2.1

Interferogram dispersion

The raw interferogram of the energy coming from a pixel of the observed scene is pre-filtered due to the finite pixel dimension p, and uniformly sampled in a region limited by the detector size D. The measured interferogram may be expressed as:

x being the pixel position. It can be easily shown that the relationship between OPD and the entering ray direction ϑ is linear, as long as the device FOV is not superior to a few degrees:

a being the proportionality constant between OPD and ϑ, and f the focal plane distance. The constant a is related to the maximum digitised optical path difference OPD_max and to the maximum angle ϑ_max.

The raw interferogram is constituted by points (x, DN(x)), which indicate the pixel position on the detector and the corresponding electronic signal expressed in digital number. According to Shannon’s theorem the interferogram sampling frequency k_s should be grater than the full bandwidth k_max of the concerned signal:

In view of Eq.3 the minimum wavelength λ_min we can reconstruct avoiding signal aliasing is λ_min = δOPD, δOPD being the optical path difference subtended by two adjacent pixels. Otherwise speaking, the greatest wavenumber (i.e. the shortest wavelength) one can observe without aliasing is that wavelength for which not less than two detector elements cover one fringe cycle. All wavelengths longer than this limit will have their fringe cycles sampled by more than two detector elements.

2.2

Spectrum recovery

As known the spectrum of any pixels have to be reconstructed by means of an inverse cosine transform; i.e. the real part of the inverse Fourier transform. A possible critical point for performing this transformation is constituted by a not perfect knowledge of the factor , which is essential for computing the OPD corresponding to a generic position x. As long as some error affects the knowledge of the factor γ the calculated inverse transform will lack of accuracy, possibily originating a wrong estimate of the pixel spectrum. A preliminary issue is allowing some theoretical modelling that can account for this type of uncertainty. Let us suppose that the value available for the factor γ be erroneous, then the retrieved spectrum differs from the true one I(k) as stated by the following equations:

being:

It can be easily shown that the:

Therefore, an error in the interferogram dispersion gives rise to a corresponding wavelength scale error in the spectrum domain. Let us note that the difference may become large around narrow absorption lines due to the atmosphere or the observed target. A second trouble that can affect the inverse transform procedure is connected with the exact knowledge of the interferogram’s centre. It is evident that an error affecting the central position of the interferogram produces a well-known cosine-like modulation of the inverse cosine transform.

Simulation of spectra retrieving is shown in Fig. 6. Estimated spectrum for a rect() source (500 nm until 800 nm) as observed with a 20 mm BK7 beamsplitter. A specific algorithm has been applied for compensating the spectral dispersion of the BK7 refraction index.

Fig. 6

Simulation of spectra retrieving

3.1

Calibration procedure

Preliminary measurements have been carried out in order to calibrate the interferometer response. Two dyed He-Ne lasers (λ^red = 632.8nm and λ^green = 543.2nm) have been employed for illuminating a double planar diffuser in order to obtain a homogeneous and isotropic radiation distribution inside the instrument FOV. Fig. 7 shows a single image-frame obtained with the green laser source. The high number of these interference fringes is related to the high intrinsic coherence-degree of the employed radiation source.

Fig. 7:

Raw image (grey scale) obtained illuminating a double planar diffuser with a green He-Ne laser. The image is filled with a pattern of across-track interference fringes of equal thickness

The pre-processed interferogram should have a null-mean, starting and ending tails approaching to zero, and any optical artefact removed. In order to achieve these characteristics, we have elaborated a general scheme which is based on the following main steps [8], [9]:

- cosine inverse transform to retrieve the un-calibrated at-sensor radiance spectrum;

The spectral dispersion of optical path differences, which is due to the dispersion law of the refractive index of the beam splitter makes the calibration of the OPD axis really a complex task. When a broad spectral range is exploited, the OPD values may significantly depend upon the spectrum of the observed target. Thus the inversion model of the interferogram as well as the spectral calibration of the sensor might be difficult and their interpretation ambiguous.

The uncalibrated radiance, retrieved applying gaussian apodization are plotted versus wavelengths in Fig. 8. Due to the circumstance that the employed spectral source may be approximated to an impulselike radiation source, this measurement is also a good test to estimate the instrument spectral resolution: roughly 23 nm at 632 nm. Let us note that the spectral resolution has been lowered due to apodization. and OPD spectral dispersion.

Fig. 8.

Uncalibrated spectra of at-sensor radiance retrieved by cosine transforming the interferograms plotted in Fig. 4.

3.2

SNR estimates

We have executed 40 measurements employing a 600W halogen lamp, each measurement being constituted by 30 frames. The mean and the standard deviation have been computed for each pixel-sample, hence obtaining a set of 1024 mean and standard deviation values interferograms. The SNR and the noise amplitude have been so estimated, and their plots are shown in Fig. 9.

Fig. 9:

Plot of SNR and noise amplitude (standard deviation) as computed from 40 interferograms collected observing the same radiation source.

2.3

Reflectance spectra reconstruction

Standard reflectance tiles together with diffusers doped with Holmium and Rare Earths have been used to check the reflectance spectra retrieved from ALISEO interferograms. For each tile a complete interferogram has been reconstructed, then the at-sensor radiance have been computed. Reflectance spectra are calculated taking as reference the radiance spectrum extracted from a Spectralon tile. Results of these measurements are plotted in Fig. 10.

Fig. 10.

Standard tiles reflectance spectra: Reconstructed spectra from measured interferograms are shown with a bold line. Labsphere reflectance spectra are plotted with a thin line.

Retrieved reflectance spectra have been used to verify the wavelength calibration of the instrument. The obtained results are in fair agreement with reference data. We suppose that partial departure of retrieved spectra from refernces is caused by the still insufficient compensation of the spectral dispersion of OPD.