## Abstract

In this paper, we propose a new method to determine the longitudinal distribution of a non-uniform transverse force applied to an optical fiber. For that purpose, we use a chirped fiber Bragg grating (CFBG) for which we monitor the polarization parameters in reflection. In particular, we demonstrate that the differential group delay (DGD) spectrum of the CFBG is an imprint of the load profile so that it can be used for the shape determination of an applied load. Thereafter, we discuss the influence of the CFBG parameters on the achievable accuracy and resolution of our technique. An experimental validation is finally reported with two 48 mm long CFBGs subject to step transverse load profiles.

© 2015 Optical Society of America

## 1. Introduction

Fiber Bragg gratings (FBGs) are key components for both optical fiber telecommunications and sensing applications. While the most straightforward applications remain temperature and axial strain sensing, FBGs are used as transversal load sensors in many domains such as structural health monitoring of composite materials [1–3] or for biomedical applications (e.g. pressure mapping in the case of knee arthroplasty [4] or force sensing robot fingers [5]).

In an optical fiber, the presence of a transversal load creates an asymmetry of the fiber transverse section. This asymmetry induces a local birefringence so that the two orthogonal polarization modes (called eigenmodes) are no longer degenerated. When an FBG is present at the load position, we observe two independent reflection responses separated in wavelength by a quantity that depends on the intensity of the load. When the wavelength separation is greater than the width of the reflection band, the two FBGs responses are clearly separated and the load value can be easily determined. However, this effect cannot be directly observed on the amplitude spectrum for limited load values (<1 N/mm). This limitation partially disappears by considering an FBG inscribed in a polarization maintaining fiber (PMF) [6] or a pi-shifted FBG [7], so that the amplitude response can still be used to monitor the load. In other hand, it has been demonstrated that the spectral response of the polarization dependent loss (PDL) [8] or the differential group delay (DGD) [9] is very sensitive to the birefringence [8, 9] so that PDL and DGD can be advantageously used to monitor weak transversal load [10,11]. It is worth mentioning that all above mentioned techniques consider the case where the load is fully and uniformly spread over the FBG, so that they work only to detect uniform transversal load.

For practical applications, it is of high importance to have the possibility to measure non-uniform load profiles with FBGs. Different techniques have been proposed so far [12–15]. In these studies, the main idea was to associate different Bragg wavelengths to different positions (by using either an array of uniform FBGs [12], a chirped FBG [13,14], or a chirped-Moiré FBG [15]) and to retrieve the load profile by using the grating amplitude response. However, these techniques suffer from either the complexity of the demodulation technique since they require to separately monitor each grating response [12,15] or the difficulty to detect weak alteration on the CFBG amplitude response [13,14].

In this paper, we present a new method able to monitor non-uniform load by using the measurement of the DGD of a CFBG. In particular, we report that it exists a direct imprint of the load profile on the DGD spectrum, so that post-processing techniques become useless. Therefore, our method offers the advantage to detect weak and non-uniform transversal load.

In the following, we first present in section 2 the effect of a transversal load on the optical and mechanical properties of optical fiber, i.e. the effect on the refractive index and on the strain in the direction of the fiber axis. In section 3, we report the results obtained when a uniform transversal load is applied to a CFBG. We then derive in section 4 the results when a non-uniform transversal load is considered. In particular, since it exists a direct impact of a local load on the DGD spectrum, we report the reconstruction method based on the DGD spectrum and show that this method is well suited for shape determination. In section 5, we then analyze the influence of the CFBG parameters on the load profile reconstruction for both sensitivity and spatial resolution. Experimental results are finally reported in section 6 for which a good agreement between numerical analysis and experimental results is obtained.

## 2. Transverse load effect on the FBG properties

Prior to study the polarization properties of CFBG under non-uniform load profile, we derive in this section the effect of a transversal load on the optical fiber properties, and, as a consequence, on the properties of an FBG located at the load position.

Let us consider an optical fiber including an FBG and subject to a transverse load as sketched in Fig. 1. The stress profile $\left({\sigma}_{x},{\sigma}_{y}\right)$ in the fiber induced by the load is well known [16]. In practice, as the diameter of the fiber core (~8μm) is small compared to the diameter of the cladding (~125μm) and as most of the optical power is confined in the fiber core, it is sufficient to know the value of the stress at the center of the fiber, which is given by [17]:

with*F*the applied force,

*b*the optical fiber radius,

*L*the length over which the force is applied, ${\sigma}_{x(y)}$the component of the stress in the

*x (y)*direction and ${\tau}_{xy}$the shear in the

*xy*plane. In the following we consider the case of plane stress $\left({\sigma}_{z}=0\right)$ applied to the cylinder (for the condition of the approximation see [18]).

Since the presence of the stress leads to an axial deformation $\left({\epsilon}_{z}\right)$ of the fiber, the periodicity of the written FBG is modified by the load.

The periodicity of the grating under load${\Lambda}_{u}$ is then given by [19]:

With ${\Lambda}_{w}$the periodicity of the grating without load and${\epsilon}_{z}$given by the Hooke law:*x*(

*y*) direction (so called x (y) eigenmode), ${p}_{11}$and${p}_{12}$the photoelastic coefficients, $\nu $the Poisson’s coefficient and

*E*the Young modulus (for the silica optical fiber$E=74.52GPa$;$\nu =0.17$;${p}_{11}=0.121$and${p}_{12}=0.270$). The Bragg wavelength of the FBG is therefore dependent on the polarization of light. Let us recall that the difference between the two effective refractive indices is called the birefringence. Using Eq. (1) and Eqs. (4)-(5), the birefringence induced by a transversal load is given by:

The Bragg wavelengths of the two eigenmodes of an FBG are therefore different when a load is applied and are given by:

These two Bragg wavelengths undergo a shift due to the modification of both parameters ${n}_{\text{eff}}$and$\Lambda $. The modification on ${n}_{\text{eff}}$is different for the two eigenmodes while the change in the grating periodicity$\Lambda $due to the longitudinal strain will be identical for all states of polarization. Note that the plane stress condition will be an accurate approximation as long as the load scale variation is large compared to the fiber diameter [20].## 3. Spectral response of a CFBG under a uniform transversal load

Before considering the case of a non-uniform load applied on the CFBG, we present in this section the amplitude and the phase responses of CFBG under a uniform load profile as well as the spectral evolution of the corresponding DGD.

Let us consider a CFBG characterized by its reflection spectrum$r\left(\lambda \right)\mathrm{exp}\left(i\theta \left(\lambda \right)\right)$.

The corresponding reflected amplitude spectrum R and group delay *τ* are then given by:

The CFBG response depends on different parameters such as the length of the grating *L _{g}*, the amplitude of the refractive index variation$\delta n$, the visibility of the refractive index variation

*v*, the chirp parameter

*C*and the initial period of the grating${\Lambda}_{0}$. ($\Lambda \left(z\right)={\Lambda}_{0}+Cz$)

In the presence of a uniform transverse load, a constant birefringence is induced along the CFBG so that the CFBG spectral response is different for the two eigenmodes *x* and *y*. Figures 2(a) and 2(b) respectively depict the reflected spectrum (*R _{x}* and

*R*) and the group delay (

_{y}*τ*and

_{x}*τ*) of the two eigenmodes for a CFBG with

_{y}*L*40 mm,

_{g}=*C =*1 nm/cm,

*δn =*4 10

^{−4},${\Lambda}_{0}$

*=*529 nm. It can be seen that the main effect of the birefringence is to induce a translation in wavelength of the CFBG responses between the two eigenmodes (Figs. 2(a)–2(b)). As in the case of a uniform FBG [17], this translation can be estimated by the modification of the Bragg wavelength (Eqs. (7)–(8)) induced by the birefringence which is given by [21]:

An important parameter used to characterize the presence of the birefringence on FBG responses is the differential group delay (DGD) [9].

The DGD is defined as the absolute value of the difference between the group delays of the two eigenmodes:

Since the main effect of the birefringence is to induce a wavelength shift of the two eigenmodes, without other major change in the spectrum, the $DGD\left(\lambda \right)$ value can be approximated by [21]:

Where ${p}_{\tau}$ is the slope of the group delay, also referred to the grating dispersion. Equation (13) thus indicates that the DGD value is proportional to the birefringence value.

We report on Fig. 2(c) the DGD spectrum for the CFBG under a uniform load. As it can be observed, the value remains constant to a few picoseconds (the exact value depends on the amount of birefringence) within the reflection band where both x and y spectra overlap. Figure 3 shows the evolution of the DGD value taken at the center of the reflection spectrum for different birefringence values. On this figure, the red line corresponds to the DGD obtained with Eq. (13). This figure shows that the DGD value in reflection is proportional to the applied load and that the value obtained with Eq. (13) accurately predicts the simulation results.

The results of this section consequently show that, under a uniform load applied on the entire CFBG, the DGD value allows a direct determination of the load value. Moreover, the DGD dependence is linear with respect to the applied force.

## 4. DGD spectral response of a CFBG under a non-uniform transversal load

Let us now consider the case of a non-uniform load applied along the grating length. This condition is the most spread practical case and we report in this section that the DGD spectrum can be advantageously used to retrieve the profile of the load.

In first approximation a CFBG can be considered as a concatenation of uniform FBGs. For our purpose, we then considered that the different FBGs can be subject to different constant load values. Since, in CFBG, the Bragg wavelengths are encoded in different positions, the spectral response of the different parts of the CFBG can be related to the different spatial positions. There is therefore a one-to-one correspondence between the reflected wavelengths and the position along the grating [15], so that the grating position and the reflected wavelength are mapped as follows:

*z*. If we apply a load only on a limited part of the CFBG, the modification of the spectral response can be observed on the corresponding wavelength band. In particular, the DGD spectrum will be only modified at the loaded part of the CFBG and will remain null elsewhere so that the DGD value at a given wavelength corresponds to the load applied at a specific location of the grating. The birefringence profile $\Delta n\left(z\right)$ is then obtained from the DGD in reflection of the CFBG:

To validate our predictive approach, simulations of the CFBG response for different load profiles have been performed with the transfer matrix method [22]. The non-uniform load is taken into account by changing the refractive index of each section according to Eq. (4)–(5) (the length of each section must be taken such that the load can be considered as constant) and, considering Eq. (2), the period of the grating is equal to${\Lambda}_{fi}={\Lambda}_{ii}\left(1+{\epsilon}_{zi}\right)$ where ${\Lambda}_{fi}$ is the period in section i when a load is present, ${\Lambda}_{ii}$ is the period of section i without any load and ${\epsilon}_{zi}$is the deformation of the section i in the z direction. The simulation can be performed separately for the two eigenmodes of polarization since no coupling is considered between the two eigenmodes. A schematic representation of this is shown in Fig. 4. A tanh profile of apodisation on the visibility is inserted in the simulation to reduce the ripple of the group delay.

Figure 5 shows the DGD in reflection for different step load profiles at different positions, for different lengths and for different amplitudes (cf. Figure 5(b)). On this figure, the amplitude (Fig. 5(a)) and the DGD (Fig. 5(c)) spectra are reported as well as the comparison between the applied and reconstructed load profiles (Fig. 5(d)).

As it can be seen, while it is difficult to identify the influence of the step load profile on the amplitude spectrum, the effect clearly appears on the DGD spectrum. In particular, it is possible to determine the position, length and amplitude of the applied force from the DGD. We observe a slight mismatch between the applied and reconstructed force profiles. This is due on one hand, to the ripple of the CFBG group delay. In practice, this ripple can be decreased by considering an apodized CFBG and by smoothing the DGD. On the other hand, a slight red-shift of the DGD shape appears. This deformation is due to the influence of the refractive index modification induced by the load on the mapping (Eq. (15)) since the wavelength of the maximum reflectivity is modified when the refractive index changes. It is possible to reduce the relative influence of this deformation (see section 5).

Finally, Fig. 6 shows a more complex load profile applied on the CFBG, the reconstructed force profile obtained from the DGD value and the relative error between these two values. The measurement is also less sensitive to the CFBG imperfections. The simulation shows that the DGD has a shape close to the load applied on the fiber and that the reconstructed force obtained using Eq. (11) is similar to the applied load profile.

## 5. Influence of the CFBG parameters on the load profile reconstruction

As in the case of a uniform FBG [11], the choice of the CFBG parameters is important since it influences the DGD value produced for a same load and, as a result, the performance of our method.

At first, it is important to notice that the proposed technique is strongly affected if the group delay created by the CFBG is not linear. Indeed, the one-to-one correspondence between the DGD spectrum and the birefringence in Eq. (13) remains true only if the slope of the group delay *p _{τ}* is a constant. It is therefore of great importance to use non-saturated CFBG and to limit the group delay ripple. In practice, it is possible to minimize the ripple of the CFBG by using an appropriate apodization profile. In particular it has been shown in [23] that the best profile is the positive hyperbolic-tangent profile. This profile has been used in our numerical analysis.

The other important parameters influencing the results are the grating length, the chirp of the CFBG and the amplitude of the index modulation *δn*.

Since the DGD spectrum is only modified by parameters that have a local influence, the grating length does not directly impact the performance of our method. However, it will determine the position range over which the load can be measured so that this parameter must be adapted regarding the measurement range needed.

We present in Figs. 7 and 8 the DGD results for three different values of the index modulation amplitude *δn* and three values of the chirp parameter, respectively.

Figure 7 presents the influence of *δn* on the DGD response for the load profile displayed in Fig. 8(a). This figure shows that the DGD response better represents the load profile when *δn* is smaller. In this case, the photonic bandgap becomes narrower and the value of the DGD at certain point is less affected by the value of the force in its direct environment.

As it is shown in Fig. 8, a small chirp value increases the sensitivity on the load amplitude while a larger chirp value increases the spatial discrimination. It is therefore possible to improve the determination of the longitudinal shape profile by increasing the chirp of the CFBG but at the expense of the amplitude load sensitivity.

The analysis of the evolution of the sensitivity and the spatial resolution with the chirp is displayed in Fig. 9. The sensitivity is determined by finding the load value that creates a DGD of 1ps (we consider that a DGD below 1ps will be difficult to measure due to both the initial DGD spectrum and the group delay ripple). For the spatial resolution of the sensor we estimate its value by determining the smallest distance between two loads such that they can be distinguished. The evolution reported in Fig. 9 speak for themselves.

## 6. Experimental validation

To validate our numerical analysis of the DGD evolution of the CFBG in the presence of a non-uniform transverse load, two experiments were conducted. In the first experiment, an increasing load is applied by putting different weights at the center of a metal plate that loads a CFBG on a portion narrower than the grating length. This experiment is used to verify the linear increase of the reconstructed load profile with the weight. In the second one, a same load is applied at three different positions on the grating in order to verify that the reconstructed load profile only shifts when the position is modified.

The CFBGs were written into stripped hydrogen-loaded standard single-mode fiber (SMF) by means of an excimer laser emitting at 248nm through a 3.92nm/cm chirped phase mask (Ibsen photonics) with a central period at 1070nm. A cylindrical lens (200 mm focal length) was used to focus the beam on the optical fiber, the adjustable width slit was placed to select a uniform energy region of the UV beam, and the translation table allowed to write FBGs with different lengths. The CFBG responses were analyzed by means of an optical vector analyzer which determines the DGD using the light reflected by the CFBG. The CFBG reflectivity spectrum and the DGD without load applied on the CFBG of the first experiment are displayed in Fig. 10. That is important to launch light in the CFBG from the small wavelength side in order to reduce the coupling between the core mode and the cladding modes [24].

The visibility and the chirp were considered as known with respectively values of 0.5 and *C* = −1.96*nm/cm*. The other CFBG parameters (the « DC » index change, the grating length and the initial period of the grating) were reconstructed from the CFBG amplitude spectrum (see Fig. 10(a)). The reconstructed values for the CFBG used in the first and second experiments are *δn* = 11 10^{−4}, *L* = 48mm, Λ_{0} = 530.79nm, *δn* = 16 10^{−4}, *L* = 48mm and Λ_{0} = 529.42nm, respectively.

For our experiments, we used U-shaped metal plates as well as a compensating fiber to avoid any tilt. The DGD value $\left|{\tau}_{x}-{\tau}_{y}\right|$was directly calculated via the Jones matrix eigenanalysis method [25]. In order to reduce the ripple observed on the DGD curve, a smoothing over a wavelength range of 0.26 nm was used before to take the absolute value and a Savitzky-Golay filter was applied on the final DGD value over a wavelength range of 1.79 nm.

In our first experiment, the part of the metal plate that stresses the CFBG is 6 mm long. Four different intensities of load (16.2N, 21.2N, 26.2 and 31.2N) have been considered. The results of the reconstructed scheme are displayed in Fig. 11. While the identification of the load profile is difficult to observe in the amplitude spectrum, the DGD well represents the expected step-shape. The reconstructed profile is relatively close to the 6mm rectangular profile applied. In addition, the reconstructed load value increases with the total applied load. Figure 12 shows the evolution of the mean value of the reconstructed load (between the two vertical dotted lines of Fig. 11) when the total load is increased. The linear fitting that is also displayed shows that the reconstructed load profile is indeed proportional to the load applied. The proposed method can therefore be used to determine the load length and amplitude.

For the second experiment, a total load of 175N was applied at two different positions on the CFBG. The two DGD obtained experimentally are displayed in plain line in Fig. 13(b). The two DGD spectra have the same profile on both length and amplitude. A numerical analysis has also been performed to validate the obtained results of the experiment. In the numerical analysis, the step profile was adjusted to obtain the best agreement between simulations and experiments. Figure 13(a) shows the force profile inserted in the simulation. The length on which the load is applied is identical for the two load positions and is equal to 10mm. Figure 13(b) also shows the DGD obtained by numerical analysis (dotted line) and Fig. 13(c) is the result of the reconstruction scheme applied on the DGD. The two reconstructed load profiles are similar as expected and the modification of the load position on the grating is visible on the reconstructed profile. A red-shifted deformation can be observed on the DGD spectrum as explained in section 4. In any case, the load profile can be determined by matching the simulation with the experimental results. It is also important to remind that the influence of this shift can be reduced by using a higher chirp as explained in section 5. Our results consequently show that the reconstruction scheme allows to determine the profile of the load applied on the CFBG.

## 7. Conclusion

In this paper, we have presented a new method able to determine a transversal load profile applied to a fiber that does not require specified post-processing.

We have first presented the modification induced by a transversal load on a CFBG. Two effects have to be taken into account to model the impact of a transversal load on a CFBG. The first is the modification of the refractive index due to the photo-elastic effect and the second is the modification of the grating periodicity variation induced by the longitudinal strain. The DGD profile has been analyzed using a numerical analysis. This analysis has proved the direct imprint of the load profile on the DGD spectrum up to a few millimeters and the validity of the formula to obtain the local value of the load amplitude with the DGD. The measurement of the DGD spectrum in reflection allows thus to determine a non-uniform transversal load on a fiber; in particular, it is possible to detect the position, the extent and the amplitude of the load. We then studied the influence of the CFBG parameters on the performance of our method. This study shows that it is possible to reduce the error on the load determination by using CFBGs with low index modulation and high chirp.

An experiment was finally carried out to validate the numerical analysis. A very good agreement was obtained between the applied force profile and the reconstructed load profile for both the length on which the force is applied as well as the linear increase of the load value. The proposed method could therefore be used practically to identify the non-uniform nature of a transverse load, which is of high importance in practical applications such structural health monitoring in composite materials or the biomedical applications listed in the introduction.

## Acknowledgment

C. Caucheteur is supported by the F.R.S.-FNRS. This research has been conducted in the framework of the European Research Council (ERC) Starting Grant PROSPER (grant agreement N° 280161).

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