Effects of Curvature on Flexible Bragg Grating in Off-Axis Core: Theory and Experiment

A high curvature flexible Bragg grating, in planar technology, is designed, fabricated and experimentally characterized by employing silica glass. The curvature sensing mechanism for slightly off-axis core is theoretically explained and experimentally demonstrated. The electromagnetic behavior of the proposed sensor is investigated through conformal mapping, modal analysis, and coupled-mode theory. The experimental measurement reveals different sensitivities to positive and negative curvatures. The reflection spectrum of the planar Bragg grating sensor and the relationship between its resonant wavelengths and the curvature is measured for a large curvature range up to $1/R = \pm \ 33\,\text{m}^{ - 1}$. The interest toward off-axis core on planar flexible platform for curvature monitoring is due to the lateral stiffness and low Young's modulus of the substrate.

optical performances have been proposed [8], [9], [10]. In particular, the development of flexible photonic structures and devices using ultrathin glasses, which are mechanically flexible, has provided a new opportunity in addition to the use of polymerbased materials [11]. Optical sensors are suitable for curvature measurements because of their key features such as light weight, dimensions, electromagnetic interference immunity, harsh environments operability, nontoxic, and high sensitivity [1], [12], [13]. The maximum curvature depends on the thickness of the glass and the presence of surface defects, which are related to the fabrication process [9].
Multidimensional curvature monitoring is spreading and multicore fibers play an important role in this context [2], [28], [29], [30], [31], [32]. Three non-aligned cores, off-axis with respect to the curvature, each of which comprising a Bragg grating, have been exploited to this aim [33], and a redundant number of cores increases the measurement reliability [29]. The overall performance of the shape change monitoring depends on the strain measurements of each Bragg grating [31]. Therefore, an in-depth study on the sensor response for slightly off-axis core position, i.e., near the "dead-zone" where small curvature strains are induced, is required.
In this paper, an accurate modeling of the curvature effect on the reflection spectrum for a few-mode, off-axis core, flexible silica planar Bragg grating is reported. The obtained results are in agreement with the experiment. A discussion on the longitudinal strain experienced by the Bragg grating explains the measured characteristics. The authors underline that, due to the curvature, the optical mode coupling within the Bragg grating is affected by the shift of the electromagnetic field profile of the propagating modes toward the region with higher refractive index. This aspect, for slightly off-axis core configuration, is essential and has to be considered in writing coupled mode theory (CMT) equations for the modelling of Bragg grating. In particular, a planar Bragg grating inscribed in a slightly off-axis core, which supports few-mode propagation, has been considered and a high curvature sensor has been designed, fabricated, and tested. The ultra-thin planar sensor consists of three doped silica layers, fabricated through flame hydrolysis deposition on a sacrificial silicon wafer. The grating is written in the central layer, making use of direct UV writing. For the first time, to the best of our knowledge, this kind of sensor is exploited for high curvature measurement. The ultra-thin platform allows to reduce the stress at surface and, thus, the possibility to extend the limited curvature measurement range of cylindrical optical fibers [20], [22], [34], [35], [36], without the use of soft materials [1], [37]. The proposed flexible planar sensors, with an off-axis core, also overcomes the limit of cylindrical optical fibers, which require that the orientation with respect to the curvature direction is guaranteed for an efficient one-dimensional curvature sensing.
The paper is organized as follows. Section I reports the introduction; Section II, the theory and simulation approach; Section III, the sensor design, fabrication, and spectrum measurement; Section IV, the comparison between numerical and experimental results; Section V, the prospects, and conclusion.

II. THEORY AND SIMULATION APPROACH
Generally, Bragg grating sensors written in standard optical fibers are practically not sensitive to curvature. In case of non-symmetrical configurations (e.g., off-axis core), the spectral properties of the Bragg grating are affected by curvature more significantly. The curvature induces a stress distribution within an optical waveguide, a variation of its refractive index distribution (due to the stress-optic effects) and a shift of the electromagnetic mode field profile. All these effects cause a change in the interaction between the optical modes coupled by the Bragg grating. The interaction between the propagating modes can be analyzed via the well-known CMT and via conformal mapping, briefly recalled in the following.
By considering the conformal mapping method, a curved optical waveguide, with a cross-sectional refractive index distribution n can be mapped into a straight one. For a large curvature radius R with respect to sensor size, the equivalent straight optical waveguide is characterized by a modified cross-sectional refractive index distribution n s [27]: (1) C = 1/R is the curvature and x proj = rcos(θ) is the projection of each point of the optical waveguide over the plane perpendicular to the neutral plane (see Fig. 1(a)). By convention, we refer to positive (negative) curvature C when the center of curvature O lies in the negative (positive) semi-axis x, see Fig. 1(a). In addition, the modified cross-sectional refractive index distribution n s is affected by strain components ( ε x = ε y = −νε z , ε z = Cx proj , where ν is the Poisson's ratio) along the three principal axes induced by the curvature C. Indeed, these strains, alter the cross-sectional refractive index distribution n which needs to be evaluated by considering the stress-optic relations [27]: (2) where n 0 is the refractive index distribution of the optical waveguide without curvature, p 11 and p 12 the components of the photo-elastic tensor [38]. For silica, p 11 = 0.121, p 12 = 0.27, v = 0.16 [39], [40]. Fig. 1(b) shows the 2D-sketch of the straight optical waveguide, in blue colored the off-axis UV written Bragg grating. In Fig. 1(c), the 2D-sketches of the curved optical waveguide for positive and negative curvature C are reported. In Fig. 1(c), the strain component along the z-axis ε z introduces a compression (yellow colored line) along the inner half with respect to the neutral plane, and tension (blue colored line) along the outer half of the device. The neutral plane (red dotted line) is not affected by strain and the refractive index distribution n does not change. These tensile and compressive strains modify the cross-sectional refractive index n according to the relation (2). The value of the refractive index increases with respect to n 0 in the outer half of the device, while decreases in the inner half [24]. The modified refractive index distribution n s is employed to calculate the electromagnetic fields distributions of the fundamental and high order propagating modes, via 2D-finite element method (2D-FEM). The transversal electromagnetic field distributions are calculated both for the straight and curved waveguide. As a result, the transversal electromagnetic field profile of the propagating modes, within the curved optical waveguide, shifts toward the region with higher refractive index [40]. The CMT can be used to explain the relationship between the spectral response of the Bragg grating and the curvature C. Before solving the coupled mode equations, the transversal coupling coefficients K t (z) are calculated for all the propagating modes travelling in the positive +z and negative −z direction [40]: where w is the angular frequency of the propagating light and Δ (x, y, z) is the dielectric perturbation which is dependent on curvature C, and related to the imposed refractive index modulation [41]; E t (x, y, z) is the transversal electromagnetic field profile depending on curvature C. The CMT longitudinal coupling coefficients K z (z), are two orders of magnitude smaller than transversal ones K t (z), therefore they are neglected. Finally, the CMT equations can be solved.
In addition, if the Bragg grating is located outside the neutral plane, the nominal Bragg grating pitch Λ 0 changes as Λ = Λ 0 (1 + ε z ). Generally, for ε z calculation, x proj corresponds to the distance between the center of the Bragg grating and the neutral plane, defined as core offset d (see Fig. 1(c)) [42], [43], [44]. This approximation is no longer true when the distance d is comparable or smaller than the core size of the optical waveguide. In this case, the shift of the electromagnetic field profile of the propagating modes toward the region with higher refractive index cannot be neglected in the mode coupling calculation [37], [42], [43], [44], [45].
The simulation approach employed for the design is summarized as follows: i) conformal mapping in order to describe the curved waveguide with an equivalent one, ii) 2D-finite element method (2D-FEM) to compute the electromagnetic mode analysis of the equivalent waveguide, iii) calculation of the coupling coefficient K t ± (z) [46], iv) implementation of the CMT as a boundary value problem to evaluate counter-propagating modes powers. The procedure is repeated for each wavelength λ in the range of interest, with a defined wavelength step Δλ, and for each curvature value C. The implemented approach is complete, accurate and applicable to any type of multimode optical waveguide. Approximated investigations consider Gaussian beam with a closed form equation [37] or adopt simplified equations without the implementation of CMT [16], [27], [34], [37].

A. Sensor Design
A design is performed with the aim of investigating the response of a curvature sensor for slightly off-axis core position. The planar technology is chosen to avoid the orientation problem typical of unidirectional curvature sensing with cylindrical optical fiber. The geometrical and physical parameters are feasible with the available technology, consisting of flame hydrolysis deposition and physical machining [47]. In order to make more evident the effect of electromagnetic modes profile shift, a fewmode waveguide is chosen. Fig. 2 reports a not to scale sketch of the designed planar Bragg grating sensor. The core thickness t c = 19.5 μm is larger than the core width w c = 14.5 μm, leading to great electromagnetic modes shift along the x axis, see Fig. 2. The other geometrical parameters are width w p = 1 mm, thickness t p = 57 μm, core offset d = 1.2 μm. The thickness t p is thin enough to reduce surface stresses and thick enough to avoid evanescent field exposure.
The underclad and overclad silica glass regions have identical composition with a refractive index n clad = 1.444 at the wavelength λ = 1570 nm.
Step-index profile is assumed for both the core layer and the waveguide. In particular, the refractive index of the core layer is n core,l = 1.4633 and of the waveguide is n w = 1.4713 at the wavelength λ = 1570 nm. The wavelength dispersion is modelled via proper Sellmeier equation [48]. The effective refractive indices and electromagnetic field profiles of all the propagating modes are simulated via FEM and considered in CMT equations. The effective refractive index of the fundamental mode, simulated via FEM, is n ef f = 1.4712 at the wavelength λ = 1570 nm when the sensor is not subjected to curvature C.
The designed uniform Bragg grating has a sinusoidal modulation with an amplitude Δ n BG = 2 × 10 −4 and a length L = 12 mm. The nominal grating period Λ 0 , when the planar sensor is not subjected to curvature C, is Λ 0 = 0.534 μm. This value is designed to obtain the mode matching between the forward and backward fundamental mode around the wavelength λ = 1570 nm.
As described in Section II, the curvature C determines a shift of the electromagnetic field of the propagating modes towards the waveguide region with higher refractive index. As  an example, Fig. 3 reports the shift of electric field norm E of the fundamental mode for curvature The simulation is carried out in the wavelength range λ from 1565 nm to 1574 nm with a wavelength step Δλ = 5 pm.
The simulated reflection spectrum when the planar Bragg grating is not curved is reported in Fig. 4. The UV written waveguide is few-mode. The spatial modes are classified as E lm modes, where l and m depends on the number of maxima and minima of the electric field distributions along the y axis and x axis of the rectangular core [49].
Four propagating modes interacts with the Bragg grating inducing multiple resonant wavelengths, marked as P1, P2, P3, P4, corresponding to the spatial modes E 11 , E 21 , E 31 , E 32 , respectively. For the sake of simplicity, only one of the almost  degenerate orthogonal polarizations E y lm is reported in the inset of Fig. 4.

B. Sensor Fabrication and Spectrum Measurement
Flame Hydrolysis Deposition (FHD) is made upon a sacrificial p-doped [1 0 0] silicon wafer with diameter d Si = 152 mm. Three doped silica layers: overclad, core and underclad are deposited and the silicon is later removed through physical machining, leaving a flexible glass substrate with a nominal thickness t p = 57 μm. Fig. 5 shows an image of the crosssection of the planar sensor captured through a CCD camera.
The overclad, core and underclad are clearly distinguishable. Underclad does not possess a constant thickness because of the removal of the silicon wafer through physical machining.
Direct UV writing is employed to inscribe the few-mode waveguide and the uniform Bragg grating in the core layer.
The planar Bragg grating reflection spectrum, measured through optical spectrum analyzer, is reported in Fig. 6; the planar sensor is butt-coupled to a cylindrical optical fiber through UV curable adhesive, as shown in Fig. 7. Four resonant wavelengths are apparent in Fig. 6 due to the few guided modes. The different reflected powers depend mainly on i) the mode power distribution in the few-mode waveguide; ii) the corresponding coupling coefficients K t ± (z). Furthermore, an unwanted Bragg grating chirping is evident, induced by UV writing. The measured resonant peaks are in good agreement with the simulated ones.

IV. CURVATURE EFFECT
Different values of curvature C are considered taking into account positive and negative curvature directions. For comparison between simulation and experiment, the considered curvature values are: In order to characterize the sensor response to curvature C, it is important to vanish the influence of crosssensitivities induced by temperature and/or spurious strain not directly related to curvature C. To this aim, the measurements are carried out in a temperature-controlled environment. A mount and a motorized stage are employed to perform curvature C measurement and particular attention is paid to avoid that the clamping could affect the planar Bragg grating spectral characteristics. Furthermore, the use of mount and motorized stage suppresses any unintentional twisting effect.
In addition, to compensate any effects of temperature, the Bragg wavelength shift Δλ B is measured as the difference between the resonance peak when the sensor is subjected to curvature C and the resonance peak when the sensor is not curved. To perform high resolution measurement the Bayspec FBGA optical spectrometer is employed instead of the optical spectrum analyzer. The spectrometer guarantees a minimum detectable wavelength change r = ±1 pm and allows the detection of peaks P3 and P4. A sketch of the adopted experimental set-up is reported in Fig. 8.
The measured Bragg wavelength shift Δλ B (black squares) as a function of curvature C are reported in Fig. 9 for peak P3 and in Fig. 10    In particular, for positive values of curvature C, the electromagnetic field shifts toward the outer half of the planar Bragg grating sensor. It is mainly subjected to tensile strain z , which linearly increases with curvature C. The linearized sensitivity for positive values of curvature C is S + P 3 = 5.9 pm/m −1 for the peak P3 and S + P 4 = 5.2 pm/m −1 for the peak P4. The reason for the dissimilar values of linearized sensitivity S + is due to the difference in the effective refractive indices between the straight and curved planar Bragg grating sensor for each mode. The difference in the effective refractive indices for peak P3 is greater than for peak P4, causing the Bragg wavelength shift of the peak P3 to be larger than that of the peak P4. This is directly influenced by the mode profile being considered. Decreasing the value of curvature C, the electromagnetic field shifts in the opposite direction toward the centre of the planar sensor. In particular, when the curvature C is slightly negative, the electromagnetic field is affected by the compressive strain z effect. By further decreasing the curvature C, the electromagnetic field intercepts the neutral plane, where no strains along the principal axes occur, and the sensitivity becomes zero, around C = −23 m −1 . As the curvature C becomes more negative, the electromagnetic field marginally extends beyond the neutral plane and tensile strain z is experienced. Therefore, the combined effect of cross-sectional refractive index change and tensile strain z on the grating period Λ, leads to a reversal of the slope. To summarize, for positive values of curvature C, the electromagnetic field shifts progressively toward the outer half of the device, where increasingly higher tensile strain z occurs; for negative values of curvature C, the electromagnetic field shifts from the inner half, where the compressive strain z occurs, toward the outer half of the device, where there is a tensile strain z , see Fig. 1(c).

V. CONCLUSION
For the first time, a high curvature sensor based on ultrathin flexible silica planar Bragg grating is proposed, designed, and experimentally characterized. The implementation of CMTbased code explains the resonant wavelengths shift for different curvatures. The experimental results agree with simulation and are reported for high curvature up to ±33 m −1 , showing an excellent mechanical robustness also with respect to the optical fiber solution. The device exhibits different sensitivities when positive and negative curvatures are applied. This is theoretically explained with a discussion on the longitudinal strain to which the Bragg grating is subjected. This concept finds applications also in curvature direction and radius sensing in multicore optical fibers. Future sensors and reconstruction algorithm can be engineered taking into account the demonstrated principle, leading to a better shape reconstruction. Lastly, the possibility to customize the geometry and the waveguides position, the absence of orientation problems typical of cylindrical optical fiber makes the adopted planar photonic platform attractive for new generation photonic sensing devices.