## Abstract

An inverse approach based on an optimization technique is proposed to characterize a fiber Bragg grating (FBG)
and the strain gauge factor (GF) when the FBG is bonded on a structure. By bonding an FBG on a substrate and simply straining this FBG into a chirped fiber Bragg grating with a predesignated strain, the proposed method, based on an optimization technique, can be used to reconstruct seven parameters of the FBG from the corresponding reflective spectrum. The parameters identified are the length of an FBG, the grating period, the average refractive index, the index modulation, the apodization coefficient, the starting point bonded on the plate, and the strain GF. The information from the predesignated strain, as well as the measured reflective spectrum, is used as the objective function during the optimal search. As a result, the design sensitivity for the optimal search is much improved compared with the design sensitivity when only the reflective spectrum is used. In particular, the strain GF, which depends on the adhesive, the bonding layer characteristics, etc., can be determined in order to provide a reference for an FBG used as a strain sensor. Results from numerical simulations and experiments show that seven parameters of an FBG can be obtained accurately and efficiently.

© 2007 Optical Society of America

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### Equations (7)

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(1)
$$\lambda \left(z\right)=2\overline{n}\left(z\right){\Lambda}_{o}\left(z\right)\left[1+{\epsilon}^{opt}\left(z\right)\right]\text{,}$$
(2)
$${\epsilon}^{opt}\left(z\right)=\left(1-\xi \right)\epsilon \left(z\right)+\frac{\delta {n}^{perm}\left(z\right)}{\overline{n}}\text{,}$$
(3)
$$\underset{\lambda \left(z=0\right)}{\overset{\lambda \left(z\right)}{\int}}\text{ln}\left[\text{1}-R\left(\lambda \prime \right)\right]\mathrm{d}\lambda \prime =\mp \frac{{\pi}^{2}}{2}}{\displaystyle \underset{0}{\overset{z}{\int}}\frac{\Delta {n}^{2}\left(z\prime \right)}{\overline{n}\left(z\prime \right)}}\text{\hspace{0.17em}}\mathrm{d}z\prime \text{,$$
(4)
$$\frac{\Delta \lambda}{\lambda}=G\left(1-\frac{{n}^{2}}{2}\left({p}_{12}+{p}_{11}\text{\hspace{0.17em}}\frac{{\epsilon}_{rr}}{{\epsilon}_{zz}}+{p}_{12}\text{\hspace{0.17em}}\frac{{\epsilon}_{\theta \theta}}{{\epsilon}_{zz}}\right)\right){\epsilon}_{zz}\text{,}$$
(5)
$$\frac{\Delta \lambda}{\lambda}=\text{GF}{\epsilon}_{zz}\text{,}$$
(6)
$$\mathrm{GF}=G\left(1-\frac{{n}^{2}}{2}\left({p}_{12}+{p}_{11}\text{\hspace{0.17em}}\frac{{\epsilon}_{rr}}{{\epsilon}_{zz}}+{p}_{12}\text{\hspace{0.17em}}\frac{{\epsilon}_{\theta \theta}}{{\epsilon}_{zz}}\right)\right)$$
(7)
$$F={\displaystyle \sum _{n=1}^{N}\left|\frac{{\epsilon}_{opt}-{\epsilon}_{obj}}{{\epsilon}_{obj}}\right|}+{\displaystyle \sum _{n=1}^{N}\left|\frac{R{\left(\lambda \right)}_{opt}-R{\left(\lambda \right)}_{obj}}{R{\left(\lambda \right)}_{obj}}\right|}\text{,}$$