1. Introduction

Wavelength encoded optical fiber sensors, such as fiber Fabry-Perot interferometer (FPI) sensors [1] and fiber Bragg grating (FBG) sensors [2], have been widely used for strain measurements in a variety of applications due to their high resolution, compact size, and electromagnetic interferences immunity. These advantages make the wavelength encoded optical fiber sensors very attractive to geophysical related applications, for example, the strain measurement in crustal deformation observation [3–5]. Generally, such applications require sensors with nano-strain level resolution to deal with the weak strain signals. Besides, response bandwidth covering the quasi-static frequency domain and long-term measurement range over milli-strain of the sensors are also necessary to obtain sufficient data during the measurements [6], which presents a challenge in the sensor system design.

In order to achieve the nano-strain level resolution, fiber Fabry-Perot interferometers (FFPI) with high finesse [7] or π-phase-shifted FBGs (π-PSFBG) [8] are usually used as the sensing elements, which have one or several narrow resonant notches in their reflection spectra. In addition, since the optical spectrum analyzers can not provide sufficient resolution when directly measure the spectra of the sensing elements with broadband probing laser, specified demodulation system is also required. In [9], a high resolution demodulation scheme has been proposed in which a narrow line-width single frequency laser is aligned to the slope of the resonant peak of a π-PSFBG, and the strain variation is directly calculated from the reflected light intensity. Such a method is a classical demodulation scheme for FBG strain sensors [10], however, its measurement range is quite small when interrogating the π-PSFBG since the slope of the resonance peak is sharp but narrow. Another solution of the high resolution demodulation is based on frequency sweeping laser [1,11]. The reflected light intensity curve in time-domain during each sweeping process is recorded and mapped to frequency domain, and then the resonance center wavelength (corresponding to the strain signal) is obtained by nonlinear fitting or peak search algorithm. These sensing schemes suffer a resolution degradation from the intensity noise in the time-domain curve and the instability of the frequency sweeping process [12]. Besides, their sampling rate is limited by the sweeping repetition rate of the narrow line-width laser, which can only reach a level of several times per second or even lower. Although the instability in laser frequency sweeping can be compensated by introducing external modulator as presented in [13], it still faces issues of low sampling rate and high cost radio frequency equipments.

The Pound-Drever-Hall (PDH) technique [14] based closed-loop interrogation method provides an alternative choice for the high resolution demodulation of FFPI or π-PSFBG [6, 15–17]. An “error signal” [18] obtained with the PDH technique can be used to indicate the frequency mismatch between the interrogating laser and the resonance center of the FFPI or π-PSFBG, and then a feedback control can be performed to lock the laser in the resonance center. For the PDH technique based quasi-static strain sensors, resolution better than 1 nε/Hz^1/2 in 0.01 to 100 Hz band has been realized [6]. However, in these works the issue of multiplexing were not mentioned because the employment of complex demodulation system leads to great difficulty for multiplexing.

For the applications in geophysical researches, there are strong demands for multi-channel high resolution sensors to acquire sufficient data, such as in 2-D or 3-D strain tensor measurements or strain field distribution observations. Considering that the cost is also one of the major concerns in practical application, it is necessary to adopt multiplexed sensor array in which the demodulation system is shared in order to reduce the total cost. In [19], a wavelength-division multiplexing scheme for the PDH technique based sensors has been proposed, in which each sensing channel occupies an independent closed-loop interrogation system. So the system configuration is still complex and the total cost can not be efficiently reduced. Recently, our group proposed a time-domain multiplexing scheme for the closed-loop high performance sensors [20]. A cyclic interrogation scheme together with sideband interrogation method is introduced, so all the sensing channels share the same demodulation system. While, to achieve large measurement range and high sampling rate, costly RF equipments and high bandwidth electro-optic modulators are still used in the system, which restricts the reduction of the device cost as well as the frequency response of the sensors.

In this paper, we propose a high-resolution multiplexed strain sensor based on the temporal responses of FFPIs. A pair of rectangular laser pulses with different frequency shifts is used to interrogate the FFPIs, and their temporal responses are employed to discriminate the frequency mismatch between the laser and the resonance of FFPIs. The FFPIs are connected in half-ladder structure by cascaded optical fiber couplers and delay fiber rolls for the time-domain multiplexing. A compact optoelectronic system, comprising a narrow line-width laser and an acousto-optic modulator (AOM), performs the closed-loop cyclic interrogation process to realize the demodulation of the FFPI sensors. In the experiment, we achieve a 3-channel strain measurement with resolution better than 0.1 nε and frequency response higher than 100 Hz.

2. Temporal behavior of fiber Fabry-Perot interferometers

It is known that a fiber Fabry-Perot interferometer has transmission peaks (and reflection notches) in its spectrum due to the constructive interference in the cavity [21]. The center frequencies of the peaks (or the notches) are sensitive to strain or temperature or other factors which affect the effective length of the fiber cavity, and hence the FFPIs can be employed for sensing purposes. For a unit amplitude incident wave and infinite build-up time, the reflection intensity of an ideal FFPI is [21]:

When a square light pulse with finite-length is incident to a high finesse FFPI for the purpose of time-domain multiplexing, the temporal response of the FFPI should be taken into consideration [21, 22] because the limited time for the interfering beams to build up. And after the incident pulse has elapsed, there is still lightwave remaining in the cavity to be released [22]. Accordingly, the temporal behavior of FFPI in reflection intensity is given by:

Further, we define E_r (ϕ) as the energy ratio of the reflected pulse and the incident pulse during 0 < t < T₀ by:

 

Fig. 1 The temporal responses of a FFPI with R = 0.99 and N_R = 313. (a) Theoretical temporal response I_r (ϕ, t) in time-domain by Eq. 2 under M₀ = 50 (solid lines) or M₀ = ∞ (dash lines) and phase mismatches ϕ = 0, ±0.005π, ±0.01π, ±0.015π, ±0.02π. (b) The pulse energy ratio E_r (ϕ) by Eq. 3 and the steady state response Ir (ϕ) by Eq. 1 near a resonance point. Curves of E_r (ϕ) with M₀ = 10, 50, 250 are separately given.

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3. Interrogation scheme

3.1. The frequency mismatch discrimination method

With the E_r (ϕ) given above and appropriate T₀, we can found the resonant frequencies of an FFPI by performing a frequency scan of the interrogating laser. However, for high finess FFPIs, a frequency scanning usually means redundant data being acquired and limited repetition rate. So, we propose a dual pulse interrogation scheme to obtain the frequency mismatch between the laser and the resonance center in a relatively short time. For convenience, we assume that the two incident pulses have the same intensity and hence their powers are equal. Figure 2 gives a diagrammatic illustration of the dual pulse interrogation method. Two short incident laser pulses with frequencies of ν′ = ν − ν_d and ν″ = ν + ν_d sequentially interrogate a resonance of FFPI, then the two reflected pulses with temporal behaviors are acquired. When the average frequency of the laser pulses ν is lower than the resonance frequency ν₀, the ν″ is closer to the resonance center than the ν′. Therefore, the power of the first reflective pulse is larger than the second one. On the contrary, if ν is higher than ν₀, the power of the first reflective pulse will be smaller than the second. Particularly, under the condition that ν is exactly aligned to ν₀, the two reflected pulses are identical, so the difference of powers of the reflected pulses is zero.

 

Fig. 2 Illustration of the dual pulse interrogation method. The left part of the figure shows the frequencies of the two laser pulses and a resonance of FFPI in frequency domain; the right part gives the diagrammatic illustration of the reflected pulse pairs under different ν and the curve of frequency deviation signal.

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The description above indicates a frequency mismatch discrimination method based on the power difference of the reflected pulses. Correspondingly, a frequency deviation signal is given by:

So, the frequency deviation δν between the interrogation laser and the resonance can be obtained by δν = K⁻¹D(ν) when the laser frequency is close to the resonance frequency. In addition, if the laser frequency ν is controllable, a closed-loop control system can be used to stabilize the D(ν) at zero, i.e., to lock the laser frequency in the resonance center. Once locked, the ν can exactly indicate the resonant frequency of FFPI and follow the variation of the resonant frequency. For FFPIs as strain sensors, the resonant frequency is sensitive to the applied axial strain, so we can get the strain signal by

3.2. The closed-loop cyclic interrogation scheme

For a closed-loop interrogation system based on a single frequency laser, it is impossible to track all the resonances of multiple FFPIs at the same time. However, if the frequency of the laser can be rapidly and arbitrarily controlled, an alternative “closed-loop cyclic interrogation” scheme can be employed to track more than one resonances by turns. For ease of description, some definitions are given below: N is the number of FFPIs; $D_i^{(k)}(v)$ represents the frequency deviation signal of the i-th FFPI in the k-th interrogation cycle with i = 1, 2, ⋯, N and k = 1, 2, ⋯; and K₁, K₂, ⋯, K_N is the slope at zero-crossing point of the corresponding D_i (ν) curve.

At the beginning, we assume that the initial resonant frequencies of each FFPI are approximately known as $v_1^0$, $v_2^0$, ⋯, $v_N^0$, which can be obtained by performing a frequency scan. In the first cycle, N interrogations are sequentially performed, getting the frequency deviation signals of all the FFPIs which are

Once the first interrogation cycle is complete, the next cycle begins immediately. This moment, the initial resonant frequencies $v_i^0$ will be replaced by the new resonant frequencies $v_i^1$ to perform the N interrogations in this round. Similarly, in the following interrogation cycles, the latest resonant frequencies $v_i^k$ are always obtained based on the resonant frequencies in the last cycle by

4. System configuration and experiment results

4.1. Experiment setup

The configuration of the sensor system is shown in Fig. 3. An 1550 nm narrow line-width laser (NKT, E15) is the light source of the system. Then, an optical circulator (CIR₁) is used to guide the laser into an acousto-optic modulator (AOM, G&H, T-M300), which generates the pulses and shifts the laser frequency at the same time. The operating frequency band of the AOM is 240–340 MHz. After the AOM, a reflection mirror (RM) is employed to make the laser beam propagate through the modulator again, which doubles the laser frequency shift by the AOM. The laser pulses returned from the AOM is then split into two beams by a fiber coupler (CP₀) with splitting ratio of 1:20. The beam with smaller intensity is directly received by a photo detector (PD₁) as local signal. At the same time, the other beam with larger intensity propagates through another optical circulator (CIR₂) and then into the sensor array.

 

Fig. 3 Schematic diagram of the experiment system. CIR: optical circulator; CP: optical coupler; AOM: acousto-optic modulator; RM: reflection mirror; PD: photo detector; A/D: analog-digital converter; SG: signal generator; RF-AMP: radio-frequency signal amplifier. Diagrammatic sketch of signals after the SG, PD₁, and PD₂ are also given below the diagram for ease of understanding.

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In the sensor array, 3 cascaded 1×2 fiber couplers (CP₁, CP₂, and CP₃) with 3 rolls of delay fiber are used to guide the probing laser pulse into 4 FFPIs. The splitting ratio of couplers are 3:1, 2:1, and 1:1 respectively, making a balanced distribution of probing laser intensity for all the channels. The directivities (reciprocal of the crosstalks) of the couplers are tested to be higher than 70 dB. The lengths of the delay fiber rolls are 100 meters, which lead to a round trip time delay t_d = 1 μs between two adjacent channels. In order to eliminate undesired reflections in the optical route which would cause crosstalk among channels, the ends of delay fibers and couplers are connected by fusion splices instead of using flexible connectors. The FFPIs, which are the sensing elements in our experiment, are composed of a pair of high reflection FBGs with spacing of L = 0.8 meter (single round-trip time t₀ = 8 ns). The reflectivities of the FBGs are R = 99%. Such a half-ladder configuration ensures that the reflection pulses from each FFPI will not overlap as long as the pulse duration is smaller than the time delay t_d caused by the delay fiber rolls. The reflected pulse trains from the sensor array propagate through the optical circulator (CIR₂) again, and then received by another photo detector (PD₂) as remote signal.

The local signal and the remote signal are converted into digital pattern by two analog-digital converters (A/D, ADI, AD9226) and then acquired by a real-time (RT) control module (Xilinx, XC6SLX16). The RT control module is programmed to process the acquired data without delay, store the results in memory, and control a signal generator (SG, ADI, AD9910) at the same time. The SG can generate pulsed RF signal with fixed duration and controllable frequency covering the operating band of the AOM. After a RF signal amplifier (RF-AMP), the pulsed signal are used to drive the AOM to generate laser pulses with arbitrary frequency shifts, as mentioned above.

With the help of the RT control module, the frequency mismatch discrimination and the closed-loop cyclic interrogation described in section 3 can be implemented. As indicated in Fig. 3, in a single detection, the SG generates two RF pulses with frequencies of ν′ = ν − ν_d and ν″ = ν + ν_d relatively. The ν_d is set to be 2 MHz in the experiment. Because the delay times in the system are known and fixed, we can take out the segments in the local signal and the remote signal which correspond to the two incident light pulses and the two reflected light pulses. Then, data in the segments are accumulated in digital domain as pulse energies $E_I^{\prime }$, $E_I^{″}$, $E_R^{\prime }$, and $E_R^{″}$ respectively, and the corresponding frequency deviation signal can be calculated as $D(v)=E_R^{\prime }/E_I^{\prime }-E_R^{″}/E_I^{″}$. The repetition rate of single detection f_s is limited by the maximum path delay L_max in the system by f_s < (2 × 2nL_max/c₀)⁻¹, which is set to be 2 × 10⁴ in the experiment. In each interrogation cycle, 4 detections are performed to interrogate all the channels. With the procedure given in section 3.2, the closed-loop cyclic interrogation can be realized. Accordingly, the repetition rate of interrogation cycle, i.e., the loop rate of the feedback system, is f_R = f_s/4 = 5 × 10³ times per second. In addition, besides of the closed-loop cyclic interrogation mode, the RT controller can also be programed in single detection mode or frequency scanning mode to acquire supportive intermediate results.

4.2. Temporal responses and frequency deviation signals of FFPI in experiment

The measured temporal responses of a FFPI used in the experiment are given in Fig. 4(a). It should be noted that there are parasitic pulses in the reflected signals after the incident pulse has elapsed. Although the parasitic pulses are not included in the calculation of the frequency deviation signal, they are possible to overlap with the next reflected pulse from other channel. In order to avoid the unexpected inter-channel crosstalk, it is necessary to retain a protective time interval between the reflected pulses. So, in the experiment, the incident pulse duration is set to be T₀ = 0.5 μs (M₀ = T₀/t₀ ≃ 63) while the time delay between two channels is t_d = 1 μs as previously given. Such a configuration can effectively prevent the crosstalk caused by overlapping of pulses.

 

Fig. 4 (a) The measured reflected pulses of FFPI 1. Incident laser pulse duration is set to 0.5 μs, and their frequency deviation to the resonance center are 0, 0.75 MHz, 1.50 MHz, 2.25 MHz, and 3.00 MHz (correspond to phase mismatches of 0, 0.006π, 0.012π, 0.018π, and 0.024π respectively). Parasitic pulses can be observed within t > 0.5μs. (b) The measured frequency deviation signal curves of FFPI 1 and FFPI 2 with laser frequency shifts from 480 MHz to 680 MHz (AOM operating frequencies from 240 MHz to 340 MHz), 3 zero-crossing points appear at 500 MHz, 580 MHz, and 630 MHz.

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With a frequency scan based on the system configuration given above, we can get the complete curves of frequency deviation signals D(ν) defined by Eq. (4). The acquired frequency deviation signal curves are shown in Fig. 4(b). The zero-crossing points in the curve correspond to the resonances of the FFPI. The sharp falling slopes at the zero-crossing points indicate a high frequency discrimination ability of the system. Due to the introduction of the reflection mirror, the frequency scanning range achieves 200 MHz which is twice of the operating range of the AOM. The FSR of the FFPIs used in the experiment is ν_FSR = c₀/2nL = 128 MHz, so it can be ensured that there is at least one zero-crossing point within the operating range, which brings great convenience to the strain measurement. In addition, if the zero-crossing point is moving out of the operating range during a long-time measurement, another zero-crossing point will certainly appear in the position with a frequency interval of ν_FSR. Under this condition, we can switch the probing laser to interrogate the adjacent resonance, and add (or subtract) the value of ν_FSR to the frequency readouts, implementing an uninterrupted, large-scale measurement.

4.3. Experiment results and discussion

When the sensor system is operating at the closed-loop cyclic interrogation mode, 4 frequency arrays $\left\{v_1^k\right\}$, $\left\{v_2^k\right\}$, $\left\{v_3^k\right\}$, and $\left\{v_4^k\right\}$ correspond to the resonant frequencies of the FFPIs can be acquired. In the experiment, FFPI 1–3 are taken as sensing channels which are stretched to apply axial strain. In order to compensate the wavelength drift of the light source and temperature fluctuation, the FFPI 4 are taken as a reference channel which is unstressed and is placed near the other FFPIs. Considering that the wavelength drift of the light source leads to the same variation in the measured frequency arrays, we use the frequency differences between the sensing channels and the reference channel to calculate the strain by ε_i = (ν_i − ν₄)/k, where i = 1, 2, 3, and k = 148.1 MHz/με (1.18 pm/με) is the tested strain sensitivity of the FFPIs. To further depress the measurement error, we take the average of every 10 frequency readouts to perform the strain calculation once. So, the sampling rate is f_R/10 = 500 samples per second.

The measured time-domain results are shown in Fig. 5. During the measurement, the FFPI 1 is stretched by a nano-positioning stage (PI, P-752.1CD) to apply controllable strain signal. Figure 5(a) gives the measured result when an 100 nε_p−p, 10 Hz strain signal are applied on FFPI 1, and a clear sinusoidal curve which is consistent well with the applied strain is observed. Figures 5(b)–5(d) show the results when constant strain are applied on the 3 FFPIs respectively. The standard deviations are calculated as the strain sensing resolution, which are δε₁ = 0.039 nε, δε₂ = 0.055 nε, and δε₃ = 0.060 nε. Because the insertion loss of the AOM at different operating frequencies is not the same, the resolutions of each sensing channel are slightly different. Generally, resolutions better than 0.1 nε for all the sensing channel is achieved in our scheme.

 

Fig. 5 Time-domain results in the experiment. (a) Measured strain on FFPI 1 when 100 nε_p−p, 10 Hz strain signal is applied. (b)–(d) Measured strain on FFPI 1, FFPI 2, and FFPI 3 respectively when constant strain are applied; the standard deviations of the results are also given.

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To verify the frequency response of the sensor, 100 nε_p−p (= 35.4 nε_rms) sinusoidal strain signals with different frequencies of 0.1 Hz, 1 Hz, 10 Hz and 100 Hz are applied on the FFPI 1. The power density spectra of the results are shown in Fig. 6(a). The sharp peaks at 0.1 Hz, 1 Hz, 10 Hz and 100 Hz are with the same amplitude, indicating that the response bandwidth of the sensor is higher than 100 Hz. Figure 6(b) gives the synchronous measurement results of all the 3 channels in order to test the crosstalk between channels. During the measurement, 1 Hz, 100 nε_p−p strain is applied on FFPI 1 while constant strain are applied on FFPI 2 and 3. In curve 1, there is a sharp peak with an amplitude 65 dB higher than the noise level at 1 Hz in channel 2 and 3. While in curve 2 and 3, there is no observable peak at 1 Hz point, which proves that the crosstalk between channels of the sensor system is better than −65 dB.

 

Fig. 6 Frequency-domain results in the experiment. (a) The power density spectra of measured strain on FFPI 1 when 0.1 Hz, 1 Hz, 10 Hz and 100 Hz strain signals are separately applied. (b) Power density spectra of simultaneous measured results from all the channels when 1 Hz strain signal is applied on FFPI 1 while constant strain on FFPI 2 and 3.

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In the experiment, we find that the resolution is mainly limited by the nonideality of the signal generator which drives the AOM. Specifically, it is because the sampling rate of signal generator is 1.0 GHz and approaches the Nyquist limit when generating signals with frequency larger than 250 MHz. So, the resolution can be further improved by employing signal generator with higher performance. In long-time measurement, the temperature fluctuation on the FFPIs will influence the sensing accuracy. So we insulate the FFPIs from the environment to ensure that the temperature on all the FFPIs are the same. Besides, the FFPIs have equal temperature sensitivity coefficients because they are written in the same type of fiber and have the same operating wavelength. Therefore, the temperature caused wavelength variation on the sensing FFPIs and the reference FFPI are equal, and hence, the temperature influence on strain measurement can be compensated. According to the results in our previous works [5, 11], nano-strain level accuracy can be achieved based on such compensation scheme. The channel number N can be further increased by appending fiber couplers, delay fiber rolls, and FFPIs. Considering that the half-ladder configuration of the sensor array leads to a power attenuation of 1/N², so light source with larger power and PD with higher gain could be used when necessary. Besides, since the sampling rate is inversely proportional to the channel number N in the cyclic interrogation scheme, the trade off between multiplexing ability and sampling rate of the sensor system should also be considered.

The proposed demodulation scheme can achieve high response bandwidth, large measurement range, and low device cost at the same time. The response frequency of the sensor is tested to be higher than 100 Hz. By comparison, the response of frequency sweeping laser based high resolution sensors [11, 13] can only reach several Hertz or even worse. The PDH demodulation scheme [6] can realize relatively high response frequency, however, its measurement range only reaches ∼2 με and it has difficulty in multiplexing. In our scheme, the high reflection band of the FBGs in the FFPI is larger than 0.25 nm. According to the discussion in section 4.2, the sensor system can work in the whole operating band of the FFPIs, corresponding to a measurement range larger than 200 με. For the issue of device cost, in one aspect all the channels share the same interrogation system, and in another aspect, the interrogation system uses only one light source, one modulator, a pair of PDs, and ordinary optical/electronic devices. So, the cost per channel is significantly lower than the single channel schemes [6, 13] or the wavelength multiplexing scheme [19], which improves the practicality of the sensor system. In addition, in the demonstration experiment we adopt an all polarization maintaining fiber (PMF) configuration to eliminate the influence of polarization change, and therefore to guarantee the long-time performance. For applications that the long-time performance is less critical, all optical devices can be replaced by ordinary single-mode fiber (SMF) devices to further reduce the system cost.

5. Conclusion

In this work, we proposed a time-division multiplexed high resolution strain sensor system based on the temporal response of FFPIs and the closed-loop cyclic interrogating scheme. In the demonstration experiment, 3-channel quasi-static strain measurement with resolution better than 0.1 nε and frequency response higher than 100 Hz is realized. Due to the characteristics of high performance and low cost, the proposed sensor system shows great potential in setting up strain sensing network for geophysical researches.