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Abstract
We demonstrated a two-dimensional vector displacement (bending) sensor with high angular resolution based on Vernier effect generated by two cascaded Fabry–Perot interferometers (FPI) in a seven-core fiber (SCF). To form the FPI, plane-shaped refractive index modulations are fabricated as the reflection mirrors in the SCF using slit-beam shaping and femtosecond laser direct writing. Three pairs of cascaded FPIs are fabricated in the center core and the two non-diagonal edge cores of the SCF and applied to the vector displacement measurement. The proposed sensor exhibits high displacement sensitivity with significant direction dependence. The magnitude and direction of the fiber displacement can be obtained via monitoring the wavelength shifts. Moreover, the source fluctuations and the temperature cross-sensitivity can be referenced out by monitoring the bending-insensitive FPI of the center core.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Displacement sensors are widely used in many fields, such as structural safety monitoring, road and bridge construction, intelligent sensing devices and mechanical deformation measurement [1–3]. Traditional electromagnetic displacement sensors suffer from the poor immunity to electromagnetic interference, high maintenance costs, etc. [4,5] that restrict their applications. Alternatively, optical fiber-bending based sensors have been developed to realize displacement measurements, and found to exhibit excellent performances. And many optical fiber sensors can simultaneously respond to both the amplitude and direction of the applied displacement which benefit from their specific asymmetric geometric structure or mode coupling methods. Importantly, the direction-dependence is an important sensing feature in real applications.
For instance, fiber interferometers based on the special asymmetric splicing configurations can achieve excellent direction-dependent responses due to their tailored coupling mechanism [6–17]. Whereas, most interferometers are subject to the strict mode coupling conditions and complicated fabrications, reducing their practicability and robustness. Besides, asymmetric fiber long period grating (LPG) that is written in D-shaped fiber is demonstrated for the direction-dependent measurement [18,19]. The serious cross-sensitivities of the LPG cannot be ignored. Meanwhile, fiber Bragg gratings (FBG) inscribed in the specially distributed fiber core [20], such as the eccentric core fiber [16,21,22] or the multicore fiber [23], can also achieve the vector measurements. The bend sensor based on FBG inscribed on a multicore fiber proposed by Hou et al. in 2018 achieved a curvature sensitivity of 59.47 $\mathrm {~pm} / \mathrm {m}^{-1}$ with an orientation test step of 15$^{\circ }$ and proposed a geometric method on orientation reconstruction [11]. In 2022 Ricardo et al. proposed a bending sensor based on multicore fiber and single-mode fiber end-face interference [24], whose curvature reconstruction method was precisely quoted from hou et al. and achieved a maximum sensitivity of 432.6 $\mathrm {~pm} / \mathrm {m}^{-1}$. Both of the above reported sensors can be temperature compensated, but their drawbacks are that the sensitivity is not high enough to achieve $0.1^{\circ }$ of test angular resolution, and the azimuthal resolution of their sensors has not been actually tested. In addition, the mechanical damage of grating-type sensors is greater than that of interferometric sensors, due to their many modulation surfaces. However, the angular resolution of the aforementioned sensors exhibits the theoretical maximum values that are calculated from the accuracy of the detection instrument combined with sensitivity, which are not sufficient to achieve a high angular resolution in practice. Moreover, the temperature cross-sensitivity would significantly affect their sensing performance.
Here we report a two-dimensional displacement vector sensor based on two cascaded Fabry–Perot interferometers (FPIs) in two non-diagonal edge cores of SCF that are fabricated by femtosecond laser direct writing. The plane-shaped refractive index modulations are fabricated as the reflection mirrors in the SCF using slit-beam shaping and femtosecond laser direct writing, which essentially form the FPI structures. Taking advantage of the unsymmetrical core distribution and the magnification of Vernier effect, the proposed sensor offers highly sensitive displacement responses with high azimuthal angle resolution (direction). The direction-dependent radial displacement responses show a perfect sinusoidal pattern in coordinate system with a maximum sensitivity of 48.03 $\mathrm {nm} \cdot \mathrm {mm}^{-1}$. Moreover, the effects of temperature and axial load can be compensated via monitoring the displacement-insensitive central core FPI wavelength responses.
2. Principle and fabrication
The principle of the proposed high-resolution displacement (bending) vector sensor is based on the Vernier effect generated by two intrinsic FPIs in a SCF. The schematic diagram of the proposed sensor is shown in Fig. 1(a). The sensor consists of two pairs of cascaded (FPIs) in the edge core of the SCF. One of the FPIs is used for sensing measurement, while another one is used for reference. The FPIs are formed by four internal reflection mirrors along the fiber core. The uniform SCF used in the preparation experiments is YOFC-fabricated and contains six edge cores and a central core, as shown in Fig. 1(b). The diameter of the cores are 8 $\mu$m and the fiber cladding diameter is 150 $\mu$m. The distance between adjacent fiber cores is about 42 $\mu$m. To form the FPI structure, the reflection mirrors are induced by the femtosecond laser pulses. The femtosecond laser pulses were generated using a regenerative amplified Ti:sapphire laser, with the central wavelength of 800 nm (TEM00 spatial mode), the pulse duration of 50 femtoseconds, and the repetition rate of 200 Hz. Here, the slit-beam shaping method was used to form a plane refractive index modified area [25] in order to obtain a reflection mirror with larger dimension and higher reflectivity. In the experiment, the fiber was held on a 3D micromachining platform (Newport Motion Controller Model XPS-D). The laser energy was tuned to 70 $\mu$W, and the laser beam was shaped via a slit (Thorlabs VA100C/M) and then focused in the fiber core by a microscope objective (ZEISS 40x NA0.75). By rotating the fiber and adjusting the focus of the objective, the reflection mirror can be written precisely into the two edge and central cores. The distance of the two mirrors can be precisely controlled by the platform. The reflection spectrum was monitored using a fan-in/out device (YOFC) and a demodulator Si255 (Luna Hyperion Si255) in the fabrication.
Fig. 1. (a) The schematic diagram of the Vernier effect in SCF inscription process using femtosecond laser direct writing technique and slit-beam shaping. (b) Microscopic image of the cross-section of the SCF.
The incident light beam can be reflected by the reflection mirrors in each core, then the reflected beams will interfere in respective cores and generate an output reflection. In the output reflection spectrum, the free spectral range (FSR) of a single FPI can be expressed as:
where $\lambda$ is the wavelength, L is the cavity length of the FPI, and n is the effective refractive index of the fiber core.The lengths of the sensing FPI cavity and the reference FPI cavity are 1000 $\mu$m and 1020 $\mu$m respectively. And the distance between the sensing cavity and the reference cavity is 10 cm (can be longer), because the distance between them does not affect their magnification factor, and secondly the reference cavity needs to remain fixed during experimental measurements, a longer spacing is necessary to avoid unexpectedly small movements of the reference cavity during the experiment. The sensing FPIs were first fabricated, shown in Fig. 2(a) and (b). It can be seen that the interference fringe is very fine and sharp, the fringe visibility is high, and the free spectral range is as small as 785 pm. The core position would be changed during moving fiber. Hence, we have to re-confirm core position for the reference FPI fabrication. In the fabrication of the reference FPI, we randomly selected an edge core, and inscribed a reflection mirror in it. The spectra of the sensing FPIs were monitoring in real time. The spectrum can be changed if one of the sensing FPIs was introduced an additional reflection mirror. Once one position of the sensing FPIs was confirmed, the spatial position of fiber cores could be confirmed as well.
Fig. 2. Schematic under microscope: (a) the sensing FPI, (c) the reference FPI. Reflection spectrum: (b) the sensing FPI, (d) after introduction of Vernier effect.
Notably, the fiber displacement induced uneven stress distribution along the cross section can change the FPI cavity length (L), and further result in the spectral responses. In addition, the spectral responses are highly direction-dependent due to the asymmetrical edge core distribution. However, the slight spectral changes that might be overwhelmed by the spontaneous noise of the sensing system. To achieve high resolution measurement, we have to further improve the sensing performance of the proposed sensor. Thus, another FPI is introduced in sensing system to generate the Vernier effect and improve the spectral responsivity.
Additional FPI was cascaded in lead-in FPI, two FPIs are served as sensing component and reference component, respectively. For cascaded FPI system, the FSR of the envelop can be written as [26]:
where $F S R_{s}=\frac {\lambda ^{2}}{2 n L_{s}}$ and $F S R_{r}=\frac {\lambda ^{2}}{2 n L_{r}}$ are FSRs of the sensing FPI and reference FPI, with Ls and Lr are the cavity lengths of the sensing FPI and the reference FPI respectively. A magnification factor is defined as the ratio between the FSR of the envelop and the FSR of the sensing FPI: It can be noticed that the amplification factor depends on the cavity length of two FPI only. The designed amplification factor can be achieved via precisely controlling of the FPI cavity length. And a negative M-factor cloud be obtained when the optical path length of the reference interferometer is greater than the optical path length of the sensor interferometer. It simply means that the Vernier envelop wavelength shift in the opposite direction of the single sensing FPI’s interferometric peak shift.In order to achieve a large amplification factor, the cavity lengths of the sensing and reference cavities should be as close as possible. However, the FSR of the envelop becomes very large when the cavity length difference is too small, which may lead to excessively envelop shift and exceeded the measurable range of the monitoring system. There is a trade-off between the amplification factor and the measurement. Thus, we selected an amplification factor of 50 according to the current properties of the demodulating system in this work. As shown in Fig. 2(c), it can be seen that the four reflective mirrors are successfully written into the edge core of the fiber. The interference spectra of the two FPIs interfere with each other twice to form an envelope, as the reflection spectrum shown in Fig. 2(d). The spectrum shows a good periodicity and the red line indicates the lower envelope, the so-called lower envelope is obtained by connecting the dips of the interference pattern. The Vernier envelope has a wide range of 38.3 nm.
3. Displacement reconstruction
The data that are monitored in this work are the envelop peak or dip wavelength shift ($\Delta \lambda$) of the sensor. The reconstruction of the orientation and amplitude of the two-dimensional displacement vector is performed. As shown in Fig. 3(a), the fiber bending can induce the radial displacement of the fiber end along the bending direction. The displacement vector includes magnitude $\left |\vec {D}\right |$ and the direction (angle) $\varphi$, which contribute towards the vector displacement measurement together. The displacement vector ($\vec {D}$) is introduced by fiber bending, and the introduction of a displacement vector ($\vec {D}$) is equivalent to the introduction of a bend. Different displacement vector values correspond to different bending states, which means that there is a correlation between displacement and curvature, so the curvature reconstruction method [11,24] is used in this design.
Fig. 3. (a) 3D diagram of an optical fiber in bent state; (b) definition of the geometrical parameters from the cross-section of the SCF.
First, it is clear that $\vec {D}$ in Fig. 3(b) is used to refer to the 2D displacement (bending) vector in the paper. According to the linear relationship $\vec {d}=\frac {\Delta \lambda }{S}$ (S is the displacement sensitivity), the projected components of the $\vec {d}$ on the cascade FPI of six edge cores can be expressed as:
where $i=2,3,{\ldots }..7..$ Subsequently, the geometric position relationship between the six edge cores and the central core is used to reconstruct the bending vector $\vec {D}$. Examples of the projected components of the bending vector $\vec {D}$ on the diagonal of core 3 and core 5 are given in Fig. 3(b), for the first quadrant. The bending components of the cascade FPI of two edge cores located on the same diagonal are equal in amplitude and opposite in orientation ($d_{2}=d_{-5}, d_{3}=d_{-6}, d_{4}=d_{-7}$), in the calculations under different quadrants, the corresponding substitutions can be made. According to Eq. (4), $\overrightarrow {d_{3}}, \overrightarrow {d_{5}}$ can be obtained from a specific envelope wavelength shift ($\Delta \lambda _{i}$) of the two cascade FPIs, so the $\overrightarrow {d_{3}}, \overrightarrow {d_{5}}$ can be expressed as follows:It is worth noting that in the actual calculation, any two non-diagonal edge cores can be selected to reconstruct the bending vector, which means that twelve combinations can be used, namely, $\left (\vec {d}_{2}, \vec {d}_{3}\right ), \left (\vec {d}_{2}, \vec {d}_{4}\right ),\left (\vec {d}_{2}, \vec {d}_{6}\right ),\left (\vec {d}_{2}, \vec {d}_{7}\right ),\left (\vec {d}_{3}, \vec {d}_{4}\right ),\left (\vec {d}_{3}, \vec {d}_{5}\right ), \left (\vec {d}_{3}, \vec {d}_{7}\right ), \left (\vec {d}_{4}, \vec {d}_{5}\right ),\left (\vec {d}_{4}, \vec {d}_{6}\right ),\\\left (\vec {d}_{5}, \vec {d}_{6}\right ),\left (\vec {d}_{5}, \vec {d}_{7}\right ),\left (\vec {d}_{6}, \vec {d}_{7}\right )$. By averaging over multiple reconstructions, a more accurate final value can be obtained.
4. Two-dimensional vector displacement (bending) experiments
Based on the principle of directional reconstruction, we selected cores 1, 3 and 5 (non-diagonal). First, the SCF with Vernier effect introduced is spliced with the fan-in/out device, and then a demodulator Si255 (Luna Hyperion Si255) is connected to measure its reflection spectrum. Then, the sensor was mounted on a pair of 3D translation tables (Zolix APFP-XYZ $\theta$L) by the fiber holders (Newport 561-FH), and Fiber Rotator (Thorlabs HFR007). The displacement vector amplitude can be changed by moving the translation table in the Y-axis direction where the free end of the fiber is located. In addition, the displacement direction can be changed by rotating the fiber rotator. Here we define the displacement azimuth angle $\varphi$ as the angle between the displacement plane in the SCF and the axes connecting core 1 and core 3, as shown in Fig. 3(b). When $\varphi$ = $0^{\circ }$ or $360^{\circ }$, the displacement direction is aligned with the axes of core 1 and core 3, and core 1 is located in the center of the bent SCF. Finally, the envelope wavelength shift of the cascade FPI on the three cores was recorded in the case of displacement (bending) measurements. It should be noted that the initial angle ($\varphi =0^{\circ }$) is not equal to the test rotation angle, in fact it is equal to the test rotation angle corresponding to the maximum slope of the fitted direction-dependent curve. The calibration of the initial angle ($\varphi =0^{\circ }$) is done by fitting the data after acquisition.
We denote the amplitude of the displacement vector in terms of Y-offset, which is the translation of the 3D stage on the left-hand side of Fig. 4 in the Y-direction with respect to the initial horizontal plane. We measured the orientation dependence of the vector displacement sensor proposed in the paper at Y-offset of 0 $\mu$m, 200 $\mu$m, 400 $\mu$m, 600 $\mu$m, 800 $\mu$m, 1000 $\mu$m and 1200 $\mu$m, respectively, in $20^{\circ }$ steps over a full range of angles from $0^{\circ }-360^{\circ }$. And the envelope wavelength shift of the cascade FPI for the three cores 1, 3, and 5 are monitored and recorded in real time. The experimental results are shown in Fig. 5 and Fig. 6. The spectral variation of fiber core 3 at 1000 $\mu$m Y-offset is given in Fig. 5, and the lower envelope peak positions corresponding to four representative special azimuthal angles of $0^{\circ }, 90^{\circ }, 180^{\circ }, 270^{\circ }, 360^{\circ }$ are marked. the sensor presents an excellent direction-dependent responses to the 2D vector displacement. We plot the wavelength difference and sensitivity as a function of magnitude and angle in a cartesian coordinates, respectively, as shown in Fig. 6(a)-(f).
Fig. 5. (a) Reflection spectrum of core 3 at $0^{\circ }, 90^{\circ }, 180^{\circ }, 270^{\circ }, 360^{\circ }$; (b) spectral responses to the azimuthal angle of the fitting lower envelope curve.
Fig. 6. Experimental results of wavelength difference for core 1, 3, and 5: (a), (c), and (e) versus Y-offset with $\varphi$ in the range from $0^{\circ }$ to $360^{\circ }$ in steps of $20^{\circ }$; (b), (d), and (f) displacement sensitivity with $\varphi$ in the range from $0^{\circ }$ to $360^{\circ }$ in steps of $20^{\circ }$.
As shown in the Fig. 6(a), the displacement sensitivity of fiber core 1 is unchanged at different direction, and the lower envelope peak wavelength shift is close to zero, making it a perfect for doing temperature compensation. Core 3 has the maximum sensitivity (48.03 $\mathrm {nm} \cdot \mathrm {mm}^{-1}$) at $0^{\circ }$ and $180^{\circ }$, and it tends to be close to 0 at $90^{\circ }$ and $270^{\circ }$. The FPIs in core 5 shows the similar displacement sensitivity and orientation dependence of core 3. Importantly, the response distribution of core 3 and 5 present a phase difference of $2 \pi / 3$.
On the basis of the above measurements, we measure the maximum angular resolution of the design. The slope at its maximum of core 3 were locked by the results of the test of the orientation dependence, which is the place at azimuthal angle of $90^{\circ }$ and $270^{\circ }$. The Y-offset was set to 1000 $\mu$m, and the test was performed at the above two slope maximum points in steps of $0.1^{\circ }$ within its left and right $1^{\circ }$, and the lower envelope peak wavelength shift of the cascade FPI of core 3 and core 5 was monitored and recorded. The results are shown in Fig. 7. It can be seen that envelope wavelength shift can still be distinguished with step of $0.1^{\circ }$. The curve showed an overall good linear trend. The sensitivity of core 3 is $4.900 \mathrm {~nm} /{ }^{\circ }$ and $-4.630 \mathrm {~nm} /{ }^{\circ }$, and that of core 5 is $-1.862 \mathrm {~nm} /{ }^{\circ }$ and $1.586 \mathrm {~nm} /{ }^{\circ }$, respectively, which is a significant increase in sensitivity and in angular resolution compared to an ordinary grating-type bending vector sensor. The above results are only from actual tests. In theory, greater resolution could be obtained by using a higher precision demodulator, however, due to the functional limitations of the test equipment, we were only able to achieve test conditions in $0.1^{\circ }$ steps.
Fig. 7. Experimental results of 0.1$^{\circ }$ steps for core 3 and 5: (a) and (c) with $\varphi$ in the range from 89$^{\circ }$ to 91$^{\circ }$; (b) and (d) with $\varphi$ in the range from 269$^{\circ }$ to 271$^{\circ }$.
According to the above experimental results, the proposed sensor is potential to realize simultaneous measurement of the displacement and the temperature because the FPIs in the central core is insensitive to the fiber bending. Hence the temperature cross-sensitivity can be compensated via monitoring the responses of the central core. The temperature characteristic of proposed sensor is investigated. In the experiment, the sensor was put into a heating oven with a resolution of 1 ${ }^{\circ } \mathrm {C}$. The temperature was gradually increased from 22 ${ }^{\circ } \mathrm {C}$ (room temperature) to 42 ${ }^{\circ } \mathrm {C}$ with a step of 2 ${ }^{\circ } \mathrm {C}$. All sensing spectra presented wavelength red-shift, and the sensitivities are 465.75 pm/${ }^{\circ } \mathrm {C}$, 532 pm/${ }^{\circ } \mathrm {C}$ and 496.5 pm/${ }^{\circ } \mathrm {C}$, respectively shown in Fig. 8. During the displacement measurement, the temperature changed can be obtained via monitoring the wavelength shifting of the central core FPIs. Therefore, the proposed sensor can provide great temperature compensation.
5. Conclusion
A two-dimensional displacement (bending) vector sensor based on the Vernier effect is proposed and demonstrated in this paper, which is fabricated by slit-beam shaping and femtosecond laser direct writing method. Taking advantage of the high-resolution azimuth response for radial 2D plane displacement, the sensor can recover the 2D information of a random displacement, including magnitude and direction, according to the displacement responses of the proposed sensor. Because of the convenient fabrication and excellent sensing performance, the proposed sensor, have great potential for many industrial applications.
Funding
National Natural Science Foundation of China (61735014, 61905160, 61927812, 62075181); Science, Technology and Innovation Commission of Shenzhen Municipality (JCYJ20200109114201731).
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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