The calculation of the rj values is performed through a Fourier transform and it is known
that Fourier transforms of bandwidth limited spectral function give some
oscillation at each discontinuity (Gibb's phenomenon). A standard way to limit
these oscillations is to window the original spectral response with an
apodization function. Another way is to reduce the resolution by averaging the
Fourier transform over several points. This windowing effect has been simulated
and the results are presented in Fig. 3-28. The solid lines for the case
without windowing clearly show the Gibb's oscillations and exhibits very good
reconstruction at the other locations. The dashed lines (which represent the
reconstruction from the spectral response windowed with a Hann function) show
that the oscillating effects at the edges are suppressed but the reconstruction
is less efficient after 7 mm due to the energy loss induced by the
windowing. We could notice this effect because the FBG1 has been design not to
be easily reconstructed.
Fig. 3-28 Reconstructed coupling coefficient amplitude (top) and local Bragg
wavelength (bottom) for the FBG1 performed with layers thickness of 20 mm, a ratio
M/N = 10 without windowing (solid lines), without windowing but with
averaging (dashed-dotted lines) and with hann windowing (dashed lines); the
curves are translated for clarity
The most interesting results are found
for the third (dashed-dotted lines) case where the spectral response is not
windowed but the resulting reconstruction is averaged over 3 points
|
(3-21) |
where f is the coupling amplitude or the
local Bragg wavelength and pj is the discrete position j. This
method offers the best reconstruction results at the price of a slightly
reduced resolution (that can be recovered by reducing the layer thickness).
|
