International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B-YF. Istanbul 2004
For OPD =0 the interferogram reaches its maximum value at
each wavelength, this property being assured by the
circumstance that the phase delay between a certain pair of
interfering ray is generated inside the semi-transparent plate.
The complete raw interferogram of the energy coming from a
certain pixel of the observed scene is not a continuous function
of OPD but it is convolved by the pixel dimension p, and
sampled with a square grid (the sampling step is considered
equal to the pixel dimension) whose extension is limited by the
detector size D . Therefore, in the X space of pixel coordinates
the measured interferogram /(x) may be expressed as:
1(x) = I(OPD(x))* reet(>) Combi en (2)
p p D
It can be easily shown that the relationship between OPD and
the entering ray direction 9 with respect to the instrument
optical axis is linear, as long as the device FOV is not superior
to a few degrees, as stated by:
OPD(S ) « a8 m (3)
7
being a the constant, which expresses the direct proportionality
between OPD and 9, and / the effective focal length of the
lens focusing the interference image. The constant a is related
to the maximum digitised optical path difference OPD,,,, and
to the maximum angle 9 max -
The raw interferogram is constituted by a pair of values
(x, DN(x)), which indicate respectively the pixel position on
the matrix and the corresponding electronic signal expressed in
digital number. :
The spatial coordinate x of a certain pixel is related to the
sampling step p we have assumed to be identical to the pixel
dimension, being / the integer index of the considered sample
and jp the position corresponding to null OPD. From
previous relationships, Eq.3 can be rewritten as:
a
OPD(« —(- fo (4)
f
The inverse cosine transformed interferogram /(k) is:
1(k) = [I(k)sinc(S3OPDk)|* comb(8OPDk) * sinc(20PD yy k) (5)
where 8OPD is the optical path difference subtended by two
adjacent pixels of the detector array.
This relationship yields an important limit for the sampling
frequency to avoid aliasing in the retrieved spectrum /(/). As
stated by Shannon’s theorem the chosen sampling frequency 4
should be grater than the bandwidth &, of the concerned
signal. In our case it should occur that:
kı=———zk (6)
280PD
From Eq.6 results that the minimum wavelength A4, we can
reconstruct without any effects of spectral aliasing is
Amin = 250PD . Obviously another limit on this quantity is
dictated by the detector sensitivity.
4. INTERFEROGRAM MEASUREMENT
In order to calibrate in wavenumber (or in OPD) the
interferometer response and to determine its spectral resolution,
an experimental activity has been carried out in our laboratory.
A double planar diffuser has been illuminated with a He-Ne
laser (Aj, — 632.8mn, output power-20mW) to obtain a
spatially coherent homogeneous and isotropic radiation field at
the interferometer entrance.
This experimental set-up allowed us to get a complete
interferogram from a single image, strongly reducing the
measurements time and the experiment complexity.
In Figure 4 a single frame obtained with the laser source is
shown.
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Figure 4. Raw image (in gray scale) obtained illuminating a
double planar diffuser with a red He-Ne laser. The image is
filled with a pattern of across-track interference fringes of
"equal thickness".
As can be seen, the interference fringes are vertical lines which
completely fill the image-frame and the high number of these
stripes is due to the high intrinsic coherence-degree of the
employed radiation source. In our case this number is, in
practice, limited by the maximum angular divergence of the
entering rays.
A. 600W halogen lamp has also been used with the
aforementioned double diffuser. The result is shown in Figure 5.
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