Effect of jitter on an imaging FTIR spectrometer Page: 4 of 13
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Real interferometers have dispersion and are not perfectly efficient, but these complications are easily dealt with, and will be
ignored in the present treatment. For a constant speed of motion of the moving mirror in the interferometer, and assuming
that t=0 corresponds to 0 optical path difference, the optical path difference is given by D = v t. The interference intensity
then becomes a function of time,
I(x, t) = J S(x, v)(1 + cos(2irvvt))dv (2)
v=O
The relation between the wavenumber v, in units of cm-1, and the modulation frequency f, in units of Hz, is given by f = v v,
where v is the rate of change of the optical path difference.
If the spectral intensity seen by a particular detector element is also explicitly a function of time, additional contributions to
the interferogram function, beyond those produced by the time dependent interferometer modulation appear. If the line of
sight of the imaging system is time dependent, so that the image position is a function of time, x= x(t), then the interferogram
function is, in leading order, given by the expression
I(x(t), t) = Io (x(0), t) + VI0 (x(0), t) * [x(t) - x(0)] . (3)
The time dependent vector x(t) is denoted as the jitter function. The next order term, beyond the leading order gradient
expansion, is given by
a2I0(x(O), t)
a~o(xo, )[x, (t) - x, (0)][xg (t) - xj (0)] ,(4)
axax(
which involves the second spatial derivatives of the un-jittered interferogram function and the second order tensor derived
from products of components of the jitter function x(t). Higher order terms will involve corresponding higher order gradients
and higher order jitter function tensors.
For a band limited spectrum, the continuous interferogram function is completely determined by a set of discrete samples. A
so-called "two sided" interferogram consists of a set of N discrete samples of the interferogram function symmetric about the
point t=0. For convenience in the formulas, the labeling of the discrete interferogram samples is such that In = I(nAt) (with
t=nA, for n e [0,N/2] and t=(n-N)At for n e [N/2,N-1]). The discrete periodogram estimate, Sk , of the continuous spectral
intensity S(f), is given by the discrete Fourier transform,
1 N-1 ckn
Sk - N n e N (5)
n=0
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Bennett, C. L., LLNL. Effect of jitter on an imaging FTIR spectrometer, article, April 1, 1997; Livermore, California. (https://digital.library.unt.edu/ark:/67531/metadc626838/m1/4/: accessed April 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.