix of the TDM1 sensor which is a complex matrix can be written as

Here ${{\Re }_{1mn}}$($mn$:$xx$, $xy$, $yy$, $yx$) are the four elements of ${{\Re }_1}$. Comparing Eq. (20), ${{\Re }_{1mn}}$ could be expressed as

Here $k_0^4k_1^2{r^2}{t^2}{e^{j[{{\phi_1} - {\phi_D}(t )} ]}}$ which contains the phase information ${\phi _1} - {\phi _D}(t )$ is the polarization-independent factor. ${k_{1mn}}$($0 \le {k_{mn}} \le 1$) and ${e^{j{\phi _{1mn}}}}$ separately represent the attenuation factors and the phase shift factors of the four polarization channels, respectively. Obviously, ${{\Re }_1}$ contains too many unknown variables to estimate in single interrogation, so the basic idea of the PS method is to estimate the elements of ${{\Re }_1}$ one by one by using four interrogation pulse pairs with different SOP combinations. As with the simulation experiment in Section 2.2, the Jones vectors of the four interrogation pulse pairs are written separately as

Bring each equation in Eq. (25) into Eq. (20) in turn, the single polarization channels’ interference intensity is expressed as

Here ${I_{DC}}$ is the DC component of the interference intensity. Next, combining Eq. (24), the interference intensity of a single polarization channel can be rewritten as

Now, let's apply PGC demodulation on Eq. (27). The Bessel expansion of Eq. (27) is multiplied by the carrier signals of $\textrm{cos}({{\omega_c}t} )$ and $\textrm{cos}({2{\omega_c}t} )$ respectively. By filtering out the frequencies above ${\omega _c}$, the two low-frequency components ${M_{1mn}}$ and ${N_{1mn}}$ can be obtained as

Here ${J_1}({\cdot} )$ and ${J_2}({\cdot} )$ are the first and second-order Bessel functions, and C is the phase modulation depth. Applying Euler's formula with Eq. (28), it is obvious that ${{\Re }_{1mn}}$ can be constructed from ${M_{1mn}}$ and ${N_{1mn}}$ as follows

Equation (29) shows that each element in ${{\Re }_1}$ can be estimated by interrogating four times. Similarly, ${{\Re }_2}$ also can be obtained simultaneously by the second TDM interference pulse.

Finally, the sensing phase of the TDM2 that is not affected by Doppler noise can be obtained as follows

The processing flow of the Doppler noise suppression method based on the PS and PGC hybrid processing is shown in Fig. 6. The four orthogonal polarization intensities of the two TDM channels are respectively mixed, lowpass filtered, and complex construction, and then demodulated to obtain a phase signal independent of the fiber birefringence, which eliminates the polarization-induced signal fading. Then, the polarization-independent phase signal is subtracted to obtain the sensing phase ${\phi _2}$, which also does not contain the Doppler phase noise. The key to this algorithm is that since the Doppler noise and birefringence are coupled with each other, it is necessary to first implement polarization synthesis and then perform Doppler noise suppression, otherwise, the suppression algorithm will be completely invalid.

 

Fig. 6. The processing flow of the Doppler noise suppression method based on the PS and PGC hybrid processing

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3.2 Numerical simulation

The polarization synthesis processing was performed on the simulation results of Section 2.2 to verify the feasibility of the suppression method. Since the interference light fields of the four polarization channels have been independently generated, the polarization synthesized signal of each TDM channel could be obtained by using the PS and PGC hybrid processing methods. Figures 7–9 show the synthesized results of the two TDM channels, colored respectively by red and blue, which correspond to the sinusoidal, multitone, and wideband white noise just as in the previous simulation.

 

Fig. 7. The suppression effect of sinusoidal leading fiber stretching excitation at (a) time domain and (b) frequency domain

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Fig. 8. The suppression effect of multitone leading fiber stretching excitation (a) time domain and (b) frequency domain

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Fig. 9. The suppression effect of uniform distributed leading fiber stretching excitation (a) time domain and (b) frequency domain

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The results reveal that the Doppler noises of TDM1 and TDM2 were nearly the same after polarization synthesis processing. Then, the synthesized result of TDM1 was subtracted from that of TDM2 to obtain the suppressed signals, as shown by the black curves in Figs. 7–9. It can be seen that the Doppler noise was effectively eliminated by the subtraction after polarization synthesis, which matched with the above analysis.

4. Experimental results and discussion

4.1 System setup

The schematic diagram of the two-TDM interferometric FBG sensor system is shown in Fig. 10.

 

Fig. 10. Schematic diagram of path-matching interferometric FBG-FP hydrophone system

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A narrow-linewidth low-noise fiber laser with a wavelength of 1539.8nm was used as the light source. The relative intensity noise of the fiber laser was below $- 120\textrm{dB}/\sqrt {\textrm{Hz}} $ above 1kHz. The continuous laser source was then pulsed by an acousto-optic modulator (AOM) with a modulation frequency of 320kHz, and the pulse width was 380ns. Whereafter, with a compensating interferometer (CIF) with imbalance arms, the AOM output pulse was converted to two cloned interrogation pulses. The arm difference (79.5m) of the CIF determined the pulse interval ${\tau _s}$, which was equal to twice the interval between adjacent FBGs in the array to achieve path matching. Meanwhile, the short arm of the CIF was wounded on a PZT to introduce the modulation signal for PGC demodulation [15,19]. The PGC modulation amplitude and frequency were set to be 2.37 and 10kHz, respectively. The demodulation software was executed based on a data block, which contained 16384 sample points for each iteration. To implement the PS method, a lithium niobate polarization modulator (LN-PM) from Photline was installed after the CIF. The SOP for each interrogation pulse could be altered orthogonally by the LN-PM. The modulation timing diagram is shown in Fig. 11, which has been demonstrated in detail in [15,20].

 

Fig. 11. Timing of the modulation and demodulation signals in the experimental system

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Three FBGs were imprinted in an SM fiber. The Bragg wavelength was all 1539.8 ± 0.01nm with a 3dB bandwidth of approximately 0.36nm, and the reflectivity of them was 0.6%. The spacing between any two adjacent FBGs was 39.75m. The left TDM channel named as the reference sensor was non-sensitive to the external environment, and the right active sensor could pick up the acoustic pressure by sensitivity enhancement. The length of the leading fiber was approximately 60m. Besides, we designed an optical fiber stretching device (see below for details) to apply longitudinal strain on the leading fiber, thereby introducing the Doppler noise.

A customized fiber stretching device (FSD) was used to introduce the leading fiber disturbances, as shown in Fig. 12. A vibration resonance speaker (VRS) was employed as a vibration source. Part of the leading fiber was bent four times and bundled into a multi-strand fiber with a length of approximately 200mm. The ends of the multi-strand fiber using Kevlar ropes were respectively fixed on the top border of the device and the vibration surface of the VRS. During the experiment, a signal generator from RIGOL was used to drive the VRS to introduce a specific disturbance signal.

 

Fig. 12. Schematic diagram of fiber stretching induced Doppler phase noise experiment

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4.2 Experiment

A sinusoidal excitation at 112Hz was imposed on the leading fiber by the FSD. The PSD in a single iteration of the demodulation results is shown in Fig. 13. The PSD curves of the four polarization channels are colored in red, blue, yellow, and green, respectively. And the synthesized results from the PS and PGC hybrid processing method are colored in black. It can be found that the Doppler noise with a frequency of 112Hz appeared in all TDM and polarization channels. The amplitudes of the four polarization channels in the reference channel were −43.32dB,−41.54dB,−44.55dB, and −40.92dB, respectively; while the ones of the active sensor were −43.05dB,−43.86dB,−45.36dB, and −45.92dB, respectively. The synthesized amplitudes of the two sensors were −43.15dB and −43.17dB. Although up to 5dB amplitude error appeared in the $yx$ polarization channel, the partial enlarged drawing in Fig. 13 indicates that the synthesized noise signal was nearly the same with an amplitude error of 0.02dB, which was consistent with the theoretical analysis.

 

Fig. 13. Phase spectral density of the demodulation results for (a) the reference sensor and (b) the active sensor

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Figure 14 demonstrates the synthesized signals of the reference and active sensors, respectively colored in red and blue. And the black curve represents the signal after subtraction suppression processing. It can be seen that the processed signal no longer contained the Doppler noise at 112Hz, which matched well with the preceding analysis.

 

Fig. 14. Time-domain signals of the polarization synthesized results for the reference and active sensor

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To evaluate the suppression of the reference sensor scheme on the Doppler noise, 30 consecutive data blocks (about 6.1s) were acquired to perform a spectral averaging operation. The average phase spectral densities (APSDs) of the polarization synthesized results are shown in Fig. 15. The red and blue curves represent the results of the reference and active sensor, respectively. The black curve represents the result by using the reference sensor subtraction method. The average amplitudes of TDM1/2 were the same at −49.3dB. After subtraction, the noise amplitude at 112Hz was reduced to −79.8dB. The reference sensor subtraction delivered 30.5dB suppression on the Doppler noise.

 

Fig. 15. Evaluation of the Doppler noise suppression based on the reference sensor subtraction

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Next, a wideband white noise excitation with a bandwidth of DC to 2kHz was transmitted to the FSD, and the RMS voltage was 4${\textrm{V}_{rms}}$. The synthesized signals of the two sensors are shown in Fig. 16.

 

Fig. 16. Time-domain signals of the polarization synthesized results for the wideband Doppler noise

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The signal after subtraction suppression processing is shown by the black curve. The APSD was used to evaluate the suppression for the wideband Doppler noise. 30 data blocks of about 6.1s were used to perform the phase spectrum averaging. The results are shown in Fig. 17.

 

Fig. 17. Evaluation of the wideband Doppler noise suppression

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Although the wideband white noise was introduced to the FSD, the Doppler noise presented a nonlinear relationship with frequency. The frequency response of the VRS was the major cause of the nonlinearity. Firstly, two extreme amplitudes of the noise appeared at the two fundamental resonance frequencies of the VRS, around 330Hz and 522Hz, which were −62.3dB and −72.3dB, respectively. Secondly, two inhibitory frequency response areas around DC to 200Hz and 1.8kHz were noticed, as indicated by the gray shadings in Fig. 17, where the Doppler noise was relatively weak.

The background noise level (BNL) was upraised by the wideband noise from −88.2dB to −68.3dB, which significantly weakened the detection performance of the sensor system. Fortunately, similar to the sinusoidal excitation experiment, by using the reference sensor subtraction method, the wideband Doppler noise could be eliminated and the BNL was recovered to −95.2dB, as shown by the black curve. The reference sensor subtraction delivered a maximum of 29.8dB suppression on the wideband Doppler noise.

It is worth noting that multiple line spectrum noises are present in Fig. 16. After analyzing, we found that the multiple line noises corresponded to multiple operating frequencies in the acquisition circuit, such as the switching power supply devices, the crystal oscillators, and the test terminal signals. Besides, when there was no leading fiber or when no leading fiber tension was applied, these line spectrum noises still existed, which also indicated that it was inherent noise of the sensing system. The leading disturbance imposed in the experiment has been clearly distinguished from these line spectra, so it does not affect the analysis of Doppler effects suppression that the research focuses on.

4.3 Discussion

The above experimental results revealed that the Doppler noise could indeed be induced by leading fiber stretching in the interferometric FBG array, which was coupled with the fiber birefringence. While up to 5dB Doppler noise difference was observed in the corresponding polarization channel, the synthesized noise signal from the PS and PGC method was nearly the same with an amplitude error of 0.02dB. Maximum suppression of about 30dB was achieved when the non-sensitive reference sensor subtraction was used to eliminate the sinusoidal or wideband Doppler noise. As shown by Fig. 17, the suppression was limited by the Doppler noise amplitude imposed in the experiment. The background noise performance of the sensing system was almost restored by the suppression processing. However, the increase in Doppler noise amplitude with frequency was not observed in the wideband experiment. The nonlinear transfer function of the excitation generator, including the VRS, the folded multi-strand fiber, and the tensional Kevlar ropes, is considered as the primary cause of it. The longitudinal stretching, in addition, causes not only the scalar Doppler effect but the coupling with the birefringence, which makes it difficult to analyze the Doppler effect separately. To induce a pure Doppler effect, a more refined stretching device is required, that is, the fiber is completely uniform and isotropic along the longitudinal axis. However, since the SM fiber is anisotropic, such a perfect stretching device cannot be realized in the experiment.

5. Conclusions

This paper focused on the leading-fiber-induced Doppler phase noise, in which a two-stage noise suppression method was demonstrated in detail. Since the random disturbance in the leading fiber length could produce the Doppler noise in the interferometric FBG array, the suppression method must be implemented to eliminate it. Otherwise, the detection performance of the sensing system could be deteriorated due to environmental disturbances. A model of the Doppler noise was presented, in which the principle of the Doppler noise generation by leading fiber stretching was demonstrated. And the simulation provided a full demonstration of the Doppler noise in all polarization and TDM channels. Both the theoretical derivation and simulation results reveal that there is a mutual coupling between the Doppler noise and the fiber birefringence state. Thus, a two-stage Doppler noise suppression method was naturally proposed, where the PS and PGC hybrid processing method and a non-sensitive sensor were combined. Experimental results showed that the synthesized Doppler noise from the PS and PGC hybrid method was nearly the same in each TDM channel. Also, the non-sensitive reference sensor subtraction could suppress both the sinusoidal and the wideband Doppler noise, where a maximum suppression of about 30 dB was achieved, which is the highest suppression for the Doppler noise of our known. Furthermore, after the suppression processing, the background noise performance of the sensing system was effectively restored. This research on the Doppler noise can not only promote the adaptability of interferometric FBG arrays in harsh environments, but also be suitable for other fiber-optic sensor arrays based on the path-matching structure [6–9,20].