4.4        Reconstruction process

4.4.2       Complex FBG spectral response calculation

The complex reflection amplitude r_fbg(n) is calculated from the slowly varying complex OLCR response through a FFT algorithm after a deconvolution from the broad-band spectral power density S(n). As the OLCR measurement is not calibrated, a subsidiary calibration of r_fbg(n) is required, using an independent transmission measurement of the grating.

a)   Fourier transform performed with the FFT algorithm

The inverse Fourier transform is performed with the FFT algorithm. Nevertheless, the spectral resolution possible with such kind of algorithm strongly depends on the OPLD range. For this reason we artificially increase the x range by padding zeros after the last measured point (a resolution of 3.6 pm is found at 1309 nm with an OPLD range of 52 cm). This zero padding is equivalent to consider the OLCR signal equal to zero for values under -120 dB. The total number of points is set to a power of two for optimal FFT algorithm use. The Fourier transform of the slowly varying OLCR function produces a frequency shifted spectral response. The frequency shift corresponds to the laser frequency.

b)   Deconvolution process

The complex slowly varying OLCR function f(x) is related to the grating complex spectral response r(n) and the effective broadband power spectral density S(n) that includes some spectral filtering of the interferometer

where the symbol "~" is used in the meaning of "proportional to" and n₀ is the laser frequency. The frequency shifted inverse Fourier transform F(n) of f(x) is proportional to S(n)×r(n). The deconvolution of the interferometer signature is in the frequency domain a simple division, i.e. F(n)/S(n). The function S(n) is obtained when the FBG is replaced by cleaved fiber end, for which the reflectivity is constant over the light source frequency range. We can then write the following relation

where r_nc represents the FBG reflection spectral response that is not calibrated and FT means "Fourier transform of" and FT^-1 "inverse Fourier transform".

We have to notice that S(n) is supposed to be a real function and that the remaining dispersion, which can appear in the interferometer is assumed to be very small and thus neglected. For this reason we use the absolute value of FT^-1(f_{cleaved fiber}(x)). In the case where the dispersion cannot be neglected, a more complicated deconvolution is required, based on equations (4-33).

c)   Normalization process

Finally, it remains to normalize the complex reflection amplitude. We have chosen not to calibrate the OLCR measurement as the polarization rotation drifts makes such calibration hard to maintain. We prefer an independent measurement of the FBG transmission intensity with the tunable laser to determine the maximal reflection intensity R_fbg,max

As long as the induced refractive index modulation of the grating is not modified (e.g. high temperature), the impulse response amplitude at the entrance |h(t=0)| remains constant for different environmental conditions of temperature and strains. This property allows to perform a single calibration of |h(0)| in a given state of strain and temperature to calibrate the grating in other environmental states.