Editing Fourier-transform spectroscopy (section)

===Extracting the spectrum===

The intensity as a function of the path length difference (also denoted as retardation) in the interferometer <math>p</math> and [[wavenumber]] <math>\tilde{\nu} = 1/\lambda</math> is <ref>Peter Atkins, Julio De Paula. 2006. ''Physical Chemistry'', 8th ed. Oxford University Press: Oxford, UK.</ref>

:<math>I(p, \tilde{\nu}) = I(\tilde{\nu})[1 + \cos\left(2\pi\tilde{\nu}p\right)],</math>

where <math>I(\tilde{\nu})</math> is the spectrum to be determined.  Note that it is not necessary for <math>I(\tilde{\nu})</math> to be modulated by the sample before the interferometer.  In fact, most [[Fourier-transform infrared spectroscopy|FTIR spectrometers]] place the sample after the interferometer in the optical path.  The total intensity at the detector is

: <math>\begin{align}
  I(p) &= \int_0^\infty I(p, \tilde{\nu}) d\tilde{\nu} \\
       &= \int_0^\infty I(\tilde{\nu})[1 + \cos(2\pi\tilde{\nu}p)] \, d\tilde{\nu}.
\end{align}</math>

This is just a [[Sine and cosine transforms|Fourier cosine transform]].  The inverse gives us our desired result in terms of the measured quantity <math>I(p)</math>:
:<math>I(\tilde{\nu}) = 4 \int_0^\infty \left[I(p) - \frac{1}{2} I(p = 0)\right] \cos(2\pi\tilde{\nu}p) \, dp. </math>