FBG_SiMul V1.0: Fibre Bragg grating signal simulation tool for finite element method models

FBG SiMul V1.0 is a tool to study and design the implementation of ﬁbre Bragg grating (FBG) sensors into any kind of structure or application. The software removes the need of an ﬁbre optic expert user, becoming more obvious the sensor response of a structural health monitoring solution using FBG sensors. The software uses a modiﬁed T-Matrix method to simulate the FBG reﬂected spectrum based on the stress and strain from a ﬁnite element method model. The article describes the theory and algorithm implementation, followed by an empirical validation.


Introduction
. This opportunity is driven by new low-cost sensors and transducers, 5 new electronics and new manufacturing techniques. In particular, the cost 6 of fibre Bragg grating (FBG) sensors has dropped over the last few years However, the sustainment of structures using these permanent on-board 10 health monitoring systems is a complex and multi-disciplinary technological 11 field that requires a holistic approach that cannot be addressed solely by ad-12 vances in the various technology platforms on which the SHM is constructed. 13 What is required is twofold; that the next generation of research scientists 14 and engineers are specifically trained with the skills, research experience, 15 and multi-disciplinary background to adopt the new structural sustainment 16 concepts. And that tools are available that enable the demanding task of 17 integrating, supporting, and maintaining an innovative holistic health man- 18 agement system and to propel its application in the aerospace, wind energy, 19 and other industries. The FBG SiMul software described here is an example of the type of 22 tool that will allow sensor simulation to become part of the design process, 23 where output is simulated and optimised to a structure. This will have The shape and response of the FBG reflected spectrum (measured signal) 35 depends on the way that the grating is deformed, i.e., the stress and strain 36 field acting along the grating will define the signal response.   Any external force/load acting in the grating region will change the effec-58 tive index and/or the period of modulation, which will create a shift in the 59 wavelength and/or modify the shape of the reflected peak. However, differ-60 ent stress and strain fields acting in the FBG sensor create different signal 61 responses [3, 11, 12, 13, 14] (see figure 1); a longitudinal uniform strain field 62 creates a wavelength shift in the reflected peak (∆λ), but its shape remains 63 unchanged; a longitudinal uniform and non-uniform strain field, acting along 64 the grating, causes an increase in the reflected peak width (∆λ W V ) and a wavelength shift (∆λ); a transverse stress field, acting along the grating, 66 causes a separation of the reflected Bragg peak due to the optical fibre bire-67 fringent behaviour, which can be described by an increase in the reflected 68 peak width (∆λ W V ) and a wavelength shift (∆λ).

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The software is divided between 4 tabs according to functionality:    In a free state, without strain and at a constant temperature, the spectral 286 response of a homogeneous FBG is a single peak centred at wavelength λ b , 287 which can be described by the Bragg condition.
The parameter n ef f is the mean effective refractive index at the location 289 of the grating, Λ 0 is the constant nominal period of the refractive index 290 modulation, and the index 0 denotes unstrained conditions (initial state).

291
The change in the grating period due to a uniform strain field is described 292 in equation (2), where the parameter p e is the photo-elastic coefficient, and ε F BG (x) is the 294 strain variation along the optical fibre direction [7]. The variation of the 295 index of refraction δn ef f of the optical fibre is described by equation (3), where ν is the fringe visibility, φ(x) is the change in the grating period along 297 the length, and δn ef f is the mean induced change in the refractive index [7].

298
By the couple-mode theory, the first order differential equations describ-299 ing the propagation mode through the grating x direction are given by equa-300 tions (5) and (5).
The parameter R(x) and S(x) are the amplitudes of the forward and 302 backward propagation modes, respectively, σ is the self-coupling coefficient 303 as function of the propagation wavelength λ, and κ is the coupling coefficient 304 between the two propagation modes [7,8,9].

305
The self-coupling coefficient σ for a uniform grating (φ(x) = 0) in function 306 of the propagation wavelength λ is described in equation (6), where the 307 parameter λ b is the FBG reflected wavelength in an unstrained state defined by the equation (1).
The coupling coefficient between the two propagation modes κ is defined 310 by equation (7), where the parameter m is the striate visibility that is ≈ 1 311 for the conventional single mode FBG [8,9].
Spectrum reconstruction

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The optical response matrix of the ith (each segment) uniform grating where R(z i ) and S(z i ) are the input light wave travelling in the positive 323 and negative directions, respectively, and R(z i+1 ) and S(z i+1 ) are the output 324 waves in the positive and negative directions, respectively. Thus, the TTM 325 matrix F x i ,x i+1 for each segment (∆x) of the grating can be calculated using 326 the equation (10) and (11).
327 F x i ,x i+1 = S 11 S 12 S 21 S 22 (10) Finally, the grating total response matrix F is obtained by multiplication of each segment response matrix, as described in equation (12).
And, the reflectance of the grating can be described by the equation (13).