Letter Vol. 48, No.

9 / 1 May 2023 / Optics Letters 2369

Remote vectorial vibration sensing based on locally

stabilized Mach–Zehnder interferometers using
multi-core fiber and optical phase-locking
Yuanshuo Bai,1 Weilin Xie,2,3,∗ Songhan Liu,2 Yinxia Meng,2 Ling Zhang,1 Wei
Wei,2,3 AND Yi Dong2,3
1
State Key Laboratory of Advanced Optical Communication Systems and Networks, Shanghai Jiao Tong University, No. 800 Dongchuan Road,
Minhang District, Shanghai 200240, China
2
Key Laboratory of Photonics Information Technology, Ministry of Industry and Information Technology, School of Optics and Photonics, Beijing
Institute of Technology, No. 5, South Street, Zhongguancun, Haidian District, Beijing 100081, China
3
Yangtze Delta Region Academy of Beijing Institute of Technology, Jiaxing 314011, China
*wlxie@bit.edu.cn

Received 7 March 2023; accepted 30 March 2023; posted 3 April 2023; published 25 April 2023

We report on remote sensing of vectorial vibration based In general, the MZI structure, including the associated cou-
on locally stabilized Mach–Zehnder interferometers (MZIs) plers, must be located at the remote end where vibration occurs
using commercial multi-core fiber (MCF). Hexa-MZIs with to prevent any potential performance deterioration due to dis-
a shared common reference arm are constructed by a 7- turbances from the remote surroundings and transmission link.
core MCF to acquire remotely vectorial vibration. A set of Despite this, to maximize sensitivity, stabilization is desirable
corresponding local receivers consisting of optical phase- to keep the operation at the quadratic point [8]. These demands,
locked loops (OPLLs) for not only eliminating the impact of however, inevitably lead to complex requirements of the remote
environmental perturbations but also maintaining the sta- sensor, giving rise to practical difficulties both in operation and
ble operation and relative stability among the MZIs, allows maintenance. To this end, in-fiber MZIs based on segmented
guaranteed stabilized remote sensing. It moreover ensures fusion of different kinds of micro-structure fibers have been
a linearized phase detection, and thus an improved sensing proposed [9]. Though they achieve a compact design by dispens-
sensitivity and dynamic range. This way, by exploiting the ing with the extra couplers, these designs could hardly address
symmetrically geometric distribution for the cores of 7-core the stabilization of the operation point or the desired vibration
MCF, the proposed all-fiber design can enable highly precise identification in a vectorial manner. Taking these into account,
remote extraction of vibration in a vectorial manner with additional designs, for instance extra interferometers with cer-
a simplified remote structure. We achieve vectorial remote tain mutual correlations, become quite popular. Nonetheless, the
sensing for vibrations with ∼0.1076° and ∼0.3603 µm pre- incorporation of these additions, not to mention the servo con-
cision for the angle and displacement, respectively, over 10 trols needed for the optimized operation point, would further
km. © 2023 Optica Publishing Group complicate the deployment, thus making the remote end more
vulnerable.
https://doi.org/10.1364/OL.489376
In this Letter, we propose remote vectorial vibration sensing
that relies on locally stabilized remote hexa-MZIs using com-
The detection and monitoring of vibration has long been a criti- mercial 7-core multi-core fiber (MCF). The shared common
cal issue in scientific, industrial, and civil scenarios [1]. Recent reference of these MZIs and their respective signal arms are
advents in relevant communities, such as structure health mon- constructed by the center and outer cores, respectively, of a 7-
itoring, seismic analysis, and industrial machines maintenance core MCF. In addition to common-mode noise rejection brought
[2], has in particular led to an urgent demand for remote sens- by 7-core MCF, a set of localized optical phase-locked loops
ing of vibrations in a vectorial fashion [3]. Notably, not only (OPLLs) is adopted for the remote stabilization of the MZIs,
the amplitude and frequency, but also the direction of vibration eliminating the impact of environmental perturbations while
is accounted for. Benefiting from the highly sensitive phase- maintaining a relative stability among the MZIs. It moreover
detection nature [4], fiber-optic Mach–Zehnder interferometers allows linearized phase detection, leading to increased sensing
(MZIs) that permit acquiring vibration of interest by interrogat- dynamic range and sensitivity. Such a design not only signif-
ing the relative delay changes integrated along their two arms, icantly simplifies the remote structure, but also allows robust
has attracted great attention [5,6]. Thanks to the advantages and precise extraction of vibration in a vectorial form directly at
inherited from optical fibers, such as light weight, low loss, the local side by exploiting the geometric symmetry and spatial
and immunity to electromagnetic interference, it has played an redundancy of 7-core MCF [10]. We realized localized remote
important role especially in fields where precise remote sensing vectorial vibration sensing including the amplitude, frequency,
over long distances and in harsh environments is necessitated [7]. and direction with high precision over 10 km.

0146-9592/23/092369-04 Journal © 2023 Optica Publishing Group

 2370 Vol. 48, No. 9 / 1 May 2023 / Optics Letters Letter

change δφi between the lights propagating in each of the outer

core and the center core is then linked to δLi in a linear manner.
By replacing the corresponding parameters with δLi , the relation
is read as

δφi = kn(1 − C2 n2 /2) · δLi = η · δLi , (3)

where k and n represent the wavenumber and refractive index,

respectively; C2 = 0.204 is a constant related to the material; and
η is regarded as a factor dictated by the characteristics of the 7-
core MCF. This way, it is possible to extract δLi by constructing
MZIs with each outer core acting as the sensing arm while the
center core plays the role of the reference arm.
Nonetheless, this acquisition suffers severely from the
Fig. 1. (a) Schematic of the proposed system. PMC: polarization unwanted environmental disturbances that induce uncorrelated
maintaining coupler, BPD: balanced photodetector, LF: loop filter, relative delay fluctuations between each of the cores. This
FS: fiber stretcher, PM: phase modulator, PT: polarization tracker. inevitably impairs the phase stability, thus hindering the extrac-
(b) Side view, and (c) cross-section view of 7-core MCF with instant tion of δφi , in particular over a long distance or in remote sensing
vibration-induced bending displacement D and central angle α.
scenarios. More seriously, the nonlinearity that strictly limits the
dynamic range, especially when δφi is not completed within the
Consider the case when a section of 7-core MCF with a linear region of a cosine function in coherent phase detection
regular hexagonal core layout and length L0 is subjected to a [14], will in turn restrict the measurement range and precision,
vibration-induced instant bending with a displacement D at a and is thus an issue requiring attention. To this end, we adopt
central bending angle α as illustrated in Fig. 1(b). According a composite OPLL consisting of two loops with distinct behav-
to material mechanics [11,12], if the angular offset from the iors: one exhibits a high loop gain and large control range but
x axis of the local coordinate system to the bending direction with a limited bandwidth to cope with the unwanted distur-
(perpendicular to the neutral plane) and that of the specific outer bances, while the other aims at high loop bandwidth to achieve
core i = 2, 3, . . . , 7 are described by θ b and θ i in Fig. 1(c), a broadband linearized phase detection with enhanced dynamic
respectively, the resulting longitudinal length variation δLi with range.
respect to the center core can be given by While these limiting factors are accounted for, by substituting
Eq. (1) and Eq. (3) into Eq. (2), it allows us to track the instan-
δLi = di · α = −d0 cos(θ b − θ i ) · α, (1) taneous bending in a time-variant manner with a high precision
enabled by the exploitation of the redundancies as
where di , related to θ b , θ i , and the core spacing d0 , is the distance
from outer core i to the neutral plane. This way, based on the δφi (t) · L0
D(t) = , (4)
relation indicated by Eq. (1), by exploiting the ratio of any two −2ηd0 cos[θ b (t) − θ i ]
δLi except for the centrosymmetric pairs, the bending direction
θ b can be obtained. which suggests that with only the measurement for δφi (t), both
To cope with the 2π phase ambiguity due to the cosine func- the direction and amplitude of the vibration can be readily
tion, a rough estimate of θ b is logically obtained by relying resolved, namely in a vectorial manner. Furthermore, six sets
on the fact that θ b should always be located between the two of D can be similarly retrieved from different δLi and a linear
cores showing the largest and second largest negative value fitting is conducted.
of δLi . Thus, it allows the derivation of a series of θ b by Experimental demonstration is carried out as shown in
exploiting the non-centrosymmetric pairs of cores, where the Fig. 1(a). In the transmitter, a fiber laser (NKT C15) oper-
redundancies—precisely a set of 12 estimates for θ b from these ating at ∼1550 nm with ∼5 kHz nominal linewidth and 10 mW
measures—can be used to reduce the uncertainties. output power serves both as the signal lights and reference
Subsequently, according to the geometrical relation suggested light. They are conveyed through the six outer cores and the
in Fig. 1(b) in connection with α and L0 , D can be resolved by center core, respectively, to the remote area in a round-trip
manner via a 20 km spool of commercial 7-core MCF (from
D = L0 · (1 − cos α)/α ≈ L0 · α/2, (2) YOFC) with fan-in and fan-out, respectively, at the two sides.
In the remote-sensing area, a piezoelectric bender with sensitiv-
where the linear approximation can be maintained provided α ity γ = 1.8 µm/V is adopted to introduce vectorial bending to
is not significantly large. More precisely, a maximum tolerable mimic the actual event.
error up to 1% would allow the preservation of the linearization The signals carrying the vibration information together with
for α ≤ 25◦ . This is quite a large range considering the vibration the returned reference are detected in their corresponding
that can be withstood by fiber from a practical viewpoint. Thus, OPLLs to extract δφi at the local receivers. In each receiver, the
the detection of an instant bending can be steadily translated into coherent beating between the signal and reference lights pro-
the measure for δLi , i.e., sensing for the relative strains experi- duces an error signal, which is successively processed by loop
enced by each of the cores. The precision for the measurement filters (LFs) to yield the feedback for the composite OPLLs asso-
of δLi hence becomes critical since it ultimately restricts that for ciated with each MZI. With a fiber stretcher (FS) (OPTIPHASE)
the vibration sensing. inserted in each signal arm before coherent detection at a 50:50
By exploiting the phase change in a bent MCF in response coupler and a balanced photodetector (BPD), one loop is used to
to both index and length variations [13], the differential phase compensate for the disturbance-induced delay variations mainly
 Letter Vol. 48, No. 9 / 1 May 2023 / Optics Letters 2371

Fig. 2. (a) Beat note spectra, and (b) output noise floors when Fig. 3. (a) Demodulated outputs from each MZI at the moment
both loops are open, only FS loop is closed, and both FS and PM marked by black dash; (b) instant θ b derived using core i, j pairs
loops are closed; (c) Time domain output signals when driving in comparison with the averaged estimation of θ b ; (c) instant D
signal is 500 Hz with 6 V amplitude (Red: only FS loop is closed; derived in core i with the averaged estimation of D; and (d) resolved
Blue: both FS and PM loops are closed). vibration vector.

For the demonstration of the remote vectorial vibration sens-

at low frequency regimes with a high loop gain and control range ing, a 500 Hz sinusoidal driving signal with 5 V amplitude is
inherited from the large tuning sensitivity of the FS. While the applied to the piezoelectric bender. The output signal Vi (t) from
other loop, with a LiNbO3 phase modulator (PM) (V π = 3.5 V) each MZI, as shown in Fig. 3(a), is linearly proportional to δφi
in tandem with the FS, is established and cooperates with the for- as δφi (t) = −πVi (t)/V π , where the negative sign stems from the
mer FS loop to accomplish the linearized phase detection. This phase-tracking principle in the PM loop. With the detected vibra-
design allows for the stabilization of the operation point and the tion frequency coinciding with that imposed on the fiber, the
linearized coherent phase detection needed for high-precision instantaneous direction can be preliminarily deduced by com-
measurement. With such a composite OPLL, the demodulated paring the sign and relative magnitude of the outputs. In detail,
phase is directly obtained at the output of the PM loop with the taking an arbitrary point as marked by the black dashed line
operation principle explained in Ref. [14]. Polarization trackers in Fig. 3(a) as an example, since V2 , V3 , and V4 are positive
(PTs) are used for the polarization alignment. This way, the sys- and in-phase while their absolute |V2 | and |V3 | are larger, it
tem is actually a structure composed by a set of locally stabilized infers that these cores are compressed with respect to the neu-
hexa-MZIs sharing a common reference, and the output of each tral plane, leading to a rough estimate of θ b that it is located
MZI is precisely a linear map of δφi . in between core 2 and 3, i.e., in the 4th quadrant. With this
To confirm the effect of the composite OPLL, the beat note pre-knowledge, a further estimate of θ b can be made using any
spectra at different loop conditions are exhibited in Figs. 2(a) non-centrosymmetric pair of cores. Ideally, the results obtained
and 2(b), showing the noise floor of the system. The band- from different pairs of cores are supposed to be equal. But even
width of the FS loop should be sufficient to effectively suppress with the common-mode noise suppression brought by the 7-
the phase noise arising from slow-variant delay perturbations core MCF and the phase-locking using the composite OPLL,
accumulated in a long transmission link, guaranteeing a stable unavoidable systematic noise, including residual phase noise and
operation for the interferometric demodulation. On this basis, other imperfections, such as the irregular vibration conduction
the PM loop signifies the tracking ability for fast-variant phase amongst the cores, would potentially bring about discrepancies.
changes, the bandwidth of which should be optimized to pro- Taking full advantage of all the outer cores allows the resolution
vide a broad detecting range. Eventually limited by the loop of a more explicit θ b using the least square (LS) method with a
delay, it is expected to be maximized in several ways, such as set of 12 results obtained from each of the non-centrosymmetric
photonic integration. Accounting for these constraints in this pairs, as shown in Fig. 3(b). With this redundancy, a more
demonstration, the bandwidth of the FS and PM loops are set to precise estimate for the direction of the actual deformation as
∼300 Hz and ∼20 MHz, respectively. Even with the inevitable θ b ≈ 292.4175◦ with a standard deviation (STD) of ∼0.1103◦ can
sacrifice in the low frequency region, this detection bandwidth be revealed as indicated by the orientation of the red arrowed
is adequate in most scenarios. Linearized demodulation is, in line, as exhibited in Fig. 3(d).
addition, verified in the time domain by the temporal waveforms With regard to the vibration amplitude at the moment, relying
attained in cases where only the FS loop or the composite loop is on such precise knowledge of θ b , a set of six estimates, Di can
closed, when a sinusoidal driving signal at 500 Hz is applied to be subsequently derived from each MZI in connection with δφi
the piezoelectric bender. As compared in Fig. 2(c), the obvious of each core according to Eq. (4), where L0 approximates to the
nonlinear distortion is eliminated with a PM loop, suggesting free length of the piezoelectric bender employed and d0 = 41
significant improvement in the linearity. It is worth noting that µm from the spec of the 7-core MCF. The linear fitting by LS
when the PM loop is further closed, the noise floor and sensing has led to an estimate of D ≈ 9.0541 µm with STD ≈ 0.3603 µm
signal are synchronously suppressed by the same factor, which as exhibited together with the results associated to each outer
implies a preserved sensitivity [15]. core in Fig. 3(c), which is also represented as the length of the
 2372 Vol. 48, No. 9 / 1 May 2023 / Optics Letters Letter

given by D = γ · Vd . The slope resulting from the linear fitting

is 1.8886, which agrees well with γ, implying excellent reliabil-
ity. Furthermore, specified rotation angles are introduced using
a fiber fusion splicer by rotating synchronously the two fiber
holders while keeping the bender fixed. After each rotation, the
7-core MCF is again stuck to the bender to maintain such a des-
ignated angle during the measurement. With a rotation step of
60.00◦ , the results covering all four quadrants with the measure-
ment errors em (m = 2, 3, . . . , 6), are given in Fig. 5(b). The
consistent STD around 0.1100◦ also confirms the reliability.
In conclusion, we report on remote sensing for vectorial
vibration using 7-core MCF-based hexa-MZIs with the stabi-
Fig. 4. Averaged θ b (t) obtained exploiting all core pairs and D(t) lization and linear phase detection established by an OPLL. We
obtained using the averaged θ b (t) and Di (t) of all the outer cores demonstrated remote vibration sensing in a vectorial manner at
(red) with the actual sinusoidal vibration applied at the remote end a distance over 10 km with testified high precision of ∼0.3603
(black). µm in displacement and ∼0.1076◦ in angle. Being attractive for
various scenarios, it confirms the simultaneous sensing of the
frequency, direction, and amplitude of remote vibrations with
high precision at long distances.
Funding. National Natural Science Foundation of China (61827807,
61805014).

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.

Fig. 5. (a) Obtained vibration amplitude versus the increased

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