We have seen in chapter 3 that the
complex coupling coefficient of the grating can be reconstructed from the
complex spectral response with a layer-peeling method. The FBG is divided in
several layers of physical thickness D where the grating is assumed uniform and
represented by a single complex reflector with reflectivity r. The required
spectral range for the layer-peeling is directly related to the thickness D
|
(4-51) |
where d = b-bd is the detuning wavelength and bd is an arbitrary design wavelength (usually set to the Bragg
wavelength). The smaller the layer is, the larger the required spectral range
is and the smaller the reconstruction errors will be. For bd = 1309 nm and D = 5 mm the spectral
range goes from 1252.8 to 1371.2 nm. The relation between the reflectivity
r
and the local complex coupling coefficient qj is given by
|
(4-52) |
From the starting reflection amplitude
r1(d)=r(d), the grating is reconstructed in an iterative way. At each step, rj for the first layer of the remaining structure at the step j is
calculated and a new reflection amplitude rj+1(d) is calculated
for the structure without the layer j (peeled off)
|
(4-53) |
where rj(m) is the
discrete form of rj(d). The number of points N for r(n) in the range of d has to be
greater than the number M of reconstructed layers (M ³ N).
The reconstruction of the grating
complex coupling coefficient is then conducted with the following procedure :
-
The layer thickness D and the design
wavelength ld are chosen, thus defining the
detuning range
-
The complex FBG spectral response is
restricted to the detuning range
-
The M complex reflection coefficients rj are calculated
-
The M complex coupling coefficient
amplitude and phase are deduced from the rj
We have seen in chapter 3 (§3.5.2) that
the complex coupling coefficient is not enough to deduce all the FBG parameters
as there are three distributions : the refractive index modulation Dnac,
the physical period L (or the period chirp q) and the refractive index chirp Dndc.
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