Flexible Photonic Sensors: Investigation of an Approach Based on Ratiometric Power in Few-Mode Waveguides for Bending Measurement

A novel approach to monitor the degree of bending via flexible photonics devices, using ratiometric power variation in a few-mode optical waveguide, is proposed. To demonstrate its feasibility, a sensor exploiting a Bragg grating, approximately aligned to the neutral axis, is designed, fabricated and characterized. The reduced thickness of the proposed planar photonic sensor is uniquely limited by optical confinement requirements, enabling for an ultra-thin and highly flexible planar device, based on all-glass platform. Finite Element Method, Beam Propagation Method and Coupled Mode Theory are employed in the design to model the electromagnetic and mechanical phenomena occurring during the three-point bending test. The experiment demonstrates that the planar device withstands tight curvature without mechanical failure. The device shows that by increasing the bending the reflected power from the fundamental mode decreases and the reflected power from the higher order mode increases. The measured ratiometric power sensitivity versus displacement is $K_{P_{R}}=-0.78 \mathrm {dB/mm}$ with negligible variation over a 40°C thermal range. Therefore, exploiting both the Bragg wavelength shift and the mode optical power change, the proposed sensor can be employed for multiparameter sensing purposes, e.g. simultaneous temperature and curvature monitoring.


I. INTRODUCTION
Flexible photonics is a rapidly advancing research field.Traditional rigid devices in planar optics can be refined The associate editor coordinating the review of this manuscript and approving it for publication was S. M. Abdur Razzak .through controlled thinning, adding the ability to flex while preserving their essential functionality.This innovative class of photonic devices has been successfully fabricated on a number of substrates, as extensively documented in various recent articles [1], [2], [3], [4], [5], [6], [7].Specifically, polymer waveguides demonstrate remarkable flexibility, enabling significant levels of curvature, twisting, and stretching [6], [8].However, it is worth noting that the refractive index contrast may not consistently meet the demands for robust optical field confinement and that polymers can exhibit poor performance when subjected to harsh environments such as elevated temperatures or exposure to solvents [6].
Recent studies have also explored the integration of glass onto polymer structures [9], [10], aligning with the growing focus on flexible photonics based on all-glass substrates and components [11], [12], [13].Glass substrates excel for their optical performance, exceptional chemical properties, thermal resilience, and superior refractive index contrast compared to polymer waveguides [12], [13].The higher elastic moduli exhibited by glass materials, as opposed to plastics and polymers, contribute to their inherent rigidity in a solid state.The bendability of all glass photonic devices significantly depends on the thickness of the material.This is the main factor affecting the ability of glass to adapt to a range of curvature configurations.Indeed, for materials that have a low tensile strength, such as doped silica, reducing thickness can also increase the total degree of flexure prior to failure.
For optical-based bend sensors, the overall thickness of the device typically arises from the established practice, wherein the waveguide is traditionally situated apart from the neutral axis.Here, a minimum thickness becomes necessary to establish the offset from the neutral axis and ensure optical modes confinement.If the neutral axis is aligned or closely aligned with the core, the sensor tends to exhibit minimal or no detectable response.
In this paper, a novel approach for curvature monitoring via all-glass flexible sensors, using ratiometric power variation between the fundamental and higher order mode in a bimodal optical waveguide, is proposed.As proof of concept, a flexible photonic bend sensor (see Fig. 1), is designed, fabricated through Flame Hydrolysis Deposition (FHD) and characterized via three-point bending test [18].Notably, the channel waveguide is deliberately aligned slightly off-axis, a departure from the conventional approach for bend sensors.The optical characteristic of the Bragg grating is comprehensively examined both theoretically and experimentally, including analyses of Bragg wavelength shifts and alterations in optical mode power, particularly under the influence of pure bending.The designed sensor has a reduced thickness that minimizes the internal strain values, allowing a strong extension of the typical sensing range of cylindrical optical fiber, up to 110 m −1 without the use of low stiffness materials [19], [20].The obtained results show that this novel approach could be exploited for the development of innovative optical sensors, allowing the simultaneous temperature and curvature monitoring, or, more in general, multiparameter monitoring.Progress in this domain is expected to influence fields including soft robotics, medicine, and the monitoring of engineered structures [4], [15], [16], [17], [21], [22].
After the Section I, the paper is organized as follows: Section II illustrates the electromagnetic design and mechanical performance investigation of the sensor; Section III describes the sensor fabrication and the comparison between numerical and experimental results; Section IV provides the conclusions and the discussion on the future prospects.

II. ELECTROMAGNETIC AND MECHANICAL MODELING A. DESIGN APPROACH
The design of the flexible photonic sensor is developed via 2D-Finite Element Method (FEM) electromagnetic modal analysis, using the COMSOL Multiphysics ® Wave Optics module.Then, the flexible photonic sensor is modeled via 3D-FEM, using the COMSOL Multiphysics ® Structural Mechanics module, to simulate the three-point bending test.The optical response of the Bragg grating, under threepoint bending, is investigated through electromagnetic and mechanical analyses.The mechanical analysis allows to determine the maximum displacement, the curvature profile, and the strain distribution.The calculated curvature profile is used in the 3D-BPM modeling (Beam Propagation Method, BeamProp ® -RSOFT Design ® ) to assess power mode mixing in addition to the Coupled Mode Theory (CMT).It is well known that in multimode waveguides, bending induces mode coupling and optical power exchange amongst other phenomena.The strain distribution, the electromagnetic field profile, and the powers of the propagating modes are employed as inputs to an in-house MATLAB ® code based on CMT [23].This combines the influence of the bending on refractive index distribution, grating period, and coupling coefficients.CMT is exploited to compute the reflection spectra and the Bragg wavelength shift of the gratings [14].
Three Bragg gratings, namely #G1, #G2, and #G3, are designed following the aforesaid approach.An exhaustive electromagnetic investigation is focused on the Bragg grating #G2, constituting the flexible sensor, which is subjected to an almost constant curvature along the entire grating length.The numerical results agree with the experiment and explain the optical response of both #G2 and the other two Bragg gratings #G1 and #G3 designed and fabricated only to better explain the sensor behavior.

B. ELECTROMAGNETIC DESIGN AND MODAL INVESTIGATION
Generally, fiber Bragg gratings can be adopted to infer the level of bending by exploiting an off-axis core and observing the Bragg wavelength shift.To investigate the feasibility of the sensor for the monitoring of two parameters, e.g. the curvature and the temperature, a channel waveguide with two propagating modes is designed.The electromagnetic design of the flexible photonic sensor with a slight off-axis core is carried out.Figure 2 shows a not-to-scale sketch of the flexible sensor cross-section.The total thickness is t fp = 60 µm and the total width is w fp = 1 mm.The thickness t fp is chosen as a trade-off, to provide both a low level of surface stress and a suitable electromagnetic field confinement when the sensor is bent.The sensor is multi-layered, comprising overclad, core, and underclad.The underclad thickness is slightly larger than the overclad so that the core layer is 1.5 µm offset from the central neutral axis of the sensor.The channel waveguide is rectangular, and it is written within the core layer.Its width is w wg = 7 µm and its thickness is identical to the one of the core layer, t co = t wg = 9 µm.The underclad and overclad layers are made of identical silica glass composition and uniform refractive index distribution n cl = 1.4452 at the wavelength λ = 1553 nm.The core layer refractive index is n co = 1.4645 at the wavelength λ = 1553 nm.At the same wavelength, the channel waveguide refractive index is n wg = 1.4695, obtainable by adopting direct UV Writing (DUW) manufacturing technique.Both the core layer and the channel waveguide have a step-index profile distribution.The refractive index wavelength dispersion is modeled via a suitable Sellmeier equation [24].The norm of the electric field E of the fundamental and high (second) order mode, i.e. the spatial modes E 11 , E 21 , is reported in Fig. 3.

C. MECHANICAL INVESTIGATION: THREE-POINT BENDING TEST
To numerically investigate the sensor response to the three-point bending test, the multi-layered glass platform described in Section II-B and having a length L fp = 60 mm, see Fig. 4 (a), is 3D-modeled.The employed mechanical parameters, such as Poisson's ratio v = 0.17 and Young's modulus E cl , have been measured through nano-indentation tests [21].The underclad and overclad compositions are identical, with Young's modulus E cl = 40 GPa.The core layer is stiffer than the cladding layers and its Young's modulus is E co = 62 GPa.The boundary conditions and the applied forces are set in the 3D-FEM model as reported in the sketch of Fig. 4   boundary of the flexible photonic sensor, while, on the right, a simply supported boundary condition is imposed.
The force F acts perpendicularly to the longitudinal axis (z-axis), at the centre of the Bragg grating #G2.A free tetrahedral mesh is employed, consisting of M 3D−FEM = 168442 domain elements.Three-point bending test is simulated by considering the force values F i = [0 mN, 3.3 mN, 6.6 mN, 9.8 mN, 12 mN] with i = 1 → 5.
The simulated maxima displacements d sim , for each force value F i , are respectively d sim = [0 mm, 3 mm, 6 mm, 9 mm, 11 mm].Figure 5 plots the simulated displacement of the flexible photonic sensor versus its length L fp due to the force F 5 , i.e. the maximum applied force value; the convention used to define the curvature sign is indicated in Figure 5.The displacement d sim obtained from the 3D-FEM model (black solid curve) perfectly agrees with that obtained by classical beam theory (red dotted markers) [25].The 3D-FEM approach allows a multiphysics investigation including the internal strain calculation.Figure 6

D. POWER MODE MIXING
The mode optical power exchange between the propagating modes in a bent optical fiber/waveguide is analysed via 3D-BPM, which exploits the conformal mapping method [26].The conformal mapping is used to map the cross-sectional refractive index n c of a curved optical waveguide, into a straight one.If the size of the flexible photonic sensor is very small compared to the applied curvature radius C R , a straight equivalent waveguide, with a novel cross-sectional refractive index n m = n c 1 + x p /C R , can be modeled [26], [27].The distance from the optical waveguide centre is indicated with x p , while n c is the cross-sectional refractive index when the sensor is unperturbed, i.e. not bent.3D-BPM approach does not consider the effect caused by the strain field on the refractive index n c .This further effect is taken into account in Section II-E for the evaluation of the Bragg grating spectra, using stress-optic relations [28].For each value of d sim , the curvature radius C R versus the cumulative arc length S is exploited by 3D-BPM to calculate the cross-sectional refractive index n m and to assess the power mode mixing.To simulate the butt coupling with a single mode optical fiber, a Gaussian beam profile is considered as an input optical field.
The grid resolution of the simulation is M BPM ,t = 0.1 µm and M BPM ,l = 0.25 µm respectively for transverse (x-y axis) and longitudinal direction (z axis).By considering the maximum displacement d sim = 11 mm, the power mode mixing investigation for the grating #G2 demonstrates (as it will be shown in Section II-E) that the simulated peak power P s,hom of the high order mode increases of P s,hom = 3.4 dBm at the expense of the peak power P s,fun of the fundamental mode.

E. BRAGG GRATING SPECTRUM EVALUATION
The 2D-FEM modal analysis is carried out by using the stress-optic relations and the strain distribution within the flexible photonic sensor, derived from the 3D-FEM mechanical investigation.To solve CMT, the transversal coupling coefficients K t (z) are calculated considering the effective refractive indices n eff and the electromagnetic field profiles E of the two modes propagating along the z axis (along both positive and negative directions) [23]: where w f is the angular frequency, E t (x, y, z) is the transversal electromagnetic field profile, ϵ (x, y, z) is the dielectric perturbation which relies on refractive index modulation.The 3D-BPM is employed to calculate the input power of the two guided modes in CMT model [23].To compute the reflection spectrum of grating #G2, the approach is executed for each displacement d sim ; the wavelength range is from λ = 1514 nm to λ = 1520 nm with a wavelength step λ step = 5pm.The effective refractive index of the fundamental mode is n eff ,fun = 1.46618 and that of the second order mode is n eff ,hom = 1.46291 at the wavelength λ = 1553 nm.The designed uniform Bragg grating has a sinusoidal modulation with an amplitude n BG = 2×10 −4 and a length L = 12 mm.The nominal grating period of #G2, when the planar sensor is not subjected to curvature, is #G2 = 0.51774 µm.This value is selected to obtain the mode matching between the forward and backward fundamental mode around the wavelength λ = 1519 nm.For completeness, the grating period of Bragg grating #G1, and #G3 were selected to obtain the resonant wavelengths of the fundamental mode at λ #G1 = 1528 nm and λ #G3 = 1588 nm.
Figure 7 reports the simulated reflection spectrum when grating #G2 is unperturbed and when a displacement d sim = 11 mm is applied, respectively.The propagating modes interact with the Bragg grating #G2 producing two resonant wavelengths.Moreover, Fig. 7 shows that the applied bending produces i) a power exchange between the two modes and ii) a blue shift of the resonant wavelengths.The results demonstrate a variation of the Bragg wavelength for both the fundamental and high order mode, reaching the maximum Bragg wavelength shift λ s,fun = −250 pm and λ s,hom = −240 pm, respectively.

III. FABRICATION AND CHARACTERIZATION A. FLEXIBLE PHOTONIC SENSOR FABRICATION
The flexible glass substrate is fabricated using FHD.This process involves the sequential deposition of three doped silica layers (overclad, core and underclad), onto a rigid sacrificial silicon wafer (diameter d sw = 152 mm, thickness t sw = 1 mm, p-doped [100]).The silicon had a thermally grown wet oxide layer, of thickness t wo = 6 µm, used to avoid chemical reaction between the deposited doped silica layers and the silicon substrate.The use of equal compositions for the underclad and overclad, balances the stress differentials on both the sides of the flexible substrate and reduces intrinsic bend when released from the sacrificial substrate.The underclad and overclad layers are processed with flow rates of SiCl 4 at 137 sccm, BCl 3 at 70 sccm and PCl 3 at 31 sccm through a hydrogen-oxygen flame with flow rate 6.5 L/min and 1.5 L/min, respectively.The fabrication of the core layer is made with flow rates of SiCl 4 at 123 sccm, GeCl 4 at 130 sccm, BCl 3 at 16 sccm through a hydrogen-oxygen flame with flow rate 5.4 L/min and 2.7 L/min.After each FHD layer, the deposited soot requires high temperature consolidation.This was achieved within a furnace, flowing O 2 at rates of 1.9 L/min and consolidating at temperatures T co = 1360 • C and T cl = 1250 • C, for the core layer and cladding layers respectively.The final processing step, to create the flexible photonic platform, is obtained by removing the rigid silicon substrate through a physical machining process [18].A key feature of the design is that the core layer is not centred on the neutral axis, but it is offset.
The measured core layer thickness t co,meas differs slightly with respect to the desired t co , due to dopant diffusion induced by the high temperature process involved.Figure 8 (b) shows the flexible photonic sensor coupled via UV-cured optical adhesive with a single mode polarization maintaining (PM) optical fiber.The refractive indices are measured, through a prism coupler, at the wavelength λ = 1553 nm.The underclad and overclad layers have a measured refractive index n cl = 1.445, while the core layer has a measured refractive index n co = 1.469.

B. BRAGG GRATINGS INSCRIPTION
The DUW technique is able to simultaneously inscribe the channel waveguide and the three Bragg gratings within the UV photosensitive core layer [29].The Bragg gratings are 29170 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
written at different positions, along the length of the device, as shown in Fig. 4 (b).They have a length L = 12 mm and a uniform apodization profile, refractive index modulation of the Bragg grating n BG = 2 × 10 −4 .Moreover, they are pseudo-randomly ordered, such that the spectral sequence does not correlate with spatial sequence.The measured resonant wavelengths of the fundamental mode are λ #G1 = 1528 nm, λ #G2 = 1519 nm and λ #G3 = 1588 nm.The reflected power versus the wavelength for the three unperturbed Bragg gratings is shown in Fig. 9.According to the electromagnetic design, the waveguide supports two modes of propagation, since two resonant peaks for each grating are evident.

C. THREE-POINT BENDING TEST AND OPTICAL MEASUREMENT
The flexible photonic sensor was loaded in a mechanical test rig to observe the optical response when subjected to large deflections.An Instron E1000 electromechanical test machine is used to apply three-point bending.The fixture comprises three steel rollers having a diameter d roll = 6 mm, each separated by a distance s = 20 mm.The upper roller is used to apply the displacement and was attached to the actuator as shown in Fig. 10.The side which is optically coupled to the fiber (left-hand side in Fig. 10) was secured using a tape to limit influence from optical fiber movement.
The flexible photonic sensor was subjected to an incremental displacement d, from d = 0 mm (i.e.unperturbed) to d = 11 mm.The optical measurements were carried out during the three-point bending test, using the set-up shown in Fig. 11.A negative Bragg wavelength shift in the pm range was  obtained [14].Grating #G2 is positioned at the loading point where the displacement and the strains are the largest and it is subjected to a curvature with negative concavity (see Fig. 5).Therefore, compared with the other gratings, it produces the greatest wavelength shifts.The maximum measured Bragg wavelength shift λ m , for the fundamental and high order mode, are λ m,fun = −264 pm and λ m,hom = −223.5pm, respectively.There is an increase of the measured peak power P m,hom = +4 dBm for the high order mode, for a displacement d = 11 mm.The optical measurements of all the gratings optical responses, for increasing displacement d = [0 − 11] mm of the central roller, are reported in Fig. 12 and Fig. 13.
Figures 12 (a) and (b) illustrate the measured Bragg wavelength shifts λ m , of the fundamental mode (triangle markers) and high order mode (cross markers) for the gratings #G1 and #G3.Figures 12 (c) and (d) report the comparison between the simulated (dotted line) and measured Bragg wavelength shifts λ, of the fundamental mode (triangle markers) and high order mode (cross markers), of the grating #G2.As previously underlined, the grating #G2 constitutes the sensor, the gratings #G1 and #G3 are considered for a critical discussion of the obtained results.It is evident that the numerical results perfectly agree with the experimental measurements.By analysing the wavelength shifts reported in Fig. 12, all the gratings and both the propagating modes, show a shift of the resonant peaks towards smaller wavelengths.Since grating #G2 lies along  the region of maximum and constant negative curvature, the Bragg wavelength shift λ m is negative with a linear trend that is larger than that of the gratings #G1 and #G3.Regarding the grating #G1, it is positioned in a region where two curvatures with opposite signs are present.This provides an explanation for the wavelength shift reaching λ m,fun = −38 pm, λ m,hom = −20.6 pm at the maximum central displacement d = 11 mm.The right end of the flexible photonic sensor is unbonded, thus also the grating #G3 is subjected to a negative curvature.In this case, the displacement and the strains are lower than that occurring at the loading point, where the grating #G2 is located.The maximum measured Bragg wavelength shift for the grating #G3 is λ m,fun = −95.2pm, λ m,hom = −112 pm.Ultimately, the grating #G2, subjected to curvature, provides Bragg wavelength shift sensitivities K λ,fun = −2.4pm/m −1 , K λ,hom = −2.1 pm/m −1 .
Figure 13 illustrates the measured peak power change P m,hom for the high order mode of the grating #G1 (triangle markers), #G2 (cross markers) and #G3 (circle markers) as a function of the displacement d.The simulated peak power change P s,hom of the high order mode of the grating #G2 (dotted line) is in good agreement with the measured value.The power trend of the high order mode of gratings #G1 and #G3 (symmetric about the loading point) is comparable but with opposite changes in power.Both the gratings reach the maximum peak power change P m,hom at a displacement of d = 6 mm.
The power exchange between the two propagating modes can be exploited to infer the level of bending of the grating #G2, by using the ratiometric power P R .Where P R is defined as the ratio between the peak power reflected by the fundamental mode P m,fun and the high order mode P m,hom .In Fig. 14

D. DISCUSSION OF RESULTS
The focus of this paper is the demonstration of a novel approach to infer level of bending, based on ratiometric power change, between the peak power reflected by fundamental and high order mode.Table 1 reports a comparison with other Bragg curvature sensors which exhibit very high performances and include different approaches: multimode interference [30], embedding in silicone substrate [31], polymeric optical fibers with eccentric core [32], multicore optical fibers [33], or tilted FBGs [34].In particular, multimode interference [30] requires a more sophisticated measurement approach, with excitation of a number of modes; embedded in silicone sensor [31] exhibits higher thickness; polymeric optical fibers [32] show higher confinement losses due to lower refractive index changes; [33], [34] show high sensitivities but smaller measurement range.The proposed proof of concept shows the following strength points: i) the minimum thickness, ii) the maximum curvature, iii) exploitation of the ratiometric power, which is immune to temperature variation, iv) immunity to power fluctuation of light source, due to the intrinsic compensation of the measurement technique, v) simultaneous measurement of bending and temperature.These results encourage further work towards the refinement in design/fabrication of other devices based on the same techniques.

IV. CONCLUSION
A new approach for bending monitoring via flexible photonics is proposed by considering both the ratiometric power between the peak power reflected by fundamental and higher order mode and Bragg wavelength shift.A sensor is fabricated to validate the approach.In particular, a comprehensive optomechanical design and characterization of the flexible photonic sensor is reported.Compared to other fiber optic curvature sensors, the proposed solution minimizes the total thickness of the substrate, enabling the achievement of higher degree of bending.The results indicate that the wavelength shift depends on the sign of applied curvature.The experiments agree with simulations and show that the proposed flexible sensor can withstand very high curvatures up to 110m −1 .The grating #G2 provides Bragg wavelength shift sensitivities K λ,fun = −2.4pm/m −1 , K λ,hom = −2.1 pm/m −1 and ratiometric power sensitivity K P R = −0.78dB/mm.Regarding thermal characterization of the sensor, negligible variation over a 40 • C range has been observed in terms of power change.The thermo-optic sensitivity is approximately K T = 11.5 pm/K.The values of Bragg wavelength shift, and temperature-independent ratiometric optical power change can be exploited for simultaneous temperature and curvature sensing, paving the way for the construction of a new class of multiparameter sensors.

FIGURE 1 .
FIGURE 1. Flexible planar photonic chip in doped silica fabricated through flame hydrolysis deposition.

FIGURE 2 .
FIGURE 2. Sketch of the flexible sensor.The channel waveguide (light gray colored), in which the gratings are inscribed, is written in the central core layer and it is slightly off-axis with respect to the neutral axis.

FIGURE 3 .
FIGURE 3. Norm of the electric field E of the fundamental E 11 and high order mode E 21 .The coordinates (0; 0) refer to the center of the waveguide.

FIGURE 4 .
FIGURE 4. (a) Sketch of the boundary conditions and applied forces to the 3D-FEM modeled flexible photonic sensor for three-point bending test simulation; (b) schematic representation of the set-up and Bragg gratings locations.
(a), (b).Moreover, Fig. 4 (b) is a schematic of the set-up, highlighting the position of the three Bragg gratings, namely #G1, #G2, #G3, and the position where the force F is applied.A fixed constraint condition is applied on the left

FIGURE 5 .
FIGURE 5. Displacement profile as a function of the flexible photonic sensor length L fp considering the applied force F 5 .
illustrates the curvature radius C R as a function of the cumulative arc length S, for the case of maximum displacement d sim = 11 mm.The curvature radius is evaluated by considering the curvature equation C R = | 1 + ẏ2 3 2 /ÿ| where ẏ and ÿ are the first and second order derivative of the displacement profile function.The values of the curvature radius C R , are then used for 3D-BPM modeling.

FIGURE 6 .
FIGURE 6. Curvature radius C R vs cumulative arch length S considering the applied force F 5 .

FIGURE 8 .
FIGURE 8. (a) Cross-section of the flexible photonic sensor captured via microscope camera; (b) flexible photonic sensor coupled to a polarization maintaining optical fiber via UV-cured optical adhesive.

Figure 8 (
Figure8(a) shows the cross-section of the fabricated device, captured via microscope camera.The flexible photonic sensor has a measured width of w fp,meas ∼ 1 mm, thickness t fp,meas ∼ 58 µm and length L fp,meas ∼ 60 mm.The thickness of the underclad is greater than that of the overclad.A key feature of the design is that the core layer is not centred on the neutral axis, but it is offset.The measured core layer thickness t co,meas differs slightly with respect to the desired t co , due to dopant diffusion induced by the high temperature process involved.Figure8 (b)shows the flexible photonic sensor coupled via UV-cured optical adhesive with a single mode polarization maintaining (PM) optical fiber.The refractive indices are measured, through a prism coupler, at the wavelength λ = 1553 nm.The underclad and overclad layers have a measured refractive index n cl = 1.445, while the core layer has a measured refractive index n co = 1.469.

FIGURE 10 .
FIGURE 10.Photograph of the experimental set-up.Centre-to-centre separation of rollers s= 20mm, diameter of rollers d roll = 6mm.

FIGURE 11 .
FIGURE 11.Experimental set-up employed for spectra measurements.

FIGURE 12 .
FIGURE 12. Measured Bragg wavelength shifts λ m , of the fundamental mode (triangle markers), high order mode (cross markers) for (a) the gratings #G1 and (b) #G3.(c) Comparison between the simulated (dotted curve) and measured (triangle markers) Bragg wavelength shift λ fun of the fundamental mode, (d) comparison between the simulated (dotted curve) and measured (cross markers) Bragg wavelength shift λ hom of the high order mode, for the grating #G2.

FIGURE 13 .
FIGURE 13.Measured peak power change P m,hom versus the displacement d for the high order mode of the grating #G1 (triangle markers), #G2 (cross markers), #G3 (circle markers) and simulated peak power change P s,hom for the grating #G2 (dotted curve).
the measured ratiometric power P R of the grating #G2 versus the displacement d (cross markers) is reported.The red dashed line is the linear fit of the experimental curve.The sensor exhibits a linear response between d = 2 mm and d = 10 mm with a ratiometric power sensitivity being K P R = −0.78dB/mm.The almost flat response between d = 0 mm and d = 2 mm, can be addressed to an unwanted initial deflection of the sensor.It is worth noting that, over a 40 • C thermal range, power change shows negligible variation, thus it is not affected by environment temperature.The measured thermo-optic Bragg shift sensitivity is approximately K T = 11.5 pm/K.

FIGURE 14 .
FIGURE 14. Ratiometric power P R versus the displacement d for the grating #G2 (cross markers).The red dashed line is the linear fit of the curve.

TABLE 1 .
State-of-art comparison between flexible sensors.