Reconstruction method and optimal design of an interferometric spectrometer.

In this paper, we present a method to estimate the power spectral distribution of a source from input data acquired by an interferometric-based spectrometer. Our spectrometer shows distortions in the fringe pattern and a lack of data, making it impossible to apply the Fourier transform approach, which is the gold standard as a spectral recovery method for interferometric spectrometers. We combined linear inverse problem solving and iterative methods instead, considering that each detector of the spectrometer has a specific and known spectral response. Iterative methods are used to overcome problems caused by lack of input data. We show that a good spectral estimation of relatively simple spectra having a resolution of 400 points is typically achieved using fewer than 10 detectors with such a method. Since the quality of spectral restitution with such an approach relies both on signal processing and an optimal selection of the detector's spectral responses, the paper also shows that some sets of spectral responses selected for the detection and consequently a spectral repartition of the detectors are more successful than others in the spectral recovery process. We chose the condition number of the inversion matrix as an optimization criterion and evaluated how this criterion can be used within this framework. We found that maximizing it achieves better spectral restitution, within a range where noise remains low.