Instrumentation & Measurement Magazine 24-5 - 58

Framework for LPFG Spectral
Reconstruction
Black-Box Modelling of the LPFG Attenuation
Spectrum
The initial approaches [1] in the spectral reconstruction of the
LPFG spectra given the DFB's information consisted of trying to
fit Gaussian, Lorentzian and Voigt profiles to the available data.
Although in some cases this provided a reasonable fit (and consequently
an acceptable determination of both amplitude and
minimum attenuation wavelength), in most cases the performance
led to a considerable error when compared to OSA data.
Given the variety of LPFG spectral shapes (which can range
from a few nanometers at full width at half maximum to almost
20 nm, and attenuation amplitude from a few dBm to ~40 dBm)
and scarcity of data provided by the low-cost system, a new
model of the LPFG's attenuation band was required to provide
a suitable fit for the spectra. Although more complicated methods
such as data fitting via Markov Chain Monte Carlo (MCMC)
or numerical simulations of LPFG spectra [5] were in principle
possible, this would prove to be a complex task to tackle, both
from a model implementation and computational performance
perspective. Hence, a black box [6] approach to the problem was
chosen, which meant the selection of some functions that could
potentially provide a good fit to the LPFG spectra comparing
their performances when fitting to real LPFG spectra.
Based on a great variety of LPFG spectra acquired with an
OSA, the performance of multiple fitting functions was tested.
Three diagnostic quantities where calculated: the amplitude
and wavelength error (comparing with OSA data) and a χ2
test (which evaluates the quality of the overall fit to the available
data). The data was fit with SciPy's (Python module)
curve fitting methods, using generic bounds such as keeping
the wavelength between 1500 and 1600 nm and at a reasonable
value for the amplitude. Inside the module, the default
method for bounded curve fitting is the Trust Region Reflective
algorithm, which was used on this setup.
The tests performed indicated that a variation of the generalized
logistic distribution provided the best fits given the
entire set of LPFG spectra available. The worst-case scenario
can be seen in Fig. 2, in which the spectrum is very wide at the
top and narrow at the bottom, causing the functions not to fit as
well as in other cases. Even in this case (which is presented as
a limiting one), the generalized logistic distribution performs
reasonably, justifying the choice as fitting function. This function
is defined:
F C A n C A sechn
 ; , ,, ,00 
     
 (1)
where λ refers to the wavelength, C is the offset power, A and
λ0
are the amplitude and central wavelength of the attenuation
band, respectively, and ω and n are parameters related to the
full width half maximum (FWHM) of the curve.
Given the fact that in LPFG sensing the main interest is
focused on the attenuation band's central wavelength and amplitude
determination (and these parameters come directly
from the curve fitting routine described), the performance of
this method can be checked to estimate the typical error.
The error for the curve fitting of multiple functions to a
variety of LPFG spectra can be observed in Fig. 3. The usual errors
for the parameters of interest (regarding the generalized
logistic function) are 0.15% or less for the case of the attenuation
band central wavelength and less than 1% in the case of
the amplitude determination, meaning that the chosen distribution
presents a good fit to the variety of available data.
In the case of the low-cost LPFG spectral analyzer, only a
few intervals of around 4 nm are available for fitting, meaning
much less information is available, which will decrease the
quality of the fitting curve and consequently the determination
of the parameters of interest.
Fig. 2. Result of the curve fitting of multiple functions to a LPFG spectrum with very narrow peak.
58
Low-cost Spectral Analyzer Simulations
Given the choice of fitting function, simulations of the lowcost
setup can be computed using the same LPFG spectral data
collected with the OSA. To
perform these simulations,
the data of the spectra corresponding
to 4 nm around
1530, 1550 and 1570 nm
were taken, and the chosen
function was used to fit this
data and compare it with
the original spectrum. This
procedure allowed for the
evaluation of the curve fitting
routine that was applied
to the monitoring platform.
Fig. 4 shows a typical
case of the spectral
reconstruction using the
available data for the lowcost
system and the chosen
curve fitting function.
IEEE Instrumentation & Measurement Magazine
August 2021

Instrumentation & Measurement Magazine 24-5

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