1. Introduction

Fiber Bragg gratings (FBG) have demonstrated their capability in applications including strain and temperature sensing [1][2]. A promising application is the embedding of these sensors into composite materials for strain and temperature monitoring [3][?]. For this application FBGs in polarization maintaining fiber (PMF) are used. FBGs possess a narrow reflection band at the “Bragg” wavelength λ_B. By using PMFs two Bragg wavelengths λ _B,p and λ _B,s for p- and s-polarized modes are obtained. The sensing principle of these devices is based on the change in the Bragg wavelength as a function of the measurand to be detected. In the case of the embedded sensors, these influences can be strain or stress and temperature. The physics behind the opto-mechanic interaction lies in the photoelasticity of the material and the change in geometry upon an applied strain field. This is represented by the position dependent strain tensor ē(x,y,z). Figure 1 shows the described scenario. The strain field is applied at the position of the sensor. The fiber, guiding the light to the position of the sensor, is considered unperturbed, although parts of this fiber may be subjected to the strain field as well. To be able to treat the problem analytically, the strain field is assumed to be homogeneous along the length of the sensor ē(x,y, z)=ē(z). This is a reasonable assumption in many cases as the length of the sensor is small, typically a few millimeters. Electromagnetic properties of the material change in a linear approximation according to Pockels law of photoelasticity. It linearly connects the change in the impermeability tensor 0394;B̄=B̄-B̄⁰ to the strains by B_ij=p_ijkle_kl, where the p_ijkl are the photoelastic constants.

 

Fig. 1. Fiber Bragg grating in polarization maintaining fiber loaded by an arbitrary strain tensor ē, leading to changes in impermeability tensor B̄. The grating is illuminated by light represented by a coherency matrix Φ. The reflected light is split into the average intensity components 〈I_p-〉 and 〈I_s-〉.

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Both, strain and impermeability tensor possess six independent entries. In an embedding application, an arbitrary strain state has to be expected in general, with each component e_{i j} being nonzero. Thus, ΔB̄ can take an arbitrary form as well. The question is, which components of the strain tensor or impermeability tensor influence the optical response of the FBG. We demonstrated, that only the components B_xx, B_yy and B_xy, or in an isotropic material the strains e_xx, e_yy, e_zz and e_xy exhibit a noteworthy influence [5].

Adding temperature as an additional parameter, this leaves, with the four strains, five parameters. A single fiber Bragg grating in PMF yields two measurable parameters, resulting in an under-determined problem. An idea by E. Udd et al. [6][3], is to superimpose a second Bragg grating at the same location of the fiber, but at a strongly different Bragg wavelength. The material properties at the second wavelength will have to differ somewhat from those at the first. Thus, in principle, four independent parameters are accessible trough measurement, leaving a still under-determined problem.

To be able to describe the exact influence of all the parameters on the reflected spectra, coupled mode theory (CMT) may be employed [7][5]. However, the emerging differential equations have to be solved numerically, yielding the required spectra, but leaving the underlying physical relations unresolved. An analytical treatment of the problem, using two fundamental mode systems for the perturbed sensor and unperturbed fiber being connected by a transformation rule that was recently derived from Maxwell’s equations by our group [8].

What can be learned from CMT, as well as from the analytical solution is that the spectral response can be used to reconstruct the strain field, as long as B_xy=(p ₁₁-p ₁₂)e_xy/2=0 is fulfilled, corresponding to an absence of shear strain. When shear strain is present, the simple reflection of one polarization mode of appropriate wavelength at the Bragg grating and its backward propagation is no longer the only effect to be considered. A coupling among the polarization modes is also observed, superimposed on the forward-backward coupling of the grating. Since both models, CMT and the analytical use monochromatic waves as a basis, only coherent effects may be described in a straight forward manner. This limits the model to using fully polarized light, viz. light with a full correlation between the fields in the two polarization modes, such as experimentally obtained by using tuneable lasers. Thus an interference effect arises, when light from one polarization mode is coherently coupled to the other mode, and in turn to complicated spectral responses, such as depicted in figure 2. We recently described a numerical model to incorporate incoherent effects such as unpolarized light within the CMT framework [9]. The obtained results showed, that the spectral response is clearly less complicated, when unpolarized light is used and may thus be accessible to some kind of measurement algorithm extracting the missing shear strain component. Figure 2 gives a comparison of a strained FBG illuminated with unpolarized and polarized light. It may be observed, that the spectral response of the FBG which is symmetric in an unloaded case becomes unsymmetrical. However, in the case of unpolarized light, both peaks show mirror symmetry. Additionally, the spectrum in the polarized case is strongly depending on the circularity, viz. the relative phase of the polarized light, which is not maintained in the PMF and thus introduces an additional free parameter.

 

Fig. 2. Reflected intensities from a shear strained FBG with a beat length of 7.5 mm, with e_xy=1·10^-3 for light linearly polarized at 45° to the principal axes and unpolarized light.

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Thus there seems to be evidence for a simplified behavior of the reflection spectrum of shear strained fiber Bragg gratings, when illuminated with unpolarized light. We now intend to obtain a simplified analytical description of the reflection spectrum for the unpolarized case by using the analytical treatment we recently presented [8], which describes the behavior of the grating for monochromatic waves of the form e^iωt and is thus not seamlessly capable of modeling unpolarized light. Our approach is to use the properties of the coherency matrix representation of light for the simplification of the model. We derive a simplification for the transformation matrix and give an approximation for the derived coefficients describing the reflected spectrum.

2. Model

Unpolarized light may be described by the Stokes vector or the coherency matrix Φ[10]. The latter may be computed from the time signals of the complex amplitudes of the two polarization modes

where the brackets denote time averaging and ε_i are the components of the instantaneous Jones vector. The Jones vector entries are given by the complex amplitudes of the two polarization modes ε={A_p,A_s}^T, if we neglect an absolute phase and only consider the relative phase of the modes [10]. Thus the entries of the coherency matrix may be represented by time integrals over the mode amplitudes. For unpolarized light, the coherencymatrix becomes diagonal, with equal entries. Thus, conclusions about the time averaged product of the complex field amplitudesmay be drawn stating

The analyticalmodel uses the amplitudes of the forward propagatingmodes A ₊={A _p+,A _s+}^T for illumination of the sensor. The reflected amplitudes are given by A-={A _p-,A _s-}^T . The monochromatic wave model connects the four amplitudes by [8]

Here Q̱ is the transformation matrix from the unperturbed mode system into the strained mode system and ρ̄=diag{ρ_p,ρ_s} summarizes the reflectivities ρ_p and ρ_s [8]. They may either be derived from the known analytical solution of the coupled mode equations [11] or in the case of a inhomogeneous grating structure from a numerical solution. Since Q̄ is non-diagonal, the reflected amplitude of one polarization mode depends upon the amplitudes of both illuminating polarization directions.

The reflected intensities detected at the measurement system in the unperturbed fiber system are computed from

The expanded equation (4) possesses “homogeneous” elements A _p/s A ^* _p/s and inhomogeneous elements A _p/s A ^* _s/p. To simplify this equation we consider the time interval T in which the intensities are measured

For intervals which are long with respect to the “coherence” time the above integral can be considered equal to the time average integrals employed in the coherency matrix. Thus the coefficients in equation (4) are reduced to integrals over “homogeneous” coefficients only by using the identity of equation (2). Using the explicit expression for Q̱ given in [8] the elements of the transformation matrix can be shown to obey

depending on the sign of the impermeability tensor entries, which will be of no further relevance. The equation only holds exactly in the case of an isotropic medium, with B_xx=B_yy.

For the considered case of an anisotropy of the order 10^-3, the arising inequality can be neglected. With the requirement for unpolarized light, i.e. 〈|A _p+|²〉=〈|A _s+|²〉=〈I ₀₊〉 and using Q ₁₁ ²+Q ₁₂ ²=1, we obtain

The expressions for the transformation matrix entries |Q ₁₁|² and |Q ₁₂|² are unconvenient expressions, that require simplification for practical use. The following definitions are used for their simplification. The effective refractive index of the fundamentalmodes of the unperturbed fiber is given by n ⁰ _eff,p and n ⁰ _eff,s, with the birefringence Δn=n ⁰ _eff,s-n ⁰ _eff,p which is usually given by the beat length L_B=2πλ/Δn. Since only the normal strains e_ii and the e_xy shear strain component show an influence on the optical response in an isotropic material, the impermeability tensor reads [12]

Furthermore, the order of magnitude of b=B_xx-B_yy and B_xy is approximately 10^-3 or less, compared to B_ii~1. Therefore we may neglect terms of the order b ², B ² _xy or higher and find for |Q ₁₁|²

and for |Q ₁₂|²

Note, that the squared terms are all not negligible since they are divided by terms of the same order.

From these analytical expressions for the transformation matrix a purely analytical description of the behavior of shear strained FBGs illuminated with unpolarized light is obtained. The reflection coefficients |ρ _p/s|² are real positive functions of the grating period Λ, the effective refractive index of the corresponding mode n _eff,p/s and the wavelength (|ρ|² _p/s=|ρ(λ,λ_B(Λ,n _eff))|²). Both, the Λ and n _eff,p/s are well known functions of the impermeability tensor [13] reading

wherein the B_ij are functions of strains given in equation (9) to (10). Thus, for unpolarized light, the reflection spectrum of the shear strained fiber Bragg grating in each polarization direction includes a linear combination of the two standard intensity reflection spectra of the unloaded FBG, shifted in wavelength by some amount.

For many practical FBG sensing applications, the absolute intensity of the reflected light is not required, since the Bragg wavelengths are determined from a normalized spectral measurement. Transferring this description to the shear strained FBG we might only be interested in the relative intensity V=|Q ₁₁|²/|Q ₁₂|² of the two Bragg peaks in one polarization mode. This parameter takes the manageable form

 

Fig. 3. Left: Intensities corresponding to figure 3 with normal strains e_xx=-2000 µm/m, e_yy=800 µm/m and e_xy=2000 µm/m. Right: Reflected intensities according to numerical and analytical description. The loads are e_xx=e_yy=0 µm/m, e_xy=2000 µm/m.

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The set of measurable parameters required for a full reconstruction of the strain tensor are thus the center wavelengths of the two Bragg peaks of the polarization modes. These can be extracted by peak search algorithms used for “conventional” strain sensing. At the wavelength of the two Bragg peaks, the intensities in the orthogonal polarization direction have to be measured, which, when divided by the intensity at the Bragg peak, yield the V-parameter. Performing this at a strongly different wavelength, as suggested by Udd et. al.[6], the required parameters for strain reconstruction are obtained.

3. Results

We now test the derived description by comparing it to the numerical solution of the tensorial coupled mode equations as described in [5], being an independent description of the phenomenon. The unpolarized light in the numerical calculation is modeled according to the procedure described in [9]. We checked the validity of the analytical model for a wide range of fiber and load parameters, all of which showed a close agreement. We used the analytical solution of ρp/s [11], as well as its numerical solution for cases of apodized FBGs, both of which agree well. The parameter set chosen for presentation here models a FBG in PMF with L_b=5 mm at 1550 nm, λ ⁰ _B,x=1550 nm and a grating length of 5 mm. For clarity of presentation, we use strongly apodized gratings since they show virtually no sidelobes. The ρ _p/s are thus found numerically.

Figure 3 left shows the results of a simulation of pure shear strain. In the absence of shear strain, both reflected intensities show a single symmetric Bragg peak. When shear strain is applied, the polarization mode coupling leads to an increase in the cross coupling parameter |Q ₁₂|² and thus to a peak at the position of the orthogonally polarized Bragg peak. The V-parameters are 3.76 and 3.74 in the analytical and the numerical model respectively, and are thus in close agreement. When normal strains are applied, the position of the Bragg peaks shift due to the change in effective refractive index and grating length, according to equations (13). This changes the difference in the impermeability tensor entries B_xx and B_yy and hence b is changed, leading to an influence in the polarization mode coupling as well. Figure 3 right shows this change in the polarization mode coupling in comparison to the numerical solution of the CMT. The cross coupled intensity almost matches that of the backward coupled intensity. Again, both results are in close agreement.

 

Fig. 4. Change of the V-parameter depending on the shear strain e_xy for different values of normal strain e_xx={-800 µm/m, …, 1000 µm/m} and a constant value of e_yy=0.

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From the intensities and wavelengths of the four maxima (two in the p-spectrum plus the two in the s-spectrum), the parameters λ _B,x, λ _B,y and V can be extracted. Assuming a load case were e_xx≠0, e_yy≠0 and e_xy≠0, we can reconstruct the full load case using the derived equations. Figure 4 shows the change in V-parameter upon loading with shear strain for different levels of normal strain e_xx. The normal strain dependency in the V-parameter can be explained by the separation of the Bragg peaks by the normal strain and thus a reduced mode coupling.

The V-parameter in figure 4, derived assuming the light to be unpolarized, only depends on the strains and temperature of the FBG sensor. Trying to define a comparable parameter using polarized light, the relative phase of the two polarization modes will have to be included, because the relative phase changes the direction of polarization mode coupling and thus the magnitude of such a parameter. Since the relative phase of the two modes is not preserved by the polarization maintaining fiber and may thus change during measurement, it is not suitable for determining the shear strain, since it introduces the phase-difference as additional parameter. Hence, the V-parameter is stable against changes in the experimental setup, or movement of the fiber and can be considered as robust a parameter as the wavelengths’ of the Bragg peaks.

Extending the presented measurement concept for load cases with e_zz≠0 and an additional change in temperature, the method of Udd et. al. [6] can be used to derive a second set of parameters, enabling the reconstruction of the full load case including shear strain and temperature.

4. Conclusions

In this work we demonstrated a simplified analytical treatment of the reflection characteristics of a shear strained fiber Bragg grating, when unpolarized light is used for illumination. We used properties of the coherency matrix of unpolarized light to simplify the projection of the unperturbed fundamental mode system onto the perturbed one. We found an intuitively appealing connection between the occurrence of shear strains and the change in polarization dependent reflection spectrum of the FBG. We derived a parameter which describes the relation between the intensity of the original Bragg peak and the incoherently coupled orthogonal Bragg peak. If, in an experiment, this parameter could be extracted from the measured spectrum together with the Bragg wavelengths λ _Bp/s, a method of reconstructing the shear strain e_xy together with two normals strains may be obtained. Furthermore, when combined with the method of Udd et. al. [6] using two superimposed Bragg gratings of strongly different wavelength, a way for obtaining the missing fifth measurement parameter is at hand. This allows to overcome the non-determinacy of the problem and hence to reconstruct the strain field at the position of the FBG. However, the performance and limitations of this method, especially when the two Bragg peaks overlap due to strong strain fields have to be evaluated.