3.4        Reconstruction examples

This section studies the principal parameters that intervene in the reconstruction of FBG by layer-peeling. The limits of the reconstruction methods are presented. The influence of the layer thickness and the number of used spectral points is evaluated. The reconstruction from a starting complex reflection response or impulse response is analyzed from the point of view of the available dynamic range and the influence of noise.

3.4.1        Reconstruction limits

We have shown in section 3.3.1 that the spectral reflectivity of homogeneous gratings can saturate when the grating length and refractive index modulation amplitude are sufficiently important. This effect is explained by the fact that the light components, for which the wavelength is in the saturation bandwidth, are completely reflected before the grating output. This effect is not limited to homogeneous FBGs.

For such gratings that have a saturation bandwidth in reflection, the reconstruction by layer-peeling reaches its limits as a part of the grating is not probed by all possible wavelengths. This can be observed in the reconstruction of homogeneous gratings with different lengths and identical refractive index modulation amplitude (Dn_ac of 2×10^-4), as presented in Fig. 3-24. The reconstruction of the 1 mm long grating is complete, but for the 10 mm grating, the last 2 mm show a small coupling amplitude decrease and also a small remaining coupling amplitude after the grating output position, indicating reconstruction errors. The layer-peeling algorithm clearly fails to reconstruct the 20 mm long FBG as we observe that the amplitude and phase information of the coupling coefficient are not reconstructed for more than half the grating length. We have performed the reconstruction of the 20 mm long FBG with two spectral resolution values and if a better reconstruction is observed for the higher spectral resolution, the complete reconstruction also failed in this case.

Fig. 3-24 Layer-peeling reconstruction with 5 mm layer thickness of the coupling coefficient amplitude (top) and local Bragg wavelength calculated from the coupling coefficient phase (bottom) for homogeneous gratings of refractive index modulation of 2×10^-4 for different lengths L and number of spectral points M (N represents the number of layers)

Fig. 3-25 Layer-peeling reconstruction of the coupling coefficient amplitude (top) and local Bragg wavelength calculated from the coupling coefficient phase (bottom) for the FBG2 (left) and a 10 mm long homogeneous FBG with a Dn_ac of 5×10^-4; the reconstruction parameters are : D = 3 mm, M = 30 N for solid lines and D = 20 mm, M = 10 N for dashed lines; the dashed lines are shifted for clarity

Another illustration of this spectral depletion effect that limits the reconstruction by layer-peeling is presented in Fig. 3-25 where the reconstruction of a 10 mm long and period chirped FBG with constant Dn_ac (FBG2) is compared to the reconstruction of a FBG with the same parameters but without the chirp. The homogenous grating reconstruction falls down after 3 to 4 mm, while the complete chirped grating can be reconstructed. The chirped grating has a smaller maximal reflection amplitude and a broader bandwidth that prevents the complete depletion of a particular spectral bandwidth.

Recently, it has been demonstrated that the worst-case error amplification factor in reconstructing a grating from its complex reflection spectrum by layer-peeling is of the order of 1/T_min, where T_min is the minimum transmission amplitude through the grating [3-20].

The limitations of the reconstruction for not too strong FBGs can be partially overcome in two ways. The first possibility is to reconstruct the grating from both sides and to combine only the first half of the reconstruction distributions. The second alternative is an experimental method that is a consequence of the results presented in Fig. 3-25, where a predefined chirp function is applied to the grating by a temperature ramp or a strain ramp. The applied chirp needs to be sufficient to reduce the maximal reflectivity of the grating.