5.2        Characterization of a Fiber Bragg Grating under Diametric Loading

5.2.4       Diametric Load of Polarization Maintaining Fiber

Diametric load has also been applied on FBG written in a PM fiber where a natural birefringence exists. The stress state in the fiber core, s, is assumed to be the superposition of stress due to the natural birefringence, , and diametric load . The natural PM fiber birefringence is described with a Plane Stress, Plain Strain model. Two distinct materials compose the fiber: Silica and borosilicate in the stress-inducing region (like the "Bow-Ties" for the used fiber). These two materials have different mechanical properties (E and n), therefore the fiber exhibits anisotropic mechanical behavior when stress is applied on the fiber surface. Even if the fiber core is isotropic, the resulting stresses at the fiber core for an external diametric load vary with the angle of applying load direction. The stress s' is a function of the load P and the angle q. To model the diametric load effect, we assume small external load leading to effective stresses.

The natural fiber birefringence, described with , is in the principal axis system given by

Fig. 5-22 : Polarization maintaining fiber geometry

In the PM fiber used, the natural birefringence is due to stress applying region in "Bow-Tie" shape (Fig. 5-22). In this case only one parameter D_p is needed to describe the natural birefringence. Using the low-birefringent fiber model, an equation for D_p is found for the case where the diametric load P=0

For a birefringence of 0.42nm at 1533.3nm, D_p = 21.5 kN/m.

The diametric load P is applied in a direction y' forming an angle q with the y-axis (Fig. 5-23). The stresses are described by a matrix :

Fig. 5-23 : Diametric load on PM Fiber

has to be transposed in in the (X,Y) reference using the rotation matrix :

We define effective stress tensor as , where the matrix considers the fiber specific anisotropy and is assumed to be independent on external load. In addition we assume that the stresses in the X and Y direction are much bigger than the shear stress. Therefore effective stresses in the X, Y directions are independent of the shear stress (a_ij =0). In addition we assume that the diagonal elements are independent of angle q. becomes:

The global stress state in the core of the fiber and in the (X, Y) reference is the superposition of the natural birefringence stress state and the stress state due to diametric load :

If the angle q is different from k×p/2, the shear stress t_xy is not zero. In this case the secondary principal stresses (p',q')_z for the light propagating in the Z direction in the fiber core have to be calculated [5-14]

Fig. 5-24 : Reference (X,Y) of the fiber, (X',Y') of the diametric load, (p',q') of the secondary principal stresses

where y : Angle between the X axis and the p' axis (Fig. 5-24).

Due to equation ((5-24), PM fibers subjected to transversal loads display a different behavior with respect to isotropic fibers. Namely, the relationship between the applied transversal strains e_x and e_y and the measured Bragg wavelength shifts is not necessarily linear, except when the strains are directed along the symmetry axes of the fiber. This theoretical development explains the experimental non-linearity observed in other works but not understood until now [5-17].

The hypothesis of fiber core isotropy and Plane Strain are also valid, then the model developed for the low-birefringent fiber can be applied, but with the secondary principal stresses (p',q') instead of s.

The variation of the Bragg wavelength with the applied load is

For q=0° or 90° and , and is therefore independent of the natural fiber birefringence. Since is independent of the load P the variation of the secondary principal stresses with external load is given by

Matrix inversion leads to

For the investigated fiber the following values are obtained:

Fig. 5-25 shows experimental and calculated values of Bragg wavelength as a function of applied load for different angle, q, between load direction and the principle axis of the fiber. Each graph represents the response along slow (p') and fast (q') fiber axis. The agreement between experiment and model is good for small load. The slope depends strongly on q.

Fig. 5-25 Bragg wavelength of a FBG written in a PM fiber under diametric load for different angle of loading (x axis : diametric load [N] and y axis : Bragg wavelength [nm]).

Fig. 5-26 Sensitivity of the FBG sensor for diametric load; circles(slow) and crosses(fast) are experimental values, dashed lines for (a₁,a₂)=(1,1) and solid line for (a₁,a₂)=(0.47,1.03).

Fig. 5-26 shows the measured slopes z_i, (FBG sensitivity) as a function of angle q. Two theoretical cases are represented, for (a₁,a₂)=(1,1) and (a₁,a₂)=(0.47,1.03). It is clear that the stress field has to be modified using effective stresses to take into account the anisotropy of the fiber. The sensitivity is a periodic function of the angle (180° period) and the response of slow and fast axis are phase shifted by 90°. The model describes clearly the non-sinusoidal behavior of the experimental data in contrast to references [5-12, 5-13], which describe similar experimental results by a sinusoidal approach.

With the analytical model, it is possible to develop a demodulation algorithm to retrieve the stresses from Bragg wavelength variations. It has to be noticed that, if we consider the shear stresses in the (X,Y) plane, we have four unknowns : s_x, s_y, t_xy, and q. With two superimposed FBG's in a PM fiber, the complete state of stresses can be obtained.