Editing Fourier-transform spectroscopy (section)

==Continuous-wave ''Michelson'' or ''Fourier-transform'' spectrograph==
[[File:Fourier transform spectrometer.png|thumb|250px|The Fourier-transform spectrometer is just a Michelson interferometer, but one of the two fully reflecting mirrors is movable, allowing a variable delay (in the travel time of the light) to be included in one of the beams.]]

The Michelson spectrograph is similar to the instrument used in the [[Michelson–Morley experiment]]. Light from the source is split into two beams by a half-silvered mirror, one is reflected off a fixed mirror and one off a movable mirror, which introduces a time delay—the Fourier-transform spectrometer is just a [[Michelson interferometer]] with a movable mirror. The beams interfere, allowing the temporal [[coherence (physics)|coherence]] of the light to be measured at each different time delay setting, effectively converting the time domain into a spatial coordinate. By making measurements of the signal at many discrete positions of the movable mirror, the spectrum can be reconstructed using a Fourier transform of the temporal [[coherence (physics)|coherence]] of the light. Michelson spectrographs are capable of very high spectral resolution observations of very bright sources.
The Michelson or Fourier-transform spectrograph was popular for infra-red applications at a time when infra-red astronomy only had single-pixel detectors. Imaging Michelson spectrometers are a possibility, but in general have been supplanted by imaging [[Fabry–Pérot]] instruments, which are easier to construct.

===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>