1. Introduction

Fiber optic displacement sensors have seen widespread application, from directing small bio-probes [1–2] to monitoring large civil structures [3]. These optical sensors offer unique advantages over their electronic or electro-mechanical counterparts owing to their compact size, flexibility, electrical and magnetic field immunity, inert and biocompatible properties, ease of system multiplexing, and overall ability to withstand high temperature or pressure [4–6].

To overcome an inherently weak photoelastic response in fiber, displacement sensing has generally required fibers modified with a multicore (MCF) [7–11], eccentric core [12–14], surface core [15], or structured cladding [16,17] that reposition the light-guiding core away from the neutral axis. Otherwise, sensing elements must be positioned asymmetrically to an off-axis position within the core or cladding of traditional fiber types, such as multimode (MMF) [18–20], multi-clad [21–23], and single-mode fiber (SMF) [1,24–27]. Various methods of micro-structuring in the fiber cladding have enabled displacement sensing through formation of tapers [28,29], tuned mass elements [30,31], and Fabry-Perot cavities [32,33], which significantly reshape the fiber at the expense of structural integrity.

Alternatively, optical sensor elements have been favorably positioned in the central core waveguide of traditional SMF through a much less invasive process, for example, by laser inscription of a fiber Bragg grating (FBG) generated through a periodic refractive index modulation. In one direction, tilted FBGs [18,22,25] or off-axis FBGs [8,23,24,27] have been used for core-to-cladding coupling to boost the displacement response, but at the cost of low return light recoupling. On the other hand, FBGs positioned away from the central axis, but at the SMF core, have provided lowest-loss probing of photoelastic responses by directly measuring the Bragg wavelength shift [34], the back-reflected power [35], or the higher-order mode spectral features [36]. Inscribing FBGs further outside the core has provided a beneficial linear rise in sensitivity, but at a high cost of an exponential drop in signal [37]. Previously, direct writing with focused ultrashort laser pulses induced gratings with phase elements localized over small cross-sectional zones of ∼1 × 2 µm² [34] or larger [35,36], over which a varying photoelastic response may be optically detected.

A reshaping of phase elements into narrow and long filaments [38] has provided a new direction for creating ultra-thin planar gratings with femtosecond lasers. Such gratings provide strong and spectrally narrow Bragg resonances in SMF [38,39], over short lengths and even when occupying a minimal cross-sectional area of the fiber core. Point-by-point, or in this case filament-by-filament writing has facilitated a flexible means to improve resolution with π-shifts or with shifted Bragg resonances for multi-grating systems [39]. This technique opens an opportunity in displacement sensing, where the photoelastic response can be measured at a single plane, offset from the neutral axis. Moreover, the geometry favors a highly contrasting azimuthal sensitivity that reaches null signal when displacement is parallel with the filament lines. Lastly, filament gratings may be overlaid with different offsets and azimuthal angles, for example, orthogonally crossing as shown in Fig. 1, to harvest more precise and complete displacement data based on vector sensing capabilities [1,27,34–36].

 

Fig. 1. Schematic of a cantilever-based FBG displacement sensor with two orthogonal sets of off-axis filaments (displacements ξ_x and ξ_y) crossed through the fiber core.

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In this paper, the laser filament writing technique was harnessed for the first time in displacement sensing, to generate off-axis FBGs in a cantilever-based sensor. The femtosecond laser inscription process was optimized for generating short length, high reflection gratings, while laterally shifted to outer zones of the fiber core. Bragg wavelength responses to bent fiber strain were recorded experimentally and compared with simulated values. These were also analyzed azimuthally to verify the directional sensitivity of the filament FBGs. Overall, the filament-by-filament writing offered a facile means for overlaying gratings to calibrate for alignment errors in the fiber manufacturing and lock-in the grating offsets during fabrication. In this way, an improved measurement calibration was obtained that is promising for azimuthally resolved displacement sensing, while complementing the existing capabilities of temperature and axial strain compensation [1,27].

2. Fabrication and characterization

The filament-array gratings, as presented in Fig. 1, were fabricated following procedures described previously [39–42]. Briefly, a frequency-doubled Yb-doped fiber laser (Amplitude Systems, Satsuma) provided light pulses of λ = 515 nm wavelength and Δτ = 250 fs duration. Pulse energies of up to ∼8 µJ were focused with an aspherical lens (0.55 NA, 4.5 mm effective focal length, New Focus, 5722-A-H) through a fused silica glass plate into the core and cladding zone of standard mechanically stripped SMF-28 telecommunication fiber (Corning, NY, USA). Refractive index matching oil (Cargille, 50350) was applied to fill the gap between the fiber and plate and remove the astigmatic aberration of the cylindrical shaped fiber. The surfaces of the glass plate induced a controlled degree of surface aberration into the beam to elongate the focal spot. As a result, a long and uniform light filament of over 50 µm in length and submicron cross-sectional diameter was generated, which induced a highly localized refractive index modification in the fiber with a single pulse exposure. Line-by-line writing of filaments was controlled with a three-axis alignment and motion stage (Aerotech Inc, Aerotech-PlanarDL-00 XY and ANT130-060-L Z). The fiber was scanned at ∼0.5 and ∼1 mm/s speed to form first and second order filament FBGs, respectively, while the laser repetition rate was down counted to 1 kHz. The filaments were assembled inside the core waveguide into a tightly packed array with a precise offset position (ξ_x or ξ_y in Fig. 1) determined from the fiber’s center axis with a precision of ±0.5 µm with respect to the cladding surface.

A pulse energy of ∼1 µJ was applied to generate a relatively strong first-order Bragg resonance with reflection peaks of -1.5 to -2 dB, targeted for a ∼1 mm grating length (ℓ). A Bragg period of Λ = 536 nm provided a first-order resonance wavelength near λ = 1550 nm. Second and third order gratings were also fabricated. The reflection spectra of the FBGs were recorded with a variety of light sources by using an optical circulator (Thorlabs, 6015-3-FC) to direct the return light to various detectors. The highest resolution spectra were recorded with a tunable external cavity laser (Photonetics, TUNICS-BT), with reflected light detected by an optical power meter (Newport, 2835-C) and acquired onto a PC through a data acquisition unit (National Instruments, NI-9206 and NI cDAQ-9188) to generate the FBG spectrum (LabVIEW NXG 5.0). In another configuration, an optical spectrum analyzer (Anritsu, MS9740B) recorded the FBG reflection spectra generated with a broadband light source (Thorlabs, ASE-FL7002). The two sources extended over the telecommunication C-band (1530–1610 nm).

Single wavelength FBGs were prepared with varying offsets, from the fiber center (ξ_x = 0) to the core-cladding interface (ξ_x = ∼4.2 µm). The FBG filaments were inspected by optical microscopy (Olympus, BX51) to verify the orientation, periodicity, and the off-axis positioning of the filament gratings with respect to the fiber core.

Figure 2(a) shows the FBG spectra recorded from single-row gratings formed with identical exposures (E_pulse = 1 µJ, ℓ = 1 mm) but in planes positioned on different offsets (ξ_x ≅ 0, 1, 2, 3 µm). The spectral responses (strength, bandwidth) of these first-order gratings ($\Lambda=536~\text{nm}$) aligned with our prior work, peaking around -1.5 to -2 dB on reflectance for similar exposure conditions [39–43]. However, the peak reflectance fell off according to -1.4, -1.8, -2.0, and -8.2 dB with increasing offset of ξ_x ≅ 0, 1, 2, and 3 µm, respectively. This falloff aligns approximately with the Gaussian mode field profile (-1.4, -1.8, -3.0 and -5.0 dB) in SMF-28 fiber at 1550 nm wavelength. The difference between the recorded and calculated reflection values is attributed to the positioning error (± 0.5 µm) of the filament array in the fiber core. The ∼1 nm bandwidth (3 dB) of the uniform gratings in Fig. 2 obscures a weak birefringence of about ∼5×10⁻⁵ as determined from a π-shifted filament Bragg grating in previous work [39]. However, an anticipated birefringence splitting of Δλ ∼ 50 pm was not spectrally resolvable in the uniform filament gratings as presented in the Fig. 2(a).

 

Fig. 2. Reflection spectra (a) recorded for uniform FBGs of 2000 filaments (E_pulse = 1 µJ, $\Lambda$ = 536 nm) at different centering offsets (ξ_x = 0, 1, 2, 3 µm) show the decay of the reflection strength with increasing offset. The optical microscope image (b) shows a side view of filaments stretching more than 40 µm to fully pass through the fiber core. Top view images show single filament-arrays with offsets ξ_x = -1 µm (c) and ξ_x = +2 µm (d) from the fiber axis.

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Figure 2(b) shows a side view image of a single-row filament grating of 1.072 µm period, generating a near-uniform contrast of straight modification lines that pass undistorted from the cladding to fully through the fiber core. Figures 2(c) and 2(d) present a top view of the fiber for single-row gratings, demonstrating a high-precision control of offsets (±0.5 µm) with intended displacements of ξ_x ≅ -1 µm and ξ_x ≅ +2 µm, respectively. Here, the end-views of the grating filaments appear as a straight line of dots, sharply resolved on the 1.072 µm grating period. However, the physical size of the filament modification cross-section is unresolved and expected to be under 200 nm based on prior studies [41–43]. Moreover, the filaments in first order FBGs were not optically resolved, but were expect to remain isolated on the $\Lambda$ = 0.536 µm grating periods as evidenced by our prior works yielding a strong AC coupling coefficient ($\mathcal{K}$_AC) between ∼0.3 to ∼3 mm^-1 [39,40].

Furthermore, pairs of gratings with different periodicities (Λ_x, Λ_y) and resonant wavelengths were prepared, overlapping in the same fiber core zone but positioned on different planes. The grating planes were either aligned parallel or orthogonal (i.e., Fig. 1) with different offsets of ξ_x1 and ξ_x2, or ξ_x and ξ_y, respectively.

For displacement sensing measurements, the FBG zone of the fiber was locked into a mounting fixture (Fig. 1) to create a cantilever with bending stress concentrated at the FBG by the support. The center Bragg resonance wavelength of the FBG was assessed (Fig. 3(a)) to a wavelength resolution of ±0.5 pm by monitoring the reflection of a broadband light source (Thorlabs, ASE-FL7002 and Ibsen Photonics, DL-BP1-1501A, 1545 to 1560 nm) through an optical circulator to a high-speed interrogator (Ibsen Photonics, I-MON 512 High Speed). Since the source was unpolarized, birefringence responses in the filament FBGs had become blended into a single peak response. The interrogator sampled spectra at a frequency of 17 kHz. The fiber displacement was periodically modulated on a low-frequency (0.33 Hz) square wave profile induced transversely by a solenoid actuator with V-grove mounting head. A duty cycle of 33% enabled unambiguous tracking of the negative or positive wavelength shifts corresponding to compressive and tensile stresses, respectively.

 

Fig. 3. (a) Schematic diagram of the optical characterization arrangement for probing the FBG reflection responses under cantilever actuation. (b) Close up view of the fiber at the V-groove clamp, imaging scattering of red waveguiding light from the filaments of the FBG. (c) Schematic of the fiber cantilever hosting a ℓ = 1 mm long grating located near the support of a L = 30 mm long cantilevered fiber (d) Cross-sectional view of two filaments arrays representing gratings FBG1 and FBG2 offset by ξ₁ and ξ₂, respectively. The fiber was rotated by θ from the horizontal neutral axis. The waveguide core is shown non-concentric with the theoretical central axis. (e) Reflection spectra recorded from a first-order FBG (ξ_x ≅ 2 µm, $\Lambda$ = 536 nm, ℓ = 1 mm) under bent (δ = 2.25 mm) and straight conditions, indicating a stress-induced wavelength shift of the Bragg resonance (inset). Note: Technical drawing of select components in the SolidWorks drawing were acquired from THORLABS, Inc. (http://www.thorlabs.com)

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For azimuthal positioning, θ, the fiber was aligned with a fiber rotator (Thorlabs, HFR007) as shown in Fig. 3(a), prior to clamping in a V-groove fiber holder (Thorlabs, HFF001). This alignment was assisted by launching visible light in the fiber and monitoring the two opposing radiation scattering modes projected orthogonally from the grating plane [40,43,44]. A telemicroscope recorded visible light scattering from filaments in the core waveguide, as shown in Fig. 3(b), permitting a precise azimuthal positioning (θ) of the filament grating plane with respect to the bending axis within ±1° tolerance. FBGs of ℓ = 1 mm length were positioned to the edge of the V-groove support using an axial translation stage (not shown), to within ±50 µm longitudinal precision. Fibers were displaced by up to δ = 2.25 mm on a cantilever length of L = 30 mm (Fig. 3(c)). This configuration provided a maximum displacement response and largest Bragg wavelength shift when the filament array was aligned horizontally (θ = 0°).

Grating offsets of |ξ_x| ≲ 2 µm were selected for generating a sufficiently strong reflectance (> -10 dB) to provide a high signal to noise ratio and sensitivity on the Bragg wavelength shifts. The positioning of the grating planes (± 0.5 µm) relative to the cladding were expected to align the offsets (ξ₁, ξ₂) closely with the neutral axis of the fiber as indicated in Fig. 3(d) due to a homogeneous fiber. However, nonconcentric errors in fiber production (≤ 0.5 µm) were found to shift the fiber core from the theoretical position as shown (Fig. 3(d)), presenting a sensor geometry with slight asymmetric displacement response on azimuthal (θ) fiber rotation.

An example of displacement sensing (Fig. 3(e)) is presented by the ∼18 pm shift in Bragg resonance, observed in the first-order reflection spectra recorded with the tunable laser from an FBG (ξ_x ≅ 2 µm, $\Lambda$ = 536 nm, ℓ = 1 mm) under fiber displacements of δ = 0 and δ = 2.5 mm. The displacement sensing experiments were supported by a 3D opto-mechanical model (ANSYS Workbench) based on finite element analysis (FEA) of a FBG under varying offsets, ξ_x, and ξ_y, azimuth angles, θ, and tip displacements, δ. The model provided the directional stresses in the fiber core, ${\sigma _x}$, ${\sigma _y},\textrm{ and }{\sigma _z}$, and the longitudinal strain, ɛ_z. These values were then applied in Eq. (1) [45,46] to calculate the corresponding wavelength shifts, Δλ_B,

The FEA also provided the fiber displacement locally along the FBG together with the radius of curvature. For example, a tip displacement of δ = 1 mm induced a vertical displacement of 1.648 µm and a radius of curvature of 310 mm at the furthest edge of the FBG positioned ℓ = 1 mm from the support.

3. Results

Figure 4(a) presents an example time history of Bragg resonance wavelength recorded for a cantilever fiber sensor with two parallel gratings, FBG1 and FBG2, fabricated with different Bragg resonances of λ_B1 = 1553.96 nm and λ_B2 = 1549.09 nm, and intended offsets of ξ_x1 ≅ +2 µm and ξ_x2 ≅ -1 µm, respectively (i.e., on opposite sides of the fiber neutral axis as shown in Fig. 4(a) inset). The gratings provided moderately strong (-1.8 and -2 dB) and narrow (∼1 nm at 3 dB) reflection peaks as typically observed (i.e., Fig. 3(e)) for the present laser writing conditions (E_pulse = 1 µJ, $\Lambda$ = 536 nm, ℓ = 1 mm) [39]. A tip displacement of δ = 2.25 mm applied at the θ = 0° azimuth induced a negative Bragg wavelength shift of Δλ_B1 = -23.7 ± 0.5 pm for FBG1 (experiencing compression) and a positive shift of Δλ_B2 = +17.6 ± 0.5 pm for FBG2 (undergoing tension). These values yielded a displacement sensitivity of 9.5 pm/mm and 7.0 pm/mm for FBG1 and FBG2, respectively, which correspond to curvature sensitivities of 2.9 pm/m^-1 and 2.2 pm/m^-1, respectively. Aside from transient ringing (∼100 ms), the Bragg shifts tracked each other with a relative amplitude ratio of -1.35:1, which aligns with the intended offset ratio of ξ_x1/ξ_x2= -2 after accounting for the ±0.5 µm uncertainty in the laser writing positions. Given a high translation precision in the sample motion stages (Aerotech Inc, Aerotech-PlanarDL-00 XY) relative grating offsets of ξ_x1 − ξ_x2 = 3.00 ± 0.05 µm could be precisely assessed, and thus provide a correction to the grating offsets values of $\mathrm{\xi }_{\textrm{x}1}^\textrm{c}$ = +1.70 µm and $\mathrm{\xi }_{\textrm{x}2}^\textrm{c}$= -1.30 µm to account for the observed -1.35:1 ratio on the Bragg wavelength shifts. The corrected offsets are given with respect to the normal axis of the fiber (Fig. 3(d)) and are independent of the nonconcentric positioning expected for the fiber core. Similar positional calibrations were applied in all subsequent data sets.

 

Fig. 4. (a) Temporal response of Bragg resonance wavelengths following two first-order FBGs with parallel and offset filaments in horizontal position (ξ_x1 = +2 µm, ξ_x2 = -1 µm, $\Lambda$ = 536 nm, ℓ = 1 mm, θ = 0°) and driven under square-wave cantilever agitation (δ = 2.25 mm). Compressive and tensile stresses induced on opposite sides of the neutral axis led opposing signs in the respective wavelength shifts, Δλ_B1 and Δλ_B2. (b) Wavelength shifts from the double grating sensor showing the azimuthal cantilever response for a tip displacement of δ = 2.25 mm and sinusoidal representations of the data (red, blue lines).

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With rotation of the fiber, Fig. 4(b) shows the azimuthal response of the double grating sensor for a tip displacement of δ = 2.5 mm observed around a full 2π azimuth. Angles of θ = 0^ο and 360^ο are seen to induce maximum compressive and tensile stresses at the top (FBG1, red) and bottom (FBG2, blue) gratings, respectively, when the filament tracks were horizontal. A rotation to θ = 180° inverted the peak tension and compression states, leading to the out-of-phase maxima in the Bragg wavelength shifts. At θ = 90° or 270°, the compressive and tensile stresses along the top and bottom halves of the filaments were anticipated to cancel out and generate a near zero net shift in the Bragg wavelength. At other angles, the intermediate values of the Bragg shift were observed to follow the sinusoidal representations (blue and red lines). However, several discrepancies arise from mechanical limitations of the present testing apparatus and from imperfections in the fiber manufacturing, explained below.

In Fig. 4(b), the magnitude of the out-of-phase peak at 180° fell below the maxima observed at the 0° and 360° positions by up to ∼29%. The fall was attributed to a residual longitudinal displacement of the FBG position in the V-groove upon fiber rotation, varying the stress focusing condition on the FBG. Another potential source of this error is misalignment in the applied load, which causes a small unintentional torsional moment or a bending moment about the transverse axis. Also, the sinusoidal curve fit for FBG1 (red line) appears shifted by ∼20° with respect to FBG2 fit (blue line). Similar offsets were also reported in [11] and may be understood to arise from imperfect centering of the core waveguide (≤0.5 µm, SMF-28, Corning, NY, USA) relative to the point of intersection of the neutral axes over different azimuths (see Fig. 3(d)). The nonconcentric core thus prevents nulling of the strain response as otherwise expected when displacements are applied on the 90° and 270° azimuths. These responses, only unveiled by overlaying two FBG sensors on opposing sides of the neutral axis (Fig. 3(d) and Fig. 4(b)), attest to the sensitivity of off-axis filament FBGs on fiber specifications, which can in turn be used to detect fiber manufacturing errors such as core-clad concentricity and cladding non-circularity.

To assess the sensing limits for small cantilever displacement, Bragg wavelength shifts were recorded over a 180° azimuth and applied in Δθ = 20° increments while reducing the tip displacements from δ = 2.25 mm to zero in Δδ = 250 µm increments. The results are shown in Fig. 5(a). A single filament array was embedded (E_pulse = 1 µJ, $\Lambda$ = 536 nm, ℓ = 1 mm) into the fiber core with an intended offset of ξ_x ≅ 2 µm. Linear fits (solid lines) of the wavelength shift data provided a good representation of the response. Mechanical limitations of the test apparatus are apparent for tip displacements below δ = 500 µm, corresponding to an estimated radius of curvature of ∼620 mm at the FBG. This limit in detecting a large bending radius compares favorably with limits of 25 mm [36] to 100 mm [34] radii as established in related FBG sensors based on SMF. For tip displacements above δ = 500 µm (i.e., radius < 620 mm), the Bragg shifts observed for tensile (θ = 90° - 180°) and compressive (θ = 0° - 90°) strain are rising nearly symmetrically from the null response condition of vertical filament position (θ = 90°). For a given displacement, the Bragg shifts for each angle are well differentiated on the Δθ = 20° increments, indicating the potential for isolating the azimuthal and displacement responses if a second grating with orthogonal alignment was also embedded for sensing as shown in Fig. 1. Moreover, multiplexing of filament gratings having different offsets at the same fiber location affords wider latitude for improving precision as well as enabling multi-dimensional temperature-compensated displacement sensing.

 

Fig. 5. (a) Bragg wavelength shifts of a first-order FBG (ξ_x ≅ 2 µm, $\Lambda$ = 536 nm, ℓ = 1 mm) for different displacements (δ = 0 to 2.25 mm) and azimuths (θ = 0° to 180°). (b) The displacement response, Δλ_Β/δ, from (a) presented on polar plot (blue dots) for varying rotation angle (θ) and the corresponding simulation data for a grating offset of ξ = 2 (red) and the variance (grey) over grating offsets of ξ = 1.5 to 2.5 µm, which is consistent with the ± 0.5 µm laser writing precision. (c) Bragg wavelength shift of FBGs at different offsets (ξ_x ≅ 0, 1, 2 µm) recorded over a 2π azimuth (green, red, blue dots) for a fixed tip displacement of δ = 2.25 mm and comparison with sinusoidal responses (solid lines) best representing the data at corrected offset values, ξ _sim.

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A displacement response, Δλ_Β/δ, was determined from the slopes of best fit lines in Fig. 5(a) for each rotation angle, θ, and considering only the data above the mechanical response level of δ > 500 µm (non-shaded zone in Fig. 5(a)). The displacement response is presented as a polar plot in Fig. 5(b) (blue dots). The data are seen to fall inside of a modelling zone (grey) for grating offsets varying from ξ = 1.5 to 2.5 µm that is consistent with the ± 0.5 µm positioning error of FBG placement in the core. The azimuthal dependency is sinusoidal-like as noted previously for the fixed displacement cases (δ = 2.25 mm) in Fig. 4(b) and approximately follow the modelling results (red curve in Fig. 5(b)) for the intended grating offset of ξ_x = 2 µm. However, the data are skewed and show a similar asymmetry (-1.23 ratio) in peak responses of +12.3 pm/mm and -9.4 pm/mm for filaments positioned above and below the neutral axis, respectively, for angles oriented at θ = 0° and 180°, respectively. This asymmetry arose from the same artifact of axial displacement as noted above (Fig. 4(b)). Mechanical improvements to the fiber mounting are necessary to ensure that the full stress concentration can be consistently applied over all fiber azimuths and fill out maximal responses matching with the modelling.

The modelling has assumed a concentric core waveguide, thus yielding null response for filaments in vertical orientation (θ = 90°). The non-zero response of +0.5 pm/mm detected with the filaments oriented near vertically (θ = 80 - 100°) arises from a non-concentric core position (Fig. 3(d)) as was noted for the double FBG in Fig. 4(b) for the 90°/180° filament positions. These low-response positions thus could provide a means for calibrating the core concentricity with respect to the neutral axes of the fiber, providing an estimated core off-centering of ∼0.5 µm for the present grating response. With further optomechanical modelling, more precise Bragg shifts may be extracted from the stress fields and resulting refractive index profiles imposed along the filaments. Such modeling may also account for mechanical limitations in components such as the V-groove clamp that together promise to improve on the precision of an azimuthally resolved FBG displacement sensor.

Under the present mechanical limitations of fabrication and sensing precision, the filament gratings have demonstrated a relative strong response to displacements that improves linearly with grating offset position, while also showing a high sensitivity to azimuthal positioning. These merits are highlighted in the polar plot of Fig. 5(c) where full azimuthal angle responses (θ = 0° to 360°) of the Bragg wavelength shifts were recorded for a fixed tip displacement of δ = 2.25 mm in Δθ = 10° increments. Identical FBGs (E_pulse = 1 µJ, $\Lambda$ = 536 nm, ℓ = 1 mm) were formed with a progression of increasing offsets, ξ_x ≅ 0, ξ_x ≅ 1, and ξ_x ≅ 2 µm. The data sets followed the expected sinusoidal-like response, and improving Bragg shift response. The displacement responses were closely matched with simulated data (solid lines) for the ideal cases of a concentric waveguide core with FBG axial positioning at the maximum stressing position on the fiber clamp. However, to account for the variance in peak amplitudes (i.e., observed Δλ_B = 4.4 pm, 14.7 pm, and 32.0 pm for expected shifts of Δλ_B = 0 pm, 12.6 pm, and 25.2 pm, respectively), the grating offsets were corrected to values of ξ_x ≅ 0.3 µm, ξ_x ≅ 1.1 µm, and ξ_x ≅ 2.5 µm. These corrections fell statistically within the fabrication precision (± 0.5 µm) of the intended ξ_x ≅ 0, ξ_x ≅ 1, and ξ_x ≅ 2 µm values, respectively. Also, the FBG wavelength shifts present a high peak-to-min contrast of ∼13, ∼12, and ∼100 for the respective ξ_x ≅ 0, ξ_x ≅ 1, and ξ_x ≅ 2 µm offsets that points to near-concentric positioning of the waveguide core here in contrast with the case presented in Fig. 4 (b).

4. Discussion

In this work, the benefit of off-axis writing of 1 mm long first-order filament gratings ($\Lambda$ = 536 nm) in SMF-28 was studied for displacement sensing. The main objective in translating filament FBGs into the field of displacement sensing was to exploit a unique planar sensing geometry. Unlike other laser fabrication techniques, the filament grating formed within a submicron thick plane to optically probe within a single layer stress-zone in the fiber cross-section. In this way, the FBGs could be precisely offset from the fiber center to scale up displacement responses in SMF and provide strong Bragg reflections that follow the mode field. Furthermore, the planar filament geometry imposed a highly sensitive azimuthal displacement response that would permit both azimuthal and displacement characterization when sensing from orthogonal and overlapping gratings (i.e., Fig. 1 inset).

The pairing of parallel filament gratings (Fig. 4) enabled a precise calibration of the grating positions to ±50 nm, permitting an accurate modeling and calibration of fiber displacement from Bragg wavelength data based on small wavelength shifts from ∼1 to 25 pm. This sensitivity extended to evaluate the small imperfections arising in fiber production, such as core eccentricity. The close sub-micron proximity of the filament pair also naturally compensated for temperature drifts in the displacement sensing (i.e., Fig. 4(a)). Such differential stress measurement may be combined with well-known techniques [1,27,47] to provide a full temperature compensation to extend the operating range of the sensor, and improve sensitivity [47] for applications such as in accelerometers [48].

The cantilever fiber design (Fig. 1 and 3) took advantage of stress concentration at the clamping support to provide a three-fold increase in displacement sensitivity compared with radial bending of the FBG that is more commonly studied [34]. Despite the non-uniform stress profile along the cantilevered fiber, the short length of the FBG probes a near-constant stress. Over the 1 mm length, there was only a 2% stress fall-off as confirmed by the opto-mechanical modelling, falling from 0.4822 MPa at the support to 0.4710 MPa at the FBG end, but decreasing significantly further to 2.2×10⁻⁴ MPa at the cantilever end. Inscribing short gratings with high reflection is a benefit of the filament technique over traditional inscription.

The Bragg resonance wavelength shift monitoring provided more stable and reliable responses compared to other intensity-based measurements techniques [24,25,35] which are susceptible to environmental factors and fiber deformation. Moreover, FBGs in a standard single-mode fiber eliminated the fabrication and optical sensing complications associated with using multimode and multicore fibers [7–11]. The detection and tracking of the single Bragg resonance response of a uniform FBG in a SMF is more practical for scaling up to distributed sensing applications, comparing with limitations from multi-Bragg resonances in other fiber types such as MMFs [49]. In this way, multi-FBGs have been tracked in parallel in a variety of single-mode waveguide configurations [1,27,35,37], such as was used here to calibrate and improve sensing precision. The SMF poses an overall disadvantage of low photoelastic response, limited by the small modal off-setting available from the neutral axis. Chen et al. [37] have pushed gratings further outside of the core to gain moderately higher responses, but resulted in low signal-to-noise due to extraordinarily weak reflection from the exponential mode fall-off. Alternatively, Waltermann et al. [35] arranged FBGs near to the cladding-core waveguide interface and used bending loss to infer fiber curvature, but with lower sensitivity than in the present work. Responses were improved in [27] by core-cladding recoupling, but with significantly higher loss and complexity in signal analysis that precludes a high-speed peak tracking interrogator as used in the present work. More complex fiber sensors are otherwise required to provide both high response and low insertion loss, for example, as shown by Lee et al. [1] in a laser-fabricated fiber circuit with multi-core waveguides, FBGs, and 3-way directional coupler.

The present filament grating fiber sensor may be readily extended to vibration sensing applications such as reported in [26], and provide both amplitude and azimuthal resolution with using multiplexed grating pairs. The further utility of π-shifted filament gratings [39,41] and the potential polarization sensitivity are promising directions to improve on the spectral resolution as well as enable biaxial sensing, but at the expense of more complex polarization control. A further advantage of the filament-by-filament writing process was a high writing speed, yielding a net 2 second exposure for each FBG (1^st order), in contrast with minutes or hours of exposure in the related studies [50,51]. Finally, such filament FBGs may be distributed over short or long fiber lengths and permit distributed sensing through wavelength-division multiplexing. The in-core sensing will permit higher sensor density and spatial acuity in contrast with the optical scattering limits occurring with the intensity-based FBG techniques [24,25].

5. Conclusion

In conclusion, off-axis filament gratings were embedded in the core of SMF for the first time to enable displacement sensing. The unique thin planar geometry of the filament array permitted a high contrast azimuthal response. Multiplexing in paired FBGs improved the precision of the multi-dimensional azimuth-displacement sensing, while also facilitating temperature compensation. This paper provides the groundwork for integrating multifunctional sensor elements in all types of optical fibers, with extension to distributed sensing that enhance portable and biocompatible micro-devices. The present sensor is suited to meet specific requirements in a wide range of application directions for millimeter-scale displacements such as structural health monitoring, material flowmeter, and accelerometer in high temperature and harsh environments.