5.1         Axial strain field distribution measurements

5.1.1         Axial stress effect on fiber Bragg gratings

We consider here the case of a fiber Bragg grating subjected to a homogeneous axial stress s_z (Fig. 5-1). In this case, the other components are null (s_x = s_y = t_xy = t_yz = t_xz = 0). In this section, the relation between the relative change of the Bragg grating and the axial stress amplitude is determined.

From the Bragg equation l_b = 2×n_eff×L, where l_b is the Bragg wavelength, n_eff the effective refractive index and L the grating period, the relative Bragg wavelength variation is given by

where n₀ = n_eff(s_z = 0). The first term, e_z, is the geometric deformation (axial strain) of the FBG and the second term, Dn_x,y/n₀, is the variation of the refractive index in the plane orthogonal to the direction of the light propagation.

Fig. 5-1 : Axial Stress

The strain components e_i are related to the stress field through the elastic tensor, and, in the simple case of axial stress, we have the following equation

where E is the Young modulus (E = 72 GPa for standard telecom fibers) and n is the Poisson ratio (n = 0.16 for standard telecom fibers).

The dielectric tensor change, De_i^-1, is related to the strain field through the elasto-optic tensor

where P_ij are the photoelastic coefficients (P₁₁ = 0.113 and P₁₂ = 0.252 for standard telecom fibers).

The refractive index change Dn_i can be approximated, using the relation with the dielectric tensor (n_i² = e_i), by

For the axial stress case, we have from equation (5-2) : e_x = e_y  =-ne_z = -n×s_z/E, and then

From equations (5-1), (5-2) and (5-5), the relative Bragg wavelength change is found as

where p_e is the effective photoelastic constant for axial stress that can be deduced from Dl(s_z)/l for homogeneous axial loading. The effective refractive index remains uniform and constant in the transverse plane of the fiber core. The relative variation of the Bragg wavelength is linear with a stress s_z. The geometric effect corresponds to the first term s_z/E and the refractive index effect corresponds to the second term - p_e×(s_z/E), which is opposite and about five times smaller than the geometrical effect.

The experimental determination of the effective photoelastic constant of the grating used in the experiment is presented in Fig. 5-2 (p_e = 0.2148). The calibration is performed applying an homogeneous axial stress field to the FBG.

Fig. 5-2 Experimental calibration of p_e : reflectivity spectrum for different axial stress (insert) and relative Bragg wavelength change