3.4        Reconstruction examples

3.4.5       Reconstruction from the complex spectral response

Here, we consider the simulation of the reconstruction from experimental complex spectral response. The influence of the available dynamic range and of the noise is analyzed.

Fig. 3-29 Reconstructed coupling coefficient amplitude (top) and local Bragg wavelength (bottom) for the FBG1 performed with layers thickness of 5 mm, a ratio M/N = 10, and for a spectral intensity range of 60 dB (solid lines), 40 dB (dashed lines) and 20 dB (dashed-dotted lines); the curves are translated for clarity

a)   Spectral response dynamic range

The dynamic range is defined in our context as the available signal range between the maximal value and the noise level. The spectral signal considered is the reflection intensity. The impact of the available dynamic range is presented in Fig. 3-29 for 20, 40 and 60 dB dynamic range. The reconstruction is relatively poor for the 20 dB limited dynamic range and quite good for the 40 dB case, except from some non compensated oscillations at the section edges. The reconstruction with a dynamic range of 60 dB or higher is very good.

b)   Spectral response noise

Noise in experimental measurement is inevitable and its influence on the reconstruction from the spectral response is simulated hereafter. For a given spectral point r = |r|×e^if, a noisy spectral point r_n = |r_n|×e^ifn is calculated :

where A_o and A_s are the noise amplitude offset and scale factor, respectively; P_o and P_s the noise phase offset and scale factor, respectively; and finally "rand" a random number between ± 0.5. The most important parameters are the noise amplitude scale factor A_s and the noise phase offset factor P_o. The results for 5, 10, 20 and 30 % scale noise and phase offset of p/100, p/50, p/20 and p/10, respectively, are presented in Fig. 3-30. It is seen that even for the smallest noise case the reconstruction encounters problems and oscillations are observed. For more noisy data, the oscillations increases and reconstruction errors becomes very important.

Fig. 3-30 Reconstructed coupling coefficient amplitude (top) and local Bragg wavelength (bottom) for the FBG1 performed with layers thickness of 5 mm, a ratio M/N = 10 for a different noisy spectral responses; solid lines : A_o = 10^-6, A_s = P_s = 5%, P_o = p/100; dashed lines : A_o = 10^-5, A_s = P_s = 10%, P_o = p/50; dashed dotted lines : A_o = 10^-4, A_s = P_s = 20%, P_o = p/20; dotted lines : A_o = 10^-3, A_s = P_s = 30%, P_o = p/10; the curves are shifted for clarity